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Studies of atomic hydrogen spin exchange collisions at 1 K and below Hayden, Michael Edward 1991

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STUDIES OF ATOMIC HYDROGEN SPIN-EXCHANGE COLLISIONSAT 1 K AND BELOWByMichael Edward HaydenMASc. (Engineering Physics), University of British ColumbiaB. Eng. (Engineering Physics), University of SaskatchewanA THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY111THE FACULTY OF GRADUATE STUDIESPHYSICSWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIA1991© Michael Edward Hayden, 1991In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.Department of iCThe University of British ColumbiaVancouver, CanadaDate /)Ec /3 /9/DE-6 (2/88)AbstractSpin-exchange interactions during collisions between hydrogen (H) atoms cause the zSF =1, Lm = 0 hyperfine transition of the H atom to be broadened and shifted from itsunperturbed frequency. A recent theory suggests that hyperfine interactions during thesecollisions lead to frequency shifts in an oscillating H maser which depend in a non-linearway upon the atomic density. These non-adiabatic contributions are of considerabletechnological concern as they may limit the ultimate frequency stability of H masersoperating at cryogenic temperatures. In the first of two experiments described here, weexamine the influence of H-H spin-exchange collisions on the oscillation frequency of aH maser operating at 0.5 K. The results of this study are consistent with the theoreticalpredictions.The second study involves several experiments with mixtures of H and deuterium (D)atoms at cryogenic temperatures. It makes use of the observation that for comparableH and D densities, the spin-exchange broadening of the LF = 1, Lmj’ = 0 hyperfinetransition of the H atom is dominated by H-D collisions. We have used hyperfine magneticresonance techniques on this transition to study interactions between H and D atomsconfined by liquid 4He (-4He) walls at 1 K and zero magnetic field. The resonance signalintensity gives a measure of the H density iii the mixture while the broadening of thetransition gives a simultaneous measure of the D density. Measurements are made ofseveral spin-exchange and recombination cross sections. An improved measurement ofthe energy required to force a D atom into £-4He is made. These results also set a lowerbound for the effective mass of a D quasiparticle in £-4He.11Table of ContentsAbstract iiList of Tables viiiList of Figures ixAcknowledgements xii1 General Introduction 12 Introduction to Zero Field Magnetic Resonance and the HydrogenMaser 72.1 Hyperfine structure 72.2 Spin analogy 92.3 Practical hyperfine magnetic resonance 142.3.1 The free induction decay 152.3.2 Power radiated by the atoms and the filling factor 162.4 The hydrogen maser 183 Introduction to Atomic Hydrogen and Deuterium at Low Temperatures 213.1 Containment with liquid 4He walls 223.1.1 Binding, sticking and thermal accommodation 243.1.2 Adsorption isotherm and the wall shift 25Hi2731313436373847475454575960606264675 The 685.1 695.2 715.3 727275783.1.3 Incomplete confinement3.2 Recombination processes3.2.1 Molecular potentials3.2.2 Recombination of atomic H, D3.3 Effects of atomic collisions on hyperfine resonance parameters3.3.1 Shift due to collisions with 4He atoms3.3.2 Spin-exchange induced broadening and frequency shifts4 Apparatus used for the 0.5 K Experiments4.1 The UBC CHM: Conventional mode of operation4.2 The UBC CHM: Modifications for spin-exchange measurements4.2.1 Electronic tuning/Q spoiler4.3 State selection4.3.1 New pumping scheme4.3.2 Frequency stability4.4 Refrigeration, temperature monitoring and control4.5 1420 MHz spectrometer4.5.1 Data acquisition4.5.2 Measurement of Q and z0.5 K H-H Spin-Exchange MeasurementsSummary of technical problemsCooldown procedure (standard)Characterization of the maser5.3.1 Magnetic relaxation5.3.2 Determination of the atomic density . . .5.3.3 Computer simulation . . .iv5.3.4 Output power of the maser 805.4 The measurements 805.4.1 Procedure and data 825.5 Analysis 835.5.1 Density independeilt frequency shift and absolute detuning . . 865.5.2 The parameters aild T0 905.5.3 The parameters Ji(Pcc + Paa) + 2 aild )‘i(Pcc + paa) + ‘2 935.5.4 Summary of measured quantities 966 Background and Technical Aspects of the 1 K Experiments 1006.1 Historical context . . . 1006.2 Experimental design . 1036.2.1 The cryostat . 1036.2.2 The resonator 1046.2.3 The discharge 1076.2.4 The bulbs 1096.2.5 Temperature measurement and regulation 1116.2.6 Data collection 1117 Experiments with H, D Mixtures at 1 K 1147.1 The basic experiment 1167.2 Analysis of the data 1187.2.1 T1 and T2 1237.2.2 The H density 1257.2.3 H-D recombination: Measurements with 11H >> nD 1317.2.4 H-D and D-D recombination: An absolute calibration of riD . . 1327.2.5 Measurement of the solvation energy 138V7.2.6 The 4He buffer gas frequency shift 1437.2.7 Frequency shift due to H-D collisions 1468 Discussion and Conclusions 1498.1 Summary 1498.2 Spin-exchange measurements 1508.3 Recombination measurements 1548.4 Solvation measurements 156Bibliography 158Appendices 172A Radiation Damping and the FID 172A.1 Electrical model for the resonator 173A.2 Radiation damping equations 176A.3 Data analysis 178B Measurement of Filling Factors 180B.1 Cavity perturbation measurements . . 181B.1.1 Type I perturbations 183B.1.2 Type II perturbations 185C Rate Equations for the Atomic Densities 189C.0.3 Simplified equations 190D Temperature Measurement and Control of a 4He Bath at 1 Kelvin 194D.1 Pressure gauge calibration 195D.2 4He wand and temperature regulation 197viD.3 Characterization of the measurement system 199D.3.1 The supercoiducting transition of Al 199D.3.2 The 4He buffer gas shift: Temperature stability 202E Molecular Energy Levels of H2, D2 and HD just below Dissociation 204viiList of TablesChapter 33.1 Theoretical energies required to force H, D, and T atoms into £-4He . . 283.2 Theoretical rate constant for gas phase H-H recombination 363.3 Selected values of the H-H spin-exchange and frequency shift cross sectionsin the DIS limit 423.4 Theoretical H-H spin-exchange frequency shift and broadening cross sections in the Verhaar theory 44Chapter 55.1 Summary of measured and theoretical spin-exchange and frequency shiftcross sections 98Appendix BB.1 Properties of liquid 02, Ar and N2 186Appendix EE.1 Dissociation energies of selected states of the H2 molecule 205E.2 Dissociation energies of selected states of the D2 molecule 205E.3 Dissociation energies of selected states of the HD molecule 206vu’List of FiguresChapter 22.1 Hyperfine energies as a function of magnetic field 10Chapter 33.1 H-H interatomic potentials 33Chapter 44.1 Schematic of a room temperature H maser 494.2 Schematic of the UBC cryogenic H maser 504.3 The CHM resonator and storage bulb region 514.4 The electronic tuning assembly 564.5 Electronic tuning and Q spoiler: range of operation 584.6 Stabilization of the microwave pump frequency 614 7 The 1420 MHz spectrometer 634 8 Data acquisition for the 0 5 K experiments 66Chapter 55 1 Relaxation of the longitudinal magnetization in the maser bulb 745 2 An example free induction decay 775 3 Power output of the UBC CHM with the I a) to d) pumping scheme 815.4 H-H spin-exchange measurements: data at constant Q 845.5 H-H spin-exchallge measurements: data at constant / 855.6 The density independent frequency fo 87ix5.7 Initial phase of a FID as a function of A 885.8 Output power of the maser as a function of A 895.9 Fits to the data at constant A 915.10 Determination of the absolute detuning A0 925.11 Fits to the data at constant Q 945.12 Determination of T0 and ) 955.13 Determination of cri(pcc + p) + cr2 and ).i(pcc + Paa) + 2 97Chapter 66.1 The split-ring resonator used to study mixtures of H and D 1056.2 Low temperature assembly for the 1 K experiments 1086.3 Data acquisition system for the 1 K experiments 113Chapter 77.1 A series of FID’s following a single discharge pulse 1177.2 A FID taken with essentially no D atoms present 1197.3 A FID with considerable broadening due to spin-exchange collisions . 1227.4 An example of the time dependeilce of T1 and T2 1247.5 The ratio T/T 1267.6 Broadening which is wholely attributed to H-D spin-exchange 1277.7 The H density as a function of time 1287.8 The rate at which T1’ decays following a discharge pulse 1337.9 The H-D recombination rate constant 1347.10 The D-D recombination rate constant 1377.11 A fit to a D atom density decay 1407.12 Comparison of the 1420 MHz and the 309 MHz solvation rate data . . 1417.13 Fit to the solvation rate data 144x7.14 The 4He buffer gas frequency shift 1457.15 Frequency shift due to H-D spin-exchange collisions 147Appendix AA.1 Lumped element circuit model for the resonator 175Appendix CC.1 An illustration of the use of the density decay rate equations 193Appendix DD.1 4He pressure measurement and regulation 198D.2 Superconducting transition of Al 201D.3 Temperature stability measurement 203xiAcknowledgementsIt is with great pleasure that I thank my friend and thesis advisor Walter Hardy, forhis support, encouragement, and enthusiasm throughout the course of my research atUBC. I was also very fortunate to have been able to work closely with Martin Hürlimannand Meritt Reynolds; two former members of the UBC atomic hydrogen group. Martintaught me the intricacies of the UBC cryogenic hydrogen maser (CHM) and was involvedwith all aspects of the H-H spin-exchange work. Meritt was involved with the earlieststudies of H-D spin-exchange and has been a constant source of encouragement andadvice throughout the remainder of that project.The H-H spin-exchange work would not have been possible without the CHM whichwas designed and built by Martin Hürlimann, Walter Hardy and Rick Cline. MarkPaetkau spent many hours assisting with the upkeep of the 1 K apparatus and with datacollection for that experiment; Dennis Chow wrote the data acquisition program. GeorgeTakis lent us the pumping station which was used during the 1 K experiments and DavidWineland lent us the rubidium standard which was used during both experiments. Inaddition, Herb Gush lent us the baratron gauge and the cathetometer used to make4He vapour pressure measurements during the 1 K experiments. Rick Cline devotedmany hours to the design and upkeep of the computer system in the lab; without hiscontribution much of this work would have been painfully slow and difficult.I would also like to thank Stuart Crampton for his continued interest and encouragement despite innumerable foiled attempts to cool the dilution refrigerator. BoudewijnVerhaar supplied us with detailed results of his H-H spin-exchange calculations, some ofxiiwhich have been reproduced here. I have certainly enjoyed and benefited from conversations with Jook Wairaven and Ted Hsu. I would also like to thank Doug Bonn for hisencouragement and friendship throughout the past few years. Finally, I want to thankmy parents Mike and Joan for the countless times that they have encouraged me. I amsure that without their faith in me, none of this would have come to be.I am grateful to the Natural Sciences and Engineering Research Council of Canadaand the I{illam Foundation for their support in the form of postgraduate scholarships.xli’Chapter 1General IntroductionAtomic hydrogen (H) is the most abundant element in the universe. Its attraction asthe simplest atom has made it the subject of innumerable experimental and theoreticalstudies over the last century. One of the most important technological triumphs to comeout of this work was the development of the atomic hydrogen maser during the 1960’s[1]. This device is essentially a storage bottle containing a gas of state selected atomichydrogen, located inside a microwave cavity tuned to a hyperfine transition of the H atom.Stimulated emission of radiation from the hydrogen produces a spectrally narrow and verystable signal at this transition frequency. The purity of this coherent radiation is due tothe combined effects of a long interaction time between the atoms and their radiation field,and the use of ‘non-stick’ teflon coatings on the storage bottle walls. The perturbationof the atomic hyperfine interaction during collisions with these walls is very weak nearroom temperature. The H maser is currently the most stable atomic clock for averagingtimes between about 102 and 106 seconds [2]. Its extraordinary frequency stability hasmade it an invaluable research tool in fields as diverse as navigation, metrology, physics,and astronomy [3].The past decade and a half has seen a resurgence of interest in the study of atomichydrogen, this time at very low temperatures [4, 5]. It was predicted as early as 1959[6] that spin-polarized atomic hydrogen would remain gaseous at temperatures down toabsolute zero. In principle, it should be possible to cool such a gas until it reaches astate of quantum mechanical degeneracy. The truly exciting feature of this system that1Chapter 1. General Introduction 2sets it apart from other examples of macroscopic quantum mechanical degeneracy, is thatthe interactions within the condensate should still be weak enough to make comparisonsbetween experiment and microscopic theories possible.Along with the intensive research effort devoted to the study of H at low temperatures,many ‘spin off’ projects of considerable interest in their own right have been undertaken.One of the most interesting of these was the development of atomic hydrogen maserswhich work at cryogenic temperatures [7, 8, 9, 10]. The intrinsic frequency stabilityof a conventional hydrogen maser is determined by the background of thermal photonspresent in the microwave cavity. The phase of the electromagnetic field associated withthese photons is random with respect to the coherent radiation produced by the maser.The frequency fluctuations caused by these photons can be reduced by going to lowtemperatures. Unfortunately, as the temperature of a gas of hydrogen is lowered, atomsbegin to stick to the walls of the storage bottle for long periods of time. It is difficult tooperate a conventional H maser below liquid nitrogen temperatures. Low temperature Hmasers did not become feasible until the development of new cryogenic wall coatings. Themost important wall coating for low temperature H research is superfluid liquid helium[11].The power output of a hydrogen maser is limited by collisions between atoms duringwhich spin-exchange interactions broaden the atomic resonance. By going to temperatures near 1 K, this broadening effect is reduced by about three orders of magnitude fromroom temperature [12]. Cryogenic H masers can thus be run at much higher densities andpower levels than conventional masers. When this effect is combined with the reductionin thermal noise, it would seem that rather dramatic improvements in frequency stabilitymight be obtained by operating H masers at low temperatures.In the time since these predictions were first made [13, 14, 15], it has been demonstrated that some improvement in the frequency stability of a hydrogen maser can indeedChapter 1. General Introduction 3be made by going to low temperatures [16, 17]. Ullfortunately the same spin-exchangecollisions which broaden the atomic resonance also cause it to be shifted from its unperturbed frequency. Unlike the broadening effect, there is essentially no reduction in thefrequency perturbation at low temperatures. In fact, since the atomic densities in a maseroperating at low temperatures are much higher, the magnitude of the spin-exchange induced frequency shifts in a cryogenic maser are typically much larger than those in aconventional maser.These spin-exchange frequency shifts depend upon the atomic collision rate and thusfluctuations in the atomic density couple directly to the oscillation frequency of the maser.This coupling imposes very strict requirements on the stability of the atomic density.A recent theory of hydrogen spin-exchange collisions1 was developed by B. J. Verhaarand several collaborators [19, 20]. This theory predicts that because of non-adiabaticeffects caused by hyperfirie interactions during these collisions, the spin-exchange inducedfrequency shifts depend not only on the atomic density but on the occupancy of theindividual hyperfine states within the maser bulb. Cryogenic hydrogen maser researchhas slowed since these predictions were made, simply because it is not obvious how toovercome this problem and make further improvements in frequency stability.Two experimental studies of hydrogen spin-exchange interactions at temperatures of1 K and below are presented in this thesis. The first is an investigation of the effect ofH-H spin-exchange collisions on the oscillation frequency of a cryogenic hydrogen maseroperating at 0.5 K. In this study the first experimental evidence that the oscillationfrequency of a cryogenic H maser does indeed contain contributions which seem to beexplained by the Verhaar theory is presented.From a technological standpoint this study is motivated by the obvious need to verify1The first theories of hydrogen spin-exchange collisions were developed over three decades ago. Inthese theories it was conventional to ignore hyperfine interactions in comparison to electron spin-exchangeinteractions [18].Chapter 1. General Introduction 4the existence of the predicted frequency shifts before further developmental work oncryogenic hydrogen masers is undertaken. A second less obvious but equally importantmotivation arises out of the sensitivity of the various collision cross sections of the theoryto the detailed form of the interatomic potentials. These parameters are particularlysensitive to the long range parts of the potentials which are otherwise difficult to test. Bymeasuring these parameters, it may be possible to test and further refine these potentials.In a sense, these measurements can be thought of as a form of spectroscopy.The second study presented in this thesis also involves measurements of hydrogenspin-exchange parameters; this time at temperatures just above 1 K. It also involvesthe use of magnetic resonance techniques on the same hyperfine transition the H maserutilizes. While spin-exchange collisions play an integral and essential role in this work,they do not form the central theme of this part of our research. We digress for a momentto put this second experiment into historical perspective.Recently we discovered that atomic deuterium (D) is not confined by liquid heliumwalls to nearly the same degree as H [21]. A deuterium atom with about 14 K of kineticenergy can penetrate a film of liquid helium. A hydrogen atom requires an estimated35 K to penetrate the same film. This discovery was made by using magnetic resonancetechniques on a hyperfine transition of the D atom at 309 MHz to study the time andthe temperature dependence of the D atom density inside a sealed2 sample bulb. It wasduring the course of this investigation that we observed a considerable broadening of theatomic resonance which could not be explained by spin-exchange collisions between Datoms. We postulated that this broadening was due to contamination of the samples withH. In a separate experiment, magnetic resonance techniques were used on a hyperfinetransition of the H atom at 1420 MHz to verify the presence of this contamination. Thesecond study presented in this thesis is an extension of this work.2The interior walls of this bulb were coated with a film of superfluid liquid 4He.Chapter 1. General Introduction 5A rather striking difference exists between the magnitude of the spin-exchange broadening cross section for collisions between two H atoms, and that for collisions between anH atom and a D atom: the H-D broadening cross section was theoretically estimated [22]to be more than two orders of magnitude larger than that for H-H collisions near 1 K.This implies that if magnetic resonance tecimiques are used to monitor the H atoms ina mixture containing comparable densities of H and D, the broadening of the transitionwill be dominated by spin-exchange collisions3with the D atoms. In this way it is possible to monitor both atomic densities simultaneously. The amplitude of the magneticresonance signal is proportional to the H density while the broadening of the transitionis proportional to the D density. These techniques have allowed us to make the firststudy of interactions between H and D atoms in zero magnetic field at temperaturesjust above 1 K. Apart from the measurement of spin-exchange parameters for collisionsbetween H and D atoms we were able to study several recombinatiori processes and tomake significant improvements in the measurement of the energy required for a D atomto penetrate a liquid helium film. All of these parameters are of considerable interestto physicists in the context of low temperature H research. They also have the samepotential as the H-H spin-exchange parameters to be used in spectroscopic refinementsof the various interatomic potentials.The body of this thesis is split into eight chapters. Chapters 2 and 3 are introductorychapters. In chapter 2 the magnetic resonance techniques used throughout this work areintroduced. Chapter 3 is a review of the properties of atomic hydrogen and deuterium atlow temperatures which are relevant to this work. This chapter includes a discussion ofspin-exchange collisions and their influence on the F = 0, LmF = 0 hyperfine transitionof the ground state of the H atom.3Other broadening mechanisms must of course be taken into consideration.Chapter 1. General Introduction 6In chapters 4 and 5 the H-H spin-exchange measurements with the cryogenic hydrogen maser at 0.5 K are discussed. Chapter 4 is an introductory chapter and chapter5 contains the results of the measurements. Chapters 6 and 7 are organized in a similar maimer. Chapter 6 introduces the study of H, D mixtures at 1 K and chapter 7contains the experimental results. Chapter 8 is the concluding chapter in which all ofthe measurements presented in the thesis are summarized. Several appendices have alsobeen included. Each one describes a technique or a measurement that was in some wayessential to the interpretation of the work presented iii the body of the thesis. Referencesto discussions in these appendices are made in the appropriate places in the main bodyof the thesis.Chapter 2Introduction to Zero Field Magnetic Resonance and the Hydrogen MaserThe experiments presented in this thesis all involve the use of magnetic resonance to studyinteractions in gaseous atomic hydrogen (H), or mixtures of H and atomic deuterium (D).In this chapter we review some the background required to understand the measurementtechniques.2.1 Hyperfine structureAt room temperature and below, gaseous atomic hydrogen (H) is found almost exclusivelyin its electronic ground state. This state is split into four magnetic sublevels by thehyperfine interaction between the proton spin i and the electron spin s. In zero magneticfield, three of these levels are degenerate and are displaced upwards from the fourth bya frequency [23]= 1420.405751773(1)MHz.Radiation from this transition is abundant in stellar environments and is commonlyreferred to by astronomers’ as the 21 cm line.The application of a magnetic field splits the threefold degeneracy of the upper hyperfine states because of the Zeeman interaction. The effective Haniiltonian for the spindegrees of freedom of a ground state H atom in a static magnetic field B0 can be written= ai S— h(7eS + 7i) B (2.1)1The reference is to the wavelength of radiation at this frequency in free space.7Chapter 2. Introduction to Zero Field Magnetic Resonance and the Hydrogen Maser 8where a = 27rhf0 and ‘Ye and -yr. are the gyromagnetic ratios of the electron and the proton.The accepted values for 7e and ‘-yr are [24]7e = 1.7608592(12) x 1011 s’Typ = 2.67522128(81) x 108s1T’The Hamiltonian 2.1 can be diagonalized in terms of a field dependent basis of states.This basis can in turn be written in terms of the spin states in3 m ), where in3 and mare respectively the electron and proton spin projections along the applied field B0. Inorder of increasing energy the eigenstates of the ground state of the H atom areIa) = cosI-) - sinI,-) (2.2)I b ) =‘ —k 1 1 1 (2.3)Ic) = sini9I—,) + cost9l,—) (2.4)Id) = I). (2.5)The parameter ‘0 is defined bya 50.6O7mTeslatan(2’0) = =11(7e +7)B0 B0and is used to indicate the degree of mixing (or the ‘admixture’) of the states —, )and -,—- ) found in the eigenstates a ) and c ). The energies which correspond tothe four eigenstates areEa= -1+ [7e +70] (2.6)Eb = (7e7p)Bo (2.7)E = —+1+[h(7e+)B0]2 (2.8)= + (7e — y)B0 (2.9)Chapter 2. Introduction to Zero Field Magnetic Resonance and the Hydrogen Maser 9The magnetic field dependence of these energy levels is shown in figure 2.1. Energies inthis diagram have been expressed in the convenient units of frequency. The conversionto proper energy units is obtained by multiplying by Planck’s constant h. Throughoutthis thesis we will also find it convenient to express energies in units of temperature. Forexample the zero field hyperfine splitting f0 of the H atom is approximately 68.2 mK. Inthis case the conversion to proper energy units is obtained by multiplying by Boltzmann’sconstant k.2.2 Spin analogyIf we superimpose an oscillating rf magnetic field 2B cos(wt) along the static magneticfield B0 B0, transitions between the a ) state and the c ) state can be induced. Infact, the matrix elements of the magnetic moment operator= h7eS + h’ypi (2.10)between the eigenstates of the H atom are such that a longitudinal field can oniy inducetransitions between these two states. In practice, as long as a small static bias field2 ispresent, the a ) state and the c ) state form an isolated two dimensional subspaceinside the iS manifold. Restricted to this subspace, the Hamiltonian for the H atom= — 2uiB cos(wt) (2.11)can be mapped onto the standard magnetic resonance problem for a spin system [25J.This mapping is quite general and can be applied to any two level system where thestates are coupled by an rfperturbation. It is a convenient procedure since it allows2The bias field is required to set a quantization axis, and to keep the neighboring (transverse) I a)to I b ) and a ) to I d ) transitions sufficiently far from the a ) to c ) transition. Cross relaxationbetween the upper three hyperfine states is also suppressed by this field. Typically oniy a few tens ofmGauss are required.Chapter 2. Introduction to Zero Field Magnetic Resonance and the Hydrogen MaserlO5 • • • • • , • • • • • . . . . .432NI0>%OIa)c—Iw—2—3—4I • I • • i • • •0.0 0.1 0.2 0.3B0 (Tesla)Figure 2.1: The energies of the four hyperfine levels of the ground state of the hydrogenatom as a function of magnetic field.Chapter 2. Introduction to Zero Field Magnetic Resonance and the Hydrogen Maserllone to use the very well documented results pertaining to NMR theory without havingto rederive any of the necessary formalism. The outline given below is only intended togive the reader a feeling for the procedure. More thorough discussions can be found inreferences [25, 26, 27].The mapping is based upon the principle that any operator A acting in the twodimensional subspace can be written in terms of the Pauli matrices u asA = Tr(A) + 2Tr(A) (2.12)where = i-cr can be associated with a fictitious spin . particle. To avoid confusion, welabel all quantities within the frame of this fictitious spin ?. particle with a tilde. Thedensity matrix and the Hamiltonian 2.11 for the H atom in a longitudinal rf field canthus be written in the form= + 2 < > . (2.13)= — h5’E . . (2.14)By comparing the matrix elements of 2.11 and 2.14 the effective gyrornagnetic ratio 5’and the effective magnetic field E can be determined. If we choose= 7e + 7p (2.15)the effective field B is given by3= 2B cos(wt) (2.16)(2.17)a++7B (2.18)h(7e+7p) 2a3Equation 2.18 is a low field approximation.Chapter 2. Introduction to Zero Field Magnetic Resonance and the Hydrogen Maserl2wherea= 50.607 mTesla.hQye+yp)Furthermore, the commutator [7-(,] can be used to determine the equation of motion forthe effective magnetic moment i = h5’ of the fictitious spin - particle with the result:(2.19)This is the classical equation of motion for a magnetic moment in a field E. In effect, themappings for and B transform the H atom problem (the time evolution of the atomicdensity matrix) onto the standard NMR problem [28] (the time evolution of the magneticmoment i). The correspondence between these two problems is obtained through theexpectation values of the effective magnetic moment operator which are related to theelements of the full 4 x 4 atomic density matrix:<x > h(e + p)(pac + pca) (2.20)<y > h(7+ 7p)(pac— pca) (2.21)< z > h(7e+ p)(paa — pcc) (2.22)Once this correspondence is established, it is only necessary to keep in the backof ones mind the transformations which lead to the analogy. In this thesis the realrf fields which are applied are longitudinal fields. The mapping 2.16 to the fictitiousspin. analogy transforms this into a transverse rf field which corresponds to the usualtransverse field in the standard NMR problem. Near zero field, the real applied staticfield B0 sets the quantization axis however it only makes a very small contribution to thetotal fictitious longitudinal field ]3 = f given by equation 2.18. 13 corresponds to thestatic longitudinal field in the conventional NMR problem.Chapter 2. Introduction to Zero Field Magnetic Resonance and the Hydrogen Maserl3The fictitious magnetization for many H atoms is simply M = nH <L > where < L>is now the average magnetic moment and n is the atomic density. The thermal equilibrium value of this magnetization I1 is related to the thermal equilibrium populationdifference between the a ) state and the c ) state of the H atom1 1 —exp(----) 1M0 = h(7e+ 7p)nH1+ 3exp (k) h(7e + 7P)4kBnH. (2.23)The transverse4components of the fictitious magnetization on the other hand, are relatedto the off diagonal elements of the density matrix via equations 2.20 and 2.21. Thefictitious transverse magnetization corresponds to a coherent superposition of the a )state and the c ) state.We illustrate the usefulness of this formalism with the simple example of a ‘tippingpulse’. In standard NMR terminology this refers to the application of a transverse field2B cos(t) for a length of timeT9 = _-_. (2.24)For the H atom problem this corresponds to a longitudinal rf field 2B cos(wt) applied atthe a-c transition frequency (the ‘Larmor frequency’) for a time=. (2.25)The fictitious spins see an oscillating rf field along the $c axis which can be decomposedin the usual way [29] into two counter rotating components in the *- plane. Whenwe transform into the reference frame ‘-‘ rotating in the same sense as the fictitiousmagnetization, the spins effectively see5 only a static field‘ = B along the ‘ axis (allquantities within this rotating frame are so indicated with a prime). The spins precess4Both the fictitious longitudinal magnetization and the fictitious transverse magnetization are directedalong the axis in real space.5The counter rotating field is displaced from the frequency range of interest by twice the Larmorfrequency. It leads to the Bloch-Siegert frequency shift [25] which can be neglected in our work.Chapter 2. Introduction to Zero Field Magnetic Resonance and the Hydrogen Maserl4about this field at an angular frequency ‘B throughout the duration of the rf pulse.After a time r0 the effective magnetization has been rotated through an angle 0 withrespect to the axis. In the lab frame the fictitious spins are left in this canted positionbut precess about the effective static field at the Larmor frequency. The real oscillatingmagnetization which lies along the axisM = flHh (yi— ‘ye5z) (2.26)is the parameter which we observe experimentally. It can conveniently be written interms of < ii,, > such thatM(t) = sin(0) sin(ct) . (2.27)A rather interesting point which is made obvious by equations 2.23 and 2.27 is that inthermal equilibrium, the fictitious magnetization fcI, is non zero even though the realmagnetization M is zero.If a K/2 pulse is applied to a sample of H atoms initially in thermal equilibrium, M0 isrotated into the transverse plane and left there as a precessing (fictitious) magnetization.The real oscillating magnetization is directed along and has amplitude M0. Physicallythis state corresponds to a coherent superposition of the a ) and the c ) states ofthe atomic sample. A r pulse on the other hand inverts the fictitious magnetization.If the system is initially in thermal equilibrium, this corresponds to an inversion of thepopulation difference between the a ) and the c ) states. The real longitudinalmagnetization associated with this state is zero. Other extensions of this analogy areobvious.2.3 Practical hyperfine magnetic resonanceIn a typical pulsed magnetic resonance experiment, rf power at the appropriate frequencyis generated by a transmitter which is coupled to some form of electromagnetic resonator.Chapter 2. Introduction to Zero Field Magnetic Resonance and the Hydrogen Maserl5The sample is placed inside the resonator in a region of relatively high rf field homogeneity.In the usual case the rf field is transverse to a large static magnetic field about whichthe spins precess. Following a tipping pulse at the Larmor frequency, the real precessingmagnetization of the sample induces a signal back into the resonator. This response canbe detected with the appropriate circuitry.Our experiments using pulsed magnetic resonance at the a-c hyperfine transition ofH are quite similar once the fictitious spin . analogy is taken into account. The real rffield is actually applied in the longitudinal direction and the bias field applied only toseparate the a-c transition from the neighboring transitions.2.3.1 The free induction decayQuite obviously the precessing (fictitious) magnetization which exists following a tippingpulse will not persist forever. Various mechanisms will cause any magnetization M torelax towards its thermal equilibrium value i’ci0. The time dependent decay of the realoscillating magnetization M(t) (equation 2.27) will thus induce a damped sinusoidalsignal in the resonator. We refer to this undriven response as a free induction decay orFID.It has been found that in many instances the relaxation of the magiletization can bedescribed very well by the phenomenological Bloch equations [25]= 7(M x B) — M*+MS — M—M0 (2.28)where T1 and T2 are relaxatioll times for the longitudinal and transverse components ofM. We have assumed that all of the relaxation mechanisms ellcountered in our studiescan be characterized in in this way and equate M with M in equations 2.28. Variousmechanisms which lead to this relaxation are discussed in the following chapter.In addition to relaxation processes it is important to consider the rf field which isChapter 2. Introduction to Zero Field Magnetic Resonance and the Hydrogen Maserl6generated by the (real) oscillating magnetization of the sample. If the coupling betweenthe magnetization and the resonator is sufficiently strong, the atomic system will precessin its own radiation field. In effect this field will cause the fictitious magnetization Mto spiral back towards II even in the absence of other relaxation mechanisms. Thiseffect is known as radiation damping [30]. The analysis of the influence of radiationdamping on the FID’s obtained in our work has played an important role in our studyof spin-exchange relaxation. This topic is discussed in more detail in appendix A.2.3.2 Power radiated by the atoms and the filling factorIn order to finally make the connection between the density of radiating spins followillga tipping pulse and the rf power which is detected, we make use of the lumped elementmodel for the resonator which is preseited in appendix A. If the resonator is tuned to theatomic transition frequency, the magnetic field H produced by the magnetization M willbe in quadrature with M. Over a period of time short compared to the time scale overwhich the atoms diffuse, the average power radiated by the oscillating magnetization isProd fH.MdVb (2.29)where the integral is carried out over the volume Vb of the sample bulb. This power mustbe equal to the power dissipated by the resonator and the external circuitryPd. JH2dV (2.30)2Qwhere Q is the loaded quality factor of the resonator and V, is the volume of the resonator. Making use of this equality and realizing that at critical coupling only half ofthe power radiated by the atoms is transmitted to the external circuitry, we findPsig =wio7lQVb2() (2.31)Chapter 2. liltroduction to Zero Field Magnetic Resonance and the Hydrogen Maserl7where the filling factor ‘i is— [j’H.Md\/b]2 232—VM(O) fH2dV,.and M(O) is the real oscillating magnetization at the center of the sample which weassume to be along . Following a K/2 pulse M(O) = M0. Equation 2.31 expresses thesignal power (which can be measured) in terms of the oscillating magnetization inside thesample bulb. In the simplest case where H is everywhere uniform within the resonator(as it would be in an infinitely long solenoidal coil for example), the filling factor j issimply given by the ratio Vb/V.For relatively homogeneous rf fields we can neglect6variations in the initial amplitudeof M over the sample volume following a ir/2 pulse. Immediately following the rf pulseand before the atoms