PENELOPE 2003 - OECD Nuclear Energy Agency

102 Chapter 3. Electron and positron interactions
(i) Sample κ from the PDF (3.118), as κ = κ c /[1 − ξ(1 − κ c )].
(ii) Generate a new random number ξ.
(iii) If ξ < κ 2 P (+)
k (κ), deliver κ.
(iv) Go to step (i).
The efficiency of this algorithm, for given values of the kinetic energy and the cutoff
reduced energy loss κ c , practically coincides with that of the algorithm for electron
collisions described above (see table 3.1).
Secondary electron emission
According to our GOS model, each oscillator W k corresponds to a shell with f k electrons
and ionization energy U k . After a hard collision with an inner-shell electron, the
primary electron/positron has kinetic energy E − W , the “secondary” electron (delta
ray) is ejected with kinetic energy E s = W −U i , and the residual ion is left in an excited
state, with a vacancy in shell i, which corresponds to an excitation energy equal to
U i . This energy is eventually released by emission of energetic x rays and Auger electrons.
However, in penelope the relaxation of ions produced in hard collisions is not
followed. The production of vacancies in inner shells and their relaxation is simulated
by an independent, more accurate, scheme (see section 3.2.6) that is free from the crude
approximations involved in our GOS model. To avoid double counting, the excitation
energy U i of the residual ion is deposited locally. On the other hand, when the impact
ionization occurs in an outer shell or in the conduction band, the initial energy of the
secondary electron is set equal to W and no fluorescent radiation from the ionized atom
is followed by the simulation program. This is equivalent to assuming that the secondary
electron carries away the excitation energy of the target atom.
To set the initial direction of the delta ray, we assume that the target electron was
initially at rest, i.e. the delta ray is emitted in the direction of the momentum transfer
q. This implies that the polar emission angle θ s (see fig. 3.1) coincides with the recoil
angle θ r [which is given by eq. (A.42)],
cos 2 θ s =
W 2 /β 2 (
1 + Q(Q + 2m ec 2 ) − W 2 ) 2
. (3.119)
Q(Q + 2m e c 2 ) 2W (E + m e c 2 )
In the case of close collisions (Q = W ), this expression simplifies to
cos θ s (Q = W ) =
( W
E
E + 2m e c 2
W + 2m e c 2 ) 1/2
, (3.120)
which agrees with the result for binary collisions with free electrons at rest, see eq.
(A.18). Since the momentum transfer lies on the scattering plane (i.e. on the plane
formed by the initial and final momenta of the projectile), the azimuthal emission angle
is φ s = π + φ.