# EFFECTS OF MICROWAVES ON THE ANNIHILATION RATES OF ORTHOPOSITRONIUM LOCALIZED IN FLUID ETHANE

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For illustrations that cannot be satisfactorily reproduced by xerographic means, photographic prints can be purchased at additional cost and inserted into your xerographic copy. These prints are available upon request from the Dissertations Customer Services Department. 5. Some pages in any document may have indistinct print. In all cases the best available copy has been filmed. University Microfilms International 300 N. Zeeb Road Ann Arbor, Ml 48106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Order N um ber 1331891 Effects o f microwaves on the annihilation rates o f orthopositronium localized in fluid ethane Arganbright, Robert Harvey, M.S. The University of Texas at Arlington, 1987 U M I 300 N. ZeebRd. Ann Arbor, M I 48106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. EFFECTS OF MICROWAVES ON THE ANNIHILATION RATES OF ORTHOPOSITRONIUM LOCALIZED IN FLUID ETHANE by ROBERT HARVEY ARGANBRIGHT Presented to the Faculty of the Graduate School of The University of Texas at Arlington in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE IN PHYSICS THE UNIVERSITY OF TEXAS AT ARLINGTON August 1987 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. EFFECTS OF MICROWAVES ON THE ANNIHILATION RATES OF ORTHOPOSITRONIUM LOCALIZED IN FLUID ETHANE APPROVED: C. (Supervising Professor) A IffllloWkljhy ' ijy Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. i DEDICATION To the loving memory of Shirley Jean and David, who instilled a love within me for travelling never ending paths to knowledge. iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGEMENTS Foremost acknowledgement for this work is due Suresh C. Sharma for providing the original concept of the experiment. Also, I am deeply grateful for his time, guidance, patience, and understanding, without which this thesis could not have been completed. I would also like to express my appreciation to the many other people who provided assistance to this work. Mike Ward provided aid in collecting and analyzing data as well as computer programming. Charley Dark also lent his skills as a programmer. Wallace Lutes and Doug Coyne provided neccessary technical advice and assistance. Dr. J. M. Kowalski provided invaluable theoretical discussions. Dr. John Owens of the UTA Department of Electrical Engineering and Elizabeth Hardin of Hewlett Packard Corp. provided for the loan of various components of the microwave system. Finally I am grateful to the Robert A. Welch Foundation in its support for this research and to A. O. Pipkin Jr. of ARL Inc. for student financial support. July 23.1987 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ABSTRACT EFFECTS OF MICROWAVES ON THE ANNIHILATION RATES OF ORTHOPOSrTRONIUM LOCALIZED IN FLUID ETHANE Robert Harvey Arganbright, M.S. The University of Texas at Arlington, 1987 Supervising Professor: Suresh C. Sharma Orthopositronium atoms localized in density fluctuations in fluid ethane have been subjected to microwaves in the 90-100 GHz range. The resulting pick-off annihilation rates of orthopositronium are significantly reduced at certain density dependent values of the incident microwave frequency. A mechanism for the absorption of microwaves by non-polar ethane molecules is described in terms of pressure induced dipoles arising through molecular quadrupole interactions. Two possible mechanisms for the resulting decrease in the annihilation rates of orthopositronium are discussed. The first model assumes a reduction in the density of electrons around a localized positronium atom as a result of the response of the fluid-compressibility to the microwave field at some resonant frequency. The second model describes the reduction of annihilation rates as due to positronium annihilation with molecules in rotationally excited states. v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OF CONTENTS ACKNOWLEDGEMENTS................................................................................................... iv ABSTRACT ...................................................................................................................... v Chapter I. IN TR O D U C TIO N .................................................................................................. II. MODELS FOR LOCALIZATION OF ORTHOPOSITRONIUM IN DENSE F L U ID S ................................................................................................................ Positronium Self-Trapping...................................................................... 1 3 3 The Simple Cavity M o d e l....................................................................... 3 The Self-consistent Field M o d e l.......................................................... 12 The Density Fluctuation M o d e l................................................................... I I I . EXPERIMENTAL D E TA ILS ................................................................................ Apparatus...................................................................................................... 14 17 17 Gas Handling S ystem ................................................................................ 17 Heating System ...................................................................................... 20 R F Generator......................................................................................... 20 . Positron S ource...................................................................................... 21 vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Lifetime Spectrometer................................................................................ Set U p ...................................................................................................... 23 Time Calibration...................................................................................... 24 Data Collection......................................................................................... 24 Analysis D etails...................................................................................... 27 Density Calculations................................................................................ 28 IV. RESULTS AND DISCUSSIO N........................................................................... V. 23 31 Orthopositronium Annihilation Rate Behavior at 98 a m ag a t................ 31 Absorption Mechanism............................................................................. 33 Mechanisms for Microwave Induced Reductions in x0.P s .................... 38 Rotionally Excited Molecules............................................................. 38 Density Fluctuation Excitations.......................................................... 39 Orthopositronium Annihilation Rate Behavior at 60,37, and 99 a m ag a t............................................................................................. 46 x0.Ps at 60 amagat and 307.15 K ................................................... 46 x0.Ps at 37 amagat and 307.15 K ................................................... 50 x0.Ps at 99 amagat and 307.15 K ................................................... 50 CONCLUSIONS............................................................................... 59 REFERENCES.......................................................................................................... 61 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UST OF ILLUSTRATIONS 2-1 V 2-2 Potential well depth versus density for ethane at 306.4 K ........................ 8 2-3 Equilibrium cavity radius versus density of ethane at 306.4 K ................. 9 2-4 Ground state binding energy versus density of ethane at 306.4 K . . . . 10 2-5 Change in the free energy versus the characteristic size ps versus density at 306.4 K .................................................................... of the wave function He at 10 K and ethane at 306.4 K ............................. 3-1 5 15 Schematic of gas handling and heating systems and the microwave system ............................................................................. 19 3-2 Decay scheme of Sodium-22........................................................................ 22 3-3 Schematic of the lifetime spectrometer........................................................... 25 3-4 Typical Lifetime Spectrum.............................................................................. 26 4-1 >.0_ps versus frequency in ethane at 98 amagat and 307.15 K ................. 