PENELOPE 2003 - OECD Nuclear Energy Agency

3.3. Bremsstrahlung emission 113
where p(Z, E, κ; cos θ) is the PDF of cos θ.
Numerical values of the “shape function” p(Z, E, κ; cos θ), calculated by partial-wave
methods, have been published by Kissel et al. (1983) for the following benchmark cases:
Z = 2, 8, 13, 47, 79, 92; E = 1, 5, 10, 50, 100, 500 keV and κ = 0, 0.6, 0.8, 0.95.
These authors also gave a parameterization of the shape function in terms of Legendre
polynomials. Unfortunately, their analytical form is not suited for random sampling
of the photon direction. In penelope we use a different parameterization that allows
the random sampling of cos θ in a simple way. Owing to the lack of numerical data for
positrons, it is assumed that the shape function for positrons is the same as for electrons.
In previous simulation studies of x-ray emission from solids bombarded by electron
beams (Acosta et al., 1998), the angular distribution of bremsstrahlung photons was
described by means of the semiempirical analytical formulae derived by Kirkpatrick
and Wiedmann (1945) [and subsequently modified by Statham (1976)]. These formulae
were obtained by fitting the bremsstrahlung DCS derived from Sommerfeld’s theory.
The shape function obtained from the Kirkpatrick-Wiedmann-Statham fit reads
p (KWS) (Z, E, κ; cos θ) = σ x(1 − cos 2 θ) + σ y (1 + cos 2 θ)
(1 − β cos θ) 2 , (3.147)
where the quantities σ x and σ y are independent of θ. Although this simple formula
predicts the global trends of the partial-wave shape functions of Kissel et al. (1983)
in certain energy and atomic number ranges, its accuracy is not sufficient for generalpurpose
simulations. In a preliminary analysis, we tried to improve this formula and
determined the parameters σ x and σ y by direct fitting to the numerical partial-wave
shape functions, but the improvement was not substantial. However, this analysis confirmed
that the analytical form (3.147) is flexible enough to approximate the “true”
(partial-wave) shape.
The analytical form (3.147) is plausible even for projectiles with relatively high
energies, say E larger than 1 MeV, for which the angular distribution of emitted photons
is peaked at forward directions. This can be understood by means of the following
classical argument (see e.g. Jackson, 1975). Assume that the incident electron is moving
in the direction of the z-axis of a reference frame K at rest with respect to the laboratory
frame. Let (θ ′ , φ ′ ) denote the polar and azimuthal angles of the direction of the emitted
photon in a reference frame K ′ that moves with the electron and whose axes are parallel
to those of K. In K ′ , we expect that the angular distribution of the emitted photons will
not depart much from the isotropic distribution. To be more specific, we consider the
following ansatz (modified dipole distribution) for the shape function in K ′ ,
p d (cos θ ′ ) = A 3 8 (1 + cos2 θ ′ ) + (1 − A) 3 4 (1 − cos2 θ ′ ), (0 ≤ A ≤ 1), (3.148)
which is motivated by the relative success of the Kirkpatrick-Wiedmann-Statham formula
at low energies (note that the projectile is at rest in K ′ ). The direction of emission
(θ, φ) in K is obtained by means of the Lorentz transformation
cos θ = cos θ′ + β
1 + β cos θ ′ , φ = φ′ . (3.149)