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A small angle neutron scattering investigation of theconcentration fluctuations in the solutions of potassium

in molten potassium bromideJ.F. Jal, P. Chieux, J. Dupuy

To cite this version:J.F. Jal, P. Chieux, J. Dupuy. A small angle neutron scattering investigation of the concentrationfluctuations in the solutions of potassium in molten potassium bromide. Journal de Physique, 1980,41 (7), pp.657-666. �10.1051/jphys:01980004107065700�. �jpa-00209292�

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A small angle neutron scattering investigation of the concentration fluctuationsin the solutions of potassium in molten potassium bromide

J. F. Jal (*) (**), P. Chieux (**), J. Dupuy (*) with the technical collaboration of J. P. Dupin (*)

(*) Département de Physique des Matériaux (~~), Université Lyon I, Villeurbanne, France(**) Institut Laue-Langevin, 156X, 38042 Grenoble Cedex, France

(Reçu le 17 décembre 1979, accepté le 13 mars 1980)

Résumé. 2014 En vue d’étudier les interactions possibles entre les fluctuations de concentration ou le phénomènede démixtion liquide-liquide et la transition non-métal métal que présentent des systèmes désordonnés tels queles solutions métaux sels, nous avons entrepris des mesures de diffusion centrale des neutrons dans une largegamme de température et de concentration autour du point critique liquide-liquide du système K dans KBr. Lesconditions expérimentales de régime critique étant respectées, on peut déterminer avec precision les indices cri-tiques 03B3 et 03BD. Tous les résultats obtenus permettent de conclure à un type de comportement Ising tridimensionnel,les fractions volumiques étant le bon paramètre d’ordre. La différence entre ce résultat et celui obtenu pour lesystème analogue que constituent les solutions de métaux dans l’ammoniac liquide où, dans la même gammede température réduite, les indices sont de type champ moyen, est interprétée à l’aide du critère de Ginzburg etdes valeurs respectives de la longueur caractéristique de fluctuation 03BE0. Cette longueur peut sans doute dépendreassez fortement du sel et du métal choisi. Dans le système K-KBr l’interaction entre les processus de localisationélectronique et les phénomènes de fluctuations de concentration semble en tout cas très faible sinon inexistante.

Abstract. 2014 In order to investigate the possible interaction between the concentration fluctuations related tothe liquid-liquid immiscibility and the non-metal to metal transition observed for the metals in molten salts sys-tems, we have undertaken a neutron small angle scattering study of the solutions of potassium in potassiumbromide over a rather large temperature and concentration range around the liquid-liquid critical point. Theexperimental conditions for the critical regime being satisfied, we have determined with accuracy the 03B3 and 03BD

critical indices. All our results are consistent with a tridimensional Ising behaviour for this system, the volumicfraction being the correct order parameter. This is different from the mean field behaviour obtained over thesame reduced temperature range for the analogous solutions of metals in liquid ammonia. We interpret thisdiscrepancy with the help of the Ginzburg criterion and the large difference in the value of the characteristiclength for the fluctuations, 03BE0. 03BE0 is probably dependent on the nature of the metal or the salt chosen. In any case,for K in KBr the interaction between electronic localization processes and concentration fluctuation is likely tobe very weak or non-existent.

J. Physique 41 (1980) 657-666 JUILLET 1980,

Classification

Physics Abstracts64.70J

Introduction. - A considerable amount of workhas already been devoted to disordered systemsundergoing transformation from non-metal to

metal [1, 2] with change of a thermodynamic variablesuch as composition, density or temperature. Amongstthose systems, we could list, by example, expandedliquid metals near the liquid-gas critical point [3],alloys of metals with the chalcogen elements (Te,Se, S) [2, 4], solutions of metals in liquid ammoniaor methylamine [5], solutions of metals in moltensalts [6]. However the detail of the interaction bet-

ween electronic and structural properties and moreprecisely the possibility of electron localization withinwell identified chemical structures or by disordereffects remains an area quite open to the investiga-tion. In particular, one still lacks structural infor-mations on cluster formation or on concentrationfluctuations in the intermediate range of concentra-tion or density where electronic localization occurs.It is known that liquid-liquid or liquid-gas phaseseparation is often observed in the above systems inthe vicinity of the electronic transition, but theexact extent of the possible interaction between theelectronic localization processes and the phase sepa-ration is not clarified. We have in the last few yearsundertaken an investigation by small angle neutron

(t) This work has received a financial support from’ the A.T.P,« matériaux ».

