Lattice vibrations constant volume

This model, the Einstein model for heat capacity, predicts that the heat capacity is reduced on cooling and that the heat capacity becomes zero at 0 K. At high temperatures the constant-volume heat capacity approaches the classical value 3R. The Einstein model represented a substantial improvement compared with the classical models. The experimental heat capacity of copper at constant pressure is compared in Figure 8.3 to Cy m calculated using the Einstein model with 0g = 244 K. The insert to the figure shows the Einstein frequency of Cu. All 3L vibrational modes have the same frequency, v = 32 THz. However, whereas Cy m is observed experimentally to vary proportionally with T3 at low temperatures, the Einstein heat capacity decreases more rapidly it is proportional to exp(0E IT) at low temperatures. In order to reproduce the observed low temperature behaviour qualitatively, one more essential factor must be taken into account the lattice vibrations of each individual atom are not independent of each other - collective lattice vibrations must be considered. 

The heat capacity models described so far were all based on a harmonic oscillator approximation. This implies that the volume of the simple crystals considered does not vary with temperature and Cy m is derived as a function of temperature for a crystal having a fixed volume. Anharmonic lattice vibrations give rise to a finite isobaric thermal expansivity. These vibrations contribute both directly and indirectly to the total heat capacity directly since the anharmonic vibrations themselves contribute, and indirectly since the volume of a real crystal increases with increasing temperature, changing all frequencies. The constant volume heat capacity derived from experimental heat capacity data is different from that for a fixed volume. The difference in heat capacity at constant volume for a crystal that is allowed to relax at each temperature and the heat capacity at constant volume for a crystal where the volume is fixed to correspond to that at the Debye temperature represents a considerable part of Cp m - Cv m. This is shown for Mo and W [6] in Figure 8.15. 

At low temperatures, almost all lattice vibrations cease to contribute, leaving the thermal excitations of the electrons dominant [5]. The electronic term contributing to the constant volume heat capacity Cv is proportional to temperature T and the vibrational term is proportional to T3. Consequently, C is expressed as [5],... 

The heat capacity and transition enthalpy data required to evaluate Sm T ) using Eq. 6.2.2 come from calorimetry. The calorimeter can be cooled to about 10 K with liquid hydrogen, but it is difficult to make measurements below this temperature. Statistical mechanical theory may be used to approximate the part of the integral in Eq. 6.2.2 between zero kelvins and the lowest temperature at which a value of Cp,m can be measured. The appropriate formula for nonmagnetic nonmetals comes from the Debye theory for the lattice vibration of a monatomic crystal. This theory predicts that at low temperatures (from 0 K to about 30 K), the molar heat capacity at constant volume is proportional to Cv,m = aT, ... 

Debye s theory of the specific heats of solids depends on the existence of a high number of standing, high frequency, elastic waves that are associated with thermal lattice vibrations. Central to his approach is the proposal that in a solid the phonon spectral density (p) increases continuously, and with a direct dependence on the square of the frequency (cutoff frequency (Q, at about 10 Hz) above which the phonon density vanishes for a solid continuum containing N atoms in a sample of volume V, the proportionality constant is 6 V/v, where V is the velocity of propagation. At a typical nuclear... 

The variation with temperature of the vibrational contribution to the heat capacity at constant volume for many relatively simple crystalline solids is shown in Figure 19.2. The C is zero at 0 K, but it rises rapidly with temperature this corresponds to an increased ability of the lattice waves to enhance their average energy with increasing temperature. At low temperatures, the relationship between C and the absolute temperature T is... 

Phonon vibration spectrum was determined from force constant k which was determined from dependence of the calculated molecule average energy on volume ( a3), i.e. from compressibility k d2Etot(a,T)lda2. The pressure in the system was determined conventionally as P(a,T) = -dF(a,T) / 8V One can determine the lattice constant a(T) for every value of (P,T) by numerical inversion of the dependence P a,T) => a(P,T) ... 

Thermal vibration of the atoms, hence the thermal energy of the polymer, increases with absolute temperature T (for gas molecules the thermal energy is 3/2RT where R is the gas constant 8.3JK mol ). When the thermal energy exceeds a critical value (at Tg), free volume is available for molecular motion. It is likely that the free volume is non-uniformly distributed in the melt, and that a number of lower density regions move rapidly through the melt (rather as dislocations move through a crystal lattice to allow plastic deformation). 

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