` TrapApp` also offers the option of simulating Monte Carlo collisions
with a background gas. This option allows the user to model, on a microscopic
scale, thermal interaction with a ``background'' gas which
is at constant temperature. The user provides the mean time between collisions and the temperature of the background gas. After each finite difference cycle, ` TrapApp` performs Monte Carlo collisions by comparing a random number with the probability that a collision should occur. In the
case of a collision, ` TrapApp` assigns to the unfortunate ion a new
velocity according to the Maxwell-Boltzmann speed distribution at the
temperature of the background gas.

To aid the number-conscious user in choosing an appropriate collision time for a desired background gas pressure, the following relations are provided:[13]

where is the mean free path length between collisions, is the mean ion velocity, and
is the mean time between collisions.
The chance of a collision occurring within a displacement **dx** is
approximately:

where **n** is the number density (per unit volume) of the background
gas, is the cross-sectional area of a molecule of the background
gas, and **dx** is an incremental distance. The above definition of
and
the ideal gas law, (where **N** is the number of
ions, **V** is the volume occupied, **p** is pressure) can be employed
to yield an expression for the mean
free path as a function of temperature, pressure, and
cross section:

For example, a plausible experimental set-up[30] might employ a background gas of helium at a background pressure mbar to cool Mg ions ( amu kg ). If we assume the radius of helium to be roughly an Angström, the mean free time between ion collisions with the helium is ms. Unfortunately, a typical Paul trap simulation uses a time step on the order of tenths to hundredths of s, so effectively cooling a collection of ions via Monte Carlo collisions at ms requires hundreds of thousands of time steps.

Fri May 12 10:36:01 EDT 1995