have had time to move, M is everywhere aligned with H and ij is[j’ HdV]= Vj H2dV(2.33)After the atoms have had time to diffuse, we assume that M is directed along ratherthan H and q is given to good approximation by the usual expression [31, 32][f HZdV]2= VbJHdVC(2.34)In our work we have used sample bulb/resonator configurations in which most of thesample is located within a region of high rf field homogeneity. We denote this volumeV’b. The sample bulbs do have small tails (Vb— V’b)/Vb << 1 which extend out ofthe resonator volume into region where the longitudinal rf fields are negligible. Only theatoms within the volume V’b contribute to the signal power (equation 2.31) and hence the‘post homogenization’ filling factor appropriate to these experiments is reduced slightly6llere we oniy consider the case of a Tr/2 pulse. A more general treatment in which field inhomogeneities lead to variations in the initial tipping angle as a function of position throughout the sampleis presented in references [21, 22, 27].Chapter 2. Introduction to Zero Field Magnetic Resonance and the Hydrogen Maserl8by the factor V’b/VbV’b LI HdV’b]2V fH2dV,• (2.35)In appendix B we discuss the measurement of the filling factors for the resonatorsand the sample bulbs used in our experiments. Both techniques which are presented areperturbation measurements in which the rf fields of the resonator are probed by puttingvarious magnetic and dielectric materials into the resonator. One of these techniqueshas been used previously; however, the second is a new technique which relies on theparamagnetic susceptibility of liquid oxygen to perturb the resonant frequency of theresonator.2.4 The hydrogen maserThe self consistent interaction of the oscillating magnetization M with its own radiationfield H inside the resonator is nothing less than ‘stimulated emission of radiation’.In 1960 Goldenberg, Kleppner and Ramsey [1] built the first atomic hydrogen maser,a device whose operation is based upon the stimulated emission of radiation from Hatoms. In a hydrogen maser a beam of H atoms is prepared in the upper hyperfine statesand injected into a sample bulb through a small orifice. The mean residency time of theatoms within the bulb is referred to as the bulb holding time Tb which is usually of theorder of 1 second. The bulb is located within the confines of an electromagnetic resonatortuned to the a-c hyperfine transition of H. Above a threshold flux of c ) state atomswhich is required to overcome losses, the population inversion 11H (pcc Paa) inside thebulb can be maintained high enough that self sustaining oscillations at the a-c transitionare obtained. Since its invention, the unparalleled frequency stability of the hydrogenmaser has made it an invaluable tool for both scientists and engineers [3]. References[33] and [31j contain excellent discussions of the theory and the operation of HChapter 2. Introduction to Zero Field Magnetic Resonance and the Hydrogen Maserl9Reference [32] presents a survey of the ‘physics of the H maser’.Without delving into the detailed theory of maser operation we can calculate a fewparameters that will be of interest later. In appendix A, a lumped element model of theresonator is used to determine the field which is generated by the oscillating magnetization. The Bloch equations 2.28 can be solved7 self consistently with this field. Thesolution is a function of the resonator detuning which is defined by=Q(E_-) 2Q’’. (2.36)To second order in , this self consistent solution (see appendix A) leads to the requirementsd1rdt = i’o1Qt ( — 2) ft + (2.37)—= —oiQL1. (2.38)The first of these equations describes the instantaneous rate at which the precessing(fictitious) transverse magnetization M. is damped by relaxation (T2 processes) andradiation of power. Under steady state operating conditions this damping must be zeroand hence equation 2.38 can be rewritten in the simple formw—w0 = - . (2.39)By changing the tuning of the resonator, the oscillation frequency of the maser is changed.This pulling effect is reduced by the ratio of the resonator quality factor Q to the ‘atomicquality factor’Q0m = -w0T2 (2.40)7The equations must be modified in order to include the flux of atoms entering the maser bulb. Theeffect of this term under steady state conditions is to constantly replenish losses in the component ofthe fictitious magnetization. As a result dM/dt = 0 and neither T1 nor the flux term appear explicitlyin the equations presented here.Chapter 2. Introduction to Zero Field Magnetic Resonance and the Hydrogen Maser20i.e.Qew —,— wo) (2.41)atomThe atomic quality factor is usually many orders of magnitude larger than Q and hencecavity pulling effects are strongly attenuated. It should be pointed out that the ‘zerodetuning’ oscillation frequency w0 is not given exactly by ‘B (equation 2.18) becauseof other effects which perturb the a-c transition frequency. Amongst these effects (to bediscussed in the next chapter) are frequency shifts due to spin-exchange collisions betweenH atoms. In chapters 4 and 5 we study the influence of spin-exchange induced frequencyshifts on w0 in a hydrogen maser operating at 0.5 K by changing the parameters, Qand 11H and observing the change in c — w0.Before leaving this topic it should be noted that the steady state solution of equation2.37 leads to an expression for the steady state value of M. With the aid of equation2.22 the population inversion inside the maser bulb can be written4(1+) 1nH (pcc — Paa) = 2 (2.42)h(7e+yp) itoiiQt 2We return to the discussion of H masers in chapter 4.Chapter 3Introduction to Atomic Hydrogen and Deuterium at Low TemperaturesIn this chapter we review the various interactions between hydrogen atoms, deuteriurnatoms, and 4He (both liquid and gas) at low temperatures. The discussion is dividedinto three main topics, each of which plays a role in the interpretation of the experimentspresented in the later part of this thesis. General overviews of much of the materialpresented here as well as other aspects of atomic hydrogen research at low temperaturescan be found in the reviews by Silvera and Walraven [4] and by Greytak and Kleppner[5].The first topic discussed is the confinement of atomic hydrogen (H) and atomic deuteriurn (D) by superfluid liquid helium (t4He) wall coatings. The use of liquid helium asa ‘non-stick’, yet impermeable, liquid coating to suppress the adsorption (and subsequentrecombination) of H atoms to the walls of an experimental chamber at low temperatureswas introduced by Silvera and Walraven [11] in 1979. This event effectively signaled thebeginning of low temperature atomic hydrogen research. Its use remains an importantelement of most experiments with H at low temperatures to this date. Recently [21] wedemonstrated that unlike the case for H, a gas of D atoms is only partially confined by-4He walls at temperatures above 1 K. In chapters 6 and 7 we discuss an experimentinvolving mixtures of H and D atoms at 1 K in which an improved measurement of theenergy required to force a D atom into £-4He is made.The second topic discussed is the recombination of H and/or D into a molecularform. During our study of interactions between H and D atoms at 1 K it proved essential21Chapter 3. Introduction to Atomic Hydrogen and Deuterium at Low Temperatures 22to determine the rates’ at which these reactions occur. Compared to most chemicalreactions, these recombination processes are simple and they can actually be calculatedfrom ab initio interatomic potentials. These potentials are already the most accuratelyknown of all atomic systems. By measuring the rates at which the various recombinationreactions occur it may be possible to further refine the potentials.Finally we present a discussion of the influence of atomic collisions on the frequencyand the width of the a-c hyperfine transition of H. These collisions are classified as‘buffer gas’ (in the case of H-4e collisions) and spin-exchange collisions (in the caseof H-H and H-D collisions). In both cases, the perturbative effects of these collisionsare calculable and provide a very sensitive means of testing the detailed form of theinteratomic potentials. A recent theory [19, 20] rigorously includes hyperfine interactionsin the calculation of the frequency shift and broadening cross sections for H-H spin-exchange collisions. This theory predicts the existence of frequency shifts in an oscillatinghydrogen maser which depend upon the occupancy of the individual hyperfine states ofthe H atoms. These shifts are of relatively little consequence for existing conventionalroom temperature H masers; however they may limit the ultimate frequency stability ofH masers operating at cryogenic temperatures. In chapters 4 and 5 we investigate theseH-H spin-exchange induced frequency shifts using a cryogenic hydrogen maser.3.1 Containment with liquid 4He wallsThroughout the work presented in this thesis, van der Waals films of superfluid liquid4He (-4He) have been used to coat the interior walls of the containers (sample bulbs) inwhich the atomic gasses are confined. These coatings are ‘self-cleaning’ in the sense thatalmost any atom or molecule other than H will penetrate the liquid and adsorb to the1The recombination of two H atoms to form H2 in zero magnetic field has been studied before[12, 34, 27]. In chapter 7 we present similar measurements for H-D and D-D recombination at 1 K.Chapter 3. Introduction to Atomic Hydrogen and Deuterium at Low Temperatures 23underlying substrate rather than being confined [35, 36]. They are also ‘self-healing’ inthe sense that any damage to the film (caused for example by firing an rf discharge todissociate molecular hydrogen) is quickly repaired. This last property is a consequenceof the liquid being a superfluid. The equilibrium thickness of the film is determined bythe van der Waals attraction of the 4He to the walls, and is independent of whether thefilm is superfluid or not.A consequence of using saturated £-4He films to line the cells is that the 4He vapourdensity above the film may in fact be much higher than the density of the hydrogen atomswhich are being studied. Typically we work with H and D densities below 103cm3.At1 K the 4He density He above a saturated film is 108cm3.This density drops rapidlyas the temperature is lowered, however at 0.5 K it is still about 3x1014cm.At 1 K the motion of H and D atoms through the 4He buffer gas is diffusive. Thediffusion coefficient can be writtenv = (3.1)32nHe Qciiffwhere v is the mean relative velocity of the colliding atoms (H and 4He or D and 4He)and Qd1ff is the effective hard core diffusion cross section. Hardy et al. [37] measured Vfor H atoms at T 1 K and inferred Q = 20(1) A2. Jochemsen et al. [38] estimatethat the corresponding cross section for D atoms is about 30 A2 at this temperature.The mean free paths of the diffusing H and D atomsdiff =nneQdjff(3.2)are thus less than 4 microns above 1 K. At 0.5 K the 4He density is much lower and themean free path of an H atom2 is comparable to the dimensions of the sample bulb.The gaseous 4He inside the sample bulbs influences our experiments in two other veryimportant ways. Collisions with 4He atoms can catalyze the recombination of H or D2QdIlf is expected to drop to about half of its value at 1 K [38] and thus dj1f 2 cm at 0.5 K.Chapter 3. Introduction to Atomic Hydrogen and Deuterium at Low Temperatures 24atoms into H2, D2 or RD. They also cause a momeutary perturbatiou of the hyperfiueiuteraction during the collision which results in a shift iu the frequency of the a-c hyperfinetrausition with which we work. Both of these points are discussed later in this chapter.We start with a discussion of the interactions between the H and/or D atoms and the£-4He walls.3.1.1 Binding, sticking and thermal accommodationAtomic H is attracted to the surface of £-4He by van der Waals forces which are weakbecause of the low polarizability [39] and the low density [40] of the liquid. There is infact only one known bound state [41] for an H atom on the liquid surface. The mostreliable measurement of this binding energy E5 = 1.011(10) K was made by Hürlimann[17, 42]. The only surface to which H is more weakly bound3 is liquid 3He for which0.4 K [44, 45]. Atomic D is more tightly bound4 to the surface of £-4He than H,the measurements of Silvera et al. [47, 48] seeming to indicate that E5 = 2.6(4) K forD on £-4He. There is some suspicion that this may only be an upper bound on the truebinding energy [49]. Atomic tritium (T) has never been successfully studied in a cell with£-4He coated walls [50].When a hydrogen atom strikes a £-4He surface it can either5 scatter from, or stickto the liquid. The sticking process is characterized by a sticking coefficient ‘s’ whichis the probability that an H atom which strikes the surface will enter the bound state.Once in the bound state, H atoms form a two-dimensional gas which can interact with3Traditionally it was thought that H would probably be bound to all known substrates. RecentlyNacher and Dupont-Roc [43] have demonstrated that £-4He will not wet solid Cs. The iaterpretation ofthis result is still open to speculation. However, if it is due to some combined effect of the low occupancyof the outermost electron shells in the Cs atom and the large de Broglie wavelength of the 4He atom atlow temperatures, a similar effect may be seen with H.4There has been some speculation that a second weakly bound state for D on £-4He may exist [46].The presence of this state has not been experimentally verified.5The issue of whether or not the atom can enter the liquid will be discussed in a later section.Chapter 3. Introduction to Atomic Hydrogen and Deuterium at Low Temperatures 25excitations (ripplons) on the free surface of the liquid and eventually thermalize [51].Collisions with energetic ripplons result in adsorbed atoms being desorbed and returningto the gas phase above the liquid. The thermal accommodation coefficient ‘a’ is a measureof the efficiency of the £-4He surface in thermalizing atoms which strike the surface. Thedetails of these processes are of considerable interest at present, especially at very lowtemperatures where quantum mechanical effects set in. The review by Berkhout andWalraven [52] and the references therein can be consulted for further details.3.1.2 Adsorption isotherm and the wall shiftIf the H atoms adsorbed on the £-4He surface are in thermal equilibrium with the Hatoms in the gas phase above the liquid, the chemical potentials of the two systems canbe equated to relate the number density of adsorbed atoms n to the bulk density n. Indoing this one obtains the relationship(EBNn nHAexp (3.3)where A (rn2T) 2 is the thermal de Broglie wavelength6of the atoms. The assumptions behind this classical formula are valid for surface densities up to about 1O’4cm2[53]. The surface densities encountered in our work are many orders of magnitude lowerthan this limit.While adsorbed to the surface an atom is distorted slightly. This distortion is due tothe combined effect of the attractive and repulsive interactions between the atom and thesurface. It causes a small change in the hyperfine interaction between the nucleus andthe electron to occur. This in turn causes the hyperfine transition frequency of each atomon the surface to be shifted from the unperturbed frequency w0 by an amount 8w. In61t is often convenient to remember that AH = 17.4 A() for atomic hydrogen and AD = 12.3A() for atomic deuterium.Chapter 3. Introduction to Atomic Hydrogen and Deuterium at Low Temperatures 26experiments with the cryogenic hydrogen maser operating at 0.5 K, this effect is manifestas a temperature dependent shift of the density independent oscillation frequency o•This effect is commonly referred to as a ‘wall shift’7 A simple theory describing this shiftis presented below. More detailed analyses can be found in references [54, 55].During each sticking event an atom picks up an average phase shift q = < >where < > is the average duration of a sticking event. The ratio of < . > to theaverage time between sticking events < Tb > is the same as the thermal equilibrium ratioof the number of atoms on the walls to the number in the bulk. In a container withsurface area A and volume V this implies that<T >< Tb> Aexp(EB/kBT). (3.4)With the assumption that both < r. > and < Tb > are distributed according to Poissonstatistics, it can be shown [25] that the atomic resonance is shifted byw =<Tb> +from the unperturbed frequency w0. The broadening of the resonance is2— <Tb> +At 0.5 K the atoms do not dwell on the surface long enough to pick up a large phaseshift and thus.< 1. In this limit the frequency shift 6w is linear inV A<Tb>wAexp jjjj . (3.7)Hürlimann used the UBC cryogenic H maser to measure this shift [17, 42]. He foundf = w/2 = —7.15(30) x iü Hz for H on £-4He. Similar measurements have not beenmade for D.71n conventional room temperature H masers a similar effect is observed as atoms collide with theteflon coatings which are used to suppress wall adsorption. The difficulty in reproducing these wallcoatings limits the accuracy of conventional masers to about 10— 12Chapter 3. Introduction to Atomic Hydrogen and Deuterium at Low Temperatures 273.1.3 Incomplete confinementLow concentrations of 3He will dissolve into £-4He accompanied by the release of energy.The physics of 3He in £-4He is a very interesting field of study and it naturally leads oneto question whether or not other impurities will also dissolve into £-4He. From our pointof view it is particularly interesting to consider the possibility of atomic hydrogen or oneof its isotopes dissolving into £-4He.Theoretical workKürten and Ristig addressed this question in a theoretical paper published in 1985 [35]They estimated the energy z which would be required to replace N+1 4He atoms (bulk£-4He) with N 4He atoms and one atom or molecule of hydrogen or its isotopes. Theyfound that the diatomic molecules of hydrogen8would enter the liquid accompanied bythe release of energy, while energy was required to force the atoms H, D, and T into£-4He. The solvation energy E required to force an atom into the £-4He is approximatelythe difference between z and the latent heat of one 4He atomE=z—L4 (3.8)Kürten and Ristig found that Li was quite strongly dependeilt upon the £-4He density.The zero pressure density of £-4He in their model was 17.2 x i0 A3 while the trueliquid density under the saturated vapour pressure is 21.7x103A3 [40]. In table 3.1 welist estimates of the solvation energies for H, D, and T made by subtracting the knownlatent heat of 4He (L4 7.2 K) from the values of z presented in figure 1 of reference[35].8The issue of whether the solvation process is exothermic or endothermic is determined by the detailedform of the interatomic potentials. The calculations of Kürten and Ristig suggest that 112 is a borderlinecase. They felt that it would enter the liquid but perhaps that it might reside in regions of low density.Chapter 3. Introduction to Atomic Hydrogen and Deuterium at Low Temperatures 28Calculated £-4He DensityIsotope 17.2x103 21.7x103atoms/As atoms/AsH 30K 65KD 11K 32KT 3.4K 20KTable 3.1: Theoretical values for the solvation energy E (in temperature units) requiredto force hydrogen (H), deuterium (D) and tritium (T) atoms into liquid 4He at two liquiddensities. These results were obtained by subtracting the latent heat of 4He (7.2 K)from the theoretical energy required to replace N+1 4He atoms with N 4He atoms andone impurity atom as calculated by Kürten and Ristig [35] .The lower density correspondsto the zero pressure density of £-4He in their model while the higher density is the trueliquid density under its saturated vapour pressure at 1 K.In their calculation of the chemical potential z Kürten and Ristig ignored multipartide correlations which lead to backfiow effects [56]. They predicted that the inclusionof these effects would lower somewhat but would not lead to a ‘substantial’ change intheir results. They concluded that H and its isotopes would not dissolve into £-4He. Ina later paper by Krotscheck et al. [57], theoretical solvation energies for H, D, and T inwere reported9 at ‘the calculated equilibrium density of bulk 4He’. These energieswere 30 K, 11.5 K and 4.3 K for H, D, and T respectively.Experimental workIn light of the calculations presented above it is not surprising that effects due to thesolvation of H into £-4He have never been observed. Most experiments with H confinedby £-4He have been performed at temperatures below 1.5 K where the probability offinding an atom with sufficient thermal energy to penetrate the liquid is extremely low.9The main emphasis of Krotscheck et al.’s paper is the study of impurity states in very thin £-4Hefilms where the substrate potential provides an attractive force which tends to draw the impurity atominto the liquid. The saturated films used in our work are much thicker and can be treated as bulk £-4He.Chapter 3. Introduction to Atomic Hydrogen and Deuterium at Low Temperatures 29In the case of tritium, it is not clear from the theory whether or not a gas of T shouldbe well confined by £-4He walls at these temperatures. The failure of Tjukanov et al.[50] to detect measurable quantities of T in a container coated with superfluid £-4He mayindicate the latter.For deuterium one might have expected a high degree of confinement. Neverthelessin a recent experiment [21] we observed for the first time the solvation of D atoms into£-4He at temperatures just above 1 K. This experiment used magnetic resonance on ahyperfine transition of the D atom to study the time and the temperature dependenceof the atomic density iiD inside cells coated with £-4He. At temperatures just above 1K the atomic density was observed to decay exponentially with time, and at a rate thatwas steeply temperature dependent. Above 1.16 K the sample lifetime was too short tomeasure. This decay could not be attributed to recombination processes and eventuallywas interpreted as being due to the solvation of D into £-4He.A simple model [22, 21] was used to describe the decay of the atom density due to thissolvation process. In this model, D atoms striking a £-4He surface were assumed to enterthe liquid with thermally averaged10 probability g1 if they had kinetic energy greaterthan E. It was then assumed that this D quasi particle traveled ballistically throughthe film with an effective mass m* and that once it encountered the substrate (solid D2)it was adsorbed. This model leads to an exponential decay of the atomic densitydnD= —)nD (3.9)where the decay rate ,\ for a cell with volume V and surface area A is given byAv_ / EN=1exp I%_j . (3.10)Here v (kI) 2 is the thermally averaged speed of the atoms striking the liquid surfaceand t is the ratio of the quasi particle effective mass m* to the bare atom mass m. The‘°This probability is likely quite angle dependent at least for kinetic energies just greater than E.Chapter 3. Introduction to Atomic Hydrogen and Deuterium at Low Temperatures 30term jg is the thermally averaged transmission probability that a quasiparticle strikingthe liquid gas interface will leave the liquid. This reverse transmission probability isrelated to gI through the expression—— / E5N= paigexp . (3.11)Analysis of the decay rate data from this experiment led to a measurement of theenergy required to force a D atom into £-4He. In this original experiment we foundE5 = 13.6(6) K. Unfortunately, due to scatter in the data and the very narrow range oftemperatures over which data could be taken, it was impossible to determine the prefactorin equation 3.10 with any precision. In chapters 6 and 7 of this thesis we present theresults of a new study of the solvation of D into £-4He. This experiment is performedusing magnetic resonance on atomic H in mixtures of H and D. This technique allows usto account for recombination processes which occur in parallel with the solvation processand thus we are able to make a considerable reduction in the scatter of the data. Theseresult include a measurement of the prefactor associated with equation 3.10.Before leaving this topic, the assumptions behind the simple solvation model presentedabove do require some comment. The assumption of ballistic transport within the filmis addressed first. The thickness ‘d’ of a saturated £-4He film on a H2 substrate is [58]d = 2.5(4) x (3.12)where ‘h’ is the height above the bulk liquid at which the observation is made. The filmthickness over a D2 substrate should be similar to this. In the original measurement11of the solvation energy for D in £-4He [21] the film thickness at the site of the resonatorwas approximately 100 A. The data in this experiment was recorded over the narrowtemperature range 1.08 K to 1.16 K. The mean free path of a 3He quasi particle in £-4He‘1n the present work, the cells are smaller and hence the films are about 50% thicker.Chapter 3. Introduction to Atomic Hydrogen and Deuterium at Low Temperatures 31due to scattering by rotons’2 ranges from about 1000 A at 1.0 K down to about 250A at 1.2 K [59]. These mean free paths are somewhat longer than the thickness ofthe £-4He films that we use. Unless the cross section for interactions between D quasiparticles and £-4He is much larger than that for 3He- £-4He interactions, no attenuationdue to diffusive transport in the film should be expected.It is somewhat more difficult to address the second assumption that every D quasiparticle becomes adsorbed to the substrate. The binding energy of D to bare D2 is about55 K [60]. Under the £-4He film it is necessary to balance the attraction of the substratefor the D atom against the increase in the chemical potential of the D atom as the £-4Hedensity increases near the wall [35, 61]. This is complicated by the fact that very close tothe substrate, the 4He tends to solidify. Krotscheck et al. [57] noted that in the case ofvery thin films, D atoms on the £-4He surface should actually be drawn into the film andbound to the substrate by van der Waals forces. A calculation of the sticking probabilityfor a D quasi particle striking this substrate would certainly be interesting.3.2 Recombination processesWe turn now to a review of the recombination of atomic hydrogen and deuterium in thecontext of the present work.3.2.1 Molecular potentialsAt low temperatures H (or D) atoms are almost always in their electronic ground state.If we neglect hyperfine interactions then the Coulombic pair interaction between twoatoms depends only upon the symmetry of the total electron spin wavefunction. Thisinteraction can be written in terms of an attractive (except at very close range) singlet‘2The dominant excitations in £-4He at this temperature are rotons.Chapter 3. Introduction to Atomic Hydrogen and Deuterium at Low Temperatures 321g potential, and an essentially repulsive triplet potential. The H-H interatomicpotential is the most accurately known potential of any atomic system. The results ofthe ab initio calculations of these potentials made by Kolos and Wolniewicz [62, 63, 64]are shown in figure 3.1. A consequence of the accuracy to which these potentials areknown is that often the cross sections for collision processes can be calculated very wellfrom first principles.Two H atoms interacting via the triplet potential’3 cannot recombine. On the otherhand the singlet potential supports many bound states. It is difficult to test the longrange part of theoretical potential energy curves at internuclear separations further thanthe classical turning point of the least bound molecular state.’4 For H2 this distancecorresponds to about 6 A for the v=14 vibrational level [62]. At low temperatures theoutcome of most scattering processes is governed primarily by the long range part of theinteratomic potentials. In a sense, careful measurements of these processes can be usedto test and refine the long range parts of calculated interatomic potentials. A tabulationof the known bound levels within 300 K of dissociation for the H2, D2 and HD moleculesis given in appendix E.As the temperature of a gas of H atoms is lowered, more and more atoms collide withenergies comparable to the hyperfine interaction. At these temperatures it is necessaryto include hyperfine interactions in the calculation of physical processes. This increasesthe complexity of the calculations considerably as the manifold of colliding states isenlarged. Calculations of this type have oniy recently been performed for H-H spin-exchange collisions in the context of the atomic H maser. These calculations are discussedin a later section. The implications of these calculations for the stability of cryogenic H‘3There is a very shallow well (about 6.5 K) deep in the triplet potential at an internuclear separationof 4.15 A. It is not deep enough to support any bound states.1i.e. spectroscopic measurements of the energy of the various molecular states can be used to testthe potentials at shorter separations.Chapter 3. Introduction to Atomic Hydrogen and Deuterium at Low Temperatures 3360000 —i-• •40000b3z200000>—20000+40000 1%—6000 . I I .0 2 4 6 8R/a0Figure 3.1: The lowest H-H interatomic singlet and triplet potentials as calculated from first principles by Kolos and Wolniewicz [62, 63, 64]. The radial separationof the nuclei has been normalized to the Bohr radius.Chapter 3. Introduction to Atomic Hydrogen and Deuterium at Low Temperatures 34inasers are of great technological concern. Measurements of the spin-exchange inducedfrequency shifts in a cryogenic H maser are presented in chapter 5.Before leaving this topic we include a comment on notation. In the case of H2, the twoatom wavefunction describing the molecule must be antisymmetric under interchange ofeither the nuclei or the electrons. That is, the sum J+I must be an even number. Twopossible types of states can be identified. By convention15 the states with 1=1 and J oddare labeled ‘ortho’ states while those with 1=0 and J even are labeled ‘para’. For the D2molecule the labeling is reversed: the states with even J are the ortho states and thosewith odd J are the para states.3.2.2 Recombination of atomic H, DThe recombination of H to form molecular hydrogen H2 (or any isotopic variation ofthis reaction) is an exothermic process which releases about 4.5 eV of energy. At lowtemperatures it is not possible to conserve both energy and momentum in this reactionunless it is catalyzed by some inert third body X. That is, recombination takes place bythe reaction’6H+H-j-X—*H2-l (3.13)This reaction leads to a decrease of the atomic density inside a sealed containerdn= —KHHn (3.14)where KHH is the rate at which the decay occurs. If the initial H density is n then thetime dependence of n11 is simply given by= i ( (3J5)15The more practical definitions ‘symmetric’ and ‘antisymmetric’ in reference to the wavefunctions arenot commonly used.16The three body reaction H + H + H — 112 + H does not contribute to the decay of the 11 densityat the low densities considered in this thesis,Chapter 3. Introduction to Atomic Hydrogen and Deuterium at Low Temperatures 35where r is a characteristic time for the decayTK (3.16)llHn11At temperatures near 1 K the dominant catalyst for this process is the ‘dense’17 4Hebuffer gas which is present as a consequence of the saturated £-4He film. In this case thedecay is also proportional to the 4He density He and so KHH kflHnH6.The recombination of two D atoms to form D2 can be described in a completelyanalogous manner by simply replacing ‘H’ with ‘D’ throughout the above discussion. Inchapter 7 we deal with mixtures of H and D at temperatures near 1 K. In appendix Cthese rate equations for the decay of the atomic densities are generalized to include H-Drecombination and tile solvation of D into £-4He as alternate density loss mechanisms.Tile recombination of H by the reaction 3.13 has been studied theoretically by Grebenet al. [65]. They found that this process could be described in terms of separate contributions from the formation of ortho and para-H2. At zero field under the assumptionthat all hyperfine states are equally populated, their results can be written1 3kHH = + jjkortho . (3.17)Their numerical results for kflH are summarized in table 3.2. The bulk of the temperaturedependence of this rate constant arises from the thermal averaging of the ortho-H2 formation rate which drops as the temperature is lowered. Measurements of kHH have beenmade previously in this laboratory. Morrow’8 reported a value of 2.0(3) x iO crn6/sat 1 K which seems to be in excellent agreement with the rate constant 1.8x1033cm6/scalculated by Greben et al. Similar calculations have not been performed for tile formation of D2 or HD. Measurements of the rate constants for D-D and H-D recombinationare presented in this thesis.1Tin the sense that 11H >> nH18A value of 2.8(3)x i0 cm6/s was originally reported in references [37, 34, 12]. It was laterdiscovered that this value was too high by a factor of [27].Chapter 3. Introduction to Atomic Hydrogen and Deuterium at Low Temperatures 36T D0 Contribution to kHH— (K) at various temperatures1K 0.1K 0.01K14 0 210.4 0.095 0.056 0.04914 1 185.2 0.183 0.01614 2 138.6 1.747 1.32 1.2914 3 73.5 14.63 1.62 0.14714 4 1.4 1.107 0.001513 7 76.2 0.025Total 17.79 3.01 1.49Table 3.2: Theoretical results of Greben et al. [65] for the contributions of different statesof the H2 molecule to the H-H recombination rate constant kHH = jkpara + kortho in unitsof 10M crn6/s. The molecular levels are labeled by their vibrational (v) and rotational(J) quantum numbers. The dissociation energies D0 of these states are included as theydiffer slightly from the values quoted in appendix E.At lower temperatures (T0.5 K or lower) where the 4He buffer gas density is reducedand the H (or D) atoms spend more time adsorbed to the liquid helium wall coatings,the recombination reaction 3.13 can be catalyzed by the walls of the container. In thiscase the reaction is characterized by a surface recombination ‘cross length’. In zero fieldthe cross length for H-H recombiuation on a £-4He surface is19 0.14(2) A3.3 Effects of atomic collisions on hyperfine resonance parametersHydrogen atoms typically undergo many collisions before finally recombining. Duringthese collisions the hyperfine interaction between the electron and the proton is momentarily perturbed. The net effect of many such collisions can be to shift and/or to broadenthe atomic resonance. These effects and their relevance to our experiments are describedbelow. They are classified as ‘buffer gas’ and ‘spin-exchange’ collisions.19A value of 0.20(3) A was reported in references [34, 44, 12]. This was later recognized to be in errorby a factor of [27].Chapter 3. Introduction to Atomic Hydrogen and Deuterium at Low Temperatures 373.3.1 Shift due to collisions with 4He atomsThe strength of the hyperfine interaction depends upon the local electron density at thesite of the nucleus. During a collision with an inert body such as 4He, polarization effectscause this density to be momentarily modified. The average effect of these perturbationsis to shift the frequency of the radiation which is observed from a hyperfine transition.They do not lead to a significant broadening of the transition. Traditionally these shiftshave been referred to as ‘hyperfine pressure shifts’ simply because the frequency shiftincreases as the density of the buffer gas is increased.The calculation of these frequency shifts is a difficult theoretical problem (see forexample references [66, 67, 68]). The N electron problem must first be solved (H-4Heis the simplest case with N=3) to determine the frequency shift as a function of theinternuclear spacing, and then the necessary thermal average must be performed. Athigh temperatures the frequency shift for the H-4e system is positive and dominatedby contributions from the short range part of the interatomic potentials. Jochemsen andBerlinsky [69] extended the calculations for this system to the low temperature regime.They found that below 10 or 20 K the shift changes sign and is dominated by the longrange part of the potential. The pressure shift coefficient Sf/nne is essentially independentof temperature over the narrow temperature range that was examined experimentally byHardy, Morrow and their collaborators [12, 27]. They measured= —11.8 x 108Hz cm3nilefor the a-c hyperfine transition of H at 1 K. We recently measured this quantity for the1, mF = 0 hyperfine transition of D in a 3.89 mTesla field [21]. The pressureshift coefficient in this case is= —3.8(1) x 1V’8Hz cm3.nileChapter 3. Introduction to Atomic Hydrogen and Deuteriurn at Low Temperatures 38It is important to note that the H-4e buffer gas shift leads to a frequency shiftwhich has the same sign as the wall shift discussed earlier in this chapter (equation 3.7).This has important consequences for any cryogenic H maser which uses £-4He to suppressadsorption (chapters 4 and 5). The combined frequency shift due to these effects passesthrough a minima as a function of the temperature. The exact temperature at which theextrernum occurs depends upon the area to volume ratio of the maser bulb. By operatingat this temperature (typically ‘‘0.55 K) the oscillation frequency of the maser becomesinsensitive to changes in the combined frequency shift. In chapter 7 we present a moreaccurate measurement of the pressure shift coefficient for H-4e collisions at 1 K.3.3.2 Spin-exchange induced