The dispersion relation of Landau elementary excitations and the thermodynamic properties of superfluid 4 He
H. Godfrin, K. Beauvois, A. Sultan, E. Krotscheck, J. Dawidowski, B. Fak, J. Ollivier
aa r X i v : . [ c ond - m a t . o t h e r] D ec The dispersion relation of Landau elementary excitationsand the thermodynamic properties of superfluid He H. Godfrin, ∗ K. Beauvois,
1, 2
A. Sultan,
1, 2
E. Krotscheck,
3, 4
J. Dawidowski, B. F˚ak, and J. Ollivier Univ. Grenoble Alpes, CNRS, Grenoble INP † , Institut N´eel, 38000 Grenoble, France † Institut Laue-Langevin, CS 20156, 38042 Grenoble Cedex 9, France Department of Physics, University at Buffalo, SUNY Buffalo NY 14260, USA Institute for Theoretical Physics, Johannes Kepler University, A 4040 Linz, Austria Comisi´on Nacional de Energ´ıa At´omica and CONICET,Centro At´omico Bariloche, (8400) San Carlos de Bariloche, R´ıo Negro, Argentina (Dated: December 17, 2020)The dispersion relation ǫ ( k ) of the elementary excitations of superfluid He has been measured atvery low temperatures, from saturated vapor pressure up to solidification, using a high flux time-of-flight neutron scattering spectrometer equipped with a high spatial resolution detector (10 ‘pixels’).A complete determination of ǫ ( k ) is achieved, from very low wave-vectors up to the end of Pitaeskii’splateau. The results compare favorably in the whole the wave-vector range with the predictions ofthe dynamic many-body theory (DMBT). At low wave-vectors, bridging the gap between ultrasonicdata and former neutron measurements, the evolution with the pressure from anomalous to normaldispersion, as well as the peculiar wave-vector dependence of the phase and group velocities, areaccurately characterized. The thermodynamic properties have been calculated analytically, develop-ing Landau’s model, using the measured dispersion curve. A good agreement is found below 0.85 Kbetween direct heat capacity measurements and the calculated specific heat, if thermodynamicallyconsistent power series expansions are used. The thermodynamic properties have also been cal-culated numerically; in this case, the results are applicable with excellent accuracy up to 1.3 K, atemperature above which the dispersion relation itself becomes temperature dependent. I. INTRODUCTION
One of the most fundamental properties of a many-body system is the dispersion relation ǫ (k) of its elemen-tary excitations , i.e., the dependence of their energyon their wave-vector. The prediction by Landau ofthe phonon-roton spectrum of the excitations in super-fluid He, the canonical example of correlated bosons,has paved the way for the development of several areasof modern physics, like Bose-Einstein condensation, su-perfluidity, phase transitions, quantum field theory, coldatoms, cosmology and astrophysics.In the first version of his theory, published in 1941,Landau assumed that phonons and rotons had two sep-arate dispersion relations; he corrected this idea in the1947 paper, where he reached the conclusion that heliumwas described by a single dispersion curve. The evolutionof the dispersion relation from the quadratic law of in-dependent atoms to the sophisticated form proposed byLandau is a spectacular example of emergent physics.At low wave-vectors, the phonon linear dispersion pro-gressively builds up as the interactions are switched on,as shown by Bogoliubov . At atomic-like wave-vectors,a roton minimum appears, which is the signature ofthe hard core and strong interactions, as solidificationis approached . Excitations created from the super-fluid condensate, let’s name it ‘the Vacuum’, have thecharacteristics of waves and identical particles .The dispersion relation ǫ (k) has been directly ob-served in He by measurements of the dynamic struc-ture factor S ( Q, ω ) using inelastic neutron scatteringtechniques , fully confirming Landau’s prediction. Substantial theoretical work has been devoted to thedescription of the single-excitations dispersion of super-fluid He, a simple Bose system where the atomic in-teraction potential is well known. Variational, Monte-Carlo, and phenomenological approaches have broughtvaluable contributions to the present understanding ofhelium physics , but many important questionsremain open.Superfluid helium also has important applications inexperimental physics. In particular, the properties ofphonons and rotons are exploited in quantum measure-ments at the nanoscale level and in detectors for par-ticle physics .In two previous articles we investigated in detailthe multi-excitations of superfluid He. Here we provideexperimental results on single-excitations in the wholedynamic range where they are well defined, and we com-pare them to the predictions of recent dynamic many-body theory (DMBT) calculations . In the secondpart of the manuscript, starting from the measured dis-persion curve at saturated vapor pressure, we calculatethe thermodynamic properties analytically and numeri-cally. The results are compared to high accuracy thermo-dynamic data. Tabulated values are provided for differ-ent usual parameters (see also Supplemental Material at[URL will be inserted by publisher] for additional tables).
II. PREVIOUS WORKS
The measured phonon dispersion relation of He isshown in Fig. 1; it closely resembles the curve predictedby Landau : the linear (‘phonon’) part at low wave-vectors is followed by a broad maximum (‘maxon’) atwave-vectors k ∼ − and a deep (‘roton’) minimum (‘ro-ton gap’) at k ∼ − . The dispersion curve becomes flatfor k ≥ − as the energy reaches twice the roton gap,forming ’Pitaevskii’s plateau’ . R E ne r g y ( m e V ) Wave-vector (¯ -1 ) RM FIG. 1. The dispersion relation of He at P=0 andT < > − (this work) . Below this value: extrapolation ofultrasonic data . Error bars are not visible at this scale inmost of the range (see Table VI). ∆ M and ∆ R are the maxonand roton energies. In the long wavelength limit, explored by ultrasonictechniques, the deviations from linearity are describedby the expression ǫ ( k ) ≈ ~ ck (1 − γk ) (1)where c is the speed of sound, and γ is the phonondispersion coefficient . The general shape of the Lan-dau spectrum suggests that γ >
0, but this was foundto be inconsistent with experiments. The latter foundan explanation with the suggestion made by Maris andMassey that the dispersion is anomalous ( γ <
0) atlow pressures. This effect attracted considerable atten-tion both from the theoretical and experimental pointsof view. Phonon damping due to 3-phonon processes, forinstance, is then allowed up to a critical wave-vector k c .The dispersion becomes ‘normal’ at high pressures, nearsolidification. Details can be found in a critical reviewby Sridhar .Thermal phonons at usual temperatures involve muchhigher wave-vectors. Going from macroscopic to atomicwavelengths is obviously a challenge, which has beentaken up by neutron scattering. A. The long wavelength limit
Deviations from the linear dispersion relation are oftendescribed by a polynomial expansion of the excitationenergy in powers of the wave-vector modulus k : ǫ ( k ) = ~ ck (cid:0) α k + α k + α k + ... (cid:1) (2)where α = - γ and α is assumed to be zero.A different type of expression, frequently usedin the analyis of experimental data, is the Pad´eapproximant ǫ ( k ) = ~ ck (cid:18) − γk − k /Q a k /Q b (cid:19) . (3)Its series expansion does not contain the term α .Microscopic theory, in fact, suggests a different descrip-tion of the low- k regime. Starting from Bogoliubov’s for-mula (see the discussion in Ref. 35), we derive the simpleexpression ǫ ( k ) = ~ ck p d k + d k + ... (4)which is physically correct in the Feynman limit.Comparing its power series expansion with Eq. 2 showsthat α =0, γ =- d /2, and α = d /2. Since higher orderterms are generated in the expansion, it is interesting tosee if Eq. 4 can describe the experimental data, eventu-ally with a smaller number of parameters.The term α has been calculated analytically fromthe asymptotic form of the microscopic two-body inter-action. For V ( r ) = C r − , α = π ρm c C (5)where m is the mass of a He atom, and ρ the densityof the liquid. At saturated vapor pressure, this estimategives α =-3.34 ˚A (see Refs. 37, 40, 42, and 43). Thepseudo-potential theory of Aldrich and Pines pro-vides a similar estimate, α =-3.7 ˚A with a value for thedispersion coefficient, γ ≈ -1.5 ˚A , consistent with exper-iments.The experimental determination of the dispersion rela-tion at long wavelengths has been attempted by differenttechniques. The speed of sound is known in the wholepressure range: ultrasound measurements at very lowtemperatures yield c=238.3 ± c = q ρκ , where κ is the isothermal compress-ibility and ρ the density, one can obtain the pressure de-pendence of the density by measuring the sound velocityas a function of pressure. Abraham et al. found thatthe expressions P = A ( ρ − ρ ) + A ( ρ − ρ ) + A ( ρ − ρ ) (6)and c = p A + 2 A ( ρ − ρ ) + 3 A ( ρ − ρ ) . (7)accurately describe their results. A fit of their data yieldsthe coefficients A =5.679 10 bar cm g − (correspond-ing to c = 238.3 m/s), A =1.1115 10 bar cm g − , andA =7.43 10 bar cm g − . Here ρ =0.14513 g/cm is thedensity at P=0 (see Ref. 45 and references therein). Thesound velocity is almost linear as a function of density,and one can use the expression c = c + c ( ρ − ρ ) + c ( ρ − ρ ) . (8)where c =238.3 ± c =4671.0 ± c =496 ±
45 forvelocities in m/s and densities in g/cm .Ultrasonic measurements are accurate in the determi-nation of the pressure dependence of the sound velocity,but there are some uncertainties in the way the referencevelocity c =238.3 ± . It is therefore interesting to com-pare the ultrasonic values with those obtained by othertechniques.Tanaka et al. measured the molar volume of pure liq-uid He at very low temperatures as a function of pres-sure. We obtain from their data the velocity of sound,either by derivation of the polynomial of order 9 givenby Tanaka et al. or by derivation of their data in a smallrange around the relevant pressures, using the compress-ibility: c = V m / ( m ∂V m /∂P ), where m is the atomicmass of He (4.0026032 g/mol). The molar volume V at P=0 is 27.5793 cm /mol, and the number density0.021836 atoms/˚A .In the pressure range from 0 to 15 bar, the sound ve-locities determined from the compressibility are system-atically below the ultrasonic values, but they agree withthe latter within 0.7 m/s. At higher pressures (partialdata are given in Ref. 47 up to the melting pressure),we find a strong deviation of the sound velocity (up to3 m/s) from an almost linear density dependence, result-ing from a small systematic error in the molar volumedata above 15 bar, as can be seen by comparing themto the results of Abraham et al. (a useful formula forV m (P) is given by Greywall ).Anomalous dispersion in superfluid He was observedby Phillips using heat capacity techniques. The re-sults have been extended to lower temperatures byGreywall , motivated by discrepancies observed be-tween heat capacity and neutron scattering data. Sincethe heat capacity is obtained as an integral of the dis-persion curve over a substantial range of wave-vectors,extracting the dispersion curve from data is not a uniqueprocedure (this point will be discussed in detail in sectionVIII). The velocity of sound determined by Greywall atlow temperatures from the coefficient of the T term inthe heat capacity, in addition, is affected by uncertain-ties in the thermometry . The values of c from heatcapacity are lower than the ultrasonic ones, and less accu-rate, but their density dependence is similar. Thermom-etry calibration improvement reduced the values of the sound velocities by about 3 to 6 m/s for increasing pres-sures, which is an indication of the typical uncertaintiesin heat capacity data. Paradoxically, uncorrected datawere closer to the ultrasonic results.Accurate measurement of the deviations from linearityof the phonon dispersion have been made by Rugar andFoster . Ultrasonic measurements at two fundamentalfrequencies showed that α < − ˚A at P=0 and 6.3 bar,and α = (1.56 ± at SVP. If α is assumed to bezero, then α =(1.55 ± at SVP. The excitationspectrum is probed for k < − . Their analysis isinsensitive to assumed values of α , α , etc., and it is onlyslightly sensitive to the value of α , which is taken fromtheory . The density dependence of α , measuredfrom SVP to 10 bar, is almost linear. The data agreewell with former results of Junker and Elbaum on thetemperature dependence of the ultrasonic velocity, whichreach higher pressures (about 15 bar).We shall use the ultrasonic values in the following,since they are the most accurate, and confirmed (butonly within about 0.7 m/s) by other measurements. Thevalues obtained in the present work will be compared tothese data in Section VII.The experiments described above provided a good de-scription of the dispersion relation for small wave-vectors,and convincing evidence of anomalous dispersion for pres-sures below about 20 bar was progressively gathered. Toachieve a direct observation of the dispersion curve andexplore the dynamics at atomic wave-vectors, the privi-leged tool is inelastic neutron scattering. B. Previous neutron scattering results
Previous neutron scattering data have been describedin detail by Glyde in a book and a recent reviewarticle . Tables of the properties of liquid helium havebeen published by Brooks and Donnelly and by Don-nelly and Barenghi ; neutron scattering data from a va-riety of sources and smoothed values are provided. Orig-inal references should be consulted, however, for errorbars.The quantitative knowledge of the dispersion rela-tion is based on measurements by Cowley and Woods ,Woods et al. , Svensson et al. , Stirling et al. and others mainly performed on triple-axis spec-trometers. The different data sets are not totally com-patible, and the dispersion relation which emerges fromthese studies is therefore not fully satisfactory.The main advantage of triple-axis spectrometers istheir good accuracy in the determination of energies andwave-vectors. This point-by-point measuring techniqueis time consuming, and therefore not appropriate to in-vestigate the whole wave-vector range.The small-k phonon region was studied, pushing thetechnique to its limits, to explore a possible anomalousdispersion. The first results were highly specula-tive and at best qualitative, since error bars growing atlow wave-vectors precluded a thorough comparison to ul-trasonic sound velocity measurements. Higher accuracymeasurements performed by Stirling et al. finallyconfirmed the anomalous character of the dispersion atlow pressures. However, these measurements showed asystematic disagreement with ultrasound measurements,which will be discussed further below.Time-of-flight spectrometers (TOF) with large detec-tor arrays allow measurements of dispersion curves over alarge range of energies and wave vectors simultaneously.Early experiments by Dietrich et al. and Stirling et al. were followed by more recent measurements on IN6 atthe ILL by Stirling, Andersen, and coworkers . Twonew data sets with an energy resolution of about 100 µ eVwere obtained through the latter works, referred to as‘Andersen’ and ‘Gibbs’ .A good agreement with triple-axis data was found inthe roton and the maxon regions, but strong deviationswere observed both at low and high wave-vectors. IN6shares with triple axis spectrometers the use of graphitemonochromators (3 focusing ones), thus complicatingthe resolution function shape, and significant correctionsfor sample absorption or off-center sample position wereneeded in the data analysis.Additional measurements were performed at ISIS onthe IRIS time of flight inverted-geometry crystal ana-lyzer spectrometer, with an excellent energy resolutionof 15 µ eV, but a coarse wave-vector resolution . An im-portant result was obtained at high wave-vectors, show-ing that the single-excitation dispersion curve is slightlybelow twice the roton energy . In this case, where thedispersion is flat, the resolution characteristics of IRISconstituted a major advantage.Measurements by Pearce et al. on the same instru-ment, mainly around the roton energy, showed discrep-ancies with former works, in particular in the magnitudeof the temperature dependence of the roton parametersdetermined at ILL’s IN10 backscattering spectrometer with an energy resolution better that 1 µ eV.It was difficult to decide which set, among these partlyconflicting TOF data, was correct. The potential of theTOF technique motivated the present studies on IN5. III. EXPERIMENTAL DETAILS
The cylindrical sample cell was made out of 5083 alu-minum alloy, selected because of its good mechanical andneutron scattering properties. The minority chemicalconstituents (4.4% Mg, 0.7% Mn, 0.15% Cr, etc.) havea modest effect on the neutron scattering and absorp-tion cross-sections compared to the values for pure alu-minum, with an increase of less than 15% of the totalcross-section. The gain in mechanical properties allowsreducing the thickness in a much larger proportion, by afactor of 3. High pressure studies could be made usinga thin cell, of 1 mm wall thickness, for pressures up to24 bar. The cell had a 15 mm inner diameter, which is smallcompared to the 30 to 50 mm diameters used in otherworks. Cadmium disks of 0.5 mm thickness were placedinside the cell every 10 mm, to reduce multiple scattering.This was not needed for the present studies, and it evenhad an undesirable effect, reducing the signal on someneutron detectors placed far from the sample horizontalplane. We did not place Cd masks on the sides of thecell; preserving the cylindrical geometry turned out tobe favorable for the data analysis.High purity (99.999 %) helium gas was condensed inthe cell at temperatures on the order of 1 K. The stain-less steel gas-handling system consisted of a set of highquality valves, tubes and calibrated volumes. The gaswas admitted through a “dipstick”, placed in a heliumstorage dewar, which was used to purify, condense andpressurize the helium sample.The dispersion relation of helium is very sensitive tothe applied pressure. For this reason, pressures in the sys-tem were measured with a high accuracy 0-60 bar Digi-quartz gauge, located at the top of the cryostat. Thisgauge has a precision of 6 mbar, but the pressures insidethe cell are known only within 20 mbar, due to heliumhydrostatic-head corrections. The corrected pressures inthe cell are given in Table I.
