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. 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

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

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

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

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

Energy E Energy

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

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

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

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