A Quadratic Diffusion Monte Carlo method has been used to obtain the equation
of state of liquid $^4$He including the negative pressure region down to the
spinodal point. The atomic interaction used is a renewed version (HFD-B(HE)) of
the Aziz potential, which reproduces quite accurately the features of the
experimental equation of state. The spinodal pressure has been calculated and
the behavior of the sound velociy around the spinodal density has been
analyzed.Comment: 10 pages, RevTex 3.0, with 4 PostScript figures include

B.M. Abraham

D.M. Jezek

H.J. Maris

J. Boronat

J. Casulleras

J. Navarro

J.A. Nissen

L. Reatto

M. Guilleumas

M.A. Solís

M.S. Pettersen

P.J. Reynolds

Q. Xiong

R. De Bruyn Ouboter

R.A. Aziz

S. Stringari

S.A. Chin

V.A. Akulichev

English

arXiv

A quadratic diffusion Monte Carlo method has been used to obtain the equation of state of liquid 4He including the negative pressure region down to the spinodal point. The atomic interaction used is a revised version of the Aziz potential, which reproduces quite accurately the features of the experimental equation of state. The spinodal pressure has been calculated and the behavior of the sound velocity around the spinodal density has been analyzed

Boronat, Jordi

Casulleras, Joaquim

Navarro Faus, Jesús

Repositori d'Objectes Digitals per a l'Ensenyament la Recerca i la Cultura

arXiv:cond-mat/9402045v1  10 Feb 1994Monte Carlo calculations for liquid 4He at negative pressureJ. Boronat and J. CasullerasDepartament de F´ısica i Enginyeria Nuclear, Campus Nord B4-B5,Universitat Polite`cnica de Catalunya, E-08028 Barcelona, SpainJ. NavarroInstituto de F´ısica Corpuscular (Centre Mixt Consejo Superior de Investigaciones Cient´ıficasUniversitat de Vale`ncia), Facultat de F´ısica, Avda. Dr. Moliner 50, E-46100 Burjassot, SpainAbstractA Quadratic Diffusion Monte Carlo method has been used to obtain theequation of state of liquid 4He including the negative pressure region down tothe spinodal point. The atomic interaction used is a renewed version (HFD-B(HE)) of the Aziz potential, which reproduces quite accurately the featuresof the experimental equation of state. The spinodal pressure has been calcu-lated and the behavior of the sound velociy around the spinodal density hasbeen analyzed.67.40.Db, 67.40.KhTypeset using REVTEX1Recent experiments [1–3] on cavitation in liquid 4He at low temperatures have motivatedthe theoretical study [4–9] of liquid helium properties at low temperature and negative pres-sure. Some interesting questions have thus been raised, as the determination of the tensilestrength (i.e., the magnitude of the negative pressure at which cavitation becomes likely),and the spinodal pressure (i.e., the pressure at which liquid helium becomes macroscopi-cally unstable against density fluctuations). The standard (or classical) theory of nucleationpredicts for liquid helium [10] a tensile strength rising from 8 atm at 2 K to 15 atm at 0.5K, and the experiment around 1.5 K reported in Ref. [1] seemed to confirm this prediction.However, Maris and Xiong estimated [4,5] the spinodal pressure to be -9 atm at 0 K, sothat the predictions of the standard theory must be incorrect (since the liquid cannot existsfor pressures lower than the spinodal one). They also carried out an experiment [2], whoseresults are in contradiction with those of Ref. [1], obtaining lower values for the tensilestrength. More recently, several experiments [3] providing information about the cavitationproblem have been performed. Unfortunately, as they do not rely on an accurate pressurecalibration, no tensile strength values are reported.In Refs. [4,5] the spinodal pressure was estimated by fitting to the measured [11] soundvelocities c as a function of pressure P several polynomial and Pade´ forms, and then extrapo-lating into the negative pressure region to determine the zero of c(P ). From a different pointof