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Brillouin Scattering In Molecular Crystals: Sym-Trichlorobenzene

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Brillouin Scattering In Molecular Crystals: Sym-Trichlorobenzene

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Content BRILLOUIN SCATTERING IN M O LE C U LA R C R YSTA LS sym -TRICHLOROBENZENE by Douglas Lynn Swanson A Dissertation Presented to the FA C U LTY O F THE G R A D U A T E S C H O O L UNIVERSITY O F S O U T H E R N CALIFORNIA In Partial Fulfillm ent of the Requirements for the Degree D O C T O R O F PHILO SO PHY (Chemistry) August 1973 INFORMATION TO USERS This material was produced from a microfilm copy of the original document. While the most advanced technological means to photograph and reproduce this document have been used, the quality is heavily dependent upon the quality of the original submitted. The following explanation of techniques is provided to help you understand markings or patterns which may appear on this reproduction. 1.The sign or “target" for pages apparently lacking from the document photographed is "Missing Page(s)". If it was possible to obtain the missing page(s) or section, they are spliced into the film along with adjacent pages. This may have necessitated cutting thru an image and duplicating adjacent pages to insure you complete continuity. 2. When an image on the film is obliterated with a large round black mark, it is an indication that the photographer suspected that the copy may have moved during exposure and thus cause a blurred image. You will find a good image of the page in the adjacent frame. 3. When a map, drawing or chart, etc., was part of the material being photographed the photographer followed a definite method in "sectioning" the material. It is customary to begin photoing at the upper left hand corner of a large sheet and to continue photoing from left to right in equal sections with a small overlap. If necessary, sectioning is continued again — beginning below the first row and continuing on until complete. 4. The majority of users indicate that the textual content is of greatest value, however, a somewhat higher quality reproduction could be made from "photographs" if essential to the understanding of the dissertation. Silver prints of "photographs" may be ordered at additional charge by writing the Order Department, giving the catalog number, title, author and specific pages you wish reproduced. 5. PLEASE NOTE: Some pages may have indistinct print. Filmed as received. Xerox University Microfilms 300 North Zeeb Road Ann Arbor, Michigan 48106 [ SWANSON, Douglas Lyim, 1946- BRILLOUIN scattering in molecular crystals SYM-TRICHLORDBENZENE. University of Southern California, Ph.D., 1974 Chemistry, general | University Microfilms, A XE R O X Company, Ann Arbor, Michigan THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED. UNIVERSITY O F S O UTHERN CALIFORNIA TH E GRADUATE SCHOOL U N IV E R S IT Y PARK LOS ANGELES. C A L IF O R N IA 9 0 0 0 7 This dissertation, written by Douglas Lynn Swanson under the direction of hXs.... Dissertation Com mittee, and approved by a ll its members, has been presented to and accepted by The Graduate School, in partial fulfillm ent of requirements of the degree of D O C T O R O F P H IL O S O P H Y Dean DISSERTATION COMMITTEE Chairman A C K N O W LE D G M E N T S I wish to express m y sincerest appreciation to Professor David D ow s for everything he has contributed to this work. He has been a source of encouragement throughout m y graduate studies. I also wish to thank Dr. Louis C. Brunei for assistance with the programs used in this work. I a m indebted to the Chemistry Department at the University of Southern California for teaching assistantships, and to the NSF and N D E A for fellowships. Fin ally, I wish to thank m y family for th eir help and understanding throughout the course of this work. tABLE O F C O N TE N TS Page A C K N O W L E D G M E N T S ............................................................................................ i i TA B LE O F FIGURES........................................................................................ iv LIST O F TABLES................................................................................................ v iii C H A P T E R I INTRODUCTION................................................................................. 1 I I TH E O R Y O F BRILLOUIN SCATTERING............................................. 6 I I I TH E O R Y O F TH E FABRY-PERO T INTERFEROMETER......................... 18 IV EXPERIMENTAL................................................................................. 39 V ASSIG NM ENT O F LATTICE FREQ UENCIES A N D C O M P A R IS O N W ITH C A LC U LA TED FREQ UENCIES ................................................ 70 VI CALCULATION O F ELASTIC C O N S TA N TS A N D VELOCITIES O F ACOUSTIC W A V E S Case 1.................................................................................... 91 Case I I ................................................................................ 95 V II C O M P A R IS O N O F EXPERIMENTAL A N D C A LC U LA TED ACOUSTIC VELOCITIES ................................................................ 108 REFERENCES............................................................... 117 APPENDIX: CALCULATION O F REFRACTIVE INDEX...................'.................... 121 TABLE O F FIGURES Figure Page 1 Representation of Bragg reflection of two monochromatic lig h t beam s from acoustic waves in a medium................................................................................ 8 2 Representation of acoustic wave velocity relation to incident and scattered lig h t, exemplifying Doppler s h ift ............................................................................ 10 3 a. Conservation of m om entum diagram for scattering experiment, k, k ', and K are respectively the wavevectors of the incident lig h t, the scattered lig h t, and the scattering phonon.................................................................................... 15 b. Wavevector balance diagram........................................... 15 4 Interference diagram for Fabry-Perot interferometer. Plates I and I I are the interferometer plates with reflecting surfaces Sj and S^ respectively, separated by distance d. 0 is the angle of incidence.................................................................................... 20 5. a. Ring pattern of Fabry-Perot interferometer when illuminated with a monochromatic frequency............................................................................ 24 b. Ring pattern of Fabry-Perot interferometer when illuminated with two frequencies x and X + AX.................................................................................... 24 iv Figure Page 6 Behavior of i ^ / i ^1) as a function of order number p at the sam e incidence angle. F is the parameter defined in equation (4 5), and R is the reflectance of the p la te s ............................................................................ 31 7 Superposition of the intensity curves for two monochromatic frequencies incident upon Fabry-Perot interferometer as a function of order number p. . . . 36 8 Crucible used for crystal growth of 1,3,5-CgHgClg . . 42 9 X-ray photograph of 1,3,5-CgHgClg used to distinguish a and b axes................................................................................ 44 10 Schematic diagram of experimental setup. Lj denotes the focussing lens, S the sample, F the iodine f il t e r , Lg the gathering lens, F-P the Fabry-Perot interferometer, Lg the collecting lens, A the aperature, the floating lens, and P M T the photomultiplier tube................................................................ 46 11 Diagram of iodine f i l t e r ........................................................ 48 12 Diagram of etalon holder........................................................ 52 13 Diagram of scattering angles in a typical scattering experiment. < j> j denotes angle of incidence, ^ the angle of refraction, e the scattering angle when crystal is in position, and ©' the scattering angle in the absence of the c r y s ta l........................................................................................ 59 V Figure 14 Spectrum of Brillouin scattering from longitudinal phonon in the (1,0,0) direction with a scattering angle of 90°. Measured frequency s h ift is 0.301 cm"*, thus the peaks shown belong to adjacent orders of in terferom eter.................................................... . . . . 61 15 Series of B rillouin spectra produced by scattering from quasi-longitudinal and quasi-transverse phonons in the (1,1,0) direction, taken at various scattering angles. Pattern shows crossover of peaks from the two different phonons. At a scattering angle of 86° the peaks from the two phonons of adjacent interferometer orders are coincident . . . . 63 16 Polarized R am an spectra; from the six independent p o la riza b ility components of l,3,5-CgH3-C l3 ................ 66 17 R am an spectra of polycrystalline sample of l,3,5-CgH3Cl3 at 20°K and 90°K. A s expected, R am an lines sharpen at low temperatures because of less rotational disorder and s h ift to higher frequency due to an increased force constant.................................... 69 ’18 Representation of a cube of volume AxAyAz acted on by a stress of -X (x) on the face at x, and a stress X of X (x + x) on the parallel face at x + A x ................ 84 X 19 Plot of t o vs. K for a model la ttic e with interactions only between nearest neighbor planes. The interplanar force constant is Cj and the interplanar spacing is a................................................................................................ 110 vt Plot of observed frequencies vs. K/K^ for the quasi-longitudinal and quasi-transverse waves along the (0,1,1) direction ............................................ Coordinate system applied to l,3,5-CgHgCl2 molecule used in calculation of refractive indices ................ LIST O F TABLES Table Page 1 Lattice vibrations of l,3,5-CgHgCl2 grouped according to species representation and p o lariza b ility components responsible for R am an a c tiv ity ............................................................................ 72 2 Comparison of calculated and observed la ttic e frequencies for l^ S -C g l^ d ^ . Calculated frequencies obtained using Bonadeo's parameter set 1 ............................................................................................ 74 3 Potential parameters. A has units of Kcal mole"1, -1 6 B has units of A , and C has units of Kcal A m o le " *......................................................................................... 77 4 Definition of parameter Q .• in terms of elastic wave IJ propagation vectors and elastic constants, e .g ., Q = CiiK^ + C c c K + CrrK^ + 2C, rK K sxx 11 x 66 y 55 z 15 z x 5 Scattering coefficients for acoustic; waves in the C l,1,0) direction, under d iffe re n t polarization configurations. ........................................................................ 96 6 Scattering coefficients for acoustic waves in the (1,0,0) direction, under different polarization configurations................................................................................ 101 v i i i Table 7 Elastic constants determined from acoustic waves in the (1 ,0 ,0 ), (0 ,1 ,0 ), and (0,0*1) directions................ 102 8 Designation of elastic constants to be used in equations (99) and (100) for acoustic waves in the (1 ,1 ,0 ), (1 ,0 ,1 ), and (0,0,1) directions................ 104 9 Measured velocities of acoustic waves in 1,3,5-C6H3C13 106 10 Elastic constants calculated from observed velo cities.................................................................................... 107 11 Comparison of calculated and experimental velo cities.................................................................................... 114 12 Comparison of elastic constants determined from calculated and experimental velo cities............................ 116 13 Bond p o la riz a b i.litie s ............................................................. 