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Magnetic resonance on atomic hydrogen confined by liquid helium walls Morrow, Michael Robert 1983

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MAGNETIC RESONANCE ON ATOMIC HYDROGEN CONFINED BY LIQUID HELIUM WALLS by MICHAEL ROBERT MORROW B.Sc, McMaster University, 1977 .Sc., University of B r i t i s h Columbia, 1979 THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS We accept th i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June, 1983 ©Michael Robert Morrow, 1983 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of /"^-/^/^ 5  The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 (3/81) ABSTRACT A gas of atomic hydrogen confined at and below 1K in a sealed c e l l has been studied by magnetic resonance at the z e r o - f i e l d hyperfine t r a n s i t i o n frequency of 1420 MHz. A review i s presented of magnetic resonance theory for a two l e v e l system, with emphasis on determination of the absolute magnetization by two methods: c a l i b r a t i o n of the spectrometer s e n s i t i v i t y and by use of the radiation damping time constant. Measurements at 1K on a gas at low density, 1 0 11 <ri|_^ <5x 1 0 1 2 cm"3, in the saturated "He vapour density have yielded the rate for the reaction H+H+He+^+He, the d i f f u s i o n constant and pressure s h i f t of the hyperfine t r a n s i t i o n for H interacting with the He gas, and the cross-section for spin exchange relaxation. At temperatures below 1K, measurements of the frequency s h i f t and e f f e c t i v e recombination rate for H adsorbed on the He fi l m have yielded values of the binding energies for H on "He and for H on 3He as well as the hyperfine t r a n s i t i o n frequency s h i f t and surface recombination rate for H adsorbed on each of these surfaces. The binding energies are found to be 1.15(5) K for H on "He and 0.42(5) K for H on 3He. Measurements have been ca r r i e d out at temperatures as low as 162 mK for H on "He and 65 mK for H on 3He. A model i s also presented for the magnetic resonance lineshape for H atoms undergoing occasional s t i c k i n g events on the helium surface. This model has been applied to frequency s h i f t and transverse relaxation data at low temperatures to y i e l d s t i c k i n g p r o b a b i l i t i e s of 0.046(5) for H on "He and 0.016(5) for H on 3He. TABLE OF CONTENTS ABSTRACT . i i LIST OF TABLES v i i LIST OF FIGURES v i i i ACKNOWLEDGEMENTS xi CHAPTER I: Introduction 1 1 .1 Background 1 CHAPTER I I : Magnetic Resonance at the Hyperfine Frequency . 7 2.1 Theoretical Background 8 2.2 Absolute Determination of the Magnetization Using Magnetic Resonance 16 2.3 Radiation Damping 20 CHAPTER I I I : Review of H Atom Properties 23 3.1 Recombination 23 3.2 Spin Exchange 34 3.3 Properties Dependent on the H-He Potential 40 3.4 Interaction with the Surface 44 CHAPTER IV: Measurements Between 1K and 1 . 3K 50 4.1 Experimental Apparatus 51 4.1a The Cryostat 51 4.1b The Spectrometer 58 4.2 Recombination Measurements 62 4.3 Dif f u s i o n Measurements 66 4.4 Spin Exchange Measurements 71 4.5 Pressure S h i f t Measurements 76 CHAPTER V: Experiments Below 1K 81 5.1 The Low Temperature Apparatus 83 5.1a The Refrigerator 83 5.1b The Sample Region 85 5.1c The Resonator Housing 92 5.1d The Spectrometer 95 5.2 Model for the Magnetic Resonance Lineshape for Atomic Hydrogen Confined by Liquid Helium Walls 97 5.3 The Binding Energy, Surface Hyperfine S h i f t , and Surface Recombination Rate for H on *He and H on 3He 110 5.3a Recombination Measurements 113 5.3b Binding Energy from Frequency S h i f t Measurements 119 5.4 Measurement of Sticking P r o b a b i l i t i e s from the Magnetic Resonance Frequency and Lineshape 128 5.4a Sticking Probability for H on 4He Coated Walls . 130 5.4b Sticking Probability for H on 3He Coated Walls . 131 5.5 Comparison to Other Results 139 CHAPTER VI: Summary * 144 REFERENCES 149 APPENDIX A: Magnetic Resonance at the Hyperfine Frequency . 154 A.1 Theoretical Background 154 v i A.2 Measurement of Diffusi o n with Zero F i e l d Hyperfine Resonance 163 APPENDIX B: Absolute Determination of the Magnetization Using Magnetic Resonance 165 APPENDIX C: Radiation Damping 176 APPENDIX D: The F i l l i n g Factor 184 APPENDIX E: Spin Exchange in Two Dimensions 194 APPENDIX F: The Thermometer Recalibration 202 v i i LIST OF TABLES TABLE I: Some Relevant Constants 8 TABLE I I : The Recombination C o e f f i c i e n t s 29 TABLE I I I : Calculations of the Recombination Rate by Greben et a l . (31 ) 33 Table IV: Thermally Averaged Surface Spin Exchange 201 v i i i LIST OF FIGURES 1 Ground state hyperfine lev e l s of a hydrogen atom in a magnetic f i e l d . a/h=1,420,405,751.768 Hz 10 2 The singlet and t r i p l e t H2 potentials 25 3 The cryostat used for zero f i e l d measurements at 1K .... 52 4a The discharge c o i l and modified Arenberg pulsed source 54 4b A s i m p l i f i e d version of the resonator tuning scheme. . 56 5 Block diagram of the 1420 MHz Spectrometer 59 6 Inverse hydrogen atom density n^" 1 vs time for three d i f f e r e n t "He gas densities n^ & 63 7 dn~|]|/dt vs. n^ e for two separate cool-downs. The slope is k 64 8a A t y p i c a l spin echo free induction decay 68 b A t y p i c a l plot of log(A(2T>/A(0)) vs T 3 68 9 Hydrogen d i f f u s i o n data plotted vs helium gas density n^ e. The v e r t i c l e axis, 37rv/32D i s equal to Qp n^ e > where i s the thermally averaged d i f f u s i o n cross-section 70 10 1/T^  vs H atom density n^ 73 11 Temperature dependence of T^ 1 due to three dimensional spin exchange scattering in the gas calculated using the improved t r i p l e t p otential of Ref.35b 75 12 Hyperfine frequency f - f Q= 1,420,405,000 Hz plotted vs. Helium gas density n(_)e« T n e slope i s -11.83x10" 1 8 Hz cm3 77 13 The thermally averaged hyperfine frequency s h i f t of atomic hydrogen vs temperature. Figure reproduced from Ref.39 78 14 The thermally averaged hyperfine s h i f t of atomic hydrogen at low temperature. Figure taken from (39) ... 79 15 Schematic diagram of the low temperature part of the apparatus 86 16 The bursting seal and room temperature gas handling system 88 17 The recombination rate for H on *He plotted as kT 1 / 2 vs T" 1. The s o l i d l i n e i s for X=0.14A and Eg=1.15K. The x's are the data of Ref.(21) and the +'s are the data of Ref. (59) 116 18 The recombination rate for H on 3He plotted as kT^ 2 vs T - 1 . The s o l i d l i n e i s for.X=0.13A and Eg=0.39K. The crosses are the results of van Yperen et al.(22) at high f i e l d for a mixture of 3He and 'He 117 19 Plot of the hyperfine frequency s h i f t for H on "He as logAf/T vs 1/T in order to exhibit the exponential dependence for small <j>0 124 20 The data of figure 19 i s replotted as Af vs T for l i q u i d aHe walls. The s o l i d l i n e i s a f i t to the data using X As=49KHz, Eg=1.l5K, and the bulk pressure s h i f t c o e f f i c i e n t measured above 1K 125 21 The frequency s h i f t in a 3He coated container plotted as logAf/f vs 1/T. The s o l i d l i n e corresponds to As=23kHz and Eg=0.43K 126 22 The s h i f t and 1/T2 for a l i q u i d "He coated container plotted vs 1/T 131 23 A C J and 1/T2 vs l/T for H on 3He 137 24 The LC equivalent to the resonant cavity c i r c u i t 166 25 Energy dependence of surface spin exchange cross sections 199 26 Temperature dependence of thermally averaged surface spin exchange cross sections 200 27 The fixed point device detection c i r c u i t 204 xi ACKNOWLEDGEMENTS I would l i k e to thank Prof. W. N. Hardy for his suggestion and supervision of thi s project. His influence at a l l stages of design and execution of thi s work has been instrumental in i t s success. I have also benefitted from many discussions with Prof. A. J . Berlinsky. His contributions to the understanding of the results are g r a t e f u l l y acknowledged. Most of the late nights associated with t h i s work were shared with Dr. R. Jochemsen. His p a r t i c i p a t i o n in this work i s also warmly acknowledged. I would also l i k e to thank Dr. A. Landesman for his work on the 1K cryostat. B. W. Statt and P. R. Kubik assisted in the i n i t i a l analysis of the 1K r e s u l t s . For this and for many helpful discussions, I thank them also. We are indebted to Prof. S. Crampton for his helpful discussions of the magnetic resonance lineshape problem and of the f i l l i n g factor problem. We also thank Profs. W. Unruh and L. Rosen for their solutions of the sum occurring in the lineshape problem and B. Shizgal for the use of a program used in the spin exchange c a l c u l a t i o n s . We are grateful to the Physics Department Electronics Shop for their help in design and construction of much of the x i i equipment; W. Walker for the loan of equipment and his assistance in maintaining i t ; R. Haering also for the loan of equipment; P. Akers for his help on the low temperature tuning linkages; R. Vessot of the Smithsonian Astrophysical Observatory for the loan of a magnetic sh i e l d ; T. Landecker of the Dominion Astrophysical Radio Observatory for several microwave o s c i l l a t o r s ; P. Venne of Societe Radio-Canada, Montreal for the loan of a rubidium atomic clock; and G. A. Gait of the Dominion Radio Astrophysical Observatory, Penticton, for his assistance in c a l i b r a t i o n of the frequency standard. I g r a t e f u l l y acknowledge the support of the Natural Sciences and Engineering Research Council in the form of a 1967 Science Scholarship and the Izaak Walton Killam Memorial Fellowships for a Predoctoral Fellowship. F i n a l l y , I would l i k e to thank my wife, Cynthia Patterson, for her patient support and encouragement. 1 CHAPTER J_ Introduction 1.1 Background In 1959, Hecht(l) proposed that atomic hydrogen, with the electronic spin polarized by a large magnetic f i e l d , would be an unusual and inter e s t i n g quantum gas. In p a r t i c u l a r , because of i t s low mass, i t would l i k e l y remain gaseous down to T=0K, and one might be able to study Bose-Einstein condensation and attendant s u p e r f l u i d i t y in a weakly interacting system. The subject apparently remained dormant u n t i l the early seventies when Hess(2) attempted to s t a b i l i z e atomic hydrogen and deuterium in large magnetic f i e l d s . The p o s s i b i l i t y of such an unusual material becoming available for study stimulated t h e o r e t i c a l interest in the properties of spin-aligned hydrogen and i t s isotopes. Etters et al.(3) used a Monte-Carlo technique to study the ground state properties of spin-aligned H, D, and T and found that H and D interacting v i a the t r i p l e t p o t e n t i a l should, indeed, remain gaseous to OK thus o f f e r i n g a chance to observe Bose-Einstein condensation in a weakly interacting system. Calculations were made of the ground state energies of l i q u i d and s o l i d spin-aligned hydrogen(4) and of the f i n i t e 2 temperature properties of the hydrogen isotopes(5). The suggestion was also made that a thin layer of spin-aligned hydrogen might be useful as a wall coating for H confinement(6). Berlinsky et al.(7) pointed out that s o l i d spin-aligned hydrogen would recombine rapidly due to i t s i n s t a b i l i t y against spontaneous magnon creation. Such high densities, however, are not needed for observation of the Bose-Einstein t r a n s i t i o n for which the theory for a non-interacting hydrogen gas predicts T. = (1 .7x 1 0" 1 * ) nf/3 ^ H where T c i s the t r a n s i t i o n temperature and n^is the density in atoms/cm3. This gives, for example, T c = 0.2K at n = 4 X I0 1 9/cm 3. There thus remained hope that a gas of n spin-aligned hydrogen would not necessarily be unstable against spin f l i p s at Bose-Einstein condensation densities.(8) Theoretical interest subsequently emphasized the properties of the r e l a t i v e l y low density atomic hydrogen gas.(9) Experimentally, the ultimate l i m i t a t i o n has been imposed by the strongly exothermic (52000 K) recombination reaction. (We w i l l normally express energy in units of K.) In experiments directed at studies with high densities of H atoms, th i s has required the use of high f i e l d s (up to 10T) to force the atoms into the two lowest hyperfine states (see figure 1) which can then recombine only as a result of the small admixture (e=a/4*xeB 3 in high f i e l d ) of the electron reversed state in the lowest hyperfine state. There are, however, a number of properties of the hydrogen gas which can be studied at low densities where recombination can be slow even in the absence of applied f i e l d . I n i t i a l attempts to cool a gas of atomic hydrogen to l i q u i d Helium temperatures were performed under such conditions. Crampton et al. ( l O ) used a l i q u i d N 2 cooled discharge to produce a flow of H atoms which entered a quartz bulb in a cavity tuned to the z e r o - f i e l d hyperfine resonance. The bulb surface was coated with s o l i d H 2 . Using pulsed magnetic resonance at the z e r o - f i e l d hyperfine frequency, they observed densities of the order of 10 1" atoms/cm3 and were able to study relaxation times and t the hyperfine s h i f t associated with the interaction of H atoms with the s o l i d H 2 surface. Hardy et a l . (11) used a room temperature discharge to produce atoms which were then cooled to between 77K and 4.2K. Magnetic resonance at the 6.5kG minimum separation of the two lowest hyperfine lev e l s was used to look at the i n t e r a c t i o n of the H atoms with buffer gases such as H 2 and He and at relaxation due to wall and spin exchange c o l l i s i o n s . The work of reference (10) in p a r t i c u l a r made clear the u n s u i t a b i l i t y of H 2 as a wall coating and, in confirmation of t h i s , Crampton(l2) was able to use the techniques of reference (10) to measure the binding energy for H on H 2 finding i t to be 38+/-5K. The next e f f o r t s were directed at confining a gas of atomic 4 hydrogen in a container with "He coated walls. S i l v e r a and Walraven(13) used magnetic compression and compression by a "He d i f f u s i o n pump to confine a sample of magnetically s t a b i l i z e d H atoms. The superfluid f i l m flow problem was solved by a separate cooling stage associated with the He d i f f u s i o n pump. The maximum density was measured to be greater than 1.8 x I0 1 f t/cm 3 by observation of the heating r e s u l t i n g when the sample was recombined. Concurrently, the experiments comprising the f i r s t part of t h i s work were carr i e d out. These experiments were directed at using magnetic resonance to observe a low density (of order 10 9 to 10 1 2 cm"3) gas of atomic H confined in a sealed glass bulb with superfluid "He coated walls between 1K and 1.3K. At these temperatures and densities i t i s interactions in the gas phase between H and He and H and H which are most important in determining the observable properties of the sample. The sealed geometry of the c e l l eliminates the complications associated with superfluid f i l m flow out of the c e l l . Atoms were produced in the c e l l by means of an r . f . discharge at 1K. The probe used was magnetic resonance at the z e r o - f i e l d hyperfine t r a n s i t i o n (F=0,Mp=0 -> F=1,Mp=0) in the hydrogen electronic ground state. With t h i s technique, one d i r e c t l y measures relaxation times, hyperfine frequency s h i f t s , and, from the signal amplitude, atom de n s i t i e s . These measurements have been used to obtain information about the recombination rate, the interactions between H atoms and He 5 atoms through the binary d i f f u s i o n constant and the hyperfine pressure s h i f t , and the H-H interation through spin-exchange relaxation. The results of these measurements were i n i t i a l l y reported in a l e t t e r ( 1 4 ) . Experiments c a r r i e d out elsewhere during t h i s period(l5) led to attainment of higher densities in f i e l d s of order lOOkG. Theoretical work, at t h i s time, was beginning to focus on the importance of the container surface. Recombination of two H atoms in the gas phase was known to require the presence of a t h i r d body (such as a He atom or another H atom) to allow conservation of energy and momentum. In the recombination of two atoms confined to a surface, the surface can play the role of the t h i r d body. It was generally recognized that i f there was s i g n i f i c a n t binding of H atoms to the container wall, surface recombination, being a two body process, could dominate the three body bulk recombination under conditions of interest in experiments directed toward Bose-Einstein condensation. Calculations of the binding energy for H on "He yielded values of 0.1K(16), greater than 0.6K(17) and less than 0.1R(1B). It was also pointed out(19) that interactions between H atoms on the surface would lead to saturation of the surface state at a density proportional to the binding energy. Measurement of the binding energies of atomic hydrogen to the helium l i q u i d s thus became an experimental p r i o r i t y . Binding energy measurements were pursued both at U.B.C. and 6 at the University of Amsterdam by measuring the temperature dependence of properties related to the surface density of H on the helium f i l m . The Amsterdam experiments were c a r r i e d out by observing the pressure of the atomic gas and thus measuring the recombination rate as a function of temperature to provide binding energies for deuterium(20) on "He, H on "He (21), and H on a 3He-"He mixture(22). In the experiments which make up the second part of this thesis, z e r o - f i e l d hyperfine resonance measurements were extended to temperatures of the order of 0.060K giving measurements of the recombination rate and the s h i f t in the hyperfine resonance frequency as functions of temperature. As w i l l be discussed in later chapters, both of these properties, at low temperatures, can be related to the density of adsorbed H thus providing two largely independent determinations of the binding energy. Letters were published presenting these results for H on 4He (23) and H on 3He (24) films. In reference (24), there i s also a discussion of how the magnetic resonance lineshape measured in these experiments has been used to obtain average s t i c k i n g p r o b a b i l i t i e s for H on the helium l i q u i d s . 7 CHAPTER II Magnetic Resonance at the Hyperfine Frequency Magnetic resonance i s well suited to the study of gaseous atomic hydrogen at low temperatures. The signal amplitude provides access to the atom density and thus the recombination rate. The shape and frequency of the free induction decay contain information about the interactions of the atoms with other atoms and with the container walls. The experiments making up t h i s work were ca r r i e d out using pulsed magnetic resonance on the zero f i e l d hyperfine t r a n s i t i o n in atomic hydrogen. In t h i s chapter, we w i l l describe some of the results of magnetic resonance theory as applied to the z e r o - f i e l d t r a n s i t i o n . These include the f i e l d dependence of the tr a n s i t i o n frequency, the absolute determination of the o s c i l l a t i n g magnetization from the signal amplitude and from the radiation damping time constant, and the use of spin echoes to measure d i f f u s i o n of atomic hydrogen in helium. The derivation of these re s u l t s i s reviewed in Appendices A, B, and C. 8 2.1 Theoretical Background The hyperfine structure of the ground ele c t r o n i c state of a hydrogen atom in a magnetic f i e l d i s determined by the spin Hamiltonian [II-1] X - -*iif 0 . ( 7 p T - 7 eS) + a(T-S) where 7 e and are the absolute values of the gyromagnetic rati o s for the electron and proton respectively, B Q i s the applied magnetic f i e l d and a i s the hyperfine coupling constant for the hydrogen atom (a/h ~ 1420 MHz; see table I for the recent values of the relevant constants). I and S are of the TABLE I Some Relevant Constants a/h : 1,420,405,751.773+/-0.001 Hz (25a):l980 : 1,420,405,751.768+/-0.002 Hz (25b):1970 7 e : 1 .760842+/-0.000026 X 10 1 1 s e c ' V T (25c) 7 D : 2.675197+/-0.000039 X 108 s e c ' / T (25c) form o*/2 where the components of o are the Pauli spin matrices and fi7e=gMB and hyp=quH so that fi7 eS and ^ 1 are magnetic 9 moment operators. In a s t a t i c f i e l d B^ >, X i s diagonalized by the hyperfine states |h> - |F,Mp> where *F=I+S i s the t o t a l spin and Mp i s i t s component along the f i e l d . In order of increasing energy, these states are [Il-2a] |1> = |0,0> = cos 0||T> - sin 0|U> [ l l - 2 b ] |2> = |1,-1> = |J*> [Il-2c] |3> = |1,0> = sin 0|lt> + cos 0|ti> [Il-2d] |4> = |1,1> = |!+> Here tan 26 = a/[fi(7 e+7p)B 0], and T and % refer to electron and proton spins in the d i r e c t i o n of B Q respectively. The energies are given by [Il-3a] E^= -a/4 • d / 2 ) [ a 2 + fc2[7e + TpPB^]" 2 [Il-3b] E2= a/4 : ( f i / 2 ) ( 7 e " 7 p ) B 0 The f a m i l i a r f i e l d dependence i s reproduced in figure 1, and the major t r a n s i t i o n of i n t e r e s t , between |1> and |3> i s indicated. For low f i e l d s , the t r a n s i t i o n frequency can be written [II-4] oj(Bo) = a + + 7p]2B(2. = a_ + 2773.05(3) B? h B — > Figure 1 Ground state hyperfine l e v e l s of a hydrogen atom in a magnetic f i e l d . a/h=1,420,405,751.768 Hz to second order in B 0(gauss). Note that the c o e f f i c i e n t d i f f e r s s l i g h t l y from the value 2750 given by Kleppner et al.(26) and a number of subsequent authors. A bias f i e l d of about 20 milligauss i s s u f f i c i e n t to is o l a t e the resonance of interest from the nearby t r a n s i t i o n s |1> to |4> and |1> to |2>. (At lower f i e l d s , we have observed shortening of the free induction decay which we at t r i b u t e to |3>**|4> and |3>**|2> t r a n s i t i o n s stimulated by the fluctuating f i e l d seen by the atoms as they traverse the c e l l . At t y p i c a l operating bias f i e l d s of 30 mgauss, these e f f e c t s are neg l i g i b l e . ) By applying such a f i e l d , we are able to do experiments on what i s e s s e n t i a l l y a two l e v e l system. The density matrix for the two l e v e l system can be expanded into terms l i n e a r in the Pauli matrices, o', operating within the manifold of states |1> and |3>. If the c o e f f i c i e n t s of each component of <T* are treated as components of the expectation value of a f i c t i t i o u s spin S'=o'/2, the density matrix becomes [II-5] p = J_ + <S'>-o' 2 It i s well known(27) that by s i m i l a r l y expanding the Hamiltonian, i t may be made analogous with that for the int e r a c t i o n , in the f i c t i t i o u s system (primed), of S' with a 12 f i e l d B* [II-6] X= 1E Q- 7'h(B'. S' ) where 7' i s a f i c t i t i o u s gyromagnetic r a t i o . The u t i l i t y of such an approach i s that for a magnetic moment, the equation of motion can be taken d i r e c t l y from spin 1/2 magnetic resonance theory. Consider the application of a small s t a t i c f i e l d , B o z, in the 2 d i r e c t i o n and an o s c i l l a t i n g f i e l d , 2B l zcoso)t, in the same d i r e c t i o n . (The factor 2 i s included in a n t i c i p a t i o n of resolving the r . f . f i e l d into two rotating components each of amplitude B,z , only one of which w i l l be important in the rotating frame). In Appendix A, i t i s shown that i d e n t i f i c a t i o n of the parameters in the f i c t i t i o u s system with the actual spins and f i e l d s leads to [II-7] m' = •h7 ,trace(pS' ) [n-83 which i s to be compared to equation [II-4] and [11-93 7'Bi = 2(7 e + 7 p) B 1 2coscut 13 We note that the actual magnetization in the z dir e c t i o n i s [11 - 1 0 ] M 2 = nHfc ( 7 e + 7 p ) ( P u + P2] >/2 whereas [ 1 1 - 1 1 ] = r^fi 7 ' (p 1 2 + p 2 1 ) / 2 and [11 - 1 2 ] M£ = n Hfi 7' (P^ - P 2 2 ) / 2 where ri|_| i s the atom density. It i s convenient to identify 7' with (7 e +7p) so that the magnitude of the f i e l d and magnetization in the z d i r e c t i o n of the actual system correspond with those in the x d i r e c t i o n of the f i c t i t i o u s system. Using the standard description of magnetic resonance on a spin 1 / 2 system(28), one may now describe the e f f e c t of a pulsed r . f . f i e l d on atomic hydrogen. If we assume that the system i s in equilibrium before the r . f . pulse, then the i n i t i a l density matrix i s [11-13] p(0) = / p , (0) 0 s { 0 p 3 ( 0 W where p^ and P 3 are the f r a c t i o n a l populations of the states | 1 > and |3> respectively. As described in Appendix A, i f a i s given by 1 4 [11-14] co = a + _a / fi B o z ( 7 e + 7n) \ 2 ft 2h \ a / then the application of an o s c i l l a t i n g f i e l d 2B ) z cos (cot) along the z axis for an int e r v a l [11-15] t = 7T * TJJ. 2(7 e+7 p)B, z '2 w i l l result in [11-16] P12(ir/2) = = ( p 1 (0) - P^(0)) Thus, immediately following the n/2 pulse, there i s an o s c i l l a t i n g magnetization along the z axis given by [11-17] M z(t) = n H f i (7 e + 7 p) (p^O) -p 3 ( 0 ) ) sin(cot) = M 0 sin(cot) For temperatures such that kT>>a, one finds (p^(O)-pg(O))~ a/4kT. The o s c i l l a t i n g magnetization w i l l induce a voltage in a c o i l aligned along the z di r e c t i o n that i s proportional to dM z(t)/dt. When the relaxation of M' i s described by the Bloch equations (see Appendix A) we may id e n t i f y T^ 1 with the rate for the diagonal elements of the density matrix to relax to their equilibrium values and T"J with 15 the rate at which the o s c i l l a t i n g magnetization along the z axis disappears. Zero f i e l d magnetic resonance on H i s thus seen to d i f f e r from standard spin 1/2 magnetic resonance in three ways: there i s no i n i t i a l s t a t i c magnetization, the frequency depends quadratically on the s t a t i c f i e l d , and the o s c i l l a t i n g f i e l d i s applied along the s t a t i c f i e l d rather than perpendicular to i t . Because of t h i s l a t t e r feature, t h i s system i s often referred to as longitudinal resonance. Aside from these points, however, the system behaves i d e n t i c a l l y to the standard spin 1/2 system. Thus results pertaining to signal amplitude, spin echoes, radiation damping, etc. can be taken over d i r e c t l y from ex i s t i n g l i t e r a t u r e . As a useful example of t h i s correspondence, we show in Appendix A, that the expression for the spin echo amplitude at time 2T following a it/2 pulse at t = 0 and a TT pulse at t=r i s , to f i r s t order in the gradient, -2r/T, - ( 2 / 3 ) D [ 7 p f f 3 B z / 3 Z ] 2 T 3 [11-18] M +(2r) = -MQe e e U where D i s the d i f f u s i o n constant, dB^Sz i s the f i e l d gradient, and 7 eff =47raB Q Z . Here, a=2773.05 Hz/gauss 2. Except for the use of 7 e f f in place of 7 , t h i s expression i s the same as that obtained for spin 1/2 magnetic resonance. 16 2.2 Absolute Determination of the Magnetization Using Magnetic Resonance To re l a t e the observed signal to the magnetization and thus the atom density, we consider the spectrometer coupled to a resonator containing a sample whose magnetization along the resonator axis i s o s c i l l a t i n g . A re s u l t i n g e.m.f. i s induced on the resonator and gives r i s e to an r . f . f i e l d which we refer to as the response f i e l d . It i s to be thought of as o s c i l l a t i n g primarily in the d i r e c t i o n of the resonator axis. The l o c a l o s c i l l a t i n g magnetization w i l l have, in addition to the expected sinusoidal time dependence, a slower variation due to e f f e c t s such as d i f f u s i o n . Accordingly, i t i s l a b e l l e d as M("r,t)sin(a>t). If the resonator i s tuned to frequency CJ, the current response w i l l be ir/2 out of phase with the o s c i l l a t i n g magnetization. The attendant response f i e l d i s then written as Ii r("r,t)cos ( u t ). It i s shown, in Appendix B, that the i n i t i a l signal power delivered to the spectrometer amplifier at c r i t i c a l coupling following a 7r/2 pulse i s 17 [ 1 1 - 1 9 ] Ps = ^ M O Q L M 0 2 V S T ? 