PHY3022S Nuclear and Particle Physics 1 - University of Cape Town

University of Cape Town
Department of Physics
PHY3022S
Nuclear and Particle Physics
Nuclear Physics Part 1
Basics
Andy Buffler
UCT Physics
andy.buffler@uct.ac.za
Room 503, RW James Building
1

Atomic Physics
(David Aschman) 20 lectures
PHY3022S Course Outline
Nuclear and Particle Physics
(Andy Buffler and Andrew Hamilton) 20 lectures
Solid State Physics
(Mark Blumenthal) 20 lectures
For nuclear and particle physics …
Prescribed book is
BR Martin, Nuclear and Particle Physics: An Introduction (Wiley, 2006)
Also used:
RJ Blin-Stoyle, Nuclear and Particle Physics (Chapman and Hall, 1991)
Tutor for Nuclear Physics: Maciej Stankiewicz
2

Conservation of energy and charge in e + e - production
3

Periodic Table of the Elements
6

The nuclear atom
Chapter 1:
RJ Blin-Stoyle,
Nuclear and Particle Physics
(Chapman and Hall, 1991)
7

Radioactivity - history
1885 JJ Thomson discovers the electron
1896 Bequerel …β-radiation from uranium salts
1896 Roentgen … X-rays
1898 Pierre and Marie Curie … α-radiation
1900 Villard … γ-radiation

Radioactivity
α-radiation:
Emission of an α-particle ( 4 He nucleus) from a nucleus.
Energy around 5 MeV (short range in matter).
β ‒ -radiation: Electron emission in nuclear decay
β + -radiation: Positron emission in nuclear decay
γ-radiation:
De-excitation of a nucleus from excited state to lower
state (keV to MeV)
10

Development of atomic theory
JJ Thomson (Nobel Prize in Physics, 1906)
“Plum pudding” model.
Poor agreement with experiment.
11

The Rutherford model
Ernest Rutherford
(Nobel Prize in Chemistry, 1908)
12

Scattering of a 4 He nucleus off …
(a) Thomson’s atom
(b) Rutherford’s atom
Simulation of Rutherford scattering
off a gold nucleus
13

Rutherford: “It was the most incredible event that has ever
happened to me in my life. It was almost as incredible as if you
fired a 15 inch shell at a piece of tissue paper and it came back
and hit you.”
Rutherford model (1911): The atom has a small hard central
core (nucleus) where all the positive charge is concentrated.
The negative charge inhabitants the nearly empty space around
the nucleus.
Thus most of the alpha particles migrate through the gold foil
with some or no (Coulomb) interaction, but some will
experience a “head-on” collision with a nucleus and return in a
backwards direction.
14

But there were still unanswered questions …
… why does the nucleus (all positive charge) not fly apart due
to Coulomb repulsion?
… why do the negative charges not radiate energy, spiral
inwards and collapse into the nucleus due to Coulomb
attraction?
… the model did also not explain existing experimental
observations.
15

Atomic size
In a solid, assume all atoms are spherical and packed in a tight grid.
For radius R, separation is 2R and volume of each atom (2R) 3 .
One mole contains N A atoms and occupies volume 8N A R 3 .
Volume also given by
Thus
R
3
M 10
…
where is density and M molar mass (in g).
1
3 3
1 M
10
2
N
A
Gold has M
…. thus
197 and 19.310 kg m
10
R 1.310 m
3 -3
All atomic radii are about the same: range
10
0.5 2.5 10
m

The radioactive decay law
Radioactive decay is a statistical (random) process.
Consider a set of identical unstable nuclei.
The set will obtain N 0 = N(0) of these nuclei at t =0 ,
and N(t) radioactive nuclei at time t .
i.e. N(t) < N(0) because of the decay.
The rate at which the nuclei decay is proportional to N(t):
dN()
t
i.e. Nt
( )
dt
is called the decay constant and depends on the nuclide.
represents the probability that any one nucleus will decay
during the next second (or any other time unit).
18