124 C H A PTER I INTRODUCTION 1 2 In 1922 (1) Brillouin predicted that i f monochromatic radiation were scattered from an optical medium a shifted frequency could arise in the scattered radiation resulting from a Doppler s h ift of the incident frequency due to scattering from sound waves in the medium. H e further predicted that the frequency s h ift would be a function of the velocity of the sound waves and the scattering angle. The observation of the shifted frequencies w as fir s t m ade by Gross (2) in 1930, and later confirmed by others (3 ). With conventional lig h t sources the B rillouin scattering in both liquids (4) and solids (5) was studied. The advent of the laser greatly enhanced the experimentalist's a b ility to observe such scattering and increased the precision with which the velocities of the scattering waves could be measured (6, 7). In liquids, the velocities of sound waves were measured and the dispersion of the velocity was used to study relaxation effects in many liquid systems (8 ). B rillouin scattering at very small angles has been detected in liquids (9) and is useful in the study of low frequency phonons. There has also been interest in the relation between the intensities of B rillouin and Rayleigh scattering in liquids as a function of thermodynamic properties of the system. The study of B rillouin scattering in crystals has mainly been centered on ionic crystals (5, 10), primarily those with cubic structures. B rillouin scattering provides a method of 3 measuring the variations of sound velocity for different directions in a crystal. From the measurement of sound velocities the elastic constants m ay be calculated and have been determined for a number of crystals (10). The width of the B rillouin components gives a measure of the attenuation of the sound waves in the scattering medium and this has also been investigated (11). Although the B rillouin scattering from ionic crystals has received quite extensive investigation, l i t t l e work has been done on the B rillouin scattering from molecular crystals (12). I t is lik e ly that such work could provide information on a variety of interesting topics. Measurement of the acoustic velocities would permit the calculation of elastic constants, m any of which are unknown, and phonon attenuation and dispersion could be investigated. Temperature dependence work could provide interesting results, especially around phase transition, and the frequencies of the acoustic waves could be related to various potential models of current interest in the fie ld of molecular crystals. In general, molecular crystals seem a logical and reasonable choice for the extension of B rillouin scattering studies. Problems in the obser vation of B rillouin scattering from molecular crystals do exist and have to be overcome. The primary problem is one of obtaining a suitable crystal. Since excessive scattering of the incident frequency is a severe problem, i t is of crucial importance that crystal imperfections are reduced to a minimum. L ittle work has 4 been done on molecular crystals that requires comparable purity and crystal perfection. l,3,5-CgH3Cl3 was chosen as an example of a molecular crystal on which the use of B rillouin scattering studies could be extended. The choice of this crystal was made for a variety of reasons. F irs t, i t is a molecular crystal typical of a class of crystals that have been of interest in the study of intermolecular potentials (13). Second, 1,3,5-CgHgClg is an orthorhombic crystal with unknown elastic constants which could be determined from a measurement of the velocities of acoustic waves. Another reason for its choice was that suitable crystals could be grown of proper size and purity to allow the observation of B rillouin scattering. Since l,3,5-CgHgCl3 is a anisotropic crystal i t is important that the formalism used to describe the scattering allows for the anisotropy of the scattering medium. Chapter I I presents a classical theory of B rillouin scattering including scattering from anisotropic media. A n experimental problem in the observation of Brillouin scattering is the separation of the incident and scattering frequencies. In order to achieve this separation an instrument with a very high resolving power is required. A Fabry-Perot interferometer satisfies this requirement and such an instrument was constructed in this laboratory for the gathering of B rillouin data. A discussion of the theory of this instrument is presented in Chapter I I I , and its 5 method of use is described in Chapter IV. Also included in Chapter IV are the problems of crystal growth, determination of scattering angle, and s ta b ility requirements of the laser used. A s previously mentioned, the measurement of the velocities of acoustic waves allows one to calculate the elastic constants of the scattering crystal. This calculation was done for 1,3,5-CgHgClg and the results are presented in Chapter VI. This section also summarizes Fabelinskii's discussion of expected intensities for various scattering configurations as applied to an orthorhombic crystal system (14). In addition to the calculation of elastic constants, the measurement of B rillouin frequencies affords us the chance to test the v a lid ity of various semi-empirical potential models of intermolecular forces in molecular crystals (13). In addition to the observation of B rillouin scattering along six different wavevector directions, the polarized R am an spectrum of ljSjB-CgHgClg was done (see Chapter IV ), and the assignment of la ttic e frequencies is discussed in Chapter V. The comparison of the calculated and experimental values of la ttic e frequencies and acoustic velocities, in Chapters V and V II, indicates that the potential models in use are able to predict experimental values to a reasonable degree of accuracy. C H A PTER I I TH E O R Y O F BRILLOUIN SCATTERING 6 Let us fir s t consider B rillouin scattering in an elementary fashion. C lassically, B rillouin scattering can be viewed as the "Bragg reflection" of lig h t waves from a periodic modulation in a medium. The periodic modulations responsible for lig h t scattering are density fluctuations. W e can write the fluctuations in density as (15): r Ap = 3p 8P A P S 3p as AS, (1) P i . e . , density w ill change with pressure, P, or with entropy, S. Entropy fluctuations do not propagate and give rise to Rayleigh scattering. W e are interested in the propagating pressure fluctuations, i.e . acoustical waves, which give rise to B rillouin scattering. In other words, B rillouin scattering is the scattering of lig h t from sound waves in a medium. Let us fir s t consider the reflection of lig h t from sound waves in an isotropic medium, e.g. a liquid. Figure 1 represents such a situation. Consider two p a ra lle l, monochromatic lig h t rays, and Rg, incident on the medium. To obtain constructive in te r ference between rays Rj and Rg-the following condition must be satisfied: b S L = 2a_ sin<e/2) (2) n s where n is the refractive index of the medium and x Q is the wavelength of the lig h t. From this equation w e can see that the 8 0/2 Figure 1 Representation of Bragg reflection of two monochromatic lig h t beam s from acoustic waves in a medium. g shortest wavelength of sound responsible for scattering lig h t is equal to one half the wavelength of lig h t in the medium, that being when 0 = 180°. By scattering through smaller angles w e observe longer wavelength sound waves. Since the sound waves responsible for scattering are not stationary, but are propagating through the medium with a velocity V g, the lig h t scattered by them undergoes a Doppler s h ift in frequency. Figure 2 represents the case of lig h t scattered from propagating sound waves in an isotropic medium. Consider a photon entering the medium from point PQ, being scattered at point P^, and detected at point Pg. The velocity of the phonon from which the photon is scattered is Vg and its direction is indicated in Figure 2. Its component in the directions toward P Q and P g is V g sin (0 /2 ). I f n is the refractive index of the medium, then c/n is the speed of the photon, in the medium. Therefore, the relative velocity of a photon, coming from P Q, and the point Pj is c/n + V s s in (0 /2 ). In this case an observer on point P^ does not see the true frequency, vQ, of the photon, but sees an apparent frequency, v ', which is given by the ratio of the relative velocity of the photon to its wavelength, i . e . , v- = vQ I + Vs sin ( 0/ 2) c/n (3) I f the scattering can now be considered as the absorption and reemission of the apparent frequency v ', undergoing a second Doppler Figure 2 Representation of acoustic wave velocity relation to incident and scattered lig h t, exemplifying Doppler s h ift. 11 s h ift due to the relative velocity of Pj and Pt he frequency seen at P g w ill be v" = v 1 1 + Vs sin (0/2) 1 c/n (4) Substituting v ' from equation (3) gives v" = v. 1 + 2V sin (0/2) Vs2 sin 2(0/2) c/n (c /n )‘ (5) Since v| is of the order of 4 x 10° m^/sec^ and (c/n)^ is of IC O o the order of 4 x 10 m /sec , the last term of equation (5) becomes vanishingly small. Therefore the frequency s h ift m ay be expressed as 6 2 , 1 + 2VS sin (e/2)] c/n ( 6) W e must also include the fact that the phonon causing the scattering m ay be moving in the opposite direction, i.e . with a velocity -Vs. In this case i t is easily seen that the frequency s h ift is expressed as: v" = V, 2VS sin (0/2)' c/n (7) From equations (6 )-and (7) i t is evident that upon analyzing the frequency of the scattered lig h t w e should observe a doublet, symmetrically surrounding the incident frequency, vQ. Equations (6) and (7) relate the frequency s h ift to the velocity of the scattering phonon. By combining the results of these equations with the restrictions of constructive interference expressed in equation (2) w e can obtain the relationship between the frequency s h ift observed and the frequency of the phonon. Defining Av as vQ - v", from equation (7 ), w e obtain: 2nV sin(0/2) A v = vo c • (8) Knowing that Vg = A svs, upon substitution of equation (2) into equation (8) w e obtain: a * = i r H - " s (9 > Since the phonon moving in the opposite direction gives a negative frequency s h ift, i t is easily shown that Av= ±vg, (10) where vg = the frequency of the sound wave. From equation (10) i t is evident that the frequency of the scattered lig h t is shifted, either up or down, by an amount equal to the frequency of the scattering sound wave. The wavelength of this sound wave can be calculated from equation (2 ), the frequency is measured experimentally according to equation (10 ), and the velocity of the sound wave can be calculated from the product of these two, i.e . 13 v_ A _ v - s o s 2n sin(e/2) (11) In m any discussions of B rillouin scattering (14) i t is co m m o n to consider the wavevectors of the incident photon, the scattered photon, and the scattering phonon. The photon can create or absorb a phonon, with the photon being scattered in the process. I f the incident photon enters with frequency t o and wavevector k, i t is scattered with frequency to1 and wavevector k 1. I f n is the refractive index of the medium, k is defined as k = 2irn > * A o (12) and the frequency is determined by the relation _ ck t o h , (13) with t o being defined as t o - 2irv. (14) The phonon created or absorbed by the photon of wavevector k is shown in Figure 3, and has wavevector K, with K being defined as i/ _ 2ir K ~ T ~ ' A s (15) Since, in the co llisio n , the wavevector and energy must be conserved, the following relationships must hold: k = k' + K (16) 14 and " fito = ‘fiio1 + tin, (17) where n is the angular frequency of the phonon, i.e . n = 2irvs. Figure 3b shows the wavevector balance of Figure 3a. I f , in Figure 3b, k = k ', (which is a very good approximation as w e shall see la te r), the triangle is isosceles, and sin (0/2) * (K /2 )/k , or K - 2k sinCe/2). (18) Multiplying equation (18) by Vs, the velocity of the phonon, and substituting for k from equation (1 2), w e obtain: 2io n V sin (e/2) VSK ----------- i _ ---------- . (19) But, V K = V„(2irA J = 2irv =n. s sv s' s Therefore, a = 2 n"sin( e/2> . (20) 0 which m ay be rewritten as 2nV sin(0/2) Av = v = v ------------ . (20) S O c 15 Figure 3 a< Conservation of m om entum diagram for scattering experiment. k,k' and K are respectively the wavevectors of the incident lig h t, the scattering lig h t, and the scattering phonon. Figure 3 b . Wavevector balance diagram. 16 So far w e have considered media with a single refractive index, n. This formalism is complete for isotropic media, e.g. liquids and isotropic solids, but for anisotropic media, e.g. a birefringent crystal, w e must account for the fact that the refractive index can be different in different directions. Let us define n ^ and ng as the refractive indices of the crystal in the directions of incidence and scattering, respectively. Figure 3 can s t ill represent the scattering process i f w e define k and Applying simple trigonometric relationships to the triangle in Figure 3b, w e obtain Substituting k and k' from equations (22a) and (22b), w e obtain Substituting for K from equation (15) and multiplying by Vg w e find k 1 as 2irn. k = x (22a) o k' = s (22b) X o K 2 = k2 + k '2 - 2 k k' cos e. (23) (24) v (25) 17 Equation (25) expresses the relationship between the frequency of the phonon, the velocity of the phonon, and the scattering angle in an anisotropic system. C H A P TE R I I I TH E O R Y O F TH E FABRY-PERO T INTERFERO M ETER 18 19 A primary experimental problem in the observation of B rillouin scattering is the separation of the shifted and incident frequencies, which, in general d iffe r by a very small amount. In order to fa c ilita te this separation an instrument with a very high resolving power is required. A commonly used instrument for this purpose is the Fabry-Perot interferom eter.