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Determination of Photoelastic Constants in the Presence of Tilt of the Axes

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DETERMINATION OF PHOTOELASTIC CONSTANTS IN THE PRESENCE OF TILT OF THE AXES

K. V. KR18HNA KAO

Physics Departm knt, Osmania ITnivkhsity, Hyderabaj)-?

(Received Jatiuarif 2, MRU)

A B S T K A C l ’ . T h is pa]>or clen cribos a d ir o ct m otJiod oI d o fo r m iiiiu ^ tlio diRV*routial st roH.s- o p t i c a l c o n s t a n t s , w h e n tlu^ a x e s o f p o la r i s a t io n in a slro sso d ( rysta.l d o not c o in c id e w ith tlu*

ji r in c ip a l d ir e c t io n s o l s tr e s s . 'I'lie m e t h o d is \ei*ifi<*d h y s tu d ie s o n b a riu m n itr a te a n d s t r o n t i u m n it r a t e c r y s t a ls .

I N T K O D UMTr ON

W ith the applieatioji o f the groii])-UieorelieaI nietliods by Blia^avaiitam (1942) to tlerive the miinber of uon-vaniHliing jihotoelastic coustaiits for diffesrent classes o f crystals and the dis(*overv of scvcmtiI errors in the seh(‘mes giveji earlier by Poekels (1S89), interest in tli(‘ photoelastie effiu t in crystals was revived and an intensive study o f the subject was undertakn by Bhagavantam and collaborators in recent years (Nye, 1957: Krishnan, 1958). During the course o f these studies, it was found (Bhagavantani and Krishna llao, 1953a) that, for some orientatJons of cubic crystals, the princi})al ax<‘s of ])olarisatiou o f the stressed crystal do not coincide with tlu* principal directions ol stress.

This phenomenon Jias been referred to as the tilt of the axes. When t-hiu’e is tilt of the axes, if the usual exjierimental nudhod for determining the dif­

ferential stress-optical (constants, is (miployed, one sliould first find the positiojis o f the axes o f polarisation o f the stressed crystal and adjust the Babinet^ compen­

sator, such that its primdpal dire(‘tions coincide with the axes of polarisation o f the stressed crystal. On the other hand, if the princijial directions of the com­

pensator are kept> vertical and horizontal, as usual, and the (crystal is stressed vertically, it has been found that the shift of the Baninet tringc is not ])roj)orti(mal to the applied stress aaid the fringe vibrates about the initial ])osition, as the stress is gradually increased. It will now be shown that, from a knowledge of the stress required to bring back the Babinet fringe to its initial ])osition, the stress-optional constant can be evaluated directly, without necessitating the determination of the tilt o f the axes.

T 11 E 0 R Y

Let OXi, (Fig. 1) be the princi])al directions o f polarisation of the stressed

crystal

in

the

X Y

plane

(normal to the direction of observation), OX^y O Y 2 the

38

341

(2)

342 K . V. Krishna Rao

principal directions o f the (umipejisator and OP the direction o f vibration o f the incident plane polarised beam o f light of amplitude a . Let a and be angles

X ^ O P and X ^ O X ^ respectively.

Fig. 1. Showing the ])riiieipal direct ions of the strortsed (’rystal (O X i, OY|), the principal directions of the comptmsator (O X j, ()Y;>) and the direction of vabration (OF) of the incident light.

On entering the stressed crystal, the incident beam will be resolved into two components, one o f amplitnde a cos a and direction o f vibration O X ^ and the other o f amplitude a sin a and direction of vibration O Y ^ . These (;omponents will have a phase difference, say 8, when they leave the crystal. When the beam enters the compensator, etK*h o f the above-mejitioned two components will split further into two components with their vibration directions along O X ^ and OTg- The amplitude of the component, with the vibration direction along OXg, is the resultant of the two components of am])litndes a cos a cos (i and — a sin a sin ft

and phase difference 8. Similarly, the amplitude of the eonipojient with vibration direction along is the resultant of two components o f amplitudes a

sin a cos ft and a cos a sin ji and phase difference 8. The amplitude A of the com ­ ponent wdth the vibration direction along O X ^ is given by

A ^ = f/‘‘^{cos2(a—//) — J sin 2a sin 2/^(1 + cos 8 ) } ,

The phase o f tliis vibration is given by

tan Aj — sin a sin ft sin 8

cos a cos sin a sin p cos 8

(1)

(2)

Similarly, the amplitude B of the component whoso vibration direction is along OFg given by

{sm^ sin 2a sin 2/?(l— cos d)} (3)

(3)

