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:
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 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.