Untitled

L,A 1 REPORT 24 

LECTURE SERTEG ON NUCBAR PHYSICS 

First Serie$: TERkINOLOGY; Lecturer: 'EIMII MCMXLLAgb 

LECTURE I: The Atom ahd the NuBkeias 

" 11: htehtidn of LIght with Matter 

It %If! Forces Holding the Nuclei Together 

{' IV: Reactions Betweeh Nuclei 

V: RadioactJvity 

VI: I) 6 -Decay; 11) Neutron Reactions 

t' VII: N~~cI.ear Fission 

Bhge 1 

1 

8 

17 

24 

' 

I,;;; 

t~.% 

32a 

g 

42 

Second Series : RADIOACTIVITY; Lecturers : E. ,Seere 

E, Teller 

F. Bloch 

LECTURE VIII: 

IX: 

X: 

The Radioactive Decay Law 

48 

48 

I, The Radioactive Decay LLPN (Cont'd). 

11. Pluctuations. 111. Passflpe Through 

Matter 55 

-Particles and Their Interactions 

with Matter 62 

XI: 

XII: 

XI'II: 

XIV: 

xv: 

XVI : 

XVI 1 : 

XVIII: 

The Theory of Stopping Power 66 

The Theory of Stopping Power (Cont'd) 72 

Interaction of Charged Particles with 

Matter 80 

Interaction of Radiations with Matter 83 

The Properties of Nuclei 

Survey of Stable Kucleus 92 

Theory of Alpha Disintegration. Theory 

of Beta Particles . 100 

I. Positron Emission and K capture. 

11. Gamma Radiation 105 

a7 

Copyrighted

. 

~ORRECTIONS 

, pI 241 - Equation 48 should read 

, .., 

p. 254 - Missing only because of an error in numbering and not 

because of censorship. 

i?-k 

'2% 

p. 276 - Equationiat top of page should read 

. .

Third Series! NEUTRON P€lS?SICS; Lecturer: J.H. Williams; Page 108 

LECTURE XIX: Discovery of the Neutron 

108 

II 

It 

XX: Neutron Sources (cont'd) 

Typical Reacttons f& Production of ~OUt 

irons 

114 

XXI2 Properties of Slow Neutrons 119 

'' XXII: Properties of Slow Neutrons (Cont'd) 125 

XXIII: 

XXIV: 

'Fast Neutrons 

Properties of Fast Nqutrons 

C r 0 s s Se c ti on Mea suremen t s 

135 

146 

Fourth Series: TWO BODY PRORLEN; Lecturer: C,L.Critchfield: 

Page 153 

LWTURn XXV: 

Nuclear Constants 

" XXVI! Neutron-Proton Interaction 

153 

162 

I' XXVII: I) Spin-Spin Forces in the Deuteron 

TI) Proton-Proton Scattering 168 

'I 

XXVIII: Three Body Problems 

178 

I' XXIX and XXX: Interaction of Neutron and Proton 

wfth Fields 

185 

Fifth Series: 

THE STATISTICAL TEEORY OF NUCLEAR REACTIONS; 

Lecturor: V, F, 4;Weisskopf Page 207 

LECTURE XXXI: Introduction 207 

'I XXXII: Qualitative Treatment 210 

" XXXIII: Cross Sections and Emission Probabilities 218 

" XXXIV: The Excited Nucleus 227 

?I 

XXXV: Resonance Processes 

237 

'' XXXVI! Non-Resonance Processes 

255 

Ju 

LECTURE XXXVII: 

Diffusion Equations and Point Source 

Solutions 287 

XXXVIIIt Point Source Problems and Albedo 292

LECTURE XXXIX! The Slowing Down of Neutrons Page 304 

11 XL: The Slowing Down of Neutrons in 

Water 315 

2 XLI: Tho Boltzrnann Equation: Carrections 

to Diffusion Theory 323

LA 24 (1) 

Sentembar 14, 1943. 

1 

LECTURE SERIES OH NUCLEAR F'HYSICS 

FXRST SERIES: TEFblIINOLOOY LECTURER: E, M. MCMILLN . 

LECTURE I: 

We consider a piece of matter, for instance copper, 

If we magntfy it, we first sea that it consists of crystallites. 

They in turn are regularly arrmged arrays of 82;oms. In the 

special. case of copper, the eeometrical arrangement i s the same 

as a close packed arrangement of spheres, the atoms in one plane 

making the following pattern: 

I 

At the center of each atom there fa a nuoleus, Of coursel the 

picture is not really like the ne drawn., 'The elsotrons fill 

the whole space, there are no botmdariea, the atoms actually 

overlap, thelr limits are fuzzy. In a metal like copper it is 

not even clear which electron belongs to which atom, In an insulator, 

f'or instance the noble gas Argon, each atam holds on to 

its own electrons and each electron belongs to 8 definite atom. 

.- 

In a chemio-al compound, f'or instance carbon dioxide, tfhe e 8 

aped by the atams belonging to one molecule, but eaahmoleas 

its own electronsr All the chemical properties of' the 

are determined by khe outer electrons, 

The question arises what arfect the nucleus has on the 

behavior af the atom, The effect af the nucleus is due to its 

This charge is an integral multiple of the

positive charge which is of the same magnitude as the charge of 

2 

the electron, 

Atoms must be neutral because a positive charge 

would result in the attraction of more electrons, whereas a 

negative charga would cause the atws to lose electrons. Each 

atom has enough (negatively charged) ezectrons to c&npensate the 

nuclear charge. 

This number of electrons 2s aZso called the atan- 

ic number and dot;eminas the chemical properties of tho atom. 

Another significant property of the atomic nucleus is 

the nucleus mass, 

This mass is much greater than the mass Of the 

electrons, Even ths lightest nucleus, namely of hydrogen, is 1840 

times heavier than that of an electron, 

Thus the mans of matter 

is essentially conoentrated in the nuclei, 

The mass of the 

nucleus depends in first approximation on the number of neutrons 

and protons within the nucleus, 

The neutrons and protons have 

approximately equal mass hut this equality does not hold as 

exactly as tho equality of positive and negative charges within 

an atm. The protons carry one unit of positive charge, The 

neutrons are uncharged, 

Tho charge of tho nucleus, that is the 

atomfc number, is equal to the numb@r of protons Sn ths nucleus, 

The total. weipht of the nucleus on tho other hand, is roughly 

eW&l to the weight of a proton (or a neutron) multiplied by the 

totah nwnbor OS neutrons and protons within the nuclcus, This 

number is callcd the mags number, 

'Jhus if two nuolei have thcj 

same charge, it docs not moan that they haw the same mass number 

because a frivcn nwnbep of protons might be associatcd with a dif- 

$'@rent nwnber of noutrons in different nucl-ei, 

If two nuclei have 

tho same nwnbor of protons thay will belong to the same chemical

dlf'fcront nwnbor of neutrons they will bo different isotopcs, 

For instance hydrogen has different ktnds af isotapes,' &very 

hydpo$on nucleus has one proton but it may havo 0, 1 or 2 noutr~ns. 

Therc cx$s.t: wany more nuclaar spcclie$ than atomic 

ordor to designata the nucloar spsoiea WQ writa the atomic symbol, 

for instanco H in thc cas0 of hydrogon, 

This alroady implias 

what tho nuclear charpa. is and thercforo shows how many protons 

aro found in thc nucleus, 

by an uppor index, 

Tho mags nwnbor of tho nuolcvs is shown 

The following tablc pivcs the numbor of n6uc 

trons and prOtons Tor tho known hydrogcn and helium isotopes: 

numbar of protons numbcr of new- 

3. trons 

B k 0 

H2 E 1 

H3 1 2 

3 

B 

r, 

2 4 

Thc? approximate S%Z?- of anatqm is IOe8 cm, 

DEfferent 

abomic spocics havo dlfforent size but the total variation in size 

is not mare than a factor of ton times. 

One may visuallza the 

size of' an atom by expanding our scale of rncasuromcnts in such a 

way that 1 om, will appear 8s ,000 milos, Then an atom will bc 

about 1 cm, in radiuls. 

Tho 8Sze of the atomic nuclous is about 

crr~.~ that is, about lo5 times smaller than the atom, 

distancc between the sun a.nd earth is about 100 timas the radius 

of the sun, 

paint of viow of 

tho earth, 

The 

Thus the yluclous is relatively much smaller from the 

its outer electrons than thE: sun as viowed from

4 

It was stated that tho lightcst nucleus is about 2,000 

timos haavior than an olsctron. /Ire know that in 1 gm, of hydropcn 

thoro arc 6 x atoms, thoroforo tho mass of one hydrogsn 

nucleus is J1/6 x 

grms, 

Bofore turning. to tho dimension of the cbctron, we 

shall d$.scuss the peculiar dirficulty whfch arises in connection 

with visualizing tho electronis within the atoms, It is ofton 

stated that an atom is likc a solar system. ThG atom of' iron for 

Instance has 26 alcctpons. Can we rnOasuro these as in motion in 

approximatcly clliptical orbits around thc nuclcus? If we should 

take a microtjcopc of vcry high powou, and rcsolution and take a 

photograph of thc inside of an atom at a ,r;rivcn instance, me would 

rind a disorderly arranprnwnt of elcotrons within tho atom. One 

might then try to tako another photograph a vemj short time later 

in order to find thc motion of' thc eloctrons within thcir orbits. 

If such a photograph is taken no relation is discovercd botwocn 

the positions in tho two consccutive pictures, hc tually if tho 

resolving powcr OS tho microscopo would have bccn infinity, the 

second picture would have boon 'blank, Thc reason fop this situstion 

is that the light which ia uacd in taking tho pictures carries 

momcntum and one cannot thcrofore takc a photograph without 

,giving a kick to the clcctrons, If good resolving power is 

wantod, you must uac light of vary short wavo longth and such 

lipht carpios particularly hiph rnomsntwn, Gnc might try to ' 

cornpromiso in following tho orbit of tho okctrons by using a 

lowcr rcsalving power and haping that in this way thc: vcloc.ity 

of the clcctrons will not bo too much disturbed in taking a

5 

J) 

0 

photograph but the quant 1.t 8 t ivc e itua t ion is such that tho 

nccossary unc c r t ain t ie s in position and momentum aro just 

to mako it impossible to definc orbits within atoms, 

This conclqsion is a consequence of quantum mechanics, 

Quantum mochanics is thc mechanics that appliss to all bodies, 

for instance to a tennis ball, but in the case of a large body 

like a tennis ball tho laws of quantum mochanics approach the laws 

of classical: mechanics so closoly that the two bscome practically 

undisbinguishablc, Thc word quantum means amount and bccum in 

the designation quantum moohanics bocauso many of thc characteristic 

laws of quantum mccharzios am as sociatcd with the fact that 

ccrta$n quantitioa appear in amounts that arc multiples of a given 

cbaracteristlc amwnt;, Tho most basic of these quantitlcs is the 

s o-cal Led "quantum of action'' which is dosienatod This 

quantity has tho dimansion of a dlstancc times tho momontm, 

h Is the measure a$ the limitation on moasuremont, If the position 

o$ a particlc (4.e. its distancc from a rcfcrcncs system) 

is well knawn, then little can ba known about its momentum and 

vice vcrsa, If tho momcntum is known tho position must bo unknown. 

The product of the uncortaintics in momonturn and positian is of 

the magnitude h, For a big mass this uncertainty docs not moan 

much, This 18 so bccause thb momcntum is a product of mass and 

velocity and if tho mass is big, the uncm-tainty in tho velocity 

bccomos correspondingly small, but for tho lightest partlcl'os, the 

electrons, $he uncortainty of momentum causcs such an uncartainty 

in velocity that to dcfins anything likc an orbit for an electron 

bocomos impossible4 The only statcrnonts that wc can mako about

ht for instance tako many dnstantanoaus photographs of a 

4 certain kind of atom in tho way'doscribod abov~, onc might then 

suporimpose those photographs and WQ thon find that tho olcctron 

position all1 distribute itself mor0 in cortain rogions (noar tho 

nucleus) than in ather rogions. 

thc probable donsity of tho c'loctrona inside tho atom, ire, the 

probability with which the c3cctrons can bc Found in tho various 

volwno elemopts insido tho atom. 

Such a plcturo gives an idea of 

Tho doscription just givon might induce onc to say that 

an olcctron is asbip 8s the wholc atom. 

6 

HOWOVOF, in an individual 

moasurcmcnt the position of an electron may be more sharply de- 

ffned and in fact electrons do havo thoir own radii which a r much ~ 

smaller than thc atomic radius. 

Tho conccpt of the electron 

radius can be undcrstood by cansidaring the clcctric field of an 

olectron. 

This clectric flold La nccQrding to tho Coulomb Law 

inversely proportional to tho squnrc of tho distance from tho 

olectron, 

If tho olootron were a point, the ffald mar the 

electron would bo infinite, It is not belicvcd that such ** in- 

finite fields can oxist and it is asswad thnt at somo givcn 

distance from tho c ctron, thc Cqulamb Law CORSCS to hold, This 

0 

distance has bocn estimated 

That is the cloc- 

tron radius4and tho nuclear radii a m of comparablc size. 

COUPSC, 

about 30-13 cm. 

tho position of tho nuclci IS not quito so hard to dsf'ino 

as tha position of olcctrons duc to tho fast that the nuclei hclvc 

much bigger mass and accordin@; to that reasoning unccrtaintics 

in their momenta, will not cause very great unccrtaintics in their 

volocitics and will not wash out thoir orbits to the seme axtent 

Of

VIQ 

7 

In some of the applications of quantum mechanics which 

shall encounter XatBr, the wave Eennths associated with par- 

ticles will play a fundamental role, 

a given particle is h/mv 

Gne lonpth associated with 

(h = quantum of action; m : mas8 of par- 

ticle; v = velocity of particle; mv = momentum of particle), 

%'his 

length is actually equal to the uncsrbainty a% position associated 

with an uncertninty in momentum of the magnitude mv. De Broglie 

was the first to postulate that a wave process is associated with 

every particle and it is shown that the wevo lencth is h/mv, This 

&$sociation between paptlcles and wave proceaaos is surprising but 

it has been actually demonstrated by interference experiments in 

which electrons or light atams (for instance hydrogen) have been 

reflected from crystal surfaces. 

The laws of' this reflection pro- 

cess are of the same kind as the laws of X-ray r'eflection and have 

, 

been described In terns of wave intcrferonca, 

It is apparant from 

the above statements that the de Sroglie wave lcnpth is closely 

related to the uncertainty of electron position and also to the 

fuzziness of atomic boundaries,

8 

LA 24 (2) 

September 265 1943 

LZCTURE SE~?XYS ON NCTCL’2AR PHYSICS 

, - 

FIRST S7RI”S : TTR!‘l INQLGGY LECTtJlBR: ?7 .I$ e MCIJ ILLAN 

LECTURE 11: , 

1WT’~RACTICN GF LIGHT TflfITK MATTER 

% 

The list of simple particles mentioned in the last Xec- 

turo comprised electrons, protons and neutrons. 

$‘fa may add to this 

1Lst light. 

It is known that,light consists of‘ eloctromagnctic 

waves, but if, as has boon mentioned, the wave process is 8580- 

ciated with electrons, n.eutrons or protons, which we ordinarily 

consider as particles, then it will, not be too suryrising to find 

that light, which ordinarily can bo thought Of‘ a8 a wRVO process, 

has certain particle properties, iillhon light bohavss like a particle, 

we talk about a Xipht quantum. 

There is an important dffforonce between light and 

proper particles, for instance tho electrons, Tho electrons have 

a strong tcmdencp to persist. fivht quanta do not. They pat absorbed 

and omitted, It 3.3 not oasy to count them and for this 

reasoh it is impracticable to consider them as constituents of 

matter, Gn tho other hand, whonever 1icht does got- absorbed or 

emitted, a chunk of anergy of a definite size disappears or is 

produced in each such process as though light did consist of pap- 

ticlcs cBrpying il certain amount of energy. 

This enerpy of a 

light quantum is hq whore zfis the frequency of the 1Pgh.t considcred 

and h 3.8 the smc quantum constant which appears iur the 

dc Broglio rolation conncctinp the momentum of a particle with the

i 

1 

I 

wave longth associated tp it. 

, 

If the light gGantwn posscsses a definite cncrgy hv, on$ 

will also expect it to bkrry a momentum. According to the theory 

1 

. of FgJ,B$ivity this momontmn is hz/Jc, where c is the light v~lo- 

city. 

bt will be noticog that thc ratio of onereg to rnomc;ntum is 

I 

'' not thc samG hs ih Nowto)-iian mobhanics. If we write for tho kino& 

1 

tic onergy mv2/2 (m is mass of gartiolo; v, its velocity) and for 

the momentum mv, thcn the ratfa of the 2 quantities is V/2, 

that ds 

one half of the velocityi of the particle rathar than the velocity 

itsclf. 

I 

1 

But vcwtonian mbchanics nppXics only as long as the velo- 

city 

v is small compared to tho velocity of light c. 

hv/c of the momentum of a light quantum can be easily soen to Ggreo 

with tho de Broglie rclaItion. 

Jcngth of light and 

I 

I 

I 

In fact, c/z/is equal to 

The value 

tho wave 

WX l momontum 

I 

is the samo equation whilch also holds for electrons, noutrons and 

protons. 

i 

I 

I 

In discussinp llight quanta, it was ncccssary to introduco 

1 

relativity, ulJhile this lbranch of physics, as wall 8s quantum 

mechanics mentioned last1 tims, have an elaborate mathematical 

foundation, we nced t9 b(c concorncd horc only with a f'ow s;lmp,lo 

consoquenccs of thesc tdoories, 

1 

I 

Nuclear physics too will bc 

troatod hero on a simildrly simple dcscriptivo plane, in which tho 

clcmontary particles, cl/cctrons, protons and ncutrons, aro intro- 

I 

duced without doscribind in detail tho evidence that thcy arc 

I 

olementary particles ani withaut discussing the difficulties to 

:I) 

whlch this concept leads. 

~ C I may ndd hcrc to tho list of tho pro- 

e 

I 

1

10 

@ 

per particles, the positron, whoso existonce was prodictod by thr: 

combined application' of quantum mechanics and rslativity, Tha 

positron has the smb proportios as an electron, exccpt that it 

rSes a positive rather than a ncgative chargc, Tho only rcason 

why its discovery occurred much latcr than that 0% tho olcctron is 

that the positron is not stable in the prosonce of olcctrons. Tho 

positron is attrnctcd by an olcctron to which it happons to comc 

elosc., The chargcs of the two particles componsato cach other and 

their energy is radiatod out in thc fom of Light, Thus an cloctron-positron 

pair is annihilated. It is possiblc to produce posi- 

trons but in tho prcscnco of clcctrons which arc found in all OPdinary 

mattcr thcy can oxist only a wry short tirnc. Actually the 

oduction of positrons i s the rcvcrse proccss of tho annihilation; 

in cach production procoss, an elactron-positron paiP appears, 

thcrcforc, whcn a productioncannihflation cycle is ondcd, we arc 

left with the original nmbcr of clcctrons, 

In addition to all thcsc particles, wo shall cncountor 

two mom half-known k5nds of pmtZclo8, thc moso- and thc 

neqtrinos, Tlno mcsotrons RPO known cxperimcntnllg, Thcy are 

creatod in spccial praccsscs obscrvcd in cosmic-ray phJTsics but 

oy have not bocn obscrvod up to now in ordinary nuclotlr roactions 

Like the positrons thcy diaappcar shortly after thcir creation. 

They arc of importancc in the theary of nuclonr forccs which at 

a 

prcsont is widely acccptod. 

Tho ncutrinos haw not bacn obscrvod. 

Thcy may bo Qonaiderod. at prcscnt RS a pu~o invcntion introduccd 

to satisfy thc conservation laws of cncrgy of momcntum in ccrtain 

wcll known nucloar roactions fn which thaso consorvation laws arc

11 

agparontly violated, 

Cnc important conscqucncc of' the thcozy of relativity is 

the conncction bctwaon maas and cnergy, A movinp particlc is heavw 

fer than the same partielc at rcst. differonce in mass is propartional 

to tha kinetic encrgy, Potcntlal cnergy has a similar 

irifluwcc on mass. Tho proportion constant batwocn oncrgy and mass 

is the square of tho lQht volocity 

%is rclatiorz fa difficult to demonstrate under ordinary conditions 

it$ which thc kinotic or patantial onurgy pcr gram of matcrial is 

rathor small. 

In thcsc cascs, duc to thz high value of thc light 

volociity, th& encrgy cha.ngas duo to thht; kinetic or patcntfal oncrgy 

are very small cornparad to the orlglnal cncrgy of' tho maasca in- 

lvcd, but in nuclcar physlca much grcatcr cncrgics are asso- 

ciatea with a pivon mass, and hcro thc chnngos in maas bcoomo de- 

toctable. 

J ~ O may conslidor thc uncrgy relations in a nuclcar roc 

aptfon in which two Doutorons (H2 

(ao" 

uclci) unit0 t o fomn a-particlo 

Tho roaction has aCtualSy not bccn observed,' Its occur- 

rbncc is vcrg improbable, bccausc conscrvatton of rnomcntun provcnts 

qmrt9cla formed from carrying away tho cncrgy rcloasad by 

action as kfnctic onorgy, 

Thorcforc, tho cncrgy. r?cloasod 

crn$tLod in Lhc form of light and all rcactions involving 

n or hbsorption of light arc impraba 

c as cornparad to thc 

rbactions in which only hoavy particles, noutrons, protons or other 

nRclc2, arc involtvcd. 

*Zt may bc of intcrcst that a collision of 

rot; on, Deuteron and W-particle, 

dd'itfon to their more systemic

two Deuterons actually Fives H3 + €I1 or He3 + n (y1 stands for 8 

neutron) with about aqua1 probability; These reactions are much 

m4re probable than the raactlon mentioned above, 

I 

I mooretically, 2H24 ~e~ .+ light, lis reaction can tse 

used as 8 very simple example for the @neFgy-mass equivalence. 

The inaas of H 

2 is 2,015 maas units while that of He4 is 4,004 mass 

uhits, Therafore, the product nucleus He4 is ,026 mass units 

j 

lightor than the two original €I2 

12 

nuclei, This d&Fferonce of mass 

corresponds to a difference of energy and since in the reaction 

thc nuclear mass and enargy have dsoroasod, the differonce in enew 

bbcomos available (in tho present case in tho fom of light) as 

1 

t&e energy of tho roaction. Tho quantitative connection betwean 

crkergy and mags works out in such a way that ,001 mass unit OT 

a 

mil 1 ima s s-unt t c orre spmd s approxima t Q ly to 1 M i 11 ion e le c t r on 

V$lts (mors accuratcly ,935 M!l[eV), 

i? an cnorpy unit 1,6 x 

orgs, 

Tho alectron Volt (sV) in turn 

It is the encqy an electron 

aequircs if it falls through a pot0ntlaL difference of 1 Volt. 

The relation between inass and energy is freqvcntly used to find 

the oxact value of a nuclotir mass, 

r 

n~.pl~ar reaction tho ~ R S S of all but one particlpants 8ro known and 

id tho snorpy of tho roactian is mcasurod; 

This can bo dono if in 8 

In practical nucloar physics intoraction of light quarita 

withmattor plays an important roJ.~. 

i$ photo ionizatiok 

In thc figure 

Cno way of such intoraction

ative ion, namely tho electron ftself, is also fomed 

photo ionization, or the praduction of ions by 138 

Ad1 actual obsorvatlon in nuclea.?? physics dcpcncls on ionization, 

A typical arranpment for such obsorvationa is sketchod in the f o l m 

I 

chamber 

1 

i 

galvanomcbor 

I 

t 

battr;ry 

The air in tho chamber is ordinarily an insulator and so no CUPrent 

can be sont by thhc bnttory thraugh the galvanomcter, but if 

ionization occurs in thc cbambcr, ions now moving up to the 

t 

clbctrodcs will carry a cortaln small amount of curront which can 

I 

bo; road ius. the galvanomctor. Thus by thc curront one can moasuro 

ionization, 

1) Ionization can bo produced by charged partiales as wall 

as: by lfpht, Actu g thc ionization by f movine electrons,

protons, Deuterons and cb-particlcs, is 8. practical basis of 

cxperimcntzl nuclcar physics, 

exert an electric ficld and if their velocity is high enough, pro- 

ducs ianization, 

14 

These particles movinp past atoms 

Such particlos may. enter a chamber like tha one 

sfiown in tho above figwro through a thin Toil 

I 

They aro stoppod by 

a 'thicker foil, Onc of' thc: simplo basic axperimcnts Of' nuoloar 

, 

physio8 is to find out what is the foil thiclcncss that will stop 

a curtain kind of particlc carrying a plvcn onorgy, 

Thcro arc other interactions bctwcon light and matt@r, 

I 

Cqc of thcm 1s the Cormpton cffcct, 

This is tho scattering of a 

light quantum on a frcc clcctron, that is, one not bound in an 

atom. 

If' 

tho clcctron is not froc but its binding energy is nogli- 

gtblc compared to the cncrgy of thc light quantum, WQ may st111 

talk of a Gompton effect, 

In this offcct;, ths behavior of light 

can be satisfactorily dcscribcd by a particle piotzlro and vw thore- 

fare draw in the following fipurc thc light quantum as a particle, 

impinpin? lipfit quantum 

0 

scattcrod light quentum 

Tho momcntwn and oncrg:y impartcd to thc clcctron by the light; 

f 

qgnn'cum may bc calculated by laws of conservation of cncrgy and 

mbrncntum in rclativistic mechanics. Thc shortor arrow on iho scat- 

r 

tOrcd lipht qLuonturn indicated that thc light quzntwn has pivon part 

of its enorgy and rnornE;ntum to tho elcctron, 

This is true if thc 

clcctron was originalby at rcst, which in przcticc may be assumed 

f

15 

almost without sxception, Tho offect was discovered by A, E. Comp- 

ton, 

EIe observsd that most of the X-ray scattered throuph differ- 

e& angles has guffered an enorpy loss which dQpend8d on the anvle. 

The enarpy losa agrees quantitativsly with the on0 derived from tho 

energy-m.orflontum conservation Zaws, The Compton effect' ea the princri'pnl 

means by which "-rays (that; is, high frequency lfpht hd#$ 

roduced in the Compton effect may be detected by thair 

ionizinp power, 

' 

Thors exfats a third kind of interaction betwocn lipht 

anh matter, this is the pair production. In this process one 

1P:Fht quantum disappears and an electron-positron pair i s created, 

T135.s cffoct can occur only for Ini7.h enor,qy 1iFht quanta, natn~fy, 

the enarc$y of tho lipht quanta must su.pply at least tho mass of 

the electron and the positron, It may in addition supply an arbi- 

I 

&aq amount of kinetic encrcy to thcse particlos, G3uantftatLvoly 

the light quantum must carry approxiaately at least 1 MeV (nom 

accurately, at least 1,024 MeV), Tho proccss can not occur in 

empty spaco. It requires tho prcsence of mattor, Tho reason for 

is is not tha6 the process requires any other raw material than 

the onerqy carried by the lipht quantum but pair production in 

empty space would violate tho law al? conssrvatfon of momontun. 

pracoss can happcn in tho nciqhborhood of 0. nuclrus. Tho 

nqcloua having a preat mass can taka up almost arbitrary amounts of 

mobontun, It thcmfarc acts like r4 pla'clarm in the n8ighborhoad of 

I 

which clcctric flolds can act to p~oducc R pair and transfer any 

oxtra nomentum to the nuclousr Tho proccas occurs with proatost 

prpbability noar to stronyly ohmgad nvclei In whose neighborhood 

i 

i

16 

hiph electrostatic fields axist, 

The highost energy lipht quanta 

formed in cosmic rays ~ O S O their energy almost completely by the 

pair production PFOCOBS, 

The hichor onergyy -rays which occur in 

ntwlear proccsses lose cncrgy primarily by the Compton effect end 

a r absorbed ~ only to a small cxtcnt; by pair production ~ ~ Q C O S I S ~ 

1-s tho energy of thoT-.rays pots ~OWOP, photo ionization plays an 

increasingly important role in tho energy loss in addition to tho 

I

17 

U 

Septambor 21, 1943 

LA 24 (3) 

LECTURE SERIES ON - NUCLEAR PmSGCS 

.,, 

st Scrios: Terminology Lecturer: 5, M, McMillan 

LECTVFG 111: FGRCES HGLPIlTC ,THE NUCLTET TOGETEER 

As tho bcst known kind of forco acting betweon particles, 

wo might first considor elcctrpa~tatic forcos. 

live know that par- 

tilcles of‘ likc charga rcpol eaoh othor and particles ol unlike 

attract each Other, tho f OPCO be in8 

to the square 05 the distanco, 

oqncept of 

from 2nf ini ty 

from each other. 

It is a 

anergy ins toad Of that 

inversoly proportionax 

simplification 

of f OFCO. 

to 

th@ 

Thc poten- 

thc amount of work naedod to bring the particlos 

whore thoy do not interact 

and potential anorgy is positive, 

dofini tc dis tancc 

If thorn is repulsion, work will nood to be done 

In casc of attraction less than 

no work has to bs donc: and tho potcntial energy is ncgativc, 

patcntial snergy is the intagral of the forcc, 

For thc invorso 

law of clectrostatics, tho potential is proportional to on0 

Tho 

ovcr thc distance, 

Thc fipurc shows such an attractiva potantgal, 

e 

t h distanoa bctwecn two particles boing applfcd a8 abscissa wid 

ttp potential as ordinate, 

Tharo is no olcctrostatic 

B neutron and a 

proton since the formcr does not carry a charge, Actually thore 

a6poass t9 bo no stzoablc forco acting botwcon thesc two particlos 

unless they apprmch to a distance comparable to 

-13 

tboir rad%$ which is ab0u.t: 10 ern, Thero a 

larpc attrac”c;ve forco suddcnly appsars as shown

18 

iguro which shows ths poto 

a1 encrgy acting between these 

twy9 particles* 

The dctailcd dspsndonco OF this pot;ontial on tho 

distance 28 npt known but the 

B u, 

A potential af this typ 

and protans, but a190 botwcon two neutrons, 

a 

ding to tho 1 

tanGOS, but 

atc shape appears to bo that 

IdGrablc, narncly 

not; only hotweon neutrons 

Two protons will repol 

cctrostatics at proat dis- 

distancos the same kind of 

attraction ~ppsaps 88 bstwocn ncutrons and pro- 

tons,. "ha goten 1 bctwcon two protons is 21- 

lustrated in th.0 graph. 

Gne can sea in this 

graph tho dSsCinctton botwson long rapgo forces 

as thc clcctrostatic forces wh 

such as those due to bhc pot 

h act at a pwatcr distancc, 

whoso effect 9s noticeable only if tho particles am closc, 

cowon cxamplo that might rnPLke this 

rcpulsivo forc:: occurring whoiz ripid bodice arc broueht 

into contaat and the more slowly va 

1 

A 

If wc now want build from sovcral ncud 

pratons, it is ncccssary at lcast thc r;d,gias of those 

should touch 80 that tho short rnnpQ nttrnctivc forces 

o upcratlvc, 

mi8 roquircrnont i s not neccsaary in the 

c olcctron Tn thc @.toms, which am held to,vather by thG 

e long rangc olcctrostatic attraction pnd in which theref'orc tho

l 

19 

avcrago distance bctwcca particles is much grcatos, than tho radii 

of thcso pnrtiolos? Anothur reason why the noutrons and protons 

c o:ns tittat ing the nuclei 

hcavy part iclc s mq7 

aan 

easily 

so close togothor is the fact that 

have srnc.llcr wavo length, 

Accord- 

ing to quantum mchmicsI it is not possible to confino 8. particle 

within a Sl?s. 11 spacc without effoctivcly pivinp it a similarly 

sTall wavc lonpth which according to thc de 3, ogl i D relation moans 

c? ,hiph momentum, 

Heavy particles may possess such R high momentum 

without thoraby obtaining an exccssivc kinctic energy which would 

surpass tho nuclmr binding( forcos and disrupt thc nuc1Cus. 

Sinco 

clcctrons confined to nuclei would have such a high kinotic cncrgy 

WCJ muat not expect to encountor than as nuclear constitucnts, 

othor words, whcn an clcctron is attractcd to 9 nuclous, thc at- 

traction will incroasc its kinetic cnergy and mcrncntum and thoro- 

by will dccroasc, its wave lcn,yth, but whcn the clcctron has ap- 

pxioachcd to within thc distance of thc nucleer radius its wave 

Longth is still not as short as the nuclaar radius and thcroforo 

one cnn not think of it as bcing confincd to thc nuclQusa++ 

nqturc operating 

cbctrons do not 

rangc forcos may 

that such short 

hccvicr psrticlos, 

mcrc is no cssentifll diffcrcnce bctwccn tho laws of 

Sn atoms and nuolci but in the ntorns thct light 

as I! rulc comG: cloec enough so that thc short 

becomo cffcctivc, 

It is not at all improbable 

rnngc forces 8C t bctwccn clcctrons as bc twecn tho 

Only cosmic ray cloctrons possessing vcry high 

may bc squcczcd Into a smnllcr spacc than. 

thcy usually occupy in atoms, "'his hEppcns in donsc stars,. The 

cncr,q which is necessary to pivc c;loctrons high rnomcntum associad 

with thcirmoro close confincrncnt Is dcrivod In this caso from 

g F avi t t! ti on. 

In 

!

energios may come clo~o enough to othcr particles and iy1 those 

cascs it is rcasonablc to look for short rRngo potmtinls, 

Thc: conclusion is that a nuclous tcnds to bc a more or 

I 

Ic(ss closcly packed assembly of ncutrbns and protons. Such an as- 

@ 

scmbly is shown schomatically, 

prossnt protons, the crirclos noutrons, 

T ~ G croaacs re- 

ThG close 

armngemcnt is due to tho short rango attractions. 

On tho othcr hand, tbr~ is also a long 'ringc re- 

pu'lsion between tho protons which tends to pull tho nuclaua apart, 

Jf'wc: consider ncutrons and protons inixcd in a oLrtain proportion 

and incrcasG tho total nutnbcr of particles, wc will cxpock that 

I 

thjc long rclngo mpulsion will cvcntual-17 pravail and dostroy thc 

I 

ndclcus, 

I 

Indocd, tlm short rangc forces may bc considcrod tho 

same kind af forcc as cohcsfion. 

a surfacc which is txs 

mattor Rnd with it tho nuclcar rndius (R), 

Thoy will tend to give thc nuclaw 

small as poxsibfc, irrcrcnsing tho amount of 

Tho stabilLzLng tcn- 

I S 

d&xy of this forcc will bc proportional to tha surfacc! and thcmfore 

to R 

2 On thc othcr hand, thc total chargo in such a nuclous 

is proporttonal to thc valumc, thRt is to R3 and thc disz.uptivc 

. Thc: 

forcc is thc squp.rc of this chnrgc, dividcd by thc squarc of tho 

drlstmcc bctvcon protons which again is R, 

Thus, tho dis- 

PUptivc f'QrCo8 Would inCPCLlSO aEl R3 X 5 - ~ 4 '?.nd tl?cy will bo.. 

R% 

comc prodominnnt with increasing R, 

, a 

Tho structuro as dcscribod abovc in a qunlitativc mnnnor 

myst not bo tckcn too literally. ,Thurc is n lot of kinctic cncrgy 

iG thc nuclcus 9,nd tho phrtichcs must not bc considcrcd as local- 

I 

izcd, protonso due to thoir ;nutunJ. rcpulsion, hpvC a tcndoncy

* This 

21 

to'bc pushed out toward thc surfacc, but &,so thc numbcr of neutrons 

and protons pcr un%t volwno tend not to dlffcir too @rcatl$. 

will bc particularly notlccd in light nuolci most of which have an 

approx9matcly equal number of tlautrons and Qrotons, Thc hcnvicst 

nublei hnvc almost twice as many ncutrons AS protons but thcrc: is 

mason to bclicvc thet q. nucleus consisting of ncutr-ons alonc (or 

clsc protons alonc) would not hold togothGP at all, For B bcttcr 

undcrstanding of the situation wc shall havc to explain thc operation 

Of' the ss+calXod cxclusion~ nri.clp,la, In order to apply this 

prfnciplc in pr'abticd. cndols, wc c?lsb hbvc to msntion thc ,snin of 

particlcs, 

Mcutrons, protons and also olcctP0,ns carry an angular 

rnqmcntwn or spin which in quantum units has thc sfzo of 8, One 

rmg think of such c pnrticlc as rotating around nn axis which might 

I 

'ab oriontcd in spncc in differcnt ways. For E completc description 

04 such e particlc, it is not sufffcicnt to say where it is or in 

which orbit it movcs, 'nut it is also nccGssary to plvc thc oricntntion 

of' its rotntioncl @xis. It is a conscqucncc of tho npplfcntion 

quantum mcchanfcs thnt for 8 pnrticlc with thc intrinsic 

3 

aigular momentum z8 the directional angular momonturn may be descTibed 

by simply stating that the spin points upward or downward, 

antization of direction is closely related to the quantiaai 

tion of? angular monentm, It also should be remarked that, where- 

l 

a$ the angular momentum of elementary particles can be dbserved, 

e 

there is no direct way to investi~ate their angular velocity* 

There is, however, a further indication or rotation of such partic- 

X~EI in their magnetic moiaent. 

1 

Sven neutrons, whose net charge is

The spin assumes its greatest importance in connection 

wi,th the exclusion principle, 

I 

particles rnuat not be in the sane state.' 

This principle states that two like 

To show how this prin- 

le operates and how it is connected with the apin, let us first 

sldetr an imapinary spinless electrdn. 

One may ~lassify the 

electron orbits around the helium nucleus by their energy, 

el+ctron will ocoupy the orbit 0% least energy. 

One 

Then tha second 

can not be assipbed the sms orbit bacausa this would 

vibrate the exclusion principle. 

The second electrm would have 

I 

tormove in a lsss stronply bound orbit and the holiw atom would 

no% have its particular stability. 

If, however, wo now remember 

that electrons have a epfn which is capable of pointing in two 

d!.,CTarant diwctlans, then it is poissibZe to put two electrons 

g in their spin dire 

tom, and a very stable arrmgament is obtained, 

The third 

alectran wotzZd have to agree in it@ spin direction with one of the 

twp ele'ctrons and tharofovle the exclusion principle provents it 

t

23 

I 

of protons, Tho simplest nucleus of this 

, consisting of two ne-utr 

tqo poasible spin orleneations in neutrons and protons, the exalu- 

I 

inoiple permits aJ.1 partiolss in tho Be4 nucleus to occu 

e (namely lowest) orbit, IT now, one mope neutron is added, 

I 

ip musrt go into a no orbit of higher energy, The resulting 

ntkleus would be He 5 but the energy of thtxt next orbit is SO high 

that it is not hcZd in at all by its conneation with the other 

PQpkicla5 and He5 does not exist, If two neutrons a m added, 

ldading to 6 the attraction bctwoen tho two additional noutrons 

i

J 

thb numbor of neutrons 

1 

1 

I 

I 

i 

-nurnbsr 

AI1 known nucZci thah fall in a rathor narrow band which 

1 

stbrts up at 85O oorrosponding to 

trbns and protons, 

Later the band 

thb inaroassd proportion fn nsutro 

oughly equal numbor of nou* 

This discussdon makos it rrlora understandablo 

number of partfclcs in nuclai is Iimitcd, 

! 

webe prclsant in a nucleus, the a1acbFostati.c forcss would disrupt 

I 

thb nucleus; if on tho othor 

If' tao many protons 

adding protons and 

incrcaso the numbcu, of' neutr s, then tho ~xclusbon princlp1.s will- 

, 

recjuf~g tho plarticlos to po intor such high orbits thab tho short; 

rahgo attract,ions aro no longor sufftalcnt; t o hold tho nuclcua 

1 

1 

1 

September 23, 1943 

LA 24 (4) 

1 

I 

I 

I 

r ' 

Fgrat Sorics: T~m~nology *N, McMbllan; 

LRCTURZ IV: 

RZACTIONS BYT'IZTN T€E f\rtTCLEI 

I 

n+lof, 

edch other, 

, 

In order to be ablo to diactxss the roactions 

we hevc to invodtipato the forces with whaoh 

These forcss may bc obtainod from their 

I 

I

which i s shown in tho figure., 

In: thia figure, tho potential cnorgy is glottcd as a Function of 

1 

th; distanco of tho nuclei ru 

static repulsion bctwccn ths nuclci is tho only forco acting, 

this, thc patsntlal incrcaacs as thc invcrsc first powor of thc 

e 

At prcatcr distancss, >tha olcctro, 

At smaller distancos, thc jfstickincss" of thc nuclei 

cokcs into play; that is, as soon as thc nuclci touch, thcy attrack 

cach othor. 

This rasults in a drop 9n potential cnorgy which wo 

have reprcscnted as a potcntfal wcll. 

tdis potcntlal well is so dcep as to ovor-ccmpcnsato thc clootro- 

stktic rcpulsion, and ih thirs ~asc tho potcntlal onorgy bccomcs 

1 

negative, as has boon actually shown in thc fPgurc. 

For not too hoavy nuclei, 

Tho strcneth of rcpulsion bctwc;cn nuclci and thcrcfcrc 

tUs height to which thc; potential barricr riscs, dopcnda on thc 

I 

prioduct OS nucloi charpcs, This product is srnR1Tcst for W 2 atoms, 

I 

I 

bu!t cvcn in this caso thc potential. barricr af a fcw hundrcd kV 5s 

to/ be cxpcctocri, 

This potcnt5al valua is of practical irnportancojh 

th;s well-known reaction botwocn two bcavy H nuclQP, which is to bc 

In 

cd below, 

For all othcr pairs of nuclcl, tho potantfa1 bar 

ovcn highor: that is, of thc ordcr of a fow million volts, 

I 

or! for two hsavy nuclei ovon considorably ~OPC* 

I 

I 

Tho stpong rcpulslion bctwoon nucloi cxplaine why nuclcar 

a raeictlons arc uncommon under ordinary circumstances, Indocd, tho 

I 

kipctic cncrgy b$ particlos due t o thcmal motion at room tomperae 

!

~ 

tube is about 1/40 of an electron volt, while energies oncounterod 

in chemical reactions arc a few clootron volts. 

arb far too littlo to allow nuclei to got closc onough to each 0 t h 

so, that thoy may touch and rsaction mey startr 

i 

These energies 

arb known to proceed howcver in the intorior of the sun, where 

I 

tokporaturcs of the order of' many million degrees arc found, 

thb laboratory, it is necossary to impart: to thc nuclei high kinc- 

bib cncrgios, in order that they should got close anough t o PBBC~;, 

28 

Nuclear roactione 

In order to accclcrato particles sufficiently, dcvfcos cf 

two esspntially different kinds have boon used. 

tioally shown in tho figure: 

In 

Tho Q ~ Q is schbma- 

c : . , - 

A' 

* ' 

+''*.*, 

[4'-&-g#Bw 

I 

-- 

c 

pabt high volocity to nuolci is tlzo USC of thc cyclotron, 

dcirico, tho cnarpy is Fivon to tho nucloi in a sorics of small 

Xn that 

pu'shcst 

The nuclci aro hcld in circular paths by a strong magnotic 

fi'old. Tho accelerating clectric ficld is actually an oscillating 

f'i'cld which exerts ita accclcrating action, always when the nuclei 

rcach a cortafn part of tho cfrclo. Ahi2,c tho particlas @et; ELCccloratod, 

they mom in incroasing circlos and arc fSnally ojocted 

.as fast particles, 

Thc tyoe of interaction betwoon nuclei doscribod in tho

* 

I 

bc@.nning 

of this lccturc Sa quite different from th@ interaction 

bo,twccn nucloi and ncutrons, 

ThZs intoraction potential i s shown 

in bha figure as a 

Thcro is af cour80 no cloctrostatic repulsion bctwocn thc neutron$ 

and 'tie nuclous, This is thc roason why noutrons, though thcy arc 

quite common particlcs in thc nuclei, have rcrnainad unknown so 

lohe]. A ncutron can approach any nuclcus f'xlccly, can rcact with 

that nuclcus (for instanc~, it can be capturod) and so it will 

stop being a frcc neutron%, Thus in tho laboratory, a. neutron 

may bo sccn just "on the fly", Gnc can obtain them by bombarding 

a isuitablc nuclous by a suitablo particlo, for inatancc, a proton. 

Thc partfclcs which hnvc bccn actually used as bombard- 

* 

ing particlcs in nuclear reactions are either uncharged or lightly 

I 

I 

charged particlcs, 

i 

hikh an cncrgy to ovcroornc tho potcntfal barrier, rind approach 

anothor nucleus sufficic,ntly close. 

flPat colzmn gfvcs tho namc of particles used in bombarding nuclci, 

If that particlo is an atomic nuclcus, its natation appcars in 

brhckofs, 

I 

A heavily charged pnrtiolc: rcquirca much too 

f'roquontly used particles arc dosignatod, 

In tho following tablc, the 

Thc sccond column gives spccinl symbols by whicrh thoso 

------- .**---------I--- 

C_f_Y._. 

Whcrc i s anothor purely thcorotfcal rca8on why froc noutrcns aro 

dif'flcult to obscrf-c,d '??:ley ere supposed to bc unstablc, md arc 

cxpcctcd to transioFm by radioactivc oction into protons ornittIng 

antclcctron at tho sarni! tho, but A long thc bcforc this radioactive 

trnnsfomation could take placc, a noutron is usually capturod;

* dcutcron 

I 28 

'r 1 

1 

proton 

d 

t% 

ncutrcm 

! radiation, oP' 7 -pays 

I 

cloctran 

:this Xist, thc nuchi El3 

P 

n 

1' 

rr3 

and He3 might bc addod, but th@y are 

i 

pr@ctically unavailable and so thq havo not bcon used In our ox- 

pcbbcnts of this khd. 

If OMC we.nta to use ncukrons ~8 bombard- 

ink particles, onc has to dorivo thorn from nuclcar reactions. 

If tkc bombarding particlo in a nuclcar ranction tzi' 

chargod, one wlll cxpoct no nuclcar rcaction to occur unlcas tho 

r 

partiolc has suf'ficiont kinctic oizcrgy to DVCPC~C tho 

haryior bctwoGn the two particles, Tbt13~ if tho yfold 

of: such a rcaction is plottcd against thc cncrg:,- CY. t?nc bombard-. 

in& pnrtfc?!~, ono cx~octs tb.c yield to remain zor? v;,3 

cngrgy V mr2 then riscj in somo manner, 

tho f?.gvi~ by thc; full ~PVC;, 

Yl".C?(1 I 

quant 

1 

I r+ 

! I 

l

., 

29 

bombarding particle extends exactly up to thb maximum of the 

potential barrier. Due to the wave nature of particles, a ceptain 

fuzziness of the orbit is unavoidable and even though the energy 

does not suffice to carry the partscle right to the top, there 

retnains a small probability that the particle may ri leak through" 

the barrier. 

limited. extent;, 

Therefore, if the particle is piven less and less 

energy and is therefore turned back according to classical rnecha- 

nlcs at preater and greater distances from the top of the poten? 

tial barrier, the probRbiLity that it can "leak through" the bar- 

rier decrerses rapidly. 

But the fuzziness in quantum mechanics is only of a 

Tl?a actual yield to be expacted accoI'dp 

ink to qwntwn mechanics is shown as a dotted line, 

At high ener- 

gies, where the particle can get easily over the potential barrier, 

the yield mwy approach a constant value or may even decrease again, 

The gield curve may show I;, sharp maximum, This is il- 

lustrated by the figure. 

I 

yie Id 

T' 

-+ enorgy 

To understand this phonoinenon, we must consider 

ing and 

bohbarded particle aftar they have touched end ha,ve formed a so+ 

* 

cazl-ed compound nucleus. This compound nucleus has quantum levels 

an@ A sharp maximum occurs in the yield curve if the energy of the 

bornbarding particle is just ripht for the formation of such 8 guan 

turn level. Thus the rnaximum is due to pn agreement between the 

I

enerpies of tha particlos bof'ore reactioh end tho energy state of 

a tho compound nucleus, Such an agreomont is called resonance. The 

phenomena due t9 such resonance are analogous to those that nre 

encounterod in clasaikal rnocknnics when the frequencies 

vibrating strings become equal to each other, 

of two 

The methemetical 

theory of those two situations are similar c?nd the shape of the 

yield curve near a resonance maximum is essentially tho same as 

, 

certzin resonp~nce curvos encountered in classical mechanics, 

It may hsppcn thRt in a reaction, the produced particles 

have greater intrinsic energy content than the initial particles. 

In'this caseJ the reaction is possible only if' this energy def'i- 

cicncy is supplied by the kinetic enorgy of the original particles, 

men a rtlinimum kinetic cnergy is required for tho reaction to pro- 

90 

ce@d, and tho yield rcmp.lns zero up to this energy, 

The energy at 

which tho reaction first becomes possible is called thc threshold 

of the reaction, 

I 

Unlikc tho case of the potential barriers, this 

threshold is sharp, and below it tho reectfon cannot proceed even 

with B small probabflity, 

As CJI 

exemplc for 1. sirnplo nuclear reaction, we may con- 

aidier the reaction between two E2 nuclei. 

There is ono possibility 

thflt the two nuclei mag stick to,?ether cnd from an CG-pQrticle. 

