On the Lifetime of Metastable Metallic Hydrogen
aa r X i v : . [ c ond - m a t . o t h e r] N ov On the Lifetime of Metastable Metallic Hydrogen
S.N. Burmistrov and L.B. Dubovskii
Kurchatov Institute, 123182 Moscow, Russia
The molecular phase of hydrogen converts to the atomic metallic phase at high pressures estimatedusually as 300 – 500 GPa. We analyze the decay of metallic phase as the pressure is relieved belowthe transition one. The metallic state is expected to be in the metastable long-lived state down toabout 10 – 20 GPa and decays instantly at the lower pressures. The pressure range of the long-livedmetastable state is directly associated with an impossibility to produce a stable hydrogen moleculeimmersed into the electron liquid of high density. For lower pressures, the nucleation of an electron-free cavity with the energetically favorable hydrogen molecule inside cannot be suppressed with thelow ambient pressure.
I. INTRODUCTION
For the first time, the question on the transition ofmolecular hydrogen into metallic phase under pressurewas apparently attempted by Wigner and Huntington [1].Later, in a great amount of papers [2, 3] (and therein)the equation of state for the metallic state as well asthe pressure of the transition into the atomic metallicstate are analyzed. In addition, there has arisen a ques-tion whether the lifetime of metastable metallic phasecould be macroscopically large in some pressure rangebelow the metal-to-molecular phase transition. Here weattempt this problem at zero temperature.In paper [2] the structure of metallic hydrogen is stud-ied in detail at zero temperature. It is shown in partic-ular that the metallic hydrogen at zero pressure is en-ergetically stable against the decay into separate atomswith the binding energy of about 1 eV per atom. As itconcerns the decay into molecules, metallic hydrogen isunstable and the energy of about 2.5 eV releases withescaping a molecule from the metal surface. The es-cape of hydrogen molecule from the surface of metallichydrogen should occur via tunneling across a potentialbarrier. Thus, in general, the lifetime of metallic phaseagainst this process may prove to be sufficiently large. Onthe other hand, this channel can be withdrawn providedthe metallic hydrogen is confined with the correspondingwalls.The formation of molecules is possible not only at thesurface of a metal but also in its bulk. The latter processcannot be eliminated. For the formation of a moleculeinside the bulk of metallic phase, it is necessary to havea cavity in which the metallic electron density is suffi-ciently small so that the molecule would be energeticallyfavorable. The inception of a cavity is always associatedwith increasing the total energy in the system due to ex-trusion of a metal from the cavity. In order to place amolecule into the cavity, the latter should have a size ofseveral interatomic distances. In general, the energy gainresulted from the formation of a single molecule cannotcompensate an increase of the total energy in the system.In any case this occurs in the pressure region near themolecular phase-metal phase transition point since thechemical potentials per atom in the metallic and molecu- lar phases are close to each other. Thus the nucleation ofthe molecular phase with the large number of moleculesin the critical nucleus becomes necessary. The large num-ber of particles in the critical nucleus results inevitably ina drastic reduction of the nucleation probability of suchnuclei because this process is a tunneling overcoming of apotential barrier and the tunneling probability dependsexponentially on the number of particles.In the present work we study in what pressure rangebelow the transition pressure the macroscopic descriptionof the nucleus dynamics is possible and how this rangedepends on the approximations chosen. As we will seelater, the lifetime of metallic phase is macroscopicallylarge and practically infinite so long as the macroscopicconsideration is possible.In the opposite case when the critical nucleus is notlarge and contains a few molecules, the lifetime of themetallic phase is small. This can be estimated as follows.The probability W for nucleating the critical nucleus aslarge as a single molecule in a specific site of a bulk isalways small W ∼ ω D exp (cid:0) − α p m/m e (cid:1) . (1)The point is that there is a large factor in the exponent,i.e. square root of a ratio of atom mass m to electronmass m e . Here ω D is a frequency of about Debye fre-quency in metallic hydrogen ( ω D ∼ s − and α is thequantity associated with the tunneling motion of hydro-gen atoms in the metallic phase in the course of nucle-ating a molecule. We estimate α . W ν is large for the inception of a singlenucleus in the bulk containing ν ∼ atoms W ν ∼ νW . (2)This gives a short lifetime for a metallic hydrogen samplewith the large ν ∼ number of atoms.Note that the same estimate for the small-sized par-ticles of about ν ∼ atoms yields a sufficiently largelifetime. For the process as an escape of molecules fromthe surface, the lifetime may also prove to be large since ν ∼ in this case. In addition, for the evaporation it FIG. 1: The potential energy U versus nucleus radius R at ambient pressure P = 40 GPa. The critical radius is R c =11 a.u. The number of particles in the critical nucleus is N c =190. is essential not the probability of a single event for theformation of a molecule but the evaporation rate deter-mined by escaping the large number of molecules.To describe a macroscopic nucleus, we employ theLifshitz-Kagan approach [4]. As a main variable in thisapproach, we take the density of the phases, i.e., stable(molecular) and metastable (metallic) ones. The poten-tial energy of the system as a function of the nucleusradius R has a typical shape given in Fig. 1. The growthof potential U at small radius R is determined by theeffective interphase surface tension and proportional tothe radius-squared, i.e. U ( R ) ∼ R as R →
0. In thecase of the junction between the metallic and molecularphases the effective surface tension is mainly associatedwith the electron liquid outflow from the metal and withthe decrease of the binding energy of a molecule in theelectron liquid.The negativity of potential energy U at large ra-dius R is due to unfavorable difference in energies ofmetastable metallic and stable molecular phases, i.e. U ( R ) ∼ − ∆ µ R where ∆ µ is a difference in the chem-ical potentials of the both phases. The transition frommetastable state R = 0 to stable state R → ∞ occursvia tunneling under potential barrier (Fig. 1) due to ki-netic energy T ( R, ˙ R depending on both radius R ( t ) andgrowth rate ˙ R ( t ) T ( R, ˙ R ) = M ( R ) ˙ R / . (3)The mass M ( R ) in the kinetic energy is associated witha difference in the densities of the metastable and sta-ble phases and results from the outflow or inflow of thesubstance during the formation of a nucleus.Besides the various densities the phase transition canbe characterized with a number of other internal vari-ables independent of density, e.g. spacing between thenuclei in the course of nucleating a molecule. These in-ternal variables are characterized with the corresponding potential barriers and kinetic energies. Below we sup-pose that the setting in equilibrium in these variables isthe faster process and we take the optimum magnitudesof those variables. In the next section we elucidate theprocedure in detail.As will be shown below, such macroscopic approach,associated with the nucleus dynamics governed with thedifferent phase densities, is possible in a wide range ofpressures below the critical one P c ∼
300 – 500 GPa downto 10 – 20 GPa. Within this pressure range the criticalnuclei have a large number of particles, resulting in along-lived stability of metallic hydrogen. Note that weunderrate the pressure range, neglecting a series of effectswhich should certainly lead to increasing the lifetime ofmetallic phase.
