14.2.Interpretation of the Heat Capacity by Various Models 275 TABLE 14.2.Debye temperatures of some materials Substance ep(K) Pb 95 Au 170 Ag 230 270 Cu 340 Fe 360 Al 375 Si 650 1850 GaAs 204 InP 162 14.2.Interpretation of the Heat Capacity by Various Models The classical (atomistic)theory for the interpretation of the heat capacity postulates that each atom in a crystal is bound to its site by a harmonic force similar to a spring.A given atom is thought to be capable of absorbing thermal energy,and in doing so it starts to vibrate about its point of rest.The amplitude of the oscillation is restricted by electrostatic repulsion forces of the nearest neighbors.The extent of this thermal vibration is there- fore not more than 5 or 10%of the interatomic spacing,de- pending on the temperature.In short,we compare an atom with a sphere which is held at its site by two springs [Figure 14.3(a)]. The thermal energy that a harmonic oscillator of this kind can absorb is proportional to the absolute temperature of the envi- ronment.The proportionality factor has been found to be the Boltzmann constant kB(see Appendix II).The average energy of the oscillator is then: E=kBT. (14.5) Now,solids are three-dimensional.Thus,a given atom in a cu- bic crystal also responds to the harmonic forces of lattice atoms in the other two directions.In other words,it is postulated that each atom in a cubic crystal represents three oscillators [Figure 14.3(b)],each of which absorbs the thermal energy kBT.There- fore,the average energy per atom is: E=3kBT. (14.6)The classical (atomistic) theory for the interpretation of the heat capacity postulates that each atom in a crystal is bound to its site by a harmonic force similar to a spring. A given atom is thought to be capable of absorbing thermal energy, and in doing so it starts to vibrate about its point of rest. The amplitude of the oscillation is restricted by electrostatic repulsion forces of the nearest neighbors. The extent of this thermal vibration is there￾fore not more than 5 or 10% of the interatomic spacing, de￾pending on the temperature. In short, we compare an atom with a sphere which is held at its site by two springs [Figure 14.3(a)]. The thermal energy that a harmonic oscillator of this kind can absorb is proportional to the absolute temperature of the envi￾ronment. The proportionality factor has been found to be the Boltzmann constant kB (see Appendix II). The average energy of the oscillator is then: E
kBT. (14.5) Now, solids are three-dimensional. Thus, a given atom in a cu￾bic crystal also responds to the harmonic forces of lattice atoms in the other two directions. In other words, it is postulated that each atom in a cubic crystal represents three oscillators [Figure 14.3(b)], each of which absorbs the thermal energy kBT. There￾fore, the average energy per atom is: E
3kBT. (14.6) 14.2 • Interpretation of the Heat Capacity by Various Models 275 TABLE 14.2. Debye temperatures of some materials Substance )D (K) Pb 95 Au 170 Ag 230 W 270 Cu 340 Fe 360 Al 375 Si 650 C 1850 GaAs 204 InP 162 14.2 • Interpretation of the Heat Capacity by Various Models