4 where the loaded Q, Q[_=o>/Aw, i s the r a t i o of the resonant frequency to the f u l l width at half maximum of the resonance in ref l e c t e d power, V s i s the sample volume, M 0 i s defined by [ 1 1 - 1 7 ] , and [ / dVs M(r\t' )-B r Cr,t' ) ] 2 [ 1 1 - 2 0 ] n = • M2 V s / | B r ( f , t ' ) | 2 dV is the f i l l i n g factor. The presence of t' in [ 1 1 - 2 0 ] i s to remind us that the question of whether or not inhomogeneities in the d i r e c t i o n of IJr("r) influence TJ depends on whether the time scale of the measurement i s fast or slow compared to that for d i f f u s i o n of the spins. As described in Appendix B, i f one i s interested in an ef f e c t manifested immediately following the n/2 pulse, the magnetization i s able to maintain i t s phase r e l a t i o n s h i p with the l o c a l f i e l d and we have "B r ("?)• MCr) « MQ | B r | . Equation [ 1 1 - 2 0 ] then becomes 18 [11-21 ] [ / d v s | B r l ] = v s / | r r | 2dv I f , however, the time scale i s long enough for d i f f u s i o n to s p a t i a l l y average the magnetization then, assuming a symmetric response f i e l d , the f i l l i n g factor becomes [11-22] T J " = M 0 2 where M Q x j s i s the average component of the magnetization along the resonator axis. In Appendix B, i t i s also shown that i?' can be determined for a p a r t i c u l a r experimental s i t u a t i o n given the power, Pj, incident during the it/2 pulse and the it/2 pulse length, , as found by varying the pulse length to maximize the response. The result i s fcjVg / it [11-23] T J * = 2*i 0Q 0Pj V < 7 e + V Tn, where Q 0 = 2Qj_ i s the unloaded q u a l i t y factor. While i t i s 17" which enters the determination of the atom density via equation [11-19], we believe that the homogeneity of bur resonator i s s u f f i c i e n t l y good that we can approximate the f i l l i n g factor by 19 TJ' and thus determine the atom density in the sample i f the spectrometer i s ca l i b r a t e d absolutely. In the next section, we w i l l discuss a means of i n f e r r i n g the atom density from radiation damping measurements, which do not depend on knowledge of the input power or ffy » in which the f i l l i n g factor enters to a d i f f e r e n t power. This provides a useful means of confirming the absolute determination of the magnetization. 20 2.3 Radiation Damping The phenomenon of radiation damping, in which the coupling of the cavity to the atomic resonance perturbs the observed atomic signal (in this case the free precession), plays an important role in the present experiments. F i r s t l y , i t damps the free precession signal by an amount proportional to the atom density and becomes the dominant "relaxation" mechanism at high de n s i t i e s . Secondly i t s h i f t s the resonant frequency i f the cavity i s mistuned. F i n a l l y , the radiation damping rate can be used to determine the H-atom density in a measurement that i s p a r t i a l l y independent of the determination via the absolute signal amplitude. In Appendix C we discuss how the usual three Bloch equations can be supplemented by a fourth which couples the f i e l d of the cavity to the spin system(29). We consider the usual case where the axis of the r . f . f i e l d i s in the x d i r e c t i o n but the important results are d i r e c t l y applicable to the longitudinal resonance case. Bloom(30) has studied the solution of the Bloch equations and obtained solutions in closed form when T^  , the longitudinal relaxation time, i s i n f i n i t e . In the treatment of Appendix C, those results are generalized for a r b i t r a r y cavity tuning. Consider the cavity tuned to frequency u>c containing, 21 immediately following the ir/2 pulse, a magnetization of amplitude M r rotating with frequency cu. If QL=W A C O, where Aco i s the f u l l width of the cav i t y resonance, and 8co=co-coc i s the cavity mistuning, then the frequency p u l l i n g e f f e c t i s shown, in Appendix C, to be [11-24] to - cuQ = T ? " ' Y O I U Q M , (t) 6co/Aco L 1 + [26co/Aco]2 and the transverse magnetization i s shown to decay according to [11-25] 2_1 dMj- = X(t) = M Q T?"7QLM z(t) 1 + 1 Mr dt 2 1 + [28CJ/ACJ]2 T 2 For T 2 = 0 0 , t h i s implies [11-26] co - coQ = X(t) 6co (Aco/2) From [11-25] we see that i f T 2 = °o and 8co=0, the asymptotic time constant i s due only to radiation damping and i s given by T«> 1=Mo*?"7Q|_M0/2. The technique used to extract r i 1 from the free induction decay i s discussed in Appendix C. For the zero f i e l d t r a n s i t i o n of interest here, 7 in t h i s discussion i s replaced by ( 7 e + 7 p ) ' These results were used to obtain M Q for actual data from the IK measurements in which d i f f u s i o n out of the resonator was li m i t e d by sample geometry thus allowing T J " of 22 [11-25] to be i d e n t i f i e d with the f i l l i n g factor of [11-19]. It is s i g n i f i c a n t that for radiation damping, M 0 depends on (rjn)'1 whereas in a determination using equation [11-19], i t i s (t}')~V2 that relates MQ and the si g n a l . Good agreement i s found between the values of M 0 calculated in these two largely independent ways which helps to confirm both our confidence in the absolute c a l i b r a t i o n of the spectrometer and in our interpretation of the f i l l i n g factor. This agreement also provides some assurance of the v a l i d i t y of approximating T J " by 7 j ' for these conditions. 23 CHAPTER III Review of H Atom Properties 3.1 Recombination The extreme i n s t a b i l i t y of atomic hydrogen against recombination into H presents the severest challenge to the success of almost any experiment involving hydrogen confined at low temperature. In t h i s section we w i l l review some of the t h e o r e t i c a l discussion of this simple chemical reaction and consider what information about i t i s accessible to z e r o - f i e l d measurements. Recombination i s a three body c o l l i s i o n process in which the t h i r d body, which i s necessary so that energy and momentum may be conserved, can be a He atom, an H atom, or the container wall. In the remainder of t h i s discussion we w i l l , i f i t i s not e x p l i c i t l y stated, assume the p a r t i c i p a t i o n of the t h i r d body in the reaction. We are thus l e f t to consider c o l l i s i o n s involving two hydrogen atoms, each in one of four hyperfine states as shown in figure 1. These interact via one of two possible p o t e n t i a l s . If the t o t a l e l e c t r o n i c spin of the pair i s S=0, the potential i s s i nglet and has an a t t r a c t i v e well supporting many bound 24 states the lowest of which has a binding energy of about 52000K (4.5 eV). In the case of a t r i p l e t , or S=1 i n t e r a c t i o n , there is no bound state and only a very weakly a t t r a c t i v e well at large distance. This i s a consequence of the Pauli exclusion p r i n c i p l e and the r e s u l t i n g s e l e c t i o n rules have been discussed by Greben et a l . O O and w i l l be reviewed below. We encounter, in the present experiments, two important recombination mechanisms. At temperatures for which the gas density, n^ g, above the l i q u i d He wall coating i s s i g n i f i c a n t , the role of the t h i r d body i s played by He atoms and the rate equation for decay of the bulk density of hydrogen, n^, i s [III-1] = - k n 2 n H e dt where k i s the rate constant. At lower temperatures, n^ e i s small and the density of hydrogen on the surface i s not n e g l i g i b l e . The dominant mechanism for loss of H atoms in the sample i s then recombination of atoms bound to the surface. Under such conditions, the role of the t h i r d body i s played by the surface and the rate equation can be written 25 26 [ 111-2 ] da = - L a 2 dt where k s i s the surface rate constant and a i s the H atom density on the surface. Because of the equilibrium between gas and surface bound H atoms, i t i s possible to rewrite equation [I I I - 2 ] to give an e f f e c t i v e bulk recombination rate due to surface recombination. This w i l l be done in a l a t e r section. Greben et a l . ( 3 l ) have calculated k from equation [111-1 ] using a f u l l y quantum mechanical treatment of the three body scattering problem presented by gas phase recombination. They present thermally averaged results for 1K, 0.1K, and 0.01K. While the d e t a i l s of the three body c a l c u l a t i o n and the numerical results are not d i r e c t l y applicable to the case of surface recombination, the general conclusions regarding the role of spin in the recombination reaction are applicable to both cases. As given in Ref. (31), we expect the rate equation to be proportional to 27 £ 1 1 1 - 3 ] I P. |T . f | 2S(p. - p f )6 (E j - E f ) i,f where P,- i s t h e p r o b a b i l i t y o f t h e i n i t i a l s t a t e o f a p a i r o f H a toms ( p l u s t h e t h i r d b o d y ) , Tjf i s t h e a m p l i t u d e f o r s c a t t e r i n g f r o m t h e i n i t i a l s t a t e t o t h e f i n a l s t a t e o f an Hp m o l e c u l e and t h e t h i r d body a n d p-^ and E ^ a r e t h e momentum and e n e r g y o f t h e i n i t i a l and f i n a l s t a t e s . The i n i t i a l s t a t e w i l l i n c l u d e two H a toms e a c h i n one o f t h e f o u r h y p e r f i n e s t a t e s g i v e n i n e q u a t i o n s [ l l - 2 a ] t o [ I I - 2 d ] . We w i l l r e f e r t o t h e h y p e r f i n e s t a t e s o f t h e c o l l i d i n g a toms by t h e l a b e l s h^  a n d h p . As n o t e d i n R e f . ( 3 1 ) , t h e h y p e r f i n e s t a t e s a r e e i g e n s t a t e s o f t h e z componen t o f t o t a l s p i n , m f = m . + m s ( a l t h o u g h a t n o n - z e r o f i e l d , n o t o f t o t a l s p i n f = i + s ) and c a n be l a b e l l e d a s | f ,m^> so t h a t [ I I I - 4 a ] |1> = | 0 , 0 > [ I I I - 4 b ] |2> = I 1 ,-1> [ I I I - 4 c ] |3> = |1 ,0> [ I I I - 4 d ] |4> = | 1 ,1> T h e f i n a l two a tom wave f u n c t i o n c a n be l a b e l l e d by t h e t o t a l e l e c t r o n i c and n u c l e a r s p i n s , S a n d I, and must be a n t i s y m m e t r i c w i t h r e g a r d t o i n t e r c h a n g e o f b o t h e l e c t r o n s a n d p r o t o n s . T h e | T j j | 2 w i l l d e p e n d on t h e p r o j e c t i o n s o f t h e i n i t i a l 28 states on the f i n a l states. We w i l l write t h i s projection as and note that conservation of the z component of spin requires that these c o e f f i c i e n t s be proportional to 5 (mf +mf^  ), (Mj +MS) Only c o l l i s i o n s in which S=0 (and thus M^=0) can result in recombination so that the relevant c o e f f i c i e n t s are of the form <f, ,mf|, f 2 ,mfJ I ,mf^ +m^,0,0> The relevant c o e f f i c i e n t s can be calculated using Clebsch-Gordon c o e f f i c i e n t s and are given in Table I I . The requirement that the two atom wavefunction be antisymmetric under interchange of the protons leads to the condition that L+I be even where L i s the rota t i o n a l quantum number of the H 2 molecule. The two possible states which result are l a b e l l e d ortho (1=1, L i s odd) and para (1=0, L i s even). The expression for the rate constant given in Ref.(31) can accordingly be separated into two terms giving the contribution to the rate from c o l l i s i o n s r esulting in 1=0 and the contribution from those r e s u l t i n g in 1=1. Within each of these terms, a l l of the spin dependence (and thus f i e l d dependence) TABLE II The Recombination C o e f f i c i e n t s < f , . m f l ( f 2 l m „ l ( M „ 0 , 0 > M i fi m„ f2 r T V j \ OO 1 -1 1 0 1 1 0 0 0 0 cose S I N e 0 0 0 0 0 1-1 0 - S I N e si 0 0 0 0 1 0 -(cosVsiN^e) 2 0 0 0 0 1 1 0 0 0 cos e 1-1 1-1 0 0 0 0 1 - 1 1 0 0 cos e si 0 0 1 - 1 1 1 v2 0 -v2 0 1 0 1 0 -cose S I N e 0 0 0 1 0 1 1 0 0 0 - S I N e /T 1 1 1 1 0 0 0 0 30 can be factored out as [III-5] F, = Z P.P |<f,,Ji, ,f?,m, |l,mf +mf ,0,0>|2 where Ph. i s the fra c t i o n of H atoms in the hyperfine state h; . The rate constant can then be written as [III-6] k = F 0 K p a r a + F^ortho Equation [111 —6] provides a means to compare rate constants obtained under d i f f e r e n t f i e l d conditions. In p a r t i c u l a r , at z e r o - f i e l d , i f one assumes =P2=P2=PZ( =0.25, the rate constant becomes [III-7] k = _L K p o r Q + _3 K o r t h o 16 16 At high f i e l d such that, from equation [II-2], cos0«1 and single, and assuming P^=P2=l/2, the rate constant becomes [III-8] k = 1 e 2 K p a r Q + 1 e 2 K o r t n o An a l t e r n a t i v e way of expressing the rate equation i s to break i t into terms l a b e l l e d by the i n i t i a l states so that the rate equation takes the form 31 [ I 1 1 " 9 1 ^-'^•VAV'tf The re l a t i o n s h i p between the rate constants in the two schemes is [ I I I _ 1 0 3 k h ! h 2 = = ,h 2|0,0,0,0>| 2 K para + \<h} ,h2|l,mfi+mf2,0,0>|2 K o r t h o At low f i e l d , one has, since sine = cos0 = 1//2", [I I I - 1 1 ] k^ = i / 4 K p a r a k 2 2= 0 k 3 3= l / 4 K p Q r a k 1 2 = l / 4 K o r t h o k 2 3= l / 4 K 0 r t h 0 k 3 A= l / 4 K o r t p o kj3 = l / 4 K o r t h o k 2 A= l / 4 K p a r a + l / 4 K o r t h o k u = l / 4 K o r t h o k u = 0 At high f i e l d , assuming c o l l i s i o n s occur which involve only states |1> and |2>, one has [ 111- 1 2 ] = e 2 K p Q r a k12 = e 2 Rortho k 2 2= 0 The t o t a l rate constants are, of course, s t i l l given by [111-7 ] and [ I I I - 8 ] . We now return to discussion of the c a l c u l a t i o n of Greben et 32 al.(31) They perform the three body c a l c u l a t i o n and present the contributions made by various h states to the gas phase rate constant at 1K, 0.1K and 0.01K. Their results are summarized in the forms of K Q r t n o , Kp Q r a , and k in Table I I I . It can be seen that, at 1K, recombination i s dominated by c o l l i s i o n s r e s u l t i n g in the formation of ortho-Hp. This contribution decreases rapidly as the temperature i s lowered and, at the lowest temperature, i s less important than the para-Hp contribution. Because, as we have seen, the rate constants at z e r o - f i e l d and at high f i e l d show a d i f f e r e n t r e l a t ive dependence on KpQj-Q and K Q ^ P Q , one should be aware of the p o s s i b i l i t y of the temperature dependence of the rate constant being affected by the presence of a large f i e l d . TABLE III Calculations of the Recombination Rate by Greben et a l . (31) 1K 0.1K 0.01K (1/16)K o r t h Q 14.84 1 .64 0. 147 ( 3 / l 6 ) K p a r Q 2.95 1.38 1 .34 k 17.79 3.02 1.49 The units are 10~3* cm6/s 34 3.2 Spin Exchange The dominant process for relaxation of the hyperfine levels of hydrogen at z e r o - f i e l d i s believed to be spin-exchange c o l l i s i o n s between H atoms. Aside from the i n t r i n s i c importance of understanding the influence of spin-exchange relaxation on the low temperature properties of atomic hydrogen, knowledge of the relaxation i s important in any magnetic resonance experiment in which an absolute c a l i b r a t i o n of signal strength to density must be made. Measurements made with TT/2 pulses separated by inte r v a l s less than a few times T^  w i l l result in the sample not returning to equilibrium between pulses. . This e f f e c t i s referred to as saturation and while i t must be avoided in measurements such as those done to study recombination, i t also affords us a means of obtaining T^  as a function of H atom density. Berlinsky and Shizgal(32) have used the results of B a l l i n g et al.(33) for the rate of change of the 4x4 single atom density matrix as the basis for t h e i r calculations of the spin exchange relaxation and frequency s h i f t cross-sections. In t h i s section we w i l l review the notation and some of the res u l t s in references (32) and (33). In appendix E we w i l l show how the resu l t s of reference (33) should be modified for c o l l i s i o n s confined to two dimensions. In a n t i c i p a t i o n of t h i s , we w i l l 35 use the subscript 3D in t h i s section to indicate results relevant to gas phase c o l l i s i o n s . The cross-sections thus l a b e l l e d have dimensions of area. We w i l l continue to use n^ as the bulk hydrogen atom density in units of atoms/cm3. We now review some of the r e s u l t s of Ref.(33) as they apply to the H-H scattering problem. The object i s to construct the time derivative of the 4x4 density matrix whose rows and columns are l a b e l l e d by the hyperfine states of the H atom. The i n d i s t i n g u i s h a b i l i t y of the H atoms i s taken into account by including, in the 16x16 density matrix operator for the 2 atom system, an operator which interchanges the H atoms. The density matrix operator i s then [111-13 ] P = l I [|k>|s>+|-k>Q|s>]P(k,s) 2 k,s x[<s|<k|+<s|Q<-k|] where s labels the spin state, k the momentum state, Q_ i s the operator to interchange the atoms, and P(k,s) i s the p r o b a b i l i t y of finding the i n i t i a l state to be |k,s>. The scattering of a plane wave state i s discussed in any quantum mechanics textbook (34). One treats the asymptotic scattered wavefunction as the sum of the incoming plane wave state plus the scattering amplitude f^^) times an outgoing spherical wave. In the H-H system, because of the p o s s i b i l i t y 36 of spin f l i p s during the scattering, B a l l i n g et a l . (33) replace the scattering amplitude by the matrix elements [111-14] M^s o U;k 0) = <s,k|f 3 k(0)P 3 + f 1 k ( e ) P 1 |s 0,k Q> where P 3 and P^  are operators which project out the t r i p l e t and singlet electronic states and the scattering amplitudes f3k(0) and ^ ( 0 ) a r e fc^e conventional forms obtained from p a r t i a l wave analysis of the scattering process with the phase s h i f t s calculated for the t r i p l e t and singlet p o t e n t i a l s . If these phase s h i f t s are written and rj^, t h i s gives CO 2i [111-15] t.u(6) = 1 Z (2l+1)(e A-1 )P, (cose) 1 K 2lk 1 = 0 1 co 2 i J [111-16] f , . ( e ) = J _ Z (2l+1)(e i-1)P,(cos0) J K 2ik 1=0 1 The projection operators are written in terms of the spin matrices for the two H atoms, which are distinguishable as H and H at t h i s point, as 37 1 (3 + o(H) • o(H)) 4 1 (1 - oTij - o i l ) ) 4 where a(H) i s the dir e c t product of the Pauli matrices operating in the space of the electron spin with the unit matrix operating in the space of the proton spin. The asymptotic form of the density matrix aft e r the c o l l i s i o n i s given by [ 111-1 9 3 p' = SpS1" where the matrix elements of the scattering matrix are given by [111-20] S s' s(k';k) = 6(k'-k)6 s, s+ 27ri (2irh2/uL3 ) 6(E-E' )M^ s(k' ,-k) where E i s the energy and u i s the reduced mass. With the asymptotic density matrix expressed in terms of the i n i t i a l matrix and the scattering amplitudes, the authors of Ref.(33) go on to derive the result for the time derivative of the single atom 4x4 density matrix in terms of cross-sections depending on k and on the difference between the singlet and t r i p l e t phase s h i f t s for each value 1 for the p a r t i a l wave angular momentum. The r e s u l t i n g expressions for the matrix elements of p are [III-17] P 3 = [ 111 -1 8 ] P, = 38 somewhat complicated. B e r l i n s k y and Shizgal(32) have l i n e a r i z e d the sub-matrix of p i n the space of hyperfine s t a t e s |1> and |3> (which they l a b e l as |4> and |2>). They assume the e q u i l i b r i u m f r a c t i o n a l population of a l l of the hyperfine s t a t e s to be 1/4 and d e f i n e [111-21] 5 h = p h h - 1/4 By l i n e a r i z i n g p with respect to 6^  and p^, they obtain [ I I I - 2 2 a ] 83 = - ( ^ D + ^ ^ D n H ( 6 3 - 6 1 ) / 4 [ I l l - 2 2 b ] 6j = -83 [ I I I - 2 2 C ] p 3 l = - [ ( o J D + ^ D ) + VV ]^DV31 / 4 where Vj^ i s the average r e l a t i v e speed of two H atoms moving i n 3 dimensions and the c r o s s - s e c t i o n s 0" and X are thermal averages of 1+1/2•1/2 [ 1 1 1-23] a ; / - ( E ) = j r _ Z ( 2 l + 1 ) s i n 2 ( 7 ? 1 S 0 ~ r ? ^ n ) [ 1 - ( - 1 ) " ] 3D k 2 1 1>3D 1/3D 1 + 1/2 * 1/2 [ 1 1 1-24] X*/_(E) = jt Z ( 2 1 + l ) s i n [ 2 ( 7 ? 1 S , N - 7 ? 1 T ) ] [ ! - ( - ! ) ' ] JL) k 2 1 1/3D 1,3D By i d e n t i f y i n g 1 as the rate at which 6^63 r e l a x e s to zero, and T"1 as the rate at which (which gives r i s e to the o s c i l l a t i n g magnetization) r e l a x e s , Ref.(32) a r r i v e s at 3 9 [111-25] T f i - v ^ n H ( a ^ + 0y/2 [111-26] T 2-i = v ^ n H ( a y ^ ^ ) / 4 [111-27] 2KLV = ^D^"^^ " V / 4 They obtained numerical results for the spin exchange cross-sections by determining the p a r t i a l wave phase s h i f t s from numerical integration of Schroedinger's equation incorporating either the singlet or t r i p l e t potentials of Kolos and Wolniewicz{35a). The most interesting result of that c a l c u l a t i o n was that o* went to a low temperature l i m i t of about 1 A 2 giving a value for ^ 7 = ( o 7 n + o~)/2 at 1K of 0.55 A 2 rather than the much lower estimate which might be obtained by extrapolating, to 1K, Allison's(36) c a l c u l a t i o n for spin exchange above 10K. It was noted, however, by the authors that an improved t r i p l e t potential had been published by Kolos and Wolniewicz(35b). When thi s potential i s used to repeat the calc u l a t i o n s of Ref.(32), o7~ i s found to be about 0.31A2. 40 3.3 Properties Dependent on the H-He Potential In t h i s section, we w i l l review, with very l i t t l e discussion, some the o r e t i c a l results l i n k i n g the d e t a i l s of the H-He potential to properties observable by magnetic resonance on H atoms in the presence of He gas. We have in mind measurements of the binary d i f f u s i o n c o e f f i c i e n t for H through He and the hyperfine frequency pressure s h i f t due to H c o l l i d i n g with He in the gas. The d i f f u s i o n constant i s one of the transport c o e f f i c i e n t s which can be formulated within the framework of the Chapman-Enskog theory for d i l u t e monatomic gases. This theory i s presented in considerable d e t a i l by Hirschfelder, Curtiss, and B i r d ( 3 7 ) . It i s based on using a perturbation theory approach to treat the c o l l i s i o n term in the Boltzmann equation and thus obtain the d i s t r i b u t i o n function for the gas in momentum and coordinate space. The re s u l t i n g lowest order approximation for the non-equilibrium d i s t r i b u t i o n function i s used to obtain expressions for the transport of mass, momentum, and energy in the gas. The proportionality constant connecting the mass flux vector and the concentration gradient of a species i s the the d i f f u s i o n constant. One result of thi s theory i s that the transport c o e f f i c i e n t s can be written in terms of integrals flfn'^ which depend on the interaction potential through the dynamics of the 41 c o l l i s i o n s . In p a r t i c u l a r , the binary d i f f u s i o n c o e f f i c i e n t for species i through species j i s [111-28] Dj- = 3TT v i i IJ 32 n J /W OTT wkT where nj i s the concentration of species j , u Is the reduced mass, and v~~ i s the r e l a t i v e thermal v e l o c i t y so that v=/8kT/7r/i. The integral $^n'^ can be thought of as a thermal average over a c o l l i s i o n cross-section c/n^  and i s given by r- (nM T "*2 2t + 3 {n) [111-29] / 2 f f M / k T 0 i n , I J = J e 7 Q d 7 where 7 2 = ft2/c2/2/ukT and K i s the r e l a t i v e wave vector. The r e l a t i v e k i n e t i c energy i s fi 2K 2/2/u. For d i f f u s i o n , the appropriate integral i s J^1'1^ and c/1^  i s given by (1) [111-30] Q = 2 2TT Z (1+1 ) s i n 2 ( n -« ) K 2 1 = 0,1,2 1*1 1 where TJ^ i s the phase s h i f t in the r a d i a l wave function obtained from p a r t i a l wave analysis of the scattering process for the p a r t i a l wave with angular momentum quantum number 1. Given the H-He potential(38), can be evaluated, for example, by using 42 a modified version of the program used by Berlinsky and Shizgal(32) for the evaluaton of the H-H spin exchange cross-section. can, of course, also be evaluated c l a s s i c a l l y and when th i s i s done for a gas of r i g i d spheres of diameter d, the result is(37) / 2 W k T ' fl^gid sphere = * d * We are thus led to define an e f f e c t i v e d i f f u s i o n cross-section Qp such that the d i f f u s i o n constant for H in He i s given by [ 111-31 ] D = 37rv/32 n H e Q D where Q^ = 7 r d 2 for r i g i d spheres. The d i f f u s i o n data obtained i s analysed in terms of t h i s expression. The hyperfine frequency s h i f t for H interacting with He has been discussed by Das and Ray(40), Davison and Liew(41), and more recently by Jochemsen and Berlinsky(39). The hyperfine s p l i t t i n g , a (see figure 1) i s proportional to the density of the electron at the nucleus. This density i s perturbed in c o l l i s i o n s with the result that the hyperfine frequency i s s h i f t e d . This s h i f t i s normally calculated as a function of the separation, R, of the H and He nuclei using f i x e d nuclei and carrying out a v a r i a t i o n a l c a l c u l a t i o n to obtain the diatomic 43 wavefunction as a lin e a r combination of products of H and He atomic o r b i t a l s . The result i s usually expressed as a fr a c t i o n a l s h i f t , Aa(R)/a(co). in the c l a s s i c a l c a l c u l a t i o n s of references (40) and (41), Aa(R)/a(<») i s then weighted by exp(-V(R)/kT), where V(R) i s the H-He interaction p o t e n t i a l , and the integral i s performed over R to give the f r a c t i o n a l s h i f t per unit density of the buffer gas at temperature T. In Ref.(39), p a r t i a l wave analysis of the H-He scattering problem i s used to construct a two atom wave function which i s then used to perform the averaging of the hyperfine s h i f t . As i s pointed out in Ref.(39), the q u a l i t a t i v e behaviour of Aa(R)/a(co) and of V(R) are similar in that both are large and positi v e for very small R and decrease sharply for increasing R. Both go through zero and exhibit a minimum. For large R, both are negative and f a l l off as R"6. At high temperatures, the c o l l i s i o n s can sample the short range repulsive part of the potential and, accordingly, the large p o s i t i v e portion of Aa(R)/a(co). The hyperfine s h i f t i s predicted to be po s i t i v e and at 323K i s found, by Pipkin and Lambert(42) to be 228x10" 1 8 Hz cm3 when expressed as the observed s h i f t divided by n ^ e • A t l ° w temperatures, the c o l l i s i o n s tend to sample the a t t r a c t i v e long range part of the pot e n t i a l , and thus the negative region of Aa(R)/a(<»). 44 3.4 Interaction with the Surface Prior to the experiments reported here, there were a number of estimates of the binding energy for H on the l i q u i d "He f i l m . These ranged from 0.1K to greater than 0.6K. The occupation of the surface state i s determined by the equilibrium between adsorbed H and gaseous H. The adsorption isotherms for H on "He have been considered by Edwards and Mantzd9) and Goldman and Silvera(43)(see also S i l v e r a and Goldman(44)). They proceed by expressing the chemical potentials for the gas and surface state in terms of the bulk and surface density. The condition for equilibrium, that the chemical potentials be equal, imposes the required r e l a t i o n s h i p between bulk and surface d e n s i t i e s . From Ref.