## PHY452H1S Basic Statistical Mechanics. Problem Set 7: BEC and phonons

Posted by peeterjoot on April 10, 2013

# Disclaimer

This is an ungraded set of answers to the problems posed.

## Question: Bose-Einstein condensation (BEC) in one and two dimensions

Obtain the density of states in one and two dimensions for a particle with an energy-momentum relation

Using this, show that for particles whose number is conserved the BEC transition temperature vanishes in these cases – so we can always pick a chemical potential which preserves a constant density at any temperature.

## Answer

We’d like to evaluate

We’ll use

where the roots of are . With

the roots of are

The derivative of evaluated at these roots are

In 2D, we can evaluate over a shell in space

or

In 1D we have

Observe that this time for 1D, unlike in 2D when we used a radial shell in space, we have contributions from both the delta function roots. Our end result is

To consider the question of the BEC temperature, we’ll need to calculate the density. For the 2D case we have

Recall for the 3D case that we had an upper bound as . We don’t have that for this 2D density, so for any value of , a corresponding value of can be found. That is

For the 1D case we have

or

See fig. 1.1 for plots of for , the respective results for the 1D, 2D and 3D densities respectively.

Fig 1.1: Density integrals for 1D, 2D and 3D cases

We’ve found that is also unbounded as , so while we cannot invert this easily as in the 2D case, we can at least say that there will be some for any value of that allows the density (and thus the number of particles) to remain fixed.

## Question: Estimating the BEC transition temperature

Find data for the atomic mass of liquid He and its density at ambient atmospheric pressure and hence estimate its BEC temperature assuming interactions are unimportant (even though this assumption is a very bad one!).

For dilute atomic gases of the sort used in Professor

Thywissen’s lab

, one typically has a cloud of atoms confined to an approximate cubic region with linear dimension 1 . Find the density – it is pretty low, so interactions can be assumed to be extremely weak. Assuming these are Rb atoms, estimate the BEC transition temperature.

## Answer

With an atomic weight of 4.0026, the mass in grams for one atom of Helium is

With the density of liquid He-4, at 5.2K (boiling point): 125 grams per liter, the number density is

In class the was found to be

So for liquid helium we have

The number density for the gas in Thywissen’s lab is

The mass of an atom of Rb is

which gives us

## Question: Phonons in two dimensions

Consider phonons (quanta of lattice vibrations) which obey a dispersion relation

for small momenta , where is the speed of sound. Assuming a two-dimensional crystal, phonons only propagate along the plane containing the atoms. Find the specific heat of this crystal due to phonons at low temperature. Recall that phonons are not conserved, so there is no chemical potential associated with maintaining a fixed phonon density.

The energy density of the system is

For the density of states we have

Plugging back into the energy density we have

where . Taking derivatives we have

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