Landau critical velocity in Helium-3

[andbe]

Hello,

If we first consider Helium-4 we can calculate the critical velocity via
$\frac{d\epsilon(p)}{dp}=\frac{\epsilon(p)}{p}$ where $\epsilon(p)=\frac{(p-p_0)^2}{2\mu}+\Delta$ is the dispersion relation for roton excitations in Helium-4.
Putting in the constants $\mu=0.164 m_4$ is the effective mass, $\Delta/k_B=8.64$K, $p_0/\hbar=19.1$nm you get roughly $v_c=59.3$m/s.

Now I want to do the same calculation for Helium-3 but can't find the values of the constants for Helium-3, if rotons even exists for Helium-3?

What is the dispersion relation for Helium-3? Taking inspiration from superconductivity and the BCS-theory I'm thinking that there will be an energy gap here as well, i.e. no phonon region as for Helium-4, but it's hard to find information about this. Can anyone point me in the correct direction? I'm mostly interested in drawing some conclusions about the critical velocity of Helium-3 from the calculation above, if it is even possible...

Regards,
Andreas

 

[eljose79]

Hello Andreas,

Thank you for your question. The dispersion relation for Helium-3 is different from that of Helium-4 because the two isotopes have different properties. Unlike Helium-4, Helium-3 does not have roton excitations. Instead, it has phonon excitations which are responsible for superfluidity in Helium-3.

The dispersion relation for phonon excitations in Helium-3 is given by \epsilon(p)=\sqrt{\frac{p^2}{2m_3}\left(\frac{p^2}{2m_3}+2\Delta\right)} where m_3 is the effective mass of Helium-3 and \Delta is the energy gap. The values for these constants can be found in various research papers and textbooks on Helium-3.

As for the critical velocity of Helium-3, it is difficult to calculate using the same method as Helium-4 because of the absence of roton excitations. However, it has been experimentally determined to be around 40 m/s. This value may vary depending on the temperature and pressure of the system.

I hope this helps in your calculations. Good luck with your research!