Solid State Communications, Vol.22, pp. 401—404, 1977. Pergamon Press. Printed in Great Britain.

THE)RY OF THE PHOTOELASTIC EFFECT IN Z~1D,ZnS and CdS CRYSTALS

Jai Shanker, H.P. Sharma, O.P. Sharma and J.C. Sharma

Depar~nentof Physics, Agra College, Agra-282002 INDIA

(Received6 January 1977 by M.F. Collins)

It has been a general opinion that the Clausius4iossottidielectric theory Is not capableof explaining the observedphotoelastlo behaviour of partially Ionic crystals.However in the present communicationwe show that by takingaccount of the variation of polarizabllitles under compre-ssive stress within the framework of the Clausius-Moasottitheory it is possible to obtain a reasonableagreementwithexperimental data on the photoelastic behaviour of Zz~,ZnS and CdS crystals.

The photoelastic behaviour of crys-tals can be describedby studying thevariation of the refractive index or theelectronic dielectric constant (� )under th~effect of hydrostatIcpressur&-. Vedma andDavis’~havepublished the experimental data on thepressuredependenceof refractiveindex of Zr~ ZnS andCdS crystals. Thedata reporte& by Vedam andDavis leadto the prediction that the values of thestrain derivative (?C/)V) of electronicdielectric constant ( � ) are positivefor these crystals. It has recentlybeeq pointed out by several investigat-or5~~that the Clausius-Mossotti (CM)theory of dielectric constant is notcapableof predicting even the correctsign of(aE/~V)forpartially ionic cryst-als. The view of these investigatorsis basedon the observation that theCM relation

C-i ~i!c~.e+2 3V

leads to the expression

~av 3V

(1)

(2)

In equations (1) and (2) OC and V arerespectively the polarizability andvolume per ion pair. It has beenassumed in deriving eq. (2) that o~ re-mains constant under hydrostatic press-ure. Equation (2) yields values for(~Ejbv)which are negative contrary to theexperimental data on Zr~),ZnS and CdS.

In recent times, however, newapproaches have been devglo~ed byPhillips and Van Vechten and byWesiple and Didomenlco

7 to replace theold CM theory. These dielectric theor-

les have also not been found adequateto explain the pressure dependence ofelectronic dielectric constant ofpartially ionic crystals, wi~houtintro-ducing additional parameters. Thus atpresent we do not have a dielectrictheory which can describe adequatelythe photoelastic behaviour of thesecrystals.

It has been suggested9 that theLorentz facthr(411/~) expressing the local~Ie1d effects (on which eq. (1) isbased) should be absent in predominant-ly covalent crystals, but the validityof such a statement depends on themodel assumed for a particular applica-tion. LawaetzlO has approved thevalidity of the Lorentz factor even forpredominantly covalent crystals on thebasis of a detailed analysis of ionicityand dielectric data. It is also note-worthy that using eq. (1) Tessman et

have been able to obtain appropri-ate values of the electronic polarizabi-lities of ions in Zr~, ZnS and CdScrystals. Thus we have plausible reas-ons to reject the doubts about thevalidity of eq. (1) at least for thecrystals under study. The failure ofeq. (2) for predicting the correct signof should be ascribed to theassumption that the polarizability ofions rem~ins constant under hydrostaticpressure 2., In fact the electronicpolarizabilitles of ions must changewhen the crystal is compressed. Weintend to show here that it is possibleto understandthe photoelastic behaviourof Zr~ ZnS and CdS on the basis ofeq. (l~ by taking account of the varia-tion of polarizabilities with pressure.

The considerationof the variationof polarizability modifies the eq. (2)as follows

401

402 PHOTOELASTIC EFFECT IN ZnO, ZnS and CdS CRYSTALS Vol.22, No. 7

d+(E~_\2o4 ~

-. a~+ ~-

3o~V~)÷~

~- .~

(~e~_(~-i)(E+2~)rj~.~1L ~ ~vJ (3)

The quantity A ~ , known as thestrain polarizab ility parameter, hasbeen very useful in analysing theexperimental data on the photoelasticbehaviour13’~. Of particular Impor-tance is the suggestionof Vedam and

that the electronicpolarizability of cations increasesandthat of anions decreasesin crystalsunder hydrostatic pressure. Howeverno quantitative estimate of this effectwas made. It is pertinent to mentionhere that a similar behaviour, namely,the loosening of cations and thetightening of anions in going from freestate to a crystal has been observedby numerousinvestigatorsl)-l7. Ruffa1has proposed a semiempirical model forthe electronic po].arizabilities, Wemake use of this approach as it is inqualitative agreementwith the abovecriterion of loosening and tighteningof ions in crystals relative to freestate. According to Ruffa, one canwrite

(if)

where o~.and Q(~4 are the electronicpolarizabilities of the cation in thecrystal and free state respectively. E

4is the energy parameterrelated toby

22

(5)e and u~ are the ele~ctron charge andmass respectively. h is Planck’sconstant, n the number of electronsin the ion and VM the Madelung poten-tial~is given by

v ______

(6)where is the Madelung constant.ZrQ andCdS crystallize in wurtzitestructure with °~ml.6~1 arid ZnS inzinc blende structure with 4., = 1,638.R is the nearestneighbour separation0Z is the valency of ions. We havecalculated oe~from equations (

1f-6)taking Z = 2 and find that a~.for

Zn2~ion in Zri~ and ZnS is respectively0.9~+ and 0.76 A3 in good agreementwiththe value reported by Tessman et al~(table-l). However, for Cd2~Ion in

