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 gNucleus
= gl +/- (gs - gl)/(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 (Dki)
= density of target atoms/unit area (z*rho*NA/A)
* cross section (Ds)
- Alternative definitions: Count rate dN/dt = L*Ds
(L = luminosity = Ibeam/e * z*rho*NA/A)
or for a single target particle dN/dt = jin*Ds
- Fully differential cross section: limit of Ds(Dki)/P(Dki)
- 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/kinkout*ds/dW
(q = kin-kout = momentum
transferred).
See slides to further elucidate scattering:
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 |Mfi|2 DFd(Ef
- E')
- Directly gives decay rate
- Cross section : ds = dP(i->f)/dt /
jin
- Matrix element Mfi = <pwf |H - H0|
pwi>
- 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 Mfi can be calculated from
Feynman
diagrams:Dirac
spinors for external legs, gamma matrices times interaction strength (e
= a1/2) 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
See above. GeV vs. MeV, mp = 1+kp
= 2.79, mn
= kn
= -1.91.
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
= [mA-1 + Eexitation + Tkin(A-1) + mp
- mA] + p2/2mp + q2/2mp
+ p.q/mp
- The first term is the missing energy Emiss (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 Q2/2mp).
- 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 Q2/2mp can be
interpreted
as average missing energy Emiss. 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 Eexitation in the formula for n
(see last lecture)
- Can solve for Eexitation or Emiss , 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).
- Emiss 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'-Ef(R))]
. Ef(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 = mA2 + 2*mA*n
- Q2 (if struck object had mass mA)
- Elastic scattering -> W = mA -> n
= Q2/2mA
- Higher excitations -> larger W
- Quasi-elastic scattering -> full range of E' (or W) centered
on (but
slightly
shifted from) elastic position for nucleon: Q2/2mp
- New cross section formula: ds/dW/dE'
= sMott[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 = mp2 + 2*mp*n
- Q2
- W2 and W1 are delta-functions at W = mp
and zero up to pion production threshold (no low-lying excitations
known)
- At W = mp + mp (pion
threshold;
mp =
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/mp
F1(x)
- W2(n,Q2)
= 1/n
F2(x)
- x = Q2/2mpn (Bjorken
scaling
variable)
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