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Thermal Properties II - Coggle Diagram
Thermal Properties II
Debye's Model
Atoms are considered as harmonic oscillators that produce elastic waves with varying frequencies from ω = 0 to ωmax
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In the 3 dimensional reciprocal space, the volume for each allowed wave vector q is:
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In the Debye approximation, the velocity of sound vs is taken as constant for each polarization type, as it would be for a classical elastic continuum. The density of states is:
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At the left of the figure, the experimental results of specific heats of four substances are plotted as a function of temperature and they look very different. But if they are scaled to T/TD, they look very similar and are very close to the Debye theory.
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The Debye model successfully describes the temperature dependence of the heat capacity not only at high temperatures but also at low temperatures
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Einstein's Model
As a model of a solid, Einstein assumed that the atoms are vibrating as harmonic oscillators, but instead of taking the classical expression for the energy of an oscillator
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He assumed Planck’s (1990) quantization rule for each oscillator, En = nhw n=0,1,2,... where n takes on all integer value from 0 to infinity.
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At low T, Einstein’s formula is essentially an exponential falloff
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