Analysis of the photoelastic effect in ionic crystals

Analysis of the photoelastic effect in ionic crystals

I. Phys. Chrm. S&s. 1976. Vol. 37, pp. 1(117-1079. Persamoo Press. Printed in Orut Britain ANALYSIS OF THE PHOTOELASTIC EFFECT IN IONIC CRYSTALS ...

399KB Sizes 0 Downloads 46 Views

I. Phys. Chrm. S&s.

1976. Vol. 37, pp. 1(117-1079.

Persamoo Press.

Printed in Orut

Britain

ANALYSIS OF THE PHOTOELASTIC EFFECT IN IONIC CRYSTALS H. P.

SHARMA,

JAI SHANKER and M. P. VU&U

Department of Physics, Agra College, Agra-282002,India (Receiued 16 February 1976;accepted 9 Aprif 1976) A~~ analysis6f the phot~I~tic effect in ionic crystafs has been presented withm the frameworkof Ciau~u~Mo~ot~ theory of the dieMric constant. The values of the strain derivative of the eIectronic dielectric constant have been calculated in alkali halides and h&O crystals by taking into account the variation of electronic polarizabilities with compressive stress. The results obtained are found closer to the experimental values. The photoelastic behaviour of MgO crystal is predicted to be of opposite nature to that of alkali halides,in conformity with the experimental observations.

The ph~toelastic effect in ionic crystals has been the subject of numerous theoretical and experimentalinves-

tigations(Vedamand Ramseshan[l]and references given therein). The photoelastic behaviour of crystals can be understood by studyingthe pressure dependence of the electronic or high frequency dielectric constant (6). The experimental values of the strain derivative of e, i.e. (dc/dV) for many crystals have been reported in the literature[2-fl. The most striking feature of the experimental results is that (dcldV) 0. There have been several theoretical attempts[%-71to calculate (de/dV) and to explain the distinct behaviour of MgObut none of them seems to be entirely satisfactory. It shouldbe mentioned that calculationsof the derivatives of l with respect to various lattice ~spla~ementsprovide a very severe test for any dielectrictheory. In the present paper we propose an analysis based on the theory of the effect of the Madelungpotential on the electronic polarizabilitiesof ions in crystals[8,9]. The values of (da/dV) have been derived from the Clausius-Mossottirelation by taking into account the variations of polarizabilities with compressive stress.

In eqn (2) V, the volume per unit cell is equal to 2R’ for the NaCl structure where R is the cation-eon separation. Assuminga to be constant the derivative of eqa (2) yields R dc rdR=-

=

R da rdR=-

(4)

It is apparent that eqn (4) reduces to eqn (3) if (du/dR) = 0. The electronic polarizabilitya per ion pair can be written as the sumof a+, cation polarizabiity and a-, the anion ~l~~b~ity. Thus one can write

E t F*

where EbFis the local atomic field, E is the macroscopic average fieid and p is the poIarization.With the help of eqn (1) one can derive the relation between the dielectric constant (6) and the polarizabilityper ion pair (a) in ionic crystals[tOl: a-l 47F e+2=3Va*

(3)

Mueller161and Bursteinand Smithf21have used eqn (3)to evaluate (RIe)(detdR). It is of priced irn~~ce to remark here that the values of (R~~~(de~~~obtained from eqn (3)for all the crystals under study disagreewith the experimentalvalues. In the case of MgO, even the predicted sign is wrong. This situation led the earlier investigatorsto conclude that the polarizabilityof ions can not be assumed constant and must change under hydrostatic pressure. This conclusion has been further substantiate by Yeasty and Kurosawa171by making an analysis of the phot~lasti~ effect within the framework of their model for the dielectric constant. However,the numericalvaluesof strainderivativesof the polarizabilitiescan not be evaluatedfrom the treatmentof Yamashitaand Kurosawa. If we take into account the variation of polarizabilities of ions with strain, we obtain from eqn (2) the following expression for solids with the NaCl structure,

The CIausiu~Mossottitheory of the dielectricconstant is based oa the assumptionthat the Lorentz local field effective in polarization at an ion site is given by

E,oc

(e + 2)(E- 1) Q ’

da da+ da -_=-$2 dR dR dR.

