Solid StateCommunications,Vol. 33,pp.785—787.PergamonPressLtd. 1980.Printedin GreatBritain.

EVALUATION OFTHE PHOTOELASTICCONSTANTSOFIONIC CRYSTALS

Jai Shanker,T.S. Verma* andRamPrakash

Departmentof Physics,AgraCollege,Agra-282002,India

(Received7 September1979byM.F. Collins)

A theoreticalmodel,for evaluatingthephotoelasticconstantsof ioniccrystals,hasbeendevelopedby consideringtheshort rangerepulsiveinter-actionsbetweennearestandnext nearestneighbours,the vanderWaalspotentialsandthreebody interactionsaswell astaking into accounttheelectronicpolarizabilitiesof cationsandanions.The modelhasbeenappliedto calculatethe photoelasticconstantsof RbCl, RbBrandRbIcrystals.The resultsobtainedin thepresentstudyarein good agreementwith experimentaldata.

STUDIESON PHOTOELASTICCONSTANTSareof crystals.In fact, the electronicpolarizabilitiesof cationsconsiderableinterestin theopticsof crystals [1] and evenin alkali halides(exceptlithium halides)arequiteparticularly usefulin predictingthe distortioneffectsof significant andcannotbeneglectedas comparedto thosehigh powerlaserbeams[2]. RecentlySrinivasanand of anions.Srinivasan[3, 4] andGoyaletal. [5, 6] havedeveloped In the presentworkwe adopta bothion polarizable

theoreticalmethodsfor evaluatingthe photoelastic shell model [10] which considersthe electronicpolar-constantsof ionic crystals.However,their methodsare izabilitiesof cationsaswell asanions.The model para-inadequatefor thefollowing reasons.Srinivasanand meters,viz., shell chargesY1 andspringconstantsk.Srinivasanhavenot takeninto accountthe second couplingthe shellandcore,are determinedfrom theneighbourand vanderWaalsinteractionswhich arevery electronicpolarizabiitiesof ions usingtheexpressionsimportantin ionic crystalsasis evidentfrom therecent and themethoddescribedearlier [4, 111. The shortstudies[7, 8] . Moreover,theyhavederivedtheshort rangeforce parametersare derivedtheoreticallyfromrangeforceparametersempiricallywithout adoptingany interionicpotentialsinsteadof empiricalfitting fromsuitablepotentialform for non-centralinteractions.The crystaldatadoneby previousworkers [3—6]. We con-shortrangepotentialparametersthusderivedappearto sidertherepulsiveinteractionsbetweennearestneigh-beunphysicaland yield thephotoelasticconstantslargely boursandnextnearestneighbours,thevan derWaalsdeviatingfrom thoseobtainedexperimentally, dipole—dipoleand dipole—qwidrupolepotentials,and

The theoreticalmethodof Goyaletal. [5, 6] is thethreebodypotential.Thustheshort rangepotentialsequally inadequateasthat of SrinivasanandSrinivasan for nearestneighbours(~~)andnext nearestneighbours[3, 4J as far asthe treatmentof the shortrangeinter- (02) canbewritten asactionsis concerned.Goyal eta!. havealso not / + —

consideredthe secondneighbourandvanderWaals Ø~(r) = bí3+ - exp r1 r2 rinteractions.It shouldbekept in mind that thephoto- 2 ‘~

elasticconstantsare,by definition, thederivativesof + e aM! — — (1)electronicdielectric constantwith respectto various a r rlatticedisplacements.Therefore,thecalculationof I ~ I2r2photoelasticconstantsis much moresensitiveto the Ø2(r) = b /3~÷exp~—)+ f3_ - expshortrangeinteractionsthanthat of dielectricconstant /

itself [9]. Since the pictureof short rangeinteractions (_r\ — (c++ + c —\ — (d++ + d. - 2adoptedby previousworkers[3—6] is inadequate,their )( exp ~ ) ~ r~ ) ‘~ r