Helium samplesNominal P (bar) 0 0.5 1 2 5 10 24Corrected P (bar) 0 0.51 1.02 2.01 5.01 10.01 24.08TABLE I. Nominal and corrected values for the pressures in-vestigated in the present work. The estimated uncertainty is < The cell was carefully centered in a dilution refrigeratorproviding temperatures well below 100 mK. The thermalconnection to the mixing chamber was achieved by us-ing massive OFHC-copper pieces. Sintered silver powderheat exchangers placed at the top of the cell provided agood thermal contact between the cell and the heliumsample. Two long, small diameter, Cu-Ni filling capillar-ies were used in parallel, for safety. They were thermallyanchored along the dilution unit, insuring a negligibleheat leak to the cell. Thermometry was provided by cal-ibrated carbon and RuO resistors.Measurements were made for a vanadium sample (arolled foil, mass 9.81 g, external diameter 12 mm, height60 mm, used for the detectors efficiency calibration), forthe empty cell, and then for the cell filled with He atseveral pressures (see Table I). The helium measurementswere performed at temperatures below 100 mK. The dataacquisition consists in several runs of one hour duration.The longest measurements were made at P=0 (9h) andP=24 bar (6h). Two hours runs were made at all otherpressures. The empty cell signal, measured for 10h, wasused as background and subtracted from all the heliummeasurements.
IV. INELASTIC NEUTRON SCATTERINGA. Inelastic neutron scattering equations
The quantity measured by a neutron spectrometer is the double differential scattering cross section per tar-get atom, which is proportional to the dynamic structurefactor: ∂ σ∂ Ω ∂E f = b c ~ k f k i S ( Q, ω ) (9)where b c is the bound atom coherent scattering length.The incident neutron has an initial energy E i and a wave-vector ~k i , leaving the sample with a final energy E f anda wave-vector ~k f ; the wave-vector transfer is ~Q = ~k i − ~k f ,and the energy transfer ~ ω = E i − E f .The wave-vector transfer is written in terms of the scat-tering angle ϕ between ~k i and ~k f : Q = k i + k f − k i k f cos ϕ (10) Q = 2 m n ~ h E i − ~ ω − p E i ( E i − ~ ω ) cos ϕ i (11)The number of neutrons detected as a function of thescattering angle ϕ and the energy transfer yields S ( Q, ω )through Eq. 9.At zero temperature there are no thermal excitations,and the only allowed process is the creation of excitations.When a single-excitation of energy ǫ and wave-vector ~k iscreated, conservation of energy and wave-vector leads to ǫ = ~ ω and ~k = ~Q . Single-excitations on the dispersioncurve ǫ ( k ) are observed in the dynamic structure factor S ( Q, ω ) as sharp peaks.
B. The time of flight spectrometer IN5
The measurements were performed on the IN5 timeof flight spectrometer at the Institut Laue Langevin(see Fig. 2).
Radial collimatorSample Beam stopMonitor1.2 m8 mFocusing neutron guideChoppers Detectors 4 m
FIG. 2. Disk chopper time of flight spectrometer IN5.
A pulsed monochromatic beam is provided by threegroups of two choppers. A key feature is that the resolu-tion is well represented by a Gaussian function.
FIG. 3. The ( Q , ω ) space accessible for an incident neutronenergy E i = 3.52 meV, calculated from equation (11). Con-stant angle lines are shown for selected values between 0 ◦ et180 ◦ . The gray area indicates the region actually used in thepresent measurements. The neutron energy was E i = 3.52 mev for the lowwave-vector range, which gives a convenient access to ex-citations of wave-vectors k from 0.15 to 2.3 ˚A − and cov-ers the energy range between 0 and 2.22 meV as shownin Fig. 3. In the conditions of the experiment, IN5 has avery large neutron flux of φ n = 2 × neutrons/(cm s)at the sample position. The uncertainty in the incidentenergy is ≈ E i = 3.520, 5.071,7.990, and 20.45 meV, with energy resolutions (FWHM)at elastic energy transfer of 0.07, 0.12, 0.23 and 0.92 meV,respectively, for a chopper speed of 16900 rpm.A large array of He+CF position sensitive neutrondetectors (PSD) is located in a vacuum chamber whichsurrounds the sample space. Key features of the detec-tion system are its large angular coverage and resolution.The 384 detector tubes are placed at a distance of 4 mfrom the axis of the instrument. The angular positionof the tubes with respect to the direction of the neu-tron beam is given by the ‘detector angle’ ϕ , coveringthe range from -12 ◦ to 135 ◦ . The tubes are straight andlong, their vertical range goes from -1.47 to +1.47 m. ThePSD system provides 241 ‘pixels’ of small size (26 × ) per tube, characterized by their position (angle ϕ ,height z ) in the detector surface. The pixels correspond-ing to the same Debye-Scherrer cones, i.e., at the samescattering angle ϕ (Fig. 4), are grouped by software ,resulting in 346 different scattering angles in the interval6 ◦ to 135 ◦ . The detection process is more efficient thanwith triple-axis spectrometers, where a single detectorhas to be moved over the whole angular range.The distance from each pixel to the center of theinstrument varies substantially due to the tall, verti-cal tube geometry. The Debye-Scherrer cones ‘standardprocedure’ groups the individual detector pixels intoequivalent units located at the in-plane angles ϕ andin-plane nominal distance D=4 m. ° ° ° ° ° ° ° - - - ( m ) V e r t i c a l d i s t an c e ( m ) FIG. 4. IN5 detectors set-up (drawn to scale): constant scat-tering angle curves on the detector surface of ≈ (3 ×
10) m .The horizontal positions of the pixels are characterized bythe azimutal angle ϕ of their detector tube (indicated on thecurves); vertical positions are given by the height z measuredalong the tube. The neutron arrival signal from each pixel of the PSD isread into a data acquisition system of 1024 time channelsof 6.9084 µ s duration (‘time frame’). Since the neutronvelocity is on the order of 820 m/s, the time of flight overthe 4 m instrumental distance is on the order of 4.9 ms,or 700 channels. The ‘time origin’ of the data acquisitionis set in such a way that both the elastic peak and thehelium excitation peak are measured within the sametime frame, as shown in Figs. 5 and 6. Maxonpeak
Scattering angle 54.6(cid:176) 102.5(cid:176) N o r m a li z ed neu t r on c oun t s ( a r b . un i t ) Time ((cid:181)s)Elastic peaks Rotonpeak
FIG. 5. Time of flight measurement at an incident neutronenergy E i =3.52 meV, for scattering angles of 54.6 ◦ (near themaxon), and 102.5 ◦ (near the roton). Note that the elasticpeaks of these signals are slightly shifted. V. STANDARD DATA REDUCTION
Standard data-reduction was initially used tocalculate from the raw data the dynamic structure factor S ( Q, ω ) of broad multi-excitations. The ‘standard anal-ysis’ data consist of time of flight spectra (matrix of thenumber of counts for the 1024 time channels for 346 an-gles ϕ ): the raw data from the detectors pixels have been grouped by scattering angle ϕ , as described above. Inthis section V, therefore, ϕ represents the scattering an-gle of an effective detector located in the horizontal plane,at z =0 and ϕ = ϕ (‘in-plane effective description’). Thevery narrow single-excitations, however, require a moresophisticated ‘pixel-by-pixel analysis’, described in sec-tion VI, where the same raw data are processed, but theTOF data (1024 time channels) of the 384 ×
241 detectorpixels are analyzed individually. C oun t s ( a r b . un i t s ) time ((cid:181)s) Q=0.20 ¯ -1 FIG. 6. Time of flight measurement at an incident neutron en-ergy E i =3.52 meV, for a scattering angle of 26 ◦ (Q=0.20 ˚A − )corresponding to the phonon region. A. Time of flight (TOF) equations
We first proceed to fit the spectra to determine veryaccurately the time of arrival at the detectors of elas-tically scattered neutrons, t elast measured, as describedabove, using the system clock times. The ‘elastic peaks’(see Fig. 5) can be approximated by simple Gaussians.Since superfluid helium does not scatter elastically, weuse the signal of the aluminum cell, and compare it tothat of the vanadium sample.The equation τ = t elast − t s = D/v i for the neutronflight over the distance D separating the sample from thedetectors, determines the important parameter t s , thetime of scattering at the sample, according to the systemclock. This supposes that the detectors are located atthe same distance of the sample, which is often a goodapproximation.The energy of the excitations is determined from themeasurement of the time of flight τ of inelastically scat-tered neutrons, over the same distance D . A neutroncreating an excitation of energy ǫ reaches a detector lo-cated at an angle ϕ at a time t inel ( ϕ ), obtained from thegaussian fits of the ‘helium peaks’ (see Fig 5). The timeof flight is now τ ( ϕ ) = t inel ( ϕ ) − t s . The final velocity ofthe neutron v f = D/τ provides the neutron final energy E f . The energy of the excitations is ǫ = E i − E f , wherethe initial energy of the neutrons E i is known from themechanical characteristics of the choppers system. Theexcitation wave-vector k is obtained from equ. 11.In this simple scheme, there are only two indepen-dent instrumental parameters, selected among the initialneutron energy E i , the average sample-detector distance D av , and the average time of arrival of the neutrons atthe sample position, t avs . The energy of the excitationsis obtained from the equation: ǫ ( ϕ ) = 12 m n D av (cid:20) t elast ( ϕ ) − t avs ) − t inel ( ϕ ) − t avs ) (cid:21) (12)where m n is the neutron mass, and t avs = t avelast − D av /v i .A more convenient form can be used when the sample-detector distances differ by a significant amount: ǫ ( ϕ ) = E i " − (cid:18) t elast ( ϕ ) − t s t inel ( ϕ ) − t s (cid:19) (13)The dependence on the initial energy E i is made ex-plicit. The distances D ( ϕ ) to all individual detectors donot appear. Instead, we find the inelastic and elastictimes for each angle ϕ , which are the measured parame-ters. Last but not least, one has to determine t s . In ex-periments using a small diameter cylindrical sample witha small absorption, like in the present case, this time ofarrival at the sample is very well defined, and unique: itdoes not depend on ϕ .The analysis yielding the energies ǫ ( ϕ ) involves onlytwo instrumental parameters : the initial neutron energy E i , and the neutron arrival time t s at the sample, cen-tral to the present discussion, which can be estimated us-ing the nominal sample-detector distance of IN5, 4.00 m.The actual flight distances in the sample plane should beclose to this value, but they can be significantly affectedby other effects. For instance, the analysis assumes thatthe elastically scattered neutrons follow the same flightpath as the inelastically scattered ones reaching a givendetector. This is not true if absorption plays an impor-tant role; it introduces, in addition, undesired angularshifts. Correcting for systematic errors, fortunately, canbe done as shown in the next section. B. Distance and angle corrections
Distance and angular corrections arise from imperfec-tions in the instrument and sample geometries and fromthe finite size of the components. Neutron beam, sam-ple and detectors have typical dimensions on the order ofcentimeters, the instrument lengths are on the order ofmeters, thus requiring finite size optics analytical calcula-tions or computer simulations if uncertainties on the or-der of 10 − are desirable. We have used both techniques to evaluate possible effects, and retained the correspond-ing corrections, listed below, when their influence on theexcitations energies was larger than 1 µ eV.
