Magnetisation formula

This is the desired result which may be substituted into the scattering amplitude formula (6). The resulting scattering formula is the same as found by other authors [5], except that in this work SI units are used. The contributions to the Fourier component of magnetic field density are seen to be the physically distinct (i) linear current JL and (ii) the magnetisation density Ms associated with the spin density. A concrete picture of the physical system has been established, in contrast to other derivations which are heavily biased toward operator representations [5]. We note in passing that the treatment here could be easily extended to inelastic scattering if transition one particle density matrices (x x ) were used in Equations (12)—(14). 

Ignoring for the moment the slightly open loop near zero applied field (Figure 3.24a), the sample appears to be antiferromagnetically ordered in zero applied field and to switch to a magnetic state at a critical field (He,) of around 3000 G. Because there are two unpaired electrons per formula unit (one on the iron and one on the organic acceptor), the saturation magnetisation is expected to be around 2 x (5585 emu G mol ). 

Statistical thermodynamics brings the result that the thermal average of the (molar) magnetisation can be evaluated through the formula... 

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. 

On substitution into the magnetisation formula, and maintaining only terms linear in the field, we get... 

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]... 

The polynomial fit formula can be applied to a more exact evaluation of the magnetisation... 

The magnetisation and magnetic susceptibility in the perpendicular direction can be treated in a similar way but, owing to square roots in the eigenvalue expressions, the final formulae will be more complex. 

Let us assume an S = 3/2 system as an example. The partition function in the perpendicular direction is defined with the help of the exact eigenvalues listed in Table 8.26. Then the corresponding magnetisation becomes as given in Table 8.28. The formula for My differs from that for Mx only in some signs. The averaged magnetisation is... 

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

The field dependence of the magnetisation is better described by applying the polynomial magnetisation formula the M versus B function is fitted to a quadratic equation... 

In evaluating the magnetic susceptibility one can apply the magnetisation formula (Chapter 2). Because the second-order derivatives are zero for the isotropic exchange model, a simplification is obtained... 

Fig. 4.21. Saturation magnetisation (in Bohr magnetons per formula unit) of various spinel ferrites at low temperature, as a function of Zn substitution (Gorier, 1954 Guillaud Sage, 1951).

A linear relationship is obtained with a slope of 7.7, predicting a magnetisation value of 10 /Ib per formula unit for total Zn substitution, X = 1, as shown by the broken lines in Fig. 4.21. This relationship is not followed over the entire composition range, however. As Zn content increases, A-O-B interactions become too weak and B-O-B interactions begin to dominate. Instead of a collinear, antiparallel alignment, a canted structure appears, where spins in B sites are no longer parallel. Fig. 4.22. Evidence of this triangular structure has been observed by neutron diffraction (Satya Murthy et al, 1969) a theoretical analysis showed that departure from collinear order depends on the ratio of the A-O-B to... 

Estimate (a) the saturation magnetisation and (b) the magnetic inductance for the cubic ferrite CoFe204 with the inverse spinel stmcture. Co " " is a d ion. The cubic unit cell has a lattice parameter of 0.8443 nm and contains eight formula units. Assume that the orbital angular momentum is completely quenched in this material. 

Derive a formula for the saturation magnetisation of a cubic ferrite A Fe204, which is only partly inverse. The fraction of Fe " " ions on tetrahedral sites is given by A, where 0 < A < 0.5 (see Section 5.3.10). [Note derivation is not given in the answers at the end of this book.]... 

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