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‫גוריון בנגב‬-‫אוניברסיטת בן‬
‫המחלקה לפיסיקה‬
1 ‫תרמודינמיקה ומכניקה סטטיסטית‬
Ben-Gurion University of the Negev
Department of Physics
Thermodynamics & Statistical Mechanics 1
Thermodynamics and Statistical Mechanics I – Home Exercise 6
1.
A little calculus
 P   V   T 
 
 
  1
 T V  P  T  V  P
Prove the following identity, for a function P(T ,V ) : 
2.
Energy fluctuations
Consider a system of fixed volume in thermal contact with a reservoir. Show that the mean
square fluctuation in the energy of the system is  2 
3.
   
2
 U 
2
 2 
   CV



V
Canonical (non-interacting) spins in magnetic field
Consider a system of N non-interacting spins of moment m in an external magnetic field H,
temperature  . The spin magnetic moment can be either 'up' or 'down'.
a. What are the possible energies of such a spin in the field H?
b. Find the partition function of one spin
c. Find the partition function of N (non-interacting) spins with fixed positions (as in a
solid)
d. Find the free energy
e. Find the magnetization using the formula  M 
F
H
f. Find the zero field (H=0) magnetization. What does it mean?
g. Find the average energy <U> and its relationship to <M>
h. What happens to <M> in the limit of high temperatures? In the limit of low
temperatures? Explain the meaning of the result physically
F
H
i.
Derive the relation  M 
j.
Derive Curie's Law of magnetic susceptibility:  0  
1
 M 
~ . Plot its

 H  H 0 
temperature dependence
k.
Find the fluctuations in the magnetic moment ( M )2 , where  M  M  M .
Derive the energy fluctuations
‫גוריון בנגב‬-‫אוניברסיטת בן‬
‫המחלקה לפיסיקה‬
1 ‫תרמודינמיקה ומכניקה סטטיסטית‬
Ben-Gurion University of the Negev
Department of Physics
Thermodynamics & Statistical Mechanics 1
4.
General spins in magnetic field
Consider a system of N fixed spins with magnetic moments in the z direction:  z  Am ,
where m   J ,  J  1,...,  J and J is given. A magnetic field H is applied at the z direction at
temperature  .
a. Find the partition function. Transform to a unitless parameter x (what should you pick
as x?)
b. Find the magnetization in the z direction
c. Write the magnetization using the Brillouin function defined by the formula1 

1   1

 1
BJ ( x)  1 
coth 
 coth 1 
 x 
 2J 
 2J
 2 J   2 J

x

d. Expand BJ ( x) for small x. Find an expression for the first order magnetization at high
temperatures and identify Curie's law
e. What happens when J=1/2? How does one adjust the result to get the known result
where  z  1, 1 ?
5.
Canonical crystal
Consider a crystal containing N=5x1023 atoms, which may be found in one of the following
states: the ground state with E0=0 and an exited state with E1=ε=4x10-20J.
a. In the beginning ¼ of the atoms were in the exited state. What is the temperature of
the crystal?
b. The crystal is placed in thermal contact with a reservoir at a temperature of 300K.
How much energy will flow between the two systems (and in which direction) until
they reach equilibrium (i.e. what is the change in the energy of the crystal)?
Ben-Gurion University of the Negev
Department of Physics
Thermodynamics & Statistical Mechanics 1
‫גוריון בנגב‬-‫אוניברסיטת בן‬
‫המחלקה לפיסיקה‬
1 ‫תרמודינמיקה ומכניקה סטטיסטית‬