(9

In order to evaluate (da+/dR) we use the theory of the effect of the crystal potential on the electronic polarizabilitiesof ions. Followingthe theory describedin Refs. [8,91one can correlate the electronic poiarizability

1077

H. P. SHARMAet al.

1078

(a,) of a cation in the crystal to its free state polarizability (trr) as

i:>=(&i>’ where E, is the free state energy parameter and V,, the Madelung potential existing in the crystal is

have been taken from Ruffa[S] and Gourary and Adrian[ll] respectively. For the MgO crystal, a, has been calculated from eqn (6), and then a. is evaluated from experimental value of l. The values of ionic radii (I+ and r-) in the MgO crystal have been taken from the recent calculations of Bisarya and Shanker [ 121. 3.REXJL'IS ANDDISCUSSION

The values of (da+/dR) and (d~../dR) in 17 crystals calcuIated from eqns (8) and (11) have been listed in Table 1. The negative values of (da+/dR) and positive values of In eqn (7), a.+, is the Madelung constant and z is the (da-/dR) suggest that under the effect of high pressure the polarizability of the cation increases and that of the valence of the ions. When a crystal is compressed under anion decreases. These loosening and tightening effects of hydrostatic pressure cu, and Ef will remain constant but VMwill vary. The derivative of eqn (4) will yieid with the cations and anions respectiveIy were long ago predicted by Fajans and 300s f13f and have recently been con~rmed use of eqn (7) by Krishnan et a/.[141 by analysing the experimental data on elasto-optical constants of ionic crystals. 2a,E; VM da+ -=..(8) In Table 2 we report the values of (R/e)(de/dR) (E, - VM)’’ 7 dR estimated from eqn (4), alongwith the experimental values A relation similar to eqn (6) can not be used for anions available for 7 crystals. For the sake of contrast, the due to a somewhat different situation. The existence of values of {R~~)(d~/dR) obtained from eqn (3) derived by excitation levels contributes to the anion polarizability in Burstein and Smith[Zl have also been included. It is the crystal which has no counterpart in the free state. In apparent from Table 2 that the values of (Rle)(dr/dR) addition, for anions the quantum states above the first calculated from eqn (4) are very close to the experimental ionization continuum contribute substantially to cr,.These values in contrast to those obtained from eqn (3). It is of cont~butions were incorporated phenomenoIogicaUy by particular importance to note here that eqn (3) yields R&a@] in his model and the agreement with experimen- negative values of (R/~)(dE/dR) even for MgO whereas tal polarizabilities could be achieved only by assuming the experimental data suggest a positive value for this arbitrary values for the interaction energy (Q,,) between crystal. In this respect eqn (4) shows a significant the free ions and crystal environment. In order to avoid improvement as it yields negative values of (Rle)(deldR) the uncertainties in the values of Q., which will of course for alkali halides and a positive value for MgO. However, be magnified in the evaluation of strain derivative of anion the magnitude of (Rlaf(dfldR) in MgO calculated from pol~~bi~ty (da.~dR), we will adopt an alternative eqn (4) is nearly half the observed vafue. It wouId be pertinent to mention here that MgO has a large Cauchy procedure. It is apparent from the detailed analysis performed by discrepancy{U], suggesting the existence of non-central Aggarwal and Szigeti[!i] that the photoelastic behaviour many-body forces in this material. A theory of the effect of ionic crystals can be explained adequately if one assumes that the electronic polarizabilities of the ions Tabie 1. Values of ~~j and &) under compression change linearly with interionic separation. In view of this analysis we assume that in P

sa,

-=-?

a+

ST+ Sam Sr-=r+ ar-

(9)

where &+y-ijSa_) and Sr+(Sr_) are respectively the changes in a&_) and r&(r-). The assumptions expressed by eqn (9) are purely empirical and their validity will be tested by a comparison with experimental data. Obviously, one can write

Equations (9) and (10) yield (11) Thus we can evaluate (da-fdR) from eqn (1 I) using the values of (da+fdR) estimated from eqn (8). The values of electronic polarizabilities and ionic radii in alkali halides

Crystal

i%)

(%)

LiF LiCf LiBr LiI

-0.007 -0.001 -0.003 -0.003

1-0.680 +i.619 +2.345 +3.306

NaF NaCf NaBr NaI

-0.035 -0.021 -0.019 -0.015

+0.77 1 i I .785 +2.547 +3.504

KF KCI KBr KI

-0.235 -0.153 -0.135 -0. I14

t-O.921

RbF RbCf RbBr Rbl

-0.299 -0.200 -0.179 -0.153

i2.800 +3.799

MgO

-0.209

+3.123

t1.985 +2.804 i-3.767 +0.928

-t1.99fJ

Analysisof tke pho~as~ effectstoic

1079

cxystals

Tabie2.Valuesof $$ f 1

ClYtd

mm

eqa

-1.89

-3.02

LiBr

-3.53

Lif

-4.27

(3)