32 4-2 versus frequency in ethane at 98 amagat and 307.15K .................. 42 4-3 A>20^>s versus frequency in ethane at 98 amagat and 307.15 K .................. 43 4-4 A?T20_ps versus q2 in ethane at 98 amagat and 307.15 K .............................. 44 4-5 A-x*20_ps versus q2 in ethane at 98 amagat and 307.15 K .............................. 45 4-6 * 0_ps versus frequency in ethane at 60 amagat and 307.15 K ........................ 47 viii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4-7 x^p, versus frequency in ethane at 60 amagat and 307.15 K ....................... 48 4-8 A * 2*-*., versus frequency in ethane at 60 amagat and 307.15 K ................. 49 4-9 A x '2,,-?, versus q2 in ethane at 60 amagat and 307.15 K ............................. 51 4-10 versus frequency in ethane at 37 amagat and 307.15K ........................... 53 4-11 A x2^ , versus frequency in ethane at 37 amagat and 307.15 K .................... 54 4-12 A x2e_p, versus frequency in ethane at 37 amagat and 307.15 K .................... 55 4-13 Xo.P, versus frequency in ethane at 99 amagat and 307.15 K ........................... 56 4-14 Ax2^ , versus frequency in ethane at 99 amagat and 307.15 K .................... 57 4-15 A x '2e_p, versus q2 in ethane at 99 amagat and 307.15 K .............................. 58 ix Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UST OF TABLES 1. Comparison between observed values of N* and R* and those calculated with the self-consistent field m o d e l................................. 2. 14 Impurities found in a representative sample of the ethane used in the experiment.............................................................................. x Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 18 CHAPTER I INTRODUCTION The localization of orthopositronium (o-Ps) in regions of lower than average density within fluids under certain PVT conditions has in recent years received considerable theoretical and experimental interest. Two models, the cavity model and the density fluctuation model have been separately successful in different fluids at different temperatures. In the cavity mode! the positronium, through exclusion forces, creates a cavity in which it resides. The density fluctuation model assumes that the positronium preferentially samples pre-existing regions of lower than average density. Both models are discussed in Chapter 2. This investigation began for the purpose of testing the simple cavity model. It was assumed that, if this model is correct, irradiation of the sample with electromagnetic quanta having energies equal to the binding energy of o-Ps in the cavity would eject the o-Ps from the cavity into the denser medium surrounding it. The delocalization of the o-Ps atom would result in an increase in the annihilation rate ( x ^ ) over that when no radiation was present. The results from this simple experiment revealed a suprising effect of the radiation upon V p s. Instead of increasing at a certain single incident energy, as expected from the simple cavity model, actually decreased below that predicted for an already localized o-Ps atom. This effect was observed for several densities for frequencies in the 90-100 GHz 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 range. This thesis is primarily concerned with the absorption of microwaves by ethane at 307.15K and the subsequent reduction of \ , _ p s . Two questions raised by the experimental results are: 1) how does the non-polar ethane molecule absorb radiation? and 2) how does the absorption of the radiation cause the o-Ps atoms to annihilate at a lower rate? Ample literature is available concerning pressure induced dipoles in non-polar molecules. The first question will be approached in terms of this model. The reduction of will be discussed in terms of two models. The first assumes that the density of electrons around a localized o-Ps atom is reduced as a result of the response of the fluid compressibilty to the microwave field at certain resonant frequencies. The second model describes the annihilation rate reduction as due to positronium annihilating with rotationally excited molecules. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTERII MODELS FOR LOCAUZATION OF ORTHOPOSITRONIUM IN DENSE FLUIDS Positronium Self-Trapping At suitable temperatures and pressures a cavity may form around Ps in a non-polar fluid. Its formation is due to short-range Pauli repulsive forces which dominate over the attractive, long-range polarization force. The size of the cavity is governed by a balance between the outward pressure of the o-Ps's zero point energy, due to its localization, and the inward pressure due to the external pressure and the surface tension of the fluid. As a result it is expected that the density of electrons within the cavity is less than the average density of the surrounding medium. This model has been used to understand the non-linear behavior of \>-ps in C2H6 and other fluids including He, Ne, and H2. The Simple Cavity Model At low densities x0_ps exhibits a linear dependence on the gas density according to the equation V p * = >v*e+ 4Trr02 cnQ% u D (2.1) where xvac is the annihilation rate of orthopositronium in vacuum, rQis the classical radius of the electron, c is the speed of light, n0 is the standard number density, % f f is the effective number of atomic electrons per molecule "seen" by o-Ps, and D is the density of the gas in 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 amagat. Figure 2-1 shows \ ^ , s versus density for ethane at 307.K. The dashed line represents Equation (2.1). The non-linear xe_pj versus density behavior for a cavity devoid of gas molecules is given by \-P s ~ 0 " P © )\/ie + P o ( W Q D ) , ( 2 .2 ) where P0 is the fraction of the positronium’s probability density extending beyond the cavity and q= 4-rr r02 cn01Z*ff. Assuming a spherical square well potential of depth V and radius R, one obtains1, P0 = sin^ / (1 -n cot n ) (2.3) and n /sin n = R (2mV/h2)1/2, (2.4) where m is the mass of the positronium atom. The simplifying assumption that the potential well of the cavity is shallow enough to allow only the ground state limits the yalue of n to the range 7t/2 to n. The total energy of the system is the sum of three terms; the zero point energy of the positronium, the work done against the external pressure in creating the cavity, and: if the fluid is a liquid, the work done against surface tension. Hence E = Ti2k2/2m +4ir/3 R3 P +4 tiR24 \ (2.5) where P is the external pressure, <f is the surface tension and k is the Ps center of mass wavenumber. The equilibrium radius of the cavity is obtained by minimizing E with respect to R, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 o o CO co csT^ co co © CO co 3 O' LU >s X) c .2: o> o o o C\J w CD C CO O) CO E • 2 * as CO o CO CD £ H c o Q CO o CO LO o o 05 6 CO c CD “O CO 3 £2 CD > If) CVJ © 3 CO O) u. o CJ o o (t . su)S d ' 0 ^ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. SE/5R « 0 (2.6) For a gaseous medium, <f<=0, and .the depth of the spherical square well potential is given by V «{2nP(ft2/2m)3/2ln2(ncotn-l )/(sin4n cosn)]}(2/5). (2.7) The ground state or zero point energy is given by E2 = V sin2n (2.8) The binding energy is therefore given by Eb = Ez-V (2.9) To apply this model to the data, we measure V p s and use Equations (2.2-2.4) to calculate n. Equations (2.5-2.9) are then used to obtain vaiues of the equilibrium radius R and the potential well depth V. Thus, Equations (2.2 and 2.3) give a value of n through the measured >>0_ps then from Equations (2.4,2.6, and 2.7) "measured" values of R and V are obtained. Assumptions implicit in this simplified model are, that the ground state is the only occupied state, and both the molecular penetration of the cavity and the diffuse nature of its surface are negligible. This model has recently been applied2,2 to \,_ps data published4 for ethane at 306.4 K as shown in Figure 2-1. At low densities up to about 11 amagat >,e^ s varies linearly with the density in accordance with Equation (2.1). For densities between 11 and 180 amagat the departure of v * , from Equation (2.1) increases with the density. At higher densities V -ps again increases linearly with the density but at a steeper rate. Up to about 100 amagat these results are qualitatively similar to those seen in He. The cavity model has been particularly Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. successful, in the case of helium, in predicting a scattering length, a=0.77 ± 0.04 A, in good agreement with theory5. For ethane at 306.4 