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Resonant recombination of atomic hydrogen and deuterium at low temperatures Reynolds, Meritt Wayne 1989

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RESONANT RECOMBINATION OF ATOMIC HYDROGEN AND DEUTERIUM AT LOW TEMPERATURES By Meritt Wayne Reynolds B. Sc. (Physics) Simon Fraser University A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA February 1989 © Meritt Wayne Reynolds, 1989 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada DE-6 (2/88) Abstract The contents of this thesis include several experiments and a theory, the unifying theme of which is resonant recombination of atomic hydrogen (H) and deuterium (D) at low temperatures. Measurements of the interaction of H and D with the l iquid helium (Z-4He) wal l coating used in low-temperature experiments are also included. Atomic hydrogen and deuterium were studied by E S R in a field of 41 k G at temper-atures below 1 K , in a second generation of experiments on an existing apparatus. The binding energy of H on / - 4 H e was measured, by direct observation of the physisorbed atoms, to be = 1.00(5) K. Extensive meaurements of recombination rate constants were made. Resonant recombination of H via the (v, J) = (14,4) level of H 2 was observed and identified. Atomic deuterium was also observed to recombine in a resonant process. A t low temperatures, resonant recombination is possible due to the predissociation of weakly bound molecular levels by the intra-atomic hyperfine interaction. The theory of resonant recombination of H and D due to hyperfine predissociation is developed in detail. In order to keep the development self-contained, it is presented as appendix D of this thesis. The theory is combined wi th our E S R data in order to predict the magnetic-field dependence of resonant recombination of D. It is shown that observation of the threshold magnetic fields for resonant recombination wi l l permit accurate measurement of the dissociation energy of the (v, J) = (21,0) and perhaps (21,1) levels of the D 2 molecule. A n apparatus was constructed to study atomic deuterium by hyperfine magnetic resonance at temperatures near 1 K. The D atoms were produced from D 2 of 99.65% isotopic purity by an R F discharge in a bulb coated with a film of superfluid / - 4 He . n Magnetic resonance in a field of 39 G on the longitudinal transition at its min imum frequency of 309 M H z was used to observe the atoms. The experiment was designed with the capability of measuring the recombination rate as a function of magnetic field for fields up to 30 k G by moving the sample between the 309 M H z resonator and a superconducting solenoid. A number of preliminary experiments were made using this apparatus. It was dis-covered that the D atom density decays away exponentially in time. The decay rate was measured as a function of temperature and found to follow an Arrhenius (thermally-activated) form, with an activation energy of about 15 K. No second-order decay was seen, permitting an upper bound to be placed on the D-D recombination rate. The ex-ponential decay of the D atom density was due to the penetration by the atoms of the / - 4 H e film coating their container. Our measurement of the temperature dependence of the density decay rate gives the energy required to dissolve a D atom in Z- 4He, Es ~ 14 K. This is the first experimental determination of this quantity. The longitudinal relaxation time Tx for the (3-8 transition was found to be too short to be due to D-D spin-exchange. It was conjectured that the sample was contaminated with substantial amounts of H , spin-exchange in H-D collisions causing the short T\. Measurements in an existing 1420 M H z apparatus showed directly, by magnetic resonance on the hydrogen zero-field transition, that H atoms were, in fact, present. B y varying the H concentration, we were able to show that the decay of the D density by was not due to recombination of H with D. Several attempts to substantially reduce the H impurity have failed. The penetration of the / - 4 H e f i lm is undoubtedly one reason why the D is so susceptible to contamination by H , since the film is permeable to D but not to H. For use in interpreting these experiments, the effects of spin-exchange relaxation on hyperfine resonance in mixtures of H and D were calculated. in Table of Contents Abstract ii List of Tables viii List of Figures ix Acknowledgement xi 1 Introduction 1 1.1 Quantum degeneracy 1 1.2 The wal l coating 2 1.3 Chemistry 4 1.4 Previous work on atomic deuterium at low temperatures 6 1.5 This thesis 7 2 High-field experiments on H and D 11 2.1 Atomic hydrogen 12 2.2 Atomic deuterium 14 3 Hyperfine resonance of deuterium 19 3.1 Hyperfine structure 19 3.2 Hyperfine resonance 22 3.2.1 Transitions 22 3.2.2 Spin 1/2 analogy 24 iv 3.3 Experimental hyperfine resonance 26 3.3.1 Decay of the free induction signal 28 4 Apparatus for recombination spectrometry of D 2 32 4.1 Top plate 32 4.2 The flow-baffle 34 4.3 The resonator assembly 35 4.3.1 Quartz tube 35 4.3.2 Shield 38 4.3.3 Coupling 38 4.3.4 Specifications 39 4.4 Bias coil 40 4.5 The discharge 40 4.6 The magnet . 42 4.7 Mobi le cell 42 4.8 F ixed cell 43 4.9 Thermometry 44 4.10 The spectrometer 44 4.10.1 The transmitter 45 4.10.2 The 309MHz preamplifier 45 4.10.3 The receiver 47 5 Experiments on deuterium in low magnetic field 48 5.1 General Procedure 48 5.2 Mobi le cell 50 5.3 F ixed cell 51 5.4 Decay measurements 53 v 5.5 Relaxation measurements 53 6 Experiments on H-D mixtures in zero-field 58 6.1 Ultra-pure deuterium 60 6.2 Adulterated deuterium 60 7 Discussion and Conclusions 62 7.1 E S R at 115GHz 62 7.2 Resonant recombination 63 7.3 The 309 M H z experiments 64 7.4 The 1420 M H z zero-field results 67 Appendices 70 A The CESR magnetometer 70 B Density decay due to solution of D in liquid helium 74 C Spin-exchange in mixtures of D and H 79 C . l Equation of motion for spin density matrices 79 C. 2 Rate constants for spin-exchange 83 D Low temperature resonant recombination of H and D 86 D. l Introduction 86 D.2 Single atom hyperfine states 88 D.3 Interacting hydrogen atoms 89 D.4 Resonant recombination of spin-polarized hydrogen 95 D.4.1 Calculation of the predissociation rate 95 D.4.2 Recombination rates 105 vi D.4.3 Stabil ization 110 D.4.4 Discussion of experimental results 118 D.5 Deuterium 120 D.5.1 The (21,0) level of D 2 122 D.5.2 The (21,1) level of D 2 125 D.5.3 Stabilization 127 D.5.4 Experimental consequences 130 D. 6 Hydrogen-Deuterium mixtures 130 D.6.1 Predissociation of H D 130 D.6.2 Rate constants 135 D.6.3 Stabil ization cross section 135 E Weakly bound levels of H 2 , H D and D 2 137 E. l H 2 137 E.2 D 2 139 E.3 H D 141 Bibliography 143 vi i List of Tables B . l Effect of solution of H , D, and T in / - 4 H e 78 D . l Levels of H 2 , D 2 and H D predissociated by fields below 200 k G , their dis-sociation energy D and threshold field B0 for hyperfine predissociation. 94 D.2 Predissociation lifetime r in fis as a function of energy E = 2fieB — D for several values of D 104 D. 3 Cross section for stabilization of H 2 in collisions with 4 H e for several inc i -dent kinetic energies. Calculated using C C with part ial waves to / = 6, and potentials to n = 6 117 E. l Dissociation energies of levels of molecular H 2 close to the dissociation threshold 138 E.2 Dissociation energies of levels of molecular D 2 close to the dissociation threshold 140 E.3 Dissociation energies of levels of molecular H D close to the dissociation threshold 142 v i i i List of Figures 2.1 Measured two-body rate constants: x - Kaa, o - Kab. Curves: solid - fit to resonant recombination form, dotted - expected from zero-field measure-ments 15 3.1 Hyperfine energies as a function of magnetic field. The zero-field splitting 2a/2h = 327.384... M H z 21 3.2 Hyperfine transition frequencies as a function of magnetic field 23 4.1 Overview of the 309 M H z cryostat 33 4.2 Cross section of the 309 M H z assembly 36 4.3 Conductor configuration on the quartz tube of the 309 M H z resonator. . 37 4.4 Discharge for production of atoms and superconducting solenoid 41 4.5 Block diagram of the 309 M H z spectrometer 46 5.1 Typ ica l free induction decay 52 5.2 Typ ica l decay of atom density after turning off the discharge 54 5.3 The deuterium density decay rate versus reciprocal temperature 55 A . l Typ ica l C E S R signal 73 D . l Ab initio singlet ( X 1 E ^ ) and triplet (6 3 E„) potentials for the interaction of two hydrogen atoms 91 D.2 Singlet and ms = — 1 triplet potentials (including Zeeman energy) for the case A ~ 2peB = 5.5 K and D = 0.7 K 92 ix D.3 Mat r i x elements of the two-atom intra-atomic hyperfine interaction in the spin-projection basis. The matr ix elements are obtained by mult iplying the sign in the table by a n / 2 96 D.4 Radia l wavefunction and effective potential of the (14,4) level of H 2 . . . 100 D.5 Singlet and triplet radial wavefunctions for (14,4) with D = 0.7 K and £ = 5 K . 101 D.6 Predissociation rate from the (14,4) state of H 2 102 D.7 Coordinate system for calculation of the cross section for stabilization of in collisions with 4 H e I l l D.8 Contributions to the two-body rate constant Kaa for recombination of \a)-state hydrogen atoms. Solid curves are for T — 0.675 K , Dotted curves are for T = 0.575 K 121 D.9 Base predissociation rate T for the (21,0) and (21,1) levels of D 2 . Hor i -zontal bars show spectroscopic uncertainty in threshold fields 123 D.10 Expected recombination of DJ, due to hyperfine predissociation of the (21.0) and (21,1) levels of D 2 . 131 D . l l Base predissociation rate TH for hyperfine predissociation of the (17,0) and (17.1) levels of H D 133 x Acknowledgement I wish to thank my supervisor, Prof. Walter Hardy for his guidance in al l phases of my research. I also thank the rest of my committee, Prof. Myer Bloom, Prof. Jesse Brewer, and Dr. Richard Cline, for their interest in my work. W i t h regard to the 115 G H z E S R experiments, I thank Dr. Bryan Statt for having constructed such a versatile apparatus. These experiments were done in collaboration with Ichiro Shinkoda. Crucia l to the success of the 309 M H z experiment was the expertise of the crews of the machine and electronic shops. Much of the apparatus was machined by Beat Meyer; the advice of Peter Haas and George Babinger was particularly useful. The labours of T o m Felton and Domenic di Tomaso of the electronics shop on the stepping motor controller and the spectrometer IF generator are appreciated. For assistance with the evaporation of the superconducting resonator, and an emergency installation of the roots blower pump, thanks are due STS technicians Anton Schreinders and J i m MacKenzie. Dr. Richard Cl ine was helpful in all stages of the work; his contribution included the setting up of a well-designed and maintained computer system, without which the experimental design, data analysis, theoretical work and the writing of this thesis would have been painfully difficult. Thanks also to Prof. Charles Schwerdtfeger (for the gift of L i F : L i ) , Prof. J i m Carolan (for the loan of the superconducting solenoid), and Prof. Myer B loom (for the loan of the N M R pulse programmer). I also wish to thank Ichiro Shinkoda, Mar t in Hi i r l imann, and M ike Hayden for much advice and assistance in all aspects of the experiments. Special thanks to M ike Hayden and Walter Hardy for their invaluable assistance collecting data. xi Chapter 1 Introduction The recombination of atomic hydrogen and deuterium at low temperatures is the single largest impediment to the attainment of quantum degeneracy in these weakly interacting gases. Recombination is, however, an interesting study on its own. This thesis is primari ly concerned with resonant recombination arising from hyperfine predissociation of weakly-bound molecular levels, and the application of this effect to molecular spectroscopy v ia recombination rate measurements. A good review of atomic hydrogen research is the article by Silvera and Walraven [1], which covers al l aspects of this interesting field. A more concise introduction is by Greytak and Kleppner [2]. 1.1 Quantum degeneracy Atomic hydrogen and its isotopes provide excellent opportunity to study gases at very low temperatures. Electron-spin-polarized atomic hydrogen and deuterium, in particular, are expected to remain gaseous to absolute zero, and it is in principle possible to cool these weakly interacting gases unt i l they are quantum mechanically degenerate. In the case of H , which is a (composite) boson, the phase transition known as Bose-Einstein Condensation ( B E C ) should occur, accompanied by the onset of superfluidity. This would be extremely interesting, as a weakly interacting gas would be far more tractable theoretically than a l iquid such as superfluid helium. The quest for B E C has been a major driving force in research on cold atomic hydrogen. Conditions for B E C wi l l 1 Chapter 1. Introduction 2 likely be acheived soon, either with wall-free magnetic confinement or in higher density compression experiments. The effect of quantum statistical mechanics is completely different for a fermion such as atomic deuterium. As the temperature is lowered towards zero a fermi surface comes into existence. The absence of an exciting phase transition and the existence of a well-studied weakly interacting degenerate fermi gas ( 3 He dissolved in 4 He) , coupled with the fact that D has proven much more difficult to work with than H, has resulted in D being far less studied than H. Interest in attaining quantum degeneracy in D has been kindled by the realization [3] that it may be much more stable than H in the quadrupole magnetic traps that have recently been used to confine H [4, 5]. The lack of experience with D is slowing efforts to trap it . Attempts to load the traps with D have failed, it appears, because of hydrogen contamination [6]. I. 2 The wall coating Crucia l to the success of experiments on atomic hydrogen has been the existence of a suitable coating for the walls confining the gas. A film of superfluid / - 4 H e is the coating most often used for experiments below I K , although it is possible to use / - 3 H e at lower temperatures [7]. Both of these coatings effectively confine H atoms without the H strongly adsorbing onto them. The binding energy 1 of H is 1.011(10) K onto / - 4 H e [8] and 0.42(5) K onto / - 3 H e [7]. These are the only known wall-coatings useful below I K . The next best wall coating is solid H 2 , onto which H physisorbs with a binding energy of about 38 K [9]. This coating is not useful at temperatures below about 4 K [10] since the adsorption and subsequent recombination prevent useful gas densities from being attained. The situation for atomic D is similar. The binding energy of D on 1We will occasionally express energies in temperature units. Just multiply by Boltzmann's constant t° find the energy. Chapter 1. Introduction 3 / - 4 H e is believed to be 2.6(4) K [11, 12]; this is based on the temperature dependence of the recombination rate, and there is a conjecture that this is only an apparent binding energy and that the true binding energy is somewhat smaller [13]. No measurements of the binding of D on / - 3 H e exist. A t higher temperatures H 2 would be the next best coating; due to recombination it would soon be coated with D 2 . The next best available wall coating is therefore solid D 2 . The binding energy of D on D 2 is roughly 55 K [10], and this coating becomes inadequate below about 4 K . The interaction of H, D, and T with / - 4 H e is of considerable interest, both practically (for sample confinement) and academically (as an example of impurities interacting with an ideal liquid). Experiments with H have permitted determination of the binding energy of H on the free surface of / - 4 H e and / - 3 H e and measurement of the sticking probability for atoms incident on the surface (which depends on the interaction of the atom with the elementary excitations of the surface). The only experimentally determined quantity for D is its binding energy on Z- 4He; only one measurement (ref. [11]) based on the temperature dependence of the recombination rate exists. Atomic t r i t ium (T) has not yet been observed at low temperatures, despite an attempt by our group using well-proven techniques [14]. A n important quantity is the energy required to dissolve a hydrogen atom into / - 4 He , as this determines how effective a barrier the / - 4 H e film presents to a hydrogen atom incident on the wall of its container. Theories indicate that the chemical potential of a hydrogen atom in the interior of the l iquid is about 30 K for H , 11 K for D and 3 K for T , al l numbers being positive. 2 A t absolute zero, then, a / - 4 H e wall coating is an effective barrier to adsorption on the underlying substrate. For H , the barrier is effective at al l temperatures of interest (up to about 1.4 K ) . This is not true for D and T. The semi-permeability of the confining film wi l l undoubtedly influence studies of these atoms, 2These theoretical values are discussed in Appendix B. Chapter 1. Introduction 4 and will permit experimental observation of the interaction of the atoms with the bulk /-4He instead of with just the surface. 1.3 Chemistry The atomic hydrogen system is also interesting from the point of view of fundamental chemistry. The recombination of atomic hydrogen is a fertile field for study, and dis-plays a wealth of phenomena, especially at low temperatures. At low atom density, the recombination involves two hydrogen atoms and a third body (an inert atom or the wall of the container), which is required to conserve energy and momentum. Recombination in the presence of a 4He atom provides a unique example of a simple gas-phase chemical reaction that can be studied experimentally and theoretically over a very wide range of temperatures. At higher temperatures (at least up to room temperature), the two-step resonant (or energy exchange) recombination process H + H ^ H* (1.1) H* + 4He -> H 2 + 4He (1.2) is the dominant mechanism [15, 16]. In this process, the intermediate reaction complexes (or metastable molecules) Hj are the orbiting resonances of the singlet potential. These complexes are stabilized in collisions with, here, 4He atoms which cause transitions to deeply bound molecular levels. At low temperatures (but not so low that the 4He vapour density is negligible) the three-body collision reaction H + H + 4He -> H 2 -f 4He (1.3) is the only effective reaction in zero magnetic field since the orbiting resonances are Chapter 1. Introduction 5 thermally inaccessible. 3 The rate of this reaction at 1 K has been calculated by Greben et al [17]. Resonant recombination is st i l l possible in a magnetic field, however, since the Zeeman energy of a pair of atoms can lie just below the energy of high-lying molecular levels, rendering them accessible even at low temperatures [18]. The formation of reaction complexes here is quite different from the higher temperature case. Since the complexes are true bound levels of the singlet potential, formation of the complex takes place with a change of electron spin-state. Resonant recombination involving high-lying molecular levels is discussed in more detail in appendix D. A t low temperaure the Zeeman energy of the reactants can completely dominate the thermal energy, resulting in electron spin polarization of the atoms and subsequent dependence of the reaction rate on magnetic field and nuclear spin polarization. In addition, at low temperature only the highest molecular rovibrational levels are involved in the recombination. The recombination rate depends on the relation between the dissociation energies of the levels (essentially independent of field) and the Zeeman energy of the atoms, and resonances in the rate occur at 'level crossings'. This opens the possibility of making measurements of the level dissociation energies by observing the recombination rate as a function of magnetic field. Resonant recombination has been observed in atomic hydrogen, both here at U B C [19, 20, 21] and elsewhere [22]. Analysis of the data provides a dissociation energy for the (u, J) = (14,4) rovibrational level of H 2 . Unfortunately, the resonance recombination is partially masked by other processes and the extracted dissociation energy is not very precise. Resonant recombination should be much more pronounced in D than in H, owing to the existence of high-lying levels of D 2 with low angular momentum. Observations 3The lowest orbiting resonances contributing to recombination are (v, J) = (14,5) for H 2 (50 K), (20,6) for D 2 (10K), and (16,6) for HD (37K). The D 2 resonance is not so inaccessible, but its high angular momentum (J — 6) renders its effect negligible in comparison with the fast processes characteristic of D. Chapter 1. Introduction 6 in easily attainable magnetic fields should permit the dissociation energy of the (v, J) = (21,0) and perhaps (21,1) levels of D 2 to be measured to higher precision than can be done with optical spectroscopy. This would be useful to clear up discrepancies that exist between spectroscopic measurements and ab initio calculations [23]. 1.4 Previous work on atomic deuterium at low temperatures Atomic deuterium was first observed below 1 K by Silvera and Walraven [11], The atoms were confined in a magnetic field of 80 k G by a combination of magnetic and 4 H e vapour compression. The atom density was determined by bolometry, measuring the heat re-leased when the sample was rapidly recombined. Measurements around 0.5 K showed that the decay of atom density after loading the cell was due to two-body recombination on the / - 4 He-coated walls. Analysis of the temperature dependence of the recombination rate indicated that the binding energy of D on / - 4 H e is 2.6(4) K , compared to 1.0 K for H. In addition, the intrinsic surface rate seemed to be much larger for D 2 formation than for H 2 . A t about the same time, atomic deuterium was studied briefly by Jochemsen et al [24] using zero-field magnetic resonance at 1 K. They found that only low densities of D could be produced (about 10 1 1 c m - 3 ) and that the sample lifetime was much shorter than that of H under similar circumstances. A thorough study of the zero-field properties of D at 1 K was not made because at that time the study of H at lower temperatures was an experimental priority. The next published attempt to observe D below 1 K was made by Mayer and Seidel [10] using E S R at 9 G H z (in a magnetic field of 3 k G ) . They were unable to generate an observable density despite having a measurable flux of atoms into their microwave cavity. The flux was deduced through the heat load due to recombination, and from Chapter 1. Introduction 7 their observations they were able to place a lower bound on the surface recombination rate. The result agreed with the surface recombination measurements of reference [11] assuming that the recombination was much higher in zero-field than in high-field, and was consistent with the recombination rate varying with magnetic field according to the 'sudden approximation' (that is, with the probability of finding a pair of atoms in the electron spin singlet state just prior to collision). Our own observations of D, using E S R at 115 G H z (in a field of 41 k G ) , confirmed the high surface recombination rate. We were also able to generate samples of D | { (electron and deuteron polarized D) with lifetimes of 30 minutes and densities above 10 1 5 c m - 3 . These experiments are described in more detail in chapter 2. 1.5 This thesis In chapter 2 we describe our experiments on H and D in a magnetic field of 41 k G . It was in these experiments that we discovered that resonant recombination was important in H , and found evidence that resonant recombination of D is fast enough to be seen even given the shorter lifetime of D samples compared to H. The remainder of the thesis describes an experiment to measure the field-dependence of resonant recombination of D, in order to determine molecular energy levels to higher precision than is possible using optical spectroscopy. A n apparatus was designed and constructed to measure the rate of recombination of atomic deuterium as a function of magnetic field between 0 and 30 k G . Since the resonant recombination of interest occurs in the gas phase it is neither necessary nor desirable to operate at temperatures below about 1 K , obviating the need for a 3 H e or 3 H e - 4 H e refrigerator. In the apparatus built , temperatures down to 1.1 K could be attained by pumping on / - 4 H e in a standard glass dewar. Chapter 1. Introduction 8 A measurement of the recombination rate of atomic deuterium requires the atom density to be measured as a function of time. Magnetic resonance lends itself readily to this, being an exceptionally sensitive and selective detection technique. In particular, under experimental conditions of interest, only low densities of D have ever been created, and in addition the density of gaseous 4 H e atoms due to the presence of the saturated f i lm is rather high. For example, at 1 K the helium density is 10 1 8 c m - 3 . A typical deuterium density of 1 0 l o c m - 3 thus corresponds to a 10 parts per bi l l ion impurity in the 4 H e . The only viable detection technique is magnetic resonance. Owing to the practical difficulties of doing magnetic resonance over a wide range of fields, 4 it is more convenient to measure the atom density at a fixed field. Thus, in order to measure the field dependence of the recombination rate, one is led to measuring the atom density in one field and leaving it to evolve (through recombination) in another. We chose to measure the atom density using pulsed magnetic resonance on the J3-8 transition in BQ ~ 38.9 G which is the field that minimizes its frequency. 