BuC such an a-particle would have a vary high energy contont, 

This enorgy might; be usod for tho omission of radiation, And if 

it 'is 80 used, the a-prticlo m2.y stick topether permnnently, 

hission of radlction takes$ howcvc~?, too lone 8 time and it is 

rl, 

more probsblc that before a? m y could bo ernittod, t;heCf;-particlc 

r 

would Ply apept again, It might fly npart into two deuterons, I.&, 

I

one would yct back the oripinal nuclei end tho net effect would bc 

rnc~oly a collision in which tho deuterium nuclei have doflccted 

cach othep, 

Gt'ner reactions c.rc also possible, 

tektims consistcd of a noutron and e, proton, 

of their colllsion, wo hnva two noutrons and two protons and cithcr 

31 

Bacli of' tho dou- 

Thus, in thc rnomont 

one of thcsc four pt-wticles mipht fly off', Thus we might got as a 

. 

result of' the reaction a proton r.nd F. H3so~ n ncutron and He 3 

Both these reactions have been obsorved, Thcty proceed with n con- 

$idcrable cvolution of onorgy but thc initial 

havc had appreciable kinetic oncrgy, othcrvuiso the reaction bccomcs 

quftc improbnblo, sincc tho pmticlcs cannot approach sufficiently 

c2ose. 

Anothcr possiblc rcaction would be tinc formation of a 

nuclcus consisting of two protons rnd of nnothcr nucleus consisting 

ofltwo noutrons, Thcsc nucloi arc not stable, Finally, throc or 

fo

m 

ovsntually by using soparntod Li smplssj in which only Lt7 or 

Li6 was prcsont, 

Thc pnrticular reaction quotod RbdTFo, 

is,/ cndothcmic, approxgmntaly 2llV kinotic cnorgy being nacdbd in 

thk prstons for the roaction to proCewd, 

in0iabn.t: grotbas over tho t oahold will turn in &qatosh part 

ihko kenetie cacrgy of khc houtx"oni 

onorg.fr ona might gat neutrons of pivan enorgics, 

Tho exccbk onoi?&y of tho 

By cbntt-ollfng $he proton 

$ ~ . q ronction is 

I 

actually tho best way to produco f'nfrly monoenergotic noutrons In 

th& p~rqa of e, fcw tonths of a MV, 

! 

' 

Tho roaction 

I ~09+ ~2- ~30-j- n 

a vary copious SOUPCC of ncutrons if deuterons of' 3 MV or rndrO 

arb umd. 

Such Routorons may easily bo producod i n cyclotrons, 

Thb roaction 'is strongly oxotbomnio e.nd givos thcrafaro noutrons 

high oncrgiea (around 10 I\!V>, Tho ncutrons so prodwcod 

, 

do'not CVOP haw a well doffnod Cnorgy, Thc 13' produccd in tho 

rohcticln may bo Lcft behind in vmious statos of Qxcitation, and 

on#WY Of 8" 

pons carry away tho varying mounts of cnargy that havo 

used vp in exciting 

is omitted In thc f'o 

at in tho d+d roaction 

VOTY light nuchk obtalncd in this 

ntually, *the axcitation 

citation occups 

tion (fop instan 

not; havc exoitad lcvela in thc ncighborhood of 3 

y avgilablc in thig rdactl~n~ This one 

4 MV, which is 

thcrcfora 

d off as kinotic onorgy of tho reaction products and one 

obtains noutrdns of wolbdcfinod oncrgZcs;

Locturor: 

Ii,M,McMillan 

I 

Historically thc first nuclcar transfomntion obscrvod 

wss radioQctivlty, Jt W R ~ naticod that Borne mincrals continunlly 

waL idmliificd with €Io* nuclei, tho B -radir.tion with olcctrons, 

r.nd ths "y -rndiction with qucnta of olcotrmagnotic radiation. 

In a radioaotivc process Q nuclrtua unclorpocs a spsntnncous chnngo, 

inlwhich it may cmlt a part of its own substcnco. 

1 

The v-rndiat;im 

ho$cvcr corrcsponds morclg to n scttlinp down of a nuclous from a 

, 

higher cnor.gT stc,t;c into P. 

lowcrI mom stablo stRtcl. 

Follows some other rcaction n8d is accompnnicd by xz'o chcrnical 

change of thc SubstFnoc. 

i 

It uszznlly 

Whcn an =-ray. is emitted, thc cjcctcd pnrticlo carricts 

twb chargos end tlic nuGLous is Icft with two chargcs loss, that is, 

it mom8 down by two placos in thc poriodic system, 

tlid ~ O Qti C on 

, Rfi226w Rag22 + Er04 

1 

An oxcrmplo is 

iniwhidh thc clcmant Radium, which is rinalngous in its proportics 

I 

@ 

to Barium, is transfornod into tho noblo gas Radon, This reaction 

is balanccd in tho sGmo mnnnor as the artificlel rcaotions mom 

tiinad aarlicr but r,n osscntinl dif'rcrcncc is that it OCCUPS spon- 

tnrilcously, 

FO1l.owing tho abovc rcnction, Radon emits in turn an

tances, thc,stickinosa of thc nuclcus will c,auso tho potential 

cqorgy t o bo loworedl md near r = Q wo find a patxntinl wall, 

Tklc h0riz;ontnl linc not far from tho top of the wcll rcprosonts 

tbic oncrgy lcvcl which tbc a,-partiolc occupics within the nua2aus. 

This oncrgy IcvcS in a radioactivo nucbus 3.s highor than tho pow 

tcntiaP cnorgy a t vcry high r valucs, 

This must bo SO bccnusc WQ 

know that at high r vaZuos thc Ot;-particlc may appear carrying 00x1, 

s&darablo khnctic cncrgy, which kinotio cnorgy must bc dcrivod 

from tho onargy that the eparticlc originally poasossod in tho 

nuolous, but in opdor that tho G-particlo should cncapo the nu- 

cious (moving .long 

tlzc dotted linc In tho figuro) it must cros8 

~l rcgion of high potantial oncrgy* According to clnssicnl mocharxtcs 

this 1s fmpossiblc, 

In qucntum mechnnios thc fuzzinoss Q$ par- 

thclcs mnkcs ponotrntion through thc potontin1 barrium? possiblc b\l; 

t1sc probability of thc particla lcakSng through is vo:-y 

small, 

This is true in pnrtfculnr if thc potcnticl barrier is high; 

, filnds that the avcragc timc which thc a-pnrticlc fs supposcd to 

spcnd in thc nuclcuhs bcforc lcalcing out by tho qumtwn rncchnnical 

mcthod is in fair aprccrncfit w ith tho tirncs actur:lly abscrvod, 

scpsitivo dcpondcnca of th5-a timc on thc hcipht of Chci potcntial 

i 

bn:rricr cxplnins why radioactivc decay times vary from a smnll 

Ono 

The 

fsaction of? Q 

sccond to scvernl billion ycmse 

e 

s 

i 

Tho rnnthomationl law of radioactive docay is basod on 

th,o fact that each radisactivu :rti;:;~~~ has a constant probability 

of! docmy, 

l 

It 26 to bc notod t;v+, this probability of dccay is 

in,dopondcnt of thc time tBo pwticuI,ar nucleus hns already livod, 

It fol,lows thnt in n bunch of radionotfvo nucloi, tho nwnbor of

, 

i 

nuclei dccaying per sccond 

proscnt. 

If ono plots tho 

35 

is proportional to tho numbor of nualci 

numbor, of nuclai as a function of tho, 

givon by tho formula 

N =: N,o 

" At 

No 53 the origiaal number of neutrons (which were present at t=G) 

nqdhis the reciprocnl of the so-called mean life, 

s$ood that some nuclei live shorter, 2nd cL few, 

I 

lqneer, th9.n 'this mem life, 

active decay by the hcllE-lffe which is piven by 

Pt is under- 

considerably 

It is more usual to dssoribe the rnc3.w 

hclf-life Z 0,6g3/'x 

The helf-lifs is the time in which hnlf of the original number of 

the nuclei hns decayed. 

I 

In discussing the ,@*decay we shall consider an artifi- 

c i R 1Xy JrPd i oac t 1.ve subs t c.nc e, flr t i Fi c 2 n 1 ly rad i oa c t ive nuc 1 e 1 

obey the spme laws as nRturnlly radioactive ones; but instead of 

being enoountsred in nature, they haw to be prOdUc8d by bombarding 

a hnturally occurring nucleus by a light nucJouB, 

In this way, 

maby 6 -active substances have been produced while artificial 

activities are very rwe, 

CG- 

A well-known RrtfficSaZ owactive sub- 

stance is obt8ined by bombarding the on0 stabla 2sotops of Sodium 

by' Deuterium 

N c L .I. ~ ~ H2 w .Na24 .f. H1 

I 

!

In: this equation we have written e" instond of 10 in order to show 

that negatlve electron is emitted, Tha nucleus produced in ths 

reiaction, Magnesium, has an8 more charge than Sodium r.nd thus the 

total charge is qQnsBrvedr 

It has beop stnted that in R nuclous thoro is no room fo$ 

electrons, How is it then possible that the nucleus should emit 

Rnt oloctran? It must be postulnted that tho oloctron is c3retlCcd 

in tho p-disintegration ~POCBBS, Cther prooosscs are indeed known 

in which pnrticlos me created, 

olsotrbn-positron pnfrs from radiationc 

pr:ocess in 

-Pctj.vJ,ty is the 

a-p++e- 

Gno mample is the crontion of 

Tho c:!smentary 

creation 

roacti~n. Tho enorgy dolivorcsd ts tho olectron is partly duo to 

tho mass dfffcrencc botwoen the proton and tho neutron end partly 

it is drawn from tho onerey store of the @-active nucleua ilyl which 

thje noutron was orig:lnnlly formedl 

ex,othormic. 

The above simpln process is 

It is ags~m~d that tho nsukron iesslf is @-activer 

This actfvity howcvor te.kOg R longror tirnc than other prosoasos 

h ncutrona can discpponr p.nd thcrefore the @ -activity of 

e 

froo nwtrona has as yot not boon obsorved, 

EC t ion 

AnoAthcr kind of 

p-nctivo nuolous is fomed in tha re- 

~14 + ~2 .-+015 + n 

The Ol5 nuclous decays according to the aquntion 

015-815 +

37 

Q 

This timc a positivc electron is emitted, This kind of' activity 

is'n littl o rarer than tho one discussed above, It does not OCCUP 

in tho natixrafly radioactive series. The olemontary reaction in 

thfs case is 

This timo thc olcmentnry roaction is endothomic, 

This monnsI 

thdt tho proton is sttlblc nnd a positron can be emitted only if 

tha ehargy storo of tho fl-active nucleus can supply an onergy sur- 

f'icriont to overcome tho mass difforonoo bctwcon proton and neutron, 

Sometimes it hr7.ppens that n nucl'cus instead OF omitting 

a positron absorbs one of its own electrons, 

tran will bc onc of' an inner layer in. an atom, tb?tz E\ 

K-clcctron, 

~'?ij s proccss requires R 

As Q rule this elac- 

so-called 

samcwhat smc.1 Icr cnergy than 

tho emission of a positron beccuao af th:: mass (an& corrcsponding 

snorgy) of the positron and eloctron. 

possible for R nucleus to ccpturo a positron, 

In principlo it would bo 

But even tho capture 

of an electron is nn improbablc procsss in spite of the fact that 

tho clactron is neap to the nucleus a11 tho tSmc, 

It practically 

ncvcr happens that 8 positron on its briof' passage nofir a nuoleus 

is absorbed, 

ns of the a-decay, 

Tho mathematical dcscription of the fl -dccay is tho smne 

sccond to hundreds of ycarsr 

Tho hnlf-livos mngo Tram R 

fraction of a 

Slownoss of thcfi3-procoss however 

ccnnot bc explained by th:: prcsoncc of a potential barrier slnco 

tho clcctron has n long cnough wave lcngth to loak thraugh nuclcar 

in exceedingly short 

long lives of B -pro- 

ccssos (oven a fraction of n sccond is 8 long time on a nuclcar

time scale) must be expla.lned by an inherent unwillingness for the 

v neutrons P M protons ~ to transform into each other, We must acm 

capt the Pact that the elementary creation processes mentioned 

above havo a low probability and the correspondine reactions pro- 

coed at a slow rate. 

This also explains the fact why bombardment 

04 nuclei by electrons does not produce typical buclear reactfons; 

The only action of electrons on nuclerl which has been observed is 

due to the electric Field of the e2ectr.on which ,$ives rise to sirni- 

lar excltation vrocessos as absorption of 

-rays. 

Btransf omations are frequently represented in. nuclear 

38 

The ordinnto gives the number of neutrons, the abscissa the number 

of ppotons. Dots in the diagram rcpresent nuclei, diagonal arrow8 

reprosant @ wtransf'omations. Tho Q -transformations actually 

limit the repion in this dia pan in which stable nuclei can be 

.$ound. 

If a nixclous contalns too many protons, it will ernit one 

or mom positrons and the nucl.eus ~1.11 thereby return to the belt 

of stable nuclei, 

sfons will accomplish the same tklinp, 

in singlo steps, 

If too many neutrons are present, electron omis 

These eixissions must happen 

Simultaneous emission of two electrons or two 

positrons is exceedinqly imprababJ.e, as will be seen from the fact 

that the probability of sucli a procsss must be the square of the

39 

e 

probability of a sinple emission process which we hRve stated is 

of itself improbable, No such double process haa ever been obsonred,. 

It, can happen howevar t1ia.t; a nucleus nay emit oither 4n 

electron or a positron, This occur$ when the nucleus can decrease 

its cnorpy efthor by tran.sfoming a neutron or by transforming a 

u 

proton into a neutron, 

Charts of the kind shown above aro useful in foliowing 

any kind of quclear reaction. In tho part of a chart shown in tho 

figure below, two reactions RPC? shown, In the first a proton is 

added; a@--activo nucleus is obtainod and a positron is emitted 

subs eq.uet?t ly 

-* 

- 

LA 24 (6) 

September 30, 1943 

Lr CTURT VI: Z) &DECAY XT, ) NETJTHCN RSAC TI OMS 

1) In a nuclear process in wh.lich a,@Hparticlo (oloctron) 

if3 emitted, the pt?rticular nuclaua goes T'rom one deflnitc enorgy 

state to another dcffnfto enorgy state. Hence ano would cmpcct 

l)i 

the omitted B-particlo to possess a defdnite kinetic B M B P ~ ~ . 

Cxperimentally, howevep, Et is found that the ,@-particles are not 

homogcyloous in enorgy 

.: 

but havo a distribution 8s shown In the 

f bllowing f ipro N 

Energy

angular rnornentw of L/2 quantum unit, 

This oxtraordlnary particlo

ducod by noutrond 

88 

(1) simplc? cnptura, usually fbilowed by $ha 

ma88 @hit Sa~gsr than the: rsrigShR3. hbeXou~3; (2) ine$&tia cob 

hielDni the ink~rnal mwgy of tho ~UC~QUS 2s changod, 

adbi0h mag bc regarded a8 an absorption of f? 

nnd a sztbsoqumt ro-omission of a nsut;ron, 

a noutron 5s absolrbod nnd 

proton is emitted.. 

This re- 

xzoutron by a nuclcus 

(3) !kc; (n,p) reaction 

Thi8 raaction pro- 

coods w$th fast neutrons, although n slow neutron reaction is known 

ogan, 

Hqro tha product nuclous has an ntornic number one 

smallor t1zr.n tho bombarded nuc1cU-8.. 

axanplos of this aro 

f -?z!c?YH 

ni 

- L 1 6 -k. n.- 

-. , &;a+ 

(4 1 The b;r:,$ cy} reaction: 

Thoso reactions proceed with slaw neutrons, For henvier nuclei, 

tho c-particPC must overcome a potontial bhrrisr; honce fast 

neutrons are requircd, 

(5) The (n, 2n) rcaction in which two 

noutrbns are Gmittod requires fast neutrons, (6) For the sake of 

complotonoss, elastic collisions are inaludcd, 

enorgy of the imadiatod nucleus is unohnngod. 

LA 24 (7) 

Hero the intarnal 

WmJor 5, 1943 

ZC'IPURE SERIES 0,N TJUCLEAR PHYSICEJ 

ries: Tmninology Lecturer: E,M, McMillan 

The word fission mean8 splbtking or brenking apart, 

We 

am concerned here with a proccse in which a nuclous splits into 

two more or 159s oqual parts, Tho product nuclei are Fairly heavy. 

h fission processes arc known In which tho initial nuclous 

is one af the hoaviost nuclci,

43 

The reason f0r the finsS-on procoss is the olectroatatio 

botwoen protons,. 

In highly charged nuclei the potential 

*his repulsion ip SQ gpeat that by breaking apart the 

nuclaua can P berate energy, 

Tg LJlustrate this, WQ plot the ma88 

defect as a function of the atomic numbor far stab 

I 

mass 

defect 

atomic number 

-* 

Fig. 3. 

dof'oct ia a rnaasuro for the potential enorgy per particle 

MUCZOUS, Ono obtalns it by subtracting from Ch.o nuc1cq~ 

o nearest intogar (which is the nm.bor of neutron 

the nucleus cnd has been cnllod tho mass x1 

0 

dividing by mass nwnbc3r). 

By going from Uranium to the SissSon pro- 

ducts, this patontial enargy per unit particle has considera 

sed and in the fiasion procees an oncrgy of tha ordor of 

be liberated, 

for a long t b o but it has been co 

i%iZ;y aP auch onorgy liberation ha b~en known 

sfon Qccurs when Uranium 28 bornbar d by neutrons, f 

ducts aro radboactivc. 

idorod4 unlikelyr, Actup~lly fig- 

ThesQ nctivitios have been observed, but for 

D. oonsldorablc t ims thcy were not Intorpretod RS due to fission pro- 

ducts bocaUsc of thc prcconcoived idea that the occu6renco of tho 

,

I 

fissiori process is practically impossible. 

44 

Finally Hahn observed 

that on@ of' the activities is definitely due to Barium. 

nucleba has only little more than half the mass of Uranium, it be- 

came evident, thgt tho Uranium nucleus did break into large frqmcn2;s 

Physicisbs the8 started to look for highly ionizfng particlast 

Since this 

GoULd be predicted thai fission di.J.1 give rise to sudh particles, 

because the great amounts of energy llberated in the process must 

evantually be dissipated in the form of ionization -processes+ The 

highly ionizing particles were easily found by letting fission prom 

ceed in an ionization chamber in which the ions formed are drawn out 

by an electric field. and collected on the electrodes. 

Et 

The pulses 

and corresponding ionizations obtained w0re m m times ~ stl?onger 

than thoso produced by =-particles 

most strongly ionizing particles, 

which up to that time were the 

A picture explaining the fission process is based on the 

droplet model ,of heavy nuclei. 

posed to be held togsther by surface tension, 

like an elastic skin. 

Liko droplets, heavy nuclei are sup- 

Surface tension acts 

It can be explained by the tendency of the 

constituent particles of the droplet to get in as close touch with 

each cther as possible, 

many contacts as they would in the interior. 

On the surf'ac'e the particles do not make as 

Thus a surface means 

ve potential energy, ancl there is consequently a tcndono-y 

to Form as small surfaces as possible. 

The charge af the nuclei acts in the opposite manner from 

surface tension in trying to gull the nucleus apart, 

As long an 

the droplet is spherical (as shown in Fig, 2) the charge, being 

symmetrical, will not be able to produce motion. 

If, however, the 

nucleus assumes an ellipsoidical shape, tho charges will accumulate

the droplqt 

sion can be 

charged nuclei 

.tron capture, 

larger tho charge, the smaller elongation w ill cause 

to break up4 

This pi tum explains therefor0 why fis- 

+ 

expected wlth greatest probability in the most highly 

In UranSum and Thorium, fission has been induced by nsu- 

If the neutron is captured, its binding energy be- 

, come@ available, This energy is a8 a rub eventually. emitted in 

the f'om of ?-radiation, For that diatlon, however, consider- 

able tbe i s needed. 

The first effect oi the binding snergy is to 

set the particles within the nucleus into motion, 

may sot in and may pivs to the nuclmm, temporarily, an elliptical 

shape which 2s well known to 

Then fission may re~ult~ 

234 

isotope u 

Uranim has two pri ipal isotopGs, 

is ve~y rare). 

An oscillation 

CUP in the oscillation of droplets. 

and'U235 (a third 

The u238 nucleus ia the parent of the 

a 

Radiuln series, 

nuclod undergo fisiion, 

u235 is that of th@ Actinium ser2~s. Both these 

In u238 the fission has a threshold of a 

littlo mope than 1 M.RcV~ The PPssSon cross-section, 

rf, varies 

with the enorgy of bombarding neutrons as shown in tb?e following

figure . 

I 

r 

4 

46 

Fig* 4 

The threshold energy Ls the amount of enorgy needed, beyond tho 

binding energy of neutron, to carry the nucleus beyond thc stability 

point, that is, to give to tho nucleus a suFFiciently ellip- 

soidfcal. 8hape. 

Since this energy I s only needed to overcome a 

potential barrier, tho questton arises why by tho leaking through 

process fission does Mot occur spontaneously. 

Tho answer is two- 

fold: 

first, the barrier I s high and broad; End socand, spontan- 

cous fissSon does occur, but at an cxcocdingly S!.OW 

Orr c 

There 18 no threshold 3.n tho fission u":'""', 

rato, 

mhis nucleus 

obtains by nautron cqturc an ovon number of neutrons and sincre 

nuclei with an even number of nwtrons have a stronger binding 

energy, mor0 enor8y is liberated in this noutron capture procmss 

than in the capture of U238. 

Therefore, enough energy is available 

to 8tmt the fission procsss ev~n if' the neutron brings no kinetic 

energy along. 

Fission in Thorium bohaves in a way which is similar to 

fission in u 238 , 

The ,@-activity of the fission products may bo explainod 

a 

by considering the plot of tho stable nuclei,

oqual fraponts, the corresponding points indicated by tho solid 

line are considerably above the roglon of stable nuclei. The fiesion 

products will then get back ta that region by a series of 

a -decays, indicated in the figure by small arr3ws- Many chains 

of this kind have been found, Most of those activftics were not 

known previously. Rowever, soma oF them near tho end of the chafns 

and 1yIng close to the region of stability are known artificial 

activitfos, These activities have holpod to identify tho mass 

numbors of these radioactive chains, The atomic numbers could, of 

course, be found by invostigating tho chemical propertios. 

While thc product nuclei in fission haw approximakclg 

equal weight, there is a marked tendency for a slight dissymmetry, 

Tn tho most probably f'lssion process, tho ratio of massc8 of' tho 

flssion products is distinctly diffcren'c from unity. In spit0 of 

many atitempta, no explanation has bsen found which is quite simple

49 

v\rh$gh the nunbsr of atoms has decreased to 1/2 of .their original value. 1% is seen 

from the radioaotiae decay formula that ecm l/~ Also one has 

from which it follows tha% 

AT z h 2 

= ,6931 . 

The designation mean life os 

is justified by the fact that it is actually the 

average of the time an atom lives before decaying. 

In fact the number of' atoms 

whidh have lived for a time between .t; and t 4- dt is J!J(t) Ad%, Multiplying by 

t, integrating from 0 to a, and dividing by the original number of %toms, one 

gets for the mean life 

a 

L/?T(O) tN(t)A.dt l/n 

0 

It; is easy to verify tihat this expression is 1/A 

3 

which by definitios is equal to 

The number of atoms decaying per unit time is defined a8 . activity. 

...i... .- 1% 

is equal %a. N(t>a , Ro-t;ivities are mensure'd in units of curies. k curie corres- 

ponds to 3.7 x lolo disintegrations per second. 

The reason for the choice of this 

numbor was that it was supposed to be tho ntxnbor of disintegrations pcr socond 

occurring in one gram of' radium. 

Re0en.t measurements seem to show that the number 

of disintegrations In radium is actually somowha% lowor (the latest valtw is 

3r46 x 1O1O dfs/sco). 

But it is better to retain tihe number 3.7 x 1O1O dis/sec as 

a fhod unit. 

In modicino a different meaning is often adop-bed for the expression 

"radium equiva1ont.l' 

Thus a milligram radium equivalent may mean an amount of! 

substance whirth gives rise bohind a 10 m load shteld to the same density of 

ionization as would be caused by 1, mg of radium. 

Tho reason fur such a dsfinit-ion 

e pOwer, 

Tho change of 

the amount of radioactive substanco with time may be quite 

is that the biological effects of radioactivo su.bstances aro dm to their ionizing 

complicated if tihe radioaotiva substanco is itself a product of radioaotiva decay

. 

~r' 

I 

Own moxe particularly if tihe radioactive, stibstance is a member of a long radio- 

'How complicated such chains may bocomo can be il.lustratod by tho 

example of bhs Urabium family* The following tablo is esseylfially takon from tho 

hemistry and Yhysios, 27Lb Ed$tion, page 315, 

Tho da6a given in *he handboak are %he results or' an intort&ianol 

agree- 

QS+ 

available information in 1950, In the meani;ime, ohangas aro 

hese chatlges have been incorporw6sd in tho following tabla. 

These are 

-P-"f - 

---- 

Uranium 

changes neodod, 

JN"W.TIQiL 

-- --CrUCC-..ICII.*rY.-"--- 

TABLE OF TEE RADIOACTIVF, ELRIIRRTS AND THRXR CONSTAPITS 

-".-*I -., +- -.-.- 

J$arne 

Series 

--I- 

Symbol 

R ~ l f period --.-.-- '$ Radiation Iso-bops -"- 

Uranium I 01. 4.56 x lo9 yrsI a, v 

Uranium X1 u41 24.6 dGyS 

49 Th 

Uranium X2 

u-xz 1,15 mine 49 PR 

Uranium IT (Brovium) 

UII 

2,68 x lO5 yrs. QL, U 

Ionium 

Io 6,9 x 104 yrs. U Th 

Radium Ra 1690 yrsr ad Ra 

Radon (Rmlium omanat%un,Niton) RQ 3.85 aaya Rn 

Radium A ' RWA 3.0 min. a Po 

Radium B Ra-B 26.8 min, 43 Pb 

' Radium C Ra-C "35 min 99.97% 4 B$. 

Radium C' Ra-C' ~4.4 10-4 s80 a Po 

Radium D (Radiolead) Ba-D 16.5 yoars 4 Pb 

Radium E Ra-R 5,O days 49 Bi 

Radiurn F (Polonium RWF 136 days Cic Po 

Radium fi 7 Ran' I O + V e * V 8 * I 

qe)e.eae Pb 

Pb206 

The ohango in tho number of' atoms N1 

Ra-C .*Cia.***. ;03'$u* Bi 

Ra-C" 1-4 mine 8 T1 

- c 

Ra SZ" 4 e e s *#e;**** Pb 

1 

of tho firs+ mornbor of p. radio- 

acrtivo sorios oboya tho s'implo differonti41 decay law monkionod in tho boginning 

af tho lecture 

1) 

wharo A1 

in 112, 

dIV1 = - XlNldt, 

is tho docay constant of the first moinbar of tho sorbs, Tho chdnga 

bho numbw of atoms of thn second member of tho series is 

dflz $Hid% - &32dt

51 

@rw of the right hand side i s the inoreaae in +he number of atoms N2 

due $0 the deaay gf tho firs% member of tha 

wrih for the khird member 

tions hold for furtiher mombare of the series. 

The system of equations 

81 4 &I-,, e" 

a33 

,-h3t 

ents all, "21, a319 etc*, may be dstermirned 

the sxpreesions for N1, N2, N3! 

into %he dif'feren-bial equa.t;ions, 

ha time rata of ohango of these quantities, and second, writing tho 

for NlI NZ, N3# @to,, for t= 0, and equating ttheso quahkities 00 tho 

a1 amounts of the radionotive subdtancas. In this way a nectoneary and 

suf'f%oleM'b numbor of squations is obtained to find all tho ooefflic 

' 

far 

%im@ dependoms of Rl, N2, @toI, ikl 6ome of tho oases of p 

tanoe in %he Ra family are tabulated in the book of 1bej. Curie. 

conseqwnce Qf' 

brim. If'., eeg,, a uranium solut 

the equations discussed is the 

is IAft standine; 90r a tilw long oompared $0 

of tho fir& 

daugh.0 

wry skate w411 establish itself in which the smo numbe 

and are formed por unS time. 

This is t 

nl, or if the atzio of the nutnbor of uranium atoms t o tho 

1 atom, 

same as tho rabio of the period of tho uranium aOom 'bo 

This simplo concop% of radioaotive equilibrium is basod on 

ho origivlal mothcr substanca U1 has tag exceedingly long xifa and i ta 

e noglootad during tho growth of the daughter subs'WwQe 

If U-XL 

is precipitatad out oP o. tiraniurn solution, all tho U-XI will bo fgrmed in the

preoipieate and none in the solution, 

52 

The solution, however, retains all of the 

~ * b substanoe r 

U1C 

a 

deony in the precipitate. 

figure, 

Subsequently, U-X~ will grow in the solu+ion, wliile it will 

This growth and decay are illustrated in the following 

U-Xl activfty inprecfpltate 

J 

U-X1 

activity in solution 

The decay of U-Xl 

-t 

in ths precipi6ate follows the simple exponential radioactive 

decay law, The grmrth of' U-XI in the solution can be obtacliimcl from the fact? 

that ohcirjcai $cy+Ir2-*

e 

53 

method is used for determining tho half-life of radium. In this case, however, 

the oldest radium samples (about 40 pars old) begin to show an adtivlty slightly 

smaller than they had originally, 

Periods in the range betweon a few seclonds and a year may bo cohven- 

ielntly dstormfned by following Srho change of actifvity with time, 

Most; of tho 

known artificial activities fall into this rango, 

This is duo to tho fat% 'chat 

activities with very short or very long lives are hard to dctoc'c, 

Rvon where 

the simplest determination of life tima is feasible, a prooiso dotormination of 

tho llfa timo is difficulfj. 

Thus tho bost known poriod ol" a naturally activo 

olcmonk, that of rad.on, is known to a precision of *05$, wliilo .the poriod oi' tho 

woll knovm artificially activo nuclcus P32 is known to .2$, 

The usual accuracy 

of half-livos is less than 1 or 2$, 

This cxporimontal unocrkainty may givo rho 

to cons5.d.erablc orrors I% amounts of a radioactivo substanoo aro to bo calculatod 

from tho docay formula from moasurcrncnts taken long aft-cr th'ni: timo in which onc 

i s intoros+odr 

Special method6 ar,i nccdod to moasuro poriods *-diioh arc shortor than a 

second. 

One motlzod axpplicablo for radioactivo gasos is to let tho gas with a 

known vclooity strcam 5 , o. ~ pipc, past a sorios of collecting clcc-brodos, as 

shown in %lis figure. 

collecting olcctrodcs 

Tho ourront collcctcd by t-hoso oloctrwclcs is proportional to -the ionization 

caused in their neighborhaod by the radioactive gas and this in turn is proportional 

to the number of radioaotive atoms remaining in tho stream. 

From this 

measurement and the known volocity of %he gas, the dQCQY period may be calculated.

A similar method has been used for known gaseous substances. 

But 

54 

instead of the velocity of the gas stream, tho recoil v9locity of the radioactive 

atoms was utilized, tho recoil being due to the a-emission by which the short 

lived subs'cance was formed, 

This method is not reljable bacauso instead of 

single atoms, groups of atoms or crystallities may recoil, thus making elre 

recoil velocity uncertain, 

A more suooessful method utilizes ooincidence counters. 

13-coun 

counter 

coi L;+----l 

cedeno6 circuit 

Let us suppose that a parent substace emits 6-rays and tlioreby transF6rms into 

ana-active substanoe of vary short periodo The substanoct i s placed nexti to an 

and 60 a &-.counter and -the counters are connocted to a coincidence oircuit 

which will give counts only if the two counters aro activated within a time t, 

apart, , Than .tho number of coinoidanco counts to be expcctod i s 

"& 

Here Cmax 9s the number of counts to bo expeotod if tic is chosen as a very long 

-At 

time. This quaritity is multipliod by tho intogral of tho activi.ty Aa Of 

unit substance owr %he period t,, Thus C gives the number of counts obbainod 

ff only tlzoso &-disintegrations are sffective which occur aikhin B time tc 

aftor t;ho @ -disintegrations. 

Rapoating this cxparimont for various times tC 

e 

one can find from .tho variation of C. This oxporimcnt has bean onrried out 

for the Ra-C -1 Ra-C' pair of radiosotivn SLibstanccs. 

It is claimod that; thSs 

mothod is capablo of mcasuring half-lives ~C~WOOM 10" and low7 scoon'&s.

$. 24 (9) 

55 

Lsoturer: 

E* Segre 

One applicaC5.m of kke radioactrive decay law is tho de'aarrninatioa of 

anium oresI If 

originally no lead was prosont in the o m and if tl?coro has remairzod undia%urbed 

shoe its form&ion, the lead formed by rildioaotivs decay will give a meamre of 

has passed since tho famation of the ore, Another similar method 

s the helium formed by raclioac;tjive dacay and remaining in *he ore From 

. 

(I 

P 

t e age of tho old@$% oras, th.e preaumablo 

Values of tho oraor of 109 years ~i.avo bQoM obtained, 

of the earth can be d&@rmlned* 

ivo moasurcmen'cs aro usually made by countj.ng disintsgratione. 

ooourring in those caun+bs in'woduao errors into bho moasuramontst 

nvasfiigato tho influanoo of tho 

Tho simplost case is if tho eubstafice undor investigation doos not 

rociably during the fimo of .the mcastro 

shall QOY18idOF suoh a 

uma khat tho aoti 

ha probab511by o 

w corresponds t 

in a second Inahad of the 

o find pn wo subdivibc tho scoond in&o 

ivision fino anough 10 th& tho probability of find'sng 

-.-IC---, 

I -.u"I 

aioulaLlon o f those probabilikios

* 

oaunb irr on0 subdivision 2s nogliglblo. To find OW oounti in such a sub- 

div5sSon has a probabili'by m/k, To ind n p~rtiolas ilz kho first n divisions 

nono Zn %ho f6Jlowfng k-n qubdivisloii has tho probabili-by 

(m/k)n (1 - 

56 

counts distributcd In any manmr Rmong tha 

e 

k intervals, wc must 

k 

is last oxprossion by 2rlio binomla1 caoffiaicht (n> i*c., tho number of 

ways in which n intQrvals m n bo choscn among k. Thus wo obtain For pn 

I 

bcaomo s 

, gn z (k) (m/k)n (1 - m/k)lrc' 

ot' k go t o infinity, t;hc first facOor in %ho abovo oxprossion 

(E) 1 

wharoas tho last wil 

-m 

k(k - L),*.(lc * ~1 1) -,kg 

nr 

k-+ ns 

c^---r-..ir*-il-rr*I .-..u--. 

-- 

tho formula simplifios to 

ynn 

Pn : - Chm 

11 s 

lcnwm as Poisson's formula, Xn thu fallowhg figure R plot of p~ 

8 dofinod only for in.togra3 

tcr p$otting thosc in"egrn1 valuas ano can Snterpola%o tho 

ro@t of' Iiho ourvo, 

It ia usual in dc+orm%nmtions of half-livos to ploG tho 

occurronco pn of a glven numbor of counts.por socand n aga:Lns% 

n and to cornparo this plot with Poisson's formula, lf n Can bo 

Sit, one lias khoroby obtained tho doshwd

value of the counts par second and has at %ha same time oheoked thu propr 

57 

1) 

sta-bistical functioning of tho coun*arr 

If m is vary large, the plot of the Poisson formula 1.001~s as IndSoa-bed 

in Glza follawing figurer 

pn 

1 

m 

The ourvo has now a rathor symmetrical sharp maximum around the value n * m 

and goes quickly to zero when the dovia-bion of n trom rn baooms appreciablev 

In Chis case, Polssonts formula may be replaced by the Gauss approxhcition 

1 . 9 *-(n - m>2/2m 

-i&Gz 

Pn a * 

This formula can be clerZved by assuming %hat 

M .. m is small compared t o m, 

using the Stirling agproxima$ion for %he fao%.t-orial, baking %he logarithm of Pn 

and expanding into a Taylor series near "Clie, maximum n me 

The wi4th of the curve in tho above figure oan be defined by choosing a 

value w. in such a way that one half the area under %he ourve lies in the reg$on 

whara In - rnl is smallor than w. This means that +he deviation O f m from n 

has the same probabili-ty of being greater or of being less than wI 

ah 

- 

be shown to be 

w 0.6745 

Ahothsr hteresbing quantiity is the average value of (D=m)2. 

The value 

Thia 5s 

* 

denoting n 

gPn (n - d2- 

m by x and %he average value of its equars by x2 and, roplaoing 

by an inttngral (which is psrmtssiblo for %he Large rn values oonsidered hare) 

II.

a 

we obtain 

a time 

t, we have an even chanoo that by repeating the measurement many Cfmes 

the kverage result will be betwen m - 0.6745Elfl;;l and. m+0.6745 4- m or outside 

that interval. 0,6745 m is called the probable error and m the standard 

4- 4- 

doviation. Tho standard doviation increases with m, the relative standard 

deviation F / m = l/llfir decreases with m. 

If wo count 100 event6 the relativo 

standard deviation of our measurements 5.s lO$, if WB count 1000 events the r o b 

tive standard dsvia%ion is 3.1$, 

cCc. 

It is sometimes important to know the probability of an error cqual to a 

multiple+ times tiha standard devia.t.ione 

Theso probabilities are tabulaked, ocg, 

in the Handbook of Physics, 27th odi-tion, page 199. 

The quostion. sometimes arises what conolusSons can be drawn about tho 

docay constant if one finas no counts during tho timo of observation. 

I+ foLlows 

from Poisson's formula that the probability of finding no counts in an interval 

ia which m 

counts should hava bocyl oxpocted is 

po = e-m 

A closoly connoctod quostion is that of the ooincidenca corrections to 

be appltsd to obscrved counting rates. 

Lek us assme that a counter takes the 

time z to rocovmo This means khat af.t;er a count, +ha oouvftar remains insen- 

.'* 

sitivt: €or a period z If' we observe a counting rate mexp , the true oounting 

the exponential factor representing the correctiotn due to the :?act thab counts 

are missed if they follou eaoh oCher at intervals smaller than z e Assuming 

that the time z is short;, so khat the probabili%y of missing any count due to 

the above qffeot is small, one finds that the correcJbd counting rate mcory 

2 

is

expresaed 'In terms of the experimental countitng rate moxp by 

59 

Mora involved prob~erns arise .if the substance decays appreciably while 

the experiment is performed, 

One praatical problem is to find which pariods of 

measurement are best suited for tihe d8tf3rmfnatiOn of the half-life of the sub- 

stanoe. 

One might guess that it will be besf to let the subsl;ance daoay for man&' 

half-lives and to take advantage of the great change in counting rate which has 

accllrred in the meantime. IIowever, if one waitis too long, there is too grsat a 

loss of accuraoy due JGo the reduoed counting rate and tho corresponding greater 

influenoo of 

fluctuations. 

In the following figure the logarithm of the oounking 

rate is plotftd against the the, 

t 

+- 

-t I 

.c. 

Acoording t o the radioac-bive deoay lavr this plot should be a stira%@t l%w, 

Zf 

one detiemines ifs value at points 1 and 3 one might expect a graater acourag. 

in the slopo. 

But this 5s true only if the moasuremont at 3 doos not bocoma too 

inaccurate as hac been ipdicatod in the figuro by the vor'cical line. 

It is found 

actually $hat i-b is bsst Go choose tho %vo points 1 and 2 at which caunting rates 

are-dstorminad one, or one and a half, half-lives apart. 

The fluotuationa whioh we have discusfwd am a consequence of tho radio- 

actim docay law, 

It is thcreforo of fundamontal intarest to sco whoChor those 

fluctuations cmform to tho predictions, This was found to be tihs aase whoncmr 

0 maasuramonts vmro porf ormod with tho nooassary prooautions. Oyto mistake whioh 

frequently has led $0 spurious doviations from tho prodictud law OS fluotuations

ia to neglock Ghe coincidence correction of the counter, 

60 

I) C-.'.---W^----- 

IIIa PASSAGE OF ?ARTICLES TFIROUGH PUTTER .--u 

--- -part iGle s 

IBasuring deflections of gE-particles in eleatric and magnetic fields 

RuOhorford had established that a-particles are helium nuclei, 

The charac.teristic property of the G-rays in which we are hsre inter- 

ested, is their defihite range. 

This mealzs thaCa-par%icles of a givon velooity 

(for instance&-particles emitted by a thin foil of polonium) w ill traverse a 

given distance in air before ba'lng stoppod. 

This can bo domonstrated in the 

7Nilson cloud chamber, 

The Wilson cloud chamber is a chamber containing satwated vapor. 

Sudden 

expansion of the chamber causes superr$atura+ion in consequence of adiabatic 

doolingc The droplets ivill then condense around ions formed by the passage of 

the &.-partiole khus marking the track of %he &-rays. 

The .tiracks seen in the 

Vilson chamber are of qukbo uniform lengthe 

The stopping 0% C;c=-particl~:s is due to collisions with elac-brotzs. 

Since 

the mass of the electron ja about 7,400 .times smaller 6han that of the 

&-parliiclo, 

tl?~ cz; -parti.clo is not deflected by such a oollision appreciably 

and its path remains visi.bly dxaigllGe Xowever, the enorw lost in such 

co$llsions accv.inll?ates. 

1% occurs somctimes tha% an c\; =-particle ooll.ides with a 

nucleus and suffers a sti-ong deflection. 

Vi1 s on chamb r 

Thcso events can be easily seen in the 

The i'ollen~ing flgum shows the density of' ionization j along the pa%h 

abssi .ssa x 

I 

is the clj.stanoe 

i ,'

travelled in air, The above curve is called the Bragg curve. 

It s last and best 

detemination 9s due to Livingston and Bollovmyz). 

Close to the 

end of the rang 

the Brag curve a%tains its mximum, making about 6,000 ion paths 

pes mm of 

air. 

At tho end of the range tho ionization declines sharply iio zoroI 

Ian$zati.on in air 5s accompanied by loss of energy. 

Tach ion path oorres- 

ponds to an energy loss of about 32 eV, This figure is fairly indoponden% of 

%ha vel.oci2;y ai' thecikpar'tiolo. 

Thus the Bragg curve gives no% only a measure 

1 

I of %ha ionization density but also of tbe onsrey lost per mm of path, 

1 It is of groat intorost %O fihd out the energy loss ofc&-purtichs in 

other w"cerials khaii air. 

Thus iZ; is important to knw %he energy lost by the 

a f s vthen eutering a chambor or oounter through a thin foil. 

The onerig loss or 

stopping power i12 various substwoas is host measured by giving the vaJ.uo of the

* 

Octobor 14, 1943 

LA 24 (10) 

62 

LECTURR SERIES ON NUCLBAR PHYSICS 

__I-- rr.-nrri..-.rr2.-"., .e. .*IIyu-II-.I- ....-- 

t 

$ocond Scries t Radioactivity Lecturer! E, Scgre 

LECTURZ X: 

n --.&--.-&"-------- 

~*PARTICJ.,ES AIfD TIiEIR Ii'JTRRACTIOM WIT73 liIATTER 

-.*-I -*--I*..-.-.--*------ 

In cloud chambcr photographs of ana-omitting source, it is ovidant that 

tho tracks oxhibit a, more or loss charactoristic longth, known as Gba range 

(v. Rusctti, p. 303). 

In tlic main, the onorgy of thc &-particlo is dissipatod 

in collisions with olactrons: tho path of liho Ctppartic3.o is not dcf'lcotQd by 

them many cncountcrs bacause o f t-ho rclatij.vely groat mass of tho a-partiolc 

comparod t o that of thc clectron. 

TJpon closo inspection of thc trsxcks, Q ~ C may 

soc tho faint- tracks of tho projoctod olcc-ixons, all having their origins in tho 

hoavy track of the drpartic1.c 

Occasiondly ancX-traclc shpws a f ork-likc 

structuro; hero t-hc a:-par%iclc has suPfcrcd a nuclcar collision nrzd my bo 

approciably doflswhcd, .tho pa'i;h of tho struck particle bcivig tho other branch of 

thu Pork, 

A monsu.rcmonti of tho anglcs and rangos (onorq) of tho various 

colliding bodics slwvrs that- tho oonsi:rva-i;ion laws of momcn-bum and cncrgy aro 

sattsfiad. 

Sometimes, one finds that kinetic enora i s apparsni;lg not conserved. 

In thoso cases an inelastic collision or nuclear reaction has taken placOr 

The many collisions of ana-particle wi%h electrons result in ion production 

along its path# The number of ions per u~I.t lcng-bh ts oalled the specific 

8ragg curve) of specific ionization VS r angs is shown in 

the follovring figure , 

* 

range

63 

The speolfic ionization inoraases with decreasing velocity of theaqartiole, 

1) 

attains a maximum near the end of the path and then drops to zeroI (See for 8 

, quanfitafive graph, Livings-ton and RQlloway, Phys, Rev, P 

54 .- 18, 1938,) 

%he 

tlze 

5 10 

range 

lL 

20 

As an example,CZ-par-ticlesfroln polonium ham an energy of 5,298 

EBV and a range 

of 3,842 cmr 

-- .. *-.- 

- 

Stragglinci If the ranges of' an lni%ially homogeneous beap of a-parfiolos are 

measured, one finds valuos dis-tributod about a certain value, the moan range R, 

with a deviation of 

one or two percent. 

A plot of the number, n(x), whioh have 

a range greator tliaiz 

x 

is shown in tho following figure. 

n(x> 

.L 

Tho value of 

x at whioh n(x) 1 1/2 n(0) is oqual to Rm The int;orsoc%ion with 

the x-axis of tho .tangent to tho curve at 3 is tho extrapolated rangob 

31) Tho distribtuion of ranges (tho phenomenon of straggliw) may bo 

oxplainad by the statistical fluctuation of tha onergy loss per ooll$sion, and

e 

64 

s% the ohmge of’ theahpartic16 varies several thousand kimss, 1.8.) 

He’, He, along its path. Those changes, however, ocour almost 

en%irely tn the last few mil[limsts~s of rangec 

\5 

$he stopping power of a substance 2fthe spaco rate of enorgy loss of the 

implngihg partioles as they travc;rse the substBnoe, io@,, if F = stopping power, 

i 

T = kinetic energy of particle, x 

P 

-dT/dx, 

space aoordina~s, 

Consequently the range R ia givon by 

R 0 TO 

R $ ilx = - jOdT/F = 

0 

The ralattvo stopping power of a su,bstanoe is dofined as 

approximately indopondonti of onorgyo 

stiopping power divided by the density of the substanco. 

mnp -I.-mCrC- in air 

rmgo in subtibst;ame’ 

Finally, ?;ho mass stopping power is tho 

The mss s2;opping pmor 

s’d 7

7 

e. 

65 

scatterer at ona fogus. The ohange from tho iniiifal direckion of approach to *he 

final dirowbion of the par”cio1a is %he so& ing angle 8, The hypo.tjh&ioal, 

distance of closest approach if the parOicJe were not deF’lected is tho so-callad 

impact parameter, b or tiha above figure,

* 

LRCTURE 

OoBober 26, 1943 

SERIES OlT NUCLEAR PIIYSIGS 

L-c- ., *- cu- 

- L 

cond Series: Rgdioaotivity Leoturerr F. Bloch 

LECTURE XLI TIT? TEEORY OF STOPPING POVP3R 

-Cw-.,+.”UlrUICUII 

_..I. I. --a- 

~ircrs*’hw.-urc-.cI 

We shall calculake the energy which oharged particles lase when pas 

tter. 

This onerm 106s detormines JGhs range of chargod particles (ergr, 

es or protons), 

Tho onergy loss 

s also olosely &nneo%od wSQh ionlza- 

d by the charged particlo in %ha substanoa through which it p1as0sC We 

shall first discuss the Cheo of st~pp!ing powor as givon by Bohr. Hi 

only tho sirnpkst comcep+s of c2a 

ioal maohagioso 

WB consider B particlo with oharge Zo and volooity Ya 1% ixzteraots 

wiOh an atom. 

llore particularly we shall examine tho interaction with ona 

shckron. ot chicgo e and mas8 M wrt.l-hin that a-koni, Tho e1oc”cron shall ’bo 

an aguil!,brium position with olaetic forces and shall haw tha vfibra- 

tionax Troqucncy &$.. 

The olootron was originally at rost. 

1’ro shall, aaloulato 

the amount of cnci’gy AT that %he s2;ron ob$ains durfng the passage of the 

lo* Tho charged partiole % thon have lost 2;ho same amount of onorgy, 

it has changod i%a onergy by -AT. 

In tho ffguro, tho pasit-ion of tho clcctron la show by tho point 

I

Tho rnin5,rp.m 

6 '7 

value of %his distance, that is, tho dis-banca of closest appfroach, is 

equal to b, JGhlB distance is called %he lision parameter. The angle 

inGluded by the path of the oharged partiale and by 

r is oalled $, Wti shall 

system whoso x direrottoll ooilzoides with the path of the 

a-partioxti and. whose y diroabion is perpendicular to it-. At ths poiht of 

olosast approach of the charged particle t o the electran wo set x s 0, 

We 

furthermore assme -khat for this point also t = 0. 

Then at all other times we 

have 

x m vfi, 

Ifire shall solve %ha problsm of tho energy %;ransfor to the elea%ron In ati 

approximate but very shpljeway. 

The approximation to be used requires; 1) that 

the eLectron be considered 88s al.ly as free during the time of collision; 

a'b the displacement of the electron during %he collision shall be amall 

to the distanoe of cLoseslt approach b. 