(tt) LA 172.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01980004107065700

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scattering, of the concentration fluctuations abovethe miscibility gap in solutions of metals in liquidammonia, and indeed the results were quite interesting[7, 24]. The Debye correlation length for the fluctua-tions was found significantly larger than the one

obtained for other metallic binary alloy systems witha liquid-liquid miscibility gap, such as the Li-Na

mixture [8]. And the critical exponents related to

those fluctuations were of mean field type up to verynear the critical point, which again is quite differentfrom most of the binary liquid mixtures investigated.We therefore decided to extend the study to the

solutions of metals in molten salts [6] which are verysimilar to the metal ammonia systems since in bothcases there exists a liquid-liquid miscibility gap inthe vicinity of the non-metal to metal transition

(see Fig. 1). Moreover the solutions of metals inmolten salts are rather simple from a theoretical ora computer simulation point ofview and their completestructure determination is possible from the isotopicsubstitution method in neutron scattering experi-ments as it has been achieved already for the puresalts [9] (1). The choice of the K in KBr system wasmade as to optimize the chances of a successfulsmall angle neutron scattering investigation and thiswas not obvious at all on the technical side, as weshall see in the next paragraph. Of course, one took

Fig. 1. - The potassium-potassium halide phase diagrams fromM. A. Bredig [6a].

(1) Recent neutron scattering experiments have revealed theexistence of very strong structural effects produced by the additionof small quantities of molten salt to the pure metal [10]. This mightbe interpreted by the formation of complex ions and is further

investigated. It shows that these systems are probably not as simpleas initially thought.

also into account all the existing informations onphase diagrams and electrical conductivity (whichare not that abundant [6]) in order to select a systemwhere the non-metal to metal transition occurs inthe vicinity of the liquid-liquid critical point. Finallya great care was taken to accumulate information onthe fluctuations as a function of temperature andconcentration on a quite large range around thecritical point. This was done in the hope of a detailedcomparison with the typical molecular liquid mixturesand with the disordered non-metal to metal transitionin metal ammonia.We will first introduce the small angle scattering

experiments and their analysis in the case of binarymixtures. In a second paragraph we will present ourresults on the determination of the critical tempera-ture, coexistence and spinodal curves, and criticalindices. These results are discussed in the final para-graph.

1. The small angle neutron scattering experimentsand their analysis. - 1.1 THE SMALL ANGLE NEU-TRON SCATTERING EXPERIMENTS, THEIR INTEREST. -

Critical fluctuations are generally observed by lightscattering. However the disordered systems under-going non-metal to metal transition have quasi-metallic reflectivity near their liquid-liquid consolutepoint, and container problems are often severe. Forthe metals in molten salts, by example, the containersmust stand the complementary corrosiveness ofalkali metals and halide ions as well as the highvapour pressure of the alkali metals. For these

practical considerations, neutron scattering is the

only adequate technique, since it allows sealed metalliccontainers to be used for experiments in transmissiongeometry. A full report has already been given [11]on the choice of the proper materials and on the

design and performance of fumace for small angleneutron scattering. Its main characteristics are a

molybdenum fumace and cell which scatter weaklyat low angles and are not sensibly affected by corro-sion problems. Temperature gradients are kept toless than 0.1 OC.The small angle neutron scattering offers moreover

some advantages of its own [14]. Since the contrastfor detecting the fluctuations depends on the diffe-rence of the neutron scattering lengths per unit

volume between the two terms of the fluctuation, onemay alter this contrast by isotopic substitution (itoften happens that isotopes have different neutroncross-sections). One could even, in principle, comple-tely separate the fluctuations of concentration fromthe fluctuations in density in binary systems. Anotherpoint of interest for critical study is the kç rangeaccessible in neutron experiments (k is the momen-

403C0tum transfer, k = 4;.1C sin 0, ç is the correlation lengthof the fluctuations). The conditions for critical

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scattering, i.e. kç> 1, are easily satisfied in practice.Most of our measurements were performed on theinstrument D17 at the Institut Laue-Langevin (Gre-noble), some on the D11 machine. On these smallangle spectrometers, one measures the scatteringintensity as a function of scattering angle withoutenergy analysis. The study is therefore strictly limitedto the static critical exponents and parameters.The use of a multidetector allows a quick coverage

of the k-range investigated [12]. An immediate

display of the results is available. Our investigationswere essentially performed at

The values obtained were between 5 A and 300 A,the highest values being limited by the thermal

gradient in the sample as well as by the technicaldifficulties [11] involved below 2 x 10-2 A-1.

