Spin-only equation

The previous section has shown that even if there is an orbital contribution to the magnetic moment it may be reduced by covalency and by distortions. What if we assume that there is no orbital contribution at all—that it is completely quenched In this section this problem is considered and we will 

We are interested in the case in which there are n unpaired electrons, each with spin j, in an isolated ion which couple to give a resultant, denoted S, where S = njl. The allowed components of S along the direction (z) of a magnetic field, when such a field is applied, are 

The average magnetization per molecule will be given by an expression of the form 

X (magnetic moment of a molecule) x (number with this moment) 

Assuming that in a macroscopic sample there is a Boltzmann distribution of ions over the various 5 levels, we have that the number of ions with moment will be e p 2S H/kT). Using this and eqn 9.6 together 

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Thus we have reduced our expression to one involving simply the number of unpaired electrons. Using this spin-only Equation (7.3), the value of pi can be readily calculated and predictions compared with actual experimental values (Table 7.8). 

The equality sign in the spin-only equation has been replaced by a proportionality because, whereas there can be no ambiguity about how a given value of S arises—one simply has to count the number of unpaired electrons and divide by two—as has been seen, a given J value can arise from a variety of different spin and orbital components. Spin magnets and orbital magnets are not immediately interchangeable and so the constant of proportionality in the J-only equation, denoted g, has to reflect the particular mix involved. The equation for g is... 

Equation (S6.1) is applicable to the salts of lanthanide ions. These have a partly filled 4f shell, and the 4f orbitals are well shielded from any interaction with the surrounding atoms by filled 5.9, 5p, and 6.9 orbitals, so that, with the notable exceptions, Eu3+ and Sm3+, they behave like isolated ions. For the transition metals, especially those of the 3d series, interaction with the surroundings is considerable. Because of this, the 3d transition-metal ions often have magnetic dipole moments corresponding only to the electron spin contribution. The orbital moment is said to be quenched. In such materials Eq. (S6.1) can then be replaced by a spin-only formula ... 

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... 

Octahedral complexes of these ions show magnetic moments somewhat lower than the spin-only value for the 4A2g term as expected from equation (73), the decrease being the greater as the value of X rises. For the ions listed, /refr 3.7, 3.7, 3.7—3.0 and 3.2 BM. For the very high values of X, equation (73) ceases to be adequate. 

Square-planar Co11 complexes also give an S = A ground term but, apparently because of low-lying excited rf-orbitals, give magnetic moments well above the spin-only value, with behaviour corresponding to equation (75). /xeff is commonly 2.2 BM, but Co(phthalocyanine) is an extreme example, with efr =2.8 BM. 

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,. 

Expanding the wave function in a linear combination of pure spin functions could yield the correct secular equations and thus correct eigenvalues. However, such spin-only wave functions could not be considered complete since complete wave functions must describe both the spatial and spin motions of electrons and must be antisymmetric under exchange of any two electrons. It would be better to rewrite the VB model (18) in the second quantization form as given in Eq. (20), in which its eigenstates can be taken as a linear combination of Slater determinants or neutral VB structures. Then... 

It can be seen that for the iron-group ions, where the 3d electrons are "exposed," the LS coupling implied by Eq. (5.12.13) overestimates the measured moment, while the modified Eq. (5.12.16) is closer to experiment this equation assumes that the orbital angular momentum quantum number L has no effect, or is "quenched," and that the observed paramagnetism for the iron-group ions is "spin-only." In contrast, for the rare-earth ions, for which the 4/... 

This last equation is the spin-only formula where the Lande g factor has been introduced to take account of any spin-orbit coupling with excited states. The Lande g factor is given by... 

The equations for other solution averaging conditions are given in references (18) and (19). In the special case of a spin-only state with an isotropic g-tensor (i.e. gy = gi and g n = gsi = 2) equation (9) reduces... 

In practice, spin-orbit coupling is small for first-row d-block metal ions, so that equation 7.1 may be applied. Furthermore, the d orbitals of a transition metal ion usually show significant interactions with the ligand orbitals. In such cases, the orbital angular momentum contribution is usually quenched so that gS(S + 1) L(L +1). Where this happens, equation 7.1 can be simplified to the spin-only formula shown in equation 7.2 ... 

In Bohr magnetons, the gyromagnetic ratio, g, is 2.00023, frequently rounded to 2. The equation for the spin-only moment p-s, then becomes... 

The EPR spectrum [17] confirms this conclusion - showing the anisotropic low-spin signal exclusively (g = 2.014, g = 2.256 see the 77K-spectrum of Cs2KNiF6 in Fig. 3 for comparison), even up to 298 K. The deviation of g from the spin-only value ((5g = 0.012) allows an estimation of the quartet-doublet separation energy for large values the following equation is valid ... 

The introduction of an equation involving quantum numbers may be daunting, but we are fortunately able to simplify this readily. Firstly, for first-row transition metal ions, the effect of L on pi is small, so a fairly valid approximation can be reached by neglecting the L component, and then our expression reduces to the so-called spin-only ... 

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