DctcTiTiiucitioTi o f P h otoeld stic GoristdTits^ €tc,

and its phase Ag is given by

ta n A2= sin a co s//sin # cos a sin /i f sin a cos // cos S Prom Eqs. (2) and (4), we get

343

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tan (Ag—Ai) sin 2 a sin

sin 2 (a + //) —2 cos 2// sin 2a . siu‘^ d/2 (•">) The fringe shift in the compensator gives (Ao^A,) vvliicb would obviously bo equal to S when the tilt o f the axes /J is zero. Kq. (5) shows that, as S is incToased, (A2—Ai) first increases, reaches a maximum and then reduces to zero when S

^ n. On a further increase o f A, (diajiges sign, reaches a maximum and again reduces to zero when S =- 2n. Thus the Ba]>inet fringe completes one oscil­

lation as the phase difference ^ iu(*reases from zei*o to 2n. The variation of A2- - A] with S, evaluated for values of a and (a | //), 30' and 45*^ res])ec1 ivel3^ using Eq. (5), is shown in Pig. 2. It is clear that the strees P, recjiiired for* one conqdete

oscillation o f the Babinct fringe, gives the stress required for a path difference A (wavelength o f the light used) between the two components with their vibration directions along the principal directions of the stressed crystal. Hence the differential stress-optical coefficient C is given by

n^Pf

where t is the thickness o f the crystal jrarallel to the direction of observation and n the refractive index o f the crystal in the unstressed state.

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3 4 4

K X P E l M E N T A L K . V. K rish n a R ao

To verify flie foregoing method, crystal pj*isms of barium nitrate and stron­

tium nitrate, with faces parallel to (111), (011) and (211) planes, have been studied applying the stress, by a lever arrangement, along [211] and making the observa­

tions along [Oil] employing the usual arraixgement (Fig. 3) for determining the differential stress-optical constants. The stress-optical constant T’ for this ori-

Fig. 3. Exporimenl,a.l for (letorminiug differential stresH-optical constants.

S—-Soxirce of 1 ight ]j—Condensing Ions P—Polarising Nicol

C— Cjystal

H—Babinet comjionsator A -Analysing Nico]

entation o f (m^) class of (*rystals, to which these two substances belong, is related to the stress-optical coetfi(‘ients r/jg, q^^ and q^^ by (Bhagavantam,

195.3) :

(! = W i I - <744)’“ (7)

where, A — ^ (2f/ii- -r/i2 “ qiz)>

In the case o f barium nitrate, the load retpiired for one oscillatioji o f the Babinet fringe is found to be 2380 grams, the me(*hani(*al advantage o f the lever arrangement being 3.992. The length o f the prism parallel to [111] direction, AVhich enters the calculations, is 0.330 cm. With these values, taking n as 1.570 (Landolt and Bornstein, 1931), the stress-optical constant is evaluated, using Eq (6). The value obtaiue<l, 10,8

x

10 vni^ dyne" is found to be in agreement with 10.2, evaluated using Eq. (7), taking the values o f A and (Bhagavantam and Krishna Bao, 1953b) as 20.60 and 1.69 respectively.

For strontium nitrate, the load for one oscillation o f the Babinet fringe is 3300 grams. The length o f the prism parallel to f i l l ] is 0.264 cm. The stress- optical constant, evaluated using Eq. (6), taking n as 1.567 (Landolt and Bornstein, 1931) is found to be 6.3, in close agreement with 6.27, evaluated using Eq (7),

(5)

Determination o f Photoelastic Constants, etc. 345

taking

A

and ((Bhagavantam and Krishna Rao, 1«54) as 13.61 and 1.38 respectively.

R E F E R E N C E S Bhagavantam, S,, 1942, Proc, Ind, Acad. Sd*y

A16,

359.

Bhagavantam, S., 1953, Proc. hid. Acad. Sci., A37, 585.

Bhagavantam. S., and Ki^ishna Rao, K. V.. 1953a, Proc. Ind. Acad. Set.,

A87,

589.

Bhagavantam, S., and Krishana Rao, K. V., 1953b, Acta CryNt.. 6, 799.

Bhagavantam, S., and Krishna Rao, K. V.. 1954, Cvrr. Sci.,

28.

257.

Krishnan, R. S., 1958, ProgrosH in Crystal Physics, Madras.

Landolt, H. H. and Bornstein, R., 1931, Physikalischo Clieinisclio Tabollon.

Nye, J. F., 1957, Physical Properties of Crystals, Oxford: Clarendon Press.

PockelB, F., 1889, Ann. der. Phja., 87, 144.

References

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