K the results for the well depth, V, the cavity radius, R, and the binding energy are shown in Figures 2-2,2-3, and 2-4 respectively. The potential barrier experienced by the oPs atom in ethane at 306.4 K varies between 0.14 and 0.56 eV over the range of densities shown. In the case of helium the density dependent potential agrees well with an optical potential given by V0 = (h2/m)27t n0aD (2.9) where a is the scattering length for o-Ps-He collisions mentioned above. Since the Ps-C2H6 scattering length was not known, an estimate was made using the known scattering length of 0.55 Afor the case of Ps-CH46 . The calculated density dependence of the potential, using Equation (2.10), is represented by the solid line in Figure 2-2. The "measured" potential does not agree with the optical potential. A value of 0.14 eV at densities approximately equal to 10 amagat is too large to describe a cavity devoid of gas molecules7. The sharp increase in the potential at densities higher than about 250 amagat is completely unaccounted for by the optical potential. A multiple scattering potential was also used2 to see if it would affect a signifigant correction. The mutiple scattering potential is given by V ms = v o I1 + a(4.5n n 0D)1/3]. This is the optical potential with a correction term for higher densities in terms of a fluid structure factor8 . This equation is represented by the dashed line in Figure 2-2. This correction hardly improves the fit of the model to the data. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.10) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.6 0.4 0.2 ms • «* ®* 0.0 0 100 200 Density (amagat) Figure 2-2. Potential well depth versus density for a spherical square-well potential in ethane at 306.4 K. The solid and the dotted lines represent the optical and multiple scattering potentials, respectively, for a=0.55 A2’3. 300 9 o o CO CO CM * <6 O CO CO CO CO O) o o © c CM CO © o CO u> E CO CO tn c o X) CO CO c 0) D o o 3 w w © > « 3 TJ «C > CO o E 3 3 cr tu co l CM © t. 3 O) U. CO CO (y ) sm peu Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.00 « ®• e® • • •% Hm -0.05 k UJ - 0.10 3 0 6 .4 -0.15 K X X 100 200 300 Density (amagat) Figure 2-4. The ground state binding energy of positronium atom to the potential well of the cavity versus the density of ethane at 306.4 K 2>3. O The measured density dependence of the equilibrium radius is shown in Figure 2-3. It differs considerably from the results obtained in He at 1.7 and 4.2K9’10. In He the cavity radius is seen to decrease exponentially with the pressure; it seems reasonable to expect the radius should decrease with increasing density. This is not at all the case for ethane up to 180 amagat. In fact the radius increases lineary with D between 50 and 180 amagat. At densities greater than 180 amagat the equilibrium radius falls rapidly with the density, approaching a value of 3.5 Afor the highest densities measured. This is approximately the size of the ethane molecule. If the cavity ever existed, it has by this point collapsed so that o-Ps samples the average density of the gas according to Equation (2.1). This fits with the fact t h a t v ^ seems to approach Equation (2.1) at high densities. In Figure 2-4 the measured binding energies are plotted against density. Even at the relative maximum occurring at 180 amagat, the binding energy of 0.15 kT is considered much too small to support a cavity should one form2. Detailed theoretical calculations of self-trapping in He have further shown that the cavity model has considerable difficulty in describing the non-linear behavior of V p s when the binding energy is less than about 6kT19. The Self-consistent Field Model; A Further Test of the Cavity Model The self-consistent field model for investigating self-trapped states of o-Ps has been summarized by lakubov and Krahpak11. A simple outline of this model is used here to calculate the transition density, D*, which is characterized as being that density at which Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 12 o-Ps makes a transition from an extended state to a localized state. The localization of the o-Ps creates a cavity within the fluid having a radius R* and binding energy, EB\ In this model interactions between the o-Ps and fluid molecules give rise to a molecular correlation in the neighborhood of o-Ps given by N(r) ■ N exp[-B V(r)], (2.11) where N is the average number density of the gas, V(r) is the interaction potential between the o-Ps and the molecules, and 13=1/kT. The change in the free energy of the system due to the trapping of a single Ps atom is given by BAF = fi2/2m/ (V'f'(r)|2dr3 - N jfd r^ e xp I-B V ^ j+ B V ^ -l} (2.12) where ^(r) = / d r ,3V(r-r,)|v * (r ) |2 (2.13) is the quantum mechanical average of the interaction potential. Assuming a psuedopotential, V(r) = 2n h 2 a S(r)/m, (2.14) where a is the scattering length for the Ps-molecule interaction, and approximating a trial wave function for the ground state, 'P(r)=(3/2 7tx3)1/2exp(-r3/x3), (2.15) the change in the free energy is obtained: BAF=[3r(1/3)/27/3n2](>>3/>.)2-N/ dr3{exp[(-3Bk2a/mx3)exp(-2r3/x3)] +(3Bk2a/mx3) expf^r3/* 3)-!} > Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ^ 16) 13 where >p= h/(3mkT)1/2 is the be Broglie wavelength of the o-Ps atom and \ characteristic range of its wavefunction. By changing the variable of integration to z=exp(-r3/v 3), and keeping the first few terms of the expansion of exp[3Bh2az/mx3], an approximate integral for Equation (2.16) is obtained giving for the change in free energy2, 8AF=[3 T(1/3)/27/3 ir2] ^ ) 2 [27N>.b4 a2/(32 A 3)](1-xB2 a/2A 3). (2.17) From Equation (2.17) and the conditions for self-trappining, namely, Ba F=0 anc 6(I3a F)/5>.=0, the transition density and characteristic wavelength can be obtained frc N*=32/27[T(1/3)/a5x B4] (2.18) and x *=(2/ti2) (a * q2)1/3- (2.19) The last three expressions, resulting from our calculations, do not agree with those given by lakubov et al. It is believed that the authors are in error due to either the actual calculation or the transcription of the results. Figure 2-5 shows the dependence of Ba F on * ' 1 for He at 10 K and ethane at 306.4 K. In the case of helium the change in the free energy is positive for all densities less than N*=4.55x1021 cm'3 so that cavity formation is unlikely. For densities greater than N*, over a certain range of x's, Ba F is less than zero and self-trapping is energetically possible. For ethane at 306.4K, Ba F is greater than zero over the entire range of densities for which annihilation rates have been measured. Only at densities greater than about 8.22x1022 cm'3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 14 (3000 amagat) is the change in free energy less than zero and in favor of cavity formation. That the localization of the positronium can occur only at such extremely large and unphysical densities above 3000 amagat demonstrates the failure of the cavity model in describing the non-linear behavior of V p s These calulations are summarized below in Table 1. Table 1. Temperature Sample Transition density, N* (K) (cm‘3) Calculated2 He 10 77 c 2h 6 Cavity radius, R* 4.55 x1021 1 .77x10 22 306.4 8.22 X1022 305.45 8.20 x1022 (A) Observed4,9 1.4 x1021 5 x 1 0 21 3 x 1 0 2° Calculated2 21.0 Observed4,9 12.7 10.7 5.97 10 Density Fluctuation Model In the density fluctuation model the reduction of the local density of electrons experienced by o-Ps is shown to result from its localization in density fluctuations2,12,13. That is, orthopositronium preferentially samples pre-existing and spontaneously forming regions of lower than average density in the fluid. According to the density fluctuation model the difference between the measured orthopositronium decay rate and the decay rate predicted Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 15 10t A/*=4.55 x1021 cm"3(~169 am agat) o .m y 1.25A/ CO. 0.02 0.06, 0.04 /V*=8.22x1022 cm“ 3(~3x103am agat) 0.75/V* 1.25/V 0.05 0.15 0.25 (A -1) Figure 2-5. The change in the free energy, as calculated from the density with the self-consistent field model, versus the inverse of the characteristic size of the wave function for (a) He at 10 K and (b) ethane at 306.4 K 2’3. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 16 by Equation (2.1), a v *>„ is related to the isothermal compressibility according to A V P s = 4jtro2cno1zeff D(kTK0/2nV0)1/2, (2.20) where K0 = V 1(3V/3P)t is the isothermal compressibility of the fluid. This equation has worked well for ethane12,12, methane14, nitrogen1® and carbon dioxide1® as long as the density fluctuations are small. A subtle but important distinction exists between these two models for localization of o-Ps. In the cavity model, the localization of the o-Ps atom is seen to take an active role in the molecular dynamics of the fluid by the creation of its own low density cavity. While on the other hand, the density fluctuation model places o-Ps in a passive role; that is, it simply falls into or migrates to pre-existing regions of lower than average electron density. In the latter view o-Ps is viewed as an excellent probe of density fluctuations in fluids. The original intent of this research was to experimentally test the simple cavity model. If o-Ps is localized in such a cavity, then it is reasonable to expect to be able to dissociate the "cavitron" by striking it with photons having energies close to its binding energy. This would result in an increase in v * ,* to a value predicted by Equation (2.1). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER I I I EXPERIMENTAL DETAILS Apparatus The Gas Handling System A schematic representation of the gas handling system is shown in Figure 3-1. The sample holder consists of a cylindrical copper block three inches long and two inches in diameter into which a cylindrical cavity 7/8 inches in diameter and 2.4 inches deep has been bored. The holder was tapped at the top in order to mount a Swage.lok plug with an entry port for the gas. The plug mounting leaves a volume 1.7 inches deep and 1 inch in diameter to be occupied by the gas sample. In order to admit microwaves into the sample chamber two optical sapphire windows were installed on opposite sides of the holder. Each window is 3/4 inch in diameter and 1/8 inch thick. When mounted on the vessel, each window has an exposed surface 1/2 inch in diameter through which the microwaves can pass. The positron source assembly was suspended by a nichrome wire from the inside of the Swag.lok plug so that it was held near the center of the holder and off to the side of the window axis. The gas used was research grade (99.99% pure) ethane. The manufacturer’s analysis of the impurities found in a representative sample of the gas is shown in Table 2. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE 2 IMPURITIES FOUND IN A REPRESENTATIVE SAMPLE OF THE ETHANE USED IN THE EXPERIMENT Impurity Oxygen Nitrogen Carbon Dioxide Hydrogen Methane Acetylene Ethylene Propane Propylene Isobutane n-Butane Water parts/million (ppm) ft 5 8it 2 5* 1* 40 h 2it 2* 2* 1* ‘ undetected, less than quantity indicated The sample holder was connected to the gas bottle and pressure gauge through stainless steel tubing. The pressure was measured with a Heise Model 7200 Bourdon type pressure gauge having a precision of ±1 psi. The pressure was manually read and recorded at the beginning and end of each run. Preparatory to introduction of the sample gas, the entire system was subjected to standard baking and flushing techniques, including evacuation of the system with a diffusion pump while heating the sample holder and tubing. After doing this Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. DC Power Supply Pressure Gauge Heise Model 7200 Apple HE PC Research Grade Ethane Hewlett Packard 8755B Sweep Amplitude Analyzer Thermistor Positro Source Nichrome Heating Wire m um Waveguide^ (Receiver) Waveguide (Transmitter) Microwave Generation System: Hewlett Packard 8350B Mainframe & Hughes 4772X Sweep Generator Figure 3-1. Schematic of the experimental set-up including the gas handling and heating systems and the microwave system. <0 20 several times over several hours, ethane gas was introduced to the system and evacuated three times to further purify the system. The sample gas was loaded and, after allowing sufficient time to reach equilibrium, the pressure was read and recorded and the valve connecting the sample chamber to the pressure guage was closed. The density of the gas was determined from the temperature and pressure of the gas just prior to isolating it from the gauge. The Heating System The heating system is shown schematically in Figure 3-1. The temperature was moniiered and controlled by an Instec Temperature Controller; a hardware/software package run by an Apple HE computer. A thermistor embedded in the copper sample holder provides the temperature, in terms of its resistance, to the computer which computes the heater power neccessary to maintain the programmed temperature setting and regulates the current from a DC power supply to the nichrome heater wire wrapped around the sample vessel. For electrical insulation Teflon tape was wrapped around the vessel before the heater wire was installed. This system provides for the control of the temperature within ± 0.05 K. RF Generator The microwave source was a Hughes 4772XH Millimeter-wave Plug-in Sweep Generator, powered and controlled by a Hewlett Packard 8350A Mainframe with a Levelling Loop. Figure 3-1 shows the arrangement for introducing microwaves into the sample holder. The radiation was transmitted through the millimeter wave guide to the transmitting horn which was pressed flush against one of the sapphire windows. There are two detectors along Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the guide; one is used to detect the original or reference signal from the generator and the other is used to detect any reflected radiation. Pressed against the opposite window and parallel to the transmitter was a receiving antenna or horn. The signal transmitted through the gas was detected here. The signals from the reference, reflection, and transmission were displayed graphically on a Hewlett Packard 8755B Sweep Amplitude Analyzer. Most of the runs were made in the center frequency mode with a frequency width of 0.10 GHz having a sweep time of 0.01 seconds. The maximum output power of the generator was about 1 mW. PQSitron.5.CMC.e The positron source was 22NaCI. Figure 3-2 shows the decay scheme for ^ N a . By the emission of a positron, ^ N a decays to 22Ne*; where the asterik indicates that 22Ne is in an excited state. In a very short time (about 10'11 seconds) 22Ne* decays to the ground state of 22Ne by the emission of a 1.275 MeV Y-ray. Because of the short time between the decay and the de-excitation, the 1.275 MeV y-ray was used to provide the START signal to the lifetime spectrometer to begin the lifetime measurement. The annihilation of a positron-electron pair produces a 0.511 MeV photon. Detection of this photon sends a STOP signal to the spectrometer. An approximately 50 jxCi source was deposited on a piece of 0.8 mg/cm2 Mylar film. The source was dryed and then covered by another piece of Mylar. The edges of the foil envelope were secured between two thin concentric rings leaving the source exposed in the center of the assembly. The diameter of the source assembly was about 15 mm. The assembly was suspended in the gas by a fine nichrome wire such that its face was parallel to and level Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 22 22Na(2.58 yrs.) EC. 9.4% (0.544 22. Ne* (1.275 Mev 9 0 .6 % / sej Mev) (1.820 Mev) 0.004 % ~100% 22. on Figure 3-2. The decay scheme of Sodium-22. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 23 with the microwave beam. Lifetime Spectrometer Setup Figure 3-3 shows a block diagram of the lifetime spectrometer. The spectrometer uses the 1.275 MeV emission y-ray and the 0.511 MeV y-ray resulting from the positron's annihilation to collect the lifetime spectrum. The two detectors for the spectrometer were RCA 8575 photomultiplier tubes(PMT) with Ortec Model 265 bases. The PMT’s were powered by an Ortec high voltage power supply. The gamma rays were detected by plastic scintillators optically coupled to the photocathodes of the PMT's. The output pulses from the PMT's were sent to an Ortec Model 583 Constant Fraction Differential Discriminator where pulse height (energy) selection was made to choose about fifty percent upper Compton distribution for the 1.275 MeV (start) and 0.511 MeV (stop) gamma rays. The output from the discriminators served as timing signals'. After energy selection, the puises were sent to an Ortec Model 457 biased time to pulse height converter (TPHC). The TPHC converts the difference in the arrival times of the two inputs to an electric pulse whose height is directly proportional to the difference in arrival times between the start and stop pulses. This pulse height spectrum is stored in a mutichannel analyzer(MCA). The lifetime spectrum analysis program POSITRONFIT-EXTENDED17 must have an accurate resolution function for measuring short lifetimes. A 60Co source was measured with the spectrometer