ia inorganic chemistry ht 2017 solid state chemistry · ia inorganic chemistry st anne’s and...

IA Inorganic ChemistrySt Anne’s and Oriel Colleges HT 2017

Solid State Chemistry

Suggested reading:

• Solid State Chemistry, Smart & Moore

• Solid State Chemistry and its Applications, West

• Chemical Bonding in Solids, Burdett

• Band Theory and Electronic Properties of Solids, Singleton

• Introduction to Solid State Physics, Kittel (not for the faint-hearted)

Questions:

(i) Attached at the end of this tute sheet is an important research paper discussing the conductivitytransition in mercury vapour. Summarise what you understand about the Mott transitionand see if you can understand the key aspects of this paper in context of this transition.MnO is often cited as an example of a Mott insulator (what does this mean?). To whatextent can your understanding of conductivity in mercury, and of the Mott transition itself,explain the prediction that MnO should be metallic at high pressures (see e.g. Fig. ?? inhttps://arxiv.org/pdf/cond-mat/0605430v2.pdf).

(ii) Polyacetylene is a one dimensional chain of covalently connected C atoms. Each atom possessesone unhybridised p orbital perpendicular to the chain with a formal occupancy of one electron.

(a) Using a molecular-orbital type approach, explain why polyacetylene might be expectedto be metallic if the C–C bond lengths are all identical. As part of your answer, derive acrude band-structure picture for polyacetylene.

(b) Sketch the molecular orbitals that correspond to the lowest- and highest-energy MOs ofthe ⇡ band in polyacetylene. How many nodes per C atom do these two MOs contain?

(c) It is known that not all C–C bond lengths are equivalent in polyacetylene, but that shortand long bonds alternate along the chain. Sketch what you would expect the HOMO ofthis chain to look like. How many nodes per C atom?

(d) On your diagram of the band structure of undistorted polyacetylene, suggest the energyof the MO that corresponds to the HOMO in distorted polyacetylene.

(e) Explain the driving force for the distortion mechanism, and hence why polyacetylene isnot metallic.

IA Inorganic ChemistrySt Anne’s and Oriel Colleges HT 2017

(iii) The following transition metal dioxides all adopt rutile-related structures. Draw a generalisedband structure diagram of relevance to conductivity in the rutile family, relating this diagramto the crystal structure. For each of the compounds below, explain why they show a greatrange of electronic properties at ambient temperatures.

TiO2 InsulatorVO2 Metal above 340K; insulator below 340KCrO2 Ferromagnetic metalMnO2 InsulatorNbO2 InsulatorRuO2 MetalSnO2 Insulator

[Note that NbO2 and the low-temperature form of VO2 have distorted variants of the rutile

structure.]

(iv) Comment briefly on the electronic properties of the following compounds.

(a) NbS2

(b) TTF:TCNQ

(c) TiO

(d) C8K

(e) K2Pt(CN)4Cl0.32·3H2O

(f) GdO

(g) GaN

(v) (a) Explain why Ca(s) might not necessarily be expected to be metallic.

(b) Why is Ca(s) metallic?

IA Inorganic ChemistrySt Anne’s and Oriel Colleges HT 2017

Past paper questions (2014 IA Examination):

XCHA 2714

3

2. Solid-state chemistry

Answer ALL parts.

(a) Identify the number of lattice points per unit cell in a ‘primitive’, ‘body-centred’ and

‘face-centred’ cubic lattice. [2]

(b) The X-ray powder patterns of CsCl and CsI (both cubic compounds) contain reflections

with the following d-spacings (Å):

CsCl Csl

d 1/d2 d 1/d2

4.120 0.0589 3.230 0.0959

2.917 0.1175 2.284 0.1917

2.380 0.1765 1.865 0.2875

2.062 0.2352 1.615 0.3834

1.844 0.2941 1.445 0.4789

1.683 0.3535 1.319 0.5748

1.457 0.4711 1.221 0.6708

Index the two diffraction patterns, identify the Bravais lattice types and calculate the unit-

cell parameters. Account for the differences between the powder patterns of CsI and

