163

総 説 Expanded Fluid Alkali Metals

Friedrich HenselA and Georg - Friedrich Hohl

Fluid alkali metals are typical examples of materials whose electronic structures depend

strongly on the thermodynamic state of the system. The most striking manifestation of this state dependence is the metal - nonmetal transition which occurs when the dense liquid evapo -rates to the dilute vapour or when the fluid is expanded by heating to its liquid - vapour critical point.

The paper discusses equation of state data, electrical, and optical properties, and neutron scattering measurements of S( Q) and S( Q, w) with special emphasis on the change in these

properties in the metal - nonmetal transition region. The shape of S( Q, w) changes consider -ably on approaching the transition from the high - density liquid side, indicating a change in the interparticle interaction and the molecular structure. S(Q, w) of rubidium in the density range between the melting point density and three times the critical density is characterized by the existence of well defined acoustic - phonon - like collective density excitations at high mo -mentum transfer, whereas S(Q, w) at a density of about twice the critical density is consistent with excitations of an optic - type mode in which two species tend to move in opposite directions.

[metal - nonmetal transition, supercritical fluid, alkali metal, static and dynamic structure, critical behaviour]

1. Introduction

The properties of fluid alkali metals have been the subject of intensive experimental and theoretical inves tigation, despite the severe ex -

perimental difficulties associated with their high -ly reactive nature. Much of this effort has been motivated by the fact that fluid alkali metals are very distinct from normal insulating fluids such as argon in that their electronic and molecular structures are strongly depending on the thermo -dynamic state of the system. It is obvious that there are substantial changes in the nature of bonding of alkali metals upon evaporation. For example, far below the liquid - vapour critical

point (e.g. near the triple point), when the density difference between the coexisting phases is large, liquid alkali metals are reasonably well de scribed by the nearly - free - electron (NFE) ap -

proximation, whereas in the vapor phase at suffi -ciently low den sities, the valence electrons occu -

py spatially localized atomic or molecular or -bitals. In such a situa tion, near the triple point, the liquid - vapor phase transition coincides pre -

cisely with a metal - nonmetal transition. Both

the density and the electronic structure change on

passing from one phase to the other. The vapour

phase is non metallic and well characterized by

highly polarizable atoms and molecules which in -

teract through weak van der Waals forces, where -

as valence elec Irons in the coexisting metallic

liquid phase are dissociated and the inter particle

interactions are thought to arise from screened

Coulomb potentials. Whilst the two limiting cas -

es of the dense liquid metal near the triple point

and the low density vapour phase are reasonably

well understood, our knowledge of the connection

between these two limits through the liquid -

vapour critical and super critical regions of the

phase diagram lags far behind. Nearer to the

critical point, the liquid density is much less and

the vapour density much greater. At what densi -

ty does the vapour become metallic, or the liquid

nonmetallic? It is this variation of the interaction

in the neighbourhood of the metal - nonmetal

transition that is difficult to deal with theoretical -

ly and which has important consequences for the

Katahira 2-1- 1, Aoba - ku, Sendai 980 Institute for Materials Research, Tohoku University

A: On leave from Fachbereich Physikalische Chemie and Zentrum fur Materialwissenschaften,

Philipps - Universitat Margurg, Hans - Meerwein - Strasse, 35032 Marburg, Federal Republic of

Germany.

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164

liquid - vapour critical point phase transition.

From another point of view, that of potential

technological applications, fluid alkali metals al -

so occupy a special place. They are considered as

ideal can didates for high temperature working

fluids because they have high latent heats of va-

porization and high heat transfer coefficients.

They may be used in nuclear electric power plants

in space and also to increase the efficiency of fos -

sil, fission and fusion power plants on earth.

They may be involved in liquid metal magneto-

hydrodynamic systems coupled to solar power.

In heat pipes heat may be transferred by passing

alkali metal vapour from hot to cool ends of a

pipe. The selection of a particular metal for such

applications requires access to a reliable body of

experimentally determined thermodynamic

properties. Safety regulations require the be -

haviour of the fluid to be known far above the

proposed operating conditions - even to beyond

the liquid - vapour critical point, where profound

changes in electronic struc ture occur.

Experimental research on such properties is

complicated, because fluid alkali metals are diffi-

cult to experiment with, because, as Table 1

shows, a combination of high temperature and

pressure is required to bring the sample any -where near its critical point.

The critical temperatures Tc and pressures pc

are low enough to be studied under static condi-

tions only for Cs, Rb, K and eventually Na. The

estimated location of the critical point of lithium is in fact outside the range of conventional mea-surements under static, equilibrium conditions.

Only transient experiments such as shock waves,

exploding wires, and laser heating are reaching

temperatures and pressures high enough to ex -

plore the critical region of lithium. But these methods are less accurate than static experi -

ments. This means that there is much interest in methods of estimating the critical conditions of

metals. The different estimation techniques are

all based on the relation between the critical data

and other thermophysical properties. The most

common basis is the principle of corre sponding

states which is well established for molecular flu -

ids, but its validity for fluid metals is open to se -

rious questions. As discussed above, the theoret -

ical assumption un der lying the principle of corre

spond ing states namely, that the nature of the in -

teraction between the particles of the fluid is in -

dependent of the thermodynamic state of the flu -

id, is violated for metals.

