Nuclear and Particle Physics SubAtomic Physics
Nuclear and Particle Physics SubAtomic Physics
Nuclear and Particle Physics SubAtomic Physics
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<strong>Nuclear</strong> <strong>SubAtomic</strong> <strong>and</strong> <strong>Physics</strong> <strong>Particle</strong><strong>Nuclear</strong> <strong>Physics</strong>Lecture 2Main points of Lecture 1As well as being an active current area of researchnuclear physics techniques are used extensively in otherdisciplines <strong>and</strong> in industrial applicationsAu foilRutherfords α scattering experiments- atoms contain an incredibly densepositively charged nucleusαDr Daniel WattsMass spectrograph - nucleus must contain unchargedneutral component – “neutron”. Subsequently discoveredby Chadwick - m n ~m pTypical energy scale in nuclei is MeV - million times largerthan energies associated with atomic systems (eV)Nucleons in the nucleus rarely collide with sufficientenergy to excite the nucleons - nucleon good d.o.fNuclei are dense objects: 1cm 3 has mass ~ 2.3x10 11 kg(equivalent to 630 empire state buildings!!)
1. Strong (nuclear) forceThe forces of nature• acts on all particles except leptons• always attractive (on average)• short range (10 -15 m)4. Gravitational force• acts between all particles with mass• always attractive• always presentbut inverse square lawshort rangeforceZAP10 -15 m2. Electromagnetic forceSTRONGEST FORCEF strong = 1m 2~ 1/R 2long rangem 1forceWEAKEST FORCEF grav ~ 10 -40 F strongbut dominant force in themacrocosmos (large masses)… binds Earth ,solar systemgalaxies…q 2~ 1/R 2long rangeq 1 force3. Weak force→ + + νproton electron(T 1/2 ~ 15 min.)neutrino• acts between all particles with charge• attractive/repulsive• always presentbut inverse square lawSECOND STRONGEST FORCEF em ~ 10 -2 F strongdominant force in atoms,molecules, solid bodies (binding)• acts on all particles• very short range (10 -18 m)• e.g. responsible for β decaySECOND WEAKEST FORCEF weak ~ 10 -7 F strongWhich forces are important to underst<strong>and</strong>ing the nucleus??Bound system of protons <strong>and</strong> neutronsInteracting under the competing influence ofAttractive nuclear (strong) <strong>and</strong> repulsiveElectromagnetic forces (between protons)Weak forces are negligible in terms of binding the nucleus,but important in that they can change neutrons ↔ protonsβ decay (see later)<strong>Nuclear</strong> force does not win over EM repulsion indefinitelyFinite number of naturally occurring atoms (Z≤ 92)Last completely stable nucleus Lead (Z=82)electromagnetic force (~Z 2 ) wins at Z≥92
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Strength of forces at different distance scalesQuantum mechanicsRevision of some basic concepts to be used later in the course• observable = physical, measurable quantity ⇔ operatorexamplestotal energy ⇔ Hamiltonianposition ⇔ xˆlinear momentum ⇔ pˆangular momentum ⇔ Lˆ2Ĥ• system described by wave function ψ(x,t) obeyingtime dependent Schrödinger equation (TDSE)Role of the different forces depends on the distance scaleExample: Which forces dominate in Rutherford’s α particlescattering measurement?Need to get within few fm of thenucleus for strong force to contributesignificantly. In Rutherford’sexperiment (~7 MeV) α’s cannot getwithin this range of the nucleus →therefore Coulomb force dominates.See Tutorial question !19779 Au(charge = +79e !)