CsCl. [5]

(c) Briefly discuss the similarities and differences in the electrical conductivity of the

following pairs of solids. In each case explain how the electrical conductivity will change

(if at all) on increasing the temperature of each solid.

i) WO3 and ReO3;

ii) VO and MnO;

iii) C (diamond) and Si (monocrystalline). [8]

Turn over (2016 Examination):

12

8. Answer ALL of parts (a) – (c).

(a) Define what is meant by the term ‘face-centred lattice’. [1]

(b) The X-ray powder diffraction pattern of KBr contains reflections with the

following d-spacings (Å):

3.8094 3.2990 2.3327 1.9894 1.9047 1.6495 1.5137

i) Index the diffraction pattern, identify the Bravais type and calculate the

unit cell parameter, a0. Propose a crystal structure for KBr. [6]

ii) Given that the ionic radius of K+ is 1.52 Å, determine whether the Br– ions

are in contact. [1]

(c) i) Define what is meant by the terms ‘band gap’, ‘Fermi level’, and ‘Mott-

Hubbard criterion’. [3]

ii) Explain why TiO2 and SrS are insulators whereas TiS2 and CdS are

semiconductors. [2]

iii) Rationalise the observation that VO2 is an insulator below 343 K whereas

CrO2 is metallic at ambient temperature. [2]

End of Paper

A10396W1

(2016 Examination):

A10396W1 10

8. Answer ALL parts (a) − (c).

(a) Explain what is meant in crystallography by the terms Primitive Bravais lattice and Face Centred Bravais lattice. [3]

(b) At 1000 °C and 25 °C pure elemental iron produces X-ray diffraction patterns with peaks at the following sets of d-spacings (given in Å). Note that reflections with d-spacings smaller than 0.8 Å were outside the measuring range of the instrument. In a separate experiment, the change in density of iron when heated from 25 °C to 1000 °C was found to be less than 2%.

1000 °C 25°C

d spacing (Å) 1/d2 (Å–2) d spacing (Å) 1/d2 (Å–2)

2.1126 0.2241 2.0407 0.2401

1.8296 0.2987 1.4430 0.4803

1.2937 0.5975 1.1782 0.7204

1.1033 0.8215 1.0204 0.9604

1.0563 0.8962 0.9126 1.2007

0.9148 1.1949 0.8331 1.4408

0.8394 1.4193

0.8183 1.4934

(i) Consider the ways in which the two diffraction patterns may be indexed using cubic unit cells. Identify possible Bravais lattice types, and determine the cell constants for each possible indexing. [5]

(ii) Suggest and sketch crystal structures for the two forms of iron which are compatible with your indexing schemes and with the density measurement. [4]

(c) Describe qualitatively why the room temperature phase of metallic elemental iron is ferromagnetic. [3]

[1 Å = 10�10 m = 100 pm]

�����������������������[End of paper]���������������������

REVIEWS OF MODERN PHYSICS VoLUME 40, NUMBER 4 0CTOB ER 1968

V. ;eta'. .-'.8'onmeta. . ..'ransition in:3ense 'V!Iercury VaporF. HENSEL, E. U. FRANCKIrtstitlt fear physihalische Chemic ttrtd Elehtrcchemie, Ursisersitat Xarlsrghe, Germarsy

This article presents the results of conductivity and density measurements of liquid, gaseous, and supercritical mercuryup to 1700'C and 2100 bar. Beyond the critical temperature of 1490'C the conductivity varies continuously from 10 4

to 102 0 ' cm ~ if the density is increased from 2 to 6 g/cm3. This variation is an example of nonmetal —metal transitionand is discussed accordingly. The activation energy of conductivity in the supercritical Quid decreases markedly withincreasing density and approaches very small values around 5 g/cm'. Conductivity data for dense gaseous mercury—cesium mixtures are given.

