Computations involving two orders of the orbit-lattice interaction

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Abstract
Scott and Jeffries (1962) used a simple form of the orbit-lattice interaction, HOL= epsilon Sigma l,m upsilon lm, in order to explain relaxation rates in rare earth salts. This simple form is a modification, by averaging the strain epsilon , of the form HOL= Sigma l,m upsilon lm epsilon lm, used by Orbach (1961). If the Scott and Jeffries modification is used to calculate effects involving HOL in second order one obtains terms which cancel identically. Such complete cancellation should not occur, and does not in calculations using the complete form of HOL. The origin of the difficulty is traced by calculating the spin-lattice relaxation rate for the Raman process. It is related to the orthogonality of terms involving different normal modes. Some other aspects of this orthogonality are discussed; in particular it is shown that the usual assumption, that one need consider second order matrix elements involving only the lowest excited state, may be misleading. It is shown that all of these orthogonality effects may be included by retaining the more general expression for HOL used by Orbach, and treating the epsilon lm as incoherent. This requires a postponement of the averaging of the strain epsilon until a late stage in the calculation.

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