# BE particle density

1. Feb 2, 2016

### ognik

1. The problem statement, all variables and given/known data
For the observation of quantum mechanical Bose-Einstein condensation, the interparticle
distance in a gas of noninteracting atoms must be comparable to the de Broglie
wavelength, or less. How high a particle density is needed to achieve these conditions
if the atoms have mass number A = 100 and are at a temperature of 100 nanokelvin?

2. Relevant equations
I found on the web that at the critical temp. for BEC, $\lambda = \sqrt{ \frac{2\pi \hbar^2}{m k_b T}}$

I also found a formula for the critical density for BEC at that temp., $n_c \approx 2.612 \frac{1}{\lambda_{T_c}^3}$

3. The attempt at a solution
I took m = particle mass $\approx 100 \times m_p = 1.67 \times 10^{-25} kg$
Mechanically plugging the numbers in I got $\lambda_{T_c} = 3.46 \times 10^{-6} m$, and $n_c = 6.3 \times 10^{16}$ particles per $m^3$

I'd appreciate if someone could tell me if I am on the right track? The numbers seem OK although I think the $\lambda$ looks high?

2. Feb 4, 2016

### Staff: Mentor

The approach is correct. I haven't checked the numbers, but they seem reasonable.