Matrix isotropic exchange

Some magnetic parameters enter the matrix elements of the Hamiltonian concerned these are magnetogyric factors (g-factors, components of the g-tensors) and the coupling constants (components of the spin-spin coupling tensors D, in the case of an isotropic exchange abbreviated as J). 

In the weak exchange limit the effect of isotropic exchange is small. This is well fulfilled when the magnetic centres are rather far from each other (say 800 pm apart). Thus the eigenvectors S, Ms) of S2 and Sz do not give a correct description of the spin states the proper molecular states are obtained by the diagonalisation of the full Hamiltonian matrix. In fact, the variation method is applied. 

As far as isotropic exchange is concerned the interaction matrix consists of all combinations of the pair-interaction matrices... 

In the case of isotropic exchange one can handle only the z-component of the Zeeman term the corresponding matrix is diagonal. 

Coefficients of the non-zero matrix elements of the isotropic exchange Hamiltonian for general triads of S, = S2 = S3... 

In order to exemplify the construction of the zero-field exchange Hamiltonian matrix let us consider the simplest case of 5, = S2 = 53 = 1 /2. Then the structure of the isotropic exchange matrix is... 

Notice that the matrix elements for the isotropic exchange do not depend upon the magnetic quantum numbers. 

Matrix elements of the isotropic exchange for a general triad... 

Let us assume an equilateral triangle of the ABA type. Then the isotropic exchange matrix is diagonal with matrix elements, , given by Tables 11.3 and 11.4. The molecular-state g-tensors, gs,sn> are easily constructed from the local g-tensors with the help of the combination coefficients collected in Tables 11.5 and 11.6. Since Sj = S2 = SA and 53 = SB, then... 

Now, in the z-direction (the spherical component q = 0) we get the total interaction matrix, involving the isotropic exchange, asymmetric exchange and the Zeeman interaction, in the form... 

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 eigenvalues of the interaction matrix are collected in Table 11.16 along with other relevant data. In the absence of isotropic exchange ... 

Owing to one zero element, all 9/-symbols collapse into 6/-symbols. The tensor rank k = 0 implies a factor s so that the isotropic exchange cannot mix states of different total spin. The simplified reduced matrix elements are collected in Table 11.17. These formulae can be simplified further by substituting the relevant 6/-symbols (Appendix 3). 

Reduced matrix elements of the isotropic exchange, Rijis,n,s,M.S -s. s s>, for with the SiS2Si2S3S4S34SAf) coupling scheme 34... 

Reduced matrix elements of isotropic exchange for finite chains... 

If instead of the isotropic exchange interaction (A, S) = (0,1) we consider the aspherical Coulomb interaction (A, 2 ) = (2,0) we obtain fig. 17.15 instead. The negative sign for y indicates pair-enhancement instead of pair-breaking. Since this is a dynamical effect it vanishes for 5 = 0. The maximum effect occurs for 8 = lOTc. It resembles the one of additional Einstein oscillators put into the matrix. The quantity t, in the definition of y has to be replaced by to which is defined in a similar way as Ts (see eq. (17.66)) but now with Hac instead of He. ... 

Cross-section structure. An anisotropic membrane (also called asymmetric ) has a thin porous or nonporous selective barrier, supported mechanically by a much thicker porous substructure. This type of morphology reduces the effective thickness of the selective barrier, and the permeate flux can be enhanced without changes in selectivity. Isotropic ( symmetric ) membrane cross-sections can be found for self-supported nonporous membranes (mainly ion-exchange) and macroporous microfiltration (MF) membranes (also often used in membrane contactors [1]). The only example for an established isotropic porous membrane for molecular separations is the case of track-etched polymer films with pore diameters down to about 10 run. All the above-mentioned membranes can in principle be made from one material. In contrast to such an integrally anisotropic membrane (homogeneous with respect to composition), a thin-film composite (TFC) membrane consists of different materials for the thin selective barrier layer and the support structure. In composite membranes in general, a combination of two (or more) materials with different characteristics is used with the aim to achieve synergetic properties. Other examples besides thin-film are pore-filled or pore surface-coated composite membranes or mixed-matrix membranes [3]. 

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

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