Table 1. Critical temperatures, pressures

and densities of alkali metals.

The most significant experiments relevant to the effect of the liquid - vapour interparticle a -symmetry and to the question of the validity of an ap proximate law of corresponding states are those on the liquid - vapour coexistence curves of the alkali metals cesium and rubidium [1] and of divalent mer curv [5].

•@ Despite extreme conditions at the critical

points of fluid metals, the liquid - vapour coexis -

tence curves have been measured to a resolution

ƒ¢T/Tc=Tc-T/Tc•¬5•E10-4

which is close enough to demon strate the impor -tant difference between metallic and insulating molecu lar fluids. Figure 1 shows a reduced plot

(p/pa versus T/Tc) of the coexisting liquid densi-ties PL and vapour densities pv for the metals ce-sium and mercury and for the inert gas argon. The comparison of the coexistence curves shows a lack of correspondence when the interatomic forces are fundamentally different. Note that even the metals cesium and mercury do not cor -respond, suggesting that inter molecular poten-

Fig.1. Comparison of T/Tc vs ƒÏ/ƒÏc coexistence

curves for argon, cesium and mercury.

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165

tials operating in these fluids differ fundamental -

ly.

The limitations of corresponding state theory

are also demonstrated by the comparison of the

critical compressibility factors

Zc=pc•EVc/R•ETC

in Table 2. The data show that as a group liquid

metals do not appear to have very similar values

for Z. The alkali metals and mercury appear to

differ significantly in their values.

Table 2. Critical compressibility factors

of different metals compared to argon.

In the absence of adequate empirical rules

and theoretical methods, the obvious alter native

is the experimental determination of the proper -

ties of fluid metals over the widest possible range

of temperatures and pressures. Careful experi-

ments on a few characteristic systems may pro-

vide the insight required for the development of a

general quantitative theory.

The purpose of the present paper is to review

recent experimental results in the liquid - vapour

critical region of alkali metals which show that

the existence of the metal - nonmetal transition

noticeably influences the electronic, thermody -

namic, structural and dynamic features of these

metals. Relatively accurate experi mental results

in the critical region can be achieved only for al -

kali metals whose liquid - vapour critical points

lie within the limits of static temperatures and

pressures available in the laboratory. Monova -

lent Cs, Rb, and K are examples of such fluids.

2. Experimental Problems

Experimentation near the critical point of

any one - component fluid is complicated by the

fact that the isobaric expansion coefficient ap and

the isothermal compressibility XT diverge. The

divergence of ap or XT implies that the fluid kept at

constant pressure p or constant temperature T

will undergo extremely large variations in densi-

ty when T or p fluctuate. Severe problems of

temperature - and pressure measurement and

control arise.

For fluid alkali metals the situation is even

more complicated because the highly reactive na-

ture of these metals makes their containment in

uncontami nated form very difficult and decreas -

es the accuracy with which properties can be

measured. Materials found to be suitable for

sample containers for Cs, Rb, and K are pure

tungsten, molybdenum, the alloys tungsten - the -

nium, and tungsten - molybdenum. An ideally

suitable material for optical measurements with

Cs is corundum single crystal up to 1200 •Ž.

These materials are chemically inert in contact

with the fluid sample; but cells from these mate -

rials cannot be constructed in such a way that

they can withstand a high internal pressure at

high tempera tares. An arrangement is therefore

used in which the different cells, together with the

necessary furnaces, are placed in larger auto -

claves, which are filled with argon or helium un -

der the same pressure as the fluid metal inside

the cell. This avoids any mechanical stress on the

cells. However, at the necessary high pressures

argon has already a large density and carries

heat about by convection currents that are diffi-

cult to eliminate. Severe problems of tempera -

ture measurement and control arise. The most

conventional way to avoid convection is to fill the

space inside the autoclave as completely as possi -

ble with a solid thermally insulating material.

High temperature - high pressure equipment of

this type have been described in the literature for

measurements of optical absorption [6, 7] and re-

flectivity [8], for NMR-experiments [9], for

measurements of the equation of state data [10]

and electrical properties [10, 11], and for neutron

scattering measurements of the static and dy-

namic structure factor [12].

A comprehensive review of the existing ex -

perimental procedures em ployed for the investi -

gation of the high temperature - high pressure

behaviour of fluid alkali metals is unnecessary

because of the surveys that exists and in the fol -

lowing we will restrict ourselves only to novel de -

velop ments which have not been adequately re -

viewed else where, e.g. measurements of density

of high accuracy. Figure 2 shows the experimen-

tal set - up which has been used to determine the

equation of state data of Cs, Rb and K up to 2000

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Fig.2. Apparatus for density measurement of alkali metals.

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167

•Ž and pressures up to 300 bars.

The high - pressure - high - temperature ap -

paratus consisted of a steel pres sure vessel with

an electric resistance furnace which is ther mally

insulated from the steel walls. The furnace con -

sisted of three independently controlled heating

elements made of rnolyb denum wire and foil.

The pressure medium was argon gas.