1. 2 DATA ANALYSIS. - The first task one encoun-ters is to extract from the raw data the coherentneutron scattering signal of interest. This amountsto correct the raw data for the background, containerand furnace scattering and remove from the signalthe multiple and incoherent neutron scattering

where B is for background, C for container, F forfurnace, m for multiple. The container and fumacescatter at small angle but their scattering is confinedto low k values (k 10 - 2 A -1 ) and is weak as

compared to the coherent signal related to the concen-tration fluctuations. Details on these correctionshave been given elsewhere [11]. The multiple andincoherent scattering terms are k-independent to afirst approximation (2).The coherent signal itself is not simple since in the

case of binary systems it is the weighted sum ofthree partial structure factors corresponding to eachpair of atoms. These partials might be redefined suchas to be simply related at the thermodynamic limit(k = 0) to the fluctuations of density and fluctua-tions of concentration which are the only two inde-pendent fluctuating terms in binary systems. Thishas been done by Bhatia and Thomton [13] for

atoms of equal size. However, even in the case ofatoms of different sizes one may obtain a one to one

correspondence between the partial structure factorsand the two types of fluctuations at the thermody-namic limit. This is achieved [14] by expressing in

(2) It would be rather interesting to have an exact treatment ofthe k-dependence of those corrections terms. The inelasticity cor-rection which is ignored in our treatment might also present aweak k-dependence. Its effect should however be very small consi-dering the low k values and very short k-range of our experiment.Computer simulation of typical experimental situations withmodel scattering laws is in the way and should clarify these points.

the. Bhatia and Thomton formalism the concentra-tions in volume fractions (pi,

(where Vi are the partial molar volume Vi = ô VIDNIand ci or NIIN is the concentration of the species i,Y- Ci = 1; L Ni = N, the number of atoms in the

i i

system) and replacing the atomic scattering lengths bl,by bi i the scattering lengths per unit volume,

One thus defines three partials SYVV(k), SQQ(k) andSycp(k) with the following thermodynamic limits

the density fluctuation term

the concentration or volume fraction fluctuations

term, and

(N is the number of atoms and V the volume of thesystem. N = Nl + N2 for binary systems. KT is theisothermal compressibility, kB, the Boltzmann cons-tant, G is the Gibbs free energy). At k = 0 and verynear the thermodynamic limit, the cohérent scat-

tering signal is simply written as [14],

where Bi and The third

partial SvQ (k) is neglected. The SQQ(k) term is the* k-0

only relevant terms for our study.Near the liquid-liquid critical point, Tc, the concen-

tration fluctuations diverge, the density fluctuationson the other hand are considered to present at mosta very weak anomaly. Typically, a few degrees awayfrom T., Syy(O)/SQQ(O) is of the order of 10-’. Athigh temperature however the contribution from

SQQ becomes very small. As a matter of fact twoneutron small angle scattering spectra taken at

Tc + 150 oC and T, + 200 OC are identical and aftercorrection for background, container and fumace,are similar to pure molten salts results, with nodetectable small angle scattering [15]. The high tem-perature spectrum offers therefore an easy way toobtain the SVV(k) term. Moreover a simple substrac-tion of the high temperature spectrum from theother data not only corrects for Svv(k), but also forfurnace and container, incoherent scattering, multiple

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scattering and inelasticity effects. The procedure is

only limited by second order correction terms dueto the temperature variation, such as the densitychange, by example. It is, for the time being, themost accurate way to extract SQQ(k) or, more preci-sely, the not yet normalized term I,,(k) from theraw data.One is left now with the evaluation of SQQ(k)

or IQQk), the concentration fluctuation part of thecoherent small angle scattering signal. FollowingOrnstein and Zernike [16], the real space profile ofthe fluctuations is a function of the type

where x = 1/ç and ç is the correlation length forthe fluctuations. More exactly [17], one should writefor the correlation function near a critical point

where d is the dimensionality of the system (d = 3 inour case) and il is a critical exponent whose value isclose to zero. The small angle partial structure fac-tor SQQ(k) can therefore be obtained [18] by Fouriertransforming the above expression as in the wellknown compressibility equation for density fluctua-tions in the monatomic case

The value of the constant A is obtained at k = 0,

and therefore

The normalization of the data is not necessary forthe analysis. One must simply define a normalizationconstant R, such as