For a particular set of nuclides, is the same for each
nucleus, at all times.
dN()
t
dN
N()
t dt
dt
N
N () t
N
0
dN()
t
Nt ()
t
0
dt
log ( ) ()
N t
t
e
N t t
N (0)
0 log ln
Nt ()
N(0)
e
t
e
Radioactive decay law:
N( t) N(0)
e t
19

Mean life of decay
: decay constant
Nt ()
N(0)
The mean life,
or lifetime,
is defined to
be 1
N(0)
e
N(0)
When t , N( t) ,
e
N()
t N e t
0
t
so the lifetime is the time taken for N(0) to drop to 1 N(0)
e
20

Half life of decay, T12
T 12
is the time taken for N(t) to drop to 1 2
N(0)
So at t T :
12
1
N( t) N( T12) N(0)
2
T
i.e. N(0) N(0)
e
1
2
T12
2 e
ln 2 T12
ln 2 0.693 T 12
0.693
T 12
is convenient, since when t = m
e.g. after 3 half-lives, N(3 T12)
12
N
N(0)
1
12
14
18
T 12
N(0)
8
1 2 3
, N(t) is reduced to
N(0)
2 m
21
m

The decay rate
often more interesting than
Write
Activity
dN()
t
Rt ()
dt
Nt ()
dN()
t
R( t) N(0)
e
dt
t
or activity of the sample is
itself.
or
R( t) R(0)
e t
where R(0) N(0)
The number of disintegrations (decays) per second also reduces
exponentially with decay constant .
SI units of activity: 1 bequerel (Bq) = 1 decay per second
Older unit: 1 curie (Ci) = 3.7 10 10 decays per second
22

For a decay chain A → B → C →
A
N ( ) (0) t
A
t N
A
e
A
At
B
For a two stage decay sequence NB( t) N
A(0) e e
B
A
For a three stage decay sequence
t
t
e e e
NC ( t)
A BN
A(0)
A B
Ct
B A C A A B C B A C B C
t

Interaction of radiation with matter
24

Electromagnetic radiation
Three primary processes of interaction with matter:
Photoelectric effect, Compton scattering, Pair production
25

7.6 cm x 7.6 cm NaI 1.9 cm x 0.5 cm HPGe
26

Probability per unit length
for removal of a photon
Linear absorption coefficient μ
Fractional loss in intensity:
I( x)
I e x
0
Mass attenuation coefficient:
27

Discovery of the neutron
James Chadwick, 1932
Outside the nucleus, free neutrons are unstable and have a half
life of 611.0 ± 1.0 s.
n p e
e
28

Neutrons
Uncharged, do not interact via Coulomb forces
Interact (via collisions or reactions)
with the nuclei of absorbing material
Secondary radiation is almost always heavy charged particles
Probability of interaction given by a cross section σ (see later)
Fractional loss in intensity of a neutron beam irradiating
a material of N nuclei per unit volume, and thickness x:
N
I( x)
I e
0
x
29

Pulse height spectrum
n
Organic scintillator
p
PMT
n
Energy spectrum
of recoil protons
30

Frank Brooks (1931 – 2012) UCT Professor of Nuclear Physics
… developed in the early 1960s the first
practical systems of pulse shape
discrimination (PSD) which allows the
identification of different types of charged
particles in certain scintillator detectors by
means of the characteristics of the
scintillation decay.
gammas
PSD in a 5 cm x 5 cm
organic liquid
scintillator
neutrons
AmBe source
Beam of 66 MeV neutrons

Charged particles
The linear stopping power (or specific energy loss) for
charged particles in a given absorber ‒dE/dx can be
calculated using the Bethe-Bloch formula …
… better via “Monte Carlo” method
… see www.srim.org
32

Specific energy loss in air
33

“Bragg peak”
34

Radiation units
Activity: the rate of spontaneous decay transitions in a sample …
SI unit:
1 bequerel (Bq) = 1 decay per second
Traditional unit: 1 curie (Ci) = 3.7 × 10 10 Bq
Exposure: quantity of radiation to which an object is
exposed which is the ionization that the radiation would
produce in dry air …
SI unit:
1 roentgen (R) = 2.58 × 10 -4 C/kg in dry air
… amount of radiation which delivers 8.78 mJ
of energy to 1 kg of dry air
“Exposure” (and roentgens) seldom used nowadays …
35

