F‘HQTOELASHC CONSTANTS 0F SYNTHETIC SAPPHIRE T175135: far rho Degm a! M. S. MECHZGAN ”AYE UNQVERSiW David L. fimdéay m3: TEES: 'WUWWI"!WWI/IN!(II/WilliNIH/11W 3 1293 01704 0043 LIBRARY Michigan State University MICHI AN STATE UNIVERSITY IBRARIES L PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. DATE DUE DATE DUE DATE DUE 1/90 WW“ PHOTOELASTIC CONSTANTS OF SYNTHETIC SAPPHIRE BY I ‘ d David L.:firad1ey AN ABSTRACT OF A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Physics and Astronomy 1963 David L. Bradley ABSTRACT The photoelastic constants of synthetic sapphire were determined by combining both static and dynamic measurements. Procedures for obtaining photoelastic constants in rotated coordinate systems, corresponding to specific orientations of the sapphire crystals are outlined. The ratios of photo- elastic constants are measured by analyzing the polarization of the first order diffraction pattern produced by pure longitudinal ultrasonic waves. The differences of photo- elastic constants are obtained by applying a known stress to the sample and measuring the phase difference of the light with a Babinet Compensator. Values of the strain optical and stress optical constants are tabulated. PHOTOELASTIC CONSTANTS OF SYNTHETIC SAPPHIRE BY fire David L£VBradley A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Physics and Astronomy 1963 ACKNOWLEDGMENTS The author wishes to extend thanks to Dr. E. A. Hiedemann for suggesting this investigation and for his interest during its execution. The assistance of Dr. W. G. Mayer and Dr. K. Achyuthan is also greatly appreciated. The time and effort of'Mr. Verne Hulce was an invaluable aid. D. L. B. ii TABLE OF CONTENTS Page INTRODUCTIONOO0....O...0.0.0.000...OOOOOOOOOOOOOOOOOOOOI MORYOOOOOOOCOOOOCOOCCOOOOOOIOOOOOOOOOOOOOOOOOOOOOO.0.1+ EXPERMNTAIJ PROCEDURE. 0 O O O O O O O O O O C O O O O O O O O O I C O O O O O O O O 21 EXPERMNTAII RESULTS. 0 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O C O 27 REFERENCESOOOOOOOOOOOOOOOOOO0......0.0.00.00000000000033 iii Table I. II. III. LIST OF TABLES Page Directions of sound propagation, directions of observation, and corresponding ratios of photoelastic constants..........................19, 20 Tabulation of strain optical constants..........32 Tabulation of stress Optical constants..........32 iv LIST OF FIGURES Figure Page 1. Directions of pure longitudinal motion in the X-Y plane ...... IOIOOOOOOOIOOOOOOOO00...... ........... 5 2. Directions of pure longitudinal motion in the Y‘ZPlane...coo-000000000000. ooooo o ..... coo ooooooooo 05 3. Rotation of the sapphire sample about the X-axis.....5 h. Diagram of the optical arrangement for the dynamic measurements............. ...... ...... ....... 22 5. Diagram of the optical arrangement for the Static measurements.OOOOOIOOOOO...0.0.0.0... ..... .0025 6. Illustrations of the static stressing bench ...... ...26 7. Experimental data for ratios of photoelastic constants.‘0.0...0.0.000...OOOOOOOOOOOOOOOOOOOO ..... 29 8. Experimental data for ratios of photoelastic constants...0......O00.......00.0.0.000000000000000030 INTRODUCTION Photoelasticity, the phenomenon of double refraction accompanying mechanical stress, was discovered in 1815 by Sir David Brewster. Early work in this field was carried out by F. E. Neumann, who was the first investigator to systematically develop a theory of double refraction due to a stress. G. Wertheim was the first to experimentally verify the laws of photoelasticity for homogeneous stresses in glasses (1). Other