Bose-Einstein condensation and the atomic momentum distribution are investigated for a simplified liquid $^{4}\mathrm{He}$ ground state obtained by antisymmetrization, with respect to the electron variables, of a product of $^{4}\mathrm{He}$ -atom wave functions, each in its individual electronic and translational ground state. The momentum distribution is defined and evaluated by the method of redundant modes and generalized Tani transformation. Bose-Einstein condensation is found, with a condensate depletion of 6% owing to the purely kinematical repulsion arising from the Pauli exclusion principle. The noncondensed atoms are spread in momentum space by an amount of a few inverse Bohr radii. Dynamical correlations would increase the depletion in a more realistic ground state.Bose-Einstein condensation and the atomic momentum distribution are investigated for a simplified liquid $^{4}\mathrm{He}$ ground state obtained by antisymmetrization, with respect to the electron variables, of a product of $^{4}\mathrm{He}$-atom wave functions, each in its individual electronic and translational ground state. The momentum distribution is defined and evaluated by the method of redundant modes and generalized Tani transformation. Bose-Einstein condensation is found, with a condensate depletion of 6% owing to the purely kinematical repulsion arising from the Pauli exclusion principle. The noncondensed atoms are spread in momentum space by an amount of a few inverse Bohr radii. Dynamical correlations would increase the depletion in a more realistic ground state.

Physical Review AOct 1, 19735 citations

Condensate fraction and atomic kinetic energy of liquid He3−He4 mixtures

Souleymane Diallo, Jonathan V Pearce, R. T. Azuah +2 more

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 $Q$ , $26\ensuremath{\leqslant}Q\ensuremath{\leqslant}29\phantom{\rule{0.3em}{0ex}}{\mathrm{\AA{}}}^{\ensuremath{-}1}$, on the MARI time-of-flight spectrometer at the ISIS Facility, Rutherford Appleton Laboratory, a spallation neutron source. Mixtures with $^{3}\mathrm{He}$ concentrations $x$ between 0 and 20% were investigated both in the superfluid and normal phases. From the data, we extract the Bose-Einstein condensate fraction ${n}_{0}$ and the momentum distributions of $^{3}\mathrm{He}$ and $^{4}\mathrm{He}$ atoms. We find that ${n}_{0}$ increases somewhat above the pure $^{4}\mathrm{He}$ value when $^{3}\mathrm{He}$ is added; e.g., from ${n}_{0}=(7.25\ifmmode\pm\else\textpm\fi{}0.75)%$ at $x=0$ to $(11\ifmmode\pm\else\textpm\fi{}3)%$ at $x=15--20\phantom{\rule{0.2em}{0ex}}%$ . This agrees with predictions but is less than the only previous measurement. We find a $^{4}\mathrm{He}$ kinetic energy ${K}_{4}$ for pure $^{4}\mathrm{He}$ that agrees with previous determinations. ${K}_{4}$ decreases somewhat with increasing $^{3}\mathrm{He}$ concentration, less than observed previously and found in early calculations but in agreement with a more recent Monte Carlo calculation. The $^{3}\mathrm{He}$ response is not well reproduced by a Fermi-gas momentum distribution, $n(\mathbf{k})$ . Rather an $n(\mathbf{k})$ 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 $n(\mathbf{k})$ is consistent with calculated $n(\mathbf{k})$ . Thus agreement between theory and experiment is obtained comparing $n(\mathbf{k})$ in contrast to earlier findings based on comparing calculated and observed $^{3}\mathrm{He}$ kinetic energies.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 $Q$, $26\ensuremath{\leqslant}Q\ensuremath{\leqslant}29\phantom{\rule{0.3em}{0ex}}{\mathrm{\AA{}}}^{\ensuremath{-}1}$, on the MARI time-of-flight spectrometer at the ISIS Facility, Rutherford Appleton Laboratory, a spallation neutron source. Mixtures with $^{3}\mathrm{He}$ concentrations $x$ between 0 and 20% were investigated both in the superfluid and normal phases. From the data, we extract the Bose-Einstein condensate fraction ${n}_{0}$ and the momentum distributions of $^{3}\mathrm{He}$ and $^{4}\mathrm{He}$ atoms. We find that ${n}_{0}$ increases somewhat above the pure $^{4}\mathrm{He}$ value when $^{3}\mathrm{He}$ is added; e.g., from ${n}_{0}=(7.25\ifmmode\pm\else\textpm\fi{}0.75)%$ at $x=0$ to $(11\ifmmode\pm\else\textpm\fi{}3)%$ at $x=15--20\phantom{\rule{0.2em}{0ex}}%$. This agrees with predictions but is less than the only previous measurement. We find a $^{4}\mathrm{He}$ kinetic energy ${K}_{4}$ for pure $^{4}\mathrm{He}$ that agrees with previous determinations. ${K}_{4}$ decreases somewhat with increasing $^{3}\mathrm{He}$ concentration, less than observed previously and found in early calculations but in agreement with a more recent Monte Carlo calculation. The $^{3}\mathrm{He}$ response is not well reproduced by a Fermi-gas momentum distribution, $n(\mathbf{k})$. Rather an $n(\mathbf{k})$ 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 $n(\mathbf{k})$ is consistent with calculated $n(\mathbf{k})$. Thus agreement between theory and experiment is obtained comparing $n(\mathbf{k})$ in contrast to earlier findings based on comparing calculated and observed $^{3}\mathrm{He}$ kinetic energies.

