Electron Scattering

1
Electron Scattering

• Electrons as probe of nuclear structure have some
  distinct advantages over other probes like
  hadrons or g-rays
• The interaction between the electron and the
  nucleus is known it is the electromagnetic
  interaction with the charge r and the current J
  of the nucleus Vint rf J.A, where f and A
  are the scalar and vector potentials generated by
  the electron.
• The interaction is weak, so that in almost all
  cases it can be treated in the one-photon
  exchange approximation (OPEA), i.e., two-step
  processes (two-photon exchange) are small. The
  exception is charge elastic scattering of the
  Coulomb field of a heavy-Z nucleus.
• The energy (w) and linear momentum (q)
  transferred to the nucleus in the scattering
  process can be varied independently from each
  other. This is very important, as for a certain
  q one effectively measures a Fourier component
  of r or J. By varying q all Fourier components
  can be determined and from these the radial
  dependence of r and J can be reconstructed.
• Because the photon has no charge, only the J.A
  interaction plays a role, leading to magnetic Ml
  and electric El transitions. In electron
  scattering, one can also have charge Cl
  transitions.

2
Electron Scattering

• Electron scattering has also some disadvantages
• The interaction is weak, so cross sections are
  small, but one can use high electron beam
  currents and thick targets.
• Neutrons are less accessible than protons, since
  they do not have a net electric charge.
• Because electrons are very light particles, they
  easily emit radiation (so-called Bremsstrahlung).
  This gives rise to radiative tails, with often
  large corrections for these processes.

3
k (E,k)
Kinematics
k (E,k)
Q
Virtual photon ? off-mass shell qmqm n2 q2 0
q (n,q)

• Define two invariants
• Q2 -qmqm -(km km)(km-km)
• -2me2 2kmkm
• (me 0) 2kmkm
• (LAB) 2(EE k.k)
• 2EE(1-cos(Q))
• 4EEsin2(Q/2)
• only assumption neglecting me2!!
• 2) 2Mn 2pmqm Q2 W2 M2

p (M,0)
Undetected final state X
p (sqrt(p2 M2),p)
Elastic scattering ? W2 M2 ? Q2 2M(E-E)
(? p2 M2 W2) !!
4
Electron Scattering at Fixed Q 2
Elastic
Quark (fictitious)
?
Proton
Elastic
Deep Inelastic
?
N
?
5
Electron Scattering at Fixed Q 2
Elastic
Nucleus
Deep Inelastic
?
Quasielastic
N
?
Proton
Elastic
Deep Inelastic
?
N
?
6
Extracting the (e,e) cross section
e'
NN (cm-2)
Ne
e
(??e, ?pe)
Scattering probability or cross section
7
Form Factor
r
k
k'
Amplitude at q
8
Electron scattering off composite target
Recall matrix element ltYOYgt Define
transition matrix element Tfi
Tfi lt f T i gt
which separates into two integrals and gives the
well-known result
where F(q) is the form factor
9
(No Transcript)
10
Form Factors of Nuclei at low energy
Elastic scattering ? W2 M2 ? Q2 2M(E-E) ? Q2
and n are correlated
ds/dW (and not ds/dWdE) sMF02(q)

• For a point charge with charge Z one has F0(q)
  Z.
• For a charge with a finite size F0(q) will be
  smaller than Z, because different parts of r(r)
  will give destructive contributions in the
  integral that constitutes F0(q).
• Often one includes the factor Z in sM and not in
  F0, such that F0(0) 1.

