Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics Condensed Matter

Measurement of the kinetic energy in through the superfluid transition by very high-energy neutron scattering

J. Mayers, C. Andreani, D. Colognesi

Published Dec 1, 1997• 32 citations
Publisher

Abstract

Measurements of the kinetic energy of atoms in liquid have been made using neutron Compton scattering at wavevector transfers between 90 and . The kinetic energy shows a marked decrease from 16.2 K at a temperature of 2.5 K, to 13.9 K at 1.3 K. Three different measurements were made using energy analysers with different resolutions. Good agreement was obtained between the measurements using different resolutions and with previous measurements and calculations of the kinetic energies. For the best-resolution data, a line shape analysis was made in order to derive the non-Gaussian components of the momentum distribution. Values of kinetic energies are consistent with a Bose condensate fraction of . The measurements show that eV neutron scattering provides an accurate method of measuring kinetic energies in quantum fluids.

DOI
10.1088/0953-8984/9/48/008

Related papers

Bose-Einstein Condensation and Kinetic energy of liquid $^3$He-$^4$He Mixtures

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

We present neutron scattering measurements of the momentum distribution of liquid ${^3}$He-${^4}$He mixtures. The experiments were performed at wavevectors $Q$, 26 $\leq$ $Q$ $\leq$ 29 {Å$^{-1}$}, on the MARI time-of-flight spectrometer at the ISIS Facility, Rutherford Appleton Laboratory, a spallation neutron source. Mixtures with $^3$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 $^3$He and $^4$He atoms. We find that $n_0$ increases somewhat above the pure $^4$He value when $^3$He is added; e.g from $n_0=(7.25\pm0.75)%$ at $x=$ 0 to $(11\pm3)%$ at $x=$ 15-20%. This agrees with predictions but is less than the only previous measurement. We find a $^4$He kinetic energy $K_4$ for pure $^4$He that agrees with previous determinations. $K_4$ decreases somewhat with increasing $^3$He concentration, less than observed previously and found in early calculations but in agreement with a more recent Monte Carlo calculation. The $^3$He response is not well reproduced by a Fermi gas momentum distribution, $n(\bf{k})$. Rather an $n(\bf{k})$ 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 $n(\bf{k})$ is consistent with calculated $n(\bf{k})$. Thus agreement between theory and experiment is obtained comparing $n(\bf{k})$ in contrast to earlier findings based on comparing calculated and observed $^3$He kinetic energies.

arXiv:cond-mat/0609529•Sep 20, 2006•0 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.

Physical Review B•Oct 5, 2006•10 citations

High-Energy Neutron Scattering from Liquid Helium in the Impulse Approximation

H. A. Gersch, Phil N. Smith

The impulse approximation is used to describe the inelastic cross section for high-energy neutron scattering from superfluid helium in terms of a ground-state momentum distribution of the helium atoms as given by existing variational calculations. The predicted shape of the cross section is compared with recent experimental data and with a previous theory which approximated the noncondensate contribution to the dynamic structure factor by a single-Gaussian function. In contrast to conclusions obtained using the single-Gaussian approximation, our predicted inelastic cross section omitting a condensate contribution provides an acceptable fit to the experimental data. This fit is worsened by introducing a contribution from a condensate fraction in the amount consistent with the variational calculations, namely 11%. If the condensate fraction is arbitrarily reduced from this value, it is found that a much smaller value, with an upper limit of about 3%, is consistent with the available experimental data. It appears that unique assignments of condensate-fraction contributions to inelastic neutron scattering, for the range of neutron energies presently available, will require an improved theory of final-state effects on the shape of the condensate contribution.

