Inelastic neutron scattering as a confirmation of a new type of gapped surface excitations in liquid helium
aa r X i v : . [ c ond - m a t . o t h e r] N ov Inelastic neutron scattering as a confirmation of a new type of gapped surfaceexcitations in liquid helium
P. D. Grigoriev , , A. D. Grigoriev , A. M. Dyugaev L.D. Landau Institute for Theoretical Physics, Chernogolovka, Russia National University of Science and Technology ”MISiS”, 119049 Moscow, Russia and Samara State Technical University, 443100, Samara, Russia (Dated: November 13, 2018)We analyze the experimental data [1] on inelastic neutron scattering by a thin 5-atomic-layer filmof liquid helium at three different temperatures: T=0.4K, 0.98K and 1.3K. These data were partiallypublished previously,[2–4] but here we present them in a better quality and at various temperatures.The neutron scattering intensity plots, in addition to the previously know dispersion of phononsand ripplons, suggest a branch of gapped surface excitations with activation energy ∼ .
5K and thedispersion similar to that expected for surfons – the bound quantum states of helium atoms aboveliquid helium surface, proposed and investigated theoretically [5, 6]. These data, probably, providethe first direct experimental confirmation of surfons. Before these surface excitations received onlyindirect experimental substantiation, based on the temperature dependence of surface tension co-efficient [5, 6] and on their interaction with surface electrons [7, 8]. The existence of surfons as anadditional type of surface excitations, although being debated yet, is very important for variousphysical properties of He surface. We also analyze previous numerical results on excitations in liq-uid helium and argue that surface excitations similar to surfons have been previously obtained bynumerical calculations and called resonance interface states [21].
I. INTRODUCTION
A deep understanding of the processes on the surface ofliquids is important for various fields of natural science:physics, chemistry, biology. The microscopic descriptionof liquid surface is a rather complicated problem, and var-ious theoretical techniques have been applied to advanceit. [9] At low temperature the quantum nature of surfaceexcitations is important, which becomes apparent in liq-uid helium and can be experimentally studied, e.g., usingthe interaction of these excitations with surface electrons[10–13]. In bulk liquid helium the excitations are wellknown both from microscopic theory [14, 15] and fromthe extensive thermodynamic and neutron scattering ex-periments. The microscopic description of the surface ex-citations in liquid helium is more complicated because ofspatial inhomogeneity of this problem. This problem wasquite successfully studied using the numerical variationmethods with the Feenberg wave function in the so-calledhypernetted-chain approximation [16–21]. These numer-ical results were used to analyze the experimental dataon inelastic neutron scattering by liquid helium films,[2–4, 22, 23] the temperature dependence of the surface ten-sion coefficient [24] and other thermodynamic properties[25].In liquid He there is only one type of gapless sur-face excitations – the quanta of surface waves, calledripplons. For the wavelength much larger than inter-atomic distance but smaller than the capillary length κ − ≈ . ω q = αρ q tanh ( qd ) , (1)where α is surface tension coefficient, ρ is the density of liquid, q is the ripplon wave number, and d is thewidth of liquid helium film. For short-wave-length rip-plons with q & − the ripplon dispersion ω ( q ) be-comes softer than in Eq. (1) and saturates at energy ~ ω D ≈ . meV ≈ K , which was obtained numerically[16–21] and observed using the inelastic neutron scatter-ing by liquid helium films. [2–4]Recently, an additional new type of surface excitationshas been proposed semi-phenomenologically to explaintoo strong temperature dependence of surface tensioncoefficient α ( T ).[5, 6] These excitations, called surfons ,can be considered as the quantum states of He atomslocalized above liquid surface. Surfons resemble the An-dreev states of He atoms in the He- He mixture [27], orthe states of He and He atoms on the surface of liquidhydrogen.[28] According to this phenomenological model[5, 6], the surfons are localized only along the z -axis per-pendicular to helium surface and can propagate along thesurface. Their dispersion is ε ( k ) ≈ ∆ + k k / M ∗ , (2)where k k is a 2D surfon momentum along the surface,∆ > k k = 0, andtheir effective mass M ∗ is of the order of the atomic Hemass M = 6 . · − g. The surfon activation energy ∆is weakly temperature-dependent,∆ ( T ) = E s − µ ( T ) , (3)where µ ( T ) is the temperature-dependent chemical po-tential of a liquid He, µ ( T = 0) ≈ − . K , and E s is the discrete energy level of He atom at liquid surface,dressed by the interaction with other atoms, ripplons andphonons. At low temperature T ≪ ∆ the surfon concen-tration is exponentially small.Currently there are several experimental facts whichcan be treated as indirect substantiation of surfons. Thefirst two are related to the interaction of surface electronswith surfons, which provides an additional temperature-dependent scattering mechanism of surface electrons.This additional scattering mechanism can considerablyimprove [7] the agreement between the observed [29] andcalculated [30] mobility of surface electrons. Surfonscan also explain the observed [11] temperature-dependentshift of the transition line between two lowest electronstates above liquid helium or solid hydrogen surface.