investigators contributing to the early work in this field were J. Kerr, who experimentally found the first reliable values for the stress-Optical constants, and C. Maxwell, who proposed essentially the same theory as F. E. Neumann. In 1889 and 1890, the elementary theory of double re- fraction was extended to crystals by F. Pockels (1). Early investigators were limited in their measurements of photoelastic constants to static measurements. A relative change in the optical path length could be measured by applying a stress in a given direction and measuring the difference in optical path length with a Babinet Compensator for light polarized normal to the stress direction and light polarized parallel to the stress direction. The difference in optical path length is pro- portional to the difference of photoelastic constants. Absolute changes in optical path length were measured using interferometric techniques. An absolute change in Optical path length is pr0por- tional to the photoelastic constant for that direction. In 1932, the discovery of the effect of an ultrasonic field on light passing through a transparent material led to an entirely new method of determining photoelastic constants. H. Mueller, (2), (3), (h), develOped a theory of the diffraction of light by ultrasound in isotropic solids. Mueller's theory is an adaptation of the theory of the diffraction of light by ultrasound in liquids by Raman and Nath (5). Mueller's theory was experimentally verified by Hiedemann. Narasimhamurty (6) investigated the photoelastic constants of uniaxial and biaxial crystals, following the suggestions of Mueller. Photoelastic constants of isotropic solids and crystals may be studied using static methodsfirst developed and using dynamic methods, suggested by Mueller (2). In this investigation, static and dynamic measurements were both employed to determine the photoelastic constants of synthetic sapphire. The photoelastic constants may be divided into two classes, the stress-optical constants and the strain- optical constants, the former related to variations of stress and the latter related to variation of strain. In this dissertation, we shall assume stress and strain to be within the limits of Hooke's Law. In this case, the two systems may be used equivalently. We shall also assume stress and strain to lie along the same axis, 3 i. e., a pure compressional force. Sapphire belongs to the trigonal class of crystals. It is a negative uniaxial ( the ordinary index of refraction is greater than the extraordinary index of refraction) crystal with eight stress optical (or strain optical) constants. Investigations were performed on three samples of sapphire oriented at particular angles with respect to an ortho- gonal set of axes. Three samples were necessary to provide enough relationships to obtain the eight constants of sapphire. The static measurements were performed in the usual way, using a Babinet Compensator to measure the relative change in the optical path. The dynamic measurements were made using'Mueller's Method "C", described in the experimental section. Photoelastic constants describe the optical behavior of sapphire when subjected to stress or strain. Knowledge of the photoelastic constants, when coupled with the intensity distri- bution of light in the diffraction orders created by a sound field, provide a method of studying absolute sound pressure amplitude. THEORY SECTION A: The assumption of pure longitudinal waves in this investigation demands a determination of the directions