1000 Solved Problems in Modern Physics

5.2 Problems 297

5.30 Einstein’s model of solids gives the expression for the specific heat

Cv= 3 N 0 k

(

θE

T

) 2

eθE/T

(eθE/T−1)^2

whereθE=hνE/k.

The factorθEis called the characteristic temperature. Show that (a) at high

temperatures Dulong Petit law is reproduced. (b) But at very low temperatures

theT^3 law is not given.

5.31 Debye’s model of solids gives the expression for specific heat

Cv= 9 N 0 k

1

x^3

∫x

0

ξ^4 eξ

(eξ−1)^2

dξ

whereξ=hν/kT,x=hνm/kTandθD=hνm/kis the Debye’s charac-

teristic temperature. Show that (a) at high temperatures Debye’s model gives

Dulong Petit law (b) at low temperatures it givesCv∝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 fromEtoE+dEis given by

n(E)dE=

2 π

h^3

(2m)^3 /^2 E^1 /^2 dE

Show that the total energy= 3 NEmax/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 × 1028 m−^3.

5.35 Calculate for silver the energy at which the probability that a conduction
electron state will be occupied is 90%. AssumeEF= 5 .52 eV for silver and
temperatureT=800 K.

5.2.4 Semiconductors...................................

5.36 An LED is constructed from aPnjunction based on a certain semi-conducting
material with energy gap of 1.55 eV. What is the wavelength of the emit-
ted 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