In ordar to soo wlmther the first assumption is valid lrm shall consider 

$he y-compcwn!; of %he force af interaction 

F = 

"& 

r2 

of this component i s zero at t E -mand t tcX) e 

It has a WXimUm at 

B width of %his maximum (%ha* is %he %imo during which ElJr is more than 

s maxArnum valuo) WB oar1 tlzs collision .time 

The condition that 

the elecr-bron may be considered as fr 

is fulfilled if Z is small compar 

*he, perrod of vibrat 

Q 

l/a 

Indeed if' tl7i.s is tho case the electron mows 

collision through a dis%anco which is small compared to 1Gs vibra- 

Gional ampli%u&, i.o., the oleotron remains close, to its equ$liW.um positioh, 

librim position tlza elastio for~os are small, *hay may be 

elQGtTQn may b9 aonaidorod a3 

TO shall diSOUS9 *he

* 

the 

68 

validity of our second assumption later, 

Tn order to caloulat-e AT we shall first compute the momentum given t-o 

eleciwon by the charged particle. According to our second assumption, %he 

alegtron has moved a small distance compared to 

b 

and the force will be prao- 

e same a6 though the eleotron hqs not moved at all, 

Under these con- 

dition@ one sees that the net change in the x-cbmponen*b of the momebtum of tho 

electron, dp, i s equal to zero, Indeed tho momentum in +tihe x direction that 

the electron has obtained while the charged particle was approaching, will be 

cancelled by a momentwn change in the opposite direc.t;ion thb$ occurs while the 

charged particle recedec. Thus the only net momentum change is in the y-dire&ion 

and after %he passage of %he charged particle *he electron is left vibrating in 

that direction. 

This momentum change 

pY 

is the time integral OS the y-oompanent 

Sub s t iky0 tlvlg 

one finds 

1 

sin2 $ 

dt* kx --- 

r = b/sin $j>~d v 

Ap, ' = ;+-Ze2 5 

Irp 

sin $ d $ = 

2Ze2 

0 

The corresponding &ange in kinetic enorgy Is 

1 2 ~ 2 ~ 4 

AT ='z;;; IPx2 + ApY2) 2 - 

lnb2v2 

\ I 

La holds as long &s b,inc: bg bmBx* If' b 

s mado above are not valid. 

is ou*side this region, *he 

We shall calculate bow the anrgy lost by tho particlo when passing 

J 

through a distance &x, Lot u.s assme that them are N cleo%rons per conti- 

motor oubed. In a cylindrical shell of length ax, inner and outor radii b and

e 

69 

b + db $here will be 23Fbdbd XN e9leotrons. Tho energy transhrred to these 

AS = Rhx 

7 --c- 

Bfl'zZOB 

db c Ndx c- 4 z%4 log % bmax $ . 

'ET7 b ' m VZ rnin 

bmin 

(i 

From th5s squal;ion tlm energy loss per aentiiineter &T/Ax 

may be obtaifiad. as 

soon as $he values of bmin and bmax arb dsbmnined. 

DiscussSon of brain 

--*- 

For b - 0 our approxbdto formulae would give AT a00 which is 

Actually tha maximum volnoity a Fxee slootron may obtain in a colli- 

v and therefore tho maxiinurn energy loss i s 

22 

I 

r 

G 

I 

ose b in suah a manner tlmt %he calaulafed onarc3 loss shall bo equal 

imal energy loss, %hen $his value ~f b 

will mark the limit 'beyand 

whiah the approximate value of the energy loss can no longor be used. 

At %his 

1 

value of b, 

the assumption that tho electron has been displaced during %he 

through a distance small oompared to b is no longer valid, This 

b we shall choose as bmtn we obtain 

2 z v 

3 

m 

(2v)2 = v- 

i 9 m b,in2v2 2 

(&vr = 

I-. 

ze 

a- 

,* binin mv 2 

, A treatmen% which is rigorous for small values of b u88s the 

d farmula for the soatitwing of the charged partiole by %he ehotron.

In the oase 

,>< 

thab 

mv 

70 

the second JGerm Liz the denominator oan bs neglsc 

and the integral reduces to 

the simpler form which has been used above. At small valuer$ of b the Inbgral 

goes to zero. 

This eives a qualitative justification for using %he simpler 

integral 

b 

for outting off the irrbegration at 

P 282 

bmih 

mv 

D ~ S C L I S ~ of' ~ bmax O ~ 

I*^.. -..-," 

c- .%-,.c** . 

-L. 

For too large values of the collision parameter 

onger be considered as free. In fa&, coll'lsl,on time is appro 

and this t5me will become great 

l/&J when b becomes suf.Piciently large, 

vibrational. period of the 

For large values of b, that 

is, when b/v))l/W 

the variation of the Toroe of interaction botwesn charged 

and eleotron is slow compared to .t; 

tion between tihe charged pa 

18 cnd %he eloctroq may 

treated by Che so-called adiabatic-, approxi.mat ion, 

Ih this approximatLon, ths 

is displaced whsn %he chargod particle approaches but whan 4.5, 

Ipeocsdes 

ron is eased baok Into its original position without- iti taking at$ 

i 

reford ju.stifio 

ur integral to the poilit where 

and to neglect oontri greater b valuos, Thus we have 

b,in 

into tho enercy loss formula 

obtained Bbove, one fhds 

* been 

c-dT ;1N 

dx 

sd out this formula fs un proximste. The o 

carried ou-b by Bohr rigorously and he obtains 

2. "N.--- 

4 e $e4 

log 

1,123 mvz 

--.r-.*C L 

dx mv2 Ze2U

0 

Wcro n donotw tho nhboq af 

YIO 

may us0 tho samo oxprossion 

73 

atoms por aubio cantimotor, For tho uppar limit 

as vms dcrivod in crlassioal fhcary, namoly, 

hmvovor, must bo kangod, This classical. lowor limit has tho‘ folloying s2gnif9- 

onaidor tho kinetic onorgy ‘of tho cloo2;ron in t hs framo of 

i 

%ho ohmgad partiolo ie at rest, thon bbin Is the distance at 

he pokential energy between the eleotron and 6he charged parflcle becomes 

o tha-t; kfnetiio energy, 

electron and charged 

Now in quant;um neohanics a closer approach 

article than the de Broglie vmv~ length is meaning- 

WQ have to use therefore @or the lover 1imi.t; 

i 

bmin (quankum) = %/m. 

’ Thi indeed so if tha lowar limit- in guanhn theory ha? a highohor value than 

the previously derived classlical lower 1j.m32;, becauss a5 “.‘1$ approaoh closer 

%han %he quantum lwier4.irni.t defraction phenomena will OCCW~ and make an effeo- 

tdve approaoh impossible. 

If hotmvor, 

%he olaaaiaal 10~1er-12rn.S;t; should %urn 0u.t; 

to ba greater, then down to %hat lower 

limit, classical theory 3s applioablo and 

be UBOdr 

Thus, %he QlaSSiCal or quantum 

sd according to afhethher tho ratio 

r small compared to uni%y. Assuming Ze2/1;% = 1, one ob%

* 

which factor does 110% 

dotailed calculation. 

74 

follon from o w argwzhnt but must bo obtained by a more 

Tho stopping power just g h n is valid If tho following throe condi- 

a 

tions are fulfilled: 

(4 viua>> ro 

(b) Z@2/*V{{ 1 * 3. 

(4"2))flOj. 

Condition (a) means that at the upper limit where wo have made use of the f'rc- 

quoncy of the oloctron, tho distanoe of' tho charged parkiclo of tho atom should 

'be great compared to the atomic radius 

ro+ 

ThSs condiAiion i s nscegsary in 

ic field of tlze charged particle should be homogeneous 

the atom whenever %he oscillakos 'urea-tmont of the elaotrou has been made use ofr 

Unless this condition is satisfied, the analogy to t5e interaction with ligh% 

rea-bmeylt by virtual oscillators breaks downo 

Co;iclition (b) mem~ thati 

%he quantum Sower-Xfrnlt i.s pater than %ha classical 1o:mr-lj.mit 

and that 

therefore Clm quan-bum 1m*mr-lImi7b should. be usedr Fina!.ly, condieion (0) 

expressss the requirement ti22a-b the upper limi.1; of 

b 

must be considerably 

greater bhan the lower limtt of b aud that therefore there shall bo a region 

in which our aBproximationi are valid* 

ConditIQna (a) and (c) bofh mean n 

ically khat *he veloai.hy of the 

harp3 particle must be great compared to the valo 

0 f -bhe 

with whlch the Gharged particle inferacts. 

For condikian (a) this Is 

fly because 

row is equal to ~ ~ 1 In , condition (c) one may replace 

'h@l(that 

is, tho sxci'bation energy of the electron) by twice %ha kinetic 

energ$ of the electron which is a guanrt,i.(;y of the same order of magnitude, 

Then 

(e) roduces to

which again means that vQ)vel. 

Condition (b) i s also satisfied if “))vel# 

IV 

ele ptrons + 

If v and 

vel become comparable, our treatment it3 no longer valid 

and an exact treatment for this case has no% yet; been carried out, 

The fhoa- 

retical difficulties are paralleled by a complicated behavior of tho charged 

particles a% relatively slow vslooitles, 

If‘ their vsloci-by bocomes equal to *he 

veloci-by of %he outermost electroiis in the sluwing dovrn medium, then the oharged 

par”cclss. my pull ouli electrons from the slovdng down modlwn, attaoh them .Do 

khernseXves, and thus reduce their affootlve charges, 

In lator collis5ons “cess 

charges may again bo lost. 

Nhilo tho chargad particle had its charge roduoed, 

itra ionizing powor and anergy Ioss will also bo smaller. 

Since various charged 

pa.rti.31~5 will spend variaus times in tho staiie of reduced oharge, tlio actual 

” r range 

of %he individual particles will dif%er, This dSff ;renco of range is 

aalled straggling, 

If an t&-particle has EVO. enorgy of 72 lcV, then its valoci%y is tho 

same ad %hit of an eleokron with 10 eV. 

This is %he avei*aqs of the ou-l;ermox+i 

electrons. 

Thus, vc must expec2; ‘chat our formula of s-bopping pwmr for 

&-particles becomes Snapplioable under 72 kV1 ActualLy, sexious deviations 

from the formula start at the somawha% higher energy of 200 kV. 

‘I/he correspond- 

ing limit for deuterons i s at 100 kV and for protons a+ 50 kV, 

From the formula of the slxpping povmr an estimate of the range of the 

particles can be obtained, For e, we replace the energy 10,s~ dT/dx 

hange in kin,e%io energy of 

iole -dBlc,n/dx. 

The formula for the

* 

This 

Forge;el%-Sng about the logarithmic factor, one ubtains 

PQIX~U~& sliws that the quantiky Elcinx ohanges at a constant rate along %he 

16 

I 

path of %lis par%icle and that therefore %ha range should be proportional to %he 

square of the original kinetic onergy. 

Actually, the range varies vti=bh a &omem 

what lower power of the original elnerpj. 

is bhe stopping power formula. 

A quantitative evaluatiion of the sDopp2~g povrer formula requires knnwl- 

edge of all absorption frequenoies and all oscillator s%rongths, 

These quan2;iWs 

iven in an analytic form for %he hydrogofi a%om and in this case the 

requirad calaula+iions have besn carrisd ouG, 

Far heavy atoms oha my ude a orudoly simpLffiec1 sbatis-bical model. 

Thia model oonsiders tlze elec%rons as a gas, %he IC 

o energy or %he e1sotro)ls 

being balanced IIT the avorag9 e1eotirosta”ciu potent 

find the oscSlLat ion frequency of th5.s gas by hydrodyix!ni-r; ca’J.oulat 

uso +ha% froquenoy instead of 

CJi and use trq Ju.,uriZ nuliber of aleow 

an atom Zo 

instead of tlzs oscillator s%reng-b 

The vibrational frequency in tho statiskical 

the ‘tsound veloc.i--by’’ divided by the a,tomlc radius, 

Tho 1),l.t;ruF(+ sacli.us can be shovn 

”. 

to vary as 2, 1/3*1 Tho sound velooity is of the saylle ordor of magialtude 8s the 

obtain 

locity of tho electrons and % 8 praporOiona1 t o Z,L/3, Thus li 

dT 4.fFe4z2 

L-rl 

ckC 7 

lag 

a 

ula, R Is the so-oalle 

Rydberg frequency, that is fqR is tho 

cw- 

*%rhhoaviar &%oms ha smaller atomic dii which %s in contradiohe 

experience about- atomic rad.ii as cus d, IIovevar, in 

the customary definit-ion the atomic radius is essentia 

the orbS.%s of‘ 

the few ou-kermosl; electrons vrhereas in our presen+ disc: 

average distaizcs of electrans from tho nucleus, the av 

eleotrons,

ionizlation energy of %ha hydrogen ationi, and 

ating it, one may 'aajust 5t in suo 

17 

is a numerical factor, Inatead 

way as t o give agreement bet;aaan 

the formula and %he oxparisnce for on0 haavy akom, f instanca for gold. Then 

la describes tho dependence of "cho st'ogping pawror on the a'ciomic number 

sfaotory vmy. 

The formula for tho stopping power of eleoGrons i s tho same a8 the for- 

mula for other charged particles rriOh the tilxooption that under 

faetor '2 cloos ndc appear : 

2 mv 

uo to %hc fact that i mining the lowr limit of b t 

mw leng.t;h musC 530 uscd vrhich corresponds to kho vclooity of' $he atomic elaotron 

e of rsfbronoa wkro tho conbo 

chargsd particlo is at rest. 

of the onooming partiole, as 

ravity of tlint ol@cCran and *ho 

olboity is practically equal to tha 

that oncoming par.l;ic;le i s heavy, 

leu$ If the part le i. s an electron its 

s one half the velocity o 

&lparticlss and of 

oharged oosnSc-ray particles the motion OS the 

ticle must be treated aclciording to rtslativis meohmlcs. This l-fill hnve %m 

) the region in whieh the elea$ria field aots is oontracfed by 

Ths Dim@' during which the 

same factor; 2) %he strength of %he electrio f is j*nQrQasQd by 

Ore, the momentun ah6 energ *trans- 

ron during the c 

airr the same as in unrelativis$ic 

ver, l3he Limits for t parameter b chakp, The vpor 

limit I s horeased by tho reciprocal of 

tion, we oar], go ta groat 

os tho period of the atornic electron. 

In %he lower 

a~8 Length must agdnba used vhich is given by

LA-24 (19) 

80 

LRCTURl3 $ERIE 

c-"L-lrr..l*..Ic 

Sooond Series : Radioaotivity Lcoturer; R, 'Pallor 

--- 

CIlARGED PARTfCLES GilITI-1 rTATTER 

*-"..v-.yL)-..i.-uyCr^w.m.--C~~ 

Tn the previous leoture it vras sh n that, apart fro 

of aner~ loss of a moving hwpd partiole ( 

nu-orsoly proporiiional to % voloci%y Y of the 

is faotor arises beonusa thc momenCum transferred lio an alaotron is 

proportional to the "kime of collision'' which In turn is inversely proportional 

to .the particle volocity. 

Ilgnca n par*iclo 

nga should therefom bo wEZO Expcrimntally, hot-ravcr, 

WE3/'. The d epanoy ma37 bo removed by a more 

tho onergy loss oquc,,tion. 

Jt will '00 raoallod that 

incldont :7nrtiolo nil1 transfor oi~orgy to tho atomic nloctrrotzs $2 i t s voloo 

or than the oloc%rron volooitios, 

Approximhly, tho numbor of olootrons 

om vhich hwo volooj.-tics swllor than v, 

and oar! thorci'oro roooivo 

onorgf from tho particle, is simply proporkfonal t o v (provided v i s in tho 

oloctron velocities). CDYW~VW 

*he ra-b of onorgy loss is n m 

go iq proportio 

0 to noto %ha% tho,ol)or@ loss per 

he, speed and charge of the 9noidelif pnrtiKle, 

Fnrticles of' the saw ' 

ere at n rate proportional *Q square of -their rsspeoCSrre Zfs, 

so 

at n 4 1vN &?partide (2 = 2) has the same range as o. 11fV p 

The enerar vrliiah is talcen from the inaiden'c parbicle my be used to 

~JI sleotron from atom (ioniza$fon), 'or rnv only produce electronic exoitation 

with subsequent om sion of a 1%

e 

81 

in detection of CC-particles), 

Andhher possdble prooesb, and of considerable 

impardanocs in contlsoCion Ivi2rh bl.dag5dn2 erfL"ec6s 02 radiation) is molecular 

n with subsequent dissooiation into atom or ibnsi Takiwrg inti0 account 

these various processes together with seoondary effects, one finds that the 

average energy per ion pair i s roughly 30 eV. 

An approximate measure of the 

biologtoal effeots of radiatiaiz on a part of the body is the amount of enerfg 

absorbed by the particular region. 

There are, of course, qunli%ntive di:?farences, 

becnuso the density of logs is qukh different along pcH~s of varioua j.i!;ipin@.ng 

pnrticles, so thnt recombin?.tion protlesses m%y be of another clznracter in the 

cam of, shy, a heavily ionized 4L-fraok fis oompared to a veak fi-track, 

Ianiz 

, --.*- 

by heavily charged partfolesb 

.---.-..-- - -I- 

We have seen thnt the Bra@ cur~@ of nn a-partiale, o. measure of the ~ 

specSf5c ionization along the pnth, increases slovrly to ~1. mnximuin and. *hen drops 

s1znrpl.y to zero+ rhe corresponding QLITYQ for fis5sion par-3.cles slions only a 

_ - 

in speolfic ionization along its pn-bh. ::hat :is the exp1:in:ci;ion 

the 

di.i.??erencol 

The ini%ial velocity of A fission pnr2;icJ.o 5.s 

apprcxi.lmt@XY 

ce 

One ccn estiinzte it.s initial ohargs because tho fisflion fragmeDt 

vi11 8hod all the electrons having veloci2;ies smaller than its ovm, 

As it pro- 

ceeds dong its pa th, %he particle vd11 pick up electrons, RS 2, result the 

speoific ionization will tend .t;o deorease, 

At npproxinctoly 1 MY, +he parthle 

becomes neueral; its pnth a6 R ohnrged pnrticla is nhou'b five to ten times 

an the distiawo it trnvels as n neutral pc.rticle, 

The range is ti'bou$ 

2 am, in air, 

of Elec-t;rons. 

*. 

ft is difficult,to asoribe a range to eleckrons traveling through 

OQ~UW of the excossive straggling. 

In contrnst to the es$enfilnlly 

il) 

strni&t; pc.th of an a-particle, an electron my be considerc2bly dcfleoted in 

n oolllsion, so that it w ill have a more or loss rnndon pnth. 

Secondly, %he

* 

energy spaatrum of #-pa clee emitted from a nucleus is not monochromatic (as 

in the aase OS W s ) but s a considerabls spread, The combined effect gives a 

range distribution which imitates an exponential absorption curv 

There are seyeral prpasslsss by wliioh the enorgy nf a fast moving elea- 

tron may be dissipabed. When ran electron passos -through the electric field of a 

nucleus, it is QGCQh3ra%0d, 

This ameleration gives riae to radiative energy 

lasses (bremsstjrahlung). 

They are appreciable for elecCron energies of several 

Mev. 

The electron-electvon radiative lossas are quite small inasmuch as there 

can be no dipolo, but oiily quadripole, radiaCion. 

The eLeotron suffers energy loases in Its collisions with eleatrons. 

An approxima'ce formula has been given for this rate of 

energy loss, and we have 

aeon it depends on the squam of the atomio numbor of thc nuclei, 

In .these 

oollislons, all of the atomic electrons par%iaipate and ionization in the trctv- 

ersed maDerial i s produced, 

Tho ratio of ionizstion loss to radiative loss is 

givah roughly by 

Rato of ionization loss I ZLmev) 

R a . t ; e T ? m L % X - 

--.I 

8 O r 

where Z is the atomic numbor of kl10 rnnt;orial, E is +he kinetic e 

million elgoZ;ron volts. 

To give an idoa of how much shielding is nocessary to stop 

one may give an axample. 

4- 

particbps, 

For aluminum 100 mg/om2 will stop 100 kv betas, and 

1 g/cm2 will be adequate for 3EW betas. 

1% is approximakely %rue that %he 

stopping power per unit muss is the 8aine for all tho olomonts. 

. .r-...u.--.--ruCu..- 

Reflectian of fi pa_r2;iales. 

.._I_ 

Coefficients of reflec.Gion bstwen 1/10 and 1/2 are, qui%e usual, 

Heavy 

nuclei by virtue of %heir charge are more Eiffoctive in producing ,largo angle 

soattaring than 'light nuclei, 

Honce pb reflects fi particles much butter than, 

I 

say,, paraffin6

e 

84 

Sn particular, tho first roaction is irnportmn-b booousc it givos information con- 

corning tho bindi.l?,g cnorgics cnd forces betwoon clomoqtary pnrticl-os, 

Gross- 

soctions for this 6ypc of' roactions arc rolakivoly small, being of tho ordor of 

m2 or smallor. 

Tho two ronatlons clitiod nbovo roquirc ralativoly low 

3. -cncrgios, i.c. about 2 IJcv, 

Elost nuclcar photoolfccts requiro 5-8 Flov 

%t-rcys 

The interac6ion OF ?-rays with rnaLter are in the maiD interactians 

with electrons. 

The phenomena may be described as the photo-electric effect, 

Compton scatCaring and pair production. 

We shall gee %hat the Parims inter- 

ac.S;ions depend difforontly on nuolear charge and the energy regions in which each 

effect becomes important is more or lesa well defined. 

Photoel@otric effect, 

.-A. ---". 

__I 

A bound electron absorbs energy and leaves the atom. 

.IC- 

Vost of the 

energ? of the %-quantum is given to thu elec.l-rrm, most of the mom@ii%uin is given 

to Che residual positive ion, 

Fach shell o:? electrons wound a nucleus will con- 

erilsute whose onera is smaller thavl the eneru of the ? quantum. 

Among them 

the shells whose ionization energy is closes% to the energy of the ?-ray 

makes 

the greatest GOYLtribLi'GiOn, 

pray energies are usually greator tlian the ioniea- 

Lion energy of the IC electrons so that the innermost; shells have the larpst 

effebt, 

For sufYicientl*y energetic ?-rays, 

tho photoelectric absorptioki 

ooefficjant goes 8s Z5/E 7/2 where Z is tho atomic number of the 

5 

and E the ?-ray enera* This formula indicates a very great differ- 

9 

ance b0.knreen the respective coefficients for Pb and C, 

Them i's actually a 

great difference in absorp6ion coefficient in the region whore photoelecbric 

effect $8 the main source of absarp'cion, 

This is the X-ray regtoi?,. 

Compkon Sffect 

_LII 

magnitude of %he scaWiering of a phokon by an elecGron can be

88 

uiicer.t;ainty of 30,000 e1eoi;ron. f~ol-bs, This aoouracy oomparcs favorably with bhat 

of good optioal spectrographs, 

The method of mass deterclinafion utilizes comparisons bs%ween ions of 

marly qual rnwses suak as1 

Differanoos of -these inasses are than measured. For acaurate measuraments, 5.C is 

best if the ion ooncen6ratioas in the discharge are so matched that %heir intensities 

are appraxim,%ely equal. 

The scale OS measuring masses used in physids is based on the iaotope 

0l6. 

Tho mass of' this isollope 3.s S Q ~ equal to accurately, 16 unl%fl, then one 

i;l&sS unit oorrosponds to 930 Ibv, 

This quantity might be compared 

with the mas8 

of the olectron whioh aorresponds to ,til Pietr, 

In ohemistry, it is usual to set @he atomic weight of the natural iso- 

topic oxygan equal t o 16, 

+,00031 x low2* grams, 

In %his c&s@, the unit of nuclear masses Is 1,66035 

In discussing a$omic masses, the mass defect is of' 

interest. 

1% is defined as 34 - A, whore I! is -the 1x83s rneacursd Ill %he phystoal 

scale, and A is the integer number closest to 11 aL?d is called tho mas8 numbor, 

A further qc'n-bi-ky frequently used is the packing frac,tiol?, which is defined as 

F'I - A 

In the following figure thb packing fraction is plot.t;od &gains% the 

nuclear chargo, 

s

The be& figure of Chis kind is found in %he ar6iole of EEahn and Mattauoh, 

a9 

Physikalische Zeitschrift, 1440, The packing fraction has a minimum a6 about 

&other 

method of determininG WSSRS is by balancing reaction energies. 

For instance, the mass of the iieu.t;ron bas bean deOarmined in thi5 manner, 

mass, of cau.rse, co~ild not be determii?ed in the mass spectrograph because neu- 

This 

trans caiinot be deflected Sn the electromagnetic fields of tha% apparatus, 

the determina+ion of the neutron mass the reactfoil 

For 

Hi2 t hv -Hll *). 11~ 0 

is used, Knowing the mass speotrographic values of the mass for M12 aiid lill 

which are 2.0147 aad 1,00812 and ~isi.ng tho minlmwn snargy 2,17 

Nev at which Gha 

light quantum can cause disiiitegratio:1, one fbds for the mass of the neutron 

1.00812. 

In many othor CBSBS, 

cross-ohecks are possible by using mass spectra- 

grapliic values of %he masses and data from nuclear reactions. 

Size of Nucleus 

e. _”I ..“ 

-I-*--- 

.,- 

One may de-bermiiie the size of’ atomic nuclei from various reaotions, 

some of these, hoviever, like reactioim of capture of c low neutrons, givs errat;ia 

values. l’he aatual size of tlza nucleus can be definod 8s the radius to vdiich 

guolear forces exteml, and is usually ob6ained by inv6stigating the 

deviatfohls from the Rutherfwd scatt-wing or by investigating the potential 

barrier effects in c&-Clecays 

and ot.her nuclear reaotiotis involving chwged par- 

ticles. OKW fj.:icis that nuclear radii arc? given approximately by 1.4 x 10-13~1/3. 

This maiis %hat %lie nuclear volume is proportionate to the iiuclear mass. 

-..- 

l!Tuolear Donsl-by 

.I-. 

different nuclei. 

It follow that nuclear densities are approximately +lie same For

first disoovered in a structure of spsctiral lires which 1s CalledAhyperfine 

90 

s%ructure. 

These structures were first observed at the end of the last. centwy, 

but their explanation by Pauli and their more dehiled study goes baolc only two 

deoades. The explanation of the hyperfine structure is as followsr The orbital 

motion of the eieotrons produce8 maglietic fields, 

magnetic moment associated with %he nuclear spins, 

These interact with the 

The interaotion elzero 

rnod5fio s the sprectral lines in a differen% way, aocording to $he orientation 

of the nuclear spin in the magnotio field of the electronic mokion; thus tho 

lines are split, 

From the limber of the Compomnks, the spin may be determined, 

while from tho sizo of the splitting, "Fie nuclear magnetic moment may be oaScu 

1 at e d 

Daiia on magne0ic spins and rnRgnetio moments have been summarized up 

to 1939 by llorsching (Zeitschrift fur PhysFk), 

A now method for the determination of the spin and magne.l;ic, moments 

has been davolopad by Xtarn and Wabi. 

This method USOS atomic or molecular 

beams passing through static and oscillating magnotio fields, 

more direct than the om involviiig the hyperfine sZiructure. 

have dotermined by this method, the magnotic moment of the noutrons which, of 

course could not be dstermined by the uso of hyperfine structure. 

The mothod is 

Bloch and Alvarez 

The following 

table suznmarizos the results for %he neut;ron, the proton and the deukeran: 

spin -- I 

magnetic moment 

.(I.-. --_11 -1 

n 1/2 ? "1.93 

HJ, 1/2 2b78 

I1 2 1 0,865 

Tlne spin of %ha iioutron has been assumed as 1/2. 

This value is probablo, but 

* 

not quite oer.l-aj.n, The magnotic momon.t.s are measured in units of 

eh 

m 

is tho mass of %he proton, 

The minus sign i n the mp;notic momoiit of

positive aharge. 

Vuo le ar s-bat 5. st i as 

t; obays Dho %om 

he premnt piattxre $hat nucloi with svon mass 

,

92 

. T11us one.muld have Qo believe by at nuolei oontain posi6rons 1 

in addl%$on to sle&ironsr 

A rather 8Z;rong argument has been given which shwrs tha% neither elao- 

011s c~ii exht in polel. 

If' a partiole is danffnsd to' a region 

lirmir d$mensiorr$ &q then the momeiihm has an uncerb;ain%y */Aq, tha% is, 

$he manreliburn may have any value from zero up 00 abou-b this latter value. l''70vr in 

%he nuoleus Aq is a few timas om. From tho GO e spoading momentum uiicsr 

tiainty one may aaloulats t 

sprea4 in klnetlo energy by the formula 

P2 

-.I 

2m 

where p is tha momoiitum, and m fs the 8s of the partxiole in question. I% 

.tihe pareicle is 8 proton or a neutmn, $hen tho uncertainty in kfrieOio energy' 1s 

of the order of the nuclear bindi 

OY~QF~IBS. 

If, hovrsvor, the pni..ti,ole v m 

Icinetig euerpg mould bs much bigger due to the 

small value of $he mass 0% the electron. Aotually, i k i -t;l?

* 

92; 

The table shovss that nuclei vri.t;h an odd nusnbor of protons ar an add numbor of 

mutrans and an odd number of pxotakis WQ particularly few. Thora &re only four 

that; these nu.oloi form a simple sories. 3; 

bo r@marked -bkx&t; not o'2&r 

arc3 

there fovmr kinds of nuclei with either an odd numbor of noukrons or an odd 

number of protons, bvA also tlicat such i:uclei aro loss abundant;. 

In connection wit& tho fa& %ha% nuoloi with an odd number of nsu%rons 

or an odd. number of protons are rolativoly rare it may b? montionod that lluclei 

vdth avr odd value of 2, i.0. 

with an odd nunibor of pi*ck~h-~ ;iovor have more than 

tno isotoopos, IZth the exception of tho ljghtas2; Q13i?E':'.n, thoso isotopos dikfor 

by two L~QLX'G~OIM. 

Similarly, if onc oansidars nuclol with a given odd number of 

'? 

sh QY?S

95 

One might imagine the figure made more complete by cadding the energy of 

the nuoleus, that is, the enera needed to pull apart all the constituent nw- 

tronl; and protons, 

One may plot this energy in the third dimension perpondia- 

ularly ta *he plane of Flgufe 2. 

The enargy surface obtained in this mannor will 

have o valley along the dotted line near which tho s%able nuclei clustcr, 

Curve as shown 

CuZ;%,Zng the enorgy gurfaco along an isobario line one obtains &n onergy 

o o ns t a nt 

The straight livlos shown in the f'ig~r~ am energies of ehe various isobario 

nuclei, 

One of thaso energies will lzavo the smallest value and a corrospondihg 

nuoleus will be the only stable isobar. The oorrosponding stablo nualwj is 

indioatwd by a dot vshilo the uylstablo ones 4.136 shown as crossesb 

shown which indioato the dkoction in which nuclear 

Arravs are also 

-transformations taka plaoo (I 

Tho situation depicted in Figwe 3 holds only for nuclei vrith an odd mss numbor. 

Isobars of such nuolsi must either contain an won nwnbor of noutrons and an odd 

number of protons os olss ~n ovon numbcr of protons and an odd number of hautrons. 

Thare Ss no systoma-tic deviahion bOtWQQn the onorgies of them two kinds of 

ass number is won, than tho nuclai my eithor contain an moM 

rons atzd an won number of protons, or else, an o4d numbor QS 

P 

neutrons and an odd number of protons. 

Tho formcr kind of nucloi have system- 

atioally 8, lawor cncrgy than thc lattor kind, 

Thus .tho nuclear onergias will 

lie on two roughly parallol curves, tho highor on0 oorrosponding to tho odd-odd 

QBSC, the 1mrcr on0 Go tho avon-ovon CRSO. This is shown in Figure 4.

96 

e 

E 

2 odd N odd 

,Z even N even 

A 

constant 

Fig* 4 

It viE1 be noticzed that In this case more than one stable nuolaus may exisf. 

In 

the figure, 

such nuclei w e shown. 

The one on the right h 

han the one to the left, But the nuoleus with the higher egergy 

oould ljransforrn into %he one of tho loiver energy only by simultaneous emission 

or two charges which is an exceedingly i d3&tble process 

Emission of one 

electron or positron) would carry us t o the upper ourvs and tvauld require 

snere rather than produce it. It will also be notiGed from 

on the highsr ourve may transform into G nuoleus of ~ Q W energy ~ F &%her 

king to the right or the left, The 

that suoh a ~iucleus emits positrons and elec%ron$. 

This has actually been 

ob served for several nuchi * 

From the sysbernatics of the nuclei, aonolusiona about nualear forces 

r nutlet. It is %Q be explained by 

wtrone and protons are similar. 

The explanation also uses the Pauli prtlnaiple, which asser.t;s that in each state 

9 

only one parkicle oan exist8 that is, in a gtvsn orbit within the nuoleus one 

may have not more than two neutrons (or 80% more thm two pra.t;ons) differing 9u 

the direcfion of their spips. From th quality of the foro s acting 012 mutrons

* 

and proGons it- follows that up to a given energy %her@ 

of orbits available far neutrovls and protons. 

97 

will be a similar number 

PF one then filla up the lowes$ 

orbits available up to a given epargy, one mi13 have to use B roughly eqLtal 

number of neutrons and probons ., 

This is t o be explained with the help of ths Coulomb law, 

It is 

easier 60 add a neutron than a proton Do a nucleus whioh is already heavily 

charged, 

In lighter nuclei, the Cablomb ~O$?O@E are considarably smaller than 

%he nuclear forges, 

In heavier nuclei Stho Coulamb energies aS the several pro- 

tons due to the longer range of 6liase forces add up and beaoma oveiitually 

comparable to the nuclsar energio s 

Mass defeob and volwne proportional to mass number, 

--I .---- -crlucI. .ua"- 

These facts ir?dicaCe forces having a saturation ahsracter, +ha% is, a 

situaWan 5.11 which one particle oon'xibutos offectivaly $0 the to'l;aX energy by 

interaq+iihg with a llhited number of &hQr particles. 

The aame situation is 

encoun-borad in C;rystalS 

Bj.nding onergies of DeuBaron and a -partiole 

-I-.-+.--.*--~ .I* * r *nrrUu.lr- -.-.- 

Tho fact +that tlia &-particle has a more khan LO times greater binding 

enera than the Douteron sliav~s that tho saturation has not been roached at the 

Zatter przrWcle. 

In the &-par.t;icle 

the saturation i s actually reaohed. 

Thls 

a% R particle, for instimice a nou 

nuoloua with another neutron and %wo otihar protona. 

an, can effcc+iv~ly intafact in a 

Tho usus1 intorprotation is 

that a par%icle oan intQracat with all oChor parfioles which aro parrnit-bod by the 

InciplQ t o move in tho same orbiti and %hat tho intorao6ion does no% 

dopond yory strongly on tho differcnk spin directions vJ'hIch tho particles my 

* 

possess in this orbit, 

The dc Dislhtogration. 

LI*-.--IUl*.-IUUC. 

The &-particI.as 

omitted by a nuc2ous have a well dcfivlsd onargy and

m 

range. A vory small fraction of tlia a;-partido has somotimos a considerably 

longor rnngo. Tho main faot of tho uniform onorEy of tho a ' s may bo o::plaiulcd 

by assuming that the &particlo has boen srni-blied by a nuoleus of R givon enorjg 

and that arnothor nuclous of a givon miergy is left bohind after OhoQCI-dooay, 

The anorgy difforonoo is thon oarriod away by tho C;C-part.iolo, 

Rnat liar important law ~f tho &wadj.oaativity is tho eonnoctioii bc%?voon 

tho range of tho par'ciolos and tho docay aons%mt, 

Thts is tho Geiger-Nuttal 

relation, 

It statos thab the logarithm 0% tho docay oonstrrlit is a linoar funm 

tion of 

tha railgo, 

Actually one finds that tho relation botiwocn tho logarithm 

of tho dacay constant and tha rango is sligJitly diffcron% for tho radium, 

aot initun, ntid trhoriuh famillo s, 

Tlzc throo straight liiio s -&pro sonting the 

relations for thoso thrm familios arc liovmvor parallel and their diatalioos from 

oach othor ara small, 

If oil0 had plottod %ho logarithm of tho doaay oonstmt 

as a fuiiotion 0% thc onorEy ratilicr than as o, function of tho rango, tho plot 

would again bc in good approximation a straight linc. 

Thc oxplana%ion of tho Goigar-1Vuth.l relation is closoly oonnoctod 

with the shape of fiho potential anergy aoting botwem tlia d-parbicle and the 

frwtion of the nucleus which +:le tr;-parbiole has Just left, At radii bigger 

than the nuclear radius ro chmm in the 'figure, the interaction is a Coulomb 

repulsion as shoTim in +he same Tigura. 

At distances closer than ro some kind 

of an at0raction must @xist, otliemvise %hers would bo no rea on why the origMal 

hualeus that ha3 emitted the &-particle 

had stayed together any length of' timer 

In the potential valley at distances smaller than 

rQ 

the &-partiole 

had in tihe 

original nucleus an en~~cy.r level at; +lie anergy value El

99 

I 

Figure 5, 

After thea-particle has moved out t$o suffioiently greati r 

values BO that all 

Znteraations including the Coulomb interaction have become negligible, the 

Cjpartiole will PcSsseS; that energy E in %he form of kinetic enera. 

If' now *lie product nucltsus is bombarded by&*particLes of energy E, 

pure Rutherford scattering is observed, iiidicatinG that at the radtus router to 

which the =-particle 

can penetrate aooording to classical mcrhanics, the unmodi- 

fied Coulomb force still holds, 

There arises the quostion: 

in what way can the 

dr-partiole OY'OSS the potential barrier bebveen rinser and router (see 

figure 5 ). 

This quss2;ion has been answered by Gurne$r and Condon, and by Garnow. 

In qu4ntwn mechanios leakage through a potenthal barrier is possiblec though %he 

probability of suoh a process becomes small when the barrier becomes high or 

broad, The time of decay may be estimated by imagining the &-partiole oscillating 

wi%hin the potential hole, by calculating the number of times the &==partiole 

bumps into the barrler per second, sand Pi)mlly, by multiplying with %he 

probability that the amparticle penetrates .the barrier in one collision. 

The facC that the a-particle can penetrate through tho barrier is 

* 

conneoted with the &partiole's wave nakuro. An illustration from optPas may 

be g5.ven. If light impingos on tho intarface between glass and air from the aide 

of the glass un4er a sufficiently glancing angle, total reflection takes place,

m Mm first glaars pLaZ;e this aeaand giargo plate will be reaaked 

exponeriCla1ly 4eoaying field 3n air and par+ of' the 13gh-b will be 6rammfttad to

* 

104 

nuolei are known whioh emit in turn, an alpha, a beba, and then another alpha. 

The 4x0 aLphas have quite definite energhs, yet the beta partiele does no$, 

Some years ago,' attempts were made to meagure the energy of the betas by using 

a very thick Pb w alhd oalorimeter, 

ft was hoped at the time tihat all the energy 

of the emitted particles would be measured. 

However, an amount, aorrespottdfng I 

o thw ama under the distribution ourve, vas found. 

Bohr made the suggestion th& perhaps energy and mornenWm vms not con.. 

served ina-4$sintegration; 

seoond partiole .IC 

still valid, 

A spsuial hypathesia made by Pauli was thak a 

the neutrine -- was emitted, 80 that aanserva'binn laws were 

Attempts ham been made to oorrelate the maximum enera E in a begs 

speofirwrn wifih the disintegration aonstantA I as was dona in the uase of 

&,-partiales (Geiger-Nutall laws), "he Sargent ourves, a plot 02 log h YS E, 

we an attempt in +his direction. 

From absorption measurements in aluminum and using a standardized 

method of extrapolating tho rangeeenorgy ourve, on0 obtains 

R 0.543 E 0.16 

, 

where E is in million electron volts and R in cmb (Fsathor rolation)r 

A satisfaotory theory of beta-disintegration must explain the oon- 

tinuous spotrum and give a oorroot rahgowenorgy rolation. 

Formi has dewlopod 

a thoosy with bho neutrino hypothosisr 

In tho nuolaus a neu~ron is aonvar%ed to 

a proton 'oogother w9.tjh tho omission of an olootron* Tho rolo of Z;b olao..t;ron 5.8 

similar to that of' photon in radiation thoory. Tho probability of omissZon OS 

tron with onor# 

in tho interval El E # AE for lfght nuole3. is gSvon by 

P{E)dB g H%7 t{U$Vmdy12 (Eo - Edl3 

whore E is in unl2;s of C the voloaity of light, E, %hc maximum

The noutirop is expe&ed to be 

astive with Q mean life of several 

hours. Experimental evide'noe 9s hckin oaqse the neutron renot6 relatSvely 

quiakly k th any nu'clsus that may be pr 

tr Hwm@r, experimefits hhvo boon 

proposed to mocasurc3 this disintogratlon oonstaht. 

Novombor 18, 1943 

LA-24 (18) 

Lectmor: E, SO~PO 

UCTURE XVXfXt 

ON AND K CAPTURE 

I 

it pr>si%rons, giving riw to 1Jsobars with one 

nuclei. 

This roaakian OOOU~S if tho avail- 

able enorgy is at loast 600 kv, tho she~gy oquivnlsnt of tho gosl.ttronz, that is, 

$1 tho mass of $he parent nuolous sxcoods thn2; of tha daugh$er by at lwst tho 

mass of tho posttron. 

If $he onwgy nvtrilnblo is lees, then tho nuolous will 

0 

capturo an oloo.t;ran fram the innormost shell and subsaquently emit an X-ray

106 

quantum, 

This so-called K (electron)-oapturs by the nucleus also ooours at $he 

higher energies but la less possible, 

2) , Gamma ,Radiat;ion 

w. , , 

I 

The eleotromgnetio radiations emitted by radioactiva wbstanoes are 

oalled ?-rays. 

The Y-rays are aharaaterized by %heir frequenoy, or 8s is more 

usual by the energy of a quaRtwn expressed in electron volts OF in unsts of 

2 The range in energy of 9 -rays from rzsturally radioaotive subs.t.anos.s is 

approximately 60,000 eV to about 3 Mev (million tslootiron volts), 

By bombarding 

Li with pro4jons more energstio ?-,rays ara obtained, values up to 17 MeV have 

been faundr 

The highest; energios In tho lwb have boen renohed with tho XersG 

betalwon, 

The theory of ?-ray omission is similar to that of the emission of 

light from atoms and moleoulad, oxoept that tho magnitudes of aomo of tho quan- 

tities involvad are muoh different, 

The probability of a transition from one 

onore fitate say m, to another lover enorgy etato n, and tho omission of Q 

light quantum is &ven by 

whore 4.d is 2fi times %ha frequonoy of the omtfkod radiation, C tho volooity of 

light. 

P i s tho so-oallod matrix olomont of tho transition and is tho ol0a.Dria 

momont avcrhgod over tho oigonfunctians of tlio two statas m and p* Equation 

(1) roprosonts tho first torm of an expcnsiqn in powars of co/x whom ro 3s 

tho nuclear ra@ius and A tho wavolongth of %he omittbd quantum, 

PhysicalLy, 

tho flrst torm in tho oxpansbon corrosponds to tho dipole radiation, %ho aoomd 

ta %ha guadrupolc radiation, otor In atornib prooossos, is 60 largo compnrod 1 

to tho radius of tho atom, SO that the soaond term is vory muohismallor than tho 

* 

first, 

For nuoloar procdsms, huwovor, )r and ro do not diffor as grocxtly; con- 

sequantly tho second term mczkos an approciable contribution, 

Tho magnitudosr of 

the matrix ~ lomont~ vary over o. oansidorably w2da rnngo corrosponding to wido

is ejeoted, then 

'kin 

Enuolsus a Eg 

This prooess is known as internal cot~version, Photographs onfl -speotrograph 

show tha2; the ihternal ootlversion electrons, sa is to be expected, form very 

homogeheous groups in contradistinotion %o disintegration -alectro~ls which have 

a oont$nuous distrj;but%on in energyr 

Usually a group of lines of varying inten- 

sity is observed cmrresponding to ejoation of electrons from different onergy 

levels of the atomv 

Oacasionally3. -radiation is acreompanted by fl-disintegration and the 

questSon arises, as to whether the radiation precedes or follows %he disintegration, 

If' inkernalfy converted eleatrons are presen4i and are analyeed to deter* 

mine the energy differences beheen the various groups, a11 anwer o m be given. 

For if the spacing corresponds to an Xnray diagram of the paront nuolsus, thovl 

%he 5 -radiation precedes %he disintegration, whereas if tho spac32ng is that of 

the dawghter nuclous, the cmnvarge is truo. 

The ratio of the number of oonvorsion sleotrons to the number of 

3, -quan.l;wn is lcnovm as the convorsion coeffiaient,

108 

reLatlvely very smdl 80 %hat the me 

Zy large, %he exaifed level $8 

r8' 

life Pf %ha% leveb S, 

be mefastable,. Le% us 

are in the ground state, %hey 

underg -disintegra+ion wi-tih a half- 18mr If on Lhw othw hmd, 

nuclei are in the first sxaited level, wbiah has a half-3iie for @)I 

L 

-eminsfan 

en the observed halfmlife of t 

sinae the rate defermining fact 

proof of suah "isarnsri.o!' nucl 

sequentfl remission &a ,found ts be 

the Blower ra2;e of 

hat they oan be produeed by exaita- 

s or e?.tzo-brons, first; t o a higher level, from wbioh tihey reaoh the 

by 3 -@mhsS$Ot1 of 

gh, ra%e r simple ohemical te thni 

rt J. 

H+ Williams 

POD Was found by Both@ . 

(1930), (Naturwiss -#39 753 (1931)). They 

published the results of tmme experiments i n whbh the 1igh.t elsmen%s L1, Be,

109 

and B were bombarded by polonium alpha particles. A wry penetrating “radiatiopf1 

was emSItted which vas more diffioule to absorb by lead than tho shorbst wav~ 

- lowlgth 1(cY)ays Ohon known, thosa of I‘hC’’, Their orlginal explanation was tha% 

the radiation was light of a vary short wavo length. 

Today wc know that ths 

inoreased absorption of wry short WWo*lQng‘ah @)f -rays by the prooass OS pair 

areaLima would alimina$e this explanation. 

They war0 lasd to *he aasurnption %ha% 

tihs Itradiation” was pho.t;ons on other grouurda, bosSdos tihat of trbsorption measure- 

mentis, on0 being that an traoks were obsorued in a cloud chamber, 

Carrying their hypothosis of hard to its logical conolusion on 

the basis of the Klein-Nishina absorption curve as a f(E >, thoso early 

”r 

axporimehters gave an enorgy for this “radia+ion” of from 7 to 55 Move Their 

wbsorpgion moasurcmants were mado wi%h Gobgar Fountor dotootors, 

In a nuoloar 

squation they pos%ula%od that the reaotlop involved was 

Supposo for an exoroiso wo oheolc tho onorgy balanco of Ghis equation 

by subeti,tuting masses in Z;ho oqua2;ion 

qBe9 9,01504 

$Io4 - 4.00389 

-1ZTiRZ 

c33 I 13,00761 

-‘TUTEZ mass uni+is 

Ono mass unit 

931 Mov 

Enorgy availablo EG-b10,5 Mov, 

The problom was next attaohad by Curia and Joliot (Comptos Rondus 194 

273, 708 (1932) )., Tho important difforonco was that in Chcso resoarohos, thin 

window ionization ohambers instoad of Goigor gountors wore usod as dotootors of 

n.lnbul’ 

tho radiation. 

Thoy found %hat if matorial aontainlng €1 were into.rposod

a 

botwoora. tho &on Ba SOU~CO and tho ionization Qhambor, tx 

'ianixa$ion was obswved. 

obsorvod, 

2arg~ incroaso in 

110 

For absorbors of othor kinds only small docroasos 'vmo 

Thay showod that this fvloraaso in iolziza~ion was duo %o the prescnco 

of H roooils leaving; thc "absorbcw" and ontoring thoir ionization ohambor. 

Cloud ohambar studias revaalad tho prosonce of those proton roooils as ~013. RS 

Ho racails in n Ha gas. 

By moasuring tho lengths of thoso roooil tracks it tlrns 

possfl~1a to de'cormine tho onora of the roooils, 

Soma of thoso rocoil protons 

had enorgias up 00 5 MOT, 

No traoks wro absckod for tho primary ltradSationtl 

which thoy assmod to bo? -rays. 

Curio and Jo1io.t; oxplninod thoso obsorva.Dions 

by saying that the onorgy lirnnsfor ocourrad in a Compton pro~~sa~ Tho? "ray 

enarm otilovlated in this way turned out to bo 50 Mov. 

Unfortunately, tho oar- 

oulations for othor roooil pnrticlos on Z;ho basis of this %hoary $avo diffaronf 

?-ray enorglrss, 

They WO~O highor for hanvior rocoil nuolai, 

Chadwiok (Prac, Roy. SOC. A* 136, 693, (1932),) solved %hJs d ilom sa 

c- 

aonvincingly tha% ho reooived tho Nobol prize, A sohomatio diagram of the 

apparatus usod follows. 

