Entropy Change at a Demagnetization Broadened First Order Transition

The literature on the thermodynamics of magnetism tends to obscure the nature of the magnetic field used in thermodynamic potentials and relations. It is often implied that H is a homogeneous internal magnetic field of a homogeneously magnetized body. In this framework, isothermal first order phase transitions are associated with a magnetization discontinuity, ΔM, at a critical field Hc with Hc = 0 for a ferromagnet and Hc \(\ne \) 0, e.g., for antiferromagnets with metamagnetic and spin flop transitions. The idealized description in terms of H is in sharp contrast to experiments performed on real samples where magnetic stray-fields are present. Here, the applied magnetic field, Ha, is experimentally controlled and the first order phase transition is continuous spreading out over a regime of coexisting phases. This work shows, with and without reference to a thermodynamic potential that, for the special case \(H={H}_{a}-{D}_{eff}M\) , the Maxwell relation \({\mu }_{0} {\left(\partial M/\partial T\right)}_{H}= {\left(\partial s/\partial H\right)}_{T}\) implies validity of \({\mu }_{0} {\left(\partial M/\partial T\right)}_{{H}_{a}}={\left(\partial s/\partial {H}_{a}\right)}_{T}\) . We show for magnetometry data of a Gd single crystal that a fictitious internal field \(H={H}_{a}-{D}_{eff}(T) M\) transforms M versu Ha isotherms to M vs H with a discontinuity at H = 0. This formal transformation comes at the price that Deff is temperatureTemperature dependent, rejecting the notion of a purely geometry dependent demagnetizing factor. The unjustified assumption of a homogeneous internal field in the mixed phase gives rise to the incorrect result \({\left(\partial s/\partial {H}_{a}\right)}_{T}=0\) . Magnetization isotherms in Gd are virtually hysteresis free and domain states of the mixed phase are equilibrium states with well-defined entropy quantifiable via \({\mu }_{0} {\left(\partial M/\partial T\right)}_{{H}_{a}}={\left(\partial s/\partial {H}_{a}\right)}_{T}\) .