the critical temperature T, the homogeneous super-critical Quid can be compressed to liquid-like densitieswithout phase separation. Thus the mean interatomicdistance can be reduced sufficiently so that merging ofthe individual electronic levels of atoms into commonbands is to be expected.Until recent years very few experimental investiga-

tions of dense metal vapors had been made. In almostall cases the substance investigated was mercury, forreasons apparent from Table I. The critical tempera-tures of metals range from 1500'C to about 20 000'C.Experimental problems arise from the necessity toapply high pressures at very high temperatures, whereonly very few materials can be utilized.The erst attempt to determine the critical point of

mercury was made by Bender' ' in 1915.The densitiesof vapor and liquid (pt;u„;eand ps„)at saturation pres-sure were measured to 1380'C using sealed quartzcapsules. Figure 2 gives the density of gaseous andliquid mercury according to Bender, together withmore recent results. " "The diagram shows that the"law of rectilinear diameters" is approximately valid;that is, the temperature dependence of —',(pt;a„;a+p„,)is almost linear. Using this linearity Grosse"" wasable to estimate the critical temperatures of numerousother metals from existing experimental data on liquiddensities of these metals at lower temperatures. Recentmeasurements of the densities of the coexisting liquidand gaseous phases of the alkali metals by Dillonet al '~'s have provided more reliable numbers for thecritical data of these metals.

I. INTRODUCTIONSince the publication by Mott in 1961,on the transi-

tion to the metallic state, the relation between themetallic and thermodynamic properties of a substancehas received increased attention. ' ' Mott presented atreatment based on overlap considerations, whichproposed the existence of a critical separation at whicha change from metallic to nonmetallic behavior shouldoccur in a system of 1.ocalized centers.The experimental veridcation of this transition,

with diverse examples, is an interesting problem. Mostof the experimental evidence has been obtained byinvestigation of nonordered heavily doped solid semi-conductors. '' Concentrated metastable solutions ofalkali metals in liquid ammonia are another group ofnonordered systems well investigated in this respect. ~ 8All of these examples, however, contain two or morecomponents. It is therefore necessary to allow for theinIIuence of the dominant solid or liquid "solvent"on the behavior of the minority component whichexhibits the transition to be investigated.It is certainly desirable to study the transition with

a one-component system also. A unique possibility isthe observation of the appearance of metallic conduc-tivity in a metal vapor, which occurs if the vapordensity is continuously increased to sufBciently highvalues. Such continuous increase of density is onlypossible, however, if the formation of a liquid phase bycondensation is avoided. This means that the compres-sion of the vapor has to be done beyond the criticaltemperature which terminates the vapor pressurecurve. Figure 1 gives a schematic illustration of theliquid —gas phase behavior of a pure substance. Above

' N. F.Mott, Phil. Mag. 6, 287 (1961).2 W. Kohn, Phys. Rev. 133, 171 (1964},s J. Hubbard, Proc. Roy. Soc. (London) A276, 238 (1963).4 J. Hubbard, Proc. Roy. Soc. (London) A277, 237 (1964);

281, 401 (1964).' H. Fritzsche, J. Phys. Chem. Solids 6, 69 (1958).e H. Fritssche and M. Curevas, Phys. Rev. 119, 1238 (1960).7 Solutions 3fetal-Ammoniac Proprietes PhysicochAniques, 6,I.epoutre and M. J. Sienko, Eds. (W. A. Benjamin, Inc., New

York, 1964).D. S. Kyser and J. C. Thompson, J. Chem. Phys. 42, 3910

(1965).

e J. Bender, Physik Z. 16, 246 (1915).se J. Bender, Physik. Z. 19, 410 (1918)."D.R. Postill, R. G. Ross, and N. E. Cusack, Advan. Phys.

16, 493 (1967)."D. R. Postill, R. G. Ross, and N. E. Cusack, Phil. Mag. (tobe published) ."K.U. Franck and F. Hensel, Phys. Rev. 147, 109 (1966)."F.Hensel and E. U. Franck, Ber. Bunsenges, Phys. Chem.70, 1154 (1966).'~ A. V. Grosse, J. Inorg. Nucl. Chem. 22, 23 (1961).'e A. V. Grosse, Rev. Hautes Temp. Refractaires 3, 115 (1966).~I. G. Dillon, P. A. Nelson, and B. S. Swanson, Rev. Sci.