Near Tc, and at small k, the terms SQQ(k) andSQQ(0) become very large. The equation (7) maythen be simplified as,

We have verified that plots of IQQ-1 (k) versus k2for temperatures near Tc give no deviation fromlinearity within our experimental conditions of sta-

tistical accuracy and the k-range investigated. There-fore the exponent 17 might be dropped from ouranalysis. It is then easy to obtain a first approxima-tive value of the correlation length ç from the slope sand the origine of the IQQ-1 (k) versus k2 graphs,

The ç are subsequently introduced in the followingexact equation, deduced from (7) and (8) with n = 0

And the parameters R, SQQ(0) and ç are computedby a least square fit and itérative procédure. Ofcourse, R could also be obtained from a calibrationwith a vanadium standard, but normalization expe-riments are quite delicate. The refinement has littleeffect on ç but is more effective on IQQ(0), especiallyat high temperature. The figure 2 displays a seriesof (I’(k»- 1 versus k2 plots, where

There is no observable deviation from linearity fortemperatures near Tc. The correction term intro-duced in (12) to define l’(k) brings the slope of thehigh temperature curves (which would otherwise besteeper) practically parallel to the low temperatureones.

Fig. 2. - A typical series of Ornstein-Zernike plots. The inten-sities have been corrected as in equation (12). As shown in the upperleft corner there is no Small Angle Scattering at high temperature.

661

2. Expérimental results. - 2.1 DETERMINATIONOF THE CRITICAL TEMPERATURE Tc, THE COEXISTENCECURVE AND THE SPINODAL. - Near the critical point,(a2 GlÔ(p2 )TPN tends to zero and SQQ(0) diverges. Aswe have -seen, the small angle signal becomes thenvery large and is correctly described by the equa-tions (9) and (10), with il = 0. The value IQQ(0) istherefore easily obtained from the data [19] and isa good test of the proximity of Tc. We represent onfigure 3 the value of (IQQ(0))-1 as a function of tempe-rature for the concentrations 44 % and 40 % of Kin KBr. The two series of data have not been nor-malized and no attention should be paid to the dif-ference of steepness between the curves. The criticaltemperature as obtained from the 44 % sample is

Fig. 3. - Determination of the critical temperature from the

extrapolated I-1(0) signal.

Fig. 4. - Determination of the phase separation temperature TDand pseudo-spinodal temperature Tsp from the I-’(0) signal.

Measurements above and below T,, were used forthis determination. In the case of the 40 % samplephase separation seems to occur slightly before

(IQQ(0))-1 reaches zero but we still are very close tothe critical concentration. On figure 4, the samecurves are displayed at a different scale and fordifferent concentrations. One notices there, for eachconcentration away from critical, the temperaturefor phase separation, TD, and the extrapolated approxi-mate value for the spinodal, Tsp.An alternative approach to determine the coexis-

tence curve is the correlation length representation.In the vicinity of Tc, the correlation lengths are

obtained from the same simplified expressions (seeequations (9) and (10) as for IQQ(0), but generallyspeaking they are obtained from equation (11)). Thefigure 5 displays the correlation lengths ç versus

temperature for different concentrations. A strikingfeature is that independently of sample concentra-tion, all the value obtained below Tc, i.e. along thecoexistence line, are identical. This effect, which aswe shall see later is probably fortuitous, is however

very helpful for locking the temperature scales ofall our experiments on the 44 % run taken as a refe-rence (3). The accuracy of the coexistence curve

determination is thus increased by a factor of five

Fig. 5. - Temperature dependence of the correlation lengthfor sample concentration around the critical composition, at

T > Tc and T Tc.

(3) The experiments having been performed months apart, it isnecessary to have some internal check of the temperature scales.In all cases, but one (the 52 % experiment, where a two degreesshift of the scale was detected), the scale drifts were only of a fewtenths of a degree at most.

662

and is evaluated to ± 0.1 oC, which corresponds tothe precision of Bredig’s [6a] phase diagram to whichit agrees within the errors.The ç-representation offers also an interesting way

to reach the spinodal curve which is not accessible

experimentally in the case of liquids. We have dis-played on figure 6 the ç values as iso-ç lines (lines ofequal correlation length) on the concentration tem-perature diagram. One notices [21] that near Tc onemay generate the iso-03BE one from another by a simpletranslation. Of course, one may in the same manner

generate the spinodal line defined [20] as the iso-çline for ç = 00. The precision on the spinodal tempe-ratures thus obtained is estimated at ± 0.5 OC forthe concentrations the furthest away from critical.

