This is to certify that the thesis entitled STUDIES IN MOLECULAR SPECTROSCOPY: I. INFRARED LASER STARK SPECTRUM OF METHYLACETYLENE; II. DETERMINATION OF ROTATIONAL RELAXATION PARAMETERS FOR OCS IN H2, C02 and CH3F; III. MICROWAVE SPECTRUM OF ISOPROPENYLCYCLOPROPANE presented by Patricia M. Thrash has been accepted towards fulfillment of the requirements for Ph 1) degree in LhwisLmL. Qfl/Waw Major professor Date July 23, 1979 0-7639 OVERDUE FINES ARE 25¢ PER DAY ‘ PER ITEM Return to book drop to remove this checkout from your record. STUDIES IN MOLECULAR SPECTROSCOPY: I. INFRARED LASER STARK SPECTRUM OF METHYLACETYLENE II. DETERMINATION OF ROTATIONAL RELAXATION PARAMETERS FOR OCS IN H2, 002 AND CH3F III. MICROWAVE SPECTRUM OF ISOPROPENYLCYCLOPROPANE By Patricia M. Thrash A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemistry 1979 ABSTRACT STUDIES IN MOLECULAR SPECTROSCOPY: I. INFRARED LASER STARK SPECTRUM OF METHYLACETYLENE II. DETERMINATION OF ROTATIONAL RELAXATION PARAMETERS FOR OCS IN H2, CO2 and CH3F III. MICROWAVE SPECTRUM OF ISOPROPENYLCYCLOPROPANE By Patricia M. Thrash The OS band Of methylacetylene was studied by microwave and laser Stark spectroscopy. The rotational constant B5 and the centrifugal distortion constants DEE) and D§5> were determined from the microwave spectrum to be 8508.119 10.003 MHz, 169:1 kHz, and 1.8t0.2 kHz, respectively. The dipole moments for the ground and 05 excited states were also determined from the microwave spectrum to be p0 = 0.783910.0010 D and “5 = 0.795u:0.0010 D. The laser Stark spectrum Of methylacetylene was Obtained by using C02 and N20 lasers. The molecular parameters were obtained from least squares fit Of the assigned laser Stark frequencies. The molecular constants Obtained include the band center v0 = 930.27H9:0.0008 cm-l; the difference in the A rota- tional constants for the v5 excited vibrational state and ground state,A5-A0 = -206i7 MHz; the corresponding Patricia M. Thrash difference in the centrifugal distortion constants, Dé5)-D§O) = H.810.7 MHz, and the difference in the dipole moments, u5—uo = 0.0110:0.0006 D. Microwave lineshapes have been studied under condi— tions of low and moderate incident power for dilute gaseous samples of OCS in H2, C02, and CH3F. The J = 2 + l transi- tion near 2N326 MHz was studied for each of the samples. From the low—power lineshapes, values of the linewidth parameters, Av/p, were obtained as follows: OCS/H2, 6.07:0.09 MHz/torr; OCS/C02, 5.26:0.08 MHz/torr; and OCS/ CH3F, ll.U5:0.l7 MHz/torr. The power-broadened lineshapes were analyzed in two ways. By fitting the lines to a super- position Of power-broadened M components values Of (Tl/T2)O = 1.05:0.10, 1.03:0.10, and 0.97:0.10 were obtained for OCS in H2, CO2, and CH3F, respectively. By fitting the line- shapes to a single Lorentzian, corresponding values Of q Tl/T2 = 1.3210.l3, l.3010.l3,and 1.23:0.12 were Obtained. The microwave spectrum of isopropenylcyclopropane was studied in the 18-U0 GHz region. Transitions were assigned to a species with rotational constants A = 6287.089 $0.600 MHz, B = 26A7.222i0.028 MHz, and C = 2235.061:0.02u MHz. Comparison of these constants with values calculated from assumed structures of isopropenylcyclopropane with a variety of torsional orientations Of the isopropenyl group relative to the cyclopropane ring showed that the rotational constants of a single species are not sensitive enough to the torsional angle to determine the torsional conformation Patricia M. Thrash responsible for the assigned species. NO assignment for another species could be Obtained in the very weak micro- wave spectrum of this compound. TO Bob 11 ACKNOWLEDGMENTS I would like to thank my advisor, Professor Richard Schwendeman for his friendship and guidance throughout the course of these studies. I wish to thank my parents for their support and en- couragement. The financial support of the National Science Founda- tion is gratefully acknowledged. iii TABLE OF CONTENTS Chapter Page LIST OF TABLES. o o o o o o o o o o o o o o o o o o Viii LIST OF FIGURES o o o o o o o o o o o o o o o o o o XiV PART I. INFRARED LASER STARK SPECTRUM OF METHYLACETYLENE. . . . . . . . . . . . CHAPTER I. INTRODUCTION. . . . . . . . . . . CHAPTER II. THEORY . . . . . . . . . . . . . . . The Molecular Hamiltonian . . . . . . . . . . Rotational Energy — Rigid Rotor . . . . . . . . \OQWWI-‘H Angular Momenta . . . . . . . . . . . . . . . Centrifugal Distortion. . . . . . . . . . . . . 1U Stark Effect. . . . . . . . . . . . . . . . . . 15 Selection Rules . . . . . . . . . . . . . . . . 22 Dipole Moments From Microwave Spectra . . . . . . . . . . . . . . . . . . . . 23 Vibration-Rotation Spectra. . . . . . . . . . . 26 Laser Stark Spectroscopy. . . . . . . . . . . . 32 Lamb Dip o o o o o o o o o o o o o o o o o o o o u 3 CHAPTER III. EXPERIMENTAL. . . . . . . . . . . . . H7 Methylacetylene . . . . . . . . . . . . . . . . A7 Microwave Experiment. . . . . . . . . . . . . . H7 Equipment . . . . . . . . . . . . . . . . . N7 Procedure . . . . . . . . . . . . . . . . . “8 Ground State Methylacetylene Dipole Moment . . . . . . . . . . . . . H8 Excited State Methylacetylene Dipole Moment . . . . . . . . . . . . . “8 Laser Stark Experiment. . . . . . . . . . . . . H9 Equipment . . . . . . . . . . . . . . . . . ug Stark Electrodes. . . . . . . . . . . . M9 iv Chapter Page Sample Cell . . . . . . . . . . . . . . 51 Assembly Of the Sample C611. 0 o o o o o o o o o o o o o o o o 51 Electronics . . . . . . . . . . . . . . 52 Polarization of Laser Light 0 O 0 0 O O O O O O O O O O O O O 55 Procedure . . . . . . . . . . . . . . . . . 55 Alignment Of the 002 Laser Beam Through the Sample Cell. . . . . . 55 Calibration of the Plate Spacing . . . . . . . . . . . . . . . . 58 CHAPTER IV. RESULTS AND DISCUSSION . . . . . . . . 63 Microwave Spectrum. . . . . . . . . . . . . . . 63 Ground State Rotational Constant, 6 B o o o o o o o o o o o o o o o o o o o o o 3 vs Excited State Rotational Constant, B'. . . . . . . . . . . . . . . . 63 Ground State Methylacetylene Dipole Moment . . . . . . . . . . . . . . . 68 The Dipole Moment of Methyl- acetylene in the v5 Excited State 0 o o o o o o o o o o o o o o o o o o 73 Laser Stark Spectrum. . . . . . . . . . . . . . 76 Calculation of the Laser Stark spectrum. 0 O O O C O I O O O O O O 0 0 O O 76 Observed Laser Stark Spectrum . . . . . . . 82 PART II. DETERMINATION OF ROTATIONAL RELAXATION PARAMETERS FOR OCS IN H2, 002, CH3F. O O O O O O O O O O O O O 0 O O O 0 914 CHAPTER V. INTRODUCTION. . . . . . . . . . . . . . 9“ CHAPTER VI. THEORY . . . . . . . . . . . . . . . . 97 Interaction Of Radiation with the Sample. . . . . . . . . . . . . . . . . . . 97 Relation Between the Absorption Coefficient and Pi' . . . . . . . . . . . . . . 99 Chapter Page The Optical Bloch Equations . . . . . . . . . . 103 Meaning of T1 and T2. . . . . . . . . . . . . . 106 Steady State Solutions to the Optical Bloch Equations . . . . . . . . . . . . 111 Low Power . . . . . . . . . . . . . . . . . 113 Moderate Power. . . . . . . . . . . . . . . llu Microwave Power . . . . . . . . . . . . . . . . 119 The Natural Linewidth . . . . . . . . . . . . . 12A Broadening by Collisions with the Walls . . . . . . . . . . . . . . . . . . . 125 Doppler Broadening. . . . . . . . . . . . . . . 126 CHAPTER VII. EXPERIMENTAL. . . . . . . . . . . . . 131 Sample. . . . . . . . . . . . . . . . . . . . . 131 Pressure Meter. . . . . . . . . . . . . . . . . 131 Microwave Spectrometer. . . . . . . . . . . . . 132 Experimental Procedure. . . . . . . . . . . . . 13A CHAPTER VIII. RESULTS. . . . . . . . . . . . . . . 136 Low Power Linewidths. . . . . . . . . . . . . . 136 Power Broadened Linewidths. . . . . . . . . . . 1A1 CHAPTER IX. DISCUSSION . . . . . . . . . . . . . . 150 PART III. MICROWAVE SPECTRUM OF ISO- PROPENYLCYCLOPROPANE . . . . . . . . . . 156 CHAPTER X 0 O O O O O O O O 0 O O I 0 O 0 O 0 O O O 156 Introduction. . . . . . . . . . . . . . . . . . 156 Theory. . . . . . . . . . . . . . . . . . . . . 157 Experimental, Results, and Discussion. . . . . . . . . . . . . . . . . . . 153 APPENDIX A. THE VOLTAGE VARIATION OF THE STARK FIELD IN THE MICROWAVE SAMPLE CELL. . . . . . . . . . . . . . 171 Low Fields. . . . . . . . . . . . . . . . . . . 171 vi Chapter High Fields . . APPENDIX B. TRANSITIONS. APPENDIX C. THE CO2 Introduction. . . . . Theory of the CO2 Laser . Description of the COZ/N2O Laser Used in this Study. . Alignment of the Laser at 10.6 n. Operation of the CO2 Laser. . Operation of the N20 Laser. Laser Stabilization . REFERENCES. . . . . vii LINEWIDTH OF THE LASER STARK AND N20 GAS LASERS . Page 17“ 176 179 179 180 183 189 190 192 193 197 Table LIST OF TABLES Molecules which have been studied by laser Stark spectroscopy Contribution Of the direction cosine matrix elements to the intensity of a vibration-rotation transition in a symmetric top molecule. All Of the intensity contributions shown are for transitions in which AK = 0. values for J and M are those of the lower state . . . . . . . . . . . The resonant voltages of several transitions Of methylfluoride mea— sured by Freund, gt a1, These transitions were used to calibrate the plate spacing. For these transitions AK = 0. . . . . . . The calculated ground state B rotational constant of methyl- acetylene . . . . . . . . . . . . Microwave transitions of methyl— acetylene. All these transitions are J = 2 + l and AK = 0. . . . . viii Page 35 42 60 6A 65 Table 10 11 Page Intensities of the rotational transi- tions in the vibrational excited states relative to the ground state . . . . . . . . 67 Frequencies (MHz) of M components Of the J = 3 + 2 transition in OCS measured as a function of dial setting on the H. P. spectrometer . . . . . 69 Frequencies (MHz) Of M components of the J = 2 + 1 transition in CH3CECH measured as a function of dial setting on the H. P. spectrometer. . . . . . . . . . . . . . . . 70 Calculated value of the electric field determined from the OCS second order Stark shifts . . . . . . . . . 72 First-order Stark effect fre- quencies (MHz) for the ground state and the v5 excited state of the kil, mil and ktl, m¥l com- ponents Of the J = 2 + 1 transitions Of methylacetylene. . . . . . . . . . . . . 7A The corrected voltages versus the first-order Stark effect frequencies of the ground state and v5 excited state of the J = 2 + 1 transition ix Table 12 13 14 15 Of methylacetylene fit to a straight line by the method of least squares The eleven parameters that can be fit from the laser Stark transitions and the derivatives of the frequency with respect tO these parameters. . . . Approximate fields Of the Observed transitions between the v5 excited state and the ground state of methyl- acetylene. The fields in parenthesis are calculated fields where a "strong" transition is to occur. The calculation includes J values up to J = 10. Transi- tions for which accurate fields have been measured and assignments made are listed in Table I” . . . . . . . . . . Assigned transitions between the v5 excited vibrational state and the ground state Of methylacetylene . . . Rotation-vibration parameters Of CH3CECH. In columns I and II are the parameters obtained from the laser Stark experiments. The parameters in column I were obtained by letting to, AA, ADK, Page 77 80 87 88 Table l6 17 18 Page and u' vary in the fitting routine while those in column II were obtained by varying v0, AA, ADK, u' and u". The primes refer to the v5 vibrational state while the double primes refer to the ground vibrational state. . . . . . . . . . 90 Comparison of the difference between the experimental and calculated fre— quencies of the methylacetylene transitions for the two cases in Table 15. . . . . . . . . . . . . . . . . . 91 Average linewidth and pressure for the J=2+1 transition of OCS in CH3F, CO2 or H2. Corrections have been made for the effects of temperature variation and the addition Of air . . . . . . . . . . . . 139 Linewidth parameters and line- widths due to wall collisions for the J=2+l transition Of OCS in a foreign gas. The slope is from the plot of linewidth versus pressure. It contains the line- width parameter due to OCS-OCS collisions and that due to xi Table 19 2O 21 22 23 Page collisions of OCS with the foreign gas. The column (Av/p) contains only the linewidth parameter due to OCS-foreign gas collisions . . . . . . . 1A2 Average values for (Tl/T2)o and (q Tl/T2) measured for the J=l+2 transition of OCS mixed with various foreign gases. All Of the parameters were Obtained at zero Stark field . . . . . . . . . . . . 1A8 Comparison Of the rotational relaxation parameters Of OCS in a mixture of OCS and CH3F found by transient effect, mol- ecular beam, and linewidth measure- ments . . . . . . . . . . . . . . . . . . . 15A Selection rules Of the asymmetric rotor for permitted changes in the values Of K_1 and K+1 . . . . . . . . . . . 16“ Comparison Of observed and calculated frequencies of isopropenylcyclo- propane . . . . . . . . . . . . . . . . . . 167 Rotational constants and centrifu- gal distortion constants of iso- propenylcyclopropane. . . . . . . . . . . . 168 xii Table Page Least squares fit to a straight line of the dial setting versus the first-order Stark effect fre- quencies of the ground state and V5 excited state of the J=2+l transition Of methylacetylene . . . . . . . 172 Voltages Of the first-order Stark effect of the J=2+1 transition of the ground and v5 excited states Of methylacetylene. The voltages have been corrected to include the Offset Of the dial reading. . . . . . . . . 173 List of the C02 laser lines Of the 001-100 band used in this experiment. Their frequencies are given in wave— numbers and megahertz . . . . . . . . . . . 18“ List Of the N20 laser lines used in this experiment. Their frequencies are given in megahertz and wave- numbers . . . . . . . . . . . . . . . . . . 185 xiii LIST OF FIGURES Figure Page 1 Plot of the rotational energy levels Of the prolate symmetric top, methyl- acetylene . . . . . . . . . . . . . . . . . 13 2 Plot Of the transitions Of the parallel band V5 in methylacetylene. The lines can be separated into sets; one for each value of K. Figure 2a shows the sets K=0 through K=A. In Figure 2b these sets are plotted together. . . . . . . . . . . . . . . . . . 31 3 A schematic plot Of the energy versus electric field for the J=l energy levels Of the ground state and an excited vibrational state Of a symmetric top molecule. The transitions shown are for the selection rules AJ=0, Am=-l. The length of the solid vertical lines correspond to the laser frequency . . . . . . . . . . . . . . 37 u Lamb dip, centered at the transition frequency v0. . . . . . . . . . . . . . . . A6 5 Block diagram of the laser Stark spectrometer. . . . . . . . . . . . . . . . 50 xiv Figure 10 11 Page Schematic diagram of the circuit used to protect the zero-crossing detector. The ten numbered resistors are 200 k9 each . . . . . . . . . . . . . . 53 The appropriate mirror configuration to rotate by 90° the electric vector of the laser radiation. The two cubes, shown by the solid and dotted lines lie directly above one another. The angles a, b, c, and d are each “5°. The arrows show the direction of the beam. . . . . . . . . . . . . . . . . . . . 56 Arrangement of the optics for the laser Stark spectrometer. . . . . . . . . . 57 Arrangement Of the optics for Lamb dip spectroscopy. . . . . . . . . . . . . . 61 Lamb dips for the J=l+l, k=1+l, m=-l+0 and m=0+-l transitions of CH3F. . . . . . . . . . . . . . . . . . . . 62 Observed laser Stark absorption lines of methylacetylene. Both transitions were shifted into resonance with the P(32) line Of the 001-100 band Of the C02 laser. The assignments of the transitions XV Figure 12 13 1A 15 16 are J=5+u, k=3, and Am=+l. For the upper trace m"=-3 while for the lower trace m"=- . . . . . . . . . . . . A Q-branch doublet of methylacetylene. The transitions are J=l, k=1, and Am=-l, with m"=0 for the lower field transition and m"=l for the higher field transition. These transitions are in resonance with the P(10) line Of the N20 laser. . . . . . . . . Rectangular hollow metal pipe wave- guide . . . . . . . . . . . . . . . . . Plot Of pressure versus time for the linewidth measurements of OCS in CH3F at a pressure Of 3A microns . . . . . . . . . . . . . . . . Plot Of effective power versus pres- sure for OCS mixed with CH3F. Three linewidths were analyzed at each pres- sure. All three values for the effec- tive power are shown. The bars indicate the size of the standard deviation of the fitted parameter . . . Plot of effective power versus pres- sure for OCS mixed with 002. Three xvi Page 8” 85 120 138 1H5 Figure 17 l8 l9 linewidths were analyzed at each pressure. All three values for the effective power are shown. The bars indicate the size Of the standard deviation of the fitted parameter . . Plot of effective power versus pressure for OCS mixed with H2. Three linewidths were analyzed at each pressure. All three values for the effective power are shown. The bars indicate the size of the standard deviation of the fitted parameter . . . . . . . . . . Possible structure for iso- propenylcyclopropane as de- termined from bond distances and bond angles of propylene and methylcyclopropane. . . . . . . . . . Rotational constants versus di- hedral angle Of isopropenylcyclo- propane. The horizontal dotted lines in the figure are the rota- tional constants Obtained from the fit Of the Observed transitions . . . . . . xvii Page 1A6 1&7 166 170 Figure 0-3 Page Diagram of the CO2 and N vibrational 2 energy levels which are of interest for the C02 laser. The energy dif- ference labeled A is W18 cm'l. The laser transitions corresponding to label B are centered around 961.0 cm-1 and those corresponding to label C are centered around 1063.8 cm"1 . . . . . . . . 181 Schematic diagram Of the CO2/N20 laser used in this study. . . . . . . . . . 186 Variation of the laser power with laser frequency for a single longi- tudinal mode. . . . . . . . . . . . . . . . 195 xviii PART I INFRARED LASER STARK SPECTRUM OF METHYLACETYLENE CHAPTER I INTRODUCTION Methylacetylene is a prolate symmetric top molecule with C3v symmetry. It has 15 normal modes: 5 nonde- generate vibrations and 5 doubly degenerate vibrations. The rotational structure of the v5 band falls in the same frequency range as the CO2 and N20 laser transitions. The v5 vibration is primarily the C-C stretch of the methylacetylene molecule. Methylacetylene has been studied both by microwave and infrared spectroscopy. Gordy and coworkers Obtained the B rotational constant and the DJ and DJK centrifugal distortion constants of the ground state.l'3 The ground state dipole moment was measured accurately by Muenter and Laurie.“ Dubrulle and coworkers determined the quartic and sextic centrifugal distortion constants.5 Thompson and coworkers made several studies of the fundamental 6-8 vibrational bands of methylacetylene, while several other investigators have looked at the v1 fundamental, hot 9,10 bands, and overtones. Duncan and coworkers have studied the Fermi and Coriolis interactions between the v“ and v7, V8+10 band system.11 Duncan, gt_§1. have also examined the harmonic force field of methylacetylene.l2 In this investigation of the ground state and V5 vibrational state of methylacetylene, both conventional microwave spectroscopy and the more recent technique Of laser Stark spectroscopy have been used. The work was undertaken in part to acquaint ourselves with the use of lasers in vibration-rotation spectroscopy. After the laser Stark spectrum of the gas molecule had been analyzed, it was hoped that several transitions might be found which would be suitable for infrared transient effect spectros- copy which could be used tO measure the relaxation param- eters T1 and T2 for vibration-rotation transitions and rotational transitions in excited vibrational states. As a result Of the extremely low absorption coefficients of the available transitions, this does not appear to be pos- sible with currently available equipment. Consequently, this work is concerned entirely with the spectroscopic analysis Of the v5 band. CHAPTER II THEORY Molecular motion can be divided into several categories. Molecules undergo translation in space and rotate end over end; the atoms within molecules vibrate; and the electrons constantly move around. A certain amount of energy is required to excite each type of motion. Energy in the microwave region with a frequency Of 1 to 1000 GHz is sufficient to excite rotational motion, whereas excita- tion of vibrational motion requires frequencies Of l x 103 - 1 x 105 GHz. An infrared laser Stark spectrum contains transitions between the rotational energy levels of two different vibrational energy levels. In this section, the molecular Hamiltonian will be discussed first. Then, rotational energy levels, vibrational energy levels, and finally laser Stark spectroscopy will be described. The Molecular Hamiltonian To find the energies of molecules, their molecular motions must be described. Equations of motion are written for the molecule and converted to the Hamiltonian function for the total energy of the system. The Hamil- tonian includes contributions from the kinetic energy and the potential energy: H(p,%,t) = T(g,g) + V(%,t). (l) The kinetic energy, T, is expressed in terms of the co- ordinates, qi, and conjugate momenta, p1, of the system; the potential energy, V, is a function of the coordinates and possibly the time.13 The Hamiltonian operator, H, is used in the SchrOdinger equation to derive the molecular energies, En, Hw=Ew. (2) Here, wn is the wavefunction which defines the state of the system in which the energy is En. The Hamiltonian contains terms involving both the nuclei and electrons of the atoms in the molecule and can be written H = TN + Te + VNe + Vee + VNN’ (3) where TN is the kinetic energy of all the nuclei and Te is the kinetic energy of the electrons. 0f the potential energy terms, VNe is the attraction Of the nuclei for the electrons, V is the electron-electron repulsion ee term, and VNN is the repulsion between the nuclei. An approximation can be made in which it is assumed that the nuclei are fixed in position relative to the motion Of the electrons. This Born—Oppenheimer approximation allows the molecular Hamiltonian to be separated into two parts, an equation for the electrons and one for the nuclei.lu Thus, the SchrOdinger equation can be written as two equa- tions, HN‘DN = EWN (I4) and (5) The electronic Hamiltonian is + <> (6) ee’ whereas the Hamiltonian for nuclear motion is HN = TN + VNN + Be, (7) and E is the total energy of the molecule. Therefore, the electronic energy, Ee, which is an eigenvalue of He for a given set of nuclear coordinates, contributes directly to the potential energy for nuclear motion. Rotational Energy - Rigid Rotor The nuclear Hamiltonian is the sum of kinetic energy 15 and potential energy terms, rot + Tvib-rot + V' (8) The kinetic energy of the molecule can be written as the sum of the translational motion Of the center of mass plus the kinetic energy of the motion of the atoms relative to the center of mass. In free space, the two motions can be treated separately. In the rigid rotor approximation there are no vibrations, and the Tvib and Tvib-rot terms are zero. In center-Of-mass coordinates, the Hamiltonian then contains only the kinetic energy of rotation, which depends on the moments of inertia and on the angular momenta of the molecule. The moment of inertia for a rigid rotor is 2 I = 2 m r , (9) i i i where m1 is the mass Of the ith atom and r1 is the distance between the center of mass and the ith atom. The moment of inertia matrix, I, may be written16 I = IXX' Ixy' IXZ' IyX' Iyy' IyZ' sz' Izy' Izz' ’ (10) where I = X m ( '2 + 2'2) and I = - X m x' ' The xx' 1 1 y: 1 xy' 1 1 iyi' other terms can be formed by a permutation of x', y' and 2'. Again, m1 is the mass of the particle and x', y' or z' are the position coordinates in a rectangular molecule- fixed coordinate system that has its origin at the center Of mass of the molecule. One orientation Of this coordinate system will cause the diagonal elements Of the moment-Of- inertia matrix to have either maximum or minimum values and the Off-diagonal elements to vanish. This is the principal inertial matrix whose diagonal elements, the principal moments of inertia, are the roots of the cubic equation, Ixx'"I Ixy' Ixz' Iyx, Iyy.