 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Natural sciences course of the University of Cambridge

Year of the natural sciences course of the University of Cambridge

Year 3 of the natural sciences course of the University of Cambridge

Physics course of the year 3 of the natural sciences course of the University of Cambridge

Thermal and Statistical Physics course of the University of Cambridge

Past exam of the thermal and statistical physics course of the University of Cambridge

2013

Paper 1

Use a three-dimensional <ideal gas> at the same <number density> n for both <isotopes>. An unpolarized <helium-3> gas has <spin degeneracy> two, hence
$$
k_BT_F=\frac{\hbar^2}{2m_3}(3\pi^2n)^{2/3}.
$$
<Helium-4> has a single <spin> state, and the ideal Bose-gas saturation condition gives
$$
k_BT_B=\frac{2\pi\hbar^2}{m_4}\left[\frac{n}{\zeta(3/2)}\right]^{2/3}.
$$
Eliminating n, using $m_3/m_4\simeq3/4$ and $\zeta(3/2)=2.612$, yields
$$
\boxed{\frac{T_B}{T_F}=\frac{4\pi}{(3\pi^2)^{2/3}\zeta(3/2)^{2/3}}\frac{m_3}{m_4}
\simeq0.519,\qquad T_B\simeq2.6\,\mathrm K.}
$$
The fermionic <spin degeneracy> must be included. This is an ideal-gas estimate; strong interactions in liquid helium prevent identifying it with an exact experimental transition <temperature>.

Solution

Past exam of the thermal and statistical physics course of the University of Cambridge / 2013 / Paper 1 / 1 / c / Solution

Past exam of the thermal and statistical physics course of the University of Cambridge / 2013 / Paper 1 / 1 / c / Solution

 Ancestors (14)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the thermal and statistical physics course of the University of Cambridge / 2013 / Paper 1 / 1 / c / Solution

 Ancestors (14)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)