isotope effects in structural and thermodynamic properties of solid neon

PHYSICAL REVIEW B, VOLUME 65, 014112

Isotope effects in structural and thermodynamic properties of solid neon

Carlos P. HerreroInstituto de Ciencia de Materiales, Consejo Superior de Investigaciones Cientı´ficas (C.S.I.C.), Campus de Cantoblanco,

28049 Madrid, Spain~Received 17 April 2001; revised manuscript received 6 July 2001; published 12 December 2001!

Path-integral Monte Carlo simulations in the isothermal-isobaric ensemble have been carried out to studyisotope effects in solid neon. This computational method allows a quantitative and nonperturbative study ofthese effects, which are associated with a strong anharmonicity of the atomic motion. The Ne atoms weretreated as quantum particles interacting via a Lennard-Jones potential. We have studied the temperature de-pendence of atomic mean-square displacements, lattice parameter, bulk modulus, and heat capacity, as well asvibrational kinetic and potential energy, for the isotopes20Ne and22Ne. The studied Lennard-Jones solids arehighly anharmonic, as measured from the ratio between vibrational potential and kinetic energies, whichamounts to about 0.8 at 4 K. At this temperature, the isothermal bulk modulus for solid22Ne is found to be 0.3kbar larger than that for20Ne and the cohesive energy is 32 J/mol larger for the former. The obtained heatcapacity, thermal expansion, and isotopic effect in the lattice parameter agree well with experimental data.

DOI: 10.1103/PhysRevB.65.014112 PACS number~s!: 67.80.2s, 05.10.Ln, 65.40.De

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I. INTRODUCTION

Anharmonic effects in solids have been studied for mayears, since they are responsible for important effects sucthermal expansion, pressure dependence of the compresity, and phonon couplings, as well as the isotope dependeof structural properties and melting temperature. For soneon, in particular, the considerable importance of suchharmonic effects has been noticed,1,2 and the associated isotopic dependence of thermodynamic and structural propehas been found to be non-negligible.2–6

The lattice parameters of two chemically identical crystwith different isotopic compositions are not equal, lightisotopes having larger vibrational amplitudes and largertice parameters. This effect is most noticeable at low teperatures, since the atoms in the solid feel the anharmonof the interatomic potential due to zero-point motion.higher temperatures, the isotope effect on the crystal voluis less relevant and disappears in the high-temperature~clas-sical! limit. At present, this isotopic effect in the lattice parameter of crystals can be measured with high precisio7

Something similar happens with the compressibility, whichdirectly related to the anharmonicity of the solid vibratioand is expected to show a maximum isotopic effect forT→0. Other quantities, such as the vibrational energy, shan isotope dependence at lowT in a harmonic approximation, due to the usual rescaling of the phonon frequencwith the isotopic massM (v}M 21/2), but this dependencecan display appreciable changes when anharmonic effare considered. All these effects are expected to be mimportant in the case of neon than for heavier rare-solids.8

From a theoretical point of view, anharmonic effectssolids have been traditionally studied by using approacsuch as the so-called quasiharmonic approximation.9,10 Inthis approach, frequencies of the vibrational modes in a sare assumed to change with the crystal volume, and for gvolume and temperature, the solid is supposed to bemonic. This procedure has been employed in the last yea

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study isotopic effects in the crystal volume of covalent solfrom ab initio density-functional-theory calculations.11,12Forsolid neon, in particular, isotopic differences in the zetemperature volume and sublimation energies have beenculated for different interatomic potentials by taking into acount cubic and quartic anharmonic corrections to the zepoint energy.5 Also, the finite-temperature properties ocrystalline natural Ne and22Ne were derived from a secondorder self-consistent phonon theory.6

An alternative procedure is the Feynman path-integ~PI! method, which is a convenient theoretical approachstudy anharmonic effects at temperatures at which the qutum nature of the atomic nuclei is relevant.13,14 The combi-nation of path integrals with Monte Carlo~MC! samplingenables us to carry out quantitative and nonperturbative sies of such anharmonic effects. The PI MC technique wapplied earlier to study several properties of solid neon.8,15 Inparticular, it has predicted kinetic-energy values in goagreement with the existing experimental data.16,17 Thismethod has been also employed to study the isotopic shithe helium melting pressure,18,19 as well as isotope effects inthe lattice constant of solid neon20 and diamond-typematerials.21,22 In the context of the PI formalism, severaauthors have developed effective~temperature-dependen!classical potentials that reproduce accurately several proties of quantum solids.23–25 More recently, Acocellaet al.26

have applied an improved effective-potential Monte Catheory27 to study thermal and elastic properties of solid neo