[8]Finally, the surfons can explain [5, 6] the long-standingpuzzle of the observed [31] too strong temperature de-pendence α ( T ) of the surface tension coefficient of liquidHe. The comparison can be used to estimate the surfonactivation energy for both He isotopes from experimentas [6] ∆ He ≈ . K, ∆ He ≈ . K (4)corresponding to E He s ≈ − . K and E He s ≈ − . K .This value ∆ He ≈ . K , obtained from the fitting of α ( T ),[6] is in a reasonable agreement with the energygap ∆ ≈ . K of resonance interface states obtainedfrom the numerical calculations in Ref. [21] (see be-low) and with ∆ He ≈ . K obtained from the semi-phenomenological description in Ref. [6]. Fitting thesurface tension of a thick He film, assuming that only rip-plons and surfons make considerable contribution, givesan upper estimate of the surfon effective mass:[6] M ∗ ≈ . M , and M ∗ ≈ . M , where M = 6 . · − gand M = 5 . · − g are the free atomic masses of Heand He correspondingly. The effective surfon mass M ∗ in its in-plane motion is, probably, renormalized by in-teraction with liquid. Note, that the inclusion of the sec-ond branch of surface excitations, obtained numericallyin Ref. [20], also considerably improves the agreementbetween experiment [31] and theory [24] on α ( T ) of thickHe films. The temperature dependence of surface tensioncoefficient α ( T ) of thin He films cannot also be fittedwithout additional type of surface excitations, thoughthe ”breathing modes” (or ”quantized bulk phonons”),obtained in Refs. [16–18, 22] make a considerable contri-bution [33] to α ( T ).The microscopic substantiation of the surfon exis-tence was provided [6] by the solution of one-particleSchr¨odinger equation for a He atom above liquid He sur-face in the effective one-dimensional potential V ( z ) cre-ated by the interaction with other He atoms in the lowerhalf-space (Hartree approximation). The correspondingone-particle Schr¨odinger equation is uniform in the x - y plane and does not take into account the correlationeffects. However, it definitively gives a discrete quasi-stationary energy level E s ≈ − . K < E s of this quasi-stationary energy level to E s ≈ − K , as was semi-phenomenologically estimated in Ref. [6], but they do not destroy these excitations. Thisquasi-stationary level also persists after the inclusion ofexchange interaction between He atoms. The latter isweak for liquid He because the wave functions of Heatoms overlap weakly due to their strong hard-core re-pulsion at distance z < . A .The surfon lifetime τ is rather short and limited mainlyby two processes: the immersion into liquid and evapo-ration due to scattering by other excitations. The sec-ond process was studied in Ref. [32]. The evaporationrate 1 /τ v of surfons depends on their initial momentumalong the surface and grows rapidly with the increaseof temperature.[32] However, below 4K it does not ex-ceed the limit ∼ E s / ~ where the surfons cannot be calledquasiparticles. The immersion rate of surfons to the liq-uid has not been calculated yet, but it should also be lessthan E s / ~ , because in order to sink into the liquid, a Heatom must overcome a potential barrier and rearrangethe surrounding atoms of the liquid.Thus, the surfons are, presumably, non-stable quasi-particle with lifetime shorter than that of long-wave-length ripplons or phonons. Nevertheless, the existenceof surfons as an additional type of surface excitations,although being debated yet, is crucial for various phys-ical properties of He surface. In addition to explainingthe strong temperature dependence of the surface ten-sion coefficient,[5, 6] they may considerably increase theevaporation rate of liquid He by adding a new evapora-tion channel via the intermediate surfon state with ac-tivation energy ∆ smaller than the evaporation energy | µ | .[32] The surfon quasi-stationary quantum states mayalso affect the reflection coefficient of He atoms by liquidHe surface [34]. Therefore, any substantiation of this newtype of excitations, experimental or theoretical, is veryimportant. At the moment there are only indirect exper-imental confirmations of the existence of surfons by theirinteraction with surface electrons [7, 8] or their contri-bution to surface tension coefficient [5, 6]. In this paperwe analyze the experimental data on inelastic neutronscattering by thin He films and study if these data canprovide a direct experimental confirmation of the exis-tence of surfons. We also summarize the available resultsof ab-initio numerical calculations of surface excitationsin liquid He, which also indicate the existence of surfons. II. EXPERIMENTAL DATA AND THEIRANALYSIS
In this section we present and analyze the experimen-tal data, obtained by the group of H. Godfrin [1] andshown in Figs. 1- 3, on inelastic neutron scattering bya thin 5-atomic-layer film of liquid helium at three dif-ferent temperatures: T=0.4K, 0.98K and 1.3K. Similarand even these data were partially published previouslyin Refs. [2–4], but here we present them at differenttemperatures, in a better quality and in color for greatervisibility and resolution [1].
