# THE THEORY OF PULSED FOURIER TRANSFORM MICROWAVE SPECTROSCOPY CARRIED OUT IN A FABRY-PEROT CAVITY

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University Microfilms International 300 N ZEEBRD ANN ARBOR, Ml 48106 8203415 CAMPBELL, EDWARD JOSEPH THE THEORY OF PULSED FOURIER TRANSFORM MICROWAVE SPECTROSCOPY CARRIED OUT IN A FABRY-PEROT CAVITY University of Illinois at Urbana-Champaign University Microfilms I n t G r n a t i O n a l 300N ZeebRoad,AnnArbor,MI48106 PH.D. 1981 THE THEORY OF PULSED FOURIER TRANSFORM MICROWAVE SPECTROSCOPY CARRIED OUT IN A FABRY-PEROT CAVITY BY EDWARD JOSEPH CAMPBELL B.A., University of Wisconsin-Madison, 1976 M.S., University of Illinois, 1977 THESIS Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics in the Graduate College of the University of Illinois at Urbana-Champaign, 1981 Urbana, Illinois •zrrr-i UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN THE GRADUATE COLLEGE June 2 2 , I98I WE HEREBY RECOMMEND THAT THE THESIS BY EDWARD JOSEPH CAMPBELL ENTITLED THL TH£0RY 0F PULSED FOURIER TRANSFORM MIGROWAVK SPECTROSCOPY CARRILD OUT IN A FABRY-PEROT CANITY BE ACCEPTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR DOCTOR OF PHILOSOPHY THE DEGREE OF. UiJ&wt/ 4iMi3iL Director of Thesis Research ttt<^Afc(jB Head of Department Committee on Final Examination! Chairman J 1 "Cfc %&AfA lUJZii^ f5 iJjo0^-^ t Required for doctor's degree but not for master's Ill The Theory of Pulsed Fourier Transform Microwave Spectroscopy Carried Out in a Fabry-Perot Cavity Edward Joseph Campbell, Ph.D. Department of Physics University of Illinois at Urbana-Champaign, 1981 A semiclassical theory has been developed to describe pulsed Fourier transform microwave spectroscopy carried out in a Fabry-Perot cavity. A density matrix formalism is used to study the interaction of a two-level quantum system with a classical standing wave electric field, appropriate for the Fabry-Perot cavity. Equations describing the polarization of, and subsequent emission of radiation by arbitrary distributions of molecules in the cavity are derived. The specific problem of a static Maxwell-Boltzmann gas is studied in detail, both theoretically and experimentally. The static gas lineshape in the power-broadened limit is described by an ordinary Doppler and pressure broadened envelope. Sensitivities of the ordinary waveguide cell and Fabry-Perot cavity pulsed Fourier transform spectrometers using static gas samples are compared. The gas dynamics of a pulsed supersonic nozzle molecular source are investigated by using the pulsed Fabry-Perot cavity microwave spectrometer to obtain free induction decay signals from rotational two-level systems in the gas expansion. An equation is derived giving the time domain emission signal IV lineshape as an integral over the active molecular distribution in the beam. detail. The Doppler splitting phenomenon is discussed in Experimental lineshapes are deconvoluted to give molecular velocities, dephasing times, and density distributions. We find that the density distribution of active mole- cules from the pulsed nozzle varies rapidly in time, starting with a depletion on the nozzle axis at short times after the nozzle is opened, and changing to on-axis concentration at longer times. Results obtained with the gas nozzle axis oriented at angles ranging from 0° to 90° with respect to the direction of propagation of the microwaves are reported. Rotational assignments are reported for 83KrH35CI, 83 KrD 35 Cl, i31 XeH 3 5 Cl, and 131 XeD 3 5 Cl. The measured rare gas nuclear quadrupole coupling constants, X , are: XR(MHz) 83 KrH 35 Cl 5.20(10) 83 KrD 35 Cl 7.19(10) 131 XeH 3 5 Cl -4.64(5) 131 XeD 3 5 Cl -5.89(20) The electric field gradient along the a-inertial axis at the rare gas nuclear site in these four molecules, and in 83 KrH(D)F, 83 KrHC 14 N, and 131 XeH(D)F is found to be directly proportional to the electric field gradient at that site calculated from the electric multipole moments of the partner hydrogen halide. The proportionality constants are, for V 83 Kr, 78.5 and for -24 and -0.12 x 10 go 131 Xe, 160, using values of 0.27 x lo" 24 cm 2 2 cm for t h e nuclear quadrupole moments of 1^1 Kr and Xe r e s p e c t i v e l y . This d i r e c t p r o p o r t i o n a l i t y i s a t t r i b u t e d t o Sternheimer-type quadrupolar s h i e l d i n g of the r a r e gas nucleus by t h e e l e c t r o n s i n the r a r e gas atom. Within t h e l i m i t s of u n c e r t a i n t y of t h i s experiment of £ 0.003 e l e c t r o n we find no evidence f o r charge t r a n s f e r from the Kr and Xe atoms. 83 The Kr n u c l e a r quadrupole coupling c o n s t a n t in 83 ^5 Kr C1F has been measured t o be 13.90 (25)MHz. Using the known 83Kr nuclear quadrupole moment, t h e f i e l d g r a d i e n t a t t h e Kr n u c l e a r s i t e i s evaluated and i n t e r p r e t e d i n terms of t h e quadrupolar s h i e l d i n g constant of Kr, the s t r u c t u r e of KrCIF, and the e l e c t r i c multipole moments of C1F. VI ACKNOWLEDGEMENT I wish to acknowledge my indebtedness to Prof. Willis H. Flygare, whose guidance and encouragement made this work possible. I would like to thank Dr. Bill Hoke, who taught me Fourier transform microwave spectroscopy and introduced me to van der Waals molecules, Dr. Terrill Balle, for the opportunity to use the Fabry-Perot spectrometer, and for discussions of many of the ideas developed here, and Dr. L. William Buxton, with whom I cooperated on these, and many other projects. The experimental measurements reported in Chapters II and III of this thesis were done in collaboration with Dr. Balle and Dr. Buxton. The microwave spectral measurements reported in Chapter IV and much of the computer programming work of Chapters II and III were done in collaboration with Dr. Buxton. I would like to thank Prof. A. C. Legon for his partici- pation in the Ar, Kr, and XeHCN work, and for many interesting discussions of the properties of weakly bound systems. I would also like to thank Dr. Michael Keenan, who participated in the XeD Cl measurements, Dr. Paul Soper, Peter Aldrich, Bill Read, Jim Shea, and Dan Wozniak for much stimulating and enjoyable interaction. The support of the Department of Chemistry through the staffs of the electronic shop, machine shop, computer facility, and student shop, and the Physical Chemistry Secretaries are gratefully acknowledged. I would also like to acknowledge vii the financial assistance of the University of Illinois. This work is based on work supported by the National Science Foundation under Grant 46-32-87-314, title NSF CHE 78-13-7719610, and by The Petroleum Research Fund, administered by the American Chemical Society. viii TABLE OF CONTENTS Chapter Page I. General Introduction II. The Theory of Pulsed Fourier Transform Microwave Spectroscopy Carried Out in a FabryPerot Cavity 10 A. B. C. D. FJ. F. G. 1 Derivation of Block-type Equations for a Standing Wave Electric Field Appropriate for the Fabry-Perot Cavity . . . 10 Solutions to the Block-type Equations for the Polarization for Near-Resonant Radiation Stimulation in the Fabry-Perot Cavity 17 Evolution of the Molecular Polarization Following the Removal of the Polarization Inducing Radiation 28 Electric Field Produced in the FabryPerot Cavity by the Molecules After the Polarization of the Gas 29 Electric Field in the Fabry-Perot Cavity Following Near-Resonant Polarization of Maxwell Boltzmann Static Gas 35 Field Coupling Out of the Fabry-Perot Cavity 45 S t a t i c Gas Experimental R e s u l t s 49 H. III. Comparison of S t a t i c Gas Fabry-Perot R e s u l t s t o Waveguide Spectrometer R e s u l t s . The Gas Dynamics of a Pulsed Supersonic Molecular Source as Observed with t h e Fabry-Perot Cavity Microwave Spectrometer . . . A. Introduction 60 65 65 IX Chapter B. IV. Page Equations Describing the P o l a r i z a t i o n and Emission of Radiation by Molecules in t h e Fabry-Perot Cavity Appropriate for t h e Pulsed Nozzle Problem 66 C. Lineshapes in t h e Pulsed-Nozzle FabryPerot Experiment 73 D. Lineshapes Taken with A l t e r n a t e Nozzle Geometries 109 Rare Gas Nuclear Quadrupole Coupling in van der Waals Molecules , 126 A. Introduction 126 B. Experimental 128 C. Rotational Spectra and Spectroscopic Constants for the Rare-Gas Hydrogen Halides 130 D. Analysis 146 83 E. Measurement and Analysis of the Kr Nuclear Quadrupolar Coupling in 83 Kr 35 ClF 165 185 Appendix A Appendix B 187 Bibliography 194 Vita 199 Chapter 1. General Introduction The study of the pure rotational spectra of small molecules in the gas phase has since the mid 194O's yielded a tremendous amount of highly detailed information about both the structural and electronic properties of these systems. This information is widely used in the study of molecular structure and chemical bonding. It has also been recognized that the shape, or envelope of a microwave resonance signal contains much information about the interactions between individual molecules in the gas phase. Lineshape studies have dealt exclusively with a source gas in thermodynamic equilibrium with its container. The fundamental aspects of this problem are now well understood, although with the introduction of sophisticated state-selective techniques and laser 2 technology, the field remains active. In recent years there have been two significant developments of a general nature that have profoundly affected both of the problems mentioned above. The first of these was the recognition that time domain spectroscopic methods offered a significant improvement in signal-to-noise over conventional steady state methods.3 The first time domain experiments in the microwave region were performed by Dicke and Romer in 1955.4 '5 The pulsed time domain microwave spectrometer as a standard laboratory instrument was developed by Flygare and co-workers, beginning in 1975. Starting from a unified theory describing coherent transient affects in the microwave 1 and infra-red regions of the spectrum, these workers investi- gated a wide variety of coherent transient effects, including transient absorption and emission, fast passage, Stark switching, and multiple pulse experiments. A pulsed Fourier Transform microwave spectrometer incorporating recent developments in microwave hardware was built and was used to show that this technique did indeed offer improvements over the steady state method. The spectrometer was used for detailed studies of 13—15 7 14 the relaxation parameters T, and T 2 in NH 3 and OCS, and it was shown that the time domain method had advantages in state-selectivity, and was superior to steady state methods for studying T-, . Advantages in sensitivity and resolution of time domain work was also illustrated by a study of the 16 molecular Zeeman effect in trans-crotonaldehyde. A second important spectroscopic development has been the recognition of the utility of using supersonic expansion of molecules in an inert, monoatomic carrier gas as a spectroscopic sample.17 These jets provide a cold (1-10 K), nearly collision free sample, resulting, in the microwave region, in the elimination of conventional Dopper broadening, of collisional broadening, and a large increase in signal resulting from the depopulation of the higher rotational levels. In addition, large numbers of weakly bound van der Waals and hydrogen bonded molecules can be formed in the jet. Conventional waveguide methods are unsuitable for studying most of these systems, except for the more strongly bound hydrogenbonded complexes,18 because of the condensation problem. 3 The first observation of microwave transitions in a van der Waals complex was that of Novick, Davies, Harris, and Klemperer,19 who measured several R-branch transitions in ArHCl in 1973. They used a Rabi-type molecular beam electric resonance (MBER) spectrometer, in which a rotational resonance is detected by observing a change in the number of particles appearing at the input to a mass spectrometer after the particles have passed through state selective electrostatic fields. The gas sample here is a highly collimated, essentially onedimensional, steady state beam that passes at a right angle to the radiation field. Background pressures ranging from -5 -9 10 to 10 torr insure that collisional effects are nonexistent. The first instrument to permit direct observation of microwave transitions in a supersonic nozzle expansion was the pulsed Fourier transform Fabry-Perot spectrometer built by Balle and Flygare in 1979.20 This device incorporates the same microwave switching and signal handling arrangements of the earlier Ekkers-Flygare waveguide instrument, but replaces the waveguide and the static Maxwell-Boltzmann gas sample with a pulsed supersonic gas expansion in a large volume FabryPerot cavity. The inherent advantages of time-domain operation, and the very important advantage of using a supersonic jet expansion are thus realized in the same instrument.21-23 A block diagram of the Balle-Flygare instrument is shown in Fig. 1. We begin by locking the master microwave 4 oscillator (MO) to a harmonic of a frequency standard. The local oscillator (LO) is locked 30 MHz from the MO and is used in the subsequent superheterodyne signal detection. The Fabry-Perot cavity consists of two dish-shaped aluminum mirrors, 14 inches in diameter, with spherical mirror surfaces. The mirrors are mounted so that their separation, and thus the resonance frequency of the cavity, can be adjusted to the molecular resonance frequency. The mirrors have small coupling holes in their exact center to allow microwave radiation to be coupled in or out of the cavity. The cavity chamber is pumped di- rectly by a 10 inch oil diffusion pump to maintain a pressure of 10-5 torr between gas pulses, so that the mean free molecular path in the cavity is determined by the dimensions of the vacuum chamber. The gas nozzle is positioned midway between the mirrors, just outside their radii, and points directly at the pump. We use a General Valve type 8-14-900 pulsed solenoid valve, with open and close times of 2-10 msec. valve The orifice is simply a flat plate with a 0.020 inch diameter hole, that is mounted in the bottom of the valve. At some arbitrary time a 50 volt pulse is used to open the valve. The valve is backed by 1-2 atmospheres of a rare gas seeded with a small percentage (0.5-3%) of another gas, HC1 for example, at room temperature. When the valve opens this mixture streams into the evacuated chamber, undergoing an adiabatic cooling process that lowers its rotational and translational temperatures to between 1 and 10 K. The gas molecules travel at approximately 5 • 10 cm/sec and take Ul Figure 1. Block diagram of the pulsed Fourier Transform microwave spectrometer with a pulsed supersonic nozzle gas expansion source. LO Frequency Stabilizer v - 3 0 MHz <gfixer 30 MHz Pin Diode £0- MO Frequency] Standard Frequency Stabilizer ® Mixer Monitor Detector D Pulsed Nozzle *^f ,e Pin Tuner Circulator U. I Diode I f DDetector e 0HfF^ [Diffusion Pump Mi Mixer A±30MHz Mixer 3 > A |Computer| - « - • jAveroger| A/D Os Display Display 7 approximately 1 msec to reach the pump. At some time during this transit interval, PIN diode switch 1 is closed for 3-6 msec, and the gas is irradiated with a IT/2 pulse from the MO. A tiny fraction of this incident microwave energy stored in a nonequilibrium population difference of the two rotational levels being probed. Equivalently, the large amplitude driving electric field in the cavity produces a coherent, macroscopic polarization of the gas. The physics of this polarization process, and of the subsequent emission process, which will be developed in full detail shortly, is formally equivalent to the process of pulsed NMR, where now the in- and out-of-phase components of polarization and the two level population differences replace the more familiar transverse and longitudinal components of magnetization. After the gas has been polarized it will retain this coherent state for a time that is long compared to the relaxation time of the cavity. After the power pulse ringing has died away,PIN diode switch 2, which has been, protecting the sensitive microwave detector, is closed. According to Maxwell's equations the polarized gas now sets up a standing wave electric field in the cavity at the exact rotational resonance frequency. This energy is coupled out of the cavity through switch 2 and is detected in the superheterodyne receiver. The signal is mixed first with the LO signal to produce a nominal 30 MHz carrier frequency, and then mixed down to near-DC with the 30 MHz intermediate frequency signal. The resulting signal at 10-500 KHz is digitized, averaged, and Fourier transformed to give the rotational spectrum. 8 The 1 MHz bandwidth of the system is determined by the quality factor, Q, of the cavity, and is sufficiently large that no distortion of the microwave signals occurs. Because we are now observing directly, for the first time, microwave transactions in a supersonic nozzle expansion, the microwave lineshapes seen in this instrument are quite unlike those studied earlier. As in the static gas, we need to relate the properties of the experimentally observed lineshapes back to the properties of the gas expansion from whence they originate. This problem is part of the more general problem of understanding the process of pulsed Fourier transform microwave spectroscopy carried out in a Fabry-Perot cavity. Aside from the unusual gas dynamics, we are dealing here with standing waves, not the traveling waves of the waveguide spectrometer, and the theory must also take into account the high-Q, narrow banded characteristics of the Fabry-Perot cavity. In Chapter II of this thesis I will develop the equations appropriate for a semiclassical description of time domain spectroscopy in the Fabry-Perot cavity.' This initial development will be quite general, and not fixed to any particular distribution of gas molecules in the cavity. Because this theory does yield specific predictions for the use of a Maxwell-Boltzman distribution of molecules in the cavity, this static gas problem will be investigated, both theoretically and experimentally. This will allow me to draw out some of the features of the high-Q, standing wave solutions in this somewhat more familiar situation, and also to develop 9 a contrast between the cavity results and the waveguide experiment. In Chapter III I will apply the cavity equations to the problem of a pulsed supersonic gas expansion. This work will establish the basic relationships between the observed 25 lineshapes and the properties of the gas expansion. The intent here is first, to understand the properties of the gas expansion, and second, to provide a systematic characterization of the spectrometer operation in order to allow us to make optimum use of it in studying chemical problems. In Chapter IV I will present measurements of the rare gas quadruple coupling constants in four rare gas-hydrogen halide systems. These results are combined with similar measurements for five similar systems to obtain a systematic explanation for the magnitude of these coupling constants in terms of the geometry of the van der Waals molecules, the electric multipole moments of the hydrogen halides, the quadrupolar shielding properties of the rare gas subunits, and the nature of the van der Waals bond. 10 Chapter I I . A. The Theory of Pulsed Fourier Transform Microwave Spectroscopy Carried Out in a Fabry-Perot Cavity. Derivation of Bloch-type equations for a standing wave electric field appropriate for the Fabry-Perot cavity Consider an ensemble of non-degenerate two-level quantum systems interacting with an electric field E(r,t) through the electric dipole interaction. In the absence of collisions, the quantum-mechanical Hamiltonian for the j ' t h molecule i s H(r.,t) = H - (-n2/2m)V2 - u • E(r.,t) (1) where r . i s t h e c e n t e r - o f - m a s s o f t h e j ' t h m o l e c u l e , H„ i s t h e t i m e - i n d e p e n d e n t f r e e m o l e c u l a r H a m i l t o n i a n whose e i g e n f u n c t i o n s i n c l u d e e n e r g y l e v e l s E a n d E, c o r r e s p o n d i n g t o t h e upper a n d lower s t a t e s r e s p e c t i v e l y , V i s t h e g r a d i e n t w i t h r e s p e c t t o r . , m i s t h e m o l e c u l a r mass, a n d y i s t h e d i p o l e moment o p e r a t o r . Since t h e e x t e r n a l f i e l d i n t e r a c t i o n does not a f f e c t t h e molecular center-of-mass motion, t h e density matrix a"1 ( r . , p . , t ) i n c l a s s i c a l phase s p a c e , where r . a n d p . a r e t h e ~j ~J ~J ~j c l a s s i c a l v a r i a b l e s f o r t h e p o s i t i o n a n d momentum of t h e j ' t h m o l e c u l e , s a t i s f i e s 26 ^ ^ " V ^ a a ' ^ j ' P j ' ^ " t V r ^ j ' ^ ^ j ' P j ' ^ a a ' ») where t h e b r a c k e t s on t h e r i g h t h a n d - s i d e i n d i c a t e t h e commutator. Provided t h a t each molecule i n t e r a c t s with t h e f i e l d , independently we may work w i t h a m a c r o s c o p i c d e n s i t y m a t r i x 26 having elements defined by ,(r,p,t) = Z/d3r.d3p aD a .