.. ,. fin“, :fiumfimm .. .9. «Pi « ‘ 7.... «at... 3H3“... .. 161...,» . “.mruw. J .. .. .r -2 32.5.35. FRVAWWH. 1. .. 1qu Gilt-1’21“ .i ‘ 3.: 0...: t : rant 1. .II‘ :3: III... - I) V )‘1 . . \a. .l x. . 51.2.. :13... - hfivfluflfi . 5. E- . V r. ‘ 1 ' gisfi'sfiit‘; BF .3.....m I} “-16le Q IGAN STATE UNNERS T Ii“ llllillllllllllllllllllllll \Mlfllll 3 1293 01387 16 This is to certify that the dissertation entitled Studies of Collisional Energy Transfer in Methyl Fluoride, Ammonia, Cyc10pr0pane and Fluoroform-D presented by Glenda M. Soriano has been accepted towards fulfillment of the requirements for Ph . D . degree in Chemi S tY‘Y Major professor Date April 20, 1995 MSU Lt an Affirmative Action/Equal Opportunity Institution 042771 LIBRARY Mlchlgan State Unlverslty PLACE N RETURN BOXtoromavothb chockomtrom your record. TO AVOID FINES Mum on or before date duo. DATE DUE DATE DUE DATE DUE JIL l: MSU IoAn Manama ActIoNEqUII Opponunlty Instltwon Wm: STUDIES OF COLLISION AL ENERGY TRANSFER IN METHYL FLUORIDE, AMMONIA, CYCLOPROPAN E AND FLUOROFORM-D By Glenda M. Son'ano A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemistry 1995 ABSTRACT STUDIES OF COLLISIONAL ENERGY TRANSFER IN METHYL FLUORIDE. AMMONIA, CYCLOPROPAN E AND FLUOROFORM-D By Glenda M. Soriano Four -1eve1 infrared-infrared double resonance studies under conditions of population and polarization modulation of the pump beam were performed for 13CH3F and 15NH3 in the absence of a homogenous field. The population modulation lineshapes consist of a narrow spike superimposed on a broad Gaussian lineshape of the expected Doppler width. The effects of collisions were described by a collisional kernel that is a sum of Keilson-Storer functions. The polarization modulation lineshapes show only a sharp spike. Sharp spikes and polarization modulation effects are observed only when the k quantum number of the pump and probe transitions are the same. A theoretical treatment shows that when a saturating pump and weak probe are used in four-level double resonance experiments under population modulation, the signals are sensitive only to the first three statistical tensor orders n = 0 (population), 1 (orientation) and 2 (alignment). For polarization modulation experiments with plane polarized beams, the signal is sensitive only to the n = 2 tensor order while for circularly polarized radiation, the signal is sensitive only to the n = 1 tensor order. The Jones calculus was used to predict the four-level double resonance absorption coefficients as functions of the velocity component of the molecules in the direction of the pump beam for different probe beam polarizations. The results of the fits to a four-level double resonance theory were much improved when the ratio of the rate constant for n > 0 to the rate constant for n = O is 2/3 for 13CH3F and 1/3 for 15NH3. Further analysis showed the importance of long-range dipole-dipole interactions, V-V mechanisms, elastic reorientation and realignment for collision-induced energy transfer in 13CH3F and 15NH3. Double resonance studies were also performed on cyclopropane and CF3D. Rotational energy transfer in CF3D clearly obeys dipole selection rules. Three level double resonance lineshapes for cyclopropane did not give the expected sharp spike. Possible reasons may include very fast V-V rates and collisional coupling of m-states. To my parents, Eduardo and Lolita iv ACKNOWLEDGMENTS I am sincerely grateful to my advisor Professor Richard H. Schwendeman for his infinite patience, valuable guidance and support throughout my stay here at Michigan State University. I am especially thankful for the numerous discussions that stimulated my enthusiasm for learning. My working with him has enabled me to see the marks of a true mentor. I am also very thankful for the kindness and support given by Professors Katharine and Paul Hunt. I am fortunate to have them as advisors and teachers. My introduction to experimental laser spectroscopy was made a lot easier and more enjoyable with the help of my former groupmate, Dr. Quan Song, who was very generous with his time and patience. The people in the Machine Shop, Glass Shop and Electronics Shop have been very helpful in keeping our laboratory in good shape. Special mention should go to Marty Rabb whose boundless patience and energy have helped us deal with a lot of our instrument problems. I also wish to thank the Hunt group--Xiaoping Li, Pao Hua Liu, Sandjaja Tjahajadiputra and Ed Tisko-- for fun times and wonderful discussions about things "science and beyond science". The comforting presence of my Arbor Drive friends (Leticia Carpio, JoAnn Palma and Sue Saguiguit), Spartan Village friends (Lilian Ungson, Rudie Altamirano, Madie and Rey Ebora, Rene and Vicky Cerdefia, Nick and Monie Uriarte) Cheribeth Tan and Manny Viray, made the long and winding road to the Ph.D. degree a lot less tortuous. I will forever be grateful for all the laughter and very happy memories. I am also thankful for the sincere friendship of Kate Noon and Teresita de Guzman. Finally, I wish to thank my family-- my parents, Eduardo and Lolita; my uncle and aunt, Carlos and Erlie; my aunt, Lucy; my cousins Zernan, Zandro and Zynthia—- for all the understanding, prayers and non-stop support they have provided me all these years. Table of Contents Contents List of Tables List of Figures Introduction Chapter One A. Infrared-infrared double resonance B. Collisional selection rules 1. A] selection rules 2. Ak selection rules 3. Am selection rules C. Reorientation of the total rotational angular momentum vector 1. Creation of anisotropic population distributions a. State selection through magnetic and electric fields b. State selection through photodissociation c. Alignment of J vector during rotational cooling d. Optical pumping of molecular states 2. Statistical tensors and tensor opacities Chapter Two Theory A. Velocity selection through pumping with an infrared laser B. Collision kernels C. Jones matrix for four-level double resonance Page 10 12 14 14 15 15 16 18 25 30 32 1. Introduction 2. Jones matrix for sample pumped by plane—polarized radiation a. Z-polarized pump and Z-polarized probe (parallel) configuration b. Z-polarized pump and X-polarized probe (perpendicular configuration) 0. Alignment modulation 3. Jones matrix for sample pumped by circularly-polarized radiation a. RCP pump and RCP probe (RCP/RCP) b. RCP pump and LCP probe (RCP/LCP) c. Orientation modulation D. Velocity dependence of absorption coefficients E. Description of four-level fitting programs Chapter Three Experiment A. Polarization experiment B. Polarization experiments with foreign gases C. Determination of precise pump offset frequencies for 15Nl-l3 Chapter Four Results A. 13CH3F-13CH3F collisions B. 13CH3F-foreign gas collisions C. 15NH3-15NH3 collisions D. Precise pump offset frequencies for 15NH3 32 32 36 37 38 38 40 41 41 42 43 45 50 50 55 79 88 101 Chapter Five Discussion A. Rate Equations B. Impact parameter dependence of rate constants C. Connection to dipolar transition rates via tensor opacities D. Connection to Am selection rules Chapter Six A. Double resonance experiments with cyclopropane (C3H6) B. Double resonance studies with deuterated fluoroform (CF3D) Chapter Seven Summary and Conclusions Appendix Bibliography 107 110 113 116 120 135 149 155 167 List of Tables Table 1 Jones Vectors and Jones Matrices 2 List of transition and laser frequencies in MHz used for the pump and probe in double resonance studies in 13CH3F. 3 Widths of the transferred spikes and corresponding for l3CHqF 4 Numerical results obtained from fitting experimental four- level double resonance lineshapes to a four-level double resonance theory 5 Pump and Probe Frequencies for 15NH; 6 Pump and Probe Transitions Used in the Determination of Precise Pump Offset Frequencies for 15NH; 7 Comparison between FTIR and IR-IR double resonance frequencies for some transitions in the v2 band of 15NH3 8 Pump and probe transitions in the vm fundamental band of cyclopropane used in the double resonance studies. 9 Comparison of calculated and observed offset frequencies of some transitions in the v10 fundamental band of cyclopropane. 10 Pump and probe transition frequencies used in the double resonance studies of CFqD. 11 Comparison of calculated and observed frequencies for the v5 fundamental band in CF3D. Page 33 57 73 81 92 105 106 122 133 136 148 List of Figures Figure Page 1.1 A four-level IR-IR double resonance energy level diagram 7 that shows two different pump-probe combinations. A transition in the fundamental band (v = 1.. v = 0) is pumped while a transition in the (a) fundamental or (b) a transition in the hot band (v = 2- v = 1) is probed. 1.2 Creation of anisotropy in the m-state populations through 19 optical pumping with polarized radiation. The rotational transition from J = 1 to J = 2 is pumped by:(a) plane- polarized radiation, and (b) circularly polarized radiation. There are (2] + 1) degenerate m-sublevels for each J. The topmost diagram in (a) shows pumping with plane- polarized radiation with its electric field parallel to the space-fixed Z-axis and obeying a Am = 0 selection rule for the transition, while the bottom diagram shows pumping with plane-polarized radiation with its electric field perpendicular to the space-fixed Z-axis and obeying a Am = :1 selection rule. The topmost diagram in (b) shows pumping with right circularly polarized radiation with Am = -1 selection rule while the bottom diagram shows pumping with left circularly polarized radiation with selection rule Am = +1. Assuming that there is no collisional transfer among the degenerate m-sublevels, the m-sublevel populations in (a) are the same for m = -m, while the m-sublevel populations in (b) are different for each m. 2.1 Saturation pumping with an infrared laser. Pumping a 29 fundamental transition with a weakly saturating IR laser creates a "Bennett hole" in the ground state velocity distribution of molecules. This hole is seen in the upper level of the pump transition as a "Bennett spike". 3.1 Block diagram of the IR-IR double resonance spectrometer used for population and alignment 46 modulation experiments. The pump and probe beams are in a counter-propagating geometry. Both lasers are frequency-stabilized by locking to a Lamb dip in the fluoresence of C02 inside a fluoresence cell (FC). The probe laser (Laser2) passes through a CdTe electro-optic crystal modulator (Mod) which is simultaneously irradiated by powerful microwaves to produce the weak sidebands that are used as the probe. The microwaves are 100% amplitude modulated at 33kHz. A plane polarizer P2 allows only the sidebands to go through, while the carrier laser, which is used in the Lamb dip, is reflected off to a mirror. A beam splitter (BS) after P2 sends a portion of the sidebands to a reference detector (RD). The other portion of the sideband goes through the sample cell into a partially—transmitting mirror which reflects the sidebands off to the signal detector (SD). The pump laser passes through a CdTe crystal modulator that acts as a N4 plate and converts plane polarized radiation to circularly-polarized radiation. Application of very high positive or negative voltage (HV) converts plane polarized radiation to either left or right circularly- polarized radiation. A Fresnel rhomb (Rh) converts circularly-polarized radiation back to plane-polarized radiation. A plane polarizer (P1) is used for population modulation because it transmits only one type of plane- polarized radiation during half of the pump modulation cycle. P1 is removed during alignment modulation experiments. 3.2 Modification of the IR-IR double resonance spectrometer 49 for use in orientation modulation studies. The Fresnel rhomb (Rh) is now found after the CdTe crystal modulator (Mod.) to produce a circularly polarized probe beam. For population modulation studies, a mechanical ch0pper that chops the pump laser is placed after the CdTe 7J4 plate. During orientation modulation, the chopper is removed, the pump laser is switched alternately between rcp and lcp through alternate application of high positive and negative voltages. The switching is done by an electronic switch designed by Martin Rabb at Michigan State University. 3.3 Arrangement of pump and probe beams for determination of precise pump offset frequency. The pump and probe beams counter-propagate and co-propagate simultaneously. A mirror was placed near the entrance of the sample cell to reflect back the pump laser. 4.1 Energy level diagram for 13CH3F. The QR(4,3) pump transition lies in the v3 fundamental band (v3 = 1°— 0), while the probe transitions are all in the 2v3~—v3 hot band. 4.2 Experimental four-level double resonance lineshapes taken under conditions of population modulation using plane polarized radiation for the pump and probe beams. The QR(4,3) transition in the v3 fundamental band of 13CH3F was pumped by a 120602 laser while the QP(6,3) transition in the 2v3--v3 hot band was scanned by a sideband system.The bigger of the two spikes was obtained when the pump and probe beams have parallel planes of polarization while the smaller spike was obtained when the beams have perpendicular planes of polarization. 4.3 Experimental four-level double resonance lineshapes taken under conditions of population modulation using plane polarized pump and probe radiation. The QR(4,3) transition in the v3 fundamental band of 13CH3F was pumped by a 120602 laser while the QP(7,3) transition in the 2v3~v3 hot band was scanned by a sideband system.The bigger of the two spikes was obtained when the pump and probe beams had parallel planes of polarization while the smaller spike was obtained when the beams had perpendicular planes of polarization. 4.4 Transferred spikes obtained from four-level double resonance experiments under conditions of population modulation using plane polarized radiation. The QR(4,3) transition in the v3 fundamental band of l3CH3F was pumped by a 12C1502 laser while the QP(8,3) transition in the 2v3o—v3 hot band was scanned by a sideband system. The bigger spike was obtained when the pump and probe beams had the same planes of polarization while the smaller spike was obtained when the beams had perpendicular planes of polarization. 52 56 59 60 61 4.5 Four-level double resonance lineshapes obtained by 62 pumping the R(4,3) transition in the v3 fundamental band of l3CI-I3F while probing the R(10,3) transition in the 2V3 -—v3 hot band region using plane-polarized radiation. The lineshape with the bigger transferred spike was obtained when the planes of polarization of the pump and probe beams were parallel, while the lineshape with the Smaller spike was obtained when the planes of polarization of the pump and probe beams were perpendicular. The R(10,0), R(10,1) and R(10,2) transitions are strongly overlapped. 4.6 Four-level double resonance lineshapes obtained under 64 population modulation using circularly-polarized radiation for the pump and probe beams. The R(4,3) transition in the v3 fundamental band of 13CH3F was used as the pump transition while the P(6,3) transition in the 2v3~v3 hot band was used as the probe. The bigger spike was obtained when the pump and probe beams had different circular polarizations (rep/lcp) while the smaller spike was obtained when the pump and probe beams had the same circular polarization. 4.7 Four-level double resonance spectra taken under 65 conditions of population modulation with circularly polarized pump and probe beams. The two spectra were obtained while pumping the R(4,3) transition in the v3 fundamental band and observing the P(7,3) transition in the 2v3~v3 transition in the hot band. The lineshape with the bigger spike amplitude was obtained when the pump and probe beams had different circular polarizations while the lineshape with the smaller spike was obtained when the pump and probe beams had the same circular polarization. xiv 4.8 Experimental four-level double resonance lineshapes 66 observed when pump and probe radiation were both circularly polarized. The R(4,3) transition in the v3 fundamental band was pumped while the P(8,3) transition in the 2v3~v3 transition in the hot band was probed. This corresponds to a AJ = 3 collision-induced rotational transition. The bigger spike was obtained when the pump and probe beams had different circular polarizations (rcp/lcp) while the smaller spike was obtained when both pump and probe beams had the same circular polarization. 4.9 Four-level double resonance spectrum of l3CHqF 68 obtained by pumping the R(4,3) transition in the v3 fundamental band while probing the P(8,3) transition in the 2v3°—v3 hot band. Results of fitting the experimental lineshape to a theoretical four-level double resonance lineshape show that the transferred spike may be represented by two components. These two components are described by two Keilson-Storer collision kernels. The difference between the experimental and calculated spectra is shown at the bottom of the curves. 4.10 Four-level double resonance spectra obtained under alignment modulation.The R(4,3) transition was pumped 70 in all of these experiments, while different probe transtions were observed. It is very evident that these lineshapes do not contain the broad components that make up the lineshapes under population modulation conditions. 4.1] Four-level double resonance spectra obtained under 71 orientation modulation. The R(4,3) transition was pumped while the P(6,3), P(7,3) and P(8,3) transitions were scanned. Superimposed on each of the spectra are the results of a theoretical fit to a single Keilson-Storer function. XV 4.12 Experimental four-level double resonance spectra 77 obtained when both pump and probe beams were plane polarized. The top spectrum shows a scan of the P(6,0) to P(6,3) transitions in the 2v3~v3 hot band of 13CH3F under population modulation while the QR(4,3) transition in the v3 fundamental band was pumped. The bottom spectrum shows a scan of the same probe transitions but taken under alignment modulation. Only the P(6,3) transition gave an alignment modulation effect. 4.13 Experimental four-level double resonance lineshapes 78 obtained with circularly polarized pump and probe beams.The top spectrum shows a scan of the P(6,0) to P(6,3) transitions in the 2v3v—v3 hot band of 13CH3F under population modulation while the QR(4,3) transition in the v3 fundamental band was pumped. The bottom spectrum, which was taken under orientation modulation is similar to the alignment modulation spectrum given in Figure 4.12 which shows that only the P(6,3) transition that corresponds to Ak = 0 shows an alignment modulation effect. 4.14 Comparisons between experimental and generated four- 80 level double resonance lineshapes. The first picture contains experimental lineshapes obtained under population modulation using plane-polarized radiation, while the second picture contains lineshapes that were generated by assuming that the rates of transfer of population, alignment and orientation are equal. The third picture which resembles the experimental picture more closely than the second, was generated by assuming that the rate constants for the transfer of alignment and orientation are 2/3 the rate of transfer of population. 4.15 Four-level double resonance spectra for 13CHqF 83 obtained under population modulation using plane polarized pump and probe beams. Each spectrum includes three other curves which are results of a theoretical fit to 3 Keilson-Storer functions. The pictures show the effect of foreign gas-CHqF collisions on the sizes of the broad Gaussian component and the transferred spike. xvi 4.16 Four-level double resonance lineshapes obtained with 85 plane polarized radiation the pump and probe beams. The topmost picture contains lineshapes that were obtained with pure 13CHqF, while the three other pictures contain lineshapes that were obtained with a mixture of l3CHqF and foreign gases. For all these spectra, the lineshapes that have the bigger transferred spike were taken when the planes of polarization of the pump and probe beams were parallel, while the lineshapes that have the smaller spike were taken when the planes of polarization were perpendicular. The R(4,3) transition in the v3 fundamental band was used as the pump, while the P(6,3) transition in the 2v3~v3 hot band was used as the probe. 4.17 Alignment modulation spectra of pure 13CH—rF and 87 13CHqF in foreign gases H2, He and Xe. The pictures include the experimental four-level double resonance lineshapes and the theoretical fit of the spectra to a single Keilson-Storer function. 4.18 Energy levels for a harmonic oscillator. The 89 wavefunctions for the vibrational energy levels from v = 0 to v = 5 are shown in this figure. The odd-numbered vibrational levels 1, 3 and 5 have nodes at x = 0, while the even-numbered energy levels, 0.2 and 4, do not have a node at this position. As a result, a potential at the x = 0 position tends to perturb the even-numbered levels more than the odd-numbered levels. This perturbation causes each even-numbered level to be pushed up to the odd- numbered level directly above it. 4.19 Four-level double resonance energy level diagrams for 91 15Ner. The asR(2,0) and asQ(5,4) pump transitions are in the v2 fundamental band while all the probe transitions are in the 2v2(—v2 hot band. For the asR(2,0) pump transition, half of the levels are missing as required by the uncertainty principle. xvii 4.20 Four-level double resonance lineshapes obtained under 93 population modulation using circularly polarized radiation for the pump and probe beams. The lineshapes in (a) were obtained while pumping the asR(2,0) transition in the v2 fundamental band and probing the saR(1,0) transition in the 2v20—v2 hot band. The lineshapes in (b) were obtained while pumping the asQ(5,4) transition in the v2 fundamental band while observing the saR(6,4) transition in the 2v2~v2 hot band. For both (a) and (b), the bigger transferred spikes are results of using different circular polarizations for pump and probe beams. 95 4.21 Four-level double resonance lineshape obtained under orientation modulation in ISNHq. The asR(2,0) transition in the v2 fundamental band was pumped while the saR(1,0) transition in the 2v2~—v2 hot band was scanned. Superimposed on the experimental lineshape is a theoretical lineshape that was a result of fitting the spectrum to a single Keilson-Storer collision kernel. 4.22 Double resonance lineshapes taken under population 97 modulation using plane polarized radiation for the pump and probe beams. The asQ(5,4) transition in the v2 fundamental band of ISNH'; was pumped while the saR(6,4) (a), and the saR(7,4) (b) transitions in the 2v2«-v2 hot band were scanned. The lineshape with the larger transferred spike was obtained when the pump and probe beams had perpendicular planes of polarization. 99 4.23 Four-level double resonance lineshapes obtained under (a) alignment modulation and (b) orientation modulation while pumping the asQ(5,4) transition in the v2 fundamental band and observing the saR(6,4) transition in the 2v2~v2 hot band. The smooth curves superimposed on the experimental lineshapes are results of fitting the experimental lineshapes to a single Keilson-Storer collision kernel. xviii 4.24 Theoretical and experimental four-level double 102 resonance lineshapes in 15NHq. The spectra in the middle are the experimental lineshapes. The top spectra are the theoretical lineshapes that were generated by assuming equal rate constants for the transfer of the n = 0, 1 and 2 spherical tensor combinations of the m-state populations, while the bottom spectra are the lineshapes that were generated by assuming that the rate of transfer of the n = l and 2 spherical tensor combinations are 1/3 the rate constant for the transfer of n = 0 spherical tensor combination. 4.25 Three-level double resonance lineshape for 103 determination of precise pump laser offset frequency. The two spikes that appeared on either side of the center frequency of the probe transition resulted from having the pump beam counter- and co-propagate sirnulataneously with the probe beam. 6.1 Folded three-level double resonance energy level diagram. 124 The pump (heavy arrow) and probe transitions have a common upper level. 6.2 Single (top) and three—level double resonance spectra of 125 cyclopropane. The single resonance lineshape was obtained by scanning the RR(9,6) transition. The three- level lineshape was obtained by pumping the RP(11,6) transition and probing the RR(9,6) transition. 127 6.3 Single resonance (top) and three level double resonance (bottom) of cyclopropane. The single resonance lineshape was obtained by scanning the RR(20,2) transition. The double resonance lineshape was obtained by pumping the RQ(21,2) transition while observing the RR(20,2) transition. 6.4 Four-level single (top) and double (bottom) resonance 128 lineshapes obtained by pumping the RQ(21,2) transition and probing the RR(14,2) and RR(16,8) transitions simultaneously. 6.5 Four-level single (top) and double (bottom) resonance lineshapes obtained by pumping the PP(23,l2) transition and probing the RR(18,8), RR(20,12) and RR(23,22) transitions. 6.6 Four-level single (top) and double (bottom) resonance spectra of the v10 fundamental band of cyclopropane. The single resonance spectrum was obtained while scanning the RR(23,20) transition. The double resonance spectrum was obtained while pumping the PP(23,12) transition and observing the RR(23,20) transition. The sample pressure was 400 mT. 6.7 Single resonance (top) and three-level double resonance in the v5 fundamental band of CF3D at 22 mTorr. The single resonance scan included the RQ(9,0) and RQ(24,l) transitions. The three-level double resonance spectrum was obtained by pumping the l’Q(24,l) while observing the RQ(24,1) transition. 6.8 Four-level double resonance lineshapes of CFqD obtained by pumping the PQ(24,1) transition while probing the RQ(23,1) transition. 6.9 Four-level double resonance in CFqD obtained by pumping the pQ(24,1) transition while observing the R(2(25,1) transition. 6.10 Four-level double resonance lineshapes of CFqD obtained by pumping the PQ(24,l) transition while probing the RQ(22, 1) transition. 6.11 Four-level double resonance in the v5 band of CF3D. This double resonance lineshape was obtained by pumping the PQ(24,1) transition and probing the RQ(24,2) transition. The collision-induced rotational energy transfer occurred with A] = 0 and A(k-l) = 1. 6.12 Two-level double resonance signal for CFqD. This lineshape was obtained by pumping and probing the RQ(30,l) transition in the v5 fundamental band. 130 131 138 139 140 141 142 144 6.13 Four-level double resonance in the v5 band of CF3D. 145 This lineshape was obtained by pumping the RQ(30,1) transition while probing the RQ(29,4) transition. The collision-induced energy transfer occurred with AJ = -1 and A(k-1) = 3. Introduction Due to the rapid advance in high resolution spectroscopic methods over the past years, much progress has been achieved in the field of collision-induced energy transfer. Knowledge of the factors that govern collision-induced processes has applications in areas like astrophysics (1), remote-sensing (2, 3), study of the earth’s climate and monitoring of air pollutants (4). The variety of experiments on collision-induced processes in the gas phase that can be performed, combined with the great computational speed attainable in many computers, has proven to be very useful in testing the numerous theoretical methods that have been developed in this particular field. The results of such experiments have led to better understanding of the interaction between molecules and have provided considerable insight into the nature of the intermolecular forces (5). A large fraction of the more recent results is due to the use of lasers as sources of very monochromatic radiation that make possible an effective and precise selection of molecular states. The ability to specify and select the internal states of the molecule prior to collision, and the accuracy with which other states can be probed, have significant implications for the type of information obtainable from the study of collision-induced processes. The number of molecules that may be studied is limited only by the type and frequency range of available laser sources. For molecules like CH3F and NH3, many strong ro-vibrational transitions are in the range of infrared laser sources such as the C02 lasers, including near coincidences with C02 laser frequencies (6). Because of this fact, there now exists a vast literature on collision dynamics studies of these two molecules. These include early experiments by Berrnan and co-workers who used a coherent optical 2 transient method to probe collisions between 13CH3F molecules (7, 8) and laser Stark spectroscopy studies by Johns and co-workers (9) of collision-induced Lamb dips in CH3F, NH3 and HZCO. In the infrared laser spectroscopy laboratory at Michigan State University (MSU IR laboratory), Matsuo and co-workers observed rotational energy transfer as a result of self collisions in 13CH3F (10) and 15NH3 (11) through the presence of "transferred spikes" obtained in four-level double resonance experiments, while collisions of l3CH3F with different foreign gases-were studied by Song and Schwendeman (12). Similar results were obtained by Shin and.co-workers (13) in their double resonance experiments with 12CH3F. The pioneering work of Flynn and co-workers culminated in a paper by Flynn and Weitz (14b) who gave a detailed vibrational energy transfer map for CH3F using laser fluorescence experiments. Flynn and Weitz (14a) also investigated the vibrational relaxation of CH3F in the presence of buffer gases. A later work by Sheorey and Flynn (14c) gave insights on the collision dynamics of intermode energy flow in CH3F. S. K. Lee et al. in our laboratory have recorded and analyzed several transitions in the v3 fundamental and 2v3~— v3 hot band transitions of 13CH3F through the method of infrared- microwave sideband laser spectroscopy (15), and recently, more accurate vibration-rotation parameters for the v3, 2v3o— v3 and v3 + v6 - v6 bands of 13CH3F were obtained by Papousek and co-workers by Fourier transform infrared spectroscopy (16). A Fourier transform infrared study of CH3F (17) by Betrencourt er al. yielded molecular constants for the v2 + v3 and v3 + v5 combination bands. The 15NH3 molecule has also been the subject of many spectroscopic studies including electric dipole moment determination for the v2 vibrational state (18), infrared- microwave two-photon spectroscopy on the v2 band (19, 20), high-resolution spectroscopy of the v2 = 2a—v2 = ls band (21), and a study by Sasada (22) of the microwave inversion frequencies of 15NH3. 