Thermodynamic Properties of Transition-Ion Impurities in Noble Metals

The low-temperature specific heat and magnetization of dilute magnetc impurities in noble metals are studied using a simple model in which a Ruderman-Kittel-Yosida (RKY) interaction is assumed between the magnetic impurities. The method used is based on, and is an extension of, the theory developed by Klein and Brout (KB) for the treatment of the $T=0\mathrm{}$ specific heat of dilute Cu-Mn. Departures from the KB treatment for the case when $T\ne 0$ are considered. The magnetic impurities are considered to be randomly and uniformly distributed over the whole solid. The thermodynamic properties of the system are obtained by performing a statistical average over the positions of the particles and an ensemble average over their spins in an Ising model. The effective field $H_0$ about an impurity is defined by the relation $H_0=\Sigma v_{0j}{\mu }_j$, where $v_{0j}$ is the RKY interaction and ${\mu }_j$ is the spin of an impurity located at site $j$. The probability distribution of this effective field as a function of the impurity concentration, the strength of the RKY interaction, and the temperature is obtained. Use of this probability distribution gives the low-temperature specific heat of magnetic impurities in noble metals over a range of temperatures and concentrations in good agreement with experiment. For each class of impurities the strength of the RKY interaction is found from the low-temperature specific-heat data. A connection between the probability distribution of $H_0$ and the hyperfine field at the nucleus, as measured by a Mössbauer experiment, is made. The recently reported Mössbauer, specific-heat, and magnetic-susceptibility measurements on gold-rich Au-Fe seem to be consistent with our model. We thus conclude that no long-range magnetic order exists in dilute Au-Fe for concentrations of less than about 5% iron.