Elastic constants, acoustic measurements

By measuring V z), which includes examining the reflectance function of solid material, measuring the phase velocity and attenuation of leaky surface acoustic waves at the liquid-specimen boundary, the SAM can be used indetermining the elastic constants of the material. 

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... 

The shear and compressional acoustic wave velocities for the inner core are the direct output parameters from seismological observations. In order to make a direct comparison between the seismic data and measured physical properties, measurements of the acoustic velocities for iron at core pressures are required. Only very recently has it become possible to measure the elastic constants of s-Fe at high pressures and room temperature (Mao etal., 1999 Lubbers etal., 2000 Fiquet et al., 2001 Anderson et at, 2001). Recent advances in theory and computational methods have also provided new tools for computing the elastic constants of s-Fe at core pressures (Stixrude and Cohen, 1995 Soderhnd et al., 1996 Cohen et al., 1997 Steinle-Neumann and Stixrude, 1999) and core conditions (Laio et al., 2000 Steinle-Neumann et al, 2001 Alfe et al., 2001). There is considerable disagreement on the elastic constants of s-Fe between experimental results and theoretical calculations. The dilferences in the aggregate shear (FJ and compressional (Vp) wave velocities are smaller (Hemley and Mao, 2001 Steinle-Neumann et ai, 2001). Further improvement of theory and experiment is required to resolve the discrepancies. 

It is important to mention that from the analysis of the elastic and dielectric measurements and of the A and B molecular field constants the relative contribution of the acoustical and optical vibrations to the structural ordering could be estimated. 

Cook RK (1957) Variation of elastic constants and static strains with hydrostatic pressure a method for calculation from ultrasonic measurements. J Acoust Soc Am 29 445-449 David WIF (1992) Transformations in neutron powder diffraction. Physica B 180 181 567-574 Decker DL (1971) High-pressure equations of state forNaCl, KCl, and CsCl. J ApplPhys 42 3239-3244 Decker DL, Petersen S, Debray D, Lambert M (1979) Pressure-induced ferroelastic phase transition in Pb3(P04)2 A neutron diffraction study. Phys Rev B 19 3552-3555 Eggert JH, Xu L-W, Che R-Z, Chen L-C, Wang J-F (1992) High-pressure refractive index measurements of 4 1 methanol ethanol. J Appl Phys 72 2453-2461... 

The realization that co-operative Jahn-Teller systems should show large acoustic anomalies led to another type of investigation. In DyV04 Melcher and Scott (1972) measured the longitudinal acoustic velocity, and confirmed that the elastic constant (cii — Cl2) fell to zero at the distortion temperature They fitted their results... 

At or near absolute zero the elastic constants determine the thermal spectrum of lattice vibrations and, consequently, the Debye temperature. Since elastic constants can be determined with high precision, it is useful to calculate the Debye temperature from acoustic wave velocities, i.e. from elastic constants measured near absolute zero. In the Debye theory the characteristic temperature at absolute zero is given (Anderson, 1963) by... 

No single crystal elastic constant data for cerium were found by the writer This probably reflects the difficulty of obtaining suitable single phase crystals. The most complete and systematic investigation of the polycrystalline elastic properties of cerium was conducted by Rosen (1969a). He used an ultrasonic pulse technique employing a frequency of 10 MHz to measure sound velocities in spectrographically pure (99.9 + %) metal and corrected the acoustic path... 

Greiner et al. (1976) also have determined the single crystal elastic constants from 6 to 300 K using the ultrasonic pulse-echo-overlap (10 MHz) method. No thermal expansion corrections for density and acoustic path length were made because no thermal expansion data were available in the experimental temperature range. In addition to the four constants measured by Lenkkeri and Palmer (1977), Greiner et al. measured C13 consequently, polycrystalline elastic... 

In comparing elastic constants measured acoustically with those obtained in a static (very low frequency) test, note that acoustic values are measured under adiabatic conditions, while static values are isothermal. The two t5q>es of bulk modulus measurements are related by the standard thermodynamic relation... 

In addition to the adiabatic or isothermal difference, acoustically determined elastic constants of polymers differ from static values because polymer moduli are frequency-dependent. The deformation produced by a given stress depends on how long the stress is applied. During the short period of a sound wave, not as much strain occurs as in a typical static measurement, and the acoustic modulus is higher than the static modulus. This effect is small for the bulk modulus (on the order of 20%), but can be significant for the shear and Young s modulus (a factor of 10 or more) (5,6). 

Strain dependence should also be considered when comparing acoustically measured elastic constants with statically measured values. As an example, for polyethylene at room temperature, the modulus is independent of strain up to a strain of about 10 (7). Beyond this point, the modulus decreases as the strain increases. Typically, acoustic measurements are made in the strain range 10 where the moduli are strain-independent, but static measin-ements... 

Figure 1. Principle of Atomic Force Acoustic Microscopy. Using the shift of the contact resonance frequencies relative to the free resonances, one can determine the local elastic constants of a material. The adhesion forces contribute to the measured vertical contact stiffness as well. Their influence can be neglected, provided the static forces are adjusted so that they are much larger than the adhesion forces.

The excitation spectrum is determined from the poles of D (q, energy states excitations). The latter enter via the frequency dependent susceptibility q V, interaction between both types of modes leads to a hybridization (Elliott et al., 1972). Due to the elastic scattering processes within a CEF-energy level there are 5 =o-contributions to the quadrupole susceptibility (see eq. 17.108). It was shown that these processes have to be included in the expressions for the elastic constants and therefore contribute to a Jahn-Teller phase transition. However they do not show up in the acoustic phonon dispersion which one would measure... 

Since the elastic constants are calculated using measurements of density and acoustic wave velocity, the individual contributions of MCN dependence of these quantities on these elastic transitions should be considered. As evident from the previous section, the density itself changes with the MCN in a manner that reflects the topological order in Ge-As-Se ternary glasses. However, the fluctuation of the density is less than 2% compared with 50%... 

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