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4.4. Green’s function solution for elastic scattering
In the case where m 1 « m 2 , the CM moves slowly in the lab
frame. Also, M ≈ m 1 . In this limit the incident particle transfers
a small fraction of its energy to the target particle. The scattering
can therefore be regarded as elastic in the lab frame, as well as
in the CM frame. This is a fair approximation for a fast electron
incident on an atomic nucleus, for example.
In the following analysis, we will calculate the differential crosssection σ(ϑ) for the equivalent one-body scattering in the CM
frame. The geometry of the scattering is shown for the equivalent
one-body problem in Figure 4.5. We assume a plane wave incident from the left, with wave vector k 0 oriented along the positive
z-axis. A scattering center is located at O, and an observation
point at P , at position x. A spherical wave with wave vector k
emanates from the scattering center O. The polar scattering angle
between the incident and scattered wave vectors k 0 and k is ϑ.
We define the normalized incident wave function u 0 (x) as the plane
wave
1 ik 0 ·x
u 0 (x) = √ e
.
(4.72)
V
We assume the observation point P is located far from the scattering center. As such, the scattered wave u(x) can be approximated
by a spherical wave,
ikr
1 e
u(x) = f (k 0 , k) √
,
(4.73)
V r
where we define r ≡ |x|, and the factor f (k 0 , k) is called the scattering amplitude.
Strictly, the incident plane wave has infinite extent. However, in
practice the incident beam is typically collimated, so that the incident and scattered waves do not interfere at the observation point
P . We write the incident flux S 0 and the scattered flux S, respectively as
S 0 = |u 0 (x)|
2 v 0 ,
S = |u(x)|
2 v
(4.74)
4.4. Green’s function solution for elastic scattering
In the case where m 1 « m 2 , the CM moves slowly in the lab
frame. Also, M ≈ m 1 . In this limit the incident particle transfers
a small fraction of its energy to the target particle. The scattering
can therefore be regarded as elastic in the lab frame, as well as
in the CM frame. This is a fair approximation for a fast electron
incident on an atomic nucleus, for example.
In the following analysis, we will calculate the differential crosssection σ(ϑ) for the equivalent one-body scattering in the CM
frame. The geometry of the scattering is shown for the equivalent
one-body problem in Figure 4.5. We assume a plane wave incident from the left, with wave vector k 0 oriented along the positive
z-axis. A scattering center is located at O, and an observation
point at P , at position x. A spherical wave with wave vector k
emanates from the scattering center O. The polar scattering angle
between the incident and scattered wave vectors k 0 and k is ϑ.
We define the normalized incident wave function u 0 (x) as the plane
wave
1 ik 0 ·x
u 0 (x) = √ e
.
(4.72)
V
We assume the observation point P is located far from the scattering center. As such, the scattered wave u(x) can be approximated
by a spherical wave,
ikr
1 e
u(x) = f (k 0 , k) √
,
(4.73)
V r
where we define r ≡ |x|, and the factor f (k 0 , k) is called the scattering amplitude.
Strictly, the incident plane wave has infinite extent. However, in
practice the incident beam is typically collimated, so that the incident and scattered waves do not interfere at the observation point
P . We write the incident flux S 0 and the scattered flux S, respectively as
S 0 = |u 0 (x)|
2 v 0 ,
S = |u(x)|
2 v
(4.74)