Surface instability and isotopic impurities in quantum solids
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] N ov Surface instability and isotopic impurities in quantum solids
E. Cappelluti , , G. Rastelli , , S. Gaudio , and L. Pietronero , SMC Research Center and ISC, INFM-CNR, v. dei Taurini 19, 00185 Rome, Italy, Dipartimento di Fisica, Universit`a “La Sapienza”, P.le A. Moro 2, 00185 Rome, Italy and Laboratoire de Physique et Mod´elisation des Milieux Condens´es, Universit´e Joseph FourierCNRS - UMR 5493, BP 166, 38042 Grenoble, France (Dated: October 29, 2018)In this paper we employ a self-consistent harmonic approximation to investigate surface meltingand local melting close to quantum impurities in quantum solids. We show that surface meltingcan occur at temperatures much lower than the critical temperature T c of the solid phase instabilityin the bulk. Similar effects are driven by the presence of an isotope substitution. In this lattercase, we show that stronger local lattice fluctuations, induced by a lighter isotope atom, can inducelocal melting of the host bulk phase. Experimental consequences and the possible relevance in solidhelium are discussed. I. INTRODUCTION
Although melting is a very common phenomenon innature, the debate about its microscopic mechanism isstill open.
The first empirical theory was advancedby Lindemann. According to this view, melting occurswhen the ratio between the root mean square (rms) u rms = p h u i of the thermally activated lattice fluctu-ations and the lattice constant a exceeds a phenomeno-logical threshold u rms /a > ∼ .
16 which is roughly materialindependent . In spite of its several flaws (melting is de-scribed in terms of the properties of only the solid phase;no cooperative process and no role of defects are con-sidered. . . ), this simple criterion seems to work reason-ably well for a variety of materials. Lindemann criterionhas been recently found to apply as well at a local levelaround crystal defects.
This large range of validity ofthe Lindemann criterion suggests thus that a microscopicmechanism is actually operative.The most simple (and employed) model to accountfor the Lindemann phenomenology is the self-consistentharmonic approximation (SCHA). This maps an anhar-monic phonon model in a harmonic one. Anharmonic-ity is, then, taken into account, at a mean-field level,through a Debye-Waller-like term which is evaluated self-consistently. The breakdown of this approach is inter-preted as a signal of solid phase instability, and hencerelated to melting. One of the strength of this theory isthat it predicts, in contrast with the Born criterion butin agreement with the experimental observation, a par-tial but not total softening of the elastic constants of thebulk.The SCHA represents moreover an efficient tool to un-derstand in a qualitative way the phenomenon of the sur-face melting (SM), as first proposed by Pietronero andTosatti (PT). In this context, the physical mechanismunderlying the surface melting is quite simple: atomsclose to the surface have larger lattice fluctuations dueto the reduced number of nearest neighbor sites, andthe SCHA breaks down consequently at smaller temper-atures than in the bulk. It is clear that this simple theorydoes not represent an exhaustive description of the sur- face melting phenomenology, which should include rough-ening, preroughening, partial wetting, the role of “crys-tallinity” etc.
In addition it should be stressed thatthe SCHA does not determine directly the melting pointbut rather the instability of the solid phase which is pre-vented by the melting process itself. In this perspectivethis criterion should not be employed at a quantitativelevel. Nevertheless, since the solid phase instability andthe actual melting process are usually related to eachother, the PT theory provides a simple and useful wayto get information about the tendency of a system to-wards melting and surface melting and their dependenceon microscopic parameters.In this paper we generalize the results of the PT ap-proach in the case of quantum solids. The Lindemanncriterion in the quantum solid is shown to be twice aslarge as the one in the classical limit, in agreement withexperimental reports. We show a phase diagram forboth the bulk and surface melting cases and we investi-gate also the local melting due to an isotopic substitution.The temperature dependence of the lattice fluctuationsfor the different classical/quantum regimes is evaluatedand also the profile of the lattice fluctuations as functionof the distance from the surface or the isotopic impurities.The paper is organized as follows: in Sect. II, we reviewthe approach of Pietronero-Tosatti for classical solids; inIII we generalize the PT approach to investigate bulkproperties in quantum solids; surface melting and solidphase instability close to a quantum isotope impurity areanalyzed respectively in Sect. IV and Sect. V. Finally,in Sect. VI, we discuss our results and draw some finalconclusions.
