The force with which the impurity acts on the system is
(7.21) |
Expanding the wave function into the sum of and and neglecting
terms of order we obtain
It is convenient to do the integration in spherical coordinates with
being angle between and . There is no dependence on the angle
and it can be immediately integrated out
The -function can be further developed
(7.25) |
The poles in the integration over appear if the square root in the
denominator is smaller than one, which leads to the restriction on the values of
momentum which contribute
Thus the energy dissipation takes place only if the impurity moves with a speed larger than the speed of sound.
Let us calculate the projection of the force to the direction of the movement of the perturbation. It means that we have to multiply formula
(7.23) on
Now we can use that square of the unperturbed wave function gives the density
and the coupling constant can be expressed as
(7.27) |
Finally, we obtain following expression for the projection of the force
The energy dissipation, , can be evaluated by measuring the heating of the gas.
For large the force is proportional to . The energy dissipation per unit time can then be presented as with the damping rate
Note in conclusion that our perturbative calculations can not describe processes involving dissipation of energy due to creation of quantized vortex rings. Such a creation is possible at but has a small probability for low velocity and for a weak point-like impurity.