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University Microfilms International 300 N ZEEBROAD, ANN ARBOR, Ml 48106 18 BEDFORD ROW, LONDON WC1 R 4EJ, ENGLAND 8108446 BALLE, TERRILL JOSEPH FABRY-PEROT CAVITY, PULSED FOURIER TRANSFORM MOLECULAR BEAM MICROWAVE SPECTROSCOPY University of Illinois at Urbana-Champaign University Microfilms I n t G r n f l t i O n S U 300N ZeebRoad,AnnArbor,MI48106 PH.D. 1980 FABRY-PEROT CAVITY, PULSED FOURIER TRANSFORM MOLECULAR BEAM MICROWAVE SPECTROSCOPY BY TERRILL JOSEPH BALLE B.S., University of Oregon, 1975 THESIS Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Chemistry in the Graduate College of the University of Illinois at Urbana-Champaign, 1980 Urbana, Illinois UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN THE GRADUATE COLLEGE May, 1980 W E HEREBY RECOMMEND T H A T T H E THESIS BY TERRILL JOSEPH BALLE F.NTTTT.FT) FABRY-PEROT CAVITY, PULSED FOURIER TRANSFORM MOLECULAR BEAM MICROWAVE SPECTROSCOPY BE ACCEPTED IN PARTIAL F U L F I L L M E N T OF T H E REOUIREMENTS FOR T H E DEGREE O F DOCTOR OF PHILOSOPHY \/Djflcctor of Thesis Research HOPCI of Department Committee on Final Examumjionf N&£ t Required for doctor's degree but not for master's iii Acknowledgments I first want to thank Dr. W. H. Flygare, for without him none of this would have happened. I thank him for his faith and support over the time it took to bring this project into fruition. I wish to thank several members of the research group after I first joined who taught me how to get things done in spite of everything. I thank the research group after the success of the spectrometer for doing all the things I couldn't do and for carrying on the work. I thank Dr. J. D. McDonald for his invaluable advice concerning the pulsed valve and many areas of spectroscopy. The help of the Materials Research Laboratory is gratefully acknowledged. They have contributed to the success of this spectrometer through support of me, through new equipment money and most importantly through their excellent machine shop. I want to thank Wayne Craig, the machine shop supervisor, for all his advice and help. Special thanks goes to Bud Dittman and Tom Koerner for all the machining and welding on the spectrometers. I wish to thank the Chemistry Machine Shop under Elmer Lash and the Student Machine Shop under Bill Hempler and Ron Harrison for teaching me everything I know about machining. Another key part of this and every project in physical chemistry is the contribution of Chuck Hawley and the Chemistry Electronics Shop. The chemistry department is truly fortunate in having two superb electrical engineers, Chuck Hawley and Carl Reiner, who designed all the special iv electronics that were needed. Special thanks also goes to Al Saldeen for all his help. I thank Art Gaylord of the Chemistry computer services for doing the initial interfacing of the experiment to the departmental VAX computer and all other help. The help of Evelyn earlier and Kim Burr is appreciated for typing this and our papers. The support of the University of Illinois through teaching and research assistantships and to Allied Chemical for a fellowship is gratefully acknowledged. Finally, I wish to thank my first wife, Andrea, for all that she taught me and for leaving me. she does. And to my second wife, Mary Beth, for doing all that V TABLE OF CONTENTS Page INTRODUCTION TO CAVITY SPECTROMETERS 1 FABRY-PEROT THEORY 3 A. B. C. D. Introduction 3 Quasi-optical Fabry-Perot Theory 4 1. Electromagnetic Fields in a Fabry-Perot Cavity. . . 4 2. Resonant Frequencies . 8 3. Q Factor 9 Circuit Theory of a Microwave Cavity io Maxwell's Equations in a Microwave Cavity 18 PULSED MOLECULAR BEAM 21 A. B. C. D. E. 21 22 25 26 28 Introduction Molecular Density Distribution Molecular Flow Velocities Numbers of Molecules Pulsed Cooling of Rotational Energy MOLECULAR POLARIZATION 31 A. B. C. D. E. 31 32 33 36 40 40 42 51 55 Introduction Bloch Equations Transient Absorption Transient Emission Signal Characteristics 1. Spontaneous Emission 2. Doppler Dephasing and Transit Time 3. Signal Detection 4. Resolution and Sensitivity THE SPECTROMETER 58 A. B. C. D. E. F. 58 58 67 70 71 75 Introduction Block Diagram Design Considerations Molecular Spectra Potential of the Method Discussion VI Page VI. VII. VIII. A JOSEPHSON JUNCTION MIXER 78 A. Introduction 78 B. Josephson Junction Detector-Mixer 78 LIST OF REFERENCES 89 VITA 95 1 I. INTRODUCTION TO CAVITY SPECTROMETERS There have been many cavity spectrometers built since the development of radar and radar techniques in World War I I . Bleaney and Penrose m e a - sured the inversion spectrum o f ammonia in a closed cavity spectrometer in 1946. The simplest w a y o f detecting a rotational transition is to note a change in the reflected o r transmitted microwave power. Due to the smallness o f this change m o s t people choose to modulate the molecular absorption with either the Stark o r Zeeman effects and detect with a phase sensitive detector. ' ' ' ' ' A superheterodyne receiver can also b e Q easily used to detect absorption lines. Scanning cavity spectrometers 6 9 have also been built. ' Closed microwave cavities are standard equip- ment in all electron spin resonance spectrometers, utilizing the effec- tive concentration of the field for small samples. Dielectric measurements have many times employed microwave cavities to obtain the real and imagin* of- the «. dielectric * i J. 4- 4- 11,12,13,14,15 ary part constant. JT Microwave cavities are 16 17 also the heart of molecular beam masers ' and beam maser spectrometers . ' Open resonators of the Fabry-Perot type have been used increasingly because of their high Q, the easy access to their interior field regions 6 1 ft and the ability to work from the microwave to optical frequencies. 20,21,22,23,24,25 Lee and White. A double resonance Fabry-Perot cell has been built by 26 Pulse techniques in microwave cavities were first discussed b y Dicke and Romer. 27 28 ' Utilizing these pulse techniques in the microwave region 2 one can create a coherent macroscopic polarization in the gas which subsequently emits radiation. This radiation is given off at a frequency that is proportional to the difference in energy between two quantized states of the molecules m the gas. This radiation is damped out by relaxation process in the gas such as collisions, Doppler dephasing and transit time effects. Because of this damping, all excitation and measurements have to be done in times short compared to the fastest relaxation process. Dicke and Romer 27 were the first to measure the emission from a low pressure gas sample of ammonia in a closed cavity. 29 from OCS in a waveguide cell was observed in 1967. Emission Laine observed emission from ammonia in a closed cavity by using fast passage techniques. A pulsed Fabry-Perot microwave spectrometer was built 32 relaxation times. 31 and used to measure The transient absorption and emission theory was derived in terms of the density matrix of the system by McGurk, et. al. •20 This theory nA ' 30 33 OC ' is fundamental to the work presented in this thesis. Pulsed time domain spectroscopy in a waveguide cell of rotational levels has been developed as a routine spectroscopic tool. 36 37 ' A further advancement combines the techniques of pulsed Fourier transform spectroscopy with a Fabry-Perot cavity and a pulsed nozzle expansion. 38,39 3 II. A. FABRY-PEROT THEORY Introduction The theory of the Fabry-Perot resonator has been developed on two fronts. Historically the Fabry-Perot interfrometer with plane reflecting mirrors has been used in the optical and infrared regions. m Connes 40 41 ' 1956 and 1958 suggested the use of spherical mirrors in the optical 42 region. In 1961, Boyd and Gordon 43 also Fox and Li, developed the theory for plane and spherical mirror Fabry-Perot cavities in the microwave region. This approach to a microwave resonator, from the high frequency limit treats the resonator as two mirrors with a wave bounding back and forth between them. Optics is used m the diffraction limit. The other approach to the theory of a Fabry-Perot resonator is from circuit theory, or from the low frequency limit. The resonator is considered to be a tuned series RLC or shunt GLC circuit, driven by an alternating voltage or current source. Closed resonators have primarily been described from this point of view. Both approaches will be used here. The quasi-optical theory will be used to obtain the field modes and resonant frequencies. This approach to find the field distribution and eigenvalues does not include the coupling to the resonator and hence does not account for power flow into and out of the resonator. This dependence needs to be known to be able to calculate magnitudes of polarizating input power and signal power coupled out into the detector. Circuit theory combined with Maxwell's equations enable one to treat these questions. 4 B. Quasi-optical Fabry-Perot Theory 1. Electromagnetic Fields in a Fabry-Perot Cavity Inherent in the assumptions of the quasi-optical treatment are that the dimensions of the mirrors are large compared to the wavelength, that the fields are T.E.M. waves and are plane polarized. 43 distribution To obtain the field in the cavity, an initial wave, u , is considered leaving one mirror, where u is P u = ik J" uaexp(-ikR) (4iTR)~ (1 + cos8)ds- This wave is the Fresnel field due to an illuminating aperature A, u is the aperture field, k is the propagation constant, R is the distance from the aperture to an observation point and 9 is the angle which R makes with respect to a unit normal to the aperture. The field at a mirror after p transits,up+1. is obtained by up by and ua by up . up is J replacing tr jr up +,. J 1 the field across the other mirror which produced the field u . A steady state is reached where the field distribution, u, between the two mirrors is identical apart from a complex constant, y, which depends on coordi- nates . up = (1/Y) P u and up+i = (1/Y) P + 1 u. When this is substituted in the above equations, we get u = Ylk A u exp(-ikR) (47rR)~ (1 + cos8)ds. This equation can then be solved iteratively. Another approach 44 45 46 47 ' ' ' uses the solution to the wave equation following the concept of wave 48,49 beams. 5 The field distribution inside the resonator is given by these methods for a TEM mode as, mnq w E(x,y,z,t) = E (t)H (»^)H (/fr— exp(-r /w 2 ) o m w n w w cos (kz + (kr2/2R) - $ - irq/2) f where 2 r W o 2 = x = W (z) ( + y 2 X 1/2 2 ? [ £ ( 2 R - *)] > = w o 1/2 X z 2 1/2 [i + ^ - ^ - r j 7TW £ o -1 2 $ = tan (Xz/7Tw ) o E (t) = E cos wt. o o Referring to Figure 1, x,y,z are the rectangular coordinates from the center of the cavity. H are Hermite polynomials of order m,n. m,n * radius of curvature of both mirrors is the same and equal to R. fundamental or TEM The For the mode the Hermite polynomials are unity and one has basically a standing wave with a gaussian fall off going away from the axis (x=y=0) of the cavity. off as a function of z, its value on axis. so that when r = w the field has dropped to 1/e of The value of w at z = 0 is w beam waist diameter. and 2w is called the o o The beam waist is the minimum diameter that the 2 beam diameter attains. wave front. The function w(z) describes the gaussian fall The factor kr /2R accounts for curvature of the The phase front has to be curved because of the curved mirrors. The only place the phase front is planar is at z = 0. The term $ is a phase shift difference between a gaussian beam and a plane wave. The mode number q is the number of nodes in the standing wave field, so that q + 1 is the number of half wavelengths between the mirrors. 6 Figure 1. The geometry and coordinate system of the Fabry-Perot cavity. R is the radius of curvature of both mirrors. between mirrors is &. The distance The center of the cavity is at x=y=z=0. The beam waste parameter w is measured at z=0, drawn in a different position for clarity. 8 The factor 7lq/2 is included to change cos(kz) into sin(kz) depending on whether the field is a maximum or zero at the center of the coordinate system. The beam waist dimension, 2w , is then the shortest distance o the molecules have to travel to enter and escape the field (assuming for convenience that the molecules only interact with the field in this region) This beam waist is a maximum when & = R, the confocal arrangement, and falls to zero when A = 0 or Ji = 2R. So that to maximize the time of flight, one would want to work near the confocal geometry. Using the radius of curvature of one of our mirrors, 84 cm, and a mirror separation of 70 cm, the beam waist at 10 GHz is 12.6 cm. 2. Resonant Frequencies The resonant frequencies V of the resonator for the TEM mode mnq are V = V [(q + 1) + (1/TT) (m + n + 1) cos" (1 - l/R)} where V = c/2Jl. Using a mirror separation of 70 cm, V that the dominant modes, TEM , is 214 MHz, so , are separated by that amount. The higher order modes m or n ^ 0 can be easily seen,as the m,n even modes are excited by our coupling iris. All modes are well separated in frequency as the width of a mode 6v = V/Q is (for Q = 10 , V = 10 GHz) 1 MHz. All spectroscopic work is therefore done single moded, either in the dominant or in a higher order mode. Experimental observations confirm the field distri45 47 50 51 butions and resonant frequencies given above. ' ' ' 9 3. Q Factor The Q of a resonant system is a very useful and important property. One of the ways to define Q is Q = CO • W/P, where to is the angular frequency of the radiation, W is the total energy stored in the cavity and P is the power dissipation. From the quasi-optical point of view of a resonator, the power in the wave bounding back and forth between two mirrors can be dissipated in two ways. The wave can diffract out the sides of the mirrors or can be dissipated in ohmic losses in the metallic mirrors. The diffraction losses can be made arbitrarily small by just making the mirror diameter large. 