In this paper, I present results for the lattice parameisothermal bulk modulus, and kinetic and potential energywell as atomic mean-square displacements of solid neonderived from PI MC simulations. These quantities are stied as functions of temperature and isotopic mass. The inatomic interaction is described by a Lennard-Jones poten

The paper is organized as follows. In Sec. II, I descrthe computational method and give some technical detailsSec. III, I present results of the MC simulations, which agiven in different subsections dealing with the energy, hcapacity, atomic mean-square displacements, lattice pa

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CARLOS P. HERRERO PHYSICAL REVIEW B65 014112

eter, and bulk modulus. Finally, Sec. IV includes a discussof the results.

II. METHOD

Equilibrium properties of solid20Ne and22Ne have beencalculated by PI MC simulations in the isothermal-isobaensemble. Quantum exchange effects were not considerethey are negligible for neon at the densities studied herethe path-integral formulation of statistical mechanics,partition function of a quantum system is evaluated througdiscretization of the density matrix along cyclic paths, coposed of a finite number of ‘‘imaginary-time’’ steps.13 In thenumerical simulations, this Trotter numberNT causes theappearance ofNT ‘‘replicas’’ for each quantum particleThus, practical implementation of this method relies onisomorphism between the quantum system and a clasone, obtained by replacing each quantum particle~here,atomic nucleus! by a cyclic chain ofNT classical particles,connected by harmonic springs with temperature-depenconstant.28 Details on this computational method canfound elsewhere.29–31

Simulations have been performed on several cubic sucells of the neon face-centered-cubic cell, with side lenLa, beinga the unit-cell parameter andL an integer in therange from 2 to 7. Thus, the largest simulation cell consered was a 73737 supercell, including 1372 Ne atomPeriodic boundary conditions were assumed. Such largepercells were necessary to study the convergence of theculated quantities with the system size, which allowed usextrapolate the obtained values to the infinite-size limit~seebelow!.

The Ne atoms were treated as quantum particles intering through a Lennard-Jones potential:V(r )54e@(s/r )12

2(s/r )6#, with e53.08431023 eV ands52.782 Å. Theparametere coincides with that employed in earlier PI Msimulations of liquid32 and solid20 Ne. However, we havetaken s to be larger than in Refs. 20 and 32 in orderreproduce the experimental lattice spacing in the limitT→0. Our value fors is close to that employed in Refs. 8, 1and 17. The dynamic effect of the interactions between nest neighbors is explicitly considered. The effect of intertions beyond nearest neighbors is taken into account bstatic-lattice approximation.17,20 This assumption was employed earlier in PI MC simulations of Lennard-Jones soligiving the same results as those obtained in simulationscluding dynamical correlations up to several neighboratom shells.15 In particular, it gives good agreement witexperiment for the kinetic energy of solid neon.17 Our simu-lations were based on the so-called ‘‘primitive’’ form of PMC, and the ‘‘crude’’ energy estimator was used.28,33

Sampling of the configuration space has been carriedby the Metropolis method34 at temperatures between 4 an24 K, and at a constant pressure of 1 atm. For given tperature and isotopic mass, a typical run consisted ofgeneration of about 104 paths per atom for system equilibration, followed by 53105 paths per atom for the calculatioof ensemble average properties. To improve the accuracthe results for the lattice parameter, atT.18 K we gener-

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ated 106 paths per atom. More details on the actual applition of this method are given in Ref. 29.

In the isothermal-isobaric ensemble, the mean-squfluctuations in the volumeV of the simulation cell are givenby

~DV!25V

BkBT, ~1!

whereB52V(]P/]V)T is the isothermal bulk modulus othe solid. Hence, the fluctuations in the lattice parameteaare

~Da!25kBT

9L3aB. ~2!

For a given temperature and simulation-cell size, the flucttions (Da)2 can be obtained from Monte Carlo simulationThus, we will employ Eq.~2! to obtain the bulk modulusBfrom the simulation results. From Eq.~1! one can see that threlative fluctuation in the volume of the simulation ceDV/V, scales asL23/2. In fact, for 20Ne at 20 K we find inthe PI MC simulations,DV/V50.023 for L52 and 631023 for L55.