The alkali metal samples were contained in a

molybdenum cell with a relatively large volume

of up to 20 cm 3 which is surrounded by the fur-

nace. The cell has a capillary tube extend ing into

a cold region of the high pressure vessel interior,

where it was connected with stainless steel bel -

lows, which were kept at about the melting point

temperature of the metal under study. The densi -

ty variation of fluid alkali metals with pressure

and tempera ture was then determined by the bel -

lows which are completely surrounded on their

outside by mercury. The expansion of the bel -

lows was monitored via the corresponding level

change of the mercury inside the level indicator.

The position of the mercury level was measured

by an inductive method. In order to minimize er -

rors arising from uncertainties in temperature

homogeneity, eight thermocouples were attached

at different positions along the sample, the capil-

lary and the bellows. For the equation of state

measurements two methods have been employed.

In the first experiment we measured the pressure

p - temperature T coordinates for a very large

number of isochores in the liquid vapour phase

close to the critical point. Whether the average

density of an isochore was greater (liquid L) or

smaller (vapour V) than critical was easily found

from the isochore slopes (•ÝP/•ÝT)L or (•Ýp/•ÝT)v,

respectively. (•Ýp/•ÝT)L or (•Ýp/•ÝT)v are greater

or smaller than the slope of the vapour pressure

curve, (•Ýp/•ÝT), at the intersection of the two

curves. From these intersections points of the co -

exis tence curve for orthobaric liquid and vapour

densities have been obtained. As the critical

point was approached with increasing temper a

ture, the slopes, (•Ýp/•ÝT)L, (•Ýp/•ÝT)v and

(•Ýp/•ÝT)sat, became indistinguishable. From this

observation the critical constants have been de -

termined.

Similarar values have been obtained from the

second experiment which measured density d -

pressure p coordinates for a series of isotherms. Approximate values of the critical data have been

deduced from an examination of the shapes of the

isotherms.

A selection of equation of state data of ce-

sium from both experiments is given in Fig.3 in form of number density n isotherms plotted ver -

sus pressure p. These data are accurate enough to yield the behaviour of the isothermal compress -ibility

X=1/n(dn/dp)T

near the critical point. The data for cesium are

displayed in Fig.4 as a function of the mass densi -

ty p at constant temperatures T. X in creases

rapidly in the density range between about 0.22

g • cm-3 (ƒÏ/ƒÏc •¬0.6) and 0.48 g • cm-3 (ƒÏ/ƒÏc•¬1.3)

for temperatures not too far from the critical

Fig.3. Isotherms of cesium at different tem -

peratures.

Fig.4. Compressibility of cesium at different supercritical temperatures.

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temperature Tc. This increase is related to the

development of critical density fluctuations which

are obviously important for cesium in the density

range 0.22 < ƒÏ < 0.48 g • cm-3 near Tc.

3. Electrical Properties

The problem of the interrelation of the met -al - nonmetal transition and the liquid - vapour

phase transition in fluid alkali metals has re -ceived considerable theoretical attention in the

past. The pioneering study of Landau and Zel -dovitch [13] suggested the possibility of separate first - order electronic and liquid - vapour transi -tions in fluid metals. Subsequent theoretical at-tempts to model the statistical mechanics of the metal - nonmetal transition in fluids reach simi -lar conclusions but are still insufficient to provide a clear - cut answer to this question from theory

[14-17]. Measurements such as those of the electrical

conductivity (Fig.5) and the equation of state

(Fig.3) clearly show that there is no sharp (first-

order) electronic transition except across the liq -

uid-vapour phase change for fluid cesium, i.e.

the liquid vapour phase separation tends to sepa -

rate the nonmetallic and metallic fluids. Near the

critical point the conductivity drops sharply, thus

showing a strong effect of the phase transition on

the electronic structure. The close correlation be -

tween the behaviour of the density and that of the

conductivity convincingly shows that the varia -

tion of density is the dominant factor governing

the metal - nonmetal transition. In practice, how

ever, very careful measurements are required to

separate the effects of density and temperature in

the critical region. Part of the difficulty arises

because not only the compressibility (Fig.4) but

also the pressure derivative of the electrical con -

ductivity becomes very large in that region. This

is demonstrated by the data in the inset in Fig.5.

1/ƒÏ(•ÝƒÏ/•ÝƒÏ)T and 1/ƒÐ(•ÝƒÐ/•Ýp)T become large in the

critical region so that small pressure errors cause

large density and conductivity errors. Conse-

quently, for a reliable correlation of a and p,

again precise temperature and pressure control is

essential.

As mentioned above, the occurrence of the metal - nonmetal transition in low density fluid metals implies that the electronic structure of the coexisting phases, liquid and vapour, are funda-mentally different. Liquid cesium just above its

melting point possesses a large degree of correla-tion in the atomic positions, and it may be con-sidered a normal liquid metal with properties

quite similar to that of the solid close to the melt -ing point. The electrical conductivity exhibits characteristic typical for materials with metallic electron concentrations. Perhaps the most obvi -ous of these is the fact that the behaviour of the electrical conductivity in this range can be ex -

plained within the framework of the Ziman theory for the nearly fee electron metal [18], (dashed

Fig.5. Electrical conductivity a of Cs as a func -tion of pressure at constant subcritical tempera -tures. The inset displays values of the critical isotherm (points) together with the pressure

(o -1 1

Fig.6. Electrical conductivity cesium as a func -

tion of temperature along the liquid - vapour

coexistence line.

