Outline of Topics discussed in the Lecture for Graduate Nuclear
Physics
(PHYS722/822 Fall 2003)
Nucleon Form Factors
Electric: GE , normalized to 1 at Q2 = 0. Magnetic: GM, normalized to µ at Q2 = 0 (µ = magnetic moment in units of nuclear magnetic moment µN).
µp = 1+kp
= 2.79, µn
= kn
= -1.91.
Form factor shape vs. Q2 roughly described by dipole parameterization.
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.
- Relativistic
(=correct) approach: Use 4-vectors for incoming electron, outgoing
electron, and 4-momentum exchanged (= 4-momentum of virtual photon q = (n,q)). Initial 4-momentum of target at rest in lab (use whole nucleus!) P = (MA,0,0,0). Final state has 2 particles (other than electron): struck nucleon (proton) with momentum P1 and residual A-1 nucleus with PA-1. Write down 4-momentum conservation for initial (q + P) = final (P1 + PA-1) momenta.
- 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
formulae.
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)
(Deep) inelastic scattering
- Formalism, variable transformation to DQ2, Dn
- W2 and W1 as functions of invariant
final
state
energy W: Elastic peak (delta-function), zero up to pion threshold
(1078
MeV), meson production, resonances, deep inelastic regime
- Production of pions (p -> n + p+
, p -> p + p0 , n -> p + p-
, n -> n + p0 )
- Pions have I=1 (I3 = -1,0,1 for p+
, p0 , p-
), mass 133-139 MeV, spin 0, parity -. Lowest mass example of a class
of
subatomic particles called "mesons". All mesons have integer spin and
interact
strongly with protons and neutrons. Other examples: Kaons (K+,
K-, K0L, K0S),
etas
(h, h'), rhos (r),
omegas (w),
.... (all the way throught the
Greek
alphabeth).
- Protons in initial state can also be converted into other
particles in
the final state: neutrons, lambdas (L),
sigmas
(S),... (upper
case Greek letters), in
accordance
with conserved quantum numbers. These particles (including the proton
and
its resonances) are called "baryons". All baryons have HALF-integer
spins
(1/2, 3/2, ...) and interact strongly with protons. Baryons are
typically
heavier than mesons, but there are some pretty heavy mesons, too.
- Resonant states are excited on top of "smooth" particle
production
spectrum:
The lowest lying one is the Delta (D), which
is a baryon with spin 3/2 and 4 different charge states: -,0,+,++
(isospin
3/2), and mass 1232 MeV. Higher lying states (P11, S11,
D13, F15, ...) are broad and overlapping, leading
to only 2 distinctive additional bumps.
- All resonance have "form factors" that fall off like the
elastic proton
form factors (leading to a decrease of W2 and W1
as functions of Q2 at fixed W).
- Deep inelastic scattering - smooth curve (sum of all resonance,
multiple
meson and baryon productions at W>2, independent of Q2
at Q2 >1...3 GeV2)
- Better interpreted as quasi-elastic scattering from point-like
dirac
particles
inside the proton (quarks)
- Interpretation: x = Q2/2mn is the
fraction of the proton momentum carried by the struck quark, as
measured
in the Breit frame (where the virtual photon carries no energy)
- F1(x) = 1/2 zq2 Pq(x),
summed
over all different kinds of quarks. Pq(x) is the probability
of finding a quark of type q carrying momentum fraction x.
- F2(x) = 2x F1(x) (Callan-Gross
relationship).
- For more details, see my writeup for the HUGS lecture series I
gave in
1997.
Return to PHYS722/822 Home
page