Effects of finite concentration in Kondo alloy systems

The thermodynamics properties of Kondo alloys based on U and Ce-based atoms behave differently as a function of either U or Ce concentration, x. In a uranium based system, the Kondo effect and the RKKY interaction compete with each other starting at low concentration ( x ≪ 1). In some cerium based system, on the other hand, the thermodynamic quantities such as specific heat and magnetic susceptibility at low temperatures (T ⩽ T _K depend linearly on x. This linear dependence implies that the contribution from independent Kondo centers is dominant and holds up to roughly 80% Ce concentration. We address the problem of finite concentration by calculating the x² correction to the thermodynamic potential at T ⩽ T _K in terms of scattering phase shifts by using the Nozières' Fermi liquid formulation of the single impurity problem. It is shown that the x² (and higher order) term lead to a small correction in the thermodynamical quantities only when the phase shift due to an impurity is close to the unitary limit at low temperatures. We also discuss the difference between Ce and U-based systems in the context of the phase shift argument.