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169

line in Fig.6). The application of this theory is

justified when the average distance between scat -tering of the electrons by the disordered liquid structure, the electron mean free path, is substan -tially larger than the average near neighbour dis -tance of the atoms. The dashed curve in Fig.6 has been calculated by employing the measured val -ues of the static structure factor [19] together with the model pseudopotential form factor given by the Ashcroft "empty core" potential [20].

Comparison between calculated and experi -

mental conductivity values shows that the nearly

free electron theory provides a good account of the

variation of the conductivity for temperatures

lower that 1100°C, i.e. at densities of the coexist-

ing liquid higher n ? 6.1021 cml At lower densi-

ties, the measured conductivity is clearly lower

than the value calculated employing the NFE -

model. The gradual failure of the NFE - model at

low densities is probably not due to the break -

down of the NFE - condition L > ƒ¿ at 6•E1021 cml

Rather, the NFE - breakdown is more likely to re -

flect the increased importance of electron correla -

tion below this density.

Magnetic susceptibility- [21], NMR - [9

and optical reflectivity [22] studies of low -densi

ty liquid cesium have yielded evidence of stronf

electron - electron - correlations in the form of

susceptibility enhancements and antiferromag-

netic spin fluctuations. The susceptibility, NMR

and optical data show a relatively sharp onset of a

correlation enhance ment of the effective mass at

a density n = 6.1021 cm1 The combined analysis

of the magnetic and NMR data implies a correla -

tion enhancement of the effective mass mdr = 5 m0,

for example, at a density of 3.6.1021cm-3 (the

density at the liquid - vapour critical point is nc =

1.72. 1021cm1 For still smaller densities (i .e.

smaller than 3.6•E1021 cm-3) the magnetic suscep -

tibility measurements [9] and quantum statisti -

cal calculations [23-25] indicate the presence of

localized species, such as Cs - atoms, Cs2- dimers

and Cs2 - dimers the concen tration of which is in -

creasing near the critical point, thus leading to

large deviations of the electrical conductivity

from NFE-behaviour.

The inference of an appreciable concen tra -

tion of associations in the critical regions of alkali

metals is also consistent with the equation of state

data of cesium, rubidium and potassium. The ef-

fect of association is vividly illustrated by the

comparison of the experimentally determined

critical compressibility factors Zc (Table 2) with

that of the hard sphere plus mean field fluid for

which Zc is 0.358 [26]. The Zc values in Table 2 are

consistent with a dimer association of about 31%

for Cs, 28% for Rb and 34% for K. These results

are consistent with the above mentioned theoreti -

cal calculation of the composition of fluid Cs and

Rb [25] which employs a quantum statistical

equation of state originally derived for partially

ionized plasmas [23] which takes into account the

interaction corrections between the various

species in a systematic way. An approximate

self - consistent solution for the system of coupled

mass action laws describes the formation of atom

M, dimer M2, and molecular ions MZ out of the el -

ementary particles, electrons e- and simple ions

M The numerical results obtained for cesium

are displayed in Fig.7. It is a plot of the compo

sition of fluid cesium as a function of density

along the liquid - vapour coexistence line . The re -

suits support the view that the atom and the spin -

paired dimer are the dominating species in the

coexisting vapour phase of Cs with the dimer con -

centration increasing sharply with increasing

density up to the critical density ƒÏc . For densities

higher than ƒÏc, a surprisingly high concentration

of positively charged molecular ions M2+ is pre -

dicted which reaches a maximum of about 35% at

about 1.5ƒÏ. For still higher densities the atoms,

neutral dimers and the positively charged molec-

ular ions M2+ disappear gradually and the metals

are predicted to reach complete ionization, i.e.

the metallic state at p >> 3

The formation of charged clusters like Cs2+ in

Fig.7. The comparison of fluid cesium as a function of density along the liquid - vapour co -existence line [25].

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170

the dense vapour phase strongly affects the densi -ty and temperature dependence of the electrical conductivity. This is seen from a glance at Fig.6 where the measured electrical conductivities of the vapour are shown in comparison with those

calculated (dotted line) for the thermal equilibri-um ionization fraction of monoatomic cesium employing the vacuum ionization potential of 3.89 eV (Saha equation). The agreement between ex -

perimental and calculated values is sufficient on-ly at very low densities where the overwhelmingly dominant species present in the vapour is the atomic monomer. At higher densities the con -ductivity can no longer be described on the basis of the assumption that the vapour is monoatomic. This is not surprising because with higher vapour densities, species like Cs2 and higher clusters will occur. These clusters have ioniza tion potentials much lower than the corre sponding atoms so that much higher degrees of ionization should be expected than those simply calculated by the Sa -ha equation. It is therefore not surprising that excellent agree ment between the experimentally observed s - values and those calculated (open cir -cles in Fig.6) can be reached by employing the electron concentrations shown in the inset of Fig. 5[27].