Finite-size effects are expected to be observed instructural and thermodynamic variables derived from PI Msimulations. Due to the finite dimensions of the simulaticell, long-wavelength acoustic phonons withl*La ~close tothe center of the Brillouin zone! are truncated, introducing aartificial low-energy cutoffv0}1/L for the solid vibrations.This causes the appearance of an effective energy gapvibrational excitations in the simulated solid that is npresent in the actual material. The dependence of sevequilibrium properties of rare-gas solids on the simulatiocell size has been studied in detail by Mu¨ser et al.20 Fromthis size dependence one can extrapolate the simulationsults to the thermodynamic limit (L→`). The finite-sizecorrections of several equilibrium properties scale as 1V}L23, as is the case of the internal energy, which converrelatively fast for our purposes.

The spatial delocalization of the neon atoms can betained directly from the quantum simulations and mayquantified by the mean-square displacement (Dr )2. Thismagnitude, however, converges especially slowly as thesize is increased. In Fig. 1 we present (Dr )2 for solid Neversus the inverse cell size. Open symbols correspond20Ne and solid symbols to22Ne. Finite-size corrections tothe atomic mean-square displacements scale linearly1/L. For L55 andT520 K, the calculated (Dr )2 is about90% of the extrapolated thermodynamic limit~the error in-troduced by the finite cell size is;10%). This contrasts withthe convergence of the kinetic energy, which for the samLand T has an error smaller than 0.1% when compared wthe corresponding extrapolated value.

For (Dr )2 the convergence with cell size becomes slowas the temperature is raised~larger slope of the dashed linein Fig. 1!. This is a consequence of the fact that at hitemperature (kBT.\v), the contribution to (Dr )2 of agiven vibrational mode with frequencyv scales as 1/v2,

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ISOTOPE EFFECTS IN STRUCTURAL AND . . . PHYSICAL REVIEW B65 014112

making very important the contribution of the low-enermodes neglected in the finite-size calculations. This conbution to the mean-square displacement (Dr )2 is given ap-proximately by20,35

~Dr !negl2 .CE

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whereC is a normalization constant. Assuming a vibrationdensity of statesr(v)}v2, which is expected to be a gooapproximation at low frequencies~Debye model!, one finds(Dr )negl

2 }v0}1/L, in agreement with the results of the PMC simulations presented in Fig. 1.

Another source of possible inaccuracies in the resultstained from PI MC simulations is the finite Trotter numbNT employed in the discretization of the path integrals. Than extrapolationNT→` has to be carried out to obtain precise values for the studied quantities.20,36 Nevertheless, theconvergence withNT of the properties studied here is mucfaster than that described above for the simulation-cell sFor example, finite-NT corrections to (Dr )2 scale as 1/NT

2 ,as for the other thermodynamic and structural properstudied in Refs. 20 and 36. For a given temperatureT, thelargest Trotter numberNT employed in our simulations wataken as the integer number givingNTT.180 K. Thismeans that, for example, a PI MC simulation on a 53535supercell (N5500 atoms! at 4 K (NT545) is equivalent incomputational effort to a classical MC simulation ofNTN522500 atoms.

III. RESULTS

A. Energy

For given volume and temperature, the internal energythe solid,E(V,T), can be written as

FIG. 1. Mean-square displacement, (Dr )2, of the Ne atoms vsthe inverse simulation-cell sizeL. Symbols represent results of PMC simulations for isotopically pure crystals of20Ne ~open sym-bols! and 22Ne ~solid symbols!. Circles correspond toT520 K andsquares toT510 K. Error bars are smaller than the symbol sizLines are least-squares fits to the data points.

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E~V,T!5E01ES~V!1Ev ib~V,T!, ~4!

whereE0 is the minimum potential energy for the~classical!crystal at T50 K, ES(V) is the elastic energy, andEv ib(V,T) is the vibrational energy. With the parameters eployed here for the Lennard-Jones potential, we findE05226.55 meV per neon atom, which corresponds to~classical! lattice parameteracl(0)54.2890 Å. This valuefor E0 translates into a cohesive energy of 2.57 kJ/mol.

For a given volumeV, the classical energy atT50 ~pointatoms on their lattice sites! increases by an amountES(V)with respect to the minimum energyE0. This elastic energyES depends only on the volume, but at finite temperatuand for the real~quantum! solid, it depends implicitly onTdue to the temperature dependence ofV ~thermal expansion!.The elastic energyES(V) represents a non-negligible part othe internal energy and for20Ne is found to be 1.21 and 2.1meV per atom at 4 and 24 K, respectively. Note thatelastic energy at 4 K is basically due to the change in volumcaused by ‘‘zero-point’’ lattice expansion.

