Fifty Years of the Simplified Pn Method

Abstract

In 1968, Ely Gelbard quipped in reference to the spherical harmonics (Pn) method, “Now we live in a period when whole fields of technology are born, mature, and, sometimes, die in a decade. In times when physics changes so quickly a 40-year-old method must be regarded as very old indeed,” (Gelbard, 1968). In this special issue we celebrate a very old method conceived by Gelbard as a simplification of the spherical harmonics method: the simplified Pn or SPn method. In a September 1960 review of nuclear reactor technology at the Bettis Atomic Power Laboratory (Gelbard, 1960), Gelbard presented a simplification to the full spherical harmonics method that greatly reduced the number of unknowns, and also provided an analysis of when the method was equivalent to the full Pn equations. A new method was born and researchers have been applying and studying it for the past 50 years. I was motivated to organize this special issue in 2009 while trying to extend the equivalence of the SPn and Pn equations to more general cases. In reading the literature I realized that the next year was the 50th anniversary of SPn. Like many in the field of transport I was initially intrigued by the simplicity of SPn, later amazed that in many cases SPn is equivalent to the more complicated Pn equations, and finally disappointed that in some problems it can give worse answers than diffusion.1 To encapsulate all of these feelings and to highlight the successes in using SPn, I decided that a special issue commemorating SPn was in order. I hope that this issue both takes a snapshot of the current uses and understanding of