An equation of state for expanded metals
aa r X i v : . [ c ond - m a t . d i s - nn ] S e p An equation of state for expanded metals
W. Schirmacher , W.-C. Pilgrim and F. Hensel Institut f¨ur Physik, Johannes-Gutenberg-Universit¨at Mainz, Staudinger Weg 9, D-55099 Mainz, Germany; Fachbereich 15,Chemie, Physikalische Chemie, Fachbereich Chemie, Philipps-Universit¨at Marburg,Hans-Meerwein-Strasse 4 D-35032 Marburg, Germany.
We present a model equation of states for expanded metals, which contains a pressure term dueto a screened-Coulomb potential with a screening parameter reflecting the Mott-Anderson metal-to-nonmetal transition. As anticipated almost 80 years ago by Zel’dovich and Landau, this termgives rise to a second coexistence line in the phase diagram, indicating a phase separation betweena metallic and a nonmetallic liquid.
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I. INTRODUCTION
For almost 80 years the relation of the metal-nonmetaland liquid-vapor transition of expanded metals is not un-derstood [1–5], despite several efforts in the last decades[4, 6].In their original paper [1] Zel’dovich and Landau (ZL)present arguments that in expanded metals, in particularmercury, in addition to the usual liquid vapor coexistenceline, a second coexistence line (with a second criticalpoint) exists in the ( p, T ) (pressure-temperature) phasediagram, which involves a phase separation between ametallic and nonmetallic liquid.Experiments in expanded mercury [4, 5, 7–9] and, morerecently in expanded rubidium [10] revealed a densityregime, in which there is evidence for an emulsion of ametallic and a nonmetallic phase, thus confirming theideas of ZL.The arguments of ZL had been based on the assump-tion that at zero temperature a discontinuous metal-nonmetal transition takes place, which continues to bepresent at elevated temperatures.At these times the only known mechanism for acrossover from a metal to a dielectric was the de-overlapping of bands. ZL argued that a continuous band-de-overlapping transition cannot take place, because inthe insulating state an excited electron across the gapinteracts via the Coulomb interaction with the hole leftbehind and thus enhances the gap, when, in the absenceof the interaction it would go continuously towards zero.The role of the electonic Coulomb and exchange in-teraction on the Metal-nonmetal transition has been ad-dressed extensively by Mott (Mott transition) [11–14].He realized that these combined interactions producetwo separate bands of electrons with opposite spins, giv-ing rise to antiferromagnetic ordering in the insulatingstate. This scenario, which can be described by the Hub-bard model [15], was called Mott transition. Mott be-lieved that the metal-nonmetal transition was discontin-uous and postulated the existence of a minimal metal-lic conductivity. These ideas have been quantified byYonezawa and Ogawa [2] for calculations of the ther-modynamic properties of expanded metals based on theHubbard model and the coherent-potential approxima-tion. In these calculations an unstable density regimedue to the metal-nonmetal transition was identified.Anderson [16] showed that disorder can be another reason for a metal-nonmetal transition. This was firstdemonstrated for non-interacting electrons (Andersontransition). It was then shown [17] that the Ander-son transition is an interference phenomenon and couldbe indentified as a second-order (i.e. continuous) phasetransition with a non-thermal control parameter, namelythe amount of spatial potential fluctuations, seen by anelectron. The Anderson scenario, i.e. an electron in arandom potential, could be mapped onto the nonlinearsigma model of planar ferromagnets [18–20], which obeysthe same scaling as the Anderson transition. In this fieldtheory the density of states at the Fermi level µ was iden-tified as the order parameter, but the critical exponent β turned out to be zero, so that in the “non-ordered state”,the nonmetallic, µ remained finite. That the Andersontransition is continuous was confirmed by experiments ondoped semiconductors [21].The nonlinear-sigma-model description was general-ized to include the electronic Coulomb interaction (Mott-Anderson transition) [22–24]. In the presence of the in-teraction the critical order-parameter exponent β becamenon-zero, so that µ acquired its usual order-parameterrole. A severe drawback of the nonlinear-sigma-modelapproach is that it is based on treating the variance ofthe potential fluctuations as small parameter, so the the-ory is restricted to the weak-disorder limit.A quite different approach, which does not suffer fromthis shortcoming, is the dynamical