(43), the chemical pote n t i a l for a single component sample i s related to the bulk and surface densities, n and a respectively, by [111-32] nA 3 = g (exp[(u-2nV(0))/kT]) + n QA 3 '2 and s [111-33] oA2 = -ln(1-exp[(u - 2aV (0) + Eg)/kT]) where A= [27rh2/mkT] ^ 2 i s the thermal de Broglie wavelength, Eg i s the binding energy, n Q i s the density of the Bose-Condensed f r a c t i o n , and V(q) and V s(q) are Fourier components of the 45 interatomic interactions respectively of bulk atoms and of atoms confined to the surface. The e f f e c t s of exchange appear as a contribution to the single p a r t i c l e energy of the form ZV(k-q)nq where nq i s the density of the f r a c t i o n of atoms with momentum ftq. Goldman and Silvera(43) argue that at low temperatures such that the range of the interatomic potential becomes small compared to the thermal wavelength, V(k-q) can be replaced by V(0) with the result that the contribution to the single p a r t i c l e energies by the interaction i s 2nV(0) for the bulk and 2aV s(0) for the atoms confined to the surface. V s(0) has been taken to be 5 X 1 0 - 1 2 mK cm 2(43). For the experiments described here, the terms in V(0) and V s(0) were always n e g l i g i b l e compared to v. Also, having been ca r r i e d out in z e r o - f i e l d , they were limited to low densities by recombination. Consequently, the f r a c t i o n of atoms on the surface was always low (less than 0.1 per cent). Under such conditions, the result of eliminating the chemical potential from equations [111-32] and [111-33] i s E R/kT [ 111-34 ] a = r^A e D By measuring the temperature dependence of properties related to the surface density, i t i s thus possible, via equation [111-34], to infer a binding energy. Part of the 46 relevance of binding energy measurements to high density atomic hydrogen experiments directed at Bose-condensation l i e s in the fact that as the bulk density i s raised, the form of equation [111-33] causes the surface density to saturate and allows the chemical potential to approach zero. The saturated surface density, s [II1-35] a S Q t = Eg/2V (0) w i l l appear in any ca l c u l a t i o n for the decay of a high density sample due to recombination or relaxation on the surface. In the experiments comprising t h i s work, two types of measurement were made to probe the surface density. If the t o t a l number of atoms on the surface i s only a small f r a c t i o n of the sample, we can write the f r a c t i o n of the sample on the surface, or equivalently the f r a c t i o n of the time an atom spends on the surface as [111-36] x = < r s > <rs > + <r B> = AA e V Eg/kT ! I§2 <Tg> where <rs> and <rg> are the average duration of interv a l s spent on the surface and in the bulk respectively. The hyperfine 47 s p l i t t i n g for an atom adsorbed on the surface i s s h i f t e d from the free atom value, a/h, by an amount a>s = 27rAs analogous to the bulk s h i f t described above. We think, here, of the atoms as t r a v e l l i n g across the c e l l and c o l l i d i n g with the walls where they can stick with a pr o b a b i l i t y which we label s. The average duration of a st i c k i n g event i s such that the f r a c t i o n of time spent on the wall i s given by equation [111-36]. While on the wall the hyperfine frequency i s shifted by A§. The general problem of magnetic resonance on a system of atoms jumping between environments with d i f f e r i n g hyperfine frequencies i s described in Chapter V. If the f r a c t i o n of time spent on the surface i s small, however, the measured hyperfine frequency i s simply sh i f t e d from the free atom value by Eg/kT [ 111 -37 ] AOJ = <T s > cos = AAo)se <Tg> V I f , as we assume for the purposes of t h i s work, us i s temperature independent over the range of temperatures used, measurement of the hyperfine s h i f t as a function of temperature provides one means of obtaining the binding energy. A second measure of the surface density i s obtained by considering recombination on the surface. Recalling equation [III-2] for the rate of change of the surface density due to recombination on the surface, we had 48 [II1-38] da = - k 5 a 2 dt Because the bulk and surface are in equilibrium, (A/V)da/dt can be thought of as an e f f e c t i v e rate of change of the bulk density. If the density of atoms in the bulk i s low, such that the rate for three body c o l l i s i o n s i s low, t h i s w i l l , in fact, be the dominant recombination mechanism. Rewriting k s as Xv s, where X i s a two-dimensional analogue of a recombination cross-section and v g ==,/«• kT/m i s the average r e l a t i v e speed on the surface, we can eliminate a with equation[III-34] to get 2E B/kT [111-39] dn H = v s X A A 2 e n 2 dt V H Measurement of the recombination rate as a function of temperature at low temperatures thus also provides a means of obtaining Eg. Another property of the interaction of the atoms with the surface which i s accessible to these experiments i s the p r o b a b i l i t y with which an H atom c o l l i d i n g with the surface w i l l be captured into the bound state. This s t i c k i n g p r o b a b i l i t y can be measured by analysis of the magnetic resonance lineshape, at very low temperatures, using the general theory to be discussed 49 in Chapter V. The r e s u l t s of such measurements, for both 3He and *He f i l m surfaces w i l l be discussed there. 50 CHAPTER rv Measurements Between 1K and 1 . 3K In t h i s chapter, we w i l l discuss magnetic resonance experiments carried out on a gas of atomic hydrogen confined by superfluid "He walls at temperatures accessible by pumping on l i q u i d helium. At these temperatures, the e f f e c t s of binding to the helium f i l m are not important and i t is the consequences of interactions in the gas which are observed. We w i l l begin by describing the 1K cryostat and the 1420 MHz spectrometer used for t h i s set of experiments. We then go on to discuss the measurements of a number of properties of the system including the recombination rate, the cross-section for d i f f u s i o n of the H atoms through the helium gas, the spin-exchange cross-section, and the hyperfine frequency s h i f t due to H atom c o l l i s i o n s with He atoms. 51 4.1 Experimental Apparatus 4.1a The Cryostat The cryostat used for the experiments discussed in t h i s chapter i s depicted schematically in figure 3. It i s suspended in a sealed dewar and the temperature of the whole "He bath i s lowered to between 1.3K and IK by pumping. The usable temperature range i s r e s t r i c t e d on the high side by the vapour density of "He in the c e l l which causes rapid recombination and, on the low side, by the capacity of the cryostat pumping system. The temperature of the bath i s measured with a 10 ohm Ohmite carbon r e s i s t o r c a l i b r a t e d in s i t u against the *He vapour pressure. The temperature inside the sample c e l l i s assumed to be equal to that of the bath. A second r e s i s t o r i s used as a heater for temperature regulation. The sample c e l l i s a sealed pyrex bulb 6.5 cm in length and 1.6 cm in inner diameter over about half that length with a narrower t a i l as shown in the figure. Some early measurements were done in a c e l l of the same length and uniform diameter of 1.6 cm. The c e l l i s f i l l e d at room temperature, through a c o n s t r i c t i o n in a 1 meter long pyrex tube, to approximately half an atmosphere each of H 2 and "He. The c o n s t r i c t i o n i s then sealed, providing the closed c e l l geometry, and a window glued to the room temperature end of the pyrex wand. This design TO SPECTROMETER F i g u r e 3 T h e c r y o s t a t u s e d f o r z e r o m e a s u r e m e n t s a t 1 K . 53 permits v i s u a l observation of the low temperature discharge. The smaller diameter t a i l extends into the region of the 50 MHz pulsed discharge c o i l . This shape does not seem to hinder atom production and i s useful in that i t reduces the f r a c t i o n of atoms outside of the microwave resonator. H atoms were produced by a pulsed r . f . discharge l a s t i n g 1 msec, or l e s s . The r . f . pulses were produced with an Arenberg Model PG-650C pulsed o s c i l l a t o r modified as shown in figure 4a to give f u l l power (about 400 watts) for the few tens of microseconds necessary to i n i t i a t e the discharge in the *He gas, and subsequently a much lower power (about 4 watts maximum) to sustain i t . It i s estimated that the t o t a l energy in the pulse i s of order 1 to 10 mJ. At low temperatures, the H 2 forms a s o l i d f i l m on the walls and i s i t s e l f covered with a Van der Waals f i l m of *He to a thickness of order 200 angstroms or more which, at the measurement temperatures, i s superfluid. Estimates show that 10 mJ i s just s u f f i c i e n t to evaporate most of the "He fi l m (this occurs at about 1.4K). At t h i s point, a small amount of H 2 i s evaporated from the walls and dissociated in the discharge to produce H. After the discharge, the undissociated H 2 refreezes on the wall and i s covered by the recondensed superfluid "He f i l m . Approximately 2 x l 0 1 2 to 5X 1 0 1 2 atoms/cm3 are l e f t trapped above the f i l m . The main body of the sample s i t s inside of a 1420 MHz s p l i t ring resonator (as described by Hardy and Whitehead(45)) 54 A R E N B E R G P G 6 5 0 C modi fed : Dpulse f o r m i n g c i r c u i t - t o a l l o w 1 m s e c pulse C-75: 0 . 0 0 2 2 p F t o O.IJJF 2) o s c i l l a t o r H.V. s u p p l y - t o a l l o w supp l y t o d r o o p t o c o n s t a n t l o w v o l t a g e a f t e r f e w j j s e c of pu lse 2 k V 2kV C-115 o s c i l l a t o r O.IJJF o s c i l l a t o r L A M B D A 2 8 2 0 0 - 3 2 5 V supply R . F O U T D I S C H A R G E C O I L 0 .5 JJH -0.01 JJH 1 4 2 0 M H z A/4 T R A P ( 3 c m of UT-081 a C O A X ) 2 0 pF Figure 4a The discharge c o i l Ahrenberg Pulsed source and modified 55 suspended from the top of the resonator housing on two teflon supports. Tuning i s accomplished by moving an aluminum plate v e r t i c a l l y along a t e f l o n spacer in front of the s l i t . (see figure 4b) The mechanical connection to room temperature i s a thin-walled stai n l e s s steel rod sealed at the top of the cryostat through a Wilson O-ring seal. Coupling to the resonator i s via a single turn loop in series with an adjustable capacitor. One plate of the capacitor i s attached to the centre conductor of a 0.141 inch s t a i n l e s s steel coaxial transmission l i n e which i s sealed to the top of the cryostat through a small brass bellows. This arrangement allows the coupling to be varied by adjustment of the capacitor gap from room temperature. The tuning and coupling are set by observing the r e f l e c t i o n of 1420 MHz pulses with the aid of an HP776D dual d i r e c t i o n a l coupler and an HP 423A c r y s t a l detector. C r i t i c a l coupling was used for e s s e n t i a l l y a l l measurements. The resonator housing i s a copper cylinder of 3 inches diameter on which i s wound an end corrected solenoid for application of uniform bias f i e l d s . The solenoid has 151.5 turns of copper wire covering a length of 4.22 inches with 2 twelve turn layers added for correction at each end. Two addi t i o n a l c o i l s provide a linear gradient across the sample for use in d i f f u s i o n and relaxation measurements to be described below. The applied bias f i e l d s and gradients were c a l i b r a t e d with a B e l l 640 Gaussmeter. Copper tuning plote Teflon spacer F i g u r e 4 b A s i m p l i f i e d v e r s i o n o f t h e r e s o n a t o r t u n i n g a r r a n g e m e n t . 57 P a r t i a l magnetic shielding was provided by a layer of 0.005 inch thick Co-Netic f o i l wrapped on the outside of the dewar. With t h i s s h i e l d in place, the ambient f i e l d was of the order of 11 mGauss in the transverse d i r e c t i o n and 30 to 40 mGauss in the longitudinal d i r e c t i o n . The ambient longitudinal f i e l d was measured by observing the applied bias f i e l d necessary to n u l l the longitudinal f i e l d and so reduce the magnetic resonance signal amplitude to zero. The bias f i e l d was adjusted so as to bring the net longitudinal f i e l d to a l e v e l appropriate to the measurement being made. 58 4.1b The Spectrometer A schematic of the 1420 MHz pulsed spectrometer designed by W. Hardy i s shown in figure 5. For t h i s series of experiments, the reference frequency was the 10 MHz timebase output from an HP5342A microwave frequency counter with high s t a b i l i t y option. This timebase was c a l i b r a t e d by comparison to the west coast ABC t e l e v i s i o n colour sub-carrier frequency which i s controlled by a Cesium atomic clock and monitored by N.B.S. in Boulder, Colorado.(46) Comparison was made by using our 10 MHz signal as the external timebase for an HP5327A timer/counter which was then used to measure the period of the colour sub-carrier frequency. In t h i s way, we were able to c a l i b r a t e the spectrometer to about 1 Hz out of 1420 MHz. For subsequent experiments at lower temperatures, to be described in the following chapter, the 10 MHz timebase was derived from a rubidium atomic clock and the c a l i b r a t i o n procedure was done by a phase comparison technique. It was then possible to c a l i b r a t e the spectrometer to a few hundredths of a Hz out of 1420 MHz. Returning to the spectrometer schematic, figure 5, the 10 MHz reference i s f i r s t m u ltiplied, amplified, and f i l t e r e d to provide a signal at 1420 MHz. A power s p l i t t e r provides part of t h i s signal for locking of a 1420.405 MHz re-entrant varactor tuned cavity o s c i l l a t o r . Part of the signal from the o s c i l l a t o r 59 LOW FREQ. SYNTHESIZER SIGNAL AV ERAGER 4 0 5 kHz VARIABLE PHASE SHIFTER MIXER AMPLIFIER x2Hx71 i MIXER & INTEGRATOR T MIXER MIXER AMPLIFIER VOLTAGE CONTROLLED OSCILLATOR G A T E 1 1420.405 10 MHz R E F E R E N C E PULSE PROGRAMMER f s i g _ f o f o =1420405MHz fsig RESONATOR Figure 5 Block Diagram of the 1420 MHz spectrometer. 6 0 i s mixed with the 1420 MHz reference in a M i n i - C i r c u i t s ZLW-2 double balance mixer. The 405 KHz intermediate frequency of this mixer i s compared with a 405 KHz signal derived from an HP 3330A automatic synthesizer using a M i n i - C i r c u i t s SRA-3 double balance mixer. The resul t i n g error signal i s integrated and fed back to the varactor diode to control the o s c i l l a t o r frequency. The other part of the o s c i l l a t o r output i s gated through a pair of microwave switches (HP33134A and HP33132A) which are controlled by a pulse programmer consisting of a Tektronix Type 162 Waveform generator and a pair of Type 163 pulse generators feeding a pulse diplexer. This signal i s then fed to the coupling transmission l i n e through d i r e c t i o n a l couplers as shown. The output power i s approximately 53 microwatts. The signal from the cavity i s returned along the same transmission l i n e and amplified in a Watkins-Johnson 7373035 1.0 to 2.0 GHz amplifier and then mixed with the 1420 MHz reference in a second ZLW-2 double balance mixer to produce a 405 KHz I.F.. This i s further amplified and mixed with the reference 405 KHz signal from the synthesizer to produce the observed free induction decays. The frequency of the observed signal i s thus the difference between the transmitted frequency, set by the HP automatic synthesizer, and the frequency radiated by the atoms. The observed beat frequency could be set appropriately for the various types of measurements c a r r i e d out. For T^ studies and spin-echo d i f f u s i o n measurements, the transmitted frequency was 61 set to produce a "zero-beat" output and a variable phase s h i f t e r , as indicated in the figure, was used to ensure that the amplitude of the signal was meaningful. The signal was averaged using a Nicolet 1170 signal averager and the averaged signal plotted on an X-Y recorder for l a t e r analysis. The absolute s e n s i t i v i t y of the spectrometer was ca l i b r a t e d using a 1420Mhz signal derived from the output of a Rhode and Schwarz Decade Signal Generator SMDW and a times four frequency m u l t i p l i e r chain. The power at the output of the chain i s set to about 50 microwatts and measured with an HP 435A Power Meter. This signal was subsequently attenuated by 70 dB using Midwest Microwave Model 263 30dB and 40 dB attenuators and then fed into the spectrometer. Attenuation in the coaxial transmission l i n e s connecting the resonator and spectrometer was measured using the same frequency source along with the previously mentioned dual d i r e c t i o n a l coupler and c r y s t a l detector. 62 4.2 Recombination Measurements For the experiments above IK, the t h i r d body p a r t i c i p a t i n g in the recombination reaction i s a He atom. Defining n^ & as the helium atom density, the relevant recombination process H + H + He -> H 2 + He proceeds with a rate [IV-1] gH- - k n f l n H e By using the c a l i b r a t i o n of the signal amplitude as discussed i n chapter I I , i t was possible to measure n|_| as a function of time as long as care was taken to avoid systematic errors in interpretation of signal amplitude due to saturation e f f e c t s . P l o t t i n g n^" 1 vs time then y i e l d s a slope of ^nne« Eight such measurements were made during two separate runs and three such plots are shown in figure 6. Using the density of helium gas derived from the 1958 "He temperature scale(47) and d i r e c t measurement of the He pressure in the pumped cryostat, a plot of kri|_je vs n^ e, figure 7, was used to obtain the rate constant k=(0.20 +/- 0 . 0 3 )X 1 0 - 3 2 cm 6sec" 1. In reference (14), the f i l l i n g factor defined by equation [11-23] was calculated 1.5 X X : 1.0 c 0.5h x X X £ x - n H e = 5x l0 , 8 c r r f 3 A - n H e =27xl0 , 8 cm" 3 © - n H e =l . lx l0 , 8 cm" 3 X x A "^1 eo o o I t ( I 0 3 S ) Figure 6 Inverse hydrogen atom density n^ 1 vs time for three d i f f e r e n t "He gas dens i t i e s n^ e. For the process H+H+He* H2+He, the slopes should be proportional to n^ e and the rate constant k. 64 . Figure 7 dn~H/dt vs. n^ e for two separate cool-downs. The slope i s k. 65 using instead of Q Q, the correct quantity, with the result that the rate constant quoted was too large by a factor of /2. We note that, in figure 6, some of the data, such as the x's, show deviation from a l i n e a r dependence of n^ 1 on t. This may be the result of a weak source of atoms as might occur i f atoms were d i f f u s i n g out of a s o l i d H 2 snow af t e r being trapped following the discharge. For the lower temperature measurements described below, for which the discharge power was considerably reduced, there was no evidence of a source of H atoms afte r the discharge. The measured rate constant i s in good agreement with the value calculated by Greben et al.(31), k=0.178x10"32 cm 6sec" 1, at 1K. 66 4.3 D i f f u s i o n Measurements Diffusi o n measurements were ca r r i e d out, as described in chapter I I , using a -n/2 pulse followed by a ir pulse, to produce spin echoes for various pulse separations, T, and helium densities, "He* Defining A(0) as the amplitude of the free induction decay following the i n i t i a l 7r/2 pulse and A(2r) as the echo amplitude, the slope of ln(A(2r)/A(0))vs r 3 was obtained by least squares f i t s to the data. The ambient longitudinal f i e l d was measured by observing the applied bias f i e l d necessary to n u l l i t and drive the signal amplitude to zero. The additional desired bias f i e l d and gradient were then applied according to the known c o i l c a l i b r a t i o n s discussed previously. It was possible to check the value of the gradient by observation of the free induction decay envelope. Carr and Purcell(48) give the shape of t h i s envelope for a f l a t f i e l d d i s t r i b u t i o n between ( B Q - b ) and (B D+b) as (sin(7bt)/7bt) for the usual case where CJ=7B. With the atomic hydrogen longitudinal resonance, for which co=a/fi + 2iraB 2, one has 67 B o Z ; b [IV-2] |M* j = Mo f cos(27ra(B| - B ^ t ) dB z 2b J In terms of the Fresnel integrals (49), C o (x) = 1 ( cos t dt = J. (x) + Jr (x) + J Q (x) Sy (x) = 1 ? sin t dt « (x) + Jj (x) + J J 1 (x) +... /2~¥ J /t l-i /2 /2 o where Jy'(x) i s a Bessel function, equation [IV-2] becomes [IV-3] |M* | = Mja j~cos(2irttE^t)y^iT | c 2(27r[B o z+b] 2at)-C 2(2ir[B o z-b] 2at ) J + sin(27raBJ.t)J~T~ | S 2 (27r[Boz+b] 2 a t ) - S 2 (2JT[B^-b] 2 a t ) | J 2at Taking b = (l/2)dB z/dz, where 1 i s the length of the sample contributing to the F.I.D., the expected shape of the decay envelope was computed. Comparison of calculated and experimental zero crossings confirmed the gradient c a l i b r a t i o n . Figure 8a shows a t y p i c a l . spin echo measurement. The spectrometer frequency has been adjusted to be equal to that of the hyperfine t r a n s i t i o n in the average f i e l d present (zero beat m I 2 \ nH, = 1/l7x10 ,8cm-3 \ 1 10msec \ n. r. 25 8 msec \ n \ ^ — ^ ^ v " ^ r • 21 msec \ n 7«19.1 msec \ Jl. r «159 msec F i g u r e 8a t y p i c a l s p i n e c h o f r e e i n d u c t i o n d e c a y . OO'6sec3) F i g u r e 8b A t y p i c a l p l o t o f l o g ( A ( 2 r ) / A ( 0 ) ) vs T 69 condition) and the shape of the free induction decay after the 7r/2 pulse i s due only to the ef f e c t of the large gradient. Figure 8b i s a t y p i c a l plot of ln(A(-2r)/A(0)) vs T 3 for a given value of n ^ . The slope of th i s type of plot can be used to calculate D, for t h i s value of n ^ , via [IV-4] d ln(A(2r)/A(0) ) = -(f^tdBz)2 2 D ^T3 V e T T 3 z ; 3 We t y p i c a l l y used B Q 2 = 0.35 gauss and 3Bz/az = 0.03 gauss/cm. Defining an e f f e c t i v e d i f f u s i o n cross-section, Q ^ , v i a [IV-5] D = 3TTV D He and assuming that Q Q i s approximately temperature independent near 1K, one can obtain Q Q = 20+/—1 A 2 from the slope of a plot of 3 7 T V/32D vs n^ e as shown in figure 9. This value of has been combined with the scattering data of Toennies et al.(38) to obtain an improved estimate of the H-He p o t e n t i a l . The new p o t e n t i a l , with a well depth of 0.6 meV i s approximately 30 per cent deeper than that given in reference (38) and i s reported by Jochemsen and Berlinsky(39) and Jochemsen and Hardy(50). 7 0 F i g u r e 9 H y d r o g e n d i f f u s i o n d a t a p l o t t e d v s h e l i u m gas d e n s i t y n H e . The v e r t i c l e a x i s , 3 i r v/32D, i s e q u a l t o Qn^He ' w h e r e °-D i s t h e t h e r m a l l y a v e r a g e d d i f f u s i o n c r o s s -s e c t i o n . 71 4.4 Spin Exchange Measurements In t h i s work, T^'s were measured using a simple n/2 - 7r/2 pulse sequence. After a ir/2 pulse at time t=0, the sample magnetization i s expected to recover according to [IV-6] M z(t) = M 0( 1 - e 1 ) (Here we are using the analogy to a spin 1/2 system; for the two leve l s under consideration there i s no magnetization.) If a second -n/2 pulse i s applied at time t, i t w i l l sample the longitudinal magnetization and the amplitude of the r e s u l t i n g free induction decay r e f l e c t s the extent to which the magnetization has recovered. Competition from radiation damping i s minimized by using a gradient f i e l d to dephase the transverse magnetization between K/2 pulses. In p r i n c i p l e , then, one should be able to measure, as a function of the delay i n t e r v a l between the two K/2 pulses, the signal amplitudes after the saturating and the probe 7r/2 pulses, A(oo) and A(t) respectively, and obtain -Tj" 1 as the slope of a plo t of ln( 1-A(t)/A(co)) vs t . In p r actice, measurements on atomic hydrogen in zero f i e l d are complicated by the fact that i s a function of n^ and n^ i s decaying due to recombination. While there i s l i t t l e recombination during the short i n t e r v a l between the saturation 72 and probe pulses, one must wait several times before applying the next saturation/probe pulse p a i r . Equation [lV-6] -can be rewritten in terms of the signal amplitude after each of the two pulses to give [IV-7] In / 1 - A(t)\ = - t ( n n f f l v + k) where k represents the rate of any other possible processes without an n^ dependence and a e x i s the t o t a l spin exchange cross-section. If n|_j i s large (and thus decaying qui c k l y ) , one can neglect k and avoid the problems associated with recombination by applying the set of pulse pairs over as short a time as p r a c t i c a l , and thus as small a range of n^ as possible, and p l o t t i n g the result as ln(1 - A(t)/A(oo)) vs tA(to). The average of A(oo) over the set i s then used to calculate the e f f e c t i v e density, n^, and the corresponding value of T^~ 1 i s taken to be A(oo) times the slope. If n^ i s small, and thus slowly changing, i t i s necessary to plot l n d - A(t)/A(<»)) vs t to obtain T j 1 d i r e c t l y from the slope. The spin exchange cross-section i s then v" 1 times the slope of a plot of T j 1 vs n^ as shown in figure 10. The success of t h i s method of compensating for the e f f e c t s of recombination r e l i e d on the a b i l i t y to reset and measure the i n t e r v a l between the probe and saturation pulse very quickly. nH (IOl2cm-3) Figure 10 1/T1 vs H atom density n H. The slope i s . equal to vo^j , where o£x i s the cross section for spin exchange and v i s the average r e l a t i v e v e l o c i t y of two H atoms. 74 To accomplish t h i s , the Tektronix pulse programmer mentioned in section 4.1b was set up so that the required pulse intervals could be preset and premeasured (with the in t e r v a l timing function of the HP 5327A counter/timer) and then switched into ef f e c t in sequence as required. It was also convenient to perform these measurements in a s u f f i c i e n t l y large gradient to dephase the o s c i l l a t i n g moments before radiation damping could s i g n i f i c a n t l y a f f e c t the longitudinal magnetization. From the T y vs n plot of figure 10, we obtain o^"= 0.43+/-0.03 A 2 at 1.1K. The o r i g i n a l c alculations of o Berlinsky and Shizgal(32) gave = 0.55 A 2 at thi s temperature. However, using an improved t r i p l e t p o t e n t i a l (Kolos and Wolniewicz(35b)) a recalculation of gave about 0.31 A 2 at th i s temperature. Thus i t i s clear that the low temperature value of a e x depends s e n s i t i v e l y on the d e t a i l s of the po t e n t i a l . The remaining discrepancy has not been resolved. Figure 11 shows a plot of T"j1 vs temperature calculated using the spin exchange cross-sections based on the improved potential(51). 