PENELOPE 2003 - OECD Nuclear Energy Agency

94 Chapter 3. Electron and positron interactions
and
J (−)
1 = ln W + E
E − W
+ (2 − a) ln(E − W ) +
aW
2
2E 2 (3.92)
J (−)
2 = (2 − a)W + 2E2 − W 2
3
aW
+ (3 − a)E ln(E − W ) +
E − W 3E . (3.93)
2
For positrons, the integrals in (3.89),
J (+)
n
∫
≡
[
( )
W n−2 W W 2 ( ) W 3 ( ) W 4
]
1 − b 1
E + b 2 − b 3 + b 4 dW, (3.94)
E E E
can also be evaluated analytically as
and
J (+)
0 = − 1 W − b ln W
1
E + b W
2
E − b W 2
2 3
2E + b W 3
3 4
3E , (3.95)
4
W 2
J (+)
W
1 = ln W − b 1
E + b 2
2E − b 2 3
3E + b 3 4
(3.96)
4E 4
W 3
J (+) W 2
2 = W − b 1
2E + b 2
3E − b 2 3
4E + b 3 4
5E . (3.97)
4
Fig. 3.9 displays total inelastic cross sections for electrons in aluminium and gold,
as well as contributions from various groups of shells, as functions of the kinetic energy
of the projectile. The curves labelled “K” and “L1+. . . ” represent cross sections for
ionization in these shells. The cross section for ionization in a bound shell decreases
rapidly with the shell ionization energy U i (since energy transfers less than U i , which
would promote the target electron to occupied states, are forbidden). As a consequence,
collisions occur preferentially with electrons in the conduction band and in outer bound
shells. Inner-shell ionization by electron/positron impact is a relatively unlikely process.
It should be noted that our GOS model is too crude to provide an accurate description
of inner-shell ionization. To illustrate this, fig. 3.9 includes K- and L-shell ionization
cross sections obtained from the optical-data model described in section 3.2.6, which are
known to agree reasonably well with experimental data (Mayol and Salvat, 1990). We
see that there are significant differences between the cross sections from the optical-data
model and the predictions of our simple GOS model, which is designed to yield accurate
stopping powers only. To get a realistic picture of inner-shell ionization, we have to rely
on much more elaborate physical schemes. In fact, even the Born approximation ceases
to be appropriate for projectiles with kinetic energies near the ionization threshold.
Collision stopping powers for electrons in aluminium, silver and gold obtained from
the present analytical model are compared with sample values from the ICRU37 (1984)
stopping power tables [given also in Berger and Seltzer (1982)] for E ≥ 10 keV in fig.
3.10. Our results practically coincide with the values in the tables of reference. In fig.
3.11, inelastic mean free paths and stopping powers for low-energy electrons (E = 100
eV to 100 keV) in aluminium and gold obtained from the present model are compared
with experimental data from several authors. We see that the theory predicts the
W 3
W 4
W 4
W 5