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Studies of spin-polarized hydrogen and deuterium at temperatures below 1 K using E.S.R. Shinkoda, Ichiro 1990

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STUDIES OF SPIN-POLARIZED H Y D R O G E N A N D D E U T E R I U M A T T E M P E R A T U R E S B E L O W 1 K U S I N G E.S.R. By Ichiro Shinkoda B.Sc, University of British Columbia, 1981 M.Sc, University of British Columbia, 1984 A T H E S I S S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E REQUIREMENTS FOR T H E DEGREE O F D O C T O R OF PHILOSOPHY  in T H E FACULTY OF GRADUATE  STUDIES  PHYSICS  We accept this thesis as conforming to the required standard  T H E U N I V E R S I T Y O F BRITISH  COLUMBIA  July 1990 © Ichiro Shinkoda , 1990  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  Physics The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5  Date:  Abstract  In this thesis we describe the results of two sets of experiments on spin-polarized atomic hydrogen at temperatures below 1 K using a 115 GHz heterodyne ESR spectrometer. First, we have made measurements of the rates of the two-body recombination processes in spin-polarized atomic hydrogen gas and spin-polarized atomic deuterium gas in a 41 kG field. In the second set of experiments, we examined the ESR absorption line associated with the spin-polarized atomic hydrogen atoms adsorbed on surfaces of liquid helium. In hydrogen, a comparision of the measured recombination rates in 41 kG with the corresponding rates measured in different magnetic fields showed that an additional recombination process is activate for the conditions of these experiments. We demonstrate that this is due to a resonant recombination process via the (i>,J) = (14,4) level of molecular hydrogen. In spin-polarized deuterium, we found that the recombination rate are much larger than the analogous rates in hydrogen, and therefore that atomic deuterium gas is much less stable than the hydrogen system under comparable conditions. We report the first observation and study of doubly spin-polarized deuterium, a gas in which both the deuteron and electron spins are aligned. At temperatures near 0.1 K, the absorption line of doubly spin-polarized H atoms changes drastically as a peak appears on both sides of the peak associated with the bulk atoms. We have shown that these new peaks are due to the atoms which are adsorbed onto the surface of the liquid helium film that coats the walls of the microwave cavity. The geometry dependent non-zero average magnetic dipolar field due to aligned spins on a planar surface results in a shift of the position of the ESR absorption lines. The ii  lineshapes of the side peaks are very unusual and are best described as ramp-like. Even after extensive attempts to explain this lineshape, we still do not know what mechanism is responsible.  iii  Table of Contents  Abstract  ii  List of Tables  viii  List of Figures  ix  Acknowledgement  xi  1 Introduction  2  1  1.1  Introduction  1  1.2  Atomic Hydrogen  2  1.3  Atomic Deuterium  4  1.4  ESR Absorption Lineshape of Adsorbed Atomic Hydrogen  5  Spin-Polarized Atomic Hydrogen and Deuterium  7  2.1  Introduction  7  2.2  Molecular Levels of Hydrogen  8  2.3  Hyperfine States of Atomic Hydrogen  10  2.3.1  Hydrogen Hyperfine States  10  2.3.2  Deuterium Hyperfine States  12  2.4  Hydrogen Atoms on /- He Surfaces  13  2.5  Recombination in Spin-Polarized Hydrogen  15  2.5.1  Resonant Recombination  15  2.5.2  Van der Waals Recombination  19  4  iv  2.6 3  5  6  21  Electron Spin Resonance  23  3.1  Introduction  23  3.2  The Technique  23  3.3  ESR Apparatus  26  3.3.1  Introduction  26  3.3.2  Cell  26  3.3.3  The Microwave Spectrometer  29  3.3.4  Data Acquisition  32  3.4 4  Spin Relaxation Mechanisms  Signal Analysis  33  Recombination Rate Studies in Spin-Polarized Hydrogen  37  4.1  Introduction  37  4.2  Recombination Rate Measurements in Spin-Polarized Hydrogen  38  4.3  Rate Equations  38  4.4  Results  40  4.5  Conclusions  49  Studies on Spin-Polarized Deuterium  53  5.1  Introduction  53  5.2  Rate Equations for Gas of Dj.  54  5.3  Results of the Experiments on D |  58  5.4  Discussion of the DJ. Results  61  Doubly Spin-Polarized Systems  64  6.1  Introduction  64  6.2  Recombination Rate Measurements  66  v  - 7 Surface Atoms  8  9  73  7.1  Introduction  73  7.2  Kinetic Properties of Hydrogen Atoms on a MHe Surface  76  7.3  Superfluid Transition in 2-D  78  Results of Experiments on Surface Atoms  80  8.1  Data  80  8.2  The Binding Energy  84  8.3  Thermodynamic Equilibrium in the System  89  8.4  Kapitza Resistivity  90  8.5  Absorption Line Data  92  8.6  Effect of the Microwave Power Level on the Line  97  The ESR Lineshape of H H Adsorbed on a Z- He Surface  107  9.1  Introduction  107  9.2  Preliminary Calculations  109  9.3  Numerical Simulations of the Absorption line  113  9.3.1  Static Distribution  114  9.3.2  Kinetic Models  115  4  10 Phenomenological Theory  123  10.1 Introduction  123  10.2 Possible T i Processes in HW at 0.1 K  125  10.3 Modified Bloch equations  127  10.4 Spin Waves  133  11 Discussions and Conclusions  136  11.1 ESR at 115 GHz in H | and Dj  136 vi  11.2 Lineshape of H | | on the Surface  138  Appendices  141  A Numerical Simulations of the ESR Response of the Bulk Atoms  141  B Exciting Atoms in a Cavity  147  Bibliography  156  vii  List of Tables  2.1  Zero-field Hyperfine Frequencies  10  2.2  Gyromagnetic Ratios  17  2.3  H2, D2, and HD levels near dissociation  18  8.1  Least Square Fits to Line Parameters  96  10.1 Time constants of decay processes  126  viii  List of Figures  2.1  The Singlet and Triplet Potentials of Hydrogen  3.1  Hyperfine energy diagram of hydrogen  25  3.2  The microwave cell  28  3.3  The present 115 GHz spectrometer  31  4.1  Measured Two-body Recombination rates  41  4.2  Zero-field Two-body Recombination Values  43  4.3  Level Shifts Due to the Zeeman Energy  45  4.4  Gamma as a function of the temperature  50  5.1  Low Gi Decay in Spin-polarized Deuterium  55  5.2  Average Recombination rates in Spin-Polarized Deuterium  60  6.1  Decay of Doubly Spin-Polarized D and H  68  6.2  Microwave induced relaxation decay of doubly spin-polarized H and D  .  