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Lecture 12b Debye Model of Solid

Lecture 12b Debye Model of Solid. Debye model - phonon density of states The partition function Thermodynamic functions Low and high temperature limits. Real crystal - waves. There are mechanical/thermal waves in a crystal since atoms are connected to each other and they all move.

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Lecture 12b Debye Model of Solid

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  1. Lecture 12b Debye Model of Solid • Debye model - phonon density of states • The partition function • Thermodynamic functions • Low and high temperature limits

  2. Real crystal - waves There are mechanical/thermal waves in a crystal since atoms are connected to each other and they all move. For long wavelengths, the frequency, v, is related to speed of sound, c, and the wavelength,  Possible wavelengths can be enumerated by integers, n by a requirement of being a standing wave in a crystal of size a

  3. Number of waves In 1 D crystal In 3 D crystal wave in an arbitrary direction As with electron gas, number of all wave wavelength greater than , G() is given by

  4. Number of waves - II In terms of frequency But there are 2 transverse waves and one longitudinal At maximum frequency, total # of waves = # degree of freedom

  5. Density of states and partition function Density of states Partition function, independent oscillators with various frequencies Where as in the Einstein model Vibrational part

  6. Partition function and F Replacing summation with integration Thus the free energy

  7. Energy E Energy

  8. Energy and heat capacity With x=hv/kT and u=hmaxv/kT And after lengthy derivation All can be expressed in terms of the Debye function, and Debye temperature

  9. Debye Temperature High Debye temperature for solids with large atomic density (N/V) and high speed of sound (high modulus and low density)

  10. High - Temperature Limit With x=hv/kT and u=hmaxv/kT T ∞ Thus heat capacity

  11. Low - Temperature Limit With x=hv/kT and u=hmaxv/kT For T 0, u  ∞ with The heat capacity is

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