Coulombic Phase Transitions in Dense Plasmas

arXiv:cond-mat/0210193v1 [cond-mat.stat-mech] 9 Oct 2002 rahi etitdt eaieywa trcieforces ap- 1 attractive Waals than weak der faster decay relatively van either to which that restricted clear of is more density became proach of it the development the by theories with strict other connection the each In below from twomolecules. predicts differ of attractive coexistence model which of Waals range phases possibility der long the temperature van as critical The well as forces. account of repulsive into model range takes physical short which simple particles a between der on Van interactions based transitions. is starting phase approach of the Waals theory as modern considered the be of may point 1873 in Waals der van trcinbtennurlmlclsadtershort- their and molecules long-range model neutral the Waals from between der result a van attraction is the transition phase to the qual- similar where The was [2,3]. picture plasma dense itative ionized be- repul- partially quantum competition the in and the sion attraction from Coulomb result effective possible tween a was as PPT authors The predicted these (PPT). 1968 transition by phase in predicted plasma Starostin transition named was order and first Norman The by transitions [2]. Coulombic given of was sys- study plasmas Coulomb systematic in first in work A transitions the metal-insulator also tems. connection on this Mott in mention points of critical to of have We vicinity the [1]. in metal-insulator fluids metal with in connected transitions transitions phase of ities forces Coulomb to applicability open. the remained Therefore ditions. ∗ eiae ote7t itdyo ihe Fisher. Michael of birthday 70th the to Dedicated h hoyo ae eeoe ntedsetto of dissertation the in developed gases of theory The n14 aduadZloihdsusdnwpossibil- new discussed Zeldovich and Landau 1943 In .FRTODRVNDRWASAND WAALS DER VAN ORDER FIRST I. 2 nttt o ihEeg este,AH fRS Izhorskay RAS, of AIHT Densities, Energy High for Institute 1 oi ytm.Fnlyw ieadsuso fsvrlnumer plasmas. several pressure of know high discussion the in a to PPT give analogy the we full to Finally in st are systems. they are (typical ionic PPT, which system hypothetical transitions one the Coulombic investi are in separated We transitions The Coulomb transition. plasmas). and Waals Waals der der van Van determi another the forces Waals and on quantu der dependence transition of van (in and curves (Coulombic shows spinodal transition which transi type the model these and theoretical of point simple properties a critical the the on of results types new several derive OLM TRANSITIONS COULOMB ntttfu hsk ubltUiesta eln Inva Humboldt-Universität Berlin, für Physik, Institut egv uvyo h rdcin fCuobcpaetransi phase Coulombic of predictions the on survey a give We olmi hs rniin nDnePlasmas Dense in Transitions Phase Coulombic /r 3 rfllteKccon- Kac the fulfil or .Ebeling W. [email protected] henry 1 [email protected] .Norman G. , 1 inta ohepesosaevldol ntelmtof limit the in only ( valid repulsion are quantum small expressions both that tion 60adwt h oeraitcnnierDebye-Hückel nonlinear value the realistic expression more the with and 2660 iaino h ey-¨ce ihteVndrWasthe- com- Waals (a der Van model Debye-Hückel the simple the with a of for develop bination we conditions reason the this For and PPT. properties a pos- the as definition, clear the as make sible to plan we experimentally identified of only study consider plasmas. will gaseous detailed we in A Here ions single-charged [16]. [15]. in given in was first He-plasmas treated were plasmas [5-14] to temperature lead critical plasmas the hydrogen of in values PPT higher the of estimates Later where rntemlwvlnt and wavelength thermal tron euso u oteucranypicpe ntelinear the temperature critical In a density principle. the quantum uncertainty in approximation the effective to an an due and described repulsion attraction expressions with Coulomb The charges effective the [4]. of Debye-Hückel corrections potential ex- quantum chemical namely the time, for that pressions func- at thermodynamic available the tions used [2] Starostin and Norman gas ideal an both as in originally [2]. present treated are were which phases, Atoms, dif- coexistent have ionization. different PPT and the of particles in degrees by charged phases of other two densities each number The in ferent from phases differ molecules. two of model the density Waals that der van remember the We repulsion. range n ml o-daiyparameter non-ideality small and ic pt o h P npamsi o e clearly yet not is plasmas in PPT the now to up Since ionized multiple in transitions phase hypothetical The ogttefis siaeo h rtcltemperature critical the of estimate first the get To iesrse10 -01 eln Germany; Berlin, D-10115 110, lidenstrasse n in.I atclrw ics several discuss we particular In tions. i eteciia on)o n Coulombic one or point) critical the ne cladeprmna eut referring results experimental and ical tet1/9 ocw151,Russia; 125412, Moscow 13/19, street a ogyiflecdb unu effects quantum by influenced rongly olmi ytm.W construct We systems. Coulombic m steinnme density, number ion the is aetecniin ofidseparate find to conditions the gate 2 olmi rniin nclassical in transitions Coulombic n aaee aus ihroealkali- one either values) parameter in ndnepams(P)and (PPT) plasmas dense in tions o yrgnadnbegas-type noble and