100
Chapter 2. Geometrical optics
suffers from two disadvantages. First, it is computationally intensive. For a given system configuration, a new simulation must be
performed for every distinct operating point, and this can be very
time-consuming. Second, it is difficult to achieve an intuitive understanding of the physical process, such as the understanding one
might obtain from an analytical theory.
Given that the N-body problem cannot be solved analytically in
closed form, it is of great interest to inquire whether a suitable analytical approximation can be found. There is a history of attempts
to express beam broadening due to stochastic Coulomb scattering
by analytic approximations. Typically these approximations yield
a simple algebraic dependence on experimental parameters such as
beam energy, system length, beam current, and numerical aperture. The reader is referred to two excellent reviews by Kruit and
Jansen [55] and by Jansen [49] for details.
This approach has the advantage that the optical properties of
a system can be estimated quickly and simply with some degree of
accuracy. This facilitates an intuitive understanding of the dependency on experimental parameters. It has the disadvantage that
the formulas depend on the specific system configuration and on
the operating point.
No simple, general formulation appears to exist. Also, it becomes
necessary to independently evaluate the accuracy of the formula
before relying on it for a detailed design of a system. Assuming the
system has yet to be built, this evaluaton relies on Monte Carlo
simulation. In practice, one uses a judicious combination of analytic approximation and Monte Carlo simulation.
In this study, we attempt an analytical analysis which does not
result in such simple formulas, but strikes at the basic underlying statistical mechanics. The remainder of this section closely
follows the earlier analysis by Groves [41]. We aim to derive a formalism which lends itself well to numerical analysis, where this
analysis is less computationally intensive than Monte Carlo sim
Chapter 2. Geometrical optics
suffers from two disadvantages. First, it is computationally intensive. For a given system configuration, a new simulation must be
performed for every distinct operating point, and this can be very
time-consuming. Second, it is difficult to achieve an intuitive understanding of the physical process, such as the understanding one
might obtain from an analytical theory.
Given that the N-body problem cannot be solved analytically in
closed form, it is of great interest to inquire whether a suitable analytical approximation can be found. There is a history of attempts
to express beam broadening due to stochastic Coulomb scattering
by analytic approximations. Typically these approximations yield
a simple algebraic dependence on experimental parameters such as
beam energy, system length, beam current, and numerical aperture. The reader is referred to two excellent reviews by Kruit and
Jansen [55] and by Jansen [49] for details.
This approach has the advantage that the optical properties of
a system can be estimated quickly and simply with some degree of
accuracy. This facilitates an intuitive understanding of the dependency on experimental parameters. It has the disadvantage that
the formulas depend on the specific system configuration and on
the operating point.
No simple, general formulation appears to exist. Also, it becomes
necessary to independently evaluate the accuracy of the formula
before relying on it for a detailed design of a system. Assuming the
system has yet to be built, this evaluaton relies on Monte Carlo
simulation. In practice, one uses a judicious combination of analytic approximation and Monte Carlo simulation.
In this study, we attempt an analytical analysis which does not
result in such simple formulas, but strikes at the basic underlying statistical mechanics. The remainder of this section closely
follows the earlier analysis by Groves [41]. We aim to derive a formalism which lends itself well to numerical analysis, where this
analysis is less computationally intensive than Monte Carlo sim