70  7.1  The square of the wavefunction of the hydrogen atom on liquid helium  .  77  8.1  Example of a "b-line" absorption spectrum  81  8.2  Example of a "d-line" absorption spectrum  82  8.3  Apparent Binding Energies of Hj to the /- He Surface  86  8.4  Density dependence of the Measured Binding Energy  87  8.5  Temperature of the /- He due to recombination heating  88  8.6  Field shifts of the <7 d  94  4  4  en  ix  9  8.7 ' "Field shifts of the cr  95  8ide  8.8  Spectrum taken with maximum microwave power level  100  8.9  Spectrum taken with half power level  101  8.10 Spectrum taken with one quarter power level  102  8.11 Composite of spectra taken with different levels  103  8.12 Effect of power levels on the observed field shift  104  8.13 Effect of levels on the observed RMS width  105  8.14 Effect of levels on the observed onset  106  9.1  9.2  Static field distribution for particles on a surface perpendicular to the polarizing field  116  Simulated Line of H | i on a surface perpendicular to B  122  10.1 Simulation of modified Bloch equations with a T i  131  10.2 Modified Bloch equations without a T i  132  A. l Calculated fractional decrease of the Magnetization  146  B. l  150  Equivalent Circuit of a Cavity  x  Acknowledgement  I would like to thank Dr. N.W. Hardy for supervising and actively participating in this work over the last seven years. I have benefited from the advice of Dr. R.W. Cline; also without his ceaseless work in maintaining and supporting the computer system, these experiments would not have been possible. I am indebted to Dr. B.W. Statt who designed and built the 115 GHz spectrometer, and who patiently explained the intricacies of his apparatus to a neophyte. The early experiments in spin-polarized hydrogen and deuterium were done in collaboration with Meritt Reynolds. The excellent work and advice of the departmental electronic shop and machine shop were invaluable during the course of this work. In particular, I would like to thank Phil Akers and Beat Meyer for their superb work; I profited greatly from the advice from Peter Haas and Jack Bosma. For the extensive advice and assistance, I would like to thank Martin Hiirlimann and Meritt Reynolds. I am grateful to the Natural Sciences and Engineering Research Council for their support in form of a postgraduate scholarship and University of British Columbia for a Graduate Fellowship during the course of this thesis.  xi  Chapter 1 Introduction  1.1 Introduction The main goal of most of the research in atomic hydrogen and atomic deuterium at very low temperatures is to observe quantum degeneracy in these weakly interacting gases. Both atomic hydrogen and deuterium are expected to remain gaseous even at 0 K. The manifestation of quantum degeneracy in a gas of atomic hydrogen (a composite boson) will result in the formation of a Bose-Einstein condensate (BEC). Similarly, atomic deuterium (a composite fermion) should undergo a phase transition. There are two major impediments to attaining quantum degeneracy in these systems: recombination processes between the atoms become more effective and the number of active processes increases as the density of the gas increases; and as the temperature is lowered, the presence of the physi-sorbed atoms on the walls of the container drastically affects the lifetime of the entire sample. These mechanisms, which are serious obstacles for experiments attempting to observe BEC, are themselves interesting physical processes. The recombination processes in the atomic systems, the formation of molecular hydrogen (deuterium) from atomic hydrogen (deuterium), are some of the simplest chemical reactions in nature, and for which the rates may be calculated using ab initio theories that start with the interactions of a system of two electrons and two nucleons! Furthermore, since many of the properties of a 2-D system of quantum particles are different from those of the bulk, it is also very  1  Chapter 1. Introduction  2  interesting to study the properties of the atomic hydrogen adsorbed on a /- He surface. 4  An indepth review of the work in atomic hydrogen and its isotopes is presented by Silvera and Walraven[l]. A more concise introduction to the field is found in the paper by Greytak and Kleppner[2]. This thesis consists of two distinct parts. The first part discusses the earlier work on the two-body recombination rates in atomic hydrogen and atomic deuterium systems. The second part is concerned with the ESR lineshape of atomic hydrogen physi-sorbed on the surface of /- He. 4  1.2  Atomic Hydrogen  Soon after the preliminary studies on atomic hydrogen at low temperatures[3, 4, 5], spinpolarized hydrogen (atomic hydrogen with the electron spins aligned) was stabilized at temperatures below 1 K using a large magnetic field and /- He covered walls to contain 4  the gas[6]. Atomic hydrogen is difficult to stabilize at low temperatures as there exists at least one bound state of atomic hydrogen on any surface. At sufficiently low temperatures, the hydrogen atoms will rapidly adsorb onto the surface, and then recombine at a much higher rate than in the bulk. A crucial step towards stabilizing spin-polarized hydrogen is to cover any surface, which is in contact with the atoms and at low temperature, with a layer of /- He, since the binding energy of atomic hydrogen to any surface, except /- He 4  4  and /- He, is larger than 10 K. If the temperature of the walls is below 1 K, then on any 3  surface other than helium, any atom that sticks to the surface virtually never comes off; in such situations, the equilibrium bulk atom density of the atomic hydrogen gas is very small. The experiments that are attempting to observe BEC continue to discover new recombination processes as the sample density is increased and processes involving many  Chapter 1. Introduction  3  body collisions become significant. With high densities and low temperatures, even the I- He covered surfaces can make major contributions to the instability of the sample. 