Physics 6011, Spring 2015, Homework 8, Version 1

Due: Monday, March 9, noon

Important note: solutions for some or all of the problems in this assignment can be found

in various sources. In particular, the paramagnet problem and the Einstein solid problem

are explicitly solved in most Statistical Mechanics textbooks. It is of course reasonable and

encouraged to read other portions of the class textbook (or other textbooks if you so desire),

to discuss with classmates, or to ask questions to the instructor. However, please resist the

temptation to read (or copy) the solutions to any of the problems, and try your best to solve

the problems by your own effort.

0. Time spent (5 pts.)

At the top of the first page of your solution, please leave 5 lines of space for the grades (and any general

comments) to be written.

Just after that, please write a table containing the following columns. Each column should report

only one time. In all cases, the times reported should refer to work done after the deadline for the

previous homework. Columns: i) One column for the total time spent on reading assignments ii) One

column for the total time spent reading/studying in preparation to work on the homework iii) One

column per problem containing the time spent working on that problem (please answer individually for

each problem), iv) One column for the total time spent on the course (in hours), including everything

you did for the course except the 3 hours of lectures and the discussion hour. This week, your table

should report 6 times: 1 for reading assignments, 1 for reading in preparation for homework, 3 for

times spent in each problem, and 1 for the total time.

1. Adiabatic Processes in Paramagnets

(In this problem, use only Thermodynamics.) Consider a Curie paramagnet, for which the magneti-

zation M is given by M = CH/T , with H the magnetic field, T the temperature, and C a constant.

Assume that the work in this case is given by dW = HdM .

(a) By using the first and second laws of thermodynamics, show that in this particular case the energy

E(M,T ) is a function of T only. Assuming the heat capacity CM at constant magnetization to

be constant, show that during a reversible adiabatic process

1

Texp

(M2

2CCM

)= constant

(b) Show that the efficiency |W |/Q> for a Carnot cycle with the magnet as the working substanceis equal to = 1 T.

2. Entropy of mixing

(a) Problem 1.11 from Pathria, 3rd edition.

(b) Problem 1.13 from Pathria, 3rd edition.

3. A molecular crystal

Consider a crystal containing N noninteracting molecules, where each molecule has r + 1 states of

energies 0, , 2, , r respectively.

(a) Compute, as functions of T and N , the canonical partition function Q(T,N), the free energy

A(T,N), the internal energy E(T,N), the heat capacity C(T,N), and the entropy S(T,N) for

this system.

(b) Why is there only one relevant specific heat here, (i.e. as opposed to having Cp 6= Cv)?

4. A paramagnet

Consider now a crystal containing N paramagnetic ions of angular momentum J in a magnetic field

B. Each ion has an energy gBBm, where m = J, , J .

(a) Draw two side-by-side energy level diagrams (i.e. a set of horizontal lines, such that the vertical

position of each of the lines represents the energy of a state), one representing the possible energies

of one of the molecules in the previous question and the other representing the possible energies

of a paramagnetic ion. Remember that the zero of energy is arbitrary, and use this to shift one

of the diagrams vertically so that the lowest state in the two diagrams correspond to each other.

Show that the paramagnet problem maps onto the problem of the molecular crystal discussed

above and find the connections between the parameters in the two problems.

(b) Use these connections, and the results you just obtained for the case of the molecules, to compute

the canonical partition function Q(T,N), the free energy A(T,N), the internal energy E(T,N),

the heat capacity C(T,N), and the entropy S(T,N) for the paramagnet.

(c) Additionally, compute the expectation value M of the magnetization M = Ni=1mi, its vari-ance (M)2, the isothermal susceptibility T = M/B|T , and find the relation between(M)2 and T .

5. An Einstein solid

Consider a crystal containing N identical atoms. As a crude approximation, assume that each atom is

under the effect of a 3D harmonic potential which keeps it localized, and that atoms do not interact

with each other. We have therefore 3N simple harmonic oscillators, all of them with the same angular

frequency .

(a) Show that this problem is equivalent to the r limit of the molecular crystal problem, andfind the relations between the parameters of the two problems.

(b) Use these connections, and the results you obtained for the case of the molecules, to compute the

canonical partition function Q(T,N), the free energy A(T,N), the internal energy E(T,N), the

heat capacity C(T,N), and the entropy S(T,N) for the Einstein solid.

(c) Additionally, compute the expectation value ni of the quantum number for each of the harmonicoscillators, and show that it is equal to the mean occupation number for photons (Chandler p. 91,

Pathria Sec. 7.2), or equivalently to the Bose-Einstein occupation number in the case when the

chemical potential is zero (Pathria Section 6.3, Chandler p. 95)

(d) Compute the variance (ni)2 of the occupation number for oscillator i (1 i 3N), and thecorrelation ninj between the occupation numbers for two oscillators i and j.

Some possibly useful formulas

Ni=1 i = N(N + 1)/2

Ni=0 z

i = zN+11z1 x x x (x)2 = (x x)2 = x2 x2