PHY455
Spring, 2015
Exam #2
Name
_______________
Total
_______________
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.
1. Assume an atom of valency 2 is in an SCC structure with a lattice constant of
2.50 Å. Calculate:
a. atomic density (SI units)
b. electron concentration (SI units)
c. Fermi wavenumber
d. Fermi energy
f. Fermi temperature
g. rs parameter
2. For an electron gas in 1D, valency of 1, in terms of the length of the crystal, L,
and the number of atoms, N, m, ∏, and h find expressions for (show your
work/derivations):
a. atomic density
b. Fermi wavenumber
c. Fermi energy
d. rs parameter (for 1D-think about it)
3. Sketch f(), D(
(
n(
,
i
) vs.  for a 3D electron gas
(indicate the location of the Fermi Energy in all cases).
a. T~100K
b. T~2000K
4. For the SCC of side a, give the G vector necessary to reduce the following k
vectors into the 1st Brillouin Zone. Give the resultant k vector. Give the k and G
vectors in terms of a and ∏. Calculate the energy of the band in each case (in
terms of ∏ a, m and h).
a. 3/a[010]
b. 7/a[110]
c. /2a[111]
d. 11/2a[122]
5. Calculate the theoretical Hall coefficient for Rubidium (SI units).
6. Suppose you calculated a theoretical value of the Hall coefficient of -1.50 (in
10-24 CGS units) assuming a valency of 1, but the experimental value was found
to be ?? in the same units. Explain what this means about the actual material and
its conductivity.
a. for a value of -3.00
b. for a value of +1.50
7. What are the 3 main contributions to the electrical resistivity of a metal?
Which depend on temperature?
8. For a large crystal of Rb at 5 K with a time constant of 5x10-9 s, estimate the
electron mean free path in m. What would be the consequence for the electrical
conductivity for a sample of 1.0 mm on a side?
9. For a 2D crystal with valency 3 and square lattice of side 2.50 Å,
a. Plot the reciprocal lattice and the 1st and 2nd Brillouin zones. Give
dimensions.
N
A
from a previous homework assignment, keeping in mind that that was derived
assuming a valency of 1.
b. Find the radius of the Fermi circle in Å-1. You can make use kF2 = 2p
c. Sketch the Fermi circle in a reduced zone scheme below.
d. What would be qualitatively different if the valency were 4?
10. For a 2D square lattice with lattice constant a, plot the lowest lying (5) bands
in the [10] direction. Find the energy of the lowest band at the zone boundary (in
terms of ∏, m, a, and h). and call that Eo. Label the various energies in terms of
Eo and the k values in terms of ∏ and a. Generate the table below first, then
sketch the bands.
Band
G vector
E(k)
E(0 0)
E(∏/a 0)
Not Used
4. Sketch the reciprocal lattice and the 1st and 2nd Brillouin zones for a 2D
crystal with sides 2.00 A and 4.00 A. Indicate dimensions on the sketch.
8. From electrical conductivity measured at 295 K for Rubidium, find the value of
the time constant that limits the conductivity at that temperature.
9. Given this rs value, backtrack to Fermi energy, wavevector, etc. What is this
closest to? Guess the valency.
Old
1. Calculate for potassium, K,
a. the Fermi wavevector in Å-1
b. the Fermi energy in eV
c. the Fermi temperature in K.
2. For an electron gas in two dimensions:
a. Derive the relationship between the electron concentration and the Fermi
wavevector. Assume a valency of 3 free electrons per atom.
b. Derive an expression for the density of levels, D().
3. For a 2D gas at T=0, a valency of 3, N atoms and area A, find the total energy
of the system in terms of A,N,m,and h.
4. Sketch the f(), D(
a. T=0
n(), vs.  for a 3D electron gas at
b. T~1000K
5. Referring to the results of #4, briefly explain the origin of the linear term in the
heat capacity for metals.
6. For the SCC of side a, give the G vector and resultant necessary to reduce
the following k vectors into the 1st Brillouin Zone:
a. 3/a[110]
b. 11/a[111]
c. 9/2a[231]
d. 3/2a[010]
e. /2a[001]
7. a. For an electron gas, find the occupation probability for a state 0.250 eV
above the chemical potential for these temperatures: 100K and 10,000K.
b. Repeat for a state 0.250 eV below the chemical potential.
c. Comment on/briefly discuss the results (do they make sense...?)
8. For a 1D crystal with valency 1 and lattice of spacing 2.0 Å,
a. Plot the real lattice, the reciprocal lattice and the 1st and 2nd Brillouin Zones.
b. Find the "radius" of the Fermi surface in Å-1 and sketch the Fermi surface in a
reduced zone scheme.
c. Find the "radius" of the Fermi surface and sketch the Fermi surface in a
reduced zone scheme assuming a valency of 3.
9. For a 2D rectangular lattice with dimensions of 2Å and 5Å
a. plot the real space lattice.
b. Plot the reciprocal lattice; give dimensions.
c. Plot the 1st and 2nd Brillouin Zones on the plot above.
10. Given the electrical conductivity of K at 295K, find a. the lifetime in s and b.
the mean free path of an electron in Å. Hint: ohm is in SI units; you just need to
covert
cm-1.
S13
1. Sketch the temperature dependence of the heat capacity for a free electron
gas and for a metal. Explain what contribution is due to electrons.
2. For an electron gas in one dimension:
a. Derive the relationship between the electron concentration and the Fermi
wavevector. Assume a valency of 1 free electron per atom.
b. Derive an expression for the density of levels, D().
3. Sketch the occupation probability, f() vs.  for an electron gas at
a. T=0, b. T~50K and c. T~2000K.
4. a. For an electron gas, find the occupation probability for a state 0.200 eV
above the chemical potential for these temperatures: 50K and 2000K.
b. Repeat for a state 0.200 eV below the chemical potential.
5. For the SCC of side a, give the G vector and resultant necessary to reduce
the following k vectors into the 1st Brillouin Zone:
a. 7/a[100]
b. 9/a[111]
c. 3/2a[231]
6. Calculate for Cs,
a. the Fermi energy in eV
b. the Fermi wavevector in cm-1
c. the Fermi temperature in K.
7. For a 2D crystal with valency 1 and square lattice of side 3.0 Å,
a. Plot the reciprocal lattice and the 1st and 2nd Brillouin zones.
b. Find the radius of the Fermi circle in Å-1. Use kF2 = 2p
N
A
from a previous homework assignment.
c. Sketch the Fermi circle in a reduced zone scheme on the plot above.
d. Find the radius of the Fermi circle and sketch the Fermi circle in a reduced
zone scheme assuming a valency of 2.
8. Consider the free electron energy bands of an SCC crystal in the reduced
zone scheme. Find the energy of the first 7 bands at the center of the zone and
at the zone boundaries in the (100) direction in terms of the lowest band energy
at the zone boundary (call it Eo); and sketch the bands in the reduced zone
scheme.
9. At 295 K, the conductivity of Na is 2.11x105 (ohm.cm)-1. Use that to estimate
the average collision time.