42 43 44 has been done by many authors. ' ' The calculation of these losses The power lost due to the finite conductivity of the metal mirrors can be calculated, although it is difficult to account for surface roughness or contamination. For a good conductor, such as the aluminum we use in our mirrors, a skin depth, 6, can be defined for O » coe, where a is the conductivity in l/(ohm m) and £ is the permittivity of the material. 52 <5 = (2/cou0) 1/2 , where y is the permeability of the material. At this distance in the con- ductor the amplitude of an electromagnetic wave has fallen to 1/e of its value at the surface. (a = 3.54 x 10 7 -5 For aluminum the skin depth is 8.5 x 10 cm mhos/m) at 10 GHz. The value of Q assuming this to be the only source of power dissipation can be calculated to be, 52 10 5 For £ = 70 cm and 6 from before, 2 * 4.1 x 10 . Q'sas high as 5 2 x 10 have been measured in our cavity. Cavities utilizing supercon- 11 53 ducting walls have reported Q's as high as 10 in the microwave region. C. Circuit Theory of a Microwave Cavity A microwave cavity connected to an oscillator and a detector through waveguide or coaxial cable can be represented by an equivalent circuit 54 as in Figure 2a. The oscillator produces a voltage v and has an internal resistance R . The total power, P , available from this oscillator 0 o 2 is v /4R . The iris couplings to the cavity and out to the load, R , are represented by transformers of turns ratio n and n„ respectively. cavity itself is the series combination of R, L, and C. The Power dissipation in the cavity is accounted for by R and energy stored in the magnetic and electric fields is accounted for by L and C, respectively. To simplify the analysis, one can transform the input and output circuits to the middle loop as m Fibure 2b. The generator and the detector load are assumed matched to the input and output waveguides, hence R = R . U The cavity L resonator is now a simple RLC tuned circuit driven by a generator. The impedance for the series circuit is, 2 2 Z = nn R + R + n„ R + i (toL - 1/coC) . 1 O i. L The resonant frequency, CO , of the circuit is defined as that frequency at which the reactance vanishes, therefore to L = 1/to C o r co = 1//LC. c c c At this frequency the cavity appears as a pure resistive load to the input circuit and the magnitude of the stored energy in the electric and magnetic fields are equal. R •vww- L C •UJUULLr 0 l:n. l:n, (2a) £ n, R 0 JWWV- R •AA/WV L "UULUlr C n|R (2b) Figure 2. Equivalent circuit of microwave cavity 12 As defined before the Q is, to (Ene-gy Stored) (Power Dissipated) Four Q's will now be defined according to this definition. The energy 2 2 stored in the inductor (1/2) LI , and in the capacitor (1/2) Cv , are equal in magnitude and equal to the total energy stored W. The first Q, the unloaded Q, Q , accounts for power dissipation in R which is (1/2) 2 RI so that, to (1/2) L I 2 to L Q = _S _ £_ . = R ° (1/2)RI2 Similarly, the input and output coupling Q's, Q .and Q to L and 2„i Q*c2 cl = ~ ^2„— n R 1 o are given by to L ° 2„ n„ RT 2 L The sum of all dissipative elements defines the loaded Q, Q , L 1/QL - d/COcL) (R + n i 2 R Q + n 2 \) = 1/QQ + V Q e l + l/Qo2- The ratio of the unloaded Q to the two coupling Q's define two coupling coefficients $ 1 and ft a 2 n, R 1 O Q *0 2 n„ R 2 L ,a xQ O The impedance of the equivalent circuit can now be written m these parameters as CO c CO terms of 13 Using the impedance, the power flow through the circuit can be calculated. For instance, to calculate the power that is coupled into the cavity from the generator, we find the power dissipated in R, P_. R P R = I 2 R = (n;Lv/|z|)2R 2 2„ n. v R 2 2w 2 2 R [a+31+e2) 2 td,_ o 2 + Q o (---^) 2 ] c 2 Dividing this by the power available from the oscillator P = v /4R gives, ffl P //P R o ~ 2w 2 < » w w c 2 * •*,<=--•#> c When the cavity is being driven on resonance (co = co ) this expression reduces to, P„/P R = 43, 1 4Q _ L ° <1+W* 2 2cl2o The amount of power dissipated in the load, P , from the generator can be derived in the same way and is, 4Q 2 43,3, X PT/P = * L ° (l + 3 1+ 3 2 ) 2 L QclQc2 Further insight into the meaning of the coupling coefficients can be gained by considering a reflection cavity. output coupling iris. This cavity has only one input- Again using the circuit theory, the amount of power coupled into the cavity on resonance is 43 P R 4Q 2 /P = = __h° <1+e>2 QCQ0 14 where 3 is the one coupling coefficient and 1/Q_ = 1/Q„ + 1/Q • If 3 = 1 L O c then PD/P^ = 1 and all the power that is coupled into the cavity is i\ o dissipated there. This is the matched load condition. The transmis- sion line is terminated m an impedance which is the complex conjugate of its own impedance. No power is reflected and the VSWR = 1. Also for this case, since 3 = Q /Q # Q = Q and Q_ = (1/2) Q . When 3 = 1 o c O C It the cavity is said to be critically coupled. undercoupled, so that Q > Q o For 3 < 1 the cavity is and more power is dissipated in the cavity than in the coupling. When 3 > 1 the cavity is called overcoupled, so that Q > Q and more power is now dissipated in the coupling than in the o c cavity. For a transmission cavity some power will always be coupled out through the coupling iris that is not feeding power into the cavity. If we call P , the power reflected from the input to the cavity then, P 0 = P F + P + R P L • or l = P„/P + PT7P F' o Using the values for P„/P R cavity, o R + PT/P o and P„/P L L • o obtained before for the transmission o we h a v e , 43, P /P F = 1 - ° 48,3 9 l (l+e^Bj) d-31+32)2 ci+e^Bg) 2 X 2 (I+SJ+BJJ) 2 15 With 3 1 = 3 2 = li one ninth of the input power is reflected and four ninths is each dissipated in the load and in the cavity. case Q Q = Q Q 1 = Q Q 2 and Q L = (1/3) Q Q . flected from the input coupling i n s , Also for this In order to have no power re1 - 3 + 3 = 0 . Another way to think about cavity coupling is to consider the electric waves or voltages traveling in the system. Voltage reflection coefficients can be obtained by taking the square root of the power reflection coefficients. When one pulse excites a transmission cavity, there will be an incident electric wave on the coupling iris. a wave will be reflected and a wave transmitted. radiate into the cavity. Upon hitting the input iris The transmitted wave will When this wave encounters the opposite mirror, part of it will be coupled out and part will be reflected with a 180° phase shift. Part of this reflected wave will be coupled out of the original mirror and part re-reflected, building up the energy in the cavity. The parts of the original wave that get coupled out the input iris after being reflected back will be 180° out of phase with the input wave that ;just gets reflected from the iris without ever entering the cavity. These two waves will destructively interfere, making it appear that nothing is coming out of the cavity input. As the energy stored in the cavity builds up, the cancelling wave builds up to a value depending upon the coupling conditions and the Q's. The totally transmitted wave also builds up with this same cavity time constant X. This time constant will be calculated next. Two other common ways of thinking about the Q factor of a cavity are related to the bandwidth and decay time constant of a cavity. To 16 derive the bandwidth of a cavity, the full expression for the power dissipated in the load P is from before, L P ^ 2 2 2 P l/ o ~ (i + 3 1 + 3 2 ) 2 u w c 2 2 + Q0 (--f)2 c This will be rewritten as , P. L "= P o 2 p 2^_VC_2 ^o VV V c where a = 4 3 , 3 , , 3 = 1 + 3, + 3 , , to = 2TTV and to = 2TW . close t o the c a v i t y frequency, 2 2 V V V -V (V+V )(V-V ) 2V Av c c_ _ c c ,, c v ~ v ~ v v ~ VV ~ VV c c c c For frequencies 2Av V so t h a t , (Av = V-V ) c p L - p a °02 2 2^2 *o v or P.L = P ° 3 2 0y 2 + Av 2 o This is the power transmitted to the load as a function of frequency. 2 When V = V ,P = P a/3 which was given previously. The frequencies above and below resonance where the power has fallen to half its peak power on resonance are given in the limit 4Q 2 2 » 8 by, 17 The bandwidth, 6v, at half transmitted power is then 6v = V /Q . C Ju A measurement of the frequencies where the power has fallen to half the maximum value is an easy way to find the loaded Q. The decay time con- stant, X, of a cavity can be obtained from the definition of the Q. The energy stored in the cavity will decay at a rate proportional to the amount already present, - d£ " W/T ' or W = W e o -t/x , ' but this rate o^ energy loss must equal the power dissipated, or as. JI P " dt ~ " QL ' so that W/x = COQW/QL , or X = QL/tOc . In summary then, Q L - Vc/<SV °-L=T(V and 6v = 1/2UX . = 1 0 4 at V For a cavity with Q Jj Usee. = 1 0 GHz, Sv = 1 MHz and x = 0.16 c 18 D. Maxwell's Equations in a Microwave Cavity To complete our description of a Fabry-Perot cavity, we will use Maxwell's equations to relate the electric field in the cavity to the Q and hence to the power flow. Maxwell's equations for a lossy and polarizable dielectric are (in cgs units) -XT V x E = c dt ->- 4ir -»• 1 3D V x H = — J + ^^-^f C C dt D= V • B = 0 -*• -> D = E + 4irP B = VH -> . J OE , Taking the curl of the first equation and substituting to get the wave equation for the electric field and the polarization, we get V 2 E -i2H. a E _ c (JL) i = c iIHp. c One way to solve this equation would be to expand the normal mode functions that we derived earlier from the quasi-optical approach. Instead of doing this, for the sake of clarity, let us expand in terms of a plane standing wave, E(r,t) = I An(t) Un(r) , 19 where we take U (r) to be simply sin k Z. n n Then k n has to satisfy the boundary and mode number requirements of the cavity so that k n where n is some integer and i is the distance between the mirrors. 2 k 2 = -5— * Also 2 = to /c . If this is done, we can get a simple equation for the time dependence of the electric field driven by the polarization, (setting u = 1) An + 4rra An + ton2 An = -47T T PUmdv. « This equation can be further simplified in two ways. The finite conductivity losses can be represented by the finite loaded Q of the cavity, so that 4TTC will be replaced by CO /Q_. The spatial part of the polarization can also n L be expanded in terms of the same set of modes as the electric field, P(?,t) = I G n (t) U n (?), n then 4/r JT P u dv = m The function G 4TT G n • is in general complex to allow for components in phase and out of phase with A(t). In all of this we have assumed the spatial modes to be orthonormal, T Un UmdV = 6 n m nm « Also since all modes are well separated in frequency we can restrict all discussion to only one mode. to A(t) + ^ A(t) Q L So that now, + CO 2A(t) = -4TT G(t). (II-D The polarization will be calculated in a following section and this equation will then be used to find the electric field. 20 The energy stored in the electric field can be calculated simply from; 2 2 W = J* (1/8TT) | E ) dV = (1/8TT) E V . (H-2) This integral, over the volume of the cavity, has been done for the gaussian standing wave given in Chapter II, section B, W = (1/8TT) E 55 so that, 2 x 7T&(w / 2 ) 2 o o where H is the distance between the mirrors, w is the maximum value of the electric field. (II-3) is the beam waist and E_ o o From before the energy stored is related to Q by, to w and the power dissipated on resonance in the cavity is given by, 48, p /p R 4Q l - = ° M1+82) 2 2 L QclQ0 " The maximum electric field can thus be calculated to be, *o= (327r VL2/Va)c2cl)1/2- For our cavity, H = 70 cm and as previously calculated w V c = 10 GHz, so that V = 2.2 x 10 3 cm 3 . Then, for Q L = 6.3 cm for = 10 , Q ci = 3 x 10 and an input power of 10 m watts, the peak electric field is 4.7 volts/cm. With a medium power TWT, input powers of 40 watts are possible, which leads to a peak electric field of almost 300 volts/cm. 21 III. A. PULSED MOLECULAR BEAM Introduction There have been many papers on the gas dynamics of molecular beams. The term molecular beam, as used here, will mean the expanding gas from the nozzle, even though there is really no collimated beam of particles. 56 57 58 Sources for much of the theory and data are in books, reviews ' or in the Rarefied Gas Dynamics series. For our microwave spectrometer, the important quantities are the spatial density of molecules in the expanding beam, the molecular velocities, the number of molecules in a pulse and finally their temperature. The molecular or dimer spatial distribution and velocity are involved with the Doppler effects which will be discussed later. The number of molecules and their temperature are used to compute the number density difference between two rotational levels at thermal equilibrium. This number density difference is used in the next chapter to compute the power emitted by the molecules. All experiments are done with a flat thin plate orifice bolted on the bottom of a solenoid pulsed valve. As used here, the gas dynamic theory is identical for the pulsed or continuous expansions. Steady state flow conditions are established quickly compared to the pulse valve open time. 61 We do not use any skimmer, the so-called free jet. all work is done with seeded beams. between 95% and 97% of a rare gas. In addition, Most expansion mixtures contain Therefore the gas dynamics will be dominated by the expansion properties of rare gases. 22 B. Molecular Density Distribution A simplified density distribution is used for distances far from ,_. 