By magnetising and demagnetising a paramagnetic sample while controlling the heat flow, we can lower its temperature.
a b: With the sample in contact with a heat bath at , we increase the magnetic field
b c: With the sample now isolated, we slowly decrease the field to again. This is the adiabatic demagnetisation step; because the process is slow and adiabatic, the entropy is unchanged.
By following these steps on a plot, we see that the second, constant entropy, step, reduces the temperature. The entropy is a function of only, not or separately (see here) so if we reduce at constant , we reduce also.
The following figure shows what is happening to the spins.
If we start with a large sample, we could repeat the process with a small sub-sample, the remaining material acting as a heat bath during the next magnetisation. By this method temperatures of a fraction of a Kelvin can be reached. However after a few steps less and less is gained each time, as the curves come together as . (Once the electron spins are all ordered, one can start to order the nuclear spins, and reach even lower temperatures--the magnetic moment of the nucleus is around a two-thousandth of that of the atom), but even that has its limits. (Wondering why we can't just take to zero? See here for the real paramagnet.)
This is an important and general result. There is always a minimum excitation energy of the system, and once there is no further way of lowering the temperature. The unattainability of absolute zero is the third law of thermodynamics.