Symmetry elastic constants

Here c are the symmetry elastic constants. They can be expressed in terms of the contracted elastic constants c°. that correspond to the Cartesian strain components... 

For TmCd and TmZn a variety of techniques have been applied to determine the important coupling constants and gj. they are listed in table 3. In addition to the temperature dependence of the symmetry elastic constant Cj (T), the parastriction method, the third-order susceptibility and the magnetic field dependence of the structural phase transition temperature Tq B) have been used. The different experimental methods have been described in sect. 2.4.1. It is seen from table 3 that the coupling constants determined with these different methods are in good agreement with each other. 

The electronic structure of rare-earth atoms with the configuration (Xe)4f"5d 6s gives rise to conduction bands with 5d and 6s character in the intermetallic compounds. The d bands can have a rather large density of states. Therefore the coupling of phonons to these itinerant electrons can be quite strong. In this chapter we review experimental effects caused by the interaction of conduction electrons with sound waves. These experiments mainly concern the anomalous temperature dependence of symmetry elastic constants and the presence of magnetoacoustic quantum oscillations observed in sound velocity and attenuation. 

For the symmetry elastic constants Cii-Ci2, Cu and Cn + 2ci2 (corresponding to tetragonal, rhombohedral and volume strain) the magnetic ion-lattice interactions are given by Mullen et al., (1974) ... 

Mechanical Properties. The hexagonal symmetry of a graphite crystal causes the elastic properties to be transversely isotropic ia the layer plane only five independent constants are necessary to define the complete set. The self-consistent set of elastic constants given ia Table 2 has been measured ia air at room temperature for highly ordered pyrolytic graphite (20). With the exception of these values are expected to be representative of... 

In solids of cubic symmetry or in isotropic, homogeneous polycrystalline solids, the lateral component of stress is related to the longitudinal component of stress through appropriate elastic constants. A representation of these uniaxial strain, hydrostatic (isotropic) and shear stress states is depicted in Fig. 2.4. Such relationships are thought to apply to many solids, but exceptions are certainly possible as in the case of vitreous silica [88C02]. 

The consequences of this approximation are well known. While E s is good enough for calculating bulk moduli it will fail for deformations of the crystal that do not preserve symmetry. So it cannot be used to calculate, for example, shear elastic constants or phonons. The reason is simple. changes little if you rotate one atomic sphere... 

It is also possible that a membrane might have an even lower symmetry than a chiral smectic-C liquid crystal in particular, it might lose the twofold rotational symmetry. This would occur if the molecular tilt defines one orientation in the membrane plane and the direction of one-dimensional chains defines another orientation. In that case, the free energy would take a form similar to Eq. (5) but with additional elastic constants favoring curvature. The argument for tubule formation presented above would still apply, but it would become more mathematically complex because of the extra elastic constants. As an approximation, we can suppose that there is one principal direction of elastic anisotropy, with some slight perturbations about the ideal twofold symmetry. In that approximation, we can use the results presented above, with 4) representing the orientation of the principal elastic anisotropy. 

It can be shown that for the cross-terms 221 = 2i2, 2si = 2b. and so on, so that of the initial 36 values, there are only 21 independent elastic constants necessary to completely define an anisotropic volume without any geometrical symmetry (not to be confused with matrix symmetry). The number of independent elastic constants decreases with increasing geometrical symmetry. For example, orthorhombic symmetry has 9 elastic constants, tetragonal 6, hexagonal 5, and cubic only 3. If the body is isotropic, the number of independent moduli can decrease even fmther, to a limiting... 

In many cases considerable simplification is possible, because of the constraints imposed on the number of independent elastic constants. For cubic symmetry, for which the elastic stiffness tensor has only three independent constants as given in (6.29), the elements of T, are given in Table 11.1(b), and for hexagonal symmetry the elements are given in Table 11.1(c). If c12 = Cn - 2c44 were to be substituted in Table 11.1(b) the isotropic elements would... 

