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Bound and Efimov states for /sup 4/He/sub 2/ and /sup 4/He/sub 3/ and Bose--Einstein condensation

Norman Henry March

Published Jul 1, 1983• 0 citations
OpenAlex

Abstract

The existence of a Bose--Einstein condensate at absolute zero is disputed. Superfluidity without a condensate is implied. Bound state of helium dimers and trimers are discussed. (AIP)

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Bound and Efimov states for 4He2 and 4He3 and Bose–Einstein condensation

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The Journal of Chemical Physics•Jul 1, 1983•12 citations

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Physical Review•Nov 1, 1956•1,570 citations
The discovery of superfluidity in liquid {sup 4}He below T{sub {lambda}} = 2.17 K, and its phenomenological characterization since then, has been one of the great success stories of condensed matter physics. The relation of superfluidity to the behavior of atoms was conjectured by F. London in 1938. Superfluidity is a manifestation of the Bose condensation of helium atoms, the extensive occupation of the zero momentum state. Ever since {sup 4}He has been the paradigm in the search for Bose condensates in other systems. At the Pune meeting scientists have heard exciting new evidence for Bose condensates of laser cooled alkali atoms in magnetic traps, of excitons in Cu{sub 2}O, and possibly pre-formed Cooper pairs of electrons in the high {Tc} perovskite superconductors. There remains the holy-grail of forming a Bose condensate in spin-polarized hydrogen. In the current excitement for new types of Bose condensates, and new phenomena such as atom lasers, it may be useful to recall the older story of the experimental verification of a relation between superfluidity and Bose condensation in {sup 4}He. This topic has been investigated over many years by neutron scattering experiments and quantum many-body theory. The authors goal is to illustrate the difficulties of establishing the existence of a Bose condensate in a strongly interacting system, even though its macroscopic effects are manifest. The author assumes readers have access to a review by Silver and Sokol which emphasizes the neutron scattering theory through 1990 and a review by Snow and Sokol of the deep inelastic neutron scattering (DINS) experiments through 1995.
It is our great good fortune that the only two substances that remain liquid down to the absolute zero obey different statistics ( 3 He: Fermi-Dirac, 4 He: Bose-Einstein) and thus allow us to study the differing effects of quantum statistics on condensed matter. An extraordinarily diverse and luxuriant field of liquid helium work has consequently developed, particularly in the last 25 years, ranging from refrigeration applications to the behaviour of pulsars, and it is surprising to realize that the ancient controversy over whether the Bose–Einstein condensation phenomenon (BEC) is necessary and sufficient for superfluidity is still not settled.
The mathematical description of B.E. (Bose-Einstein) condensation is generalized so as to be applicable to a system of interacting particles. B.E. condensation is said to be present whenever the largest eigenvalue of the one-particle reduced density matrix is an extensive rather than an intensive quantity. Some transformations facilitating the practical use of this definition are given.An argument based on first principles is given, indicating that liquid belium II in equilibrium shows B.E. condensation. For absolute zero, the argument is based on properties of the ground-state wave function derived from the assumption that there is no "long-range configurational order." A crude estimate indicates that roughly 8% of the atoms are "condensed" (note that the fraction of condensed particles need not be identified with $\frac{{\ensuremath{\rho}}_{s}}{\ensuremath{\rho}}$). Conversely, it is shown why one would not expect B.E. condensation in a solid. For finite temperatures Feynman's theory of the lambda-transition is applied: Feynman's approximations are shown to imply that our criterion of B.E. condensation is satisfied below the lambda-transition but not above it.