(r.,p.,t)6(r-r.)5(p-p.). ( Using v = p/m, Eq. (2) becomes m t + v y ) a m l ( r , v , t ) = [H -y-E (r,t) ,0 (r,v,t) ] ,, which we take as the starting point for this work. Before taking up the standing wave solutions to this equation it will be useful to review the traveling wave reg suits. The development here will differ only slightly from that in the McGurk, Schmalz, and Flygare paper. We take E(r,t) = 2z ecos(cot-ky) , a plane polarized traveling wave, where z is a unit vector perpendicular to the direction of propagation, and where £ may in general have the form e(r,t). Using this, Eq. (4) becomes iii( !t + r ! ) a a a > " ^ ^ b a ^ b ^ a b ^ b a 5 coa( U t-ky) ±R(jL. + Y - ^ a b ^ a b ' W (5) + 2lJ itt( lt + Y*! )a bb = 2£ ab e ^ a a - c W cos(ut-ky) ^ab a ba"^ba°ab ) coB< M t-ky). Following McGurk, et al., we transform to the density matrix in the interaction representation defined by p = exp(^(t-J)aexp(-^- (t-£)), (6) ( 12 where E. (7) S » 0 E& +hu Substituting p for a in Eq. (5) and making the rotating wave approximation, 6 one obtains 9t + Y-!>Paa = e i,h{ 3t + Y'! } Pab = -*AwPab ^ a b ^ a a - P b b * + Y'^Pbb = ^Wba-lWW (10) - w(l-v/c), (11) ^It ^baPab^abPba5 (8) i<ft( (9) where Aw = u and VBa w o'T - ' the two-level resonance frequency. relationship k = u/c. in McGurk et al. dependence. vV We have assumed a dispersion To make the connection with Eqns. (17-19) we specify e(r,t) = e (t), with no spatial With no r dependence in Eqns. (8-10) above, the terms may be handled trivially by specifying that p = p. , (v,t) . 0606 ** One is left with , (r,v,t) 13 3p a a ^ T T " ^baPab'Wba* 3p 411 x ab TF" - * T t "= ^ab^aa'Pbb^^Pab (12) e (y abPba"libaPab> * These may be compared to Eqns. (17-19) and (65) in McGurk et al. Consider now the standing wave case. The electric fields 27 in the cavity in the TEM modes used in this experiment will be polarized Gaussian standing waves of the form (See Fig. 2) E(r,t) = 2ze(r,t) cos wt . (13) We transform to the density matrix in the interaction representation defined by p = exp(^p t)a e x p ( ~ t ) , where S i s defined in Eq. (7). (14) Again making the rotating wave approximation one obtains ifl( lt + Y'^Paa = - e ( £ ' t } ^abPba^baPab 5 (ft + Y'^Pab = ^'^ab'Paa-Pbb* <ft + Y-^Pbb = - e ^ ' f c ) ^baPab^abPba) ift ift " *Aupab <15> where Aw = w0-w, (16) Figure 2. Geometry and coordinate system used for the Fabry-Perot cavity. The distance at which the electric field falls to 1/e of its on-axis value is w(y), with w(y = 0) = w , the minimum value. The radius of curvature of both mirrors is R = 83.8 cm. 15 16 with no v/c term, since the transformation a+p involves no spatial dependence. The effect of molecular motion on the standing wave solutions enters via the v'V terms. This point will be discussed later. Following closely the development in McGurk et al. for the traveling wave we rewrite Eq. (15) in terms of the polarization and population difference of the gas. The polari- zation is given by P = Tr(ya), (17) where Tr represents the trace of the matrix representation. Using Eq. (14) gives P = Tr(y exp(-|£ t)p exp(^- t)) . (18) Expanding this, one obtains * - ^baPab ^ + »abPba ^ = (Pr+iP.) e la)t + (Pr-iPi) e" i w t , <19> (20) with ( ViPi) = ^baPab by definition, and where P r and P. are real quantities. ing that N a = p <21) Not- , N b = P b h , the number densities of molecules in the a and b states respectively, from Eqns. (15), (19), and (20) one obtains the Bloch-type equations for the standing wave case: 17 P 3 ( ft + ( ft + )P +AwP Y'! r i + r T^ = °' (22) pt a )p 2 Y*! i " ^wP r +K e(r,t) (^M) + ^ i = 0, *AM Here AN = Na-Nj3, AN (23) * (AN-AN ) is the equilibrium value of AN in the absence of an electric field, and K = 2fi |<a|u |b>|. The first-order relaxation terms involving T, and T 2 have been introg duced phenomenologically in the usual way. B. Solutions to the Bloch-type equations for the polarization for near-resonant radiation stimulation in the Fabry-Perot cavity Our problem now is to solve Eqns. (22-24) for P.,Pr, and N using functional forms of e(r,t) appropriate for the pulsed Fourier transform experiment carried out in a FabryPerot cavity. work, ' The reader is referred to previously published detailing the pulsed Fourier transform technique. We are interested in applying these equations to two different kinds of problems. nozzle experiment. Consider first the pulsed In this case it is possible to define for each coordinate r a unique velocity v(r), since it is known that all molecules travel at constant speed along straight line paths originating at the nozzle opening. Eq. (22) by dv, and integrating, one has Multiplying 18 IT- /dv Pr(r,v,t) + / dv v V P(r,v,t) + Aw ; dv P.(r,v,t) + -L / dv P„(r,v,t) = 0. »V J. rv INS A »\ ^* i *M *« This is 9P (r,t) ——n + V • / dv v P r (r,v,t) + Aw P, (r,t) + ^- P r (r,t) = 0. l ~ r ,, i2 Because each coordinate (25) has a unique velocity associated with it, the second term is V • / dv v P„(r,v,t) = V • (v(r) P^ (r,t)) . Now define Pr(r,t) = N(r,t) pr(r,t) P. (r,t) = N(r,t) p. (r,t) (26) AN(r,t) = N(r,t) An(r,t) where N (r,t) = Z/dv p a - aa (r,v,t) , ~ -v the number density of two-level systems at coordinate r at time t. Eq. (25) becomes 9p r 3N N -rr + P„ |z- + P- V • (v(r)N) + v(r)N • Vp„ Np r + AwNp. + -==• - 0. (27) We will assume a conservation law for N of the standard form 19 9N( f l t ) + V • (v(r)N (r,t)) = 0. Dividing the remaining terms in Eq. (27) by N(r,t), and repeating this entire procedure for Eqns. (23) and (24), one obtains ( ( p 9 3t + ( } p Y 5 'V 3t + Y ( £ } *!) p i " a r + Awp Awp r i + T^ = ° ' r + <2 e( £' t) ^x L + r 1 ,((«« = 0/ (28) -n(An-An ) < J L + y ( r > . v ) * A n - E ( r , t ) p . + — T ^ - 2 - - 0, where An for a given molecular transition will depend only on the Boltmann factors, and will be assumed to be a constant. Now consider the static gas problem. a constant. Here N(r,t) = N, We may cast Eqns. (22-24) into a form similar to Eq. (28) by defining P„(r,v,t) = NW(v) p„(r,v,t) , •*• «S* *V Al X. (29) *s, <\« with analogous expressions for p. and An. The normalized Boltzmann distribution of velocities is given by \ w W{ 1v = l(_JL_)V2 Y' 2irkT; e -mv2/2kT ~ Using Eq. (29), Eqns. (22-24) are, for the static gas problem < £ + v -V) pr + Awp. + |j= 0 , ( jt * + Y 'V p i " ,»«„ Aa)p r + <2 e ^'t)1T1 + *r = ° ' «fi(An-An ) (30) 20 This is identical to Eq. (28), except v(r) is replaced by the phase-space vector v, and pr(r,v (r),t) is replaced by P r (r,v,t), with similar exchanges for p. and An. For the so- lutions to Eqns. (28) and (30) to be developed here it is useful to note that for any velocity field satisfying the criterion t v(r) • V v. = 0,i = x, y, z (31) solutions to Eqns. (28) and (30) may be exchanged by making a formal substitution p„(r,v,t) ->- pv.(r,v(r) ,t) , etc. Since the pulsed nozzle molecular velocity field satisfies Eq. (31), the terms v and v(r) in the solutions p , p., and An to be «- ~ •v r i developed below may be freely interchanged according to whether one is dealing with the static gas or with the pulsed nozzle. We first note the existence of formal solutions to Eqns. (28) and (30). One may combine the first two equations in (30), for example, as ( 3t + VV ^r"*"1?^ " 1 Aw(p r +ip i ) (32) 2 ttAn <?_+!?;,) r 1 + iK 2 e<r,t)^p + = 0. The formal solution to this is 28 (pr+ip.) (r,v,t) = / d t - ( - i 4 J W i A t t " l A 2 M t - t , > x 1 ~ ~ -co 4 e(r-v(t-t') ,t')An(r-v(t-t') ,v,t') , See Appendix A (33) 21 with a similar result (An(r,v,t) - AnJ = / dt' (i)e" (t_t ' ) / T l x (34) e(r-y(t-t'),t') pi(r-v(t-t'y, ,v,t') , for the population difference. Using Eqns. (33) and (34), the solutions for the polarization and population differences may be developed iteratively to any desired degree of accuracy, a process entailing considerable mathematical complexity. 29 It will be possible to adopt here a simpler approach still capable of explaining the phenomena occurring in the Fabry-Perot cavity experiment. Consider an experiment in which the molecules in the cavity are brought into contact with a radiation field 2e(r,t)cos wt for a duration T , where e(r,t) satisfies the cavity boundry conditions. We consider two sets of solutions to Eq. (30). In the case where KET <ff/2throughout the central part of the cavity, and where kvi << 1, where v is a characteristic molecular velocity component along the cavity axis, a typical molecule during the pulse time T travels only a short distance compared to the characteristic distance k" of changes in the intensity of the standing wave electric field. obtain One may then ignore the v • V terms in Eq. (30) to 22 3P r Pr -gt+AWPi + ^ ^ O , P i . „ . 2 . . . «nAn . P i _ n - g £ " • AooPr + K e ( r , t ) - j - + ^ - = 0, 3tT" e(r,t)Pi + — ^ ,, CN (35) =0. These equations are similar in form to Eq. (46) in McGurk et al. , except t h a t now p_, p . , and — X n are functions of r as X *v well as t, and Aw here contains no velocity dependence. When the two-level system is brought instantaneously into contact with a radiation field of constant amplitude, so that e(r,t) may be written e(r), one has p i(^ 2 K *ne(r)An t) = —w: , ,. -K2 ,fled:)An . P r (-,t) = —^ £ f r 4i ~ 2 " An(r,t) = An ° _t/T e 2 (cosftt - ftT^sinftt)-1 7TZT72 2 d/T2r + n e" ^ 2 (OT0cosftt + sinftt)-OT0 \ ± 2 (1/T 2 )' + or (ice (r)) 2 e"" t/ ^2 (cosJ2t+(OT2) _ 1 sin^t) + (1/T2) 2+Aw2 * 5 (1A2> + « where 2 ft = (assuming T,=T_) 2 2 , v 2 Ke ( r ) + Aw (36) 23 These expressions can be simplified under certain experimental conditions. The electric field e(£) inside the cavity appearing in these expressions can be related to the input power to the cavity by making use of bulk circuit concepts. The quality factor Q is defined for the Fabry-Perot cavity by Q _ 2TT (total energy stored) (energy dissipated per cycle) . Four Q's are defined for the cavity: ,3?* ' Q , Q - / Q c 5' a c c o u n t i n g for power dissipation in the cavity alone, in the input coupling, and in the output coupling, respectively, and Q_, which combines all the power dissipating elements, where 1/QL = 1/QQ + 1/Q cl + V Q c 2 . We will assume that Q = Q j = Q„2' (38) I n t h e cav one n a s ^Y a maximum electric field strength given by 21 1/2 -2 i f (39) ,u Q cl d w o where R = total available power at the input coupling, d = mirror spacing w = beam waist parameter. Taking R = lmW, Q L = 1-104, w = 2Tr'12«109s_1, Q ± = 3«10 4 , d = 47 cm, and w = 6 cm, values typical for our experiment, _3 one o b t a i n s E = 6-10 e s u , o r , p i c k i n g up t h e f a c t o r o f 2 from Eq. ( 1 3 ) , e Q = 3 ' 1 0 ~ 3 e s u , and Ke o /2ir = 1*10 Hz f o r 24 a typical 1 debye electric dipole moment. The bandwidth that one can work with in the Fabry-Perot experiment is limited by 4 the cavity bandwidth, Av c . For a loaded Q of 10 at 12 GHz one h a s . v _ 12*10 9 Av_ = pr- = 2 C U L = , 0.,n6 „ 1«2'10 Hz. 10* This f i x e s a maximum Av of 0.6*10 Hz. Our experimental work i s a c t u a l l y generally l i m i t e d t o frequency o f f s e t s Av < 0.3*10 Hz, and t h i s w i l l always be the c a s e in experiments discussed in t h i s paper. Over most of the c e n t r a l region of t h e cavity then, excluding the nodal surfaces of the standing wave f i e l d , one may invoke t h e condition (Aw = 2irAv) A 2 ^—5- « 1. (40) I t w i l l be shown l a t e r t h a t most of the molecular emission i s produced in t h e h i g h - f i e l d , a n t i n o d a l , c e n t r a l region of the c a v i t y , so t h i s approximation i s a good one. 2 2 2 >> T —2 , and In the c a s e where ( K C o J >> Aw , U Oo ^9 where t h e p o l a r i z i n g p u l s e width T s a t i s f i e s P T K< K< T (41) P 7"f ' T P 2 ' r ^ Aw conditions easily met in the pulsed nozzle experiment, Eq. (36) reduces to 25 P i(5't) Pr(r,t) = = K-nAn 4~^ sin(< e (r)t) K-RAn . j-°- £ | [cos(Ke(r)t)-l] (42) A n ( r , t ) = AnQ c o s ( < e ( r ) t ) . If -^— << 1, then p may be neglected in comparison to p.. Ke o The resulting expressions for p , p., and An satisfy X 3Pi + K 3t 3 2 e& (v r ) hAn * t' 4 / »nAn ^ = 0 ,. e X (43) n p at I T - ; - <£) i - °' with the initial conditions p <r,t=0) = p.(r,t=0) = 0, An(r,t=0) = An . X *v X »« -« (44) o Following the same considerations that led to Eq. (43), a second set of solutions to the polarization Eqns. (28) and (30) are obtained by setting Aw equal to zero, and dropping the terms involving T, and T 2 . The resulting equations (tr: + v • V)p r = 0 , (ft + v * V)p. + K2e(r,t) ^f- ( (45) = 0, ft + Y ' V T 2, - e^'t^i = °' (46) <47> 26 2 will be valid for (KeQ) 2 —2 (KG 0 ) >> T 2 , and T 2 >> Aw , T << T 2 . 2 (<e )/Aw , and when « Alternately, Eqns. (45-47) may be regarded as a microscopic description of on-resonant polarization. The exact solutions to Eqns. (45-47) with the initial conditions in Eq. (44) are Pr(r,y,t) = 0 (48) iChAn t — T - 2 sin<K:/e(r-v(r)(t-t'),t')dt'), P,(r,v,t)=X «v <v " , t t An(r,v,t) = An cos (K / t r%, «%» O *v *v (49) *v O e (r-v(r) ( t - t ' ) , t • )dt') . *V *V (50) *W Here to is defined as the time at which the electric field e(r,t) is switched on. We require that the velocity field v(r) satisfy v • Vv- = 0, where i = x,y,z. The compatibility of solutions (48-50) to the general result in Eqns. (33) and (34) may be verified by direct substitution. Comparison of Eqns. (48-50) to Eq. (42) shows that the terms e(r)t in Eq. (42) have been replaced by / e(r-v(r)(t-t«),t')dt'. % " - (51) - When e(r,t) = e(r), and a molecule travels only a short distance compared to the distance scale of changes in e(r), Eq. (51) does reduce to the simpler expression. r See Appendix A In general, 27 subject to the previously listed conditions, the quantities p. (r,v,t) and An(r,y,t) depend only on the total integrated electric field envelope experienced by the molecules as they move through the spatially complex, time varying fields of the .. 30 cavity. We will not work here with that part of the polarization and population expressions (49) and (50) resulting specifically from the v • V terms in Eqns. (45-47). Experimentally, we find no unusual behavior in the emission line shapes obtained from long polarization times (i.e., under conditions where v • V may not be set to zero in Eqns. (45-47)). Because it is al- ways possible to work with optimum signal-to-noise in the short pulse limit, and because the additional algebraic complications of any long pulse results, which must be handled numerically, will not provide a significantly more helpful description of the operation of the spectrometer, we omit any attempt to provide a description of the long pulse limit. Accordingly, Eqns. (49) and (50) are simplified to KftAn t j - 2 s i n (</e(r,t')dt') , c o p. (r,t) = An(r,t) = An Q t cos UJe(r,t')dt'), fc o (52) (53) strictly valid in the short pulse limit defined by kvx << 1 , f o r t h e c a s e when KE„T„ < o p ~ i p <<(kvKe0)~1/2 (54a) 1 , o r by •* (54b) 28 when ice T > 1. Relationship (54a) has already been discussed in connection with Eq. (31). Equation (54b) follows from the fact that we are working here with standing waves, and will be derived in Part III. We mention finally, that the on-resonant solutions to Eqns. (28) or (30) in the presence of a simple first-order relaxation mechanism with T. = T» may be recovered from Eqns. (48-50) by regarding t as a time of last collision and per- forming a weighted average over all possible such collision times. C. Evolution of the molecular polarization following the removal of the polarization inducing radiation After the system has been polarized, the input radiation is switched off and the molecules emit. Let the polarizing radiation at frequency w be removed at time t, . At that instant one has a polarization (from Eq. (20)) given by P(r,v,t,) = (P(r,v,t,) + iP. (r,v,tn) )eltotl + c.c. , «w «u X X * * * s * X X * v * w (55) X where 'c.c' stands for complex conjugate. The values P (r,v,t,) and P.(r,v,t,) are determined by solving Eqns. (28) or (30) 1 *v <v X for t < t , and then using either Eq. (26) or Eq. (29). At times greater than t., the evolution of the quantitites p., p , and An is described by Eqns. (28) or (30) with e(r,t) in those expressions set to zero. ( These are ft + Y"! )p r + A w p i + T" = °' (56) 29 p ,a (ft + •s )p Y'! i ' AwP J«.A„ r + i T^ = °' (57) 'MAn-AnJ <h+ rz> Y + —wf 2 - - °- <58> The solutions to these equations with arbitrary initial conditions at time t, are: p (r, v, t) = e" ( t - t l ) /*!{* (r-v (r) (t-t,), tn) cosAw (t-t,) (59) Pite-vlx)(t-tx),t1)sinAw(t-t1)>, - P4(r,v,t) = e " (t-t l )/lr 2 {p. (r-v(r)(t-t1),t1}ooBflw(i?-t1) X *v <**» X «* *v *w X X X (60) + pr(r-v(r) (tr-t1),t1)sinAw(t--t;L)}, An (r,v,t) = An + e"( t ~ t i^ T i {An (r-v(r) (t-t.), t,) -An }. (61) These expressions may be compared to Eq. (132) in McGurk et a l . Again w e require that the velocity field satisfy Eq. (31). D. Electric field produced in the Fabry-Perot cavity by the molecules after the polarization of the gas The wave equation for the electric field in the cavity produced by the polarized molecules is, from Maxwell's equations , (V2 - iy-4 " H ydZ or at* cT 8 = *Z^4-4 " c* St* 4uV(V'P), ~ ~~ (62) 30 where a is the conductivity of the medium in the cavity, and P is given by expressions (20), (59), and (60), along with Eqns. (26) or (29) , depending on which problem is being considered. Consider first the V(V#P) term. radiation of the TEM For the z-polarized cavity modes (Eq. (13) and Fig. 1 ) , where P = P z, this is !«!•!> " l i f e ; + wfs" ^ + &-; • (63) The electric field arising from the x and y terms in Eq. (63) is not coupled out of the cavity, so those terms are ignored. 2 2 -2 2 2 The final 9 P/3z term will be small compared to c 3 P/3t as long as P varies slowly along the transverse, z-axis in the central region of the cavity. For the TEM modes to be used here this requires that the applied polarizing electric field not be so large that molecules in the center of the cavity are driven past the condition for optimum emission of radiation. Because this requirement is met when the cavity is operated at optimum signal-to-noise, and since we observe no unusual effects in lineshapes obtained under conditions of either high polarizing electric fields, or operation in modes other than the TEM , we will ignore this last term as well. The remaining equation must be solved with the boundary conditions appropriate for the Fabry-Perot cavity, and in this experiment, with a set to zero. Rather than follow this last procedure however, we 29 will adopt the method of Lamb and adjust the a term in Eq. (62) to account for the cavity damping. The cavity time 31 c o n s t a n t Tc i s defined by (64) Tc = QL/w, where Q includes coupling, reflection, and diffraction losses in the cavity. Then <v* - h —2 c* 3tz h — > E = H- ~~2 • xcc 3t ~ < 65 > c* 3t z Rewriting Eq. (20) using Eqns. (59), (60), (26), and (16), one has (choosing the pulsed nozzle case) P = N(r,t) {(pr • + ipl ")e i ( w o t + ( u " a , o ) t l > + c.c.