hydrogen for T cr 04 a band emen- We obtained. was 10640 = λ/r D ∗ r ) D << γ steDberadius), Debye the is T = (here 1 stetemperature. the is e 2 n 1 i / λ 3 k << /kT steelec- the is T cr = 1, ory) which shows a separate Coulombic transition - which tions in the regions where - according to the first es- is a PPT. Other related phenomena in plasma systems timates - the phase transition is to be expected. As are discussed only in brief. we have learned from the theory of phase transitions in Plasma-like phase transitions in the optically excited neutral gases, already simple expressions as the van der electron-hole system in semiconductors were studied with Waals equation might give qualitatively correct results. similar methods [5,6,17,18,19]. The experimental obser- We will show here that the Debye-Hückel expressions for vations seem to point out that these instabilities of the the thermodynamic functions together with the mass ac- theory correspond to a real phase transition in semicon- tion law provide an approximation which might serve a ductors, this topic will not be considered here in detail. a zeroth approximation for the description of Coulom- In more detail we will discuss comparison of the PPT bic phase transitions. In their first work Norman and to the known transitions in classical Coulombic systems. Starostin [2] proposed for a first estimate of the criti- The first results on phase transitions in ionic systems cal temperature, to use Debye-Hückel-type expressions go back to the late sixties. In 1970 Voronzov et al. [20] (1 cλ/r ) for the chemical potential. Here λ is electron − D found a coexistence line and a critical point in the course thermal wavelength, r is a Debye radius, c 0.1 is a D ≃ of Monte Carlo studies of charged hard spheres imbed- numerical coefficient. The Debye-Hückel term describes ded into a dielectric medium. An analytical estimate of an effective Coulomb attraction, the (λ/rD)-term rep- the critical point and the coexistence line for electrolytes resents in this approach an effective quantum repulsion based on the Debye-Hückel theory was given in a short due to the uncertainty principle. Norman and Starostin note by one of the authors [21]. However a systematic [3] used also the original expression of Debye- Hückel for −1 theory of classical Coulombic phase transitions including the chemical potential (1+cλ/rD) which has the same a comparison with numerical and experimental data was accuracy if (λ/rD) tends to zero. given only in the pioneering work of Michael Fisher and We will give now a systematic discussion of the Debye- his coworkers [22]. These authors have also made an ex- Hückel approximation for the classical and quantum tensive investigation of the state of art in this field [23]. cases. We consider a binary Coulomb system with n+ Summarizing these results for classical systems we may positive ions (cations) and n− negative charges (anions or say that theory and experiment (Monte Carlo data as electrons) per cubic centimeter n+ = n−. The density of well as measurements on electrolytes) are in quite good neutrals is n0. The total density is n = n++n0 = n−+n0. agreement. There is no doubt any more that a Coulom- In the following we will use the plasma notations, i.e. bic phase transition in classical systems exists. We will we call the negative charges ”electrons” and the posi- not repeat here all the arguments but refer to the careful tive charges simply ”ions”. In the Debye-Hückel theory investigations of Fisher et al. [22,23]. for charged hard spheres with the diameter a the excess The classical Coulombic transition is due to a balance chemical potential of the charges of species i is given by between a hard-core repulsion and a Coulombic attrac- the simple expression tion. We will show here that the PPT is a balance 2 between quantum repulsion between point charges and ex ei µi = (1) Coulombic attractions. In this respect PPT is a kind of −D0kT (rD + a) quantum variant of the classical Coulombic transition. The densities of free and bound particles are connected If on one hand the existence of a Coulombic transition by the mass action law (Saha equation) in ionic systems is now well confirmed, there is on the other hand no proof yet for the existence of a PPT in n0 µex plasmas. At present the problem of the existence of a = K(T )exp (2) n n k T PPT is still open. From the point of view of the theory i e B this is connected with the difficulties to derive an ac- where K(T ) is the mass action constant which is given curate equation of state for nonideal quantum plasmas. by Bjerrum’s formula or certain modifications [5,24] in From the experimental point of view the difficulties are the classical case. The osmotic pressure is given by the connected with the very high pressures where the PPT formula could occur. However as we will point out here there are now several experimental and numerical data which 1 3 P/kT = ni + ne + n0 κ φ(κa) (3) point to the existence of a PPT in real plasmas. − 24π Here the Debye-Hückel function φ(x) is defined by ¨ II. DEBYE HUCKEL APPROXIMATION FOR 3 1 CLASSICAL AND QUANTUM SYSTEMS φ(x)= 1+ x 2 log(1 + x) (4) x3 − 1+ x − In order to describe Coulombic phase transitions one The easiest way to transfer the Debye-Hückel approxi- needs good approximations for the thermodynamic func- mation to quantum plasmas is the replacement of the