(16). The Fabry-Perot interferometer consists of two f la t , parallel plates of glass or quartz, separated by a distance d, as represented in Figure 4. The inner surfaces of the plates, and Sg, are coated with a p a rtia lly transparent film of high re fle c tiv ity . The plates are positioned such that the surfaces S. and S are p a ra lle l, while the individual plates themselves 1 2 are s lig h tly prismatic in order to avoid unwanted effects due to reflections at the outer surfaces. Let us consider "a plane wave, R, incident upon the fir s t plate at point A, with an angle of incidence < j » . After refraction through plate I and partial reflection at point B, the ray enters the space between the pilates and fa lls upon point C on the surface S£. A portion is refracted through plate I I and emerges as ray 1, while the remaining portion is reflected at point C, p a rtia lly reflected at points D and E, and refracted through plate I I as ray 2. Due to sim ilar multiple reflections from the surfaces Sj and Sg, this process is continued to form rays 3, 4, 5, . . . , a ll emerging parallel to each other, with th eir intensities decreasing due to an 20 Figure 4 Interference diagram for Fabry-Perot interferometer. Plates I and I I are the interferometer plates with reflecting surfaces Sj and S2 respectively, separated by distance d. < ( > is the angle of incidence. 21 increasing number of reflections. Since rays 1, 2, 3...................m ay undergo interference processes, i t is important to know th eir relative phase difference in a co m m o n plane perpendicular to th e ir direction. Considering only rays 1 and 2, le t us take this plane through the points E and F on Figure 4, Starting from point A, ray 1 has a path length of A B + B C + CF, (26) while ray 2 has a path length of A B + B C + C D + DE. (27) Subtracting equation (26) from equation (27) ..yields the path difference: path difference = C D + dT + CF. (28) The phase difference between these two waves can be found by multiplying the path difference by 2-ir/x^, where x^ is the wavelength of lig h t in the medium i . I f w e le t x, be the wavelength a in a ir , i . e . , between the plates, and X g be the wavelength in the plates, w e find the phase difference rays 1 and 2 to be: phase difference = 2tf C D + D E C F a 9 (29) 22 The term in brackets in equation (29) is usually called (17) the order number, p, and its value gives the number of wavelengths in the path difference. Recalling that a^ = * a/ ng> where n^ = sin (j>/sin i p , from geometrical considerations of Figure 4 w e can see that: p = d/cos < j > + d/cos < j > _ 2(d/cos j>) sin<j) sin i f r ^ o ) a g Substituting for g w e obtain: p . z i cos t _ (31) a I f w e le t n be the refractive index between the plates w e see that P = 2 4 " c°. s ♦ (32) A where A is the wavelength of the incoming ray. I t is interesting to note that in equation (32) the refractive index and thickness of the plates are not functions of the order number. Disregarding losses due to internal reflections within the glass plates then, the Fabry-Perot interferometer actually acts as a plane, parallel plate of a ir , the faces of which are defined by high re fle c tiv ity . W e are now ready to consider interference effects between rays 1 and 2 . A phase difference equal to an integral multiple of 2ir w ill give rise to constructive interference, while a phase difference equal to a multiple of ir w ill produce destructive interference. M axim um intensity, therefore, corresponds to integral values of 23 the order number p, while minimum intensity corresponds to values of p equal to an integer + h . The pattern of this intensity distribution must now be determined. I f a lens is placed in a position to collect the lig h t emerging from the plates, the rays 1,2,3. . ., which were a ll s p lit o ff the original ray R, w ill a ll be focused to a point in the focal plane of the lens. I f < j > is an angle such that p is an integer, this point w ill correspond to an intensity maximum. All rays incident at angle $ w ill then produce, in the focal plane of the lens, bright points, these points forming a c irc le , with the radius of the circle being a function of $ and the focal length of the lens. According to equation (32) there can be m any angles, < j> ^ , $ 2* ^3» ••• » f ° r which p w ill be an integer. Again, a ll rays incident under an angle < | > . such that p is an integer w ill form a circle of bright points in the focal plane of the lens, with radius rU ..). The center ring is formed by the lig h t entering the plates at the angle < ( > • nearest zero. The ring system obtained when the interferometer is illuminated with monochromatic lig h t is illu strated in Figure 5a. W e must now ask, how would the interference pattern change i f the wavelength of the monochromatic lig h t incident upon the plates changes by a small amount, ax? B y differentiation of equation (32) w e obtain: p ax = -2 d n sin < j > A < j > (33) Figure 0 8 .. Ring pattern of Fabry-Perot interferometer when illuminated with monochromatic frequency. A A + A A Figure 5 b . Ring pattern of Fabry-Perot interferometer when illuminated with two frequencies, x and x + a x. 25 which shows that an increase in wavelength produces a decrease in the angle < f > which corresponds to a m axim um in intensity. Therefore, an increase in A produces a shrinking of the ring pattern, while a decrease in A produces a swelling of the ring pattern. I f both A and a + a a were incident upon the plates, each would have its o w n ring pattern, with the A ring pattern between the A + a a pattern, as shown in Figure 5b. Now, i f a a could be shifted continuously, what would w e observe? Let the fixed wavelength be Aj, and the variable wavelength be a^. Starting at A ^ - a2 w e would see only one set of rings. With a gradual increase in a2, the A 2 rings would separate from the Aj rings, slowly decrease in size, and fin a lly com e to coincide with the Aj rings again. In this case, a a2 ring of order p would separate from a Aj ring of order p and decrease in size until i t coincided with a A ^ ring of order p+1. At this point, w e can no longer distinguish between Aj and a2. The change in wavelength necessary to s h ift the ring system to this extent is called the free spectral range, a a . In this case, equation (32) te lls us that (p + 1) Aj = 2 d n cos <p, and p A 2 = 2 d n cos < ) > . 26 For the sam e angle of incidence, w e have (p +1) = p A s Therefore, X x 2 AX * X ? - X = = y j -n j '- v . • 2 i p 2d n cos < | > Replacing x^ by x, for the general case, w e obtain: x2 Free spectral range = ax = ^ - cq- s—- . (34) Since, in practice, the orders used have < j > = 0, w e can approximate the free spectral range as: I 2d AA = h • (35) ■ ? To express this in wave numbers w e divide by x to obtain A v = ^2* = 23* * A n important factor in determining the resolution of a Fabry-Perot interferometer is the re fle c tiv ity of the plates. To determine the optimum re fle c tiv ity w e must inspect the intensity as a function of re fle c tiv ity . Let us again consider a lig h t wave incident upon the plates as depicted in Figure 4. For a wave traveling through the plates, incident upon or Sg, le t r be the reflection coefficient (ratio of reflected and incident amplitudes) and t be the transmission coefficient (ra tio of transmitted and incident amplitudes). For a wave traveling between 27 the plates, le t r 1 be the reflection coefficient and t 1 be the transmission coefficient, at the surfaces Sj or Sg. Recalling that the consecutive rays emerging from Sg d iffe r in phase by 2 p, w e can now calculate the complex amplitudes of the waves transmitted through the plates. Ray 1, which has undergone transmission through Sj and $2 , would emerge with a wave amplitude of: t t 1 A ^ where A ^ is the amplitude of the incident wave. Ray 2, which has undergone the sam e transmissions as ray 1 , plus two reflections, would emerge with a wave amplitude of: t r ' r ' t ' A^> e"i27rp. Likewise, ray 3 w ill emerge with a wave amplitude of: t r ' r V r ' t ’ A ^ e"i4lTp. (These amplitudes a ll disregard an unimportant constant phase factor com m on to a ll rays s p lit o ff from the original ray R.) Thus, for a ll rays 1, 2, 3, . . . k, originating from ray R, w e have a systematic decrease of wave amplitude; t t ' A ^ , t t ' r ' V ^ e ' 2^ , t t ' r ' V ^ e " 4*1 3 , . . . , t t V ^ ' ^ A ^ V 1 ' 2^ - 1). 28 I f the transmitted waves are superimposed, as in the focal plane of a projecting lens, A ^ , the amplitude of the transmitted lig h t, for k waves, is given by the following expression: A ^ ( k ) = t t ' { l+ r ,2ei27rp + r ,4e"*4lTp+ . . . + r ,2 (k -l)e-i2 n p (k -l)}A( i) (37) From classical optics w e know that: t t ' = T, and, r ' 2 = R where R and T are the re fle c tiv ity and transmissivity, respectively, of the plate surfaces, and are related by R + T = 1. (38) Substituting for t t - and r 'r ' in equation (37), w e find: A ^ O O = T {1 + Re“l27rp + R2e~l4lTp + . . . + R( k - l) e- i 2T rp ( k - l ) } A( i) (39) This sum m ay be rewritten as: AW (k) = T A( i) 1-Rke“i2irp - i 2irp (40) > R e I f the number of waves is large, as k », since R < 1, equation (4(J) reduces to 29 A ^ ( k ) = T A( l ) -i2irp (41) [1 - R e and the corresponding intensity of the transmitted lig h t is 1 or, j ( t ) = f S V h W * = r(t) = tV 1 V 1)* 1+R -2R cos;.2irp (42) (1-R)^ + 4R s in \p (43) Since A ^ A ^ * = 1 ^ , the intensity of the incident lig h t, using equation (38) w e obtain: i d ) r(t) = 1 + — — y Sin2(irp) tt-Rr (44) I t is convenient to define a parameter, F, by the formula: F = 4R (1-R) 2 • (45) Substituting equation (45) into equation (44), w e fin a lly arrive at: ,(t) . ____ L(1> r 2 1 + F sin up (46) (t) A s before, m axim um intensity, I vwmax> occurs when p has integral values, and minimum intensity, occurs when p equals an integer + h . Therefore, 30 j ( t ) = j ( i ) m ax (47) and (48) The behavior of as a function of order number at the sam e incident angle, for various values of F, is shown in Figure 6 . From Figure 6 i t is apparent that as R is increased the intensity of the minima of the transmitted pattern fa lls , and the maxima becom e sharper u n til, when R approaches unity, so that F becomes very large, the intensity of the transmitted lig h t is very small except in the immediate neighborhood of the maxima. At this point, the pattern of transmitted lig h t consists of narrow, bright rings on an almost completely dark background. The sharpness of the rings is measured in terms of the finesse, F, of the interferometer which is defined as: where the half-width is the width between points on either side of a maxima, where the intensity fa lls to half its m axim um value. The finesse, F, then is the ratio of adjacent ring separation to the half-width. In order to calculate the finesse in terms of re fle c tiv ity , w e should consider the case in which the points of half m axim um intensity, for order p, fa ll at p ± e/2. Hence, according to equations (46) and (47) w e have: v = free spectral range half-width (49) 31 I (t) ( 0 ^F=0.2 (R-0.046) //|Y F = 2 (R.-0.27) ' / | U f=20 (R-0.64) P + 1 *rF=200 (R=0.87) - > p Figure 6 Behavior of as a function of order number p at the sam e incident angle. F is the parameter defined in equation (45), and R is the reflectance of the plates. 32 I(t) a i[l ) _________ ?_______ (50\ 7 ^ I ( t ) m ax 1 + F Sin27r(p + e/2) ' Since p + e/2 is the point of half m axim um intensity, for integral values of p, equation (50) reduces to: J s = ---------------------- • (51) 1 + F sin2 lo using the approximation for small e, that sin e = e, w e obtain 1 V.