4  Conditions on the surface are more favourable for recombination to occur, and the atoms in the surface state are compressed to an effective density much higher than the bulk density of the sample, thus further enhancing the contribution of the surface recombination processes over that of the corresponding bulk processes. The energy of recombination due to the formation of molecular hydrogen is usually deposited into the /- He and increases the temperature of the film, since there is a large 4  Kapitza resistance between the liquid and the walls of the container. At most temperatures, since the gas temperature is determined by the surface temperature, the gas is also warmed by the recombination energy. Recent experiments attempting to attain BEC either use magnetic quadrupole confinement[7, 8], which avoids the use of walls altogether, or confine the gas as a bubble in /- He[9], which decreases the area of the surface in contact with the gas. 4  Since the recombination processes and the properties of the surface state are also interesting problems, we have concentrated our efforts in these areas. In a second generation of experiments using an existing 115 GHz ESR apparatus, we have accurately measured the two-body recombination rates, amongst others, for spin-polarized hydrogen in a 41 kGfieldfor the temperature range 0.25 to 0.8 K. The most interesting result of these measurements is the discovery that at the warmer temperatures, the decay rate is dominated by the contribution from the resonant recombination of two hydrogen atoms via the  = (14,4) molecular level. This process was previously thought to have a  negligible rate[10], but we found that this is not the case. The intrinsic surface recombination rates for the formation of ortho- and para-hydrogen by the Van der Waals process were measured, as well as the two-body surface spin relaxation rate.  Chapter 1. Introduction  4  We describe the experimental technique in chapter 3, and present the results in chapter 4.  1.3  Atomic Deuterium  Spin polarized deuterium at temperatures below 1 K was first observed by Silvera and Walraven[ll] in the same apparatus they used to observe spin-polarized hydrogen. They found that the temperature dependence of the measured two-body decay rate was indicative of a decay which is dominated by surface processes. An analysis of the temperature dependence of the recombination rates indicated that the binding energy of D on Z- He is 4  2.6(4) K, much larger than the 1.0 K for H. The maximum density of 1 0 c m 14  in the atomic deuterium experiments was much lower than the 10 c m 16  - 3  -3  obtained  density in their  atomic hydrogen experiments, and under similar conditions, the samples of deuterium decayed more rapidly than the hydrogen samples. At about the same time, Jochemsen et al.[12] observed atomic deuterium at 1 K in 1  a Z- He coated cell in a 327 MHz zero-field hyperfine resonance experiment. They found 4  that only low densities of D could be produced (about 10 cm ) and that the sample n  -3  lifetime was much shorter than that of H under similar circumstances. Since at that time the lower temperature measurements on atomic hydrogen was an experimental priority, a thorough study of atomic deuterium was not completed. All subsequent attempts to observe atomic deuterium at low temperatures, including one by the Amsterdam group in a different apparatus[13], have been unsuccessful. In the only published account of such an attempt, Mayer and Seidel[14] could not detect atomic deuterium under conditions where atomic hydrogen was readily observed. From measurements of the incident atom flux and the level of heating in the cell due to the ^his work is unpublished.  Chapter 1. Introduction  5  recombination heating, they determined a lower limit for the surface recombination rate in atomic deuterium which is consistent with the results of Silvera and Walraven. We have observed spin-polarized atomic deuterium in a 41 kG field at temperatures around 0.7 K with the ESR apparatus used in the spin-polarized atomic hydrogen experiments. From the analysis of the decay curve measurements, we have determined the surface two-body recombination rates for the formation of ortho- and para-deuterium. We were also able to generate samples of doubly spin-polarized deuterium (both electron and deuteron spins aligned) at densities up to 10 c m 15  - 3  and with lifetimes of 30 min-  utes. The results of our measurements are presented in chapter 5. Some of the samples of doubly spin-polarized deuterium also contained double spin-polarized hydrogen. The results of the decay analysis of this unusual system are discussed in chapter 6.  1.4  ESR Absorption Lineshape of Adsorbed Atomic Hydrogen  The observation of the absorption line of hydrogen atoms adsorbed on the /- He covered 4  cavity walls permitted the direct measurement of the binding energy of atomic hydrogen to the /- He surface[15]. At that time it was noticed that, although the lineshape of the 4  bulk atom absorption line was determined by the inhomogeneity in the polarizing field, the lineshape of the atoms on the surface appeared to be due to intrinsic properties of the system. The last half of this thesis is concerned with the study of the ESR line of hydrogen atoms which are adsorbed on the /- He film that coats the microscopically flat surfaces 4  of the cell. We have observed the intrinsic lineshape of the atoms, but unfortunately, the measured lineshapes are not the small-signal reponse lines, but include some non-linear effects. The reason for the early onset of the non-linear effects in this spin system is not understood. We have attempted to develop a simple theory to explain the early  Chapter 1. Introduction  6  onset of the saturation effects, but have not found any encouraging results. Although the absorption lineshapes are very distinctive, we have not been able to construct a model which simulates the observed line. An introduction to the surface state of hydrogen adsorbed onto the /- He surface can 4  be found in chapter 7. The results of the measurements are presented in chapter 8 and the simulations are discussed in chapters 9 and 10.  