II. PROBLEM STATEMENT
Let spherical molecular nucleus of radius R be in themetallic hydrogen at the ambient pressure P . The po-tential energy U ( R ) of a nucleus can be written as (A1) U ( R ) = 4 π Z R n ( r, R ) (cid:18) ε (cid:0) n ( r, R ) , R − r (cid:1) − µ ( P ) (cid:19) r dr +(4 π/ P R + 4 πσR . (4)Here µ ( P ) is the chemical potential of metallic hydro-gen, n ( r, R ) is the density of the molecular phase at point r of the nucleus with radius R , ε (cid:0) n ( r, R ) , R − r (cid:1) is theenergy density of the molecular phase, and σ ( P ) is thesurface tension of the interface.The magnitude of surface tension σ and the behavior ofenergy density ε (cid:0) n ( r, R ) , R − r (cid:1) , as a function of the dis-tance from the boundary with the metal, are determinedwith extending the electron liquid outside the metal intothe near-surface region of about Wigner-Seitz radius r s in size [5]. For the energy density ε of molecular phase,depending on the molecular phase density n and the dis-tance from the metallic hydrogen boundary, we employsimplest approximation ε ( n, x ) = ε ( n ) + h ( x ) . (5)The first term here corresponds to the energy density ofmolecular phase for the given density n in the lack of themetal electron density. The second term h ( x ) impliesthat the energy of a molecule beside the metal boundarydiffers significantly from energy 4.7 eV in vacuum takenfrom the energy of two separate atoms due to dipping amolecule into electron liquid of a metal. The term h ( x )can be represented as an external potential affecting themolecule as a result of extending the electron liquid out-side the metal into the boundary region of about r s insize. Thus, potential h ( x ) as well as surface tension σ de-pend on the Wigner-Seitz radius r s or, correspondingly,on the pressure P inside the metal.In addition, these both quantities, σ and h ( x ), dependon the nucleus size R as well. We will neglect this depen-dence since we are interested in the macroscopic R ≫ r s nuclei and the dependence of σ and h ( x ) on radius R becomes insignificant as R & r s .Relation (5) corresponds to the gas approximation inthe density of molecular phase, meaning a possibility toneglect dependence of h ( x ) on density n . The approxi-mation can be used while the density of molecular phaseat the boundary is much smaller than the density of theadjacent metal phase. The point is that potential h ( x )is governed with the outflow of electron liquid from themetal, which is almost independent of the strongly lo-calized electron density at the molecule [6, 7]. In thenear-surface region, where the magnitude h ( x ) is large,the density of molecular phase takes the smaller valuecompared with that at the nucleus center since such den-sity distribution corresponds to the minimum of poten-tial energy U ( R ). At low P .
100 GPa pressures thereappears a gap of about r s between the molecular andmetallic phases. Inside the gap the density of hydrogenatoms vanishes. For the higher pressures, it is impossi-ble to assert that the density of molecular phase besidethe nucleus boundary is much smaller than the density ofmetallic phase. However, even near the molecular phase-to-metallic phase transition point P c the density of molec-ular phase differs from that of metallic phase by a factorof 2. In the next section we discuss function h ( x ) indetail since this quantity governs mainly the nucleationprobability.Due to the same reason we will neglect the dependenceof surface tension σ on the molecular phase density at thenucleus boundary, i.e. we put the surface tension equal toits magnitude for the vacuum-metal boundary. For thedependence of the energy density of molecular phase ε ( n )upon n , we apply the local approximation ε = ε (cid:0) n ( r, R ) (cid:1) since the involvement of, e.g., density gradient in ε ( n ),corresponds to considering the quantities in such scalewhich we neglect in describing the potential h ( x ). Thisis also associated with the smallness of density gradientin the nucleus due to large nucleus radius R ≫ r s .The expression (4) for the potential energy of a nucleusassumes that the internal energy of the system dependson the densities of phases alone. On the whole, this im-plies the liquid-like description of the both phases. Theinvolvement that the both phase are the crystalline onesincreases the potential energy. The liquid-like descriptionof the system means neglecting the shear energy com-pared with the energy of the bulk compressibility whichis completely taken into account in Eq. (4). (See Ap-pendix A.) Such neglect in metallic hydrogen is alwaysjustified since the shear modulus is small as comparedwith the bulk modulus [2, 3]. In molecular hydrogen un-der low P .