2 hard sphere diameter by an effective quantum diameter In the case of classical ionic solutions the parameter a is of charges. According to the Heisenberg principle a point to be identified with the diameter of ions. In the case of particle with the thermal momentum dense quantum plasmas we have to introduce the quantum effective diameter a(T )=Λ/8. This way we have 8mkT shown that the PPT is the analogue of the known clas- pˆ = (5) sical Coulombic transition in ionic solutions. At least in r π the simplified theory given above the classical transition corresponds to a wave packet of size in ionic solutions and the quantum PPT in dense plas- ¯h π¯h mas are in full analogy. In both cases the spinodal curve δx = = . (6) is given by pˆ 4√2πmkT 1/2 2 2 2 Identifying this with the effective diameter of the charges b b 2 µ1 2 = b b b (13) we get , 8 − ± 8 − − " # Λ h 2 2 a a(T )= = (7) where b = e /kBTa(T ) and µ = e κ/kBT . In the quan- → 8 8√2πmekT tum case we find the corresponding estimate for the critical temperature In this so-called Lambda approximation [7,10] quantum effects are expressed by just one characteristic length, the Tcr 12570K (14) thermal de Broglie wavelength Λ in combination with ≃ Debye-Hückel type approximations. The Lambda ap- In the case of electrolytic solutions or semiconductors the proximation agrees very good with the exact quantum- critical temperature is much lower due to the appearance 5 statistical results for small densities for T < 10 K [7,9]. of the dielectric constant in the denominator. Another This approximation requires that the plasma is nonde- effect which decreases Tcr is the finite size of ions. In or- ∗ 3 ∗ generate i.e. n Λ << 1 (where n denotes the density der to describe the size of the ion cores in alkali or noble of free particles). Furthermore only a small fraction of gas plasmas we introduce an effective ion radius R, e.g. the atoms should be bound in molecules β2 << 0.5. The we have R 1.69A for Cs ions. Further we replace standard choice of the mass action constant for quantum Λ/8 by ≃ − plasmas is the Brillouin-Planck-Larkin expression a(T )=Λ/8+ R (15) K(T )=Λ3σ(T ) (8) Assuming that R is temperature-independent we find where now from Eq.(15) ∞ 2 2 σ(T )= s (exp( βE ) 1+ βE ) (9) Tcr = 12570K (16) − s − s 1/2 s=1 1+(1+(6R/πaB)) X 2 In this way the classical and the quantum case might be where aB =h/e ¯ me is the Bohr radius. We see that a treated in the present rough approximation by the same finite ion size may lead to an essential decrease of the crit- procedure. The easiest way to check for stability of the ical temperature to values far below 10000K. For Cs plasmas this estimate gives e.g. T 6000K. Since− system is the investigation of the sign of the derivative of cr ≃ the pressure. The region where this value is still far above the observed critical point Tcr 2000K we have to search now for additional ef- ∂βP fects,≃ which might reduce the critical temperature. < 0 (10) ∂n corresponds to thermodynamic instability. Equivalent is III. COMBINATION OF VAN DER WAALS AND the condition [2,3] DEBYE-HUCKEL¨ APPROXIMATION