to produce a "prompt" peak. Using the peak's full width at half maximum (FWHM), the time resolution was determined by fitting a sum of three Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 24 gaussians to the data. A typical lifetime spectrum is shown in Figure 3-4. Time Calibration Each channel of the MCA corresponds to a certain time interval. The interval or, the time-per-channel(A) was determined with an Ortec Model 462 Time Calibrator module. The time calibrator produces start and stop pulses at fixed intervals over an adjustable time range, generating multiple peaks simultaneously. The multiple peak spectrum was collected and transferred to an IBM computer where a least-squares fitting program was used to determine a and its error. Data Collection After thoroughly purging the system, testing for pressure leaks, and adjusting the temperature control, a large amount of ethane gas was transferred from the gas bottle to the experimental vessel and the valve between them closed. The system was allowed at least four hours to reach equilibrium; following which, the pressure was read and recorded and the valve connecting the test vessel to the pressure guage was closed. The microwave frequency, band width, and sweep time were adjusted and the antennae aligned to the vessel's windows. The positron lifetime spectrum was collected on the MCA and, after a sufficient number of total counts, was transferred from the MCA to the IBM computer for analysis by POSITRONFITEXTENDED. To begin the next run a different microwave frequency was selected and the temperature control verified. Usually, five minutes were allowed between changing the RF frequency and beginning a new run. A series of runs consisted of measuring the lifetime Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 25 High Voltage Power supply PM Tube PM Tube Source CFDD CFDD S ta rt Delay Box TPHC MCA Macintosh 512 K Computer P lo tte r IBM 4341 Computer Figure 3-3. Schematic of the lifetime spectrometer. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 26 o o °O CO 0) *£ E 3 z a> c c ca :js. :.i x: O o o CM Figure 3-4. A typical lifetime spectrum. in O O ujii.Lt in co CM o o o jduueqo jad siunoQ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 27 spectrum for a roughly even spread of frequencies between 90 and 100 GHz. To begin a new series of runs the gas density was changed by either admitting gas to, or releasing it from the vessel. In this fashion V p s was measured as a function of incident microwave frequency in the range of 90 to 100 GHz for gas densities of 37,6 0,9 8 , and 99 amagat. The resolution function and timing calibration were checked periodically throughout each series of experimental runs. Analysis Details The lifetime spectra were analyzed with the standard computer program POSITRONFITEXTENDED. The program employs a standard non-linear least squares analysis method which assumes that the spectrum is a sum of decaying exponentials convoluted with a gaussian resolution function with a constant background added. The analysis applied in this thesis used three exponential terms given by, N{ (t) = Nf(0) exp(-Xf t) V sW = N q-Ps W = No-Ps(°) e x P f-V p s 1* where Nj{0) is the number of species j and Xj annihilation rate of the species j. The resolution function is given by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. {3-1) 28 ^ R(,,' v f r i t - ath 2 -» 6xp{ - [ — } «»« where the </oj are the relative intensities of the gaussians such that Zwj = 1, T0 is the center of the primary gaussian, and AT; is the displacement of the centroid of the ith gaussian from T0. The <fj's are related to the the full width at half maximum by FWHM = 2(ln2)1/2<T. (3.3) The spectrometer set-up and the data analysis are components of the standard technique employed for the determination of Ps lifetimes. More complete details of this method can be found in numerous works (see, for example, References 1,3,7, and 12). Density Calculations The density of the gas was calculated with an iterative computer program which uses both the vir'al and the Benedict, Webb, Rubin (BWR) equations of state18. The vir'al equation of state is given by 19 PV/RT = 1 + B(T)<f + C(T)<f2 (3.4) <r=D/Dc (3.5) where is the reduced density. B(T) and C(T) are the dimensionless second and third virial coeffecients. B(T) is given by B(T)=B1+ B^x + Bg/X2 + B4/ x4-5, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.6) 29 where x= T/T* is the reduced temperature, and Tc=305.32 B1=0.522671 B2=-1.106244 B3=-0.592947 B4=-0.041944 Dc= 6.74 mole/liter (150.1 amagat)20 C(T) is given by C(T)=(C1/x +C2/X3 +C3/x5)(1-To/T) (3.7) where T q =217.8K C 1=0.24423 C2=0.83253 C3=0.53488 The virial equation of state works well in the realm of low to moderate densities. The BWR equation of state, which was designed to fit P-V-T data in hydrocarbons up to Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 30 twice the critical density, is given by18 P-RTD+(B0 RT-A0 -Cc/ T 2)D2+(bRT-a)D3+ aaD+(cD3/T2)(1+Y D 2 )exp(- y D2 ). (3.8) where P is the absolute pressure in atmospheres, D is the density in moles/liter, T is the absolute temperature in Kelvin, and Ao=4.15556 liter2 atm/mole B0= 0.0627724 liter/mole C0=1.79592 x 105 liter2 atm/mole2 K2 a=0.345160 liter3 atm/mole3 b=0.0111220 lite^/mole2 c=3.27676 x 104 liter3 atm K2/mole3 a =2.43389 x 10'4 lite^/mole3 y = 1 .18006 x 1 0 '2 litei^/mole2 R= 0.08207 liter atm ./mole K . The computer program that calculates the gas density uses the measured values of the temperature and pressure to calculate a density with the viral equation of state. This initial estimate of the density is then used in the BWR equation of state to calculate a pressure. This pressure is compared with the measured pressure. If the calculated pressure does not fall within a tolerance of 0.005 PSI, the density is iterated and a new BWR pressure is calculated. The iterative process continues until the measured and the BWR pressures are within the prescribed tolerance. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER IV RESULTS AND DISCUSSION Orthopositronium Annihilation Rate Behavior Figure 4-1 shows the orthopositronium decay rate, %0_ps, in fluid ethane at 307.15 K and at a constant average density of 98 amagat as a function of microwave frequency. In this series of measurements the RF generator was operated in the center frequency (CF) mode having a band width of ±0.200 GHz and a sweep time of 0.01 second. The dashed horizontal line represents a decay rate of 0.04234 ± 0.0012 ns'1 measured with no microwaves incident on the fluid. This value is in close agreement with the previously published experimental value of 0.03963±0.0005 ns'1 for ethane at 98 amagat and 307.15 K with no microwaves present20. Comparison of the "off' value of to that predicted by Equation (3.1) shows the o-Ps is localized in a region of lower than average density. The decay rate is thus seen as a measure of the time-averaged local density of the fluid experienced by orthopositronium during its lifetime. The data in Figure 4-1 shows two sharp, resonance-like "dips" in \^ > s centered at 92.0 and 96.2 GHz. The indication is that microwaves are absorbed by ethane at certain frequencies and this absorption induces local density fluctuations and thereby reduces the density of electrons in the neighborhood of o-Ps during its lifetime. Two questions are 31 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.044 0.043- 0.042 0.041 0.040 98 amagat 0.039 0.038 89 91 93 95 97 99 101 Frequency (GHz) Figure 4-1. >,0_r>f versus frequency in ethane at 98 amagat and 307.15 K. The solid curve hand drawn to guide the eye. The dotted horizontal line represents the decay rate with no microwaves. “ 33 immediately raised by these data: 1) by what mechanism is radiation absorbed by ethane molecules having no permanent dipole moment? and 2) how does the absorbed radiation induce local density fluctuations? The first question is addressed in terms of pressure induced molecular dipoles. Then, once we have allowed for the fact that microwaves are indeed absorbed by the ethane molecule, two possible reasons for the resulting increase in the lifetime of o-Ps are discussed. The first involves an analysis of theoverlap integrals between the wave functions of o-Ps and the ethane molecule in the ground state and then, in the first few rotationally excitedstates.The second possibility explored is that of density fluctuation excitations wherein the bulk compressibility of the fluid is related to the frequency of the incident radiation. Absorption Mechanism Consider a molecule whose unperturbed