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Physica Scripta

Hole–hole scattering in spin-polarized3He–HeII mixtures

Ayman S. Sandouqa, Mohamed K. Al-Sugheir, H. B. Ghassib

Published Jun 16, 2006• 11 citations
Publisher

Abstract

In this paper the Galitskii–Migdal–Feynman (GMF) formalism is applied to spin-polarized 3 He–HeII mixtures with special emphasis on the effect of the magnetic field on hole–hole scattering. For comparison purposes, the relative phase shifts for S- and P-waves, in both GMF and Brueckner–Bethe–Goldstone (BBG) formalisms, are calculated and analysed. Also, the corresponding scattering lengths are evaluated. In polarized mixtures, the S-wave pairing decreases because of the flip of one spin in the pair; whereas the magnetic field has no effect on P-wave pairing.

DOI
10.1088/0031-8949/74/1/002

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The Green's function method is applied to investigate the magnetic spin resonance spectra of three-dimensional and two-dimensional spin-polarized quantum gases. The Hartree-Fock approximation is employed to calculate the one-particle Green's function of the atoms, then this one-particle Green's function is used for the calculation of the vertex part of the Green's function. Such a combination yields a self-consistent result. The absorption spectra are obtained from the calculation of the susceptibility in terms of the two-particle Green's function (bubble diagram). Some general expressions for the dispersion relation, for the effective mass of a spin wave, and for the dipolar frequency shift are given in the calculation. In order to estimate the shift of the electron-spin-resonance (ESR) frequency, the effective dipole-dipole interactions among the hydrogen atoms are included in the calculation. These effective interactions are deduced from the ladder approximation, and hence are characterized by the scattering amplitude. The scattering amplitude is calculated numerically. The result shows that the theoretical value of the shift is smaller than the experimentally observed value by about 35%.