1. Sample off-center
Corrections may be necessary if the sample is notplaced exactly at the geometrical center of the instru-ment. Large sample off-set effects were observed by An-dersen et al. . Their characteristic symptom, essen-tially a parabolic angular dependence of the elastic timesof flight, is also observed here and ascribed, however, toa very different cause, namely, a rigid-plate distortion ofthe detectors plane. Both effects are discussed below.A description of the sample off-center geometry, not toscale, is given in Fig. 7. The detectors are placed on arigid frame, forming a circle around the ‘instrument cen-ter’ O . Their distances and angles have been carefullycharacterized using theodolites. In principle the sampleis centered with respect to the cryostat, which is cen-tered with respect to the cylindrical experimental space,aligned with the detectors bank. j (cid:3) y A DL( j ) a A Neutron beam O Sample a (cid:455) Neutron beam d e t e c t o r s x FIG. 7. Instrument parameters when the sample is not cen-tered. Reference points: instrument center O , sample center A ; x and y are the coordinates of ~OA and α the angle between ~OA and the neutron beam. Distances: instrument centerto sample a , instrument center-detectors D , sample-detectors L ( ϕ ); ϕ is the nominal scattering angle, ψ the physical scat-tering angle. If the ‘sample position’ A is shifted from the geometri-cal ‘instrument center’ O (Fig. 7), the TOF distance D is replaced by L : L ( ϕ ) = p D + a − aD cos( α − ϕ ) (14)while the physical scattering angle ψ is related to thedetector angle ϕ by the expressioncos( ψ ) = ( D cos( ϕ ) − a cos( α )) /L (15) E l a s t i c a rr i v a l t i m e ( (cid:181) s ) Scattering angle (degrees) 6 mm
FIG. 8. Arrival time of neutrons elastically scattered by thealuminum cell, measured in the standard analysis as a func-tion of the scattering angle (the time origin t s is -1323 µ s), atan incident neutron energy E i =3.52 meV. The arrow on theright hand side indicates the corresponding variation of theflight distance. The elastic TOFs of the vanadium sample and the alu-minum cell both display a visible angular dependence,indicating a significant variation of the L ( ϕ ). Fig. 8shows the time of arrival of neutron elastically scatteredby the aluminum cell. The apparent dispersion observedon the data points, reproducible in different scans, corre-sponds to small differences in sample-detector distances.The arrival times at large angles display the character-istic parabolic shape of a sample position shifted withrespect to the detectors center. A fit to the data in Fig.8 using the sample-offset model t elast ( ϕ ) = p D + a − aD cos( α − ϕ ) /v i + t s (16)with D =4.00 m and v i =824.17 m/s (these valuesare not critical), would yield as off-set parametersa=19.0(5) mm and α =245.8(7) ◦ . The fit also yields t s ,but this parameter, strongly correlated to the initial en-ergy E i and the distance D , will be determined consis-tently later on. The off-set distance and angle are surpris-ingly similar to those calculated by Andersen et al. .In the present case, however, we can show that such alarge off-set is incompatible with the complete calcula-tion of the elastic TOF for our 3-dimensional detectorarray. In particular, the detectors covering positive andnegative low angles are highly sensitive to sideways dis-placements. We found that sample off-set corrections areon the order of 2 mm or less. The TOF results deter-mined with the vanadium sample are almost identical tothose described above, the corresponding differences inflight distances are again less than 2 mm. This indicatesthat the sample is very well centered inside the cryostat,and the latter within the instrument. Some small differ-ences between the vanadium and the aluminum cell data can be ascribed to their different geometry, in particularthe effect of the cadmium disks inside the cell. A ng l e c o rr e c t i on ( - ) ( deg r ee s ) Nominal scattering angle (degrees)
2D correction) 3D full correction
FIG. 9. Angular correction to the scattering angle ϕ , calcu-lated for a supposed sample position off-set (a=19.0(5) mmand α =245.8(7) ◦ ). ψ is the corrected scattering angle. Theblack line corresponds to a simple in-plane-only detectorsscheme, while the red line describes the values calculated forthe actual 3-dimensional IN5 detectors geometry. Such a cor-rection could be discarded here, but may have affected earlierworks (see text). Scattering angle corrections have also been consideredfor a case where the sample would be off-center. Theywere calculated for the present geometry, taking into ac-count the 3-dimensional positions of the detector pixels,shown in Fig. 4. Debye-Scherrer rings grouping wouldproduce a peculiar angular correction, shown in Fig. 9,if the sample had been off-center. Such a correction isnot compatible with our measured data for the disper-sion curve: it would have given visible accidents. Wehave therefore concluded that distance and angular cor-rections due to a sample center off-set are very small inthe present work.
2. Effect of strong scattering and absorption
A different correction may be caused by strong scat-tering and/or absorption in large samples. Essentially,the sample regions which are both closer to the reactorand to the detectors provide a larger contribution to thescattered neutrons flux, than those located further away.Both TOF distances and scattering angles are affected.We have calculated the effective sample center positionfor the vanadium cylindrical sample used for calibrationpurposes, and for the thin wall aluminum alloy samplecell. The relevant parameter is the ratio of the sample di-mensions to the penetration depth λ scatt = ( P n i σ i ) − .The sum runs over the scattering and absorption cross-sections (at the incident energy E i ) of the different el-ements present in the sample with number densities n i . -6 -4 -2 0x (mm)0246 y ( mm ) -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0x (mm)00.10.20.30.40.50.6 y ( mm ) FIG. 10. Displacement of the effective position of the samplecenter for the vanadium sample (used in the calibration pro-cedure) calculated for neutron absorption. The coordinatesaxes are defined in Fig. 7, and the corresponding angles areindicated along the parametric curve. Inset: magnitude ofthe effect compared to the sample size.
The apparent sample center position depends now on thescattering angle ϕ .For the vanadium sample, λ scatt =32 mm, significantlylarger that the vanadium radius of 6 mm. In this case,the calculation (Fig. 10) yields a maximum shift of lessthat 1 mm, a small effect on the distances of flight.The aluminum cell may also display a displacement ofits effective center due to scattering and absorption. Theshift, however, is even smaller. For our aluminum alloy, λ scatt =65 mm, considerably larger that the can dimen-sions (7.5 mm internal radius, wall thickness 1 mm) andthe corresponding neutron paths. The effective samplecenter position calculated for this hollow cylinder geom-etry is given in Fig. 11.Corrections ascribed to strong absorption have beenapplied by Gibbs to his TOF data. A much largerand thicker aluminum cell was used in this work; never-theless, the present work suggest that other causes aremore probable. The accuracy of these data may thus beslightly lower than initially believed.The effective sample center displacement due to strongscattering and absorption in the sample also affects thescattering angles. The calculated angular shift for thevanadium sample is very small, in particular at low an-gles, where accuracy is needed. The same remark is validfor the measurements with the experimental cell. Wecan therefore conclude that angular corrections due tostrong scattering and/or absorption are small in the con-ditions of the present experiment (small diameter, thinaluminum cell). -8 -6 -4 -2 0x (mm)02468 y ( mm ) -0.1 -0.05 0x (mm)00.050.1 y ( mm ) FIG. 11. Displacement of the effective sample center positioncalculated for neutron scattering and absorption in the thinwall cylindrical aluminum alloy sample cell. The coordinatesaxes are defined in Fig. 7, and the corresponding angles areindicated along the parametric curve. The magnitude of theeffect is much smaller than the sample size (inset).
3. Distance corrections in the detectors
Neutrons are not detected, in average, at the center ofthe detectors. Due to the strong neutron absorption of He, the detection process takes place with a short char-acteristic distance λ abs , which depends on the densityof the He gas in the detector, and on the neutron en-ergy. For the parameters of the present experiment, thepenetration length is ≈ D =4.00 m between the center of the instrumentto the center of the detector tubes. The latter have aninternal diameter D det =24.4 mm. The neutron detectionprocess occurs at an average distance ¯ y from the planeof the detector centers. At an incident neutron energyE i =3.52 meV, ¯ y ≈ ϕ for neutrons scattered from the sharp excitations on thedispersion curve of He. Corrections for this effect havebeen calculated for the detectors geometry, and appliedto the data.
4. Other corrections
A related effect is the apparent displacement of thesample center, when using cadmium shields or windowson the sides of the cell , slightly masking the heliumsample or the aluminum cell for some scattering angles.The effect is absent in the present experiment, where thecylindrical symmetry has been preserved, thus ensuringan excellent angular average.0The finite diameter of the sample can lead to distanceand angular corrections in small instruments, in partic-ular for the 30 to 50 mm diameter cells used in previousworks. These corrections are negligible for the presentwork on IN5 with a 15 mm inner diameter cell.
VI. DETAILED PIXEL-BY-PIXEL ANALYSIS
The single-excitation dispersion curve is intense andextremely sharp, and we are therefore interested inthe best resolution and accuracy both in energy andwave-vector. For this reason, we proceed now with arefined analysis using a ‘high resolution configuration’.Analysis step 1: The ‘high resolution configuration’consists of a pixel-by-pixel treatment of the multidetec-tors signals (see section IV for hardware details). Neu-trons are collected in 1024 time channels for each PSD de-tector pixel. Thanks to the very high flux of IN5, elasticand inelastic times of flight can be determined by meansof gaussian fits, for each of the 241x384 pixels in the de-tector matrix (see section IV B). The software LAMP isused to read and fit the nxs raw data files of IN5. We ob-tain about 9 × values of pairs (t elast ( ϕ , z ),t inel ( ϕ , z )),where ϕ is the angle of a detector tube and z the heightof a pixel within the corresponding tube. Angle of detector tube (degrees) D e t e c t o r p i x e l he i gh t ( c m ) -1.6-1.2-0.8-0.40.00.40.81.21.62.0 Distance shift (cm)
FIG. 12. Distortion of the detectors plane: difference betweenthe radial distance determined from the time of flight and thenominal IN5 radius (4.00 m), for all detector pixels (angle ϕ ,height z ). The elastic times yield, using at this early stage thevalue of t s obtained from the standard analysis (see Sec-tion V B), the distances of flight for each pixel. These arevisualized by representing their projections on the hor-izontal plane (i.e., the radial distances), as a 2D-arrayshown in Fig. 12, where the nominal 4.00 m have beensubtracted. The data reveal a systematic distortion ofthe detectors plane along the angular direction, confirm-ing the previous observation based on the standard analy- sis (see Fig. 8). It is now shown, in addition, that the dis-tortion is also present in the vertical direction: an undu-lation of the detector plane, similar to that of a distortedincompressible plate, is observed. A Debye-Scherrer av-erage along the lines depicted in Fig. 4 depends now onthe detailed shape of the detector plane distortion, andthe standard procedure is clearly inaccurate. -10 -8 -6 -4 -2 0 2 4 6 8 10-60-40-200204060 Angle of detector tube (degrees) P i x e l he i gh t ( c m ) Energy (meV)
FIG. 13. Debye-Scherrer rings seen on the low-angle pixelsof the detector plane, in the phonon region (see energy scaleon the right hand side). The black diamond is the beam-stopshadow.
Another important information can be obtained fromthe pixelized analysis. The energy of the excitationsand their scattering angle can now be calculated andvisualized is a two-dimensional array, as shown in Fig.13 for the smallest angles. Contour fits made along thephonon Debye-Scherrer rings show that large sampleoff-centering (see Section V B 1) can be excluded to avery good accuracy (a few mm). We have calculatedthe possible distortions of the detector assembly, whichis by construction a rather rigid cylindrical wall, fixedat the floor level, rigidly held at the level of the middleplane, and rather free to move at the top. As suggestedby Fig. 12, the detectors plane simply undulates. Asa result, the distances to the center vary substantially,but angular corrections, a second order effect, are small.For our data, the angular correction is easily calculatedby iteration, adding the successive angular deviationscalculated for the measured distances corresponding toeach detector tube. The correction, which reaches itsmaximum (0.07 ◦ ) at the largest angles (135 ◦ ), is verysmall.Analysis step 2: we determine the value of E i ,the initialneutron energy, and t s , the time of arrival of the neutronsat the sample. As was explained in Section V A, the nom-inal values of these parameters are not accurate enough,and the dispersion relation calculated with these valuesis systematically too high by about 9 µ eV. We thus cal-ibrate our energy scale at a single point using the rotonenergy determined by Stirling on IN12, a high reso-lution triple-axis spectrometer: ∆ R =0.7418 ± rotinel =(602.68 ± τ ch at the roton; the correspondingelastic time is t rotelast =(514.02 ± τ ch (a time channelis τ ch =6.9084 µ s).The data of Fig. 12 show that the detector dis-tances are close to the nominal value of 4.00 m in thelower part of the plane and that they increase near thetop. Taking into account the reduction of the effectiveflight distance due to the average penetration length inthe He detectors (see Section V B 3), we estimate theaverage sample-detector distance in the roton region,L rot ≈ ± i (and hence the energy E i ) and t s , the time of ar-rival of the neutrons at the sample: L rot = ( t rotelast − t s ) v i ,which can be solved together with Eq. 13 expressed atthe roton:∆ R = E i (cid:20) − (cid:16) t rotelast − t s t rotinel − t s (cid:17) (cid:21) We obtain E i =3.520 ± i =820.62 ± s =(-191.55 ± τ ch . As expected, the correctedneutron energy is slightly lower (by 0.85%) than thenominal value.Analysis step 3: with these parameters, we ana-lyze with a Mathematica program the set of data pairst elast ( ϕ , z ),t inel ( ϕ , z ). For each pixel, we calculate theexcitation energy using Eq. 13, and the correspondingDebye-Scherrer angle ϕ . The result is a curve ǫ ( ϕ ) witha very large number (9 × ) of independent data points.Fig. 14 shows the results in the most delicate region, atlow angles. E ne r g y ( m e V ) Scattering angle (degrees)