Scatter from uniform sphere with radius R at low
q sin(qr) qr (1/6)(qr)3
1st term disappears (charge normalization) 2nd
term gives direct RRMS measurement (for q low
enough) At higher q pattern looks like slit
scattering with radius R
11
History - Charge Distributions
In 70s large data set was acquired on elastic
electron scattering (mainly from Saclay) over
large Q2-range and for variety of
nuclei Model-independent analysis provided
accurate results on charge distribution well
described by mean-field Density-Dependent
Hartree-Fock calculations
12
Electron Scattering off point particles
k (E,k)
k (E,k)
1
3
2
4
p (M,0)
p
Discuss handout of 10 pages on formalism here
13
Elastic Scattering from a Proton at Rest
(m,0)
(?,q)
Before
After
Proton is on-shell ?
(? m)2 ? q2 m2 ?2 2m? m2 ? q2 m2 ?
Q2 ? 2m
14
Scattering from a Proton , contd.



structure/anomalous moment
15
Scattering from a Proton , contd.
Vertex fcn
Dirac FF
Pauli FF
Sachs FFs
GE and GM are the Fourier transforms of the
charge and magnetization densities in the Breit
frame.
16
Cross section for ep elastic
17
Proton charge and magnetism in 2006
2-g exchange important
Hall A
smaller distance ?
charge depletion in interior of proton
Orbital motion of quarks play a key role
(Belitsky, Ji Yuan PRL 91 (2003) 092003)
18
What about the neutron?
Neutron has no charge, but does have a charge
distributions n p p-, n ddu. Use
polarization and 2H(e,en) to access. Guarantee
that electron hits a neutron AND electron
transfers its polarization to this neutron.
charge and magnetization density
J. J. Kelly, PRC 66 (2002) 065203
19
JLab Data Reveal Deuterons Size and Shape
Combined Data -gt Deuterons Intrinsic Shape
The nucleon-based description works down to lt 0.5
fm
20
(No Transcript)
21
Elastic Scattering from a Moving Proton
Before
(? E)2 (qp)2 m2 ?2 2E? E2 ? q2 ?2pq
? p2 m2 Q2 2E? ?2pq ? (E/m) (Q2 ? 2m)
pq ? m
22
Quasielastic Scattering
For E ? m ? ? (Q2 ? 2m) pq ? m
If we quasielastically scatter from nucleons
within nucleus
Expect peak at ? ? (Q2 ?
2m) Broadened by Fermi motion pq ? m
23
Quasielastic Electron Scattering
R.R. Whitney et al., Phys. Rev. C 9, 2230 (1974).
24
A(e,e'p)B
p
e'
scattering plane
?pq
reaction plane
(?,q)
e
pA1
?x
out-of-plane angle
In ERLe Q2 ? q?q ? q2 ? 2 4ee'
sin2?/2 Known e and A Detect e and p ?
Infer Missing momentum pm q p
pA1 Missing mass ?m ? Tp
TA1
25
accidental (uncorrelated)
e
e'
real (correlated)
p
e
26
events
?r
?a
relative time te tp
27
Accidentals Rate Re ? Rp ? ??/DF ? I 2 ??/DF
Reals Rate Reep ? I
SN Reals/Accidentals ? DF /(???I)
Compromise Optimize SN and Reep
28
Extracting the (e,ep) cross section
e'
NN (cm-2)
Ne
e
(??e, ?pe)
(??p, ?pp)
p
29
Cross section for ep elastic
However, (e,e'p) on a nucleus involves scattering
from moving protons, i.e. Fermi motion.
30
Cross Section for A(e,e'p)B in OPEA
A-1
where
Current-Current Interaction
31
Square of Matrix Element
???
W??
32
Cross Section in terms of Tensors
Mott cross section
Electron tensor
Nuclear tensor
33
Consider Unpolarized Case Lorentz Vectors/Scalars
34
Nuclear Response Tensor
Xi are the response functions
35
Impose Current Conservation
Get 6 equations in 10 unknowns 4 independent
response functions
36
Putting it all together
37
The Response Functions
Use spherical basis with z-axis along q
38
Response functions depend on scalar quantities
Note no ?x dependence in response functions
39
Including electron and recoil proton polarizations
40
Extracting Response Functions For instance RLT
and A? (A LT)
41
Cross section for ep elastic scattering off
moving protons
Follow same procedure as for unpolarized (e,e'p)
from nucleus
We get same form for cross section, with 4
response functions
42
Plane Wave Impulse Approximation (PWIA)
spectator
A-1
q p pA-1 pm p0
43
The Spectral Function
In nonrelativistic PWIA
e-p cross section
nuclear spectral function
For bound state of recoil system
proton momentum distribution
44
The Spectral Function, contd.
Note S is not an observable!
45
Recall, in nonrelativistic PWIA
p
A1
FSI
where q p pm p0
e'
FSI destroys simple connection between the
measured pm and the proton initial momentum (not
an observable).
p0'
q
e
p0
A
46
Distorted Wave Impulse Approximation (DWIA)
Treat outgoing proton distorted waves in presence
of potential produced by residual nucleus
(optical potential).
Distorted spectral function
47
(e,e'p) advantages over (p,2p)