•Jan 1, 2010•1 citation
In this paper the Galitskii–Migdal–Feynman (GMF) formalism is applied to spin-polarized 3 He–HeII mixtures with special emphasis on the effect of the magnetic field on hole–hole scattering. For comparison purposes, the relative phase shifts for S- and P-waves, in both GMF and Brueckner–Bethe–Goldstone (BBG) formalisms, are calculated and analysed. Also, the corresponding scattering lengths are evaluated. In polarized mixtures, the S-wave pairing decreases because of the flip of one spin in the pair; whereas the magnetic field has no effect on P-wave pairing.
In this paper a theoretical study of a possible phase transition in dilute 3 He-He(II) mixtures is presented using the Galitskii-Migdal-Feynman formalism.The effective scattering length is calculated from the Galitskii-Migdal-Feynman T -matrix, which is essentially the effective scattering amplitude dependent on the medium.It is found that at very low 3 He concentrations the s-wave effective scattering length for 3 He-He(II) varies discontinuously from positive to negative values at some critical concentration.This indicates a crossover from a regime with dimers to another with the Cooper pairs.The binding energy of the weakly-bound dimers 3 He2 is computed.The effective p-wave scattering lengths are calculated and compared to the effective s-wave scattering lengths at low and high concentrations.It is found that p-scattering has an important effect on the instability of these mixtures at concentrations x > 1%.Finally, the transport coefficients are computed and compared to the theoretical predictions of Fu and Pethick and the experimental results of König and Pobell.
The cross sections and the man free path of 3He? atoms in HeII background are calculated. The Ramsauer-Townsend effect in the cross sections is found in these mixtures at low temperatures. In the low energy limit the cross sections are dominated by the S-wave scattering. The influence of S-scattering decreases with increasing magnetic field. With increasing k, S-wave scattering tends to decrease; whereas the contribution of the higher angular-momentum waves (especially P-wave) to the scattering increases. Our results of the mean free path exhibit the 3He? atoms propagate through the HeII background with an exceedingly long mean free path and without friction.
We report measurements of the transverse-spin-diffusion coefficient ${\mathit{D}}_{\mathrm{\ensuremath{\perp}}}$ and the spin-rotation parameter \ensuremath{\Omega}${\mathrm{\ensuremath{\tau}}}_{\mathrm{\ensuremath{\perp}}}$ for two very dilute $^{3}\mathrm{\ensuremath{-}}^{4}$He mixtures (x3{\mathit{x}}_{3}x3​=1.82\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}3}$ and 6.26\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}4}$) spin polarized by an 8-T magnetic field. Brute-force spin polarization up to 40% was achieved at the lowest temperature, 6 mK. We find that \ensuremath{\Omega}${\mathrm{\ensuremath{\tau}}}_{\mathrm{\ensuremath{\perp}}}$ increases monotonically as the temperature is reduced through the Fermi temperature TF{\mathit{T}}_{\mathit{F}}TF​, in disagreement with the only previous experiment but in good agreement with recent theory. Unlike the earlier experiment, which measured spin echoes, the present experiments employed a spin-wave technique that avoids nonlinear excitation of the spin field. We compare our results with the recent calculations of Jeon and Mullin for spin transport in dilute gases with arbitrary polarization and degeneracy. The best fit to the data is obtained by scaling the quasiparticle interaction V(q) proposed by Ebner by a modest factor, 1.07. The corresponding s-wave scattering length is a=-1.21 \AA{}. Good agreement is found for \ensuremath{\Omega}${\mathrm{\ensuremath{\tau}}}_{\mathrm{\ensuremath{\perp}}}$(T) at both concentrations and all temperatures, and for ${\mathit{D}}_{\mathrm{\ensuremath{\perp}}}$/\ensuremath{\Omega}${\mathrm{\ensuremath{\tau}}}_{\mathrm{\ensuremath{\perp}}}$(T) apart from the lower concentration at T20 mK. The discrepancy in ${\mathit{D}}_{\mathrm{\ensuremath{\perp}}}$/\ensuremath{\Omega}${\mathrm{\ensuremath{\tau}}}_{\mathrm{\ensuremath{\perp}}}$ at the lowest temperatures and x3{\mathit{x}}_{3}x3​ could be explained by an unanticipated polarization dependence or by modification of the spin-wave boundary condition by processes occurring at the interface between the mixture and the silica cavity wall.
In the strong-coupling Bose-Einstein condensation (BEC) region where a Feshbach resonance gives rise to tightly bound dimer molecules, we show that a spin-polarized Fermi superfluid reduces to a simple Bose-Fermi mixture of Bose-condensed dimers and the leftover unpaired fermions. Using a many-body functional integral formalism, the Gaussian fluctuations give rise to an induced dimer-dimer interaction mediated by the unpaired fermions, with the dimer-fermion vertex being given by the (mean-field) Born approximation. Treating the pairing fluctuations to quartic order, we show how the action for a spin-polarized Fermi superfluid reduces to one for a Bose-Fermi mixture. This Bose-Fermi action includes an expression for the effective dimer-unpaired fermion interaction in a spin-polarized Fermi superfluid beyond the Born approximation, in the BEC limit. In the low-density limit, we show how this dimer-fermion interaction gives the sss-wave scattering length aBF=1.18aF{a}_{\mathit{BF}}=1.18{a}_{F}aBF​=1.18aF​ (aF{a}_{F}aF​ is the sss-wave fermion scattering length), a result first derived by Skorniakov and Ter-Martirosian in 1957 for three interacting fermions.
We present a quantum phase-shift calculation of the two-dimensional second virial coefficient B22D{B}_{2}^{2\mathrm{D}}B22D​ and spin-wave characteristic parameters for spin-polarized quantum gases H\ensuremath{\uparrow}, D\ensuremath{\uparrow}, and 3He^{3}\mathrm{He}3He. Emphasis is put on the study of quantum exchange effects which become essential in the 1-K range and below. The dependence of B22D{B}_{2}^{2\mathrm{D}}B22D​ upon the nuclear polarization is analyzed. The calculation of statistics-independent coefficients is extended up to 50 K. The effect of a reduced dimensionality on the second virial and spin-wave coefficients is discussed and demonstrated by a comparison between 2D and 3D results. Two-dimensional bound states of helium dimers are estimated with the best present potential of Aziz et al. to be EB{E}_{B}EB​(3{(}^{3}(3He${\mathrm{\ensuremath{-}}}^{3}$He; 2D)=-0.88\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}5}$ K and EB{E}_{B}EB​(3{(}^{3}(3He${\mathrm{\ensuremath{-}}}^{4}$He; 2D)=-3.38 mK.
The Green's function method is applied to investigate the magnetic spin resonance spectra of three-dimensional and two-dimensional spin-polarized quantum gases. The Hartree-Fock approximation is employed to calculate the one-particle Green's function of the atoms, then this one-particle Green's function is used for the calculation of the vertex part of the Green's function. Such a combination yields a self-consistent result. The absorption spectra are obtained from the calculation of the susceptibility in terms of the two-particle Green's function (bubble diagram). Some general expressions for the dispersion relation, for the effective mass of a spin wave, and for the dipolar frequency shift are given in the calculation. In order to estimate the shift of the electron-spin-resonance (ESR) frequency, the effective dipole-dipole interactions among the hydrogen atoms are included in the calculation. These effective interactions are deduced from the ladder approximation, and hence are characterized by the scattering amplitude. The scattering amplitude is calculated numerically. The result shows that the theoretical value of the shift is smaller than the experimentally observed value by about 35%.