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Physical Review Letters

Evidence for Different Interaction Potentials for He4 - He4 and He3 - He3 from Scattering Cross-Section Measurements

H. G. Bennewitz, Harald Busse, H. D. Dohmann, Daniel E. Oates, W. Schrader

Published Aug 28, 1972• 37 citations
Publisher

Abstract

By means of low-energy cross-section measurements on ${\mathrm{He}}^{3}$-${\mathrm{He}}^{3}$, we derive an interatomic potential which has a minimum 3.4% deeper than that of ${\mathrm{He}}^{4}$-${\mathrm{He}}^{4}$. We demonstrate that the difference in the potentials is significant. The atomic-mass polarization effect is proposed as a possible explanation for the different potentials.

DOI
10.1103/physrevlett.29.533

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By means of low-energy cross-section measurements on He3{\mathrm{He}}^{3}He3-He3{\mathrm{He}}^{3}He3, we derive an interatomic potential which has a minimum 3.4% deeper than that of He4{\mathrm{He}}^{4}He4-He4{\mathrm{He}}^{4}He4. We demonstrate that the difference in the potentials is significant. The atomic-mass polarization effect is proposed as a possible explanation for the different potentials.
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{}${10}^{\ensuremath{-}6}$ \ifmmode^\circ\else\textdegree\fi{}K.
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.
The modifications of effective-range theory, necessitated by the long-range nature of interatomic dispersion and relativistic retardation forces, are discussed. A numerical comparison is given of parameters describing $^{3}\mathrm{\ensuremath{-}}^{3}$He, $^{3}\mathrm{\ensuremath{-}}^{4}$He, and $^{4}\mathrm{\ensuremath{-}}^{4}$He scattering at very low energies calculated with several recently published interaction potentials. The effect of relativistic retardation on these parameters and on the calculated binding energies of the $^{4}\mathrm{\ensuremath{-}}^{4}$He dimer is shown. It is demonstrated that all of these quantities are extremely sensitive to the potential. Scattering lengths and effective ranges for the lowest triplet interactions of H-H and D-D are also reported.
We predict sss-wave elastic cross-sections $\ensuremath{\sigma}$ for low-energy atom-molecule collisions with kinetic energies up to 40 mK for the 4He^{4}\mathrm{He}4He collision with weakly bound diatomic molecules formed by 4He^{4}\mathrm{He}4He with 7Li,6Li^{7}\mathrm{Li},\phantom{\rule{0.28em}{0ex}}^{6}\mathrm{Li}7Li,6Li, and 23Na^{23}\mathrm{Na}23Na. Our scattering calculations are performed by using diatomic and triatomic molecular binding energies obtained from several available realistic models as input in a renormalized zero-range model as well as a finite-range one-term separable potential in order to quantify the relevance of range corrections to our predictions. Of particular relevance for possible experimental realization, we show the occurrence of a zero in $\ensuremath{\sigma}$ for the collision of cold 4He^{4}\mathrm{He}4He on a 4He23Na^{4}\mathrm{He}\phantom{\rule{0.16em}{0ex}}^{23}\mathrm{Na}4He23Na molecule below 20 mK. Also our results for the elastic collision 4He^{4}\mathrm{He}4He on 4He6,7Li^{4}\mathrm{He}\phantom{\rule{0.16em}{0ex}}^{6,7}\mathrm{Li}4He6,7Li molecules suggest that $\ensuremath{\sigma}$ varies considerably for the realistic models studied. As the chosen molecules are weakly bound and the scattering energies are very low, our results are interpreted on the light of the Efimov physics, which explains the model independence and robustness of our predictions despite some sensitivity on the potential range.
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