From eon (4) -0.57 -i.it

Fromeqn(13) Experimentit -1.24

-1.Mf

-1.54 -2.14 -0.61 -0.91

-1.81 -1.16

[email protected]

-1.71

NaF

-1.59

N&l

-2.47 -2.83 -3.35 -1X

-1.21

-1.64

-1.96 -1.33

-1.76

-2.25

-1.09

W RbF Rtrcf

-2.51 -2.90 -1.89 -2.26

-1.20 -t.39 -1.74 -I.30

-1.49 -1.59 -1.72 -1.35

-0.93 -1.35 -1.57

-f.fo

-1.44

RbBr

-2.49

-1.34

RbI

-2.80

-1.47

-i.b - 1,6Q

MN

-3.39

+0.46

-1.48

NaBr

NaI

RF KQ KRr

-Is3

-0.95

-1.28

t1.M

ICited in Ref. [17].

of May-~dy forces on the po~~~~~ties of ions has been developedby Vermaand AgarwalI:161.If we use the modified ~Ia~~t~~ from Verma and Agarwal we obtain fR IP)(defdR ) = + 1.02,whichis in excellent agreement with the experimentalvahm i-137, In crystals where experimeutaldata are not availableit is dilftcult to test our cak~~Iatious.However, Van Vechtonr~~~has recently dev~~o~d a rn~b~ for calculati% (R~~)(d~~dR)based on the Phiips-Van Vechten dielectrictheory. The vakresof fR~e~de~~ ) in aI1the crystals can be evaluated by assuming Rdc 0 CSZ” as suggested by Van Vechten. In eqn (12) c is the elec~o~eEativi~ parameter defining the beteropolar ~o~~bu~o~to the optical energy gap.Usingeqn (12)Vaa Vechten obta~ed from the Penn rnodelj‘l81the fo~ow~~ expression R de -..-t edR

2E-I y

t

The v&es of ~R~~~d~~~) from eqn (13) have been Listedin TabIe 2. It is interests to observe from Table 2 that the values of lRlr)(doJdR) obtained from eqn (13) are closer to those obtainedfrom eqn (4)In contrast to the values obtained from eqn (3). In view of the exigent resuhs one should have (da IdR) < @a/R) in alkalihalidesand IduIdR) > (3~/R)

in MgUas is evident from equ (4).The values of (da IdR) in LiF, NaC1,KCI,KBr, ICIand MgOcrystals determined by A~~ and &i&i as wefl as those in the present study satisfy this requirement. Acknowf~~~en~s-me authorsare thankfti to Dr, S. C. Goyal for useful disc~sio~. The rein assistance received from U.G.C.,New De&i-i is stinky ac~w~e~

1. Vedam K. and Ramsheshan, Progress in Crystal Physics (Editedby R. S. Kirshnan),Vol. I. Wiley-~~terscience, New York 0958). 2, Rurstein B. and smith P. L., Pkys. R&x74,22Q(1948~ 3. VedamK. and SchmidtE. D. D.,Phys.Reu.146,584(1966). 4. VednmK. and SehrnidtE*a. D., Phys.Rev.158,766(@66), 5. AggarwalK. G. and SzigetiB., J. Pkys. C3, lOQ7(iY?O), 6. MuellerB., Pkys. Rev. 47,947 (1935). 7. YamashitaJ. and KurosawaT., J. Pkys. Sot. &purr ltr, 6l6 (1955). 8. RuffaA. R,, P&r. Rev. 138, 1412(f%3& 9. Banker $. and Verma M. P., Pkys. &a. BfZ3449 (1975). 10. Tessm~ J. R., K&m A. H. and ShockleyW., Pkys. Reo.92, 890(1953). (@jOI. 12. Bisarva S. D. and ShankerJ., Prornoaa 2, 1% (1974). I3. Faja& K, and foes G., 2. Pkysik 23, 1 (19241. 14. KrishaanR. S.. Vasudev~ T. N. and NarayauanP. S., Nficf.

P& S&ii &te Pfiys.@r&a)IX, 1%@?4). X5.~rne~c~ I&#& o~~k~~icsff~b#~ 2nd Edn. McGrawHill, New Ytxk (1963). 16. Venus N. P. and AganvalL. D,,Phys.Rev.B9,1958(1974), 17. Van Vechten J. A., Pkys, J&e. lsz, 89f (19693. 18. Penn D,, Pkys. Reu. 128,2093(1962).