5.2 Problems
297
5.30 Einstein’s model of solids gives the expression for the specific heat
C v = 3N 0 k
θ E
T
2
e
θE /T
(e θ E /T − 1) 2
where θ E = hν E /k.
The factor θ E is called the characteristic temperature. Show that (a) at high
temperatures Dulong Petit law is reproduced. (b) But at very low temperatures
the T
3 law is not given.
5.31 Debye’s model of solids gives the expression for specific heat
C v = 9N 0 k
1
x 3
x
0
ξ
4 e
ξ
(e ξ − 1) 2 dξ
where ξ = hν/kT , x = hν m /kT and θ D = hν m /k is the Debye’s characteristic temperature. Show that (a) at high temperatures Debye’s model gives
Dulong Petit law (b) at low temperatures it gives C v ∝ T
3 in agreement with
the experiment.
5.32 For a free electron gas in a metal, the number of states per unit volume with
energies from E to E + dE is given by
n(E)dE =
2π
h 3 (2m)
3/2 E
1/2 dE
Show that the total energy = 3NE max /5.
5.33 Assuming that the conduction electrons in a cube of a metal on edge 1 cm
behave as a free quantized gas, calculate the number of states that are available
in the energy interval 4.00–4.01 eV, per unit volume.
5.34 Calculate the Fermi energy for silver given that the number of conduction
electrons per unit volume is 5.86 × 10
28 m
−3 .
5.35 Calculate for silver the energy at which the probability that a conduction
electron state will be occupied is 90%. Assume E F = 5.52 eV for silver and
temperature T = 800 K.
5.2.4 Semiconductors
5.36 An LED is constructed from a Pn junction based on a certain semi-conducting
material with energy gap of 1.55 eV. What is the wavelength of the emitted light?
5.37 Suppose that the Fermi level in a semiconductor lies more than a few kT below
the bottom of the conduction band and more than a few kT above the top of
the valence band, then show that the product of the number of free electrons
and the number of free holes per cm
3 is given by
297
5.30 Einstein’s model of solids gives the expression for the specific heat
C v = 3N 0 k
θ E
T
2
e
θE /T
(e θ E /T − 1) 2
where θ E = hν E /k.
The factor θ E is called the characteristic temperature. Show that (a) at high
temperatures Dulong Petit law is reproduced. (b) But at very low temperatures
the T
3 law is not given.
5.31 Debye’s model of solids gives the expression for specific heat
C v = 9N 0 k
1
x 3
x
0
ξ
4 e
ξ
(e ξ − 1) 2 dξ
where ξ = hν/kT , x = hν m /kT and θ D = hν m /k is the Debye’s characteristic temperature. Show that (a) at high temperatures Debye’s model gives
Dulong Petit law (b) at low temperatures it gives C v ∝ T
3 in agreement with
the experiment.
5.32 For a free electron gas in a metal, the number of states per unit volume with
energies from E to E + dE is given by
n(E)dE =
2π
h 3 (2m)
3/2 E
1/2 dE
Show that the total energy = 3NE max /5.
5.33 Assuming that the conduction electrons in a cube of a metal on edge 1 cm
behave as a free quantized gas, calculate the number of states that are available
in the energy interval 4.00–4.01 eV, per unit volume.
5.34 Calculate the Fermi energy for silver given that the number of conduction
electrons per unit volume is 5.86 × 10
28 m
−3 .
5.35 Calculate for silver the energy at which the probability that a conduction
electron state will be occupied is 90%. Assume E F = 5.52 eV for silver and
temperature T = 800 K.
5.2.4 Semiconductors
5.36 An LED is constructed from a Pn junction based on a certain semi-conducting
material with energy gap of 1.55 eV. What is the wavelength of the emitted light?
5.37 Suppose that the Fermi level in a semiconductor lies more than a few kT below
the bottom of the conduction band and more than a few kT above the top of
the valence band, then show that the product of the number of free electrons
and the number of free holes per cm
3 is given by