Instr. 37, 614 (1966).I. G. Dillon, P. A. Nelson, and B. S. Swanson, J. Chem.Phys. 44, 4229 (1966).

697

698 Rzvrzws oz MoozRN PHvsIcs ~ OcToazR 1968

super-criticalfluid

Tc

super-criticalfluid

from 1 to 1000 bar. Over this range the conductivitydecreases by about a factor 4 and the density changesby about 20% of its room-temperature value. Kikoinand Sechenkov have published two papers' "of meas-ured densities and electrical conductivities as a func-tion of temperature (0'-2000'C) and pressure (1—5000bars). The conductivity in this range varies from 10'to 10 ' 0 ' cm ' while the density changes from 13.5to 1.0 g/cm'.Hensel and Franck'3" have measured the electrical

conductivity and the density of liquid and super-critical mercury to temperatures of 1700'C and pres-sures to 2100 bars. Under these conditions the conduc-tivity varies from 10' to 10 4 Q ' cm ' and the densitydecreases from 13.5 to 1.5 g/cms.In the present paper the emphasis is on the inter-

mediate range of conductance between the nonmetallicand metallic states. Therefore, special consideration is

12 ~

Tc

FrG. 1. The liquid —gas equilibrium and the critical point CP,pressure-temperature (p—T) and density —temperature (p- T)diagrams.

ThaLE I. Critical temperatures and pressures of some metals.(The values for cesium and mercury have been measured, theother data have been estimated. )

Metal

Cs

Hg

Pb

Criticaltemperature('I)

2 0572 5731 7331 7531 .763

8 720

5 400

23 000

Criticalpressure(bar)

145

3501 6101 5201 5102 100850

y10 000

References

17, 18

17, 18192113, 1415

"F.Birch, Phys. Rev. 41, 641 (1932).

Birch" in 1931, was the 6rst to demonstrate experi-mentally the occurrence of metallic conduction in adense gas using supercritical mercury. He derivedcritical data for mercury from these conduction experi-ments which are in good agreement with recent results(viz. Table I). Unfortunately only five conductancevalues .and no densities could be determined at thattime. Since 1963 at least three groups have developednew techniques to investigate liquid and supercriticalQuid metals at high pressures.Postill, . Ross, and Cusack" " described measure-

ments of the density and electrical conductivity ofliquid mercury in the range from 20' to 1100'C and

10 ~ 0

g8 ~

' V

~~lO

4sA4 s

I

liquid + +gag2."

0 8ender (1918)Hensel and Franc% 09/5)Postill, Ross, and CUsack

C.P.

01500 1000 1500

Temperature{ oC )FIG. -2. Density p of liquid and gaseous mercury as a function

of temperature up to the critical point CP.

"I.K. Kikoin, A. P. Sechenkov, E. V. Gel'Man, M. M. Kor-sunski, and S. P. Naurzakov, Zh. Kksperim. i Teor. , Fiz. 49, 124(1965) LEnglish transl. : Soviet Phys.—JETP 22, 89 (1966)j.~' I. K. Kikoin and A. P. Sechenkov, Order of Lenin Institute

of Atomic Energy, Moshow (1967).

given to the properties of mercury at densities betweenabout 20% and 60% of the normal liquid density, thatis, at critical or subcritical pressures. This range is notcovered by the experiments of Kikoin et al. New dataare presented and the conductance is discussed as afunction of density and mean interatomic distance aswell as a function of temperature at constant densities.Activation energies are derived for the conductance ofmercury.