The vibrational energyEv ib(V,T) depends explicitly onboth V andT and can be obtained by subtracting the elasenergy from the internal energy. Path-integral Monte Casimulations allow us to obtain separately the kineticEk andpotential energyEp associated with the lattice vibrations, aa function of temperature.30 Both energies are shown in Fig2 for 20Ne ~open symbols! and 22Ne ~solid symbols!.Squares and circles correspond to the vibrational kineticpotential energy, respectively. Our results for the kineticergy are close to those derived earlier from PI MC simu

.

FIG. 2. Temperature dependence of the vibrational energysolid neon. Squares and circles correspond to kinetic and poteenergy, respectively, as derived from PI MC simulations. Opsymbols, 20Ne; solid symbols,22Ne. Error bars of the simulationresults are less than the symbol size. The dashed and dash-dlines correspond to a Debye model for20Ne and22Ne, respectively,with QD(20Ne)570 K. Diamonds are results for the kinetic energof 20Ne, obtained by Timmset al. ~Ref. 16! from inelastic neutronscattering in solid neon with natural isotopic composition.

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tions with Lennard-Jones8,16,17and Aziz16 interatomic poten-tials. For comparison, we present values of the kinetic eneof 20Ne, derived by Timmset al.16 from inelastic neutronscattering in solid neon with natural isotopic compositi~diamonds!.37 The kinetic energy of21Ne ~natural abundance0.27%) in solid neon has been more recently measuredusing resonant nuclear reactions,38 and a value of 3.2(60.6) meV/atom was found atT58 K ~not shown in thefigure for the sake of clarity!. The results of our PI MCsimulations indicate that the vibrational potential energyclearly smaller than the kinetic energy, for both solid20Neand 22Ne. As a result,Ep is about 20% lower thanEk in thewhole temperature range under consideration.

For comparison, we also present in Fig. 2 the kinetic apotential energy obtained for20Ne and 22Ne from a Debyemodel for the lattice vibrations~dashed and dash-dottelines, respectively!. In this approximation, the kinetic~poten-tial! energy per atom at temperatureT is given by

EkD5Ep

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vD3 E0

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whereb51/(kBT) andvD is the Debye frequency~see, e.g.,Ref. 38 for an equivalent expression forEk

D). For 20Ne wehave takenvD(20Ne)548.7 cm21, which corresponds to aDebye temperatureQD570 K.39 The Debye frequency isassumed to scale with the isotopic massM as vD}M 21/2,and thus we have takenvD(22Ne)546.4 cm21. The kineticenergy derived from the PI MC simulations is higher ththat found from this Debye approximation, but the potenenergy obtained in the simulations is clearly lower for boneon isotopes.

A quantitative evaluation of the overall anharmonicitythe atom vibrations can be obtained from the PI MC simutions by calculating the ratio between potential and kineenergy at different temperatures. This ratio should be 1 foharmonic solid at any temperature, as follows from the virtheorem. In Fig. 3 we display the temperature dependencthe potential-to-kinetic energy ratio for20Ne ~squares! and22Ne ~circles!, derived from our quantum simulations. Thratio is found to be of the order of 0.8 for both isotopeshowing a clear departure from the harmonic expectancdecreases as temperature rises and is higher for22Ne than for20Ne, as expected from the fact that the heavier isotshould be closer to the classical limit, for which one hasT50, Ep /Ek51.

The departure from harmonicity of the solid vibratiocan be also analyzed by studying the ratio between kinetipotential energies of both neon isotopes,20Ne and22Ne. Thisis shown in Fig. 4, where we present the temperature dedence for the ratioEp(22Ne)/Ep(20Ne) between potential energies~circles! and for the ratioEk(

22Ne)/Ek(20Ne) between

kinetic energies of both isotopes~squares!. The dashed linecorresponds to the Debye model discussed above, whicT50 converges to the valueA20/2250.953 expected in aharmonic approximation. Both potential and kinetic eneratios derived from the PI MC simulations are larger thanDebye harmonic approximation in the whole temperat

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range considered. The anharmonicity of the vibratiomodes affects the potential energy more strongly thankinetic energy of the neon atoms. The actual trend ofDebye curve in Fig. 4 can be changed by modifying tDebye temperatureQD , and thus this curve could approacthe PI MC results~e.g., by taking smaller values forQD!.However, the low-temperature limit of the energy ratio pr

FIG. 3. Potential-to-kinetic energy ratioEp /Ek for solid 20Ne~squares! and 22Ne ~circles!, as derived from PI MC simulationsError bars are on the order of the symbol size. Dotted linesguides to the eye.