5 The transition moment is essentially that of a free electron, so the magnetic resonance is quite sensitive. There are several reasons for the choice of such a low-field transition over, say, E S R at some high field. The most important advantage is that operation at a field where the transition frequency is (to first order) independent of field allows a relatively large cell to be used without field inhomogeneities degrading the lineshape. In addition, a high Q superconducting resonator can be employed, and the resonator region can be shielded from stray magnetic fields, by using a simple shield made of a type I superconductor such as lead. Another reason is more pragmatic: the /3-S min imum transition frequency is 309 M H z . Construction of a spectrometer at this frequency is rather more straightforward than at, say, 72 G H z and more in line with our existing stock of microwave components. 4Zero to 30 kG corresponds to a frequency range 0.3 to 86 GHz for those transitions with large (i.e. electronic) moments. 5The hyperfine structure of D is discussed in chapter 3. Chapter 1. Introduction 9 The 327 M H z field-independent point at 30 m G was also a possibility, but there is an interfering nearby transition [24]. Final ly, to the best of our knowledge the /3-S transition has never been used for magnetic resonance on atomic deuterium and this provided an added attraction. In order that the field could be varied quickly, we settled on a design wherein the cell containing the deuterium is moved bodily from the resonator to a high-field region gen-erated by a superconducting solenoid. Several alternatives to this design were considered and rejected. One possibility, having regions of differing field communicating through a tube, was ruled out by the high density of helium vapour above 1 K , through which the D atoms diffuse only slowly. Another, having the resonator inside a solenoid and varying the current, was deemed unsuitable because of eddy current heating as well as because of the difficulty of fitting a resonator into a solenoid which in turn must fit into a small dewar. The apparatus is discussed in more detail in a later chapter. First a brief discussion of the overall experiment is in order. The apparatus consists of three modules: the discharge, the resonator, and the magnet. In the intended experiment, a sealed pyrex bulb containing helium and D 2 was to have been first positioned in the discharge region, where a strong R F pulse would dissociate the D 2 and produce atomic deuterium. The bulb would have then been moved between the resonator (to measure the atom density) and the magnet (to allow the atoms to recombine in a magnetic field). The recombination rate as a function of magnetic field could have then been mapped out by repeating the experiment with different magnetic fields. In fact, the intended measurements could not be made because of an unforeseen difficulty: the D penetrates the / - 4 H e film too readily, resulting in a relatively short sample lifetime. On the other hand, we were able to use the apparatus to study D in low magnetic field, and were able to infer its solvation energy in / - 4 He. Chapter 1. Introduction 10 After no signals could be seen on the first few attempts, the movable cell was removed and was replaced by a long cell spanning the discharge and resonator regions. This improved the filling factor somewhat, avoided the complication of a moving cell, and permitted signal averaging with the discharge running. At the same time we improved the coupling to the discharge coil, as described below. After these modifications were made, deuterium signals were finally seen, and we realized that the mobile cell experiments would be far more difficult than had been anticipated. The decay of the atom density upon turning off the discharge was observed to be discouragingly fast and exponential in time, obscuring the decay mechanism of D recombining with D. We were, as a result, unable to perform experiments to measure the recombination rate as a function of field. As will be described later, we discovered that the atomic D gas was contaminated with H atoms but that, unfortunately, recombination of H with D was not responsible for the decay of the D density. Instead, the exponential density decay was due to the deuterium atoms dissolving in the helium wall coating and subsequently sticking to the underlying solid D 2 layer. Undoubtedly these atoms would eventually recombine with each other, given some reasonable surface mobility. Chapter 2 High-field experiments on H and D In conjunction with I. Shinkoda and other members of the U B C atomic hydrogen group, observations of atomic hydrogen and deuterium were made with the 115 G H z E S R appa-ratus developed by B. W . Statt. The apparatus has been described in detail elsewhere [25, 26]. Briefly, the apparatus consists of a sensitive fixed-frequency heterodyne spec-trometer, an atom source and a microwave cavity cooled to below 1 K by a 3 H e - 4 H e dilution refrigerator. The microwave cavity is in a magnetic field of 41 k G provided by a superconducting solenoid. Atoms are produced in a weak R F discharge at about 0.5 K and are subsequently confined to the microwave cavity by a liquid helium valve operated v ia the fountain effect. The superfluid / - 4 H e f i lm coats the walls of the entire apparatus. The valve eliminates the problem of thermal evaporation of the atoms out of the compressing magnetic field, and makes recombination data much more reliable since the atoms are confined to a closed volume whose temperature and magnetic field are well-defined. A t di lution refrigerator temperatures the atoms entering the cavity are electron-spin polarized. B y varying the magnetic field with a sweep coil, the various E S R transitions are brought into resonance in turn. From the resonance intensities, and given that the gas under such conditions of field and temperature is essentially electron spin polarized, the density of the 'species' a and b of H atoms and a , /?, and 7 of D atoms can be determined. 1 Recombination measurements are made by observing the atom densities as functions of time after the microwave cavity is sealed off. 1The hyperfine eigenstates of H and D are given in appendix D. In order of increasing energy in low field, the states of H are a, b, c, and d, and the states of D are a, j3, 7, 6, e, and £. 11 Chapter 2. High-field experiments on H and D 12 The experiment is extremely computer-intensive and has required massive program-ming efforts. The data collection is automated, necessarily given the quantities of data involved. The raw data is the (complex) microwave voltage reflected from the cavity as a function of the sweep coil current. This is corrected for variations in phase and trans-formed via a non-linear complex function2 (equation 1 of reference [25]) to absorption and dispersion as a function of magnetic field. The absorption is then integrated to yield the density; the density can also be calculated from the asymptotic form of the dispersion, which gives a more reliable value for high densities, where the cavity reflection coefficient is near unity when the absorption is non-zero. Once we have the density as a function of time, we choose a model for the recom-bination and relaxation and determine the rate constants by fitting the decay to the model rate equations. This usually involves the numerical integration of two (three for D) coupled non-linear differential equations. In some cases, a correction must be made for the electron spins that are flipped during the measurement.3 Relaxation rates are taken from theory for the most part, although under some conditions it is possible to measure selected rates. 2.1 Atomic hydrogen The two most interesting results of the second-generation experiments on atomic hydro-gen using the high-field ESR apparatus were the direct observation of H adsorbed on the /-4He film coating the cavity walls, and the discovery of resonance recombination due to the hyperfine predissociation of the (14,4) level of H 2 . It is worth noting that we would 2In most of the data, the atoms seriously perturb the frequency and Q of the cavity and small-signal approximations can not be used. 3 In our experiments, ESR is a completely destructive technique. Every photon absorbed from the mi-crowave field flips the electron spin of an atom, resulting in the loss of two atoms through recombination. Which species of atoms are lost depends on spin-exchange and recombination rates. Chapter 2. High-held experiments on H and D 13 have missed both of these milestones had we employed an open geometry. We have never observed H on the walls when the valve was open, probably due to the presence of a atoms, which recombine quickly in the cold microwave cavity and heat the walls. Only once the cell was closed and the H became fully (electron and proton) polarized could physisorbed atoms be seen. Resonance recombination, at the higher temperatures, would have been masked by thermal evaporation of the atoms back to the low field discharge region, where they would recombine. A t around 0.1 K , a sizable fraction of the atoms in the cavity are physisorbed onto the / - 4 H e f i lm. The atoms on the walls see a magnetic field shifted from the applied field by the magnetic moments of the other adsorbed atoms (the dipole-dipole interaction does not average to zero in 2-d as it does in 3-d). Consequently, the microwave absorption (as a function of magnetic field) acquires 'sidebands': a peak shifted to higher field due to atoms on the surfaces of the cavity perpendicular to the applied field and a peak shifted to lower field due to atoms on the surfaces of the cavity parallel to the applied field. From the strength of the absorption in these two peaks we were able to determine the fraction of the atoms on the wal l , and hence the binding energy of H on / - 4 He. The result, EB = 1.00(5) K , was the first direct determination of this quantity. Previous determinations were made through the temperature dependence of recombination (under conditions where the surface recombination dominated), which required an assumption about the (unknown) temperature dependence of the intrinsic surface recombination rate. A t higher temperatures (between about 0.5 K and 0.7 K ) we observed and identified resonance recombination. Measuring the a and b atom densities na and n j , and fitting to the phenomenological equations na = -Kabnanb - 2Kaan2a (2.1) hb = —KabUaUb (2.2) Chapter 2. High-field experiments on H and D 14 (including corrections for electron spin relaxation, microwave-induced recombination, etc.) we obtained the data shown in figure 2.1. Below 0 .5K, both Kaa and Ka\, are due to recombination on the surface of the / - 4 H e film,4 and decrease with increasing temperature as atoms spend less time on the surface. Above 0.5 K , the Kaa rate constant increases with temperature, well before expected from zero-field measurements. We explained this upturn in the Kaa rate constant above 0.5 K by resonance recombination involving the (14,4) rovibrational level of H2, which is predissociated by the hyperfine interaction. Resonance recombination due to hyperfine predissociation is discussed in more detail in appendix D. Many of our observations on atomic hydrogen have been published, as references [27], [19], [20] and [21]. The theory of the resonance recombination of HJ, was published as reference [21]. 2.2 Atomic deuterium The high-field E S R apparatus was also used to study spin-polarized atomic deuterium (DJ.). Deuterium proved much more difficult to work with than hydrogen. We made measurements of the recombination of DJ, as a function of temperature. The total atom density n decayed essentially as a two-body process n = —Kn2, although there were some complications due to the development of partial nuclear polarization during the course of the decay. The range of temperatures over which we could confine a sample of reasonable density was quite narrow (0.6 to 0.8 K ) , but we were st i l l able to observe two distinct behaviors. For the lower temperatures the recombination rate constant K increases with decreasing T, due to the increasing adsorption of atoms onto 4In connection with this mechanism, it should be mentioned the the area to volume ratio of the cavity was A/V ~ 30 cm - 1 . Chapter 2. High-field experiments on H and D 15 1 / T (K) Figure 2.1: Measured two-body rate constants: x - Kaa, o - Kab. Curves: solid - fit to resonant recombination form, dotted - expected from zero-field measurements. Chapter 2. High-field experiments on H and D 16 the l iquid helium f i lm lining the cavity as the temperature is lowered. A t higher tempera-ture, however, K is an increasing function of temperature, indicating either electron-spin relaxation or a gas-phase recombination process. A rate increasing with temperature can be indicative of electron-spin relaxation, which has an activation energy equal to the Zeeman splitting (5.5 K in our field of 41 kG ) , because electron-spin polarization is suppressing recombination, which is fast for unpolarized atoms. However, calculations indicate that both spin-exchange and dipolar relaxation are much too slow. The rate constant (subtracting off the surface contribution) is K ~ 1 0 - 1 5 c m 3 s _ 1 at 0.7 K. As -suming a recombination process with K = knue we obtain k ~ 1 0 _ 3 2 c m 6 s _ 1 . If this is translated to zero field, by dividing by the recombination-suppression factor (in the sudden approximation) e2 ~ 2 x I O - 6 , and used to predict the rate constant at 1 K , we find K(l K ) — 6 x 1 0 - 9 c m 3 s - 1 ; this implies a 2 ms lifetime for a density n = 10 1 1 c m - 3 . Since the observed lifetime is of order 30 s (and appears to be exponential in time, ru l -ing out two-body recombination altogether), something is amiss. It turns out that the increase in the rate constant with temperature can be attributed to a breakdown of the sudden approximation because of a resonance arising from the hyperfine predissociation of the (21,0) level of D 2 . The predissociation lifetime, calculated in appendix D, is short enough that an equil ibrium population of the (21,0) level exists, and the recombination is due to collisions of the metastable molecules with 4 H e which cause transitions to bound molecular levels. The data imply a cross section for the de-excitation collision D(21,0) + 4 H e ^ Dlower + 4R& ( 3 3 ) of about 10 A 2 , which seems eminently reasonable. W i t h this cross-section, the field dependence of the gas-phase recombination should show marked threshold behavior at about 22 k G . A measurement of this threshold field should yield a value for the dis-sociation energy of the (21,0) level, and hence the D 2 molecule, better than has been Chapter 2. High-field experiments on H and D 17 obtained through optical measurements. This observation was one of the incentives for the construction of the 309 M H z experiment described later in this thesis. We note that the solvation of D into the / - 4 H e was probably not important in these high-field low-temperature experiments, although it may not have been completely absent either. The highest temperatures were 1 / T ~ 1.3 K _ 1 , for which the solvation lifetime should have been about 200 s using Ea = 12.5 K and A/V = 30 c m - 1 . For comparison, the observed density decays were clearly due to two-body recombination, with time constant (Kn)-1 ~ 30 s. In later experiments we were able to produce doubly spin polarized atoms, DJ.1. This is a gas in the 7 hyperfine state, which does not recombine in two-body collisions since collisions are always in the electron-spin triplet state. Densities of more than 10 1 5 c m - 3 could be attained by flushing the contents of the discharging atom source into the microwave cavity with a burst of helium gas, which also caused the l iquid helium valve to close. The gas was observed to decay away with a rate that decreased with increasing temperature up to the highest temperature, 0.71 K. This decay was attributed to nuclear spin relaxation of atoms while adsorbed on the / - 4 H e wall due to interaction with magnetic impurities in the copper substrate. Once relaxed, the atom quickly recombines. The relaxation is the rate-l imiting step and so the decay is exponential in time. Often, the DJ.^ contained a sizable HJ, J. impurity. In this case, the decay of the DJ. J; was accompanied by a reduction in the H J . | density, after which the HJ. i was very long-lived. These data allow us to place a bound on the importance of f i lm penetration. The highest temperature data set with H J . | impurity, at 0.68 K , is consistent with the DJ.J. penetrating the film wi th a lifetime of longer than 25 minutes. This implies, v ia the theory developed in appendix B , that the energy Es required to dissolve a D into / - 4 H e is no smaller than about 12.7 K. We conclude that the upturn in the recombination data at higher temperatures could not have been due to film penetration. Chapter 2. High-held experiments on H and D 18 The recombination and relaxation measurements on DJ. were published as reference [28]; the experiments on mixtures of DJ.J. and HJ.J. were discussed in [29]. Our calculations on resonance recombination of D appear in [30]. Chapter 3 Hyperfine resonance of deuterium 3.1 Hyperfine structure Deuterium is a very simple atom, a single electron orbiting a deuteron. For our purposes we can neglect the existence of electronic excited states, and can treat the atom as a spherical object whose only degrees of freedom are the orientation of the electron and deuteron spins. As the electron has spin 1/2, and the deuteron has spin 1, the state space is six dimensional. The spin Hamiltonian of the deuterium atom in an applied magnetic field B i s 1 H = -7 d a i - B + 7e^s-B + a i - s (3.1) where i and s are the deuteron and electron vector spin operators, 7d = 4.106628(4) x l O ^ ^ G " 1 7 e = 1.7608592(16) x l O V ^ - 1 are the well-known gyromagnetic ratios [31] and the zero-field hyperfine splitting, mea-sured using a maser [32], is fo = = 327.3843525222(17) M H z The spin Hamiltonian is readily diagonalized. Since it commutes with sz + iz for B in the ^-direction, the problem reduces to, at most, diagonalization of 2 x 2 matrices. The lThis neglects the small nuclear electric quadrupole moment. 19 Chapter 3. Hyperfine resonance of deuterium 20 eigenstates of (3.1) (listed in order of decreasing energy in low field) are 10 (3.2) = . 7/+ | i ,0) + e + | - (3.3) l*> l ~ , ° > ' (3-4) l7> - i - i - » (3.5) 1 0 = l y - l - i 0) - e_ | i , - D (3.6) l«> = >7+ 1 - 2 ' ) ~ C + l|,o> (3.7) where \ms,rrii) is the state with electron and deuteron spin projections ma and m,- along the applied field, and the coefficients rj± = cos(0±) and e± — s in(0±) . The mixing angles 9± are given by tan(0±) = (3.8) with 1 1 At = -^(7e + 7 d ) i 3 ± - a The energies of the hyperfine states, shown in figure 3.1, are given by Ec = +h(le-~fd)B/2 + a/2 (3.9) Ec = +y/[fi(7e + 7 d ) # / 2 + a/4] 2 + a 2 /2 - a 7 d J 5 / 2 - a/4 (3.10) £ 5 = - ^ ( 7 e + 7 d ) 5 / 2 + a/4] 2 + a 2 /2 - %ldB/2 - a/4 (3.11) E1 = +^/[ft(7 e + 7 d ) # / 2 - a/4] 2 + a 2 /2 + hldB/2 - a/4 (3.12) Ep = -^/[h(% + 7d)B/2 - a/A}2 + a2/2 + hldB/2 - a/4 (3.13) Ea = - A ( 7 e - 7 d ) 5 / 2 + a/2 (3.14) Chapter 3. Hyperfine resonance of deuterium 21 Figure 3.1: Hyperfine energies as a function of magnetic field. The zero-field splitting 3a /2 / i = 327.384... M H z . Chapter 3. Hyperfine resonance of deuterium 22 3.2 Hyperfine resonance 3.2.1 Transitions The application of an oscillating (RF) magnetic field causes transitions between the hy-perfine eigenstates. Treating the R F field as a perturbation we determine the allowed transitions and their moments (strengths) by linearizing the Hamiltonian (3.1) with re-spect to variations in the magnetic field. Al lowed transitions have Amp — 0, ±1, where mp = ms + mi is the projection of the atomic angular momentum along the z axis, the direction of the bias magnetic field. The transition frequencies 2 are shown in figure 3.2 as a function of field. The transitions are denoted 7 r + , 7r~, or cr, depending on which components of the oscillating field cause them. W i t h the bias field along the z axis, transitions denoted 7 r + (ir~) are those with Amp = +1 (Amp = —1), and are driven by a magnetic field rotating clockwise (counterclockwise) in the x-y plane viewed from above. The o transitions have Amp = 0 and are driven by a magnetic field oscillating along the bias field. The frequency of the /3-S o transition passes through a min imum at an applied field of BQ = 38.931 G. The min imum frequency is / - = ^ / o - 308.6609275 MHz. In the immediate vicinity of the minimum the frequency assumes a quadratic dependence on the applied field: Us c- + J ^ ( 7 e + 7 d ) » ( 2 ? - f t ) a (3.15) * /min + 12.7 kHz G - 2 ( B - B 0 ) 2 (3.16) A t the field BQ, ??- = e_ = l/y/2, and the transition moment is ^ ( 7 e + 7a) /2, essentially that of a free electron. We find here the happy situation of being able to do electron spin 2Given by / = \Ei — Ej \/h for transition between levels i and j. Chapter 3. Hyperfine resonance of deuterium 600 fl(G) Figure 3.2: Hyperfine transition frequencies as a function of magnetic field. Chapter 3. Hyperfine resonance of deuterium 24 resonance on deuterium without requiring the field homogeneity that would be necessary to do resonance on free electrons. In addition, figure 3.2 shows that this transition is well separated from any others. This avoids the problem of exciting multiple transitions which can be troublesome when working in very low fields. The R-t ir+ transition, in particular, also has a field-independent point, at about 30 m G . In this field, however, the ct-8 TT~ transition is only about 40 Hz away, which confounds magnetic resonance signals [24]. 