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Unformatted text preview: CHEM 356, Lecture 4, Fall 2009 1 4. Heat Capacity of Monatomic Crystals . Classical Physics: Recall that from classical physics the heat capacity at constant volume for a monatomic crystal was given by the law of Dulong and Petit as = 3 at all temperatures. the atoms of the crystal are assumed to vibrate independently in single har- monic motion (SHM) in all three spatial directions if we apply the classical principle of equipartition of energy, i.e., that the classical energy will have a contribution of 1 2 B for each quadratic term in classical expression for the energy SHM = 1 2 2 + 1 2 2 B for each spatial direction; hence = 3 , = ( ) = 3 In the 1890s first C (in the form of diamond), and then a number of metals were found to have ( ) differing from 3 , especially at low temperatures. CHEM 356, Lecture 4, Fall 2009 2 Einstein (1907) : Einstein followed the classical physics argument in treating the atoms as SHOs un- dertaking SHM about their equilibrium positions in the crystal lattice. However, unlike the classical treatment, by treating the oscillator motions as quantized, i.e., by assuming that the energies of all oscillators were given by the Planck relation = osc , he obtained for the expression = 3 ( E ) 2 e E / [e E / 1] 2 , in which (in modern terminology) E osc / B is known as the Einstein char- acteristic temperature. This expression for actually decreases too rapidly at low temperatures, behaving as E 3 ( E ) 2 e E / . Debye (1912): Debye recognized that the incorrect behaviour of at low temperatures was likely due to Einsteins assumption in his model that all oscillators have the same fundamental frequency of oscillation, osc = E . He replaced the single oscilla- tor fundamental frequency E by the RayleighJeans distribution of frequencies, namely class...
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