1    (a)  i. Write down an expression for the entropy S in a microcanonical ensem-

ble, in terms of the number of microstates, Ω .

ii.  Consider a system with three degenerate energy levels, containing in total N  distinguishable bosons.   Calculate the entropy of the system as a function of N .

(b)  i.  Define, and briefly explain the significance of the partition function in statistical mechanics.

ii.  Show that the partition function, z, for a single quantum harmonic oscillator may be written as:

e-βhω/2  

z = 1 - e-βhω    ,

where β =  and ω, T have their usual meanings.

iii.  Hence, using the relation F = -kBT ln z, find an expression for the free energy,  F ,  of a single quantum harmonic oscillator.   Comment on its temperature dependence.

YОU haì USX t)X rXSU之t

o

xn =    , 】Оr x < 1  .

(c)  In a Fermi-Dirac gas the average occupation number of a level with energy

e is given by:

n(e) =         1       

where µ is the chemical potential.  Sketch n as a function of e for T = 0 and for T slightly above zero, where 0 < kBT ≤ eF , indicating the location of the Fermi energy eF . Give a physical interpretation of the shapes of the occupation number distributions for T = 0 and T above zero.

2A    (a)  Without performing detailed calculations, discuss the extent to which the       heat capacity predicted by the Einstein model of a solid agrees with experi-       ment, and how this model can be improved.                                                      [4]

For the rest of this question, consider a dielectric crystal consisting of layers of atoms, with rigid coupling between layers so that atoms are restricted to moving in the plane of the layer. A layer can be modelled as a large number N of atoms placed in a two-dimensional array of area A = L2 . The total internal energy can be calculated as:

D

U =         g(ω)u(ω)dω ,

0

hω          hω                            1  

(b)  Show that the density of states g(ω)dω can be expressed as:

Aω 

g(ω)dω = G        dω ,

where c is the speed of sound in the material and G is the number of polarisa-       tion states.                                                                                                       [3]

(c)  By setting the number of available modes to 2N and assuming that two polarisation states are available, show that:

ωD  = 2c^πn ,

(d)  Hence, determine the energy density U/A at T = 0 as a function of n.        [4]

(e)  Show that the heat capacity of the system is given by:

(f)   Show that at high temperature, CV  = 2NkB . Obtain a low-temperature expression for CV  and comment on its temperature dependence.

YОU can aSSUme

o    x3 ex dx  

0      (ex - 1)2  = 6ξ(3) ≈ 7.2123 .

[5]

2B   Consider a gas of N spin-zero, non-relativistic, identical, non-interacting bosons

of mass m at temperature T in volume V .  The corresponding distribution function fBE  in energy e is:

fBE  =       1      

where β =  and B is determined from the total number of particles.

(a)  Show that in wavevector (k) space, the density of states in three dimen- sions is given by:

g(k)dk = 4πk2 dk .                                              [2]

(b)  Show that N = ZF (B), where:

Z = V ╱ 、3/2    ,  F (B) = 4^π    o o    ,  and   y2 =  .

(c)   Make a physical argument for the allowed values of B .                                 [1]

(d)  Hence, by considering the behaviour of the function F (B) and noting that

F (1) ≈ 2.6, explain what is missing from this treatment of the Bose-Einstein

gas in the limit T → 0 and outline an improved treatment.                                 [5]

(e)   Consider the case where the same gas is confined to two dimensions. By following similar steps to those of parts  (a)-(d), explain whether the same problem arises.

YОU haì USX t)X rXSU之t:

o     ydy          1

0      Bey2  - 1     2

0 o   =  log ╱ 、  .