# Notes on Negative Temperature ```PHYS-4420 THERMODYNAMICS &amp; STATISTICAL MECHANICS
SPRING 2006
Negative Absolute Temperature
Nuclei, with nuclear magnetic moments, in a magnetic field provide an interesting system to
study statistically. The Hamiltonian for the interaction between a magnetic moment and a
magnetic field is, H = – &middot;H. If the magnetic field is in the z direction, and has a magnitude of H,
this can be written as H   HI z , where  is the gyromagnetic ratio, and Iz is the z component
of the spin operator. The energy levels for this system are given by,
Em   Hm ,
where m can have values of I, I – 1, I – 2, … – I, and I is the spin of the nucleus under
examination. Transitions can be induced between two adjacent states when radiation is applied
with a the photon energy that is equal to the energy difference between those states.Then,
  Em 1  Em   H (m  1  m)   H .
Then, the transition frequency is,
  H .
The probability of finding a nucleus in a state of energy Em is,
Pm 
 m
 Em
e
e kT
.

 E
 m '
e
kT
m'
e
m'
m'
Temperature = 300 K
If the direction of the magnetic field could be reversed, or if every spin could be flipped, the
energies of every state would change sign. In fact, it is possible to flip the spins. In that case, the
entire population is inverted. The higher energy states are more populated than the lower energy
states. Then, if the system can come to equilibrium in that configuration, it can be described by a
temperature. It is found that the system does come to equilibrium, for the nuclear case, and the
1
temperature that describes the system is negative. At negative temperature the system has more
energy (it is hotter) than at positive temperature.
Temperature = – 300 K
It is correct to describe the system as having a negative temperature because the system achieves
a local equilibrium. Eventually, the system will return to the lower energy positive temperature,
but it can take several minutes. During that time, the nuclei are able to exchange energy among
themselves by a mutual spin flip mechanism, so they truly can be described by a negative
temperature.
These temperatures can be added to the range that we have been using. They must be added as
temperatures that are above an infinite temperature. At an infinite temperature, all states would
be equally populated. Then if the energy of the system can be increased further, the temperature
of the system would change to a negative, infinite value. As the energy of the system increases
further, the magnitude of the negative temperature decreases, until the energy approaches a
maximum at a temperature of negative infinity. Thus the progression of temperature from lowest
to highest energy is:
T = 0  –   – (0 + ).
Interestingly, the quantity that increases monotonically as the energy of the system increases is
1
 .
T
1
1
1
At T = 0,    ;
at T = ,   0 ;
and at T = – 0,    .
T
T
T
1
were the quantity used instead of T, it would seem reasonable that 0 K is unattainable,
T
1
since    .
T
If 
2