-= __________________ » * T 2 T 1 + F which yields: e - t/F ir (52) Since the separation of adjacent rings corresponds to a change of p by 1, the finesse is then: F = . (53) Substituting for F from equation (45), w e find: 1 -R (54) 33 In order to optimize the Fabry-Perot interferometer w e want to maximize the finesse and the m axim um transmission. So fa r, this appears to be no problem; simply le t R approach unity. U p to this point, however, w e have neglected the absorption of the reflecting film on the plates. I f w e define the fraction of lig h t absorbed by the reflecting film as A = 1 -R - T, (55) and using equations (43) and (45) w e find T ( t ) l_ r a 1 -R 4 — 5— ‘ <56) 1+ F sin irp At peak transmission, w e see that 1 ^ = 1 - f — - — I 1 (57) L m ax 1 [ 1 - Rj 1 • I t is clear from equation (54) that increasing R w ill increase the finesse, but from equation (57) w e see that increasing R w ill decrease the m axim um transmitted lig h t. Thus, maximizing both T and I ^ m ax are incompatible requirements and a compromise must be reached. I t must be mentioned here that a phase change upon reflection at the coated surfaces has not been included in the preceding discussion. The introduction of this phase change adds a constant to the phase for each order and is equivalent to 34 increasing the optical distance between the plates (38). For our purposes this term can be conveniently neglected. The preceding discussion has assumed that the reflecting surfaces are perfectly fla t and p ara lle l. In practice, of course, the flatness of the surfaces, and consequently the separation of the plates, varies by a small amount. The effect of such variations has been studied by Dufour and Picca (19) w ho found that as R -* 1, the finesse, F, approaches a lim it, Fd, which depends only on the surface defects of the plates. Increasing R such that theoretically F > Fd results in a decrease in peak transmission with l i t t l e compensating increase in finesse. The value of Fj depends on the particular form of the surface defects, but for the case where there is a slight spherical curvature of the plates such that d changes by an amount A / m , i t is found that Fd = m/2 (58) The resolving power of a Fabry-Perot interferometer is generally expressed in terms of its finesse. In order to derive the relationship between the resolving power and the finesse w e must consider the case of two "just resolved" lines. I f two wavelengths of lig h t "just resolved" by an instrument are a q and Aq + A A , the quantity a q / a a is called the resolving power, i.e., R _ » (59) AA 35 where R is the resolving power. The criterion of being "just resolved" was introduced by Lord Rayleigh, w ho proposed the two components, of equal intensity, should be considered "just resolved" when th eir m axim um intensity and minimum intensity between them have the ratio of 1:0.81. Figure 7 shows the superposition of the intensity curves for two monochromatic wavelengths as a function of p. Using the designation of Figure 7 w e can see that l(t>max= «Po>+ I (Po + e>- <«> and, t ( t ) min ' « P 0 + e/2) + I <P0+£- e/2> =2I(p 0 +e/2). (61) Substituting equations (60) and (61) into equation (46), w e obtain, when p is an integer, I W = i W = l - - - y - ■ , (62) m ax 1+ F sin t t e and, I ( t ' , = . (63) m 1n 1 + F sin2 I m m Figure 7 Superposition of the intensity curves for two monochromatic frequencies incident upon Fabry-Perot interferometer as a function of order number p .. 37 Since by Rayleigh's criterion w e know 1^ ) . = 0.81 min max’ (64) by substitution of equations (62) and (63) into equation (64), w e have 21 (i) . . r _ * 2 TT6 1 + F sin —g = 0.81 ( 1) + I ( 1) 1 + F sin^ire (65) Approximating as before, sin e = e , equation (65) reduces to F^ir^e^ - 3.88 Fir^e^ - 1.88 = 0, which yields: e - 2.08 vF t t (66) I t is now convenient to express the resolving power as a function of p. At small angles of incidence, equation (32) becomes px = 2dn, which upon differentiation gives PAX + X A p = 0, (67) which y ield s : . _ E . AX A p * (68) Combining equations (59), (66) and ( 68), w e obtain: R - * £ £ - . (6 9 ) Substituting equation (53) into equation (69), w e obtain: R = 0.96 p F * pF. (70) For near normal incidence, equation (70) reduces to R = ^SLf. , (71) which relates the resolving power of an instrument to the finesse. C H A P TE R IV EXPERIMENTAL 39 40 Single crystals of l,3,5-CgH3Cl3 were grown by a Bridgeman-Stockbarger (20) technique in an easily constructed oven. The oven consisted of two liquids, mineral o il and ethylene glycol, in a large, three inch diameter tube. These liquids, being immiscible, form a clearly defined interface. The top of the tube was wrapped with heating tape and a moderate voltage was applied to maintain the upper liquid at a temperature of approximately 80°C, or som e 15° above the melting point of the sample. L ittle thermal transfer occurred at the interface and the bottom liquid maintained a temperature well below the melting point of the sample with no external cooling. The l,3,5-CgH3Cl3 was obtained commercially and was of unknown purity. Purification was needed and was done by dissolving the solid in hot methanol, filte r in g , and recrystallizing. This process was done at least two times and a noticeable amount of impurity was removed. These crystals were then zone refined for approximately three days, during which time the melted zone traversed the column som e th irty times. This process produced a marked change in appearance and the crystals appeared to be free of impurities. N o test of purity w as performed. These purified crystals were then transferred to the crucible used to grow single crystals. The crucible was m ade from a section of 13 m m pyrex tubing, a part of which was hand blown to a larger diameter and then drawn to a capillary tip . Various types of tips were tried until a suitable type was found. A typical crucible is shown in 41 Figure 8. This crucible w as fille d with a purified sample, evacuated, sealed and placed in the oven. Care was taken see that the melted sample completely fille d the capillary tip . The crucible was then lowered through the interface at the rate of 1 m m per hour. Large single cyrstals were thus obtained. Although this method proved to be suitable for the growth of single crystals, i t w as somewhat unpredictable. Under seemingly identical conditions, one crucible would yield a single crystal, while another would produce a crystal internally fractured and completely unuseable. Single crystals, approximately cone shaped, measuring 3 c m in length, were obtained. l,3,5-CgHgCl3 is known to crystal lize elongated on the c axis (21). Thus, this axis was easily determined. The a and b axes were easily seen under the polarizing microscope but were not uniquely identified by such a procedure. Unambiguous assignment was done by x-ray analysis on a p r e c e s s i o n camera instrument which clearly distinguished the axes. Figure 9 is a photograph taken on the instrument from which the assignment was made. (After the polarized R am an spectra of an oriented crystal were known, this method proved to be a convenient way to assign the axes and m uch more expedient than doing an x-ray analysis on every sample.) From the cone shaped crystal, sections with various faces were needed for R am an and B rillouin scattering. In general, to obtain the needed crystal, a section of the cone was fir s t cut Figure 8 Crucible used for crystal growth of-l,3,5-CgH3Cl3 Figure 9 X-ray photograph of l,3,5~CgH3CI3 used to distinguish a and b axes, 43 45 perpendicular to the c axis. A wire saw was used for this purpose and proved to be an excellent device. Attempts to cleave the crystal resulted in fractures, while use of the wire saw did l i t t l e i f any damage to the bulk crystal. This section w as then cut as desired and the axes assigned via polarized R am an spectra. Polishing of the faces w as done on f ilt e r paper using #400 Alundum abrasive moistened with methanol. This procedure le ft the surfaces with a somewhat "foggy" appearance, which w as removed by blowing dry a ir on the surfaces. This was done only for a matter of seconds and polished the surfaces to a glass like appearance. The crystal was then glued to a brass rod and mounted in a goniometer. The B rillouin spectra were obtained with a Fabry-Perot interferometer. Figure 10 shows a schematic diagram of the exper imental setup. The laser used was a Coherent Radiation Model 53 Argon ion laser equipped with a Model 434 Wavelength Selector and a Model 423 Oven Stabilized Etalon. These features allowed o single longitudinal m ode operation at 5145 A. This was of crucial importance since the multi longitudinal m ode line-width is approximately 5300 M H z (0.175 cm"*) and would be unuseable when trying to resolve a signal less than 0.1 cm"* from the exciting lin e . The addition of the etalon in the cavity produces a very narrow lin e with frequency s ta b ility of ± 75 M H z over a 10 hour period. Choice of individual m odes is also permitted and is of great importance, as we shall see la te r. The laser was monitored PMT Figure 10 Schematic diagram of experimental setup. denotes the focussing lens, S the sample, F the iodine f il t e r , Lg the gathering lens, F-P the Fabry-Perot interferometer, Lg the collecting lens, A the aperature, the floating lens, and P M T the photomultiplier tube. 47 with a scanning interferometer to check the frequency s ta b ility . A laser output of 650 m w w as routinely obtained. The beam was directed parallel to the optic axis of the system into mirror M in Figure 10. The mirror was mounted in a precision mirror mount which was placed on an optical bench, allowing a continuous change in scattering angle. The beam was focussed into the crystal with a plano-convex lens with a 10 c m focal length. The crystal was mounted on a goniometer which was mounted on an X-Y-Z translator. The goniometer allows precise angle orientation in order to maintain the proper scattering wave- vector direction at various scattering angles. The X-Y-Z translator allows for precision sample positioning. Both adjustments were found to be c r itic a l. The scattered lig h t is then passed through an iodine f il t e r . This f i l t e r is shown in greater detail in Figure 11. I t consisted of a two inch diameter pyrex tube with four windows fused in place to form three chambers. The two end chambers were sealed under vacuum and provided insulation for the middle chamber. Crystals of iodine were placed inside the middle chamber and i t too was sealed o ff under vacuum. The entire tube, except for the windows, was wrapped with aluminum fo il and a heating tape to which a moderate voltage was applied. By changing the voltage the vapor pressure of the iodine in the middle chamber could be changed. The purpose of the f il t e r was to absorb the incident laser frequency scattered from the crystal. Though, in E V A C U A T E D C H A M B E R S K C R Y S T A L S Figure 11 Diagram of iodine f ilt e r . 49 general, the crystals were well polished and appeared to be free from defects, the problem of unwanted scattered lig h t proved to be quite severe. The use of the iodine f il t e r is based on the fact that two very strongly absorbing lines in the iodine spectrum fa ll within the tuneable range of the 5145 A laser line (22). The linewidth of the absorbing lines is very narrow, approximately 900 MHz, and the laser can be tuned to the appropriate single m ode that is strongly absorbed while the scattered lig h t of nearly the sam e frequency is transmitted with l i t t l e loss. Devlin, et a l. (23) reported an attenuation of the unshifted frequency by a O factor of 10 while attenuating the shifted frequency by a factor of 10, using a f il t e r of sim ilar design. In these experiments i t is estimated that the unshifted frequency was attenuated by about a factor of 10^ and the shifted frequency by a factor of 3. The voltage was adjusted until an appropriate setting was found such that the absorption was strong for the exciting frequency but not broadened enough to interfere with the scattered lig h t. After passing through the iodine f il t e r the scattered lig h t is collected by a 18 c m focal length, f/3 .5 photographic enlarging lens. This lens, L2 in Figure 10, was mounted on an X-Y-Z translator and placed such that its focal point coincided with the beam position in the crystal. The scattered lig h t collected at the focal point is sent parallel through the Fabry-Perot etalon. The Fabry-Perot interferometer consists of two f la t parallel plates and its operation was described in Chapter I I I . 50 The interferometer used in these experiments used two inch diameter plates of ground quartz with a d ielectric reflection coating on the inner surfaces with a re fle c tiv ity of 98%. The plates were ground to a x/200 wave flatness. Adding the uncertainty in flatness for the two plates gives an uncertainty of X/100 in d, the distance between the plates. According to equation (58) this would give a lim iting finesse of 100. In practice, the instrument was used at a finesse of 35 - 40. Using equation (71), this would correspond to a resolving power of approximately 1.5 x 106 . In the determination of aperature size to be used, a compromise must be m ade between the finesse attainable and the amount of intensity passed. The aperature size used in these experiments was 300p and proved to be satisfactory. A n aperature of 100p was tested and was found to greatly reduce the signal with very l i t t l e compensating increase in finesse. A quartz spacer w as used to control the separation of the plates. The spacer was 1.000 ± 0.001 c m thick, which according to equation (36), would correspond to a free spectral range of 0.5 cm ”*. The estimated accuracy in the measurement of the frequency shifts is ± 0.005 cm ”*. The plates and spacer were placed in a holder patterned a fter one used by Stoicheff (6) . In the assembly used, the plates and spacer were held in position between two stainless steel discs with three invar rods connecting them. O ne plate was positioned against three precision set screws, while pressure was applied to the other plate by means of three adjustable leaf springs. 