FIG. 4. Energy ratio between solid22Ne and20Ne. Circles cor-respond to the ratio between vibrational potential energEp(22Ne)/Ep(20Ne), and squares to the ratio between kinetic engies,Ek(

22Ne)/Ek(20Ne). Error bars are on the order of the symb

size. Dotted lines on the simulation data are guides to the eye.dashed line is the ratio obtained for both kinetic and potentialergies in a Debye model withQD(20Ne)570 K and QD(22Ne)566.7 K.

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sented in this figure is independent ofQD , and the differentvalues obtained from PI MC simulations for the ratioskinetic and potential energies at lowT actually reflect theanharmonicity of the zero-point vibrations in the crystal.

Summarizing our energy results, we find that the cohesenergy for 20Ne at 4 K amounts to 18.98 meV/atom, whicmeans an important decrease of 7.57 meV/atom with resto the classical result atT50. According to Eq.~4!, thischange in internal energy of the solid can be split into thcontributions: elastic energy originating from lattice expasion ~1.21 meV/atom! and kinetic and potential vibrationaenergies~3.51 and 2.85 meV/atom, respectively!. For solid22Ne we find at 4 K a cohesive energy of 19.31 meV/atoi.e., 0.33 meV/atom higher~more stable! than for solid20Ne~a difference of about2%). At 24 K, thecohesive energydecreases to 16.75 and 16.96 meV/atom for20Ne and 22Ne,respectively. Our result for the cohesive energy of20Ne atT50 is about 1 meV/atom lower than that found by Acoceet al. from an improved effective-potential theory.26 This dif-ference seems to be due to the different potential parame~and not to the different computational procedures! em-ployed in both calculations.

B. Heat capacity

The heat capacityCP for 20Ne and 22Ne has been calculated from the MC simulations atP51 atm as a numericaderivative of the enthalpy,H5E1PV, with respect to thetemperature. Our results for20Ne are shown in Fig. 5~a! asopen squares. The data points found for22Ne lie slightlyhigher than those corresponding to20Ne and are not shownfor the sake of clarity. In this figure we also present texperimental results obtained by Somoza and Fenichel4 for20Ne ~solid line!, as well as those found by Fenichel anSerin40 for natural neon~dashed line!. Taking into accountthe average isotopic mass of natural neon (Mav520.18 amu), the latter results should be slightly highthan the former, contrary to the trend observed in the dactually measured, indicating that the difference betwboth sets of data atT.16 K is caused by experimental uncertainty. This is in line with the indications by Somoza aFenichel,4 in the sense that atT*18 K the error introducedby uncertainties in the temperature and to the partial vapization of the solid increases appreciably. Our results follclosely the experimental data up toT518 K, and atT>18 K they seem to be slightly lower.

In Fig. 5~b! we present the relative change in heat capity, c5(CP

222CP20)/CP

20, as a function of temperature. The PMC results are shown as open squares, and the solidrepresents the data obtained by Somoza and Fenichel.4 Ourresults follow closely the experimental data atT.8 K. Atlower T, the error in thec values derived from the simulations increases very fast asCP approaches zero. An arrow iFig. 5~b! indicates the zero-temperature value expected focin a Debye~harmonic! approximation, irrespective of the assumed Debye temperature. In fact, atT!QD , CP}(T/QD)3, whence we haveCP}T3M3/2, andc→0.154 atlow T in this approximation. Values ofc derived from ex-

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neperiment atT,6 K decrease as the temperature is lowercontrary to the expectancy of any harmonic model forlattice vibrations.

C. Atomic mean-square displacements

The mean-square displacements (Dr )2 of the neon atomsare displayed in Fig. 6~a! for temperatures between 4 and 2K. In this figure, symbols are data points obtained from oPI MC simulations: squares,20Ne; circles,22Ne. At low tem-

FIG. 5. ~a! Heat capacityCP of solid neon as a function otemperature. Squares indicate results of PI MC simulations20Ne. The solid and dashed lines are experimental results for s20Ne ~Ref. 4! and for solid neon with natural isotopic compositio~Ref. 40!, respectively. Error bars are given for three experimenpoints, indicated by black dots.~b! Relative change in heat capacitc5(CP

222CP20)/CP

20, vs temperature. Symbols represent resultsour PI MC simulations, and the solid line indicates the experimendata from Somoza and Fenichel~Ref. 4!. Two dotted lines aboveand below the experimental results correspond to the error bathese data. An arrow shows the value ofc obtained in a Debyemodel atT50 K.

