Biquadratic exchange

To model several other electronic situations (e.g., in mixed-valence compounds), the Heisenberg Hamiltonian can be augmented by specific exchange operators that accommodate, for example, double exchange, antisymmetric exchange, anisotropic exchange, or biquadratic exchange. 

E. A. Harris and J. Owen, Biquadratic exchange between Mn2+ ions in MgO. Phys. Rev. Lett., 1963,... 

In fact, on close examination of the susceptibility variation, slight discrepancies appear at low temperatures, so that a model involving a biquadratic contribution was postulated in a second step [54]. In this assumption, the exchange Hamiltonian is written as ... 

Fig. 17. Experimental and theoretical magnetic susceptibilities of Ba3CaRu20, showing the influence of a biquadratic exchange coupling...

The spin-spin interaction is not necessarily restricted to quadratic terms in the spin operators. Higher-order terms could be considered, among which the (isotropic) biquadratic exchange reads... 

OTHER INTERACTIONS IN DINUCLEAR SYSTEMS 10.4.1 Biquadratic exchange... 

The operator of the biquadratic exchange for a dinuclear system is written as follows... 

Evidently, the matrix elements fill only the diagonal of the interaction matrix and consequently the biquadratic exchange is a kind of isotropic interaction. 

The biquadratic exchange applies side by side with the bilinear exchange, hence... 

The biquadratic exchange coupling constants adopt rather small values the experimental data show that exchange coupling constants obey j/J 0.05. The separation of the energy levels, however, alters significantly as the deviation from the Lande interval rule is dramatic for higher spin values [19]... 

Zero-field energy levels for biquadratic exchange ... 

Fig. 10.13. Temperature variation of the magnetic susceptibility (a) and the effective magnetic moment (b) for the SA = SB = 5/2 dimen solid—bilinear exchange with J/hc = —100 K long dashed—biquadratic exchange with additional j = — 1 K medium dashed—biquadratic exchange with additional j = +1 K.

Several kinds of exchange interactions, which couple the magnetic centres, are distinguished the (bilinear) isotropic, asymmetric, antisymmetric, biquadratic and double exchange. Various fine-structure Hamiltonian terms enter the spin-spin coupling tensor D these were derived from the relativistic Dirac and Breit equation. 

The operator of biquadratic exchange involves not only the pair-interaction terms but also the triad-interaction ones... 

Nevertheless, biquadratic exchange spans the class of isotropic exchange. The individual terms are of the general type... 

Fig. 11.9. Effect of biquadratic exchange on the temperature dependence of the effective magnetic moment for the 5 = 52 = 53 = 1 system J/k = —100 K, individual curves (from bottom to top) correspond to j/k = 0, 10, 15, 20 and 25 K.

Matrix elements of the biquadratic exchange for the Si = S2 = S3 = 1 system (an isosceles triangle)... 

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