AN INVESTIGATION OF BRILLOUIN LIGHT SCATTERING AND THERMAL RELAXATION IN CF 3 CC] 3 USING A COMPUTER INTERFACED SPECTROMET ER Dissertation for the Degree of Ph. D. ‘ MICHIGAN STATE UNIVERSITY WILLIAM JOSEPH TOTH, In 1974 LIBRARY Michigan 7.7 .1526 University This is to certify that the thesis entitled AN INVESTIGATION OF BRILLOUIN LIGHT SCATTERING AND THERMAL RELAXATION IN CFSCCIS USING A COMPUTER INTERFACED SPECTROMETER presented by Willian Joseph Toth, Jr. has been accepted towards fulfillment of the requirements for Ph .D . Chemistry degree in M 6 M Date 4/ 19/ 74 0-7639 ABSTRACT INVESTIGATION OF BRILLOUIN LIGHT SCATTERING AND THERMAL RELAXATION IN CF3CC13 USING A COMPUTER INTERFACED SPECTROMETER BY William Joseph Toth, Jr. Brillouin light scattering spectra of Freon 113a (CF3CC13) were examined as a function of angle, temperature, and wavelength. The light scattering and acoustical proper- ties studied were the Brillouin frequency shift, the Bril- louin linewidth, the spatial and temporal attenuation coef- ficients, the sound velocity and its dispersion, and the thermal relaxation linewidth. The depolarization ratios were determined at room temperature, and the density and refrac- tive index were measured as a function of temperature. A pdp 8/e mini-computer was interfaced to the spectro- meter and programmed to allow faster and more accurate data acquisition and analysis. In order to correct for the dis- tortion of the spectrum by the presence of a thermal relaxa- tion line, a series of programs was used to submit and analyze the mini-computer data on a CDC 6500 computer. Curve fitting the data allowed the accurate determination of the spectral parameters and permitted an approximation of the thermal relaxation time. w I a} William Joseph Toth, Jr. Freon 113a was found to have a thermal relaxation time of about 2 X 10-10 seconds and showed a sound velocity dis- persion that decreased with temperature from about six percent to four percent in the frequency range probed. The sound velocity temperature dependence was found to be frequency dependent, ranging from 3.7 to 4.2 m/sec/deg, and the spatial and temporal attenuation coefficients were evaluated. These results were compared with the values for CC14, and several conclusions were drawn about the differences in the liquid structure, molecular interactions, and relaxation processes of the two compounds. INVESTIGATION OF BRILLOUIN LIGHT SCATTERING AND THERMAL RELAXATION IN CPSCCl3 USING A COMPUTER INTERFACED SPECTROMETER By William Joseph Toth, Jr. A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemistry 1974 To My Wife Ruth ii ACKNOWLEDGMENTS The author wishes to express his gratitude to the Chemistry Department of Michigan State University for the instructorship and assistantships which permitted him to pursue this degree without interruption, and to that depart- ment's staff, glass sh0p, and machine shop for their assistance. Appreciation is expressed to Professor J. B. Kinsinger for his encouragement and guidance during the course of this study. The friendly encouragement of Henry Yuen is also greatly appreciated. The author is also indebted to Professor G. Leroi, Pro- fessor R. Cukier and Edwin Doak for their valuable assistance and discussions during the writing of this manuscript. He also wishes to thank Brian Hahn for his suggestions on computer interfacing. iii TABLE OF CONTENTS Page LIST OF TABLES ......................................... vii LIST OF FIGURES ....................................... viii CHAPTER I. INTRODUCTION.................. ...... .............1 General Background ...... ................... ...... 1 Purpose...................... ........ ............8 II. THEORIES OF LIGHT SCATTERING... ..... ............11 Introduction ..... ...............................11 Thermodynamic Theory............................15 Hydrodynamic Theory.............................18 The Frequency Spectrum .......................... 26 The Relaxation Process ........................ ..38 Dispersion ...................................... 42 Depolarization Ratios .......... . ............. ...45 III. EXPERIMENTAL EQUIPMENT AND PROCEDURE............49 The Spectrometer................. ........ .......49 The Laser... ..... ..........................49 Optical Rails. ....... . ..... ................56 Mirror Mount................... ...... ......58 Alignment Laser. ...... .....................58 Bridgeport Lens ...... ......................59 iv TABLE OF CONTENTS (Cont.) CHAPTER Page Bridgeport Mirror .......................... 59 Light Collection Tube... ......... ..........59 Interferometer. .......... ..... ............. 61 Photomultiplier Shutter ....... . .......... ..62 Sample Cells .................................... 64 Density Cell .................................... 68 Sample Preparation ................. . ............ 70 Alignment Procedure...... .................. .....75 Depolarization Measurements............ ......... 76 IV. MINI-COMPUTER INTERFACE AND PROGRAMS..... ....... 79 Hardware ........................................ 79 Programming ....... . .......... .... ............... 83 Data Acquisition and Display...............83 Data Listing..................... ....... ...85 Paper Tape Punching ................... .....86 Messages .................. . ................ 86 Vertical Cursors ........................... 87 Data Smoothing ................... ..........87 Data AnaIYSiS.I...OOOOOOOOOOOOOO0.00.0.000090 Data Acquisition and Analysis Procedure.. ...... .93 V. DATA ANALYSIS..... .............................. 98 Programming ....... .. ........................ ....98 Program LPRATIO............... ..... ........98 V TABLE OF CONTENTS (Cont.) CHAPTER VI. VII. Page Program LIST.............. ......... .......101 Program CONPUN ........... . .............. ..101 Program KINET ........................... ..102 Program BRILL ............... . ............. 103 Program WAVENUM ......................... ..104 Analy5is Procedure...0.000000000000000000......104 EXPERIMENTAL RESULTS ............ . ............. .108 PrOperties of CF3CC13 .......................... 108 Refractive Index.... ....... .. ...... . .......... .110 Density ......... . ........... . .................. 119 Depolarization Ratios......... .............. ...119 Brillouin Shifts...............................126 KINET Results ................................ ..138 BRILL Results.. ........... . ................... .164 CONCLUSIONS AND RECOMMENDATIONS................172 BIBLIOGRAPHY ............... . ................... 181 vi TABLE 3. 0000‘ 6.10 7. .11 .12 .13 .14 .15 .16 .17 .18 .19 .20 1 LIST OF TABLES Page Argon ion laser wavelengths and output powers ............. ... ..... .. .................... 52 The physical prOperties of Freon 113a ........... 109 Refractometer readings............. ...... .......112 Corrected refractive indices..... ..... ..........112 Program WAVENUM results.................. ..... ..114 Program WAVENUM results................... ...... 115 Program WAVENUM results ...... ... ................ 116 Program WAVENUM results ....... . ............... ..117 Refractive index and wavenumbers................118 Density cell volume calibration.................118 Weights and densities of CF3CC13................120 Depolarization ratios of CF3CC13 at 22.80°C ..... 122 Depolarization ratios of some organic liQUids ........... ......OOOOIOI ...... 0.0.0000000125 Brillouin shifts and sound velocities.........127-8 Comparison of the three analysis techniques.....129 Examples of KINET results....................139-40 Pair correlation coefficients...................142 Average KINET results........... ..... ..... ..... .152 Corrected KINET results.........................155 Velocity of sound and attenuation coeffi- CientSoooooooooooooooooooooooooooooooooooo000000163 Intensity ratios from program BRILL.............171 Comparison of some properties of CC14 and Freon 113a............................... ...... .173 vii mum 000-1> .10 .11 .12 LIST OF FIGURES Page Field diagram... ....... . ...... ...... ....... ......20 Scattering diagram ........ . ..................... .20 The light scattering spectrometer................50 The detection optics ..... .....OOOOOOOOOOOOOOOOOO.51 Line structure and multimode laser operation.....52 Doppler envelope for single mode operation (5145 A°) ................ . ...... .................54 The two overlapping envelopes of the 4880 A° line ................. ........... ........ .........54 Double mode Operation at 4880 A°........ ....... ..55 The light collection pipe........................60 The photomultiplier shutter.... ............... ...63 A constant pressure scattering cell..............65 The sealed scattering cell.......................67 The density cell .......... . ...................... 69 19F NMR spectrum of Freon 113a...................72 The data acquisition circuit ........... ..........80 The gating circuit .................... ...........82 The computer recorder circuit....................82 Plot of refractive index versus temperature.....113 Plot of density versus temperature.... .......... 121 Depolarization ratio as a function of wave- length...‘ ooooooooooooooooooo coo-00000000000000.0124 Frequency shift as a function of sin 6/2 at 39.SS°C ..... . ...... .............................131 viii TABLE OF FIGURES (Cont.) FIGURE 6. 0000000 5 .6 .10 .11 .12 .13 .14 .15 .16 .17 .18 Page Velocity of sound versus frequency ............. .133 Velocity of sound versus frequency..............134 Frequency shift versus temperature..............l36 Velocity of sound versus temperature ........... .137 KINET plot of a very good curve fit spectrum....150 KINET plot of a fair curve fit spectrum.........151 KINET plot of a poor curve fit spectrum... ...... 153 Plot of corrected and uncorrected Brillouin half widths versus temperature....... ........... 157 Plot of corrected and uncorrected Mountain half widths versus temperature ..... ..... ........ 158 Plot of corrected Brillouin half width versus the square of frequency............ ..... ........165 BRILL plot of a high finesse spectrum ....... ....166 BRILL plot of a poor finesse spectrum...........167 Actual recording of a spectrum with good finesseOOOOOOO............OOOCIOOOOOOOOOOOO0.00.168 Actual recording of a spectrum with poor finesse.........................................169 ix CHAPTER I INTRODUCTION General Background The scattering of light by matter has long been of great interest to physicists and has now become a very powerful tool for the study of molecular interactions. When light passes through matter, a portion of the light is scattered by random, local fluctuations in the refractive index. These fluctuations are now known to be caused by a variety of phe- nomena, e.g., molecular vibrations, sound waves or phonons, thermal fluctuations, concentration fluctuations, rotons, excitons, etc. The detailed study of many of these phenomena by light scattering has been possible only since the develop- ment of the laser in the early 1960's. Light scattering became of interest to science relatively early. Leonardo da Vinci in the early sixteenth century sug- gested with remarkable insight that the blue color of the sky was due to the scattering of light by particIes in the air. This suggestion was investigated further by many scientists, including Newton and Tyndall, in attempts to identify the par— ticles responsible for scattering. A satisfactory explanation was not proposed until 1871 by Lord Rayleigh. Following a suggestion by Maxwell, Lord Rayleigh finally deduced that the air molecules themselves were responsible for the scattering 1 2 and further proposed that the scattering intensity was pro- portional to the inverse fourth power of the wavelength of light, provided that the scatterers were small compared to the wavelength. Therefore, the more intense scattering at the shorter wavelengths of visible light explains why the sky is blue and not black, as would be expected in the absence of light scattering. Likewise, the reddish color of the horizon at sunrise and sunset is due to the removal of the blue por- tion of visible light by scattering and the transmission of the longer wavelengths of the spectrum. Since Lord Rayleigh's eXplanation of light scattering seemed complete, little new work was undertaken in this area until Brillouin8 predicted in 1922 that scattered light should consist of frequencies other than that incident on the medium. He reasoned that incident monochromatic light would be scat- tered by periodic density fluctuations, vis., sound waves, which are constantly traversing the medium. Since light scattered by the moving periodic density fluctuations under- goes a D0ppler frequency shift from the incident frequency, and since there is statistically the same number of sound waves traversing the medium in every direction, then the scattered light consists of two components with the same intensity and equally shifted to higher and lower frequencies. Since peri- odic density fluctuations are quite analogous to the periodic atomic potentials which scatter x-rays in crystals, the light 3 scattered from sound waves obeys the Bragg diffraction law: 2 A sin 6/2 = A0 (1) The wavelength A of the sound wave which scatters the incident light A0 is only one of a whole continuum of wavelengths present in the medium, but it is selectively chosen by the scattering angle 9. Converting (l) to a frequency equation, f 2 w (VS/c) n sin 9/2, (2) 0 Am = ms = “B Brillouin predicted that the frequency shift of each peak, Aw, from the incident frequency w would be the same, forming a O doublet centered about the incident frequency. Furthermore, this frequency shift is exactly the frequency of the sound wave, ms or “B, which travels at the speed of sound VS through a medium whose refractive index is n. The velocity of light in the medium is given by c/n. Moreover, the frequency observed can be varied by changing either the angle of observation 6, that is, the angle between the direction of propagation of the incident light beam and the ray of observation, or the velocity of sound, which changes with temperature and pressure. At about this same time, similar investigations were performed in Russia by Mandelstam.4S Since he derived essen- tially the same predictions as Brillouin for the scattering of light from acoustic phonons, this effect is sometimes also called Mandelstam-Brillouin scattering. Smekal's64 proposal in 1923 of frequency shifted light due to a system of quantized levels was experimentally confirmed 4 in 1928 byRaman,60 and led to a great interest in light scattering by others and, finally, to Gross'27 work of 1930 which experimentally verified Brillouin's predictions. However, in addition to the doublet predicted, Gross and other workers also found a third unshifted component. Since the theory at this time did not account for this component in any manner other than Tyndall scattering, this line was thought to be the result of stray light reflections from cell walls or Tyndall scattering from dust particles contaminating the sample. New cells were constructed and the samples were purified to remove dust; however, these endeavors failed to remove this central component, which remained unexplained until the work of Landau and Placzek37 in 1934. These re- searchers reasoned that the light scattering system must be treated as a thermodynamic system. As such, a one-component system can be fully described by any two independent state variables, which they chose to be entropy and pressure. Sound waves in the medium were then described simply as pres- sure fluctuations at constant entropy, (as/3P)S. The third component of the scattered light spectrum must, therefore, be caused by entropy fluctuations at constant pressure, (Be/BS)p. In a pure one-component system, entropy fluctuations take the form of thermal fluctuations which are dissipated by thermal diffusion and have lifetimes dependent upon the thermal diffusivity of the medium. For this reason, the 5 central line is sometimes called the "heat conduction mode," or, more commonly, the Rayleigh line. Unlike pressure fluctuations, entrOpy fluctuations are non-prOpagating. Therefore, the Rayleigh scattered light suffers no DOppler shift in its central frequency, but is Doppler broadened by thermal diffusion. The frequency ana- lyzed Brillouin spectrum thus appears as a symmetric triplet centered at the incident frequency. Since the actual Rayleigh line is extremely narrow, the width of the central line, as observed with a Fabry-Perot interferometer, is due to the instrumental response function. Acoustic sound waves sometimes are referred to as phonons, i.e., discrete energy packets carried by longitudinal acous- tic waves, somewhat analogous to photons, which are discrete energy packets carried by electromagnetic waves. As a phonon traverses the medium, energy is drained from it into other modes of motion and to overcome the frictional forces between molecules. Thus, the phonon is rapidly damped, and its life- time is determined by the rate of this process. In turn, the rate of damping is a function of the sound absorption coef- ficient. Because the lifetimes of phonons are very short, the Brillouin peaks are frequency broadened and thus are much broader than the Rayleigh peak. Landau and Placzek also were able to predict the relative intensities of the Rayleigh and Brillouin peaks, since the intensity of the scattered light from each of the two types of fluctuations is dependent on the size of the fluctuations 6 of the thermodynamic variables. The ratio of the two thermo- dynamic fluctuations reduces to the following expression, now called the Landau-Placzek ratio: IR C -C BT-BS —— P V = Y‘]. = B , (3) 213 CV S where CP and CV-are the specific heats at constant pressure and volume, 8T and 38 are the isothermal and adiabatic com- pressibilities, and y is the specific heat ratio CP/Cvu In the 1960's with the invention of the laser”:49 with its high intensity, monochromaticity, and directionality, interest in light scattering again rose rapidly. The theories of Placzek, Rayleigh and Brillouin were rigorously tested and effects previously hidden by the broad linewidth of conven- tional light sources were observed in both Raman and Brillouin scattering. For example, it was soon noticed that the