1. Thermal Energy & Heat Capacity:
The Debye Model

2. Debye Model Density of States
According to Quantum Mechanics, if a particle is constrained; the energy of particle can only have discrete values. It has to go up in steps.
A simple example of this is a finite potential well.

3. g(k)  Number of discrete k
For systems with huge numbers of atoms, such as solids, these steps can be so small that the energy can be approximated as continuous. That is, the energy levels are so close together that it is a good approximation to ignore the discreteness & treat the energy as continuous.
For the case of the classical normal mode frequencies (k), we know that k in the first Brillouin Zone is discrete, but because the steps in these discrete k are so small, we can treat k as continuous when we want to do sums of some funtions over all k in the Brillouin Zone.
To do this accurately requires the introduction of a function g(k), which is called the density of states.
g(k)  Number of discrete k
values between k & k + dk .

4. We’ve seen that the thermal average energy of a quantum oscillator of frequency  in equilibrium with a heat reservoir at temperature T is:
We’ve seen that the Einstein Model assumes that all frequencies are the same,
 = E, so that the thermal energy for the solid with N atoms is just N times the above expression.

5. The functional form of g(k) depends on the form of the function (k).
The thermal average energy of a quantum oscillator of frequency  in equilibrium with a heat reservoir at temperature T is:
(1)
A more accurate treatment should take into account the fact that the frequencies are wavevector dependent  = (k) so that the thermal energy of the vibrating solid is a sum over all wavevectors in the first Brillouin zone of the expression in (1).
It is convenient to change this sum to an integral in 3 dimensional k space by use of the density of states g(k).
The functional form of g(k) depends on the form of the function (k).

6. g()  Number of discrete 
Furthermore, it is even more convenient to change the 3 dimensional wavevector integral of
into an integral over frequencies  by use of the frequency dependent density of states g().
g()  Number of discrete 
values between  &  + d

7. Choose standing waves to obtain;
Choose standing waves to obtain
Let’s remember dispertion relation for 1D monoatomic lattice

8. Multibly and divide
Let’s remember:
True density of states

9. True DOS(density of states) tends to infinity at ,
True density of states by means of above equation
constant density of states
True DOS(density of states) tends to infinity at ,
since the group velocity goes to zero at this value of .
Constant density of states can be obtained by ignoring the dispersion of sound at wavelengths comparable to atomic spacing.

10. One can obtain same expression of by means of using running waves.
The energy of lattice vibrations will then be found by integrating the energy of single oscillator over the distribution of vibration frequencies. Thus
for 1D
Mean energy of a harmonic oscillator
One can obtain same expression of by means of using running waves.
It should be better to find 3D DOS in order to compare the results with experiment.

11. Debye Model Let’s do it first for 2D, then for 3D.
Consider a crystal in the shape of 2D box with crystal sides L. y + - L - + + - x L
Standing wave pattern for a 2D box
Configuration in k-space

12. Let’s calculate the number of modes within a range of wavevector k.
Standing waves are choosen but running waves will lead same expressions.
Standing waves will be of the form
Assuming the boundary conditions of
Vibration amplitude should vanish at edges of
Choosing
positive integer

13. Standing wave pattern for a 2D box Configuration in k-spacey + - L - + + - x L
Standing wave pattern for a 2D box
Configuration in k-space
The allowed k values lie on a square lattice of side in the positive quadrant of k-space.
These values will so be distributed uniformly with a density of per unit area.
This result can be extended to 3D.

14. kx,ky,kz(all have positive values)
Octant of the crystal:
kx,ky,kz(all have positive values)
The number of standing waves; L L

15. is a new density of states defined as the number of states per unit magnitude of in 3D.This eqn can be obtained by using running waves as well.
(frequency) space can be related to k-space:
Let’s find C at low and high temperature by means of using the expression of

16. High & Low Temperature Limits
Each of the 3N lattice modes of a crystal containing N atoms =
This result is true only if at low T’s only lattice modes having low frequencies can be excited from their ground states;
Low frequency
long  w
sound waves k

17. Velocities of sound in longitudinal and transverse direction
at low T
depends on the direction and there are two transverse, one longitudinal acoustic branch:
Debye Model
DOS
Velocities of sound in longitudinal and transverse direction

18. Discuss!
Zero point energy =
at low temperatures

19. Debye Model of Heat Capacity of Solids

20. How good is the Debye Approximation at low T?
The lattice heat capacity of solids thus varies as T3 at low temperatures; this is referred to as the Debye T3 law. The figure illustrates the excellent agreement of this prediction with experiment for a non-magnetic insulator. The heat capacity vanishes more slowly than the exponential behaviour of a single harmonic oscillator because the vibration spectrum extends down to zero frequency.

21. The Debye interpolation scheme
The accurate calculation of must be done numerically in 3D.
Debye obtained a good approximation by neglecting the dispersion of the acoustic waves, i.e. assuming
for arbitrary wavenumber.. Debye’s approximation gives the correct answer in either the high and low temperature limits, and the language associated with it is still widely used today.

22. The Debye approximation has two main steps:
1. Approximate the dispersion relation of any branch by a linear extrapolation of the small k behaviour:
Debye approximation to the dispersion
Einstein approximation to the dispersion

23. Debye cut-off frequency
2. E nsure the correct number of modes by imposing a cut-off frequency , above which there are no modes. The cut-off freqency is chosen to make the total number of lattice modes correct. Since there are 3N lattice vibration modes in a crystal having N atoms, we choose so that

24. The lattice vibration energy of
becomes
and,
First term is the estimate of the zero point energy, and all T dependence is in the second term. The heat capacity is obtained by differentiating above eqn wrt temperature.

25. The heat capacity is
Let’s convert this complicated integral into an expression for the specific heat changing variables to
and define the Debye temperature

26. The Debye prediction for lattice specific heat
where

27. How does limit at high and low temperatures?
High temperature 
x is always small 

28. We obtain the Debye law in the form
How does limit at high and low temperatures?
Low temperature
For low temperature the upper limit of the integral is infinite; the integral is then a known integral of
T < <
We obtain the Debye law in the form

29. Lattice heat capacity due to Debye interpolation scheme
Figure shows the heat capacity between the two limits of high and low T as predicted by the Debye interpolation formula. 1
Because it is exact in both high and low T limits the Debye formula gives quite a good representation of the heat capacity of most solids, even though the actual phonon-density of states curve may differ appreciably from the Debye assumption.
Lattice heat capacity of a solid as predicted by the Debye interpolation scheme 1
Debye frequency and Debye temperature scale with the velocity of sound in the solid. So solids with low densities and large elastic moduli have high Values of for various solids is given in table. Debye energy can be used to estimate the maximum phonon energy in a solid.
Solid Ar Na Cs Fe Cu Pb C KCl

30. total number of states for a
maximum |k|
so with this we have the density of states

31. The Debye model for the heat capacity
substitute
low temperature, large
at low T, x is big. We can integrate to infinity, then we differentiate and get the desired T^3 law.
for one mole

32. Comparison Debye model - experiment
there is again an important temperature, the Debye temperature which corresponds to the Debye angular frequency.
It is different from the Einstein frequency. It is the highest frequency which would be reached, if we were to continue the acoustic dispersion.
But it also gives a feeling for the vibrational frequencies.
why diamond?

33. Why does the Debye model work better at low T than the Einstein model?
The Debye model gives a better representation for the very low energy vibrations.
At low temperatures, these vibrations matter most.

34. Limits of the Debye model
phonon density of states from copper. Not very realistic but good at low T