Kinetic energy and momentum distribution of isotopic liquid helium mixtures
KKinetic energy and momentum distribution of isotopic liquid helium mixtures
Kinetic energy and momentum distribution of isotopic liquid helium mixtures
Massimo Boninsegni a) Department of Physics, University of Alberta, Edmonton, Alberta, Canada T6G 2G7 (Dated: 8 October 2018)
The momentum distribution and atomic kinetic energy of the two isotopes of helium in aliquid mixture at temperature T =2 K are computed by quantum Monte Carlo simulations.Quantum statistics is fully included for He, whereas He atoms are treated as distinguish-able. Comparison of theoretical estimates with a collection of the most recent experimentalmeasurements shows reasonable agreement for the energetics of He and pure He. On theother hand, a significant discrepancy (already observed in previous works) is reported be-tween computed and measured values of the He kinetic energy in the mixture, especiallyin the limit of low He concentration. We assess quantitatively the importance of Fermistatistics and find it to be negligible for a He concentration (cid:46) He kinetic energy contributionassociated with the tail of the experimentally measured momentum distribution.
I. INTRODUCTION
Liquid mixtures of the two isotopes of helium have longbeen regarded as an interesting playground for quantummany-body physics. For example, in the limit of low He concentration ( x ), in which it remains homogeneousas the temperature T →
0, such a mixture is perhaps thecleanest and most easily controlled experimental realiza-tion of a Bose superfluid ( He) in the presence of mobileimpurities. In that limit, He behaves very nearly asan ideal, essentially non-interacting Fermi gas whose de-generacy can be tuned by varying x . As x is increased,both the interaction of He quasiparticles and the effectof He Fermi statistics become more and more significant,as quantitatively expressed by higher values of the Fermimomentum and temperature. Indeed, Fermi statisticsdecisively contributes to shaping the experimental phasediagram of the mixture at temperatures below ∼ State-of-the art theoretical calculations based on re-alistic interatomic potentials have provided considerablequalitative and quantitative insight into the physics ofthe mixtures. For example, Path Integral Monte Carlo(PIMC) simulations have yielded definite predictionsfor the effective mass and chemical potential of a single He atom dissolved in superfluid He, and quantitativelyreproduced the experimentally observed, monotonic de-crease of the He superfluid response with increased x ,in the miscibility region.The same agreement between theory and experimenthas been lacking, however, for the single-particle atomickinetic energy, which can be obtained as the secondmoment of the momentum distribution f α ( k ), α =3 ,
4, in turn measurable by means of neutron scatter-ing experiments.
While there is reasonable quantita-tive agreement between the experimental and theoretical a) Electronic mail: [email protected] estimates of the He kinetic energy per atom ( K ), re-ported values of the corresponding He quantity ( K )have been consistently below the theoretical ones, byamounts worth as much as several K (peaking at around50% of the experimental value in the x → K on x at low T , experimental data showvirtually no dependence of K on x , despite the substan-tial ( ∼ x = 0 (pure He) and x = 1 (pure He) limits; thissurprising observation was made in different independentmeasurements.