3 One of the most powerful tools that is being utilized in the study of collision- induced rotational energy transfer is the technique of double resonance. This method utilizes two sources of radiation: one moderately powerful source pumps molecules from one state to another while a second weaker source probes the resulting population changes (23). The main concern of the study reported in this thesis is to relate collision- induced rotational energy transfer to velocity changes that occur upon collision. ChapterOne of this thesis gives a brief account of the collisional selection rules that were found for rotational energy transfer in systems involving polar molecules. Our laboratory has utilized the technique of infrared-infrared double resonance for monitoring collision-induced rotational transitions in gas phase molecules, particularly 12CH3F, 13CH3F and 15NH3. Chapter Two describes the processes of optical pumping and velocity selection through pumping with infrared radiation. More recently, there has been an increasing number of studies aimed toward observation of the correlation between energy transfer and reorientation of the molecules during collisions. Reorientation here refers to the change in the spatial orientation of the total rotational angular momentum vector with respect to a space-fixed axis. In quantum mechanics, this refers to changes in the value of the m quantum number, which describes the projection of the total rotational angular momentum vector on a space-fixed Z axis. Studies have sought methods by which to obtain rate constants (24, 25) and cross sections (26) for this reorientation process. Among these are experiments in which a specific m state within a specific rotational level was selected, either through optical pumping (27) or through the Stark effect (28). The nature of the double resonance experiments in our laboratory is such that all the m sublevels of a particular ro-vibrational state are degenerate. So, we need to have a convenient means to describe the population in these degenerate m sublevels and to relate these populations to quantities that describe the orientation of the J vector. One way of doing this is by writing the density matrix for the sytem that is being investigated. In Chapter Two, we use the concept of a multipole 4 expansion of the density matrix of the system. In that chapter there are two types of multipole expansions that are useful for characterization of the reorientation process. One is an expansion using "statistical tensors" and another is an expansion using "tensor opacities". The statistical tensor expansion describes the reorientation process in terms of different tensor orders that relax independently, while the tensor opacities give the reorientation picture in terms of multipolar relaxation rates (25). The infrared-infrared double resonance spectrometer and other experimental details are described in Chapter Three. A brief discussion of the optical elements used in the polarization studies is also included in this chapter. Chapter Four is divided into two main sections. The first section gives the results from double resonance studies for 13CH3F while the second section discusses the results for 15NH3. A discussion of the results for NH3 and CH3F will be given in Chapter Five. Two other molecules, cyclopropane (C3H6) and CF3D, were the subject of rotational energy transfer studies in our laboratory. The recent unavailability of certain isotopic C02 gases used in our semi-sealed C02 laser, particularly the mixed isotope gas, C150180, prevented us from doing more detailed experiments on the two molecules. The experimental results that were obtained for these two molecules will be given in Chapter Six. Finally, a summary of the results of this thesis and recommendations for future study will be presented in Chapter Seven. Chapter One A. Infrared-infrared double resonance In double resonance experiments, two sources of radiation interact simultaneously with a molecule. One radiation source acts as the pump while the other source acts as the probe. Early double resonance experiments for relaxation studies included microwave- microwave double resonance, which was pioneered by Oka (1) and used to study collision dynamics in molecules such as NH3 (1, 29, 30) and ethylene oxide (31). This technique was also applied to the measurement of rotational relaxation in the OCS molecule (32). Later on, different combinations of radiation for double resonance were used, such as infrared-microwave double resonance (33, 34, 35), which was used to study molecules like N H3 (33, 36, 37) and CH3OH (34); infrared-radio frequency double resonance (38, 39, 40); and infrared-ultraviolet double resonance, which was used to study vibrational and rotational relaxation mechanisms in D2CO (41). The method of infrared-infrared (IR- IR) double resonance has been used extensively in the study of collision-induced rotational transitions, particularly in molecules such as CH3F (10, 12, 13, 38, 42-44,), NH3 (1, 5, 11, 37, 38, 45-48) , HZCO (38), CO (49), and the non-polar molecule C02 (50). Through the use of double resonance methods, much has been learned about selection rules underlying collision-induced processes. From these selection rules, the type of intermolecular forces that exist between the molecules may be inferred (l, 5). Rotational relaxation times and rate constants for rotational, vibration-rotation, and vibration-vibration (V-V) energy transfers may be extracted from double resonance experimental data. Double resonance methods have also been used to confirm the 6 correctness of transition assignments (34), and the method of IR-[R double resonance was used in the determination of the electric dipole moment for the v2 vibrational band in 15NH3 (18). The use of infrared radiation has increased the number of molecules that may be studied by double resonance methods because there is no longer a requirement for a permanent dipole moment. Because the higher frequency leads to greater thermal equilibrium population differences, infrared pumping creates much greater deviations from the thermal equilibrium populations of the pumped levels than pumping with microwaves. Also, since the Doppler profile in the infrared region is wider than the inhomogenous broadening for low pressure gases, it is possible to restrict the pumping to molecules belonging to a narrow velocity group. As will be explained in Chapter Two (velocity selection through optical pumping), only molecules that are travelling in the direction of the pump beam in a specific velocity group are selected by the pumping process. Figure 1.1 shows a typical four-level double resonance energy level diagram. Direct information on population and velocity changes in the v = 1 state may be obtained in four-level double resonance experiments when the pumped transition lies in the fundamental band (v = 1(— 0) and the probe transition lies in the hot band (v = 2<—l). This is a result of the fact that collisionally-induced vibrational energy transfer is slower than collisionally-induced rotational energy transfer. Thus, in a fundamental pump-hot band probe combination, only collisionally-induced transitions between rotational levels in v = 1 affect the intensity of the probe; the number of molecules in v = 2 is usually negligible. By contrast, in a fundamental pump-fundamental probe combination, the four-level double resonance probe signal will contain effects of rotational energy transfer from the rotational states in the ground vibrational state as well as in v = 1. In a four-level double resonance experiment, the probed levels do not share a common level with the pump transition, and are coupled to the pump transitions only through collisions. The velocity distribution in these rotational states can be probed by (b) I (a) v = o —— Figure 1.1. A four-level IR-IR double resonance energy level diagram that shows two different pump-probe combinations. A transition in the fundamental band (v = 1~ v = 0) is pumped while a transition in the (a) fundamental or (b) a transition in the hot band (v = 2—— v = l) is probed. 8 recording the spectrum of a vibration-rotation transition originating in one of the states with weak, non-saturating infrared radiation. It is shown in the section on collision kernels in Chapter Two that the velocity distribution in the probed state reflects the pumped upper state velocity distribution as it was modified by collisions. The lineshapes obtained from this form of infrared-infrared four level double resonance (strong pump beam/weak probe beam) with low pressure gas samples have two components: a broad Gaussian lineshape and a sharp transferred spike superimposed on the Gaussian (10, 11). The Gaussian component has the expected Doppler width and a center frequency equal to the resonance frequency of the probe transition, while the transferred spike has a center frequency that is offset from this resonance frequency. This makes it relatively easy to separate the two components. The results of analyses of the four-level double resonance lineshapes provide evidence that the Gaussian components come from near resonant vibrational-vibrational (V-V) energy transfer between unpumped ground state molecules and molecules in the upper level of the pump transition. Thus, the molecules that reach the v = 1 level by V-V transfer are not part of the population originally pumped by the radiation. Since these molecules originated from a Gaussian distribution in the lower state of the pump transition, it is expected that they maintain this distribution. Experiments with CH3F show that the appearance of the broad Gaussian components does not follow any selection rules. On the other hand, the transferred spikes appear to result from molecules that were originally part of the pumped population and underwent rotational energy changes with little change in velocity. It is readily apparent that these transferred spikes obey certain selection rules. B. Collisional selection rules Oka's pioneering work in microwave-microwave double resonance (1) established the selection rules for collision-induced energy transfer which may be discussed as follows. 1. A] selection rules For molecular collisions in systems involving polar molecules, the long-range dipole-type selection rules , A] = 0 or :1, appear to be dominant. Here, J is the total rotational angular momentum quantum number, and A] is the difference in the J values of one of the levels pumped and one of the levels probed. In our laboratory, most of the work has been performed with a fundamental vibration-rotation transition (v = l— 0) as the pump transition and a hot band transition (v = 2— 1) as the probe. Because of the very small initial population of v = 2 levels and the slow rate of collisionally-induced transfer of vibrational energy, the v = 2 populations are usually ignored. Studies performed in this laboratory have shown that the sharpest transferred spikes are observed when A] = :I: 1, or when dipole-type selection rules are followed. However, transferred spikes have been observed for AJ as high as 17, with the widths of the spikes increasing with AJ (10). A study performed elsewhere on fluoroform-d self collisions and collisions with buffer gases shows a propensity for the A] = :t: 1 selection rule (51). Collisions that change the value of J by one unit are interpreted as being due to long-range dipole-dipole interactions that can be characterized by a dispersive force with a r3 dependence. Transitions that occur with A] = :2 have also been observed, and in time- resolved infrared-infrared double resonance studies with both NH3-NH3 and NH3-Ar systems, collision-induced transitions were observed with Ak = 0, AJ = 2-3, and AR = 3, A] = 1-4 (5). These A] > 1 transitions may be attributed to higher order terms in the multipole expansion of the intermolecular potential. Detailed kinetic analysis of the collision-induced processes showed that the rate constants obtained for the AJ= 2-3 transitions are not equivalent to those predicted for sequential A] = 1 transitions (5). Simulations of infrared-microwave double resonance effects in DeLucia's laboratory (52) favor a strong contribution from Al >1 collisions. A microwave triple resonance experiment by Lees and Oka (1, 53) provided direct evidence of single A] = 2 transitions 10 in the NH3-H2 and CH3OH-He systems. No such transitions were observed for NH3-NH3 and CH3OH-CH3OH collisions. The A] > 1 transitions may be the result of successive collisions that change I by one unit or the result of one or more collisions that change J by several units. Because of the observation that the width of the transferred spikes increases with increasing value of AJ (10), work done in our laboratory on transferred spikes in l3CH3F suggests that AJ > 1 spikes are the result of a cascade of A] = 21:1 transitions or the result of smaller impact parameter collisions than AI = j: ltransitions. Intensity measurements of collision-induced signals with A] up to 9, from various kinds of infrared-radio frequency double resonances in CH3127I, suggest that the observed A] > 1 transitions are results of multiple collisions rather than a single collision (38). In the same study, it was argued that the estimated duration of collisions is too small to allow direct A] > 1 transitions to take place. 2. Ak selection rules For CH3F, NH3 and other C3,, molecules, collision-induced rotational transitions were shown by Oka and co-workers to follow definite selection rules for the change in k, the quantum number that describes the projection of the total rotational angular momentum vector J on the molecule-fixed axis of highest rotational symmetry. By convention, k is a signed integer (including zero) while K = Ikl. The selection rule shown by Oka et al. for C3,, molecules is Ak = 0 or 3n, where n is a positive or negative integer. The Ak = 3n selection rule for collision-induced rotational transitions in C3,, molecules is rigorous and holds even for strong collisions (1). This is because the rule is based on the symmetries of permuting identical nuclei and the nuclear spin state of a molecule cannot be changed even by hard collisions that change the other quantum numbers. For a C3,, symmetric top molecule with only 3 protons off axis, two symmetry species exist. In a totally symmetric vibrational level, the A symmetry species (ortho) 11 corresponds to k = 3n states and the E symmetry species (para) corresponds to k at 3n states. After a molecule has undergone a collision, its nuclear spin state remains the same. Therefore, its change of state will be governed by the following selection rules: Ak=3n, or A¢4>AandE®E This selection rule mixes by collisions the ortho states with k = ...-6, -3, 0, +3, +6, or the para states k = ...-T—5, :2, i 1, :4,... Another way of saying this is that in the absence of an external inhomogenous magnetic field, there can be no interconversion between the ortho and para states of a molecule as a result of binary collisions. Wilson was the first to give a general method for calculating the statistical weights of the energy levels of some polyatomic molecules (54). A description of the method based on modern molecular symmetry considerations is given by Bunker (55). An apparent violation of the spin statistics-allowed Ak = 3n processes was observed when energy transfer occurred via the near resonant V-V swapping mechanism.When energy transfer occurs via the near resonant V-V swapping mechanism, apparent transfer of population between states of opposite symmetry in the same vibrational level can occur. This apparent transfer between the A and E symmetry species occurs because of the manner in which vibrational energy is exchanged between two molecules via a V—V swapping mechanism as given, for example, by the following process in CH3F: CH3F(A,v3 = 1) + CH3F(E,v3 == 0) t—V—‘V—t CH3F(A,V3 = 0) + CH3F(E,V3 = 1). The most probable collision partner for a molecule in the pumped upper state is a molecule in the ground vibrational state. Therefore, in a V-V energy transfer process, a molecule in the pumped upper state returns to the ground state by dumping its excess energy to the ground state molecule. The excess vibrational energy that is absorbed by the 12 ground state molecule causes it to reach the excited state. However, this process preserves the symmetry of each collision partner (52) so that an excited state molecule belonging to an A symmetry species will decay back to a ground state that has A symmetry. The apparent A to E transfer is observed because the molecule that is now in the lower level of the probe was not the molecule that was originally pumped into the upper state of the pump transition. 3. Am selection rules In order to determine the collisional selection rules on the m quantum number, which describes the projection of J on a space-fixed Z axis, the degeneracy of the m levels must be removed by an external field. Experiments performed with NH3 (1), where the (J , k) = (3, 3) inversion levels were split by an applied electric field, showed that the m selection rules were of the dipole-dipole type, or Am = O, :1: 1. The same observations in 14NH3 and CH3F were reported by Johns and co—workers (9) from Lamb dip measurements using the technique of laser Stark spectroscopy, but their results also show the importance of Am > 1 transitions. Their studies reveal that these transitions may occur without appreciable changes in the velocity and without changes in the total angular momentum. Experiments with H2CO (56) gave collision-induced signals that occurred with AJ = 1 and with Am values up to :6. These results indicate that large reorientations of the total angular momentum vector can take place without changing the magnitude of J and without an appreciable change in the velocity of the molecule. The technique of polarization-detected transient gain spectroscopy (PTGS) (56), which is a time resolved version of polarization spectroscopy, was used to obtain rotational state-to-state rates in v4 = 1 of the first excited singlet state (A1A2) of formaldehyde. A linearly polarized pulsed laser was used to pump the J1,a Jic = 413 rotational level in the ground electronic state (1X) to the J kaJtc = 40,4 level of the first excited electronic state (A). As a result of pumping 13 with polarized radiation, an anisotropy in the distribution of populations in the magnetic m sublevels was created. A cw probe laser, linearly polarized at an angle of 45° with respect to the vertical was then tuned into resonance with the A~X234L 40’4-41,3 transition. Results from this study suggested that elastic reorientation (AJ = 0) of the angular momentum, i.e., pure m-changing collisions, occurs much less frequently than other collision-induced mechanisms such as rotational energy transfer with and without changes in the m quantum number. State-to—state experiments on HZCO have shown that there is a substantial tendency for the preservation of the anisotropic molecular population distributions. Hence, there must be a Am selection rule that accompanies A] 2 i1 inelastic processes, because the distribution of population in the magnetic m sublevels is not completely randomized even after state-changing collisions. Silvers and co-workers (57) used optical-optical double resonance to look at m changes that BaO molecules undergo during collisions with Ar and C02. Here, a linearly polarized dye laser prepares A122+ BaO in the single J = 1, m = 0, sublevel in the v = 1 vibrational state, and then a second polarized laser probes other m sublevels in the J = 1 (AJ = 0) and J = 2 (AJ = 1) states by inducing fluorescence from the C122+ state. The change in the intensity of the fluorescence was detected when the probe beam was polarized parallel and then perpendicular with respect to the pmnp beam. For elastic collisions (AJ = 0) with C02, direct evidence was obtained for transfer from m = O to m = :1. No such transfer was seen in collisions with Ar. For inelastic collisions (AJ at 0), direct evidence for transfer from m = 0 to other m sublevels was also observed. This transfer of m state populations occurred for both C02 and Ar as collision partners. This double resonance technique is very useful for studying m transitions and has the added advantage of allowing the selection of a particular velocity group through optical pumping. 14 Experiments on Liz that used the method of high resolution circularly polarized laser fluorescence (58) to study collisions with buffer gases suggested that for A] = :6 rotationally inelastic collisions, the final accessible m channels are restricted, though there were significant deviations from Am = 0 behavior for Liz-Ar interactions. The change in the m quantum number therefore describes how the J vector is reoriented in space. In the past years, considerable effort has been devoted to studying reorientation collisions (24-28, 43, 49, 59-63). This is due in part to the growing interest in the orientation and alignment of molecules during rotational cooling in supersonic molecular beams (64). It is also known that products of chemical reactions and photo- fragmentation processes are present in the nascent state with anisotropic distributions of the angular momentum (65-69, 74). Changes in the m quantum number during collisions also provide a measure of the anisotropy of the potential well (70). The different processes that affect the changes in the J, k, and m quantum numbers of a molecule during collisions are each sensitive to the forces that govern the interactions between molecules. So, by studying reorientation phenomena, one can expect to gain deeper insight into the nature and dynamics of chemical reactions. C. Reorientation of the total rotational angular momentum vector 1. Creation of anisotropic population distributions The spatial orientation of the total rotational angular momentum vector, J, of an atom or molecule may change as a result of many processes, such as collisions and photodissociation reactions. This reorientation of the angular momentum vector can be studied by preparing initial states in which the atoms or molecules have been preferentially aligned or oriented. The population distribution in such states will no longer be isotropic. Anisotropic population distributions may be created through the interaction of molecules with an electric field (28, 71, 72, 73), through selective photodissociation techniques (74- 76), through rotational cooling in a molecular beam (64), and through optical 15 pumping (23, 28, 77-79, 80,). Examples of the different methods used in preparing states with anisotropic population distributions are given in the next paragraphs. a. State selection through magnetic and electric fields Selection of atoms with or spins or with m, = +l/2 (ms = Z-axis component of the electron spin) was achieved through the use of a magnetic field and a motional electric field as shown by Lamb and Retherford (71) in connection with their study of the fine structure of the H atom. In the presence of a magnetic field strength of around 540 gauss, the two metastable states, or and B (m, = -1/2), of the 2281/2 states have very different lifetimes. When a perturbing electric field of ~ 4 V/cm is turned on perpendicular to the magnetic field, the lifetime of the (1 atoms is about 1600 times that of the [3 atoms. Therefore, after travelling a length of 6 cm to a detector only the or atoms will be able to keep their excitation and a beam consisting of only or atoms is obtained. In a reactive scattering experiment with K atoms and CH3I molecules, oriented molecules of CH3] were produced from the combined action of electric fields (73). In this experiment, a molecular beam of CH3I molecules was passed through an electric hexapole field created by rods charged to 10 kilovolts d.c. After this initial exposure to an electric field, spatial orientation of the CH3I molecules was achieved as the beam passed through a uniform electric field generated by charged parallel plates. b. state selection through photodissociation Dehmelt and co-workers (75,76) succeeded in aligning H2+ ions through the use of the technique of selective photodissociation. In their experiment, Hf ions were created by electron bombardment and then subsequently trapped in a radio frequency quadrupole field. A fraction of the created positive ions photodissociated upon interaction with linearly polarized light. Since the probability of photodissociation depends on the absolute 16 value of the magnetic quantum number, m, the population of Hf ions in different m sublevels decreased at different rates. As the photodissociation process proceeded, the remaining ions were preferentially aligned. This method has been applied to studies which involved the reaction of excited Xe atoms with molecules of IBr to produce excited XeI* and XeBr* molecules (78). c. Alignment of the J vector during rotational cooling Anisotropic distribution of the rotational angular momentum may be achieved through rotational cooling in molecular beams as was shown by the group of Friedrich et al. (64) in which alignment of the J vector was observed in experiments with 12 seeded in carrier gases. d. Optical pumping of molecular states It has been known for a long time that pumping with polarized radiation leads to anisotropic population distributions in the pumped levels (23, 77, 81, 80). Since the transition probability depends on the m quantum number, an anisotropic population distribution in the m levels equivalent to an anisotropy in the spatial orientation of J can be produced by optical pumping (23). Optical pumping with polarized radiation is being used increasingly because of the development of laser systems, especially single-mode lasers. Estler and Zare (82) used a linearly polarized argon ion laser beam to prepare aligned excited 12 molecules, 12* . The excited 12* molecules reacted with a beam of metal atoms, M (M is either Tl or In atom), in a vacuum chamber to form the exchange products I and MI*. Loesch and co-workers (83) used linearly polarized infrared radiation to align HF molecules in order to see the effect of reagent and product orientation on the reaction of K atoms with HF molecules. In the case of sodium atoms, a two-step excitation method with linearly polarized pulsed lasers was used to create aligned atomic Rydberg states (79). Rotational alignment in the v2" = l vibration in acetylene (84) was produced through the method of stimulated Raman pumping. This alignment was monitored via l7 laser induced fluorescence. The fluorescence was studied after polarization either perpendicular or parallel relative to the polarization of the pumping beam. McCaffery and his group (85), in a study of inelastic collisions between Liz molecules and Xe atoms, used a circularly-polarized pump laser to prepare oriented molecules. Work by Band and Julienne (59) shows that it is possible to achieve total transfer of population of a molecular vibronic level to another optically accessible level by chirped pulse absorption with polarized radiation. Saturation pumping of a particular vibration-rotation transition v' ,J‘ , K' (— v",J",K" causes a change in the population of the degenerate m sublevels in each of the pumped levels. The factors that can affect the degree of change in the population include pump power, the absorption cross section, and relaxation processes that tend to repopulate the v", J ", K", and collisions with the walls of the cell and with molecules in other states. The population distribution in the m sublevels created by continuous wave saturation pumping also depends on the polarization of the pumping radiation because the selection rules for the transitions are different for different polarizations of the absorbed radiation (86). For example, a pump beam whose electric field is parallel to the space-fixed Z axis induces Am = 0 transitions among the m sublevels for J ', m' o—J", m" while a pump beam that is perpendicular to the space-fixed Z-axis induces Am=il transitions. When the pump beam has right circular polarization, the selection rule for the J', m' o-J", m" transition is Am = -1 and when the pump beam has left circular polarization, the selection rule for the transition is Am = +1. Thus, pumping with polarized radiation prepares molecules with particular m state distributions. Pumping with linearly polarized radiation creates a distribution that is termed "alignment", while pumping with circularly polarized radiation creates a distribution that is called "orientation". It is important to realize that this "alignment" or "orientation" is an alignment of the angular momentum of the molecules and not necessarily the spatial arrangement of the molecules. 18 Figure 1.2 shows how different distributions of m-state populations are created by interaction of molecules with polarized radiation. Strictly speaking, an accurate definition of population, orientation and alignment comes from the spherical tensor expansion of the m state populations, as will be described below. 