Absorbed dose: a measure of the energy deposited in a medium
by ionizing radiation per unit mass …
SI unit: 1 gray (Gy) = 1 J kg -1
Traditional unit: 1 rad = 0.01 gray
The absorbed dose depends not only on the incident radiation but
also on the absorbing material: a soft X-ray beam may deposit four
times more dose in bone than in air, or none at all in a vacuum.
Kerma: “kinetic energy released in matter” … the sum of the
initial kinetic energies of all the charged particles liberated by
uncharged radiation (i.e., indirectly ionizing radiation such
as photons and neutrons) in a sample of matter, per unit mass of
the sample … kerma always greater than absorbed dose since
some of the energy escapes from the absorbing volume in the
form of bremsstrahlung x-rays or fast moving electrons.
SI unit: grey (G)
36

Equivalent dose: a computed average measure of the radiation
absorbed by a fixed mass of biological tissue, that attempts to
account for the different biological damage potential of different
types of ionizing radiation. It is therefore a less fundamental
quantity than the total radiation energy absorbed per mass (the
absorbed dose), but is a more significant quantity for assessing
the health risk of radiation exposure.
SI unit: 1 sievert (Sv) = 1 J kg -1
Traditional unit: 1 rem = 0.01 Sv
equivalent dose (H) = absorbed dose (D)
× radiation weighting factor (W R )
37

equivalent dose (H) = absorbed dose (D)
× radiation weighting factor (W R )
Radiation weighting factors (legislated):
Radiation
W R
X-rays, gamma rays, electrons, muons 1
Neutrons: thermal 5
Neutrons: < 0.1 MeV 10
Neutrons: < 2 MeV 20
Neutrons: > 2 MeV 10
High energy protons 5
Alpha particles, fission fragments, heavy nuclei 20
One sievert of two different radiations produces the same
biological effect (more or less).
“Dose rate” also useful … equivalent dose per unit time.
38

http://xkcd.com/radiation/
39

Nuclear phenomenology
nucleus positive core of the atom
discovered by Rutherford in -scattering experiments
… almost all the mass of the atom concentrated in the nucleus
… the nucleus consists of
A nucleons Z protons N neutrons
held together by
strong nuclear forces
positively
charged
no charge
43

• nuclide: a particular combination of protons and neutrons
• nuclides are characterized and represented by:
A A A
AZ , X X X X A
atomic number
mass number
Z Z N
60 60 60
27 27 33
X: chemical symbol
e.g. 60,27 Co Co Co Co 60
A Z N
neutron number
• isotopes: Nuclides with the same Z
• isotones: Nuclides with the same N
• isobars: Nuclides with the same A
35 36
e.g.
17Cl 17Cl
40 39
e.g.
20Ca 19K
20 20
e.g.
9F 10
Ne
44

Nuclear and atomic masses
exactly
Mass of 1 neutral atom of 12 6 C
12.0000
u
One mole of a substance contains Avogadro’s number,
N A = 6.02310 23 atoms (or molecules)
… giving 1 u = 1.66110 -27
unified mass unit
One mole of a substance of molecular weight M u is M 10 -3 kg
Then 0.012 kg of 12 C contains N A atoms of mass 12 u …
kg
Then
Eu
27
8 -1
1.66110 kg 310 m s
19 -1
1.610 J eV
2
u c
931.5 MeV
2
So we can write 1 u = 931.5 MeV/c 2
46

Energy equivalent masses:
M p = 1.007276 u = 938 MeV
M n = 1.008665 u = 940 MeV
M e = 0.0005486 u = 0.511 MeV
47

Nuclear sizes and shapes
Size and shape of a nucleus can be found from scattering experiments
… use electrons to learn about charge distribution
… and hadrons (often neutrons) to learn about matter distribution
Charge distribution
In principle the charge distribution of a nucleus may be
obtained from the measurements of the differential cross
section of electron scattering, but more reliable
determinations are obtained from numerical solutions of the
Dirac equation fitted to experimental data …
48

