Layered structure of superfluid4He at supercritical motion

Landau’s criterion plays an important role in the theory of superfluidity. According to this criterion, superfluid motion is possible if \(\tilde \varepsilon \left( p \right) \equiv \varepsilon \left( p \right) + pV > 0\) along the curve of the spectrum?(p) of excitations. For⁴He it means thatv<v _c,v _c≈60 m/sec.v _s is equal to the tangent of the slope to the roton part of the spectrum. The question of what happens to the liquid when this velocity is exceeded, as far as we know, remains unclear. We shall show that for small excesses abovev _c a one-dimensional periodic structure appears in the helium. A wave vector of this structure oriented opposite to the flow and equal toρ _c/h whereρ _c is the momentum at the tangent point. The quantity \(\tilde \varepsilon \left( p \right)\) is the energy of excitation in the liquid moving with velocity v. Inequality of Landau ensures that \(\tilde \varepsilon \) is positive. If \(\tilde \varepsilon \) becomes negative, then the boson distribution function \(n\left( {\tilde \varepsilon } \right)\) becomes negative, indicating the impossibility of thermodynamic equilibrium of the ideal gas of rotons; therefore the interaction between them must be taken into account. The final form of the energy operator is $$\hat H = \int {\left\{ {\hat \psi + \tilde \varepsilon \left( p \right)\hat \psi + \tfrac{g}{2}\hat \psi + \hat \psi + \hat \psi \hat \psi } \right\}} d^3 x, g \sim 2 \cdot 10^{ - 38} erg.cm.$$ Then we can seek the rotonψ-operator in the formψ=ηexp(i p _c r/h), determiningη from the condition that the energy is minimized. The result is (η)²=(v?v _c)ρ _c/g, forv>v _c. The plane waveψ corresponds to a uniform distribution of rotons. It leads, however, to a spatial modulation of the density of the helium, since the density operator \(\hat n\) contains a term which is linear in the operator \(\psi :\hat n = n_0 + \left( {n_0 } \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}} {A \mathord{\left/ {\vphantom {A {\hat \psi \to \hat \psi ^ + }}} \right. \kern-0em} {\hat \psi \to \hat \psi ^ + }}\) ), where |A|²～ρ _c ² /2m?(ρ _c). Finally we find that the density of helium is modulated according to the law $$\frac{{n - n_0 }}{{n_0 }} = \left {\frac{{\left| A \right|^2 \left( {\nu - \nu _c } \right)\rho _c }}{{n_0 g}}} \right]^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \sin \rho _c x \approx 2,6\left {\frac{{\nu - \nu _c }}{{\nu _c }}} \right]^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \sin \rho _c x$$ . This phenomenon can be observed, in principle, in the experiments on scattering ofx-rays in moving helium.