The Ising Chain

It is always useful to check for the number of independent parameters. For the Brownian chain there is one parameter only, namely Rq. For the expanded chain in general, we find two parameters, Rp and but in the Kuhnian limit 0 we return again to the simple one-parameter case. 

Here (pi denotes the state of particle i and u (pi-i,(pi) is the pair interaction energy. 

The main task in the computation of thermodynamic functions is the calculation of the partition function, denoted Z. In our case, its basic form can be formulated directly, as 

The summation includes all conformational states, here shortly designated by each state being determined by specifying the conformations of all bonds 

The energy for each conformational state of the chain follows from Eq. (2.119). 

Different from the ideal chain description, the persistent chain model is not only able to deal with short chains, but also addresses the effect of temperature. Another model, which will introduced as the Ising chain in the next section, accomplishes these tasks in an even more detailed, rather perfect manner. 

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An exact solution has also been obtained for the Ising chain model, for which the spins are assumed to be aligned along a given z axis, hi this limit, the spins possess an infinite uni-axial magnetic anisotropy with an easy z axis. The corresponding Hamiltonian reads ... 

When the Ising chain (SY spins) interacts substantially with another one (5, spins), forming thus a ladder-type structure, the model Hamiltonian is extended for an interchain coupling parameter. In the simplest case of the S) = Si = 1/2 system the representation of the transfer matrix has the dimension k = 4 x 4. Its eigenvalues are then obtained only numerically and thus the magnetisation and the magnetic susceptibility cannot be represented by an analytic function. However, numerical solution is accessible using computers [17]. 

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