PENELOPE combines numerical and analytical total and differential cross sections (DCS) to describe the different interaction mechanisms. These cross sections are the result from approximate physical models and, therefore, are affected by systematic uncertainties. In particular, the DCSs in the database correspond to interactions with free, isolated atoms. DCSs for compounds and mixtures are derived by using Bragg’s additivity rule, i.e., the molecular DCS is approximated by the sum of atomic DCSs of the atoms in a molecule. Aggregation effects are therefore disregarded. These effects are known to be appreciable, e.g. for elastic scattering of electrons with energies less than about 1 keV. Aggregation also affects the photoelectric cross section, causing extended x-ray absorption fine structure effects. On the other hand, the adopted x-ray energies and transition probabilities of excited atoms with inner-shell vacancies pertain to free atoms with a single vacancy. Nonetheless, the structure of the code is flexible enough to enable the use of alternative, more elaborate physical models.
The interaction mechanisms considered in PENELOPE, and the corresponding DCSs, are the following:
Particle tracks are simulated from the initial energy down to the absorption energies selected by the user, at which particles are considered to be effectively absorbed in the medium. Secondary particles emitted with initial energy larger than the corresponding absorption energy are simulated after completion of each primary track. Secondary particles are produced in direct interactions (inelastic collisions, bremsstrahlung emission, positron annihilation, Compton scattering, photoelectric absorption and pair production) and as radiation (characteristic x-rays and Auger electrons) following inner-shell ionisation. PENELOPE simulates the emission of characteristic x-rays and Auger electrons that result from vacancies produced in K-shells and L- and M-subshells by photoelectric absorption and Compton scattering of photons and by electron or positron impact. The relaxation of these vacancies is followed until all vacancies have migrated to N and outer shells. The adopted transition probabilities have been extracted from the LLNL Evaluated Atomic Data Library .
The simulation of photon tracks follows the usual detailed procedure, i.e., all the interaction events in a photon history are simulated in chronological order. Electrons can also be simulated in detail, or alternatively, by using a mixed (class II) simulation algorithm . Mixed simulation algorithms are useful when the electrons undergo a very large number of interactions and detailed simulation becomes impractical (e.g., for primary electron energies above 100 keV). In a mixed algorithm, the user defines cut-off values of the polar scattering angle () and the energy loss () in single interactions. Hard interactions, with scattering angle or energy loss , are simulated individually according to the adopted single-scattering DCSs. These interactions are responsible for large lateral displacements or large energy-loss fluctuations, which detailed simulation describes faithfully. The rest of interactions, with and , are classified as soft interactions; they have a mild effect on the electron trajectories and are simulated by using simple, but still fairly accurate, multiple scattering approximations. With a judicious selection of the cut-off parameters and , mixed simulation algorithms yield reliable results in moderate computation times, even for electrons with very high energies.
The straight simulation of EPMA spectra may be very inefficient, even when using mixed simulation algorithms for electron transport. The reason is that the two processes leading to the emission of x-rays (inner-shell ionisation and bremsstrahlung emission) occur with exceedingly small probabilities. That is, on average, we need to generate many electron histories to obtain one emerging photon. Moreover, as the solid angle subtended by the detector is usually very small, only a tiny fraction of the emerging photons hits the detector and contributes to the spectrum. The method adopted in PENEPMA to cope with this difficulty is to use the variance-reduction technique of interaction forcing (also known as the method of weights), which consists of artificially increasing the probability of inner-shell ionisation and bremsstrahlung emission and, at the same time, assigning appropriate statistical weights to the generated secondary particles in such a way that the simulation results remain unbiased. Interaction forcing is also advantageous in studies of fluorescence corrections; in this case the interactions that should be forced are photoelectric absorption and Compton scattering of photons. PENELOPE includes a flexible implementation of interaction forcing, which allows x-ray spectra to be computed in reasonable times (of the order of 1 hour, on modest personal computers).
Notice that PENELOPE tracks primary electrons and all generated particles (electrons and photons), that is, it generates the whole electron-photon shower produced by each primary electron. Therefore, fluorescence effects are consistently described.
The practical usefulness of Monte Carlo simulation lies in its ability to handle complex geometrical structures. PENELOPE contains the subroutine package PENGEOM which automatically tracks particles through arbitrary material systems consisting of homogeneous bodies limited by quadric surfaces. Limiting surfaces can be specified either through their implicit equation, , or by means of their reduced form and a few simple geometrical transformations. A quadric surface divides the space into two exclusive regions; the region where () is assigned the side pointer -1 (+1). A body is defined by giving its limiting quadric surfaces and corresponding side pointers. Previously defined bodies can be used to define new bodies. To speed up the geometry operations, the bodies of the material system may be grouped into modules (connected volumes, limited by quadric surfaces that contain one or several bodies), which in turn can form part of larger modules. This modular structure permits a substantial reduction of the work of the geometry routines, and allows very complicated structures to be considered (the current version of PENGEOM can handle systems with up to 9,999 surfaces and 5,000 bodies and modules). Repetitive structures can also be easily defined by using the “clone” operation.