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Physical Review Letters

Interactions Between He3 Atoms in Dilute Solutions of He3 in Superfluid He4

J. Bardeen, Gordon Baym, David Pines

Published Aug 15, 1966• 105 citations
Publisher

Abstract

The magnitude, sign, and momentum dependence of the effective interaction between ${\mathrm{He}}^{3}$ atoms in superfluid ${\mathrm{He}}^{4}$ are derived from the experimental data. Calculations of the Fermi-liquid parameters for the normal state of ${\mathrm{He}}^{3}$ in solution yield results in agreement with experiment; the temperature for a superfluid transition associated with the ${\mathrm{He}}^{3}$ is estimated to be \ensuremath{\sim}2\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}6}$ \ifmmode^\circ\else\textdegree\fi{}K.

DOI
10.1103/physrevlett.17.372

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Liquid helium and Bose-gas

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This book is about a certain class of superfluids, namely the high temperature superconductors. Perhaps the simplest of the superfluids is liquid He4. Since the very early days of quantum mechanics it has been known that the helium atom He4 is a boson; it is built of an even number of particles (two electrons, two neutrons and two protons), so that a wave function ^(ri,F2) must be symmetrical in FI — r? where FI, r-z are the electron's coordinates. This has the result that half the rotational states of the molecule H62 are absent and that the scattering of alpha particles by helium for low energies deviates from the Rutherford scattering law. It also has the remarkable result that this is not so for the light isotope He3, which shows that the number of neutrons in the nucleus affects properties depending otherwise only on the electrons and nuclear charge.

