PENELOPE 2003 - OECD Nuclear Energy Agency

72 Chapter 3. Electron and positron interactions
are the direct and spin-flip scattering amplitudes, respectively.
k ≡ p¯h = 1 [
E(E + 2me c 2 ) ] 1/2
¯hc
(3.7)
is the wave number of the projectile, P l (cos θ) are Legendre polynomials, Pl 1 (cos θ) are
associated Legendre functions and δ l± are the phase shifts. These are determined from
the asymptotic behaviour of the Dirac radial functions for large r (Walker, 1971). Thus,
to determine each phase shift we must solve the radial Dirac equations for the potential
V (r). The convergence of the partial-wave series (3.6) slows down when the energy of the
projectile increases. This makes the calculation difficult for energies larger than a few
MeV (in the case of scattering by gold atoms, about 10,000 phase shifts are required at
E = 10 MeV). The partial-wave DCS, eq. (3.5), rigourously accounts for spin and other
relativistic effects, as well as finite nuclear size effects. A computer code to calculate
elastic scattering cross sections by this method has been written by Salvat (2000).
Single elastic collisions are determined by the values of the polar and azimuthal
scattering angles, θ and φ, respectively. Owing to the assumed spherical symmetry of
the scattering centres, single and multiple scattering angular distributions are axially
symmetrical about the direction of incidence, i.e. they are independent of the azimuthal
scattering angle φ. For simulation purposes, it is convenient to measure polar angular
deflections produced by single scattering events in terms of the variable [see eq. (1.61)]
µ = (1 − cos θ)/2 (3.8)
instead of the scattering angle θ. Notice that µ varies from 0 (forward scattering) to 1
(backward scattering). The DCS per unit angular deflection is
dσ el
dµ = 4π dσ el
dΩ . (3.9)
Fig. 3.2 displays DCSs for elastic scattering of electrons and positrons of various energies
by aluminium and gold atoms. These numerical results illustrate the variation
of the DCS with the atomic number Z, the charge of the projectile and the energy E.
Since the interaction V (r) is attractive for electrons and repulsive for positrons, the
scattering is more intense for electrons (which can fall deeply into the potential well of
the atom) than for positrons (which are repelled from the nucleus and cannot “see” the
inner part of the atom). The DCS for low-energy electrons exhibits a diffraction-like
structure, while the DCS for positrons decreases monotonously with the deflection µ.
The Born approximation (see e.g. Mott and Massey, 1965) predicts a structureless DCS
that decreases with µ and is proportional to the squared charge of the projectile (i.e. the
same for electrons and positrons). This approximation considers the scattering field as a
perturbation (to first order) and, hence, it is valid only for weak fields (low-Z elements).
The difference between the (partial wave) DCSs for electrons and positrons gives a clear
indication of the applicability of the Born approximation.
The total elastic cross section is given by
∫
∫ dσel
σ el =
dΩ dΩ =
dσel
dµ
dµ. (3.10)