Elastic constants cubic crystals

Table 1 Hsts the properties of several semiconductors relevant to device design and epitaxy. The properties are appropriate to the 2incblende crystal stmcture in those cases where hexagonal polytypes exist, ie, ZnS and ZnSe. This first group of crystal parameters appHes to the growth of epitaxial heterostmctures the cubic lattice constant, a the elastic constants, congment sublimation temperature, T. Eor growth of defect-free...

In table 2 and 3 we present our results for the elastic constants and bulk moduli of the above metals and compare with experiment and first-principles calculations. The elastic constants are calculated by imposing an external strain on the crystal, relaxing any internal parameters (case of hep crystals) to obtain the energy as a function of the strain[8]. These calculations are also an output of onr TB approach, and especially for the hep materials, they would be very costly to be performed from first-principles. For the cubic materials the elastic constants are consistent with the LAPW values and are to within 1.5% of experiment. This is the accepted standard of comparison between first-principles calculations and experiment. An exception is Sr which has a very soft lattice and the accurate determination of elastic constants is problematic. For the hep materials our results are less accurate and specifically in Zr the is seriously underestimated. ... 

For interpreting indentation behavior, a useful parameter is the ratio of the hardness number, H to the shear modulus. For cubic crystals the latter is the elastic constant, C44. This ratio was used by Gilman (1973) and was used more generally by Chin (1975) who showed that it varies systematically with the type of chemical bonding in crystals. It has become known as the Chin-Gilman parameter (H/C44). Some average values for the three main classes of cubic crystals are given in Table 2.1. 

However, in a cubic structure the value of G will be equal to C44 only when slip is on the 110 <001> slip system (Kelly et al 2000). In rocksalt-structured nitrides and carbides, slip in indentation at room temperature occurs on the 110 <110> slip system (Williams and Schaal, 1962 Molina-Aldareguia etal., 2002). The appropriate value of 6 is related to the different single crystal elastic constants, cy, by... 

A body will obey Young s modulus only if it is stretched or compressed within its elastic limit if this limit is exceeded, structural failure ensues. For a one-dimensional system, or for a cubic crystal, Young s modulus reduces to the Hooke s law constant kH ... 

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. 

We have considered MOFs with face-centered cubic (FCC), simple cubic (SC) or body-centered cubic (BCC catenated) crystal structures. The electronic properties have been analyzed for the FCC unit cells only. The bulk moduli B were calculated from the elastic constants, which have been obtained by calculating the total energy change after applying a suitable strain to the system. Furthermore, we have employed molecular dynamics (MD) simulations to check the thermal stability of the MOFs. 

The anomalous temperature dependence of elastic constants found from the CJTE calculations is shown for transition metal compounds of the NiCr204 and CuCr204 type crystals on the Fig. 4. As in this case the soft acoustic mode is double degenerate there is a splitting of the high symmetry modulus of elasticity Ci of the cubic crystal phase in C2 and C3 constants of the tetragonal phase. 

Fig. 4 Temperature dependence of the elastic constants Cj, C2, and C3 in crystals with the cubic to tetragonal phase transition (0 = go, Td = A [15])...

Thus one would expect from a (6x6) matrix of the elastic stiffness coefficients (c,y) or compliance coefficients (sy) that there are 36 elastic constants. By the application of thermodynamic equilibrium criteria, cy (or Sjj) matrix can be shown to be symmetrical cy =cji and sy=Sji). Therefore there can be only 21 independent elastic constants for a completely anisotropic solid. These are known as first order elastic constants. For a crystalline material, periodicity brings in elements of symmetry. Therefore symmetry operation on a given crystal must be consistent with the representation of the elastic quantities. Thus for example in a cubic crystal the existence of 3C4 and 4C3 axes makes several of the elastic constants equal to each other or zero (zero when under symmetry operation cy becomes -cy,). As a result, cubic crystal has only three independent elastic constants (cu== C22=C33, C44= css= and Ci2=ci3= C2i=C23=C3i=C32). Cubic Symmetry is the highest that can be attained in a crystalline solid but a glass is even more symmetrical in the sense that it is completely isotropic. Therefore the independent elastic constants reduce further to only two, because C44=( c - C i)l2. 