10 GPa pressures the both energies, bulkcompressibility energy and shear energy, are small com-pared with the energy of formation of molecules and areinessential in the expression for potential energy. Forlarger P &
10 GPa pressures, there occurs the same situ-ation as in metallic hydrogen. The shear energy is smallcompared with the bulk compressibility energy and canbe neglected as before. We will suppose that the nucleus grows slowly, namely,the growth rate ˙ R of nucleus boundary is much less thanthe sound velocity s ˙ R ≪ s. In the case of such quasistationary nucleus growth thereis a sufficient time to set the mechanical equilibrium inthe bulk of the both phases. An existence of mechanicalequilibrium in the metastable metallic phase is taken inEq. (4) into account since the metallic density n as wellas pressure P are assumed to be constant in the deriva-tion of Eq. (4). As it concerns a nucleus, the equilibriumover the nucleus corresponds to constancy of the totalchemical potential µ (cid:0) n ( r, R ) (cid:1) + h ( R − r ) = C. (6)Here C is a constant to be determined with the conditionsat the nucleus boundary.Equation (6) can be obtained with varying the poten-tial energy U ( R ) over nucleus density n ( r, R ) in (4). Onthe other hand, this equation is a result of the hydrody-namical Euler equation [8] ∂ v ∂t + ( v ∇ ) v = − n ∇ P − ∇ h. (7)Putting v →
0, we arrive at Eq. (6) since ∇ P/n = ∇ ( ε + n∂ε/∂n ) resulted from relation P = n ∂ε/∂n . BesidesEq. (6) the minimum in energy U can also be associatedwith n ≡ n > n = 0 allow us to determine the den-sity distribution n ( r, R ) in the nucleus bulk. Constant C in (6) can be found using the condition of mechani-cal equilibrium between the nucleus and the metastablephase. The condition of mechanical equilibrium betweenthe phases is obtained with varying the potential energy U in the nucleus radius R under invariant total number N of particles in the nucleus (cid:0) ∂U/∂R (cid:1) N = 0 , (8) N = 4 π Z R n ( r, R ) r dr. (9)Equation (8) means the following. As the nucleusradius varies, the system keeps the state of the mini-mum potential energy but has no sufficient time to per-form the transition of particles from one phase to theother. Emphasize that this condition holds due to as-sumption about the quasistationary nucleus growth whenthe growth rate is small compared with the sound veloc-ity s . The latter velocity characterizes the rate of settingthe mechanical equilibrium. Equation (8) determines therelation between constant C (6) and pressure P in themetallic phase (Appendix B) P = − σ ( P ) R − R Z R n ( r ) h ′ ( R − r ) r dr + n ( R ) (cid:2) C − ε (cid:0) n ( R ) (cid:1) − h (0) (cid:3) . (10)Here n ( r ) = n ( r, R ) and n ( R ) = n ( R, R ).Equations (6), (10) and n = 0 determine the den-sity distribution n ( r, R ) in the nucleus at the ambientpressure P . If we substitute the above distribution intoEqs. (4) and (9), we obtain the dependence of energy U and particle number N upon nucleus radius R . The typi-cal behavior of energy U as a function of radius R is givenin Fig. 1. The point R c at which U ( R c ) = 0 correspondsto the critical nucleus radius.The role of temperature in the quantum transitionfrom the metastable phase with R = 0 to the state ofovercritical nucleus with R > R c is replaced with thekinetic energy in variables R and ˙ R resulting from thedifferent densities of the phases and outflow of a matterfrom the nucleus [4]. The kinetic energy of a nucleus isgiven by T = T + T , (11) T = m π Z R n ( r, R ) v ( r ) r dr,T = m π Z ∞ R n ( P ) v ( r ) r dr. Here m is the hydrogen atom mass, kinetic energy T isassociated with the motion of particles in the nucleus,and T is due to the motion of the mass in the metallicphase. Velocity v ( r ) inside the nucleus is determined bythe continuity equation ∂n ( r, R ) ∂t + 1 r ∂∂r (cid:0) r v ( r ) n ( r, R ) (cid:1) = 0 . (12)Remind that density n depends on time t via variable R = R ( t ) alone and r is a running coordinate. Thus, wehave ∂n∂t = ∂n∂R ˙ R. (13)The solution of the above two equations for velocity v ( r )at a given distribution n ( r, R ) can be written as v ( r ) = ˙ Rr n ( r, R ) Z r ∂n ( r ′ , R ) ∂R r ′ dr ′ . (14)The substitution of Eq. (14) into (7) shows that theleft-hand side of Eq. (7) is small on the scale of a ra-tio ˙ R/s ≪
1. So, we have a quasistationary growth ofa nucleus and, at first, we can find distribution n ( r, R )obeying (6). Then, we substitute the distribution ob-tained into Eq. (14) to find the velocity distribution v ( r )and avoid the combined solution of the Euler equation(7) and continuity equation (12). Substituting (14) intothe equation for T yields T = m π ˙ R Z R drr n ( r, R ) (cid:18)Z r ∂n ( r ′ , R ) ∂R r ′ dr (cid:19) . The velocity distribution v ( r ) in the metastable phaseobeys the continuity equation (12) as well. However, un-like (13) the term with the time derivative of the den-sity vanishes since the density in the metastable phase is constant for all R and is determined with the ambientpressure P . This entails the following behavior [4] v ( r ) = A ˙ RR /r . (15)The dimensionless factor A can be found using the con-dition of conserving the total number of particles in thesystem dN/dt = 4 πR n ( P ) (cid:0) ˙ R − v ( R ) (cid:1) . (16)The left-hand side equals the rate of varying the parti-cle number N in a nucleus and the right-hand side doesthe incoming flow of particles from the metastable phase.Since the time dependence in (9) enters via