Landau- Placzek intensity ratios of carbon disulfide and carbon tetra- chloride differed significantly from predictions. Closer examination of the spectrum of carbon tetrachloride revealed a small but real intensity excess between the peaks that was too large to be attributed to peak overlap. Therefore, new ideas of sound propagation in liquids had to be formulated, and new theoretical studies were undertaken. To date, several theories have been proposed, but all agree that the additional intensity is due to the coupling of the internal vibrational modes of the molecules with the external translational modes. Energy absorbed from the 7 sound waves as they pass through the medium vibrationally excites the molecules. Later this energy is "relaxed" back from the molecules into the system, giving rise to a fourth component in the Rayleigh-Brillouin spectrum. This component is called the "relaxation" line or the "Mountain" line after Raymond Mountain, who presented one of the first viable theo- ries for this phenomenon. Since it provides a complete description of the decay of thermal fluctuations, the spectrum of scattered light con- tains much information about molecular interactions in sim- ple fluids. For example, the Landau-Placzek intensity ratio of the Rayleigh and Brillouin lines gives both the ratio of the compressibilities and the ratio of the specific heat capacities. These are macroscopic quantities of the liquid, and the magnitude of the ratio describes the preferred mech- anism of energy transfer. In associated liquids such as water and alcohols, the intensity ratio is small, implying that thermal fluctuations decay primarily through propagation of sound waves, since entrOpy fluctuations are reduced by the hydrogen bonding. In normal liquids the intensity ratio is large, implying that thermal fluctuations decay primarily through thermal diffusion, since the molecular attractions are too small to allow efficient conversion of the thermal energy to sound waves. For simple liquids, i.e., those in which the relaxation line is absent, the total intensity of the scattered light allows the evaluation of the isothermal compressibility, and, 8 therefore, Cp. With these values and the Landau-Placqek ratio, BS and Cv.can be found. If light beating spectrosc0py is employed to determine the Rayleigh linewidth, the thermal dif- fusivity can be measured. The shift of the Brillouin line gives the sound velocity, and the width of this line allows the determination of the sound attenuation coefficients. In binary systems, concentration fluctuations, which are another form of entropy fluctuations, contribute scattered light to the in— tensity of the Rayleigh line, allowing the binary diffusion coefficient and the activity coefficients to be determined. In macromolecular solutions, the intensity ratio allows the evaluation of molecular weights, and the Rayleigh line shape can provide an estimate of the macromolecular shape. The dis- persion of the sound velocity can be used for locating phase transitions and determining ultrafast reaction rates.21 Purpose Since data acquisition and the analysis of Brillouin spectra involve a large amount of data reduction, the aid of a mini-computer is highly desirable. Usual data acquisition involves scanning the spectrum at the slowest possible speed while running a strip chart recorder as fast as possible. This enlarges the spectrum as much as possible and reduces analysis errors. A planimeter is used to find areas and a ruler is used to measure line separations and widths. Sizable errors are involved in this analysis due to l) the nonlinearity of the recorder drive, 2) the limited accuracy of ruler 9 measurements, and 3) system dynamics, which cause changes in instrumental response while the spectrum is recorded. The pdp 8/e mini-computer that has been interfaced to our spectrometer allows more rapid data acquisition, and spectrum analysis is much more accurate because of the greater resolution realized in the data acquisition. Furthermore, the digitized data can be used to produce an expanded recor- der output, or it can be converted and read into the CDC 6500 computer for more complex data analysis involving curve fitting. The second purpose for this research was to use this improved system to perform a complete analysis of spectral changes for a single liquid. Since Brillouin scattering had been used solely as a tool for proving the validity of theo- retical improvements, the spectral data on pure compounds was only partially complete. Data had been taken at a 90° angle for a series of temperatures or pure liquids, or at various angles for a single liquid at one temperature. For a complete analysis, it was decided that data should be taken as a func- tion of all the variables, i.e., temperature, angle, wave- length and polarization of the incident light. This required a large number of spectra, but with computer assistance the collection and analysis of these spectra were feasible. The fluid selected was l,l,1-trichloro-2,2,2-trif1uoro- ethane. It was of interest because its narrow liquid range of about thirty Centigrade degrees is in the region of room temperature and allows data acquisition over the entire 10 liquid range without the need of elaborate high and low tem- perature equipment. In addition, this compound was expected to have optical properties similar to those of carbon tetra- chloride, since the only structural difference was the sub- stitution of a compact CF3 group for a chlorine atom. The isotrOpy of the spherically symmetric CCl4 molecule is some- what reduced by the larger CF3 group in CF3CC13. There was more similarity in the optical properties of the two compounds than expected, since both show the Mountain or relaxation line. This discovery led to further investi- gation of data analysis techniques to ascertain the feasi- bility of determining the parameters associated with this component and to note its effects on the total spectrum. CHAPTER II THEORIES OF LIGHT SCATTERING Introduction The theory of light scattering has been developed to a remarkable degree of complexity. While many phenomena have been explained theoretically, a complete, unified theory has yet to be developed. In his original work, Lord Rayleigh considered as a model light scattered from non-interacting dielectric spheres which are much smaller than the wavelength of light A0, but are separated by distance much larger than A0. For a gas of density p whose molecules have a polarizability a, Rayleigh predicted the scattered light intensity would vary as pa ( l + cos a) A 4 r2 (4) 0 where e is the scattering angle and r is the distance from the scattering volume to the observer. However, this rela- tion cannot be expected to hold in dense media, i.e., solids, liquids or gases at atmospheric pressure, since the intermole- cular distances are much shorter than A0 and it is no longer valid to assume that the molecular light scattering is inde- pendent of the neighboring molecules. When an electric field E is applied to a medium, a polari- zation P is induced in each molecule. Therefore, an oscil- lating electric field (such as that present in light incident 11 12 ‘ on a medium) will induce in each molecule an oscillating polarization, which appears as an oscillating dipole. As long as the electric field is not too large, i.e., the light is not too intense, the oscillating dipole at a point 1 will follow the electric field linearly in time t as B (Lt) = .9. (1.11) ' Einc (Lt) (5) where the molecular polarizability a is a tensor. According to electromagnetic theory a permanent polarization or dipole will not interact with radiation, but an oscillating dipole will absorb and re-radiate the incident light in all direc- tions. Thus, the scattered light intensity at any point is simply the vector sum of the fields produced by each oscil- lating dipole. In dense media, where the molecules are small compared to A0, the Born approximation, which says the electric field can be assumed constant or slowly varying over the dimensions of the molecules, is applied. Moreover, for small molecules (IO-100A°), the field can be assumed to be constant over many molecular dimensions so that the medium can be treated as a homogeneous system; and, further, the scattered light is actually a measure of the average polarizability of all the scattering molecules. For this reason, the molecules cannot be treated as independent scatterers as in the Rayleigh theory. Furthermore, the light scattered from each molecule is phase related to that of the neighboring molecules (by the Born approximation), and the light scattered from a given molecule 13 is said to be coherent with the light scattered by the neighboring molecules. If the induced polarization is constant throughout a dense medium (one that has a perfectly uniform density), the medium is said to be perfectly homogeneous. In this case, the vector sum of the scattered fields at any point will be zero except at points along the direction of propagation of the incident beam. In contrast to this theoretical predic- tion, experimental results show that there is a nonzero scattering intensity at points other than in the forward direction; and, therefore, real media are not totally homo- geneous, but have small density fluctuations throughout caused by local thermal fluctuations. These density fluctuations take the form of sound waves, and the heat content which drives them can be viewed as a "thermal exciter," quite anal- ogous to an acoustic exciter which produces sound waves of all frequencies from very small to very large. Since the density fluctuations are not static, their observation will lead to an insight of the temporal behavior of the liquid. Brillouin's suggestion of light scattering is ideally suited to detect and determine the properties of these thermally-produced sound waves. To interpret the light scat- tering spectrum, Onsager's regression hypothesis is assumed to be valid, i.e., the equations used to describe the relaxa- tion of externally produced fluctuations from equilibrium can be used to describe the relaxation of the natural, thermally- produced fluctuations already in the system. 14 The problem of theoretically predicting the features of the Rayleigh-Brillouin light scattering spectrum has been approached on three fronts. The first is thermodynamic and was originally begun by Einstein17 in 1910 and continued by other workers up to the 1960's. Only a brief discussion of this theory will be given. While the thermodynamic approach gives a fairly satis- factory description of light scattering as arising from entrOpy and pressure fluctuations in fluids, it leaves several scat- tering phenomena unexplained. For example, depolarized light scattering from anisotrOpic molecules and the relaxation effects due to molecules with internal degrees of freedom are completely neglected and cannot be predicted by a purely ther- modynamic approach. The second approach is based on hydrodynamic theory. This, like the thermodynamic theory, is a macroscopic approach, but it differs significantly from the thermodynamic theory in that it allows the introduction of frequency-dependent parameters to account for any additional degrees of freedom that the sys- tem may have. This approach began in 1957 with Rytov's phe— nomenological theory61 in which the density parameter is re- placed by a strain tensor whose elements are frequency depen- dent. The off-diagonal elements go to zero at zero frequency, allowing for dispersion, and the diagonal elements have a frequency dependence that allows for coupling between the in- ternal vibrational and the external translational degrees of 15 freedom. Mountain49»51 extended this theory and his approach will be discussed in detail as it relates to the present study. At present it appears that the hydrodynamic theory is quite successful in predicting various light scattering phe- nomena. However, a third approach is available, differing from both the thermodynamic and hydrodynamic theories in that it treats the problem from a microscopic approach. While this method is the most rigorous because it predicts the scattering in terms of fundamental correlation functions, it suffers a serious drawback-—the correlation functions cannot be properly evaluated in dense media. In attempts to salvage this approach, a phenomenology must be prOposed in order to obtain numerical results that can be compared with experimental values. This procedure is not satisfactory because the values obtained strongly depend on the phenomenology that is supposed. Several attempts“:69 are now being made to correct this problem. Since the hydrodynamic theory seems at present to be the most reliable, the microscopic approach will not be presented. Thermodynamic Theory The elementary analysis of the light scattering spectrum by Brillouin is well known. The most important aspect of this work is the prediction of the frequency shift of the Brillouin 'components by (1). With the values of the scattering angle and the Brillouin frequency shift, the velocity of sound can be calculated as a function of angle, and a dispersion relation 16 can be found. The scattering process itself can be viewed either classically, as a reflection from a moving grating, or quantum mechanically, as an inelastic photon-phonon col- lision. These descriptions are both correct and lead to the same scattering equations.12 Since the Optical inhomogeneities which give rise to light scattering are coupled to fluctuations of the dielec- tric constant of the medium and, therefore, its refractive index, we can express fluctuations in the dielectric constant in terms of fluctuations of the two state variables, 6e=(§—;-)86P+(%§-)P'As. (6) Since the intensity of the scattered light depends on the mean square dielectric constant fluctuation, 2 2 2 ((6a)2) = (-3-; )S ((613%) + (%§ )P ((68) ), (7) where the cross terms are zero because the average fluctua- tions of P and S are uncorrelated and zero. From thermo- dynamic identities, we find that 2 2 ((513) ) = Ego— and ((55) > = CPVOpk, (8) where BS, V0, CP and p are the adiabatic compressibility, volume, specific heat at constant pressure, and the density, respectively, and k is Boltzmann's constant. Since it is easier experimentally to measure (ac/3T)P than.(ae/BS)P, it is convenient to change variables to obtain 3e: = T as . g (3):» onVO (HIP () 17 Thus, by substitution, (7) becomes 1“ + (if-)2 1“” . (10) 2 _ a 2 Equation (10) identifies the two sources of scattering. The first term is the pressure fluctuation term for the Brillouin peaks. The second is the entrOpy fluctuation term giving rise to the Rayleigh peak. It is noted that the Landau- Placzek intensity ratio may be obtained by taking the ratio of the second term over the first, with the use of the appropriate thermodynamic identities. Obtaining an explicit spectral distribution function from (10) is not possible from a strictly thermodynamic approach; however, some conclusions about thefrequency spec- trum can be made.3’21 As mentioned in the introduction, the sound waves will give rise to two lines at 3 ms. By assuming that the sound waves are damped exponentially in time by the sound absorption coefficient, the Fourier transform of this exponential decay produces Lorentzian line shapes for the Brillouin peaks. The Rayleigh peak will be unshifted in frequency due to the nonprOpagating entropy fluctuations, but will be broadened in proportion to the thermal diffusivity. Other than these qualitative predictions, which are described in more detail by Benedek,3 the spectral distribu- tion function cannot be evaluated in terms of frequency without stepping out of the thermodynamic realm. In the fol- lowing section, the hydrodynamic approach will be shown to 18 give explicit functions, as well as account for other com- ponents in the frequency spectrum. Hydrodynamic Theory We begin with a classical approach to the scattered field intensity. Using (5) to describe the interaction of the in- cident field of the light beam with the molecule at point r and time t, the incident electric field is put in the form of a plane polarized light wave. Einc(£’t) = Eoexp[i(k;£-mot)] (11) The oscillating dipole, induced by this field, spherically radiates energy from its volume element at 3. At some point 3, shown in Fig. 2.1, the scattered electric field has a value given by dEscatt(B_.t)= first; IEZTIQI (12) where sin e is the angle between Einc(£,t), the polarization vector, and R, the direction of propagation of the scattered wave, and t' is given by t-|R;£|/cm, where cm is the velocity of light in the medium, i.e., cm=c/n. The polarization at .time t' is given from (5) and (6) as P(£,t') = a(_1;,t')Eoexp[i_k_-_r_-iwot + i 9% lg-g). (13) To find the second derivative, we neglect the time dependence of the polarizability, since it changes slowly compared to the rapidly oscillating field. Substituting §(£,t') into (12) 19 and integrating we get m z exp[i(k'fiR-m t)] sin ¢ Escattcg’t)” 50(‘33 R o I xfa(z_.t)exp[i(_l_<_-1<_')‘I_]|d_1;l . (14) V . where V is the scattering volume, the R in the denominator replaces R-r on the assumption that 3525, and, from Fig. 2.1, ,1_<_' = 5k.[2w/(Ao/n)] . w ‘(15) Escatt(R,t) is a spherically spreading field whose ampli- tude is modulated by interference, as given by the inter- ference integral 1 I = Jra(£,t)exp[-i§-r]|d£| . (16) where the scattering veXtor E is shown in Fig. 2.2 to be igiven by K. = k'-k_ (17) To evaluate I, we first separate the polarizability into its mean value plus a term that represents the fluctuations about the mean. A a(£,t) = + 6a(£,t) (18) Substituting this into equation (11), I - fexp[-il_(_°£]|d£| + f6a(_1‘_,t)exp[-i§_-r]|d_r_| V V . A (19) The first term defines a delta function which is zero unless E = 0. This implies that a constant polarizability will not scatter light in any direction except the forward direction. This is true in a totally homogeneous medium. However, the second term does not go to zero, and the scattering in a 20 3 ield Point R-r ' Source \ 3- w’lw IH Point rwr '/ ’ . It A ; EoexP[1(]_(.o°£"”otJ-V / I l / / Q 4 V. . Origin Figure 2.1 Field diagram. l7< \ y \ ['5‘ Figure 2.2 Scattering diagram. 