Page 294 - Biomedical Engineering and Design Handbook Volume 2, Applications

272 DIAGNOSTIC EQUIPMENT DESIGN

FIGURE 10.4 Geometry of Compton scattering.

reduced energy hn ′, and the electron recoils through the angle f, with increased kinetic energy E
k
(Fig. 10.4). The fact that the electrons are free and not bound to an atom reduces the complexity of
the problem. As a consequence, the principles of conservation of momentum can be invoked to deter-
mine the division of energy between the colliding particles. For the conservation of energy we have
hn − hn′= E . For conservation of momentum we have in the horizontal direction
0 k
hv hv′
0
= cosφ + 2 mE cosθ (10.7a)
0
k
c c
and in the vertical direction
hv ′
0 = sinφ + 2mE sinθ (10.7b)
0
k
c
where m is the rest mass of the electron. Combining these relations, we obtain an expression for the
0
energy of the scattered photon in terms of the scattering angle:
hv
hv′ = 0 (10.8)
1 +α( 1 − cos θ)
and the angle of the scattered photon in terms of the electron recoil angle:
⎛ θ ⎞
cotφ = ( + α )tan (10.9)
1
⎝ ⎠ 2
2
2
where a = hn /m c and m c = 0.511 MeV. It is convenient to rewrite Eq. (10.8) in terms of wave-
0 0 0
length, so that
c − c = ′ − λ = h
λ
v′ v 0 mc 2 1 ( − cos θ) (10.10)
0
2
where h/m c = 2.246 × 10 −12 m is the Compton wavelength. This expression indicates that the
0
change in wavelength, for a given angle of scattering q, is independent of the incident photon energy.
The collision differential cross section is given by the Klein-Nishina relation:
r ⎡ ⎧ α 1( − cos θ) 2 ⎫ 1 + cos θ ⎤
2
2
2
dσ = e 1 ⎨ ⎨ ⎢ + 2 ⎬ 2 ⎥ dΩ (10.11)
2 ⎩ ⎣ 1 ( + cos θ 1)[ + ( 1 θ)] 1[ + (α 1 − cos )]θ ⎦
α − cos
⎭
2
2
where the classical electron radius r = (1/4p )(e /mc ). The scattered radiation is strongly peaked
0
e
in the forward direction, where the cross section falls with increasing angle q and falls off more