II. SCHA AND SOLID PHASE INSTABILITY INBULK AND ON SURFACES
Let us consider for simplicity a one-dimensional chainof atoms. At the harmonic level, we can write the equa-tions of motion for the lattice displacement u n : M ¨ u n + k n,n +1 u n − u n +1 ) + k n,n − u n − u n − ) = 0 , (1)where M is the atomic mass, n denotes the site in-dex. The constant forces k n,n ′ , at the harmonic level,are related to the inter-atom potential V n,n ′ through therelation k n,n ′ = ∂ V n,n ′ /∂u n | { u n } =0 . Writing the po-tential V n,n ′ in terms of a Fourier expansion, V n,n ′ = P q V q exp[ iq ( u n − u n ′ )], we have thus, at the harmoniclevel, k n,n ′ = k = − P q q V q .In the spirit of SCHA, anharmonic terms can be takeninto account, by replacing the constant forces k n,n ′ , eval-uated at the lattice equilibrium, with their expectationvalue ˜ k n,n ′ averaged over the lattice fluctuations. Wehave thus explicitly:˜ k n,n ′ = (cid:28) ∂ V n,n ′ ∂u n (cid:29) = − X q q V q exp[ − q h| u n − u n ′ | i / ≃ k exp[ − λ h u n i / − λ h u n ′ i / , (2)where in the last line we have neglected the cross termsand we have replaced the dependence on the momenta inthe exponential with an effective parameter λ .By inserting (2) in Eq. (1) and considering the motionof each atom as an Einstein oscillator we have: M ¨ u n + 12 h ˜ k n,n +1 + ˜ k n,n − i u n = 0 , (3)where anharmonic effects are taken into account in theself-consistent renormalization of the elastic constants˜ k n,n ′ . Note that ˜ k n,n ′ depends on the expectation valueof the quadratic lattice fluctuations on both sites n , n ′ .It follows that the atomic motion described in Eq. (3)is ruled by the lattice fluctuations of the lattice environ-ment . In a bulk system h u n i = h u n ′ i = h u i , then˜ k n,n ′ = ˜ k = k exp (cid:2) − λ h u i (cid:3) . (4)and we get an unique self-consistent equation h u i = k B T ˜ k = k B Tk exp (cid:2) λ h u i (cid:3) , (5)where k B is the Boltzmann constant. In similar way,the SCHA phonon frequency is given by ˜ ω = q ˜ k/M = ω exp[ − λ h u i / ω = p k /M is the barephonon frequency at the purely harmonic level. It isconvenient to rewrite Eq. (5) by introducing the dimen-sionless quantities y = λ h u i , τ cl = λk B T /k : y ( τ cl ) = τ cl e y ( τ cl ) . (6)Eq. (6) has no solution for τ cl ≥ τ maxcl = 1 / e =0 . k B T c =0 . k /λ . At this value y ( τ maxcl ) = 1 and the maximummagnitude of the allowed lattice fluctuations above whichthe solid phase is unstable is h u i max = 1 /λ . Note that h u i max depends neither on the atomic mass nor on the force constant k , in agreement with the observation ofa material independent Lindemann criterion.Eq. (3) represents also the starting point to apply theSCHA to surface melting. In this case, one defines a localaverage lattice fluctuation h u n i which depends on the siteindex n . In the same spirit one can define a local elasticconstant:˜ k n,n − ,n +1 = h ˜ k n,n +1 + ˜ k n,n − i = k e − λ h u n i / h e −h u n − i / +e −h u n +1 i / i (7)We can write thus a set of recursive equations where thelattice fluctuations of the atom n depend on the latticefluctuations of the n − n + 1 atoms. The recur-sion is truncated at the atom n = 1 which represents theouter atom close to the free surface. This atom probesan effective harmonic potential smaller than the bulk,which increases its tendency towards melting. A numer-ical solution shows that the solid phase for the surfaceatoms becomes unstable at τ SMcl = 0 . Note that, althoughthe temperature of surface melting is smaller than in thebulk, local lattice fluctuations of the outer atoms can be larger than the ones in the bulk, violating locally theLindemann criterion. This is also in agreement with Ref.[9]. For instance, for the outer atoms n = 1 one finds y SM1 = 1 .