3.2.2 Spin 1/2 analogy A well-isolated transition between two levels is called a 'simple line', and the dynamics of interaction with the R F driving field can be mapped onto standard spin 1/2 transverse N M R theory [33]. The procedure is to restrict consideration to the two-dimensional subspace spanned by | 8) and | 6) and expand the 2x2 density matrix in this subspace in terms of Pau l i matrices in the usual way: p = 1/2 + (S') • o- (3.17) where the polarization (S') is the expectation value of the vector spin of a fictitious spin 1/2 particle. The Hamiltonian (3.1), linearized with respect to the addition of the perturbing R F field b to the bias field Bo and restricted to the subspace spanned by | /3) and | 6), can be written as a linear function of the vector operator S'. This linearized Hamiltonian is Hx = huS'z - h(le + ld)bzS'x (3.18) This is the Hamiltonian for a spin 1/2 particle with gyromagnetic ratio 7 = 7 e + 7d in a (fictitious) magnetic field Bx = bz Chapter 3. Hyperfine resonance of deuterium 25 By = 0 BZ = %LO/I Our longitudinal transition is thus mapped onto transverse resonance of a spin 1/2 par-ticle, where the transverse fictitious R F field is the z component bz of the real R F field. The interaction of the system of atoms with the R F field follows immediately from the theory of magnetic resonance of spin 1/2 particles. The (fictitious) spins have an equil ibrium longitudinal magnetization Mo = n(p0p- pSs)hll1 (3.19) where n is the density of D atoms in the subspace and p is the density matrix in this subspace. In future we wi l l use for n the total D atom density and for p the ful l 6 x 6 density matrix. The same expression for M0 then applies. For temperatures around 1 K , a <C k&T (high temperature l imit) and we can approximate ppp-pss * hu/6kBT (3.20) The equil ibrium magnetization is, in this case, M0 = n(h*f/2)(hu;/6kBT) (3.21) The effect upon the atoms of an R F magnetic field b cos(wi) is found by analogy to the spin 1/2 case. The 'spins', with Larmor frequency to, see an R F field along the fictitious x' axis BX = bz cos(u>t). This field can be decomposed in the usual way into two components counter-rotating in the fictitious x-y plane. Transforming to the reference frame (x'-y1) rotating at the Larmor frequency, the spins effectively see only a static field Bxi = bz/2 and precess about it with angular freqency jbz/2. After a time 2Qj^bz the spins have been rotated through angle 9 (the 'tipping angle'). If the R F field is then removed, the Chapter 3. Hyperfine resonance of deuterium 26 spins are stationary in the rotating frame. In the non-rotating frame, then, there is a rotating transverse magnetization. If a 7r /2 pulse is applied to a sample in equil ibrium, then the fictitious magnetization M0 is tipped into the x-y plane. Experimentally we observe the real magnetization, which is along the z axis (the Hamiltonian Hi couples only to bz) and is given by Mz(t) = M0 sin(0) sin(wi) (3.22) 3.3 Experimental hyperfine resonance The heart of any experimental arrangement for pulsed hyperfine resonance of atomic hydrogen (or its isotopes) is an electromagnetic resonator whose resonant frequency co-incides with the frequency of the transition used to probe the atoms. The resonator is coupled to a transmitter and a receiver. To minimize the noise introduced from room temperature electronics, and to maximize the sensitivity, the transmitter is weakly cou-pled and the receiver is critically coupled. The (gaseous) sample is confined to a glass bulb in the resonator. For a longitudinal resonance, the atoms respond to the compo-nent of the resonator (RF) magnetic field parallel to the externally applied bias field. As described in the previous section, longitudinal resonance on a two-level system can be mapped onto a standard spin 1 / 2 transverse resonance. W i t h each atom we associate a (fictitious) spin. The spins are tipped by turning on the transmitter for some short time r. Generally one wants r <C T 2 , the relaxation time for the oscillating magnetization, so that relaxation can be neglected during the tipping pulse. After the pulse the spins pre-cess; in the real system of atoms there is an oscillating magnetization along the bias field which induces a response in the resonator which in turn is detected by the receiver. After the ringing of the resonator has died out and the detection electronics recover from the pulse, the observed signal is due to the voltage induced in the resonator by the oscillating Chapter 3. Hyperfine resonance of deuterium 27 magnetization. For spin systems obeying the Bloch equations the magnetization decays exponentially in time, resulting in a signal which is an exponentially damped sinusoid. Since the spins are precessing in the absence of a driving input, the signal is called a free induction decay (FID). The coupling of the transmitter and receiver to the atoms via the resonator is discussed by analogy to the case of transverse magnetic resonance where the resonator is an L C resonant circuit. W i t h power P0 coming out of the critically-coupled receiver line during the tipping pulse, the amplitude bz of the R F field in the inductor is given by (SI units) p° = \ h v - k ( 3 - 2 3 ) where / i 0 is the permeability of free space, Vies is the volume of the inductor, and QL is the loaded cavity quality factor. Since the fictitious spins t ip with angular frequency "fbz/2, the duration 7^/2 of a 7r /2 pulse is given by T*/2 = 57 U s o z J ( 3 - 2 4 ) For an arbitrary resonator, with an inhomogeneous R F magnetic field, we can generalize the notation and use the same expression. We consider a spin placed at some reference point, the centre of the resonator for instance, which we take to be the origin of our coordinate system. We assume that the R F field at the origin is along the z axis. It is convenient to define the field (3(r) = b(r)/6,(0) (3.25) which is the R F field of the resonator normalized to the R F field at the origin. We define VTes (the effective resonator volume) to be the volume of a long solenoid which would have the same magnetic energy as the resonator if energized to the magnetic field found at the centre of the resonator. The effective resonator volume is given by Vres = jjj d3r\/3(r)\2 (3.26) Chapter 3. Hyperfine resonance of deuterium 28 Using this quantity in equation (3.24) we find the pulse time to tip a small sample situated at the origin through an angle TT/2. For a resonator tuned to the Larmor frequency u0, the signal power out of the critically coupled receiver line after a pulse which tips the spins at the origin through an angle 9 is given by In standard theory, a fil l ing factor r] is used to take into account the fact that the sample is not fi l l ing completely a long solenoid. In this case (3.27) is used for TT/2 pulses by replacing Jeff by rjVs where V$ is the sample volume. As we are interested in samples whose size is ill-defined since they may extend well beyond the resonator, it is more appropriate to use Kfr as the primary quantity. For a small sample situated in a region of uniform R F field around the the origin, Ven = sin(0)?7Vs, with the filling factor r) — Vs/VTes-Equation (3.28) may be used, if the R F fields are known, to determine the signal power as a function of the pulse length. For a sample extending beyond the central (high R F ) region of the resonator, the pulse length corresponding to the maximum signal wi l l be longer than the ir/2 pulse given by (3.24). In order to relate the signal power to the atom density, an accurate estimate of Veff is essential. 3.3.1 Decay of the free induction signal Equation (3.27) gives the signal power immediately following a tipping pulse. This signal wi l l not persist forever, but wi l l decay away due to a variety of mechanisms. We wi l l discuss only those that are (or could have been) relevant to our experiments. (3.27) where the effective sample volume is defined by (3.28) Chapter 3. Hyperfine resonance of deuterium 29 The largest contribution to the decay of the free induction signal in our experiments on deuterium is spin-exchange. This is in contrast to experiments on H , where radiation damping dominates despite lower resonator quality factors [34]. The reason for this is that spin-exchange in D-D collisions is 300 times faster than in H - H collisions. In appendix C we have calculated the effect of D-D spin-exchange on the free induction decay (FID) and the corresponding rate constant (JDD- Spin-exchange causes an exponential decay of the signal amplitude. The time constant T 2 is given, in terms of the rate constant GX>D and the deuterium atom density n, by T 2 - x = ^ G m n (3.29) Using the results of appendix C , the relaxation time for a typical D atom density n = 5 x l O 1 0 c m - 3 is 140 ms. Spin-exchange is the only effective mechanism for establishment of thermal equilib-r ium of the hyperfine level populations, with a longitudinal time constant T\ given by Ti1 = GDDn (3.30) Interaction with magnetic impurities in the container wall is inhibited by the slow diffu-sion of D atoms in 4 He above 1 K , and the effect of diffusion in any field inhomogeneities across the sample is completely negligible. There are several mechanisms beside spin-exchange that can influence the F ID . Most important in experiments with hydrogen was radiation damping [34]. Radiation damping is due to the back-reaction of the cavity on the atoms, which results in the precessing magnetization spiraling back to its equil ibrium posit ion 3 . Radiation damping does not play a large role in our experiments on deuterium due to the large spin-exchange cross-section. Both the spin-exchange and radiation damping rates are linear in the atom 3This mechanism is required by energy conservation. Since the precessing magnetization induces an oscillating voltage in the resonator, which dissipates energy, the magnetization must tend towards its equilibrium value. Chapter 3. Hyperfine resonance of deuterium 30 density, and so are conveniently expressed in terms of rate constants. The asymptotic (large time) free induction decay rate due to radiation damping is given by [35] r " 1 = fi0r]QLlM0. (3.31) If we define a rate constant GT such that r^ 1 = Grn, then for r}QL = 10 4 (the value appropriate to our experiments on D) and T ~ 1K we find Gr ~ 2 x 1 0 - 1 1 c m 3 s - 1 . This is substantially less than the spin-exchange rate constant GDD = 2.5 x I O - 1 0 c m 3 s " 1 calculated in Appendix C. Diffusion of the D atoms is another possible perturbation of the F ID . This is especially important in our experiments where the sample cell extended well beyond the resonator; i.e., after the tipping pulse the magnetization diffuses out of the resonator, decreasing the signal amplitude. The cross-section QD for diffusion of D in 4 H e vapour has been calculated by Jochemsen and Berlinsky [36] to be about 30 A 2 at 1 K . Since the diffusion constant D = (3.32) varies inversely with the 4 H e vapour density, the fastest diffusion takes place at the lowest temperatures. A t I K , wi th n n e 1 0 1 8 c m - 3 we have D ~ 1.2 c m 2 s - 1 . For our cylindrical resonator and cell geometry, a simple model for diffusion out of the resonator region (open only at one end) depends on a characteristic time r given by * = s <3-3> For L ~ 6 cm, we find that at 1 K the time T ~ 7 s. Diffusion causes a non-exponential decay; in the worst case scenario, where the atoms no longer contribute to the signal once they pass beyond the end of the resonator, the signal would be decreased to 90% of its in i t ia l amplitude at approximately O . l r = 0.7 s. This is much longer than the spin-exchange relaxation times for reasonable densities. The diffusion is even slower at Chapter 3. Hyperfine resonance of deuterium 31 higher temperatures. We conclude that diffusion has a negligible effect on the deuterium FIDs observed in the present work. Chapter 4 Apparatus for recombination spectrometry of D 2 In this chapter the apparatus we constructed to study the field-dependence of the gas-phase recombination of atomic deuterium is described. The primary objective of the experiment was to use resonant recombination to give an improved value for the dissoci-ation energy of the (v, J) = (21,0) rovibrational level of D 2 . The design philosophy was given in chapter 1. Here we present the design. A n overview of the apparatus is shown in figure 4.1. The apparatus is constructed to fit into a standard glass dewar, 10 cm inside diameter and a clear inside length of 120 cm. The dewar is surrounded by a bath of l iquid nitrogen. In operation the dewar is filled with l iquid helium ( / - 4 He). B y pumping on this helium bath, the temperature of the bath, and hence the apparatus, could be lowered to about 1.1 K. 4.1 Top plate Access to the experiment is only possible through the top of the dewar. A n integral part of the apparatus is a brass plate which seals against an O-ring flange. This plate has al l of the vacuum-tight electrical and mechanical connections necessary to operate the experiment, as well as ports for filling the dewar, etc. The low-temperature end of the apparatus is suspended from the top-plate by three stainless steel tubes. A t intervals down these tubes are placed radiation baffles of 1/16 inch copper plate. 32 Chapter 4. Apparatus for recombination spectrometry of D 2 33 vacuum i n s u l a t e d p a r t of de.war bias solenoid t o p p l a t e V f l o w - b a f f l e 309MHz r e s o n a t o r d i s c h a r g e r e s o n a t o r i i magnet % J Figure 4.1: Overview of the 309 MHz cryostat. Chapter 4, Apparatus for recombination spectrometry of D 2 34 4.2 The flow-baffle The lowest radiation baffle is more than just a plate. We call it the flow-baffle, for reasons which wi l l become apparent. A n important constraint in the design of the experiment A was the length of the dewar. The part of the apparatus that must be kept cold extends from the bottom of the dewar to only about 20 cm below the top of the vacuum insulated section, above which the temperature of the dewar walls is 77 K ( / -N 2 at one atmosphere). In pumping a helium bath down from 4.2 K to below 1.6 K , less than 60% of the helium is left [37]. This means that it is necessary to f i l l the dewar with / - 4 He , pump it down, and then fi l l and pump it down at least once more without warming up, in order to have a sufficiently deep bath of / - 4 He. In order to permit / - 4 H e transfer without pressurizing the dewar, the flow-baffle is equipped with a needle valve. A / - 4 H e transfer siphon is inserted down one of the support pipes to the baffle, and when a refill is desired, the needle valve is opened. The needle valve is operated v ia a stainless steel tube inside another of the support tubes. 1 Another important feature of the flow-baffle is that a flow of cold helium from the transfer siphon may be vented up a tube inside the remaining support tube. The flow-baffle may thus be kept colder than it would be otherwise, which reduces the heat load on the main bath. The operation is much like a flow-cryostat. In addition, the leads for the superconducting solenoid (described later) enter the cryostat through this vent tube. The cold helium gas diverts much of the heat load conducted down and generated in the leads. Below the flow-baffle, the remainder of the experiment is suspended from three th in -wal l stainless steel tubes, to reduce the conduction heat load. The apparatus can be considered as three modules: the resonator assembly, the discharge, and the magnet. *As built, the thread of the valve is at the top plate. This caused problems, as the valve would jam, probably due to differential thermal contraction. A better design would be to have the thread down with the needle. Chapter 4. Apparatus for recombination spectrometry of D 2 35 4.3 The resonator assembly The resonator is used to measure the D atom density, by using pulsed magnetic resonance on the (3-6 transition at 309 M H z . This is a longitudinal transition, responding to an R F field along the bias field of 38.9 G. Since a fundamental-mode cavity at 309 M H z would be too large, a lumped constant type resonator was used. In order to obtain a good signal to noise ratio, the resonator was fabricated out of superconducting materials. A cross section of the resonator assembly is shown in figure 4.2 4.3.1 Quartz tube The 309 M H z resonator is of the split ring type [38]. The conductors are 6 cm long films of lead evaporated on the inner and outer surfaces of a 50 m m o.d. 2 m m wall fused quartz tube. On the outer surface the film forms a ring with a longitudinal gap approximately 1 m m wide. On the inner surface the film is a plate bridging this gap. Figure 4.3 shows the layout of the lead film on the quartz tube. The lead films are approximately 1 fxm thick, and so do not distort the bias magnetic field which is parallel to the axis of the tube. In the lumped circuit approximation, the ring is a single turn inductor shunted by two capacitors in series. In addition to small size (and resulting higher filling factor) and the possibility of a uniform bias field along the R F field, this type of resonator has good separation of electric and magnetic energy, most of the electric energy residing in the quartz. This reduces frequency and Q shifts caused by motion of the cell, changes in helium level, etc. The quartz tube is held at the bottom by a ring of flat brass fingers epoxied to a region around the quartz roughened by sandblasting. This results in a very rigid mount, but one that can also accomodate the differential thermal contraction without breaking the quartz. Chapter 4. Apparatus for recombination spectrometry of D 2 Figure 4.2: Cross section of the 309 MHz assembly. Figure 4.3: Conductor configuration on the quartz tube of the 309 MHz resonator. Chapter 4. Apparatus for recombination spectrometry of D 2 38 4.3.2 Shield The resonator sits at the centre of a hollow lead cylinder. This superconducting tube has three functions. Firstly, it serves as the housing for the resonator, confining the R F fields while not degrading the Q. Secondly, it provides the uniform bias field of 39 G in which the 8-6 transition frequency is minimized. This field is trapped in by allowing the tube to go superconducting with an external field applied. The external field is generated by a coil immersed in the l iquid N 2 jacket around the dewar and is set to zero once the tube is fully superconducting. Thirdly , the tube shields the resonator from the stray magnetic fields of the high-field solenoid, which are estimated to be about 100 G at the end of the shield when the solenoid is energized to 30 k G . The lead tube is 30 cm long, 8.0 cm inside diameter with walls 3 m m thick. The thickness was chosen, on the basis of a simple energy argument, to be able to withstand the magnetic pressures involved without the destruction of the superconductivity. The lead was cast between aluminum (on the inside) and brass (on the outside) tubes, in the presence of copius amounts of standard plumbing flux so that the lead would bond to the brass. The aluminum tube was subsequently bored out to leave a bare lead inner surface. The brass tube was left on to make the shield stiffer and facilitate mounting in external support structures. 4.3.3 Coupling The resonator is coupled by separate transmitting and receiving loops to 50 0 coaxial transmission lines. The larger receiving loop is critically coupled, while the smaller transmitting loop is coupled at about —20 dB . This is both to maximize the signal and to minimize the noise from room temperature which would otherwise be added to the signal from the atoms. To vary the coupling, the transmission lines and loops could be Chapter 4. Apparatus for recombination spectrometry of D 2 39 moved vertically through sliding seal and screw arrangements on the cryostat top flange. The sizes of the loops were chosen on the basis of the quasi-static model for the resonator magnetic field; no modifications were necessary. 4.3.4 Specifications The unloaded quality factor Q0 of the resonator at room temperature is quite low, ap-proximately 30. In l iquid helium at 4.2 K , the lead is superconducting and Q0 ~ 3 x 10 4 . Upon pumping the helium bath down to 1.25 K , Q0 improves to 1.4 x 10 s . Since the loss tangent for fused quartz below 4 K is approximately 1 0 - 6 [39], an upper bound of Qo ~ 10 6 can be expected. Coarse tuning of the resonance frequency was done by warming up and tr imming the inner lead plate. Starting from 276 M H z , the first iteration took it to 301 M H z and the second took it to 311 M H z . Unfortunately this last step overshot the desired frequency of 308.661 M H z more than the fine tuning could compensate, and so it was necessary to add a small copper foil on the quartz tube just below the gap in order to increase the capacitance. Fine tuning is done in situ by moving a fused quartz rod in the fringing electric fields near the gap. This can lower the frequency by about 1 M H z . This rod was the main source of sensitivity of the resonance frequency to vibrations, and had to be secured carefully. The effective volume of the resonator was estimated using the quasi-static model for the magnetic fields. The result was Ves = 232 c m 3 . Treating the ring as an inductor, we calculate L = 21 nH . After the coarse tuning, the inner lead plate overlapped each end of the ring by 20 mm. W i t h each of the overlapping sections treated as capacitors, the net capacitance shunting the inductor is (using e = 3.78 for fused quartz) C = 10 pF . The resulting resonance frequency is 347 M H z , somewhat higher than the experimental value (without the copper t r im foil) of 311 M H z . This is probably due to fringing electric Chapter 4. Apparatus for recombination spectrometry of D 2 40 fields, which would tend to lower the frequency. 4.4 Bias coil In order to provide a fairly uniform bias field of 39 G to trap into the superconducting shield, a solenoid was placed in the l iquid nitrogen jacket surrounding the Z- 4He dewar. Having the coil in the l iquid nitrogen-as opposed to around the outside of the dewar-reduced considerably the power and copper requirements. The coil was wound from A W G 16 copper magnet wire on a 5 inch diameter lucite form which was then slipped onto the / - 4 H e dewar and suspended from a lucite plate sitting on the r im of the surrounding / - N 2 dewar. The single layer coil is 50 cm long, the central 30 cm wound at a density of 1 turn per cm, the remainder wound at 2 c m - 1 as an end-correction. To trap 39 G into the lead shield, a current of 31 A was passed through the bias coil while the shield went superconducting. 4.5 The discharge The discharge and magnet modules are shown in figure 4.4. Atoms are produced by an R F discharge which dissociates the D 2 frozen on the cell walls. The discharge is a helical A / 4 resonator with lucite supports. The R F fields are confined by a copper can of 6 cm inside diameter, open at both ends to allow the cell to pass through. The helix consists of 14 turns of 1.7 m m copper wire on a 2 cm radius in a 5 cm length; the resonance frequency is 46 M H z . The resonator was init ial ly coupled to a 50 coaxial cable through a 3.3 p F capacitor three turns from the shorted end. This gave near-critical coupling in l iquid helium, wi th a loaded Q ~ 2700. It was found that the init iation of a discharge changed the resonance, cutting off the R F power to the discharge and squelching atom production. The capacitor was subsequently increased to 33 pF , making the loaded Q ~ 150. After Chapter 4. Apparatus for recombination spectrometry of D 2 ,1/ y A s \ •- d i s c h a r g e housing d i s c h a r g e helix 2L3 ^- magnet windings c e l l s e a t Figure 4.4: Discharge for production of atoms and superconducting solenoid. Chapter 4. Apparatus for recombination spectrometry of D 2 42 this modification atoms were observed. The discharge pulses were generated by an Arenberg Model PG -650C pulsed oscillator. Pulse duration was 10 fxs; typical power levels were 400 W . 4.6 The magnet The magnet is a 2 inch bore superconducting solenoid 2 with a correction notch on the inside. The current supply 3 is connected to the magnet through a bundle of 14 1/16 inch brass rods inside a 3 /8 inch stainless steel tube. When the current is flowing, the heat load on the cryostat is reduced by the cold helium gas flowing (from a storage dewar) up this tube. The brass rods are tinned with solder so that at and below the lowest baffle (where the helium flow is supplied) they are superconducting. The rods end just above the resonator shield and the current is carried below by copper clad N b T i wire. Connection is made to the magnet by clamping the supply leads to tinned copper tabs which are soldered onto the magnet wire. A pair of diodes were placed across this joint to protect the magnet from voltage breakdown in case of lead burnout. As mentioned previously, the magnet was not used in the present experiments. 4.7 Mobile cell The mobile sample cell is a cylindrical pyrex bulb, whose volume is about 20 c m 3 . On the top of the bulb is a pyrex cone tapering to a stalk with a loop at its tip. The cell is suspended from a windlass by a double strand of fine stainless steel wire passing through the loop. The windlass is mounted on the shaft of a stepping motor so that the cell can be moved vertically (under computer control) to any desired altitude. Posit ion feedback is provided by an arrangement of pulleys and switches which monitors the tension in the 2Borrowed from Prof. James Carolan. 3 A Hewlett-Packard 6260B DC Power Supply Chapter 4. Apparatus for recombination spectrometry of D 2 43 suspending wire and indicates whether the cell is ful ly raised (seated in the resonator) or is fully lowered (seated in the magnet). Intermediate positions, such as the discharge resonator, are reached by counting stepping motor steps. In the resonator, the cone on the top of the cell seats in the mouth of a pyrex tube so that the cell would not swing to and fro and cause the resonant frequency to vary. The frequency shift of the resonator when the cell entered was 0.1 M H z , and the reproducibility was such that the resonator d id not need re-tuning when the cell was moved to the discharge and returned. The cell was filled with 100 torr D 2 and 500 torr 4 H e , and was then sealed by melting and pinching off the f i l l line on the bottom of the cell. This amount of 4 H e wi l l give a saturated Z- 4He f i lm for temperatures below 1.4 K. The effective sample volume for this cell is given by the small sample form: Veff = rjVs, where Vs = 20 c m 3 and TJ = Vs/VTes = 0.09. For this cell, then, Ven ^  1.7 c m 3 . 4.8 Fixed cell A long cell spanning the resonator and discharge regions was used to observe D without having to move the cell between the regions. The cell was 3 0 m m o.d. 2 m m wall pyrex tubing sealed at the ends, with total length about 35 cm. The bottom of the tube sat just below the discharge, the top reached the top of the split-ring resonator. Atoms are produced by the dissociation of D 2 in the discharge and reach the resonator by diffusing through the 4 H e vapour, possibly assisted by helium refluxing driven by the discharge. The effective sample volume for the long cell was found by a Monte Carlo integration of the appropriate function of the magnetic field within the quasi-static approximation. For the first maximum in the signal amplitude as a function of pulse duration, Veff = 4.5 c m 3 , which is what one would find using the small sample approximation with Vs equal to the volume of a 6.5 cm length of the sample tube. Recall that the split ring Chapter 4. Apparatus for recombination spectrometry of D 2 44 is 6 cm long. The filling factor appropriate to use in estimates of radiation damping is 17 = 7 ( ^ ^ ) ^ 0 . 1 4 . 