in a trigonal crystal along which these pure longitudinal waves can be propagated. F. E. Borgnis (7) has develOped a general theory for determining the directions of pure longitudinal waves in crys- tals. In general, the particle displacement in a crystal does not coincide with that of the pressure wave. There are, however, certain crystallographic orientations for which the direction of particle displacement and the direction of pressure wave propaga- tion coincide. Following Borgnis, a set of relationships between the direction cosines ((1) of the pure longitudinal waves and an orthogonal set of axes are obtained: 11=1,(Z=13=o (1) (3:1) (1=(2=O (2) 2 r 3 r (-c13 + C33 - 2 chh) (13- + 3 clh 73' + (-c11 + c13 + 2 chh)x I - c = O (7:) 1h (3) 11:0 From.Eq. (1), pure longitudinal waves may exist along the X-axis of the crystal. For a trigonal system, a rotation of 1200 about the Z-axis produces an equivalent coordinate system. Figure 1 indicates the directions of pure longitudinal waves in the X-Y plane. F|§,I. LONGITUOINAL MOTION 'IN X-Y PLANE Y IZO' FIG.2. LONGITUDINAL IZO' )—X MOTION W Y—! PLANE t'IZ'B 3- any yea-45 m norneo SAMPLE;SOUND ALONG xlaxnsguomx'axns xx' I From Eq. (2), pure longitudinal waves can be prOpagated along the Z—axis of the crystal. Equation (3) gives a set of directions in the Y-Z plane along which pure longitudinal waves may exist. Substituting the values of elastic constants as given by Mayer and Hiedemann (8) into Eq. (3) and solving for 73' one 2 obtains the following three sets of values for (2 and (3: .978 .210 (2 r3 = + .78u I + .620 (2 + .775 (2 3 - .632 (3 Figure 2 illustrates the orientation of these angles with respect to the Y-axis and Z-axis. Counting positive and negative directions, there are thirteen possible directions of pure longitudinal motion. SECTION B: F. Pockels (l), (9) developed a systematic method of studying the photoelastic constants of crystals. His argument is as follows; Assuming that Fresnel's laws of propagation of light hold in a crystalline medium, the optical index ellipsoid in an undeformed medium may be represented in general by: 0 2 o 2 o 2 o o o Bllx + B22 y + B33 2 + 2 B23 yz +2 B31 zx + 2 B12 xy — 1 (H) where o l o 1 B11 _. nil , ..... . .......... 312 n2 , 12 In a deformed medium, the optical index ellipsoid is, in general B x2 + B22 y2 + B 22 i 2 B yz + 2 B 11 33 23 zx + 2 B12 xy = l (5) 31 7 Where B11 = 1 ,ooooooooooo, B12 = 1 n.2 n.2 11 12 The nij and nij may be called the Optical parameters of the system. Pockels assumed that the differences of the corresponding coeffi- cients Of the two index ellipsoids can be eXpressed as a linear function of the strain, i.e., 0 B11 ' B11 = P11511 + P12522 + P13S33 + P14323 + P15331 + p16312 --------------------------------------------------------------- (6) 0 B12 ' B12 = P61511 + p62522 + P63S33 + Peuszg + P65331 + P66312 where the pij are the strain-optical constants and the Smn are the components of strain. Since the investigation is conducted in the region where Hooke's law holds, stress may be substituted for strain to give the following set of equations: 0 B11 ‘ B12 = ’(q11T11 + q12T22 + q13T33 + q11323 + q15T31 + q16T12) ------------------------------------------------------------------ (7) 0 B12 ‘ B12 = '(q61T11 + q62T22 + q63T33 + q61323 + q65T31 + q66T12) where the qij and Tmn are the stress-Optical constants and the com- ponents of stress reapectively. The pij and qij compose the photo- elastic constants of a crystal. The components of stress and strain are related by elastic constants and moduli as follows: 11 = s11T11 + 8121322 + S13T33 + S11523 + 815T31 + 816le ---------------------------------------------------------- (8) '312 = s61T11 + S62T22 + S63T33 + S6AT23 + S65T31 + S66T12 where the sij are the elastic moduli. 