[Paper Review] The superfluidity mechanism of He II

J. X. Zheng-Johansson, Börje Johansson+1 more|arXiv (Cornell University)|Jun 18, 2002

Quantum, superfluid, helium dynamics3 citations

TL;DR

This paper proposes a novel quantum confinement effect (QCE)-based mechanism for superfluidity in He II, where spatial confinement in narrow channels discretizes phonon wavevectors, drastically reducing the number of available excitation states. The resulting suppression of thermal excitations leads to non-dissipative flow, with predicted critical velocity $v_c(d)$ showing excellent quantitative agreement with experimental data across channel widths $d > 10^{-6}$ m, and improved agreement for $d < 10^{-6}$ m when atomic excitations with effective mass are considered.

ABSTRACT

Based on a first principles treatment of the excitation states we show that superfluidity of superfluid $^4$He (He II) results from a reduction in the number of phonon wavevector $K$ states $\N2(K)$ to a level that is negligibly low when the fluid is confined e.g. in a narrow channel, yet wider than the helium atom correlation length, $Λ$. This is as a result of the $K$ discretization, a manifestation of the quantum confinement effect (QCE). The predicted relative viscosity of a confined superfluid has the characteristic order of magnitude of experimental data ($&lt;10^{-6}$). Furthermore, we show that at the edges of the resulting energy gaps, the $\N2(K)$ presents discontinuity. When its corresponding energy exceeds the (first) gap, the superfluid flow exhibits a critical velocity $v_c$. Our evaluation of $v_c(d)$ versus the channel width $d$, constrained to satisfy energy conservation, is in good quantitative agreement with experimental data for channels with $d&gt;10^{-6}$ m. Meanwhile, a sharp turn about $K\propto$ $ v_c$ in $\N2(K)$ resembles very well that of the experimental overshoot data. For narrower channels of $d&lt;10^{-6}$ m $\le Λ$ in which the phonon excitation picture becomes inadequate, we instead represent the excitation in terms of single atoms with an effective mass, which yields a $v_c(d)$ in close agreement with experiment. Accordingly, the reduction in the number of atomic states results in superfluidity. The theoretical finding in this work, which can be termed the {\bf QCE superfluidity mechanism}, provides a consistent explanation for this puzzling phenomenon, the non-dissipative, superfluidity motion, of He II and could have a significant impact also on the understanding of other superfluids.

Motivation & Objective

• To resolve the long-standing puzzle of non-dissipative superfluid flow in He II by identifying a microscopic mechanism rooted in quantum confinement.
• To address the failure of existing theories (e.g., Landau and Feynman) that neglect low-energy phonon excitations.
• To provide a first-principles explanation of superfluidity based on experimental data on phonon dispersion and neutron scattering.
• To quantitatively reproduce the experimentally observed $v_c(d)$ dependence in narrow channels, including the critical velocity's width dependence.
• To extend the model to atomic-scale confinement where phonon picture fails, using effective mass for single atoms.

Proposed method

• Derives the phonon excitation energy $\mathcal{E}(K) = \hbar c_1 K$ from experimental neutron scattering data, establishing a first-principles basis for phonon states.
• Applies quantum confinement to a 2D channel, leading to discrete wavevectors $K_n = n\pi/d$, which reduces the number of available $K$ states $\mathcal{N}_2(K)$.
• Uses the condition $\mathcal{N}_2(K) \ll 1$ to define superfluidity, where suppression of excitation states prevents energy dissipation.
• Derives the critical velocity $v_c(d)$ from energy conservation when flow energy exceeds the threshold excitation energy $\Delta_{\text{ph}} = \frac{3h^2}{8m_{\text{ph}} d^2}$.
• For $d < \Lambda$, replaces phonon picture with single-atom excitations using effective mass $m^*$, leading to $v_c(d) \propto h / (m^* d)$.
• Solves the Schr"{o}dinger equation for phonons and atoms in a 1D box to model quantized energy levels and wavefunctions.

Experimental results

Research questions

• RQ1 How does spatial confinement in narrow channels lead to superfluidity in He II, despite the presence of thermal excitations?
• RQ2 Why does the critical velocity $v_c$ depend on channel width $d$, and can this dependence be quantitatively derived from first principles?
• RQ3 What happens to the superfluid mechanism when the channel width $d$ becomes comparable to or smaller than the helium atom correlation length $\Lambda$?
• RQ4 Why do Landau and Feynman's theories fail to explain the $d$-dependence and magnitude of $v_c$?
• RQ5 Can the observed experimental overshoot in $v_c(d)$ be explained by a discontinuity in the number of available $K$ states?

Key findings

• The superfluidity mechanism arises from quantum confinement-induced reduction in the number of phonon wavevector states $\mathcal{N}_2(K)$, which becomes negligibly small in narrow channels.
• The predicted relative viscosity of a confined superfluid is of order $<10^{-6}$, matching experimental observations.
• A discontinuity in $\mathcal{N}_2(K)$ at the edges of energy gaps leads to a critical velocity $v_c(d)$ that agrees quantitatively with experimental data for $d > 10^{-6}$ m.
• The critical velocity follows $v_c(d) \approx 5.1 \times 10^{-8} / d$ m/s for $d > 10^{-6}$ m, derived from phonon energy gaps.
• For $d < 10^{-6}$ m, the model switches to single-atom excitations with effective mass $m^*$, yielding $v_c(d) \approx 5.1 \times 10^{-8} / d$ m/s, showing improved agreement with experiment.
• The model explains the experimental overshoot in $v_c(d)$ as a sharp turn in $\mathcal{N}_2(K)$ at $K \propto v_c$, matching observed data.