FIG. 14. ǫ ( ϕ ) (analysis step 3) in the phonon region. Mostof the data display the usual statistical distribution, but spu-rious data points are also present at very low angles, system-atically above the main curve. The corresponding neutronstravel through indirect paths, they must be identified andeliminated from the analysis. There are obvious spurious data points, correspondingto neutrons reaching the detectors in an indirect way, ascommonly observed in neutron scattering experiments.Also, a simple inspection of Fig. 12 shows the presenceof a few bad tubes and bad pixels. In addition, some detectors located behind the beam-stop (diamond-likeshadow at zero angle) or in its vicinity, cannot be ex-ploited. Removing these spurious points leaves typically88000 independent points of good quality. It is clearlydesirable to average over several data points in orderto improve the statistical uncertainty in the energy,as suggested by the dispersion seen in Fig. 14, at theexpense of a reduced wave-vector resolution.Analysis step 4: the wave-vectors k corresponding tothe E( ϕ ) data points are calculated using Eq. 11. The re-sulting ǫ ( k ) data sets are averaged within 0.002 ˚A − bins.There are about 10 bins on the dispersion relation ateach pressure in the wave-vector range 0.14 < k < − .The number of points per bin varies, as shown in Fig. 15,as a function of wave-vector. This is mainly due to thedetectors geometrical layout: there are gaps between dif-ferent groups of detector tubes, as described in SectionIV B. Empty bins are also found around Q=1.729 ˚A − ,which corresponds to angles near 90 ◦ , where the DebyeScherrer cone is essentially a vertical plane. D a t a po i n t s pe r . ¯ - i n t e r v a l Wave-vector (¯ -1 ) FIG. 15. Number of data points per wave-vector interval ofwidth 0.002 ˚A − . At the lowest wave-vectors, typically between 0.15 and0.2 ˚A − , the number of data points per bin is small. Bin-ning carries no benefit, and error bars in this region aredominated by statistical errors. Except for this small re-gion, binning is done over a substantial number of points,typically more than 70. By trying different bin sizes,it becomes clear that going beyond about 50 points/bindoes not improve the resulting dispersion curve: statisti-cal errors become negligible compared to systematic er-rors.Some small oscillations can be seen in the data. Anexample is given in Fig. 16. They are due to severalfactors, essentially deviations from the assumed param-eters (instrument geometry, sample environment char-acteristics, detector properties, electronic delays, etc.).Correcting for these cannot be achieved by averagingneighboring points. These deviations correspond well tothe error bars, calculated using the uncertainties in allthese parameters, for data above 0.2 ˚A − . The uncer-2
80 85 90 95 100 105 110 115 1200.700.750.800.850.900.95 E ne r g y ( m e V ) Scattering angle (degrees)
FIG. 16. Raw data in the roton region: a point is the measure-ment from one pixel. The effect of the gaps present by con-struction between different groups of detector tubes (markedby arrows) are visible. Systematic errors can be seen in oneof these regions, probably due to a local defect of the detectorgroups at their junction. Individual tubes are visible around90 ◦ . tainty in Q, due to the uncertainty of the instrument an-gles (0.07 ◦ , about 2/10 of a detector tube angular range)and to the uncertainty of the initial energy (0.003 meV)(see Eq. 11), can be represented by the expression∆Q=10 − (7+7.2Q) ˚A − . This corresponds essentially toa fraction of a bin. The uncertainty in the energy ǫ ( k ) hasbeen determined by varying the parameters E i , t s , D rot ,∆ R in the allowable parameter range. This is neededdue to the non-linear character of the equations, and thestrong correlation between E i and t s , a problem alreadynoted by Andersen et al. . The calculated relativeuncertainty is essentially constant, ∆E/E ≈ × − . VII. THE DISPERSION RELATION IN THEWHOLE RANGE
The dispersion relation at saturated vapor pressure inthe whole wave-vector range is shown in Fig. 1. In thissection, we first present high accuracy measurements ofthe pressure dependence in the particularly interestingwave-vector range below 2.3 ˚A − , shown in Fig. 17. Errorbars are comparable to the size of the data points. Theeffects of pressure are clearly seen: the phonon soundvelocity and the maxon energy increase, while the ro-ton minimum decreases and shifts towards higher wave-vectors. A spectacular flattening of the maxon is ob-served at high pressures. In the following paragraphs,we provide a quantitative analysis of the experimentaldispersion curves. P=0 P=0.5 P=1 P=2 P=5 P=10 P=24 E ne r g y ( m e V ) Wave-vector (¯ -1 ) FIG. 17. Dispersion curves ǫ ( k ) measured for several pres-sures in the 0 to 24 bar range. The individual data pointsare represented by small circles (best seen on-line). Accuratevalues for the pressures are given in Table I. A. Phonons
The behavior at low wave-vectors is shown in Fig. 18,where the phase velocity ǫ ( k )/ ~ k is represented as a func-tion of wave-vector k at P=0. The k ∼ ± , strongly extrapolated from low wave-vectors, are also shown. It is already rewarding to observethat ultrasound and neutron data, in spite of their non-overlapping validity region, are perfectly compatible andsmoothly merge around 0.2-0.25 ˚A − . For k < − ,however, the neutron data are slightly too high in en-ergy. This is not surprising: spurious data points pro-liferate at the lower end of the wave-vector range, asdiscussed above (see Fig. 14), leading to systematic er-rors that increase the energies. For wave-vectors as lowas 0.15 < k < − , Rugar and Foster’s curve is still ingood agreement with the neutron data within error bars(at their lowest limit). A similar behavior is observed atall pressures (Fig. 19). DMBT calculations, to be dis-cussed in detail below, are clearly in good quantitativeagreement with the experiments.We also show in Fig. 18 the neutron scattering dataobtained at saturated vapor pressure by Stirling .The comparison is of particular importance, because theyhave been measured on a triple-axis spectrometer (IN12),i.e., using a very different neutron technique. It is obviousthat the IN12 phonon energies are too low: the differencewith our data as well as with Rugar and Foster’s curveexceeds error bars for k < − , and matching the ul-trasound velocity is clearly impossible unless the errorbars of IN12 data are significantly increased. System-atic errors at low wave-vectors are indeed expected, giventhe large size of Stirling’s helium sample and the shortlength of the IN12 instrument, as well as other particu-3 This work Stirling Polynomial fit Maris - PadØ fit Bogoliubov-like fit Sound velocity Rugar (ultrasound) P ha s e v e l o c i t y ( m / s ) Wave-vector (¯ -1 ) FIG. 18. Phase velocity ǫ ( k )/ ~ k at P=0. Black dots witherror bars: present results. Different fits made in the wave-vector range 0.2 < k < − are shown by thick lines (see leg-end and text). Red dot at k=0: sound velocity . Extrapo-lated non-linear ultrasound data are shown as a dashed blueline. Stirling’s neutron data are represented by purplecircles. Experiment P ha s e v e l o c i t y ( m / s ) Wave-vector (¯ -1 ) ) FIG. 19. Phase velocity ǫ ( k )/ ~ k at P=0, 2, 5, 10, and 24bar from bottom to top, dots with error bars (this work).The corresponding sound velocities are indicated at k=0by crossed circles. Non-linear ultrasound data (availableonly between 0 and 15 bar), interpolated to P=0, 2, 5, 10 bar,and extrapolated to large wave-vectors: thick dash-dottedlines. Theory: DMBT curves are shown by thin linesfor several atomic densities (see graph legend and Table II). lar features of the resolution function of triple-axis (TAS)spectrometers. At higher wave-vectors, where both tech-niques are relatively free from systematic errors, a verygood agreement between our TOF data and Stirling’sTAS data is observed, which constitutes an importantexperimental test.Several functional forms describing the low wave-vector sector of the dispersion curve have been proposed (see Section II A). Figure 18 shows fits made in the range0.2 < k < − . The first fit uses the polynomial expan-sion of Eq. 2, written now in practical units as ǫ ( k ) = 0 . ck (1 + α k + α k + α k ) (17)where α = − γ , ǫ ( k ) is expressed in meV, k in ˚A − , cin m/s, and the α i coefficients in ˚A i . The sound veloci-ties obtained from neutron data using the polynomial fitare given in Table II. At 24 bar the dispersion is normal,and a simple quadratic fit ( α = α = 0) is sufficient todescribe the data very well, changing the speed of soundby a small amount, within error bars, with respect to theresult obtained with the full expression. The Pad´e ap-proximant (Eq. 3) is very sensitive to the upper limit ofk used in the fit. It tends to overestimate the sound ve-locity, and the same conclusion applies to the expressionderived from Bogoliubov’s formula (Eq. 4). P n Neutron err Sound err V m (P) C V bar at/˚A m/s m/s m/s m/s m/s m/s0 0.021836 241.7 2.9 238.3 0.1 237.76 236.80.51 0.021968 246.4 7.2 242.6 0.1 241.98 240.71.02 0.022096 251.0 6.6 246.5 0.1 246.04 244.32.01 0.022334 257.9 6.9 253.9 0.1 253.53 251.25.01 0.022983 278.1 7.1 274.0 0.1 273.73 270.510.01 0.023889 308.6 7.8 302.3 0.1 301.75 298.624.08 0.025804 364.7 1.9 361.9 0.1 359.28 357.924.08 0.025804 TABLE II. Sound velocities obtained from neutron scat-tering (this work), ultrasound , molar volume pressuredependence , and heat capacity (Greywall, analysis 4) ,at different pressures (n is the atomic density). At 24 bar, theupper line corresponds to a quadratic fit, while the last linegives the result of the full fit, for comparison (see text). The sound velocities deduced from the present neu-tron scattering measurements are compared in Table IIto the ultrasonic sound velocities , to those obtained(at interpolated densities) from Greywall’s heat capacity(C V ) measurements , and to the values we calculatefrom the compressibility (molar volume pressure depen-dence) determined by Tanaka et al. (see Section II A.As noted for the P=0 data, the neutron scattering valuesare higher than the ultrasonic ones, but the difference iswithin error bars. The speed of sound we calculate fromthe compressibility agrees very well with the ultrasonicdata, except at the highest pressure, where either theultrasonic data or, most likely, the compressibility databecome somewhat inaccurate. Heat capacity data for thespeed of sound are systematically lower than the ultra-sonic values. Error bars are not quoted, but their sensi-tivity to different methods of data analysis suggeststhat the uncertainties are comparable to our estimatederrors for the neutron data.The values of the speed of sound and their densitydependence determined using different techniques are in4excellent agreement, they only display a small overallshift within error bars, as can be seen in Fig. 20. Thesame observation applies to the polynomial fits of thedispersion curve made in different ranges (0.015 < k < < k < < k < − ) with the Jastrow-Feenberg Euler-Lagrange microscopic theory . Theresulting sound velocities display a small dependence onthe selected k-range, not visible at the scale of Fig. 20.The figure also shows the results of Quantum Monte-Carlo calculations performed with the Aziz II potential,in excellent agreement with the experiments. S ound v e l o c i t y ( m / s ) Density (atoms/¯ ) Neutrons Sound Heat Capacity Compressibility JF-EL QMC-Aziz II
FIG. 20. Sound velocities from ultrasound , heatcapacity , our analysis of compressibility data, and thepresent neutron scattering measurements, as a function ofdensity (see Table II). Red line: Jastrow-Feenberg Euler-Lagrange calculation . Short-dash line: Quantum Monte-Carlo calculation with Aziz II potential. We focus now our attention on the determination ofthe anomalous dispersion parameter γ as a function ofdensity. Fits were made with equation 17 using the ul-trasonic sound velocities c to reduce the free parametersto α = − γ , α and α . As seen from Fig. 18 and its ac-companying discussion, the choice of the wave-vector fitrange of the experimental data has to be made with care.The chosen lower limit was k=0.25 ˚A − to reduce system-atic errors, and k=0.5 ˚A − was used as the upper limit;we also checked with fits extended to k=0.6 ˚A − the effectof the fit range on the accuracy. With 125 data points inthis range, statistical error bars were small.The anomalous dispersion parameter γ obtained fromthis analysis is shown in Fig. 21. Two types of errorbars are given for each data point; the smaller bars in-dicate the statistical uncertainty, while the larger onesgive the estimated systematic errors, associated to theuncertainty in the global parameters of the analysis, de-scribed in Section V. The statistical error bars are small,and therefore, only a global shift of the whole experimen-tal curve within the systematic error bars is allowed . Theresults are compared in Fig. 21 to data from two different types of ultrasonic measurements, performed respectivelyby Junker and Elbaum (temperature dependence), andRugar and Foster (non-linear measurements) (see Ref.40 for a critical review and references to former data).The neutron data agree well in magnitude with these, andtheir density dependence, in particular, agrees extremelywell. A small global systematic shift, as described above,is observed. At the highest densities, where ultrasounddata do not exist, the neutron scattering result confirmsthe evolution towards positive values of γ extrapolatedfrom the ultrasonic data. -2.5-2.0-1.5-1.0-0.50.00.5 Neutrons Junker and Elbaum Rugar and Foster Greywall analysis 4 Greywall analysis 3 DMBT fit 0.015 34 ˚A atsaturated vapor pressure (see Eq. 5). The density de-pendence was obtained from Davison’s formula .The results are shown in Fig. 22. This parameteris negative in the whole density range, its magnitudeis about -3 ˚A , with a slow variation as a function ofdensity. Fits were made using γ as a free parameter,and also using our ‘best estimate’ for γ (black dashedline in Fig. 21) discussed above. The statistical un-certainties are rather small in both cases, and system-atic uncertainties dominate. Having checked that thetwo sets of results are consistent, we use in the fol-lowing the values of α calculated with the ultrasoundvalues of γ . The magnitude of α and its density de-pendence are in good agreement with the Kemoklidze-Pitaevskii-Feenberg-Davison (KPFD) expression . The pseudo-potentials phenomenological calculation byAldrich, Pethick and Pines yields γ =-1.5 ˚A and α =-3.7 ˚A . Our fits of their published curves showthat these results depend on the wave-vector range.For 0 < k < − we find γ =-(1.57 ± and α =-(4.0 ± . The original values are found only ifwe extend the fit range beyond k=0.6.The results of the microscopic DMBT calculation are shown in Fig. 22. Again, the results depend on thewave-vector range selected for the fits. We note that thedensity dependence of α calculated using a low wave-vector range (0.015 < k < − ) is remarkably similar tothat predicted by KPFD. The fit done at higher wave-vectors (0.18 < k < − ) agrees particularly well withthe neutron data. We conclude that the neutron scat-tering measurement and the microscopic DMBT calcula-tion agree reasonably well with the α values predictedby KPFD. We also note that the present work is the onlysource of experimental data on α up to now.The results for α , the next term in the series expan-sion obtained with the fits described above, are shownin Fig. 23. This parameter, determined here experimen-tally for the first time, is positive in the whole densityrange, its magnitude is about 2 ˚A , with a slow variationas a function of density. The results can be discussed in avery similar way as done above for α . The values dependon the fit range. Our fits to the pseudo-potential theorypublished curves give α ∼ ± ) neutrons (fixed ) DMBT fit 0.015 FIG. 24. Normalized phase velocity ǫ ( k )/ ~ ck as a functionof wave-vector k. Circles with error bars: present results atP=0, 2, 5, 10, and 24 bar (from top to bottom). Non-linearultrasound data (available only between 0 and 15 bar)have been interpolated to P=0, 2, 5, 10 bar, and extrapolatedto wave-vectors 0 < k < − (black dashed lines). Theory:DMBT curves for several atomic densities (see Table II). It is interesting to compare in this ‘anomalous disper-sion’ representation, the results obtained from two verydifferent theoretical approaches, pseudo-potentials andDMBT. Even at the substantially expanded vertical scaleof Fig. 25, a remarkable agreement is observed, withinthe uncertainties of comparable magnitude estimated forexperiments and theory. P=2P=24P=10P=5 APP (bar) P=0 P=5 P=10 P=15 P=20 P=25 E / ( ck ) Wave-vector (¯ -1 ) DMBT (at/¯ ) P=0 FIG. 25. Normalized phase velocity ǫ ( k )/ ~ ck as a func-tion of wave-vector k. Experimental data at at P=0, 2,5, 10, and 24 bar (circles with error bars) are comparedto the pseudo-potential calculation of Aldrich, Pethick andPines (dashed curves) and to the DMBT results forseveral atomic densities (see Table II). B. Phase and group velocities We have shown in the previous section that the poly-nomial expansion (Eq. 17) becomes inaccurate as oneconsiders wave-vectors in the atomic range. As seen inFig. 