FIG. 1: (Color on-line) Experimental data on neutron inelastic scattering intensity by liquid helium film, containing only 5atomic layers at T = 0 . K, as a function of the in-plane momentum Q = q k and of the energy transfer. The solid whitelines along the intensity maxima mark the phonon and ripplon spectra. The white dashed line corresponds to the expecteddispersion of surfons. The experimental setup and method were describedpreviously in detail.[2–4] Helium was adsorbed onto asubstrate of exfoliated graphite. The He film of the thick-ness of approximately 5 atomic layers has been used, be-cause for thicker films the contribution of surface excita-tions is too weak as compared to the dominant contribu-tion from the bulk excitations (phonons). Thinner filmsalso have drawbacks for the study of surfons. First, the2-3 atomic layers adjacent to the substrate are solid andtheir structure differs considerably from that in the bulkHe. Second, due to the dimensional quantization alongz-axis, the bulk excitations in too thin films may also con-tain the energy gap ∼ K and resemple the surfons. Theinelastic neutron-scattering experiments were performedat the ILL on the time-of-flight spectrometer IN6 with anincident wavelength of 5.12˚A. The detectors were locatedin an angular range, corresponding to momentum trans-fers between 0 . − and 2 . − for elastically scat-tered neutrons. The energy resolution was ∼ . − . q || > . A − ofin-plane wave-vector. In this paper we present these ex-perimental data in a full available wave-vector interval q || > .
25 ˚ A − and concentrate on the momentum-energyregion corresponding to the expected dispersion of sur-fons. In addition, we provide partially unpublished data[1] at three different temperatures of liquid He, namely, T = 0 . K , 0 . K and 1 . K , while in Refs. [2–4] only thedata at T = 0 . K are given.The intensity of neutron scattering in Figs. 1-3 as afunction of the energy ~ ω and in-plane wave-vector q k of induced excitations is given by color (brightness ingreyscale) as shown on the left panel of Fig. 1. The brightareas form thick lines in the q k − ω coordinates, whichgive the dispersion relation of excitations. In all Figs.1-3 one can easily distinguish the ripplon branch (lowestcurve) and the phonon branch (upper curve), marked bywhite thin curves. The phonon and ripplon modes agree FIG. 2: (Color on-line) Experimental data on neutron inelas-tic scattering intensity by liquid helium film, containing only5 atomic layers at T = 0 . K, as function of the in-plane mo-mentum Q = q k and of energy transfer. The white dashedline corresponds to the expected dispersion of surfons. with those in Refs. [2–4]. The phonon branch has a rotonminimum at q k ≈ A − , which gives a strong intensitymaximum of inelastic neutron scattering. In additionto these two well-know excitation branches, on each ofFigs. 1-3 one can distinguish another curve of intensitymaxima, located at 0 .
