Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit

Problem Set 4

PHYC4230

Due:  Monday Mar.  6, 2023

1    Debye model in 2D

(a) State the assumptions of the Debye model.

(b) Determine the energy of a two-dimensional solid as a function of T using the Debye approximation. You do not have to solve the integral.

(c) Calculate the heat capacity in the high T limit

(d) At low T, show that

Cv  = KTn                                                                                                       (1)

Find n. Express K as an indefinite integral

2    What is a Phonon?

(a) Explain in your own words what is a “normal mode” and what is a “phonon”.

(b) Consider a vibrational mode with wavevector k, which has a crystal momentum ℏk. Show that this is not the momentum of the atoms by calculating the total momentum  mvi .

(c) Derive the phase velocity and group velocity for a monatomic chain and sketch their dependence on k .

(d) The following problem explores how to construct a localized phonon in order to examine their particle- like nature. Consider a gaussian wavepacket:

π/a

A(x) =      exp( −ξ(k k0 )2 )exp(i(kx ωt)).                                      (2)

k= −π/a

Use this construction to demonstrate that the packet travels at a velocity equal to the group velocity. Assume that the packet has a very narrow wavevector distribution about the value k  = k0 .  This assumption affords you two approximations:  Firstly, is allows you to convert the sum over the first Brillouin zone to an integral over k from −∞ → ∞ because the gaussian envelop will be assumed to be very small outside of the Brillouin zone.  Secondly, it permits you expand the frequency ω(k) in a Taylor series.

3    Density of State of a Monoatomic Chain

In class we showed that the dispersion relation for a monoatomic chain is

ω(k) = 24  'sin ( ) ' .                                                              (3)

(a) Generate a plot of this dispersion relation in appropriate reduced units over the first Brioullin zone. Note that ω(k) is actually not a continuous curve except in the limit of an infinite sample but rather is a collection of discrete points corresponding to the k-points determined by the boundary conditions. Choose an appropriate chain length so that the individual points are visible in your plot.

(b) Show that the density of states is given by,

g(ω) = 6N          1        

π   ^ωM(2)  ω 2 ,

where ωM  is the maximum frequency.

(c) Compare the above g(ω) with the density of states derived from Debye’s model. Make use of the group velocity expression for small k .

(d) Notice that the density of states in Eq. 4 diverges as the frequency approaches this maximum. Super- impose a graph of ω vs g(ω) next to your plot in part (a). Use this plot to explain to the origin of the divergence (think carefully about what density of states means).

4    Diatomic chain with different masses

Suppose we have a vibrating 1D atomic chain with two types of atoms in the basis. The masses of the two atoms are M1  and M2  (M1  > M2 ). There is only one spring constant, G. The lattice constant is a.

(a) Make a sketch of this chain and indicate the length of the unit cell.

(b) Derive the equations of motion for this chain.

(c) By filling in the trial solutions into the equations of motion (which should be similar to Ansazt used in the lecture), write the equations as an eigenvalue problem (matrix form).

(d) Find the eigenvalues and eigenvectors at k = 0 and k = π/a (four solutions in total). 

(e) Describe the vibration patterns corresponding to the above four eigenvectors. 

(f) What will happen to the periodicity of the band structure if M1  = M2 .