2. Statistical tensors and tensor opacities Work in our laboratory on collision—induced rotational transitions was done in the absence of any static field, so the 2] +1 m components of the total rotational angular momentum J were all degenerate. It is, therefore, difficult if not impossible to look at each individual m state and determine how each state is changed during a collision. In this case, it is useful and convenient to carry out a spherical tensor expansion of the density matrix elements for the system of gas molecules. We will be most interested in such an expansion for the diagonal or "population" elements of the density matrix as follows: 6(J,n)=Z(—l)J-m(2n+1)1/2(:1 _Jm 3)p(1m,Jm). (1.1) m Here n is the tensor order and the quantity in the brackets is a Wigner 3-j symbol (87). The form of the 3-j symbol shows that n ranges from 0 to 2J, and as a result of the properties of the 3-j symbols, the set of 6(J, n) represents an orthogonal linear transformation of the p(Jm, Jm). The impetus for this expansion is that the populations of molecules in various m states are proportional to density matrix elements p(Jm, Jm). In this equation, the quantum number J in O'(J, n) and p(Jm, Jm) stands for all of the quantum numbers except m, whereas it is just the numerical value of the total angular momentum quantum number elsewhere. The spherical tensor expansion of the density matrix in Equation (1.1) is called the "statistical tensor expansion" (88). It is defined so that when n = O, the sum is called the l9 Ti iii ‘3: ‘27; (a) (b) Figure 1.2. Creation of anisotropy in the m-state populations through optical pumping with polarized radiation. The rotational transition from J = 1 to J = 2 is pumped by: (a) plane-polarized radiation, and (b) circularly polarized radiation. There are (2] + 1) degenerate m-sublevels for each J. The topmost diagram in (a) shows pumping with plane- polarized radiation with its electric field parallel to the space-fixed Z-axis and obeying a Am = 0 selection rule for the transition, while the bottom diagram shows pumping with plane-polarized radiation with its electric field perpendicular to the space-fixed Z-axis and obeying a Am = :1 selection rule. The topmost diagram in (b) shows pumping with right circularly polarized radiation with Am = -1 selection rule while the bottom diagram shows pumping with left circularly polarized radiation with selection rule Am = +1. Assuming that there is no collisional transfer among the degenerate m—sublevels, the m-sublevel populations in (a) are the same for m = -m, while the m-sublevel populations in (b) are different for each m. 20 "population"; when n = 1, it is called the "orientation"; and when n = 2, it is the "alignment". In a qualitative sense, an "alignment" can be considered to be a symmetric distribution of m—state populations wherein the fractions of molecules in states with m = -m are equal, while an "orientation" can be considered as an asymmetric distribution where the fractions of molecules in states with m = -m are not equal (89). At thermal equilibrium, only the n = O tensor order, the "population", is non-zero. By using the algebraic expressions given by Edmonds (90) for calculating the numerical values of the 3-j symbols, the spherical tensor expansion in Equation (1.1) for n = 0, l and 2 becomes: 1. n = 0 (population) ZpUme) 0(J’0)=.E(1.27+_1)172_ (1.23) 2. n = 1 (orientation) 2 mp(Jm, Jm) ,1 = m . 0(1) [.I(J+1)(2J+1)/3]"2 (12b) 3. n = 2 (alignment) 2m? -J(J+1)/3)]p(Jm,Jm) = 11L OU’Z) [J(J+ l)(21+1)(21-1)(2]+3)/45]m (12¢) From Equation (1.2a) it is easy to see that when n = 0, CU, n) is proportional to the m-state populations p(Jm, Jm). It is clear from Figure 1.2 that "orientation" is created in the pumped levels when the pumping radiation is circularly polarized, while "alignment is created in the pumped levels when the pumping radiation is plane polarized. More 21 precisely, pumping with linearly polarized radiation creates only the even order tensor combinations (n = 0 for population, n = 2 for alignment and other higher order terms), while pumping with circularly polarized radiation creates all the tensor orders (91). This point has not always been recognized, even recently (25). By contrast, probing with weak, non-saturating radiation samples only the n = 0, 1 and 2 tensor orders. In an isotropic collisional environment, the different multipoles or tensor orders of the statistical tensor expansion relax independently with different relaxation rates, which is why this tensor formalism is very useful. As long as the collision environment is spherically symmetric, there is no interconversion between the different tensor orders. However, there are perturbations that can cause conversion of one form of tensor order to another. For example, excited-state alignment can be converted into orientation during the time between excitation and decay to the ground state when internal interactions within the excited atom (e. g. interactions due to hyperfine structures) combine with external perturbations such as those created by a magnetic field (92). It is known that anisotropic collisions and the presence of external fields can cause partial transformation of alignment into orientation in atomic systems. In an experiment involving Ne atoms that were excited by linearly polarized radiation, an applied constant magnetic field inclined the axis of alignment created by the radiation so that the anisotropic collisions between excited and ground state Ne atoms caused a fraction of the alignment to be transformed to orientation (24). Another tensor formalism is also useful. In this case, the m-dependence of collisional transfer rates can be expressed in terms of tensor opacities (25, 93). These factors relate the collision-induced transitions that the molecule undergoes to relaxation rates that are in turn related to multipolar transition rates (e.g., dipole-dipole or dipole- quadrupole rates). The general form of the tensor opacity decomposition is given by Coy, et al. (25) and is rewritten here for diagaonal density matrix elements as 22 (JJ'A kéfe—aj ___ (2A+1) Z(_1)f+m'-J-m -m m, p 2 ) ku'J'm'e—orfln' (13) all m Here, IQ 1H,] is the tensor opacity of rank A, and is a spherical tensor combination of rate coefficients Ira. fm'(_a_]m for population transfer. The letters J and J ' refer to the total rotational quantum number; or and or' refer to all other quantum numbers except J and m. The converse of the above equation may be written as f+m'—m—J J J. A 2 ka'J‘m‘t—oer = (4) 2m +1) , lei...) (1.4) A,p -m m p Equation (1.3) gives the rate coefficients either for population or coherence transfer for a specific A . A particular value of A is related to a multipolar transition rate (e.g., dipole- dipole, etc.). Therefore, Equation (1.3) may be used in calculating rate constants relative to a specific process, e.g. dipole-dipole. Equation (1.4) gives the rate coefficients for population transfer in terms of the contributions of different multipolar transitions with different A values. In other words, Equation (1.4) contains contributions from different multipolar transition rates. The quantities k9} t—aJ are called the tensor opacities for the (X'J'(—(XJ transition. The rates predicted by the statistical tensor formalism and the rates given by the tensor opacities are related. The equation for interconversion of the rate constants in the two forms of tensor expansions has been given by Coy and co-workers (25), and is given here as Equation (1.5). J J A 23 The quantity in curly brackets is a Wigner 6-j symbol, and n is the statistical tensor rank. Equation (1.5) may be used for population transfer rates. For population transfer between two levels, J and J ' are used. The above equation expresses the rate coefficient for transfer of a specific statistical tensor combination of m—state populations in terms of the contributions from different multipolar rates with different ranks or different values of A. The rate coefficients in the tensor opacity picture are given in terms of the statistical tensors as J J n [414}e—(1J = 2(_1)J+f+n+A(2n +1){J' J' A}k1(1"l.)r(—GJ . (1.6) A The tensor opacity decomposition contains information about the interaction potential and the collision energy of a system (93). The tensor opacity rank, A, is related to m conservation during collisions, and it was experimentally found that increasing values of A correspond to an increasing range of Am values for transfer of population (25). A value of A equal to 1 is equivalent to dipole-dipole interaction, and this predicts the greatest persistence of m (Am = O, i 1) during collision-induced energy transfer. The use of polarized pump and probe beams in four-level double resonance studies makes it possible to see what happens to the population distribution in the various m states during collision-induced rotational transitions. A sample that has been pumped with polarized radiation is no longer isotropic, so the absorption of a probe beam that passes through the sample will depend on how the sample was polarized by the pumping radiation and on how the collisions affected the distribution of population. If the effect of the pumping can be predicted, the effect of collisions can be extracted from the experimental data. The effect of a sample pumped by polarized radiation on the absorption of a probe beam can be predicted by use of Jones (94) matrices. Here the electric field of the 24 radiation is represented by a vector and the sample is represented by a 2 x 2 matrix. The product of the matrix and the vector gives the effect of a polarized sample on the absorption of a probe beam. The Jones vectors and matrices for a sample pumped and probed with radiation in three-level double resonance with different combinations of polarizations were given in a paper by Shin and Schwendeman (87). The corresponding vectors and matrices for f our-level double resonance will be reported in this thesis. Chapter Two Theory A. Velocity selection through pumping with an infrared laser Velocity selection is achieved by tuning highly monochromatic, moderate power radiation to a particular vibration-rotation transition. The frequency of the laser as seen by the molecules is equal to v: tap-Xi] (2.1) C where v = laser frequency as seen by molecules, v I = laser frequency, v2 = component of the velocity of the molecules in the direction of the pumping beam, and c = speed of light. The above equation can be solved for v2 to give v2 = vii“, — v) . (2.2) For low pressure gases, only molecules that see v very close to v0, the resonance frequency of a transition, are pumped by the laser. Therefore, only molecules having 25 26 velocity components very close to vz, which is given by the following expression, are pumped by the radiation: vz = c(1- v0 / w) (2.3) The fractional number of molecules with a given component of molecular velocity between v2 and v2 + dvz is given by the Boltzmann distribution (23), f (vz)alvz = fiexpkvz / u)2dvz. (2.4) In this equation, the most probable speed is u = J22!“- (where k3 is the Boltzmann m constant, T is the absolute temperature, and m is the molar mass). The number of molecules per unit volume in a given energy state that have velocities in the range V2 to V2 + dvz is equal to: N. III-(v2)de = find—(v2 / 02]de (2.5) where N, is the total number of molecules per unit volume in the state i. For the case in which the pressure broadening parameter is small compared to u, and for which saturation broadening and Dicke narrowing (95) are negligible, the population distribution in Equation (2.5) may be shown to lead to a Gaussian lineshape for low-power radiation, as follows, ot(v) = aoexp{-[(v - v0)/ kulz} (2.6) where 010 is the peak absorption coefficient and k = v0 / c. For this lineshape, the halfwidth at half height is called the Doppler width, AVD, and is given by the expression 27 AvD=flQ 1n2. (2.7) C The difference in the populations of the pumped levels can be calculated by using an equation that is similar to the Karplus-Schwinger equation (96): (Vp " Vba - kpvz)2 + 7% — (2.8) (vp “Vba —kpvz)2 +16 “3%:- n _ In this equation, n,- d‘vZ is the number of molecules per unit volume (population) of the pumped level i, and 11,9 dvzis the thermal equilibrium population of level i in velocity group V2 to vZ + dvz. The quantity 71 is the relaxation rate for the difference in the population of the lower and upper levels of the pump transition, while 72 is the relaxation rate for the coherence between these two levels. The Rabi frequency, xp, is equal to 111,28 / h , where 11.220) and 8 are the transition moment for the pumped levels and the electric field of the pumping radiation, respectively; kp is kp = Vba / c; and vp and Vba are the frequency of the pump laser and the resonance frequency of the pump transition, respectively. In the absence of pumping radiation, xp is equal to zero, r is equal to one, and no - "b is equal to n2 - n2 , which is the difference in the thermal equilibrium population densities between the pumped levels. At steady state, for very large pump powers (i.e.,very large values of xp), the populations in the pumped levels become equal and r —-> O. The sum of the population densities, na + rib, and the sum of the thermal equilibrium population densities, n2 + n3 , can be taken to be approximately equal: "a + "b = n3 + tr? (2.9) This equation may be combined with Equation (2.8) to describe the difference in populations n2 — n0, 28 1 ng-na=-2-(n2-ng)(l-r). (2.10) When the pump power is very large, i.e., when xp >> 1, r becomes equal to zero, and Equation (2.10) reduces to ng-na=-;-(ng-n8) or na=%(n2+ng). (2.11) The quantity n3 -— na represents the depth of the "burnt hole" in the lower level of the transition. This means that even for very large pumping power, the change in the population of the pumped lower level cannot exceed half the value of the thermal equilibrium population difference of the pumped levels. The "burnt hole" that was created in the ground state by the pumping process is seen in the upper level of the pump transition as a sharp spike ("Bennett spike" (23)). It is easily seen that It), - n2 = r12 — no , so that Equations (2.10) and (2.11) also apply to n, - n2. Plots of the lower and upper state velocity distributions for saturation pumping are shown in Figure 2.1. If the pumped transition is simultaneously scanned with a low power tunable laser, the resulting lineshape is Gaussian with a Lorentz-shaped "Bennett hole" (23) in the absorption whose halfwidth at half-maximum is the following: mm: ’fiufifi. (2.12) Equation (2.12) shows that an increase in the pumping power results in a broadening of the width of the burnt hole and, by inference, of the Bennett spike. By means of one or several collisions, this spike gets modified and transferred to other rotational states. In IR-IR four-level double resonance experiments, for which the probe transition lies in the hot band region, the velocity distribution seen in the lower level of the probe is essentially the pumped upper level velocity distribution modified by 29 Figure 2.1. Saturation pumping with an infrared laser. Pumping a fundamental transition with a weakly saturating IR laser creates a "Bennett hole" in the ground state velocity distribution of molecules. This hole is seen in the upper level of the pump transition as a "Bennett spike". 30 collisions. In mathematical terms, the velocity distribution in the lower level of the probe is a collision kernel convolution of the velocity distribution in the upper level of the pump transition. B. Collision kernels As already mentioned, the lineshape obtained in high resolution infrared-infrared double resonance of gases is a superposition of a broad Gaussian component and a sharp transferred spike, and we assume that the Gaussian portion of the lineshape results from near resonant vibrational-vibrational energy transfer between molecules in v = 1 of the pumped vibrational level and the molecules in the ground vibrational level, while the transferred spikes are assumed to result from direct rotational energy transfer and therefore represent the contribution to the absorption by molecules that were part of the pumped population. Previous work has shown that the transferred spikes are well represented by convolutions with a collision kernel that is the sum of two Keilson-Storer functions (10, 11, 97). Each Keilson-Storer function has the following form: Aas(v.V') = fi‘fiexi’i‘“ - (NV/132] . (2.13) Here, Am(v,v' )dv is the rate constant for transitions in which a molecule in a particular state a with velocity equal to v' goes to state 3 with a final velocity between v and v + dv. Keilson and Storer have shown that the parameter B is directly proportional to the root- mean-square change in the velocity as a result of collisions and is given by the expression 1/ 2 B = J2<(Av)2) . The parameter B can be treated qualitatively as a measure of the spread in the distribution of velocities as a result of collisions. To ensure that the distribution returns to thermal equilibrium after collisions, at and B are related by: 31 a2 =1—(B/u)2 (2.14) where, as above, u is the most probable speed. The parameter or varies from 1 when B = 0, to 0 when B = u. The effect of representing the collision kernels as Keilson-Storer functions is seen in the lineshape function F(v) for the transferred spikes. The form of this lineshape function was given previously in References 10 and 11 as: F(v):Jdv'f(v')exp{—[(v-orv')/B]2} . (2.15) The function f( v ') describes a power-broadened pump transition, expi—(v' /u)2] (VI-Vo-kv' )2+Y%+Y%J%/Yr . f(V' ) = (2.16) When the form of the Keilson-Storer kernel is such that B = u, or is equal to zero, the lineshape function F(v) describes a Gaussian characterized by a Boltzmann distribution of velocities, there is complete randomization of velocities after collisions. This Gaussian portion in the four-level double resonance lineshape represents molecules that are part of a near-resonant V-V swapping mechanism. When or is equal to l (B = 0), the collision kernel A(v,v') is a delta function that is centered about v = v'. It was found in previous studies (10) and also confirmed in the present study that the two Keilson-Storer functions which describe the transferred spikes have different B values. Since the parameter B is a measure of the spread of the distribution of velocities after a collision, it may be used to assess the effectiveness of certain collisions in causing larger changes in the relative initial velocities of the molecules. If one assumes that smaller impact parameters lead to larger 32 sz, then information on the relative sizes of the impact parameters may be inferred from the value of B. C. Jones matrix for four-level double resonance 1. Introduction In the Jones calculus (93), a beam of radiation is represented by a two-dimensional column matrix 13%;) in which E0 f and E0 f ' are the complex amplitudes of the electric field of the radiation in mutually perpendicular directions that are perpendicular to the direction of travel of the beam. The actual electric field of the radiation is the real part of E where new) in which a) is the frequency of the radiation. In this theory, optical elements are represented by two-dimensional square matrices such that the Jones vector of the beam leaving the element is given by the matrix product of a Jones matrix for the element by the Jones vector of the incoming beam. Jones vectors for some polarized beams are given in Table 1. A more extensive table of vectors and matrices are given elsewhere (87). In the Appendix, equations for the elements of Jones matrices for four-level double resonance in an optically pumped sample are derived for the cases of plane-polarized and circulary-polarized pumping. 2. Jones matrix for sample pumped by plane-polarized radiation When plane-polarized radiation is used as the pump, the space-fixed Z axis is chosen to be in the direction of the electric field of the pumping radiation and the space- fixed Y axis is chosen to be the direction of travel of the beam. Then, the F and F' axes are 33 Table 1. Jones Vectors and Jones Matrices (87) 1 ZY plane-polarized beama £2 = 50(0) , 0 XY plane-polarized beama EX = 150(1) n ht circularl olarized beama E, — 1 15" l g y p 75 i left circular] olarized beam“ 15, - 1 5° 1 y p 72' -i aDirection of propagation of the beam is Y for plane polarization and Z for circular polarization 34 the Z and X axes, respectively. It is shown in the Appendix that the Jones matrix for this case can be conveniently expressed in the following form: 2 pZ 0 M = . 2.17 a! (0 px] ( ) The matrix elements pZ and p, are defined as p2: e a M (2.18) and pJr = e-(le/z (2.19) where L is the path length through the sample. The values of or, and orJlr are the complex absorption coefficients for a Z- or X- plane polarized probe beam and are given by the following equations: 41t(DN .= n. (L..— D.)(S‘°’- SE”) (2.20) . = 4’;°’N(L. -iD..)%(Si"’ +51, -s: -s:) (2.21) C These absorption coefficients consist of a real Lorentzian part ch and an imaginary dispersion term, ich, where 72 = 2.22 t... m ‘ ’ 35 8cb D = . 2.23 .. m ‘ ’ The term 7, is the relaxation rate for the coherence of the levels connected by the probe beam and Sch is the difference between the frequency of the probe radiation and the resonant frequency of the probe transition. The Sf,“ are defined as Sf,“ = Zifiqkn), in which the sgq)(n) are given by the following equation: 1 l n l l n q) = lJb—Jc_q-2 2 4,1“2 ”l; (n) () Hbc( ’1 ) _q q 0 Jr, Jb 1. }O(J,,,n) (2.24) Similar expressions hold for 55.4) and hymn). In these equations, the lower and upper level of the probe transition are labelled by the letters b and c, respectively. The quantities inside the ( ) and the { } are Wigner 3-j and 6-j symbols, respectively, and Eb. is a reduced transition dipole moment matrix element that is independent of m. The 3-j coefficients are non-zero only for n equal to O, 1 or 2. This means that because of the nature of the four- level double resonance scheme (strong pump/weak probe) only the O, l and 2 tensor orders of the spherical tensor expansion of the m state populations can be accessed by the experiment I We show in the Appendix that because of the definition of the Sf,” terms and the properties of the Wigner 3-j coefficients that the absorption coefficients or, and or, contain terms that depend only on the even tensor orders, n = 0 and 2. The important consequence of this result is that pumping with saturating plane-polarized radiation and probing with a weak, plane-polarized radiation samples only the n = 0 (population) and n = 2 (alignment) tensor orders. 36 Population modulation in double resonance experiments is perfomed by chopping the pump beam at a fixed rate and by coherent demodulation of the signal from the probe beam detector at the pump chopping frequency. The output of the coherent demodulator is then proportional to the intensity of the probe beam leaving the sample while the pump beam is on minus the corresponding intensity while the pump beam is off. Population modulation can be performed with the planes of polarization of the pump and probe beams either parallel or perpendicular. Alignment modulation is performed in our laboratory by switching the plane of polarization of the pump beam between horizontal and vertical planes while keeping the plane of polarization of the probe beam vertical. The output of the coherent demodulator for alignment modulation is proportional to the difference between the probe intensity for parallel pump and probe beam planes of polarization the intensity for perpendicular planes of polarization. a. Z-polarized pump and Z-polarized probe (parallel configuration) The Jones vector for ZY-polarized radiation is written as: 1 E. = 50(0) . (2.25) Then, the electric field at the detector is obtained by multiplying the Jones matrix for the sample, Ma, , by the Jones vector for the probe beam, E2. From the previous equation for Ma, given in this section, we find that the electric field of the probe at the detector is equal to E; = E°(I:)z) . We are interested in the intensity of the double resonance signal at the detector which is proportional to the absolute square of the electric field, 1,2 = 125 E; = 15°21;ng . (2.26) 37 The complex absoption coefficient a can be expressed as a sum of real and imaginary terms, for example, or, = 2: +182 . (2.27) After working through the algebra, the intensity of a four-level double resonance signal taken under the parallel configuration of pump and probe beams becomes 1zz = isozte‘izL s 1302(1- 521.), (2.28) 41th L..(Si°’ -s:°>) hc where 8, = Here, we have used the fact that for our experiments, ezL << 1. For population modulation experiments, the pump beam is chopped mechanically or electronically so that it is off during one half of the cycle. Therefore, the intensity of the double resonance signal seen by the detector is actually the difference in the intensities of the signals when the pump beam is off and when the pump beam is on, or 1:101,-IO,=E°2-£°2(1-e,L)=eZLE"2. (2.29) b. Z-polarized pump and X-polarized probe (perpendicular configuration). From Table l, the Jones vector for XY-polarized radiation is written as 12, 40(2)). (2.30) By taking the Jones matrix for the sample Ma, , the electric field at the detector is 0 now equal to E; = E°( x J. Then, the intensity of a four-level double resonance signal 38 taken when the pump and probe beams have perpendicular planes of polarization is expressed as 1,, = e'ExL = 1502 (1 — st) , (2.31) 4mm...[ogsmsi-a-(sr‘)+4“)- where a, = c. Alignment modulation The intensity seen by the detector under alignment modulation is given by: I =E"2(e -e al 2 x )L. (2.32) where, 82 and 8, are given after Equations (2.28) and (2.31), respectively. The difference in Equation (2.32) is shown in the Appendix to depend only on the n = 2 tensor order (alignment) of the spherical tensor expansion of the m state populations: _ _ 4nmN 3 . . i. L..-2-(~i°’(2)-~£°’(2)). (2.33) The definition of the quantity 5%") (n) was stated in Equation (2.24) where it may be verified that 82-6,, is proportional to the alignment of the populations, o(n=2), only. And the intensity seen by the detector is equal to l = 2,1102 . (2.34) 3. Jones matrix for sample pumped by circularly polarized radiation It is shown in the Appendix that the Jones matrix for a sample that has been pumped with circularly polarized radiation is given as 39 (2.35) ~ (n+1); -i(p,-pi)) 07 i(prfli-pl) pr+pl) . In the above expression , p, and p, are defined as: -or L/2 p. =6 ' and Pr = e-or,L/2 . For circularly polarized pumping radiation, the F and F ' axes are most conveniently chosen to be the X and Y axes, respectively. The terms or, and or, are the complex absorption coefficients for a probe beam that has right circular or left circular polarization. The form of these coefficients is similar to the expressions given for plane-polarized radiation: or, = 1— 4’2"” (ch —ich)(S,(,+” 45‘”) (2.36) C and , =1- 4’2?” (ch -iDc,)(si,") -S§‘”) . (2.37) The terms ch and Dd, are defined in Equations (2.22) and (2.23), and SS” which involves a summation over tensor orders, is given just before Equation (2.24). For four-level double resonance studies that involve circularly polarized pump and probe beams, it is shown in the Appendix that the elements of the Jones matrix, Mo, , depend only on the three tensor orders, n = 0, l and 2. Population modulation experiments can be performed by having different pump- probe combinations and chopping the pump beam, as in the previous case where the pumping radiation was plane-polarized. For example, the polarization of the pump beam can be right circular (RCP) all the time while the probe beam can have either right or left 40 circular polarization (LCP). The Jones matrix that is given in Equation (2.35) is written for right circularly polarized pumping radiation. The matrix for a sample pumped with left circularly polarized pumping radiation is obtained by interchanging the positions of p, and p,. In this discussion we will assume that the pump beam is right circularly polarized. a. RCP pump and RCP probe (RCP/RCP) The Jones vector for right circularly polarized radiation is given in Table 1 as 1 1 E = 15° — . 2.38 The electric field of the probe at the detector then becomes: ._E°p. 1 Ea- J5 (I) (2.39) The intensity of the double resonance signal is equal to the absolute square of the amplitude of the electric field of the probe at the detector, and if we recall that the complex absorption coefficient or is written as a sum of a real and imaginary terms, or = e + i8 , then the expression for the intensity of the signal is: I = 2.1.1502 (2.40) where e, = 4—-—1;leLb(S(“)- SP”). C Again this equation assumes that the actual intensity of the double resonance signal is equal to I = 1017 - 10,, and that a weak probe beam was used, e,L <<1. 41 b. RCP pump and LCP probe (RCP/LCP) From Table l, the Jones vector of left circularly polarized radiation is given as “ET—(3:) <24” . By using the same method as in part 3a, the actual intensity seen by the detector is 1 = 9,1202 (2.42) where e, = 412%, L AS; I) —S(‘”). c c. Orientation modulation For four-level double resonance spectra taken under orientation modulation of the probe beam, the intensity seen by the detector is equal to 10, = 1502(2, - e,)L . (2.43) Here, 8, and e, are the real parts of the complex absorption coefficients for a probe beam that has either right or left circular polarization. It is shown in the Appendix that this difference contains only terms that are sensitive to the n = 1 tensor order, or to the orientation, such that =4’m’N ch22(3,‘,”(1)—3§”(1)) (2.44) 8-81 r For a hot band probe transition, the population in the upper level is negligible so that the term séq)(n) has very little effect on the double resonance signal and may be ignored. 