∂ni < 0 (11) In the Debye-Hückel type approximation derived in the ∂n T previous section the interaction of charges with neutrals which is valid only if atoms are treated as ideal gas. In- and the neutral-neutral interaction was not taken into ac- stability is restricted to temperatures below the critical count. This model might fail if the interactions with neu- one which is defined by trals have an essential influence. In the following we consider for concreteness a plasma system. We treated the e2 neutrals as ideal particles in the simple approach given T Tcr = 16 (12) ≤ D0kBa above which is analytically solvable. The system of mass

3 action laws might become rather difficult if one includes for hydrogen. The degree of ionization is practically zero the interactions with neutrals or/and chemical equilibria in this region, i.e. there is no interference with the PPT. between neutrals as e.g. between atoms and molecules. On the other hand, at least for hydrogen, the PPT oc- A better starting point for such more complicated cases curs in a region where the number density of neutrals is is as a rule the expression for the free energy, including rather low. We may assume therefore that the assump- a minimization procedure. tion a =0,b = 0 is justified in this case. Then, as shown Combining the Debye-Hückel model with a van der above, the PPT may be treated in an analytical way. In Waals expression and a contribution which takes into ac- the Lambda-approximation we got in the previous sec- count the polarization of neutrals we get for the free en- tion an analytical expressions for the critical point of the ergy PPT [7,19]. By introducing the Bohr radius aB we may transform the expression for the critical temperature to 3 3 n0Λ0 niΛi the form F = n0kBT ln 1 + nikBT ln 1 σ(T ) − g − 2 i e n Λ3 κ3 κΛ Tppt = (21) + n k T ln e e 1 k T V τ 8kBaB e B g − − B 12π 8 e Further the critical density of the free electrons may be 2 k Tn0 ln(1 n0B) An Wnn0 (17) − B − − 0 − written as −3 Here the last three terms denote the contributions from nppt = aB (22) the short range van der Waals interactions and the contri- What happens in a system where both types of in- butions from polarization terms. The polarization terms teractions are present? Evidently now besides the are of particular importance for the description of alkali classical first-order phase transition typical for neu- and mercury plasmas [25,26]. tral gases a second first-order phase transitions due Three free constants characterize now our model to Coulomb forces may appear. The second one - the repulsion of atoms : B which might occur only at rather high temperatures - the attraction of atoms : A T > 104K is the PPT. In Fig. 1 we give the pres- - the strength of atomic polarizability : W sure isotherm of hydrogen plasmas at T = 10000K. In order to reduce the number of free parameters we assume in the following W = 0. We note that the free en- 0.0030 ergy in this form defines a model plasma with properties intermediate between van der Waals gas and a Debye- Hückel plasma. The equilibrium composition is given by 0.0020 the nonideal Saha equation

0.0010 1 α 3 ∆I(α) −2 = nΛ σ(T )exp (18)

α − k T p [a.u.] B 0.0000 Here e2κ√α (1 α)nB ∆I(α)= − −0.0010 k T (1 + κa(T )√α) − 1 (1 α)nB B − − + log(1 (1 α)nB) −0.0020 − − 1.0 2.0 3.0 4.0 + (1 α)nA Wn(2α 1) (19) lg ( v [a.u.]) − − − FIG. 1. The isotherm T = 10000K for hydrogen plas- is the lowering of the ionization energy. mas showing the PPT instability. In the unstable region of the isotherm the plasma is As well known, real gases with attractive interactions separated into two phases. The dense phase is highly show a ﬁrst-order phase transition described in the p T − ionized and the Coulomb interaction dominates over the plane by a critical point C1 and a coexistence line ending van der Waals forces. The less dense phase is only weakly in C1. Assuming that the Coulombic terms are omitted ionized and van der Waals interactions dominate. A in our expressions for the free energy a simple van der more realistic calculation of the coexistence line for hy- Waals type expression is obtained with the critical point drogen plasmas was given recently [28]. A coexistence 8a 1 pressure of p 115GP a was found in the cited work for T = ,n = (20) ≃ vdW 27k b vdW 3b T = 10000K; the corresponding mass densities of the co- B 3 3 existing phases are ρ1 0.62g/cm and ρ2 0.82g/cm . 1 3 ≃ ≃ The critical temperature is about 10

