4. METHOD OF SOLUTION
A programme called DATAPAC-4 takes data for a particular material from a library tape and further processes them. The function of DATAPAC-4 is to produce single-scattering and multiple-scattering data in the form of tabular arrays (again stored on magnetic tape) which facilitate the rapid sampling of electron and photon Monte Carlo histories in ETRAN.
The photon component of the electron-photon cascade is calculated by conventional random sampling that imitates the physical processes of compton scattering, photon electric absorption, and pair production. In the calculation of the electron component, no attempt is made to follow successive individual interactions with atoms and atomic electrons because these are too numerous. Instead, a Monte Carlo model is used in which attention is focused on the effect of groups of successive collisions.
The electron tracks to be sampled are divided into a large number of short segments, and the energy loss and angular deflection in each segment are sampled from pertinent theoretical multiple scattering distributions. At the end of each short step, the direction of motion of the electron is changed by a multiple scattering angular deflection that is sampled from the Goudsmit-Saunderson distribution. This distribution is assumed to be the same for all short steps lying within a given step. The energy loss in a step, resulting from the cumulative effect of many inelastic collisions, is sampled from a distribution that is a convolution of a Landau distribution with a Gaussian. An option is also provided for using the continuous-slowing-down approximation in which energy-loss fluctuations are disregarded and the energy loss by collisions is simply computed with the use of the stopping power formula.
The production of knock-on electrons is sampled in each short step with the use of a probability distribution derived from the Moller cross section for collisions between free electrons (binding effects are disregarded). Histories of these particles are then followed by procedures identical with those used for the primary electrons.
The production of continuous bremsstrahlung photons is sampled in each short step with the use of a probability distribution derived from the bremsstrahlung cross section (Bethe-Heitler theory with modifications taking into account the correct high-frequency limit, empirical corrections, etc.). The probability is usually quite small that more than one bremsstrahlung photon will be produced in a single short step. Allowance is made for such a contingency by sampling the frequency of bremsstrahlung production events from a Poisson distribution. The energy of the secondary bremsstrahlung photons is subtracted from the energy of the electrons producing them. Thus photon emission contributes to the energy-loss straggling of the electrons. The photons are started out at a random position in the short step in a direction relative to that of the primary electron specified by the sampled intrinsic bremsstrahlung emission angle. For problems in which the production of the thick-target bremsstrahlung is of prime interest, there is an option to increase the rate of occurrence of bremsstrahlung events artificially by a specified factor.
The production of secondary characteristic x-rays in each short step is sampled with the use of the k-ionization cross sections of Arthurs and Moiseiwitsch and Kolbenstvedt. The programme is arranged so as to treat simultaneously many slab targets with different thicknesses.
Boundary crossings (transmission and reflection) usually occur in the middle of a short step. The energy with which the electron crosses the border is determined by subtracting from the energy at the beginning of the step an energy loss sampled from the Landau- Blunck-Leisegang distribution for the fraction of the step taken to the boundary. The direction at the time of crossing is determinedby changing the direction of motion at the beginning of the short step involved, using a deflection sampled from an exponential approximation to the Goudsmit-Saunderson distribution for the fraction of the short step to the boundary.
The target is subdivided into many thin sublayers of equal thickness, and the energy deposited in each sublayer is recorded for each sampled track. The energy allowed to be deposited is that dissipated by electrons in inelastic collisions resulting in the production of slow secondary electrons with energies below the chosen cut-off-value. The energy given to fast secondary electrons with energies above the cut-off is not immediately scored, because the histories of these electrons are followed further so that the energy may eventually be deposited in a sublayer different from the one in which the electrons were produced. Bremsstrahlung losses are similarly not scored immediately. Photons are allowed first to penetrate further through the medium so that the energy of the electrons set in motion by them may eventually be deposited in a different sublayer.
The treatment of charge deposition is quite similar to that of energy deposition, involving the scoring of charge deposited in sublayers. A track is assumed to "end" when the residual range of the electron is so small compared to the size of the sublayers that escape to a different sublayer is no longer possible. When secondary electrons are produced, either as the result of a knock-on collision or as the result of Compton scattering or photon-electric absorption, a unit charge is withdrawn from the sublayer in which the electron is born. The charge is then allowed to be carried to a different sublayer. Electron-positron pairs are excluded from this scheme because on the average their production does not lead to a net transfer of charge.
The electron flux is computed in ETRAN as a quantity differential in energy but integrated over all directions. (*) A Monte Carlo estimate of the flux is obtained by dividing the target into many sublayers and scoring the tracklength of electrons with specified energies in each of the sublayers. The average tracklength per incident electron divided by the thickness of the sublayer provides an estimate of the average flux in the sublayer.
The flux calculation includes primary as well as secondary electrons with energies down to some cut-off value which is chosen so that the electron is effectively trapped in the sublayer in which it finds itself, because its residual range is smaller than the distance to the nearest sublayer boundaries.
(*) In the IBM 360/75 version, the flux differential in angle is also computed, and a distinction is made between the flux in the forward and backward directions.