8resultsshowlargedeviationsfrom experimentaldata.Inadditionto theinadequatepictureof short rangeinter- whereb andp are repulsivestrengthandhardnesspara-actions,themethodof Goyaleta!, considersthe one ion meters,respectively.r

1 andr2 are ionic radii. f3~~are

polarizablemodelwhich is inappropriatefor mostof the Pauling’scoefficients[121. CU andd~are thevan derWaalsdipole—dipoleanddipole—quadrupolecoefficients.e is theelectroncharge,aM theMadelungconstantanda

* PermanentAddress:B.S.A.College,Mathura(U.P.), is the equilibriumnearestneighbourdistance.fis theIndia.

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786 EVALUATION OFTHE PHOTOELASTICCONSTANTSOF IONIC CRYSTALS Vol. 33,No.7

Table1. Calculatedvaluesof themodelparameters Y2 [3x, 3x] — Y~[3x, 4x]P2 =

Parameter RbCl RbBr RbIY?[4x, 4x] + Y~[3x, 3x] + 2Y1 Y2[3x, 4x]

Y1 —4.95 —4.95 —4.95 P3 =—1.60 —1.88 —2.20

K1 1219.86 1387.10 1670.50 andK2 45.65 52.84 57.57

= [3x,3x][4x,4x]—[3x,4x]2.A

1 15.08 15.62 16.35B1 — 1.44 — 1.46 — 1.50 The elementsrepresentedby squarebracketscanbeC1 — 149.70 — 157.40 — 167.41 expressedin termsof the shellmodel andshort rangeA2 —0.48 -—0.63 —0.71 potentialparameters[3,4, 17].B2 0.07 0.09 0.12 To demonstratetheapplicability of themodel

Table2. Valuesofphotoelasticconstantsp~ (a) calculatedin thepresentstudy,(b) calculatedby SrinivasanandSrinivasan[4], (c) calculatedbyAggarwalandSzigeti[91and (d) experimentalvalues[19/

p71,11 P1122 P12,12Crystal(a) (b) (c) (d) (a) (b) (c) (d) (a) (b) (c) (d)

RbC1 0.289 0.40 0.63 0.288 0.187 0.18 0.20 0.172 —0.054 —0.07 —0.05 —0.042RbBr 0.293 0.37 0.64 0.293 0.189 0.16 0.21 0.185 —0.053 —0.07 —0.04 —0.034Rb I 0.265 0.32 0.65 0.262 0.202 0.21 0.22 0.167 —0.043 —0.02 —0.02 —0.024

threebody force parameterintroducedby Lundqvist describedabove,we evaluatethephotoelasticconstants[13]. Following thesuggestionof Cochran[14] we can of RbCl, RbBrandRb! crystals.Thesecrystalsarewrite: selectedmainly for two reasons.Firstly thesecrystals

1 r could not be studiedon thebasisof oneion polarizablef = f~exp ——). (3) model [3,5,6]. The electronicpolarizability of Rb~

\ ~i ion is comparablewith the polarizabilitiesof halogenionsIt shouldbe mentionedthat relation(3) hasbeenfound [181.Therefore,the rubidiumhalidesshouldbestudiedto be consistentwith the elasticbehaviourof ionic on thebasisof a both ion polarizableshellmodel assolids [IS] . p is derivedfromthe overlapintegralsfor adoptedhere.Secondly,theexperimentaldata onneighbouringionsandf0 from the Cauchydeviation photoelasticconstantsP~ki for thesecrystalshave(C12 — C~)usingthe expressionsfor elasticconstants recentlybeenreported [19].obtainedby Gargetal. [16]. Theshell model parametershavebeencalculated

The generalexpressionsfor thephotoelastic usingthe electronicpolarizabilitiesof ionsobtainedbyconstants,usedin the presentpaper,areas follows [41: PirenneandKartheuser[18]. The shortrangeforce

parametersA1 , B1 , C1 , A2 andB2,which are relatedto

= {[3x, 3x,xx]P~4- [4x, 4x,xx]P~ thederivativesof shortrange potentialenergy [4, 161havebeencalculatedusinginterionic potentialsgivenby

+ 2[3x, 4x,xx]P1P2 + 3J~} (4) equations(1)and (2). The inputdatausedareas follows.Thevan derWaalscoefficientsCU andd11 are takenfromtherecentwork basedon thevariationalmethod [20].