Fig. 6. - Phase diagram and iso-ç lines for the K in KBr system.The coexistence curve as determined from the small angle scatteringexperiments agrees with the data of figure 1. The pseudo-spinodalline (see text) has also been drawn.

Finally the coexistence and spinodal curves indicatea critical concentration slightly different from 44 %say 43 % ± 0.5 %. (This does not affect the abovequoted error for the Tc value neither the precisiongiven for the critical exponents in the next paragraph.)The values obtained for the coexistence curve and

spinodal are summarized in the table 1.

2.2 THE CRITICAL EXPONENTS AND PARAMETERS. -

a) Along the critical isochore. - Several critical expo-nents and parameters [17b], [19] are directly obtainedfrom our small angle measurements. Along the cri-

tical isochore, the temperature dependence of thecorrelation length ç is described by a, power lawwith exponent v,

and the temperature dependence of the thermodyna-mic limit l’(0) (see equation (12)) is described by,

(The constants Ço and A being characteristics of thesystem.)These power laws describe the experimental data

over a large temperature range, but a correct valueof the exponents is reached only at the limit e - 0,i.e. if data are accumulated very near Tc and if Tcis accurately determined. This is self explanatory onfigures 3 and 4 where one sees that Tc is not correctly

Fig. 7. - a) Different temperatures and k-range conditions of oursmall angle neutron scattering experiments ; b) Schematic repre-sentation of the temperature and k-range used for data analysiswhile following the kç > 1 criterion for critical regime.

Table 1.

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obtained from an extrapolation of the high tempe-rature data. (As we have seen previously the relativeaccuracy of the Tc determination in our experimentshas been evaluated to ± 1 x 10-4.)Moreover, other constraints have to be imposed

on the data analysis. Modem theories of criticalphenomena [22] distinguish two different regimesfor the fluctuations, the hydrodynamic regime wherek ç 1 and the critical regime where k ç > 1. In ourcase, considering the momentum transfer range inves-tigated (2 x 10-’ A -1 k 0.1 A -1 ) one may bein either of the two regimes depending on the ç value.This is displayed on figures 7 which represent theexperimental k-range and reduced temperature range e.One notices that different series of experiments havebeen performed, always with k in = 0.029 A - 1 butwith kmax at either 0.068 or 0.103 A-1. At each tem-perature one takes into account the data from thefull k-range studied in order to obtain a first estimateof ç, this allows to construct the k ç = 1 line (Fig. 7a).The analysis of the data is then repeated, while

restricting the k-range at each temperature to the

k ç > 1 condition. Of course, as seen from the shadedarea in figure 7b, this means that low ç values (hightemperature) cannot be included in the treatment ifwe do not extend the k-range on the high k-side. Inpractice, for kmaX = U.103 A-1 one is limited to ç= 15 A.Moreover the data collected in the high k-range(0.068 k 0.103 A -1 ) are less accurate since theyare statistically less precise (low intensity) and subjectto systematic errors such as introduced by densitydependent terms in the corrections. The table IIsummarizes (4) the values for the v index obtainedfor specified conditions on k and e.

Table II.

(*) gmax is limited in the case of the k03BE > 1 condition at valuessuch as to obtain a k- range sufficient to compute an accu rate 03BE.

It tums out that the effect of the k03BE > 1 conditionis not sensible at very low k values (compare b)andb’) in table II) and is only detectable if we take intoaccount the larger k values (compare a) and a’) intable II). Since, as we have said, larger k values intro-

(4) All the numerical values of table II have been obtained forthe 40 % concentration with a Tsp value 0.3 OC below Tc and deter-mined to ± 0.1 °C. These values are identical within the quotederrors to the values obtained at 44 %.

duce probably more systematic errors, one must

be cautious. We therefore choose the final value of the

exponent v, half-way between the condition a) and a’).The same conditions of analysis cannot be appliedin the case of the exponent y since it is obtained atk = 0. It is clear however that the precision of theextrapolation to k = 0 will be the best if the experi-ment is extended to very low k.One has also investigated the effect of the tempera-

ture range on the analysis. As long as the critical

temperature Tc is correctly defined, it is interestingto note that even the data taken far from Tc, fit wellthe same power law. There- is however a slight ten-dency to deviation (see d), c’), d’), in table II) whichgives an idea of the order of magnitude of the syste-matic errors introduced at high temperature.The table III summarizes our results, including the

value of the parameter 03BEo.Table III.

