Interaction biquadratic

Here Eh and a are the parameters of a normal Morse potential, and m = mh mi in the framework of the model discussed). Although this combined potential (4.3.26) provides plausible estimates, it can hardly be substantiated in terms of the theory of intermolecular interactions and contains, in addition, only biquadratic anharmonic... 

Now we can see that the spin states S = 2, Ms = 2) are coupled through the biquadratic spin-spin interaction parameter, a, giving rise to two slightly separated roots by the amount of a. 

The existence of a small energy gap between six members of the S,Ms) manifold (S = 5/2) can be explained by introducing a biquadratic spin-spin interaction ... 

Fig. 66 Energy levels of an S = 5/2 system in the cubic symmetry by means the biquadratic spin-spin interaction Hamiltonian (numerical diagonalization) a/k = 0.1 K (left) and 1 K (right)...

Case 2 On increasing the CF strength 10Dq (which reduces A0) the biquadratic spin-spin interaction takes on significance. The zero-field energy levels lie at (/7) = -2a (doublet) and e(rs) = +a (quartet). The ZFS is S = 3a. 

We suggest here that the non-nearest center interaction can be regarded as having indirect nature via intercentric biquadratic terms of the form (S) Sj)(St Sk) and... 

Higher terms, biquadratic in the spin operators opS, also can occur.18 Spin-spin anisotropic interactions of the type D [S S2] are known for electrons,19 where D is a parameter vector this expression is the antisymmetric part of the most general bilinear spin-spin interaction. Observation of a field-induced magnetization in SrCu2(B03)2 via 11B NMR has revealed the presence of such an interaction therein.20... 

Now we can see that for 5 > 2, in addition to the second-order equivalent operators, the fourth-order equivalent operators can contribute. These, in fact, correspond to higher-order spin-spin interaction—a biquadratic spin-spin interaction, like... 

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... 

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

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... 

Pair-interaction reduced matrix elements of the biquadratic exchange for a general triad3... 

Another approach is to start with a large set of initial tilts and to run the minimization process munerically. The risk is that the minimization procedure will lock onto metastable solutions due to the energetic barriers between structures with similar periods. When biquadratic interactions are important, the procedure may produce artificial structures. 

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