Operator isotropic exchange

The isotropic exchange discussed so far represents only a portion of the total exchange interaction operating in binuclear and oligonuclear systems. 

The scalar product in the isotropic exchange operator can be expressed by using the shift (step-up and step-down) operators... 

The isotropic exchange operator does not contain the extra electron so that it projects only to the subspace of genuine spin functions /0) = SaSbScSabSM) with the usual matrix elements as listed in Table 11.4. In the case of an isosceles triangle the eigenvalues reduce to a simple formula... 

The conventional interaction which is usually assumed to take place between RE-ions and conduction electrons is the isotropic exchange interaction of the form -IJnsSii. Here s, S t denote the conduction electron and RE-ion spin. In terms of electron creation and annihilation operators c, Cka the interaction hamiltonian is written as... 

The B% can be directly expressed in terms of /-operator products. Within the formalism just outlined the spherical Coulomb interaction is specified by (A, S) = (0, 0) with / = /. The isotropic exchange interaction with a q-independent /ex is characterized by A, S) = (0, 1) and / = / = 0. yl = 0 implies that it is not depending on the orbital moment and 2 = 1 states that the spin part transforms like a vector (tensor of rank equal to one). 

The physical meaning of and f L.., is obvious they govern the relaxation of rotational energy and angular momentum, respectively. The former is also an operator of the spectral exchange between the components of the isotropic Raman Q-branch. So, equality (7.94a) holds, as the probability conservation law. In contrast, the second one, Eq. (7.94b), is wrong, because, after substitution into the definition of the angular momentum correlation time... 

In noncubic materials, A must be replaced by the 3x3 exchange-stiffness tensor A v, and the energy is X J A v dM/dxy dM/dxv dV. Here the indices ju and v denote the spatial coordinates x, y, and z of the bonds. The energy is anisotropic with respect to the nabla operator = d/d (bond anisotropy) but isotropic with respect to the magnetization M. By contrast, the relativistic anisotropic exchange Xa i Ha(i VMa VMP dV is isotropic with respect to V but anisotropic with respect to M. 

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... 

The expressions assumed for the angular dependence of the free energy and the simple relations (eq. 6.42) between the coefficients k/" and Bf" are strictly valid only in the limit where the anisotropy is much smaller than the isotropic two-ion exchange interaction. Under these conditions the expectation values of the Stevens operators renormalize with temperature in the straight-forward manner calculated by Callen and Callen (1%5, 1966) ... 

Eu +, Gd +, Tb + ions have a half-filled 4f-shell with zero total orbital moment and spin 5 = I in the ground state. The pure spin magnetic moment of 5-ions is represented by an operator is =gfJ,BS with the g-value close to 2. The exchange interaction of 5-ions is supposed to be isotropic and is presented in the Heisenberg form ... 

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