42 D. Velocity dependence of the absorption coefficients The velocity dependence of the four-level double resonance signal is built into the expressions for the absorption coefficients. For plane polarized pumping radiation and for a ZY- polarized probe beam, the equation for 82 after Equation (2.28) may be used to show that the velocity-averaged absorption coefficient may be written as e, = j [eO 6(1”, 0, vy) + c2 6(J”,2,vy)] 1...,(vy) dvy (2.45) in which c0 and c2 are constants, and o(J",n,vy) is the nth order statistical tensor combination of the populations of molecules with vy between vy and vy + dvy for the lower level of the probe transition (the population of the upper level of the probe is assumed to be negligible). Only the n = 0 (population) and n = 2 (alignment) combinations of the populations contribute to the absorption coefficient. From Equations (2.8) and (2.22), 72 L ( )= vay (Vt-Vb‘kvy)2+7% The factor L(vy) represents a Lorentzian absorption profile for a beam in the +Y direction. For circularly polarized pump and probe radiation, the absorption coefficient contains contributions from the first three tensor orders it = O, l (orientation) and 2, as seen in the following expression: e, = “c0 o(J”,0,vz)+clo(J”,1,vz)+ c2 O'(J”, 2, v2)] L(vz) dvz. (2.46) Here, L(vz) represents a Lorentzian absorption profile for a beam in the +2 direction (convenient direction for circular polarization). The absorption coefficients in Equations (2.45) and (2.46) are each an integral over all of the velocity groups in the lower level of 43 the probe transition. The velocity groups in the lower level of the probe transition are actually collision kernel convolutions of the velocity groups in the upper level of the pump transition, so the quantity 0(J",n,vy) is given by O(J”,n,vy) = j A(vy,v'y) 6(J,n,v'y) dv'y. (2.47) Here, GO, n, v',) is the nth order statistical tensor combination of the populations of the upper level of the pump. The collision kernels A(vy,v'y) are represented by a sum of three Keilson-Storer functions given by Equation (2.13). The sum of two kernels represents the transferred spike, while the third kernel (with B = it) represents the Gaussian contribution from the V-V collisions. E. Description of four-level double resonance fitting programs Calculation of theoretical four-level double resonance lineshapes for an iterative non-linear least squares fit of the experimentally obtained lineshapes required the calculation of the populations in the pumped upper state as a function of initial velocity vy (for plane-polarized pumping radiation) or v2 (for circularly-polarized pumping radiation) and Rabi frequency of the pump laser. Equation (2.8) shows how this is done. Determination of the population for the pumped upper state is greatly simplified by using a Am = 0 selection rule for the pump transition for plane-polarized radiation, and either a Am = +1 or -1 selection rule for the pump transition for circularly-polarized radiation. It is also assumed in the fitting programs that collisions do not couple the degenerate m sublevels of the same rotational J state. Therefore, each pair of m transitions is considered "as an isolated two-level system. The validity of this assumption is discussed by Schwendeman (91) in his treatment of the lineshape of power-broadened microwave transitions, where conditions are given for which a lineshape described as a sum of power- 44 broadened m components is the same as the exact expression given in terms of continued fractions. The populations in the pumped upper state were then expressed as spherical tensor combinations of the m-state populations. For plane polarized pumping, only even order tensor combinations are non-zero, while for circularly-polarized pumping, combinations of all tensor orders are non-zero. The values of the tensor order combinations are then transferred intact to the lower level of the probe transition and subsequently multiplied by the n—dependent absorption coefficients of the probe transition. As shown above, the coefficients are non-zero for n s 2. The resulting n = 0, 1 and 2 spherical tensor combinations for the probed level are then multiplied by the appropriate Keilson-Storer functions and integrated over all the contributing velocity groups in the lower level of the probe transition. The relative values of the rates of transfer of the n = 0, l and 2 spherical tensor combinations and the offset frequency for the pump laser are specified in the input file. Adjustable parameters in the least squares fit include values for the spectroscopic background (slope and intercept), resonance frequency of the probe transition, amplitude of the broad Gaussian component, and amplitude and width of the transferred spikes. The width of the transferred spike is expressed as kB (in MHz), where k is equal to the value of the probe transition resonance frequency divided by the speed of light in vacuum. The Rabi frequency of the probe is usually set at a very small value, around 0.01 MHz. This choice of value for the probe Rabi frequency is based on the assumption that the weak probe beam can not cause transfer of population from the lower to the upper level of the probe transition. An increase in the Rabi frequency of the probe is seen as an increase in the amplitude of the double resonance signal. Chapter Three Experiment A. Polarization experiment The block diagram of the spectrometer that was used for the IR-IR double resonance studies is shown in Figure 3.1. The pump and probe lasers are C02 lasers that were frequency stabilized by locking to a Lamb dip in the fluorescence from a sample of C02 in an external cell. A portion of the radiation from each laser was sent through the fluorescence cell and reflected back on itself to create the Lamb dip in the fluorescence. The laser frequency was continuously adjusted to keep it at the peak of the fluorescence. By this means, the laser frequency was stabilized to 1:015 MHz. The pump and probe beams were arranged in a counter-propagating geometry. The frequency of the pump laser was set at a frequency that is nearly coincident with a vibration-rotation transition frequency of the 13CH3F sample. For this particular study, the QR(4,3) transition in the v3 fundamental band of 13CH3F was chosen as the pump transitionl. The pump laser was tuned to the 9P(32) 120502 laser line whose frequency is 24.2 MHz above the center frequency of the pump transition (98). i In double resonance studies with 15NH3, the asR(2,0) and the asQ(5,4) transitions in the v2 fundamental band were chosen as the pump transitions? For the asR(2,0) pump 1The nomenclature for transitions in simple symmetric tops like C11312 is discussed in Chapter Four. 2The nomenclature for NH3 transitions is different from that used for simple symmetric tops because of the possibility of inversion. This will be discussed briefly in Chapter Four. 45 46 Figure 3.1. Block diagram of the IR-IR double resonance spectrometer used for population and alignment modulation experiments. The pump and probe beams are in a counter-propagating geometry. Both lasers are frequency-stabilized by locking to a Lamb dip in the fluoresence of C02 inside a fluoresence cell (FC). The probe laser (Laser2) passes through a CdTe, electro-optic crystal modulator (Mod) which is simultaneously irradiated by powerful microwaves to produce the weak sidebands that are used as the probe. The microwaves are 100% amplitude modulated at 33kHz. A plane polarizer P2 allows only the sidebands to go through, while the carrier laser, which is used in the Lamb dip, is reflected off to a mirror. A beam splitter (BS) after P2 sends a portion of .the sidebands to a reference detector (RD). The other portion of the sideband goes through the sample cell into a partially-transmitting mirror which reflects the sidebands off to the signal detector (SD). The pump laser passes through a CdTe crystal modulator that acts as a N4 plate and converts plane polarized radiation to circularly-polarized radiation. Application of very high positive or negative voltage (HV) converts plane polarized radiation to either left or right circularly-polarized radiation. A Fresnel rhomb (Rh) converts circularly-polarized radiation back to plane-polarized radiation. A plane polarizer (P1) is used for population modulation because it transmits only one type of plane- polarized radiation during half of the pump modulation cycle. P1 is removed during alignment modulation experiments. 47 transition, the pump laser was tuned to the 10R(42) 120602 laser line whose frequency that is about 50 MHz below the center frequency of the pump transition, while for the asQ(5,4) pump transition, the laser was tuned to the 10R(18) 13C1602 laser line whose frequency is about 13 MHz above the center frequency of the pump transition. The precise offset frequencies for the 15NH3 pump transitions were determined in this work and will be reported in Chapter Four. The probe laser is passed through a CdTe electro-optic crystal modulator that was simultaneously irradiated by powerful microwaves (IO-20 W). The IR and the microwave radiation are mixed in the crystal and the resulting weak sidebands (99) have frequencies that are equal to v L i VMW, where v L and VMW are the laser and microwave frequencies, respectively. For the experiments reported here, only one of the sidebands (+ or -) is absorbed by the sample. The amplitudes of the sidebands are square-wave modulated at 33 kHz by turning the microwaves on and off by means of a PIN diode. In addition, the amplitudes of the sidebands during the "on" portion of the modulation cycle are stabilized by controlling the transmission of the PIN diode during this portion of the cycle. For the polarization studies, the pump laser was passed through a CdTe electro- optic crystal modulator that is switched alternately between a high positive and negative voltage. The switching rate was chosen to be 150 Hz. The crystal acts as a quarter wave plate and converts plane to circularly-polarized radiation. For alignment modulation studies, where the pump laser is switched between vertical and horizontal planes of polarization, a Fresnel rhomb was placed after the crystal. The rhomb acts as a quarter wave plate that converts the alternately right and left circular to alternately vertical and horizontal plane polarization. A plane polarizer placed after the rhomb blocks the pump beam during one half of the modulation cycle which produces population (or intensity) modulation of the sample. For either of these experiments, the probe beam is vertically polarized. 48 For orientation modulation studies, the Fresnel rhomb is removed from the path of the pump beam and placed in the path of the probe radiation, as shown in Figure 3.2. The probe beam is then circularly polarized while the pump beam switches between right and left circular polarization. For population modulation studies with circular polarization, a mechanical chopper operating at 150 Hz is placed in the pump beam path and only one of the high voltages is applied to the crystal to produce the desired circular polarization. For all of the experiments, measurements were obtained to make sure that the pump power does not change when the pump polarization is changed. For example, the horizontally-polarized pump should have the same power as the vertically-polarized pump beam. The same should be true for right and left circular polarization. Intensity measurements were also made with a Fresnel rhomb and a linear polarizer to test the purity of the circular polarization. Results of this test show that the intensity of the pump laser stayed the same as the polarization was changed from right to left circular polarization, and also there was no difference (< 1%) in the pump intensity for the vertically and the horizontally-polarized pump laser. The signal and reference detectors are HngTe detectors cooled by liquid nitrogen. A double demodulation scheme was used in the detection. The output of the signal detector was sent to a lock-in amplifier where it was demodulated at the probe modulation frequency of 33 kHz. The output from this phase sensitive detector was processed by a second phase-sensitive detector and demodulated at the pump modulation frequency of 150 Hz. The output from this second detector consists entirely of double resonance effects and is proportional to the difference in the double resonance effects during each half of the modulation cycle. The 13CH3F sample was obtained from Merck. The 15NH3 sample was 99% pure and was obtained from Cambridge Isotopes. The samples were not purified further except for the usual freeze-pump-thaw cycle. Sample pressures were from 8-20 mTorr and all experiments were done at room temperature. 49 Figure 3.2. Modification of the IR-IR double resonance spectrometer for use in orientation modulation studies. The Fresnel rhomb (Rh) is now found after the CdTe crystal modulator (Mod) to produce a circularly polarized probe beam. For population modulation studies, a mechanical chopper that chops the pump laser is placed after the CdTe N4 plate. During orientation modulation, the chopper is removed, the pump laser is switched alternately between rep and lcp through alternate application of high positive and negative voltages. The switching is done by an electronic switch designed by Martin Rabb at Michigan State University. 50 The sample cell was a l m. long Pyrex glass tube with 2 cm. inner diameter, and the NaCl cell windows were tilted by a few degrees to reduce etalon effects. B. Polarization experiments with foreign gases Both population and alignment modulation experiments were done with 13CH3F in the presence of foreign gases He, H2 and Xe. The double resonance spectrometer for these experiments was described in Section A of this chapter. Foreign gas pressures of 50 mTorr were added to a 2 m. sample cell that was first filled with 3 mTorr of 13CH3F. The CH3F was then allowed to freeze using liquid N2 before mixing in the foreign gas. All experiments were done at room temperature. He and Xe were obtained from AGA Gas, Inc. and The Matheson Company, Inc., respectively. C. Determination of precise pump offset frequencies for 15NH3 The method employed by Song and Schwendeman for the determination of precise pump offset frequencies for CH3F (98) was used in this work. The double resonance spectrometer that was used was discussed above and shown in Figure 3.1 except for a slight modification as shown in Figure 3.3. The key difference is that for this work, both co-propagating and counter-propagating pump-probe geometries were used simultaneously. In this case, a focusing mirror was placed after the sample cell to reflect the pump beam back into the sample cell. The resulting double resonance lineshape has two sharp transferred spikes, one spike on either side of the resonance frequency of the probe transition. When a transition is pumped off resonance, the pump laser frequency and the pump transition frequency are related, as shown in Equation (2.2b), by 51 The sample cell was a l m. long Pyrex glass tube with 2 cm. inner diameter, and the N aCl cell windows were tilted by a few degrees to reduce etalon effects. B. Polarization experiments with foreign gases Both population and alignment modulation experiments were done with 13CH3F in the presence of foreign gases He, H2 and Xe. The double resonance spectrometer for these experiments was described in Section A of this chapter. Foreign gas pressures of 50 mTorr were added to a 2 m. sample cell that was first filled with 3 mTorr of 13CH3F. The CH3F was then allowed to freeze using liquid N2 before mixing in the foreign gas. All experiments were done at room temperature. He and Xe were obtained from AGA Gas, Inc. and The Matheson Company, Inc., respectively. C. Determination of precise pump offset frequencies for 15N113 The method employed by Song and Schwendeman for the determination of precise pump offset frequencies for CH3F (98) was used in this work. The double resonance spectrometer that was used was discussed above and shown in Figure 3.1 except for a slight modification as shown in Figure 3.3. The key difference is that for this work, both co-propagating and counter-propagating pump-probe geometries were used simultaneously. In this case, a focusing mirror was placed after the sample cell to reflect the pump beam back into the sample cell. The resulting double resonance lineshape has two sharp transferred spikes, one spike on either side of the resonance frequency of the probe transition. When a transition is pumped off resonance, the pump laser frequency and the pump transition frequency are related, as shown in Equation (2.2b), by 52 PUMP PROBE eemple cell "til Figure 3.3. Arrangement of pump and probe beams for determination of precise pump offset frequency. The pump and probe beams counter-propagate and co-propagate simultaneously. A mirror was placed near the entrance of the sample cell to reflect back the pump laser. 53 V, V Vp'V0p=Vp—C"5V0p_cz‘ (3.1) where VI) = pump laser frequency in MHz vop = resonance frequency of pump transition in MHz vZ = component of velocity in Z direction 0 = speed of light . The second equality is only approximate but is sufficiently accurate (i.e., 1 ppm) for our purposes. The probe beam may be set up to co-propagate and counter-propagate with the pump laser. In that case, the center frequencies for the transferred spikes become V: = v0(l i vz / c) (3.2) where v: = spike center frequency [(+) for co- and (~) for counter-propagating beams)] v0 = resonance frequency of the probe transition . Equation (3.2) may be solved for vZ to give 1’; = M . (3 3) C 2V0 . This equation may be substituted for v2 in Equation 3.1 to give VOp(V+ — —) V - V0 = P P 2,,0 (3.4) where vp-vop is the pump offset frequency. In terms of the Doppler widths of the pump and probe transitions, Equation 3.4 may be written as Av — D,0p (v+ V_) (3.5) V -VO = a p p AVD,O 2 54 where Avm)p = Doppler width of the pump transition, and AVD,0 = Doppler width of the probe transition . The center frequency, v0, of the probe transition is halfway between the two spike frequencies, v+ and v_, or, 2 (3.6) V0 Chapter Four Results A. 13CH3F-13CH3F collisions Methyl fluoride is a prolate symmetric top that has C3,, point group symmetry. Both 12CH3F and 13CH3F molecules have strong vibration-rotation transitions that are within the 9-11 urn range of the C02 lasers. As mentioned earlier, for a C3,, symmetric top, two symmetry species exist that are labelled A (ortho, k = 3n) and E (para, k at 3n), according to the spins of the three identical H protons. The QR(4,3) pump transition and the hot band probe transitions P(6,3) to P(8,3) all belong to the A species. Figure 4.1 shows the energy level diagram for four-level double resonance studies with l3CH3F. Table 2 lists the pump and probe transitions that were used in the double resonance experiments with 13CH3F. The QR(4,3) pump transition is in the v3 fundamental band (v3 = l -— v3 = 0) while the probe transitions are all in the 2V3 ~ v3 hot band (v3 = 2 -— v3 = 1) of 13CH3F. The nomenclature for transitions in simple symmetric top molecules like CH3F is AkAJ (J ,k), where Ak = kupper - klower’ AJ = J upper - J lower’ and J = 110%,, and k = klower- The symbols P, Q, and R are used for Ak or A] = -l, 0 and +1, respectively. Thus the QR(4,3) transition is a transition from J = 4, k = 3 in the ground vibrational state, to J = 5, k = 3 in the upper vibrational state. The v3 fundamental band (C-F stretching mode) has v3 = 0 in the ground state and v3 = 1 in the upper state. We have shown in Chapter Two that we can write Jones matrices for a sample that has been pumped and probed with either plane or circularly-polarized radiation in four- level double resonance experiments. Four-level double resonance spectra taken under 55 56 V3: 11.3 7.3 5.3 6.3 I i 7 3 8,3 10,3 6.3 Figure 4.1. Energy level diagram for 13CH3F. The QR(4,3) pump transition lies in the v3 fundamental band (v3 = 1-— 0), while the probe transitions are all in the 2v3+-v3 hot band. 57 Table 2. List of transition and laser frequencies in MHz used for the pump and probe in double resonance studies in 13CH3F. Pump v0 Offseta Laser line Band QR(4,3) 31 042 693.82 24.25b 12C1602,9P(32) v3 Probe QP(5,3) 30 092 980.45b 14 572.61 130602, 9P(16) 2V3<—-V3 QP(6,3) 30 040 821.15b 12 707.54 13C1602,9P(18) 2v3<—v3 QP(6,0) 30 040 540.10 12 988.6 130602, 9P(18) 2V3é—V3 QP(7,3) 29 988 049.52b 10 600.52 13C1602,9P(20) 2v3<—v3 QP(8,3) 29 934 666.9 c 8 253.8 13C1602,9P(22) 2V3(—V3 QR(10,3) 30 843 131.4C -9 119.5 13C1602,9R(14) 2v3<-—v3 a (Vlaser'VO) in MHz bQuan Song and R. H. Schwendeman, J. Mol. Spectrosc. 165, 277-282 (1994) (Reference 100). CD. Papou§ek, R. Tesar, P. Pracna, J. Kauppinen, S. P. Belov, and M. Yu. Tretyakov, J. Mol. Spectrosc. 146, 127-134 (1991) (Reference 101). 58 conditions of population modulation using either vertical or horizontal planes of polarization of the pump beam are shown in Figure 4.2. These are similar to the double resonance spectra reported by Shin and Schwendeman (87). For all the probe transitions studied, it was shown that the spike intensity for the case where the pump and probe beams have parallel planes of polarization is greater than the spike intensity for the case where the pump and probe beams have perpendicular planes of polarization. Figures 4.2, 4.3 and 4.4 show this effect for the P(6,3), P(7,3) and P(8,3) probe transitions. The double resonance lineshapes obtained when the R(10,3) transition was observed are given in Figure 4.5. The R(10,0), R(10,1) and R(10,2) transitions are very close to the probed R( 10,3) transition as can be seen in the figure. The difference in amplitudes of the transferred spikes under different polarization conditions is very interesting. If a diffference occurs, that the amplitude of the sharp transferred spike obtained when the beams have parallel planes of polarization is greater than the amplitude of the spike obtained when the beams are perpendicular is not surprising. This is because for an R-branch pump and P-branch probe combination, the pump transitions with the greatest intensities couple with the probe transitions that have the greatest intensities when the two beams have the same planes of polarization. However, for perpendicular combinations of pump and probe polarizations, it can be shown by calculating the intensities for the transitions that in this case, the weakest pump intensifies couple with the weakest probe intensities. What makes the existence of the difference in amplitudes interesting is that in four-level double resonance the probe levels are only collisionally coupled to the levels of the pump. The fact that this difference in intensity survives even during collision-induced rotational energy transfer, as shown in the four-level double resonance spectra, means that the spherical tensor combinations n (in the case of plane-polarized pump-weak probe radiation, n = O and 2 only), which were initially 59 130H; Plane Polarization Pump: R(4,3) Probe: P(6,3) Figure 4.2: Experimental four-level double resonance lineshapes taken under conditions of population modulation using plane polarized radiation for the pump and probe beams. The QR(4,3) transition in the v3 fundamental band of l3CH3F was pumped by a 12C1502 laser while the QP(6,3) transition in the 2v3~v3 hot band was scanned by a sideband system.The bigger of the two spikes was obtained when the pump and probe beams had parallel planes of polarization while the smaller spike was obtained when the beams had perpendicular planes of polarization. 60 13CH3F Plane Polarization Pump:R(4,3) Probe:P(7,3) I r I I -570 -620 Offset frequency (MHz) 1 I I I Figure 4.3. Experimental four-level double resonance lineshapes taken under conditions of population modulation using plane polarized pump and probe radiation. The QR(4,3) transition in the v3 fundamental band of l3CH3F was pumped by a 12C1502 laser while the QP(7 ,3) transition in the 2v3—v3 hot band was scanned by a sideband system.The bigger of the two spikes was obtained when the pump and probe beams had parallel planes of polarization while the smaller spike was obtained when the beams had perpendicular planes of polarization. 61 ”CH3F A Plane Polarization Pump: R(4,3) ii Probe: P(8,3) I I I I I I I I I I -2l86.I —Zl36. Offset frequency (MHZ) —336. Figure 4.4. Transferred spikes obtained from four-level double resonance experiments under conditions of population modulation using plane polarized radiation. The QR(4,3) transition in the v3 fundamental band of l3CH3F was pumped by a 120602 laser while the QP(8,3) transition in the 2v3-—v3 hot band was scanned by a sideband system. The bigger spike was obtained when the pump and probe beams had the same planes of polarization while the smaller spike was obtained when the beams had perpendicular planes of polarization. 62 ”CHJF Pump: R(4,3) Probe: R(lO,k) IIIFIIIIIIIIITITj’IIIIIIIIIFIIIIITFIITjI]IIII 850 950 1050 1150 1250 Offset frequency (MHZ) Figure 4.5. Four-level double resonance lineshapes obtained by pumping the R(4,3) transition in the v3 fundamental band of 13CH3F while probing the R(10,3) transition in the 2v3~v3 hot band region using plane-polarized radiation. The lineshape with the bigger transferred spike was obtained when the planes of polarization of the pump and probe beams were parallel, while the lineshape with the smaller spike was obtained when the planes of polarization of the pump and probe beams were perpendicular. The R(10,0), R(10,1) and R(10,2) transitions are strongly overlapped. 63 created in the pumped levels as a result of pumping with polarized radiation, are not totally scrambled during collision-induced rotational transitions. This shows that some fraction of the initial alignment of the rotational angular momentum that was created in the upper level of the pump transition by polarized radiation was transferred intact to the lower level of the probe transition during collisions. As will be shown later in this section, conservation of the polarization of the angular momentum during collisions can be related to the velocity changes that occurred during collisions. The four-level double resonance spectra shown in Figure 4.6 were taken when the pump and probe beams were both circularly polarized. Similar to the case for plane polarized pump and probe beams, two types of population modulation experiments were performed. One experiment involved the same circular polarization for the pump and probe radiation while another experiment involved different circular polarizations for the pump and probe beams.The difference in the amplitudes between the sharp transferred spikes, which was evident in the experiments using plane-polarized radiation, can also be seen in the double resonance spectra using circularly polarized radiation for the pump and probe beams as shown in Figure 4.6. In this case, for an R branch pump and a P branch probe, the bigger spike intensities occur when the pump and probe beams have different circular polarizations, whereas the smaller spike intensities result when the pump and probe beams have the same circular polarization. Again, given the fact that a difference occurs, this result is not surprising when one examines the dependence on m of the transition probabilities when both the pump and probe sources are circularly polarized. Because only the lower level of the probe transition is collisionally coupled to the pumped Upper level, the difference in the amplitudes is due to the partial preservation of the original orientation that was created in the upper level of the pump transition by the circularly polarized pumping beam. That this condition persists for AJ > 1 collisionally- induced rotational transitions as shown by the four-level double resonance spectra in Figures 4.7 and 4.8, means that the collisions were not strong enough to completely 64 ”on; Circular Polarization Pump: R(4,3) Probe: P(6,3) I I I I I I I -7[30 Offset frequency (MHz) Figure 4.6. Four-level double resonance lineshapes obtained under population modulation using circularly-polarized radiation for the pump and probe beams. The R(4,3) transition in the v3 fundamental band of 13CH3F was used as the pump transition while the P(6,3) transition in the 2v3~v3 hot band was used as the probe. The bigger spike was obtained when the pump and probe beams had different circular polarizations (rcp/lcp) while the smaller spike was obtained when the pump and probe beams had the same circular polarization. 65 13 CH3F Circular Polarization Pump:R(4,3) Probe:P(7,3) w l \p Q I“ 4 M “W A ‘ . ,I ‘ ‘ ‘ u‘Wl’“ I “I ‘mb v I I I I l I I I I I I I —670 —620 —570 Offset frequency (MHZ) Figure 4.7. Four-level double resonance spectra taken under conditions of population modulation with circularly polarized pump and probe beams. The two spectra were obtained while pumping the R(4,3) transition in the v3 fundamental band and observing the P(7,3) transition in the 2v3~v3 transition in the hot band. The lineshape with the bigger spike amplitude was obtained when the pump and probe beams had different circular polarizations while the lineshape with the smaller spike was obtained when the pump and probe beams had the same circular polarization. 66 13C H31: Circular Polarization Pump: R(4,3) Probe: P(8,3) the ' 1"" N l i" \ll m‘ 1‘: M‘ Ti)” A) Matt "1"" .\ ‘q, t... I I I I I I I I I T I I I I I 1 I I --350. —300. —250. —200. Offset frequency (MHZ) Figure 4.8. Experimental four-level double resonance lineshapes observed when pump and probe radiation were both circularly polarized. The R(4,3) transition in the v3 fundamental band was pumped while the P(8,3) transition in the 2v3o—v3 transition in the hot band was probed. This corresponds to a A] = 3 collision-induced rotational transition. The bigger spike was obtained when the pump and probe beams had different circular polarizations (rcp/lcp) while the smaller spike was obtained when both pump and probe beams had the same circular polarization. 67 destroy the anisotropy in the spatial orientation of the total rotational angular momentum vector. Also, this implies that rotational state changes of more than one I unit, which we know can take place without appreciable changes in the initial relative velocities, also occur with very little reorientation of the rotational angular momentum vector. Figure 4.9 shows an experimental lineshape taken under population modulation that was fit to a theoretical lineshape using a four-level double resonance program that was written by Schwendeman. The four-level double resonance fitting program that was used to fit the lineshapes was discussed previously in Section 7 of Chapter Two. Included in this picture are three other curves that resulted from the theoretical fit of the spectrum to a sum of three Keilson-Storer functions. One Keilson-Storer function is used to describe the broad Gaussian component while the transferred spike itself is represented by a sum of two spikes of different widths and amplitudes. Each of the two spikes is described by a Keilson-Storer function with different values for the parameter B. The parameter B that was used in the fitting program is actually kB, where k = v0,p/c (where v04, = resonance frequency of the probe transition) so that the width has units of MHz. It was mentioned earlier in the introduction that the Keilson-Storer kernels relate the widths of the transferred spikes to the change in the relative initial velocities during collisions through the variable B, where B/fi is the root mean square change in velocity during collisions. Results from the fits of the spectra to a theoretical lineshape show that the narrower spike has a smaller value for B which means that the relative initial velocities did not change much during collision-induced rotational energy transfer. The broader component of the transferred spike which fit to a larger B value then represents collisions that caused larger changes in the velocity, which is why there is a larger spread in the distribution of velocities after collision. An obvious interpretation of these results is that those collisions which were more effective at causing greater changes in the relative initial velocities occurred with small impact parameters and those that did not cause large Changes in the velocities occurred with large impact parameters. 