Zeeman coefficient

The modeling of the effective magnetic moment is shown in Fig. 15. It can be seen that for a compressed bipyramid (v = AaxA < 0) the effective magnetic moment nearly follows the octahedral pattern. Since the ground state r6 is nonmagnetic in the z-direction (which results from the calculated Zee-man coefficients Zz(r6) = 0 and Zz(r7) = 1.82), the z-component of the susceptibility should change to zero when the temperatureis lowered. In the x-direction, however, a small Zeeman coefficient appears (Zx(/),) = 0.05 and ZX(17) = 0.26) so that the x-componenl of the susceptibility rises when the temperature is lowered (Fig. 16). 

The calculations in the complete d2 space spanned by 45 functions show (Fig. 121) that the lowest multiplet is a nonmagnetic singlet A separated from another nonmagnetic singlet A by an energy gap 834/hc = 13 cm 1 (Fig. 27). This has no parallel to the traditional SH ZFS. Both these singlets, however, possess some quadratic Zeeman coefficients, and the susceptibility components show a complex temperature dependence (Fig. 29). 

With a weaker axial CF the ground multiplet is quasidegenerate, 7T, r2, with the linear Zeeman coefficient Zz = gzMj = 1.71. This is followed by a true Kramers doublet r5 separated by 5 = - 3D. The nonmagnetic singlet /3 lies at 513 = -4D above the ground multiplet. 

The lowest crystal-field multiplets IR-Bethe(Mulliken) x degeneracy, energy (Zeeman coefficient gzMj) ... 

IR-Bethe(Mulliken)xdegeneracy, energy/cm (Zeeman coefficient gz-Mj) ... 

Following the procedures outlined in the calculations presented above, substitution of the Zeeman coefficients and zero-field energies into the Van Vleck equation yields the following expression 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. 

Since the ground state second order Zeeman coefficient is proportional to — 1/A, this vibronic interaction has a profound influence on the magnetic properties of the [Ti(OH2)6] + cation [50, 54, 55], With just the ground state populated, the susceptibility and effective magnetic moment have the form ... 

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. 

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. 

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... 

The terms and in the Van Vleck equation are the first- and second-order Zeeman coefficients obtained from the expansion of the magnetic moment in terms of a power series of the magnetic field. 

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