Introduction to Quantum Ideas
41
I
0 2.4
( – ) Å 10
l l
´
0
2
(a)
hn 0
hg
q
f
p
(b)
Fig. 2.3 (a) The intensity of the scattered x-rays as a function of (λ – λ 0 ) Å × 10
2
, for
scattering angle θ = 90°. (b) Compton scattering as particle-particle scattering.
If the initial and final photons have energies hv 0 and hv, and momenta
0
hv
c
n 0
and
hv
c
n, respectively, where n 0 and n are unit vectors in the directions of
propagation and the momentum of scattered electron is p, then from momentum
and energy conservation [Fig. 2.3 (b)],
0
cos
cos
ν
ν
=
θ+
φ
h
h
p
c
c
(2.28)
0
sin
sin
ν
=
θ−
φ
h
p
c
(2.29)
hv 0 + mc
2
= hν + (p
2
c
2
+ m
2
c
4
)
1/2
(2.30)
Eliminating φ from the first two equations,
2
2
2
2
0
0
2
(
2
c o s )
h
p
c
=
ν +ν − νν
θ
(2.31)
Also from Eq. (2.30)
2
2
2
2
2 2
0
2
(
/ )
h
p
m c h
mc
c
=
ν −ν+
−
(2.32)
Equating the two expressions for p
2
leads to
0
0
2
(1 cos )
ν ν
ν − ν =
−
θ
h
mc
(2.33)