San José State University

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Thayer Watkins
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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

z² = 2R² − 2R²cos(θ) = 2R²(1 − cos(θ))
and hence
cos(θ) = 1 − ½(z/R)²

What is desired is the probability density of z, g(z). The relationship that prevails is g(z)dz=f(θ)dθ and thus

g(z) = f(θ(z)|dθ/dz|

From the expression for cos(θ)

cos²(θ) = 1 − (z/R)² + ¼(z/R)4
and hence
sin²(θ) = (z/R)² − ¼(z/R)4

Also from the expression for cos(θ)

−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.

(To be continued.)


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