4. The Coexisting Vapour Phase

The inference of a high concentration of

dimers in the saturated vapour phase of alkali metals close to the critical point is also consistent with the behaviour of the magnetic susceptibility

[21]. Figure 8 shows for example, the tempera-

ture dependence of the total magnetic mass sus -ceptibility, expressed as c, i.e. susceptibility per

gram, for cesium in the vapour phase along the coexistence curve (the critical temperature T° = 1651 °C). The full curve in Fig.8 gives the Curie

paramagnetism calculated for a purely atomic cesium vapour. The comparison clearly indicates the development of spin - paired species like Cs2 as already pointed out by Freyland [21]. He inter -

preted his susceptibility data of the vapour phase in terms of a crude model in which the total vapour susceptibilities are split into a mixture of

paramagnetic monoatomic Cs and diamagnetic Cs2 molecules without consid ering nonideality correlations. His estimate of more than 25% dimer concentration at a temperature of about 1600°C and a corresponding density of 0.15 g cm-3 is in close agreement with the values shown in Fig.7 which have been calculated with a model

[25] which takes into account the interaction cor -rections between the various species.

Fig.8. Susceptibility of Cs vapour along the liquid - vapour coexistence line (points). The solid line shows the curie - law behaviour for purely monoatomic vapour.

It has been known for some time that dimer -

ization is common in alkali metal vapours. The

most direct evidence for this stems from optical

absorption experiments in the dilute vapour

phase [28]. An example of such data is shown in

Fig.9 for the relatively low temperature of 420 °C

(the corresponding density is N = 4.1017cm-3 )

[29]. The dominant stable species at this condi-

tion is the neutral atom Cs whereas the dimer

mole fraction is only about 0.001. The atom con -

tributes a series of fundamental electron absorp -

tion lines due to the transition 62S1/2•¨n2 P1/2, 3/2. In

addition to this dipole allowed s - p transitions,

the quadrupol allowed 62S1,2 -1.52 D ,2,512 lines are al -

so detectable in the absorption spectrum. Despite

of the very small dimer concentration, the low -

energy part of the spectrum is dominated by a

number of broad Cs2- molecular bands with max -

ima at about 2.6 eV, 2.1 eV, and 1.9 eV which ex -

tend roughly over the range

1.8eV•¬hw•¬2.8eV.

However, the most intensive Cs2 - absorption

bands (not shown in Fig.9) are those in the ener-

gy range between 0.9 eV and 1.65 eV with maxi -

ma around 1.0 eV (a3ƒ®-X1‡”g+), around 1.2 eV

(A1ƒ® -X1Eg+ ), and around 1.6 eV (B1ƒ®u-X1‡”g+).

The intensity of these bands increases very rapid -

ly with increasing density as is revealed by the

data displayed in Fig.10 [7]. It is obvious that the

major effect of heating the vapour to higher

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171

Fig.9. Absorption spectrum of Cs vapour: 1) Quadrupol transitions 62S1/2 •¨ 52D3/2 and 62S1/2 •¨ 52D5/2, 2) C1 ‡”u

- X1‡”g+ , 3)D1‡”u+ - X1‡”g+ , 4) A1ƒ®+p - X1 Eg+ , 5) 62S‚P/2 •¨ 72P1/2, 6) 62S1/2 •¨ 72Py3/2

, 7) unknown, 8) 6251/2 •¨ 82P1/2

,3/2 (dublett unresolved), 9) 62S1/2•¨92P1/2 ,3/2 10) 62S1/2•¨102P1/2,3/2,11) 62S1/2•¨112P1/2 ,3/2,12) ioniza-tion continuum. The inset shows the photoionization cross section as a function of the energy

.

Fig.10. Absorption spectra of Cs vapour at different temperature. The inset displays the density

dependence of the absorption spectrum at the high end of and above the spectral series of atomic t

ransitions.

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172

temperatures and densities (i.e. in the direction toward the critical region) is a change in the rela-tive concentration of the neutral dimer.

Another marked effect of increasing the

vapour density is a reduction of the n - value of the

highest resolvable 62S1/2 •¨ n2P1/2, 3/2-transition, in

effect, lowering the ionization energy of the iso-

lated atom of 3.89 eV. It can be seen from a

glance at Fig.9 that even at the low vapour density

of 4.1017 cm-3 the atomic transitions at the high

energy end of the spectral series, i.e. near the ion -

ization continuum, are superimposed on an ab -

sorption background which has the shape of the

photo - ionization cross section for the processes

which lead to the formation of the positively

charged dimer ion Cs2+ (see inset of Fig.9). Two

processes are mainly involved. The first is the

ionization process

Cs2 + hw •¨ Cs2 + e

It involves ionization of a stable Cs2- molecule in which the Cs - atom is closer to its molecular

partner than to any other atom in the vapour, and the paired electrons are confined to their home molecules. The second process is that of pair-absorption where two atoms absorb one photon in the state of collision (quasimolecule). The idea and the theoretical analysis of this quasimolecule absorption is quite old and dates back to the year 1972 [30]. Continuum - state pair - absorption bands, i.e. pair-absorption bands in the pho-toionization continuum of the atoms involved, are generally very broad compared with the bound -state pair - absorption bands, for which the sum of the excitation energy of both excited species does not exceed the ionization energy of the atoms involved. The large width or diffusiveness of the continuum - state pair - absorption is a clear indi -cation that the ionization probability of the excit -ed quasimolecule or collision complex is very high, i.e. closee to unity [31].

