Page 324 - Biomedical Engineering and Design Handbook Volume 2, Applications

302 DIAGNOSTIC EQUIPMENT DESIGN

This is a very inefficient process with most of the energy appearing as heat. As a consequence, the
temperature of the target material can reach melting point. The liquid metal is subsequently removed
by electron beam pressure, resulting in the excavation of a pit or groove. This degradation of the tar-
get material creates an enlarged focal spot. Therefore, it is important to determine the maximum
power density that a target material can safely accommodate in order to produce maximum x-ray
flux to satisfy the threshold of detection condition [Eq. (10.91)].
For the large-focal-diameter electron beams
associated with medical x-ray sources, the heat
dissipation is considered to occur at the surface
and the analytical geometry is that of a disk-
heating model. 28 However, this model is inaccu-
rate for focal diameters of small extent where the
volume dissipation of heat is an important factor.
In this respect the penetration of the beam elec-
trons into the target material must be taken into
account (Fig. 10.37). Furthermore, the extent of
the electron distribution will contribute to the
enlargement of the focal spot, since it represents
FIGURE 10.37 Heat transfer models.
the volume of x-ray generation within the bulk of
the material. For example, at 50 keV the electron
penetration depth is ~10 mm which is a significant amount in comparison with a typical microfocal
electron beam focal diameter of ~5 mm.
A simple model for the electron motion in the target is to assume that the electrons travel in
straight lines to a depth of complete diffusion, after which they diffuse evenly in all directions, to
29
cover a total distance called the range. Along the range an electron is assumed to loose energy
according to the Thomson-Whiddington law, written as
2
2
V − V = kx (10.113)
0
where V is the energy after the electron travels a distance x and the constant k is given by
ρ
bZ
k = (10.114)
A
where r, Z, and A are density, atomic number, and
atomic weight, respectively. The parameter b varies
slightly with the type of target material but may be
assumed to have the constant value b = 7.75 ×
2
2
10 10 eV ⋅ m /kg. From Eq. (10.113), the electron
range x is given by
0
V 2
x = 0 (10.115)
0
k
29
and the depth of diffusion may be taken to be
40 x
z = 0 (10.116)
d
7 Z
Hence, the diffusion center is nearer to the surface
the larger the atomic number of the target material
(Fig. 10.38).
The volume in which heat is dissipated is the
sphere of radius x − z centered at D, which is the
0
d
center of complete diffusion. The shaded region
FIGURE 10.38 Model of electron penetration. contains the backscattered beam power. For a small