quantum mechanical Hamiltonian operator is H(°). The corresponding unperturbed time dependent wave function, satisfies the Schroedinger equation: - E f f ^ - t i / i (d*|<°)/dt) (4.1) where E, refers to the energy of the Ith levei. The wave function can be factored into time dependendent and time independent parts; <°> -= )*, (°)exp(-i E, t/h), where (4.2) is the time independent wave function of the unperturbed molecule. Let us assume that the molecule is subjected to electromagnetic radiation of frequency Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 34 v0 = c/x0 propogating in the z-direction, and polarized in the x-z plane. Further assuming the wavelength of the electromagnetic radiation to be large compared to the size of the molecule, the electric field component of the radiation can be considered constant over the molecule. Also, since the speeds of the molecular electrons are small compared to the speed of light, the magnetic field intensity of the radiation field can be ignored. Under these assumptions the only component of the electromagnetic field remaining for consideration is EX=E0 sin2nv0t, so that the total Hamiltonian for the molecule subjected to the radiation is H= H<0>-(E0 sin2jtvot ) I iqixj, (4.3) where the qj and Xj are the charge and x-coordinate of the ith particle in the molecule. The Schrodinger equation for the perturbed molecule is {H<°) -(E0 sin2nvot ) I jqjxj}'i'| = -fi/i (9 */a t). The wave function of the perturbed system, (4.4) can be expanded in terms of the unperturbed wavefunction, (4.5) Substituting this equation into Equation (4.4), multiplying by and integrating the resulting expression over all space, the following differential equations for the expansion coeffecients (fi/i) are obtained: “ (E0 Sin27tv0 ) . | ^ ai j ( f l x |nj exp[i(Em -Ej -hv0)t/h]} where Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.6) 35 I qi Xi (4-7) are the matrix elements of the x-component of the molecule's dipole moment. If at a time t, the molecule is in a state I, a^ =1 and all other a,m=0. During a short time interval the probabability of a transition taking place is so small that the values aM, a^, a^,. .. are negligibly small compared to a« so that Equation (4.6) becomes (h/i) da|m/dt = E0 sin 2ji v0( ^ ) m| exp[i(Em-E,)t/h] ■cEo ( ^ m |( 'i/2)(exPli(Em-El+hvo)t/hI‘ expri(Em*Er hv,o)t/hl} {4>8) Direct integration of this equation gives aim= E0(px)m|(-i/2){[exp[i(Em-El+hvo)t^]-1V(Em-El+hvo) [exp[i(Em-E,-hv0)tm]-1]/ (Em-E,-hv0). (4.9) The quantity |atm|2= a ^ * a ^ is the transition probability for the molecule to go to the state m from the state I during a short time under the influence of polarized light of frequency v0. If |Em-E|U v0, the problem is one of resonance absorption. Equation (4.9) was derived on the basis that the radiation field was monochromatic. In practice this is never the case; so we assume the frequency covers a narrow band from v, to v2 including |Em-E1|/h. i Then by taking the absolute value of the square of Equation (4.9) and integrating from v, to v2, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. is obtained, where the integration variable is ?= (lE^-Ed-hvJt/'h. If the frequency width is large compared to the natural width of the absorption line, not too much error is introduced by letting $(v,) and s(v2) go to plus and minus infinity. Integration then gives (4.11) where I is the intensity of the radiation. The transition probability is thus proportional to the intensity of the incident radiation, the time and to the square of the matrix component of the dipole moment. This relation has been derived for polarized radiation. For an isotropic electromagnetic field the relation above becomes (4.12) where M 2 - l<M*U2+ l<M,>n/+ IW n /' (4.13) For an isolated symmetric top molecule such as ethane the matrix components of the dipole moment vanish and thus absorption of electromagnetic radiation is forbidden according to Equation (4.12). However a colliding pair of molecules may produce a transient dipole during the collision. If the gas is dense enough these collisions can occur with sufficient Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 37 frequency to produce an induced dipole moment time-correlation function owing, primarily, to quadrapole interactions between pairs of molecules. At higher densities one must consider shorter ranging octapole and hexadecapole interactions as well. The dipole moment time correlation function gives rise to an absorption coeffecient per unit path length in a fluid of volume V in thermal equilibrium at temperature T given by21 oc[ uj ) = (4 rr2 w/3hcV)(1-exp(-Bfi<Aj))G(t/o), (4.14) where w is the angular frequency and B'1=kT. The spectral density, G(w), is given by G (w ) = (2n)'1 / exp(-i wt)4>(t)dt, (4.15) where $(t) is the quantum-mechanical correlation function defined by $(t)= (N2/2)<fL(0)-p(t)>, (4.16) where N is the number of molecules contained in V and p. is the pair moment due to bimolecular interactions. For the past few decades the study of molecular absorption in the far-infrared and microwave bands has been a valuable tool in determining molecular intereactions and structure. As an obvious example, the quadrapole and higher order multipoles of several molecules, including CH41( C 0 22, and H23 , have been measured from such data. To our knowledge only two groups have investigated microwave absorption in compressed ethane. Dagg et al, using a microwave frequency of 137 GHz, measured slight absorption that decreased with density in the 20-50 amagat range 23. Birnbaum and Maryott had earlier reported negligible absorption at 24.35 GHz in ethane at 298.15 K in the range of 0- 38 amagat24 To our knowlwdge no one has investigated the absorption of microwaves by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 38 ethane in the 90-100 GHz range. However, the collision induced spectrum of methane exhibits absorption at about 95 GHz.25. We do not have a measure of the relative absorption of microwaves at 92 and 96.2 GHz as we were not equipped to measure this directly. Nonetheless, the observed changes in the decay rates over the range 90-100 GHz indicate an absorption of microwaves at these two frequencies. Mechanisms for Microwave-Induced Reductions in > , Pr Rotation ally Excited Molecules It is possible that the absorbed microwave radiation contributes to the rotational degrees of freedom of the ethane molecules. Quantum mechanically, the probability of positronium undergoing pick off annihilation within a molecule's electron cloud is proportional to the value of the overlap integral defined as / l * where vPnim and n J 2 l * n J 2 < * - <4 - 1 7 > are the wave functions of positronium and the molecule respectively. A simple model for investigating the effect of rotationally excited molecules on the annihilation rate of o-Ps assumes the molecule to be hydrogen-like. Evaluation of the overlap integrals of the ground state of orthopositronium with the hydrogen atom in different I states shows that as the hydrogen is excited to higher rotational states the value of the overlap integral and hence, the annihilation rate, decreases. Thus o-Ps is expected to have a longer lifetime in the fields of rotationally excited molecules. Assuming that the microwave energy absorbed at 92.0 and 96.2 GHz., as seen in Figure Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 39 4-1, raises the ethane molecule to a rotationally excited state, we obtain an estimate for the quantum number J and the moment of inertia, I, by using, v ■ 2B(J+1) -4D(J+1)3, (4.18) where v is the frequency of the absorbed radiation, B=h/(8n2 I) is the rotational constant for the ethane molecule with a moment of inertia I, and D is the centrifugal stretching constant whose estimated value for ethane is „10® Hz26. Using Equation (4.17) and the two absorption frequencies from Figure 4-1 we obtain J = 15, and l= 2.4 x10'45 kg m2. The moment of inertia is in close agreement with that of 1= 0.422 x 10*43 kg m2 reported by Dagg, Smith, and Read23. There are no data available for microwave absorption in ethane in the 90-100 GHz range. However, transitions between such high quantum numbers have been reported for methane 25. Density Fluctuation Excitations Measurements of in ethane, methane, nitrogen, and carbon dioxide as functions of density and temperature have shown positronium to be localized in density fluctuations. The density fluctuation model relates the difference between the measured orthopositronium decay rate and the decay rate predicted by Equation (3-1) to the isothermal compressibility of the fluid according