4.9 Thermometry A germanium resistance thermometer 4 is attached to the flange at the bottom of the resonator shield, where it is always immersed in the superfluid helium. The helium, when superfluid, is a good enough thermal conductor that it may be assumed isothermal. This thermometer was calibrated in situ against the helium vapour pressure, measured at the top of the dewar using a McLeod gauge. Auxi l l iary thermometers were Al len-Bradley 200 0 1/8 Watt resistors. These were placed on the flowbaffle and on the resonator shield (as a diagnostic during the setting of the shield). 4.10 The spectrometer For pulsed magnetic resonance at 309 M H z , a homebuilt spectrometer is used. The spectrometer is of the superheterodyne type, with an IF of 20 M H z . A block diagram of the spectrometer is shown in figure 4.5. The 329 M H z L O (at +10.6 dBm) is provided by a HP8662A synthesized signal generator, which also has a 10 M H z reference signal which is doubled, amplified and split to produce the three 20 M H z L O signals (at +7 dBm) used by the transmitter (one signal at 0 degrees) and receiver (signals at 0 and 90 degrees). The synthesizer is locked to a 10 M H z Rb clock whose calibration is available through online monitoring by N B S of a Global Positioning System (GPS) receiver. 5 Using relays, the spectrometer can easily be switched into a tuning mode, where the synthesizer output (now 309 M H z at low level) is fed down the transmitting line and the output of the 4Lakeshore GR-200A-1500 #21177. 5On loan from the US National Bureau of Standards in Boulder, Colorado. Chapter 4. Apparatus for recombination spectrometry of D 2 45 amplifiers on the receiving line (with the preamplifier turned off) is fed into a diode detector. 4.10.1 The transmitter The tipping pulses are generated by mixing the 329 M H z L O and the 20 M H z L O , pro-ducing sidebands at 309 and 349 M H z . The upper sideband is reflected by the resonator. The lower sideband drives the resonator to tip the atoms. In order not to have any power at al l in the sidebands after the pulse, both LO 's are gated wi th F E T switches before entering the mixer. During the pulse, the power fed into a 50 f l load is — 1 9 d B m (12.6/ /W); half of this is in each sideband. The T T L pulse signals are provided by a Bruker pulse programmer. 6 4.10.2 The 309MHz preamplifier A sensitive magnetic resonance system requires a good low-noise preamplifier. For this we constructed a single stage amplifier using a Mitsubishi MGF1412 GaAs F E T . The amplifier is capable of being cooled; below 20 K , its noise temperature should be less than 10 K. In the work described in this thesis, the amplifier was kept at room temperature. This avoided complications with the setting of crit ical coupling, self-oscillation of the amplifier, etc. The noise temperature, including the semi-rigid co-axial line from the resonator, was then roughly 90 K. The power gain of the amplifier was 22 dB at 309 M H z and its bandwidth was 50 M H z . The F E T was biased 7 to ID = 10 m A and VD = 2 V . 6Borrowed from Prof. Myer Bloom. 7The bias circuit was a Berkshire Technologies PS-3B power supply, which adjusts the gate voltage to maintain a constant drain current. Chapter 4. Apparatus for recombination spectrometry of D2 46 T T L p u l s e P u l s e d s i d e b a n d g e n e r a t o r eOMHz L O 0 d e g r e e s I n a g A 2 0 M H z L Q _ 90 d e g r e e s Z I 3 2 9 M H z L D N o t e s ' A l l d i r e c t i o n a l c o u p l e r s M l n l - C i r c u l t s Z F S C - 1 0 - 2 1 <10dB) ^.ejmtnlxer M l n l - C i r c u l t s Z F M - 1 5 D C+ lOdBn L D ) ^o»Wnixers M l n l - C i r c u l t s Z F M - 3 <+7dBn L O ) 3 2 9 M H z L D I n p u t a t +10 .6dBn 2 0 M H z L D I n p u t s o t + 7 d B n transmit loop A R e a l A 2 0 M H z L D 0 d e g r e e s T u n i n g o u t A A resonator receive loop Figure 4.5: Block diagram of the 309 MHz spectrometer Chapter 4. Apparatus for recombination spectrometry of D 2 47 4.10.3 The receiver The preamplifier is followed by 44 dB of gain, a bandpass filter about 20 M H z wide 8 and another 24 dB of gain. The signal is then mixed with the 329 M H z L O to translate it from 309 M H z to 20 M H z , filtered at 20 M H z and amplified 9 dB. It is then split and mixed against the 0 and 90 degree 20 M H z LO 's to produce in-phase and quadrature signals about D C . These signals pass through 100 kHz low pass filters and are amplified with a voltage gain of 10. Due to technical problems with our existing computer driven data collection system, only one of the phases was digitized. The signal was fed through an oscilloscope preamplifier into a Nicolet 1170 signal averager. The observed free induction decays could then be transferred to computer v ia an RS-232 connection. The receiver is calibrated using a Rhode and Schwartz synthesizer, the ouput of which is measured with a HP435B power meter and then fed through calibrated attenuators into the preamplifier input. The overall (power) gain of the spectrometer, into 501) and including the preamplifier, is 94 dB. The bandwidth is approximately 5 kHz , determined primari ly by the D C amplifiers. As mentioned previously, the noise temperature of the preamplifier, which determines the sensitivity of the spectrometer, was about 90 K. The sensitivity could be improved for future experiments by cooling the preamplifier, providing the problem of amplifier self-oscillation can be avoided. 8This filter is necessary to reduce the total noise power, lest subsequent amplifiers be saturated. It also gives a factor of two in the signal to noise by eliminating the noise in the sideband not containing the signal. Chapter 5 Experiments on deuterium in low magnetic field In this chapter, the experiments on deuterium using magnetic resonance at 309 M H z are described. The apparatus was described in the preceding chapter. After discussing the general procedure, and some measurements of the properties of the apparatus, we wi l l present the experiments more or less chronologically. 5.1 General Procedure Once the insert is assembled and the electrical connections checked, the insert is placed into the dewar. The dewar and its vacuum jacket are then evacuated, and the dewar is filled with dry nitrogen gas. The l iquid nitrogen jacket is filled and kept topped off for about 6 hours, after which time the experiment is essentially at 77 K. After this pre-cooling, the nitrogen exchange gas is replaced by helium, and the transfer of l iquid helium into the dewar is begun. When the temperature of the lead shield reaches 20 K , the bias coil in the l iquid nitrogen jacket is energized. The field at the resonator is monitored using the home-made C E S R magnetometer described in appendix A . The shield goes superconducting at the bottom first, 1 trapping in the flux that wi l l determine the final field. As the transition zone passes the magnetometer the field is observed to fluctuate and then fal l back to about 1% higher than the field due to the bias coils. When l iquid helium has been transferred so that the shield is completely submerged, the bias coil is JThe superconducting transition temperature of Pb in a 39 G field is essentially its zero field value of 7.2 K. 48 Chapter 5. Experiments on deuterium in low magnetic field 49 de-energized. No effect is observed on the magnetometer. The profile of the trapped-in field was measured using the magnetometer and found to be more inhomogeneous than the field generated by the bias coil alone. This was attributed to the fact that the dimensions of the shield are not very uniform. The observed inhomogeneity in the vicinity of the resonator (but between the resonator and the lead shield) was of the order of 0.05 G e m - 1 , consistent with the 0 .1mm distortions measured in the shield dimensions. This requirement of high uniformity was overlooked in the design and construction of the shield. In the future a shield could be made to much closer tolerance should it be desired. In any case, field inhomogeneities had no effect on the observed signals, due mainly to the short spin-exchange relaxation times. After verifying that the correct field had been trapped into the shield, the magnetome-ter is replaced by the quartz fine-tuning rod and the frequency and Q of the resonator are checked. The helium is then pumped down, first using a Stokes pump and finally using a Roots blower backed by the Stokes pump. The pumpdown lowers the helium level to be-low the resonator, and it is necessary to transfer helium at least twice through the needle valve on the flowbaffle in order to top off the level sufficiently to give reasonable running times. Typical ly the experiment could run for 3 hours before needing a refill. Whi le running at the lowest temperatures, a constant flow of helium from a storage dewar was maintained through the flowbaffle to keep its temperature about 15 K. This was neces-sary to get the bath temperature below 1.2 K. A faster flow of helium was too expensive, and had l itt le effect on the base temperature. The base temperature obtainable was just below 1.1 K , varying somewhat with the helium level. A more relevant quantity is the helium vapour pressure, since the recombination rates vary directly with helium density. W i t h the level above the top of the shield, we could reach Pue — 0.30 torr (1.103 K ) . W i t h the level just above the split -r ing of the resonator, we could reach Pue = 0.23 torr (1.071 K ) . The highest operating temperature was about Pne = 0.5 torr (1.17 K ) , at Chapter 5. Experiments on deuterium in low magnetic held 50 higher temperatures the D atom density was inconveniently low, and the sample lifetime too short. Once at operating temperatures, the receiver loop is moved so as to crit ically couple to the resonator and the loaded quality factor Qi measured. Typical ly QL — 7 x 10 4 . The transmission of power from the transmitter to receiver loop is measured using the Rhode and Schwartz synthesizer and HP435B power meter. F rom the power output P0 from the critically-coupled receive line during the tipping pulse we can calculate the duration 7V/2 of a 7r /2 pulse using equation (3.24). Typical ly P0 ~ —45 d B m , giving r T / 2 — 100 /is. Once the pulse width is set, the discharge is fired to dissociate D 2 and the search for atoms begins. 5.2 Mobile cell Our first experiments were made with the mobile sample cell. The cell could be moved when cold, without noticeably heating the bath, between the resonator, discharge and magnet regions. W i t h the cell in the discharge, the discharge could be fired. The glow was visible through the glass dewar if all the room lights were turned off. After firing the discharge, the cell was returned to the resonator. No signal from atoms was observed, even at the base temperature of about 1.2 K. We decided that it would be wise, in these preliminary experiments, to use a cell extending from the discharge to the resonator, so that opt imum discharge and resonance parameters could be determined without the difficulties associated with moving the cell back and forth. Whi le the experiment was warmed to replace the cell, the pumping system was modified to incorporate the Roots blower in an attempt to lower the base temperature. Chapter 5. Experiments on deuterium in low magnetic field 51 5.3 Fixed cell The fixed cell, spanning the discharge and resonator regions, was used in the next cool-down. The Roots blower gave a base temperature P n e — 0.2torr (1.06K). St i l l , no signal from atoms was seen, and we began to fear that the bias field was not being set correctly. A careful investigation of the discharge behavior revealed that the init iation of a discharge altered the characteristics of the discharge resonator sufficiently that the almost al l of the pulse energy was reflected back into the generator. This 'self- l imiting' was due to the high coupled Q (2700) of the resonator. A t the next cool-down, with the resonator overcoupled to a Q of 150, much more of the power was absorbed in the discharge and atoms were observed. Use of the fixed cell permitted signal averaging in a steady state situation, where the discharge was firing repeatedly. Typical ly the discharge was synchronized with the tipping pulses, the firing of the discharge occurring a few tens of milliseconds before the tipping pulse. As it took some time to bui ld up an observable atom density (the diffusion time from the discharge to the resonator region is many seconds), the exact time delay is not important. A typical free induction decay (FID) following a 7r /2 pulse is shown in figure 5.1. The tipping pulses were repeating every 0.1s and the F ID is the sum of the response to 32 pulses. The amplitude decays exponentially in time, and so the fid can be characterized by an in i t ia l amplitude A(0), a frequency / , and a decay time constant T 2 . The in it ia l amplitude is related to the density of atoms using the spectrometer calibration and equation (3.27). For the F ID of figure 5.1 the density is 5 x 1 0 1 0 c m - 3 . The relaxation time constant T 2 ~ 12 ms. Chapter 5. Experiments on deuterium in low magnetic Held 52 Figure 5.1: Typical free induction decay. Chapter 5. Experiments on deuterium in low magnetic Held 53 5.4 Decay measurements Recombination of atomic deuterium is observed by turning off the discharge and observing the in it ia l amplitudes of the FIDs as a function of time. Due to the non-automation of the data collection and the fast decay of the density, only a few points could be collected before the atoms were gone. A typical result is shown in figure 5.2. Each point entailed some signal averaging, typically 32 sweeps. Provided that the change of density during the averaging is essentially linear in time then the appropriate time to use is the middle of the measurement. The temperature of this run is 1.133 K , as determined from the helium vapour pressure Pue = 0.38 torr. The decay is exponential in time, with a time constant r ~ 15s. The decay of the deuterium density after turning off the discharge was observed as a function of temperature. The decay rate r - 1 is plotted logarithmically versus 1 / T in figure 5.3. The decay rate increases with increasing temperature. A n Arrhenius interpretation of the data yields an activation energy of about 15.2 K (the line drawn in the figure). 5.5 Relaxation measurements From our fits to the FIDs to determine the D atom density, we obtained the transverse relaxation time T 2 . For al l of the FIDs, we found T 2 ~ 10 ms. This is very short, even given the large cross-section for D-D spin-exchange. In order to have T 2 = 10 ms, the deuterium density would have to be 7 x 1 0 n c m - 3 , more than 10 times higher than it typical ly was. Also, the T 2 was nearly constant as the D density decayed away. It was tempting to ascribe the short T 2 to spin-dephasing due to magnetic field inhomogeneities. However, the F ID fit the exponentially damped sinusoid quite well, which one would not expect of a field effect. Chapter 5. Experiments on deuterium in low magnetic field Figure 5.2: Typ ica l decay of atom density after turning off the discharge. Chapter 5. Experiments on deuterium in low magnetic held Figure 5.3: The deuterium density decay rate versus reciprocal temperature. Chapter 5. Experiments on deuterium in low magnetic field 56 In an attempt to find an explanation for the short T 2 , we we also measured the longitudinal relaxation time T\. In order not to be bothered with the decay of the atom density with time, the T\ measurements were made with the discharge and tipping pulses firing repeatedly. The most successful technique was to use a ir — TT/2 pulse sequence and look for the pulse separation for which the amplitude of the F ID after the TT/2 pulse vanishes. This occurs when the pulse separation is T i In 2. What we found was that the Ti was also roughly 10 ms. Diffusion or radiation damping could give comparable T i and T 2 , but, as discussed in chapter 3, both of these mechanisms are not effective on these short timescales. In any case, they do not give exponential FIDs. The only possibility remaining was that there was a sizable H atom concentration. The short T i and T 2 would then be due to spin-exchange in H-D collisions. Using the results of Appendix C, the hydrogen density required to give I i = 10 ms is approximately 4 x 10 1 1 c m - 3 , an order of magnitude higher than the D density, but not atypical for a hydrogen experiment. This raised the possibility that the exponential decay of the D density after turning off the discharge was due to the scavenging of the D by the H in the reaction H -f D + 4 H e —> H D + 4 H e (5.1) The rate of this reaction increases with increasing temperature, as the 4 H e vapour density rises. This would tend to explain the shortening of the D lifetime at higher temperatures. However, assuming a constant H density and a rate constant proportional to « H e (the 4 H e vapour density), the effective activation energy should be L^/k^ + ST/2 ~ 9 K. The observed energy, 15 K , is somewhat higher. We attempted to prepare an ultra-pure sample of D 2 by adsorbing the gas onto activated alumina in a pot held above l iquid 4 H e , raising the pot away from the l iquid unt i l the D 2 vapour pressure above it was about 1 torr, and then pumping off 93% of Chapter 5. Experiments on deuterium in low magnetic held 57 the gas. The gas remaining on the alumina was to be our pure D 2 . Unfortunately a mass-spectrometer analysis of the gas indicated that it was no purer than the original. This may be due to a distribution of binding energies on alumina and perhaps using a more uniform substrate such as grafoil (exfoliated graphite) would have given better results. We mention this failure lest others suffer the same fate. The 'purified' gas was used in a new cell (before we knew its purity). The D density decay time constant was the same as before, although T i and T 2 were shorter, about 5 ms, indicating that the hydrogen density had been doubled. This showed that although the hydrogen was shortening the FIDs, it was not causing the decay of the D density. In the next chapter, experiments that reinforced this conclusion are described. The exponential decay of the deuterium density is due to the penetration of the / - 4 H e f i lm coating the walls of the cell. Once the atoms reach the D 2 substrate they are strongly bound there and do not return. The interpretation of the data in terms of this mechanism is discussed later in the thesis. Chapter 6 Experiments on H - D mixtures in zero-field In order to ascertain the effect of hydrogen contamination of the D 2 in our low tempera-ture experiments, we used zero-field magnetic resonance at 1420 M H z to look for atomic hydrogen. The apparatus was essentially that used by Morrow in his studies of atomic hydrogen at 1 K [34]. A mixture of D 2 (99.65% isotopic purity) and 4 H e was sealed into a pyrex bulb. The bulb and resonator were placed in a / - 4 H e bath pumped down to about 1.1 K . After an R F discharge, a signal from H atoms was observed. In steady state, with the discharge pulsing, the H atom density was approximately 1.5 x 1 0 l o c m - 3 as determined by an absolute calibration of the 1420 M H z spectrometer. The free in -duction decay time constant was very short for a hydrogen signal, about 40 ms. This got rather longer after the discharge had been off a short time. We attribute the excess relaxation ( T 2 - 1 ) to spin-exchange in H-D collisions. Using the result from appendix C for the contribution to the relaxation rate due to H-D spin-exchange, T , - 1 = ^ G H ^ D (6.1) with GDD — 2.4 x I O - 1 0 c m 3 s - 1 , the deuterium density corresponding to T 2 = 40 ms is about 1.4 x 10 1 1 c m - 3 . The H atoms thus made up about 10% of the atoms in the gas. For comparison, the H atom impurity level in the D 2 used to fill the cell is less than 0.35%. Upon turning off the discharge the relaxation rate T 2 _ 1 was observed to decay quickly as the D density dropped. After making a small correction to T 2 - 1 due to radiation 58 Chapter 6. Experiments on H-D mixtures in zero-field 59 damping of the H signal, the remaining relaxation, proportional to the D atom density, was seen to decay exponentially with a time constant similar to that observed in our 309 M H z apparatus. The hydrogen atom density did not decrease appreciably during this time, although there was init ial ly much more D than H. Initially we thought that this ruled out recombination of H with D as the cause of the decay of the D density. However, on several occasions, after the D atoms had gone, the H atom density increased slowly to several times its in it ia l value. A plausible explanation of this is that the D atoms penetrate the / - 4 H e f i lm and adsorb onto the D 2 substrate. They then migrate about on the surface and eventually encounter a H D impurity molecule. The exothermic exchange reaction D + H D -+ D 2 + H (6.2) then takes place, liberating an H atom and about 500 K of energy, 1 which kicks the H atom back through the f i lm into the gas. Therefore, after the discharge is off and the D atoms have all gone to the substrate, the production of H continues unt i l the D atoms have all found H D to exchange with or other D atoms to recombine with. The increase in H atom density was essentially exponential, with a time constant of 200 s. For the exchange of D with H D on the surface to be this slow, the D atoms would have to be diffusing slowly, for example by hopping or tunneling from site to site. A free 2-d gas would be inconsistent with such a long time constant; the D atoms would find H D or another D atom very quickly in that case. The final decay of the H atom density was consistent with gas-phase recombination of H with H, and permitted a check of the density calibration by comparing the decay rate to the rates measured by Jochemsen et al [41]. JThis reaction occurs in solid D2-HD mixtures when atoms are produced by irradiation. [40] Chapter 6. Experiments on H-D mixtures in zero-held 60 6.1 Ultra-pure deuterium In an attempt to remove entirely the complication of hydrogen contamination, we ob-tained a sample of very pure D 2 . This was pure o r t / i o -D 2 prepared about 10 years ago, and the ortho-para separation would have eliminated any H D impurity. A cell was loaded with this D 2 and enough 4 H e to give a saturated f i lm, and placed into the 1420 M H z ap-paratus. Again, a signal from H was seen, with a short T 2 due to spin-exchange with D. The H atom density was comparable to the previous cell, which contained D 2 of only 99.65% isotopic purity. The time constant for the decay of the D density (inferred from the T 2 ) was somewhat longer than in the previous (impure) cell. 6.2 Adulterated deuterium Since it appeared that the H contamination was unavoidable, we decided to investigate just what effect the contamination might have had on the decay of the D density in the data collected so far. To do this, a sample of D 2 was prepared wi th a 0.9% admixture of H 2 . In this cell, the H atom density after discharge was about 5 times as high as in the cell containing normal purity D 2 (99.65%). Once again, the D atom density was observable through an increase in T 2 _ 1 , and the decay of the D after the discharge was measured. As the hydrogen density was rather high, the T 2 had to be corrected for radiation damping. Encouragingly, the D atom lifetime was not any shorter than in the 'normal purity' , where the H atom density was much lower. This conclusively ruled out recombination of D with H as the cause of the D atom decay. In order to be sure, a further sample was prepared, with a 5.7% H 2 impurity. The H atom density was the highest yet, 1.8 x 1 0 1 2 c m - 3 , and the shortening of the T 2 due to the D was too small compared to radiation damping for the D density decay to be extracted in the usual way. We were able to measure the D decay using a trick that Chapter 6. Experiments on H-D mixtures in zero-field 61 eliminated radiation damping altogether. Rather than use T2, we keyed on T i to give us the D density. The technique we used to measure the time dependence of T i was based on a 7r — TT/2 pulse sequence. After a TT pulse, there is no precessing magnetization because the spins are neatly inverted. If a 7r /2 pulse is applied a time r = T i m 2 after the 7r pulse then, again, there is no signal because at this time the relaxing longitudinal magnetization, on its way to its equil ibrium value, is passing through zero. The decay of T f 1 is obtained by observing the increase of the pulse-spacing time r required for null-signal as a function of the time after the discharge is turned off. Practical ly, we set r , waited for the D density to decay through the null-signal condition, set r longer, waited for the D density to catch up, and so on. The amplitude of the response to the 7 r /2 pulse was caught by a box-car integrator, whose output as a function of time was recorded using a slowly sweeping signal averager. Since we measure a quantity at nul l -signal, radiation damping is absent. A series of decays were observed at a temperature of 1.055 K. The D density just after the discharge was turned off was 8 x 1 0 1 0 c m - 3 . The density decay time constant was 5.0(2) s, much shorter than in the cells with lower H densities. This decay can be attributed to recombination of D with H . Chapter 7 Discussion and Conclusions 7.1 ESR at 115GHz Experiments were made on atomic hydrogen and deuterium using E S R at 115 G H z . These were the second generation of experiments using an apparatus constructed by Bryan Statt for his doctoral work. The atoms are confined to a microwave cavity in a field of 41 k G and cooled to below 1 K by a dilution refrigerator. Under such conditions the atoms are electron-spin polarized. We were able to observe, for the first time, the E S R signal from hydrogen atoms physisorbed on the / - 4 H e f i lm confining the sample. This enabled us to determine the binding energy of H on / - 4 H e to be = 1.00(5) K , the first direct measurement of this quantity. Atomic hydrogen is unstable against recombination into diatomic molecules, a process which is of interest from both technological and academic viewpoints. The recombina-tion measurements of Statt were extended in our experiments, both by increasing the range of temperatures and by more careful analysis of systematic effects. A particu-larly interesting result arose from the measurement of the temperature dependence of the phenomenological rate constants Kaa and Ka\> (for the formation of para and ortho H 2 in collision of two H atoms). A t the highest temperatures (above about 0.5 K ) , we found that Kaa was increasing with temperature, in stark disagreement wi th the expected behaviour obtained by extrapolating zero-field results. We were able to show that this anomaly was due to resonant recombination v ia the (14,4) level of H 2 . This level does 62 Chapter 7. Discussion and Conclusions 63 not play an important role in recombination in zero magnetic field. In our magnetic field of 41 k G , however, the level is predissociated by the intra-atomic hyperfine interaction, and makes a rather large contribution to recombination. In order to explain the anoma-lous temperature dependence of Kaa, it was necessary to develop the theory of resonance recombination of H due to hyperfine predissociation of weakly-bound molecular levels. We also used the E S R apparatus to study atomic deuterium. As previous workers had discovered, D is much more difficult to work with than H. It recombines much more quickly on the / - 4 H e surface, which restricts densities to very low values. Nevertheless, we discovered that resonant recombination was occuring in D, overtaking surface recom-bination at temperatures above about 0.7 K. We were also able to generate the first samples of DJ,!., in which both electron and deuteron spins are aligned. 