74. This is 74 % larger than the value in thebulk.
III. BULK PROPERTIES OF QUANTUMSOLIDS
We generalize now the above theory to the case ofquantum solids. In the following we shall assume a one-particle picture to be still valid, because of the smallnessof the exchange terms in the solid phase ( J max ∼ . He, J max ∼ µ K in He) with respect to the melt-ing temperatures T m > ∼ On the other hand amajor role in our approach will be played by the quan-tum fluctuations which dominate at low temperature inthe quantum regime. According to this perspective, theatomic motion of the atom n is described in terms of theSCHA Hamiltonian of the quantum oscillator: (cid:20) − ¯ h ∇ u M + 14 ˜ k n,n − ,n +1 u n (cid:21) Ψ ( u n ) = E Ψ ( u n ) , (8)where the self-consistent expression for the local potential˜ k n,n − ,n +1 is reported in Eq. (7).We consider first the melting properties of bulk sys-tems (˜ k n,n − ,n +1 = 2˜ k ). In this SCHA quantum modelthe total amount of lattice fluctuations is now easily com-puted as: h u i = ¯ h M ˜ ω (cid:20) n (cid:18) ¯ h ˜ ω k B T (cid:19)(cid:21) , (9)where n ( x ) = 1 / [e x −
1] is the Bose factor and where weremind ˜ ω = q ˜ k/M and ˜ k is given by Eq.4. In the classiclimit k B T ≫ ¯ h ˜ ω , n ( x ) ≃ /x ≫
1, and we recover theclassical result of Eq. (5). On the other hand, in thezero temperature limit, lattice fluctuations are due onlyto zero point quantum motion. In this case n ( x ) = 0 andEq.(9) reads: h u i = ¯ h p M ˜ k = ¯ h √ M k exp (cid:2) λ h u i / (cid:3) , (10)which, introducing the variable τ Q = λ ¯ h/ √ k M , can bewritten in the dimensionless form: y ( τ Q ) = τ Q e y ( τ Q ) / . (11)Eq. (11) represents the quantum generalization of Eq.(6) where the instability of the solid phase is now trig-gered by the magnitude of the quantum lattice fluctua-tions. This occurs for τ Q ≥ τ maxQ = 2 / e = 0 . y ( τ maxQ ) = 2, two times larger thanfor classical solids. This behavior is indeed in agreementwith the report of the Lindemann ratio u rms /a ≃ .
28 inhelium solids to compare with u rms /a ≃ .