Generally, the elastic properties of crystals should be described by 36 elasticity constants Cit but usually a proportion of them are equal to zero or are interrelated. It follows that in crystals, the tensors (2.6) and (2.7) are symmetric tensors, owing to which the number of elastic compliance coefficients is reduced, e.g., in the triclinic configuration, from 36 to 21 (Table 2.1). With increasing symmetry, the number of independent co-... 

Group Configuration (type of symmetry) Tensor symmetry Number of elastic constants... 

We may then write the arrays for the elastic constants for various symmetries, the two most useful being hexagonal (also transverse isotropy as in fibre symmetry) and isotropic. Hexagonal gives ... 

The elastic constants derived by Van Fo Fy and Savin are as follows. (The symmetry axis is 3, c is the concentration of the circular reinforcing phase in a hexagonal array. The compliance constants Sy are quoted)... 

The strength of the thermal fluctuations t is only rescaled, since there is no non-trivial renormalization of t (i.e., of the elastic constant c) because of a statistical tilt symmetry [39], Note, that (18) is written in rescaled dimensionless parameters and the different renormalization of the kinetic and elastic term is reflected in the different renormalization of v and c, i.e., K and t, respectively. 

Thus it may produce the transitional densities of the alg, egc, and tluz symmetries. At this point selection rules pertinent to the frontier orbitals approximation enter for the 12-electron complexes the symmetries of the frontier orbitals are Th = eg and Tl = ai3, the tensor product Th <8> TL = eg aig = eg contains only the irreducible representation eg so that the selection rules allow only the density component of the egc symmetry to appear. In its turn this density induces additional deformation of the same symmetry. That means that in the frontier orbitals approximation, only the elastic constant for the vibration modes of the symmetry eg is renormalized. This result is to be understood in terms of individual nuclear shifts of the ligands in the trans- and cis-positions relative to the apical one. They, respectively, are ... 

By contrast, for the 14-electron complexes (nontransition nonmetals) the symmetries of the frontier orbitals are I n = aig and rL = l u and the tensor product Th rL = a g 11 = 11 so that only the transition density corresponding to the representation t survive. Analogous moves allow us to conclude that the off-diagonal elastic constant for stretching the /.ran.s-bonds has the form ... 

By means of these elastic constants and the orientation distribution of the symmetry axes with respect to the fibre axis, the Compliance (S = 1/E = reciprocal modulus) of the fibre could be calculated. 

Amorphous solids and polycrystalline substances composed of crystals of arbitrary symmetry arranged with a perfectly disordered or random orientation are elastically isotropic macroscopically (taken as a whole). They may be described by nine elastic constants, which may be reduced to two independent (effective) elastic constants. 

With elastically anisotropic materials the elastic behavior varies with the crystallographic axes. The elastic properties of these materials are completely characterized only by the specification of several elastic constants. For example, it can be seen from Table 10.3 that for a cubic monocrystal, the highest symmetry class, there are three independent elastic-stiffness constants, namely, Cn, C12, and C44. By contrast, polycrystalline aggregates, with random or perfectly disordered crystallite orientation and amorphous solids, are elastically isotropic, as a whole, and only two independent elastic-stiffness coefficients, C44 and C12, need be specified to fully describe their elastic response. In other words, the fourth-order elastic modulus tensor for an isotropic body has only two independent constants. These are often referred to as the Lame constants, /r and A, named after French mathematician Gabriel Lame (1795-1870) ... 

For a crystal having the symmetry of diamond or /.incblende (thus having cubic elasticity), there are three independent clastic constants, c, t 12, and C4.4. The bulk modulus that was discussed in Chapter 7 is B = (c, + 2c,2)/3. We can discuss the bulk modulus, and the combination c, — c,2, entirely in terms of rigid hybrids, and therefore the two elastic constants c, and c,2 do not require deviations from this simple picture. This will not be true for the strain, which is relevant to c 44, and this is a complication of some importance. 

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