Weiss field

The calculated energy of interaction of an atomic moment and the Weiss field (0.26 uncoupled conduction electrons per atom) for magnetic saturation is 0.135 ev, or 3070 cal. mole-1. According to the Weiss theory the Curie temperature is equal to this energy of interaction divided by 3k, where k is Boltzmann s constant. The effect of spatial quantization of the atomic moment, with spin quantum number S, is to introduce the factor (S + 1)/S that is, the Curie temperature is equal to nt S + l)/3Sk. For iron, with 5 = 1, the predicted value for the Curie constant is 1350°K, in rough agreement with the experimental value, 1043°K. 

In the following discussion, we treat the surface effect on the basis of the Weiss field (molecular field) approximation (17-19), assuming no relaxation (fluctuation of the electron spins). In the treatment, the reduced magnetization m (magnetization at a certain temperature divided by that at OK) of the surface ferric ions at temperature T K is described by... 

The results of Weiss field calculation on ferric ions at the surface metal ion sites are given in Figure 6 of ref 4, and the values for room temperature are shown in Figure 10. Since both ferric and pentavalent Sb ions can occupy octahedral or distorted octahedral sites with six ligand oxide ions and bulk hematite is considered to accommodate pentavalent Sb—119 ions in the metal ion sites (3 ), we can estimate STHF interactions on tetravalent Sn-119 ions at the surface metal ion sites of hematite. Using the magnetization of surface ferric ions at room temperature, the STHF magnetic fields on tetravalent Sn-119 ions at the surface sites are calculated to be... 

Fig. 15. Spontaneous magnetization vs. temperature, according to Weiss (655) theory, (a) Graphical solution. The value of M,/Mo is given by the intersection of the two curves, (b see facing page) Reduced-scale plot. The solid lines represent the Weiss field theory for J — 1/2, 1. The experimental (dashed) curves for iron and nickel fit more closely the theoretical curve for J = 1/2. (After Bozorth (92)).

From the Weiss field model, the relation between Jex, the effective exchange integral of equation 90, and Te for the case L — 0 can be derived from the fact that the energy of interaction with the Weiss field, — fx w = —gSfjLBHw, is equal to the exchange energy, —Jex Si-S, = — 2zJ xS2, where z is the number of nearest neigh-... 

It should be realized, however, that the Weiss field model first con-... 

For Jij < 0, the Wa < 0 and since action is equal to reaction, Wij = Wji. Since all interactions between sublattices are assumed to be contained within the Weiss field, it follows that the magnetization and the susceptibility of each sublattice are described by equations 91 and 95 provided the Weiss fields of equation 100 are used. 

Substitution of equation 100 for the Weiss field in equation 95 gives the set of equations... 

Formation of the angle 20 between Mi and M2 is resisted by the Weiss field. At equilibrium, the magnitude of the component of the Weiss field that opposes H is equal to H, so that for small 0... 

Those curves that do not approach T = 0°K with zero slope are not realized in nature. The N6el model is a molecular field model, and is subject to the same criticisms as the Weiss field model for ferromagnets. Kaplan (325) has applied spin wave theory to ferri-magnets and worked out a Bloch Tz/2 law, similar to equation 98, for low temperatures. In this approximation M /M% remains constant,... 

Before consideration is given to the experimental data, it is of interest to consider the connection between the Weiss field constants used by N6el and the Heisenberg exchange integrals. In the non-collinear models, the Heisenberg formalism is generally used. 

The energy of an atom of moment in a Weiss field Hm is equal to the sum of the exchange interactions with its near neighbors. Therefore from equation 90 it follows that... 

It was pointed out in Chapter II that the Heisenberg exchange Hamiltonian of equation 90, which can be directly related to the Weiss field parameters at T = 0°K by equation 94, is an excellent formal expression for the interactions between atomic spins (or moments) of neighboring atoms. There remains the problem of establishing the various spin-dependent mechanisms that contribute to the Jij. In general, there are two types of interaction cation- -cation and cation-anion-cation (or even cation-anion-anion-cation) interactions. 

Besides the external field, B, a mean internal field, BM, is introduced. This is called the Weiss field ... 

It has been pointed out that any relationship between the exchange integral and the Weiss field is only valid at 0 K, since the former considers magnetic coupling in a pair-wise manner and the latter results from a mean-field theory (Goodenough, 1966). Finally, it is also essential to understand that Eq. 8.43 is strictly valid only for localized moments (in the context of the Heitler-London model). One might wonder then whether the Weiss model is applicable to the ferromagnetic metals, in which the electrons are in delocalized Bloch states, for example, Fe, Co, and Ni. This will be taken up later. 

The discussion of the preceding two sections relied on the presumption that localized (atomic-like) moments were present. However, valence s and p electrons are always best described by Bloch fimctions, while 4/electrons are localized and 5/are intermediate. Valence d electrons, depending on the intemuclear distance, are also intermediate -neither free nor atomic-Uke. In such cases, the dilemma is that the Heisenberg exchange interaction of Eq. 8.43, which is the physical basis for the Weiss field, is not strictly applicable in the case of delocalized electrons in metallic systems, in spite of the success of the Weiss model. 

The sublattice magnetisation has been studied as a function of temperature up to the Neel temperature, and good agreement can be obtained with a Weiss-field model if a strong positive B-B interaction between the Ni + and... 

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