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Effective mass and normal fraction

If $V\neq 0$ a quadratic term in the impurity contribution to the energy is present. It can be denoted as $\chi m^* V^2/2$ with

\begin{displaymath}
m^*=\frac{2\sqrt{\pi}}3(na^3)^{1/2}\left(\frac{b}{a}\right)^2m
\end{displaymath} (7.19)

being the induced mass, i. e. the mass of particles dragged by an impurity [Ast01]. Applicability of the perturbation theory demands $m^*$ to be small compared to $m.$ This gives the condition $(na^3)^{1/2}\left(\frac{b}{a}\right)^2\ll
1$. At zero temperature the interaction between particles does not lead to depletion of the superfluid density and the suppression of the superfluidity comes only from the interaction of particles with impurities. Thus % latex2html id marker 16030
$\left(\ref{mstar}\right)$ defines the normal density
$\displaystyle \frac{\rho_n}{\rho} = \frac{m^*}{m} \chi =
\frac{2\sqrt{\pi}}{3} (na^3)^{1/2} \chi\left(\frac{b}{a}\right)^2$     (7.20)

This result is in agreement with the one obtained by the means of Bogoliubov transformation starting from the Hamiltonian written in the second-quantized form in the presence of disorder[HM92,ABCG02]. The normal density of a superfluid is an observable quantity. It was evaluated in liquid $^4$He by measuring of the moment of inertia of a rotating liquid or by measuring of the second sound velocity. Both methods can be, in principle, developed for BEC gases.


next up previous contents
Next: Drag force and energy Up: Three-dimensional system Previous: Total energy   Contents
G.E. Astrakharchik 15-th of December 2004