16 forclassical solids.We also consider now the general case where both ther-mal and quantum fluctuations are important. From Eq.(9), after few straightforward passages, we get y ( τ Q , τ cl ) = τ Q e y ( τ Q ,τ cl ) / (cid:20) n (cid:18) τ Q τ cl e − y ( τ Q ,τ cl ) / (cid:19)(cid:21) . (12)Eq. (12) generalizes the stability criterion based on theSCHA in the full quantum-thermal case. As a generalrule we can expect that the classical regime is relevant inthe empirical range k B T / ¯ hω > ∼ /
4, which correspondsto τ Q < ∼ τ cl , while in the opposite regime τ Q > ∼ τ cl quantum effects are dominant.In Fig. 1, we show the phase diagram in the full τ Q - τ cl space where the instability of the SCHA occurs.Along the boundary line, the critical lattice fluctuationsincrease smoothly from y = 1 in the τ Q = 0 case to y = 2 in the τ cl = 0 case. Also interesting is the depen-dence of the lattice fluctuations as function of τ cl , namelythe temperature (Fig. 1, bottom panel). In the classicalcase, τ Q = 0, the quadratic fluctuations y ∝ h u i increaselinearly with τ cl until anharmonic effects take place. An-harmonicity is reflected in a upturn of the temperaturedependence of y ( τ cl ) and eventually in the breakdown of τ Q e00.20.40.60.81 τ c l e classical quantumy=2y=1 τ cl e00.511.52 y τ Q e=0 τ Q e=1.9 τ Q e=1.8 FIG. 1: (top panel) Phase boundary of the SCHA in the τ Q - τ cl space; (bottom panel) Lattice fluctuations y = λ h u i asfunction of the classical parameter τ cl for (from the bottomto the top) τ Q e = 0 . , . , . , . . . , . , . , . τ maxQ = 2 / e ). the solid phase for τ cl = 1 / e and y = 1. Increasing τ Q leads not only to the presence of zero point motion quan-tum fluctuations at τ cl = 0, but also to an overall changeof the temperature dependence of y . In particular, therange of the linear temperature dependence, character-istic of classical harmonic solids, is rapidly reduced andfor strongly quantum solids it disappears. Lattice fluc-tuations are large already at T = 0 and they are almostconstants in a wide temperature range (note that in thisregime anharmonic effects are in any case present due toquantum fluctuations) until an abrupt upturn with thetemperature leads to the breakdown of the solid phase.This trends is in good qualitative agreement with re-cent experimental measurements and Quantum MonteCarlo calculations. We shall discuss them in details inSect. VI.
IV. SURFACE MELTING OF QUANTUMSOLIDS
After having investigated the bulk properties of quan-tum solids, we analyze now the occurrence the role ofquantum fluctuations on the surface melting.We can write a recursive set of equations by consideringthe quantum/thermal SCHA solution of the n -th atom h u n i = ¯ h M ˜ ω n (cid:20) n (cid:18) ¯ h ˜ ω n k B T (cid:19)(cid:21) , (13)where ˜ ω n = q ˜ k n,n − ,n +1 / M and where the local elasticconstant ˜ k n,n − ,n +1 is still given by Eq. (7). Employingthe usual dimensionless variables τ Q , τ cl , y n , we can thuswrite: y n = √ τ Q e y n / √ e − y n − / + e − y n +1 / × " n τ Q τ cl √ e − y n − / + e − y n +1 / √ y n / ! , (14)which is valid for any n ≥
2, while the outer atom n = 1obeys the relation y = √ τ Q e ( y + y ) / (cid:20) n (cid:18) τ Q √ τ cl e − ( y + y ) / (cid:19)(cid:21) . (15)In order to obtain a numerical solution of Eqs. (14)-(15) for given τ Q , τ cl in the stable solid phase, we startby choosing a trial value of y . The full set of { y n } isthus obtained by Eqs. (14)-(15). The initial trial valueof y is thus varied until y n = ∞ converges to its bulk value.Typically, this is the only physical solution, since y n = ∞ diverges for larger values of y while it becomes rapidlynegative for smaller values of y . For τ Q , τ cl larger thansome critical value, the procedure does not converge for any value of y , signalizing that the solid phase of thesurface atom, described by the SCHA, is unstable.The resulting phase diagram, in the full τ Q - τ cl space,is shown in Fig. 2 (top panel), where we compare theboundary of the surface melting instability (dashed line)with the one of the bulk melting (solid line). For the purequantum case, τ cl = 0, at zero temperature the surface in-stability occurs for τ SMQ = 0 .