II. EXPERIMENTALIn order to study the conductivity and density of

mercury and cesium within a not too limited super-critical temperature range it is desirable to extend themeasurements to 1800' or 2000'C and to at least 2000

F. HENszz. AND E. U. FRANCE Metal-Sonmeta/ Transition in Dense Memlry Vapor 699

bar with mercury and to about 500 bar with cesium.Pure molybdenum and pure tungsten were found satis-factory materials for the cells to contain the super-critical metals. Thin tubes of sintered, nonporousalumina or thoria were used as insulators for the con-ductance experiments. Since the molybdenum andtungsten cells could not withstand high internal pres-sures at high temperatures, they were placed inside aninternally heated autoclave, using argon as the pres-surizing medium. The pressures of argon and liquidmetal were equilibrated outside the autoclave at lowtemperature within a stainless steel "buret. "Resistanceheaters surrounded the cell inside the autoclave.Figure 3 shows the assembly schematically. Details aredescribed elsewhere. "' "To obtain the conductivity, the potential diGer-

ences at various currents between two molybdenumelectrodes in the mercury were determined. Sincespecific resistivities of the Quid metal varj. ed from 10 'to 104 0 cm, according to temperature and pressure,several diferent shapes of cells and electrodes had tobe used. A similar ce11. made from tungsten and withinsulating tubes of thoria served for the cesiummeasurements.To determine the density of mercury the conductance

cell inside the autoclave was replaced by a molybdenumcylinder of 5 cm' internal volume, closed at one endand connected by a narrow capillary to the lower endof the buret, mentioned above, which was partly filledwith mercury. Since the mercury level in the buretcouM be determined by the resistance of a thin cen-trally located platinum wire, the amount of mercurywithin the heated molybdenum cell could be evaluated.Temperatures were measured by platinum —rhodiumor tungsten —rhenium thermocouples inserted into holesat diferent positions in the walls of the molybdenumand tungsten cells. Pressures were determined bycalibrated precision Bourdon gauges connected to theargon-filled branch of the assembly. Usually the pres-sure was held constant and the temperature changedslowly while the resistances were recorded. Cesium

Vacuumi~ ~i

MercuryI f-l H

Recorder'i r

l

Buret forI

r,Pressure Equi=

Molybdenum librationCell

Resistance Two- Chamber&/+7+&+ H t „--Compressor

AutoclaveTransformer

I

Argon

= Argon

Pump

Mercury

Fro. 3, Assembly for conductivity and density measurementsvrith mercury at high temperatures and pressures.

"R. Schmutzler, thesis, Karlsruhe (1967), Ber. Bunsenges.Phys. Chem. (ta be published).

1500-

Mercury1000- g rm ee MerCury

.1/ eli'

~~15

500-

0500 1000 1500 Q,5f Og

y

1.01/7 1Q ('I& ')

{a) (b)Pro. 4. (a) Vapor pressure curve of liquid mercury up to the

critical point. (b) Log P as a function of 1/T. ~=experimentalpoints, this work. $=values of Cailletet, Collardeau, and Riviere(1926).+=values of Douglas, Ball, and Ginnings (1951).

metal was purchased in evacuated glass ampules. Afterthe ampules were cracked under pure argon gas themetal was dropped into the cell, or for preparing amal-gams, into the mixing vessel. The concentrations of theamalgams were measured by weighing the amounts ofcesium and mercury. A detailed description of thepreparation of the amalgams is given elsewhere. "

III. RESULTS

B. Cond. uctivity of Mercury

Figure 6 gives isotherms of the specific conductivityof liquid, gaseous, and supercritical mercury as a func-tion of pressure. Each of the points is an average of fivediferent determinations of temperature and conduc-tivity. Only a part of the experimental points is in-dicated in the diagram. At slightly supercritical tem-peratures a continuous but very steep rise of conductance

"H. Renkert, thesis, Karlsruhe (1968).

A. Vapor Pressure Curves

At fixed pressures the temperature of the liquid metalin the cell was raised and the slowly increasing resist-ance observed. With subcritical pressures a discontinu-ous change to very large resistances was recorded atcertain temperatures, which was taken as indicationfor the replacement of the liquid by a gas phase in thecell. The respective pair of values for temperature andpressure were taken as points of the vapor pressurecurve. Figures 4 and 5 give the vapor pressure curvesfor mercury and cesium thus obtained. Boiling andcondensation temperatures at constant pressure co-incided within 1%. Above 1510+30bar, discontinuousresistance changes were no longer observed with mer-cury. This value was assumed to be the critical pressureI', with a corresponding critical temperature of 2;=1490&15'C. The values for cesium are T,=1750&30'C and P,=110~10bar."