Physical Review BOct 5, 200610 citations

The Momentum Distribution of Liquid

^4\hbox {He}

Bose-Einstein condensation and the atomic momentum distribution are investigated for a simplified liquid

^{4}\mathrm{He}

ground state obtained by antisymmetrization, with respect to the electron variables, of a product of

^{4}\mathrm{He}

-atom wave functions, each in its individual electronic and translational ground state. The momentum distribution is defined and evaluated by the method of redundant modes and generalized Tani transformation. Bose-Einstein condensation is found, with a condensate depletion of 6% owing to the purely kinematical repulsion arising from the Pauli exclusion principle. The noncondensed atoms are spread in momentum space by an amount of a few inverse Bohr radii. Dynamical correlations would increase the depletion in a more realistic ground state.

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

Q

, $26\ensuremath{\leqslant}Q\ensuremath{\leqslant}29\phantom{\rule{0.3em}{0ex}}{\mathrm{\AA{}}}^{\ensuremath{-}1}$, on the MARI time-of-flight spectrometer at the ISIS Facility, Rutherford Appleton Laboratory, a spallation neutron source. Mixtures with

^{3}\mathrm{He}

concentrations

x

between 0 and 20% were investigated both in the superfluid and normal phases. From the data, we extract the Bose-Einstein condensate fraction

{n}_{0}

and the momentum distributions of

^{3}\mathrm{He}

and

^{4}\mathrm{He}

atoms. We find that

{n}_{0}

increases somewhat above the pure

^{4}\mathrm{He}

value when

^{3}\mathrm{He}

is added; e.g., from ${n}_{0}=(7.25\ifmmode\pm\else\textpm\fi{}0.75)%$ at

x=0

to $(11\ifmmode\pm\else\textpm\fi{}3)%$ at

x=15--20\phantom{\rule{0.2em}{0ex}}%

. This agrees with predictions but is less than the only previous measurement. We find a

^{4}\mathrm{He}

kinetic energy

{K}_{4}

for pure

^{4}\mathrm{He}

that agrees with previous determinations.

{K}_{4}

decreases somewhat with increasing

^{3}\mathrm{He}

concentration, less than observed previously and found in early calculations but in agreement with a more recent Monte Carlo calculation. The

^{3}\mathrm{He}

response is not well reproduced by a Fermi-gas momentum distribution,

n(\mathbf{k})

. Rather an

n(\mathbf{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(\mathbf{k})

is consistent with calculated

n(\mathbf{k})

. Thus agreement between theory and experiment is obtained comparing

n(\mathbf{k})

in contrast to earlier findings based on comparing calculated and observed

^{3}\mathrm{He}

kinetic energies.