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The Distribution of Separation Distances of Particles in
a Spherical Shell

If particles are located at the vertices of a polyhedron such as a tetrahedron, cube, octahedraon,
dodecahedron or icosahedron the distances can be computed but it is a very tedious process. This material
works out the continuous distribution of separation distances.

Consider a sphere of radius R. It has an area of 4πR². Now consider the north pole of that
sphere and a point at an angle of θ from the north pole. The radius of the circle whose points
are all at angle θ from the north pole is r=Rsin(θ). For a band of width Rdθ the area
is

dA = [2πR·sin(θ)]sin(θ)dθ = 2πRsin²(θ)dθ

This can be converted into a probability density f(θ) by dividing by the area of the sphere 4πR²; i.e.,

f(θ)dθ= 2πRsin²(θ)dθ/(4πR²) and therefore
f(θ) = sin²(θ)/(2R)

The straight line distance z between the north pole and a point at angle of θ from the north pole
is

−sin(θ)(dθ/dz) = −z/R²
and hence
dθ/dz = z/(sin(θ)R²)

Thus

g(z) = [sin²(θ)/(2R)][z/(sin(θ)R²)]
which reduces to
g(z) = z·sin(θ)/(2R³)
and further to
g(z) = z[(z/R)² − ¼(z/R)^{4}]^{½}/(2R³)
and still further to
g(z) = (z/R)²[1 − ¼(z/R)²]^{½}/(2R²)

This distribution has the shape shown below. The average value of (z/R) is 1.35.