mean-field theory(DMFT) based on the Hubbard model [25–27] and turnedout to be a reliable means for treating correlated elec-tronic systems and the Mott transition.By including disorder into the Hubbard model itproved possible to treat the Mott-Anderson transition bymeans of the DFMT [28–31]. These developments showedthat the local single-site density of states µ i of the disor-dered interacting electron system exhibits a very broaddistribution. As a consequence the arithmetic mean h µ i and the geometric one h µ i g was shown to become verydifferent in the limit of strong disorder. It was shown,that, in fact h µ i g is critical at the Anderson-Mott tran-sition, even in the Anderson case, whereas h µ i is not.[28–31].In the present contribution we shall show that sucha continuous Mott-Anderson transition of the electronsproduces in an expanded metal an instability in the den-sity regime near the transition, and thus a second phase-separation line between a metallic and nonmetallic liquidphase, as anticipated by ZL.We argue that in expanded metals the transition sce-nario for the electrons is a qualitatively different onefrom that for the metallic atoms/ions. For the electronsthe transition is one with increasing spatial disorder inthe presence of the electron-electron interaction. Thismeans that for the electrons the disorder is of quenchedtype. This is so because of the adiabatic principle: Ontheir time scale the electrons experience a snapshot ofthe atomic arrangements. These arrangements produceincreasing spatial potential fluctuations with decreasingdensity, so that at a critical density the Anderson-Motttransition takes place.On the other hand, for the atoms/ions the transitionscenario is not governed by quenched disorder but byequilibrium thermodynamics. The electronic degrees offreedom provide a density dependent interaction. Usingthe standard expression for the equation of states for asimple liquid with a potential, which includes a density-dependent screened Coulomb term, we demonstrate thatan unstable density interval appears, which, in turn, pro-duces the metal-nonmetal separation in the liquid state.This mechanism of phase separation into a metal-rich andmetal-depleted liquid is very similar to that suggestedsome time ago [32–34] for solutions of metals in moltensalts [35, 36].In II. we introduce our model and present the resultingequation of state. In section III. we show isotherm calcu-lations, which are used to calculate a phase diagram. Weconclude with discussing achievements and shortcomingsof our approach. II. MODEL
1. General formalism
We start with the expression for the pressure equationof states [34, 37] P ( V, T ) = k B TV (cid:18) −
16 1
V k B T Z d r rφ ′ ( r ) g ( r ) (cid:19) (1)where g ( r ) is the radial distribution function, T the tem-perature, k B is Boltzmann’s constant, V = M/ρ M theatomic volume, ρ M the mass density, and M the atomicmass.We now assume that the interatomic potential is com-posed of three contributions:( i ) A hard-sphere contribution φ hs ;( ii ) an attractive contribution φ att .( iii ) a screened-Coulomb contribution φ sc ;We now lump the free-gas contribution to the pressureand the hard-sphere potential together to a hard-spherepressure P hs and write P ( V, T ) = P hs ( V, T ) − V Z d r r [ φ ′ att ( r ) + φ ′ sc ( r )] g ( r )= P hs ( V, T ) + P att ( V ) + P sc ( V ) (2) For the hard-sphere pressure we use the Van-der-Waalsrepulsion term P hs ( V, T ) = k B TV − B (3)with B ≈ d , where d is the distance of nearest approachor effective hard-sphere diameter. B is also of the orderof the atomic volume at melting.Because the radial distribution function is stronglypeaked near the nearest-neighbour distance d , and thepotential contributions vanish for r ≫ d we may replace φ ′ ( r ) g ( r ) by a delta function and approximately write1 V Z d r rφ ′ att;sc ( r ) g ( r ) ≈ Z ( V ) dφ ′ att;sc ( d ) (4)with the coordination number Z ( V ) = 1 V Z | r |≤ r Z d r g ( r ) (5)where r Z is taken to be at the first minimum of g ( r ). Ithas been found experimetally [38] that in some expandedmetals Z increases linearly with density Z ( V ) = Z /V (6)with Z ≈ V M , mp for both expanded Rb and Cs, where1 /V M , mp is the density at the melting point. So we maywrite Z d r rφ ′ att;sc ( r ) g ( r ) ≈ Z dφ ′ att;sc ( d ) (7)As generally the minimum of the attractive potential con-tribution is located at r min > d , φ ′ att ( d ) < P att ( V ) = − A v . (8)with A = Z r | φ att ( d ) | . (9)Without the screening term the equation of states p hs ( v ) + p att ( v ) becomes the van-der-Waals equation ofstates, which gives the usual liquid-vapour transition sce-nario.
