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Peeter’s lecture notes from class. May not be entirely coherent.
In order to maintain a conservation of particles in a Bose condensate as we decrease temperature, we are forced to change the chemical potential to compensate. This is illustrated in fig. 1.1.
Fig 1.1: Chemical potential in Bose condensation region
Bose condensatation occurs for . At this point our number density becomes (except at )
Except for , is well defined, and not described by this distribution. We are forced to say that
Introducing the density of states, our density is
We worked out last time that
This is plotted in fig. 1.2.
Fig 1.2: Density variation with temperature for Bosons
For , we have . This condensation temperature is
This is plotted in fig. 1.3.
Fig 1.3: Temperature vs pressure demarkation by T_BEC curve
There is a line for each density that marks the boundary temperature for which we have or do not have this condensation phenomina where states start filling up.
Compare this to the classical and Fermionic specific heat as plotted in fig. 1.4.
Fig 1.4: Specific heat for Bosons, Fermions, and classical ideal gases
One can measure the specific heat in this Bose condensation phenomina for materials such as Helium-4 (spin 0). However, it turns out that Helium-4 is actually quite far from an ideal Bose gas.
A system that is much closer to an ideal Bose gas is that of a gas of photons. To a large extent, photons do not interact with each other. This allows us to calculate black body phenomina and the low temperature (cosmic) background radiation in the universe.
An important distinction between a photon sea and some of these other systems is that the photon number is actually not fixed.
Photon numbers are not “conserved”.
If a photon interacts with an atom, it can impart energy and disappear. An excited atom can emit a photon and change its energy level. In a thermodynamic system we can generally expect that introducing heat will generate more photons, whereas a cold sink will tend to generate fewer photons.
We have a few special details that distinguish photons that we’ll have to consider.
- spin 1.
- massless, moving at the speed of light.
- have two polarization states.
Because we do not have a constraint on the number of particles, we essentially have no chemical potential, even in the grand canonical scheme.
Our number density, since we have no chemical potential, is of the form
Observe that the average number of photons in this system is temperature dependent. Because this chemical potential is not there, it can be quite easy to work out a number of the thermodynamic results.
Photon average energy density
We’ll now calculate the average energy density of the photons. The energy of a single photon is
so that the average energy density is
Mathematica tells us that this integral is
for an end result of
Phonons and other systems
There is a very similar phenomina in matter. We can discuss lattice vibrations in a solid. These are called phonon modes, and will have the same distribution function where the only difference is that the speed of light is replaced by the speed of the sound wave in the solid. Once we understand the photon system, we are able to look at other Bose distributions such as these phonon systems. We’ll touch on this very briefly next time.