0PHY455
Spring, 2014
Exam #2
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.