FFg ' 

1 

-** 

7- 

I 

I 

7 

VQCUUm paraffin ionization ahambas 

l 

amp3iFiQr 

Tho measured rocoil onorgios of H' and 

nualai vmro 5.7 M w nnd 1.2 

Mov 

cospoctivoly. 

On the basis of tho phofon hypotiho sis tho oalculatad ltradlatioatl 

onoF6;y would bo 55 MQV and 90 Mov rospocti-valy. 

Thus 'Gho photon bypathosis was inconsistant with tho obsarvntionsr 

Either tho laws of aonservation of onorgy and momantum or the hypothosis as to 

* 

the naturo of tho pherzomnon was 'Go bo rojeotod, 

Chadwick thorefam assumed 

that tho "radiation" was a new pnr%iolo of mas8Nmss of proton arzd having Zero

mutual forces between nuclei are fhe subdoof of Mr. 

Artifioial Souroos of Neutrons 

3.12 

Critohfieldrs leoturesr 

Not a source of monoonorgotic neutrons, 

This mode of disintegration withdj.particlas as tho bombardikg particle 

1 

which results in neutron produotioh has beon obsorvod for sovcwal light oloments, 

Li7, 1311, NI4, 

FL9, Na23, Mg24, A1Z7, but is moa6 intenso from Be, then 3 and Li.' 

Those souroes af neuJcrons am assentially v&y woak, ralativo Do 

sources of whioh we shall speak lator, 

Tt was impoosiblo up bo 1942 to get 

sourco8 in OhSa manner using Po as tho G-sourco which give mor0 than ,405 r/goo 

or oxpressod in anathor way tho yield is 60 n/lOs stoppod, On the o6hor 

hand, if tho &,$s &re obtainod from R, which is intimately mixed with Bo, the 

numbor of noutsrons em$tt.l-od per mor par mq. of Rn is of tho ordor of 104/soa. 

With a curb of Ra on0 thwofora has an appreciably notivt7 sourcoo 

Unfortunalioly 

' 3( -rays are prasont hero whore %boy aro abscn9 in tho Po &~ourcos, except 

tw 1 h ?.',ha 

A complete disousaion of @Len roac%Sons, %ha enorgies of tho neutrons 

and of .Oxlo Z X S S O G ~ ?*rays ~ ~ Q ~ 

is given in Livingstion and Bcthe, Rev', Mod, Phya', 

9, pagee 303 to 308, 1937, 

*- 

Photoe1oct;rio Produa@ion of Nautrons, 

With two of the li&% elaman4x H12 and Bo noutrons oan bo produoed 

4 ' 

by tho 7 nmys of' ThC" or aqy ?trays whose onergy OXoQeds fjhg bi'nding enorgy 

of tha neutron in tha primary nuolOuar 

The probability of $his process is low 

and as a oonsequonco tilio yield of those SOUYGOS 

- 

Examplos - H2+ hW-+H1 .t n'+ 

Q 

is oorrospondingly' lawc 

Chadwfak and Goldhabar, {Procr Royv SOC, 151, 479 (1435),) usad tho ihCtt +L*rays 

of anergy 2,62 

Mov and munsumd tho aa.t;ual mrxgnitudo of tho ionization of the

protons formod in tho above reaction in ordor to dohrmi'ne thoir onorgy. 

113 

Sinoo E, E ~8 oonsaquonoo of their having approximately cqual 

P 

0 masses, tho olnergy equation moms 

The analysis of %he data gav0 Q "-2.2 

MCIV 

This moans tho naukm is bound fn*tho H2 nudous wi%h an sncrgy of 

212 MWVI 

Knowledgo of this binding anorgy is of groat thoorotical importance 

sinoo it allows us to dskormine the maso of tho nautwon, 

Mass of Nou6ron 

.-...I 

RowrlOing the pho.i;ordisintagrat;ion equatlon 

Mn a 1,00893 f?OOOO5 

A tnblo in Livingston and Botho, page 353, swnnarizos tho avidmoo on 

?-araac%;iQns up to 39370 

A modem usoful SOUFOO of noukroyls usiylg ?-rays on light elemonts 

illustrchtas tho advanoos in toohniquo mada in nuolsnr physics in rooenf yomsu 

As we vrS11 dlsou~ls IcL~Q~, It is possiblo to produoo an isotopo of 

by dou4xwon bombarbont of Br vfhioh has a l/Z Iifo of 80 days nnd omits a "')+-ray 

of suffid.en6 anorg to dlssoclake Bo9 and givo nautrons of 16Q-k~ onergyt 

This souras hnq boon extromoly usoful RS Q SQUTOO of poln%ively monoonorgo'eio 

Y 

e

134 

December 

2, 1.943 

Third Series: Neutron Physics Lecturer : J. H. Williams 

Neutron Sources (Continued). 

Neutrons may he produced by 

the bombardment af certain nuclei with protons, deuterons or alpha 

particles, that is, a nuclear faaction takes place and @ 

emitted. 

neutron is 

To initiate the re~ctim, the bombarding particle mu8t 

penetrate the potential barrlar surrotmding the target nucleus 

In other words9 .the projectiles must have a sufficiently high 

kinetic energy 

We may distinguish two methods used to e.ccelerate such 

particles, namely direct and resonance acceleratcrs, 

Tne first 

is long vacuum tube one end of which is at a very 

high potential and contains a sourca OS ionsS and the other i s 

usually at ground potential and contains the target. 

The ions 

traverse the vacuum tube acquiring kinetic energy came sponding to 

the drop in potential, 

The resonance acceleratorg or cyclotron, consists cf a 

flat, hollow cylinder 

resembling the letter D. An 

slepara ted into two sections each 

oscilla tine voltage difference 

applied between the ~ W O “doesft OS the order of. 

and &t a Frequency of several rnapacycles. 

a hundred kilovolts 

Along the axis, a 

‘magnetic field of several thousand. gauss is applied, 

The ion 

i

* 

115 

source is on the axis, Under the Froper conditions, the ions 

travel in circular orbits of increasing radii and Increase tketir 

kinetic energy during each passage from one af the dees to the 

ather. The sucees~ of the cyclotron is due to the fact that the 

angular frequency of the motion is independent of the energy (ieeo 

of the radius of curvature of the orbit). When the voltage frequency 

is equal to the angular frequency, resonance conditions 

obtain. 1t'l.s possible by such resonance means to attain energies 

of the order of several million electron voltso 

In this type, the Incomlne; particle is a proton and the emitted 

one a neutron. Written in detail: 

ZA + HI-+ (Z + I)* + nt 

where A is the mass number and 2 the atomic number, Q is the 

energy of the reaction, It turns out that Q in th

116 

and 1.86 MEV. Thus this method is very convenient for producing 

mono-enerpetic neutrons simply by controlling the bombarding voltage. 

It is clear from consideration of momentum and energy conservation 

that Tor a given bombarding voltage, the energy of the emitted, 

neutron depends on the direction of emission, so that neutron6 

I 

observed in the backward direction of the beam will have less energy 

than those in the forward direction, SQ that sne may obtain a range 

of neutron energies as a funct$on of angle. Practically, there is 

an effective spread of bombarding energies because of the fact that 

some of the neutrons will be slowed down in the finite thickness of 

the tarpet before they initiate a nuclear reaction. This variation 

can be made arbitrarily small- but is usually of the order of 5 to 

50 kv for intensity reasons, 

I_di,_n_) reaction: Here the typo is 

ZA .c ~ 2 - (Z I 1W 4 n1 I Q 

The so-called (d,d) reaction is an example: 

H2 4 H2 .crlt He3 4 nt + 3.2 MEV 

This reaction provides a copioits source of neutrons. Yields of the 

order 09 IO8 neutrons/sec are possible with bombarding currents of 

the order of lOrA impinging on heavy ice targets. The efficiencies 

are of the order of lnh05 deuteronso A desirable feature is that 

the reaction has an appreciable cross-section at low bombarding 

energies. One difficulty is the preparation of thin tarpets. Often 

gas targets are used but the yields are correspondingly smaller. 

Another example is 

c612 + HI2 

~*lb* N13 + n' + Q

117 

e 

with Q t -,25 lvaEV9 which means that the reaction is endoergic. 

The energy spectrum of the emitted neutrons is not simple because 

of the C13(d,n)N14 reaction. 

ut_eRra_tions muted by neut= &a) re&ctiongA 

An example of this type is 

n' + N14-He 

4 

+ B 11 

One can obtain cloud chamber photoeraphs of this process to 

establish its identity beyond question, but it is a somewhat 

tedtous procedure since many photographs must be taken before one 

finds evidence for the reaction. Another well known reaction is 

and also 

nf C Li6- H3 9 He4 

330th of these reactions have relatiy?ely very large cross-sections 

for thermal neutrons and are very useful for the detection of 

neutrons by detecting the ionization produced by the emitted 

cparticles. A common procedure 3.3 to line the walls of the 

ionization chamber with a layer of boron. Often proportional 

counters are used which contain gasious boron trifluoride and the 

individual particle pulsas are detected. 

&D) _r_q&tian: Symbolically this reaction is 

* 

This reaction is usually endoergic, the case ZA c,~Nl4 being an 

exception. 

This type is the simple capture 

process in which a neutron is absorbed and a gamma quantvm

I 

118 

e 

omitted. 

Process was first observed by Fermi in the production of 

artificially radioactive isotopes by slow neutron irradiation, 

&.$I 

r3Gion: 

ZA + nt ...lp ~A-1-t 2n 

It 9s clear that this react.ion requires at least an amount of 

energy corresponding to the binding energy of a neutron in the 

nucleus. Hence fast neutrons are required to make reaction go. 

This process ban only be reliably identified if the resulting 

nucleus is radioactive,

* 

LA-24 

December 7, 

1843 

(21) 

119 

Third Series: Neutron Physics Lecturer:: J, H, Williams 

Slow neutrons are produced by allowing fast neutrons to 

pass through paraffin, The collisions of the neutrons with the 

protons in paraffin result in the initial energy of the neutron 

being divided up between the neutron and the proton. Thus In 

successive collisions the neutron loses energy until it finally 

winds up with thermal energies I) 

The process of slowing down of the neutrons depends on 

the collision cross-section between neutrons and protons, We shall 

discuss the concept of a collision cross-section, as it has been 

introduced in the kinetic theory OF gases. 

Let us assume that two gas atoms collWe if they come 

closer to each other than the distance s, Assume further that the 

relative velocity of the two particles is V. Then, %he particle 

under consideration wllZ sweep out per unit time a. volume Ss*V and 

the particle will coll%de with all collision particles within this 

volume, Therefore the number of collisions par second is $T.s2Vn 

where n is the number of collision partners per unit volume. 

Assumln'g that a11 of the relative velocity Y is due to the motion 

of the incident particle, the distance through which this particle 

0 travels between colllsions, that is the mean free path, is given by

1) 

I 

,&=l/fls%. (in the kinetic energy of gases the incident particle 

and its collision partners move with the same average veloqity 

which introduces into the farmula or the mean free path a factor 

1/42.) 

distance x 

The probability that a particle should move through a 

without making a csllision is given by e -x/k re; 

e -xm% Here the colZislon cross-section o t.f$s' has 

been introduced, 

This cross-section i s in gas kinetic theory the 

geometric area within which collisions take place. 

the above exponentSal formula may be used to meaaure the cross- 

-- 

.L 

La 

Experimentally, 

section, For this purpose a $'lab of the material made aut of the 

collision particles and having thickness x 

is placed into a beam 

of incident neutrons with intensity Io, A smaller intensity I: will 

emerge from the slab since the neutrons which made the collision 

are missing from the beam, Then the ratio of the two intensities 

Ss given by the formula I 

=cox 

,/ro 

X 

I 

0 

Nuclear cross-sections have usually the order of 

magnitude 10 -24,,2 

This quantity has been introduced as a unit 

of nuclear cross-sections. It is called the barn.

3.21 

e 

The collislon cross-section and mean free path o f neutrons 

in paraffin vary a great deal. with tha energy of the neutrons. 

crawwmction Is a few barn9 above a My whereas it i o fifty barna 

for thermal energies, 

few cm, while thermal neutrons have a mean free path af a few mm. 

The 

At the hSgh energies the maan free path is a 

ThB ~nergy loss d h natttybn in a collision with a probdti 

depends on the angle of deflecti6M. rf the ifiitial energy of the 

neutrons is E, 

collisions i s Eo/en. 

then the average energy of the neutrons after 

to.reduce fast neutrons to thermal energy. 

Thus apnroximately 20 collisions are necessary 

The number of slow neutrons that can be obtained in a 

given volurqe of paraffin depends on the rate of production of sluw 

neutrons and on the rate of absorption of slow neutrons by the 

hydrogen in the paraffin, 

H' + n'4H2 + htb 

This reaction is only one example of the very 

common process in which a slow neutron is absorbed by a nucleus and 

a )t 

active. 

-ray is emitted, 

This absorption is due to the process 

The resulting nucleus Is frequently rad5o- 

(This is not the case when neutrons are absorbed by HI 

The production of radioactive nuclei provides us with an easy means 

of measuring the capture cro9s-section9 iqes9 the part of the calli-. 

sion cross-section which is duo to the capture process. 

n

* 

122 

To perform the measurement we use a thin sheat containing n absorb- 

ing atoms per om2, 

and serr~nd. 

We let N nuclei impinpapan the sheet per cm2 

c , 

Then the number of r%dio-ac$ivs nuclsP fo~r~lad, per cm2 

aqd second. is given by nNo,, 

Thus the capture croasmxtlon may be 

measured by determining the radio-activity formed b 

Some nuclei have very high thermar capture orosstsections. 

For instance, the capture cross-sectlan of cadmium is 3,000 barns, 

While this cross-section I s very muoh greater than the geometric 

cross-gection of the nucleus, tha scattering cross-8ection at car- 

responding energies i s only a fev barns, that is, comparable to the 

geometric cross-section, 

A further peculiarity of the capture cross-sections at law 

energies was found by Moon and Tillman (Proc. Roy. Soc, 

A =_.467 

(1936) ) They observed that the capture cross-section changes 

rapidly with the neutron energy showing maxima as can be ~een in 

the followinp figure e 

The maxima are separated bp distances of the order of 10 electron 

0 . vults on the enerpy scale.

123 

The crude explanatfan of this phenomenon is resonance. 

The word resonknee suggests the analogy with the great energy trans- 

Pers which become possible 

quencies are equal to each other, 

between vibrating systems If the fre- 

If the rest energy o f the initial 

nucleus, plus the rest energy of the neutron, plus the kinetic 

enemy carried in by the netttron, happens to agree with the energy 

of a compound nucleus made up from the Initial nucleus and the 

neutron, then the rearbIan becomes more probable 

For instance I ths 

above reaction of neutron cap turs by hydrogen may be written in 

form HI .I?' n? ++H$-r-)-H 2 $I hz/ where the carnpound nucleus H2* is 

an excited state of the nucleus H2, 

Not only capture but a .1~0 

scatter5hg may be describd with the help of compound nuclei, 

iristance, the scattering of neutrons by protons may be written in 

the fallowing form H1 4- h'+H 2 jc -.-E1 .f. nf 

For 

The various means by 

* 

which the compaund nucleus may disappear, that is, emission of a 

quantum or emission of a neutron, or possibly emission of another 

particle, compete with each other, At low naut;'ron energies emission 

of a 9 -quantum and consequent neutron capture predominates for 

most nuclei. 

The first theory of resonmce sarture of neutrons was 

given by Breit and Wignsr (Phys. Rev. s...T19 (1936) >; Rather then 

discuss the complete theory we shall describe the behavior of the 

neutrons 

A) if their energy is very close to a resonance 

B) if their enorgy is very low. 

In this way, we shalL understand the peneral appearance of the 

curve for at which has been given above, that is, the general rise 

I

1) 

of with demeasine energy (8) and the superposed resonances (A). 

If condition [A) is PulfllLed, the collision cross-section 

is deteranlned by the sinple neiphboring resonance with energy E,. 

Then, according to Breit and Wlgner a neutron with energy E . 

possesses a collision cross-section 

' 

Here k is the de Broglie wave length divided by 2'R of the neutron 

of energy E and, fir is the same quantity for the case that the 

neutron possesses the resonance energy E,, f is the width ~f the 

resonance and Fn is that part of the width which is due to the 

neutron emission, A datafled explanation of these quantities will 

be given in a later lecture. 

If E - E, 

is great compared to r the co?-.iision cross- 

section w ill fall off inve r s e ly as the square 

of the resonance. 

In exact resollitnc~ only the 

sz both sides 

the denominatar and at this point (or rather cl-ose tc this point) 

the cross-section attains its maximum. 

7"-r 

2 t~m remains in 

IF on the cther hand, the energy osf the neutron.is small 

compared to all resonance energies (assump K!.[J~ B?, then the energy 

dependence of the cross-section is determined by the variation of 

5 with energy. In this region, the cross-section varies as 1 

over the velocity of the neutron. 

on the spacing of the resonance levels. 

level is reached the l/v law breaks down. 

The size of this region depends 

When the first resonance

December 9, 

1943 

LA-24 

Third Series: Neutron Physics 

Lecturer: 

Lf the width of the lowest level 

is greater than the 

energy E, of the lowest level, then the Z/V Law will break clown when 

the neutron energy becomes comparable t o the width r 

The above statements concerning the region of 

the l/v law can be proved from the energy dap-endence of 

cross-se~tion as given by the Breit-Wkgner formula (see 

lecture). 

This energy dependence has the form 

validity of 

the capture 

end of last 

One sees that Cr, 

d/dE 

will behave as 

-1/2 )) 

log E 

l/v if the condition 

r *} 

is 

satisfied, This in turn is proved if either 

ar

126 

One very important substance in slow neutron physics is 

wdmiun. Its absorption crossIsection as a function of energy is 

roughly represented by Fig. 1, 

Its strong absorption for thermal neutrons is due to a resonance 

lying very close to thermal energies, A few tenths of a millimeter 

of Cd shielding is sufficient to remove from a distribution of 

neutrons the thermal neutrons. 

The first experlments, by which information on the Cd 

absorption and on the distribution of thermal neutrons was obtafned, 

were performed at Columbia, The experimental arrangement is shown 

in Fig. 2. 

'B 

Ra+ Be 

Cd

* 

e 

127 

On the left side of' this figure a paraffin I1howitzerlr 28 shown, 

which is a block of paraffin shaped as shoyn in the figure and containing 

a radium-beryllium scmrce at the indicated positjon, The 

paraffin blook emits S ~ Q W neutrons from a11 of its surfaces hut the 

arrows indicate a particularly stronp and somewhat better directed 

beam of slow neutrons ppoduced by the peculiar shape of the howitzal;, 

This neutron beam impinges upon a velocity selector which is showp 

in the figure to the right of the howitzer.' This velocity SeZectar 

consists of two rotating discs mounted on an axis at a distance D 

from each other and two stationary discs located before the first 

and behind the second rotating disc, On each of the Tour discs, Cd 

segments are mounted in such a way that there ar0 on every disc 50 

Cd segments and 50 segments free of Cd, This arrangement is shown 

schematically,below the velocity selector in the figure, To the far 

right of the figure a boron trifluoride counter is shown which 

detects the slow neutrons that have passed through the velocity 

selector. 

The velocity selector is rotated with a velocity of n 

revolutions per second, The Cd strips of the rotating and the 

adjaxent stationary discs wSll at a given time owrlap, so that the 

, 

neutrons can pet through the Cd-free parts of the disc. Shortly 

thereafter, the Cd segments on the rotating and stationary discs 

will alternate so that no neutrons can get through. IT the velocity 

of the neutrons is such that they will find either the flrst or the 

second pair of rotating and stationary discs closed, then a minimum 

of transmitted neutron intensity will he observed, Assuming that 

all neutrons have the same velooity v, this minimum should occur 

I

when the relation 

r*l 

v E lOOn/D 

is satisfied, It has been found that the neutrons omitted by the 

howitzer have an approximate velocity of 2,500 meters per second, 

This corresponds to thermal velocities, 

A somewhat more detailed 

1 

analysis showed thgt the neutron velocities have approximately a 

Maxwelltan distribution shown in the figure. 

In this figure, the number of neutrons having a given anergy E is 

plotted against that energy, The maximum of that curve shifts to 

higher energies when tho temperature is raised, 

An InvestigatSon of the resonances occurring at a few eV 

was first carried out by a somewhat indirect method, This method 

takes advantage of the fact that the reaction 

has a cross soctian obeying the l/v law up to quite high energies. 

The experimental arrangement is shown in Fig. 4. 

7°F" 

13 absorber 

Source * 

Cd box 

Detector fotl

129 

On the left hand side is indicated a neutron source (this might be 

a howitzor or another arrangement)# The neutrons emitted by this 

source pass through a boron absorber of thickness x, 

are carried out with various thicknesses X. 

Experiments 

On the right hand side 

of the figure is seen 8 detector foil which i s made of a substance 

mes radioactive under neutron bombardment. 

It is in thisz 

substance that we want to study tha neutron capture r@sonances. 

Tho detoctor foil is enclosed in a Cd box. 

. ment are made! 

Four kinds of measure- 

I. no boron absorber present, no aadmim present, 

2. no boron absorber present, cadmium present, 

3. boron absorber present, no cadmium present, 

4. boron absorber presant, cadmium present. 

Comparing measurements 5 and 3rl one Finds what fraction 

of the thermal and higher energy neutrona have been absorbed by the 

boron, 

Comparing measurements 2 and 4, one finds what fraction 

* 

higher energy neutrons have beon absorbed by tho boron (the , 

thermal neutrons are stopped by the Cd box), 

Comparing tho d$fforenco of 1. and 2 with tho difference of 

3 and 4, one finds what Traction of the thermal ne'utrons haw been

130 

absorbed by the boron, 

We plot in Flg. 5 the quantity log Ax, that is, the 

logarithm of thc intensity observed in the detector foil as a 

of' the thickness of the boron absorber, The lower one of 

the two stralght lines in the figure 

"j A 

l - ~ . , 

-7t 

is abtained by taking the ratio of msasurements 4 and 2. 

curve, therofcrre, refers to high energy neutronso The higher lino 

is obtained by. dividing (3 - r> by (1 - 2), 

thoreEdre to thermal noutrons, 

The intensity va~5'itions with boron 

absorber thickness may be represented by formulae of tho type 

-Kx -noox 

A, s AOo =IO 

ThSa 

This curve refers 

Tho slope of the curws 6ivl;s the absorption co9fficiont X, The 

coef%lc?.cct j . boron ~ is proportion41 to -&ho-1/2 powor 

rgy of t3:e :ioii$r0111s, Therofore, one obtains for the 

ratio of rssonanco oncrgy to thermal onorgy tho expression 

o m can calculate the resonance energy, 

b

One limgta an of the mothod $uaL 

cribed f s that It; 

assumes that neutron 

the neutron beam only by 

absorption in .boron a ing in boron. This 3.8 a valid 

asmmption for small neutron energies where tho capture cross- 

Bection in boFon is la~g6, For em ies above a kw, soattaring by 

boron disturbs the me8$ur@mentd 

1 In this manner, the relsonance captures fop v&rious 

have bosn explored, For medium and heavy 

found that redonances spacod Irrsgularly at 8, dista 

of 10 volts from each other are present and that these! POS 

have wzdths of the order of 1 volt. In light elements no such 

found, Theso elements shau show m'uoln mom broa 

i 

ea which are much more widely spaced, if In fact they do not 

ovexz 

ft i s more difficult ta study tho resonances o f an isatape 

which doas not; becoma radioactfve when It captures a neutron but 

' 

s Into anothor stable isotope, 

A rather inadequate mothod 

which has been applied in such cases is, based on the comparison of 

* 

The neutrons originating from tho SOUPGO 

and then after having flown a ~ Q C Q distcznce ~ D 

132 

fall on n noutron absorber 

dstoctod in a 

nautmn Gountoq for instnnke in BF chamber, Tb souma is not 

3 

g noutsons a13 tho Limo bot'bnaY dtirimg shopt Sntsrvals of 

'clrn~~ let us say of 10 micromconds duration, sepamacd by longer 

time interval$, of far instance 100 microsecond$, 

The neutron 

pulees arnitted by the SQU~CQ are shown in tho uppar half of Fig. 7. 

*-* . I C - P U l B @ Of SOUPCe 

Tho lower half of this flguro shows the time intorvnls during whYch 

tha rroutron detector $s sensitive, It is seen that the ddmtar 

will not roact to neutrons unless* they arrive t after they have 

been-emittad, Thus only neutrons with 8 voloclty 

e 

are observed. By varying tho thickness of tho absorber, ona can 

find the abssrpt;,ion coefficient for those partlcular neutrons in 

the material of the absorber, By varying the time delay A t om 

can investigate the depondenco of Po upon neutron anergy,

133 

sensitive time of tho datcctor which CQ~QS 

L period Jator, 

This is 

called recycling, 

Thls difficulty can bo minZmieod by choosSng a 

long rctpeat period for the emission pulscs and the corresponding 

sansitive intervals 

By tho mothad descrtbed bbovo it WBS possible to explore 

the propedies of' $IuW kc&aibns dj? ho abduk $0 cv, 

We shall no summarize briefly tha intoraction of neuktons 

with some element@. This will bo done in connection with a table 

which in part has bson taken from A, Both@, Rev, Mod, Phys. 29

I) a 

$> 

1 

H1 

ETa s tic soatt ering 20 1.7 Efficient neutron 

slow-up &orL 

of thermal 

rleutrons 

Elastic scattering 4 

1.7 

3x6 

Neutron absorbed 

alcpha emitted 

goo 

310 

tl 

C= Elastic Scattering 

3000 

H3-4 Elastic scattesing 10 

and neutron absopbed 

proton emitted 

st? Elastic scattering 12 

and radiative capture 

n 

5 

6000 Produces active 

ZSQtOpe

December 24, 1943 

~~-24 (23) 

UTURE $E 1 ;HY IC 

Third Series: Neutron Physics Lecturer f J ,H, Willlams 

Generally speaking, neutrons are produced as fast; neutrons, 

There ape three principal react2ons in which fast neutrons are 

produced, In the first one an alpha particle enters the nucleus 

and a neutron is emitted, The reaction is symbolized by (a& 

This reac.tion has mostly historical interest since it was the first 

reaction in which the production of neutrons was obsemved. 

A second type of reaction in which neutrons are produced 

is the reaction in which a deuteron impinges on the nucleus and a 

nrsutroi? is emitted, The symbol for this reaction is (d,n>, Twa 

reactions of this type have become of great practical importance. 

One is the Be(d,n) reactton in which'the Be is the bombarded 

ntlcLeus and which gives a very copious source of neutrons. Another 

is the D(d,n) in which the deuteron hits a stationary deuteron in 

the target. In this reaction rnonoenergeliz neutyons can bs produced. 

The energy of the neutrons depends ?yi the energy of the 

impinging dm'teron, Nsutronv between 2 1vle!v .?,:i2 7 Mev may be pro- 

The third important source of neutrons is. the (p,n) reactions 

and in particular the Li(p,n). In this reaction an Li 

nucleus is bombarded by protons and neutrons emerge leaving the Be

* 

nucleus behind. 

This reaction i a endoenergetic, that is, the 

protons musk have a certain mlnimm energy in order that the 

136 

r~ectlpn ahouXd proceed at a18. Monoene~ket3.c neutrons between 10 

kflsvolts and 2 million volts may ba produced by this reaction for 

the similar range of incident particle energies, $.eoo up to 3.5 

Mev. 

The Li(p,n) reactlon may be written in somewhat mors 

detail ixl the following form 

** 

~17 

where the energy of reaction Q 

suspect that instead of the above rea 

takes place 

where Se7',9 

+ HI n1+ ~ e+ 7 Q 

is a negative quantity, one might 

Li7 H1- n I). Be7'4- Q1 

&n excited form of the ;Be nucleus and Q1 Is 8 greater 

negative reaction energy, If this second reaction proceeds, the 

neutron produced would bQ no Langer monoenergetic. 

this possibiliey has been found, 

ion the fdllowing rsaction 

No evidence of 

It is true that the Li7 nucleus 

(which i s quite analcgms to the Be nucleus having as many neutY'ons 

as the Bo7 'ia:3 ~)cckons 3iiC RS many protons as ths Be7 has neutrons) 

has an excitaii atnm of 485 kilovolt 

hat the Be'? h 

There is some reason to 

a similar excited state and the presenoe of 

such a state might giv rise to S~OWQT nautrons whenever the Li(p,n) 

reaction is used to produce neutrohs ab e 3.1'2 

.Measurest of. Fast Nqutrqu 

milaion volts, 

Ths first method of measuring the energy of fast neutron8 

made use of the Wilson Cloud chamber, 

The arrangement by which this 

be perfosmed is shown In the following f igura i

137 

, 

On the left hand. side the position of the neutron source i s indi- 

cated, 

shown. 

Qn the right hand side, a Wilson Chamber 3s schematically 

The chamber $8 fillad with a gas of small atomic weight 

such as hydrogen or helium, The path of the neutron is also ihdicated 

in'the f$gure. The heavy line in the chamber is the track of 

a recoiling particle, If this track makes the angle B with the 

continuation with the path of the neutron then from this angle and 

from the energy of the recoiling parlicle, the original energy of 

ran may be calculated, In partlcular, if the recoiling 

particle is a proton then the energy of the neutron is given by the 

2 

E, t E, cos Q 

1) 

Where Er is the energy of the recoiling particle and En is the 

energg of the neutron, it is usual to count only the tracks 

which go in the forward direction and make a small angle 0 (for 

smaller than 10') wi$h the direction of the incident 

he factor cos2 Q in the above formula may be 

and the energy of' the recoiling proton is aqua1 to the 

energy of tha inoident neutron, 

One danger of this pmcedure is that the neutron might 

have bean scattered In the wall of the chamber and while the protan 

seems to go farward, it actually does no$ lie in the continuation

2.38 

sf the neut;son which hag hit it, 

I 

Another difficulty is 

cks of the recoiling particfe are rather lo~g and they 

d d;n the ch er. The anergy 04 such recoils cannot be 

chamber measur 

A fwkhor diff icu 

tn eva2mtin.g results of cloud 

fits as walZ as of other meksuremants of fast 

noutrona i s tha* a pr0vious bowledge of the dependende of the 

callision cross section on tho anergy f the incLdent neutron is, 

A typtoal depsndende of thi8 c 

olLowirtg figure t 

SS sectlaxi 6 (wh 

mtia cross selStion) on the neutrqn energy E, 

.li_ 

E, 

Umfortwately sometimes, as far instance, in the case of helltun 

ndence is mors involvrad EL^ 

shown in the figurb: 

is shown 

e 

resonance is nd 2 MeV, ThSs resonance 

to d great: 

between J and 2 rn112l.m 

of this resonance them? many tracks were e 

8hQWing the presonce o 

Recently pho 

ha incident neutron has 

, Previous t o the disc 

cluud ohamber-s In measuring the energy aP neutrons, 

the phQtographic plates eon$cd.ns hydrogen, 

ous3.y interpreted as 

and 2 MeV, 

n USQ~ 

instead of 

cause fani;zat$on and leave @evelopable grains in the waka 

The emulslon of 

The rocoil protons

, 

I 

Wilson Chamber only those tracks are $0 be counted which make a 

g2e with the direction of the neutron, 

The photographic 

technique has tho Pollawing advantages a$ aomparod $0 tho Wilson 

chamber* 

First9 the stopping power of the omuJ-sion is g?eatar9 the 

lsnbth of the tr~cks is only a few thoasandths of an Inch and the 

.danger that ths tsack IoaVtss the emulsion bafore ending is smaller', 

S@c?ond, in tho photo$ra$hi@ plate %Hers heod be much less material 

than there is in a Wilson chcmbar; consequently the danger of a 

scattered neutron causing Q recoil. i s decreased. 

Thirdly, the 

photographic plate is continuously sonsitive while the sensftivity 

of a cloud chambcr I s rostr$ctcd to the moments of expansson, The 

photographic technique has howavar the disndvantage that it becomes 

Jnaccurabe in couqting slowor neutrons, 

The recoils produosd by 

utrans make few grains and since the number of the grains 

produced is used in determining the neutron energy, fluctuations 

in the number of grains mako such anergy doterminations aifl$icult, 

The photographla mothod must be calibrated by using R reaction of 

known enorgy 

A less straightforward, but perhaps mme trlxstworthy 

mothod of rncaszxring nautron enargios makes uso of the ionization 

chambers, 

Such a chamber is shown 6chemnticnlZy In the following

filled with hydrogen. 

ians produced by tho rocolling protons. 

140 

"fie field wlthln the chamber collects the 

0111.the sight hand side the 

eonsloction to tho ampltfying grid is shown in which the charge pro- 

duced by the ;loris is amplified SQ as to give a measurable effect, 

The following figuro shows these amplified current pulsesp each of 

which corrosponds t o one recoiling particle according to the angle 

which tho recoiling proton makos with tho neutron. 

The rocoiling 

particle will carry more or Xess energy and the size of the pulse 

will vary come rspondingly e 

Amp 1. iff e, P 

output 

I 

9 

lJlilLnii 

l-cI3L"..b 

\ 

O m may plot tho number of pulsos which have more cne'rgy than a 

given value E agninst srmplifior output 

b?mber 

of 

pu 3.8 e s 

pulse size or energy, 

Very small puZsos cannot be counted bacause they arc? obscured by 

%ha background or noise oE tho nmplifiey, 

By dffforontiating this 

curve9 one obtains the number of r(ccoil!-i?g particles of a glven 

encrgy. 

This I s shown in the Tollawing grnpht 

I 

Number of recoils 

If. tho gas in the ionization chamber was hydrogen then a second 

dlfforentiation gives tho number of primary neutrons as plotted 

against their energy.

* 

Ember of privariea 

The figure$ given above refer to the case of monoenergetic incident 

neutrons. 

If a distribution of neutrons with wrious energias 

impinges OR the ionization chamber, figures of the following kind 

may be obtained: 

Number of pulsos 

above a given energy 

I 

Tn w 

Ember of primary 

neutrons 

I 

Inswoad of recording the pulses of various hoig,,t, 

one mighu use an 

anplificr wfth a bias such as to cut out all pulses las~ 

than a 

certein energy. 

The sonsitAvity of such a counter is shown below: 

S0nsdtivity 

-E 

Dslow the bias the 

amplifier-dotactor will not respond at all. At

142 

higher ener(gies the sensitivity will rise to a plateau and than 

docrease. 

Instend of using an ionization chamber fillod with hydrogen 

gas, one mSght use a cbambor in which the side facing the * 

neutron SQU~CO is coated with a thick layer of paraffin. 

Such chambers are then fillod with argon, 

sufficiently heavy so that argon recoils carry rolatlvely little 

energy and tho bias may be tzd3ustod in such il way as to count only 

the proton rocoils originating in the paraffin, 

such a chamber is shown in tho following figure: 

Sen s i t ivity 

L..=L 

---- - %2 

The argon nuclei are 

The sensitivity of 

Th..ere 2s this time an inzraasing ,rise o f sensitivity above the bias 

value, 

This is duo to tho fact that hfghor e~iergy neutrons produce 

protons which hcive R longer range llsn the paraf?'.k ad. have therefore 

'

14 3 

0 

a 

The coun$er Is filleti with hydrogen, 

Instead of a ~oll&ting plate 

it has a collecting wire near which Parge electric fields are 

present, 

In this field electrons are aocehrat~d towards the wire 

causing further ionization and in thts way greater- pulse result, 

With such proportional cQunter8, pulses as srilall as 10 kv may be 

detected whllle the ionization chamber described above is insensitive 

up to 100 kv or higher. 

Ionization chambers also may be filled with BF,, 

chambers the neutron reacts with the Bra 

products of total kinetic energy about 3 Mav. 

pulses are produced. 

3 

In such 

nucleus giving rise to end 

Thus sufficiently big 

Not only the measurement of the energy of fast neutrons 

but also the measurement of their number is a diSficuXt question, 

In one method due to Fermi and Amaldi 

the neutrons are slowed down 

in it water bath of sufficiently big volume surrounding the neutron 

source on all sides. 

After. beinp slowed down, the neutrons am 

detected by rhodium or indium foils distributed in the bath. 

From 

the known cross sections of thewe foils for slow rleutrons, one may 

determine the original number of fanSt neutron?, In order to do that 

one has to integrate over the 

the bath, 

the detectors in 

This last operation may be avoided if instead of the 

foils the slow neutrons are detected by manganese dissolved in the 

bath, BY stir Ping this solution and taking sample afterwards, 

one may find the integral of neutron absorption in the bath and from 

this quantity the original number of fast neutrons may be calculated. 

A theoretically very effective way of counting fast 

neutrons and measuring their energy is shown in the following *sketah:

.. 

On the left hand side, an incident neutron beam is shown. 

This 

ges on an Iankmtion chamber which an the side of the" 

neutron beam has a thin paraffin film. This film is folhwed by a 

collimator which" is a plate transverged by naxvowe1: channels point- 

' ing in the original direction of the neutron, These channeLs will 

A 

let through only such protons whose direction colncidss with the 

direction of the neutrons, From the number of protons in 

the thin paraffin layer, from the colllsion cross section between 

protons and neutrons and from the number of pulses of various siaes 

izat3.m chamber the number of neutrons with various 

._ 

- 

not work, The reason 

-c--_m 

is that there is too much hydrdgen in other places than in the thin 

rotons corne from 

We shall discuss the following processes ich occur when 

ring, ilzelastic 

8, (nJn> process, 

processes, These 

rmation abaut these 

een the neutron

14 5 

0 

$bt$xbs and thk neutran detector. 

if the scatterer is smdll enough 

and if thb neub!dn has beed ikfiected in the scatterfhg pyoeess by a 

I 

big enough angle9 then this neutron in the originlra bbam will miss 

the detector and no neutron outside the original beam will have an 

appreciable chance to be scattered into the detector. 

In this wag 

the major p’orlion of the elastically scattered neutrons w ill be 

missing from the beam, 

neutrons vdll be only partially observed, 

Similarly the fnalastically scattered 

In the case of the 

ally scattered neutrons an argument may be given which 

t these neutrons are distributed spherically after the 

z 

scattering process. In fact the scattering process may be described 

by the following equation: 

- - Q1+ ZA z(A3.1)’ zA*+ n1 

This equalton shows that the first act in an anelastic scattering 

probess is the formation of a compound nucleus with the atomic 

number A 4-1, 

neutron will have forgotim about its original direction of ink- 

dence, 

to be scattered 

When this compound nualeus emits the neutron, the 

Thus there is no reason for inelastically scattered neutrons 

part Icularly small angles and the scattering 

geometry described above will cause most inelastically scattered 

neutrons to miss the de te c t OT o 

(with It 

The ne u t r oqs which are captured 

gamma emission) or whrich undergo a (n,&dg or (n,p) reaction 

be naturally missing from the original beam. 

In the 

process the neutrons emitted are again spherically distributed and 

will therefore in all probability miss the detector. 

Thus practic- 

ally wha t eve r reactfon a neutron hag made with the scattering 

material the result will be that the neutron w ill not appear in the

I C 

14 6 

detector 

rblated to 

The intensity I transmitted by the bcatterer will be 

the intensity Xd 

df the 6rdgiba3. beam by the formula: 

Here small 

scatterer, 

4n%d 

I/J, 0 8 

n is the numbep of nuclei 

x is the thickness of the 

per cubic 

scatterer 

aJ1 C~QSS sections or the total cross section. 

centimeter In the 

and d is the sw of 


LA+24 (24) 

Third Series Neutron Physics 

--.- LECTURE SJRIES OcN_ELEAR, PHYSIC$ 

LECTURE XXIVt PROPF,RTIE$ OF' 

CROSS SECTION 

The total. cross section of a substance for fast neutrons 

can be measured by a direct transmission experiment In which the 

source,. scatterer, and detector are situated in what is known as a 

"good geometry". This good geometry is so defined that the solid 

angle subtended by the scatterer at elther the position of ,the 

source or of the detector is small.

* 

14 7 

In 

scatterer at 

by a neutron 

removal from 

the detector, 

this case the neutrons from the source strike the 

practically normal incidence and any process undergone. 

in its passage through the scatterer will result in its 

the-beam and its consequent failwe to be recorded by 

The neutrons counted by the detector will consis$ 

almost entirely of those which have had no interaction with the 

atoms of the scatterer. 

A very small percentage of the detected 

neutrons wfll consist- of those scattered (elastically or inslasti- 

@ally) through very small angles, but in a really good geometry 

they will constttute a negligible fraction. 

A transmission experi- 

ment in such a good geometry wi.11 therefore measure the total cross 

section for all processes. 

If the incident neutrons are normal to 

the scc7.ttf;ererS which may be supposed to have 8 thickness Ax and a 

number t~1 atoms per cm3, the ratio of the detected intensities with 

and without scatterer will be given by 

- 14, 

-nom 

0 

where 6“ 2s the total CTOSS section. 

In contrast to this measurement with good geometry, trans- 

Mission experiments in poor geometry are performed in order to find 

the elastic scattering cross section. 

In this case the solid angle 

subtended by the scatterer at the positfon of the detector is very

14 8 

Q 

large, 

This makes It possible for neutrons scattered through large 

angles to be scattered into the detector, thus compensating 

.---- 

Sauroo 

De t e c t ox- 

*-.-.- _._"" 

Scat t oyer 

PGGR GCGMSTR'SI 

for those scattered neutrons which miss the detector. 

I 

Since this 

compensation is almost exact, scattering processes alone would have 

' 

j 

if the detector is biased so as not to record neutrons below a 

cartarn energy, the neutrons which are &elasticallyl scattered will 

still contribute to the measured reductions in beam intensity along 

with all the other processes with the exception of elastic ssattering 

This experiment with poor geometry and biased detector 

therefore measures the sum of all cross sections with the exception 

of. tke cross section for elastic scattering, To find the elastic 

scattering cross section alone, w0 therefore subtract the cross 

section as measured in poor geometry with a blased detector From 

that measured in good g'eometry, 

Capture cross sections are easily measured if the nucleus 

resuZt$np from the addition'of the neutron 1s radioactive, The 

* 

C~OSS section can than be obtained from a measurement of the 

!I inttuced activity. When the resulting nuofaus is not radioactive

149 

there is na easy means for measurinp the capture CPOSS section, X t 

sometimes happens that one isotope of a given element will become 

radioaofive upon capture of a neutron while 30me other isotope of 

the same element will not, However9 it is found empirically that 

the capture cross sections of isotopes of a given element are of the 

same order of magnitude, Therefore one can sometimes estimate the 

captum cross section of an isotope which does not gfve a radioactive 

product. 

Dividing the energy range Snto roughly three regions, the 

resonance Pegion from 0 to 10 kv, the medium fast region from 10 kv 

to 1/2 Mev, and the region of fast neutrons from 1/2 Mev on up, one 

gets a behavior of the capture cross section with energy which looks 

like this: 

crc 

0 10 kv 

In the resonance repian there are sharp resonance peaks in the cross 

sockLon superposed on a l/v background. Tn the medium fast region 

the resonances are much broader and are discrete only for( the 

lighter elementsp say up to oxygen, For the heavier elements the 

broad levels overlap and give a smooth behavior something like l/v, 

In the fast region above 1/2 Mev the behavior is more like l/E. 

For the very fast neutrons the cross section is determined principally 

by the area of the nucleus and the competition with other

e 

possible processes like inelastic scattering which becomes more 

probsbla 

the energy of the neutron$ increases. 

The cross section for an (n,& or an(n,p) reaction can be 

measured by putting a sample of the dement in question inside an 

ionization chamber which w ill count the alpha particles or the 

pr~tom directly, when the chamber is placed in a neutron flux. 

cross section for an (n,2n) reaction is extr)emely difficult to 

pt in the case that the residual. nucleus Os radioactive. 

Inelastic Scattering of a neutron is characterized by a 

reduction in the energy of the neutron as a result of the scatterXng 

process, 

In light elements liko hydrogen or helium or carbon this 

reduction in energy comes from the no 

a1 classical collision of 

two particles of comparable maw and. such collisions should really 

The 

be called elastic, However for the heavier elements lika lead OT 

gdd, the neutron enters the nucleus and emerges with reduced 

enorgy leaving tho residual nucleus in an axcited state, 

excited n~clous then falls down into the ground state with the 

emissdan of a @pray, Thus a nuclenr inelastic scattering is always 

nied by one or more 'prays, 

The 

The probability for inaxastic 

scattering I~CPQ~S~S 

with increasing atomic weight sinco the 

nu~loi, havlng many constit nt particles, have a large 

variety of closely spaced energy 1 

els into which the nucleus can 

be raised by the addition af the energy of the incident neutron, 

Qualitative fnformation about Inelastic scattering can be 

obtained by measuring the activity induced by the inelastically 

* 

scattered neutrons in various detectors whlch respond to neutrons in 

different energy ranges. For example, siLver or rhodium can be used

as radioactive detectors of slow neutrons and aluminum ox silicon 

ns detoctars of the f3st neutrons. 

By measuring the activities in 

thcse Botectors with and without the scatterer one can get some idea 

of how the SneLastically scattered neutkons have to be dogrnded in 

cnergy. Experiments have also been performed in which theq-ray 

intonsdty which accompanies any inelastic scattor$ng is rnm6urad. 

The magnitude OS this intensity gives an indication of the cross 

section for the inelastic scattering. However the experiment is 

complicated by the possible presenco of 3. -rays in the source, Not 

much Information about the energy of the inelastically scattered 

neutrons can be obtained from the energy of the? -rays since the 

excited nucleus will in general emft a variety of ?-rays of 

different energy corresponding to the many states into which the 

nucleus can fall before reaching the ground state. 

Complete infor- 

mation would be given by n measurement 00 the spectrm of the 

inelastically scattered neutrons but this is F? 

very difficult thing 

to do. 

@ & m C T S RADIATION PROTECTION 

Electromagnetic radiotion like x-rays or 3. -rays damage 

tissue by praducing fast electrons (through the photoelectric 

effect or the Coripton effect) which cause lonization and disruptfon 

of the molecular structure of' the tissue, These electrons have R 

relatively small ionization per unit; path length and ix correspondingly 

long range. Neutrons will in many cases give rtse to 

energetic heavy chargod particxes like protons or bclrparticles which 

will pave R vory high specific ionimtion but R short range. Fast 

protons cuuld be produced by collision of a fast neutron wSLh a

162 

r, 

hydrogen atom in one of the molecules. Slow neutrons may bo 

m@urod nnd give rise to q&tys or in some cases, like those of 

Agatnst 3/ -rays, lead Or cancre te walls of suff iciene 

thickness give good protection. Lead is of no value against 

neutrons, however I Materials aontalning hydrogen, like water, 

paraffin, or c~ncrete, are ganerally used bec&use of their effoativeness 

in slowing the noutrons down to 8 point where they can be

I L A 24 625) 153 

, \ 

'.. .:, * 


LEC'IWE SERIZS ON NUCLEAR PHYSIC$ 

. . , . . . , 

Fourth Series : Tm-Bod.y Problem Lecturer: C. L. Critchfield 

..). c -. LXC'MJRE XXV: . NUCLRAR CObJSThflTS . ' 

,. , 

11.1(,~ 

, ' %, 

,-> 

I:+ 

. . . , 

There aye two kind; qf nuclear quanti t3e.s: 

'5 

Quantize& and 

., not qvantized. The quantized' numbers are: mass number,- charge, 

',* L ' 

',, spin, statistics and parity, 

The qtla'ntiths that are not quan- 

.Fized are the nuclear radius, .the nuclear mass, the magnetic mom- 

\ 

eht and the electrical quadruple moment. 

.~ 

It is believed. that eventually there will exl st a thevy 

of ithe quantized nuclear numbers correl.ating these to each ' ather. 

SUCH, n theory shall link the varjous numbers t.2 each other and 

:, ' posslbly to other 'physical qiiantltl.es. IJp ti nw there has been 

: I ). 

. ,.d 

., 

(,'.*'W.' 

. . 

I $,'. 

, G I 

'\i 

' \,, . . 

( ' 

, , 

' i 

I 

mly one sipnificant attempt in thl.s direction, 

'. 

PaiJ-li tried to 

cwrehte spin and sttitfstics by statlng that particles wTth a 

half ,integral spin have Fermi-Qirac statist1 cs while particles 

vj th an i-Qteger spin have Pose-Einstein statistics. 

\, The nani.ng of Fermi-Dirac statist! cs mag he explained 

with $he help tj? the example ~f the H2 molecule. 

'that the slsins 

tho Value 1/25 are Rarallel (trlplet state). 

StatLstics, which ap&.es 

Let us assu.me 

"?,the two protons of^ this mqlecu.lo (which have 

Then Fermi-Dirac 

to the protons, postulates that an: 

'intprchange of the posl$ions of th,e, protons will came the wave 

. . 

\, \\\ " . 

fundtions tq change sign, 1 , -

154 

The cmsequence is that a rqtational state mi.th no n0d.e (J = 0) 

d?es not occi?r simultaneously. 

A31 even Values ~f the rotational 

qvantim number J are exclud.ed. Odd J values are alloved and 

the lovest state correspmds to 3 .= 1. This state of hydrQgen 

is called orthohydrogen. 

Let u.s now cmsider the D2 molecule. 

Let 'tis again assme 

that the spin 0.p the devtrons (which have the value 1) are para- 

llel. Then tlie Bose-Einstein statistics whJ,ch applies t? the 

deuterons 'pmtijlates that the even'rotatiqnal states OF the rnole- 

&le, are alone pTesent while the odd. rqtational states do &?& 

xcvr. 

d eu. t er on. 

