Cubic elasticity

The group-theoretical stiffness parameters can be expressed in the conventional (cubic) elastic stiffness coefficients ... 

The same analysis that has been made here for the zincblende semiconductors can of course be carried out for the wurtzitc structure. For comparison with experiment, another approach is simpler that is to compare the formulae obtained here with effective cubic elastic constants, which were obtained by Marlin (1972b). These were estimates of what the constants of wurtzitc compounds would... 

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. 

Al—Li. Ahoys containing about two to three percent lithium [7439-93-2] Li, (Fig. 15) received much attention in the 1980s because of their low density and high elastic modulus. Each weight percent of lithium in aluminum ahoys decreases density by about three percent and increases elastic modulus by about six percent. The system is characteri2ed by a eutectic reaction at 8.1% Li at 579°C. The maximum soHd solubiHty is 4.7% Li. The strengthening precipitate in binary Al—Li ahoys is metastable Al Li [12359-85-2] having the cubic LI2 crystal stmcture, and the equhibrium precipitate is complex cubic... 

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...

Mechanical Properties. Measuremeat of the mechanical properties of diamoad is compHcated, and references should be consulted for the vahous qualifications (7,34). Table 1 compares the theoretical and experimental bulk modulus of diamond to that for cubic BN and for SiC (29) and compares the compressive strength of diamond to that for cemented WC, and the values for the modulus of elasticity E to those for cemented WC and cubic BN. 

J.N. Johnson, Wave Velocities in Shock-Compressed Cubic and Hexagonal Single Crystals Above the Elastic Limit, J. Phys. Chem. Solids 35, 609-616 (1974). 

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]. 

Table 2 Elastic constants and bulk moduli for 4d cubic elements. Comparison is made between the results of our tight-binding parametrization (TB), first-principles full potential LAP., results (LAPW), where available, and experiment (Exp.). Calculations were performed at the experimental volume.

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. ... 

Beck, et al. have used the permeation technique to study the effect of uniaxial tensile stresses in the elastic region on hydrogen permeation through pure iron, and have shown that it increases with increase in stress. The partial molar volume of hydrogen (cubic centimetres of hydrogen per mole of iron) in ferrous alloys can be evaluated from the variation of permeation with applied stress, and from the relationship... 

Also known for some time is a phase transition at low temperature (111K), observed in studies with various methods (NQR, elasticity measurement by ultrasound, Raman spectrometry) 112 temperature-dependent neutron diffraction showed the phase transition to be caused by an antiphase rotation of adjacent anions around the threefold axis ([111] in the cubic cell) and to lower the symmetry from cubic to rhombohedral (Ric). As shown by inelastic neutron scattering, this phase transition is driven by a low-frequency rotatory soft mode (0.288 THz 9.61 cm / 298 K) 113 a more recent NQR study revealed a small hysteresis and hence first-order character of this transition.114 This rhombohedral structure is adopted by Rb2Hg(CN)4 already at room temperature (rav(Hg—C) 218.6, rav(C—N) 114.0 pm for two independent cyano groups), and the analogous phase transition to the cubic structure occurs at 398 K.115... 

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. 

H. Ledbetter and S. Kim, Monocrystal Elastic Constants and Derived Properties of the Cubic and Hexagonal Elements, in Handbook of Elastic Properties of Solids, Liquids and Gases. 2, 97 (2001). 

A few examples of the moduli of systems with simple symmetry will be discussed. Figure 1A illustrates one type of anisotropic system, known as uniaxial orthotropic. The lines in the figure could represent oriented segments of polymer chains, or they could be fibers in a composite material. This uniaxially oriented system has five independent elastic moduli if the lines (or fibers) ara randomly spaced when viewed from the end. Uniaxial systems have six moduli if the ends of the fibers arc packed in a pattern such as cubic or hexagonal packing. The five engineering moduli are il-... 

Equation (5.2) also implies that a crystalline solid becomes mechanically unstable when an elastic constant vanishes. Explicitly, for a three-dimensional cubic solid the stability conditions can be expressed in terms of the elastic stiffness coefficients of the substance [9] as... 

---