} *\# (66) u / / )/T w i t h p ' = e —v (t—t l i 2 p _ ( r - v ( r ) ( t - t , ) , t . ) and an analogous X JL ~ expression for P.;'• ~ ~ X X , We will assume a solution to Eq. (65) of the form E(r,t) = (e +ie,)u(r)e i ( w o t + ( w - w o ) t l ) + c.c. -v X X (67) ~ where e , e . , and u(r) a r e r e a l , and e and e. a r e slowly vary- ing functions of time, (V2 + (-£> 2 u(r) = 0, (68) where w„ i s t h e resonant frequency of the c a v i t y , ' c a v i t y *(Vu<VdV - l> and u(r) s a t i s f i e s the c a v i t y boundary c o n d i t i o n s . i n g Eq. (67) i n t o Eq. (65) and ignoring a l l terms <69> Substitut- 32 9e r 3 2 e r 3e i 3 2 e i 3p' r 3 2 p' r 3pi« 3 ^ ' 3N 32N 3t, 3t 2 , 3t ,3t 2 , 3t , 3t 2 , 3t, 3t 2 , 3t, 3t 2 , one obtains (er + i e ^ V ^ r ) + (-§)2u(r) (e r +i £i ) iw_ ^2 4TTW (er+ i e i> u ( r > T ~ (p r' + iPi')' = ~ TCC < 7 °) c plus the corresponding complex conjugate equation. Equating the real and imaginary part of this expression, multiplying through by u(r), and integrating over the cavity, one has, using Eqns. (64), (68), and (69), w c 2 'u(r)N(r,t)dV /p^utoNOr^dV - 0^ (l-(^n/p r e = 47TQ. r - L 2 2„ ,V2,2 OL^ST* > o l—I , (7i) . (72) + 1 w Jpr 'u(r)N(r,t)dV + Q.ii ( l - w^c n2/ p ,i 'u(r)N(r,t)dV ~ ~ ~ ~ e = -4-nQ, 1 2 •" ? w c 2 2 Q 2 a-(-fn 2 + i L wQ In the1'Fabry-Perot cavity the frequency width of any single mode is always much less than the distance to the nearest nondegenerate mode. Our work here will therefore be confined to single-mode operation. The solution for u(r) in the 27 TEM^ ooq mode is U( ^ = u w 2 2 2 2 2 e ( X + Z > / W {y) x o w7yT " cos(ky+k( ^-)-<(,-Trq/2), (73) 33 where we use the coordinate system of Fig. 2, and w c k = — c , ' wrt = (=^(d(2R - d ) ) 1 / 2 ) 1 / 2 , O O 2lT ' (74) 2 1 2 w(y) = w 0 ( l + ( - A Y _ ) ) / , 7TW 0 <>j = t a n " 1 (-^-%) , 7TW0 A is the free space radiation wavelength, d is the mirror separation, R is the radius of curvature of either mirror, q + 1 the number of half-wavelengths between the mirrors, and u is determined by the normalization condition in Eq. (69). The beam waist, w , is the distance from the center of the cavity to the 1/e points of the field strength. The Gaussian beam diameter expands from a minimum value of 2w at y = 0 to 2w(y = d/2) at the mirror, typically an increase of 20%. For our purposes here the form 2 2 u(r) = u 0 e" p / w o cos (ky- -rrq/2) 2 will suffice. Here p 2 = x (75) 2 + z , and w(y) has been set to w . Substituting Eq. (75) into Eqns. (71) and (72) and using Eqns. (59), (60), and (48) one obtains for the pulsed nozzle in this case: 34 2 2 e"P/wo u 2 i— 2_ 2 2 2 a (i-(-£) ) + i "L wQ STTO. E(r,t) = cos(ky-ir(q/2)) e -(t-tO/To x ** x / v d 3 r ' p ( r ' - v t r ' J t t - t , ) , ^ ) exp(-p' 2 /V 2 ) cos(ky'^Tq/2)N(r',t) x {ooBfc^fcffcaTo^t^ + Q j U - C - ^ n sin(w 0 t+(w-to Q )t 1 )} (76) . For the static gas problem, replace v(r') by v, and perform an additional integration over dv, weighted by the normalized Boltzmann distribution, W(v). Boundary conditions for the cavity are incorporated in this result in the normal mode form u(r) outside the r' integral, and in the Q_, cos(ky'—rrq/2), and exp(-p'2/w 2) terms inside the integral. A radiating dipole placed between two mirrors can build up an electric field enhanced by a factor of Q_ only if it is positioned so that successive reflections of the emitted field do not destructively interfere. As the dipole moves off axis the effectiveness of the mirrors in 2 2 H /w intercepting the emitted field is reduced by the e-o' ' o term. Notice also that the presence of the normal mode over the integral in Eq. (76) weights the polarization in favor of that produced in the central, antinodal, and thus high-field portions of the cavity. Comparing Eq. (76) to Eq. (138) in McGurk et al. we find that the argument w ^ t - t ^ + w(v/c) (t-t^-kz + wtj^ (77) 35 in Eq. (138) has become separated i n t o w ( t - t . ) + wt, o u t s i d e t h e i n t e g r a l i n Eq. (76), and [k] (r'-v(r')(t-t,)) * * * * * * (78) X inside the polarization term. This separation is analogous to, and results from the separation of cos(wt-kz) into cos wt cos kz that occurs in going from a traveling wave to a standing wave electric field. Expression (77) may be understood as a Doppler shifted radiation frequency of m + in (v/c) , and expres- sion (78) as a phase shift resulting from the movement of a dipole from a region excited with one phase to another region 5 where the dipoles have a different phase. E. Electric field in the Fabry-Perot cavity following nearresonant polarization of a Maxwell Boltzman Static Gas We are now ready to discuss pulsed time-domain spectro- scopy carried out in a Fabry-Perot cavity. The work in this section will be limited to the static gas problem. This will allow us to draw out some features of the high-Q, standing wave solutions from the previous sections common to both the static-gas and pulsed nozzle experiments while still treating a familiar problem. The pulsed nozzle and the resultant gas dynamics will be taken up in the following chapter. Consider now the static gas experiment. microwave pulse of typical duration T A high power «, 1 us is applied to a tr gas sample in the cavity at thermodynamic equilibrium and at a pressure of several millitorr or less. Our discussion is 36 of course limited to the region Aw <<<e as discussed earlier, although this restriction does not appear to be important experimentally. To help simplify some later results, we will for convenience also specify w = w and Aw <<Aw_, that is, c c all polarization and emission processes are to be carried out well within the cavity bandwidth, with the cavity tuned to the carrier. Polarization of the gas is described by Eqns. (48) and (52) . Using the assumption that w = w„, we take e(r,t) (Eq. (13)) to be -(x2+z2)/w e(r,t) = e e ° 2 -t/x (1 - e ) sin ky (79) for 0 <_ t <_ T , P and £(r,t) = £ y (x2+22)/w o2(l-e*Tp/Tc)exp(„(t.Tp)/T^sin ky f for t > T - P Without any loss of generality we have chosen q + 1 to be even. All terms in this expression have been previously de- fined and we use the geometry of Fig. 2. As an aside, we mention that the particular form of the electric field used here may need modification if thew= w condition is relaxed. c For example, if the cavity is tuned to the molecular frequency, w = w , and then the carrier frequency is swept across the cavity while monitoring the emission signal, one can observe a modulation of the emission as the cavity selects 37 those Fourier components of the polarization pulse that fall within its bandwidth at any particular frequency offset w - wc. The effect can be quite large for the short, square pulses used here, with the amplitude of the emission signal effectively tracing out the frequency envelope of the pulse, completely disappearing when w - w_ is an exact multiple of 2TT/Tp. The polarizing electric field envelope in the cavity as described by Eq. (79) as a function of time is shown in Fig. 3. The finite rise- and fall-times encountered in this high-Q system have been explicitly accounted for in the timedependent exponential terms in Eq. (79). Remembering from Eq. (51) that the polarization is determined by the area under the e(t) curve, and noting that areas A and A' in Fig. 3 are equal, we find that the finite rise and fall times do not, to first order, produce any changes in the polarization components compared to that produced by the square pulse also shown in the figure. Defining AN = An N, a constant, and using Eq. (52) one obtains P±(r,t1) = KftAN ~ - sin(Ke0T -(x2+z2)/w 2 e ° sin ky) . (80) Considering this functional form, note that, whereas in the waveguide (traveling wave) experiment the quantity P. (but not the polarization itself of course) is nearly independent of r, in the standing wave experiment P. may exhibit the CD Figure 3. Electric field amplitude of the polarization pulse in the cavity as a function of time. The square input microwave pulse of duration x with 10 ns rise and fall times is P distorted by the long time constant of the cavity. Since the molecular polarization depends only on the area of these curves, region A here being equal in area to region A', this distortion does not affect our results. 39 0) E IxJ - * 40 feature of spatial variations on a distance scale of less than one wavelength of radiation. scale of k" If a characteristic distance is assigned for changes in the amplitude of the standing wave electric field along the cavity axis, then the corresponding distance scale for changes in P. and AN along that axis will be approximately (kice x ) ~ for KG. X > 1. Solutions to Eqns. (22-24) without the v • V terms are therefore strictly valid only for times x such that a typical mole- P cule can move only a short distance along the cavity axis com-1 -1/2 pared to (k<e x„) , or x « (kv<e ) ' , which is expression (54b) . Using the result in Eq. (80) for P. in Eq. (76), evalu2 ating u from Eqns. (69) and (75), and using the condition w = w , one o b t a i n s -(x2+z2)/wo2 E ( r , v , t ) = - 8TTQTe ° s i n ky e —=__ x - ( t - t , ) / T 9 KftAn •* —^—— cos(w ( t - t ^ + w t ^ x /d r ' s i n ky' e s i n {(<e x ) x (81) e x p ( - ( ( x - v x ( t - t 1 ) ) 2 + (z - v z ( t - t 1 ) ) 2 ) / w Q 2 ) x sin k(y' - v ^ t - t ^ ) } . Modifications of this expression for the case where the mode number q+1 is odd are straightforward. Our concern right now is with the shape of the emitted signal. We temporarily drop 41 the time-independent terms in front of the integrals. Integrating the remaining expression over a Maxwell distribution of velocities for the static gas, one obtains -t/T 2 E(t) " e cos(w t + wt,) o i x 2 mv / 2 k T /dvdvdv, x y z e "" /dxdydz sinky e 2 ° x (82) s m { (Ke0xp) exp(-((x-v x t) 2 + (z-vzt)2)/wQ2) sin k(y-v t) } Here t - t , has been replaced by t, measuring now from the end of the polarization pulse, and the primes inside the spatial integral have been dropped. Molecular emission occurs at the transition frequency w , modified by an envelope with a T, exponential decay and a time and velocity dependent sixdimensional integral containing all the Doppler dephasmg information, as determined from the movement of molecules through the cell. This general arrangement of terms is inde- pendent of any approximations made in the expression for the cavity normal mode. Because of the simple form of the normal mode in Eq. (75) used here, it is possible to continue to evaluate Eq. (82) analytically. Making a change of variables k(y - v t ) •*• u, the integrals over y and v are: 42 -1 k w ~mvv / 2 k T ff(q+D/2 / dv„e * / s i n ( u + kv t ) x y -oo y -ir(q+D/2 -((x-v t)2+(z-vzt)2)/w sin(<e T „ e ' o p Here we assume that (¥} (83) 2 sinu)du 1/2 t « d (84) so that we may ignore edge effects due to the mirrors. Using sin(u + kv t) = sin u cos k v t + cos u sin kv t and the symy y y metry properties of the two integrals one obtains 2 , oo -mv /2kT y 4k~ / dv e cos kv t x o (85) 1T(q+1,/2 2 2 2 -((x-vt) +(z-vt) )/w / sin u sin (KGOT e p o sin u) du. Using the fact that q+1 is even, the integral over u is a representation of the ordinary Bessel function of order one, J,. This last expression becomes , 2ir(q+l)k 2 -mv /2kT oo y / dv e o " cos kv t (86) 2 2 -((x-v t) +(z-v t) )/w J l (Ke T o p e x y 2 > • 7r(q+l)k v is just the cavity length d. The integral over g is standard (McGurk et al. Eq. (145)), giving \ fc2 4s2 2ks / e- / , (87) 43 where s = U n 2 ) 1 / 2 Aw J"1, D and ^ D - £ (^82,1/2 , the Doppler half-width. (88) An entirely conventional Doppler- broadened envelope is obtained. Numerical studies carried out with less approximate forms of the cavity normal mode Eq. (75) and including the movement of molecules during the polarization pulse as in Eq. (51) showed no significant changes from this result under the conditions of pulse lengths and input powers normally used in our experiments. It follows from Eq. (87) that condition (84) may be written „ (,2kT.l/2 ,c. , m ,1/2 <^ , < d , or N _ ) v(—) (v.. .I,') m w 2kT' ' 2 (89) ^ << d , easily true for all cavity modes except possibly a fundamental TEM 0 mode. Implicit in Eq. (87) is the necessity of having polarized and entire Doppler envelope of the transition of interest, and this can indeed be shown to follow from the conditions imposed in deriving that result. (3kT/m) ' Choosing v_j.s = for a Maxwell-Boltzmann gas, from Eq. (88) one obtains AW D " IT V RMS " kV RMS * (90) Now if AwD < Aw, so that the carrier falls outside the Doppler envelope, then condition (40) guarantees that the entire line 44 profile lies within Ke . However, if Aw D > Aw, so that the carrier lies within the envelope, Eq. (54) gives kV K< X RMS T p ~ A t V p ' or (91) Aw D « l/xp , which, when combined with Aw < Aw D , ensures that the Fourier components of the carrier cover the entire line profile, even if Ke Q <_ AwD« Using Eqns. (86) and (87) in Eq. (82), and replacing the terms from Eq. (81), one has 2 2 KftAN -p /w E(r,t) = 16 Tr Q. —r-°- sin ky e ° x ~ -t/T 2 t 2/ 4 s 2 e e u -wrdvdv {{ 2H^ ff dvx^z Ht h e e C0S ^ (W t 2 ) o (92) + wt,) x i / M 2 2 ; / dxdz -(x2+z2)/w2 o x e O - ( ( x - v x t ) 2 + (z-v z t) 2 )/w Q 2 J (K£ l oTp e )} ' still measuring from the end of the polarization pulse. If Q_ is written as «L-5f ' <93) where a is the fraction of energy lost by a wave in one transit of the cavity, then the effective length of the cavity may be taken to be d , typically 10 2 d for a Q of 10 at 10 GHz. 45 The four-dimensional integral in Eq. (92) contains two kinds of information. First, there is the slowly varying time dependent envelope resulting from the overlap of the exponen2 2 2 2 tial x , z , and (x-vxt) , (z-vzt) terms. Physically, this describes the gain or loss in signal as molecules more through the cavity beam waist. Second, one finds that the concept of a TT/2 pulse, generally obtained by adjusting ice x maximum of J,, to the first is not directly applicable to the cavity because of significant transverse variations in the electric field amplitude. Computer calculations carried out using more exact forms of the cavity normal modes show no changes in the lineshapes from those predicted by this analytically derived result. F. Field coupling out of the Fabry-Perot cavity Expression (92) is sufficient for analysis of lineshapes seen at the detector, but does not of course represent the actual field at the detector. To estimate that quantity with- out actually solving a complicated boundary value problem we resort again to the standard methods of cavity circuit theory. The energy stored in the electromagnetic fields in the cavity after the molecules begin emitting is (in cgs units) U = ^r / v (E2 + B 2 ) dV = i- / v E 2 dv , to be averaged over one cycle of the radiation fields. refer to Eq. (92) for the field. We drop the v„ and v_ (94) We 46 dependence inside the J, term in Eq. (92) so that we may perform the integral over dv dv„. Substituting Eq. (92) into x Z 2 Eq. (94) and replacing cos (w t + wt,) by its average value of 1/2, one obtains ~ /tcttAN \ 2 l ^ , , -2t2/4s2 -2t/T~ x ™o I (95) 2 (SI dxdz ^ ( K E Q T 2 2 2 -p /w -p /w e °)e ° ) ** . If the cavity is coupled so that Q = Q , = Q 2 , the power reaching the detector will be p „„* = S- w~ • out Q_ o Using Eq. (95) this is p out = ( 8 / 9 ) vQ o (96) o (dropping the terms in t) w o(l<al^Zlb>lANo)2lTWo 2d x (97) 2, 2 2 - 2 -p / w 1 ( ^ - y / / dxdz J irw/ (<e T_ e ° p - p /w ° ) e 2 ° P or, Ke P out = (8/9) ^ o ( 2 2 I <a I Mz I b> IANQ) TTWO d ( ^ \ - ) I oTp J (u) d u ) 2 (98) o po This may be compared to the expression for P . for a reflec5 tion cavity following a TT/2 pulse: P out = b Q ow o (|<a|uz|b>|AN0)2 V . (99) 47 Here V is the cavity volume. The corresponding waveguide ex- pression in the dominant mode far above cutoff is P out * b (T) % ( | < a | " z M A N o ) 2 Ji 2 (Ke0Tp) V , where % is the cell length, and V its volume. (100) In comparing the waveguide experiment, Eq. (100), to the cavity experiment, Eq. (98), it is useful to refer to Eq. (93). Experimentally, we measure not the emitted power, but the electric field, which may be obtained from Eq. (98) by using the waveguide formula. One thus obtains a Q1/2 ' dependence in the signal from the cavity. The presence of a radiation- damping time constant directly proportional to Q ensures that energy is conserved. That is, the higher radiation energy densities in the cavity shorten the radiative lifetimes of the polarized molecules, causing them to release their energy to the detector at a rate higher than they would in a waveguide, and before collision and Doppler dephasing processes take over 32 and dissipate that energy in experimentally nonuseful ways. Given a sufficiently high Q and number density of molecules, this effect will result in an effective line broadening. Since we have no evidence that radiation damping is affecting our lineshapes, we have omitted this effect in our theoretical treatment. Referring again to Eq. (98) we see that the measured signal is proportional to the first power of the dipole moment u, provided that one is working somewhere near the first 48 maximum of J,, this being characteristic of time-domain spectroscopy. We also mention that the terms Q_w , or (wd/ac) w , when combined with the w dependence of AN for a gas described 2 at some temperature, lead to the w dependence in signal, this also characteristic of the time domain experiment. The presence of the functional form J, in Eqns. (98) and (100) deserves some comment. In the asymptotic limit x •+ co J, (x) •*• (-^r)1^2 (sinx + cosx) 1 , (101) 1TX resulting in a loss of signal for large K E T the condition x << T„). (subject still to In the waveguide experiment Jid^e T J replaces sin (ice x ) when the variation of the electric field across the broad dimension of the waveguide is taken into account.12 We should expect m general that sin(KG X ) will be replaced by some function of KG; X that is zero when (<e x )=0, damped in the limit of large argument, and in between displays a structure at least reminiscent of sine, whenever a pure traveling wave electric field is replaced by an electric field such that spatially separated molecules experience electric fields that differ by more than just a phase factor. This is certainly the case in the cavity. Physically this damping results from an increasingly severe dephasing of adjacent dipoles as the total time integrated electric field envelopes that they experience become different by larger and larger amounts. The dephasing is closely 49 related to the development of spatial variations in the polarization in the cavity on a distance scale much shorter than k as discussed in connection with Eq. (80). Experimentally this effect is readily observable, both in the static gas and in the pulsed nozzle molecular source, particularly with 33 large dipole moment molecules such as HFHCN (5.6 debye), where we can completely destroy the signal by increasing the input microwave power above a very low level. G. Static Gas Experimental Results In Sections E and F two results were derived for the static gas experiment: Eq. (92) for the emission envelope, describing the product of an exponential e-t/T ' 2 relaxation with a conventional Gaussian time decay in the short pulse, power broadened polarization limit, and Eq. (98) for the amplitude of the emitted signal as a function of the parameter xeo x_, again valid only in the short pulse limit described by p Eq. (54). In this section we examine the experimental results as a function of t and Ke xDSome details of the experimental apparatus are shown in Fig. 4. Microwave power at frequency v from the phase stabi- lized master oscillator (MO) is shaped into short, square pulses of typical duration 1 us by PIN diode switch 1. In the lineshape measurements discussed in this section, and for most molecules having an electric dipole moment of 1 debye or greater, this pulse passes directly into the cavity. For Monitor Detector Pin Diode SI Pin Diode /Impedance SI / Match OJ Pin Diode S2 { Variable Attenuator Circulator Detector Figure 4. Part of the experimental apparatus used to make the static gas measurements reported in Section G. o 51 small dipole moment molecules, and in the measurement discussed in the next section, a TWT amplifier is used to obtain pulses with peak powers of 2 Watts or more. In this case switch Si' is used to block the TWT noise during the detection sequence. Switch 2 is used to protect the detector crystal from the high power pulse. The cavity input is critically coupled by adjusting a waveguide tuner until the power reflected from the input coupling is zero. When the microwave energy enters the cavity it creates a coherent, macroscopic polarization of the gas as discussed in Section II. The subsequent molecular emission is detected by a balanced mixer with a 30 MHz i.f. amplifier. This signal band is mixed again with the 30 MHz i.f. signal, amplified, digitized, and recorded. In Fig. 5 we show the time domain signal from the OCS J = 0-*-l transition at 12163 MHz. Data was taken at a gas pressure of less than 1 mtorr, with a polarization time x 2.5 us, and digitized at the rate of 0.5 us per point. = Digi- tal points have been connected by straight lines in Fig. 4. Switch S2 was closed at time t on the figure, 0.35 us after s the p o l a r i z a t i o n pulse had ended. 