68 ”CH3F Pump:R(4,3)(u,=1*‘0) Probe:P(8,3)(u,=2“1) T I I I I I I I I I I I I —350 ~330 —310 —290 —270 —250 —230 —210 —190 Offset Frequency (MHz.) Figure 4.9. Four-level double resonance spectrum of l3CH3F obtained by pumping the R(4,3) transition in the v3 fundamental band while probing the P(8,3) transition in the 2V3 ~—v3 hot band. Results of fitting the experimental lineshape to a theoretical four-level double resonance lineshape show that the transferred spike may be represented by two components. These two components are described by two Keilson-Storer collision kernels. The difference between the experimental and calculated spectra is shown at the bottom of the curves. 69 We have shown in Chapter Two that it is possible to take the difference between double resonance signals obtained when the pump and probe beams have parallel planes of polarization and when the pump and probe beams have perpendicular planes of polarization (alignment modulation). The resulting difference is proportional to pure alignment (n = 2) of the angular momentum in the lower level of the probe transition. Similarly, one can take the difference between double resonance signals when the pump and probe beams have opposite circular polarizations (e. g. rcp-lcp pump-probe) and when the beams have the same circular polarization (either both rcp or both lcp). It was shown that this difference is proportional to pure orientation (n = 1) of the angular momentum vector in the lower level of the probe transition. The double resonance spectra taken under these conditions of polarization modulation have lineshapes that are very different from the lineshapes taken under conditions of population or intensity modulation. As shown in Figures 4.10 and 4.11, the alignment and orientation modulation spectra do not contain any broad Gaussian components. That the broad Gaussian components cancel out for the two halves of the modulation cycle means that for both types of pump-probe combinations, the broad Gaussian components have equal intensities. This should be true because the molecules that make up the broad Gaussian part of the four-level double resonance lineshape are the results of V-V swapping mechanism and are not part of the directly pumped population. Upon fitting the experimental four-level double resonance spectra taken under polarization modulation of the pump beam, it was found in this study that one collision kernel is sufficient to represent the spike as opposed to the two kernels that are needed to represent the transferred spike obtained under population modulation. The Keilson-Storer kernel that was needed to describe the spike obtained under alignment or orientation modulation gave a B value that is close to the B value of the narrower of the two 70 (a) R(4,3)—P(6,3) R(4,3)—P(7,3) I I I I I I T I I f I T I I I I 1 I I 1 I I I —650 — 40 —630 -620 —610 —600 R(4,3)—P(8,3) I I I I I I I I I I —300' - [90' - [so -27o — '60' -250 Offset frequency (MHZ) Figure 4.10. Four-level double resonance spectra obtained under alignment modulation. The R(4,3) transition was pumped in all of these experiments, while different probe transtions were observed. It is very evident that these lineshapes do not contain the broad components that make up the lineshapes under population modulation conditions. 71 (a Orientation Mod. ”CH3F R(m3)-P013) I I I I j I I I I I I I I T I I I T l ' l —7ao —755 —730 —705 — 80 — 55 (b) R(4,3)—P(7,3) R(4,3)—P(8,3) Ifi I I I I I I I I I I I I I I I I I I I I _335 —3os —280 -255 —'30 —'05 CHfsetfrequency OWHZ) Figure 4.11. Four-level double resonance spectra obtained under orientation modulation. The R(4,3) transition was pumped while the P(6,3), P(7,3) and P(8,3) transitions were scanned. Superimposed on each of the spectra are the results of a theoretical fit to a single Keilson-Storer function. 72 components of the transferred spike obtained from population modulation experiments. This is true for all probe transitions with Ak = 0. Therefore, there is reason to believe that only those molecules that reached the lower level of the probe from the upper level of the pump with very little change in the relative initial velocities contributed to the lineshapes from polarization modulation experiments. Since there is very little change in the initial relative velocities of these molecules, we conclude that the collision responsible for the rotational transitions seen under the polarization modulation experiments are weak collisions that occur with large impact parameters. The results of the fitting program yielded values for the widths of the transferred spikes in terms of RB (k = V0,probe/C) for each of the two components of the transferred spike. From the kB values, we can calculate the root mean square change of velocities as a result of collisions for each of the probe transitions that was studied. Table 3 gives the values for kB and the corresponding root mean square change of the velocities of the molecules (). An increase in the value of AJ resulted in an increase of the widths of the transferred spikes, which in turn corresponds to an increase in the value of kB and an increase in the during collisions. This result agrees with the previous observations of Matsuo et a.l (10) and Shin et al. (13) that the transferred spikes get broader as A] increases. In Table 3, RB] describes the width of the narrow component of the transferred spike while sz describes the width of the broader component. These two spike components represent molecules that were part of the pumped population, but for four-level double resonance experiments performed under alignment or orientation modulation, only the narrow component of the transferred spike was present. All of the probe transitions listed in Table 3 gave orientation and alignment modulation effects in the four-level double resonance experiments. Clearly, the molecules that contribute to the transferred spikes undergo collisions With a continuous distribution of impact parameters and a corresponding continuous distribution of changes in velocity. We model these distributions with a sum of Keilson- 73 Table 3. Widths of the transferred spikes and corresponding for 13CH3F. Probe M 14310)) lro .4320») (Aymara transition“) Population Modulation P(6,3) 1 0.82 6 9.13 64 P(7,3) 2 1.42 10 12.69 90 P(8,3) 3 2.10 15 12.89 91 Alignment Modulation P(6,3) 1 0.48 3 .......... P(7,3) 2 1.08 8 .......... P(8,3) 3 0.94 7 .......... Orientation Modulation P(6,3) l 1.15 8 .......... P(7,3) 2 1.58 ' 11 .......... P(8,3) 3 3.58 25 .......... (‘9 Probe transitions are in the 2v2o—v2 hot band region. The QR(4,3) transition in the v2 fundamental band was pumped. 0’) kin kB (MHz) is equal to the frequency of probe transition divided by the speed of light in vacuum. kB] describes the width of the narrow spike while sz describes the width of the broad spike. (C) 1 and 2 are the root mean-square velocity changes for the molecules that contribute to the narrow and broad spikes, respectively. Values are in m - s‘l. 74 Storer collision kernels and find that the spikes that occur in population modulation experiments are adequately represented by a sum of two kernels. It is convenient then to describe the rotational energy transfer process as involving two groups of molecules, one which transfers with little change in velocity and one which transfers with substantial change in velocity. It is important to remember that this is a discussion in terms of a convenient model, that in fact the two groups overlap to a considerable extent. In Table 3, these two groups of molecules are characterized by kBl and the corresponding 1, and by sz and the corresponding 2. The lower portion of Table 3 gives values for RB] and 1 for the only component of the double resonance lineshapes obtained under alignment and orientation modulation. The lineshapes obtained under alignment modulation for the probe transitions P(6,3) to P(8,3) have kBl values that are smaller than the kBl values for the same probe transitions under population modulation, as shown in Table 3. This was not unexpected because the lineshapes obtained under alignment modulation should describe only the pure alignment or n = 2 spherical tensor combination of the m-state populations.The width of the spike for the P(6,3) probe transition (0.48 MHz) is only about half of the widths of the spikes for the P(7,3) and P(8,3) probe transitions. Our values, as given in Table 3 also show that while the widths of the spikes under population modulation were significantly different for the P(7,3) and P(8,3) probe transitions, the widths of the spikes for these transitions did not vary significantly under alignment modulation. It should be recalled that the narrow component of the transferred spikes in four-level IR-IR double resonance experiments obtained under population or intensity modulation using plane polarized radiation contains information on both the population (n = 0) and alignment (n = 2) tensors while the transferred spikes obtained under alignment modulation contain information only on the alignment tensor. The narrow spike for the population modulation lineshape contains contributions from molecules that have a wider spread in their than molecules that make up the spike for the 75 alignment modulation lineshape. Therefore, one would expect that the values for kBl for population modulation lineshapes increase as one goes up in A]. In contrast, the kBl values for spikes obtained under alignment modulation should not vary much as one goes up in A] for A] > :1. One explanation for this very small change in the widths of the spikes for AJ > :1 is that while collisionally induced rotational transition into the (J = 6, k = 3) level occurred with A] = 1 which is dipole-allowed, the transitions into the (J = 7, k = 3) and (J = 8, k = 3) states occurred with A] = 2 and 3, respectively, both of which are not dipole-allowed transitions. Since collision-induced transitions that are not dipole- allowed require harder collisions than transitions that are dipole-allowed, the change in the initial velocities of the molecules is expected to be greater for the former. The results in Table 3 for alignment modulation also confirm our earlier interpretation that only those molecules that experienced very little change in their initial velocities are represented by the alignment modulation transferred spike. Alignment modulation effects were observed for rotational transitions that took place even with A] > 1 (still Ak = 0). The observed double resonance signal for polarization modulation decreased with increasing values of AJ. The reason for this decrease is that as one goes up in AJ, the transferred spikes become broader, which means that the change in the relative initial velocities of the molecules is also greater. Therefore, in collision-induced rotational energy transfers that occurred with A] > 1, fewer molecules contribute to the narrow component of the transferred spike of the double resonance lineshape obtained under population modulation. Upon performing alignment or orientation modulation, the double resonance signal for A] > 1 transitions is consequently smaller than the signal obtained for A] = :1 transitions. One interpretation of these A] > 1 transitions is that they were brought about by a cascade of AJ = 1 collisions rather than by a single collision that changed J by more than one unit. If this interpretation were correct, the alignment modulation effects that were observed when A] > 1 are due to a series of single-collision events. This interpretation is 76 probably correct because a single A] > 1 collision-induced transition is not dipole-allowed and most probably occurred with too small an impact parameter to cause a change of J by more than one unit in a single collision. Also, the observation that only the rotational transitions that were caused by collisions with large impact parameter and those that caused only small changes in the initial velocities of the pumped molecules preserved the initial alignment or orientation of the angular momentum suggested that a long-range dipole-dipole interaction was predominant. It was previously reported (87) and later confirmed in this study that only collision-induced rotational transitions that have Ak = 0 give alignment or orientation modulation spectra as shown in Figures 4.12 and 4.13. There is no experimental evidence for preservation of alignment or orientation for rotational transitions that occurred with Ak = 3n other than for Ak = 0 transitions. Previous workers (11,13) in our laboratory have studied collision-induced transitions that occurred with Ak = 3 and 6. In four-level double resonance studies with NH3 (11), the lineshapes obtained with Ak = 3 showed an asymmetry which indicated the presence of a broad transferred spike. The lineshapes for 12CH3F (13) and 13CH3F (10) do not show obvious sharp transferred spikes for these values of Ak. The interpretation for these lineshapes is that collisions that change the k quantum number by 3n units (n =1: 0) were strong enough to cause randomization of the initial velocity distribution upon transfer to the probed level, and that such collisions occurred with small impact parameters. Thus, collisions that lead to Ak = 3n (n > 0) are not expected to preserve the intial alignment or orientation. Direct rotational energy transfers that take place with Ak at 3n are forbidden by spin statistics, which is presumably the reason why no transferred spikes are observed. The results presented in Table 3 were obtained from a four-level double resonance fitting program that allowed different rate constants for the transfer of population, orientation, and alignment from the upper level of the pump transition to the lower level of 77 ”CH3F Plane polarization Pump: R(4,3) P(6,3) P(6,0), (6,1) P(6.2) I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I Ij I I I I I I I I I —l 100 —lOOO —900 —800 -—7OO Offset frequency (MHZ) Figure 4.12. Experimental four-level double resonance spectra obtained when both pump and probe beams were plane polarized. The top spectrum shows a scan of the P(6,0) to P(6,3) transitions in the 2v3—v3 hot band of 13CI-I3F under population modulation while the QR(4,3) transition in the v3 fundamental band was pumped. The bottom spectrum shows a scan of the same probe transitions but taken under alignment modulation. Only the P(6,3) transition gave an alignment modulation effect 78 Circular Polarization Pump: R(4,3) P(6,3) P(6,0), (6,1) IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII -—l 100 -lOOO —900 -800 —700 Offset frequency (MHZ) Figure 4.13. Experimental four-level double resonance lineshapes obtained with circularly polarized pump and probe beams.The top spectrum shows a scan of the P(6,0) to P(6,3) transitions in the 2v3o—v3 hot band of 13CH3F under population modulation while the QR(4,3) transition in the v3 fundamental band was pumped. The bottom spectrum, which was taken under orientation modulation is similar to the alignment modulation spectrum given in Figure 4.12 which shows that only the P(6,3) transition that corresponds to Ak = 0 shows an alignment modulation effect. 79 the probe transition. This was done after we realized that the theoretical four-level double resonance lineshapes resemble the experimental lineshapes more closely when different rate constants were used for the transfer of the different spherical tensor combinations. For l3CH3F, our results show that the rate of transfer of the population tensor (n = 0) is about 3/2 the rate of transfer of the orientation (n = 1) and alignment (n: 2) tensors. Figure 4.14 shows a comparisons between an experimental four-level double resonance lineshape and generated lineshapes. The observation that the broad Gaussian components of the four-level double resonance lineshapes cancel out during polarization modulation experiments led us to believe that these Gaussians should have equal areas for the two types of population modulation experiments. For example, the area of the Gaussian obtained when the pump and probe beams have parallel planes of polarization should be equal to the area of the Gaussian when the pump and probe beams have perpendicular planes of polarization. However, the experimental data showed that the Gaussian components are not exactly equal. The difference was attributed to possible changes in the pump and probe powers during the course of the experiment. A typical experimental run takes approximately 15 min to get good signal to noise ratio. It is possible for fluctuations in the laser or microwave powers to occur within that period. These fluctuations affect not only the Gaussian components, but also the amplitudes of the transferred spike. Therefore, in order to account for the difference between the two experimental runs, the size of the Gaussian component was used as a scale factor. An example of how this scaling affected the results of the theoretical fits is shown in Table 4. B. 13CH3F-foreign gas collisions Foreign gas pressures of 50 mTorr were added to 3 mTorr of 13CH3F in order to ensure that there were more foreign gas-methyl fluoride collisions than methyl fluoride self 13C HJF Pump:R(4.3)(u.' l *0) Probe:P(6,3)(v.=2"1) Gaussians removed iJ-‘ L __. __ A A‘ A‘. WW“ “'7 WV wvv— V-rvvv k(0)=k(2) 2/3k(0)-I<(2) I I I I I I T I I I I I I I I I T I I I—I I I I -770 -745 —720 —695 —670 —645 Offset frequency(MHz.) Figure 4.14. Comparisons between experimental and generated four-level double resonance lineshapes. The first picture contains experimental lineshapes obtained under population modulation using plane-polarized radiation, while the second picture contains lineshapes that were generated by assuming that the rates of transfer of population, alignment and orientation are equal. The third picture which resembles the experimental picture more closely than the second, was generated by assuming that the rate constants for the transfer of alignment and orientation are 2/3 the rate of transfer of population. 81 Table 4. Numerical results obtained from fitting experimental four-level double resonance lineshapes to a four-level double resonance theory ParallelW Perpendicular“) Perpendicularla),(b) Gaussianlc) 3759.7 3423.6 A1 13.65 11.94 13.11 kB1/MHz 0.725 0.650 A20) 7.28 5.31 5.83 kB2/MHz 8.529 7.955 (a) Data taken with pump and probe beam polarizations parallel or perpendicular (b) These areas were obtained after multiplying A1 and A2 by the ratio 3759.7/3423.6. (C) Height of the Gaussian contribution to the four-level double resonance in arbitrary units (‘1) A1 and A2 are the areas under the narrow and broad components, respectively, of the transferred spikes in arbitrary units. 82 collisions. A four-level double resonance lineshape was also obtained with pure 13CH3F, and this lineshape was compared to the four-level double resonance lineshapes obtained in the presence of foreign gases. The QR(4,3) transition in the v3 fundamental band of l3CH3F was used as the pump while the P(6,3) transition in the 2v3+-v3 hot band was chosen as the probe transition. Figure 4.15 shows four-level double resonance lineshapes that were obtained with just pure l3CH3F, and with mixtures of 13CH3F and foreign gases, H2, He and Xe. The lineshapes obtained in this work are similar to the ones that were obtained by Song and Schwendeman (12) in our laboratory. The main difference between the features of the lineshape obtained with pure 13CH3F and the lineshapes obtained with 13CH3F in foreign gases is the relative size of the sharp transferred spike and the broad Gaussian component. It is obvious from Figure 4.15 that the ratio of spike to Gaussian for pure 13CH3F is much greater than the ratio obtained for l3CH3F-foreign gas mixtures. The apparent reason for this difference is that for 13CH3F self collisions, long-range dipole-dipole interactions are dominant. For collisions with foreign gases like He, H2, and Xe which do not possess a permanent dipole moment, dipole-dipole interactions no longer play a major role. Other interactions which might be of shorter range than the dipole-dipole interaction come into play. For collisions induced by short-range interactions, the impact parameter must be smaller, which leads to larger deviations from the intial velocities of the l3CH3F molecules. The results of fitting the four-level double resonance lineshapes to a sum of Keilson-Storer functions reveal that the widths (value of the B parameter) of the transferred spikes for pure 13CH3F are smaller than the widths of the spikes for the 13CH3F-foreign gas systems. It is also evident from Figure 4.15 that the spike-to-Gaussian ratio is bigger when He is the collision partner than when Xe is the collision partner. This is presumably because during collisions, a heavy atom like Xe can cause greater changes in the initial relative velocities of the 13CH3F molecules than alight atom like He. However, the spike-to-Gaussian ratio when He was the foreign gas present is bigger than the ratio 83 3 mT ”CH3F R(4,3)—P(6,3) 3 mT "CH,F + SOmT Xe A I j I I I I I T T j I I I I r I I l I h 435 —no —715 -eeo -ee5 Offset frequency (MHz.) Figure 4.15 . Four-level double resonance spectra for 13CH3F obtained under population modulation using plane polarized pump and probe beams. Each spectrum includes three other curves which are results of a theoretical fit to 3 Keilson-Storer functions. The pictures show the effect of foreign gas-CH3Fcollisions on the sizes of the broad Gaussian component and the transferred spike. 84 when the foreign gas was H2 even if H2 is lighter than He. Other factors can account for this difference, e.g., the CH3F-H2 interaction may be stronger than CH3F-He, and the approach geometry might also be important For methyl fluoride self collisions, the broad Gaussian component was interpreted as resulting from a near resonant V-V swap. In the case of the atomic foreign gases like He and Xe, this V-V swapping mechanism is absent and so the broad Gaussian component of the four-level double resonance lineshape is a result of rotational energy transfer between 13CH3F and the foreign gases. This rotational energy transfer is a result of collisions that are strong enough to cause a complete therrnalization of the velocity distribution in the lower level of the probe transition. In situations in which the effects of V-V transfer may be ignored and where interactions other than the dipole-dipole interaction are important, it is useful to see how the reorientation of the total angular momentum vector is affected. Figure 4.16 shows four-level double resonance lineshapes obtained under population modulation for pure l3CH3F , as well as for 13CH3F-foreign gas systems. Each of the four pictures shown in Figure 4.16 include lineshapes that were obtained when the planes of polarization of the pump and probe beams were parallel and perpendicular. Similar to the results for pure 13CH3F, the amplitude of the transferred spike for the parallel configuration is bigger than the spike obtained for the perpendicular case. This difference can still be seen even for Xe as the collision partner. The same argument that involves the intensities of vibration- rotation transitions mentioned in the first section of this chapter may be used to account for this difference in amplitudes. That this difference in amplitude is still evident in systems with foreign gases may mean two things: 1) not all of the l3CH3F-foreign gas collisions are sufficiently strong to destroy the initial alignment that was created in the upper level of the pump transition, or 2) molecules that contributed to the transferred spike include some 13CH3F-13CH3F collisions rather than being entirely due to 13CH3F-foreign gas collisions. However, the width of the transferred spike for pure 13CH3F is considerably smaller than 85 3 mT. tier-1,1? R(4,3) —P<6.3) .3 mT. "Cl-13F + SOmT Ha £5 mT "Cl—13F + 50mT He .3 mT "Cl—13F + SOmT X- . . . v j Y . . . . f r . , . . v . , . . —765 —740 —7’15 —690 —665 Offset frequency (MI-12.) Figure 4.16. Four-level double resonance lineshapes obtained with plane polarized radiation the pump and probe beams. The topmost picture contains lineshapes that were obtained with pure 13CH3F, while the three other pictures contain lineshapes that were obtained with a mixture of 13CH3F and foreign gases. For all these spectra, the lineshapes that have the bigger transferred spike were taken when the planes of polarization of the pump and probe beams were parallel, while the lineshapes that have the smaller spike were taken when the planes of polarization were perpendicular. The R(4,3) transition in the v3 fundamental band was used as the pump, while the P(6,3) transition in the 2v3-—v3 hot band was used as the probe. 86 the widths of the spikes for each of the 13CH3F-foreign gas systems. In addition, the widths of the transferred spikes are different for each of the foreign gas partners. We conclude that the transferred spikes for the 13CH3F-foreign gas systems are the results of 13CH3F-foreign gas collisions and that some orientation or alignment is preserved by these collisions. Four-level double resonance experiments under alignment modulation were also performed for the 13CH3F-foreign gas systems. Figure 4.17 shows the alignment modulation spectra for these systems, including a spectrum for pure 13CH3F. The gain settings for these curves were taken into account in plotting the spectra to reflect the relative sizes of the spikes. The amplitude of the alignment modulation signal when He or H2 were the foreign gas partners was about 4 times smaller than the alignment modulation signal when only pure 13CH3F was present. In the case of Xe as the foreign gas, the amplitude of the alignment modulation signal was about 5 times smaller than the signal for pure 13CH3F. The halfwidths of the lineshapes obtained with foreign gases are larger than the halfwidths of the lineshapes obtained with only pure 13CH3F in the sample cell. Lineshapes for Xe and He have almost the same halfwidth, while the halfwidth for H2 is slightly larger. This suggest that collisions with H2 are slightly less effective in destroying alignment than collision with He or Xe. These observations show the dependence of the halfwidths of the alignment modulation spikes on the type of foreign gas present and thereby confirm that the transferred spikes for the foreign gas sytems are mainly due to 13CH3F-foreign gas collisions, and that contribution from pure 13CH3F self collisions are probably small. If the collisions were due mostly to 13CH3F-13CH3F collisions, there should be little or no difference in the halfwidths of the spikes for the different 13CH3F- foreign gas systems. 87 3 mT "CHJF R(4,3)—P(6,3) 3mT ”Cl-if + 50mT Ha 3mT "Cl-13F + SOmT Xe W T I V I I V 1 Ti I I Y I ‘r Y I 7 V V V 1 f I V I I V —730 -750 -740 —7so -7zo -7ro Offset frequency (MHZ) Figure 4.17. Alignment modulation spectra of pure 13CH3F and 13CH3F in foreign gases H2, He and Xe. The pictures include the experimental four-level double resonance lineshapes and the theoretical fit of the spectra to a single Keilson-Storer function. 88 C. l5NH3-15NH3 collisions The energy level diagram of N H3 differs from the energy level diagram of CH3F because of the presence of inversion splittings in the NH3 spectrum. Inversion here refers to the umbrella motion of the NH3 molecule as the N atom appears on either side of the plane of the three H atoms. The presence of the inversion partners for each vibration-rotation level in NH3 is best illustrated by looking at the energy levels of a harmonic oscillator (86, 102) as shown in Figure 4.18. The harmonic oscillator wavefunctions for odd values of v (vibrational quantum number) each have a node located at the center of the potential well, whereas wavefunctions for even v do not have a node at this position. If a potential hump, which represents the energy barrier to this umbrella motion, is raised at the center of the potential well, the energy levels for even values of v are perturbed more than the odd energy levels. As a result, each even energy level is pushed up closer to the odd energy level directly above it (102). For CH3F and other molecules that are thought to invert only very slowly, the potential hump is so high that the odd and even levels coincide. In the case of NH3, the hump is low enough that the inversion partners for v = 0 are separted by 0.8 cm‘1 and the inversion partners for v = 1 are separated by about 36 cm]. As expected the energy separation between inversion partners increases as one goes up in the vibrational energy levels. For our purposes, the v = 0 and 1 vibrational energy levels will be called v = 0s, andv=0a, thev =2and3 levels will becalledv=ls andv=1a, andthev=4and5 as v = 2s and v = 2a.. This notation is often used because the separations between the energy levels are small enough that each odd and even pair may be considered as belonging to one vibrational energy level. Each inversion level is labelled symmetric (s) or anti-symmetric (a) with respect to the inversion motion of the molecule (86). For k = 0, half of the inversion partners are missing, as required by the Exclusion Principle. 89 Figure 4.18. Energy levels for a harmonic oscillator. The wavefunctions for the vibrational energy levels from v = 0 to v = 5 are shown in this figure. The odd-numbered vibrational levels 1, 3 and 5 have nodes at x = 0, while the even-numbered energy levels, 0,2 and 4, do not have a node at this position. As a result, a potential at the x = 0 position tends to perturb the even-numbered levels more than the odd-numbered levels. This perturbation causes each even-numbered level to be pushed up to the odd-numbered level directly above it 90 The four-level double resonance energy level diagram that shows the different pump-probe combinations is given in Figure 4.19. Table 5 lists the different pump and probe frequencies. Only the a-s transitions in the v2 fundamental band of 15NH3 have near coincidences with the C02 lasers, so that the pump transitions that were used in this study are of the a—°s type. The probe transitions in the 2v2—v2 hot band region that are accessible with our laser-sideband system are of the s-a type. Thus, in our four-level double resonance experiments, both the upper level of the pump transition and the lower level of the probe transition have s-type symmetries with respect to inversion. Results for the asR(2,0) and asQ(5,4) pump transitions will be presented separately. 