1. Heat capacity at constant volume
Thermal Properties of Crystal Lattices
Remember the concept of
heat
capacity
Temperature increase
Amount of heat
Heat capacity at constant volume

2. ? dV=0 constant volume Internal energy
Click for details about differentials
dV=0
constant volume
Internal energy
How to calculate the vibrational energy of a crystal ?
Classical approach
In the qm description
approach of independent
oscillators with single
frequency  is called
Einstein model x
In classical approach
details of Epot irrelevant

3. Average thermal energy
Let us calculate the thermal average of the vibrational energy
Classical:
Energy can change continuously to arbitrary value
defines state (point in phase space)
Boltzman factor, where
Average thermal energy
of one oscillator

7. The same applies for the second integral

8. Classical value of the thermal average of the vibrational energy
Understanding in the framework of: Theorem of equipartition of energy
every degree of freedom
Example: diatomic molecule
Vibration involves
kinetic+pot. energy
Only rotations relevant where
moment of inertia

9. Solid:
N atoms
3N vibrational modes
with
= n 24.94J/(mol K)
# of moles
Gas constant R=kB NA = J/(mol K)
Classical limit
Requires quantum mechanics

10. 1D Quantum mechanical harmonic oscillator
Schrödinger equation:
Solution:
Quantized energy x x

11. CORRESPONDENCE PRINCIPLE
Large quantum numbers:
correspondence between qm an classical system x
Classical probability density
Classical point of reversial

12. Quantum mechanical thermal average of the vibrational energy
Einstein model:
N independent 3D harmonic oscillators
Energies labeled by discrete quantum number n
Boltzman factor weighting every
energy value
Probability to find oscillator in state n

13. Let us introduce
partition function
therefore calculate Z

14. Bose-Einstein
distribution
where

15. and
With
In the Einstein model where
for all oscillators
zero point energy
Heat capacity:
Classical limit
Note: typing error in
Eq.(2-57) in
J.S. Blakemore, p123

16. 1 for
for
good news: Einstein model
explains decrease of Cv for T->0
bad news: Experiments show
for T->0

17. Assumption that all modes have the same frequency
unrealistic
refinement
Debye Model
We know already: 1) 2)
wave vector k labels particular phonon mode 3)
total # of modes = # of translational degrees of freedom
3Nmodes in 3 dimensions
N modes in 1 dimension
Let us remind to dispersion relation of monatomic linear chain
N atoms
N phonon modes
labeled by equidistant k values within the
1st Brillouin zone of width
distance between adjacent k-values

18. A more detailed look to the origin of k-quantization
Quantization is always the result of the boundary conditions
Let’s consider periodic boundary conditions
Atom position n characterized by
After N lattice constants a we end up again at atom n

19. ? In 3D we have: and One phonon mode occupies k-space volume
Volume of the crystal ?
How to calculate the # of modes in a given frequency interval
Density of states
Blakemore calls
it g(), I prefer D()

20. total # of phonon modes
In a 3D crystal
Let us consider dispersion of elastic isotropic medium
Particular branch i: vL
vT,1=vT,2=vT
here

21. ? Taking into account all 3 acoustic branches
What is the density of states D(ω) good for
Calculate the internal energy U
# of modes in
temperature independent
zero point energy
Energy of a mode
= phonon energy
# of excited phonons

22. ? How to determine the cut off frequency max
Density of states of Cu
determined from neutron scattering
also called Debye frequency D
choose D such that both curves
enclose the same area

23. v T U C ÷ ø ö ç è æ =
with

24. Debye temperature energy temperature Substitution:
Click for a table of Debye temperatures