664 where the lattice fluctu-ations of the outer atoms become as large as y SM1 , Q = 3 . . < τ Q < . even at zero temperature whereasthe bulk solid phase is always stable up to a finite tem-perature range. The ratio T SM c /T c between the surfacemelting temperature and the temperature of bulk melt-ing is shown in the bottom panel of FIG. 2 showing thatthe critical temperature of surface melting can be signif-icantly lower than the bulk one in quantum solids.Before concluding this section, we would like to brieflycompare the melting occurring at a free surface withother cases such as grain boundaries. In the case of a τ Q e00.20.40.60.81 τ c l e bulksurface y =3.21y =1.74 melting instabilitystable τ Q e00.20.40.60.8 T c S M / T c bu l k quantumregimeclassical regime i n s t ab ili t y FIG. 2: (top panel) Phase boundary for the surface meltinginstability (dashed line) compared with the bulk instability(solid line) in the τ Q - τ cl space; (bottom panel) Ratio betweensurface melting temperature T SM c and bulk melting temper-ature T c as function of τ Q . For τ Q e even the bulk phase isunstable. For 1 . < τ Q e < k B T SM c / ¯ h ˜ ω < ∼ / free surface, in going from Eq. (14) to Eq. (15), we havedropped in Eq. (15) the contribution of the n = 0 atom.We note that the same results would be obtained in Eq.(14) considering n = 1 and assuming the lattice fluctua-tions at the site n = 0 to be infinite, namely y n =0 = ∞ .This latter condition would be obtained by the harmonicoscillator solution of (13) at the site n = 0 with a van-ishing elastic constant ˜ k n,n − ,n +1 , and it express nothingmore than the condition that atoms for n < n = 1 of a grain would not probe afree surface at the site n = 0, but it will interact with alattice environment with a different arrangement. Thesetwo situations can be described by a similar set of re-cursion relations (14) but with different boundary con-ditions: in the free surface case boundary conditions atsite n = 0 will be described by a completely soft oscillator˜ k n,n − ,n +1 = 0, signalizing that bulk solid is interfacedwith a free gaseous phase; on the other hand, in the caseof grain boundaries, the outer atom n = 1 will still probea crystal structure for n ≤
0, although with a differentarrangement. The boundary conditions at site n = 0will be still described thus by Eq. (13), but with a notcompletely soft mode. We expect thus that melting pro-cesses occur as well at grain boundaries as in the case offree surface. From the mathematical point of view, thissituation is identical to the case of quantum isotopic sub-stitutions, and it will be discussed in details in the nextsection. V. QUANTUM MELTING DRIVEN BYISOTOPIC IMPURITIES
In this section we address the problem of the solidphase stability close to a single local isotopic substitutionembedded in a perfect lattice structure. In the SCHA ap-proach, local stability of the solid phase is given by thesolution of Eq. (13). It is easy to check that, in the clas-sical limit k B T ≫ ¯ h ˜ ω n , the dependence on the atomicmass M in Eq. (13) drops out, so that different isotopesolids should probe the same stability conditions. On theother hand, the mere observation of a different meltingline for He and He is a direct evidence that helium isin a quantum regime.
Different isotopes are thus ex-pected to affect the bulk solid phase stability. We expectthe same at the local level.In the following we shall consider the case of a isolatesubstitution with a lighter isotope in a host matrix ofheavier atoms. Quantum fluctuations in the two caseswill be ruled locally by the parameters τ L = λ ¯ h/ √ k M L τ H = λ ¯ h/ √ k M H , respectively for the lighter (L) and forthe heavier (H) atoms. To study the stability of the solidphase close to this isotopic quantum impurity, we can stillemploy the recursive relations (14), namely for n ≤ − n ≥ τ Q = τ H , whereas for n = 0 (quantumisotope impurity) we have τ Q = τ L . We shall considerthe representative case of a He impurity embedded in He solid. In this case τ L /τ H = p / He solid close to thequantum isotopic He impurity. It is instructive to com-pare the classical limit τ Q = 0 with the pure quantumone τ cl = 0. In the first case lattice fluctuations of theguest atom, as well as of the host atoms, are indepen-dent on the relative atomic mass and