1. asR(2,0) pump transition Since half of the inversion partners are missing when k = 0, for the asR(2,0) pump transition in the v2 fundamental band of 15NH3, there was really no choice but to pump the (v = ls, J = 3)o—(v = 0a, J = 2) transition. Similarly, for the probe transition, saR(1,0) in the 2v2~v2 hot band, only the (v = 2, J = 2, a)~—(v = l, J = l, 5) transition was scanned by the probe beam. Therefore, only rotational energy transfer from the (v = 1, J = 3, 3) level to the (v = 1, J = 1, 5) level was observed in this study. This collision-induced transition occurs with A] = 2, Ak = 0 and with s~s, but since the dipole selection rule allows only transitions accompanied by swa-type processes and with AJ = 0 or :1, there is reason to believe that the observed double resonance signals were results of energy transfer that may have been caused by more than one collision or may have been caused by a single collision that occurred as a result of a higher multipole mechanism. Figure 4.20(a) shows four-level double resonance lineshapes that were obtained when circularly polarized pump and probe beams were used. The lineshapes show that the transferred spike is larger when the pump and probe beams have the same circular polarization (both rcp or both lcp), than when the pump and probe beams have different 91 1. Pump: asR(2,0) a V=lJ=2 ———-— s ______ a v=l,J=3 —""- a v=1J=l S ———S a a 2. Pump: asQ(5,4) I v=2,J=7 I “121:8 V=l,]=5 v=1,J=6 v=l,1=7 V=QI=5 Figure 4.19. Four-level double resonance energy level diagrams for 15NH3. The asR(2,0) and asQ(S ,4) pump transitions are in the v2 fundamental band while all the probe transitions are in the 2v2<—v2 hot hand. For the asR(2,0) pump transition, half of the levels are missing as required by the uncertainty principle. 92 «$22 Co was: 8: N>l~>~ 05 5 8a 25.5%wa 38¢ a: 283 £5 E 35590 82a> .o>-.o§> 0 ~22 E moses—out gumbo 8v .mmZE he was 3:08.253 N> 05 E as 33653 ESE 3v am: a 86:83.. =< 3 «OBOE v63 2 ~ch 3v mm Gaza cubism NOEUS Nome—US We? 2. Zach So am 655 $6523 3.2 szme 2:2. m3. 5 @503 NOEUS mm: : 32 9% mm 3.9:: 3.003. NOEUQ «DRUN— mwon c. VNVN com om Awmvme 8:553 Zumm- A935— mwflma wmo om amine @3th mecca—5.5 .393 A339:— Auvuombc hummu— moaesweum Eva—E:— EmIZS e8 momoaezveun— 38m new 9E5 .w 28$. 93 IOfiJImI3 Circular pol. Pump:osR(2.0) Probe:soR(l .0) I I I T I I I I I I I I I I I I —900 —850 —800 —750 —700 (21) “NH, CII'CUIOI' pol. Pump:osQ(5.4) Probezsc R (6.4) I I 1 I I h I I I I I I I I I I I I I I I 350 410 450 510 560 Offset frequency (Ml-l2.) (b) Figure 4.20. Four-level double resonance lineshapes obtained under population modulation using circularly polarized radiation for the pump and probe beams. The lineshapes in (a) were obtained while pumping the asR(2,0) transition in the v2 fundamental band and probing the saR(1,0) transition in the 2v2-—v2 hot band. The lineshapes in (b) were obtained while pumping the asQ(S ,4) transition in the v2 fundamental band while observing the saR(6,4) transition in the 2v2~v2 hot band. For both (a) and (b), the bigger transferred spikes are results of using different circular polarizations for pump and probe beams. 94 circular polarizations (rep/lcp). Similar to the results obtained for 13CH3F, the explanation for this difference lies in the transition intensities when circularly polarized radiation is used. For right circularly polarized radiation, the Am selection rule for transitions is Am = -1, while for left circularly polarized radiation, the selection rule is Am = +1. Then, for R branch pump and probe beams, the predicted intensities are greater when both pump and probe beams have the same circular polarization, which is exactly what was observed in this experiment. That this difference is still evident in four-level double resonance lineshapes where the pumped and probed levels are only collisionally coupled, suggests that there is at least a partial preservation of the initial angular momentum polarization that was created in the upper level of the pump transition. In order to test this hypothesis, four-level double resonance experiments under conditions of orientation modulation were performed for the asR(2,0) pump-saR(1,0) probe combination. The resulting spectrum is shown in Figure 4.21 which also includes a theoretical lineshape that was obtained from fitting the experimental spectrum to a single Keilson-Storer kernel. The orientation modulation spectrum, which is sensitive only to the orientation (n = 1) of the angular momentum, is proof that indeed, at least a portion of the initial orientation of the angular momentum survives collisionally-induced rotational energy transfer. Unfortunately, for the asR(2,0)-saR(1,0) pump-probe combination, only orientation modulation experiments are possible because the transition probablilities are such that predicted intensities are almost the same for both parallel and perpendicular polarizations of pump and probe beams. Any difference in the amplitudes of the transferred spikes for the two polarization arrangements is too small to be seen experimentally. This is the reason why no four-level double resonance signal under alignment modulation was observed for this particular pump-probe combination. When the asQ(3,3) transition in the 2v2o-v2 hot band was probed while the asR(2,0) transition was pumped, we obtained only population modulation lineshapes. 95 'ISN'._+3 Circular pol. Pump:osR(2,0) Probe:soR(1,0) I I I r I I I I I I T I I l I I I I r I I —80C'3 —775 —750 —725 —700 Offset frequency (Ml-12.) Figure 4.21. Four-level double resonance lineshape obtained under orientation modulation in 15NH3. The asR(2,0) transition in the v2 fundamental band was pumped while the saR(1,0) transition in the szo—vz hot band was scanned. Superimposed on the experimental lineshape is a theoretical lineshape that was a result of fitting the spectrum to a single Keilson-Storer collision kernel. 96 There was no evidence of an orientation modulation lineshape even though this pump- probe combination represented a A] = 0 collision—induced transition. However, the k quantum number changed by 3 units, and in experiments done with CH3F in this study and in a previous study (87), there is evidence that transitions that correspond to Ak = 3 did not preserve the intial alignment or orientation. Upon fitting the saQ(3,3) double resonance lineshape to a theoretical four-level double resonance lineshape in a previous work by Matsuo and co-workers (11), it was found that this lineshape contained a spike, or a velocity-conserving component However, this spike was very broad which means that the collisions that induced the transition were strong enough to cause a large change in the relative initial velocities of the molecules. 2. asQ(5,4) pump For the asQ(5,4) pump transition, where k = 4, all the rotation-inversion partners are present. However, the same restrictions with regards to transitions that are accessible by the C02 lasers and the sideband system still apply; i.e., only absorptions resulting from collisionally-induced transitions between two s states were scanned by our probe laser. For the asQ(5,4) pump, two probe transitions, saR(6,4) and saR(7,4) in the 2vzo—v2 hot band, were found to be suitable for our four-level double resonance studies. Table 5 gives details for the different pump and probe transitions for 15NH3 that were used in this study. Four-level double resonance experiments under conditions of population modulation with either horizontally or vertically-polarized radiation were performed for these two probe transitions. Figure 4.22(a) shows four-level double resonance lineshapes obtained when the asQ(5,4) transition was pumped while saR(6,4) transition was observed. The four-level double resonance lineshapes show that the amplitude of the transferred spike when the pump and probe beams have perpendicular planes of polarization is slightly larger than the spike when the beams have parallel planes of polarization. This is because the predicted intensity for a Q branch pump and an R branch probe that are both plane polarized is a little greater when the pump and probe beams have 97 15NH3 . Pump:osQ(5.4) Plane DOI' Probe:soR(6,4) 360 410I ' I ‘4g01 I ' '5‘IO‘ Offset frequency (MHz.) (a) 15NH3 Pu mp:oaQ(5.4)(v.- 1 *0) Preheat: R(7.4)(v.-2" 1) ~r, I T T fI I I I I I I I I IfiT I fiI I I I I I I I --590 —540 —490 —440 —:590 —:540 Offset frequency (Ml-12.) (b) Figure 4.22. Double resonance lineshapes taken under population modulation using plane polarized radiation for the pump and probe beams. The asQ(5,4) transition in the v2 fundamental band of 15NH3 was pumped while the saR(6,4) (a), and the saR(7,4) (b) transitions in the 2v2o—v2 hot band were scanned. The lineshape with the larger transferred spike was obtained when the pump and probe beams had perpendicular planes of polarization. 98 perpendicular planes of polarization than when they are parallel. It is interesting to note that this small difference in intensities between the transferred spikes has survived during collision-induced rotational energy transfer also in NH3. Figure 4.22(b) shows this effect for the saR(7,4) probe transition, for which the difference between the intensities for parallel and perpendicular pump-probe combinations is barely noticeable. Molecules that reached the (J = 6, k = 4, 8) level of the probe transition from the (J = 5, k = 4, 3) level of the pump transition through direct rotational energy transfer by dipole-allowed collisions must go through an intervening state, because sws transitions are not dipole-allowed. All of the pump-probe combinations in this work involved s~s collisionally-induced transitions.This is partly the reason why the transferred spikes that were observed in this present work and in a previous work (11) for 15NH3 are always broader. (and therefore showed larger values of ) than the transferred spikes in 13CH3F with the same A] and Ak. As explained earlier, since N H3 is a polar molecule, dipole-type interactions are dominant. Hence, the rotational energy transfer that was observed in this study and in an earlier work (11), might have occurred in a series of two dipole-allowed collision-induced energy transfers that took the molecules first from the (v = ls, J = 5) level of the pump transition to (v =1a, J = 5 or 6) and then to (v =1s, J = 6) of the probe transition. Also, because the spike to Gaussian ratio is much smaller in NH3 than in CH3F, we can conclude that the ratio of the rate of V-V transfer to direct collisional energy transfer must be greater in N H3 than in CH3F. Since linewidth data show that the total rate of collisional energy transfer in v2 = 1 of NH3 is comparable to that in V3 = 1 of CH3F, the absolute rate of V-V transfer must be greater in NH3 than in CH3F (103-105). Alignment modulation experiments were done with the asQ(5,4) pump-saR(6,4) probe transitions. A spectrum from this experiment is shown in Figure 4.23(a). Relative to the signals obtained for 13'CH3F, the alignment modulation signals for 15NH3 are much smaller. The existence of the alignment modulation signal for NH3 is evidence that the 99 15N H5 Alignment mod. Pump:asQ(5.4)(V.=1"0) Prabe:saR(6,4)(V.-2"1) I I If I I I I I I I I T I T I T I I I I 400 425 450 475 500 Offset frequency (MHZ.) (a) ISNHJ Orientation rnod. Pump:asQ(5.4)(u.=1‘-0) Probe:saR(6,4)(V.-2" 1) I I I I I I I I I I I Ifii I j I I I I 400 425 450 475 500 Offset frequency (Ml-l2.) (b) Figure 4.23. Four-level double resonance lineshapes obtained under (a) alignment modulation and (b) orientation modulation while pumping the asQ(5,4) transition in the v2 fundamental band and observing the saR(6,4) transition in the 2v2~v2 hot band. The smooth curves superimposed on the experimental lineshapes are results of fitting the experimental lineshapes to a single Keilson-Storer collision kernel. 100 initial alignment that was created in the upper level of the pump transition is conserved partially during collision-induced rotational energy transfer. That there is still a preservation of the initial alignment even for NH3 collisions which are thought to be harder than CH3F collisions, means that the alignment of the angular momentum is not easily destroyed during collisions. The Gaussian to spike ratio for the double resonance lineshapes when the saR(7,4) transition was probed is larger than the ratio when the saR(6,4) transition was probed. This is now considerably larger than the Gaussian to spike ratio observed in CH3F systems. Furthermore, collisions that took molecules from the (v2 = ls, J = 5) pumped upper state to the (v2 = ls, J = 7) probed lower state are stronger, and of closer range than the collisions that took molecules from the (v2 = l, J = 5, s) pumped upper state to the (v2 = l, J = 6, s) probed lower state. Population modulation experiments with circularly polarized radiation were performed for the asQ(5,4)-saR(6,4) pump-probe combination. The spectra are shown in Figure 4.20(b). The corresponding orientation modulation spectrum is shown in Figure 4.23(b). Similar experiments were also performed for the asQ(5,4) transition as the pump, while scanning the saR(7,4) transition, but as with the spectra with plane-polarized beams, there was very little difference in intensity in the double resonance lineshape taken when both pump and probe beams have the same circular polarizations and the lineshape taken when the pump and probe beams have different circular polarizations. The double resonance intensities for alignment or orientation modulation spectra are marginal when either the pump or the probe transition (or both) are Q branch, simply because of the algebra of the direction cosines. The fitting program that was used in the analysis of the data for NH3 assumed that the rate constants for the transfer of alignment and orientation are only 1/3 the rate constant for the transfer of population from the upper level of the pump to the lower level of the probe transition. This ratio was determined after generating lineshapes using the 101 four-level fitting program and finding that the generated lineshapes resembled the experimental lineshapes more closely when the ratio of the rate constant for the transfer of population was set equal three times the rate constant for the transfer of orientation or alignment. Comparisons between two sets of generated lineshapes and a set of experimental lineshape are shown in Figure 4.24. D. Precise pump offset frequencies for l5NH3. Precise offset frequencies for the asQ(5,4) and asR(2,0) pump transitions in the v2 fundamental band of 15NH3 were determined in this work. For the asQ(5,4) pump transition, both two-level and three-level combinations with probe transitions are possible. Table 6 lists the pump transitions and the corresponding probe transitions used for this particular experiment. For the asQ(5,4)/saR(5,4) and asR(2,0)/asP(2,0) combinations, three-level double resonance experiments with both counter- and co-propagating pump and probe beams were performed. As shown in Figure 4.25, a sharp spike was obtained on either side of the center frequency of the probe transition. One spike resulted from the counter-propagating pump-probe geometry, while the other spike resulted from the co-propagating pump- probe geometry. The smaller spike was obtained under co-propagating conditions because the beam that was reflected back to the sample cell was considerably weaker than the incident beam. The experimental lineshapes from the three-level double resonance experiments were then fit to Lorentz linehapes to get the center frequencies of the two spikes. These two values plus the resonance frequencies of the pump and probe transitions were used as input into Equation (3.5) to determine the precise pump offset frequencies. After the pump offset frequencies were obtained, the precise pump transition frequencies were determined from the offsets and the known C02 laser (106) frequencies. Equation (3.6) was then used to determine the microwave portion of the center frequency 102 k(1)=k(2)=k(0) 'SNHJ Pump:osR(2.0)(l/,=1"0) Probe:soR(1,O)(u,=2"1) 1/3k(0)=k(l )=k(2) IITIIlfiIIIIII‘IfiIIIIIIIIIIITITIIIl -950 —900 —850 —800 -—750 —700 -650 -600 Offset frequency (MHZ) Figure 4.24. Theoretical and experimental four-level double resonance lineshapes in 15NH3. The spectra in the middle are the experimental lineshapes. The top spectra are the theoretical lineshapes that were generated by assuming equal rate constants for the transfer of the n = 0. 1 and 2 spherical tensor combinations of the m-state populations, while the bottom spectra are the lineshapes that were generated by assuming that the rate of transfer of the n = 1 and 2 spherical tensor combinations are 1/3 the rate constant for the transfer of n = 0 spherical tensor combination. 103 15rq*_.':5 Pump;asQ(5,4)(u.=1"0) cou ntor Probe;saR(5.4) (u."2"1) I I I I f I I I —910 — I85 Offset frequency (Ml-l2.) i l —860 —835 Figure 4.25. Three-level double resonance lineshape for determination of precise pump laser offset frequency. The two spikes that appeared on either side of the center frequency of the probe transition resulted from having the pump beam counter- and co-propagate simulataneously with the probe beam. 104 of the probe transition which was finally used together with the known C02 laser frequencies to get the center frequency of the probe transitions. For a two-level system, where the transition pumped was also the transition probed, the double resonance lineshape was fit to a theoretical lineshape by means of the three-level double resonance equations. A pump offset frequency were used as input to the fitting program whose output included the resonance frequency of the transition. The pump offset frequency was adjusted until the resulting resonance frequency matched the resonance frequency that was used to compute the offset for input in the fitting program. The resulting frequencies are given in Table 6 while comparisons between our values and FTIR values (109, 110) are given in Table 7. 105 .mmza co can 5.. miss 05 5 Sugars £22 .6 EB 38.523 N> 2: e Saucers .953; - coma; n ~52 5 35:68“ Homage are? £5 E 85308 ~32 E momoaosqoa .8288ka NOo_Um_ «Ow—Um— mw.mmm :. .SmEE 3.me :6 cm 38.8%.“ .ANEMA: Kdm- qwqu wmc cm 8.8%? NOEUS «0.3.x 3- .Amzmq emcee 33 _m 83.2%; mafus N029: KdE. w- .93va 2:2. a2. R @9003 szmo— .2 .2 coup o2. R @503 3&8qu nomad Gemuaoaueuh one; .823 Swamp—O Aachen—vacuum 9:5.— .anm— .5. mogoauouh com—«O 9:5.— ommuoum ma 55555.6qu 05 512.5 98532.! one...— 6—5 95:— .e 93.»! 106 Ag: 2:50 .0 .3 mIZQ we was ~>~ 2: E x53 ME Ea... 82.9» 320300 AS: .3 B «EEO .Q mIZB .3 co acme 358523 ~> 05 E x83 .mE Eat 8:?» 33330.. £83 £5 E 8:58 ~32 E 36532”? cam. new OS 3“ mo. hm“. 02. RN 2. Omo wmc am no. _mo :0 ON ARE NM. new CG 3 09. EN. a2. 5N ow. mmo wmc mm mm. mwc :6 cm amméz N>N N> N> N> “Ham Q.mv~_em SkCOmw 8.3Mme AQNvmmw :oEmnmur—t .mIZfl mo acme N> 05 5 3556wa 2:8 c8 mosaosvné 855mm: 055% 5-5 can ME 59582 eaten—Sou .5 2an Chapter Five Discussion A. Rate equations The results from the four-level IR-IR double resonance experiments have shown that it is possible to see how the polarization of the rotational angular momentum is affected during collision-induced transitions. Through the use of polarized pump and probe beams, it is possible to look at the different statistical tensors in the lower level of the probe transition and see how these tensors change during rotational energy transfer. Using the orthogonality properties of the Wigner 3—j symbols (90) , Equation (1.1) may be inverted to give J n -m0 p(Jm,Jm) = Z(-l)J—m(2n +1)‘“2[1 m )6(J,n) (5.1) n where p(Jm,Jm) is the population of the J,m rotational level, and 6(J,n) is a statistical tensor expansion of the p(Jm,Jm). In this manner, the population of a polarized ensemble of molecules p(im,jm) is a linear combination of the different statistical tensors, a(J,n). For double resonance lineshapes obtained under conditions of population modulation with a saturating pump and a non-saturating probe. the theory in the Appendix shows that information can be obtained only for the first three terms in this expansion. The results of the theory and double resonance experiments with the gases 13CH3F and 15NH3 show that when circularly polarized radiation is used for pumping and probing under conditions of population modulation, the narrow component of the transferred spike with width equal to kBl, contains information on all of the first three terms - population, orientation and alignment. When plane-polarized pump and probe beams are used, the narrow component of the transferred spike contains information only on the population and alignment tensors . In contrast, the broad component of the transferred spike with 107 108 width equal to sz, contains information only on the population tensor because lineshapes obtained from polarization modulation experiments, which are sensitive to either the n = l or n = 2 tensor only, do not contain this broad component. This is true for either polarization of pump and probe beams. Thus, by performing both population and polarization modulation experiments. the n = 0, 1, and 2 tensors may be characterized. In the discussion below, the molecules that contribute to the sharp spike are assumed to have reached the lower level of the probe transition due to collisions that occurred with large impact parameter and involved primarily dipole-dipole interaction. The molecules that contribute to the broad spike are assumed to have reached the lower level of the probe transition due to collisions that occurred with small impact parameter and are due to both dipole-dipole and many higher-order interactions. A kinetic master equation may be written to describe the rates of collisional transfer of the different statistical tensors from the upper level of the pump transition to the lower level of the probe transition as follows: (Wyn) = —k§;’)o(J,,n) + Zk§g’o(JJ-,n) (5.2) 1 where 6(Js,n) = rate of change of the statistical tensor, n, in the lower level (15) of the probe transition; a(Js,n) and o( J j,n) = spherical tensor combinations of m-state populations in the lower level of the probe transition (13) and another level (1]), respectively; kg” = rate constant for the loss of statistical tensor n for the lower level of the probe transition by any process; and kg” = rate constant for the transfer of the statistical tensor n, from the level j to the lower level of the probe, 8. Equation (5.2) assumes no mixing of spherical tensor combinations by collisions. It should be recalled that for our experimental conditions, direct collisional transfer from one 109 vibrational state to another is very slow. Also, our double resonance experiments involved a fundamental band pump transition and a hot band probe transition. As a result of the combination of these factors, the contribution to the rate equation from the molecules in vibrational states other than v = 1 may be ignored. The a(Js,n) refer only to the spherical tensor combinations of the m-state populations that contribute to the transferred spike . The rate constant kg) includes first order rate constants for wall collisions and molecules leaving the beam as well as products of second-order rate constants times the total population for collisionally-induced processes such as direct rotational energy transfer and V-V swapping. Under conditions of steady-state pumping, 6(Js,n) = 0. With the assumption that molecules contributing to the transferred spike found in the lower level of the probe transition all came originally from the upper level of the pump transition (JP)’ Equation (5.2) may be simplified as k(") 6(Js,n) = 7%ch). (5.3) SS Because of the fact that the different statistical tensors (population, orientation and alignment) relax independently with different rate constants, Equation (5.3) may be used to write separate equations for n = 0, 1 and 2, as follows: km) for n = 0 (population), a(Js,0) = figure), SS km for n = 1 (orientation), 0(Js,1) = fioUpJ), and SS (2) for n = 2 (alignment), o(Js,2) = figupz). (5.3a) SS 110 When plane-polarized pump and probe beams are used in the double resonance experiments, only the first and third equations are needed. Unfortunately, our experiments do not allow for the determination of the individual rate constants kg) and kg.) . As far as k(") our experiment is concerned, the ratio #57 reduces to an effective ratio of rate constants, SS (n) kefl. The results of the four-level fitting program gave the following relative values for kg"; for the molecules that contributed to the sharp spike in samples of pure CH3F and e pure NH3: .(1) (0)- (2) ,(O) eff/ke- —keff/keff 13CH3F 2/3 15NH3 1/3 Within a Ak = 0 stack, the results of the theoretical fits show that the relative values of kg}? / kg) for CH3F and NH3 do not depend on the value of A]. For example, for 13CH3F collisions, the rates of transfer of the alignment and orientation tensors are always 2/3 the rate of transfer of the population tensor, regardless of the value of A]. B. Impact parameter dependence of the rate constants Double resonance experiments on NH3 and CH3F point to an impact parameter dependence of the rate of collisional energy transfer from the upper level of the pump transition to the lower level of the probe transition (in the case of a hot band probe transition). The most important observation that supports the previous statement is that rotational energy transfers that proceed with Ak = 3 produce double resonance lineshapes that are characterized by broad transferred spikes, whereas energy transfers that occurred 111 with Ak = 0 give lineshapes that have narrow and sharp transferred spikes. The widths of the spikes are a measure of the change in the initial velocities as a result of collisions. Therefore, the wider the spike, the larger the change in the , the smaller the impact parameter of the collisions. Using Equation (5.3a), the spherical tensor combination of m-state populations in the transferred spikes for Ak = O and Ak = 3n (n ¢ 0) may be described by: k(n.b) 6(Js,n) = fioupm) for Ak = 0, and SS 0(JS' ,n) = gggoUpm), for Ak = 3n (5.3b) where the rate constants now contain a dependence on the impact parameter, b. The primed quantities are rate constants for Ak = 3n processes, and Js and JD here each refer to the J and k for the particular level. The R values for .1s and II) are the same whereas the k values for 15' and JD differ by 3n. Consideration of the dominant terms in kg’b) and k'gg’b) leads to the conclusion that these rate constants are of comparable size. The transferred spikes are much sharper and larger for Ak = 0 than for Ak = 3n collision-induced (rub) [)3 ' transitions, and so it must be true that kg’b) >> k' If we make the reasonable assumption that small impact parameters lead to large values of the nns change in velocity on collision and, conversely, small changes in velocity mean large impact parameters, the occurrence of sharp spikes leads to an intersting conclusion about the dependence of the rates on impact parameter. As shown in Equations (5.3b), our observations for Ak = 0 depend on the ratio of rate constants, kg’b) / kgf’b) . In order to observe spikes, this ratio must first increase with increasing impact parameter (b) and then decrease. The decrease at large impact parameter is the natural result of a decrease in the interaction between two molecules as b increases, whatever the interaction. 112 The decrease at small impact parameter, however, requires a relative increase in the denominator compared to the numerator. This is most likely the result of an increased contribution to kmb) 53 from V-V energy transfer at small b. In systems that consist solely of polar molecules, the main form of collisional interaction would be the long-range dipole-dipole interactions. The rates that describe dipole-allowed processes should be the fastest. Rates for the different A] = 0, i1, Ak = 0 and the A] = n (n > 1), Ak = 0 collision—induced processes for both 12CH3F and 13CH3F have been determined by DeLucia and co-workers (52) through time-resolved millimeter/submillimeter—IR double resonance. The fastest rate was seen for the dipole- allowed A] = 0, :t 1, Ak = 0 process which occurs at a rate equal to 63:6 mS'1 mTorrl. For the A] = i 2 process (single collision) which is not dipole-allowed, the collisional transfer rate was found to be much smaller (10001-0089 mS‘1 mTorr‘l) than the value for A] = 0, i 1. Therefore, for A] = O, :1, the rate constant, kg’b), for the transfer of spherical tensor combinations from the upper level of the pump transition to the lower level of the probe transition should reflect values that correspond to dipole-dipole rates. (This relation to dipole processes will be considered below in the light of the tensor opacities.) However, at close range or for collisions that occur with Ak = 3n, which are much slower than for Ak = 0, the dipole-dipole type of interactions are relatively less important and therefore, it is not surprising that the spectrum is broader as a result of collisions with smaller impact parameter. For the polar molecules CH3F and NH3, the rate of transfer from the upper level of the pump transition to the lower level of the probe transition of the n = 0 or population tensor is faster than the rate of transfer of the n = l and n = 2 tensors. This also means that the alignment and orientation decay rates are larger than the population decay rates, especially in NH3 (because a slower rate gives more time for destructive processes to occur). Observations from our experiments support the above statement, namely: 113 1. Only collisions that do not change the k quantum number show tendencies to preserve the alignment or orientation, for both CH3F and NH3. 2. Only collisions that cause little change in the initial velocities of the molecules tend to preserve the alignment or orientation. The broad components of the transferred spikes contain information only on the n = 0 or population tensor. 3. The alignment and orientation modulation signals decrease faster than the population modulation signals with increasing values of AJ (i.e., the ratio of the area under the sharp spike to the area under the broad spike decreases as A] increases). 