This stake of the deuteron molecule is. cn1.led ortho- 

There also exists a secsnd stnte fw- both the hyd.rqpen 

and. the. deu.term in which tho spins o?' the nuclei are nut parallel 

and in nhtch those rdational states are allowed which did nqt 

Occur in the ortho states and t.hose rotational states are missing 

whjch were present in the ortho-molecil.1-es. 

called pnrahydrQFen and. paradeuteriurn. 

These states are 

The experience ab711.t the 

statistics qf nuclei is a strmg.argument j.n favw of the nuclei 

heing cmstitu.ted frm neutrons and. protqns rather than from 

prqtons and electrons. 

Assuming Fermi-Dirac statistics f w all 

these elementary par'tj.cles (prqtons, electrqns and neu.trms) ,it 

follows that a nucleus cmtajning an qdd number of slvh particles 

behaves accwdinp to the Fermi-Di.ra,c statistics whilo a nuclew 

cqntaining an even number qf particl4s has Rose-Tinstein statis- 

tics. 

NTW the deuteron, i-? it vere cmstitu.ted from pr3tons and 

electrons, would have to contain two protms and one electron, 

that is, an ogd number of particles and. should, therefwe, have

155 

Fermi-Dirac statistics, wliereas In reality it, has Bosa-Einstein 

es. Xf on the 7 t h ~ hand, prrltqns and neutrons are the 

ents of the sru.c$ei, a 4euterqn Is made up of ane pratm 

and onemntron, that, of an even number qf particles which 

edicts Rose-Finstein statistles, 

ong the non-uixmtizec! qii 

e radius can hot be determined v 

e values have been qbtai 

ItJ es characteristie of 

accurately. 

t an early date in the 

nbcl.ear physics by scattering 04' alpha particles. 

The pPTncLlple of' this determination I s that when the alpha parr 

ticle ap3roaches to within the s vm o 

he radii of the scattepinfr 

nr: the alpha particle then tho Rutharfwd scatterinp Inw 

ceases Ita hold, From thls anomalous catt,er.ing csf alpha particlos 

4 2 

adfi in tho order of 10 CM have kecm qbtained. 

Phe electric quadruple moment of a raucleus for a sharply 

deftned qmntity has not Fe~n 

determfned up tq now wtth v&y 

high 

* 

Tt has been first dis 

d rneaswOd by 1 vest i Pa t Sng 

errine strvctvre qf' atgntc spectrum. This hyperfine slrucdire 

to the intaractton of' the magnekic moment of the ebecw 

e ta the spin 

ita1 moment sf' the 

e nucleus Differant 

cl.ear and eLectTica1 angular morn 

wont energies and cavse Q nRrrryc~ 

s which is as a rille less than one 

on descr-1.bed abme gives a 

fine strvcturee Small deviations 

hcavy nuclei, These 

additional term in the ener

expl-anation which depends qn the? sqvare of tho cqsine of the 

angle Inclvded between dlroctims of the nuclear ancl el-ectric 

956 

angvlnr moments, PhysicntLy thl.5 Lima1 term is due to 

an olectrlc qiiadruolp moment of tho nucleus whl ch interacts with 

the inhqmqgeneity q? thr. electric field prqd,tlced hv t h elcctr-+Is. 

~ 

Thcro is also me rothpr direct detwmj,natjm of the quadruple 

moment qf the deilteron which IW 

The masses 9" 

mns8 spectr9gravhic wwk. 

shall Clscuss later. 

the nuclei arc knwm qllttp accllr~t~1.y frw~ 

and megnotic flc3,ds inrhlch focus ions T? a riven !.veJght into a 

given pqsi ti qn, 

this methqd have been carried qvt, he Ralnbrldgc; fw lipht nticloi 

and by Dmqpster for hcrvy nucLoi, 

This method uses crqsscd el(3CtTic 

Rcccnt acvyate determjnatj 3ns of masses by 

BainbridEe was able to obtain 

high intensities by f'xx~slnq a beam of groat angular deviations, 

In hi)s sot-up the positi9n af the fwvs dopends on the maSs qf 

the nucleus vcry accurately as a linear function, 

In this v~ny 

it was possible to obtain mass determinations Isf high accuracy 

by comparing the 'ocus Q? ions OF nzarly the same wei-ght, 

Thus In his car1,y 141vk he cwparod the mPss of the hellurn 1qn 

1R4.t;h the MBSS qf the triraatqm1.c DHFI. 

The actual value deduced 

frqm this measurement for the mass of thc dovterm titrncd out to 

to oxygen masses g?vcn by Astgn. 

carried out vith abixdant deutarium s~urces and evaluntcd. 1NitdZ 

the correct helium to qxygen ratio gave very 8ccvratc deutaron 

masses. 

gics. 

More rccent work of Rainbridgo 

NL~CIOQT rnaSses aro ~llso used lin balancing reactlon oncr- 

A systematic sti4y of" thcsc rmctiqn cnergles lcad Bcthc

n 

H 

1,60893 

1 * 09812 ' He3

158 

It is svTp3:ising that by morely d.ovbling thc numbor gf particles 

within tho nvC&%s tho bindlng rnergy should be Increased by a 

fnotor of 13. 

Thc magnotic moment is also knmn vith very hirh accuracy. 

The beet detorminations in! 

by Rahi nnd his cqllahwfltwq. 

e carried out at Cplinmbin University 

They ueed an ingenious method. 

Tlic meam.vemsnts wwe carriad out on rnolccvlcs like H2 or D2 in 

vhich the olqctrona nre paired and the effect ~f tha electron spins 

cancel each other. 

through 3 magnetic fields, 

gmous and d,ePLccts thc: molocular be~m, 

genotis. 

motion. 

h beam of the moloct~le passes c-msequcntly 

Thcs first 9.F these fio'tds is inhomo- 

Arovnd this field the nuclear spins perform a prwessing 

The t!i%rd fScld is Inhoni.1 nous again and brings the 

molecular beam back t~ c", 

The second field is horn&- 

fxus in Tvhich thc molecv1.@s moct inde- 

penden$ of tho spin qrientztim Q-' the nuclei, In addftim ts 

this arrmgewnt an osci?I.ntj,nr;t mngnetic, fi ~ l is d si1pcrp~secl on 

the hamogcnws mngnotic flr57-8. Th$a wcill ating fYo1.d iy.parts 

energy to the nucl.ci crlusinp thc nucZear soin tr, jump from ono 

stcte into c.nd,he,r. 

If this happons thc socmd. inhomogcnavs field 

will no lonpar compensate thc crleflnctfm cclv.sct3 by the Fimt in- 

11s flcld and any chanpc in winntption cmased by the 

oscil-lating f2GI.d Tlrj.11, glvo rlsc to EI diminishwl numbor of mole- 

cules arriving in thct "(3c1.t~~ A rhanyc: In tho nuclear spin ~rfcn- 

tRtfon w111 occw howevw mly i.p the frcqvency of the mcillating 

magnetic field is in resonance with the precession of thc nucl.oi?.r 

spin. 

This precession depends on the s'tl?cngth of the hr>mogonaus 

fiold, on the magnetic moment $.nd on tha value of the nuclear 

spin. The last quantity may bo detclrmincd from 

for instancc by tho analysis of tho spcctrttm. 

other oxpcTiment S,

159 

fn this way th.en, tho mngnetic rnomont mag be dc&rminod to a high 

accuracy, 

Actually *’he procisiqn of tho oxpor2ments is liniitod 

only by the 2cngth of tho rcpim of tho hqmogcnous electric fidd 

in thEt even the tins of flight of the moleculo in this region is 

short. 

T ~ P accuracy of sesonmce betwan an, oscillating magnetic 

fiold and precession frcqucntly is limited. 

Vory accwi?.to values 

haye been obtained- by the rs?.ztiva megnitvdes of rnagnct5.c momcnts 

proccssitm frcqvcncies of tho H and D nuc1ou.s. 

is used., 

A rm1”e complex plctvre l.6 Ilstained I$ the> H2 molecule 

Here thE C rotcltjnnal stnte can not bc vsed br.cnuse in 

this stc?.tc thc nuclear spins hwe opposite directions and the 

magnetic moments cmoonsntce 

state (J u 1) is t o be used. 

Tkorcfwo thc first rqtatlonal 

‘In this state also thc spin of the 

, ntzcloi add up to mc and th2To ~ ~ rcsult ~ ~ altqgothcr 1 1 9 mtnt1ona.l 

and spin stntcs, 

Thc em-rglc3s of them states me nf”ectcd by 

tho magnetic Interoeticm hctwwn thc prolma and ’nv tho h13tir)n of 

the ~l~ctrons induced by rnoLwvlnr rotatlm. 

Retweon the 9 statos 

mcntioncd nhovc, two sots of‘ 6 transitions are possihlt? and cor- 

respondingly ono dme~ves tvto sets of 6 dips of intensity, 

order to oxpl.ain thr. oxnct frcqiii’ncy at iKihJ.ch these dips occur, 

assumptjons have to bc! made abmt the intcractiqn hetirvem thc mag- 

netic rnommts of the nuc2.ci c7nA t;hc rnngnotic fic??.d pr?dt”.ccd by 

In

tho rotntkng molecvla. 

16 0 

Giving apprqprinte Values to thaso cqndi- 

tions the position of. .the 6 dips can be explained., 

Hnvfng deterrnincd the molocvLwr qilantitios that have an 

effect on tho nuclear mnkncts thc! position Q? 

rnay ba predicted.. 

in ivhjeh J 3 1 and also thc: spin fs equal to 

the dips In D2 

The ppodictions 

hawever did not cbock wit19 the experimental rasults me agroclroi.nt 

could. he obtqincd only bv assuninp an adcljtional term in thc enor- 

gy which depands m the sqilare of‘ thc cminc? Inclilc?cd by the nux- 

lcar rnqment and the mQlecular crlxis. This fs the kind c>f term that 

should be if thr? detttcrm h2.d a quadruple morwnt. 

Rabi, Rnrnscy 

and. othcrs have indeed beon led to as~rno an electric qvadruple 

mmnent for the deutrriurn; 

Par this mcslccir3,e /)ne may lnvostipato thc. St2.t~ 

Tho discovcry or the a1indrupl.e rnowmt 

Q?? thr deuterium 

csmo as a svrprise bocnusc cal..culntiwls on the htncllng cnorgy 

find other properties .of the deu.term were based r>n R 

symmetric model of tho deuteron whjclz did not porm!t 

of any ailndrvple mnmont. 

s.r’.mple 

the -xistance 

The mt:sciwe sf such t? moment shows 

that evon. in the simplest two-hqdv problem r7f nuclcar physics 

quadrup1.e f’orces appear, 

moments of thc simplest niiclei mmsured in units of nuclear 

magnltrons 

Thc: follqwinp tahLr? gives thc magnetic 

H! 2.785 

1 

n -1. (73 

D2 

@ 

855 

Tho negative side attczchcd to the magnetic mmcnt of thc neutron 

Is naccssary in qrdor that the pTotqn ann’ the neutron rnagncts 

lincd up in a parc?l-lcl way shovld give Dt least npnroximat*:ly 

the magnetic momcnt of the deuteron.

L A 24 (26) 

162 

Fourth Series:! "vc, Body Pmhlem Lecturer: L. C, CritchfiePd 

* 

The earliest atterpts to bitild a mwe detailed thesrv 9f 

orces were cmcerned dith explaining the binding ener- 

pies of the lirpht nuclei, particularly of the deuteron. 

It was 

nded that the bindinp energy of the helium nv.cl.eus was thirteen 

ti.ws that ~f the deuterm elthwph tlie nvv?"nr of attractive links 

Pmir particles is only six, a t mosts av?pared wjth one 

between twg particles, 

"ripner minted o3.t that the extraordinar- 

ily large bqn.ding in the alpha particle ccxzpared with that of tho 

indicated that the kinetic energy of the partlc1.e~ in the 

c?evtelt.lon almost compensptes the mutual attractim an@ that thls can 

be theGcase if the attractive f'wces have 3. very shwt ranFe. ioerq 

the ranpe of Forces is srrlal.ler than the wcive length of the gar- 

ticlea. 

In the alpha particle, on the Qther hand, the number of 

attrinctive bmds is prqportimately preatsr than the increase in 

kinetic energy, and a 1mer average energy Is possihle. 

EhQrt ranpe fwces between nuclear part4 cles makos possible 

many simplifiqatims in the tlieory of the deuteron. 

Essentially, 

we may say t!vt if ncvtrm and wotm are farther apart than a 

distance they d . not ~ inf'ltience each other ap:-reciably, but at 

separations less than g strxtFr; attractive ~orces act and there is 

.

the expct s%pe of the reeion cannot he important tq the des- 

criptisn of the stater Accmd.fnpXg, it my. he nssl?msd that the 

patenticll energy is a c?nstc?nt Vas inside the rad.ivs 

outside it. 

Let r be tho rolative cmrdimte of tlic urqtcm. ?v!th 

3rd zero 

resmct to the n.evtron, 14 the mass of" &?tm m novtron, znd E 

TrTc accordingly let

The solution Is then, 

164 

their 

smoothly 

I 

I 

a8 Vo 

Thus the 

the 

by protons, It i.s tlwn assvmcd thzt i s t1-m

* 

been considered, 

165 

So far, only tho space coomjinates af the particles have 

It is well established that; ths ground ntat@ of 

the deukepon i s a triplet, i&@J tho spins of neutron and proton 

a m parallel, 

let state. 

Thus the above rough calaW,ation applies to the trip- 

Furthermore, the wave fu.nction asawned i s spherical-ly 

symmetrical SQ that no quadrupole moment is obtained, 

In spite of 

these defectsl, however, tha scattering induced by the forces thus 

derived will be calculated rand cornparad with experiment. 

oomputa 

Scattering by short range force$ bS particularly easy to 

especially if' the wavewlcngth of th3 colliding particxes 9s 

1ong'e.r than tho range of forces, 

Under these conditions a flnite 

orbital mgular momentum will prohibit the particles from coming 

close enough together to be akttracted and there is no scattering 

if the particles pass each other at a distance larger than a wave- 

length. 

Consider a plana wave for the neutron incident upon a pro- 

tan. This wave represents a aefinito relative velocity v of 

ftcollision" but for all possible values of Z;ho impact parameter and 

hsnce all angular rnorqent. 

According to the foregoing argument de- 

flection will bo expcrdcnced only by those collisions of ZDPO 

angu- 

lar mmcmtum, ire, in ths spherically symmetric state. h t the in- 

cident beam be ropresented by 9.. ikr cos0 & 

Tho spherical part 

oS'qin absencc of a potential well. can be determined by averaging 

over all 0, 

yoo = sin kr 

-E?-- 

In tho presence of a field the wave will be strongly refracted fn- 

side P = a and, so far as the wavo at 2aqp distavlcos i s con- 

Lc 

carried, tho effect will be to introduce R phase shift e The 

e 

general form of tho spherical part wlth potential woll is thon

166 

ab 

with 0 and 8 to be doteminedl, The wave yv" must represent tho 

spherical part of the inoidsnt beam plus a scattered wave, and the 

Zatter must have the form Seikr/kr. kence 

The number of scattored neutrons per unit volma i s then 

2 

sin 6 

--zET-- 

k r 

and the total number of particles mattered by the proton per second 

k 

The incidtont cii.n~:=lt eensity postulated iu v klence the effective 

cross-seck3.cn :xaaoni;ed for scattorlng 9s 

k 

Dekormination of .5 can be made approximately a8 follows: 

the banbarding energy of the neutron8 is aaawnod to be small campared 

with depth of the potential wall, hence the wave function 

inside the range of' forces, 3, is substantially the same a8 far 

the itable deuteron, To join functions smoothly? therefore 

+-

e 

d' tho bltidihg energy bf t;ha dezrt6,rop 

Thus the binding dnergg of the deuteeon determhed the scattterihg 

crosarsection, at least if the callision is suffidiantly slow4 conid 

' 

parison with experiment, howevsr, showed that tho predictsd CPOSBsection 

is a little hi'gh at high enepgy (E>2Mv) and about a factor 

10 too small at low ehe~?gyi To explain the discrepancy Wignsr 

pointed out this cross-section 19 calbulated kroni what is known 

absbt tkie b9ipkek btate of the deutehn Only and that it is posd 

il 

sible to ohooao a I'biilldlhg'' dnergy, for die sihglat skate ih 

such a way as to account for the experimontai $esult$r Since Lhi-oe 

out of four collisions between neutron and proton ape triplet colj 

lisions and on8 is singlet the result for the complete cross-sac.- 

tion is 

E is tho energy of collision in ths center of mass system and hence 

is onowhalf the bombarding energy if the prokons may bo considered 

at mst. Thc value of ET needed to fit experiments is of thhe order 

of 0.10 MeV but the sign of it is not determined becauso sin 

2 

6 

is determined from the aquaro of tho logarithmic derivative at a, 

By taking into account the effect of the finite kinetic 

encrgy inside the distance a the expression for fbecomes, in 

next approximation

168 

Al%hough the spherically symmetrical solution for the 

1) 

deuteron must be incomplete because of the existence of the quadru- 

pale moment it represqnts the grosg propertias quite well, 

Bof'oro 

considering refinements of tho theory, therefore, it ia worthwhile 

to gtudy, the implicd dependence of tho force between neutron and 

proton upon their relative spin orientation, With parallal spins 

thsrc is a binding energy, 2.16 Mov and with opposite spins 

(singlet state) the binding enorgy is very close t o zero. Thus the 

relation between V, and a2 obtained in an approximate way abovo applies 

vsry closely for the singlet state, For tho triplet the cxact 

relation determining V 5 V1 is, in the units used above, 

Thus for 2 = 1 the potential wolf. i s almost twice as doop for the 

triplet state as for the singlot. This indicates that the spins 

of the nuclear particles play a dominating role in the attraction 

between neutron and proton, 

LA 24 (27) 


LECTURE SERXES ON NUCLEAR PHYSICS 

Fourth Series: Two Body Problem Leoturer: C,L. Critchfield 

LECTUm XXVII: 

I) SPIN-SPIN FORCES IN THE DEUTER 

ON 

PRO,TCN-PROTON SCATTERING 

rx) 

@* 

T. 

The theory that has been developed for the deutei-on is 

generally satisfactory except for describing the electrical quadrupolo 

moment. In order to aocount for the quadrupole moment Sbhwinger 

used a spln-dependent interaction similar to that between mag-

netic dipoles, 

The wave equation then becomes 

169 

whore 

+ $1 

SN is the neutron spin operator (Pauli matrix), 6p tho pro- 

ton spin operator andvdetomines the amount of the spin-spin 

potential, 

;I 

Sinco the additional forco introduces components of ? ' 

In a rion*spherical combination it $s in general impossible t o aasme 

spherical symmetry f or the solixtion, 

For the singlet staee of' tho 

dstiterond however, the dipole-dipole interaction cannot affect the 

motion of th.e papticles and its average value is zero BO the same 

equation as before ie obtained and the same relation betweon Vo and 

a2 holds, 

For the triplet state the intoraction does not vanish 

but rather tends to change the spin directions of the particles and 

hence a380 thoir orbital motion, 

The intoraction is invariant to 

reflection in thc center of symmetry so the orbital stato into which 

the dcuteran might go is limitod to even parity and the only pos83.7 

bility l a a D stater Symmetry of the operator In 6j, and bp shows 

that tha spins of the neutron and protan remain parallel and tho 

solution will bo a mixture of 3S, and 3D, states. 

Tha mount of 

3D, that i s mixod with the 3S, to obtain the lowost snorgy is detsr- 

minod by the zliee of "p which is, in turn, choson to give the Correct 

value of the quadrupole rnmont, which is 2.73 x 10 -27 2 om 

Thc observed quadrupole mmont is positive and that means 

that the charge distribution is "cigar shapedVrs Mathematically the 

quadrupole moment is defined as tho average of (1/4#8 2 - r 2 ) over 

the charge distribution ob.t.ained from the c~olutlon of tho wave

~ 

e, 

equatlsn. 

170 

Qualitatively it is readfly demonstrated that when two 

parallel dipoles interact the energy will be lowest; when they lie 

on sn axis and that suoh a ,coqffguratlon 5s brought about by in* 

em6 between D waves and S w8vea. The pwt of the D 

wave that varies a8 3 cos2 62 

*I 1 will interfere with the S wave 

becaqse the spins are tho setme for the two waves, 

Along tha axis, 

ther8fOm, amplitudes add whereas the parts of the D wave that 

are large near 0 z (H/2)(mL>O) 

netic quantum numbers and intensities add, 

one phase of 

do not have identical spin mag- 

The result is, with 

D wave, a prolate charge distribution about the 

spin :axis and with the other phase an oblate charge distribution, 

The sign of" is thus chosen to glve bQe prolate distribution and 

-13 

the valus of y appropriate to a range of force 2,80 x 10 om 

(a 1) is Ot775, The corresponding depth of potential well is 

v, 8 27 mo2 which is close to that for the depth of the gislglet; 

2 

W e l l Vor 25 m@ 

In faot Schwingar points out that if the range 

of forces had been taken 2.7 x lo -13 cm the two potentials would 

have been the same and the entire dependence upon spin of the 

neutron and praton is imparted by the dipole-dipole Interaction, 

It may be noted hers that if' this dipole interaction had been due 

to the magnetic moments it would be of the opposite sign and its 

ed value much smaller than that found above. 

3 

The amount of D-wave in tho lovest state ia found to be only 

4 percent, The charactoristic influence of the dipole forcos on 

scattaring, capture and photoelectric disintegration are corroa- 

pandih@y mall, mounting in general to 2 percent corrections, 

Formulas obtainod in the simple 

S wave theoyy are thsrafore

3-71 

0 

adeqyate for most calcu5ations and fail only 18 the true intcrpretation 

of the forces and in predicting the quadrupole moment, The 

ble effeot; of :'this evidently vory anal1 adrnlxturo of or'bfion 

i>s that gsssntl8lly the same depth of potential well 

obtain8 for bgth triplet and singlot state. In other words, the 

difference in depth found necessary in the spherical states is 

just made up by tho strength of the spin coupling but the '13 state 

to which tho 3S is coupled is very pd t;o excite, . 

A reasonable thaoq and method of calculation is then available 

for the deuteron provided that the energies are not too high. 

At high energy of collisian the potential well can no longer be 

oonnidered sna1.l compared with a wave length and the shape of the 

potential CUTW will matter, This energy is roughly equal to the 

depths of the potential wells, say 10 MeV, in the canter of mass 

Actually, the noutron-proton cross sectim has been obt 

24 Mov (12 Mev in the c tep of mass frame) by Sharp 

results are fairly well accounted for by tho simpla theory 

g P- waves, Abovo t the effects ?f the shape of 

and' of tho response t ions with highor angular 

momontum should bo in convinci 

\ 

.II) 

TI 

The range of forces bbtwepn heutron and proton h8s been taken 

to be 2.8 x cm (a l)'* The ape two methods of establishing 

this value For the range, The first. to be oonB€dered is the scattering 

of protons by protons and detection of deviations from the 

Rutherford law. 

tho range of foroes fn the deuteron

from khese results assumes that the rangas are the same, 

second method 

and Ha4 

As in 

involves 

the 

the cam Q f 

onergies available are 

making estlmates 

O f the deuteron, 

The 

172 

of the masses of H3, He 3 

neutron-proton forces the experimental 

capable of f inding onSy the nuclear fQPCeS 

exerted between two protons when they collide Zn an S-state, 

interaction may therefore again be represented by a "square well" 

but calculation of the influence of such a ffw~llt' on the scattorad 

distribution I s BQmQWhat more complicated for two charged particles 

There are three reasons for the complication : The first is that 

the 908. t te ring affect Of the "we 11 i s superposed on that of the 

electric fields so that intorference terms appear in addition to 

thoss due to nuclear forces alone; 

The 

the socomd roason is that the 

Coulomb farces are long range and modify the inciden2; wave even 

at infinite distanco so that the simple cxponentials are not solu- 

tions at any distance from the point of collision, 

In addi ti on, 

the collision of two protons if influenced by the statistics of 

the particles and a lsecond klnd of interference is introduced, 

The Fermi-Dirac statistics admit collision in a IS stat6 and ox- 

3 

clude the SI so we are here concerned only with the nuclear forces 

between protons Of opposite spin, 

is - 

The wave equation of the reduced system in the Coulomb field 

vIr,e;y)=u . . 

M 

where v is the relative velocity at infinit@ separation. Let 

3 

metrical, part of the solutionr Thon the equation for u(r) becomos 

dr

have the same asymptotic exponential form RS vi except for a 

IE 

ohan$Q in sign, 

This is the case if 

Tho sign and the phase,B are not &yon by tho considorations 

buts is a function of velooity only, The canplebe theory shows 

The spherical portion of the wavo in a Coulomb field ha3 tho 

asymptotic $om 

The effect of 

tb.e forces between protons will be to intPOdUGe a 

constant phaae shift Ko 2n such a way as to leave the incoming part 

of u unchanged, 

“he tom OP the wave that has been influenced 

by tkie nuclear forces is therefore 

Uf t ,u3iiK, 

The asymptotic form of the perturbed wave then becomes 

e 

If the particles were not identical tho crogs section would 

be gtven by the absolute square *of the curly brackets, 

In classi-

caL theory, &he cross section for identical particles is the sum 

of the c,ross, seckians a('@) + de+ 0). 

is an additianal ponsideration to be mader 

175 

But in quantum theozy there 

If one praton is at ra 

and the other at rb the wave function describing the state is 

~~('a) *JrG(rblG yl(rb) yz(Pa). 

in the triplet, 

Therefore (since rate) = r,@+@)) 

tho saattering 

at R-8 which is thesame as at%+B interferes constructively with 

that at 8 in the singlet colldsLons, i,e, 1/4 of the'time, and 

interferes aastructively in triplet collisions (314). If we call 

the quantity in curly brackots f',(0) the cross section for scattering 

in unit aslid ang3.e at 8 then becom s 

This effect w88 first discussed by Mott, 

(g). 

+ ' ~ ' 

The first term is the pure CouZomb scattering# the second the pure 

0os(&a,0s2(1/2)0+ 

cos2(i/z)e 

potential scattering and the last the interference tam. TO get 

the woss*8ection for scattering between @ and 84-68 multiply by 

2flsiln 8dB; and to obtain the cross saction in the laboratory 

Ko) 

, - 

j 

1

c 

system replace every 0 by 20, 

taryf and a$ this angle 

176 

The largcst effect of the potehtial 

At a million volts colllsion energy? is about 1/6 and it becomes 

smaller as the energy increasos. The effect of a $ma91 phase 

shXft due to nuclear forcos is therefore greatly amplified at 

these energies,' Further, due to interference, the sign of KO is 

determined .'I 

Experiments on proton-proton scattering have been carried out 

with good precision by Tuve, Hafstad, Hoydenburg and Herb, Kerst, 

Parkinsdn and Plain.. 

The latter have carried the collisiovl energy 

up to 2.4 Mev where the ratio OF cross soctions is 43 and KO is 

480* At 1 MeV KO 2 33O,, Both experimental groups accelerated 

protons by electrostatic,generators a d scattered them in hydrogen 

gas rrlaking counts at several angles to the beam, 

Comparison of the result8 obtained wieh those that are expected 

frm a square we31 can now be made,: For this purpose,, 

however, it i s necessary., in general,, to know both the bounded and 

* - 

unbounded solutions t o the wave equation in the Coulomb field,.. 

These solutions are called 

F and G respectively, 

The phase 

shift, KO, has been defined to relate to the asymptotio foms of 

Fo and W0 for Sdw'aves which are taken t~ be */2 out of phase at 

.-

* 

inside 

177 

large r;. At the boundary of the.well the combination Fo COS 

sin KO must fit onto the wave function, Fir that applies 

the well 

and this determines KO& 

Fy = bit G& tan K, 

'";L T+ cfo tan K, 

If we g.ssume that the Coulomb field has 

not affected Fb and a, 

greatly we gat tho usual ralation 

9.k.Q Mevk 

Breit and others have shown that about 0.8 Msv sh6txld be 

added as an average effect of the Coulomb repulsion insicto thc WGLL 

The more exact calculation gives 11L3 Mev as the dspth of the well 

and also shows that good agreement w ith results at all energies is 

obtained, 

The rough method used would be applicable only at 

energies very high compared with the Coulomb rapulsion at 

Po = 62/rnC 2 4 

Using the exact wave functions a. search for the best value of 

Po has been made by Breit et el and the decision roached that 

2 2 

0 /mc is the most acceptable (a = l), A value of B 0 cy5 

gives a noticeably Inferior fit as ala0 does a = 1425, The depth 

of welS pertaining to a = 1 is 11,3 Mov in good aereenaent with the 

most camful estimates of the depth of tha singlst deuteron well 

of the same width, 11,6 MeV;

178 

e 

December 301 1943 

LA 24 (28) 

LECTUFG3 SERIES ON NUCLEAR PHYSICS 

Fourth Series! Two Body Problem Lecturart C. Lc Critchffeld 

pwrum xxvIIr: THREE BODY PROBLEMS 

A second method of determining the range of huclear forces is 

obtaihed in estimating the binding energy of the nuclei containing 

three partic&esBH3 and He3, 

The binding energies of these nuclei 

are approximately equal, the difference being caused 

tho electro 

statlc repulsion of the two protons in He 3 +* 

Since an attractive 

force bet'ween two protons hag been demonstrated in scattoring experiments, 

it follows from the equality of bindlng in H3 and Ha3 

'chat the attractive force exists between neutrons,, Attractive 

forces between all particles in these nuchi and in He4 are 

also found necessary in getting agreement between observation and 

the estimates, 

The method of 

c alculn ti ng 

binding energies that we shall ap- 

PlY 

is to 

problem" 

reduce the 'chreerbody problem to an "equivalent 

certain as sump tions and avo raging over tho 

two body 

third 

particlo. 

The method was first used in this connection by Feon-' 

berg, Consider the nucleus H3 which has a binding energy 16.3 mc2. 

Assume that the wave function that describes any two particles in 

H3 is the same as for the deuteron* In order to exprcss the douteron 

wave function as a single algebraic function tho form of the 

two particle function will be assumed to be 

whore (2/d)3/2 is essentially the radius of the wave of the deu-

179 

tsron, 

&rCher asmane that the wave fixnation for a19 three par- 

tiales is a symmetric product of throe such deuteron functions in 

the three separations r 328 q3 and pZ2;: 

PO8 

ten 

one 

hihe 

I 

%e potential eneygy af' eaah particle ~$32 depend upon the 

ition of each of tho other two) but an effective two body PO- 

Carrying out tho integral OVBP dT3 (the volma elomont of particle 

particles is the sarne as in the deuteron except that the range of' 

fomas is a larger. 

The total potential is three times that 

between any two particlea, 

In Q similar calculation the kinetic 

sner$y may be averaged and the result is that the fotal kinetic 

is three thes that in the d'euterqn, 

We shall apply the results obtained above to determining the 

equivalent solpkion with a ttsquaro" potential well. Let be 

the binding energy of I?; the wave squat$.& far the two body

problem "equivalent" to H3 is then 

180 

a r 1 

- 

M 

Ar -- 

-I 

mixture of singlet and triplet potentials, The two neutrons is 

€I3 are certainly in 8 singlet state and if' the spin of the proton 

ia parallel to ons of the neutron spina it forms the triplet with 

that neutron and half triplet and half singlet wlth the other. 

The expected relatkon between vT and a can be determined from the 

deuteron calculations and is a simple avorage of singlet and trip 

let relations 

a- a 

This average is based on the assumption that th8 affect Qf the I3 

3 

wave in i~l the same bctwoen poirs of particlea in tie triplet 

state as in the dmateron. 

The other relation between vT and 

from the oquivalent two body equation for H 3 

a may be obtained 

That equation may 

bo mad@ formally the sme QS for the triplet deuteron by substi- 

tuting 

* 

wheree is the binding energy of the triplet deuteron. Then the 

D 

relation between X~V a is the Sam8 as botweon vo and 

T

a for the triplet deuteron,,% i . e . q 

The approprlato VdUe Of x2 is 0.78 and 

This, combinad with the relation obtained from tho deutoron (H2) 

has the solution 1 

a = le08 

for equal d@ptbs of well4 vT, The valus for the range of forcos 

thus obtained is hiphcr than *hat; indicated by proton-proton sent- 

taring but only by 8 porcsnt, 

and the result obtained i s satisfactory, 

The method is very rough, of COUPSO, 

If a neutron-neutron 

force had nob been postulated the agromont would be unacceptable, 

There is a dif'feroncso in binding betwaen He3 and H3 

about '76 Mov that should bo duo to the CouLomb repulsion of the 

ppotons. 

DE : i. 

We may estimate tho expected rspulsion as follows: 

, 

oO-(3/2) ( drBf 

( Q2/r 1 ( r2dr ) 

.& , , , ~,, : , , , 

$m-(3/2,(t/re) Q 

dr 

."(%) 

, , 

2 represento the reoiprocal square o f the cxtent of tho wave 

1/2 

of 

o! 

f'unctfon and mag be taken approximately aquaa t o MG/fi2 

= 0,43 in 

units of m 2C,/4.. 

4 

e 

This givas Or.65 Mev for the calculated Coulomb 

energy in satisfactory apoomont with the oxperimsntal value con- 

.sldoring; the approximations mado., 

Similar throe and four body onloulationa have been made

y tho variational method with substantia y the sane result for 

102 

oarroot accounting Qf the bi 3ng ensrgios or H3, Ha3 and, 

There is another ty 

tal interest to the thoory of' 

i 

a-body problem O f fundman- 

on, This is the scatter- 

neutrons by hydrogen rnoleoufes* The theory bas been worked 

out 'by Schwingor and TB Ejr and haa ahown that the scattering cross 

of tho molcculs 2s oxtranslg sensitive to whother the sing- 

to of tho deuteron i8 real or ''vtrtual". 

Experiments on 

scRttw4ng of liquid-air notltrons havc been intorproted by tho 

thoory to prove concZusivaly that the singlet state is virtual, 

ieO*, thePo is bawd SlnglOt Stat@' 

Tho groat decSsiv 

cause af' an accidental near cnncollation of 

tion, 

tho exprimant come8 about be- 

tams in tho aa1cul.a- 

Details of tho cnlculntions will not bo given here but tho 

principal feature of them Is adily deacribed. The cnlculnted 

aross section for noutron-proton scattering duo to the triplot 

-24 2 

wo11 aXona is 3*50 x ZQ at low enovlgy whoreas the observed 

-24 2 

La 53 x 10 ern From Wignor's hy.po$hosis the singlot 

soattaring must have a CPO~SS saotion 41,5 x lo2* om2, 

If 

ia 

* orda 

tho triplet and q tho singlet CF 

0; 11*8 [r; 

Scattoring ?Prom the hydrogon molecule will show inter.. 

etwoon spharical waves coming from tho two nuclei if tho 

th of tho neutron Is long compared with tho internuclsar 

separatian, At room temporaturs tho wave length i s of the sana 

do 8s the scgnratlan. Calcvlation of ths CPOS8

and 

mounts to 41 if tho singlet stab i s stablo, ire', bo and 

havo tho smo sign, and only to 0,38 if the singlet is unr 

ro is thus d otor of ov@r 200 in cross section de- 

pending upon tho atabilitg 

of tho factor is due to tho 

the singlet stnte and thc large size 

act that sin6 2s so nearly equal to 

0 

-3 times sinal* catterinp; fj=* 3 = +J. 

5 . 2 

The right; side is 53 for tho stablo singlet nnd 20 for an unstGabSe 

3 irqb t 

Tho oxporimontal proce ro i s to cornpara 

Of para-hydrogen with that of orthohydrogen (or rather the USURZ. 

1 for llquid air 

to the furtho?? oomplicat 

ion is subject 

tpons may convert 

one kind of hydro@n infm the othbr, fn this cam the wavo 

function of the final molecule has ono si n if tlzo spin of' one prochanged 

and the opposito sign if the spin of the other i s 

The interforone 8 then dsstructivo and the CTQSB 880- 

tion propoptional to 

O"conv rrd 

(SW 6, - sin0,) 2

185 

09 Qourso the oonvorsion is accompanied by n. change in xwtational 

state because of the atatfstics, 

%onv 

Sclzwirrger an4 Teller have shown 

is of the 883110 order of magnitude as 

0" 

when tho parahydrogen is mostly in the ground state, J 

whm tho nautrons have su 

state i s *023 

00 

so that only 

0, and 

low energy w i l l its CrodS see- 

so oxtrornely small for tho cas8 of an unstable singlot 

n, %e energy required tb rniso 8 para molecule to ortho 

e unablo to oonvort 

octron volts, hence liquid air neutrons (,012 ev) 

pa to orbho 

perimsnt is Chat the acattering cros~ seotion OF ortho-H 2 

much larger than that of para hydrogen. This provos conaluaivdly 

thak tho singlet deutaron is not stabla, 

Exautly the same kind of considerations have boon mado by 

Schwingor Tor the possibility that tha noutron has spin f3/2jf\ in- 

stead of (L/2;)* * 

onergy would then bo a quintet, 

%e Icv of the deuteron that lies near zero 

In this case however, the fortul- 

tous oancekxation of toms goas not OCCUP cnd all ross s~ctfons 

in molecular hydrogen am a$' ths same order of' magnitude, 

the experiments prove not only that the singlet; state of thc deu- 

teron is unstable but ~lso that ths ncuhron spin $8 (1/2)fi a 

24 (29 & 30) 

Thus 

& XXX: INTERAC F NEUTRON AND PROTON WITH 

Neutrons and protom interact with the fields that hnve 

tablOshQd by classical physies nameiy, tho gravi tati ow~l

1 

sB Oravitattonal phenomena am 

ar procemes. 

zhe 

hand, am very 3ngaFtan 

captum, emhaion of gnmma 

G dissociation are 

Trs. addition to thsse we 

s. One of thsao fa 

its rnagnotlc moment and th& 

ts magnetic moment, 

s there are 

fields that influance tho nut- 

e~vatione on th. 

tho field of moson8, 

o of this thoory ia to m- 

e nature of 

beta activity is cirawnacrfbcxl to sornc oxtant, Some fundamental , 

sa of the neutran is a PO- 

tho deutoran. 

Also, RccuraCe

I 180 

tea particlea, l/fs is the WBVB function of the ground state 

of the deuteron and e, FQP s;tmpLlicity we shaS1 assws 

q8 in tho f ~ m 

Y?

189 

is 

1 = c(E2$, H2/8m~) e , E Z / ~ * ~ ~ 

and the CYOSSI section for photo electrtc disintegration (only one 

polarization is effective) 

If we had normali2ad the neutron-proton Pmwave function8 to unit 

@ndrgY the factor pp would have been unity, 

Calculation of M is straightforward. Consider thO die 

E 

pols moment In the direction, B:: Q, 

ME sJy8(@I'/z C O s @ ) ~ d ~ 

D 

neutron and grolj'on and T, oxpressed as a function of E andE 

1 

becomes 

% = /8T/3)(@2/Cio)( fi*/Ite ) 

(@* 1 -t E/€" )3 

For the 3. -rays of Th C ', used by Chadwick and GoSdhaber, the cal- 

culated CTOSS soction is 6.7 x lon2' cm2 

To the CrOB8 section rE must be added the O~QGS soction 

due to elfact of the magnetic v@CtQP,CM, The perturbing enerey 

is then p. €i, and the matrix olomont, Mmj contains not only tan

190 

s bat a sun over t spina, n o 

ts to Et 1s state or poai.t;ive 

vel of the doukeron in tho poaj, 

is very Large 

at low onorgy 

relative weakness of t 

ron magnetic mamonts to bo

I) 

neutron by proton, 

198 

* (€3, / $B) ( h2 BA/ 2 

of photo disintegration ilq simple radiative capture of 

Syatam A is composod of one photon and tho 

stable deuteron; %he photon is capable a$ two polarizwtLons and tho 

n of three, hence eA 9 6, 

ono proton cacb of 

sect 

apturo 18 than 

8gstsm B aontains on@ neutron and 

Tho mom 

3/2 ( 2*d/kc ' (a;.,+ % ) 

ectxio and magnotic effects have dilfsront dependenciss on 

tho onsrgy of the nsutron-proton gygtern. 

At htgh energy $he mag- 

y low energy,. howover, tho 

gnotic offact ppedominatad and 

rc: 1,4 x 10 -28 6 E fE? 

utron at. 1-00112 tompernture, I/~U c,v,, E 2 X[~O Q~v,. 

*24 2 

% = .,I8 x 10 Gm 

lrYhcn the noutron was diacravared thc way wa8 opened for 

ament of a field thoory of beta radiation, 

Foml In analogy to a&ocCromagnotie themy, 

mi# was 

!bo fundamsn- 

ptions am:" . 

nuc lo on ) ., 

ron a,nd proton 8ro two 

(2). Energyr spin and stati6tic.s am oansorvod in bsta 

111) pRd€Gtlon by tho introdpotio f the Pauli mutrino, 

($1 T ~ Q rest maeg OS the nautrino is zero (or nsarly

its spin is + 

t with the' 

o are radiated when neutron 

1 

and a neutrino are radi 

n.gw to proton an6 8 positron 

hangoa to nazltron. 

(6) Elslotran and neutrkno sharo tka availab onorgy in 

a11 

sible pr6partions. 

Xn analogy t o the inte 

n botween chargad particle8 

and light quanta, tha 

(26) 

numbor of 

w = g2 

matrix element c 

e 

The interaction eon8 

dypam the amp 

at thg YlU(J2OU8, p ep%ikkoa 

*iC 

and/0,, are tha

194 

and, as in tho case of 

dou tar on 

the photoolcctric disintegration of thc 

Tho shapo of the electron spoctrum is thus determinod,, 

Exporimen- 

tally. tho nwnbor per unit enorgy (or unit momontum) is ma8surods 

divided by pE or its equivnlcnt in the Coulomb field and tho square 

root of the resulting nmbor plottad against E, 

In tho carefully 

done work with nuclei that hnvo only ono beta activity tho plot is 

a good straight line, tha Intercept of which W, Thc thoorg Sa 

thoraforo substantinted by tho exporimcnts, 

If the nuclonr charge is lcrgcr than, say 20, or if tho 

electron encrgy is particularly low the sine wavc 'Is not a. good 

2 

approximation for tho clcctron W~VC,. 11 mom oxact value of c 

is plus the charge nwnbqr of' tho residual nucleus If a positron 

9s emittad and minus the chargo number for cloctrun emission, 

Tho probability of omission of an olsctron with energy 

botwcon E End E -k dE is

4 

195 

The total probebi 

trino per saoond is gftren by the integral OF P($)dE OV8r @l@ctPoll 

'energies 

EI ma2 to E = W, . For ~>>mt@ th$s integral 5s 

czosaly equal to 

+ 

h z 60Q c fi 

w5 

Tho decay constant, A, $a proporblonal to the fifth power Of 

total, energy relaased by the tranaftion, 

Moasuremgnts 

the half livas of the light radioaotive aloments makes 

a detomination of the ''Fermi conatan 

mare'is a class of positron mittera "chat 

larlv well suited far deternining gr 

JZght radioaotive yzucolol. c 

i s p 

This clasa oomprisoa the 

tha 

taining one mope proton than neutron, 

ra proton turns into a neutron with the emission of a posI* 

tron and, since the nuclear forces am avidontap the sms between 

all pairs of paPticlos, -bho wave Punotion. of the neutron in the 

final nuoleus should be very neBr3y tho same as the wave funo.t;ion 

of tho Initial proton, 

In thia 

nu~lei, their half lives and cncrglbs roleasad am give 

on tho following pago, 

nttng two*pOssiblo final statoar 

e I.I" 3' 

A table of thsse 

In some Instances two oncrgiQs are givsn 

g the Wansltlon mloasing tho most energy, 

The life time Is dotor- 

tion goes to tho excited etate ths nucleurj subsequent1 

IF tho transir

I 

196 

MAXIMUM KINETXC 

sR$xa 

1# I6 

93 m , 092, 2.20 

52'6.0 8 ;LJ 

? b 1 

2; 20 

% 

11.6 I 2m82 

1.7, 3,o 

3.54 

3,. 69 

J a b 2 8 fl 3 ,,87 

2,4 I 4.33 

3188 s 4.43. 

e07 s 4.94 

We have assmod thkt klucbron and neutrino are emitted 

lizLi angular momsntum, 

the omitted park5 

The rnaxbwn amount OS spin that 

8s is, t,horoforo,% . Zn order 

the nucleus bs ohmged 1 

WOOM rloutro 

lsld contain tha 

tog, 

Rmnl'a oI?t?igi 

trnn@tion from o m spin t o anot 

locttxon or boCh 

to bo am$tted with orbital. 

il) 

e 

93% amp5iLudtm of wavos of highar wngulor momcn-%wn 

knallar at'tha ntzozoar radiua than the amplitude of tho

e 

s 

I 

and such transitions are tsmed "forbidden". 

197 

The half-ligo 

of a forbiddon transition is therefore much longer than that of an 

allow'ad 

appears 

transition of tho same cnorgy release, 

Certain nuclei produce allowed transitions alkhbugh it 

certain that the nuclear apiM changes by$). 

This is parti- 

cula~3y true of He6 which must have spin 0 but decays at an "al- 

lowed'' rate to LIB with spin%,, Xn Fermi's theory such a transim 

tion would be forbidden, 

To acco 

of span k l# Gmow and Teller postulated a spin dependant interac- 

tion with tho electron-neutrino fiQld, 

t fop transitions with cha.nge 

The colculatlons made thus far apply only to allowed 

transitions in light nuclei. 

To oxtend thorn to heavy nuolQi and 

bidden transitions more exaot accouet must be takon of the 

wave functions for relativistic motion in a Coulomb fiold and there 

are aevcral interesting fcaburas of the solution that are Drnpha- 

sized in betadecay theopy. 

FJELQ 

THEORIES OF NUCLEAR FqRCES 

The SUCCBSS of Femfrsr theory of beta decay promptsd an 

a't;taqst to explain the forces between nuclear pccrrticles by the 

same interaction. 

Thus the olectron-neutrino field should play 

the same role in t h ~ attraction between nuchar particles as tho 

eloctromagnotic field plays in the force8 batween ohargsd particles 

At the same time ths anomalous charaoter of the rnagnebtc moments 

of neutron and proton might also bo explained by the interaction 

with this field (~Viok). 

The attempt was not successful, howevc 

For two very good reasons, either OF which fs dufficicnt, 

the forces calqulated are much too weak t o account for the ob- 

First,

sorved strength an6 range of 

198 

nwlear f'oroes; seaondly, ths theory 

gives attraction, In first approximation, only bekwcen neutron and 

proton, 

The forces betwoen like particles nppaar Zn second appro.. 

ximsltion and are repulalve, 

Two linos of endmtvor have been pwausd in tryZng to con* 

struct a field theory of nuclear forces. 

One generaliaod the beea 

bntoractionr 

, 

Thia was first don@ by Garnow and Teller, and by 

Wentzol, who postulated an intoraction with pure ePoOtran fields, 

80 that neutron and praton would be able to croate an 6leotron- 

positron pair as well a8 an electron-neutrino pair, 

This same 

type of "pair emission" forces waa lator extonded to meson pairs 

(Mllarshak) on ?;ho asaumption that tho meson has spin 

On acw 

count of the flexibility in SntroducZng new fields th.ose theories 

are able to account for ths nuclear forces, 

jsctionablo features of the results, howevor, 

There aro aavoral ab- 

In the electron* 

positron patr tbeory, for exwpl8, special forms of tho states into 

which tho cloctrons are arnittod have to be assmod in order to 

avoid scattering of slow neutrona by the atomic electrons* In the 

meson-pair theory the cnlculatod CPO~S section for scnttaring of 

cosmic ray mesons by nuclei is higkor than that observcd excopt 

For very weak interaction Eorcos (and a rather long range for the 

forms), Them is R general objectton to the mson pair theory 

that obsarvatiohs ahow about 30% mare positive mesons at sea SevoZ 

than negative onc8, 

If they were croatod in pairs tho numbor 

might be expected to bo more nearly oqual, 

Pending further oxparl- 

mental ovidenco of 

the spin of tho meson, howeivw, thc possibibity 

of a nmson-qpair f

e 

Tho othor class of attqnpts to describe nuclear forces 

by n field tlzeory is typifiod by more direct ganeralizations of 

electramagnotic theory, 

sented by Yukawa, in 1935, before: the meson was dlhcavered. 

Yukawa found that the range and strength of nuclear forces could. 

199 

Th@ First theory of this class was pre- 

be understood if tho nuclear particles omitted "hsavy quanta" hw- 

ing a finito rest ma98 of aboub 200 times the electron rest mass. 

This quantum was conoeived as having Bossdstatistics, the sgm~ a8 

a light quantum, but for simpl;lcStg the spin was assumed to bo 

zerod 

hrthermoro tho heavy quantum could have one unit of elac- 

tric charas posittve or nogativs and. Yukawa suggested that the 

heavy quantum could di sintearate intg elaatron and neutrino glvivlg 

beta radiation. 

The discovery of tho realmeson, of about tho samo mawt 

by Neddormeyor and Anderson stimulated interest in Yukawafs theory, 

Then it was proveh that chargod quanta of zwo spin gave repul- 

sion between neutron and proton in the 33 deuteron, 

To correct 

this obvious defect the spin of the quantum was as$umed tic) be 

unity, 

The free apace wave equation9 of tho meson of spin one are 

therefores very sim91ar to those of a light quantum, ike. 