7.2 Resonant recombination A major contribution of this thesis is a detailed theory of resonant recombination of atomic hydrogen and deuterium due to the hyperfine predissociation of weakly-bound molecular levels. This theory was developed in order to interpret the results of our E S R experiments, and is included in this thesis as an appendix (appendix D). As already noted, this theory was used to explain the temperature dependence of the recombina-tion rate in atomic hydrogen. It was also used, together with our E S R data, to predict the magnetic field dependence of resonant recombination of deuterium. It appears l ikely that a measurement of deuterium recombination as a function of magnetic field wi l l reveal strong threshold behaviour, permitting 'recombination molecular spectrometry'. For example, the threshold magnetic field for resonant recombination involving the (21,0) level of D 2 is 22 k G . This value comes from a spectroscopic determination of the level dissociation energy, and is uncertain by 2 k G . Our theory indicates that as the magnetic Chapter 7. Discussion and Conclusions 64 field is increased past threshold, the recombination rate should rise dramatically, indicat-ing unambigously and to high precision the dissociation energy of the (21,0) level. The success of this method wi l l depend, in part, on the size of non-resonant contributions to recombination, which are not known very well. The more weakly bound (21,1) level may also be a suitable candidate for recombination spectrometry, although its threshold could well be masked by other recombination processes. 7.3 The 309 MHz experiments The results of our deuterium experiments in the high-field E S R apparatus and their ex-trapolation to other magnetic fields v ia our theory of resonant recombination were very encouraging. We decided to improve on spectroscopic determinations of D 2 level energies by measuring the magnetic field dependence of resonant recombination of D, or at least lay the foundations for such measurements by observing the zero-field (non-resonant) recombination rates, which might obscure resonant processes. A n apparatus was con-structed to study atomic deuterium by hyperfine magnetic resonance at temperatures near 1 K. The D atoms were produced from D 2 by an R F discharge in a bulb coated with a film of superfluid / - 4 He. Magnetic resonance in a field of 39 G on the longitudinal (3-8 transition at its min imum frequency of 309 M H z was used to observe the atoms. The apparatus has the capability of measuring the recombination rate as a function of mag-netic field for fields up to 30 k G by moving the sample between the 309 M H z resonator and a superconducting solenoid. This capability was not used in the preliminary exper-iments described in this thesis. The magnetic resonance system incorporated a number of innovations in order to overcome the signal-to-noise problems inherent in working wi th D. A superconducting split ring resonator consisting of lead films evaporated onto a quartz substrate, and a low-noise pre-amplifier resulted in enhanced sensitivity. The Chapter 7. Discussion and Conclusions 65 bias magnetic field was trapped into the resonator housing, which is superconducting. The housing wi l l also shield the resonator region from the stray magnetic fields of the superconducting solenoid. The apparatus was used to characterize the behaviour of D in low-field. A n important discovery was that the D atoms penetrate the / - 4 H e fi lm. This caused the D density to decay exponentially with a time constant ranging from 10 s at 1.16 K to 27 s at 1.08 K . From the temperature dependence of the density decay rate, we were able to infer a value for the energy Es required to dissolve a D atom into / - 4 He. A simple kinetic theory was derived, assuming that the only rate l imit ing process is the solvation of the D atoms into the / - 4 H e film (fast diffusion in the vapour and in the film and instant adsorption on the substrate once in the film). In this case, the decay time constant r is given by where fi is the ratio of the D mass in the film to its mass in the vapour, cxiv is the probabil ity that a D atom in the l iquid incident on the liquid-vapour interface passes into the vapour, and Es is the energy required to dissolve the atom into the l iquid. The details can be found in appendix B. The result depended somewhat on the model we chose: Es ~ 14.6 K for ucxiv constant, 1 K lower if potiv ~ T , as it should in the low temperature l imit . These results must be regarded as preliminary, because of possible complications arising from the substantial H atom impurity present in the D sample. The presence (and amount) of H was inferred from its contribution to relaxation of the [3-8 transition v ia spin-exchange in H-D collisions. B y varying the H density, we were able to show that the bulk of the sample decay rate was film penetration, not recombination of D with H. This was done accidentally; a supposedly pure sample of D 2 was prepared; the H impurity density was found to be double what it was before, but the life of the D was Avficxiv E,/kBT (7.1) 4 V Chapter 7. Discussion and Conclusions 66 not appreciably shortened. The short sample lifetimes resulting from the penetration of the / - 4 H e f i lm prevented the measurement of the magnetic field dependence of D-D recombination, and in fact the recombination of D with D was not observed at al l in the present experiments; it was completely masked by the exponential decay. The lowest temperature point, with time constant 27 s and init ial D density n = 6 x 1 0 l o c m - 3 , implies that the zero-field D-D recombination rate constant K at 1.08 K is rather less than 6 x 1 0 - 1 3 c m 3 s _ 1 . This is consistent with the lower-bound result of Mayer and Seidel [42] for zero-field recombination on the / - 4 H e surface. Their result, extrapolated to our conditions by assuming a binding energy of 2.6 K for D on / - 4 He, is K > 7 x 1 0 " 1 4 c m 3 s _ 1 . Because the film-penetration rate decreases very rapidly with the temperature, it may be possible to increase the sample lifetime sufficiently to observe D-D recombination by working at lower temperatures; unfortunately, the rapid increase of surface recombination due to the large binding energy may result in the suitable temperature window being rather narrow. The present execution of the recombination spectrometry experiment would not be easily adapted to a dilution refrigerator. However, it is possible that use of larger 4 H e pumps may achieve a low enough temperature directly. It is useful to speculate on the origin of the H atom impurity. The semi-permeability of the / - 4 H e f i lm (passing D but not H) makes the D especially susceptible to contamination by H if any H are produced by the discharge. In addition, the D atoms penetrate the f i lm and adsorb on the D 2 substrate. There they may encounter H D impurity molecules. The exchange reaction (6.2) wi l l then occur, generating an H atom with sufficient kinetic energy to return, through the f i lm, to the gas. This production of H could be very efficient, one H for every D passing through the fi lm. In connection with this reaction, and with the production of atoms by the discharge, it is worth noting that the H D concentration may be greatly enhanced at the surface of the D 2 substrate by fractionation during the Chapter 7. Discussion and Conclusions 67 cool-down [43]. This might be defeated by using only a l imited quantity of D2, so that the total amount of H D present is small. However, then the D 2 can be easily contaminated by small quantities of H D or H 2 , released, for example, from the glass cell during the sealing-off procedure. 7.4 The 1420 MHz zero-field results It proved possible to use hyperfine resonance on H to observe D, through the effect of H-D spin-exchange collisions. Experiments were made using D 2 charged cells in an apparatus previously develped for the study of H using zero-field magnetic resonance at 1420 M H z . Always, observable H densities were generated by the R F discharge. The T 2 of the hydrogen F ID depended strongly on the D density, and the decay of the D density was measured by measuring the decay of T 2 - 1 . The hydrogen density was varied, in different cells, by doping the D 2 with H 2 . For the highest hydrogen densities, the H signal was so large that radiation damping masked the spin-exchange shortening of T 2 . However, it was sti l l possible to observe the D decay by measuring the longitudinal relaxation rate T i - 1 in a way that eliminated radiation damping. The high H density data implies that the rate constant for H -D recombination KUD — 1.1 x 1 0 ~ 1 3 c m 3 s - 1 at 1.055 K. The temperature dependence of this rate was measured, and it was found that J^HD was essentially proportional to the 4 H e vapour density, indi -cating that the recombination is due to the three-body process H + D + 4 H e -+ H D + 4 H e (7.2) Wr i t ing AHD = &HD nHe) w e find &HD = 6 X 10 - 3 2 c m 6 s _ 1 . This can be compared to fcHH = 0.28 x l O - ^ c m S " 1 for H -H -He recombination [41, 17]. A curious discrepancy arose when we attempted to apply the measured H-D recom-bination rate to the data obtained in the 309 M H z experiments. The hydrogen density Chapter 7. Discussion and Conclusions 68 in most of those experiments, inferred from the shortening of T i and T 2 due to H-D spin-exchange collisions, was about nn = 6 x 10 1 1 c m - 3 . A t this density, the recombination of D with H should have resulted in sample lifetimes even shorter than those that were observed. A n d yet, further measurments in the 309 M H z apparatus had showed that doubling the H atom density did not significantly shorten the lifetime. Also, the density of H inferred from the T 2 is more than an order of magnitude larger than the density directly measured in the 1420 M H z apparatus with a D 2 sample of the same purity. We have considered several possible explanations for the discrepancy. First ly, the calibration of the 1420 M H z spectrometer may have been in error. This does not seem likely, as we have carefully checked the calibration and errors in our estimation of relevant parame-ters, such as the filling factor, could not possibly be large enough. Secondly, there may have been another mechanism causing the short T i and T 2 in the 309 M H z measurements. We have carefully considered and rejected various mechanisms, and have been unable to come up with anything reasonable. The final (and most likely) conjecture is that the H-D spin-exchange rates calculated in appendix C are too small, resulting in a system-atic overestimation of the H atom density. This could be due to the use of adiabatic ab initio potentials in the spin-exchange calculations. Since the (17,0) level of H D hovers just around the dissociation l imit , the calculated spin-exchange rates would depend sensi-tively on just where the level was. In fact, spectroscopic measurements do not determine whether or not the (17,1) level of H D is bound and it may be a scattering resonance. Further cogitation, and perhaps a direct measurement of the H-D spin-exchange rate, wi l l be necessary before this discrepancy between the two methods of determining the H density can be eliminated. Since both the D 2 and H D molecules have low J levels close to dissociation, the D-D and H-D spin-exchange rates may be substantially different from those calculated in appendix C. This does not weaken the interpretation of the D decay as being due to Chapter 7. Discussion and Conclusions penetration of the / - 4 He film. Appendix A The CESR magnetometer In order to be able to accurately set the bias field in the resonator to the required 39 G a magnetometer that would function at 4 K was required. A n electron spin resonance magnetometer turned out to be ideal for the purpose, having good signal-to-noise at low fields and, of course, being self-calibrating. The magnetometer head is a small L C circuit resonant at a frequency near the elec-tron larmor frequency in the desired field, approximately 107 M H z . The coil of the L C circuit is filled with L i F i L i 1 . This substance is made by subjecting L i F to heavy neutron irradiation. The resulting mobile L i atoms diffuse around and coalesce to form small inclusions of the pure metal. The Conduction Electron Spin Resonance ( C E S R ) signal in this material is relatively strong and quite narrow (AJB = 0.65 G) , and as a result L i F : L i is widely used as a '^marker ' [44]. In addition, the Paul i susceptibility (which determines the signal strength) is essentially temperature independent, which permitted development at room temperature of a magnetometer that would work when immersed in l iquid helium. The capacitor was a 9 cm length of coaxial cable with copper inner and outer con-ductors and teflon dielectric. In order that the head be nonmagnetic at 4 K , pure indium solder was used. The L C circuit is coupled to a 50 f i transmission line by a variable capacitor which is set to give nearly crit ical coupling in l iquid helium. The reflection of microwave power incident on the L C circuit down this line is the signal used to determine 1 The L i F : L i was supplied by Prof. C. F . Schwerdtfeger, who also suggested this as a suitable magne-tometer material. 70 Appendix A. The CESR magnetometer 71 the magnetic field. The magnetometer works as follows. The coil is at the centre of a solenoidal sweep coil. Using the sweep coi l , the field at the L i F : L i is adjusted to make the Larmor frequency of the electron spins coincide with the resonance frequency of the L C circuit, about 107 M H z . From the sweep coil current required to do this the externally applied magnetic field is inferred. The resonant frequency of the L C circuit is fixed, and was chosen so that the sweep coil current would be small when measuring fields close to the 39G required for deuterium hyperfine resonance (this was done in order to reduce systematic effects such as reflection of the sweep coil fields in the surrounding superconductors.) The sweep coil field is sufficiently homogeneous that external fields from zero to 80 G can be measured with ease, although such large sweep currents (200 mA) cannot be employed at room temperature for long without the coil becoming hot. The L C circuit and sweep coil fit inside a 7 m m o.d. pyrex tube, filled with helium gas, that can be inserted into an access port on the cryostat. The spectrometer used to observe the C E S R signal is the same as that used for the deuterium hyperfine resonance at 309 M H z , suitably modified to measure the on-resonance C W reflection coefficient of the magnetometer head. 2 The transmitter is set to run continuously, and the two sidebands are fed through a filter which passes only the one at 107 M H z , the resonant frequency of the L C circuit. This filter also acts as a variable phase shifter. The power incident on the magnetometer head is about —25 d B m , low enough to avoid any saturation effects. The signal reflected from the head is fed into the spectrometer input, whose gain before the first mixer is reduced to 24 dB for this application. The phase of the transmitted signal is adjusted using the sideband filter in order to null either the 'in-phase' or 'quadrature' spectrometer outputs. This 2 A simpler spectrometer could have been employed. During prototyping the reflected 107 MHz was amplified and mixed directly to D.C., which worked satisfactorily. In the present experiments it is convenient, and ties up fewer components, to use the main spectrometer to do the magnetometry. Appendix A. The CESR magnetometer 72 output is fed into an oscilloscope preamplifier, whose output in turn is digitized. The magnetometer is operated through a microcomputer, which communicates v ia a G P I B data bus with the D / A setting the sweep coil current and the A / D digitizing the signal. The sweep coil current is varied linearly in time to pass through the spin resonance. A typical signal is shown in figure A . l . The center of the signal is found by subtracting the average value and looking for the zero crossing. The corresponding current is used to calculate the external field, based on a calibration of the sweep coils done in a mu-metal shield. The field measurement, accurate to ±0.05 G , is updated about once a second. Appendix A. The CESR magnetometer 73 Figure A.l: Typical CESR signal. Appendix B Density decay due to solution of D in liquid helium In this appendix we develop the theory for the decay of the deuterium density due to the penetration of the l iquid helium ( / - 4He) f i lm lining the container. Since the properties of the saturated f i lm are essentially those of the bulk l iquid, the problem we consider is that of D atoms dissolving into / - 4 He. Calculations indicate that D, like H, does not dissolve into / - 4 H e at zero temperature. A t finite temperatures, the quantity determining the importance of solution is Es, the energy necessary to dissolve a D atom in Z- 4He. No experimental determinations of Ea exist for any hydrogen isotope, and calculation is very difficult. For the purposes of this discussion, we wi l l take a calculated value Ea/k^ = 11 K [45, 46]; the theoretical uncertainty is at least a few K. This value of Es/ks, and the values used later for H and T , are obtained from figure 1 of reference [46] by subtracting from fj, (the chemical potential for replacing one 4 H e atom by an impurity atom) the energy L 4 ~ 7 K required to remove a 4 H e atom from the l iquid. We use values for /J, appropriate for / - 4 H e at the theoretical zero-pressure density of p = 0.0172 A 3 . A t the experimental density p = 0.02185 A 3 , Es/k^ is about 12 K larger. It is not obvious a priori which density one should use in order to obtain the best theoretical value, although our choice appears to be the customary one. We shall examine how the solution of D into the f i lm causes the density to decay away and how diffusion both in the vapour and in the f i lm may be expected to modify the results. Consider a gas of D above the surface of l iquid helium at a finite temperature T. Assume for now that the diffusion of atoms is fast enough in the vapour that the density 74 Appendix B. Density decay due to solution of D in liquid helium 75 n of D in the vapour is uniform. The flux of atoms entering the film is given by $vi = -^nvcxvi (B.l) where v — yj8k&T'/nm is the mean thermal velocity of D atoms in the vapour, and cxvi is the thermally averaged probability that an atom in the vapour incident on the surface will enter the liquid. If the density of D just below the surface is n' then the flux of atoms out of the film is $/v = ^n'vfi-1^ (B.2) fi is the ratio of the effective mass of D in /-4He to the mass m of a D atom in the vapour and otiv is the probability that an atom incident on the surface of the liquid from below passes into the vapour. In the absence of recombination, an equilibrium would eventually be reached and we would have $iv = $„; and (equating ideal-gas chemical potentials of D in the vapour and in the film) n' — nu3/2e~E*/kBT'. This implies that the transition probabilities are related by a„/ = ua\v e~Es^kBT. In practice, the /-4He film is coating a solid D 2 substrate. The binding energy of D on bare D 2 is quite high, about 55 K [10]. We assume that the D atoms reaching the substrate are instantly and irrevocably adsorbed. If we further assume that motion within the film is fast enough, then n' ~ 0 and there is only the flux $„/ into the film. The density of a sample of D in a finite volume V and confined by a film of area A then decays away exponentially in time, h = —An, with an inverse time constant A = i & e-B.W ( B > 3 ) Putting in T = 1.1 K, A/V ~ 2/R ~ 1.5 c m - 1 , //aj„ = 1, and the theoretical Ea/k& ~ 11 K we obtain T = A - 1 ~ 6 s. The observed time constant r ~ 20 s would result from Es/k = 12.3 K. Given the approximate nature of the theoretical value, this should be considered quite good agreement. Appendix B. Density decay due to solution of D in liquid helium 76 The temperature dependence of the observed lifetimes were analyzed assuming ficnv is independent of temperature. In this case it can be shown that an Arrhenius plot (In r versus 1 /T ) over a narrow range of temperature gives a straight line of slope Es/&B + T / 2 . The data is plotted in figure 5.3 is fit by the form (B.3) with Ea/k-B = 14.6 K and }iauv ~ 12. 