21 medium is due entirely to the fluctuations in the polari- zability, i.e., the scattering is due to inhomogeneities in the medium giving rise to the fluctuations in c. There- fore, any variable that is coupled to the polarizability will cause light scattering if that variable is not constant throughout the medium. One such variable is density. To find the fluctuation responsible for the scattering in the direction of k', a Fourier decomposition of 6a(£,t) is performed, giving 6a(£»t) = f5a(g.t)explig-z_1 qul. (20) Substituting (20) into (14) we see that I = -/Ba(q,t)|dqj—[;exp[i(qf§)'g]|d£|. (21) The second integral here defines the delta function GCQfE). This implies that the fluctuation giving rise to scattering in the k' direction must have its wave vector equal to E. In other words, to observe scattering at any point, the Bragg reflection law must be satisfied for the classical descrip- tion, or the conservation of momentum condition must be met in the quantum mechanical description.12 Since the polarizability actually sampled is the average over many molecules, and the dielectric constant is a more readily measured quantity, a change was made using 6a= 66/4n. This assumes that the molecules have a dielectric constant close to unity and no permanent dipole moment. In addition, the induced dipole moment from neighboring molecules has 22 been neglected. The result is I = 6a(1(_,t) = Ge(_l_(_,t)/41r, (22) and the scattered field is given by Escatt(B-’t) = 130(232 exp[i(1_g'_-%;wot)] Sin (1’ 662%,”.(23) Benedek and Greytak4 proceed by replacing Ge(§,t) by an expression depending on the fluctuations in density. Then using the continuity equation from hydrodynamics, they are able to relate the density fluctuations to the displacement function U(£,t). This leads to a derivation of a time cor- relation function on which they perform a Fourier transform to get a spectral distribution function. This method is short, but the spectral distribution function obtained pre- dicts only the two Lorentzians of the Brillouin components. It is more illustrative to follow the method of Mountain to 'get the exact function which also gives the thermal relaxa- tion line, and to reduce this to a more general form. The total scattered intensity is given by4 * N<§scatt(3't) ' Escatt(5.!t)) ~ 2 NIOI9-g—I4 {—a‘7fi((6eq<_.t))2>. (24) where E* is the complex conjugate of‘E (thus removing the 1(B.t) exponential), I0 is E02 (the incident intensity), and N is the total number of molecules contributing to the scattered light intensity from the scattering volume. Since we wish 23 to calculate the frequency spectrum and not the time spec- trum, it is necessary to perform the Fourier transform of (24), which gives ng 16112c4R2 1(3,m) = 1, sin2 I» ((ce(1<_.w))2) . (25) This is the same equation derived classically by Einstein and Smoluchowski, 17,65 where the frequency Inis the fre- quency shift from (m) The dielectric constant at a given point r and time t can be separated into three parts,42 e(£,t) =‘E + Ge(£,t) + AS(£,t). (26) The first term is the mean dielectric tensor, which does not give rise to scattering, as noted previously in (19). The second term is the scalar departure from the mean dielec— tric constant and is caused predominantly by density fluc- tuations. In general, fluctuations which are scalar in nature give rise to scattering whose polarization is parallel to that of the incident light. Therefore, the second term should fully describe the polarized spectrum except for con- tributions from the third term, which is a symmetric tensor whose trace is zero. Fluctuations caused by shear stresses, strains and reorientation of non-spherical molecules (which have a polarizability tensor with non-zero off-diagonal elements) give rise to the third term. The light scattered from this term is depolarized due to the zero-trace nature of the tensor. It should be noted that the analogous term has been omitted from (18), since it was assumed that 24 spherically symmetric molecules were being considered. Since we are considering only the polarized spectrum, only effects of the second term will be considered. The dielectric constant fluctuations are usually ex- pressed in terms of density and temperature fluctuations, given by 85 MM) = (33% min») + (gga— 0 misc»). (27) Usually the temperature fluctuations are ignored by assuming that (Be/Bp)T>>(Bc/8T)p. The contribution of the temperature fluctuations to the total integrated scattered light inten- sity for most organic liquids is thought to be about 1%, much smaller than experimental error.40 However, Piercy and HanesS7 dispute the validity of neglecting these fluctuations. Their argument is that a 1% contribution to the integrated intensity implies that the relative magnitude of the two terms in (27) is 10:1. For the (65(£,w))2 portion of I(R,w) in (25), (27) is squared and the cross terms are zero (if p and T are uncorrelated) or very small (if p and T are correlated, which should always be the case3). A contribution of this relative magnitude from temperature fluc- tuations could result in as much as a 20% contribution to the intensity of the frequency components, since they are added and then squared; therefore, they argue, the tempera— ture fluctuations should be included in this analysis. Al- though this concept has been treated theoretically by Piercy and Hanes, their final results are essentially the same as Mountain's. 25 Neglecting the second term, we substitute (27) into To evaluate <[6p(§,m)]2) in dense media, the most approp- riate model to use, as suggested by Onsager's regression theorem, is a hydrodynamic model, which is described by the linearized hydrodynamic equations of irreversible thermo- dynamics. It has been assumed in (28) that (as/3p)T is independent of w, the frequency shift from mo; however, this does not exclude an intrinsic dependence on “o itself, which would reflect the frequency dependence of the dielec- tric constant. Thus far, this approach is equivalent to Einstein's, in that it considers a volume element with linear dimensions that are small compared to A0, but still large enough to contain many molecules. In addition, the mean square fluc- tuating dipole moment of this volume element is non-vanishing. However, a very important consideration is neglected, i.e., the interference of scattered radiation from two neighboring volume elements. When the fluctuations between two such volume elements are correlated, the density fluctuations in (28) no longer have the correct form. The correction is made by noting that the electric field at-R from two in- duced, oscillating dipoles at points 11 and £3 will be the 26 sum of the individual fields, modified by the interference factor exp[ik°(£j-£i)].46 Thus, (28) becomes 4 _ Nwo . 2 as 2 1(3’w) ' Iol6fl2c4R2 Sln ¢(3°) X Jfi/(dp(£i,t)6p(£i,o)) CXP[1£.(£j-£i)]d£id£ji (29) where the double integral implies an integration over all pairs, t is the variation in time and the average ( ) is taken over all initial states of the system. Noting that the Fourier component of the density fluctuations is given by p(§.t) = J[p(g.t) ei Kri dr. (so) (29) can be rewritten as ng . 2 as 2 1(3)”) = 1016flzc4R2 Sln ¢ (530T (31) where47 (o(£.w)o(-£)> = f(o(l<_.t)o(-_19) e-iwtdt (32) and ' p(§) = p(5.t)lt.o (33) The Frequency Spectrum Kamarov and Fisher33 have adapted Van Hove's neutron scattering theory68 and derived (31). They also noted that (as/3p)T can be replaced by me, the effective polarizability of the molecules. Their final expression for the scattered intensity was49 2 Nag mo 4 . 2 1(8)“) = 10W (C—-) 5111 ¢ 3(5)”) (34) 27 where S(§,w) is called the generalized structure factor and is the space and time Fourier transform of the two— body correlation function, which is defined by Van Hove to be COW) = N'1( LL. 6TB. + 11(o)-_r_]. (35) i,j=l where £i(t) is the Hiesenberg Operator.68 For sufficiently long times and large R, G(R,t) reduces to the autocorrelated density given by49 _ G(B_.t) = N'1 f ° (38) This implies, then, that (34) and (28) predict the same fre- quency distribution of the scattered light by density fluc- tuations. The generalized and ordinary structure factors are related by the sum rule 3(5) = 1“J[' 5(ng)dw = (o(§)o(-§)) = (Io(§)|2) -(39) Furthermore, S(§,w) is an even function of m at normal tem- peratures, where w is the frequency shift from mo; thus, the integral reduces to twice the integral from zero to infinity. 28 Since the compound used in this work is a Kneser type liquid, i.e., the internal vibrational degrees of freedom are weakly coupled to the translational degrees of freedom, we shall follow Mountain's method of evaluating the gener- alized structure factor, S(§,m).49 Furthermore, we will also assume that one relaxation time is sufficient to describe the transfer of energy from internal degrees of freedom to the external degrees of freedom. In keeping with previous equations, we will use density and tempera- ture as the independent state variables. The first linearized hydrodynamic equation for a non- relaxing liquid325’31 is the conservation of mass or con- tinuity equation. It is given by apl 3t + po div 1 = 0. (40) The second equation describes the conservation of momentum, and is called the Navier-Stokes equation. Since the momen- tum is a vector quantity which can be separated into three components, the Navier-Stokes equation is a vector equation given by 3v v: o d. + v3 ,390 grad T1 (41) pO 3?- + Y gra pl Y ”(€45 n5+TIB) grad div 1:0. The third and final equation is the conservation of energy equation, given by .aT_C -1 a -2 _ pOCv at) v(; ) 3:1 Av T1 - 0. (42) 29 This equation is known as the energy-transport equation, or the Generalized Heat Flow equation. In these equations, 0 = p0 + p1 is the number density, po being the equilibrium and p1 the departure from equilibrium. The instantaneous velocity of'a small volume element of fluid is y. The low frequency (adiabatic) velocity of sound is denoted by Vs,o’ and t is the time. The ratio of Cp to CV, the specific heats at constant pressure and volume, is given by y. The thermal coefficient of expansion is B, and T = To + T1 is the temperature, where the subscripts have the same meanings as used above for the density. The shear viscosity is as and ”B is the bulk or volume viscosity, and both are assumed non-relaxing, i.e., frequency independent. Finally, A is the thermal conductivity which describes the heat flow re- sulting from the dissipation of thermal fluctuations. Since the above equations have no provision for thermal relaxation, it is necessary to modify them. The most common approach is to separate the bulk viscosity into relaxing and non-relaxing components.29’48'49’71 The bulk viscosity is chosen because it can be used to describe any loss mechan- ism other than loss due to shear viscosity. In polyatomic molecular fluids, it usually is not possible to identify the internal motions which are responsible for the deviations from classical behavior. Since the bulk viscosity does not require specification of the internal relaxation process, it is a general procedure to use a relaxing bulk viscosity. This changes only the Navier-Stokes equation, and the 30 relaxing, linearized equation is thereby given as 31 + V:,o V + V§,ono VT _ ( 4n + n ) 6(6 . V) pofif’ Y p1 ‘-—;_‘_ l j s B _ t —— -JI n'(t-t') V(v ~x(t')) dt' = 0. (43) 0 where n'(t) is the Fourier transform of the frequency de- pendent part of ”B' The next step is to take the divergence of (43), solve (40) for VT! and substitute this into (43) to remove the velocity from these equations. Then the Fourier transform is taken of the resulting equation as well as (42) to yield the following two equations, 32 2 a V§,oK2 n' t 2 a [512‘t bK 3? + y + p, K '5?1 01(E4t) V e K2 + I £40400 1T1(£.t) = 0. (44) -1 3 3 _ and I -1%;;l 5; 191(£.t) + [ 3; +aKZIT1(£.t) — 0. (45) where the Fourier transform of Vzpl is -K2p1(£,t) and a s A/(pOCV) and b a ( éns + nB)/po. The Laplace transforms of (44) and (45) are performed, carefully noting the trans- formation of derivatives, to give: K2 2 2 M-I- szs + b'(S)KZS]p1(§_,S) + [V5,OBDOK Y Y Is2 + ]T1(£.S) =(s + bK2)pl(§.0) + 51(§.0) (46) and Iléégill—Io1(5,s) + [s+aK2]T1(£.s) = illllol(§.0)+rl(§.0) O 800 (47) where b'(s) = bl/(l+sr), I being the relaxation time. From 31 ultrasonics,29 b1=(Vg’,-V2 )T, V being the velocity of sound extrapolated to infinite frequency. For thermal relaxation, Zwanzig71 has found b = (C -C )CI v2 T 48 1 I (C3-C¥)CP ] 5,0 ( ) where CI is the specific heat of the vibrational degrees of freedom, which will be discussed later. Since we took the Laplace transform to t and not the Fourier transform, 5 is the complex frequency, i.e., s = im. Thus, 01(£,w) is re- lated to p1(§,s) by 91(£.w) 2 Re Jf 01(£,t)e'iwtdt O 2 Re JI e‘iwtdtjr eTStp1(£,s)ds. (49) 0 0 By finding 01(£,S) we will then be in a position to solve for the density-density correlation function of (38). Solving (46) and (47), we find 2 2 2 S+bK p109 VS’onoK /Y ‘(Y-l)/Bpo p1(§) s+aK2 01(E,S) 52+V3’0K2/y+(b+b'(s))Kzs V§,onoK2/Y -S(Y'1)/Bpo s+aK2 (50) which reduCes to _ F s p(§.s) - p(i)§%;} . (51) where F(s) = 153 + $2[1 + (a+b)Kzr] + s[(a+b+b1)K2 + abK4r + v: OK2(I-1/y)r] ’ + V§,oK2(1—1/y) + a(b+b1)K4. (52) 32 and 0(5) = 154 + s3[l + (a+b)Kzr] + sZ[(a+b+b1)K2 + abK4r + VS’OKZT] + s[V§’OK2 + a(b+b1)K4 + V§,oaK4/Y]+V§,oaK4/y. (53) Using (38), (39), (49), and (50), we see that SQSM) =(p(_l£.t)o(-£)) = 2(|S(5)I2) ReIF(im)/G(iw)1. (54) The derivation of (54) shows the advantage of choosing the temperature as the second independent thermodynamic variable?6 since >FB, and w::wB in the vicinity of the peak. Therefore, wgzgwfi, and, by factoring (wz-w§)2 to (w-wB)2(w+wB)z, we get IB a wfiFB = 2 F .1 B . (61) 4 4w§(w-mB)2+4FBZwB FBZ+(w'wB)Z 36 If the above derivation is performed without assuming relaxa- tion, (59) has the form of only the Rayleigh and the two Brillouin lines. Thus, vibrational coupling manifests itself as a new component, Im, whose half-width is (V§,o/V§)l/r. Since the integrated intensity of Im changes markedly as the product ”BI approaches unity, three regions of interest be- come apparent in the relaxation contribution to the spectrum. 1) 1/T>>wB: This implies that the relaxation frequency is much greater than the Brillouin shift, which is the usual case for simple liquids. The Mountain line then is very broad and of such low intensity that it appears as a flat background. Therefore, its intensity is negligible and the spectrum reduces simply to the three normal modes. The Ray- leigh line is Lorentzian (actually an Airy function that falls off so rapidly in intensity that a Lorentzian is sufficient 38), and the two Bril- to describe the noticeable intensity louin lines form a resonance line as described above. An example of this is the spectrum of carbon disulfide.24 The Landau-Placzek ratio is given by the simple form y-l. This result corresponds to earlier predictions that this vibra- tional relaxation component would be unobservable because the frequencies usually associated with this phenomenon would be too high to be observed by optical means. 2) 1/t<>FB, w§>>abemfi>>ab1K4 and l>>aKzr, which are valid for many liquids, Stegeman £3 31.66 have . Iderived the following coupled equations for V5 and PB: z 2 IB[3-VsLO- b1K T ] = bK2+aK2[1-v§ o/(vv§)] ’ v: (1-rBr)2+v§K212 + b1K2(l-aKzr) (l'FBT)2+V§K2T2 (66) 44 and 2 2 _ 2 2 2_ _ 2 2 2 VSK T — 1/2[Vs’mK T (1 FBI) + 3TB T ] + 1/2 [v2 K212 + 3r 212 - (1-r r)2 5,00 B . B +4(l-FBT)2(V§ OKZI + 3FB212) - 2b1K21]1/2 (67) These equations define the dispersion of the velocity and of the temporal attenuation coefficient, PB. When there is no overlap of the Mountain and Brillouin lines, i.e., in the two frequency regions 1/T<>wB, the center of the Brillouin peak is the Brillouin frequency. This is not so for the region of overlap because of the fre- quency pulling effect; however, if a suitable curve fitting procedure is utilized to separate the Mountain and Brillouin components, ”B still corresponds to the center of the actual Brillouin peaks. Once “B is known, the velocity of sound can be calculated from VS = VS(K) = mB/K (68) The sound absorption coefficient, a' (sometimes called the' spatial attenuation coefficient), and the Brillouin half- width (or the temporal attenuation coefficient) are related by 0' = rB/V5(K) (69) 45 Depolarization Ratios When polarized incident light is scattered from a medium, the scattered light can be separated into two com- ponents, i.e., vertically and horizontally polarized light. From (26), the dielectric constant contains two terms that determine the relative magnitudes of these two components. The scalar departure from the mean dielectric constant leads to totally polarized scattered light. However, the third term, AS(£,t), is a symmetric matrix with zero trace. In a liquid, this term arises mainly from the reorientation of anisotropic (non-spherically symmetric) molecules. The lar- ger the anisotropy (the less spherical the polarizability) of the molecule, the larger this term is. Moreover, for an isotropic molecule, AS(£,t) will be small, but not zero, be- cause molecular collisions and vibrations, as well as the I incident electric field, induce anisotropy in the molecules. It must be recalled that stresses and shear waves also make contributions to AS(£,t); however, these are absent in most liquids. Since AS(£,t) is a matrix, the incident polarization vec- tor is rotated, giving rise to depolarized scattering. Since the polarization vector is not rotated a full 90 degrees, the rotated electric field vector is resolved into vertical and horizontal components when the scattered light is analyzed by a polarizer. While this term is the only source of de- polarized scattering, the AS(£,t) term also contributes to the polarized light scattered from the scalar term. This fact 46 must be taken into consideration when the polarized fre- quency spectrum is analyzed. The total intensity of the horizontal and vertical scattered light is usually denoted by an H or a V. The polarization of the incident field is denoted by a subscript V or H. Thus, four components of the scattered light can be measured, i.e., VV (6), VH (e), HV (9), and HH (9), where e is the scattering angle and represents the angular dependence of these intensities. Kielich34 expressed the Rayleigh Ratio (the ratio of the scattered light and the incident light intensities) in terms of Fis and Fanis: which describe the isotropic and an- isotrOpic scattering respectively