TD8 Statistical Physics (M1)
Phonons
Exercise 1: The Debye Model (1912) for specific heat in solids.
1) Vibrational eigenmodes.
Let’s consider first the case of 2 atoms with mass m, coupled through a spring with stiffness
K, and connected through an identical spring to fixed boundaries. u1 and u2 are the displacements of
the particles 1 and 2 respectively.
a) Prove that the equations of motion can be written as a set of 2 decoupled equations:
..
X  12 X
..
Y   22 Y
where X and Y are linear combinations of u1 and u2. X(u1,u2) and Y(u1,u2) are called the
vibrational eigenmodes, and 1 and 2 are the corresponding eigenfrequencies.
b) Use a quantum description of the vibrational energy to compute the canonical partition
function, and the Heat capacity of the system with 2 atoms. Compare the temperature
dependence of the Heat capacity obtained for a system with only one oscillator
(Einstein’s model) and the Heat capacity obtained in this question.
2) One-dimensional chain.
a) Write the equations of motion for a one-dimensional chain of atoms coupled through
springs with stiffness K. The positions of the atoms are noted xn=n.d+un where d is the
interatomic distance at rest, and un is the displacement of atom n.
b) In case of periodic boundary conditions, the solution can be obtained with the help of a
~
Fourier transform: u n 
u q .e iqn.d i .t . Write the resulting relation of dispersion (q).

q
Use the periodic boundary conditions to get a quantification for the wavevector q.
How many distinct eigenmodes must be used? What is the density g() of vibrational states
with frequency ?
c) Write the expression of the Heat capacity, first as a summation over all the eigenmodes,
and then as an integral over . Compute the Heat capacity in the high and in the low
temperature limit, with a linear approximation of the dispersion relation. Comment.
3) Three-dimensional monoatomic solid.
In a 3D solid the relation between ω and k (the angular frequency and the wave vector of a
lattice vibration) can be described by 3 branches, the first is associated with longitudinal while
the two others correspond to transversal modes.
For each mode, the energy associated to one ”phonon” is simply : = hν = h ω/2π


The momentum of this phonon as a function of k is obtained by p  k
For each branch the relation of dispersion at large λ is supposed to be linear: k = ω/cL for the
longitudinal branch, and k = ω/cT for the transversal branches.
a) For large λ and in the approximation that k and p are continuous, calculate g(p)dp the
density of states as a function of p for one branch only.
b) In the Debye model, we suppose that the relation of dispersion at large λ holds for the all
range of available energies. Calculate the total density of states g()d as a function of
the energy . To do this use the effective velocity defined by :
3
1
2
 3  3
3
c
cL cT
c) Given that the total number of states is 3N, where N is the number of atoms, define D
and θD (Debye energy and Debye temperature).
d) Find the analytic expressions for U (energy) and Cv (Heat capacity).
Compute Cv in the limit of low and high temperatures.
e) Experimental measurements give an effective velocity c = 1.84 103 m.s−1 for Ag. Is this
consistent with the Heat capacity measurements that give θD =220°K?
Exercise 2: Three-dimensional diatomic solid.
In a 3D solid composed by two atomic species of different masses we suppose that the
transversal and the longitudinal velocities are identical, and that the density of states g() is
schematically given by the figure below:
The two peaks refer to acoustic and optical branches that extend on different energy
domains ( 1 >> 3 − 2 ).
The total number of atoms is 2N (N for each specie) and the sum over the modes for each
peak is 3N.
a) Use simultaneously the Debye and Einstein models for the phonons to define g0() an
approximate model representation of g(). To do this the only parameters that you will
introduce are : N, 1 and 0 = (3+2)/2.
Schematically draw g0() on a graph.
b) Briefly explain why the knowledge of g() is especially important at low . Experiments
show that for low energies g() can be written as : g() = b2 (where b takes a precise
value given by the experiments). This leads to a modification of the previous model.
Write g1() the density of states of the modified model that takes into account the
experimental data. To do this you will need to introduce a new energy reference 4.
c) Calculate the energy and the heat capacity Cv by using g1(). Give the limit of Cv at high
and low temperature.
Datas:

x 2 .e x
 e
0
x

1
2
dx 
2
3
h /2 ≈ 10−34 J.s
NAvogadro ≈ 1024
kB ≈ 10−23 JK
ρAg = 10.53 × 103 Kg.m−3
MAg = 108 g.mol-1