1 In the low-temperature l imit one might expect that the situation is analogous to a particle incident on a potential step. In that case one can show that the transmission probabil ity a/„ ~ T. If at 1 K we are already in this low-temperature l imit , then we would infer that Es/kB = 13.6 K. The above treatment has assumed that the motion of the atoms is everywhere ballistic, or that the diffusion is fast enough that the atom density is uniform in the vapour and zero inside the fi lm. We wi l l examine this more closely. The characteristic time for diffusion to erase radial inhomogeneities in the vapour is given by TE1 = DxlJR2 (B.4) where D is the diffusion constant, Xi.i — 3.83 is the first zero of the bessel function Jx(x), and R ~ 1.3 cm is the radius of the cylindrical cell. Using equation (3.32) for D we see that Try ranges from 70 ms at 1 K to 300 ms at 1.2 K. As TD is always much shorter than the sample decay time, the assumption that diffusion in the vapour maintains a uniform density across the cell is a good one. Diffusion along the long cell extending from the discharge to the resonator also should not influence the exponential decay. This is because the cell ends at the resonator, so that there is, even with the discharge firing, no flux of atoms through the resonator. Diffusion within the film is more uncertain, as no data for the behaviour of D atoms within superfluid / - 4 H e exists. We wi l l be able to make some estimates by analogy to 1If this value for paiv seems absurdly large, just keep in mind that it includes the (unknown) increase in A/V due to surface roughness. In addition, the zero-point motion of the D in the liquid carves out a little bubble. The effective mass might be much larger than the bare D mass as a result of the backflow of the heavy 4He atoms around the moving bubble. Appendix B. Density decay due to solution of D in liquid helium 77 the case of 3 H e in / - 4 He. Assuming that the D atom density within the film decreases linearly from the film surface to the substrate, one can show that the decay rate of the atom density in the vapour is reduced to A e f f = Xil+Do/D)-1 (B.5) where A is given by equation (B.3), D is the diffusion constant within the film, and the characteristic scale for the diffusion constant, Do, is given by D0 = vdfil/2alv/4 (B.6) with d the film thickness. Taking as typical values d •— 200 A , v — 1 0 4 c m s _ 1 and Hll2aiv = 1 we find D0 = 5 x 1 0 ~ 3 c m 2 s _ 1 . The experiments of P tukha [47] for 3 H e in / - 4 H e gave D ~ 4 x 10~ 3 c m 2 s _ 1 for T = 1.2 K , with D increasing rapidly as the temperature dropped, to a value about 30 times higher at 1.1 K. This must be used with care as the mean free path of 3 H e in / - 4 H e at 1.2K is about 200 A , a typical film thickness. The scattering of 3 H e is due primari ly to collision with rotons, and the mean free path increases rapidly with decreasing temperature. Assuming that the mean free path for D atoms is similar then we can expect that over the experimental temperature range the motion of D in the film is ballistic. It must be kept in mind, though, that D may not be like 3 H e . If diffusion within the film were important and the data was analyzed neglecting it , the value of Es inferred from the data would be a lower bound on the true value. It is interesting to ap^ply equation (B.3) to hydrogen and t r i t ium as well. The results for al l three isotopes for 'A/V = 1 c m - 1 are summarized in table B . l , which gives the expected sample lifetimes (which must be taken as order of magnitude estimates) at 1 K and the temperature at which the lifetime is 100 s. Hydrogen is safe from any observable effect due to this mechanism, the lifetime at the highest accessible temperatures (about Appendix B. Density decay due to solution of D in liquid helium 78 Table B . l : Effect of solution of H, D, and T in / - 4 He. atom Ea/kB r ( I K ) T ( r = 100 s) H 30 K 95 y 2.3 K D U K 24 s 0.9 K T 3 K 0.01s 0.3 K 1.4 K ) being many days. Deuterium is the borderline case, with a lifetime slow enough to permit creation of a sample of D, but fast enough to be an important mechanism for sample decay. The result for t r i t ium probably explains the failure of recent experiments of Tjukanov et al [14] to observe t r i t ium in zero field experiments around 1 K , although the sample lifetime would not be as short as given in the table due to the finite time an atom takes to diffuse to the wall . Appendix C Spin-exchange in mixtures of D and H Spin-exchange scattering is the most important spin relaxation mechanism in our low field experiments. In experiments where the D is contaminated with H, but only one isotope is observed directly (by magnetic resonance), it is possible to infer the density of the unobserved isotope by measuring the transverse relaxation time in a free induction decay. In this appendix the cross-sections for D-D and H-D spin exchange are calculated and applied to the magnetic resonance of mixtures of H and D. C.l Equation of motion for spin density matrices Around 1 K the hyperfine level splittings can be neglected compared to the thermal energies, and we can use the theory of Bal l ing, Hanson, and P ipk in (BHP ) [48]. This theory assumes that the spin-exchange collisions are elastic and that there is no spin-orbit coupling; it has been useful in the description of spin-exchange in atomic hydrogen at low temperatures [49]. In collisions of atoms of type 1 (D for example) with distinguishable atoms of type 2 (H), the equation of motion of the spin density matr ix px of the type 1 atoms is given by (cf. B H P equation (27)) pi = Gn2 ^ T r 2 [-3 /3x2 + (1 + 2i/c)(cr 1 • a2)pi2 + (1 - 2i/c) /9 1 2(o~i • <r2) + (o-i • o-2)/o12(<r1 • <r2)] (C.l) 79 Appendix C. Spin-exchange in mixtures of D and H 80 where G is the rate constant (described later), K is the frequency shift parameter (see B H P for details), n2 is the density of the type 2 atom, p\2 — p\®pi (C.2) is the two-particle spin density matrix, and o~\ and o~2 are the electron spin Paul i oper-ators for atoms 1 and 2 respectively. In the case where the collisions are between two identical atoms (D with D for example) px = p2 = p and the equation of motion is modified to (cf. B H P equation (B2)) p = Gn^ T r 2 [-3/3i2 + (1 + 2iit)(o-i • cr2)pl2 + (l - 2iK)p12(o~1 • o~2) + {ori • o2)p12(o-1 • o2)] + G'n J T r 2 [-3Pl2 Q + (l + 2iK,)(a1-o-2)Qp12 + (1 - 2iK,)puQ(a1 • cr2) + (<r1-<r2)Qp12(a1-o-2)} (C.3) where the operator Q interchanges the atoms. It is apparent that p is given by products of rates Gn and G'n and matrices depending only on the spin density matrices of the colliding atoms. It proved convenient for us to evaluate the matrices which mult ip ly the rates numerically, rather than performing the algebra. The procedure is to make small perturbations to the density matrix of the colliding atoms and look at the part ial traces in the equations of motion. Essentially, one is linearizing with respect to deviations of the spin density matrices from equilibrium. A t around 1 K the effect of a t ipping pulse on the density matr ix is small , since the hyperfine level populations are essentially identical, and Appendix C. Spin-exchange in mixtures of D and H 81 therefore this linearization of the equations of motion wi l l be a good approximation. The numerical coefficients we obtain in this manner are readily expressed as simple fractions. The spin-exchange frequency shifts, which should be not be important in our experiments, were neglected. We shall describe our numerical procedure for the case of D-D collisions. We start with the 6 x 6 single atom spin density matr ix corresponding to equil ibrium (diagonal elements only, al l equal to 1/6 in the high temperature approximation). The equation of motion (C.3) gives for this case p = 0. We then change one element of the density matr ix by a small amount x and calculate p/x. The input to the calculation is the magnetic field and the element of the density matrix that is perturbed. We begin with expressions for the electron Paul i operators with respect to the basis of electron and deuteron spin projections. The unitary transformation to the hyperfine eigenstate basis, the outer products from one-atom to 36x36 two-atom matrices, the matr ix products and the trace over the second atom to give p, are al l done numerically. We are interested in the [3-8 transition in the magnetic field which minimizes its frequency. In order to determine the response to a tipping pulse, we consider (separately) three perturbations: 1. Increasing the /3 population by setting ppp — 1/6 + x. 2. Increasing the 8 population by setting pss = 1/6 + x. 3. Introducing an off-diagonal element, pp$ = x. Perturbations (1) and (2) yield diagonal p whose difference indicates that PPP ~ Pss = -Gn(ppp - pss). (CA) Perturbation (3) gives many non-zero elements of p, but following the argument of B H P al l elements other than pp$ can be neglected. The reason is that the pij are oscillating Appendix C. Spin-exchange in mixtures of D and H 82 rapidly, at angular frequency u>ij = (Ei — Ej)/%, and unless u>ij = tops the element /?,_/ (which is oscillating at u)p$) wi l l not contribute to the change of pij in a time-average. Since the 8-6 transition is well separated from any others, only pps wi l l feel the effect of p. We find that 7 fas = -j^Gnpps (C.5) The effect of D-D spin exchange on the f3-8 transition is to cause both the on-diagonal and off-diagonal elements of the density matr ix to recover their equil ibrium values expo-nentially in time. The rate constant G' describing atom-identity effects does not enter into the result, for this transition. We have used our numerical perturbation technique to explore: • The effect on the deuterium /3-8 transition at its min imum frequency of spin-exchange in D-D collisions (described above). • The effect on the /3-8 transition of spin-exchange in H-D collisions. • The effect on the hydrogen u-c transition in zero magnetic field of spin-exchange in H-D collisions. • The effect on the a-c transition of spin-exchange in H - H collisions (much studied in connection with cryogenic hydrogen masers). We shall now summarize the results. In resonance experiments on deuterium using the fl-8 transition at its min imum fre-quency, both the on-diagonal (longitudinal) and off-diagonal (transverse) elements of the D atom density matrix return to their equil ibrium values exponentially in time after a t ipping pulse. The longitudinal and transverse relaxation times TID and T 2 D are given by Tw = G D D " D + G H D « H (C.6) Appendix C. Spin-exchange in mixtures of D and H 83 7 3 T2D = Y 2 ^ D E > r a D + 4&HDftH (C.7) where no and n-u are the D and H atom densities. This result can be applied to a sample of D uncontaminated with H by setting nu = 0. In experiments on hydrogen using the a-c transition in zero-field, the longitudinal and transverse relaxation times T in and T2u are given by 1 1 3 r 2 H = 2 G l i H n i i + 4 GHD^D, (C.9) which agrees with the result of reference [49] for the case n-Q = 0. C.2 Rate constants for spin-exchange In the above, we have expressed the spin-exchange rates in terms of rate constants. The primed rate constants in the identical particle expression (C.3) do not enter into the expressions for the transitions considered here. The unprimed rate constants are defined by G = vo where v is the thermally averaged relative speed of the colliding atoms, and <7 is the thermally averaged cross-section. This is given by [49] o = (kBT)-2 / Eo(E)exp(-E/kBT)dE, (C.IO) Jo where kB is the Boltzmann constant and T is the temperature. The quantity cr(E) is the mono-energetic spin-flip cross-section (cf. BHP equation (25)) _ oo CT = T S E ( 2 ' + 1 ) s i n 2 (^ 3 -^ 1 ) ( C H ) fc 1=0 where 6? and 8* are the triplet and singlet phase shifts for collision with angular mo-mentum hi and relative kinetic energy E = fi2k2/m (m denotes twice the reduced mass), ^ote that G H D = G D H -Appendix C. Spin-exchange in mixtures of D and H 84 Subscripts on the rate constants denote the colliding atoms: for example GDD is the rate constant for spin-exchange due to D-D collisions. We have calculated values for the spin-exchange rate constants. The cross section cr is a function of the triplet and singlet phase shifts. In order to evaluate these phase shifts, we integrated the radial Schrodinger equation ' ( ' + 1 ) + > ( r ) - 2 S ) / (C12) r 2 h2 using ab initio triplet and singlet potentials for V(r). We started from hard cores at r 0 = 0.2a o for the singlet potential and r 0 = 1.0a o for the triplet potential: boundary conditions / ( ro ) = 0 and drf(r0) = 1. The equation was then integrated using a Numerov method with step size 0.005ao to a matching radius rm = 100a o . Here the function is matched to the free particle form / ( r ) oc cos(<5) kr ji(kr) — s'm(8) kr yi(kr), (C.13) where ji and yi are spherical bessel functions and %2k2/m = E, in order to determine the phase shift 8. This was done as a function of energy to 100 K and for / = 0 to 4, which gives the bulk of the partial wave contributions. This gave us a as a function of collision energy. The integral (C.10) was then performed in order to find the rate constant at 1 K. The calculation was done for D-D, H-D and H - H collisions, with the following results: G D D = 2.5 x l O - ^ c r n ^ - 1 , G H D = 2.4 x 1 0 - l o c m 3 s - 1 , and GHH = 7.8 x l O ^ c i ^ s " 1 . These rate constants are essentially independent of temperature between 1 K and 1.5 K . As a check on our procedure, the effective H - H cross-section obtained from our H - H rates, C T H H = 0.39 A , was compared to the coupled-channels calculations of Koelman and Verhaar [50] which gave 0.40 A ; the agreement is good. Surprisingly the D-D and H-D rates are nearly equal, but over 300 times larger than the H - H rate. The large D-D and H-D rates are due to the presence of weakly-bound Appendix C. Spin-exchange in mixtures of D and H 85 J = 0 and J = 1 molecular levels; this might make the rates very sensitive to the precise level energies. Appendix D Low temperature resonant recombination of H and D In a magnetic field, resonant recombination of spin-polarized atomic hydrogen and deu-terium can occur even at low temperatures (around 1 K ) . This is due to the predissoci-ation of weakly bound levels of molecular hydrogen by the hyperfine interaction. In this appendix we describe our investigations into the chemical reactions that occur in a gas of spin-polarized atoms as a result of the hyperfine predissociation of levels of H2, D 2 and H D in fields below 100 k G . D. l Introduction Resonant recombination of H is the formation of H 2 v ia the two-step process H + H ^ H* (D . l ) H* + X -* H 2 + X (D.2) where H 2 is a metastable molecule, or reaction complex, and X is a body which collides with H j to stabilize it and effect the recombination. In conventional resonant recombi-nation of H , the reaction complexes are the quasi-bound levels of H 2 which disintegrate when the atoms tunnel through the centrifugal barrier which binds them. Recombination v ia these quasi-bound levels has been investigated theoretically by Roberts et al [15] and by Whit lock et al [16]. A l l of these 'levels' have energies of more than 50 K , and at low temperatures, resonant recombination v ia these levels is frozen out. In zero magnetic field, and in the presence of 4 H e atoms, the recombination of atomic hydrogen proceeds 86 Appendix D. Low temperature resonant recombination of H and D 87 through the direct (three-body) process H + H + 4 H e —> H 2 + 4 H e (D.3) whose rate has been calculated by Greben et al [17]. For deuterium the situation is similar, although, to the best of our knowledge, no detailed theoretical study of the reaction mechanisms exists. The lowest quasi-bound level of D 2 has an energy of 10 K , and standard resonant recombination is frozen out by 1 K. The resonant formation of H D in mixtures of H and D is likewise suppressed at low temperature. In a magnetic field, resonant recombination can be effective even below 1 K. The res-onances involved are formed from levels of electronic ground state (X 1 SJ") H 2 , D 2 and H D , which are bound in zero magnetic field. In a sufficiently strong magnetic field, these levels are predissociated; the molecule disintegrates into spin-polarized atoms owing to the singlet-triplet coupling provided by the hyperfine interaction. This is energetically possible when the total Zeeman energy of the spin-polarized atoms produced lies below the nonmagnetic molecular level; the kinetic energy of the atoms produced then makes up the difference. That such predissociations should occur and contribute to the recombina-tion of spin-polarized atomic hydrogen (HJ.) and deuterium (DJ.) through the formation of reaction complexes has been pointed out by Stwalley [18]. Predissociation-enhanced recombination of HJ. was first observed and studied as a function of temperature in exper-iments in our lab [19]. In our experiments on spin-polarized atomic deuterium, we also saw resonant recombination [28], although the data did not permit a quantitative study. More recently, Silvera and co-workers have measured the field dependence of resonant recombination in HJ. [22]. In this appendix we consider the hyperfine predissociation of weakly bound levels of H 2 , D 2 and H D and its contribution to the recombination of HJ, and DJ.. In DJ, and in Appendix D. Low temperature resonant recombination of H and D 88 mixtures of D j wi th H j , resonant nuclear spin relaxation also occurs. We concentrate on the experimental situations encountered in spin-polarized hydrogen research: high magnetic fields, temperatures around 1 K and a saturated 4 H e vapour from the superfluid f i lm lining the sample container. A discussion of available experimental results wi l l be made. Much of this appendix, without the deuterium results, has been published [21]. D.2 Single atom hyperfine states The hyperfine states of a solitary hydrogen atom in a magnetic field B are \d) = \\,\) (D.4) Ic) = , | i , - I ) + e | - i , I) (D.5) \b) = \-\,-\) (D.6) l«> = ~ *\h-k) (D-7) where | m s , m , ) is the state with electron and proton spin projections ms and m,- along the applied field, r] = cos(0), and e = sin(0). The mixing angle 9 is given by tan(20) = ° H / ( ^ (7e + 7p)-^)> where an/h ~ 1420 M H z is the zero-field hyperfine splitting. In high field TJ ~ 1 and e ~ (253G) / i ? <C 1. In this case the two states | a) and | b) are predominantly electron spin down, with energy ~ —fieB. A gas of atoms in these two states is spin-polarized and is denoted by H | . For deuterium, the nucleus has spin 1 and the hyperfine states are 10 = 1^,1) (D.S) I*) = rj+\\,Q) + (D.9) l*> = ^ - l ^ - 1 ) + e - ! - i ° ) (D.10) |7> = h | , - l > ( D .H) Appendix D. Low temperature resonant recombination of H and D 89 10 = V-\~\,0) - - 1 ) (D.12) k ) = i?+| - i , l> - e + | | ,0> (D.13) where | m s , m t ) is the state with electron and deuteron spin projections ras and ra,- along the applied field, r]± = cos(9±), and e± = sm(6±). In magnetic field B the mixing angles 6± are given by tan(20±) = ar)/(%(je±i<\)B), where fan/h ~ 327 M H z is the zero-field hyperfine splitting. In high field rj± ~ 1 and e ± ~ (55 G ) / J5 <C 1. The spin-polarized gas DJ. is composed of \a), \/3) and | 7 ) state atoms. The energies of the hyperfine states of H and D can be found in the review article of Silvera and Walraven [1]. Note that their hyperfine states differ from ours by an unimportant phase factor. D.3 Interacting hydrogen atoms The interaction of two hydrogen atoms (H or D) is well described within the Born -Oppenheimer approximation, treating the motion of the electrons adiabatically with respect to the slower nuclear motion. In the center of mass frame, the degrees of freedom are the relative motion of the atoms and the electron and nucleon spins. The Hamiltonian is taken to be H = Ho + Hz + Hhf . (D.14) The first term #o = - + V,{r)P0 + VJ(r)P! (D.15) m is the Hamiltonian of the (relative) atom motion, ra is twice the reduced mass of the two atoms (ra = ran, the mass of a hydrogen atom, for H 2 ) , and PQ and Px are the projection operators onto subspaces of total electron spin 0 and 1 respectively. We use the atomic mass rather than the more customary proton mass since, for the extended Appendix D. Low temperature resonant recombination of H and D 90 molecular levels we are interested in , the atom moves as a unit. The Born-Oppenheimer singlet and triplet potentials Vs(r) and Vt(r), shown in figure D . l , have been accurately calculated from first principles [23]. The second term is the Zeeman Harniltonian of the electrons in the magnetic field B. We omit the nucleon Zeeman effect as it gives only an small shift of the energy levels. The final term is the intra-atomic hyperfine interaction, the only term in this Harniltonian not commut-ing with the total electronic spin S. It is this term which predissociates the molecule and gives rise to resonant recombination. We have neglected the inter-atomic hyperfine and dipole-dipole interactions; al l of these have a negligible effect on the phenomena in which we are interested. The singlet potential Vs(r) supports a large number of bound states. These are specified by a vibrational quantum number v and the orbital angular momentum J , usually written (v,J). Because singlet states have S = 0, their energies are nearly independent of magnetic field (the nuclear Zeeman effect is small). The triplet potential Vt(r) is repulsive, except for the long range Van der Waals part which produces a well too shallow to support any bound states. Triplet states (S = 1) are magnetic, and those with ms = — 1 are lowered in energy by an amount 2ueB if a magnetic field is applied. As a result, a singlet bound state with dissociation energy D is higher in energy than two spin-polarized atoms (with ms = —1) when 2fieB > D. To illustrate this, the potential-energy curves in the situation where 2fieB = 5.5 K and D = 0.7 K are shown in figure D.2.. Terms in the Harniltonian that do not commute with S cause singlet-triplet transitions, and the state is predissociated; a molecule prepared in this state disintegrates into a pair Hz = 2 A i e(s 1+s 2)-B (D.16) Hhf = • Si + a 2 i 2 • s 2 (D.17) Figure D . l : Ab initio singlet ( X X E + ) and triplet (b 3 £+) potentials for the interaction of two hydrogen atoms. Figure D.2: Singlet and ms = — 1 triplet potentials (including Zeeman energy) for the case A ~ 2fiEB = 5.5 K and D = 0.7 K. Appendix D. Low temperature resonant recombination of H and D 93 of atoms which recede to infinity with relative kinetic energy 2fieB — D. The inverse process also occurs, and metastable molecules may be formed from spin-polarized atoms if they collide with the requisite energy. If these molecules are stabilized, for instance by collisions, with atoms of a buffer gas, that cause transitions to lower (nonpredissociated) levels, then the atoms are lost. The two step process of inverse predissociation followed by a stabilizing collision is a resonant-recombination process. Table D . l lists levels of H 2 , H D and D 2 that are predissociated by the application of fields below 100 k G and gives their dissociation energies as determined spectroscopically. 1 We shall consider each molecular species in turn. See Appendix E for references. Appendix D. Low temperature resonant recombination of H and D 94 Table D . l : Levels of H 2 , D 2 and H D predissociated by fields below 200 k G , their dissoci-ation energy D and threshold field BQ for hyperfine predissociation. molecule V J D(K) B0 (kG) H 2 14 4 0.7(9) 5(7) D 2 21 0 2.9(3) 22(2) 21 1 0.2(5) 2(4) H D 17 0 6.9(7) 51(5) 17 1 0.1(7) 1(5) Appendix D. Low temperature resonant recombination of H and D 95 D.4 Resonant recombination of spin-polarized hydrogen From table D . l , the only level of the hydrogen molecule predissociated in fields below 100kG is (14,4). We shall assume for now that it is a true bound level, with D > 0, although the uncertainty in the spectroscopic dissociation energy permits it to be un-bound. In fact, the level is predissociated by the hyperfine interaction whether or not it is bound; we wi l l come back to this later. We shall first calculate the predissociation rate (or inverse lifetime) due to the hy-perfine interaction, and then relate it to the reaction rates in a gas of HJ.. A comparison with experiment wi l l then be made. D.4.1 Calculation of the predissociation rate It is convenient to work with the basis vectors | S ms I mi) of definite total electron and proton spin S and I and their projections ms and mi along the applied magnetic field. Figure D.3 gives the hyperfine Hamiltonian of two H atoms with respect to this basis. 