and are prOportional to the mean-square isotropic and anisotrOpic fluctuations in the dielectric constant. Dezelic15 has used this notation in his derivation of the magnitudes of the four components, which are given by VV (6) = («Z/SA§)(SF,, + 4Fanis)4 vH (e) = HV (6) = («Z/5A3) '3Fanis, and HH (9) = (“z/512)[5 F15 C052 9 + (3 + C0526)Fanis (70) Note that only HH (6) is angular dependent, since Fis and F are not. This angular dependence leads to some inter- anis esting results. When 6 is equal to ninety degrees, HH (6) is equal to VH (e) and HV (6). This implies that light can- not be isotrOpically scattered in a direction which is 47 parallel to the electric field vector, a result that is in agreement with the basic laws of Optics. However, rotations of anisotropic molecules can change the incident polarization and allow scattering so that HH (6) is not zero unless Fanis is. The equality is to be expected since the scattering mechanism for the HH (6) component at ninety degrees is the same as for the HV (6) and VH (6) components. When a is equal to zero or 180 degrees, HH (6) is seen to be equal to VV (6). At intermediate values of 6, HH (6) varies from VH (e) to VV (6) according to its cos2 dependence. Measurements of absolute intensities are of theoretical and practical use, especially in the determination of mole- cular weights Of macromolecules. However, the literature shows large disagreements in values because of the difficulty in measuring absolute intensities and the lack Of a uniform calibration technique.20 Therefore, relative intensities are generally used because the ratios of these values give meaningful information on the nature of the molecules and are independent Of the absolute intensity. The intensity ratios most commonly used are called "depolarization ratios," which are given by pV = HV (9) lVV (e) = 3 Fanis/ (5 Fis + 4 Fanis). pH = VH (6)/HH (e) = 3 Fanis/[S Fis cosze+ (3 + cos2 6)Fanis] pu = Hu (6)/Vu (e) = [5 Fis cosze + (6 + cos2 6) X Fanis/ (5 Fis + 7 Fanis)] (71) 48 where mu (6) HH (9) + HV (6), Vu (6) vv (4) + VH (e) . and the subscript u denotes unpolarized incident light. At ninety degrees, pH is equal to unity for small anisotrOpic molecules. In solutions, pH can exceed unity as the critical temperature Of mixing is approaches, but is equal to unity well below this temperature. The situation in which pH is not unity is called the Krishnan effect,18 and is due to the formation Of molecular clusters. The quantities pv and pu are related by on = 2/(1 + pv‘l) (72) Of the three ratios, pH, 6V and pu, the most useful for the detection of molecular anisotropy is pv. It is not angular dependent, but it is wavelength dependent, which is not yet accounted for by theory.20 The molecular anisotropy is directly related to pv, since pv goes to zero as the mole- cular anisotrOpy decreases, i.e., as the molecular polariz- ability becomes spherically symmetric. Thus, will be of DV principle concern in this study. CHAPTER III EXPERIMENTAL EQUIPMENT AND PROCEDURE The Spectrometer The Brillouin spectrometer, as shown schematically in Figure 3.1, consists of three basic parts. The first is the light source, which is a laser for the obvious reasons Of high intensity, coherence, and monochromaticity. The second part is the sample cell, its temperature-controlled jacket, and the rotating table, which allows angular measurements. The third part consists of the detection Optics, which are shown schematically in Figure 3.2. These optics collect and spectrally analyze the scattered light. The details of the design and construction Of the Brillouin spectrometer and Optical table have been presented elsewhere;23 however, several modifications Of the system were necessary during the course of this study. 1) The Laser. A Spectra-Physics Model 165 Argon Ion laser was installed to replace the low power, multimode HeNe laser as the Optical source. This laser has a total output power Of four watts and emits at eight visible wavelengths, which can be selected individually by a prism assembly lo- cated in the rear Of the laser cavity. Table 3.1 lists these eight wavelengths and their maximum output power, which con- sists Of many cavity modes lasing simultaneously, as shown in Figure 3.3. The laser is equipped with an etalon ( a small 49 .Hou0Eouqumm mcfiuouumum unmfia Ogh monk ¢MDLEUDSOPOIauHZa mOH4aa3m mmBOm kin—"mm >§.m mwBOm buxos. o H¢m¢o_-9_mmEm>zoo o2 21.. m dxnd CXFO.N .5. 3d .. 5.2.50 I - - IL ood re locations. TO obtain a plot free of these markers, core 1 o cation 37558 is changed to a NOP instruction (70008). 8) Since a wired plug-in board for the department key- punch would allow the reading of hexadecimal paper tapes and the punching of computer cards, it was believed that this would Prc>‘\r:ide an excellent means of transferring data to the CDC 6500 foz~ (czonversion to a format suitable for program KINET. There- f"1“3 , a hexadecimal paper tape punching program was written. \3- r‘¢E=sides in core locations 208 to 1648 in the upper field. Leade r tape is punched, followed by the data and trailer tape. The h exadecimal format allows a parity check, which halts the keYpl-lil'1ch when either a punching or a reading error appears. 93 Since the hexadecimal format requires three paper tape rows per data word instead of the two required by the binary for- mat, one half more time is needed to punch a hexadecimal data tape, i.e., about 10 to 11 minutes for 3840 data points. The programs written above are quite versatile and can be easily modified for new applications. Several modifica- tions can be made to make them more impressive, but these modifications were not performed, since all programming had to be done with a teletype using paper tape, instead of the much faster and easier methods available with other computers, e.g., a high speed paper tape reader and magnetic tape sys- ‘tems. Thus, a simple modification that could be performed *vvith a magnetic tape system in less than five minutes often t:<30k three hours or more with the teletype paper tape reader, I:>1? a display on the oscilloscope. If the data is satisfac- ‘t:c>ry, the program is halted, and the data punch routine is 11:5zaseline measured between the Rayleigh and Brillouin peaks. When the width at half intensity is measured with the cursors, the left cursor is moved to the right a distance equal to the half width at half intensity. If good peak symmetry is Present, the left cursor will lie on the visual center of the Rayleigh peak. The "IR" command is given, not for the a11"e<'=:1 but for the Rayleigh peak core location. When the 10- cat ion of the Rayleigh of the next order is found similarly, the S e locations are subtracted for the interorder spacing. Ne): 1:1 , the cursors are moved equal distances from the Rayleigh to JF>‘<3$it10nS Of minimum intensity between the Rayleigh and 96 Brillouin peaks. The baseline is set and the peak is inte- grated. The same procedure is used for finding the Bril- louin peak areas and locations. From these data, the fre- quency shifts, interorder spacings, intensity ratios, finesse, sound velocity, etc., can be calculated using pro- gram LPRATIO, described in the next chapter. When no Mountain line is present, the above procedure works well and analysis times are on the order of 10 to 15 minutes per order, much faster than chart paper analysis. The accuracy is at least an order of magnitude greater becau$e of the enlargement capability of the oscilloscope, the absence of distortion from a nonlinear chart drive, and the precise integration of the areas. When the Mountain line is present and distorts the Bril- louin peaks, the above method of finding peak locations is still the best. However, peak areas need not be measured, because of theoretical uncertainty of the meaning of the in- tensity ratios. If the areas are to be measured in spite of this, the instrumental response function must be very narrow, i.e., high finesse values must be obtained with the spec- trometer alignment. Otherwise, severe errors occur in the determination of the baseline between the Rayleigh and lBrillouin peaks, because a broad spectral response function vvill cause much greater overlap, significantly raising the Ulinimum intensity between peaks. The approximation made when meaasuring the Rayleigh area using the minimum intensity be- tuveen peaks as a baseline is that the Mountain line is broad 97 and appears as a constant background in the frequency range of a narrow Rayleigh peak. This approximation is prone to error, especially when the response function is poor or asymmetric. However, it will give approximate values for the intensity ratio R/(ZB + M). After the Rayleigh area is measured with this method, the area of the total order is measured, and the Rayleigh area is subtracted to give the value of (2B + M). An alternate method is to measure the Rayleigh and the order half—areas, after the Rayleigh center has been determined. When the Rayleigh half-area is sub- tracted from the order half-area, the intensity ratio can be calculated for each half of the order, and the two results averaged. CHAPTER V DATA ANALYSIS Programming The mini-computer provides fast and accurate data acquisition and preliminary analysis. When the Mountain line is absent, the results are meaningful and allow a number of quantities to be calculated directly. However, the presence of the Mountain line distorts the spectrum, and the prelim- inary results from the mini-computer must be carefully scru- tinized, because the Brillouin frequency shift, peak width, and peak area have been altered. In order to decompose these spectra into their four components, a large computer (e.g., the CDC 6500) is necessary to reduce the data. Consequently, several computer programs were used to perform the complete analysis. These programs are discussed below in the order in which they would most likely be employed. 1) Program LPRATIO. Even in the absence of the Mountain line, a large computer is useful for the large number of cal- <:ulations required. Since the mini-computer results are given in octal format, it is convenient to convert them to deecimal before calculating the sound velocity, the frequency sluift, the Brillouin half widths, and the other spectral parameters. Therefore, a program was written to accept the 0C1:a1 mini-computer results and perform these calculations. The: data input consists of an identification card, the number 98 99 of full or partial orders in the spectrum, the Brillouin and Rayleigh peak areas, the left and right Brillouin peak separa- tions, the interorder spacings, the mirror separation of the Fabry-Perot interferometer, the Brillouin and Rayleigh peak heights and full widths, the refractive index, the laser line wavelength and the scattering angle. If any of the above quantities are not known, a value of zero is given and the program automatically sets to zero any calculation results depending on these unknown quantities, and omits them from the calculated averages. After the spectrum identification is printed at the top of the page, the computer lists the results as follows. The octal areas and their decimal equivalents are listed for each peak, and the Landau-Placzek ratios are calculated from the peak areas. Three ratios corresponding to R/(ZB + M) are calculated for each order. The first uses the Rayleigh area with two times the area of the left Brillouin peak, and the second uses twice the right Brillouin area. The disagreement between these two values is an indication of instrumental misalignment. The third ratio uses the Rayleigh area and the sum of the left and right Brillouin areas. Next, the octal aind decimal equivalents of the splittings, interorder spacings, {weak widths and peak heights are computed. The mirror slpacing given to the program is corrected for the mirror hC>1der lip thicknesses, and the free spectral range is cal- cullated. Using this value, the average value of the Brillouin freequency shift is calculated. A table is given which lists 100 the values of the average splittings, the corrected Bril- louin widths at half intensity in computer units and in giga- hertz, and the ratio of the Brillouin and Rayleigh peak heights. The corrected widths at half height are calculated assuming that the Rayleigh peak is Lorentzian. (Since the Brillouin peaks are Lorentzian, the observed peak is a con- volution of the real Brillouin and the instrumental response function, which is given by the Rayleigh line. The convolu- tion of two Lorentzian profiles is again a Lorentzian, whose width is the sum of the widths of the original two profiles, and, therefore, the Rayleigh width is subtracted from the Brillouin peaks to give the real Brillouin width.) The instrumental finesse is calculated by dividing the interorder spacing by the Rayleigh widths. These values are listed next to the ratios of the Brillouin peak separations and interorder spacings. The average of the latter values is multiplied by the free spectral range to give the average Brillouin shift. Finally, the velocity of sound is calculated from the scattering equation using the measured values of the frequency shift and the supplied values of refractive index, incident wavelength and scattering angle. The sound veloci- ties calculated for each order are listed, followed by the re- fractive index and scattering angle. Consequently, the print- <3ut contains all the information available from the spectrum. This program works well when no Mountain line is present, alnd the calculated values are of use to characterize the Sp>ectrum. However, when the Mountain line is present, neither 101 the mini-computer analysis, nor this program are of great value, since only the frequency shifts and sound velocities have meaning. The peak areas vary greatly with finesse, be- cause of peak overlap, and the intensity ratios are difficult to interpret in this case. Furthermore, the Brillouin peak— widths are not the correct values, because no corrections have been made for distortion from the relaxation line. There— fore, a more elaborate analysis, involving curve fitting is necessary. 2) Program LIST. In order to list the data on the paper tapes directly, a program was written for a departmental pdp 8/I computer. This computer is interfaced to a line printer and is equipped with a high speed optical paper tape reader. The tape is read, and the data is listed 20 columns per line, preceeded by the core address of the data point in the first column. The total listing time for each tape is about thirty seconds. 3) Program CONPUN. Since the mini-computer analysis was not useful for spectra with a thermal relaxation component, and no direct method of submitting the data to the CDC 6500 was available, a program was written to convert the pdp octal data to decimal and punch it onto cards for the curve-fitting program. The data listing from program LIST was used to manually punch the octal data on computer cards, twenty data points to a card, starting at the center of a Rayleigh peak. The total input to this program consists of the interferometer mirror spacing, the number of points per interorder spacing, 102 the data cards, and the maximum intensity. Using the mirror spacing, the program calculates the free spectral range and the average frequency interval per data point. The data is then normalized by dividing each point by the maximum inten- sity. The x-axis (frequency) variance was calculated assuming a one-point uncertainty, and the y-axis (intensity) variance was set equal to one percent of the maximum intensity for normalized intensities less than .5 and to 2 percent of the normalized intensity when it exceeded .5. The frequency shift from the Rayleigh peak center, the frequency variance, the normalized intensity and its variance were then punched on data cards, two sets per card. These values are listed in the same format, followed by the values of the free spectral range, the frequency increment per point, the maximum intensity, the mirror spacing, the number of points per order, and the number of data points used. The decimal data deck, which is produced by the University compu- ter center, is then used for the curve fitting program. 4) Program KINET. In order to obtain usable data from spectra containing the thermal relaxation component, program KINET was obtained from the Michigan State University Computer Center. This program does nonlinear least-squares curve- fitting on the data supplied by program CONPUN, and its de- tails and merits have been discussed elsewhere.16 The data input consists of a control card to specify which program options are desired, an identification card for the data set, initial guesses for the parameters of the fitting equation, 103 the number of points to be fit, and, finally, the CONPUN data deck. The output consists of a data listing, the values of the adjusted parameters after each iteration, the final values of the parameters, their standard deviations, various statistical parameters to indicate the quality of the fit and the relations between parameters, the differences between the calculated and experimental points, and a plot of the calculated and experimental points. if fewer than 100 points are used in the fit. The equation used in the fit, and the results and limitations of this program are discussed in later sections. This program was initially tested on ideal data generated by program BRILL and was found to give very satisfactory re- sults. The data submitted is for one half of an order and cannot exceed 500 points. The full weighting option was used to allow for noise in the spectral data. 5) Program BRILL. When the parameters of the spectral distribution function are known (or found with program KINET), these parameters can be submitted to program BRILL, which calculates the values of the total spectrum and of the four individual components of the spectrum. These values can be listed, punched on cards, and plotted. The plot provides an excellent medium for the visual inspection of the contribution of each component to the total spectrum. In addition, the I program numerically integrates each component and calculates the intensity ratios, listing these values with the values of the initial parameters. Examples of the utility of this program are provided in a later section. 