variable R ,we arrive at dNdt = 4 πR n ( R ) ˙ R + 4 π ˙ R Z R ∂n ( r, R ) ∂R r dr. (17)Using Eqs. (15) – (17), we obtain for AA = 1 − n ( R ) n ( P − R n ( P ) Z R ∂n ( r, R ) ∂R r dr. (18)Putting Eq. (15) into the expression for T and calcu-lating the integral in r , we represent kinetic energy T in(11) as T = M ( R ) ˙ R / , (19) M ( R ) = 4 πm (cid:20) n ( P ) R (cid:18) − n ( R ) n ( P ) − R n ( P ) Z R ∂n ( r, R ) ∂R r dr (cid:19) + Z R drr n ( r, R ) (cid:18)Z r ∂n ( r ′ , R ) ∂R r ′ dr ′ (cid:19) (cid:21) . This expression differs from the corresponding one in [4]because the compressibility of the stable phase is takeninto account. In our case the involvement of compress-ibility is essential since the density of molecular hydrogenvaries by a factor of 10 from zero pressure to the 100 GPapressure region.The next analysis of nucleation kinetics is based onthe hamiltonian with the potential energy (4) and kineticenergy (19). The density distribution in the nucleus bulkis governed with Eqs. (6) and n = 0 and related with theambient pressure via Eq. (10).Deriving the hamiltonian, we have assumed that thestate of the system is completely determined with thedensities of the phases and the other physical quanti-ties are adjusted adiabatically and unambiguously to themagnitudes of the densities. As an example of such quan-tities, we can mention the spacing between two atoms inthe hydrogen molecule, symmetry of the crystalline lat-tice in the both phases, and electron density tracing adia-batically the nuclear motion. The adiabatical adjustmentof these quantities supposes the slow and quasistationarynucleus growth when the setting of all processes in equi-librium occurs faster than the nucleus growth, i.e., transi-tion of particles from the metallic phase to the molecularone. In particular, this implies the mechanical equilib-rium between phases (8). The adiabatical relaxation ofthese parameters means the neglect of contributions ofthese parameters both to the potential and to the kineticenergies. Such optimization underrates the lifetime of themetastable phase. Setting the equilibrium in the theseparameters occurs at about sound velocity or faster as,for example, in the case of adiabatic relaxation of elec-trons to the motion of nuclei. Hence, we suppose thesmallness of nucleus growth rate ˙ R compared with thesound velocity s . Adiabaticity and relaxation of all pa-rameters are equivalent to the fact that the frequency ofoscillations, associated with the underbarrier motion, ismuch smaller than all other frequencies in the system.The frequency ω b , determining the nucleus underbarrierevolution, reduces as the nucleus radius R c grows ω b ∼ (cid:18) U max M ( R c ) R c (cid:19) / ∼ ω D (cid:18) r s R c (cid:19) / . (20)Here U max ∼ σR c and M ( R c ) ∼ mn R c where m is anatom mass and ω D is of the order of the Debye frequency.For the nucleus of large radius R c , frequency ω b is small,entailing a correctness of the quasistationary application.In the case when the size of the critical nucleus is aboutseveral interatomic distances it is necessary to take thelack of quasistationary approximation into account. III. DISCUSSION OF THE PARAMETERS INTHE HAMILTONIAN
The potential energy U ( R ) (4) and kinetic energy T ( R, ˙ R ) (19) are governed with the following parameterssuch as: ( i ) chemical potential of metallic phase µ ( P )depending on the pressure, ( ii ) energy ε ( n ) and chemi-cal potential µ ( n ) depending on the density, ( iii ) energyvariation h ( x ) of a molecule beside the metal surface, and( iv ) surface tension σ ( P ) of a metal. Below we considerthe consistent description of the above quantities.The behavior of energy and chemical potentials asa function of pressure are given for the molecular andmetallic phases in Fig. 2. Figure 3 plots the equations ofstate for the both phases.We take the results for the metallic phase after [3]. Forthe molecular phases, we use the results from the samework [3] and also after [9]. Note that the functions givenfor metallic hydrogen are obtained with high accuracyat high P &
100 GPa pressures and, correspondingly, atsmall r s < .
45. In zero pressure region of r s ≈ . ± FIG. 2: The chemical potentials of metallic ( I ) and molecular( II ) phases as a function of pressure. Curve III is the energyof molecular phase.FIG. 3: The equations of state for metallic (I) and molecular(II) phases. P .
10 GPa. This is associated with the existence ofprecise hydrostatic measurements and with the relativelyexact description since it is sufficient mainly to considerthe pair interactions alone. For higher P &
10 GPapressures, the consideration of pair interactions alone be-comes insufficient [9] and the theoretical description ofthe equation of state has a worse accuracy. The same isreferred to the experiments in this region. An uncertaintyin the data for the equation of state results in a disper-sion of the phase transition pressure P c [3]. However,for our purposes such dispersion has no principal mean-ing since the pressure region where the metallic phaseexists practically the infinite time is wide and extendsdown to pressures of about 10 GPa. An uncertainty ofthe equations of states for the phases shifts the pressureboundary by ± β . This entails that the small varia-tions of the exponent results in a strong variation of the FIG. 4: The functions h ( x ) and ˜ h ( x ) as a function of distancefrom the metal surface. The metallic phase is approximatedwith the jellium model of r s = 1 .