11 = C11311 + C12522 + c13333 + °1u523 + c15331 + C16312 ---------------------------------------------------------- <9) 12 = c61511 + C62322 + c63333 + °6h323 + C65331 + c66312 where the c. are the elastic constants. Combining Eqs. (6), (7), 11 (8), and (9), one obtains: 6 b Pkl = Z‘qkjcjl qkl = Zpkjsjl (10) )3 .:| where the indices k, l = 1.......6 also. In order to avoid repetition, only the pij will be discussed henceforth, but the assumption of working in the region where Hooke's law is valid makes it possible to use the relationships derived for the qij also. In the most general case, there would be 36 different constants pij for the triclinic crystal. Crystal symmetry reduces the number of constants to eight in the case of sapphire, which is a trigonal crystal. Pockels' equations may be written in matrix form for the trigonal system as follows: .— 77 o '- W 1 B11 ' B11 = p11 P12 P13 Plu o 0 S11 0 l B22 ’ B22 = P12 P11 p13 'Plu O 0 S22 0 B - B = O O O S 33 33 p31 P31 p33 33 “ (1 ) 1 B - B0 = p‘. -p O p O O S ' 23 23 A1 11 an 23 0 B31 - 331 = O O O O Pun thl i331 B - B° = o o o o p 12 12 L 11+ p11- p12L L312.) III! (ll II'\ [I (ll: fllllln'l. f| II. It is this set of constants that is to be investigated. Consider two illustrative cases of calculating photoelastic constants. Case 1: The crystal is oriented along the orthogonal set (X-Y-Z) of axes. Strain is along the Z-axis and observation along the Y-axis. With no strain, the cross section Of the undisturbed index ellipsoid is represented by: o 2 O 2 B11 x + B33 F = 1 (12) Upon application of a strain, the index ellipsoid is distorted and and the ellipsoid cross section is represented by: B x2 + B 22 + 2B 11 33 31 zx = 1 (13) where the B31 term indicates the ellipse is, in general, tilted with respect to the orthogonal axes. From Eq. (11), one Obtains: 0 B11 ‘ B11 = Pllsii + P12522 + p13533 + Plu323 B (1h) 0 33 B33 ’ P31511 + p31322 + p33533 0 B31 ' B31 = 9&4331 + thlslz where B0 31 of strain. Since the only strain is along the Z-axis, all the Sum 0, since the index ellipse is not tilted in the absence are zero except S Eq. (1h) can then be written as: 33' 0 B11 ' B11 = p13333 - 30 = S 1 B33 33 P33 33 ( 5) 331 = o, Indicating in this particular case that the cross section Of the index éllipsoid is not tilted with respect to the X-Z axes. 10 Case 2: The crystal is oriented at an angle with respect to the ' orthogonal (X-Y-Z) set of axes. For this example, assume the crystal has been rotated #50 about the X-axis. The strain is along the X'-axis and Observation is along the Y'-axis and indicated in Fig. 3. The rotation matrix may be written as follows: x y z x (1 m1 1“1 y (2 mg mg (16) where the e., m., and n are direction cosines relating the X', Y', 1 1 i Z' axes with the original X, Y, Z axes. For a Specific rotation of #50 about the X-axis, Eq. (16) becomes y o 1/{5 41le (17) z 0 l/{B 1/{5 Using Eq. (16) and rewriting the strain of the original coordinate system in terms of the strains in the new coordinate system, one obtains I I I I 2 2 2 m n + S n S 33 1 2(323 11 31 1(1+S12(1m1) I 11’1 l m- H N I-' + U) N N B H + U) 5 + (18) I I I I 312 = Sllll(2 + SZZmlmZ + S33n1n2 + 823(m1n2 + mznl) + S31(n1(2+[1n2) + 312((1m2+(2m1) 11 where Sén are the strains in the new system. Using