51 Adjustment of the tension exerted by these springs allowed for alignment of the plates to a parallel position A diagram of this holder is shown in Figure 12. This entire assembly was enclosed in an aluminum cylinder with anti reflection coated windows, sealed with 0-rings, on either end. The cylinder and etalon holder, along with various lens holders and other items, were constructed at the machine shop at the University of Southern C alifornia. The cylinder was a ir tig h t and connected to a vacuum line and could be evacuated. A slow leak w as connected to the vacuum line and allowed for a slow change in the refractive index inside the cylinder which effectively changed the spacing between the plates and thus scanned the interferometer over several orders. This method of scanning the interferometer was chosen over other methods, such as thermal expansion of the spacer (24), electro magnetic attraction (25), piezo electricity, precision screw drives, and other methods of mechanically varying the optical path. Effectiyely changing the optical path by a change in the refractive index of the medium between the plates was thought to be a more convenient method of scanning the interferometer. O ne problem to be solved when pressure scanning is that of linearizing the scan. This problem has been dealt with in m any ways, including the use of a d ifferen tia l feedback valve in the g ai> flow c irc u it (26), a method described by Dufour (27) using a manometer to monitor and control gas flow, or using a pressure- O P T I C A L P L A T E S bv 7 22) Figure 12 Diagram of etalon holder. tn ro 53 voltage transducer in the Fabry-Perot chamber to drive the frequency scan on an xy recorder. Rank and Shearer (28) developed a nearly linear m ass flow system using specially constructed capillaries as leaks to change the pressure, and found the system to be very nearly linear. A capillary leak of sim ilar design was used in this system and deviations from lin e a rity were not found to be a problem. A fter passing through the Fabry-Perot, the scattered lig h t was focussed with a 13.5 c m focal length f/2 .8 camera lens, denoted L3 in Figure 10. This lens w as mounted on a stationary base connected to its focussing ring. This allowed for the translation along the optic axis of the system. The rings passed by the Fabry-Perot were focussed in the focal plane of lens L3 and an sperature was mounted on an X-Y-Z translator which allowed for translation in directions perpendicular to the optic axis. This motion, coupled with the translative a b ility of lens l _ 3 provided enough m obility to fa c ilita te the alignment of the system. After the aperature was positioned to pass the center ring, the scattered lig h t was focussed by lens on the photomultiplier tube. The photomultiplier used was an FW-130 placed in a thermoelectrically cooled housing, cooled to -30°C. A b rief summary of the alignment procedures w ill undoubted prove helpful in:understanding the use of the system. O riginally, a ll parts of the system were removed except for the plates and spacer. A He-Ne laser, with 1 m w power was directed 54 into the plates and aligned such that the portion of the beam reflected from the front surface of the second plate was directed back dow n the incident beam. This beam then defined the optic axis of the system. A diverging lens was then placed in front of the plates and the He-Ne laser passed through i t . The beam, diverged enough to completely cover the plates, was attenuated with a piece of white paper to permit viewing the transmitted rings with the naked eye. Looking through the plates, i t is possible to see interference fringes with one's eye focussed at in fin ity . Moving one's eye across the ring system allows one to check the diameters of the rings. Adjustment of the plates is done in the following manner. I f the diameter of the rings appears to change, say increase, as your eye is moved from le ft to rig h t, this indicates that the distance between the plates is too great on the right side. The appropriate springs are adjusted to correct this, until no change in the ring diameters can be seen. The sam e test is then made by moving the eye along a vertical diameter and the tension of the springs adjusted accordingly. Each direction was tested several times until no change in ring diameter appeared in any direction. This simple te s t, co m m o n in interferometry (17) allowed for rapid and precise adjustment of the Fabry-Perot etalon. After adjustment of the plates, the diverging lens was removed and the focussing lens, Lg in Figure 10, was positioned on the optic axis by passing the He-Ne laser directly through its 55 center. Adjustment of the aperature presented som e d iffic u lty until a suitable method of alignment was developed. F irs t, the collection lens, Lg, was placed on the optic axis and a sheet of teflon was placed at its focal point. The 5145 A line of the Argon laser was m ade to be incident upon this point and the teflon scattered lig h t effectively into the interferometer. I t was then possible, through the aperature, which had been placed as nearly as possible on the optic axis, to view the ring system 1f lens Lg was moved near the plates. The cylinder around the etalon was put into position and the pressure was adjusted so that the laser frequency was passed in a small center ring. Viewing this center ring through the aperature, lens Lg was moved back toward the aperature while the necessary adjustments were m ade to keep the center ring in view. Lens Lg was moved until its focal point coincided with the aperature position. Lens L^ and the photo m ultiplier were then attached and the system was in operating order. A fter alignment of the instrument, the spectra for a given crystal was taken in the following manner. Mirror M in Figure 10 was positioned such that the laser beam was incident upon the focal point of lens Lg and was at right angles to the optic axis of the instrument. In this configuration the scattering angle, 0 , was equal to 90°. A typical crystal was generally cut in the shape of a cube with 1 c m edges. For 90° scattering the beam entered normal to one face and emerged normal to the other. In this case, the scattering wavevector is in the direction bisecting the angle 56 between the incident beam and the optic axis of the instrument. (This is true only when the polarization of the incident and scattered lig h t is in the sam e direction.) For a ll angles of e for which spectra were taken, the crystal was always positioned such that this sam e wavevector bisected this angle. After aligning the wavevector, the translational adjustments on the crystal were adjusted to direct the beam into a portion of the crystal where there was not excessive scattering from imperfections. Lens Lg was adjusted for m axim um signal and the Fabry-Perot chamber was pressure scanned with the photomultiplier output recorded on an xy recorder. To change to a new value of 0 , the crystal was removed, mirror M m oved to a different position and aligned so that the beam w as s t ill incident upon the focal point of lens Lg. The crystal was then repositioned, rotated by the proper amount to maintain the correct scattering wavevector, and the spectrum was taken for this new value of 0 . This procedure was repeated for 17 different values of 0 from 51° to 129°. (For som e crystals, data for the extreme values of 0 were not obtained due to excessive scattering of the incident frequency from internal reflections in the crystal.) For each value of 0 i t was necessary to consider the refractive index for that specific polarization and direction of the incident and scattered lig h t. Since 0 is easily measured when the crystal was not in position, i t w as necessary to find a 57 relationship between 0 when the crystal was not in position and the actual value of 0 when the crystal was in position. The angles to be determined are shown in Figure 13. In Figure 13, 0 ' denotes the scattering angle when the crystal is not in position, 0 is the actual scattering angle when the crystal is in position, and ^ and ^ the angles between the beam and the normal to the surface outside and inside the crystal, respectively. Since, for each value of 0 ', the crystal was always rotated to maintain the proper scattering wavevector, can be expressed as: . . (72) I f n is the refractive index of the crystal, $2 is determined by the relationship rsin < j > 2 = Arcsin — ------- . (73) For the case shown in Figure 13, with 0 ' < 90, an investigation of the geometry of the system gives 0 = 90° - 2 $ 2 ’ (74) Substituting equations (72) and (73) into equation (7 4 ), one can obtain the relationship between 0 ', which is easily measured, and the actual scattering angle 0 . Such a substitution yields Figure 13 Diagram of scattering angles in a typical scattering experiment. < j > ^ denotes angle of incidence, ^ the angle of refraction, 0 the scattering angle when crystal is in position, and 0 ' the scattering angle in the absence of the crystal. 58 59 / / / e v f * » f / i * « » • ■ •N_!a « • ( 2^1 - N O R M A L B E A M W I T H O U T C R Y S T A L B E A M W I T H C R Y S T A L 60 0 = 90° - 2 Arcsin (75) n for ©' < 90°, and 0 = 90° + 2 Arcsin (76) n for 0 ' > 90°. For each value of 0 at least two orders of the in te rfe r ometer were scanned and the polarization of the scattered lig h t determined. The frequency s h ift was measured for each recording using the free spectral range of the interferometer as a calibration standard. were not available in the lite ra tu re and had to be calculated. The method of calculation is summarized in Appendix A. The values obtained were na = 1.76, nb = 1.82, and nc = 1.49. In a ll, crystals were cut such that six different wavevector directions could be observed, those being: K = (1 ,0 ,0 ), K = (0 ,1 ,0 ), K = (0 , 0, 1 ), K = (1 , 1, 0 ), K = (1, 0, 1) and K = (0 , 1, 1). Figure 14 shows a typical spectrum for K = (1, 0, 0 ), and Figure 15 shows a series of spectra for K = ( 1, 1, 0) , at various values of 0 . The values of the refractive index used for 1,3,5-CgHgClg 61 x 10 .25 .25 Figure 14 Spectrum of B rillouin scattering from longitudinal phonon in the (1,0,0) direction with a scattering angle of 90°. Measured frequency s h ift is 0.301 cm"*, thus the peaks shown belong to adjacent orders of interferometer. Figure 15 Series of B rillouin spectra produced by scattering from quasi longitudinal and quasi-transverse phonons in the ( 1 , 1 , 0) direction, taken at various scattering angles. Pattern shows crossover of peaks from the two different phonons. At a scattering angle of 86° the peaks from the two phonons of adjacent interferometer orders are coincident. 62 63 0=104 0= 98 0=94 0 = 90' 0=86 0=79 0 = 6 7 * .7 0 .1 64 The R am an spectra of l,3,5-CgH3Cl3 were taken with a Spex 1401 double monochromator with an ITT FW-130 photomultiplier having an S-20 response and operated in a thermoelectrically cooled housing maintained at -30°C. Canberra photon counting electronics were used and the spectra were recorded on an xy recorder. The estimated accuracy of the frequency measurements was ± 1 cm ”*. A Coherent Radiation Model 52 Argon ion laser was used which could o be run either multimode or on a single longitudinal mode, at 5145 A. To obtain polarized R am an spectra, a crystal of 1,3,5- C 6H3C13 was cut in the shape of a cube with edges along the principal crystal axes. D ifferent orientations of the beam and crystal allowed for scattering from the six independent p o larizab ility components. The laser, in this case, was run multimode and no iodine f il t e r was necessary because the crystal was relatively free from imperfections, and excessive Rayleigh scattering w as not a problem. The monochromator was run with a spectral s lit -1 -1 width of less than 2 c m and i t w as possible to work within 10 c m of the exciting frequency. Figure 16 shows the R am an spectra obtained for six d ifferen t scattering configurations. Low temperature R am an spectra were obtained using the sam e equipment but using an iodine f il t e r and the laser operating in a single longitudinal mode. The sample of l,3,5-CgH3Cl3 was a polycrystalline film made by vapor deposition onto a copper cold finger. The copper finger was attached to the base of a "Cryo-tip" Joule-Thompson cryostat. With this cryostat the temperature, Figure 16 Polarized R am an spectra from the six independent p o larizab ility components of ljS^CgHgClg. 