Solid StateCommunications, Vol. 13, pp. 1385—1387,1973. PergamonPress. Printedin GreatBritain

THE DIELECTRIC CONSTANTOF LIQUID NOBLE METALS AND THEIR ALLOYS*t

GerardoCisneros

ESFM,Instituto PolitécnicoNacional,Mexico 14, D.F. Mexico

and

J.S.Helman

Dep. de Fisica. CIEA, Instituto PolitCcnicoNacional,Mexico14, D.F.,Mexico

(Received10Juli’ 1973 bi’ J.L. Olsen)

Thenearly-free-electronpart of the dielectric constantof liquid Cu, Ag andAu arecalculatedand theresultsare comparedwith availableexperimentaldata.The formulaeare generalizedfor thecaseof liquid alloys.

THE AVAILABILITY of pseudopotentialsfor liquid Table1. ValuesofkF, EF and Ed (Fermimomentum,noble metalst4allows now the calculationof the Fermi energyand topofthe d-bandof thesolidnearly-free-electronpart of their dielectric Constant respecnvely)in a.u., andcalculatedandempirical

resistivities(Pcaic and Pem~ respective/i’)in 4�—cm.(NFEDC).We havecalculatedtheNFEDC of liquid _____________________________________________Cu, Ag andAu for different pseudopotentials24 Cu Ag Auusing the formulaeof reference5, and theresultswere

*

comparedwith availableexperimentaldata.Thesame kF 0.6915 0.6097 0.6143formulaecanalsobe usedfor liquid alloys if IVQI2S(q) EF* 0.4782 0.3717 0.3774is substitutedby Ed + 0.324 0.077 0.193

~ P V~~ + ~ [S~V(q)— I] } Pcaic~ 22.6 30.0 55.0~ QM.vPcaic~ 22.1 19.4 38.5

where V~is the pseudopotentialof the alloy component Pemp~ 20.6 17.5 31.31.1, c~,its concentration,andS~(q)are the usual alloystructurefactors.6 * Reference2.

In the calculationwe haveusedtheempirical ~ SMITH N.V.,Phys.Rev. B3, 1862 (1971).structurefactorsgiven in reference7,which weredeterminedat 1125°Cfor Cu and 1050°Cfor Ag, and ~ Usingthe pseudopotentialsof Moriarty.23the pseudopotentialscalculatedby Moriarty23 andBorchi andDeGennaro4for comparison.8In TableI § Usingthepseudopotentialsof Borchi andDeGennaro.4we give therelevantparametersusedin thecalculationand thestaticresistivity obtainedwith eachmodel ~1Reference9.togetherwith its measuredvalue.9The resultsfor the

* Work partially supportedby ResearchCorporation. conductivitya and the real part of the dielectric con-stante,in theform i — e,versusthe photonenergy

t Thiswork wassubmittedto the ESFM, IPN, as a areshown in Figs. 1—3 togetherwith thosegivenbyprofessionalthesisby G.C. Drude’smodel involving theempiricalstatic resistivity.

1385

1386 DiELECTRICCONSTANT OFLIQUID NOBLE METALS Vol. 13,No. 9

0.OI0r~

~ F ~j c1’~° 1 0_0l0~ ~ ‘—ito

oMeratyb\ OMQruarly

(I4&ci.~ra Ci b.B~cht&0.G~maro b b8adi~DGmore

300

00~ ~ 2000.005 (I/~~) E a C Oruds.~00

,—

00 ‘ 00

0 r ~ F ~3 I 2 ‘3 40 I 2t~E3 4 5 6 1EF Er

~u ~V) p.~~

FiG. I. Dielectric constantof liquid Cu. a: solid line, FIG. 3. Dielectricconstantof liquid Au. a: solid line.Calculated:v, experimental.I — e: dotted line, calcu- calculated.I — e: dotted line, calculated.lated: EJ,experimental.

_______________________________ NFEconductivity obtainedwith the theoryofOOIC C

1 c~iIO reference5 usingany oneof the pseudopotentialsoM~arty

Ag b•Borchr &D.Gsrmorotestedhereare in reasonableagreementwith exper.iment. Thesevaluesare larger thanDrude’s by a factor

(I/1~Clcm) aL o\

~200 of the order of 2 in the range3—5 eV.