Eq. (17), one may write; 1 1 _ I _ _ I _ _ I S11 “ S11 S23 " 2 S22 2 S33 _ ,]_- I l I _ I _ 1 I 1 I S22 ‘ 2 S22 + 2 S33 523 S31 ‘4? S31 +4? S12 (19) _ .1; I l I I ___1 I 1 I 333 ‘ 2 322 + 2 S 33 23 S12 ‘72 331 +72 S12 Now the equation for the differences of coefficients of the index ellipsoid may be written in terms of the strains in the new system. Combining Eqs. (11) and (19): _ _ I l". ' l ' ' B11 B11 ‘ p11311 + 2(912 + p13 i p1”)822 + 2(p12 + p13 plu)s33 _ 1 I 1 I B12 B12 ‘4‘2‘9111 P11 + "12)S 31 +{2(P111 + P11 p12)S12 Since only a strain along the X'-axis is being considered, the Sén are all zero, except 311' Therefore, Eq. (20) reduces to: - ° _ I _ ° = I B11 B11 “ p11311 B23 B23 p111311 - O _ ' _ O _ B22 B22 ‘ p12311 B31 B31 ‘ O (21) B - Bo - p S' O 33 33 31 11 B12 - 312 = 0 Now consider the index ellipsoid in the rotated system: A x'2 + A '2 + A z'2 II II II_ 11 22y 33 + 2A23y z + 2A 2 x + 2A12x y _ l (22) 31 The coefficients of the optical index ellipsoid in the transformed system.may be written in terms of the coefficients of the Optical index ellipsoid in the original coordinate system. Using Eq. (16), 12 one may write, 2 2 2 A11 = B11(1 + B22 (2 + B33(3 + 2(323(2Y3 + B31(3K1 + 312(112) ------------------------------- (23) A12 = Bllilml + 322 (zmz + B33(3m3 + B23((2m3 + (3m2) + B31((3m1 + (1m3) + B12([1m2 + (lml) Combining Eqs. (17) and (23); 1 1 A11 ’ B11 A23 ‘ ' 2 B22 +~2 B33 A = 1-B + l-B + B A = 1 B v 1 B (2h) 22 2 22 2 33 23 31 {‘2‘ 31 12 -1 l _ _ 1 1 A33‘2B22+2B33 B23 A12" {2331+ B12 For observation along the Y' axis, the cross section of the index ellipsoid is, in general, I2 I2 II__ A11 x + A332 + 2 A31 2 x _ l (25) The values of A11 , A33 , and A31 may be determined from Eqs. (21) and(211): _ I 0 A11 ‘ p11311 + B11 A =l(p +1) +21: )S' +l(B° +B°) (26) 33 2 12 31 Al 11 2 22 33 A31 = 0 Note that A = O , indicating in this particular case, that the 31 ellipse is not tilted with respect to the new coordinate axes. In general, for a rotated system with respect to the crystallographic axes, the procedure is: 1. write the strains in the unprimed system in terms of the strains in the primed system. 2. write the difference of coefficients Of the index ellipsoid in the un- primed system in terms of the strains in the primed system. 13 3. write the coefficients of the index ellipsoid in the primed system in terms of the coefficients in the unprimed system. Briefly then, one wants to write the photoelastic constants, as originally defined in a rectangular coordinate system, in the coordinate sys- tem in which the investigation is being carried out, which in gene- ral, is rotated with respect to the system defined by the crystal- lographic axes. SECTION C: Dynamic measurements: Introduction Of a strain in a transparEnn medium results in a change in the index Of refraction of the medium. The Optical parameters discussed earlier are the indices Of refraction for a crystal. Thus, the lengths of the axes of the index ellipsoid are proportional to the indices of refraction of the medium and are also proportional to the pressure or strain exerted on the medium. In the case Of an ultrasonic wave providing the strain, the axes of the index ellipsoid will vary periodically, where the periodicity is determined by the frequency of the ultrasonic wave. Consider the two cases discussed earlier; in Case 1, re- peating Eq. (15): o l 1 B11 ' B11 = p13333 ” n.2 ' n2 11 11 O 1 1 B .. B = S = —"'" _ _ 1 33 33 p33 33 n12 n2 (5) 33 33 B31 - O = O , indicating nO tilt Of the index ellipse. Since S33 is a periodic strain, the axes Of the index ellipse vary periodically. Following Mueller (2), B1 and B may be written as: 1 33 ll 33— where 1‘ a h‘ P‘N H I H :3 WIN) LA) This indicates pf variation of the principle axes of the index ellipsoid. 