Self-consistent model of a solid for the description of lattice and magnetic properties

Abstract

In the paper a self-consistent theoretical description of the lattice and magnetic properties of a model system with magnetoelastic interaction is presented. The dependence of magnetic exchange integrals on the distance between interacting spins is assumed, which couples the magnetic and the lattice subsystem. The framework is based on summation of the Gibbs free energies for the lattice subsystem and magnetic subsystem. On the basis of minimization principle for the Gibbs energy, a set of equations of state for the system is derived. These equations of state combine the parameters describing the elastic properties (relative volume deformation) and the magnetic properties (magnetization changes).The formalism is extensively illustrated with the numerical calculations performed for a system of ferromagnetically coupled spins S =1/2 localized at the sites of simple cubic lattice. In particular, the significant influence of the magnetic subsystem on the elastic properties is demonstrated. It manifests itself in significant modification of such quantities as the relative volume deformation, thermal expansion coefficient or isothermal compressibility, in particular, in the vicinity of the magnetic phase transition. On the other hand, the influence of lattice subsystem on the magnetic one is also evident. It takes, for example, the form of dependence of the critical (Curie) temperature and magnetization itself on the external pressure, which is thoroughly investigated.V, considered at some temperature T (see for example Ref. [2]). Therefore, the magnetic equation of state involves three variables: h, m and T .In majority of cases both the subsystems of solid state (magnetic and lattice one) are described separately, with no coupling between them. In such situation, the lattice-related properties of the magnetic solid are considered as fully independent on magnetic properties, leading to another equation of state interrelating p, V and T . However, this approach neglects the magnetoelastic interactions which occur between these subsystems. The simplest source of this kind of coupling is the fact that the magnetic exchange integral between magnetic moments depends on their mutual distance, thus is a volume-dependent quantity. The magnetoelastic interactions are basis for such effects as the magnetostriction and piezomagnetism, which are important from the point of view of possible applications. They are also responsible for sensitivity of any magnetic properties (for example, the critical temperature) to external pressure.Among the literature concerning the studies of magneto-elastic interactions, many particular contributions can be mentioned. One of the subjects of intensive studies was compressible Ising model of various dimensionalities [3][4][5][6][7][8][9][10][11][12], including the diluted case [13,14] and the quantum versions [15]. The studies involved also Heisenberg model [16][17][18] or spin glasses [19] or highly interesting frustrated magnetic systems [20][21][22]. Some of the results also incorporate magnet...

Hubbard pair cluster with elastic interactions. Studies of thermal expansion, magnetostriction and electrostriction

“…In the Hubbard Hamiltonian we select the exponent n in the hopping integral (8) equal to n = 6 in order to describe a rapid variability of the function around d 0 . It is in analogy with the possible dependence of the exchange integral vs. the distance [64] and refers to the attractive part of the Lennard-Jones potential. A more accurate description of the hopping integrals dependence on the bond length can be given by the Goodwin, Skinner and Pettifor function [67].…”

Thermodynamic model of a solid with RKKY interaction and magnetoelastic coupling

“…(A.12), is assumed with the value n = 6, analogously to Ref. [40]. As far as the electron subsystem is concerned, the energy parameter E 0 from Eq.…”

“…For isotropic system the sum can be conveniently performed over the coordination zones with radii r k,0 and the coordination numbers z k . Finally, the elastic energy can be presented in the form of [40]…”

“…In particular, by reduction of the long-range RKKY interaction and by leaving only NN magnetic interaction, as well as without energy of electronic subsystem the formalism will be equivalent to that developed in our previous papers (Refs. [40,41]). Such a method could be applicable to magnetic insulators.…”

“…In our previous papers [40,41], the thermodynamic model for the magnetoelastic couplings was presented, for the simplest case when the magnetic interaction between localized spins was of Heisenberg type. The energy of itinerant electrons was not considered in that approach, thus restricting the model to magnetic insulators.…”