MaxwoLZ's 

equations, Thus thore will be the vector field quantities and 

I H and potentials for tlilose fi@ldsyand &, 

The inclusion of a 

finite rest mass) however, introduces a characteristic length whioh 

we shall denote by l/&where 

This Lengthl may be. usad to generalize Maxwell's squat$.ons for froe

Thrw ctlffkwent assumptions about the tma electric 

202 

charge on the sqson8 are generally 

sidered, 

These are (3.) Utn- 

charged, (2) either positive or neg6ttive (+e), 

(3) both charged 

and unehapged. 

The theories developed on thess assumptions are 

referred to as neukra'l, chargad and symmetrical theories raspec- 

tiuelg. 

Tho requirements on the. 8's and f*s for neutron aad pro;. 

ton a$@ somewhat different $'or the different theorlea# 

Tt is nscesaary to account for oquality of neutronmneuc 

tmn, proton-proton, and neutron roton forGes In the stnglot state; 

In the neutral theoryI whiah does not; distinguish in any other way 

between neutron and proton W(3 have Ql 2 g2 and fl k f2 for any 

nrxcleon. 

Consoquently the potential function for a singlat: 

q' T2"2 ,3, and the third term v&nighes, Sa 

The, potential funotion Po& the triplet state is 

vl(P) = (e2 3- 2/3 f2I (@*"/P) 

4- dipole-dipale tern, 

The ertatlc, g 2 

term is repulsive for both states and must be ovw- 

come by the =2f2 in the singlet and by the dipOlQAdlpbl@ tsrm in 

the triplet, 

In the triplet the repulsion is increased by 2/3 f2 

Bethe has shown that g m 

le6 and triplet binding energle 

taken to vanish ana the &in$* 

orreot--y ab- 

tained by c 

ipoke term has been discunssed unrjer the theary of the 

deuteron, 

In that discussion an attractive atatic potential was

203 

e 

mcnt with ouscrvations on the deuteron with f = 3,246. 

In tho charged meson theory there are. no neutral mesons 

postulated and the meson must have either &e4 A neutron can emit 

only a negative meson and turn into a proton, and @ proton can 

emit only a positive meson turning into a neutronr This has an 

important effect on the nrathre of the forces. The neutron and the 

proton are Considered to be two states of the same particle, The 

property of being one or the'othsr is called the isotopic spin in 

analogy to the ordinary spin. A nucleon may have m$= -9 in which 

case it is a proton or m ~ & = for a neutron, 

analogous to ms * -8 and fig f 8 

quantum numbers of the nucleon or eloctron, 

This is directly 

For the two possible rnapnetic 

The significance of 

the formulation of isotopic spin is evidenced in constructing 

rnanyFperticls wave functlons in aocordance with the Pauli exclub 

slon principle, 

By thfs principle, and the assumption that neu- 

tron and proton reprosent two states of the same particle, the 

wave function of the deuteron, for example, must be antisymmetric, 

Thus the '3 

stat6 of deuteron, which is obviously symmetric in 

spin and coordinate space, must be antisymmetric in isotopic spin 

spaoe; this means that if' the two particles exchangc charge, as in 

the emission and absorption of charged mesons, the complete wave 

funotion of' tho 3S changes sign on each exchanp. 

hand the complete wave function of the 'S 

on the other 

will not change sign, 

sincc it is already antisymmetric duc,to the spin state and the 

isotopic spin function must be symmetric, 

The nature of charge exchange is thereforc 

the two particle potential function by giving a plus 

manifested in 

sign to states

204 

* 

with’even 3 L and a minus slgn to states with 0d.d S +Le The 

static Forces are thus attractgve far some states and repulsive lor 

others, instead of being alwetys repulsive as in the neutral theory,, 

Unfortunately the char$ed kheory gives no forces between like parbecause 

of the impossibility of exohange In that case and 

cann9.t bs seriously considered 8s a theory of nuclear forces, It 

represents the simplest case of QxOhanegs fOPCeB, I’rowever, and we 

shall need the, description of such forccsa in the sgrmmetricsal 

theoqh 

The symnmtrical theory postulates that both charged and 

ged mesons are represented by ths field.. Slnce the chargad 

part contributes nothing to the forces between like particles the 

ohoics Of g, = gp and fY1 * f for the neutral mesons is rulsd out 

P 

$or, then Che forces between unlike particles and batweon like 

Ica in the ’S 

would bo equal bQcRusc 0‘2 the neutral rnesoYla 

Theladditionnl charged field between unlike particles 

wowlD then make them unaqual, 

neutron-neutron and nsutron-proton states are 

Tho potentials in the singlet 

v0 (n,n) = (gn2 - 2fn 2)(e “EZr/p) 

= (gngp 2fnfp) + 2(gn 

1 2 . cc 2f lf 1) (e-KT/r) 

n P 

rl, 1 c fn 1 etc,, apply to c d meaoylsl The factor 2 in 

ed Bxgression takes acc 

The unprirned g and F re 

the two kinds of cJhapged 

o neutral rn@sms1 as before, 

We can equate Vo (n,nS, V, (p,p) and V,(n,p) by taking 

= -gp n 3 gpl = @, 2 * gp2 

f, = mf rnl rgl fn2 M f 2 

0 P P

.I, 

205 

which has the symmetrical solution: all lg/ are equal and ~ l l 

if1 are equal. Only tho phases thesf3 mlmbers diJY@r* 

Tho ralative merits of tho neutral and symmetrical 

I 

theories have been determined by Bathe $0 whom thfs semi-classical, 

description of nuclear fields is also due. Bothe Finds that tho 

neutral theory with Q = o and f 0 3r24a can account for Che two 

body problems satisfactorlky. 

Thars is one artifioe neaessary, 

however, and this arises becauee of the 8ivergenoe of the dipoledipole 

forces at r s 0, Ra the wave f'unction of a particle is 

concentrated 1x1 a region of radius r tho rninhum kinetic energy 1s 

of the order K2/2mr2. 

The averagB dSpole-dipole potontlal, on ths 

other hand, is proportional to l/P3* 

the radius of the wave function the lower the energy and cohse- 

quently there is no finfte state of lowest enwgy, 

it is customary tolassme thBt the l/r3 law or attraction cease3 

at a certain "cut off" radiuq, 

This shows that the smallar 

To remedy this 

In the neutral theory Bethe finds 

it nwessary to cut off at 0,32 the range of force (,32/K) if it 

is assumed that the potential remains covlstant at smaller r and 

equal to the value at cut off, 

The theor$ "can give8 the carreot 

upole mmont and also the correct S-states and Slscattering* 

The same method applied to the symmotrioa 

to a cut off at (1.3) (l&) 

heory laada 

and to a quadrupole moment of the, 

signa Thus tho num"e5cal results, at least for the 

D states, are hlghly unsatiafacto 

and 

Qualitatively tho symmetri- 

ea1 theory is to be preferred because the charged meson6 appear 

-* 

in cosmic rays; they might reaeonably be supposed to disintagpats 

into electron and neutrino; and they mako possible, in princip3eI

206 

an understanding of Lhs anornolous magnatiQ moments of neutron and 

@ 

proton, 

In case the mesons are supposed to emit beta particles 

the Fern% constant is a product of the intoraction constants per- 

taining to nucleon-meson interaction and meson-beta interactions, 

The decay period of a free meson, which has been accurately mea- 

sured by Rossl, ’ should depend only on the meson-beta interaction 

constant. 

There RP~, therefore, thr.ae determinations of the two 

constants and they are in violent disagreement, 

all tihis the calculated burat producing cross section of ohargsd 

mesons in the Rtmosphere is muoh JaPger than the observed oross 

see t 1 on, 

In addition to 

From 8 quantitrtive point of view, the neutral meson 

theory is the most promZalng analog of the eloctromagnetio theoq, 

Perhaps part; of the reason for this l a that it need not explain 

the prop0re%e~ of 

the obwrved, chargod mesons4 

There i’a some 

indication Pram the scattering of High energy neutrons by probons, 

Ler, tho cas@ in which the P-wave bocomes important, that the. 

nsutral theory predicts 8. too large cross sootion, 

The symmetri- 

cal theory accounts for tho exparbents on fast neutrons satis- 

fczctorily. 

s 

Zn conclusion it O m on3.y bc said that it is hoped that 

further work on scattering CTOSB aectlons at higher energlea, 

angular distributions in scattering and photoeloctrlc procsasos, 

the naturo, particularly the spin, of the mesons# @to., 

will guide 

us in tomulnting n fiold theory of nuclear forces that will ex- 

plaSn everything.

LECTURE SE2zLE;S ON NUCEEAR PHY 

Lecturer: 

V, F,: Weisskopf 

The ca>cuI.ation and the theoretic prediction of the properties of 

is made difficult chiefly by two reasons: our ignorance as to the 

e of $the nuclear 

inability to solve the complicated, 

bbms which we face in the quantum mechanical treatment of almost 

us except the deuteron, 

Some attempts have been made, however, to 

abt$in Qualitati,ve results from the theory in order to interpret the vast ex- 

perhankel material on nucleap reactions which been collected so Tar, In this 

, 

of nuclali, 

account is given of the statistical metbods of describing the behavior 

These methods use aa few as possible actual assumptions concerning 

ar forces or tha nuclear structure. 

marized in the following pointsr 

Their main assumptions may be sum- 

A) Thb nucleus can be considered as a itcondensed phase" of the neutron- 

proton system in the tharaodynainical aense. 

"he neutrons and protons form a 

* 

ich they are dsnsely 

ed, so that tihe nuclear matter has definite 

t the volume lis proportional to the number of constituent8, 

By a@- 

aripal form, one can, erefore, introduce a nuclear' radius 

-til 

where A Is the number of constibuents and ro s -I:*# ko em. 

The results from this formula are in fair agroement with values Pram~experbnts 

in which the si~s of the nucleus is involved, 

Furthermore - and this is the 

of assumption A - the characteristic psopsrties of nuclear matter,

208 

especially the close proximity of all constituents, are maintained even if the 

* 

nucleus is excited to energies high enough SQ that St rdght emit one ar more 

constituents, This fs valid beclause the bability of emission of the donstituents 

is very small, 90 that the nucleus is in a wt3,J.l defhHed state before 

having emitted the particle, 

It is in a stehe which has essentialky a e s m 

properties as .the lower excited states which do not emit partioles, 

This is in definite contrast to the situation with excited atmns,, An 

e4eotron escapes very rapidky if excited above Sonisa.t,ion snsrgy and no states. of 

tb atoms in which the electron is still oloss to the atom could possible be de- 

fi#ed in this regiov of excitation, 

'Ilia nuclear conditions however, are similar 

to that of a liquid drop or a solid body, dnce khe heat energy of such systems 

b 

may weal be much higher than the work necassary to evaporate one singbe molecub., 

Tho state befare emission of this molccul@ hiis a life tba lQng enough to bo well 

dofined, 

Emission af a constituent by means of nuclear excitation may then be 

divided into two steps: 

first the oxcitation to the excited state, wd then the 

emission of the particle from the excited state. 

The analogous proces# in atoms, 

. 

however, must be describcd in one single step, since the time interval until the 

electron leavus the atom is not long enough to define a regular quantum statc 

in which the electron is sti4.k within the atom. 

The assumption A is no longer fulfilled for very high excitation 

anergies of the nucleus, 

as in the condensed phwe 

The range of anergy in which the nucleus can be con- 

ria from nucleus to nuclaus, but it is a&f@ 

to assume that the assumption is true f ~ a r rang@ of excitation up to a large 

fraction of its total binding energy? 

E) According to Ikshn, a nuclear reaotion can be divided into two well 

od states. 

The first is the formtion by the incident particle of a 

compound nucleus in a well defined state; the second is the disintegration of, 

the compound system into tha product nucleus and an emitted particle, 

The second 

I) stagu can be treated as independent of the first atago of the processe 'She basis

209 

of this assumption is the classical picture of a nucleus as a system of pasthies 

. 

with very strong interaction and short range forces, 

If the incident partido 

comes within the range of the forces, its unsrgy is quickly shared among all 

constituents we1L before any r'cemission can ocmr., The state of the compound 

nucleus is now no longer dependent on the way it was formed and could hava been 

produced by any other particle with corresponding mer& incident on a suitable 

nucleus, 

Its decay into an emitted particlo and a residual nucleus is thus 

independmt of what happened before 4 

Bohr's assumption is certainly right for a oase in which them is only 

one single quantum stato of: tha compound nucleus which can be formed by tho in- 

cident particle conaidering its energy and othdr circumstances @ 

Then, evidcntly, 

the properties of this state are well defined and independent of the way it has 

been excited, 

It is very probably not a good assumption if tha state of the 

cirmpound nucleus which is formed by the incident particles consists of a super- 

position of several stationary status, 

'fhe course of the process then depends 

tgtrongly on thc relativo phases of tho states, and is thmofora not independent 

af the initial procoss of excitation. 

In the case, howeverl that the density 

bf states of the compound nucleus is very high, so that their widths overlap 

each otha strongly, a groat many states can bo excited simultaneously by the 

incident particle, 

This phase relation my be at rdfidom znd the resulting 

pPocesscs, themfore, independent of the way of excitation. 

Bohrls assumption regains approximate validity for this case, 

It is possible that 

'his is made more 

than probable by the classical picture of strong binding forces used above. 

Actually, classical considerations are justified in the region of high level 

density wit'n overlapping lcvols where no quantum selectivit, v occur.s4 

C) The statistical method assums the existence of averagl: values of 

certain mvgnitudes averaged over stctes within R not too wide t3xcitation energy 

interval. 

energy, 

The ?"vexages are supposed to be slowly varying functions of that 

The ~gnitudes concerned here are level distances, transition probabili-

tain statas, etc~ The validit 

ful applications.*, Xf the energ 

210 

is assumption m n onZy be proved 

rvals of averaging must be taken 

LECTURE SERIMS QN 

V, %, WGisskopf 

Ad 

Tbr: garieral chhractar af the energy states of a nucleus om be described 

lowing' menner~ Thcre isj 50 far, no definite regularity found estabtho 

spacing of tha onorgy Levels, The distance between Lha lowost 

of the order of ma. 

1 Mdv* Solsttimes they lie Qloser and in 

groups. Information ?'bout 1, 

ecently measworn@ 

sible to detarmin B directly, It is interesting 

t there is no i 

est levels on 

dependence of the average dis- 

alight tendency to smaller disi 

tances for highctr weight, 

* 

It is 'safe to assw~o that tht, level dclnsity increases with higher 

energy as is oxpdCtod of elmost eny mechanical system, 

tha stronger the mow particlas the nuolcus consists of. 

o sxpox'imental muterial mailable 

l'hb inorease 

Thdre is, 

the slow neutron capture oxporim@nta provide some very soant

evidence on the Xsvsl de& 

The states of th nuckeus, W i t 

a finite lifetim bacaGsc. of the possibi 

231 

citation as is explained below, 

ception of th ground stat@, have 

f radiabivs transitions to lower 

states and of cjoction of particles, 

(The emission of@rays 

aan be naglmtud 

hero since its probaki&ity is extremely smalli) 

l'he states thus hnva a finite 

which is connected with the lifetbe %hof .tho nth skate according t o 

the rclakion 

the total width 

I$! the finite li5etSmo is due to different emissions, 

he sm or partial widths# which themselves are propor- 

tional to the specific ern$ision 

-.. A" 

' 2s the radiation width and 

litLos* We therefore can write : 

he partial width corresponding to 

of a particle ci2 whish may be o, proton, a noutron, or anaek-;partiole, 

tho emission probabil 

per second of a partiole & by the 

Thle width of the law lykng levels is purely a radiation width as long 

tfon energy is lower thnn the Lowost binding enorgy of a particle,, 

The bindbg anorgy Ba of a p&rtieAe 5 i s the energy ncoessar-y to remove it from 

the ground state and have the residua9 ru~?leus in its ground stab,, 

above I 

t&e gram4 state. 

The binding energy B, 

We there- 

ciLation energy Qf Lhs nth love1 

of e. proton or a nuutron is found 

to be about 8 Mav for nucJ,ei which are not taa near %he lower or upper end of 

ic tab&, @hs vor olernents about up t o oxyg do not show .my 

I From then OM)- 6 rcge hut thz value my be several 

B heavy and of the table (A320Q) 

g snergy is smlle 

af the weight, of'Uc 

arags of 6 MeV for the elemen58 

Thsse figures are taken from the total binding onergg Wo of the nucleus; 

he eno~gy neoessary to d~ornpoa~~ the nucleus into neutrans and protons, 

which is given By tho mss defect* The toba 

inding energy 'No is roughLy

212 

proporbion&. to *the number A of constitucnka and it is therefare justified to 

assume that the 'binding energy BB, OS one particle is W,/A, 

which leads to the 

figure s rrlant iane,d above & 

We know, howcverj from the &-values of p-n reactions that it requires 

a8 much as 4 Mav to replaoe n neutron by a proton4 

This showe thaC 

large flVctuatSans in Sa from the averng,s value must be expected. 

With sufficient excitation energy En, a pwticle 8 can be emitted with 

different energies, corresponding to the differont states in which the residual 

nucleus can bo left behind, 

We thsn can subdivide ran in2;o differant partial 

widths: 

(1) 

s the width corresponding to an ornissj-on bf 

a with the spatlid 

hat the residual nucleus should bs: left in the statead 

The widths 4" increase w5th higher excitation cnorgy since, firstly, 

icles can be emitted and secondly, mors dfffuront states #&of the 

nuclei are possible, so that the number .of t'esms in the bum (1) in* 

creases, and finally, it is expected that the probabuity of ekssion of a 

partScle increases with its velocity. 

At a certain excitation enorgy E 

the 

width A" becoms larger thsn the level distance and the ldvels overlap, 

The nuclear levc?&s in tho region abavc En= 3, can be invostfgatcd by 

the bombardment of nuclei with pastLclos 

of a given an~x'gyg 

If the particle 

hits the nucleuy and is c;hsosbod, it forms a compound nuclcus with &m GxCitation 

energy Ba-kc J This absorption should be ap ociable only if the energy BzI 44 

0, or within thu width of, an excitation level of the wmpound nuclsus,' 

monoenergdtic partiole beams, it is possible to investigate, by varying 

I 

peotrum of the compound nuclsus from 3, upwards, 

We expect to 

obwrvs ltr@sonanccil absorption when Sa 

Resonance levels have been observed wlth 

6 aqua1 to an excitation enorgy E,; 

tich and protQn bombrzrdmmts on 

averago dlstanau of LsvcLs

213 

with an excitation energy above Ba becomas vary sntall with increaszng atomic 

number so that the energy of the bombarding particle beam cannot be dofined 

e 

sharply enough to separate them in heavier nuclei. No regonance has been observed 

with beams of protons, deuterons, oreparticlcs for elements heavier 

than zinc (2 qO),A sr66, 

Although it is even more difficult Lo produce monoenergetic neutron 

beams of arbitrary energy, there is one energy region in which neutrons can be 

used with great success; naniely the rogion of wry low thermtt energies. 

aan be slowed down until they rcach thermal or nearly thermal energies, 

Nautrons 

Their 

energy is then very woll defined compared to the poor energy definition of beams 

of fast particles. 

Their anergy can also be measurad to a high accuracy by means 

of absorbers which have, in that region, n known snorgy dependent absorption 

(baron) + 

Recently methods have been developed to produce neutron bums with very 

sharp energy up to 5 eve 

Resonance absorption with slow neutrons can, therefore, be expccted even 

for heavy nuclei. 

Although it is not difficult to produce similar low anergy 

beams of chnrged particles, they are useless bceausa they do not ponetretc the 

Coulomb bcrrier of the nucleus. 

The resonance capture experiments with slow natrons give informtion 

as to the shape and width of the sesoniince levels of the compound nucleus and will 

be discussed in connoction wfth the quantitative expre~sions. It is found that 

thu main contribution to thc width of thus@ lcvels is the radiation width. 

This 

means that a captured slow neutron will in all probability stay within thc com- 

pound nuclcus, sincc the most probably effect is the wdssion of’),-quantwn after 

which tho excitation dnolrgy sinks below 3, 

and no particle omission is possiblec 

Some evidence as to the level density of the compound nucleus cm be 

obtained from the number of nuclei which are found to show resonanccj cilpture of 

slow neutronsI 

This nwnbor indicates the probability that a level lies in c? 

re- 

gion of a few volts above Be+, 

One obtains by that method p, lsvel distance of

2 14 

roughly of the order 10 ov at the excitation energy Ba for dements whose atomic 

c) 

number A is about 100 or higher, It is much larger for lighter elsmcntsj in the 

region of Ad60 it is probably of the order of 100 ev. 

We now proceed to tho discussion of the cboss swtions of mtud nuclcw 

reactions, According to assumption B, we may write for the cram section 6(a,b) 

of a nuclear reaction in which a nucl.ous is bombarded wfth a particle g and a 

pwtlcle .- b is e jectod: 

, 

. (2) 

Here fa the woss section for the formation of the compmnd nucleus, 

and 'Qp the relative probability that the particle b is emitted, 

The cross section 6 can bi. written in the following form8 

a 

ba55i&;l 

Here Sa is tho maximum value (g can possibly assum and e 1s tho 

a 

''sticking probability" of a which necassarily is smaller than unity. For 

neutrons whose wave length 

is large compared to tho nuclear radius R Sn?$Tx ;B 

c* 

For charged particles, Sa can bo roughly defined as the cross section for penc- 

trstion to the surface of the nucleus and is strongly dependent on the onorgy in 

1 

the familiar way of the Gamow-penetration of Coulomb barriers, 

discussion is found in Section XXXIII* 

The quantitative 

6 2 3 tlic $robability that a formtion of the compound nucleus takes 

plnca, if the particle has reached the nucleus. 

As a function of the onergy,e 

1 

will display the reeonance propartizs, mentioned before. 

It w ill be small betwoen 

tho resonances and large when B +e is equal or near to an excitation levo1 of the 

Q 

compound nucleus, From a oertain value 4 on, tho widths of the levels overlap 

and 6 becomas a smooth function which, however, must be smaller or oqual to 

unity. 

Naturally, an average value of f over the resonance is needed if the 

e 

t 

to 

incident bdam is not sharp enough in onesgy to separate tho ~JV&, 

It is reasonable to assume that e becoms unity if & is small compared 

a 

the distance between thc? nuclear constituents. Then, classical considorations

215 

can be applied, which make it a~em vo y plausible that every ppxticle which comcs 

bange of ths nuclear $orccs is incorporatad into a cornpaurld nucleus, 

th 

is SuAfilled for energies abovs 10 &7t, 

I 

Thu'second factor in (2 

clo is anittad by the compound nucILe 

ivc probdbility that a pnrti- 

s evidently given by 

re p"k i+ the; satin 

/ 

of a particle b, amraged 5ver the compound st~tes 

whigh arc exci 

atage of the process, 

particles c which me ejected, 

TLo sun in tho denofibn8.tor shau 

c 

a 

(3 ) 

ound stabs, for the omission 

be extended Over aJJ, 

TLG wjig~t&i~n of t k . ~ la furdG2gr:t ;.ob different; from tho oal- 

culation of the a&, TLa 

fornation of a aompaund nuc3eus and there 

are tho probabilities of tho opposite gpoaoss to the 

a fundaraefltql relet2on be tw@@n 

in 

A few qualitative conclusions can be drawn without using the formul2s. 

The for neu6t.m emission is always much grsa'csr than the 

emission, or the rb for any ChargBd particle, because of the Coulo 

barrier, except under unusual condiblons 'chat the bin 

is considerably smaller than that om Of bhe neutron, $0 that tho proton has much 

more energy. at its disposal; Thus, if a reaction with neubron omissi 

geticakxy possible, this is gene 

probable than all. the otha 

are found only Just abbve the threshold of w 

with neutron 

when the neutron has very smal energy. me 

energy dependent and beconie smaller with larger 2, Nuclear re 

s therefore are very Pare far heavr olemcnts, 

P 

f 

Deuteron emissions have never bwn observed, 

This, is becauge OS tho 

to remove a deuteron from the compound nucleus; it ia equal to the 

in4 two constitumts &nus the small. amount of the detu

216 

energy* The emission of a neutron alone is therefore by far much more probable, 

What can be predicted as to the energy distribution of the emitted 

particles? Aftar emission of a particle, the residual nucleus is left in on@ of 

its excited states, say with an excitation energy EB or in its ground statat 

The energy 

where E is the sxcitaticm energy of the compould nucleus which 1s in turn dete?r- 

mined by the energy e of the incident particle sr producing the compound 

;a 

nucle;us: E z Ela.CQaI To every level E of the residual nucleus corresponds 

k9 

and energy value e Thus tho energy distribution of the particles omitted 

bfi 

consists of a series of peaks which are a picture of the spectrum of the residual 

nucleus, thQ highest energy corrosponding to tho lowest level of the residual 

nucleus, 

In the special case that the incident particle is the sanie a8 the 

emitted one - (n,n) - (p,p) process - we thus obtain 

I 

B 

A - Graund level of initial nucleus 

B - Ground level of final nucleus 

C - Ground level of compound nucleus 

The highest energy corresponds to the elastic scattering, This 

b 

peak wilb be the strongest, especially in the case of charged particles, sinct:

it contains also those partjdes which are scattered by the outside fields, 

217 

Actually it is impossible to distinguish between the folLowiizg fQur contributions 

to the elastic scattering, which give rise to scattered wave-functlohs which &re 

superposed coherentlp) 

a) Scattqrbg by the Coulomb field if any 

t 

b) Scattering.jby the &ape of the nucleus in analogy to the 

optical diffractAon 

c) Penetration to tho surface of the nucleus and reflection 

there 

d) The forming of a o 

emission with squa 

und nucleus wit 

It is, howover., important to distinguish d) from the other contributions in ordor 

which cqrresponds to the fsrmtgon of the compound nucleus can be dcdined by con- 

wctring it with

general (apart from individual fluctuations) proportional to the energ;y.Q 

218 

and 

to the penetration probability through the potential barrier in case of oharged 

1 a decreasing function of (Eb. 

p&rtiCluS. P(cb) is an increasing function,CJ(E 

P 

We thdrafore obtain cn energy diatribution in the region in which the 10~018 of 

tho residual nucleus become undissolved, with a definite maximum which is in the 

case or" uncherged pnrticles very similar to a Maxwellinn distribution, 

discussed quantitatively in Section XXufVI. 

This is 

January 20, 194.4 

LECTUW SdRIES ON NUCLEAR PHYSICS 

Fifth Series: The Statistical l'hcory Lecturer: V. F, Weisskopf 

of Nuclear Reactions 

UCTUW xXxI11, gross' Seqtions and Emission Probabilities 

Tho values of the cross sections of the formation of a compound nucleus 

by impact of a particlo with a nucleus, and the probabilities of tho inverse 

process, tha emission of a particle by a nucleus, obey certain general rulss, 

which are derived from the geome~rical'propertics of the problem, 

Let us consider the cross section $('!of 

the formation of a compound 

nucleus. 

It is appropriate to decompose my cross section 6 of a nuclear process 

R 

with an angular momQntum fi 1 in raspect to the sccttering center: 

into tho partial cross sections @ belonging to tho contributions of the particles 

)a-, 

The angular monieqtwn in respect to the scattering center of a classical particle 

_." (6) 

* 

moving in a straight beam is given by m v d whore d is the nearest distance of a 

straight directisn of motion to the center, Quantum theory admits only integer 

multiples of % as values for the angylar momentum and it is justifiod to 'say

IJI 

219 

that, in a straight beam, ell particles nzoving along lines, whose distnnca d from 

.ChQ center lies between 

$ave an angular morrantwn .kR. 

us all particles of angular momentum 

asmhn area perpendicular to the beam 

of the size ($?gf'i)$%! A nucloar rtlaction which removes particles of a given 

angular mornentum,frorn the beam, cannot remove more thsn come in end thus never 

A 

ss section 6 larger than that area: 

I- .-.". (7) 

A more exact derivation of this relation is given below, 

It is to bs expected that partioles, whose nearest approach on o. 

'. ? 

straight lina, d, is largljr than the range W of any forces coming from the -scat- 

terer, wul not be scattered at all, Thus 6 150 if d >I2 

R 

$%4 

We obtain the maxjmwn possible cross section of a system if we assume tha$ 6" 

d" 

has its m;sxhm value for .h

* 

that every subwmm consists of a superposition of i331 incoming and n.n outgoing 

spherical wave of equal intensity (first and second tvrm in tho sqtaaro brccl,ceb){ 

If a perticle wave of this form impinges upon a nuclcfus, the superr 

position is changed in two ways: 

2) the intensity of tho outgois spherical. W&VP 

is diminished due to absorption of the parkiclos (the latter may or may not be 

reemittcd); b) the phase of the remaining outgoing witv~j is changed, 

Both effects 

give rise to a scattered wave since tho outggolng wave is na longur ablo to inter- 

fere with thc. incoming wave in tho sam way as in (y), 

thd only way ,j.n which there 

is no radiation from the center. 

Thus tha aatual wave function ft has not tho 

asymptotic form (LO) but 

taken place, *q is srnaller than unity, 2.b 

waves b 

ht us, docompose (loa) into the incoming subwavrts: 

.: f'g,e*.~rjs~ 

and the outgoing ones 

$,; 3 *YE '\/2~ 

% kr 

1 

is the phase shift of tho outgoing 

il @-~(~~+L.N.,J-*(O? .-__ 

4 

--,.---

c wave : 

The d 

to a plme wave 

221 

nce betwoen (loa) wd (10) is the wava which one has to add 

) $0 get the actual wav9 

/f.'i 

* I 

it id therefore the scattered 

The crcm aectian for elastic scattering, for a single d, can be obtained 

by calculeting the current of the 4th suiwe.ve. of #-- )I$ .through a large 

sphere and dividing'it by the current of the incident wave. 

1 

' f ci-qe rig) d(&+i) " .. 

This cen also be writthn in tho form 

This gives 

(14) 

It is seen froni (12) and (14) that curhain relations hold bctwaen the 

Capture cross section is given by (7). 

"he elastic cross soctbn, however, has 

a higher upper limit, which is reached if r)'sl and a g 2R/#: 

--% 

This mnxhm can only be reached if 7 S 1 or if 6')s # 

c 

This can also be understood in tho? following wzy: 

. , 

the maximum effect 

of elastic sez*ttering is obtained by a change of phase of #' of tho second term 

in the square bracket of (10) (tho outgoing subwave) without reducing its 

This is identical, with adding to the plana wc.vy just twice the outavd. 

Thus the cross section corresponding to this, is 22 times larger 

than the one resulting from absorbing thc outgoing subwave, Hence we gat a 

maxhurn cross section far scattering Pour times largar than for absorption, 

* 

given value of@ "I. 

ljJXitL3 

1' 

**f 

'9 

The expressions (12) and (14) limit the possible values of @ R (4 for a 

4 

This is indicatod in Pig. 1, whoro thc upper nnd lower 

6''. flf" ,.is plottad against It is seen that it is nccessarily 

IC) assues its msurimunl value(& A + i ) ~fi F . 

djl 

We foLlowing theorem is OS interest, since it connccts the total cross 

ith the amplitude of the elastic scattering; the total cross section is

222 

e which also can be writton in tho farm: 

1 

q ftd -.-%J.&”pAfl RX%R+I) 

wheso An(1-W aigJ] is the value by which ono has to multiply tho outgoing 

subwave of the undisturbed plane wave in order to get the scattored subwavo(b3), 

i 

Fig, 1: Upper and Jowor limit of ths elastiu cross section for given capture 

cross section. 

If the scsttering object (nctuczlly tho range of the scattering forces) 

is large coni9ared , ~ll’wc?vc?s for which kfi(.gcan be absorbed, but the wava 

for which 1 - R/n )) 1 vanish for r 5 R and thus arc not influancud and should 

a have gicl ;31 0 If R/j( is a large number, it is a goad approximction to 

L

the,fact that a 

interference effect Pf a ac 

kon out by tho sh&daw, 

becomes com- 

value vnly for vvry few values Q 

cind 5t is masthy lower

a 

224 

nucleus in the state a; b) the umission by tho compound nuclcus, of a particld 

a with an engular mornenturn$,!,, and Iswing the residual nucleus in the stato G. 

The two processes a) and b) arc invurse. The first procuss is rmasurod by E! cross 

section 6aaf and the secmd by Bn emission probability Gak Wo should c?lso 

ifiod the direction of khc apin of the emitted particlo in respocL to 

some arbitrary mist We omit this and assume in the following that the spin is 

given together with tho angular momentum A e 

In the definition of CaGAit i s 

assumed that the incoming beam is spread over an energy intervd largo enough 

to uxcita many component levels, T;*,B is the average value of the emission 

probability, averaged over thsse.lavds which can be excited by the first process 

within a snmll. anergy region, 

In order to siivd indices, we replace the triple index (ad) byeand 

writs and T;r/ , wherever it is practicable, The result of the following 

considerations will be the relation 

whae D is the avcmgo distcnca between the levels, which can be oxcited, Before 

c 

going into the detailed dcrivntion, it is useful to understand, why a rslation of 

this chwacter is expected to exist, 

Let us first assurm that no other process but the one charnctcrized by 

can follow the c@porption given by 

In this case, the level width is 

equal to r& It is to be expected that $a is proportional to the ratio p' 

(524 

of the width to the distance of lwmls, becam this magnitude measures the rela- 

tive portions of the spectrum in which absorption is possiblo. 

It is plausible 

m&ximum absorption is reached, ifr/ is of the order of unity, The 

mimwn value of rf is fik '(2ktl)and 

d'D 

WQ may expoct P, relation of the typQ 

where the constant $s of the order unity, The detailed calculation shows that it 

II 

is 2rt/f,k+ 1). 

This considsration also holds true if tha coriipound nucleus 

created by the absorption of the particlo a. ccvl decay in suvoral WF.YS. 

- 

This ef-

228 

fect would broaden the lines but would not change the clbsorption avurege ovw 

the lines, 

9 Let us now oomparo the probabilities of the two invorsb processes, 

The probabilities should be equal if they cndud in states of eque-1 weight. 

Since 

this is not the cnse, they should ba egucQ aftzr dividing thm by the stntjstical 

weight af tho ebd stQteS, 

Tho pyobability Wl per second of the creation of tho compound nucleus 

by tho particle a if it is enclosed in n big sphsro of radius R is giwn by 

_. 4" V (15) 

W i " ~ f h .. ~ n ~ 

whore v is the velocity of the particle and g its wave length. 

understood in tho following wey: 

This can bo 

if 1IJZgCc'aaumos its maxiiiiw valw (2k+f)fl 

the particle with an angulzr niorientum )iRis sur@ to be captured on its wcy to 

the contw of the big sphere, 

The averago probability par unit kbm Is bhen 

v/2R shoe it nlay be fowxl anywhere along (z diamter, {Its angular momentum 

buing fixed, it mves on lines which havc a given distance ,(,% from thc centor, 

The length of these lines is na&y 2R, 

In C~SQ @&is 

invarsu ~TOCQSSOIS. 

6 

since R i s assumed very large: a >)R& 

snxillsr than its ~.ililximrUn value this probebility 5,s reducod in tho 

hg ratio and (15)results. 

\Ye now compme the statisticr,l wights ,of tht! end status of the two 

If the anergy of tho incident particles lies botween Q and 

4 

the statistical woight of the end state is -Uc(E) dE, hhoro U,(E) is the 

level density of the compound nucleus at tho excitation energy E which will. be 

sclcction rules,) 

(Only thess Iuvcls are couqtod, which can be excited owing to angular 

r/, 

Tho probability of the inverso process is ~;t 

and the statistical 

weighb of its end state is thr! stetistical waigbt of a fro@ particle with the 

L 

* 

cnorgy between 6 and 

We thus get 

+dIE,and the angulcr nornentm 6 in a sphere of a radius 

.- - 

-

By expressing cross sootion ca by Q dimensionloss mgnitudc .Q s 

which never can be larger than unity, an 

Ita, 

y introducing the a.vcrngo love1 dis- 

momentum and with the rasidual nucleus left in a &finit@ state a, is thus bound 

by an upper limit (Q 4 L) : 

distance D. 

6 (16b) 

A few remarks are necessary concorning the significance of the lovol 

nuoleus, frorn which the particlc can be omitted. 

the levels which con bs exciked by the particle .- 

Q wi$h tho angular riionentwn 

(and given spin), hitting a nuclous in the state Crj, On the other hand, Lis 

tho pnrtial width for the snlissfon of a, 

the lovols descrihd cbove (whose nverngo distcnco is D) within a cortdn sml1 

energy ragion, 

D is defined as the distcncs bQtwccn tho levels of the coritpound 

Zt is tho diatcww betwoen 

with the SCZI:IC? properties awragod ovar 

Expression (160) is not bound to this definition of Db 

Far 

m%r@le, D could bG dafined as tha tweragu distance between 

levels of tho 

coiuponont nucleus in this energy region; is then understood 11s the averngu 

over all thGse levels, (for ciost of whiohra So), tho rolation (ab&) is then 

d, In Cha following we will, howuver understend by D End ra 

average distance and width of the levels which man emit tho particle 8 with 

tho properties specified above.

* 

227 

LECTURE 3E3S1ES ON NUCLEAH PHYSICS 

s: The Statistical Theory Lecturer; V, F. W&iskopf 

gf Nuclear Reaction8 

LECTUM XXXZVe Th~Clted NQCZ~US 

st us consider a vluc1tzus in an excited state n in which it i s able t o 

icles. 

The st&.Ce tz has a finite Llfebsule and the probability of finding 

rn is the width of 

eus in $his state decays accordinG to the law e cf”t e 

the state, In general bhe state n can emit different particles B and these 

y be emitted with different anguLar momenta Asand the residual 

be left behind in different states&. 

!Therefore tho width Tncan 

be subddvided 

where 

%a, 

3 

the following way: 

rn, $ (18) 

B ‘the partial width for the n of a with an angular rnomentw 

fng the resfdual nucleus in the state ‘@L, 

The value of 

is determined by intranuclear and extranuclear 

and it is useful t o separate them into the following ones: the probabil- 

of emission of a particle a is proportional to the fol- 

1) me final velocity which it acquires after having left the 

ftdr having penetrated the p entia1 barrier if there i3 any; 

&) Q$ bhe barrier which fs defined as the 

f intensity of an outgoing spherical wave, corresponding to 

face through the potential barrier to the 

t is useful tQ consider as PO 

barxier, not only the Coulombfield 

ged particles, but also the potential of the Csntrlfugal force 

i 

;*

e 

$'A2& 

A. 

228 

(4 *I)&-# the case of an emission of a particle with an angular momentum 

This force is described by a repulsive potential, inversely proportional to 

the square of the distance, and acts on a particle in the same way as the Coulomb 

repulsion. T,fi&)is a function of the charge and of the angular niomentum of a 

and of the energy

e 

229 

being emittod by the original nucleus as, for example, a statu in which tho 

particle has a very high angular momentum in respect to the residual nuclous. In 

order to describe the excited status of the nuclexs zbove $which aro defined by 

. 

our assumption A, 

% 

it is necessary to introduce a device to oxcludc thu solutions 

of (2Q) which correspond to a residual nucleus plus a fmdy moving indqwniient 

particle which is not created by an emission of the compound nucleus. 

This can be 

done by mans of boundary conditions for 

which should express tho Pact Lhct, 

in case the state n includes frec particles outside the nucleus, thoy must be 

represented by outgoing spherical 'CIICVI?~. With these conditions the equation (20) 

no longer has a continuous spectrum above E. It then hcs discrete Gigenvalues 

which belong to quantum states of finite lifctime, becouso of their c;bility to 

onli t particles , 

Let us introduce a radius 8, the nuclaar radius, which is the short-est 

distance from the nucioar center at which prnctieslly no nuclear force acts upon 

vn 

a nuclear particle. 

(The particle might, however, at that distance bo under 

influence of, say, the Coulomb force, ) Tha uigenfunctions yn haw the following 

form for ra>R: 

Here X are the eigenfunctions of the states 4: of the residual nuclaus whioh is 

a# 

- 

left behind, if tho pnrticlc a is remowd from tho nucleus; pn Is the wave 

function of the particle R outside the nucleus, 4; (Ea) is a solution of tha 

following one body equation of the pcrticle a in tho space outside of the nucleus: 

A

where YQ is e lcrge distilncc! zt which Ghl: kinotic onorgy of c".lJ,'.snrticlos mitt4 

is inuck lt?rger than tho potontieil crwrgy Vpw[rF.) cnd thu cuntrifugnl forcl: 

. Tho swlu can b;: expresslud by boundcry conditions: 

-c 1 =O 

n 

24) k,, is givw by (a) 

Per Pa =% (24) 

n 

wiiicti 1 ~ 0 ~ G A 

part of kmis positive,,) 

aid thct sign of thc squcrc, root must bo 

Thew Bonditions and, of courset compctiblu with thu ~YEV~ oquation (20) 

Tho equation (20) with thc (camplux) boundary conditions (23) or (24) has soh- 

tions fop which '& is nucossarily comp2c.x: 

wn =E,- irya ' (25) 

oigenvc?.luo rizcan~ thrlt tht? st&b decays dxponentially e~oordi& to 

Ft , 

,

so that rn is the rociprocal half life OS the width of the st&e n. 

23 1 

(Actually kn c..lso is coinplox because of thc fact that (21) cdntakis the 

complex VJn. It ~xprossus the fact 'chat the outgoing wave (23) changes in intensity 

with re, It increases due to the fact, that the nuclear stato decays expanan+ 

that the parts farther away aro emitted in an earlier stago. This 

change of iintansity czlong thd outgoing wave, howvor, is only a very sn~ll effect 

na long a? thc width r" of the levo1 is srimu cornpcr'$d ta tht, 3 t?nyy of thu 

eniittod pcrticla, llnc ratio bQtwoen rwal and ink7ghary parts of k, is equal 

to tho distance trcmllsd by the emit ed pwLic3.e within tho lifotinie of thc level 

ed in wavdungthsX'(k *)-' . This ratio will be wry lar o in all 

intarost and we ncgltlct?frorn no n the imginnry part of ka A completely 

axcopt, of courss the C~SB TIn < Ba { itntion energy less than boundary 

anergy) in which 

rn,s 

is purely hginary nnd r".: 0,) 

ih now express tho lwd wtdth r A in term of't.heQq, n . The veluo of 

the rcciprocal life time of the level n and tlwreforc equal to the numbor 

of palkticlcs emittQd by the nucleus per second. 

following way: 

We first subdivide rnin the 

rn = E r;&, (26) 

a,a,a 

whero &is thc: partial width for, tho emgssian of 2 with an angular momentum 1 ; 

leaving the nucleus in the state a. Ue surround the nucleus by a sphere wikh (he 

radius ro and dotermine the number of pcrticlos 2 wilt% tho above proporties crossn, 

ing that sphurv por second, which in turn is equal toGL& Thus WI.? obtain (the 

index a d is repincod bycx:) 

undor the condition : 

9n thz limits 1 e' 1 < r. 

Q 

(27) 

owr all vnricblos 

*) If pn is 2, slowly deccying stctc3, so thnt thc 

particle dsnsiJdy 'wtxwn'thc boundary R of thu nucleus and ro is wry srrtall com- 

pemd to the ddnsik,y hsidt! of tho, nuclwa, wo c m also norrmlizo 

in tugral 

by the 

n 

In the oxpression (27) the v;?luc. of *-,n is still dt.pcndont on tha field outside 

of thc nuc!-a~s. Since thG latter is wvll known it is usoful to show its influence 

1 2 

is the vdocit of tho pnYklcle at ro~~~(?~,l 

is, in the 

usud htiro, x-3 t$m~s tho probability of finding a particlo Q in 

om volwnu unit at ro. 

'

explicitly in the expression for tha width, ?% thoroforc express n (ro)l 

232 

by tha value of the s m function at the nuclusr radius Re Vh introducit thd 

magnitude Tasg(€)which 

is tho ratAo by which the "(intensity x r2)1f diminishes 

in an outgoin; wave of a particle 2 with an onwgy 6 and an angular momntum 16, 

if it pehotratos thQ field around the nucleus from R to roe ro had boon chosen 

outside of dl fictlds of forcrs so that the 1l(intsnsity x r 2 )It is coxistant outside 

r,. T,g&)can 

be cnllcd the transmission coefficient of thc potuntial barrior. 

It cnn be calculntod from tho wave cquation of i? parliclt: with an onorgy and 

&n angular momntum .46 in the fiold outsi.de of the nuclcus, whew solution wc 

will call Fa 

0. 

r(r) for this purposu: 

[& ." 

If we choosu .the p&rticular solution which 

has tho asymptotic form 

wc get 

T ddpend9 on the naturi: of 

F-e 

F

233 

far xcc 1 

with x k,R. e v&lue of Ta kcG> for chargod particles is diffic 

put* sxactly, ~GC~USQ of tho wry SLQW ca 

enoe of the expressions for t 

enfunction of the Coulomb tiel 0 region considerad horc. 2t 2s 

enorgy E. 

e8 stronglr with 

An approximate cxprwersion, c 

char chargo 2 and with fa2S2ng 

cd by using tho N.X.B. 

method, 5s 

(28) 

1 

is the charge of the pnrticla 

is the anergy barrier height for 

The valuas of c 

mwgy E of the

* 

Expression 

254 

The Pactw G depends on the value of the aigenfunction C& at the boundary af the 

nucleus and thus reprosents tho influenee at the intranuclear structum. 

(29) soparntes the offects of tho forcos outside of tho 

could therefore ba called partial wgdth witho-ut 'outsidc potential barrier, 

This 

separation of the barrim effect and eho nuclear effwt is by no mucins completx!, 

n 

It is to be expected that the value ~f..~~~~(R)&I.so depaqds on the €arm of the 

WEV~ outside of Rk sincc tho Inside and the outside p ~ of ~ the t function must 

join smoothly, 

Under certain conditions, howovw, thore is good reason to expect that 

the valuds cP0lf\(@do not dopwd on tho bahwior outsidc of the nucleus, 130- 

cause of tho strong forces inside of the nucleus, the averagw momentum pi of a 

particle inside ,of the nucleus is very high and corresponds to an unwgy of 20 to 

30 Lev. 

The average monientum is an indication in what space intcrvnl the wave 

functSon chenges ita veluc considorably. :!e dofinr: an c2Verc?gu wL?v'c length 

xi = 

i 

and wo can say that bhc? W~VB function insidc the nuclous chcngus its 

value by an cmount of the ordm of its avcrco.gc size, if tho coordimtc, of one 

portich is chnngvd by h+ 

1' 

much smallm than pi: 

If the momontum po outside of the nuclom surfacu is 

tho wave function insidw the nucluus must be joined at t h nucltmr ~ surface to 

e, tvaw function, whoso villuo chnngos wry slowly cornpard to conditions insido, 

ft 1s thOn a good approxhtian in thu duturmination of tho TGVQ function inside 

to mswrs: that 

t 

for all pmtiolos. The vcluGs for 

aa.4 

arc not very difforunt from thc: truu on@* 

guts with this boundary condi4ion 

Sinco this condition does not contain

any reforanoe to the properties outaide, tho genwal behavior of 4, 

(R) is to 

oc" i" 235 

I 

I 

be oxpsdtod the sane for chargod and uncharged particles and also the szmc for 

ciiffsranC Rt s, 

Particularly, the aqprago vnlucs 

neighboring luvols 

should bo practically indopndcnt of tho energy 6. a of tho outgoing particlu. 

It 

will depend only on the properties OS tho excited nuclous, whosl: excitation 

enmgy is Ba+f4, an enorgy 

- c. 

valud 1 aaLof 

validity aP (31) end docs not 

the pnrtizl 

which is much lcrgcr then 

chcngo ap2wclablgr in thct region. 

in the region of 

'l'hi. avcrcgi: 

width OVM neighboring 1C;v@ls, is thcn Given by 

whcre C(3) is a funation of tho total excitation energy i= Ijk 4- ea only, It 

dQcs not vary strongly in an onorgy intaval which is sm11 compnrod to 38a . 

Furthergoru , At is c.,pproximRtuly uqu&Q for tho different pcrticles which can bd 

arnittcd by Chc! conpound nucleus, 

and for t hG diffsront vnLuos of o.ngular monicntum 

of thew pwtic1c.s. 

The uaofulnoss of this approximation con bc illustratod in n few 

oxahplea: 

thc herap ixirtial width for the emission of a neutron with 

is gj.von by: 

F = c(EC}dq (33) 

which is ~ssuntidly proportional to the square root of thc outgoing noutron 

mwgy in the rGgion in which (31) is valid, 

Tm ?.wrap cross scction over neighboring lcvols for the capture of s- 

n~utrons can be obtained from (16): 

%= ---- D 

This relation ShQw thu wll lcnotvn 

slow s-natrons ( 1x0). 

(34) 

1 l ( a ' 

'/u law for thc ccvpturo cross sdction of 

Thc particl width for thG emission of neutrons with 1 # O is givcn by

236 

It is thus proportional ta e itg for mill cmrgies. The emission 

prababllitics of chargud pcrticlcs haw tho trmsnzission cosffiaicnt TaL(G) 

as thG dominwring factor. 

z'hu factor C appccring in tha ncutron widths cannot bo datmniiiud but an 

.c... 

uppor limit can bo sat: WG know from (17) that % 5 ?&T 

Since (33) should bc valid up to cbout 

1 bkv wo obtain 

for noutrons &to). 

