PHY455
Spring, 2015
Exam #3
Name
_______________
Total
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All answers should be given in eV or Å and related units where appropriate, and
SI unless otherwise specified. All answers should be given numerically wherever
possible unless otherwise stated.
Show your work.
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1. Sketch the temperature dependence of the heat capacity from 0K to room
temperature for a nonmetal and indicate the functional dependence on T at low
and high T.
2. Sketch the temperature dependence of the thermal conductivity for a large
crystal and for a very small crystal.
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3. If you increase or decrease the wavevector of a phonon mode by a reciprocal
lattice vector, what happens to
a. the energy/frequency?
b. the displacement of the atom?
4. For a 2D lattice with N atoms, mass M, 1 atom per unit cell, and side length L,
a. Find the expression for the phonon density of states, D(𝜔).
b. Using the result of part a., find an expression for the energy of the system.
c. The answer in part b. should include a sum over polarizations. How many
polarizations are there and what are they?
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d. Find an expression for the density of states using the Debye Model in terms of
vs and the parameters given at the beginning of the problem.
e. Find an expression for the Debye frequency in terms of w , vs and the
parameters given at the beginning of the problem.
f. Find an expression for the density of states in the Einstein Model in terms of
w , the Einstein frequency w o , and the parameters given at the beginning of the
problem.
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g. Find an expression for the energy in the Einstein Model in terms of the
Einstein temperature TE.
h. Find an expression for the heat capacity in the Einstein Model.
i. Find the high T limit of the heat capacity in the Einstein Model.
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5. For a 1D lattice with N atoms, 1 atom per unit cell, mass M, force constant C,
and lattice constant a, write out the equation of motion and solve for the
dispersion assuming solutions of the form:
un = Aexp[iw t + ikna]
Plot the  vs. k dispersion in the first Brillouin Zone and find the value of  at the
zone boundary.
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6. For the 1D lattice with mass M, lattice spacing a, and 1 atom per unit cell,
assume the 𝜔 and k are related by 𝜔 = 𝐴𝑘 3 . Find an expression for the density
of states.
7. Given a velocity of sound for Al of 5000 m/s, calculate (SI units):
a. The density of Al atoms.
b. The Debye frequency.
c. The Debye temperature.
8. A beam of wavelength 4.00 Å neutrons is incident normally on a cube face
(call this the x-direction) of a monoatomic SCC crystal with cube edge 3.25 Å.
Some neutrons are scattered in a single phonon event and exit along a diagonal
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of a cube face (in the x-y direction) with wavelength 2.50 Å. Find the energy (eV)
and wavevector (Å -1)of the phonon involved in this process.
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Old
1. For a 1D chain of 4.0x1010 atoms spaced 2.50 Å apart, find
all of the allowed k values in the 1st Brillouin zone (in SI units). Assume periodic
boundary conditions.
2. A beam of neutrons with wavelength of 3.00 Å is incident normally on a cube
face of a monoatomic SCC crystal with cube edge 4.50 Å. Call the incident
direction the x-direction. Some neutrons are scattered in a single phonon event
and exit along the z-direction with wavelength of 4.00 Å. All answers should be in
eV or Å -1 .
a. Find the incident neutron energy and wave vector (in vector notation).
b. Find the outgoing neutron energy and wave vector (in vector notation).
c. Find the energy and wave vector (in vector notation) of the phonon involved in
this process.
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3. Derive D(for a 1D chain of N atoms, mass M, lattice constant a.
4. Derive D(for a 2D square lattice of N atoms, mass M, lattice constant a.
5. Sketch the temperature dependence of the heat capacity for a non-metal and
give the T dependence at high and low T.
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6. Sketch vs. k for
a. 1D chain of atoms; 1 atom/unit cell
b. 1D chain of atoms; 2 atoms/unit cell
c. 3D lattice of atoms; 1 atom/unit cell
Indicate longitudinal, transverse, acoustical , and optical modes where
appropriate. Assume a general case (no high degree of symmetry).
7. In the Einstein model, find the average over 2 (3D case). Answer should be
in terms of N and o.
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8. The velocity of sound in Aluminum is 6420 m/s. Find the force constant in N/m.
9. In the Debye model, calculate the Debye frequency and Debye temperature
assuming 1 mole of atoms in a volume of 1 cm3. Assume a velocity of sound of
5000 m/s.
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10. Sketch the thermal conductivity vs. T for a typical non-metal vs. T, give the T
dependence at high and low T and describe the contributions from phonon and
lattice scattering at these limits.
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7. In the Debye model, find the average over 2. (3D case) Answer should be in
terms of V, vs, and D.
10. Explain the modes and differences in number of modes in the Figure above.
8. Estimate what is reasonable for low and high T
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S13
1. For a 1D chain of 5.0x1010 atoms spaced 3.00 Å apart, find
all of the allowed k values in the 1st Brillouin zone. Assume periodic boundary
conditions.
2. A beam of neutrons with wavelength of 4.00 Å is incident normally on a cube
face of a monoatomic SCC crystal with cube edge 3.50 Å. Call the incident
direction the x-direction. Some neutrons are scattered in a single phonon event
and exit along the diagonal of an (x-y) face with wavelength 5.00 Å. All answers
should be in SI units.
a. Find the incident neutron energy and momentum (in vector notation).
b. Find the outgoing neutron energy and momentum (in vector notation).
c. Find the energy and momentum (in vector notation) of the phonon involved in
this process.
3. For a 1D monoatomic chain of 4.0x1010 atoms with lattice constant 2.50 Å,
find the density of states in the Debye approximation assuming a velocity of
sound of 5000 m/s (approximately the speed of sound in iron).
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4. For a 1D chain of atoms with 2 different masses alternating along the chain,
sketch a typical dispersion curve ( vs. k) for phonons in the 1st BZ. Include the
different polarizations.
5. On a linear temperature scale, sketch the temperature dependence of the
heat capacity of a non-metal as a function of T. Give the T dependence at low T
and include the limiting value at high T.
6. Sketch a typical dispersion curve ( vs. k) for phonons in the Debye
approximation.
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7. Sketch a typical dispersion curve ( vs. k) for phonons in the Einstein
approximation.
8. Plot the density of states, D(), vs.  in the Einstein model. Assume a 1D
monoatomic chain of 3.0x1010 atoms with lattice constant 4.00 Å and a
characteristic Einstein temperature of 1500K. Provide numbers for Do and o.
9. For a 3D monoatomic SCC lattice of 2.0x107 atoms on a side with lattice
constant 4.00 Å, find the Debye frequency and the Debye temperature assuming
a velocity of sound of 300 m/s.
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10. Give some  vs. k relation, find D(). Plot it.
11. Find frequency for M, 3M or C, 2C, or M or C(1+); let  go to 0
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