60,61 n the nozzle, p(r,e> = P n (D 2 /r 2 ) cos m B , (III-I) where D is the nozzle diameter, r and 9 are the radial distance and polar angle from the nozzle orifice, as in Figure 3. The density of molecules at the nozzle is given by p . For an effusive source it can be shown by the kinetic theory of gases that the power of cos 9 is m = 1. occurs m Effusive flow through a nozzle the realm where each molecule's motion is independent of all other molecules. The condition for effusive flow through a circular nozzle of diameter D is, X'/D » 1 , where X' is the mean free path, and X'/D is called the Knudsen number. The mean free path is given by, X' = (/2~TT p n d2)"1 , where d is the molecular diameter. For an effusive source p = p , the n o —8 source or stagnation molecular number density. Using d = 4 x 10 cm and -5 1 atmosphere, the mean free path is 4.3 x 10 mm. The smallest nozzle diameters we have used are ~.l mm. For practically all our experiments we use source pressures greater than one atmosphere, therefore we are far from the effusive limit. NOZZLE A •". v, Figure 3. Geometry of the nozzle relative to the Fabry-Perot cavity. The radial distance and velocity from the nozzle are r and v respectively. The polar angle from the nozzle is 0. The projection of v on the z axis is v = v sin 8. o z o 24 For a supersonic jet expansion, the basic isentropic relations for a perfect gas are p/Po = (1 + 3ti. M 2 ) " 1 / ^ - ! ) (III-2) = (T/T ) 1 / 2 = (1 + ^ ) " (III-3) a/a O O 1 / 2 2 v = Ma. (III-4) In these equations, the o subscript denotes source conditions, M is the Mach number, y is the ratio of specific heats (y = 5/3 for a rare gas), a is the local speed of sound and v is the flow velocity. The definition of a is a i = /vkT /m. i Returning to equation (III-2), we can now compute p by noting that the Mach number is one at the nozzle plane, v n P o v 2 ' , (III-5) and for a rare gas p = p (0.65). With 1 atmosphere and 300°K as our n o 19 3 19 source conditions rp = 2.69 x 10 molecules/cm , so that rp = 1.7 x 10 o n 3 molecules/cm . Using a nozzle with a diameter of 1 mm, the number densaty on the nozzle axis and at the cavity center is p(r = 17 cm, 6 = 0 ) = 14 3 6 x 10 molecules/cm . This number density would correspond to a static pressure of 17 microns. We will discuss later, in the section on Doppler broadening and in two upcoming publications, the ability to use the observed time domain rotational line shape to measure the power of the cosine distribution in equation (III-l). 25 Another question is whether or not the dimers follow this same distribution. Mass separation effects have been observed in beam expansions. Condensation reactions that produce not only dimers and trimers but clusters containing thousands of molecules have been observed and studied. ' An experiment could be done to measure the power of the cosine for a monomer and a dimer in the same gas mixture to see if the dimers were distributed differently. C. Molecular Flow Velocities In the reservoir the static gas behind the nozzle has, of course, zero flow velocity. A static gas has a Maxwellian distribution around zero velocity and an average speed, v , of, 3i ~kJF a VTT m During the expansion the random translational kinetic energy and any internally stored energy (rotations and vibrations of polyatomics) is converted into mass flow through binary collisions. The effective rotational and vibrational temperature drops, the velocity distribution narrows and moves out along the flow velocity axis. Since local thermo- dynamic equilibrium was assumed to exist at all times m the derivation of equations (111-2,3) that describe this process, we can calculate the flow velocity as a function of beam temperature. From equation (III-3) and (III-4) we have v = !L ^o a- f ) Y-l m o 1/2 (III-6) 26 If the expansion were to convert all internal energy to directed mass flow the temperature would be zero and equation (III-6) simplifies to 2Y kT . v m = (—T — - ) /Z , T y-1 m (III-7) the terminal velocity. Note that if the temperatures were zero, the sonic velocity would be zero and since the flow velocity is still finite the Mach number is infinity. Also, the terminal velocity is only a factor of 1.4 larger, for a rare gas, than the average velocity in the source. For Kr 4 gas, the terminal velocity is 3.8 x 10 cm/sec. D. Numbers of Molecules Pulsed The number of molecules that are released, N, in a single pulse of our nozzle is, N = Cp v A t , n n n v where v is the flow velocity at the nozzle, A is the nozzle area, t n * n v eg is the valve pulse time and C is a discharge coefficient Y = 5/3). (C = 0.55 for At the nozzle, the Mach number is unity so that from equations (III-3) vn = an = ao ,Y+1x-1/2 (Jr 2r-) This and Equation (III-5) gives , N = p a A t (0.31) . *o o n v (III-8) For a pulse time of 3 msec, a circular nozzle of diameter 1 mm, 1 atmosphere 18 and 300°K source conditions, there will be 4 x 10 particles released. 4 Since the particles are traveling around v = 3.8 x 10 cm/sec, there are 27 about one tenth of this total inside the beam waist at any one time. Also, because we use a seeded beam only 2%-5% of this number are dipolar molecules. If one is interested in the molecular dimers, the number of these dimers formed is then again some fraction of the number of molecules. Just what fraction of the molecules are dimers has been the subject e i, 4.T. * • .- 64,65,66,67,68 _. . .. . , of much theory and experiment. It is generally accepted that dimer formation proceeds by a three-body collision, k l M + M + M^ D + M, k„ where D is the dimer. The third body is needed to carry off excess kinetic energy, allowing the complex to fall into the shallow potential well. The forward rate constant is proportional to the termolecular collision rate and the back reaction is proportional to the bimolecular collision rate. The termolecular collision frequency is proportional 2 to p D, where p is the pressure or number density at the nozzle and D, as before, is the nozzle diameter. Although this simple model would predict a plot of dimer concentration versus pressure and diameter to fit as pD where q is 0.5, measurements are close. Reports of measureg*7 £C CD ments on dimer concentrations find q = 0 . 5 5 , q=0.63, q = 0.5 and 69 q = 0.55. Maximum mole fractions of dimers have been reported to be 65 67 as high as 0.1 for certain expansion conditions. ' Other factors af- fecting dimer concentration are the source temperature, the geometry of the nozzle ' and clearly the species of the molecule involved. Just as every molecule has a different boiling temperature depending on its polarizability, multipole moments, mass, etc., so every molecule has different expansion dynamics. 28 E. Cooling of Rotational Energy As mentioned before, energy in the internal degrees of freedom is converted into increased mass flow by binary collisions. collision frequency is proportional to pD. The binary Because energy exchange be- tween translation and rotation is very efficient, rotational temperatures should be equivalent to translational temperatures. An expression giving the terminal translational temperature of a beam can be derived from equation (III-3). The terminal Mach number can be given by 71 72 75 ' ' MT=£(X'/D)(1-^, where e is a collisional effectiveness constant. given as to be M For Argon this was = 133 (pD) ' so that equation (III-3) can be written 71 T = T [1 + 5896 (pD) 0 - 8 ]" 1 , where p is in atmospheres and D m cm. Using this formula and for our conditions, 1 atmosphere, 1 mm and 300"K,. the terminal translational temperature calculates to be 0.3°K. tures for various expansions Measurements of rotational tempera- 71 73 74 75 ' ' ' are higher but still in the 1°-4°K range. A study of the vibrational and rotational relaxation of I„ in 75 seeded beams shows a way to control the rotational temperature. Rota- tional temperatures from 3°K, I- seeded in Ar, to 66°K, I_ seeded in n-butane, were measured. Because the processes of dimer formation and rotational cooling cannot be separated, one will always have to trade off dimer concentration and beam temperature. A higher beam temperature 29 for dimer studies is needed to populate vibrational levels of the dimer. If a higher than ground vibrational level is populated enough to observe and assign rotational transitions in that state, this would enable us to get a far better idea of what the true dimer potential energy well is like. No one has seen any of these vibrational satellites for rare gas small molecule van der Waals complexes although they have been seen in hydrogen bonded type complexes such as HCN-HF. 76 Presumably the reason no one has seen these vibrational satellites is that the beam is too cold; the HCN-HF complex was observed in a cooled static gas cell. Vibrational cooling proceeds with a slower rate and hence final vibrational temperatures are always higher than rotational temperatures. 75 One can bracket the rotational temperature in the cavity experiment by measuring several rotational transitions. A molecule can be picked that has several transitions at nearly the same frequency but one that has the energy levels of these transitions spanning a large energy range. Since the molecules populate the rotational levels by the Boltzmann factor, one can observe various transitions whose energy levels are at successively higher energy levels until one can no longer observe the transition. states of a symmetric top molecule could be conventionally used. m K Even a beam expansion, molecules are observed to follow the Boltzmann dis- tribution. For later reference, the number density difference between two rotational levels will be calculated. j th The number density of molecules in the .. state is, N where p = PJ 211 exp(-EVkT) is the total molecular number density, g is the degeneracy of 30 the level, q the partition function and E_ the energy of the level. The u number density difference of molecules available for the transition J •*• J + 1 is then AN = N_ - N - O J J*rl = (p /q)(2J+1)(e" E J /kT -e" E J+l /kT) (III-9) O As an example, consider the pseudo-diatomic van der waals molecule KrHCl with a rotational constant B a 1200 MHz and energy levels E_ = u hB J(J+1). is The rotational partition function for a diatomic molecule 77 q = | (1 + 1/3 <f> + ^ ( | ) 2 + . - . . ) , where 9 = hB/k, the equivalent rotational temperature. valid only for T > 6, for our case 9 = 0.06°K. This formula is To show the effect on the number density difference that temperature makes, we will calculate AN /p for several values of J and two temperatures. 0.5°K 300 °K ANQ(0,l)/po 2.3 x 10~ 2 7.4 x 10~ 8 ANo(3,4)/pQ 1.2 x 1 0 _ 1 2.1 X 10""5 ANo(9,10)/Po 5.9 x 1 0 - 5 1.4 x 10~ 5 31 IV. MOLECULAR POLARIZATION A. Introduction The last of the three main elements of the spectrometer will be discussed m this chapter. The groundwork and introductions to the theory as OO "3/ 3C we use it have been done. ' ' This theory was developed for pulse Fourier Transform Spectroscopy in a waveguide cell with a static gas. The extension of the theory to a Fabry-Perot cavity with a molecular beam is the subject of a forthcoming publication. Although this chapter will ignore most of the subtle changes that occur, it will give most major effects and point out the kinds of differences one would expect. are presented and discussed first. P The Bloch equations is re-expressed as the solution to a driven and damped harmonic oscillator equation. Since P is mainly respon- sible for the detected signal, its time behavior is followed from pulse preparation to emission. The solutions to the Bloch equations used in this chapter are for traveling waves and have been given before. ' ' Although a standing wave can be written as the sum of two traveling waves oppositely directed, the polarization for the standing wave in the cavity cannot be analytically re-expressed as the sum of two polarizations from two traveling waves. Even though the polarization cannot be exactly obtained from the traveling wave solution, it is a good approximation and will be used'here. Other approximations used m this chapter are that the molecules are consider- ed to be stationary during their excitation and emission, and that the fields are of uniform strength inside the beam waist diameter. 32 A. Bloch Equation's A derivation of the molecular polarization that is produced by pulse excitation of microwave rotational transitions was first done by Dicke. 27 Derivations similar to those done in nuclear magnetic resonance have been developed in this group. ' ' 78 theory These same types of equations are used in the infrared and optical regions of the spectrum as well. 79 80 ' The three coupled differential equations that describe the coupling of the molecular dipoles to the radiation field are P P + Ato P. + ~ 1 P. - ACOP i r 2 +K E 33 = 0 (IV-1) 2 fejd +I±-o o \ 4/ T2 (1V-2) , . /AN-AN \ S AN - E P. + T ° = 0 m 4 o i 4 \ T, / (IV-3) These equations are written in a reference frame rotating at the applied frequence so that there is no time dependence of the electric field. K is 2y. E /h, r ij o u r i;j is the dipole moment matrix element and Ato is c to - CO. to is the rotational transition angular frequency. AN is the population density difference between two rotational states and AN is that difference at thermal equilibrium. T1 and T_ are phenomenological relaxation times. T accounts for the relaxation of the population difference to equilibrium. the loss of the coherent polarization. T„ accounts for There are many mechanisms for these relaxations, the most common one being collisions. Because kT at room temper- ature is so much larger than hv at microwave frequencies, most collisions have enough energy to change the rotational state of the molecules involved. 