99
2.6. Stochastic Coulomb scattering
fore effectively infinite. It follows from these considerations that
the sample length must be much greater than the diameter of the
beam for accurate simulation. This can lead to a very large number of particles, and correspondingly long computation time. The
stochastic contribution to the displacement is expected to have
relatively short range, however, since the fluctuations average out
over long distances. A technique exists to take advantage of this by
separating the effects of the average and stochastic contributions
[40]. The reader is referred to this reference for details.
The above procedure applies to a drift length, with no external
fields present. In principle, one can add these fields into the expression for the Lorentz force. For many applications, a thin lens
approximation suffices, in which the direction of the velocity is
shifted toward the optic axis by an amount proportional to the radial distance off axis. The magnitude of the velocity is unchanged.
In this way, complex systems can be analyzed by Monte Carlo simulation of a series of drift lengths separated by thin lenses. Such
simulations have shown close agreement with experiment [51, 70].
Monte Carlo simulation thus offers a powerful predictive tool in
the design of practical systems.
Problem
Calculate an analytic expression for the time rate of change a ˙ i
in terms of the position r i , velocity v i , and acceleration a i .
2.6.2 Analytical approximation by Markov’s
method of random flights
Monte Carlo simulation is inherently accurate, due to the minimal
assumptions needed to describe the physical process. However, it
2.6. Stochastic Coulomb scattering
fore effectively infinite. It follows from these considerations that
the sample length must be much greater than the diameter of the
beam for accurate simulation. This can lead to a very large number of particles, and correspondingly long computation time. The
stochastic contribution to the displacement is expected to have
relatively short range, however, since the fluctuations average out
over long distances. A technique exists to take advantage of this by
separating the effects of the average and stochastic contributions
[40]. The reader is referred to this reference for details.
The above procedure applies to a drift length, with no external
fields present. In principle, one can add these fields into the expression for the Lorentz force. For many applications, a thin lens
approximation suffices, in which the direction of the velocity is
shifted toward the optic axis by an amount proportional to the radial distance off axis. The magnitude of the velocity is unchanged.
In this way, complex systems can be analyzed by Monte Carlo simulation of a series of drift lengths separated by thin lenses. Such
simulations have shown close agreement with experiment [51, 70].
Monte Carlo simulation thus offers a powerful predictive tool in
the design of practical systems.
Problem
Calculate an analytic expression for the time rate of change a ˙ i
in terms of the position r i , velocity v i , and acceleration a i .
2.6.2 Analytical approximation by Markov’s
method of random flights
Monte Carlo simulation is inherently accurate, due to the minimal
assumptions needed to describe the physical process. However, it