65 ^UUOV jJJ-IOV O D 0L 09 09 Of O E 02 01 OB O E 01 01 BE 67 measured with a chrome!-constantan thermocouple, could be maintained at any value between 20°K and room temperature. The laser beam was directed into the sample at a glancing angle and the scattered lig h t collected at 90°. Figure 17 shows the R am an spectra of the polycrystalline sample at 20° and 90° K. Figure 17 R am an spectra of polycrystalline sample of l,3,5-CgH3Cl3 at 20°K and 90°K. A s expected, R am an lines sharpen at low temperatures because of less rotational disorder and s h ift to higher frequency due to an increased force constant. 68 69 O J O) o c o o o o N1 o C N •c. I... I ■ ? _ o o _ o -N _ o CN C H A P TE R V ASSIG NM ENT O F LATTICE FREQ UENCIES A N D C O M P A R IS O N W ITH C A LC U LA TED FREQ UENCIES 70 71 In order to make any assignments of la ttic e frequencies or spectral predictions for a crystal i t is fir s t necessary that the crystal structure be known. The crystal structure of l,3,5-CgH3Cl3 was determined at 20°C and -183°C, and at both temperatures the crystal was found to belong to the orthorhombic space group P2^2j2j (Dg^) with four molecules per unit cell (29). This group contains three independent screw axes with a corresponding factor group of D 3 and a site group of C^. The translational and rotational degrees of freedom of the four molecules per unit cell generate three acoustic m odes and 21 optical la ttic e modes. All la ttic e m odes are expected to be R am an active. Table 1 presents the number of lines expected for each symmetry species and the p o larizab ility components responsible for scattering. The designation of species representations follows that of Wilson, Decius and Cross (30). A s previously mentioned, the crystal used for room temperature polarized R am an scattering was in the shape of a cube with edges cut parallel to the principal crystal axes. Appropriate orientations of the crystal and beam polarization allowed for scattering from a ll six independent p o larizab ility components. The spectra obtained are shown in Figure 16. The scattering due to each p o la rizab ility component was observed with at least two d ifferen t orientations of the crystal, e .g ., for observation of the “ac P°lan zab ility component both a(ca)c and c(ac)c were run. 72 Table 1 Lattice vibrations of 1,3,5-CgHgClg grouped according to species representation and p o larizab ility components responsible for R am an a c tiv ity . Species # of Modes # of Acoustic Modes # of Optical Modes R am an Acti vi ty A 6 0 6 “aa* “bb, “cc, B i 6 1 5 “ab B 2 6 1 5 “ca B3 6 1 5 “be 73 Table 2 lis ts the frequencies observed for each symmetry species. The estimated accuracy of the frequency measurements is ± 1 cm “*. In a ll, 17 of the expected 21 la ttic e m odes were observed. There is current interest in the fie ld of la ttic e vibrations of molecular solids, particularly in the use of semi- empirical potential models to explain and predict la ttic e dynamic properties (13, 31). A number of studies have been directed to simple molecular systems of hydrogen (32), nitrogen (33, 34), and carbon dioxide (35), and extended to aromatic compounds such as benzene (36, 37, 38) and naphthalene (39). l,3,5-CgH3Clg w as thought to be an interesting sample on which to extend these types of studies since we are able to measure the frequencies of the acoustic modes, as well as the la ttic e modes, which would provide a further test of the potential model used. A n atom-atom potential model was used to calculate the la ttic e frequencies. The use of this model has been described in detail elsewhere (13, 35, 36), and only a b rief summary w ill be given here. The intermolecular potential energy, V, of the crystal is expressed as a sum of interactions between molecules, i.e . V=Z Vkk, (77) k,k' 74 Table 2 Comparison of calculated and observed la ttic e frequencies for IjS.S-CgHgClg. Calculated frequencies obtained using Bonadeo's parameter set 1, (see Table 3). Species Calculated Frequencies Observed Frequencies 18 25 23 A 30 30 H 42 34 45 46 50 57 28 38 34 B i 41 ' 46 47 54 54 61 21 22 31 32 B! > 33 c 39 44 54 56 27 32 31 39 36 6 43 48 57 58 where k and k' denote s p e c ific molecules. For the atom-atom model (78) where R .. is the separation between the two atoms i and j , ■ J belonging to the molecules k and k' respectively. W e have chosen to express V ..(R ..) as a Buckingham type * J * J potential in which A, B, and C are parameters depending on the nature of the atoms i and j . The energy of the crystal, V, is the su m over k ,k ', i , and j limited to those R .. within a chosen distance R • J called the radius of interaction (36). The lim it chosen for these calculations w as 6 A. The methods used to calculate the la ttic e frequencies with the use of this or sim ilar potentials have been discussed elsewhere (13, 35, 36). What is essentially done is the solution of a secular equation of the form + B exp(-C R i j ) (79) o (80) Where is defined as (36): where 3 labels the unit cells , p and v molecules within a unit c e ll, and z and m normal m odes of motion taken as a basis for the crystal calculation, k is the wavevector and r Q the vector position of the 3-th unit c e ll. x( k ) and x° are the wave lengths of the crystal m odes and the free molecule normal m odes respectively. Three sets of parameters, A, B, and C were obtained from Bonadeo (40), and are listed in Table 3. Also listed is the root m ean square deviation between the observed frequencies and the frequencies calculated using each set of parameters. The values of the parameters for carbon and hydrogen interactions were obtained from Williams (41) using his parameter set IV in the paper listed . 77 Table 3 Potential parameters. A has units of Kcalj mole"*, B has units -1 1 of A , and C has units of Kcal A mole . Atom-atom interaction Parameter Set 1 Set 2 Set 3 A 3650 2320 3030 Cl-Cl B 263000 243000 264000 C 3.51 3.51 3.51 A -631 808 500 Cl-C B 44200 128000 127000 C 3.653 3.653 3.653 A 1005 247 400 Cl -H B 33300 15900 18600 C 3.623 3.623 3.623 Root mean square deviation 4.5 4.4 4.4 chapter VI CALCULATION O F ELASTIC C O N S TA N TS A N D VELOCITIES O F ACOUSTIC W A V E S 78 7? O ne of the important quantities that can be calculated from the frequencies of the acoustic waves in a solid, which are measured experimentally, is the elastic constants of the crystal. In order to calculate the elastic constants for a solid, the basic relationships between an elastic wave and the elastic constants must be explored. The following discussion w ill be based on a treatment of the elastic constants by K ittel (42), and a procedure devised by Fabelinskii (14) to calculate the intensity of acoustic waves. In this discussion only infinitesim al strains w ill be considered. AAA Consider three orthogonal vectors, x, y, z, of unit length, securely imbedded in an unstrained solid. After a uniform deformation the axes are distorted in orientation and length, into the new axes x ', y ', and z ', defined by the relations X' = Cl + X = Exy y + sxz z y' - cyx X + (1 + eyy) y + eyz z (81) A A A Z 1 = e X + e y+ (l + e ) Z zx zy J v zz7 The coefficients e .. define the deformation, are dimensionless, and ■ J are very small for infinitesim al strains. Under this deformation, the point ^ A A A r = x x + y y + z z 80 is taken into the point r ' = x x ' + y y ‘ + zz' The displacement, R, of this deformation is defined as R r' - r = x (x 1 - x) + y(y - y) + z(z ' -z) (82) Substituting equation (81) into equation (82), w e obtain: R( r ) = (x£j(x + yeyx + z tzx) x + y6yy) + ezy) y * <Xexz + yeyz + zezz> z ’ which m ay be rewritten as R ( r ) = ux(r) x + ut (r) y + uz(r) z (83) with u^(r) clearly defined. I f equation (83) is compared with a Taylor series expansion of R (r) using R (0) = 0, i t is found that 9U (r ) 9U (r) xexx “ x “ m - 5 ycyz “ y " l y - 5 etc- Instead of using the e^-'s i t is co m m on practice to define strain * J components by the relations e He XX XX eyy = eyy 3 u x ( r ) 9u x ( r ) 3 u v ( r ) + — 1 ----- 3X ’ c x y 3 y 3X 3 u v ( r ) = -----i -------- • 3.U„(r) e = — *■-------- 3u ( r ) + — ±----- 3 y ’ y z 3 z 3 y 3 u z ( r ) 3 u y ( r ) e = — ---------- 3 u 7 ( r ) + r (84) ZZ " zz 3Z ZX 3Z 3X 81 In order to define the elastic constants, i t is necessary to explain and define stress components. Consider the forces, acting on a unit area in the solid, causing the deformation. These forces are defined as stress. The six independent stress components are denoted as: X , Y , Z , Y , Z , X . In this notation, the capital x y z z x y le tte r indicates the direction of the force, and the subscript indicates the normal to the plane to which the force is applied, e.g. X is a force in the x direction on a yz plane; Y is a X z force in the y direction on a xy plane, etc. Hooke's law states that for su fficien tly small deformations the strain is directly proportional to the stress, so that the stress components are linear functions of the strain components, i . e . , Xx = C11 e x x + C 12 ey y + C13 e z z + C 14 ey z + C15 e z x + C 16 e x y Yy = C 21 e x x + C 22 ey y + C 23 e z z + C 24 ey z + C 25 e z x + C 15 e x y Z z = C 31 e x x + C 32 ey y + C 33 e z z + C34 ey z + C 35 e z x + C 36 e x y Yz = C 41 e x x + C 42 ey y + C 43 e z z + C 44 ey z + C 45 e z x + C 46 e x y (85) Zx = C 51 exx + C 52 eyy + C 53 ezz + C 54 eyz + C 55 ezx + C 56 exy X y = C 51 exx + C 62 eyy + C 63 ezz + C 64 eyz + C 65 ezx + C 66 exy The quantities C j j , C ••• are called the elastic stiffness constants. 82 With these quantities defined, i t is now possible to derive the equations of motion for a solid. Consider the cube, in Figure 18, of volume A X A y A z. A stress of -X ( x ) is applied on the X face at x , and a stress of X ( x + a x ) on the parallel face at X X + A X . I f 9XX X x (x + A X ) * Xx(x) + A X , fax. then the net force is x AX ax A yA z . Other stress in the x direction, X^ and X z , also contribute to the total force in the x direction. The combination of force in the x direction can easily be shown to be: Fx = f8Xx * * “ s1 3X 3y 3Z A x A y A z . ( 8 6 ) Since i t is known that this force w ill equal the m ass of the cube times the component of acceleration in the x direction, w e m ay write a2u (r ) aX ax ax < ■ - f c - s - i f + - £ + i f - (87) Let us now consider a plane, ultrasonic wave propagation through the solid. The displacement, u ( r ) , of this wave is given by the expression: u(r) > A e1( st ' { •'*> (88) Figure 18 Representation of a cube of volume A x A y A z acted on b y a stress of -X ( x ) on the face at x , and a stress of X ( x + A x ) of the parallel X X face at x + A x . 83 84 X X X < + x 85 where: A = Ay, the amplitude of the displacement; ^ A A A ' y = yx + y^y + yzz > the polarization vector of the wave; It = K k , the wavevector of the wave; ^ A A A k = <xx + K ^ y + kzz, the propagation vector of the wave; n = 2irv, the frequency of the wave. In equation (88), y and k are unit vectors, i . e . , 2 X 2 4. 2 - 1 Y + Y TV = 1 . x ry rz * 2 2 2 + + = 1. K X <y K Z Equation (88) m ay be rewritten as u (r) = ux(r)x + u y(r)y + uz(r)z = (89) A(YXX ♦ V + where u (r) = A y e ^ Bt " K^ kxx + K yy + K zz^ a a J Substituting equation (90) into the le ft side of equation (87), w e find: where V = ^ = the velocity of the wave. Substituting equations (84) and (85) into the right side of equation (9 7 ), and dividing by K 2Aei(sit - K ( kx x + cyy + kz z ) ) _ a fter som e laborious rearranging, w e obtain: 3XV 3XV 3X —— + — + — -— = Q y + Q y + Q y (92) 3x 3y 3z M xxrx ^xy’y Xxz’z ' ' with the values of Q .. taken from Table 4. * J Equating equations (91) and (92), w e obtain: o pV Y = Q Y + Q Y + Q Y ^ T x sxxT x H xy'y vxzT z Following a sim ilar procedure for the equations of motion in the y and z directions w e obtain a set of three equations from which the velocities of the elastic waves may be calculated, i . e . , Table 4 Definition of parameter Q.