Chapter 2  Spin-Polarized Atomic Hydrogen and Deuterium  2.1  Introduction  This chapter is mainly concerned with introducing the dominant recombination and spinrelaxation processes that occur in a gas of spin-polarized atomic hydrogen in the regime of densities and temperatures encountered during the ESR measurements. Only a summary of the various relevant processes and their physical mechanisms is presented. The reader may refer to the review by Silvera and Walraven (SW) [1] and to the references within for the precise details and for a more complete compilation. We will use the notation of SW, who designate recombination rates by  K^JJ^J  and  relaxation rates by G^-j, where "c" may be a "v" or a "s" denoting a volume or a surface process respectively. If the rate is a recombination rate, then the indices {ijk} are replaced by the initial states of the atoms involved in the collision; if the rate is a relaxation rate, then the indices are replaced by the initial and final states of the spinflipped atoms. For example, K  v b b  is a recombination process in the bulk gas involving  an | a) and two |b)-state atoms. Also included in this chapter, for convenience, are discussions of some the physical properties of atomic hydrogen that can indirectly affect the decay rates in a gas of atomic hydrogen.  7  8  Chapter 2. Spin-Polarized Atomic Hydrogen and Deuterium  2.2  Molecular Levels of Hydrogen  The low temperature molecular recombination rates of atomic hydrogen are determined by just a few of the multitude of bound states of molecular hydrogen [10]. The bound states are classified by their orbital angular momentum J and vibrational v quantum numbers. They are usually denoted as rl2(v, J). If the hyperfine interactions are neglected, then when each of the hydrogen atoms is in the electron ground state (as is usually the case at temperatures below 1 K), the Coulombic pair interaction between the atoms depends only on the symmetry of the electron spin wavefunction of the pair, and can be described with only two interatomic potential curves. The potential curves are the singlet, 5 = 0, and the triplet, 5 = 1, where S = « i + «2 and a,- is the vector spin operator of the electron on atom i. The hydrogen interatomic potentials are the most accurately known of all atomic systems. The results of the ab initio calculations by Kolos and Wolniewicz [16, 17, 18] are shown in Figure 2.1. On the scale of Figure 2.1, the triplet potential curve 6 E^ appears to be a purely 3  repulsive one; however, there exists a very shallow well 6.5 K deep at r  n u c  i  e a r  = 4.15A.  Due to the light mass and consequent large zero-point motion of the hydrogen atoms, there exists no bound state for two hydrogen atoms interacting via the triplet potential; there are, however, many bound states in the singlet potential curve [19]. The energy required to dissociate an H2 molecule in its vibrational-rotational ground state is 51,967K [20], and for D it is 52,874K [21]. 2  Unless the gaseous hydrogen atoms always interact via the triplet potential, the gas of atomic hydrogen will be unstable and the density will decay. The electron spins of non-interacting atoms can be aligned parallel to each other by applying a large enough polarizing magnetic field: this ensures that the atoms interact via the triplet potential.  Chapter 2. Spin-Polarized Atomic Hydrogen and Deuterium  9  Figure 2.1: The singlet and triplet 6 £ + interatomic potentials of hydrogen calculated from first principles by Kolos and Wolniewicz [17, 18]. The radial separation is in units of Bohr radii. 3  10  Chapter 2. Spin-Polarized Atomic Hydrogen and Deuterium  However, any mechanism that introduces interactions via the singlet potential will eventually destroy the sample. From energy considerations, it is apparent that, even at low temperatures, any practical barrier used to prevent the recombination process in a high density gas of atomic hydrogen will be only partially successful and eventually the atoms will recombine.  2.3  Hyperfine States of Atomic Hydrogen  2.3.1  Hydrogen Hyperfine States  The Hamiltonian appropriate for describing the spin states of a hydrogen atom in its electronic ground state and in an applied magnetic field Bo is  H = ai-s - ft(- s + 7 i)-Bo, 7e  P  (2.1)  where i (s) is the proton (electron) vector spin operator, a is the hyperfine constant, and 7 (7e) is the gyromagnetic ratio of the proton (electron). The most recent values of the P  constants can be obtained from Table 2.1 and Table 2.2. Species H D  Zero-field Frequency 1420.405751773(l)MHz 327.3843525222(17)MHz  Reference [22] [23]  Table 2.1: The most recently measured zero-field hyperfine frequencies in hydrogen. In the basis set of | m TO,-), where m (mi) is the electron (proton) spin projection a  s  along the applied field Bo, the eigenstates of (2.1), in order of decreasing energy, are Id) =  \\,\),  |c> =  sin.0|-i  (2.2)  |) +  cos0||,-|),  (2.3)  11  Chapter 2. Spin-Polarized Atomic Hydrogen and Deuterium  l )  =  |-^-|>»  |a)  =  cos0|-|,i)  b  a  n  d  -  (2.4)  11,-|>,  sin<9  (2.5)  where tan(20) =  506.07 G  (2.6)  B  Hie + Kp)Bo  The corresponding energies are a h .  .  (2.7) ^(7e +  E  b  =  a h .  1V)BQ  „ i 2^ 2  (2.8)  .  (2.9)  - - ~(7e - 7P)-DO, and  „  a  a  ^(7e  +  7 )#o  2^  (2.10)  P  Since the energies of the | b)- and | a)-states decrease with increasing magnetic field, atoms in these states are attracted to regions of high field. If the temperature of a gas 1  of atomic hydrogen is much less than the energy difference between the upper and lower energy states, then only the lower energy states are appreciably populated. A gas of atomic hydrogen which contains mostly | b) and | a)-states atoms is called spin-polarized hydrogen H.J.. A gas containing only atoms in the |b)-state is called doubly spin-polarized hydrogen H||., and in such a gas both the nuclear and electron spins are anti-parallel to the applied field. For | B o |> 500 G, equation 2.6 implies that cos 6 « 1 and e  H  = sin0 K 0 = 253G/B . 0  We will often use temperature as an energy unit, it being understood that the actual energy requires multiplication by Boltzmann's constant. 1  12  Chapter 2. Spin-Polarized Atomic Hydrogen and Deuterium  In the present experiments, the applied magnetic field is 41 kG, and consequently, the admixture coefficient en = 6.17 x 10~ . Although the admixture of the 1 1 , — |) in the 3  | a)-state is quite small, it is not insignificant and dramatically reduces the lifetime of a sample of Hj. 2.3.2  Deuterium Hyperfine States  The hyperfine structure of deuterium is very similar to that of hydrogen except for the differences due to the spin 1 of the deuteron. The spin Hamiltonian, in a magnetic field Bo, i s  2  H = a i-s  - h(-j s  d  + 7di)-B .  