On the Isospin Crossing Matrix

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Abstract
New properties and relations to Clebsch-Gordan coefficients have been found and proved for a certain subset of isospin crossing matrix elements. These properties, coupled with the well-known fact that the crossing matrix is simply related to a real orthogonal matrix, provide a quick method to calculate crossing matrices of rank ≤ 4. Procedures of construction and tables are given for all crossing matrices involving I ≤ 32. For the sake of completeness as well as clarification of the confusing point about phases, we include a systematic discussion of crossing relations and a general expression for crossing matrices applicable to arbitrary phase conventions. For the crossing matrix in which both direct and crossed reactions are elastic, some interesting inequalities among elements in the first and last columns are noticed and their physical implication is briefly discussed.