-I Iyz. = 0. (ll) sz' Izy' Izz"I If the three roots are designated Ix, Iy, and 12, the rotational Hamiltonian becomes 2 2 2 P P P H=%(T{+T¥'+T§')g (12) X y Z where Px’ Py, and P2 are components of the rotational angular momentum projected on the molecule-fixed principal axes Of inertia. The Hamiltonian and hence the energy equations are different for different types of molecules. If all three moments of inertia are the same, the molecule has very high symmetry and is called a spherical rotor. If only two Of the moments are the same, it is a symmetric rotor, and if all three moments of inertia are different and none equals zero, the molecule is an asymmetric rotor. Methylacetylene is a symmetric top molecule because two Of its principal moments Of inertia are equal. A symmetric top has one rotational axis Of C3 or higher symmetry. One of the principal inertial axes always lies along this axis, and corresponds with the unique moment Of inertia. The three principal axes are mutually perpen- dicular and are conventionally labelled Ia, Ib, and Ic' Two cases exist for symmetric tops: the moment of inertia about the symmetry axis is either larger or smaller than the other two moments. If it is larger (Ia = ID < Ic)’ the molecule is an oblate symmetric top. If it is smaller (Ia < Ib = 10), the molecule is a prolate symmetric top. Methylacetylene is a prolate symmetric top. The Hamiltonian operator for a prolate rigid rotor 1316 H = _P_2_+l(1_._1_)p2 (13) r 2 b 2 I;' Ib a ’ where 2 _ 2 2 2 P - Pa + Pb + Pc . (1A) TO find the energies of the rigid rotor, the SchrO- dinger equation for Hr must be solved. The Hamiltonian Operator for the symmetric top contains constants and angular momentum operators. Since the matrix elements of the angular momentum operators are well known, the matrix elements of the Hamiltonian operator can be easily found. Angular Momenta The angular momentum Of a single particle is P = F x p, where P is the vector from the origin and p is the linear momentum of the particle. The angular momentum can be expanded and written in terms Of its components PX’ PY’ and P2 which lie along the direction of the space-fixed axes. These components can be expressed as follows:16 PX = sz - Zpy. PY = pr ’ sz: (15) P = XPY ' pr, 10 and 2 2 P = PX + P + P 2 Z. (16) 2 Y It is easily shown that P2 commutes with PX’ PY’ and P2; P2PX — PXP2 = 0, etc. (17) However, the components Of the angular momenta do not com- mute among themselves; PXPY — PYP PYPZ - PZPY ihPX, (l8) PZPX - PXPZ = ihP The angular momentum can also be expanded and written in terms of its components PX, P and P2, which lie along y, the molecule-fixed axes. Again, 2 = 2 2 P Px + Py + Pz, (l9) and P2 commutes with Px’ Py, and Pz; they do not, however, commute among themselves, 11 PxPy - Pny = —ihPZ, Psz — PzPy = -ihPx, (20) PZPX — P PZ = -ihPy. With the aid Of these commutation rules it is possible to derive eigenvalue equations for angular momenta as follows:17 2 2 PZkam =‘meka (21) Pszkm ='fikakm’ The square of the total rotational angular momentum is J(J+l)h2 where J = 0, 1, 2, 3,..., while the projection of the angular momentum on the space-fixed Z axis is mfi; m is an integer with permitted values J Z.m :_-J. The projection of the total angular momentum on the molecule- fixed 2 axis is kfi where k is an integer equal to or between -J and J. Left multiplication Of both sides Of the eigenvalue equations by an eigenfunction and integra— tion yields the non-zero matrix elements of the angular momenta in the eigenfunction basis, as follows: 12 n2J(J+1) ='fim (22) = hk. The rigid rotor Hamiltonian for the prolate symmetric top is given by Equation (13). The Hamiltonian depends upon only the molecule-fixed coordinates, so the energies have no m dependence. The m degeneracy is removed, however, when a space—fixed electric field is applied. The en- ergies for the rigid rotor are the eigenvalues of the Hamiltonian operator, 2 2 - -1; 2.1.12 Er - - 21b J(J+1) + 2, (Ia - TEJK , (23) where K = |k|. Generally, 8 2 is written as a rotational w I constant B and 2 as A; therefore, 8n I a Er = h[BJ(J+l) + (A-B)K2]. (2A) Figure 1 shows a plot of the energy levels of methylacetylene, a prolate symmetric top. Since K is squared in the energy expression, all of the energy levels except K = 0 are doubly degenerate. Perturbations outside the molecule, such as application Of an electric field, cannot lift this K degeneracy. Normally, the Hamiltonian does not include an angular momentum term that depends on the 13 J J J 9___ 8 u _ 700 _ 3 __ 2 -— 7 — eoo . 8— 500 L 6 '— 7—— :‘PS 1:00 _ 5 _ (D 1, Sc 6— £4 a u ._ m 300 . 5— 3— 200 - 2 ____ 14— __ 100 - 3.— 2— l— 0 - O— K-O K=l K=2 Figure 1. Plot of the rotational energy levels of the pro- late symmetric top, methylacetylene. 1A relative orientation Of the molecule-fixed and space fixed— axes. Therefore, the energy levels also possess a (2J+l)- fold degeneracy in m. This degeneracy is lifted in the presence of an electric field and will be discussed in the Stark effect section. Centrifugal Distortion A molecule is not really rigid. As it spins about an axis the atoms are pushed out away from that axis, chang- ing the moments Of inertia. The effect of centrifugal distortion can be added to the rigid rotor Hamiltonian as a perturbation. The centrifugal distortion Hamiltonian is given by18 Hd = IV'.3§5 Tasyé PaPBPYPG, (25) where O,B,y, and 6 refer to the principal inertial axes x, y, and z. The centrifugal distortion is determined by the angular momenta and the constants TOBYG’ which are functions Of the force constants and the moments Of inertia Of the molecule.16 The centrifugal distortion energies can be found by using first-order perturbation theory, in which the energy is expressed as Ed = . (26) 15 The centrifugal distortion Hamiltonian is averaged over the rigid rotor wave functions. Terms that contain odd powers of the angular momentum average to zero. The centrifugal distortion Hamiltonian can be simplified by use of the commutation rules and the fact that many of the T constants are equivalent.19 By using these a8y5 relations and first-order perturbation theory, the cen- trifugal distortion energy of a symmetric rotor is found to be16,20 _ 2 2 2 u Ed - -DJJ (J+1) - DJKJ(J+1)K - DKK , (27) where D = (- fiy11 J 77' xxxx h“ DJK = 'ZDJ ' <'22—){Txxzz + 2szxz} (28) flu DK = 'DJ ' DJK ' (7r)Tzzzz' Stark Effect When a rotating molecule is placed in an electric field, the molecular dipole moment interacts with the field, removing the m degeneracy of the energy levels. This splitting Of the rotational energy levels by an electric field is known as the Stark effect. The Stark effect causes additional lines to be seen in the l6 rotational spectrum of a molecule and hence is useful in identifying transitions. It is also used to determine the value Of the molecular dipole moment. Generally, the electric field is held constant and fixed in space. For Stark effect calculations the electric dipole moment, u, is assumed to be constant in the mole- cule-fixed coordinate system. The interaction between the dipole moment and the electric field is expressed as Hs = -¥.€' (29) Here, H8 is the Hamiltonian for the Stark effect and E is the electric field vector, which is normally in a constant direction and defined in the space-fixed co- ordinate system. TO express both Of these vectors in the same coordinate system, a means of relating the molecule- fixed and space-fixed systems is needed. This relation is given by the cosines of the angles between the two systems. If the electric field is along the space-fixed Z-axis, the Stark Hamiltonian becomes H = -8 Z IJ (DZ 9 (30) s s=X.y.z g E where s is the magnitude of the electric field and the TZ are direction cosines of the molecule-fixed x, y, and 16 8 z axes relative to the space-fixed Z axis. Only the l7 dipole interaction will be considered here. The inter- action of the field and the polarizability will be neglect- ed since its effect is quite small. For the symmetric rotor, the Stark Hamiltonian simplifies to S = -6uZ¢Zz, (31) because Ifil = “z and “x = 0. Exact expressions for “y the eigenvalues Of the combined Hamiltonian Hr + H8 can- not be Obtained. However, the eigenvalues can be deter- mined by numerical methods. The matrix elements for ¢Zz can be Obtained from commutation rules between the angular momentum Operators and the direction cosines.22 For a symmetric rotor, the diagonal elements of the Stark effect matrix are = -u€ (32) ekm = _ flJuJfl , (33) while the matrix elements off-diagonal in J are = -ue (3“) ue [2-K21[|2 S J'fJ , (38) EJ,k " EJ',k which may be shown to be16 E(2) = u2e2((J2-K2)(J2-M2) _ [2-M2] (39) S 2hB J3<2J-1)(2J+1) (J+1)3<2J+1)(2J+3) The total rotational energy in a Stark field is Er + Ed + Eél) + Eéz). As before, the frequencies of the transi- tions are equal to differences in this total energy. A plot of the frequency of one Of the Stark transitions against the electric field will result in a straight line for symmetric rotor molecules if one of the levels in the transition has k and m different from zero. These molecules are said to exhibit a first-order Stark effect. If k or m is equal to zero for both levels in the transition, the leading field-dependent term is proportional to :2. Such transitions are said to have a second-order Stark effect. At the same field, first-order Stark shifts are much larger than second-order ones. At low pressures, it takes only a few tens of volts per centimeter to separate the m components Of a transition with a first-order Stark effect, while a field Of several hundred volts per centi- meter is needed tO separate the components when the Stark shift is second-order. Second-order Stark shifts are 2O superimposed upon first-order shifts. However, since they are much smaller at low fields than first-order shifts, they are usually not noticed unless the field is very high. The J and k selection rules for a rotational transition in an electric field are determined by the type Of molecule. For a symmetric top molecule, AJ = 0, :1 and Ak = 0. The values allowed for Am are 0 or :1. If the electric vector Of the radiation is parallel to the applied electric field, Am = 0 selection rules apply, and if the electric vector Of the radiation is perpendicular to the electric field, Am = :1. In microwave spectroscopy the usual arrangement of the sample cell employs a metal septum running down the length of the cell in its center, and parallel to its broad face. The electric vector Of the radiation is then parallel to the Stark field and the allowed transitions are those for which Am = 0. These spectra are much easier to interpret than Am = :1 spectra. Both Am = 0 and Am = :1 transitions are not normally seen in the same spectrum, because different experimental setups are needed for each. In the microwave region a parallel plate sample cell or a split waveguide sample cell is used to Obtain Am :1 spectra. In laser Stark spectroscopy a parallel plate sample cell is used and the plane of polarization of the laser light may be selected to Obtain either Am 8 0 or Am = :1 transitions. When the Stark effect is first order, the levels for +k and +m have the same energy as those 21 with -k and -m. When the Stark effect is second order, the levels for ik and tm are degenerate, except for m = 0, for which only the 1k degeneracy holds, or for k = 0 for which only the 1m degeneracy holds. The levels with k = 0 and m = 0 are non-degenerate. The number of Stark components, spacing between the components, and their relative intensities can be used to assign the transitions. The intensity of each Stark com- ponent is proportional to the square Of the appropriate direction cosine matrix element. For Am = 0 and AJ = 0, the intensity of each Stark peak is proportional to M2, so the peaks with the largest intensity are those with the greatest absolute value of m. The m = 0 component will be missing. For Am = 0 and J + l + J, the Opposite is true. The largest values of |m| have the smallest intensity, since the intensity is proportional to [(J+l)2-M2]- In this case, no component has zero intensity since J is always at least as large as M. When looking at the relative intensities of second-order Stark components, the calculated value for the M # 0 transitions should be multiplied by 2, since the degeneracy Of these levels is twice that for M = 0. The frequency spacing between the Stark components of a transition increases with increasing value of M. Various patterns of spacing and intensity Of Stark components are shown in reference 16. 22 Selection Rules Molecules can undergo only certain transitions as a result of interaction with radiation. To see when stimu- lated absorption will occur, the time dependence of the interaction of the radiation with the molecule must be examined. By using time-dependent perturbation theory, it may be shown that the probability Of a transition per unit time to state n by a molecule in state m 1523 2 2p(vmn) (”0) for unpolarized radiation. Here, is the dipole moment matrix element Of the molecule, and p(vmn) is the radiation density. An allowed transition, then, is one for which the square of the dipole moment matrix element does not vanish. After expansion, = uxfwchandI + uyfwg¢wandI + uszchzwndt, (41) where ¢Fg are the direction cosines. In a rigid symmetric top molecule, the dipole moment lies along the symmetry axis so that u = “z and “x = “y = 0. Therefore the general dipole moment matrix element becomes 23 * fwgqundT = uszmTszndI . (”2) From the table of direction cosine matrix elements, it can be seen that the dipole moment matrix elements of a sym- metric rotor are non-zero when AJ = 0, :1; Ak = 0; and Am = 0, :1. Which Am selection rule is observed depends upon the situation. In the absence of a Stark field, the direction of the electric vector Of the exciting radia- tion defines the space-fixed F direction. Then, F may be set equal to Z and the dipole moment matrix element is non—zero only if Am = 0. If an applied Stark field is also in the Z direction, Am = 0 selection rules again apply. However, if the applied Stark field is reasonably large and perpendicular to the electric vector of the radiation, then F = X or Y and the dipole moment matrix element is non-zero only if Am = i1. Dipole Moments From Microwave Spectra The measurement of Stark shifts in microwave spectros- copy is one of the most accurate methods for the determina- tion Of molecular dipole moments. Some problems encountered in other methods for determining dipole moments are not sources of difficulty in microwave spectroscopy. These dipole moments are unaffected by impurities in the sample since measurements are made on known microwave transitions. 2“ Thus, the dipole moments of molecules which decompose rapidly may be Obtained since the decomposition rates will not affect the dipole determination. Solids and liquids with sufficient vapor pressure can be studied in the gas phase by microwave spectroscopy, since so little sample is needed. Dipole moments Obtained from these measure— ments will be independent of solvent effects which would otherwise affect them. Dipole moments are obtained for distinct vibrational and rotational states, rather than averaged over all states as in other methods. And, finally, small dipole moments can be measured with almost the same accuracy as large dipole moments.zu The best microwave transitions to use are those with a Stark shift which is as large and purely first- or second— order as possible. The lines should be sharp and should not be complicated with hyperfine structure. The electric field employed must be uniform to Obtain sharp Stark com- ponents and its value must be well known in order to Obtain accurate dipole moments. The first-order Stark energy is given in Equation (37). Thus, the first-order Stark shift in frequency is given by _ 2pc km A"1 ' " Ti ' J—T—J+1‘T(J"—7+2 ' (“3) for the case J + l + J, Ak = 0, and Am = 0. Also, a is 25 the electric field, H is the dipole moment, and h is Planck's constant. The second-order Stark energy is shown in Equation (39) and the second-order Stark shift in frequency for a symmetric top, Ak = 0, with k = 0, J + l + J, and Am = 0 1525 A _ p252 3M2(16J2+32J+10) - 8J(J+l)2(J+2) (uu) V2 ‘ h2 J(J+2)(2J-l)(2J+l)(2J+3)(2J+5) V 0 where to is the zero-field frequency and Av2 is the dif— ference between the Stark component frequency and Do. This equation can also be used for a linear molecule since a linear molecule can be thought Of as the special case Of a symmetric top with k = 0. The Am 0 selection rule implies that the electric vector of the microwave radia- tion is parallel to the electric field. This is the case for most microwave spectrometers in which the sample cell waveguide contains a septum as one of the electrodes. The cell used in this investigation is of this type. When the Stark effect measurements are made at high electric field strengths, higher order corrections to the energy must be included.)4 For a first-order Stark effect the energy is then E = E(l) + E(2) + E(3) + ... . (A5) 26 For a second-order Stark shift - i.e., when the first-order effect is zero - all higher odd-order terms vanish,26 so the energy becomes: E = E(2) + E(u) + ... (A6) and the Stark shift is of the form Av = ADu2e2 + BDuuz-tll + ... . (A7) In Equation (A7) AD and BD are constants for a microwave transition and can be found from perturbation energy expres- sions. For example, AD may be Obtained from Equation (AA). Vibration—Rotation Spectra Molecular vibrational spectra usually occur in the infrared region between 10 and 10,000 cm'l. However, overtones may be found in the visible region. With the laser Stark technique the rotational fine structure Of the vibrational bands can be observed. This work is done with gaseous samples. Infrared spectra of liquids and solids do not show rotational structure. In the simplest approxi- mation, the energy Of a vibrating and rotating molecule is just the sum of the vibrational energy and the rotational energy. In the next approximation, the interaction between 27 the vibration and rotation is taken into account. Since the moments of inertia of the molecule change during the vibration, the vibrationally-averaged rotational constants A, B, C, etc. are slightly different in each vibrational state of the molecule. Even the constants in the lowest vibrational state differ from the equilibrium values, since the energy Of the lowest vibrational level does not equal zero. It may be shown that the energy associated with each vibrational mode Of the molecule is approximately equal to that of a harmonic oscillator, _ 1 EV - (V + §)hvo , ((48) where v = 0, l, 2,..., v0 is the classical frequency of the system, and h is Planck's constant. A molecule with several classical frequencies has a vibrational energy Of15 Ev = (v1+%)hvl + (v2+%)hv2 + ... + (V3N-6+%)hV3N-6’ (A9) where a quantum number Vk and classical frequency Vk are associated with each normal coordinate Qk Of the molecule. Anharmonicities and other perturbations are added to the vibrational energy of the simple harmonic oscillator to approximate the energy levels of a real molecule. These coordinates will not be discussed; in this investigation of CH3CECH only the fundamental (v = 1 + v = 0) of a single 28 vibration-rotation band was studied. As a result, the only purely vibrational parameter determined was the band center, v0. More data would be needed to determine the difference between V0 and the vibrational frequency, v. The fine structure of a vibration-rotation band can form several patterns. A symmetric top molecule which has a symmetry axis of c3 or higher can have two types Of fundamental vibrational bands. When the change in the electric moment of the molecule during vibration is parallel to the symmetry axis, the vibrational band is termed a parallel band. When k does not equal zero, the selection rules for parallel bands are k g 0: AJ = 0, :1 Ak = O. (50) k = 0: AJ = :1 AR = O. (51) For a perpendicular band, the change in the electric moment is perpendicular to the symmetry axis. These transitions are allowed when27 AJ = 0, :1 Ak = :1. (52) 29 An accidentally symmetric top or some overtone or combina- tion gives rise to a hybrid band. These are bands in which both components of the electric moment exist, so the hybrid band has characteristics of parallel and perpen- dicular bands. The band Observed in this study Of methylacetylene was a parallel band. The band structure Of a parallel band is simpler than that of perpendicular or hybrid bands. The energy levels for a prolate symmetric top in zero Stark field were found to be A _ 2_ 2 2_ 2_ Er - BJ(J+1) + (A—B)K DJJ (J+1) DJKJ(J+1)K DKK . (53) By using the selection rules for a parallel band and add— ing the vibrational band center frequency we, the fre- quency for each line of the P and R branches is written as27 P,R n "' v_ n_ n "2 _ v n ‘3 v + (B'+B )m + (B B D3+DJ)m 2(DJ+DJ)m V0 - (Dj-ngu + {[(A'-A") - (Er-Em - (DJIK-Dgxm n "2 2 _ n Ll where fi'is a running index with fi’= J + l for the R branch and 5' -J for the P branch. In Equation (5A) J refers to 30 the value of J in the lower energy level and the single prime indicates an excited state parameter while the double prime indicates a lower state parameter. The frequency of a Q branch transition is given by27 Q _ A \) - \) '_DH t 1 u u 2 + [(A -B ) - (A -B )]K - (DK K)K 0 + (B'-B")J(J+l) - (Di—D3)J2(J+l)2 - (DijDEK)J(J+l)K2. (55) The transitions of the parallel band can be separated into sets - each set containing lines with the same value of K. Figure 2a shows the sets K = 0 through K = A for the v5 band of methylacetylene. In Figure 2b, these sets are plotted together as actually seen in the spectrum. Notice that for K = 0 there is no Q branch line. Also, for sets of increasing values of K, lines are missing from the beginning of the branches since J must be greater than or equal to K for any energy level. In Figure 2b, where all the sets are plotted together, it can be seen that the lines from each set do not lie exactly on top of one another. Thus the K components are resolved in the P, Q, and R branches. The lines in each set do not exactly coincide, because the spacings between the energy levels Of the upper state are different from the spacings between the lower state energy levels. 31 K=OIIIIIIIIIII lIIIIIIIIII (=lu1l, I I I I I I I l I I I I I I I I I 1 1 K=2 L11 1 I I I II I I I I I I I I l I l buJJfllJIIJIIILILIM 1 1 11 928 930 93A Frequency/cm"l Figure 2. Plot of the transitions Of the parallel band v in methylacetylene. The lines can be separated into sets; one for each value of K. Figure 2a shows the sets K=0 through K=A. In Figure 2b these sets are plotted together. 32 Laser Stark Spectroscopy The first working laser was built in 1960 by T. H. Maiman.