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peratures, (Dr )2 goes to the value corresponding to the zepoint motion of the atoms in the crystal, and we find (Dr )0

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values between 7 and 831022 Å 2. The atomic mean-squardisplacements change with the isotopic massM, isotopeswith larger mass having a lower spatial delocalization. Tmass dependence is most important at low temperaturesdecreases as the temperature goes up. In the case of(Dr )2 for 20Ne is still clearly larger than that for22Ne attemperatures close to the melting temperature, sinceTm(524.5 K for natural Ne! is much lower than the Debytemperature of the solid (QD;70 K). For comparison, wealso present in this figure the atomic mean-square displ

FIG. 6. ~a! Temperature dependence of the mean-squareplacement, (Dr )2, for 20Ne ~squares! and 22Ne ~circles!, as derivedfrom PI MC simulations. Error bars are less than the symbol sDashed and dash-dotted lines: Debye model for solid20Ne and22Ne, respectively, withQD(20Ne)570 K. ~b! Ratio betweenmean-square displacements of22Ne and 20Ne: symbols, results ofthe PI MC simulations; dashed line, Debye model. Dotted lines~a! and ~b! are guides to the eye.

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ment obtained from a Debye model for the lattice vibratioIn this approximation, the atomic spatial delocalizationtemperatureT is given by22

~Dr !D2 5

9

2

\

MvD3 E0

vDv cothS 1

2b\v Ddv. ~6!

With QD(20Ne)570 K, we find for 20Ne and 22Ne thedashed and dash-dotted lines shown in Fig. 6~a!, respec-tively. At temperaturesT,10 K this approach gives valuefor (Dr )2 slightly larger than our PI MC results, but thsimulation data become higher as temperature rises.

In Fig. 6~b! we display the ratio between mean-squadisplacements (Dr )2 for isotopically pure crystals of22Neand 20Ne ~squares!. For comparison, we also present in thfigure the delocalization ratio obtained from a Debye mofor the lattice vibrations~dashed line!. This approach givesresults that deviate appreciably from those yielded byMC simulations, indicating a strong effect of the anharmnicity of the vibrational modes. In fact, such a simple hamonic approach gives for the delocalization ratio results vclose to those found from MC simulations in covalesolids,22 where the anharmonicity is much smaller than in tpresent case. At low temperatures, the delocalization robtained in the Debye model for solid Ne convergesA20/22, the expected value in a harmonic model atT50.The low-temperature limit for the results of our PI MC simlations is clearly lower than the harmonic expectation. Astemperature rises, this ratio increases and should approaat temperaturesT.QD .

D. Lattice parameter

The effect of anharmonicity appears clearly in the averaNe-Ne distance~or in the lattice parameter! at low tempera-ture, which is larger than that corresponding to the minimpotential energy of the~classical! crystal.20 Results for thelattice parameter derived from our PI MC simulations folloclosely the experimental data for20Ne and22Ne up to 24 K,as displayed in Fig. 7~a!. Classical MC simulations give anearly linear temperature dependence for the interatodistance,20 which converges at lowT to the value corre-sponding to the minimum potential energy of the crys@here, dcl(0)53.0328 Å#. The increase in20Ne-20Ne dis-tance obtained in the quantum simulations, with respecthat found in the classical ones at low temperature, amouto 0.1237 Å. This quantum effect on the nearest-neighatom distance atT50 corresponds to an increase in lattiparameter of 0.1749 Å.~Note that for a face-centered-cublattice a5A2d). In fact, we find for the quantum crystalzero-temperature lattice parametera20(0)54.4639(2) Å vsacl(0)54.2890 Å for the classical one. This means ancrease of 4.1% in the lattice constanta due to anharmonicityof the zero-point atomic motion, which amounts to abotwice the change ina due to thermal expansion between 0and the melting temperature of neon.

The relative change in lattice parameter between ncrystals with different isotopic composition can be measuby the quantityx5(a202a22)/a22, which is shown in Fig.