156.5 kHz. The c a r r i e r o f f s e t i s Av = A low-Q mode, Qr < 5000 was used in t h i s measure- ment t o minimize r i n g i n g from the high power p u l s e . Care was taken t o o b t a i n an emission envelope free from a m p l i f i e r filter distortion. From Eq. (92) t h e envelope should be combination of Gaussian and e x p o n e n t i a l decays. If or in to Figure 5. Transient emission signal obtained from the OCS J = 0 -> 1 transition at 12163 MHz taken in the TEMonr-7 cavity mode with a static gas sample. The switch S2 blocking the detector was opened at time t_. t' is the first point at which the signal envelope can be measured. The signal was digitized at the rate of 0.5 ys/point and the points connected by straight lines. t s f 60 80 time {ftseconds) in to 54 ,-*„ J i f ^ i ) 1 ' ' 2 (102) is plotted as a function of t-t', where I is the envelope amplitude, and t* occurs a short time after t_, one obtains a curve with a straight line asymptote having slope (2s) and intercept sT 2 ~ , at t = t', where s is defined by Eq. (88). This is shown in Fig. 6 choosing t' to be the first available maximum in the signal. We obtain s = 13.12 ys, compared to an expected result of 13.7 ys for OCS at 293K, and a T 2 of 21 42 ys, corresponding to a pressure of about 1/2 mtorr. Al- though Eq. (92) has been derived subject to various restrictions on pulse length and carrier offset, we have never observed any significant deviations from this result over a wide range of variations of these parameters. We now take up the question of emitted signal as a function of the parameter KC x . We have already derived one result in Eq. (98) and have discussed the general features expected in this experiment. The emission signal for the OCS J = 0-*l transition as a function of the parameter <e x was measured for pulse lengths x The of 0.5, 0.8, and 1.0 ys. XT experimental apparatus is shown in Fig. 4. A continuous flow system employing a standard gas regulation valve was used to introduce the gas into the cell. The flow rate was adjusted once at the beginning of the experiment and then monitored during the measurements on a gauge capable of registering a 5% change in pressure at the pressure of one mtorr used here. in Figure 6. Plot of (-In I(t - t')/I(t*)) ' versus t - t ' for the signal in Fig. 4. Here I(t) is the signal envelope amplitude. The solid line is (t - t'J/26.24 ys + 0.320, and may be used to determine the Doppler and T 2 dephasing times. 56 V ( (.4)1 (.4-4)1 U|-) 57 No change was detected. The cavity quality factor was deter- mined by measuring the frequency position of the half-power points of the transmitted power as the carrier was detuned from the cavity. For a half-power width of 1.38 MHz at 12163 MHz, one obtains a Q T of Q_ = L Ju , ,» = 8814. A precision rotary If J O vane attenuator was used to adjust the input power to the TWT amplifier. Input and transmitted power levels were later measured directly with a power meter. Equations (13) and (39) were used to calculate e , where d = 57 cm, and Q , was set to 3QL« We estimate this absolute electric field calculation to be in error by less than 50 percent. Experimental results are shown in Fig. 7. All measurements were taken with Av = 90 KHz. On plot (p-jnDUt)1/2 T , which is measured which is directly proportional to Ke x maxima near 1.4 mW 1/2 ' ys correspond to the horizontal axis we experimentally, and . Ths pronounced input powers of 5.5 mW, 2.3 mW, and 2.1 mW, for pulse lengths of 0.5 ys, 0.8 ys, and 1.0 ys, respectively, giving calculated values of K E T_ or 2.9, 3.0, and 3.6. The signal is reduced by no more than a factor of four from its maximum over a range of a factor of sixteen in electric field, or about 24 dB in input power. The short pulse result 34'35 /E0Tp ; Ke T o p ° 1 - J 0 (K£ X ) J. X (u) du = Ke ° ° oTp p (103) in oo Figure 7. Experimentally measured amplitude of the OCS J = 0 -*• 1 static gas emission signal as a function of the polarizing electric field for the TEM_n._ cavity mode. Intensity (Arbitrary Units) N & 3'rS 3 5 co l> D ro O) 6S O - o p o oo o» T:T=T= </* tn tn 60 taken from Eq. (98) for the static gas signal as a function of <eQx is shown in Fig. 8. at Ke0T The peak value of 4.2 • 10 occurs = 2.8. Although the width of the peak in the calcu- lated curve is larger than the experimentally measured results, overall agreement between the two curves is good. Using condition (54b) one would expect the short pulse expression Eq. (103) to be valid for xp « (kv K e 0 )" 1/2 # or (104) 1 KE~T_ << KVT °P D tr Taking v = 3 * 104 cm s-1 for OCS at 293 K, this is KCoXp « 26 and 13 for pulse times x of 0.5 ys and 1.0 ys, respectively. kr H. Comparison of Static Gas Fabry-Perot Results to Waveguide Spectrometer Results To conclude our discussion of the static gas, we use Eq. (98) for the cavity experiment, and the corresponding waveguide experiment result in Eq. (100) to compare the strength of the static-gas signal from these two types of time-domain 4 spectrometers. For the cavity, take Q = 3 ' 10 , <£_.?„ = 2.8, wo = 5.4 cm, and d = 47.5 cm for 12.2 GHz radiation; for 3 the waveguide, % = 6 m, V = 1400 cm set J, to its maximum of 0.6. obtains with these values: for an X-band guide, and From Eqns. (98) and (100) one Figure 8. Calculated amplitude of the static gas emission signal as a function of KG x mode. for an arbitrary TEM00cr cavity Intensity (Arbitrary Units) Z9 63 P out (cavity) , / 2 <Pou^waveguide)) ' " 10 <105> ' Any discussion of the ratio of signal-to-noise in the two experiments must include several other considerations. The cavity is obviously a narrow-banded instrument, and although this makes searches for new resonances more difficult, it also reduces the noise power included in its normal working bandwidth. The bandwidth of the waveguide experiment however, depends on a relationship similar to Eq. (4 0), and is generally made as large as possible by using a high-power TWT amplifier and wideband signal processing components, reaching values as large as 50 MHz for molecules with transition moments on the order of 1 debye. In the Doppler broadened limit, from Fig. 5, signal relaxation times will limit both experiments to pulse repetition rates of perhaps 20 kHz or less. Wall relaxation, definitely not a problem for the cavity experiment, may limit decay times in the waveguide cell to much less than that shown in Fig. 5, particularly if a small cell cross section is used to increase the microwave power density, and thus the bandwidth. In a carefully designed 20 foot waveguide system the power-pulse ringtime can be made as short as 200 ns, which, when combined with power pulse lengths of 10 - 30 ns, make it possible to work well within the pressure broadened regime without much loss in signal. The larger number of molecules approximately compensate for the shorter duration of the emission signal. In this case repetition rates of 300 kHz are routinely possible. This 64 approximately even trade-off between the number density and the signal duration changes if one is attempting to work with weakly bound dimer systems in a static gas. Here the signal duration is still inversely proportioned to the first power of the gas pressure, but the number of species of interest, which we assume to depend on the product of the partial pressures of the two monomer components, can be made to be proportional to the square of the total pressure. Having worked with both types of pulsed spectrometers, we find that Eq. (105) probably overestimates the ratio of sensitivities of these two instruments. The two spectrometers are sufficiently different that a choice for a static gas experiment should be heavily weighted by the other listed considerations. Of course, the tremendous increases in sensi- tivity and resolution of the Fabry-Perot instrument come when the large volume of the cavity is combined with the pulsed nozzle molecular source, and it is this combination that is taken up in the following chapter. 65 Chapter III. A. The Gas Dynamics of a Pulsed Supersonic Nozzle Molecular Source as Observed with the Fabry-Perot Cavity Microwave Spectrometer Introduction Having developed the theory of pulsed Fourier transform spectroscopy carried out in a Fabry-Perot cavity in the previous chapter, and having treated many of the details of this type of spectroscopy in the static gas discussion, our emphasis in this work will be on the properties of the gas nozzle expansion itself. In Part B, we review the derivations and results from the previous chapter that are applicable to arbitrary distributions of molecules in the cavity. In Part C, these general results are used to obtain the equations describing the emission lineshapes produced specifically by a gas m the nozzle expansion. The characteristics of experi- mentally observed lineshapes are analyzed according to these results. From this we obtain information about the speeds, density distributions, and dephasing properties of molecules in the expansion. All work done in this section is with the nozzle positioned at its normal location midway between the mirrors, pointed directly at the center of the cavity. In Part D we discuss nozzle configurations in which the nozzle is either tilted or relocated from its normal position. These experiments provide additional information about the beam, but may also be of use in choosing a geometry for the nozzle and mirror system. 66 B. Equations Describing the Polarization and Emission of Radiation by Molecules in the Fabry-Perot Cavity Appropriate for the Pulsed Nozzle Problem The theory describing the absorption and emission of rad- iation by molecules in the standing wave electric field of the Fabry-Perot cavity has been developed in detail in the previous chapter. We summarize here the results necessary for the pres- ent study. When a number of non-degenerate two-level quantum systems are brought into contact with a standing wave electric field E(r,t) = 2 z e(r,t) cos wt (we use the geometry of Fig. ** fV *** f* 9) it can create a macroscopic polarization, P, of the gas of the form P(r,v,t) = (P„(r,v,t) + i P. (r,v,t) )ela)t + c.c. , where P (106) and P. are real-valued functions of the six phase-space coordinates r, v, and of t, and 'c.c.' indicates complex conjugate. The polarization components P and P., along with the two-level population difference per unit volume associated with the velocity v, AN(r,v,t), satisfy the Bloch-type coupled partial differential equations: (ft + Y • V)P r + A ^ + ^ = 0 , (ft + v • V)P ± - Ao>Pr + K 2 e»'r,t) ^ ( |_ + ^- = 0 , (107) *AN , _ , *(AN-ANQ) + v . V) ^ - e(r,t)P. + = 0 , 4Ti - where V is the gradient operator, A<D = to -w, the difference between the molecular transition angular frequency w , and the 6? Figure 9. Geometry and coordinate system used for the gas nozzle and Fabry-Perot cavity. The density of the gas expan- sion is parameterized as (cosp 0)/r . The microwave radiation travels along the y-axis, perpendicular to the nozzle axis. 68 carrier angular frequency y, T-, and T 2 are phenomenologically introduced first-order relaxation terms, AN is the equilibrium value of AN in the absence of an electric field, and K = 2ft"1|<a|yz|b>| , (108) where <a|yz|b> is the electric dipole transition moment connecting the two levels. When the gas is introduced into the cavity via a pulsed nozzle it is possible to assign to each coordinate r in the cavity an unique velocity v(r). It is con- venient to define new quantities p (r,t), p.(r,t), and An(r,t), by N(r,t)pr(r,t) = / dv Pr(r,v,t) N(r,t)p (r,t) = / dv P.(r,v,t) (109) N(r,t)An(r,t) = / dv AN(r,v,t). Here N(r,t) is the number of two-level systems per unit volume. We will assume that the electric field interaction does not affect the center-of-mass motion of the molecules. We will also assume that N(r,t) satisfies the conservation equation 3N(r,t) + v • (v(r)N(r,t)) = 0 . 3t The quantitites p , p., and An, then satisfy p ( ( ft + 1 )p + Y -* ' ! r A w P i + f £ = 0 ft + Y(5> * !>?! ~ Aw P r + e(r,t) *|B. + _i = o, 2 K (110> ' P< (111) 69 a ( *Ar, + ( ) } • e( -nfAn-AnJ t)p + ft Y 5 ' ! ^ T ~ 5' i — 4 T — ~ where An = °' (112) may be taken to be a constant. The Fabry-Perot cavity is inherently a narrow-bandwidth, high electric field instrument, so we concentrate on solving the on-resonance, Aw = 0 form of these equations. When the polarization pulse length x is sufficiently short that a kr typical molecule travels only a short distance compared to the characteristic distance of variations in p , p., and An, and provided that x_ >> T,,T2, we may ignore the v • V and T^, T 2 terms in Eqns. (110-112). The solutions to the resulting equations (with Aw set to zero) are pr(r,t) = 0, (113) K&An j-S. sin (</ e(r,t')dt'), p. (r,t) = An(r,t) = An t cos (K / e(r,t')dt') . % (114) (115) ~ The conditions necessary for these solutions to be strictly mathematically valid are: 2 2 i. (K£Q) >> Aw , ii. iii. iv. x /* 2 , << xe /Aw x -1/2 << (kvice ) ' , (KE 0 ) 2 >> T 2 -2 , x (116) « T2 , 70 where z is one-half the maximum electric field amplitude in the cavity, and v is a typical molecular velocity component along the cavity axis. To obtain an appropriate form for e(r,t') in expressions (114) and (115), we note that the frequency spacing between successive nondegenerate modes in the Fabry-Perot cavity is much greater than the width of any single mode. We will work in this chapter exclusively with the TEM modes. Experi- mentally, we find no significant variation in lineshapes taken in different cavity modes, and the approach here using TEM modes may easily be generalized. The electric field e(r,t') is separable as e(r)f(t'), where, for the TEM £{ V with p = £ w 2, 2 , . e P o wlyT " cos(ky-7rq/2) , = x + z /STT (117) (see Fig. 9), and w^ = (^ (d(2R -d)) 1 / 2 O mode: O 1/2 ) , (118) W(5 = " "° (1 + fr) : w c , k = -2 c ' \ is the free-space radiation wavelength, d is the mirror separation, R the radius of curvature of either mirror, and q + 1 the number of half-wavelengths between the mirrors. The beam waist, w , is the distance from the center of the cavity to the 1/e points of the field strength. The normal mode from Eq. (117) has been simplified from the more exact expression (Eq. (73) in chapter II by neglecting certain terms in the cosine argument. To be consistent, we will use in this cahpter a simplified equation to relate the frequency and mode number to the mirror separation. The polarization pulse inside the cavity is assumed to begin at time t' = 0 and end at t' = t-.. As explained in chapter II, although the narrow-band cavity may severely distort the shape of the square microwave pulse of duration x P incident upon it, the pulse area remains unchanged, so the integral in Eq. (114) may be immediately done to obtain, at the end of the polarization pulse inside the cavity: p ( t i £' l ) = K*iAn_ w 4~^ sinl<e o T p |^) e 2 .2 cos (ky-Trq/2)}, (119) P /W with a similar result for An. After the polarizing radiation has been removed, p , p , and n continue to evolve as described by Eqns. (110-112) without the e(r,t) terms. The polarized gas now produces an electric field in the cavity. Using exact solutions to Eqns. (110-112) without the e(r,t) terms, and using Maxwell's equations to couple the polarization to the emitted electric field, one obtains for the emitted electric field in the cavity: 72 2, E ( r , t ) - 8TTQ_ ( W » e " /w p 2 cos(ky-irq/2) x e" t / / T 2 c o s UOt + ( w - w j t , ) — V2-, O 1 7TW x d (120) 3 / d r« p (r'-v(r')t,t )(w/w(y')) V 1 ~ ~ ~ < p N(r',t) e" 2 / /w 2 X . x o I, lY ; cos(ky'-7rq/2) . This has been simplified from the more general result Eq. (76) in chapter II by setting the cavity resonance angular frequency w c equal to w . Time is measured from the end of the polarization pulse at t,. The quantity p.(r'-v(r')t,t,) is to be ob- i i — — ~ i tained from Eq. (119) . As discussed in detail in chapter II, Eq. (120) has been derived under the assumption that the velocity field v(r) satisfies v(r) * V v, = 0 , i = x,y,z , (121) which may be verified directly for the pulsed nozzle velocity field to be given below. Physically, we require that all molecules travel at constant speeds along straight line paths. The normal mode form (w /w) e"p ' w cos (ky-irq/2) outside the integral in Eq. (120) simply describes amplitude variations in the emitted electric field signal characteristic of the TEM mode. The essential lineshape information is contained in the time-dependent e~ ' 2 cos(wQt + (w-w )t.) terms, and 73 in the spatial integral, which contains all information related to the molecular motion, including Doppler dephasing, and dephasing due to the movement of the molecules through the cavity. To proceed further we need to consider the exact form of the pulsed-nozzle molecular velocity field. C. Lineshapes in the Pulsed-Nozzle Fabry-Perot Experiment To complete the derivation of the functional form of the time-domain lineshapes seen in the pulsed-nozzle experiment we consider certain characteristics of the pulsed nozzle gas expansion. These may be summarized as follows: 1. All molecules travel at constant speed v on radial paths originating at the nozzle. Using the geometry of Fig. 9, the velocity field v(r) may be written xx + yy - (h-z)z v(r) = v - ^ ^ YT72 ' (x^ + y* + (122) {h-z)*)*-'* which satisfies Eq. (121). 2. To begin our analysis of the gas expansion we will assume a molecular density distribution N(r,t) of the form H(r,t> c go4i= 2 r (h-s)P .ov2J.„2a.,._ .2,p/2+l ' (x'+y"+(h-z)') (123) where r and 6 are the radius and polar angle from the nozzle. The exponent p will be determined experimentally. 3. As has already been noted, in the expanding gas T 2 > 100 ysec, justifying our neglect of relaxation processes 74 during the polarization. Using Eqns. (119), (120), (122), and (123), and dropping the normal mode form and numerical constants from in front of the integral in Eq. (120), one ob- tains for the emitted electric field lineshape: E(t) = e""t/,T2 cos(w0t + (w-wo)t1) 'v x d 3 r sin k y s i n ( K e T ( o p w<y-5 t)> x (124) 2 exp( 2 -(x-v t) -(z-v t ) s ) sin k(y-v t) } x y w (y-v t) , Wo x ,-x -z . cos p 0 ( ) exp(—^ * 2— w(y) w (y) r where v , v , and v ' are functions of x, y, and z. We have chosen q + 1 to be even, and have dropped the primes inside the integral. Note that the integrand contains two gaussian waist terms, one inside the sine expression and time dependent, and one outside the sine and time independent. Physically these describe the loss of signal as the molecules move out of the beam waist. Because of the complex algebraic form of Eq. (124) it is not possible to simplify this result analytically. It is useful, however, to consider two special cases that can be treated analytically, these being tightly collimated beams moving at uniform speed v Q either perpendicular or parallel to the cavity axis. In the perpendicular case v x = v y = 0, v z = v Q , 75 and the number density N(r,t) is N(r,t) = N 6(x)5 (y) , (125) where 6 is Dirac's delta function. Using Eq. (124) (with q+1 odd in this case) and Eq. (125) one has -t/T 2 E(t) = e cos(wQt + (w-w0)tx) /"sin {Ke0x /z-v t\ 2 /z_\ 2 e " ! - ^ - ) } e " U 0 ) Az , (126) which is a single resonance centered at w , decaying with an envelope combining T 2 dephasing with movement of the molecules out of the cavity. In the parallel case vx = v z = 0. ' vy — v o', and N(r,t) = N Q 6(x)S(z) , (127) for -d/2 + v t <_ y <_ d/2, and zero otherwise. Using Eqns. (124) and (127) with v > 0, one has o _t T E(t) = e / 2 cos(wQt + (w-wQ)t1) d/2 / dy sin ky sin{xe_x_ sin k(y-v t)} , (128) -d/2+v t o p o o dropping the w(y) dependences in Eq. (124) . Making a change of variables ky - kvQt ->- u, the integral is k-l;7r(q+l)/2-kvot d u s i n ( u + k v t ) B i n ( K e -TT(q+l)/2 ° - S in u ) . (129) ° p 76 The -kvQt term in the upper limit is of minor interest here, so we ignore it for v Q t << d. Expanding sin (u + kv t) and remembering that q + 1 is even, one obtains a lineshape -t/T2 E(t) = e cos(wQt + (w-wQ)t1)d x (130) cos ( w o "ft) J l< Ke oV ' where we have used (q+1)(X/2) = d, and J, is the ordinary Bessel function of order one. The resulting line E(t) consists of the two frequencies wQ(l+v /c) and wQ(l-v / c ) , with an obvious physical interpretation. By including the -kv t term in the upper limit of Eq. (129) , it can be shown that there will be no Doppler splitting in the TEM mode. We return now to the full pulsed nozzle expression Eq. (124), and consider the properties of this result in comparison to experimentally observed lineshapes. For the purpose of introducing the unusual features of these time-domain spectra we refer to Figs. 10 and 6. Figure 10 is the time domain 16 12 32 record for the J=0 ->• 1 transition of O C S , which is known to consist of a single line at 12163 MHz. The spectrum was recorded by pulsing a 4% mixture of OCS in Ar through the nozzle. We use a thin plate flat orifice bolted to