‘28’29 It was a three-level ruby laser, pumped by a xenon flash lamp. In the following years, the number and types of lasers greatly increased and the uses to which they were put covered many fields. They soon made their way into spectroscopy, where they were used as light sources in the infrared and visible regions. Lasers can be made monochromatic, unlike conventional light sources. Some lasers, such as dye lasers, have continuously variable frequencies, while others can be made to oscillate only at a number Of particular frequencies. Most lasers which operate in the infrared are not con- tinuously tunable. It was found that the application of a magnetic field to the He-Ne and He-Xe lasers could shift their energy levels by means of the Zeeman effect and cause the laser to be tunable over several gigahertz.30 It is possible to observe the rotation-vibration transitions of a number of molecules in this limited range. Recently, tunable diode lasers, color-center lasers, and difference dye lasers have been used as tunable laser sources in the infrared region. Molecular lasers in the infrared, however, cannot be tuned over a very large range. An individual CO2 line can only be tuned about 50 MHz across its linewidth. Since the laser frequency cannot be tuned, the molecular frequency 33 is tuned to bring it into resonance with the laser line. In 1968 Uehara, Shimizu,and Shimoda reported the use Of the Stark effect to tune the molecular frequency.31 This technique is known as laser Stark spectroscopy or laser electric resonance. As described earlier,the application Of an electric field to the sample splits the m degeneracy Of the molecular energy levels, separates the Stark com— ponents, and may bring one Of the Stark components into resonance with the laser frequency. In laser Stark spec- troscopy, fields as high as 70 kV/cm have been used. Some first-order Stark components have been shifted as much as 1. The separation between adjacent 1 28,990 MHz, or about 1 cm- N20 laser lines is about 1 cm' and that between two adjacent CO2 laser lines is about 2 cm-l. The information gained from laser Stark spectroscopy includes molecular parameters such as the band origin, excited state rotational constants, centrifugal distortion constants, upper and lower state dipole moments,and, for perpendicular transitions, Coriolis coupling constants and z-type doubling constants. The technique of laser Stark spectroscopy is limited to molecules with a significant permanent dipole moment. Molecules most suited to laser Stark spectroscopy with CO2 or N20 lasers are those which have a sizable Stark shift since the laser lines are one or two wavenumbers apart. At lower fields, first-order Stark shifts are much faster than second-order Stark shifts, 3A so symmetric top or near symmetric top molecules are likely candidates. Of course, the molecule must have an infrared active band in the range of the laser frequencies, which for the N20 and CO2 lasers is about 900 cm"1 to 1100 em‘l. Table 1 lists the molecules which have been studied to date by laser Stark spectroscopy. Although laser Stark spectroscopy is not applicable to all molecules, it has several advantages over conventional spectroscopy. The laser is used as the frequency standard, so there is no frequency calibration problem. Since the frequencies Of the laser are well known and the instability of the laser is only a few megahertz, the technique is remarkably accurate and precise. Laser Stark spectroscopy is quite sensitive. Shimoda has estimated that the minimum detectable number of molecules in the beam is 102 to 105.72 For H 00, Johns and McKellar estimate that less than 107 2 molecules are involved in the transitions they observed, and that 106 seen.50 This sensitivity allows detection Of weak transi- molecules are responsible for the Lamb dips tions. The technique is also useful for looking at short- lived molecules such as HCO.37 Because Of the sensitivity, sample pressures as low as several mtorr can be used so that the transition linewidth is not broadened by collisions. The resolution of laser Stark spectroscopy is limited only by the Doppler width of the molecule. The laser has enough power to saturate the transition, so that Lamb-dip and level- 35 Table l. Molecules which have been studied by laser Stark spectroscopy. OCS 32, 33a CH“ 55 HCN 3A CH3Br 56 FCN 35 CH3C1 57 020 36 CD3Cl 58 HDO 36 CH3F 59, 6O HCO 37 13CH3F 59 NH3 31, 38—A3 CD3F 61, 62 15 NH3 A3 CD3I 63, 6A NH2D A5, A6 CH2F2 65 PH3 A7 SiHu 66 H2CO A8-51 POP3 67 HDCO 52 CH3OH 68 D200 A9, S3 C2H301 69 FCECH 5A CH2CF2 70,71 8The numbers refer to references in the list at the end of this thesis. 36 crossing spectroscopy can be done. The linewidth of these transitions may be less than the Doppler width.58 Lamb- dip studies have enabledihnTStryland and Shoemaker to observe nuclear quadrupole coupling in an excited vibra- tional state of NH 13,”5 with linewidths as small as 700 kHz. 2 As in microwave spectroscopy, the position and intensity of the Stark components can aid in identifying the transitions. And, finally, information is obtained about the dipole moment of both the lower and excited states of the molecule. A symmetric top such as methylacetylene exhibits pre- dominately a first—order Stark effect. A schematic plot of the energy versus electric field of the J - 1 energy levels Of the ground state and an excited vibrational state of a symmetric top molecule is shown in Figure 3. The rotational constants Of the upper and lower vibrational levels are different, so the splitting of the lower and upper state energy levels is different. The transitions shown are for the selection rules AJ = 0, Am = -1. The difference between an m level in a lower and upper vibra- tional state corresponds to the frequency of the transi— tion. It is easily seen that scanning across the electric field strength causes the frequency of the molecular transi— tion to vary. It may be possible to find a particular electric field strength for which the molecular transition frequency equals the laser frequency. The length of the solid vertical lines in the figure correspond to the 37 R: O Frequency+ )) (L _)1 I? )L Electric Field+ Figure 3. A schematic plot of the energy versus electric field for the J=l energy levels of the ground state and an excited vibrational state Of a symmetric top molecule. The transitions shown are for the selection rules AJ=0, Am=-1. The length of the solid vertical lines correspond to the laser frequency. 38 laser frequency. The difference in electric field strength between the two lines corresponds to the difference in dipole moments of the upper and lower vibrational states. Energy level diagrams of this type may be calculated and drawn for all of the transitions. Clues such as relative intensity, linewidth, and the Sign of 3V/3E are useful in the assignment of laser Stark transitions. The rules for identifying Stark components are also helpful. When a first-order Stark effect is expected and the Am = 0 selection rule holds, m-component transitions from the same parent transition usually occur in a series at electric fields proportional to m-l. In other words, the electric field times the value of m is equal to a constant for transitions with the same J and k quantum numbers. For Am = :1 transitions, patterns are more difficult to see. Although the absolute intensities of the transitions are difficult to determine experimentally, the relative intensities of the transitions seen with a single laser line can be measured. The relative transition intensity is proportional to the square of the dipole moment matrix element and can be calculated. If the rotational and vibrational motions are assumed to be independent, the component of the electric dipole moment along the molecule- fixed x-axis can be expressed as27 39 aux “X _ ux + Z an Qk=o Qk + ... (56) where U? is the permanent electric dipole moment along the molecule-fixed x-axis and Qk is the kth normal coordinate of vibration. The components Of the electric dipole moment along the molecule-fixed y and z axes can be expressed in a similar manner. As discussed in the section on the Stark effect, the direction cosines relate the molecule-fixed and space—fixed coordinate systems, so that the dipole moment along the space-fixed axis is written If the rotational and vibrational motions are indepen- dent, the Hamiltonian can be written H = Hr + Hv' Thus, the energy can be written in two parts, E = Br + Ev’ and the wavefunction is expressed as the product of the rota- tional and vibrational wavefunctions, w = wrwv. The electric dipole moment matrix element is v is " g: I it v as vi 1! . fw qu dT fwv Ar qurwvdTrdTv (58) After substitution Of Equation (56) and Equation (57) into Equation (58), the dipole moment matrix element becomes A0 1 uérw:*iagw;drrfw:*w3drv Bu + g ;IPA* (EEiIOkavdva¢r*¢Fgwngr ° (59) This equation has two independent terms;27 because of the orthonormality of vibrational wave functions in a given electronic state, the first term is for a rotational transition occurring within one vibrational energy state: 2 uofw'*¢ w"dr (60) g g r Fg r r’ The second term is for a transition between two rotational states, each in a different vibrational state: 3n g i (56:)0Qkfwv*w3dTVIW}*TFgwngr. (61) Equation (61) is of interest in laser Stark spectroscopy, while Equation (60) applies to microwave spectroscopy. The intensities of rotational spectra are proportional 0)2, whereas the intensities of vibration rotation 8n ——5 2. The laser Stark BQk spectrum generally contains transitions between many to (NS transitions are proportional to ( rotational energy levels, but only two vibrational levels. Thus the vibrational part Of the matrix element is a constant for all the rotational transitions in a particular vibrational band. Therefore, the relative intensities of Al the laser Stark transitions are given by the squares of the direction cosine matrix elements and any symmetry considera— tions. Table 2 shows the contribution of the direction cosine matrix elements to the intensity. Because of the K degeneracy in symmetric top molecules, the transitions arising from the K ¢ 0 energy levels must have their intensity contributions from the direction cosines multi- plied by 2. It is useful to know whether the frequency Of the laser line lies above or below the zero field frequency Of the transition. In other words, is the slope of the transition frequency with respect to the electric field, av/ae, posi- tive or negative? It should, of course, correspond in sign to the experimental slope. The experimental linewidths can be compared to the calculated linewidth of the transition. An expression for the linewidth is derived in Appendix IL The full width Of the line at half height is expressed as 2 )1/2 AvD (62) A” = (EEE' (av/3e)E where AvD is the Doppler width of the transition and (av/Be)e is the slope, at an electric field value e, of the transi- tion frequency plotted against electric field. Table 2. A2 Contribution of the direction cosine matrix elements to the intensity Of a vibration-rotation transition in a symmetric top molecule. All Of the intensity contributions shown are for transi- tions in which AK = 0. The values for J and M are those Of the lower state. AJ AM Square of the Direction Cosine Matrix Element +1 +1 K2M2 J2(J+l)2 K2(J¥m)(J:m+1) AJ2(J+1)2 (J2-K2)(J2-M2) J2(AJ2-l) (JZ—K2)(J$m)(sz—i) AJ2(AJ2-1) (J+k+l)(J-k+l)(J+m+1)(J-m+l) (J+l)2(2J+l)(2J+3) (J+k+l)(Jrk+1)(J:m+l)(J:m+2) A(J+1)2(2J+l)(2J+3) aP. C. Cross, R. M. Hainer, and G. c. King, J. Chem. Phys., 1g, 210 (19AA). “3 Lamb Dip At the very low pressures used in laser Stark spectros- copy, the transition linewidth is due mainly to Doppler broadening. This type Of broadening occurs because mole— cules traveling at different velocities absorb radiation corresponding to a particular transition at slightly dif- ferent frequencies. The gas molecules do not all travel at the same velocity, but rather a Maxwell velocity distribution exists. The molecular velocity vectors can be thought of as having two components, one parallel to the propagation of the laser radiation in the z direction and another component perpendicular to it. Since the molecules are moving with respect to the incident radiation, the frequency at which the molecule appears to absorb will be v Doppler shifted by an amount i—E, where v is the molecular c z velocity component in the z direction. Each group of molecules with velocity Vi will then absorb radiation v v centered about either v0 + —£§ or V0 - -$E-depending whether c c the molecule is traveling toward or away from the radia- tion, respectively. When the narrow Lorentzian lineshapes of each velocity group are superimposed, an inhomogeneous Doppler-broadened lineshape is the result. In the infrared region, the Doppler broadening is tens of megahertz. Under certain conditions, a narrow dip can be seen in the center of a Doppler broadened lineshape. These "Lamb dips" may occur in atomic as well as molecular gas systems. They AA were first Observed73’7u in 1963 and were named after Willis E. Lamb who investigated the dips mathematically.75 For a Lamb dip to occur, the sample must be simul- taneously irradiated from both the forward and reverse directions. These conditions occur inside a laser cavity. The sample gas may also be placed in a cell outside the laser cavity. In this case, a mirror is placed at the end Of the sample cell so the laser radiation traverses the sample cell and is reflected back through the sample cell by the mirror. The Lamb dip is a saturation effect of an inhomogeneously broadened line. Saturation occurs when the power of the radiation is so high that the molecules absorb power and undergo a transition to an excited state at a faster rate than they can return to the lower state by means of avail— able relaxation processes. Thus, the population dif- ference between the two states becomes smaller than it would be at thermal equilibrium. As there are now fewer molecules available in the lower state to absorb the radia- tion and more in the upper state to emit radiation, the intensity of the peak decreases. When monochromatic radiation of high power traverses forward and back through the sample cell, those molecules in the velocity group v = 0 (those molecules traveling Z perpendicular to the direction of the propagation Of the radiation) will be saturated by both radiation beams. “5 Therefore, the incident power appears doubled to this velocity group and these molecules will experience more saturation than the rest of the molecules in the sample. Thus a dip appears in the lineshape as shown in Figure A. Furthermore, because molecules with a velocity component vZ = 0 absorb radiation at frequency v0, the center fre- quency of the transition, the Lamb dip will appear at the center frequency of the transition. A6 1+ h——--—-—-—-‘ C (D C I Figure A. Lamb dip, centered at the transition frequency v 00 CHAPTER III EXPERIMENTAL Methylacetylene The methylacetylene was purchased from the Linde Division of Union Carbide Corporation, New York, NY. An infrared spectrum Of the sample was Obtained which showed some acetylene to be present. After vacuum distillation, the very strong 730 cm-1 line of acetylene could no longer be seen in the methylacetylene infrared spectrum. Even so, the v5 vibrational band of methylacetylene studied absorbs at about 930 cm'l; acetylene does not absorb between about 825 and 1200 cm'l. The infrared spectra of methylacetylene and acetylene have been published.76 Microwave Experiment Equipment The Hewlett Packard MRR spectrometer used to take the microwave measurements is described in Part II of this thesis. A7 A8 Procedure Ground State Methylacetylene Dipole Moment - Rather than measuring the voltage and the spacing between the electrodes each time a dipole moment is to be obtained, a molecule with a well-known dipole moment is used to calibrate the field. In the R band where the methylacetylene spectrum was obtained, OCS is generally used as the cali- brant. Second-order Stark shifts of the J = 3 + 2 transition Of OCS and the J = 2 + 1 transition of methylacetylene were obtained at five different fields with a sample which con- tained approximately 50% OCS and 50% CH3C5CH. The fre— quencies of the OCS transitions at one electric field set- ting were measured. Then, without changing the field, the frequencies Of the methylacetylene transitions were obtained. The field was changed and the next set of OCS and methylacetylene frequencies were obtained, and so on. Excited State Methylacety1ene Dipole Moment — The first- order Stark shifts of the J = 2 + 1 transitions of methyl- acetylene in the ground state and v5 = l excited vibra- tional state were Obtained at 15 different fields. The sample contained pure methylacetylene. Again, all Of the ground state and excited state transition frequencies were Obtained at one field setting; then, the field setting was A9 changed and the next group of transition frequencies were measured. Laser Stark Experiment Eqpipment In the laser Stark experiments, a C02 or N20 plane- polarized laser beam was passed between parallel electrodes inside a sample cell and onto an infrared detector. A block diagram of the apparatus is shown in Figure 5. A discussion of the laser used in this experiment can be found in Appendix C. Stark Electrodes - The sample cell contains two parallel electrodes. These electrodes are nickel plates, A0 cm long, 5 cm wide, and 2.5 cm thick. The plates are flat to better than :0.5 microns, and all the edges and corners are rounded. Five quartz spacers, one-quarter inch in diameter, are used to separate the electrodes. The spacers are about 1 mm thick and were ground to be flat, parallel and of equal thickness to within $0.2 microns. The Stark plates and spacers are surrounded on four sides by glass plates. There are no glass plates on the two sides where the laser beam enters and exits the sample cell. The elec- trodes are held in an aluminum cradle which rests on the bottom of the sample cell. 50 I [Stabilizer [ Laser Power Supply Laser Beam 0-3 kV Power Supply Figure 5. r——————" L"1 Grating I \ l / | Laser Plasma Tube \ I / 0-10 kV Power Supply [Sample Cell Pb-Sn-Te l I Detector L , I- 1 Audio 10 kHz Amplifier pre-amp 1 10 kHz Oscillator Zero-Crossing Detector Marker Ramp Voltage Generator [ Lock-in-Amplifier l Oscilloscope or Chart Recorder Block diagram of the laser Stark spectrometer. 51 Sample Cell - The sample cell is a six inch diameter glass "T", the outside of which is covered with wire mesh. Aluminum plates are clamped to the three Openings of the cell. Large rubber gaskets are used between the metal and glass. The two side plates have holes in the center, covered with NaCl windows, to pass the laser light. The top plate contains the gas inlet and exit, electrical con- nections for the electrodes, and a Hastings DV-6M pressure gauge. Inside the cell, about one foot Of #20 braided wire connects a separate high voltage bnc connector on the cover plate to each electrode. A brass plug was soldered to one end of each wire. This plug was inserted snuggly into a hole in the electrode to make electrical contact. The end Of the high voltage, high vacuum bnc adaptor was covered with heat shrink teflon tubing, and the whole length Of the wire was covered with teflon tubing tO help eliminate electrical discharge upon application Of high voltage. Assembly of the Sample Cell — When the sample cell is assembled, everything must be very clean. Any dust, grease, or fingerprints may cause discharge paths to form under application Of high voltage. Care must be taken not to scratch the surface of the Stark plates. To assemble the Stark plates, the cradle is laid on its side, with the five screws on top. Two pieces of glass and an electrode are 52 set in the cradle. A drOp of methanol is placed directly beneath each screw and a spacer is placed on each methanol drOp. This procedure floats away minute particles Of dust. A drop of methanol is placed on top of each spacer and the other electrode is eased on. Another glass plate is placed on the electrode. A piece of 2 mm thick teflon is placed between each screw tip and the glass plate. The five screws are tightened to hold the assembly together. Electronics - A block diagram of the laser Stark spectrom- eter is shown in Figure 5. A high voltage Fluke Model AlOB power supply applies a DC voltage to one Of the electrodes. This voltage can be changed in varying steps from 0 to 10,000 volts and is stable to 10.005% per hour after warm up and to 10.02% per day. A ramp voltage is applied to the other electrode by a Kepco Model 2000 operational power supply (OPS). The width Of the ramp can be varied from 0 to 1500 volts. A Wavetek Model 112 function gen- erator is used to drive the ramp output of the OPS. The ramp voltage and the voltage from another high voltage power supply, a Fluke Model A12B, are compared in a zero- crossing detector circuit. The large voltages of the OPS and Fluke A12B power supply are reduced in magnitude by five 200 k0 resistors. For calibration purposes, the 20 k9 potentiometer shown in Figure 6 is set so that the resistance across the OPS equals the resistance across the 53 Output of operational C) power supply R ¢ Zero—crossing detector 20 k0 .‘ <:) R' '5 A12B Fluke power supp 1y 6.) Figure 6. Schematic diagram of the circuit used to protect the zero-crossing detector. The ten numbered resistors are 200 k0 each. 5A Fluke A12B power supply when their voltages are equal in magnitude (but Opposite in sign). For good precision, the 200 k0 resistors have low and equal temperature varia- tion. The two diodes keep the voltage entering the zero- crossing detector within the range of :0.5 volts tO protect it. When the output voltage of the OPS equals the Fluke A12B voltage, a marker is generated. It is possible to change the time width of the marker. If this is done, the zero difference of the voltages is at the leading edge of the marker. The field across the electrodes at the time of a marker can be Obtained from the sum Of the voltage settings of the Fluke AlOB and the Fluke A12B power supplies. An amplified 10 kHz sine wave voltage from a Hewlett Packard Model 651A Test Oscillator is used to modulate the ramp voltage. The sine wave voltage is amplified by a Bogen Sound Systems Model DBlO audio amplifier. A Cal— electro Corporation G. C. Electronics transformer steps up the voltage from the audio amplifier. The laser beam is detected by a Barnes Pb-Sn-Te photo— voltaic detector. The output voltage of the detector is amplified by a 10 kHz tuned amplifier. A Keithley Model 8A0 Autoloc Amplifier is used as a phase sensitive de- tector. The output is seen as a first derivative line- shape On an oscilloscope or chart recorder. 55 Polarization of Laser Light - The infrared radiation leaving the laser is plane polarized with its electric vector parallel to the floor. Since the electric field across the Stark electrodes is also parallel to the floor, spectra with the selection rule Am = 0 can be Observed. To Observe spectra under the selection rule Am = :1, the polarization of the electric vector Of the radiation must be perpendicular to the electric field between the Stark plates. The electric vector of the laser light can be rotated 90° by reflection between two appropriately placed mirrors. This mirror configuration is shown in Figure 7. The net movement of the laser beam is to raise the beam up and to turn a corner 90°. Procedure Alignment of the CO2 Laser Beam Through the Sample QEll.‘ A He-Ne laser is used to help in the position- ing of mirrors to send the infrared laser light through the sample cell. The two irises within the CO2 laser should be positioned in the apparent center Of the plasma tube. The position Of the CO2 beam varies slightly, de- pending upon which laser line is oscillating and whether or not it is oscillating at the top Of its gain profile. Thus, the conditions for the particular C02 laser line must be Optimized. If none of the mirrors in Figure 8 are in position Figure 7. 