The quoted errors include the uncertainty intro-duced by the conditions of the analysis (kç > 1

criterion for v) and are larger than those based on astrict least square fit. Systematic errors such as com-ing from the high temperature data substraction pro-cedure, or the neglect of the temperature dependenceof the S., partial and any other weakly temperaturedependent effect, are not included. We believe howeverthat they should not exceed 3 % of the exponentvalues and become negligible if one works at tempera-tures near Tc only.

fi) At concentrations away from critical and alongthe coexistence curve. - We have investigated severalconcentrations different from the critical one, and thiseven for temperatures below Tc, along the coexistencecurve. An analysis of the data with a power lawdescription for T > TD is still adequate as long as oneintroduces a new definition [20] for the reduced

temperature e,

where T,,, is the spinodal temperature correspondingto the concentration studied. A coherent set of resultsis plausible if one considers the procedure by whichTsp is reached (see details in paragraph 2.1), but itstill has to be proved. As a matter of fact, all the y and vexponents obtained are identical within errors to

those found on the critical isochore. Of course, theresults obtained at the concentrations 40 % and 44 %are the most accurate since it is near the criticalconcentration that the smallest e (or largest 03BE) valuescan be reached (see Fig. 5). We must however empha-size that no detectable deviation from the quotedcritical law has been observed over the whole investi-

gated area of the phase diagram, which would undoub-

664

tedly have been the case if clustering or any large sizestructural anomaly had occurred at some concentra-tion.The figure 8 summarizes the data analysis of the

différent concentrations for the y and v exponentsdetermination. No attention should be paid to theordinates of the y plots since the intensities were,for the time being, not normalized.At temperatures along the coexistence line, all then

values are describing a unique curve (see Fig. 5),which we might express as [21],

One finds v’ = 0.64 + 0.02 and 03BEo’ = 0.9 Â + 0.2 Â.It seems therefore that the relation v = v’ is verified.These values are however possibly subject to largersystematic errors (than at temperature above Tc)since we have no control of the position of the meniscusneither of the volume ratio of the two phases in theneutron beam for these different concentrations.

Fig. 8. - Determination of the y and v critical indices. One has also

displayed the values for T Tc, i.e. the v’ index determinationfor the near critical 40 % concentration.

Finally, the description of the coexistence and

spinodal or iso-ç lines (see paragraph 2.1) as achievedby the experiments at different concentrations givesaccess to another critical exponent, fi, which is definedas

or

03A6 is the order parameter of the system, i.e. the para-meter which describes the diverging fluctuations at Tc.For binary liquid mixtures, it is the concentration.The equation (17) expresses the concentrations or

strictly speaking, the order parameters as measuredon the coexistence curve at a temperature TD as afunction of the reduced temperature E ; a is a constant.The equation (18) expresses the concentrations asmeasured on an iso-03BE line at a temperature T as afunction of the reduced temperature.

Tc03BE and 03A6 cç are defined for one iso-03BE as the criticalparameters Tc and 0,,, are defined for the coexistencecurve. They are obtained from the analytical expres-sion of the iso-03BE lines.The figure 9 displays the coexistence curve and

iso-03BE lines as expressed in mole fraction units. All thecurves have been obtained by polynomial fit of theexisting data. A unique rectilinear ,diameter may betraced for all the iso-03BE lines and is, within errors,aligned with the rectilinear diameter of the coexistencecurve. It is obvious however, that the curves are notsymmetrical with respect to an iso-concentration line.Onç might wonder then if the experimentally deter-nfined volume fractions would symmetrize the repre-sentation with respect to the critical concentration,since the volume fractions should be the appropriateorder parameter in the case of binary mixtures [23, 24].The question is of interest since the y and v exponentsmust, strictly speaking, be measured for binarysystems, along iso-0 lines, 0 being the order parameterwhich is also required for a correct measurement ofthe exponent 3.

Fig. 9. - The iso-03BE lines as expressed in mole fraction units.