2. Screening length and Metal-nonmetal transition
We now turn to the main object of the present exercise,namely the screened Coulomb potential.As indicated in the introduction we rely on the adia-batic principle, from which follows that in the situation See e.g. Ref. [34] for the identification of Wan der Waals’s k B T/ ( V − A ) term with the repulsive pressure. of an expanded metal the (interacting) electrons expe-rience a strongly spatially fluctuating external potentialdue to the ion cores. These fluctuations are “frozen”on the time scale of the electrons ( ∼ φ ( r ) = Q r + 1(2 π ) Z d q e i qr | v ps ( q ) | χ e ( q ) (10)Here Q = N e e is the ionic charge, e is the elementarycharge, N e is the ionic relative charge (number of elec-trons per ion or valence), χ ( q ) is the electronic suscep-tibility and v ps ( q ) the electron-ion (pseudo) potential. χ e ( q ) is the electronic susceptibility, which in the Hartreeapproximation can be written as χ e ( q ) = q πe − ǫ ( q ) ǫ ( q ) (11) ǫ ( q ) is the Lindhard dielectric function of the free electrongas [41], which can be simplified using the Thomas-Fermiapproximation [40, 41] ǫ ( q ) = 1 + λ T F q (12)with the Thomas-Fermi screening parameter (inversesquared screening length) λ T F = 4 πe µ F (13)Here µ F = 4 k F /a B e is the free-electron density ofstates at the Fermi level, a B = ~ me the Bohr radius, m the electronic mass and k F the Fermi wavenumber k F = p π N e /V .Using this approximation for ǫ ( q ) and the empty-corepseudopotential of Ashcroft [42], which is a Coulomb po-tential v ps = − Q/r outside of the ionic radius R c andzero for r < R c , one obtains [40] φ ( r ) = c ( R c ) Q r e − λ TF r (14)where c ( R c ) is a prefactor related to R c [40].For the potential derivative we obtain r ddr φ ( r ) = − c ( R c ) Q r [1 + λ ( V ) r ] e − λ ( V ) r (15)As mentioned in the introduction, the density of statescan be considered as the order parameter for the Mott-Anderson metal-nonmetal transition, i.e. the transition density x= V c /V - 1 f ( x ) = µ ( V ) / µ ( ) FIG. 1: The function µ ( v ) /µ (0) f of Eq. (17) for smoothingparameters s = 0.0, 0.01, 0.02, 0.03, 0.04 from bottom (blue)to top (red). of interacting electrons in the presence of quenched disor-der [23, 28–31]. In the typical-medium DMFT treatmentof the Mott-Anderson transition the geometically aver-aged local density of states µ vanishes linearly with thecontrol parameter, which is the width of the distributionof the fluctuating local potentials, divided by the bandwidth. As the latter is strongly density dependent, wemake the following ansatz for the local density of states µ ( V ) = µ F f (cid:0) x ( V ) (cid:1) (16)with the normalized density x ( V ) = V MNM V − f ( x ) = xθ ( x ) (17)where θ ( x ) is the step function and V MNM is the criticalatomic volume of the metal-nonmetal transition.So we have λ ( V ) = λ ( V ) f / [ x ( V )] (18)with λ ( V ) = 4 πe µ F ( V ). As the density, viz, volumedependence of λ is considerably weaker than the criticalone we set λ constant, i.e λ ( V ) = λ ( V mp ). Finally wemay write P sc ( V ) = CV (cid:2) λ ( V ) d (cid:3) e − λ ( V ) d (19)with C = c ( R c ) Q Z / d .Collecting all the terms contributing to the pressurewe obtain our central result P ( V, T ) = k B TV − B − AV + CV (cid:2) λ ( V ) d (cid:3) e − λ ( V ) d . (20)Beyond the Anderson-Mott transition ( V > V
MNM ) wehave P sc ( V ) = CV (21)so that in this limit we obtain an effective Van-der-Waalsequation of states with A eff = A − C (22) volume v p r e ss u r e p temperature t p r e ss u r e p v c LV v MNM s=0.02ML s=0.02Liquid-vapour CP
FIG. 2: p − v isotherms according to our equation of states(25) for zero smoothing parameter s (thin lines) and s =0 .