56 The appropriate mirror configuration to rotate by 90° the electric vector of the laser radiation. The two cubes, shown by the solid and dotted lines lie directly above one another. The angles a, b, c, and d are each A5°. The arrows show the direc- tion of the beam. 57 .pouoEoppoodm xpmpm commfi on» pom moapdo on» do psoEmwchL< .m opsmfim opmo xmocfi wLOLgHE umam momHLH .6 “mamas: : c m loo whospfie pm Hosea w .0 Loop Q Hm m to H .3 Ha m Emma EsficmEme whoapfie HtOfipondm w . r—INm momma ozumm _ HHOQ OHaEmm . a N _ H comma moo 58 yet, mirror 3 is replaced by an iris and an infrared detector is placed behind it. The position Of the He-Ne laser is adjusted until its beam is superimposed upon the CO2 laser beam. The He-Ne laser beam can now be used to position the mirrors and irises in Figure 8. With the exception of mirror 8, all irises and mirrors should be mounted on translation stages and secured to the table. (Room should be saved to place a detector between iris 5 and mirror 6.) Alignment of the CO beam through the sample cell is 2 begun by placing a detector between iris 5 and mirror 6. The two irises are centered about the CO2 beam and the position Of the He-Ne laser is adjusted until its beam is colinear with the CO2 laser beam. The detector is removed and a mirror is placed at position 8. Mirrors 6 and 7 are adjusted until the He-Ne beam passes through the sample cell and makes one small symmetric spot on the index card, 9. Mirror 8 is replaced by a detector. The 002 laser beam should pass through the sample cell and strike it. Calibration of the Plate Spacing The quartz spacers in our sample cell were ground at the same time, so they would be the same thickness. Since the plate spacing may change slightly each time the plates are put together, the plate separation must be determined each time this happens. The resonant voltages Of Lamb 59 dips Observed for several lines of methylfluoride were measured and compared to voltages Obtained by Freund, _£’a1.59 The transitions compared were a doublet and a quartet which absorb the P(18) line of the 9.6 u band Of the CO2 laser when the electric vector Of the radiation is aligned to be perpendicular to the electric field. These transitions and the voltages Obtained by Freund, 33 31. are given in Table 3. The average of the ratio of our voltage to the voltage of Freund, g£_a1. for the above six transitions was then multiplied by their reported plate spacing of 0.1002851 cm tO Obtain our plate spacing. In order to Obtain these voltages to the accuracy required, the technique of Lamb dip spectroscopy was employed. The mirrors and detector were set as shown in Figure 9. In this arrangement, the path of the returning beam was about 3 mm below the incoming beam so that it could be reflected from the edge of a flat rectangular mirror into the detector. The modulation voltage was very low, about 3 volts out of the audio amplifier. With the irises wide Open to saturate the transition, a dip was seen in the CH3F transition. Because of the small ampli- tude Stark modulation, the dip appeared as a first deriva- tive lineshape that was 180° out Of phase with respect to the main transition. Figure 10 shows Lamb dips for the J = 1 + l, k = l + l, m = -l + 0 and m = 0 + 1 transitions of CH3F. 60 Table 3. The resonant voltages Of several transitions Of methylfluoride measured by Freund, et al.a These transitions were used to caliEFaEE the plate spacing. For these transitions Ak=0. Quantum Numbers Voltage J' J" k m‘ m" AAA.82 l l l -l 0 A59.96 l 1 l 0 1 1019.13 2 2 2 -2 -1 1050.75 2 2 2 -l 0 108A.l2 2 2 2 0 1 1119.29 2 2 2 l 2 as. M. Freund, G. Duxbury, M. Rdmheld, J. T. Tiedje and T. Oka, J. Mol. Spectrosc. 52, 38 (197A). 61 1 C02 Laser I Sample Cell] A 1 _ ______27 I 7“———-—L* I 6 I5 2 T3 1, A spherical mirrors 2 l or 2 mirrors depending upon polarization 3, 5 flat mirrors 6 flat rectangular mirror 7 detector Figure 9. Arrangement of the optics for Lamb dip spec— troscopy. 62 I l A.A A.6 Electric Field/(kV/cm) Figure 10. Lamb dips for the J=l+l, k=l+l, m=-1+0 and m= 0+-l transitions of CH3F. CHAPTER IV RESULTS AND DISCUSSION Microwave Spectrum Ground State Rotational Constant, B" The frequency of a microwave transition in a symmetric tOp molecule in a vibrational state of A symmetry is v = 2B(J+l) — ADJ(J+1)3 - 2DJK(J+1)K2 , (63) where J is the angular momentum value of the lower state. For a J = 2 + 1 transition, this equation becomes _ 2 v — AB - 32 DJ — ADJKK . (6A) Since the centrifugal distortion constants are known, the ground state B rotational constant can be calculated and is shown in Table A. 15 Excited State Rotational Constant1 B' A microwave spectrum of methylacetylene was taken between 33800 and 3A550 MHz. Table 5 lists the frequen- cies of those lines which were assigned. It was of 63 6A Table A. The calculated ground state B rotational constant of methylacetylene. K v/MHz B"/MHz avg B"/MHz Other B" O 3A183.A13 85A5.877(1)b 85A5.877 1 3A182.762 85A5.877(1)b 0.00298a MHz H based on DJ H DJK 0.1629a MHz 85A5.877a aC. A. Burrus and W. Gordy, J. Chem. Phys. g9, 391 (1957). bThe number in parenthesis shows the error of the last significant figure. 65 Table 5. Microwave transitions of methylacetylene. All these transitions are J = 2 + l and AK = 0. v/MHz Assignment 3A031.7 95:1 K=l 3A032.3 v5=l K=0 3A168.6 CH31300H K=1a 3A169.1 OH313CCH K=Oa 3A182.7 K-la 3A183.3 K=0a 3A187.2 V9=l K=:l 2-+1b 3A202.9 v9=l K=:l £=ilb 3A20A.1 v9=l K=0 2=:1b 3A223.3 v9=l K=:l 2=+1° 3A2A6.2 910-1 K=:1 2:11a 3A277.1 v10: K=:l 2=:1a 3A279.1 vlo-l K=0 i=:1a 3A313.3 910—1 K=:l t—Tla 3A386.9 vu=1 K=:1 3A387.8 v“: K=0 aR. Trambarulo and W. Gordy, J. Chem. Phys. 18, 1613 (1950). bFrequency difference between K=£=:l lines m2q(J+l) where where q N 2B2/w. 66 particular interest to identify the vibrationally excited v5 J = 2 + 1 line. The intensity of a vibrationally ex- cited transition relative to the corresponding ground state transition, is given by = -—e g (65) where gex is the degeneracy of the excited state, ggr is the degeneracy of the ground state, AB is the energy difference between the vibrationally excited state and the ground vibrational state, kB is Boltzmann's constant, and T is the absolute temperature. Table 6 lists the intensities of the rotational transitions in the vibra- tional excited states relative to the ground state. The v5 excited state transition is about 100 times less intense than the ground state. However, transitions of methyl- acetylene with one 13C are also 100 times less intense 12C. To be certain that the than the species with only transition assigned to us was not one due to methyl— acetylene with 13C, the spectrum was taken under dry ice conditions. The relative intensity Of the 130 peaks re- mained the same, while the relative intensities of peaks due to excited vibrational states decreased. The frequen- cies of the rotational transitions in the v5 excited state were found to be as follows: 67 Table 6. Intensities of the rotational transitions in the vibrational excited states relative to the ground state. Vibrationally _1a Egg Excited State AE/cm Igr v1 3335.1 1x10"7 v2 29Al 7x10"7 v3 21A2 3x10.-5 vu 1390.6 1x10"3 95 930.1 1x10“2 06 2980.8 lxlo‘8 V7 1A50.9 2x10"3 v8 1036.0 1x10”2 69 638.6 1:10"1 -1 010 329.2 AxlO aT. Shimanouchi, N5RDS-NBS 39, Tables of Molecular Vibra- tional Freqpencies Consolidated Volume I. (U.S. Government Printing Office, 1972.) bT = 298°K. 68 c_, ll 1 + 0 K II O C II 17016.231 MHz J = 2 + l K 3A032.A16 MHZ II 0 C II J = 2 + 1 K = 1 v 3A031.7AA MHz. Theoretical expressions for these transitions are as follows: C... II H 1. o 13‘: II o C II AB — 32 D c4 II M t [—1 73 II o C II J AB - 32 DJ — ADJK. L, II M f H 7: ll ,4 C II By the appropriate combination of these equations, the v5 vibrationally excited state B, DJ, and DJK are cal- culated to be: B 8508.119 : 0.003 MHz 5 = D§5> = 1.8 : 0.2 kHz D§§) = 169 : 1 kHz. Ground State Methylacetylene Dipole Moment Tables 7 and 8 contain the second-order Stark shifts for OCS and CH3CECH. The Stark voltage was found to differ from the voltage indicated on the dial by an amount that was not constant as a function of dial reading. The 69 .UHOfih onom Ca ocm UHOHQ on» Ca mocozvopm UOLSmmoE one coozuoo mm: CH monopommao map ma >prHop mmfiocosvopm oopsmmoe on» Ca mpcfimppoocs boomefipmo ones .Nmz moo.OHmaw.mm=mm ma :oHuHmcmpp on» now mocosuopm oaoamuopou woodmmoe onem mom.m oeo.:oaom ooe.oa mmo.oozom HHo.mu moo.oozom oooH moo.: ooo.mozom oeo.oa mom.oo=om oee.mu mmo.oo:om ooze amo.m eqo.fio:om oma.ou emm.oooom Hmo.en moa.eoaom oomH oae.m moo.ooaom Hmm.oa moa.ooaom msa.eu oeo.eoaom oooH mom.H oom.oo:om oem.on moo.oo=om mae.on ooo.ooeom ooo o>< > o>< > 094 9 Ampao>v wceooom Hoao o.m.po»oEoppOOdm .m .m ecu co wcfippom Hmfio mo cofiuocsm m mm woodmmoe moo :fi coapfimcmou m+muh on» go mpCOCOQEoo 2 no anmzv moaocosconm .o canoe Table 8. Frequencies (MHz) of M components of the J=2+l transition in CH CECH measured as a function of dial setting on he H. P. spectrometerfi,b Dial Setting K = O, M = 0 K = 0, M = l (Volts) 0 Av0 v Av0 800 3A179.596 -3.8l7 3A186.51A 3.101 1000 3A177.573 -5.8A0 3A188.161 A.7A8 1200 3Al75.lA5 -8.268 3A190.13A 6.721 1A00 3A172.322 -ll.091 3A192.AA7 9.03A 1600 3A169.08A -1A.329 3Al95.067 11.65A aThe measured zero-field frequency for the transition is 3A183.A13:0.005 MHz. b The estimated uncertainty in the measured frequencies relative to the zero-field frequency is 10.005 MHz. cAv is the difference in MHz between the measured frequency in the field and in zero field. 71 variation of the Stark voltage is discussed in Appendix A. Since the OCS and CH3CECH frequencies were measured at exactly the same field, the square of the field was cal- culated from Equation (AA), the second-order Stark shift equation for a linear molecule, by using the OCS Stark shifts and the known OCS dipole moment of 0.71519 D.77 These results are given in Table 9. The M = l Stark lobe of OCS was not included since the line is asymmetric, due to its proximity to the zero-field line. The asymmetry of the line makes the determination of its center fre- quency difficult and uncertain. A second-order Stark shift with higher order correc- tions can be expressed as 2 2 A A AV = ADUDE + BDUDE + co. (66) where e is the electric field and AD and BD are constants. After dividing by ADEZ, the equation becomes, A 2 Av - 2 BD“DE 2 - “D + T— . (67) ADE D Plotting Av/ADe2 versus 52 yields a straight line with us as the intercept. To Obtain the methylacetylene dipole moment, the fields calculated from the OCS M = 0 transi- tions were averaged with the fields calculated from the OCS M = 2 transitions. The methylacetylene Stark shifts were 72 Table 9. Calculated value of the electric field determined from the OCS second-order Stark shifts. Dial Setting M = O M = 2 (Volts) Av/MHz 82/(kVolt/cm) Av/MHz 82/(kVOlt/Cm) 800 - .795 2.752 1.395 2.7A9 1000 -1.1A3 A.223 2.190 A.216 1200 -1.621 5.988 3.03A 5.978 lA00 -2.l78 8.0A6 A.083 8.0A5 1600 -2.811 10.384 5.263 10.369 73 used for Au, and AD was calculated from Equation (AA), the equation for the second-order Stark shift. The pairs of Av/ADe2 versus :2 were fit by the method of least squares to a straight line for the K = 0, M = 0 Stark shifts and for the K = 0, M = l Stark shifts of methyl- acetylene. Thus, the methylacetylene ground state dipole moments for the J 2 + 1 transitions were: for K 0.78395 D; - 0.78381 D II o ‘3 II o T.‘ D II for K ll 0 3 II I-J v 1: U I with the average dipole moment equal to 0.7839 : 0.0010 D. This may be compared to “D = 0.78A0 : 0.0020 D derived from A after correction 77 the value found by Muenter and Laurie for the more recent value of the dipole moment of OCS. The Dipole Moment of Methylacetylene in the v: Excited J State First-order Stark effect frequencies for the ground state and the V5 = 1 excited state of the k = :l, m = :1 and k = :1, m = :1 transitions of methylacetylene are shown in Table 10. The transitions in a Stark field, 6, are found at frequencies of _ _ km _ nka V 68 V'VO m moaocosvoam mumpm ocsouo oe.oeomm oo.ooo=m so.omflsm oo.ommam ooa oo.osomm nunuuuau am.amazm mm.=mmom owe om.mmomm nuununun mo.mmaam Insults: ooH no.3womm :e.mso:m mm.omaom mo.omm:m omH os.soomm oo.mso:m ae.omeam om.omw=m osH oa.ooomm Hm.mooom Hw.HoH=m oe.mmm:m omH oo.moomm :o.oeozm oa.=sflsm ma.ammzm one mm.mmmmm mo.ooozm ma.soazm mo.mamzm oHH mm.mommm Hm.mooam ms.oaflom sw.mamom ooH oo.ooo:m om.moosm sa.mmeam oe.mfimom oo mm.moo:m om.omozm oo.amaom om.oam:m om mm.ooosm om.smosm Ho.emeam oo.eomam oe Ho.oooam mo.smosm om.ooeam mm.mom:m oo mm.aaosm oo.amoom mo.mofiam ao.mom=m om se.Hmozm ae.emo:m os.moesm oe.moeom o Hfius .Hsux Hens .Hsux Hans .Hau: Hens .Hsux Anoeo>v moaooom Haao m .ocoampoowamnuoe mo mCOfip Homopp H+mnw map mo mpCOCOQEoo HHS .Hax ocm Has .Hux on» no oomph umpfioxo 2 on» cam oomph ossopm on» com Aumzv moaocoscomm poommm xEMum poonoupmpam .OH canoe 75 where v0 is the zero—field frequency of the transition, V is the applied voltage and S is the cell spacing. A straight line is obtained when u versus V is plotted. However, in our instrument the voltage obtained from the dial setting is not the true voltage - there is an Offset which appears to be constant at these low fields. (The offset is discussed in Appendix A.) In this case, the Stark field can be written as e = (V9883) (69) __§_ , where Vd is the voltage from the dial, V0 is the voltage offset, and S is the cell spacing. Thus the first-order Stark frequency is u km V +V v = v _ D d o (70) 0 J(J+15 S ' This equation can be rearranged to u km V n km = - D ° - D v (71) " V0 J(J+I) ‘S’ 'J'(J+l)s d ° Thus, when v is plotted versus Vd a straight line is still Obtained but the y-intercept is now uka V0 V0 " +1 (I) 76 The frequencies and dial voltages were fit to a straight line by the method of least squares. These reSults are shown in Table 11. The excited state dipole moment of methylacetylene can be found from the ratio of the slopes of the ground and excited states and from the ground state dipole moment as follows: slope ———ex) (72) 11 =11 ex SP (slopegr The dipole moment for the v5 state of methylacetylene was found to be 0.795A : 0.0010 Debye. Laser Stark Spectrum Calculation of the Laser Stark Spectrum In order to calculate the frequencies of the laser Stark transitions, the vibration-rotation energy levels of the molecule must be calculated. Then, the Stark splitting of each energy level must be taken into account. The transition frequency is given by the difference between two molecular energy levels. The vibration-rotation transi- tion frequencies (1;g;, the transition frequencies at zero field) are given for a symmetric top in Equation (5A) and Equation (55). As mentioned before, when the fields are low, the Stark shifts of the transitions can be found by Table 11. 77 The corrected voltages versus the first-order Stark effect frequencies of the ground state and v5 excited state of the J = 2 + 1 transition of methylacetylene fit to a straight line by the method of least squares. Slope Zero Field Frequency/MHz ground state k = :1, m = 11 0.263516 3A182.76 k = :1, m = :l -0.263A19 3A182.76 v5 excited state k = :1, m = 11 0.267AA9 3A03l.7A k = :1, m = :1 -0.267219 3A031.7A 78 perturbation methods. Since the electric fields used in laser Stark spectroscopy can be very high, the Stark shifts approach the magnitude of the spacing between the rotational energy levels, and the perturbation equations are not suf- ficiently accurate. Therefore, the rotational energies, including the Stark shifts, were calculated by diagonaliz- ing the matrix of the Hamiltonian, H = Hr - p'g . (73) The diagonal matrix elements are22 =BJ(J+l) + (A—E)K2 - DKK” - DJKK2J(J+1) _ DJJ2(J+1)2 — [km/J(J+1)]uDe, (7A) while the off-diagonal matrix elements are l ([(J+1)2—M2][(J+l)2-K2])l/2 = ‘(J+1) (2J+15(2J+3) UDE . (75) A program written for the CDC 6500 computer calculates the laser Stark transition frequencies at nine different electric field values. Since the Hamiltonian matrix is diagonal in k and m, a matrix was set up for every m value 79 for each value of k 3’0. One set of matrices was generated for the lower vibrational state and one set for the excited vibrational state. The matrices were truncated at Jmatrix = J + 5 and diagonalized. The laser Stark fre- transition quencies are then equal to v = to + E'(J',k',m') - E"(J",k",m") (76) where E' and E" are eigenvalues of the matrices for the upper and lower states, respectively, and to is the fre- quency of the vibrational band center. Another program written for the CDC 6500 computer uses a least squares routine to fit frequencies of experimental laser Stark transitions to calculated frequencies and to determine the band center and rotational constants. The equation 3v (e) _ (C) = (e) _ (C) up up IIEE ) c (Oi1 0L1 ) (77) is used to fit the transitions and determine the parameters. The experimental frequency Of a transition is vée) and the calculated frequency of that transition is véc). The fitting parameters are a1. There are eleven parameters in this case, which are listed in Table 12. Since the energy is found from Equation (7A) and Equation (75) it can be seen that the derivative of the 80 Table 12. The eleven parameters that can be fit from the laser Stark transitions and the derivatives of the frequency with respect to these parameters. a1 Parameter Sup/Bai a1 00 l 62 B' T'EJ'(J'+1) - K21T' a3 B" EHEJH(J"+1) _ KZJT" cu AA K2 A as ADK -K a6 DJK, T'E-K2J'(J'+1)]T' O‘7 DJKn T"[-K2J"(J"+l)1T" a8 DJ. T'E-J'2(J'+l)21T' OL9 DJ" T"[-J"2(J"+1)21T" “10 U' T'U'T' 0‘11 U" T"u"T" 81 frequency with respect to B for a particular value of the quantum numbers is given by the matrix element, 30 _E .-. {E‘[J(J+1)—K2]T}pp. (78) B In Equation (78), [J(J+l)-K2] is used to denote a diagonal matrix with diagonal values given by the expression in brackets; T is the matrix used to diagonalize the Hamil- tonian for the total rotational energy. The derivative sup/BB is the p,p matrix element of the matrix product indicated. The derivatives Sup/3AA, avp/BADK, and avp/BvO can be computed without matrix transformation. By contrast, the derivatives with respect to the dipole moments require transformation of a matrix that has both diagonal and off-diagonal elements. All of these deriva- tives are listed in Table 12. To Obtain the fitted parameters, a set of parameters £0) are calcu- is assumed. The transition frequencies v lated from these assumed parameters. For each transition, two calculations are made. The difference vée) - véc) is computed and the set of derivatives (avp/aai) is com- puted for each parameter c1. The linear system of Equa- tions (77) is solved for (a§e) - aic)) = A1. The values for the experimental parameters are found from die) = die) + A1. (79) 82 The set of parameters a§e) is used in place of the assumed £0) and the procedure is repeated. The itera- §e> _ a§°1 values of a tion is stopped when the a are all below some preset tolerance. Observed Laser Stark Spectrum The v5 band of the methylacetylene spectrum is very weak. It is weaker than the hot band 2v3 + v3 of CH3F observed by Freund, eg’a1.59 Recently, the infrared ab- sorption intensities of methylacetylene have been studied.78 The integrated intensity for the 05 band of methylacetylene was measured to be 1 km mOl-l as compared to 95 km mol-l for the v3 band of CH3F.79 Since the v5 laser Stark Spectrum of methylacetylene is sparse in the area of the laser lines and very weak, few lines were seen at all. The methylacetylene v5 transitions were much too weak to be able to use the Lamb dip technique to help locate the line center. The transitions were assigned by observing patterns for the case of the Q-branch lines by comparing the ob- served and calculated spectrum and by comparing the Ob- served and calculated linewidths and relative intensities. For some of the transitions, it was possible to watch the line move to a higher or lower voltage as the cavity length, and hence the laser frequency, was changed slightly. If 83 the cavity length is increased and the line moves to a lower field, the zero-field frequency of the transition lies below the frequency of the laser line. However, if they both increase, the zero-field frequency of the transi- tion lies above the frequency of the laser line. The fields and quantum numbers of the assigned lines are then fit to Equations (5A), (55), (7A), and (75) to obtain the molecular parameters. Several Observed laser Stark absorption lines of methyl- acetylene are shown in Figures 11 and 12. In Figure 11 both transitions were shifted into resonance with the P(32) line of the 001-100 band of the CO2 laser. Both transitions have J = 5 + A, k = 3, and Am = +1. For the upper trace m" = -3 while for the lower trace m" = -2. A Q—branch doublet of methylacetylene is shown in Figure 12. The transitions are J = l, k = l, and Am = —l, with m" = 0 for the lower field transition and m" = l for the higher field transition. These transitions are in resonance with the P(lO) line of the N20 laser. The spectra in Figures 11 and 12 were taken with a time constant of 1 sec and a sample pressure of 7 mtorr, although lower pressures some- times were used at high fields to avoid electrical dis- charge in the sample gas. The scanning rate (determined by the cycle rate of the ramp and its maximum amplitude) was adjusted in conjunction with the time constant to optimize the appearance of the spectra. A modulation amplitude of 8A L l 19AAO 203A0 I I 2AA30 25110 Electric Field/(V/cm) Figure 11. Observed laser Stark absorption lines of methyl- acetylene. Both transitions were shifted into resonance with the P(32) line of the 001-100 band of the C0 laser. The assignments of the transitions are J=5+A, k=3, and Am=+l. For the upper trace m"=-3 while for the lower trace m'=-2. 85 l 1 13321 139A6 Electric Field/(v/cm) A Q-branch doublet of methylacetylene. The transitions are J=l, k=l, and Am=-1, with m"= for the lower field transition and m"=l for the higher field transition. These transitions are in resonance with the P(lO) line of the N20 laser. 86 A0 to 60 volts was used to observe the spectra. With the configuration for Am = 0 selection rules, lines P(2A) through P(A6) of the 001-100 band of the CO2 laser were scanned between 0 and 32,000 v/cm. In addition, lines P(5), P(7), P(12), P(15), and P(l7) of the N20 laser were scanned up to 22,000 v/cm. With the configuration for Am = :1 selection rules, P(30) through P(AO) of the 001-100 band of the C02 laser and lines P(6) through P(l6) of the N20 laser were scanned. The upper limit for scanning the electric field, which varied between 30,000 and 50,000 v/cm, was determined by the field strength which caused electric discharge through the sample. Tables 13 and 1A list the observed transitions for each laser line. The assigned transitions are listed in Table 1A, and lines for which approximate fields have been determined are listed in Table 13. At most, only one or two transitions were observed with each laser line. The calculated spectra for each of these laser lines show only one or two transitions with intensities strong enough to be likely to be observed. The calculated spectra only include values of J up to 10. Table 13 also lists the calculated field and probable assignment of these transitions. (To ascertain that the assignment is correct, more precise fields for the transitions need to be obtained.) Table 1A lists the assignment of twenty lines and includes the laser lines they absorbed, and the fields needed to Stark shift the transitions into resonance Table 13. 