0 1 I 2 3 T ( K ) 4 Figure 11 Temperature dependence of T j ' due to three dimensional spin exchange scattering in the gas calculated using the improved t r i p l e t p otential of Ref.35b. The c i r c l e obtained using the experimentally measured cross-section from the present work. 76 4.5 Pressure S h i f t Measurements During c o l l i s i o n s between H and He atoms the electron density at the proton i s perturbed, with a corresponding perturbation of the z e r o - f i e l d hyperfine frequency as described in section 3.3. This results in an average s h i f t in the z e r o - f i e l d t r a n s i t i o n frequency proportional to the buffer gas density, n^ e. In figure 12, we have plotted the t r a n s i t i o n frequency versus n ^ for measurements made in a constant longitudinal f i e l d . The slope i s -11.8 x 10" 1 8 Hz cm3. The zero density l i m i t , 1,420,405,760.1 Hz, i s about 0.8 Hz above the free atom frequency calculated for the 53 mGauss f i e l d present during these measurements. This discrepancy i s most l i k e l y due to an error in the time base c a l i b r a t i o n which was r e l a t i v e l y crude at th i s stage of the experiment. The observed s h i f t i s about 16 times smaller than and of opposite sign to that for the same density of He gas at room temperature(42). The pressure s h i f t c a l c u l a t i o n s of Jochemsen and Berlinsky(39) are shown in figures 13 and 14 along with the higher temperature and 1K pressure s h i f t measurements. They have chosen to use their empirical potential in conjunction with the averages of Aa(R)/a(«) from references (40) and (41) to perform the quantum mechanical thermal average for Aa. Also shown are the pressure s h i f t c a l c u l a t i o n s of references (40) and Figure 12 Hyperfine frequency f - f</» 1 ,420,405 kHz) plotted vs. helium gas density n H e . The slope i s -11.83x10" 1 8 Hz cm3 T (K) Figure 13 The thermally averaged hyperfine frequency s h i f t of atomic hydrogen vs. temperature. The dashed and dotted l i n e s show the results of Ref.(39) derived from the c a l c u l a t i o n s of Davison and Liew(41) and of Ray(40) respectively. The s o l i d l i n e represents the re s u l t s of Ref.(39) with an empirical potential based on the d i f f u s i o n result reported here. The crosses show the c l a s s i c a l average res u l t at T=300K and T=100K for the three d i f f e r e n t c a l c u l a t i o n s . The dash-dot l i n e shows the experimental res u l t of Wright et al.(42b) and the tri a n g l e i s a measurement by Pipkin and Lambert(42a) at T=45°C. The square at 1.15K i s the re s u l t reported in t h i s work. This figure i s reproduced from Ref.(39) Figure 14 The thermally averaged hyperfine s h i f t of atomic hydrogen at low temperature. The s o l i d , dashed, and dotted l i n e are explained in the caption of figure 13. The square represents the experimental result at 1.15K of the present work. The c i r c l e indicates an unpublished measurement of the hyperfine s h i f t in 3He by the authours of Ref.(24). The star shows the calculated hyperfine s h i f t at T=0.5K for H in 3He using the empirical p o t e n t i a l of ref.(39) and the average of Aa(R)/a(<») from Refs.(40) and (41). This figure i s taken from Ref.(39) 80 (41). The authors of reference (39) f i n d that with proper quantum mechanical averaging the hyperfine s h i f t i s calculated to change sign at low temperatures r e f l e c t i n g the fact that, in low energy c o l l i s i o n s , the atoms are sampling predominantly the long range a t t r a c t i v e part of the p o t e n t i a l . They calculate c o e f f i c i e n t s of -7 x 10' 1 8 Hz cm3 for H with "He at 1K and -6 x 10" 1 8 Hz cm3 for H with 3He at 0.5K. As can be seen from figure 14, the calculated c o e f f i c i e n t for H with "He displays only weak temperature dependence at these temperatures. 81 CHAPTER V Experiments Below 1 K As discussed in chapter I I I , the existence of a bound state for H atoms on the walls of their container leads to the p o s s i b i l i t y that the dominant loss mechanism for atoms in the bulk, at high densities and low temperatures, might be adsorption and subsequent recombination on the container surface. Two of the important parameters for determining the ef f e c t of th i s loss mechanism, the surface state binding energy, Eg, and the recombination rate constant on the surface, k s, are accessible to a low temperature extension of the z e r o - f i e l d magnetic resonance studies of the type described in the previous chapter. These experiments were done using an apparatus similar to that described in Chapter II but adapted for use with a commercial d i l u t i o n r e f r i g e r a t o r . We w i l l begin t h i s chapter by describing the equipment used for the low temperature experiments. We w i l l also discuss a model for the magnetic resonance lineshape for hydrogen atoms confined by walls on which they can undergo occasional s t i c k i n g events. We w i l l then describe the results obtained f i r s t using wall coatings of 82 l i q u i d "He and, l a t e r , l i q u i d 3He. These results include measurements of the decay rate of the atom density as a function of temperature, which are used to extract the binding energy and recombination constants. Also included are measurements of the frequency s h i f t and free induction decay time as a function of temperature. For the most general s i t u a t i o n , the model describing the magnetic resonance parameters must incorporate the p r o b a b i l i t y for a wall c o l l i s i o n r e s u l t i n g in a s t i c k i n g event, as well as the s t a t i s i c s governing the duration and separation in time of such events. For the purposes of obtaining the binding energy, however, i t i s s u f f i c i e n t to take measurements in the temperature regime for which the observed hyperfine frequency i s the average of the bulk and surface frequencies weighted by the average time spent by an atom in each environment. Accordingly, discussion of the determination of the binding energy using the frequency s h i f t w i l l appear in the same section as the complementary recombination measurements. Discussion of the application of the complete model to the determination of the st i c k i n g p r o b a b i l i t i e s w i l l be postponed to the f i n a l section of th i s chapter. 83 5.1 The Low Temperature Apparatus 5.1a The Refrigerator Cooling for the experiments to be described below was provided by an S.H.E. Model 420 3He/ 9He d i l u t i o n r e f r i g e r a t o r which i s capable of cooling as low as 7 mK but was used only down to 60 mK in these experiments. The p r i n c i p l e of operation of such a r e f r i g e r a t o r i s described by Lounasmaa(52) and w i l l not be repeated in d e t a i l here. In describing the i n s t a l l a t i o n of the 1420 MHz resonator and the sample c e l l in the cryostat, i t w i l l , however, be helpful to have reviewed the nomenclature associated with the d i l u t i o n r e f r i g e r a t o r . Operation of the re f r i g e r a t o r e n t a i l s the c i r c u l a t i o n of about 0.55 moles of 3He using an S.H.E. G.H.S. 2F gas handling system which includes a sealed c i r c u l a t i n g pump and l i q u i d N 2 cooled molecular sieve f i l t e r and o i l trap. Prior to reaching the low temperture section of the r e f r i g e r a t o r , the incoming 3He i s cooled, through the c a p i l l a r y wall, by the 4.2K l i q u i d "He in the dewar. The low temperature stages are housed in a four inch diameter by twenty-five inch long vacuum can immersed in l i q u i d *He at 4.2K. The vacuum can allows for necessary thermal i s o l a t i o n of the fi v e basic stages of the ref r i g e r a t o r which, in operation, are maintained at temperatures between about 1.5K and the temperature of the mixing chamber and provide progressive 84 cooling of the incoming 3He. The f i r s t stage of cooling within the vacuum can i s a pumped "He coldplate. It operates at about 1.5K, and draws i t s "He through a c a p i l l a r y impedance from the main l i q u i d helium reservoir. The lowest stage of the r e f r i g e r a t o r , both phy s i c a l l y and in temperature, i s the mixing chamber in which 3He atoms dissolve across the phase boundary between a 3He r i c h phase and an approximately seven per cent solution of 3He in superfluid "He. The dissolved 3He d i f f u s e s through a stationary column of *He extracting heat from the incoming 3He in a number of l e v e l s of step and continuous heat exchangers. Reaching the s t i l l , at about 0.8K, i t evaporates and i s pumped away by the c i r c u l a t i n g system. A tee at the top of the vacuum can pumping l i n e allows a d i r e c t l i n e of access from room temperature to below the s t i l l . 85 5.1b The Sample Region The mounting of the sample and resonator housing below the mixing chamber is shown in figure 15. The thin-walled pyrex sample c e l l i s surrounded by a superfluid "He bath in contact with the mixing chamber. It contains approximately 10 kPa of Hp and 50 kPa of the appropriate He isotope sealed in at room temperature. The helium i s s u f f i c i e n t to form a saturated f i l m (200A2 to 300A2) at low temperatures. The outer pyrex tube terminates with a glass to copper seal and i s attached, with epoxy, to a copper flange. This can be sealed with an indium O-ring, to a copper disc which i s in strong thermal contact with the mixing chamber. The space surrounding the sealed c e l l i s f i l l e d with superfluid "He through a pair of 0.018 inch diameter x 0.004 inch wall copper/nickel c a p i l l a r i e s i n s t a l l e d and heat sunk by S.H.E.. At room temperature, the c a p i l l a r i e s connect to a simple gas handling system which allows an approximately known amount of "He gas, from a 16 l i t r e aluminum tank, to be introduced into the c e l l through a l i q u i d Np cooled molecular sieve trap. Because of the danger of trapping l i q u i d helium in the sample bath by blocking of the f i l l c a p i l l a r i e s , the outer pyrex tube i s also connected, v i a two c a p i l l a r i e s , to a bursting seal. This device consists of a 12.5 mm diameter disc of 0.025 mm thick aluminum f o i l sealed, with an indium O-ring, across an Grophite Filled Polyimide Support Adjustable Coupling Resonator Mixing Chamber I—Fill Capillary Ge Thermometer Glass/Metal Seal Sealed Pyrex Cel l Tuning Plate Discharge Co i l Figure 15 Schematic diagram of the low temperature part of the apparatus. The sample c e l l i s cooled by contact with l i q u i d "He in contact with the mixing chamber of a commercial d i l u t i o n r e f r i g e r a t o r . The resonator and housing are cooled only to the s t i l l temperature (0.5K). 87 approximately 8 mm diameter cavity in a brass d i s c . The t i p of a hypodermic needle is positioned, by t r i a l and error, so as to puncture the f o i l when the pressure in the c e l l r i s e s above about 200 kPa. The bursting seal and the room temperature gas handling system are shown schematically in figure 16. The superfluid f i l l i n g the outer pyrex tube makes thermal contact with the mixing chamber via a cylinder of powdered copper sintered to a copper post which is screwed into the copper disc below the mixing chamber. The sin t e r i n g was done with a powder of about 0.025 cm diameter copper grains at 860°C in an H 2 atmosphere for about 20 minutes. Radial force was applied using a s p l i t s t a i n l e s s steel tube closed by a standard hose clamp. The s i n t e r i n g conditions were based on the method used by Enholm and Gylling(53) in construction of their large cooling power d i l u t i o n r e f r i g e r a t o r . The e f f e c t i v e area, assuming a packing fr a c t i o n of 0.5, was estimated to be greater that 600 cm2. The temperature of the superfluid bath i s measured using a germanium resistance thermometer clamped to the free end of the sintered post. The c a l i b r a t i o n supplied by the manufacturer, Lakeshore Cryotronics, was found to be in substantial error following completion of these experiments. The thermometer was recalibrated against an NBS SRM 768 set of f i v e superconducting fixed point samples and the c a l i b r a t i o n was interpolated using a thermometer based on the s u s c e p t i b i l i t y of cerium magnesium n i t r a t e . At the completion of t h i s 88 BURSTING S E A L II needle to cell 16 I tank to He supply to pump I - N 2 cooled trap Figure 16 The bursting seal and room temperature gas handling system 89 c a l i b r a t i o n , however, the d i l u t i o n r e f r i g e r a t o r was no longer available for repetition of these experiments and the temperatures used in the analysis below are corrections of the o r i g i n a l l y measured temperatures. The r e c a l i b r a t i o n i s discussed in Appendix F. The l i k l i h o o d that the c a l i b r a t i o n had not changed during the o r i g i n a l experiments was supported by examination of the relationship between the germanium resistance thermometer and a carbon resistance thermometer which was used to monitor mixing chamber temperatures, for regulation purposes, during the o r i g i n a l experimental runs. The corrected c a l i b r a t i o n has subsequently been confirmed with a r e c a l i b r a t i o n by the manufacturer. The adequacy of the thermal contact provided by the superfluid bath and the copper sinter has been considered using simple models for the heat transport in the helium column and the Kapitza thermal resistance across the superfluid/copper interface. We estimate the maximum energy delivered to the c e l l by the discharge to be less than 1 m i l l i j o u l e and take 10 m i l l i j o u l e s to be a worst case parameter for design purposes. Taking the volume of the c e l l bath to be 20 cm3, t h i s energy corresponds, in the l i m i t of i n f i n i t e resistance to thermal transport out of the c e l l bath, to a temperature r i s e of about 1K. The necessity for good thermal contact to the c e l l i s apparent. There are two thermal resistances which play a role in the 90 transport of heat away from the c e l l . The f i r s t of these i s the Kapitza thermal resistance between the copper sinter and the superfluid and the second i s the thermal resistance of the annular column of superfluid between the c e l l and the outer tube. The time constant for decay of a temperature difference between an i n f i n i t e thermal bath and a body of heat capacity C separated by an area A of Kapitza resistance R i s simply RC/A. Using the heat capacity given by Keller(54) for the superfluid, the Kapitza resistance given by Lounasmaa(52) at about 0.5K, and an estimate for the bath volume of 20 cm3 and for the sinter area of 687 cm2, we estimate a time constant of order 4 msec. Thermal transport along a column of area A and of heat capacity per unit length with a thermal conductivity of K is governed by the equation dT = Kh_ d 2T dt Cx dx 2 This type of problem has been treated by Carslaw and Jaeger(55). The time constant for decay of an i n i t i a l temperature p r o f i l e can be estimated by considering a column of length 21 with the ends, x=0 and x=21, fixed at some temperature. We w i l l consider the decay of the f i r s t fourier component of the temperature p r o f i l e . We thus take the difference in temperature between a point x and the end to be proportional to sin(irx/21). Since 91 there i s no heat transport across the center of the model column, the symmetry of the problem allows us to associate the experimental situation with the length 1 corresponding to one half of the column. The time constant for decay of the i n i t i a l temperature p r o f i l e i s then 41 2Cj_//cA7r 2. Using the heat capacity for the superfluid(54) and the thermal conductivity(52) of a column of superfluid equal in area to the cross section between the glass walls, we estimate a time constant of less than 10 msec. The smallness of these estimated time constants provides some confidence in the a b i l i t y of the sample region to recover quickly following a discharge. In practice, our best indication of the recovery of the temperature .in the c e l l comes from the i n s e n s i t i v i t y of the magnetic resonance signal frequency to the i n t e r v a l elapsed following the discharge. 92 5.1c The Resonator Housing The resonator housing i s thermally isolated from the mixing chamber by three graphite f i l l e d polyimide rods (VESPEL) as shown in figure 15. The resonator and discharge c o i l are similar to those used at 1K. The housing i s thermally linked to the s t i l l temperature through the copper outer conductors of the magnetic resonance and discharge coaxial transmission l i n e s . Between the s t i l l and the vacuum can flange at 4.2K, the magnetic resonance signal and discharge pulse are transmitted through special low thermal conductivity coaxial l i n e s . These sections are fabricated from a standard 0.141 inch diameter coaxial l i n e with s t a i n l e s s steel outer conductor(56), by removing the steel center conductor and replacing i t with a tinned s t a i n l e s s s t e e l conductor. This assembly i s heatsunk, halfway between the ends, to the coldplate. Thermal contact to the center conductor i s accomplished with crimps in the outer conductor which compress the teflon insulation against the inner conductor. The crimping i s done with a modified tube cutter using a disc with rounded edges in place of the standard cutting wheel. The connection to room temperature i s made via 0.141 inch diameter stain l e s s s t e e l outer conductor coaxial l i n e s sealed to brass plugs on the vacuum can flange. With t h i s type of coaxial l i n e , leakage into the vacuum can through the space between the teflon insulator and the outer conductor was not 93 observed. This was, however, a problem with a smaller coaxial l i n e of diameter 0.080 inches and t h i s problem had to be dealt with by crimping the outer conductor onto the inner te f l o n conductor using the tool described above. Problems were s t i l l encountered with perforations in the outer conductor at the crimp and ultimately any of the small diameter coaxial l i n e used was connected, at the vacuum flange, to special sealed feed-throughs in which the space between the inner and outer conductors was f i l l e d with an epoxy such as Stycast 2850FT(Emerson and Cummings). Adjustment of the coupling and tuning for the resonator i s accomplished using mechanical connections through bellows mounted on the room temperature end of the vacuum can pumping l i n e . Threaded posts sealed into the bellows may be positioned using captured nuts above the bellows. Inside the pumping l i n e , below the bellows, s t a i n l e s s s t e e l wires are attached to the moveable posts. These wires pass through the radiation b a f f l e at the vacuum can flange and continue through central access ports in the coldplate and s t i l l . The wires are clamped to f l e x i b l e copper braid at the vacuum can, coldplate and s t i l l to provide heat sinking. Below the s t i l l , the wires connect to lengths of unwaxed dental f l o s s fed through pairs of te f l o n pulleys in order to pass outside the diameter of the heat exchangers and mixing chamber. Below the pulleys, the force i s again transmitted by thin s t a i n l e s s steel wires. At the 94 resonator housing the wires attach to the ends of fibreglass lever arms and are kept in tension by phosphor bronze return springs as shown for the tuning assembly in figure 15. This assembly consists of a s l i d i n g tuning plate linked to the end of the lever arm via a r i g i d f i b r e glass rod. The coupling capacitor assembly consists of an upper fixed plate attached to the centre conductor of the magnetic resonance coaxial l i n e and a moveable bottom plate supported on a phosphor bronze loop attached to the resonator, as in the 1K design. The moveable plate i s attached, via a s t r i n g , to the end of a second fi b r e g l a s s lever arm inside the resonator housing. In t h i s assembly, the phosphor bronze coupling loop also acts as a spring to return the lower plate of the coupling adjustment capacitor. 95 5.1d The Spectrometer The spectrometer for these experiments i s e s s e n t i a l l y i d e n t i c a l to that used for the 1K experiments with the exception that for the experiments of thi s chapter, the 10 MHz reference signal was provided by a Tracor Rubidium Standard. As for the 1K experiments, th i s standard was cal i b r a t e d in s i t u using, primarily, the N.B.S. monitored t e l e v i s i o n colour sub-carrier. The c a l i b r a t i o n was confirmed by comparison to signals from LORAN-C navigational transmitters using an Austron LORAN-C reciever. It i s estimated that with the rubidium standard ca l i b r a t e d , i t i s possible to measure frequencies from the spectrometer to 3 parts in 10 1 1 or about 0.04 Hz out of 1420 MHz. Control of the magnetic f i e l d for the experiments below 1K is via a bias c o i l wound on an aluminum former. The c o i l region has diameter 10.95 cm and length 20.3 cm. The bias c o i l consists of 349 turns of 24AWG copper wire with a pair of 42 turn end correction c o i l s . Gradient c o i l s were also i n s t a l l e d but not used in these experiments. The aluminum former i s wrapped with a Co-Netic f o i l magnetic s h i e l d . This assembly i s mounted around the d i l u t i o n r e f r i g e r a t o r vacuum can and i s in contact with l i q u i d helium. In the experiments with 3He coated walls, provision was made to degauss the magnetic shi e l d using 7 turns of 24AWG copper wire wound t o r o i d a l l y on the sh i e l d . 96 Degaussing was done with the sh i e l d at room temperature by slowly reducing a 60 Hz alternating current from about 8 amps. This procedure resulted in a longitudinal f i e l d at the resonator of about 4 mGauss and a transverse f i e l d of about 2 mGauss. 97 5.2 Model for the Magnetic Resonance Lineshape for Atomic Hydrogen Confined by Liquid Helium Walls Recall that the magnetic resonance experiments described here are done at the (F=1,Mp = 0—F=0,Mp = 0) hyperfine t r a n s i t i o n , which in very low magnetic f i e l d has frequency o)0/2i:= 1 , 420, 405, 751 sec" 1. (see F i g . 1). This resonance i s observed by applying a pulse of r . f . magnetic f i e l d along the d i r e c t i o n of a small s t a t i c f i e l d . The o s c i l l a t i n g magnetization i s then detected along the same axis. This d i f f e r s from the usual si t u a t i o n for spin 1/2 p a r t i c l e s where an r . f . pulse of the appropriate length and magnitude, the TT/2 pulse, is applied perpendicular to the s t a t i c f i e l d . This results in the magnetization being rotated by TT/2 after which i t precesses in the plane perpendicular to the s t a t i c f i e l d . As discussed in chapter II, however, i t can be shown (27) that a general two l e v e l system can be represented by a f i c t i t i o u s spin 1/2 system. Because of t h i s correspondence, the evolution of the free induction decay (FID) following the -n/2 pulse can be discussed in terms of the familiar magnetic resonance terminology. Correspondingly, the system of interest here can be modelled by an ensemble of precessing spins, which are in phase 98 immediately following a K/2 pulse. The experimentally accessible variables are the amplitude and instantaneous phase of the precessing magnetization which i s the vector sum of contributions from a l l of the spins. These spins are confined by a closed surface with which they can c o l l i d e . The average s t i c k i n g p r o b a b i l i t y , s, i s equal to the f r a c t i o n of c o l l i s i o n s which result in the atom being stuck to the surface. There are thus three c h a r a c t e r i s t i c times which describe the system: the average time between c o l l i s i o n s with the container walls, < T C > » t n e average time between s t i c k i n g , <Tg>=<rc >/s, and an average duration of the s t i c k i n g events, <r s>. During a s t i c k i n g event the precession frequency i s s h i f t e d by C J s . We neglect dephasing of the spins due to inhomogeneities in the ambient f i e l d or c o l l i s i o n s in the gas. We also neglect any longitudinal relaxation (T^ processes). The instantaneous rates of change of the phase and magnitude of the magnetization (Acu and 1/Tp) are then determined by the s t a t i s t i c s of adsorption and desorption. The f r a c t i o n of time an atom spends on the surface, <r s>/(<r s>+<Tg>), i s determined in t h i s simple system by the binding energy, Eg, and the surface to volume r a t i o of the container, A/V. For a l l conditions of interest to us, t h i s f r a c t i o n i s less than 1 0 " 3 and given, to a good approximation, by 99 [V-1] = AA e V E g / k T < T g > <TC > + < T n > where A=17.4x10"8//T cm i s the H atom thermal de Broglie wavelength. Before dealing with the way in which the current model relates the FID to the s t a t i s t i c s of adsorption and desorption, i t i s helpful to consider the result which would be obtained for t h i s system by st a r t i n g with the type of multiple residency model, due to Anderson and Weiss (57), which i s discussed by Abragam (27). That model, as presented in reference (27), contains the i m p l i c i t assumption that the p r o b a b i l i t y of no jump -AtTr(°°- z 0 0 - ) taking place in an i n t e r v a l At i s given by e 1 1 where (tr(o>| , C J . ))" 1 can be thought of as some kind of average residency time at s i t e i . This i s equivalent to the assumption that residency times, T and r D for example, are d i s t r i b u t e d according to Poisson s t a t i s t i c s . However, there i s no requirement that the duration of residency at any s i t e be n e g l i g i b l e . If the jump rate i s less than the s p l i t t i n g , the res u l t i s a l i n e associated with each of the s i t e s . On the other hand, when one s t a r t s from t h i s model and makes the assumption that the duration of residency at one of two s i t e s (the s t i c k i n g event) i s very short, the result i s a dominant l i n e s h i f t e d from the main frequency by 100 [V-2a] ACQ = 1 <p0 < r B > and with decay rate, 1/Tp , of the FID of [V-2b] J _ = J _ <f>n2 T 2 <rB> where <J>Q=<TS>U)S. There i s , in addition, a broad weak l i n e associated with the surface residency. We note here that under the conditions of our present experiment, t h i s second l i n e would never be seen. In fact, expressions [V-2a] and [V-2b] display most of the q u a l i t a t i v e features which are observed in the data. For small 0O , one finds Aco= #Q/< Tg >. As <t>0 increases, corresponding to lower temperatures and longer s t i c k i n g times, the s h i f t , Aw, goes through a maximum value of l/(2<Tg>) from which s can be inferred i f one knows <r c>. As 4>Q becomes larger s t i l l , only a small number of s t i c k i n g events are required for atoms to get out of phase with each other; thus the signal i s dominated by atoms which have undergone few or no s t i c k i n g events since the TT/2 pulse. Accordingly, A C J approaches zero and 1/Tp goes to l/<Tg>. While the d e t a i l s may change, t h i s behaviour i s expected to be q u a l i t a t i v e l y correct for any 'reasonable' d i s t r i b u t i o n functions for r 0 and T„ . 