Coefficient van Vleck

Having determined the van Vleck coefficients, the molar magnetization... 

Unlike Van Vleck coefficients, the expression within the brackets in the above equation, i.e., the expansion coefficient corresponding to uniform strains, is independent from n. This results from translational invariance of the coefficients c jr,(An/ta) and the obvious relations ... 

In this equation only Van Vleck coefficients corresponding to the point k = 0 are involved, for which they have the same values at all unit cells. 

Examples of other polynuclear clusters that have more metals, dissimilar metals and/or higher spin metals will be of greater complexity than the simple example above, as they will have a larger number of energy levels to consider and it may be more laborious to specify the correct Van Vleck coefficients for each. The resulting expressions may then involve multiple g values, many exponential terms in the denominator and numerator and appear superficially complex however, the principles for the derivation of such equations are the same as those presented here. Those interested in the methods required to treat the general case are referred to the literature. ... 

This allows to us write approximate formulae where terms proportional to B1 appear. Such expressions are particularly useful in applying the van Vleck equation since the identification of the van Vleck coefficients is an easy task. 

The most frequent treatment is that the components of the magnetic susceptibility are derived for the case of D 0 and E = 0 by utilising the van Vleck equation. For this purpose we need the van Vleck coefficients as they result from the approximate expansion of magnetic energy levels. Based on Table 8.22, the non-zero van Vleck coefficients are arranged in Table 8.23 notice that... 

With the van Vleck coefficients determined, their substitution into the van Vleck equation yields the analytic formulae for the parallel and perpendicular components of the magnetic susceptibility, respectively (Table 8.24). However, one should keep in mind the above-mentioned restriction under which the expansion of the square roots is applicable. 

Van Vleck coefficients for zero-field splitting systems (E = 0)... 

The zero-order van Vleck coefficients are determined as the relative energies of levels of the 25+1 Lj multiplet, as... 

The determination of the second-order van Vleck coefficients E M is the most laborious step. The magnetic field gives rise to matrix elements between the different multiplets of a term the non-vanishing matrix elements are of the form... 

The derived expressions for the van Vleck coefficients allow us to construct the particular susceptibility function for a multiplet... 

It leads to the following zero-order van Vleck coefficient... 

Consequently the first-order van Vleck coefficients become... 

Notice that in the transformed basis set the z-component of the Zeeman operator is no longer diagonal there are off-diagonal matrix elements of the type P i Hf Pj).) The full spin Hamiltonian matrix Hy is then diag-onalised. The obtained eigenvalues ,-(/ ) are expanded in powers of B, yielding the van Vleck coefficients. 

In the presence of the magnetic field the Zeeman matrices are added to the zero-field Hamiltonian matrix. As the z-component of the Zeeman interaction enters the diagonal, factorisation to the 2 x 2 secular equations is possible and from the analytical roots the identification of the van Vleck coefficients is possible... 

An approximate derivation utilises the van Vleck equation in which the van Vleck coefficients occur... 

The application of the van Vleck formula is based on the van Vleck coefficients which are simply... 

However, some second-order van Vleck coefficients are non-zero due to the fact that certain magnetic states are coupled through the Zeeman operator... 

With the van Vleck coefficients determined, the construction of the susceptibility function is an easy task... 

The other second-order van Vleck coefficients can be enumerated analogously (Table 10.13). Then the magnetic susceptibility becomes expressed as... 

The corresponding secular equation has the set of algebraic roots from which the van Vleck coefficients can be identified individually for the parallel and perpendicular directions, respectively. These are collected in Table 10.17. With the van Vleck coefficients evaluated, then the components of the magnetic susceptibility, as they result from the van Vleck formula, are... 

Assuming only axial anisotropy Ex = E2 = 0), the eigenvalues, together with the van Vleck coefficients, are readily obtained (Table 10.18). Substitution... 

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