broadening and frequency shiftsCollisions between paramagnetic atoms such as H and D also lead to perturbations of thehyperfine interaction and can lead to shifts of atomic hyperflne resonances. The effectswhich are induced by these collisions are dominated by electron-exchange interactions [18]and thus they are labeled ‘spin-exchange’ collisions. This exchange mechanism is a formof spin relaxation and hence spin-exchange collisions can broaden an atomic hyperfineresonance. Many discussions of spin-exchange can be found in the literature, howeverPiuard and Lalöe [70] give a particularly nice interpretation of these collisions in termsof the Pauli exclusion principle. This paper highlights the importance of spin identity indetermining the outcome of a collision. Other important papers are referred to below.Much of the theory of spin-exchange collisions as they pertain to atomic hydrogenwas developed during the 1960’s in the context of the atomic hydrogen maser [71, 72, 73,74]. In the semi-classical formalism which was adopted for these studies the hyperfineinteraction was deemed to play an insignificant role in determining the outcome of acollision. The frequency shift of the a-c hyperfine transition of H predicted by this theoryis proportional only to the rate at which H-H collisions occur and the level populationChapter 3. Introduciion to Atomic Hydrogen and Deuterium at Low Temperatures 39difference between these two states. This theory has been quite successful at describingphenomena induced by H-H spin-exchange collisions at energies much higher than thehyperfine splitting of the H atom.Recently the first fully quantum mechanical treatment of the H-H spin-exchangecollision problem was published [19, 20]. The non-adiabatic effects introduced by theinclusion of hyperfine interactions result in frequency shifts and broadening of the ac hyperfine transition of H which depend on the detailed occupancy of the individualhyperfine states.In this thesis we will be concerned with the influence of both H-H and H-D spin-exchange collisions on the a-c hyperfine transition in a gas of atomic hydrogen at lowtemperatures. We begin with a review of the frequency shift and the broadening of thistransition which are predicted by the semi-classical treatment of spin-exchange collisions.This is followed by a summary of the predictions which are made by the fully quantummechanical theory [19, 20] for H-H spin-exchange collisions in an oscillating H maser. Ananalogous theory has not yet been developed for H-D spin-exchange collisions.Semi-classical treatmentIn the semi-classical picture used by Crampton [71, 72] and others (for example references[73, 74]) to describe H-H spin-exchange collisions, all interactions except electron spin-exchange are neglected and the initial states of the colliding atoms are considered to beidentical. This is often referred to as the degenerate internal states or DIS approximation.By neglecting spin orbit coupling the (elastic) scattering process can be described in termsof independent phase shifts from the singlet and the triplet potentials of the transientmolecule. Following this procedure it is possible to write down the equation of motionfor the spin density matrices of the colliding atoms (see for example reference [74]). Thisequation depends upon whether or not the colliding atoms are identical.Chapter 3. Introduction to Atomic Hydrogen and Deuterium at Low Temperatures 40In the context of a hyperfine magnetic resonance experiment, a tipping pulse (de—scribed in the previous chapter) is a small perturbation which modifies the density matrixof the atomic system. Spin-exchange collisions will cause these matrix elements to relaxback to their thermal equilibrium values. We only consider the effects of spin-exchangecollisions on the a-c hyperfine transition of the H atom in zero field in the discussion thatfollows.In the case where the colliding atoms are identical (two H atoms), the spin-exchangerelaxation rate for the diagonal elements of the H atom density matrix can be written[71, 72]Ise = flHVHHnH (3.18)1 HRwhere T;HH is the longitudinal relaxation time due just to H-H spin-exchange collisions.The spin-exchange relaxation rate for the off diagonal elements of the density matrix is-H = flllVHHUHH (3.19)2where T HR is the corresponding transverse relaxation time. In these equations VHHis the mean relative speed of the colliding H atoms and flll is the thermally averagedspin-exchange (or ‘spin flip’) cross section for the collision. The product flH = vHH isoften referred to as the thermally averaged spin-exchange broadening parameter or rateconstant. It should be noted that in the DIS theory the ratio= (3.20)is independent of the atomic density. This relationship has been verified experimentallyat temperatures as low as 77 K by Desaitfuscien and Audoin [75].If an H atom collides with heteronuclear atoms such as D rather than other H atoms,these equations become [71, 22]:Ise= 11flvDUHD (3.21)1 HDChapter 3. Introduction to Atomic Hydrogen and Deuterium at Low Temperatures 41—:-; = }IflDHD (3.22)2 HDwhere the notation should be obvious. In this case the ratio Tje/T is also independentof the atomic densities but equal to . This relationship has been verified near roomtemperature by Berg [76].The thermally averaged20 cross sections for these collisions can be written in terms ofthe singlet and triplet phase shifts (6 and 6) for collisions involving angular momentum£h and relative linear momentum hkij = (2+ 1)sin2(6_6)) (3.23)/L1 £=0 kwhere denotes the reduced mass of the colliding atoms (not to be confused with theeffective mass ratio t introduced earlier in this chapter) and= 8kT (3.24)7rII1is their mean relative speed.H-H spin-exchange collisions also lead to a shift in the frequency of the a-c hyperfinetransition of H. This shift is proportional to the population difference between the a )state and the c ) state1——= pcc — Paa)flHVHHHH (3.25)where o is the unperturbed transition frequency and )HH is the thermally averagedfrequency shift cross section. XHH can be calculated in a similar manner to HH [72, 71]= -4[i +(_l)](2+1)sin2(6_)) (3.26)VHH Ii £=0 kThe product VHH?HH is often referred to as the thermally averaged frequency shift parameter or rate constant. Crampton has shown [71] that in the DIS limit, H-D spin-exchangecollisions do not lead to a similar shift of the a-c hyperfine transition of H.20The symbols (•. .)k are used to indicate the thermal averaging.Chapter 3. Introduction to Atomic Hydrogen and Deuterium at Low Temperatures 42frnperature HH (A2) HH (A2)(K) (calculated) (measured) (calculated) (measured)0.5 47.2 [77] 0.526 [77]55 [78] 0.75 [78]1.1 25.9 [77] 0.387 [77] 0.43 ± 0.03 [27]0.55 [78]0.31*300 3.6 [77] 4.1 ± 1.0 [79] 23.4 [77] 23.1 + 2.8 [75]3.8 + 0.4 [80] 23.5 [80]Table 3.3: Summary of measured and calculated values for the thermally averaged H-Hspin-exchange frequency shift \HH) and broadening (HH) cross sections. The value ofindicated with a * was reported in reference [27] after the result calculated in reference[78] was reanalyzed using the improved potentials of reference [63].A compilation of measured and theoretical values of‘HH and HH for H-H spin-exchange collisions at temperatures relevant to the present work can be found in table 3.3.Room temperature values are also given for comparison. The theoretical results obtainedby several authors have been included to illustrate the sensitivity of these parameters tothe detailed form of the interatomic potentials which are used in the calculations. Thecalculations of Verhaar et al. [19, 20, 77] are believed to be the most reliable as theyhave used the most modern potentials.After examining the measured broadening cross sections presented in this table, itbecomes obvious why very enthusiastic predictions about the potential frequency stabilityof cryogenic hydrogen masers were once made [81, 15, 14]. The important parameter forthis discussion is the spin-exchange broadening rate constant VHHJHH. This parameter isnearly three orders of magnitude smaller at 1 K than it is at 300 K. If we only considerfrequency instabilities due to the thermal photons in the maser resonator,2’then theintrinsic instability of a maser Sf/f is proportional to () where P is the outputpower of the maser and r is the averaging time for the measurement. The stability of21The phase of these photons will be at random with respect to the phase of the atomic coherence.Chapter 3. Introduction to Atomic Hydrogen and Deuterium at Low Temperatures 43a maser obviously increases as the temperature of the resonator is lowered but it alsoincreases as the power output is increased. The power output of a maser is limited byspin-exchange broadening and thus by going to low temperatures it appears that fantasticimprovements in the intrinsic stability of the maser might be made.This analysis is insufficient as it assumes that any frequency instabilities which arecoupled to the atomic density can be neglected. At room temperature it is possible to tuneout these density dependent shifts to a high degree of accuracy [72]. Unfortullately, asthe temperature of the maser is lowered it becomes important to consider spin-exchangeeffects which are induced by the hyperfine interaction. These effects couple the oscillationfrequency of the maser to the population of the individual hyperfine level populations ina way that cannot be simply eliminated. This point is illustrated in the next section.Before leaving the semi-classical treatment of the hydrogen spin-exchange problemwe note that Reynolds [22] has calculated the H-D spin-exchange broadening parameterVnDOHD in the DIS limit for temperatures in the range 1.0 to 1.5 K. He found that thisrate constant was essentially independent of temperature over this narrow range andequal to22= 2.4 x 10cm3/s . (3.27)This implies that the H-D spin-exchange cross section flD = 140 A2 at 1 K. No measuremeilts of this quantity at cryogenic temperatures have been made prior to the workpresented in chapter 7.Fuiiy quantum mechanical pictureIn the time since the first H-H spin-exchange theories were developed it was realizedthat non-adiabatic effects introduced by the hyperfine interaction could lead to changes22There is a significant difference between the H-H and the H-D spin-exchange cross sections near 1K. It is the H-H spin-exchange cross section which is anomalously small.Chapter 3. Introduction to Atomic Hydrogen and Deuterium at Low Temperatures 44rTemp ir0 A2 00 Oi[ (K) II (A2) (A2) (A2) (A2) (A2) (A2)0.39 -13.8 —2.41 x 10—2 —1.06 x 10—2 5.49 x 10—2 0.560 0.554 x 1O0.50 -11.8 —1.77 x 10—2 —1.15 x 10—2 4.09 x 10—2 0.524 0.748 x iO0.63 -10.1 —1.29 x 10_2 —1.25 x 10_2 3.02 x 10_2 0.484 0.972 x iO0.79 -8.46 —0.927 x 10—2 —1.34 x 10—2 2.21 x 10—2 0.442 1.22 x 1O1.00 -7.00 —0.650 x 10—2 —1.41 x 10—2 1.60 x 10_2 0.400 1.49 x 1O300 -0.90 0.36 x 10—2 —0.67 x 10—2 —0.26 x 10—2 -0.60 12Table 3.4: Calculated values of the thermally averaged H-H spin-exchange frequency shiftand broadening cross sections in the Verhaar theory [77].in the frequency shifts described in the previous section [82]. It was not until recentlyhowever that the fully quantum mechanical problem was addressed. B. J. Verhaar andhis collaborators [19, 20] included the effects of interatomic and intraatomic hyperfineinteractions as well as dipole terms in the H-H interaction Hamiltonian. Their resultsfor spin-exchange collisions between H atoms display fundamental differences from thesemiclassical picture presented above. These differences are particularly significant whenapplied to the operation of H masers at cryogenic temperatures. They shoved thatthe frequency shift (equation 3.25) and the line broadening (equation 3.19) due to H-Hspin-exchange collisions are actually of the form23— = [VO(pcc— paa) + )i(pcc + p) + (3.28)[o(pcc— paa) + i(Pcc + p) +2]VHHnH (3.29)Table 3.4 contains a summary of calculated values for the thermally averaged frequencyshift () and broadening (j) cross sections at temperatures relevant to room temperatureand cryogenic H masers [77].To assess the influence of spin-exchange collisions on the oscillation frequency of a231n reference [83] it is shown that in the presence of a strong magnetic field further terms ) andwhich couple to pbb and pdd must be added to these expressions. These additional terms are of noconsequence for a hydrogen maser operating near zero field.Chapter 3. Introduction to Atomic Hydrogen and Deuterium at Low Temperatures 45hydrogen maser it is necessary to add the effects of cavity pulling which were describedin the previous chapter (equation 2.39) to the spin-exchange frequency shift 3.28. Thecombined frequency shift is— wo = + [XO(pcc— paa) + Xi(pcc + paa) + VHHIIH (3.30)where zi is the detuning of the resonator from the atomic resonance defined in equation2.36. If we define4VHH3 31— h(7e + 7p)2/1Oh1Qthen the population inversion IIH (pcc paa) inside the maser bulb (equation 2.42) can bewritten(pcc — p) vHHnH = (i + 2) . (3.32)We now assume that the oniy density dependent part of the linewidth is that due tospin-exchange collisions, i.e.= + [o(pcc — paa) + i(pcc + Paa) + 2] VJ{HnH. (3.33)All other contributions to the Iinewidth are contained in the term T9. As a result of thedynamics of an oscillating maser the difference (pcc — Paa) is much smaller than the sum(Pcc + p). The broadening term proportional to (pcc — p) is thus very small and canbe neglected. The full expression describing the combined spin-exchange/cavity pullingfrequency shift of an oscillating H maser can thus be written=+ { [ + i3)o(1 + 2)] [i(pcc + paa) + 21 + {i(pcc + p) +X2]}VHHnH.(3.34)In general the dependence of this frequency shift on the atomic density is quite complicated. The essential point which should be noted is that it depends upon the density inChapter 3. Introduction to Atomic Hydrogen and Deuterium at Low Temperatures 46a nonlinear way. Various schemes which minimize the coupling between the oscillationfrequency and the atomic density have been discussed [20, 84, 17]. The details of thevarious strategies are beyond the scope of this thesis.The corrections to the semi-classical H-H spin-exchange frequency shift parametersare small. It is doubtful that these parameters could be measured without utilizing thenarrow linewidth and the frequency stability of a hydrogen maser. In a maser operatingat room temperature, the absolute spin-exchange induced frequency shifts are typicallyless than 1 mHz. A measurement of the various spin-exchange cross sections at roomtemperature by detecting changes in these shifts as parameters such as the tuning ofthe resonator are varied would be a very difficult task. At low temperatures one gainssomewhat as the atomic densities are increased dramatically over those in a conventionalroom temperature maser. in the following two chapters we describe such a measurementusing a cryogenic hydrogen maser.Chapter 4Apparatus used for the 0.5 K ExperimentsIn this chapter we describe the apparatus used to explore the influence of H-H spin-exchange collisions on the operation of the cryogenic hydrogen maser (CHM). In particular, these experiments were designed to test the H-H spin-exchange theory developed byVerhaar et al. [19, 20] and its applicability to the CHM. On occasion we refer to themas the ‘0.5 K experiments’ in contrast to the higher temperature experiments presentedlater in the thesis.The maser to which we refer is the CHM designed and built at the University ofBritish Columbia (UBC) by Drs. Martin Hürlimann, Walter Hardy and Richard Cline. Acomplete description of the design, construction and operation of this device would be fartoo lengthy to be presented here. Several papers [9, 85, 42, 16] containing this, and relatedinformation have been published but by far the most complete and informative documentis the PhD thesis of Martin Hürlirnann [17]. In this chapter the only aspects of themaser which are addressed are those directly related to the spin-exchange measurementspresented in the following chapter.4.1 The UBC CHM: Conventional mode of operationA conventional room temperature H maser requires a source of atomic hydrogen, a stateselector (magnet) which directs atoms in the c ) state (often both the I c ) and d )states) into a storage bulb located within a high Q microwave cavity tuned to the ac hyperfine transition of H. These regions are shown schematically in figure 4.1. A47Chapter 4. Apparatus used for the 0.5 K Experiments 48minimum (threshold) flux of H atoms in the I c) state is required for stimulated emissionof radiation to occur1. Some of the power radiated by the atoms can be extracted fromthe cavity and fed to external monitoring circuitry. The atoms eventually wander out ofthe storage bulb with a time constant typically of order a fraction of a second, and arepumped out of the system.The UBC CHM is unlike conventional H masers in that it recirculates the H atoms. Itmakes use of the fact that the recombination lifetime of H atoms confined by £-4He wallsat 0.5 K is of order hundreds of minutes at typical masing densities of 1012 cm3. Ratherthan being pumped away, atoms which have left the maser storage bulb are allowed toreturn to the state selection region of the maser where they are pumped to the upper(I c )) state. In this way, the same atoms are repeatedly cycled through the maser.There are 5 basic regions inside the UBC CHM (see figure 4.2). All internal surfacesare coated with a film of superfluid 4He in order to suppress the recombination of atomicH. A consequence of this film being present is that the maser volume is filled with abuffer gas of 4He whose density is a strong function of temperature. A low temperatureatom source [27] uses a pulsed 50 MHz discharge to provide bursts of atomic H. Thesource contains a small Co6° source whose /3 radiation provides excess free charges whichgreatly facilitate initiation of the discharge. Typically, only a few pulses (of roughly 10zs duration, 0.4W peak power) are required to obtain a H density high enough to operatethe maser. A large buffer volume2 acts as a low pass filter to ‘smooth out’ the bursts ofatomic H produced by the source. It also reduces the area/volume ratio of the assemblywhich in turn increases tile lifetime of the H atoms. The density within the maser can bemaintained at any desired value within its operating region by modifying the dischargepulse duration and repetition rate3. Atoms from the source/buffer volume region are bled1i.e. dissipative losses in the cavity must be overcome.2This volume was added for the present experiments.3A constant trickle of H is required to compensate for the loss of hydrogen due to recombination.Chapter 4. Apparatus used for the 0.5 K ExperimentsadC49Figure 4.1: A schematic representation of the basic components of a room temperatureH maser showing (a) the rf discharge/atomic source, (b) state selector, (c) storage bulb,(d) microwave cavity and (e) coupling loop.e_bHH2Chapter 4. Apparatus used for the 0.5 K Experiments 50Figure 4.2: A schematic representation of the UBC cryogenic hydrogen maser showing (a)the low temperature atomic source (discharge), (b) buffer volume (added for the presentexperiments), (c) atom tube, (d) state selector region, and (e) the storage bulb inside the1420 MHz split-ring resonator. The atom tube and the storage bulb are shown in moredetail in figure 4.3.Chapter 4. Apparatus used for the 0.5 K Experiments 51deCagb hFigure 4.3: Details of the split-ring resonator and maser storage bulb region showing(a) the inner (storage) bulb, (b) outer bulb, (c) split-ring resonator, (d) atom tube andbrass cone, (e) orifice, aild (f) the mechanical tuning and coupling assembly (coaxialfeed line not shown). Also shown is (g) the placement of the electronic Q spoiler/tuningarrailgement which is detailed in figure 4.4 and (h) the bias leads for this assembly.Chapter 4. Apparatus used for the 0.5 K Experiments 52into the maser proper which consists of an additional three volumes. The maser bulb isa 5 cm3 cylindrical pyrex bulb placed concentrically inside a slightly larger pyrex bulbwhich is filled with £-4He (see figure 4.3 for details of the maser resonator and storage bulbregions). This helium bath provides good thermal contact between the inner bulb andthe resonator housing. Atoms enter the inner bulb (storage bulb) through a small orificewhich results in a mean atom residency time inside the bulb of about 0.8 seconds at 0.5K. The two bulbs are axially located inside a split-ring resonator [86] tined to the zerofield hyperfine transition of atomic hydrogen at 1420 MHz. The resonator is capacitivelytuned by moving a teflon/Cu plate near the gap in the ring. This assembly is housedwithin a cylindrical Cu can with sealed ends to eliminate radiative electromagnetic lossesfrom the resonator and thus maintain its Q. The temperature of this housing is regulatedabout 100 mK higher than the mixing chamber temperature.The state selector region (not shown in figure 4.3) contains a 39.46 0Hz cylindricalcavity operating on the TM010 mode. It is located inside a superconducting solenoidmade from Cu clad NbTi wire which produces the loilgitudinal field (about 1.5 Tesla)necessary to bring the desired ( b ) to c ) or a ) to d )) hyperfine transition intoresonance at this frequency. The fringing fields from this solenoid draw high field seekingatoms (I a ) and b )) into the cavity and sweep away atoms in the low field seekingstates (j c ) and d )). Atoms entering this cavity must pass near thin sheets of metallicfoil impregnated with magnetic impurities having a mean spacing of a few hundred A[17, 87, 88]. This spacing is chosen such that at typical thermal velocities, the H atomssee time varying magnetic fields with a substantial Fourier component at the a) to b)hyperfine transition frequency in the local field. In effect the magnetic impurities act asa relaxing mechanism which tries equalize the populations of the a ) and b ) statesemmtering the state selector. By tuning the magnetic field to 1.43 Tesla, b ) state atomsentering the cavity can be subsequently pumped up to the c ) state. The geometry ofChapter 4. Apparatus used. for the 0.5 K Experiments 53the entrance to the cavity is designed such that when atoms drift out of the state selectorregion, they are less likely to pass near the relaxing foil. Once out of the fringing fieldsof the solenoid, the c ) state atoms drift down the atom tube towards the storage bulb.The lower end of the tube and the storage bulb/resonator assembly are shielded from thesolenoid fields by a superconducting Pb shield (not shown in figure 4.3).Atoms entering the storage bulb eventually wander out again and, if they are in thehigh field seeking states, can be drawn back into the state selector and recycled. Themean free path of these atoms is determined by scattering from the 4He gas within themaser. The atom tube and state selector are typically run about 100 mK colder thanthe maser bulb. This temperature gradient is maintained across a short, conical lengthof brass tubing located at the end of the atom tube where it constricts to form the orificeleading to the maser bulb (figure 4.3). As a result, H atoms leaving the maser bulbexperience a mean free path which increases as they move away from the orifice. Thisreduces the likelihood of an atom leaving the storage bulb and reentering within a timeperiod comparable to the mean bulb holding time.The UBC CHM operates with atomic densities in the l0”cm3 to 102cm3range.Just above threshold the maser power increases linearly with 11H• As the density isincreased, spin exchange collisions both broaden the resonance and reduce the populationinversion (T2 and T1 processes). Eventually the output power of the maser reaches amaximum and then decreases to zero. The maser has been run at temperatures spanning230 inK to 660 mK. The fractional frequency stability of the device has been measured[17] to be 6.3 ± 3.7 x l0-’ over an averaging time of 1 second. This is better than theperformance of the best conventional masers.Chapter 4. Apparatus used for the 0.5 K Experiments 544.2 The UBC CHM: Modifications for spin-exchange measurementsThe spin-exchange frequency shifts and line broadening of the cryogenic hydrogen maserare characterized by 6 parameters (X1 and j) in the theory developed by Verhaar etal. [19, 20]. In order to measure these parameters (or combinations thereof) at a giventemperature, it is necessary to independently vary the atomic density, the cavity Q andthe cavity tuning. The H density is easily varied4 by changing the repetition rate andthe power of the rf discharge. The buffer volume (260 cm3) mentioned in the last sectionwas added for the present measurements to filter out bursts of H from the source andthus to help maintain a constant 11H• The most important modifications to the maser forthe purposes of making these measurements were the addition of electronic tuning andQ spoiling devices. These modifications are described below.4.2.1 Electronic tuning/Q spoilerThe split-ring assembly is located inside a cylindrical Cu housing with sealed ends tokeep electromagnetic radiative losses from degrading the resonator Q. The unloaded Q ofthis system was about 2800 [17] prior to the installation of the tuning/Q spoiling systemdescribed below.For the spin-exchange measurements a 2.591 cm o.d., 1.956 cm i.d. ring with a squarecross section was fabricated from OFHC Cu. This ring was cut into two semicircularhalves. Enough material was removed from each face of the cut so as to form a small gapin the annulus when reassembled. The four corners of each face were rounded off in orderto reduce the gap capacitance. On one half of the ring, holes were drilled and tappedon both faces in order to mount two Frequency Sources GC51105-57 varactor diodes5.4Note that while n can be varied, there is no control over the occupancy of the individual hyperfinestates.5Frequency Sources, 16 Maple Road, Chelmsford MA 01824. These devices have a nominal capacitance of 0.6 pF with a bias potential of-4 V, and 1.6 pF at -15 V.Chapter 4. Apparatus used, for the 0.5 K Experiments 55The other half annulus was split into two equal pieces along the radius and this gapwas filled with a 0.015 cm teflon spacer. Mounting holes were drilled in each Cu pieceso that the ring could be reassembled on a teflon holder. This support was designed tohold the axis of the ring concentric with the split-ring and 0.874 cm below its lower elld.Prior to the final assembly of this device all Cu pieces were gold plated. Small flexiblestrips of Cu sheet were silver epoxied between the diodes and the other two quarters ofthe tuning ring. These metallic strips accommodate the motion of the ring due to thethermal contraction of the teflon holder, and also provide electrical contact to the diodesso that bias potentials can be applied. This assembly is shown in figure 4.4.The capacitive gap between the two quarter sections of the ring was bridged betweenthe drain and the source of an NEC 720 FET packaged in a hermetically sealed PLCCpackage.6 Prepackaged devices contain enough magnetic material that it was necessaryto purchase the dies and do our own packaging. This assembly was then mounted on theteflon support structure adjacent to the tuning ring.Electrical contact was made to each of the Cu section of the annulus and the gateof the FET using Fluorosint graphite leads7 fed up from beneath the teflon supportstructure. These leads have a resistivity of about 2 kQ per cm at 300 K and 3.5 k1 percm at 4.2 K. We have coaxially encased them in Cu braids so that they can be broughtinto the resonator volume without seriously degrading its Q.The ring geometry is chosen such that the structure has its lowest resonance around1900 MHz. By changing the bias potential V which is applied to the varactor diodes, theresonant frequency of the loop is changed, pulling the frequency of the split-ring resonator;the maximum attainable tuning range for our geometry is about 500 kHz (about 30% ofthe FWHM of the cavity resonance) at 0.5 K. By changing the bias potential Vq which6Kyrocera America, 5701 NE Fourth Plain Blvd. Vancouver WA 98661 : part numbers PB45238 andKE77004-17The Polymer Corporation, 2140 Fairmont Aye, Reading PA 19603Chapter 4. Apparatus used for the 0.5 K Experiments 56ab0 C000deFigure 4.4: The tuning ring/Q spoiler shown mounted on the base of the Cu can whichencases the split-ring structure (see figure 4.3). The Cu can below the ring is simply ahousing for the graphite leads used for biasing the semiconductor devices and a thermalshield. Details show (a) the Cu ring (3 sections), (b) the varactor diodes (two), (c) theFET bridging a teflon filled capacitive gap, (d) the teflon support structure for the ring,and (e) the Cu base. The graphite bias leads individually encased ill Cu braid are notshown.Chapter 4. Apparatus used for the 0.5 K Experiments 57is applied to the FET (essentially a variable resistance in the tuning ring) the Q of thetuning structure can be changed. This Q spoiler serves to reduce the Q of the split-ring from about 820 (the maximum low temperature Q after the modifications describedabove) to a minimum value around 570.The tuning and Q spoiling functions of the ring are not completely independent ofeach other. By measuring both the resonator Q and tuning as a function of the twobias potentials a two dimensional mapping can be made from which lines of constantQ or constant tuning can be derived. A typical mapping is shown in figure 4.5. Thereproducibility of a particular setting is better than the detection limit for both theresonator Q and the resonant frequency.4.3 State selectionIn all of the work reported prior to the present experiments the a ) to b ) to c )pumping scheme (reliant upon the relaxing foil) described earlier was used. Computersimulations have indicated [17] that the efficiency of the state selection process was infact bottlenecked by the production of b ) state atoms by the relaxing foil. As a result,the flux of c) state atoms entering the maser bulb was a rather insensitive function ofthe tuning of the microwave radiation used to pump I b) state atoms into the I c) state.In order to measure the atomic density inside the maser bulb8, the pumping microwaves were swept off resonance, stopping the flux of c ) state atoms into the maserbulb and causing a cessation of maser action. Magnetic resonance techniques were thenused to determine the atomic density inside the storage bulb. After this the microwavepump was swept back onto resonance and maser operation reestablished. Because of theaforementioned bottleneck, the reproducibility of the initial conditions was not limited8This measurement is discussed in more detail in the next chapter.Chapter 4. Apparatus used for the 0.5 K Experiments 58900__8009In700->c3600-500 I I1420 1420.2 1420.4 1420.6 1420.8Frequency (MHz)Figure 4.5: A typical mapping of the resonant frequency and the quality factor of the1420 MHz split-ring resonator as a function of the bias potentials on the varactor diodes(Vi) and the Q spoiling FET (Vq). ‘Horizontal’ lines correspond to lines of constant Vqand ‘vertical’ lines to constant V. Interpolation allows one to derive combinations of Vand Vq which produce lines of constant tuning or constant Q.Chapter 4. Apparatus used for the 0.5 K Experiments 59by either the stability or the reproducibility of the frequency and power of the microwavepump.Prior to the present experiments new relaxing foil was manufactured and installedin the hopes of improving the efficiency of the state selection process. The modificationindeed seemed to remove the bottleneck, however it was found that the operating conditions were now much more sensitive to the stability and the tuning of the microwavepump9. Several modifications were necessary in order to improve the reproducibility ofthe maser operating conditions after making a measurement of These modificationsare outlined below.4.3.1 New pumping schemeIt has been known for a considerable length of time [17] that the present version of theUBC CHM has considerable cross relaxation between the upper three hyperfine statesdue to some relaxing mechanism (magnetic impurity) believed to be located near theorifice connecting the maser bulb to the atom tube. Repeated attempts to find andeliminate the source of this relaxation have failed. In this work we have adopted a newpumping scheme which makes use of this cross relaxation. Rather than pumping theI b ) to c ) transition in the state selector we have decreased the magnetic field slightlyin order to pump the a ) to d ) transition. This procedure circumvents the relaxingfoil as a state selection mechanism. In this situation, atoms which approach the storagebulb are predominately d ) state atoms, however the strong cross relaxation near theentrance to the bulb ensures that the c ) state is populated inside the bulb.9The measured sensitivity is consistent with the estimated field inhomogeneities in the solenoidal fieldof the NbTi magnet ( 7 x 10”).Chapter 4. Apparatus used for the 0.5 K Experiments 604.3.2 Frequency stabilityThe microwaves used in the state selector are generated by a Micro-Power model 221backwards wave oscillator (BWO). Instead of letting this device operate under free running conditions as in previous work’°, we have frequency stabilized it using a high quality5 MHz quartz crystal oscillator” (see figure 4.6). This was done by generating a 2.47GHz signal (LO) with a HP 8662A frequency synthesizer phase locked to the output ofthe quartz crystal and then mixing this with the output of the BWO at 39.5 GHz (RF).The 16th harmonic of the LO signal is offset from the RF by 50 MHz and produces anIF signal which is fed into a HP 5342A microwave frequency counter with a digital toanalogue converter. The analogue output of the counter was used to lock the outputfrequency of the BWO to within a few tens of kHz. The output power of the BWO wasmonitored using a directional coupler and a diode detector at the top of the cryostat.Power levels were maintained constant to within about 1% throughout the experiments.Rather than stopping the maser by sweeping the microwaves off resonance as donepreviously, in the present work a resistive attenuator card could be inserted through aslot in the broad wall of the Ka band waveguide leading down into the refrigerator. Thisgave about 20 dB of attenuation which is more than sufficient to stop the pumping actionin the state selector. The insertion of this card was actuated by a mechanical relay.4.4 Refrigeration, temperature monitoring and controlThe UBC CHM is mounted inside an Oxford Instruments (Special) Model 400 dilutionrefrigerator with a home made dilution unit. Typically it runs at temperatures around0.5 Kelvin (at or near the temperature at which the frequency shifts due to the wall and10Previously this procedure was not necessary as the state selection process was bottlenecked by therelaxing foil and thus nK was insensitive to the frequency of the pump microwaves.11Oscilloquartz OSA model 8600.03 BVA very high stability modelChapter 4. Apparatus used for the 0.5 K Experiments0—byouttuningcontrolFMin IHout 39.5 GHzSwitch A7LctorPower Monitor61MixerFigure 4.6: Method used to stabilize the microwave pump frequency used in the atomicstate selection process. Details are described in the text.2.47 GHzrf1.5VI resetAAAAvvvy5V,uWaveSweepOscillatorto State SelectorCavityChapter 4. Apparatus used for the 0.5 K Experiments 62the buffer gas shifts pass through a minima). The atom source is thermally attachedto a special heat exchanger and run at temperatures near 0.4 K. The state selector ismounted on the mixing chamber (MC) of the refrigeration unit and run at temperaturesabout 100 mK lower than the actual maser bulb (discussed previously). The temperatureof the state selector is monitored by a carbon resistance thermometer. The temperatureof the state selector is also regulated using this resistor. The temperature of the MCis monitored and regulated using a calibrated germanium resistance thermometer. Thetemperature of the inner maser bulb is governed by the temperature of the 4He bath iiithe outer bulb which is in turn thermally linked to the Cu resonator and housing via avolume of Ag sinter. This temperature is also monitored and regulated with a calibratedgermanium resistance thermometer.4.5 1420 MHz spectrometerThe magnetic resonance experiments described in this thesis were all performed usinga two stage heterodyne detection system combined with a gated pulse generator. Thissystem was originally designed by W. N. Hardy and has been used in other experimeiltswith atomic H [17, 27]. The spectrometer design is sketched in figure 4.7. We describethe basic principle of operation in the text below. Details specific to the experiments performed at 0.5 K are described imm the next section. Details pertaining to the experimeiltswith H, D mixtures near 1 K are given in chapter 6.The system is based upon two local oscillator signals derived from either a 10 MHzRb frequency standard’2 or a high quality quartz crystal oscillator’3. A signal at 1420MHz is obtained by direct frequency multiplication (x2, x 71) of the frequency reference.The 405 kHz signal is synthesized from the same reference using a phase locked HP3330A12Efratom model FRK-L Rb frequency standard, on loan from D. Wineland, NIST Boulder CO.13Oscilloquartz OSA model 8600.03 BVA ultra high stabilityChapter 4. Apparatus used for the 0.5 K Experiments 631420.405 MHz resonatorFigure 4.7: A schematic diagram of the 1420 MHz pulsed magnetic resonance spectrometer used in these experiments. ‘F’ is used to label the various filters. The operation ofthis device is described in the text.Output to datacollection systemfromresonatorSweeper toChapter 4. Apparatus used for the 0.5 K Experiments 64frequency synthesizer.An rf signal at the hyperfine frequency of atomic H is derived from these two localoscillator signals using a single sideband generator. Externally generated pulses of varyingwidths are then multiplexed and used to gate this signal. In order to avoid the possibilityof interference at the sum frequency (1420.405 MHz) the phase shifter which is used toproduce the two phases of the 405 kllz for the SSB is also gated. As a result, the 1420.405MHz signal is only produced during the application of the external pulses. This systemallows one to produce coherent pulse sequences such as ir/2 pulses or sequences.Rf power emitted by the atomic system’4 is first amplified and then mixed with theLO at 1420 MHz in an image rejection mixer15. This signal is then filtered, amplified,and mixed with the second LO at 405 kllz. The frequency of this LO is adjusted toproduce a convenient beat frequency (near 10 Hz) which is fed to the data acquisitionsystem.In order to set the tuning and the coupling of the resonator, provision is made toinject a swept frequency signal into the cavity. The reflected signal is monitored usinga crystal detector. Absolute power calibrations are performed using a HP435B powermeter4.5.1 Data acquisitionIn the experiments performed at 0.5 K the reference frequency we used was derivedfrom a second 5 MHz high quality quartz crystal oscillator’6 loosely phase locked to aRb time standard. The crystal oscillators are (commercially) packaged in temperatureregulated dewar flasks and have fractional frequency stabilities better than 5 x14This is either the free induction response of the atoms to some pulse sequence, or a cw maser signal.15Using this procedure the response signal at 405 kllz does not contain noise from the rf signal at1419.595 MHz.16Oscilloquartz OSA model 8600.03 BVA ultra high stabilityChapter 4. Apparatus used for the 0.5 K Experiments 65for averaging times in the 0.1 to 10 second range. We have packaged these devices ina second temperature regulated/insulated Al can which can be evacuated and sealed inorder to pressure regulate the devices. Cooling and temperature regulation is providedby a temperature regulated water bath which circulates water through a heat exchanger.By using the Rb time standard as a flywheel we were able to improve the long termstability of the combination without seriously degrading the short term stability of thequartz crystal. The stability of the combination is about 1 part in 1012 over time periodscomparable to the duration of the experiments and hence limits our practical frequencyresolution to about 1 mHz.The gated output from the spectrometer is derived from the crystal oscillator/Rbclock combination. The pulsed signal is fed down a coaxial line to a 20 dB directionalcoupler mounted on top of the main vacuum can of the dilution refrigerator (T4.2 K).This coupler feeds the signal down to the maser cavity through another coaxial line. Thereflected and/or emitted signal from the maser cavity is amplified using a GaAs basedpreamplifier17 operating at 4.2 K before being brought up to room temperature. Thegain of this device is 18.1 dB at 4.2 K with a 3 dB bandwidth of 395 MHz and a noisetemperature of 20 K. A second directional coupler is used to inject calibrated signals forabsolute power calibrations of the system. These components are shown in figure 4.8The output of the final mixer stage of the spectrometer is fed to a Nicolet 1170 signalaverager, an HP 5345A frequency counter and an HP 3478A ac voltmeter. The signal averager is used for general purpose data acquisition including FID’s from pulsed magneticresonance experiments and Q curves from swept frequency reflection measurements ofthe 1420 MHz resonator response. Both types of signals are subsequently transferred toa computer for further analysis. The maser output power is determined from the amplitude of the ( 10 Hz) beat frequency between the maser signal and the two LO’s in the17based on a design by Williams e at. [891.Chapter 4. Apparatus used for the 0.5 K Experiments 66CalibratedSignal1.