On Suhl's Theory of Exchange Scattering in Metals

Abstract

A closed-form solution to Suhl's integral equations for the T-matrixes for exchange scattering in metals is obtained which is valid at all the temperatures for both signs of J and is essentially the same as Suhl and Wong's solution. The solution turns out to be manifestly the same as that obtained by Bloomfield and Hamann except minor differences. It enables us to calculate the free-energy shift which turns out to be essentially the same as the exact perturbation expansion 'at high temperatures at least up to fourth order, including the T log T term obtained in the preceding paper. The free energy does not show any discontinuous change across T K although both the normal part and the non-analytic part of it change discontinuously. As the temperature goes to zero, it develops a binding energy corresponding to the reduction of the entropy of the conduction electrons and tends to the exact ground-state energy which we obtained previously. These results support Suhl's assertion that the ground state of the s-d problem is reached by analytically continuing the high-temperature expansion across T K. In this paper we investigate the logarithmic anomalyl) associated, with the s-d exchange model of a localized spin in metals. SuhP)-~) pointed out that the spin-flip component of the T-matrixes for electron scattering develops a pair of inadmissible poles for negative exchange interaction as the temperature becomes lower than T K' Opinions of people about this difficulty may roughly be devided into two. The first opinion is that in the low-temperature region perturbation theory is not valid and there realizes a bound state which cannot be obtained by perturbation theory. 01~ the other hand there prevails the other opinion that the poles reflect inadequacy of the calculation and may be removed by a more refined theory which becomes essentially the same as the perturbation theory at high temperatures. The question of which is right may be phrased in another way: It is now well established that the ground-state energy involves a binding energy which is not analytic in J, the s-d exchange integra1. 6 )-9) Thus the question may be expressed as whether the ground state with the binding energy is reached by the perturbation theory (or analytic continuation of it across T K ) or the true ground state is essentially foreign to the perturbation theory. There have been two contradictory results which are related to this ques­ tion. Yosida and Miwa 10 ) calculated the free-energy shift by high-temperature expansion up to fourth order and found that it is practically temperature-inde­ pendent. This means that the extrapolation of the free energy of perturbation