Physics 7440
Assignment 9
Due Friday April 11, 2003
Reading:
1. Sections on phonons, Marder Chapter 13.
Problems:
1. Molecular and crystal binding potentials and phonons.
In Problem Set 7, we solved worked with the singly ionized hydrogen molecule to
see an example of a variational calculation of the electronic energies. We found that
the energies are approximately given by:
 2  Z 
1
E Z, R 

  1  I 2  
1  S  2m  a0 

2


e2 Z
1  I 4  2 I 5 
4 0 a0

where
2
 ZR   ZR 1  ZR  
S  Z , R   exp  
 
 1 
 
a0 3  a0  
 a0  

I1  1


 ZR  
ZR 
I 2    S  2 1 
 exp  
 
a0 

 a0  

 a
 ZR  
a0  
I 4     0  exp  2

 1 
a0  
ZR  

 ZR
 ZR  ZR 
I5  exp  
1 

a0 
 a0 
The total energy for the system is then just the sum of the electronic and protonproton potential energy:
Etotal  Z , R  
Physics 7440
e2 1
 E Z, R
4 0 R
H9.1
Spring 2003
Physics 7440
Assignment 9
Due Friday April 11, 2003
In this form, we treated the proton-proton separation, R, and the effective charge, Z,
as variational parameters used to minimize the total energy. We found a minimum
energy located at R  2a0 and Z  1.23 .
To determine the vibrational modes of this system, we would treat the total energy as
an R-dependent potential and put it into the Schrodinger Eq. along with proton
kinetic energy terms and solve for the eigen-energies of proton motion.
Even for this simple problem, solving for the eigen values is difficult because the
‘potential’ is not a simple function. For example, Z is still a variable parameter. If
you like, you can think of it as some unknown function of R, which must be
determined before the potential can be written in terms of just R.
a) Use Mathematica or some other program to plot the potential above at ten
different R positions and graphically find the minimizing values of Z. Plot the
approximate values of Z(R) and produce a table of the minimum potential energy
vs. proton-proton separation. This table is a numerical approximation to the
actual effective proton-proton potential.
Often, the use of such numerical potentials is considered not worth the effort for the
precision desired. An alternative approach is to use a simple function that reproduces
the numerical data reasonably well. The Lennard-Jones 6-12 potential is one
example. Another is the Morse potential, which looks like this:


V  r   V0 exp 2  r     2 exp   r   
One of the cool things about this potential, the main reason that it is used, is that you
can actually solve the Schrodinger Eq. with the Morse potential exactly!
b) Use Mathematica, etc. to plot the Morse potential along with the points you found
above for the numerical approximation to the actual effective proton-proton
potential. Adjust and list the parameters of the potential to get close to the data. If
you go nuts, you can use the NonlinearFit function of Mathematica to optimize
the parameters. Now you have an approximate potential in closed form
The Morse potential sees a fair amount of use in this role. In the next few parts, let’s
imagine that we use it to model the binding potentials in a simple cubic solid.
Assume that each atom only interacts with its closest neighbor atoms.
c) Assume that the parameters of the Morse potential are known for some system of
interest. Given an assumed simple cubic lattice, calculate the equilibrium lattice
spacing, a, for the system and the total binding energy of the crystal (assuming N
atoms and zero kinetic energy) as functions of the Morse parameters.
d) Do a second order expansion of the total potential about the equilibrium lattice
position. Determine how the spring constant, C, which describes harmonic
Physics 7440
H9.2
Spring 2003
Physics 7440
Assignment 9
Due Friday April 11, 2003
motion of the atoms depends upon the parameters of Morse potential. (Recall that
C is the coefficient in Hooke's Law for a spring such that the force on a spring
when stretched to position, x, is F=Cx i.e., the potential is Cx2/2.)
e) If we did the normal modes problem for this system, we would find that the small
k-vector modes for the phonons would describe sound waves i.e.,
  cs k
where cs is the sound velocity. Typically, the sound velocity is related to the
spring constant, lattice spacing and the mass of the atoms by:
cs 
Ca 2
M
Assume that the interacting atoms are Sodium atoms: Then, the mass is
MNa=3.7x10-23 gm. Further, Sodium has a sound velocity of roughly
100,000 cm/sec. and a lattice spacing of 3.5x10-8 cm. Estimate the spring
constant, C.
2. Density of states for various situations (from Kittel 5.1.)
a) In class, we found the dispersion relation for a monatomic linear chain of N
atoms or mass, M, with harmonic nearest neighbor interactions determined by
spring constants, C, and with lattice spacing, a, to be given by:
 2 k  
2C
1 coska
M
or
 k  
4C
 ka 
sin  
M
 2 
Plot this dispersion relation in the first Brillouin zone. Then, show that the
density of phonon states is given by:
D   
Physics 7440
2N

1

2
M
2 
H9.3
Spring 2003
Physics 7440
Assignment 9
Due Friday April 11, 2003
where M is the maximum frequency in the zone. This result shows that there
is a very large (infinite) number of modes near the maximum frequency. Can
you see why? Plot the density of states result too.
b) In the last problem set, we also found the optical modes for the diatomic linear
chain. In that case, we considered the masses to be equal along the chain, but
the spring constants alternate between K and G in magnitude. The dispersion
relation we found is:
 2 k  
K  G
M

1
M
K 2  G 2  2 KG cos  ka 
where the + gives the optical modes and the - gives the acoustic modes. In
class, I also pointed out that for K>>G, the optical mode would be very flat.
Show that in this limit, and for small ka, that the optical dispersion relation
can be approximated as:
  k   M  Ak 2
Write down an explicit form for the maximum frequency and for the constant,
A. This form is also found to hold in 3-D. Show that for the 3-D case, that the
density of states is:
 L 3 2
 M       M

D     2  A3 / 2

0
  M

Here, you find a discontinuous density of states.
3. Heat capacity of layered systems.
a) If we generalize the 1-D monatomic chain to 2-D, then we have a two
dimensional square lattice of atoms, each of mass, M, with a square lattice
constant of, a, and with harmonic nearest neighbor interactions determined by
spring constants, C. Then dispersion relation is then very similar to the case
for 1-D (see above), except that the frequency now depends on components of
the k-vector along the x and y-directions. It looks like this:
 
2 k 

2C
2  cos  k x a   cos  k y a 
M

Plot this dispersion relation in the 1st Brillouin Zone.
Physics 7440
H9.4
Spring 2003
Physics 7440
Assignment 9
Due Friday April 11, 2003
b) Generalize the Debye Model to two-dimensional systems and show that the
low temperature heat capacity for the 2-D system is proportional to T2.
c) Suppose we consider a more realistic case, such as graphite. Then the system
is composed of atomic layers (roughly 2-D systems), which have weak
interactions between layers. How would you expect the heat capacity to
behave as you decrease the system temperature? Sketch the heat capacity and
label the various temperature regions.
d) Go to the library and find some heat capacity data for the graphite form of
carbon and compare it to your expectations from c). Try to estimate the
strength of the springs in the graphite planes from this data.
Physics 7440
H9.5
Spring 2003