700 REvIEws oz MQDERN PHYsIcs OcTQBER l968

p (bar)

o c.p.3r

80--(5--

60--

40--

log P FIG. 5. (a) Vapor pressure curveof liquid cesium up to the criticalpoint. (b) Log p as a function of1/ T. X=experimental points,this work. =values of Stoneet al.

20--

500 1000 5 G 7 810 /T('K)

with pressure is observed. Here every point is an aver-age of up to twenty independent determinations. Amongthe possible sources of error a conceivable differencein temperature between the thermocouple junctionsin the cell wall and the Quid metal within the cell wouldbe important. The agreement, however, of the vaporpressure curve of mercury determined here with earlier,reliable data reported in the literature'4 '~ which extendto 860'C for mercury and to 1200'C for cesium shouldrestrict the uncertainty of the temperature values ofthis work to &20'C.

C. Density

The variation of the molar volume of liquid andsupercritical mercury with pressure is demonstratedby the isotherms of Fig. 7. The dotted line gives theassumed values for the subcritical liquid at equilibriumvapor pressure. In order to interpret the conductancedata below about 10 ' 0 ' cm ' knowledge of themercury density below 3.5 g/cm' as a function oftemperature and pressure are desirable. For the purposeof the discussion following below these density datawere obtained by short extrapolations. They were basedon an empirical equation of state" in polynomial formderived from the supercritical isotherms of Fig. 7.

10

10

10

10

(Ohm Cm )-o- O'C

~ ~

——~~~- 1200'C~~ 1400'C~~ab

~~',~~+~~ c~~ o~~+~ 1520'C

0'C

580 'C

1600'C

1620 'C

1650 'C

NO'C

D. AmalgamsIt is interesting to study the inQuence of the small

addition of an element with a low ionization energy onthe conductivity of a dense metal vapor. Figure 8 shows'4 L. Cailletet, E. Collardeau, and C. A. Riviere, Compt. Rend.

130, 1585 (1900)."T.B.Douglas, A. F. Ball, and J. Ginnings, J. Res. Natl. Bur.Std. 46, 334 (1951)."C.F. Bonilla, D. L. Sawkney, and M. M. Makansi, Trans.Am. Soc. Metals SS, No. 3, 877 (1962).

2 J. P. Stone, C. T. Ewing, J. R. Spann, E. W. Steinkuller,D. D. Williams, rind R. R. Miller, J. Chem. Eng. Data 11, 315(1966l.

10—(bar)

I I I I I I I I I Il200 1400 1600 1800 2000 2200

Fre. 6. Electrical conductivity of mercury at subcritical andsupercritical conditions. ———Boundary of the liquid —gas two-phase region (a=1450'C, b=1480'C, c=1490'C, d=1500'C).

F. HENsEL AND E. U. FRANGK Metal-Nonmetal Transition in Dense 3IIerclry Vapor 701

the eGect of small additions of cesium on the conduc-tivity of supercritical mercury at 1600'C as a functionof the density of pure mercury. These data were ob-tained by using the same experimental arrangement andmethod that has been described for measuring theconductivity of pure mercury. The concentrations inFig. 8 are those of the liquid amalgams at 25'C. Withthe experimental conditions prevailing, this compositionis certainly not maintained in the Quid phase in thecell at 1600'C. The actual cesium concentration. mustbe lower. It is quite evident, however, that the additionof a small amount of cesium to supercritical mercurycan increase the conductance by a factor of 10 to 100.

