The Van Vleck Equation

For less than half-filled subshells, the ground state is given by / = L - S, whereas for more than half-filled subshells, J = L + S is the ground state. This leads to predicted values of xT as given in Table 3.2. As a consequence of the coupling rules for /, the largest are associated with greater than half-filled shells. 

Because of the significant orbital contribution, in general, rare earth ions will not obey the Curie law and analytical expressions for x a function of T are rare. Nevertheless, the limiting value of %T at room temperature should approach the values given in Table 3.2. 

To derive expressions for the magnetic susceptibility as a function of temperature for a wide range of situations, including the interactions involved in exchange-coupled systems and zero field splitting, an expression 

The Van Vleck equation then expresses the temperature-dependent magnetic susceptibility as a function of the various E as  

The terms are the energy of the each of the spin levels of the system before application of a magnetic field. The terms and E the first- and second-order Zeeman terms, respectively, quantify the response of the system to the applied magnetic field to the first and second orders. The exponential terms represent the relative population of each of these levels as a function of temperature according to Boltzmann statistics. The derivation of and Tor complex systems using perturbation theory is beyond the scope of this introductory chapter however, for illustration, the equation is applied to three cases to demonstrate its general use. 

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The zero-order ground state spin-orbit wave functions for these systems, obtained as previously described (41), have therefore been used to calculate the magnetic susceptibilities via the Van Vleck equation... 

Eq. (1.36) is called the Van Vleck equation. In that form the Zeeman operator operates only to first order. Indeed, we should include the second order Zeeman term, which allows the interaction between the ground S multiplet (j functions) with all the excited ones ([Pg.18]

Several simple models exist5 that approximately describe the temperature dependence of x for transition metal cations that do not represent spin-only centers. As one example that is applicable to coordination complexes at low temperatures, the Kotani theory6 incorporates the effects of spin-orbit coupling into the Van Vleck equation and describes y(T) as a function of the spin-orbit coupling energy C,. 

While the singlet ground state will be unaffected by an external magnetic field, the S 1 state will become Zeeman split into the Ms =—1,0, 1 sublevels. Thus, the energies of the four possible 15, Ms) states are known (10, 0)) = Jex (ll, + 1)) = —g/rB77 (ll, 0)) = 0 (ll, —1))= I /ab//. When these four expressions are introduced into the Van Vleck equation, the Bleaney-Bowers equation7 is obtained. This equation describes the temperature dependence of the susceptibility (for the zero-field limit), independent of the sign of Jcx ... 

A more general approach43) to the calculation of x which does not rely on the assumption of the Van Vleck equation uses the general definition (2)... 

If a straight line is not obtained from the Curie-Weiss plot, then it is clear that more detailed theory is necessary for data analysis, and the Van Vleck equation or other more sophisticated theory must be used. 

If second-order effects are ignored, substitution into the Van Vleck equation yields for the magnetic susceptibility... 

The steps to be followed may be summarized. Secular determinants must be constructed for each of the doubly degenerate levels in both directions. First-order Zeeman coefficients must be evaluated for each direction. Matrix elements connecting the three secular determinants must be evaluated to yield second-order Zeeman coefficients. The first-and second-order Zeeman coefficients must be substituted into the Van Vleck equation to yield the anisotropic magnetic susceptibilities x and x - Generally, anisotropic magnetic properties are discussed in terms of /x and n since the variation of these anisotropic components are much more easily visuaUzed. 

The Van-Vleck equation [78] and the vibronic levels are used for the calculation of the magnetic moment of the valence tautomeric molecule. The following values of the key parameters were utilized in calculations U3 = 1,050 cm U4 = l,575cm /3 = 40cm J4 = 30cm A3 = 1,145cm A4 = 2,018cm ... 

For the purposes of illustrating the Van Vleck equation, the effects of zero-field splitting for S = 1 can be examined as shown in Figure 3.10. In this case, it is no longer true that all of the terms are equal to 0 and so the Curie law will not be obeyed. With the applied field applied along the z direction, the = 0 state will be unaffected, but the 1 states will be Zeeman split and will vary linearly with applied field. 

It is a relatively simple matter of plugging in these six values into the Van Vleck equation to obtain Details of the derivation of the perpendicular... 

The three principal magnetic susceptibilities Xx, Xy and %z can then be calculated through the Van Vleck equation, which requires the eigenvalues and eigenfunctions of % defined in Eq. (55), and the first and second order Zeeman coefficients. 

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. 

Fig. 8.31. Temperature dependence of magnetic susceptibilities for the 4TXg term at different applied fields m—mean magnetic susceptibility d—differential magnetic susceptibility (solid) a— approximate magnetic susceptibility based on the van Vleck equation.

Fig. 8.32. Temperature dependence of effective magnetic moments for the 4Tlg term at different applied fields m—based on the mean magnetic susceptibility d—based on the differential magnetic susceptibility (solid) a—based on the approximate magnetic susceptibility via the van Vleck equation.

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

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