1. Lattice Vibrations Part III
Solid State Physics
355

2. Back to Dispersion Curves
We know we can measure the phonon dispersion curves - the dependence of the phonon frequencies upon the wavevector q.
To calculate the heat capacity, we begin by summing over all the energies of all the possible phonon modes, multiplied by the Planck Distribution.
number of polarizations is 3 for three dimensions
Planck distribution gives us the number of phonons at a given temperature and energy
h bar omega – energy of a phonon of polarization p and wavevector q
Planck
Distribution
sum over all
wavevectors
sum over all
polarizations

3. Density of States D() What’s this good for?
For photons, there are an infinite number of modes for each energy, but not so far solids. We are working with a discrete system – within which the atomic spacing puts limits.
number of modes
unit frequency
D()

4. Density of States: One Dimension
determined by the dispersion relation
Each normal vibrational mode of polarization p has the form of a standing wave.
If the ends are fixed, what modes, or wavelengths, are allowed?

5. Density of States: One Dimension
# of wavelengths
wavelength
wavevector
0.5 2L
/L 1 L
2/L
1.5
2L/3
3/L 2
L/2
4/L
If you try and put larger values of the wavenumber ...
This means that the atoms cannot move if the wavelength is shorter than 2a – the minimum you can have for a wave in a discrete system.
Each value of q corresponds to a different standing wave and there is only one mode for each interval
q=/L. Thus, there are N-1 possible modes.

6. Density of States: One Dimension
To calculate the density of states, use
number of modes
unit frequency
D()
There is one mode per interval  q =  / L with allowed values...
So, the number of modes per unit range of q is L / .

7. Density of States: One Dimension
There is one mode for each mobile atom.
To generalize this, go back to the definition...the number of modes is the product of the density of states and the frequency unit.

8. Density of States: One Dimension
monatomic lattice
diatomic lattice
Knowing the dispersion curve we can calculate the group velocity, d/dq.
Near the zone boundaries, the group velocity goes to zero and the density of states goes to infinity. This is called a singularity.

9. Periodic Boundary Conditions
No fixed atoms – just require that u(na) = u(na + L).
This is the periodic condition.
The solution for the displacements is
The allowed q values are then,
This method results in the same number of modes: one per mobile atom, but now we have both positive and negative values of q and q = 2/L.

10. Density of States: 3 Dimensions
Let’s say we have a cube with sides of length L.
Apply the periodic boundary condition for N3 primitive cells:

11. Density of States: 3 Dimensionsqz
There is one allowed value of q per volume (2/L3) in q space or
allowed values of q per unit volume of q space, for each polarization, and for each branch.
The total number of modes for each polarization with wavevector less than q is qy qx
In q-space, we are looking at all possible modes from q = 0 to maximum value of q. Since we are looking in three dimensions, this is a sphere. We are talking about an infinite cubic lattice within which there is this sphere.
How many q values are possible within this sphere?

12. Debye Model for Heat Capacity
Debye Approximation:
For small values of q, there is a linear relationship =vq, where v is the speed of sound.
...true for lowest energies, long wavelengths
This will allow us to calculate the density of states.
Another assumption Debye made was that this relationship is only true up to a certain cut-off frequency D. There can only be N acoustic modes for the N atoms in a crystal. Once you run out of possible modes, there are no more q values or wavelengths to be concerned with.

13. Debye Model for Heat Capacity

14. Debye Model for Heat CapacityqD
For the energy...
For simplicity, assume that all three polarizations have the same energy dependence – not true, but close enough.

15. Debye Model for Heat Capacity

16. Debye Model for Heat Capacity
Debye Temperature is related to
1. The stiffness of the bonds between atoms
2. The velocity of sound in a material, v
3. The density of the material, because we can write the Debye Temperature as:

17. Debye Model for Heat Capacity
How did Debye do??

18. Debye Model for Heat Capacity
High T limit
In the high-T limit:
kT >> ħωD,
so T >> θD
( or x = θD/T << 1

19. Debye Model for Heat Capacity
Low T limit
No closed form solution – need to solve numerically.
At low temperatures, x = θD/T becomes large, so ex becomes even larger. In the low T limit, the denominator gets large for large x. This means that the contribution to the integral isn’t very large at large values of x (the exp part kills it off). We will make the approximation that we can integrate to infinity because the intregrand is dead in this limit anyways (the contributions are ~ 0).

20. Debye Model for Heat Capacity
Low T limit

21. Debye Model for Heat Capacity
This is best seen in materials which have no free electrons to absorb heat (like solid Ar, as opposed to say metals which have free electrons which can move in the lattice and absorb heat)
Why does this work so well?
At low T, we only have acoustic modes, where ω= vK approximation works really well.
The Debye Model actually works over a wide range of temperatures(and is a better model than Einstein.s)
.It is difficult to calculate the integral over a wide range of temperatures though

22. Debye Model for Heat Capacity
Einstein's oscillator treatment of specific heat gave qualitative agreement with experiment and gave the correct high temperature limit (the Law of Dulong and Petit).
The quantitative fit to experiment was improved by Debye's recognition that there was a maximum number of modes of vibration in a solid.
He pictured the vibrations as standing wave modes in the crystal, similar to the electromagnetic modes in a cavity which successfully explained blackbody radiation.

23. ωD represents the maximum frequency of a normal mode in this model.
ωD is the energy level spacing of the oscillator of maximum frequency (or the maximum energy of a phonon).
It is to be expected that the quantum nature of the system will continue to be evident as long as
The temperature in gives a rough demarcation between quantum mechanical regime and the classical regime for the lattice.

24. Typical Debye frequency:
(a) Typical speed of sound in a solid ~ 5×103 m/s. A simple cubic lattice, with side a = 0.3 nm, gives
ωD ≈ 5×1013 rad/s.
(b) We could assume that kmax ≈ /a, and use the linear approximation to get
ωD ≈ vsound kmax ≈ 5×1013 rad/s.
A typical Debye temperature:
θD ≈ 450 K
Most elemental solids have θD somewhat below this.

26. Measuring Specific Heat Capacity
Differential scanning calorimetry (DSC) is a relatively fast and reliable method for measuring the enthalpy and heat capacity for a wide range of materials. The temperature differential between an empty pan and the pan containing the sample is monitored while the furnace follows a fixed rate of temperature increase/decrease. The sample results are then compared with a known material undergoing the same temperature program. 

27. Measuring Specific Heat Capacity
The instrument is programmed to heat between the initial and final isothermal stages at Ti and Tf, first with empty pans. A perfect instrument would show no deflection from the isothermal baseline, but this is never the case, and a small signal is given which can be used to correctly measure the deflections given by the sample and sapphire standard when run under precisely the same conditions. The deflections obtained are directly dependent on the heating rate; linearity and reproducibility of the temperature program are therefore essential for accurate work.