Debye temperature elastic

Table 3 Room-temperature elastic constants, density, and the Debye temperature p of a number of Pd-Ni-P and Pd-Cu-P bulk amorphous alloys. The elastic moduli are in units of GPa and the density p is in units of g/cm. ...

The calculated Debye temperatures are also listed in Table 3. From this table, it is clear that the elastic properties of the bulk amorphous Pd-Ni-P and Pd-Cu-P alloys change little with changing composition. The elastic moduli of the Pd-Cu-P alloys are slightly lower than those for the Pd-Ni-P alloys. 

The Debye temperature, can be calculated from the elastic properties of the solid. Required are the molecular weight, molar volume, compressibility, and Poisson s ratio.11 More commonly, do is obtained from a fit of experimental heat capacity results to the Debye equation as shown above. Representative values for 9o are as follows ... 

The three inelastic processes (flow, twinning, and phase changes) all require the shearing of atomic neighbors, so they all tend to occur at the same critical elastic strain (at low temperatures i.e., temperatures below the Debye temperature of the specimen material). As they occur, they interfere with one another, thereby increasing the stress needed for further deformation. 

For covalent crystals temperature has little effect on hardness (except for the relatively small effect of decreasing the elastic shear stiffness) until the Debye temperature is reached (Gilman, 1995). Then the hardness begins to decrease exponentially (Figure 5.14). Since the Debye temperature is related to the shear stiffness (Ledbetter, 1982) this softening temperature is proportional to C44 (Feltham and Banerjee, 1992). 

U. Piesbergen, Heat Capacity and Debye Temperatures G.Giesecke, Lattice Constants J.R. Drabble, Elastic Properties... 

Table 8.3 Comparison of Debye temperatures derived from heat capacity data and from elastic properties.

Even better agreement is observed between calorimetric and elastic Debye temperatures. The Debye temperature is based on a continuum model for long wavelengths, and hence the discrete nature of the atoms is neglected. The wave velocity is constant and the Debye temperature can be expressed through the average speed of sound in longitudinal and transverse directions (parallel and normal to the wave vector). Calorimetric and elastic Debye temperatures are compared in Table 8.3 for some selected elements and compounds. 

Similar expressions can be generated for holes simply by letting coc - — relaxation time xB needs justification, which will not be attempted here. Suffice it to say that this assumption is not bad for elastic scattering processes, which include most of the important mechanisms. A well-known exception is polar optical-phonon scattering, at temperatures below the Debye temperature (Putley, 1968, p. 138). We have further assumed here that t is independent of energy, although this condition will be relaxed later. 

The Debye temperature can also be obtained from the elastic constants. The measurement of the elastic constants of polycrystalline AIN was used by Slack et al [8] to derive the Debye temperature, giving 0D = 950 K. Therefore, Slack et al have criticised the value of the AIN Debye temperature 0D = 800 + 2 K, derived from the heat capacity measurements by Koshchenko et al [6], as too low. Also, Slack s value differs considerably from Meng s result [7]. Since the cubic dependence T3 approximates the Debye specific heat well in the temperature range below T = 0d/5O [9], it is likely that the upper temperature limit used by Meng is too high and led to error and the difference from the results of Slack et al [8],... 

XIV, wc have information from which the Debye temperature can be calculated from the elastic constants and the density. These constants are known for NaCl and KC1, and in the table we also give the calculated Debye temperature found from the elastic constants. Finally, in Chap. 

Debye temperature are obtained from elastic data using the mean sound velocity and mean atomic volume in the relation... 

Where v/ and v/- are the longitudinal and transverse velocities. A comparison of the elastic Debye temperature and calorimetric Debye temperature gives a measure of the contribution of the acoustic modes to heat capacity. Since no optic modes are excited at low temperatures, the elastic and calorimetric Debye temperatures are expected to be comparable which in the case of vitreous silica is not obeyed (see chapter 12). 

Although Si02 is a typical inorganic glass, it is also atypical in many ways. Several properties of vitreous silica are known to vary anomalously at low temperatures. Anomaly in the low temperature specific heat is the most notable and well investigated. This is reflected in serious disagreement between Debye temperatures calculated from thermal and acoustic measurements. (thermal) and(elastic) are respectively given by (Anderson and Dienes, 1960),... 

It should be noted that despite the similarity between the different estimates, there is still no perfect agreement with all experimental data. The total cross section, the low energy virial coefficient, the elastic constant and the Debye-temperature of the solid are examples of such deviations. Furthermore the question of the three-body forces is not solved completely. The fact that contributions of higher-order non-additive multipole forces were found to be... 

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