____i1420MHz__________1Spectrometer4.2K0.5KSignalAveragerFrequencyCounterComputerMaser CavityFigure 4.8: A schematic diagram of the rf wiring and data acquisition system for the 0.5K experiments.Chapter 4. Apparatus used for the 0.5 K Experiments 671420 MHz spectrometer. This measurement is made with the ac voltmeter. This beatsignal is also fed to the frequency counter in order to determine the oscillation frequencyof the maser.4.5.2 Measurement of Q and LDue to limited access to the maser we were forced to measure the resonator Q and resonant frequency in reflection rather than in transmission. Swept frequency measurementsof the reflected signal were fed to a HP 8473B crystal detector operated in the square lawregion. The curves were numerically fit to a single Lorentzian lineshape with a slopingbackground, the width and position of which gave the Q of the cavity and its detuningfrom the atomic resonance.This technique is inherently susceptible to distortion induced by standing waves inthe external circuitry. Ill order to evaluate these measurements we have performed extensive computer simulations of the measurement technique using a commercial simulationprogram18. The s parameters of all components in the circuit were measured using anHP 8754A/H26 RF Network Analyzer and used as input parameters to the models. Carewas taken to evaluate effects due to the changing £-4He level in the main bath of therefrigerator as well as the effects of room temperature changes of the effective length of(teflon dielectric) coaxial cables in the measurement circuit. The results of this studysuggest that our measurements of Q are accurate to about 1%. The reproducibility ofindividual measurements is considerably better than this. Estimates of the accuracy andthe precision of the measurement of the cavity tuning are discussed in the next chapter.‘8EESOF Inc. (5795 Lindero Canyon Rd, Westlake Village, CA 91362 USA) ‘Touchstone’ and ‘Libra’microwave simulation programs.Chapter 5The 0.5 K H-H Spin-Exchange MeasurementsThe results described in this chapter represent the culmination of a considerable effortat UBC to attempt to test the predictions of the H-H spin-exchange theory proposedby B. J. Verhaar et al. [19, 20]. These predictions impose rather severe limits on theultimate frequency stability of the CHM. At the same time, the sensitivity of the predicted shifts to the details of the H-H interaction potential suggests that by studyingthe shifts, delicate refinements to the potentials might be possible. For these reasons anexperimental verification of the sensitivity of the oscillation frequency of the CHM to theatomic density is of considerable technological importance as well as theoretical interest.Unfortunately much of this work was frustrated by poor workmanship in the construction of the commercial dilution refrigerator in which the CHM was housed. Severalattempts to study these frequency shifts were made over a period of a few years arid ineach case the experimental runs were thwarted by problems associated with the refrigeration system. After several rather drastic measures aimed at repairing the malfunctioningrefrigerator, it was possible to stage a one month concentrated effort to obtain experimental data. This run was not carried out without techilical difficulties (which will bediscussed in the next section), however at the time it was feared that attempts to correctthese problems might cause further damage and bring further delays to the program.Rather than risk never obtaining data1, we made every attempt to obtain the best datapossible given the circumstances.1This run effectively signaled the end of the CHM program at UBC due to the lack of manpower andresources.68Chapter 5. The 0.5 K H-H Spin-Exchange Measurements 69In the time since this work was carried out, techniques developed in the lab andadditional studies have given us far more confidence in the original work. The results ofthat investigation are presented here in light of these new findings. The measurementsare significant in that they demonstrate that that formalism which Verhaar et al. haveintroduced appears to be valid and certainly merits further evaluation. This is, to ourknowledge, the only experimental verification of this theory to date. Unfortunately asa consequence of these frequency shifts, it is not likely that the CHM will ever attainfrequency stabilities in the 10—18 range as was once predicted [15].5.1 Summary of technical problemsWe begin the discussion of these measurements by quickly outlining a few of the technicaldifficulties which were encountered during the experimental run. These points initiallyled to some concern about the interpretation of the data and have delayed its presentationuntil now.1. The cooling power and the base temperature of the refrigerator were limited by athermal ‘touch’ between one of the Cu braids used to bias the semiconductor devicesin the tuning/Q spoiling system, and a thermal shield surrounding the maser cavity.This touch also initially led to some concern as to the actual temperature of themaser bulb. During the run it was discovered that the heat leak could be alleviatedsomewhat by forcing the main vacuum jacket of the refrigerator off centre, causingit to push against the vacuum can which housed the maser and ultimately relievingsome of tile pressure between the Cu braid and tile thermal shield. It was notpossible to completely alleviate the problem. Eventually it was decided that anytemperature gradients which might have been introduced by this heat leak weresmall enough to be neglected. The temperatures which are reported here are thoseChapter 5. The 0.5 K H-H Spin-Exchange Measurements 70which were measured during the experiment. No corrections have been applied.2. As mentioned in the last chapter, we were forced to measure the resonator Q andcentre frequency in the reflection mode. This technique is inherently quite sensitiveto the standing wave patterns in the microwave circuitry. These standing wavesled to some concern about the absolute accuracy of the measured Q’s and cavityfrequencies. This problem was compounded during the experimental run by thechanging -4He level2 in the main bath of the refrigerator. The resulting changein the temperature gradient along the coaxial lines leading to the resonator causesshifts to occur in the standing wave pattern via the change in the electrical lengthof the lines. Our concerns surrounding these measurements were alleviated afteradditional measurements of the electrical parameters of the system were made andused in a computer model of the measurement technique. This additiollal workhas allowed us to place reasonable limits on the uncertainties incurred during themeasurements.3. The measurements of the maser output power as a function of density made during this run did not initially seem to be consistent with earlier measurements. Inparticular the threshold density for maser oscillations seemed to be too low. Thisobservation led us to question the absolute power/density calibration measurementsand the absolute measurement of the resonator Q. Since that time the filling factorof the maser bulb inside the split-ring resonator has been carefully measured (appendix B). In addition, the techniques outlined in appendix A were used to includeradiation damping in the numerical fit to the FID data used for determining thewas not possible to regulate the £-4He level during the spin-exchange measurements because ofthe sensitivity of the maser to mechanical vibrations. This sensitivity was not normal and was relatedto the thermal touch.Chapter 5. The 0.5 K H-H Spin-Exchange Measurements 71atomic densities. The extent to which radiation damping influences the data provides a check on the power calibration. No discrepancy between these methods ofdetermining nH was observed during the data analysis. The result of the numericalfits to the FID’s are entirely consistent with the measured power/density calibration. They are also consistent with results obtained later on a different system (seechapters 6 and 7). As a result, we are confident in the measurements reported here;any discrepancies between this run and earlier reports must be due to the use of thenew pumping scheme used in the present work and the fact that the filling factorfor the maser bulb with the modified tuning arrangement was actually higher thanfor previous work. This last point is discussed in the section on the determinationof the atomic density.5.2 Co oldown procedure (standard)After the system is cooled below 1 K for the first time, a small quantity of 112 is admittedto the source region through a heated capillary. Sufficient 4He to coat the internal walls ofthe maser is then admitted via the same capillary. The discharge is fired several times tocreate a sample of H gas from the frozen H2, and pulsed magnetic resonance techniques(usually 7r/2 pulses at the a-c hyperfine transition frequency) are used to detect thepresence of these atoms.The maser cavity is axially located inside a cylindrical superconducting Pb shieldwhich itself is located inside a high i metal shield (‘Co-netic foil’3). The Co-netic foil isdemagnetized at room temperature and again at 77 K. Using a solenoid located betweenthe two shields a small longitudinal bias field at the site of the maser bulb is then appliedto set the quantization axis and to split the degeneracy of the upper three hyperfine3Perfection Mica Company- Magnetic Shield Division, 740 Thomas Dr, Bensenville Ill.Chapter 5. The 0.5 K H-H Spin-Exchange Measurements 72levels of the H atoms. As this field is increased, cross relaxation between the upper threehyperfine levels is reduced. At the same time the sensitivity4of the oscillation frequencyof the maser to fluctuations iu the bias field is increased. The maguitude of the biasfield is chosen as a compromise between these effects. The field is set while the Pb shieldis driven normal. The shield is then cooled in order to trap the field. In the presentexperiments, a field of 1 x iO Tesla was applied, resulting in an upwards shift of about30 Hz in the oscillation frequency of the maser.5.3 Characterization of the maser5.3.1 Magnetic relaxationDuring earlier studies [17] it was established that H atoms in the UBC CHM are subjected to strong magnetic relaxation somewhere near the orifice which leads to the maserbulb. This relaxation is predominantly cross relaxation between the upper three hyperfine levels5. This relaxation affects the evolution of the longitudinal magnetization ofthe H atoms during magnetic resonance experiments. It is necessary to characterize thisrelaxation if the techniques outlined in appendix A are to yield reliable fits to free induction decays. We proceed by assuming that the relaxation rates between the c) andthe d ) and between the c ) and the b ) states are equal and denote this rate by T’,.Longitudinal relaxation between the upper three states and the a) state is considerablyless efficient. Again we assume that all three of these relaxation processes occur at thesame rate, which is denoted by P.At low densities where spin-exchange relaxation can be be neglected, the relaxation4The oscillation frequency is essentially independent of the bias field when the field is zero andincreases as the bias field is increased.51n fact, the new pumping scheme which we have used in these experiments is made possible becauseof this relaxation.Chapter 5. The 0.5 K H-H Spin-Exchange Measurements 73of the longitudinal component of the magnetization (using the spin analogy introducedin chapter 2) towards its thermal equilibrium value1 i1L40IIH(Paa— Pcc)eq flHVT (5.1)following a ir-ir/2 pulse sequence, is such that the time dependent deviation SflH(Paa —p)(t) from this value is represented very well by a sum of exponeiitials. When thisdeviatioll is normalized to the thermal equilibrium value given by equation 5.1 we canwriteSM(t) 2 1SM(0) = —2 ( exp(—Pit) + exp(_F2t)) (5.2)where P1 = Pb + 4’a P. and T’2 = ‘b + 1a + 3P 1’ + 3P. Here Pb represents therate at which atoms are lost from the maser bulb due to effusion from the orifice. Infigure 5.1 we show the measured amplitude of the atomic response to a ir/2 pulse whichoccurs at a time t following a r pulse. The signals have been normalized to an initialamplitude of 1 so that the data is proportional to 2M(0)• The measurements were madeat a density nH = 4.9 x 1010 cm3 where contributions due to spin-exchange are expectedto be small.When the amplitude of these decays is fitted to the form given in equation 5.2 weobtain 1’= 1.14 s_i and F2 = 7.73 With the assumption that Pa 0 [17] we obtain= 2.20 s_i and Pb = 1.14 s. These values are consistent with earlier observations[17].In appendix A, the phenomenological differential equations describing tile evolutionof the spin magnetization are presented along with a discussion of the procedure usedto fit these equations to magnetic resonance data. These equations are presented onlyfor the case in which there is a single T1 process. It is a relatively simple matter tomodify these equations to include the additional relaxation processes described above.That is, the relaxation of the z component is assumed to occur at a rate which is theChapter 5. The 0.5 K H-H Spin-Exchange Measurements 741 . . • • • • • •.CN-INCo0.1I I I I I t . I I i0.0 0.5 1.0 1.5Time (s)Figure 5.1: The normalized amplitude recovery of the longitudinal magnetization following a ir — ir/2 pulse sequence, as a function of the interpulse spacing. This information isused in the fits which are made to the FID’s using the techniques described in appendixA. The atomic density is nH = 4.9 x 1010 cm3 and the temperature is 0.500(3) K.Chapter 5. The 0.5 K H-H Spin-Exchange Measurements 75sum of the processes described by Pb and f, and an additional, as yet unspecified T1’which accounts for additional relaxation due to processes such as spin-exchange. Thefree induction decays obtained during the course of the experiment have all been fittedusing this modification to the radiation damping equations.5.3.2 Determination of the atomic densityIn order to measure the spin-exchange frequency shift parameters it is necessary to havean absolute calibration of the atomic density. For a collection of spins in thermal equilibrium this measurement can be made using magnetic resonance techniques as outlined inchapter 2 and appendix A. Under thermal equilibrium conditions the static (longitudinal)magnetization M0 is proportional to the H density6. Following a r/2 pulse this magnetization is converted into a precessing (transverse) magnetization IIr which (initially) hasthe same amplitude. The time evolution of this response gives rise to the free inductiondecay. It is observed experimentally as a damped oscillatory signal which is comparedwith the local oscillator of the 1420 MHz spectrometer. The power emitted by the atomsimmediately following the pulse is given by[25, 27]P — fLoL.’oVb7Q 2 — jiowoVb7]Q h2(7e + 7’o22 3— 2 r— 2 8kBT IIH. (.)Half of this power is absorbed by the detection circuitry when it is critically coupled to theresonator. The gain of the detection system is carefully ineasuredso that absolute powermeasurements can be made. The FID data is fit to the numerically integrated differentialequations outlined in appendix A, with the free parameters T1,T2, nH, frequency, andan initial phase angle. Measured quantities such as the power calibration factor, bulbholding time, cross relaxation rate, cell volume and filling factor are included as knownparameters in the appropriate equations.6We are using the fictitious spin analogy again.Chapter 5. The 0.5 K H-H Spin-Exchange Measurements 76An example of a fit to a FID is shown in figure 5.2. The H density determined fromthe power calibration is 7.1 xlO’1 cn3. The decay is not obviously non-exponential;however radiation damping does account for about one quarter of the observed damping.Radiation damping also influences the data in more subtle ways. In this particularexample, a fit to a single damped sinusoid would overestimate the frequency of the signalby 9 mHz and underestimate the atomic density by 4%. These effects have been studiedin detail by Bloom [30].An essential part of this procedure is to have a proper measurement of the fillingfactor . Prior to the present work the filling factor for the UBC CHM was only knownindirectly. Estimates placed this value at 0.2 [17] with a rather large uncertainty. Inappendix B of this thesis we present a new technique for measuring magnetic fillingfactors which is based on the use of the pararnagnetic properties of liquid 02. We haveused this technique to measure the filling factor of the unmodified CHM resonator andfind i=0.24(1). The modified resonator used in the present experiments contains aconsiderable amount of teflon located below the actual split-ring resonator. This dielectricmaterial increases the effective length of the resonator can. At the same time it tends toenhance the longitudinal fields near the lower part of the maser bulb and thus increases. The filling factor of the modified resonator assembly used in this work is 0.27(1).There is some corroborative evidence that our measurements of the filling factor arequite reliable. Throughout this work radiation damping makes a significant contributionto the FID envelope. If we try to fit the differential equations given in appendix A tothe FID’s at high n using larger values of m, we find that the damping quickly becomesexcessive and that a good fit is not possible. That is, the measured values of i appearto be the largest values consistent with the observed damping. Decreasing the value ofand accounting for the additional damping with a smaller value of T leads to a moreexponential decay. If we use as a measure of the goodness of the fit to the data, weChapter 5. The 0.5 K H-H Spin-Exchange Measurements 77I I I I I I ICl,CL0‘I,-oI I I I0.0 0.2 0.4 0.6Time (s)Figure 5.2: An example FID and the numerical fit to the data using the techniquesoutlined in appendix A. The decay was taken with nH = 7.1 x lOll cm3 at a temperatureof 0.50 K. Radiation damping accounts for about one quarter of the observed damping.The residual T due to processes other than radiation damping is 0.18 s.Chapter 5. The 0.5 K H-H Spin-Exchange Measurements 78find that x2 doubles if is decreased to 0.22. This is a somewhat less stringent lowerbound on but still rather suggestive.It should be noted that implicit in the power/density calibration is the assumptionof thermal equilibrium. The translational degrees of freedom for the H atoms are veryquickly thermalized by collisions with 4He atoms and the £-4He walls. The spin degreesof freedom on the other hand are only thermalized by spin exchange interactions andmagnetic field gradients. At low H densities, magnetic relaxation is likely to be thedominant thermalizing process. During the normal operation of the CHM the atomsare in contact both with maser bulb volume and with the slightly cooler state selectorregion. At the lowest densities where spin-exchange relaxation is not efficient, the spintemperature in the maser bulb may be lower than the measured temperature. This effectwould lead to an overestimation of the atomic density inside the maser bulb for lowdensities.5.3.3 Computer simulationThe dynamics of the operation of the maser are quite complicated and cannot be directlymonitored. In particular, it is not possible to measure (or control) the occupation of anyone (or any combination) of the hyperfine states at any point within the maser while it isoperating. To determine the atomic density inside the maser bulb, it is necessary to stopthe maser oscillations and then probe the system with magnetic resonance techniques.A prerequisite for the use of these techniques7is that the system be in a state of thermalequilibrium (including the spin degrees of freedom which are thermalized predominantlyby spin exchange and magnetic relaxation). While the maser is operating, there is aconstant flux of low field seeking atoms approaching the storage bulb. When the state7A r/2 pulse measures the population difference between the I a) and I c ) states. In the derivationof equation 3.30 it is explicitly assumed that the atomic system is in thermal equilibrium.Chapter 5. The 0.5 K H-H Spin-Exchange Measurements 79selector is turned off, this flux is stopped almost completely, causing the net H atomdensity inside the bulb to decrease. As a result, the density measured when the maser isnot operating is less than that when it is masing.In order to overcome this obstacle we rely upon the result of a computer simulationof the quasi-steady state operation of the UBC CHM which was developed by M. D.Hürlimann [17, 16, 90] who looked at the dependence of the atomic density in the variousmaser volumes as a function of numerous factors such as the efficiency of the relaxingfoil. The results of his study suggests that the ‘correction’ or ‘density enhancement’factor which must be applied in order to relate the non-operating and operating atomicdensities is about 1.4. There is of course some uncertainty in this value. Howeverreasonable assumptions [17], suggest that this uncertainty is likely of the order of 5%. Inthe present work we have made no attempt to refine the results of this study and use thefactor of 1.4 for determining 11H in the operating maser. We also assume that this factoris independent of density within the 5% uncertainty. This is again consistent with thecomputer simulation results.The simulation also allows one to predict the density dependence of the populationsof the various hyperfine levels. Of particular interest for the measurement of the spinexchange frequency shifts in the CHM is the dependence of Pcc+Paa on the atomic densityinside the maser bulb. Hürlimann finds that changes in this sum are not expected to bemore than about 5% across the operating range of the maser. The uncertainties associatedwith the measurement of the spin-exchange parameters are too large to warrant the useof this model to attempt to extract further information. For our purposes it can beassumed that Pcc + Paa 0.5 inside the maser bulb [17].Chapter 5. The 0.5 K H-H Spin-Exchange Measurements 805.3.4 Output power of the maserThe output power of the CHM was measured as a function of the atomic density insidethe maser bulb using the new pumping scheme outlined in the previous chapter. Theresults of this measurement are shown in figure 5.3. The resonator Q was 809(8) andthe bulb temperature was 0.500(3) K. The threshold for masing is slightly lower thanexpected when compared with earlier experiments [17]. This discrepancy may indicatean error in the density enhancement factor determined from the computer simulation,either in this work or the previous work.5.4 The measurementsThe equation describing the combined frequency shifts due to spin-exchange and cavitydetuning for the CHM as predicted by Verhaar et al. [19, 20] is repeated here for thesake of convenience:f - fo 1[L + +2ir T0+{[+(i+2)][i(pcc+paa) +2] + [i(pcc+paa)+2]}VHHnH(5.4)where4VRH 55and the detuning(5.6)is a measure of the offset between the resonator (we) and atomic (w) resonances. It ispossible to identify eight types of shift terms in this equation which depend in differentways upon the resonator quality factor Q, detuning , and the atomic density H• Four ofChapter 5. The 0.5 K H-H Spin-Exchange Measurements 81I .5e——13 I I I I , •DUUI.Oe—13U00-I-- U-Io 5.Oe—14UUU0 .Oe—I—0OI I I I IIIII••I0.Oe+00 1.Oe+12 2.Oe+12 3.Oe+12H Density (cm3)Figure 5.3: The output power of the UBC CHM utilizing the new pumping scheme ( a)to I d )) which relies on cross relaxation near the bulb orifice. Measurements were madewith a resonator Q of 809(8) and a bulb temperature of 0.500(3) K.Chapter 5. The 0.5 K H-H Spin-Exchange Measurements 82these terms are independent of density while four are proportional to the density. Of thedensity independent terms there is one which depends solely on , one solely on Q, andone on both L and Q. There are three density dependent terms with similar dependenciesoii Q and Li. In addition there is one density independent shift fo (due to factors such asthe wall shift, buffer gas shift and the bias field), and a density dependent shift (involving) and )2), both of which are independent of Q and zS. In this experiment we aim toidentify and measure each of these shifts. The most important shift from the point ofview of trying to establish the influence of hyperfine interactions during spin-exchangecollisions is the shift which depends upon and )2 (see chapter 3, equations 3.25 and3.28). In the absence of these effects we do not expect to find a density dependent shiftwhich is independent of both Q and L. The contribution of the hyperfine interaction tothe total frequency shift can be measured with the dimensionless ratio [20]:1(Pcc+P)+\2 (57)—5.4.1 Procedure and dataPrior to making the frequency shift measurements, the resonator Q and tuning weremapped out as a function of the bias potentials Vt and Vq as described in the lastchapter. From these measurements, a series of potentials were derived which enabledus to change the resonator tuning at a constant Q (near the maximum attainable Q800) and to change the Q at a constant detuning (near L = 0). The Q’s and tuningsat each of these settings were then individually measured. Note that unless explicitlystated otherwise, the quality factors we report refer to the loaded values. All work wasperformed under conditions of critical coupling.Atomic densities were set by varying the discharge pulse duration and repetition rate.The pulse widths were typically 1 iisec and the repetition rates were of order 1 everyChapter 5. The 0.5 K H-H Spin-Exchange Measurements 8310 seconds. Densities produced in this manner did not fluctuate measurably within thedetection noise limit of about 1% of the measured density.The maser output was beat down to a 10 Hz signal in the 1420 MHz spectrometer8asdescribed in the previous chapter. Once stable maser operation was established at a givendensity, this frequency was measured repeatedly with the frequency counter using a 10second averaging time. This frequency measurement was then repeated as a function ofthe preset Q’s and z’s. At each setting the frequency was measured a minimum of threetimes. Multiple checks of the unperturbed oscillation frequency (maximum Q, =0) weremade throughout the measurements. Finally the microwaves used to pump the a ) tod) transition were stopped by inserting the resistive card switch and the atomic densitywithin the maser bulb was measured using several ir/2 pulses. The original operatingcondition of the maser was reestablished before moving on to a new density. At the endof the run the Q’s and ‘s at each of the canonical settings were remeasured.The maser oscillation frequency measured in this way is shown as a function of densityat constant Q in figure 5.4 and and as a function of Q’ at constant z 0 in fig 5.5.It should be noted that a small but measurable shift in the absolute detuning occurredpart of the way through the measurements. This effect is most obvious in the data atconstant (figure 5.5) where the data appears to be grouped into two clusters. Thisshift is presumably due to an abrupt change in the mechanical tuning.5.5 AnalysisIn the following sections we present our analysis of the frequency shift data presented infigures 5.4 and 5.5.8The LO of this spectrometer was derived from one of the quartz crystal oscillators. The long termstability of this oscillator was enhanced by using the Rb clock as a flywheel.Chapter 5. The 0.5 K H-H Spin-Exchange Measurements 840.1 — i • i I I I I I I I I I I I I I I I I J I I I I I I I I I..0.0... .* ** * *>-1)C * * * *G-0—0.3-D-DD—0.4. I • . . I , . . • .O.Oe+00 1.Oe+12 2.Oe+12 3.Oe+12Density (cm3)Figure 5.4: The measured frequency shift of the oscillation frequency of the UBC CHMas a fullction of the H density inside the maser bulb for various cavity detunings. Thefrequencies have been converted to absolute shifts from the density independent oscillation frequency fo. The resoiator Q=809(8) and the temperature of the maser bulbT=0.500(3) K were held constant through these measurements. The lowest set of datacorresponds to Li —0.2 and the highest set to L 0.2.Chapter 5. The 0.5 K H-H Spin-Exchange Measurements 85II I I I F I I I I I I I I I I I I I—0.151*ZIZB A>% BC.)*V—0.20 A -AB—0.25 ‘ i . • I II .2e—03 I .4e—03 1 .6e—03 I .8e—03Q-1Figure 5.5: The measured frequency shift of the oscillation frequency of the UBC CHMas a function of Q’ for various H densities. The frequencies have been converted toabsolute shifts from the density independent oscillation frequency fo. The resonatordetuning was set near zero and the temperature of the maser bulb was held constant at0.500(3) K. The lowest set of data corresponds to nH 2.6 x 102cm3while the highestset corresponds to H = 2.0 x 10”cm3.Chapter 5. The 0.5 K H-H Spin-Exchange Measurements 865.5.1 Density independent frequency shift and absolute detuningThe analysis of the frequency shifts requires a careful establishment of the density independent oscillation frequency fo and the absolute resonator detuning. In figure 5.6 weplot the frequency of the FID used to determine the H density during the spin-exchangemeasurements, as a function of n11. The data extrapolates to a zero density frequencyshift9 fo = 1420.405 783 434 (1) MHz. All other frequency shifts mentioned in thischapter are reported with respect to this value.The cavity tuning is determined by fitting the cavity resonance to a single Lorentzianlineshape with a sloping background. Since this measurement is made in reflection it isvery susceptible to apparent shifts caused by standing waves in the external circuitry. Infigure 5.7 we show the initial phase of several FID’s as a function of the measured cavitydetuning from the atomic resonance. Changing the cavity tuning by should lead toa change of 2 in this phase. The experimentally determined slope obtained from thisdata is 1.983(23) which indicates that our measurements of relative detunings are goodto about 1%.We have employed two independent methods to determine the absolute detuning ofthe resonator from the atomic resonance. In figure 5.8 we show the normalized measuredpower output of the maser as a function of the measured detuning. This data has beenfit’° to a quadratic of the form-=1 —a(L—Lo)2 (5.8)with the result that Lo = -0.014(30).9This frequency is measured against the combination of the quartz crystal oscillator and the Rb clock.We are only interested in relative shifts from this value. The absolute frequency measured directly againstthe calibrated Rb clock is about 400 mHz lower. This offest is due to the oscillation frequency of thex-tal oscillator and is of no consequence in these measurements.cv, zi variable.Chapter 5. The 0.5 K H-H Spin-Exchange Measurements 870.4.6 . • . . i . ._-, 045N>%C)Ca):5c3- 0a)Li0.44-00.43 • . I • • . I . .0.Oe+0O 1.Oe+12 2.Oe+12 3.Oe+12Density (cm3)Figure 5.6: The frequency of the free induction decays used to determine the atomicdensity during the spin-exchange measurements, as a function of n. A frequency of1420.405 783 MHz has been subtracted from this data. The zero density frequency foobtained from this plot is 1420.405 783 434 (1) MHz.