5 (Qhmcm )

jo--

-210--C

-310--

&600 'Cdiatom%

C, = 02 atom%

IV. DISCUSSION

y (i

(cm'IMolj

20- ~O ~O-S~

(9l'cm )- l40oc

800 'C 12

1200oC1400OCacb -915100C1530oC

30-1550oC

15800C

40. 1610 4C

1630DC

16500C

1680oC

-3,5

l200 1400 l800 2000(bai )——

2200

FM. 7. Volume or density of mercury at subcritical and super-critical temperatures in dependence of the pressure:Boundary oi the hquid~as two-phase region (a= 1450'C,b= ~48O'C, c= I5OO'C),

The results presented in the last section demonstratethat by compression a metal vapor which is an insulatorat low pressure can be transformed continuously intoa dense, Quid, disordered system exhibiting metallicconductance. This requires a simultaneous change ofthe interatomic forces and the interatomic cohesion. Itis of interest to examine the macroscopic thermody-

J (g/cm')2,5 3.0

F&G. 8. The increase of the speci6c conductivity of super-critical mercury caused by small additions of cesium. Co=composition of the liquid amalgam at 25'C.

namic properties of the Quid in the same range ofconditions.Figures 4 and 5 show that for mercury and cesium,

over the entire range of temperature from their meltingpoints to their critical points, the logarithm of thesaturation vapor pressure is a linear function of thereciprocal temperature. This is the normal behaviorof most substances. From the slopes of the logarithmicplots the enthalpies of vaporization can be derived. Formercury the result is 14.1 kcal/mole and for cesium15.7 kcal/mole, which is in good agreement with datareported in the literature. "'8The "law of rectilinear diameters" referred to in the

introduction has until now been tested only for a fewmetals"'r " (Fig. 1) . Using it Cusack et al "estimateda critical density of 5.3 g/cm for mercury, which ishigher than the value of 4.2 g/cm' proposed earlierby the present authors" using the same data and an-other method. The value of 5.3 g/cm' gives a criticalcompressibility factor RT,/p, V,=2.5. This value islower than that for most other substances. It is sug-gested, however, that this low value may be a pecu-liarity of mercury, because the estimated values forthe alk.ali metals" are much higher.A comparison of the mercury conductivity and den-

sity isotherms of Figs. 6 and 8 demonstrates that largechanges in conductivity are primarily caused by varia-tions of molar volume. The data allow a separation ofthe eGects of temperature and volume expansion.Figure 9 shows the conductivity of supercritical mer-cury at 1550'C as a function of the density. (The upperscale gives the cube root of relative volume per atom. )A change in the density from about 6 to 2 g/cms de-creases the conductivity by 10. This large change incr is evidence of the metal to nonmetal transition for

702 REVIEWws oz MoDERN PHYslcs ~HYSICS ~ OCTOSER j.968

20 1S

dfohmcm l

10

1,2 (V/V, )"'

10

1550 oC

101

102

10

A

10 —- '2 4

[g /cm3l

. Specific conductiviFIG. 9. ~6 8

unction of densitivity 0- of supe

f (V/V)'" Vand 1 bar.

0=molar volas a

vo ume of liquid

present eexperiments with mercuol t}1

yp y. At a density of 10 g cm'ropsra idl .

an ree

ree path h 1 d f 1bqu de sit th

d t dt}10

e uctivitA odi p opos ton of Mott

er ensities it is ext is and

be e

i nsing

excitation of nens. is is w

ew con uc-y o

h o' d bdbed by o a of t

ersus the reciprocal ternogarithmic plot of a.

line. Fi 11 ho ~ h lo — '" ~ ga emperature sh oui

densities.e og o—1/T plot at different

The form of the curves obviouo viously gives evidse

' 'g eiavior. At the 1

s rang variation with te lowest densit

increemperature

'a

'y e slope of the curve ds ecreases

(Ohm'cmi)

10-

10-

10-2

10 "

10-

100 bar000 bar900 bar800 bar

10

00 bar10-

600 bar500 bar2„0g/cg

1/00

1300 bar

1500I

mercur atepen ence of the s ec' ivispeci c conductivit

sures and constant dens sties,

I

16001300

Quid divalent me s described bcurs in the vi

y ott in ~966

t d tdfb K h P'f o yn c considerati

pe to e held on a latt T

rom insulator to mice: he

pe avior as th

rough

At 1istance

e y an eneras energy.ll...d 1...1„„.. W

g should dec daexist. With an increa '

y e

b dbt d d of th

constant

t high supercritic 1 dperature depend

a an atli quid densities th tviy co s nt

bl b Z' ' hive y

~8N. F.M, '. aott, Phil. Maiquid metals. "In thi . n e

~9 N. F. M tt, Ady, |son, and J. M. Z'oman, FIG.