02 (thick lines) for the temperature range 0.2 ≤ t ≤ v MNM = 1.3 and λ d = 8.The dashed lines are the equilibrium pressures calculatedwith the Maxwell and double-tangent construcion.Inset: p - t phase diagram.The dots indicate the critical points.It should be noted that the equation of states (20)interpolates [43] between that of a liquid metal (small v )and a free gas (large v ).
3. Inhomogeneities and smoothing of the metal-nonmetaltransition
Relation (18) together with (17) describes (at zero tem-perature) a rather sharp transition between the metallicand nonmetallic state. Such a transition is predicted bythe generalized nonlinear sigma model, which is based onweak disorder [22–24]. At elevated temperature one mayexpect this transition to be somewhat smoothed.On the other hand, the alternative theory of the Mott-Anderson transition, tailored for the case of strong cor-relations and strong disorder [28–31], predicts a verybroad distribution of local densities of state µ i . A sharptransition is found for the geometric average h µ i g =exp {h ln µ i i} , wheras the arithmetic average h µ i i is non-critical.We now phenomenologically introduce a smoothing ofthe critical law [44]. We replace the function f ( x ) in Eq.(16), which is the antiderivative of the step function θ ( x ),by the antiderivative e f ( x ) of the complementary Fermifunction [1 + e − x/s ] − : µ ( V ) = µ (0) e f (cid:0) x ( V ) (cid:1) (23)with e f ( x ) = s ln (cid:2) e x/s (cid:3) (24)where s is the smoothing parameter. For s → s on the critical law.As intended by construction the curves become increas-ingly smoother with increasing s . temperature t den s i t y / v -3 -2 -1 p r e ss u r e p MLNLNVMLNLNVs=0.02 s=0s=0.02 s=0
FIG. 3: The pressure ( p , t ) and density ( v − , t ) phase diagramscorresponding to the isotherms of Fig. 2. The dots indicatethe critical points. ML = metallic liquid, NL = nonmetallicliquid, NV = nonmetallic vapour. III. RESULTS
We now use dimensionless units v = V /B , t = k B T B/A eff and p = P B /A eff . In these units the equa-tion of states takes the form p ( v, t ) = tv − − cv + cv (cid:2) λ ( v ) d (cid:3) e − λ ( v ) d . (25)with c = C/A eff In these units the critical liquid-vapour quantities aregiven by t LVc = 827 ≈ . v LVMNM = 3 p LVc = 127 ≈ . v MNM .We have calculated the equilibrium volumes, pressuresand temperatures using both the Maxwell constructionand the double-tangent method [45]. For the regimeabove v MNM we used Gibbs’ parametric solution of theVan-der-Waals coexistence problem [46]. Below v MNM we implemented a grapical double-tangent construction.In the immediate vicinity of the critical point we useda numerical Maxwell construction, i.e. equating the vol-umes above and below the coexistence pressure. Theresulting phase diagrams are shown in Fig. 2 (dashedlines) and in Fig. 3.It is remarkable that the smoothing of the electronicmetal-nonmetal transition results in a strong reduction ofthe critical endpoint of the corresponding atomic transi-tion.Let us now consider the situation in expanded Hg andRb. We chose our parameter v MNM , which is equal to3 ρ c /ρ MNM to be equal to 1.3, which corresponds to thevalue of ρ MNM /ρ c = 2.3 in expanded Rb [10]. Pilgrimet al. [10] have evidence by inelastic neutron scatteringthat in the density range around 2 . ρ c a micro-emulsionof two liquids is present. Similar evidence has been pre-sented earlier by Ruland and Hensel [9] for expanded Hgin the range around 1 . ρ c by analyzing published small-angle scattering data [7, 8]. This would correspond to v MNM = 2.5. It has been pointed out in Refs. [9, 10],that the Coulomb interaction between the metallic micro-droplets, surrounded by the second-phase nonmetallicmaterial prevents a complete demixing and establishesthe micro emulsion.