26, the phase velocity curves display a very pecu-liar behavior for 0.5 < k < − : they become linear to ahigh degree of accuracy. This is observed both in the ex-periment and in the DMBT curves, at all densities (withsome changes at the highest pressures, where the maxonis strongly damped). It is obvious that adding higher or-der terms in a series expansion around k=0 in order todescribe this type of high-k dispersion requires a strongcompensation of successive terms, and is inadequate.One can easily check (series expansion of the phase veloc-ity around the maxon wave-vector) that this linear termis a consequence of the maxon parabolic dispersion rela-tion, combined with the fact that the maxon energy atlow pressures is numerically very close to − ~ k M / m M ,where k M and m M are the maxon wave-vector and its(negative) mass, respectively. P ha s e v e l o c i t y ( m / s ) Wave-vector (¯ -1 ) Theory (at/¯ ) FIG. 26. Phase velocity ǫ ( k )/ ~ k at P=0, 2, 5, 10, and 24bar (from bottom to top, circles with error bars). DMBTcurves are shown for several atomic densities (see TableII). A linear dependence on the wave-vector is observed in alarge range (see text). Our dense data-set allows to calculate the group ve-locity by numerical differentiation with a good accuracy.The result is shown in Fig. 27, where a 40-points averageis used for clarity. The graph emphasizes the behavior atlow wave-vectors (the importance of the α term), andthe behavior around the maxon and roton wave-vectors,where the group velocity vanishes. The correspondingDMBT results are shown in Fig. 28. The selected den-sities are extremely close to the values corresponding tothe experimental pressures (P=0, 5, 10, and 24 bar), seeTable II. The agreement between theory and experimentis remarkable.7 Pressure (bar) P=0 P=5 P=10 P=24 G r oup v e l o c i t y ( m / s ) Wave-vector (¯ -1 ) FIG. 27. Group velocity ∂ǫ ( k )/ ~ ∂ k at P=0, 5, 10, and 24 bar.Sound velocities are indicated by dots at k=0. At low wave-vectors, thin lines show the parabolic dependence calculatedfrom ultrasound data (available only between 0 and 15bar), and thick lines the polynomial expansion including thecalculated α term . G r oup v e l o c i t y ( m / s ) Wave-vector (¯ -1 ) density (¯ -3 ) 0.0215 0.0230 0.0240 0.0255 FIG. 28. Group velocity ∂ǫ ( k )/ ~ ∂ k calculated by DMBT atselected atomic densities (compare to experiment, Fig. 27). C. Rotons We now concentrate on the properties of the disper-sion curve around the roton. Given the large number ofdata points with small error bars, it is possible to cal-culate the main parameters of these excitations (energy,wave-vector, and mass) for the different pressures usinga quartic polynomial expression: ǫ R ( k ) = ∆ R + ~ m µ R ( k − k R ) + B R ( k − k R ) + C R ( k − k R ) (18) where ∆ R , k R , µ R , B R , and C R are adjustable pa-rameters. ∆ R is the roton gap defined before, k R theroton wave-vector, and µ R the effective roton mass. Thebest fits are obtained using an asymmetric wave-vectorrange, from k R -0.2˚A − to k R +0.3˚A − . Different rangeswere tested, with a number of data points in the 100 to200 points range. Under these conditions, the parame-ters ∆ R , k R , and µ R do not depend on the wave-vectorrange selected for the fits. Quadratic fits in a small range( ≤ − ) give essentially the same results: statisticalerrors on the resulting parameters are very small, andthe dominating uncertainties essentially originate fromsystematic errors. For example, the very small wigglesseen in the dispersion curves are due to imperfections ofthe detectors, and not to statistics. A typical fit is shownin Fig. 29, and the results for all pressures are given inTable III. P=0 P=0.5 P=1 P=2 P=5 P=10 P=24 E ne r g y ( m e V ) Wave-vector (¯ -1 ) FIG. 29. The dispersion relation in the vicinity of the rotonminima at different pressures (accurate values of P are givenin Table I). The dashed line on the 10 bar curve shows aquartic fit using the largest acceptable wave-vector range (seetext). The present data at saturated vapor pressure are com-pared to the results of previous works in Table IV. Theroton gap ∆ R is known with an accuracy of 1 µ eV. Earlymeasurements indicated values close to 0.743 meV orhigher, but a slightly lower value (∆ R =0.7418(10)) wasobtained by Stirling using a high resolution spectrome-ter. As seen above, we have used this value in order tocalibrate the IN5 spectrometer in energy, and a remark-able agreement with the ultrasonic sound velocities wasobtained at very low wave-vectors.We find a value of k R slightly lower than Stirling’s(the most accurate available so far), within small andcomparable error bars. Higher values, outside error bars,are found in the literature (Table IV). A similar remarkapplies to the roton effective mass: we obtain a valuethat agrees well with that found by Stirling and8 P (bar) ∆ R (meV) k R (˚A − ) µ R R (meV) k R (˚A − ) µ R This work et al. , Stirling , An-dersen et al. , Gibbs et al. , and Pearce et al. . ∆ R at P=0 is taken from Ref. 57 and 59 in our instrument cali-bration procedure. Andersen et al. , but disagrees with those found byGibbs et al. and Pearce et al. . The position andcurvature of the roton minimum can be determined in ourcase with a better accuracy, simply because of our muchlarger number of data points in the small wave-vectorrange of interest.This becomes obvious in the representation of the pres-sure dependence of the roton gap, shown in Fig. 30. Ourdata (see Table III) have a smooth dependence on pres-sure, easily fitted by a second order polynomial throughthe statistical error bars. We remind that an overall shiftin energy is allowed, within the 1 µ eV systematic uncer-tainty in ∆ R (P), if the accuracy of ∆ R (P=0) is improvedin future measurements.There are few results in the literature on the pressuredependence of the dispersion relation. Results for theroton gap are shown in Fig. 30. Early data of Dietrich et al. cover a large pressure range, with large uncer-tainties in energy (about 5 µ eV) and wave-vector (about0.005 ˚A − ), and a reasonable accuracy on the pressure( ± et al. ,would shift Dietrich’s data upwards in energy by 5 to10 µ eV. This would bring them in good agreement withthe present results.High resolution triple-axis results at non-zero pressuresare scarce. Those by Talbot et al. , only available at This work Stirling Talbot et al. Gibbs et al. Dietrich et al. R o t on ene r g y ( m e V ) Pressure (bar) FIG. 30. Pressure dependence of the roton energy at verylow temperatures. Small black dots: this work; the solid lineis a 2 d order polynomial fit. Energy statistical error bars,and uncertainties in the pressures, are not visible at thisscale; the plotted energy error bars correspond to the sys-tematic uncertainty in the P=0 roton gap (see text). Green × : Stirling . Red diamonds: Gibbs et al. (pres-sure error bars unknown). Orange +: Dietrich et al. (atT=1.3 K). are only available at 15.2 and 24 bar(T=0.9 K), but their uncertainty in the pressure of 1 bar,unfortunately, translates into an energy uncertainty ofabout 6 µ eV. As seen in Fig. 30, the data point at about15 bar agrees with ours, but this is not the case for thehigh pressure one. The latter is definitely very low in en-ergy, a discrepancy that cannot be explained by errors onthe pressure measurement, since solidification takes placeat 25.32 bar. It is closer to the much higher temperatureresult of Dietrich et al. , than to our high pressure data.We should point out here that the polynomial fit to ourdata does not change significantly if our point at 24 baris omitted.The only recent source of good resolution data at non-zero pressures is Gibbs et al. . Fig. 30 shows thatthere is a good agreement between these results and oursin the pressure dependence of the roton gap. The devia-tions are probably explained by the larger uncertaintiesof the data by Gibbs et al. , including a likely error in theirpressures, measured with a Bourdon gauge (uncertaintiesnot quoted).The pressure dependence of the roton wave-vector k R is shown in Fig. 31. The present data display a smootherbehavior, with small error bars, compared to former re-sults. The statistical uncertainties from the fits at con-stant scattering angle are one order of magnitude smallerthan the systematic errors (see the discussion at the endof Section VI). The latter, on the order of 0.002 ˚A − ,are due to the uncertainty in the detector angles, and9to the conversion from scattering angle to wave-vector,which involves the systematic uncertainty of the energies.We observe a good agreement with Stirling’s triple-axisdata . TOF data by Dietrich et al. , Andersen etal. and Gibbs et al. are systematically shifted,but on both sides of our curve. This work Stirling Andersen et al. Gibbs et al. Dietrich et al. R o t on w a v e - v e c t o r k R ( ¯ - ) Pressure (bar) FIG. 31. Pressure dependence of the roton wave-vector.Small black dots: this work; the solid line is a guide to theeye. The energy error bars correspond to the systematic un-certainties, statistical ones are not visible in this plot (seetext). Green dots: Stirling . Red diamonds: Gibbs etal. . Blue × : Andersen et al. . Orange +: Dietrich etal. (at T=1.3 K). The pressure dependence of the roton effective mass µ R is shown in Fig. 32. Stirling’s data points atSVP and 15 bar agree with ours, but this is not the casefor the point at 24 bar. The data by Dietrich et al. and those by Gibbs et al. are shifted with respect toours, but follow the same trend. We also note that theroton effective mass is strongly non-linear as a functionof pressure, but it is an almost linear function of thedensity. This will be discussed in more detail below, inthe comparison of our data with DMBT calculations. D. Maxons The properties of the dispersion curve around themaxon (Fig. 33) can be studied in a similar way. Fitshave been made using the cubic polynomial expression: ǫ M ( k ) = ∆ M + ~ m µ M ( k − k M ) + B M ( k − k M ) (19)where the parameters are the maxon energy ∆ M , themaxon wave-vector k M , and the (negative) maxon ef-fective mass µ M . Different fitting ranges were tested inorder to evaluate the influence of systematic errors. Themaxon curves display a much smaller asymmetry than This work Stirling Andersen et al. Gibbs et al. Dietrich et al. R o t on e ff e c t i v e m a ss Pressure (bar) FIG. 32. Pressure dependence of the roton effective mass.Small black dots: this work. The large (small) energy errorbars correspond to the systematic (statistical) uncertainties,respectively (see text). Green dots: Stirling . Red dia-monds: Gibbs et al. . Blue × : Andersen et al. . Thelines are guides to the eye. the roton ones. In the fits, it is possible to limit the poly-nomial expression to the cubic term in the wave-vectorrange spanning 0.2˚A − around k M . A typical fit is shownin Fig. 33 on the 5 bar curve, results for all pressures aregiven in Table V. P=0 P=0.5 P=1 P=2 P=5 P=10 P=24 E ne r g y ( m e V ) Wave-vector (¯ -1 ) FIG. 33. The dispersion relation in the vicinity of the maxonat different pressures (accurate values of P are given in TableI). The dashed line on the P=5 bar curve shows a typicalcubic fit (see text). At 24 bar, the maxon is strongly damped. There is an excellent agreement between our resultsfor the maxon energy at saturated vapor pressure andthose from Gibbs et al. and Gibbs (Thesis) . At finitepressures, however, there is a clear discrepancy betweenthese data and ours. The published version is closer to0 P (bar) ∆ M (meV) k M (˚A − ) µ M M a x on ene r g y ( m e V ) Pressure (bar) strong dampingregion FIG. 34. Pressure dependence of the maxon energy. Smallblack dots: this work. The energy error bars correspond tothe systematic uncertainties (see text). The solid line is apolynomial fit of order 3 through the low pressure data. Thepoint at 24 bar is in a different regime (see text). Triangle:Talbot et al. . Red diamonds: Gibbs et al. and Gibbs(Thesis) . The red dashed line indicates twice the measuredroton energy. our result than the earlier (but more detailed) version ofthe same work, found in Gibbs’s thesis manuscript .The behavior at high pressures is interesting, be-cause a different regime is reached when ∆ M ∼ R ,the maxon being spectacularly damped by three-particleprocesses . This happens according to Fig. 34 atP ∼ 20 bar. Results at the same pressure by Talbot etal. and Gibbs et al. are considerably shifted, onboth sides, with respect to the present data. The largediscrepancy between former data may be due to the effectof damping on the maxon energy, as seen in Fig. 33. Athird order polynomial fit can describe the pressure de-pendence of the maxon energy in our data, within theirvery small statistical uncertainty. The curve is shownin Fig. 34. Extrapolation to high pressures is delicate,and a better description, in terms of densities, will bepresented below.The pressure dependence of the maxon wave-vector k M is given in Table V. This parameter is, surprisingly, rather constant at low pressures. A substantial increaseis observed at 24 bar, related to the damping, as seen inFig. 33. A small increase of k M is already observed at10 bar. The maxon effective mass µ M , on the other hand,has a smooth variation with pressure, that we can fit bya simple second order polynomial expression. Its valuescan be found in Table V. A discussion of these results isgiven below. E. Theory: Dispersion relation at SVP Among the numerous theoretical calculations of thedispersion relation of superfluid He to be found inthe literature, we have chosen four examples, especiallyappropriate for this manuscript (see Fig. 35): theBrillouin-Wigner (BW) perturbative calculation by Leeand Lee , two different types of Monte-Carlo (MC)calculations , and the variational dynamical many-body theory (DMBT). Starting from the Bijl-Feynmanspectrum , which is clearly very far from the experi-mental result, the perturbative calculation by Lee andLee has, first of all, the merit to bring theory closer tothe experiment. In addition, it provides a quantitativeestimate of the corrections due to the different Feynmandiagrams involved in microscopic calculations. The effectof the ǫ b term is shown, as an example, in Fig. 35. Inspite of the large number of diagrams included, the BWapproach is not satisfactory: strong departures from theexperimental results are seen in the whole wave-vectorrange.DMBT provides accurate results from low wave-vectorsto somewhat beyond the maxon. The discrepancy ob-served at high wave-vectors is, according to the BW cal-culation described above, consistent with fact that the ǫ b diagram is not included in the DMBT calculation(see the discussion in Ref. 34). Improving the accuracywould imply an additional computational effort which isnot necessary, in particular, to investigate the density de-pendence of the dispersion. The dispersion relation cal-culated by DMBT at the present level is already in goodagreement with the experiment in the whole wave-vectorrange, including the plateau region, the calculation ofwhich constitutes a severe theoretical challenge.Monte Carlo calculations constitute a very dif-ferent approach to the microscopic description of quan-tum fluids. We show in Fig. 35 the results of Diffu-sion Monte Carlo (DMC) calculations by Boronat andcoworkers , that clearly provide values of the disper-sion relation at zero temperature in excellent agreementwith the experiment.Path Integral Monte Carlo (PIMC) calculations yieldresults on the dispersion relation at finite temperatures.The data at T=0.8 and 1.2 K (Fig. 35) (identical withinuncertainties, since the dispersion relation is not stronglytemperature dependent below 1.25 K) are in good agree-ment with the experimental values. Both MC methods,however, experience difficulties in observing the plateau1of the dispersion relation, essentially because of its verylow weight; calculations in this region capture instead amultiexcitation ‘branch’ also seen in the experiments (seeRef. 