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arXiv:cond-mat/9802122 · Physical Review Letters

Concentration Dependence of the Effective Mass of 3He Atoms in 3He−4He Mixtures

E. Krotscheck, M. Saarela, K. Schörkhuber, Robert E. Zillich

Published May 25, 1998• 62 citations
arXivHTMLPDF

Abstract

Recent measurements by Yorozu et al. [S. Yorozu, H. Fukuyama, and H. Ishimoto, Phys. Rev. B 48, 9660 (1993)] as well as by Simons and Mueller [R. Simons and R. M. Mueller, Czech. J. Phys. Suppl. 46, 201 (1996)] have determined the effective mass of ${}^{3}\mathrm{He}$ atoms in a ${}^{3}\mathrm{He}{\ensuremath{-}}^{4}\mathrm{He}$ mixture with great accuracy. We here report theoretical calculations for the dependence of that effective mass on the ${}^{3}\mathrm{He}$ concentration. Using correlated basis function perturbation theory to infinite order to compute effective interactions in the appropriate channels, we obtain good agreement between theory and experiment.

DOI
10.1103/physrevlett.80.4709

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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.
The integral cross sections of large 3He{}^{3}\mathrm{He}3He droplets and the number of atoms in the corresponding droplets $(\overline{N}=2\ifmmode\times\else\texttimes\fi{}{10}^{3}--2\ifmmode\times\else\texttimes\fi{}{10}^{4}\mathrm{atoms})$ have been measured in molecular-beam scattering experiments. The experimental results are in very good agreement with integral cross sections calculated from the radial density distributions predicted from density-functional theory calculations. The experimentally confirmed theoretical 10--90% surface thicknesses vary between 8.0 \AA{} (N‾=103)(\overline{N}{=10}^{3})(N=103) and 7.6 \AA{} $(\overline{N}=4\ifmmode\times\else\texttimes\fi{}{10}^{4})$ and are about 30% larger than calculated for 4He{}^{4}\mathrm{He}4He droplets of similar sizes.
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.
The Green function is derived for a small concentration of fermions in a weakly interacting Bose gas using mass operator perturbation theory. For small momentum it is found that the spectral weight function of the system consists of a quasi-particle peak associated with an effective mass m* as well as some weight at higher energies The momentum dependence of both parts of the spectral weight function is determined. The implications of the weight at higher energies in the case of dilute 3He-4He mixtures are discussed.