104 6) Program WAVENUM. In order to calculate the sound velocity and its temporal attenuation coefficient, P/KZ, it is convenient to have a table of values of the wavenumber of the scattering phonons, K, and its square. The scattering equation gives K as IE] 2|_l_<_| sin (6/2) (4nn/AO)Sin (6/2). (73) The only data required is the refractive index, n, at the wave- lengths and temperatures used. The program calculates K and K2 for five degree increments of the scattering angle from 20 to 165 degrees at four temperatures and wavelengths (5145 A°, 5017 A°, 4965 A°, and 4880 A°). It can be modified easily to allow these calculations to be performed at other angles, temperatures and wavelengths. Examples of these tables are given in the data results section. Analysis Procedure The spectral distribution function was given in the theory section and has the form P I‘ I‘ R M , B 50") ‘ ”7:7 " B'TT * C 2 r ’ (74) 0) TR (0 +PM _ (N'NB) +TB where A', B', and C' are the large expressions for the coef-' ficients of the Lorentzian profiles and w is the frequency shift from the Rayleigh center frequency. When a spectrum is taken, the observed Rayleigh line is not the real Rayleigh line but the instrumental response function, since the actual Rayleigh line can be approximated by a delta function when 105 compared to the spectrometer resolution. However, it was shown elsewhere67 that the instrumental response function can be approximated closely by a Lorentzian. When the above expression is convoluted with this response function, the peak profiles are all still Lorentzian, with peak widths equal to the actual peak width plus that of the convoluted Lorentzian response function. Therefore, the above expression retains the same form. Since the data from CONPUN were scaled for unit intensity at S( i.e., at the tOp of the Rayleigh ”)max: peak, the spectral distribution function is similarly scaled by the scaling factor S(w)fi;x = S(o)'1. To use this scaled spectral distribution function, it is necessary to evaluate the coefficients A', B' and C' from the theory, because good approximations of these quantities are needed for KINET. Since many of the constants required for this evaluation do not appear in the literature, the values of these coefficients cannot be found. However, if normalized Lorentzians are used, the coefficients then correspond direct- ly to the peak amplitudes. To normalize the Lorentzians, the coefficients are assumed to be equal to the peak half- width times a new coefficient, i.e., A' = RPR, B' = MPM and C' = BPB. The relative values of R, M, and B are then easily found by the relative peak heights in the spectrum. In addition, since the equation then corresponds to a sum of three Lorentzians times a constant, the spectral distribution function is divided by the Rayleigh coefficient. The final 106 expression is given by S(w) rRZ S'(w) = II + :5| 3 x (75) l + M/R + (B/R)(I‘B2/(wB2 + P323) This equation was used in KINET to fit the data of one half of an order. The parameters that were evaluated by KINET were FR, PM, M/R, and B/R. The initial guesses of TB, wB, these parameters were supplied by the mini-computer analysis. On a given spectral half-order, two computer fits were performed. A preliminary fit, using every fifth point of the data and the initial guesses from the mini-computer, gave improved values for the parameters, and a plot of the calcu- lated and experimental points permitted visual inspection of the quality of the fit. The improved values from the preliminary fit were then used as the initial guesses for the second fit, which used all the data points of the half order. This procedure was followed, because fitting up to 500 points with six parameters is a complex task and KINET would either require a large amount of time to perform a large number of iterations, or it would fail to arrive at a satisfactory result and abort. The preliminary fitting values thus insured a good second fit in a shorter time. Occasionally, a mis-punched data point or a large noise spike would appear in the data. This occurrence was usually quite obvious, since KINET would abort, being unable to find 107 satisfactory values for the parameters, or the parameter values and quality-Of-fit values would be inordinately large. A search of the data would usually reveal the offending point or points, which were either discarded or replaced with values comparable to the surrounding data points. The quality of the fits was generally quite good. The only spectra in which the fits were consistently unacceptable were low angle spectra (50° - 70°) where poor response func- tions caused large Rayleigh-Brillouin peak overlap. In this situation, the Mountain line parameters were found with standard deviations on the order of 100 percent or more of the parameter value. Fortunately, this situation arose only twice, and the fitting procedure was in general quite trouble free. CHAPTER VI EXPERIMENTAL RESULTS PrOperties of CFSCCI3 As a result of its molecular shape, 1,1,1-trichloro- trifluoroethane (Freon 113a) has several unusual physical properties. It is almost spherical, differing from carbon tetrachloride (CC14) in structure only by the substitution of a CF3 group for a chlorine atom. The CF3 group is small and compact because of the strong C-F bonds, but it is slightly larger than a chlorine atom. Therefore, Freon 113a should resemble CCl4 in many of its properties. Several physical properties of Freon 113a are listed in Table 6.1. Since the other isomer of Freon 113a, Freon 113 (CFZClCFClz), is more readily available and has been used as a refrigerant for many years, physical data on Freon 113a has been hard to find in the literature, and most data reported are from NMR29, ir10:43:53:54, and Raman 25:43:53,54 studies of homologous series to locate trends in vibrational, torsional13 and coupling constants of halocarbons. The vibrational energy level scheme of Freon 113a is of impor- tance in this study, as will be discussed later. On freezing, Freon 113a forms a solid that appears to have a large number of bubbles and fractures. On standing, the solid appears to creep up the container walls and form a clear, isotrOpic solid at the tOp of the container. This 108 109 Table 6.1. The physical properties of Freon 113a. Property Molecular Weight (g/mole) Melting Point (°C) Boiling Point (°C) Refractive Index, ”D Density (g/ml) CP° (Cal/deg-mole) Viscosity (centipoise) Solubilities Water Alcohol Ether * Manufacturer's value 221.19.; 187.38 14.2 45.9(760mm Hg) 1.3599 1.3596 1.5790420 1.5702 20 24.39 (25°C) .720:2% Insoluble Soluble Soluble ** Calculated from ir and Raman studies Reference (1) (11) (11) (i) (1) t (43)** (9) (11) 110 phenomenon is caused by the solid subliming and recondensing at the tOp, indicating a high vapor pressure for the solid. This is confirmed by the high vapor pressure of Freon 113a at its freezing point, 203.1 mm Hg.30 The other isomer, Freon 113, melts at -35° C and boils at 47.6° C, showing a liquid range of over eighty degrees compared to 31.6 degrees for Freon 113a. The narrow liquid range, the high vapor pressure of the solid, and the spherical nature of the molecule indicate that Freon 113a should form a plastic crystal, as does CC14. This has_been shown to be true, and the glass transition temperature has been found to be at about -30i4° C by NMR spin echo measurements of the transverse relaxation time.70 It may be noted that the glass transition temperature of Freon 113a is very close to the freezing point of the isomer, Freon 113. Refractive Index In order to calculate the velocity of sound from the equation “’B = izwo(y%)" sin (3), (76) it is necessary to have the refractive index as a function of temperature. We have measured n(T) with a Bausch and Lomb Abbe 3-L Refractometer, which was calibrated using the glass plate described in the polarization calibration section. The refractometer gives the values of the refractive index and dispersion for the S890 A°, sodium D line. From the equations and tables supplied with the refractometer, the lll refractive index can be corrected to the argon-ion laser wavelengths with an accuracy of 3.0005 units. The values for the measured refractive index and average dispersion are listed in Table 6.2. The temperature was mea- sured to the nearest .05 Centigrade degree. The value at 20.50°C agrees with a literature value at 20°C of 1.3599. These values, which are plotted in Figure 6.1, were fit by the least squares procedure of KINET to the straight line n(T) = mT + b, (77) where T is the temperature in Centigrade degrees, m is the slope and b is the intercept. The slope (an/3T) was found to be 5.611 X 10'4 and the intercept was found to be 1.3720, with standard deviations of .90 percent and .01 percent, respectively. The calculated values of n for the temperatures used for the light scattering measurements are listed in Table 6.3. Also listed are the refractive indices corrected to the four exciting lines used from the Argon-ion laser. The refractive indices are used by program WAVENUM to calculate the wavenumber of the scattering phonons, K. Tables 6.4-6.7 list these values for the 4880 and 5145 A° lines at the four temperatures studied. In addition, Table 6.8 lists the values for the 4965 and 5017 A° lines for 90° angles, since this was the only angle studied for these lines. T (°C) 24.75 29.55 34.55 39.60 Table 6.2. T (°C) 20.45 25.20 29.90 35.10 40.10 Table 6.3. nD 1.3604 1.3579 1.3552 1.3523 1.3494 Corrected refractive indices. n5145 1.3581 1.3554 1.3526 1.3498 1.3607 1.3578 1.3555 1.3525 112 n5017 1.3612 1.3583 1.3560 1.3530 Dispe 16. 17. 16. 16. 16. Refractometer readings. rsion 6 0 8 n4965 1.3615 1.3585 1.3562 1.3532 n4830 1.3619 1.3589 1.3566 1.3536 .opspmyomaop m3mpo> xowcfi o>fiuompmoh mo poam .H.o ohsmfim 36:. 0.0¢ 06» . 0.0» QnN 0 .ON led d 1 4 d 10000.. .Oflnn; 3 o 1 .9151” _ 1 my 4 ~h 1033 O 13”.. 503.. Table 6.4. RFFROCTIVF INDFK 3 1.3619 IHFTA 2.00000F001 2.50000E001 1.000005001 '1.90000F°Ol “.OOOOOFOOI «.SOOOOEOOI 5.00000F001 §.50000F001 A.OOOOOEOOI 6.50000F601 7.OOOOOE°OI 7.50000F901 8.00000F001 n69000017901 9.00000E901 HAVFNUNRCP K 6.039816006 76590516004 9.07677E004 1.09457E005 ].|Q°46E00§ 1.1hPOTE90§ 1.48212F909 1.61Q3RFOOS 1.793505009 1.934316005 2.011916005 2.13499E005 2.?54256005 2.16929E905 2.4798?E909 QFFQACTIVE INDEX 3 1.1589 THETA 2.00000F001 2.50000F001 1.00000F001 3.50000E001 6.00000E001 4.500006901 5.00000E001 5.50000E001 6.00000E001 6.50000F001 7.00000E001 7.50000E’01 RoOOOOOEOOI 3.500005001 9.00000E001 HAVFNUNBFP K 6.0764?F00b~ 7.571816006 96056706004 1.05225E005 1.]Q6826005 1.13911E005 1.47386E005 1.6ISTREOOS 16749646905 1.88016E005 2.00710E009' 2.1302?Eoos 2.249296oos 2.16407E005 26b7$36E005 LAMBOI I 6880. K SQUARE 3.708615009 5.76161E009 8.21878E609 1.112135010 loh3fl71E010 1.80115F010 26196686010 2.6?230E010 3.07475E010 3.55061E010 6.04625E910 6.557905010 5.08166E010 5.61154E910 6.1b951E010 LAHIDA I #380. K SQUARE 3.69229E009 5.716265o09 3.2ozszeoo9 1.107235olo 1.4azascolo 1.793225oao 2.1n7015o1o 2.610165olo 3.06122Eo10 3.53409E.Io 4.023455+1o 6.517866010 s.osaao£o10 5.58886E010 6.12245Eoio 114 Program WAVENUM results. TEU'EPITURE I Zbo7fi THETA 9.500006001 1.00000E902 1.05000E002 1.100006002 1.15000E002 1.200006002 1625000600? 1.30000E002 1.35000E002 1.60000E002 16650006002 1.500006002 1.55000E002 1.600008002 1.650006602 HAVFNUMIEP I 26535636905 2.6A6S1E605 2.78229F005 2.37276E005 2.95777F605 3.0371SFOOS 3.11076F605 36178626009 1.26006F905 3.20550F005 3.344686005 3.187%0E005 3.4?3RTEOOS 1.653726005 3.576Q95005 TEIDERATURE C 29.59 THFTA 9.500oocool 1.ooooo£ooz 1.050006002 1.10o0oEooz 1.xsoooEooz 1.zoooo£ooz 1.250005o02 1.aoooocooz 1.3soooEooz 1.60000E002 1.4sooosooz 1.500005002 1.550oocooz 1.6oooocoo2 1.6sooncooz UAVFNUNIER K 2.51QQJEOOS 2.68060E005 26776165005 2.36663E005 2.991?SEOOS 3.03OQ6F005 3.10389E905 36171525005 3.232°0F005 3.28924E905 36137315005 36380065005 3.61632E005 3.666115009 3.469135oos K SQUARF 6.68567E010 7.?17366010 7.74112E010 8.?52776010 R.758¢1E910 9.?2426E010 9.6767PE010 1.0102366]! 1.04979F911 1.086036911 1.1186960]! 1.16791E011 1.172?9€011 16192826011 1.?089SE911 K SOUAPF 6.65605F010 7.19560E010 7.70705E010 R.?16¢5€¢10 8.709916010 9.18367E010 96636145010 1.00579E011 1.06517E011 16031756011 1.11377E911 1.1“?“66011 16167135011 1.13757E011 1.?03635011 Table 6.5. RFFR‘CTIVE INDEX 3 1.3966 THPTA ?.00000€001 P.50000E001 1.000005001 '1.90000F901 4.00000F001 4.50000F601 9.00000E001 5.50000F901 6.000005001 6.500005001 7.00000E°OI 7.50000F001 3.00000F001 9.90000E001 0.00000F001 HAVFNUNRER K 6.066116004 7.9609QE004 9.041496000 1.0%067E009 1.106806605 1.33689E005 1.4763SE005 1.61305E009 1.76667Eooq 1.376Q7E005 2.00370E009 2.126676005 2.?456RF005 2.360076009 2.47017EOOS RFFPACTIVE INDEX - 1.3936 IHFTA 2.00000E001 2.§0000€901 3.000005001 3.50000F001 6.000005901 6.50000F001 9.00000F901 5.50000F001 6.00000E001 6.50000F901 7.00000F901 7.900005001 8.00000E001 9.50000F001 °.00000F001 HAVFNUHRER K 6.0%???F004 7.94427E006 c.021465ooa 1.oanlssoos 1.192165oos 1.1wanEoos 1.473095oos 1.6ooanEoos 1.142015oos 1.37297soos 1.099278005 2.12191E005 2.240575oos 2.3saaqcoos 2.5647IE005 LAMBDA . 4880. K SQUARE 3.679805009 5.716865009 8.174786609 1.10369E010 1.62754E010 1.787168010 2.17962E010 2.601935010 3.09087E910 3.52303E010 4.016826010 4.5?2495o10 5.04218E010 5.569966010 6.10174E010 LAMBDA - 4980. K SQUARE 3.66356F009 5.69160E009 8.138675009 1.09861E010 1.421236010 1.779265010 2.169996010 2.59043E010 3.03719E010 3.507676610 3.99108E910 6.502516010 5.01991E010 5.56513E010 6.0767BE010 211.5 Program WAVENUM results. TEMPERATURE I 36.59 TEN-ERATURE 3 39.60 THETA 9.500005001 1.00000E’02 1.050006002 1.10000E002 1.15000E00? 1.20000E00? 1.25000E902 1.30000E00? 1.350005002 1.60000590? 1.65000E002 1.500005002 1.550005002 1.60000E002 1.650005002 THETA 9.sooooEool 1.ooooo£ooz 1.osooo£ooz 1.100005oo2 I.lsooo£ooz 1.7doooEooa 1.2sooosooz 1.3ooooso02 1.asoooEooz 1..ooooEooz 1.4soooEooz 1.soooo£oo2 1.ssoooEooz 1.6oooocooz 1.6sooosooz UAVFNUHUER K 2.57597E005 P.67606EOOS 2.771‘65005 P.86198F005 2.966P6F009 3.0?933E005 3.098666005 3.16605E605 3.?2743E009 3.23267F905 3.331676005 3.37632E005 3.61054E005 3.660286005 3.663k66005 UAVFNnuIER I‘ 2.56QR7E005 ?.5701b6005 2.76533E005 2.85525F005 2.01916F005 . 3.01864E005 3.09179E905 3.15905E005 3.22030F005 3.279616005 1.3?61OE005 3.366855005 3.50300E005 3.93?67F005 3.63%ROE005 x §OUAOE 6.63354£.vn 7.16130ro10 7.68099E910 8.138665010 s.onoaseolo 9.15261E910 9.601556010 1.00239r.ll 1.04161E.]1 1.01759r.11 1.110006011 1.13860F911 1.16318E011 1.18359Eo11 1.199566011 K SOUARF 6.60623E010 7.129666010 7.64705E610 9.15268F010 8.64210E010 0.11217E010 9.559166010 9.979586010 1.0310380]! 1.07283F011 1.10510E011 1.133575011 1.158066011 1.1783260]! 1.194?6E011 11.6 Table 6.6. Program WAVENUM results. RFFDACTIVE INDEX 8 1.3607 LAMBDA I 9155. TEHRERATURE l 26.73 RFFDACTIVE INDEK 3 1.3S7B LAMROA . 5195. TENREDATURE ' 2Q.S§ THETA HAVENUNRER K K SQUARE THETA UAVFNUMUER K K SQUARE P.00000E001 5.771ORE00h 3.33OSAEOOQ 9.50000E601 ?.h§0?9F005 6.00391E010 7.90000E001 7.10327F004 5.176?5F009 1.00000E00? 2.5afiROFOOS 6.48159F010 1.00000E901 8.6016RF004 7.39388E009 1.05000E00? 2.61666E005 5.99196E010 '1.50000F001 9.99179E006 9.98791F009 1.]0000E902 2.???605005 7.h1164E010 4.00000F°01 1.11669E005 1.29POhE010 1.15000600? ?.R0?Q§F005 1.89659F010 4.30000F001 1.?7197F009 1.61751E010 1.20000E00? ?.87818E60S 8.28390E010 S.00000E901 1.40656E005 1.97274F010 1.25000E00? ?.°h7°?FOOS 8.6ROPhE010 q.‘SnOOOFOOl 1.51550E005 2.35497E010 1.30000E00? 3.01?09E005 9.07?#6E910 6.00000F001 l.6AI7?FOOS 2.76110E010 1.35000E002 3.07055F005 9.42767F010 A.SOOOOE'OI 1.7R469F005 3.18865E010 1.90000E00? 3.1?300F005 °.7§316E010 7.00000E*01 1.90626E009 3.63176F010 1.45000E002 1.16661F605 1.00665E011 7.50000E001 2.0?113E005 4.09175E010 1.50000E00? 1.21019F005 1.01OS3E011 n.OOOOOEOOI 2.11626E009 b.56351E910 1.55000E002 1.2b669E005 1.0%278E911 8.50000E001 2.74929E005 5.0“]?BE010 1.60000E00? 3.?7296E005 1.07121E011 q.00000F‘01 2.1900?E005 5.52260E010 1.65000E‘02 1.?9500F005 1.08570F011 YHFTA HAVENUNPER K K SQUARE THETA UAVFNUI'FR K K QQUARF P.00000F001 5.7savnrooa ' 3.31635roo9 9.soooosoo| 2.4asovroos 5.97836F010 7.900006001 7.171Aorooa 5.152226009 1.oooooEooz 2.94641rons 6.45399rono 1.00000F001 8.58336F004 7.367166oo9 1.050005ooz 2.611o45o05 6.922355olo 3.5noooro01 9.972456004 9.94499roo9 1.100006002 2.71699E005 7.37oans.;o 4.000606o01 1.114265oos 1.29654E010 1.15000660? 2.1969asoos 7.82310:o10 4.500006001 1.769115oos 1.61065E010 1.zbooocoo2 2.872066005 0.248615o10 s.nnonoro01 1.4oxsscoos 1.964345o10 1.250006002 9.94164r.os n.6sazasolo 5.soooorool 1.631326oos 2.34.945o10 1.3oooocooz 3.66563Eoos 9.03361Eoto 6.00000F001 1.6salvsoos 2.149546o10 1.asooo£oo2 3.063916o05 9.397sarovo 6.soonoroon 1.79137Eoos- 3.17sovsoio 1.60000E00? 3.116asr.os 9.711636610 7.000006001 1.9021nsoos 3.6lazesolo 1.4soooEoo2 3.16?H6EOOS 1.00037ro11 7.snooorool 2.olnavsoos 4.07snzsolo 1.soooo£ooz 3.203355oos 1.0261aro11 n.ooonocool 2.11171Eoos 4.544105o10 1.ssooo£ooz 3.2311aroos 1.643365o11 n.sooooEoon 2.7aoaosoos 5.01.915olo i.eooooconz 3.26597F605 1.06666ronl 9.000006001 2.145015oos 5.490o96o10 1.6soooEooz 3.2379nroos |.oannnr.11 .Table 6.7. RFFPACTIVE INDEX 8 1.3SSS THFTA P.00000EoOI P.500006001 1.00000Eo01 5.900006o01 4.000005o01 4.50000F001 6.00000Fo01 9.50000E001 6.000006001 6.600006o01 7.00000F601 7.900006001 9.000006001 8.50000F901 °.00000E001 unvrnuunEn w 5.7aooweooa 7.16571E004 3.96880Eona 9.0SSSfiE°Oh 1.13234560s 1.?6696F005 1.19019Foos 1.5?n7wroos 1.6s537Eoos 1.77R89E60S 1.RQ896E00S 2.0)5455006 2.1?8106005 2.?3670E009 2.14104Eo05 REFRACTIVF INDEX 8 1.3S2S THFTA P.00000F001 P.50000E001 1.00000F001 3.50000E001 4.00000E001 4.50000E001 S.00000F901 S.SOOOOFOOI 6.00000F001 6.50000E001 7.00000E‘01 7.50000E001 “.OOOOOEOOI R.S0000F001 °.00000F001 HAVFNUHRFR K 5.716306004 7.149886004 8.SAQRAE904 9.91351E904 1.12981E006 1.?64166005 1.19609E005 1.R2SIAEOOS 1.6S170E005 1.77697E005 1.a0416E~05 2.010996005 2.1?33QEOOS 2.?317SE005 2.31S86E00S LAMRDA = 9145. K SQUARE 3.30913E909 S.13678E.99 7.34244F.o9 9.911326009 1.28?1°E*10 1.60S196010 1.9S769Eo10 2.11700F010 P.740?4F010 3.16612F010 3.606066010 6.06202E010 4.578806610 5.00282E010 S.480h7E010 LAMBDA 8 S168. K SQUARE 3.29052F009 5.112076009 7.30998E009 9.867SOE‘09 1.27652E010 1.59810E010 1.94904E010 2.32667E010 2.72812E010 3.15033E010 3.59010E010 6.046068010 4.50877E010 4.98070E010 5.456?AE010 11.7 Program WAVENUM results. TEHRFRATURF I 36.5S THETA 9.50000E001 1.00000E00? 