101 In the more general model to be formulated below, we are going to relax the requirement that T d have a Poisson D d i s t r i b u t i o n , but retain a Poisson d i s t r i b u t i o n for T s . This r e s t r i c t i o n , which makes the analysis tractable, can be shown to correspond closely to the actual experimental s i t u a t i o n . The argument i s as follows. For H on the He l i q u i d s , the surface state has a binding energy somewhat greater than kT at the temperatures of inte r e s t . Therefore once an atom has entered the bound state, i t i s l i k e l y to undergo a number of c o l l i s i o n s before i t experiences a s u f f i c i e n t l y energetic c o l l i s i o n to eject i t from the surface. This implies that, with respect to i t s motion along the surface, i t i s l i k e l y to come into thermal equilibrium in a time short compared to the average residency time. The pr o b a b i l i t y of desorption in a given time in t e r v a l w i l l then be independent of the time elapsed since the st i c k i n g event. Thus, the d i s t r i b u t i o n of surface residency times i s given by the Poisson d i s t r i b u t i o n 102 - T / < T > [V-3] P-(i- s) = J _ e < r s > I t i s not,however, obvious that the d i s t r i b u t i o n of times between s t i c k i n g event Pg(rg)» need be Poisson. Indeed, we f i n d that the r e s u l t s f o r H confined by l i q u i d 3He coated w a l l s cannot be described by equations [V-2a] and [V-2b], We begin the d e s c r i p t i o n of our model by noting that <7_>/<r > i s s u f f i c i e n t l y small that we may neglect the passage of time during a s t i c k i n g event. This i s equivalent to d i s r e g a r d i n g the presence of the weak, broad resonance l i n e a s s o c i a t e d with r a d i a t i o n from atoms while they are on the surface. The d i s t r i b u t i o n P S ( T S ) w i l l then serve only to determine the d i s t r i b u t i o n of e x t r a phase, <p, accumulated during the s t i c k i n g events which are otherwise taken to be of n e g l i g i b l e d u r a t i o n . A c c o r d i n g l y , the d i s t r i b u t i o n of <t> i s given by [ V - 4 ] pU) = where <f>0 = <rs>cos. We continue by noting that the most general expression f o r S 103 the free induction signal w i l l be i * ( t ) [V-5] S(t) = S0<e > where 0(t) i s the accumulated phase of a given precessing spin and the average i s taken over a l l of the atoms in the sample. S Q i s the i n i t i a l amplitude. A given phase, 4>(t), can be accumulated by one of an e s s e n t i a l l y i n f i n i t e number of possible sequences, of s t i c k i n g events. A given sequence of n s t i c k i n g events giving a phase n [V-6] 0(t) = u0t + L 0. i=1 1 w i l l occur with a p r o b a b i l i t y which i s the product of the p r o b a b i l i t y for the n s t i c k i n g events to occur and the p r o b a b i l i t i e s for each to have contributed the appropriate excess phase 0j . This p r o b a b i l i t y has the form n [V-7] p(n,t) n pU- ) i = 1 where p(n,t) i s the p r o b a b i l i t y that n s t i c k i n g events w i l l occur within a time t of some a r b i t r a r y s t a r t i n g point which we 104 w i l l i d e n t i f y with the time of the TT/2 pulse. The p(0j ) are given by equation [V-4]. The average of equation [V-5] can then be done by summing over and performing the n i d e n t i c a l integrals over <f> to give i"ot oo f -<t>/4>o i<t> n [V-8] S(t) = S Qe 2 p(n,t) [ J j _ e e d*] n=0 0 4>0 icu 0t oo p(n,t) [V-9] = S e Z n . n=0 (1 - i 0 O ) In order to express p(n,t) in terms of the d i s t r i b u t i o n of times between st i c k i n g events, Pg(r), we define the conditional p r o b a b i l i t y , P'n ( t ) , that a s t i c k i n g event at t=0 w i l l be followed by n events in the time t or have been preceded by n events in the time - t . We can then construct p(n,t) as follows. We place the f i r s t s t i c k i n g event aft e r the TT/2 pulse at time t j . The p r o b a b i l i t y that no event occurred in the time leading up to the f i r s t event i s i t y ) . The p r o b a b i l i t y of an event occurring in an interval dt 1 at time t 1 after the ir/2 pulse i s simply d t ^ / < T Q > since there i s no c o r r e l a t i o n between st i c k i n g events and the TT/2 pulse. F i n a l l y , the pr o b a b i l i t y of the f i r s t event having been followed by (n-1) more in the time ( t - t ^ ) remaining i s the conditional p r o b a b i l i t y P' (t-t«). We thus 105 have [V-10] p(n,t) = _J_ 00 ( < TB > dt, P G ( t 1 ) P ^ ( t - t 1 ) CD where P n ( t ) i s defined to be P' (t) for p o s i t i v e arguments and zero for t<0 since for t,>t or t,<0, there are only n-1 events from zero to time t and t h i s s i t u a t i o n should not be included in the i n t e g r a l . By similar reasoning, one obtains CD [V-11] p(0,t) = 1 / df P Q(t+r) - 0(-t) <rB> 0 CO = _ i _ f 6t] p ^ t - t p e(-t 1 ) - e(-t) <Tg> -co where the second term, 0 ( - t ) , sets p(0,t)=0 for negative t . In turn, the functions P n ( t ) can be constructed as follows: [V-12] P Q ( t ) = ( J dr e ( r - t ) p ^ r ) ) - 0(-t) - C D where again the step function, 6 ( - t ) , ensures the desired behaviour for negative argument, and 1 06 C O [V-13] P n ( t ) = | P B ( T ) Pp., (t-r) dr - C O In equation [V-12], we make use of the property that CO ^jpg(r)dT = 1 so that the f i r s t term i s interpreted as the pr o b a b i l i t y that given an event at t = 0, the next event w i l l occur at r > t. In the integrand of equation [V-13], Pg(r)dr i s the p r o b a b i l i t y that the f i r s t event w i l l follow the event at t = 0 by a time T and P n_.,(t-T) i s the p r o b a b i l i t y that the event at time T w i l l be followed by (n-1) events in the remaining time ( t - r ) . With these p r o b a b i l i t i e s in place, we can begin to construct S(t) by taking the fourier transform of p(n,t) to be 7r(n,x), (not to be confused with TT (<*>- rco -) of Ref. (27)), and writing - i x t co ir(n,x) e I n . n=0 ( 1 - i 0 o ) Defining the fourier transform of p a ( r ) to be J T ( X ) , and of e(-t) to be [V-14] S(t) = S De iw 0t CO dx 2ir J •co V 107 6(-x) = lim - i e-»o x - i e we can use the convolution theorem to get CD l X t [V-15a] J d t e P Q ( t ) = n o(x) = e(-x)(ir(x) - 1) -co CO l X t I l X t [V-!5b] j d t e P n ( t ) = n n ( x ) = ir(x) n ^ t x ) -co n = 7 T ( X ) ( T T ( X ) - 1 ) 0(-x) We thus have, for n > 0, 2 2 n-1 [V-16a] 7r(n,x) = J _ [0(-x)] (TT(X)-1) T T ( X ) <r B> and for n = 0 2 [V-I6b] T T ( 0 ,X ) = ]_ [6(-x)] ( T T ( X ) - 1 ) - e(-x) <r B> which, when substituted into equation [V-14] gives, after summation, V 108 [V-17] CO ( T T ( X ) - I ) ^"ty^i < T D > Ti ff(x) ico Dt /" - i x t S(t)=S ne [dx e 2TT ; -CD -e(-x) + e ( - x ) 2 ( 7 T ( x ) - i ) O'ty^ fa;)) We have thus constructed the free induction decay from two quant i t i e s : the average phase s h i f t per s t i c k i n g event, <f>0, and the d i s t r i b u t i o n of times between s t i c k i n g events p D ( r ) . When P the Poisson d i s t r i b u t i o n , p D ( r ) = J e - r / < r B > 8 <r 0> i s used, equation [V-17] y i e l d s ico nt iAcot-t/T 0 S(t) = S Q e e 2 where Aw and l/Tp are given by equations [V-2a] and [V-2b] as expected. Equation [V-17] i s very s i m i l a r in form to that recently derived by Crampton et al.(58) in the i r analysis of magnetic resonance for H confined by s o l i d Hp walls. They have defined a multiple bounce c o e f f i c i e n t because of the possible roughness of the s o l i d Hp surface. We have neglected such an ef f e c t in l i g h t 1 0 9 of the presumably smoother He surface. With t h i s difference aside, the model of Ref.(58) d i f f e r s from that presented here by working in terms of the times between c o l l i s i o n s rather than in terms of the times between s t i c k i n g events. They also define an average phase accumulated per c o l l i s i o n whereas in the current model i t i s the average phase accumulated per st i c k i n g event which appears. Consequently, the stic k i n g p r o b a b i l i t y appears e x p l i c i t l y in the model of Ref.(58) with the i m p l i c i t requirement that the d i s t r i b u t i o n s of times between st i c k i n g events and of times between c o l l i s i o n s be the same except for scaling of the times by the stic k i n g p r o b a b i l i t y . We have chosen to avoid such a requirement in l i g h t of the p o s s i b i l i t y , due to the energy .and angle dependence of the st i c k i n g p r o b a b i l i t y , that the d i s t r i b u t i o n s might not be so simply related. In thi s event, i t would then be the d i s t r i b u t i o n of times between st i c k i n g events which would determine the free induction decay. As s h a l l be discussed below, there are conditions under which th i s d i s t r i b u t i o n can, in p r i n c i p l e , be measured d i r e c t l y . As i s expected, when the st i c k i n g p r o b a b i l i t y of Ref. (58) i s set equal to 1, so that the d i s t i n c t i o n between c o l l i s i o n s and stic k i n g events in that model i s removed, the resulting expression for the FID i s equivalent to equation [V-17]. 110 5.3 The Binding Energy, Surface Hyperfine S h i f t , and Surface Recombination Rate for H on "He and H on 3He We proceed, in t h i s and the following section, to describe work which was i n i t i a l l y reported in references (23) and (24). One feature of both the recombination rate and observed hyperfine frequency s h i f t below IK i s that they display a pronounced minimum as they pass from bulk dominated to surface dominated behaviour. This comes about as a result of the very strong temperature dependence of the helium vapour pressure and the exponential increase, with T" 1, of the f r a c t i o n on the surface.The consequences of such behaviour, p a r t i c u l a r l y with regard to the hyperfine s h i f t , are discussed below. Most of the measurements described in t h i s chapter have been c a r r i e d out at temperatures for which the vapour pressure of the l i q u i d helium wall coating i s s u f f i c i e n t l y small that the contribution, from atoms in the bulk, to the recombination rate and the observed hyperfine frequency s h i f t can be neglected. Even at the lowest temperatures and highest densities used in t h i s work, however, the f r a c t i o n of atoms on the surface remains less than 0.1 per cent and so can be approximated by equation [111-36]. In t h i s section, we w i l l be concerned with measurements for which the d e t a i l s of the s t i c k i n g s t a t i s t i c s average out so that the r e s u l t s are sensitive only to the f r a c t i o n of time spent by 111 an atom on the surface. The observed bulk recombination rate, under such conditions, r e f l e c t s the rate at which atoms from the bulk are lost to the surface in order to maintain equilibrium as the adsorbed atoms recombine. Assuming a temperature independent recombination cross-length, the e f f e c t i v e bulk recombination rate i s given by equation [111-39], By p l o t t i n g In k/T~vs. T" 1, one can extract the surface recombination rate and binding energy Eg. As i s apparent from the discussion of the magnetic resonance lineshape, somewhat more care must be taken in choosing the temperature regime for which the hyperfine frequency s h i f t r e f l e c t s only the average occupation of the surface state. From equation [V-1], one sees that the average phase s h i f t per s t i c k i n g event, 0 O , i s strongly temperature dependent: Eg/kT 8 V It i s at higher temperatures, for which # Q « 1 , that the d e t a i l s of the adsorption and desorption s t a t i s t i c s drop out. Equation [V-17] then y i e l d s a frequency s h i f t of the form of equation [V-2a] which, for small 0 O , becomes 1 12 Eg/kT Aw = <f>Q = a>s AA_ e <rB> V This i s equivalent to equation [111-37], In t h i s l i m i t , A C J i s just the product of the surface frequency s h i f t , cos, and the fr a c t i o n of time the atom spends on the surface. Noting that A i s proportional to T" 1 / 2, one can plot In Aco/T" V S . T"1 and obtain C J s and Eg in analogy with the case for the recombination rate measurements. We now go on to discuss the result of applying these two types of experiment to the measurement of the binding energies. 113 5.3a Recombination Measurements As was the case for the measurements ca r r i e d out at IK, the measurements of recombination rates required accurate determinations of absolute atom densi t i e s . This problem has been considered in some d e t a i l in appendix B. Accurate measurements of atom densities require that the resonator be tuned and the spectrometer be accurately c a l i b r a t e d . The s t a t i c f i e l d in the resonator should be predominantly along the resonator axis, and the atoms must be allowed to relax between 7r/2 pulses in order that the f u l l signal be sampled. It i s also necessary that the f i l l i n g factor be known and, again, the i n i t i a l published results(23)(24), obtained using in place of QQ, overestimate the recombination rate by a factor of ,/2~• The low temperatures at which these measurements were car r i e d out led to two additional complications. The f i r s t arose because the population difference between the Mp=0 hyperfine l e v e l s , at these temperatures, i s not well approximated by the simple T" 1 dependence and must be given by the f u l l expression of equation [A-12]. At the lowest temperatures used, the difference between the exact and approximate expressions i s about 20 per cent. At 0.5K, i t i s about 3 per cent. The second complication i s attributed to faster d i f f u s i o n of H atoms as the density of the He vapour above the saturated 1 1 4 f i l m becomes very small. Under such conditions, the atoms in the resonator at the time of the 7r/2 pulse may d i f f u s e into the rest of the c e l l in a time short compared to the duration of the free induction decay. The r e s u l t i n g signal displays a very fast i n i t i a l decay, of order a few milliseconds long, due to d i f f u s i o n out of the resonator. The amplitude of the remainder of the free induction decay i s then c h a r a c t e r i s t i c of the true density d i l u t e d by the r a t i o of the volume of the c e l l within the resonator to the t o t a l c e l l volume. Under most circumstances, the signal amplitude i s found by extrapolating the signal height measured at the maxima of the 10 to 100 Hz beats obtained by mixing the free induction signal with the spectrometer frequency. For the recombination measurements (with H confined by both "He and 3He), at temperatures below those for which bulk recombination i s observed, such plots cannot display the rapid i n i t i a l decay and thus underestimate the t o t a l density. The necessary correction factor has been obtained by working at exactly the H atom resonant frequency (zero-beat condition) and observing the f i r s t few tens of milliseconds of the r e s u l t i n g free induction decay. This allows us to obtain the r a t i o of the true i n i t i a l signal amplitude to that obtained by extrapolation of the free induction envelope following the i n i t i a l decay. This r a t i o i s expected to be dependent on the shape of the c e l l used. For the c e l l used with a "He wall coating, the r a t i o was 1 . 5 9 and, for the c e l l used 115 with 3He, 1.44. It was noted that in some of the recombination measurements carried out on the high temperature side of the recombination minimum, the decay time of the i n i t i a l decay was long enough to be observed and the true signal amplitude could be obtained d i r e c t l y . The surface recombination results for H on "He and on 3He are shown in figures 17 and 18 respectively. Recall, from chapter I I , that the simple theory for the e f f e c t i v e decay rate of the bulk density due to two body recombination on the surface y i e l d s f£H = "keffnH dt where 2Eg/kT K p f f = XvsAA!e v Here X i s interpreted as a surface recombination "cross-length" and v s i s the average r e l a t i v e speed in the two dimensional gas of H atoms. The temperature dependence of v s A 2 can be taken into account by p l o t t i n g In k/f versus T" 1. If X i s temperature independent, the re s u l t i n g plot has a slope of twice the binding energy. Figure 17 The recombination rate plotted as kT 1 / 2 vs T" l i n e i s for X=0.14A and x's are the data of Ref, 8T, appropriately scaled zero f i e l d and the A/V experiment reported here from Ref.(59) taken at s i m i l a r l y scaled. for H on "He The s o l i d Eg=1.15K. The (21), taken at to correspond to ra t i o of the The +'s are 11T and are £ — i — i — i i i i . i 2 4 6 6 1 0 1 2 1 4 1 6 l/T < K - ' ) Figure 18 The recombination rate for H on l i q u i d 3He plotted as kT^ 2 vs T" 1. The l i n e i s for fixed X=0.13A and Eg=0.39K. The crosses are the results of van Yperen et al.(22) at high magnetic f i e l d scaled by the expected r a t i o of the cross-sections 2c 2 and also adjusted for the d i f f e r e n t A/V r a t i o of the two experiments. The increase in recombination at high temperature i s due to rapidly increasing He vapour density. 118 F o r H on " H e , t h e b i n d i n g e n e r g y m e a s u r e d i n t h i s way i s ( 1 . 1 5 + / - 0 . 0 5 ) K w i t h X = ( 0 . 1 4 + / - 0 . 0 2 ) A . " T h e r e i s no a p p a r e n t t e m p e r a t u r e d e p e n d e n c e t o t h e c r o s s - l e n g t h . F o r H on 3 H e , t h e d a t a do n o t f i t a c o n s t a n t s l o p e a s w e l l . The b e s t e s t i m a t e f o r t h e b i n d i n g e n e r g y and X, t a k e n t o be E g = 0 . 3 9 K a n d X = 0 . 1 3 A , c o r r e s p o n d s t o t h e s o l i d l i n e o f f i g u r e 18 . The d e v i a t i o n f r o m a s t r a i g h t l i n e s u g g e s t s a t e m p e r a t u r e d e p e n d e n c e o f X w h i c h m i g h t n o t be u n r e a s o n a b l e i n l i g h t o f t h e s t r o n g t e m p e r a t u r e d e p e n d e n c e f o r t h e b u l k r e c o m b i n a t i o n r a t e c a l c u l a t e d by G r e b e n e t a l . ( 3 l ) a n d d i s c u s s e d i n c h a p t e r I I I . A t s u c h low t e m p e r a t u r e s , h o w e v e r , t h e p o s s i b i l i t y o f t h e r m o m e t r y e r r o r must a l s o be c o n s i d e r e d , p a r t i c u l a r l y i n l i g h t o f t h e o r i g i n a l t h e r m o m e t r y p r o b l e m . The r e c a l i b r a t i o n o f t h e ge rman ium t h e r m o m e t e r i s d i s c u s s e d i n A p p e n d i x F . We b e l i e v e t h a t t h e e r r o r b a r s on t h e l o w e s t t e m p e r a t u r e p o i n t s o f f i g u r e 18 f a i r l y r e f l e c t t h e a c c u r a c y o f t h e r e c a l i b r a t i o n . 119 5.3b Binding Energy from Frequency S h i f t Measurements Accurate measurements of surface hyperfine frequency s h i f t s require that some care be taken to avoid the e f f e c t s of cavity p u l l i n g as described in the t h i r d section of chapter I I . Recall, from equations [11-25] and [11-26], that the amount by which the frequency of the o s c i l l a t i n g magnetization i s pulled i s proportional to the f r a c t i o n a l mistuning of the cavity and the instantaneous magnitude of the magnetization. Accordingly, the procedure during frequency measurement runs included tuning of the resonator by observation of the ref l e c t e d 7r/2 pulse using a d i r e c t i o n a l coupler inserted between the spectrometer and resonator. A -free induction decay was then recorded for a small atom density after which the discharge was f i r e d to achieve a high atom density and another free induction decay recorded. Agreement of the frequencies for the two atom densities was taken as confirmation of proper tuning. Following reduction of the atom density, usually by warming to temperatures where bulk recombination sets i n , the frequency versus temperature data were c o l l e c t e d . With the cavity mistuned, large amplitude signals might be expected to display cavity p u l l i n g as a va r i a t i o n of the frequency of the free induction signal across the free induction decay. The deviation of the observed frequency from the true signal frequency would then be smallest at the beginning of the 1 2 0 decay. A variation in frequency across the free induction decay might also occur i f some transient response of the spectrometer had not died away before recording of the i n i t i a l portions of the free induction decay. In l i g h t of the l a t t e r consideration , the f i r s t portion of the free induction decay was normally discarded for purposes of frequency determination and, for the H on 3He measurements, the power in the v/2 pulse was reduced to less than 1 0 microwatts. A more i n t r i n s i c complication arises in the separation of the hyperfine frequency s h i f t , due to surface adsorption, from the s h i f t due to other sources including the residual longitudinal and transverse magnetic f i e l d s , the atomic clock c a l i b r a t i o n o f f s e t , and the pressure s h i f t due to interactions in the bulk gas. It i s found, for both 3He and "He wall coatings, that the pressure s h i f t due to H atom interactions with the gas above the saturated helium f i l m was of the same sign as the s h i f t for atoms adsorbed on the surface. The result i s that as the saturated helium vapour pressure decreases with decreasing temperature, the observed frequency approaches the free atom hyperfine frequency u n t i l the wall s h i f t intervenes and the observed frequency s h i f t increases with decreasing temperature. The r e s u l t i n g minimum in the frequency s h i f t thus gives a r e l a t i v e l y temperature independent benchmark from which frequency s h i f t s may be measured. Accordingly, an observation of the frequency at the minimum was included with any set of 121 frequency s h i f t measurements taken for a p a r t i c u l a r magnetic f i e l d and clock c a l i b r a t i o n . The remaining problem was then to determine absolutely the hyperfine frequency s h i f t , A f m , at the minimum. The observed frequency at the minimum, fobs » m a v ke written t v " l 8 J fobs " fo + a t B , i + Bx 23 + A f m - Af c where f Q= a/h, a=2773 Hz/gauss 2, B and BL are the longitudinal and transverse components of the magnetic f i e l d , and Af c i s the atomic clock offset as measured by comparison to the N.B.S. monitored t e l e v i s i o n colour sub-carrier and scaled to the spectrometer frequency of 1420.405 MHz. The longitudinal magnetic f i e l d i s the sum of the ambient residual f i e l d l e f t by the magnetic shield at the resonator, B|| , and the applied bias f i e l d , 01 where 0 has units of f i e l d / c u r r e n t . The frequency s h i f t , at the minimum, w i l l contain a contribution from the bulk s h i f t as well as the surface s h i f t . We can rewrite equation [V-18] to e x p l i c i t l y show the current dependence. We then have 122 fobs- f o + •- a B j r e s + 2 t t P B||res 1 + a ^ 1 2 + ( f l B* + ^ m ] where f 0 , Af c , and a are known independently. By f i t t i n g the current dependence of the frequency at the minimum to a quadratic, i t i s possible to determine B|| res a n c ^ ^* T n e remaining parameter of the quadratic f i t then determines (aB 2 + A f m ). The u t i l i t y of t h i s analysis depends on Bx being independently measureable and of small enough magnitude that the uncertainty in oBf be i n s i g n i f i c a n t compared to A f m . The measurements with "He as the wall coating were done under conditions for which neither c r i t e r i o n was s a t i s f i e d and we w i l l return to these below. By the time that 3He was used as the wall coating, provisions had been made for degaussing of the magnetic shi e l d and the p r i n c i p l e of using the transverse resonances, |1> to |2> and |1> to |4>, to determine Bx, as described in Appendix A, had been established. The minimum s h i f t with 3He coated walls was found to occur at 0.234 K. Frequency versus current analysis yielded aB 2 + A f m = -0.25 Hz and the transverse resonance frequency implied B^  * 2 mGauss so that aB 2 = 0.01 Hz. The determination o f A f m was thus large l y i n s e n s i t i v e to uncertainties in B A . The frequency s h i f t s for H confined by 3He walls were thus translated so as to 1 23 be consistent with A f m = - 0 . 2 4 + / - 0 . 0 3 Hz. The measurements with "He as the wall coating were performed within a magnetic s h i e l d but without a means to degauss the s h i e l d . Furthermore, the transverse resonance had not yet been recognized as a useful means of determining B^  . Fortunately, the hyperfine s h i f t for H atoms interacting with 'He atoms in the bulk gas was known from the 1K experiments described in chapter IV. It was thus possible to determine the minimum s h i f t i t e r a t i v e l y by estimating a minimum s h i f t and using i t to adjust the other s h i f t s which were then plotted as ln(Acj/T) versus T~1 . By f i t t i n g the re s u l t s to equation [ 1 1 1 - 3 7 ] , Eg/kT A C J = cosAA e V a surface s h i f t , cos= 27rAs and binding energy. Eg, were extracted and then used to calculate the contribution to the s h i f t , at the minimum, from the adsorbed atoms. With t h i s and the known saturated 'He vapour density at the temperature of the minimum and the known bulk pressure s h i f t , the t o t a l s h i f t at the minimum was calculated and used to readjust the observed frequency s h i f t s . These were again plotted and the binding energy and surface s h i f t re-evaluated. The result for H on 'He i s shown in figure 19. The frequency s h i f t has a minimum value 1 3 Figure 19 Plot of the hyperfine frequency s h i f t for H on "He as logAf/T" vs 1/T in order to exhibit the exponential dependence for small <t>0. The theo r e t i c a l l i n e (As=49kHz and EB=1.15K) i s curved s l i g h t l y at the lower temperatures due to breakdown of the small <f>0 approximation. Figure 20 The data of figure 19 for H on "He walls replotted as Af vs T. The s o l i d l i n e i s a f i t using As=49kHz, Eg=1.15K, and the bulk pressure s h i f t measured above 1K. The dashed l i n e shows the contribution from the bulk pressure s h i f t . ( K - « ) Figure 21 The frequency s h i f t in a 3He coated container plotted as log(Af/T) vs 1/T. The s o l i d l i n e corresponds to As=23kHz and ED=0.43K. 127 of -0.335 Hz at T=0.6K. The best f i t to the data i s obtained for Eg= 1.15K and A s = 49 kHz. The s o l i d l i n e i s drawn using the complete expression for the frequency s h i f t , equation [V-2a], and the s t i c k i n g p r o b a b i l i t y , the determination of which i s described below. In figure 20, the same data, along with some measurements above the temperature of the minimum, are plotted as f ~ f Q versus T. The dashed l i n e shows the contribution from the bulk pressure s h i f t . The r e s u l t s with 3He wall coating, adjusted to be consistent with -0.24 Hz as the minimum frequency s h i f t , are shown in figure 21. In t h i s p l o t , the s o l i d l i n e corresponds to Eg = 0.43K and A g = 23 kHz. 128 5.4 Measurement of Sticking P r o b a b i l i t i e s from the Magnetic Resonance Frequency and Lineshape At very low temperatures, the slope of In Aa>/T versus T" 1 deviates from Eg because <J>0 i s no longer small. It i s then possible, in p r i n c i p l e , to extract information about Pgff) from the experimentally observed S ( t ) . As a f i r s t step, we assume that C J s and Eg are temperature independent. This allows us to extrapolate <t>0(7) to the lower temperatures and avoids having both <f>0 (T) and Pg(r) unknown. Direct application of equation [V-17] to the f i t t i n g of S(t) i s c l e a r l y inconvenient and we have chosen to calculate S(t) using a t r i a l function for Pg( T) which i s then adjusted so as to f i t the data. One i s greatly aided in t h i s task by the following observation. As the temperature i s lowered, the average surface residency time, and thus <t>Q, increases proportionally with [T~ 1 e ^ B ^ j . For large 0 O, equation [V-8] implies that the only contribution to S(t) i s from the n=0 term so that iw Qt [V-19] lim S(t) = S Q e p(0,t) T*0 This i s p a r t i c u l a r l y s i g n i f i c a n t in l i g h t of equations [V-11] and [V-12] the l a t t e r of which, for t>0, can be rewritten 129 t [V-20aJ p c ( t ) = 1 - / p (r) dr Equation [V-11] then becomes f t + T [V-20b] p(0,t) = 1_ J dr [1 - / p R ( r ' ) dr'] <Tg> O O ° Equation [V-20b] can be d i f f e r e n t i a t e d twice to y i e l d [V-21] d j _ [p(0,t)] = _ L _ p D ( r ) d t 2 <T g> y The importance of equation [V-21] l i e s in the fact that i t allows one to estimate both the form and the parameters of Pg(r) simply by taking the second derivative of the envelope of the free induction decay under conditions where S(t) i s proportional to p(0,t) (low temperatures). In practice, the short decay times and poor signal to noise r a t i o associated with the large 4>0 signals lead to substantial uncertainty in the determination of the parameters when equation [V-21] i s used alone. Consequently, the parameters were refined by comparison of the simulated and experimental signals at intermediate values of 4>0. This procedure i s discussed in more d e t a i l below. 