the bottom of the pulsed valve; we do not use a skimmer. The signal was digitized at the rate of 0.5 ysec/point and the points connected by straight lines. The corresponding power spectrum is shown in Fig. 11 and has a frequency resolution of 3.90625 kHz/ point. The most prominent characteristic of this spectrum is Figure 10. Time domain signal from the O C S J = 0 -*- 1 transi- tion, converted at 0.5 sec/point. The points have been connected by straight lines. Gas mixture is 4% OCS in Ar. 78 o CM o CO o <v tn 8 o 8 Ld vo Figure 11. Power spectrum obtained by adding 256 zeros to the data in Fig. 10 and Fourier transforming. The resolution is 3.90625 kHz/point, with a splitting of 36.3 kHz. —»•»—•••••••• FREQUENCY 00 o 81 the symmetric splitting apparent in both figures. This split- ting is a general feature of all spectra taken with the pulsed nozzle in the Fabry-Perot cavity. We have found that the position of the center of this pattern is invariant under all changes in spectrometer operating conditions, and corresponds to the molecular resonance frequency. The lineshape itself provides several different kinds of information about the gas expansion, and will be our main concern in the remainder of this paper. It is convenient to separate Eq. (124) into two parts. The cos(w t + (w-w ) t.,)exp(-t/T2) terms center the emission signal at w and provide a routine exponential damping. The interesting lineshape information is contained in the remaining integral expression, which we denote I(t) . This envelope function I(t) contains two types of signal damping. As dis- cussed in the special case of a narrow beam traveling through the cavity perpendicular to the axis, there is a fall-off in signal due to the transverse motion of the molecules out of the cavity, mathematically expressed as the overlap of the 2 2 2 2 (x-v t) , (z-v t) and x , z terms in the exponentials. A x z second type of damping, Doppler dephasing, results from the movement of molecules from the region where they were polarized with one phase, to regions where they would have been polarized with a different phase, mathematically appearing in the k(y-v t) argument. Numerical studies show that in the pulsed beam, the damping due to transit of the molecules out 82 of the cavity is negligible in comparison to the Doppler dephasing. Because inclusion of this transit-time damping in our analysis would provide at best only a marginal improvement in our description of the gas expansion, we will not include this effect in our analysis. Accordingly, we will use here the lineshape expression w 3 J I(t) = /'cavity .. d r " sin sin {ice ° " ky ""•" ^ o 'x, p w(y) exp exp 3111k v t > } "HrM ^- y * wTyT \ w (y) / x (131) cos p 8 r2 \i/(y) / In Fig. 12, we show the triple integral Eq. (131) calculated for the TEM Q Q 3 1 cavity mode as a function of t, for p = 0.5, corresponding to a cos0 "56 density distribution, a frequency v cm/sec. = 10 GHz, and a molecular speed v Q = 4-10 No relaxation processes except Doppler dephasing are included in this calculation. The time, L , from the maximum of the curve at t = 0 to the first zero crossing, and the distance L-, i = 1, 2, ..., between successive zero crossings are, in microseconds, 41.8, 47.0, 46.6, 46.4, and 46.5, respectively. These spacings conform to a general pattern, independent of any of the parameters in Eq. (131), in which the spacings L. following the first zero are nearly constant, 00 Figure 12. The envelope function Eq. (131) calculated for a density 0 5 2 distribution (cos " 6)/r , v = 1 0 GHz, molecular speed 4 v = 4-10 cm/sec, and a mirror spacing of 48 cm. No T~ is included in this calculation. 84 but possibly quite different from the f i r s t spacing, L . The r a t i o L Q / L i i s an important parameter, independent, as we will show, of v and p. v , depending only on the coefficient I t varies monotonically from 0.85 for an isotropic dis- tribution with p = 0, to 1.07 for p = 2, t o a r b i t r a r i l y large values as p increases. As p increases t h e f i r s t zero crossing moves away from the origin, and the heights of the successive maxima in the curve become smaller in r e l a t i o n to t h e height of the curve at t = 0. A calculation for p = 4, where a l l other parameters are held fixed from the calculation in Fig. 12 is shown in Fig. 13. In the limit of l a r g e p then, the envelope will show no o s c i l l a t i o n s , as i s required for a t i g h t l y collimated beam traveling perpendicular to t h e cavity axis, as has already been discussed. Figures 12 and 13 i l l u s t r a t e the very important result t h a t the spacings L-, i > 0, are independent of p. This r e - s u l t is useful in the present analysis because i t w i l l permit us to consider the velocity and frequency dependence of the Doppler s p l i t t i n g independently from the s p a t i a l d i s t r i b u t i o n of the gas expansion. In the frequency domain t h i s inde- pendence i s not as clear-cut, since the i n i t i a l part of the time domain spectrum up to the f i r s t zero crossing may change the frequency s p l i t t i n g , or even eliminate i t altogether, as in the case of large p . Calculations show that the frequency s p l i t t i n g in the power spectrum agrees to within 10% with the inverse average zero spacing 1/L of the time domain of p < 2. To f i r s t order for p < 2 the i n i t i a l time domain lobe only 00 Figure 13. The envelope function Eq. (131) calculated for a density 4 0 2 distribution (cos ' 0)/r and otherwise the same parameters as Fig. 12. 87 O Ixl 2 88 fills in the center of the power spectrum pattern. For strong, well resolved spectra, this splitting invariance is unimportant, since the peak midpoint position is always invariant under all changes in spectrometer conditions. For very weak, complex spectra, however, where it may be difficult to deconvolute the Doppler splitting except after taking many spectra, during which time the nozzle source conditions may change, it is important. In Fig. 14 we show the power spectrum of the curve in Fig. 12. The envelope in Fig. 12 was multiplied by an arbi- trary cos wt term before it was transformed, to displace the power spectrum along the frequency axis. The peaks in Fig. 14 are separated by 2 x 10.23 kHz, which may be compared to the value 2 w o / c of 2 x 13.33 kHz for v = 10 GHz, and v = 4*10 cm/sec. The splitting may be understood intuitively as follows. Consider the electric field emitted by a single narrow beam traveling on a radial path centered in the zy-plane, away from the nozzle at angle 0 with respect to the z-axis. lineshape The in this case may be obtained from Eq. (131) by dropping the cosp9 term in that expression and restricting the angular integration to the differential solid angle occupied by the beam. The lineshape for the special case of 0 = 0 ° has already been given in Eq. (126). When 0 = 90° we obtain a result similar to Eq. (130). Numerical studies show that for arbitrary angle 6 the lineshape is closely 00 Figure 14. Power spectrum obtained by multiplying the envelope in Fig. 12 by an arbitrary term cos w^t and Fourier transforming. The peak separation is 20.46 kHz. FREQUENCY O 91 approximated by the expression vQsin6 t J E(t,0) cc cos(wQt + (w-wQ)t1) cos(w Q — ) i('ce0Tp^ i where the v (132) term in Eq. (130) has been replaced by the velo- city projection v sin0 onto the cavity axis. Here we omit the T 2 dependence, and any loss of signal that occurs as molecules move out of the cell. Expression (132) is valid for values of the parameter K E T <_ 4 and is not affected by removal of the beam waist terms in Eq. (131) . In a time domain experiment we excite and detect the emission from all molecules in the cell simultaneously. The total emitted field is obtained by adding contributions from all 0, weighted by the angular density distribution. We continue to deal with the problem where all molecules lie near the zy-plane, since inclusion of an azimuthal angle makes the subsequent algebra impossible to carry through analytically. One then obtains for the total emitted electric field E ( t ) <* dJ^KCQXp) C O s ( w o t + (w-W Q )t ; L ) X //2coB(a»0V° 0 = d(j) J-^KEQX IP" 1 ? 11 p/2 ( ^ *) S ^n 9 t) cosp 0 d 0 ) cos(wQt + (u-w0)t1) J D / ? U n ^ t) p / 2 ° c . (133) x (134) 92 Note t h a t t h e p a r a m e t e r KE-T a f f e c t s o n l y t h e a m p l i t u d e , t h e s h a p e , of t h i s r e s u l t . The c h a r a c t e r i s t i c not functional form J (X) D P // ^2 xP/2 (135} U 3 5 ) containing a l l the information r e l a t i n g t h e s p a t i a l t i o n of m o l e c u l e s i n t h e e x p a n s i o n t o t h e o b s e r v e d distribulineshape i s i n e x c e l l e n t q u a l i t a t i v e agreement w i t h t h e more e x a c t , 3-dimensional numerical r e s u l t already discussed. We a l s o mention t h a t t h e c u r v e i n F i g . 12 b e a r s a c l o s e r e s e m b l a n c e t o ( s i n x ) / x , t h e F o u r i e r t r a n s f o r m of a s q u a r e a b s o r p t i o n curve in frequency space. We h a v e s t u d i e d t h e b e h a v i o r of t h i s f r e q u e n c y separation Av as a f u n c t i o n of t h e gas m i x t u r e s and of t h e f r e q u e n c y of the molecular t r a n s i t i o n . I n F i g . 15 and Table 1 we show r e s u l t s f o r t h e s p l i t t i n g of t h e J = 0 •> 1 l i n e of OCS a t 12163 MHz i n v a r i o u s m i x t u r e s o f He, Ar, and Kr. The m o l e - c u l a r s p e e d v for a g i v e n m i x t u r e should b e d i r e c t l y -1/2 tional to m propor- ' , where m is the concentration weighted mole- cular weight of the mixture. The observed peak separations -1/2 for the four different gas mixtures follow the Av = m ' rule quite closely. This behavior can be determined from the integral expression in Eq. (124) by inspection, since v Q and t never occur in the integral in Eq. (124) except as the product v t. 93 i 1 i 1 T 100 d/ N V X 50 < o7 0,0 i 1 1 1 1 0.1 0,2 0.3 0.4 0.5 /*/m Figure 15. Splitting of the OCS J = 0 -*• 1 transition at 12163 MHz as a function of the inverse square root of the concentration weighted molecular weight of the gas mixture for four different mixtures. splitting values are given in Table 1. The Table 1. Splitting of the OCS J = 0 -> 1 transition at 12163 MHz as a function of gas mixture.7 Mixture Av (kHz) a. 96% Kr, 4% OCS 27.7 + 2 b. 96% Ar, 4% OCS 35.2 + 1 c. 71% He, 25% Ar, 4% OCS 60.5 + 2 d. 96% He, 4% OCS 89.5 + 2 mirror separation 47.3 cm. 95 The complementary experiment is to study the splitting as a function of frequency for a given molecule in a constant gas mixture. Some experimental results are shown in Fig. 16. We find that Av is directly proportional to v. lated from Eq. (131) are listed in Table 2. Results calcu- In addition to adjusting the frequency in this computation, it was necessary to adjust the mode number q so that the mirror separation remained approximately constant. As the mirrors are brought closer together, with the nozzle remaining positioned at their midpoint, they constrain the geometry of the expanding gas in such a way that the splitting becomes dependent on that separation, falling to zero as the TEM Q 0 0 mode is approached, and asymptotically approaching a constant value in the limit of large separation. Because of the direct proportionality between Av and v , and between Av and v, and because the frequency splitting is nearly independent of p, as has been noted above, we may use the Av values from Table 1 to determine the speed v of OCS in the various rare gas carriers. From our discussion of Fig. 14, the relationship is „ V _ Ave 13.33 n (136) o " T V 10723 ' from a 48 cm mirror separation. ,M Values for speed v obtained in this manner are listed in Table 3. For comparison we also list values for the terminal speed v_, obtained from the . 23 expression 96 N X < Figure 16. Doppler splitting as a function of frequency for several different molecules. Gas mixture in each case was 4% halogen ocid in the corresponding rare gas carrier. 97 Table 2 Calculated frequency splitting Av as a function of v. number Frequency v (GHz) Av (kHz) Av/v d (cm) 17 6 12.45 2.08 45.0 18 6 13.30 2.22 47.5 24 8 16.95 2.12 46.9 25 8 17.60 2.20 48.8 30 10 22.00 2.20 46.5 31 10 21.75 2.18 48.0 37 12 26.15 2.18 47.5 44 14 29.75 2.13 48.2 <3 a Calculated for v Q = 4*10 4 cm/sec. 98 Table 3 Speed of OCS at 4% concentration in various carrier gases. Carrier Gas v„, (cal'd)a ,*,«4 (xlO cm/sec) Speed (expt'l) _/_x (xlO,4 cm/sec) He 14.38 + 1.5 13.9 Ar 5.66+0.6 5.4 Kr 4.45 + 0.5 3.8 r = fe_ kT\ 1/2 , where Y = T I^Y-1 m J T = 293K. c p/ c v = 5/3, a n d 99 V ( T Y -1 "m} (137) , where v-, is the speed far from the nozzle that would be attained in the limit of zero temperature of the expanding gas. Here T is the gas temperature before it reaches the nozzle, andy = C p / C v = 5 / 3 . Values for v Q and v are nearly identi- cal, indicating that the OCS molecules apparently travel at the same speed as the rare gas atoms. Systematic errors in this calculation include neglect of transit through the cavity, the simplified cavity normal mode form, and small differences in the apparent splitting as measured in the time and frequency domains due to the initial lobe in the time domain envelope. Having established that Eq. (131) is capable of explaining certain general properties of the experimentally observed lineshapes, we now analyze some specific time domain spectra. The following data was taken in the TEM Q0 __ cavity mode at a Q of 6300. A four-percent mixture of OCS in Ar at room temp- erature and at a source pressure of between 2.13 atm. and 2.07 atm. was expanded through a 0.020" diameter nozzle opening. Input microwave power was 1.5 mW. Using the cavity 23 equation E =1 128 R Q_2 \ 1/2 %- , (138) • ( 2 when R = 1.5 mW, Q L = 6300, V = 12163 MHZ, Q c l = 3 Q^, W Q 2 -3 30 cm , and d = 46 cm, one has E Q = 5.9*10 esu, and eQ = 100 —3 fi —1 E Q /2 = 3*10 esu. This gives KBQ - 2.8*10 sec . Data was taken at a frequency offset Av = 182 kHz, giving Aw = 1.1*106 sec , and 2 =0.167 « (£)- 1. (139) For the 0.5 usee pulse time used here, <eQT 4 v = 6*10 = 1.4, and using -1/2 cm/sec from Table 2 for Ar, (kvice ) = 1.52 ys. We are therefore operating strictly within the boundaries set by Eq. (116). The pulsed valve is opened by a square voltage pulse that operates a solenoid mechanism. We define the delay time A as the interval from the time the leading edge of this voltage pulse reaches the valve, to the time that microwave power is applied to polarize the gas. A plot of observed signal-to- noise for the ArHCl van der Waals complex as a function of the delay A, is shown in Fig. 17. After a short interval, during which the valve opens and molecules travel to the center of the cavity, the signal rises sharply, then trails off slowly after the main group of molecules has passed through the cavity. From this graph it is apparent that for short times after molecules reach the cavity, the properties of the expansion are probably undergoing major changes on a timescale of milliseconds or less. The spectra are shown in Figs. 18-21. are connected by straight lines. Digital points Digitization time is 128 I-" o Figure 17. Observed signal from an ArH CI J = 2 •>• 3 quadrupole hyperfine resonance at 10069 MHz as a function of the delay A. between the leading edge of the square voltage pulse that opens the nozzle value, and the microwave polarization pulse. SIGNAL (arbitrary) o m 5 Cfl CD O 30T o Figure 18. OCS J = 0 -*• 1 signal taken at a delay A of 3.6 msec. A 4% mixture of OCS in Ar was expanded through a 0.020" diameter flat plate orifice. The superimposed curve is calculated from the envelope function Eq. (131), multiplied by an exponential e~ ' 2. For A = 3.6 msec we find COS-0 '50 and T 2 = 160 ysec. The signals shown in Figs. 19-21 were taken under identical conditions to this one, except at different delays. 10-+ 105 106 107 ysec/point. The four spectra were taken at delay times A 3.6 msec, 4.2 msec, 4.5 msec, and 5.0 msec. of Lineshape dis- tortion was minimized by using a low-Q mode to eliminate ringing, and by using wideband signal processing components. Superimposed on each signal trace is a curve calculated from Eq. (131). Each fit has three adjustable parameters: p, which is determined solely by the ratio L /L, , v , and T 2 < In all cases we obtain v ~6*10 cm/sec, except Fig. 21 which was arbitrarily fit using this value. We find that the molecular distribution in the pulsed beam, as characterized by the single parameter p, varies from -0.5 at A. = 3.5 ms, to p >_ 3 in the later stages of the expansion. A negative exponent indicates a depletion of signal from the axis of the beam. We measure only the population difference density AN between the OCS J = 0 and J = 1 rotational levels, of course, and not the particle density. Since the molecular distribu- tion appears in Eq. (131) only in the integrand, the observed lineshapes cannot be regarded as an extremely precise test of that distribution. Nevertheless, the calculated curves do reproduce the data quite well, and the general trend in the distribution from on-axis depletion at early times, to high directivity at large times is well established. The values for T 2 obtained here must be regarded as a lower bound for this parameter because of the omission of the transverse motion terms in going from Eq. (124) to Eq. (13]). Note that T 2 remains nearly constant for all four spectra despite the severe damping apparent in the late spectra. 109 D. Lineshapes Taken with Alternate Nozzle Geometries Up to now all discussions of the pulsed beam experi- ments refer to the standard spectrometer configuration with the gas nozzle positioned at z=h (see Fig. 9 ) , exactly midway between the mirrors and pointed directly at the center of the cavity. We now consider experiments in which the position and orientation of the nozzle is changed. The first set of experiments involve a rotation of the nozzle in the zy-plane through an angle 9 with respect to the (-z) axis, keeping the nozzle opening fixed in space at (0,0,h), as shown in Fig. 22. Signals were recorded for the J = 0 -»• 1 transition in OCS at angles ranging from 8 = 0°, the standard configuration, to 90°, for three different rare gas carriers. Source and nozzle conditions were 2.4 atm of 4% OCS at 293K expanded through the 0.020 inch diameter flat plate orifice. Results are summarized in Table 4. Referring to the top row of figures for each gas, note that the Doppler splitting is virtually independent of nozzle orientation for all angles 0° — < 0o — < 90°. We have made modifications to the density term in Eq. (29) appropriate for the tipped nozzle experiment and find that the calculated Doppler splitting for distributions cosp(0-0 ) up to at least p - 2 vary by less than 10% in this range 0° <- 0 <_ 90°. The calculations are therefore consistent with the experimental results, which were taken with short delay times A*. t o keep the Doppler peaks well resolved. For discussing the lower set of Geometry and coordinates used for the tipped nozzle experiments. Our results show that the gas density distribution for OCS in He can include a cone structure superimposed on the near-isotropic background. Table 4 Tipped Nozzle Doppler Splitting Data for OCS J = 0 •> 1 Transition at 12163 MHz. * Angle 8 Splitting (kHz) 50° 70° 0° 37° Gas carrier Kr 24.9 + 1 27.7 + 1 Ar 40.0 + 1 42.7 + 1 „ T 3 12 He 90° O 9 0 + 4 4 2 + 1 4 0 + 1 40+3 90+4 92+3 99+5 +3 102 +8 35 +6 4% OCS mixtures at 2.36 atm. source pressure. 0 + 8 112 splitting values listed at 37° for Ar and He, and at 50° for He, we refer to Figs. 23 and 24. In Fig. 23 we show a power spectrum taken for OCS in He at a nozzle angle of 37°. The exact 0 -»• 1 frequency, v , is indicated at the mid-point of the two sets of peaks. The outer two peaks, separated by 102 kHz, result from the isotropic, or near isotropic cosp(0-6 ) distribution in the beam, as has already been discussed. The inner peaks, separated by 33 kHz, and again symmetric about v are only observed in He at this 37° nozzle orientation. In Fig. 24 we show a power spectrum taken under conditions identical to those of Fig. 23, except with the nozzle now at 50°. Evidently the two inner peaks from Fig. 23 have merged. The two outer peaks remain stationary with a 103 kHz separation. No sets of inner peaks have ever been observed with the nozzle at 0°. The obvious explanation for the orientationdependent inner structures, considering the azimuthal symmetry of the nozzle about its own axis, is a cone with apex halfangle a, chosen such that 2 sin(a-37°)xl3.7 ' 10 cm/sec x 12163 • 106sec 3 . 