they depend onlyon the temperature. As a consequence, the solid phaseclose to the guest atom is completely unaffected by theisotopic substitution. A quite different situation occursin the highly quantum regime τ cl = 0. In this case lo-cal quantum lattice fluctuations of the lighter guest atomcan be significantly enhanced due to its reduced atomicmass, and they can be sufficiently large to induce a local melting of the host solid phase. At τ cl = 0 this occursfor τ H > . τ Q > . τ H e00.20.40.60.81 τ c l e bulkquantum impuritymeltinginstabilitystable τ H e00.20.40.60.81 T c Q i m p / T c bu l k quantumregimeclassical regime i n s t ab ili t y FIG. 3: (top panel) Phase boundary for the lattice instabilityaround a quantum isotopic substitution with τ L /τ H = p / T Qimp c around the quantum impurity and bulk melting tem-perature T c as function of host quantum parameter τ H . In thequantum regime 1 . < τ H e <
2, where k B T Qimp c / ¯ h ˜ ω < ∼ / . < τ H e < T = 0. defines a region (quantum impurity melting) where solidphase is still stable in the bulk but local quantum lat-tice fluctuations break down the solid phase close to theisotopic substitution. On the physical ground we can ex-pect liquid bubbles of host atoms to appear close to theguest isotope. Unfortunately, since the present analy-sis is only related to the stability condition of the solidphase, we are not able to estimate the size of the liquidbubble, and more sophisticated approaches are needed.It is interesting to note that, for quantum solids, the crit-ical temperature T Qimp c for the local stability of the solidphase close to the quantum isotope impurity is reducedwith respect to the bulk T c . This is shown in the bottompanel of Fig. 3 where the ratio between the local T c closeto the impurity and the bulk T c is plotted as function ofthe quantum degree of the system, parametrized by τ H .In the quantum regime, where T Qimp c < ∼ ¯ h ˜ ω /
4, the lo-cal melting temperature T Qimp c can be significantly lowerthan the one in the bulk T c , and, for 1 . < τ H e < T = 0, although the bulk phase is stillstable. VI. DISCUSSION AND CONCLUSIONS
In this paper we have investigated the stability of quan-tum solids with respect to surface melting and to isotopicquantum substitutions. Both these phenomena can be es-sentially related to the amount of lattice fluctuations, andthey can be driven thus by thermal fluctuations as well asby the zero point quantum motion. We have shown thatthe effects of isotopic impurities and surface melting arestrongly enhanced in quantum solids. In particular weshow that when quantum fluctuations are dominant inquantum solids the solid phase can be rapidly destroyedon the surface and close to quantum impurities at tem-peratures much smaller than for the bulk melting.Helium solids are the natural candidates where thequantum instabilities of surface or interface can occur.The actual relevance of these quantum melting effectsare of course ruled by the magnitude of the quantum lat-tice fluctuations which are parametrized in our model bythe quantity τ Q . An accurate calculation of the quan-tum lattice fluctuations as a function of the tempera-ture in He and He solids has been provided recently,by using of Quantum Monte Carlo (QMC) techniques,by Draeger and Ceperley in Ref. 20, in excellent agree-ment with the experimental data. Quite interestingly,they find that the mean square lattice displacement h u i T does not follow at low temperature an harmonic behavior h u i T ≃ h u i T =0 + αT , but rather a more shallow one h u i T ≃ h u i T =0 + βT .Ref. 20 represents a suitable source to estimate an ef-fective value of τ Q representative of solid helium. To thisaim we fit the temperature dependence of the QMC dataof Ref. 20 with our quantum SCHA model described byEq. (12), where only two independent fitting parametersappear, namely λ and k (remind that τ cl = λk B T /k , τ Q = λ ¯ h/ √ k M ). The fit of our quantum SCHA [Eq.