130 5.4a Sticking P r o b a b i l i t y for H on "He Coated Walls It was found that for the lowest temperature to which the H on l i q u i d "He studies were extended, the free induction decay displayed an exponential envelope with the implication that Poisson s t a t i s t i c s were s u f f i c i e n t to describe the si t u a t i o n for "He walls. Accordingly, Aco and 1/Tp were analysed using equations [V-2a] and [V-2b]. The data in figure 22 have been f i t t e d taking Eg=1.15 K, ws = 2ir x 49 x 103 s e c - 1 and <TQ> = 1.6 x 10"3 T- V* sec. The value <Tc> = 7.4 x 10"5 T" 1 / 2 was obtained from computer simulation of the motion of the atoms in our c e l l using the assumption that, following each c o l l i s i o n with a wall, the atom was reemitted according to the equilibrium v e l o c i t y d i s t r i b u t i o n . With these results we obtain s = 0.046+/-0.004. (The T" V* dependence of <rg> and <Tc > i s included to allow for the T ^ dependence of the thermal v e l o c i t y . ) The smaller value for s reported in Ref.(24) was the result of an incorrect i d e n t i f i c a t i o n of the bias magnetic f i e l d used in the measurement of the fi v e lowest temperature points of figure 22. i Figure 22 The frequency s h i f t and 1/T2 for a l i q u i d "He coated container plotted vs 1/T. The s o l i d l i n e s are calculated from equations [V-4a] and [V-4b] using Eg=1.l5K, and L>S = 2J T X49X 10 3sec _ 1, as determined from figure 19, and a s t i c k i n g p r o b a b i l i t y of 0.046. 132 5.4b Sticking Probability for H on 3He Coated Walls For H confined by l i q u i d 3He walls, the res u l t s are more complicated. At the lowest temperatures, we fi n d that the envelope of the free induction decay resembles the sum of two exponentials. Unfortunately, the signal i s l o s t in the noise soon a f t e r the second exponential has become apparent. This has forced us to carry out the refinement of the parameters for Pg(r) by comparison of calculated and experimental FID's at higher temperature. Before considering the f i t t i n g of equation [V-17] to the actual H on 3He data, i t i s he l p f u l to consider some of the general features of S(t) when P g ( r ) i s of the form -ar -br [V-22] Pgd") = A e + Be where, again, the thermal v e l o c i t y i s assumed to contribute a TV 2 dependence to A, B, a, and b. For convenience, we define CO [V-23] r " 1 = < T b > = J r P g U ) dr o so that for P Q ( T ) given by equation [V-22], f 1=(A/a 2)+(B/b 2). Equation [V-22] implies 133 [V-24 ] 7r (x ) = i ( A + B ) x + i b a [(x + i a ) ( x + i b ) s o t h a t , f r o m e q u a t i o n [ V - 1 7 ] , [V-25 ] S ( t ) = S 0 e i co 0 t CD f - i x t dx e T I T J -co ( x - i e ) ( x + i ( a + b - A - B ) ) L i - i 0 o i xz + ix •A+B _ a+b-- -1 -100 . m i 0 o a b 1- i0 c Our p r o b l e m i s t h e n t o s o l v e t h e q u a d r a t i c e q u a t i o n i n t h e d e n o m i n a t o r a n d c a l c u l a t e r e s i d u e s f o r t h e two r o o t s w h i c h we l a b e l and x _ . We g e t some i n s i g h t i n t o t h e n a t u r e o f t h e p o l e s o f e q u a t i o n [V-25 ] i f we c o n s i d e r t h e l i m i t i n g c a s e o f P o i s s o n s t a t i s t i c s by l e t t i n g a , b , and A+B t e n d t o r . The two r o o t s a r e t h e n g i v e n by x . = 0 - i r w h i c h g i v e s a r e s i d u e e q u a l t o z e r o , and - 0 9 r *«. = 1-i0o w h i c h g i v e s t h e u s u a l r e s u l t f o r A C J and 1/Tp f o r P o i s s o n s t a t i s t i c s . As p D ( r ) becomes l e s s ' P o i s s o n - l i k e ' , we f i n d t h a t P t h e g e n e r a l n a t u r e o f t h e s e r o o t s p e r s i s t s . In p a r t i c u l a r , f o r a > b and A > B , we f i n d 134 x . * 0 - i b and f o r s m a l l 4>0 -r<t>g ~ ir<t>Z w i t h r t e n d i n g t o be r e p l a c e d by a a s 0 O becomes l a r g e . The s h o r t t i m e b e h a v i o u r o f S ( t ) i s a l w a y s f o u n d t o be ' P o i s s o n - 1 i k e * w i t h Aa> and 1 / T 2 g i v e n by e q u a t i o n s [ V-2a] and [ V - 2 b ] . The l o n g t i m e b e h a v i o u r o f S ( t ) i s d o m i n a t e d by w h i c h e v e r o f t h e r o o t s has t h e s m a l l e r i m a g i n a r y p a r t ; f o r s m a l l 0 O t h i s i s x „ a n d , f o r l a r g e 4>0, x . . By f a c t o r i n g S ( t ) i n t o a t i m e d e p e n d e n t a m p l i t u d e and p h a s e f a c t o r , i w Q t i * ( t ) S ( t ) = S 0 r e A ( t ) e we c a n d e s c r i b e i t s g e n e r a l b e h a v i o u r i n t e r m s o f In A ( t ) v s t and * ( t ) v s . t . When 0 o i s s m a l l , d l n ( A ( t ) ) / d t i s i n i t i a l l y - r o ^ / f 1 +4>l) and becomes more n e g a t i v e a s t i n c r e a s e s . The f r e q u e n c y s h i f t , d * ( t ) / d t , i s c o n s t a n t and g i v e n by r 0 o / ( 1+ c 4 2 ) . F o r l a r g e # 0 , on t h e o t h e r h a n d , t h e i m a g i n a r y p a r t o f x . i s s m a l l e r a n d b o t h In A ( t ) a n d * ( t ) s t a r t w i t h a ' P o i s s o n - l i k e ' s l o p e a n d t h e n go on t o b e h a v i o u r d e t e r m i n e d by x_. In t h i s l a r g e 0 O r e g i m e , d * ( t ) / d t t e n d s t o a p p r o a c h z e r o a c r o s s t h e f r e e 135 induction decay u n t i l the x_ behaviour takes over. In the intermediate <t>Q regime, where the imaginary parts of the roots are comparable, there may be variations in dlnA(t)/dt and d*(t)/dt across a given free induction decay but the mean 1/T2 and Aco w i l l be roughly constant. For the parameters used in the f i t t i n g of the H on 3He data, t h i s type of behaviour i s not as pronounced as i t might be for a case where a and b d i f f e r e d to a greater extent. In order to compare the observed signals to the calculated signals for H on 3He, i t i s necessary to analyse them in the same way. Experimentally, A C J was measured by counting zero crossings of the FID within the region where the signal f e l l from about 0.75 of i t s i n i t i a l amplitude to about 0.1 of i t s i n i t i a l amplitude. The i n i t i a l part of the signal was not used as a precaution against transient electronic e f f e c t s . The f i n a l part of the signal was unusable because of the d i f f i c u l t y of determining zero crossings as the signal amplitude approached the noise l e v e l . Applying t h i s analysis in a consistent way to the calculated signals amounted to the following procedure. The times were found for which the signal dropped to 0.75 and 0.1 of i t s i n i t i a l amplitude and then the slope of the l i n e between these two points on the * ( t ) vs t curve was taken to represent Aw. The uncertainty was estimated by drawing the chords over the i n i t i a l and f i n a l 0.75 of t h i s time i n t e r v a l . One result of t h i s type of analysis, both experimentally and for the 136 calculated signals, i s that for cases where the slope of * ( t ) vs. t i s tending to zero near the end of the measurement i n t e r v a l , the value of Aco estimated w i l l be somewhat less than the i n i t i a l value of d*(t)/dt. When comparing the observed 1/Tp to the results from equation [V-17] i t was the i n i t i a l decay rate, both experimental and calculated, which was used for l/Tp since t h i s was predicted to go as r <j>£ / ('\ + 4>£ ) for a l l values of 0 O . For small #0 , the decay was limited by radiation damping and the observed values of 1/Tp were not used for f i t t i n g purposes. A reasonable approximation to the H on 3He data was obtained with the following set of parameters : E R = 0.43K cos = 2ff x 23 x 1 0 3 sec' 1 A//T = 280 sec' 1 K- 1/2 B//T = 19 s e c 1 K" 1^ a//T = 357 sec" 1 K-b//T" = 90 sec - 1 K" V2 The experimental data and calculated Aco and 1/Tp for t h i s set of parameters are displayed in figure 23. With <*g> as calculated for t h i s d i s t r i b u t i o n and <Tc > as found from a computer simulation, we obtain s = 0.016 with an uncertainty of about 30 per cent. Figure 23 Aw and I/T2 vs 1/T for H on 3He. The dashed l i n e uses the parameters of figure 21 and a value of S=0.011 (determined from the peak value of Au>) in the model for which PQ(T) i s taken to be Poisson. The s o l i d l i n e corresponds to a f i t using more general PQ(T) obtained by f i t t i n g low temperature FID's to a sum of two exponentials. The s t i c k i n g p r o b a b i l i t y i s s=0.016(5) for t h i s set of parameters. 138 It should be noted that because r, and thus <TQ>, can be estimated d i r e c t l y from the i n i t i a l values of 1/Tp at the lowest temperature, s may be determined even i f the rest of the d i s t r i b u t i o n parameters are less well known. This i s , to some extent, the case here. While the experimental A(t) i s reasonably well approximated by the calculated function for some temperatures, there are other temperatures for which the model is less s a t i s f a c t o r y . Because i t i s the long time t a i l of the FID which i s most sensitive to the form of pg(r), i t was f e l t that answers to questions about the form and temperature dependence of th i s d i s t r i b u t i o n must await an improvement in the signal to noise r a t i o of our 1420 MHz spectrometer. Work i s currently underway to bring about t h i s improvement through i n s t a l l a t i o n of a cooled Ga-As FET pre-amp, the replacement of the continuously running 1420 MHz o s c i l l a t o r by a pulsed single side-band generator, and the replacement of the input mixer with an image rejection mixer. It may also prove desirable to analyse the frequency s h i f t data in such a way that the change in Aco across a free induction decay can be measured accurately thus removing the somewhat a r b i t r a r y assignment of a window across which the continuously changing Aco, from equation [V-17], i s averaged. 139 5.5 Comparison to Other Results It i s possible to compare some of the re s u l t s described in t h i s chapter with measurements done elsewhere on samples of H s t a b i l i z e d by large magnetic f i e l d s . Results for the binding energy and surface recombination rate for H on "He, by measurement of the H gas pressure in large f i e l d , have been reported by Matthey et a l . ( 2 l ) and Cline et al.(59). The measured rates from Ref.(21), scaled by the area to volume ra t i o s and 2e 2 have been plotted as x's in figure 17. The value for the binding energy reported for t h i s data i s Eg=0.89(7) K. It can be seen that the corresponding recombination rate i s larger for the high f i e l d measurements by a factor of 2.5 (after scaling by 2e 2). In Ref.(59), a binding energy of 1.01(6)K i s reported. This data, again scaled for f i e l d and A/V r a t i o , i s plotted as +'s in figure 17. The corresponding rate i s about 25 per cent lower than the zero f i e l d result a f t e r s c a l i n g . A number of comments can be made. The difference in the positioning of the three sets of data suggest an error in the scaling factors used to translate to zero f i e l d and consistent A/V although i t i s d i f f i c u l t to imagine any of the estimates being off by factors approaching 2. It i s important to note that while we have scaled the high f i e l d 140 results by 2e 2, the true r a t i o , in terms of k p a r a and k o r^ n o , as described in chapter I I I , i s R=4e 2(1+k p Q r a / k o r t h o )/(3 + k p a r Q / k o r t n o ) which can be rearranged to give k p a r a / k o r t h o = <3R-4e2 )/(4e 2-R) At f i r s t glance, the apparent agreement of the high and low f i e l d r e s u l t s , when scaled with 2e 2, might suggest that k p a r a / k o r t h o * s a D O u t unity. In fa c t , however, the factor of 2.5x2e2 between the x's and the low f i e l d results i s outside of the maximum r a t i o of 2x2e 2 expected when kp Q r a/k 0 (-^ n 0 i s very large. On the other hand, the r a t i o of about 0.8x2e2 for the + *s to the zero f i e l d r e s u l t s implies a k p a r Q / k o r ^ n o of about 1/3. It i s thus clear that without the a b i l i t y to almost completely eliminate uncertainties in the r e l a t i v e A/V r a t i o s , comparison of high and low f i e l d r e s u l t s are not l i k e l y to y i e l d any useful information about the kp Q r a / k o r t n o r a t i o . The differences in the reported binding energies merit somewhat more attention. In t h i s work, the binding energies reported from the recombination and frequency measurements for H on *He are in good agreement. This implies that any systematic temperature dependence of the surface recombination rate or frequency s h i f t at zero f i e l d must be small enough that the 141 influence on the binding energy measurement be n e g l i g i b l e . The s l i g h t disagreement in temperature dependence with the high f i e l d r esults does raise the p o s s i b i l i t y that the r e c a l i b r a t i o n of the germanium thermometer, c a r r i e d out after the experiments were fi n i s h e d , might s t i l l be in error. This r e c a l i b r a t i o n i s discussed in d e t a i l in Appendix F. We believe that the c a l i b r a t i o n was done c o r r e c t l y and, on the basis of comparison with a carbon resistance thermometer in place during the actual experiments, i s transferrable to the i n i t i a l measurements. Resolution of doubts about the c a l i b r a t i o n , however, w i l l l i k e l y await r e p e t i t i o n of the experiments with provisions for concurrent c a l i b r a t i o n . Setting aside questions of thermometry, one may ask what information might be contained in the d i f f e r e n t temperature dependences should they turn out to be r e a l . It i s tempting to consider the p o s s i b i l i t y that the difference might arise because of a temperature dependence of the kparo. /^or tho r a t i o . The apparent r a t i o of the rates decreases by a factor of about 2 between 0.5K and 0.2K. If the difference were due only to kpara /^or tho , i t would be necessary that t h i s r a t i o decrease with decreasing temperature. While the t h e o r e t i c a l r e s u l t s of Greben et al.(31) might not be applicable to recombination on the surface, they do imply that k p a r a / k o r t h o should increase as the temperature i s lowered. The allowed range of the r a t i o of high to low f i e l d rates goes from 4e 2 for k p a r a A o r t h o very 142 large to (4/3) e 2 for k p a r a / k o r ^ n o =0. Thus, while a change in the high to low f i e l d r a t i o by a factor of two i s possible, i t would demand a large change in k p Q r a / k o r t n o and, accordingly, the zero f i e l d cross-length. This i s inconsistent with the agreement of the binding energy as found from recombination and frequency s h i f t . It would appear, then, that at least some of the disagreement in the temperature dependence of recombination must be attributed to causes other than a temperature dependence of the k p a r a A o r t h o r a t i o . It i s also possible to compare the recombination for H on 3He with results obtained at high f i e l d . The measurments of van Yperen et al.(22) are plotted on figure 18 with the scaling being done as described above. They obtain Eg=0.34(3)K as compared to ED=0.39K used to draw the straight l i n e of figure 18. The zero f i e l d r e s u l t s for H on 3He do not f i t the model for a constant surface rate as well as for H on 4He and a precise measurement of the binding energy using frequency s h i f t measurements may allow the temperature dependence of the recombination cross-length to be extracted. We note, however, that the temperatures for t h i s data have also been recalibrated a f t e r the o r i g i n a l experiments were completed and that questions of the accuracy of the c a l i b r a t i o n , p a r t i c u l a r l y at the lowest temperatures, cannot be over-ruled. F i n a l l y , we mention the r e s u l t s of Salonen et al.(60) for the f r a c t i o n of energy transferred to the *He surface by 143 c o l l i d i n g H atoms. They fi n d that 20+/-10 per cent of the energy c a r r i e d by a b a l l s t i c pulse of H atoms i s transferred to the helium surface on c o l l i s i o n . We have found that 4.6 per cent of the c o l l i s i o n s result in the atom being stuck. The recent c a l c u l a t i o n by Zimmerman and Berlinsky(61) of the st i c k i n g p r o b a b i l i t y on the helium surface at OK has given S(E,0) = 0.055( 1 + 1 .25E)E 1/ 2cos0 where E i s the energy in K and 6 i s the angle of incidence. We can estimate the fr a c t i o n of energy transferred to the surface by s t i c k i n g events alone by averaging the product of E and S(E,0) over a Boltzmann d i s t r i b u t i o n and div i d i n g by the average energy at temperature T. We f i n d that the f r a c t i o n of energy transferred by s t i c k i n g for the temperature range 0.2K to 0.4K is about twice the fr a c t i o n of atoms s t i c k i n g . Since both fractions display the same angular dependence, the i r r a t i o i s independent of 8. It would thus appear that while i n e l a s t i c c o l l i s i o n s are l i k e l y important in the experiments of ref.(60), transfer of energy to the surface by stic k i n g c o l l i s i o n s might be of similar importance. 144 CHAPTER VI Summary Almost a l l of the experiments undertaken on atomic hydrogen in the past few years have been motivated by the hope of being able to use a weakly interacting gas of bosons as a test of theory e s p e c i a l l y with regard to Bose-Einstein condensation. Indeed, the i n i t i a l aims of the experiments described here were, primarily, to study the properties of a confined gas of atomic hydrogen, p a r t i c u l a r y i t s l i f e t i m e , as a preliminary step towards confining atomic hydrogen at high d e n s i t i e s . In t h i s regard, the importance of t h i s work i s twofold. Any experiment with high density atomic hydrogen w i l l be done in the presence of a saturated surface layer of atomic hydrogen. Knowledge of the recombination rate and of the binding energy, and thus the saturated surface density, are e s s e n t i a l i f r e a l i s t i c estimates of the l i f e t i m e and heating due to recombination are to be made. The sig n i f i c a n c e of the recombination rate measurements has been diminished somewhat by the r e a l i z a t i o n that the l i f e t i m e of samples at high density i s l i m i t e d by relaxation between the two lowest hyperfine states rather than by the recombination rate. It has become increasingly c l e a r , however, that atomic hydrogen 1 4 5 provides f e r t i l e ground for the comparison of theory and experiment even under r e l a t i v e l y e a s i l y accessible conditions of temperature and density. In t h i s respect, we have demonstrated that zero f i e l d magnetic resonance i s p a r t i c u l a r l y well suited to studies of the interactions of atomic hydrogen with helium gas, the l i q u i d surface, and i t s e l f . Accordingly, we w i l l now proceed to review and discuss the re s u l t s which have been presented here. The experiments at 1K were sensitive to interactions of the H atoms with other atoms, H or He, in the gas. Information about the H-He interatomic potential was accessible through the d i f f u s i o n cross-section, Q^=20 (1)A 2, and through the hyperfine pressure s h i f t , -11.8x10" 1 8Hz cm3. These results have been subsequently used to construct and test an empirical H-He potential ( 39 ). This pot e n t i a l has been used to calculate the temperature dependence of Q Q , ( 6 2 ) and ( 5 0 ) , and i t i s expected that below 1K, the d i f f u s i o n cross-section w i l l decrease almost l i n e a r l y with temperature. It would thus be of some interest to extend the d i f f u s i o n constant measurements to lower temperatures in "He and, for even lower temperatures, 3He as the buffer gas. Measurement of the longitudinal relaxation time, T, , gave the spin exchange cross-section, =0.43(3)A 2, at 1.1K. In p r i n c i p l e , t h i s measurement can be extended to lower temperatures although care may have to be taken to separate out the contribution due to relaxation on the surface. It would 146 also be interesting to test the relat i o n s h i p Tp =2T^  for spin exchange. This w i l l require separation of the radiation damping contribution to transverse relaxation. The measured rate constant for H+H+He+Hp+He, k=(0.20+/-0.03)xl0 3 z cm 6sec _ 1, i s in good agreement with the ca l c u l a t i o n of ref(31). In l i g h t of the strong temperature dependence predicted by theory, i t would be interesting to extend bulk recombination measurements to lower temperatures. This has been done with 3He as the buffer gas (62) giving k=0.09xl0 3 2cm 6sec _ 1 at about 0.5K. Unfortunately, the next highest temperature for which the bulk rate has been calculated in ref.(31) i s 0.1K and surface recombination i s t o t a l l y dominant for both 4He and 3He surfaces at th i s temperature. The extension of zero f i e l d measurements to lower temperatures yielded information about the interaction of H with the l i q u i d helium surface. Assuming the hyperfine frequency s h i f t for atoms on the surface and the surface recombination rates to be temperature independent, we obtained values for both of these parameters as well as p a r t i a l l y independent measurements of the binding energies. For H on "He, we found the surface s h i f t to be 49 kHz, the surface recombination cross-length to be 0.14(2)A, and the binding energy to be 1.15(5)K. Because of the thermometry d i f f i c u l t i e s discussed previously, i t would be useful to repeat these measurements with provisions made for concurrent c a l i b r a t i o n of the thermometer. 147 There i s no evidence in the H on "He data for a temperature dependent recombination cross-length. For H on 3He, the surface s h i f t i s found to be 23kHz. The recombination cross-length i s 0.13A, and the binding energy i s measured to be 0.39K from recombination and 0.43K from the hyperfine frequency s h i f t . Again, i t would be worthwhile to improve the confidence in the thermometry for these measurements. The assumption of a temperature independent surface hyperfine frequency s h i f t seems to be supported by the data but the case for a temperature independent recombination cross-length, excepting the agreement of the values for H on "He and H on 3He, i s less convincing. A better determination of the binding energy using the frequency s h i f t measurements and better thermometry would allow the temperature dependence of the cross-length to be extracted. For recombination and frequency s h i f t measurements, with both 3He and "He walls, i t was observed that the rapid drop in the saturated helium vapour pressure, coupled with the r i s e in the fractio n of atoms on the surface for decreasing temperature, resulted in a minimum frequency s h i f t and recombination rate. This should prove helpful in the construction of low temperature atomic hydrogen masers since operation at the minimum frequency should render such a device less sensitive to temperature fluctuations. F i n a l l y , the frequency s h i f t and transverse relaxation time 148 at low temperatures have been used to extract the average s t i c k i n g p r o b a b i l i t i e s . For H on "He, i t was found that s=0.046(5) at 0.2K and, for H on 3He, s=0.016(5) at 0.09K. 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Thesis, U.B.C., (1982) 62. W. N. Hardy, M. Morrow, R. Jochemsen, and A. J . Berlinsky, Physica J_09 and 1 10B , 1964, ( 1982) 63. R. A. Waldron, Theory of Electromagnetic Waves , Van Nostrand Reinhold Company, Toronto, (1969) 64. B. Shizgal, J . Phys. B_1_2 , 361 1 , (1979) 65. R. J . Soulen and R. B. Dove, NBS Special Publication 260-62, Washington, (1979) 66. W. R. Abel, A. C. Anderson, and J . C. Wheatley, Rev. S c i . Instru. 35 , 444, (1964) 154 APPENDIX A Magnetic Resonance at the Hyperfine Frequency A.1 Theoretical Background As discussed in Chapter II, we may treat atomic hydrogen in zero f i e l d as a two l e v e l system. If the density matrix for the two l e v e l system i s expanded into terms linear in the Pauli matrices, the c o e f f i c i e n t s of each component of a can be treated as components of the expectation value of a f i c t i t i o u s spin S'=a'/2. (With exceptions to be noted below, we w i l l use primes to denote operators within the manifold of states |1> and |3>.) The form of the density matrix i s then [A-1 ] p = J_ + <!