4 increasing values of the parameters a and b. Fig. 2 shows the isotherm T = 0.07 a.u. and a = b = 30 a.u.. 0.0030

0.0030 0.0020

0.0020 0.0010

0.0010 p [a.u.] 0.0000 p [a.u.] 0.0000 −0.0010

−0.0010 −0.0020 1.0 2.0 3.0 4.0 5.0 lg ( v [a.u.]) −0.0020 FIG. 3. The isotherm T = 0.07 a.u. for a noble 0.0 2.0 4.0 6.0 8.0 lg ( v [a.u.]) gas-like model plasma with a PPT and a separate van FIG. 2. The isotherm T = 0.07 a.u. for an alkali-like der Waals transition (van der Waals parameters of the model plasma with one transition of mixed PPT-van model: a = 100 a.u., b = 100 a.u.). der Waals type (van der Waals parametersof the model: a = 30 a.u., b = 30 a.u.). In spite of the fact that these values are rather large we IV. DISCUSSION OF PLASMA PHASE cannot detect a separate van der Waals wiggle. Both at- TRANSITIONS tracting forces support each other and lead to one phase transition approximately in the same density region as Considerable efforts were applied to obtain better the PPT. This is the situation we observe in nature for plasma thermodynamic functions for larger range of the alkali plasmas. At further increase of the values of the nonideality parameter. We will not go into the details of van der Waals parameters to a = b = 100 a.u. we ob- the theory here which are explained elsewhere [7,9,10,11]. serve two separate van der Waals loops as shown in Fig. The critical temperatures derived from more refined ver- 3. This is the situation we observe for real noble gas sions of the theory lay in the region T = 14000 19000K plasmas. We are to emphasize, that Fig. 2 and Fig. 3 cr [7,8,11]. There is a scatter in the values of T −, p and have just illustrative character; in reality we do not see cr cr n , since the procedure of getting the thermodynamic two wiggles in one isotherm due to the drastic difference cr functions for large nonideality is not a unique one. In- between the critical temperatures. stead of going into details on the several theoretical ap- Summarizing these findings we may state that in de- proaches we concentrate here on the physics and the re- pendence on the values of the van der Waals parameters lation to experiments, to numerical simulations and to we may obtain either a phase diagram with two first order related phenomena. transitions (hydrogen and noble gas plasmas) or a phase Let us start with the experimental aspects. The exis- diagram where both transitions fuse to just one (alkali tence of a PPT in dense plasmas is still less clear than plasmas). The existence of a phase diagram including in classical Coulombic systems. This is mainly due to a van der Waals type transition and a separate metal- the fact that pressures predicted for the coexistence line insulator phase transition was discussed for the first time are above 1Mbar what is still hardly reachable in ex- in 1944 by Landau and Zeldovich [1]. We repeat that the perimental situations. However in the last time more existence of two separate phase transitions requires the and more experiments were performed which cover the validity of the inequalities region of interest [29,30,31,32,33,34,35,36]. Experiments with shock-compressed hydrogen and deuterium plasmas have verified that around 140GP a and 3000K a transi- TvdW << Tppt; ρvdW << ρppt (23) tion to a highly conducting state occurs [29,31]. This is approximately the pressure region where the coexistence line of the PPT is expected [28]. A corresponding behav- This is fulfilled for hydrogen and for noble gas plasmas. ior of conductivity data has been reported [30]. Pressure In the case of alkali plasmas the van der Waals and dissociation and ionization, which almost do not depend the PPT transition have about the same critical den- on the temperature, become a dominant factor at 104K sity and temperature. Therefore both transitions fuse and below in hydrogen and deuterium fluid studied ex- just to one with very specific properties [14,25,26,27]. perimentally at high pressures [31,34,35,36].