e  0  (2.11)  Again, using as the basis set of | m , m,), where m (m,) is the electron (deuteron) spin s  s  projection along the polarizing field, the eigenstates of 2.11 (listed in order of decreasing energy ) are 3  (2.12)  ,1)  10  •|.>  (2.13)  l-i,0)  (2.14)  = "!+ |i,0) + « | +  = VIT)  = 1-  (2.15)  \fi) = V- l ~ , ° >  "  \a)  -  = 1+ | - i , D  £  " |i,-D  (2.16)  e |i,0)  (2.17)  +  where the coefficients T)± = cos(0±) and e± = sin(0±). The mixing angles 0± are given by taa(  ±)  . vva +p - *  Since I = 1, the zero-field hyperfine frequency given in Table 2.1 is actually If B Q > 167 kG, then |C) and \ 6) are interchanged.  2 3  *  ,  (2 8)  13  Chapter 2. Spin-Polarized Atomic Hydrogen and Deuterium  with  <A± =  ^ft(7e + 7d)#o ± 1  In general, two mixing angles are required to describe the hyperfine states of deuterium, but in the high field limit  \/2(7e +  7d)fiB ' 0  The corresponding energies of the hyperfine states are given by £  =  +ft( - 7d)-Bo/2 + a/2  =  +v [M7e + 7d)B /2 + a/4p + a /2 -  /J7 B /2 -  a/4  (2.20)  E  6  =  - v ^ f r e + 7d)5 /2 + a/4] + a /2 -  h B /2  -  a/4  (2.21)  £  7  =  + [^(7e +  a/4  (2.22)  Ep  =  -y/[h(l* + 7d)5o/2 - a/4] + a /2 + h^ B /2 "  /  (2-23)  E  =  -ft(7e -  c  £  £  a  (2.19)  7e  /  2  0  2  2  0  /  v  0  dJ  ld  0  7 d ) £ / 2 - a/4] + a /2 + ft7d#o/2 2  2  0  2  2  d  7d)B /2 0  a  0  a  4  (2.24)  Analogous to the hydrogen case, spin-polarized deuterium DJ. is a gas composed of atoms in the | a), | 3) and | 7)-states; doubly spin-polarized deuterium D|j. is a gas containing only |7)-state atoms.  2.4  Hydrogen Atoms on /- He Surfaces 4  The Van der Waals attractive potential between a hydrogen atom and any known surface is strong enough that there exists at least one bound state of hydrogen on any surface [6]. Any recombination or relaxation process that occurs in the bulk gas, can also be active in the 2-D gas adsorbed on a surface. Since the contribution of the surface processes to  Chapter 2. Spin-Polarized Atomic Hydrogen and Deuterium  14  the total rate increases as the temperature is lowered, and since the kinetic behaviour of the surface state strongly influences the size of the contributions, it is important to understand the nature of the surface state of H | . The size of the adsorption energy of Hj to a surface is the central factor in determining how effectively the surface processes contribute to the total rates. At temperatures below 1 K, unless the binding energy is significantly smaller than 1 K, a significant number of the atoms will be adsorbed on the surface . The smallest adsorption energy E B for atomic 4  hydrogen on known surfaces is 0.3QK [25] on the surface of /- He. The next smallest is 3  1.010 ± 0.010 K [26] for a Z- He surface. Accurately measuring the binding energy of HJ, 4  on /- He is not an easy task. Although almost every group working in the field of Hj 4  has made measurements of the binding energy [27], a general consensus on the value of EB[27] has emerged only recently. In Chapter 8, we discuss the binding energy results from our ESR measurements. Goldman and Silvera [28] have shown that for surface densities less than 1 x 10 c m , the classical adsorption isotherm formula is applicable. If the two systems 14  -2  are in thermodynamic equilibrium, then the density of adsorbed atoms a is related to the bulk atom density n by cr = n\ exp—— th  where A /i t  =  (^f^f)^  1 S  kB 1  (2.25)  the thermal deBroglie wavelength, with m the mass of the  hydrogen atom and T the temperature. At temperatures below 0.3 K, the calculated effective mass of HJ. adsorbed on a /- He 4  surface is within a few percent of the bare mass [29]. The calculated surface mean free path[29] of the H | due to interactions with excitations on the helium surface is long enough that the atoms adsorbed on the surface behave like an ideal 2-D gas. The geometric factors must also be considered.  4  Chapter 2. Spin-Polarized Atomic Hydrogen and Deuterium  2.5  15  Recombination in Spin-Polarized Hydrogen  One of the simplest chemical reactions, the reaction H + H —• H is interesting to study 2  because it is one of the few chemical processes for which the reaction rate may be calculated from first principles. All recombination processes in Hj at low temperatures must involve three bodies in order to conserve energy and momentum. Resonance recombination and Van der Waals recombination processes were observed in the ESR experiments; both processes require only two hydrogen atoms plus a third chemically inert body, which may be a helium atom, H | , or a wall of the container. In small magnetic fields, the Van der Waals process [10] is always the dominant recombination mechanism in a low density gas of HJ. at low temperatures. If a sufficiently large magnetic field is applied, then a resonant recombination process is activated and its contribution to the total recombination rate may dominate over that of the Van der Waals process.  2.5.1  Resonant Recombination  The high temperature resonance recombination mechanism in atomic hydrogen proceeds in two steps: a pair of hydrogen atoms collide and form a quasi-bound state of the X E g potential. These quasi-bound states are positive energy states in which the atoms 1  are temporarily trapped by the orbital angular momentum barrier. If the quasi-bound molecule collides with a third body, then it may make a transition into a truly bound state. At ambient temperatures, this is the dominant recombination process [30], but since all the quasi-bound states have energies of more than 50 K, resonant recombination through these channels is frozen out at temperatures below 1 K. The situation changes when a sufficiently large magnetic field is applied. The resonance recombination process can now proceed via the highest energy bound states: the H.2(14,4) state in hydrogen, the D (21,l) and D2(21,0) states in deuterium, and the 2  Chapter 2. Spin-Polarized Atomic Hydrogen and Deuterium  HD(17,1) state in the mixture.  