^ in terms of elastic wave propagation vectors and elastic constants. e -g - QXX = C11KX2 + C66kv2 + C55K72 + 2C5fiKy,cz + + 2C16KyKv_______________________ K ? K Z^X KxK -y 'XX '11 '66 '5 5 2C 56 2C 15 2C 16 '66 '22 '44 2C 24 2C 46 2C 26 'zz '55 '44 '33 2C 34 2C 35 2C 45 ^yz '56 '24 '34 C 23 + C 44 C 36 + C 45 C 46 + C 25 'zx '15 '46 '35 C 45+ C 36 C + C °13 °55 C 14 + C 56 xy '16 '26 '45 C 25 + C 46 C 14 + C 56 C12 + C 66 “ 2 where X = p V . Using this set of equations and a procedure by Fabelinskii w e can now calculate the velocity and polarization of elastic waves seen experimentally. Since 1,3,5-CgHgClg is an orthorhombic crystal w e must see the matrix of elastic constants for an orthorhombic system whih w e know to be C 11 C 12 C13 0 0 0 C 12 C 22 C 23 0 0 0 C 13 C 23 C 33 0 0 0 0 0 0 C 44 1 0 0 0 0 0 0 C 55 0 0 0 0 0 0 C 66 From Table 4, the coefficients Q.. are found to be: • J 89 Q xx = Cl l K x2 + C 66K y2 + C5 5 k x 2 ; Q yz = ( C 23+W K yK z Q yy = C 66kx2 + C 22K y2 + C 44K z2; Q zx= ( C 13+(W K xK z Q zz = C5 5 k x 2 + C 44K y2 + C33kz2; Q xy = (C12+C66)KxK y Let us now consider the situation where a lig h t wave of frequency u > and wavevector k is incident upon the crystal, is scattered by a phonon of frequency t o and wavevector K, and emerges as a wave of frequency to' and wavevector k '. The incident lig h t wave m ay be written as t <r) = E0 ie H “t -'< • " > , where . A A A I = ex* + Byy + ezz. The scattered wave m ay be written as E (r) = EQ ae1^ ' k ' * r ) where a = a xX + cyy + az Z . Fabelinskii finds that the light-scattering coefficient for B rillouin components, for a sample irradiated by lin early polarized lig h t, is proportional to < | > OTo t B where 90 d > a 3 = F d > a 3 yctt a x L Vc a i ot a x a,x=X,y,Z 3t and a a are the components of the polarization vectors of the incident and scattered lig h t waves, respectively. < j > is a function of the refractive index and the elasto-optical constants of the medium, and the propagation and polarization vectors of the acoustic waves, i . e . , and k^.. The elasto-optical constants describe the effect of an elastic strain on the d ielectric tensor. The matrix of elasto-optical constants i f found to have non-zero values only in the positions of non-zero values in the matrix of elastic constants, which for our case is shown in equation (94). is defined as: ox *01 ■ "4p0i i j Ki<i' using the sam e summation notation as equation (95). Here, n is the m ean refractive index and p .. are the elastic-optical rOXlJ r constants. For convenience, four indices are used, running over values from 1 to 3, in place of two indices running over values from 1 to 6. Therefore, pl m = pn ; pU 22 = p12; p2231 = p25; etc. Since as in the matrix of elastic constants, pn m = pm n > therefore p •• = p ' • . = p = p „ . , • > thus < t > = < t > . r a t l J r Ta1J * a T j l t c t Yxa For our case, defining ifr' = < | > aT/n^, w e find that 91 * 'x x = pl l KxTx + p1 2 V y + p13Kz V * 'y z = p44( V z +KzTy ) * 'y y = P12lcxvx + P2 2 V y + P23“ z V * 'z x = " ssI V j V i 1 (9 6 ) + 'z z = p 1 3 k x y x + p2 3 V y f p33Kz V + 'x y = W V y V x 1 ' Let us now consider two cases to illu s tra te Fabelinskii's procedure applied to l,3,5-CgHgCl3. Case I The lig h t beam is incident along the -x direction and scattered along the +y direction. For this situation w e have the wavevectors given as k = (-k ,0 ,0 ) for the incident wave; k'= (0 ,k ,0) for the scattered wave. For the elastic wave, K = k - k' = (k ,k ,0 ). Thus, K is in the (1,1,0) direction and this m ay be expressed as K = K ( kx x + K yy + <zz) = K ( / £ / 2 x + /2 /2 y + 0 z) i . e . , kx = J 2 / 2 \ Ky = t / 2 / 2 \ < z = 0 . From Table 4 w e find: Substituting these values into equation (93) w e obtain: (Q - X)y + Q y + 0 y = 0 w xx / r x w xy y z Q xy Yx + (Q yy - x )ly + 0 y z ‘ 0 0 Yx + 0 Y y + ^ z z " x^Yz“ 0 For a nontrivial solution, i t is required that X satisfy the equation: « x x “ X> xy xy CQ yy - x) o 0 (Qzz - x) = 0 W e have three solutions: xi /= Q zz> fo r which Yx = Y y = 0; yz = 1. Thus, this solution describes a transverse wave with displacement perpendicular to the scattering plane. j ------------------------------------ This solution describes a quasi-longitudinal wave. W e must now consider the possible cases of the polarizations of the incident and scattered waves to determine the expected intensity for each case. (a) The incident lig h t is polarized along z, and the scattered lig h t is polarized along z. Thus, ex = 0, 3y = 0, 3Z = 1 3X = o , ey = o , ez = i . Therefore: < j > ' a 3 = <(>__ a, 3, y a x a t ZZ Z rz Substituting the values of from equation (97) into equation (96), w e obtain: ^ 'z z ^ z = T (p13Yx + P 2 3 V * For the Xj solution: = 0. This means that w e expect no intensity from this elastic wave for this configuration. n r For the Xn and Xm solutions: 4>'z z = ~2 (P i3Yx + P23Y y^ • For this case, < j > ‘ w ill have a non-zero value, i . e . , we expect som e intensity of scattering from these waves. 94 (b) The incident lig h t is polarized along z, and the scattered lig h t is polarized along x. Thus, Px = 0, B = 0, ez = 1, “x = 1 ’ ay = 0’ az = 0 - Therefore: * ^ 0 , ^ = * ' zx< x zB x - P5 6 V For the Xj solution: < | > ' = P55• For the Xjj and Xjjj solutions: (ji1 = 0. (c) The incident lig h t is polarized along y , and the scattered lig h t is polarized along z. Thus, 3X = 0, 3y = 3Z = 0, “x = ° ’ ay = ° ’ “z = X- Therefore: t ' a x * a Z x = * ' y z « / z = 4 P 44V For the Xj solution: * yz 2 p44‘ For the Xjj and Xjjj solutions: < f > ' = 0. (d) The incident lig h t is polarized along y , and the scattered lig h t is polarized along x. Thus, 3X = 0, 3y = 1. 3Z = 0, < * X = 1 * “y = 0 » az = 0 * 95 Therefore: < f > ‘ a g = < j > T rrn- rr r ' For the XT solution: < ) > ' = 0. i xy For the Xjj and Xjjj solutions: ' These results are summarized in Table 5, where R.^ is 1 J proportional to the intensity scattered from a given wave with the incident beam polarized along the i axis and the scattered beam polarized along the j axis. Remembering that K = 2ksin(e/2) w e can see th a t.fo r 0 = 90°, Thus, Since Xjjj = Xjj w e expect the frequency of the quasi- longitudinal wave to be greater than that of the quasi-transverse wave. This procedure, when applied to acoustic phonons in the C l,0,1) and (0,1,1) directions gives sim ilar results. W e w ill now consider phonons along the principal axes, using the (1,0,0) direction as an illu s tra tio n . Case I I The lig h t beam is incident in the (1,1,0) direction, and is scattered in the (1,1,0) direction. For this situation w e have wavevectors given as: Therefore, /2X/p k. (98) Table 5 Scattering coefficients for acoustic waves in the (1,0,0) direction, under different polarization configurations. R zz R zx R yz V Xr: Transverse wave 0 2 5 5 f L P 2 4 4 0 Xj j : Quasi-transverse wave T ( P 1 3 y x + P 2 3 V 0 0 J l 2 ^P66Yx f P66Y y Xjt,: Quasi-longitudinal wave T ^P 1 3 y x + P2 3 Yy ) 0 0 / I 2 ° W x + P66Y y ) to CD 97 k = (-t/2/2k, /2/2k, 0 ), for the k' = = (^ /2 k , i/2/2k, 0 ), for the For the elastic wave: K = k ‘ - k = (v/2 k, Thus, k = 1. K y = ° , and kz = 0,» From Table 1 w e find: Q xx = Cn ; Q = 0 w yz . Q y y = C 66’ Q = 0 w zx Qzz = C 55’ Q = 0. w xy The determinant to be solved for this case is: - X> 0 0 ( Q y y - x ) 0 0 (Qzz - X> = 0 . The solutions are: XI = Qxx’ for which Yx = lf Y y = Yz = ° ’ i,e’ a longitudinal wave. XI I = Qyy’ for which Yx = °* Y y = 1* Yz = °* i *e' transverse wave. XIU = Q zz’ for which Yx = Y y = °* Yz = * ’ i,e-’ wave, a a transverse 98 Again, there are four cases of polarizations. (a) The incident lig h t is polarized along z, i.e. J_, and the scattered lig h t is polarized along z, i.e. ]_. Thus, ex = 0, B = 0, B z * 1, “x = 0* “y = 0* “z = 1* Therefore.♦'0Tc .0St - + ' „ = P 13yx For the Xj sol ution: < j)1 = p13< For the Xjj solution: < ( > ' = 0. For the Xjjj solution: < j> 'zz = 0. (b) The incident lig h t is polarized along z, i.e., and the scattered lig h t is polarized in the xy plane, i.e., ||. Thus, 3X = 0, 3y = 0, 3Z = 1 - ^ - — - n “x • V “y " " 2 ,otz * Therefore, T <*'zx ' +V = 4 P55V For the Xj solution: < j> 'zx = 0. For the X^ solution: < )> 'zx = 0. 2 For the XX II solution: = - y pg5. 99 (c) The incident lig h t is polarized in the xy plane, i.e., 11, and the scattered lig h t is polarized along the z axis, i.e., J _ . IK IK Thus, 3X = 2’ 3y = ~2 > 3^ = 0» a = 0, a. = 0, a =1. x y z Therefore, ^ o t V t = f t ^'xz + ^yz) = 4" p55yz* For the Xj solution: < } > 1 zx = 0. For the Xjj solution:<j>'zx = 0. IK For the Xjjj solution: < /> ' = p5g. (d) The incident lig h t is polarized in the xy plane, i.e., | | , and the .scattered lig h t is polarized in the xy plane, i.e., ||. “x f t a ~ & “ 2 ’ 3y - £ * _ f t _ f t 2 ’ (X y - T 2 , Therefore: + 'o u ao eT = ^ ♦ ' x x " ♦ V = hzyy) For the Xj solution: h (<|>'xx - < |> 'yy) = % (pn - p12) For the X ^ j solution: h ( < i> 'xx - < |> 'yy) = 0. 100 These results are summarized in Table 6, where R .. is * J proportional to the intensity scattered from a given wave with the incident beam polarized i , and the scattered beam polarized j , where i and j can be || or J _ to the scattering plane. Using this procedure for waves in the (0,1,0) and (0,0,1) directions, sim ilar results are obtained. Going through this procedure for a ll scattering configurations measured experimentally serves two purposes. F irs t, for a given wavevector, w e are able to determine which acoustic waves w e can expect intensity from and thus determine the type of wave w e are actually viewing. Secondly, w e can relate the frequency of the wave to the elastic constants of the solid and in this way calculate the elastic constants. In order to calculate the diagonal elastic constants only scattering along the axes need to be considered. Using the 2 relationship X = pV for the various cases of scattering, one can derive the elastic constants for the various acoustic waves. The results of this investigation can be found in Table 7. For B rillouin scattering from l,3,5-CgH3Cl3 in the directions listed in Table 7, only the longitudinal waves were observed. Therefore, only C ^ , Cg2 and C 33 could be calculated from the velocities measured. In order to calculate the remaining diagonal element i t is necessary to consider the scattering from acoustic waves traveling in the (1 ,1 ,0 ), (1 ,0 ,1 ), and Table 6 Scattering coefficients for acoustic waves in the (1,0,0) direction, under different polarization coefficients. R 1 1 R l l l Rll i Rll II Xj : Longitudinal wave P13 oc 0 (PU - P12) Xj j : Transverse wave 0 0 0 0 Xu j : Transverse wave 0 ^ n 2 P55 T P55 0 Table 7 Elastic constants determined from acoustic waves in the (1 ,0 ,0 ), (0,1,0) and (0,0,1) directions. Direction of wave Direction of propagation particle displacement Type of wave Coeffi ci ent determi ned 100 Longitudinal C 11 100 010 Transverse C 66 001 Transverse C 55 100 Transverse C 66 010 010 Longitudinal C 22 110 Transverse C 44 100 Transverse C 55 001 010 Transverse C 44 001 Longitudinal ^33 o PO 103 (0,1,1) directions. In these cases the velocities were measured for both waves with expected non-zero intensity when the incident and scattered waves were polarized perpendicular to the scattering plane. For these waves, le t us denote Xy and as the solutions for the transverse and longitudinal waves respectively. Upon substitution of the proper values of Q. • from Table 1 for the various ' J scattering cases, i t is found that X T = % Jg(C. -+ C . . + 2 C , , ) - ^ % (C . - - C . •) + ( C . . + C , , ) n JJ kk; 2 4 V n j j ' U kk' (99) and X , =% ^ c i f + c j j t 2 c k k > ] + ‘* K r c « > 2 + ( c i J + c k k > 2 ) ’ s < 1 0 0 > The appropriate values of i , j , and k are given in Table 8 for the various wave directions. I t can be seen that by the addition of equations (99) and (100) that C kk VT2p + VL2p " ** ^C i i +C j j ^ (101) where Vj and are the velocities of the transverse and longitudinal waves, respectively. Use of equation (101) enables one to calculate the remaining diagonal elements in the matrix of elastic constants. After a ll the diagonal elements have been calculated, the off-diagonal elements can be calculated using equation (100), which, upon rearrangement yields: 104 Table 8 Designation of elastic constants to be used in equations (99) and (100) for acoustic waves in the (1 ,1 ,0 ), (1 ,0 ,1 ), and (0,0,1) directions. Direction of wave propagation i J k 110 1 2 6 101 1 3 5 011 2 3 4 A s previously mentioned, for scattering from acoustic waves in the (1 ,0 ,0 ), (0 ,1 ,0 ), and (0,0,1) directions, only scattering from longitudinal waves was observed. For scattering from waves in the (1 ,1 ,0 ), (1,0,1) and (0 ,1,1) directions, the quasi longitudinal wave and one of the quasi-transverse waves were observed. The measured velocities for these waves are listed in Table 9. The uncertainty listed is the root m ean square deviation of the velocities measured at different scattering angles for each acoustic wave. Using the velocities listed in Table 9, and the method described in the preceding section, the elastic constants of ljS.S-CgHgClg m ay be calculated. The elastic constants are listed in Table 10, with the uncertainty listed as that which is propagated through the calculations due to the uncertainty in the velocity. 