A mathematical model, the spherical model, of a ferromagnet is described. The model is a generalization of the Ising model; and one-, two-, and three-dimensional lattices of infinite extent can be extensively discussed. A three-dimensional lattice shows ferromagnetic behavior and provides a statistical model of the Weiss phenomenological theory. The limiting free energy appears in a form which contains two of the essential features of the exactly known Ising model results in one and two dimensions. This suggests the probable form of the limiting free energy for the three-dimensional Ising model. A simplified model, the Gaussian model, is briefly discussed because this model also contains some of the significant features of the Ising model. However, the Gaussian model, unlike the spherical model, is not defined for all temperatures.
The behavior of the spherical model of a ferromagnet with an interaction energy between the magnetic spins which varies with distance as $\frac{1}{{r}^{d+\ensuremath{\sigma}}}$ (where
d is the dimensionality of the lattice and $\ensuremath{\sigma}>0$) is analyzed. It is shown that the model exhibits a ferromagnetic transition in one and two dimensions, providing $0<\ensuremath{\sigma}<d$. (The usual spherical model with nearest-neighbor interactions does not have a transition in one and two dimensions.) The critical-point behavior is investigated. It is found that the singularities in the specific heat and susceptibility are dependent on $\ensuremath{\sigma}$ and
d, but the behavior of the magnetization is independent of $\ensuremath{\sigma}$ and
d. In three dimensions the susceptibility diverges as ${(T\ensuremath{-}{T}_{c})}^{\ensuremath{-}\ensuremath{\gamma}}$, where $\ensuremath{\gamma}=1$ for $0<\ensuremath{\sigma}<\frac{3}{2}$, $\ensuremath{\gamma}=\frac{\ensuremath{\sigma}}{(3\ensuremath{-}\ensuremath{\sigma})}$ for $\frac{3}{2}<\ensuremath{\sigma}<2$ and $\ensuremath{\gamma}=2$ for $\ensuremath{\sigma}>2$. The asymptotic form of the spin-spin correlation function $\ensuremath{\Gamma}(\mathrm{r})$ is studied in the neighborhood of the critical temperature
Tc. At
T=Tc, $\ensuremath{\Gamma}(\mathrm{r})$ decays for large
r as $\frac{1}{{r}^{d\ensuremath{-}\ensuremath{\sigma}}}$. Several two-dimensional models with long-range interactions falling off as
r21 in certain directions only are also investigated.
The partition function of the Ising model of ferromagnetism is examined in the limit of high density in the anticipation that in the limit of infinite density one recovers the Weiss molecular field. The formal parameter of expansion is
z1 where
z is the number of spins in the range of the exchange potential (not restricted to nearest neighbor interactions). In the absence of long-range order, only ring diagrams in the cluster expansion contribute. These give a divergence in the specific heat at $k{T}_{c}=\ensuremath{\Sigma}{j\ensuremath{\ne}i}^{}{v}_{\mathrm{ij}}$ where
vij is the exchange potential. This is the molecular field value for the Curie point
Tc. In the presence of a magnetic field the partition function is evaluated for fixed magnetic moment
M in the same approximation,
M being determined by minimization. This results in a susceptibility differing from the molecular field theory and hence an inconsistency in the theory.The inconsistency is traced back to the observation that the acceptance of ring diagrams is equivalent to the gaussian model of Kac and Berlin which violates the sum rule ${{\ensuremath{\Sigma}}_{i=1}}^{N}{{\ensuremath{\mu}}_{i}}^{2}=N$. Here ${\ensuremath{\mu}}_{i}$ is the "spin" per particle and
N is the total number of particles. This condition is reinstated by insuring the sum rule. The result leads to the spherical model. Thus, a consistent high density approximation to the Ising model is the spherical model. Below the Curie point or for fixed magnetic field,
M is again held fixed and only the Fourier components of the spin density with nonvanishing wave vector are "sphericalized." The result leads to a physically acceptable model which becomes the molecular field theory at low temperatures or high fields and deviates in
O(z1) in general. Formally, the results are simply expressed in terms of a temperature dependent Weiss field. These results differ from the ordinary spherical model which is physically unacceptable below the Curie point. However, a molecular field modification of the spherical model due to M. Lax yields the same result when properly interpreted.It is shown that the above results are also valid (to the same approximation) in the quantum mechanical Heisenberg model, for temperatures above the Curie point.
The Curie–Weiss model is an exactly solvable model of ferromagnetism that allows one to study thermodynamic functions in detail, in particular their properties near the critical temperature. In this model every magnetic moment interacts with every other magnetic moment. Because of its simplicity and the correctness of at least some of its predictions, the Curie–Weiss model occupies an important place in the statistical mechanics literature and its application to information theory. It is frequently presented as an introduction to the Ising model or to spin-glass models, and usually only general features of the Curie–Weiss model are presented. In this paper, we discuss the properties of this model in a rather detailed way. We present the exact, approximate and numerical results for this particular model. The exact expression for the limiting magnetic field is derived.
An infinite set of coupled integral equations is found, with the use of a spectral theorem for the correlation functions of the spin-1/2 three-dimensional Ising model. The second equation is linearised to yield the two spin correlation function. The Ward identity is then satisfied. Those approximations which give the Weiss and spherical model results are indicated. The transition temperature is calculated for SC, BCC and FCC lattices using a product average decomposition of the higher-order correlation functions. The scheme is seen to converge toward the respective values 0.738, 0.789 and 0.823 of 4/ beta c J(0) which are within 2% of the exact values 0.752, 0.794 and 0.816.
For pt. I see abstr. A37614 of 1967. High temperature series expansions for the specific heat of the Ising model of a ferromagnet are given for the face-centred cubic, body-centred cubic, and simple cubic lattices. From a numerical study it is concluded that the critical index ( alpha ) is lattice independent and that in three dimensions alpha approximately=1/8. A numerical representation of the specific heat in the range T C <or=T<or= infinity is given in each case, together with estimates of the critical energy, entropy, and free energy. Asymptotic forms for the specific heat and energy are given.
The authors present the results of a study of the free energy and specific heat of the classical vector model of ferromagnetism. High-temperature series for the free energy are presented as far as the 12th term in general spin dimensionality (D) and the 13th term for the case D=3. The techniques used to derive these series are discussed in some detail. The general series is shown to reduce to that for the Ising Model for D=1, and to agree with the expansion of the exact result for the spherical model for the case D= infinity . The cases D=2 and D=3 correspond to the classical planar and classical Heisenberg models, respectively. It is demonstrated that the general series reproduces previous results for both of these models, with the exception of a small disagreement with the calculated 11th terms of Ferer et al. (1971, 1973). In addition, the general series extends the series for each of these models by one term, and independent work for D=3 provides one further term.