F. HENSEL AND E. U. I'RANCK Metal-Sonmeta/ Iransition in Dense 3ferclry VaPor

tag g

10

2ppp oK 1500o K

3,7 g/cm'

g3 g/cm~

10

2P g/cm'

100,50 0,52 0,54

I

0,56 10 1/T

FTG. 11. Temperature dependence of the specific conductivity ofmercury at constant densities.

indicating that the activation energy approaches zero.Unfortunately, because of experimental dif5culties withinsulation only a relatively small range of temperaturewas accessible. There are not yet enough data availableon the variation of conductivity with temperature atdensities above 4 g/cms to determine in what mannerand at what density the activation energy vanishes.Figure 12 shows the activation energy of expanded

mercury multiplied by a factor of 2 as a function of thedensity (in the upper scale vs the cube root of therelative volume). This energy is a measure of theeAective energy gap which appears when a disorderedmercury lattice is expanded. At zero density the activa-tion energy is half of the ionization energy of the isolatedmercury atom with a value of 10.4 eV. From this valuethe observed activation energy drops with decreasinginteratomic distance to 1.0 eV at a density of 3.5 g/cm'.This is in good agreement with an estimation of Mott"which gives an activation energy of 1.1 eV or an "eGec-tive ionization energy" of 2.2 eV for this density.An unexpected behavior is observed in the neighbor-

hood of the critical point. At temperatures near thecritical temperature and densities in the range of thecritical density the conductivity decreases with in-creasing temperature, at constant density. It is supposedthat density Quctuations associated with the criticalrange ir8uence the conductivity. In this region thederivatives (Bo/BT) p, (r)o/r)p) s, (BV/Bp) s, (r)V/~'T) phave large values indicating that errors in the tem-

1,6

5 ~

OO 4 5 6P (g/crn )

FIG. 12. "Eftective ionisation energy" AE of mercury as afunction of density p and {V/Vs)'ls. Vo=volume of liquid Hgat O'C and 1 bar.

perature and pressure significantly aGect the values ofconductivity and volume.

Discussion of Hensel and Franck's Payer

M. PoLr.AK {University of California, Riverside): My questionis not really directly related to the Mott transition. I want toinquire about the region where the conductivity decreases withincreasing temperature near the critical point. You made astatement that the change might be due to the decrease of thesize of the clusters with increasing temperature, and I wonderif you would like to comment on the following question. If thesize of the clusters increases then the average separation betweenthe c',usters will decrease. One would then oGhand think that theconductivity would decrease when the temperature decreases.E. U. FRANcK: I really cannot give an explanation for this

eGect. I tried to stress that the experiments in that range werecertainly difBcult because very slight variations or inaccuraciesof the temperature have a very pronounced eGect on the density.Actually to investigate that range one should construct a dif-ferent apparatus which is only used for that particular rangeand gives accurate results there. Some recent mass spectometryresults show that for one of the alkali metals the effective ioniza-tion energy {or the energy to free electrons from clusters) ismarkedly decreased if you proceed from monomers, to dimers,to trimers, and so on. Now, if we should have a considerableamount of trimers, or hexamers, or so, here in the critical rangethen they may contribute to the ease of setting the electronsfree. The size of these clusters should be very temperaturedependent and I should think that an increase of temperatureshould destroy these clusters and thus might have an e8ect inthis range. The energy of interaction of two mercury atoms ortwo potassium atoms in the gas phase is not very low. Theseare energies in the order of 10-20 heal/mole so it may well bethat duster formation should be very temperature dependent.But I should not give too much weight on these results beforethey have been verified.