Mean inter-particle distance

Mean inter-particle distance (or mean inter-particle separation) is the mean distance between microscopic particles (usually atoms or molecules) in a macroscopic body.

From the very general considerations, the mean inter-particle distance is proportional to the size of the per-particle volume

is the particle density.

However, barring a few simple cases such as the ideal gas model, precise calculations of the proportionality factor are impossible analytically.

One such estimation is the Wigner–Seitz radius which corresponds to the radius of a sphere having per-particle volume

Another popular definition is corresponding to the length of the edge of the cube with the per-particle volume

The two definitions differ by a factor of approximately

, so one has to exercise care if an article fails to define the parameter exactly.

On the other hand, it is often used in qualitative statements where such a numeric factor is either irrelevant or plays an insignificant role, e.g., We want to calculate probability distribution function of distance to the nearest neighbor (NN) particle.

(The problem was first considered by Paul Hertz;[1] for a modern derivation see, e.g.,.

particles inside a sphere having volume

Note that since the particles in the ideal gas are non-interacting, the probability of finding a particle at a certain distance from another particle is the same as the probability of finding a particle at the same distance from any other point; we shall use the center of the sphere.

An NN particle at a distance

particles resides at that distance while the rest

particles are at larger distances, i.e., they are somewhere outside the sphere with radius

The probability to find a particle at the distance from the origin between

kinds of way to choose which particle, while the probability to find a particle outside that sphere is

The sought-for expression is then where we substituted Note that

, we obtain One can immediately check that The distribution peaks at or, using the

is the gamma function.