87 Approximate fields of the observed transitions between the v5 excited state and the ground state of methylacetylene. The fields in parenthesis are calculated fields where a "strong" transition is to occur. The calculation includes J values up to J=10. Transitions for which accurate fields have been measured and assignment made are listed in Table 1A. Electric Field Value of Laser Line Am Observed Transitions/(kV/cm) N20 P(ll) :1 7.7 N20 P(12) :1 39.0 (38.96) CO2 P(30) :1 many lines 002 p(32) :1 A6.6 52.0(51.91) 55.0 co2 P(3A) :1 31.A 33.9 52.7 co2 P(AO) :1 A6.2 CO2 P(30) 0 28.6 002 P(32) 0 19.5 26.0(26.05) co2 P(3A) o 17.3 CO2 P(36) 0 22.A(22.5A) CO2 P(38) 0 28.7 CO2 P(AO) 0 21.A(21.3l) Table 1A. 88 Assigned transitions between the v5 excited vibrational state and the ground state of methylacetylene. Laser Line J'K'M'+J"K"M" Field/(Volt/cm) (vg—ve)/MH2 N20 P(10) l 1-l+1 1 0 13321 -2689.93 1 1 0+1 1 l 139A6 2 2-2+2 2-1 2A601 -3A05.5A 2 2—l+2 2 O 25A5A 2 2 0+2 2 1 26337 2 2 1+2 2 2 272Al 3 3-3+3 3-2 A5163 -A788.65 3 3-2+3 3-1 A650A 3 3-1*3 3 0 97923 3 3 0+3 3 1 A9372 3 3 1+3 3 2 50856 3 3 2+3 3 3 52365 002 P(32) 5 3-2+A 3-3 19925 1922.97 5 3-l+A 3-2 2A771 5 3 0+A 3-1 33135 002 P(36) l 1 0+2 1-1 32527 3A33.A3 l 1 1+2 1 2 A1685 1 0 1+2 0 O 50A13 3266.8 002 P(AO) 7 7 1+8 7 2 25A83 -778.63 7 7-5+8 7-6 3A066 89 with the laser. The differences between the frequencies of the laser line and the zero-field frequency of the transi- tion are also listed. Table 15 compares the molecular parameters obtained by fitting the laser Stark frequencies to two sets of parameters: set I includes the values of the band center, difference in A rotational constants, difference in DK centrifugal distortion constants, and the excited state dipole moment; set II includes the parameters of set I and the ground state dipole moment. The values of the other parameters were assumed from the microwave study. The errors listed for the parameters of column I and II are twice the standard deviation of each parameter obtained from the fit. They reflect the precision but not accuracy of the numbers. Although the parameters in column II are more precise than those in column I, the ground state dipole moment given in column II differs from that obtained from the microwave data. Hence, the param- eters in column II are considered to be less accurate than those in column I. The difference between the experimental and calculated frequencies of methylacetylene for these two cases are compared in Table 16. The band center obtained from the laser Stark spectrum is 930.27u9to.ooou cm’l. A value of -2o6:7 MHz was ob- tained for the difference between the ground state and v5 = l excited state A rotational constant. The ground state rotational constant AO has been measured by Thomas 90 .a mflmfis.o u Amoco: 6» meomppoo ”Ammmflv mmm .wm .mmsm .5620 .h .mfipsmq .3 .> 8:6 66626:: .m .56 .Asmmfiv Ham .mm .msnm .5620 .m .mepou .3 cam magnum .< .09 .Awmmav nmma .¢=m .mpo¢ .Efinooppomam .comQEone .3 .m cam mmEone .x .mm oo=m>.o Acavmmm~.o havmmm~.o Q\=: AOHvamms.o Aflvmmms.o Amvmzms.o a\.: no.moa umxxxmo AHV.meH me\ewa amm.m me\mo Ame m.H umxxwa AHV :.m lav m.: Nmz\ma:mn Aofiv o.mmHI Aonv m.mom| um2\= pmnuo m>wzopoaz Ahnou HH H nonmempmm Imponmq manev .opmpm Hmcofipmpnfi> 6:30pm map on Momma mmefipo mansop on» mafia: opmum chofipmpnfi> m9 on» on pouch mmefipa one .=: cam .: .xa< .<< .09 wsfimnm> an cocfimpno who: HH cesaoo ca omonp mafia: mCHpSOh wcfippfim map 2H mpm> .n can .xa< .<< .09 wcfipuma an cocfimppo who: H casHoo CH mpmpoEmpma one .wpcoefipooxo xpmpm momma on» song cocfimpao mpouo lempmm on» mum HH com H mcESHoo :H .momommo mo mpmmempmo coapmnnfi>ucofipmpom .mH magma 91 Table 16. Comparison of the difference between the ex- perimental and calculated frequencies of the methylacetylene transitions for the two cases in Table 15. Transition Ia IIa J'K'M'+J"K"M" (ve-vC)/MHZ (ve-vc)/MHz l l-l*1 l O -8.H —O.7 1 1 0+1 1 1 -9.0 -1.1 2 2-2‘2 2-1 6.8 .1 2 2-l+2 2 0 “.0 .6 2 2 0*2 2 1 0.6 -1.5 2 2 1+2 2 2 -2.0 -3.8 3 3-3+3 3-2 —u.u -l.A 3 3-2+3 3-1 +1.9 1.“ 3 3-l+3 3 O -3.M O.A 3 3 0+3 3 l -3.8 0.5 3 3 1+3 3 2 -3.U 1.u 3 3 2+3 3 3 -l.3 “.0 5 3-2+u 3-3 10.8 5.7 5 3-l+u 3-2 2.5 -5.5 5 3 0+“ 3-1 3.1 -u.8 1 1 0+2 1-1 -1.7 -6.6 1 1 1+2 1 2 -u.4 -6.6 1 0 1+2 0 O 15.6 12.8 7 7 1+8 7 2 1.2 7 7-5+8 7-6 -l.3 Standard Deviation 9.8 6.6 aThe parameters included in sets I and II are listed in Table 15. 92 and Thompson to be 5.31 cm-1 (159,000 MHz).8 The laser Stark experiments also lead to a value of 4.810.? MHz for the difference between the ground state and v5 = l excited state centrifugal distortion constant DK' Duncan has calculated the ground state centrifugal distortion con- stant DK of methylacetylene to be 2.98 MHz.12 The ground state dipole moment of 0.7839:0.001 Debye determined from the microwave measurements of the J = 2 + 1 transition is in excellent agreement with that of Muenter and Laurie.“ The difference between the ground state and excited state dipole moments obtained from the laser Stark spectrum compares very well with that obtained from the microwave measurements. This good agreement supports the quality of the other constants determined from the laser Stark spectrum The weakness of the laser Stark spectrum of the v5 band of methylacetylene made it difficult at times to obtain accurate resonant fields for the transitions. It also led us to question for some time whether the observed transitions were assigned correctly to the fundamental v5 band. In support of this assignment is the unmistak- able appearance of the Q-branch multiplets and the strong correlation between the infrared laser Stark and the micro- wave data. Also, as already mentioned, conventional in- frared intensity measurements have shown that this band is extremely weak. 93 The transitions of the v5 band of methylacetylene are too weak to use for good linewidth measurements. It should be possible, though, to interface the ramp voltage of the laser Stark spectrometer with a minicomputer in order to average a large number of scans of each transi- tion. In this way, it might be possible to obtain the molecular parameters more accurately. PART II DETERMINATION OF ROTATIONAL RELAXATION PARAMETERS FOR OCS IN H2, C02, AND CH3F CHAPTER V INTRODUCTION All spectroscopic transitions have a finite line- width that is a result of many different effects. Con- tributions to the linewidth include the natural line- width, as well as 1ine broadening due to the Doppler ef- fect, pressure broadening, saturation broadening, wall collision broadening, and instrumentation problems. For gaseous samples at pressures above ~20 mtorr, the major contribution to the linewidth of microwave transitions is the collisions between the molecules. Perturbation of the rotational energy levels of molecules by collisions (or near collisions) with other molecules causes an increase in the width and a shift in the frequency of the molecular transitions. Thus, the linewidth of a rotational transi- tion can provide information about intermolecular col- lisions. In addition to the investigation of the physical phenomena responsible for them, lineshapes are studied in several ways for a number of reasons. Astronomers need to know the effects of line broadening since spectral lines are collision broadened in the atmosphere of some planets. In the atmosphere of Jupiter, the linewidth of methane 9A 95 is broadened by other methane molecules and by ammonia,80-83 and in the atmosphere of Mars, C0 is broadened by C02.8u Lineshape data are also useful for developing measurement techniques for atmospheric pollutants,85 and knowledge of the pressure-induced linewidths and frequency shifts are needed for the design, operation, and frequency determina- tion of gas lasers.86 Microwave lineshapes have been studied both experi- mentally and theoretically for many years. The theoretical approach is to identify and use the forces between the radiating molecule and the molecules colliding with it to account for the molecular lineshape. Review articles by 87 and by Birnbaum88 discuss the results of theo- Rabitz retical calculations of microwave linewidths. For treat- ment of the experimental data, it is sufficient to assume that the collisions act as a first—order relaxation process to return the populations of the states and the polariza- tion of the molecules which have absorbed radiation back to their equilibrium values. The lineshape expected as a result of this assumption will be described in the next chapter. The studies in this thesis were undertaken to apply the experimental methods recently developed at Michigan State University for the determination of the population relaxation time, T1’ and the polarization relaxation time, T2. The system of OCS in CH3F was studied to try to resolve 96 apparently conflicting values of the ratio of the relaxa- tion parameters. A transient effect experiment by Mader, 8 - l. 9 showed that T2/Tl was almost 1 (T11 = 0.079i0.009 usec”l mtorr-1, T51 = 0.07710.002 usecml (D ('7' mtorr-1), whereas a molecular beam experiment by Wang, g£_§1.90 showed that the cross sections CI and OII’ which are closely related to T11 and T51, respectively, were significantly different (oI = 990 32, oII = 750 AZ). The system OCS—CH3F is that of a linear molecule perturbed by a polar symmetric tOp molecule. For comparison purposes, it was decided to study OCS-H2, a linear molecule perturbed by a non-polar diatomic molecule, and OCS-C02, a linear molecule perturbed by a non-polar linear molecule. As indicated, the ulti- mate goal of measurements of the relaxation times is two- fold. The data can be used to test theories of inter- molecular interactions and also for theoretical treatments of interstellar, atmospheric, or laser phenomena. CHAPTER VI THEORY In these experiments, the values of the relaxation parameters, T1 and T2, were desired. The following sections describe how these parameters can be obtained from the width of the molecular transition. Interaction of Radiation with the Sample The microwave radiation used to excite the sample is a beam of electromagnetic radiation that is monochromatic (to better than 1 part in 108), coherent, and plane polarized. The electric field of the radiation may be expressed as E(z) = 60 cos(wt-kz), (80) where 50 is the amplitude of the field, w is the angular frequency, and k is the wave number. The beam travels in the +z space-fixed direction and the direction of the field will be taken to be the x direction. This electric field induces an oscillating polarization in the sample mole- cules of the form 97 98 P = Pr cos (wt-kz) —,Pi sin (mt-kz). (81) In this expression Pr and P1 are the amplitudes of the components of the polarization that are in phase and in quadrature with the radiation, respectively. The polariza— tion oscillates at the same frequency as the radiation, but with a phase shift when Pi # 0. The electric polariza- tion is the macroscopic dipole density of the sample. It is the result of molecular dipole moments tending to align themselves with the electric field. The electric field creates a stress in the molecules as a result of their non- uniform charge distribution. This stress manifests itself as an orientation of the molecules in space or as a stretch- ing and bending of the molecular bonds, all of which con- tributes to the polarization. The electric field of the radiation can also alter the charge distribution of the electrons relative to the nuclei. The amplitude of the electric field of the incident radiation is assumed to be constant in space along its direction of propagation and therefore the polarization is uniform throughout the sample. However, it oscillates at the same frequency as the electric field. The oscillat- ing polarization of the sample creates an oscillating electric field that has the same phase relative to the exciting field through the sample. Thus, the intensity of the induced electric field appears to increase with distance 99 through the sample, since each volume of sample dx dy dz contributes to it. At the end of the sample the detector monitors the change in intensity of a combination of the incident and induced electric fields. In this picture, absorption or emission of the radiation depends on the phase of the induced electric field. If the phase of the induced field is opposite that of the incident radiation, the ampli- tude of the combined electric field at the exit of the cell will be less than at the entrance. Then, absorption is said to have occurred. Emission occurs when the induced and incident fields have the same phase. Relation Between the Absorption Coefficient and Pi If the radiation is strong enough to be described as a classical wave, Maxwell's wave equations can be used to explain its behavior. These equations will be written in the Gaussian system of units. In a non-conducting, non-magnetic medium, the electric field, E, and the mag- netic induction, B, are related as follows:91 1 8B =___ N X X E c 56 ’ (82) where 3 3 3 X=ta§*iw*§t§’ (83) 100 c is the speed of light, and t is time. Thus, v x (gxlé) = - 8'??? (57x13). (8A) '\.o To find an expression for 9x8, substitute g = Hg (85) and D = E + “WP (86) 'b "b 'b into Maxwell's equation, 0: (+80 3 (87) 8< >3 8:13 ll 0 ||-‘ to obtain one 332m 8 x x g = + An 5g) . (88) (3 In Equation (88) the permeability of the gas sample, u, is assumed to be unity. Also, D is the electric displace- ment and P is the polarization of the molecular sample; H is the magnetic field of the radiation. Therefore 2 x (XXE) = - iL-(-——-+ Aw ——— . (89) 101 Since microwave radiation in a rectangular waveguide is plane polarized, let 1; =1E(z) (90) so that the direction of propagation is along the z axis and the electric field vector oscillates in the xz plane. Then, zx8=afl%fl 9“ and 32E(z) X x (2X8) = -% -;—§——u (92) 2 Upon comparing Equations (89) and (92), it is apparent that R = $P : (93) the other components of B being zero. With this in mind, Equations (89) and (92) are combined to obtain 2E 1 2 2 "‘2’”? az c o: 02 4a I I (DI-l: N a: J P . (91:) Q) ('1‘ Q) t The electric field of the incident radiation induces a polarization in the sample as expressed in Equation (81). 102 If the change in Pr and P1 with time is very slow com- pared to the frequency of the radiation, differentiation of Equation (81) gives 313- -”-= .82? . (95) 3t Equation (9A) then becomes 3213 1 321»: 1477sz "—2 " "'2' ""2' = " 2 ' (96) 82 c at c The next step is to evaluate the derivatives of E, as given by Equation (80) with respect to t and z. The left and right sides of Equation (96) then each become a sum of a sine and a cosine term with slowly varying coefficients. After equating the coefficients of the sine terms, it is found that as 36 0 1 0 2wn az c t c Pi' (97) However, as shown by Flygare and his coworkers,92 as as l O O E" 513' << 82 9 (98) so that de ; 21$.p dz (99) 0 c i ' 103 The absorption coefficient is defined in terms of the fractional energy lost to the system by the radiation, _ w de - - w . (100) Here, y is the absorption coefficient and W is the energy. The energy per unit area of the radiation flowing into the system can be calculated from the Poynting vector to be eozc w = (101) 8n Hence, 2de dw _ 0 W" 80 . (102) By combining Equations (99) and (102) it can be seen that the absorption coefficient is proportional to the ampli- tude of the in-quadrature part of the polarization,92 P A i Y = _ _%E.E_., (103) 0 The Optical Bloch Equations The optical Bloch equations have been derived for a two-level system in the presence of radiation.92 The 10A Hamiltonian for the two-level system without collisions is H=H°'}d.§3 (10”) where H° is the Hamiltonian for the field-free system and the interaction between the radiation and the sample is given by the dot product of the molecular dipole moment and the electric field of the radiation. The normalized th wavefunction for the a molecule of a system of N mole- cules is expanded in terms of eigenfunctions of H° to give (a) _ (a) o w - 2 0a 0a . (105) a Then, the population of state a is defined to be ( )* ( ) Ca“ ca“ , (106) 2 II IIMZ a 1 whereas the contribution to the polarization from the states a and b is N _ ( ) )* ( ) ( )* ( ) Pab ' aZl ”a8 (Céa Cba + de Caa ) (107) if the electric field of the radiation is in the x direc- tion; uég) has been assumed to be real. If the energy 105 difference between states a and b is approximately‘hw, and if there are no other pairs of states for which this is true, then to a good approximation P = Pab‘ It may be shown that with suitable approximations the time-de— pendent Schradinger equation for the system of N molecules is equivalent to the following set of equations:92 é%-(fiAN) + eOP1 = O (108) dPr 7fi7'- 5 Pi = o (109) 9P1 :8) IE7 - |,h I eocnaw) + 6 Pr = 0 . (110) In these equations AN is the difference in populations of states a and b, 6 is equal to the resonant frequency of the system minus the driving frequency of the electric field, and uég) is the dipole moment matrix element which connects states a and b. Equations (108)-(110) are similar to the Bloch equations for NMR.93 To account for the effect of collisions, phenomenological first-order relaxa- tion terms are added so that the optical Bloch equations for a set of two-level systems with collisions are as follows:92 (AN-AN°) d _ a? onAN) + EOPi +«n T1 = 0 (111) 106 0 P -P d r r _ d “88) 2 Pi-Pio a—E' Pi - I ’h I 80(fiAN) + 6 PI’ + T = O . (113) The phenomenological relaxation terms T1 and T2 will be discussed in the next section. The quantities P10, Pro, and AN0 are the equilibrium values of P1, Pr and AN; 0 _ 0 = ordinarily, Pi - Pr 0. Meaning of T1 and T2 Originally, T1 and T2 were introduced into the optical Bloch equations phenomenologically. In the two level system, T1 is the relaxation time for the difference in the populations of the two energy levels of the molecules under study to return to its equilibrium value. Thus, Tl’l is a measure of the rate at which the population dif- ference between the two levels returns to its equilibrium value after the radiation perturbation is introduced. The relaxation rate T2"1 is a measure of the rate at which the components of the polarization return to their equilibrium values. It is of interest to determine what kinds of collisions contribute to the relaxation of the population difference and polarization. The relaxation times have been inter- 9A preted in several ways. Liu and Marcus obtain expressions 107 for T1 and T2 in terms of the scattering matrix elements. Expressions for T1 and T2 can also be obtained by applica- tion of the Bloch-Wangsness-Redfield theory95 that has been used to characterize relaxation in NMR. In this theory the relaxation times are interpreted as combina- tions of time correlation functions of a random perturba- tion H(r) that is introduced to describe the collisional 96 processes. It turns out that for a two-state system, -1 _ ' T1 - 21‘12 + r , (11A) and T'1 = r + r' + r (115) 2 12 a ’ where _ J. w (P) (r) (r) (r) ra - SE-fo [Hll (t)-H22 (t)J[Hll (t-T)-H22 (t-T)]dT, (116) r = l(w + w ) (117) 12 2 12 21 ’ r' = %’ 2 (W1k + w2k) , (118) k¢1,2 - .1. ... (r) (r) 1‘”an wmn - 152);0° HInn (t)Hnm (t—I) e dT (119) and in which 108 wmn = (E; - E;)/fi These equations were derived for non—degenerate energy levels where levels 1 and 2 are connected by the radiation; E; and E; are unperturbed energies of levels m and n, respectively. Expressed in this way, T1 and T2 have a physical inter- pretation. The constant Whn can be thought of as the rate constant for collision-induced transitions from level m to level n. Thus, F12 is the mean rate constant for col- lision-induced transitions between states 1 and 2, while P' is the combined rate constant for collision-induced transitions between states 1 or 2 and any other state. The constant ra can be thought of as the rate constant for adiabatic processes such as reorientation or phase changing collisions. These adiabatic collisions do not change the populations of the levels, so they are not included in the expression for the relaxation time T1. Also, it should be noted that a transition between levels 1 and 2 causes the population difference between the two states to increase by two or decrease by two. Thus, one transition between levels 1 and 2 should cause the popula- tion difference to change by two and the polarization to change by one. From Equations (11A) and (115) it is easy to see that transitions between levels 1 and 2 are twice as effective in changing Til than in changing T31. 109 The ratio of the relaxation parameters can be ex- pressed in terms of the time correlation functions as follows: T 2P12+F' Tl T12+T'+Ta (120) The variation of T2/Tl for different values of the time correlation functions can be investigated.96 If Fa is very small, the value of T2/Tl will be between 1 and 2. If Ta is small and F12 >> I', as would be the case when collisional transitions involving molecules in states 1 or 2 are due predominately to transitions between states 1 and 2, then T2/Tl m 2. However, if Ta is small and the collisional transitions which involve molecules in states 1 and 2 are mostly those between states 1 or 2 and some other state, as is the case when 7' >> 712, then T2/Tl m 1. So far it has been assumed that I the rate constant for a’ adiabatic relaxation processes, is small. If Pa is very large, T2/Tl m 0. Thus, T2/Tl can range in value from 0 through 2, depending upon which type of relaxation process predominates. The expressions Just given for Til and Tgl for non- degenerate levels have been extended to the case of m degeneracy, which always occurs in the absence of Stark or Zeeman fields. For this case, 110 + Z w im' JfiJi’Jf Jm'Jim + ————— 2w + w (121) 2Jf+l 1 Jimem' JgJZ J Jm'me’ i’ f and TE1=RelZf(m)ZX(W +w ) 2 m J, m' J'm'me J'm'Jim J J l J J l t f i -32 Z <-l>’“+m ( )( f 1 )9; . , m m' HI -m 0 UN —m' 0 fm Jim (122) where W = Jimimef co -1W T r Jme f_we 1 i f f dT, (123) - w (r ¢me,J*m - f_wdT, (12A) and = (m) (m) inmimef (JimilH |J1m1> ‘ (err'H lerr>° (125) The factor f(m) is three times a Wigner 3-J coefficient;97 i.e., 111 J J l f(m) = 3 ( f. j' ) (126) 01 -m 0 The wJ J can be interpreted as the rate constants for rmr 1m1 collisionally induced transitions from state Jimi to state mef. Thus, all the contributions to T11 are rate constants for a change of state. As before, any collisions between level J1 and Jf contribute to Til approximately twice as much as collisions between level Ji or level Jr and any other J level. The relaxation term T21, however, contains contribu- tions from collisions which cause a change in state as well as contributions from adiabatic collisions. In this case collisions between level J1 and level Jf, collisions between level Ji or Jr and any other level J, and simple reorienta— tion collisions me + me' or Jim + Jim', occur with the same weight factor. The second term in Equation (122) for Tgl contains adiabatic contributions. Steady State Solutions to the Optical Bloch Equations A long time (several relaxation times) after the onset of the radiation, it is assumed that steady state condi- tions have been reached. In the steady state, d(AN)/dt = 0, dPi/dt = 0, and dPr/dt = 0. Under these circumstances, the Bloch equations become + h (AN-AN°) 60 Pi T1 = O , (127a) Pr — 6 P1 + T;.