Radial charge distributions () r of various nuclei, in units
of e fm −3 ch
; the thickness of the curves near r = 0 is a measure
of the uncertainty in () r ch .
0
() ch
ch r
ra
b
1
e
“Saxon Woods”
For medium
and heavy nuclei …
a
13
1.07 A fm
b 0.54 fm
0
ch 0.06 to 0.08
49

Mass distribution
Almost identical nuclear density in the nuclear interior of all
nuclei.
ch
0
… the decrease in with increasing A is compensated by
the increase in A/Z with increasing A
Interior nuclear density then
nuclear 0.17 nucleons / fm
3
and for medium and heavy nuclei …
R
13
nuclear
1.2 A fm
50

Elastic differential cross-sections for 52 MeV deuterons on 54 Fe
51

Differential cross-sections
(normalized to the
Rutherford cross-section)
for the elastic scattering of
30.3 MeV protons, for a
range of nuclei compared
with optical model
calculations; the solid and
dashed lines represent the
results using two different
potentials.
52

Periodic Table of the Elements
53

Nuclear binding energy
Use atomic masses to define the mass deficit / defect / excess
M ( Z, A) M ( Z, A) Z( M m ) NM
Define the nuclear binding energy to be:
p e n
B
M ( Z, A)
c
2
Measure nuclear masses
with a mass spectrometer
54

… B/A is the binding energy per nucleon for a particular nuclide.
Plotting B/A versus A gives the binding energy curve...
Average
binding
energy
per
nucleon
(MeV)
Mass number A
Note that the region of greatest stability is at a maximum of
8.7 MeV per nucleon, occurring at A = 50 – 80.
55

Example: Binding energy of the deuteron.
What is the binding energy of the deuteron if
B
D
M c
where
D
2
M D = 2.013553 u
M p = 1.007276 u
M n = 1.008665 u
M M M M 0.002388 u
D p n D
MeV
B u
c
uc
2
D
0.002388 931.5 2.224 MeV
2
n – p capture:
n + p d + g
2.224 MeV
57

Stability of nuclides
Not all nuclides are stable. Unstable nuclei decay spontaneously
to become stable.
i.e. emit particles
e.g. Gold has 30 isotopes ( 175 Au 204 Au). Only 197 Au is stable.
The other 29 isotopes are radioactive nuclides (radionuclides)
Organising the nuclides
The periodic table shows the most common or most stable isotope
of each element. Usefulness limited since different isotopes have
very different nuclear properties.
The nuclidic chart shows all known nuclides
… is a plot of Z vs. N …
58

Stable nuclides shaded dark
Radioactive nuclides
shaded light
For a stable nuclides
with N 20, N Z
This increases to N 1.5 Z,
for N 120
59

“Valley of stability”
61

–decay
Nuclear decay
4
helium nucleus
2He2
… the unstable “parent” nucleus emits an -particle.
A
Z
X Y He
A4 4
Z2 2
b –decay
b
an electron
… occurs when n p inside a neutron-rich nuclide
X
Y e
A
A
Z Z1 N1
e
Possible when M ( Z, A) M ( Z 1,
A)
63

–decay
b
X
a positron
… occurs when p n inside a proton-rich nuclide
Y e
A
A
Z Z1 N1
e
Possible when M ( Z, A) M ( Z 1, A) 2m
e
Electron capture
A nucleus captures an atomic electron which has ventured
too close to the nucleus
X e Y
A
A
Z N Z1 N1
e
Possible when M ( Z, A) M ( Z 1, A)
where is the excitation energy of the atomic shell
of the daughter nucleus
64