view, the spinodal pressure was calculated in Ref. [7] using two different phenomenolog-ical models that reproduce the equation of state in the measured positive pressure region.Although an overall agreement between the phenomenological calculations and the empiricalresults was obtained, some questions arise, as for instance to what extent the extrapolatedresults depend on the form used in the fit, or on the density functional used in the calcula-tions. It is therefore necessary to handle with a precise equation of state for liquid 4He validin the full range of pressure, down to the spinodal one.Many-body techniques have achieved a high level of accuracy in the description of liquidhelium. In particular, Monte Carlo (MC) methods [12] give exact information, apart fromstatistical uncertainties, on the ground state of bosonic systems both at zero and finite2temperature. The well known interatomic interaction for helium has been a key ingredientto reach an excellent agreement between the theoretical results and the experimental data.Recently, we have used a Quadratic Diffusion Monte Carlo (QDMC) method to calculatethe equation of state in the positive pressure region [13]. One of the main conclusions of Ref.[13] is that the HFD-B(HE) potential suggested by Aziz et al. [14], hereafter referred to asAziz II potential, improves the results obtained with the Aziz potential [15], especially whenthe density dependence of derivative magnitudes of the energy is considered. In this work,we have extended those calculations to lower densities, with the hope of studying withoutambiguity the zero-temperature properties of homogeneous liquid 4He in this zone.The QDMC method solves stochastically the Schro¨dinger equation in imaginary timeassuming a short-time approximation form for the Green’s function [16]. The ground-statewave function is sampled in an iterative process after a time larger enough to project outhigher energy components. Rigorously, the exact ground-state energy is obtained when thelimit ∆t → 0 is considered. In linear DMC algorithms one has to perform calculations atseveral time steps, and then extrapolate to the exact value. The QDMC method, whichhas evidenced a quadratic dependence on ∆t [13,17], improves the efficiency of the diffusionalgorithm making feasible to use larger time steps than in DMC and avoiding the necessityof the extrapolation to ∆t = 0.In order to guide the diffusion process a Jastrow trial wave function of the form [18]ΨT (R) =∏i<jexp−12(brij)5−L2exp−(rij − λΛ)2 (1)has been used in our calculations. The values of the parameters appearing in Eq. (1)(L = 0.2, λ = 2.0 σ, Λ = 0.6 σ, b = 1.20 σ , where σ = 2.556 A˚) have been fixed to optimizethe Variational Monte Carlo (VMC) estimation of the energy at the equilibrium density(ρ0 = 0.365 σ−3). The Monte Carlo simulation has been carried out with 108 particles forρ < 0.328 σ−3 and 128 particles for ρ ≥ 0.328 σ−3, maintaining an asymptotic populationof 400 walkers. The errors associated to the use of both a finite volume simulation box anda finite population have been analyzed and, in all cases, are smaller than the size of the3inherent statistical fluctuations.In Table I are reported the total energies (in K) obtained with the Aziz II potential, fordifferent values of the density (in units of σ−3). The experimental values of Ref. [19] are alsodisplayed. The origin of the slight differences observed between theory and experiment wasdiscussed in Ref. [13].Derived quantities of the energy such as the pressure or the sound velocity have beenobtained through a third and fourth degree interpolation, with unnoticeable changes whenlarger degrees were introduced in the interpolation method. The QDMC prediction of P (ρ)is shown in Fig. 1 (solid line) for the whole range of densities, in comparison with experi-mental data for positive pressures [19]. The agreement between the Aziz II results and theexperimental data