# Collisionality

In a plasma, the collision time τcoll is defined as the time in which the trajectory of a (charged) particle undergoes a change of direction of 90 degrees. Due to the long range of the Coulomb force, Coulomb interactions are typically small angle scattering events, so that this direction change typically requires a large number of interactions.

Consider a test particle with charge q, mass m, and velocity v colliding with bulk particles with charge q*, mass m*, and thermal velocity v*. Then the collision frequency ν = 1/τcoll is given by [1]

${\displaystyle \nu =\left({\frac {qq^{*}n^{*}}{\varepsilon _{0}m}}\right)^{2}{\frac {\ln \Lambda }{4\pi (m_{r}/m)v^{3}n^{*}}}}$

assuming v > v*, where mr = mm*/(m+m*) is the reduced mass and n* the bulk particle density. The factor ln Λ appears due to the accumulation of many small-angle collisions within a Debye sphere.

## Dimensionless collisionality

The dimensionless collisionality ν* is defined as [2]

${\displaystyle \nu ^{*}={\frac {\rm {connection~length}}{\rm {trapped~particle~mean~free~path}}}}$

See Connection length. The mean free path is estimated by the thermal velocity divided by the collision frequency, vth / ν.

## References

1. K. Miyamoto, Plasma Physics and Controlled Nuclear Fusion, Springer-Verlag (2005) ISBN 3540242171
2. ITER Physics Basis, Nucl. Fusion 39 (1999) 2137