33 This change of rotational state thus contributes to the T relaxation of the population difference and also to the T 2 relaxation of the coherent polarization, as the relative phase of the rotational oscillations is randomized. For our nozzle expansion to first order the collisions are drastically reduced, so that other types of relaxation processes are important. These processes are Doppler dephasing, spontaneous emission and transit time effects, which will be discussed more fully later. P ization. and P P are the real and imaginary parts of the macroscopic polar- is the in phase (in phase with the driving electric field) part and allows for dispersive effects such as wavelength changes in a dielectric. Just letting the cavity fill with air up to atmospheric pres- sure, from 0.01 microns, casues the cavity fill with air up to atmospheric pressure, from 0.01 microns, causes the cavity resonant frequency to change 6 MHz. This means that the change m the cavity resonance frequency due to a gas pulse at about 10-20 microns average pressure would be around 2 x 10 times the 6 MHz or around 100 Hz. This change is well below our normal line position uncertainty ( 1 - 5 kHz) and is ignored. P is the out of phase component and accounts for power flow from the field to the molecules and back. It can be simply related to the absorption coefficient of a gas, Y, by T B. p. _ 4irto i " " c E o Transient absorption The three coupled differential equations are usually solved by Laplace transforms for P., P and AN. l r 33 For the case that T. = T 0 , a simple second 1 2 ^ order differential equation can be written for P . This is accomplished by 34 differentiating again with respect to time equation (IV-2) and substituting into this the three original equations, to give P i + ( 2/T 2) P i + (AW + 1/T 2 + o ' Pi = 4T"" 2 ' (IV ~ 4) Because this equation is also written in the rotating frame, the right hand side has no time dependence. In this frame, when the electric field is turned on it looks like a step function. Initially, at time zero, P and P are both zero and AN is at its equilibrium value, so that, P.(0) = 0 P (0) = -K2£ o (^ • from eq. (IV-2). When these initial conditions are used to solve the driven and damped harmonic oscillator equation for P, we get l ilK2AN E ( l / T J [ e P (t) = o o 1 1 -t/T 2 (costO't-tO'T n sintO't)-l] 1 4[(l/T2)2+tO'2] where co' 2 = Ato2 + K2E 2 . o In the pulse Fourier transform experiment one naturally wants to maximize P and due to the exponential decay this only happens at times very short compared to T„. In the limit that T. = T 2 = °° equation (IV-4) reduces to, P. + (Ato + K E i o )P.=0 i , and using the same initial conditions the solutipn is , hK2E A N P i(t) = - 40)' ° s i n M>t ' 2 2 2 2 where CO' = Aco + K E . in addition, the experiment is operated so that o Kff » Ato, which further simplifies the result to be 35 P. (t) = i KE ftKAN r-£ sin KE t. 4 (IV-5) o is called the Rabi frequency. For microwave transitions, where the dipole matrix element can easily be 1 Debye and the electric field easily 5 volts/cm the Rabi frequency is, Kff = 2\X E fa = 31.6 MHz . O "1J o " Since the bandwidth of our cavity is around 1 MHz, if the oscillator is exactly on the cavity resonance the molecular transition has to be less than 500 kHz away, which makes Aco = 2TTAV around 3 MHz. A similar derivation for AN can be done using the same assumptions, AN(t) = AN cos Kff t o o . (IV-6) At time equal to zero, the polarization is zero and the population difference is at its equilibrium point. When the electric field is turned on the polarization starts to build and AN falls, at Kff t = TT/2 the polarization is maximized and the population difference is zero. This 7T/2 pulse is the normal operating condition for pulsed Fourier transform NMR, microwave and optical spectroscopy. In a waveguide microwave experiment where the elec- tric field is fairly uniform this maximum polarization condition is easily seen, for constant input power the coherent emission signal magnitude follows the sinusoidal behavior in equation (IV-5) as one varies the pulse length. In the cavity experiment where there is a standing wave field the electric field goes from zero to its maximum in 1/4 of a wavelength. Mole- cules that are at a anti-node see the large electric field and are rotated t through the angle 0 = KZ? t or more correctly 0 = J KB dt. Molecules at a o 36 node see zero electric field. So that for a static gas one has a distri- bution of polarized molecules from zero to a maximum. The signal from this distribution of polarized molecules in the cavity increases as one increases the pulse length, goes to a maximum but generally levels off as the pulse length is increased further. Increasing the pulse length past the IT/2 point, according to our simple equations, should result in a decrease in the magnitude of the polarization. At KE" t = TT the polarization should be zero and the population difo ference should be the negative of its original value. Since the population is now inverted one can give an average time it takes a molecule to absorb a photon. TT/KB If every molecule were in the lower level at time zero, at time every molecule is in the upper level. All of the previous discussion was in the rotating frame. In the lab frame the electric field does not appear as D.C. so that the driving term m equation (IV-4) will be oscillating at to. The polarization will be oscillating at to and modulated by Kff . D. Transient emission The effect of the microwave pulse is to create a macroscopic polarization in the gas sample. The molecules in each successive field anti-node will be polarized X/2 wavelength apart and 180° out of phase. Classically such a phased array of dipoles will coherently radiate in a direction along the line of the anti-nodes. Each molecule is phased in such a way that all emissions add up (in the absence of movement) together. This coherent emission from the molecules after the microwave pulse has passed occurs at the rotational transition frequency. Also the power due to the coherent emission is proportional to the number density difference squared. 37 Returning to equation (IV-4) to obtain the time evolution of P. after the pulse we set E = 0 and get P\ + (2/T2) P ± + (Ato2 + 1/T22) P x = 0 . (IV-7) Equations (IV-5) and (IV-6) are used as the solutions for the polarization and population difference that describe the preparation of the system prior to pulse cut-off. The pulse length, X , is chosen to be of a length ir such that KE" X = Tr/2. With this idealized situation the values of P , op i' Pr'. A N and Pi at t = xp will be, hKAN o 4 P (X ) 1 p P v(x ); * 0 r p AN(x ) = 0 P vv = h.KAN o 4T„ The last equation follows from equation (IV-2). The solution to the damped harmonic oscillator equation (IV-7) for times t > X using these initial IT c o n d i t i o n s i s then, -t/T P (t) = e hKAN ——• cos AtOt . (IV-3) In the lab frame Ato is replaced by to , the rotational transition frequency. This polarization is what we use to drive the electric field in the cavity. The spatial part of this polarization is given by the field modes for a Fabry-Perot cavity. This part is not needed to obtain the signal that is coupled out and detected. The magnitude of the time dependent part of the electric field is obtained by solving equation (II-l), 38 to A(t) + -~ W L A(t) + W 2 A(t) = - 4TT G(t) , C where P.(t), in equation (IV-8), will be used for G(t). P (t) as explained before, will be ignored. In taking the second derivative with respect 2 to time, of P., only the term with to will be retained as to » 1/T„ 10 4 (10 versus 10 ). Substituting this into equation (II-l) gives to A ( t ) + ~ A ( t ) + CO Q_ c -t/T A ( t ) = 47T tO o e 2 = to P ,. c o s to t O lM o hKAN r-24 COS to t o . The transient solution to this differential equation will have the term, exp-(to t/Q ) , in it, the buildup or decay of the fields in the cavity. Because this time Q_/to ~ 0.2 ysec is short compared to the exponential fall off of the driving fields T„ ~ 100 ysec, the cavity fields will be considered to be steady state. In the steady state solution to A(t) we will take cos to t as the only time dependent driving term because to » 1/T„. With these assumptions, the well known solution with initial conditions A(0) = A(0) = 0 is, to A(t) = - where 2 P„ 'M 2 _ .-XM_ /co to\ 2 1/2 / 2 2,2_, (to -to ) + c o 39 If the molecular emission signal is exactly on the cavity resonance, to = to , t h e n , c o ' 6 - TT/2 , and A ( t ) = p . M Q„ s m to t iM L o The electric field is E(r,t) = U(r) P.„ QT sin CO t lM L o The amount of power delivered to the cavity, P , by this process can R be obtained by integrating over the volume, that the molecules occupy, V , the time averaged product of the electric field and P (t), P R = J" < E *Pi> dV V s W o P = —— R 2 2 P V I> iM s 20 Since this is the only source of power we can calculate the energy stored in the cavity from the definition of the Q, W = Q^/tO = 1/2 Q 2 P 2 V o . L R c L lM s From the definition of the coupling Q we can find the amount of power coupled out of the cavity, P_, and available to our detector by substituting L in the value of W, 40 L ~ 2 c2 " to Q 2 PT (t) = — — — L 2 Qc2 2 2c2 L E. s , ~2t/T (TlhKAN ) e o B P (t) iM 2 = T 2L i + g ^ V s ,„_.. 4 „ . 2 " e (27r A H* V (IV-9) 2 t / T 2 v s Signal Characteristics 1. Spontaneous Emission One feature of the output power calculated in the last section is seemingly very unphysical, that being the dependence of the output power on Q. From conservation of energy the integral of the power over the time of the decay should be independent of Q. If we could collect all the energy emitted by the molecules and if there were only this one power dissipation process, then this integral should equal Nhto where N is the number of molecules emitting a photon of energy h.to. If there were only this one relax- ation process its rate would have to be proportional to Q so that the Q's would cancel upon taking the integral. The decay process has to depend on the molecules being in the high Q cavity and on the coherence between the molecules. It has been shown that the spontaneous emission rate is enhanced in a cavity. ESR theory. 84 This phenomena has been used in NMR theory 78 81 82 83 ' ' ' and in The Einstein A coefficient for spontaneous emission can be 85 written as A = B W(to) (exp (hCoAT) - 1) , where B is the Einstein B coefficient for stimulated emission and W(to) is 41 the electromagnetic energy density per frequency interval. The B coeffi- cient is obtained from quantum mechanics and is , B = iry2 ./3h2 • ij The energy density per unit frequency for free space is 3 W(C0) = "l^y (exp (hto/kT) - l ) " 1 . TT C When these two results are substituted in for A one obtains the usual expression for the spontaneous emission rate. But the molecules are not in free space, they are in a cavity with highly reflecting mirrors. The energy density per unit frequency in the cavity is enhanced by Q. W(to) for the 7 8 «. is • given • u cavity by 4Q Lh W(co) = — — -1 (exp (hto/kT) - 1) The other factor that leads to an enhanced spontaneous emission rate is the coherence established between molecular oscillations by the pulse. This coherence causes the emission rate to be faster by the factor N, the number 81 86 of oscillators. ' This enhancement in spontaneous emission rate by N occurs for molecules in free space as well. Combining all these factors gives for the A coefficient, Try? A = N A N A = 4Q h =J2 — v 3h 15lK2AN Q T V * o L s 3V where N was replaced by AN V . V o s s is the volume that the molecules (sample) occupy, and V is the field volume of the cavity, see equations (II-2) and (II-3). This additional decay process has to be combined with equation 42 (IV-9) for the power coupled out of the cavity. Lumping the molecular con- stants together, the equation is now, P T Q2 = *,n-FT- (t) L -2t/T e e , (IV-10) 10 Q c 2 where p . o = (1/2) (oovs (TftKAN/ . There are two more important effects that limit the time behavior of the emitted power. 2. Doppler Dephasing and Transit Time The Doppler effect in microwave spectroscopy causes a shift in the apparent resonant frequency of a molecule due to its translational motion. In a laboratory fixed frame, where the radiation is propagating along the +z axis the emission frequency of a molecule moving with a component of velocity along the +z axis will be higher than co by kv . Likewise a molecule that is emitting radiation along the +z axis but is traveling with a velocity component in the -z direction will have the emission frequency shifted down by kv to the fixed observer. As shown in chapter III section B there is some distribution of molecules leaving the nozzle p(r,0). This distribution will be simplified to D(0) and will be normalized so as to be dimensionless, D(0) = D cos 2n 9 . o Figure 3 shows v and 0, where v is the velocity of the molecules coming out of the nozzle, assumed here to be a constant, and v = v z sin0. o We now define a polarization for molecules at each angle 9 and both propagation directions, 43 -t/T P.(0,t) = e i hKAN — j - 2 - cos (to ± kv sin9)t . 