A discrepancy of this magnitude for a quantity like thekinetic energy, in a relatively simple system like the oneconsidered here, could possibly point to some significantgap in the present understanding of the physics of themixture, specifically the local environment experiencedby a single He atom dissolved in superfluid He. It wassuggested in Ref. 8 that the disagreement may point to“effects of Fermi statistics” as the (unexplained) causeof the departure from the expected density dependenceof the single-particle mean kinetic energy. While it iscertainly true that the Fermi component of the mixtureis that for which the disagreement between theory andexperiment is quantitatively most important, one is hardpressed thinking of a physical mechanism underlain byFermi statistics whose overall result would be that of low-ering the kinetic energy. This seems especially the casein the low x limit, and at a temperature as high as T =2K, where effects of Fermi statistics should be relativelysmall. For example, assuming a He effective mass of theorder of 2.3 times the bare mass one can estimate thedegeneracy temperature of the He fluid in the mixturefor x = 0 . (cid:46) . is that the root of the dis-crepancy may lie in the (possibly large) contribution tothe kinetic energy per particle from the tail of the mo-mentum distribution, the estimate being quite sensitiveto the model function utilized to fit the experimentaldata, particularly for the Fermi component. Specifically,it was contended that the experimental underestima- a r X i v : . [ c ond - m a t . o t h e r] O c t inetic energy and momentum distribution of isotopic liquid helium mixtures 2tion of the K may stem from the use of a free Fermigas type model for the n ( k ), inadequate to describe thesignificant high-momentum tail of the observed distribu-tion. Indeed, is was proposed therein that a more reli-able comparison may be between the calculated and ob-served n ( k ) rather than their second moment (namely K ) which is not well determined experimentally.This is the kind of quantitative, well-defined questionsthat computer (QMC) simulations can usually address ef-fectively. Unfortunately, numerically exact results for the n at finite x are difficult to obtain, due to the well knownfermion “sign” problem, plaguing any quantum MonteCarlo technique, including PIMC. In Ref. 4, use wasmade of the so-called restricted path integral (RPIMC)technique, which removes the sign instability at thecost of making an uncontrolled approximation, namelyrestricting paths to regions in which a trial many-fermiondensity matrix (in this specific case that of a system offree fermions) is positive. The use of this approximationcan be justified in the x → He shouldbehave as an ideal Fermi gas. On the other hand, inthe opposite (pure He) limit the results afforded by thisapproach are only semi-quantitative. In any case, noRPIMC results have been reported to date of the mo-mentum distribution of any Fermi system.At least at T =2 K, however, it is conceivable thatone may obtain reliable results by neglecting He Fermistatistics altogether, i.e., by regarding He atoms as dis-tinguishable . This approximation removes the sign prob-lem and allows one to compute by QMC the momentumdistribution for both components, affording a direct com-parison of theoretical and experimental results. This isthe computational strategy adopted in this work.This paper reports results of QMC simulations of themixture in the 0 ≤ x ≤ T =2K, although a few simulations at T =1 K were carriedout as well for comparison. The main physical quan-tities of interest are the momentum distributions f α ( k )and the atomic kinetic energies K α ( x ). Quantum statis-tics is fully included for the Bose component, namely He, whereas as stated above quantum exchanges are ex-cluded for the He fluid (i.e., He atoms are assumed toobey Boltzmann statistics). Quantitative arguments arefurnished to the effect that this is indeed an excellent ap-proximation at T =2 K and for x (cid:46) . K at low x , while for K the agreement with experiment in the whole x range,while not impressive, seems satisfactory. It is worth men-tioning that the calculation carried out here yields a ki-netic energy values in reasonable agreement with exper-iment at x = 1, i.e., for pure He, where effects of Fermistatistics should be most important. Actually, the totalenergy value computed at this temperature for pure Heat its equilibrium density seems to be in closer agree-ment with experiment than the RPIMC one from Ref.10. Whether that is the result of a fortuitous compen-sation of error, or whether maybe it points to exchangesbeing less important in fluid He at T =2 K than previ-ously thought is unclear, but certainly worthy of furtherinvestigation.The one-body density matrix