Solid StateCommunications,Vol. 20,PP.725—727,1976. PergamonPress. Printedin GreatBritain

THEORYOF STRAIN DERWAT1VE OFELECTRONIC DIELECTRIC CONSTANTOFIONIC CRYSTALS

O.P.Sharma,*H.P. Sharmaand Jal Shanker

Departmentof Physics,AgraCollege,Agra-282002,India

(Received16 July 1976 byMF. Collins)

A theoreticalmethodfor evaluatingthe strainderivativeof the electronicdielectricconstantof ionic crystalshasbeendeveloped.The analysispre-sentedhereis basedonthe shell model andtakesaccountof the exchangechargepolarizations.Valuesof strainderivativeof the electronicconstantdielectriccalculatedfor 6 alkali halidesandMgO showa remarkableagree-mentwith experimentaldataon photoelasticconstants.

A THEORETICALanalysisof strainderivativeof the R de = — 2ir(e + 2)2 [3a dal(2)

electronicdielectricconstant(e) yieldsvaluableinfor- � dR 9R2e [~

mationregardingthe photoelasticbehaviourof ionic wheresolids.1’2The experimentalvaluesof strainderivativeof a = a.

1.+ a_ (3)� canbe obtainedfrom thephotoelasticconstants(Pu Subscripts“+“ and“—“ representfor cationandanionandP12) usingthe following relation’ respectively.BursteinandSmith

8 havecalculated

R de — dc/dRby assumingthatpolarizabilitiesof ionsdonot—— — e(~u+ 2pj~) (I) changeunderhydrostaticpressurei.e. da/dR= 0. In6 dR — this caseequation(2) will bereducedto thefollowing

whereR is the interionicseparationchangingunder formhydrostaticpressure.The moststriking featureof the ~ = — (~+ 2)(�— 1) (4)experimentaldataon photoelasticconstantsavailable � dR �

formanyionic crystalsis that the experimentalvalues It shouldbementionedherethat thevaluesof dc/dRof dc/dR are negativefor alkali halidesandpositive for obtainedby BursteinandSmith from equation(4)differMgO. Therehavebeenseveraltheoreticalattempts35 from theexperimentalvalues.ForMgO eventhe sign ofto evaluatedc/dR andto explainthe distinct behaviour dc/dR predictedfrom equation(4)is wrong.This illus-of MgO but noneof themhasbeencapableforunder- stratesthefailure of the assumptionthata doesnotstandingthe photoelasticbehaviourof MgO. Recently changeundercompression.In factashouldchangeSharmaeta!.6 haveproposeda semiempiricalanalysis when thecrystalis compressedandits variationda/dRof the photoelasticeffectof ionic crystals.An interesting canbecalculatedin a mannerdescribedbelow.andremarkablefeatureof their analysisis the investi- Accordingto thetheoryof exchangechargepolar-gation of thefact thatunderhydrostaticcompression izationsandshell modelanalysisdevelopedby DO, onetheelectronicpolarizabilityof cationincreasesandthat cancorrelatethecrystallinestatepolarizabiities(ar)of aniondecreases.In thepresentpapera theoretical with the correspondingfree statevalues(af) asfollows:basishasbeenprovidedfor theselooseningandtighten-

= a~+ i+—I (5)ingeffectsof cationsandanionsrespectively.Wepro- { D 12posea methodbasedon the shell modelanalysisfor cal- n+ejculatingdc/dR. Theeffect of exchangechargepolariz-ationshasalsobeentakeninto accountfollowing the a~_= af — __~ 2 (6)

n..e]model of Dick andOverhauser7(hereafterto bereferredto as DO).

The strain derivativeof electronicdielectric con- whereD is the exchangechargepolarizationparameterstantcanbe obtainedby differentiatingthe Lorentz— andn÷e(n_e)representstheshell chargeof cation(anion).Lorenz relationwith respectto R. Thus,for NaCI Now differentiatingequations(5) and(6)with respectstructure,onefinds4 toR andrememberingthatD cx exp(—Rip)assuggested

by DO andHardy,9we obtain

* P.G.College,Ambah(M.P.) India. = — ~ (1 + P...) (7)dR n+ep \ n~e

725

726 THEORYOF STRAIN DERWATIVE Vol. 20,No.7

Table1. ValuesofD/n+e,D/n_e,da+/dR(in A2) alternatively be determinedfrom equations(5) and(6)andda_/dR(in A2) if we know the ratio ar/ar for the polarizabilities of ions.