4. The rate constant ratios, kg}; and kg?) are 2/3k22f) for CH3F but only U362?) for NH3 These observations present a different point of view concerning reorientation and realignment. If the rates of orientation-preserving and alignment-preserving collision are slower than the rates of the p0pulation-changing collisions, then V-V processes and wall collisions, both of which are true reorientation and realignment processes, can occur before the collisions that preserve the orientation or alignment, in which case it would appear that collisions destroy the n = l and 2 statistical tensor components. Even if the rates of n = 0, 1, and 2 processes are practically the same, then an adiabatic reorientation or realignment process, either in the upper state of the pump or the lower state of the probe or both, could make it appear that collisions destroy the n = l and 2 statistical tensor components. Adiabatic m-changing collisions are reorientation and realignment processes whose contributions to the sharp spikes in CH3F and NH3 are discussed in the next section. C. Connection to dipolar transition rates via tensor opacities The connection of the rate of transfer of the different statistical tensors to multipolar transition rates becomes clear in the tensor opacity picture as suggested by Coy and colleagues (25). Equation (1.5) may be rewritten to express the rate cofficient for 114 transfer of a particular statistical tensor in terms of the contributions from different multipolar transitions with different ranks as follows: J J n ,(n) _ _ J+J‘ +n+A A ' LPS - A2 ( ) (2A + l){f J, A}kaJ<-—a f. (1.5) For A] = O, :1, and for polar molecules such as CH3F and NH3, the greatest contribution to kg) should come from the tensor opacities kzha. J with rank A = 1 which correspond to dipole-type processes, so that Equation (1.5) may be written only in terms of rate constants that have A = l, 1]): k(n) = _1 J+J‘+1+n 3 PS ( ) J' J' 1 }ktli(—or'f- The rate constant kg) is related to the effective rate constant kg? for the transfer of statistical tensor components. For the collision-induced transition from the (v =1, J = 5, k = 3) level to the (v = 1, J = 6, k = 3) level in 13CH3F, the relative values for kgpfor A = l (dipole—dipole) in terms of the 6—j symbols and tensor opacities are given below. k‘p’s” (J = 6~5) 6 0 n = 0 (population) 3{5 1}k(116(_a' 5 = 0.25 ”3164—065 . . 6 1 n = 1 (orrentatron) 3{ 1 l 5 }ka6<—a'5 = O- 247 kt1x6<—0t'5 MONUIONLIIG 6 n = 2 (alignment) 3{5 2 1 1 1 }ka6<—or’ 5 = 0- 240 ka6e—a'5 The tensor opacities $69055 reflect dipolar rate constants. In calculating the rate constants for the three statistical tensors using Equation (1.7), it is seen that the difference in rate constants comes from the numerical values of the 6-j symbols. The numerical 115 66n values of {5 5 I} vary only slightly from n = 0 to n = 2; i.e., the transfer rates are almost independent of the statistical tensor rank, n. Under the assumption that a dipole- dipole type of interaction is dominant, the rate constants for the transfer of population, orientation and alignment are the same. That the rate constants for the three statistical tensors are the same, implies very little realignment or reorientation of the angular momentum for this combination of J values, for collision-induced transitions that are governed mainly by a dipole-dipole type of interaction. However, this does not agree with our experimental results in that the values of kg} and k? are smaller than the value of kg), for both CH3F and NH3. The dependence of the transfer rates for the sharp spikes on the statistical tensor rank may be partially accounted for by considering the effects of direct elastic collisions, which are collisions that change only the m-state of the molecule without changing the rotational state. If these direct elastic collisions occur before a rotational energy change, they contribute to a loss in the alignment or orientation in the upper level of the pump transition, which appears as a corresponding loss in the lower level of the probe transition. Similarly, when realignment or reorientation occurs after a rotational state change has taken place, then it appears as a loss in the alignment or orientation in the lower level of the probe transition. This reasoning was used to explain the fact that the decay rate of the alignment in HZCO was seen to be always greater than the decay rate for the population (25). To see the effect of elastic m-changing collisions on the k8? for the sharp spikes in CH3F and NH3, Equation (A.l6) in Reference 25 may be used to derive the following: J J J k(")— km): 2 2 [—2] 1—(— 1)’"d""r(2n+1)[_ "I J "H 0- o mam”, mane meme thmedeaer (5.4) 116 where kmhrajmc is a rate constant for an elastic process in which all quantum numbers except m stay the same. From Equation (5.4), if the kaJmahaJmc are very close to zero, kgnzkm) SS , which is independent of n. On the other hand, if the kaJmar-almc are non-zero, then kg” for n > O is almost certainly greater than kg”. This seems necessary given the definition of the J ma~Jmc processes, which are by their nature, reorientations. Coy, et al. (25) showed this to be true for J = 1 by evaluation of the 3—j symbols and recognition that the kaJmar-almc are necessarily positive. A calculation for J = 5 and J = 6 and n = l and n = 2 based on Equation (5.4) shows that nearly all of the terms in the sum are positive, so that if the kalma~almc do not vary drastically for different m3 and me, kg) and kg) are both greater than kg?) . A calculation with the assumption that the kaJmahaJmc follow dipole selection rules also shows that kg” > kg?) for n = 1 and 2. If the kg.) for n = o, 1, and 2 are all about the same, but the 14;!) for n > 0 are larger than kgO)’ then the kg? = kg) / kg) for n > 0 will be smaller than kg), as found experimentally. Therefore, consideration of elastic m-changing collisions can in principle explain the factors 2/3 for CH3F and 1/3 for NH3 found for the ratios of kg"; / kg}? for e km) n = 1 and 2. However, consideration of the strong contributions of V-V processes to SS , especially for NH3, makes it unlikely that the entire deviation from unity of the ratios of kg}? to kg? can be explained by elastic reorientations. There are probably some additional contributions from other than dipole-dipole interactions that make the kg.) < kg) for n>0. D. Connection to Am selection rules When an ensemble of molecules is excited by polarized radiation, the distribution of populations among the degenerate m-sublevels becomes anisotropic. As defined in 117 Equation (5.1), the distribution is now a sum of the contributing 0(J, n)'s. During rotational energy transfer, these anisotropies may be transferred to other rotational levels. The extent of transfer depends on the strength of the collision and the nature of the intermolecular forces among the colliding species. Reorientation or realignment describes how the J vector is tilted with respect to a space-fixed Z-axis. The degree of reorientation or realignment therefore provides a good measure of the changes in the m quantum numbers of molecules during collision-induced rotational energy transfer. The extent of reorientation is directly related to the strength of the collision and gives an insight into the kind of interactions that governed the collision. Results in l3CH3F show that realignment and reorientation of the rotational angular momentum are not a significant part of the rotational energy transfer process for long-range collisions because for this species, kg} or k? equals 2/3 kg); for the molecules that contribute to the sharp spike. A substantial fraction of the state-changing CH3F-CH3F collisions that preserve the k quantum number (Ak = 0) are not sufficiently strong to destroy entirely the initial alignment or orientation that was created by the pumping process. In systems where CH3F molecules collide with inert gases, the experimental results also show a partial preservation of the alignment during collision-induced rotational energy transfer. These results reveal that as long as one of the collision partners is polar, and as long as the k quantum number is unchanged, collision-induced rotational energy transfer preserves, at least partially, the initial polarization of the angular momentum vector. In N H3 self collisions, where the resulting transferred spikes were broader than for CH3F, significant realignment and reorientation occurred(keff km- —k(127)- — 1/3k k(0ff)). However, since N H3 is also a polar molecule, for Ak = 0 there was still a partial preservation of the initial alignment and orientation during collision-induced energy transfer. 118 The observation that for certain collision-induced rotational transitions, there exixts at least a partial preservation of the initial alignment or orientation of the m-state populations, means that a Am selection rule is obeyed for collision-induced rotational energy transfer. Unfortunately, under field-free conditions it was not possible to monitor the flow of population from each m-sublevel. The various Am selection nrles must be inferred from the results of the experiments and from the results of the fit to a four-level double resonance theory. Am = 0 vs. Am > 0 selection rule Because of the possibility of realignment or reorientation occurring without a change in the rotational state, it is difficult to say whether the collisional energy transfer was due entirely to Am = 0 selection rule or to contributions from various |Am| > 0 selection rules. A problem also arises because of the difficulty of probing how much realignment occurred before and after a collision-induced transition. There is reason to believe that in both CH3F and NH3, for A] = :1, there are significant contributions from the dipole-allowed Am = :1 collision-induced transitions to the four-level double resonance lineshapes. As the value of AJ gets larger or as the probed level gets farther away from the pumped level, the molecules have more time to realign or reorient their J vectors so that contributions from higher ranges of Am start to become important in the rotational energy transfer process. As Coy and colleagues (25) pointed out, dipolar collisions (A = 1) show the greatest propensity to m-conservation. Alexander and Davis (93) gave an expression for the cross section of the J ‘m'o—Jm processes in a cell experiment which allows for the existence of several Am selection rules. Polarization transient gain spectroscopy of HZCO (56) have also shown that rotational energy transfer is a more probable pathway for energy transfer than just pure m-changing collisions. The results of the experiments reveal that for systems where one or both collision partners are polar, collision-induced energy transfers occur with definite Am selection 119 rules as long as the Ak = 0 selection rule is obeyed. Alignment and orientation-conserving effects were seen for transitions as high as A] = 5 for pure CH3F and A] = 2 for pure NH3. The anisotropies initially created by the pumping process are transferred at least partially to neighboring rotational levels during rotational energy transfer. Collisions that tend to change the projection of J on the molecule-fixed Z-axis (Ak at 0) tend to destroy the alignment and orientation completely. Chapter Six A. Double resonance experiments with cyclopropane (C3H6) Cyclopropane is a symmetric top, molecule belonging to the D311 point group. Its perpendicular bands of E' vibrations, v9 and v10, have been analyzed extensively by many authors (110-112). The v9 band lies near 1438 cm'l, while the v10 band lies near 1028 cm'l. The v9 vibrations correspond to the CH2 scissor-type motion while the v10 vibrations correspond to the CH2 wag (113). The v10 vibrations are in the range of our C02 lasers and were thus chosen for double resonance studies. Very little work has been done on collision-induced rotational energy transfer in non-polar molecules. A notable work is the double resonance study on C02 performed by Bischel and Rhodes (50). In non-polar molecules the long-range dipole-dipole interaction, which was found to be dominant in systems with polar molecules. is absent and replaced by other forms of interactions such as induced dipole-induced dipole interactions. These interactions are characterized by shorter range and steeper potentials than dipole-dipole interactions. Therefore, collision-induced rotational state changes in these molecules are expected to be accompanied by large changes in the initial velocities. The rotational wavefunctions of cyclopropane have A (ortho) and E (para) symmetries so that the collisional selection rule for Ak is Ak = 0, i3n. The pump and probe transition frequencies for our experiments were calculated by using the spectroscopic constants given by Plr’va and Johns (110, 111). By contrast to the experiments with CH3F and N H3, the pump and probe transitions used in this study are both in the fundamental band (v10 in this case). This was a result of the fact that no hot band assignments are available for this molecule. 120 Experiment The four-level double resonance spectrometer used for this study is the same as the one described in Chapter Three. For cyclopropane, pressures of 100-400 mTorr (considerably higher than those used for CH3F and N H3) were used in a 1 1n sample cell. The gas sample was obtained from the Matheson Company and was not purified further except through the usual freeze-pump-thaw cycle. Results Table 8 gives the pump and probe transitions used in the double resonance studies. All the transitions are in the vm fundamental band of cyclopropane. The double resonance effects for the different pump and probe combinations will be presented below. A. RP(11,6) pump The RP( 1 1,6) pump transition frequency was calculated to be about 44 MHz below the frequency of the 9P(20) line of the 120602 laser. This pump transition and the RR(9,6) probe transition share a common upper level so that a folded three-level pump- probe combination is formed as shown in Figure 6.1. The resulting three-level double resonance spectrum is shown in Figure 6.2. The three-level double resonance lineshape that was obtained was not one which we expected because there is no evidence of a sharp spike. Only a broad Gaussian was observed. The spike was predicted to occur at a sideband frequency equal to 13869.67 MHz. Several spectra were taken for this three-level combination but there was still no evidence of a spike. There are a number of possible reasons for this unexpected finding. One possibility is that we were pumping or probing the wrong transitions in which case 122 33mg S. 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The pump (heavy arrow) and probe transitions have a common upper level. 125 Cyclopropane(z/m=1“0) PumszP(11,6) ProbezRR(9,6) I 1 I I T I I T 1 T r l T 1 1 -925 —875 -825 -775 -725 Offset frequency (MHZ) Figure 6.2. Single (top) and three-level double resonance spectra of cyclopropane. The single resonance lineshape was obtained by scanning the RR(9,6) transition. The three- level lineshape was obtained by pumping the RP(l 1,6) transition and probing the RR(9,6) transition. - 126 we could be observing a four-level rather than a three-level double resonance. In order to make sure that we were really pumping the RP(11,6) transition, we searched the list of calculated frequencies for other potential three-level double resonance combinations. We found one combination, the RQ(21,2) pump and RR(20,2) probe described in the next paragraph and two additional combinations that require a 12C160130 laser. Unfortunately, our supply of 12C160180 gas was not enough to fill the semi-sealed laser tube, and as a result of a world-wide shortage of 130, commercial sources were no longer able to supply us with the needed gas. B. RQ(21,2) An additional three-level combination was predicted with the RQ(21,2) transition as a pump and the RR(20,2) transition as a probe. The resulting lineshape is given in Figure 6.3. This lineshape is similar to the three-level double resonance lineshape shown in Figure 6.2 in that the three level spike is absent. A four-level double resonance lineshape obtained by pumping the RQ(21,2) transition and probing the RR(14,2) transition is given in Figure 6.4 The presence of a nearby transition is indicated by an asymmetry of the lineshapes. The RR(l4,2) transition and the RR( 16,8) transition in the vm band are so close in frequency that the two transitions were assigned to the same frequency by Plfva and Johns (100). However, the single and double resonance lineshapes that we obtained clearly showed two peaks for the two transitions. Calculations of the frequencies using the constants taken from Reference 102 are reported below. The offset frequencies are vo-vl, where v0 is the center frequency of the transition and V] is the frequency of the probe laser. The experimental lineshapes were fit to a Gaussian to obtain the experimental offset frequencies which are also reported below. Transition Frequency (MHz) Offset (MHz) Offset (MHz) (calculated) (observed) RR(14,2) 31 399 160 .613 15 260 .205 15 234 .508 RR(16,8) 31 399 204 .383 15 303 .975 15 297 .901 Cyclopropor)e(1/10=Il <“0) PumszQ(21,2) ProbezRR(20,2) T T T r l 1 T T T T T T —eoo —550 —Iod —Isd Offset frequency (MHZ) T T I I I T I T — 00 —350 Figure 6.3 Single resonance (top) and three level double resonance (bottom) of cyclopropane. The single resonance lineshape was obtained by scanning the RR(20,2) transition. The double resonance lineshape was obtained by pumping the RQ(21,2) transition while observing the RR(2O,2) transition. 128 CycIOpropcne(z/10=1 (’0) PumszQ(21,2) RR(i 4,2) RR(i 6,8) T T I I 1 1 1 1 I l 1 1 150 zoo I230 Offset frequency (MHZ) T I I T I T T TI T 1 300 350 400 Figure 6.4. Four-level single (top) and double (bottom) resonance lineshapes obtained by pumping the RQ(2l,2) transition and probing the RR(l4,2) and RR(l6,8) transitions simultaneously. 129 The pP(23,12) pump transition has a near coincidence with the 9P( 18) line of the 130602 laser. Because only one of our carbon dioxide lasers is a semi-sealed type that may be used for isotopic gases , the probe transitions must be in the range of the 120602 laser. Several probe transitions were found that were suitable for four-level double resonance studies and are listed in Table 8. The three probe transitions RR( 18,8), RR(2(),12) and RR(23,22) were scanned simultaneously by the sideband system. The resulting single and four-level double resonance signals are shown in Figure 6.5. The four-level double resonance lineshapes consist of broad Gaussians, which are most likely due to a near resonant V-V swapping mechanism. Collision-induced direct rotational energy transitions to the RR(18,8) and RR(23,22) levels presumably do not occur because such transitions are accompanied by A(k-1) = 4 and 10, respectively. For the RR(20,12) transition, in which A(k-l) = 0, direct rotational transitions are allowed. However, no obvious velocity-conserving component was observed even with this probe transition. Figure 6.6 shows a four-level double resonance lineshape when the RR(23,20) transition was probed while pumping the l’P(23,12). The resulting double resonance lineshape is a broad Gaussian with the expected Doppler width of the probe transition. A number of two-level and three-level double resonance experiments with cyclopropane were planned but all of them required either 12C1802 or 12060130 gas, neither of which were commercially available. A sample of 120302 purchased before the 18O shortage turned out to be so severely contaminated that it shut down the laser which then required extensive cleaning. C. Comparison of FTIR and IR-IR double resonance frequencies The offset frequency, VO-Vlaser, was obtained for each of the probe transitions used in the double resonance studies by fitting the experimental spectra to Gaussian lineshapes. 130 CycIOpropcne (1402190) Pump:PP(23,12) RR(18,8) RR(20,12) RR(23,22) TTTIITTTITIIIIIITTTTIITTIITJITIITTTIIIllITjIIIIIIIIIIIITTIIIITj 120 220 320 420 520 620 720 Offset frequency (MHZ.) Figure 6.5. Four-level single (top) and double (bottom) resonance lineshapes obtained by pumping the PP(23,12) transition and probing the RR(18,8), RR(20,12) and RR(23,22) transitions. 131 Cyclopropane(vw=1"0) Pump2PP(23,12) ProbezRR(23,20) Offset frequency (MHZ) Figure 6.6. Four-level single (top) and double (bottom) resonance spectra of the v10 fundamental band of cyclopropane. The single resonance spectrum was obtained while scanning the RR(23,20) transition. The double resonance spectrum was obtained while pumping the I’P(23,12) transition and observing the RR(23,2O) transition. The sample pressure was 400 mT. 132 The offset frequencies that were obtained from the fit were used together with the known frequencies of the 120602 laser lines (99) to obtain the center frequencies of the probe transitions. A comparison between our frequencies and the values reported by Plfva and Johns (103) is given in Table 9. Discussion The most unusual feature of the double resonance experiments with cyclopropane is the absence of a sharp spike in either of the two apparent three-level double resonance spectra observed. As mentioned, one possibility is that a four-level rather than three-level double resonance was being observed in each case. This could result from experimental misidentification of one or both laser lines, from miscalculation of the spectra, or from a complete misassignment by Plr’va and Johns. All of these possibilities are unlikely. The fact that all of our single resonance spectra, including those of the probe transitions in the three-level double resonance combinations, occur very close to the frequencies measured by Plfva and Johns by FTIR spectroscopy argues against wrong laser lines. The fact that the frequencies we calculated agree with their reported frequencies argues against miscalculation. And, finally, it seems highly improbable that an assignment in which hundereds of frequencies can be fit to within the very small uncertainty of the FTIR frequencies could be in error. As a result of these conclusions, it seems necessary to ask some questions about the three-level double resonance theory as it has been applied to an infrared-infrared double resonance. These experiments with cyclopropane seem to be the first very high resolution studies of a molecule that not only has no permanent dipole moment (except a very small vibrationally-induced moment), but also has rather non-polar bonds. The v10 band is an allowed transition with significant intensity, so there is a substantial transition moment, and that is the only molecular parameter (aside from the molecular mass for 133 Table 9. Comparison of calculated and observed offset frequencies of some transitions in the vm fundamental band of cyclopropane. Transition FrequencyalMHz Frequencyb/MHz AC/MHz (FTIR) (IR-IR DR) RR(9,6) 31 145 681 .29 31 145 680 .65 -O.64 RR(14,2) 31 399 182.80 31 399134.70 -48.10d RR(16,8) 31 399 182 .80 31 399198.30 15.50d RR(18,8) 31 479 263 .36 31 479 263 .72 0.36 R1209,11) 31 477 839 .34 31 477 836 .96 -2.38 RR(20,2) 31 636 393 .58 31 636 384 .44 -9.14 RR(20,12) 31 503 861 .33 31 503 862 .27 1.39 RR(22,12) 31 583 890 .93 31 583 892 .72 1.79 RR(22,19) 31 483 274.58 31 483 274 .97 0.39 RR(23,20) 31 508 628 .03 31 508 623 .21 -4.82 RR(23,22) 31 478 768 .70 31 478 762 .25 -6.45 aObserved FTIR frequencies reported by Pliva and Johns (110). l3Frequencies obtained through IR-IR double resonance in this work. cCalculated difference between FTIR and IR-IR double resonance values. dThese transitions were overlapped in the FTIR spectra and were assigned to a single frequency by Pliva and Johns (110). 134 Doppler averaging) that enters in the calculation of the three-level double resonance lineshape. The double resonance experiments for cyclopropane were carried out with higher pressure samples (0.1-0.4 Torr) than for CH3F or N H3 (~ 0.01 Torr), so there could be some unforeseen effect of the pressure. At 0.1 Torr, the pressur broadening is probably ~l MHz (the pressure broadening parameter is not known), whereas the Doppler width is ~ 30 MHz. Therefore, a spike should be observed. The only other explanation that has occurred to us concerns the effect of the dominant relaxation mechanism or mechanisms on the spatial m degeneracy of the states. Our calculations of three-level double resonance assume that the pumped transition is a superposition of independent two-level systems. For example, for plane-polarized pumping, we assume Am = 0 selection rules and each m state in the lower level of the transition is coupled to the same m state in the upper level to form the two-level system. The overall pumping is assumed to be a simple sum of the pumping in each two-level system. Although this is probably not correct, even for CH3F and N H3 where it gives superb agreement with experiment, it is hard to see why a problem with this assumpiton would lead to broadening of the spectral line, because all the two-level frequencies are the same. It is not impossible that the dominant relaxation mechanism in cyclopropane is V-V transfer. This mechanism replaces molecules in the velocity-selected pumped upper state with molecules from the random velocity populations of the ground state. If the dynamics work out such that even three level experiments are dominated by V-V transfer processes, then a broad three-level double resonance lineshape would be observed. This is an interesting theoretical problem that has not been solved to our knowledge. 135 B. Double resonance studies with deuterated fluoroform or CF 3D The interest in deuterated fluoroform or CF 3D stems from its importance as a medium for an optically-pumped submillimeter wave laser (114, 51). Population inversion between two rotational levels in the same vibrational state is achieved by pumping with a C02 laser. It is known that collisions may induce further population inversion between other rotational levels which are not directly connected to the pumped levels. CF3D is an oblate symmetric top molecule with C3v symmetry. The symmetry axis passes through the OD bond. The v5 perpendicular band. which is the CF3 degenerate stretch (113), lies in the 975 cm'1 region and is therefore within the frequency range of our C02 lasers. Table 10 contains the frequencies that were used in this study. The frequencies of the transitions in the v5 fundamental band were calculated by using the constants given in a paper by Leavitt et al. (114). Since CF3D is a polar molecule, collision-induced rotational energy transfer is expected to be dominated by long-range dipole type processes. The dipole-allowed Ak = 0, A] = O, :1 processes are expected to show the greatest propensity for velocity conservation during collisions. For C3,, symmetric top molecules, the collisional selection rule for Ak is Ak = 3n. Experiment The double resonance spectrometer has been described in Chapter Three. The sample was obtained from MSD Isotopes. N 0 further purification was done except for the freeze, pump and thaw cycle. Sample pressures of 2()- 100 mTorr were used. .c—SB m2. 5 355.8% 5:258. .339. .NIE E o>-.oma_>o N22 5 83.5 A3: 3 .e .3334 ..o €82..qu 05 53. 33.323 35:35.. .an—U ..o 9:... 35.58:... n> on. E a... 33.65... one... 9... 9.5% 136 $3.... menses- Noe.uN. 8.3.2.3 $.39. 33%. 83%. 5%.er N. 8202 3.838 %N 2539. enema- No.28. 2.338 %N .339. GEM—o— mmm. new NT Nave—UN. m3. mmo .9 am 3680: 9.3%. :98: Noe.uN. SwNe 8N oN 2.2.9. 9.3%. .323. Noses 8.4%...82 2.2.9. $3.5. Ens... Noe.uN. :QeNSNeN 2.89. $3.... 33”.... wveaweo. NcebN. ENEBNeN ASN.9. o3..- NoauN. SNNNSNaN 2.3.0.. 9.8.:9 comma— ..mucoavouu «3e...— o.om&O humus— ...mugavouh «9:...— dmmu .o 3:58 85:58. 03.5.. on. 5 new: 3.2.2.95 5:55... one... 3.... 9...... .o. 2an 137 Results A. PQ(24,1) pump As shown in Table 10, the l’Q(24,1) pump transition is about 14.6 MHz above the frequency of the 9R(12) 120302 laser line. Four probe transitions were chosen for this double resonance study of CF3D, and will be given below. a. Three-level double resonance study Figure 6.7 shows the three-level double resonance spectrum that was obtained by pumping the PQ(24,1) transition while observing the RQ(24,1) transition in 22 mTorr CF3D. b. Four-level double resonance studies The four-level double resonance lineshape obtained by pumping the PQ(24,1) transition while probing the RQ(23,1) transition is given in Figure 6.8. This rotational energy transfer occurred with the dipole-allowed collisional selection rules A] = -1 and A(k-l) = 0. The lineshape is characterized by a sharp transferred spike, which is evidence of velocity conservation during rotational energy transfer. This sharp spike is superimposed on a broad Gaussian lineshape which has the expected Doppler width and resonance frequency of the probe transition. Figure 6.9 shows the four-level double resonance lineshape obtained when the RQ(25.1) transition was scanned. The resulting lineshape also contains a sharp transferred spike that is superimposed on a broad Gaussian. A similar result was observed when the RQ(22.l) transition was probed, as shown in Figure 6.10. In this case, the rotational energy transfer took place with A] = 2 and A(k-l) =0. When the RQ(24,2) transition was probed, however, the resulting lineshape did not contain a velocity-conserving transferred spike as shown in Figure 6.11. This is not CF30 @5160) Pump:pQ(24,1) RQ(9,O) RQ(24,1) [TllllTlllTllTTTllllIllllllllTllllllllIT —788 —738 —688 ~638 —588 Offset frequency (MHZ) Figure 6.7. Single resonance (top) and three-level double resonance in the v5 fundamental band of CF3D at 22 mTorr. The single resonance scan included the RQ(9,O) and RQ(24,1) transitions. The three-level double resonance spectrum was obtained by pumping the PQ(24,1) while observing the RQ(24,1) transition. 139 CF30 (u.=1¢-O) Pump:'Q(24,1) Probe:"Q(23,1) 22 mTorr Offset frequency (MHZ) (a) CF30 (v.=1"0> Pump:'Q(24,1) Probe:"Q(2.3,1) 100 mTorr ......... , . Y r T . - - , . I —255 —235 —155 Offset frequency (MHZ) (b) Figure 6.8. Four-level double resonance lineshapes of CF3D obtained by pumping the I’Q(24,1) transition while probing the RQ(23.l) transition. 1 40 (a) can (u.=1"0) Pump:'0(24.1) Probe:'Q(25,1) 22 mTorr .................. r . . . . . . . . . , -225 -175 —125 -75 Offset frequency (MHZ) (b) CFJD (u.=1“0) Pump:'Q(24,1) Probe:“Q(25,l) 100 mTorr “22:5 ....... L19; ....... .1215" ”UH—1’s . Offset frequency (MHZ) Figure 6.9. Four-level double resonance in CF3D obtained by pumping the PQ(24,1) transition while observing the RQ(25 ,1) transition. 141 (a) can (u.=1"0) Pump:'Q(24.1) Prober'Q(22,1) 22 mTorr -655........._8,05.......-ins Offset frequency (MHZ) (b) can (v.31“0) Pump:'Q(24,1) probe:'Q(22,1) 100 mTorr -ass 7.....fif-g05.,......._755 Offset frequency (MHZ) Figure 6.10. Four-level double resonance lineshapes of CF3D obtained by pumping the PQ(24,1) transition while probing the RQ(22,1) transition. 142 CFJD (V5=1"'O) Pumpsz(24,1) ProbezRQ(24,2) lOO mTorr l r T l l T T l l I f l I l l l I l I I l 520 570 620 Offset frequency (MHZ) Figure 6.11. Four-level double resonance in the v5 band of CF3D. This double resonance lineshape was obtained by pumping the PQ(24,1) transition and probing the RQ(24,2) transition. The collision-induced rotational energy transfer occurred with A] = O and A(k-l) = 1. 