This analysis using the Bogoliubov model does not apply to liquid helium-4, because it is a strongly interacting system. Nevertheless, we can estimate the critical velocity for the superfluid by the following approximation:

(a) The Born approximation, also known as the s-wave scattering approximation, relates the scattering potential to the scattering cross-section as follows (e.g., Cohen- Tannoudji et al. 1977: s. VIII.B.4):

Assuming that the interaction is a contact potential U(r(vector)) = U₀δ(r(vector)), show that for bosons this implies U₀ = 4πh̄²a/m, where a is the s-wave scattering length and U₀ is the potential U(0) that appears in the Bogoliubov formulas.

(b) Assuming that the s-wave scattering length a is comparable to the helium atom size, and that the superfluid density is 10²²cm⁻³, use the interaction potential of (a) to determine the critical velocity.

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