for the He component in the x → He atomdissolved in superfluid He as penetrating a potential bar-rier as it moves past the surrounding He atoms. The He momentum distribution deviates significantly from aGaussian, displaying both a low momenta enhancement,as well as a slowly decaying tail at high momenta, al-together supporting the contention of Ref. 3 and 9, andsuggesting that the disagreement between theoretical andexperimental estimates may be removed by the use of anappropriate model for n , featuring a high momentumtail, to fit the experimental data.The remainder of this paper is organized as follows:in Sec. II the model of the system is introduced, andthe computational methodology briefly reviewed; the re-sults are presented in detail in Sec. III; conclusions areoutlined in Sec. IV. II. MODEL AND METHODOLOGY
The mixture is described as an ensemble of N pointlikeparticles, of which N x are He, which are regarded asdistinguishable, the rest N (1 − x ) He atoms, obeyingBose statistics. The system is enclosed in a cubic cell withperiodic boundary conditions in the three directions.The quantum-mechanical many-body Hamiltonian ofthe system reads as follows:ˆ H = − (cid:88) iα λ α ∇ iα + (cid:88) i Bosons , with the inclusion ofquantum-mechanical exchanges. This point will be dis-cussed in depth in Sec. III.inetic energy and momentum distribution of isotopic liquid helium mixtures 3 TABLE I. Theoretically computed (columns marked with(T)) kinetic energy per particle K ( x ) of Boltzmann He, and K ( x ) of Bose He for mixtures of different concentration x ,at a temperature T =2 K. Results are in K. Included are alsoexperimental results (columns marked with (E)) from Refs.5–8, as reported in Ref. 8 (Table I). Statistical uncertainties,in parentheses, are on the last digit(s). The results at x = 0(100)% refer to a single He ( He) atom in bulk He ( He). x (%) ρ (˚A − ) K (T) K (E) K (T) K (E)0 0.02187 18.4 (2) 15.0(1) 16.0(5)10 0.02140 17.96(8) 12.1(4) 14.87(6) 13.8(6)20 0.02090 17.54(6) 10(2) 14.89(4)35 0.01995 16.56(8) 10.4(3) 14.20(5) 12.0(6)65 0.01822 14.52(4) 11.8(7) 12.62(4)100 0.01550 12.42(6) 12(1) 10.0(1) Details of the simulation are standard; for instance, theshort-time approximation to the imaginary-time propa-gator used here is accurate to fourth order in the timestep τ (see, for instance, Ref. 17). All of the resultspresented here are extrapolated to the τ → τ = (1 / − are in-distinguishable from the extrapolated ones, within thestatistical uncertainties of the calculation. We carriedout simulations of mixtures comprising N = 256 parti-cles altogether.The physical quantity of interest, besides the usual en-ergetic and structural ones, as well as the He super-fluid fraction (computed using the well-known windingnumber estimator), is the one-body density matrix n α ( r , r (cid:48) ) = (cid:104) ˆ ψ † α ( r (cid:48) ) ˆ ψ α ( r ) (cid:105) (2)where (cid:104)· · · (cid:105) stands for thermal expectation value, andˆ ψ α , ˆ ψ † α are field operators for the two components. For atranslationally invariant and isotropic system like a ho-mogeneous fluid, it is n α ( r , r (cid:48) ) ≡ n α ( | r − r (cid:48) | ). The one-body density matrix is easily accessible for both compo-nents, using the worm algorithm. The momentum distri-bution is obtained as a three-dimensional Fourier trans-form, namely f α ( k ) ≡ f α ( k ) = 4 πk (cid:90) ∞ dr r sin( kr ) n α ( r ) (3)with the normalization1(2 π ) (cid:90) d k f α ( k ) = 1 (4)which fixes to unity the value of n α ( r ) at the origin. III. RESULTS Our results for the kinetic energy per helium atom K α ( x ) at a temperature T =2 K, are shown in Table I.The values of the density of the liquid mixture at whichcalculations were carried out are taken from Table I of Ref. 8. It is worth mentioning again that at this temper-ature the mixture is homogeneous , i.e., no phase separa-tion takes place at any x . The results presented in TableI are consistent with those of the previous calculations, taking into account slight differences in density, for Heconcentrations below ∼ x , the He ki-netic energy is underestimated in this work, as effects ofquantum statistics become important; for example, for x = 35% the value of K reported here is as much as 1 Klower than that of Ref. 4, in which Fermi statistics is atleast in part included through the nodal restriction. Onthe other hand, the He kinetic energy obtained here isconsistent with that of Ref. 4.The first immediate observation is the large discrep-ancy between theoretical and experimental estimates of K , in line with what reported in previous works. Thedeviation is largest in the limit x → 0, whereas in the op-posite limit, i.e., x → 1, the theoretical estimate for K is actually consistent with experiment, obviously makingallowance for the relatively large uncertainty quoted inRef. 