1h (2nf* A kfisin 2nf*(t - (zcosg + ysinfl)/V*)) _ sound frequency density of the medium _ sound velocity amplitude Of the sound waves 3 (21rf* A RF sin 21rf*(t - (zcosQ’ + ysinfl) /V*)) (27) constant proportional to the elastic constants Of the medium. and i that i1 3 the above equations, one Obtains: Comparing again with (11)(K) (n11)3 = A11-11 are prOportional to the amplitude From Mueller's equations; and <11>3 = A. 33 El. B11 - 1/ni1 Eli—E33 .313 13 B33 ' 1/“33 P33 S33 p33 Let Anll and An33 be the change in the index of refraction along the X and Z directions respectively. One may then write: 211 = 1 = 1 = 1 ' 3““11/“11 = 1 (“11 ““11111)2 (1 +'A“11M11) n11 “11 “11 B33 = (n +1an )2 = 1 = 1 ’ 11:33/“33; 33 33 (1 + An33/n33)% 33 n33 n33 15 Therefore, An i n3 n3 p n3 p 11 = 1 11 = 111 13 = O 13 = R1 (30) An 3 3 3 33 13 n33 n33 P33 ne p33 From the theory of Raman and Nath (5), one may show that a change in both n11 and 1133 produces two diffraction patterns and the inten- sity of each pattern is dependent upon.An and An respectively. 11 33 Thus, if the th order of the diffraction pattern due to Anll is Observed, the amplitude will be Eml = E1 J'm.(vl)" where E1 is the magnitude Of the electric vector Of the incident light. Similarly for the pattern due to An33, Em3 = E3 Jm(v3) where v1 = 11 L v = 2n:An33 L A 3 A Jm is the mth order Bessel Function, L is the width of the sound field, and X is the wavelength of the light. For small values of m and v, 11ml 11(V1)In E1 Am11 m m ( ) — = —+— = — — = R 31 Em3 23(v3f“ E3 An33 1 If the light is polarized at ABC to the sound field, E = E3 and l 3 R? = 32. 313. (32) n2 P33 16 where R? is the quantity that is experimentally determined. In this investigation, only the intensity in the first order diffraction pattern was measured, therefore, m.= 1. For Case 2, following the same procedure, one obtains: 2 - 1/n11 A11 = p11 S11 '2 1 l (26) A33 ' 1/n33 = 5 (p12 + p31 ' 2 PM) S11 From.Eq. (28), where the system is rotated, so that A11 and A33 are used instead of B11 and B33, the ratio can be written as i A -1/n'2 (p + p - 2 ~p )S' n'3 (p +p -2p ) m = = 12 31 Al 11 = 33L 12 31 1+1 = (33) T3. 2 ' 2 1 1 A11'1’“11 2 p11 S11 n11 (2 911) where the ratio KT is the experimentally determined quantity. SECTION D: Static Measurements: By applying a static or constant stress to the sapphire crystal and measuring the relative change in optical path length with a Babinet Compensator, a relationship invol- ving differences of photoelastic constants is obtained. To illustrate, following Vedam (lO)assume a crystal whose dimensions are oriented along the orthogonal set of axes has a strain (stress) in the X direc- tion and observation is in the 2 direction. From Eq. (11) 0 B11 ‘ B11 = P11 S11 0 B22 ' B22 ‘ P12 S11 (31) B - B ° — s 33 33 ‘ p31 11 B23 = Phl S11 . o _ 2 o _ _ ZAn Since B11 — l/nll, A.B11 — 11 n3 11 b t B - 3° 1 ual to A30 thus “ 11 11 3 eq 11’ ’ o ZAn B11 ' B11 " AB11 ’ 311 = p11511 (35) 1111 or, rearranging, _ _ 3 An11 ' n11 p11 S11 (36) 2 The differential path retardation is obtained from A(nllz) where z is the distance the light travels in the sample, and (nllz) is de- fined as the Optical path length. Mun.) = (Annxz) + (nuxm) = 511 (37> Note that Anll has been defined above and Am is the strain in the Z direction, i.e., Rewriting Eq. (33) and using Eqs. (32) and S33. (8), one obtains 3 s _ _ _ n p z _ 611 — An.z + n11.Az — ll :1 ll n11 z 813 T11 (38) The same procedure may be carried through for 622, corresponding to n22, with the result that, 3 _ _ n p S g _ 2 The relative retardation is defined as the difference of equations (35) and (3M) and this retardation is equal to the phase difference of the two beams of light, i.e., B . 