Susceptibility Curves

The quantity x is dimensionless (since M and H have the same units) but is usually reported in relation to the sample volume and, hence, is reported as volume susceptibility in cgs units of emu/cm, where emu is an abbreviation for electromagnetic system of units or, in reahty, gauss cm per gram. The corresponding value for the volume suscep-tibUity in SI units is 47t times larger than the value in cgs units. The molar susceptibility, Xm, may be obtained by multiplying x by the molar volume. 

In a single crystal, M is, in general, not parallel to H. The susceptibility, therefore, must be defined by the magnitudes and directions of its principal susceptibilities Xx, Xy, Xz- Hence, is a second-rank tensor, which hnearly relates the two vectors, M and H. 

Write the expression for the eiiipsoid representing the magnetic susceptibility for a triciinic crystai referred to an orthogonai system of coordinates x, y, and z. 

However, only isotropic solids will be considered such as cubic crystals or randomly textured polycrystals, for which the magnetic susceptibility becomes equivalent to a scalar. In a powder with random crystallite orientation, M of the aggregate is in the direction of H, since those components of M transverse to H in the individual crystallites cancel on average. However, the mean value of M for a powder is the mean value of the principal susceptibilities, which are equal in magnitude. Hence, in a powder, x = ( t. + A + Tz)(Nye, 1957). 

Equation 8.7 is the equation of a line passing through the origin (zero intercept) with x plotted on the ordinate and 1 /T plotted on the abscissa. The slope of the line is C, the Curie constant. Equivalently, a plot of 1 /x versus T yields a straight line with slope 1/C (Fig. 8.3b). For the case of classical mechanics, Paul Langevin (1872-1946) later showed that the exact form of the equation for the magnetic susceptibility, in fields insufficient to cause saturation, is  

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Fig. 5.7 (A) TEM micrograph of silica-alginate-Co hybrid nanocomposites and (B) corresponding Zero Field Cool-Field Cool (ZFC/FC) susceptibility curves (H = 500 0e) indicating superparamagnetic behavior (Adapted from [34]).

For the Ag(ll at% Mn) sample, the ac susceptibility curve measured on cooling lies below the curve subsequently measured on heating, except close to the lowest temperature (see Fig. 3.12). If the aging at different temperatures is accumulative, the heating curve would appear older than the cooling curve and therefore lower in amplitude. For the Feo.soMno.soTiOs sample this is indeed the case (see Fig. 3.13). The nonaccumulative behavior observed in the... 

The magnetic susceptibility curves presented by Korsukova et al. (1989) show some unusual curvatures and it may be worthwhile to investigate the behavior in more detail, applying different magnetic fields. 

In the case of antiferromagnetic Ising chains, there is a great difference between the parallel and perpendicular components of the susceptibility. The susceptibility curves of a ferromagnetic and an antiferromagnetic interaction are illustrated in Figs. 3 and 4. 

The extension to S > A has been carried out by Weng21 and the resulting susceptibility curves are shown in Fig. 5. 

According to Eibschutz et al. the susceptibility curves of BaFeF4115 show anisotropic behavior (Fig. 16) in the three directions of the crystal and indicate that the moments are... 

In the paramagnetic state, the susceptibility curves of BaMnAIFy (Fig. 10) and BaMnGaFy are extremely similar, obeying in most of the temperature range a Curie-Weiss law C/(T-9) with Curie constants of 4.4 for both compounds and 9 values respectively equal to —35 and -31 K. Both are nicely fitted using... 

The susceptibility curve of Ba2MnFeAl2Fi4 [18] is given in Fig. 19. Above 50 K, the behavior is paramagnetic, the Curie constant C=8.2 allowing to calculate a contribution of 3.8 for the divalent iron. 

The susceptibility curve of Ba2CaCuFe2FI4, given in Fig. 27, is well described down to 25 K by a model considering discrete CuFe2 ferrimagnetic trimers the exchange interactions are simply described by an Heisenberg hamiltonian, but more troubles are encountered for the Zeeman hamiltonian which must account for the g-factor anisotropy of Cu2+ it is then written down as ... 

Figure 9 Calculated reduced susceptibility curves for the one-dimensional Heisenberg model for various S. (Reprinted from Ref. 2 with permission from Springer Verlag)...

Flg. 1. Magnetic susceptibility curves x(T) for three samples of powder... 

Figure 8. Experimental (solid line with solid circles, ) and computed magnetic susceptibility curves (the inset expands the low temperature region). The computed curves are for model spaces 2mbb (+), 3mbb ( ), 4mbb (A), 2mbb-(mbb) (x), and model 2mbb with a scaling factor 1.1 ( ), which reproduces best the experimental curve.

Figure 5.20 Magnetic susceptibility curve for a ferromagnetic material...

Figure 5.21 Magnetic susceptibility curve for an antiferromagnetic material...

Low-temperature turnover in the susceptibility curves required incorporation of a (T - d) term in Van Vleck s equation (11), as well as a small component of paramagnetic impurity. Use of variable field experimental data in this low-temperature region, combined with the thermodynamic form of x, Eq. (12), would have been desirable. Exploration of wide parameter space in the fitting process and use of best-fit contour maps of e/ M vs JcrM yielded the following parameter values ... 

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