‘4-0a)C,)a-c0Chapter 5. The 0.5 K H-H Spin-Exchange Measurements 88//—1.6—1.8—2.0—2.2—2.4—2.6—0.2 —0.1 0.0 0.1 0.2 0.3p I I I p I I I I i p p pAFigure 5.7: The initial phase of a free induction decay as a function of the measuredresonator deturiing. The slope of this plot is 1.983(23) (theoretically 2) which indicatesthat relative changes in detuning measured in reflection agree with the changes felt bythe H atoms to within 1%.Chapter 5. The 0.5 K H-H Spin-Exchange Measurements 891.05 • • • • . . . . . . • •Do0.95D090 • I • I • • •—0.2 —0.1 0.0 0.1 0.2 0.3Figure 5.8: The output power of the maser arbitrarily normalized to a value at Li = 0.019as a function of the measured resonator detuning. When fit to a quadratic of the formin equation 5.8 it appears that there is an offset Lio=-0.014(30) in the absolute measureof L.Chapter 5. The 0.5 K H-H Spin-Exchange Measurements 90The second determination of the absolute detuning is based upon an analysis of thedata shown in figure 5.5 which was taken at a constant L 0. According to equation5.4 we would expect this data to depend linearly upon /9 cx Q1 at constant nH. In figure5.9 we reproduce this data along with the results of a linear fit to each data set. In thelimit of zero detuning (we ignore terms in 2) and infinite Q, the residual frequency shiftis expected to bef—f0 = { + V1 ([(pcc + paa) + 2] + [i(pcc + paa) + X2i) } (5.9)The data in figure 5.9 has been extrapolated to Q’=O and the residual shift plotted asa function of H in figure 5.10. The shift in the absolute detunirig which occurred part ofthe way through the run is quite obvious as a discontinuity in the data. In the followingsection we determine a value for T0 (which is independent of the absolute value of L tofirst order). As we have measured values for z before and after the shift, we can correctfor it. This correction has been applied to the data marked by the triangles. The residualshift does not have an obvious dependence upon density and hence we attribute it solelyto an offset z between our measured values of L and the absolute detuning”. Usingthe value of T0 determined in the next section we find zo = -0.046(1). This value is inagreement with the value determined from the tuning dependence of the maser powerand is the value that we have chosen to use throughout the remainder of this analysis.5.5.2 The parameters and T0The frequency shifts measured at constant Q (figure 5.4) have been replotted in figure5.11. In this figure a straight line has been fit to each data set (lines of constant detuning)and used to determine the zero density residual shift and the rate of change of this shift11Note that neither the numerical results calculated by Verhaar ci al. [19, 20], nor the measuredvalues for the spin-exchange parameters that we present later in this chapter suggest that a significantcontribution from the density dependent shifts should be expected in figure 5.10.Chapter 5. The 0.5 K H-H Spin-Exchange Measurements 91I I I I I I I I I I I I I I I I I I I 1 I I I I—0.15N:i:>%0C00I—0.20—0.25i a a a a i •- a . a r iI .2e—03 I .4e—03 1 .6e—03 I .8e—03Q—1Figure 5.9: The data taken near Li = 0 as a function of Q’ reproduced from figure 5.5.A straight line has been fitted to the data set at each density in order to extrapolate tothe residual shift at infinite Q.Chapter 5. The 0.5 K H-H Spin-Exchange Measurements 920.000—0.010N—0.020________________________________________>-C.) -Ca)D I]0w —0.030LLL—0.040—0.0500.Oe+00 1.Oe+12 2.Oe+12Density (cm3)Figure 5.10: The Q’=O intercept of the data taken near zero detuning plotted as afunction of the H density (squares). The shift which occurred in the absolute detuning isquite obvious. This measured shift has been corrected for and the shifted data indicatedby the triangles. The residual shift is fo = —0.021 (4) Hz which corresponds to a detuningoffset Lo = -0.046(1).Chapter 5. The 0.5 K H-H Spin-Exchange Measurements 93with density for each detuning. The zero density intercept is quadratic in L (see equation5.4) and should be given byurn f—fo= —-- [+,9Ao(1+z2)j ± (5.10)11H+O 2K T0The intercept obtained from the data in figure 5.11 has been plotted in figure 5.12 alongwith a fit to the quadratic a0 + a1z + a2. Even though it is slight, the curvaturein this plot is significant and indicates that the sign of A0 is negative. The linear termin this fit depends only upon the relative measure of to first order and can be usedto determine T0. The value ai=0.850(19) Hz yields a valueT0=0.187(4) seconds. Thisvalue was used in the previous section to establish the absolute cavity detuning.The ratio of the linear term to the constant term can be used to determine A0:A0=(5.11)We obtain a value A0 = -21.7(2.8) A2 where this error reflects all of the experimentaluncertainties including the uncertainty in the absolute detuning. We note that bothparameters determined in this way (T0 and A0) are quite insensitive to the absoluteatomic densities, however A0 is sensitive to the absolute detuning.5.5.3 The parameters o1(pcc + Paa) + o2 and Ai(pcc + Paa) + A2The rate of change of the frequency shift with respect to the H density predicted byequation 5.4 is= { [ + + )] [‘(p + paa) + g2] + [Ai(p + p) + A2]} (5.12)In figure 5.13 we have plotted the slope of the data shown in figure 5.11 along with a fit’2to12A similar analysis can be performed by fitting the data to a quadratic however the error bars arelarge enough so as to make the determination of the term b2 quite uncertain.Chapter 5. The 0.5 K H-H SpinExchange Measurements 940.1 , I I I I I I I I0.0F’--—0.3—0.4 I0.Oe+00 1.Oe+12 2.Oe+12 3.Oe+12Density (cm3)Figure 5.11: The data taken at constant Q reproduced from figure 5.4. A straight linehas been fit to the data set at each detuning in order to determine the zero densityintercept (figure 5.12) and the rate of change of the frequency shift as a function of n(figure 5.13).Chapter 5. The 0.5 K H-H Spin-Exchange Measurements 95Figure 5.12: The zero density frequency shift as a function of the absolute detuningat Q=809(8). Error bars are smaller that the symbols used to indicate the data. Thenegative curvature is significant and indicates that is negative. This data has been fitto the quadratic a0 + a1z + a2/. to determine T0 and0.10.0N>.0C00_I.—0.3—0.4I I I I I I I I I I I I I I I I I/.I I I I I I I I I I i i I i i I I—0.25 —0.15 —0.05 0.05 0.15 0.25AChapter 5. The 0.5 K H-H Spin-Exchange Measurements 96b0 + b1z. Here b0 is simply the L = 0 intercept of equation 5.12= = {o [Ui(pcc + paa) + U2] + [i(pcc + Paa) + 21} (5.13)and if terms in are neglectedö Ia(f—f0) ‘ Vb1 =8r ) j [ai(Pcc + Paa) + U2] . (5.14)The slope of the data in figure 5.13 gives a direct measure of °i(pcc + p) + a2 which issensitive to the absolute atomic density but depends only upon the relative measure ofzS. We find Ui(Pcc + Paa) + a2 = 0.385(30) A2. The uncertainty which is reported hereincludes a relative error of 5% which was added because of the uncertainty to due withthe choice of the density enhancement factor (determined by computer simulation [17]).Finally we note that the residual shift due to the term \i(Pcc + paa) + )‘2 (if present)is given by the differenceXi(pcc + paa) + 2 = b0 — (5.15)vHHWe find from this data that si(pcc+paa)+A2= 0.021(14) A2. The error in this quantity islarge as it depends upon the difference between two frequency shifts. The result dependsboth upon the absolute atomic density and the absolute resonator detuning. It doeshowever appear to make a significant contribution to the total observed frequency shift.This (density dependent) frequency shift (which does not depend on either L or Q) isthe first indication that hyperfine interactions infiuellce the oscillation frequency of theCHM as predicted by Verhaar et al. [19, 20].5.5.4 Summary of measured quantitiesA global fit to all of the data taking illto account the small (measured) deviations fromconstant Q and L does not lead to an improvement of the uncertainties associated withChapter 5. The 0.5 K H-H Spin-Exchange Measurements 97Figure 5.13: The rate of change of the observed frequency shift with H as a functionof the absolute resonator detuning. The data has been fit to the straight line b0 + b1zwith b0 = -7.56(8)x10’Hzcm3 and b1 = 8.87(24)x10’4Hzcm3. This data is used todetermine the parameters O1(cc + p) + o2 and Ai(p + Paa) +E1’4=0C‘0LI‘02.01.00.0—1.0—2.0—3.0I I I I I • I I IIIIIIIIIIIII I I I I . I i a I a a I I a a a a—0.25 —0.15 —0.05 0.05 0.15 0.25Chapter 5. The 0.5 K H-H Spin-Exchange Measurements 98Theoretical MeasuredParameter value [77] valueA2 A2A0 —11.8 —21.7(2.8)Ui(pcc + paa) + ‘2 0.262 0.385(30)Ai(pcc + paa) + A2. —0.0204 0.021(14)l 0.0779 —0.055(35)Table 5.1: Summary of measured and calculated [77] spin-exchange frequency shift andbroadening cross sections at a temperature of 0.5 K. We have assumed that Pcc + Paa =0.5 in reporting the theoretical values.the quantities A0,T0,Ui(Pcc + Paa) + 02, and Ai(pcc + Paa) + A2. The measured valuesdetermined in the previous sections are summarized in table 5.1 along with a few of thenumerical results of Verhaar et al. [77].We note several significant discrepancies between the measured values and the calculated values. The measured amplitude of A0 is nearly twice as big as that of thecorresponding theoretical value. As determined, this quantity does not depend upon theabsolute density calibration and hence should be quite reliable. The value determinedfor the combination of the ‘s is also somewhat larger than the numerical value. Thisvalue does depend upon the absolute density calibration, however a 5% uncertainty hasalready been added due to the uncertainty regarding the density enhancement factor.The possible density dependence of this factor has not been included in the analysis.The numerical results [20] suggest that u2 < o. If we assume Pcc + p=0.5 as predicted[17] for the UBC CHM, we obtain o-i=0.770(60) A2. There are however no experimentalresults which allow us to distinguish the relative sizes of and o2 in this way.Finally we note that the term depending upon the combination of A1 and A2 appears tobe significantly different from zero but that it has the opposite sign from the numericallydetermined value. The calculations of Verhaar et al, [77] yield a negative value for thisChapter 5. The 0.5 K H-H Spin-Exchange Measurements 99combination of terms for all temperatures in the range i0 K to iO K if we assume thatPcc + Paa = 0.5. In this experiment we observed a positive frequency shift. If we use thedimensionless ratio (equation 5.7) as a measure of the contribution of the contributionof the hyperfine interaction to the total spin exchange induced frequency shift we findthat = -0.055(35) at a temperature of 0.500(3) K rather than the value 0.078 predictedin references [20, 77]. The theoretical spin-exchange cross sections are quite sensitive tothe detailed form of the interatomic potentials used in the calculation. It is difficult toinfer what effect the discrep3ncies between our measurements and the theoretical resultslisted in table 5.1 might have on the interatomic potentials. This is a question that couldbe best answered by the Eindhoven group led by Professor Verhaar.Chapter 6Background and Technical Aspects of the 1 K ExperimentsWe turn now to a study of mixtures of atomic hydrogen (H) and deuterium (D) confinedby superfluid £-4He walls at cryogenic temperatures. This study comprises the first investigation of interactions between two different isotopes of hydrogen in zero field at thesetemperatures. At the same time we have also been able to make significant improvementsin our understanding of the solvation of D into £-4He.In this chapter we describe the apparatus used to perform the experiments. Theresults which were obtained are presented in the following chapter. The historical basisfor the design of this experiment played an important role in the evolution of these studiesand warrants mention before further details are discussed.6.1 Historical contextRecently we observed for the first time the solvation of atomic D into £-4He [21, 22]. Theseexperiments were performed using magnetic resonance on the 3- hyperfine transition’ ofthe D atom at 309 MHz in a 39 Gauss magnetic field. Atomic D densities were observed todecay exponentially with time which was quite unlike the two body decay rate (equation3.14) which had been expected by analogy with the gas phase or surface recombinationof two H atoms [37, 12]. The rate at which the atomic density decayed was also a verysteep function of temperature, making the observation of D nearly impossible above 1.161This is the zF 1, zmF = 0 hyperfine transition. It passes through an extremum as a function offield at 39 Gauss.100Chapter 6. Background and Technical Aspects of the 1 K Experiments 101K. On the other hand, the base temperature of the (pumped 4He) cryostat was 1.08 K,so that the range of observation was very narrow. The decay of the D atom density wasattributed to the solvation of D into the £-4He film coating the walls and subsequentlyadsorbing to the underlying substrate which was composed of solid D2. The data wasanalyzed in terms of the simple theory presented in chapter 3 whereby the sample decayrate is:A__ (E’ajg1Uvexp\kT) (6.1)In equation 6.1, E is the energy required to force a D atom into the 4He liquid and guis the ratio of the effective mass m* of the D quasi particle inside the £-4He to the bareD mass m. g is the thermally averaged probability that a D quasi particle striking the£-4He surface will leave the liquid and V is the mean thermal speed of the D atoms inthe gas phase. This study indicated that E=13.6(6) K. Unfortunately the very narrowtemperature range and the limited accuracy of the data made the determination of theprefactor to the exponential uncertain to more than an order of magnitude.During these studies we observed startlingly strong broadening of the D atom magnetic resonance line: under some circumstances it could be more more than an orderof magnitude larger than expected for a D atom gas. It was postulated that this excessbroadening was due to spin exchange collisions between the D atoms, and H atoms whichwere present in the sample as a contaminant. Reynolds [22] calculated numerically therelevant broadening cross sections in the DIS limit (see chapter 3) and used this to inferthe density of H atoms which were in the cell. In a separate experiment we used magneticresonance on the a-c hyperfine transition of atomic H at 1420 MHz in order to verify thiscontamination by direct observation of the H. This work was done with a similar butconsiderably smaller cell filled with the same D2 gas. The levels of contamination whichwere observed were roughly consistent with the broadening which had been observedChapter 6. Background and Technical Aspects of the I K Experiments 102during the 309 MHz experiment. Work up to this point has already been reported inreferences [21] and [22].Over time it was realized that a better measurement of the solvation energy opposingthe penetration of £-4He by D might in fact be made by using the 1420 MHz apparatusto monitor the H atoms inside the cells (and indirectly the D atoms). The key to thisexperiment is the fact that at 1 K the thermally averaged spin-exchange broadeningcross section for H-D collisions is more than two orders of magnitude larger2 than thecorresponding quantity for H-H collisions. As a result we found that it was possible tomonitor both the H and the D densities simultaneously: following a 7r/2 tipping pulse,the initial power radiated by the spin system is related to the H density whereas thebroadening of the transition is related to the D density3. Effectively we obtain the sametype of information that a double resonance experiment4would yield. Note that it ismore difficult to perform the analogous experiment (inferring the H density from thebroadening of the j9-8 transition of D) at 309 MHz since the broadening cross section forD-D collisions is expected to be comparable to that for H-D collisions [22].The motivation for pursuing the 1420 MHz experiments further was twofold. Initiallywe hoped to reproduce and perhaps improve upon the original measurement of the solvation energy for D in £-4He. The 309 MHz experiment had originally been designedto study resonance recombination of D in magnetic fields [21], not to study the solvation of D into £-4He! Apart from this fact, the 309 MHz experiments were technicallyvery demanding and the amount of attention which could be devoted to actually makingmeasurements was rather limited. It was quite obvious that the 309 MHz apparatus2This was predicted by Reynolds [22]. It has been verified by the experimental results presented inthe next chapter.3The determination of the absolute D density was initially dependent on the numerical calculations[22] for the broadening cross section. Later we were able to measure this cross section and thus obtainan absolute calibration.142O MHz and 309 MHz simultaneously. An experiment of this type would be a rather dauntingundertaking.Chapter 6. Background and Technical Aspects of the 1 K Experiments 103would have to be redesigned if further studies of the solvation of D into £-4He were to beperformed with it. Modifications to the 1420 MHz apparatus were much easier to make.As time passed, it became apparent that the 1420 MHz experiment would also yietda much richer variety of information about interactions between H and D atoms. Inparticular it was possible to study the various processes by which the D atoms in thesample recombine. By accounting for these alternate decay mechanisms which occur inparallel with the solvation process, it was possible to make a much ‘cleaner’ measurementof the solvation energy. It was also possible to make absolute measurements of the H-Dspin exchange broadening and frequency shifts parameters. These measurements forman important complement to the study of H-H spin exchange presented earlier.6.2 Experimental designThe apparatus and techniques used to carry out the experiments reported here weresimilar in many ways to earlier experiments performed in this laboratory using hyperfinemagnetic resonance techniques [27, 22, 37]. These types of experiments have been summarized in reference [12]. In the following discussion we concentrate on the elements ofthe experimental design and the data analysis which were most essential to the studiespresented in the next chapter.6.2.1 The cryostatThese experiments were performed in a 8.9 cm inner diameter (i.d.) glass dewar with asecond glass 1-N2 jacket. The experimental cell and resonator housing were suspended inthe main £-4He bath from above by a stainless steel tube incorporating several radiationbaffles. During the experiments the entire resonator volume was flooded with £-4He fromthe main bath. This bath was cooled below 4.2 K by evaporative cooling. A rootsChapter 6. Background and Technical Aspects of the 1 K Experiments 104blower5 backed by a rotary piston pump6 were combined to achieve a pumping speed ofabout 600 cfm near the base temperature of the system. The dewar was elevated andnioved to within 1.5 m of the pumping system in order to make maximum use of theavailable pumping speed. Bellows were used to mechanically decouple the experimentfrom the pumping system. The lowest temperature obtained in this way was 0.990(3)K. This represents a significant improvement7over the base temperature obtained in the309 MHz experiments.Two concentric high permeability metal shields8were wrapped around the outer dewarin order to shield out the earth’s magnetic field. These shields were demagnetized usinga 10 turn coil and AC currents up to 10 A. The residual field was subsequently mappedout using a Hall probe. Typically it was less than 5 x iO Tesla. Fluctuations in thisresidual over the resonator volume were less than 1 x iO Tesla.6.2.2 The resonatorThe 1420 MHz resonator used in this work was a Cu split-ring resonator [86]. It isillustrated in figure 6.1. It was 4.6 cm high and had an outer diameter of 2.54 cm.Two overlapping holes with different diameters were bored parallel to (but offset from)the axis of this cylinder. The larger hole (1.8 cm diameter) was used to contain thepyrex experimental cells (bulbs) described later. The second hole formed a relief volumebetween the capacitive gap in the split-ring and the sample bulb. The idea behind thisdesign is to keep the fringing E fields near the gap away from the pyrex ( 5) walls ofthe cells.This assembly was located inside a 3.8 cm i.d. Cu rf shield such that the axis of the5Leybold-Heraeus WA 1000: 685 cfm displacement.6Leybold-Heraeus E250: 162 cfm displacement.7The effective range over which data could be taken was more than doubled.8Co-netic foil: Perfection Mica Company-Magnetic Shield Division, 740 Thomas Dr, Bensenville III.Chapter 6. Background and Technical Aspects of the 1 K Experiments 105badfCgFigure 6.1: The 1420 MHz resonator used in order to study H, D mixtures at 1 K. Detailsshow (a) the split-ring resonator [86] into which the sample bulb is inserted, (b) the teflontuning bar and, (c) its tracking grooves, (d) the rf coupling loop and , (e) the adjustablecoupling mechanism (schematic), (f) the teflon supports for the resonator, and (g) theresonator housing (rf shield). Further details are shown in figure 6.2.Chapter 6. Background and Technical Aspecis of the 1 K Experiments 106bulb is aligned with the axis of the shield. This shield acts as a barrier to electromagneticradiation and confines the rf energy to the resonator volume. The resonator was heldin place by two teflon supports. It should be noted that a parasitic TEM mode is oftenencountered when using split-ring resonators. It is believed that some of the earlierstudies of atomic H carried out in this lab [27] may have been affected by this problem.The much narrower shielding can used here ensures that this mode is pushed far fromthe frequencies of interest.Grooves on either side of the 2.3 mm gap in the split-ring acted as a guide for theteflon slab which is used to capacitively tune the resonator. This bar was made slightlythicker than the available distance between the resonator and the shield. A cut in thisbar made perpendicular to the split-ring gap allowed it to be squeezed into position andbe held positively against the resonator. The tuning bar was moved along the length ofthe resonator by a mechanical linkage driven by a micrometer at room temperature.A single rf coupling loop was brought into the resonator volume opposite the split-ringgap. A mechanically adjusted capacitance (shown schematically in figure 6.1) allowedadjustments to the coupling between the resonator and the external circuitry to be made.All work reported here was performed under conditions of critical coupling. Correspondingly, all reported quality factors are equal to one half the unloaded values (Q°).The quality factors of similar split-ring resonators at this frequency are generallyabout 6 x iO at critical coupling. The Q of this resonator was intentionally lowered toabout 1.5x103 to reduce the effects of radiation damping on the envelope of the FID’s.Prior to its use in these experiments the filling factor of the resonator was measuredusing a type I perturbation measurement as described in appendix B. A pyrex sleeve wasplaced inside the resonator during these measurements to mimic the walls of the samplebulb. This measurement was essential if the techniques outlined in appendix A were tobe used to analyze the broadening of FID’s. A filling factor i = 0.37(4) was measuredChapter 6. Background and Technical Aspects of the 1 K Experiments 107for this geometry. The parameter of interest for determining the influence of radiationdamping on a FID at a given atomic density is the quantity9 This parameter wasabout 25% larger for this experiment than it was for the H-H spin-exchange experimentsdescribed in chapters 4 and 5.6.2.3 The dischargeA cross sectional view of the resonator and its housing is shown in figure 6.2 with thesample bulb in place. The rf discharge is located in a separate encasement below theresonator volume. These regions are connected via a short Cu tube. This tube acts asa waveguide beyond cutoff and effectively isolates the two chambers. The discharge isa simple LC parallel resonant circuit tuned to 50 MHz. A coaxial line is tapped intothe inductor about 1 turn away from its lower end (which is grounded) in order to forma step-up voltage transformer. The impedance of this circuit is roughly matched tothe impedance of the external circuitry. RF power is supplied by an Arenberg pulsedoscillator’0.Typical power levels of 400 W were used with 10 to 20 usec discharge pulsedurations.A central access port is located above the sample bulb and opens into the resonatorhousing. This port allows for visual confirmation of the firing of the discharge. Duringseveral runs a ‘stinger’1’was passed down through this access hole and moved near to thesample bulbs in order to help initiate the first discharge inside the cell. On other occasionsa 60Co 3 source was used for this purpose. Generally no difficulties were encountered infiring the discharge after it had been fired for the first time.9In the high temperature limit the thermal equilibrium population difference between the a ) stateand the c ) state scales as In the spin 4 analogy this difference is related to the effectivemagnetization of the radiating spins.‘°Arenberg Ultrasonic Laboratories, Boston MA. Model PG-650C‘1A tungsten wire axially encased in an evacuated pyrex tube. Both ends of the wire pass throughthe sealed ends of the tube. High voltages are applied to the wire using a Tesla coil.Chapter 6. Background and Technical Aspects of the 1 K Experiments 108Figure 6.2: A cross sectional view of the low temperature experimental assembly whichis submerged in the main £-4He bath. Details show (a) the Cu split-ring resonator (seefigure 6.1), (b) the pyrex bulb containing the atomic samples, (c) the rf coupling loop,(d) the 50 MHz discharge, the coaxial lines leading to (e) the 1420 MHz spectrometer,and (f) the pulsed oscillator, (g) the bias solenoid and gradient coils, (h) the encasementfor the variable tuning and coupling linkages (which are not shown), and (i) the supporttube and central access port.1hgabfCdChapter 6. Background and Technical Aspects of the 1 K Experiments 1096.2.4 The bulbsThe cells used to confine the gases in this work were pyrex cylinders typically 1.7 cm indiameter and 4.6 cm long with a 0.5 mm wall thickness. One end of each tube was sealedand the other necked down to form a narrow tail. The main body of the cells filled thecentre bore of the split-ring resonator while the tail extended into the rf discharge region(figure 6.2). The bulbs were held in place with nylon set screws (Ilot shown).Prior to filling the cells they were heated to 200 C and pumped on through a cold trapfor 24 hours with a diffusion pump. The bulbs were then filled with 99.65% isotopicailypure D2 and UHP 4He and sealed with a torch. The D2 gas was from the same bottle12used in the 309 MHz experiments [21]. The hydrogen which is the central object inthe present magiletic resonance experiments, either enters the cells as a contaminant inthe D2 gas or is liberated during the sealing process (exchange between D and someproton bearing material in the glass). On one occasion the H level inside the bulb wasintentionally raised by adding some H2 gas before sealing the cell. Typically cells werefilled to a room temperature pressure of between 20 and 100 Torr of D2 and 500 Torr of4He. At the temperatures of interest the D2 forms a solid which coats the pyrex substrate.The 4He forms a saturated superfluid £-4He layer which coats the inner walls of the cell.Any excess forms a pooi in the tail of the bulb.The A/V ratio of a sealed bulb was typically 3.5 cm. This is about double thecorresponding factor for the 309 MHz experiments. The tails comprised only about 5%of the total cell volume. It is important to realize that diffusion of atoms into the tailvolume following a ir/2 tipping pulse leads to a time dependent decrease in the effectivefilling factor for tile cell. Radial diffusion times for both H and D [12, 38] lie in therange 30 to 150 msec in this experiment depending 011 the 4He density. Longitudinal‘2Bio-Rad Laboratories, Richmond CAChapter 6. Background and Technical Aspects of the 1 K Experiments 110diffusion times are somewhat longer. Immediately following a 7r/2 pulse and before theatoms have had time to move, the appropriate filling factor is the full measured value.At times much longer than the spin homogeiization time, the filling factor must bereduced by the appropriate ratio of volumes (see chapter 2). While this correction wasminimized by keeping the tail volumes small, we felt it necessary to examine the effect ofspin homogenization in the final analysis. All data was analyzed with both the pre- andpost-homogenization values for i. Iii general, pre-homogellization values were used forwork involving short T2’s and post homogenization values for work involving long T2’s.Uncertainties have been reported accordingly.Once the cells have been cooled to 1 K, the thermal link betweell the cooling bath andthe gas of H and D atoms passes through the glass walls, the solid D2 substrate and the4He film. The dominant time constant for thermal diffusion through a slab of materialof thickness L, density p, specific heat C, and thermal conductivity tc is given by= CpL2 (6.2)K2Using typical values for pyrex (C 3.1 x 106J/g K [91], , = 1.2 x l04W/cm K [92],and p 2.5g/cm3)we find that r < 20 iisec for our cells. The implication of this isthat we can consider the inner wall of the pyrex cell to be at the temperature of themain £-4He (superfluid) cooling bath. The thermal time constant associated with theKapitza resistances and thermal conduction through the £-4He film are about the sameas r. Radial thermal diffusion times in the 4He buffer gas inside the cells are slightlylonger but are only about 60 msec. All of these time constants are short and thus we didnot expect to observe any heating phenomena associated with firing the discharge. Nonewere observed.Chapter 6. Background and Technical Aspects of the 1 K Experiments 1116.2.5 Temperature measurement and regulationThroughout this work temperatures were inferred from a measurement of the 4He pressureabove the main bath. In the first studies we report, pressures were measured usinga McLeod gauge. Temperatures were maintained simply by regulating the pumpingspeed of the 4He evaporation system. This was sufficient for measurements of relativelytemperature independent quantities such as the H-D recombination cross section. Overtime the vapour pressure measurement evolved into a much more accurate and precisemeasurement centred around the use of a capacitance pressure gauge. This techniquealso made it possible to implemeilt temperature regulation based upon the 4He vapourpressure above the main bath. This system proved indispensable in the final studies ofthe solvation of D into £-4He and the H-D spin-exchange frequency shift cross section.The details and characterization of this system are described in appendix D.All temperature uncertainties which are reported are uncertainties in absolute temperature. Vapour pressure measurements were made through open stainless steel tubesextending to just above the £-4He bath. Where necessary small corrections due to thermomolecular pressure gradients [93] have been applied. In the later work, the uncertaintiesin relative temperatures (important in the measurement of the solvation energy and the4He buffer gas shift) are typically less than 100 jiK.6.2.6 Data collectionThe rf electronics associated with this experiment are essentially the same as the systemdescribed in chapter 5. The data acquisition system is diagramed in figure 6.3. The1420 MHz spectrometer was operated using the Rb frequency standard as the frequencyreference. The low frequency beat signal between the atomic FID and the spectrometerLO’s was fed to an analogue to digital converter and then passed to a computer forChapter 6. Background and Technical Aspects of the 1 K Experiments 112storage. This system allowed us to accumulate data at sample rates up to one per 75tsec. Typically a single FID was sampled 1000 times and then plotted on the computerscreen along with simple diagnostics. The rate at which data (FID’s) could be acquiredwas set by the thermal equilibration time for the spin degrees of freedom of the H system.The dominant equilibration process in these experiments was H-D spin exchange andhence T2 for the FID, (related to T1) gave a rough measure of this time. This was ofcourse lot true in experiments where the D density was low and H-H spin-exchange andmagnetic relaxation were the dominant spin relaxation mechanisms. Detailed analysis ofthe FID’s was performed after the experiment was completed.Resonator Q’s were measured in reflection as described in chapter 4. Absolute powercalibrations of the detection circuitry were also made during each run.Chapter 6. Background and Technical Aspects of the 1 K Experiments 11316 bit A/D 1420 MHzConverter Spectrometerdc to gate rf in rf out100 Hz__A/D—ControllerPulseGenerator(trigger)‘1KILW]Split RingResonatorFigure 6.3: An outline of the data acquisition system used to generate trains of 7172 tipping pulses and to record the ensuing free induction decays. The 1420 MHz spectrometeris described in chapter 4.Chapter 7Experiments with H, D Mixtures at 1 KIn this chapter we summarize the experimental results of our studies of atomic hydrogen(H) and deuterium (D) confined by liquid helium (t4He) walls at 1 K. They are carriedout using magnetic resonance at the a-c hyperfine transition of atomic H at 1420 MHz.The evolution of these experiments from first being a means of verifying the contamination of samples of D with H, to their present state was described in the previous chapter.It was noted that by using magnetic resonance techniques to observe the H contaminationin our D samples, it was possible to monitor, and to make absolute measurements of boththe H and the D densities simultaneously and with a single resonator. This procedure,which is much less complicated than a double resonance technique, allowed us to studyfor the first time, interactions between two hydrogen isotopes confined by £-4He walls inzero field at 1 K.The key to this experiment is the fact that the thermally averaged spin exchangebroadening cross section’ of the a-c hyperfine transition2of H due to H-D spin exchangecollisions is more than two orders of magnitude larger than the corresponding cross sectionfor H-H collisions. The broadening of the a-c hyperfine transition of aton,ic H due to1Throughout this chapter we refer to spin exchange cross sections and rate constants without specifically indicating that these are thermally averaged quantities. All thermally averaged quantities areindicated with a line drawn above the appropriate symbol.2The phrase ‘of the a-c hyperfine transition of H (at 1420 MHz)’ will often be omitted. It shouldbe understood that all spin exchange parameters which are referred to here are with respect to thistransition at zero magnetic field.114Chapter 7. Experiments with H, D Mixtures at 1 K 115both H-H and H-D spin exchange collisions is [71, 22]1 1— 3—= G11n+7GHDUD (7.1)t2 ‘= On + GHD11D. (7.2)Reynolds [22] has calculated theoretical values for the broadening rate constants O inthe degenerate internal states (DIS) approximation at 1 KGHH = 7.8 x 10’3cm/s (7.3)= 2.4 x 10’°cm3/s , (7.4)values essentially independent of temperature between 1.0 and 1.5 K. Dividing by theappropriate mean relative speeds for the colliding atoms (v or V0) at 1 K, we obtainHH = 0.38A2 (7.5)HD = 140A2 . (7.6)The result for H-H collisions is in agreement with the more sophisticated coupled channelcalculations by Verhaar et al. [20] which yield HH = 0.40 A2. The only experimentallydetermined value3 for HH at 1.1 K is 0.43(3) A2 [27].The result of this large difference in spin-exchange cross sections is that for comparable H and D densities, the broadening of the a-c hyperfine transition of H is completelydominated4 by H-D spin exchange collisions. Consequently the amplitude of a free induction decay (FID) following a 7r/2 tipping pulse at the a-c transition frequency givesa measure of the H density in the cell (fl11) while the decay time of the envelope providesa simultaneous measure of the D density (nD). Under the right conditions G0 can be3The value reported in references [37, 12] was in error (high) by a factor of4Other broadening mechanisms being neglected.Chapter 7. Experiments with H, D Mixtures at 1 K 116measured and thus it is not necessary to rely on the numerical prediction (equation 7.4)to calibrate n.It is important to realize throughout the presentation that follows, that the conclusions which are drawn were often based upon inferences from several experiments,including our earlier studies of D at 309 MHz [21] and studies of pure H at 1420 MHz[12].7.1 The basic experimentThe sample bulbs are filled with a mixture5 of D2 and 4He gases as described in theprevious chapter, and cooled to 1 K. As the temperature is lowered the D2 first forms asolid on the inner walls of the bulb and is then covered by a layer of superfluid £-4He.We start the experiment by firing the rf discharge. This liberates both H and D fromthe molecular ice underlying the £-4He film. These atoms interact with each other andwith the £-4He coated walls: recombination processes (H-H, H-D, and D-D) reduce theatomic densities inside the cell; the D atom density also decreases due to the solvationof D atoms into the £-4He film and subsequent adsorption to the substrate.At any instant in time the fraction of D atoms in the cell with sufficient energyto penetrate the film is low; however collisions with 4He atoms ensures that a thermaldistribution of kinetic energies is maintained. As a result there is a constant flux of Datoms passing through the film. At high temperatures (T > 1.2 K) essentially all of theD atoms penetrate the film within seconds of the discharge. At lower temperatures thisdecay rate is slower but solvation is usually the dominant mechanism by which the Ddensity decays in the experiments reported here.In their simplest form, our measurements consist of the application of a train of5Generally H is present in the cells only as a contaminant.Chapter 7. Experiments with H, D Mixtures at 1 K 1171111111 iiliiitli, till.... IllIJ iiiiiiIillli iii,,,,I 2s24s0.0 0.2 0.4 0.6 0.8 1.0Time (s)— iv/2 pulseFigure 7.1: The FID’s resulting from the application of a series of ir/2 tipping pulse atthe a-c hyperfine transition of H (1420 MHz) to a mixture of H and D at 1.001(3) K.The labels on the right hand side of the decays indicate the time following the dischargepulse at which the 7r/2 pulse was applied. An arbitrary constant has been added to eachdecay in order to display the data in this fashion. The beat frequency is arranged to beat a convenient value via the synthesizer setting.Chapter 7. Experiments with H, D Mixtures at 1 K 118K/2 tipping pulses at the a-c hyperfine transition of H to the sample. These pulsesare separated by a variable time interval. Each pulse interrogates the atomic system,returning information about both the number of H and D atoms in the cell at that time.In figure 7.1 we show an example of the FID’s which result from such a pulse train. Thisparticular data was taken at a temperature of 1.001(3) K where the lifetime of the Datoms in this cell due to solvation is about 30 seconds. An arbitrary constant has beenadded to each decay in order to display them in this fashion. The label on the righthand side of each decay indicates the time at which the Tr/2 pulse was applied followingthe discharge pulse. The most obvious change in the data over this period of time isthe increase in the apparent. decay time T associated with the envelope of the FID. Itwas this type of observation which initially piqued our interest in this system and led tothe preseit studies. We attribute this time dependent broadening to the presence of Datoms in the cell. A simple analysis shows that the rate at which the D atoms disappearis roughly exponential and that the temperature dependence of this rate is about whatone would expect due to the solvation of D in the £-4He film [21]. A more detailed analysisindicates that the data also contains contributions from recombination processes.7.2 Analysis of the dataThe notation in this chapter deviates somewhat from that in the magnetic resonanceliterature and thus requires special mention at this point. We refer to the apparentdecay time of the FID as T. T2 is used for the effective relaxation time once theeffects of radiation damping have been accounted for. In the strictest sense this latterquantity should be denoted T as it may still contain a contribution from magnetic fieldinhomogeneities. We reserve the notation T;e for the true spin-spin relaxation time dueto spin exchange collisions, once all other effects have been accounted for.Chapter 7. Experiments with H, D Mixtures at 1 K 1191 .5•1.00.500.0 \/\—0.5-—1.0-—1.5 I • I • I I0.0 0.2 0.4 0.6 0.8 1.0Time (s)Figure 7.2: A FID taken at 1.000(3) K and the numerical fit to the data. The H densityis nH = 8.6 x 10cm3. Radiation damping accounts for about one third of the of thedecay of the FID envelope. In the example chosen here, T21 is approximately equal tothe residual broadening which, as explained in the text, sets the detection threshold forthe D atom density. The normalization is such that the amplitude of the fit to the datais initially one.Chapter 7. Experiments with H, D Mixtures at 1 K 120Immediately following the discharge pulse T is dominated by H-D spin exchangecollisions and the FID envelope is roughly exponential. As time progresses and thebroadening due to H-D collisions decreases, radiation damping begins to play an increasingly more important role in determining T. We have analyzed all of the FID databy fitting the decays to the numerically integrated differential equations outlined in appendix A in order to properly account for the effects of radiation damping. In thesefits, the filling factor, resonator Q, and the power calibration of the signal are entered asmeasured quantities. The free parameters in the fits are T2, T1, the beat frequency, theH density (signal amplitude) and an initial phase angle. If the data is not analyzed inthis way it becomes difficult to determine nH and T2 accurately when the D density islow and radiation damping is beginning to influence the FID’sIn figures 7 2 and 7 3 we show examples of the numerical fits to the FID’s in differentregimes In figure 7 2 nH = 8 6 x 1010 cni3 at a temperature of 1 000(3) K Radiationdamping accounts for about one third of the observed damping The residual T2 ( 1 s)is very close to the ‘infinite time’ relaxation time T2°° (z e long after the discharge pulsewhen all of the D atoms are expected to have left the gas phase) The observed residualbroadening is in general greater than that expected from H-H spin exchange alone [27].The excess is due to factors such as field inhomogeneities, magnetic relaxation, and thediffusion6 of H atoms out of the resonator volume. In general this residual broadening isnot well reproduced between cooldowns, however it is reproducible throughout a givenrun. It is a weak function of the H density. In essence, the combined effects of theprocesses which lead to this residual broadening set the practical detection threshold onthe D atom density. At this level, 10% changes in the broadening are easily detected.Using the numerical results [22] for HD and equation 7.1 we infer a threshold detection6Diffusion does not strictly give rise to an exponential FID; however it does lead to a shortening ofthe effective T2 determined in the fit.Chapter 7. Experiments with H, D Mixtures at 1 K 121limit near71 4 T)—’thresh.