7. The dash line correspondsto the energy of a hydrogen atom far from the metal surface. exponential expression.The main parameter, which determines the regionof the long-lived metastable state, is the energy of amolecule h ( x ) varying beside the metal surface due tospreading the electron density outside the metal. Themagnitude h ( x ) can be obtained with the direct calcu-lation of the behavior of the energy of a molecule as afunction of the distance taken from the metal surface.Such calculation is performed in [6] using the model ofjellium. Figure 4 shows the dependence h ( x ) extrapo-lated from the data after [6] to r s = 1 . r s = 1 . h ( x ).Minimum A at the curve ˜ h ( x ) (Fig. 4) corresponds tothe energy position of the chemical potential for atoms inmetallic hydrogen. At this point there occurs a chemicalsorption of hydrogen molecule at the metallic hydrogensurface. This means a possibility of adding new layerto the metallic hydrogen surface. Minimum A of curve h ( x ) lies higher than the magnitude h ( x ) at x → ∞ .Otherwise, there occurs an associative chemical sorptionat the surface, i.e. chemical sorption with releasing theenergy at the transition of a particle from the surface tothe infinity. If atoms are located far from the surface ascompared with the intersection point x c of curves h ( x )and ˜ h ( x ) (Fig. 4), the existence of a molecule becomespossible. For the distances closer than x c , the separateatoms are more energetically favorable.The energy h ( x ) of a molecule in the field of electronliquid can be subdivided into the energy of atoms in themolecule and the binding energy of a molecule h ( x ) = 12 (cid:2) ˜ h ( x − R /
2) + ˜ h ( x + R / (cid:3) + H. (21)Here R is the equilibrium distance between the nucleiin the molecule. The binding energy H of a molecule [7] proves to be slightly affected with the orientation of amolecule. Thus, the local approximation governed bythe electron liquid density ρ ( x ) is well adequate, i.e., H = h (cid:0) ρ ( x ) (cid:1) . (22)The behavior H ( ρ ), as a function of r s , is shown in Fig. 5and ρ − = 4 πr s /
3. The different curves in Fig. 5 corre-spond to various methods of calculating the function H .Curves 1, 2 and 3 are obtained from the data [6] onthe behavior of a hydrogen molecule beside the surfaceof metallic jellium with various r s of a metal, namely,2.07, 2.65 and 4. Subscript 0 in r s differs r s of metallicsubstrate from r s determining the magnitude of electrondensity with the aid of ρ − = 4 πr s /
3. One can see thatthe curves, obtained with the different ways, are veryclose to each other and one may say about the universalbehavior.In order to derive the behavior of energy of a moleculeas a function of the distance from the metal surface ifone knows behavior H ( ρ ), it is sufficient to apply thebehavior ρ ( x ) beside the surface [5]. The latter problemis one-dimensional and, therefore, is simpler.One can see from Fig. 5 that the binding energy of amolecule vanishes at the electron density r s = 4 .
6. Forsmaller r s , molecule is energetically unfavorable and dis-sociates into atoms.Note that the behavior of h ( x ) and ˜ h ( x ) in Fig. 4 cor-relates well with the data [2] on metallic hydrogen. Theasymptotic behavior h ( x ) for x → ∞ is determined withthe binding energy of hydrogen molecule and the positionof minimum A is governed by the binding energy of anatom in metallic hydrogen. The accuracy of coincidencebetween the minimum at curve ˜ h ( x ) and the position ofthe chemical potential for atoms in metallic hydrogen isdetermined by the neglect of the coupling between atomsin the surface layer. This approximation is analogous toneglecting the dependence h ( x ) on the density of molec-ular phase n .While obtaining the plots given in Figs. 4 and 5, thegenuine metal with the discrete structure is replaced withthe jellium model. This approximation is well justified atthe large distances from the metal. In essence, if x > r s ,the discreteness of the crystal lattice becomes insignifi-cant. In addition, function h ( x ) at such large distancesis of most interest. The point is that in the range of rel-atively low pressures of about 10 GPa, there appears aspacing between the metal and molecular phases in whichthe density of molecular phase vanishes. Thus, for suchpressures an uncertainty due to inaccurate determina-tion of function h ( x ) at small distances is negligible. Forhigher pressures, the lifetime grows more and an addi-tional specifying h ( x ) in this range becomes inessential.The growth of h ( x ) at small x < r s distances in the metalwith the discrete lattice is reduced as compared with thejellium model since the discrete ions are located from themolecule farther on than for the smoothed background ofthe jellium. The discrete ions attract electrons strongerand, therefore, electron liquid spreads at smaller distance FIG. 5: The plot of binding energy H for a molecule immersedinto an electron liquid. from the surface as compared with the jellium model.This results in the slower enhancement of function h ( x )in the metal with the discrete lattice than that in thejellium model. Below we take this fact into account andvary function h ( x ) at small distances in order to clarifyits effect on the lifetime of the metallic phase.While calculating h ( x ) in [6], the axis of a moleculeis assumed to be normal to the metal surface. Providedone neglects the effect of the molecule orientation on thedistribution of electron liquid beside the metal surfaceand takes into account that the binding energy H of amolecule depends only on the magnitude of the electronliquid density and is independent of the molecule orienta-tion, one can obtain information about h ( x ) for an arbi-trary orientation using the data on h ( x ) for the normal-to-surface orientation of a molecule. For example, in thecase when the molecule axis is parallel to the surface thefollowing relation is valid h k ( x ) = ˜ h ( x ) + H ( x ) . (23)In the calculation [6] of function h ( x ) the parameter R , distance between the nuclei in the molecule, keepsunvaried as the molecule approaches the metal surface.This approximation is well justified due to slight depen-dence of the binding energy on R .To conclude the discussion of functions h ( x ) and ˜ h ( x ),we note a few aspects. First, the binding energy of amolecule is positive starting from the electron densitieswith r s = 4 .