The inset of Fig.10 displays the density de -

pendence of the absorption spectrum at the high end of and above the spectral series of the atomic transitions. Comparison with the inset in Fig. 9

clearly shows that at the highest densities the

photoionization processes which lead to the for-mation of Csa are dominating the absorption

spectrum.

5. Static and Dynamic Structure

It is evident from the foregoing that there are substantial changes in the nature of bonding of alkali metals upon evaporation, at which point there is a transition from a metallic bonding state to an insulating covalently bonded state. The liq -uid just above its melting point is most often treated as a monoatomic state. Its structure is regarded as built up of single screened ions, each diffusively uncoupled from every other with in -teractions described by an effective density de-

pending pair potential. This approach is essen -tially based on the experimental observation that relatively little change in the local atomic ar -rangement occurs during melting. For cesium, for example, the molar volume increases by only about 2.5 %, and the average near - neighbour dis -tance, given by the position of the first peak of the radial distribution function g(R), is almost iden-tical to the near neighbour distance in the bcc -crystalline structure close to melting. Conse -

quently, the liquid metallic phase is most often treated as a monoatomic state which typifies the solid structure.

However, as noted above, this correspon dence between liquid and solid applies only at high densities. When the liquid evaporates, the low density vapour cannot be regarded as purely monoatomic. In particular, in view of the argu -ments given in section 4., it can be suggested that the diatomic structure of the cesium vapour per -sists up to relatively high densities. The question to emerge, therefore is up to what density and temperature the pairing mechanism present in the vapour can survive the passage to the metallic liquid [32]. This question is based on the view that the mechanism for the nonmetal - to - metal transition which occurs in diatomic molecules with increasing density, may begin with metal -lization of the diatomic system by an overlap of the valence - and conduction bands which at higher density is followed by a gradual dissocia-tion into a monoatomic state. Thus, while liquid alkali metals might be regarded as monoatomic near the melting point, the viewpoint might be rather different for the lower density liquid regime where dimers, dimer ions or higher clus -ters can be the aDnronriate dynamic subunits.

This problem has been a primary moti vation for neutron scattering measure ments of the stat -ic structure factors of fluid Cs and Rb up to the critical region [33]. Typical results are presented in Fig.ll for Cs in form of the Fourier - Trans -

高圧力の科学 と技術Vol.3,　 No.2(1994)

173

Fig.ll. Pair correlation function of Cs at three

different temperatures and densities.

forms of the static structure factors S(Q), i.e. the

pair correlation functions g(R), at three different temperatures and densities. With decreasing den -sity or increasing temperature a number of changes in g(R) are noteworthy.

•@ Now the pair correlation function g(R) is re -

lated to the radial distribution function

N(R) = 4ƒÎR2ng(R)

which determines the number of neighbouring atoms N(R)dR in a spherical shell of radius R and thickness dR centered on a particular atom of interest. The average coordination number N, is determined by the area under the first peak in

g(R), whereas the average nearest neighbour -distance R, is given by the position of this peak. Analysis of data such as those of Fig.ll shows that for Cs, N, tends to decrease as the density is decreased bythermal expansion. It is particular-ly note worthy that the position of the first peak in

g(R), namely R,, remains virtually constant while N decreases by about the same factor as the density. This indicates that clustering occurs as the density decreases, keeping many atoms high -ly coordinated. The average coordination num -

ber N, then decreases because of an increase in

the number of atoms on the surface of clusters.

The density dependence of N, and R, reflects the

fundamentally different character of thermal ex -

pansion in the solid and liquid. In connection with the question whether the

dense vapour phase of alkali metals, can survive

the condensation to the liquid state it is also in -

teresting to consider the coherent dynamic struc -

ture factor S( Q, w) which can be measured by in -

elastic neutron scattering. It has been known for

some time that liquid metals (i.e. rubidium), un -

like dense Lennard - Jones (inert gas) systems,

exhibit distinct collective excitations over a large

region of the Q -ƒÖ plane [34]. Recent measure-

ments of S(Q, w) for expanded liquid rubidium

[35, 36] extending nearly to the critical point have

shown that these collective excitations can be ob -

served over a very wide temperature range up to

1400•Ž.

Samples of the large number of data are

shown in Fig.12 for a constant momentum trans -

Fig.12. The dynamic structure factor S(Q, ƒÖ)

of expanded liquid Rb at a constant momentum

transfer Q =1 A-1 and at selected temperatures

near the liquid - vapour coexsistence line.

高圧力の科学と技術Vol.3,NO.2(1994)

174

fer Q = 1 A-1 which corresponds to a wavelength

27r/Q comparable in the rubidium case to the in -

teratomic spacing. The central peaks in S(Q, ƒÖ)

versusw curves which have a width of a few

meV, are usually called quasi - elastic peaks.