1.05000E602 1.10000E002 1.15000E002 1.20000E00? 1.25000E002 1.30000E00? 1.35000E00? 1.60000E002 1.65000E902 1.50000E‘0? 1.55000E002 1.60000E90? 1.65000E00? HAVFNUNIER K P.46093E005 2.S1517F605 2.6?6985005 2.71199E009 ?.7°??4E90S ?.86718F00S 2.9166SE605 1.000S4E60S 3.0S87?EOOS 3.11107E005 3.1STSOE60S 3.10797E00S 3.232?SFO0S 3.26043F60S 3.?8261EOOS TEM'EPATURE I 30.60 THEYA q.sooooco01 1.ooooo£o02 1.050006o02 1.100005oo2 1.150005e02 1.20000Eo02 1.2soooco02 1.aoooocooz 1.3sooncoo? 1:4ooooEooz 1.4soooEooz l.soooo£ooz l.ssooo£ooz 1.6ooooEooz i.esooocooa vavruuunrn K 2.4wssaroos 2.s30ssroos 2.670775o05 2.1os¢9€.os 2.76606Foos 2.96onaroos 2.930166o05 2.oqaoos.os 3.051956o05 3.104186005 3.150s1fioos 3.190846605 3.2?sIoEoos 3.2s3226o05 1.?7S16F005 K SQUARE S.9SRI3EO1D 6.61215E010 6.99892E010 7.3saooroln 7.7966?F010 8.27071E010 8.6?394F010 0.001ZSF010 9.1557SF010 9.67876F010 9.96981E010 1.02267E011 1.0647SF011 1.06304Fol] 1.07742F011 K SQUARE 9.93178F010 6.403716010 6.8684?F910 7.3?238E010 1.7621$Poln 3.18636F010 3.58581E010 8.96366E010 9.31439E010 9.63596F010 0.9?S73F610 1.018156011 1.06013F011 1.0S8366011 1.07266E011 0 (A0 4965 4965 4965 4965 5017 5017 5017 5017 Table 6.8. ) T(°C) 24.75 29.55 34.55 39.60 24.75 29.55 34.55 39.60 Table 6.9. T(°C) Weight(g) 23.90 4.1969 27.35 4.1932 30.95 4.1879 34.40 4.1834 36.95 4.1802 39.05 4.1771 118 Refractive index and wavenumbers. n 1.3615 1.3585 1.3562 1.3532 1.3612 1.3583 1.3560 1.3530 K(105cm’1) 2.437 2.431 2.427 2.422 2.411 2.406 2.402 2.396 Density cell volume calibration. (g/ml) .99735 .99644 .99538 .99426 .99337 .99260 Ave. St. dev. (m1) 4.2081 4.2082 4.2073 4.2076 4.2081 4.2082 4.2079 K2(1010cm'2) 5.937 5.911 5.891 5.865 5.812 5.788 5.768 5.742 .0003 m1 119 Density The density of Freon 113a was measured for the calculation of the adiabatic compressibility. The cell described in Chap- ter III was calibrated using distilled, de-ionized water that was boiled before use to remove carbon dioxide. The volume was calculated at several temperatures over the range of 20 to 40°C. These values are presented in Table 6.9. The density of water was taken from the International Critical Tables of Numerical Data, Volume III. As can be seen from Table 6.9, the volume did not change to any noticeable degree over this range and was assumed constant. Table 6.10 lists the weights ‘ and densities for Freon 113a. The densitydecreases mono- tonically and is plotted in Figure 6.2. The value at 21.00°C is in very good agreement with the values listed in Table 6.1. Depolarization Ratios The measured values for VH and VV at 22.80°C and the cal- culated depolarization ratios, 0V and Du, are listed in Table 6.11. The horizontal depolarization ratio, 9H= VH/HH’ was always found to be unity, within experimental error. Since HV was found to be in error due to leakage of VV through the polaroid, the Rayleigh reciprocity law was used to calculate pv, as given by v V" V— . (78) Since pH was found to be unity, the depolarization ratio for unpolarized light was calculated, using Krishnan's expression, Nfimm.fi Nmmm.H Aoem.fi omem.a emem.H emem.fi eoem.H efimm.fi ommm.H eemm.H Aae\mv ooov.o mmwv.© mHom.o omom.o omam.© wmam.o wwmm.o mwmm.o sawm.o Amveemfiez .mfiuumau we meeuemeee eee ”pause; .oH.e efieee mm.mm mn.nm mm.mm om.mm ow.mm mv.Nm mo.Nm mo.Hm wmmm.a womm.H mwmm.H Boom.H cmom.H whom.a momm.H monm.a oanm.a mmmm.~ Afle\mv moqm.c oomm.o momm.o mmom.o mmnm.o woom.o mwoo.o Hofic.o mofio.o Humo.o AwseemHez mm.wm mv.wm om.mm mn.om oo.vm oo.mm oo.m~ mm.mm oo.HN AUOVH .oHSHmHomEou mamgo> xufimdop mo uoam .N. o 853.... 121 8 .E. 8W1 . 0.341 Odn SN 0.0N / 1 . no.- 0/ (D.- o /o/ ./// // 0099. ./. lolo «gay ON 008.. ./ 5.. 00.— 122 Table 6.11. Depolarization ratios of CFSCC13 at 22.80°C. A(A°) VV VH oV* on 5145 '443:5 26:2 .059:.005 .lll:.009 458:5 26:2 .057:.003 .108:.006 227:3 14:2 .062:.009 .117:.017 448:4 25:2 .056:.005 .106:.009 427:4 24:2 .056:.005 .106:.009 428:3 24:2 .056:.005 .106:.009 248:3 15:1 .06l:.004 .115:.008 Ave..058:.005 .110:.009 5017 305:4 19:2 .062:.007 .117:.013 295:7 19:2 .064:.007 .120:.Ol3 Ave..063:.007 .ll9:.013 4965 167:3 11:1 .066:.006 .124:.011- 353:4 23:2 .06S:.006 .122:.011 352:5 24:2 .068:.006 .127:.011 Ave..066:.006 .124:.Oll 4880 443:4 30:2 .068:.005 .127:.009 436:4 30:2 .069:.005 .129:.009 Ave..068:.005 .127:.009 * Slope = (-4.02:.40) x 10-5 A°‘1 Intercept= .265 + .020 123 which is given by20 2 01,1 = - . (79) (1 + 9v 1) The values listed in the table were measured at the four wavelengths used in the Brillouin experiments, and the average values of the depolarization ratios are calculated for each wavelength. Figure 6.3 is a plot of pv versus 10 and shows an inverse dependence of pv on 10. This trend can- not be verified theoretically; however, it agrees with most published data for other simple fluids. Kratohvil £3 31.35 have tabulated long lists of data for benzene, toluene, car- bon tetrachloride and carbon disulfide, and in every case, except carbon tetrachloride which shows no changes at all, pv decreases with increasing wavelength. Finnigan and Jacobs20 have measured pv at 4880 A° for the four compounds mentioned above, and their results are listed in Table 6.12, along with results at other wavelengths by other authors. All values were taken near room temperature. It should be noted from these results that as the molecular shape becomes less spherical, pv becomes larger. The value of pv obtained here at 4880 A° was 0.068:0.005. Therefore, it can be seen that Freon 113a is quite spherical, but less isotropic than carbon tetrachloride. This anisotropy is seen to be small, confirming the conclusion that the CF group is 3 compact and only slightly larger than a chlorine atom, and that Freon 113a is quite similar to carbon tetrachloride in shape and scattering properties. 124 .auwnoao>m3 mo aofluonsm m we ofiumu coflpwufiamaomom .m.e ohnmfim .5 .« 0:0 :00 nwm¢ 009v . - . 4. . 430.0 In .0000 / 11 .v 1.. .rOW0.0 Ir. 0 I... Q / 000.0 2.0.0 L [nil {Eli ’Il... . 125 Hoe.ew Nem.o mwe.e mee.e4eom.e moe.e Afie.e0 emm.e mem.o Ade.owoem.o mum.e Hoo.e. HmN.o Am~.o woe.efifiw~.o oum.e mooe.owmmfio.o eNe.e moo.e.e.e.e eme.e ammo< wmme.>e mmn0< Heem.>e eNflo< ewwe.>e mmfio< wmmev>e .mpflsvfia oficwmuo 050m mo mofiumh :ofiummflhmaomoo .NH.o vague Nmu oaoSHoH oceanom eaou pasvaq 126 Brillouin Shifts The average Brillouin shifts measured on the mini- computer and those obtained from KINET are listed in Table 6.13. The mini-computer results are shown for one complete scan, usually three to four full orders. The KINET results are from one to three of the best orders from the data set. The errors shown are determined by the range of the data for each spectrum. These errors are much larger than the actual errors involved in determining the frequency shifts. On the mini-computer, the shift for each order can be found to with- in one data point, which usually corresponds to about 0.01 GHz. Using KINET, the Brillouin shift can be found to the nearest .08% or about 0.002 GHz. KINET is more accurate because it uses the whole data profile to locate the peak, whereas the mini-computer method depends only on the points at half in- tensity and does not account for the distortion due to the relaxation line. However, the range of the data is much larger than these errors because the interferometer piezoelec- tric scan is nonlinear. As a result, the inherent high accuracy of the computer analysis cannot be fully realized. In addition to the values in Table 6.13, additional spectra were taken and analyzed before the half-width method of analysis was used. The values for these spectra are listed in Table 6.14 under the heading "Old." The method of analy- 'sis for these spectra was the visual sighting of the center of the top of the Brillouin peaks. At a later time, several of these spectra were reanalyzed by the "new” half-width 127 Table 6.13. Brillouin shifts and sound velocities. pdp KINET T(°C) A(A°) e wB(GHz) VS(m/sec) mB(GHz) VS(m/sec) 24.75 5145 60 1.84:.03 696:11 70 2.12: 02 699: 7 ' 2.13:.03 702:10 90 2.64:.03 706: 8 2.62:.04 700:11 105 2.98:.03 710: 7 2.99:.02 713: 5 4880 60 1.93:.02 692: 7 70 2.23:.03 697: 9 90 2.78:.03 704: 8 2.77:.01 702: 3 105 3.15:.02 711: 5 5015 90 2.73:.03 711:11 2.71:.03 706: 8 4965 90 2.75:.03 709: 8 2.75:.01 709: 3 29.55 5145 60 1.79:.02 678: 8 70 2.07:.02 684: 7 90 2.58:.04 691:11 2.54:.09 681:24 105 2.91:.03 695: 7 120 3.23:.02 707: 4 3.23:.03 707: 7 4880 60 1.88:.03 675:11 70 2.18:.02 682: 6 90 2.70:.02 686: 5 2.81:.17 714:43 105 3.08:.03 697: 7 ' 120 3.38:.05 701:10 5017 90 2.66:.03 695: 8 4965 90 2.64:.05 682:13 2.64:.01 683: 3 34.55 5145 60 1.75:.03 664111 128 Table 6.13. (Continued) pdp KINET T(°C) A(A°) wB(GHz) VS(m/sec) wB(GHz) Vs(m/sec) 34.55 5145 90 2.51:.01 674: 3 2.50:.02 671: 5 105 2.82:.03 675: 7 4880 60 1.84:.03 662:11 90 2.67:.01 679: 3 2.69:.02 684: 5 105 2.98:.03 676: 7 2.98:.02 676: S 5017 90 2.52:.02 659: 5 4965 90 2.58:.02 668: 5 39.60 5145 50 1.42:.02 639: 9 1.38* 621* 60 1.69:.03 643:11 70 1.96:.02 650: 7 1.94* 643* 80 2.21:.01 654: 3 90 2.42:.01 651: 3 ‘ 2.41:.04 648:11 100 2.66:.05 660:12 110 2.84:.01 659: 2 2.84:.05 659:12 120 3.00:.02 659: 4 4880 50 1.47:.03 627:13 60 1.79:.02 645: 7 70 2.08:.02 654: 6 80 2.33:.03 653: 8 90 2.56:.01 653: 3 2.55:.01 ‘650: 3 100 2.81:.01 661: 2 110 3.00:.03 660: 7 120(vv) 3.18:.02 662: 4 120(HH) 3.18:.03 662: 6 5017 90 2.49:.03 653: 8 2.49:.03 653: 8 4965 90 2.51:.01 651: 3 2.52:.03 654: S * Results from partial fit 129 m ammo QNHHwo Nanmmo Hauwvo Ho.Hvo.N mo.H> Hmsz maumwc w “moo HHHHmo e “Noe v Home N Hmmo oaflooo m «Hmo mo.wvo.~ mo.woo.~ co.uwm.N No.HwH.m No.Hoo.m Ho.uvw.m mo.woo.m Ho.HN¢.N >> Bmz (\NQ'CDOCVO mo.w-.m No.HNm.m mo.HvN.m mo.ww~.m vo.HHN.m No.HOH.m Ho.uvo.m mo.umm.~ mm mHHmmo m ammo mawemo 0 None N “N50 HNHmoo “who “moo “moo Hose Heme “coo “coo “moo “Noe wml-OLDBBNOO fluem\a.m> ”Nau.me fleem\evm> Aumuvme fleem\e.m> humuUmemeem\e.m> GAO .moscficzuou mwmzamcm mess» 0:» mo cemwummsoo > mo.won.m No.wmc.m mo.Hmm.N mo.HN~.m Ho.nmm.m oa.wm~.m mo.nn~.m mo.HmH.m Ho.HNH.m mo.nmo.m mo.wmm.~ No.85w.m No.wnn.m No.Hno.N mo.uo¢.~ flNmuvme > .eH.o manmh om om om oNH oea mma mma omH mNH QNH mad OHH moa OOH om owwe naom meam omwe mvam mm.m~ oo.mm flo<.. floove 130 technique, and some of these were analyzed with KINET. As can be seen from the table, the visual technique is less accurate because noise can distort the top of the peak and personal bias enters into the analysis. The half-width method and KINET are seen to agree nicely. From depolarization ratio theory, the “H component varies in value from HV to VV because of the cos2 0 dependence. It was speculated that at 90°, HH would consist of a flat back- ground when analyzed, and at angles other than 90°, it would consist of the VV triplet with an intensity that would be angle dependent, with HH giving exactly the same spectrum as VV at zero and 180 degrees. In order to prove this, several spectra were taken of both the HH and VV components. At ninety degrees, the background was indeed flat, and at other angles, a triplet was present, its intensity varying with angle. The reduced intensity of the triplet decreased the signal-to-noise ratio to the point that the spectra became unusable at angles between 75 and 105 degrees. The HH spec- tra were analyzed with the visual method, and the results, shown in Table 6.14, agree within experimental error for the “H and VV components. Since the scattering equation gives the frequency shift as V5 0 2 V5 71 0 01B = :2000 ("5)n sin (E) = :(T) sin ('2'), (80) a plot of “B versus sin (6/2) should give a straight line if V5 is constant. Figure 6.4 shows the Brillouin shifts at 131 3,40: 3.20? I 300. I/////f//// 2. - I//// 2.60r 638 I (GHZ) ‘88 : 2.40: 5145 2.20- I I 2.00» I Leo~ I 1.60” 1.401;: . . 1 1 . . . - .420 .480 .540 .600 .660 .720 .780 .840 .900 SIN (9/2) Figure 6.4. Frequency shift as a function of sin 6/2 at 39.55°C. 132 39.60°C for 5145 and 4880 A° plotted in this manner. Within experimental limits, the plots are linear, implying that V5 is constant. However, when V5 is calculated from VS = wB = ZflwB , (81) (Zn/A°) sin (0/2) K where K is the wavenumber of the scattering phonon as listed in Tables 6.4 through 6.8, V5 is seen to change by as much as thirty-five m/sec (about 5 percent) throughout one set of angles. By plotting Vs against “B’ as shown in Figures 6.5 and 6.6, it is clear that the velocity of sound is frequency dependent. (The error bars on these and following plots are sometimes quite large and overlap. Therefore, they are omitted when they lead to confusion.) Plotting Vs versus ”B in this manner should give a sine curve as shown by the scattering equation above. Since the frequency region shown here is limited, the plots appear almost linear. The normal procedure for determining the velocity dis- persion is to fit the VS versus ”B data with the following dispersion relation66 5,0 m P CP 2 (l-rBt)+[Cv/(Cv-CI)](TBzrz-PBr+w2T ) (1-rB:)2 + 02:2 (82) Since little research has been done on Freon 113a, the values of CV,CP and VS 0 are not available in the literature. , . 133 .xocoscoum mamuo> endow mo xpwooao> .m.o ohsmfim m ANT—ow 3 ON“ 00...... ond 00d 0 SN CNN 00d 00.. 00.. 0S. ‘1 d - 1 4 d1 W q u 4 82... con... to». I at... on. em 00 I 1 . . W 90......» Ohm / . 3 k1 3 \\.\4fl¢\~\ TS. L 0. h .zucoscoum msmwo> wasom mo xufiooHo> .o.o ohsmflm m ANIO. 3 on.» ond on.“ 03 otu on.“ 8... 8.. o... o... 1— — _ S A _ _ _ — _ no... .0 on... I I to. .d ...... 0 68 0.3.3 \0\ 6...... I1. \\\I\ .03 \\\\m\ 1P.- o I I\\\.OI 0 .08 M \\\\\\\\\\\ fin 1 [Ohm xl I\\ m \ 0 \\ -083 .9 I \ 3 I \0 0 0 and... 1o$( \ \I\\. 4 e \I\\\\\ 1 2. 0 -oz. ONK 135 Therefore, this procedure cannot be followed here, and the dispersion of the sound velocity is unobtainable, except to say that it is about three to six percent over the frequency range studied. However, from the temperature data, one can observe how the dispersion changes with temperature. On close inspec- tion of Figures 6.5 and 6.6, it is observed that the velocity dispersion tends to get smaller as the temperature increases. This effect is more obvious when the observed frequency shift at a given angle and wavelength is plotted against the tem- perature, as is done in Figure 6.7. As can be seen, the frequency range is larger at 24.75°C than at 39.60°C by about 100 MHz. This decrease in the frequency range is also shown by the slopes of these lines, which decrease monotonically from -.018 GHz/deg to -.010 GHz/deg. Thus, the available frequency range narrows as it shifts to lower frequencies when the temperature rises—-i.e., the high frequency end falls off faster than the low frequency end. This implies that not only does the velocity of sound fall as the temperature rises, as seen in Figures 6.5 and 6.6, but also the velocity dis- persion is smaller at higher temperatures. Using the Slope 0f ”B versus T, one can calculate approxi- mately the value of st/dT by differentiating the scattering equation. The value of st/dT was found to be between 4.2 and 3.7 m/sec/deg. A second and more direct method of find- ing dVS/dT is to plot the velocity of sound against the tem- perature, evaluating dVS/dT by taking the slope. This 2.80 2.60 2.40 (GHZ) 2.20 2.00 .24 3.001- r- h 136 I I\ ‘I\1%I\~ ‘1 I\ \I 00.90- I " .f_~g_-‘-~‘§§.T~AE“~“‘~—-._____~‘~‘~§~‘§ 1%, e . ‘ \I~ II\\I 4880,? o' I. {\IW 4.807: \IWI\I~ I\I~ 1.6T 1 1 1 2:75 29.55 54.55 39.00 T (°C) Figure 6.7. Frequency shift versus temperature. 700 (NI/SEC) V5 2 :9 . “of 137 710 690 Q G O I Issuauov 0=4aao,|oo' - .\ A=5u45 , 90° - 5501- =4aeo,90° Q .‘5145, 70' A=4550.7o- ’\ $5145.50 . 54o 884850.60- 24.75 29.55 34.55 55.50 T(‘CJ Figure 6.8. Velocity of sound versus temperature. 138 procedure was followed in Figure 6.8. The large scatter in the data points made it difficult to find a difference in the slopes; however, an average value of 3.3 m/sec/deg was found for dVS/dT. KINET Results Since the results from the mini-computer for the Bril- louin half-width and peak areas are obviously in error for the reasons given in the data analysis section, the only use- ful values for these quantities were those obtained from KINET. Even with the help of KINET, some of the values ob- tained are of questionable accuracy because noise spikes, ramp non-linearity, and slight misalignment can result in very poor fits from KINET; however, in most cases satisfac- tory results were obtained. The effects of ramp non-linearity and peak asymmetry can be seen in the results listed in Table 6.15. Included in this table are the six parameters used in KINET, their standard deviations, the percent standard deviation and the multiple correlation coefficient, which is a measure of the dependence of a parameter on the other 'parameters in the equation. The Brillouin shift is seen to have a low multiple correlation coefficient, which means the frequency shift is only slightly correlated to the other pa- rameters. The Rayleigh half-width has a moderate dependence on the other parameters, and the remaining four parameters show a high correlation with each other. KINET also gives the pair correlation coefficients, and a typical set is given 139 mow. moo. Hum. woo. ohm. mum. How. ooo. Vow. mmo. mmo. Now. .u.2 Nomv. homo. memo. ooav. oomm. nNNm. omvm. momm. oNoN. unnm. NNBN. Hoom. m\m oomo. ooHo. ono. noHo. memo. oomo. momo. onmo. sumo. mvNo. ommo. mnNo. m\z NNNNOOOOOIOHO I I I I I I I I I I HHHHHHHHOHHH omoo. mvoo. mmoo. mmoo. umoo. mmoo. mmoo. omoo. oeoo. mmoo. Heoo. ovoo. .m.mm.. mov.m v~o.~ mnm.~ vm¢.N omo.~ Noo.N mun.N vnh.N oHo.N omo.~ ohm.~ mmo.m monm. nmwm. Nvmm. ommm. ofinm. moom. oHom. whom. Homm. onmm. oon. omNm. Hmo. mmo. omo. «Ho. ovo. mmo. Nmo. omm. ooo. Now. ooo. moo. .u.2 woo.~ oem.~ ooo.~ ovo.m omv.N omv.m oNo.N omm.~ omn.m oNo.m mHo.N man.