Download PHYS 355 Thermal Physics Problem Set #8

PHYS 355 Thermal Physics
Problem Set #8
After you finish this problem set, you should be familiar with:
Microcanonical Ensemble
Ising model
Paramagnet
1.
One-Dimensional Ising Model.
The system represented by this model is a linear closed chain of N quantized spins  i (  i  1 or - 1
only). The spins are coupled to their nearest neighbors and hence the Hamiltonian (or total energy) is
given by
N
H   J  i i 1 .
i 1
(Note that when i  N  1 , this corresponds to i  1 .) If all spins are parallel the energy is -NJ. Let Na
be the number of neighbor pairs having antiparallel spin, and Np be the number of parallel spin neighbor
pairs.
a
p
a
p
a
a
p
p
a
a
Example:
N  10
Na  6
Np  4
Na  N p  N
a)
Show that the value of the energy E is given by
E   J ( N p  N a )   J N  2 N a  .
b)
For a given value of Na and Np show find the number of distinct microstates of this system. (This
is the degree of degeneracy of this energy level.)
c)
Obtain and expression for the microcanonical entropy,
S  k lnE, N 
Show that the entropy is extensive and may be written as
S  kN 12  x ln  12  x    12  x ln  12  x 
where x  E 2 NJ .
2.
Paramagnetic Solid
A paramagnetic solid is one example of a general class of systems in which each particle can be in one
of two possible energy states,   0 and   0 . Let N  and N  be the number of particles, respectively,
in the states   0 and   0 . The total energy is E  N   N   0 .In the paramagnet, the total
magnetization M  N   N   is the net number of spins whose moment is along the direction of the

magnetic field H . Also remember that  0  H and E   MH .
a)
From the microcanonical entropy calculate the form of the relationship between the total energy
E and the temperature T. Show that
 H 
M  N tanh
.
 kT 
b)
Obtain the magnetic susceptibility  T  
M
H
versus T. Obtain expression for  T  accurate
T
in the limits H kT  1 and H kT  1 .
c)
d)
e)
Obtain the specific heat C versus temperature for this two level system. Here, C 
dE
.
dT
Make a graph of C N   N   k versus kT  0 for this system. Show that the specific heat has
an anomalous peak near kT  0  1 . Make a graph of this specific heat C Nk versus kT  0 over
the range, and find where C Nk is a maximum. This specific heat curve is called the “Schottky
anomaly”.
Each spin can be regarded as residing in either the   0 or   0 energy state with probabilities
P  N  N and P  N  N . Show that
e  H
e  H
P 
and P 
2 coshH 
2 coshH 
where   1 kT .
f)
The mean moment M calculated in part (a) can be regarded as the net magnetic moment of all the
spins taken over the distribution shown below:
P
PP+

+
-
2
That is,
N
M i  Pi  Pi
and
M   Mi .
i 1
Since Pi is the same for all spins show in this way that
 H 
M  N tanh

 kT 
as before.
g)
Use the fact that the definition of the second moment of the distribution is
Mi
2
  2 Pi   2 Pi
to calculate the mean square fluctuation in the total magnetic moment about its mean value,
M2  M
2

N 2
.
cosh 2 H 
Hint:
M i  M j  M i  M j and M i M j  M i M j if and only if M i and M j are independent.
h)
Examine the result found in (g) in the two limits H kT   and H kT  0 and give a
physical explanation for your findings in the two limits.
i)
Plot M 2  M
2
versus H over the range 0  H  10 .
M2  M
j)
Obtain a formula for the relative fluctuation:
M
3
2
2
.