2 The spin state of the (14,4) level is 10 0 0 0) since the overall state must be symmetric under atom exchange. We specify a particular value of M = mj to select one of the 2J + 1 degenerate states making up the level. This discrete state can be written \<f>) = ^±YJM(r) |0 0 0 0). (D.18) Since J , M and ms + m j are good quantum numbers, the continuum of states predis-sociating | <j>) must have the same angular dependence, and wi l l correspond (at infinite separation) to two atoms in the | a) hyperfine state. Measuring energy from the zero kinetic energy of this continuum, the energy of | <f>) is A — D, where — A / 2 ~ — (ieB is the energy of an | a) state hydrogen atom and D is the dissociation energy of the 2Different from reference [1] and correct. Appendix D. Low temperature resonant recombination of H and D 96 0 - 0 1 0 1 1 1 0 1 0 -0 1 1 1 0 0 1 0 0 0 0 1 0 1 1 1 1 0 1 1 1 0 1 0 0 1 0 1 - 0 1 1 0 -1 1 - 0 1 0 0 -0 1 1 1 1 1 1 1 1 o + 1 1 1 + o O 1 + 1 0 0 0 0 1 1 1 -1 0 1 0 1 - 1 1 0 + + 0 0 0 1 0 1 0 0 0 0 + + 0 1 1 1 0 1 0 1 1 0 + + 0 0 0 1 1 1 1 0 0 0 + + 0 1 0 1 -1 - 1 0 0 + + 0 0 0 1 -1 - 0 0 + 1 1 1 1 + 1 - 1 -S T7l^ / TTlj Figure D.3: Mat r ix elements of the two-atom intra-atomic hyperfine interaction in the spin-projection basis. The matr ix elements are obtained by mult iplying the sign in the table by an/2. Appendix D. Low temperature resonant recombination of H and D 97 level. We assume that the field B 253 G so that the two atom hyperfine state | aa) is approximately the pure spin state 11 — 1 1 1). We write the continuum as \*E) = ^ - Y J M ( r ) \aa) (D.19) r where tpE(r) is the radial wavefunction for atoms of relative kinetic energy E interacting through the triplet potential. In doing so we are ignoring the evolution of the spin state during the collision of two atoms in the absence of any coupling to the discrete state. The predissociation rate T is given by Fermi's golden rule [51, 52] r = ( 2 7 r / f t ) \{<f>\H\ipE) | 2 (D.20) where E = A — D is the relative kinetic energy of the two | a) state hydrogen atoms produced and the normalization of \ I(>E) is to unity density of states: (%pE | ipE') = 6(E — E'). A molecule prepared in the state | <^>), denoted r l j , disintegrates at rate T into two | a) atoms. The reverse process also occurs, giving rise to the chemical reaction H* ^ a + a, (D.21) which is equivalent to saying that the discrete state becomes a resonance in the scattering of | a) state atoms; the resonance width e is related to the disintegration rate T by e = fiF. W i t h the high field approximation and the fact that Hhj does not depend on the relative motion of the atoms, equation (D.20) becomes I /•oo 2 r = (27r /a)(a H /2) 2 / <f>(r)^E(r)dr . (D.22) |./o To calculate Y we must evaluate the overlap integral between the (bound) singlet wave-function and the (free) triplet wavefunction of the same energy. The triplet wavefunction ipE(r) is found by solving the radial Schrodinger equation . J(J + 1) m ,T., A r - ^ r ^ - p ( V J ( r ) - E) $j ( r ) = 0 (D.23) Appendix D. Low temperature resonant recombination of H and D 98 and choosing the normalization so that the asymptotic (r —> oo) form is ( m X 1 / 2 7r s i n ( K r - - J + 6) ; K = ^(mE)/h (D.24) n 7TK/ I which gives the unity density of states implied by (D.20). We start from a hard core at r — 1.0a 0 (#o *s the Bohr radius, approximately 0.529 A ) and integrate equation (D.23), using the Numerov method [53] with step size 0.005a o, out to r = 20a o . Here the potential (which decreases as r - 6 for large r) is negligible, but the centrifugal potential (decreasing as r~2) is not. Rather than integrate further, unt i l the centrifugal potential is negligible and the solution has attained the form (D.24), we match the solution to the known form in the centrifugal potential v/4(r) ~ cos(6) Kr JJ(KT) — s'm(6) nr yj(nr) (D.25) (jj and yj are spherical bessel functions of order J) and scale it appropriately. The discrete state wave function <f>(r) is found by solving r 2 K with normalization condition • / ( J + 1 ) -S (V . (r ) + D) <f>(r) = 0 (D.26) / I Mr) |2 dr = 1 . (D.27) Jo For D > 0, equation (D.26) has square integrable solutions only for discrete values of D. We integrate outwards from a hard core at r = 0.2a o , and inwards from the free particle form at r = 20ao. The Numerov method is used, with step size 0.005a o. A t r = 1.4a 0 (the min imum in the singlet potential) we compare the logarithmic derivatives of the outward and inward integration results. B y varying D (or the potential) we can converge on the solution <f)(r) which is continuous and smooth and has the desired number of zeroes. For the (14,4) level we set J = 4 and look for the solution with 14 zeroes. Using Appendix D. Low temperature resonant recombination of H and D 99 the Born-Oppenheimer singlet potential of reference [23], we can find no such solution: (14,4) is not a bound level. It is not a bound level of even the best ab initio adiabatic Hamiltonians, and if, as spectroscopic measurements indicate, (14,4) is in fact bound then nonadiabatic effects are probably the cause. In order to have (14,4) be bound and sti l l work within our adiabatic framework, we scaled Vs(r) by a factor very close to unity in order to give the level a positive dissociation energy D. For example, a scaling factor 1.000835 results in D = 0.7 K. Although this procedure is clearly arbitrary, we believe that the predissociation process is accurately described provided the scale factor gives the true value of D. The bound-state radial wavefunction <j){r) for D = 0.7 K is shown in figure D.4 to-gether with the effective potential that was used. The mean atom separation in the (14,4) state, (r) ~ 6.2a 0 , is much larger than the ground state value (r) ~ 1.45a 0. It is for this reason that the interatomic spin spin interactions can be neglected. Both <j>(r) and ipE\[r) for D = 0.7 K and E = 2fieB — D = 5 K are shown in figure D.5. The result-ing predissociation rate T is plotted in figure D.6 as a function of the kinetic energy E for D = 0.7 K. The lifetime for values of E relevant to experiments below 1 K is given in table D.2 for values of D at the centre and edges of the spectroscopic uncertainty. The calculation for the unbound D = — 0 . 2K case is described below. For fields just above the predissociation threshold (2fieB = D), V is an increasing function of field as the triplet wavefunction penetrates into the centrifugal barrier and the phase space available to the disintegration products grows. B y considering the form of the triplet wavefunction for low kinetic energy, one can show that the threshold behaviour is, for any J , Y ~ £'( 2 J+ 1)/ 2. For high enough fields Y is a decreasing function of field since the final state wavefunction becomes too oscillatory and the overlap integral drops; this is known as momentum mismatch. For the (14,4) level of H 2 the maximum predissociation l inewidth (occurring at E ~ 70 K, corresponding to an applied field of nearly 500 kG) is Appendix D. Low temperature resonant recombination of H and D Figure D.4: Radia l wavefunction and effective potential of the (14,4) level of H Appendix D. Low temperature resonant recombination of H and D 101 r/a0 Figure D.5: Singlet and triplet radial wavefunctions for (14,4) with D E = 5K. = 0.7 K and Appendix D. Low temperature resonant recombination of H and D 102 Figure D.6: Predissociation rate from the (14,4) state of H2. Appendix D. Low temperature resonant recombination of H and D 103 only 20 fiK. Thus, the l inewidth is always very small and is l ikely to be observed only through the effect of the resulting scattering resonance on the recombination rate. We have also evaluated the shift A D in the dissociation energy of the (14,4) level due to the hyperfine interaction [51], and we find that it is negligible. For D = 0.7 K , A D is essentially constant at about 30 for fields at least as high as 100 k G , and is smaller in higher fields. It is worth pointing out that the above analysis can be applied to the case where the (14,4) level is not bound. In this case the 'level' has a finite width due to the disintegration of the molecule by tunneling through the centrifugal barrier at a rate TT- The level wi l l also be predissociated by the hyperfine interaction, with a rate T that can be calculated as follows. W i t h turned off, we can think of (14,4) as a singlet discrete state | <f>) which has been dissolved into a singlet continuum, where it is a narrow resonance. In the vicinity of the resonance the overlap of the resulting continuum and a triplet continuum is where P(E') is a Lorentzian of width hT^. We have numerically determined the overlap of singlet and triplet continua and found that it is well described by the first term alone. A fit gives both the tunneling rate Ti and the overlap /0°° <j>(r)ipE(r) dr needed to find T using equation (D.22). For D = —0 .2K , we find that the tunneling lifetime T^ 1 = 2ms; the predissociation lifetime T - 1 is given in table D.2 as a function of E. The tunneling lifetime was also estimated using the approximate W K B method [54]; we obtained Tx = (1 /J S K 4 , 5 ) | D | - 4 , 5 for a resonance in the singlet continuum at — D. This gives f x — 1.4 ms for D = — 0 .2K. smaller terms (D.28) Appendix D. Low temperature resonant recombination of H and D 104 Table D.2: Predissociation lifetime r in fis as a function of energy E = 2fieB — D for several values of D. D(K) E(K) -0.2 0.7 1.6 1 833.3 3640.6 8486.8 2 190.5 414.9 734.6 3 76.0 128.4 197.6 4 39.2 58.0 81.9 5 23.4 31.9 42.6 6 15.3 19.8 25.4 7 10.7 13.3 16.6 8 7.9 9.5 11.5 9 6.0 7.1 8.4 10 4.8 5.5 6.4 11 3.8 4.4 5.0 12 3.2 3.6 4.1 13 2.7 3.0 3.3 14 2.3 2.5 2.8 15 2.0 2.1 2.4 Appendix D. Low temperature resonant recombination of H and D 105 D.4.2 Recombination rates To relate the predissociation lifetime to the recombination of spin-polarized atomic hy-drogen, we find it convenient to consider H 2 in the (14,4) state as a distinct chemical species and denote it by H 2 . Assume for now that (14,4) is bound. The relevant reactions are then H* ^ a + a (D.29) H* + X -> R2 + X (D.30) Reaction (D.29) is the formation of | a) state hydrogen atoms by predissociation of H 2 with rate T, and its inverse, while (D.30) is the stabilization of H 2 in inelastic collisions resulting in transitions to lower (bound) levels. A t low temperatures, the bound levels involved are those lying close to the dissociation threshold, because lower levels are inaccessible due to the large recoil kinetic energy (momentum mismatch). The stabilizing agent X is an atom or the wall of the container. We shall assume for now that the hydrogen atom density is sufficiently low that X is not a hydrogen atom. In the absence of loss of H 2 by reaction (D.30) the equil ibrium of reaction (D.29) is found by satisfying [55] P* = 2 / i a (D.31) where fi* is the chemical potential of H j , and / j a is that of | a) state hydrogen atoms. The ideal gas expressions are Ha = kBT l n ( n a A 3 ) - A / 2 (D.32) and A** = ksT In C^fA ~ D (D.33) Appendix D. Low temperature resonant recombination of H and D 106 where the thermal de Broglie wavelengths are / « )_*2 \ 1/2 A=(^) =^ 2A* <D'34> and na and n* are the number densities of | a) state H and Hg respectively. Substituting into (D.31) we obtain n? = n2(2J+l)Aje-*'*Br (D.35) where A M = A /2A is the de Broglie wavelength for a particle of reduced mass (i = mn/2 and, again, E = A — D. If the predissociation proceeds at rate T then the number of Hj disintegrating per unit time per unit volume is Tn*, and in equil ibrium the formation-rate density is the same. F rom the principle of detailed balance, the formation-rate density depends only on na and is given by Yn^9 (where n£9 is the function of na given by (D.35)). In general we have time derivatives of na and n*, due to reactions (D.29) and (D.30), given by dtna = 2r(n* - < 9 ) (D.36) dtn* = - r ( n * - <«) - T c n * (D.37) where YQ is the stabilization rate. There is a clear separation of timescales: the predis-sociation lifetime is much shorter than the characteristic time for evolution of the atom density (minutes). Consequently the density n* rapidly reaches its steady state value < = r r r ^ n * ( D , 3 8 ) for which the | a) atom density changes at rate dtna = - — „ ? (D.39) 1 + 1 c obtained by substituting n" into (D.36). Defining the phenomenological rate constant K o a by [1] dtna = -2Kaan2a (D.40) Appendix D. Low temperature resonant recombination of H and D 107 we have Kaa = .T^V + IK''*"*1, (D.41) 1 + l c or, in terms of lifetimes r = V 1 and TC = T c /<„ = W + W * ' " ' (D.42) T + TQ This rate constant is applicable to experiments in which the hydrogen atom density is too low to contribute to the stabilization of the metastable H 2 . We wi l l now consider what happens when the hydrogen atom density is not so low. Stabil ization reactions involving atoms of spin-polarized hydrogen are H* + a H 2 + a (D.43) H* + b -> H 2 + b (D.44) H 2 + a —> H 2 + 6 (D.45) H* + b -» H 2 + a (D.46) Insofar as the evolution of the proton spins can be neglected during these collisions, the final nuclear state of the molecule (and hence the parity of J ) can be inferred from the conservation of m / + m,-, the sum of the proton spin projections. Reactions (D.43) and (D.44) do not alter the nuclear state of the molecule and so the final state wi l l have J even. The most probable final state is (14,2) and the rate constant K2 wi l l be the same for both reactions. Reactions (D.45) and (D.46) involve the exchange of a proton between the atom and the molecule during the stabilization, and the final state has J odd. The most probable final state is (14,3) and both reactions wi l l have the same rate constant /c 3 . The rate constants can be written in terms of cross sections: /c 2 = vo2 and K 3 = vo~3, where v is the mean relative speed of H j and H atoms. W i t h reactions (D.29) and (D.30) occurring as well, the density time derivatives are dtna = / c 3 » * ( n 6 - na) + 2T(n* - < 9 ) (D.47) Appendix D. Low temperature resonant recombination of H and D 108 dtnb = - K 3 n * ( n 6 - na) (D.48) dm = - r ( n , - <») - T c n , (D.49) where now T c = (K 3 + K2)(nb + na) + Tge (D.50) is the total stabilization rate (l?Qe is the contribution due to 4 He) . Solving (D.49) for the steady state density and substituting it for n * in (D.47) and (D.48) we obtain equations for the decay of the observable densities dm = +£r-Kaa(nb - na)n2a - 2Kaan2a (D.51) 1 c dm = I<aa(nb - na)n2a (D.52) 1 c where Kaa is again given by expression (D.41). The effect of the exchange stabilization (D.45) and (D.46) is negligible if TQC >^ K3(nb + na). In terms of stabilization cross sections this requirement is nb + na < ^ n H e (D.53) 2CT 3 The cross sections a2 and <r3 would be difficult to calculate: owing to the strength of the interaction potentials, it is essentially a three-body problem. Our expectation is that they are larger then o~ne (which we calculate in a later subsection) but not by a large factor. Apart from stabilizing the metastable H2, collisions may disintegrate it. For example E2 + 4 H e -» H + H + 4 H e . (D.54) This results in a resonant electronic relaxation, since the pair of atoms produced wi l l be unpolarized. The above reaction is the inverse of the recombination process calculated by Greben et al [17], and their results can be inverted to yield a cross section of less Appendix D. Low temperature resonant recombination of H and D 109 than 0.01 A at 1 K , much smaller (as we shall show) than the cross section for transi-tion to more deeply bound molecular states. It is likely that the dissociation of H 2 is always slower than its stabilization, even in collisions with hydrogen atoms, and as a first approximation can be neglected. This is not the case at higher temperatures; there dissociation is an important outcome of collision with 4 H e [16]. We shall now discuss how the (14,4) level contributes to the reaction kinetics if it is not bound in zero field. If the (14,4) level of H 2 is a positive energy 'orbiting resonance,' then it can disintegrate by tunneling through the centrifugal barrier. In addition to the hyperfine predissociation H* ^ a + a (D.55) with forward rate T, we also have E*2 ^ a + c (D.56) H* ^ b + d (D.57) each with forward rate T x / 2 . The hyperfine predissociation of an unbound (14,4) wi l l result in a resonant electron spin relaxation, since H 2 can be formed from spin-polarized hydrogen and then disintegrate into a pair of atoms with no electronic polarization. If the stabilization of the metastable molecules is sufficiently fast (Fc >^ Tx) , then the resonant relaxation is suppressed since the molecules are stabilized before they tunnel apart. For D within the range of spectroscopic uncertainty, Tx is small enough that resonant relaxation is negligible. In summary, for hydrogen atom densities well below the 4 H e vapour density, the predissociation of (14,4) contributes to a recombination of the form (D.40) while for higher densities its effect is given by equations (D.51) and (D.52), involving both \ a) and | b) species. Appendix D. Low temperature resonant recombination of H and D 110 D.4.3 Stabilization Here we estimate the cross section for stabilization of H 2 in collisions with atoms of the 4 H e buffer gas: H* + 4 H e -> H 2 + 4 He. (D.58) This is the dominant stabilization mechanism for low hydrogen-atom density. From (14,4), transitions are primari ly to lower (14, J) levels since more deeply bound levels are inaccessible due to momentum mismatch 3 and other nearby levels have large J and so are inaccessible due to the large difference in angular momentum. Because the nuclear spin is not affected by the collision (no ortho-para conversion), transitions wi l l be to even J levels. As al l of the levels under consideration ((14,4), (14,2) and (14,0)) have almost identical radial wavefunctions corresponding fairly closely to atoms fixed at a separation R ~ 6 a 0 , we treat the molecule as a rigid rotator. The centre-of-mass Hamiltonian is then (with coordinates defined in figure D.7) h2 H = - — L\T -f- V(r, cos7) + ej (D.59) where the reduced mass p, ~ 1.33raH. The energy levels ej are taken to be those de-termined by spectroscopic measurements [56]. We expand the three-body potential in Legendre polynomials V(r, c o s 7 ) = £ K(r)P„(cos 7) (D.60) n where Vn(r) =(" + !) / V(r, x)Pn{x) dx (D.61) are the partial potentials. B y symmetry, only those with even n wi l l be nonzero. The two atoms are assumed to interact independently with the 4 He: V(r, cos 7) = V(r- ( i ? / 2 ) c o s 7 ) + V(r + (R/2) cos 7) (D.62) 3For low incident energy, the overlap between the final and initial states decreases dramatically as the final state wavefunction is made more oscillatory. Appendix D. Low temperature resonant recombination of H and D 111 Figure D.7: Coordinate system for calculation of the cross section for stabilization of in collisions with 4 H e . Appendix D. Low temperature resonant recombination of H and D 112 where V ( r ) is the H-He interaction used in reference [17]. We first calculate the relaxation cross section in the Distorted Wave Born Approxi -mation ( D W B A ) , where the distorting potential is Vo(r). Using the results and notation of Messiah [57], the differential cross section is given by da i a2 V -W = 4 & T 1 r - | 2 ( D - 6 3 ) where a = (a, k) is the init ia l state (a denotes the internal degrees of freedom | JM) and and k is the incident wavevector) and b = (8, k') is the final state. The total reaction cross section for final state pi f rom init ial state a is don The transition matr ix in the D W B A is °~> = h h - l " ' ^ ( D - 6 4 ) r.irA = l i t + P f j - ' IWIXW) (D.65) where T^b is the transition matrix due to the isotropic Hamiltonian %2 # i = ~ 2 j j A r + V0(r) (D.66) | X^) and are outgoing and incoming spherical waves associated with Hamilto-nian Hi and W is the anisotropic part of the potential W = £ K ( r ) P „ ( c o s 7 ) (D.67) 7l>0 We can write |X<+>) = A n ^ i ' e ^ Y ^ Y U ^ ^ ^ l J M ) (D.68) lm K r \XP) = 4 ^ ^ ^ ) ^ ) ^ ! ^ ' ) (D.69) Appendix D. Low temperature resonant recombination of H and D 113 where the radial wavefunction Fi(k; r) is the solution of F,(*;r) = 0 (D.70) with asymptotic (large r) behaviour 7T . Fi(k; r) ~ sin(&r — — / + r]i) Li (D.71) The incident kinetic energy is %2k2/2n and the in i t ia l and final wavevectors k and k' are related by w+ej = —+ej' (D72) Since the total stabilization cross section a from level | JM) does not depend on M (since it does not matter which way the molecule is tumbling) we can write (D.64) as J'<J I V where the part ial cross sections can be shown to be (for J' < J) a{Jl-+J'V) = T ^ T ^ T ( 2 ; + 1 ) ( 2 / ' + 1 ) ( 2 J ' + 1) (D.73) h\k'k)2 k n>0 1 2n + 1 I J' n J 0 0 0 0 0 0 Cnm(k'; k) with Cnlll(k'-k) = / drF^k'-^V^F^r) Jo (D.74) (D.75) (D.76) Since only Vn(r) wi th n even are nonzero, transitions only occur if J + J' and / + /' are even. For transitions from J = 4 t o J = 0 o r 2 only n — 2, 4 and 6 contribute, and because of this, | / — /' | is not larger than 6. The evaluation of the partial cross sections is done as follows. The Legendre de-composition of the potential is performed for atoms separated by 3.35 A by integrating Appendix D. Low temperature resonant recombination of H and D 114 (D.61) for values of r from 1.7 to 10 A in steps of 0.05 A . The radial functions Fi(k; r) are generated by integrating the differential equation (D.70) from a hard core at 1.7 A using a Runge -Kut ta technique. The normalization is effected by matching to spherical Bessel functions at 10 A . The matrix elements Cn///(A;'; k) are then formed, and finally the sum-mation (D.75) is done to yield the cross sections. Intermediate results are examined to ensure that nothing untoward happens. Most of the integrand in (D.76) is concentrated around r ~ 4 A , well beyond the hard core. The total cross section for incident kinetic energy 0.5 K and with part ial waves to / = 6 is 114 A 2 . The final state is (14,2) 90% of the time. The bulk of the cross section is obtained by summing incident part ial waves / = 0 and / = 1 since at these low energies the centrifugal potentials for higher / are effective at keeping the 4 H e away from the molecule. In the D W B A we can distinguish the contributions from the part ial potentials: the n = 2 partial potential is responsible for 80% of the stabilization. The D W B A is based on the smallness of the matrix elements of the anisotropic po-tential W. Here the anisotropy is not small. For the scattering matr ix to be unitary, the total inelastic cross section for incident partial wave /, ff|=E&(J'-^ (D-77) j<j' v must be less than crrx = ^ ( 2 / + l ) (D.78) For the D W B A to be accurate, o~\ must be much smaller than c 7 , m a x . For / = 0 and an incident kinetic energy of 0.5 K we calculate <r0 = 66 A 2 , which is not much smaller than A2 . This indicates that the true cross section is probably not tiny, but that its exact size cannot be determined within the D W B A . To get a better idea of the size of the stabilization cross section we performed a Appendix D. Low temperature resonant recombination of H and D 115 coupled-channels (CC) calculation. This involves integrating the set of coupled equations &JAB = J2uBCIAC (D.79) c which arise when Schrodinger's equation is decomposed with respect to both the internal state | JM) of the molecule and the state | Im) of angular motion of the 4 H e about the molecule [58]. Here A , B and C denote states of the form | ImJM), and /AB(0 is the behaviour of the 5 t h channel in the Ath. solution. W i t h the Harniltonian given by equation (D.59), the coefficient matr ix is UBc(r) = ^VBC(r) + [lB(lBr2+l) - k% ) 6BO (D.80) where k\ = (2fi/%2)(E — £B) and VBC is the interaction potential matrix. The potential (D.60) has matr ix elements (I'm'J'M'l V \lmJM) = £ (l'm'J'M'\ P„(cos7) \ lmJM)Vn(r) (D.81) n where the VN (r) are determined as for the D W B A calculations and the matr ix elements of the Legendre polynomial are readily expressed in terms of Wigner 3j symbols. We restrict J to the values 0, 2 and 4. Starting from init ial condition JAB^O) = 0 and drfAB(fo) — &AB (corresponding to a hard core at r 0 = 1.7 A ) the coupled equations (D.79) are integrated using a Runge-K u t t a technique. To avoid catastrophic loss of linear independence we reorthogonalize the solutions every 0.02 A . A t a radius of 10 A , the resulting /AB(T) are matched to their free particle forms, and linear combinations Us(r) = YlaAcfcB{r) (D.82) c are formed with OLAC chosen so that the asymptotic form of (AB is (since al l channels are open) £AB(T) ~ ^ s e x p -i (kAr - ^IA) - SAB exp i (kBr - (D.83) Appendix D. Low temperature resonant recombination of H and D 116 The matr ix SAB obtained in this way is the (unitary) scattering matrix, related to the cross sections by <W = E ' £ ' I T I SAB - SAB | 2 (D.84) A B A where the primed sums are over all A and B wi th internal states a and pi respectively. We tested our C C software by reducing the anisotropic potential W by a factor of 100 and comparing the results with those obtained in the D W B A , which should be good for such weak anisotropy - the results were essentially the same. The part ial cross sections with ful l scale anisotropy were calculated with partial waves with / < 8 and potentials to n = 6. As in the D W B A , the bulk of the cross section is obtained from incident waves / = 0 and. / = 1. The total stabilization cross section for an incident energy of 0.5 K is 32 A , to be compared with 114 A from the D W B A . The total cross section does not change by more than a few A 2 if we restrict the calculation to waves with / < 6 or if we include the n — 8 partial potential. The energy dependence of o is investigated restricting / < 6, which greatly reduces the computation time; table D.3 gives o for several incident energies. The thermally averaged cross section o should be well approximated by the cross section for incident energy kBT since o decreases only slowly with increasing E, essentially as 1 jE. Appendix D. Low temperature resonant recombination of H and D 117 Table D.3: Cross section for stabilization of in collisions with 4 H e for several incident kinetic energies. Calculated using C C with partial waves to / = 6, and potentials to n — 6. E(K) o(K2) 0.5 31.2 1.0 21.4 1.5 16.0 2.0 12.4 Appendix D. Low temperature resonant recombination of H and D 118 The stabilization rate Tc is related to the cross section o by Tc = o^v^nue, where Vfj, = (8kBT/nfi)1/2 is the average relative speed of the H$ and the 4 H e , and nue is the 4 H e density. Assuming that the 4 H e vapour is saturated, the stabilization rate is much larger than (more than 10 times) the calculated predissociation rate in a field of 41 k G if the temperature is above 0.6 K. In this case the resonant recombination is 'formation bottlenecked.' The formation of metastable molecules is the rate-l imiting step and so the exact stabilization rate is unimportant. The threshold temperature changes negligibly from 0.6 K if we use the larger D W B A cross section, due to the extremely fast variation of the 4 H e vapour pressure with temperature. D.4.4 Discussion of experimental results Measurements of the recombination of spin-polarized atomic hydrogen were made in a field of 41 k G using E S R at 115 GHz . The data displaying resonant recombination is shown in figure 2.1, where the phenomenological rate constants Kaa and Kab are plotted as functions of temperature. The data was fit to sum of surface recombination wi th constant cross-lengths and resonant recombination of the form (D.40). Assuming that the resonant recombination