3 A value of 1.68 g/cm was used for the density of 1.S.S-CgHgCl^. 106 Table 9 Measured velocities of acoustic waves in l,3,5-CgHgCl3. Direction of propagati on Type of wave Velocity (m/sec) 100 Longitudinal 2186 ± 15 010 Longitudinal 2556 ± 29 001 Longitudinal 2168 ± 41 110 Quasi-longitudinal 2550 ± 34 110 Quasi-longitudinal 1208 ± 50 101 Quasi-longitudinal 2397 ± 35 101 Quasi-longitudinal 1106 ± 31 011 Quasi-longitudinal 2467 ± 33 011 Quasi-longitudinal 1267 ± 31 107 Table 10 Elastic constants calculated from observed velocities. 10 2 Elastic constants in units of 10 dynes/cm Cn 8.03 ± 0.11 C 22 10.98 ± 0.25 C 33 7.89 ± 0.31 C44 3.49 ± 0.36 C g5 3.75 ± 0.35 C g6 3.88 ± 0.38 C12 4.47 ± 0.80 C 13 3.85 ± 0.69 C 23 3.88 ± 0.78 C H A PTER V II C O M P A R IS O N O F EXPERIMENTAL A N D C A LC U LA TED VELOCITIES 108 109 By examination of equation (88) i t is seen that a range of 2it for K • r covers a ll independent values of exp(-iK • r ). The range of independent values of K in the (1,0,0) direction can be specified as - | < K a < l (103) where a is the length of the unit cell in the (1,0,0) direction. The range of K in the other unit directions m ay be expressed sim ilarly, i . e . , - l < K b i £ (104a) and - ! < K c < f . (104b) This range of values of K is called the fir s t B rillouin zone (42). By plotting the frequency of an acoustic wave vs. the value of its wavevector one obtains the dispersion curve for that wavevector direction. Figure 19 shows a plot of w vs. K for a model la ttic e with interactions only between nearest neighbor planes, with the interplanar force constant Cj and interplanar spacing a (42). By allowing the wavevector k in equation (80) to take on various values the dispersion curve for 1,3,5-CgHgClg could be obtained. Although, for this system, w e cannot assume that the dispersion curve w ill exactly duplicate the model in Figure. 19, w e can expect the general form to be sim ilar. The model shown in 110 FIRST BR1LLOUIN ZONE Figure 19 Plot of in vs. K for a model la ttic e with interactions only between nearest neighbor planes. The interplanar force constant is and the interplanar spacing is a. I l l Figure 19 is very nearly linear at low values of K, indicating t h a t t h e v e l o c i t y , 2 ttv/ K , i s i n d e p e n d e n t o f K i n t h i s r e g i o n . For K % 0, round-off errors m ay arise in the calculations and therefore K = 0.1 K „, was a rb itra rily chosen as a reasonable m ax point of calculation. Since the experimental values of K/Km ax observed varied from approximately 0.002 to 0.013, and the calculations were done at 0.1 w e are particularly interested in the lin e a rity of the dispersion curve between 0 and 0.1 K/K . m ax For the acoustic waves along the principal crystal axes the gradient of the curve was checked at K/Km ax = 0 .1 , and for the waves observed, was found to be equal to v/K which would indicate that the frequency was a linear function of wavevector in this region. Figure 20 shows a plot of observed frequencies vs. K/Km ax for the quasi- longitudinal and quasi-transverse waves along the (0,1,1) direction. This curve also seem s to indicate that the dispersion curve is linear in this range. From the frequencies calculated at 0.1 K/Km ax w e can calculate the velocity of the phonon at this point. A comparison of calculated and experimental velocities is shown in Table 11. With the exception of the longitudinal wave in the (0,1,0) direction, agreement is quite good and would seem to indicate the va lid ity of the potential model used. The elastic constants were calculated using the values of the calculated velocities listed in Table 11. The method of Figure 20 Plot of observed frequencies vs. K/Km ax for the quasi-longitudinal and quasi-transverse waves along the (0,1,1) direction. 112 1 A-J (cm ) .001 K/Kmax .002 LONGITUDINAL Table 11 Comparison of calculated and experimental velocities Direction of propagation Type of wave Experimental velocity (m/sec) Calculated velocity (m/sec) 100 Longitudinal 2186 ± 15 2202 010 Longitudinal 2556 ± 29 1924 001 a Longitudinal 2168 ± 42 2140 110 Quasi-longitudinal 2550 ± 34 2385 110 Quasi-transverse 1208 ± 50 1118 101 Quasi-longitudinal 2397 ± 35 2205 101 Quasi-transverse 1106 ± 31 1123 Oil Quasi-longitudinal 2467 ± 33 2086 Oil Quasi-transverse 1267 ± 31 1258 115 calculation described in Chapter VI was used. Table 12 shows a comparison of the elastic constants obtained from the observed and calculated velocities. I t is obvious from Table 12 that the potential model used to calculate the wave velocities is not good enough to accurately predict the elastic constants. The large error in the calculated velocity of the longitudinal wave in the (0,1,0) direction is propagated into larger errors in som e of the elastic constants, prim arily C C ^ , and C F u r t h e r studies on sim ilar systems m ay provide information useful in the refinement of the potential model. Table 12 Comparison of elastic constants determined from calculated and experimental velocities. Elastic Constant Value from experimental velocity Value from calculated velocity C 11 8.03 + 0.11 8:15 C 22 10.98 + 0.25 6.22 C 33 7.89 + 0.31 7.69 C 44 3.49 + 0.36 3.01 C 55 3.75 + 0.35 2.36 C 66 3.88 + 0.38 4.47 C 12 4.47 + 0.80 2.93 C13 3.85 + 0.69 3.70 C 23 3.88 + 0.78 1.59 R E FE R E N C E S 117 118 1. L. B rillo uin . Ann.Phys., (P a ris ). 17:88 (1922). 2. E. Gross. Nature, 126:201 (1930). 3. E.H.L. Meyer and W . R a m m . Physik. Z ., 33:270 (1932). 4. D.H. Rank, E.R. Shull and D.W. Axford. Nature, 164:67 (1949). 5: R.S. Krishnan. Proc. Indian Acad. S c i., A41:91 (1955). 6. R.Y. Chiao and B. Stoicheff. J. Opt. Soc. Am., 54:1286 (1964). 7. R.Y. Chiao and P. Fleury. Physics of Quantum Mechanics, N ew York: McGraw-Hill Book Publishers, Inc. (1966). 8. D.H. Rank, E.M. Kiess, and U. Fink. J . Opt. Soc. Am., 56:163 (1966). 9. G.B. Benedek. Opt. Soc. A m . (San Francisco), October (1966). 10. G.B. Benedek and K. Fritsch. Phys. Rev., 149:647 (1966). 11. J.B. Lastovka and G.B, Benedek. Phys. Rev. L e tt., 17:1039 (1966). 12. A.J. Hyde, J. Kervorkian, and J.N. Sherwood. Disc. Far. Soc., 48;19 C1969). 13. D.A. Dows. Proceedings of the Scuolas Internazionale di Fisica Enrico Fermi (Varenna), N ew York: Academic Press (1972) (in press). 14. I.L . Fabelinskii. Molecular Scattering of Light, N ew York: Plenum Press (1968). 15. S.P.S. Porto. Proceedings of the International Conference on Light Scattering Spectra of Solids, Edited by G.B. Wright, N ew York: University Press (1968). 16. C. Fabry and A. Perot. Ann. Chim. Phys., 16:115 (1899). 17. K.W. Meissner. J. Opt. Soc. Am., 31:405 (1941). 119 18. M . Born and E. Wolf. Principles of Optics, 3rd Edition, N ew York: Pitman Press, 1965. 19. C. Dufour and R. Picca. Rev. d'O pt., 24:19 (1945). 20. R.A. Laudise. The Growth of Single Crystals, Englewood C liffs , N ew Jersey: Prentice-Hall, Inc. (1970). 21. D.E. Muller, T. Inoue, R.H. Larkin and H.D. Stidham. Spectrochimica Acta, 27A:405 (1971). 22. D.Swanson and M . K roll. Chem . Phys. L e tt., 9:115 (1971). 23. G.E. Devlin, J.L. Davis, L. Chase, and S. Geschwind. Appl. Phys. L e tt., 19:138 (1971). 24. H.C. Burger and P.H. van C litte rt. Physica, 2:87 (1935). 25. C.F. Bruce and R.M. H ill. Australian' J. Phys., 14:64 (1961). 26. P.A. Fleury and R.Y. Chiao. J. Acous. Soc. Am., 39:751 (1966). 27. Charles Dufour, thesis, University of Paris (1950), 28. D.H. Rank and J.N. Shearer. J. Opt. Soc. Am.. 46:463 (1956). 29. H.J. Milledge and L. Pant. Acta C ryst., 13:285 (1960). 30. E.B. Wilson, J.C. Decius, and P.C. Cross. Molecular Vibrations, N ew York: McGraw-Hill Book Co., Inc. (1955). 31. 0. Schnepp and N. Jacobi. Advan. Chem . Phys., 22:205 (1972). 32. W.N. Hardy, I.F . Silvera, K.N. Klump and 0. Schnepp. Phys. Rev. L e tt., 21:291 (1968). 33. A. R o n and 0. Schnepp. J. Chem . Phys., 46:3991 (1967). 34. J.E. Cahill and G.E. Leroi. J. Chem . Phys., 51:1324 (1969). 35. A. Anderson and T.S. Sun. Chem . Phys. L e tt., 8:537 (1971). 120 36. G. Taddei, H. Bonadeo, M.P. Marzocchi and S. Califano. J. Chem . Phys., 58:4299 (1973). 37. H. Bonadeo and G. Taddei. J. Chem . Phys., 58:979 (1973). 38. G.R. E llio t and G.E. Leroi. J. Chem . Phys.. 58:1253 (1973). 39. G.S. Pawley and S.J. Cyvin. J. Chem . Phys., 52:4073 (1970). 40. H. Bonadeo and E. D'Alessio. Chem . Phys. L e tt., 19:117 (1973). 41. D.E. Williams. J. Chem . Phys., 47:4680 (1967). 42. Charles K itte l. Introduction to Solid State Physics, N ew York: John Wiley and Sons, Inc. (1966). 43. J.0. Hirschfelder, C.F. Curtiss and R.B. Bird. Molecular Theory of Gases and Liquids, N ew York: John Wiley and Sons, Inc. (1954). 44. Landolt-Borstein. L Band, 3 T e il, Molekeln, 2^:510 (1951). 45. E. Manghi, C.A. DeCaroni, M.R. DeBenyacar, and M.J. DeAbeledo. Acta. C ryst., 23:205 (1967). APPENDIX CALCULATION O F REFRACTIVE INDEX 121 122 Since the refractive indices of l.SjS-CgHgClg are not known experimentally i t was necessary to calculate them. The refractive index of a medium is closely related to the p o larizab ility of its constituent atoms. Knowing the individual bond p o larizab ilities of the molecule i t is possible to add these p o la rizab ilities to obtain the molecular p o la riza b ility , from which the refractive indices m ay be calculated. The quantum mechanical theory of this calculation is given by Hirschfelder, Curtis and Bird (43). AUbrief summary of the method used w ill be given here along with the results of the calculation for 1,3,5-CgHgClg- The coordinate system used is shown in Figure 21. The bond p o la riza b ilitie s used are shown in Table 12. The molecular p o la riza b ilitie s are calculated using the relationships: C l H H C l H Figure 21 Coordinate system applied to l,3,5-CgH3Cl3 molecule used in calculation of refractive indices. 124 Table 13 Bond po larizab ilities Bond c t| | x 1025 (cm3) a^ x 1025 (cm3) C-C 22.5 4.8 C-H 7.9 5.8 C-Cl 36.7 20.8 125 where: (°t| j) n = the bond p o larizab ility parallel to the n bond; (a^n = the bond p o larizab ility perpendicular to the n bond; 0ni = t * le an9^e between bond n and the i axis; N = the number of bonds. The molecular po larizab ilities found by this method are: az = 108.6 x 1025 cm 3 av = 188.7 x 1025 cm 3 A ay = 188.7 x 1025 cm 3 These values of the molecular p o larizab ilities can now be used to calculate the po larizab ility of the unit c e ll. The unit cell p o la rizab ility m ay be found by the addition of the p o larizab ilities of the four molecules in the unit c e ll. The components of each molecule along the unit cell axis m ay be found through the use of the rotation matrix R which takes the molecular axis coordinate system into the unit cell frame. The contributions of molecules correct position in the unit c e ll. The contributions of the molecules to the p o la rizab ility of the unit cell m ay be expressed as: C = I a R R (105) oe L 33 n3 eB 3 126 where: C is a matrix representing the p o lariza b ility of the unit c e ll; is a diagonal matrix of the molecular p o lariza b ilitie s ; R is the rotational matrix for a specific molecule. For example, the R matrix for molecule 1 in the unit cell is -0.764 -0.491 0.428' R = 0.497 -0.850 -0.089 (106) 0.412 0.149 0.900 The rotation matrices for molecules 2, 3, and 4 m ay be found by multiplying the matrix in equation (106) by the appropriate symmetry matrix to obtain matrices that transform the rotation of a molecule on the origin of the unit cell to its rotation in positions 2, 3 and 4. Substituting the appropriate R matrix for each molecule in the unit cell into equation (105) and adding the results w e obtain: “a 701.6 X L O CM O r - H c m ii 3 744.8 X IQ25 c m ' “c 497.2 X 1025 . c m where is actually on the on-diagonal p o larizab ility tensor component, a ^ , of the unit c e ll. The off-diagonal elements are 127 found to be zero as expected from symmetry considerations. The principal refractive indices were calculated by substituting those values into the Lorentz-Lorenz formula: where: n. = the refractive index for lig h t with its ele c tric The results obtained were: n = 1.76 a nb = 1.82 nc = ! - 49* Manghi, et a l. (45) calculated the refractive indices of p-dichlorobenzene in this way and found very good agreement with experimental results. (107) vector along axis i; = Avogadro's number; p = the density M = the molecular weight

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Creator Swanson, Douglas Lynn (author)

Core Title Brillouin Scattering In Molecular Crystals: Sym-Trichlorobenzene

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Advisor Dows, David A. (committee chair), Porto, Sergio P.S. (committee member), Segal, Gerald A. (committee member)

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