Monte-Carlo Model for Radiation Transport in Solid X-Ray Targets
X-rays represent high-energy electromagnetic radiation capable to penetrate matter and are therefore widely used in medical diagnostics, material testing and various other applications. They are typically produced by vacuum tubes with electrons emitted from a hot cathode and accelerated towards an anode by an applied voltage. Upon their impact on the metal target and subsequent penetration, electrons are decelerated by the electric field near the nuclei leading to emission of X-ray photons, termed as Bremsstrahlung. Characteristic X-rays, on the other hand, are produced by radiative transitions during relaxation of atoms excited by atomic electron and photon interactions.
Modeling of X-radiation transport through matter gives valuable quantitative insight into the performance of a target and emitted spectrum. Particle-based approaches in the framework of the Monte Carlo method are well suited for this purpose by resolving the cascade (shower) of photons and secondary electrons from electrons impinging the target and produced by atomic interactions.
In the present work, the implementation of a Monte Carlo model portraying electron and photon transport in the target and substrate of an X-ray anode around its focal spot within COMSOL Multiphysics is presented. The Particle Tracing Module is employed to sample electron and photon tracks and atomic excitation. With showers assumed as independent from each other, the particle histories are integrated along their tracks as random sequences of free flights and subsequent atomic interactions. The model accounts for Bremsstrahlung emission, elastic and inelastic scattering of electrons and photo-electric absorption, coherent, and incoherent scattering of photons. The required collision cross sections are compiled for tungsten, rhenium, and molybdenum – materials commonly employed for X-ray anodes – from the EPICS-database published by the IAEA.
A large set of functions and algorithms is implemented to provide the random sampling methods required to determine the characteristics of free flights, atomic interactions, and secondary emissions. Particles feature continuously updated state variables allowing detailed statistical evaluations such as spectrum computation. Boundary and domain accumulators are used to determine particle and energy fluence as well as dose distributions, respectively.
Computation is performed for an ideal monoenergetic electron pencil beam for a given number of incident electrons. Assuming an isotropic solid medium, the obtained results are transformed to an arbitrary finite e-beam profile using the convolution theorem. Particle population sizes are governed by shower evolution requiring allocation of numerous secondary particles making integration computationally expensive even for a moderate number of incident electrons. Due to independence of showers, however, an arbitrary combination of parallel and consecutive runs can be performed to establish statistically representative results.
Exemplary application is demonstrated for an inclined tungsten target for a tube potential of 50 kV. Evaluation of utilizable X-radiation is accounted for by filtering photon (energy) fluence using a detector plane. Results of the model are found to be plausible yielding efficiency factors in the order of magnitude of 0.1 to 1%. In the future, the model will be further enhanced and applied to parametric assessments such as the influence of tube potential, target thickness and angle on dosimetry quantities.