25 ˚ A − < q k < . A − betweenthe phonon and ripplon branches and approximately co-inciding with the white dashed line of expected surfondispersion. The intensity of neutron scattering (bright-ness) of this curve is beyond the error-bar [1] and at smallwave-vector is even stronger than that of ripplons on allthree Figs. 1-3. This intermediate dispersion curve givesa gapped excitation and, possibly, originates from sur-fons, because the white dashed lines in Figs. 1-3 showthe expected surfon dispersion given by Eq. (2) with theactivation energy ∆ ≈ . K . The corresponding effec-tive mass of surfon branch coincides with the mass of afree He atom, M ∗ = M , because the inelastic neutronscattering is a process of short time ∼ ~ /ε < ~ / ∆, whilethe ”dressing” of surfons by the formation of the rip-plonic polaron (dimple), leading to the increase of surfoneffective mass,[6] requires longer time. Note that sometraces of this additional branch of surface excitations areseen already in Fig. 1 of Ref. [2], but in the Figs. 1-3of current paper this branch is clearer. Thus, the avail-able experimental data on inelastic neutron scattering bythin He films support the existence of surfons [5–8] as anadditional type of surface excitations. FIG. 3: (Color on-line) The same as in Figs. 1-2 but atdifferent temperature T = 1 . K and rotated by 90 ◦ . III. COMPARISON OF VARIOUSTHEORETICAL CALCULATIONS OF SURFACEEXCITATIONS IN HELIUM AND DISCUSSION
The microscopic numerical calculations also proposeseveral types of surface excitations in addition toripplons.[16–22] These numerical calculations apply thecorrelated basis function (CBF) method [15] for inho-mogeneous liquid, using some additional approximations.These CBF calculations are based on the Feenberg wavefunction with only pair correlations and performed inthe hypernetted-chain approximation. These calcula-tions also assume that only one-body component of theFeenberg function is affected by excitations and by ex-ternal perturbations.[16] This assumption of static two-body correlations limits the regime of validity of thistheory to wavelengths longer than the average distancebetween two particles. It also may restrict the theoryto small deviations from the equilibrium (ground-state)density of liquid He. The backflow effects [14] are also ig-nored in these numerical calculations. Therefore, the ob-tained theoretical excitation energies calculated at largewave numbers q k are substantially higher than the ex-perimental results.[20] Since the applied Jastrow vari-ational treatment of the bulk liquid does not producea self-bound system at saturation He density, an exter-nal potential is introduced phenomenologically in thesenumerical calculations to stabilize the surface.[20, 21]The strength of this additional phenomenological poten-tial is adjusted so that the calculated chemical potentialmatches the experimental saturation value.[20] Finally,the standard numerical computations assume that sur-face excitations do not violate the translational symme-try along the surface, which may not describe the caseof a single surfon with zero in-plane momentum. Nev-ertheless, the comparison of the surface excitations pro-posed by these approximate numerical calculations withthe surfons proposed semi-phenomenologically [5, 6] isquite useful.In Refs. [16–18, 22] the ground state and excitationsin thin He film of several atomic layers on a substrate ofvarious materials were investigated, and several types ofexcitations in He films with a non-zero gap were found.The analysis of the particle currents and transition den-sities, in addition to the dispersion relation of these ex-citations, allowed to describe their nature:[22] they wereattributed to the ripplon mode on the He-substrate inter-face and to the so-called ”breathing mode”. The latterdescribes the standing wave in the z -direction perpendic-ular to the film,[22] similar to the volume phonon, whichmay propagate along the film. The energy gap of this”breathing mode” reduces with the increase of the filmthickness. Thus, no one of the excitations found in Refs.[16–18, 22] can be attributed to surfons. The reason isthat these calculations [16–18, 22] include only excita-tions inside the liquid, neglecting He vapor and the statesabove liquid He. This restriction was eliminated in Refs.[20, 21], where a free boundary between a deep liquid Heand saturated vapor was investigated by the similar nu-merical microscopic CBF approach, and notably differentresults were obtained. This limit of deep liquid He, fillinga half-space instead of atomically thin film, is closer tothe model of semi-phenomenological description of sur-fons in Ref. [6]. Again several gapped surface excitationswere obtained in deep liquid He [20, 21], but the structureof these excitations is completely different from those inthin films [16–18, 22]. First, no signature of ”breathingmode” was found at the free surface of bulk He,[20, 21]which is natural because this mode was found to spreadalong the whole He film thickness [22], being rather thebulk excitation. Nevertheless, two new types of gappedsurface excitations in deep He were found.[21] The firsttype has a large excitation energy above ∆ ≈ . K andis interpreted as a bound roton trapped in the interfaceregion.