They are interpreted as a result of energy losses

or gains by the neutrons when they exchange

translational or rotational energy with diffusing

nuclei. The separate, or inelastic, side peaks at

larger hw represent the collective excitations. It

is interesting that plotting the peak position

hw(Q) as a function of Q, as is done in Fig.13,

gives a dispersion relation which resembles very

much that of the well - ordered density fluctua -

tions, the so - called longitudinal phonons, in

crystalline rubidium. Comparison shows that the

dispersion relation of the solid has a maximum

frequency which is very similar to that observed

for the liquid. This shows that the restoring

forces are essentially the same in liquid and solid

rubidium. In both cases the cages of nearest

neighbours define the crucial dynamic unity: The

Einstein frequency is the same in the two sys -

tems. However, it must be emphasized, that this

does not imply that phonons of long lifetime are

propagating through the liquid. The broadening

of the side peaks indicates strong damping with

increasing temperature. The contrast with

phonon spectra in solids is obvious. If the inelas -

tic peaks make just a perceptible shoulder on the

central peak, as it is the case at high tempera -

tures (see Fig.12), it becomes difficult to decom -

pose the observed curves into central and inelastic

components. One way to avoid this difficulty is to

choose the longitudinal current fluctuations J(w,

Q) = ƒÖ2•ES(Q, w)/Q2 as the dynamical variable in -

stead of the density fluctuations S( Q, £0). Selected

J,(w, Q) curves for liquid rubidium at Q =1.3 A-1

at high temperatures are plotted in Fig.14. It is

obvious that J(w, Q) has the advantage that there

is no central peak and that the high frequency

part of the spectrum is enhanced, thus allowing

the study of weak high frequency excitations. Al -

though these frequencies cannot be identified with

the true physical frequency determined from S(Q,

Fig.13. Dispersion curves obtained from the

maxima in the longitudinal current correlation

function for liquid Rb at conditions along the

liquid - vapour coexistence line. The data at 45 °C

are from Ref .35.

Fig. 14. The experimental longitudinal current correlation functions of liquid Rb at a constant wavevector Q = 1.3 A-1 for different tempera-tures along the liquid - vapour coexisting line.

高圧力の科学 と技術Vol.3,　 No.2(1994)

175

w), they are frequently used to establish the main features of the dispersion. The corresponding dispersion relations for the high temperature curves are also shown in Fig.13. The characteris -tic features of these curves reflect the structural changes with temperature and density discussed above for g(R). The minima of the dispersion

curves shift only slightly to lower Q. At high Q-values and high temperatures, where the oscilla -tions in S(Q) and g(R) are much more strongly damped, the free particle dispersion becomes more pronounced. The latter is simply given by the maxima in

which are found at

(hƒÖ) )free ges = h/2QV

with Vo the thermal velocity.

In terms of our interest in the changes in in -teratomic interactions occurring in the course of the metal - nonmetal transition, it is a most inter -esting fact that the results of these experimental observations, namely the exis tence of well de -fined collective excitations in liquid metallic ru -bidium at high momentum transfer and their ab -sence in the inert gas liquids, are well reproduced by molecular dynamics calculations. For the in -ert gas liquids the molecular dynamics calcula-tions used Lennard - Jones potentials [37, 38] and for rubidium long-range oscillatory potentials characteristically for metals which are mediated by the itinerant metal electrons [39-42]. This in -dicates that the pair potential in rubidium is re -sponsible for damped collec tive motions with wavelength comparable with the interatomic spacing. The quite different Lennard - Jones po -tential does not lead to such phenomena.

Interestingly, the molecular dynamics calcu -

lations (employing a typical metallic interaction

potential) are in good agreement with the exper -

imental results up to 1400 •Ž and a density of

about three times the critical density which indi -

cates that the mean forces given by the metallic

binding and the screening are still controlling the

dynamics of liquid rubidium at these conditions.

This is no longer the case, if the temperature is

increased still further (see e.g. in Fig.13, T=1600

•Ž , the corresponding density is about twice the

critical density). A clear change in the shape of J,

(Q, w) is observed. J(Q, ƒÖ) at Q = 1.3 A-1 ex-

hibits well defined excitation peaks around hw about 3.2, 6.4 and 9.6 meV. One way of interpret -ing this observation is to propose that the peaks are due to optic - type modes, i.e. optic vibrations , in which two species tend to move in opposite di -rections. The features of the J,(Q, c)) curve are consistent with the model for scattering from a

particle executing harmonic oscillations about a center which is diffusing [36]. This obser vation is consistent with the view that intra molecular dynamic effects are present in liquid expanded rubidium under these conditions.

In view of the possibility that for alkali met -als the dynamic units can change signifi cantly throughout the liquid range both with density and temperature, it is clear that the practice of invert -ing measured structure factor data.to get effective

pair potentials [43, 44] or to model the structure factor of expanded metals [45-50] by employing

pair potentials mediated by the itinerant metal electrons hinges critically on insight into the den -sity range in which alkali metals can be viewed as monoatomic metallic fluids.

6. Critical Behaviour of Alkali Metals

As we have seen in the foregoing sections, liquid and gaseous metals do not behave like simple insulating substances like argon for which the interatomic potential function may be consid -ered independent of density to a good approxima -tion. By contrast, the electronic and molecular structures of fluid metals differ with density. In -deed, at ordinary conditions far from the critical

point, the vapour phase is non metallic and con -tains an appreciable fraction of dimers whereas the liquid phase is metallic and the interaction is strongly affected by the presence and number of the valence electrons. The fact that the cohesion in metallic systems consists of a sum of long-range Coulomb interactions has led to the specu -lation [51] that critical points of metals could be in a different universality class than that of insu -lating fluids with potentials which decay with in -teratomic separation as RI This is clearly not the case [5] for the alkali metals cesium [1], ru -bidium [1], and potassium [52] and the divalent metal mercury [11]. The shapes of their coexis -tence curves in the critical region can be de -scribed with high accuracy by the scaling law

高圧 力の科学 と技 術Vol.3,No.2(1994)

176

Fig.15. The density of the diameter as a function of the reduced temperature. The insets show in comparison the diameter of Hg [11] and the diameter of several nonmetallic fluids [53, 55].