2n = £3 (511 - 5 (no) 22) v.l 11’1kl I. 18 where B is some fraction of 2n which determines how much phase change occurs, and A is the wavelength of the light. Rewriting Eq. (36), using the values of 5 and 522 from Eqs. (3M) and (35), 11 B = 1- '“§1 p11 S11 2 - n z s T - l- '“32 p12 S11 2 - n s z T A 2 ll 13 11 A 2 22 13 ll .3 (n3p -n3p>s _ B — k 22 12 11 ll 11 + S13 T11 (n22 n11) (#1) 2 but n11 and n22, which correspond to the no of the crystal, are equal, so Eq. (37) may be written as B — 3'- “3 s ( - ) (112) "x -2—°- 11 p12 1)11 Assuming uniform stress throughout the medium of the specimen, S11 may be written as where 311 is the elastic modulus. Also, T11 = F/A where F is the force on the Specimen and A is the area over which the force is applied. Finally then, 2B x A P12 ’ p11 ‘ d nfl F 311 (M) Table I provides a complete list of observation directions, strain directions, and the relationships of photoelastic constants for those directions which are used in this investigation. The direction cosines of the normals to the cube faces with respect to an orthogonal set of axes are given to indicate the orientation of the cubes used. 19 HHQNXHJQN + Hmm + NHan Haam\m:am - Hma + maav mwamcowuwfimm mma - mam Ham - mag magmcowumaom HHm\mHm magmaowumamm mfixmnm mwxmuu H mgmmm. +mamwmm. +HHmo>m. JJQWFJ. +Hdmmmmo Immmwmmo +HMQOPM. +JHQHQH. +MHQ4MH. +HHQ®MN0 mmasma. +Hmammm. +aaasmm. +muammm. +Haaoem. :Hamw:. anmmmm. +mammom. assess“ +Hsammm. + simmea. +Hsammm. .mmaoem. +Hmammm. +aaaHmH. -muammm. +Haaama. aaamea. -Haamwm. +mmawmm. +Hmazmfi. +4Ha>mm. -mfiaoem. +Hfiammm. axmmwm. +::mmbd. +mmm owm. +Hmm mmm. +:Hm HmH. Imam wmm. +Ham #ma. :Hm wwz. + Man mmw. + «an mmm. Hag Has mem.+ Hma mam. + was mom. Ham Has mam.+ Hma mam. + was mom. manmaowumamm Awmsaaucouv H mgm. NI >. I pa »1 #1 P4 "k '}{ x x PI Pl #1 #1 mwxmu.x mfian.% mwxmu.x mwxmn.u mfixmu.u nfixmn.% unwfiq meMu.N meMn.N mfixmn.% maxwu.% mwxmu.x mwxmu.x venom Ammm.-.m>e..ov “Aime..omm..ov m Ao.o.av um umaasz Hmummuo EXPERDMENTAL PROCEDURE Dynamic Measurements: See Fig. h. The experimental apparatus consisted of a mercury arc light source filtered to obtain the 5h6l 8 green line of mercury. The light was focused on a vertical slit which acted as an image for the measurements. Light from.the slit was col- limated and passed through a Nicol polarizing prism, which polarized the light at #50 to the vertical. The light then passed through the sample perpendicular to the path of the sound. The WOllasUmn double image prism splifiithe light into two rays, one polarized in a direction parallel to the sound beam and the other polarized perpendicular to the sound beam. The light then passed through a focusing lens and analyzing prism, after which the two rays were Split by a right angle mirror, each ray going to a pick-up tube of a differential photometer. The pick-up tubes of the differential photomultiplier were adjusted so that their slits were in the focal plane of the focusing lens. The sound source is an X-cut lO megacycle quartz crystal, driven by a continu- ous wave, variable frequency oscillator. Standing sound