Physical Review A•Jul 1, 1971•15 citations

Bose-Einstein condensation in liquid 4 He near the liquid-solid transition line

Souleymane Diallo, R. T. Azuah, Douglas L. Abernathy +3 more

We present precision neutron scattering measurements of the Bose-Einstein condensate (BEC) fraction ${n}_{0}(T)$ and the atomic momentum distribution ${n}^{\ensuremath{\star}}(\mathbf{k})$ of liquid ${}^{4}$He as a function of temperature at pressure $p=24$ bars. Both the temperature dependence of ${n}_{0}(T)$ and of the width of ${n}^{\ensuremath{\star}}(\mathbf{k})$ are determined. The ${n}_{0}(T)$ can be represented by ${n}_{0}(T)={n}_{0}(0)[1\ensuremath{-}{(T/{T}_{\ensuremath{\lambda}})}^{\ensuremath{\gamma}}]$ with a small ${n}_{0}(0)=2.80\ifmmode\pm\else\textpm\fi{}0.20$$%$ and large $\ensuremath{\gamma}=13\ifmmode\pm\else\textpm\fi{}2$ for $T<{T}_{\ensuremath{\lambda}}$, indicating a strong interaction. The onset of BEC is accompanied by a significant narrowing of the ${n}^{\ensuremath{\star}}(\mathbf{k})$. The narrowing accounts for 65$%$ of the drop in kinetic energy below ${T}_{\ensuremath{\lambda}}$ and reveals an important coupling between BEC and $k>0$ states. The experimental results are well reproduced by path integral Monte Carlo calculations.

arXiv:1111.2284•Physical Review B•Apr 9, 2012•27 citations

Bose-Einstein Condensation in Solid He4

Souleymane Diallo, Jonathan V Pearce, R. T. Azuah +3 more

We present neutron scattering measurements of the atomic momentum distribution $n(\mathbf{k})$ in solid helium under a pressure $p=41\text{ }\mathrm{bar}$ (molar volume ${V}_{m}=20.01\ifmmode\pm\else\textpm\fi{}0.02\text{ }\text{ }{\mathrm{cm}}^{3}/\mathrm{mol}$) and at temperatures between 80 and 500 mK. The aim is to determine whether there is Bose-Einstein condensation (BEC) below the critical temperature, ${T}_{c}=200\text{ }\text{ }\mathrm{mK}$, where a superfluid density has been observed. Assuming BEC appears as a macroscopic occupation of the $k=0$ state below ${T}_{c}$, we find a condensate fraction of ${n}_{0}=(\ensuremath{-}0.10\ifmmode\pm\else\textpm\fi{}1.20)%$ at $T=80\text{ }\text{ }\mathrm{mK}$ and ${n}_{0}=(0.08\ifmmode\pm\else\textpm\fi{}0.78)%$ at $T=120\text{ }\text{ }\mathrm{mK}$, consistent with zero. The shape of $n(\mathbf{k})$ also does not change on crossing ${T}_{c}$ within measurement precision.

arXiv:cond-mat/0702347•Physical Review Letters•May 17, 2007•45 citations

Momentum distributions in superfluid helium: bose condensate fraction

Paul E. Sokol, Ralph O. Simmons, D. L. Price +1 more

We have measured the momentum distributions in superfluid and normal Helium 4 at identical densities using inelastic neutron scattering as a function of temperature and density. At large enough momentum transfers Q the Impulse Approximation is valid and the single particle momentum distribution n(anti p) can be extracted from the inelastic scattering spectrum. Results are reported. (WHK)