P71,22 = —i-- {[3x, 3x,yy]P~+ [4x, 4x,yy]P~ Valuesof ionic radii usedare thosebasedon theelectron+ 2[3x, 4x,yy]P1P2+ ~ } (5) densitymeasurementscompliedby Sysio [21]. The

repulsivehardnessparametersderivedfrom the overlap4ir integrals[22] are usedin thepresentcalculations.The

P72,12 = —j- {[3x, 3y,xyJP~+ [4x, 4y,xy]P~ repulsive strengthparameterb is evaluatedfrom thecrystalequilibrium condition[161. Calculatedvaluesof

+ [3x, 4y,xy]P1P2+ P3 } (6) the modelparametersarereportedin Table 1. Thesearewheree... is thehigh frequencydielectricconstant, usedto obtain thephotoelasticconstantswith thehelp

of equations(4—6).Y1 [4x, 4x] — Y2 [3x, 4x] A comparisonof thecalculatedphotoelasticP1 =

Vol. 33,No.7 EVALUATION OFTHE PHOTOELASTICCONSTANTSOFIONIC CRYSTALS 787

constantswith recentexperimentalvaluesis presentedin To conclude,the presentmodel properlyaccountsTable2. Theagreementbetweentheoryand experiment for theshort rangepolarizabilityeffectwhich is veryis verygood.The modelusedin thepresentwork not importantin the evaluationof photoelasticconstantsofonly predictsthe correctsign but alsothe magnitudesof ionic crystals.photoelasticconstantsin close agreementwith exper-imental values.The goodagreementbetweentheoryandexperimentshouldbe ascribedto the adequateand REFERENCESexplicit form for interionic potentialsusedin thepresentmodel.For thesakeof comparisonwe haveincluded 1. M. Born & E.Wolf,PrinciplesofOptics,

Macmillan,New York, 1964.inTable 2 the resultsobtainedby Srinivasanand 2. C.A. Pitha,.&oc.of 1972 Conf on High-PowerJRSrinivasan(SS) [4] andby Aggarwal and Szigeti(AS) LaserWindowMaterials,(Editedby AFCRL-TR-[9].It is seenthat thevaluesof p7122 andp7212 calcu- 73-0372AFCRI), Bedford, Mass.(1973).lated in thepresentstudyarecloseto thoseobtainedby 3. R. Srinivasan& K. Srinivasan,J.Phys~Chem.AS and SS.On theotherhand,our calculatedp7111 are Solids33, 1079 (1972).4. R. Srinivasan& K. Srinivasan,Phys.StatusSolidimuchsmaller thanthose of AS andSS.In particularthe (b) 57,757 (1973).P7111 calculatedby AS largelydeviatefrom the exper- 5. S.C. Goyal,R. Prakash& S.P. Tripathi,SolidStateimental values.The causefor this largediscrepancyis Commun.25, 1105 (1978).that AS did notconsiderthe short rangepolarizability 6. S.C. Goyal,R. Prakash& S.P. Tripathi,Phys.Status

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17. R. Sninivasan,Phys.Rev. 165, 1041 (1968).asa functionof interionicseparation,equations(1—3). 18. J.Pirenne& E. Kartheuser,Physica30,2005Thethreebody force parameterfappearingin equation (1964)(1) is involved in the calculationsof shortrange force 19. B. Bendow,P.D.Gianino,Y. Tsay& S.S. Mitra,parametersA1,B1 andC1 whichare usedto determine AppL Optics 13,2382(1974).the photoelasticconstantsP

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