wherec.c.stands for the complex conjugate of the preceding term. In carrying out the∇derivative
with respect to the variablex, several terms emerge. In view of the derivatives
∇i(
1
r)
=−
xi
r^3∇i(
xj
r)
=
r^2 δji−xjxi
r^3(12.42)
we see that the∇derivatives of the 1/rfactor and of the outgoing momentumk′=kx/rwill lead
to terms behaving like 1/r^3 injwhich will not contribute to the differential fluxdφ/dΩ since this
involves taking the largerlimit. The only term that contributes is thus obtained by differentiating
the exponential factor, which gives,
jscattered∼k′
mr^2|f(k′,k)|^2
(2π)^3(12.43)
The scattered differential flux is then found to be
dφ
dΩ=
|v|
(2π)^3
|f(k′,k)|^2 (12.44)The differential cross section is defined as the ratio of the differential flux by the total current
density. The factors of(2|vπ|) 3 cancel in both quantities and we are left with
dσ
dΩ
=|f(k′,k)|^2 (12.45)Note that, becausef has dimensions of length, the differential cross section has dimensions of
length square. The physical interpretation is that the cross section represents an effective aspect
area for the target. This interpretation will be borne out insome simple examples that we shall
study. The total cross section of the process is the integralof the differential cross section over the
full range 4πof solid angle,
σ=∫4 πdΩdσ
dΩ(12.46)
It again has dimension of length square.
A final remark is in order. In evaluating the incoming and outgoing fluxes, we seem to have
neglected the cross term between the incoming wave and the outgoing one. While this term would
contribute to the current density at any finite distancex, in the limit of larger, its oscillatory
behavior, given byeik·x−ikr, is responsible for the vanishing of its contribution to thedifferential
flux.
12.8 The Born approximation
The Born approximation consists in retaining only a finite number of terms in the expansion in
powers ofλ. The first Born term is the one linear inλ, i.e. linear in the potentialV. It is obtained
by settingψ±k(x) =φk(x) in the expression forf(k′,k) in (12.31), which yields,
f(1)(k′,k) =−1
4 π∫
d^3 ye−i(k
′−k)·y
U(y) (12.47)