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7~b!. Solid squares indicate results of the present PI Msimulations, which are compared with those obtained frx-ray diffraction experiments3 ~solid line!. We also presenresults derived from PI MC simulations by Mu¨ser et al.20

~open circles!. For increasing temperature, the relatichangex decreases, since quantum effects become lessevant to describe the atomic motion. However, the expmental data show an increase inx at temperatures highethan 20 K, close to the melting temperature of neon. Oresults do not reproduce this increase inx, which was appar-ently obtained in the PI MC simulations of Ref. 20@see open

FIG. 7. ~a! Temperature dependence of the lattice parameteisotopically pure crystals of solid Ne. Squares20Ne; circles,22Ne.Error bars are less than the symbol size. Solid lines represent rederived from x-ray diffraction experiments by Batchelderet al.~Ref. 3!. Error bars of the experimental data are less than the lwidth. ~b! Relative change in the lattice parameter,x5(a20

2a22)/a22, vs the temperature. Solid squares, results of the prePI MC simulations; open circles, results from Mu¨ser et al. ~Ref.20!; solid line, experimental data from Ref. 3. Two dotted linabove and below the experimental results correspond to thebars of these data.

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circles in Fig. 7~b!#. We have carried out some simulationwith the same potential parameters and cutoff distancedynamic correlations employed in Ref. 20~up to the fourthneighboring shell!, but did not find such an increase inx atT.20 K. A possible explanation for this increase in isotopeffect in the experimental data is related to the generationthermal equilibrium vacancies,3 which are not allowed in ourMC simulations. Moreover, the experimental data atT.18 K could be affected by the problems discussed abin connection with the heat capacity results~partial vaporiza-tion of the solid and temperature uncertainties!.

The temperature dependence of the lattice constannatural neon and22Ne was calculated by Goldmanet al.6

from a self-consistent phonon theory. These authors fofor a 6-12 Lennard-Jones potential atT50 a relative isotopicdifference x(0)52.031023, close to the extrapolation oexperimental data3 @xexpt(0)51.9131023# and to our low-temperature results. This method gavea values differingfrom experiment by 1.431023 Å at T50, to be comparedwith a difference of 531024 Å found for our simulationresults. This difference with experiment increases as teperature is raised, and at 18 K, we found 1.331023 Å vs;331023 Å for the results in Ref. 6.

E. Bulk modulus

The isothermal bulk modulus of solid neon has been cculated from the mean-square fluctuations of the latticerameter by using Eq.~2!. The results are plotted in Fig. 8where squares and circles correspond to20Ne and 22Ne, re-spectively. At 4 K we findB59.88(3) kbar for 20Ne and10.19~3! kbar for 22Ne. These values are very close to thofound by Goldmanet al.6 from their self-consistent phonotheory atT50: B59.94 and 10.22 kbar, respectively. Thuwe find a difference of 0.31 kbar between both isotopes~a

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FIG. 8. Isothermal bulk modulus as a function of temperatuOpen symbols represent results of PI MC simulations: Squares20Ne and circles for22Ne. Error bars are of the order of the symbsize. Dotted lines are guides to the eye. Solid squares indicateperimental results from Batchelderet al. ~Ref. 42!.

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3% of the calculated values!. This difference is reduced to;0.1 kbar at 24 K. We note that the classical expecta~which in our context corresponds to the limitM→`) at T50 is Bcl(0)575e/s3517.2 kbar~see, e.g., Ref. 41!. Thismeans that quantum effects cause an important decreaseB,which for 20Ne amounts to 42% of the classical value atT→0.

Experimental results for natural neon obtained by Batelderet al.42 are displayed in Fig. 8 as solid symbols. At lotemperatures (T,18 K) these data are larger than ourMC results, and at higherT, the experimental data decreafaster than those derived from the simulations. Our resfor the bulk modulus of solid neon are very close to thofound by Acocellaet al.26 by using an improved effectivepotential Monte Carlo theory with Lennard-Jones aTang-Toennis-type43 interatomic potentials. This methogives results for the lattice parametera and heat capacityCVin good agreement with experimental data, but similarlyour PI MC simulations, predict values for the bulk modulB lower than the experimental findings atT,18 K. Thus,interatomic potentials that reproduce well several structuand thermodynamic properties of solid neon give too smvalues for the low-temperature bulk modulus.

IV. DISCUSSION

From the results of the quantum simulations presenabove, one can study the delocalization of the atomic nuin phase space as a function of temperature. With thispose, we calculate the productDrDp between the root-mean-square fluctuations of the position and momentumthe neon atoms. The latter are related to the kinetic energ

FIG. 9. Temperature dependence of the productDrDp for solidneon. Symbols indicate results of PI MC simulations: squares20Ne and circles for22Ne. Dotted lines are guides to the eye. Tdashed and dash-dotted lines were derived from a Debye modethe lattice vibrations with QD(20Ne)570 K and QD(22Ne)566.7 K.