~ OMLOVQ'VOQ .oH 000300005:ow o\° mmoo. mama. oomo. oomH. mmoo. mmoo. mmoo. mmoo. oooo. mono. vaoo. mnmo. Aezoome flemoome flezooze Aemuom. ooN. mmm. oNN. ouo. ooa. mod. mom. ooa. mmN. omN. ohm. mam. .a.m2.. oov ooe moo moo oov omw oov oov mow owe owe oov .mpmu ooh. Hun. moo. Hmn. ooo. Hoo. mmo. Hoo. How. moo. mum. ohm. .U.Z mm AN mm 4N mN AN mm AN me go mm AN Hoouo .meflsmee emsz we mefieeexm .m..e 6.5.9 oo oo oo oo oo o BBVBMVMVOBNU} I I I I I HHHHHHHHHHHVH w oo.om oo.om mm.mN mn.vm me.e~ noooe vaoo. .NH oHoo. mHoo. «Hoo. HHoo. HHoo. NHoo. NHoo. HHoo. HHoo. HHoo. oHoo. ammo-monomer! v-Ir-l .a.mme .NH ooov .HH HNMQ’LHOBWQO I-l mvam owwv owwv mefim nooo onmocmpm u .a.m eeeeeemmeou enmeee emee eeeofleeem u m\m ucoflofimmooo unmfion xmom ammucsoz u m z wwwnm xoaosvoum cwsoaafihm n 3 xufimcoucfi Esefixms mow Mam: pm xmom afisoaafium mo :uofi3-mHm: n ma xuwmcounfi ESEwaE mum mam: um xmom :flwpcsoz mo nuoH3-mHmn u 2p zufimcoucw Esaflxme mum Mam: um Mmoa :mfioaxmm mo :uow3-men u mg m 1 mmm. o.o eNoo. mum. H.m oNoo. omm. o.o mmoo. Hom. e.mH mmoo. mun. o.o omoo. mem. o.o mHoo. eon. m.o Hmoo. mmm. o.oH mHoo. omm. m.o mmoo. mom. o.e eHoo. mun. m.o uaoo. mmm. H.e mHoo. men. m.o mHoo. mem. m.m eHoo. mmn. m.o mHoo. emm. n.m eHoo. Hen. n.o mHoo. eNm. m.e NHoo. omn. o.o mHoo. mmm. n.e NHoo. mmn. n.o mHoo. mmm. ~.m NHoo. oeh. n.o mHoo. mum. H.e HHoo. .u.2 m .o.m m\m .u.z m .o.m m\2 fleeseeueeuo .m..e emcee HoH. HNN. ona. omm. oeH. onH. mma. emH. omH. moH. eoH. ond. .U.2 no. mo. no. no. oo. no. no. no. mo. mo. oo. mo. m moo. .NH moo. .HH moo. .oH moo. .o moo. .m Noo. .5 moo. .o moo. .m Noo. .e Noo. .m Noo. .N Noo. .H .Q.mm3 141 in Table 6.16. The pair correlation coefficients give an in- dication of how strong a correlation two parameters have with each other, when all other parameters are held constant. If the sign of the coefficient is positive, the parameters are inversely correlated, i.e., as one grows larger, the other grows smaller; and, if the parameter is negative, they are directly correlated, i.e., they grow smaller or larger together. Unity implies that the parameters cannot be independent in the equation used in the fit. As can be seen from the table, ”B is only slightly correlated with the other five parameters, implying that its value is not strongly dependent on the other parameters of the equation. On the other hand PM is highly correlated to the values of PB, M/R and B/R, a result that is not unexpected since the Mountain line gives only a small contribution to the overall spectrum and a small change in the area will significantly alter the value of PM. Thus, a poor overall fit will give poor values for PM, as was fre- quently observed. The Brillouin heighth coefficient, B/R, is highly dependent on the Mountain line parameters and, of course, TB. Other correlations can easily be seen in this table. Returning to Table 6.15, it is seen that the standard deviation for ”B is consistently .002 GHz, which is about one- fifth of the deviation found with the mini-computer. This accuracy, however, is meaningless when “B is compared from one order to the next, or even from one half of an order to the 142 ooo.H- oom.o- mmo.o ono.o- on.o- mmo.o- m\m oom.o- mmo.o ono.o- omm.o- ooo.H- emo.o oeo.o- oan.o- emo.o ooo.H- mHo.o- mmo.o- oeo.o- mHo.o- ooo.H- ooo.o- oau.o- mmo.o- omo.o- ooo.H- ome.o- mHo.o mmo.o- noo.o- m\z m3 mo 2% mucofiufiwwmou :ofiumaoupoo ammo .oH.o ofiome mmo.o- ome.o- mHo.o moo.o- noo.o- ooo.H- 5 m3 ~12 m3 m. 2.. N... 143 other half of the same order. There are two reasons for the size of the variations in ”B' The first is that very slight misalignment of the interferometer leads to an asymmetric Rayleigh line, i.e., the instrumental response function. On the recorded spectrum this asymmetry is noticeable only under close examination of the peak, but it is very evident on the mini-computer oscillosc0pe and in the KINET results. The asymmetry always manifests itself as a slow-rising, fast- falling Rayleigh line. As a result, the apparent center of the Rayleigh peak is shifted to the right by as much as .03 GHz in the least acceptable spectra used here. The first con- sideration made was the possibility that the high-frequency filter RC time constant was distorting the peak. Since a resistance of 3300 ohms and a capacitor of .l microfarad was used, the time constant for the signal to change to 63.2 percent of the difference of a sudden jump in the signal is T = 3300 X 10‘7 or 3.3 XlO'4 seconds. Since it is usually assumed that the new value will be attained in 5 T, i.e., that the capacitor will be charged completely to the new value in that time, the response time of the filter is 1.65 milli- seconds. The points obtained from the eight-point-average routine were taken at 14 millisecond intervals, far longer than the response time. In addition, eight of these points are averaged, and, thus, 112 milliseconds are required for an averaged point to be recorded, indicating that the RC time constant has a negligible effect on the value of this point. Therefore, the asymmetry is probably due to misalignments 144 in the optics or to distortion from the picoammeter due to a long RC time constant from the variable damping knob. The RC time constant of the total electronics detection system can be checked by building an electrically-switched (for faster switching times than mechanical switches), shielded (to avoid external fields from distorting the current signal), picoampere source for the generation of a step-signal or square wave. The signal can be put on a storage oscillosc0pe, or the mini-computer can be used, without the eight-point: average routine, to sample the signal every millisecond (if the data points always are taken at times longer than one millisecond). The rise and fall time can be measured and com- pared to the sampling time interval. If the rise or fall time exceeds 25 percent of the sampling interval, steps have to be taken to decrease the RC time constant or to increase the sampling interval. The second reason for large variations in ”B lies in the nonlinearity of the piezoelectric scan. To determine if the non-linearity originated in the voltage ramp generator or in the piezoelectric drive, the voltage output of the ramp gen- erator was measured several times at various ramp speeds. Within the experimental error of these measurements, the ramp was found to be linear in the region of -300 to -1200 volts. It is, therefore, believed that the nonlinearity arises from the piezoelectric drive, and is caused by hysteresis effects in the crystals. In addition to nonlinearity, there is another source of error from the piezoelectric drive. Occasionally, 145 thile running a spectrum, the output signal suddenly seems to back up and retrace a portion of the spectrum. The jump is always in the range of .30 to .35 GHz and is evident only when it occurs on the side of a peak as a large noise spike or a shoulder. When this was found to happen, interorder spacings and frequency shifts (when the jump occurred between a Brillouin and Rayleigh peak) would be increased by about .30 GHz. These spectra were discarded, and new spectra were taken. Since this jump always resulted in a portion of the spectrum being retraced, and the piezoelectric drive decreases the mirror spacing of the interferometer, this jump seems to indicate a sudden increase of the mirror spacing that could possibly be caused by a slippage of one of the piezoelectric crystals. The change in the mirror spacing required for a scan of one complete order is 1/2 or 2573 A° for the 5145 A° line. One order is equal to one free spectral range, which is given by Af = C fsr 2nd (83) where c is the speed of light in vacuum; 0 is the refractive index of the medium between the mirrors, viz., air; and d is the mirror separation. A typical mirror separation is 1.5 cm and gives a free spectral range of about 10 GHz. Therefore, a frequency change of .30 to .35 GHz corresponds to about an 80 to 90 A° slippage of the piezoelectric crystal. Since a complete scan of 1600 volts gives about five full interfero- meter orders and causes a change in the mirror separation of 146 about 12,700 A°, a slippage of 80 to 90 A° (less than one per- cent of the total expansion) seems a very real possibility. From Table 6.15, the effects of asymmetry and nonline- arity on the parameters can be seen. The first three data sets show no asymmetry that can visually be detected, even on the mini-computer oscillosc0pe. The values of FR are about the same and reflect the best symmetry obtainable with this interferometer. The last two data sets are examples of the worst asymmetry accepted. Visually, the centers of these peaks, as found by the half-width method, appear to be only about .03 GHz from the center of the top of the peak. This value agrees well with the difference of the KINET values for PR. The total width of the instrumental response func- tion, 2 PR, is seen to be much larger for these two sets and reflects the poorer finesse of these spectra. The differences in the values of ”B on the first three data sets show the effects of nonlinear scanning when asym- metry is at a minimum. Typical differences in mB are shown in the first data set. The second and third are exceptional cases, showing the best and worst linearity that is obtained. The difference in the ”B values in the fourth set is mainly ‘due to the asymmetry; however, the fifth set is an example of both large asymmetry and large nonlinearity. The values of PB and PM are affected more by the asym- metry of the instrumental response function than by the non- linearity of the scan, as can be seen in a comparison of the 147 last two sets with the first three. When the symmetry is good, the values of r agree within the standard deviation, B even when the nonlinearity is large. However, when the symmetry is poor, the values of PB from one order can diverge significantly, the left Brillouin always being broader than the right. As mentioned earlier, the asymmetry of the response function is always the same--slow-rising and fast- falling. Since the actual Brillouin peak has an asymmetry due to distortion by the Mountain line, the asymmetrical response function interacts with the two Brillouin peaks in different manners. The left Brillouin asymmetry is opposite that of the response function, so that convolution with the response function produces a more symmetrical peak, which KINET fits with a broader Lorentzian. Part of the intensity contribution from the Mountain line is included with the Brillouin fit, and the Mountain line appears as a low-inten- sity, broad peak, as shown by PM and M/R in the table. On the other hand, the right Brillouin asymmetry is similar to that of the response function, and the resulting peak has an accentuated asymmetry. This results in a KINET fit that ascribes a narrower Lorentzian to the sharper peak fall-off, leaving some of the Brillouin intensity to be accounted for by the Mountain line. Therefore, r is smaller and M/R is M larger, producing a taller, narrower Lorentzian for the right half of the spectrum than for the left half. This problem can be somewhat alleviated by folding the spectrum and aver- aging the two halves prior to fitting. Of course, the best 148 solution is to achieve more stable alignments to obtain better symmetry, which is a difficult task on the present interferometer. As discussed above, the Mountain line parameters are severely influenced by an asymmetrical response function interacting with the Brillouin line. However, the task of finding the Mountain line is difficult in any case, since the Mountain line has a small overall intensity contribution to the total spectrum. Thus, KINET finds the Brillouin and Rayleigh line parameters and ascribes any intensity left over to the Mountain line. Even in the best of circumstances, as in the second data set, large fluctuations can occur from one half of an order to the other, and from one order to the next. Thus, care must be taken when interpreting these results. The results listed in Table 6.15 are from fits using 460 to 500 data points. As mentioned earlier, a preliminary fit, using every fifth point from each half order, was performed to permit better initial guesses of the six parameters used in the equation. As expected from statistics, the standard deviations from these fits were larger, by about the square root of five, than those listed in Table 6.15. In addition, these preliminary fits allowed a visual inspection of the quality of the agreement between the experimental and cal- culated values. This visual inspection cannot be made on the larger fits because the KINET plotting routine cannot be used for more than 100 points. Examples of the best and 149 worst fits used are shown in Figures 6.9 and 6.10, respec- tively. As can be seen, even the worst fits appear to be good, the biggest deviations between the experimental and calculated points appearing on the Rayleigh peak. The average values of the six parameters and their errors are given in Table 6.17 for all the data sets used with KINET. As before, the errors are calculated from the range of the values and not the standard deviations. The two data sets at 50 and 70 degree scattering angles for 39.60° have no errors listed because only one half of an order from each could be fit, and the values of the parameters are questionable. Figure 6.11 shows that the fit for the 50 degree spectrum is poor, especially on the Brillouin peak, giving a narrower Brillouin which appears to have a greater frequency shift than the experimental data indicates. The wB values obtained from KINET have previously been shown to agree well with the values obtained with the mini- computer, and the information available from the trends in “B have been discussed in detail. The values of the height co- efficients, M/R and B/R, contain little information in them- selves, except to help characterize the changes in the appear- ance of the spectrum. The Mountain line coefficient shows a slight tendency to increase with angle, but little can be inferred from this at present. However, the Brillouin height coefficient shows a strong angle dependence. Since Brillouin's original prediction states that the integrated intensity of the Brillouin peaks is constant with angle, the decrease in 150 .Esouuoom pew o>psu ooom xpo> m we “can Hmzoz .o.o ossmflm »mllllmcOI1U!!!IWIIIIUIIIIU!IIIUIIIIVIIIIUIlllrlIIIWIIIImIIIImlIUImIIIIWIIIIUOIIIUOIIDUIIIlmllllmtlli— Ulflfl "nunuflflflflflflnnnHHHHHHUHHHHNVxx u m — CCCHH I C "unnuunuuuuufluunucuuunuu — — ”NIX nu flu "I ~ _ CH CH" H a b 5x ux x ~ ..u C I C m. o I u ~ — n u n _ u I ~ _ c x _ m. x c o m — 5 xu _ a u C x H _ In ~ _ — mu m. — ~ _ a _ _ ~ ~ m. w _ n h _ _ ~ A c ~ .... m — — o _ h — _ a m. m u _ — x u _ a h — U m. _ _ _ ~ _ ~ ~ _ U m _ _ — ~ _ _ u m m. m. _ n _ ~ _ u _ u— ymilIlUllIlmllllmilllUIltlrlIllvllllmllIIUllllmllllmllllmlltlmlllIUIOIlmullIlmllalrollcwllllwtllimllil— ecoEoosmm ooumflsofimo one Hmpcoeficomxm .n osfim> ooumfisuamo no osam> prcmeflpomxm “x 151 .ESHHuomm “Hm o>H:o HHmH m we uoHH HHZHH .e..e eczema HmnoitmtuoImnluomcnlomclutmuooumloctmcannmuutcmococmuccumulunmutlcmuuoamulnnmanormonucmuoocmllIlmcocIH mun" nun unuuuuuxxx 00000000 x m H ooounxxx OOOnOHOHHnnunxxxxxxxxu"unenunuxxx H H OOOux Connuxxx x x OOOXX H H oux Ouxx C H H oux oux o H m o nnx x m H u u o H H u u H H n u H H o n o H m o u x m H xxo xo H H xu xo H H , o H H H m m H H H H H H H H m x m H H H H H H H H m m H H H H H H H H m w H H H H H H H H m m H H H H H H H H m m H H H H H H H H m m H H H H H XH H oH HmooaomonucmucnumuutnmttulmiloomnaatmntttmtootmtlutmncIlmtuocmIncumltuImoouamunu-mnluumtnnumunllmUIulH peoEoonm eonHsono ecm HmpcoeHHomxm . osHm> eonHsonu ” osHm> HmpcoeHHmmxm H >< CD 152 Hoouo mHm: oco mo muHH oHOHHEoocH « Hooo.HeHme. mmoo.HmmNe. emoo.Hmoee. Hooo.mmmmm. mmoo.HHNHo. Noeo.Hmnme. Hemm. moms. Hooo.Honm. onoo.momHm. oemo.Hmoee. oNoo.HoeNm. mmoo.wmoom. ooNo.HmeHN. Nomo.nomom. eHoo.Hnoem. Hmoo.HNooN. momo.HomNm. momo.HonN. omoo.HmeHN. Hmmo.HHooe. m\m ooHo.Hmm~o. onoo.neomo. omoo.uoe~o. oeoo.HHmHo. moNo.Hmo~o. HoHo.HNHNo. moHo. eHHo. omoo.emoeo. Nooo.moomo. mooo.HHHmo. HHoo.Hono. mmoo.nmemo. Hmoo.HmmNo. oHoo.HeoHo. nooo.HNHmo. mmoo.eommo. mmoo.uoomo. mmoo.wmnmo. mmoo.uommo. emoo.Hemmo. m\2 ooo.Hmmm.~ mHo.HmHm.N omo.uume.m mmo.HmNN.m meo.womm.~ oeo.Hmoe.N eeo.H mmm.H mHo.Honm.N emo.umoo.m mmo.Hnom.N NNH.HeHm.N ooo.e~eo.~ ooH.HmH~.m moo.Hon.N Hoo.HeHH.N eoo.HNmH.N Heo.HNHH.N emo.Homo.N Hmo.HoNo.N nmo.HoNH.N Hezuome enoo.mmmnm. emoo.Hmmom. mmoo.«mmmm. eeHo.HHone. mHHo.HHoHe. omHo.HHeem. HNoN. HooN. mmoo.HHooe. Hmoo.womme. HoHo.HmHmm. mmoo.HHoom. Hooo.uneom. ano.HHoee. omoo.HNeHm. mmoo.enemm. emoo.nooom. mmHo.Hmoom. mmoo.HnHoe. oHoo.HHmNm. meHo.HoHeN. HNzuome .muHSmoH HmzHH ommho>< HNH.eme~.N HH~.eeem.N oee.emmm.~ ewH.HmAH.m oee.eHHm.N emN.HNee.~ me.H emm.H mmo.eeme.m omm.eefi~.m Nme.eama.m HHe.Hmme.~ emo.eewe.~ eao.eoem.m eeH.emem.N eme.emwm.m meH.eNeA.N wme.eeNN.m mNN.emem.m wmfi.eamm.m NNm.HOAN.m Hemooza .AH.e eaeee mflmo.emo~H. wmm¢.eHNHH. em~o.HQGOH. weflo.emwafi. moeo.eflemo. Hemo.HHwHH. Newo. ease. emHo.eemHH. ammo.HHNeH. mHNo.HoeHH. Hooo.ewwwo. emoo.ee~mo. aeHo.eemwe. mmoo.emmee. wooo.ememo. omoo.eNmao. owfio.4mfiwo. moao.emowo. eaoo.ewmeo. omflo.eemeo. HNmoom. oo oo oo mmH oHH oo one omu moH oo oo oo oo oNH .oo oo oo oo moH oo on o omme mooe HHom meHm omme meHm omme mooe meHm omme mooe HHom meHm Hopsu been a mo HoHQ Hmsz .HH.o oHSmHm .m----m----m----m----m----m----m----m----m----m----m----m----m----m----m----m----m----m----m----m----. m" IflflflflflfluflflflflHflflflflflflflflflflflflflfl " u H OOOCOO KXXXXXWHHWHHNX ll 11 II II >< >< mHHHH-«m >< O O O>< > ooumHsono osHm> Hmucoawpomxm >—nn—nmu—-H—4Hmu—tu—It—tu—lmD—d—nI—tp—mn—qt—oHHmHmu—dn—umH—n—n—tmn—nu—du—or—nmt—th—uu—I y >< O .—pug—.—p—tmu—tu—QHHLnH5—1t—nHmHHHt—dLnHHH—flmH—‘HHmHHHHmHH—‘HmH—‘HH m I a I m l I I mmmmmmmmmmmmmmmmmm 154 B/R with angle must be accompanied by an increase in PB, which is what actually happens. Therefore, as 0 decreases, the Brillouin peaks are seen to increase in height, while becoming narrower. The values for PM and PB in Table 6.17 are the experi- mental values and must be corrected for the instrumental re- sponse function. Since it was previously found that instru- mental response function approximates a Lorentzian,67 the Rayleigh peaks were to fit to Lorentzians. The observed spec- trum is thus a convolution of the real spectrum and the in— 38 Since the Brillouin and strumental response function. Mountain lines are also Lorentzian, we can utilize the well- known mathematical result that convolution of two Lorentzians is again a Lorentzian whose width is the sum of the widths of the original two Lorentzians, i.e., Pobs ='I‘real + rRF (84) Thus, the corrected half-widths are listed in Table 6.18. Also listed here are the corrected and uncorrected values of the reduced coefficients for the spectral response function. In order to demonstrate the importance of correcting for the instrumental response function, Figure 6.12 shows the plots of TB and rB-PR versus temperature. The upper set of curves are the uncorrected, measured values of PB, and, as can be seen, no definite trends can be inferred. When the data is corrected for the instrumental response function, the data becomes very well behaved, as shown by the lower set of 155 moHo.HmNHH. mHHo.HmHoH. mNHo.H~moH. omoo.HHmoH. ooHo.Hommo. Nomo.HmmoH. memo. Homo. Hmoo.HHmNH. enHo.HHmmH. mmHo.HmmoH. HHoo.HmNmo. mHoo.Hommo. HmHo.HNomo. omoo.m~eno. oooo.HoHoH. oNoo.HHmmo. mHHo.Hmomo. HoHo.Hm~mo. mmoo.uomno. HNHo.HHoHo. Hme-meo x m\m pooho HHm: oco mo muHH ouoHHEoocH . mmoo.ummoH. mmoo.ummmH. mmoo.