As was discussed in some detail in chap. 2, the notion of an elastic solid is a powerful idealization in which the action of the entirety of microscopic degrees of freedom are subsumed into but a few material parameters known as the elastic constants. Depending upon the material symmetry, the number of independent elastic constants can vary. For example, as is well known, a cubic crystal has three independent elastic moduli. For crystals with lower symmetry, the number of elastic constants is larger. The aim of the present section is first to examine the physical origins of the elastic moduli and how they can be obtained on the basis of microscopic reasoning, and then to consider the nonlinear generalization of the ideas of linear elasticity for the consideration of nonlinear stored energy functions. 

The nickel aluminide NijAl - known as the y phase - crystallizes with the cubic LI2 structure (CujAu-type) which results from the fc.c. structure by ordering (see Fig. 1). Deviations from stoichiometry are accommodated primarily by antisite defects (Lin and Sun, 1993). The density of NijAl is 7.50 g/cm (see Liu et al., 1990) and thus is only slightly lower than that of the superalloys (see Table 2) which, however, is still of interest. The elastic constants have been studied experimentally and theoretically by various authors (e.g. Davies and Stoloff, 1965 Dickson et al., 1969 Kayser and Stassis, 1969 Foiles and Daw, 1987 Wallow et al., 1987 Yoo and Fu, 1991, 1993 Yasuda et al., 1991a, 1992). Young s modulus of cast polycrystalline NijAl at room temperature is about the same as that of pure Ni with a weaker temperature dependence (Stoloff, 1989),... 

All these results apply to a completely general triclinic crystal system whose elastic properties are expressed by the twenty-one independent quantities Cy or Jy. For crystals of higher symmetry there are further relations between the Cy or 5y which reduce their number still further. For the hexagonal and cubic systems these relations are illustrated in fig. 8.1, together with similar relations for a completely isotropic, non-crystalline material. It can be seen that for a hexagonal crystal like ice there are only five non-zero independent elastic constants Jn, i3> % and 44 or the corresponding Cy. 

Let us consider cubic crystals in some more detail. From Table 2.3, it is found that three elastic constants are needed for cubic crystals, c, c,2and c. This is a result of the three independent modes of deformation in this crystal system. The first mode is dilatation by a hydrostatic stress (o-=cT =a-2=o-3). For this case, using Eq. (2.53) and Table 2.3, Hooke s Law for cr, is given by... 

As with other anisotropic materials, Hooke s Law for cubic crystals may be expressed in terms of the engineering elastic constants. Equation (2.57) can be written as... 

Table 2.4 Elastic stiffness constants of various cubic crystals... 

If an accurate equation for the interatomic potential is known, the elastic constants can be calculated from first principles. This analysis is straightforward for cubic ionic crystals. The potential for a pair of positive and negative ions is often written in the form... 

How could you recognize from the array of elastic constants that a single crystal would produce a shear strain in response to a normal stress The Zener ratio is the ratio of the shear moduli for deformation on which crystallographic planes in a cubic crystal (include direction of displacement) ... 

This simplified equilibration process holds strictly for a cubic crystal in a general non-cubic case, the volume, static pressure and bulk modulus have to be replaced by the strain components (related to unit-cell parameters), stress components and elastic constants, respectively. The computer program PARAPOCS[17] performs the lattice-dynamical, thermodynamical and quasi-harmonic calculations in the general tensorial formalism, and has been used to obtain all results reported below for calcite and aragonite. 

With few exceptions, we shall idealize the elasticity of solids as isotropic, as stated earlier, so as not to burden the discussion of the physical mechanisms with inessential operational detail. We note here, however, that many cubic crystals are quite anisotropic. Tungsten, W, which is often cited as being isotropic, is so only at room temperature. Thus, we shall make use principally of the elastic relations in eqs. (4.15) and (4.16), unless we are specifically interested in anisotropic solids such as some polymer product that had undergone deformation processing. The relationships among various combinations of elastic constants of isotropic elasticity are listed in Table 4.1 for ready reference. 

The number of the elastic constants decreases for crystals with the cubic lattice due to the symmetry to three independent components Qi, Cu and C44. 

For single crystals with transverse dimensions large enough to permit a plane wave condition to be attained, the results are unambiguous and virtually free from theoretical assumptions. Five independent elastic constants (stiffnesses or compliances) are required to describe the linear elastic stress-strain relations for hexagonal materials. Only three independent constants are required for cubic (y-Ce, Eu, Yb) materials. Since there are no single crystal elastic constant data for the cubic rare earth metals, this discussion will concentrate on the relationships for hexagonal symmetry. 

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