Computing initial energy distributions for field emission sources

In Monte Carlo simulations of electron sources, random initial conditions of the particles' velocity components at the cathode need to be generated: these velocities are governed by the initial energy and angular distribution of the source. To generate the distributions numerically, the probability densities must be integrated so that the distribution can be inverted. Unlike for thermionic sources, where these integrals can be performed using simple trigonometric functions, for field emission or thermal field emission sources it is not so straightforward to integrate the probability distributions analytically. In this paper, we will show how we have derived analytic formulæ for the integrals of the energy probability distributions for cold and thermal field emission sources. These formulæ involve the Gaussian hypergeometric function, which can be evaluated numerically using established computational techniques. Hence, the random energy, and therefore velocity components, can be generated efficiently for the many thousands of particles typically needed for a Monte Carlo analysis of field emission sources. We have implemented these integrals in a computer program and give results of the energy distribution for field emission sources generated by these techniques.