?Would need higher incidentα energies to see effectsof the strong force in thescattering → large deviationsfrom simple Coulombscattering2 2⎧ h ∂⎨ −⎩ 2m∂x2⎫ ∂+ V(x,t)⎬Ψ(x,t)= i h Ψ(x,t)⎭ ∂t• static potential - stationary states exist. Described by spatial wavefunction ψ(x) obeying time independent Schrödinger equation (TISE)⎧ 2 2h d⎨ −⎩ 2mdxeigenfunction2⎫+ V(x) ⎬ψE(x) = EψE(x)⎭Ĥψ (x) = Eψ (x)Eeigenvalue equation only satisfied by certain values of Eenergy is QUANTISEDmeasurements of total energy can only yield values which areeigenvalues of Hamiltonian operatorEeigenvalue
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2x2Orbital angular momentum ⇔ Lˆ2Lˆ = Lˆ + Lˆ + Lˆ square of magnitude of angular momentum2y2zIn spherical polar coordinates:2[ Lˆ ,Lˆ ] 0iLˆ2= −hImportant property:Lˆ2Choose:2⎡⎢⎣1sin θ∂ ⎛ ∂ ⎞ 1⎜sin θ ⎟ +∂θ⎝ ∂θ⎠ sin2∂2θ ∂φcommutes with any Cartesian component of angular momentum= both operators have simultaneous eigenfunctionsspherical harmonics Ylm (θ,φ)∂Lˆz = −ih z-component of angular momentum vector∂φ2• Eigenvalue equation for Lˆ2Lˆ Yml2 ml(θ,φ) = l ( l + 1) h Y (θ,φ) l = 0,1,2,3 ,...• Eigenvalue equation for LˆzmmLˆ Y (θ,φ) = mhY(θ, φ) − l ≤ m ≤ l in integer stepszlangular momentum is QUANTISEDlEigenstates of Lˆ2are DEGENERATE ⇔ (2l+1) possible valuesexperimental evidence: Stern <strong>and</strong> Gerlach experiment (spatial quantisation)2⎤⎥⎦orbital angular momentumquantum numbermagnetic momentumquantum numberN.B. we refer to a particle in a state of angular momentum l meaningl ( l + 1)hIntrinsic angular momentum ⇔No good classical analogueElectrons, protons, neutrons all have half integer spin: FERMIONSŜŜzhas eigenvalue s = ½has eigenvalue m s = ± ½Ŝspin upspin downAddition of angular momentum vectors in quantum mechanicswhen two states of angular momentum quantum numbers j 1 <strong>and</strong> j 2add (or couple) together,they form a state with definite TOTAL angular momentum j <strong>and</strong>definite j z component⏐j 1 -j 2 ⏐ ≤ j ≤ j 1 + j 2-j ≤ m ≤ jthese states can be expressed as linear combination of statesof the uncoupled basis {⏐j 1 ,m 1 ,j 2 ,m 2 >} with coefficients known• Eigenvalue equation for2ĴĴas Clebsch-Gordan coefficients2Total angular momentum JSum of orbital angular momentum <strong>and</strong> spin of nucleon J = L + S• Eigenvalue equation for2j,m = j(j + 1) hĴ zĴzj,mj,m = mhj,m − j ≤ m ≤ j1 3j = 0, ,1, ,2,...2 2in integer steps(2j+1) possible projectionsset of (2j+1) states { j,m } called MULTIPLET
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Central symmetric potentialPotential energy function V=V(r) is function of r only, not of θ <strong>and</strong> ϕSolution of TISE:• radial wavefunction R(r) solution of radial TISE:2( l 1)⎡ h 1 d ⎛ 2 d ⎞ l + h⎢−r + V(r) +2µ2⎜ ⎟r dr dr2⎣ ⎝ ⎠ 2µr2centrifugal potential(or barrier)⎤⎥R( r) = ER( r)⎦• spherical harmonics Y(θ,ϕ) solution of angular TISE:⎡− ⎢⎣1sin θu(r,θ, φ) = R(r)Y(θ,φθ,φ)∂ ⎛ ∂ 1sin θ⎞⎜ ⎟ +∂θ⎝ ∂θ⎠ sin2∂2θ ∂φ2⎤⎥Y θ,φ⎦( ) = λY( θ,φ)Parity πParity operator ⇒ inversion in the origin coordinatesPˆ uIn polar coordinates:nlmm( r, θ, φ) = Pˆ R( r) Y ( θ,φ)r → rθ → π − θφ → π + φr r→ −Behaviour of eigenfunction under parity transformation determinedby properties of spherical harmonics.ml m[ ] = R( r) Y ( π − θ, π + φ) = R( r)( − 1) Y ( θ,φ)lPˆunlmEigenfunctions have definite parity:ll( r, θ,φ) = ( − 1) u ( r, θ,φ)nlmEVEN (or positive) for even lODD (or negative) for odd llN.B. potential V(r) does not appear in angular equation⇒always solutions, no matter what form for V(r)Ylm ( θ,φ)Very general result following from spherical symmetry of potentialm| ( θ,φ)Y l2|Distance of eachpoint from the originproportional to themagnitude in thatdirectionPolar plots of sections at y=0 through spherical harmonicszz+0 2| Y0 ( θ , φ)|Even parityy+-0 2| Y1 ( θ,φ)|Odd parityWe will see in later lectures how conservation of total angular momentum <strong>and</strong>parity are of crucial importance in underst<strong>and</strong>ing nuclear structure <strong>and</strong>reactions This overview deals with the QM of stationary states produced in astatic potential. To calculate the probability of transitions between thesestates we need more advanced QM. (We will touch a little on this later whenwe look at electron scattering -Fermi’s golden rule)y+-+ -0 2| Y2 ( θ,φ)|Even parity+, - sign offunction ineach region
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The nucleus <strong>and</strong> its propertiesTime evolution of the nuclear chartNucleus = central part of an atomit contains A nucleonsA = Z (protons) + N (neutrons)atomic numbermass number(nucleon = proton or neutron)AXZ NZ ≡ X chemical symbol)Proton nmber (Z)A bit of nomenclature…NUCLIDE element with given N <strong>and</strong> ZISOTOPES elements with same Z but different NISOTONES elements with same N but different ZISOBARS elements with same AISOMERSelements in metastable (i.e. very long-lived) stateChart of nuclidesNeutron number (N)isotonesisobarsChart still being added to as experimental facilities improveWe are still quite some way from reaching the limits of nuclearexistence – particularly on the neutron rich side.New facilities coming online in the near future like FAIR inGermany (http://www.gsi.de/fair/) should remedy this!isotopes
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External nuclear propertiesCharge Ze: protons have +ve charge e = 1.6022x10 -19 Cneutrons have zero chargeneutral atom: (A,Z)contains Z electrons orbiting around nucleussymbolically:AZXN(Z ≡ X chemical symbol)Mass M:nuclear <strong>and</strong> atomic masses are expressed inATOMIC MASS UNITS (u)definition: 1/12 of mass of neutral 12 C ⇒ M( 12 C) = 12 u1u = 1.6605x10 -27 kg or 931.494 MeV/c 2 (E=mc 2 )M(A,Z) < Zm p + Nm ndifference:∆M = Zm p + Nm n - M(A,Z) ⇒ mass defect (excess)accounts for BINDING ENERGY of nuclei (see later)N.B. we typically use ATOMIC <strong>and</strong> not NUCLEAR masses⇒ mass of electrons also includedSize R: nuclear radii are expressed in fermis (fm) 1 fm = 10 -15 mcompare with atomic dimensions1 Å = 10 -10 mmatteressentiallyEMPTY SPACE!
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… more on nuclear sizeWhat do we define as nuclear size?NotesConsider the following:• the nucleus has a net positive charge Ze (Z protons)• take into account Coulomb + nuclear forceVBResulting potentialextends to ∞as 1/R 2Coulombrepulsivehas short(~10 -15 m) rangeDefine:barrier height B at a distancefrom centre R:B = Zze 24πε 00 RRrfor incident charge ze-V 0 nuclear attractiveR = POTENTIAL RADIUS ⇒ related to distribution of protons &neutrons AND the range of nuclear forcepotential radius > charge (or mass) radiusCHARGE RADIUS⇒ related to charge (proton) distribution
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