The density measurements which are necessaryfor the volume fractions determination are not at all

easy to be made in those systems with high alkali

665

vapour pressure. Preliminary data [28] at a 2 % accu-racy show the existence of a strong negative volumeexcess in the vicinity of the miscibility gap. Thesedata significantly straighten up the tilt of the rectilineardiameter and give the best agreement between the f3value obtained from equations (17) and (18) and whatis expected from the scaling laws [27], consideringwhat hasBbeen obtained already for y and v, i.e.

f3 = 0.30 ± 0.05. (An analysis based on the molefraction representation gave fi = 0.40 ± 0.05.)One should not also that the relation [21] ]

(where a’ /b’=_ (b/a)1/B, a, band P being parametersand exponent of the equations of the coexistence curveand iso-03BE line for 03BE = 50 A (17) and (18) is verifiedwithin errors by the previously quoted values for thecharacteristic correlation lengths of the system Çoand 03BE’ 0 measured respectively above and below Tc,and by the quoted value for the exponent v. The

result is independent of the choice of representation(mole fraction, volume fraction) for the coexistencecurve and iso-03BE

3. Discussion. - From all the above results we

may unambiguously conclude for a tridimensionalIsing behaviour of the critical parameters and expo-nents for the K in KBr system. This behaviour isfollowed by the data over a quite large temperaturerange (3 x 10-4 e or (T - Tc)Tc 7 x 10-2).These results are similar to the best data obtained formolecular liquid mixtures [23]. In this respect, thenon-metal to metal or Mott transition displayed bythe metals in molten salts does not introduce anypeculiar critical behaviour for the phase separation.This is different from the Metal Ammonia systemwhere, over a similar temperature range, mean fieldexponents were obtained for the liquid-liquid phaseseparation. The mean field description is however

always inadequate at temperatures very close to Tc’The Ginzburg criterion which gives some indicationabout the temperature range AT near Tc where themean field will fail gives for Metal Ammonia a ATvalue of a few degrees which is consistent with theexperimental results showing deviation from meanfield near Tc [24]. In the case of metal in moltensalts, a tentative computation gives a AT of the orderof 104 degrees, which again means that a typicaltridimensional Ising behaviour should be observedeven for rather large e values. A fundamental diffe-rence between the two systems is the value of thecharacteristic length Ço which is more than threetimes shorter for K in KBr than for Metal Ammonia.The potassium in potassium bromide is in this respect

similar to the metallic binary Li-Na mixture wherecritical tridimensional Ising has also been found withshort values for 03BEo (jo = 1.56 ± 0.20 À).We may now ask ourselves on the existence of an

interaction between the electronic transport propertiesand the concentration fluctuations for K in KBr.Since the critical properties are not described by meanfield equations, one should observe some anomalyin the properties while approaching the critical point.In particular, as it has been shown for the Li-Namixture [25], [26], the temperature coefficient of theelectronic transport properties should be affected.

However, this has nothing to do with the non-metalto metal transition. As a matter of fact a closer lookat the electronic transport properties shows (Fig. 10)that the non-metal to metal transition as estimated

by Mott’s criterion occurs at a concentration of about30 % in potassium, i.e. more dilute than the critical(42 %) concentration for phase separation. Thismeans that most of the concentration fluctuations

appear here in a system which is already metallic.All these features indicate that there is no strongcoupling and may be no coupling at all betweenelectronic localization and concentration fluctuations.On the other hand structural anomalies such as

clustering which could enhance the electron localiza-tion have not been detected even in the more dilute

(30 %) concentration. One might then wonder ifthere are cases where a weak coupling between concen-tration fluctuations and electron localization exists.If one looks, for example, at the solutions of potassium

Fig. 10. - The electrical conductivity (from H. R. Bronstein,A. S. Dworking and M. A. Bredig, J. Chem. Phys. 34 (1961) 1843).The Mott’s [1, 2] different conduction regimes have been quoted.

666

in potassium fluoride one notices that the concentra-tion for the non-metal to metal transition is nearthe liquid-liquid critical concentration which is at

significantly lower concentration than for K in KBr.A critical concentration quite different from the molefraction value of 0.5 reveals a strong volume difl’e-rençe between the constituants, as is the case, onenotices, in Metal Ammonia systems. It would be

interesting to figure out if it also leads to a muchlarger Ço value and a tendency to mean field exponents.

Acknowledgments. - We would like to thankR. Serve and M. Roth for their help during the smallangle neutron measurements and G. Guiraud for histechnical collaboration.

References

[1] MOTT, N. F. and DAVIS, E. A., in Electronic Processes in noncrystalline materials (Clarendon Press) 1971.

[2] MOTT, N. F., in Metal insulator transitions (Taylor and FrancisLondon) 1974.