[21] The wave function of this excitation is mostlyinside liquid He (see Fig. 2 of Ref. [21]), so it cannot beinterpreted as the surfon.The second type of gapped surface excitations, foundin Ref. [21] and called resonance interface state (RIS),has the structure and properties very similar to those ofsurfons. First, RIS correspond to a peak of He densityjust above the liquid surface, as shown in Fig. 6 of Ref.[21]. This is very similar to the wave function of surfons,shown in Fig. 1 of Ref. [32]. Second, RIS in-plane disper-sion is very similar to that of surfons (see Figs. 7 and 9 inRef. [21]): their energy gap ∆ ≈ . K , and at in-plane momentum k k < . A − they have almost a quadraticdispersion in Eq. (2) with effective mass M ∗ close to theatomic He mass M = 6 . · − g. Third, similar to sur-fons, these surface excitations are interpreted as the Hevapor atoms with wave function having large peak justabove liquid surface [21], forming a bound state at thesurface at zero temperature. At finite temperature theseatoms in the bound surface states are quasi-stationary,i.e. they have small finite probability to become delocal-ized, similar to surfon evaporation at finite temperaturestudied in Ref. [32]. Thus we suggest that the resonanceinterface states, obtained numerically in Ref. [21], andthe surfons, proposed in Refs. [5, 6], describe the sametype of surface excitations with two different approximateapproaches.[35]Therefore, the experimental investigation of the disper-sion law of surface excitations, provided by inelastic neu-tron scattering on He films, is very helpful for detectingsurfons and studing their properties. These propertiesmay somewhat differ from those predicted by the semi-phenomenological approach of Refs. [5, 6] or approximatenumerical calculations of Ref. [21]. The numerical cal-culations of the dynamics of one He atom approachingthe surface and interacting with nearest atoms from theliquid could additionally prove the existence of surfonsand even estimate their lifetime. Taking into accountthe important role of surfon excitations in the physicalproperties of the surface of liquid helium and, possibly,of other cryogenic liquids, further numerical calculationson this problem are highly need.The observed additional branch of intensity maxima,giving the in-plane dispersion of surface excitations andapproximately coinciding with the dashed line in Figs.1-3 of possible surfon spectrum, give a strong supportof the existence of surfons and suggest their dispersionlaw. Alternatively, this additional branch could be dueto the ”breathing mode”, obtained in Refs. [16–18, 22]for thin He films. This breathing mode has a differentin-plane dispersion, closer to linear rather than quadraticas for surfon. In addition, for thick films there should beseveral such modes, corresponding to different quantumnumbers of dimensional quantization along the z-axis.The monitoring of the evolution of the activation energyof this mode with the change of He film thickness couldelucidate the nature of this excitation and completelyrule out (or confirm) its breathing-mode origin, but this,probably, requires experimental data with higher energyresolution.To summarize, we analyze the experimental data on in-elastic neutron scattering by thin ∼ ∼ .
5K and dispersion similar to that expected for sur-fons, proposed and investigated semi-phenomenologicallyin Refs. [5–8]. Surface excitations with very similarstructure and properties were also obtained by numeri-cal calculations and called resonance interface states.[21]Before there were only indirect experimental substantia-tions of surfons, based on temperature dependence of sur-face tension coefficient [5, 6] and on interaction of surfonswith surface electrons [7, 8]. The shown data on inelasticneutron scattering, probably, provide the first direct ob-servation of surfons. However, further experimental andtheoretical study is need for the undoubted confirmationof surfons as surface excitations and for the quantitative analysis of their properties.The authors thank H. Godfrin for providing thepartially unpublished experimental data [1] and E.Krotscheck for useful discussions. The work was sup-ported by the program 0033-2018-0001 “Condensed Mat-ter Physics” by the FASO of Russia. A.D.G. thanksRFBR grant [1] H.J. Lauter and H. Godfrin, private communication.[2] H. J. Lauter, H. Godfrin, V. L. P. Frank, and P. Leiderer,Phys. Rev. Lett. , 2484 (1992).[3] H. J. Lauter, H. Godfrin, and P. Leiderer, J. Low Temp.Phys. , 425 (1992).[4] B. E. Clements, H. Godfrin, E. Krotscheck, H. J. Lauter,P. Leiderer, V. Passiouk, and C. J. Tymczak, Phys. Rev.B , 12242 (1996).[5] A.M. Dyugaev, P.D. Grigoriev, JETP Lett. , 466(2003) [Pisma v ZhETF , 935 (2003)].[6] A.D. Grigoriev, P.D. Grigoriev, A.M. Dyugaev, J. LowTemp. Phys. , 131 (2011); arXiv:0905.2306.[7] P.D. Grigoriev, A.M. Dyugaev, E.V. Lebedeva, JETP (2), 316 (2008).[8] P. D. Grigor’ev, A. M. Dyugaev and E. V. Lebedeva,JETP Letters , 106 (2008) [Pisma v ZhETF , 114(2008)].[9] J.S.Rowlinson and B. Widom, Molecular Theory of cap-pilarity , Dover Publications, Mineola NY, 2002.[10] D.O. Edwards and W.F. Saam, Chapter 4 in
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