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KEELE UNIVERSITY

DEGREE EXAMINATIONS 2008

Level 2 (PRINCIPAL COURSE)

Friday 16th May 2008, 09:30 – 11:30

PHYSICS

PHY-20026

STATISTICAL MECHANICS AND SOLID STATE PHYSICS

Candidates should attempt to answer FOUR questions,

TWO from section A and TWO from section B of the paper.

Tables of physical and mathematical data may be obtained from the

invigilator.

/Cont’d

1

SECTION A: STATISTICAL MECHANICS (Answer TWO questions)

1. (a) Explain the meaning of microstate and macrostate. As an example, give one

macrostate of the “tossing a coin three times” system, and write down all

corresponding microstates. [10]

(b) Explain why the expression

Ω = N !

∏

ni!

gives the number of microstates for a given macrostate with ni particles in the

non-degenerate energy level Ei . [10]

(c) Which macrostate will be observed in a real system? [5]

(d) Starting with the above expression for Ω, derive the expression

ni = A exp (−βEi)

for the number of particles in the energy level Ei. [30]

(e) For the following assume β = 1 kBT

. Show that the normalization constant is

A = N

ZSP

where ZSP is the single particle partition function, and N the number of parti-

cles. [10]

(f) Hence, show that the internal energy is given by

U = NkBT 2 ∂ ln ZSP

∂T

where T is the temperature, and kB is the Boltzmann constant. [35]

/Cont’d

2

2. (a) Describe the properties of the spin- 1 2

paramagnet. Derive and simplify the

corresponding partition function. [15]

(b) From the partition function show that the internal energy of the spin- 1 2

para-

magnet is

U = −NµBB tanh (

µBB

kBT

)

where N is the number of particles, µB Bohr’s magneton, B the magnetic field

strength, T the temperature and kB Boltzmann’s constant. [30]

(c) Discuss this result, both as a function of T for a given magnetic field strength B

as well as a function of B for a given T . Include both mathematical and physical

arguments in your discussion. [40]

(d) The internal energy can be expressed in terms of the magnetisation M : U = −MB.

In the weak-field limit (B → 0) derive Curie’s law of paramagnetism. [15]

3. (a) Explain the difference between quantum statistics and classical statistics. In

which situation do we have to use quantum statistics? [15]

(b) Write down the distribution function for Bosons. There are two types of

Bosons. What are they? How does the distribution function differ for the

two types, and why? [15]

(c) In the context of the photon gas show that the density of states in k-space is

g(k)dk = dG(k)

dk dk =

V k2

2π2 dk

where k2 = k2x + k 2

y + k 2

z is the magnitude of the wave number vector (check

information sheet). [25]

(d) Express this density of states in angular frequency ω-space. [10]

(e) Derive an expression for the spectral energy distribution U(ω)dω for blackbody

radiation. What is the total energy in the box? How does it depend on the

temperature T ? (See information sheet for a hint!) [35]

/Cont’d

3

SECTION B: SOLID STATE PHYSICS (Answer TWO questions)

4. (a) State what is meant by the following terms:

i. lattice [5]

ii. basis [5]

iii. unit cell [5]

iv. lattice vector [5]

(b) A certain lattice is described by the following vectors:

a = a i

b = a (i + j)

c = 2ak

where a is a constant.

i. Make a scale sketch of the orientation and length of the vectors a,b,c. [10]

ii. Find the volume of the unit cell. [10]

iii. Find the reciprocal lattice vectors a∗, b∗, c∗. [40]

iv. Find the volume of the unit cell in the reciprocal lattice. [10]

v. Make a scale sketch of the orientation and length of the vectors a∗, b∗, c∗.

[10]

5. (a) Outline, in non-mathematical terms, the Drude theory of the electrical con-

ductivity of metals. [20]

(b) What are the successes and what are the shortcomings of the Drude theory?

[15]

(c) If τ is the mean time between electron and ion collisions in a metal, show that

the Drude theory leads to Ohm’s law for a conductor

j = ne2τ

m E

where j is the current density, E is the applied electric field, n is the number

of electrons per unit volume, and e and m are the electron charge and mass

respectively. [40]

(d) Sodium has density 968 kg m−3 and resistivity 4.7× 10−8 Ωm. Estimate τ for

sodium. [25]

/Cont’d

4

6. (a) Sketch the occupancy of a Fermi–Dirac gas as a function of energy for the case

of (i) a gas at absolute zero and (ii) a gas at temperature 1000K. Include in

your sketch the Fermi energy. [30]

(b) The density of states is given by

g() d = 1

2π2

(

2m

h̄2

)3/2

1/2 d .

Show that, for a Fermi-Dirac gas at absolute zero, the Fermi energy is given

by

F = [

3π2n ]2/3 h̄2

2m

where n is the number of particles per unit volume and m is the mass of each

particle. [30]

(c) Silver has density 10490 kg m−3. Calculate

i. the number density of conduction electrons per m3 [15]

ii. the Fermi energy. [10]

iii. the temperature of the classical gas in which the mean particle energy

would be the same as your answer to part (c)ii. [15]

/Cont’d

6

Information Sheet

1. The Schrödinger equation for a free particle in a 3-dimensional infinite square well

has the solutions

Ψ = A sin(kxx) sin(kyy) sin(kzz)

where a is the length of one edge of the box and

kx = nxπ

a , ky =

nyπ

a and kz =

nzπ

a

to satisfy the boundary conditions (Ψ = 0 at edges of the box).

2.

∫

∞

0

x3dx

exp x − 1 =

π4

15