Magnetization of Slow Electrons in a Polar Crystal

Abstract

An approximate expression for the free energy of an electron which is coupled to a phonon field (a "polaron") and is in a uniform magnetic field is given. This expression is obtained by an extension of Feynman's path integral variational calculation of the polaron binding energy and is valid for all values of the field, the temperature and the electron-lattice coupling strength. The explicit form of the result is given in terms of a model system with an action functional different from that of the actual problem. A minimum-variational method is given for determining the best model to employ in calculations. We have evaluated the magnetic free energy at small magnetic fields using a model of a simple form. The two variable parameters of this model have already been optimized by Feynman and Schultz for Fr\"ohlich's Hamiltonian and zero field. The result is applicable at all coupling strengths and carried to the first two terms in a Laurent series in the temperature. The first (inverse temperature) term yields an approximation to the effective polaron mass that varies, as the coupling is varied, by at most 1.5% from that calculated by Feynman and others. The second (temperature independent) term reflects the internal structure (atomic diamagnetism) of the polaron at large coupling and also the temperature variation of effective mass. The difference between our magnetic mass and that calculated by Feynman is shown to disappear when the model system is fully optimized.