Elastic - constants modes

Kieffer has estimated the heat capacity of a large number of minerals from readily available data [8], The model, which may be used for many kinds of materials, consists of three parts. There are three acoustic branches whose maximum cut-off frequencies are determined from speed of sound data or from elastic constants. The corresponding heat capacity contributions are calculated using a modified Debye model where dispersion is taken into account. High-frequency optic modes are determined from specific localized internal vibrations (Si-O, C-0 and O-H stretches in different groups of atoms) as observed by IR and Raman spectroscopy. The heat capacity contributions are here calculated using the Einstein model. The remaining modes are ascribed to an optic continuum, where the density of states is constant in an interval from vl to vp and where the frequency limits Vy and Vp are estimated from Raman and IR spectra. 

In SmA and lamellar phases, all twist modes are forbidden due to the relatively large values of twist elastic constant (K22), resulting in a different frequency dependence for 2-dimensional DF. Thus, the spectral density J u>) due to 2-dimensional DF (or layer undulations) is given ( T33 [Pg.102]

The position of Ti and Zr is again important in this context. While the b.c.c. phase in these elements has long been known to indicate mechanical instability at 0 K, detailed calculations for Ti (Petty 1991) and Zr (Ho and Harmon 1990) show tiiat it is stabilised at high temperatures by additional entropy contributions arising from low values of the elastic constants (soft modes) in specific crystal directions. This concept had already been raised in a qualitative way by Zener (1967), but the... 

If one side of the quartz is coated with material, the spectrum of the resonances is shifted to lower frequencies. It has been observed that the three above mentioned modes have a somewhat differing mass sensitivity and thus experience somewhat different frequency shifts. This difference is utilized to determine the Z value of the material. By using the equations for the individual modes and observing the frequencies for the (100) and the (102) mode, one can calculate the ratio of the two elastic constants Cgg and C55. These two elastic constants are based on the shear motion. The key element in Wajid s theory is the following equation ... 

Bari Sivardiere (1972) had proposed a model for spin-state transitions based on Chestnut s model for the singlet-triplet transition in radical ion salts (Chestnut, 1964), which included the displacement of the breathing mode, Q, and the elastic constant in the Hamiltonian ... 

Surface acoustic-wave (SAW) elements Plate-mode oscillators Interface impedance elements Fiber optic elements sensitive to elastic constants... 

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 ... 

In our discussion of elastic constants we have imagined uniform distortions of the crystal. An elastic medium can sustain vibrations, which at any instant consist of nonuniform distortions. The normal modes of an elastic continuum arc sound waves (longitudinal and transverse) propagating in the medium, and these normal modes will also exist in the crystal. Indeed, viewing the crystal as an elastic... 

We could have used particular frequencies instead of the elastic constants to determine the force constants in the model that alternative is of some interest, We have used, for reasons to be discussed, the measured transverse optical frequency at /( = 0 and the transverse acoustical mode at k — Inja to obtain alternate values of C,) and C, which are listed in Table 9-1. The differences from the values given... 

For wave numbers in this direction, there arc also transverse modes in which the motion is perpendicular to k. It is clear from symmetry that the two modes are of the same frequency. The calculation of these modes is more complicated. At long wavelengths these modes arc describable by the elastic constant C44 and are seen therefore to have internal displacements. Thus we must introduce displacement amplitudes in the z-direction as well as in the y-direction. This is not a major... 

To determine the vibrational modes of the system we use an extension of the simple Keating model [10]. We start with the isolated Cao for which we use two nearest-neighbor on-ball elastic constants a,/3 with a bond-... 

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. 

Hill R (1952) The elastic behaviour of a crystalline aggregate. Proc Phys Soc Lond A 65 349-354 Hochli UT (1972) Elastic constants and soft optical modes in gadolinium molybdate. Phys Rev B 6 1814-1823... 

Elastic constants depend on pressure and temperature because of the anharmonicity of the interatomic potentials. From the dependence of bulk and shear moduli on hydrostatic and uniaxial pressure, third order elastic constants and Griineisen parameters may be determined. Griineisen parameter shows the effect of changing volume, V, on the phonon mode frequencies, co. 

Material Constants, Elastic wave velocities have been obtained for oil shale by ultrasonic methods for various modes of propagation. Elastic constants can be inferred from these data if the oil shale is assumed to be a transversely isotropic solid (9). This is a reasonable approximation considering the bedded nature of the rock. Many of the properties of oil shale depend on the grade (kerogen content), which in turn is correlated with the density ( 10). The high pressure behavior of oil shale under shock loading has been studied in gas-gun impact experiments (11). 

An elastic constant predicted by using LCAO bands and the special points method, compared with experiment also, calculated and experimental values of a zone-boundary transverse acoustic mode frequency, cota(- )> to be discussed in Chapter 9. 

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