6. For lower r s , molecule is energeticallyunfavorable. This specific magnitude r s is three times aslarger than r s of metallic hydrogen. Correspondingly, theelectron liquid density at the center of cavity, in which themolecule could be placed, should be at least ≈
30 times assmaller if compared with the electron density of metallichydrogen. Thus, to nucleate a single molecule inside themetallic phase, it is necessary to produce a large cavitywith the radius of a few r s . This fact correlates with theassumption in the previous sections that the nucleationof the molecular phase requires the outflow of a matter in the metallic phase and that the radius of the criticalnucleus should significantly exceed r s .The next point to be mentioned is that the hydrogenatom escaping from the metallic phase and traveling frompoint A (Fig. 4) along curves h ( x ) and ˜ h ( x ) should over-come an energy barrier. In our model (Sec. II) we neglecta possibility for reflection of hydrogen atom from the en-ergy barrier in the course of quantum tunneling. Thisimplies that we underrate the lifetime of the metastablemetallic phase.Here we emphasize also that the real crossover betweenthe curves h ( x ) and ˜ h ( x ) is smooth-like since an addi-tional parameter R , distance between the nuclei in themolecule, varies with the distance from the metal sur-face. In principle, there are possible two situations forthe transition from the molecular to metallic phase.First, R varies smoothly from the typical spacing be-tween the nuclei in a molecule to that in the metallicphase. Second, variation R is a jump-like one at thephase interface. Below, as in the previous section, weimply that distance R follows adiabatically the density.Then a single distinction between these two cases is thefollowing. As distance R varies continuously, function h ( x ) behaves more smoothly in the narrow transient re-gion between the phases. So, from this viewpoint it isuseful to vary h ( x ) at distance of about r s in the tran-sient region between the phases. This will be done belowwith the analysis of the data.Finally, we discuss the surface tension σ ( P ) of metallichydrogen for the vacuum-metal boundary. The calcula-tion of the surface tension at the vacuum-metal bound-ary and comparison with the experimental data has beentreated in a large number of works [5], [11] and [12].In all these papers the consideration is based on theHohenberg-Kohn-Sham density functional in which thekinetic, exchange and correlation energies of nonuniformelectron gas are described as a functional of electron den-sity ρ ( r ). The surface tension of a metal results from theredistribution of electrons and ions beside the metal sur-face as compared with the bulk distribution. We hereemploy the simplest version of Ref. [12] when the elec-tron distribution is assumed to be homogeneous in themetal bulk. In this case the surface tension σ as a func-tion of pressure is given with curve I in Fig. 6. The curveis well fitted with the relation σ I ( P ) = − P in erg/cm . (24)Here pressure P is given in GPa. This approximationneglects a series of contributions to the surface tension.The main contribution neglected is that the density ofexchange and correlation energies in nonuniform electrongas has a nonlocal relation with the electron liquid den-sity. Provided this contribution is taken into account as asimple gradient correction, we obtain the surface tension-pressure plot as a curve II in Fig. 6. The curve is welldescribed with σ II ( P ) = − P + 520 in erg/cm . (25) FIG. 6: The surface tension-pressure dependence.
Here we do not discuss the finer effects associated, e.g.,with a shift of the edge ion planes beside the metal surface[11] or with nonuniform distribution of electrons in themetal bulk [12] since these contributions are smaller thanthe term resulted from the gradient exchange-correlationenergy. In addition, these corrections become insignifi-cant due to uncertainty of the equation of state in the zeropressure range. Emphasize that the shift of the equationof state with about 5 GPa results in varying the surfacetension by about 100 erg/cm within the zero pressurerange. In what follows, we use mainly expression (24)for the surface tension since this expression results inthe stricter condition for the lifetime of the metastablemetallic phase.Note that the final result for the lifetime of metastablemetallic hydrogen is not noticeably sensitive whether onetakes Eq. (24) or Eq. (25) for the surface tension. Thepoint is that the function h ( x ) itself results in the effectivesurface tension which exceeds surface tension σ ( P ) by afew times for the moderate pressures of about 10 GPa.For the larger pressures, the effect of surface tension σ ( P )is weaker due to effect of the bulk term 4 πρR / IV. QUANTUM NUCLEATION OF THEMOLECULAR PHASE. THE DISCUSSION OFRESULTS
Like [4], we perform the semiclassical analysis of thetunneling transition between the phases. The classicalLagrangian of the system reads L ( R, ˙ R ) = M ( R ) ˙ R / − U ( R ) . Here effective mass M ( R ) is determined with Eq. (19)and energy U ( R ) is given by Eq. (4). In Appendix C thederivative ∂n ( r, R ) /∂R in the equation for mass M ( R )is transformed to the ambient pressure-fixed expression.The hamiltonian corresponding to the above Lagrangianreads H = p R / M ( R ) + U ( R ) . (26) Within our approximation the dynamic description ofthe system during phase transition is governed with thesingle principle variable R = R ( t ) and correspondingmomentum p R . The initial state of the system is ametastable state. Provided a possibility of the tunnelingtransition is ignored, the ground state in potential U ( R )for radius R close to zero (1) can be estimated with usingthe uncertainty principle as p R · R typ ∼ ~ . Since radius R is not large, one can approximate potential U ( R ) as U ( R ) = 4 πR . Denoting the ground state energy as E = ~ ω , we have ω ∼ πσR / ~ . Due to p R M ( R typ ) ∼ πσR and using M ( R ) = 4 πmn ( P ) R , we obtain R typ ∼ (cid:2) ~ / (cid:0) π σmn ( P ) (cid:1)(cid:3) / and ω ∼ (16 π ) / σ / ~ − / (cid:0) mn ( P ) (cid:1) − / . (27)The estimate for ω coincides with the semiclassical ex-pression in [4]. Note that unlike ω b (20), frequency ω isindependent of critical radius R c . The point is that fre-quency ω is associated with the heterophase quantumfluctuations in the homogeneous metastable phase andis insensitive to critical radius R c . In the semiclassicalapproximation the probability for the quantum transi-tion from level ~ ω to nucleation of the critical nucleusis given by W = νω exp (cid:0) − β (cid:1) , β = 2 ~ Z R c | p R | dR. (28)Equation (27) yields ω ∼ ω D ( m/m e ) / ∼ s − . (29)Here m/m e is a ratio of proton mass to electron one and ν is the number of virtual nucleation centers of new phase.The latter is of the order of the number of particles in thesystem. Thus, the preexponential factor coincides withEqs. (1) and (2) and is about 10 particle/s.Momentum p R is determined semiclassically for thestate of energy E close to zero, ~ ω ≪ U max , as | p R | = p M ( R ) U ( R ) . FIG. 7: The critical radius R c versus ambient pressure.FIG. 8: The number of particles N c in the critical nucleusas a function of ambient pressure (logarithmic scale). The integration in (28) is performed over the positiveregion of potential energy U ( R ) >
0. To have a macro-scopically long-lived state of metastable metallic phase,the exponent β in (28) should be large and not smallerthan 80 - 100. For the smaller exponents, the large pre-exponential factor in (28) compensates the effect of theexponent, resulting in the large decay probability.The main parameter determining the exponent is thecritical nucleus radius R c or critical number of particles N c . The R c - P dependence is plotted in Fig. 7. Curve I corresponds to choice h ( x ) obtained by extrapolatingthe data to metallic hydrogen r s at zero pressure r s = 1 . I in Fig. 6. Curve II in Fig. 7 is obtained with thesame surface tension but with the function h ( x ) changedfor x < . h ( x ) is truncated and putequal to h ( x = 1 . a.u. ) (Fig. 4). Such variation h ( x )remains the critical radius R c unchanged (Sec. III) atlow pressures of about .