D D da da A methodfor evaluatinga~/a~hasrecentlybeenpro-Crystal — — __± — posedby Jai Shanker andVerma’~ using the polariz-

fl+� fl_C dR dR ability-radiuscuberelation. Accordingto this relation

LiF 0.90 0.19 —0.35 + 1.12 we canwrite for a given ionNaC1 0.37 0.13 —0.60 +2.74 a / \3

KC1 0.19 0.13 —1.11 + 2.50 -I = (9)KBr 0.19 0.11 —1.12 +2.92 a~ \rfJIU 0.19 0.10 — 1.06 + 3.71 Valuesof a~/afcanbe estimatedfrom equation(9)RbC1 0.24 0A8 —2.69 + 3.52 usingfree ion radii (re) from Pauling’3andcrystalradiiMgO 0.58 0.26 — 0.52 + 4.47 (re,) correspondingto electrondensitymeasurements

citedby Sysio.’4 ForMgO, thecrystal radii of ionsweretakenfrom BisaryaandJai Shanker.’5Valuesof

Table2. Valuesof(R/e)(dc/dR)__________________________________________________ D/n.

1.eandDin_c are thencalculatedfrom equations(5)

Present Burstein Van Expen- and(6)andhavebeenlistedin Table1. InsertingtheseCrystal study & Smith

8 Vechten’8 mental valuesofD/n+c andD/n...e in equations(7) and(8)and

takingp from Tosi” anda~form Pauling’6we haveLiF —0.57 — 1.89 — 1.24 — 0.53k estimateddajdRandda_/dR(Table 1). In view ofNaC1 — 0.82 — 2.47 — 1.53 — 0.97a equation(3), we canwriteKC1 — 1.33 —2.25 — 1.49 — o.91~KBr —1.38 —2.51 —1.59 — 1.33a = ~ (10)KI — 1.47 —2.90 — 1.72 — 1.4s~ dR dR dRRbCl — 1.73 — 2.26 — 1.50 — 1 •441) Fromequations(2) and(10)wehaveevaluatedMgO + 0.69 —3.39 — 1.48 + 0.86’~ (R/c)(de/dR)for LiF, NaC1,KU, KBr, K!, RbCl and~

a Fromreference8. MgO takingvaluesof � from LowndesandMartin.’7b From reference18. Thesevaluesof (R/c)(dc/dR)havebeenlistedin Table2

Fromreference4. andcomparedwith experimentalvaluesobtainedfromthe photoelasticconstants.Forthe sakeof contrastwe

and da 2aD / D havealsoincludedin Table2 thevaluesof (R/e)(dc/dR)— = + ~ ~i——I (8) calculatedby BursteinandSmith8 andby VanVechten.’8dR n_ep ‘~ n_c,

It is interestingto observefromTable2 that theEquations(7) and(8) havebeenderivedconsideringa~ valuesof (R/c)(de/dR)calculatedin thepresentstudyandnc to beindependentofR and thereforewill not are verycloseto the experimentalvaluesin contrasttochangeunderthe effectof pressureasalsosuggestedby thoseobtainedby BursteinandSmith. In caseof MgOHavinga.’°p is therepulsivehardnessparameterandits wehaveobtaineda positivevalue for (R/c)(dc/dR)invaluesfor alkali halideshavebeenlistedby Tosi.11 It is conformitywith experimentaldata.In thecaseof alkalipredictedfrom equations(7) and(8) that (da+/dR)is halidesour valuesof (R/c)(dc/dR)are closerto thosenegativeandda_/dRispositive. This impliesthatpolar- obtainedby VanVechtenfrom Phillips—VanVechtenizability of cation increasesandthatof anion decreases dielectrictheory.’9 However,it shouldbementionedwhenthe crystalis compressed.In orderto evaluate thateventhemethodof VanVechtenis not capableofda+/dRandda_/dRfrom equations(7) and(8)weneed explainingthe positivevalueof (R/c)(de/dR)in MgO. Inthevaluesof D andne.DO evaluatedthemagnitudesof this respect,the methoddescribedin this papershowsathesequantitiesbut thevaluesobtainedby them are significant improvementas it yieldsnegativevaluesofsupposedto behighly uncertainfor the reasonsdis- (R/e)(de/dR)for alkali halidesanda positivevalue forcussedat lengthby Havinga.’°ValuesofD/ne can MgO in accordwith theexperimentalobservations.

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Vol. 20, No.7 THEORYOF STRAIN DERIVATWE 727

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