*'>• a' 2 This analogy of a f i c t i t i o u s spin 1/2 system for a two l e v e l system i s well known(27). The Hamiltonian operating in the space of the two leve l s can also be expanded into terms lin e a r in the components of S'. By treating the c o e f f i c i e n t s of SjJ , S y , and S z as the components of f^'B' , where 7' i s a f i c t i t i o u s gyromagnetic r a t i o , the Hamiltonian can be made 1 5 5 a n a l o g o u s w i t h t h a t f o r t h e i n t e r a c t i o n o f t h e f i c t i t i o u s s p i n S ' w i t h t h e f i c t i t i o u s f i e l d B' t o g i v e V = J E Q - 7 ' f t ( B ' - S ' ) 2 A f i c t i t i o u s m a g n e t i c moment , m' =^17' t r a c e ( pS ' ) c a n t h e n be shown , u s i n g t h e a b o v e e x p r e s s i o n f o r p a n d H , t o have an e q u a t i o n o f m o t i o n o f t h e u s u a l f o r m [A-2] dm'= 7 ' m ' x B' d t To draw t h e c o n n e c t i o n be tween t h e a c t u a l s p i n s a n d f i e l d s and t h o s e i n t h e f i c t i t i o u s s y s t e m , we w i l l c o n s i d e r t h e a p p l i c a t i o n o f a s m a l l s t a t i c f i e l d , B o z , i n t h e z d i r e c t i o n and an o s c i l l a t i n g f i e l d , 2 B ) Z c o s o > t , i n t h e same d i r e c t i o n . (The f a c t o r 2 i s i n c l u d e d i n a n t i c i p a t i o n o f r e s o l v i n g t h e r . f . f i e l d i n t o two r o t a t i n g c o m p o n e n t s e a c h o f a m p l i t u d e B, z , o n l y one o f w h i c h w i l l be i m p o r t a n t i n t h e r o t a t i n g f r a m e ) . The H a m i l t o n i a n o f e q u a t i o n [ 1 1 — 13 becomes 156 [A-3 ] H= a(T-r> - f i B o z ( 7 p I z - 7 e S z ) - 2 f t B e c o s w t ( 7 p I 2 - 7 e S z ) I f we t r e a t t h e o s c i l l a t i n g f i e l d a s a s m a l l p e r t u r b a t i o n , we c a n t a k e t h e m a t r i x e l e m e n t s o f H i n t h e s u b s p a c e o f s t a t e s | l > a n d |3> ( s t i l l t a k i n g t a n 26 t o be a s d e f i n e d a b o v e ) . E x p a n d i n g t h e r e s u l t i n g H a m i l t o n i a n i n t e r m s o f t h e p r e v i o u s l y d e f i n e d c o m p o n e n t s o f S ' y i e l d s H = ~-| ~ ( a + | ( h B o z ( 7 e * Tp) ) 2 ) S Z - 2 B | 2 ft c o s w t ( 7 e + 7 p ) ( c o s 2 0 - s i n 2 0 ) S z - 4 B , z f t coscot s i n e c o s e ( 7 e + 7 p ) S x w h i c h , f o r s m a l l f i e l d s s u c h t h a t cos6ssin#-1 //2 " , becomes K = - A " ( a + | ( ^ B o z ( 7 e + 7p) ) 2 ) S Z ~ 2 < 7 e + 7p ) f i B | Z c o s w t S x T h e i d e n t i f i c a t i o n w i t h t h e p a r a m e t e r s o f t h e f i c t i t i o u s s p i n 1/2 s y s t e m t h e n l e a d s t o [ A - 4 ] ft7'Bz = a + a ^ f t B o z ( 7 e + 7p> ) 2 w h i c h i s t o be c o m p a r e d t o e q u a t i o n [ I I -4 ] a n d 1 57 [A-5] 7'B' = 2 ( 7 e + 7 ) B ( z coscot The application of a f i e l d in the x or y d i r e c t i o n has no ef f e c t to f i r s t order in B x or By but does enter in second order through the tipping of the quantization axis. We can now see how to do a pulsed magnetic resonance experiment on atomic hydrogen. If we assume that the system i s in equilibrium before the r.f. pulse, then the i n i t i a l density matrix i s [A-6] p(0) = / p, (0) O N * 0 p 3(0)J where p, and p 3 are the f r a c t i o n a l populations of the states |1> and |3> respectively. Recalling that, in the f i c t i t i o u s system, m'= 7'fiTrace(pS*' ), one has that the i n i t i a l z magnetization i s [A-7] M 0 = n^'-f^p, (0) - P T ( Q ) N , where n^ i s the atom density. By analogy with the usual spin 1/2 system, we consider the transformation to the frame rotating with angular frequency [A-8] co = a. + a / ft B o z ( 7 e + 7 D ) \ 2 ft 2h ^ a K — / 158 In t h i s frame, the o s c i l l a t i n g f i e l d can be resolved into two components of amplitude Bx/2 one of which i s stationary in the rotating frame. The f i c t i t i o u s magnetization w i l l rotate about the axis of t h i s f i e l d with angular v e l o c i t y 7'Bx/2 = ( 7 e + 7 p ) B | z . The appl i c a t i o n , therefore, of the o s c i l l a t i n g f i e l d 2B | Z cos(cot) along the real z axis for an in t e r v a l given by [A-9] t = I » r w i l l rotate the f i c t i t i o u s magnetization into the f i c t i t i o u s x-y plane. In terms of the density matrix, t h i s implies that, after t h i s so-called TT/2 pulse, we have P 1 2 U/2) = P 2 1 U/2) = P 1 ( 0 ) - p^(O) Transforming back to the non-rotating f i c t i t i o u s frame, we thus f i n d , immediately after the ir/2 pulse, [A-10] M x = sin(cot) n H7'fi ( Pi<0) ~ P^(0) ) To relate t h i s to an observable s i g n a l , we note that the real magnetization along the z axis, 159 [A-1 1 ] M z = n H f i ( 7 p I z - 7 e S z ) has the representation n^fi ( 7 e + 7 p ) S x in terms of the f i c t i t i o u s spins. With the system i n i t i a l l y in equilibrium, a high temperature approximation gives -a/kT [A-12] P l ( 0 ) - p 3(0) = 1 - e -a/kT 4kT 1 + 3e so that the o s c i l l a t i n g longitudinal magnetization af t e r a TT/2 pulse i s [A-13] M z(t) ^  n Hfi ( 7 e + 7p) a sin(cjt) 8kT This w i l l induce a voltage in a c o i l aligned along the z di r e c t i o n that i s proportional to dM z(t)/dt. The f i c t i t i o u s spin analogy also allows one to characterize the relaxtion times for M' in terms of the phenomenological parameters of the Bloch equations, [A-14a] dM7 = M'p - M'z + 7' (M'x B' ) 7 dt T, [A-14b] dMj= 7' (M'x B' ) v ~ MV dt T2 [A-14c] dMy= 7' (M'x B' ) y - M^ y dt T 2 160 A c c o r d i n g l y , 1/T^ i s i d e n t i f i e d w i t h t h e r a t e f o r t h e d e n s i t y m a t r i x t o r e l a x t o i t s e q u i l i b r i u m v a l u e ( s e e e q u a t i o n [ A - 7 ] ) , a n d 1/Tp i s r e l a t e d t o t h e r a t e a t w h i c h t h e o s c i l l a t i n g m a g n e t i z a t i o n a l o n g t h e z a x i s d i s a p p e a r s . E x c e p t f o r t h e f a c t t h a t t h e l o n g i t u d i n a l r e l a x a t i o n i s n o t a c c o m p a n i e d by t h e r e c o v e r y o f a s t a t i c m a g n e t i z a t i o n , t h i s i s c o m p l e t e l y a n a l o g o u s t o t h e p h e n o m e n o l o g i c a l i n t e r p r e t a t i o n o f t h e u s u a l s p i n 1/2 r e s o n a n c e . I t c a n be s e e n , t h e r e f o r e , t h a t , e x c e p t f o r t h e s e c o n d o r d e r d e p e n d e n c e o f t h e f r e q u e n c y on t h e s t a t i c f i e l d and t h e f a c t t h a t t h e o s c i l l a t i n g f i e l d i s a p p l i e d a l o n g t h e s t a t i c f i e l d r a t h e r t h a n p e r p e n d i c u l a r t o i t , t h e s y s t e m b e h a v e s i d e n t i c a l l y t o t h e s t a n d a r d m a g n e t i c r e s o n a n c e s y s t e m . In p a r t i c u l a r , a l l o f t h e r e s u l t s p e r t a i n i n g t o s i g n a l s t r e n g t h , s p i n - e c h o e s , r a d i a t i o n d a m p i n g , e t c . c a n be t a k e n o v e r d i r e c t l y f r o m e x i s t i n g l i t e r a t u r e . F o r c o m p l e t e n e s s , we i n c l u d e t h e a n a l o g o u s r e s u l t s f o r t h e n e a r b y t r a n s i t i o n s |1> t o |2> and |1> t o | 4 > . We c o n s i d e r t h e c a s e o f a s m a l l s t a t i c f i e l d i n t h e z d i r e c t i o n a n d a s m a l l f i e l d , w h i c h may be o s c i l l a t i n g , i n t h e x d i r e c t i o n . In t h e m a n i f o l d o f s t a t e s |1> a n d |2> t h e h a m i l t o n i a n becomes 161 H = Za -M7 e-7 D)B Z + (-a+fi(7 e-7 D)B z)S z - fi ( 7 P + 7P ) By SI where the primed operators operate in the manifold of states |1> and |2> so that [A-I5a] 7'B Z = a - ( 7 e ~ 7 p ) B 2 •fi 2 and [A-I5b] 7'B' = ( 7 n + 7») B x The magnetization in the x d i r e c t i o n , n ^ t i ( 7 p l x - 7 e S x ) has the representation n^fi(7 p + 7 e )S x//2~. The hamiltonian may s i m i l a r l y be written in the manifold of states |1> and |4> giving [A-16a] 7'B Z = a + ( 7e ~ 7 p ) B z f i 2 [A-16b] 7'B' = - ( T o + T f i J BX with the representation of the magnetization in the x di r e c t i o n being - n H h ( 7 p + 7 e ) S x / / 2 . A f i e l d o s c i l l a t i n g perpendicular to a small s t a t i c f i e l d , B z, can thus leave components of magnetization precessing about the s t a t i c f i e l d with angular frequencies 162 [A-17] a «. (7 e - 7n) B z -h " 2— K" In the present experiment, t h i s e f f e c t i s observed when the ambient longitudinal f i e l d i s nulled by the applied bias f i e l d leaving a residual s t a t i c f i e l d perpendicular to the resonator. C a l l i n g t h i s f i e l d B z, the o s c i l l a t i n g f i e l d , B x, may be applied with the resonator. Normally, these t r a n s i t i o n s are excited simultaneously with approximately equal amplitude. If the detection frequency i s close to the zero f i e l d t r a n s i t i o n frequency a/ft, one observes a single beat frequency of magnitude ((7 e -7p)/2)B z where B z i s the residual transverse s t a t i c f i e l d . This e f f e c t has been used as the basis of one method of measuring the residual magnetic f i e l d s in magnetic shielding used in t h i s experiment. 163 A.2 Measurement of Diffusion with Zero F i e l d Hyperfine Resonance The dependence of spin echo amplitudes on the d i f f u s i o n constant for pulsed magnetic resonance in a gradient i s well known(48). Slichter(28) treats t h i s for the usual case for which to = 7B Z. We b r i e f l y review these re s u l t s to provide a background for our discussion of how to deal with the quadratic f i e l d dependence of co in the current experiments. When d i f f u s i o n i s present in a uniform f i e l d , Bloch's equations must be supplemented by d i f f u s i o n terms. With the usual notation, M+ = M ^ iMy» where Mx'and My'are components of the magnetization in the rotating frame, one has [A-18] 3M» Cr,t) = -M+Cr,t) + DV 2M +0r,t) at T 2 where D i s the d i f f u s i o n constant. If the f i e l d i s inhomogeneous, so that B z = B o z+ z d B z / a z , the spreading term, - i 7 ( B z -B^jM* Or,t), i s included to give [A-19] 8M+ Cr ,t) = -i7zdB2M + Cr ,t) - M* Cr ,t) + DV 2M +0r,t) a t d z T 2 From t h i s , the amplitude of an echo at time 2T following a ir/2 pulse at t = 0 and a rr pulse at t = r can be shown to be 164 - 2 T / T 2 - ( 2 / 3 ) D [ 7 a B z / a 2 ] 2 r 3 [A-20] M + ( 2 T ) = - M ( 0 ) e e As discussed above, the f i e l d dependence of the (F=0,MFr=0 •» F=1,M F=0) t r a n s i t i o n for H in low f i e l d s i s u = a/h + 2jraB z 2 where a = 2773.05 Hz/Gauss 2. In the f i c t i t i o u s spin 1/2 formalism discussed previously, one can write the equation of motion for the f i c t i t i o u s magnetic moment as [A-21] 8M' = 7 1 M' X B' a t where 7'BZ = a/h + 2iraB*. For the inhomogeneous f i e l d B z = B o z+z3B z/az, we can write, to f i r s t order in the gradient, 7'B Z = a / f i + 27ra(B02z + 2B Q Z z3B z/8z ) . The transformation to the rotating frame in the f i c t i t i o u s system, with d i f f u s i o n present, then gives [A-22] 3M1 * = i47raB z e B z - M ' + + DV2 M ' + dt 02 air T 2 One i s then able to carry the usual spin echo result over d i r e c t l y i f we i d e n t i f y an e f f e c t i v e gyromagnetic r a t i o yeff = 4 7 r a B o z a n d u s e i t f o r 7 equation [A-20]. Note that t h i s approximation i s v a l i d only for B o z » zdB z/dz. 165 APPENDIX B Absolute Determination of the Magnetization Using Magnetic Resonance In the experiments reported here, the determination of the atom densities r e l i e s d i r e c t l y on a detailed knowledge of the microwave c i r c u i t and i t s interaction with the spin system. This section deals with the equivalent c i r c u i t of the cavity and an expression i s obtained for the power incident on the input amplifier in terms of the sample magnetization and the f i l l i n g factor, T J , appropriate to the present s i t u a t i o n . A convenient equivalent c i r c u i t that corresponds rather c l o s e l y to the experimental s i t u a t i o n of a r e f l e c t i o n cavity i s shown in figure 24. The impedance looking into the transformer, Z L, i s [B-1] Z L = n 2 [ r + j(a>L - 1/coC)] where n i s the turns r a t i o of the transformer. If Z j_ i s terminating a transmission l i n e of c h a r a c t e r i s t i c impedance Z Q, then the voltage r e f l e c t i o n c o e f f i c i e n t i s 1 6 6 n 1 L z; coupl i ng C c a v i t y o r lumped constant r e s o n a t o r Figure 24 The LC equivalent to the resonant cavity c i r c u i t . 167 [B-2] p = Z L ~ Z 0 z L + Z 0 and the f r a c t i o n a l r e f l e c t e d power i s P r/Pj=|p 2| where Pr and Pj are the ref l e c t e d and incident power respectively. Under conditions of c r i t i c a l coupling, P r/P ( goes to zero on resonance. It can thus be seen that, for the equivalent c i r c u i t , c r i t i c a l coupling corresponds to the case [B-3] n 2 = Z Q / r Defining C J c = (LC)" ^ , Q 0=to cL/r, and Q c=n 2 c o cL / Z 0 , and using Z[_ as above, the f r a c t i o n a l r e f l e c t e d power for u a u c can be written — ( — ~ — *\ + ( u ~ N 2 4 I Qo Qc{ I cur J 1_/J_ + j_ ^2 +/co - coc ^ 2 " 4 \ Q 0 Q C ) K coc ) The f u l l width at half maximum of the resonance in r e f l e c t e d power i s given by [B-5] Aco = u c ( Q 0 1 + Q c 1 ) = "c/Q-L Q|_, referred to as the loaded Q of the system, i s what is normally measured in practice. One sees that the condition n 2 = Z Q / r corresponds to Q c , the "coupling Q" of the c i r c u i t being equal to Q Q f the unloaded Q of the resonator so that, at [B-4] Pj = 168 c r i t i c a l c o u p l i n g , Q[_=Q 0 /2. The f o r m o f e q u a t i o n [B-4] i s , i n f a c t , g e n e r a l l y a p p l i c a b l e t o any r e s o n a n t c i r c u i t f o r w h i c h t h e i n c i d e n t and r e f l e c t e d power a r e c o u p l e d i n and o u t o f t h e r e s o n a t o r v i a t h e same c o u p l i n g c i r c u i t . I t t h u s p r o v i d e s an i d e n t i f i c a t i o n be tween t h e l o a d e d a n d u n l o a d e d Q ' s o f t h e a c t u a l m i c r o w a v e c i r c u i t a n d t h e s i m p l e c o m p o n e n t s o f an e q u i v a l e n t c i r c u i t . In p a r t i c u l a r , i t i s c o n v e n i e n t t o t r e a t t h e i n d u c t o r o f t h e e q u i v a l e n t c i r c u i t a s a l o n g s o l e n o i d s u c h t h a t t h e i n d u c t a n c e i s L ~ M 0 N 2 A / 1 a n d t h e f i e l d i n t h e s o l e n o i d i s B =<z 0 i N / 1 where N i s t h e number o f t u r n s , A i s t h e a r e a , a n d 1 i s t h e l e n g t h o f t h e s o l e n o i d . By u s i n g t h i s s i m p l e a n a l o g y t o o u r r e s o n a n t c i r c u i t i t i s p o s s i b l e t o c h e c k t h e g e n e r a l r e l a t i o n s h i p s , d i s c u s s e d b e l o w , be tween t h e f i e l d s i n t h e r e s o n a t o r a n d t h e i n c i d e n t , r e f l e c t e d , a n d e x t r a c t e d p o w e r . C o n s i d e r t h e s i g n a l power r e a c h i n g t h e a m p l i f i e r a f t e r t h e s a m p l e e q u i l i b r i u m h a s been d i s t u r b e d by t h e a p p l i c a t i o n o f an r . f . p u l s e . F o r t h e c u r r e n t s y s t e m o f i n t e r e s t i t was d e m o n s t r a t e d i n t h e p r e v i o u s s e c t i o n t h a t i m m e d i a t e l y f o l l o w i n g 169 a 7r/2 pulse, an o s c i l l a t i n g magnetization i s set up along the resonator axis. In a standard magnetic resonance system, the 7r/2 pulse results in a rotating magnetization which can be decomposed into two o s c i l l a t i n g components one of which can couple to the r . f . c o i l . By considering only t h i s component, we can draw a direc t analogy with the longitudinal s i t u a t i o n . With some care, then, the following arguments may be applied to either case. We are dealing with an imposed o s c i l l a t i n g l o c a l magnetization and an r . f . f i e l d which we take to be the steady state response of the resonator in the si t u a t i o n where the response time of the resonator i s much shorter than the decay of the s i g n a l . This response f i e l d i s to be thought of as o s c i l l a t i n g primarily in the d i r e c t i o n of the resonator ax i s . The l o c a l o s c i l l a t i n g magnetization w i l l have, in addition to the expected sinusoidal time dependence, a time dependence due to e f f e c t s such as d i f f u s i o n . Accordingly, i t i s l a b e l l e d as MCr,t)sin ( c o t ). If the resonator i s tuned to frequency to, the current response w i l l be tt/2 out of phase with the o s c i l l a t i n g magnetization. The attendant response f i e l d i s then written as B r (r", t )cos (cot) and the instantaneous power extracted from the spin system i s [B-6] P = j B r ( r , t ) cos (tot) • d_[M(r* ,t)sin(tot) ] dV dt 170 where the integral i s over the sample volume V s. Neglecting the contributions to dM/dt from the nonsinusoidal time dependance and averaging over one cycle, we have [B-7] P = C J fdV s [ I r (r\t)'M(r*,t) ] 2 J Noting that t h i s must be equal to the t o t a l power dissipated in the combined c i r c u i t r y (using SI units) [B-8] Pc = oj EstoreH= &> rB r 2 dV where the int e g r a l i s over a l l space, we have [B-9] J [ B r ( r \ t ) - M ( ? , t ) ] dV s = _L_ fBr 2 dV We can use equation [B-9] to rewrite [B-7] as [B-10] P = <JU^Q|[/dV«; Br(r.t)'M(r.t) I 2 2 / B r 2 dV Under conditions of c r i t i c a l coupling, half of t h i s power i s dissipated in the resonator and the other half i s absorbed by the amp l i f i e r . C a l l i n g the l a t t e r power the signal power, P s, we have 171 [B-11] P S = CJMOQI t/dVs Br ( r . t ) - M ( ? . t ) 1 2 4 | B r 2 dV At t h i s p o i n t , i t i s necessary to d i s c u s s the meaning of the time dependence i n equation [B-11], We are a c t u a l l y i n t e r e s t e d i n t h i s expression p r i m a r i l y as a means of r e l a t i n g the i n i t i a l s i g n a l power to an average i n i t i a l magnetization M D which, as suggested by equation [11-17], we can then r e l a t e to the atom d e n s i t y through M 0 = n^ft(-y e + 7pHp-|( n) - p^(0)]/2. I t i s thus c l e a r that the i n t e g r a l s must be evaluated at a time, t ' , a p p r opriate to the measurement being done. By d e f i n i n g a f i l l i n g f a c t o r 77 as [B-12] rj = r / d V s M ^ t ' )'Br ( r , f ) I 2 M o 2 V sJ|B r ( r , f ) | 2 dV and r e w r i t i n g equation [B-11] as [B-13] P S = COMQQLMQ 2 V ST? 4 the question i s reduced to one of f i n d i n g the f i l l i n g f a c t o r a ppropriate to various experimental s i t u a t i o n s . The simplest case a r i s e s i f the resonator i s a very long s o l e n o i d . B r i s then everywhere uniform and, f o l l o w i n g a JT/2 p u l s e , the l o c a l magnetization w i l l be everywhere equal to M 0. Equation [B-12] then becomes 77=VS/Vc where V c i s the volume of 172 the long solenoid. An inhomogeneous B r may involve inhomogeneities both in the magnitude and d i r e c t i o n of the f i e l d . If the magnitude i s inhomogeneous, there w i l l be a d i s t r i b u t i o n of angles 8(r), close to T T/2, through which the magnetization (real or f i c t i t i o u s depending on the type of resonance) has been rotated so that the l o c a l magnetization w i l l be M osin(0(r)) where sin(0 ( r ) ) i s close to a turning point. If the inhomogeneity i s everywhere small enough, the component of the l o c a l magnetization in the d i r e c t i o n of Br(r*) may be taken as MQ. We w i l l l i m i t our consideration to such cases. In fact, t h i s i s not a p a r t i c u l a r l y severe constraint on |Br| and i t seems to be s a t i s f i e d in our s i t u a t i o n . Inhmogeneities, however, do r e s t r i c t the accuracy with which the f i l l i n g factor may be estimated and, in Appendix D, we discuss the extent to which the f i l l i n g factor determination for the current experiments might be influenced by an inhomogeneous |Br|. Whether or not inhomogeneities in the d i r e c t i o n of Br0r) enter depends on whether the time scale of the measurements i s fast or slow compared to d i f f u s i o n of the spins. If the measurement involves a short time, such as i s the case for the determination of the v/2 pulse length by maximization of the i n i t i a l signal amplitude, the l o c a l magnetization w i l l be able to maintain i t s phase r e l a t i o n s h i p with the l o c a l f i e l d and we have Br(r)«M0r) = MQ|Br |. Equation [B-12] then becomes 173 2dV On the other hand, the determination of the density from the signal amplitude or radiation damping time under most conditions in the present experiments w i l l involve time scales long compared to the time for d i f f u s i o n to s p a t i a l l y average the magnetization. After a short transient period, the magnetization w i l l be s p a t i a l l y uniform with M Q x j s = ( 1/VS )JM q x| s dV where M Q X j s i s the component of M along the axis of the resonator and the integral i s c a r r i e d out immediately after the TT/2 pulse. Assuming a symmetrical response f i e l d , there w i l l be no component of magnetization other than along the axis and equation [B-12] w i l l take the form [B-15] 77" = Woloi2 [ / Brn*ic;dV ] 2 v s M o 2 / l B r l 2 d V which leads to an e f f e c t i v e reduction in signal strength. We can construct a p r a c t i c a l expression for the f i l l i n g factor by making use of the fact that the response f i e l d of the resonator, B r(r*), i s everywhere proportional to the applied r . f . f i e l d of the ir/2 pulse. In the discussion to follow, we take the amplitude of the o s c i l l a t i n g applied f i e l d to be 2B|Z. The arguments are d i r e c t l y applicable to both the usual magnetic 174 resonance sit u a t i o n and the longitudinal resonance. Because of the shortness of the 7r/2 pulse, there is es s e n t i a l l y no d i f f u s i o n during i t s a p p l i c a t i o n . Thus, while the inhomogeneity in the d i r e c t i o n of the applied r . f . f i e l d influences the value of the maximum power observed a f t e r the ir/2 pulse, i t does not enter into the determination of the length of the 7r/2 pulse. If the maximum signal i s obtained by varying the pulse length to obtain then i t i s the magnitude of the applied r . f . f i e l d which i s probed. The functional form of the |Br| dependence of the apparent i s discussed in Appendix D. If |Br| i s not too inhomogeneous, the r e l a t i o n s h i p can be written [B-16] 2B | Z « where BJ^ = (1 /Vs )J | BiZ | dV s. This i s , in fa c t , the type of average which enters in equation [B -14] for T J ' . If the power incident on the c r i t i c a l l y coupled cavity during the r . f . pulse is Pj , then we also have, since the amplitude of the o s c i l l a t i n g r . f . f i e l d i s 2 B | Z , [ B-17] f[2B, Z ] 2 dV = 2 M OQOP J ) CO Recognizing the proportionality of B R and B | Z , we can write 175 equation [B-14] in terms of equations [ B-16] and [ B-17] to give [B-18] T J ' = coVs / ir \2 2 M 0 Q 0 P J V ( 7 P + 7 P > V 72 A point that i s not so obvious at f i r s t glance i s that i t i s the unloaded q u a l i t y factor, QQ = 2Q^ , that enters in equation [ B - 1 7 ] . This i s because a l l of the incident power i s dissipated in the cavity under matched conditions. This has been confirmed by di r e c t measurement of B l z in a lower frequency resonator of the type used in these experiments. 176 APPENDIX C Radiation Damping Spin systems obeying the Bloch equations in the absence of cavity coupling can be described by four coupled equations; the usual three Bloch equations plus a fourth which couples the f i e l d of the cavity to the spin system(29). In t h i s treatment, we w i l l consider the usual case where the axis of the r . f . f i e l d i s in the x d i r e c t i o n . Again, the important results w i l l be d i r e c t l y applicable to the longitudinal resonance case. The fourth equation takes the form ( r e f e r r i n g to the equivalent c i r c u i t of figure 24) [C-1] -7?"u^d2Mv = d 2B* + r' dB x + 1_BX d t 2 d t 2 ^ L dt LC where the prime on r' i s to c a l l attention to the fact that i t must include the losses of the external c i r c u i t (r'=2r at c r i t i c a l coupling). The f i l l i n g factor, T J " , i s given by equation [B-15]. Because radiation damping manifests i t s e l f in the decay of the s i g n a l , the time scale i s considerably longer than that associated with determining the i n i t i a l signal amplitude. We had previously chosen between 17" and rj' on the 177 basis of whether or not d i f f u s i o n within the resonator was s i g n i f i c a n t within the time scale of the measurement. In treating radiation damping, we are discussing time scales over which d i f f u s i o n out of the resonator altogether can be s i g n i f i c a n t i f the sample c e l l i s longer than the resonator. Accordingly, measurements of radiation damping time constants had to be done with c e l l s in which the volume outside of the resonator was minimized by extending only a narrow t a i l into the discharge region. The d e t a i l s of such measurements w i l l be discussed below. With d i f f u s i o n out of the resonator r e s t r i c t e d , the same f i l l i n g factor, T J " , i s appropriate to measurements over any time scale longer than the c h a r a c t e r i s t i c time for d i f f u s i o n within the resonator. In p a r t i c u l a r , i t i s 77" which enters in the determination of the atom density by both absolute c a l i b r a t i o n of the i n i t i a l signal strength and by measurement of the radiation damping time. In general, the solutions to the three Bloch equations plus equation [C-1] w i l l be very complicated i f the c i r c u i t damping time constant i s comparable to the damping time of the spin system. For the 1420MHz resonator, the c i r c u i t damping time, Q/jrf, i s about 0.2 jxsec. which i s much shorter than any of the resonance damping times. One i s thus j u s t i f i e d in taking the steady state solution of equation [C-1] which, using the complex notation M x(t) = M re- | u , t i s 178 [C-2] B x ( t ) = Re(K M x(t)) where K = T ? " = K R + iK, [u'LCl-'-l-ir'/wL and K R = Re(K) and Kj = Im(K). Bloom(30) has studied the solution of the Bloch equations with the added constraint of equation [C-2] for conditions of exact c i r c u i t resonance, and obtained soutions in closed form when , the longitudinal relaxation time, was i n f i n i t e . Here, these results are generalized for a r b i t r a r y cavity tuning. Following the application of a ir/2 pulse, there exists a rotating magnetization M = M rcos(cot)x - Mjsin(cot)y. The ringing f i e l d , B x, i s related to Mx v i a equation [C-2] which can be written Bx= KpMrcos(cot) + Kj Mr sin (cot). B x can be decomposed into components rotating and counter-rotating r e l a t i v e to M. Only the component rotating in the same sense as M can be important in radiation damping and t h i s component i s given by [C-3] B4= Kt|Mr (cos(cot)x - sin(cot)y) + K^Mr (sin (cot) x + cos (cot)y) It i s now convenient and conventional to transform to the frame 179 rotating with angular v e l o c i t y co0 = 7 B 0 * T ^ e a x e s °^ t h i s frame are denoted x' f y', z and we define co' = co - coQ. In the rotating frame we thus have [C-4a] Mx< (t) = Mrcos(o>'t) [C-4b] My (t) = -M rsin(u't) [C-4c] B + X'(t) = Mj (K Rcos(w't) + KjSin(aj't)) = J_ (KjM x' (t) - KRM y» (t) ) [C-4d] B + y'(t) = M £(-K Rsin(cj't) + Kjcos(w't)) = J_ (KjM x. (t) + KpMy» (t)) The Bloch equations in the rotating frame are then e a s i l y shown to be, for T, =co, [C-5a] dM i = j L K , M r 2 dt 2 [C-5b] dMx' = zMx: - 7MZ_(KTM x' + KRMy- ) dT~ T 2 [C-5c] dMv'= + T MZ _ ( K R M x ' - KjMy' ) « T2 2 But from [C-4a] and [C-4b], we see d i r e c t l y that, [C-6a] dMy' = J_ dMr My + to'My' dt Mr dt 180 [C-6b] d M Y ' = J _ CiMj- M y ' - co' M X ' dt M r dt Defining A 1_ dMj- + J_ + x M z K l M r d t T 2 2 and B to' + 2. M z K r 2 equations [C-5b], [C-5c], [C-6a], and [C-6b] can be combined to give two equations A M x . + B M y , = 0 and A My» - B M x / = 0 which have n o n - t r i v i a l solutions only i f A=B=0, i e , Equations [C-7a] and [C-7b] are e f f e c t i v e l y those used by Bloom(30) to describe the free precession except that his coupling constant k has been replaced by ( 1 / 2 )K J . Equation [C-7b] gives the frequency p u l l i n g e f f e c t [C-7a] 1 d M r = -1 - 7 M 7 K r [C-7b] M r d t T 2 2 co' = -7 M Z K R 181 [ C - 8 ] co - u_ = r)"yO, u rM T(t) Sco/Aco L 1 + [26co/Aco] 2 where Aw i s the f u l l width of the loaded cavity resonance (Q|_= 1 / ( / L C AC O ) » coL/r') and 8co = co - 1//LC i s the mistuning of the cavit y . One can also write out equation [C-7a] to obtain the instantaneous damping c o e f f i c i e n t X(t) = (-l/Mr)dMr/dT, [ C - 9 ] X ( t ) = M 0 T ? " 7 Q l M z ( t ) 1 + _ J _ -TT 1 + [26co/Aco] 2 T 2 so that for <= co t [ C - 1 0 ] co - co0 = X(t) ^ (Aco/2) This says that the frequency p u l l i n g i s just the damping rate times the f r a c t i o n a l mistuning. (To convert equations [ C - 8 ] to [ C - 1 0 ] to cgs units, u0 has to be replaced by 1 / 4 T T . ) In the experiments c a r r i e d out at IK, the transverse relaxation rate i s normally dominated by spin exchange and i s therefore proportional to n^. As a r e s u l t , the r a t i o of the radiation damping (also proportional to n^) to transverse relaxation i s independent of n^. The numerical value of the r a t i o , which depends on the experimental arrangement through the factor 7?"QL , was about 5 for those experiments. 