Chapter 2. Spin-Polarized Atomic Hydrogen and Deuterium  Species electron proton deuteron  Gyromagnetic Ratio s *G 7e 7 7d P  1.7608592(16) x 10 2.67522128(81) x 10 4.106628(4) x 10  1  7  4  3  Table 2.2: List of the accepted gyromagnetic ratios [24].  17  Chapter 2. Spin-Polarized Atomic Hydrogen and Deuterium  molecule H D 2  2  HD  V J 14 4 21 0 21 1 17 0 17 1  D(K) 0.7(9) 2.9(3) 0.2(5) 6.9(7) 0.1(7)  18  B (kG) 5(7) 22(2) 2(4) 51(5) 1(5) 0  Table 2.3: Levels of H2, D and HD predissociated by fields below 200 kG, their dissociation energy D and threshold field BQ for hyperfine predissociation. 2  Chapter 2. Spin-Polarized Atomic Hydrogen and Deuterium  19  As all the bound states have zero total electron spin, the energies of these states are not shifted by an applied magnetic field (except for the small nuclear Zeeman shift). On the other hand, in a magneticfieldB the triplet ms = —1 potential is shifted by — j hB; e  thus it is possible to tune the asymptotic energy of the triplet potential below that of a bound state of the singlet potential. In this situation, the bound level is predissociated: the intra-hyperfine interaction, which mixes the S = 0 and S = 1 states, causes a molecule in that bound level to disintegrate into polarized atoms. The reverse reaction produces metastable molecules, which may then be stabilized in collisions. The contribution of the inverse-predissociation process to the total recombination rate in H | at low temperatures was first considered by Stwalley [31], who deemed it to be insignificant due to the large, 1 = 4, rotational barrier, which prevents appreciable occupation of the H2(14,4) state. Recent calculations by Reynolds and Hardy [32] suggest that the predissociation contribution to the total recombination rate can be significant. The results of those calculations and their implications for the results of our measurements are discussed in Chapter 4. 2.5.2  V a n der Waals Recombination  The Van der Waals recombination process is a direct process which precedes as follows: H | + Hj + X  H + X, 2  where X is an inert third body. Grebene* al. [10] have studied this process in the bulk using the "sudden approximation" for the evolution of the spin wavefunction of the hydrogen atoms. The "sudden approximation" is quite commonly used in the study of processes when the exact behaviour of the spins during the collisions is not important. When the approximation is used to simplify electronic spin interactions this approximation is referred to as the Wigner rule, which may be invalid at very low temperatures  Chapter 2. Spin-Polarized Atomic Hydrogen and Deuterium  20  [33].  .,Greben. e£ al. found that, for the processes between the various hyperfine states of hydrogen, every recombination rate constant can be written as a linear combination of Kortho and K a , where K tho is the zero-field formation rate of ortho-hydrogen (I = 1), par  or  and Kpara is the zero-field formation rate of para-hydrogen (1 = 0). The recombination rates in the high magnetic field B limit are =  K  cc  =  K*K t  =  K^ =  ad  =  =  Kbb  =  K K  a b  d  I Y  (2.27)  K tho)  d  IQ  (2.26)  2 H para, or  (2.28)  ^(K tho "1" Kpara)) or  (2.29)  = o,  where en oc 1/B is the admixture coefficient defined in subsection 2.3.1. Since the recombination constants for a gas of H | scale as  oc 1/B , the decay rates 2  in the gas can be suppressed by applying a large polarizing magnetic field; although,there is a practical limit on how small the admixture coefficient can be made by increasing B. However, the | 6)-state has zero admixture of the electron spin up wavefunction, and K  b b  = 0 in the "sudden approximation"; thus H j | is much more stable than H | . Although Greben et al. calculated the recombination rates in bulk H | , it is straight-  forward to show that the results (2.26-2.29) are applicable to the surface recombination processes. Analogous results for the recombination rates in DJ, and mixtures of Dj and HJ. can be found. Those results will be presented as they are needed. Of course, the  scaling of the recombination rates is valid only if no resonant re-  combination mechanism is activated by the magnetic field.  Chapter 2. Spin-Polarized Atomic Hydrogen and Deuterium  2.6  21  Spin Relaxation Mechanisms  Spin relaxation mechanisms can depolarize the Hj atoms and contribute to the instability of the gas. The three processes considered here are spin-exchange, intrinsic dipolar relaxation, and extrinsic dipolar relaxation. While spin exchange relaxation is the dominant process in zero-field, in high magnetic fields the intrinsic and extrinsic dipolar processes must also be considered. Since the spin exchange processes set a limit on the accuracy and precision of the atomic hydrogen maser as a time standard, these processes have been extensively studied [34, 35, 36]. The electron-nucleon (-electron) dipole-dipole interaction between pairs of atoms can result in nuclear (electron) spin relaxation in the 3-D or 2-D gas. The relaxation processes resulting from the interatomic interactions are labelled intrinsic. The extrinsic nuclear relaxation mechanism, with rate Gi, is the result of NMR transitions induced in a Hj atom as it moves through varying magnetic fields caused by randomly distributed magnetic impurities in the cell walls [37, 38, 39]. The calculations of Berlinsky et al. [39] indicate that, for the conditions of our experiment, Gi oc exp (—ah), where a is the hyperfine frequency. Thus the rate in DJ. should be larger than the rate in Hj. It is possible to suppress the extrinsic nuclear relaxation rate by coating the cell walls with molecular hydrogen [37]. The spin exchange relaxation mechanism is due to the interchange of indistinguishable constituent spin particles during the collision of a pair of atoms. The spin exchange calculations for a low temperature hydrogen system were first done by Berlinsky and Shizgal [40], and the magnetic field dependence first analyzed by Statt [41]. In the "sudden approximation" theory for Hj in a high magnetic field and at low temperatures, only the ac *-* bd exchange process is not inhibited by the reduced admixture coefficient and the Zeeman barrier for a spin flip. Under the same conditions,  Chapter 2. Spin-Polarized Atomic Hydrogen and Deuterium  the important exchanges processes in DJ. are the ct6  7C, ae *-*  22  and /?