PHY455
Spring, 2014
Exam #3
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.
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.
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.
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.
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.
Not Used
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.
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).
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.
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.
Not Used
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