Outline of Topics discussed in the Lecture for Graduate Nuclear Physics 
(PHYS722/822 Fall 2003)

Shell model wrap-up

• l^.s coupling: leads to splitting of states with given n, l into j=l+1/2 and j=l-1/2. Splitting proportional to (2l+1) -> rearranged level sequence -> magic numbers.
• "Doubly magic" nuclei (both Z and N fills completely top shell) -> Jp = 0+, no deformation.
• Single nucleon outside closed shells or single hole in closed shell: Jp given by l and j for that nucleon (shell). Magnetic moment given by g_Nucleus = g_l +/- (g_s - g_l)/(2l+1)
• Can also explain first excited state(s) for these nuclei
• More realistic picture (especially for higher excitations and occupation numbers: Take basis set of states made by Slater determinants of single-particle states in model potential. "Residual" potential (especially 2-nucleon interaction vs. average) introduces perturbations - "true" states are superpositions including higher states than just the lowest A. Gives occupation numbers which extend beyond Fermi surface. Need large Hilbert space for accurate calculations!
• Wave functions for single nucleons can be used to predict probability distribution in space (radial and angular); addition of all filled nucleon states gives shape of nucleus.
• Results for nuclei further away from closed shells: even-even nuclei tend to have Jp = 0+ (pairing force). Even-odd nuclei quantum numbers tend to be determined by quantum numbers nlj of highest state occupied by the odd nucleon. Odd-odd nuclei are more difficult; however, tendency for the two odd nucleons to combine to total spin of 1 (quasi-deuteron).

Scattering

• Beams: Photons, electrons, muons, neutrinos, pions, kaons, protons, hyperons, antiprotons, nuclei (from deuterium to uranium)
• Accelerators:
• radioactive elements, cosmic rays
• electrostatic (van de Graaf, Cockroft-Walton,...)
• radio-frequency linear (SLAC, CEBAF)
• circular (synchrotrons, storage rings - CERN, HERA,...; cyclotrons - TRIUMF, PSI, MSU,...)
• Targets: Liquids (H, D, He, waterfall ...), solids (all metals, carbon, ice, ...) and gases (high-pressure, gas jets, internal targets in storage rings). Also: countercirculating beams (from positrons over protons to gold).
• Detectors:
• Tracking devices: Wire chambers, Silicon Pixel, GEMs, microMEGAS, TPC, ...
  
• Non-magnetic (Plastic and NaI/CsI Scintillators, Lead glass etc. calorimeters, Geiger-Mueller counters, Si or GeLi photon counters)
• Magnetic spectrometers (quadrupole magnets to focus scattered particles, dipoles to fix momentum, detector hut with cerenkov tanks, scintillator hodoscopes, wire chambers, transition radiation detectors and electromagnetic and hadronic calorimeters to detect and identify particles- > see SLAC end station A, CEBAF halls A and C)
• large acceptance spectrometers (open geometry, cylindrical or toroidal magnetic field, large detectors - drift chambers, TPCs; HERA Zeus, CERN, CEBAF CLAS).
• Data acquisition (recording), conversion to Physics observables, binning in kinematic bins.

Scattering cont'd

• Classification  a (+ b) -> c + d + ...:
• Decay (only one object a in initial state)
• elastic scattering (a = c and b = d)
• inelastic scattering (d is excited state of b)
• production (additional particles e, f,... in final state)
• general reaction (all outgoing particles different from incoming ones)
• Exclusive: all final state particles determined, complete set of kinematic variables measured (3*N - 4 for N final state particles)
• Inclusive: only one particle measured (or none at all)
• Semi-inclusive: part of the final state measured, integrate over unobserved rest.
• Beam and target: target is of finite size and fully traversed by beam
• Definition of cross section: Probability (P) for event into kinematic bin (Dk_i) = density of target atoms/unit area (z*rho*N_A/A) * cross section (Ds)
• Alternative definitions: Count rate dN/dt = L*Ds (L = luminosity = I_beam/e * z*rho*N_A/A) or for a single target particle dN/dt = j_in*Ds
• Fully differential cross section: limit of Ds(Dk_i)/P(Dk_i)
• Example for elastic scattering: ds/dW = lim Ds/Dcosq/Df
• Partial (or semi-inclusive) cross section: Integrate over unobserved or uninteresting variable: ds/dq = 2p sinq ds/dW
• Conversion of kinematic variables using Jacobians. Example: ds/dq² = p/k_ink_out*ds/dW (q = k_in-k_out = momentum transferred).

Born Approximation - First order Perturbation Theory

• Fermi's Golden Rule: Probability for transition from initial plane wave to final plane wave state per unit time is given by  dP(i->f)/dt = 2p/hbar |M_fi|² DFd(Ef - E')
• Directly gives decay rate
• Cross section : ds = dP(i->f)/dt / j_in
• Matrix element M_fi = <pw_f |H - H₀| pw_i>
• DF = V*D³p / h⁶

Lecture 9

Elastic electromagnetic scattering - First order Perturbation Theory

• Cross section from Fermi's Golden Rule (see lecture 8)
• Matrix element M_fi  can be calculated from Feynman diagrams:Dirac spinors for external legs, gamma matrices times interaction strength (e = a^1/2) for vertices, 1/(M²+Q²) for propagators of virtual particles of mass M.
• Alternative method: Describe nucleus via electrostatic potential, incoming and outgoing plane waves. Get Rutherford cross section times form factor F(q²) (Fourier transform of charge density distribution).
• Important properties: F(q²) = 1 if wave length is much larger than nuclear size; next-order approximation is 1-<r²>q²/6 -> rms charger radius. In general can extract charge distribution.
• Necessary improvements for high-energy electron scattering: replace q² with Q²; electron spin contribution cos²(q/2) => Mott cross section; recoil factor E'/E => Point cross section. Electric and magnetic form factors (Ge and Gm). See homework and references.