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the relation (Dp)252MEk . In Fig. 9 we present the producDrDp as a function of temperature for20Ne ~squares! and22Ne ~circles!, as derived from the PI MC simulations. Thdashed and dash-dotted lines correspond to the Debye mdiscussed above for20Ne and 22Ne, respectively. In this approximation, the productDrDp at T50 amounts to 1.59\,irrespective of vD . Note that for an isotropic threedimensional~3D! harmonic oscillator this product takesT50 the value 3\/2. At T54 K, we find, for both 20Neand 22Ne, DrDp51.62\, very close to the value expecteda Debye model. Differences between both neon isotopethe productDrDp increase as temperature rises,DrDp beinglarger for 22Ne. In fact, in a Debye model, at high tempertures (kBT.\vD) one hasDrDp53A3kBT/vD , and henceDrDp}TM1/2.

In the presentation of our simulation results, we hacompared the MC data with those predicted by a Debmodel with a constant~temperature-independent! QD . Themain purpose of this comparison was to assess the imtance of anharmonic affects in the various quantities conered. Usually,QD is assumed to change with temperature,order to fit experimental heat capacity data. Something silar could be done with our simulation results, by assumintemperature-dependentQD(T). For example, one could fithe MC data for the kinetic energyEk presented in Fig. 2 byincreasingQD . However, thisad hocfit would not match thepotential energy, which is clearly smaller thanEk ~an anhar-monic effect that cannot be taken into account by changQD). Also, to fit other quantities, a different functionQD(T)is necessary, as is the case of (Dr )2, for whichQD should beat T.12 K lower than the value employed above to calclate the lines shown in Fig. 6~a!.

Anharmonic effects in solid Ar have been quantified elier by comparing the kinetic and potential energies derivfrom PI MC simulations.20 In the limit T→0 the potentialenergy was found to be 7% lower than the kinetic enerindicating an important anharmonicity of the zero-point mtion in this solid. Such an anharmonicity is expected tolarger for Ne. In fact, from the results shown in Fig. 3, wfind that at low temperaturesEp is 19% and 18% lower thanEk for 20Ne and 22Ne, respectively. These values are velarge when compared with those found for covalent solidslow temperatures. For example, for diamond, silicon, agermanium one finds at lowT differences betweenEp andEksmaller than 1%, and at the Debye temperature of eachterial, they are less than 3%.21,22,44Another interesting pointis that for these covalent materials the vibrationalEp is largerthan Ek , just the opposite to the trend found for rare-gsolids. Thus, the fact thatEp,Ek , found for Ne and Ar, isnot general in solids and can be associated with the particcharacter of the interatomic interactions present in rare-~Lennard-Jones! solids.

The low-temperature changes in the lattice parameteadue to isotopic mass can be explained quantitatively fromzero-point lattice expansionaref(0)2acl(0) for a crystalwith a reference isotopic massM ref . In fact, in a quasihar-monic approach the differenceDa(0)[aM(0)2aref(0) atT50 can be expressed as22,45

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Da~0!5@aref~0!2acl~0!#~AM ref /M21!. ~7!

Taking 20Ne as our reference, we havea20(0)2acl(0)50.1749 Å ~see above!, and for the isotopic differencea22(0)2a20(0), Eq.~7! givesDa(0)528.131023 Å. Thisvalue is close to the low-temperature results found fromrect calculation of the lattice parameters correspondingdifferent atomic masses@squares and circles in Fig. 7~a!#.

In summary, we have studied the dependence of structand thermodynamic properties of solid neon on isotomass by path-integral Monte Carlo simulations. These qutum simulations allow us to study phonon-related properof solids, without the assumption of a harmonic or quasihmonic approximation, usually employed in theoretical stuies. In this way, we could separate the kinetic and potencontributions to the vibrational energy of the solid and fouthat their relative values depart appreciably from those

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pected in harmonic approaches. We have found quantitaagreement with experimental data for the isotope effecthe heat capacityCP and lattice parameter. The calculateisothermal bulk modulus is smaller than the experimendata at low temperatures, but the relative difference of 3betweenB(22Ne) andB(20Ne) is expected to be close to thactual one. More experimental and theoretical work willnecessary to understand the difference between calcuand experimental results for the bulk modulus of solid ne

ACKNOWLEDGMENTS

The author benefited from discussions with R. Ramı´rez, T.Lopez-Ciudad, and L. M. Sese´. This work was supported byCICYT ~Spain! through Grant No. BFM2000-1318 and bDGESIC through Project No. 1FD97-1358.

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isotope effects in structural and thermodynamic properties of solid neon

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