5 In order to compare theory and experiment in a more experiments are available which give reliable information quantitative way refined treatments of the phase tran- about the critical properties of hydrogen and noble gas sition induced by pressure dissociation and ionization plasmas. Experimental studies of classical electrolyte so- were given [7,8,11,13,14,28]. Most of the possible species lutions revealed that Coulomb liquids characteristics dif- and various interactions were taken into account to cal- fer from those of simple liquids. Even if crossover was ob- culate the equations of state. There is a certain dis- served from classical scaling laws to Ising scaling, it takes cussion whether the model equations of state reproduce place much closer to the critical point. The Ginzburg pa- reasonably well the recent experiments. However the rameter, which characterizes the size of the Ising region, theory predicts in all variants a phase transition. with turns out to be of 1-2 and more orders of magnitude less Tcr = 14000 19000K. Excited states do not contribute than for simple liquids. So there is a remarkable dif- significantly− to the hydrogen partition function in the in- ference in crossover range between simple and Coulomb termediate range of temperatures. liquids [23,49,50]. Experimental study of crossover phe- Path integral Monte Carlo calculations and wave nomena in real plasmas and in particular in cesium plas- packet dynamics also give some hints to the existence of mas might help to find out if the critical point is of the the plasma phase transition [37,38,39,40,41,42,43]. For plasma or gas-liquid kind. Additional factors, which do example the validity of the thermodynamic expressions not exist in electrolytes, namely, quantum effects and used was tested by comparison with molecular simulation charge-atom interaction can influence the crossover phe- and experimental data for nonideal plasmas and limiting nomena in PPT. expansions for weakly nonideal plasmas [43]. The prob- Michael Fisher [49] formulated clearly the challenge lem continues to attract remarkable attention, for exam- to the critical phenomena theory: ”Critical behavior is ple at the recent (June 2-7, 2002) “International Confer- thoroughly understood in Ising models alias lattice gases: ence on Warm Dense Matter and FEL Experiments” in But how far does that help our insight into the critical Hamburg, Filinov et al [44,45] claim that they “present region of continuum models and real fluids? In particu- results of direct path integral Monte Carlo simulations lar, what causes some ionic solutions to exhibit van der which, for the first time, provide first-principle support of Waals or ”classical” critical exponents, while other, so- such a phase transition in dense hydrogen”. It should be called ”solvophobic” systems are of Ising - type?”. It noted, however, that these particular PIMC simulations is evident that the difference between properties of var- yield rather low energy values for the “droplets” that ious liquids under similar external conditions can be at- appear in the simulations. Kohanoff and Scandolo [46] tributed principally only to the interaction potentials be- discussed on this workshop the same problem in a softer tween the particles. However the Kadanoff-Wilson theory manner and presented their results of ab initio molecular of critical phenomena, which is based on the universal- dynamics simulations. We mention these works as part ity hypothesis, neglects interaction potential peculiari- of the ongoing effort to simulate the PPT. ties and appeals only to large-scale properties, such as Let us discuss now in brief the phase transitions ob- system dimension and Hamiltonian symmetry. So other served in alkali-plasmas. As pointed out in previous approaches are needed. sections the theory predicts for this case that the PPT and the van der Waals transition fuse just to one transi- First of all Fisher points to the integral equation ap- tion. Near to the critical point the degree of ionization is proach. But most of the standard approaches to the low but different from zero. Coulombic interactions are theory of critical phenomena based on integral equa- present and influence the critical behavior, in spite of the tion method fail [49]. Though both the Kadanoff-Wilson fact that they do not play the dominant role. The area theory and the integral equation approach are on equal of parameters close to the critical point of cesium was rights, the first is based on global treatment whereas carefully investigated [25]. It was shown that near to the the latter is related to the local Ornstein-Zernike equa- critical point the ionization landscape changes quickly tion. For this reason Fisher puts the integral equation and has a quite complicated structure. The theoretical approach on the first place in his hope to solve the prob- predictions which take into account the charge-atom in- lem of Coulomb criticality. However there is a specific teraction (polarization effects) are in rather good agree- difficulty in the local approach which is connected with ment with the data [26,14]. the choice of closing relation in the integral equation ap- The existing experiments for argon plasmas based on proach. This difficulty was attacked recently by Mar- shock wave data are also in satisfactory agreement with tynov [51]. He suggested an integral equation which de- the theory [34,47,48]. pends on the interaction potential and permits in prin- We will discuss now the problem of critical behavior ciple to define the critical indices and their dependence and the critical indices. For the classical case much atten- on the interaction potential. Martynov [51] applied his tion has been devoted to this problem [23,49,50]. For the local approach to the system of charged particles. The PPT where quantum effects influence the critical point, critical indices obtained differ almost as much as twice the problem of critical indices is still open. So far no from van der Waals and Kadanoff-Wilson predictions.

