Cauchy relation

The failure of the Cauchy relations derives from the three- and four-body interactions, which stem from the overlap terms. The description of the properties of ionic crystals was brought to a new and improved level by Per-Olov s thesis and he developed an arsenal of tools, which were sharpened and extended, throughout his career. 

Finally, substitute the numbers into Eq. 10.53 to obtain 75.2 GPa (1 GPa = 10 Pa). This result is noticeably larger than the experimental value (percent error = 88.2 percent). The difference implies that a pairwise potential is not entirely adequate for describing NaCI, even though the Cauchy relation holds. 

Show, for an isotropic cubic polycrystal for which the Cauchy relations (cn = 3ci2 C44 = C12) hold, that the Voigt and Reuss approximations of the shear modulus reduce to ... 

For most distortions, both overlap interactions and Madelung terms contribute to the elastic constants. However, for the distortion associated with C44 in the rocksalt structure, only the Madelung term enters. (There are no changes in nearest-neighbor distance to first order in the strain.) Thus the experimental values can be compared with the Madelung term, and by virtue of the Cauchy relation, c, 2 should take the same value. That contribution has been calculated by Kellerman (1940) and is Values of this expression arc given in Table... 

Notice that though the energy is written as a central force interaction of two bodies, there are additional terms depending upon volume, so that the system is not in equilibrium under the influence of these interactions alone. Thus the Cauchy relations discussed in Chapter 13 are not expected to be obeyed, and indeed they are not. 

This gives a deviation from the Cauchy relations. 

These are the familiar recurrence relations originally derived by Wagner [11] and Ansbacher [12], Closed formula can also be obtained by means of these techniques, for which we need to introduce the well-known Cauchy relation in the complex variable theory... 

So the FC integral is added to the very few physical systems [18] which are realizations of this particular algebra. Using the Taylor theorem for shift operators due to Sack [19], and the Cauchy relation mentioned above, we can apply this very general idea to the specific case of the harmonic oscillator to obtain the closed formula (5). Recurrence relations can also be obtained by noticing that O is in reality a superoperator which maps normal ladder operators by the canonical transformation ... 

For cases in which central forces exist between atoms in a solid and the atoms are at the crystallographic centers of symmetry, the Cauchy relation, Cj2=c, should be satisfied. In cubic solids, this relation is satisfied only when the Poisson s ratio is exactly 0.25. The Cauchy relationship holds fairly well for many ionic crystals but fails for more covalent materials, such as MgO, Si and Ge. The ratio is sometimes used to estimate the degree of covalency in the atomic bonding. 

Models that include ionic distortions are much more accurate than those based on rigid ions [26].In particular, the many-hody effects introduced by the distortions are necessary to account for the ohserved violations of the Cauchy relations among elastic constants [27, 28]. Since the form of lT(r) is model dependent, we defer its discussion until the following section. 

In the case of FCC transition and noble metals one has to introduce non central Interactions. In fact for central forces should hold the Cauchy relation between the elastic constants c... 

In the case of cubic crystals, the Cauchy relations require that C " 44 so that there are only two independent elastic constants, namely, C 3ind... 

The experimental values of the elastic constants for most alkali halides satisfy the Cauchy relations moderately well (Table 3.2). However, from Table 3.2 it is seen that this is not the case for metals. In an alkali metal such as sodium, it is true that the ions are at centers of inversion symmetry and that the screened Coulomb interaction between the ions is a central force however, the crystal is not in equilibrium under the action... 

Besides the discrepancies between calculated and observed dispersion curves as illustrated in Fig.4.5, there are other deficiencies of the rigid-ion model. Since the model is based on central forces, it predicts the Cauchy relations which for a cubic crystal are C 2 = 44- glance at Table 3.2 shows that these relations are only approximately satisfied for the alkali halides. 

An interesting extension of the shell model is the deformable shell model or breathing shell model introduced by SCHRODER [4.16]. This model allows for radial deformations of the shells in the course of lattice vibrations which leads to three-body interactions and, correspondingly, the model does not predict the Cauchy relations [4.46]. 

Current theoretical research on ionic crystals includes treatment of noncentral or three-body forces and their effects on shear forces (e.g., deviations from the Cauchy relation for elastic constants). Such calculations are of secondary interest from a chemical and structural point of view. 

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