Zeeman effect 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 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... 

For two of the lanthanoid M ions, equations (68) and (69) do not hold. They are Eu, and Sm, / ( if2 /,)> where another state lies fairly close above the ground state which then cannot be considered in isolation. For the Fq state, / = 0 and fEu and /u,ed are predicted to be zero. In fact, xeu 100 mm mor independent of temperature. This effect arises because the Fi state, a few hundred cm higher, interacts with the ground state to give substantial second-order Zeeman effect terms in equations (64) and (65) and, as there is no first-order term and the higher... 

Components of the Q tensor along the principal inertial axes a, b, c (b bisects the bond angle, c 1 molecular plane) from the first- and second-order Zeeman effect of several pure rotational transitions [26, 27,32,37] areQaa = -1.6 1.4, Qbb= + 2.1 1.1, and Qcc = -0.5 1.9. 

The magnetic susceptibility anisotropies 2xaa Xbb Xcc = 8.8 1.4 and 2xbb Xaa Xcc = - 4.4 0.7 follow directly from the second-order Zeeman effect of several rotational transitions [1,26, 27], see also the microwave spectral tables [3]. The theoretical anisotropies -12.26 and -2.03 have been derived from the respective components [28]. 

In these expressions, w is a numerical coefficient which is a complicated function of both C and t. H is the field strength—that is, the first-order Zeeman splitting is proportional to the field strength—and P is the Bohr magneton. The first-order Zeeman effect has removed all degeneracies. Inclusion of the second-order Zeeman effect gives... 

These equations are quite horrific in appearance and we have considered a particularly simple case The important thing to remember is that the second-order Zeeman effect is characterized by terms which depend on the square of the field strength. The splitting of the six t2g functions we have just discussed are shown schematically in Fig. AlO.l, which has previously appeared as Fig. 9.2. 

If we consider a temperature range where only the ground crystal field state is populated and there is no second-order Zeeman effect, the molar susceptibility in the z-direction may be written as ... 

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