«oomH. Neoo.HmHmH. mmoo.HoomH. omHo.HeHmH. moeH. mNmH. omoo.HmmnH. mmoo.HeNNN. mmoo.uman. NHoo.HmmHH. mHoo.HeHmH. mNHo.wHoNH. omoo.«ommo. HHoo.HHHmH. mHoo.HomoH. mmHo.umHHH. oHHo.HomHH. Hmoo.HHomo. ooHo.Heomo. me x m\m Nflmo.eooeo. mmmo.emmeo. meflo.emwmo. mAHo.emHeo. HAHo.H~Nmo. meflo.ememo. mmHo.eomeo. meo.emmeo. wmmo.eeamo. eemo.eemmo. ee~o.emmeo. mmoo.ewfimo. ewmo. memo. HAHO. omHo. wHHo.eHmeH. mwfio.eowea. mm~o.eomHH. Hemo.emmNH. Hemo.ee~HH. owmo.eoeHH. oeoo.eweeo. Neoo.emmeo. oeoo.eewmo. mmoo.nmoeo. eaoo.ememo. Heoo.eowmo. Hmoo.eeeeo. Hmoo.ewmeo. mmoo.efimmo. mmoo.eweeo. maeo.eoaao. meoo.emmao. meHo.ememo. NHHo.eeemo. Hefio.emomo. meHo.emNeo. ewoo.emHHo. Hweo.eomeo. OHHo.Hoon. QOHo.emer. HmH-2Ho x m\z 2. x m\z .mqumoH HmzHH oouuonpoo Hmmo.umnmm. NHNo.HHHm~. Hmmo.HomeN. Nmmo.Honm. emmo.moeHm. Hmmo.woomm. moHH. HHmH. mmNo.Hmoem. mmmo.emon~. mHmo.Hmoe~. mmoo.«mom~. Heoo.HHNHN. emmo.HHomm. mmHo.HmmeN. omoo.HNmmN. emoo.uenmm. mmmo.«meu~. mmHo.HNHNm. mmoo.nmeo~. momo.ummnH. .mHumH .mH.o oHomH meH. mmN. mmo. mmH. Hmm. mHm. Hmo. onm. ewe. HHo. Hmo. emo. mmH. oeo. oHH. one. mmm. moH. emm. ammo.m “NmN.N HmNH.N uomo.m HmHm.N Hemm.m NeH.H mom.H Hemm.m emmo.m HeHo.m Homm.~ “Hmm.m «Hmm.m “mm~.N “mom.m Hmmo.N “NeH.m anm.m “mmH.N Heom.m 2 .HuH moH om on omme mome HHom meHm omme meHm omme mome meHm omme mome HHom meHm Ho .mucoHoHHHooo aoHuwscouum ocm endow Ho HuHooHo> omme mome HHom meHm omme meHm omme mome meHm omme mome HHom meHm Hoeoe .mH.o oHLMH oo.mm mm.em mm.m~ mm.e~ Hoooe 164 The values of these constants seem to increase with tempera- ture as expected; however, the errors make this trend un- certain. Therefore, it was felt that a plot of rBR versus 8%, whose slope is proportional to a, would reveal changes of a with temperature. This plot is shown in Figure 6.14, where the error bars are omitted to avoid confusion. As can be seen, all the experimental points fall on the same line, within experimental error. This indicates that a is constant with temperature, at least over the temperature range used. BRILL Results The six parameters for the spectral distribution function, as found by KINET, are used in program BRILL. The results from this program are a composite plot of the total spectrum and of the four lines. In addition, the printout includes the numerically integrated intensity of each peak and the two intensity ratios, R/(2B + M) and (R + M)/2B. Figures 6.15 and 6.16 show the computer plots of spectra with very good and poor symmetry, respectively. The actual spectra are shown in Figures 6.17 and 6.18, and it is seen that the fitted parameters reproduce the spectra very well, except that the asymmetry is gone, since the computer plot produces the full spectrum by reflecting the mirror image of one half of the. order. 165 .Honosdoum Ho ohmscm one mnmho> :uon «Hm; cHsoHHHHm oouuohhoo mo HOHm .eH.o ohsmwm ...N: 20.. m3 ONO- 09.9 00.0 00.5 OOH ON.0 0*.0 00.? 00.” \ 2... SN. .00“. .NIQ ...... OB. oo. o o o a oo-oo0.500000oooooooooooooooooooooooo ..........oooooo000000oooooooooooooouoooooooooococoooooooooooooooooococoouoooooooooo... o .00 00.000000 0050000. 55.00. .....oooo 166 C' ' I’I'IIIIIIIII I IIIIII’IIIIIIIII'I Afifliflfllfllflflfll.‘- BRILL plot of a high finesse spectrum. Figure 6.15. 167 II... II. 'IIUO 'I. [flu-o 0.. $05.. 0.. II... .‘I II... .99 2 Ohio (I. K .5.. II. 2' DD. 0.. I I I. II. IE. I I I I O O - Or I I I o o s -- .....9..... 00.0.0.0... II... 0". A BRILL plot of a poor finesse spectrum. Figure 6.16. 168 .ommoaHm ooom aqu Enuwoomm 8 mo mnHoHoooH Hanuu< .AH.e oeomHm 169 .mmmocHH goon :HHZ Eshuoonm a mo mcmououou Hmsuu< .wH.e oesmfim 170 The intensity ratios, listed in Table 6.20, are given by66 = W -1 21B CV - CI and IR Cp = ‘T‘ -1 (87) 21B + IM CV As can be seen from the data, the scatter closely follows the uncertainty in the PM values, which is to be expected. The over- all trend with temperature seems to be decreasing for the first and increasing for the second, but no conclusions can be drawn. Since both PM and PM(M/R), the coefficient of the Mountain line, are decreasing with temperature, the Mountain line becomes less of a contribution at higher temperatures. This result might also be expected because as the temperature rises, so does the popu- lation of the lowest excited vibrational state. Therefore, a larger percent of the molecules will naturally occupy this state, and the relaxation will diminish. As commonly stated, 66 few conclusions can actually be drawn from the intensity ratios in the region where the relaxa- tion frequency is of comparable magnitude to the phonon frequency, wB’ since the coupling between the modes is not a simple one. The above data is presented here to show that values can be obtained from BRILL in the event that T and wB are greatly different in M magnitude. 171 Table 6.20. Intensity ratios from program BRILL. (IR + IM)/ IR/ T(°C) 1(Ao) 0 213 (ZIB+IM) 24.75 5145 70 .33i.03 .17i.02 90 .35i.03 .17i.03 105 .34i.01 .17i.02 5017 90 .35i.04 .17i.01 4965 90 .35i.01 .18i.01 4880 90 .33i.01 .163.01 29.55 5145 90 .30i.03 .181.02 120 .34i.02 .19i.01 4965 90 .29i.01 .18i.01 4880 90 .33i.01 .17i.01 34.55 5145 90 .32i.03 .19i.03 4880 90 .31i.03 .192,03 105 .34i.01 .17i.03 39.60 5145 *50 .16 .14 *70 .21 .16 90 .27i.03 .20i.05 110 .293.05 .201.09 135 .31i.02 .23i.03 5017 90 .27i.02 .181.04 4965 90 .28i.02 .18i.04 4880 90 .281.02 ,19i,05 * Incomplete fits of one half order CHAPTER VII CONCLUSIONS AND RECOMMENDATIONS By interfacing a mini-computer to a light scattering spectrometer, data acquisition times and uncertainties have been greatly reduced, and the accuracy of data analysis has been improved by at least an order of magnitude, further im- provements being limited only by the spectrometer. Moreover, the use of the digitized data in a large computer has enabled the analysis of spectra containing a thermal relaxation line. The analysis of the light scattering characteristics of Freon 113a has led to the discovery of the second compound known to exhibit a thermal relaxation line with a width on the order of the Brillouin frequency shift. Since Freon 1138 has been compared to CCl several times, 4 it is appropriate to summarize some of their properties. Table 7.1 lists the literature values of several properties of CC14 and compares them to the values obtained in this study for Freon 113a. From these values, the polarizability and molar volume can be calculated. For CC14, the polarizability is 1.046 X 10'23 cm3 and the molar volume is 96.6 cmS/mole, while the values for Freon 113a are 1.038 X 10'23 cm3 and 118.7 cm3/ mole, respectively. Since the polarizability is a rough meas- ure of the molecular volume, we see that the two molecules are about the same size; however, the CF3 group will have a higher charge density than a Cl atom, and a small dipole moment will 172 173 Table 7.1. Comparison of some properties of CC14 and Freon 113a. Constant CC14 Freon 113a Refractive Index 1.459 1.360 (20°C) Density 1.595 g/cm3 1.580 g/cm3 Sound Velocity, VS 1030 m/sec 711 m/sec (20°C, 3.37 GHz) (25°C, 3.15 GHz) Depolarization Ratio 0.016 (4880 A°, 0.068 (4880 A°, 25°C) 25°C) a/fz 5.20 x lO'lscm'l/ 4229 x 10'15cm'1/ H22 (20°C) Hz (25°C) r/K2 5.8 x 10‘3cm2-Hz 9.6 x 10‘3cm2-Hz (20°C, 2.05 x (25°C, 2.55 x 105cm-l) 105cm'1) exist in Freon 113a, whereas none is present in CC14. The depolarization ratio implies that Freon 1133 is less spheri- cally symmetrical than CC14, as expected. These two facts indicate that the molecules of Freon 113a should have a greater molecular repulsion due to the small dipole, and the shape should not allow as efficient packing of molecules as in CC14. The molar volume confirms these suppositions, showing that one mele of Freon 113a occupies significantly more volume. The decrease in the sound velocity also indicates that molecules of Freon 113a are less closely packed, even though the weight densities are of the same magnitude. Finally, the temporal and Spatial attenuations of Freon 113a are larger, indicating I! l. l (‘l‘lll llllll (' ' 174 that the phonons are damped faster and travel shorter distances than in CC14. This implies that more molecular friction must be overcome, or the exchange of energy between the internal and external degrees of freedom is more efficient. It is probable that both of these phenomena are responsible, due to the presence of increased molecular repulsions and new vibra- tional modes in Freon 113a, as compared to CCl4. The vibrational energy level schemes of the two com- pounds are markedly different. The lowest lying vibrational level of CCl4 consists of a doubly degenerate deformation mode, which is a scissoring motion of the C1 atoms. In Freon 1133, three levels lie below these deformation modes, which are shifted to higher energies. The lowest lying state is a tor- sion mode, i.e., the rotation of the CC13 and CF3 groups in opposite directions around the C-C bond. The next two levels are due to rocking modes of the C013 and CF3 groups. The Raman spectrum shows a weak line for the torsion mode and strong lines for the rocking modes; however, the CC13 deforma- tion mode gives the strongest peak of the spectrum. Therefore, the deformation modes in both molecules may be the primary vibrational levels involved in the relaxation process, and the usual assumption that the relaxation process involves the ex- change of energy through only the lOwest lying vibrational state may have to be reconsidered. In addition, the normal Boltzmann population distribution can be disrupted by pressure waves, since they are assumed adiabatic at high frequencies. The passage of a pressure wave can cause a local temperature 175 jump which, in turn, can cause the higher vibrational levels to become more highly populated, allowing thermal relaxation to then proceed through the most favored path. This mechanism would imply that molecules in a homologous series can have the same relaxation mechanism, even though some other part of the molecule is altered and the vibrational modes are shifted in energy. Consequently, further studies should be done on other substituted CCl4 compounds. Correlations or similarities be- tween the vibrational energy level schemes of the compounds exhibiting a thermal relaxation line should produce new in- sights into the mechanism of the relaxation process. Since both CCl4 and Freon 113a exhibit plastic crystalline phases, a unique opportunity is available for the study of the thermal relaxation line in a phase in which translations are severely restricted, but rotations and vibrations are relative- ly unhindered. Since some molecular ordering exists in the plastic crystalline phase, the preferred orientation of molec— ular collisions for vibrational excitation could reveal the particular vibrational mode or modes involved in thermal relaxa- tion. In addition, the change in the intensity ratios would assist in accurately locating the glass transition tempera— ture and understanding the nature of structural changes occur- ring in this phase change. It has been suggested that there may be more than one relaxation time needed to describe thermal relaxation,6 and that the Mountain line actually is due to a sum of Lorentzians, 176 one from each of the vibrational levels involved in the relaxa- tion process. It might be beneficial to investigate this possibility; however, the use of more than one relaxation time would prove meaningful only when the interferometer stability and finesse are improved since the results for PM have shown themselves to be greatly dependent on the shape of the instru- mental response function. The problem of the actual mechanism of thermal relaxation is one that requires both more theoretical and experimental research. If the velocity of sound and the density of the liquid is known, the usual procedure in light scattering studies, and, especially, ultrasonic studies, has been to calculate the adia- batic compressibility, which is given by B§(pV§)’1. However, the velocity of sound is assumed to be the low-frequency velocity, Vs,o: and the significance of this calculation is questionable since we are in a dispersion region and VS changes with frequency. For low frequencies, well away from the dis- persion.region, the velocity of sound is given by52 b K2 2 1/2 ° )1 . (88) wherebo is the kinematic viscosity, and K is the scattering waverrnnnber. Squaring both sides of (88) and expanding the expression.on the right, we have the result = -22. 89 vs vs,o bo K /4 ( ) thnn this expression, Vi can be plotted linearly against Kz/4, 177 if bO is a constant, and the slope and intercept will give values for b0 and Vg’o, respectively. This would seem an ideal method for obtaining Vs,o for Freon 113a, since this value has not been published; however, the dispersion region of these studies also renders this calculation meaningless because the slope changes rapidly in this region. Therefore, VS,O can only be obtained by direct ultrasonic measurements. The advantages of the computer interfaced spectrometer can be enhanced with some improvements in data handling. The greatest difficulty at this time is the transfer of data from the pdp 8/e to the CDC 6500 computer for analysis by KINET. At least two solutions to this problem can be proposed. With the extra core space (12K) and the magnetic tape drive of the newly acquired pdp 8/e, it may be possible to rewrite KINET to work on a mini-computer. This task may prove to be a major undertaking, but certainly one worthwhile, since data can be taken, stored, and analyzed immediately, allowing poor data to be discarded and better spectra to be taken. The tape drive (xvi be used to call into core only the subroutines being used, alltnving KINET to occupy less memory. Alignment techniques also can.be improved by noting the asymmetry and distortion from the response function. The second solution consists of using the tape drive on the rufiv pdp to store data on magnetic tape and converting this tape 11) one compatible with the CDC 6500. As of this writing, the enithor knows of no method or facilities available at this Luriversity'to perform this tape conversion directly. It can 178 be done indirectly in at least two ways using paper tape, ' which is undesirable because of the large number of errors that occur at random in the punching and reading processes. The first method involves writing programs enabling the pdp 9 in the cyclotron building to read binary paper tapes and store the data on nine track magnetic tape in EBCDIC. This tape could then be converted to seven track magnetic tape using the Michigan Treasury Department facilities. The seven track tape, which is compatible on the 6500, only needs to be read, and the data converted to KINET format and restored on magnetic tape. The second way employs a high speed paper tape reader in con- junction with the CDC 3600 computer in the Michigan State Uni- versity Computer Center. Programs are available to store the data on magnetic tape. The data then can be converted to KINET format and restored. This method proved impractical be- cause the poorly maintained paper tape reader made numerous reading errors. In addition, this reader is soon to be removed from the Computer Center, and no plans have been made to replace it. Another possibility for the transmission of data to the CDC 6500 lies in the current plans to interface the pdp 11/45 of the Computer Center with a regular computer access port. This would allow direct communication between the pdp 8/e and the CDC 6500 through the pdp 11/45. This link would also be extremely valuable for data analysis, because the 6500 could store the data directly and analyze it while the pdp 8/e is being used to collect new data. This capability and a more 179 stable interferometer would make the recording and analysis of Brillouin spectra extremely fast and reliable. In this manner, the usefulness of Brillouin spectrosc0py could be more fully realized. As mentioned previously, KINET is very sensitive to response function asymmetry and large noise spikes. The first can be minimized by proper alignment, and the second can be minimized by using the smoothing routine before submitting the data to KINET. Since PM is such a difficult value to extract accurately, a better analysis technique might consist of fitting the Rayleigh peak to a Voigt function,38 Airy func- tion, or a polynomial,66 and then deconvoluting the spectrum before a KINET fit is performed. Special care will have to be taken to allow the Rayleigh removal during the deconvolu- tion, because a delta function is produced when a peak is de- convoluted with itself, and this is what causes most decon- volution programs to fail. The mini-computer programs for data acquisition and dis- play are very flexible and can be adapted quite easily for use in light beating measurements. Since the wave analyzer used in self-beating spectroscopy provides a zero-frequency ref- erence point, time averaging can be performed to reduce the errors caused by the large amount of random noise present in these spectra. The use.of available programs to fit the data to one or more Lorentzians would permit the in-lab analysis of these spectra, thereby reducing analysis time from days to minutes. 180 The spectrum of scattered light contains a wealth of information for the chemist, since it provides a complete picture of the decay of thermodynamic fluctuations. Since this decay is dependent on the nature of the molecular inter- actions and short-range order of the liquid, qualitative infor- mation on these properties and the nature of the preferred mechanism of decay can be obtained from the magnitude of the intensity ratios. 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