to Equation (3.20), Ax0.Ps“4"ro2crV)1zeffD{kTKo/2,tVo)1/2' where K0=\T^ (dN/dP)x is the isothermal compressibility of the fluid. Positronium decay Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4‘19) 40 rates can therefore provide information about the equilibrium properties of a fluid. At temperatures close to the liquid-vapor equilibrium point, the isothermal compressibility of the fluid becomes infinitely large and, therefore, highly succeptable to external pertubations. In this analaysis an analogy is made between the highly compressable fluid subjected to microwaves and the classic harmonic oscillator set into motion by an external driving force. In the latter case the vibration of a system of particles is described by a linear combination of normal modes with each normal oscillation occurring at the frequency of the driving force. In the following we consider the possibility that the changes seen in the decay rates of orthopositronium in ethane subjected to microwaves are a result of the response of the fluid's compressibility to the incident radiation. In a simple fluid, the Ornstein-Zernike form of the susceptibility (isothermal compressibility) is given by K (q )« C p /(U $ 2 q2 ), (4.20) where Cp is the specific heat at constant pressure, $ is the correlation length and q is the wavenumber of the scattered radiation. The wavenumber is given by q(0) =(47in/>.) sin(0/2), (4.21) where n is the index of refraction, > is the wavelength of the incident radiation, and 0 is the scattering angle. Relating the positronium decay rates to the wave number dependence of the compressibility by combining Equations (4.19) and (4.20), we obtain K (q)« A * 20_Psa Cp/(1+$2 q2 ). Therefore, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.22) 41 AVPs'2 - ( 0/Cp)(1+*2q2)’ where a is a constant. The measured values of a v (4'23) * , 2 versus frequency at 98 amagat are plotted in Figures 4-2 and 4-3. The curves are obtained by a computer program that fits the data to a Lorentzian peak. These data refelect sharp increases in the fluid’s compressibility at 92.0 and 96.2 GHz. Following Equation (4.23), A x ^ '2 is plotted against q2 at 98 amagat in Figures 4-4 and 4-5. An average value of q =4nA , obtained by integrating q(6) over all possible scattering angles, was assumed for these calculations. Also the index of refraction, n, was assumed to be 1. These data are clearly in agreement with Equation (4.23). The straight line is obtained from a linear least squares fit of the data. From the absolute values of the slopes and intercepts of these data a correlation length $=(m/b)1/2 is obtained; m and b are the slope and intercepts of the straight lines given by Equation (4.23). From the four straight line segments in Figures 4-4 and 4-5 we obtain an average value of ?=(2.5 ± 0.1) x106 A . This correlation length is considerably larger than the expected value of a few Angstroms for correlation lengths measured in fluids such as CO2 in light scattering experiments27. The reason for this discrepency is not immediately clear. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 98 amagat 3.5 or ■ o © V) 2.5 « a. ©I o /C < 90.9 91.9 92.9 Frequency (GHz) Figure 4-2. & \ 20+ s versus frequency in ethane at 98 amagat and 307.15 K for the peak centered at 92.0 GHz. The solid curve is hand drawn to guide the eye. ■rw ro Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.5 - 2.5 - 98 amagat cT• o s n ■o ~ M a ■ CM O /< < 1.5 95.95 95.55 96.35 Frequency (G Hz) Figure 4-3. versus frequency in ethane at 98 amagat and 307.15 K for the peak centered at 96.2 GHz. The solid curve is hand drawn to guide the eye. CO Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 98 amagat « o ® n e co o T « a CM 1.47 1.45 1.49 q2 (10 3cm * 2 Figure 4-4. a* 1.51 ) -2^ , versus q2 in ethane at 98 amagat and 307.15 K for the peak centered at 92.0 GHz. The solid lines represent the results of a weighted least-squares fit. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CM O © co c M O CO a Vc < 98 amagat 1.61 1.62 1.63 1.64 q 2 (10 3cm * 2) Figure 4-5. versus q2 in ethane at 98 amagat and 307.15 K for the peak centered at 96.2 GHz. The solid lines represent the results of a weighted least-squares fit. cn 46 Orthopositronium Annihilation Rate Behavior at 99.60. and 37 Amaaat In the following, data similar to that shown for 98 amagat are presented for ethane densities of 99,60, and 37 amagat at 307.15 K. For each of these densities, although not as clearly, the annihilation rates seemed to exhibit "dips" at certain density-dependent values of the microwave frequency. 60 Amaaat Figure 4-6 shows V p s as a function of frequency at 60 amagat and 307.15 K. The data were taken in order of increasing microwave frequency over a period of two days. A dip in the decay rates is suggested at 94.0 GHz. Figure 4-7 represents data taken over the following week in an attempt to duplicate the data in Fig 4-6. Once again a characteristic dip in the annihilation rate is suggested, but the dip at 94.0 GHz in Figure 4-6 has shifted to 93.5 GHz in Figure 4-7. These data were collected with the microwave generator in the continuous wave (CW) mode where the RF power is essentially monochromatic having a small band width of only ±0.00038% of the selected frequency. The results using microwaves in the CF mode, at 98 amagat, seem to give better resolution of the dips than when operated in the CW mode at 60 amagat. This might be attributable to the fact that the transition probability is proportional to the integral in Equation (4.10) which decreases with decreasing band width. Figure 4-8 shows versus frequency at 60 amagat for the five frequencies centered around and including 94.0 GHz. The curve is a computer fit of the data to a Lorentzian. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.034 60 amagat 0.032 .o © 0) c m a 0.030 0.028 i 89 91 93 95 97 99 Frequency (GHz) Figure 4-6. versus frequency in ethane at 60 amagat and 307.15 K. The solid curve is hand drawn to guide the eye. The dotted horizontal line represents the decay rate with no microwaves. 101 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.034 - 60 amagat o ® » c n a 0.030 _L 0.026 88 90 92 94 96 98 100 Frequency Figure 4-7. versus frequency in ethane at 60 amagat and 307.15 K. The solid curve is hand drawn to guide the eye. These data were collected in the week following the data in Figure. 4-6. The dotted horizontal line represents the decay rate with no microwaves. 03 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.0 60 amagat 0.8 - 0.6 _ 0.4 _ CM i O © CO in c ■o n Q. CM O 0.2 _ 0.0 93 92 94 95 96 Frequency (GHz) Figure 4-8. a * 20-p* versus frequency in ethane at 60 amagat and 307.15 K for the peak centered at 94.0 GHz shown in Figure 4-6. The solid curve is hand drawn to guide the eye. -U CO In Figure 4-9 A V p / 2 is plotted against q2 for the data shown in Figure 4-8. The straight lines are linear least square fits to the data. From the slopes and intercepts of the two lines an average value of the correlation length $=(2.5 ± 0.05) x106 A is obtained. Once again this is much larger than expected, but it is in excellent agreement with the correlation length obtained from the 98 amagat data. 37 Amaoat The frequency dependance of V p s for 37 amagat at 307.15 K exhibited dips at three frequencies; 93.0,95.5, and 99.0 GHz as shown in Figure 4-10. The horizontal dashed line in the figure represents the measured decay rate of 25.70 ±0.86 ps'1 with no microwaves. This is close to the value of 26.1 ps*1 predicted by the linear Equation (3-1) which indicates the density to be near the point, below which localization of o-Ps will not occur. Figures 4-11 and 4-12 show A \0_ps2 versus frequency for the dips centered at 93.0 and 95.5 GHz. The solid curves are Lorentzians fitted to the data. The dip at 99.0 GHz did not provide a good fit to a Lorentzian. When subjected to the same analysis as for the 98,60, and 99 amagat the data for 37 amagat gave poor results with regard to showing the dips to be Lorentzian and in determining correlation lengths. 