[3] a) See references in :HENSEL, F., in Liquid metals, Ed. Evans, R. and Greenwood,

D. A., Conference series no. 30, The Institute of Physics,Bristol and London (1976), p. 372.

b) CUSAK, N. E., in Expanded liquid metals, Ed. Friedmann,L. R., Tunstall, D. P., Proceedings of the 19th ScottishUniversities Summer School in Physics, NATO SummerSchool (1978), p. 455.

[4] ALEXANDER, M. N. and HOLCOMB, D. P., Rev. Mod. Phys. 40(1978)815.

[5] a) THOMPSON, J. C., Electrons in liquid ammonia (ClarendonPress) 1976.

b) Solutions Metal Ammoniac : propriétés physico-chimiques.Colloque Weyl I, Ed. Lepoutre, G., Sienko, M. J. (W. A.Benjamin) 1963.

c) Metal ammoniac solutions. Colloque Weyl II, Ed. Lagowsky,J. J., Sienko, M. J. (Butterworths, London) 1969.

d) Electron in fluids. Colloque Weyl III, Ed. Jortner, J.,Kestner, N. R. (Springer Verlag) 1973.

[6] a) BREDIG, M. A., in Molten salt chemistry, Ed. Blander,M. (Wiley Interscience, New York) 1964.

b) NACHTRIEB, N. H., in Advances in chemical physics, vol. 31Non Simple Liquids, Ed. Prigogine, I., Rice, S. A. (1975).

c) WARREN, W. V., in Advances in molten salts chemistry,vol. 4, Ed. Braunstein, J., Mamanton, G., Smith, G. P.(Plenum Press) 1979.

[7] CHIEUX, P., J. Phys. Chem. 79 (1975) 2891.[8] Wu, E. S. and BRUMBERGER, H., Phys. Lett. 53A (1975) 475.[9] DERRIEN, J., DUPUY, J., J. Physique 36 (1975) 191.

EDWARDS, F. G., ENDERBY, J. E., RowE, R. A., PAGE, D. I.,J. Phys. C 8 (1975) 3483.

MITCHELL, E. W., PONCET, P. F. J., STEWART, R. J. Philos.Mag. 34 (1976) 721.

[10] JAL, J. F., CHIEUX, P., DUPUY, J., unpublished.[11] JAL, J. F., GUIRAUD, G., CHIEUX, P., DUPUY, J., J. Phys. E 10

(1977) 977.[12] IBEL, K., J. Appl. Crystallogr. 9 (1976) 296.[13] BHATIA, A. B., THORNTON, D. E., Phys. Rev. B 2 (1970) 3004.[14] DAMAY, P., CHIEUX, P., Z. Naturforsch. A 34 (1979) 804.

[15] JAL, J. F., CHIEUX, P., DUPUY, J., Ber. Bunsenges. Phys. Chem.(1976) 820.

[16] ORNSTEIN, L. S., ZERNIKE, F., Physik. Z.19 (1918) 134.[17] a) FISHER, M. F., J. Math. Phys. 5 (1964) 944.

b) STANLEY, H. E., in Introduction to phase transitions andcritical phenomena (Clarendon Press, Oxford) 1971.

[18] EGELSTAFF, P. A., WIGNALL, G. D., AERE Report R. 5627(1967).

[19] JAL, J. F., CHIEUX, P., DUPUY, J., J. Appl. Crystallogr. 11

(1971) 610.[20] CHU, B., SHOENES, F. J., FISHER, M. E., Phys. Rev. 185 (1969)

219.

[21] DAMAY, P., CHIEUX, P., to appear in J. Chim. Physique.[22] a) HOHENBERG, P. in Microscopic Structure and Dynamic of

Liquids, Ed. Dupuy, J. and Dianoux, J. (Plenum Press),NATO Summer School (1977).

b) HALPERIN, B. I., HOHENBERG, P. C., Phys. Rev. Lett. 9(1967) 700.

[23] GREER, S., Phys. Rev. A 14 (1976) 1770.[24] CHIEUX, P., DAMAY, P., DUPUY, J., JAL, J. F., Fifth Intern.

Conf. on Excess Electrons and Metal Ammonia Solutions

Colloque Veyl IV (1979) to be published in J. Phys. Chem.[25] SCHURMANN, H. K., PARKS, R. D., Phys. Rev. Lett. 27 (1971)

1790.

[26] RUPPERSBERG, H., KNOLL, W., Z. Naturforsch. 32 (1977) 1374.[27] STEIN, A., ALLEN, G. F., J. Phys. Chem. 2 (1973) 443.[28] KADANOFF, L., J. Stat. Phys. 15 (1976) 263.