= 0 , (127b) 112 and 2 P1 8 P1 - K eo(nAN) + T” = 0, (1270) 2 where |u(X)| K =-—————- . (128) -n These equations may be solved to yield KZEOfiANO/T2 . (129) 62 + (1/T2)2 + K 250(T1/T2 ) Pi = It was shown earlier, (Equation (103)) that the in—quadra- ture component of the polarizability is related to the absorption coefficient, Therefore, by combination of Equations (103) and (129), K2hAN0/T2 C 2 . (130) 5 + (l/T2)2 + K 2cowl/1‘2) = Avw 113 Low Power When the power of the radiation is very low, that is, as 60 + 0, Equation (130) becomes Y = UnwKafiANO 1/T2 c 62 + (1/T2)2 (131) This is the equation for a Lorentzian line shape centered about 5 = 0 and with the half-width at half—height given by 1 Aw = —— , (132) T2 or _ 1 AV - W . (133) Hence, Equation (131) can be expressed as _ AwwKZfiANO ( Am (13”) _ c 2 2 ' (w-wo) + Aw Thus, under steady state conditions, the relaxation time T2 can be obtained from the linewidth of the transition when the radiation power is low. 11A Moderate Power Under conditions of moderate radiation power, the term proportional to s2 is not negligible and the absorption 0 coefficient is expressed by Equation (130). Under condi— tions of uniform moderate power, Equation (130) still represents a Lorentzian line shape centered about 6 = O and with a half-width at half-height of A0) = \/(1/T2)2 + K223(T1/T2) . (135) If T2 is known, the ratio Tl/T2 can be obtained from this equation. The signal at the input of the preamplifier of a microwave spectrometer is proportional to the mean dif- ference in the current of the detector crystal in the presence and in the absence of sample. The change in the current at any time is proportional to the change in the microwave power, W, as seen at the detector. It may be shown that the change in the microwave power is propor+ 2 i;92 hence, it is proportional to yeo. Un— fortunately, so is not constant throughout the sample. tional to 50? Therefore, to obtain dW it is necessary to integrate over an assumed distribution for so so that 2 2 _._ ANQhw K s0 dxdydz dw ‘ 2T 2 2 2 2 (136) 2 5 + K 80 Tl/T2 +(1/T2) 115 In Equation (136) dW is the contribution to the change in power that results from absorption in a box of volume dxdydz located at x, y, z where s = s0 cos(wt - kz). The power distribution in the sample cell is assumed to be the power of a TB microwave mode, attenuated by ohmic 10 losses in the walls of the cell; thus, as discussed more fully in the next section, s3 = s2 sin2 (1X) 4E3.BZ (137) where B is the cell attenuation coefficient and sm is the peak electric field of the radiation at the entrance of the cell. After substituting for the power distribution and integrating over x, y, and 2, Equation (136) becomes 2 — AND no ab \lco + Cl + C0 dW = T 8 in 2 2 , (138) 2 -B£ \fiOJ'C1e +0 0 where a and b are the width and height of the sample cell, A is its length, AW is the difference in the power at the inlet and outlet of the sample cell, 2 _ 2 116 and = ——’K s . (1A0) In the presence of an electric field many molecular transitions are split into several components. Equation (138) is the power broadened line shape of one such m component. At zero Stark field, when the transition is degenerate in m, the usual procedure has been to follow the work of Karplus and Schwinger and consider the line- shape to be a simple superposition of power broadened line shapes.98 Recently, it has been shown by a number of workersgu’ 99-103 that under conditions of partial saturation the m components of the transition are not com- pletely uncoupled in the zero field experiment. Liu and Marcus911 have shown that the power broadened line shape collapses to a single Lorentzian when certain assumptions are made about the populations of the m states. Schwende- 10” have shown that the assumption Liu and mand and Amano Marcus made about the population of the m states was in- adequate and made a different assumption. These results still lead to a line shape for the power broadened transi- tion that is approximately a single Lorentzian. In the work by Schwendeman and Amano the absorption coefficient times the power was found to be 117 K s2 YW = 2 2m 2 . (1A1) (w-wo) + (l/T2) + Ys Tl/T2 where K is a constant. In this expression Ys is defined as —2 y: = 0 X ufi siflh2 , (192) where J + J + 1 3(2Ji+1)(2Jf+1) and J1 and Jr are the J values for the lower and upper states, respectively; q is a parameter that will be dis- cussed below. For a transition of the type Jf = J + l + J1 = J, 2 (J+1) (1AA) x=- . 3 (2J+l)(2J+3) The averaged dipole moment matrix element is expressed as —2 2 “f1 = 3 X l‘meIU |J1m>| , (1A5) m z and for the J + l + J transition, (J+l)2-M2 (2J+1)(2J+3) 02 . (1A6) 2 - [(J+1amIUZIJsm>I — 0 118 Thus, the averaged dipole moment matrix element becomes, 2 _2 _ (2J+l)(J+l)2 5M 2 (2J+1)(2J+3) (2J+1)(2J+3) Since J 2 1 Z M = -3—J(J+l)(2J+l) , (1A8) m=-J the averaged dipole moment matrix element is equal to ii, = u§ . <1u9> The parameter q in Equation (1A2) is defined as follows: 2 f(m)£Ap> 3>y <———b > Figure 13. Rectangular hollow metal pipe waveguide. 121 where a is the length of the waveguide along the x axis and b is its length along the y axis. The radiation propa- gates along the z axis. Wave equations for the electric and magnetic part of the radiation can be derived from Max- well's equations. After solving the wave equations for a lossless waveguide, the components of the electric field and the magnetic field for the TElO mode are written105 i(wt-z/X ) 60(2) = em sin (%¥) e g (155) sy = sx = Hz = 0 (156) s 1(wt-Z/k ) Hy = 3173*; sin (3‘51) e 8 (157) s w n iEwt—l/Xg(z+Ag/A)] X = wuob COS (I?) e (158) where X x = 0 (159) g (1-(%%>2}172 In these equations s is the amplitude of the electric field, m w is the angular velocity of the radiation, no is the per— meability of a vacuum, 10 is the wavelength of the radiation in free space, A is the guided wavelength of the radiation, g and Kg = Ag/Zw. When the free-space wavelength of the 122 radiation is greater than 2b, the radiation will not propa- gate along the waveguide. Thus, Ac = 2b is the cut-off wavelength. The propagation of radiation in a hollow rectangular waveguide is discussed in books on microwave theory and on electromagnetic theory.106 It can be shown that the amount of energy passing through unit area per unit time can be obtained from the real part of the Poynting vector. The time average of the Poynting vector in mks units is ex- pressed éav = %(E X E*)av , (160) where E is the electric field vector and H* is the complex conjugate of the magnetic field vector. Thus, in a rec- tangular waveguide, the real part of the time averaged Poynting vector105 is _ l x * Re(8av) - 2 Re('Etz>1 + ExHyghv ’ (161) or 2 R ( ) = ——:9—— i 2(EX-)k (162) e §av S n b ,b ° To obtain the transmitted power, W, the time averaged Poynting vector must be integrated over the surface area, 123 x=a y=b s3 W = f f -————— sin2(%¥)dx dy (163) X=0 y=0 2wuokg soab w = —— . (16“) Amuox Since w = 5L, *0 e0 ab 10 2 1/2 W = (l-(f-) (165) C110 C Thus A 1/2 -1/A _ 1/2 Cu0 _ A0 2 and -1/A 1/2 “on In I A Keo - wl/2( abo) T” (no—2)?) . (167) The dipole moment matrix element, luifl, can be expressed as the dipole moment “D times the direction cosine matrix element. For the J + 1 + J, AM = 0 transition of a linear molecule, the direction cosine matrix element is expressed 2 1/2 2 rM = ( (J+1) - M . (168) (2J+l)(2J+3) 12“ Thus, in an X-band waveguide in which a = 1.016 cm and b = 2.286 cm, Kso in rad/usec is given by -l/A w K€O(Pad/ sec) = 2.5A8 pD(Debye)fM[W(mW)Jl/2(l-(—£-2) , “’0 (169) when the dipole moment “D is expressed in Debye and the power is given in milliwatts. The Natural Linewidth The smallest contribution to the width of a microwave spectral line is called the natural linewidth. It arises from the uncertainty principle, At - AB :36 . (170) Thus there is an energy spread and a spread in the lifetime associated with each excited state of the molecule. Since frequency is equal to energy divided by Planck's constant, a spread in the frequency of each transition also exists. This is the natural linewidth. The frequency spread is expressed as107 3 3 320 v Av = mn 2 , (171) 3hc3 125 where vmn is the transition frequency and is the dipole moment matrix element from state m to state n. For a frequency of 30 GHz and a dipole moment matrix element of l Debye, the natural linewidth is roughly 10"7 Hz. Such a linewidth is negligible compared to the other linewidth contributions. Most of the other line broadening mechanisms increase the linewidth by amounts on the order of kHz. Broadeninggpy Collisions with the Walls Collisions of sample molecules with the cell walls cause a broadening of spectral transitions which becomes a significant factor when the sample pressure is so low that the mean free path of the molecules approximates the cell dimensions. It is assumed that a collision between a sample molecule and a cell wall interrupts the radiation process. The collisions return the molecules to a Boltzmann distribu- tion. According to theory, the contribution to the total lineshape differs slightly from a Lorentz line shape.108 The full width at half-maximumixlkHz is given approximately by107 1 21w . 1.16 ( + B- + -§-) (1%)“? , (172) mll—J where a, b, and c are the dimensions of the sample cell in cm, T is the absolute temperature of the gas, and M is 126 its molecular mass. In an X-band waveguide at room tempera- ture, the half-width at half maximum due to wall collision broadening is about 7 kHz for OCS. Doppler Broadening At thermal equilibrium gas molecules in the sample cell are moving in random directions and have a Maxwell—Boltzmann distribution of velocities. In each molecule-fixed frame of reference the frequency of a beam of radiation differs from the laboratory frequency m. If a molecule has a component of its velocity in the direction of the radiation, V the radiation appears to have a frequency w = w(l — 3), a where v is the velocity component in the direction of propa- gation of the radiation and c is the velocity of the radia— tion. Thus, if the molecule is moving with the radiation, its frequency appears to be lower than the laboratory fre- quency. A molecule moving towards the radiation sees a higher frequency, while a molecule moving in a direction perpendicular to the direction of propagation of the radia- tion sees a frequency equal to the laboratory value. To see the effect of Doppler broadening on the line- shape, it is useful to consider Equations (lll)-(113) from which it is seen that AN, P1, and Pr depend on the velocity of the molecules only in the frequency difference, 6 = w—wo. The molecules can be divided into velocity 127 groups each of which contains molecules with velocities between v and v + dv. The fractional number of molecules in each group is 2 -Mv /2kBT M )l/2 e dv , (173) F(v)dv = (m where M is the molecular mass, kB is the Boltzmann constant and T is the absolute temperature. The frequency of the radiation seen by the molecules in a given group is m(l - %) so that - V - 0 - w(1 - -c) mo (17“) for each group. Thus the velocity—averaged Pi is found from 21 = fmei(v)F(v)dv. (175) Similar equations can be written for Pr and AN. It has already been shown in Equation (103) that the absorption coefficient is proportional to Pi’ so that ;’= ffwy(v)F(v)dv . (176) From Equations (130), and (173), the average absorption co- efficient is found to be 128 7 = LHTwIC hAN I M )l/Zfoo e dv __ -m 2 2 0T2 \2kaT M 2 1 2 K soTl (w - —- - 0) ) +(--) —— c 0 T 2 T2 (177) To integrate Equation (177), let - fl - Then, dx = - gidv , (179) and v2 = (w-wo-x)2(%)2 . (180) If the microwave power and the sample pressure are very low, 2 2 K e T A2 =(T1_)2 +—Q_]_'. 2 T2 is small. In this case, the Lorentzian in the denominator will fall and rise much faster than the Gaussian in the numerator. Under these circumstances, x can be set equal 129 to zero in the numerator and removed from the integral, which becomes MC (LU-U.) )2 - 0 dx — 0 2k T7112 no (181) _ _ B y w e f_co x2+A2 ’ or Mc2(w-wo)2 ' 2 -’_ lg 2kBTw y wA e . (182) The Doppler half-width occurs when Mc2(w-wo)2 2kBTw2 e = 1/2 (183) or (1 2) 2k T 2 2 n B m (w-wo) = . (18A) Mc2 Thus, the Doppler half-width is 2k T£n2 16D ”in—)1” -‘*’-. (185) 130 The other limiting case occurs when A is much larger than AmD. In this case the Gaussian numerator rises and falls much faster than the Lorentzian denominator. It is then sufficient to set v = 0 in the denominator of Equation (177) and remove the denominator from the integral. The integral then becomes equal to l and ? reverts to the expression for y given by Equation (130). For inter- mediate cases, 112;: when A N AwD, Equation (177) must be integrated numerically, as there are no suitable analytical expressions or approximations. For the OCS J = 2 + 1 transition at 298°K, AmD W 19 kHz. Since the linewidths in this work are all greater than N200 kHz, the extra broaden- ing due to the Doppler effect is negligible and has been ignored. CHAPTER VII EXPERIMENTAL Sample The OCS sample was obtained from the Linde Division of Union Carbide Corporation, the H2 and CO2 were obtained from Matheson Chemical Co., and CH3F was obtained from Peninsular Chemical Research Company. The OCS was analyzed at room temperature by means of a Beckman GC-2 gas chromato- graph with a six foot silicone column. The sample was found to contain 1.8% C02. The sample was not purified, but corrections for the CO2 impurity were calculated by assuming that Av/P for OCS/CO2 is 5.3 MHz/torr.109 The correction was necessary in the case of a pure OCS sample, but was insignificant for the gas mixture which contained about 3% OCS. Gas chromatography applied to the C02 and CH F samples showed no impurities. The H2 sample was 3 advertised as being 99.95% pure and was not checked. Pressure Meter The sample pressures were measured by an MKS Baratron 77M-XR pressure meter with a 77Hl pressure measuring head; the Baratron is a capacitance manometer. With this system, 131 132 sample pressures can be measured to 0.01 mtorr and the ac- curacy of pressure measurements above 10 mtorr appears to be of the order of 1%. Some of the transitions were recorded over a period of about A minutes, during which the pressure varied by a maximum of 0.0A mtorr. Others were recorded over a 7 minute period with a maximum pres- sure variation of 0.09 mtorr. The majority of the pres— sure variation was attributed to a slow leak in the sample cell. Microwave Spectrometer The spectrometer used for the linewidth measurements was a Hewlett-Packard 8A60A Molecular Rotational Resonance (MRR) spectrometer. Its radiation source is a backward wave oscillator, the frequency of which is stabilized to harmonics of the output of a frequency synthesizer by means of phase—lock loops. Stark modulation of 33.33 kHz and phase sensitive detection are used to increase the sensi- tivity. Just before and just after the sample cell, the radia- tion power was sampled through 20 db directional couplers. Microwave power measurements were obtained by using two Hewlett-Packard Model KA86A thermister mounts with a Hewlett-Packard Model A32A power meter. This combina- tion is capable of measuring power between lpW and 10 mW with a maximum uncertainty of about A% in the 0.03 mW 133 range. The power meter and thermister mount are designed so that the power measurements are free from drifts due to changes in the ambient temperature. The sample cell was X-band waveguide, 6 feet long, which encloses a metal septum. The microwave radiation was rectified and filtered by a crystal detector, and the resulting DC crystal current was passed through a resistor in a preamplifier. A low pass filter was placed in front of the detector to remove the effects of the J = A + 3 transition of 008. The frequency of this transition is approximately twice that of the J = 2 + l OCS transition and the transition occurs when the molecules absorb the second harmonic of the microwave radiation. The line- shape was observed under conditions of low and moderate radiant power. To insure that the crystal detector would always sense the same microwave power, attenuators were placed before and after the sample cell. The two attenua— tors were adjusted so that the sum of their attenuations was always the same. The lineshapes were acquired in digital form by means of a Digital Equipment Corporation PDP-8/E computer. The computer was interfaced to the spectrometer through a Heath EU-801E analog-digital-designer and an interface card designed by Steven Brown. The analog-to-digital (A/D) converter was a Heath EU-900-EB 10 bit system. The computer program averaged the results of an optional number of 13A readings of the A/D converter at each frequency for an optional number of sweeps through the spectrum. The time delay before making the first reading at each frequency and the time between readings at each frequency could also be selected by the operator. The results of thorough tests of the apparatus with pure OCS as a sample have been 0 described.lo’ Experimental Procedure Linewidth measurements were made for OCS mixed with various other gases. The samples were 3.1% OCS in CH3F, 1.6% OCS in CO2, and 1.9% OCS in H2. The relative amounts of the different gases were determined by pressure measure- ments with a mercury manometer in a particular section of a vacuum line. Before the linewidths were obtained, the best Stark voltage for recording the OCS transition had to be found. A Stark voltage of 1600 volts was used because at this voltage the Stark components do not overlap the zero-field transition. The lineshape of the J = 2 + 1 transition in the ground state of OCS was scanned from 2A32l MHz to 2A331 MHz at 50 kHz intervals. When CH3F was used as the perturb- ing gas, the A/D converter was read 10 times at each fre- quency with a 200 msec pause between readings. The initial reading at each frequency was made after a pause of 100 msec from the last reading at the previous frequency. 135 The spectrum was scanned once and the average readings were recorded on magnetic tape. When CO2 or H was used as 2 the perturber, the spectrum was scanned once, the A/D converter was read 20 times at 50 msec intervals, and there was a 200 msec pause before the initial readings at each frequency. For all of the systems, the time constant of the lock-in amplifier was 10 msec. Each of the three samples was studied at various total pressures between 20 and 120 mtorr. The OCS transition was recorded seven times at each pressure. The second, fourth, and sixth recordings were taken with approximately 10 m watts of microwave power entering the sample cell. The other four recordings were obtained under conditions of low incident power, about 22 db below the power level of the three high power recordings. For each lineshape recorded, the time of day, sample cell temperature, sample pres- sure, and temperature of the Baratron head were obtained. Before the linewidths of the transition were analyzed, some properties of the spectrometer were checked. The linearity of the attenuator and synchronous detector was checked. For the high power measurements it was necessary to determine the attenuation of the sample cell and other components of the spectrometer waveguide. This was done by sampling the microwave power before and after the wave- guide components by means of directional couplers. The ratios of the input and output microwave power with and without the element under consideration were compared. CHAPTER VIII RESULTS Low Power Linewidths The low-power lineshapes were fit to a Lorentz function of the form A(A\))2 + (710)2 + Bo + Bl(v-vo) , (186) where A, Bo, B1, V0 and Av are the adjustable parameters. In Equation (186), A is the amplitude of the transition, V is the microwave frequency, V0 is the center frequency of the transition, B0 is the background voltage, and B1 corrects for any slope of the background. The background corrections are introduced to account for variations in the output of the lock-in amplifier, which result from dif- ferences in electronic pickup at various microwave fre- quencies. Variation of the crystal current with frequency could also contribute to the background, but this effect was found to be negligible for our instrument.109 It has been found that the linewidth of the OCS J = 2 + 1 transition is nearly inversely proportional to temperature near 298 K.109 All of the lineshapes were 136 137 recorded at room temperature and throughout the day the temperatures varied by several degrees. Therefore, all linewidths were corrected to 298°K by multiplying the room temperature linewidth by T/298, where T was the measured temperature in degrees Kelvin. The sample cell had a slow leak of less than one micron per hour. Figure 1A shows a plot of pressure versus time for the linewidth measurements of OCS in CH3F at a total pressure of about 3A microns. For each set of linewidth measurements at one pressure, the pressure was plotted against the time and, by using the method of least squares, the sample pressure was found at the time the sample cell was closed (and hence no longer pumped on). This pres- sure was subtracted from the measured pressure at any time to yield the pressure of air in the sample cell at that time. The contribution to the linewidth from the air was found by multiplying the pressure of air by A.Al MHz/torr,88 the linewidth parameter for N2, since nitrogen is the main component of air. The linewidth of OCS in the perturbing gas was found by subtracting the linewidth due to air from the measured, temperature-corrected linewidth. Four low-power linewidths were observed at each pres- sure. After temperature corrections and corrections for air were made, the four linewidths were averaged. Table 17 lists the corrected linewidth and pressure without air for OCS in CH3F, OCS in C02 and OCS in H2. These points 3A.70 ‘11 3A.50 O 4,2 3 CD :1 :3 U) 8 311.30 1. LL, 3L).10 Figure 1A. 138 l l l l 50 7O 90 ’11 F4 O time/minutes Plot of pressure versus time for the line- width measurements of OCS in CH3F at a pres- sure of 3A microns. 139 Table 17. Average linewidth and pressure for the J = 2 + 1 transition of OCS in CH3F, CO2 or H2. Corrections have been made for the effects of temperature variation and the addition of air. OCS in CH3F Linewidth/kHz Pressure/u 212.6 17.9” 229.A 19.38 322.9 27.70 393.2 33.59 A80.9 Al.80 577.9 50.52 668.6 58.30 760.1 66.71 962.2 8A.02 OCS in CO2 Linewidth/kHz Pressure/u 210.6 38.29 261.8 A8.07 3A2.7 63.38 A52.9 8A.33 506.6 9A.6A 582.1 108.86 633.9 118.u7 OCS in H2 Linewidth/kHz Pressure/u 175.9 26.8A 236.6 36.98 281.5 AA.58 362.7 58.10 398.A 6A.02 A53.0 72.87 531.3 85.69 659.8 106.60 1A0 were least squares fit to a straight line, Av = Avo + (Av/p)p , (187) where A00 is the limiting linewidth at zero pressure and Av/p is the linewidth parameter. One way to perform this experiment would be to use a constant pressure of OCS and to vary the pressure of the foreign gas. The linewidth of the transition can then be plotted against the pressure of the foreign gas. This will result in a straight line, the slope of which is the line- width parameter of OCS with this collision partner. The intercept of the line will be the linewidth due to colli- sions of OCS with the wall and the linewidth contribution due to OCS colliding with other OCS molecules. However, in our experiment the sample always had the same percentage of OCS mixed with the foreign gas. Changing the pressure of the sample would change the amount of OCS as well as the amount of the foreign gas. In this case, a plot of the line- width against the total pressure would again yield a straight line. The intercept, however, is the linewidth due only to collisions of OCS with the wall. Therefore, the slope of the line in this case, will contain contributions both from the linewidth parameter due to OCS-OCS collisions and the linewidth parameter due to collisions between OCS and the foreign gas. Thus, the contribution due to OCS-OCS 1A1 collisions must be removed from the linewidth parameter of the gas mixture. To remove the effect of self—broadening, the linewidth is expressed as follows: _ A1 A1 AvT - Avwall + p1(p)l + p2(p)2 (188) where pl is the pressure of the OCS, p2 is the pressure of the foreign gas, (%¥)l is the linewidth parameter for col— lisions between OCS molecules and (%¥)2 is the linewidth parameter for collisions between OCS and the foreign gas. Equation (188) can be rearranged to Av -Av = T wall E; 93 _ _A_v_ AVT-Avwall pT of the linewidth against total pressure. ,Table 18 lists the where pT = pl + p2 and is the slope from the plot linewidth parameters taken from the slopes, the corrected linewidth parameters for collisions between OCS and the foreign gas, and the linewidth contribution due to col- lisions of OCS with the wall. Power-Broadened Linewidths The power-broadened transitions were fit by the two different methods which were discussed in the theory chapter. In one method the power broadened lineshapes were 1A2 Table 18. Linewidth parameters and linewidths due to wall collisions for the J = 2 + 1 transition of OCS in a foreign gas. The slope is from the plot of linewidth versus pressure. It contains the line- width parameter due to OCS-OCS collisions and that due to collisions of OCS with the foreign gas. The column (Av/p) contains only the line- width parameter due to OCS-foreign gas collisions. Perturber Slope/(MHz/torr)a (Av/p)/(MHz/torr) Avw/kHza CH3F 11.29 11.A5(17)a 9.9 11.9A 002 5.27 5.26(8)a 8.u 5.13d a H2 6.07 6.07(9) 11.2 6.10d OCS 6.03 aThis work. b H. Mader, J. Ekkers, W. Hoke, and W. H. Flygare, J. Chem. Phys., 62, A380 (1975). 