Ising chain model

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 ... 

To obtain a more quantitative picture of the temperature dependence of tr and R IR, specimens of TPBIO were dissolved in a proprietary mixture of low-molar-mass nematics, designated E5, which has a broad nematic temperature range [Chiang et al., 2000]. Experimental results for two TPBIO specimens having DP= 18 and 63 are shown in Eigure 1.10. Eirst, the temperature dependence of R and R may be considered in the context of pertinent statistical theories. An analysis by Halperin and William [1992] of the conformation of main-chain LCPs using an Ising chain model led to the conclusion that the chain dimensions, and R , each exhibit exponential temperature dependence ... 

If the ordered, crystalline regions are cross sections of bundles of chains and the chains go from one bundle to the next (although not necessarily in the same plane), this is the older fringe-micelle model. If the emerging chains repeatedly fold buck and reenter the same bundle in this or a different plane, this is the folded-chain model. In either case the mechanical deformation behavior of such complex structures is varied and difficult to unravel unambiguously on a molecular or microscopic scale. In many respects the behavior of crystalline polymers is like that of two-ph ise systems as predicted by the fringed-micelle- model illustrated in Figure 7, in which there is a distinct crystalline phase embedded in an amorphous phase (134). 

Curie-Weiss behaviour from 309 to 83 K with a Weiss constant 6 s 10 K. The 3-carboxylate was studied in the range 275-4.2 K and the data were interpreted in terms of an antiferromagnetic interaction between silver ions using the Ising linear chain model. This gave an exchange energy of / = —30.8 1.0 cm-1 and g = 2.5 0.02 which was somewhat smaller than the experimentally determined (g) of 2.08.513... 

In the following sections, we will see that even the simplest presently known SCM system can not be considered as a regular ferromagnetic Ising chain. Therefore, some improvements of the previous models are essential to describe experimental magnetic properties. 

Fig. 30. Magnetic susceptibility of Cu3(P04)2. The full lines correspond to the best fits from (1) Heisenberg chain model, (2) Ising model of linearly connected trimers...

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]. 

Although at first sight they appear different, ID Markov chain models are formally equivalent to Ising models. In the Ising formulation, it is more convenient to change from the (0,1) variables to <7, (— 1. +1) variables, where... 

This simple pair-interaction Ising model is equivalent to the Markov chain model of Eq. 9 for mA=0.5. The interaction parameter J is related to the correlation [i by the following ... 

Fig. 179. Ni(CHjCSNHj)j(SCN-)2. Temperature dependence of 1/Xm- Broken curve Ising model. Solid curve Coupled double-chain model, J/k=4.6K, — J /J=0.033, g=2.18 t71E5].

The dissipative transverse-field Ising chain consists of a one-dimensional transverse-field Ising model, as discussed in the first example above, with each spin coupled to a heat bath of harmonic oscillators. The Hamiltonian reads... 

In considering the cooperative behavior of linear chain molecules consisting of identical monomeric units, a particularly simple situation arises if cooperative interactions occur only between immediately adjacent monomeric units. This corresponds to the linear Ising lattice model [18] which is the basis for the Zimm-Bragg thermodynamic theory of the cooperative helix coil conformational transition of linear polymers [3]. Considering the processes in a simple chain of = 4... 

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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