182 The use of the radiation damping time as an independent test of the absolute c a l i b r a t i o n of the spectrometer s e n s i t i v i t y was based on Bloom's solution to equations analogous to [C-5a] and [C-7a] with the assumption T^ = 0 0 . From t h i s solution, one finds that the signal voltage after a it/2 pulse should be proportional to ( l / 7 r ) s e c h ( ( t - t M ) / T ) where T i s the assymptotic long time decay time and t m i s given by 2 t m A [C-11 ] e = T 2 - T T 2 + T for a TT/2 pulse. If T 2 = ° ° and 6to=0, the assymptotic time constant, , is due only to radiation damping and given by T^, 1 =M 0»?"7QL MO/ 2' Equation [ C - 1 1 ] i s used to separate T 2 from r by noting that an extrapolation of the assymptotic decay back to the time of the rr/2 pulse w i l l give a voltage intercept greater than the actual i n i t i a l signal by 2 t * n A 1 + e For Tj =00, the i n i t i a l condition on M r can be shown to give T 2=T 2 2 + T^2. However, since T^  = T 2 /2 for spin exchange in low f i e l d s , the condition Tj = co i s not a very good approximation. For f i n i t e T^  , the Bloch equations give \ 183 lim M z(t) = M 0 t * CD so that equation [C-9], for 6u> set to zero, suggests [C-12] J _ = ^ Q T ? " T Q LM Q " 1 - 1 TA 2 r T 2 Numerical integration of equation [C-7a] and the f i n i t e Tj analogue of equation [C-5a], [C-13] dM2 = 7 K TM r 2 + M2 - M Q ar~ 2 1 — r p ^ were used to produce model free induction decays with known values of M0, Kj , and the appropriate Tj and T j expected for spin exchange. These models were then analysed using the assumption of equation [C-12] and compared to the results using 1/T 2 = 1/Tj + Equation [C-12] was indeed found to y i e l d a s l i g h t l y better approximation to the value of M c used to generate the model free induction decay. Equations [C-11] and [C-12] were used to obtain M Q for actual data from the 1K measurements in which d i f f u s i o n out of the resonator was l i m i t e d by sample geometry thus allowing T J " of equation [C-12] to be i d e n t i f i e d with the f i l l i n g factor of equation [11-19]. 184 APPENDIX D The F i l l i n g Factor The expression for the f i l l i n g factor used in t h i s work i s given in equation [11-23]. In the s p l i t ring resonator used here, the e l e c t r i c and magnetic f i e l d s are expected to be separated in space with the e l e c t r i c f i e l d s largely confined to the gap. Because of the small skin depth at 1 GHz, the magnetic f i e l d i s expected to be uniform and p a r a l l e l to the axis within the resonator and small outside of the resonator. These considerations would lead one to expect that the f i l l i n g factor should reduce to the r a t i o of the sample volume, V s, to the resonator volume, V c. We w i l l refer to t h i s r a t i o as the geometric f i l l i n g factor. If the magnetic f i e l d i s largely confined uniformly within the diameter of the resonator then one would expect the f i e l d p r o f i l e to be well approximated by the s t a t i c f i e l d from a c y l i n d r i c a l current sheet with the a x i a l current d i s t r i b u t i o n adjusted so as to set the transverse f i e l d to zero at the resonator wall in approximation of the skin depth e f f e c t . We have used a program which approximates the 3.2 cm long resonator 185 a s 321 c u r r e n t l o o p s . The d i s t r i b u t i o n o f c u r r e n t i n t h e l o o p s i s a d j u s t e d t o m i n i m i z e t h e t r a n s v e r s e f i e l d a t t h e w a l l o f t h e r e s o n a t o r and u s e d t o c a l c u l a t e t h e f i e l d i n a n d a r o u n d t h e r e s o n a t o r . T a k i n g a s a m p l e v o l u m e c o m p a r a b l e t o t h e e x p e r i m e n t a l c e l l , we c a n c a l c u l a t e a q u a s i - s t a t i c f i l l i n g f a c t o r f r o m e q u a t i o n [11-21 ] . The r e s u l t i s a b o u t 95 p e r c e n t o f t h e g e o m e t r i c f i l l i n g f a c t o r . When e q u a t i o n [ 11-23 ] i s u s e d t o c a l c u l a t e t h e a c t u a l f i l l i n g f a c t o r f r o m Q 0 , Pj , a n d t h e TT/2 p u l s e l e n g t h , we o b t a i n a r e s u l t w h i c h i s a b o u t 1/3 o f t h e g e o m e t r i c f i l l i n g f a c t o r f o r t h e 1K e x p e r i m e n t and 0 .42 t i m e s t h e g e o m e t r i c f i l l i n g f a c t o r f o r t h e l o w e r t e m p e r a t u r e e x p e r i m e n t s . I t seems c l e a r t h a t t h e f i l l i n g f a c t o r a t 1.4GHz w i t h t h e r e a l r e s o n a t o r , i n c l u d i n g t h e c a p a c i t i v e g a p , i s n o t w e l l a p p r o x i m a t e d by t h e q u a s i - s t a t i c p i c t u r e . A l l o f t h e r e l a t i o n s h i p s , be tween t h e f i e l d s a n d p o w e r , u s e d t o d e r i v e e q u a t i o n [ 11-23 ] h a v e been v e r i f i e d by a n a l o g y t o t h e c a s e o f a l o n g s o l e n o i d r e s o n a t o r . I t was f e l t , h o w e v e r , t h a t an i n d e p e n d e n t v e r i f i c a t i o n o f t h e s u r p r i s i n g l y low e x p e r i m e n t a l f i l l i n g f a c t o r was n e e d e d . A f i r s t a t t e m p t a t m e a s u r i n g t h e m a g n e t i c f i e l d s i n t h e r e s o n a t o r was c a r r i e d o u t w i t h a 3 . 7 5 cm d i a m e t e r by 10 cm l o n g s p l i t r i n g r e s o n a t o r moun ted i n s i d e o f an 8 . 6 cm d i a m e t e r by 21 cm l o n g a l u m i n u m t u b e . Power was m a g n e t i c a l l y c o u p l e d i n t o t h e r e s o n a t o r by a f o u r t u r n l o o p . T h e c o u p l i n g was a d j u s t e d by c h a n g i n g t h e l o o p t o r e s o n a t o r d i s t a n c e . T h e f i e l d s i n t h e 186 resonator were probed in one of two ways. The homogeneity of the magnetic f i e l d could be studied by introducing a lossy f e r r i t e bead and observing the change in Q as a function of p o s i t i o n . A l t e r n a t i v e l y , a small loop, with i t s axis p a r a l l e l to the resonator axis, was used to d i r e c t l y measure the o s c i l l a t i n g magnetic f i e l d in the resonator. It was found that at frequencies above a few hundred MHz, the perturbation of the resonance by the probes rendered measurements unreliable. With a wrapping of insulated metal f o i l to increase the gap capacitance and lower the resonator frequency to about 100 MHz, the e f f e c t of the f e r r i t e on the resonance was lar g e l y uniform throughout the resonator. The implication i s that at t h i s frequency, at least, the magnetic f i e l d s in the resonator are uniform. By measuring the power extracted by the small loop, i t i s possible to infer the magnitude of the longitudinal magnetic f i e l d . These measurements tended to confirm the r e l a t i o n s h i p of equation [B-17] with corrections made for the power coupled out by the probe. This agreement i s s i g n i f i c a n t in that i t confirms the use of Q Q rather than Qj_ in equation [B-17]. Because the f i e l d s are uniform and confined to the resonator at t h i s frequency, t h i s i s also a demonstration that the f i l l i n g factor i s approximately equal to the geometric f i l l i n g factor under these conditions. This measurement was repeated at 300 MHz. Assuming equation [B-17] to have been proven, we took the i n t e g r a l of the squared f i e l d to be proportional to the incident 187 power as in equation [B-17] and measured the a x i a l f i e l d in the resonator d i r e c t l y . This gave a f i l l i n g factor that was about 75 per cent of the geometric f i l l i n g factor. Comparison of these results with the f i l l i n g factor measured at 1420 MHz suggests that at higher frequencies the magnetic f i e l d might not be as well confined in the resonator. Measurement of the f i e l d s by cavity perturbation was used as a means to study t h i s hypothesis. The application of perturbation theory to microwave c a v i t i e s i s discussed by Waldron(63). Employing the well known depolarizing factor, N e, and demagnetizing factor, N m, i t i s possible to derive the expression for the f r a c t i o n a l frequency s h i f t of a cavity of volume Vc perturbed by a body of d i e l e c t r i c constant e, pe r m i t t i v i t y u, and volume V s to be J/fe0E20 dV s 6CJ = -(c-1) ~ co 1 + N e(e-1) J J j ( E 0 - D 0 - H 0-B 0) dV /JJnoH% dVs 1 + N m U " 1 ) ///(E 0.D 0 - H 0-B 0) dV where the subscript o labels f i e l d s in the unperturbed c a v i t y . We define a reduced magnetic f i l l i n g factor to be 188 V s / j / (E Q. D 0 - H0- B 0) dV and a reduced e l e c t r i c f i l l i n g factor to be V s /// (E0. D Q - H0' B 0) dV If the unperturbed f i e l d s are constant over the region of a small sample, then the f i l l i n g factor of equation [B-17] i s seen to be well approximated by i? m rV s/V c . The demagnetizing and depolarizing factors for a sphere are both 1/3. For a metal sample i t i s appropriate to use u=0 and e=-jco so that for a metal sphere of volume V s m we have ?M = ZA %rIsm - _3_ T j m rVsrji co 2 r V c 4 V c One sees that in order to extract the magnetic f i l l i n g factor from a cavity perturbation measurement, i t i s necessary to separate out the contribution to the s h i f t from perturbation of the e l e c t r i c f i e l d s . This can be done by measuring the perturbation with a glass sphere of d i e l e c t r i c constant e, v = \ , and volume V 5 r l for which we have 189 is? = - U - O Her. Ysg CJ 1 + N e(e-1 ) 2 V c For samples of similar s i z e , r j e r i s the same for a glass or metal sphere. These expressions were tested using a 1420 MHz cavity operating in the TM 0 1 0 mode for which the f i e l d s are known. The samples were placed on a styrofoam support at the magnetic f i e l d maximum. Unfortunately, the e l e c t r i c f i e l d i s changing quickly with r a d i a l distance so that the s h i f t i s very sensitive to po s i t i o n . The measured s h i f t s for a glass and a metal sphere did, however, f a l l within the range predicted for the estimated uncertainty in the pos i t i o n and d i e l e c t r i c constant of the glass with the implication that the expressions for the s h i f t are correct. The measurements of the reduced f i l l i n g factors for the actual d i l u t i o n r e f r i g e r a t o r resonator and housing were done with a pyrex sphere of volume 0.042 cm3 and a metal sphere of volume 0.056 cm3. The volume of the resonator i s 10.74 cm3. The samples were introduced into the resonator on a s l i d i n g styrofoam support oriented so as to minimally perturb the e l e c t r i c f i e l d s near the gap. The d i e l e c t r i c constant of the glass sphere was taken to be 4.9+/-0.2. For t h i s resonator, fortunately, the reduced e l e c t r i c f i l l i n g factor was found to be 190 small enough that the uncertainty in e contributed only a small error in T J _ , . •mr A single measurement consisted of recording the resonance several times while c y c l i n g the metal sphere, glass sphere, and empty holder into the resonator in order to check for r e p r o d u c i b i l i t y of p o s i t i o n . With the spheres at the center of the c a vity, t j e r was found to be about 0.09 giving T j m r to be about 0.54+/-0.01. When the spheres were moved to a position midway between the center and end of the resonator, i ? m r was lowered to about 0.4. For the work below IK, the f i l l i n g factor calculated from the ir/2 pulse was about 0.42 of the geometric f i l l i n g factor implying an average of T j m r over the sample of 0.42. While the values of r ? m r measured by cavity perturbation were not th i s small, the fact that T ) m r i s dropping with distance from the center, along the axis, i s suggestive. At any rate, the measurements do not seem to admit the p o s s i b i l i t y of the average r j m r being as large as 0.84 as would be the case i f , for example, the r i g ht hand side of equation [B-17] were a factor of 2 too large. One must ask, at t h i s point, to what extent these measurements d i s c r e d i t our i n i t i a l requirement that |Br| be s u f f i c i e n t l y homogeneous that the component of l o c a l magnetization along B r ( r ) be approximately M Q immediately following the TT/2 pulse. We have addressed t h i s question by 191 assuming a model inhomogeneous d i s t r i b u t i o n for |Br| and then c a l c u l a t i n g the exact f i l l i n g factor from equation [11-20] and comparing the result with the f i l l i n g factors calculated using equations [11-21 ] and [ 1 1 - 2 3 ] , We have chosen to model the inhomogeneity in |B*r| as a quadratic decrease in magnitude with distance along the axis from the center. We work to f i r s t order in the c o e f f i c i e n t of the quadratic term. It i s found that even for a |Br| which i s twice as large at the center as at the ends of the resonator, the f i l l i n g factor of equation [11-21 ] i s smaller than the exact f i l l i n g factor by only about 10 per cent providing some j u s t i f i c a t i o n for our o r i g i n a l assertion that immediately following the rr/2 pulse, MCr, t' ) • B r ("r, t' ) may be replaced by M0|Br|. This r e l a t i v e i n s e n s i t i v i t y to inhomogeneities in |Br| does not, however, extend to our determination of the JT/2 pulse length and the f i l l i n g factor of equation [11-23] derived from i t . We normally estimate t^. by adjusting the pulse length to maximize the amplitude of the signal immediately following the pulse. When one analyses the problem more c l o s e l y , one finds that rather than being determined by the average magnitude of the o s c i l l a t i n g f i e l d , as in [B-16], the apparent tVl i s proportional to / | i f r | 2dV //|§*r | 3dV . For the case of |B*r| decreasing quadratically such that i t s value at the ends of the resonator i s 7 5 per cent of that at the center, we f i n d that 192 equation [11-23] underestimates the true f i l l i n g factor by about 6 per cent. If we increase the inhomogeneity such that |B r| at the center i s twice |B r| at the ends, the discrepancy in the f i l l i n g factor r i s e s to 26 per cent. If the response f i e l d of the resonator i s inhomogeneous, then, there seems to be a danger of underestimating the f i l l i n g factor and thus overestimating the atom density. It i s interesting to determine what the implications of t h i s model would be i f a l l of the difference between the measured f i l l i n g factors were due to a quadratic v a r i a t i o n of |B r|. For T? m r=0.54 as measured by cavity perturbation at the center of the resonator and Tj m r=0.42 for the whole sample volume as estimated from the jr/2 pulse length, the model requires that |B r| at the center be greater than at the ends by about 30 per cent. When t h i s f i e l d p r o f i l e i s used, the f i l l i n g factor estimated from the T/2 pulse length i s smaller than the exact f i l l i n g factor by about 7 per cent. The atom density would be overestimated by about 4 per cent. It would appear that while we may be observing e f f e c t s due to an inhomogeneous |Br|, the consequences for our determination of the atom density are not too severe. F i n a l l y , i t i s of interest to re-examine the comparison of the atom de n s i t i e s , as obtained from the signal strength and radiation damping, using the exact f i l l i n g factor estimated by t h i s model. It i s found that these measurements s t i l l agree to within about 15 per cent. The remaining 193 discrepancy i s in the d i r e c t i o n that would be expected i f we were s l i g h t l y overestimating the f i l l i n g factor by using T J ' in place of T J " . In the absence of more complete knowledge of the response f i e l d s in the resonator, we have chosen to use the f i l l i n g factors as given in Chapter II with the expectation that any systematic error introduced into our estimate of n^ by inhomogeneities w i l l not be unreasonably large. 194 APPENDIX E Spin Exchange in Two Dimensions In section 3.2, the derivation of the spin exchange cross-sections for H-H c o l l i s i o n s in the gas was discussed. This treatment can e a s i l y be extended to c o l l i s i o n s confined to two dimensions. In three dimensions, expressions for Tj" 1 , T^ 1 , and Lv may be written in terms of cross-sections which are sums over angular momentum states of functions of the difference of p a r t i a l wave s h i f t s for scattering from the singlet and t r i p l e t H p o t e n t i a l s . The analogous expressions for c o l l i s i o n s confined to two dimensions were obtained by solving the two dimensional scattering problem and using the resu l t to derive the equation of motion for the density matrix. The derivation follows c l o s e l y that of reference (33). By suitable d e f i n i t i o n of the two dimensional analogues of the cross-sections and atom dens i t i e s , i t i s possible to a r r i v e at the same expression for the rate of change of the density matrix. The expressions for the frequency s h i f t and relaxation rates from ref.(32) can then be used d i r e c t l y with substitution of the appropriate two 195 dimensional analogues as follows: [E-1a] _J_ = vs n s (o* + lv)/2 T1s [E-lb] _1_ = 1 1 T 2 s 2 Tj s [E-1c] 2irAvs = -Vg n s X* (63 - 6^/4 where the subscript s labels two dimensional qua n t i t i e s . Here, v s =/7rkT/m i s the average r e l a t i v e speed of two atoms of mass m moving fr e e l y on the surface, n s i s the surface density in atoms/cm2, and ( 6 3 - 6 ^ ) i s the difference in the f r a c t i o n a l populations of the F=1,Mp=0 and F=0,Mp=0 hyperfine states. The cross-sections 0 *, a", and X*, to be discussed below, are thermally averaged two dimensional analogues of the cross-sections calculated in Ref.(32). The derivation of the cross-sections for the two dimensional case departs from the treatment of Ref.(33) only in the form of the S matrix. A treatment of scattering such as that found in Merzbacher (34) can be applied in two dimensions to show that an incident plane wave, L _ 1e ,' <o' r , i s scattered asymptotically to 196 ik0« r ikr [E-2a] (r) = J_ [e +e f u(c6) ] ° L / F K where co im<6 2 i r? [E-2b] f.(tf) = I 1 e (e - 1) m=-co /27rik <6 i s the angle between k~o and k where k = k 0 r and T ? m i s the phase s h i f t for the mth p a r t i a l wave. The i n i t i a l and f i n a l density matrices, p and p' respectively, are related through the S matrix v i a p' = SpS* where S i s of the form for spinless p a r t i c l e s . We obtain S for scattering in two dimensions by r e l a t i n g the T matrix, tE-4] T j - - = J ( -ik«r e V(r) * V (r) d 2 r to the scattering amplitude using a Green's function analysis of the scattering problem as given, for example, by Merzbacher(34). The r e s u l t , d i f f e r i n g s l i g h t l y from that for three dimensions, i s 197 i 7 r / 4 r [E-5] T r r = i e / 2 fi" Trk £u(<t>) k>ko / L " u2 where u Is the reduced mass and the wave function i s normalized on a square of side L. As in Ref.(33), the e f f e c t on the S matrix of including spin i s to replace t^(<j>) in equation [E-5] by the sum of scattering amplitude times the projection operator for each of the possible interactions, singlet and t r i p l e t . I n d i s t i n g u i s h a b i l i t y of the H atoms i s taken into account by including in the density matrix an operator to exchange H atoms. The remainder of the derivation i s unchanged and res u l t s in an equation of motion of the form given in Table X of Ref.(33) with the bulk density and cross-sections replaced by the surface density and cross-sections which, when reformulated in the notation of Ref.(32), are [E-6a] ol co m t s [E-6b] X- = J _ Z [1 t (-1) ] s i n [ 2 ( 7 ? m - 7 ? m ) ] k m=-oo The phase s h i f t s , rj^ and r^, are calculated for scattering, by the t r i p l e t and singlet H-H potentials respectively, of atoms confined to move in two dimensions. The thermal averages of o o m t s = 1 Z [ 1 t (-1) ] s i n 2 (r? m - n m > k m = - c o 198 these cross-sections for motion on the surface take the form f -E/kgT [E-7] c7 = 7T-V2 [k T ] " V 2 j 2E1^2 o(E) e dE o where kg i s the Boltzmann constant. The calculations of the phase s h i f t s were performed using the program of Berlinsky and Shizgal (32)(64) modified to calculate wavefunctions in two dimensions. The singlet p o t e n t i a l , the same as that used in Ref.(32), was based on the 1975 Kolos and Wolniewicz potential(35a). The t r i p l e t potential was taken from Ref.(35b) as suggested by the note added in proof of Ref.(32). The energy dependence for 0<E<10 meV of o~, a s +, and X* i s shown in Fig.25. The major contribution to o's i s from the m=5, v=14 p a r t i a l wave which exhibits a resonance around 3 meV. The results of thermally averaging the cross-sections for 0<T<10K are given in Table IV and Figure 26. 199 Figure 25 Energy dependence of surface spin exchange cross sections Og a n d as« The contribution of to 0 5 i s shown by the dashed curve. Figure 26 Temperature dependence of thermally averaged surface spin exchange cross sections. 201 T A B L E I V T h e r m a l l y A v e r a g e d S p i n E x c h a n g e C r o s s S e c t i o n s T(K) o+(A°) o-(A') \+(A°) T(K) o+(A°) o-(A') X+(A°) 0.05 0.084 0.000 4.764 3.500 0.044 0.014 0.791 0.06 0.082 0.000 4.526 3.75 0.044 0.016 0.727 0.07 0.080 0.000 4.333 4.00 0.043 0.018 0.665 0.08 0.079 0.000 4.173 4.25 0.043 0.022 0.606 0.09 0.078 0.001 4.036 4.50 0.043 0.025 0.550 0.100 0.077 0.001 3.918 4.75 0.043 0.030 0.495 0.120 0.076 0.001 3.721 5.00 0.044 0.036 0.442 0.140 0.075 0.001 3.561 5.25 0.044 0.042 0.390 0.160 0.074 0.001 3.427 5.50 0.045 0.049 0.340 0.180 0.073 0.001 3.311 5.75 0.045 0.057 0.291 0.200 0.072 0.001 3.210 6.00 0.046 0.066 0.243 0.225 0.072 0.002 3.099 6.25 0.047 0.075 0.197 0.250 0.071 0.002 3.001 6.50 0.048 0.086 0.151 0.300 0.070 0.002 2.835 6.75 0.049 0.097 0.107 0.350 0.069 0.003 2.698 7.00 0.051 0.108 0.063 0.400 0.068 0.003 2.581 7.25 0.052 0.120 0.021 0.5 0.066 0.004 2.389 7.500 0.054 0.133 -0.021 0.750 0.062 0.006 2.057 7.75 0.055 0.146 -0.062 1.000 0.059 0.007 1.828 8.00 0.057 0.160 -0.102 1.250 0.056 0.008 1.651 8.25 0.059 0.173 -0.141 1.500 0.054 0.009 1.507 8.50 0.061 0.188 -0.179 1.75 0.052 0.009 1.383 8.75 0.063 0.202 -0.217 2.0 0.050 0.010 1.274 9.00 0.066 0.216 -0.254 2.25 0.048 0.010 1.177 9.25 0.068 0.231 -0.290 2.50 0.047 0.010 1.088 9.50 0.071 0.246 -0.326 2.75 0.046 0.011 1.006 9.75 0.073 0.261 -0.360 3.00 0.045 0.012 0.930 10.00 0.076 0.276 -0.394 3.25 0.045 0.013 0.859 202 APPENDIX F The Thermometer Recalibration The main thermometer for a l l of the d i l u t i o n r e f r i g e r a t o r experiments was a germanium resistance thermometer. It was mounted in the superfluid bath on the sintered post which provides thermal contact between the bath and the mixing chamber. Unfortunately, at the conclusion of the experiments, the thermometer c a l i b r a t i o n , as supplied by the manufacturer, was found to be in substantial error. The r e c a l i b r a t i o n was ultimately c a r r i e d out by using an NBS-SRM 768 superconducting fixed point device (65) to c a l i b r a t e a Cerium Magnesium Nitrate magnetic thermometer. The c a l i b r a t i o n was then transferred to the germanium resistance and the temperatures for the experiments reassigned. The fixed point device consists of a mutual inductance with two pairs of primary and secondary c o i l s . The primary c o i l s are connected in seri e s . The secondary c o i l s are in series but with opposing sense with respect to the primaries in order to minimize the t o t a l mutual inductance of the device. The f i v e superconducting samples are situated in one or the other secondary so as to displace the mutual inductance in alternate 203 directions as the t r a n s i t i o n s are encountered in sequence. The samples and corresponding t r a n s i t i o n s are: Auln 2 at 205.74 mK, AuAl 2 at 160 mK, Ir at 99.02 mK, Be at 23.01 mK, and W at about 16 mK. The detection c i r c u i t i s e s s e n t i a l l y that given in Ref.(65) and consists of a " l o c k - i n " amplifier and 0.22mH reference mutual inductance at room temperature. The primaries of the thermometer and reference mutual inductances are connected in series with a 10 kohm r e s i s t o r and the reference output of the loc k - i n . The device and reference secondaries are also in series. The tap of a 100 ohm potentiometer, in p a r a l l e l with the reference secondary, connects to the input of the phase sensitive detector. This c i r c u i t i s i l l u s t r a t e d in figure 27. In practice, the device was operated with an input current of 29M A, set by the reference l e v e l and the 10 kohm r e s i s t o r , providing 0.46M T peak to peak at the samples. This f i e l d i s prescribed by the manufacturer. The 100 ohm potentiometer provides adjustment for the n u l l of the mutual inductance. Degaussing of the room temperature magnetic s h i e l d provided the required reduction in the ambient f i e l d to below 1 jxT. The fixed point device was mounted below the the mixing chamber on a gold-plated post in good thermal contact with the mixing chamber. The magnetic thermometer was based on the design by Abel et al.(66). The primary c o i l i s wrapped over a former supporting 204 10 kO STANDARD MUTUAL INDUCTANCE PRIMARY REFERENCE OUT LOCK-IN AMPLIFIER SIGNAL IN FIXED POINT DEVICE PRIMARY SECONDARY The fixed c i r c u i t . Figure point 27 device detection 205 two secondary c o i l s wound in opposition. The mutual inductance between the primary and one of the secondaries i s of the order of 10 mH. The powdered CMN i s sealed within one of the secondaries. The mutual inductance i s measured using an SHE model RBU Precision Low Level A.C. Impedance Bridge operated at 160 Hz. The bridge outputs a signal to the thermometer primary in series with a 10 kohm r e s i s t o r . The secondary of the thermometer i s connected in series with the secondary of a 0.2 mH standard mutual inductance and the input of a PAR 190 10:1000 transformer. The output of the transformer goes to a PAR type A preamp and HR-8 lock-in a m p l i f i e r . The primary of the standard inductance i s connected to the n u l l i n g output of the bridge. In practice, the signal to the lock-in i s nulled by adjustment of the bridge settings which can then be read to give a number proportional to the mutual inductance and thus the s u s c e p t i b i l i t y of the CMN. The thermometer i t s e l f i s mounted in the superfluid bath below the mixing chamber in place of the sealed c e l l of figure 15. Using the four highest fixed points, the mutual inductance was f i t to the form m=A+B/(T-T0) where A, B, and T 0 are parameters of the f i t . With the germanium thermometer positioned as for the experiments, the CMN thermometer was used to compare germanium resistance and temperature. This 206 c a l i b r a t i o n was then checked by comparison of the germanium resistance to the fixed point device. This procedure was repeated with the germanium r e s i s t o r in good thermal contact with the fixed point device mount. These procedures d i f f e r e d in result by about 3 mK at 63 mK and by about 2 mK at 80 mK. They were in good agreement at 90 mK. The f i n a l c a l i b r a t i o n was taken to be the average of these two with the error given by about half of the difference. This error i s displayed in the lowest temperature points of figure 18. It was determined that the c a l i b r a t i o n of the germanium had not changed over the course of the H on "He and H on 3He experiments by comparing the rel a t i o n s h i p of the germanium thermometer to a carbon r e s i s t o r mounted on the mixing chamber during a l l of the experiments. The new c a l i b r a t i o n was thus used to recalibr a t e a l l of the results obtained below 1K Following t h i s c a l i b r a t i o n , the germanium thermometer was returned to the manufacturer for r e c a l i b r a t i o n . Their revised c a l i b r a t i o n i s in good agreement with our c a l i b r a t i o n down to about 87 mK. Below t h i s temperature i t would appear that our resistance bridge i s introducing some heating due to excessive noise on the drive s i g n a l . At any rate, the experiment and c a l i b r a t i o n , having been performed under the same conditions, were taken to be consistent. 

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