ƒÏL-ƒÏv=B(ƒ¢T/T)ƒÀ

with the same exponent ,3 as molecular fluids,

where OT = Tc - T and B is a constant. The ex -

perimentally determined exponents ,6 for cesium,

rubidium and mercury lie between 0.35 - 0.36, a

value only slightly higher than that found for the

three dimensional Ising - model.

The difference between the coexistence curves

of molecular and metallic fluids be comes visi -

ble mainly in the behaviour of the diameter which

seems to be strongly affected by the strong ther -

modynamic state - dependence of the effective par-

ticle interaction in metallic fluids as the critical

region is traversed.

A careful analysis of the coexistence curves demonstrates that fluid metals violate the hun-dred year old empirical law of rectilinear diame -ter over a surprisingly large temperature range. By contrast, the deviations from this law are ex -tremely (mostly immeasurably) small for the co-existing curves of essentially all nonmetallic one -component fluids [53]. The law states that the to -cus of the tie - line mid - points

ƒÏd = 1/2 (ƒÏL + ƒÏv)

2

is a linear function of T. Since both PL and pv ap -

proach the limiting density ƒÏc at the liquid -

vapour critical point, the law can be written

ƒÏd-ƒÏc=D1ƒ¢T/Tc

Modern theory of liquid - vapour critical phenom -

ena based on renormalization group studies [54]

permits calculation of the diameter anomaly in -

cluding effects of large scale density fluctuations.

The theory predicts that the temperature deriva -

tive of the diameter dƒÏd/dT, diverges at least as

fast as the constant - volume specific heat Cv.

That is, as the reduced temperature ĢT/Tc goes to

zero. the diameter varies as

ƒÏd-ƒÏc=(ƒ¢T-Tc)1-ƒ¿+D1(ƒ¢T/Tc)+•c

where a = 0.11 is the same exponent that de -scribes the behaviour of the constant - volume specific heat Cv. Since 1- a = 0.89 is not very dif -ferent from unity, the true singularity is difficult to separate from the analytic tem per a ture term. The coefficient D, does not even have to be much larger that Do for the analytic term to dominate the entire range accessible to experimentation. The latter causes the difficulty in observing the 1- a singularity for nonmetallic fluids. The inset in Fig.15 shows high precision experiments for Ne, N2, C2H4 [53] and SF6 [55]. It is evident that the diameter anomalies in these fluids extend on -ly over a very small portion of the temperature

高圧力の科学 と技術Vol.3,　 No.2(1994)

177

range close to the critical point.

Analysis of these data [56] has led to the suggestion that many - body interactions lead to the anomalous term

(ƒ¢T/Tc)1-ƒ¿

in these fluids. In particular, it is believed that

the symmetry - breaking present in these fluids

due to many body dispersion forces may be un -derstood in terms of a thermodynamic state de -

pendent effective pair interaction. Consequently, there seems to be a natural connection between

this explanation and the observation of very large

amplitudes of the diameter anomalies in fluid metals where the occurrence of the metal - non -metal transition implies a strong variation of the

interparticle interaction. It is evident from a glance at Fig.15 that the

diameter anomaly in the alkali metal rubidium is

so strong that it is seen over several decades in the

reduced temperature ĢT/Tc. It is certainly

tempting to speculate [57] that the strong depen -

dence on density of the effective interparticle po -

tentials in the range of the gradual metal - insula -

tor is responsible for the large amplitude of the

term

(ƒ¢T/Tc)1-ƒ¿

in cesium and rubidium.

This view, that the changes in the electronic

structure and the accompanying changes in the

interparticle interaction are responsible for the

coexistence curves asymmetries in fluid metals is supported by the diameter behaviour of mercury

which is also shown in an inset of Fig.15. Far be -low the critical temperature, the diameter has a

positive slope as is observed for molecular fluids and for the alkali metals. In this region of tern -

perature, when the density difference between the coexisting phases is large, the coexisting liquid is metallic (pd >_ 11 g cm-3), while the insulating va-

pour consists of atoms interacting through weak van der Waals forces. At higher temperatures,

where the liquid is in the electronic transition

range, the diameter actually slopes towards higher densities, opposite to the behaviour of

molecular fluids and the alkali metals. Close to

the critical point of mercury the behaviour of the

diameter is consistent with the term

D0(ƒ¢T/Tc)1-ƒ¿

with a positive Do. The competing variations of

the electronic structure of the coexisting liquid

and vapour phases of mercury with density and

temperature cause a strong wiggle at intermedi -

ate values of the reduced temperature.

Acknowledgment

The hospitality of Prof. Dr.Kenji Suzuki, Insti-tute for Materials Research, Tohoku University, is gratefully acknowledged.

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〔1993年12月20日 受 理 〕

高圧力の科学 と技術Vol.3,　 No.2(1994)