6 However the results cannot be applied directly to elec- the PPT is an analogue of the known Coulombic transi- trolyte solutions since they do not take into account sol- tion in classical ionic solutions. In both cases the mecha- vent molecule influence, neither it can be applied directly nisms are the same, the transition is due to the interfer- to dense plasma, since quantum effects are not taken into ence between a Coulombic attraction of the charges and account. repulsive forces. The classical repulsion of ions at small Let us finally discuss several special problems con- distances is in the quantum case replaced by Heisenberg nected with Coulombic phase transitions. At first we and Pauli repulsion effects. We have shown above for mention the magnetic field influence. It was shown re- dense gaseous plasmas that in dependence on the values cently that strong magnetic fields increase the critical of the van der Waals parameters of the neutral compo- temperature [52,12]. Another important effect is con- nent we may obtain either a phase diagram wi two first nected with nonequilibrium phenomena. It was noted order transitions (hydrogen and noble gas plasmas) or a [53] that nonideal plasmas are usually generated in non- phase diagram where both transitions fuse to just one (al- equilibrium state with respect to plasma wave excitation. kali plasmas). In spite of considerable efforts there is no This might complicate the theory [14] and the analysis final proof yet that the PPT really exists, however new of experiments. theoretical and experimental results seem to confirm this At third we would like to discuss a new class of phe- hypothesis. One strong theoretical argument in favor of nomena observed in Cs-plasmas [54]. Though the stan- the PPT is the analogy to the Coulombic phase transition dard cesium phase diagram is well known [27], Recently in classical systems. For ionic solutions theory and ex- a new phase state of Cs was observed by Holmlid et periment (Monte Carlo data as well as measurements on al [54,55,56]. In experiments with thermoionic convert- electrolytes) show a corresponding Coulomb transition. ers clusters were observed first at 1300K. These cluster Therefore there is no doubt any more that a Coulombic were cooled and associated in microdroplets. The micro- phase transition in classical systems exists [22,23,49]. For droplets formed were fixed and measured at 70K. The quantum systems as e.g. hydrogen plasmas the problem number density obtained was about 1018 atoms/cm3. of the existence of a PPT is still open. The main reason Holmlid [54] considered his results as an observation of for still unsolved problems are the Rydberg matter predicted by Manykin et al [55,56]. the theory of the PPT requires an accurate theory Generally speaking Rydberg matter exists only at 0K • of dense quantum plasmas, and temperature. At higher temperatures Manykin’s ap- the pressures where the PPT occurs are in the proach can and should be combined with our non-ideal • plasma treatment. In fact both approaches enhance each Mbar region what leads to very difficult experimen- other. The phenomenon might be connected with the tal problems. idea of isolated segment of metastable nonideal plasmas We discussed here several results of theory and experi- which was introduced into the plasma thermodynamics ment which point to the existence of a PPT in real plas- long ago [57]. Supercooled nonideal plasma states were mas and discussed the conditions and parameter regions a direct consequence of the PPT [58]. Manykin’s Ry- where the PPT is expected. dberg matter [55,56], Holmlid’s microdroplets [54] and plasma phase transitions [2,3,59] are various facets of Coulombic phenomena. The existence of isolated regions ACKNOWLEDGEMENTS of metastable nonideal plasmas is not an exceptional feature of cesium. Several authors [57,58,55,56] suggested We thank V.E. Fortov, A.S. Kaklyugin, E.A.Manykin that this might be a more general feature. In fact Holm- and two referees for valuable discussions. The work is lid [54] observed his clusters in many other substances as supported by RFBS, grant 00-02-16310a. well. The isolated region of metastable nonideal plasmas is possibly an inherent part of the phase diagram. This part complements the standard diagram and is superim- posed on it.

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