(12)] compared with the QMC data is shown in Fig. 4 forthree representative cases where the number of numeri-cal data is larger than the number of independent fittingparameters to guarantee the significance of the fittingprocedure. Also shown is the fit with a purely harmonicmodel obtained by setting λ = 0. The extracted valuesof λ and k , as well as of the corresponding τ Q and of theanharmonic renormalized phonon frequency at T = 0 ˜ ω are reported in Table I, where also we report the criticaltemperature T c for the solid phase bulk instability eval-uated within the SCHA and the experimental meltingtemperature T expm . It is worth to comment about the temperature behav-ior of the QMC data compared with the harmonic ( λ = 0) < u > ( A ) < u > ( A ) SCHAQMC [Ref. 20]EinsteinDebye0 5 10 15 20 25T (K)0.090.100.11 < u > ( A ) fcc Hefcc Hehcp He FIG. 4: Lattice fluctuations h u i evaluated within the SCHA(solid lines) as function of temperature for different heliumsolid conditions compared with Quantum Monte Carlo dataof Ref. 20. Values of k and λ in SCHA obtained by fittingQMC data are reported in Table I. Also shown are the purelyharmonic fitting of the QMC data with a Einstein and a Debyemodel. and anharmonic SCHA fit. An important point to behere underlined is that QMC results show a large meansquare lattice displacement at zero temperature all to-gether with a rapidly turn up of h u i close to the solidbulk instability. As we have discussed in Sect. III, thisis a characteristic trend of highly quantum solids. Onthe other hand, this behavior is poorly reproduced bya purely harmonic model where the amount of the lat- hcp He fcc He fcc He V (cm /mole) 12.12 10.98 11.54 k (meV/˚A ) 110 ±
10 140 ±
10 150 ± λ (˚A − ) 14 ± . ± . . ± . τ Q . ± .
08 0 . ± .
06 0 . ± . ω (meV) 4 . ± . . ± . . ± . T c (K) 14 ± ± ± T exp m (K) ∼ ∼ ∼ k and λ in SCHA obtained by fitting theQMC data of Ref. 20 for three representative helium solids,namely: hcp He at molar volume V = 12 .
12 cm /mole,fcc He at molar volume V = 10 .
98 cm /mole, and fcc Heat molar volume V = 11 .
54 cm /mole. Also reported arethe corresponding values of τ Q , the renormalized phonon fre-quency ˜ ω and the predicted critical temperature T c of thesolid phase bulk instability compared with the experimentalmelting temperature T expm . tice fluctuations at T = 0 is inversely proportional tothe temperature dependence. This is even more true if aDebye model would be employed since the temperaturedependence of a Debye model is even more shallow thanin the Einstein case.The strong quantum degree of solid helium, qualita-tively predicted by these arguments, is confirmed by thenumerical analysis of the SCHA fit which predicts a quan-tum parameter τ Q in the range τ Q ≃ . − . τ Q ≃ .
69, for the low pressure/ highmolar volume V = 12 .
12 cm /mole, is safely larger thanthe value τ SMQ ≃ .
664 where surface melting occurs atzero temperature, and also or the same order and slightlylarger even than τ SMQ ≃ .
681 where isotopic impurityinduced melting also occurs at zero temperature. Al-though these estimates have to be meant only indicativeof the quantum degree of helium solid, they clearly pointout that quantum anharmonic effects are large enoughin solid helium, for these or larger molar volumes, to en-force surface melting and local melting close to quantumimpurities down to zero temperature . Quantum MonteCarlo simulations have actually confirmed premelting at surface between helium solid and Vycor walls and in-ternal interfaces of a pure helium system , although notall possible interfaces undergoes a solid/liquid transition.These results shed an interesting light also on therecent report of the Non-Classical Rotational Iner-tia (NCRI) observed in He.
While it was ini-tially claimed to be an evidence of a supersolid (SS)phase, subsequent experiments showed a strong de-pendence of the NCRI on the annealing process, onthe presence of grain boundaries, on the amount of He concentration as well as on the freezingprocedure.
These observations give rise to an alter-native hypothesis to the SS phase, namely, that a liquidphase is confined at the grain boundaries and that massflow is related to superfluidity of the liquid component. Our results confirm this scenario and shed new perspec-tives about the role of disorder/grain boundaries in solidhelium. In particular we provide a natural explanationfor the existence of a liquid (and thus probably super-fluid) phase at the grain boundaries and we predict a localliquid phase also around He impurities. Local meltingclose to isotopic He impurities should be thus explicitlyconsidered.
Acknowledgments
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