Energy released in radioactive decay
In nuclear decay, binding energy is released, becoming
kinetic energy of the decay products…
Q
energy released in the decay
mass mass
c
2
before
Q > 0 is a condition for the decay to occur.
A
A4 4
For example, consider the –decay: X Y He
Z
after
Z2 2
The Q–value or disintegration energy is
2 2 2
Q M c ( M c M c )
This can be shown to be the same as:
Q B B B
X
Y
B B B
( 4) 4
A A
4 4
X
Y
A A
X
Y
65

Ground and excited states of nuclei
66

Gamma decay
Decay is a ”step” towards stability. Nuclei, like atoms have
discrete energy states. Excited nuclei decay by transitions in
which high energy photons (g-rays) are emitted.
A *
A
ZX
ZX
g
excited state
e.g. the decay of 60
27 Co:
b
g
g
2.50 MeV
(1.17 MeV)
1.33 MeV
(1.33 MeV)
0
60
Co is therefore a source of both g s and 27 b– s,
both radiation exhibiting a half life of 5.3 years.
67

Natural radioactivity
69

Angular momentum of a nucleon
1
Nucleons are fermions, therefore spin quantum number s 2
Eigenvalue of
Eigenvalue of
2 1 1
s is 2
2 21
sz
is
1
2
Nucleon has orbital angular momentum quantum number 0,1,2,...
Eigenvalue of
Eigenvalue of
Add spin and orbital angular momentum vectorially to get
total angular momentum j l s
Eigenvalue of
Eigenvalue of
is
2
l 2
z
is
is
m
1
j j1
2
j 2
jz
is
m
j
m l,...,0,..., 1,
m j,...,0,..., j 1,
j
j
71

Angular momentum of a nucleus
Add all spins of nucleons vectorially to get total nuclear spin S.
Add all orbital angular momenta vectorially to get total
orbital angular momentum L.
Add total spin and total orbital angular momentum vectorially
to get total angular momentum (or nuclear spin)
J L S
Write nucleus wavefunction as
JM
Therefore
J 2 1
2
JM
J J JM
and JzJM M JM
where
M J, J 1,...,0,..., J 1,
J
72

Parity
Parity is the transformation
r
r
x
x
Pˆ , t P , t
The intrinsic parity of a single proton and a single neutron
is defined as +1
Under the parity transformation … PY ˆ , 1 Y ,
Therefore for a single particle nuclear state P 1
Total parity of a multiparticle state is the product of parities of
individual particles.
m
m
A pair of nucleons with the same l have combined parity of +1
… therefore parity of a nucleus depends on the parity of the last
unpaired proton and/or neutron.
73

Spin and Parity
Individual and/or collective motions of nucleons (determined
by nuclear structure and dynamics) determine symmetry and
other properties of the nuclear wave function.
A nuclear state is labelled with both its spin and parity in the
form J e.g.
1
2
,1 ,0 ,...
Some observations for ground states:
Even A nuclei have
J 0,1,2,...
Odd A nuclei have
J
1 3 5
2
, 2
, 2
,...
All even-even nuclei have J 0
74

Recall … the magnetic moment of the electron
Classically for an electron of charge e and mass m orbiting with
angular momentum L … associated orbital magnetic moment is
e
μL
L
2m
And similarly define an intrinsic (spin) magnetic moment
e
μ g s
2 m
where g-factor = 2.0023192 (or 2.000000 from Dirac)
Actual value of electron intrinsic magnetic moment is an
eigenvalue of when electron is in m substate.
1
z
s 2
e e
1 1 1
2 g 2 g 2
gB
2m
2m
where the Bohr magneton B e 2m
= 9.274 × 10 -24 J T -1
75

Nuclear magnetic dipole moments
Both the proton and the neutron have an intrinsic magnetic
moment … and since the proton is charged, it can also produce a
magnetic moment when in orbital motion.
Write the magnetic moments for spin up protons and neutrons as
or
Measure
Then
p
e
1 1
p
2
g
p
n
2
gn
2M
p
2
g
1
p 2 p N
g
e
M
p
1
n 2 n N
where the nuclear magneton e 2M
= 5.05078× 10 -27 J T -1
g 5.5856 and g 3.8262
p
Note that
2.7928
and 1.9131
p
N
32
n
N
n
n
p
(predicted by quark model)
N
76

