Specific heat three-dimensional crystals

We now consider exchange interactions which extend throughout a three-dimensional crystal lattice. Obviously, the transition to this long-range order is a phase transition and as such it is characterized by a specific heat anomaly and peculiar behaviour of the magnetic susceptibility. As in section 1.1.3.7, the interaction may be correctly described by the Hamiltonian Eq. (49). However, there exists no exact solution to this Hamiltonian for a three-dimensional problem, although numerical calculations are available [163]. It is therefore customary to discuss the properties of three-dimensionally ordered substances on a classical basis. 

In Chap.5, anharmonic effects are considered. After an illustration of anharmonicity with the help of the diatomic molecule, we derive the free energy of the anharmonic linear chain and discuss the equation of state and the specific heat. The quasi-harmonic approximation" worked out in detail for the linear chain is then applied to three-dimensional crystals to obtain the equation of state and thermal expansion. The self-consistent harmonic approximation" is the basis for treating the effects of strong anharmonicity. At the end of this chapter we give a qualitative discussion of the response... 

We have derived an expression for the specific heat of the anharmonic linear chain which is valid at high temperatures (5.65) due to anharmonicity, Cj (T) is proportional to T at high temperatures. The same result also applies for the specific heat c (T) of three-dimensional crystals [5.31,32]. As in (5.65), the coefficient of proportionality is determined by the third and fourth-order anharmonic coupling constants. In addition, the generalization of the expression (5.67) for Cj - Cj to cubic crystals is straightforward, namely,... 

Tarassov (1955) and also Desorbo (1953) have considered these ideas in relation to a onedimensional crystal in which case the one-dimensional frequency distribution function predicts a T dependence of the specific heat at low temperatures. In the case of crystalline selenium, however, it has been found necessary to combine the one-dimensional theory with the three-dimensional Debye continuum model in order to obtain quantitative agreement with the data below about 40° K. Tem-perley (1956) has also concluded that the one-dimensional specific heat theory for high polymers would have to be combined with a three-dimensional Debye spectrum proportional to T3 at low temperatures. For a further discussion of one-dimensional models see Sochava and TRAPEZNrKOVA (1957). 

Molecular sieve zeolites constitute a class of stationary phase that combines exclusion with specific adsorption properties. These materials, which are crystalline aluminum silicates (commonly sodium or calcium aluminum silicates), have rigid, highly uniform three-dimensional porous structures containing up to 0.5ml/g of free pore volume, resulting when water of crystallization is removed by heating. Although munerous natural zeolites are known, most practical work is done with... 

As discussed in Section 3.5 for tridimensional lattices, v 0 for k —> 0 for the three acoustical branches with a positive slope and with a discontinuity at k = 0 (T). The shape of g(v) 0 has been matter of strong interest in physics for the calculation of specific heat and related thermodynamic quantities. As already mentioned for one-dimensional lattices in vacuo, some of the acoustical branches tend asymptotically to zero. The derived calculated g(v) shows a very strong singularity at v = 0. Such a singularity is meaningless and only due to the limitation of the molecular modes adopted in the calculations (ID lattice) which is unable to account for intermolecular forces. Also, for real polymer samples the experimental g(v) —> 0 for k 0 as expected for classical crystals since they do interact with the neighboring chains with very weak intermolecular forces. 

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