High energy neutron recoil scattering from liquid 4He

Volume 126, number 5,6

PHYSICS LETTERS A

11 January 1988

HIGH ENERGY NEUTRON RECOIL SCAUERING FROM LIQUID 4He R.S. HOLT, L.M. NEEDHAM’ and M.P. PAOLI RutherfordAppleton Laboratory, Neutron Division, Chilion, Didcot, Oxon. OXII OQX, UK Received 5 October 1987; revised manuscript received 9 November 1987; accepted for publication 12 November 1987 Communicated by D. Bloch

The neutron recoil scattering from liquid 4He at 4.2 and 1.6 K has been observed for a momentum transfer of 150 A I using the electron volt spectrometer (eVS) on the pulsed neutron source ISIS. The experiment yielded mean atomic kinetic energy values = 14.8±3Kat4.2 K and
There is considerable interest in studying the quantum properties of liquid 4He because of the occurrence of a superfluid phase below 2.17 K (T

and 1.6 K were measured with this technique in order to determinethe mean kineticenergies of the He

2).

atoms above and below the 2 point and to look for evidence of the Bose condensate. The basis of many ofthe experiments on liquid He is the impulse approximation which states that for measurements at high momentum transfer the scattering function depends only on the initial atomic state with the final state bning considered as free [2—5].Within this approximation the incoherent neutron scattering function S( Q, ) is related to the momentum distribution n(p) of the target atoms of mass M by 2 Q2 hQ.p\ h S( Q, ~)= j n(p) dp / —k--), (1)

London [1] postulated that the presence of this superfluid phase was associated with a Bose condensation of the He atoms into the zero momentum ground state. Hohenberg and Platzman [2] suggested that the fraction of particles in this ground state, the Bose condensate fraction, n0, could be measured using high momentum transfer inelastic neutron scattering. At large momentum transfers, Q, the impulse approximation, which assumes that the atoms scatter as though their final states were free, becomes valid. measuredrelated neutron scattering function S( Q, E) isThe thendirectly to the ground state momentum distribution of the recoiling atoms n(p).

With the advent of accelerator based pulsed neutron sources high fluxes of epithermal neutrons are nowavailable which allow measurements to be made at sufficiently high Q to satisfy the impulse approximation. The experiments reported here have used the electron volt spectrometer (eVS) on the spallation neutron source ISIS. The eVS is specifically designed to exploit neutrons with energies in the range 1—20 eV by using the sharp neutron absorption resonances to monochromate the scattered neutron beam. The scattering functions of liquid 4He at 4.2 Permanent address: Clarendon Laboratory, University of Oxford, Parks Road, Oxford, UK.

~.

—

where ~is the energy loss of the neutron. The delta function represents the conservation of energy and momentum in the scattering process. The recoil scattering function is centred at the recoil energy, given by Er = h 2Q 2/2M, and the width is proportionalto Q. Above the lambda transition S(Q, ) should reflect the momentum distribution of particles in the normal fluid. Hohenberg and Platzman pointed out that below the lambda point the distribution corresponding to the normal fluid should still persist but that the condensatefraction would appear as a spike on top of this distribution and broadened by a small “lifetime” effect. At low momentum transfer the effects of final state interactions and atomic interferenceon the broadening are important [6—11]. Indeed

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the majority of experimental work in determining recoil spectra for 4He carried out on reactor and pulsed neutron sources have been undertaken in the low momentum transfer regime, where Q values be-

11 January 1988

He cryostat

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low 20 A have been used [12—151.Data analysis of these results have therefore been complicated and have included corrections to the impulse approximation to take account of final state interactions. In the present set of experiments Q values in

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excess of 150 A have been obtained which yield results which are less influenced by final state interactions and atomic interference and where the impulse approximation is applicable. The mean atomic kinetic energy can therefore be calculated directly from the recoil width at a constant Q ~(~ given by 21f1\_4E ‘K.E~

lOm - I 11.6m I Fig. 1. Schematic illustration ofthe resonance detector system on the electron volt spectrometer (eVS) on the pulsed neutron source ISIS. This indirect geometry spectrometer is based on the resonance neutron absoiption and prompt gamma emission to define the scattered neutron energy. Time-of-flight techniques are used to determine the incident neutron energy.