Nucleon Form Factors continued

Quasi-elastic scattering

• Impulse approximation (incoherent scattering of single nucleons inside nuclei)
• Scatter off proton moving inside nucleus A with initial momentum p, leading to a final state nucleus (A-1) moving with momentum -p and a final state proton moving with momentum p+q.
• Nonrelativistic approximation: Energy transfer n = [m_A-1 + E_exitation + T_kin(A-1) + m_p - m_A] + p²/2m_p + q²/2m_p + p^.q/m_p
• The first term is the missing energy E_miss (sum of separation energy, excitation energy and kinetic energy for the residual nucleus), typically of the order of 10's of MeV
• The second term is the part of the kinetic energy of the knocked-out proton (again from zero to 30 MeV)
• The third term is the result for free scattering from an unbound, stationary proton (more properly Q²/2m_p).
• The last term depends on the initial momentum of the proton and its direction relative to q. It can assume fairly large values, leading to a "broadening" of the quasi-elastic peak of several 100's of MeV at high energy.
• Cross section becomes ds/dW/dE' = ds/dW(free)*P(E'), where the probability distribution P(E') describes the likelihood of finding a proton with initial momentum p leading to a particular value of E' = E - n  according to above formula. The width of this distribution can be used to extract an "experimental" value for the Fermi-momentum (largest possible momentum) for a given nucleus, and the shift from the free value Q²/2m_p can be interpreted as average missing energy E_miss. Early results from quasi-elastic inclusive scattering show good agreement with Fermi model and Shell model expectations.

Quasi-elastic scattering cont'd

• Measure ejected proton together with electron in final state (exclusive instead of inclusive):
• Completely determine initial momentum p, and therefore all terms except E_exitation in the formula for n (see last lecture)
• Can solve for E_exitation or E_miss , so we have complete kinematic information on initial momentum and energy of struck proton (this is unfortunately spoiled in the real world due to final state interactions - the proton does not escape the nucleus completely unperturbed).
• E_miss spectra show which "shell" the proton got knocked out of (sharp peaks for uppermost occupied shells, broad bumps for lower shells)
• Broadening due to correlations within nuclei: The total A-nucleon wave function is NOT a simple product of singe-nucleon energy eigenstates, but a superposition.
• Cut on different shells, study momentum distribution for protons in those shells. Findings: s-shell has maximum likelihood for zero momentum, while p-shell has zero likelihood for zero momentum (because of parity symmetry - wave function inverts sign under reflections).

Inelastic scattering

• Range of possible final states (ground state, excited states/resonances, break-up states) leading to range of possible final state energies for scattered electron => cross section ds/dW/dE'
• Isolated resonances: ds/dW/dE' = ds/dW*Sum[d(E'-E_f(R))] . E_f(R) is the final state energy of the electron, calculated using 4-momentum conservation, for the proper specific value of the resonance energy for a given resonance R.
• In reality: distribution can be broader if resonance has finite lifetime and therefore (Uncertainty principle) finite width.
• New kinematic variable: W = invariant mass (energy) of final state of the struck object (nucleon, nucleus) in its own center-of-mass frame.
• Formula: W² = m_A² + 2*m_A*n - Q² (if struck object had mass m_A)
• Elastic scattering ->  W = m_A -> n = Q²/2m_A
• Higher excitations -> larger W
• Quasi-elastic scattering -> full range of E' (or W) centered on (but slightly shifted from) elastic position for nucleon: Q²/2m_p
• New cross section formula: ds/dW/dE' = s_Mott[W₂(n,Q²) + 2tan²q/2 W₁(n,Q²)] (no recoil factor - included in W₂ and  W₁)
• Structure function W₁ contains magnetic/transverse excitation strength (think of wiggling a magnet with another magnet)
• Structure function W₂ contains both transverse and longitudinal (Coulomb) excitation strength

Inelastic scattering on the nucleon

• Same cross section formula
• W² = m_p² + 2*m_p*n - Q²
• W₂ and  W₁ are delta-functions at W = m_p and zero up to pion production threshold (no low-lying excitations known)
• At W = m_p + m_p (pion threshold; m_p = 0.14 GeV), W₂ and  W₁ begin to rise.
• Maxima at several resonance peaks (3 clearly visible in electron scattering)
• At even higher energy transfer (W > 2 GeV), smooth "scaling" curve ("quasielastic scattering from quarks")
• In the scaling region (W > 2 GeV, Q² > 1 GeV²), structure functions become functions of one variable only:
• W₁(n,Q²) = 1/m_p F₁(x)
• W₂(n,Q²) = 1/n F₂(x)
• x = Q²/2m_pn (Bjorken scaling variable)