4 o o The polarization is obtained by integrating this from 0 to TT/2 , Tf/2 P (t) = 21* P. (0,t) D(0) d9 . i « i o Substituting in and dropping the zero terms gives, -t/T P. (t) = e faiKAN 1 TT/2 )D cos to t P cos (kv t sin0) cos O O O «J o o 0d0 . This is a known integral and can be expressed as , -t/T P (t) = e where J (hKAN )D J (kv t) cos CO t — -— ° (kvt)n o is a Bessel function of integer order n. , (IV-11) The time domain signal from the J=0 to J=l OCS transition is shown in figure 4. There is only one transition but the beating due to the Doppler effect is clearly seen. Bessel functions are similar to sinusoidal functions in that both oscillate about zero. Bessel functions, apart from their first zero, have almost equal differences between zero crossings to all orders. Although J starts at one while all others start at zero, the major distinction between different orders is the location of the first non trivial zero. This property can be used to try to obtain the power of the cosine distribution from fitting observer time domain signals to equation (IV-11). Another result of this slow oscillation is that when the data is Fourier transformed there will be two peaks in the frequency domain. The peak sepa- ration in the frequency domain is then a measure of the velocity v . We have recently measured the microwave spectrum of OCS and the dimer Ar'OCS under identical gas dynamic expansion conditions. We found the ratio of the doublet frequency splittings to be equal to the inverse ratio of the- 44 Figure 4. The time domain digitized emission signal from the J=0 to J=l rotational transition in O C S . There are 256 points digitized at a rate of 1 ysec per point. The gas mixture is 4% OCS in Kr. ^VWW^'^W./W r 0 32 64 96 128 160 ^92 DIGITAL POINTS 224 256 46 masses of the two species. This is in complete agreement with equation (III-6) given for the flow velocity of particles in a nozzle expansion. The time domain and Fourier transform of the Ar'DBr, J=3, F=9/2 to J=4 F=9/2 transition is shown in figures 5 and 6 respectively. It is easier to explain a doublet if the molecules were pulsed down the cavity axis. Let us assume for the present that there is no angular distribution to these molecules and therefore v = v . z o The standing wave fields in the cavity can be considered a superposition of two oppositely directed traveling waves. The molecules in their own moving reference frame are emitting at one frequency but this frequency is Doppler shifted up by kv to the wave traveling in the direction of the molecules and shifted down to the other wave. There are therefore two emission frequencies present in the cavity which beat together and are transformed to a doublet. The peak separation in the frequency domain would be just 2kv , which for Kr velocities and at 10 GHz gives the separation to be 25 kHz. Adding in an angular distribution to this beam would just broaden out the two peaks. The fact that pulsing the gas perpendicular to the cavity axis also produces an effective beat (the Bessel function) is remarkable. The measured doublet separation for the perpendicular beam is about 80% of 2kv . This doubling is a great hindrance to the measurement of transition frequencies when the different transitions are separated by frequencies of the order of the Doppler splitting. The third effect of equation (IV-11) on the signal is the decay of the polarization produced by the term (kv t) . This decay is due to the distribution of the emission frequencies destructively interfering and is the dominant emission decay process. 87 of a forthcoming publication. The lineshape problem is the subject 47 Figure 5. The time domain digitized signal from the J=3, F=9/2 to J=4, 79 F=9/2 transition in ArD Br. There are 256 points at 0.5 ysec per point. The gas mixture is 4% DBr in Ar. 96 128 160 DIGITAL POINTS 192 224 256 CO The frequency domain Fourier transform of the data in figure 5. This spectrum is the sum of the squares of the real and imaginary parts of the transform. Also 256 zeros were added to the time domain data before transforming. The frequency axis is in MHz and the center of the dip gives the transition frequency. two peaks are caused by the Doppler effect. The \,V^/W, ^d 8764.9508765.000 8765.050 8765.100 8765.15 FREQUENCY o 51 The final limit to the duration of the emission signal is the finite time the molecules spend in the field regions of the cavity. Although it is not clear at which point to consider the molecules in or out of the field we will utilize the beam waist 2w to compute the transit time X . o T Use of the terminal velocity will give a minimum transit time, x m = 2w /vm . T o T For Krypton v 4 ~ 3.8 x 10 cm/sec and for 2w = 12 cm this minimum transit time is 316 ysec. For molecules that have strong signals such as monomer OCS seeded in Kr as shown in figure 4, the maximum time the emission signal lasts is around 250 ysec. To overcome the 300 ysec transit time limitation mirrors with larger beam waists could be used or the nozzle could be directed down the cavity axis. 3. Signal Detection The detectors used in these experiments are double balanced mixers. To describe what is detected by these diode mixers, consider the currentvoltage plot of one of these diodes. To explain the conversion of the microwave frequency to an intermediate frequency (IF), we expand the current in a Taylor series around zero voltage v , / N / i ^ /dA i(v) = i(v ) + (—J ° W v 0 ^ •> /-> A 2 A 2 . i A3 A 3 . v + 1/2 — ^ v + — —^-j v + ... , 6 \dv2/vo \dv3/Vo where di/dv has the dimensions of a conductance, g, and d i/dv will be written g . There are two inputs to the mixer, the signal coming from the cavity, v = v.. cos (to. t + <J).) and the local oscillator (LO) signal s i l l vT„ = v„ cos(co„t + d>-), so that v = v + v__. The term linear in v will 52 give frequency components to the current that are just the input ones. The quadratic term gives, (v +vr/J S IiO 2 2 2 2 2 = vn cos (to.t+tj).) + v_ cos (C0ot+tj>_) 1 1 + 2 v v - 1 costco^+^J /. <£ £• cos(to 2 t+^ 2 ) (v 2 / 2 ) (l+cos(2C0]t+2!J)1)) + (v 2 2 /2) (l+cos(2C02t+2(J>2)) + v v [cos ((to -to2)t+cj) -t|> 2 )+cos((to +to2)t+(jJ1+(j)2)] . There are current frequency components at 2to1» 2co2,to-^-to,,a n d W l+aV Similarly the cubic term will give current frequency components at a^, to„, 2C0n -to„, 2co. +to„, 2C0.-C0 , 2co +to , 3co and 3to . 2 1 2 1 2 2 1 *21 1 & So that in general, one can obtain all frequencies nto +mco where n and m can be any positive or negative integer or zero. The power in the higher harmonics will fall off depending on the diode but one can easily produce and use the n, m = 30 to 40 harmonics. The circuit of the mixer filters out all frequency components except to -to , the intermediate or difference frequency. The current output of the mixer is then, l = 1/2 g v x v 2 cos (to -to )t. Depending on the impedance in the output circuit a voltage will be produced by this current that is amplifier, mixed down again, and digitized. fore, it is the voltage or electric field that is detected. There- The electric field in the cavity output waveguide is related to the power in the dominant mode by -ST. 53 where P is the power coupled out of the cavity, a and b are the inside dimensions of the waveguide and Z is the waveguide impedance, _ 7 n ^r • In this formula r| is the impedance of free space, 377 ohms, and X is the c cutoff wavelength for the guide (A = 2a, TE mode). Combining these c xu results, the digitized signal is proportional to the square root of P as L given in equation (IV-9). In order to successfully digitize and Fourier transform the signal emitted from the molecules, the most important parameter that has to be optimized is the signal to noise ratio. The power signal to noise ratio is used because the noise figure is defined in terms of powers. The voltage signal to noise ratio is the square root of the power signal to noise ratio. Since we do not record the signal directly but mix it down, additional noise is introduced by the mixers and amplifiers. The ratio of the signal power to noise power, at the input of the mixer, to the signal power to noise power at the output, is defined as the noise figure of the mixer, F, S S./N - x x ~ S /N o o The noise figure of a mixer is a function of the internal parameters of the mixer, the losses in the circuitry, the quality of the nonlinear device, matching problems and local oscillator bias. For the input signal power, S , we take the power coupled out of the davity as given in equation (IV-9). The input noise power is given by kTB, where k is Boltzmann's constant and B is the noise bandwidth. then, The signal to noise ratio out of the mixer is 54 S./N. P N (IV"12> V o " "V^ - kTiF • The mixer is just the first stage of the detection circuit. through n devices in cascade the total noise figure F (F - 1 ) B T? F - w F T" I 4. + 2 ^i^ 1 l,n where F i (F,-l)B, 3 3,n 2,n + 1 2 l,n 88 (F - 1 ) B n n + ^G^ is For one •" * £ — i 7 ~ ' l,n /Tt7 n_x (IV 13) " l,n is the noise figure of the first device, B, is the bandwidth of l,n the first through nth devices and G is the gain of the first through ±,n nth device. The gain of several devices in cascade is just the product of the gains for each device alone. For a passive device like a mixer the gain is equal to 1/L where L is the conversion loss, L = S /S . i o As can be seen from equation (IV-13) if the g a m of the first element is high the total noise figure will be essentially the noise figure of the first element. The double balanced mixer we use has a low noise IF preamplifier built into it and has an overall noise figure of 6dB, or in other words a signal to noise degradation of four. This mixer also has a RF/IF g a m of 20 dB so that the gain of the IF amplifier divided by the conversion loss of the mixer is 100. Considering this mixer-IF amplifier as the first device, the second device is another amplifier with at most a lOdB noise figure and a gain of 27 dB. The bandwidth of the second through last device is essentially the bandwidth of the entire cascade because the narrowest bandwidth device is last. Using just these numbers the first term in equation (IV-13) is four, the second term 0.09 and the third term even smaller. 55 4. Resolution and Sensitivity The resolution, R, in frequency interval per point of a Fourier transformed time domain signal is just the reciprocal of the total time of the record, so that, R = (time per point x total number of points) One can add zeros onto the end of a time record and increase the resolution but of course no new infromation is gained. In order to observe smaller splittings the time that one observes the molecular emission has to increase. If the emission from the molecules were a simple exponentially damped, exp(-t/T 2 ), sinusoidal then the half width at half maximum of the transformed line would be l/(2Tf T_). The basic limitation is the transit time through the cell, although the Doppler dephasing is now the major limitation. If we could attain a 300 ysec T„ which would entail say a 500 ysec record length then the frequency per point would be 2 kHz. We mostly operate at 0.5 ysec per point and take 256 points, giving a 128 ysec record length. Two definitions of sensitivity will be used here, the first in terms of a minimum number of molecules and the second in terms of the smallest dipole moment matrix element detectable. These calculations will assume the power signal to noise ratio, single shot, to be one, at a time t = T„. Equation (IV-9) with the simple exponential decay will be used for P_, a similar calculation could easily be done using P_ as given in equation L (IV-11) with a similar result. Use of the time domain signal to noise, single shot is in keeping with the way signals were usually found. The signal to noise improvement by averaging many scans and by Fourier trans89 .. forming with filter functions has been given before. Combining equations 56 (IV-12) and (IV-9) and solving for the product of the dipole moment matrix element times the number density gives, F u AN 11 where Q /Q =( 2 ™ ° UA-VB^L (1+0 +B >\l/2 r-*-\ (IV 14) ' ~ 2 = 32/(l + 3, + 3-) and K = 2y. fa were also used. A numerical value of the right hand side of equation (IV-14) can be obtained by substituting in T = 300°K, B (the overall bandwidth) = 300 kHz, 4 F = 4, to = 2TTV = 2TT x 10 GHz and XQ_ = 10 . For V_, the sample volume, c o o L S we use the expression given in equation (II-2) and (II-3), the field volume, V s = V = TfJl (w / 2 ) 2 . o 3 3 Using A = 70 cm and w = 6 cm the volume is 1.9 x 10 cm . One would like to minimize the function (1 + 3, + 3-)/32, the smallest it gets is unity, when 3 2 = °°- For this calculation we will take 3T = 3~ = 1 which makes the function equal to three. Using the above values we get, yH ij -13 2 AN = 2.16 x 10 statcoul/cm . o If we assume that the molecules are evenly divided through the volume V, then the number of molecules, N, needed for a given rotational temperature can be calculated from N = p V, and equation (III-9). Using y.. = 1 Debye and AN(0,1) at 0.5°K and 300°K from the calculations following equation (III-9) we get for a J=0 to J=l rotational transition, N = 1.7 x 10 molecules at 0.5°K 15 N = 5.4 x 10 molecules at 300°K . 57 Going back to the calculations following Equation (III-5), we gave the 14 3 number density at the cavity center to be 6 x 10 particles /cm . If we 13 -3 are using a seeded beam of 2% molecules we have p = 1.3 x 10 cm . Then using AN(0,1) at 0.5°K and 300°K as before the minimum dipole moment matrix element required is, y.. = 7.8 x 10~ 7 Debye at 0.5°K ID y =0.24 Debye at 300°K . ID In order to have a polarization bandwidth B = 2y. E fa equal to the -7 cavity bandwidth, 1 MHz,with a 7.8 x 10 Debye matrix element one would need an electric field of around 200 k volts/cm. We have previously calculated a peak electric field of 300 volts/cm with 40 watts input power. With this electric field the smallest matrix element that gives a polariza-4 tion bandwidth the same as the cavities bandwidth would be 5 x 10 Debye. These calculations show that if the rotational temperature is 0.5°K we need only 1.7 x 10 molecules in the cell to observe a signal with the 5 given S/N ratio. We would need about 5 x 10 more molecules if the temperature were 300°K, to get the same S/N. The calculation of the smallest molecular dipole matrix element required to observe a transition shows our ability to study small dipole moment species such as Ar'Kr. 58 V. A. THE SPECTROMETER Introduction This chapter discusses the synthesis of the cavity, the pulse Fourier transform and the pulsed molecular nozzle into a working spectrometer. Underlying the feasibility of the experiment, x, x and P T„ have to be ordered as T_ > X > X. The polarization has to remain c 2 2 p coherent for a time, T„, longer than the microwave pulse time, x , 2 p which has to be longer than the cavity decay time X. The block diagram is discussed first, so that how the spectra are measured should be clear. This microwave circuit is an analogy of the 90 classical Rayleigh refractometer. Several of the features that govern the design and construction are discussed next. The section on molecular spectra traces the major results that have been generated since the first observation of a van der Waals molecule. In the potential of the method, some possible future experiments that seem feasible will be outlined. Several areas are mentioned where application of this method could contribute to further chemical insight. The last section discusses the general advantages and disadvantages of the three techniques, whose combination has yielded this new spectrometer. B. Block Diagram The block diagram of the spectrometer is shown in Figure 7. The master oscillator (MO) is frequency stabilized by the lock box (LB) to The block diagram of the spectrometer. (MO) master oscillator, (1GHz) a one GHz signal, (VFO) variable frequency oscillator, (COUNT) frequency counter, (LB) lock box, (LO) local oscillator, (Ml) harmonic mixer 1, (ATT) attenuator, (AMP) amplifier, (ISO) isolator, (M2) mixer 2, (PIN1) PIN diode 1, (C) circulator, (SST) slide screw tuner, (DET) diode detector, (DRV) valve driver, (PUL BOX) pulse box, (A/D) analog to digital converter, (DIS) display, (AVE) averager, (COM) computer. 