On the Half-range Orthogonality Theorem Appropriate to the Scattering of Polarized Light

ONE OF the principal merits of the singular-eigenfunction-expansion technique, introduced initially by CASE(‘) in regard to one-speed neutron transport theory, is the systematic and classical manner in which solutions to boundary-value problems in particle transport theory are constructed. In general, we first construct a sufficiently general set of solutions, denoted as normal modes, to the homogeneous equation of transfer. To a resulting superposition (with arbitrary expansion coefficients) we add a particular solution to account for any inhomogeneous source term, and we then constrain the complete solution to meet the boundary conditions of a specified problem. At this point, a completeness theorem (either full or half range) is proved, which in fact ensures that a sufficiently general set of normal modes has been obtained. In many cases, (2) the various completeness proofs are actually constructive, in that, in addition to proving an expansion theorem, we can construct analytical results for the desired expansion coefficients. KUSCER et d.‘3’ were the first to observe that the results of the half-range completeness theorem related to one-speed theory could be expressed quite naturally in terms of scalar products based on the proof of a half-range orthogonality theorem. Though the results so expressed were, of course, identical to those patiently deduced from the completeness theorem, the use of orthogonality relations provided a significant impetus to a more universal appeal of the singular-eigenfunction method. We should like to demonstrate here the manner in which an orthogonality theorem in two-vector transport theory can be used to codify the results of half-range expansions in normal modes. Half-range orthogonality relations related to the generalized picketfence model in radiative transfer, a special case of multi-group neutron transport theory, have been reported by SIEWERT and ZWEIFBL. (4) There, because of the rather special structure of the picket-fence model, SIEWERT and ZWEIFEL (4) obtained exact closed-form results and were able to express the half-range weight matrix solely in terms of scalar Xor Hfunctions.‘5) Here the weight function is written in terms of an H-matrix for which no