- Spherical model
The spherical model in
statistical mechanicsis a model of ferromagnetismsimilar to the Ising model, which was solved in 1952by T.H. Berlin and M. Kac. It has the remarkable property that when applied to systems of dimension "d" greater than four, the critical exponentswhich govern the behaviour of the system near the critical pointare 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.
The model describes a set of particles on a lattice which contains "N" sites. For each site "j" of , a spin which interacts only with its nearest neighbours and an external field "H". It differs from the Ising model in that the 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.
partition functiongeneralizes from that of the Ising modelto
where is the
Dirac delta function, are the edges of the lattice, and and , where "T" is the temperature of the system, "k" is Boltzmann's constantand "J" the coupling constant of the nearest-neighbour interactions.
Berlin and Kac saw this as an approximation to the usual Ising model, arguing that the -summation in the Ising model can be viewed as a sum over all corners of an "N"-dimensional
hypercubein -space. 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
Equation of state
Solving 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
internal energyper site is given by
an exact relation relating internal energy and magnetization.
critical temperatureoccurs 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 temperaturewhere ferromagnetism ceases. The critical behaviour of the spherical model was derived in the completely general circumstances that the dimension "d" may be a real non-integer dimension.
The critical exponents and in the zero-field 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.
* R.J. Baxter, "Exactly solved models in statistical mechanics", London, Academic Press, 1982
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