60 UJ •**• 8 d5 Q- QQ ro CVI > cr T\1 J E o CO —*" CVI z Q. 8 61 two lower frequency standards. A one GHz signal is produced by multi- plying up a 20 MHz oven controlled crystal oscillator. This crystal has -9 a frequency stability of 1 x 10 parts per day so that after multiplying it up by 50, the 1 GHz should be stable to Hertz. A variable frequency oscillator (VFO) is also used for continuous frequency adjustment within its range. We use a 10 MHz to 480 MHz (in 5 bands) tube oscillator that is stable to hundreds of Hertz over minutes. This VFO is counted (COUNT) by a standard seven digit frequency counter. The mixer (Ml) mixes the first harmonic of the VFO and the n oscillator. harmonic of the 1 GHz with the master An IF frequency of 30 MHz is produced that is fed into the frequency stabilizer. This frequency stabilizer compares the IF frequency with its own 30 MHz crystal and gives an error voltage whose magnitude is proportional to the frequency difference and whose polarity depends on whether the IF frequency is below or above 30 MHz. When one inputs the same frequency as the internal crystal, the error signal is zero. If the VFO's frequency is changed so that the mixed down frequency is no longer 30 MHz, the frequency stabilizer will apply a voltage to the MO, changing its output frequency so as to cause the mixed down signal to once again be 30 MHz. Now we can not only control the MO's frequency, but by counting the VFO's frequency and by knowing which harmonic of the 1 GHz we are using, we can calculate the MO's frequency to hundreds of Hertz. The local oscillator (LO) is locked by another frequency stabilizer to the master oscillator through mixer (M2). Part of the 30 MHz signal that is fed into the second frequency stabilizer is split off to be used as a phase coherent reference signal. Both the master and local oscillator are backward wave oscillators (BWO) and when locked, they have the 62 frequency stability of the VFO. BWO's have good bandwidth, a relatively level power output and can be easily swept. This last feature is convenient when working with cavities, as the cavity resonant shape and position are often adjusted. Cavity coupling information can also be obtained by sweeping across the cavity resonance. Since both the master and local oscillator are running continuously, the pulse to the cavity is formed by opening PIN diode 1, (PIN 1) for a time, t . This pulse of microwave energy, typically t_ = 6 ysec, goes through the circulator (C), slide screw tuner (SST) and impinges on the input coupling iris of the Fabry-Perot cavity. Any energy that is reflected from the cavity travels back up the waveguide and is routed to a detector (DET) by the circulator. The output of this diode rectifier is fed into an oscilloscope which is triggered in synchronism with the PIN diode pulses. By looking at the reflected power as displayed by the oscilloscope, several things can be learned. If the frequency of the MO is not the cavity resonant frequency, all the power incident upon the cavity is reflected and the output of the detector traces out the microwave pulse envelope. The PIN diodes we use have rise times (10-90%) of 10 nsec and if one uses a fast detector such as a tunnel diode that can follow this rise time, the pulse envelope shape will almost be rectangular. When the MO or cavity resonant frequency is changed so that one is driving the cavity on resonance, the pulse envelope shape is different. Since the cavity energy builds up and decays with a time constant x = 160 nsec, as given before, we can see the process described in Chapter II, Section C. When the input wave hits the iris, there is a reflected wave causing a large signal at the detector. As the cavity energy builds up, this reflected wave is increasingly 63 cancelled by the fields coupled out of the cavity so that a spike with an exponential decay is seen at the detector. The voltage that the decay levels off to is indicative of the coupling. When the input energy is switched off, the input wave reflecting from the iris is gone, so that only the wave coupled out is seen. This causes another spike with an exponential fall off as the energy dissipates from the cavity. Energy during the pulse is coupled out the opposite mirror. To protect our detector mixer (M3), we have another PIN diode (PIN 2) that blocks this energy. diodes are shown m The TTL trigger pulses that operate these two PIN Figure 8. The diodes conduct when the TTL level is +5 volts and have a 80 dB isolation when the level is zero. As seen in Figure 8, PIN diode 2 reflects any power coupled out of the cavity when PIN diode 1 is conducting. The time t, is adjustable and the time t . is equal to t_ plus another adjustable amount to make sure all of the original power is dissipated. The pulse box (PUL BOX) forms these pulses and the pulse to the driver (DRV) that operates the molecular pulse valve. The pulse length to the molecular valve, out of the driver, is about t. = 3 msec. This valve is a commercial m-line solenoid valve. valves have been described in the literature improve on ours. Faster 91 92 ' and we should eventually The time t. + t 2 is an important parameter, in this duration, the mechanical valve has to open and the molecules have to travel to the field regions of the cavity. If you look at the molecular signal as a function of this time interval, at short times you will see nothing, then there will be a fast rise in signal amplitude and a fast fall to about half height m 1.5 msec, then a slow tail for about 5-10 msec. This delay, t. + t„, has to be adjusted for each different carrier gas and many times for each gas mixture, to optimize the signal amplitude. *3 +5 — 0 — (I) +5— (2) 0 — *2 <—: >• +5 — (3) 0 Figure 8. The timing sequence of PIN diode 1(1), PIN diode 2(2), and the drive to the molecular valve (3). en 65 The pulse to the molecular valve is repeated at a rate from one to ten Hertz, depending on the carrier gas, back pressure and nozzle size. The number of particles released was given by equation (III-8) and as an 18 example was calculated to be 4 x 10 particles per gas pulse. ten inch diffusion pump that can pump around 4 x 10 19 We use a particles per second, hence the ten Hertz repetition rate. The pulse sequence to the PIN diodes occurs at twice the solenoid valve rate. One microwave pulse interacts with the molecules in the free expansion and the next microwave pulse interacts with the evacuated cavity. If we call the frequency that the master oscillator is locked at, V, then the local oscillator frequency is V - 30 MHz. The pulse of microwave energy creates in the expanding molecules the macroscopic polarization which emits radiation at the rotational transition frequency, v . Both of the frequencies, V and V , have to be within the bandwidth of the cavity, so that |v -vj = A « 1 MHz. Usually A is between 10 kHz and 600 kHz. After the power pulse dies away, PIN diode switch 2 opens and the molecular emission mixes with the local osciallator m mixer (M3). The signal out of mixer (M3) is at a frequency 30 MHz ± A, depending on whether v > V V < V . or This is amplified and mixed down again in mixer (M4). The other input to mixer (M4) is the 30 MHz signal we obtained before by mixing the master and local oscillators. If we did not use such a reference signal here, the decaying emissions from the molecules in the cavity from pulse to pulse would be arbitrarily phase shifted relative to one another. means we could not average or add up all the molecular emissions. This The signal out of mixer (M4) is at the frequency A, or the offset between the master oscillator and the molecular transition frequency. This signal is filtered and amplified and then fed into our analog to digital converter 66 (A/D)• The (A/D) is triggered by the pulse box to digitize a signal every time the microwaves are pulsed. The digitizer we currently use is only a 6 bit converter so that our dynamic range per pulse is only one in 64. This will be shortly replaced by a 10 bit A/D that gives a one in 1024 dynamic range. After the signal has been digitized, the data is transmitted to an averager (AVE). This averager adds and subtracts alternate pulses so that all molecular emissions are added together and all alternate background scans are subtracted. As mentioned before, in the section on resolution in Chapter IV, the digitizer we currently use has 256 points and by using 0.5 ysec per point, the entire data scan takes 128 ysec. Since the molecular pulse valve provides observable molecules in the cell for times longer than this, we also have the option of taking n adds and n subtracts in one molecular valve period. After the averager memory fills, the data is sent to the departmental VAX 11/780 computer which does a fast Fourier transform and returns this data to the averager. Either the time domain or Fourier transformed data can be displayed from the averager. Because of the way we mix down the molecular emission signal, the start of the frequency domain data is at the master oscillator frequency. By counting the points over to a molecular peak we find A and therefore we know the transition frequency V . Searching for unknown molecular lines consists of stepping the cavity mirror separation, hence its resonance frequency, and following along with the microwave oscillators. Since the cavity bandwidth is around 1 MHz, the step size is 500 kHz or less. Searches may easily involve going over a GHz or more so that many, many steps are taken. on ways to automate this stepping procedure. We are presently working 67 C. Design Considerations The two mirrors used are solid aluminum type 6061 disks with a spherical concave mirror surface. It is very easy to machine a spherical surface in a plate and mirror alignment is not critical. The radius of curvature of the mirror is found by making the Fresnel number unity for the lowest frequency, 2 RX lf where a is the mirror radius, R the radius of curvature and X is the wavelength. This one requirement insures a good Q because the mirror then captures over 95% of the wave amplitude at any point. In Chapter II, Section A, we gave the complete expression for the Gaussian standing wave. We also learned that at the confocal arrangement of the mirrors, R = SL, the beam waist, 2w is maximized. o So the worst case to consider for diffraction losses is the confocal arrangement. For this spacing, the formula for w(z=£/2) and w w o simplify to, = (RX/2ir)1/2 w(z=V2)= /2"w . o " 2 The electric field falls off as exp - (r/w(z)) as a function of z, the cavity axis and r, the radial direction. z = %/2. At the mirror surface 2 By using the requirement a /RX = 1, one is requiring, a = RX , or a = J^RT = W Q / 2 ? = w(z=V2)/F. 68 Since r = a at the mirror edge and z = Si/2, exp-(r/w(z)) 2 = exp(-ir). The field amplitude is 0.043 times what it is on the cavity axis. Ex- perimentally when the field amplitude is greater than 0.13, one can notice effects of the field spilling out of the mirrors and reflecting around the vacuum chamber. We have used a set of mirrors designed for 8 GHz from 4.5 to 18 GHz. The mirrors could be used at even higher frequencies but other microwave components of the spectrometer will not operate there. Flat mirrors have also been tried but due to the very critical parallelism adjustment, high Q's were never achieved. The coupling iris is a round hole centered along the waveguide axis. Part of the back of the mirrors are machined out so that the waveguide can butt up against an area that is from ten to twenty thousands of an inch from the front mirror surface. The coupling hole is centered in this area and centered in the front mirror surface. The inside dimensions of X-band (8-12 GHz) waveguide are 0.4" x 0.9" and the coupling hole is about 0.38" in diameter. For lower frequencies, a coaxial connector device replaces the waveguide section. Then antennas are inserted from the inside of the cavity into this coaxial connector. Simple straight wire configurations with the wire coming out of the connector and bending 90° along the mirror surface work well. The length of the wire and its distance from the mirror surface are adjusted to control the coupling, just as the circular diameter size controls the coupling in the waveguide feed mentioned first. The mirrors are 14 inches in diameter and are suspended on four oneinch diameter stainless steel rods. The rods run the entire length of the vacuum chamber, which is an 18-inch outside diameter tube, 41 inches long. 69 The rods connect two end plates that are bolted onto the main tube. There is a ten-inch port coming off at the middle of the main tube that a teninch diffusion pump is connected to. Directly above the ten-inch port is a six-inch flange where the pulse valve and associated electrical and gas feedthroughs are located. The mirrors ride on the four rods and are connected to the outside by waveguides that feed through end caps of the 18-inch tube. One mirror is held fixed while the position of the other mirror is adjusted by a rack and pinion and gear reduction mechanism. This mirror translation mechanism has to be made with some care m order to step the cavity resonant frequency 500 kHz as explained before. Considering the dominant mode, the resonant frequency was given as V = (c/2A) [ (q+1) + (l/7r)cos"1(l-VR)]. The difference between two closely space frequencies Av = V.