0005, J = 1 + 0 transition. d B. Th. Berendts and A. Dymanus, J. Chem. Phys., 38, 1361 (1968). eR. A. Creswell, S. R. Brown, R. H. Schwendeman, J. Chem. Phys., 63, 1820 (1976). 1A3 fit to the following equation, which assumes that the absorp- tion coefficient is the sum of power broadened Lorentzian lineshapes for each m component of the transition: ‘/C§+c§ + CO S(w) = NA 2 1n 2 82 + Bl(w-w0) + BO , m \/ C0+C§e- + C 0 (190) where l -2 2 —1 T2 +01 + T2 -1 N = (2 2n , (191) "1 «022+Cie‘82 + T21 C8 = (w’w0)2 + T52 9 (192) ci = Tlmgluzegnz , (193) and the sum is over all the m components of the transition. Thevalueikn'Té is obtained from the average of the low power linewidth just before and just after the high power recording. The quantities A, B0, B1’ mo, and Tl/T2 are the adjustable parameters. As before, A is the amplitude of the transition, w is the angular frequency of the micro— wave radiation, wo is the center frequency of the transition, B0 is the background value, and B1 is the slope of the 1AA background. The sample cell length is 2 and its attenuation is 8. Inclusion of the factor N improved the convergence and stability of the iteration procedure. Both the low- power and moderate-power transitions were analyzed on the PDP8 minicomputer. At each pressure, three moderate-power transitions were recorded. Each transition was fit to the power broad— ened lineshape. The ratio Tl/T2 was obtained from the fitted parameter, Ep, defined as follows: E [(J+l)2—M2] 2 _ p 1 ’ [(2J+1)(2J+3)]2 T1 C = 0263 T; ' (19A) The quantity E is called the "effective power". p Figures 15, 16, and 17 show plots of the effective power versus pressure for OCS mixed with CH3F, OCS mixed with C02, and OCS mixed with H2. As mentioned, three line- widths were analyzed at each pressure. All three values for the effective power are shown. The bars indicate the size of the standard deviations of the fitted parameter. As there is no clear indication of a pressure dependence, all of the values of the effective power were averaged for each of the three gas mixtures and Tl/T2 was determined for each sample. The results are shown in Table 19. The same power broadened lines were also fit to a single Lorentzian lineshape from which the parameter 1A5 12.0_. I o S. 0» l 3 . O l C. I ' a) g _ a l > 11.0 | «H . 1.) c; ' ° | (D Q4 ‘ CH ‘1 L11 0 | O l 1 10.07' 1 l I l 20 A0 60 80 Pressure/mtorr Figure 15. Plot of effective power versus pressure for OCS mixed with CH3F. Three linewidths were analyzed at each pressure. All three values for the effective power are shown. The bars indicate the size of the standard deviation of the fitted parameter. 1A6 12.0 '- g; I : I I ‘ 3 o o ('1 O I o 11.0 " ° 1 ‘ A I i: I I I 4-) I C) o (1) CH CH I 10.0 I“ I 11, L, I l A0 60 80 100 120 Pressure/mtorr Figure 16. Plot of effective power versus pressure for OCS mixed with 002. Three linewidths were analyzed at each pressure. All three values for the effective power are shown. The bars indicate the size of the standard deviation of the fitted parameter. Effective Power 1A7 I‘ 13.0 __ I I I I 1 I .. I : 1 I 1 | ’ I I 2 12.0 " . I I 11 0 “- l l I I J A0 60 80 100 120 Pressure/mtorr Figure 17. Plot of effective power versus pressure for OCS mixed with H2. Three linewidths were analyzed at each pressure. All three values for the ef- fective power are shown. The bars indicate the size of the standard deviation of the fitted parameter. 1A8 Table 19. Average values for (Tl/T2)o and (q Tl/T2) measured for the J = 2 + 1 transition of OCS mixed with various foreign gases. All of the parameters were obtained at zero Stark field. Perturber (Tl/T2)o (q Tl/T2) CH3F 0.97:0.10 1.23:0.12 CO2 1.0310.10 1.30:0.13 H2 1.05:0.10 1.32:0.13 1A9 (q Tl/Tz) was obtained. In this case the power broadened lineshapes are fit to a function of the form I 2 2 CO+Cl + CO 2 ~82 ‘ICS+Cle + CO 11. T S(w) = AN in + Bl(w-wo) + Bo . (195) , or In Equation (195) CE = y: 2 2 2 u E T 2 (J+1) 1: 0 (q l) . (196) 3 (2J+1)(2J+3) 2 T— l—‘M The results for Tl/T2 and q Tl/T2 for the various collision partners of OCS, taken at zero field, are given in Table 19. Neither the power broadened lines which were fit to a single Lorentzian nor those fit to a sum of Lorentzian line- shapes showed a significant shift in the center frequency with pressure. CHAPTER IX DISCUSSION The values of T2 obtained from the low—power linewidth measurements of the J = 2 + 1 transition of OCS in a sample of OCS mixed with foreign gases compare favorably with those measured previously in the same mixtures by Berendts. These T2 values are known slightly less well than the T relaxa- 2 tion constant for a pure OCS sample, since the mixtures contain only 3% or less of OCS and the linewidths were ob- served in the same pressure range. The results of the low-power linewidth study of these same mixtures is shown in Table 18. The uncertainties in the linewidth parameter are probably dominated by an esti— mated il.5% uncertainty in the measurement of the absolute pressures. The linewidth parameter for the OCS—H2 mixture is not much different from the linewidth parameter for OCS self broadening. The same result was found for the linewidth of another linear molecule, N20.88 The linewidth parameter for the NZO-H2 mixture was found to be 5.15 MHz/torr, and for N20 self broadening it was found to be 5.22 MHz/torr. When H2 was used as the perturbing gas for non-linear molecules, however, the linewidth parameter was found to be much different for the mixture and for self broaden- ing.88 Also, our value for the linewidth parameter for 150 151 the OCS-CH3F mixture lies between that for pure OCS at 6.03 MHz/torr and that for pure CH F at about 20 MHz/ 88 torr. 3 Effective cross-sections for collisions, o, and effective collision diameters, b, were computed using the expressions107 0 = 2nAv/N VEel , (197) and b2 = o/w . (198) 3 In Equation (197), N is the number of molecules per cm rel is the mean relative velocity of OCS and the per- and turbing molecule. The effective collision diameter for 005 perturbed by H2 is A.56 K, b for 003 perturbed by 002 is 8.07 A, b for OCS perturbed by OCS is 9.01 A and b for OCS perturbed by CH3F is 11.A5 A. As expected, the effective collision diameters increase for increasing size and increasing dipole moment of the perturber. The high-power linewidths yield the ratio T2/Tl. In Chapter VI, TZ/Tl was shown to be related to collisional rate constants, as follows: 152 As mentioned previously, the rate constant due to adia- batic processes, r is expected to be small for micro- a, wave transitions. Therefore, since the experimental values of T2/T1 for the gas mixtures studied here are approxi- mately one, collisional transitions involving the states J = l and J = 2 are dominated by transitions to and from other states. This might well be expected for a molecule such as OCS, in which the energy separations of the rota- tional levels are small compared to the collisional energy. This may be contrasted to a system such as NH3, in which pairs of rotational energy levels, the inversion doublets, are separated by energies corresponding to microwave radia— tion, and are separated from other pairs of levels by energies corresponding to far-infrared radiation. Work done in this laboratory has shown that for NH3, Tl # T2. In fact, 1 < T2/Tl < 2. It would be useful for future work to examine the rotational relaxation constants T1 and T2 for other quasi two-level systems such as formaldehyde or HCN as well as for a molecule with a typical level struc- ture such as 802, to see if the value of T2/T1 is near 2 for two level systems and near 1 for multilevel systems. Mader £3 31.89 found that Tl/T2 was almost equal to l for the J = 1 + 0 transition of OCS in a mixture of OCS and CH3F. This result was obtained from a transient experi- ment in which a w-T—w/2 microwave pulse sequence was applied to the sample. The ratio of the peak absorption at the 153 end of the n/2 pulse to the peak absorption in the middle of the w pulse was plotted-against T to obtain T The 1. rotational relaxation parameter T2 was obtained from the envelope of the decay of the transient emission following a w/2 pulse. In a molecular beam experiment, Wang £3 £1.90 found the cross sections “1 and °II (which are thought to be closely related to Til and T52, respectively) to be quite different for the J = 2 + 1 transition of OCS in a mixture of OCS and CH3F. Our value of Tl/T2 for the J = 2 + 1 transition of OCS in a mixture of OCS and CH3F was very close to Flygare's value - near 1. The values of Tl/T2 found by Mader, gt_al,, by Wang, g£_§l., and in our laboratory are listed in Table 20. Upon comparison of the Tl/T2 value from our linewidth measurements of the J = 2 + 1 transition of OCS for the OCS—CH3F mixture to the OII/OI value from the molecular beam experiment, it is apparent that two different quanti- ties are being measured. Apparently, the ratio of the molecular beam cross sections, oII/oI, can be significantly different from Tl/TZ' The zero field linewidth value of (Tl/T2)O obtained here for the J = 2 + 1 transition is, however, essentially the same as the Tl/T2 value obtained by the transient effect measurements for the J = 1 + 0 transition. A comparison of the (Tl/T2)o values for the mixtures OCS-H OCS-002, and OCS-CH F is shown in Table 19. 2’ 3 15A .3t63 nurse I). .Amemav mbmm .mm ..nsnm .sano .e .BOHHoxsx .u .m can .cbesbmuctm .< .nauso .m .o .wsn3 .m .3 .en .Amsmav omms .mm ..nssm .sano .e .btnw3H3 .3 .3 but .6363 .3 .naaxxm .e .taemz .mn no.0 os.o ems omm sm.o sso.o mso.o oflmexflev stHHb HHe H6 waxes awe Hme mw\o HILLOpE HIoow:\HIB A | A 6H+mne rueaeasaq bH+NIe seam ssflsoafloz no+Hue .ueauum Antananae . w CmfimhdmmmE Quofizocfifi ocm .Emon amazooaoe .poommo pcofimcmop 39 UCBOM m mo new moo mo oLprHE m CH moo mo whoposmnmo cowpmxmaoo HMCOfiomuop on» mo comfimmano .om canoe 155 The uncertainty in the values of (Tl/T2)O from our moderate-power linewidth studies is estimated to be 110%. Most of the error is due to uncertainty in the magnitude and distribution of the microwave power. The values of (Tl/T2)o are nearly equal for all three of the mixtures. The trend in (Tl/T2) however, might be more accurate than 0’ the absolute magnitudes of the ratios since the power un- certainties are unaffected by the gas mixture and are thereforetflmasame for all three gases. Hence, the slight differences in (Tl/T2)o may be real. For the same experimental lineshapes of the three dif- ferent mixtures, the ratios of the rotational relaxation times were calculated in two different ways. The discus- sion just given referred to the calculation of (Tl/T2)o, the value of Tl/T2 at zero field when the lineshape is fit to a sum of power—broadened Lorentzian lines. This cal- culation followed the procedure first described by Karplus and Schwinger. Recently, it has been shown that it is more correct to fit the lineshapes to a single power-broadened Lorentzian, from which it is possible to extract the single parameter q Tl/T2' This calculation has been performed and the values of q Tl/T2 for each gas mixture are shown in Table 19. No method of interpretation of q Tl/T2 exists as yet; however, C. Bottcher has stated privately that he has estimated q to be about 1.3, which is what we obtain if Tl/T2 % 1. More theoretical work must be done before experi- mental q Tl/T2 values can be interpreted. PART III MICROWAVE SPECTRUM OF ISOPROPENYLCYCLOPROPANE CHAPTER X Introduction The internal rotation of various groups attached to cyclopropane has been studied by microwave spectroscopy in this laboratory and elsewhere. These compounds have also been studied by electron diffraction. As an extension of this work, we wanted to compare the internal rotation of the isopropenyl group with the internal rotation in vinyl- 110 111 cyclopropanecarboxaldehyde, cyclo- 112 cyclopropane, propanecarboxylic acid fluoride, and cyclopropylmethyl- ketone.113 Two conformations were found by electron dif- fractionllu and by microwave spectroscopy for cyclopropane- 111 and by microwave spectroscopy for cyclo— 112 carboxaldehyde propanecarboxylic acid fluoride. In one configuration the oxygen atom is trans to the cyclopropane ring, and in the other the oxygen is cis to the ring. It has been hypothesized that trans and cis conformers, rather than the expected trans and gauche conformers, are found in these compounds because of conjugation across the C-C bond between the ring and the carbonyl double bond. Vinylcyclopropane 156 157 was studied to see if the conjugation to the ring would occur when the attached n-system was a -C=C- group.110 Unfortunately, only the trans species of Vinylcyclopropane could be assigned in the microwave spectrum. However, an electron diffraction investigation has indicated that vinyl- cyclopropane occurs as trans and gauche conformers,115 and this result has been confirmed by ab_initio calculations.116 A recent study of 3-methyl-l-butene has shown that the con- formers in this molecule have the vinyl group trans and gauche to the plane of the isopropane carbons.117 Hence, it was of some interest to try to determine the conforma— tions in isopropenylcyclopropane. Unfortunately, as des- cribed below, the study of the microwave spectrum of this molecule proved even less conclusive than that of vinyl— cycloprOpane. Only one species could be assigned in the very weak spectrum obtained, and it does not appear to be possible to identify the conformation in this species. Because of the overall weakness of the spectrum, it did not appear to be worthwhile to continue this investigation further. The results obtained prior to termination of the study are reported in this chapter. Theory When none of the three principal moments of inertia of a molecule is equal to zero - and no two of the moments are equal - the molecule is an asymmetric rotor. 158 Isopropenylcyclopropane is such a molecule. As for the symmetric top case described previously, the moments of inertia are ordered such that Ia < Ib < Ic' In the asym- metric rotor there are two limiting cases: that of the near— prolate symmetric top when Ib approaches I and that of c, the near-oblate symmetric top when Ib approaches Ia“ The closer Ib approaches either Ia or I the more symmetric C, the molecule is. A parameter K is used to quantitatively describe the amount of asymmetry;l6 K = 2B - A - C ’ (199) A - C where A, B, and C are the rotational constants of the molecule. The parameter K + :1 as the molecules approach the limiting cases of an oblate or prolate top, respec- tively; K = 0 for the case of maximum asymmetry. In a sym- metric rotor, the energy levels are doubly degenerate in K. There is no such degeneracy in an asymmetric rotor, so that there are (2J+l) rotational energy levels for each value of J. Since K is not a good quantum number for an asymmetric rotor, it is necessary to find a way to identify these energy levels. Two systems are now in use. King, 118 Hainer, and Cross designate each level by the symbol J . The subscript K is the K value the energy level K_1K+1 -1 would have in the limiting case of the prolate symmetric rotor, while the subscript K+1 is the K value the energy 159 level would have in the limiting case of the oblate sym- metric top. The energy levels can also be identified by another system where each energy level is designated by JT where —J i’T i.J and I increases in the order of increas- ing energy of the levels for a given J. It may be shown that the two quantum number schemes are connected by the relation T = K-l - K+1° The Hamiltonian for the asymmetric top molecule in the rigid rotor approximation is H = APa + BP + CPc , (200) where Pa, Pb, and Pc are the components of the rotational angular momentum projected on the molecule-fixed principal axes of inertia. For an asymmetric rotor, it is not in general possible to obtain direct solutions to the Schrhdinger equa- tion. The procedure usually followed is to expand the eigen— functions in terms of an orthogonal set of wavefunctions such as those of the symmetric top. In this procedure, the asymmetric top wavefunction, wJTM, is a linear combination of the symmetric rotor wavefunctions wJKM’ = a (201) WJTM JAM JKM wJKM ’ where the aJKM'S are numerical constants. Substitution of Equation (201) into the SchrBdinger equation yields 160 (202) 23. HIP =EZaJM‘I’ ° JKM JKM JKM JKM K JKM After left multiplying both sides of the equation by the complex conjugate of wJK,M,integrating over all the co- ordinates, and rearranging, one obtains J K=-J Here K' = —J,...,+J; HK'K = , GK'K is the Kronecker delta, and A is an eigenvalue. The non-zero matrix elements in the symmetric top basis are found to be16 2 n 1 1 2 2 1 1 = [J(J+1)(—+ )+ K (—- —- -—-)l ’ ’ ’ ’ T Ia 1'; 1c Ia 1b (20A) and 2 (JgKgMIHIJ,Ki’2,M> = {1.8.— [J(J+l)-K(Kil)]l/2 x [J(J+l)-(Kil)(Ki2)]l/2(Tl; - 12L) . (205) a Since the Hamiltonian for the asymmetric rotor does not commute with P it is not diagonal in the symmetric top 2’ JMK representation. However, the asymmetric rotor Hamiltonian 161 does commute with P2 and P2; hence, the Hamiltonian matrix elenmnts are diagonal in J and M. The set of equations, ZEquation (203), can be solved for A, the allowed energy levels of an asymmetric rotor, by solving the secular determinant. Note that this matrix has elements on the diagonal and two off the diagonal. The matrix is infinite because of the infinite possible values of J. However, since it is diagonal in J, it can be factored into blocks along the diagonal, one for each value of J. Each submatrix has (2J+l) rows and columns, but it is possible to factor the matrix further from knowledge of symmetry properties of the asymmetric rotor. The ellipsoid of inertia of an asymmetric rotor is symmetric with respect to the identity operator E, of course, but also to a C2 symmetry operation about each of its principal axes of rotation. These symmetry operations: E, C3, Cg, Cg form a group known as the Four-group. As a result of these properties of the ellipsoid of inertia, the Hamiltonian for an asymmetric rotor commutes with the opera- tions of the Four-group. If the wavefunctions used to cal- culate the rotational energies belonged to this group, each one would be classified according to the way it behaves under the four symmetry operations. Since the Four-group contains four irreducible representations, there are four different symmetry species that the wavefunction can be classified under. Since the Hamiltonian operator is 162 invariant under all four symmetry operations, the only non- zero matrix elements of the Hamiltonian, , allowed by symmetry are those for which the two wavefunctions, mi and 73: are of the same species. Therefore the matrix can be factored into four parts, one for each symmetry species. If this is done, there are four matrices for each J; the order of each is about J/2. The symmetric rotor wavefunctions, wJKM’ do not belong to the Four-group; however, the Wang linear combinations of 16 symmetric rotor wavefunctions have this symmetry. Hence, before diagonalization of the Hamiltonian matrix, the sym- metric rotor wavefunctions are usually transformed to a new basis, as follows:16 8 = Xv . (206) where S is the new set of wavefunctions, w is the set of symmetric rotor wavefunctions, and the Wang transformation is 0-1 010 _ -1.1 X ‘ X ‘ "' (207) ./2 0 0.600 O H O H O 163 The four submatrices which result from the Wang transforma— tion of a submatrix for a given J are labeled E+, 0+, 0', and E", and the eigenvalues of each of these submatrices are often labeled accordingly. Allowed transitions are those for which the dipole moment matrix element does not vanish. As for the sym- metric rotor, the allowed changes in J are AJ = 0, :1 . (208) However, in the case of the asymmetric rotor, all three of these changes may cause absorption of radiation. The usefulness of the K-1’ K+l’ notation is that the changes in K__l and K+1 for allowed transitions are easily stated for each component of the dipole moment. Table 21 shows the selection rules in terms of changes in the subscripts K_l and K+1. Experimental, Results, and Discussion The sample of isopropenylcyclopropane was obtained from Chemical Samples Company, Columbus, OH,and was used as received. The spectra were obtained with the 33 kHz Stark-modulated Hewlett Packard Model 8A60A microwave spectrometer discussed in Part II of this thesis. The 119 reported molecular structures of propylene and methyl- 120 cycloprOpane were used as a basis to guess a probable 16A Table 21. Selection rules of the asymmetric rotor for permitted changes in the values of K-l and K+l° Type of Dipole Moment Transition Componentb AK_1 AK+1 a ua#0 0,12,... il,13,... b ub#0 i1,13,... 11,13,... C uc#0 il,13,... 0,i2,... aW. Gordy and R. L. Cook, Microwave Molecular Spectra: Chemical Applications of Spectroscopy, Part II in Tech- nique of Organic Chemistry, A.Weissberger, editor (Inter- science Publishers, New York, 1970). b ua where Ia < Ib < 10' lies along Ia’ ”b lies along Ib, and be lies along Ic, 165 structure for isopropenylcyclopropane; the structure ob- tained is shown in Figure 18. The microwave spectrum was observed in the R band between 30000 and A0000 MHz. By comparison with calculated spectra, it was possible to assign a number of R-branch transitions, all of which were a-type transitions. The frequencies of the assigned transi- tions are listed in Table 22. The frequencies were fit to the eigenvalues of the following non-rigid rotor Hamiltonian given by Watson:121 H = % (3+0)?2 + [A - NIH (B+C)]P: 1 2_ 2 __ A + 2 (B-C)(Pb PC) AJP 2 2 A 2 2 2 AJKP Pa - AKPa - 26JP (Pb-PC) 2 2 2 2 2 2 AKEPa(Pb-PC) + (Pb—PC)Pa] . (209) In this equation A, B, and C are the rotational constants, and AJ, AK, AJK’ SJ, and 6K are quartic centrifugal distor— tion constants. Table 22 also shows the difference in the observed and calculated frequencies from this fit. The rotational constants and centrifugal distortion constants obtained from the fit to the Watson Hamiltonian are shown in Table 23. No transitions from another conformation were assigned. 166 .1] ’Jo gure 18. Possible structure for isopropenylcyclopropane as determined from bond distances and bond angles of propylene and methylcyclopropane. 167 Table 22. Comparison of observed and calculated frequencies of isopropenylcyclopropane. Observed Transition Frequencya/MHZ Obs-calc/MHz 616 + 515 27887.90 -0.01 606 + 505 28362.50 -0.0l 651 + 550 295N9.27 -0.05 652 + 551 295u9.27 -0.01 6A3 + 5“2 29582.05 0.10 6142 + SUl 29585.21 0.02 615 + 51“ 30ul3.33 -0.02 717 + 616 32““5.01 0.01 707 + 606 32807.n6 0.01 726 + 625 3U07u.3l O 761 + 660 3““69.u0 0.03 762 + 661 3UH69.HO 0.03 753 + 652 3MH97.70 -0.06 7““ + 643 3u5u7-32 0.08 735 + 63“ 3u563.67 -o.05 7N3 * 6u2 3M557,95 0.01 73“ + 633 3h802.99 -0.02 716 + 615 35280.09 0.02 725 + 62“ 35620.86 -0.0l 818 + 717 36977.95 0 808 + 707 37233.36 -0.01 :71 + 770 39329.69 0.02 72 + 771 393 7.69 0.02 862 + 761 39ulh.8l —0.0U 863 + 762 39u1h.81 -0.03 8“5 + 7U” 39525.39 0.02 8““ 7&3 39554.50 0.03 aEstimated accuracy of 0.05 MHz. 168 Table 23. Rotational constants and centrifugal distortion constantsa of isopropenylcyclopropane. Parameter Value Standard Error A/MHz 6287.089 0.30 B/MHz 267u.222 0.01M C/MHz 2235.061 0.012 AJ/kHz 0.02 0.06 AJK/kHz u.o9 0.29 AK/kHZ 29. 