San José State University

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Thayer Watkins
Silicon Valley
USA

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

Also from the expression for cos(θ)

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.)