-V„ is given by Av = (c/2)(l/£1-l/£2)(q+1). The movement of the mirrors AS. = &2~^-\ ^s 2 ^1^2 ~ ^ ^ AO. - 2Ava2 * ~ c(q+l) ' A where & is the distance between the mirrors and q + 1 is the number of half wavelengths between the mirrors. As an example, for % = 70 cm, X = 3.5 cm, so that q + 1 = 40 and Av = 500 kHz, the mirror movement is 41 ym. This points to another concern: mechanical vibration can vary the cavity resonant frequency and thereby can amplitude modulate the output signal. An amplitude modulated signal is demodulated in the mixer 70 and contributes to low frequency noise in the receiver. The diffusion pump with its boiling silicone oil contributes to noise in this way. The filter amplifier after mixer (M4) has a low frequency roll-off of about 1 kHz and blocks most of this noise. D. Molecular Spectra The first van der Waals molecule to be seen was Ar*HCl which had 93 94 been assigned before. ' The only technique for studying rotational transitions in van der Waals molecules up to the development of this spectrometer was the molecular beam electric resonance spectrometer. This technique is severely limited in the number of possible transitions it can observe because of the use of Stark focusing. While only one of the quad- rupole components of the J=2 to J=3 transitions were observed with the electric resonance machine, we could easily see the full seven-line multiplet. The next van der Waals complex to be observed was Kr*HCl. molecule had never been assigned before. This The complete structure and molecular constants of this molecule were published along with an introduc39 tion to this new method. Within two months, two more molecular complexes were observed and assigned. The rotational spectra, molecular constants and equilibrium structure of Ar«HBr and Kr*HBr was the subject of this next paper. 95 A number of rotational lines of other complexes had also been observed but never assigned. These include the van der Waals molecules Kr'CHJF, Ar«CH3F, Xe'HBr and Ar«CF,Br. The next paper 96 van der Waals bond. attempts to g a m insight into the nature of the In this paper, the Kr*HF complex was assigned and for the first time the quadrupole splitting caused by a rare gas atom was 71 observed. spins. The 82, 84 and 86 isotopes of the Kr atom have zero nuclear The fluorine and hydrogen atoms have nuclear spins of 1/2 and therefore have no quadrupole splitting. The 83 isotope of Kr has a nuclear spin of 9/2 and so will couple to the rotational angular momentum to give a quadrupole multiplet. This multiplet splitting is proportional to the coupling constant which in turn is proportional to the product of the nuclear quadrupole moment (a known constant) and the field gradient at the nucleus due to the electrons. In the free Kr atom this field gradient is zero and therefore the coupling constant is zero. But, due to the bonding in the complex, the field gradient at the nucleus is distorted and this results in a measureable quadrupole coupling constant. What this changed field gradient means in terms of bonding is discussed. Further work done in this connection is presented in a paper just submitted 97 for publication. Here the two quadrupole complex 1 "^1 Xe'HCl is analyzed. The series of complexes, CO bonded to HF, HC1 and HBr has also been observed and assigned. 98 These molecules are interesting because it is the carbon of the carbon monoxide that binds to the HX as OC"**HX. Although the hydrogen has a vibrationally averaged position not colmear with the OC---X, its potential minimum is colinear. Work is continuing on these three complexes to completely elucidate their structure. Other molecules that are being studied included HF'HCN and Ar-HCN, as well as others. E. Potential of the Method One point that I think should be made clear is that all the work described in the previous section has been done in less than one year. There is considerable work to be done on just refining and developing 72 new instrumental techniques to make this method even faster, easier and more versatile. The major contribution of this spectrometer so far and for some time to come is in the structure and bonding of weak molecular complexes. The vibrationally averaged bond distances and angles are obtained from the rotational constants. fitted to a distortion constant. The rotational lines are also Since all the spectra are taken in the lowest vibrational level, the harmonic approximation to the potential is reasonable. In this approximation, the distortion constant can easily be related to the vibrational stretching frequency of the van der Waals bond. Using this stretching frequency and the reduced mass, an effective force constant can be calculated. Assuming a form for the potential and these parameters, one can get an estimation of the well depth of the complex. 39 95 ' The measured quadrupole multiplet and coupling constant gives structural and bonding information. Other interactions such as spin-spin and spin-rotation also give information into the nature of the bond. As well as dimer complexes of van der Waals or hydrogen bonded molecules, one could look for trimers or n-mers. seen by laser fluorescence techniques 99 Higher complexes have been and in mass spectroscopy. Although we have concentrated on molecular complexes, monomers can obviously be studied. Due to the sensitivity, one can observe most isotopes in natural abundance. Although one would want to discourage dimer formation, by using He or Ne, use of a seeded beam still provides a larger signal for some cases. It has been shown experimentally, for a beam expansion employing pure molecules, the beam temperature was measured to be 30°K. For an expansion using 5% molecules seeded in Ar, the beam temperature was measured to be 3°K. If we use these two tempera- tures for a diatomic molecule with a rotational constant of 1200 MHz, we 73 can compare the values of AN the two cases. and therefore the signal strength in For a J=0 to J=l rotational transition, we gain a factor of 20 in the number density due to the pure molecular expansion, but we lose by a factor of 100 in the partition function times the Boltzmann factors due to the higher temperature. The temperature effects nearly cancel for a J=9 to J=10 transition. Spectroscopy on heavy molecules that have low vapor pressures could be done by vaporizing them in a high temperature oven that could easily be placed in the vacuum chamber. Another possibility that is being pursued is to use a high temperature nozzle source to form rare gas metal dimers. Some of these types of complexes have been observed by molecular beam magnetic resonance spectroscopy and by laser fluorescence. By crossing the nozzle expansion with some excitation source such as a laser, electron beam or plasma, one might be able to see rotational transitions in excited states. As mentioned before, for the weaker com- plexes no rotational lines have been seen in even a vibrationally excited state. Other types of nozzle sources might allow one to observe combus- tion or explosion products. Molecular radicals or ions would also be interesting species to study m the large cell. A Start cell to measure dipole moments would be a useful addition to the cavity spectrometer. Due to the large extent of the fields, plates cannot be placed very near to one another to insure a high homogeneous electric field. expansion. Another problem is the large divergence of the nozzle A solution to the first problem is to use a combined parallel plate, Fabry-Perot cavity. This type of cavity m skimmed beam would be worth trying. in our laboratory. conjunction with a A Zeeman cell is under construction We have purchased a superconducting solenoid magnet 74 with a 12-inch diameter bore. A Fabry-Perot cavity and nozzle source will be placed in this bore with a diffusion pump underneath. The nozzle source will pulse molecules along the axis of the bore while the cavity axis will be perpendicular to the bore. The electric field polarization vector of the microwaves can then be rotated either parallel or perpendicular to the magnetic field lines to observe all AM transitions. Double Fabry-Perot cells have been constructed to do microwavemicrowave double resonance and double resonance modulation. Other types of double resonance spectroscopy could also be done combining the microwaves with a laser or a radio frequency field. sequences used in NMR tried in the cavity. 78 Multiple pulse and in microwave spectroscopy have never been And non-linear effects such as two-photon transient spectroscopy which has been done in optics and in NMR should be observed in the microwave region. Another area that this new method will hopefully contribute to is better understanding of the gas dynamics of nozzle expansions. It was shown in Chapter IV, Section E-2, within the approximations given, how the time domain signal can be used to find the power of the cosine distribution. Finally, knowledge about condensation or dimerization processes may be discerned. Once the reproducibility of the pulse nozzle is improved so that relative concentrations can be inferred from signal intensities, three component mixtures could be studied. The signal intensities from two van der Waals complexes, obtained from a single rare gas and two polar molecules, could be observed. By varying the concentration of the two polar molecules, one might learn about the competing kinetic processes that form the dimers. 75 F. Discussion The last section will go over the main features of the three elements that make up the spectrometer, the Fabry-Perot cavity, pulsed Fourier transform, and the pulsed nozzle source. The advantages and disadvantages mentioned are not exclusive to this method but they all contribute to this method. The real advantages of the Fabry-Perot cavity are its high Q, and therefore its high field strengths and its large open structure. The large electric fields ensure that one can polarize a band of frequencies that is at least as large as the cavity bandwidth. lend to studying non-lmear effects. The high fields also The large open structure provides a large volume for the nozzle expansion. There is room in the cavity to place pulse valves, hot ovens, other mirrors or whatever, in an area readily accessible to the field-molecule interaction region. The disad- vantages of the Fabry-Perot are the non-uniform complex field distribution and the necessity to operate in a high mode to have the large volume. Even though the Q and the bandwidth cannot be separately adjusted, the 1 MHz window we have to operate in is a disadvantage. Finally, as with all cavity spectrometers, mechanical vibrations have to be isolated to reduce noise. The classic advantage to using pulse Fourier transform techniques is in the signal to noise enhancement of the square root of the ratio of the polarization bandwidth to the line width. For the cavity, this polarization bandwidth is limited by the cavity bandwidth. By working in the time domain, it is very easy and fast to add and subtract successive scans, thus simulating phase sensitive detection. With this type of microwave set up, it is easy to use superheterodyne detection. A 76 superheterodyne detector eliminates 1/f noise by converting the microwave signal to the IF frequency where this noise is small. frequency is easy to amplify and filter. The signal at this Finally, because the IF is the result of the mix between the molecular signal and the LO signal, much weaker molecular signals are observable. Since the frequency is fixed when the spectra are recorded, the frequency dependence of the microwave components is less trouble. The fact that one is observing the signal at different times than when one is exciting the sample, like a radar, eliminates having to deal with the exciting power. You observe a signal with theoretically only thermal noise in the background instead of trying to observe a small change on top of a much larger microwave background. Also, since you are observing the molecules when the only field they experience is their own emitted field, there are no power saturation problems or Stark shifts. The m a m disadvantage to the pulsed Fourier transform method is that it is expensive. One has to have two oscillators with lock boxes, switches and timing controls, several mixers, a digitizer, and a computer, as well as all the microwave plumbing. The advantage, in signal to noise, of using a pulsed nozzle source as compared to a continuous source is now considered. There are a fixed number of molecules that can be pumped through the cell per unit time. This fixed number of molecules can either be pulsed all at once into the cavity or be strung out over the unit of time and be observed a number of times, m. The same number of molecules are observed in both situations so the signal produced is the same (assuming nozzle dynamics are the same). But every time a scan is taken, noise proportional to vST is also recorded. The signal to noise ratio for the pulsed experiment is proportional to the number of molecules, N, but the signal to noise ratio for the continuous 77 flow is N/v^n. The number of shots, m to see all the molecules would be, m = t/t = d/2w , w ' o' where t is the unit of time, one second, t is the time it takes the w molecules to travel through the beam waist diameter 2w and d is the total distance traveled by the molecules in the unit of time. For Kr 4 -4 where v. = 3.8 x 10 cm/sec and 2w = 12 cm, t = 3 x 10 sec so that t o w m is around 3.3 x 10 or the S/N is down by a factor 58 after 1 second. The rotational cooling, as could be seen by the difference it made in the calculated values of AN is a great advantage for looking at low rotational levels. species. The cooling is crucial to the observation of dimer Finally, once any molecule or complex makes it through the region of the expansion where all the energy exchange takes place, and passes into the more or less collionless flow, the complexes or molecules are available for observation for a long time. The disadvantages of the nozzle expansion are that the beam is too cold, that it is hard to control the temperature, and that the expansion dynamics are not completely understood. The expansion dynamics are not understood in the sense that experimental conditions such as source temperature, pressure, nozzle diameter or geometry are not known beforehand to achieve a maximum dimer concentration. Finally, the manifestation of the Doppler effect that doubles the number of lines and causes the signal to decay more rapidly is a definite disadvantage. 