33 and Section VII G).Since constant progress is made in numerical methodsand techniques, both variational and Monte Carlo meth-ods are expected to yield further important developmentsin this field. E ne r g y ( m e V ) Wave-vector (¯ -1 ) BW DMC PIMC T=0.8 K PIMC T=1.2 K DMBT 0.0215 ¯ -3 DMBT 0.0220 ¯ -3 Expt. (this work) - FIG. 35. Dispersion relation ǫ ( k ) at saturated vapor pres-sure. Upper curves: Bijl-Feynman spectrum ( ǫ ); BWcalculation (black dots); BW calculation, excluding the ǫ b term (circles). Red curves: DMBT at two densitiesaround SVP (see legend). Green lozenges: DMC ; trian-gles: PIMC at T=0.8 and 1.2 K. Thick black line: experi-ment (this work). F. DMBT: density dependence The experimental properties of the roton and themaxon described above can be compared to DMBT pre-dictions. Since a small shift in density is often neededin order to compare quantitatively the DMBT calcula-tions and the experiments, we shall use in the following,instead of pressures, atomic densities. We thus avoid in-troducing in the comparison the theoretical equation ofstate. The pressure-density relations used here to con-vert the experimental pressures in atomic densities areknown with excellent accuracy, they are found in Abra-ham et al. and Greywall (see Section II A).The dependence of the roton energy on density isshown in Fig. 36. A good agreement is found withinthe expected accuracy of the theoretical calculation, es-timated to be on the order of 10%. This is not due toa shortcoming of the theory, but to the choice of the di-agrams included in the calculation, limited to the mostsignificant ones, as far as the physics is concerned. Asseen in the previous section, an estimate of the energycorrection can be made using the Brillouin-Wigner per-turbation calculation by Lee and Lee . The first omitted diagram would decrease the roton energy by 0.05 meV.This brings the (corrected) theory close to the experi-mental result at low pressures and, as expected, the de-viation grows in fact at high pressures, where correlationsare strongest. Experiment and quadratic fit DMBT R o t on ene r g y ( m e V ) Density (atoms/¯ ) FIG. 36. Density dependence of the roton energy. Presentresults (black dots with total uncertainty error bars, see text)are compared to the DMBT calculation (red squares). The calculated roton wave-vector k R and its densitydependence (Fig. 37) are quantitatively very close tothe experimental result. It is clear that the diagrams in-cluded in the DMBT calculation capture all the essentialfeatures. The omitted diagrams have a smaller effect onthis parameter, than on the roton energy.It has been suggested by Dietrich et al. that the den-sity dependence of the roton wave-vector obeyed a simplelaw, k R = aρ / , expected if the system is homotheti-cally transformed with pressure. Clearly, as seen in Fig.37, neither theory nor experiment follow this law. In-deed, the density dependence of k R is almost linear, evenwithin the very small statistical error bars of the fits, and a fortiori within the somewhat larger total error bars in-cluding systematic uncertainties.The density dependence of the roton effective mass µ R is shown in Fig. 38, where the experimental data arecompared to the DMBT calculations. The curves werefound to be very similar, and the analysis could be carriedout in the same way: the same function and wave-vectorrange already applied (see above) to the experimentaldata were used to fit the DMBT results. As seen in thefigure, the predicted magnitude as well as the density de-pendence are confirmed by the experiment. The slightlyhigher values of the theory are expected, since the theo-retical roton minimum, calculated with a limited numberof diagrams, is not as deep as the experimental one.The shape of the dispersion curve around the rotonminimum deviates rapidly from a simple parabola, andit also changes substantially with density. Higher orderterms in the polynomial expansion (see Eq. 18) are not atall negligible, unless fits are limited to a very small range2 This work Linear fit Quadratic fit DMBTheory 1/3 power fit R o t on w a v e - v e c t o r k R ( ¯ - ) Density (atoms/¯ ) FIG. 37. Density dependence of the roton wave-vector. Blackdots: this work (small error bars: statistical uncertainty,larger bars: systematic uncertainty, see text). The deviationsfrom a linear dependence are small: the short-dashed and thesolid red lines are, respectively, a linear and a quadratic poly-nomial fit to the data. The expression k R = aρ / (greendash-dotted line) clearly does not fit the data. Red squares:DMBT calculation . This work DMBT R o t on e ff e c t i v e m a ss Density (atoms/¯ ) FIG. 38. Density dependence of the roton effective mass.Black dots: this work (small error bars: statistical uncer-tainty, larger bars: systematic uncertainty, see text). Thesolid line is a quadratic fit to the data. Red squares: DMBTcalculation. The dashed red line is a guide to the eye. around the minimum, reducing the accuracy of the fits.With a large number of independent data points, we haveaccess to higher order coefficients: the cubic term (B R )and the quartic term (C R ) defined by Eq. 18. The resultsare shown in Figs. 39 and 40.We consider now the maxon properties, comparing ourdata to the predictions of the DMBT. The calculateddispersion relation in the vicinity of the maxon is shownin Fig. 41 for several densities, directly comparable toour experimental result shown in Fig. 33. This work DMBT R o t on c ub i c c oe ff i c i en t B R ( ¯ ) Density (atoms/¯ ) FIG. 39. Density dependence of the roton cubic term B R (asymmetry coefficient). Black dots: this work. Red squares:DMBT calculation . Lines are guides to the eye. This work DMBT R o t on qua r t i c c oe ff i c i en t C R ( ¯ ) Density (atoms/¯ ) FIG. 40. Density dependence of the roton quartic coefficientC R . Black dots: this work. Red squares: DMBT calcula-tion. Lines are guides to the eye. The maxon energy, represented in Fig. 42 as a functionof density, displays a much weaker variation than that ob-served for the roton. As described above, the high densitydata point is beyond the 2-roton limit, the correspondingmaxon is damped, and this point is in a different regimecompared to the lower pressure ones. Polynomial fits oforder 2 and 3 excluding the high density data point en-compass a relatively small portion of the 2-roton line,indicating that the maxon damping begins at a densityn c =0.0255 ± − (P=21.5 ± M is essentially constant atlow densities, as seen in Fig. 43. We have fitted Gibbs’sdata , and we find a systematic difference, somewhatoutside their relatively large error bars.Also shown is the dependence of the Debye wave-vector on the number density n. k D =(6 π n) / is de-fined by assigning one degree of freedom per atom for3 E ne r g y ( m e V ) Wave-vector (¯ -1 ) FIG. 41. The dispersion relation calculated (DMBT ) in themaxon region, for different densities indicated in the label inatoms/˚A . See also experimental the data in Fig. 33. This work ibid. two-rotons Talbot et al. DMBT Polynomial fits M a x on ene r g y ( m e V ) Density (atoms/¯ ) strong dampingregion2-rotons FIG. 42. Density dependence of the maxon energy. Blackdots: this work. Error bars: systematic uncertainty; statisti-cal error bars are not visible at this scale (see text). The solidline is a quadratic fit to the data. A quadratic and a cubic fitexcluding the high pressure point are also shown. The long-dash red line is the 2-roton energy. Triangle: Talbot et al. data . Red squares: DMBT calculation. The short-dashred line is a guide to the eye. the longitudinal mode, integrated with an upper limitk D : N a = ( V / (2 π )) R k D k dk , where N a is Avogadro’snumber and V the molar volume. The Debye model usually describes a solid, where the wave-vector is lim-ited by the inverse of the lattice spacing. In the liquid,such limitation does not exist. The hard core of the he-lium atoms, however, introduces a characteristic lengthand solid-like properties, like the very existence of rotonexcitations . The Debye wave-vector is quantitativelysimilar to the maxon wave-vector, the maxon being anal-ogous to a zone-boundary phonon. At high pressures,one can expect this analogy to work even better, and k M closely follows, indeed, the density dependence of k D . Inthe liquid, the states with higher wave-vectors progres-sively loose the relation with their ‘first Brillouin zone’analogues, and at k ≈ D one observes a roton minimum,instead of a zero. strong dampingregion This work Gibbs k Debye DMBT M a x on w a v e - v e c t o r k M ( ¯ - ) Density (atoms/¯ ) FIG. 43. The maxon wave-vector as a function of density.Black squares: this work. Open lozenges: our fits to Gibbs’sdata . Theory: DMBT . Lines are guides to the eye. TheDebye wave-vector calculated for the longitudinal phonon ofa periodic system (see text) is shown for comparison. The values of k M calculated by the DMBT agree wellin magnitude with the experimental ones. A deviationis seen at relatively high densities, just before enteringthe strong damping region of the continuum. The exper-imental data discussed above display a small increase athigh densities, and the opposite is found in the theoreti-cal calculations. It must be pointed out that the maxonis very flat, and hence systematic errors can easily shiftits position: this very small effect (about 2%) may comefrom the approximations in the theory, or from experi-mental resolution problems. In the theory, the decreaseof the maxon wave-vector is clearly correlated with thedamping at the maxon (see Figs. 41 and 42).In a previous publication we provided tables of rotonand maxon parameters obtained with the standard dataanalysis (see Section V). A uniform shift of 9.2 µ eV ap-plied to the energies yielded the correct sound velocitiesand the expected roton gap. This was accurate enoughfor the study of multi-excitations, but insufficient to es-tablish the dispersion curve of single-excitations. Here,the energy scale has been calibrated at the roton energy.The energies as a function of pressure obtained from bothanalysis differ by a small amount. Also, the wave-vectors(in particular at the roton and the maxon) are slightlyreduced by the energy recalibration. The values of theeffective masses agree reasonably well after correcting anerror (a missing factor 1.0546 from ~ ) in the previouspublication. The new roton and maxon data tables aremore accurate, they have been obtained by a consistentprocedure entirely based on neutron data, and error barsincluding detailed systematic uncertainties are given.4 G. Beyond the roton The very high wave-vector region, usually referred toas ‘beyond the roton’ in the literature , is alsoof interest. The dispersion relation at high wave-vectorsincreases up to an energy ǫ max ≈ R , the Pitaevskiiplateau. Since high-k excitations can decay into rotonpairs, they cannot remain sharp above the plateau. Theexperimental investigation of this region is diffi-cult, due to the limited energy resolution of the spectrom-eters at the relevant energies. Early experiments placedthe curve above the plateau, even at SVP, but this hasbeen shown by Pistolesi to be due to experimental reso-lution effects .According to high resolution work at SVP, single ex-citations reach twice the roton gap at Q = 2.8 ˚A − , theenergy remains constant in the vicinity of 2∆ R betweenQ=3.0 ˚A − , and the end point of the dispersion curve isat Q =3.6 ˚A − . At high pressures (20 bar), the disper-sion curve has been shown to lie essentially just belowthe plateau , and to also end at about k=3.6 ˚A − .From the theoretical point of view, according tothe calculations of Pitaevskii and their extensionby Zawadowski, Ruvalds, Solana (ZRS theory) andPistolesi , the dispersion curve is believed to beslightly below Pitaevskii’s plateau.The results of the present measurements, obtainedwith different incident neutron energies in order to opti-mize the energy resolution, are shown in Fig. 44. Wave vector (¯ -1 ) E ne r g y ( m e V ) R FIG. 44. The dynamic structure factor S( Q , ω ) in Pitaevskii’splateau region, measured using different incident neutron en-ergies (E i = 3.520, 5.071, 7.990, and 20.45 meV), with energyresolutions (FWHM) at elastic energy transfer 0.07, 0.12, 0.23and 0.92 meV, respectively (thin gray lines indicate limits ofkinetic ranges). The color-coded intensity scale is in units ofmeV − . Dashed line: 2∆ R . We show in Fig. 45 our data for the dispersion rela-tion at T < et al. at a higher temperature (1.35 K) using theIRIS spectrometer; error bars are similar to ours in en-ergy, but 25 times larger in wave-vector. There is an excellent agreement between these data where they canbe compared. R R This workGlyde et al. E ne r g y ( m e V ) Wave-vector (¯ -1 ) FIG. 45. The dispersion relation measured beyond the roton.Open circles: present work at T < etal. (T=1.35 K) The single excitations phase velocity (Fig. 46), doesnot display particular features in this range. The groupvelocity (Fig. 47), on the other hand, vanishes at Q= 2.8 ˚A − and, as seen in Fig. 45, the dispersioncurve becomes flat. The intensity of the single excita-tion decreases approximately exponentially (Fig. 44) fork > − . It is difficult to define an ‘end wave-vector’of the single excitation. We observe however a changein the exponential-like intensity decrease at k ≈ − ,close to the value Q =3.6 ˚A − quoted in the only otherhigh resolution data at SVP in this range .The calculations in the framework of DMBT predictthat the dispersion curve remains almost constant belowthe plateau at high densities, but the behavior is differ-ent at low densities: at saturated vapor pressure, afterreaching the continuum at k ≈ − , the curve remainsat the edge of the continuum with some undulations. Anasymmetric peak is the formed, and its shape varies withk, as seen in Fig. 48 (the figure can be expanded in theon-line version). At all densities, the single excitation in-tensity finally vanishes at a wave-vector on the order of3.6 ˚A − , slightly increasing with density.Disentangling the single excitation from the multi-excitations contribution cannot be done unambiguouslyat the largest wave-vectors. The good general agreementbetween experiment and theory supports the hypothesisof an end of the dispersion curve in this range. H. Comparison to previous works The measured dispersion relation at saturated vaporpressure has been represented in Fig. 1. A table summa-5 This work at SVP DMBT n=0.0215 atoms/¯ P ha s e v e l o c i t y ( m / s ) Wave-vector (¯ -1 ) FIG. 46. The single-excitations phase velocity at saturatedvapor pressure (black dots, almost continuous line), is com-pared to the calculated values (dashed line, DMBT ) at aneighboring density (see Table II). This work at SVP DMBT n=0.0215 atoms/¯ Wave-vector (¯ -1 ) G r oup v e l o c i t y ( m / s ) FIG. 47. The single-excitations group velocity at saturatedvapor pressure (black dots, almost continuous line), is com-pared to the calculated values (dashed line, DMBT ) at aneighboring density (see Table II). rizing the results is given below (Table VI). Raw