8. For He the agreement is better but not en-tirely satisfactory either; specifically, in at least one case( x =35%) the difference between experimental and theo-retical estimate is well outside the quoted statistical un-certainties.The limit of low He concentration x is that in whichthe disagreement between theory and experiment, re-garding the quantitative determination of K , is great-est in magnitude, and as mentioned above the sugges-tion was made that poorly understood effects of Fermistatistics may be responsible for it. Because in thiswork exchanges of indistinguishable He atoms are ne-glected, it seems appropriate to offer a quantitative jus-tification for this approximation, which is crucial in or-der to carry out the QMC simulation without incur-ring into the ‘sign” problem. One may begin by notingthat, in order for exchanges of identical particles to occursufficiently frequently, the characteristic spatial exten-sion Λ T ≡ (2 λ/T ) / of a single-particle “path” shouldbe of the order of the average distance d between twosuch particles. For, the relative probability for an ex-change including n particles to take place, is roughlyproportional to γ n , where γ = exp[ − d / Λ T ]. Considerfor definiteness a x = 20% mixture at a temperature T =1K; assuming an unpolarized He component, the averagedistance of two He atoms with parallel spin projectionsis ∼ . T ∼ . He effec-tive mass equal to twice the bare mass is assumed; thus, γ ∼ × − , which can be compared, for example, to thevalue ∼ − for liquid He at the superfluid transitiontemperature. Thus, one may expect exchanges of Heatoms to be strongly suppressed in this system, at leastdown to this temperature. A similar analysis shows thatthis conclusion holds a fortiori for mixtures with lower x . Direct, quantitative support for the above conclusionis offered by simulations of a fictitious mixture in which both He and He atoms are assumed to be spin-zeroBosons, with exchanges allowed for both components. At T =2 K and x = 20% it is found that exchanges of Bose He atoms are exceedingly infrequent; specifically, over99% of all single-particle paths close onto themselves,inetic energy and momentum distribution of isotopic liquid helium mixtures 4and this percentage remains above 90% as the temper-ature is lowered to 1 K. Moreover, the rare exchangesthat occur mainly involve relatively few particles (of theorder of five). As expected, exchanges occur even moreinfrequently at lower He concentration. Because ex-changes of identical particles are sampled exactly in thesame way in Fermi or Bose systems (the difference beingrather in how contributions to physical observables as-sociated to exchange paths are added to averages), onecan a fortiori conclude that effects of Fermi statistics inan isotopic helium mixture are negligible for concentra-tions below (cid:46) 20% at least down to temperature T =1 K;in other words, regarding He atoms as distinguishableparticles is an excellent approximation. Thus, one mayconfidently expect that in this region of the phase dia-gram, estimates of most structural and energetic prop-erties of the mixtures computed in this way should befairly accurate. However, special care must be exercisedwhen it comes to the one-body density matrix and themomentum distribution, which, as the example of liquidparahydrogen near freezing shows, can display importantsignatures of quantum statistics, virtually undetectablein all other observables. At greater He concentrations effects of Fermi statis-tics are expected to become increasingly important; cu-riously, however the disagreement between theory andexperiment regarding K is quantitatively smaller in thislimit. For example, for pure liquid He at T =2 K theatomic kinetic energy at the experimental density quotedin Ref. 8 is consistent with experiment. It is also worthmentioning that a calculation carried out in this work atdensity ρ = 0 . − for pure liquid He at T =2 Kyields an energy per particle equal to -1.51(4) K, actu-ally in rather good agreement with experiment, at leastcomparable to (if not better than) that afforded by theoriginal RPIMC calculation for normal He, yieldingapproximately -1.3 K. While this could be the result of afortuitous compensation of error, it suggests that quan-tum exchanges may be perhaps quantitatively less impor-tant than expected for this system, at this temperature.One way to test this hypothesis may be that of incorpo-rating the He effective mass enhancement in the nodalrestriction of the calculation of Ref. 10 (also based onthe free Fermi gas approximation); this has the resultof suppressing in part exchanges, because of the shorteratomic thermal wavelength.We now illustrate our results for a mixture with x =10%, for which the