is quite impressive.At this point, we would like to draw the attention on the quality of the extrapola-tions coming from the previously available data, laying mainly in the positive pressureregion. In the majority of microscopic calculations on liquid helium the energy per particleis parametrized using a polynomial formE/N = e0 +B(ρ− ρ0ρ0)2+ C(ρ− ρ0ρ0)3. (2)On the other hand, in calculations based on Density Functional Theory, the formE/N = b ρ+ c ρ1+γ (3)proposed by Stringari [20], has proved to be very efficient in describing properties of ho-mogeneous and inhomogeneous (including an additional surface term) liquid 4He. We haveused both forms to fit our previous QDMC results [13], which included only one point belowthe equilibrium density. Proceeding in this way (i.e., considering only the five last densitiesof Table I), we have found that although both fits are compatible with the previous resultsof the energy, only the second form, see Eq. (3), predicts the new QDMC results at densitieslower than 0.328 σ−3. This fact is reflected in Fig. 1, where the pressure derived from thesefits is plotted as a function of density. The short-dashed line corresponds to the fit (2) and4the long-dashed line to the fit (3). The starting points from the left of all curves depicted inFig. 1 correspond to the location of the spinodal point. The QDMC result for the spinodalpressure is Pc = −9.30±0.15 atm, corresponding to a density ρc = 0.264±0.002 σ−3. The fit(3) predicts (ρc, Pc) = (0.266,−9.08) whereas the fit (2) gives (ρc, Pc) = (0.292,−6.66). ThisFigure illustrates that the extrapolation from the positive pressure region to the negative oneis quite sensitive to the form used in the fit. One can also see the capability of the built-indensity functional given in Eq. (3) to reproduce quite accurately the equation of state ofliquid 4He at any density, even when only positive pressure values are used as input in thenumerical fit. It is also noticeable the stability of Eq. (3) when all the QDMC energy results(Table I) are used to fix the optimal parameters b, c and γ, being their relative changes lessthan 5 %. In this case, the spinodal point turns out to be (ρc, Pc) = (0.267,−9.16). If the fit(2) is extended to all the energy values, the spinodal point (ρc, Pc) = (0.267,−9.24) comesclose to the Monte Carlo prediction.In Figure 2 is displayed the sound velocity c as a function of pressure P . The points arethe experimental values of Ref. [11], and the solid line corresponds to the QDMC results.The accuracy provided by the Aziz II potential is again remarkable, giving results for thesound velocity in close agreement with the experiment. It can be seen that c drops to zerovery fast when approaching the spinodal point. The behavior of c near Pc is expected tobe of the form c ∝ (P − Pc)ν , being ν the critical exponent. It is known [7] that ν = 1/4provided that the quantityP ′′c =(∂2P∂ρ2)ρc(4)be different from zero. Our QDMC estimation of P ′′c is 1200±100 Kσ3. Therefore, the cleardeparture from zero of P ′′c guarantees that ν = 1/4, in disagreement with the Maris’ model[6] which, explicitly taking the value P ′′c = 0, predicts ν = 1/3. Nevertheless, in the positivepressure region the experimental values are very well reproduced by the above form with νclose to 1/3. In fact, we have found that for pressures only slightly higher than Pc and upto almost solidification pressure, the behavior is also c ∝ (P −Pc)ν with the exponent given5by [7]ν = ν0 ≡ −Pc κ0 u0 , (5)where κ0 is the isothermal compressibilityκ =1ρ(∂ρ∂P)T(6)and u0 is the Gru¨neisen constantu =ρc∂c∂ρ, (7)both quantities evaluated at the saturation density ρ0. Our QDMC results give for thesequantities the values κ0 = 0.01231±0.00005 atm−1 and u0 = 2.81±0.04 with ρ0 = 0.3645±0.0003 σ−3, to be compared with the experimental values [11] κexp0 = 0.01230 atm−1, uexp0 =2.84 and ρexp0 = 0.3646 σ−3. The QDMC result for the exponent (5) is ν0 = 0.322± 0.007,close to the value 1/3 pointed out in Refs. [5,6].The power-law behavior of c is shown in Figures 3 and 4. The quantity (c/c0)1/ν , wherec0 is the sound velocity at the equilibirium density, is displayed in Figure 3 against 1−P/Pc.When ν = ν0 a straigth line is obtained in a very wide range of pressures (Fig. 3), whereasthe case ν = 1/4 manifests a nearly quadratic behavior. However, for pressures close to thespinodal pressure one can see (Fig. 4) that the linear behavior is obtained taking the criticalexponent as 1/4.To summarize, we have calculated the equation of state of liquid 4He in a QDMC frame-work using the Aziz II potential from the spinodal point up to the solidification point. Thecharacteristics of the spinodal point have been evaluated without the uncertainty of the ex-trapolation from the positive pressure data. As a byproduct, we have noted that our QDMCresults are very well fitted by the form (3), giving thus additional support to this purelyphenomenological density functional.This work has been partly supported by DGICyT (Spain) Grant Nos. PB90-06131,PB92-0761 and PB92-082. Most of the simulations were performed on a multiprocessorCM2 of the CEPBA (Centre Europeu de Paral·lelisme de Barcelona).6REFERENCES[1] J.A. Nissen, E. Bodegom, L.C. Brodie, and J.S. Semura, Phys. Rev. B 40, 6617 (1989).[2] Q. Xiong and H.J. Maris, J. Low Temp. Phys. 82, 105 (1991).[3] M.S. Pettersen, S. Balibar and H.J. Maris, Phys. Rev. B, submitted.[4] H.J. Maris and Q. Xiong, Phys. Rev. Lett. 63, 1078 (1989).[5] Q. Xiong and H.J. Maris, J. Low Temp. Phys. 77, 347 (1989).[6] H.J. Maris, Phys. Rev. Lett. 66, 45 (1991).[7] M.A. Sol´ıs and J. Navarro, Phys. Rev. B 45, 13080 (1992).[8] M. Guilleumas, M. Pi, M. Barranco, J. Navarro, and M.A. Sol´ıs, Phys. Rev. B 47, 9116(1993).[9] D.M. Jezek, M. Guilleumas, M. Pi, M. Barranco, and J. Navarro, Phys. Rev. B 48,16582 (1993).[10] V.A. Akulichev and V.A. Bulanov, Sov. Phys. Acoust. 20, 501 (1975).[11] B.M. Abraham, Y Eckstein, J.B. Ketterson, M. Kuchnir, and P.R. Roach, Phys. Rev.A 1, 250 (1970).[12] A modern review of Monte Carlo methods can be found in Monte Carlo methods inTheoretical Physics, eds. S. Caracciolo and A. Fabrocini (ETS Editrice, Pisa 1991).[13] J. Boronat and J. Casulleras, Phys. Rev. B, in press.[14] R.A. Aziz, F.R.W. Court and C.C.K. Wong, Molec. Phys. 61, 1487 (1987).[15] R.A. Aziz, V.P.S. Nain, J.S. Carley, W.L. Taylor, and G.T. McConville, J. Chem. Phys.70, 4330 (1979).[16] P.J. Reynolds, D.M. Ceperley, B.J. Alder, and W.A. Lester, J. Chem. Phys. 77, 55937(1982).[17] S.A. Chin, Phys. Rev. B 42, 6991 (1990).[18] L. Reatto, Nucl. Phys. A328, 253 (1979).[19] R. De Bruyn Ouboter and C.N. Yang, Physica B 44, 127 (1987).[20] S. Stringari, Phys. Lett. A 106, 267 (1984).8FIGURESFIG. 1. Pressure as a function of density. Points: experimental results [19], solid line: QDMCresult, short-dashed line: fit (2), and long-dashed line: fit (3). The fits (2) and (3) include onlythe results from .328 to .438 σ−3.FIG. 2. The sound velocity as a function of pressure. The experimental points are taken fromRef. [11], and the solid line corresponds to the QDMC results.FIG. 3. Power-law behavior of the sound velocity using as a critical exponent ν0 (5) and 1/4.FIG. 4. Same as Figure 3 for pressures close to the spinodal one. To illustrate more clearly thepower-law behavior, the quantity (c/c0)1/ν / (1− P/Pc) has been plotted against 1− P/Pc for thevalues ν = ν0 and 1/4.9TABLESTABLE I. Results for the total energies (in K) at several densities (in σ−3) from the QDMCcalculations with the Aziz II potential and experiment [19].ρ E/N E/N (exp)0.264 -6.376 ± 0.0100.267 -6.441 ± 0.0100.285 -6.692 ± 0.0100.300 -6.892 ± 0.0100.310 -7.011 ± 0.0100.328 -7.150 ± 0.0100.365 -7.267 ± 0.013 -7.170.401 -7.150 ± 0.016 -7.030.424 -6.877 ± 0.022 -6.770.438 -6.660 ± 0.017 -6.5510

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