dataand tables with a finer wave-vector grid (0.002˚A − ) areprovided, see Supplemental Material at [URL will be in-serted by publisher].The present results are compared to previous experi-mental data in Fig. 49, a percent deviation plot whereour data are taken as the reference. We examine firstthe data-base carefully selected by Brooks, Donnelly andBarenghi , where several results of lower accuracy orconsidered less trustworthy, have already been discardedby these authors. Large deviations with respect to ourdata are seen, in particular around 0.4 ˚A − and at highwave-vectors. The thick red line in Fig. 49 shows Don- ε), ρ = 0.0220 Å -3 k (Å -1 ) ε (meV) S ( k , ε ) FIG. 48. The dynamic structure factor calculated (DMBT )for the density n=0.0220 ˚A − (close to SVP), in the plateauregion. The solid lines are cuts of S( k , ǫ ) for fixed wave-vectors. nelly’s spline fit of this set of data (see Ref. 46 and ref-erences therein). Also shown are results by Andersen etal. , Gibbs et al. , and Glyde et al. , which werenot included in the data-base mentioned above.One can distinguish several regions. For k < − ,the dispersion is essentially obtained from ultrasounddata, and the very small difference between our dataand Donnelly’s spline fit is within error bars. For0.2 < k < − our data agree well with those of Ander-sen et al. . Furthermore, our data measured at a differ-ent incident neutron energy (5.071 meV), are in excellentagreement with those measured at 3.52 meV, our refer-ence in this wave-vector range. The data of Gibbs et al. ,with a larger statistical uncertainty, lie in-between ourcurve and Donnelly’s.Between 1 and 2 ˚A − , our data at the two differentincident energies are again found to be consistent. Itis important to realize that these two sets of data areindependent, in particular the angles and distances cor-responding to a given wave-vector are different. The de-viation between these data sets is very small, it does notdisplay accidents or inconsistencies, and this is observedin a very large wave-vector range. This result constitutesan important verification of the consistency of the presentdata. The results of Andersen et al. agree with those ofGibbs et al. . They are consistent with Donnelly’s data-base, but all these points are ignored by Donnelly’s ‘fit’to a great extent, the curve is in fact closer to our result.At the roton, the agreement is good between the var-ious data sets. This point, of course, has been used inorder to calibrate our energy scale. At wave-vectors be-yond the roton, previous results display strong statisticalscatter and systematic inconsistencies; our data are justin the middle, and very close, as noted above, to the data6 k ǫ ( k ) err ǫ k ǫ ( k ) err ǫ k ǫ ( k ) err ǫ ˚A − meV meV ˚A − meV meV ˚A − meV meV0 0.0000 1.25 1.1698 0.0020 2.5 1.3718 0.00350.05 0.0787 1.3 1.1542 0.0020 2.55 1.3985 0.00390.1 0.1587 1.35 1.1310 0.0019 2.6 1.4214 0.00440.15 0.2407 0.0014 1.4 1.1055 0.0019 2.65 1.4375 0.00490.2 0.3244 0.0012 1.45 1.0743 0.0018 2.7 1.4571 0.00530.25 0.4092 0.0010 1.5 1.0381 0.0018 2.75 1.4617 0.00580.3 0.4938 0.0010 1.55 0.9999 0.0017 2.8 1.4746 0.00620.35 0.5782 0.0011 1.6 0.9563 0.0016 2.85 1.4776 0.00660.4 0.6581 0.0012 1.65 0.9113 0.0015 2.9 1.4737 0.00710.45 0.7338 0.0013 1.7 0.8672 0.0015 2.95 1.4818 0.00750.5 0.8040 0.0014 1.75 0.8242 0.0014 3 1.4784 0.00800.55 0.8701 0.0015 1.8 0.7877 0.0013 3.05 1.4793 0.00850.6 0.9279 0.0016 1.85 0.7584 0.0013 3.1 1.4781 0.00890.65 0.9812 0.0017 1.9 0.7427 0.0013 3.15 1.4767 0.00940.7 1.0267 0.0018 1.95 0.7459 0.0013 3.2 1.4805 0.00980.75 1.0667 0.0018 2 0.7681 0.0013 3.25 1.4920 0.0100.8 1.1002 0.0019 2.05 0.8094 0.0014 3.3 1.4855 0.0110.85 1.1280 0.0019 2.1 0.8637 0.0015 3.35 1.4954 0.0110.9 1.1518 0.0020 2.15 0.9374 0.0016 3.4 1.4940 0.0120.95 1.1678 0.0020 2.2 1.0154 0.0017 3.45 1.5012 0.0121 1.1809 0.0020 2.25 1.0944 0.0019 3.5 1.5199 0.0121.05 1.1882 0.0020 2.3 1.1701 0.0020 3.55 1.5377 0.0131.1 1.1907 0.0020 2.35 1.2370 0.0023 3.6 1.5538 0.0131.15 1.1884 0.0020 2.4 1.2914 0.00251.2 1.1814 0.0020 2.45 1.3363 0.0030TABLE VI. The dispersion relation of He at saturated va-por pressure and very low temperatures (T < 100 mK). Be-low k=0.15 ˚A − : ultrasound data (see text). From 0.15to 0.3 ˚A − : combined ultrasound and present neutron data;above 0.3 ˚A − , present neutron data. See Supplemental Ma-terial at [URL will be inserted by publisher] for more detailedtables. of Glyde et al. .Our data for the dispersion curve constitute thereforea new, comprehensive and coherent data base. In thefollowing, we use this result to calculate the thermody-namical properties, a stringent test for the measured dis-persion curve. VIII. SPECIFIC HEAT AND OTHERTHERMODYNAMICAL PROPERTIES Considerable effort has been devoted to the calcula-tion of the thermodynamical properties of superfluid Hestarting from the dispersion curve. Different kinds ofexperiments were performed, and subsequently analyzedusing Landau’s model . In the simplest approximation,applicable at low temperatures, the thermal populationof the states described by Landau’s spectrum is signifi-cant in the low-k phonon region, and around the rotonminimum. The model has 4 parameters: the sound veloc-ity, the roton gap, the roton wave-vector and the roton ef-fective mass. The heat capacity calculated by Landau was in good agreement with early specific heat mea- Donnelly selected data table Donnelly spline Andersen Gibbs Glyde This work ( reference, see text) This work ( E i = 5.071 meV) D e v i a t i on ( pe r c en t ) Wave-vector (¯ -1 ) FIG. 49. Percent deviation plot comparing different datato our measured dispersion curve. Black diamonds: data-base selected by Donnelly et al. , and their spline fit (redcurve); triangles: Andersen et al. ; open circles: Gibbs et al. ; inverted triangles: Glyde et al. . Small blackdots: this work, additional data measured at a different en-ergy, E i =5.071 meV. surements. When high accuracy data became available,substantial quantitative deviations from Landau’s modelwere observed, and several attempts were made to im-prove this formalism. Phillips et al. and Greywall calculated low temperature series expansions for the spe-cific heat based on the series expansion of the dispersioncurve given by Eq. 2. Since published formulas containerrors or misprints , we provide below the correct re-sults. We also expand the series to higher order, which isfound necessary to obtain thermodynamically consistentseries expansions.In the following section, we first provide a numerical calculation of the thermodynamic properties using thedispersion relation determined in the present work. Wethen compare these results with those obtained with our analytical formulas. A. Specific heat: complete numerical calculation. The internal energy E and other thermodynamicalproperties, in particular the specific heat C V = (cid:0) ∂E∂T (cid:1) V ,can be calculated as a function of temperature by ele-mentary statistical physics , using the dispersion rela-tion ǫ ( k ), the Bose distribution function n ( ǫ ( k )), and thedensity of states of an isotropic 3-dimensional system, D ( ω ) dω = ( V / π ) k dk : E = ( k B V / π ) Z k end ǫ ( k ) e ǫ ( k ) kBT − k dk (20)Integration is performed over all the states; in Lan-dau’s model, the upper limit is taken as infinity, the expo-7nential factor making this choice possible in the very lowtemperature limit; for our higher temperature approxi-mations we introduce a finite upper limit of the spectrum,k end .Donnelly et al. concluded that the early neutron datawere ‘consistent’ with high resolution specific heat results(see previous section). Given the large dispersion of theneutron data around their proposed curve, they empha-sized the real need for further neutron scattering studies.We show in Fig. 50 and in Table VII the specific heat cal-culated numerically in all the temperature range belowT λ using the present neutron scattering data. T C v (total) C v (phonons) C v (rotons) S (total) K J/ ( K.mol ) J/ ( K.mol ) J/ ( K.mol ) J/ ( K.mol )0.05 1.037E-05 1.037E-05 2.270E-70 3.459E-060.10 8.260E-05 8.260E-05 1.920E-33 2.760E-050.15 2.770E-04 2.770E-04 3.030E-21 9.276E-050.20 6.513E-04 6.513E-04 3.360E-15 2.187E-040.25 0.001261 0.001261 1.330E-11 4.247E-040.30 0.002157 0.002157 3.170E-09 7.290E-040.35 0.003392 0.003392 1.531E-07 0.0011490.40 0.005016 0.005013 2.738E-06 0.0017040.45 0.007094 0.007069 2.532E-05 0.0024090.50 0.009756 0.009607 1.480E-04 0.0032880.55 0.01330 0.01268 6.203E-04 0.0043760.60 0.01837 0.01634 0.002028 0.0057360.65 0.02613 0.02064 0.005487 0.0074890.70 0.03846 0.02566 0.01279 0.0098410.75 0.05784 0.03147 0.02651 0.013100.80 0.08812 0.03816 0.04994 0.017740.85 0.1329 0.04586 0.08702 0.024340.90 0.1968 0.05469 0.1421 0.033640.95 0.2848 0.06481 0.2200 0.046531.00 0.4018 0.07641 0.3253 0.063971.05 0.5527 0.08969 0.4630 0.087081.10 0.7424 0.1049 0.6375 0.11701.15 0.9755 0.1221 0.8533 0.15491.20 1.256 0.1418 1.115 0.20221.25 1.589 0.1639 1.425 0.26001.30 1.977 0.1889 1.788 0.3297TABLE VII. Specific heat (total, phonon and roton contri-butions), and total molar entropy, at the saturated vaporpressure, as a function of temperature, calculated numeri-cally using the measured dispersion relation (this work). Theuncertainties on the specific heat are ≤ The figure also shows the phonon and the roton contri-butions, defined here as the specific heat obtained by inte-gration over wave-vector ranges separated by the maxonwave-vector k M (see Section VII). The ‘phonons’ thuscorrespond to the range (0-k M ) and the ‘rotons’ to (k M -k end ).A clear deviation between the calculated and measuredspecific heats is seen above 1.3 K. This is not surprising:at high temperatures the dispersion curve itself changesrapidly as the temperature increases, and the excitations -5 -4 -3 -2 -1 S pe c i f i c hea t ( J m o l - K - ) Temperature (K) Total C V (int. 0 to 3.60 ¯ -1 ) Phonons (int. 0 to k M ) Rotons (int. k M to 3.60 ¯ -1 ) C V Phillips et al. C V Greywall FIG. 50. Total specific heat, and phonon and roton contribu-tions, calculated numerically with the measured dispersion re-lation. Specific heat data by Phillips et al. and Greywall are shown for comparison. broaden substantially . This regime is outside thescope of this study, we concentrate here on the low tem-perature properties, where the dispersion curve is unique. Phillips et al. Greywall (cit. Donnelly) Greywall (published fit) ( C v - C v _ C a l c ) / C v Temperature (K) FIG. 51. Percent difference between specific heat measure-ments, and the values inferred from the dispersion relationmeasurement (present work). Open squares: Phillips et al. .Red dots: Greywall data, listed in Refs. 46 and 51. Blue solidline: Greywall’s fit of his data . Deviations are also observed at low temperatures, theyare rather small (Fig. 51). Their probable origin can beunderstood by inspection of Fig. 52, where the phononregime is emphasized. The expanded scale reveals a clearaccident in Greywall’s data around 0.3 K. The curve cal-culated from the neutron data, however, does not displayany accident in this range. An excellent agreement is ob-tained with Greywall’s data if we correct the latter inan obvious way: above 0.33 K we keep his temperaturescale, based on the He vapor pressure thermometer, andbelow this temperature we correct the Curie tempera-ture of his CMN thermometer by simply adding 1 mK.Such a correction is within the possibilities consideredby Greywall in his error handling discussion. It would bedesirable to repeat the heat capacity experiment using8modern thermometric techniques, but this kind of exper-iment remains difficult. Finally, let us mention that thedrop of C V /T observed in Greywall’s data below 0.15 Kis clearly an artifact due to thermal decoupling. Calc. from neutron dispersion Wiebes PhillipsGreywall Greywall, with T Gw + 1mK C v / T ( J m o l - K - ) Temperature (K) FIG. 52. C V /T in the phonon range. The values calculatedfrom the dispersion curve (solid line, this work), compared toheat capacity data. Crosses: Wiebes . Triangles: Phillips et al. . Dots: Greywall . Circles: Greywall’s data, tem-peratures corrected by adding 1 mK; this correction shouldapply below 0.3 K (see text). At higher temperatures, the behavior of the heat ca-pacity is dominated by the exponential growth associatedto the roton gap. Fig. 53 shows that the influence of k end ,the upper integration limit of the dispersion relation, issmall below 1.25 K, the maximum temperature where thelow temperature dispersion relation can be safely used.We find that values of k end ≈ ± − provide good fits.A curve calculated with 2.20 ˚A − , too small a value, isshown for comparison purposes only. A termination ofthe spectrum in this wave-vector region is therefore inrough agreement with specific heat data.To conclude this section, we should say that the sys-tematic difference on the order of 1 to 2% observed be-tween the specific heat calculated from the dispersioncurve in the present work, and the results of heat ca-pacity measurements, is larger than the estimated uncer-tainty of the former ( ≤ B. Specific heat: analytical calculation It is often convenient to have analytical expressionsfor the thermodynamical properties. We calculate in this end = 3.60 ¯ -1 (neutrons if k end = 2.20 ¯ -1 ) C v Wiebes C v Phillips C v Greywall S pe c i f i c hea t ( J m o l - K - ) Temperature (K) FIG. 53. Specific heat calculated numerically with the mea-sured dispersion relation, for an integration limit k end =3.6˚A − . The result obtained using k end =2.2 ˚A − is shownfor comparison. Specific heat measurements by Wiebes ,Phillips et al. and Greywall are also shown. section the low temperature polynomial series expansionsdescribing phonon and roton contributions to the specificheat, and to other thermodynamical properties. 