99 Amaaat Figure 4-13 shows V p s versus frequency at 99 amagat and 307.15 K. We consider the data at this density as distinct from those at 98 amagat because of the differences in pressure Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.0 60 amagat CM O 3.0 © « e o 0) CL CM o ’/< 2.0 < A 1.0 1.50 1.52 A 1.54 A 1.56 1.58 1.60 q 2 (10 3cm ‘ 2 ) Figure 4-9. a * " 2o-p * versus q2 in ethane at 60 amagat and 307.15 K for the peak centered at 94.0 GHz shown in Figure 4-6. The solid lines represent the results of a weighted leastsquares fit. cn 52 readings at the times each experiment was run. The pressure reading taken at the time the vessel was sealed prior to collecting the data for 98 amagat was 695.0 PSIG. After collecting the data ethane was removed from the vessel for a series of runs made at a lower density. After this, ethane was readmitted to the vessel in an attempt to get as close as possible to 98 amagat again. The resulting pressure turned out to be 696.5 PSIG. Within the precision of the Heisse Gauge, there is signifigant difference between the two pressures measured and therefore, the densities calculated are distinct. The data in Figure 4-13 shows a "closeup" of a dip in the decay rate centered at 96.4 GHz. Since 99 amagat is very close to 98 amagat one would expect that this is the same dip as the one exhibited at 96.2 GHz in the 98 amagat data. Assuming this to be the case, the shift to the higher frequency reflects a decrease with the density either in the molecule’s moment of inertia or in its distortion constant according to Equation (4.4). In Figure 4-14 A>0_ps2 is plotted against the frequency. The curve is a Lorentzian fit to the data. Figure 4-15 shows A * ^ ' 2 for this density. From the slopes and intercepts of the straight line segments a correlation length, £=(2.48 ± 0.02) x106 A, is obtained. Which is, once again, in excellent agreement with those obtained at 98 and 60 amagat, but is much larger than expected. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 37 amagat 0.028 .o © n c « CL 0.024 0.020 89 91 93 95 97 99 101 Frequency (GHz) Figure 4-10. versus frequency in ethane at 37 amagat and 307.15 K. The solid curve is hand drawn to guide the eye. The dotted horizontal line represents the decay rate with no microwaves. cn CO Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 37 amagat CM O I 94.5 Frequency (GHz) Figure 4-11. A>>20^ , versus frequency in ethane at 37 amagat and 307.15 K for the peak centered at 93.0 GHz shown in Figure 4-10. These data provide a poor lorentzian fit. C7I Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 25 - 37 amagat CM •o ffl 0) c 15 - o> a ■ CM O 5 - 4 -5 93 94 95 96 Frequency (GHz) Figure 4-12. a x 20_j>, versus frequency in ethane at 37 amagat and 307.15 K for the peak centered at 95.5 GHz shown in Figure 4-10. These data provide a poor lorentzian fit. Ol cn Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.044 ■u ® V) c « Q. 0.042 99 amagat 0.040 95.9 96.1 96.3 96.5 96.7 96.9 Frequency (GHz) Figure 4-13. >>0 ^ , versus frequency in ethane at 99 amagat and 307.15 K. The solid curve is hand drawn to guide the eye. The dotted horizontal line represents the decay rate with no microwaves. Ol o> Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.3 99 amagat e* 'o 2.8 © 0) C i o <i 0) Q. • <M O 2.3 X 1.8 95.8 9G.0 X X 96.2 96.4 X 96.6 96.8 97.0 Frequency (GHz.) Figure 4-14. a >,20_j>s versus frequency in ethane at 99 amagat and 307.15 K for the data shown in Figure 4-10. The solid cun/e is hand drawn to guide the eye. tn Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.0 do © CO c o o CO CL 4.0 V 99 amagat 3.0 i 1.61 1 . 62 1.63 1.64 1.65 q 2 (10 3cm ' 2 ) Figure 4-15. a >t 2o_p, versus q2 in ethane at 99 amagat and 307.15 K for the data presented in Figure 4-14. The solid lines represent the results of a weighted least-squares fit. cn CO CHAPTER V CONCLUSIONS The experiment performed for this thesis provided for observations of the changes in the decay rates of o-Ps in fluid ethane subjected to microwave radiation. The sharp dips seen in the decay rates at certain frequencies were totally new and unexpected. These resonant-like dips indicate some absorption of the microwaves incident upon the ethane molecules. This absorption is described in terms of transient dipoles induced through quadrapole-quadrapole interactions during collisions between molecules. Two possible mechanisms for the decrease in resulting from the absorption of microwave energy are proposed. The first describes the decrease in V p s as being due to the interaction between the localized positronium and rotationally excited molecules, i. e. The overlap integral between the o-Ps atom and a molecule in the state J=1 is smaller than when the molecule is in the state J+1. Applying this idea to the data for ethane at 98 amagat and 307.4 K show that the transitions in the 90 to 100 GHz range occur in the neighborhood of J=14. Also the moment of inertia for the ethane molecule was calculated to be 1= 2.4 x10‘45 kg m2, in close agreement with published values. The second mechanism proposed describes the frequency dependent dips in >0_ps in terms of density fluctuation excitations which involve the response of the fluid's compressibility to the microwave field at some resonant frequency. The dips in the annihilation rates thus 59 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 60 reflect sharp increases in the compressibility at certain microwave frequencies. The peaks observed in the 98,60, and 99 amagat were shown to be Lorentzians centered about the resonant frequency. The fit for the peaks at 37 amagat was not as good. In this analysis the density fluctuation model for localization of o-Ps in ethane relates the incident radiation to the Ornstein-Zernike form of the isothermal compressibility of the fluid. This relation predicts that, near the resonant frequency, a v p ,'2 is proportional to the the square of the frequency. The data for all densities investigated, except at 37 amagat, did yield this result. Furthermore the slopes of the straight lines from the data gave correlation lengths which were all in the range of 2.45 to 2.55 x 106 A. The consistency of the calculated correlation lengths is satisfying but we have yet to reconcile the discrepency between the value of around a million Angstroms measured in this experiment and published values of a few Angstroms in other non-polar gases. The data obtained at densities other than 98 amagat suggest problems in controlling the density of the gas. It is hoped that increased accuracy in the the control of the pressure, temperature and hence, the density, will provide future researchers with more convincing evidence for microwave induced density fluctuations. If such is the case Ps may prove to to be an effective probe of the molecular dynamics of fluids. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. REFERENCES 1. K. F. Canter, J. D. McNutt, and L. 0 . Roellig, Phys. Rev. A 12,375 (1975) 2. E. M. Juengerman, R. H. Arganbright, M. H. Ward, and S. C. Sharma, J. Phys. B: At. Mol. Phys. 2Q, 867 (1987) 3. E. M. Juengerman, Masters Thesis, Univ. of Texas at Arlington, 1982 (Unpublished) 4. S. C. Sharma, A. Eftekhari, and J. D. McNutt, Phys. Rev. Lett. 45, 953 (1982) 5. R. J. Drachman and S. K. Houston, J. Phys. B: At. Mol. Phys. 2.1657 (1970) 6. E. Bruche, Ann. Phys. 4 , 387 (1930) 7. J. D. McNutt, S. C. Sharma, M. H. Franklin, and M. A. Woodall II, Phys. Rev. A 2<2.357 (1979) 8. L. L. Tankersley, J. Low Temp. Phys. H , 451 (1973) 9. P. Hautojarvi, K. Rytsola, P. Tuovinen, and P. Jauho, Phys. Lett. 57A. 175 (1976) 10. J.P. Hernandez, Phys. Rev. A 14,1579 (1976) 11. I. T. lakubov and A. G. Khrapak, Rep. Prog. Phys. 4 5 , 697 (1982) 12. S. C. Sharma, A. Eftekhari, J. D. McNutt, and R. A. Hejl, J. Chem. Phys. Z 5 , 1226 (1981) 14. J. D. McNutt and S. C. Sharma, J. Chem. Phys. 68.130 (1978) 15. T. Kawatarani, Y. Nakayama, andT. Mizogawa, Phys. Lett. A 122,75 (1985) 16. G. L. Wright, M. Charlton, G. Clark, T. C. Griffith, and G. R. Heyland, J. Phys. B: At. Mol. Phys. 12,4065 (1983) 17. P. Kirkegaard and M. Eldrup, Comput. Phys. Commun.2,401 (1974) 18. M. Benedict, G. B. Webb, and L. C. Rubin, Chem. Eng. Prog. U47U, 419 (1951) 19. D. A. Gyrog and E. F. Ohert, AlchE J. IQ, 621 (1964) 20. A. Eftekhari, Ph.D Dissertation, Univ. of Texas at Arlington, 1982 (Unpublished) 61 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 62 21. G. Birnbaum. Intermolecular Spectroscopy and Dynamical Proprties of Dense Systems. Proceddinas of the International School of Phvsics. Enrico Fermi. (1978) 22. G. Birnbaum, J. Chem. Phys. £2,59 (1975) 23. I. R. Dagg, W. Smith, and L. A. A. Read, Can. J. Phys. £Q, 16 (1982) 24. A. A. Maryott and G. Birnbaum, J. Chem. Phys. 3£, 2026 (1962) 25. I. Ozler, M. C. L. Gerry, and A. G. Roblette, J. Phys. Chem. Ref. Data IQ, 1085 (1981) 26. C. H. Townes and A. L. Schawlow, Microwave Spectroscopy. McGraw-Hill. New York, (1959), p. 78 27. H. L. Swinney and D. L Henry, Phys. Rev. A £ , 2586 (1973) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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