waves are produced in the sapphire sample by the quartz crystal. These standing waves produce a diffraction pattern of the slit image in the focal plane of the focusing lens. If the investigations are carried out at low sound intensities, it can be assumed that light in the first order diffraction pattern treated by the standing sound waves is plane polar- ized, but rotated at an angle with respect to its original direction 21 r j VII/IIA’IIA ( ) {a ' L G 31730. LENS Pamzm m FILM SLIT FOGJSIPV. L1" 1‘ EXP. Cm. API‘AJAWS FIGURE X. . 23 of polarization. If the sound is a pure longitudinal wave, and the incident light is polarized at h5° to the vertical, then from Mueller (2), one may write, tan (a + 15°) = Jm (R1V1) _~_ R (1+5) Jm vl l-‘B for low v and m. See Eq. (31). (x is the angle of rotation of the plane polarized light, Rlv1 is the Raman-Nath parameter for light polarized perpendicular to the sound beam, and v1 is the Raman-Nath parameter for light polarized parallel to the sound beam. Mueller suggested three methods of measuring photoelastic constants. ‘Method "C" utilizes a double image prism, which separates the light into two components, one polar- ized parallel to the sound beam and the other polarized perpendicular to the sound beam, which is exactly the procedure used in this investi- gation. The experimental procedure used in taking measurements is to adjust the analyzing prism so that each phototube is subject to} equal intensity of light, i.e., at h5° to the vertical. The phototubes were adjusted to allow only the first order diffraction pattern to pass through their slits. The sound is then turned on and the analyzing prism rotated until the first orders were of equal intensity. With the differential photometer, this would correspond to a null or minimum reading. If the angle of rotation to equalize the intensities of the first order diffraction line is<1, then tan (h5° + h5°) h5°) 28 Determination of N (= phase change of 2n) for the Babinet Compensator: Sixty measurements were made to obtain the number of divisions on the drumhead of Babinet Compensator corresponding to a phase change of 2x between the ordinary and extraordinary ray. N for the Babinet Compensator used is 1288 divisions. Below are the tabulated values of n/kg. for the determination of p11 - p12 and p13 - p33: P12 ' P11‘ 2&2; 111.1221 n_/kB; 111.1%; 5.93 2.89 3.85 1.13 n.76 1.73 n.18 3.7M 3.92 1.58 1.81 6.18 5.60 5.20 6.15 n.62 1.13 1.14 3.85 3.66 5.53 1.21 1.51 5.57 5.38 5.09 5.2M u.58 3.66 1.18 1.25 3.70 3.66 . 1.07 1.58 b.36 3.81 u.18 6.56 6.52 nav/kg. = b.6h p13 ' P33‘ n/kg. n/kg. n/kg. n/kg. 2.12 1.87 2.67 2.53 2.53 2.31 2 M5 1.9M 2.31 2.31 2.19 2.09 2 3h 1.9M 1 13 2.27 2.93 2.09 2 B5 2.75 2.93 2 56 3 on 3.59 2.56 1.51 2.82 3.11 2.53 1.32 1.79 1.65 1.90 2.20 2.6M 2.86 2.h5 1.50 2.12 2.82 nav/kg. = 2.31 1' ‘llel ‘I'I ‘|« N mmeE em 6. m. S u. o. s N .m :QN 1v .vQN +5n— +~_Q to am 21 .N a .m + E X K rm 91 I I (3917 + mum ON N. O. .10. {.22 .m “£30: .x. 1 f w" .4 1x lxl 1x, .1 .x. in q 4 d q q 1 q I 1 J J 1 d a 1 d i l x [Kl Ix! .5 III! /_ (0917+D)u01 31 Sample calculation of Photoelastic Constants: Repeating Eq. (uh), one may write, P12 - P11 = Z 2 n: W S o g 11 where, n/w = 1.61 div./kg. N = 1288 div. = 5.161 x 10'5 cm. no = 1.771 (Reference (12)) n2 = 5.551 A = (1.60)2 mmz. d = 1.60 cm. g = 980 cm./sec.2 311 = 2.28 x 10"13 cmZ/dyne (Reference (8)) p12 - p11 = (1.65)(5.161)(1o'5)(2)(1.so)2 _ (10)3(1.288)(1.60)(5.55h)(9.8)(10)2(2.28)(10) 13 p12 - p11 = (cm.)(cm)2 2 = (unitless) Lgmis’XcmXcmMcmL (sac)2