Page 292 - Biomedical Engineering and Design Handbook Volume 2, Applications

270 DIAGNOSTIC EQUIPMENT DESIGN

FIGURE 10.2 Photons scattered into solid angle dΩ.

t
written as P = dΦ/Φ= n s dx, where n is the number density of material atomic or electron par-
t p p
ticles. Hence, the number of photons dN /dV removed per unit volume for event t is given by
s
dN s =− dΦ = Φ n σ τ (10.1)
dV dx p

The angular distribution of photons for scattering events is determined by considering the number of
photons scattered at an angle q into an elemental solid angle dΩ. This leads to the concept of the dif-
ferential collision cross section (ds/dΩ) , which can be defined by differentiating Eq. (10.1) with
q
respect to dΩ to give
2
dN s = Φ ⎛ dσ τ ⎞
dV dΩ n p ⎜ ⎝ dΩ ⎠ ⎟ (10.2)
θ

Here, ds is the probability that the incident photon will be deflected into the elemental solid angle
2
dΩ, so that d N is the number of photons scattered into dΩ from volume element dV, for incidence
s
flux Φ (Fig. 10.2).
If the radiation is unpolarized, the differential collision cross section depends only on the angle
of deflection q, so the cross section s can be determined by integrating (ds/dΩ) over a hemisphere,
q
according to
π d σ ⎞
⎛
σ = ∫ 2sin θ θ (10.3)
π
d
⎠
⎝ dΩ
0
θ
t
Since we may interpret the cross section s as the probability that a photon will be removed from the
T
primary beam, for a single t event collision site per unit area, the total cross section s for all events
can be expressed as
T
s =∑ s t (10.4)
t
t
where the s are now considered as components to the overall attenuation process.
10.2.2 Principal Photon Interactions
The principal collision processes observed for x-rays, in the low- to medium-energy range 0 to
200 keV, are photoelectric absorption and Compton scattering. Absorption losses refer to events that
remove the incident photon from the primary beam by the complete conversion of its energy.
Scattering losses refer to events that remove the incident photon from the primary beam by a