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!

Qni!

gives the number of microstates for a given macrostate with niparticles 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=Aexp (−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=NkBT2∂ln ZSP

∂T

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

/Cont’d

2

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

2paramagnet. Derive and simplify the

corresponding partition function. [15]

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

2para-

magnet is

U=−NµBBtanh µBB

kBT

where Nis the number of particles, µBBohr’s magneton, Bthe magnetic ﬁeld

strength, Tthe temperature and kBBoltzmann’s constant. [30]

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

as well as a function of Bfor 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)

dkdk=V k2

2π2dk

where k2=k2

x+k2

y+k2

zis 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)dωfor 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=ai

b=a(i+j)

c= 2ak

where ais 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 τis 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

mE

where jis the current density, Eis the applied electric ﬁeld, nis the number

of electrons per unit volume, and eand mare the electron charge and mass

respectively. [40]

(d) Sodium has density 968 kg m−3and resistivity 4.7×10−8Ω m. Estimate τfor

sodium. [25]

/Cont’d

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