•Aug 9, 1984•0 citations
Measurements of the kinetic energy of atoms in liquid have been made using neutron Compton scattering at wavevector transfers between 90 and . The kinetic energy shows a marked decrease from 16.2 K at a temperature of 2.5 K, to 13.9 K at 1.3 K. Three different measurements were made using energy analysers with different resolutions. Good agreement was obtained between the measurements using different resolutions and with previous measurements and calculations of the kinetic energies. For the best-resolution data, a line shape analysis was made in order to derive the non-Gaussian components of the momentum distribution. Values of kinetic energies are consistent with a Bose condensate fraction of . The measurements show that eV neutron scattering provides an accurate method of measuring kinetic energies in quantum fluids.
Bose-Einstein Condensation and Kinetic energy of liquid 3^33He-4^44He Mixtures
We present neutron scattering measurements of the momentum distribution of liquid 3{^3}3He-4{^4}4He mixtures. The experiments were performed at wavevectors QQQ, 26 ≤\leq≤ QQQ ≤\leq≤ 29 {Å−1^{-1}−1}, on the MARI time-of-flight spectrometer at the ISIS Facility, Rutherford Appleton Laboratory, a spallation neutron source. Mixtures with 3^33He 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 n0n_0n0​ and the momentum distributions of 3^33He and 4^44He atoms. We find that n0n_0n0​ increases somewhat above the pure 4^44He value when 3^33He is added; e.g from n0=(7.25±0.75)n_0=(7.25\pm0.75)%n0​=(7.25±0.75) at x=x=x= 0 to (11±3)(11\pm3)%(11±3) at x=x=x= 15-20%. This agrees with predictions but is less than the only previous measurement. We find a 4^44He kinetic energy K4K_4K4​ for pure 4^44He that agrees with previous determinations. K4K_4K4​ decreases somewhat with increasing 3^33He concentration, less than observed previously and found in early calculations but in agreement with a more recent Monte Carlo calculation. The 3^33He response is not well reproduced by a Fermi gas momentum distribution, n(k)n(\bf{k})n(k). Rather an n(k)n(\bf{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(\bf{k})n(k) is consistent with calculated n(k)n(\bf{k})n(k). Thus agreement between theory and experiment is obtained comparing n(k)n(\bf{k})n(k) in contrast to earlier findings based on comparing calculated and observed 3^33He 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 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 impulse approximation is used to describe the inelastic cross section for high-energy neutron scattering from superfluid helium in terms of a ground-state momentum distribution of the helium atoms as given by existing variational calculations. The predicted shape of the cross section is compared with recent experimental data and with a previous theory which approximated the noncondensate contribution to the dynamic structure factor by a single-Gaussian function. In contrast to conclusions obtained using the single-Gaussian approximation, our predicted inelastic cross section omitting a condensate contribution provides an acceptable fit to the experimental data. This fit is worsened by introducing a contribution from a condensate fraction in the amount consistent with the variational calculations, namely 11%. If the condensate fraction is arbitrarily reduced from this value, it is found that a much smaller value, with an upper limit of about 3%, is consistent with the available experimental data. It appears that unique assignments of condensate-fraction contributions to inelastic neutron scattering, for the range of neutron energies presently available, will require an improved theory of final-state effects on the shape of the condensate contribution.
We present precision neutron scattering measurements of the Bose-Einstein condensate (BEC) fraction n0(T){n}_{0}(T)n0​(T) and the atomic momentum distribution ${n}^{\ensuremath{\star}}(\mathbf{k})$ of liquid 4{}^{4}4He as a function of temperature at pressure p=24p=24p=24 bars. Both the temperature dependence of n0(T){n}_{0}(T)n0​(T) and of the width of ${n}^{\ensuremath{\star}}(\mathbf{k})$ are determined. The n0(T){n}_{0}(T)n0​(T) can be represented by ${n}_{0}(T)={n}_{0}(0)[1\ensuremath{-}{(T/{T}_{\ensuremath{\lambda}})}^{\ensuremath{\gamma}}]$ with a small ${n}_{0}(0)=2.80\ifmmode\pm\else\textpm\fi{}0.20$% and large $\ensuremath{\gamma}=13\ifmmode\pm\else\textpm\fi{}2$ for $T<{T}_{\ensuremath{\lambda}}$, indicating a strong interaction. The onset of BEC is accompanied by a significant narrowing of the ${n}^{\ensuremath{\star}}(\mathbf{k})$. The narrowing accounts for 65% of the drop in kinetic energy below ${T}_{\ensuremath{\lambda}}$ and reveals an important coupling between BEC and k>0k>0k>0 states. The experimental results are well reproduced by path integral Monte Carlo calculations.
We present neutron scattering measurements of the atomic momentum distribution n(k)n(\mathbf{k})n(k) in solid helium under a pressure p=41 barp=41\text{ }\mathrm{bar}p=41 bar (molar volume ${V}_{m}=20.01\ifmmode\pm\else\textpm\fi{}0.02\text{ }\text{ }{\mathrm{cm}}^{3}/\mathrm{mol}$) and at temperatures between 80 and 500 mK. The aim is to determine whether there is Bose-Einstein condensation (BEC) below the critical temperature, Tc=200  mK{T}_{c}=200\text{ }\text{ }\mathrm{mK}Tc​=200  mK, where a superfluid density has been observed. Assuming BEC appears as a macroscopic occupation of the k=0k=0k=0 state below Tc{T}_{c}Tc​, we find a condensate fraction of ${n}_{0}=(\ensuremath{-}0.10\ifmmode\pm\else\textpm\fi{}1.20)%$ at T=80  mKT=80\text{ }\text{ }\mathrm{mK}T=80  mK and ${n}_{0}=(0.08\ifmmode\pm\else\textpm\fi{}0.78)%$ at T=120  mKT=120\text{ }\text{ }\mathrm{mK}T=120  mK, consistent with zero. The shape of n(k)n(\mathbf{k})n(k) also does not change on crossing Tc{T}_{c}Tc​ within measurement precision.