= () 2 x 109cm3. (7.7)This is comparable to the H atom detection limit with the present system.In the second example of a fit to a FID, shown in figure 7.3, the H density has beenincreased to l.8x10”cm3but more importantly there is considerable broadening dueto H-D spin-exchange collisions. The FID envelope yields T2 = 60 ms. This broadening(T2’) is well above the detection limit and we infer8 that n, 9 x 10’°cm3. In thefollowing sections we examine the various parameters determined from these numericalfits in more detail.The dominant decay of the FID’s is set by T2 however T1 also makes a contributionand can be determined from the numerical fits to the data. This procedure is outlinedin appendix A. T is a measure of the time taken for the longitudinal component of the(fictitious) magnetization to relax to its thermal equilibrium value following a tippingpulse. Information about T1 is contained9in the rate at which the instantaneous dampingof the FID changes as a function of time. This relationship is described explicitly inappendix A.The parameter T1 is not determined to the same precision as T2 in this type ofanalysis. In particular the relative error in T1 is large both when the number of zerocrossings in the FID’s is small, and when radiation damping dominates the FID. The firstsituation occurs when the dominant decay time of the FID (i.e. T2) is comparable to,or shorter than, the oscillation period of the beat frequency between the radiating atomsand the local oscillator of the spectrometer. T1 is determined from the rate at which theinstantaneous damping of the FID changes. Without any zero crossings in the data thePH-H spin exchange broadening is completely neglected in this simple estimate.8Again we have used the numerical values for GHD.9Radiation damping also causes the longitudinal component of the magnetization to relax and andmust be taken into account.Chapter 7. Experiments with H, D Mixtures at 1 K 1221.5 • • • • •1.00.500.0--- -_S?-—0.5-—1.0-—15 • I • I • I I • I I • I0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40Time (s)Figure 7.3: A FID taken at 1.001(3) K shown with the numerical fit to the data. Thedata has been normalized to the fit amplitude at t=0. Considerable broadening due toH-D spin exchange collisions is apparent. The numerical results [22] for GHD imply a Datom density n, 9 x 1010 crn3. The H density is n11 = 1.8 x 1011 cm3.Chapter 7. Experiments with H, D Mixtures at 1 K 123damping of the FID is easily confused with an apparent frequency shift. On the otherhand when radiation damping dominates the FID, T1 makes very little contribution tothe relaxation of the longitudinal magnetization and thus it is only poorly determined.7.2.1 T1 and T2An example of the simultaneous time evolution of T1 (squares) and T2 (triangles) asdetermined from the fits to a series of FID’s following a single discharge pulse is shownin figure 7.4. Initially the FID decay time is much shorter than that caused by radiationdamping. In this case T2 is set quite accurately by the envelope decay time. The relativeerror in T1 on the other hand can be large if the number of zero crossings in the data isof order 1.. Care has been taken in the analysis of all data in this regime to ensure thatthe zero crossings are well represented by the fits.Both T1 and T2 increase with time and eventually ‘saturate’. This saturation occursat different times for the two parameters. Just as was the case with T2, the saturationlimit for T1 sets a threshold limit on the lowest D densities which can be inferred usingequation 7.2. In general we find that the practical range over which T1 data can be usedto infer nD is less than the useful range over which T2 can be used.A Tr-Tr/2 pulse sequence can be used to verify the determination of T1 from the FIDdata. In the spin analogy, a r pulse inverts J’1 which subsequently relaxes back toits thermal equilibrium value by T processes. Radiation damping does not play a rolein this relaxation since the transverse magnetization remains zero. A r/2 pulse whichinterrogates the relaxing magnetization after a time r yields a measure of the amplitudeof M at that time. If r = Ti/ln (2) then fI = 0 and no FID is observed after the 7r/2pulse. By changing r until a null signal is observed, a measurement of T1 can be made.We have measured T1 in this way following several ir/2 pulse trains. In each case theresults are in agreement with the values of T1 determined from the FID data.Chapter 7. Experiments with H, D Mixtures at I K 1241.25 I ) I I ‘ I ‘LitLiLi1.00 Li -LiE 0.75I—CF-0.50 LiLi• Li0.25- LiLiQD-LiQD• • •0 50 100 150Time (s)Figure 7.4: The fit parameters T1 (squares) and T2 (triangles) determined from a sequence of FID’s following a single discharge pulse at t=O. The results for T1 are inagreement with measurements made using a ?r-lr/2 pulse sequence in the saturated limit.This data was taken at 1.001(3) K.Chapter 7. Experiments with H, D Mixtures at 1 K 125The long time limiting values of these relaxation rates (l/T10°and 1/T2°°) are subtracted from the relaxation rate data obtained from the fit to the FID’s (1/T and 1/T2)and the residual relaxation is attributed to H-D spin-exchange collisions. The residualbroadening rates are denoted 1/Tv and 1/Tv. The ratio of T to T;e determined in thisway is essentially ; the ratio which is expected due to spin-exchange collisions betweenheteronuclear species of H (see chapter 3). A typical example of this ratio immediatelyfollowing a single discharge pulse1°is shown in figure 7.5. This is an important check thatthe residual broadening is indeed due to H-D spin-exchange collisions. Repeated measurements of this ratio indicate that T/T = 0.74(2). This is the only measurement ofthis ratio for H-D spin-exchange collisions that we are aware of below room temperature[76].An example of the broadening 1/Tv which remains after the subtraction of the residual 1/T20°, and which is attributed wholely to H-D spin-exchange, is shown in figure7.6. This broadening is proportional to the D atom density nD inside the cell: using thenumerical result 7.4 for GHD we infer an initial D density of 6.6x1O° cm3. This is about40% of the H density in the cell at the same time.7.2.2 The H densityAn example of the time dependence of the H density inside the cell as determined fromthe power emitted by the H atoms following a series 7r/2 pulses is shown in figure 7.7.This data is from the same decay as that from which the broadening data of figure 7.6is derived. It has been plotted separately in order to expand the vertical scale.Before proceeding further we want to point out the existence of three different timescales. The first of these is associated with the drop observed in the H density immediatelyfollowing the discharge. The amplitude of this drop depends upon the D density inside‘°In this example the relative error in T1 becomes too large to determine Tr/T at longer times.Chapter 7. Experiments with H, D Mixtures at 1 K 1261.0 . I I I I I I I I Iu0.6‘.4I-0.40.20.0I I . I I I I • I • •0 20 40 60 80Time (s)Figure 7.5: The ratio Tr/T as a function of time following the discharge pulse. Thedashed line indicates the ratio which is expected due to spin-exchange collisions betweenH and D in the DIS limit. The average value of this ratio determined from repeatedmeasurements is 0.74(2).Chapter 7. Experiments with H, D Mixtures at 1 K 12720 • • • • I • I I I I15-ci010ci5cicicici0 • . . .0 25 50 75 100 125Time (s)Figure 7.6: An example of the time dependence of the broadening 1/Tv which is wholelyattributed to H-D spin exchange collisions (i.e. residuals have been subtracted). Thedecay of 1/Tv (which is proportional to the D atom density) is approximately exponential.Using the theoretical result GHD = 2.4 x 10’° cm3/s [22] the initial D density is 6.6x10’°cm3. The measured H density is about 9x10’° cm3 (see figure 7.7). The temperatureof the cell is 1.001(3) K.Chapter 7. Experiments with H, D Mixtures at 1 1< 1282.Oe+11 • • • . . •1.8e+11-DD-DQQQDQDD-El>%t 1.6e+11 --C,)C0-D1.4e+11 --I .2e—f—I1 i I I I • • I I I I • I I I I • • I I0 25 50 75 100 125Time (s)Figure 7.7: The H density in the sample bulb during the broadening measurementsshown in figure 7.6. Three time scales are distinguished. The first period lasts tensof seconds and is associated with the recombination of H and D immediately followingthe discharge. The second time scale lasts hundreds of seconds and is associated withthe delayed source of H atoms. This ‘peak’ is thought to be due to H liberated in achemical exchange reaction occurring on the substrate. The third time scale is governedby H-H gas phase recombination. The characteristic time for this reaction is measuredin thousands of seconds.Chapter 7. Experiments with H, D Mixtures at 1 K 129the cell and we attribute it to the recombination of H and D to form molecular HD. Thisreaction is discussed in the next section. The second time scale is associated with theincrease in“H, which follows the initial decrease. Iii figure 7.7 this increase begins about50 s after the discharge. During other cool downs, the time at which this increase beginsis delayed for as long as 200 s. This unusual feature was quite unexpected. It seems to bea source of atomic H which is delayed from the initial discharge. A analogous feature isnot observed in samples containing only H2. The third time scale is set by the gas phaserecombination of two H atoms to form H2. This process has been studied previously byHardy et al.” and is known to occur on a time scale of thousands of seconds at thesetemperatures and densities. For a short period of time following the apparent source ofH atoms, the decay of the H atom density is somewhat faster than expected based onthe rate reported by Hardy et al. After a few hundred seconds the decay rate slows downand is consistent with the earlier report. We have not studied this part of the decay indetail: it is not understood why the decay of the H atom density initially seems to befaster over this time period.The bulk of the H impurity in our samples is likely present as HD. In addition, the gasphase recombiriation of H and D ensures that the surface of the substrate 12 underlyingthe £-4He film is covered with HD. We postulate that D atoms, which penetrate the filmand adsorb to the surface, encounter aid react chemically with this HDHD+D—iD2+H. (7.8)A similar (exothermic) reaction has been observed before in D2-HD mixtures when atomsare produced by irradiation [94, 95]. In this picture the H atom which is liberated issubsequently ejected from beneath the film and returned to the gas phase. These atoms11The recombination rate constant kHH determined by Hardy and his coworkers was O.20x1032cm6/sat 1 K. The value reported in [37] and [12] was in error by a factor of The proper value was reportedin [27].12All molecular isotopes of hydrogen pass easily through the £-4He film unlike B [35].Chapter 7. Experiments with H, D Mixtures at 1 K 130appear in our data as a delayed source of H. Simple estimates made by balancing thevan der Waals attraction of the D2 substrate to a D atom [96] against the repulsive forcedue to the increasing £-4He density near the substrate [35, 61] seem to indicate that theD atom is bound to the surface. Similar estimates made for H atoms suggest that the Hatom is not bound. These calculations are rather speculative since the solidification ofthe £-4He near the substrate is likely to play an important role. For this reason we donot present the details of this calculation here.Details of the diffusion of D on the substrate have not been studied systematicallyhowever a few observations are worth noting. If the sample bulb is cooled slowly (10 to20 K/hour) below 20 K, the delayed source of H atoms occurs within 50 to 100 secondsof the discharge. We also note that the lower the temperature at which the data is taken(and hence the slower the D solvation process) the further the peak is delayed. Fastercooling of the bulb seems to delay the release of H from the surface for up to severalhundred seconds. In the later case the fast cooling rates may result in a more convolutedsubstrate. If a ‘cold spot’ were to develop at some point on the walls of the cell duringthe cooling procedure, a large fraction of the D2 might concentrate at this point. Fastercooling rates also tend to result in lower (maximum) H densities at a given dischargepower; this may be a result of less fractionation of the HD during cooling. Under theseconditions we have been able to obtain n >> nH and we find that the integrated numberof atoms from the delayed source is equal to the number of H atoms which are lost dueto H-D recombiriation. This seems to suggest several things. First of all, it would appearthat each H atom which is scavenged from the gas phase is eventually liberated from thesurface by a D atom (in this case). As there are many D atoms which reach the surfaceand yet only a limited number of H atoms which return to the gas phase, it is likelythat there are very few H atoms near the substrate surface immediately following thedischarge. Finally, as the source of H atoms appear to be delayed from the time at whichChapter 7. Experiments with H, D Mixtures at 1 K 131the D atoms pelletrate the liquid, it would seem that the D atoms are at least somewhatmobile on the surface of the substrate. In situations in which the sample was cooledslowly, we do not see the same agreement between the number of H atoms scavenged byD and the number which are eventually released from the surface. Likely there is muchmore HD on or near the substrate in this case. The observation of this delayed source ofH gives important support to our picture of the solvation process in that it shows thatD atoms and not D2 molecules have actually penetrated the film.7.2.3 H-D recombination: Measurements with nH >> nDThe decay of the D atom density due to H-D recombiriation in the cell occurs at the rate:nIJ—= —KHDnH (7.9)If this reaction is predominantly catalyzed by gaseous 4He atoms in the cell (as opposedto atoms in the £-4He walls) then we expect KHD = kHDnHe. In order to study thisrecombination process a bulb was prepared with additional hydrogen added in the formof H2 gas. The isotopic ratio of H atoms to D atoms in the resulting mixture was 0.06.Using this cell it was found that following a discharge pulse at low temperatures, 11H >> nD.Typical densities were nH 5 x 1012 cm3 and n 6 x 1010 cm3. Measurements ofT1 for the H atoms using a r-r/2 pulse sequence were made as a function of time afterfiring the discharge. Data for each series of T1 Ineasuremeilts was fit to the form=aexp(-(t)+ (7.10)In effect the first term in this equation represents 1/Tv while the second is 1/T00. Infigure 7.8 the exponential decay rate is plotted as a function of the H density in thecell for a series of measurements made at constant temperature (and hence constant 4Hedensity). In a later section it is shown that these decay rates are much faster than theChapter 7. Experiments with H, D Mixtures at 1 K 132rate at which D atoms are lost due solvation at this temperature. A straight line fit tothe data gives an H-D recombination rate constant KHD 4.0(2) x 10—14 cm3/s.By changing the temperature of the atomic gas only slightly, the 4He density abovethe £-4He film can be changed quite dramatically. In figure 7.9 the ratio C/nH is plottedas a function of the 4He density inside the cell. Again a linear relationship is observedwhich further indicates that this recombination is occurring in the gas phase catalyzedby 4He atoms (H + D + 4He —* RD + 4Re). From the slope of this plot we findkHD = 2.5(1) x 10_32 cm6/s. We have assumed that kuD is independent of temperatureover the narrow range of temperatures at which data was taken. The value obtainedfor k0 is more than an order of magnitude larger than the corresponding value kHH =0.20(3) x 10_32 cm6/s for H-H gas phase recombination at this temperature [27]. TheH-H-4e recombination problem has been tackled theoretically by Greben et al. [65] withremarkable success.13 No calculations of the H-D rate constant have been made.7.2.4 H-D and D-D recombination: An absolute calibration of riDThe measurement of kuD presented in the previous section was independent of the absoluteD density inside the cell. In this section a second measurement of kuD which does dependon nD is presented. By comparing these two measurements we arrive at an absolutecalibration for the D density or in effect, a measurement of the R-D spin exchangebroadening rate constant UHD.This study was carried out on a cell which was cooled quickly below 20 K. Thisresulted in a much lower H density and further delayed the source of H atoms froni thechemical reaction of equation 7.8 occurring on the substrate. Throughout this experimentwe operated with n >> n where n is tile D density immediately following the discharge.After a typical discharge pulse n 5 x 1011 cm3 while n 5 x 1010 cm3. Tile observed13They calculated a value kHH = 0.18 x 1032cm6/sat this temperature.Chapter 7. Experiments with H, D Mixtures at 1 K 133J .30 i I I I • , •0.28--0.26--.—.. 0.24 --0.22--020--0 18 ---I . I • I I I4e+12 5e+12 6e+12 7e+12H density (cm3)Figure 7.8: The rate constant with which T1 decays following a discharge pulse underthe condition nH >> nD. The data has been plotted as a function of the H density insidethe cell. The slope of the data gives an H-D recombination rate KHH = 4.0(2) x iO’cm3/s at this temperature (1.043(3) K).Chapter 7. Experiments with H, D Mixtures at 1 K 134IllIllIll 111111 IIIIIIII II I4IIIyII I 11111111111 III. I1.2e—13-E 8.Oe—14-C,0C--4.Oe—14-0 I I I . . I a a I • . . . iOe+OO 2e+18 4e+18 6e+18Helium density (cmjFigure 7.9: The rate constant for the D atom density decay, normalized to the Hdensity n11 in the cell and plotted as a function of the 4He density. The fact that thedata lies along a straight line indicates that H-D recombination is occurring in the gasphase catalyzed by 4He atoms (i.e. H + D + 4He — RD + 4He). The rate constantfor this process kHD = 2.5(1) x 10—32 cm6/s is more than an order of magnitude largerthan the corresponding rate constant kHH for H-H recombination reported in [27]. Herecontributions due to the solvation of D into the £-4He are small and can be neglected.Chapter 7. Experiments with H, D Mixtures at I K 135initial drop in nH was more pronounced than in the example shown in figure 7.7, withup to half’4 of the H atoms being lost due to H-D recombination over the first 10 to 30seconds following the discharge. The H density then remained relatively constant untila time of about 200 seconds when the delayed source of H began to appear. n on theother hand initially decayed more rapidly than expected simply due to solvation. Thisexcess decay was correlated strongly with n0 (rather than nH) and was attributed to D-Drecombination.In appendix C the rate equations for the decay of both the H and the D densities inthe presence of solvation (D only) are presented. In the limit where H-D recombinationcan be neglected as a means of reducing n (i.e. KDDnD >> Kn and ,\ >> KHDnH) andH-H recombination can be neglected as a means of reducing 11H (i.e. KHDIIH >. K1111n),these rate equations can be solved analytically. The equations do not include a term forthe delayed source of H atoms; however at times short compared to the reappearance ofthese atoms, the use of this formalism is justified. The key point to be made here is thatin this limit the H and the D densities are analytically related viaKg0n0(t) = (nH(t))KHDexp(—t) . (7.11)This relationship is independent of the absolute atomic densities. Furthermore, the initialdrop in the H density due to the scavenging of H by D is given by= ( (7.12)flwhere TDD is a cl,aracteristic time for D-D recombination:— 1—4GRDT;e(0)Tgg_V o_ V (. )DDIn equation 7.13 we have made use of the fact that ng is proportional to (Tv)’. T(0)is the value of T;e immediately following the discharge. It is evident that the drop in14Less than 5% of the D atoms were lost due to H-D recombination during this period.Chapter 7. Experiments with H, D Mixtures at 1 K 136the initial H density contains information about the number of D atoms which are lostdue to H-D recombination and in effect gives a measure of the calibration between theD density and T’. The simultaneous decays of nH and nD immediately following thedischarge (where the use of the formalism outlined above is justified) were used in orderto determine 1DD and the ratio KHD/KDD as a function of temperature’5.In figure 7.10the ratio T;’(O)/rflD is plotted as a function of the 4He density in the cell. From equation7.13 we see thatT;e(0) 3KDD (7.14)0HD is expected to be essentially independent of temperature near 1 K and should notbe related to the 4He density in any way. Quite obviously KDD is proportional to nHethus D-D recombination is occurring in the gas phase, just as was the case for the H-Drecornbination observed in the last experiment. From the slope of this plot we obtain= 1.9(1) x 1022cm3. (7.15)In addition the analysis yields the ratio KHD/KDD = knD/kDD = 1.03(4) (independent oftemperature). Combining these results with the measured value of kHD, we find kDD =2.4(2) x 1032 cm6/s and HD = 1.7(2) x 10’s cm3/s.The value for the D-D gas phase recombination cross section is essentially the sameas that for H-D recombination. Just as was the case for H-D recombination, there areno known calculations of this quantity. The value GHD = 1.7(2) x 10’° cm3/s is 30%smaller than the value calculated by Reynolds in the DIS limit [22]. If we take theexperimental values as being the correct one, all D densities which were inferred fromthe theoretical result and mentioned up to this point should be increased by 40%. Thethermally averaged H-D spin exchange broadening cross section at 1 K which is implied‘5The solvation rate A is also determined in this analysis. We delay the presentation of these resultsuntil the next section.Chapter 7. Experiments with H, D Mixtures at I K 137I e—’O.3 • • • • . • • • •8e—046e—041—04e—042e—04Oe—t—’OO • • • I • I • • I • I •Oe+OO Ie+18 2e+18 3e+18 4e+18 5e+18Helium Density (cm3)Figure 7.10: The ratio of the H-D spin exchange broadening T immediately following thedischarge pulse to the characteristic time TDD for D-D recombination plotted as a functionof the 4He density inside the cell. T(0)/rDD is proportional to the D-D recombinationrate KOD. The linear relationship between KDD and nHe indicates that D-D recombillationis taking place in the gas phase rather than on the £-4He surfaces. The slope of thisplot relates the the D-D recombination rate to the H-D spin exchange broadening rateconstant. We find 4kDD/3011 = 1.9(1) x 10—22 cm3.Chapter 7. Experiments with H, D Mixtures at I K 138by this result is obtained by dividing by the mean collision speed of the colliding atoms.We find HD = 96(11) A2. This is the only known measurement of this quantity. Notethat the experimentally determined ratio E/ = 2.2(4) x 102 is indeed large as waspredicted in reference [22].7.2.5 Measurement of the solvation energyUp to this point we have dealt with situations in which the H and the D densities werequite different. In this section we present results obtained in the intermediate regimewhere these densities are comparable. The cell used in this work was prepared in thestandard way but cooled slower than in the previous section. Typically n n 10”cm3 following a discharge pulse. In this regime the decay of the D atom density containssmall contributions due to both H-D and D-D recombination which make the decay non-exponential. In order to analyze the data we make use of the measured recombinationrate constants kuD and kDD, the measured spin exchange broadening rate constant GHD,and the H and 4He densities. The rate equation for the decay of the D density (seeappendix C)rnD(t) = —kHDnHenH(t)nn(t) — kDDnHen(t) — AnD(t) (7.16)or equivalently the H-D spin exchange broadening seen by the H atomsd( 1 ( 1 ‘\ 4kDDnHe( 1 N2 1 1 NdtkTy(t)} = —kHDnHenH(t) T(t))—T(t)} — A T(t)} (7.17)is then numerically integrated and fit to the data with the solvation rate A as the onlyfree parameter.In figure 7.11 an example of an extreme case in which both H-D and H-H recombination play an important role in determining the rate at which T changes is shown.The initial D density’6implid by the broadening is n = 2.5 x 1011 cm3. The H densitytm6This data could have equivalently been plotted as n. We choose to plot the measured broadening.Chapter 7. Experiments with H, D Mixtures at 1 K 139is about 1011 cm3 throughout the decay. The dashed line is drawn such that it has thesame slope as the decay due to solvation plus the average rate’7 due to H-D recombination. The amplitude of this line has no significance. It is simply intended as a guidefor the eye. Initially the slope of the decay is steeper than the dashed line due primarilyto D-D recombination. As‘1D drops, only H-D recombination and solvation contributeto the decay of riD and the two lines are essentially parallel. Near the end of this timeperiod the H density drops below its average value and H-D recombination contributesslightly less to the decay of riD. The slope indicated by the data is correspondingly lesssteep than that of the dashed line.Tile solvation rate A determined in this manner for a series of decays spanning the fullrange of accessible temperatures is shown in figure 7.12 (squares). In general the H andD densities were kept lower than tile example outlined in figure 7.11 to try to minimizethe decay of the D density due to H-D and D-D recombination. Also plotted ii this figure(triangles) are the results of our earlier measurements of A made by direct observationof tile D atoms [21]. This earlier set of data has been scaled by the appropriate ratio toaccount for the fact that the A/V ratio of the two cells were different.The first point to be noted is that the agreement between the absolute values of thetwo sets of data is quite good. We find similar agreement between measurements madeon the same cell during different cooldowns and also between different cells. Throughoutthis work we were concerned that the actual area of the £-4He film seen by the D atomscould have been substantially larger than the geometrical area of the bulb. This wouldoccur if the D2 substrate underlying the film had a ‘snow-like’ texture. The data shownin figure 7.12 indicates that the ratio of the area of the £-4He film to the geometrical areaof the bulb appears to have remained constant throughout these studies. Variations in17Recall that the 11 density is also changing due to fl-D recombination and due the delayed source ofH atoms.Chapter 7. Experiments with H, D Mixtures at 1 K 140I I I I I I I I I j I I I I I I I I I I I10.0 —.— S(I)—— S—S—F- SS1.0 —-. .. -.0.1 . I I i . , I . i I a • a I I • i0 25 50 75 100 125Time (s)Figure 7.11: A measurement of T’ as a function of time following a discharge pulse ata temperature of 1.000(3) K. The line passing through the data is a fit to a numericalintegration of equation 7.17 with the solvation rate ) as the only free parameter. Thedashed line indicates the rate at which T’ (or nD) changes due to solvation and theaverage H-D recombination rate. Immediately following the discharge D-D recombinationalso contributes to the decay. Near the end of this time period nH is decreasing and theH-D recombination rate drops below its average value.Chapter 7. Experiments with H, D Mixtures at 1 K 1410.30 —D0.20DU,i:00.10C0.02— • . • . • . .0.82 0.86 0.90 0.94 0.98 1.02I /T (ic1)Figure 7.12: The solvation rate ). at which the D atoms penetrate the £-4He lined wallsof the sample bulb plotted as a function of the inverse temperature. Squares indicatedata obtained in this work using a bulb with A/V = 3.4 cm1. The triangles representdata obtained earlier using magnetic resonance on the /3-6 hyperfine transition of D [21].This earlier data has been scaled to the appropriate A/V ratio.Chapter 7. Experiments with H, D Mixtures at I K 142the quantity of D2 admitted to the cells does not appear to change the absolute value ofthe solvation rate and thus we equate the the area of the £-4He film with the geometricalarea of our cells. The second observation which is readily visible in figure 7.12 is thatslopes of the two sets of data are consistent. Oii the other hand the scatter18 iii thepresent data is much less.The solvation rate data shown in figure 7.12 has been fit to the form’9 (refer to chapter2 for details)A__ (E5N= aig[tvexpkT) (7.18)and replotted in figure 7.13. The free parameters in the fit were E5 and the productIf we assume that t1g is independent of temperature we obtain a value E = 14.0(1)K for the energy required to force a D atom into the £-4He film coating the inner wallsof our cells. This value is consistent with our earlier [21] measurement E5 = 13.6(6) K.The experimental uncertainty associated with E has been reduced considerably in thepresent work. We also find fig = 4.0(5). This represents a remarkable2°improvementover the earlier experiment in which this quantity was uncertain to more than an orderof magnitude. This last result is quite interesting as it sets a lower limit on the effectivemass m* of a D quasi particle in£-4He.2’ The thermally averaged transmission probabilityfor ejecting a D quasi particle from the liquid 1g can be at most 1 and thus m* must beat least 4.0+0.5 times the bare D mass or about 8 mass units. This is the first knownmeasurement of the effective mass of a massive neutral particle inside-4He other than3He. The experimental value for for 3He in £-4He extrapolated to the zero concentration18n light of the recombination rate constants measured during these studies it seems fair to say thatmuch of the scatter in the 309 MHz data was due to H-D and D-D recombination. The remaining scatterwas likely due to fluctuations in the temperature of the cell.