The variation in between the normal fluid and the superfluid is small and arises mainly from the Bose condensation rather than from a change in ternperature since the zero point motion of the atoms dominates, The eVS, described in detail elsewhere [16], has been designed to use the intense epithermal neutron flux available from ISIS and is based on neutron resonance absorption by nuclei in the 1—20 eV region. The resonance energy defines the scattered neutron energy; the incident energy is determined by time-offlight (TOF). Recoil scattering measurements were undertaken with the resonance detector spectrometer (RDS) section of the eVS which uses a bismuth germanate detector (BGO) to observe the prompt gamma-rays emitted by the neutron resonance capture. The BGO detector consists of a cylindrical scintillator crystal 5 cm long and 5 cm in diameter. The RDS geometry for recoil scattering from 4He is shown schematically in fig. 1. The scattering angle was 1750 and the incident and final flight paths were 12.6 m and 1.0 m respectively. The 6.67 eV resonance in a uranium foil, 0.02 5 mm thick, was used to define the final neutron energy. This energy taken together with the large scattering angle defines the Q value in excess of 150 A—i and an incident neutron energy of 18.7 eV at the

below a 10 litre main helium reservoir. The sample cell can be isolated to maintain constant sample density and cooled further by pumping on the 300 cm3 pot. The outer tail of the aluminium cryostat was thinned to reduce scattering. The temperature ofthe sample cell was determined using both a calibrated carbon resistance thermometer and by a digital pressure gauge attached to the cell. For both measurements the temperature was maintained to within ±0.1 K throughout the duration of the measurements. Measurements were made with and without the analysing foil (the latter gives the background spectrum) and at two temperatures, 4.2 K and 1.6K. Approximately equal beam times were assigned to each data set. A neutron monitor in the incident beam was used to normalise the data. All sets of data were collected in 0.25 ~.tstime channels. Before an accurate value of the mean kinetic energy can be obtained a careful analysis must be carned out of the various components which contribute to the energy transfer resolution. The main component arises from the energy width of the resonance absorption. The width was measured to be 133 meV (FWHM) by placing the uranium analyser foil directly in the main neutron beam. The energy transfer resolution component from this width is magnified

recoil peak. The 4He sample was maintained in a purpose built cryostat designed at the Clarendon Laboratory, Oxford. The 10 cmx 5 cmx 1 cm sam-

due to the dispersive nature of recoil scattering. The recoil energy is a function of Q and an uncertainty in the final neutron energy leads to a larger uncer-

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ple cell was filled via a 300 cm3 pot which was itself

Volume 126, number 5,6

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11 January 1988

tainty in the excitation energy. Such an effect is sim-

scattering function at constant 0 was also evaluated

ilar to the resolution considerations made when measuring dispersion relations such as those for

using a more accurate computer generation of S0(Q, a). The predicted widths were very similar for both calculations. The observed distributions forrepresents liquid He are shown in fig. 2. recoil The continuous line the best gaussian fit to the data which gives values for of 14.8±3.0 K (at T=4.2 K) and 14.6±3.2 K (at T= 1.6 K). These values are in agreement with those of Sears [20] of 15.3 K (at T= 4.2 K) and 13.8 K (at T= 1.6 K) derived from a collection of experirnental data obtained at lower Q. They are also

phonons. This amplification is largest for light atoms and high scattering angles. The4He effect, therefore, is at 1750 and the large for recoil scattering from analyser energy width gives rise to a resolution cornponent with FWHM=417 meV. Another variation in Q for the observed scattering arises from the spread in the scattering angle for the measurement. The latter occurs because of the finite sizes of the moderator, sample and detector foil. Such time-focussing effects have already been studied in detail for recoil scattering experiments [17,18]. For the current spectrometer geometry the secondary flight path was large and the scattering angle close to

Momentum

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backscattering giving rise to a minimal time focussing resolution component. Adding in the remaining components to the resolution, including the small

time-focussing contribution, moderator pulse length and flight path uncertainties, gives a total resolution of 442 meV (FWHM). This compares with an expected recoil width of 560 meV for a mean atomic kinetic energy of 15 K. The validity of these contributions tothe energy transfer resolution were checked

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_______________________________________

from other recoil measurements carried out on graphite, lead and lithium. An indirect geometry spectrometer, like eVS, determines the scattering function S( Q, a) at constant O rather than constant Q. The method of Hilleke et al. [15] was therefore adapted to the analysis of an indirect geometry spectrometer which gives rise to a

correction factor of the form (3) where a( 0) and a( Q) represent the standard deviations for the distributions at constant angle and constant momentum transfer, respectively, and where C(0) is given by

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4He at (a) 4.2 K and (b)

k0, k1 arerespectively, the incidentand andMscattered neutron wavevectors, is the mass of the re-

