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!
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=NkBT2ln 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
(d) The internal energy can be expressed in terms of the magnetisation M:U=MB.
In the weak-ﬁeld limit (B0) 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 m3and resistivity 4.7×108Ω m. Estimate τfor
sodium. [25]
/Cont’d