10 GPa when the region for along-lived existence of the metastable metallic hydrogenis determined. Thus for determining the boundaries of
FIG. 9: The exponent β (28) versus ambient pressure (loga-rithmic scale).FIG. 10: The density distribution in the nucleus of radius R = 9 at pressures 20 GPa (curve I ) and 75 GPa (curve II ). stability of stable existence of metallic state, the behav-ior of function h ( x ) is essential only at large distancesfrom the surface. The behavior at large distances, asnoted in Sec. III, is well-known. For the higher pressureswhen critical radius R c is already large, the truncationmentioned results in some reduction of critical radius R c .In Figs. 8 and 9 we give the plot of the number ofparticles N c in the critical nucleus and the plot of theexponent β in Eq. (28) for different functions h ( x ). FromFigs. 8 and 9 we can obtain the relation between β and N c . It proves to be that in a wide pressure range thisrelation can approximately be described with the linearlaw β = αN c , α being 200 as N c .
100 and α being 120as N c . .In Fig. 10 we plot the typical distribution for the molec-ular phase density n ( r, R ) inside the nucleus at variouspressures. It is seen that, for the relatively low pressures,there is a spacing d where the density of the molecularphase vanishes. For the pressures larger than 75 GPa, thedensity of molecular phase does not vanish everywhere.0 FIG. 11: The spacing d c = d ( R c ) for the critical nucleus as afunction of pressure for various functions h ( x ) (curves I and II ).FIG. 12: The dependence of the minimum magnitude ofdensity n ( R c ) of molecular phase in the critical nucleus forvarious functions h ( x ) (curves I and II ). The density ofmetallic phase versus pressure is shown with curve III . For
P < P I , n ( R c ) = 0 (curve I ) and for P < P II , n ( R c ) = 0(curve II ). In Fig. 11 we show the dependence of spacing d c = d ( R c )as a function of pressure P for the critical nucleus.The plot in Fig. 12 demonstrates the dependence of theminimum magnitude of density n ( R c ) of molecular phasein the critical nucleus. Within the whole pressure rangethe density at the boundary of molecular phase is signif-icantly smaller as compared with the density of metallicphase (curve III in Fig. 12). The latter, as is noted inSec. II, makes it possible to neglect the dependence offunction h ( x ) on the density of molecular phase.As the data show, the variation of function h ( x ) withinthe reasonable limits affects insignificantly the main re-sult, i.e. long-lived stability of existing the metastablemetallic phase within the wide pressure range below thetransition point P c ∼
300 GPa down to pressure ∼
10 GPa.The variation of surface tension σ ( P ) affects the resultsto slight degree. Emphasize that we have used the sur- face tension (Fig. 6, curve I ), resulting in the minimummagnitude of the lifetime. V. SUMMARY
We have analyzed stability of the hydrogen metallicstate against nucleation of the stable molecular phase be-low the transition pressure P c ∼
300 – 500 GPa. The nu-cleation dynamics is governed by the tunneling of a crit-ical molecular nucleus through a potential barrier in thelow-temperature region and by thermal activation mech-anism at high temperatures. In a wide 0 . P c . P P c pressure region below the phase transition pressure P c the critical nucleus of the molecular phase contains alarge number of particles and has, correspondingly, alarge critical radius as compared with the interatomicspacing. The main reason for the large critical nucleuslies in the impossibility to form a bound state of twohydrogen atoms under high extrinsic electron density ofthe metallic phase r s ∼ .