Thc: exprimentally d~tr:r&od vnluos (see Scotion 5)$ indiontc that C is not much 

smeller then D ea, if the energy units arc chooson in 1ievtss 

/ 

So fc?r tho radiation of the nuclear statos has beon neglectod. It intron 

, which we call the rodiation wid?&, !he 

duccls another additional width 

n 

tatcl width A of the lcvol is thm giwn by 

rn 

Tho radtction width consists of a sum af partial. widthsr 

transition8 to spacial hvels f9 below n: 

fP 

corresponding to 

u; is thc! matrix dcmont of the from n t o . P 

Tho spoctrwn of khc nuclous can be dividd a) into 2" tlstc?bJ.a" region 

from 0 to 

to rndiatiori; b) in E 'lroson~.nco~J rvgjon abovc 3 whwe tho stntas arc able 'ca 

(36 1 

where the lvvcls omit light quanta only and the width is moroly duo 

Qmit ptzrticlt\s, but still aro smrt from each otlicr by mor3 then their width so 

th2.t thoy form c? discrot

c 

January 27, 1944 

LA 24 (35) 

237 

UCTUiU 3El.&3 ON NUCUAd PHYSIC3 

Fifth Series: The Statistical Thecrry Leeturcr: V. F, Ykisskopf 

of Nuclear Reactions 

A) The Brcit-WiPntlr Formu2 

The cross sections of nucloar rmctions, which wuro cnlculated from the 

emission probabilitios in Section WIII, woro definQd as avorag“ cross sections 

over 8 region oI enurgy of the incaming particle, which includas mny resonance 

levels. 

This suction is dovotod to tho study of t h crosq section BS a function 

of energy, if the anergy is skierply definod. 

phcnomenc. a3 deskxibed qualitativcby in Zection %XII. 

In this casu vro oxpuct resonance 

:‘io first investigP.tc tho onmgy relations: if eoCia tho energy of the 

incoming particle, the onurgyeb of an emittdd particle is givon by: 

% = Ea+D,--Bb+E,-Eg (38 1 

whcre E 

aru the excitation enorgios of the initial and final nucluus 

cb and %3 

respcctiwly, an4 Ba and i$, we tho binding energius of a to the initial 

nucleus or b b0 the final nuclcus rospoctivafy. 

The cnorgyca is thus dependent 

on €a d i s in general c.l- not equal to thr: onergye,$ 

of a particlo b umittcd by 

any of the excited statm n of tho compound nucleu$, excqt if is just in tho 

middlo of n rosclnancu; 

This is tho rxsa if Fa+*Ba is just equal to tho 

oxcitation energy of thc compound stat3 n, 

The expression of ths cross section of a nuclear reaction in the 

od by meit and ?fignor” and later by Both@ and 

1) Broit and iligner, Phys, Rw. 519 (1936) 

I

e 

238 

Placeek2), by Peierls and Kr.pur3) and: by Sicgwt4), In tho following it is not 

attempted to give an exact dcrivation of it, The anelogy to the rosonancc of 

dmped oscillators is uscd, to mkc tho main foaturus of tho formula as plcusible 

as possibluc 

!le first introduce thr? concept of effective wid6hs. According to (29) 

22 

tho partial widths depend on the energy of the outgoing particlo by ntonns of 

n w 

the fsctor kd Tb (Cpg Since the oncrgy E 

nuclezr reaction is not nocdsserily equcl.1 t oe 

@ 

fin 

of thd outgoing parkicls in a 

, it is usoful to introduce the 

uffectivo width') (cfj) which is a function of It is obtainod from ths 

fd 

B' 

actuel width by tcking thc energy dcpundant factor at th+ value cfi instead of 

This rolntion is not quitit exact 

I 

bucczuao of tha slight energy depundsnco of the 

other factors in (281, namLy 

I . 

(F) l2 ; however, the rosonanc~ rugion dous 

not uxtand furthm than the rugion of validity of (31), in which I 

not cnergy dopcndcnt, 

fflfl(R)/:s 

ht us assume that tho energy af an incoming particle a lies within an 

intcrvcil d which i s much s m l l G r thnn thv width of a rosonnnco lcv~l of the 

compound nucleug. Furthwnorc we ossumo thct thu wwgy is nmr to the onwgy 

En at which thc particle would bo at thd mmir;ium of a rosonmcG5), and compara- 

tively far 2.~2.~ from othor rusonenoes so that wc do not wed to consider the 

influonco of othm lwds. The cross wction c(E)as 2. function of thd wiergyt$, 

B 

has than thd cheractwistic L*Gscxw.iicd dqwndonco : 

(1937 1 

3) Poiorls and K~pur, Boy. Soc, Proc. 166, 2'7'7 (1938), 

4) SiegGrt, Phys, f2cv, s, 750 (2.939). 

. 

5) en should, in accordanco with Suction XXXIV, bo writton a," , cnd 

shouLd 

haw an indox Ea. In this section we omit this indcx in tho cnljrgy of the 

incident particlo and the cnGrgy ab which it is at rusonancG with thG nth 

IWCZ.

23 9 

1) 

a(+ F(0 (43 1 

wherc d(E)is thc effective totc21 width of the resonance, which $6 thu sum of 

all effuctivc: partial widths: 

6&1 E E? (43d 

B& 

is the energy (given by (38) ) with which tho particle is omitted. n e sum 

c,, 

is taken over a11 possible index triples b41' 

F(6) is 3. slowly varying 

function ofe, which we determine later, If is in resonance, 6 ran thc 

\ rB 

energies E arz tho sarie as they would be if th3 particles wcm cmlttod by the 

resonance level n indopondently. Thus 6(gn) =An, 

Exprcssion (43) cnn bo understood by analogy Lo the caw of forced 

vibrations of 3. dcrriiped oscillator with cz proper frequoncy V 

~nce of a pirriodic forcc F cos trt with a frequency v 

to bt: due to a fruquoncy dcpendunt friction forcc -mb(v)X 

undur the influ- 

The damping is asawd 

(x is thu dis- 

placcmnent of the oscillator), 

On0 obtains for thc: displacommt xt 

And f.or the avorage energy cbsorbud by the oscillator per unit thw: 

In this oxpression, tarms smller than of the order (%-Pv(q 

me negluct- 

cd (mcond term in (44s)). The timo duoendance of the frw vibration of this 

oscillator would bc: x hl COS % ; its intonsity in this case 

docays wpponontinlly with thu dscny constcnt o(t/,). 

From this wc: cm undmstcnd dxprljssion (43) for thd cross sGct$on, by 

* 

rdpllacing in (Skb) fraqucncics by cnorgios. d(t/)goi!s ov~r into thc: &fm$ivu 

total width o(c)of thc: lmd, whoso v2.1~~: for &= krn is rlso (aftvr division 

by 'h ,) the actual dr3wy conatcnt of thc: 1 ~ ~ just 1 , as &(%) is thc: dl;cay

constant of the intensity of the oscillator. 

The function F(€) in (43) is an 

240 

analogue to a frequency dependent 'force F, acting on the oscillator. 

with its inverse: 

We deterdne the function F( ) in (43) by comparing this cross section 

particle with the energy 

the deoay of the compound nucleus with an emission of a 

within the small interval de . 'ilne probability of 

this emission P(e)dG must be proportional to the same resonance factor, which is 

contained in c(€), but should give, after integration over the total width of the 

level, the total emission probability Tn (the partial width for the emission of 

a, 

the particle a> 

c, is an energy interval large compared to the width. This is fulfilled by 

since 

6@)/2?T 

?an(e) &py (bp + 

)e 

be. (45 1 

This relation is right if "'&n(e)and 6" (&)are sufficiently slowly variable 

over the resonance region. 

It &so would be fulfilled by &tin;S the aotual partial width Fz 

instead of the effective partial width ?*?&)as the first factor in (45). It 

is evident, however, from the derivation of the energy dependence of 9 "(e) in 

4x4 

(42) that the emission of a particle with an energy e is proportional to?n(C) 

rather than to 

:;?e now apply the expression of detailed balance (16) to calcuiate the 

4% 

* where 

function F(4) in (43) from (45). Since the end state of $he'capture process has 

the weight unity in this case, dG(E)dE must be replaced in (16) by unity, 

obtain instead of (16): 

is the wave length corresponding to 6 , From this we get by using (45): 

We

U 

(6) &"(E) 24 1 

4 = ?cs2 (@)P.+ [-Y6&]2 (46) 

This is the cross section for capture of the particle 

a regardless 

what reactiok should follow, Since the bn in the numberator of (461 was writ- 

ten in (k3a) as a sum of the partial widths belonging to all processes which can 

be initiated by the capture of 

a, it is self evident that the cross section for 

a special nuclear reaction 5 4 6 is given byz 

e q$*?ye)Z(kJ- 

-Q(aLlfl) =fix (c-gn)a*l*n(G&gJF (47) 

If the resonance en~rgies f" of several levels are near enough to e , 

the different contributions add up coherently; 

tensitiea must be sunmd, 

be understood from the analogy with a forced oscillation. 

to (h4a) is proportional to the complex Factor 1 

pears in the form ,/?+-E" +ib% ) ' 

The cross sectioli of a6 f b3,a:, 1 

the wplitudes and not the in- 

One then obtains an expression, whose form again can 

Tha amplitude according 

+ix which here 

,.@%P,k)reaction with the initial particle 

l 

a incident with an energy +Z. and a given angular mornentum1-h upon the initial 

nucleus in the state Q,, resl~lting in an emission of b with an angular momentum 

1' and with the final nucleus in 10 , is given by: 

* 

Here f;, are the resonance values:-&n = z,yI--*Ba 

Ea , i4n is $he 

f the excited state n OS the compound nucleus acting as intermediate 

state, &% is the energy of the statem of the initial nucleus; 

/d.crs 

phase difference of the initial and the final sta,to and it vanishes if cX;.""fl 

and 3.' t,jp 

is the 

are the effective partial widths of n, corresponding to the 

emission of a and b respectively, with the residual nucleus left in a, or fl . 

f(€) is a slowly varying function of 

ea which is small compared to the r@son- 

ance contributions of the first terra, . It is introduced to replace tho contribu- 

tions of the second term in the amplitude (4ka) and to inc;lude the contributions

states n and 

z 

(%md n 

k, and kb are the wave numbers correspo 

8 a or b respectively, Hare the 

abed in the *119. E' (6) is the @me 

aftm division bY the factors 

The numeratar of the ~8sonance terms consists here only of nuclear 

s independent of the cnargies af the particles, 

In formias (48) and (49) dJ ,stata;s 6) 

6) 

n of tho nuclei are con* 

sidered aa of weight unity. Zf song$ of 'thew stabw are degenerated it is so01'3- 

re useful to suni over all substates, Tho tots1 cross section is the sum 

of all crow sections from every single qubstate of $% to every one of c In 

cwe 03 angular rmmentum degenaracy these summations can be performd and Bethe: 

te A. 

Tho initial and final states are no denoted with capitallotters 

stead of the greek letters (C~fl> in order to indicate that they cow 

prisa all magnetic substates. The compound state is called C, I/ 1' andjlj/ 

are the arbitaL and tba total angular molllenturn of the incident or emitted parti- 

ectively; 

J is tho total angular momentum of' incoining particle and 

cleus and therefore also the total spin of the coirrpound state C, 

is the, amplibuds 

of the compound state C in regpect tc) the emiesion 

tide a, with orbital and total angular momentum L and 3 and with the 

residual nucleus fn A, 

The sumnation sign 1' 

C with the spin J, 

c 

It is evident that @&A 

=o wess/j++j)l j-i( 

Indicates that the sum should be taken only over states 

The sumr.;stion over J 1s outside dF the absolute square signs 

'

which 'chat thd cantr cPlCtefjng with Egerent total 

moment 

In most cases in which (50) Ls appl%ed, the contributi 

4 

of B 

with a 

e exfendsd t 

utie glso rgdiative phenGmena sutsh 

ar reaction in wkii&i 

ia particle 

ent upon a nucleus in the srta 

and a light quantum 2s wdtted 

ending in the stla,

m B) 

A fm general conclusions cats be drawn from the ono-Zeva1 formula (47); 

If there is only ano einission procoss poadible from tha corrlpound 

nucleus, which thi! 

can put oh 

cwsaPi2.y is the reamissios of the absorbed particle, we 

and we get; Par the only poslsibls process, the elastic scat- 

elastic scattering. 

It thura i.3 bne eorty3uCing ~FQCQSS, say the ofixl-ssion of b, we obtain 

which i s off reBo 

but, in resonance, weakm

consideration: 

246 

if the proton would be emitted with highor energy than tho one of 

1) 

thb ndutron, the energy of the residual nucleus would bG lowsr than thc ona of 

the initial nucleus, and the two nuclei would be isobars diffGring by onv unit of 

charger nus the initial nucleus would bop-Unstable, which is impossible:. If 

tho emitted proton had equal or ldss energy than thc nautron it wwld not be able 

to pcnetrat,: the potential barrier, 

1% first assumo that thdm is only one level sufficiently near to tho 

onurgy to which the com2ound nuclas can be excrited by thd neutron. \'TU then apply 

(5b) and uso (33) for thc: partial. widthTaA of the neutron, F'urtherrnoru, 

' 1 1 

JaIkE since tho noutron can contribute only its intrinsic spin, 932, and 

The cross 

is thcn: 

(55) 

which is obtaincd by putting ?( s C&" also according to (33) .! 

60 

The cross sdction for emissioii of a light quantum or anothm particle 

This cross section shows the characteristic €-* dapndmae on the neutron 

t 

ondrgg (tho so-called l/v law), If thhi? wicith bc is wry Xarg6, tho e'g-factor 

is the doterraining factor for tho energy doyndonce. This is the casu obgf,) in 

thu boron reaction: 

where thc total width is datmnjuld minly by th3 ~Z-entission width, 

Tho &-pBr- 

ticbs are mittod with an urisrgy of about 2.7 MeV, and am able to pass above 

the barrier, The corrwsponding width 72 is therefom vory large and was found 

to bt: sbout 200 Kay. 

Thus the boron raaction follows tho

of a light quantum, (as in triost Dams of he 

246 

nuclei) the capture OPOQ~ section 

C) Wdrimctnt- 

A number of resananc; phonomma haw bwn obsorvod with bombardribnts of 

oi, 

Staub and Tatc17) haw obasrvGd a strong resonance in th\: elastic 

.sf neutrons by helium, Thus rQmni&nce could be assign9d to a doubbt 

(54). 

itud by p-neutrons and was ana 

Hcrb and collaborators kn Wiscons 

ed by mdans of a formula sinxLlar to 

and Tuve et aJ9) havc found row- 

nanco~ by inasuring ' thQ emitted ?*r&y intunsZt7 as a function of thu enorgy of 

tho bomlwrdGd protons on Li, F, a. Tho? prays @re cithur crnittod by tho coni- 

pound nucl~~ il;scLf or by th\: oxcitud fSna.1 nuclwus which is loft over afta 

omission of an &-particlo (thu lattur pos$$blo occurs in F). The obsurve 

9 dray intensity shows sharp maxhti if the enorgy of thG bombarding protons is 

such that B love1 is a citod in thu COLipQUnd nucl@us. Ttiua tho resonances giva 

a picburu of tha syct/rurrri of 

leus abovc! the proton binding 

nd which, bQoawc! af ssldction rubs, 

emit GL-particlc:s are so broad 

8) Herb, Kurst, McKibbtm, Phys, itev 

&irnot, Hwb, Parkinson, Phys, 1% 

Plain, Hwb, Hudson, Ijlarro

.- 

247 

resonanw. The most striking rcsult of a comparhon batweon thc thrw o;tc.mcnts 

is the grcatdr leu4 density in the ha.vi+r nuclei. Tho avoraec love1 distanco 

is about 500 Kciv in Li, 200 Kcv in F, and 30 Kev in Al. 

thc levds arc? a sum of particle: and radiation widths. 

The obsurvad widths of 

Thuy ari: 11 Jim in Li, 

about 8 Kcv in F and of tho order of on3 or two K m in Al, 

Tho luwl density in A1 is so high that ono would not expoct to bo able 

to find resalvabld rJsonancas in dumcnts much havier than Al with chargGd 

partiall;: beam. 

Hdsonanws in huaviw nuchi w~ro found with slow neutrons only, 

The formula (56) for thu capture of slow nwtrons can thon be tcsted, and tho 

diffmant menitudes: ec +ec and C; can be dotm&wdr 

nQutron width C& is usually much smalldr than thd radiation width; b;r, thurc- 

ford put d"W3i.c 

and thG forriiula ($6) rGduces thcn to: 

In this caw, tho 

In thc swmd hcpression thc dnurgy f is exprdmUd in GV. 

c 

In this formula tho Dopplor cffdct has bdzn nugl0ctt.d. &causu of tho 

thmnal motion of thc: nucllzi, tho rtilative energy batwcwi neutron and nuclleus is 

not uxactly cqugl to thc! noutron cnwgy. 

around ths nuutron unargy E witih r?. Gaussian distribution: $,(/2x e*pE(E-!$v&@ 

whum the Doppld width K is giwn by 

K=2(mekT/M) A 

The rolatiw wmgy E. is distributed 

and m/N is thu ratio of thu msscs af tht, nJutron and tho nucleusI 

Sinw (56a) givvs th3 dopondoncr: of d(e)on thu relative enurgy, thc3 

cross wction as a function of thct actual. c3nmgy 3 i s givdn by: 

c 

If thd Dopplm width K is Si&U corAipnrr;rd to thc: natural. line width 6 

Doppler 4fmt is nGgligiblt: and (568) can bu us& with as thf: actual nGwtron 

(57) 

the

c 

endrgyr If K i s Larp compared to 6 the ahapo of the absorption 

rely by th3 Dopplvr dffwt and WF: gat: 

24 8 

lino nuar 

Howover if the anorgy 

is sufficiently far away from rosonznce, G-G>> K 

c ,C End d of a reswmnce levul: 

1) Thd diruct wiwthod of mnquring bh+ absorption of, or bhG radioactivity 

0 

produced by, a mor?ochrorm,tic nautron beam with vnriablr: unurgy. 

fixpurhmts with 

rilonoenergljtic ndutrans hi;w been first performed by AlvarGz by mums of a i,iodu- 

lated nwtron SQU~C~ and by a suitably aodulatud amplifier, which registers only 

nvutrons which havti arrived at the counter jXr s1 spucificd thzu intvrval sftur 

uction, 

I3zche-r and Baker have jpiprovod this mthod and wcru abll; to 

thc nuutrons, 

rang@ of encrghs invdstigstod Pram 'two to ebout 5 cv. 

This rwthod gives $&,turi3 

diroctly as a functAon of thc anlilrgy e of 

Thc quzntity mcaaurdd in an absorption r-iuasuretmnt is actually tho 

SWJ of ccepturL. and the scathring cross suction, 

For low enorgius and n~ar 

2) T~L: boron absorption mthQd, Thu boron absorption 1;idLhod is used for 

dupendonct: of the cross soction of th\: (na$ proccss in baron, which is 

bld foY thu absorption of ndutrons boronu ThG PrinciPlQ of this 

cuthod consists of thd ~follov~ing: n ~,lc?,terial is r.ctivnt;rd by zbsorption of 

nc;utrons from p. bJ.m contcining a continuous spcrctrw of neutron ohcr@Gs. 

Thd 

of thd activation is r&nsurod afto covding the matd%d. 

with boron. 

ction is pToportionc1 to thc absorption In boron and thdr$:foru pro- 

to @;' if f?,c is thu wmgy of thu nwtrons at msonanca. As shbwn 

bufom,, JVUY imteriab CapturUs to i! curtain atmt very slow n3utm-M of thwwl

important, and tha intensity is given by: 

&lation (58) is! corrdct if C!&n$))W , which ;,i&ns that tho andrgy rc:gions 

sffi;ct\zd by thc: Dopplvr uffdot $.r$ coriiplcrtuly e.bsorwd. 

4) Thd activity mthodq Tho activity producud by a nuutron barn of 

given spoctrurn is riiaasurod, aftJr shiolding off tho 'thzrmal nuutrans. 

ndutron spoctrurii is usually obtainud by slowing dovfn high unorgy noutrons in 

light mc.terial as carbon or pp.raffin, 

thm proportional to 

y& 

4 

This Method givds c mI?surc, 

sy 

of 

de= mr*+~ AC 

inddpcndcint of thr; Dapplor df&%, 

Tho 

TI~G intmsity p~r' ondrgy intdrval de is 

5) Tho mcsur,JrtiGnt of thG thdmo.1 ?.bsorption cross scction. 'he thurml 

absorption ccn bd usGd for thc: dwtGrrlninntion of f+:vc?l constcat8 only if tho 

oncrgy of all highdr bwls is lnrg\: cofiipcrod to thu lowost onQ, and if no luvd 

* 

bdow zero unurgy is of cny influmct.. 

Tht: lowst 3 ~ vnust d b6 ~ OQJ onough so 

that thu absorption at thmml cinurgids is taking placu in the I'wingN of the 

rus0nrZncc: ti&" 

In cava the one-levo1 formnula is applicable, WJ obtain:

261 

I 

-11) 

from niethod 2): ec . 

frois niuthod 3); Q"~ if ac >I> K or from thick Iqws, 

Thus thc! throd unknown qupntitius E. C I &, c cen bd dotwrmindd, JnconsiEstcs 

bwxusu of tht, contribution of othJr hvds. 

1Lthods 3) 2nd 4) may giw too 

largd vnlues bduause thay includc: thc: uffuct of higher 3.~vds4 

A tab3.u of thu charactGristic vc+lucis far 6011id lGw3.s of diffurcnt ulu- 

iwnts is givsn below* 

Only dmonts arc: includcd whurc: the ttviduncc soms to bu 

not too contradictory. 

Thu nuinburs in peronthcsis e.t tho r&.wncos 

indicek thu 

muthods usud, as list4 pruviously, 

C~SOS w h v tho spin of thu nvutron is addud to, or subtmctud f.rorsi, thu spin oft 

tho initial nucleus, 

Uopplor width in this cnso is dominmt, 

C 

Thu value of A in As cannot bw doturriinod bocausc thu 

The radiation width, rF which, in thwu cases, is practiocl2.y Gqual to 

C 

thQ total width A , is vury nd?.rly Qque.1 in all c8.swsc It sham uriusunlly 

little fluctuction. 

?%is r~letivt! constancy is connuctod with thG fkct thct 

rtzprdsants a sum of wqy partial widths corrcsponding to r-zdictive transitions 

tQ n gr6F.t nuGibor of lowor stctos. 

Tho fhctuntions of single trnnsition proba- 

* 

bilitias wc nv3ragJd Out, Thd valws of C show somewhat gruahr fluctuations, 

"h~y rc(prt.amt e singlir transition probability pad my dlspund mor0 critic,n.lly on 

thG propwtius of thu individud ldvobr 

Thu vnlu~s C' m-wIi to fulfill the, candition (36) fairly vrLLT, 

Tho vc?J.ul:

Bombarded Life time ~,Ccvl Ref . Tr Ref. Enx10 * CW* Ref. 

Element of product 

AS76 27 h 40 1(2,3 1 32 5 1I3) 

Aul'* 2.7 days 

d - 

669 (1937) ' 

Phys. Rev. 53, 18 (19401 

Phys. Rev, 59; 154 (1.941) 

- 

* *

* 

2k (341 

255 

UCTWl38 8FiRI;E;S ON NUCLEAR PHYSICS 

ies: lzse Statistical Theory Lecturer: V. F. VJeFsskopf 

of Nuclear React ions 

L!ECTU.dE XXXVI:, 

Non Reaonance Processes 

AI 

If the energy of the incfdent particle is higher than a certain limit, 

resontanbes do no longer occur. This can 

anergy cannot be made mOnQChrOmatic enough to resolve the levels; it can be due 

to the Doppler effect; the D~Qpler width reaches, for example, 50 ev for an 

inddent particle energy of 250,000 ev, if the nucleus is 100 th3s heavier than 

41s; 50 ev is of the order of or 

sonames $or heavy nuclei. Furthermare, at energies of several Mav the 

actual width of the level reache8 the order of level distances, This can be seen 

in the following way: according to (17) the partial width for the endssSon af 8. 

fker leaving the nuoleus in a given state, reaches the value (2k+l)D/211; 

&dual nucLeus can be le& 

in several excited statas, the corresponding 

tance D, 

d up and gLve rise to a total width, equal or larger than the level dis- 

Thus,, even under ideal conditions of' a monochromatic beam and zero 

m, rescmances dbappear at atr energy of several &vts. 

In order to compute the cross qrsctions of non-resonance processes, we 

use Elohr's assumption 3 (see introductiofi). The CPQSS section c3n then be ex* 

* where 

pressed by (2) 

t~ (a, 

b) +2= ra *tlb (2) 

is the cross section of tha for,mtion of the compound nuO&eUS, and qb

is the relative probability of the emission of the particle b: 

256 

Here 

is the partial width for the emission of b irrespective of in what 

- - 

state the residua;l nucleus is left: r averaged over the levels 

Ak bB/t 

of the compound states excited by a, A- is the averaged total width. 

- 

The partial widths 7~ (we use again the abbreviated index&; see 

page '7, part XXXIII) are connected with the cross section Caof the inverse 

process: the formation of a compound nucleus by the particle a of angular 

momentum 1 and a nucleus in the state 05. This relation is given by the 

expression (16): 

The cross section Sa appearing in (2) is Gal with 6 being the ground 

state of the initial nucleus. Thus we get for (2): 

(2a) 

where the summation over 1 and 1' refers to the angular momentum index in the 

abbreviated indexa, or,@ 

respectively. 

The consistency of the expressions (2), (3), (16), with the resonarlce 

formula (48) is shown by averaging (48) over an energy regionbe, which contain9 

many resonances. 

If the level distances are large compared to' the widths, this 

integration is a sum over the contributions of the energy regions around every 

single level. 

One then obtains: 

The sum is extended over all levels within the energy interval Ae . 

This relation is right if the term f(g) in (4.8) is sufficiently small compared to 

the resonance contributions. 

After introducing the average width 2; over neigh- 

boring levels, we obtain: 

-1 

where D= [c+ (E,)] . is the average level distance. After sumriiing over k! and 1'

* 

We 

e57 

and / j we obtain the sane expression as (2a). This shows that expression (2) for 

a non-resonance reaction is in agreement with the resonance formula in the region 

in which the latter one is valid (D )) )# 

now proceed to the quantitative discussion of the magnitudes occur- 

. 

ring in (2). We make use of the intimate connection between partial widths r& 

and cross sections which is given in (16a). 

In some energy regions the cross 

sections are better known, in others the widths; either magnitude can be used to 

determine the other, 

absorbed) particle, 

We distinguish three energy regions for the emitted (or 

The characteristic magnitude to distinguish the three regionp 

is Kawhich is the wave number of the particle near the nuclear surface and which 

is defined as follows: 

M a 

kt, 

2 2 

11, LOW energies: /Kat .C h, . This region includes all energies 

near to the one for which it just passes over the barrier or, if there is no bar- 

2 

rier, all energies smaller than% k:/zm . It includes also energies which are 

2 

not sufficient t~ pass over the barrier. In this case ff a is negative . 

2 2 

111. Penetration region: K, < - k, . This includes the cases of 

penetration of barriers with energies much lower than the barrier height. 

In the first region, claasical considerations are justified since the 

wave length of the particles is small compared to the nucleus and of the order or 

* 

Smallw of the distance between constituents. 1Ve may therefore assume that every 

particle that comes to the surface is absorbed. 

nearly equal to its maximum value and get: 

Hence we put 8&,,eaqual or

258 

The second expression follows from (16a). is a pure number smaller than 

unity which goes to unity for high energies and is introduced to take care of the 

transition regions, 

The second energy region is the same region in which condition (31) in 

Section XXXIV is valid. 

The value of fia can therefore be obtained from the 

expression (28) in a simple way. 

the form (32): 

It was shown there that (28) can be written in 

where C(E) is a function of the excitation energy E of the compound nucleus, 

which is independent of the nature (and angular momentum) of the emitted particle, 

In case of an s-neutron (1 = 0, T = 1) this expression should join (59a) at ka=k, 

and it is therefore tempting to put 

where 

cCFJ=t(D&t)(P1+1)(1 / k 0 ) 

may be some function of E only. Since, in the extent of the Region 11, 

E does not vary by much, we may assume that the value of is approximately 

constant in this region and of the order unity as in Region I. We then get: 

Region SI (593) 

The second equation comes from (16a). 

In the third region (high barrier), the wave length x, will be always 

ao small that the potential energy outside the nucleus does not change appreciably 

over a distance %a. Thus the wmcalculation is . applicable and it yields

259 

Here Pa is the penetration of the barrier, which is, in WKB approximation, given 

Here rl and r2 are the radii between which the potential energy V is larger khan 

the total energy E. 

we obtain Pa= (ka/&) 

By comparison of this with the VJKB expression (29) for Ta 

Ta. 

Thus the regions I1 and I11 join smoothly. 

It is interesting to investigate the Region 11 by means of the VKB 

method of approximation. 

'Illis region is defined by the condition that the wave 

length ,&of tho particle outside of the nucleus is long cornpared to the distance 

between tho nuclear constituents. 

If we assume that 5 is still small enough to 

assume the potential fields outside the nuclear surface (centrifugal or Coulomb 

force) as slowly varying, it is allowed to apply tho TiKB approximation for the 

wave function from the outside up to the 

nuclear surface, There however, conditions 

change abruptly in distances small compared 

to 'f; . It seems to be appropriate to as- 

sume that, once a particle has panetrated 

this surface, it will not return without 

change in energy; in other words, it will 

It 

v 

have formed a compound nucleus, 

Thus the 

penetration process is equivalent to one 

dimensional problem with a potential energy 

of the type indicated in Fig. 2. 

Here a 

particle comes from the left side over a 

smooth potential barrier. 

At x=O, the 

potential abruptly drops to a value which 

would give to the particle a wave length% 

comparable to the wave length inside of the 

nucleus, which is of the order of the Fig. 2

260 

distance butween nuclear particles , (Vie call the corresponding wave number: EG) 

* 

This region oxtends to tho infinite, so that the particle does not return, If a 

wave of particles with an energy E (A particles per 'second) comes from the left, 

it is partially reflected (even if there is no barrier, because the potential 

energy changes abruptly at, x = 0). 

The current which is not reflected but pene- 

trates to the right (B 

particles per second) can be calculated easily by 'vbKI3 

where XI 

is the point at which the potcwtial energy V becomes larger than the 

total energy E of the particle, and Vo is the value of' V at x=O. B/A is tho 

ratio of the penetrating particles to thc incident ones, 

According to this ex- 

pression, the cross section for the formation of the compound nucleus is given by 

the maximum possible cross section, multiplied with B/A: 

sinc*$

Region I: 

The energy is more than 1 or 2 Mev above the barrier, 

261 

ier . 

Region 11: 

Region XII: 

The energy is less than 1 or 2 Mev above or below the bar- 

The energy is more than 1 or 2 Mev below the barrier. 

We now apply these considerations to the calculation of neutron cross 

sections, The total cross section cn for the formation of a compound nucleus by 

neutrons, is given by 

momentum 6 1 

Cn= 

where 6" is the contribution from the angular 

f 9, L- 

We first consider neutrons of low energy E {

(N 

Fi9.3 Neutron CI-GSS Section vs. Erlerjy

as functions of the energy e. 

k K 

B is given by 0 - 2 ( 21 

1 263 

The energy is given in units of (kR)&* The curve 

, This sum converges rapidly for the values 

L=O 

of E far which it is plotted so that the upper limit of the sum does not enter. 

B is proportional to C,V, and A is proportional to the cross section itself. 

It is seen, that, with the plausible value for koR ~ ' 3 , the cross saction(Jln goes 

smoothly over into its asymptotic value RR 2 

. 

For a heavy nucleus of Afi.200, the radius R is near to 9 x SO'13 

cm, 

so that (kR)2 is a measure of the energy in units of 250 Kev, 

Hence we must ex- 

pect cn to follow a l/v law for low energies up to about 130 Kev and to assume a 

value near flR2S2 3 x low% above that energy, 

The cross section 0"for charged particles can be computed as follows: 

it is given by (60) as long as their energy is appreciably lower than the barrier. 

The IVKB method can be applied to calculate the penetration factor, if the barrier 

extends over a region which is large enough, so that the potential energy does not 

change appreciably over one wavelungth of the particle, 

This is the case for 

higher nucloar charges (2 } 20). 

If the energy is well abovc the barrier, the 

expressions (59a) give the cross sections directly, since there is then no pene- 

tration factor to calculate, 

sions (59b) should be used. 

For energies near the barrier, however, the expres- 

The VfKB method in this region is no longer reliable, 

since the remaining barrier is necessarily very low and extends over a narrow 

region,' In the following comptttations, this region has been left out completely. 

The validity of the WKB solution, applied at low energies, has boon sxtonded up 

to energies equal to the barrier, where it was joined with the solution valid at 

energies much higher than the barrier. 

These two solutions join smoothly at that 

energy since the WKB expression for the penetrability P of the barrier becomes 

unity at the barrier. 

The justification df this procedure lies essuntially in the 

fact that, the barrier heights for different angular moments of the incident 

particle differ by amounts of the order of 1 Mev, so that, for a given energy of 

the incident particle, only one or two angular moments lead to a case of a Ftegion 

,

.:c) 264 

11, Most of the partial cro8s sections gare . thus computed correctly . 

The cross section for the ponetration through the barrier is a very 

sensitive function of the nuclear radius. 

The following tables give tho values 

for two sets of nuclear radii, one givon by d * 1.5 x A 113 , the other by R=L3 

x A l/3 

It seems that the second set with its steeper excitation functions, fits 

better to the erpx-imental material available. 

- 

the rb for dl particles b emitted by the compound nucleus. 

the sum f" =x 

I3 

The calculation of q of expression (2) involves the computation of 

A? kP1 

They are given by 

each term of which can be computed from (59a), (59b), 

(60) with reasmatle accuracy, apart from the factor 4 which does not differ much 

from unity. 

L.? thl final nilclew can be left in many oxcitdd statesfl, the sum 

can be exgresscd PS Pri inkegred and we obtain from (16a):

.4 

05 

06 

.7 

-8 

09 

1.0 

1.2 

124 

1.6 1;8 

2.0 

1 

40 

i Ca 

2s55 

6,37 

12.3 

18.3 

26.5 

35.1 

40,3 

50,o 

56,2 

60.4 

64,O 

66,2 

B 1 6.4'7 I 10.06 15.60 

120 

a 

0 

0 

,0239 

. 

298 

1.72 

6.13 

15-7 

27.6 

53.6 

71.7 

83.4 

93,o 

i02, 

8-72 

6.0 

7.1 74,6 

106 145 

I 18.731 9.70 116.24 

d 

r 

L 

LI 

.84E 

3.42 

8.15 

15.1 

23.2 

33.7 

4613 

54.8 

63.8 

70.2 

73.7 

78,3 

64 

Zn 

30 

8,28 x5&74 - 

I 

90 

Zr 

- - I 

TT 

Gc/ I P 

0 

0 

. 00468 

.099E 

. 871 

3.90 

11,2 

20,6 

44.7 

61.9 

71-4 

€33.3 

86.9 

14 2 

Ed 

60 

,00433 

32;o 7 i.85 

.02 

I1 

19 

12,67 I 121.45 

"-- 

73c 

3,04 

8,23 

16.1 

26.7 

38,8 

49.3 

69. 9 

79.3 

86,9 

3.03. 

110 

7,17 

- 

---- 

_13 

* 0499 

, 531 

2.88 

8.79 

17,3 

38, I 

88.8 

91-6 

.15 

.37 

.41 

50 

10- 98 

.04 

.14 

I 

26.5 44;6 

173 

v 

1 70" 

5L 

27,O 

79.4 67.5 

96.9 94.4 

I 

I 

4 

0 

0 

0 

0223 

,410 

2.87 

11.7 

I 

1)

266 

where the sua in the denominator is extended over all particles c wh$ch ccqn be 

emitted in this reaction. It is evident that the integrand in (64) 

TABLE: I1 

The cross sections 6 and Ca @re given for 8 typical elements as 

P 

function of the energy : 

cp 3 @ = n(x) '10-26 CWl* 

n is the number given in the tables, x is a masure of the energy in units of the 

barrier hekht B given in each colwnn: e== x.B . The columns headed by I corres- 

pond to a nuclear radius R = ro All3 with ro = 1.3 x lOe13 cm, the colwnns 

headed by XI, correspond to ro = 1.5 x 

cm, 

The cross section q(6~) for the penetration of the deuteron is equal 

to the one of the proton for en element with a charge s'mller by a factor (2) 1 413 

and with an energy smaller by (h) h 

is the probability of emission of b with an energy between 

and e +dk. It 

describes the energy distribution for the outgoing particle as a function of its

energy e . (This is identical with expression (4) ), 

267 

* 

It is interesting to discuss the connection between the distribution 

(67) and the Maxwell distribution of particles evaporating from a hot sphere, as 

indicated in the introduction. 

The distribution I(E, ) ch be written 

According to statistical mechanics, $(E) is the entropy of the residual nucleus 

with the energy E. 

By writing 

and by using the well-kncwmstatistical expression 

we obtain 

=i/T 

Mere T is the temperature at which, according to (67), the residual nucleus has 

the average energy E, -. If Cb is a slowly varying function of the energy, 

as it is in the case of the emission of neutrons, the emitted particles have a 

bxwellian energy distribution, just as particles would have, that are evaporated 

by a material with a temperature T. 

This toniperature is, however, not the 

ntemperature of the compound nucleus before emission11 , but the ftternprature of 

a nucleus after emissionfl. Here we understand by Ittemperature of a nucleus in a 

state of energy Elf the temperature at which it has an average excitation energy E. 

This comes from the fact that the Itevaporation” of one particle constitutes a 

relatively large loss of energy which reduces the temperature considerably. 

The 

* 

Maxwell distribution of the emitted particles is determined by the temperature 

- 

after the emission. 

If 6b is a strongly varying function of the energy, as it is in case of 

charged particles emitted, the Maxwell distribdtion is deformed by the factor a“b 

in (68). Since cT~ is usually a strongly increasing function for charged parti-

e 

268 

cles, the maxhwn of the distribution will be shifted to higher enqrgies.. 

The limitations of tho validity of the Maxsv 11 distribution in ni clear 

emissions are obvious from the method of derivation: it is assumed (a) that the 

level density of the residual nucleus in the region of possible excitation is 

high enough to permit statistical treatment, (b) that e (( Eb m, which means 

that the energy of the emitted neutron is small compared to its mcurimum available 

energy, 

Since the level density of nuclei is unknown to a great extent, it is 

hard to predict the limits of validity in terns of Mevs, 

3t ) 

In recent experiments 

it seems that the level 

- 

density of the lower lyhg levels is rather small, viz., 

about one or two levels per MeV in the first or second Idlev of epitation energy. 

The maximum energy Eb p robably must be at luast 5 - 7 MeV in order to expect 

a shape similar to a Mawhian distribution for the emitted neutrons by elements 

in the middle or at the heavy end of the periodic system. 

The values of tho temperature T, which are expected in nuclear 

reactions, can be estimated in the following way: 

the average energy E of a 

system is a monotonically increasing function of the temperature T, 

If the temp- 

erature is very high, so that all degrees of freedom are excited, E is propor- 

tional to T, 

This is certainly not the case in a nucleus, when it is excited by 

a normal nuclear reaction, If one considers the nucleus very roughly as n gas of 

A particles concentrated in a voluma (G R3), one finds that this ttgasll is de- 

genersted if the excitation energy is of the order or less 

.3As Meve Heavy nuclei should therefore be considered as highly de- 

J 

generated tlgasoslt 

The stroilg interaction between the particle does not affect 

r) 

this cor,c!-usion which also holds for an electron gas in metals in spite of their 

--u_ 

-L-.-,uru-I-uI-.c- 

WFlkins, T, R, and Kuerti, G., Phys, ibv, z, 1134 (1939). 

Wilkina, T. R., Phys, Rev..@, 365 (1941). 

Dicke, R, H. and Nhrshall, J., Phys. Rev. Q, 86 (1943). 

Dunlsp, H, F, and Little, R, N., 

Phys. Rev, 60, 693 (194l)*

interackion, 

269 

The function E(T) ha8 ta vanishing derivative$ ( &E/ 4 T)T=o = 0 for 

c T= 0. (This means that the specifio heat is aero for T*O.aocording to the third 

law of thermodynamics,) The expansion of E(T) near T*O etar‘ts therefore with a 

quadratic term, If a system is highly degenerate, we may write 

E = bT2 (70 1 

and neglect the higher powers of T, which are small in this case, From %his we 

and finally for the level density: 7 

pE 

W(ES =C (711 

One can try to adjust the constants a and C, 80 that this equation 

reproduces our rather scant knowledge on level densities. A fair choice for A } 60 

is e*g,, 

cad = [;Z/(WOjJ (MeV)”’, Ceven = 9 Codd? a = 3 35 (A-40)* (MeV )”’ 

if thB eXCitatiOn energies E are measured in Mev, 

Codd and Ceven refers to 

nuclei with odd’ or even A respectively. 

The expression (71) is adjusted to our knowledge of level densities for 

very low excitation energies and for excitations in the region of capture of slow 

1 

neutrons. In the latter case only levels, whose spin differs by kzfi 

1 

from 

the initial nucleus Eire observed. The level density in the expresslon (711, 

however, should include all levels which can emit or absorb the particles b with 

the energy Eb Our knowledge of level densities and of the distribution of 

angular momenta among the levels is insufficient to make any distinction of that 

kind at the present %be, 

From (70) an estimate can be given of the nuclear temperature which 

Ir 

appears In expression (69) and which determines the Maxwell distribution of *he 

outgoing neutrons. If the maaimum possible energy of the outgoing neutrons is . 

Eo, the temperatures are given in the following table in Mev’s according to

I- 

60 150 

-1 

5 1w-15 76 

10 1.63 1.07 

15 

2.00 le31 

I- 

230 

- 

66 

.93 

1.14 

These temperatures can be obtained by bombarding the 

nucleus with neutrons of an energy Eo, or by any process leading 

to the same excitation energy of the compound nucleus. 

270 

Expression 

(69) shows that the maximum intensity is emitted with an energy 

2kT, if'ris a slowly varying fuiiction of E 

to be in case of neutron ernissfon. 

:i/ 

which serve to compute the factor 'Y\baccording to 

are shown in Fig. 4. 

107 

c- 

LO6 

102 

J Y. -/- 

as it is expected 

With the expresston (71) for the level density It is 

possible to calculate the functions f ( ebmax defined in (65) 

(65). Examples 

101 ' 

1 

.rl) 

/I I I I 

0 2 4 6 8 10 12 14 

Fig.4. The functions fn,fp and fvare given as functions of energy 

for the compound nucloi Cu, Zr and Sn, These values must be multiplied 

by 2 or 0.5 if the residual nuclous is odd-bdd or even-even, 

. 

respectively. Expression ('71) for the level density Is used in the 

c ornput a ti on

271 

* 

The reaction types, discussed in this section, are ordered aocordtng 

to the nature of the incoming particle, 

Resonance reactions are excluded, since 

they were discussed in Section V, 

1. Neutron Reactions 

The cross section %(e) for the formation of a compound nucleus for 

energies e above the resonance region is given by (63a) and (63b), and is represented 

in Fig. 2, 

in ,n ) Reactions, This is not a real nuclear reaction. eince the bom- 

barded nucleus remains unchanged, It may however be left in an excited state 

after reemission of the neutron, in which oase we speak of inelastic scattering 

of neutrons by the nucleus, 

Once the compount nucleus is formed, every other 

particle except the neutron has to penetrate a potential barrier to get out. 

The 

neutron width, therefore, is much larger than any other particle width, and also 

larger than the radiation or fission width if E is sufficiently high. Under these 

conditions. 7n N t and the cross section for the (n,n) reaction is given by OB., 

its elf. 

There is a difficulty in interpreting the observed (n,n) cross section 

by means of the expression (2), due to the presence of elastic scattering. 

This 

is the part of the n,n reaction in which the outcoming neutron has the same energy 

a8 the incornbg one. 

As mentioned in Section XXXI, page 8, part of the elastic 

scattering is due to effects in which the neutron is deflected by the nuclear 

potentials, and thus does not penetrate and form a compound nucleus. 

of the scattering is evidently not ineluded in the expression (2). 

This part 

The observed 

cross aection of an (n,n) reaction therefore contains a part which has nothing to 

do with the formation of a compound nucleus, 

The waves of the scattered neutrons 

due to external effects (in which no compound nucleus is formed) combine coheront- 

ly with the neutrons, emitted after the formation of the compound nucleus if they

are re-emitted by leaving the nucleus in its original state, 

2 72 

These two effects 

II, 

together give the elastic scattering. 

The observed cross section of an (n,n) reaction therefore contains a 

part which has nothing to do with the formation of a compound nucleus. The ob- 

served C(n,n) can be much larger than Cr,, especially at low energies, when the 

elastic scattering is a relatively large part of the total scattering. It is, of 

course, possible to separate tho elastic from the inelastic scattering by dis- 

kinguishing between the scattered neutrons of different energy. 

The cross section 

qn for inelastic scattering, which Is due to the neutrons which have left the 

nucleus in an excited state, is certainly lower than CJI,, because it does not 

contain the contribution from the elastic scattering, which corresponds to a real 

penek-sfion into the nucleus and a suhsequent re-erni$sion by the compound nucleus 

wtLh kko sarw emrgj as it came in. 

The pax% cf the elastic scattering coming from the ren1,formation of the 

csnpovnd ste

15 rn 

2 73 

There is only very 

little experimental material available on the spQctrum 

of the scattered neutrons by heavier elements, Dunlap and Little*) measured the 

spactrum of neutrons scattered by Pb with an initial energy of 2.5 MeV and found an 

inelastic scattering which indicates several excitation levels of Pb. 

The h.2n)-reaction occurs if the energy of the incident neutron is sufficiently 

high, the residual nucleus after the re-emission of the incident neutron, 

. may still be excited highly enough to emit a second neutron. 

This is possible only 

if en- Gib is larger than tho binding energy E, of a neutron to the bombarded 

nucleus where Gn is the energy of the incident neutron and et,, is the energy of 

the first emitted neutron. The probability of (n,2n) reaction is therefore given 

by: 

where I(€ )de 

given by (67). 

is the probability of emission of a neutron with the energy f. , 

For energies en which make a (n,2n) reaction possible, the energy 

distribution can be well approximated by 8 Maxwell distribution (69) with a temperr 

ature T, with which one obtains (with. q, =c rCg2 .): 

If A 

AE-Gf) - B, 

-0 the energy surplus over tho threshold energy of the reaction I- is large 

compared to tho temperature T, the (n,2n) reaction should be the dominant reaction 

. 

and its cross section is then nearly equal to 6,. The (n,n) reaction cross section 

should become correspondingly smallerr This should occur for A,E 9 2 

or 3 Mev for 

e 

elements in the middle of the periodic systems. For very high primary enorgies -0 

e > 2B 

n 

(n,3n) reactions will occur. 

qunlap and 'Little, Phys. Rev.

2 74 

The fn.p) reactions naturally are loss probable than the (n,n) reactions, 

c 

because of the potential barrier which prevents the proton from leaving the 

nucleus as easily as a neutron. 

The cross section is given by: 

where the fts are given by (65). This cross section assumes measurable values 

only if the energy of the incident neutron is several Mev above the reaction 

threshold, beoause the outgoing proton needs that much energy to penetrate the 

barrier, 

The rather steep dependence of C(n,p) on the energy gives rise to an 

apparent threshold which is much higher than the actual threshold. 

If the nucleus which islcrsated by the (n,p) reaction is a negative elec- 

tron emitter, the final product of the,@-dewy is identical with tho initially 

bombardeq nucleus. 

In this case the reaction 

energy Q = Cp - en of tho (n,p) 

reaction can be calculated by the energy law. 

The energy balance in the nuclear 

react ion 

n+X=Y+p 

is given by 

En+ EX = Ey+ Ep 

where 

and Ey are the binding energies of the nuclei X and Y in their ground 

states, The ,&-decay has the following energy balance: 

%= %---mn+m c e-+ ht/ 

P 

where my, and mp arc the mass energies of the proton and the neutron and 

Eo in- 

cludes the mass energy mc2 and the kinetic energy CGin of the electron emiteed; 

h U is. the energy of a 7 -ray which may accompany the ,@ =decay. 

Be thus get for 

the +value: 

Q =%- %=mn-mp- e-- hn/ = 0.7 Mev - hiV - G& (72 ) 

In,&) reactions have an extremely small cross section in heavy nuclei . 

]I, 

because of the potential barrier, which makes it very improbable that the 

&-particle leaves the nucleus, If the neutron energias are extremely high,

is 

however, (nt&) reactions may occur in heavy nuclei. 

275 

There will be more chance 

to observe these reactions at very heavy elements, where the binding energy of %he 

f -particle is sqallcr , 

The (n,’)) 

reaction is usually called radiative neutron capture and was 

disausscd extensively in the r~~onanc~ region in Section XXXV. 

energy region the average value over many resonances is observed. 

a.-(n 99 = q (T!, 

In the higher 

The processes which compete in heavy elements with the radiation, are the re- 

emission of the neutron and, in some elements, the nuclear fission. 

the latter from these considerat5ons. 

of a neutron with an angular inomentum .,df~ is given by 

. . 