78 VI. A. A JOSEPHSON JUNCTION MIXER Introduction We currently use superheterodyne detection in a balanced mixer using a backward wave local oscillator and the difference frequency of 30 MHz. The balanced mixer has a noise figure of 6.5 db relative to 300°K which is equivalent to a mixer noise power generation at about T = 1000°K (T = (F - 1)T ) . There are two ways to reduce the effects of this noisy e o balanced mixer. First, we could amplify our transient microwave signal (from the molecular coherent emission) before the mixer. This is expen- sive, but more importantly, wideband low noise amplifiers in the 4-12.4 GHz range currently have noise figures around 5dB so not much is gained. second alternative is to develop a low noise mixer system. The A Josephson Junction (JJ) metal-metal oxide superconducting detector could replace our balanced mixer described above. The noise figure of the JJ detector should be equivalent to noise power in the 50-100 K temperature range, thereby improving greatly the sensitivity of our system. The JJ detector will work effectively in low power applications which is ideal for our superheterodyne detection system used here in the Fourier transform cavity system. B. Josephson Junction Detector-Mixer The mixing and detecting properties of point contact Josephson Junctions have been investigated since the late sixties. We would like to 79 utilize the low noise temperature (as low as 50°K), high sensitivity -14 (10 -15 -10 12 watt/ Hz NEP), and large bandwidths (10 cal detection system for microwave spectroscopy. Hz) in a practi- Josephson devices are typically one of three basic configurations, thin films, sold blobs, or point contacts (or arrays of point contacts). Thin films and point con- tact devices have been used to detect microwave radiation, although the theory is the same for all types. A Josephson Junction is a "weak link" between similar or dissimilar 107 metals with each in the superconducting state. The Ginzberg-Landau complex wavefunction for a superconductor is of the form lp = n = |I(J| n e , where is the density of Cooper pairs and 9 is the phase, in general a function of space and time. The Cooper pairs comprise the superconducting media arising through phonon interaction between electrons of opposite spin and different wavevectors. As the metal becomes superconducting, an energy gap appears, A, symmetrical about the Fermi level, which is a function of the temperature. The gap is zero at T , the transition temperature, and is a maximum at 0°K. The value of the energy gap is dependent on the particular material. The bound Cooper pairs are lower in energy by an amount 2A from the normal electrons (quasiparticles). 2A is 3.05 meV for Nb, which has the largest gap and therefore the largest T conducting elements. tacts. of all super- Nb is the material most commonly used for point con- When two superconductors are brought close to one another but not touching, ~ 10 A, phase coherence can take place that lowers the free energy sufficiently to exceed the thermal fluctuations, thus weakened superconducting order extends across the gap. In a regular superconductor there is long-range order that fixes the phase over the entire macroscopic length of the material. Due to this fixed relative phase or order, if one tries to establish a voltage drop across a section of superconducting wire, a 80 supercurrent will be induced. If the voltage is of such a magnitude as to cause a current that qualitatively corresponds to a kinetic energy equal to 2A, the superconductivity is destroyed and the wire acts as a normal conductor. The value of this current is called the critical current and it plays an important part in the Josephson theory. The critical cur- rent for a typical Josephson point contact is in the range of 1-500 y amps. The sharpened end of a point contact is on the order of microns in diameter although the exact geometry of a point contact is open to question. The critical current of a superconducting wire 1 ym in diameter is ~ 3 y amps -7 2 assuming a critical current density of 10 amps/cm . Josephson derived the relations that govern the behavior of loosely 108 coupled superconductors in his original papers. The required set of equations for a flat barrier in the z plane are: 3H _^_ 3x 3H . „ *„,., 4TTJ X_4TTC3VjtL=_J.z. 9y c t c where H is the tengential magnetic field in the barrier, C is the selfcapacitance, V is the potential difference across the barrier, and j is z the current density through the barrier. |i-fv<t, . (VX-2, where <|> is the change in phase of the wavefunctions across the barrier, (J> = 8 - 9 , and 2e is the charge of a Cooper pair. M = 2ed 3x ~ ilc y (VI-3) M = _ 2ed 3y he z where d is the effective thickness of magnetic field in the barrier. 81 J z where j = j sin<|>(r,t) c , is the critical current density. (VI-4) Since we will not be considering any magnetic phenomenon, Eqs. (VI-2) and (VI-4) are sufficient to describe the Josephson junction used as a detector-mixer. Eq. (VI-4) is a simplified form neglecting contributions from quasiparticle currents. Eq. (VI-2) can be rewritten by setting 3<j)/3t = to; then h.C0 = 2eV(t) and the Cooper pair can radiate a photon of energy hto as it falls through the potential V(t). radiation has been detected experimentally. ' This At small values of impressed potential across the barrier, the junction acts as a single superconductor and there is a current through the barrier with no potential difference across it. When the external potential is sufficient for the cur- rent to exceed j , a potential difference develops across the barrier with c the two metals remaining superconducting. for V(t) = V By solving Eq. (VI-2) and (VI-4) (a constant) gives, considering the current instead of the current density, 1=1 sin en or 1 = 1 s i n to t c D , where 2eV w j = DC -F- is called the Josephson frequency. exceed and V = 3cj> at Therefore, when the critical current is is nonzero, an oscillating current of 483.6 MHz/y volt (i.e., 2e/2l) flows in the circuit. This relation is exact and forms the basis for a new measurement of the constant lefr.111 82 112 The Resistively Shunted Junction (RSJ) model considers an ideal JJ shunted by a real resistance R, as in figure 9 where I__ = I_ + I_ DC R J and V D C = <V(t)> = (h/2e)<d<j>/dt). Then for I D C < I c , V D C = 0 and for T DC > I V C DC = R(I 2 2 1/2 J > " C ' I n t h S l i m i t ° fIDC > > IC ' DC the slope of the current voltage (I-V) plot goes to 1/R as in figure 10. Many point contact systems can usefully be described by this approach. Considering now incident microwave radiation on a DC biased point contact, Josephsen has given for a R.F. voltage source, V(t> "V D C + V BPcoa(WRPt) or « (to1 -/' t)) ua 3t = tr h v(V DC + V RF l-cos ^RF Solving for I gives I = I 0 s m [ f VDCt + g ^ - . smfco^t + * )] RF or 1 = 1 c /2SVRFA 2 J 1-T1 sin n \ hco„„/ m n=-°° \ RF/ to. + ntor,_)t + r <H • jJ RF o This predicts that there will be DC spikes when co = nto J , where n i s RF an integer. Usually the impedance of a JJ is smaller than that of the RF source so the junction sees more nearly a current source than a voltage source. Taking account of the shunt resistance, current steps at constant voltage instead of spikes should result. Experimentally 113 if a fixed microwave signal is applied to a JJ as one varies the D.C. bias, current steps will occur in the I-V plot whenever the bias voltage produces Josepheson oscillation at co. which satisfies V = n&co /2e or co. = 2eV „fa = nto . The slope of the line between 83 ! Figure 9. R+ I,} The resistively shunted junction model. 84 Figure 10. The current versus voltage plot of a Josephson junction with no applied field. I is the critical current. 85 current steps is related to the dynamic resistance, R^ = (dl/dV) as in figure 11. The first step height varies from I to zero as the microwave power c is increased. With high power this and all other steps are smoothed out, giving a straight line I-V plot. The JJ used as a video detector utilizes this change in step height. Consider a junction biased at a point of high R^ (point A on the figure) with a constant current DC bias. change from V to V The voltage through the circuit will as the bias point changes from A to B with the appli- cation of microwave power that depresses the I-V plot. 114 Kanter and Vernon See figure 12. 3 have calculated the current response to be ~ 3 x 10 amps/watt at 10 GHz and the NEP ~ 5 x 10 watt/ Hz at 80 GHz with point contacts. The application that we are primarily interested in would be to use a JJ device as a mixer with an external local oscillator (LO). Grimes 113 and Shapiro analyzed the mixing of microwave signals at 23 and 72 GHz. Proceeding in an analogous manner to the video detector for an R.F. voltage source, V(t) = V D C + V cos (co t+6) + V2cos(co.t+62) where to and co. a r e t h e two microwave f r e q u e n c i e s , 00 °° /2eV.\ /2eV2\ + £(to 2 t+9 2 ) where k and & are integers, <f> is the initial phase difference across the junction, and 9. and 9 are phase factors for the incidence radiations. 86 Figure 11. The current versus voltage plot of a Josephson junction with an applied microwave signal. 87 V Figure 12. A V B The current versus voltage plot of a Josephson junction showing the change in the step height of one step as the applied microwave power is varied. 88 Current steps are observed at co. = kco. + &C0,,. The DC current that flows in the step at zero voltage should then go as I J I -r——J J j -r- jsin The intermediate frequency (IF) current at Aco = to -to would go as 1 WW As pointed out by Grimes and Shapiro for the case of to ~ CO- » to.,-CO and the amplitude of one signal much larger than the other, the signals would add to produce a beat. This is equivalent to a signal which is ampli- tude modulated at the I.F. frequency. In this case the detection mechanism is essentially the same as for the video detector, only now the bias point will oscillate from A to B at the I.F. frequency. in the detection circuit. This is thus the signal I.F. frequencies as high as 9.2 GHz have been used. The JJ used as a mixer has advantages over its use as a video detector. Since the mechanism of detection is the same both should have similar sensitivities but for the mixer, not only can the bias current be optimized but the L.O. power also. 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Lett., 9_, 353 (1966). 110. D. N. Langenberg, D. J. Scalapmo, B. N. Taylor, R. E. Eck, Phys. Rev. Lett., 15, 294 (1965). 94 111. W. H. Parker, B. N. Taylor, D. N. Langenberg, Phys. Rev. Lett., 18, 287 (1967). 112. P. L. Richards, F. Auracher, T. Van Duzer, Proc. IEEE, 61, 36 (1973). 113. C. C. Grimers, S. Shapiro, Phys. Rev., 169, 397 (1968). 114. H. Kantor, F. L. Vernon, Jr., J. Appl. Phys. 43_, 3174 (1972). 95 VIII. VITA Terrill Joseph Balle was born on October 22, 1947 in Eugene, Oregon. After graduating from high school in 1965 he attended the University of Oregon for 2 years majoring in sculpture. He enlisted in the U.S. Army in 1967 and spent a year in Viet Nam. After separation from the army in 1970 he worked several years as a welder and millwright for Pierce Corporation in Eugene, Oregon. In 1971 he re-enrolled at the University of Oregon and received a B.S. in chemistry in 1975. He started graduate school in 1975 and will receive his Ph.D. in Physical Chemistry in 1980. He is co-author of the following publications: J. S. Wieczorek, T. Koemg, T. Balle, The He (I) Photoelectron Spectra of Amine N-Oxides, J. Electron Spect. S Rel. Phenon. , 6^, 215 (1975) . T. Koenig, R. Wielesek, W. Snell, T. Balle, Helium(I) Photoelectron Spectrum of p-Qumodimethane, J. Amer. Chem. Soc, 97_, 3225 (1975). T. Koenig, T. Balle, W. Snell, Helium(I) Photoelectron Spectra of Organic Radicals, J. Amer. Chem. Soc, 97_, 662 (1975). T. Balle, E. Campbell, M. Keenan, W. H. Flygare, A New Method for Observing the Rotational Spectra of Weak Molecular Complexes; KrHCl, J. Chem. Phys., 71, 2723 (1979) (Communication) T. Balle, E. Campbell, M. Keenan, W. H. Flygare, A. New Method for Observing the Rotational Spectra of Weak Molecular Complexes; KrHCl, J. Chem. Phys. 72, 922 (1980). K. F. Gebhardt, P. D. Soper, J. Merski, T. J. Balle, W. H. Flygare, Conductivity of a-Silver Iodide in the Microwave Range, J. Chem. Phys. 72, 272 (1980). M. R. Keenan, E. J. Campbell, T. J. Balle, L. W. Buxton, T. K. Minton, P. D. Soper, W. H. Flygare, Rotational Spectra and Molecular Structure of ArHBr and KrHBr, J. Chem. Phys. 72, 3070 (1980). 96 E. J. Campbell, M. R. Keenan, L. W. Buxton, T. J. Balle, P. D. Soper, A. C. Legon, W. H. Flygare, Chem. Phys. Lett. 70/ 420 (1980). A. C. Legon, P. D. Soper, M. R. Keenan, T. K. Minton, T. J. Balle, W. H. Flygare, to be published J. Chem. Phys. M. R. Keenan, L. W. Buxton, E. J. Campbell, T. J. Balle, W. H. Flygare, submitted J. Chem. Phys. E. J. Campbell, L. W. Buxton, T. J. Balle, W. H. Flygare, submitted Chem. Phys. T. J. Balle, W. H. Flygare, submitted Rev. Sci. Inst. J.