Fig. Recoil scattering spectra from 1.6 K2. obtained using a uranium analysing foil at roomtempera-

coiling atom. This is a first order correction which assumes that S 9( Q, a) is a gaussian. The width ofthe

atoms can be estimated.

ture. Also shown are the least squares fit to a gaussian momentum distribution from whichthe meankinetic energy ofthe helium

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PHYSICSLETTERSA

slightly lower than the experimental results of Ikeda and Watanabe [19] who obtained values for (KE> between 15 and 21 K using a similar resonance technique over a similar Q range. At the present level of experimental accuracy it is not possible to distinguish differences in the line shape between the two recoil spectra, i.e., no condensate peak is observed. Indeed, computer simulations of the normal and superfluid components suggest that measurements with a much greater statistical accuracy are required before a distinct condensate peak would be visible above the normal distribution. A careful analysis of the recoil data at T 1.6 K and the simulated spectra for condensate fractions of 0%, 20% and 40% (as shown in fig. 3), assuming an average value for the kinetic energy of 14.6 K, indicates that the standard deviation on a measurement of n0 is approximately 12%. Such an upper limit is in reasonable agreement with the values normally quoted in the literature and which vary between 9% and 12%. This is in contrast with the results of Ikeda and Watanabe [19] where a value of 30% was determined by a two term fit to the data using n0 and as variable parameters. Such an approach was not adopted here. Indeed a recent calculation by Olintopotential [21] based angiven analysis of liq4He chemical dataonhas a value of uid flo( T= 0) as low as 6.2%. This is approximately 50% Momentum Transfer IA1 I 150

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11 January 1988

lower than the values given above and indicates that data with greater statistical accuracy and higher resolution are required before the Bose condensate fraction can be accurately determined by inelastic neutron scattering techniques. We would like to thank Mr. Bill Turner and members of the Clarendon workshop for the construction of the helium cryostat. We would also like to thank Dr. M.J.M. Leask, Professor E.W.J. Mitchell, Professor P. Sokol and Dr. A.D. Taylor for their interest and support.

References [1] F. London, Superfluids, Vol. 2 (Wiley, New York, 1954). [2] P.C. Hohenberg and P.M. Platzman, Phys. Rev. 152 (1966) [3] P.M. Platzman and N. Tzoar, Phys. Rev. B 30 (1984) 6397. [4] B. Tanatar, G.C. Lefever and H.R. Glyde, Physica B 136 (1986) 187, and references therein. [5] J.M.F. Gunn, C. Anreani and J. Mayers, J. Phys. C 19(1986) L835. [6] H.A. Gerschand L.J. Rodnguez, Phys. Rev. A 8(1973) 905. [71L.J. Rodriguez, H.A. Gersch and H.A. Mook; Phys. Rev. A 9 (1974) 2085. [8] Phys. Rev. 30 (1984) 44. [9] V.F. T.R. Sears, Kirkpatrick, Phys.B Rev. B 30 (1984) 1266.

~~ ~

[12] P. Martel, E.C. Svensson, A.D.B. Woods, V.F. Searsand R.A. Cowley, J. Low. Temp. Phys. 23(1976) 285, and references therein. [13] H.A. Mook, Phys. Rev. Lett. 51(1983)1454. [14] V.F. Seears, E.C. Svensson, P. Martel and A.D.B. Woods, Phys. Rev. Leti. 49 (1982) 279. [15] R.O. Hilleke, P. Chaddah, R.O. Simmons, D.L. Price and S.K. Sinha, Phys. Rev. Lett. 52 (1984) 847. [161 R.J. Newport, M.P. Paoli, V.T. Pugh, R.N. Sinclair, A.D. Taylor and W.G. Williams, Proc. 8th Meeting mt. Collaboration on Advanced neutron sources (ICANS-VIII),

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___________________________________________ 10

11

12

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Energy Transfer 4He LeVI at 1.6 K compared with expected recoil linespectra shape from calculated using condensate Fig. 3.the The recoil scattering fractions, n 0, of0% (—), 20% (———) and 40% ( )

376

[17]

14

[18] [19] [20]

Rutherford Appleton Laboratory, July 1984 (RAL-85-110) p.562. J.M. Carpenter and N. Watanabe, NucI. Instrum. Methods 213(1983)311. H. Rauch, S. Ikeda and N. Watanabe, Instrum. MethS. and N. Watanabe, Phys. Lett.Nucl. A 121(1987) 34. odsIkeda 224 (1984) 469. V.F. Sears,Phys. Rev. B 28 (1983) 5109.

[21] A.C. Olinto, Phys. Rev. B 35 (1987) 4771.

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