<5  7c  transitions (but not the a6 +-> /3e transition, as one might expect). For the reasons given in the previous paragraphs, the much weaker dipolar relaxation processes are significant paths for spin relaxation at high fields, since the spin exchange processes become ineffective channels for thermalizing the hyperfine states. Both the surface and bulk dipole processes have been well studied. The electron-nucleon dipole interaction between colliding atoms thermalizes the | a) and | 6)-states via nuclear spin relaxation, with rate G - The nuclear relaxation rate is small enough so that (if Gi is aD  negligible), the preferential recombination of |a)-state atoms [42] (see (2.26-2.29)) in H | results in a gas composed mainly of | b)-atoms—the B.H gas[42] (see (2.26-2.29)). In principle, the electron-electron dipole interaction can induce electronic spin flips, which would relax the electron polarization of the HJ, gas [43]. However, when T << IK, the rate of the spin flip process is greatly suppressed, since the energy required to flip a spin is the Zeeman energy of 5.5 K in a field of 41 kG.  Chapter 3 Electron Spin Resonance  3.1  Introduction  In collaboration with M. Reynolds and other members of the U.B.C. HJ. group, we have measured the various recombination and relaxation rates in HJ. using a 115 GHz ESR spectrometer. We have, throughout the course of the various experiments, refined and improved every facet of the ESR experiment that was originally built by B. Statt as his Ph.D work [41]. It has evolved from an experiment measuring the recombination rates in a gas of HJ,, into one measuring the intrinsic ESR absorption lineshape of HJ.J. adsorbed on the surface of /- He. Detailed descriptions of the past 115 GHz ESR experiment have 4  been published elsewhere [44, 37, 41]. In the rest of the Chapter, we will first describe the ESR technique as applied to Hj, and then discuss the apparatus and the improvements that have been made to it. We will conclude with an explanation of the method used to determine the density of HJ. atoms from the field dependent microwave reflection coefficient of the cell.  3.2  The Technique  For practical reasons, the ESR apparatus uses afixedtuned resonant cavity in conjunction with a fixed frequency microwave spectrometer.  The susceptibility is measured as a  function of the applied magnetic field. There are two allowed ESR transverse magnetic field transitions between the hyperfine states of hydrogen in a large magnetic field. These  23  Chapter  3:  Electron  Spin  Resonance  24  are the a ^ d and b ^ c transitions, which are shown in Figure 3.1. In a gas of DJ., there are three transitions:, these are the a <r± £, j3 ^ e, and 7 ^ 6  transitions. At  temperatures below 1 K and in a 41 kG field, the higher energy Zeeman states are unpopulated, so that the integral of the resonance absorption curve is proportional to the number of atoms in the lower energy state coupled by the transition. In high magnetic fields, for all practical purposes, the strengths of the transitions are determined by the bare electron magnetic moments // , and so e  (3.1) where n is the density of atoms in the appropriate initial state. The absorption x"(H), as a function of the applied field, is the imaginary part of the complex magnetic susceptibility X = x' ~ x"> with x' the dispersion. ?  The shape of the ESR absorption curve of the HJ, atoms in the 3-D gas is determined by the gradients in the applied magnetic field, since the intrinsic broadening mechanisms are negligible at the densities encountered. Given that the width of the line is determined by the field inhomogeneity, the absorption is essentially zero outside of a narrow region ( « 0.3 Gauss). In Section 3.4, we exploit this localization property when processing the signal. Since the energy splitting between the states connected by the ESR transitions depends almost entirely on the Zeeman energy of the electrons on the hydrogen atoms, this technique can also be used to study spin-polarized deuterium, and in principle, even spin-polarized tritium. The basic ideal of the recombination rate experiments is to monitor the density of the different states of HJ. gas in the cell by repeatedly and sequentially sweeping the magnetic field through the resonance condition for each of the species. The disadvantage of using ESR is that each measurement depolarizes the electron spins on some fraction of the atoms, and thus partially destroys the sample. This problem  Chapter 3. Electron Spin Resonance  25  Figure 3.1: Hyperfine energy diagram of hydrogen. The ESR transitions are shown by the lines between the upper and lower states.  Chapter  3.  Electron  Spin  26  Resonance  is alleviated by using as low excitation power as possible (~ pW), but even at these low levels, the fate of the spin-flipped atoms  1  must be taken into account when fitting rate  equations to the decay data.  3.3 3.3.1  ESR Apparatus Introduction  The experimental apparatus divides naturally into three sections. One part consists of: a microwave cavity which is attached to the mixing chamber of a dilution refrigerator, a superconducting magnet that applies the biasing magnetic field, and a sweep coil that ramps the field through the resonance condition of the various ESR transitions. The second section consists of a 115 GHz heterodyne detection system, and the various phase locked loops associated with the detection system. The data acquisition components consist of a dual 16-bit D / A convertor and a computer, which controls the experiment and stores the data. Since many of the components have been described by Statt et al. [44], the following subsections will only slightly fill-in the sketched descriptions, and will concentrate on describing the modifications to the system. 3.3.2  Cell  The cell, a right cylindrical copper microwave cavity with its axis aligned parallel to the polarizing magnetic field, is attached to the bottom end of the mixing chamber on a commercial dilution refrigerator (which has been modified to increase its cooling power). The cell is connected by a fill line to a low temperature r.f. hydrogen discharge source, which is attached to the base plate of the dilution unit. During the experiment, all the 2  Usually less than 0.1 % of the atoms areflippedduring a sweep. The base plate is an intermediate heat sink, which is situated between the continuous and the step heat exchangers. 