We exclude 

The cross section for the radiative capture 

r- 

c 

I 

is the average radiation width and -1 r, 

the neutron width for the compound 

states, which are excited by a neutron with an angular momentm . 1% . If the 

primary neutron energy is not too high (less than 1 Mev), the probability of in- 

elaotic scattering is relatively low, so that the re-omission of the neutron is 

R 

essentially tho process inverse to the capture”). stands than in the relation 

(16) to 5(”), and me get for the total capture cross section: 

This can be calculated with the help of (59b). For small energies ( 

A!= 0 contributes and we obtain fkom (73) and (59b) by assuming 

>> rn 

of the incoming neutron, 

>> R) only 

(Ql 

a value of 3*corresponding to 2 MeV, one obtains 

-----.--- 

$E the angular momentum of the bombarded nucleus is difforent from zero, the 

neutron can bo re-omitted with an angular momontum different from th 

angular monenbum of thE Cap%U&Qd one, ’ The psrfcty rule cal~~ows only ld’-/’I = 

2, 4 etc., and a closer investigation shows that the probability of /--/’#O is 

;1)1 rather small. 

(74 1 

With

cT" 

* if 

6 (n 

P) =t$"dz--) v 

io+ (&< r& 

E is measured in ev. is & magnitude which should be not far from unity:) 

An example of the energy dependence of cr(n,y) is given in Fig, 5, which 

276 

shows expression (73a) with the following choice of constants: 

R = 9 x 

k$= 3, D/2fl= 45F ) 

high. 

278 

Then the advantage of the neutron emission, which comes from the absence 

of a barrier is made up by the high energy of the competing proton re-emission, 

e 

In this case qr, can be smaller than 1 and is determined by 'qn Z= fn/(fn+fp). 

The f's are functions of the maximum energy Emax of the respective particles as 

described previously. If E, m a f,, 

P 

particularly if the nuclear charge Z is not too high, (See Fig. 4.) A case of 

this sort is found in the Ni62 (p,n) Cu6' reaction which has a threshold of 4.7 

%- 1 

MeV , 

In general, however, the cross section of a (p,n)-reaction 

should be 

very closely equal to 3 in Table I1 for different nuclei. Comparison with experiments") 

have shown that a nuclear radius R = ro A u3 with ro N 1.3 x lom8 cm 

fits better for elements in the middle of the periodic system between Ca and Ag, 

The energy distribution of the emitted neutrons, is similar to the 

distribution in an (n,n)-reaction. 

The highest possible energy occurs when the 

final nucleus is left in its ground state, and the emitted spectrum corresponds 

to the excitation spectrum of the final nucleus in the way that lower energies in 

the spectrum differ fram,the highest by an amount equal. to an excitation energy 

of the nucleus. 

For high initial. proton energies, this spectrum becomes s ~ l a r 

to a i'ulaxwell distribution. 

If the mrni;nwn energy E, 

of the neutrons of a (p,n) reaction is 

larger than the binding energy of a neutron to the final nucleus, the latter can 

be left in an excited state high enough to emit a second neutron, 

We then obtain 

a b,2n)-react=. The probability of this event is given by a srimilar expression 

to (71a); rn has to be replaced by WP. The energy distribution I(€) of 

the emitted neutrons can be represented here also by a illaxwell distribution and 

* 'TWaisskopf 

aid Euving, P~Ys. Bev. d, 472, (1940).

&e is the excess energy of the proton over the (p,2n) threshold. 

279 

The (pa?) reaction is the most important resction in case the proton 

energy is below the (p,n) threshold. 

The only competing process is then Che re- 

emission of the proton r(p,p) reaction] , which is not probable since the 

- 

proton has to penetrate the barrier again. The cross section is given by: 

n 

and is vanishingly small above the (p,n) threshold because fn rises rapidly and 

becomes larger than all other fts, No resonance effects should be expected in 

the (p,?) 

reactions since in heavy elements, in the resonance region, the energy 

of theof the proton is much too small to penetrate the barrier to an observable 

extent, 

The (p,p) reaction can be observed if the proton energy is not sufficient 

to emit a neutron with appreciable probability, 

Since the Butherford scattering 

by the Goulomb field is always present and usualiy much stronger than any nuclear 

reaction, a (p,p) reaction can only by observed either by registering scattered 

+t 1 

protons with less energy than the incoming ones or by finding the bombarded 

dH:) 

nucleus in an exLited state 

order to be observed. 

which must be a metastable (isomeric) state in 

The cross section of a (p,p) reaction is given by: 

fl- (p, p) = "r -*r+--$--li.K 

s 

It rises steeper than cp with increasing energy ep of the incoming proton, as 

long as fp < f3, since f 

P 

also is a strongly rising function of ep* qP,P) 

is then essentially proportional to the square of the penetration of the barrier. 

3. GC-Particle Reactions 

These reactions are very similar &o the proton reactions. 

The (CC,n) 

II) 

3t) R. Wilkins, Phys, Rev. 60, 365 (1941). 

R. Dicke and J,. IvIarshall, Phys. Eev, Q, 86 (1943). 

3Ht) S, Barnes, Phys, Rev, $6, 414 (1939).

B O 

reaction is the most probable one and its cross section is given by an expression 

like (75) with Crcr, instead of 5, 

threshold energy. 

after replacing 

(77). 

T~ is almost unity except below or near the 

The (G,2n) reaction is given by the same expression as (76) 

by Q, and so is the (a??) reaction cross section given by 

The latter one is unobssrvably small sime, below thd (cX,n) threshold, 

the a-particle hardly can ponetrate the barrier. 

The (CL$p) reaction has a cross section of the form 

- tions especially with low Z and E,.p > en -, 

It is hard to estimate the threshold energies for the different 

and is always smaller than the (CL,n) r&ction but can have observable cross sac- 

a-reactions, because of tho fact that the binding enerm of tP.e a-particle to 

the compound nucleus is not wall known, but is neaded to determine the excitation 

energy of the compound nucleus. 

The binding energy Badcan bo estimated as fol- 

lows: 

four nuclear particles are added, which have had in the a-particle a 

binding energy of 28 Mev. 

Thus the .binding energy of the =-particle 

should be 

BG= 4B - 28 MeV, where B is the average binding energy of the two protons and 

two neutrons added. B is not well enough known to make any guess about BG. Recent 

rssulis on (CL,Zn) 

have shown that, with 15 &lev&-particles, 

(G2n) reactions were not observable. This shows that, at least in the investi- 

Bnl + %2 - B -2 15 Mev, where B 

gated elements: 

are the binding energies 

"1,2 

of the first and second neutron of the (dX;,2n) reaction, 

4* Deuteron Reactions 

The d-reactions follow in general the same laws as the proton raactions. 

They'distinguish themselves from all other reactions because of the very high ex- 

* +b) 

citation of the cornpcmnd nucleus, The excitation energy is equal to the kinetic 

energy of the deuteron plus the binding energies of two particles reduced by the 

R. N, Smith, Dissertation, Purdue University, unpublished

0 

281 

relatively small internal binding energy of the deuteron of 2.15 Niv. It amounts 

to about 14-16 MeV plus the kinetic energy, This is one of the reasons why 

deuteron-reactions have a relatively large yield. 

The (d,n) and (d,2n) reactions obey the same laws as (75) and (76). 

bd is naturally smaller than 5 at ths same energy. The ~iiaximum energy €n - 

of the outgoing neutron is usually much higher than the deuteron energy because 

of the high excitation of the compound nucleus. 

Thus, there is no positive 

threshold for the (d,n) reaction, and Ykn is always nearly unity. 

The threshold of the (d,2n) reaction can be calculated from the 

.&'energy 

of the produced radioactive nucleus if it is a positron emitter, Ac- 

cording to the same principles as in (72), we find 

Here ED L 2.15 

(d,2n) threshold = €+ $. h t/ + m ,-'InpfED 

-. 4. 

= %n+ hu+2.85 ~ i v 

Ivkv is the internal binding energy of the deuteron. 

The (d,p) reaction should be much less probable than the (d,n) reaction. 

Actually, however, (d,p) reactions are observed with equal yield as (d,n) 

reactions. 

This has been explained by Oppenhcimer and Phillipz) and is due to 

tho following process: the distance between the proton and the neutron within 

the deuteron is relatively large, narnely -3.5 x cp.1 and its binding force 

60 of 2 bv is un*mxxily small, 

This is why the deuteron becomes strongly 

"polarizodlt when it approaches a nucleus; the proton goes to the farther end and 

.the neutron to the nearer end, 

If the neutron arrives at the nucloar surface, 

the proton is skill some distance away and has still a potential barrier to penu- 

%rate, The nuclear forces, which then act upon the neutron are much larger than 

ED, and the deuteron is likely to break up, when the proton still is relatively 

far away from the surface. After breaking up, the proton is only under the infbuence 

of the repulsive Coulomb force and leaves the nucleus. The actual

nuclear reaction is a capture of a neutron after the deuteron has been broken 282 up 

into its parts by the Coulomb field of the nuclear charge. 

The apparent effect, 

however, is a (d,p) reaction. 

The cross section of the Oppenheimer-Phillips process is much larger 

than 6d3 since the latter is given by the probability that the charge reaches 

the nuclear surface, whereas, in the Oppenkeimer-Phillips process, it reaches a 

point some distance outside of the surface. 

(d,/) ) processes do not occur with measurable yield since the compound 

nucleuB always is able to emit a neutron with considerable energy, 

5 Radiative Processes 

Nuclear rwctions can be initiated by the absorption of a ?-ray. 

It 

is evident that the energy ht., must be larger than the lowest binding energy of 

a particle. 

The compound state is, in this case, a highly excited state of the 

initial nucleus. 

The levels are then RO close that no resonance can be expected. 

The cross section for the cliffwent processes is given by 

o-(?,b) = (T+(fb/t 

a 

where b is the emitted particle and the sum is extended over a11 competitors. 

The most cofmon reaction of this type is the (T,n) reaction. 

art3 possible, but LL’F/ 

pete with either FJ:’Ct:J’*& 

mcii venker. 

3: :i?utron emission. 

fa 

(9,~) reactions 

Any other particle WGU~! not be able to com- 

Exceptj-or.:: ta tAis are found in 

very heavy elements where photo-fission occurs. 

The excited state is then un- 

stable against sp3..tttting into two fragments 

The VL.?.W 

of G;, Is connected with the transitim probabilities between 

nuclear states, and before describing its properties, it is necessary to discuss 

other radiative nuclear processes. 

* 

Radiative trnnsitions between low-lying excited states have been ob- 

served in great number in the ? -ray emissions accompanyingB-decay andGdecay, 

In analyzing term systems, it was found out, that, transitions occur between

283 

states which differ by two units of angular momenta, Aj =. 2 and not only between 

I) 

states differing by one or no unit, as it is the case with atomic spoctra. It is 

known that the change of two units corresponds to a qundrupole radiation which 

2 

should be smaller in the ratio (R/%) compared to the dipole radiation emitted 

in the transitions A j=l,O. 

This ratio is about 1000 in tho observed region of 

hU-5 x 105 ev whereas the observed quadrupole transitions were about equally 

probable as the dipole trcnsitions, 

This has been explained by the fact that, 

dipole radiation only occurs if the center of charge is moving relative to the 

center of mass, whereas quadrupole and higher pole radiation is emitted also in 

motions where the two centers stay coincidente In a nucleus, the charge is nearly 

evenly distributed over the mass, since protons and neutrons are not distinguished 

in their nuclaar motion. Thus the center of mass and charge stay practically 

together and this strongly seduces the dipole radiation. According to 

the observations, thds reduction is strong enough to reduce it to tho strength 

of the quadrugole rp? ie t icn , 

We themfoYY exile^::, that the absorption of 3/ -rays by nuclei, also 

follows the law nf quidrdy&3 3r dipole absorption, 

The ratio of the transition 

probabilities in qur-tGr.:po3.e and dipole radiation is pr o?ot*tJi.oml to 2 1 ', and we 

expect therefore ~.,zhl-y qmL*ip.de transitions for the a!uxption of T-rays of 

both types of radisiio:i. vjre cqually probable for lejs 3v.a 3 

bkv. 

Quadrupole rric'.itlt,ior, corresponds in the partick ;-ict,ure of light, to 

the emission or ~~JT',~:.LoI: ol' a photon with an orbital arigller momentum of one 

unit, whereas in tk- d;.r;~:~e ?%diation, the photon only posses3es an intrinsic 

tfsphll of one unit and no orbital momentum. 

sponds to the absorption of a p-garticle, 

Thus quadrupole absorption corre- 

The cross section Tv ,md its energy dependence can then bo estimated. 

We use the fact that tho wavelength 'x , even for energies as high as 30 k v is 

still larger than nuclear dimensions. 

The absorption probability should then be

284 

just proportional to the intensity E2 of tho electric field 6 at the nucleus. 

* 

(The same principle, namely that the probability should be proportional to the 

neutron density at tho nucleus, lt3& 

to the l/v law for neutron capture,) 

intensity for a photon is, firstly, proportional to tho frequency, since E2fi~ 11%'. 

Secondly, because of the fact that it is a "p-photon" with en angular momentum 

one, its intensity at thc nucleus is proportional to (R/?i2 

) (cf, page 7, Section 

XWrIV) 

This 

Thus the probo"bi1ity of absorption is proportional to U3. The cross 

section 67 is l/v times the probability which gives here, because of v = c: 

0;z = C (hGy 

.3k) 

where C is a constant 

e 

This relation is found to be well fulfilled according to (y,n) experi- 

ments made by Bethe 2nd @ntner'"*) 

who bombardad a number of elements withy-rays 

of 17 iv1ev aid of 12 lev. 

The constcnt C shows surprisingly little change ovor 

the psriodic system and is C = 0.65 x cm2/(Mev)3, This value can also be 

brought into connection with the observations on T-rays between low-lying ele- 

ments: by cpplying expression (161, the width for the emission of a ?-ray can 

D is the level distance in the region of the upper level. 

If this is applied to 

low-lying lovels of h t/ FJ 1 lvlev with a level distcnce of 0 ~ 0 . 5 hv we obtain 

rpw10-3 ev, 

This is in good agreement with the known transition probabilities 

among low-lying nuclear levels 4 , 

The r?.dini;ion width 

.--- 

++) Weisskopf, Phys. Rev, 3, 318, (1941). 

3Mt) Bethe and Gentner, Z. fur Phys. 112, 45, (1939). 

of levels of the compound nucleus created by 

) H. A, &the, Rav. Nod, Fhys. 2, 229 (1937), see also experiments bylialdmn, 

Phys. Rev.

285 

neutron cbsorption is a composite hzr-gnitude and canQot be compared directly with 

rv . rr represents the sum of all transitions from the level of the energy E 

to lower lying levels, whose number is very high. 

Formula (78) gives the radia- 

tion widtb for a transition from an excited state to the ground stat2 only. 

one applies (78) &so to transitions to excited states, the radiation width 'F 

If 

is givcn by: 

PE/* 

where r7 is the width 2s a function of the emitted rcdiation and Uis the level 

density and E tho excitation energy of the level, whose r; is calculated. If E 

is assumed 9 Mev and the expressions (71) are used for the level densities (A-1 

100) one obtajae from (79): f, *uO.3 Nev which is rathar close to the observed 

values in Table I, uut somewhat large, 

The applicetion of (78) to transitions to excited stF.tes is question- 

is slow~y *;nriable where D is the leva1 distance, if one varies either the upper 

level or the lower levo1 of the transition. 

In this case one would get instead 

whereW(0) is the level density necr the ground state, which is of the order of 

3.5 

One then gets for Em9 MEW: 

r;?= 2.0 x loa4 D' where D' is the 

level distance at the upper level, which is of the order of 40 ev. This gives 

definitely too srmll vclues. 

The truth lies probably soucwhere batween (79) and 

(80) 

The dependence of r;l on the excitation energy E is given by (80) as 

proportional to E6 and by (79) as very slowly indroasing function of E, Hence it 

* 

is very difficult to predict the dependence of rr on the excitation energy, 

Tho expected spectrum of the emitted radiation fron a state with the 

excitation energy E depends nlso on the assumptions made. The intensity I(z/)dv

* energies; 

286 

as a function of frequency is given by the integrand of (79) or (80). If (80) 

is valid, thc distribution is proportional tdd and has a maximum at high 

3 

the transitions to the lowest stntas nre the most probable ones. If 

(79) is valid, the intensity has a maximum soniewhere at intermediate oncrgies; it 

is proportional tov5 and also to thG level density at the end state, which de- 

creases sharply for the transitions of larger energyr

287 

February 15, 1944 

LECTURE SERIES ON NUCLEAR PHYSICS 

Sixth Series: Diffusion Theory Lecturer: R,F. Christy 

LECTURE 37: 

DIFFUSION EQUATION AND POINT SOURCE 

SOLUTIONS 

In the first lectures we will discuss the diffusion of 

neutrons which have come into energy equilibrium with their surroundings 

(thermal neutrons). Later we will discuss the slowing 

down and diffusion of fast neutrons and corrections to the diffusion 

theory of thermal neutrons, 

The Diffusion Equation 

The flux of thermal neutrons is --Dan where n is the 

thermal neutron density and D the diffusion constant, 

D can be 

obtained from simple kinetic theory arguments and equals xv/3 

where xis the mean free path and v the neutron velocity, 

"standard" velocity of themal neutrons 1s 2.2 h/sec. 

related to the total cross-sectiorrU-by 

where N is the nwnber of nuclei per cc, 

sec, = 

is 

The 

the relation 1: 1/N< 

Tho diffusion equation can then be written 

div (D grad n) 4- productfon/cc sec. - absorption/cc 

$n/ At 

The production/cc sec. 

position, 

is usually denoted by q, a function of 

If O-, is the absorption cross-section, the absorption/ 

cc sec. is equal to Nd'nv; a or, writingh= 1/N Ta G the mean 

free path for absorption, the absorption/cc sec equals (v/A)n=dT 

where 

is the mean life of the neutrouis,

This 1ead.s t o the diffusion equation in the following 

288 

forms* 

If we consider the time independent diffusion equation 

- xv CJ2n 

3 

D# n - t q = o 

- A nfq = O 

2 

where L 

=AM3 = DT* 

Plane Source in an Infinite Medium 

L is called the diffusion length, 

Let the source be of strength Qn/cm2 and occupy tho 

plane 2; = 0, With q = 0 everywhere except 2; = 0, the equation 

I 

reads 

-L.- d2n n = 

dz 2 

'zz 

which has solutions e and Since the neutron density 

must approach zero for z = 

n=Ae 

fa the correct solution must be 

-.. 

In order to relate A to Q, we note that the flow out from the 

-- 

SOUPC8 per cm2 is 

2 xv 

r &.v A sQ 

3 g)z P O - 3 Ti; 

so A = SV 

and n = 3LQ 0 - 1 

ZG 

This equation ,can be used to determine L experimentally

289 

r) 

for such materials as paraffin or water whero LX3 cm is small 

compared to convcnient transverse dimensions of about 1 mater, 

A source or” thermal neutrons is not directly available but can be 

obtained as follows. A fast neutron source is placed below the 

plane z = 0 in some slowing medium. At Z = 0 a removable sheet 

of Cd (which selectively absorbs thermal neutrons) is placcd, 

Above z = 0, data on n is obtained for the Cd in place and with 

no Cd, The difference represents the effect of the Cd which is a 

thermal source or sink and follows the above equation. 

If the diffusing medium cannot be considered infinite in 

the transverse direction and if the source can be considered to 

have the transverse distribution cos *3c;/a cos Ty/a where a is 

the transverse dimension, then the solution is 

Substitution in the diffusion equation 

- d2n 2 2 

i--- 

d n d n - n 

dx 2 2+---2 ;2 0 

dY dz 

leads to 

r e.- 

g+.- 

2 

a 

1 

- 1 

a b 7 

= o 

which determines b, 

This relation 9s frequently used for the 

determination of L in media whore L is relatively long, of ordor 

Point Source in an Infinite Medium 

Here we write tho diffusion equation with g = 0 in

290 

* 

spherical coordinatos, considering only spherically symmetric solu* 

tions. 

if u = rn then u satisfies the 

d2u - 

7 

2 

dr 

e qu a ti on 

= o 

The condition that n be finite at r -&gives 

and 

,=A 8 -r/L 

r 

The source strength 

Doprassion of Neutron Density Dear a Fofl 

d simple cxprcssion for the depression 6f the neutron 

density near a foil can be obtained only if we assume the foil to 

have thc form of a spherical shell, Let its thickness b? t, ab- 

sorption cross-section Fa, and its nuclear density be N. 

The diffusion equation is 

Let n = qT A 

I2 

- e? -r/L which satisfios the equation and has in 

r 

general the right form. The number of neutrons absorbed per 

second by a foil of radius ro is

291 

0 

The fractional depression in neutron density at the foil is 

i 

- 

A 

e 

-r/L 

r 

0 - 

1 

qT 

14- +Po+ VL 

3NCr t 

a 

which can be considerably larger than the fraction of neutrons 

absorbed in a single traversal of the foil which is N C t r 

a 

a 

r

292 


In the idsf; I.eci;;;:e 

iwo considered the effect; of a detector, 

such as an indrivm .Y9il7 or? tho neutrcm density in a medium in which 

theilmal neut.ron..; we~cj c‘,irCi:!sing. It was shown how, as a first 

ap~rzximation, t h i‘a?.l ~ coiild. be replacod by a sphei’r? and fairly 

simple roslG.ts obt3t:md. 

ta the prohi em. 

Let us now consider a mo~e exact approach 

S.c,-gpose we ccjimider the element of area (dA) of the fail 

surface as a paint absaorber, If p is the coordinate of the 

element r3A and r ta the cccrdinate of the point at which we want to 

know the neutron densj.tyy, then the effect of a point sinh such RS dA 

is given by: 

To evaluate G 

- L 

Ce 

- Ir -‘TI 

we constdor the following: 

The number of netltrnns absorbed per second may be set equal to the 

number passing thrmgh the surface of a very small sphere surround- 

ing our point sinh, fiToa;r the number of neutrons absorbed per second 

equals Ntqnv6A wbers : 

N : number of atoms/cs in foil

c cfa 

t thickness of foil 

= absorption cross-section of foil material 

v~ : neutron velocity 

The number of neutrons passing into a very, very small sphere is 

given by: 

293 

Therefore : 

1_3 4ffXv C = -Nt%n 

3 

v dA 

The effect of the whole foil is then the sum of the effects of each 

element. On adding these effects and passing to the limit as the 

number of subdivisions is increased we get the total effect of the 

foil at any point r to be 

If we assume our source to be such that in the absence of the foil 

the neutron density in the region in which we are interested is 

everywhere constant and equal to one, then the neutron density at 

any point is merely given by the sum of the contributions of the 

foil and the source without the foil 

Thus 

* 

The more pFeclse approach to our problem therefore is seen to lead

2 94 

* 

to an integral equation, Since this equation is difficult, if not 

impossible, to solve exactly the usual procedure is to have recourse 

to approximating the foil by a sphere, 

Point Source in a Finitesphere 

As another example of the way in which the diffusian 

equation may be appliQd Let us consider the problem of a point source 

in a finite spherical medium. The general solution in such a case 

is: 

r r 

where A and B are arbitrary constants to be determined, B may be 

found ira terns rf A by ems of the relation; n(R) z 0 where R is 

the outpi’ ~ai!?-us of’ ’the sp1iere. 

4R/L 

R t’ -A e 

anl! the solution becomes: 

The cmstkirt A may then bs determined by .equating tha 

* As 

Up until now the solutions to the diffusion equation that 

we have found have involved only a few terms, In general such representations 

will not satisfy the boundary conditions, Instead we 

must have recourse to aore complex methods, (e.g. the use of Fourier 

Series). an illustration of suoh a problem we can take the case 

of a point source embedded in a rectangular medium of length a 

in

295 

0 

both x and y directions but infinite in extent in the and - 3 

directions, We take axes such that the source which is at the center 

of the medium is at the point (0, 0, 0). The general solution of the 

equation c7 2 n- 0 is: 

3 

To determine bmn we can use the fact that each term separately must 

satisfy the differential equation. Hence b, is given by the rela- 

tion: 

To find the coefficients A, we consider the fact that we have a 

point source, 

Such a smrce can be represented as 6 (x,y) where 6 

stands for the Dirac delta function, Were we are assuming a source 

of slow neutrons, though in practice a fast neutron source is used. 

Let us expand 6(xpy) in a Fourier series. 

We determine the Bmn’S as follows:

296 

e 

To relate the A's and Bls we must use the fact that the number of 

neutrons flowing out of the plane z = 0 per second in a glven mode 

must equal the number produced per second in that mode. The latter 

is equal to Bmn 

2xv JK,,, 2 xu m% nKy 

Number flowing out ==T A, COS 7 cos 

Hence our solution is: 

If the so'u:~ce stxngth were Q instead of 1 the above solution must 

be multfplied by a factor of Q. We note that B, decreases rapidly 

with increasing m and n and so at great distances the only significant 

term in our series will be the first. Thus at great distances 

from our source: 

This presents a very convenient method of measuring L since the 

decrease of n with z enables us to findbll and from bll we can 

readily compute L, 

solution of the Diffusion Equation in an Iqfinite Medium with a as a 

Function of Position. 

As another type problem suppose we consider the solution

* 

of the diffusion equation with a term due to the production of 

neutrons (9) by a rather special method. The equation involved is: 

297 

If g P 0, the solution is (assuming an infinite medium spherical 

symmetry) 

-'IL 

n=Ae where A 

r 

is some. constant. 

As shown in a previous lecture A can be related to the source 

strength Q by: 

A .P 

4.R Av 

Hence : 

- 3Q e 

-r/L 

n-- - 

r 

4T xv 

Suppose now q f (x, y, 2). 

We can solve the problem as follows: The solution for a1 point source 

has the form given above. Our f'unction q can then be treated as an 

infinite number of point sources continuously distributed throughout 

the medium. The total effect will then be the sum of the effects 

due to each point. If ;r is the point at which we want to how the 

neutpon density and rf the point where our source is located we 

have : 

c 

Sometimes the above integral can be readily evaluated and 

under such conditions represents a very great simpliEication of the 

problem. In utilizing this method it is very important to take 

advantage of whatever symmetry the problem offers. Thus if plane

298 

e 

symmetry were exhibited the appropriate plane solution of the diffusion 

equation, should be chosen. If cylindrical symmetry were the 

case it would be well to find the solution for such conditions and 

then apply the method here followed of integrating to gat the contributions 

of the many point sources making up the q function. 

A quantity of considerable utility in neutron diffusion is 

the albedo or reflection coefficient, 

If a finite medium with a 

point source were left in empty space, the neutrons reaching the 

bounding surface would diffuse out and be lost, Now, if another 

medium were surrounding the first one, some of the neutrons reaching 

the boundary would be reflected back in, thus increasing the neutron 

density in our original medium, 

The magnitude of the reflection 

effect is measured by the albedo? of the, in this case, outside 

medium. 

Actually the albedo of a medium will depend, not only on 

the medium, but also on the angular distribution of neutrons falling 

on the medium. 

Albedo of Various Media' 

\fire w ill neglect this effect. 

For this purpose we must consider the angular distribution 

of neutrons at a point. We let be the cosine of the angle 

between the velocity of the neutron and the radius vector and write 

I 

' 

nv(rp) = + LAW +p~(rg 

rl 

*

299 

\ ,and nv(r,p) = $A(r) -phA'(r) 

Tho total flow in the positive radial direction is called the flow 

out 

flow out = P nv(r,p),pdr = [Aw - ij x A ~ (rj 

The total flow in 

the negative &dial direction is called the flow in. 

')=flow 

in z 

Now the albedo of 

the inside surface R of the medium is 

1 -r/L 

We can take n = c. e 

r 

do - - dn z - 1 e 

dr r 

($ + 

The larger is R the better the albedo and for R 

the albedo 

approaches that of a plane 

2 1 

= I -5 

l + Z & 

3 L

0 

If in addition we make the medium non-absorbing, L =a,

in diffusion theory nl n2 at the boundary 

Ap dnl Ag 

and - ,-, = 

3 dr 

dn2 

3 dr 

Instead of one af these we can use the ratio 

- - 

at the boundary 

301 

NOW this ratio for a medium can be written in terns of the albedo so 

is another possible form of the boundary condition, 

Extrapolated End Point 

Let us examine the neutron densltg. in a plane semi-infinit6 

non-absorbing slab, "e will show that if the neutron density 

inside the slab is continued to the edge and beyond it would vanish 

at a distance 

3 

Xbeyond the ed&. 

n 

/ 

/ 

/ 

/ 

- / 

x- 

We have, as before

* 

302 

The solution deep in the slab w i l l be nv = A(x) = a(l 4- t) where L 

is the extrapolated end point. Shce A(x) = 0 for x = -L 

At(x) z $ 

nv(x,r.) = 2 

The total flow in the +x direction is 

J 

whichmust vavlibh at x = 0 since there are no neutrons entering the 

slab. 

An exact solution of this problem leads to L z .7104x actually. 

Angular Distribution of neutrons Emerging from a Surface 

As above we may take nv(x) = a(l -k ) 

I 

We may consider these neutrons uniformly distri- 

buted in angle and calculate the probability of 

escape from the surface at angle p (per unit 

range of 

nv (p) 

).“I 00 

= 1, kk+$ a) e-’pwx 

= i{y+gy2) 

So then the flux from the surface -cos 0 4 cos2 8 

z 

X

303 

Actual Distribution Near a Plane Boundary of a Non-absorbing Medium 

Actual angular distribution in n -p +@y* , 0 0 m ; 

Tho extrapolation of' the asynptotic solution vanishes at 

,7104Xfrom the surface, 

Very close (in a mean free path) to the 

surface the neutron density decreases and has a singulasitg in 

slope at the surface, The value of n at the surface is 

of tihe value of the straight line solution at the su~face. 

- 

.*n 

a

304 



Sixth Series: Diffusion Theory Lecturer: R, F. Christy 

SECTURE 39: THE SLOWING DOWN 0.F NEUTRONS 

Fermi’s ARe Eauation 

The preceding lectures have all dealt with the diffusion 

of thermal neutronsn Lot us now consider how neutrons coming from 

a fast neutron soui“c”o :ce s!-med 5~vm- In the discussion the 

quantity q will kij *JG::I’~ cox~ve~;.1.eni:, :-I, 

1s defined as follows: 

It fs obvious that the -i;c-ta_? EL,. 

must equal the noo produced per second, 

of 

q we must have: 

o+’ neuti-cnr: c:rs.;sing energy E 

Hence from our definition 

\ 

* the 

where Q is the total number of neutrons produced per second. 

The mechanism by which the neutrons are slowed down is 

that of collision with other particles. The amount of kinetic 

energy lost by a neutron in a given encounter may be computed by the 

methoas of classical mechanics as applied to elastic collisions. In 

general, if a neutron hits a particle of mass M the ratio of its 

opaiginal en@l”gy (B) to its final energy (E’) is a fWLction both of 

mass M and the angle between the orfginal path of the neutron

305 

1) 

and its path after the collision. 

angles we can conclude that 

By appropriate averaging over all 

where C is a constant which depends only on M. 

usually written as: 

This relation is 

where is the average of . 

It can be shown that: 

where the approximation is better for heavier nuclei (i.e. for 

larger MI* A few sample values are: 

for hydrogen: 5 P 1 

for deuterium: 5 = 0,725 

for carbon: 5 = 0.158 

for oxygen: 5 0 = 0.120 

Using the above we may find the mean number of collisions 

suffered by a neutron in slowing down, Suppose the neutrons start 

off with energies of 2 Mev and end up at thermal energies (about 

1/30 ev), Then the mean number of collisions:

* 

The 

treatment now to be presented for slowing down is 

somewhat limited in its application - even more so than is ordinary 

diffusion theory. In the latter case our only important approximation 

was that grad n be small, In slowing down theory it will 

be demanded that the number of collisions be large and also that the 

mean free path (A) not vary much between collisions, These 

assumptions are necessary if we are to be able to replace the essentially 

discontinuous process of energy loss by collision by a continuous 

process, (It may be noted that the most common slowing dom 

substances, such as paraffin and water, fail to satisfy the conditions 

very well and so any treatment of paraffin, water, and 

similar materials by the methods given here may result in considerable 

error.) 

306 

The starting point in treating slowing down phenomena is 

the so-called Fermi Age Equation. We shall. give here a non-rigorous 

.justification rather than a formal derivation of the equation. Our 

standard diffusion equation was: 

By analogy we write: 

* 

If we consider t in this last equation to be a variable depending 

on the past history of the neutrons we can relate this to the neu- 

tron energy. Thus: - AhE per collision h number of 

3 

dt 

collisions per second.

307 

* 

But Adn E per collision =k 

V 

and number of collisions per second =x 

Hence : 

Then our original equation becomes: 

We introduce the quantity 2: (the ttAgeft) by the ,relation: 

On substituting we get: 

(Fermi's Age Equation) 

We note that the tlAgefl has the dimensions of length squared. It 

plays very much the same role in slowing down as the quantity L 

2 

did 

in thermal neutron diffusion, 

is a function of the energy (E) to 

which the neutrons have been slowed and may be found from the 

* 

where Eois the energy with which the neutrons leave the aource. 

Point Source Solution 

As our first example of the use of the Age Equation let us

308 

* 

calculate the distribution of neutrons In space and energy coming 

from a point source. We shall consider an infinite medium with 

spherical symmetry. Our equations are: 

The equation involving the Laplacian reduces to: 

We shall solve this equation by the standard method of separation of 

variables : 

Let 

Then 

rq = f(r) 0 (TI 

s=fd2 

f d%/ 

fl dT 

Since the left hand side is a function of r alone and the right of 

7 alone we must have: 

Thus we get the two equations: 

d'f 

ti> +-'k2f = o

309 

* 

The solution of (2) is: 121 = e -k2”C 

The solution of (I) Is:: f = A sin kr+ B cos kr 

t Solution is: q D, 5 

m e 

(A sin kr J- 3 cos kr) e 

r 

-k2y 

where A and B are arbitrary constants depending on k. 

stay flnite at r = 0, we must have B z 0. 

Since q must 

Hence: 

q z & sin kr e 

r 

-k2”r: , 

Since we have no boundary conditions to be satisfied, any real value 

of k will satisfy our equations. 

The general solution, which is the 

sum of‘ all possible solutions, is then found by by multiplying the 

above expression by dk and integrating from - fl to -COO 

Using the fact that we have a point source we can determine the 

‘ function A(k). We know that; 

q( T = 0) =: c 6(r> 

where C is a constant to be determined later whiczh depends only on 

the source strength &. b(r) is a delta function such that: 

b(r) =: 0, r # 0 and 

c 

30 

r2 h (r)dr t 1 

Substituting in the above we get: 

500 

r Ch(r) P A(k) sin kr dk 

-00 

As the integrand is symmetric (as will be seen from later results) 

written as: 

I

3 10 

rCb(r) 

a 2 5 

cx;, 

A(k)sin kr dk 

0 

Multiplying by 

- 1 yields 

Now the Fourier Sine Transform tells us that 

fla 

If f(x) =fi pI (u) sin xu du 

then: @(u) =fi 0 

\oof(x) sin ux dx 

30 

Applying this to our problem: 

i' 

By the definition of the delta function this becomes 

Hence 

dividing by 

r94"C; 

e 

y i e Ids'*

311 

To evaluate the first integral let 

Then; 

The first term on the right yd6lds zero while the second becomes: 

Similarly 

Thus : 

To find C we remember that: 

Lo1 

qdvol=Q 

Let u 

I-~/~~C' then:

* 

Hence : 

Qe 

3" (4mp 

That this result is what we should expect is easily seen from the 

3 12 

following argument, Since at r s 2fi q has reduced to l/e of its 

value at the origin, we see that 

is a measure of the width of 

curve representing q. If were very small our curve should be 

very narrow (a) . 

* 

This is ph..~s?c~l?~y plausible, since small 

in dicates fast 

neutrons and ther;rl we would expect to be grouped near the source. 

On the other hand large means slower neutrons. These 

would be expected to be more uniformly distributed than the faster 

ones, That Is just what our formula leads us to conclude.

I 

0 -1” 

Plane Source Representations, 

If instead of a point 

source the solution would be of 

+ 

-z2/42 

q = Q e 

pflT 

source we have an infinite plane 

the form: 

Q = neuts/cm 

2 

sec from 

source 

Let us suppose now that we have a medium finite and of 

width (a) in both x and y directions and infinite in the - and - z 

directlons. 

e quat ion : 

Assume a point source at (0, 0, 0). A solution of the 

which satisfies the boundary conditions at x 

- a/3, y o L a/2 is: 

..- 

where A? and n are positive integers, C(k) is a constant depending 

on k, and

3 14 

p2 s (d2 +, m2j,, q2+ k2, 

2 

a 

(k is any real number) 

The general solution i s then found by adding up all possible solutions. 

This means swnming over m and A? -and integrating over k. 

Thus 

2 

C(k) >: A cos& cos& cos kz e **-P z, 

m,k 1,m a 8 

Applying the orthogonality principle, integrating, and using the 

fact that a point source may be represented by a delta function we 

gat: 

We note that for >) a (i,e, for the lower energy neutron) the 

higher harmonics die out. Hence, with this zipproximation our 

expression becomes:

315 

LA-24 (40) 

February 29th, 1944 

LECTmE SERIES ON NUCLEAR PHYSICS 

Sixth Series: Diffusion Theory 

Lecturer: R, F. Christy 

LECTURE Vz 

THE SLOWING DOWN OF NEUTRONS IN WAX% 

In this lecture we shall consider the f,ollowing question, 

What happens when a Ra-Be source of neutrons is placed in a large 

water tank? What is the distribution of neutrons in space and 

energy? 

Suppose we were to meazure the distribution by means of a 

cadmium covered indium foil. Such a counter records the number of 

neutrons of the indium resonance energy (slightly above 1 ev). 

Plotting 

fi(r2A) against r (where r is the distance from the 

source and A is the activity measured) we obtain results such as 

those in the diagram, 

. 

\ L. 

\ 

\ 

\ d-- Gauss i a n 

\

cally straight, 

We note that at large values of 

This implies that in that region: 

/ 

316 

r the curve is practi- 

$I 

We also note that the curve of measured neutron density differs 

considePBbly at large r from what it would look like if it 

followed a Gaussian distrfbutlon (dotted line), 

The above distributlon of neutrons is readily recognized 

as being the same as that of neutrons from a point source which have 

not yet had any collisions. That this is so is seen from the fact 

that the probability that a neutron will go a distance 

a collision is 

r without 

2 

to 

e -vi and that we must have a factor Qf 1/4‘Kr 

satisfy geometrical considerations, 

Now it is easy to demonstrate using classfcal mechanics 

that in an elastic collision with a proton B neutron can never be 

scstt,c:wd at an angle greater than go0. Hence the average scattering 

angle will be between 0’ and 90’. live see, therefore, that on 

coLlic?ing with protons the directton of the neutrons is not much 

changed and so we would expect the neutron density to follow a law 

similar to the above (i.e, we expect A 0’ (l/r2)enr/LX where -)-L 

is actually somewhat greater than the true mean (free path), A 

further reason why we should expect such a distribution is seen when 

we remember that the scattering cross-section for hydrogen decreases 

rapidly with increasing neutron energy, Thus the mean frea path is 

largest at higher energies and so most of the distance traveled by 

the neutrons is covered in the first few collisions. But in the 

first few collisions the (l/r*)e-”klaw is followed, IVe conclude

317 

m 

from this argument also that the last is the law we should expect to 

find for the neutron density for large r. 

A concept which we shall find useful in considering our 

problem is that of the mean square distance r2 traveled by a 

neutron. 

This quantity is defined by: 

where f(r) is the kxnctional dependence 0.f the neutron density on 

the distance (r) from the source. 

__L 

As an example let us calculate r2 for the case where 

our neutron density follows the (l/r2>e-r’h 

law to the first 

collision and then follows an age diffusion law (i.e, proportional 

to e -r2/47=). Let us denote the first portion by the subscript (1) 

an?. the second by the subscript (2). 

Then: 

.r 2 =:

1 

3J8 

Substituting 

Hence : 

pre-age 

dif fusioq 

we 

diffusion 

In order to calaulate the distribution of ne 

k 

consider the following! 

From the poipt source the neutrons go to a ppi$t p 

following the (l/r?)ae-”.h 

I , ] ’ , 1, 

law. From rl they go a 

I 

2 

age dZTTusion (i,e, following e*r /4z). We can 4 

6 

pofnt rl as a source for the age diffusion which pr 

‘1 < ~ 

neutron reaohes r TnCagrating per the whole volwe2 

I 

4 

contribution of egch su h source we get: 

I 

” k b i 1

319 

Substituting 

we get 

c03 B 

?-

320 

* 

This integration with respect to rl cannot, unfortunately, 

be done analytically, 

To get an approximate analytic solution we 

can do the following I Approximate the distribution of (l/r2}e*r/)L 

by (l/~>e-~/~. In this form we can do the integral. L should be 

2 

chosen so that the mean square distance traveled (T- ) comes out the 

2 

same. Originally Pr2 was 2x so we must have: 

We note paFt$cul&dj? hbw analogous L2 is to "r: in the slowing 

down equation. There we found 

-2 I 

3: 

carried quite far, 

slowing down equation by a diff'usion equation. 

- 6T, This analogy can be 

approximation is good only over a small energy range and for 

not too large compared to 

Under certain conditions we can even replace a 

However9 such an 

fiT. 

In our water tank problem it is fairly obvious that a 

certain percentage of the neutrons emitted by the source per second 

(Q) will escape from the tank without ever undergoing any collisions 

and so will not be adequately described in terms of a diffusion 

equation. 

To correct for this we can replace OUT 

strength Q by one of strength QJ- such that: 

source of 

r

* 

We can *en 

321 

find the distribution by assuming It to be of the form$! 

- 4 L -(2R - rP/L 

c - e 

4nrL2 e 

C is de‘term’ined fmm the ?elation: 

s”, 

2 

C - (,@*’/L c ,e-(2R - r)/l) 4”lr dr s Q‘ 

4XrL2 

6till another, though mor8 foTrna1, way of treating the 

problem ‘3% as follows; 

First consider slowing down and then thermal 

dif ftlsi0t1.r 

The standard age equlati6n la: 

With spherical symmetry, this reduces tbO 

where R*= ra4ius of tank. 

The coefficients (Ah) can be determine& as follows: 

T= 0 we know the distribution is 

For 

Hence;

ll: 

..*L- 

--. 

322 

To find the number of thermal neutrons escaping from the 

tank we consider the ordinary diffusion equation and assume each 

mode, which we shall denote by the subscript n, satisfies the 

equation. Thus: 

As a solution assume: 

Substituting in our equation we get: 

Hence : 

Naw for the number of neutrons captured per second is: 

s 

’V f. 

-0ct vol= 1 .+ $pfl”Ln 

7 

B7j.l *hhg g d vol is equal to the number of thermal 

-. Svnl L3 Pa vos Sdvol 

I vol 

neutrons prDC!i(:ed ps second. 

just equal to the riumber produced minus the number absorbed is: 

Hence, the number escaping Which I S

323 

LA-24 (41) 

March 2, 1944 


Sixth Series: Diffusion Theory 

Lecturer: R, F. Christy 

_LECTURE VI: THE BOLTZMANN EXJUATJON: 

- CORRECTIONS TOXFFUSION THEORY 

Boltzmann Equation_ 

In this lecture we shall consider some corrections to the 

elementary diffusion theory previously treated, 

More exact methods 

depend on either the Boltzmann equation or an integral equation, 

shall make our approach through the former. 

To derive the Boltzmann equation for diffusing neutrons, 

consider the following: 

Let f(x9 y, z9 vx, vy9 va> be the density 

of neutrons at the point x,y,z having velocity components between: 

vx and vx+ dv,, v Y and and vz and vd + dv,. It is 

readily seen that: 

c 

\ 

&e 7.0 '3 ity 

f d vel 

ordinary neutron denkllty 

We 

Now the tot,3:L rate o:? change of 

f with time, which we denote by

324 

* 

But this total rate of change must be equal to the number of neutrons 

entering the given velocity range per second minus the number lost 

by collisions, Let 4T= ea+ c$where sc$ is the absorption cross 

section per unit volume and cK3 is the scattering cross section. In 

terms of these symbols the number of neutrons lost per second is: 

vof, The number entering the velocity range is found by averaging 

vo f over all angles. Thus, 

where 

due element of solid angler 

When the above are substituted, the Boltzmann equation becomes: 

where w0 substitute 

* 

In passing it is interesting to observe that this 

equation is considerably simpler than the ordinary Boltzmann 

equation of kinetic theory. This is due to our neglecting collisions 

between neutrons and the fact that we have considered the 

objects hit by the neutrons to remain stationary. The great simplification 

is that the ordinary Bol'czmann equation Is non-linear 

while our above derived equation is linear and so is considerably 

more tractable 

Let us consider the Boltzmann equation in the Steady

325 

e 

state, Then it reduces to: 

7 * ef =-vrf 

- Assume further that we have an infinite medium, that f e kx and is 

independent of y and z. (This is one of the few instances in 

which the Boltzmann equation may be solved.) 

Our equation becomes: 

vx kf = -vsf -1- VGT 

Averaging over all angles and substituting .s g vx/v z Goa 6, we have: 

Hence 

This equation serves to determine p( 

The first approximation for 

small x/r and 

diffusion theory result L2 

Tdrgives K2 = 35% which is the same as the 

l/3%~ , However, If is not much 

less than 05 the BoZtzmann equation shows that we must correct our 

value for 6/ 

A rather good second order approximation is: 

2 

e Hence the value of L 

that should be used for more accurate work is:

326 

3 i-7n 2h 

x* 

Another important instance in which we can correct 

ordinary diffusion theory through use of the Boltzmann equation is 

in the matter of neutron flux. 

Until now we have used -( )~v/3)Vn 

for the flux (io@. we have said the diffusion constant D, was 

+Xv/3). 

A more accurate D can be obtained as follows: 

Let us find 

7 

vxfo This is: 

From this we see that a more accurate diffusion constant is: 

Thus, while 

the exact relat2on is: 

- 

In all, the above work when averaging over angles to find 

9 we tacftly assumed to be independent of the angle of scatter- 

,-I 

* -

~ 

327 

-41 

tfig (f.e. Isotropic scattering). If le aid floe have sphsyidal 

symmetry we could have expanded SS in a power series in cos 8, 

Thus 

The effect would be to give us a dlighkly different value for ? and 

would serve to complicate the problem. 

Assuming such an expansion neatjrd#ktr& i.f; hbrns ofit that 

still an additional correction for & is necessqryi de get 

I / 

Let US Introduce a quptity called the trandbdvt 6Ybrtd lagCkib$S 

defining it by the relation: 

o&, .= b(F-cos e) 

where a is the average of cos 0 over all collisions. 

*ly we obtain the equation: 

Then for 

where M 

is the mass of the scattering nucleus, Then 

We see that for light elements, such as hydrogen, the correction is 

quite important, On the ather hand, for heavy elements the correction 

is negligible. 

StlZl another way in which the Boltzmann equation can be

328 

4- 

of use to us is in helping u8 to derive the Age equation, We shall 

not carry this derivation through here, but will merely indicate the 

method to be followed, 

First assume a certain energy change per scattering. 

follows that the number of neutrons entering the given velocity 

It 

range is -f(vl), where vr represents the neutron velocity at a 

somewhat higher energy level. Then we can represent f(v’ by a 

Taylor series and break off the series after the first two terms. 

Thus: 

f(v’) =f(v) +(v’--v) $$ 

This is then substituted in the Boltzmam equation I 

and the same 

general procedure is followed as previously, We obtain a relation 

between and a’I;/dv The quantity ‘T is then related to V2q and 

and 8/Av to &q/&?; 

Application to Water: 

BfndinP: Corrections: 

To illustrate the application of the formulas we have 

developed, let us consider the apparent; paradox that arises in com- 

puting the diffusion length for water. 

Since the cross sections 

for hydrogen are much larger than those for oxygen, we need only 

take hydrogen cross sections into account. for hydrogen when 

measured turns out to depend on V. At thermal velocities it is 

about 40 barns. Hence : 40/3. for hydrogen is 0.33. When 

* 

we substitute in our Formulas for 

differs greatly from the value L2 

L2 for water we get a value that 

2 

8.3 cm which has been measured. 

Here it seems that our formulas are wrong and so must be thrown 

away. 

Fairly good agreement, though, can be reached by using the

329 

following argument: 

I 

In the calculations it was assumed that the mass of the 

hydrogen atom is 1. Actually it is effectively greater than this, 

since the hydrogens are bound to a greater or lesser extent to the 

heavy oxygen atoms. Actually it is found that for'snergies greater 

than 1 ev the hydrogen atom is practically free and so has an effective 

mass of about 1, For energies less than this, the effective 

mass is somewhat more than 1, 

For epithermal energies (Le. energies slightly over 

thermal) Cr, for hydrogen is approximately 20,. For hydrogen eom- 

pletely bound (this corresponds to 0 energy) theory leads us to 

2 

g 80, For thermal energies we should 

expect CS to be 20 (w) 

then expect fl5 to be somewhere between 20 and 80, The observed 

cross section is the geometric meanp 40. It corresponds to a 

hydrogen binding that is about halfway complete. 

By analogy we expact the 1 - cos 0 

term also to be same- 

where between its two extrerrleei of l/3 and I, If we again take the 

Using this value we get rather good agreement in the measured and 

computed values for I,'.

Untitled

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?