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10 Chapter 2. Reaction Observables and Kinematics
for longitudinally polarized electron beams, but are suppressed by a factor m e /ɛ.
Accordingly, these terms can be safely ignored. The various R K fi are the nuclear
response functions which contain all of the nuclear structure and dynamics information;
the factor v 0 ≡ (ɛ+ɛ ′ ) 2 −q 2 and the V K are electron kinematics and polarization
factors. In the ERL for the electrons, it can be shown [25, 26] that the differential
cross section for the scattering of longitudinally polarized electrons from nuclei is
given by
(
d 5 ) h
σ
dɛ ′ dΩ e ′dΩ f
fi
= MM A−1k f
8π 3 f
M
recσ −1 M
[(v L R L + v T R T + v T T R T T + v T L R T L )
A
]
+ h (v T ′R T ′ + v T L ′R T L ′) , (2.13)
where σ M is the well known Mott cross section
( ) 2
α cos θe /2
σ M =
2ɛ sin 2 , (2.14)
θ e /2
with θ e the angle between the incident and the scattered electron. The electron
kinematics is contained in the kinematical factors
( )
Q
2 2
v L =
q 2 , (2.15)
v T = 1 ( )
Q
2
2 q 2 + tan 2 θ e
2 , (2.16)
v T T = − 1 ( )
Q
2
2 q 2 , (2.17)
v T L = −√ 1 ( ) √
Q
2 (Q
2 )
2 q 2 q 2
v T ′ = tan θ e
2
+ tan 2 θ e
2 , (2.18)
√ (Q
2
q 2 )
+ tan 2 θ e
2 , (2.19)
v T L ′ = − 1 √
2
(
Q
2
q 2 )
tan θ e
2 . (2.20)
As for the contraction of the nuclear tensor with the electron tensor, we have that
J 0 (⃗q) fi = ρ(⃗q) fi , the Fourier transform of the transition charge density < f|ˆρ(⃗r)|i >,
while
⃗J(⃗q) fi =
∑
J(⃗q; m) fi ⃗e ⋆ (⃗q; 1, m) , (2.21)
m=0,±1