10 cm/sec _ 35 kHz (140) giving a = 55°. At 0 = 50° then, we would expect a splitting of 10 kHz, which would appear as a single peak at v , as in Fig. 24. It is useful to note that, referring to Eq. (136), 81x1-1 TJtH " 50 ° ' <141> to Figure 23. Power spectrum obtained from OCS J = 0 •*• 1 with 4% OCS in He at the nozzle angle 9 = 3 7 ° . o frequency v The OCS transition is located at the exact center of the peaks. The outer two peaks, separated by 102 kHz, are always present for OCS in He at any nozzle angle 0 . The inner set, separated here by 33 kHz, take on this splitting value only for 9Q = 37°. FREQUENCY 115 V 0 FREQUENCY Figure 24. Same gas mixture as that used for spectrum in Fig. 23, except now the nozzle is tipped at angle 8 = 50°. The two outer peaks are still present, separated by 102 kHz, but the inner two peaks are merged. 116 so that when present, the cone does not produce additional complication in Doppler pattern when the nozzle is positioned at 0°. This cone structure is most pronounced in He, although even in He not present in every gas pulse. We have observed weak inner structures in Ar at 9 = 37° as indicated in Table 3, but these are small in comparison to the outer set. No such structures have ever been observed in Kr. We have modified expression (131) to calculate lineshapes for a thin cone with a = 50° at various nozzle orientations 0 . We show one such time-domain envelope for 0 = 4 0° in Fig. 25. Here v Q = 4 • 10 cm/sec, v = 10 GHz, and q+1 = 30. The period of 49.6 ysec is nearly identical to the 48.9 ysec expected for a narrow beam traveling at this speed at 50° with respect to the vertical axis. Consideration of the beam waist terms in Eq. (131) indicates that the cavity discriminates against signal from the sides of the cone, effectively producing a geometry not greatly different from a narrow beam. Note that L /L, = 0.5 in Fig. 25. We find that when a = 50° the Doppler splitting scales with 0 Q as sin(ct-0o)/ as has been assumed in Eq. (140). In summary, the experiments listed in Table 3 indicate that the Doppler splitting is due primarily to an isotropic or near isotropic background, but that a cone structure with apex half-angle 55° may be formed in He, and to a much lesser extend, in Ar. This cone structure should not affect the splitting pattern in the normal 0 = 0° configuration. Figure 25. Envelope function Eq. (131) calculated for a cone gas 4 distribution with 8^ = 0° and a — 50°, v = 4*10 cm/sec, o o v Q = 10 GHz. 118 We finish by considering a second alternate nozzle configuration, this one with the nozzle placed just behind the center of one mirror at (0, -d/2, 0 ) , with the beam directed out along the cavity axis. In Fig. 26 we show an envelope calculated for p = 0.5, q+1 = 32, v = 10 GHz, and v = 4*10 cm/sec. This should be compared to Fig. 12 which was calcu- lated using these same four parameter values, but with the nozzle at its normal position. We find that in going from the perpendicular to the on-axis position the splitting changes from 20.5 kHz to 26.7 kHz, which is just equal to 2 v v/c. The dephasing time is greater in the on-axis case, as is expected from considering again the effect of the beam waist. Experimentally, for 84 KrH 35 Cl J = 1 -> 2 at 4802.9 MHz we find splittings of (10.2 + 2) kHz and (11.7 + 2) kHz for the perpendicular and on-axis experiments, respectively. Although the on-axis arrangement should provide resolution superior to the perpendicular case, we have not implemented it because of practical problems related to the input coupling of the microwave radiation. Up to now we have treated the Doppler splitting as a tool for studying the gas dynamics. The molecular spectro- scopist however, is more intent on eliminating the phenomenon rather than using it for this purpose. Several suggestions for this have already been mentioned in the paper. TEM In the cavity mode, with only a half-wavelength between the mirrors, the splitting will disappear. Implementing this to o Figure 26. Envelope function Eq. (131) calculated for a gas distribution (cos0 5 0)/r2 with the nozzle positioned at (0, -d/2,0) and pointed along the cavity axis. The envelope was calculated for the same set of parameters as used for Fig. 4. 121 122 method in the Fabry-Perot cavity at microwave frequencies, where the dispersion relationship is given in Eq. (118), is obviously impractical. The key idea here however, is the half-wavelength, and by choosing a different type of cavity that could be operated near cutoff, the splitting could be 36 eliminated. By considering the discussion leading to Eq. (134) it is clear that the splitting is not restricted to standing wave structures, and cannot be removed by carrying out the experiment in a traveling wave structure, as for example, by using a set of microwave horns. The idea of restricting the beam so that all molecules travel perpendicularly to the radiation field has been suggested in several discussions. We have tried this experiment by placing a skimmer in front of the nozzle. Our skimmer was a metal sheet with a rectangular slot that permitted molecules to travel only in the xz-plane (see Fig. 9 ) . An OCS J = 0 •*• 1 line obtained with this arrangement is shown in Fig. 27, taken under conditions nearly identical to those used for the spectrum in Fig. 12, except that a 20% OCS mixture in Ar was needed just to obtain a reasonably strong signal. The severe loss in signal prevented us from ever using a skimmer while studying new resonances. Although we have been unable to eliminate the Doppler splitting in any general manner, the direct proportionality between Av and v can be exploited by taking rotational spectra at sufficiently low frequencies that the two Doppler peaks l-> to Figure 27. Time domain signal from OCS J = 0 •»• 1 converted at 0.5 ysec/point. The spectrum was obtained with the geometry of Fig. 1, but with a skimmer in place that nearly eliminated molecular motion along the cavity axis. Very little doubling structure remains on this line. The envelope is instrumentally distorted at short times. 124 s . o o o 00 o CO O w <0 JJJ ? a I 125 merge. As will be discussed in the next chapter, this technique, which has several other important advantages, has enabled us to assign rotational transitions for approximately ten different molecules whose spectra are complicated by the presence of two nuclear quadrupole coupling interactions. In those cases when Doppler doubling is unavoidable, it is possible, as shown in Figs. 18-21, to select a beam dis tribution that gives two very sharp, well resolved peaks. For the reasons discussed above, these peaks can be regarded as fixed. 126 Chapter IV. A. Rare Gas Nuclear Quadrupole Coupling in van der Waals Molecules Introduction Van der Waals molecules are combinations of atoms and ordinary small molecules held together in part by atom-atom or atom-molecule attractions that are hundreds of times weaker than ordinary chemical bonds.37 '38 Some examples are Ar 2 , ArXe, or any combination of two rare gas atoms, pairs of ordinary molecules such as (H 2 ) 2 , (N0) 2 , and rare gasmolecules pairs, all of which have well defined electronic, vibrational, and rotational states that can be characterized by experimental and theoretical techniques. of the thermodynamic and transport Although studies properties of gases have for many decades given indirect evidence for the existence of many van der Waals complexes, it has only been within the last fifteen years or so that detailed structural and physical information about these complexes have become available. Most of this information comes from scattering experiments, infrared studies, and most importantly, pure rotational, or microwave spectroscopy. Experience indicates that it is possible to form a van der Waals molecule from virtually any two ordinary molecules simply by mixing the two species together in the gas phase, along with a rare gas propellant if necessary, and expanding the mixture through a small orifice into a vacuum. The measurement of nuclear quadrupole coupling constants has long been recognized to be a powerful tool for obtaining 127 i n f o r m a t i o n a b o u t t h e d e t a i l e d a r r a n g e m e n t s of e l e c t r o n i c w a v e f u n c t i o n s n e a r t h e n u c l e i b e i n g s t u d i e d . 39-42 We r e p o r t 83 t h e measurement and a n a l y s i s of t h e 83 quadrupole coupling constants in 1 31 3^ XeD Kr and 131 Xe n u c l e a r 35 83 KrH Cl, K D 35 c, d Cl. The rare gas nuclear quadrupole coupling is ob- served through the coupling of the nuclear angular momentum to the rotational angular momentum of the molecule as measured in the vibrational ground state, pure rotational transitions of KrH(D)ci and XeDCl. The rotational spectrum is observed by using the newly developed experimental method of pulsed Fourier transform microwave spectroscopy in a Fabry-Perot cavity with a pulsed supersonic gas expansion. Because the nuclear quadrupole coupling constant of the closed shell free rare gas atom is identically zero, the effect measured here arises from a distortion of the spherical symmetry of the atom upon complexation. We thus have a sensitive probe for studying small changes in the electronic environment of the rare gas nucleus that occur when the rare gas atom binds to the hydrogen halide molecule. Since our report 23 of the first measurement of a rare gas nuclear quadrupole coupling constant in the 83KrHE van der Waals molecule, we have reported similar measurements in "^eH^Cl J and °KrDF. These studies have established, based on calculated values of the Sternheimer quadrupole shielding constants 45-47 for Kr and Xe, that most of the magnitude of the rare gas quadrupole coupling constant in 128 each of these molecules can be attributed to quadrupole shielding effects occurring in the rare gas atom in the presence of the electron field gradient of the partner hydrogen halide 83 molecule. By combining our experimental results for KrH(D)F with the 83KrH(D)35Cl measurements reported here, we establish this result empirically for Kr, thereby obtaining an experimental estimate for the Sternheimer parameter for this atom. We also report an improved measurement of the Xe nuclear quad131 3^ rupole coupling constant in sults for XeH(D) XeH Cl and show that our re- Cl are consistent with the interpretation used for the Kr containing species. B. Experimental The experimental technique used here is the method of pulsed Fourier-transform microwave spectroscopy carried out in a Fabry-Perot cavity with a pulsed supersonic nozzle gas ex20 pansion. The source gas for the nozzle consisted of mix- tures of 0.3% to 1.1% HC1 or DC1 with 70% to 80% He and 20% to 30% Xe or Kr at pressures ranging from 0.17 to 2.5 atmospheres. A pulsed solenoid valve with a 0.040 inch diameter orifice plate was used to pulse the gas into the cavity at repetition rates of about 1 Hz. Careful optimization of the gas mixture and source conditions was crucial for obtaining adequate signal-to-noise. When the expanding gas, which is now at temperatures of 1-10K and contains large numbers of weekly bound molecular complexes, passes between the FabryPerot mirrors a TT/2 microwave pulse is used to polarize all 129 rotational transitions within the bandwidth of the cavity. After the polarizing radiation dies away these polarized rotational transitions emit coherently at their resonance frequencies. This coherence decays because of Doppler-effects, molecular collisions, and the molecular transit out of the cavity. The decaying molecular emission is coupled out of the cavity and detected in a superheterodyne receiver by mixing with a local oscillator. The signal is digitized and stored to be averaged with the signals from repeating gas pulses. After an adequate signal-to-noise ratio is obtained the time domain signal is Fourier transformed into the frequency domain. In the case of a single resonance frequency, the gas flow from the nozzle and the resultant Doppler effect gives rise to a symmetric doublet in the frequency domain. The gas dynamics of the pulsed nozzle, the molecular polarization and subsequent emission processes, and the characteristic Doppler doubling phenomenon have been described in detail in Chapters II and III. Most of the data presented here for molecules containing two interacting nuclear quadrupole moments was taken at 48 frequencies below 8 GHz. Because the Doppler splitting in the frequency domain is proportional to the resonance frequency, at these lower frequencies the two Doppler peaks become merged. This simplifies the spectrometer signals and doubles the signal-to-noise ratio. Furthermore, the quad- rupole splitting patterns are simplified and better resolved 130 at lower values of the angular momentum quantum number J. The deuterium nuclear quadrupole splittings remained unresolved, causing the resonance lines to broaden. In Fig. 28 we show a power spectrum containing the J = 1 -> 2, F^ = 5/2, F 2 = 7, F = 8 -»• P1* = 7/2, F2* = 8, F' = 9, and F1 = 5/2, F 2 = 6, F = 5 + F ^ = 7/2, F 2 ' = 7, F* = 8 lines of 83 KrD 3 5 Cl. The spectrum was obtained by averaging 30 emission signals, weighting the resultant time domain record with a digial exponential filter, and taking the power spectrum. In Fig. 29 we show a portion of the calculated pattern in the frequency 83 3 *5 domain of the KrD Cl J = 1 •*• 2 transition. The envelope 83 35 was generated from the measured values of the Kr and Cl nuclear quadrupole coupling constants, and the calculated projection of the deuterium quadrupole coupling constant in free D 35Cl. The nine resonance lines reported here are represented as vertical bars. The assignments are identified in Table 6. C. Rotational Spectra and Spectroscopic Constants for the Rare Gas Hydrogen Halides 83 35 The rotational energy levels for KrH(D) Cl and 131 35 XeH(D) Cl are determined by the K=0 symmetric top Hamil- tonian E H(J ' XCV XR' V •*--*• 2 I F l ' F 2 'F ) = B o J 3 -»• " V , •*••*• 2-> 2. 2li(2Ii-l)(2J-1)(2J+3) (142) ' H Figure 28. A power spectrum showing the J = 1 •*• 2, F, = 5/2, F 2 = 7, F = 8 -*- F^ = 7/2, F2' = 8, F' = 9, and F^^ = 5/2, F 2 = 6, F = 5 -*• F^^' = 7/2, F2' = 7, F' = 8 lines of 83 KrD35Cl at 4.7 GHz. Both the deuterium nuclear quadrupole splittings and the Doppler splittings are unresolved. This spectrum was taken in approximately one minute. T 4752.700 • n 4752.800 1 r 4752.900 4753.000 (MHz) to to CO Figure 29. A calculated envelope and the measured transitions for the 83KrD35Cl J = 1 + 2 multiplet. The assignments are identified in Table 6. 134 N X 135 where B~o = 7? z (B„ o + C O) is the rotational constant, DJ is the centrifugal distortion constant, and X-j and !••» i = 1, 2, 3, are the chlorine, rare-gas, and deuterium nuclear quadrupole coupling constants and nuclear spins, as necessary. Matrix elements of this Hamiltonian were calculated to first order in the basis * + Icl = F r F-L + I R = P2/ •*• •*• (143) -*• F 2 + I D = F, where the subscript "R" indicates either Kr or Xe. I i*j1 XeH Cl data was fit with energy levels calculated from Eq. (142) only to first order. Small second-order chlorine nuclear quadrupole effects in the and The O C XeD KrH(D) Cl and 131 XeD 3 5 Cl Cl spectra were taken into account by adjusting the appropriate energy levels obtained from the first-order evaluation of Eq. (142) by amounts corresponding to the second order corrections obtained by direct diagonalization of the single chlorine quadrupole Hamiltonian. Except in the 83 35 case of the F,=3/2->-F1,=3/2 series of transitions in KrH Cl, where a second order shift of 9 kHz did effect the fit to Cl X / second order shifts were 4 kHz or less. Calculated line positions for all deuterium containing molecules were obtained as weighted averages of all unresolved components estimated to contribute to the corresponding 136 measured frequencies. Since some of these lines consisted of up to ten unresolved hyperfine lines, the assignments are in certain cases identified only by the first, last, and largest contributing components. The observed spectra, their assignments, calculated frequencies, and frequency differences, are listed in Tables 5-8. The spectroscopic constants used to fit the data are shown in Tables 9 and 10. of The spectroscopic constants KrHC N, taken from a report on the microwave spectrum and structure of this molecule,49 are listed in Table 11. The structural parameters, R , the distance between the rare gas nucleus and the center-of-mass of the hydrogen halide, and 0, the rare gas—center of mass of the hydrogen-halide—hydrogen (deuterium) angle, are listed for KrKCl and XeDCl in Table 12. These structural parameters are shown in Fig. 30. The angle Y between the a-inertial axis and the hydrogen-halide bond is 23 obtained in the usual way by assuming that the measured value of xCl is given by the projection of the chlorine nuclear quadrupole coupling constant x Q Cl of free H 35 Cl onto the a- inertial axis according to X C1 . X Q C1 ( 3 C O S ^ , ( 1 4 4 ) where the brackets indicate averaging over the ground vibrational state of the complex. If the hydrogen halide bond length r remains unchanged upon complexation, then R Q and 0 may be obtained from y and the rotational constant B . The assumption here regarding r, and the assumption that the 137 Table 5 Observed and Calculated Frequencies for J+J' 2 2F 1 2 1 10 1 10' 1 10 1 8 I± 2F ' 2F' Observed (MHz) 83 35 KrH Cl Calculated (MHz) Difference (kHz) f 4819.3741 4819.3689 5.2 1 8 1 1 8 1 3 12 5 14 4819.6873 4819.6885 -1.2 5 14 7 16 4819.9086 4819.9061 2.5 5 6 7 • 4819.9687 4819.9723 -3.6 5 12 7 14 5 10 7 12 4820.1545 4820.1524 2.1 5 8 7 10 4820.2222 4820.2217 0.5 3 8 5 • 4820.2581 4820.2459 12.2 5 6 7 8 3 12 3 10 4825.0474 4825.0454 2.0 3 12 3 12 4825.0872 4825.0878 -0.6 3 8 3 10 4825.4199 4825.4184 1.5 3 8 3 6 4825.4681 4825.4714 -3.3 3 10 3 10 4825.5646 4725.5705 -5.9 3 10 3 8 4825.5912 4825.5857 5.5 10 8 - 8^ 138 Table 6 Observed and C a l c u l a t e d F r e q u e n c i e s f o r J->J' U-7-U 2F 2F 2F 2F ' 2F ' 2F _1 _ ! —A " 2 1 2 a o b s e r v e d (MHz) 83 35 KrD' C l Calculated (MHz) Difference (kHz) 3 6 4 5 6 4 4752.2616 4752.2591 2.5 b 3 12 14 5 14 16 4752.4514 4752.4565 -5.1 c 5 4 6 7 2 4 4752.6426 4752.6366 6.0 d 5 14 16 7 16 18 4752.7641 4752.7656 -1.5 5 12 10 7 14 12l 5 12 14 7 14 16 ' 4 7 5 2 . 8 4 9 2 4752.8545 -5.3 5 10 10 7 14 12, 3 8 8 5 6 6> 5 10 8 7 10 4753.0087 -2.4 5 12 12 7 10 12. 5 10 12 7 12 U 5 10 8 7 12 10 ' 4 7 5 3 . 1 1 0 6 4753.1061 4.5 5 10 10 7 12 12, h 5 8 10 7 10 12 4753.1979 4753.2012 -3.3 i 3 10 12 5 12 14 4753.5214 4753.5171 4.3 e f g 8 • 4753.0063 139 Table 7 Observed and C a l c u l a t e d Frequencies for 131 XeH35Cl 2F X 2F 2 V 2F' Observed (MHz) Calculated (MHz) Difference (kHz) 5 6 7 8 3963.9586 3963.9587 -0.1 5 4 7 6 3964.0333 3964.0335 -0.2 5 8 7 10 3964.1764 3964.1749 1.5 3 6 5 8 3964.3899 3964.3911 -1.2 5 6 7 8 5944.9615 5944.9625 -1.0 7 6 9 5945.0682 5945.0703 -2.1 5 4 7 7 8 9 5945.1482 -0.7 4 9 "} 5945.1475 7 7 10 9 12 5945.2383 5945.2373 1.0 5 8 7 10 5945.3294 5945.3297 -0.3 : ) 140 Table 8 131 35 Observed and Calculated Frequencies for XeD Cl J-+J' 2F 1 2F 2 2F 2Fn ' 2F 0 ' 2F' Observed Calculated Difference (MHz) 1 (MHz) (kHz) 5 2 3 6 -4.0 8 5856.5740 5856.5780 7 7 6 4 9 7 6 -2.3 8 5856.7137 5856.7160 9 5 4 4 7 7 8 10 9 7 -3.7 8 5856.8119 5856.8156 8 9 7 4 4 9 7 10 -0.5 5856.9279 5856.9284 8 9 5 8 6 7 0.3 5857.0395 5857.0392 5 8 10 7 5 12.8 5857.0665 5857.0537 8 8 7 5 -5.6 5857.1040 5857.1096 2 4 7 5 3 4 8 6 7 5 -0.3 7807.2187 7807.2190 8 10 7 5 2 4 7 7 6 4 9 7 -5.5 7808.2831 7808.2886 8 10 9 7 8 8 9 9 8 10 11 6.5 7808.3202 7808.3137 9 8 8 11 9 6 8 11 -8.2 7808.3538 7808.3620 9 6 6 11 9 10 12 11 0.5 7808.3722 7808.3717 9 10 10 11 -1.2 7808.4308 7808.4320 9 12 10 11 -1.5 7808.4901 7808.4916 7 10 12 9 7 5 6 7 11 9 4 5 10 8 14 12 9 7 13 11 9758.9873 9758.9855 1.8 13660.7125 13660.7133 -0.8 Table 9 Spectroscopic Constants of KrHCl Isotope 83 KrH 35 Cl 83 K r D 35 c l B0-8Dj(MHz) Cl x (MHz) xKr(MHz) D X <MHz> 1204.84742(40) -29.238(45) 5.200(100) 1188.01053(60) -40.824a 7.192(100) 0.1135b a 83 3K 84 3^ Fixed at average of the KrD Cl and KrD Cl values from Ref. [23]. b3Fixed at t) 35 Fixed at the projection of the free D Cl deuterium quadrupole coupling constant. Table 10 Spectroscopic Constants of XeHCl Isotope B v(MHz) o ~ "' DT(kHz) ~J x (MHz) x (MHz) 132 XeD 3 5 Cl 974.50768(35) 3.4209(20) -44.800(40) 131 XeH 3 5 Cl 990.86302(32) 3.796(20) -34.76b -4.641(50) 131 XeD 3 5 Cl 976.11556(40) 3.4285(67) -44.780(200) -5.89(20) X (MHz) 0.1245* 0.1245* a 3s Fixed at the projection of the free D Cl deuterium quadrupole coupling constant. b Fixed at the value for C1 X in 129 XeH 3 5 Cl from Ref. [43]. 