Microstate and Macrostate - Statistical Mechanics - Past Exam, Exams for Statistics . Alliance University

Statistics

Description: This is the Past Exam of Statistical Mechanics which includes Reciprocal Lattice Vector, Primitive Translation Vectors, Miller Indices, Cartesian Unit Vectors, Volume of Unit Cell, Equilibrium Distance, Angular Frequency, Optical Vibration etc. Key important points are: Microstate and Macrostate, Corresponding Microstates, Number of Microstates, Non-Degenerate Energy Level, Normalization Constant, Particle Partition Function, Internal Energy, Boltzmann Constant
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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

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