Formula van Vleck

The field dependence of the energy levels of magneto-active particles (atoms, molecules, clusters) can be expressed through a Taylor expansion. This leads to the appearance of several terms linear in B—the first-order Zeeman term, quadratic in B—the second-order Zeeman term, etc. [2] 

To this approximation the thermal average magnetisation becomes expressed as follows 

This derivation, evidently, is well valid for substances obeying a linear dependence of the magnetisation upon the applied magnetic field (paramagnets), and this cannot be overemphasised. Now the expression for the mean magnetic susceptibility is straightforward 

The van Vleck formula can be improved by maintaining some more terms in the numerator as well as in the denominator in the expression for the magnetisation. The Zeeman coefficients, however, should be accessible and the derivative of the magnetisation provided numerically. Then the magnetic susceptibility becomes a function of the magnetic field. 

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The 7Li resonance in zeolites is also difficult to interpret, even though the quadrupole moment is much lower. Lechert et al. (227) believe that the 7Li linewidth is controlled by the dipole-dipole interaction with 27A1 nuclei in the aluminosilicate framework. According to Herden et al. (232) the increase of 7Li frequency from 9 to 21 MHz does not affect the second moment of the spectra in zeolites Li-X and Li-Y, which means that the quadrupolar interaction is small. The second moment was also independent of the Si/Al ratio. The mean Li-Al distance calculated from the van Vleck formula was 2.35 A. Small amounts of divalent cations reduce the movement of Li + considerably, with the activation energy for this process increasing from 30 to 60 kJ/mol. 

Then the van Vleck formula applies for the magnetization of a paramagnetic system in the absence of a permanent magnetic moment ... 

The situation with a pair of spins is illustrated in Fig. 1, where the static field B0 is along the z-axis. To first order and neglecting terms in the expansion of Eq. (2) which lead to a change in nuclear quantum numbers of 1 or 2 (i.e. Am — 1 and + 2 transitions), the effect of is to split the Zeeman levels into many closely spaced energy levels, thereby causing a distribution of resonant frequencies and consequently a broad line. Eq. (2) has been simplified by the van Vleck formula 2... 

Application of the Van Vleck Formula to an Isolated, Spin-Only Metal Complex... 

For many (but not all) first row metal ions, A is very small and the spin and orbital angular momenta of the electrons operate independently. For this case, the van Vleck formula (equation 20.16) has been derived strictly, equation 20.16 applies to free ions but, in a complex ion, the crystal field partly or fully quenches the orbital angular momentum. Data in Tables 20.7 and 20.8 reveal a poor fit between observed values of p.eff and those calculated from equation 20.16. 

The van Vleck formula presupposes knowledge of the orientation of the principal molecular susceptibilities. For molecules lacking the symmetry a generalised expression has been derived [3]. 

It has been claimed that the residual magnetisation disappears in the zero field and coefficients of higher powers of the field are negligible. The generalised van Vleck formula for the magnetic susceptibility is [3]... 

This reduces to the common van Vleck formula when applied to a diagonal element of the susceptibility tensor... 

Only the c(0), c(l) and c(2) coefficients enter the van Vleck formula for the components of the molar paramagnetic susceptibility which relaxes to the polynomial fit formula of the form [5]... 

This is how the limitations of the van Vleck formula can be overcome. The evaluation of the mean magnetic susceptibility is, as above,... 

Another elegant derivation of the Curie law is based on the van Vleck formula (introduced in Section 6.1). The Zeeman coefficients for this particular case are... 

Consequently the magnetic susceptibility is evaluated either via the van Vleck formula (mean magnetic susceptibility) or by differentiating the molar magnetisation (differential magnetic susceptibility). 

In this most realistic case the van Vleck formula adopts the form of... 

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... 

With the van Vleck coefficients determined, one can apply the van Vleck formula to the parallel and perpendicular components of the magnetic susceptibility, hence... 

The magnetic susceptibility, resulting from the van Vleck formula, is... 

If the first excited states are assumed to be too high in energy to couple with the ground state, g is in principle isotropic and equal to g, . As E, is linear in H, the above Van Vleck formula [Eq. (12)] with = 0 can be used. 

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