disagreement between experimentallydetermined and theoretically computed K is rather large(Table I). Fig. 1 shows the pair correlation functions g ( r )computed at T =2 K, both that for two He atoms as wellas that between a He and a He atom; although thereare some detectable differences, it is clear that the localenvironment experienced by a He atom in the mixture isessentially the same as that experienced by a He atom.There is no evidence that He atoms push He atomsfurther away, in order to reduce their kinetic energy ofconfinement, as speculated, for instance, in Ref. 8. Thus,since the interatomic potential is the same for all pairs,and since as stated above effects of Fermi statistics arenegligible, one can account for the He kinetic energy in-crease with respect to the pure He case (an increase of FIG. 1. Color online. Pair correlation functions g ( r ) for aliquid helium mixture with a He concentration x = 10% attemperature T =2 K, computed by QMC simulation. Thedensity of the mixture is 0.0214 ˚A − . Shown are the resultsfor the He- He (triangles) and He- He correlation functions.Statistical errors are smaller than the sizes of the symbols. roughly roughly 6 K) simply based on the higher equilib-rium density of the mixture. FIG. 2. Color online. One-particle density matrix (log scale,base 10) computed by QMC for the He (diamonds) and He(circles) components of a mixture qith x = 10%, at T =2 K.The density of the mixture is 0.0214 ˚A − . When not shown,statistical errors are smaller than the sizes of the symbols.Dashed line is an exponential fit to the He result for r > Because the kinetic energy is experimentally deter-mined through a measurement of the momentum distri-bution, we now turn to the discussion of this quantityin the mixture, specifically beginning with the one-bodydensity matrix. Fig. 2 shows the result for the samethermodynamic conditions of Fig. 1. The two curvesare nearly indistinguishable up to a distance of the orderof the diameter of the repulsive core of the interatomicinetic energy and momentum distribution of isotopic liquid helium mixtures 5potential, as expected displaying markedly different be-havior at long distances.The He density matrix plateaus at long distance to avalue slightly above 3%, which is the estimate of the con-densate fraction. This is approximately 25% lower thanthe value in pure He at T =2 K, at the considerablyhigher equilibrium density ρ = 0 . − . The de-crease of the condensate fraction is due to the presence ofthe He impurities, which have the effect of inhibiting inpart long exchanges of He atoms, which underlie bothBose condensation as well as the superfluid response. The He superfluid fraction ρ S is 0.17(3), which is consistentwith the result quoted in Ref. 4, where the RPIMC wasused. Its value in the pure He system at this tempera-ture (at the aove mentioned density ρ ) is close to 0.48.At these low x , the He density matrix and momentumdistributions largely reproduce those for pure bulk He,extensively discussed elsewhere. We therefore now fo-cus on the He component.The n ( r ) shown in Fig. 2 has obviously a very differ-ent behavior from the n ; the first obvious thing to no-tice is that, despite the neglect of quantum statistics, itis very different from a Gaussian, which is what it wouldbe for a fluid of distinguishable quantum particles. It ismonotonically decreasing, in a way that at large distancesis consistent with an exponential decay, within the un-certainties of the calculation. Although a similar decaycan be observed in the one-body density matrix of liq-uid parahydrogen at freezing, in that context it is due toquantum-mechanical exchanges; in this case, on the otherhand, He atoms are regarded as truly distinguishable.Rather, the exponential decay at long distances of the n is consistent with the Landau-Pomeranchuk notion ofa He atom penetrating a potential barrier, representedby the surrounding, nearly homogeneous superfluid Hemedium.As noted above, the one-body density matrix oftendisplays effects of quantum statistics that do not showup (as obviously) in structural or energetic properties ofthe system; thus, one need asses the possible effect ofthe neglect of quantum statistics on the results shownin Fig. 2. One may expect deviations between the one-body density matrix computed by treating particles asdistinguishable, and one in which Fermi statistics weretaken into account, to appear at a distance of the orderof the average separation between two exchange mates,i.e., two He atoms with parallel spin projections, whichis close to 10 ˚A at the physical conditions of the resultsof Fig. 2; noting the exponentially decreasing behavior,and considering that the long-range part of the n ( r ) af-fects the low-k part of the f ( k ), we may conclude thatwhatever change