Quadrupole Interactions in Rare-earth Intermetallic Compounds

A survey of the role played by the one-ion magnetoelastic and two-ion quadrupolar interactions on the magnetic and elastic properties of the rare-earth intermetallic compounds is given. These couplings are present in numerous systems and manifest themselves through a great variety of effects. The study of the multipolar interactions concerning the 4f shell has known a great development about rare earth intermetallic compounds [I]. It involves the complex cohabitation of i) the 4f electron-lattice coupling which gives rise to the usual crystal field acting on the rare-earth, as well as to the one-ion magnetoelastic coupling; ii) the bilinear and quadrupolar pair interactions between different sites leading to possible long-range ordering The aim of this review is to present a survey of the experimental manifestations of the one-ion magnetoelastic and the two-ion quadrupolar interactions; both couplings involve the orbital character of the 4f shell through its quadrupolar components. 1. The Hamiltonian neglect any anharmonic or two-ion magnetoelastic coupling which may however play a role in some particular cases. The next term in 'FI involves the total angular momentum J through the Zeeman coupling -gj pg H . J as well as the isotropic bilinear Heisenberg-type Hamiltonian: 2 'FIB = ( g ~ pg) n (J) . J = -gj p~ n M . J . (3) Within the mean field approximation (MF-4), n is the bilinear exchange parameter. In the same way, the 4f quadrupoles may be coupled between each other through direct or indirect mechanisms (see paragraph 4), leading to the general expression in the MFA: The existence of non quenched orbital moments in where the KP'S are the two-ion quadrupolar paramethe rare earth ions leads to anisotropic magnetic propters. erties. A classical description of these properties is no longer valid. In a auantum formalism, the 4f shell is represented by wavifunctions expanded in the I J, MJ) basis. The appropriate Harniltonian, 7i which deter2. Treatment of the Hamiltonian mines these wavefunctions, includes several terms associated with the various interactions in presence. The first term to be considered is the Crystalline Electric Field (CEF) Hamiltonian which may be expressed as a function of linear combinations 0, of the Stevens operators [2] : where the AP1s are the CEF parameters. The number and the detailed form of the 0,'s depend on the point symmetry of the rare earth site under consideration. This CEF term may be modulated by the strain through the one-ion Hamiltonian : In this expression, the Q,'s represent proper combinations of the second-order Stevens operators, which are coupled with the corresponding symmetrized strain E, through the magnetoelastic coefficients Bp. Here we There is two ways to use the above Hamiltonian to describe the magnetic properties. First a diagonalization can be performed in a self-consistent manner. This allows us to determine the magnetization and magnetostriction processes, in particular in the ordered phase or in large external stresses (magnetic field or external pressure). The second possibility is to apply perturbation theory in order to describe both magnetic and magnetoelastic properties in the nonordered phase and in small external stresses. An analytical expansion of the free energy F p is then derived as a function of the magnetic field H and of the strain E,, for a given symmetry: 1 F p = F; 1x0 (H + n ~ ) ~ SX, (Bp E, + Kp Q,)? 2 ~ f ) (B" E, + Kp Q,) (H + n ~ ) ~