96 dJ/kHz 0.03 .05 SK/kHz -l8.0 3.6 aDetermined in an IR axis representation: Aez, sex, CeY. 169 In order to identify the conformation observed, the rota— tional constants for the assumed structure were calculated for every ten degrees of rotation about the dihedral angle. Figure 19 shows a plot of the rotational constants versus the dihedral angle. The horizontal dotted lines in the figure are the fitted rotational constants obtained from the observed transitions. It is apparent that the varia- tion in rotational constants with angle and the derived constants are such that it is not possible to determine the configuration of the molecule from these data alone. 170 6800 — 6300 6000 Rotational Constants/MHZ C \ 22cc -_ """""""""""" _';:_:—=—=-=' 0 NO 80 120 160 Dihedral angle/degrees Figure 19. Rotational constants versus dihedral angle of isopropenylcyclopropane. The horizontal dotted lines in the figure are the rotational constants obtained from the fit of the observed transi- tions. APPENDICES APPENDIX A THE VOLTAGE VARIATION OF THE STARK FIELD IN THE MICROWAVE SAMPLE CELL The Stark voltage was found to differ from the voltage indicated on the dial by an amount which was not constant as a function of the dial reading. This voltage variation will be discussed for both high and low Stark fields. Low Fields First-order Stark effects for the ground state and v5 = l excited state of methylacetylene were measured at Stark fields corresponding to voltages - as read from the dial - between 0 and 180 volts. To calculate the correct voltage, the first-order Stark effect transitions were fit to a straight line by the method of least squares. If the x-axis is the dial voltage and the y axis is the transition frequency, the intercept and slope are shown in Table A-1. These slopes along with the measured frequencies and a zero-field frequency of 3A182.78 MHz for the ground state and 3A031.7A MHz for the v5 excited state were used in the straight line equation to calculate the voltages. These calculated voltages are shown in Table A-2. The actual voltage appears to be greater than the dial voltage 171 172 Table A-1. Least squares fit to a straight line of the dial setting versus the first-order Stark effect frequencies of the ground state and v5 excited state of the J = 2 + 1 transition of methyl- acetylene. Frequency Slope Intercept/MHz ground state k=il, m=¥1 0.263519 3u189.u8 k=il, m=i1 -o.263u2o 3u176.08 v5 excited state k=il, m=11 0.267u51 3ho38.5u k=il, m=tl -o.267213 3uo2u.95 173 Table A-2. Voltages of the first-order Stark effect of the J = 2 + 1 transition of the ground and v5 excited states of methylacetylene. The voltages have been corrected to include the offset of the dial reading. Calculated Voltages/Volts Dial Ground State vs Excited State Setting _ _ (Volts) k=t1, m=+1 k=il, m=i1 k=il, m=+l k=il, m=i1 50 75.AA 75.32 75.38 75.A5 60 85.61 85.A2 85.55 85.AA 70 95.63 95.U7 95.39 95.58 80 105.61 105.A2 105.AA 105.38 90 115.AA 115.10 115.31 115.19 100 125.6A l25.b3 125.52 125.AA 110 135.AA 135.26 135.31 135.32 120 1A5.61 1A5.A0 1A5.A5 1A5.A3 130 155.59 155.A6 155.43 155.A1 1A0 165.A5 165.59 165.3A 165.60 150 175.55 175.A2 175.73 175.51 160 ------ 185.A1 ------ 185.2A 170 195.70 195.20 ------ 195.12 180 205.A5 205.3A 205.35 205.6“ acaic' d 25.55 25.u3 25.uu 25.u1 17“ by a near constant 25.A volts. The measured frequencies and voltages of the dial reading plus 25.A volts were then fit to a straight line with the results shown in Table 11. Again, the v5 excited state dipole moment is calculated to be 0.7954 Debye. High Fields Second—order Stark effect measurements were obtained with dial voltages between 800 and 1600 volts. In this range the offset is no longer constant, but appears to increase with increasing field. The frequency shift for a second-order Stark effect can be expressed as _ 2 2 Av2 - “DADS (A-l) where AD depends upon the quantum numbers and rotational constants. After expressing the electric field as the total voltage divided by the cell spacing, the frequency shift becomes (A-Z) OI" 175 2 S Av 2 _ 2 2 2 - vd + 2vdvO + vO (A—3) ADub For the molecule OCS, the dipole moment is known and AD can be calculated. The Stark shifts were measured at dif- ferent dial voltages. The only unknowns are the offset voltage and the cell spacing. By using the M = 0 Stark shifts for dial voltages of 800 volts and 1000 volts, the two simultaneous equations can be solved to yield an off— set voltage of 38 volts. Repeating the process for lAOO volts and 1600 volts yields an offset voltage of 71 volts. It is apparent that at high Stark fields the difference between the dial setting and the apparent voltage is not constant. APPENDIX B LINEWIDTH OF THE LASER STARK TRANSITIONS A Doppler-broadened Gaussian lineshape, I, can be expressed as -Y(Vi-V€)2 I = Ae where A and y are constants for each transition. width at half height, AVD, can be found from 2 2 ' e ’ which can be rewritten as 2n 2 = ‘Y(AVD)2 (B-l) The half (B-2) (B—3) The frequency of a transition at an electric field a can be expressed as avo v = v0 + (jfir)€(€-€O) , (B-“) where v0 is the transition frequency at electric field 80' Therefore 176 177 (vi-v6)2 =[vi- vo- (— v) (e-eo)]2 as e If V0 is chosen to be equal to Vi, 2 = 80 2 _ 2 (Vi-V8) (FE-)8“: 80) The lineshape function, Equation (B-l), then becomes 3 2 2 -Y(3:)E(€-EO) I = Ae or —8(€-eo)2 I = Ae , where B = figs)E Thus, 1 —BAeg 2‘9 ’ where AeD is found to be AvD (B-5) (B-6) (B-7) (B-8) (B-9) (B-lO) (B-ll) 178 A first—derivative lineshape was used in the laser Stark experiment. The first derivative of I is given by BI —B(€-EO)2 SE = - 28(e—eO)Ae . (B-l2) The points of maximum and minimum intensity, 6 and e' 2 p p’ can be found by setting the derivative §—% equal to zero, Be 2 2 -B(e-e ) I g_§ = [-2B+u82(€'80)2)]Ae O 3 (B-l3) as 32 and ——§ equals zero when Be —28 + AB2(e-eo)2 = o (B—lu) Therefore 6: -e =+ —1— (13.15) p 0 ‘ 2B ' Substitution of Equation (B-3) into Equation (B-9) Yields 2 22.? _ 1n (ae)e 2 (B—16) (BVD) So that the full linewidth is expressed by e _ 5' = (_§_ql/2 .3:£L. (3-17) F p 2n2 ' APPENDIX C THE 002 AND N20 GAS LASERS Introduction Lasers that have been used to excite sample gases in laser Stark spectroscopy include as the active gas: C0, C02, N20, D20, HCN, He-Xe, He—Ne, and some carbon—l3 species. In this study a C02 laser and an N20 laser were used. 122’123 consists of two basic components, the A laser active medium and an optical cavity. The active medium in this case is the C02 or N20 gas and the optical cavity is made of two reflectors, one mounted on either side of the active medium. Initially, the CO2 or N20 molecules are excited by an external source of energy. Photons are emitted which in turn stimulate the emission and absorption of photons by the gas molecules. If the external energy source excites more molecules to an upper state than a lower state, population inversion is achieved and laser action occurs. Because of the design of the laser cavity and because stimulated emission is involved, laser radiation has some rather unique properties. The C02 and N20 lasers used emit power continuously and can be made to be mono- chromatic - i.e., emit radiation at one frequency. Also, the laser emission is coherent so that the divergence of 179 180 the output beam is very small and the radiation is all in phase. Hence, a small beam of radiation can be passed through a long distance and accurately aimed at a small target. The laser can be designed so that the radiation is plane polarized. Theory of the C09 Laser The active medium of a C02 laser is a gas mixture consisting of about 5% 002, 15% N2, and 80% He. The nitrogen gas is used to excite the carbon dioxide. A partial energy level diagram for CO2 and N2 is shown in Figure C-l. The rotational levels have been left out for simplicity. An electric discharge raises the N2 molecules to excited vibrational levels. Because N2 has no oscillating dipole moment, vibrationally-excited N2 in the ground electronic state cannot decay through electric dipole radiation. Deactivation must occur through collisions with other mole— cules and with the walls. The energy difference between the 001 level of CO2 and the v = 1 level of N2 is only “18 cm-l, so the excited N2 molecules can easily transfer their energy to the C02 molecules upon collision. Any C02 molecules in the 001 level may decay with emission of radiation to the 100 or 020 energy level and from there to the 010 level. Collisions with helium help depopulate the 010 level. With enough excitation it is possible to have more molecules in the 001 state than in either the 181 001 1 C02 ground state N2 ground state Diagram of the C02 and N2 vibrational energy levels which are of interest for the C02 laser. The energy difference labeled A i m18 cm'l. The laser transitions corresponding to label B are centered around 961.0 cm-1 and those corresponding to label C are centered around 1063.8 cm'l. 182 100 or the 020 states. At this point, population inversion is achieved. When a C02 molecule undergoes spontaneous emission from the 001 to the 100 level, the photon given off stimulates another molecule to undergo this transition and the effect snowballs. The laser transitions are from individual rotational levels in the 001 vibrational state to individual rotational levels in the 100 state at a 1 or from a rotational level in the frequency near 961 cm- 001 state to a level in the 020 state at a frequency near 106A cm'l. The transitions near 961 cm'1 have higher gain and will undergo laser action unless conditions are made to 1 transition. favor a 1064 cm- Stimulated emission is in the same direction and has the same phase as the stimulating radiation. Photons which hit one of the mirrors at either end of the laser may be reflected back into the optical cavity. Photons which never strike a mirror eventually leave the optical cavity and are lost. Thus, the only photons which survive to stimulate a large number of new photons are those which travel back and forth between the mirrors. Output from the laser is achieved when some of these photons are allowed to escape through a small hole at the center of one mirror or through one mirror which is partially transmitting. The output of the laser Just described would probably not be monochromatic, since the radiation would arise from transitions between many rotational energy levels. To 183 obtain monochromatic laser radiation, one of the mirrors may be replaced by a diffraction grating. By changing the angle of the grating, different laser frequencies can be selected. The CO2 and N20 lasers have transitions spaced 1 covering most of the 9-11 u region. The every 1 or 2 cm- intensities of the transitions vary according to the Boltz— mann distribution of the rotational levels. The most intense transitions are those near P(18) or R(l8) of each band. Tables C-1 and C-2 contain lists of the CO2 and N20 laser lines used in this investigation. Their frequencies are given in megahertz and wavenumbers. By adjusting the length of the optical cavity, an individual CO2 or N20 laser line can be continuously tuned by about 125 MHz. This is the only manner in which these lasers are continuously tunable. Description of the CO2/N20 Laser Used in this Study Figure C-2 is a schematic diagram of the laser used in this investigation. The laser may be used with either CO2 or N20 as the active gas. The gas mixture, pressure, and the method of stabilization employed are different for the two gases. In either case, the laser gas mixture flows through the system and is rapidly pumped away. The plasma tube confining the gases is made of pyrex, 2.8 cm in diameter and 2.6 m long. It is surrounded by a water Jacket with a 18A Table C-l. List of the CO2 laser lines of the 001-100 band used in this experiment. Their frequencies are given in wavenumbersa and megahertz.b Laser Line Frequency/MHz Frequency/cm-l P(2A) 28196922.58A 9H0.5A81026 P(26) 281A1165.95A 938.6882616 P(28) 2808A669.777 936.8037519 P(30) 28027U31.849 93U.8935002 P(32) 27969AA9.737 932.960A251 P(3A) 27910720.770 931.001A376 P(36) 278512A2.035 929.017AA05 P(38) 27791010.378 927.0083287 P(AO) 27730022.39u 92u.9739885 P(A2) 2766827A.U28 922.9lu298l P(AA) 27605762.565 920.8291267 P(A6) 275A2782.627 918.7183352 ac = 299,792,A56.2 i 1.1 m/sec. bThe frequencies 0f the 002 laser were calculated with the parameters determined by K. M. Evenson, J. S. Wells, R. F. Petersen, B. L. Danielson and W. G. Day, Appl. Phys. Lett. 22, 192 (1973) and F. R. Petersen, D. G. McDonald, J. D. Cupp, and B. L. Danielson, Phys. Rev. Lett. 31, 573 (1973). 185 Table C—2. List of the N20 laser lines used in this experi- ment. Their frequencies are given in mega- hertza and wavenumbers.b Laser Line Frequency/MHZ Frequency/cm-l P(5) 28020012.99A 934.6u703u P(6) 2799AA88.53A 933.795629 P(7) 27968863.122 932.9u0858 P(8) 279U3l36.871 932.082722 2(9) 27917309.89o 931.221227 P(lO) 27891382.28u 930.356375 P(ll) 2786535A.160 929.A88170 P(12) 27839225.620 928.616616 P(l3) 27812996.763 927.7“1716 P(lA) 27786667.689 926.863A72 P(15) 27760238.A93 925.981889 P(l6) 27733709.268 925.096969 P(17) 27707080.105 92A.208716 as. G. Whitford, K. J. Siemsen, H. D. Riccius, and G. R. Hanes, Opt. Comm. 133 70 (1975). be = 299,792,u56.2 :1.1 m/sec. 186 .mUSpm mane CH pow: momma 0m2\moo one no Empmmfio ofipwEmnom .mlo madman mcfipmgm :ofiuomnmmfio w m wHLH s pfixm was mocmspcm waspxfie mam m mmNHqum J CH Hmcwfim m \xw _ maddsm Lozom 187 5 cm outside diameter. Attached to the plasma tube about 20 cm from each end are ground glass Joints to which the electrode compartments are sealed with black wax. Each electrode is a hollow stainless steel cylinder attached to a 1.3 mm tungsten rod. The tungsten rod carries the elec- trical connection through the glass wall of the electrode compartment. Attached to each end of the plasma tube are NaCl windows, purchased from the Harshaw Company. The windows are 5.1 cm in diameter, 6 mm thick, flat to within 11 um and the faces on each are parallel to 30 seconds of arc. These windows are oriented at Brewster's angle,l2u so that the output radiation is plane polarized with the plane of the electric vector parallel to the floor. At one end of the optical cavity is a PTR Optics ML 303 orig- inal diffraction grating that is flat, has 150 grooves/mm, and has first order reflective efficiency of about 95%. A partially-transmitting germanium mirror is at the other end of the optical cavity. This mirror has a 10 m radius of curvature, a front surface which is 80% reflecting at a wavelength of 10.6 um, and an antireflection-coated rear surface. (A 95% reflecting mirror is also available.) The length of the optical cavity is 398 cm and can be changed slightly by applying a voltage to a piezoelectric crystal which can be controlled by a Lansing Research Model 80—21Alaser'stabilizer. The laser is supported by three “.1” m long Invar rods which are 3.2 cm in diameter. The 188 rods are held in place at each end by "U"-shaped aluminum blocks. Three sets of smaller aluminum blocks support the Invar rods at one-quarter, one-half, and three-quarters of their length. The power supply for the laser is a 30 kV, 50 ma unit manufactured by the Megavolt Corporation. The current regulator was designed for our use by Martin Rabb. The laser may be run on a commercially available gas mixture which contains 5% of either CO2 or N20, 15% N2, 80% He, and in the case of the N20 laser, a small amount of CO. However, the intensity of a laser line depends in part upon the gas mixture. Consequently, the gases were mixed at the inlet to the laser tube. By mixing the gases ourselves, it was possible to obtain laser output on a weak line which would not lase with premixed gases. The laser was designed as a free-flowing system. Copper tubing con- nects the gas tanks to fine metering valves, one each for N2, He, and CO, and one for either CO2 or N20. A fifth metering valve can close the plasma tube to all the gases. The gas mixture enters the plasma tube at one electrode and leaves at the other. A Precision Scientific Company Vac Torr 1000 (lOOOl/sec) pump was used to keep the gas flowing. PVC reinforced tubing and Pyrex tubing is used for all the connections between the micrometer valves and the laser. 189 Alignment of the Laser at 10.6u* Alignment of the laser is begun by insuring that the centers of both irises inside the cavity lie along the apparent center of the plasma tube. To do this, one iris is closed, covered with a Kimwipe,and illuminated from behind by a flashlight. A mirror is placed at the opposite end of the laser in order to be able to look down the plasma tube. The position of the second iris is changed until it is centered around the white circle of light. The same procedure is then used to center the other iris. Once the irises are centered, a He-Ne laser is used to aligntflmemirror and diffraction grating. The partially transmitting mirror of the C02 laser is removed. The He-Ne laser is set outside the optical cavity of the CO2 laser and adJusted so that the He-Ne radiation shines down the center of the plasma tube, through both irises, and onto the diffraction grating. The lines of the diffraction grating should be vertical and the grating itself placed at such an angle that the reflection of the He-Ne beam off the grating forms a horizontal row of spots. The brightest spot is the thirteenth-order reflection. The grating is rotated until the seventeenth—order reflection is *In the operation of any laser, it should be remembered that laser radiation can be harmful to human tissue, es- pecially the eye. Therefore suitable precautions must be taken. 190 superimposed upon the incoming He—Ne laser beam. The radia- tion from the seventeenth-order reflection now travels back toward the He-Ne laser, strikes its mirror and is reflected back toward the diffraction grating. If the alignment has been well done, this very faint spot can be seen on the diffraction grating and is moved on top of the primary spot. The partially transmitting mirror of the CO2 laser is replaced without disturbing the position of the He—Ne laser. Two spots from the He-Ne laser can be seen on the back of the CO2 mirror. The strong spot is the He-Ne beam strik- ing the mirror. The other is the light from the He—Ne laser reflected from the C02 mirror to the He-Ne mirror and back. The partially transmitting mirror of the CO2 laser is adJusted until the weak spot is superimposed upon the stronger one. Operation of the COD Laser The alignment for the seventeenth line of the He-Ne laser is very close to the alignment needed to cause lasing on the P(36) line of the 001-100 band of the CO2 laser. Before the laser is started, the irises within the optical cavity should be open and the stabilizer set to fast sweep. Slightly more than one torr CO2 and about 6 torr He are added to the plasma tube. (The pressure meter 191 is between the plasma tube and the pump, so the pressures given may not be those in the plasma tube.) The high vol- tage power supply is turned up to 20 k volts and the current to 20 m amps. As the pressure of N2 is slowly increased to a few torr, the plasma color changes from violet to rose and lasing should occur. The output beam of the laser is conveniently observed by chopping the beam and observing the detected beam by means of an oscilloscope. Sometimes a slight displacement of the diffraction grating is needed for the laser to lase. The PZT voltage is manually adJusted to the top of the gain profile and He is added to the plasma tube until the total pressure is about 10 torr. The CO2, N2, and He pressures are adJusted to give the maximum laser power. The intracavity irises should be closed as small as possible and the diffraction grating and partially- transmitting mirror adJusted to give the maximum signal. This process is repeated until the irises can be closed no farther without losing the signal, even after the grat— ing and mirror are adJusted. The cross—sectional shape of the laser beam should be the simple circle of the TEM00 mode. The following is a sample of different mode patterns: ' 0 u :: 192 If one of the other modes is obtained, the simple mode may be restored by decreasing the size of the irises. If necessary, the laser power can be increased by increasing the gas pressure. Operation of the N20 Laser It is slightly more difficult to operate the N O 2 laser than the CO2 laser. The frequency of the P(32) line of the 001-100 band of the CO2 laser at 932.9569 cm"1 is almost the same as the frequency of the P(7) line of the 1 001-100 band of the N20 laser at 932.9368 cm- Hence, this is an easy place to change over from the CO2 to the N20 laser. To operate the N20 laser, the irises should be com- pletely open. A very small amount of N20 and about 5 torr of He are added to the plasma tube. The high voltage of the power supply is turned up to 20 kV, while the current is increased to 10 ma. The discharge should glow blue if there is little or no N20. More N20 is added until the plasma is deep purple. If too much N20 has been added, a gap - a clear space - forms in the plasma near the grounded electrode. (Lasing will not occur when too much N20 is present.) The addition of a little N2 will cause the N O 2 laser to lase. The amounts of N20, N2 and He are adJusted until the maximum signal is obtained with the total pres- sure about 7 torr. Then CO is added until the signal is 193 maximized; it will Just about double in size. Again, the laser is aligned by adJusting the diffraction grating, mirror, and irises. Laser Stabilization The frequency of a laser is determined by the active medium — in this case, the CO2 or N20 gas - and the cavity length. When lasing occurs, a standing wave pattern is set up between the mirror and the diffraction grating. Excluding the end surfaces, there are n-l nodal planes in the standing wave pattern. Large changes in the cavity length will change the number of nodes in the standing wave. When this happens the laser is said to change modes. The frequency within a given mode is inversely proportional to the cavity length, £L 2L ’ g:- = (C-l) where v is the frequency, n-l the number of nodes (exclud- ing those at the boundary), 0 is the speed of light, and L is the cavity length. When the cavity length is slightly changed, the frequency changes in inverse proportion. Therefore, to stabilize the laser, the cavity length must be kept such that the laser oscillates at the top of the gain profile for the mode that it is in. 19A The stabilization system is shown in Figure C-2. The length of the cavity is varied by sinusoidally modulating the position of the mirror. As the cavity length varies, the laser frequency and hence the laser power varies as shown in Figure C—3. The modulated component of the laser power is detected by a pyroelectric detector and enters the Lansing stabilizer where it is preamplified and ob- served by means of phase sensitive detection. This provides a discriminator signal which is zero when the cavity length allows the laser to oscillate at the top of its gain pro- file; that is, at its maximum power. This occurs at fre- quency v0, when the slope of the laser power gs frequency equals zero. The discriminator signal is appreciably large at frequencies larger or smaller than V0’ and the slope of power Xi frequency is positive for frequencies smaller than v0 and negative for frequencies larger than v0. The stabi- lizer produces a correction voltage to be applied to the piezoelectric crystal to change the length of the laser cavity. The result is to stabilize the cavity length so that lasing occurs at the top of the gain profile at fre— quency v0. A slight variation of this method can also be used to stabilize the CO2 laser. Modulation of the cavity length causes a modulation in the output power of the laser, which causes the impedance of the plasma tube to oscillate. 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