(PHYS722/822 Fall 2003)

- 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).

- 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.

- 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/d
**q**^{2}= p/k_{in}k_{out}*ds/dW (**q**=**k**_{in}-**k**_{out}= momentum transferred).

- 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 - H_{0}| pw_{i}> - DF = V*D
^{3}p / h^{6}

- 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/(M^{2}+Q^{2}) 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(q
^{2}) (Fourier transform of charge density distribution). - Important properties: F(q
^{2}) = 1 if wave length is much larger than nuclear size; next-order approximation is 1-<r^{2}>q^{2}/6 -> rms charger radius. In general can extract charge distribution. - Necessary improvements for high-energy electron scattering:
replace q
^{2}with Q^{2}; electron spin contribution cos^{2}(q/2) => Mott cross section; recoil factor E'/E => Point cross section. Electric and magnetic form factors (Ge and Gm). See homework and references.

- 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}] +**p**^{2}/2m_{p}+ q^{2}/2m_{p}+**p**/m^{.}q_{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 Q
^{2}/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 Q^{2}/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.

- 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).

- 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: W
^{2}= m_{A}^{2}+ 2*m_{A}*n - Q^{2}(if struck object had mass m_{A}) - Elastic scattering -> W = m
_{A}-> n = Q^{2}/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**: Q^{2}/2m_{p} - New cross section formula: ds/dW/dE'
= s
_{Mott}[W_{2}(n,Q^{2}) + 2tan^{2}q/2 W_{1}(n,Q^{2})] (no recoil factor - included in W_{2}and W_{1}) - Structure function W
_{1}contains magnetic/transverse excitation strength (think of wiggling a magnet with another magnet) - Structure function W
_{2}contains both transverse and longitudinal (Coulomb) excitation strength

- Same cross section formula
- W
^{2}= m_{p}^{2}+ 2*m_{p}*n - Q^{2} - W
_{2}and W_{1}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), W_{2}and W_{1}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, Q
^{2}> 1 GeV^{2}), structure functions become functions of one variable only: - W
_{1}(n,Q^{2}) = 1/m_{p}F_{1}(x) - W
_{2}(n,Q^{2}) = 1/n F_{2}(x) - x = Q
^{2}/2m_{p}n (Bjorken scaling variable)