Griineisen coefficient

Second, the rapid laser-induced thermal expansion causes [103] a pressure rise, given by the product of the Griineisen coefficient, T, and the absorbed energy density... 

Eq is the lattice energy at zero temperature and r5(F) is the volume-dependent Griineisen coefficient for the solid phase. 

Furthermore, stresses were calculated as functions of strain and temperature. For each temperature, each component of stress was fit to a second order Taylor series expansion in terms of the strains, about the P = 1 atm reference volume Vq(T) at each specific temperature. Based on the stresses, the elastic moduli Cij and the Griineisen coefficients ji (i = 1,2,3) of the non-crystalline interlamellar phase were calculated using... 

The approximate steady-state profiles computed in the above manner for nitromethane at the BKW C-J state are shown in Figure 1.3 for a Griineisen coefficient, 7, of 1.7 and 0.68. The partially reacted Hugoniots are very sensitive to the equation of state used for the mixture of undecomposed explosive and detonation products. A 7 of 1.70 in addition to 0.68 was used to illustrate the magnitude of this effect. 

The expansion coefficient of a solid can be estimated with the aid of an approximate thermodynamic equation of state for solids which equates the thermal expansion coefficient with the quantity where yis the Griineisen dimensionless ratio, C, is the specific heat of the solid, p is the density of the material, and B is the bulk modulus. For fee metals the average value of the Griineisen constant is near 2.3. However, there is a tendency for this constant to increase with atomic number. 

The experimental data usually give the specific heat at constant pressure cP. Theories usually refer to the specific heat at constant volume cv. The specific heat cP is greater than cv by a factor (1 + jgT), where f5 is the volumetric coefficient of thermal expansion and yG is the so-called Griineisen parameter ... 

This isothermal bulk modulus (Kj) measured by static compression differs slightly from the aforementioned adiabatic bulk modulus (X5) defining seismic velocities in that the former (Kj) describes resistance to compression at constant temperature, such as is the case in a laboratory device in which a sample is slowly compressed in contact with a large thermal reservoir such as the atmosphere. The latter (X5), alternatively describes resistance to compression under adiabatic conditions, such as those pertaining when passage of a seismic wave causes compression (and relaxation) on a time-scale that is short compared to that of thermal conduction. Thus, the adiabatic bulk modulus generally exceeds the isothermal value (usually by a few percent), because it is more difihcult to compress a material whose temperature rises upon compression than one which is allowed to conduct away any such excess heat, as described by a simple multiplicative factor Kg = Kp(l + Tay), where a is the volumetric coefficient of thermal expansion and y is the thermodynamic Griineisen parameter. 

The parameter Kp is also known as the isothermal bulk modulus, and a as the volume expansion coefficient. They are interrelated by means of the Griineisen constant, y ... 

Mitra [86] determined the pressure dependence of the Raman- and infrared-active lattice modes in KN3 and found that the librational mode is highly pressure dependent (2 cm"Vkbar pressure). The Raman-active translational mode involving motions of the ions only appears to be insensitive to pressure. The infrared-active modes, however, show a pressure coefficient of 0.5-0.7 cm Vkbar. Raman measurements at high temperature on KN3 by Iqbal [87] indicate that the librational modes are also very temperature dependent (Figure 17). Within the quasiharmonic approximation, correlation of the data with thermal expansion and compressibility measurements indicates a sizable anharmonic or selfenergy contribution to the librational mode. The mode Griineisen parameter 7y(q) of mode cjy(q), defined as... 

There are some physical generalities concerning thermal expansion coefficients. One empirical correlation is that is constant for a wide range of cubic and close-packed compounds, where T is the melting point and is the volume coefficient of thermal expansion. The Griineisen equation relates ol to the compressibility Kq, the heat capacity c , and the molar volume V here y is the Gruneisen constant, a proportionality constant of first order ... 

The linear thermal expansion coefficient a can be determined from the derive of the temperature dependence of the cell dimension as a = d ln L(T) /dT. Here, L(T) is the cell dimension at the corresponding temperature T. The volumetric TEC can be expressed via fundamental parameters, such as the Griineisen parameter Yv, the bulk modulus K, the molar voliune Vm, the heat capacity C, phonon frequencies o), and the internal energy U ... 

Senyshyn et al. (2005b) calculated the above-mentioned properties and determined the thermal expansion coefficient of rare earth gallates using a semi-classical approach. Ideal (X-ray) density, Griineisen parameter, isohoric heat capacity Cy, bulk and shear moduli, and thermal expansion coefficient were calculated for RGaOa (R = La-Gd) at 300 K are listed in Table 47. 

We note that through the Griineisen parameters anharmonic corrections of the potential energy are involved in the thermal expansion coefficient. The approximation that the vibrational free-energy function depends on the volume of the crystal through the change of the phonon frequencies described by the first-order approximation... 

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