Fermi statistics may impart to the n ( r )computed here, it is likely to have very little effect on the He kinetic energy.Fig. 3 shows the resulting. theoretically predicted mo-mentum distribution f ( k ) for the He component, ob-tained from the computed n ( r ) through Eq. 3; specif-ically, a numerical integration was performed based ondata up to a distance r = 7 . FIG. 3. Color online. He momentum distribution f ( k )(solid line, log scale, base 10) for a liquid helium mixture witha He concentration x = 10% at temperature T =2 K (solidline). The function f ( k ) is obtained using Eq. 3 from theone-body density matrix n ( r ) (shown in Fig. 2) computedby QMC. The density of the mixture is 0.0214 ˚A − . Statis-tical errors are not visible on the scale of the curve. Dashedline represents a Gaussian momentum distribution yieldingthe same value of the He kinetic energy per atom, namely18.1(1) K. long-range part of the n ( r ) is not visible on the scale ofthe curve as shown in Fig. 3. Also shown in Fig. 3, forcomparison, is a Gaussian model momentum distributionyielding the same kinetic energy per particle as the com-puted f ( k ).Obtaining accurate estimates for f for momentagreater than k ∼ . − using the above procedure,is rendered problematic by the discretization of n ( r ).However, the result shown in Fig. 3 suffices to illustratethe main physical conclusions. The first obvious observa-tion is significant deviation from a simple Gaussian, bothat low momenta, where f gains strength due to the en-hanced delocalization of a He atom in superfluid He,as well as at high momenta, as a result of hard core re-pulsive interaction with nearby, heavy He atoms, whichimpart to the dissolved He atoms its renormalized mass.It is precisely the presence of this long, non-Gaussiantail in the f ( k ) (note the logarithmic scale Fig. 3), thatrenders the extraction of the kinetic energy from f quitedelicate, as already suggested by several authors. For,a substantial contribution to the kinetic energy comesfrom the tail; taking for example the two distributionsshown in Fig. 3, in the case of the Gaussian the por-tion of the distribution for momenta greater than 3 ˚A − contributes a mere 3% of the total kinetic energy, butclose to 20% for the computed f . Thus, the use of aninadequate model to fit the experimentally measured dis-tribution, especially one that does not properly describethe tail, can easily lead to an underestimation of K .inetic energy and momentum distribution of isotopic liquid helium mixtures 6 IV. CONCLUSIONS We have carried out a computational study of isotopicliquid helium mixtures at a temperature T =2 K, with theaim of possibly shedding light on a present disagreementbetween theoretically computed and experimentally mea-sured atomic kinetic energies. Our study is based on firstprinciple computer simulations, whose only input is theinteratomic pair potential; the only approximation builtinto our simulations is the neglect of quantum (Fermi)statistics for the He component. The results of the sim-ulation confirm basic theoretical arguments suggestingthat this is an excellent approximation for low He con-centration (less than ∼ He limit.The momentum distribution for He at low x , wherethe disagreement between theory and experiment is mostsubstantial, displays a slowly decaying tail at high mo-menta, arising from the short-range, repulsive interactionof a light He atom with the surrounding cage compris-ing heavy He atoms. This effect is expected to becomeprogressively less important as the He concentration in-creases (and the equilibrium density correspondingly de-creases).It is worth noting that the non-condensate part of themomentum distribution of superfluid He, which is theone that contributes to the He kinetic energy, does notfeature the same kind of long range tail, , but can ac-tually fairly well be approximated by a Gaussian. Thisis why the determination of the kinetic energy is moreaccurate than for He in the low concentration mixture.On the other hand, as x increases the He componentturns normal, and concurrently the agreement betweenthe computed and experimentally measured K worsens(see result at x = 35% in Table I), the experimental es-timate again falling below the theoretical one.In conclusions, this work supports the hypothesis firstproposed in Ref. 3, subsequently expounded on in Ref. 9,that the disagreement between reported theoretical andexperimental estimates of the kinetic energy of He inthe mixture at low temperature, in the limit of low Heconcentration, may be the result of the model utilized tofit the measured momentum distribution. ACKNOWLEDGMENTS This work was supported in part by the NaturalSciences and Engineering Research Council of Canada (NSERC). 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