Equation van Vleck

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]

Derivation van Vleck equation (linear magnetics) or expansion of Brillouin function Restrictions n Bg /kT < 1 - low fields and higher temperatures Formula Curie law Mean magnetic susceptibility... 

Derivation perturbation theory for eigenvalues (except S = 1, where the variation method is applied), van Vleck equation (linear magnetics)... 

Derivation perturbation theory for eigenvalues, van Vleck equation Restrictions / gusB - well fulfilled X = /J2S Average molar magnetic susceptibility... 

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 ... 

Penney and Schlapp (35) many years ago discussed the influence of crystal field on the temperature dependence of magnetic susceptibility x in paramagnetic materials. X may be computed from the fundamental Van Vleck equation (36)... 

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

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... 

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