ENGR-1100 Introduction to Engineering Analysis

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 xhv/kT and uhmaxv/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 xhv/kT and uhmaxv/kT T? 8 Thus
heat capacity
11
Low - Temperature Limit
With xhv/kT and uhmaxv/kT For T? 0, u ? 8
with The heat capacity is