Zeeman second-order

First-order means that we consider nothing beyond that described here. In second-order , we would include the effects of mixing between ground and excited states brought about by the magnetic field. This is briefly discussed under second-order Zeeman effects later. 

The perturbing Zeeman interaction has no elements on the diagonal, so there is no first-order correction. The second-order corrections are... 

For / = 0 there is obviously no first-order Zeeman splitting however, application of a magnetic field can result in second-order splitting. As such, it is necessary to evaluate the corresponding factor, which isg(j = 2 + 7,(2 + S). 

Let us calculate the frequencies of transitions between Zeeman eigenstates s) and r), assuming that the nuclei are only subjected to an isotropic chemical shift and the first- and second-order quadrupolar interaction. As seen in Sect. 2.1, the Hamiltonian that governs the spin system in the frame of the Zeeman interaction (the rotating frame) is... 

The temperature dependence (296-330 K) of the ring methyl proton resonances of these monomeric heme complexes in the hydrophobic micellar cavity shows [22] a small deviation from the Curie law as in the low-spin complexes in organic and simple aqueous solvents [1, 52]. The origin of such deviation has been variously ascribed [3, 1, 53] either to aggregation or second order Zeeman (SOZ) effect or presence of low-lying spin-quartet state. Since these low-spin hemes in micellar solutions are in deaggregated form, the deviation may be due to the SOZ and/or presence of low-lying excited state. 

The explanation of this contradiction, which seemed unlikely at the time it was proposed, is that the energetically distant levels 2Z , and 2At must be included in the calculation they contribute through spin-orbit coupling and in the second-order Zeeman effect. Their effect (22), though small, is important and can be seen from Fig. 6. 

Figure 27 Splitting of a free-ion state into 27+1 components by a magnetic field (a) for J even, (b) for J odd (i) for the first-order Zeeman effect (ii) including the second-order Zeeman effect...

The first- and second-order Zeeman effect coefficients in the expansion of equation (62) are defined by the quantum numbers which specify the atomic energy level. They are in general a function of the direction of the magnetic field with respect to the axis of quantization of the wave functions. They are obtained by the use of the magnetic moment operator for the appropriate direction, q = x,y ox z ... 

Note that if L = 0 in equation (67) then J = S and g = 2, the spin-only-value . The energy pattern of Figure 27 is linear in H there is no second-order Zeeman effect unless other states are considered. Application of equation (64) to this system is fairly straightforward since it yields... 

For state separations of the order of kT the first- and second-order Zeeman effect coefficients for equation (65) are both important and the exponents vary with temperature there results complicated expressions for the magnetic behaviour for which a general expression has been developed.2-28... 

Fig. 3. Energy level diagram for a spin f nucleus showing the effect of the first-order quadrupolar interaction on the Zeeman energy levels. Frequency of the central transition (shown in bold lines) is independent of the quadrupolar interaction to first order, but is subject to second-order quadrupolar effects (see text).

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]

The procedure based on the direct use of the g tensor anisotropy and Eq. (2.24) is quite common for S = xh systems, since g values from frozen solutions are easily obtainable. In this case, both the second order Zeeman contributions and possibly the effects of temperature on the g values are neglected. Furthermore, the directions of the molecular axes are arbitrarily assumed unless single-crystal data are available. Attempts are available in the literature regarding low spin cobalt(II) [77] and copper(II) [61]. 

The last term in Equation (2.36), Agoz/soc, is a second order contribution arising from the coupling of the orbital Zeeman (OZ) and the spin-orbit coupling (SOC) operators. The OZ contribution in the system Hamiltonian is ... 

The Zeeman effect must be mentioned in the case of nitrogen it behaves normally when 77 = 0 but, in the general case of a non-zero asymmetry, the Zee-man part of the hamiltonian no longer commutes with the quadrupolar part and there appears to be no first-order Zeeman effect, The second-order treatment of the perturbation yields the following values for the transition frequen cies 81 ... 

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