143 T a b l e 11 S p e c t r o s c o p i c C o n s t a n t s of KrHCNa Isotope 83 a KrHC l 4 N Ref. [49]. B -8DT(MHz) o J 1184.60700(20) XN(MHz) -3.2630(60) xKr(MHz) 7.457(50) Table 12 Structural constants of KrHCl, KrHF, KrHCN, XeHCl, and XeHF Isotope RQ(A) 0 (deg 83 KrH 35 Cl 4. 0824 38.07 83 KrD 35 Cl 4. 0652 31.03 83KrHF 3.6076 39.17 83KrDF 3. 5575 31.27 83 4. 5203 27.50 KrHC 14 N 131 XeH 3 5 Cl 4.2456 34.76 131 xeD 3 5 Cl 4. 2259 28.37 131 XeHF a 3..7772 35.7 131 XeDF a 3..7339 29.55 a Ref. [52]. H center-of-mass a-axis Cl A = Kr, Xe Figure 30. Structural parameters for the rare-gas hydrogen chloride series of van der Waals molecules. 146 change in x Cl from x 0 Cl is purely a geometrical effect and does not involve any changes in the electronic environment of the 50 51 chlorine nucleus can be tested experimentally, ' and are found to be correct for the rare gas hydrogen halides within experimental error. . . The structural analysis for KrHCN is 14 similar, using the projection of the coupling constant. N nuclear quadrupole The structural parameters for this mole- cule are shown in Fig. 31. D. Analysis The rare gas nuclear quadrupole coupling constant, X/ is determined by the product of the quadrupole moment Q of the rare gas nucleus and the electric field gradient, q, at the nuclear site along the molecular a-inertial axis due to all charges outside the nucleus, x isigiven by where e is the proton charge and h is Planck's constant. Since the quadrupole moments of the 83Kr and 131Xe nuclei have been determined to approximately 15% accuracy by atomic hyperfine structure measurements, the problem of evaluating x reduces to determining q. Direct calculations of this quantity from first principles are difficult, particularly in molecules with such small binding energies, and do not give a very physical picture of the origin of the results. We will follow here 3 9 '40 of trying to the more useful and conventional approach 1 ' understand the origin of the field gradient by using a few H a-axis oKr Figure 31. 9 A center-of-mass Structural parameters for KrHCN. 0 and y differ by 0.6 degrees. The angles 148 simple parameters characterizing the structural and electronic properties of the Kr and hydrogen halide subunits, and the structure of the van der Walls complex. The sources of possible contributions to q may be divided into the following categories: 1. Valence electrons of the rare gas atom, 2. Distortion of the closed shells of the rare gas atom by the hydrogen halide partner, 3. Charge distributions lying outside the rare gas atom. For atoms participating in ordinary chemical bonds, for example the halogens Cl, Br, and I in ordinary molecules, contributions from electrons in the uncompleted valence shell account for most of the observed field gradients. Because this contribution is identically zero in a free rare gas atom, we can expect it to be small in the very weakly bound van der Waals molecules, so that the ordinarily negligible effects of the second and third type can become important. Foley, Sternheimer, and Tycko have shown47 that the field gradient at the nuclear site in a closed shell system resulting from an external charge 2e e at a distance R from the nucleus can be written — j (!"¥„,) i R where R must exceed the radius of the atom or ion. The pro- portionality factor Yoo is specific to each atom or ion, and results from the interaction of the electrons in the closed shells with the perturbing charge. Since this is a first- order effect, this same direct proportionality will hold for an electric field gradient arising from any system of external charges. 149 83 23.44 131 35 43 In view of our results for ^KrHfDjF^'** and XeH JD Cl, we will begin by analyzing contributions to field gradients at the rare gas nuclear sites due to the direct and shielding enhanced field gradients arising from the first few electric multipole moments of the hydrogen halide bonding partners. The electric field gradient, q , at a point (R,0) along the radial direction outside a neutral cylindrically symmetric molecule is given by /cos9\ yP2(cos8)v ,P3(cos0)i qo = ^ 2 2 J i ) - 12Q ( - ^ 5 — ) - 20^-1-^—) (146) ,P 4 (COS0) y - 30$/ ? ) - ... where u, Q, J3, and $ are the electric dipole, quadrupole, etc. moments of the molecule. The brackets here indicate the ex- pectation values in the vibrational ground state. All coordi- nates and moments are referred to the molecular center of mass. Provided that the electronic wavefunctions of the rare gas atom and hydrogen halide do not overlap, the field gradient at the rare gas nuclear site can be written q - q0 ( 1 - Y J • (147) The contribution to x i s then given by X = ~ eq n Q(l-Yj w ° (148) The series for q in Eq. (146) was calculated out to the third term for KrHCl, KrHF, and KrHCN using the values of R and 0 listed for each molecule in Table 12, and the 150 multipole moments for H 35 14 Cl, HC N, and HF listed in Table 13, 14 and 15. The calculated values of -eqQ/h and the measured P X values are listed in Table 16. In Fig. 31 we have plotted Kr X as a function of -eq /h. We find a direct proportionality Kr "* Kr between x and -eq /h, with a predicted value of x of near zero in the absence of an external field gradient. The most immediate problem with this analysis is the question of the convergence of the multipole expansion. In these molecules the center-of-mass separations across the van der Waals bonds are sufficiently large that the respective electronic wavefunctions barely overlap. In 83KrD 35Cl for o example, the R of 4.06A compares to the van der Waals radii o o of 2.0 A and 1.80 A for Kr and Cl, respectively. That part of the perturbing Hamiltonian giving rise to the effect parameterized by Eq. (147) is given by 53 2 H pert = e( - R-> V cos9 a> 10OT (cos0 K ) 3ucos0, ^ — ^ + 6QP2(cos0j;)) — ^ T (149) 15$P,(cos0 K ) 4 R* 5 R° •*•.../ for an e l e c t r o n i n a r a r e gas sheet a t c o o r d i n a t e s ( r , 0 , a ij) ) (See Fig. 33) in the presence of a cylindrically symmetric charge distribution at a center-of-mass distance R, and o orientation 0fa. Taking R = 4.1 A, 0. = 31°, and using the 35 multipole moments for D Cl, we obtain from Eq. (149), in relative importance 151 Table 13 Molecular Properties of H 35 C1 and D 35 C1 H 35 C1 D X (MH Z ) a D 35 C1 0.18736 li(D) a 1.1085 1.1033 Q(DA) 3.74 (12) b 3.74d £2(DA2) 2.446° 2.446d $(DA3) 4.704° 4.704d a E. W. Kaiser, J. Chem. Phys. 53_, 1686 (1970). F. H. deLeeuw and A. Dymanus, J. Mol. Spect. 48, 427 (1973) C D. Maillard and B. Silvi, Mol. Phys. 40, 933 (1980). d 35 Fixed at corresponding H Cl value. 152 Table 14 Molecular Properties of HC u(D) a 2.9846 Q(DA) b 2.42(60) fi(DA2)C 4>(DA3)° a 14 N " 6 '366 6.422 A. Maki, J. Phys. Chem. Ref. Data 3_, 231 (1974) . b ° Average of calculated values of 2.12 DA, J. Tyrrell, J. Phys. Chem. 83, 2907 (1979), 2.03 D&, ref. [54], and an experimental value of 3.1(6) DA" measured for Hcl5N by S, L. Hartford, W. C. Allen, C. L. Norris, E. F. Pearson, and W. H. Flygare, Chem. Phys. Lett. 18, 153 (1968). c Ref. [54] . 153 Table 15 Molecular Properties of HF and DF a HF DF y(D) a 1.8265 1.8188 Q(DA) a 2.36(3) 2.32 n(DA2) 1.699b 1.699° °3 $(DA°) 1.804b 1.804° See Ref. [44]. *u D. Maillard and B. Silvi, Mol. Phys. 4_0, 933 (1980). °Values for DF assumed to be identical to those for HF. Table 16 Summary of Measured and Calculated Quantitites Related to the Rare Gas Coupling Constants in KrHF, KrHCl, KrHCN, XeHCl, and XeHF Isotope XR(MHz) eq(calc'd)/h(MHz/b) 83 KrH 35 Cl 5.20(10) 0.2629(402) 83 KrD 35 Cl 7.19(10) 0.3494(570) 83KrHF 10.23(8) 0.4987(575) 83KrDF 13.83(13) 0.6706(880) 7.46(5) 0.3971(1013) 83 KrHC 14 N 131 XeH 3 5 Cl -4.64(5) 0.2522(388) 131 XeD 3 5 Cl -5.89(20) 0.3166(504) 131XeHF -8.59 a 0.4578(570) 131XeDF -10.57 a 0.5616(710) a Ref. [52]. Ol Figure 32. The measured Kr nuclear quadrupole coupling constant values plotted as a function of the electric field gradient at the Kr nuclear site calculated from the first few electric multipole moments of the hydrogen halide bonding partner. The slope of this line is related to the product of the quadrupole shielding of the rare gas nucleus, and the nuclear quadrupole moment, and the intercept can be related to the amount of charge transfer occurring in these systems. 16 14 KrDF 12 10 83 8 X X KrHC,4N 83 6 KrD35C. 83 KrH35CI 4 0 -2 0.0 1 0.1 0.2 0.3 0.4 - e q (calc'cD/h 1 0.5 0.6 (MHz/b) 0.7 0.8 •A e center-of-mass -H R Figure 33. The coordinate system used to parameterize the multipole expansion of the energy of an electron in a rare gas shell at coordinate (r, 0 , 4> ) in the field of a cylindrically cl cl symmetric system of charge at a center-of-mass distance R. j- 1 tn 158 H pert = °* 86 + 1 '° + °* 13 - No experimental results are available for /p.(cos0)\ for i >^ 3, and algebraic calculation of these terms using arc 9 I/a cos ( (cos 0)0/ is not useful for i > 4. The behavior of these higher order expectation values can probably be estimated using hindered rotor wavefunctions 55 for the hydrogen halide portion of the molecule. When the dominant term in the angular potential in the region being sampled by the hydrogen atom is of the form l-cos0, the expectation values for P.(cos8) decrease monotonically to zero as i increases. using a potential for the form 113 cm For example, in KrD Cl, (l-cos0) with hindered 35 rotor wavefunctions to describe the D Cl subunit, those authors calculate P-(cos0) = 0.85, P,(cos0) 1 = 0.61, P,(cos0) = 0.37, P4(cos0) = 0.19. (151) The experimental results for the first two expectation values P, and P 2 are 0.88 and 0.61. Algebraic expressions for P, through P 4 yield 0.86, 0.60, 0.29, and -0.02, respectively for 0 = 31.0°. The behavior of the <P (cos9)/ values as suggested by Eq. (151), which contrasts sharply with the behavior of algebraically evaluated values of P.(cos6), i >_ 4, which oscillate between +1 and -1, is likely to make an important contribution to the very rapid convergence of the 159 series in Eqns. (146) and (149) . Using the value from (151) for ^P^tcosO)^ to calculate the next term of the series in Eq. (150) we obtain 0.012. Other uncertainties in applying Eq. (146) can be estimated more directly, and these are indicated as error bars in Fig. 32. Because the a-inertial axis in these molecules is nearly aligned with the center-of-mass axis, errors in ^P2(cos0)S will be negligible. An experimental value for ^P,(cos0)/ can be obtained from electric dipole measurements a s 5 6 ' 5 7 M(complex) = u(diatom){P±(cos0)) (1 + 2a, =|) , R o (152) where the hydrogen halide dipole moment is projected onto the a-axis, with a correction for the polarizability, a, of the rare gas atom. Because (/cos Y ) ) from Eq. (144) and (COSY/ as obtained from Eq. (151) differ by no more than 2-3% in KrH(D)Cl, and XeHCl, we have chosen to calculate (cos0/ algebraically from ( P 2 ( C O S Y ) } . Except in the case of KrHCN, errors in the hydrogen halide molecular quadrupole moment contribute uncertainties in q of 1-2%. Errors arising from the neglect of terms higher than PgtcosO) when calculating Eq. (146) have been estimated as 2OJ2<P2(cos0)) R6 o (153) With these qualifications considered we take up the physical interpretation of the curve in Fig. 32. Within the 160 uncertainties assigned to the calculated values of -eq /h, the effect is first order, as is expected from the estimated importance of dipole polarization 43 '47 and second-order quadrupole shielding effects.47 A linear least squares fit to the data in Fig. 32 yields a slope of (21.2+4) barns, and a frequency axis intercept of (0.376+1.6) MHz. The estimated uncertainties in the slope and intercept include the uncertainties in the values of -eq /h. The direct proportionality observed here between q and q can readily be understood qualitatively as arising from Sternheimer-type quadrupolar shielding occurring in the Kr atom. We emphasize however, that we do not know enough about the very short parts of the Kr and hydrogen halide interaction to be able to reliably convert this constant into Ye f° r K r * It is well known47 that in the idealized case of a closed shell system perturbed by a point charge at distance R, the shielding parameter has a functional form Y ( R ) / with limiting Y^* F o r moderately heavy + + — 47 systems such as Cs , Rb , and Cl , Sternheimer et al. found that 1~Y(R) attains approximately 70-80% of its asymptotic values of y(Rr=Q)-0, and Y^"* 00 ) = o value of 1~Y for R - 2 A, with most of the contribution at o R > 2A coming from the valence p shell of the ion. Although our proportionality constant is empirically well established, it is probably related in a complicated way to the wavefunctions of the hydrogen halide, and to the functional form Y ( R ) for the rare gas atom. It is still interesting to convert our slope into an effective shielding constant for Kr in the Kr-hydrogen halide systems studied here. Using Eq. (148) and the nuclear quad59 60 83 rupole moment of 0.27b for the Kr nucleus, we obtain an estimate of (-77.5+15) for the Kr shielding parameter, t subject to the serious qualifications outlined above, and not 83 including uncertainties of perhaps 15% or less in the Kr Estimates of Yoo f o r 44 Kr include a non-relativistic variational calculation including exchange effects for radial perturbations, giving ym = 62 nuclear quadrupole moment measurement. -68, a non-relativistic frozen-core calculation giving Yoo = -67, estimated by those authors to be in error by perhaps 15%, not including errors arising from neglect of relativistic fi 3 effects, and a calculation based on relativistic HartreeFock-Slater electron theory, giving Yro = -84. Relativistic 64 effects have been estimated elsewhere to account for approximately 7% of this last result. The vertical axis intercept of the line in Fig. 32 (0.376+1.6) MHz indicates that the combined effects of orbi65 tal overlap and charge transfer from the Kr valence p orbitals is small. We may obtain an estimated upper limit to the amount of charge transferred from the Kr 4p orbital aligned with the molecular axis by noting that the transfer of one electron out of this orbital would result in an electric field gradient at the Kr nuclear site along the 44 a-inertial axis of magnitude 750 MHz. The 1.6 MHz error 162 bound on the intercept then corresponds to a fractional elec_3 tron transfer of 3*10 , or less. This result is consistent with our previous estimation 43 '44 that charge transfer makes a negligible contribution to the quadrupole coupling constant in KrH(D)F. In Fig. 34 we have repeated the analysis used for the krypton data. We have included here measurement of the 131Xe nuclear quadrupole coupling constant reported by Baiocchi et al. for Table 16. XeHF and XeDF, which are also listed in Fitting a line to the data shown in Fig. 34 we obtain a slope of 19.17 + 4 barns, and a frequency axis intercept of -0.183 + 1 . 4 MHz. Using the 131Xe nuclear quad6 6fi7 rupole moment of -0.12b, ' we obtain an effective Xe shielding parameter of -158.8 + 33. for Xe are -138, 4 3 -130, and -177, Calculated value of Y ^ with relativistic effects accounting for about 19% of this last value. Using a conversion of 374 MHz/electron appropriate for Xe,43 we obtain an estimated upper bound of 0.004 electrons transferred from the Xe atom to the hydrogen halide, consistent with our measurement for the Kr system. The validity of these conclusions regarding change transfer rely heavily on the empirically observed linear relationship between the measured coupling constants and the multiple expression for the electric field gradient. Although the reasons for the apparent usefulness of a long range expansion at these very short distances between the rare gas and hydrogen halide molecules have never been Figure 34. The measured Xe nuclear quadrupole coupling constant values plotted as a function of the electric field gradient at the Xe nuclear site calculated from the first few electric multipole moments of the hydrogen chloride bonding partner. The XeHF and XeDF values are taken from Ref. [52] . 164 i T mm 1 ' "| 1 ^w _ O m O — a0> - <P — - .O •**< — x * • o X •**^ o o ^-** OJm o \ > X — ro | I i 5 «*—" ro v. -o d o «° K 3*P\ g X £ X JO CvJ *—. GO i (0 I I ( z HIAI) 9 x X CVJ I OJ O" 1 165 investigated, here or elsewhere, it is known that similar long range expressions are valid in two other cases. The values of electric dipole moments in these hydrogen-halide rare gas can be explained in a consistent number by projecting the dipole moment of the free hydrogen halide onto the a-inertial 19 68 axis of the complex, ' and including terms to account for the dipole and quadrupole induced electric dipole moment of the rare gas atom. Electric dipole moments calculated this way appear to be consistent with projection angles obtained from measurements of the halogen coupling constant Eq. (144). More evidence for the validity of applying ]ong-range expansions to calculate the properties of these systems comes from the calculations reported by Keenan et al. on the bending modes of the hydrogen halides. ' Expressions giving the attractive interaction between a polarizable sphere and a polarizable linear molecule are shown to give accurate values for the experimentally measured values of ^?2(cos0)} in these systems. E. 83 Measurement and Analysis of the Kr Nuclear 11 Quadrupole Coupling in 3c go Kr C1F The change transfer limits of less than 0.002 to 0.004 electrons reported here, and the long range electron dipole and bending mode analyses mentioned above have a direct bearing on a fundamental question underlying the studies of van der Walls molecules, namely "What determines the structure of a given van der Waals molecule?" Can the structures of these 166 systems be understood entirely in terms of what are normally regarded as long range interactions - electrostatic, induction, dispersion and exchange repulsion forces - or do some of the concepts applicable to more strongly bound molecules still remain important even in these very weakly bound systems? The strongest evidence suggesting that ordinary chemical interaction may be important in rare gas containing van der Waals systems come from the apparent tendency of the rare gas atom to act as a weak Lewis base when it interacts with a strong Lewis-acid. The most clear cut examples of this tendency 69 70 are the complexes between Ar and BFo and SO , in which the Ar binds to the B or S atom, giving C3v- symmetry. This 71 interpretation of van der Waals bonding has been proposed to provide a qualitative explanation for a large number of van der Waals complexes, although it is certain that in many of these cases the detailed calculations and measurements needed to establish the validity of the polarization explanations do not yet exist. In an attempt to exploit the empirical relationships 83 obtained in our study of the Kr nuclear quadrupole interaction in the rare gas-hydrogen halide system to a molecule 83 not a member of this class, we have measured the Kr 83 3 ^ quadrupole coupling constant in Kr ClF. Assignments of 82 35 the microwave and radiofrequency spectra of the Kr ClF, 84 T% 8fi "^R 84 37 Kr ClF, Kr ClF, and Kr 'ClF isotopic forms of this molecule have been reported by Klemperer and coworkers. 72 167 83 Kr containing isotopic forms of this mole72 cule were reported in that earlier work. The present study No results on any is particularly interesting because although ArUlF and KrClF are structurally very similar to the rare gas hydrogen halides (see Fig. 37), with the equilibrium position of the diatom at 0 = 0, and with the more electropositive Cl atom located at the bonding site, the force constant, k b , for the bending modes in both cases are from 5 to 7 times the force constants in the rare gas hydrogen halide system (see Table 17). Considered in terms of its polarizabilities and electric multipole momentum however, ClF is quite similar to the hydrogen halides. This apparent discrepancy has prompted suggestions 71 that this difference in bending behavior results from the substitution of a highly directional, primarily Cl 3p z , accepting orbital m ClF for the more isotropic, hydrogen ls-like accepting orbital in the hydrogen halides. Although our quadrupole coupling measurement does not rule out the possibility of a electron donor-acceptor interaction, it does show that any charge transfer in KrClF is probably below the level of a few thousandths of an electron, at least a factor of ten smaller than ordinary chemical amounts. 