1. Applications of the Canonical Ensemble:
Simple Models
of Paramagnetism
Section 6.3 (Spin = ½)
Section 7.8 (Spin = S)

2. Paramagnetic Materials
The following is a simple model to develop our understanding of Statistical Mechanics in The Canonical Ensemble.
Paramagnets contain atoms which have magnetic dipole moments (). These do not interact with each other but can respond to an applied external magnetic (B) field.
The dipoles can be crudely thought of as independent (atomic) bar magnets arranged on a crystal lattice.

3. Crude Picture: 
The magnetic dipoles can be crudely thought of as bar magnets arranged on a crystal lattice.
Crude Picture:

In an external magnetic field B each dipole can exist in one of two states – aligned with the field (spin up) or anti aligned (spin down).
Spin up dipoles have energy -B, spin down +B.
We want to calculate dependence of the magnetization of the material on the temperature T & the applied field B.

4. Curie Model of Paramagnetism
Energy level diagram of an S = ½ spin in an
external magnetic field along the z-axis
Note: B = 0H, E = gmBH
Typically g = 2, which corresponds to
E ~ 1 cm-1 at H = 104 G

5. All dipoles are independent, so we only need to look at the average properties of one dipole. We can use all the other dipoles as the heat bath in the
Canonical Ensemble.
There are only two possible microstates and we know their energies already, so its easy to get the Partition Function Z1 for a single dipole.

6. Spin Down Probability:
In the Canonical Ensemble, it’s easy to calculate the probabilities of spin up & spin down:
Spin Up Probability:
Spin Down Probability:

7. Calculate the mean magnetic moment of one individual dipole:
So the mean energy of an individual dipole is:

8. These results are for one individual dipole, but because the dipoles do not interact, all other dipoles behave similarly.
So, a solid of N dipoles has average energy:
Similarly, the mean magnetic moment (magnetization) in the direction of the applied field is:
Note that
U = -MB
which is the classical E & M result

9. So, the magnetization M or the magnetic moment per unit volume L has the form:
In the limit of weak B fields or high temperatures T,
x<<1  tanh x  x

10. Results

11. “Curie’s Law” of Paramagnetism
Reif, Figure 6-3-1
“Saturation
Magnetization”
“Curie’s Law” of Paramagnetism
M0  χH, χ  (N0μ2)/(kT)
χ  “Curie Susceptibility”

12. Curie’s “Law”
The susceptibility  is the magnetization per applied field intensity which for small magnetizations is given by H = B/0.
This is Curie’s “Law”. It holds well for many paramagnetic materials. It works so well that it can be used for temperature calibration!
For example Cerium Magnesium Nitrate obeys Curie’s “Law” down to T = 0.01K!

13. Experimental
 vs. T for
Cu(CH3CN)4(BF4)2
1/ = T/C is a straight line of slope C-1 & intercept zero
T = C is a straight line parallel to the x-axis at
constant T, showing the temperature independence
of the magnetic moment.

14. Curie’s “Law” Because single electrons have magnetic dipole
moments, if they are placed them in a magnetic
field, they’ll align with the field. However the
energy difference between dipoles aligned with
the field & against it is much smaller than the
thermal energy at room temperature: 2μB << kBT
So, there are random orientations, with equal
populations aligned with & against the field.
As T is lowered, this energy difference becomes
more important & the population changes, with
more becoming aligned anti-parallel to the field.

15. “Magnetic Susceptibility”, χ C = “The Curie Constant”
To explain this behavior, Curie invented a
parameter – called the
“Magnetic Susceptibility”, χ
which is a measure of how attracted a sample
is to a magnetic field. This is normally
measured as an apparent mass increase. As
more electrons align anti-parallel to the field
at low temperature, χ increases. In fact, χ is
inversely proportional to the temperature.
This is the
Curie “Law”: (1/χ) = CT
C = “The Curie Constant”

16. Magnetic Heat Capacity
A paramagnetic solid has an energy that depends on temperature. So, it has a magnetic heat capacity.
Experimentally one measures the heat
capacity at constant magnetic field intensity H.
The paramagnetic compound is weakly magnetic, so
B = 0H
so B is also constant.