IV. DISCUSSION
By combining the standard expression for the pressureequation of state of a simple liquid with a hard-core re-pulsion, a short-range attraction and a screened Coulombpotential reflecting the Mott-Anderson transition via adensity-dependent screening length we have constructedan equation of state, which gives rise to a second co-existence line in the phase diagram. As postulated byZel’dovich and Landau [1] we obtain a separation into ametalic and nonmetallic liquid phase. Contrary to theirideas and the ideas of Mott [11–14], we show that also a continuous
Mott-Anderson type metal-nonmetal transi-tion of the electrons gives rise to a discontinuous liquid-liquid phase separation of the ions/atoms. We have in-troduced a phenomenologic model for the density depen-dence of the screening length including the possibility ofa smoothed transition. We modeled the smoothing orrounding of this transition [30, 44] by means of the anti-derivative of the Fermi function featuring a smoothingparameter.We find that the smoothing results in a reduction ofthe length of the coexistence line. Let us consider againthe reasons for such a smoothing to happen. First ofall, the electronic transition does not take place at zerotemperature but at a temperature approaching the Fermi temperature. Secondly, for T = 0 we assumed the crit-ical exponent of the Mott-Anderson transition to be 1.If it would be larger than one the curve would look likea rounded transition. Thirdly, as mentioned before, thelocal density of electronic states in the Mott-Andersonscenario as given by the dynamical mean-field theory [28–31] is known to exhibit strong spatial fluctuations, so thisas well will effectively lead to a smoothing of the transi-tion. So a more detailed experimental investigation of theliquid-liquid separation line will shed light on the detailsof the mechanism of the Mott-Anderson transition.Finally we would like to discuss a point concerningthe temperature dependence of our model. Our equationof states (20) has a linear temperature dependence likethe van-der-Waals one. By elementary thermodynamicrelations one can show that if the second temperaturederivative (at constant volume) of the pressure is zero,such is the first volume derivative of the specific heat(at constant temperature). This implies that the spe-cific heat does not depend on the volume. Of course,in a material, in which the electronic degrees of freedomplay a dominant role, the linear-temperature term of thespecific heat should be present, which is proportional tothe density of states at the Fermi level and, hence, shouldexhibit the same volume dependence as the screening pa-rameter λ ( V ) . This is not included in our rather crudemodel. The model has mainly been introduced in orderto demonstrate, how a smooth metal-nonmetal transitioncan lead to a first-order phase transition in an expandedmetal and a second coexistence line, as anticipated byZeldovich and Landau [1]. A more refined version shouldinclude a term quadratic in the temperature, which isthen related to the specific heat. This term will be acorrection of the order of ( k B T /E F ) , where E F is theFermi energy. In a future publication we shall present amore refined version of our equation of states, in whichthe delta-function approximation of (4) will not be made,and the T term, related to the electronic specific heat,will be included. [1] L. Landau and J. Zeldovich, Acta Phys. Chim. USSR ,194 (1943).[2] F. Yonezawa and T. Ogawa, Suppl. Progr. Theor. Phys. , 1 (1982).[3] F. Hensel and H. Uchtmann, Annu. Rev. Phys. Chem. , 61 (1989).[4] F. Hensel and W. W. Warren, Fluid Matals: The Liquid-Vapor Transition of Metal (Princeton University, Prince-ton, NJ, 1999).[5] F. Hensel, in
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