CRC Press eBooks•Nov 24, 1994•0 citations
The magnitude, sign, and momentum dependence of the effective interaction between He3{\mathrm{He}}^{3}He3 atoms in superfluid He4{\mathrm{He}}^{4}He4 are derived from the experimental data. Calculations of the Fermi-liquid parameters for the normal state of He3{\mathrm{He}}^{3}He3 in solution yield results in agreement with experiment; the temperature for a superfluid transition associated with the He3{\mathrm{He}}^{3}He3 is estimated to be \ensuremath{\sim}2\ifmmode\times\else\texttimes\fi{}10\ensuremath−6{10}^{\ensuremath{-}6}10\ensuremath−6 \ifmmode^\circ\else\textdegree\fi{}K.
The authors present for the first time an effective interaction between two 4He^{4}\mathrm{He}4He atoms in a bath of liquid 3He^{3}\mathrm{He}3He at temperatures below 0.1 K. This interaction turns out to be, in its long-range limit, an oscillatory function of the 4He^{4}\mathrm{He}4He interatomic separation, the oscillations arising essentially from the sharp discontinuity in the momentum distribution at the Fermi surface. In addition, an expression is formulated for the oscillations in the density of the 3He^{3}\mathrm{He}3He bath induced by one 4He^{4}\mathrm{He}4He impurity, as well as a modified Friedel sum rule, which relates the relative amount of matter displaced by this impurity to the phase shifts pertaining to the impurity-3He^{3}\mathrm{He}3He quasiparticle scattering at the Fermi surface. Finally, the dependence of the results on temperature and on the 3He^{3}\mathrm{He}3He superfluidity gap is discussed.
Dilute mixtures of He3{\mathrm{He}}^{3}He3 in liquid He4{\mathrm{He}}^{4}He4 are regarded as a set of He3{\mathrm{He}}^{3}He3 atoms intereacting with elementary excitations of the He4{\mathrm{He}}^{4}He4. The coupling between the He3{\mathrm{He}}^{3}He3 and the He4{\mathrm{He}}^{4}He4 is chosen phenomenologically and it is not weak but, for low temperatures, its main effects may be eliminated by means of two canonical transformations. The equations which determine the transformations describe the motion of a He4{\mathrm{He}}^{4}He4 excitation in the presence of one or two He3{\mathrm{He}}^{3}He3 atoms and, in the latter case, they resemble Faddeev's equations in the three-body problem. At low temperatures, there are few He4{\mathrm{He}}^{4}He4 excitations and many of the properties of the mixture are determined by the effective He3{\mathrm{He}}^{3}He3 Hamiltonian produced by the transformation. In this respect, the system behaves like a Fermi liquid. The He3{\mathrm{He}}^{3}He3-He3{\mathrm{He}}^{3}He3 potential is essentially momentum-independent and consists of the Van der Waals interaction together with an effectively repulsive potential induced by the He4{\mathrm{He}}^{4}He4. It is suggested that a Fermi-liquid analysis is the best way of obtaining information about this potential, and the possible existence of a fermion superfluid phase transition is discussed from that point of view.
We present neutron-scattering measurements of the momentum distribution of liquid $^{3}\mathrm{He}\text{\ensuremath{-}}^{4}\mathrm{He}$ mixtures. The experiments were performed at wave vectors QQQ, $26\ensuremath{\leqslant}Q\ensuremath{\leqslant}29\phantom{\rule{0.3em}{0ex}}{\mathrm{\AA{}}}^{\ensuremath{-}1}$, on the MARI time-of-flight spectrometer at the ISIS Facility, Rutherford Appleton Laboratory, a spallation neutron source. Mixtures with 3He^{3}\mathrm{He}3He concentrations xxx between 0 and 20% were investigated both in the superfluid and normal phases. From the data, we extract the Bose-Einstein condensate fraction n0{n}_{0}n0​ and the momentum distributions of 3He^{3}\mathrm{He}3He and 4He^{4}\mathrm{He}4He atoms. We find that n0{n}_{0}n0​ increases somewhat above the pure 4He^{4}\mathrm{He}4He value when 3He^{3}\mathrm{He}3He is added; e.g., from ${n}_{0}=(7.25\ifmmode\pm\else\textpm\fi{}0.75)%$ at x=0x=0x=0 to $(11\ifmmode\pm\else\textpm\fi{}3)%$ at x=15−−20x=15--20\phantom{\rule{0.2em}{0ex}}%x=15−−20. This agrees with predictions but is less than the only previous measurement. We find a 4He^{4}\mathrm{He}4He kinetic energy K4{K}_{4}K4​ for pure 4He^{4}\mathrm{He}4He that agrees with previous determinations. K4{K}_{4}K4​ decreases somewhat with increasing 3He^{3}\mathrm{He}3He concentration, less than observed previously and found in early calculations but in agreement with a more recent Monte Carlo calculation. The 3He^{3}\mathrm{He}3He response is not well reproduced by a Fermi-gas momentum distribution, n(k)n(\mathbf{k})n(k). Rather an n(k)n(\mathbf{k})n(k) having a small step height at the Fermi surface and a substantial high-momentum tail characteristic of a strongly interacting Fermi liquid provides a good fit. This n(k)n(\mathbf{k})n(k) is consistent with calculated n(k)n(\mathbf{k})n(k). Thus agreement between theory and experiment is obtained comparing n(k)n(\mathbf{k})n(k) in contrast to earlier findings based on comparing calculated and observed 3He^{3}\mathrm{He}3He kinetic energies.
This book is about a certain class of superfluids, namely the high temperature superconductors. Perhaps the simplest of the superfluids is liquid He4. Since the very early days of quantum mechanics it has been known that the helium atom He4 is a boson; it is built of an even number of particles (two electrons, two neutrons and two protons), so that a wave function ^(ri,F2) must be symmetrical in FI — r? where FI, r-z are the electron's coordinates. This has the result that half the rotational states of the molecule H62 are absent and that the scattering of alpha particles by helium for low energies deviates from the Rutherford scattering law. It also has the remarkable result that this is not so for the light isotope He3, which shows that the number of neutrons in the nucleus affects properties depending otherwise only on the electrons and nuclear charge.