‘9Remember that V is a function of temperature.20The relatively small temperature range of the data makes it quite difficult to determine the prefactoraccurately.21The films used in this work are likely thick enough to be treated as bulk £-4He.Chapter 7. Experiments with H, D Mixtures at 1 K 143limit is 2.34(18) [97] (m* for 3He is about 7 mass units).7.2.6 The 4He buffer gas frequency shiftThe 4He buffer gas shift of the a-c hyperfine transition of atomic hydrogen has beenmeasured previously in this laboratory. This shift is more than an order of magnitudesmaller and has the opposite sign from the room temperature value [98]. Hardy andcoworkers report [12]6f —(1.18 x 10’7Hzcm3)nHe . (7.19)This measurement was made using a McLeod gauge to measure the pressure of the £-4He bath. We expect our measurements to be considerably more accurate because ofthe method in which the 4He density was determined (see appendix D). In figure 7.14the frequency of a FID is plotted as a function of‘1He inside the cell. The frequencymeasurement is made against a Rb frequency standard. The slope of this plot indicatesthatSf = —(1.150(2) x107Hzcm3)nue (7.20)which is 3% less than reported in [12, 27]. If we assume that the higher 4He pressuresin the earlier work were correct this discrepancy suggests that the measured values ofpressure may have been out by as much as 10% near 1 K.22The buffer gas shift is quite sensitive to the detailed form of the H-4e potential.Jochemsen and Berlinsky [69] used existing H-4e potentials in order to calculate thisshift and found moderate agreement between their numerical result (0.7 x 10’7Hz cm3)and the experimental value at 1 K. There is certainly room for improvement in thesecalculations.22A temperature error of about 10 mK.Chapter 7. Experiments with H, D Mixtures at 1 K 1440.300.20‘000.080.060.040.020.82 1.021/T (K1)Figure 7.13: The solvation rate data obtained in this work. Uncertainties in the solvationrate are typically about half the height of the symbols. The data has been fit to the form7.18 with E and [t as free parameters. If /1g is assumed to be independent oftemperature we find E 14.0(1) K arid it = 4.0(5). This measurement solvationenergy E is in agreement with our earlier result [21]. If we assume that the ejectionprobability 1g is 1 the effective mass of the D quasi particle implied by this data is 4times the bare D mass.0.86 0.90 0.94 0.98Chapter 7. Experiments with H, D Mixtures at I K 145760 , • • . . . . • . . , • . . . . • • • • ‘ • • • • • • •750-740-NZIZ>%0Ca)0 -a)710-700-690 • • . . I . r i I . i I . . . I . . . • .0 1 2 3 4 5Helium Density (lO18cmjFigure 7.14: The 4He buffer gas shift measured in this work. Uncertainties in the frequency and the 4He density are discussed in appendix D. They are insignificant on thisscale. The slope of this plot implies a pressure shift coefficient of -1.150(2)x io’ Hzcm3. A frequency of 1420.405 MHz has been subtracted from the measured frequency ofthe FID’s.Chapter 7. Experiments with H, D Mixtures at 1 K 1467.2.7 Frequency shift due to H-D collisionsDuring the study of H, D mixtures at 1 K, a shift in the frequency of the a-c hyperfinetransition of H proportional to the D density in the cell was observed. In figure 7.15 thisfrequency shift is plotted as a function of the absolute D density in the cell (determinedfrom the measured value of‘HD). The maximum shift due to H-D collisions observed hereis very small in comparison to the temperature dependent (i.e. 4He density dependent)buffer gas shift. The error bar which is drawn on this plot indicates the frequency shiftwhich would be expected from a 1 mK change in the temperature of the cell. The successof this measurement relied heavily on the temperature stability of our experimental cell.It has already been mentioned that when the number of zero crossings in a FIDbecome small, there exists the potential to misinterpret the broadening due to Tje as afrequency shift during the numerical fitting procedure. This point is discussed furtherin appendix A. The data presented in figure 7.15 has been checked carefully to correlatethe frequency shift with the zero crossings in the frequency data. This frequency shift isnot an artifact of the fitting procedure.Crampton [71] has shown that in the DIS limit, the a-c hyperfirie transition of H shouldnot be shifted by H-D spin-exchange collisions. This does not mean that a fully quantummechanical treatment of this problem will not lead to such a shift. Unfortunately thesolution of this problem is rather involved. It has only been performed for the H-Hspin-exchange problem [19, 20] and thus there is no framework in which to analyze thisdata.To proceed further we define an H-D spin-exchange frequency shift cross section bySf )iHDVIIDnD (7.21)where VHD is the mean relative velocity between H and D atoms in the gas. The slopeof the data plotted in figure 7.15 is —1.15(3) x 102Hz cm3 at 1.017(3) K. This in turnChapter 7. Experiments with H, D Mixtures at 1 K 1470 . • . . . . • . • • • • •—50NE—200—250i . I • I • • • • I • • •0.0e00 5.Oe+10 1.Oe+11 1.5e+11 2.Oe+11D Density (cm3)Figure 7.15: The frequency shift of the a-c hyperfine transition frequency of H attributedto H-D spin exchange collisions at 1.017(3) K. The frequency shift has been plotted as afunction of the D density inside the cell inferred from the measured H-D spin exchangebroadening rate constant. The slope of the data indicates that the thermally averagedspin exchange frequency shift cross section (defined by equation 7.21) is )HD = —1.1(2)A2. The ratio‘HD/HD = —1.21(3) x 10—2 is determined independently of any othermeasurement. The error bar drawn on the right hand side of the plot indicates themagnitude of the 4He buffer gas frequency shift that would be expected from a 1 inKchange in temperature.Chapter 7. Experiments with H, D Mixtures at 1 K 148implies that )HD -1.1(2) A2. Note that in essence, the data in figure 7.15 is a measureof the ratio of the H-D spin exchange frequency shift cross section for the a-c transitionof H (defined by equation 7.21) to the corresponding broadening cross section. This ratiois determined independently of any other parameters. We find that at 1.017(3) K, thisratio is:= 1.21(3) x 10_2 (7.22)HDChapter 8Discussion and Conclusions8.1 SummaryThe unifying theme throughout this thesis has been the study of spin-exchange collisionsand their perturbing influence on the a-c hyperfine transition of atomic hydrogen (H)at 1420 MHz in zero magnetic field. During the first set of experiments we used acryogenic hydrogen maser (CHM) as a very sensitive probe of this transition, to studyspin-exchange collisions between H atoms at 0.5 K. This technique allowed us to observe,effects induced by hyperfiie interactions acting during these collisions. The implicationsof this first study are of considerable technological interest since the frequency shiftsinduced by these collisions will most certainly impose very strict design constraints onattempts to improve the frequency stability of the CHM. At the same time, discrepanciesbetweell our results and a theory [19, 20] which attempts to predict these frequency shiftsmay suggest that a reevaluation of the H-H iiteratomic potentials is necessary.During the second set of experiments we examined the influence of spin-exchangecollisions between hydrogen and deuterium (D) atoms on the a-c hyperfine transition ofH at temperatures around 1 K. The key to the success of these experiments was the factthat the broadening cross section for H-D collisions is more than two orders of magnitudelarger than the corresponding cross section for H-H collisions at this temperature. Pulsedmagnetic resonance techniques were used to monitor the density of H atoms in a gaseousmixture of H and D. The broadening of the transition was almost wholly determined by149Chapter 8. Discussion and Conclusions 150H-D spin-exchange collisions and in effect gave a simultaneous measure of the density ofD atoms in the mixture. This allowed us to study for the first time interactions betweentwo isotopes of H in zero field at cryogenic temperatures. Measurements were made of thebroadening and frequency shift cross sections for H-D spin-exchange collisions as well asthe gas phase recombination rate constants for H-D and D-D recombination. All of theseparameters are quite sensitive to the detailed form of the interatomic potentials. Theyshould certainly be useful in any future attempts to reevaluate the various potentialsinvolved. These techniques also allowed us to study the solvation of D into £-4He. Ourfindings make important contributions to the experimental picture of solvation [22, 21]and provide an improved measurement of the energy required to force a D atom into£-4He. The results also impose a lower bound on the effective mass of a D quasi particlein £-4He.8.2 Spin-exchange measurementsThe study of H-H spin-exchange induced frequency shifts in the CHM was made by independently varying the resonator Q, detuning, and the atomic density, while monitoringthe oscillation frequency of the maser. The deviation of this frequency from the densityindependent frequency of the hyperfine transition was analyzed in the context of a theorydeveloped by B. J. Verhaar and his collaborators [19, 20]. In this theory the influenceof the hyperfine interaction has been rigorously included in the calculation of the H-Hspin-exchange frequency shifts and broadening parameters for this transition. We had nomeans by which to independently vary or control the occupation of individual hyperfinestates inside the maser bulb. As a consequence we were able to distinguish only three’parameters (or combinations of parameters) of the theory. Experimental and theoretical1out of a total of 5 parameters which are believed to be of any consequence for our CHMChapter 8. Discussion and Conclusions 151results for the parameters2Xi (Pcc + paa) + 2i(Pcc+paa) +O2are summarized in table 5.1 of chapter 5. Implicit in our analysis is the assumption thatPcc + p is independent of the atomic density in the maser bulb.In the absence of any hyperfine induced effects during H-H spin-exchange collisions,ii (pcc + p) + ).2 is expected to be zero. The dimensionless ratio;\1(pcc+p)+’\21(Pcc+Paa)+2is a measure of the contribution of hyperfine interactions to the total spin-exchangefrequency shift. Our measurements indicate that Q = —0.055(35). The magnitude ofthe observed effect is in agreement with the theoretical predictions made by Verhaar etal. We interpret this as a signature of hyperfine interactions during H-H spin-exchangecollisions. This measurement represents the first observation of this effect at cryogenictemperatures.The technological implication of this measurement is rather sobering. The early 1980’ssaw the development of a good deal of enthusiasm based on the potential frequencystability of a H maser operating at cryogenic temperatures. The theory developed byVerhaar et al. was first published in 1987 [19], soon after the first CHM’s were realized[8, 9, 10]. It bore with it the unsettling revelation that hyperfine interactions coupledthe oscillation frequency of a H maser to the atomic density in a way that dependsupon the individual3hyperfine level populations. Theoretically the dominant thermally2Recall that Paa and Pcc are two of the diagonal elements of the H density matrix.3Jn the semi-classical approach to this problem in which hyperfine effects are neglected, the oscillationfrequency is only coupled to the population difference between the I a) and the c ) states.Chapter 8. Discussion and Conclusions 152averaged frequency shift rate constant ).OVflH for H-H spin-exchange collisions [19, 20]is smaller at 0.5 K than it is at 300 K by a factor of two. On the other hand, thetypical atomic densities in a CHM are several orders of magnitude higher than those in aconventional room temperature H maser. The resulting potential frequency instabilitiesdue to fluctuations in the level populations are correspondingly greater at 0.5 K thanat 300 K. Our measurements form an experimental verification of the effects predictedby Verhaar et al. It remains to be seen whether or not any new and innovative CHMdesigns4 can attain significant improvements in frequency stability.The results summarized in table 5.1 are interesting from another point of view aswell. While there is qualitative agreement between the magnitudes of the measured andthe theoretical values for the various parameters, discrepancies are obvious. The moststriking of these is the fact that the measured sign of the term >i (Pcc + Paa) + ‘2 ispositive while the numerical calculations5indicate that it is negative. The parameters ofthe theory are quite sensitive to the detailed form of the atomic potentials used in thecalculations. These discrepancies will likely form an important indicator in any futurerefinements of these potentials. They also may be an indication that a reevaluation ofthe role of hyperfine interactions during spin-exchange collisions is warranted. Certainlycorroborative experimental evidence6 is required before such steps are taken.A few words of caution are required at this point. The most serious drawback of ourexperiment was the fact that we simply had no wayto monitor the occupancy of theindividual hyperfine states in the maser bulb nor the actual atomic density while themaser was oscillating. Our results rely on a computer simulation [17, 90] in order to4A full treatment of the design constraints imposed by these frequency shifts is far beyond the scopeof this thesis. Cursory discussions of these requirements can be found in references [19, 20, 17].5The numerical calculations indicate that this term is negative over a very broad range of temperaturesregardless of the value chosen for p + Paa.6The only other study of fl-H spin-exchange at these temperatures was a measurement of the broadening cross section at 1.1 K made in this laboratory[27]. No attempt was made to investigate hyperfineinduced effects. The results of this earlier study are consistent with our findings.Chapter 8. Discussion and Conclusions 153infer this information from measurements made while the maser was not operating. Wehave tried to place reasonable uncertainties on these estimates; however their accuracyis difficult to verify. We do note that unlike the other combinations of parameters, thedetermination of ) is only weakly dependent upon the actual atomic density in themaser. The second uncertainty which was of some concern during this experiment wasthe establishment of the absolute resonator detuning. We have placed what we believeto be generous limits on this uncertainty.Currently we are aware of two proposals to study the role of the hyperfine interactionduring H-H spin-exchange collisions in a similar manner. If successful, these investigationswill provide additional information near7 10 K and near8 300 K. The results of thesestudies are certainly eagerly awaited. They should form an essential complement to ourwork.During our second set of experiments with H, D mixtures at 1 K we were able tomeasure the frequency shift and the broadening of the a-c hyperfine transition of Hdue to H-D spin-exchange collisions. The sensitivity of this experiment was in no waycomparable to that of the CHM experiment and hence no attempt was made to directlyaccount for effects induced by hyperfine interactions. On the other hand, these resultsdo represent an interesting complement to the H-H spin-exchange measurements.The H-D spin-exchange broadening cross section is 2.2(4) x 102 times as big as thecorresponding cross section for H-H collisions9 [27] at 1.1 K. In chapter 7 we notedthat a discrepancy between the experimental and the only theoretical value [22] for the7Dr S. B. Crampton (Williams College, Williamstown MA, USA) has proposed a study of spin-exchange collisions in a H maser with neon coated walls operating near 10 K.8Drs R. L. Walsworth and R. F. C. Vessot (Harvard-Smithsonian Center for Astrophysics, CambridgeMA, USA) have proposed a new study of H-H spin-exchange collisions using room temperature H masers.9The amplitudes of the spin-exchange cross sections depend upon the phase shifts introduced by thesinglet and the triplet molecular potentials. It is the H-H cross section which is anomalously small atlow temperatures. For convenience a tabulation of the ro-vibrational levels of the H2 and RD moleculesclose to the dissociation limit is given in appendix E. The calculation of the spin-exchange cross sectionsare sensitive to the exact placement of these levels.Chapter 8. Discussion and Conclusions 154broadening cross section does exist. This is likely an indication that the long range partof the potential used in the calculation of the cross section needs to be modified. Themeasurement of a frequency shift due to H-D spin-exchange collisions is an interestingobservation. This shift is not predicted by the semi-classical theories of spin-exchange[71]. Unfortunately the H-D spin-exchange problem has not been treated in the samefully quantum mechanical manner as the H-H spin-exchange problem and there are notheoretical estimates with which to compare our result. The H, D system examined inthis thesis may prove to be’° an important testing ground for the formalism used byVerhaar et al. [19, 20] to calculate the H-H cross sections.8.3 Recombination measurementsDuring the experiments at 1 K, measurements were made of the rate constants for HD and D-D recombination catalyzed by 4He atoms. Interestingly enough, both rateconstants are about an order of magnitude larger than the rate constant for the H-Hreaction [27]. This observation is consistent with the relative abundance of low angularmomentum molecular levels close to dissociation (the spectrum of the D2 molecule neardissociation is also tabulated in appendix E). The H-H recombination rate constant atthis temperature has been calculated with remarkable success11 [65]. Similar calculationsfor the H-D and D-D reactions have not been performed though they would certainly beenlightening. This is especially so given the apparent discrepancy between the calculatedand the measured H-D spin-exchange broadening cross section discussed earlier.A rather interesting comparison can be made between our results for D-D recombination and those of Mayer and Seidel [60]. Using ESR they set a lower bound on the surface10lnterest in examining the H-D system theoretically has been expressed by the Eindhoven group ledby Professor Verhaar : private communication.11The rate constant reported in references [37] and [12] were too high by a factor of The properexperimental value is reported in reference [27].Chapter 8. Discussion and Conclusions 155D-D recombination rate at 0.5 K in a magnetic field of 0.3 Tesla. They then scaled theirresult to set a lower bound of 300 A on the effective cross length for the surface catalyzedtwo body D-D recombination reaction in zero magnetic field. Their result is consistentwith the data of Silvera and Walraven [47] which was scaled in the same manner. Thisscaling procedure’2has been used moderately successfully in the case of H to relate highfield recombination data [100, 101, 102] to low field data. The recombination cross lengthobtained for D however, seems anomalously large in comparison to the corresponding H-H cross length. In zero field the H-H cross length is 0.14(3) A [27]. There has been somecontroversy regarding the reason for this difference with speculations as to the possibilityof additional recombination channels for D [49]. If we scale Mayer and Seidel’s resultsto our conditions using a binding energy of 2.6(4) K for D on £-4He we would expectto observe a two body recombination rate constant of > 3 x i0—’ cm3/s at 1 Kdue just to recombination on the surface. We measure13 KDD = 3 x cm3/s at thistemperature. This is an order of magnitude smaller than expected based on Mayer andSeidel’s data! Furthermore we find that the D-D recombination above 1 K appears tobe occurring in the gas phase rather than on the surface. This result either implies thatthe surface recombination cross length is very temperature dependent or it casts furtherdoubt on the validity of scaling high field surface recombination rates to zero field in thecase of D. Applying the converse argument to the comparison of these two experimentsour results place an upper limit of 30 A on the D-D surface recombination cross lengthin zero field. As we have not seen an obvious indication of surface recombination in ourdata, the actual zero field cross length is likely to be somewhat shorter than this. Thislimit is much more in keeping with the H-H cross length.‘21n the case of H, the recombination rate scales as the admixture coefficient between the a ) stateand the I b ) state in high field [99]. For 0 there are two such coefficients due to the larger nuclear spin.‘3This information is taken from figure 7.10 of chapter 7 where we have used the measured H-D spinexchange rate constant GHD to obtain the absolute rate constant.Chapter 8. Discussion and Conclusions 1568.4 Solvation measurementsOur study of H, D mixtures at 1 K has allowed us to make an improved measurement ofthe energy required to force a D atom into £-4He. We determined E = 14.0(1) K. Theimprovement over our earlier report E = 13.6(6) K [21] made by direct observation of Dusing magnetic resonance on the D atom, is due primarily to the fact that we were able toaccount for H-D and D-D recombinatiori as alternate mechanisms by which the D atomdensity inside our cells decayed. The theoretical calculation of this quantity by Kürtenand Ristig [35] is in fair agreement with our results if we assume that the appropriate£-4He density is that corresponding to zero pressure in their model [21]. If we use thetrue £-4He density, their calculation results in a value of E which is too high by morethan a factor of two.Two ancillary observations which were made during our studies are of considerableinterest. The first of these was the inferred chemical-exchange reactionD-j-HD--* D2-j-Hoccurring on the substrate underneath the £-4He films in our cells. This is an indicationthat the D atoms which ‘disappear’ during the pulsed magnetic resonance experimentactually reach the substrate: The second observation was that the roughness of the D2substrate did not appear to enhance the effective area to volume ratio of our cells. Thissecond observation combined with our improved solvation rate data has allowed us toset a lower bound on the effective mass of a D quasi particle inside liquid helium. Wefind m*/m > 8(1) (m is the proton mass) which can be compared to a value of m*/mp= 7.0(5) for 3He in £-4He in the limit of zero concentration [97]. This is the first timeinformation regarding the effective mass of a neutral atom in £-4He other than 3He hasbeen obtained. We only claim that our measurement is a lower bound as we do not knowthe actual thermally averaged probability that a D atom with kinetic energy> E strikingChapter 8. Discussion and Conclusions 157the £-4He surface will enter the liquid (likely an angle dependent problem). 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Allison [80] has calculated)(308K) = 4.0A2[80] A. C. Allison. Spin-change frequency shifts ill H-H collisions. Phys. Rev. A,5(6):2695, (1972).[81] W. N. Hardy and M. Morrow. Prospects for low temperature h masers using liquidhelium coated walls. Journal de Physique, 42(12):C8—171, (1981).[82] S. B. Crampton and H. T. M. Wang. Duration of hydrogen-atom spin-exchangecollisions. Phys. Rev. A, 12(4):1305, (1975).[83] A. C. Maan, H. T. C. Stoof, and B. J. Verhaar. The cryogenic H maser in a strongB field. Phys, Rev. A, 41(5):2614, (1989).Bibliography 167[84] J. M. V. A. Koelman, S. B. Crampton, H. T. C. Stoof, and 0. J. Luiten. Frequencyinstability of cryogenic and room temperature hydrogen masers. In Proc. 3rd Jut.Conf on Spin Polarized Quantum Systems, (1989).[85] Martin D. Hürliamnn,A. John Berlinsky, Richard W. Cline, and Walter N. Hardy.A recirculating cryogenic hydrogen maser. IEEE Trans. on Instr. and Measurement, IM-36(2):584, (1987).[86] W. N. Hardy and L. A. Whitehead. 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Rev., 102(2):304, (1956).Bibliography 168[94] Tetsuo Miyazaki and Kwang-Pill Lee. Direct evidence for the tunneling reactionHD + D —* H + D2 in the radiolysis of a D2-HD mixture at 4.2 K. J. Phys. Chem.,90:400, (1986).[95] Kwang-Pill Lee, Tetsuo Miyazaki, Kenji Fueki, and Kenji Gotoh. Rate constantfor tunneling reaction D2 + D —* D + D2 in radiolysis of D2-HD mixtures at 4.2and 1.9 K. J. Phys. Chem., 91:180, (1987).[96] L. Pierre, H. Guignes, and C. Lhuillier. Adsorption states of light atoms (H,D,He)on quantum crystals(H2,DHe,Ne). J. Chem. Phys., 82(1):496, (1985).[97] Ray Radebaugh. Thermodynamic properties of HE3-H4 solutions with applications to the HE3-H4 dilution refrigerator. In NBS Technical Note 362. USDepartment of Commerce, (1967).[98] 0. Das, A. F. Wagner, and A. C. Whal. Calculated longe-range interactions andlow energy scattering in He+H, Ne+H, Ar+H, Kr+H and Xe+H. J. Chem. Phys.,68:4917, (1978).[99] Isaac F. Silvera. Spin-polarized hydrogen and deuterium: Quantum gases. InPhysica, volume 109 & 11OB, page 1499, (1982). Proc. 16th mt. Coiif. on LowTemperature Physics.[100] R. W. Cline, T. J. Greytak, and D. Kleppner. Nuclear polarization of spin-polarizedhydrogen. Phys. Rev. Lett., 47:1195, (1981).[101] R. Sprik, J. T. M. Walraven, G. H. van Ypereii, and Isaac F. Silvera. State dependent recombination and suppressed imclear relaxation in atomic hydrogen. Phys.Rev. Lett., 49:153, (1982).Bibliography 169[102] B. Yurke, J. S. Denker, B. R. Johnson, N. Bigelow, L. P. Levy, D. M. Lee, andJ. H. Freed. NMR-induced recombinatiori of spin polarized hydrogen. Phys. Rev.Lett., 50:1137, (1983).[103] J. C. Slater. Microwave electronics. Rev. Mod. Phys., 18:441, (1946).[104] D. J. Wineland and N. F. Ramsey. Atomic deuterium maser. Phys. Rev. A,5(2):821, (1972).[105] R. A. Waidron. The Theory of Waveguides and Cavities. Gordon and BreachScience Publishers, New York, 1967.[106] J. C. Arnato and H. Herrmann. Improved method for measuring the electric fieldsin microwave cavity resonators. Rev. Sci. Instrum., 56(5):696, (1985).[107] L. C. Maier and J. C. Slater. Field strength measurements in resonant cavities. J.Appl. Phys., 23(1):68, (1952).[108] C. Kittel. Introduction to Solid State Physics. John Wiley and Sons, 5th edition,1976.[109] J. A. Osborne. Demagnetizing factors of the general ellipsoid. Phys. Rev., 67(11and 12):351, (1945).[110] J. Wong. Filling Factor of a Microwave Resonant Cavity. Undergraduate thesis,University of British Columbia, (1989).[111] H. M. Roder, R. D. McCarty, and V. J. Johnson. Liquid densities of oxygen,nitrogen, argon and para hydrogen. In NBS Technical Note 361 - Revised. USDepartment of Commerce, (1972).Bibliography 170[112] Victor J. Johnson, editor. US National Bureau of Standards: The Properties ofMaterials at Low Temperatures. Pergammon Press, (1959).[113] G. Föex. Diamagnétisme et paramagnétisme. In Constantes Se1ectioné’es Volume7. Masson & Cie, Paris, (1957).[114] B. A. Younglove. Measurements of the dielectric constant of liquid 02. In K. D.Timmerhaus, editor, Advances in Cryogenic Engineering: Proc. 1969 Conf. onGryogenic Engineering, volume 15, New York, (1989). Plenum.[115] A. P. Brodyanskii and Yu A. Freiman. The short range structure and properties ofliquid oxygen. Soy. J. Low Temp. Phys., 12(11):684, (1986).[116] I. S. Gradshteyn and I. M. Ryzhik. Table of Integrals, Series, and Products. Academic Press, Toronto, 1980.[117] D. A. Bonn, D. C. Morgan, and W. N. Hardy. Split-ring resonators for measuring microwave surface resistance of oxide superconductors. Rev. Sci. Instrum.,62(7):1819, (1991).[118] John F. Cochrane and D. E. Mathoper. Superconducting transition in aluminum.Phys. Rev., 111(1):132, (1958).[119] J. Bardeen, L. N. Cooper, and J. R. Schrieffer. Theory of superconductivity. Phys.Rev., 108:1175, (1957).[120] G. Herzberg. The dissociation energy of the hydrogen molecule. J. Mol. Spectry.,33:147, (1970).[121] Robert J. Le Roy and Margaret G. Barwell. Ground state D2 dissociation energyfrom the near-dissociation behavior of rotational level spacings. Can. J. Phys.,BibJiogrpy 17153:1983, (1975)f122j I. DabrowskI The Ly11 and Werner bands of H2 Can. J. PhyS. 62:1639, (1984)(123] H. Bredohj and 0. Rerzberg The Lynan and Werner bands of deuterium Can.J. Phys, 1(9):867, (1973)1124] 1. Dabrowskj and C. Rerzberg The absorptjo and emission spectra of RD in thevacuum ultraviolet Ca Phys 54:525, (1976)(125] W. C. Stwally The dissociation energy of the hydrogen molecule Using longraIgforces Chem Phys Lett., 6(3):241 (1970).Appendix ARadiation Damping and the FIDWe have made repeated use of the free induction decay (FID) response of a gas of hydrogen atoms (H) to a 7r/2 tipping pulse at the a-c hyperfine transition of H to measurethe spin relaxation times of the atomic system. In this appendix we discuss the formalism used to model the evolution of the effective magnetization of the atomic system.Throughout this discussion we make use of the spin - analogy introduced in chapter 2.The free evolution of the net magnetization of a spin system in a magnetic field is astandard topic in most books on magnetic resonance [25]. One finds that in many situations the magnetization M evolves according to the phenomenological Bloch equations:d Mk + — M0M=-y(MxB)—m— m (A.1)12 11where M0 is the thermal equilibrium magnetization in the static magnetic field and 7 isthe gyromagnetic ratio of the spins. In the spin . analogy for H described in chapter 2,7e = 7e + or and M is equated with M. T2 aild T1 are the well known transverseand longitudinal (‘spin-lattice’) relaxation times. T21 characterizes the rate at whichspins become dephased with respect to each other. T1’ is a measure of the couplillgbetween the spin system and some external bath of spins (the ‘outside world’).If we start with a system in thermal equilibrium and momentarily perturb it byapplying a ir/2 tipping pulse, the thermal equilibrium magnetization M0 is rotated intothe c — plane and becomes a precessing transverse magnetizationMr = M* + (A.2)172Appendix A. Radiation Damping and the FID 173withM = Mr cos(L)t)M = Mr sin(wt)which eventually relaxes back to M0 according to equations A.1. These precessing atomsradiate power which can be detected by coupling the atomic system to an electromagneticresonator. In the regime of atomic densities and resonator Q’s encountered ill our work,the coupling between the radiating atoms and the resonator is strong enough that theatomic precession following a ‘r/2 pulse can no longer be considered ‘free’.’ That is, thespins feel, and begin to precess in, their own radiation field. This effect was studied indetail by Bloom [301 and is commonly referred to as radiation damping.A.1 Electrical model for the resonatorThe calculation of the rf magnetic field produced by an oscillating magnetization inside aresonant cavity was first done by Slater [103]. The analysis of the coupling between thesespins and the resonator is considerably more transpareilt than the original derivation ifwe use lumped circuit elements to model the resonator. This procedure was first outlinedby Wiieland et al. [104]. The results are equivalent to those obtained by Slater.We assume that near resonance the resonator and the exterilal electronics (as seenby the atomic system) can be modeled by the series LCR circuit shown in figure A.1.The precessing magnetization of the atomic system is coupled to the oscillating magneticfield in the inductor L. Physically the oscillating field in the inductor (or the split-ringresonator) is loiigitudinal. In the spin analogy for the H atom this field is mappedonto the transverse plane and is written H. The external circuitry (with characteristic‘In spite of this coupling we continue to refer to the response of the atomic system following a ir/2pulse as a FID.Appendix A. Radiation Damping and the FID 174impedance Z0) is coupled to the resonator via an ideal transformer. The impedance Zseen looking into the transformer is= 2 [R+J (WL- -)j. (A.3)The coupling parameter= n2R(A.4)is equal to one at critical coupling.Using Faraday’s law and Ampere’s law the voltage induced across the inductor canbe written2:V = L (H + M) (A.5)where is the filling factorfHdVbfH2dV (A.6)and Vb and V are the volumes of the sample bulb and the resonator. When the voltagesaround around the circuit in figure A.1 are summed up and differentiated with respectto time we obtain—--M + + wH (A.7)where1= (A.8)wcA(A.iO)This is the classical equation which couples the field in the resonator to the magnetizationof the spin system and is essentially the result derived by Slater.2L is assumed to remain unchanged by the presence of the atomic sample.Appendix A. Radiation Damping and the FID 175Cz1 H______L_n:1__turnsratioFigure A.1: The lumped parameter circuit used to model the coupling between the atomicsystem and the detection circuitry.Appendix A. Radiation Damping and the FID 176The response time of the resonators used in this work are much shorter thanany of the relaxation times associated with the spin system. Thus the ringing field Bdue to the radiating spins can be assumed to respond instantaneously to changes in themagnetization. The steady state solution to equation A.7 for the oscillating field (ane_jt time dependence is implicit) due to the spins isB = = Re{KM} (A.11)where the complex coupling constant KK = K’+jK”is given byK= [oQt()] (z- j) (A.12)aid= Q ( - 2Q L) (A.13)is a measure of the detuning of the resonator from the free angular precession frequencyof the magnetization.A.2 Radiation damping equationsFollowing a ir/2 pulse the precessing transverse magnetization A.2 produces a ringingfield B which is related to the magnetization via equation A.11. This field can can bedecomposed into components rotating and counter rotating with respect to Mr. Thecounter rotatiig field does not make a contribution to radiation damping and can beneglected. The next step is to transform into the frame ‘-‘ rotating with angularvelocity w0 = ‘-yB0 and in the same sense as Mr, and use the constraint A.11 to eliminatethe magnetic field in the transformed equivalent of equations A.1. The algebra involvedAppendix A. Radiation Damping and the FID 177in this transformation and substitution is straightforward. We omit the details. Theresulting equations are:= -K”M— MZTM0 (A.14)Mt = — -M [K”M + K’M1] (A.15)M1=—+ -M [K’M’ — K”Mt] (A.16)whereM1 Mr C05 [(Ci) — wo)t] (A.17)= Mr 5fl [(w — wo)t] (A.18)are the transverse components of M in the rotating frame. These equations describethe evolution of the spin magnetization in the presence of radiation damping. Limitingapproximations to these equations (such as the case where both T1 and T2 are infinitelylong) have been studied previously [27]. In general, for work with H in which spin-exchange collisions cause T1 and T2 to be comparable to the apparent decay time of theFID, these simplifications are not justified.It is worth examining these equations further in order to get a feeling for the therelevant parameters of the theory. Substituting A.17 into A.15 and A.18 into A.16 theequations involving the transverse magnetization can be rewritten31 17MK” A19Mrdt r T2 2w—w0=—-MK’. (A.20)When the resonator is tuned close to the atomic resonance (i < 1) K’ is small and thereare essentially 110 frequency pulling effects due to radiation damping. The precessionis fully described by the remaining equations A.14 and A.19. These are in effect the31n writing these equations we make the assumption that 0.Appendix A. Radiation Damping and the FID 178equations that were studied by Bloom [30]. The instantaneous damping rate for thetransverse magnetization= ldMr (A.21)is obtained from equation A.19. If the coupling K” between the resonator and the spinsis turned off, T2 processes lead to an exponential decay of Mr following a ir/2 pulse. Thisdecay is accentuated as the coupling is turned back on. This additional damping is dueto the precession of the magnetization about its own radiation field. The rate at whichthis additional damping is incurred depends upon the coupling, 1/T2 and in part 1/T.A.3 Data analysisKeep in mind that the quantity which is measured in these experiments is the voltageamplitude of the rf power coupled from the atomic system, via the resonator, to theexternal circuitry. It is relatively simple to show [25] that the power which is detected is— ttowQoVbM2 (A 22)- 4(1+) rThe FID amplitude is proportional to P and is thus proportional to Mr. The constant ofproportionality consists solely of measured parameters and hence the FID represents anabsolute measurement of Mr cos [(w—Ll.’Lo)t + ] where WLO w0 is the angular frequencyof the local oscillator (LO) of the spectrometer against which the FID is compared. 475 isa phase angle which accounts for the relative phase between the precessing spins and theLO.The FIll data is fit to a numerically integrated solution of equations A.14, A.15, andA.16 with the initial conditions appropriate to a ir/2 pulseM’(0) = M0 cos(gS)M(0) = —M0sin(475)Appendix A. Radiation Damping and the FID 179M(O) = 0where the thermal equilibrium magnetization isM0 = h(’ye + ‘yp)n. (A.23)The precession frequency of the rotating frame is that of the LO rather than in thiscase. The free parameters of the fit are T1, T2, w , and the amplitude of the FID(or in effect via equations A.22 and A.23, the H density).A note of caution is required regarding the determination of T1 from the FID datain this manner. As the effective damping time of the FID becomes comparable to, orshorter than the period of the beat frequency between the LO and the atomic precessionit becomes difficult to determine T1 precisely. The T1 information is in the data aspart of the rate at which the instantaneous damping of the FID is changing. This rateinformation is easily confused with an apparent frequency shift as the number of zerocrossings in the FID data becomes small. Care has been taken throughout the dataanalysis to ensure that the zero crossings are well represented by the fits.Appendix BMeasurement of Filling FactorsIn this appendix we discuss the measurement of the filling factor which was introducedin chapter 2. Two techniques for measuring this quantity are discussed; both are cavityperturbation measurements in which some perturbing body is introduced into the electromagnetic resonator so as to distort the rf field pattern. The change in the resonantfrequency of the resonator is then used to infer the rf field strength at the site of the perturbation. A general discussion of these types of measurements can be found in reference[105].The first technique has been used previously to study rf electric fields ill microwavecavity resonators for particle accelerator applications [106] and rf magnetic fields in split-ring resonators [27]. It is based on a technique developed by Maier and Slater [107]. Thesecond is a new technique which we have developed in the context of the low temperaturestudies of atomic hydrogen presented in this thesis. This measurement is introduced inthe second part of this appendix. A detailed description of this technique including adiscussion of its adaptability to other applications will be reported elsewhere.Before reviewing cavity perturbation measurements, an important distinction betweendielectric media and magnetic media must be pointed out. Dielectric materials can beused to distort rf electric fields without distorting the rf magnetic fields. In general it isdifficult to find materials which will distort rf magnetic fields and not rf electric fields.Consequently it is easier to study rf electric fields by cavity perturbation methods thanit is to study rf magnetic fields. The latter measurement must be performed in two180Appendix B. Measurement of Filling Factors 181steps. First a magnetic substance must be introduced to the cavity to distort both therf magnetic and electric fields. A second purely dielectric body must then be introducedto determine the contribution of the dielectric properties of the first body to the totalperturbation.B.1 Cavity perturbation measurementsCavity perturbation measurements usually fall into one of two categories [105]. If thelocal change in the material parameters (electric and magnetic susceptibilities) during themeasurement is large, the volume of the perturbing bodies must be small. If on the otherhand the change in these parameters is small, the total volume which the bodies mayoccupy can be large. We refer to the first type of measurement as a type I perturbationand the latter as a type II measurement. Morrow’s [27] determination of magnetic fillingfactors falls into the class of type I cavity perturbation measurements. The new techniquewhich we describe later in this appendix is a type II measurement.The distortion of the rf fields inside the resonator due to the insertion of the perturbingmedium must be small enough that the electromagnetic mode structure is not changedsignificantly. In effect, we require the energy which is stored inside the perturbing bodyto be small in comparison to the total energy stored in the resonator in its perturbedcondition. The perturbation condition can be stated in terms of the shift 6w which isobserved in the resonant frequency w of the resonator when the perturbing body isintroduced. In general we require << 1 for the perturbation assumptions to be valid.For both type I and type II perturbation measurements the changes in the fields (E1and H1) and their inductions (D1 and B1) due to the perturbation must be small incomparison to the unperturbed quantities (E0,HD and B0) over most of the volumeAppendix B. Measurement of Filling Factors 182V of the cavity.1 The fractional frequency shift which is observed when a dielectric withmagnetic properties is used to perturb the cavity resonance is2 [105]6w— fv. [(E1 D0 —E0.D1).