PHYS 4310
Assignment #4-5
Due 28 Mar 2025 (end of the day)
Question#1
This question is more like a review:
Heat Capacity:
Prove that the heat capacity of a one-dimensional classical harmonic oscillator at temperature T
is kB. Include in your solution answers to the following questions:
(a) What is the energy of the system?
(b) What are the variables which define the state of the system?
(c) What is the probability that a particle will have a particular velocity and a particular
displacement?
(d) What is the constant of proportionality in (c)?
(e) What is the average velocity of the oscillator?
(f) What is the average displacement of the oscillator?
(g) What is the average energy of the oscillator?
(h) What is the heat capacity of the oscillator?
Question #2: Kohn Anomaly:
Let us suppose that the interplanar force constant K is of the form, πΎ =
π΄ sin(ππ0 π)
ππ
, where A and
π0 are constants and n runs over all integers. Such a form is usually expected in metals.
Use this in the Force equation
(i) Find ω2 in terms of k and k0
ππ 2
(ii) Find ππ
(iii) Plot ω2 vs k
(iv) So far in this problem, we have considered the role of displacements of atoms immediately
prior and next to the u(n). However, this can be extended to the other planes n’ and the total
force on the atom in the plane where we located u(n) can be calculated. In that case, the force
will be:
PHYS 4310
Assignment #4-5
Due 28 Mar 2025 (end of the day)
πΉπ = ∑ πΎπ′ (π’π+π′ − π’π′ )
π′
Obtain (i) and (ii) for the above force with πΎπ =
π΄ sin(ππ0 π)
ππ
(In the materials exhibiting charge density waves (basically extreme cases) the phonon energy
even goes to zero and Kohn anomaly helps us to understand the phonon behavior. Also, see an
interesting article on Palladium (Fig. 2). (https://iopscience.iop.org/article/10.1088/13672630/10/4/043025/pdf)
Question #3:
Write a note on the neutron scattering experiments to detect the phonons in a crystal. Your
answer can be 10 sentences long and you can use a sketch if you think it is required. Cite the
source for your answer. (your sources can be one or more of the following: (i) websites of
neutron scattering centers (ii) books, (iii) review articles. Not general websites which could not
be used in scientific literature)
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