Probability distribution of distance in a uniform ellipsoid: Theory and applications to physics

A number of authors have previously found the probability P-n(r) that two p oints uniformly distributed in an n-dimensional sphere are separated by a d istance r. This result greatly facilitates the calculation of self-energies of spherically symmetric matter distributions interacting by means of an a rbitrary radially symmetric two-body potential. We present here the analogo us results for P-2(r ; epsilon) and P-3(r ; epsilon) which respectively des cribe an ellipse and an ellipsoid whose major and minor axes are 2a and 2b. It is shown that for epsilon = (1 - b(2)/a(2))(1/2) less than or equal to 1, P-2(r ; epsilon) and P-3(r ; epsilon) can be obtained as an expansion in powers of epsilon, and our results are valid through order epsilon(4). As an application of these results we calculate the Coulomb energy of an ellip soidal nucleus, and compare our result to an earlier result quoted in the l iterature. (C) 2000 American Institute of Physics. [S0022-2488(00)04304-8].