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Spherical modelThe spherical model in statistical mechanics is a model of ferromagnetism similar to the Ising model, which was solved in 1952 by T.H. Berlin and M. Kac. It has the remarkable property that when applied to systems of dimension d greater than four, the critical exponents which govern the behaviour of the system near the critical point are independent of d and the geometry of the system. It is one of the few models of ferromagnetism that can be solved exactly in the presence of an external field. Additional recommended knowledge
FormulationThe model describes a set of particles on a lattice which contains N sites. For each site j of , a spin σ_{j} which interacts only with its nearest neighbours and an external field H. It differs from the Ising model in that the σ_{j} are no longer restricted to , but can take all real values, subject to the constraint that which in a homogenous system ensures that the average of the square of any spin is one, as in the usual Ising model. The partition function generalizes from that of the Ising model to where δ is the Dirac delta function, are the edges of the lattice, and K = J / kT and h = H / kT, where T is the temperature of the system, k is Boltzmann's constant and J the coupling constant of the nearestneighbour interactions. Berlin and Kac saw this as an approximation to the usual Ising model, arguing that the σsummation in the Ising model can be viewed as a sum over all corners of an Ndimensional hypercube in σspace. The becomes an integration over the surface of a hypersphere passing through all such corners. It was rigorously proved by Kac and C.J. Thompson that the spherical model is a limiting case of the Nvector model. Equation of stateSolving the partition function and using a calculation of the free energy yields an equation describing the magnetization M of the system
for the function g defined as The internal energy per site is given by an exact relation relating internal energy and magnetization. Critical behaviourFor the critical temperature occurs at absolute zero, resulting in no phase transition for the spherical model. For d greater than 2, the spherical model exhibits the typical ferromagnetic behaviour, with a finite Curie temperature where ferromagnetism ceases. The critical behaviour of the spherical model was derived in the completely general circumstances that the dimension d may be a real noninteger dimension. The critical exponents α,β,γ and γ' in the zerofield case which dictate the behaviour of the system close to were derived to be which are independent of the dimension of d when it is greater than four, the dimension being able to take any real value. References


This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Spherical_model". A list of authors is available in Wikipedia. 