Magnetic moments of nuclei involve spin and orbital
component, and are indictors of structure.
For a nucleus … the intrinsic magnetic moments of the
constituent protons and neutrons will contribute to the total
magnetic moment … with further contributions from any
orbital motion of the protons.
Then
Jg
J J N
g J
where is the nuclear g-factor … the ratio of the magnetic
moment in nuclear magnetons to the total angular momentum
quantum number for the nuclear (or particle) state.
Find
0
for all J = 0 ground states (why?)
77

Nuclear electric quadrupole moments
Define quadrupole moment as ( ) 3 2 2
0 ch
r
Q z r dV
… with z-axis along symmetry axis defined by nuclear spin.
Then can write
2 2
Q0 Z 3 z r and has units of barns.
2 2 2 2
z
where r x y z
For a spherical nucleus:
z
r and thus Q0 0
2 1 2
3
For a non-spherical nucleus:
z
r
2 1 2
3
z
z
r Prolate ellipsoid Q0 0
2 1 2
3
z
z
r Oblate ellipsoid Q0 0
2 1 2
3
Find Q0 0
1
for all J = 0 and J 2
ground states (why?)
78

A nucleus may be deformed
away from spherical into a
(a) prolate shape (rugby ball)
or
(b) (b) oblate (flying saucer)
79

8 May 2013
A representation of the radium-224 nucleus. The
atomic nucleus is a many-body quantum system with a
shape determined by the number of nucleons that it
contains and the interactions between them. Most of
the several thousand known stable and radioactive
atomic nuclei, with differing numbers of protons and
neutrons, are spherical or rugby-ball shaped. But there
is circumstantial evidence that some heavy, unstable
nuclides are distorted into a pear shape through the
phenomenon of octupole deformation. Samples of
these rare atomic species can be accelerated to 8% of
the speed of light in the REX-ISOLDE facility at
CERN, and now Coulomb excitation experiments on
beams of the short-lived isotopes radium-224 and
radon-220 have demonstrated clear octupole
deformation in the former. The results make it possible
to discriminate between the various theoretical models
of octupole-deformed nuclei, and are also relevant to
the pursuit of physics beyond the standard model.
80

The nucleon-nucleon potential
Martin
7.1
We infer that the N-N force must be …
1. Short range (from Rutherford scattering)
2. Attractive at short range (to bind nucleons in
nuclei)
3. Strong relative to Coulomb (to bind protons)
4. Repulsive at very short range (i.e. must
“saturate” (to prevent collapse of all nuclei to
the dimension of the force range)
V()
r
F r
81

Electrostatic and gravitational potential is long range V 1 r .
Near constancy of nuclear binding energy per nucleon B/A means
that each nucleon feels only the effect of a few neighbours.
This is called saturation … implies the strong inter-nucleon
potential is short range.
Range is of order of the 1.8 fm inter-nucleon separation.
Since volume A , nuclei do not collapse, there is a very
short range repulsive component. Reminiscent of interatomic
potential in molecules. Is nuclear physics just quark chemistry?
Depth of potential is of order of binding energy,
perhaps tens of MeV.
82

The deuteron
The deuteron 2 H is the bound state of a neutron and a proton.
Experimental properties:
Binding energy B = 2.2245 ± 0.0002 MeV
Angular momentum and parity J π = 1 +
Magnetic dipole moment μ D = 0.8575 nuclear magnetons
Electric quadrupole moment Q D = 0.00282 × 10 -28 m 2
Deuteron has very small binding energy, and no excited states.
Also ( 1.91 2.79) 0.88
n p N N
And with very small Q D , infer that deuteron is in L = 0 state
where L l p
ln
83

Deuteron ground state L = 0
Then since J L S
… J = 1 (measured)
… implies proton and neutron spins are aligned (S = 1)
Thus using the standard spectroscopic notation 2S
1 L
J
where L = S, P, D, F, G, H
for L = 0, 1, 2, 3, 4 …
… the deuteron has a triplet ground state 3 S1
84