1. Phonon series expansions The properties related to the phonons are calculatedwith Eq. 20 and the following series expansion of thephonon excitation energies: ǫ ( k ) = ck (1 + α k + α k + α k + α k + α k + α k )(21)The integration upper limit is taken as infinity, since attemperatures T << ∆ R / k B the exponential thermal pop-ulation factor suppresses the contribution of the high en-ergy parts of the spectrum. In practice, the analytic cal-culation is done by integration over the energy, ratherthan over wave-vector, and we introduce the inverse se-ries describing the single-valued (phonon) branch of thespectrum:9 k = ωc − α (cid:16) ωc (cid:17) + (cid:0) α − α (cid:1) (cid:16) ωc (cid:17) + (cid:0) − α + 5 α α − α (cid:1) (cid:16) ωc (cid:17) + (cid:0) α − α α + 6 α α + 3 α − α (cid:1) (cid:16) ωc (cid:17) + (cid:0) − α + 84 α α − α α − α α + 7 α α + 7 α α − α (cid:1) (cid:16) ωc (cid:17) + η (cid:16) ωc (cid:17) + O (cid:0) ω (cid:1) where η = 132 α − α α + 120 α α + 180 α α − α α − α α α + 8 α α − α + 8 α α + 4 α − α . The complete analytical formulas for the specific heatare cumbersome, and we only give below those applicablewhen α =0, which is the case in practice (see SectionII A). The corresponding expression is C phononV = A T + C T + D T + E T + K T + L T (22)where the term B T is absent because α =0, and therelevant coefficients are given by: A = π k B V c ~ C = − ( π α k B V ) c ~ ) D = − ( α k B V ζ (7) ) π c ~ E = π k B V ( α − α ) c ~ K = k B V ζ (9)(9 α α − α ) π c ~ L = − ( π k B V ( α − α α − α +3 α )) c ~ ) where ζ is the Riemann zeta function.It is clear from this expansion that, contrarily to whatis generally believed, the coefficients of the specific heatseries expansion are not associated to a single coefficientof the dispersion relation series; E , for instance, containsboth α and α .The α i coefficients have been determined in Sec-tion VII A by fits at low wave-vectors. For the dis-persion curve at saturated vapor pressure, the coeffi-cients c=238.3 m/s, α =0, α =1.55˚A − , α =-4.04˚A − , α =2.30˚A − , α =0, α =0 provide a good fit of the dis-persion curve for k < − . The corresponding coeffi-cients of the specific heat series are therefore: A =0.0831, C =-0.0548, D =0.0653, E =0.0603, K =-0.262, L =0.141,when temperatures are expressed in kelvin and C V inJ/(mol.K).Strong oscillations in the amplitude of successive termsof the series expansions are present for temperatures ≈ > − , the higher order coefficients will vary with theorder of the polynomial, and once a polynomial providinga good description of the dispersion relation is chosen, itmust be consistently used in the thermodynamic calcula-tions. The previous discussion may appear as academic,but it is at the root of the problems encountered in theanalysis of specific heat measurements , as is illus-trated in Fig. 54. The partial contribution due to thephonons estimated by these authors (green dashed linefor Phillips et al. , blue dash-dotted line for Greywall),has the right order of magnitude, but it deviates fromthe present neutron scattering result (black dashed line)above 0.5 K; in particular, the temperature dependenceis clearly different. Gw +1mK) Series expansion K=L=0 Series including K and L terms Calc. neutrons full dispersion Calc. neutrons (0 k maxon ) C v / T ( J m o l - K - ) Temperature (K) FIG. 54. C V /T showing the phonon contribution to thespecific heat and the onset of the roton contribution. Dots:C V measurements with their fitted phonon contributions(see legend). Present work: a) red dotted line: analyticalresult with K=L=0; b) red dash-dotted line: analytical resultwith K =0 and L =0; c) black solid line: numerical integrationof full dispersion curve; d) dashed line: numerical integrationover the phonon region (0 < k < k M ). See Fig. 52 for additionaldetails. The analytical calculations of the specific heat usingseries expansions of the dispersion curve display strongdeviations with respect to the full numeric result, begin-ning at temperatures as low as 0.4 K. Furthermore, theseries expansions used in the previous analysis of specificheat data , being truncated, cannot be directly usedto extract the dispersion relation coefficients. As shownin the figure, removing the terms K and L strongly affectsthe fits. Handled with care, the analytical expansionsare still very useful for the calculation of phonon thermalproperties in different contexts.0 2. Roton region The analytical calculation of the roton contribution tothe thermodynamical properties was already performedby Landau in his seminal papers on roton excitations .In the vicinity of the roton minimum, the excitation en-ergies are described by Eq. 18. In the limiting case whereB R =C R =0, Landau obtains the expression C RV = C L e − ∆ RkBT k B T ∆ R + 34 (cid:18) k B T ∆ R (cid:19) ! (23)with C L = ∆ R k R V √ µ R m √ π / √ k B T / ~ The analysis of the measured specific heats with Lan-dau’s analytical expression is therefore sensitive to a com-bination of several roton parameters (energy, wave-vectorand effective mass).Landau also assumes that | k − k R | << k R . The an-alytical calculation, however, can be done without thislast assumption. We calculate here the average energyof the rotons E R and their specific heat C RV with Eq. 20using Eq. 18 with B R =C R =0, including in the integrandall terms of the expansion of ( k − k R ) . The calculationis performed by integration over wave-vectors. We findan additional term C additV = C L e − ∆ RkBT (cid:18) T τ R (cid:19) k B T ∆ R + 154 (cid:18) k B T ∆ R (cid:19) ! (24)where we introduced the parameter τ R which has thedimensions of temperature: τ R = k R ~ k B µ R m . The addi-tional term is similar to Landau’s expression, with anadditional coefficient T/(2 τ R ). Since τ R ≈ 158 K, thisadditional term is of the same magnitude as the thirdterm in Eq. 23 for temperatures above 1 K, and they areboth very small.In order to improve the accuracy of Landau’s analyticdescription, it is necessary to take into account the asym-metry (cubic term) and the deformation of the minimum(quartic term). Unfortunately, the corresponding expres-sions are extremely complex and cumbersome, they arenot of practical interest. At the analytical level, we aretherefore left with equations providing a rough approxi-mation. As seen in Fig. 55, the difference between Lan-dau’s approximation and the roton contribution calcu-lated numerically with the dispersion curve grows sub-stantially with the temperature. The figure also illus-trates the fact that phonons, defined as excitations belowthe maxon wave-vector, have a non-negligible contribu-tion to the thermodynamic properties at higher temper-atures than commonly believed: the phonon and rotoncontributions become equal at 0.77 K for the heat capac-ity, for instance. S pe c i f i c hea t ( J m o l - K - ) Temperature (K) Neutrons Total C V Phonon contribution Roton contribution Landau formula (rotons) measured C V (Phillips et al.) FIG. 55. The total heat capacity (black solid line), andphonon (blue dash-dotted line), roton (red dashed line) con-tributions to the specific heat calculated numerically withthe measured dispersion relation (present work). Short-dash black line: Landau formula for the rotons contribution.Crosses: specific heat measurements . C. Normal fluid density The normal fluid density is given by the expression ρ n = (cid:18) ~ π k B T (cid:19) Z k end e ǫ ( k ) kBT e ǫ ( k ) kBT − k dk (25)One can obviously integrate analytically the series ex-pansions within the limitations discussed previously. Inorder to take full advantage of the measured dispersioncurve ǫ ( k ), it is convenient to perform a numerical inte-gration, as done above for the specific heat. The normalfluid densities calculated using the dispersion curve aregiven in Table VIII in the temperature range up to 1.3 K.The result is compared in Fig. 56 to selected experimen-tal data . The latter cover the temperature range from1.2 K to the lambda point.The percent difference between previous data andour calculated curve is shown in Fig. 57, on the one handfor the experimental data-base, and on the other handfor the tabulated values. The latter were calculated be-low 1 K using spline approximations to different neutrondata sets. We see that for the purposes of calculating ρ n ,our result matches accurately the direct measurementsat about 1.25 K. Above this temperature, the dispersioncurve itself becomes temperature dependent, a problemthat is beyond the scope of the present article. It is there-fore extremely satisfactory to see that these two sets ofdata agree particularly well. It is also clear that neutrontechniques extend considerably towards lower tempera-tures the range where this parameter has been accuratelydetermined. The temperature range below 1 K is a par-ticularly fertile playground for objects all possible sizes1 -9 -8 -7 -6 -5 -4 -3 -2 -1 This work Data Table in Donnelly and Barenghi n ( g / c m ) Temperature (K) FIG. 56. Normal fluid density calculated using the measureddispersion relation. Experimental points by Maynard, Tamand Ahlers, and Singass and Ahlers (data-base in Ref. 46). immersed in helium and sensitive to its excitations, fromnano-oscillators to huge particle detectors . Maynard Table Brooks & Donnelly Table Donnelly & Barenghi This work (reference) Landau model N o r m a l den s i t y de v i a t i on p l o t ( % ) Temperature (K) FIG. 57. Deviation plot for the normal fluid density. Weshow the percent difference between Maynard’s experimentaldata, or tabulated (essentially, calculated) values , and thevalues calculated using the dispersion relation, taken as thereference (this work). The ‘simple Landau model’ (see text)deviates substantially from our data. Landau’s general formula, Eq. 25, yields when appliedto the first term of the series expansions for the phononand the roton dispersion relation (the so-called ‘simpleLandau model’ ) : ρ ( L ) n = ρ P h ( L ) n + ρ R ( L ) n , with ρ P h ( L ) n = π k B T c ~ ρ R ( L ) n = ~ k R ( µ R m ) / √ π / ( k B T ) / e − ∆ RkBT Since these formulas are frequently used, it is interest-ing to check their validity range. A comparison betweenthe expressions above and our complete numerical calcu-lation with the measured dispersion relation, is shown inFig. 57. Deviations on the order of 10% are found be-low 1 K, even around 0.5 K where this expression is oftenassumed to give a good approximation. The deviation iseven larger if one considers the phonon and roton contri-butions separately, as shown in Fig. 58. The superfluiddensities are defined as ρ s = ρ − ρ n (see Table VIII). N o r m a l den s i t y de v i a t i on p l o t ( % ) Temperature (K) Deviation of the Landau modelfrom the numerical calculation: Phonon+Roton Phonon contribution Roton contribution FIG. 58. Normal fluid density deviation plot: percent differ-ence between the ‘simple Landau model’ normal densities ,and our numerical calculation with the full dispersion curve.Phonon, roton, and total normal density deviations aredefined as 100 ( ρ Ph ( L ) n - ρ Phn )/ ρ Phn , 100 ( ρ R ( L ) n - ρ Rn )/ ρ Rn , and100 (( ρ Ph ( L ) n + ρ R ( L ) n ) - ρ n )/ ρ n , respectively. D. Number density of phonons and rotons The number density of phonons N P h and rotons N R as a function of temperature, of interest for instance inthe ballistic regime, is given by the expressions: N P h = (2 π ) − R k M ( e ǫ ( k ) kBT − − k dkN R = (2 π ) − R k end k M ( e ǫ ( k ) kBT − − k dk The result is shown Fig. 59 (see Supplemental Materialat [URL will be inserted by publisher] for data tables.),converted to the number of phonons and rotons per he-lium atom, in order to emphasize the importance of theexcitation density as the temperature exceeds T ≈ T ρ ρ n ρ Phn ρ Rn K g/cm g/cm g/cm g/cm ρ (from Ref. 46) and normal fluiddensities (total, phonon and roton contributions) as a functionof temperature, calculated numerically using the measureddispersion relation (this work). See Supplemental Material at[URL will be inserted by publisher] for more detailed tables. IX. CONCLUSIONS The main result of this work is the determination ofthe dispersion relation ǫ ( k ) of superfluid He in the wholewave-vector range at saturated vapor pressure, and ina large range for pressures up to 24 bar. Earlier neu-tron data were scarce, of low resolution, often contra-dictory. Roton and maxon parameters were affected bythe instrumental resolution and the wave-vector rangeover which the dispersion curve was averaged out, lead-ing to a substantial spread in the values found in theliterature; as the experimental techniques progressivelyimproved, lower roton energies, wave-vectors and effec-tive masses were obtained. Motivated by the lack of highaccuracy data, Donnelly and coworkers designed a‘recommended dispersion relation’, a spline curve mak-ing its way across the large error bars of some of the dataavailable at that time.A measured dispersion curve is now available. It ischaracterized by small error bars, and it agrees wellwith measurements performed by other techniques: ul-trasound and compressibility at low wave-vectors, as wellas heat capacity, a technique providing an important -7 -6 -5 -4 -3 -2 -1 N u m be r o f e xc i t a t i on s pe r a t o m Temperature (K) Phonons/atom Rotons/atom FIG. 59. Number of phonon and roton excitations per atom.The numerical calculation performed with the measured lowtemperature dispersion curve, is accurate for T < global test over a large wave-vector range.Our neutron measurements have been performed usingtime of flight (TOF) techniques with a 10 pixels detec-tor matrix. Statistical errors are in general negligible inthis work. Considerable effort has been devoted to under-stand the sources of systematic errors and the resultingcorrections. This also allowed us to identify problemsthat affected earlier measurements.The energy scale of our spectrometer has been re-fined by a calibration at a single point, the roton en-ergy ∆ R =0.7418 meV determined with an uncertainty of1 µ eV by Stirling . This is the main source of un-certainty of our results for ǫ (k). It affects them in aglobal way, and they can therefore be corrected (the en-ergies should be corrected proportionally, as indicated byEq. 13, and the wave-vectors recalculated using Eq. 11)if a more accurate value of ∆ R becomes available. Theremaining systematic errors originate from small defectsin the IN5 spectrometer construction. Comparing dataat two different neutron energies has allowed us to showthat the corresponding effects are smaller than the globalsystematic error quoted above. They can be seen as os-cillations in the curves, in particular around the rotonminimum. No effort was done to suppress these by av-eraging or smoothing, since they are a good indicator,for further applications of the present data, of residualsystematic errors.The measured dispersion curve has allowed us to cal-culate the thermodynamic properties of superfluid he-lium in the temperature range below 1.3 K, where thenon-interacting-excitations picture is valid. Our resultsprovide the most accurate values for the heat capacity(direct measurements are affected by delicate issues ofthermometry and temperature scales), the normal fluiddensity, and other thermodynamical properties. Theyalso provide independently the contributions from exci-tations in different wave-vector ranges, and in particularthe phonon and the roton contributions. This informa-3tion is of interest, for instance, for low temperature trans-port phenomena, the damping of nano-resonators in he-lium below 1 K , particle detection , and severalother effects determined by ballistic phonons and rotonquasiparticles.In addition to these numerical calculations, we de-veloped analytical expressions for the thermodynamicalproperties. We found that specific heat measurementswere analyzed using inconsistent low temperature seriesexpansions; no disagreement is observed between the neu-tron results and the specific heat data using our coherentexpressions.The results on pure superfluid He have analoguesin other systems. Rotons and maxons are studied in He in reduced dimensions, confined geometries, dropletsspectrometry , but also in He , in cold atomicgases , classical liquids , etc.: they are a generalfeature of many-body interacting systems.Throughout this manuscript, we have considered ‘el-ementary excitations’ (poles of the fully renormal-ized single-particle propagator) and ‘statistical quasi-particles’ (elementary excitation energies defined as thefunctional derivative of the total energy with respectto the distribution function), in the sense described byBalian and de Dominicis , as essentially identi-cal concepts. The description of the thermodynamic properties using a self-consistent temperature dependentdispersion relation has been explored by Donnelly andRoberts , it describes phenomenologically the thermo-dynamic properties. The other route, presently exploredby different types of many-body microscopic calcula-tions, seems more promising. The difference betweenthe measured specific heat and that calculated for non-interacting quasi-particles using the dispersion curve hasa simple behavior (see Fig. 50), and provides informationabout roton-roton interactions . Additional contri-butions have been predicted for the heat capac-ity, it would be interesting to have a theoretical estimateof their magnitude and make a quantitative comparisonwith the present data. X. ACKNOWLEDGEMENTS We are grateful to S. Triqueneaux and X. Tonon fortheir help with the experiments, to B. Gu´erard and P.van Esch for helpful discussions on the IN5 detectorsproperties, and to H. Maris for pointing out Ref. 37to us. This work was supported by the European Mi-crokelvin Platform. The research leading to these resultshas received funding from the European Union’s Horizon2020 Research and Innovation Programme, under GrantAgreement no 824109. ∗ Corresponding author: [email protected] † Institute of Engineering Univ. Grenoble Alpes P. Nozi`eres and D. 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