-(H B0 —H0.B1)]dV (B 1)wo Ivc(Eoofoo)dVwhere V. is the volume occupied by the perturbing body. The first term in this expression represents the dielectric contribution to the frequency shift while the second termrepresents tile magnetic contribution. It is the second term upon which we will focusmost of our attention. Under the proper conditions it can be related to the filling factorwhich was introduced in chapter 2.We begin by writing equation B.1 in terms of the unperturbed fields. It is assumedthat the dielectric constant€ and the permeability i of the perturbing body are knownand homogeneous throughout the material. This allows one to rewrite the electric displacement D and the magnetic induction B in terms of tile fields E and H. The depolarizing field E1 and the demagnetizing field H1 can also be written in terms of E0and H0 with the aid of a well known property of homogeneous ellipsoidal bodies; namelythat exposure to a uniform polarizing or magnetizing field will in turn produce a uniformpolarization or magnetization within that body. If the polarization (or magnetization)along some axis within the body is P (M), then the depolarization (demagnetization)field is related to it by [108, 1091NeP NmMH1=— . (B.2)f[Lowhere Ne (N1) is the depolarizing (demagnetizing) factor in that direction. The materialpolarization (magnetization) on the other hand, exists as a consequence of the total field1For example the perturbed electric field Eo + E1 must be very nearly equal to the unperturbed fieldBo most places within the resonator, and likewise for the other electromagnetic quantities.2The convention that E = E0 exp (jwt) and H = H0 exp (jwt) has been adopted and thus the termsD0 and —H0 . B0 have the same sign.Appendix B. Measurement of Filling Factors 183E=Eo+E1(orH=Ho+Hi)andthus:Ne(1) Nm(ti1) B= + ( — 1) E0 H1 = + N — 1) H0. ( .3)Using these relationships the fractional frequency shift B.1 cab be rewritten in the moreuseful form6w——— 1) f, E . EdV—•(t — 1) f, Ho HdV (B 4)l+Ne(_1)fE0EdV 1+Nin(tt_1)fvHoHjdVAs would be expected, an increase in either of the material parameters E or ,u results ina drop in the resonant frequency of the cavity which is being investigated.B.1.1 Type I perturbationsFor a type I perturbation measurement where the volume of the perturbing body V. ismuch smaller than the volume Vb of the sample bulb used to confine the gasses of atomicH, we define a reduced magnetic filling factor i such that—V, f.,, H . HdVii ( .)vs JV -‘O hoand— fi1(r)dV7m . (.)The rf magnetic fields inside the split-ring resonators used in our work with atomic gassesare homogeneous and uniform3 over the sample volume. In this situation we are justifiedin associating tile true filling factor i with i.Similar quantities er and e can be defined for the electric fields. The frequency shift(equation B.4) can thus be written:6w (E1)77er— (ii1)ii (B7)w0 1+Ne(l) 1+Nm(il)3The sample bulbs consist of two distinct volumes. The main body of the bulb is located withinthe bore of the split-ring resonator. The fields here are uniform and longitudinal. The tails of thebulb extend into regions where the rf field intensity is essentially zero. The volume associated with thetransition region is small and can be neglected.Appendix B. Measurement of Filling Factors 184Both 7ler and i, can be determined at a given position within the resonator by measuringthe fractional frequency shifts for two different perturbing bodies placed in that location.The bodies must have well known but different dielectric and magnetic properties. Bypassing these bodies throughout the volume of interest, the filling factors 1e and m areeasily determined.This technique was used to measure the filling factor of the resonator used in ourstudies of mixtures of H and D described in chapters 6 and 7 [110]. The measurementswere made using small teflon ( 2, = 1) sphere as the dielectric body and a similaraluminum sphere ( = —joo, = 0) as the other. Each sphere was suspended froma fine silk thread which was held taught between mechanical locating devices. Thedielectric constant of the teflon sphere was first measured by perturbing the resonantfrequency of a microwave cavity with a known mode structure. The two spheres were thenpassed (individually) throughout the entire volume of the split-ring resonator normallyoccupied by the sample bulb. During this measuremeilt a cylindrical pyrex sleeve wasused to mimic the walls of the sample bulb. The shift & of the resonant frequency ofthe split-ring resonator was recorded as a function of the position of the spheres. Asthe demagnetizing and the depolarizing factors for a sphere are both equal to it is asimple matter to determine 7ler and at each positioll inside the resonator at whichmeasurements were made. The filling factor is determined by performing the sumindicated by equation B.6.The main drawbacks associated with this technique are that it is quite time consumingto perform the measurement and that often (substantial) changes need to be made to theresonator to allow access to the sample volume. The accuracy of the technique is limitedby the reproducibility of the sample placement within the resonator volume. The newtechnique which is introduced in the following section was developed to eliminate someof these problems.Appendix B. Measurement of Filling Factors 185B.1.2 Type II perturbationsIn contrast to a type I measurement, the perturbing body in a type II measurementoccupies the entire volume of interest.4 This means that detailed information about thelocal field homogeneity is not available. On the other hand the complexity of the sampleand the resonator geometries can be quite intricate.To interpret the frequency shift predicted by equation B.4 in terms of a magneticfilling factor, we again assume that the rf magnetic fields are homogeneous and uniform5over the sample bulb volume. The resulting frequency shift is—•(e— 1) JVb E0 EdV — (B 8)Wo1+Ne(1)JcEOEdV1+Nm(1)We do not make any assumptions about the electric field homogeneity and leave the firstterm in this equation unevaluated. We have implicitly assumed that the frequency shiftsscale as if effective depolarizing and demagnetizing factors6 can be associated with thesample volume.The choice of perturbing media for a type H measurement is very limited. The firsttrick is to find a material with a relatively large permeability it and at the same time arelatively low dielectric constant e within the constraint that both of these parametersmust be close to unity. That is, the frequency shift due to the magnetic properties shouldnot be swamped by the dielectric contribution to the shift. The total shift must of coursebe small. The second trick is to find another material which has essentially the samedielectric properties as the first but which is non-magnetic. The measurement is thenmade by filling the entire sample bulb alternately with each material and measuring thefractional frequency shift of the cavity resonance. As long as the dielectric constants4i. e. the sample bulb volume5As mentioned previously the sample volume can be thought of as being composed of two distinctvolumes. The assumption of field. homogeneity in both regions is confirmed by the type I measurementsdescribed above.6Strictly speaking these quantities are only defined for ellipsoidal bodies.Appendix B. Measurement of Filling Factors 186Temperature Liquid Density [111] — 1— 1(K) (g/cm3)77.35 N2 0.80739 0.4430 [112] -8.7x106 [113](liquid N2) 02 1.2027 0.51716 [114] 3.93x10 [113, 115]87.28 Ar 1.3939 0.519 [112] -8.65x10 [113](liquid Ar) 02 1.1553 0.49411 [114] 3.55x10 [113, 115]Table B.1: Various material parameters of liquid oxygen, argon, and nitrogen at the boiling points of the liquid N2 and liquid Ar under 1 atmosphere of pressure. Permeabilitiesand permittivities are in MKS units.of the two media are matched7 the ‘depolarizing factor’ in equation B.8 is irrelevant. Inaddition, for practical materials i — 1 <<1 and demagnetizing effects can be ignored altogether. The difference between the two fractional frequency shifts thus gives a measureof the filling factor 1rn.We have made use of the rather unique properties of liquid 02, and Ar in our measurement of the filling factor of the UBC cryogenic hydrogen maser bulb. Liquid oxygenis very strongly paramagnetic and at the same time has a relatively low dielectric constant. Liquid Ar has very nearly the same dielectric constant as liquid 02. As our workwith atomic gasses is cryogenic by nature, the adaptations required to use liquid 02 andAr were minimal. Furthermore, as both of these liquids have been studied in detail byothers, no measurements of the relevant material properties had to be made. The permeabilities, permittivities and densities of liquid 02, Ar and N2 are presented in table B.1alollg with references to the source of the data. The reason for including the propertiesof liquid N2 will become obvious below.The measurement of was made by admitting the various cryogens to the maserbulb one at a time and observing the shift in the resonant frequency of the split-ring71f they are not perfectly matched then it is possible to measure the effective depolarizing factor.This procedure is described below.Appendix B. Measurement of Filling Factors 187resonator. To make use of the temperature dependence of the dielectric properties of theliquids, the liquid 02 was condensed into the maser bulb at the temperature of boilingliquid N2 and the liquid Ar at the temperature of boiling Ar. The difference in e — 1 forthe two liquids under these conditions is only 0.4%. The fractional shift of the cavityresonance which is measured when the resonator is loaded with liquid 02 is larger thanwhen it is loaded with liquid Ar due to the paramagnetic susceptibility of the 02. Thedifference between the two shifts is essentially that due to the magnetic properties of the02.If the small difference between the dielectric constants of liquid 02 and Ar is takeninto account the dielectric contribution to the liquid 02 shift is— (e02—1) 1+Ne(€Ar1) (B9)\ W0) dielectric — (EAr — 1) 1 + Ne(€2 — 1) kL4’O)where Ne is the effective depolarizing factor for the particular geometry. This factorcan be estimated by also filling the cell with liquid N2 at its boiling temperature. Thedielectric shift of the resonance when the cell is loaded with liquid N2 is somewhat smallerthan the dielectric shift caused by either of the other cryogens. The ratio of two purelydielectric shifts (N2 and Ar) can be used to determine the effective Ne from equation B.8.We findNe=1 ( r 1) (B.10)1 —r CN2 —1 QArwherer= (wo)tN2 <1 (B.11)W LAris the ratio of the fractional shifts in the cavity resonance when the cell is filled withliquid N2 and with liquid Ar.The Ar and N2 frequency shifts determine Ne and ultimately the dielectric contribution to the liquid 02 shift (equation B.8). The magnetic contribution to this shift isAppendix B. Measurement of Filling Factors 188given by(i =—(B.12)W0 J £02 magnetic “ A)0 J £02 “ W0 J £02 dielectricand hence the filling factor im can be written8= 2 () . (B.13)(i — 1) L4’J £02 magneticAgain because of the homogeneity of the magnetic fields in the particular geometries wehave used, can be associated with the filling factor .8Nm has been neglected as — 1 << 1 for liquid 02.Appendix CRate Equations for the Atomic DensitiesIn chapter 3 we discussed the recombination of two hydrogen atoms (H) catalyzed by athird bodyH+H+X—*H2+ . (C.1)The H atom density (flH) inside a sealed container filled with a gas of H decays as= (C.2)HHwhere KHH is the rate constant for this reaction. The recombination of two D atoms toform D2 can be described in a completely analogous manner by simply replacing ‘H’ with‘D’ in the above equations. In both cases, if no other processes lead to a change in theatomic density n, the time dependence of n, is given byn(t)=—( 1 ) (C.3)where n is the initial atomic density andT= K11n (C.4)is a characteristic time for the recombination process. When a gas consisting of a mixtureof H and D is considered, the additional reactionH+D+X—HD+X (C.5)also contributes to the decay of both the H and the D densitiesd d(nH) = —(nD) = —KHDnHnD (C.6)lID recomb. HID recomb.189Appendix C. Rate Equations for the Atomic Densities 190The symmetry of these rate equations is broken by the fact that the solvation of Datoms through the £-4He coated walls of a cell at temperatures near 1 K [21] leads to anadditional decay mechanism= —ll (C.7)so1vtionfor the D atoms. In this equation ) is the solvation rate constant. The complete rateequations for the atomic densities inside a sealed bulb containing a mixture of both Hand D near 1 K are thus= —KHHn— KEOnHuD (C.8)= —KDDn,— KHDIIHIID — )n0 (C.9)where the dots indicate differentiation with respect to time. In general there is no knownanalytic solution to these equations and they must be integrated numerically.C.O.3 Simplified equationsIn chapter 7 we encountered a situation’ in which two simplifying assumptions in therate equations C.9 and C.8 were justified. The solution to the rate equations C.8 andC.9 with these approximations is discussed below.The first assumption2we make is that H-D recombination makes a negligible contribution to the decay of the D atom density. That is, we assume KDDnD >> andthat \ >> KHDnH. In this approximation equation C.9 becomes= —K,n—(C.10)‘This situation occurred during the studies with ng >> H•2There is in fact no need to make any assumptions at this point. The derivation of equation C.12 isequally valid without this assumption except that A needs to be replaced everywhere with KHDnH + A.The solution of equation C.13 however does require this assumption to be made. We make it here onlyto simplify the notation.Appendix C. Rate Equations for the Atomic Densities 191This equation has the same form as the Ricatti equation [116]a(t)z + b(t)z2 + c(t) (0.11)with c(t) 0. With the initial condition n0(0) = n we obtainArexp(—At)nD(t) = nD (1 + Ar) — exp (—At) (0.12)where r is the characteristic time for D-D recombination given by 0.4. The decay of n0is exponential in the limit r1 —* 0 as it should be due to solvation. On the other handat high D densities r’ >> A and equation 0.12 reduces to equation 0.3 as it should forD-D recombination.The second assumption which we make is that H-H recombination makes a negligiblecontribution to the decay of the H density. That is, we assume that KHDnD >> KHHnR.Substituting equation 0.12 into 0.8 with this approximation, the differential equationArexp(—At)nH(t) = —KHDnD (1 + Ar) — exp (—At) (0.13)for H is obtained. This equation can be solved analytically with the initial conditionnH(0) = n. The result isKHflAT KDDnu(t) nH [(1 + Ar) — exp (—At)] (0.14)At short times (t A1) the H density is a decreasing function of time as some H atomsare scavenged from the gas phase by D atoms. On a longer time scale the H densitylevels off at a fractionlim nH(t)= [ Ar ] KDD (0.15)t—oo (1 + Ar)of its initial value. This ratio is valid only for as long as any other processes which causen to change can be neglected. Eventually H-H recombination must reduce nH further.Appendix C. Rate Equations for the Atomic Densities 192An interesting point which should be noted is that the H and the D densities arerelatedKDDnD(t) (nH(t))KHDexp(—Ar) (C.16)This relationship is independent of any calibration of the H or D densities and can providea means of measuring the ratio of the D-D and H-il recombination rates.The approximations discussed in this appendix are illustrated in figure C.1. We havechosen to set KHD = KDD and A = = 0.1s1 The initial conditions are such thatn = 1 On. The functions describing the approximate H and D densities (equationsC.14 and C.12) are plotted, as solid lines in this figure. The H density is plotted ona linear scale while the D density is plotted on a logarithmic scale. The dashed lines(short dashes) adjacent to the solid lines indicate the densities obtained by numericalintegration of equations C.8 and C.9 with KUR = 0. The discrepancies between thesepairs of curves are due to the neglect of H-D recombination as a means of reducing the Ddensity. The relative error in D gets progressively worse with time as the decay rate fornD is underestimated by equation C.12. The error in the effective solvation rate which isinferred from these decays is 4%. The approximations also lead to an underestimation ofthe residual H density by about 3%. The third dashed line which is plotted in figure C.1(long dashes) indicates the decay that would be expected for 11D neglecting recombination(solvation only).Appendix C. Rate Equations for the Atomic Densities 1931.0 • • p p i • • • I I I I I I I I I 10.0NNNN-NN D density (no recomb.)-0.8- NNNN-NNo______NN 0C N z_1.Oc- H density - N>ND densitySS0.2 01110 10 20 30 40Time (s)Figure C.1: An illustration of the approximations used for the H and the D densities.We have chosen to set KHD = KDD and ) = = 0.1s1 with the initial condition= 1On. The solid lines indicate the approximations C.12 and C.14 for nD and nH.The short dashed lines are obtained by numerical integration of equations C.9 and C.8with KHH = 0. The long dashed line indicates the decay of nD which is expected due tosolvation alone. Note that the H density is plotted on a linear scale while the D densityis plotted on a logarithmic scale.Appendix DTemperature Measurement and Control of a 4He Bath at 1 KelvinThe measurement and control of absolute temperatures near 1 K played an importantrole in the studies of H, D mixtures presented in chapters 6 and 7 of this thesis.Accurate temperature measurements were needed to reliably determine the solvation energy required to force a D atom into a £-4He film. This is a consequence of the narrow(170 mK) temperature range over which measurements could be made’. Precise temperature control was needed in order to measure the H-D spin-exchange induced frequencyshift of the a-c hyperfine transition of atomic hydrogen. This stability requirement wasimposed by the strong temperature dependence of the 4He buffer gas shift upon whichthe spin-exchange shift is superimposed2.In order to meet these requirements we used a commercial capacitance pressure gaugecalibrated against a column of oil to measure the 4He vapour pressure at the surface of the£-4He cooling bath in which the experiment was staged. This measurement was madethrough an open stainless steel tube placed just above the liquid surface; this avoidserrors due to pressure gradients in the gas above the bath. The tube was equipped witha ‘film burner’ aud a ‘knife edge’ in order to inhibit the flow of superfluid 4He up itsinner walls. The vapour pressure reading was then converted to a temperature using the‘The upper temperature limit is set by the rapid solvation of the D atoms into the £-4Re film whilethe lower limit is set by the base temperature of the 41-le evaporation system.2The change in the 4He buffer gas shift which results from a 1 mK change in temperature is comparableto the maximum observed H-D spin-exchange shifts.194Appendix D. Temperature Measurement and Control of a 4He Bath at 1 Kelvinapproximation [81]T 7.16888KP4He(T) = 155.827 Torr x () exp (— T )195(D.1)Finally the voltage output of the capacitance gauge was used to temperature regulatethe bath using a heater immersed in the bath.D.1 Pressure gauge calibrationThis work was performed using a 1 Torr full scale MKS Baratron gauge3. We encasedthis device in 5 cm of RiO styrofoam insulation to enhance its temperature stability.Prior to making 4He vapour pressure measurements, the baratron was calibrated againsta column of Inland TW vacuum pump fluid4. This oil was chosen simply because of itsrelatively low vapour pressure and its availability. Before using the oil it was vacuumeddegassed at 90 C by pumping5 on it through a liquid N2 cold trap. After one day theresidual vapour pressure was below iO Torr.The density of the oil was measured to be 0.8571(7) g/cm3 at 22.7 C using a volumetricflask. The volume of the flask was measured independently using demineralized distilledH20 which had been allowed to come to equilibrium (in air) at the same temperature. Inboth cases care was taken to account for weight changes due to the adsorption of moistureon the flask. The error in this density measurement due to gasses which dissolved intothe oil during the measurement is estimated to be considerably less than the reporteduncertainty.The inner surfaces of the manometer were first cleaned with hydrofluoric acid and3Model 220-2A1-1. MKS Instruments Ltd. Nepean Ont. Canada4lnland Vacuum Co. New York, USA.5Heating the fluid reduces its surface tension and expedites the formation of gas bubbles. A magneticstirrer was also used to ensure that all of the oil reached the free surface where the pressure head due tothe depth of the reservoir does not inhibit bubble formation. Note that by causing the stirrer to ‘vibrate’rather than rotate smoothly one can actually assist in the nucleation of gas bubbles.Appendix D. Temperature Measurement and Control of a 4He Bath at 1 Kelvin 196then filled with oil to a depth of 15 cm, with care being taken to keep oil off the upperwalls of the tube. It was immediately evacuated and reheated to ensure that any gassesthat had dissolved during the transfer were removed.The temperature dependence of the density of the oil was measured by observing theheight of the oil column (no pressure difference between the two tubes) as a function oftemperature. This dependence6 is less than 1 part in iO per degree C. 4He pressuresup to 10 Torr were admitted to the manometer but no detectable change in the densityof the oil due to the dissolution of gas was observed. At no time during or after thebaratron calibration was the pressure allowed to rise more than 40% above full scale. Nogasses other than 4He were admitted to this system during these experiments.The baratron gauge was allowed to reach thermal equilibrium and then calibratedagainst the height of the column of oil in the manometer as different pressures of 4Hegas were admitted to the system. Measurements of the height of the oil column weremade with a Geartner Scientific Corporation cathetometer with an accuracy of 0 05 mmPressures were determined using the local acceleration due to gravity Care was takento investigate possible errors due to wetting of the glass surfaces by the oil as well ashysteretic effects: none were observed. A linear7 least squares fit (60 data points) to thecalibration data over the entire operating range of the Baratron was performed in orderto determine the calibration between the capacitance gauge output voltage VB and themeasured pressure P:P = ic x VB + Constant (D.2)6Throughout the calibration run the temperature of the oil was maintained constant to within 0.1 Cof the temperature at which the density of the oil was measured,7The zero of this type of gauge drifts with time, however the calibration constant (ic) does not appearto be measurably affected. During each experimental run the zero reading of the Baratron output wasmeasured and subtracted from the data before using ic to convert to the true pressure.Appendix D. Temperature Measurement and Control of a 4He Bath at I Kelvin 197No systematic deviations from this fit were observed and thus fits to higher order poiynomials were not used. The rms deviation of the data from the fit was 4 mTorr. Thisis less than the measurement error associated with the height of the oil column. Therelative error in the calibration constant t due to statistical fluctuations was 0.16%.D.2 4He wand and temperature regulationPressure measurements above the 4He bath were made through a thin wall 0.953 cmdiameter stainless steel tube. This diameter is sufficient to limit temperature correctionsdue to thermomolecular pressure gradients[93] to 0.6% at 1.0 K. This correction drops to0.2% at 1.1 K and 0.05% at 1.2 K. A sliding 0-ring seal allowed the wand to be moved toarbitrary heights above the liquid 4He bath. Typically the wand was placed within 2 cmof the £-4He surface. No corrections have been made for the pressure head of £-4He abovethe experimental cell. The lower end of the tube was tapered down to a short 0.635 cmdiameter section terminating in a knife edge in order to inhibit superfluid film flow (seefigure D .1). Several turns of AWG 40 manganin wire were wrapped around the outsideof the tube just above the knife edge and held in place with GE varnish. The free ends ofthis heater wire were passed carefully back over the knife edge and 50 cm up the insideof the wand where a transition was made to AWG 32 brass wire. A small current passedthrough this ‘film burner’ during pressure measurements was sufficient to stop the flowof superfluid 4He up the inside of the wand.The analogue output of the Baratron gauge was fed to a differential amplifier witha variable reference potential. The error signal from the output of this amplifier wasthen fed to a PID temperature regulator which in turn drove a resistive heater inside the£-4He bath. This system allowed us to set and regulate the pressure reading to within 0.1mTorr of a desired value. At 1 K this translates into a temperature stability of +0.1 mK.Appendix D. Temperature Measurement and Control of a 4He Bath at 1 Kelvin 198PumpOut_____DC PowerSupply toThermocouple Pump1 Torr_____Gauge BaratronGauge- S.S. TubeTeflon GuardFilm Burner— Knife FdgehquidhehumZN_. see detailFigure D.1: A schematic diagram detai1iig the method used to measure 4He vapourpressures above the £-4He cooling bath used in the experiments with H, D mixtures.Details are discussed in the text.Appendix D. Temperature Measurement and Control of a 4He Bath at 1 Kelvin 199At 1.2 K the implied stability is about 5 times better due to the increased 4He density.The film burner was tested by applying varying amounts of power and monitoringthe (unregulated) baratron reading. As the power was increased an abrupt drop in thepressure reading was observed at around 0.5 mW. This pressure drop was typically lessthan 1 mTorr or about 1 mK at 1K. Further increases in power did not influence thereading measurably until considerably higher power levels were reached and the entire£-4He bath was warmed. Under all operating conditions 1 mW of power was sufficient toburn off the film without causing a significant heat load.D.3 Characterization of the measurement systemIn order to establish the practical limitations of this method of measuring and regulatingtemperatures we undertook two experiments using the split-ring resonator described inchapter 6. These measurements are described below.D.3.1 The superconducting transition of AlAl undergoes a superconducting transition at a temperature T near 1.2 K and is thusa useful fixed point temperature reference for the work described in chapters 6 and 7.Microwave absorption measurements on superconducting materials using split-ring resonators are routine in this laboratory[l 17]. These measurements can be used to determineT. A 99.9999% pure Al disc (0.6 cm diameter, 1 mm thick) containing several large crystalline grains was attached to a thin strip of Cu with GE Varnish. This strip was thenattached to the inner wall of the Cu housing for the 1.4 GHz split-ring resonator suchthat the clean face of the sample was normal to the radius of the split-ring and locatedabout 3 mm away from its outer wall. The azimuthal location of the sample was about90 degrees away from the gap in the split-ring. This location and means of mountingAppendix D. Temperature Measurement and Control of a 4He Bath at 1 Kelvin 200the sample was chosen simply to make the mounting procedure as simple as possiblewithout disturbing any components of the resonator and housing. No attempt was madeto optimize the filling factor for the sample or the sensitivity of the measurement.This assembly8 was cooled to temperatures around 1.2 K and the resonator Q wasmeasured in reflection as a function of the £-4He bath vapour pressure. Values of Q weredetermined by fitting the reflected power from a swept frequency measurement of thecavity resonance to a single Lorentzian lineshape with a second order background. Infigure D.2 we plot the inverse quality factor of the resonator and sample as a function ofthe 4He bath vapour pressure (squares). This data has been normalized to an arbitraryvalue Qo determined as the average Q ‘far’ from the data near 600 mTorr. The dataconsists of two temperature scans in opposite directions. No evidence of hysteresis isvisible. The data in the wings9 is quite flat in comparison to the feature near 600 mTorr.As the temperature of the sample is decreased from above 1.2 K, the microwave loss dueto the combined resonator and sample first drops by a few percent before rising to a valuewhich is slightly above the high temperature value. The decrease in absorption occursat the onset of the superconducting transition. The sample and the sample mount werecapacitively coupled to the resonator in such a way, that as the loss associated with theAl dropped, the microwave currents flowing through the sample holder increased. This isreflected in the data as an increase in the microwave loss part way through the transition.Also shown in figure D.2 are two results (diamonds) from a measurement of T forAl by Cochrane and Mathoper [118]. The point at lower pressure represents their determination of T = 1.196(4) K based upon a fit of their measurement of the criticalfield H as a function of temperature to the BCS theory [119]. We have plotted thisvalue in equivalent 4He vapour pressure units. Near T their fit deviates from their data.8The Al sample was immersed in the £-4He bath in this geometry.9Additional measurements extend to higher and lower pressures but are not shown.Appendix D. Temperature Measurement and Control of a 4He Bath at 1 Kelvin 2011.04.I 0 II I1.03 -1.02 -1.01 -00.990980.97-0.96I • I • • • • I • I • • • I i •500 550 600 650 700 750 800‘He Pressure (mTorr)Figure D.2: The inverse quality factor (1/Q) of the 1420 MHz split-ring resonator loadedwith an Al sample (squares) as a function of the vapour pressure of the 4He coolingbath. This data has been normalized to an arbitrary value Qo=4l5. The onset of thesuperconducting transition appears to be at about 640(10) mTorr (1.203(3) mK). Alsoshown (diamonds) are two determinations of T for Al using critical field data obtainedby Cochrane and Mathoper [118]. The lower value (1.196(4) K) is obtained from a fit totheir data. The upper value (1.205(4) K) is the actual temperature to which their dataextrapolates.Appendix D. Temperature Measurement and Control of a 4He Bath at 1 Kelvin 202The second point indicated by a diamond (at higher pressure) is our extrapolation oftheir actual data to zero field in order to determine T = 1.203(4) K. Our results are inexcellent agreement with these earlier results.D.3.2 The 4He buffer gas shift: Temperature stabilityIn chapter 7 we presented a new measurement of the 4He buffer gas shift of the a-chyperfine transition of atomic hydrogen. This shift is proportional to the 4He densityand is thus a strong function of temperature near 1 K. In order to place a practical limiton the temperature stability of the actual experimental bulb containing the atomic gasses,we monitored the frequency of a series of FID’s taken at a constant 4He pressure over aperiod of 40 minutes. This measurement was done at a pressure of 144.0(2) mTorr. Thetechniques outlined in Appendix A were used to determine the frequency of the FID’swith respect to the Rb frequency reference. The results of this measurement are shownin figure D.3. No detectable drift in the frequency of the FID is seen above the scatter inthe data. The standard deviation of the frequency from the mean is 13.5 mHz which iscomparable to the fluctuations we would expect due to the fractional frequency stabilityof the Rb clock. The rate of change of the 4He buffer gas shift with respect to temperatureat this pressure is about -80 Hz/K. This data indicates that the pressure regulation ofthe system was stable enough to maintain the temperature of the experimental bulbcontaining the H constant to within better than 0.2 mK during the measurement. Thisresult is in agreement with the measured fluctuations in pressure over this period.Appendix D. Temperature Measurement and Control of a 4He Bath at 1 Kelvin 203I I I I I I I I I I I I J I I I6.22N.Ia • aIa61?..I•.•iZ 6.18 • . a • aI. • . a • •1a•I• •I••a• a a..•‘.4— II•0 •• • Ia••..>_.1•.1.1 a• ?••0•a • ••l a. aSI•C•a• • •••I Ia) .a a a•‘‘a.aIa aa II.• •1)L6146 ( i I I i I I • I • • .0 500 1000 1500 2000 2500Time (s)Figure D.3: The frequency of a FID with respect to a local oscillator (LO) near the a-chyperfine transition of atomic hydrogen as a function of time at a 4He pressure of 144.0(2)mTorr. The scatter in the data is consistent with the fractional frequency stability of theRb frequency reference used to generate the LO. The 4He buffer gas shift is expected tochange by -80 Hz/K at this temperature. This implies that the temperature stability iiithe experimental cell is better than 0.2 mK.Appendix EMolecular Energy Levels of H2, D2 and HD just below DissociationSpectroscopic measurements of the weakly bound levels of the various molecules of H andits isotopes have been made by several authors. These molecular states play an importantrole in determining the strength of various interatomic interactions such as molecularrecombination and spin-exchange at low temperatures. Reynolds [22] has compiled tablesfrom the literature listing all known levels within 300 K of the dissociation energy for112, D2 and HD. We reproduce his tabulation here for the sake of completeness.In each case the original data was reported in energy units of cm1. Here we usethe more convenient temperature units where 1 K is equivalent to 0.69502 cm1. Thedissociation energy of the homonuclear species 112 and D2 were measured by Herzberg[120] who obtained values of 51967.0(4) K and 52381.2(6) K respectively. LeRoy andBarwell [121] measured a value of 52874.29(43) K for the HD molecule. The energies of thevarious molecular levels with respect to the ground state were measured by Dabrowski (112[122]), Bredohl and Herzberg (D2 [123]), and Dabrowski and Herzberg (HD [124]). Theseresults are combined and tabulated in tables E.1, E.2, and E.3. in order of increasingbinding energy. In these tables v refers to the vibrational quantum number of the statewhile J refers to the angular momentum quantum number. Note that for the (14,4) levelof the 112 molecule which lies extremely close to dissociation, a refined dissociation energyof 51967.4(7) K was used. This value was obtained by Stwalley [125] who reanalyzed thedata of Herzberg [120] paying special attention to the weakly bound levels.204Appendix E. Molecular Energy Levels of H2, D2 and HD just below Dissociation 205v J D0(K)14 4 0.712 10 19.73 27 29.26 22 62.014 3 72.513 7 74.014 2 136.914 1 183.914 0 208.511 12 208.913 6 286.7Table E. 1: Dissociation energies of the rovibrational levels of molecular H2 close to thedissociation threshold.v J D0(K)21 1 0.221 0 2.920 5 41.820 4 91.520 3 135.220 2 170.220 1 194.320 0 206.8Table E.2: Dissociation energies of the rovibrational levels of molecular D2 close to thedissociation threshold.Appendix E. Molecular Energy Levels of H2, D2 and HD just below Dissociation 206v J D0(K)17 1 0.1417 0 6.916 5 50.916 4 135.915 8 155.116 3 212.116 2 273.5Table E.3: Dissociation energies of the rovibrational levels of molecular HD close to thedissociation threshold.

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