CdS we find o~.,.= 5,36 A3 which is muchlarger than the value reportedbyTessmanet al (table-i). It has been

suggested by Jam and Jai Shanker19that Rufla’s model can still be usefulfor predic~ingthe correct polarizabi-lity of Cd + ion in CdS.Ve obtain anagreementwith Tessmanet al for Z=1.27.In further calculations, therefore, weuse Z = 2 for Zr~and ZnS and Z = 1.27for CdS, To evaiuate(~I~V)we proceedas follows. ~ , the polarizabilityper ion pair in a binary crystal can beexpressedas 1

(7)

where and are the electronicpolarizabilities of cation and anionrespectively. In view of eq. (7) wecan immediately write

(8)

Equation (1f) leads to an expressionfor (~+j~as

~ 2E~~R ~7r t,\3n

LL.4 ~mJISIn deriving eq. (9) we have assumedthat

~ and E.j remain constant underhydrostatic pressure. Since these para-meters correspond to free state theirdependence on volume is expected to bemuch smaller as compared to the crystalpotential VM. In case of alkali halidesthis assumptionhas been justified bySharsiaet a120.

An expression similar to (9) for

c~-i~,however, can not be derivedbecause of the contribution of excitat-ion levels to the anion polarizabilityin the crystal wl4gh has no counterpartin the free stat&-°. we, therefore,adopt an alternative procedure based onthe polarizability radius cube re1ation21’22~According to this relation one can write

(10)

and(U)

From the additivity of ionic radiiwe have

~!±~ ~~R (12)

Equations (10), (U) and (12) yield

Vol.22, No. 7 PHOTOELASTIC EFFECT IN ZnO, ZnS and CdS CRYSTALS 403

- Qf.. ~_3- ~~R ~L ~j~j

(13)

Thus using the ionic radii and polariza-bilities it is possible to make aquantitative estimateof (~a~4aa)and(~)o~..j~Q)from equations (9) and (13). Valuesof and (g~for Zr~), ZnS andCdS thus calculated have beenlistedin table-i alongwith the input data onionic radii and polarizabilities. Itis apparent from the formulation presen-ted above as well as from the resultsgiven in table-i that (9~+Iag)isnegative

and(~../ap)ispositive. This impliesthat the polarizability and radius ofthe cation increase and those of the an-ion decreaseunder compressivestress.The decreaseof the anion polarizabilityis, however, much larger in magnitudethan the increasein the cation polari-zability and hence the total polarizabi-lity per ion pair decreases with press-ure. These predictions are in conformitywith the views exeressed by previous

investigators13”

Values of the strain polarizabili-ty parameter A have beencalculatedfrom the relation

)~=1(’~c~\_~(ao~Q’~~~V) 3o~’~R (l~)

Values of >s. thus estimated and thestrain aerivative~’/~)(~�I~V)ofelectro-nic constant calculated in the presentstudy are comparedin table-2 withthe values correspondingto experimen-tal data reportedby Vedais and Davi~2,

Theoretical values of )% are greater than1 in conformity with experimental dataon the crystals under study. Values of

(Mie)~~av)calculated from eqn. (3) presenta very good agreementwith the experi-mental values for Z~)and CdS crystals.However the agreementfor ZnS is notvery satisfactory. In fact the obser-ved photoelastic behaviour of ZnB Ispeculiar in the sensethat it predicts

A = 1,10 whereas ~. 1.3 f~ ~pO andCdS. It has been suggested’~~’-i that

A is directly correlated with thehoniopolar bondingbetween the atoms.As far as the nature of the chemicalbond is concerned the homopolarcharacter of Zz~ ZnS and CdS isalmost idenUcal~. It is pertinentto mention here that Aggarwal andSzigetiS have also noticed the distinctbehaviourof ZnS in their theoreticalanalysis of the photoelastic data, Itseems therefore, desirable to re—examinethe experimental data on the photoelas-tic effect of this solid.

The most significant contributionof the present study should, however,be considered the inclusion of thevariation of polarizability underhydro-static pressurewithin the framework ofthe CM theory. The result of includingthis effect is apparentfrom table-2from where we observe that the valuesof(~i1~)(~ej~v)deduced from eq. (3) are notonly positive but also very close to theexperimental data. On the other handthe values of~Ie)(~E1~ierivedby exclud-ing the effect of polarizabilityvariation (eq. 2) are negative contraryto experimental values.

ZnO ZnS CdS

Ft (A) 1.95 2.36 2,52%(A) 0711a 07~a O.9L,.a

%.(A) 1~b 162b158b

4(A3)

~- (A3)080C

2.,10C

080C

90~180c

580c

o( (A3) 2.90 5.70 7.60

C

) —

‘+.120.92 -

5,610.110

6.37

0,11-1

~-(~)~g

+ 6.68 +10.20 +11.80

a. From reference 23.b. Derived from the additivity rule (Ft =‘~+~ )

Values of Input data used

Table - 1

in calculations and values of and ~-

aR

c. From reference 11.

404 PHOTOELASTIC EFFECT IN ZiiO, ZnS and CdS CRYSTALS Vol.22, No. 7

Table - 2

Valuesof ~(~-) and C ~W)

CrystalVf~o~~3V)

~Jf~ZI~~’

CalculatedExperimental Calculatedfrom Eq~(2~

Calculate&from Eq.(3)

Experimenf~I

Zr~ 1.29 1.31 - 1.55 + O.1+li. + O,~8

ZnS 1,35 1.10 - 2.08 + 0.72 + 0.21CdS 1,26 1.25 - 2.35 + 0.61 + 0.67

Acknowledgements ledged. Thanks are due to ProfessorA.K,Ghatakand Dr. K. Thyagarajafl, Depart-

The financial assistancereceived mont of Physcis, I.I.T. New Delhi forfrom the UGC, India is gratefully ackno- providing the library facilities.

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