7. This entails the necessityto produce a cavity inside the metallic phase with thelow electron density in the center insomuch that the for-mation of molecules would become energetically favor-able. The nucleation dynamics of molecular nuclei atboth low and high temperatures can be described withinthe framework of the macroscopic approach. Within thementioned 0 . P c . P P c pressure region the lifetime ofthe metallic hydrogen phase is macroscopically large andthe metallic state is practically stable, i.e. long-lived.In the low pressure region P . . P c the inception ofa cavity in the metallic state cannot be suppressed withthe applied external pressure P and the critical nucleusamounts to a few particles or less as the external pressure P vanishes. Thus, we expect the opposite behavior withtoo small lifetime of the metastable metallic state, result-ing in practically instant decay of the metallic phase. Appendix A:
Let V and n be volume and density of the metastablephase. After the nucleation of stable phase of volume V ′ and density n ′ the volume of metastable phase becomes V and density does n . The energy of nucleus can bewritten as U = Z V ε ( n ) n d r + Z V ′ ε ′ ( n ′ ) n ′ d r + Z V ′ σ dS ′ − Z V ε ( n ) n d r. Here σ is the surface tension, ε ( n ) and ε ′ ( n ′ ) are theenergy density of the metastable and stable phase, re-spectively. Expanding ε ( n ) in small ( n − n ) as ε ( n ) = ε ( n ) + ( n − n ) P /n U = − Z V + V ′ ε ( n ) n d r + Z V ′ ε ′ ( n ′ ) n ′ d r + Z V ε ( n ) (cid:2) n + ( n − n ) (cid:3) d r + P n Z V ( n − n ) d r + σdS ′ = Z V ′ ε ′ ( n ′ ) n ′ d r − Z V ′ ε ( n ) n d r + Z V (cid:18) ε ( n ) + P n (cid:19) ( n − n ) d r + σ Z V ′ dS ′ . Since Z V ( n − n ) d r = Z V n d r − Z V + V ′ n d r + Z V ′ n d r = Z V ′ n d r − Z V ′ n ′ d r, we have finally U = Z V ′ (cid:2) ε ′ ( n ′ ) − µ ( P ) (cid:3) n ′ d r + P Z V ′ d r + σ Z V ′ dS ′ (A1)taking ε ( n ) + P /n = µ ( P ) into account. Appendix B:
It follows from Eq. (6) that density n ( r, R ) of molecularphase depends on C as a parameter n = n ( r, R, C ) . Therefore the potential energy U of a nucleus and thenumber of particles N depend also on C as a parameter U = U ( R, C ) , (B1) N = N ( R, C ) . (B2)Using the last relation, parameter C can be expressed viathe total number of particles C = C ( R, N ) . (B3)Substituting Eq. (B3) into (B1), we find U as a functionof R and N U = U (cid:0) R, C ( R, N ) (cid:1) . The condition of mechanical equilibrium (8) reads (cid:18) ∂U∂R (cid:19) N = 0 . Then we have4 πR (cid:18) n ( R ) (cid:2) ε (cid:0) n ( R ) (cid:1) + h (0) (cid:3) − µ ( P ) n ( R ) + P (cid:19) +8 πσ ( P ) R + Z R n ( r ) h ′ ( R − r )4 πr dr + Z R (cid:18) µ (cid:0) n ( r ) (cid:1) + h ( R − r ) − µ ( P ) (cid:19) × (cid:20)(cid:18) ∂n∂R (cid:19) C + ∂n∂C (cid:18) ∂C∂R (cid:19) N (cid:21) πr dr = 0 where n ( r ) ≡ n ( r, R ) and n ( R ) = n ( R, R ). In the lastintegral the expression in the parentheses is constant dueto (6) and can be put in the front of integral. The mag-nitude of the remaining integral can be found with dif-ferentiating Eq. (B2) in R Z R (cid:20)(cid:18) ∂n∂R (cid:19) C + ∂n∂C (cid:18) ∂C∂R (cid:19) N (cid:21) πr dr + 4 πR n ( R ) = 0 . Then we obtain Eq. (10) for pressure P which can berewritten in the convenient form for numerics P = − σ ( P ) R − R Z R (cid:2) r n ( r ) − R n ( R ) (cid:3) h ′ ( R − r ) dr + n ( R ) (cid:2) C − ε (cid:0) n ( R ) (cid:1) − h ( R ) (cid:3) . Appendix C:
In the kinetic energy (19) the nucleus mass depends onderivative ∂n ( r, R ) /∂R , ambient pressure P being fixed.In Eq. (6) the density is directly expressed in terms of C related to pressure P . So, it is necessary to transform ∂n/∂R from one variable to another (cid:18) ∂n∂R (cid:19) P = ∂ ( n, P ) ∂ ( R, P ) = ∂ ( n, P ) ∂ ( R, C ) / ∂ ( P, R ) ∂ ( C, R )= (cid:20)(cid:18) ∂n∂R (cid:19) C (cid:18) ∂P∂C (cid:19) R − (cid:18) ∂n∂C (cid:19) R (cid:18) ∂P∂R (cid:19) C (cid:21) / (cid:18) ∂P∂C (cid:19) R = (cid:18) ∂n∂R (cid:19) C − (cid:18) ∂n∂C (cid:19) R (cid:18) ∂P∂R (cid:19) C / (cid:18) ∂P∂C (cid:19) R . (C1)Differentiating Eq. (6) in R under fixed C and then in C under fixed R , we have (cid:18) ∂n ( r, R ) ∂R (cid:19) C = − h ′ ( R − r ) µ ′ ( n ) , (cid:18) ∂n ( r, R ) ∂C (cid:19) R = 1 µ ′ ( n ) . Derivatives (cid:0) ∂P/∂R (cid:1) C and (cid:0) ∂P/∂C (cid:1) R are found withdifferentiating Eq. (10). Though σ and h depend on P ,the derivatives of σ and h in P do not enter the ratio (cid:0) ∂P/∂R (cid:1) C / (cid:0) ∂P/∂C (cid:1) R . One can see this directly usingthe cumbersome calculation2 (cid:0) ∂P/∂R (cid:1) C (cid:0) ∂P/∂C (cid:1) R = (cid:26) σ ( P ) − R n ( R ) h ′ ( R ) + 2 Rn ( R ) (cid:2) h ( R ) − h (0) (cid:3) − Z R (cid:2) r n ( r ) − R n ( R ) (cid:3) h ′′ ( R − r ) dr + 2 R Z R (cid:2) r n ( r ) − R n ( R ) (cid:3) h ′ ( R − r ) dr + Z R h ′ ( R − r ) µ ′ ( n ) r dr (cid:27) × (cid:18) R n ( R ) − Z R h ′ ( R − r ) µ ′ (cid:0) n ( r, R ) (cid:1) r dr (cid:19) − . Substituting the above three relations for the deriva-tives into Eq. (C1), we obtain the relation for ∂n/∂R which should be employed for calculating M ( R ) (19) un-der fixed pressure P . [1] E. Wigner and H. B. Huntington, J. Chem. Phys. , 764(1935).[2] E. G. Brovman, Yu. Kagan, and A. Kholas, ZhETF ,2429 (1971) [ Sov. Phys.
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