is formation bottlenecked (TC <C T) expression (D.42) described the high-temperature data with D = 0.7(1) K and r — 62(9) / is; the fit is shown in the figure. The lifetime we calculate for this D in a field of 41 k G is r = 36(3) fis, somewhat smaller than the measured value. It was recognized at the time that the 4 H e vapour was not dense enough to maintain Tc < T over the entire experimental temperature range, and that a sudden decrease of Kaa with the 4 H e vapour pressure should have been observed below 0.6 K. Given that no deviation from the formation bottlenecked form was observed, it was assumed that stabilization of H 2 must also have been occurring in collisions with hydrogen atoms. This was not an unreasonable assumption, as the hydrogen atom density was approximately 10 1 5 c m - 3 (the same as the 4 H e vapour density Appendix D. Low temperature resonant recombination of H and D 119 at about 0.6 K ) so that if the H stabilization cross section is as large as the 4 H e one, then the resonant recombination would have been formation bottlenecked over the entire temperature range. Unfortunately, we did not understand properly the contribution of exchange stabilization (reactions (D.45) and (D.46)) and so did not consider the effect it might have had on the analysis of the data. Exchange stabilization may, in fact, have been a major contributor to the stabilization of H^. We re-analyzed our experiments (including some unpublished data) and find that the density decay due to the resonance was better fitted by equations (D.51) and (D.52) than by equation (D.40) as first assumed. This makes the determination of D f rom the data more doubtful, as it folds into the analysis the cross sections o2 and o~3 for direct and exchange stabilization due to H , which are difficult to estimate. Assuming that cr3 ^ > o2 and that the stabilization was due entirely to H we obtain better fits to the density decays, and fitting the temperature dependence of the resulting Kaa we obtain D = 0.2(1) K and r = 20(5) fxs. For comparison, the calculated lifetime for D = 0.2 K is 23 fis. The init ial ly published fit [19] corresponds to the opposite l imit , 03 <C o2. Unfortunately, the data does not indicate how dominant exchange stabilization is; some estimate of cr3 and o2 is required. The data has also been re-analyzed assuming that the ab initio predissociation lifetime r calculated in this work is accurate. The result of the fit is D = 0.56(10) K if exchange stabilization dominates, and D = 0.06(9) K if it is negligible. It is now clear that the dissociation energy D of the (14,4) level is not as well de-termined by the our recombination measurements as we first thought. This is due to stabilization of the metastable molecule in collisions with H atoms, with the possibil ity of exchange. Another problem is that in a gas of spin-polarized hydrogen many recombi-nation and relaxation processes may occur simultaneously, introducing into the analysis quantities which are known only approximately (or not at all). In order to make accurate measurements of rate constants, it is therefore important to work at fields, temperatures Appendix D. Low temperature resonant recombination of H and D 120 and densities where the processes of interest are dominant. The most important con-founding influence is exchange stabilization. Although it may be possible to carefully correct for it , it is probably best to eliminate it altogether by either setting n& = 0 (hard to do) or rtf, = na (easier). Figure D.8 shows the largest contributions to the two body rate constant Kaa as a function of magnetic field for typical temperatures of 0.675 K (solid) and 0.575 K (dotted). The labels on the curves: resonance (formation-bottlenecked res-onance recombination with D — 0.7 K ) , electronic (electronic spin relaxation), surface (surface recombination for an area-to-volume ratio A/V = 30 c m - 1 ) , and bulk (three-body recombination a + a + 4 H e —• H 2 + 4 He) . One can see that there is a range of field (and temperature) for which the resonant recombination is the dominant contribu-tion. In a carefully designed experiment it should be possible to correct for the smaller contributions, take into account (or eliminate) exchange stabilization and determine the dissociation energy of the (14,4) level to high precision, despite the fact that the predis-sociation threshold itself is not very sharp and in any case is hidden by other processes. Regretfully, our existing data is just not good enough. D.5 Deuterium Resonance phenomena involving deuterium are more complicated, since the deuteron has spin 1. The decomposition of the intra-atomic hyperfine interaction Hhf and the two-atom hyperfine eigenstates with respect to the two atom spin states | S ms I mj) can be found in the review article of Silvera and Walraven [1]. As indicated in table D . l , there are two levels that can be predissociated in laboratory fields: (21,0) and (21,1). For each of these levels is it necessary to consider the predissociation of the nuclear sublevels separately. Appendix D. Low temperature resonant recombination of H and D 121 Figure D.8: Contributions to the two-body rate constant Kaa for recombination of | a)-state hydrogen atoms. Solid curves are for T = 0.675 K, Dotted curves are for T = 0.575 K. Appendix D. Low temperature resonant recombination of H and D 122 D.5.1 The (21,0) level of D 2 Since J is even, / must be even in the electronic singlet state in order that the overall state be antisymmetric under atom exchange (D is a composite fermion). The spin states of the molecular level are therefore \S ms I mi) = |0 0 0 0) (D.85) |0 0 2 mi) ; m 7 = 0,±1,±2 (D.86) It is best to consider separately the subspaces specified by ms + m / . As in the case of H 2 , we assume the field is high enough that we can write the continuum states degenerate with (21,0) as products of a triplet spatial part and spin parts which are eigenstates of Hhj (but are essentially pure ms = —1). This wi l l be a good approximation for B 55 G. We wi l l express the predissociation rates in the subspaces in terms of a 'base rate' Y given by (D.87) which in turn is calculated as for H 2 . Figure D.9 shows T as a function of the relative k i -netic energy E of the disintegration products. The predissociation turns on quite sharply as a function of field (r ~ Ell2 for small E) and the maximum predissociation rate occurs for E ~ 0.2 K , which corresponds to only 1.5 k G above threshold. The detailed balancing of chemical reactions due to predissociation and collisional stabilization parallels that for H 2 . . _ In the m s + raj = —2 subspace the singlet level | 0 0 2 — 2) is degenerate wi th a continuum with asymptotic spin part (|/?7) — \^B))/^2\ the reaction is D* ^ 0 + 7 (D.88) with forward rate Y^ = (1/2)T. Including a stabilization at rate T c , one can show that Y = (27r/h)a2D / flr)$j(r)dr Jo Appendix D. Low temperature resonant recombination of H and D 123 Figure D.9: Base predissociation rate T for the (21,0) and (21,1) levels of D 2 . Horizontal bars show spectroscopic uncertainty in threshold fields. Appendix D. Low temperature resonant recombination of H and D 124 the resonant recombination rate constant is = M^§jre~E^T (D.89) 1 (3y + 1 C In the ms + mi = — 1 subspace the singlet level |0 0 2 — 1) is degenerate with the continuum with (asymptotic) spin part (\aj) — \ ja))/^/2. The reaction is D* ^ a + 7 (D.90) with forward rate r a 7 = (1/4)T. The rate constant for resonant recombination v ia this subspace is K»r = A»r r°i r Cr e ~ E / k B T • (D.91) L ay + 1 C The ms + mi — 0 subspace contains two discrete levels, | 0 0 2 0) and | 0 0 0 0). These levels are, even with the inclusion of the nuclear Zeeman effect, degenerate. Only one of the continua in this subspace can be made degenerate with these two states; its asymptotic spin part is (| a/3)— | /?a))/>/2. Using the results of Fano [51] for two (degenerate) discrete states predissociated by a single continuum, one can show that one state (a linear combination of the original two) is predissociated, with a width that is the sum of the widths that would be calculated for the two discrete states above. We have D* ^ a + 3 (D.92) with forward rate Tap = (3/4)T. This gives a recombination rate constant Ka0 = Al *k££_e-W ( D > 9 3 ) 1 a/3 + 1 C The relative sizes of these rate constants depend upon the ratio of F c to T, which wi l l make extraction of the rates from density decay data difficult except in two l imit ing cases. In the formation bottlenecked regime (Fc ^ T) these rates contribute as an increase in the KOTtho impulse approximation rate constant [28] and Kap : Kai : = 3:1:2. In Appendix D. Low temperature resonant recombination of H and D 125 the stabilization bottlenecked regime (Tc <C T) we have Kap : Ka~, : = 1 : 1 : 1 , the ratio of the number of nuclear sub-levels involved. Due to the short predissociation lifetimes, experiments below 1 K are likely to be in the stabilization bottlenecked regime. D.5.2 The (21,1) level of D 2 We next consider the (21,1) level of D 2 . We assume that it is bound. There are three spin states of the molecule: \S ms I mj) = |0 0 1 m 7 ) ; m 7 = 0,±l . (D.94) We consider each subspace labelled by ms + mi separately, and express predissociation rates in the various subspaces in terms of the base rate T defined by equation (D.87). Figure D.9 shows the calculated dependence of T on the relative kinetic energy E of the disintegration products. The ms + mi = 0 subspace is interesting: there are two continua degenerate with the single discrete level. The (symmetrized) two free atom hyperfine spin states are | flpl) and (|cry)+ \^a))/y/2. We have two disintegration channels TJ*^=/3 + /3 (D.95) and D* ^ a + 7 (D.96) wi th forward rates Tnp = (1/2)T and Tai — (1/4)T respectively. The metastable molecule can be formed in a collision of either pair of atoms, and disintegrates with both outcomes possible. As a result, there is a nuclear spin relaxation reaction 3+8^a+7 (D.97) Appendix D. Low temperature resonant recombination of H and D 126 whose rate wi l l depend on the predissociation parameters and on the presence of collisions which stabilize the molecules before they disintegrate. We wi l l now relate the recombina-tion and relaxation rate constants to the predissociation and stabilization rates. Include the (irreversible) stabilization reaction D*+ X T)2 + X (D.98) with rate Tc due to collisions with , for example, atoms of the saturated 4 H e vapour present in low temperature experiments. W i t h both reaction (D.95) and (D.96) in equi-l ibr ium (Tc = 0) the density n* of D 2 would be equal to both < ^ = A J ( 2 J + I ) e - W n 2 ( D > 9 9 ) and < l y = A ' ( 2 J + l ) e " ^ B T n a n 7 (D.100) and away from equil ibrium the rates of change of atom densities are (applying the prin-ciple of detailed balance to both (D.95) and (D.96)) dm = T a 7 ( n ^ - n ^ ) . (D.101) dtn0 = 2TPp(n> - (D.102) dm = T a 7 ( n * - nHJ. (D.103) Including collisions (D.98) the rate of change of D 2 density is given by dm = - r > K - <p0) - r a 7 K - <'Q7) - r c n* (D.104) As in the case of H 2 , due to the large timescale separation, we can set dm = 0 and solve (D.104) for the steady state density n" which can then be substituted for n* in equations (D.101), (D.102), and (D.103). Defining phenomenological rate constants Kai, Kpp, and Appendix D. Low temperature resonant recombination of H and D 127 dtna — GpptQ1(np — nan^) — I ^ a 7 n a n 7 (D.105) dtnp = -2Gppia^(nl-nan1)-2Kppn23 (D.106) d t n 7 = GpptCtl(nl - n a n 7 ) - Kainan^ (D.107) we obtain, after some manipulation, the contribution of (21,1) |0 0 1 — 1) to the reac-tion rates: KPp = 3A» If* e-Elk°T (D.108) 1 a-y + 1 pp + 1 C Ka, = 3A» lf\v e~E^T (D.109) 1 c*7 + t pp + 1 C 1 7^ + 1 /3/3 + 1 C If there are no stabilizing collisions (Tc = 0), there is no recombination and the relax-ation process in unhindered. If collisions are frequent, then the relaxation is suppressed as molecules are stabilized before they can fall apart. Returning now to the other subspaces, ms + mi — 0 and 1, the situation is the, by now, familiar single discrete state predissociated by a single continuum. From the ms + mi = 0 subspace comes the recombination Kap = ZAlJ^Oe-E/ksT ( D i n ) 1 ap + 1 C with Tap = (1/4)T, and from the ms + m j = 1 subspace Kaa = Mlrraa^Cv e-E'k»T (D.112) 1 aa + 1 C with r a a = (i /2)r. D.5.3 Stabilization Now we turn to the stabil ization (rovibrational de-excitation) of D | (either (21,0) or (21,1)) in collisions with 4 He. The nuclear state of the molecule should be unaffected and Appendix D. Low temperature resonant recombination of H and D 128 the stabilization rate can be estimated ignoring the spin degrees of freedom. Previously we estimated the cross section for the stabilization of the (14,4) level of H 2 by treating the molecule as a rigid rotator. This approach cannot be applied directly here because there are no purely rotational transitions from either (21,0) or (21,1) to lower molecular levels. We have generalized our D W B A treatment of H 2 — 4 H e collisions to the present problem by including the separation of the D atoms as a dynamical variable rather than as a parameter. The Harniltonian (D.59) is generalized to H = ^ + V(R,r,cos7) + evJ (D.113) where now the level energies evj depend on both the vibrational and rotational quantum numbers. We again write the interaction in terms of partial potentials V(R,r, c o s 7 ) = £ Vn(R, r)P„(cos7) (D.114) n The matrix element (I'm'v'J'M'| V |ImvJM) = ^ ( i / J ' | Vn \ vJ)(l'm'J'M'\ P n ( c o s 7 ) \lmJM) (D.115) n where ' <j>*vlJI(R)Vn(R,r)<f>vJ(R)dR (D.116) o and <f)vj(R) is the radial wavefunction of the (v, J) molecular level, obtained as described earlier for the levels of H 2 . Note that we recover the rigid rotator treatment of H 2 if the products 4>*,j,(R)<f)vj(R) are strongly concentrated around R ~ RQ, the mean separation of (all) the final states. In this case we could approximate (v'J'\Vn(R,r)\vJ) ~ (v'J'\vJ)Vn(Ro,r) (D.117) with (for radial wavefunctions that were similar) (v'J'\vJ) ~ 1. For the (21,0) and (21,1) levels this approximation is patently bad. We do not make the approximation of similar Appendix D. Low temperature resonant recombination of H and D 129 radial wavefunctions, but instead use the part of the potential U = E \vJ){vJ\V \vJ)(vJ\ vJ (D.118) that does not couple molecular levels as the distorting potential; the remainder is, of course, the perturbation. The appropriate generalization of expression (D.75) for the part ial cross sections is n>0 1 2n + 1 / n I A 0 0 0 ( v J' n J [ 0 0 0) Cnvij'i'vJi(k'; k) (D.119) with Cnv.j.i.vji(k';k) = rdrFll(k';r)(v'J'\Vn\vJ)Fl{k;r) (D.120) Jo The radial functions Fi(k;r) for the outgoing waves | X^) is found for the potential (vj\ Vn \ vJ); the incoming Fii(k';r) are found for (v'J'\ Vn \ v'J'). The transitions are probably to levels with v = 20. Levels with even lower values of v are either too deeply bound (momentum mismatch) or have excessive angular momenta. We find that o is negligible owing to the fact that the v — 21 levels are spatially so much more extended than the lower levels. This is because (21,0) and (21,1) have such small binding energies and angular momenta. In a low-energy collision, a 4 H e and a D 2 molecule with v = 21 do not come close enough for there to be substantial coupling to a v = 20 level (or any lower level). As a result, o is negligible. It could be that a more exact calculation, including v irtual dissociations (which may be important for such long-range molecular levels), would give a larger o. Such a calculation would be quite difficult and we have not attempted it . As discussed below, there is experimental evidence that o ~ 10 A 2 . Appendix D. Low temperature resonant recombination of H and D 130 D.5.4 Experimental consequences Using our ab initio predissociation lifetimes, we have calculated the rate constants that one should observe assuming stabilization is entirely due to collisions with atoms of the saturated 4 H e vapour. For the stabilization cross-section we took a = 10 A 2 , which gives good agreement of the theory with our experimental observations in a field of 41 k G which were published as reference [28]. Figure D.10 shows the expected contributions to recombination due to the (21,0) and (21,1) levels of D 2 as a function of magnetic field at a temperature of 1 K. The resonant recombination is stabilization bottlenecked (Fc <C T) and so the rate constant is, like the 4 H e density, a rapidly increasing function of temperature. The threshold behaviour of the (21,0) resonant recombination is quite striking. The spectroscopically determined dissociation energy (table D. l ) corresponds to a threshold field of 21(2) k G . The uncertainty in field is enormous, and even a rough measure of the recombination rate as a function of magnetic field should accurately determine the dissociation energy of the (21,0) level, as Stwalley realized long ago [18]. Of course, if the stabilization cross section a is small, the (21,0) and (21,1) levels wi l l not make much contribution to recombination (resonant or otherwise) and it wi l l be impossible to use recombination measurements to do molecular spectroscopy. D.6 Hydrogen-Deuterium mixtures D.6.1 Predissociation of HD Lastly we turn to the heteronuclear molecule H D , with two low J states predissociated in laboratory fields: (17,0) and (17,1). Since the nuclei are not identical, we work with Appendix D. Low temperature resonant recombination of H and D 131 Figure D.10: Expected recombination of DJ. due to hyperfine predissociation of the (21,0) and (21,1) levels of D 2 . Appendix D. Low temperature resonant recombination of H and D 132 spin states \ S ms mf mf). Each level of the singlet molecule has six spin states \Smsmfmf) = |0 0m? mf) ; mf = ±^ mf = 0,±1. (D.121) Since the atoms are distinguishable, we can treat both levels simultaneously, and the results will be applicable to any molecular level. We will consider each ms + mi subspace separately. It is convenient to define the base predissociation rates r» - T t e ) 2 i r « r » ^ r » * i 2 <d-i22> and T D = y ( y ) 2 | / o °° <j>{r)xbE{r)dr | 2 ~ T H / 25 (D.123) due to the hyperfine interaction within the H and D atom respectively. The overlap integral is calculated as for H 2 , and the resulting TH is shown in figure D . l l as a function of kinetic energy E of the disintegration products. We will discuss the nature of the subspaces, and the reactions that result, but will not present the rate constants till later. Subspace (1): ms + mi = —3/2. A single discrete level is predissociated by two continua. The predissociations are HD* ^ a+ 7 (D.124) HD* ^ b + 3 (D.125) with forward rates T 0 7 = TH and Tbp = respectively. In addition to resonant recom-bination, there is a resonant nuclear spin relaxation reaction b + 6 ^ a+ 7 (D.126) with rate constant Gbptl Appendix D. Low temperature resonant recombination of H and D 133 Figure D. l l : Base predissociation rate TH for hyperfine predissociation of the (17,0) and (17,1) levels of HD. Appendix D. Low temperature resonant recombination of H and D 134 Subspace (2): ms+mj = —1/2. Two discrete states are degenerate with two continua. Due to the nuclear Zeeman effect, the discrete states are sufficiently well separated (at least for fields predissociating the (17,0) level of HD) that the interaction of each of the states with the continua can be considered separately. The predissociations involving |0 0 -1/2 0) are HD* ^ a + 3 (D.127) HD* ^ b + a (D.128) with rates Tap = TH and Tba = respectively. This gives rise to a relaxation reaction a + 3 ^ b + a (D.129) The disintegration of 10 0 1/2 —1) (denoted •*) is just to one continuum: HD** ^ a + 3 (D.130) with rate T'ap — Trj. The calculated lifetime for predissociation into the \ aB) continuum in a field of 100 kG, T~p ~ 70 ns, is in close agreement with the value of 83 ns calculated by Uang and Stwalley [59] using a rather different coupled channels approach. Subspace (3): ms + mi = 1/2. Two discrete states are degenerate with a singlet continuum. Since the states are not degenerate (nuclear Zeeman splitting) their contri-butions to resonant recombination just sum. We have HD* ^ a + a (D.131) HD** ^ a + a (D.132) with forward rates r a a = To and T'aa = TH. Here the symbols * and denote the spin states |0 0 1/2 0) and |0 0-1/2 1) respectively. Subspace (4): ms + mi = 3/2. There is no predissociation in this subspace since going from ms = 0 to ms = — 1 would require a change Am/ = 1, which is not possible. Appendix D. Low temperature resonant recombination of H and D 135 D.6.2 Rate constants We now present the rates of reaction due to the hyperfine predissociation of any given level of H D . The nuclear spin relaxations are a + 7 ^ b+6 a + d ^ b + cx both with the same rate constant Gbp,ay = Gba,af3 = ( 2 J + 1)A 3 The recombination rate constants are r H r D " r H + r D + r 0 Kaa = ( 2 J + 1)A 3 Kap = (2J + 1)AJ Kay = ( 2 J + 1)A 3 Kba = ( 2 J + 1 ) A 3 LrH + r c r D + r c -E/kBT r H r c + r n r DJ- C -E/kBT r H r c " r H + r D + r c  r D r c r H + r D + r c ,-E/kBT -E/kBT -E/kBT D.6.3 Stabilization cross section (D.133) (D.134) (D.135) (D.136) (D.137) (D.138) (D.139) (D.140) The cross section for stabilization of H D * can be estimated in the same manner as for stabilization of D2, provided we make allowance for the fact that H D is heteronuclear. Since the atoms are not the same mass, partial potentials wi l l exist with odd n and transitions w i l l occur between J of different parity. We have not made any estimates of the cross-sections, since the situation is so similar to the D 2 case discussed earlier that the calculation would probably lead to the same conclusion: no stabilization in collisions with 4 H e . Appendix D. Low temperature resonant recombination of H and D 136 Since the predissociation rates are so large, recombination is l ikely to be in the sta-bi l ization bottlenecked regime. In this case the relaxation proceeds essentially with its ful l ( rc = 0) rate; it may be possible to use the field dependence of the relaxation rates to obtain an accurate determination of the dissociation energy of the (17,0) level even if resonant recombination is negligible. For example, in our experiments on (stable) mix-tures of | b) state H and I7) state D , the probability that the relaxation of a 7 (due to interactions with magnetic impurities) results in the loss of a 6 atom may be influenced by the existence of resonant relaxation. Experiments like ours (published as reference [29]) may give an unambiguous signal when the magnetic field passes threshold. The threshold field, 51(5) k G , could be attained in transient excursions from the operating field (41 kG ) of the existing spectrometer used at U B C . Appendix E Weakly bound levels of H 2 , H D and D 2 Molecular levels involved in recombination are those close the the dissociation threshold. We have examined the literature for experimental data on levels with dissociation energies of less than 300 K . The results are summarized in the following sections. The unit of energy used in molecular spectroscopy is c m - 1 , with 1 c m - 1 ~ 1.4388 K. E . l H 2 The dissociation energy of the ground state of the H 2 molecule was measured by Herzberg [60], who obtained D® — 36118.3(3) c m - 1 . His data was re-analyzed by Stwalley [61] who focussed on the weakly bound states whose energies vary with quantum number in a manner determined by the known form of the long-range part of the singlet potential; he found DQ = 36118.6(5) c m - 1 . The energies of the molecular levels were measured by Dabrowski [56], who gives the energies with respect to the ground state, accurate to 0.1 c m - 1 . Combining the dissociation energy of reference [60] with these energies we obtain the results summarized in table E . l . The exception is that for the (14,4) level we use the 'refined' estimate of reference [61], since there it makes a difference. 137 Appendix E. Weakly bound levels of H 2 , HD and D 2 138 Table E. l : Dissociation energies of levels of molecular H 2 close to the dissociation thresh-old. V J D(K) 14 4 0.7 12 10 19.7 3 27 29.2 6 22 62.0 14 3 72.5 13 7 74.0 14 2 136.9 14 1 183.9 14 0 208.5 11 12 208.9 13 6 286.7 Appendix E. Weakly bound levels of H 2 , H D and D 2 139 E . 2 D 2 Le Roy and Barwell [62] measured the U V spectrum of D 2 , and found for the dissociation energy (of the ground state) D = 36748.88(30) c m - 1 . They also were able to state that the dissociation energy of the (21,0) level was D - G(21,0) = 2.04(22) c m - 1 and the splitting between the (21,0) and the (21,1) levels was G(21,1) - G(21,0) = 1.89(10) c m " 1 . The energies of the lower levels (with respect to the ground state) were measured by Bredohl and Herzberg [63]. Table E.2 gives the dissociation energies of the levels within 300 K of the continuum. Appendix E. Weakly bound levels of H 2 , H D and D 2 140 Table E.2: Dissociation energies of levels of molecular D 2 close to the dissociation thresh-old. V J D(K) 21 1 0.2 21 0 2.9 20 5 41.8 20 4 91.5 20 3 135.2 20 2 170.2 20 1 194.3 20 0 206.8 Appendix E. 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