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Using topology, fractal analysis and investigation of lattice formation process we find two types of equivalence transformations among Ising models: topological equivalence transformation and formation equivalence transformation. With the help of the transformations and the known data of the critical points of simple cubic (sc) lattice and planar square (sq) lattice we get directly the critical points for face-centered cubic (fcc) lattice, body-centered cubic (bcc) lattice and diamond (d) lattice. The transformation itself results no error in the calculation. Other than Monte Carlo method and series expansion approach the equivalence transformations help us simplify much more greatly the calculation of the critical points for the three-dimensional models and understand much more deeply the structural connection among Ising models.

Fractal structure is a class of complex ordered structures in nature, which exhibits not simply a higher degree but an altogether different level of complexity. During the 1980s physicists tried to describe phenomena on fractal, they succeeded in calculating some of physical characteristics of fractals [

Mentioning the equivalence transformation we may trace back to the 1940s, Kramers and Wannier discovered a transformation to enable them to get the critical point for square lattice [

Solving of three-dimensional Ising models has more far-reaching significance because the actual ferromagnetic elements have different crystal textures; for example iron is body-centered cubic (bcc) while nickel is face- centered cubic (fcc). The investigation of their structures will help us deeply understand general laws of ferromagnetic. According to our investigation in the 3-dimensional models there is a unique irreducible lattice: the tetrahedron lattice [

In this paper we find two types of equivalence transformations: topological equivalence transformation and formation equivalence transformation, by means of which we get directly the critical points for the fcc, bcc and d lattice systems. In Section 2, we first introduce some new concepts then derive the two types of equivalence transformations. In Section 3, the two transformations are tested and verified and we further compare different theories of studying Ising models making use of the obtained data. The critical exponents are discussed simply. Section 4 is conclusion remark.

In order to designate the relationship between the structure of a particular model and its critical temperature, and to compare the critical points for different models, it is convenient to unify their coupling constants, the applying of the normalized coupling constant is a wisdom choice. In terms of quantum mechanics the coupling constant is the exchange energy [

The

A normalized critical point is defined as

For the fcc,

For the bcc,

For the sc,

It can be seen that different coupling constants provide different critical points for the same configuration, which critical temperature is unique. The normalized critical point for the sc lattice is just its equivalent critical point.

The fcc lattice is a familiar structure for us, its structural diagram is often shown in the books on solid state physics. If we consider merely the nearest neighbor interaction the lattice can be considered a structure made up of infinite parallelepipeds, each of which is a primitive cell for the fcc. According to topology such structure is equivalence to the sc, we may call the fcc an equivalent sc. In a mathematics sense the topological equivalence model has the same fractals as the sc, which means that the sub-block and the block of the fcc are just the sc ones. Therefore, the equivalent critical point for the fcc can be represented by Equation (16) of the reference [

In a similar way, the equivalent critical point for the bcc is given by

Substitution of Equations (4) and (5) puts Equations (7) and (8) into

In the sc system there are infinite horizontal planes parallel to one another and infinite vertical planes relatively parallel, each lattice belongs to not only one horizontal plane but also one vertical plane. Such structure means that the sc is a direct sum of the square lattices (sq) [

A composite lattice cannot be homeomorphism to a simple lattice such as the sc, so we should seek another way. The d lattice as a composite structure can be described as two interpenetrating fcc lattices displaced along the body diagonal of the conventional cube by one-fourth of the diagonal length. The nearest neighbor lattices make up a diamond primitive cell, which is a simple cubic with one lattice at the cube’s center and the rest four lattices at its vertices, two of them on the top surface and the others on the bottom as shown by the figure 23 in the chapter 1 of the reference [

There has been no way to solve exactly the three-dimensional Ising models so far, except our theory [

The magnitudes of the critical points in Equation (12) are greater than that they themselves compute, which may be related to the value of

The magnitudes of the data in Equation (13) are also lager than the author’s. Using our theoretical results

The behavior of the huge fluctuations attracted by a critical point shows the phase transition is irreversible, and the critical point is stable as being a minimum like a valley bottom between mountains. This critical property rules the principle of the method of series expansion in that the asymptotic value should finally go to a minimum after infinite iterating calculation. Such calculation, however, never been met in the practice, since the terms number in all of series expansions are always limited providing that the obtained values have to be regarded as results by man-made extension. This may be the cause that there are slight differences between the magnitudes of the critical points in Equation (14) from our theory and the ones out of the series expansions. An obvious example is the value of the

An equivalence transformation itself does not result in any error, which comes from the initial values of

The formation equivalence makes a composite lattice be simultaneously consistent of two or more lattice systems. Equations (9)-(11) lead to an algebraic expression

Finally, we discuss simply about the critical exponents, which are relative to the series expansions. They should be regarded as the variables describing phenomenologically the critical behaviors without referring to the critical fluctuation mechanism. As we have found out that the heat capacity of a three-dimensional Ising model at the critical temperature is attributed to four types of spin phonons originating in the sub-blocks, the ordered blocks, the lattices in the sub-blocks, and the lattices in the ordered blocks [

We find two types of equivalence transformations among Ising models: the topological equivalence transformation and the formation equivalence transformation. These transformations make us investigate effectively more Ising models in structures, especially for the three-dimensional ones. With the help of our approach we have obtained exact critical points for the bcc, fcc, and d lattice spin systems.