0.005

The resultsfor theNFEDC are expectedto bedirectly comparablewith experimentonly for photonenergieshw ~ E~-— Ed whereEF is the FermienergyandEd theenergycorrespondingto thetop of theJ d.band’.

0 2 3 ~FEd

5EF

~U2 IsV)It is importantto point out that thephaseshifts

FIG. 2. Dielectric constantof liquid Ag. a: solid line, techniqueusedto describetheelectronscatteringincalculated:G, experimental.I — e: dottedline, calcu. the calculationof transportpropertiesof liquidlated;~, experimental, metals9is not adequateherebecauseinelasticelectron

scatteringprocessesare important.Thereforeapseudopotentialapproximationmust be used.

Someavailableexperimentalvalues,determinedat1083°Cfor Cu and 1000°Cfor Ag. are also included.10 Acknowledgements— We want to thank Drs.W.

Baltensperger,R. Evans,H.J. Guntherodtand F.It is known thatDrude’s formulaincorporating Sánchez.Sinenciofor fruitful discussions.J.S.H.

d.c. parametersdoesnot fit properly measured acknowledgesa grant from theConsejoNacional deopticaldata.11The valuesfor the frequency-dependent Ciencia~‘ Tecnologia(CONACYT).

REFERENCES

1. HARRiSON W.A.,Ph~’s.Rev. 181. 1036(1969).

2. MORIARTY J.A.,Phvs.Rev. B 1.1363(1970).

3. MORIARTY J.A.,Phj’s. Rev. B 6. 1239 (1972).

Vol. 13,No.9 DIELECTRIC CONSTANTOF LIQUID NOBLE METALS 1387

4. BORCHI E. andDEGENNARO S.,Thys.Rev.B 5,4761(1972).

5. HELMAN J.S.andBALTENSPERGERW., Phys.kondens.Materie 5, 60(1966).Weusedformulae(2), (18),(19), (34) and(35)of this reference.In that paperthereis an error.To obtain thecorrectformulae thefirstterm within bracketsin the integrandof(18)hasto be deleted.Seeerratapublishedin thesamejournal,15,346(1973).

6. FABER I.E. andZIMAN J.M.,Phil. Mag. 11, 153(1965). ForbinaryalloysS~v(q)cansometimesbe obtainedempirically,or theycanbecalculatedtheoretically(ASHCROFTN.W. andLANGRETH D.C., Phys.Rev. 156.685 (1967)).

7. WAGNER C.N.J.,OCKEN H. andJOSHIM.L., Z. Naturf 20a,325 (1965).We usedfor Au the samestructurefactor reportedfor Ag.

8. In the caseof Moriarty’s pseudopotentials.forq < 2k,,.we usedthevaluesof reference2, andfor q> 2kFthoseof reference3 with the correspondingkF for the liquid. Borchi andDeGennarop~udopotentials4areadjustedto fit theempirical resistivity whenusedtogetherwith the correspondingtheoreticalstructurefactors.Sincewe useempiricalstructurefactors in all cases,a smalldiscrepancyis foundbetweenthe resistivity calcu-latedwith thesepseudopotentialsand themeasuredvalue(Table 1).

9. DREIRACH 0., EVANS R.. GUNTHERODTH.J. and KUNZI H.U..J.Phys. F.’ MetalPhj’s. 2. 709 (1972,1.

10. HODGSONJ.N..Phil. Mag. 5, 272 (1960).

11. FABER T.E..Introduction to theTheoryofLiquid Metals,CambridgeUniversity Press(1972). p. 390. Fabetwrites. ‘We havestill to relate the optical propertiesof liquid metalsto their measuredd.c. conductivities.andit is at this stagethat discrepanciesappear’.andlater in thesamepage,‘Thesediscrepanciesare still notunderstood’.We would like to point outthat a theoryof the dielectricconstantof liquid metalswhich includesthed.c. conductivity asthe limiting caseh~ 0 is given in reference5. Thistheor~accountsfor theabovementioneddiscrepancies.

Wir berechnenden Beitrag quasi-freierElektronenzur Dielektrizitatskonstantenvon flussigemCu, Ag und Au und vergleichendie Resultatemit verfugbarenexperimentellenDaten. Die Formeinwerdenfur den Fall flussigerVerbindungenverallgemeinert.