• Electron interaction relatively weak OPEA is
  reasonably accurate.
• Nucleus is very transparent to electrons Can
  probe deeply bound orbits.

However ejected proton is strongly interacting.
The cleanness of the electron probe is
somewhat sacrificed. FSI must be taken into
account.
48
At pm?160 MeV/c, wf is probed in nuclear interior.
J.W.A. den Herder, et al., Phys. Lett. B 184, 11
(1987).
49
Saclay Linac, France
12C(e,e'p)11B
J. Mougey et al., Nucl. Phys. A262, 461 (1976).
50
12C(e,e'p)11B
p-shell l1
Saclay Linac, France
s-shell l0
J. Mougey et al., Nucl. Phys. A262, 461 (1976).
51
12C(e,e'p)11B
NIKHEF-K Amsterdam
DWIA calculations fit data reasonably well.
Missing strength observed however.
G. van der Steenhoven, et al., Nucl. Phys. A480,
547 (1988).
52
History - Proton Knock-out (NIKHEF)
53
Correlated Strength in Nuclear Spectral Function
- History
Electron-induced proton knock-out has been
studied systematically since high duty-factor
electron beams became available, first at Saclay
(70s), then at NIKHEF (80s) with 100 keV
energy resolution. For complex (Agt4) nuclei the
spectroscopic strength S for valence protons was
found to be 60-65 of the IPSM value
Long-range correlations account for about 10,
but the rest was ascribed to short-range N-N
correlations, by which strength was distributed
at energies well above the Fermi edge
54

• Some of the lessons learned
• (e,e'p) sensitive probe of single-particle
  orbits.
• Proton distortions (FSI) must be accounted for
  to reproduce shape of spectral function.
• Missing strength in valence orbits, even after
  accounting for FSI

55
Where does the missing strength go?
One possibility
Detected
populates high ?m
recoils
56
History - Charge Distributions
Discrepancy??
In 70s large data set was acquired on elastic
electron scattering (mainly from Saclay) over
large Q2-range and for variety of
nuclei Model-independent analysis provided
accurate results on charge distribution well
described by mean-field Density-Dependent
Hartree-Fock calculations
57
Proton Momenta in the nucleus
Similar shapes for few-body nuclei and nuclear
matter at high k (pm).
Short-range repulsive core gives rise to high
proton momenta
0 200 400 600 800 1000
P (MeV/c)
58
3He(e,e'p)d
3He(e,e'p)np
3BBU similar to d?np
C. Marchand et al., Phys. Rev. Lett. 60, 1703
(1988).
59
Short Range Correlations in Nuclei
CLAS
Hall B
K. Egiyan, et al, (CLAS), PRC 68014313,2003
PRL 96,
082501,2006.
A(e,e)X, A 3He, 4He, 12C, 56Fe
Measured Composition ( )
Q2/2Mn
60
Are correlated nucleons indeed the cause for the
missing strength? Locate the strength ? JLab Hall
C
Strengths (in ) from CBF Theory
Correlated strength, integrated over shaded area
accessed by experiment (in of total strength)
Cuts
61
Summary - II

• (e,e'p) sensitive to single-particle aspects of
  nucleus, but
• More complicated physics is clearly important.
• Spectroscopic factors reduced compared to naïve
  shell model (including FSI corrections).
• Missing strength at least partly due to
  interaction currents direct interaction with
  with exchanged mesons or interaction with
  correlated pairs (spreads strength over ?m).