|San José State University|
& Tornado Alley
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
This can be converted into a probability density f(θ) by dividing by the area of the sphere 4πR²; i.e.,
The straight line distance z between the north pole and a point at angle of θ from the north pole is
What is desired is the probability density of z, g(z). The relationship that prevails is g(z)dz=f(θ)dθ and thus
From the expression for cos(θ)
Also from the expression for cos(θ)
This distribution has the shape shown below. The average value of (z/R) is 1.35.
(To be continued.)
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