143 surprising because the rotational energy transfer to these rotational levels occurred with A(k-l) at 3n. B. RQ(30,l) pump The RQ(30.l) transition was found to be nearly coincident with the 10R(10) line of the 120802 laser, and to be about 13.8 GI-lz from the 10R(18) line of the 120602 laser. As a result, a two-level double resonance study of the RQ(30,l) transition could be performed and a precise offsetfrequency for this transition could be measured. The precise offset frequency for this transition was then determined by fitting the experimental lineshapes to a three-level double resonance theory. Since in a two-level system, the transition pumped is also the transition probed, the pump offset frequency should be equal to the probe offset frequency. In the fitting program, a reasonable resonance frequency for the transition was assumed. The value of the offset frequency was changed before each fit until the resulting resonance frequency of the probe transition matched the assumed resonance frequency of the pump. a. Two-level double resonance studies The two-level double resonance lineshape was obtained by pumping and probing the RQ(30,l) transition, as shown in Figure 6.12. The precise offset frequency obtained for this transition is given in Table 10. lb. Four-level double resonance studies Unfortunately, only one probe transition was used in the four-level double resonance studies with the RQ(30,1) pump transition. Figure 6.13 shows the four-level double resonance lineshape that was obtained by probing the RQ(29,4) transition. The collision—induced rotational energy transfer to the probed levels occurred with A(k-l) = 3. The resulting four-level double resonance lineshape contained only a broad Gaussian component. No obvious velocity-conserving transferred spike was observed. 144 CFJD (V5=1"O) PumszQ(30,1) Probe:RO(30,1) 4O mTorr I l l l l l l f T —856 —§oe Offset frequency (MHZ) l l l I l l l l l Figure 6.12. Two-level double resonance signal for CF3D. This lineshape was obtained by pumping and probing the RQ(30,1) transition in the v5 fundamental band. 145 CFSD (V5=1‘-O) PumszO(I’>O,l) Probe:RO(29.4) 7O mTorr l 1 l l 420 Offset frequency (MHZ) Figure 6.13. Four-level double resonance in the v5 band of CF3D. This lineshape was obtained by pumping the RQ(30,1) transition while probing the RQ(29,4) transition. The collision-induced energy transfer occurred with A] = -l and A(k-l) = 3. G 146 Comparison of calculated and observed frequencies in CF3D Comparisons of calculated and observed frequencies for some RQ(J,k) lines in the v5 fundamental band of CF3D are given in Table 11. The calculated frequencies were obtained from the parameters determined by Leavitt and co-workers (114) by means of a heterodyne technique. The diode-laser-heterodyne method that was employed by Leavitt's group allowed the measurement of frequencies of several lines in the v5 band of CF3D up to 5 GHz away from the frequencies of the C02 laser. In our laboratory, a sideband system that generates a frequency equal to V1356, i- va was used to scan each transition listed in Table 11. This sideband system allows for the measurement of transition frequencies that are 8 - 18 GHz offset from the C02 laser frequencies. Because of this reason. the v5 fundamental band transitions that are accessible with our spectrometer are different from the transitions that were measured by Leavitt's group. As a result, we cannot compare the values for the transition frequencies that we measured with any of the values given by Leavitt and co-workers. The offset frequency ,iva, for each transition was obtained by fitting the experimental single resonance spectra to a Gaussian lineshape. The observed frequencies were then obtained by adding or subtracting va from the known frequency of the C02 laser line that was used as the carrier for the microwave for the particular transition that was scanned. Discussion The results of previous analyses of four-level double resonance lineshapes in CH3F (10, 12) and NH3 (11) indicate that sharp transferred spikes were observed only for rotational energy transitions that were induced by weak collisions, or collisions that have large impact parameters. These weak collisions do not change the projection of the total angular momentum vector on the molecule fixed axis (Ak = O). 147 The results of the four-level double resonance experiments with CF3D also show that rotational energy transfer can occur without changing the k quantum number as shown in Figures 6.8 and 6.9 where A] = :1 and A(k-l) = 0. Rotational transitions that change J by more than one unit can occur. as shown in Figure 6.10 where A] = -2 and A(k-l) = 0. That sharp transferred spikes were observed for these rotational transitions indicates that the collisions occurred at long range with large impact parameters.There is no evidence of a sharp transferred spike when A(k-l) =1: 0. These observations suggest that for CF3D, the long-range dipole-dipole type of interaction is important in the rotational energy transfer process. This is not surprising given the fact that CF3D is a polar molecule. However, since CF3D is an oblate top, it is of interest to us to see how its rotational relaxation mechanisms compare to that of a prolate symmetric top like CH3F. A time resolved infrared double resonance study of CF3D by Harradine and co-workers (51) suggests that collisions that change the k (Ak = 3n, n at O) quantum number can occur faster than collisions that change J. Our preliminary results do not support their findings because no transferred spike was observed when the RQ(30,1) transition was pumped while the RQ(29,4) transition was observed. In this case, collision-induced transition to the Q(29,4) levels occurred with A] = -1 and A(k-l) = 3. Observation of a transferred spike would mean that k-changing collisions occur faster than J-changing collisions. For prolate tops like CH3F and N H3 a wider range of A] than Ak values occur as a result of collisions. Unfortunately, only one set of observations was made so that we cannot comment on the difference in the rotational relaxation between the oblate top CF3D and the prolate tops NH3 and CH3F. 148 Table 11. Comparison of calculated and observed frequencies for the v5 fundamental band in CF3D. Transition Frequency/MHZa Frequency/Msz (Obs-Calc)/MHz (calculated) (this work) RQ(22,1) 29 208 628.07 29 208 627.18 -().18 RQ(23,1) 29 208 194.90 29 208 197.11 2.21 RQ(24,1) 29 207 742.04 29 207 743.33 1.29 R(260,1) 29 204 605.76 29 204 620.00 14.24 RQ(24,2) 29 191 023.03 29 191 032.31 9.28 RQ(29,4) 29 155 144.05 29 155 130.93 -13.12 aValues calculated from parameters given by Leavitt et al.(114). bValues obtained from VCOZiVMW The values for va were obtained by fitting the experimental single resonance spectra to Gaussian lineshapes. Chapter Seven Summary and Conclusions We have derived expressions for the elements of Jones matrices to describe the spectroscopic properties of a sample under different polarizations of pump and probe radiation (Chapter Two and Appendix). The elements of the Jones matrices can be used to predict double resonance absorption coefficients that are functions of the velocity component in the direction of the pump beam for different polarizations of the probe beam. These absorption coefficients are shown to depend on the three statistical tensor ranks, n = 0 (population), 1 (orientation) and 2 (alignment). It is also pointed out that polarization modulation experiments can yield information on either pure alignment or pure orientation of the m-state populations. Knowledge of the change in the alignment or orientation of the m-state populations as a result of collisions can give insights into the Am selection rules The predictions of the four-level double resonance theory are as follows: 1. When the experimental arrangement includes a saturating pump with a weak probe beam, information on the n = 0 (population), 1 (orientation) and 2 (alignment) spherical tensor combinations of the m-state populations are obtained from four-level IR-IR double resonance lineshapes taken under conditions of population modulation using circularly polarized pump and probe radiation. 2. When both the saturating pump and the weak probe radiation are plane-polarized, the four-level double resonance lineshapes yield information only on the even tensor components, n = 0 and 2. 3. Four-level double resonance lineshapes with strong pump and weak probe obtained under polarization modulation by using circularly polarized radiation (orientation modulation) are sensitive only to pure orientation (n = l) of the m-state populations. 149 150 4. Four-level double resonance lineshapes with strong pump and weak probe obtained under polarization modulation by using plane polarized radiation (alignment modulation) are sensitive only to pure alignment (n = 2) of the m-‘state populations. Analysis of experimental lineshapes of four-level double resonance spectra of 13CH3F and 15NH3 obtained under a variety of polarization conditions with a saturating pump beam and a weak probe beam lead to the following: a. Four—level double resonance lineshapes for these molecules obtained by means of alignment and orientation modulation are very different from the lineshapes obtained by means of population modulation. The transferred spike for the population modulation lineshape is best fit to a sum of a narrow and a broad component each of which is described by a Keilson-Storer kernel with width equal to kB. In contrast. the spike for either alignment or orientation modulation has only a narrow component. b. Neither the alignment modulation nor orientation modulation lineshape contains the broad Gaussian component that is present in all population modulation lineshapes. c. The four-level double resonance lineshapes have been analyzed by a fitting program that calculates the n = t). l and 2 spherical tensor combinations of the m- state populations in the upper level of the pump transition. The spherical tensor combinations are then transferred intact to the lower level of the probe transition after multiplication by an n-dependent relative intensity of the probe transition. The key results of the fittings are as follows: i. A single Keilson-Storer collision kernel is sufficient to describe the experimental four-level double resonance lineshapes from either alignment or orientation modulation. ii. The single Keilson-Storer kernel has a width kBl that is close in value to the width of the narrow component of the population modulation lineshape. 151 iii. The theoretical fits were much improved when different rate constants for the transfer of the n = 0 and n =1, 2 spherical tensor orders were assumed. The best values for the molecules that contribute to the sharp transferred spikes in 13CH3F and 15NH3 are: 13CH3F k5,}; = g? = ugly/3 15NH3 k9) = kg}? = 5.22/3 For the broad transferred spikes, the best values are kg} = kg]? = O for both molecules. Thus, the rate constants for the transfer of orientation or alignment are smaller than the rate constant for the transfer of population for both 13CH3F and 15NH3. Also, the difference between the rate constants, k(2.)- 5,}; e (0)_ km or k efi‘ , is apparently impact-parameter dependent. The difference is large e when the collisions are strong, or when the collisions have small impact parameters. 5. Alignment and orientation modulation signals were observed only for Ak = 0 collision- induced rotational energy transfers for both CH3F and NH3. Our results demonstrate that the spherical tensor combinations that are created in the upper level of the pump transition as a result of pumping with either plane or circularly-polarized radiation can be transferred to the lower level of the probe transition during collision-induced rotational transitions. For CH3F and NH3 this transfer obeys collisional selection rules that are dominated by the dipole-type selection rules, A] = :t 1, Ak = 0. Because the collision environment is still isotropic (only a small fraction of the molecules in the ground vibrational state are pumped in our experiment), the different spherical tensor combinations presumably relax independently. This means that an alignment tensor that was created in the upper level of the pump transition relaxes out of this level to the probed level as an alignment 152 tensor. There is no interconversion among the different tensor orders (An = O). For example, the initial alignment cannot be detected as an orientation in the lower level of the probe transition. The existence of alignment or orientation modulation effects in the lineshapes of transitions whose J values differ by more than 1 from the upper state of the pump is interpreted as arising mainly from multiple collisions that consist of a series of A] = :1 steps, although collisions that lead to A] > 1 are also possible. The work of De Lucia and co-workers (52) pointed to a much faster rate for the dipole-allowed A] = :1 (63:6 rns‘1 mTorr‘l) as compared to other A] > 1 transitions. For example, A] = :2 transitions were reported to occur at about 1 mTorrl mS'l (52). The processes with lower rate constants probably require smaller impact parameters, and our experiments show the orientation or alignment transfer requires collisions of large impact parameters. Our results show that in order to observe alignment and orientation modulation effects, there should only be small changes in the initial velocities of the molecules during rotational energy transfer which in turn requires collisions of large impact parameters. Therefore, dipole-type collisional selection rules (AJ = 0, :1) are preferred. The Am selection rules are more difficult to probe because of the degeneracy of the 2] + 1 m states. However, as indicated by the presence of alignment and orientation modulation signals, substantial collision-induced rotational energy transfer proceeds by processes that tend to preserve m. For systems that are made up of polar molecules, the dipole-allowed Am = O, :1 selection rule should be dominant, but contributions from higher values of Am may also be significant. The degree of preservation of the alignment and orientation of the m-state populations during collisions depends on the strength of the collisions. This is directly related to the size of the impact parameter. As our results indicate, small impact parameters cause a greater degree of realignment and reorientation of the angular 153 momentum vector. This means that the torques exerted on the J vector are greater and this causes a greater tilt of the vector with respect to a space—fixed Z axis and also with respect to the molecule—fixed z axis. We have shown that all collisions that change the quantum number k are sufficiently strong to randomize the distribution of the m-state populations. These results show that in systems in which at least one collision partner is polar, dipole type selection rules predominate during collision-induced rotational energy transfer and that the rates of transfer of the alignment and orientation tensors are comparable to the rate of transfer of the population tensor. Apparent violations of the An = 0 collisional selection rule can occur due to various thermal processes such as the V-V swapping mechanism, wall collisions, and exit of molecules from the beam. The V-V processes tend to take away a pumped molecule from the upper level of the pump transition or lower level of the probe transition, and replace that molecule with a molecule from v = 0. Since 0(J, n) at 0 only for n = 0 in v = 0, an apparent n = 1(or 2)—~ n = 0 event occurs. What happens in this case is that the probed molecule is not the molecule that was originally pumped. Wall collisions and beam transit effects lead to a thermalized population for which 6(J, n) = 0 for n > 0. These thermal processes provide a time scale for all events that occur during collision-induced rotational energy transfer. Only processes that occur on a time scale that is at least comparable to the V-V rate lead to results that can be observed as part of the double resonance lineshape in our experiment. Thus, only pumped molecules that reach the lower level of the probe transition through direct rotational energy transfer that takes place faster than the thermal processes go into the transferred spike of the double resonance lineshape. Therefore, the molecules that make up the sharp transferred spike for lineshapes obtained either under population or polarization 154 modulation conditions are molecules that underwent fast direct rotational energy transfer. . The three-level double resonance lineshapes for the non-polar molecule cyclopropane are described by broad Gaussians. This observation is contrary to the predictions of our current three-level double resonance theory which predicts that the three-level double resonance signal should be a sharp spike. A possible reason for this observation is that the relaxation mechanism in cyc10propane is dominated by fast V-V transfer which replaces molecules in the velocity-selected pumped upper state with molecules from the ground state. If this is the case, a broad three-level double resonance lineshape will be observed. . The four-level double resonance studies on fluorofonn-d (CF3D) show that collision- induced rotational energy nansfer in this molecule occurs with dipole-like collisional selection rules, and that sharp transferred spikes are observed only for transitions that occur with Ak = 0. APPENDIX Appendix Jones matrix for an optically-pumped sample The purpose of this appendix is to outline the derivation of a Jones matrix for an optically-pumped sample in a four-level double-resonance experiment. This matrix converts the Jones vector for an incoming probe beam to that for an outgoing probe beam. It, therefore, describes the absorption or emission by the sample at the frequency of the probe beam. The derivation will follow the procedure used by Shin and Schwendeman (87) in which the Jones matrix for three-level double resonance was obtained. In the previous work, the matrix had to be left in a somewhat unsatisfactory state, because there does not as yet exist a full treatment of three-level double resonance including all of the effects of spatial degeneracy. For the case of four-level double resonance treated here, the situation is quite different, at least for plane-polarized pumping beams, because for this case there exists a full treatment for saturation of a single spectroscopic transition that can be used for the pumping radiation if sufficient information is available concerning the collisional relaxation rates. If the collisional data are not available, the theory can be used to determine the validity of approximations. The probe beam in many four-level double resonance experiments is assumed to be non-saturating and, as will be seen below, for this situation a completely satisfactory Jones matrix can be derived and used to design experiments that provide useful information about collisonal interactions for different tensor orders of the populations. For our purposes, the electric fields for the probe beam entering and leaving the sample, Em and Bout, are the real parts of vectors E and Ed, respectively, which may be defined in terms of unnorrnalized Jones vectors, as follows: E . E = (E :i§ )e'm‘ (A1) 155 156 (d) E . and E“) =[ fd)]e’w‘ (A2) where the quantities in brackets are the Jones vectors. In Eq.(A1), (Dis the frequency of the radiation, BF and Epeig are the components of E in the directions of the unit vectors e and e', respectively, which are perpendicular to each other and to the direction of the probe beam. The choice of axes and values of the real quantities EF, Ep, and E, for plane or circular polarization are given in Table A]. In Eq.(A2), Eff!) and Eff!) are components of E“) in the directions of e and e', respectively. The Jones vectors for the two fields are related by a Jones matrix M, as follows: Efed) : MFF MFF’) Er J (A3) Egg) MFF M F’F’ Eye'g ' The incoming beam induces a macroscopic polarization that can be represented as the real part of p . P=£ FJe’m’ . (A4) PF, The equations of McGurk et al. (115) can be manipulated to show that the induced polarization is accompanied by an electromagnetic field that is the real part of (p) , Em) =[EF )elmt (A5) 157 Table A1. Values for Em, Em) and E for different polarizations of the beam (Reference 87) E03) Em) é Polarization“,b £0 0 0 ZY plane 0 E0 0 XY plane E0 / J2 E0 / J2 1t / 2 right circular 130 / J2 130 / .5 -1t / 2 left circular 3130:), Em), E are defined in Equation (Al). bDirection of propagation of the beam is assumed to be Y for plane polarization and Z for circular polarization. 158 where E(Gp) : _inPG (A6) in which G = F or F’ and n = 2ndec where c is the speed of light and dL is the optically- thin path length through the sample. The field of the incident radiation and the induced field add to form the field at the exit of the sample cell, as follows: E“) = E + 18‘!” = E - mp . (A7) The induced polarization is the trace of the product of the density matrix p and transition moment matrix [.1 multiplied by N, the number of molecules per unit volume. For a two- level system with spatially-degenerate m states, we find RC(P) z szlpmbmcumcmb + pmcmbUmme] - (A8) mb Inc The subscripts mb and me stand for all of the quantum numbers for a given m state in levels b and c, respectively, while the sums are run only over the m values for each level (elements of the dipole moment matrix connecting m states in the same level, if present, are ignored, because they do not lead to terms with the right time dependence). If, in anticipation of application of the rotating-wave approximation, we write pmbmc = dmbmceiwt 9 (A9) then PG = 2N22dmbmcu§$nb (A10) m m where G = F or F'. 159 To obtain an expression for dmbmc we use the usual density matrix equation augmented by relaxation terms: Q dt 1' z'ilHipl-fip-p") (All) where y is the relaxation superrnatrix, p0 is the thermal equilibrium value of p, and H = Hm) + H“) with Hm) the Hamiltonian in the absence of the radiation and Hm = — rt'Em. The square brackets in Eq. (A11) indicate a commutator. Manipulation of Eq. (A11), including application of the rotating wave approximation, leads in the steady state to the following: (1,,me = -xmmeAmbmc (ch + ich) / 2 (A12) where xmbmc = (ufiiincEp + lag-(:1 EF'eié) / h , (A13) Ambmc = pram, —pm,m, . (A14) ch= 5:1” 2 , (A15) and LC), = 72 . (A16) 555% These equations were derived by assuming no coherence transfer among the m states and a common relaxation rate 72 for all of the off-diagonal density matrix elements connecting a state in level b to a state in level c. The quantity 8C), is the difference between the (angular) frequency of the radiation and the (angular) resonance frequency of the 160 transition, including the appropriate Doppler shift. For the work considered here, the m states for a given level are degenerate, so all of the 5's are the same. After substitution from Eq. (A12) and (A13) into Eq. (A10), we find N . - , , PF = —-;(ch +1ch)[EF(S,()F F ’ —S§F F ))+ e'iEFngFF ) 4:.” U] (A17) and N . I I ' I I I I PF, = -7“ch +rch)[EF(s,§”) -s§F F’)+e'5EF,(s,§F F ) —5§F F 0]. (A18) In these equations, (GG') _ E: 2 (G) (G) S!) _ l‘lm Hmbumbm pmbmb (A19) and GG’ G G’) a >- 2211:. has). p... (A20) where GG' can be FF, F'F', FF', or PE Substitution from Eq.(Al7) and (A18) into Eq.(A7) produces an expression that looks like the expanded form of Eq.(A3) in which the elements of the M matrix have the form M GG’ = 500' ‘ 1960' (A21) in which 566' is the Kronecker delta and 1266, = nN(ch — ibcb)(s,§GG') — 520(7)) / rt . (A22) 161 In these equations, G and G' are any combination of F and F'. The sums in Eq. (A19) and (A20) can be evaluated in closed form by using spherical tensors. We follow Edmonds (90) and define 11(0)=11(Z); u‘i1)=Tiu(X)iiu(Y))/~5 _ (A23) Then, _ J J 1 __ 4-1) a . . )1... C mr'an where the quantity in brackets is a Wigner 3-j symbol and Hbc is a reduced matrix element that depends on all of the quantum numbers for the states b and c except mb and me; Jb and Jc are the rotational angular momentum quantum numbers for the states b and c, respectively. Similarly, we can define statistical tensor combinations 6(J, n) of the diagonal (population) density matrix elements, as follows: J_ J J n pm,m zinx-l) m(2n+1)1/2[m-m0)6(1’n) . (A25) As discussed in the text, the statistical tensor combinations of the populations have the property that collisions in an isotropic environment relax the CU, n) for a given It independently. Inversion of Eq. (A23) and substitution into Eq. (A19) and (A20) leads to (XX)- (+1) (-1) . (XY)_- (+1) (—1) Sb _(sb +51) )/2 , 5b —l(Sb —sb )/2, 162 08).. . (1) (—1) . (Yr)_ (+1) (-1) Sb _—r(sb+ —Sb )/2 , 5b _(5b +Sb )/2, (A26) and (ZZ)_ (0) Sb "Sb . The Sign”) with FF' = X2, Y2, ZX, or ZY all vanish. Also, a similar set of equations holds for the Sin”) in terms of the S éq) . In all of these equations, 2 S134) = 22(“%)mc) Pmbmb "lb m and 2 Séq)=22(11$ni)mc) pmcmc . (A27) m), m To evaluate the sums in Eq. (A27) we substitute from Eq. (A24) and (A25) and then use a familiar equation for sums of products of three Wigner 3-j coefficients given, e. g., by Edmonds. The results are sgq)=2s,§4)(n) and s§4>=zsgq>o> (A28) n n inwhich 1 1n 1 1 n (q) = -1 Jb-JC—q '2 2 4.11/2 o' J A29 Sb ('1) ( ) libc( n ) _qq0 JbeJc (bat) ( ) and 1 1n 1 1 n .(q) = _1 Jb_Jc_q -2 2 +1 1’2 a J A30 56 (r1) ( ) Hbc( n ) _qq0 JchJb ( an) .( ) As a result of the properties of the 3-j symbols in Eq. (A29) and (A30), the sums in Eq. (A28) are limited to n = 0, 1, and 2. 163 To proceed further it is necessary to consider separately the situations in which the pumping radiation is plane polarized and circularly polarized. For plane -polarized pumping radiation it is most convenient to choose F = Z and F’ = X. In this case the Jones matrix, which we call M31, is diagonal, as follows: P2 0 Ma,=[0 px) . (A31) Combination of Eq. (A21), (A22), and (A26) leads to P2 =1-11N(ch -iDcr)(S§,°) -S§°’)/h (A32) and [)X =1— 118/(ch - ich)(s,§” + 5);” — s51)- Sg‘1))/2ri . (A33) Upon consideration of the properties of the 3-j symbols in Eq. (A29) and (A30), we find that 5,30) = 3,30) (0) + s,§0)(2) (A34) and that l — 1 5(Sé1)+SI() 1)) = if?) (0) --2—sgo)(2) . (A35) Similar equations hold for 5:0) and as?) + 55‘”). Consequently, if 001,, 2) and one, 2) are both zero (as they would be at thermal equilibrium, e. g.), then pz = px. We will see shortly thath and px are related to the absorption coefficients for vertically and horizontally plane-polarized radiation, so it is not surprising that these two quantities 164 should be the same in the absence of optical pumping. In the presence of optical pumping, the n = 2 statistical tensor elements are not zero, and we find that the difference in the M matrix elements is 122 - px °c $020) (2) - 3(0) (2)) (A36) so that this difference is directly proportional to the n = 2 (alignment) combination of the diagonal density matrix elements. The form of this difference is dictated by the fact that the Z axis has been chosen to be the axis of defrnition of the quantum number m. For a probe beam of amplitude E0 polarized in the Z direction, the amplitude of the beam leaving the sample is PZEO- This is the situation for an optically-thin sample (1] contains the path length dL). According to the theory of Jones matrices, if we stack 11 samples of thickness dL, the Jones matrix for the stack is M g, , which is a diagonal matrix with diagonal elements p% and p3}. If we take the limit as n —) oo while dL —) 0 with ndL = L = constant, we find pZ=e—azL/2 and px=e’°‘XL/2 (A37) where 41:00N . oz: he (L6,,—tDC,,)(s§,°)(0)—s§°)(0)+s§,°)(2)—s§0)(2)) (A38) and 41ttuN . 1 1 x= he (ch—tD,b)(s§,°)(0)—s§°)(0)—5sg°)(2)+§rg°)(2)) .(A39) The quantities OLZ and ax are the complex absorption coefficients for ZY and XY plane- polarized radiation, respectively, for a sample that is optically-pumped by plane-polarized radiation. 165 For a sample pumped by circularly-polarized radiation, it is most convenient to choose F = X and F' = Y. Then, combination of Eq. (A21), (A22), and (A26) leads to Jones matrix that we call Mor that is of the form + -i — Mm =l[. pr Pl (pr 171)) (A40) 2 1(Pr *Pz) Pr +Pl where pr =1— nN(ch _ ich)(Sl(,+D -S£+l)) / h P1 =1-11N(ch - ich)(Sr(,—1) — S§‘1))/rt . (A41) Consideration of the properties of the 3-j coefficients in Eq. (A28)-(A30) shows that 5,?“ = sg‘)(0):s,§1)(l)+sg1)(2) . (A42) As with the Mal matrix, Mor in Eq. (A40) is for an optically-thin sample since 11 contains the optically-thin path length dL. We again divide the sample into n successive samples each with path length dL. The effect of such a sample on radiation passing through is represented by a product of the Jones matrices, or by M3, , where, surprisingly, it n _- n_ n Mg, :1 Prn+Pln “I”; {:1 ) . (A43) 2 i(Pr ’Pl) Pr +Pl If we let it —> 00 while dL -> 0 with ndL = L = constant, we find that that for finite path length, Mor still has the form of Eq. (A40), but that now Pr = e-or,L/2 and P1 = e—a,L/2 (A44) 166 where 41th . ot,= he (ch-tch)(As(0)(0)+ A5'(1)(1)+As(1)(2)) and 41:0)N . ot,= he (ch—tch)(As(0)(0)-As(1)(1)+As(1)(2)). (A45) In Eq. (A45), As(q)(n) = 3,390) — s§4)(n). It is apparent that or and 0t], which are the complex absorption coefficients for right and left circularly polarized probe radiation, respectively, differ only in the sign of the n = l (orientation) contribution. Also, as expected for an isotropic sample, when the n = 1 and n = 2 components of the statistical tensor combinations of the populations vanish, all of the absorption coefficients are the same. To fully appreciate the results of this analysis, it is necessary to perform matrix multiplications similar to those of Eq. (A3) with M = Mal and Mor and with the incoming probe E of the forms shown in Table A1. The two M matrices found in this work for four- level double resonance have the same form as the corresponding matrices for three-level double resonance presented in Reference 87. Therefore, the results of such matrix multiplications shown in Table III of that work are valid here as well. As pointed out in the Introduction, optical pumping with plane-polarized radiation induces non-zero statistical tensor combinations of the populations of all even orders. In the isotropic sample, each statistical tensor combination of the population relaxes independently. 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