17. Magnetic Heat Capacity

18. This is often called The Schottky Heat Capacity
In fact, this is a general result for the heat capacity in any two level system.

19. Isolated Paramagnetic Solid
Now consider a very similar problem to the one just discussed. Constrain (fix) the total energy U of the isolated system.
N total dipoles, n spin-up aligned with the applied B field: 
U is obviously a function of n. A
A given energy U(n) corresponds to a given
number of n spin up atoms with statistical weight:

20. So, the Entropy is given by:
For large N (~1023) use Stirling’s Approximation:
So The Temperature of the spin system is:

21. Further manipulation gives:
Also, we know that:
So that:
Now, solve for n, to find the number density of spin up atoms:

22. Negative Temperatures?!!
From the previous derivation,
Manipulate this to obtain:
Note that, if n < N/2 then more than half the dipoles are anti-parallel to the field & we also get the surprising result that
T becomes negative!

23. What is a Negative Temperature?
First note that, as the temperature T , the populations of spin-up & spin-down particles become equal!
We just saw that if
n < N/2 (more spins are down than up),
T becomes negative!
This is called a
“population
inversion”!

24. A “population inversion”.
If n < N/2 (more spins down than up),
T becomes negative!
A “population inversion”.
A negative temperature state must therefore be “hotter” than T , because it is a higher energy state of the system than T  !
For T < 0, the entropy & statistical weight must be decreasing functions of E.

25. What is Negative Temperature?
For a negative temperature to occur, the entropy and statistical weight must be decreasing functions of E.
This can happen if the system possess a state of finite maximum energy – such as the paramagnet with U = NB.
No systems exist where this happens for all degrees of freedom! (i.e. vibrational, electronic & magnetic energies).

26. What is Negative Temperature?
No systems exist where this happens for all degrees of freedom! (i.e. vibrational, electronic & magnetic energies).
However, if one subsystem is effectively
decoupled from the others, so they do not interact, that subsystem may be considered to reach internal equilibrium without being in equilibrium with the others.
This is the case for magnetic systems where the relaxation times between atomic spins is much quicker than the relaxation between spins and the vibrational modes of the lattice.

27. Negative Temperature In the paramagnet, the lowest possible energy is
U = -NB and the highest U = +NB. These are both non-degenerate microstates so entropy S = 0.
In between, we can only reach states with positive energy with a negative temperature!!!

28. A More General Discussion of Negative Temperature
In physics, certain systems, such as the magnetic system just discussed can achieve a negative temperature on the Kelvin scale!
A system with a truly negative temperature on the Kelvin scale is hotter than any system with a positive temperature!
If a negative-temperature system and a positive-temperature system come into contact, heat will flow from the negative temperature system to the positive-temperature system.

29. That a system at negative temperature is hotter than any system at positive temperature seems paradoxical if absolute temperature is interpreted as an average kinetic energy of the system.
The paradox is resolved by understanding temperature through its more rigorous definition as the tradeoff between energy and entropy, with the reciprocal of the temperature, thermodynamic parameter
 = [1/(kBT)], as the more fundamental quantity.
Systems with a positive temperature will increase in entropy as one adds energy to the system. Systems with a negative temperature will decrease in entropy as one adds energy to the system.[3]

30. Most familiar systems cannot achieve negative temperatures, because adding energy always increases their entropy.
The possibility of decreasing in entropy with increasing energy requires the system to "saturate" in entropy, with the number of high energy states being small. These kinds of systems, bounded by a maximum amount of energy, are generally forbidden classically.

31. So, a negative temperature is strictly a quantum phenomenon
So, a negative temperature is strictly a quantum phenomenon. The possibility of decreasing entropy with increasing energy requires a system to “saturate” in entropy, with the number of high energy states being small.
Most such systems, such as the paramagnetic system just discussed, have a maximum amount of energy that they can hold.
So, as they approach that maximum energy their entropy actually begins to decrease.