83 35 The Kr ClF spectra reported here were obtained with a 3.4% mixture of ClF in Kr expanded through a 0.040 inch diameter orifice. We use a Model 8-14-900 pulsed solenoid valve manufactured by General Valve Corp. Careful optimiza- tion of the gas mixture, source pressure, and orifice size Table 17 Comparison of Bending Force Constants for Several Rare-Gas Hydrogen Halide Molecules and Ar and KrClF kb (mdyne A) KrHF 0.0039 ArHCl 0.0015 KrHCl 0.0022 ArClF 0.022 KrClF 0.031 169 were crucial to obtaining adequate signal-to-noise. By working at 5.7 GHz we eliminated the Doppler splitting effect and obtained simplified quadrupole splitting patterns. In Figure 35, we show a power spectrum of the J = 2 ->• 3 , F, = 7/2 •*• F = 8 •+ F. = 9/2, F' = 9, and F± = 7/2, 7/2, F = 7, 2 •+ F.^ = 9/2, 9/2, F' = 8, 2 resonances at 5576.8 MHz. This spectrum is the transformed average of approximately twenty time domain records taken in about one minute. The rotational energy levels for 83 35 Kr ClF are determined by the K = 0 symmetric top Hamiltonian 1 ,-> ->- E H(J I -»•-)•-»• I P ' Cl' Kr l' _ -*2 P) °V -*4 D " JJ 2 yCl[3(fcl.3) +|(fcl*^) - ^ 2 ] (154) 2IC1(2IC1-1)(2J-1)(2J+3) + | d K r - J ) - lgrJ2] 2IRr(2IKr-l)(2J-1)(2J+3) y Kr[3(I K r .J) 2 , where B = 2"(B +C ) is the rotational constant, D is the disci Kr tortion constant, and x / X / an(^ I ci' '''Kr a r e t n e chlor:i-ne and krypton nuclear quadrupole coupling constants and nuclear spins. Matrix elements of this Hamiltonian were calculated to first order in the basis ->• ->- ->• J + I cl = FV (155) •+ F l ->+ X -+• F Kr - ' Second order chlorine nuclear quadropole effects were taken into account by adjusting the appropriate energy levels obtained 170 from the first-order evaluation of Eq. (153) by amounts corresponding to the second order corrections obtained by direct diagonalization of the single chlorine quadrupole Hamiltonian. The observed spectra, their assignments, calculated frequencies, frequency differences, and second-order corrections are listed in Table 18. The spectroscopic constants used to fit the data are listed in Table 19. Also listed here is the value for B obtained from B - 18 D_ by assuming a D_ of 1.86 kHz as determined earlier. 72 In Figure 36 we show a portion of the calculated pattern in the frequency domain of the 83Kr 35ClF Cl Kr — J = 2 -> 3 transition using the values of x / X / an<3 B 18 D_. Measured resonance lines are represented as vertical bars. The structure and force field of KrClF are discussed in detail in reference [72]. In Table 20 we summarize the struc83 tural information needed to analyze the Kr nuclear quadrapole coupling constant. The vibrationally averaged structure of KrClF in the vibrational ground state is nearly linear, as shown in Figure 37, with the chlorine atom lying between the other two atoms. The angle Y between the a-inertial axis and Cl —1 2 1/2 the ClF axis is obtained from the usual way. B~0 and Y are x as cos ((cos Y ) ) in then used to obtain the center- of-mass separation R and the structural angle 0. The Cl-F distance is assumed to be unchanged upon complexation from its value in free ClF. Table 18 Observed and Calculated Frequencies for 83 35 Kr ClF 2F -> 2FJ 2F' Observed (MHz) Calculated (MHz) Difference (kHz) 3 12 5 14 5567.6165 5567.6133 3.2 -94.9 1 10 3 12 5567.7178 5567.7197 1.9 48.2 5 14 7 16 5576.5876 5576.5874 0.2 5.5 7 16 9 18 5576.8415 5576.8457 4.2 21.9 7 14 9 16 5576.8721 5576.8681 4.0 21.9 7 4 9 4 7 12 9 12 7 12 9 14 5577.0151 5577.0103 4.8 18.9 5 8 7 8 5 12 7 14 5577.0715 5577.0776 -6.1 5.5 2F 2 } Second-order correction (kHz) to Figure 35. A power spectrum showing the J = 2 •*- 3, F. = 7/2, F = 8 -»• F^ = 9/2, F2" = 9 , and Fx = 7/2, 7/2, F = 7, 2 •»• F^ = 9/2, 9/2, F* = 8, 2 lines. The envelope is the transformed average of approximately twenty time domain records. Resolution in 3.906 kHz per point. The vertical bars represent the appropriate frequency measurements taken from Table 18. b 5576.800 5576.900 5577.000 (MHz) ^1 Table 19 Spectroscopic Constants for B Q - 18Dj (MHz) x C 1 (MHz) 929.2174(10) -141.519a a b x K r (MHz) 13.90(25) Fixed at the average of values for and 86 83 35 Kr ClF 82 Kr 35 ClF, B~o (MHz) 929.2509b 84 Kr 3 5 ClF, Kr 3 5 ClF from Ref. [72]. Obtained from B" - 18D_ using D_ = 1.86 kHz from Ref. [72]. o J J tn Figure 36. A calculated envelope and five of the seven assigned transition frequencies for the J = 2 -»- 3 transition in 83 KT35C1F. 5576.500 5576.700 5576.900 (MHz) 5577.300 177 Table 20 Structural Constants of 83 35 Kr ClF o R cm (A) 3.95525 0 (deg.) 8.62 Y (deg.) 8.11 a-axis center of mass R Figure 37. Structural parameters for 83 35 Kr ClF. The angle y has been shown in Ref. [72] to be acute. 00 179 The measured rare gas nuclear quadrupole coupling constant, x / i s related to the electric field gradient, q, along the inertial a-axis at the rare gas site by XR = - ^ , (156) where e is the proton charge, Q is the electric quadrupole moment of the rare gas nucleus, and h is Planck's constant. Kr 83 35 The present measurement of x in Kr ClF•follows previous determinations of x K r in 83 KrHC 1 4 N, 4 9 83 83 KrH(D)F, 25 ' 44 KrHC 1 5 N. 4 9 83 KrH(D) 35 C1, 73 We have also measured the nuclear quadrupole coupling constants in XeH(D) 131 Cl. Xe ' These previous measurements, all involving a rare gas atom bound to a hydrogen halide, have established that q, calculi lated from the known values of x a nd Q, is directly propor- tional to the electric field gradient at the rare gas site calculated from the electric multipole moments of the hydrogen halide bonding partner and the geometry of the van der Waals molecule. We may write q = qoK, (157) where K for Kr has been determined to be 78.5(15) 73 using —24 2 83 0.27*10 cm for the Kr nuclear quadrupole moment. Here q is given by /cose\ ^.P,,(cos8) (eose).i •^js—) " R - 30 .P4(cos0) » R R ^/P,(cos8) (cosen 20n ^ R (158) 180 where u, Q, etc. are the electric dipole, quadrupole, and higher moments of the hydrogen halide change distribution. All coor- dinates, moments, and distances are referred to the symmetry axis and center-of-mass of the hydrogen halide. The series in Eq. (158) appears to be converged by the octupole term for all eight molecules listed above. The non-unity value of the pro- portionality constant K may be understood qualitatively as resulting from Sternheimer type shielding [47] of the rare gas nucleus by the atomic electrons. Comparison of our value for K to calculated values of the Sternheimer shielding parameter for Kr indicate that the effect seen here is of the correct magnitude. 43 ' 44 ' 73 The series in Eq. (158) was calculated out to the third 35 term using the values of ]i, Q, and Q, listed for ClF in Table 21. The values of the hexadecapole and higher moments of this molecule are not known. We obtain contributions to q of -2.15, -1.84, and -2.80, in units of 1012 statcoulomb _3 cm , from the first three terms in Eq. (158). Using Eqns. (156) and (157) with K = 78.5, the estimated contribution Kr to x is 10.83 MHz. The 3.1 MHz difference between this value and the measured value of 13.9 MHz could be due to neglected terms in the clearly non-convergent series in Eq. (158), to Lewis acid-base type charge transfer from the Kr atom, or to electronic orbital overlap effects. Consid- ering the first possibility, if the discrepancy in x results from a neglected contribution of q of -2.0 • 1012 sc cm-3, 181 Table 21 Properties of 35ClF a r(Cl - F) (A) a 1.63178 u(D) b 0.8881 Q(DA) C 1.54(7) fi(DA2)d 5.83 See Ref. [74]. R. Davis and J. Muenter, J. Chem. Phys. 57 (1972) 2836. C B . Fabricant and J. Muenter, J. Chem. Phys. 66 (1977) 5274. S. Green, a private communication quoted in Ref. [74]. 182 then at least two additional terms in the series in Eq. (158) would be needed to establish a converging trend in q and bring it into agreement with the model used to explain the krypton-hydrogen halide systems. The largest discrepancy in 12 —3 83 14 73 q Q in those systems was 0.6 • 10 sc cm for KrHC N. Because the higher moments of the ClF molecule are increasingly dependent on the outer portions of the ClF wavefunction, the validity of using free ClF hexadecapole and higher moments in Eq. (158) becomes questionable. We have no way of making accurate estimates of overlap effects for this complex. Dipole polarization of the Kr atom by the electric field of the ClF molecule is a second order effect and can be estimated to be of negligible importance.43 Experimentally, we find no evidence that electric field induced field gradients or second order quadrupolar shielding effects have any importance in the Kr-hydrogen halide systems. Kr If the 3.1 MHz difference in x values is attributed entirely to charge transfer, we obtain an approximate upper bound of 0.004 for the fractional vacancy in the Kr 4p orbital aligned 39 along the molecular axis, using the Townes-Dailey theory with a conversion factor of 750 MHz per electron.44 If this transferred charge were to be located at the fluorine nuclear 74 site, as has been suggested by analogy to the IC1 2 system, it would contribute to q by q ~ -2 x 0.004 e R~3 (78.5), or about 3-1012 sc cm-3 , a negligible amount. The limit of 0.004 electron is less than a factor of two greater than the 73 183 more reliable upper bound of > 0.003 electron established for 44 73 the krypton-hydrogen halide systems. ' The present study of the 83Kr nuclear quadrupole coupling constant in KrClF was motivated in large part by the unusual properties of this complex in comparison to the hydrogen halide series of molecules, as determined from the study of the normal isotopic forms of KrClF by Klemperer and co-workers. These include bonding and stretching force constants 72 that exceed the corresponding force constants in the Kr-HX series by factors of approximately 10 and 2, respectively, and a van der Waals well depth that may be as large as 930 cm-1 75 estimated from a Born-Oppenheimer angular-radial separation analysis 76 of the vibrational ground state rotational spectroscopic constant. 24 Comparable estimates of well depths of KrHF, KrHCl, and —1 —1 83 KrHBr range from 150 cm to 250 cm . Our studies of Kr nuclear quadrupole coupling constants in five krypton-hydrogen halide molecules have established a baseline for specific comparison to 3 KrClF. In 83 Kr 35 ClF we find that the first three terms of the electric multipole expansion for the field gradient at the Kr nuclear site, combined with a quadrupolar shielding factor for Kr of 78.5, determined by these earlier studies, can account for 75% of the observed 83Kr coupling constant m this molecule. This gives an estimate of 0.004 for the maximum fractional transfer of an electron from the Kr 4p orbital. This series is not converged at the third term, and neglected higher terms could easily account for the 184 remaining 25%. This result appears to be consistent with the 72 electric dipole measurement reported earlier for KrClF. Using polarization of the Kr atom by the first three nonzero moments of ClF and back polarization of the ClF by Kr, 72 those authors found that the calculated KrClF dipole moment falls short of the measured value by 0.07 D. Again, the series does not converge, and they were unable to distinguish between polarization effects and charge transfer. The transfer of 0.003 electrons from the Kr atom to F would suffice to make up the dipole moment difference. It appears that computa- tional techniques may have to be used to obtain transfer estimates for KrClF more precise than those reported here. 185 Appendix A Here e is a well-defined function of three spatial and one time coordinate (x., x„, x_, x . ) . e(r - v(r)(t-t'),t') is formed by taking the composition of e with the functions x 1 = x - vx(t-t') x 22 = y - vy (t-f) (Al) x3 = z - vz(t-f) x 4 = t», where v , v , and v may themselves be functions of x, y, and z. We write Eq. (49) as KfcAn 2-2. sine, p. (r, v, t) = where 9 = K/fc 1 at e (r - v(r) (t-f) ,t ')dt'. K 2 *An — 2 3p. -. (A2) Then cos6{e(r, t) (A3) ,t ,3e „ ~ f. ( 3x7 V x o + K2*iAn 4 3p ± 3x + , 9e V „ + , 3eV „+ , 3e -i-.i 3x7 y 3x7 z 3x7 ' °> d t } ' COse t a £ ^x, t o 9v 9x, 3x a l X (A4) |e_^+|§_Z)x ( t - t') }df , 3x 2 3x 3x 3 3x ' 3p. 3p. with similar expressions for ^-— , -g— . Taking (,-3g-r- + v • V)p. K AAn Q yields - — j cos6e(r, t) 186 2 < AAn t - — 4 - ° - cos0 / { a_ a_ giS. v • Vv x + §§- V * Vvy + (A5) |§- v • Vv x (t - f )}df , 2= - ^ e(r, t) An(r, v, t) . (A6) The results in Eqns. (33) and (34) with the functional form v(r) replacing the independent variable v, and in Eqns. (50) and (59)-(61) may be demonstrated in the same way. The physical significance of the criterion in Eq. (31) may be determined as follows. A particle at coordinate r with velocity v(r) at time t will at later time t + 6t have moved to r + v(r)6t. Then v v (r + v(r)6t) = v ( r ) + v(r)St ' Vv v + .... (higher order terms) = v x (r). Since the term v(r) • Vv„ in this expansion is zero everywhere in the cavity, all particles must be traveling on straightline, constant speed trajectories. 187 Appendix B Solution of Eq. (62) without the phenomenologically introduced a term is useful for illustrating the physics of the result for the electric field. Setting a equal to zero for a neutral gas, and agreeing to explicitly solve the boundary value problem, Eq. (62) becomes (again ignoring the 4irV(V*P) term) (v2 - 4 ^-T)-3 = % —y c* Zt* ~ c <B1> * 9f To keep the problem manageable without losing any important features we restrict to the y-dimension. Consider first a small volume of gas located at y' with polarization P(y) = 2Pr(y) cos u 0 t 6(y - y 1 ) dy -2P±(y)sin u 0 t 6(y - y')dy where 6 is Dirac's delta function. (B2) We take E(y,y\ t) = 2A cos ojQ(t - ly ~ y ' 1 ) + 2B sin w0(t - ly ~c Y ' I) , where A and B are undetermined constants. (Bl) becomes (B3) When y ^ y', Eq. 188 satisfied by Eq. (B3). For y = y' take y = y' + e 2 2 i-K / y = y' - e - -A -4)Edy ay c^ af 4ir 2 Y •= y ' + e = ~ (-w 0 ) / {2Pr(y)cos u 0 t 6(y - y ' ) c y - y' - e - 2 P i ( y ) s i n w Q t 6(y - y ' ) } dy For e-K) t h e n 3y y' + e -^P^y') Using Eq. A = 4lT0) 3E 3y 3E (2P ( y ' ) c o s to t y' - e sin a)Qt). (B3) on t h e r i g h t hand s i d e o n e o b t a i n s 2iro)_ — c- Pi . (Jy ' ) (B4) 2TTO). B = pr(y') Boundry surfaces have so far been neglected. Assume now that the dipole in Eq. (B2) is placed at coordinate y' between Fabry-Perot mirror surfaces located at y + d/2. The resultant electric field is observed at - d/2 < y <. d/2. To obtain the required boundry condition E(y = + d/2) = 0 we use the method of images as illustrated in Fig. 38. Here we have the real dipole at y' with image dipoles labelled Fn , where F is the d I F2 i F' > / > . • /* 1 1 / / / '• - 2 d +»' -1 F° £F' -d y / / / / / y -y , F3 L r 0 / i F* -d- *' " 1 -y 2d+y 3ci-y' Figure 38. A radiating electric dipole at coordinate y' and its associated images from two mirrors placed at y = + d/2. The resultant electric field is observed at y. The reflection coefficient of the mirrors is F, assumed to be near unity. CD 190 reflection coefficient for either mirror, and n is the number of reflections corresponding to that particular image. One has: real dipole, 2Pr cosu)0t - 2P. sin (jjQt, at y' , and image dipoles F2lnl (2Pr cos u t - 2P. sin <0ot) at (B5) 2nd +y', n = + 1, + 2 , ..., and -F' 2 1 1 " 1 ' (2P,. cos a) t - 2P. r o i sin coot) at (2n - l)d-y', n = 0 , + 1 , + 2 , ... From Eqns. (B3) and (B5), and writing down only the P. terms to same space, E(y, y', t) = — ° - P.(y') c x {E F^'nl n=-» cos Uo (t - ly - (2nd + y ) |} -E FI^^'COS ~. _ (B6) co (t - ly - (2m-l)nd + y'|)} ° c m=-<» + Pr(y') terms. Assuming the cavity to be tuned to the radiation frequency to and using the fact that F is nearly unity, one obtains 191 4TTIO_ I y E(y, y', t) = — ° - (P. (y') cos m (t - l ~ c 1 *•* ' o c 11 y I) + Pr(y') sin w0(t - ly^y'l)) 47TW_ __o + 4p (B7) 2 -E-g. (p^y*) cos w 0 t + Pr(y')sin w Q t) x 1—F cos(kQy - IT (q/2))cos(koy' - ir (q/2)) . Here k = w./c. We consider expression (B7) in two cases, When F = 0, as in a waveguide experiment, this is E(y, y', t) = — 3 ° - p.(y')cos wo(t - l y c y 1) (B8) 4TTW IV V I| + -3-°- P r ( y ' ) s i n u 0 (t - l y c y ') . We could have written Eq. (B2) including an arbitrary phase term <j> so that u> t is replaced by w t + <|> in all following results. In the waveguide an appropriate value for <}> is <\> = k y 1 (wave traveling from y' to y for y* ^ y) . (B9) In the no-boundry case then Eq. (B7) becomes 4frw E(y, y', t) = —f- P i cos (w0t + kQy' - kQy' + k Q y) (BIO) 4TTCO + —^P- P r s i n U 0 t + k Q y ' - k Q y ' + k Q y ) , 4irw_ = -o5- (P i » B ( t t o t + k Q y) (Bll) + Pr sin (w Q t + k Q y ) ) , 192 with no y' dependence. wave experiment P., P We use the fact that in a traveling are independent of y', which follows from the derivation leading to Eq. (12) in Chapter II. To obtain the electric field seen at the end of the cell at y = I, take I E(y = l, 4TTW t) = / - c -°o (P± cos (wot + kQy) + P r sin((oQt + kQy)) dy» , integrating over the source coordinate. (B12) The absence of this coordinate in the integrand means that the emitted fields add up coherently along the guide. Now consider the F-dependent terms in Eq. (B7). These are 47TW o E(y, y \ t) = — - ° - 4F2 ^ - 5 - (P. (y')cos u> t c (B13) l-F + Pr(y')sin u>0t)cos(k0y - TT (q/2) )cos (kQy' - 7r(q/2)). Notice that E does satisfy the boundry requirements at y, y1 = + d/2. Comparison of Eq. (B13) with Eq. 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Klemperer, Farad. Disc. Chem. Soc. 62 (1977) 179. S. L. Holmgrin, M. Waldman, and W. Klemperer, J. Chem. Phys. 67 (1977) 4414; ibid. 69 (1978) 1661. VITA Edward J. Campbell was born on June 28, 1954, in Milwaukee, Wisconsin. He graduated from the University of Wisconsin at Madison, receiving a B.A. degree in Mathematics and Physics in June, 1976. He entered graduate school in physics at the University of Illinois in June, 1976. During his graduate studies he has been a University Fellow, a teaching assistant, and a research assistant. ber of Phi Beta Kappa. He is a mem- He is co-author of the following publications: W. E. Hoke, H. L. Voss, E. J. Campbell, and W. H. Flygare, "The Rotational Zeeman Effect in trans-Crotonaldehyde," Chem. Phys. Lett. 58, 441 (1978). T. J. Balle, E. J. Campbell, M. R. Keenan, and W. H. Flygare, "A New Method for Observing the Rotational Spectra of Weak Molecular Complexes: KrHCl" J. Chem. Phys. 71, 2723 (1979) and 72, 922 (1980). E. J. Campbell, M. R. Keenan, L. W. Buxton, T. J. Balle, P. D. Soper, A. C. Legon, and W. H. Flygare, " 83 Kr Nuclear Quadrupole Coupling in KrHF: Evidence for Charge Transfer," Chem. Phys. Lett. 1Q_, 420 (1980). M. R. Keenan, E. J. Campbell, T. J. Balle, L. W. Buxton, T. K. Minton, P. D. Soper, and W. H. Flygare, "Rotational Spectra and Molecular Structures of ArHBr and KrHBr," J. Chem. Phys. 12, 3070 (1980). M. R. Keenan, L. W. Buxton, E. J. Campbell, T. J. Balle, and W. H. Flygare, " 1 3 1 Xe Nuclear Quadrupole Coupling and the Rotational Spectrum of XeHCl," J. Chem. Phys. 73, 3523 (1980). L. W. Buxton, E. J. Campbell, M. R. Keenan, T. J. Balle, and W. H. Flygare, "The Rotational Spectrum, Nuclear Spin-Spin Coupling, Nuclear Quadrupole Coupling, and Molecular Structure of KrHF," Chem. Phys. 54_, 173 (1981). 200 E. J. Campbell, L. W. Buxton, T. J. Balle, and W. H. Flygare, "The Theory of Pulsed Fourier Transform Microwave Spectroscopy Carried Out in a Fabry-Perot Cavity: Static Gas," J. Chem. Phys. 74., 813 (1981). E. J. Campbell, L. W. Buxton, T. J. Balle, M. R. Keenan, and W. H. Flygare, "The Gas Dynamics of a Pulsed Supersonic Nozzle Molecular Source as Observed with a Fabry-Perot Cavity Microwave Spectrometer," J. Chem. Phys. 74_, 829 (1981). M. R. Keenan, L. W. Buxton, E. J. Campbell, A. C. Legon, and W. H. Flygare, "Molecular Structure of ArDF: An Analysis of the Bending Mode in the Rare Gas-Hydrogen Halides," J. Chem. Phys. 74, 2133 (1981) . L. W. Buxton, E. J. Campbell, and W. H. Flygare, "The Vibrational Ground State Rotational Spectroscopic Constants and Structure of the HCN Dimer," Chem. Phys. 56, 399 (1981). E. J. Campbell, L. W. Buxton, M. R. Keenan, and W. H. Flygare, n83 K r an{j 131xe Nuclear Quadrupole Coupling and Quadrupolar Shielding in KrHCl and XeDCl," Phys. Rev. A. (in press). 83 L. W. Buxton, E. J. Campbell, and W. H. Flygare, " Kr Nuclear Quadrupole Coupling in KrClF," Chem. Phys. (in press).

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