1  2  Chapter 3. Electron Spin Resonance  27  interiors of the apparatus are coated with a saturated film of superfluid helium to prevent the HJ. atoms from immediately adsorbing on the walls and not reaching the cell . The 3  cell sits in the center of a superconducting magnet with a 40.5 kG field. The magnetic field serves three purposes: it polarizes the spins on the hydrogen atoms and prevents them from recombining; it draws in only the "highfieldseeking" | a) and 16)-state atoms and compresses them; and it sets the ESR resonance frequency of the atoms. The temperature of the cell is measured by a specially prepared Matsushita carbon composition resistor attached to the copper cell. The resistor is calibrated against a He 4  3  melting curve thermometer[45] at the beginning of the experiment. A fountain effect valve[46] opens and closes the cell. This valve uses the fountain effect to transport /- He between the two reservoirs which are separated by a superleak 4  (see Fiqure 3.2). When the fill line is uncovered, the atoms can flow into the cell; when the fill line is submerged, the atoms are trapped in the cell. For practical reasons it is convenient to use the TMon cavity mode; it was chosen to be at a frequency of 115 GHz. The cavity is coupled to the waveguide through an off-center mylar covered orifice at the top end. In cylindrical coordinates, only He, the transverse component of the a.c. magnetic field, is non-zero. The basic design of the cell has not changed from that of Statt, but the new cell and the fill line were constructed and assembled differently in order to minimize the amount of magnetic "dirt" in the walls and to thereby reduce the one-body relaxation rate Gi in the system. The fill line was modified in order to test the feasibility of building a high flux HJ.J. source for the next generation of experiments. The degree of contamination with magnetic "dirt" and the effectiveness of the "dirt" were decreased by chemically etching all the copper pieces used and coating every piece of stainless steel tubing with a thin See the discussion in Section 1.4 An ERC-18GJ lOOfi, 1/8 W resistor.  3  4  Chapter  3.  Electron  Spin  Resonance  28  WAVEGUIDE  r5mm  MICROWAVE CAVITY  -0  BALLAST RESERVOIR  FOUNTAIN PUMP RESERVOIR  LEVEL A \ , LEVEL B  HEATER WIRE SUPER LEAK  LIQUID HELIUM VALVE THERMOMETER  SINTER PLUGS  Figure 3.2: The microwave cell.  Chapter  3.  Electron  Spin  29  Resonance  film of polystrene. The connections between the tubes are made with either clear epoxy  5  .or: very pure indium solder. For soldered joints the connecting parts were designed to minimize the flow of solder to the inner surfaces. All the recombination data were taken with the old cell, whose walls had only been chemically polished. The new cell, built with well defined surfaces, was used to study the intrinsic lineshape of surface bound atoms. The surfaces were first uniformly polished with decreasing grades of diamond dust and then electropolished. After the polishing, a scanning electron microscope study of the surfaces showed that the surfaces were basically flat and featureless on scales from 0.05 cm down to at least 10~ cm. Since the capillary 6  radius for superfluid /- He is 50 pm when the free surface is 1 cm below, the /- He coated 4  4  surfaces are effectively atomically flat . There should not be any broadening of the 6  lineshape due to variations in the orientation of the surface normal vector. The old cell had a loaded quality factor of 3635(12) and a coupling constant B of 0.518(5); the new cell has a loaded Q of 3600(20) and a coupling constant of 0.903(8). The unloaded Qo, given by QL(1 + /?), is 25% larger for the new cell than for the old cell, which is consistent with the better finish of the cell walls. 3.3.3  The Microwave Spectrometer  Improvements to the original apparatus have greatly enhanced the accuracy of the density determinations. The improvements signficantly increased the accuracy of the determined recombination rates, since the density dependence of the contributions to the decay rates from the various processes are not drastically different and the fitting procedures require accurate values of the densities in order to separate out the contributions from the different processes.  That the densities can be measured accurately is due to the  The resin used is thefilledepoxy, Emerson and Cumming 2850 FT. At T < 0.1K, the thermal ripplon population is small, and gives negligible contribution to surface roughness. 5  6  Chapter  3. Electron  Spin  Resonance  30  combination of a sensitive bi-phase spectrometer and a closed geometry for the atom cell, which prevents the leakage of atoms. Since the spectrometer has been described elsewhere, we will only outline the important parts sufficiently so that the modifications can be discussed. Figure 3.3 shows a schematic diagram of the present 115 GHz ESR spectrometer. The spectrometer consists of three parts: the phase lock loop chains which stabilizes the klystron output to the reference 10 MHz output of a rubidium clock, the 115 GHz heterodyne receiver which measures the reflected voltage from the cavity cell, and the new 1.480 GHz spectrometer that feeds into a 50 kHz bi-phase lock-in detector. Only the first part of the spectrometer is unchanged. The rest of the parts have been modified to varying degrees. The major improvement in the ESR apparatus is that now both the real and imaginary parts of the signal are measured. The importance of this has already been mentioned, and it will be discussed further in the next section. A dedicated 1.480 GHz spectrometer, built by W. Hardy, and a dual phase 50 kHz lock-in amplifier have replaced the old monophase detection system. The 115 GHz spectometer required modification when the old cavity was replaced. The new cell turned out to be nearly critically coupled to the incident waveguide, whereas the old cell was substantially undercoupled. The reflected voltage, at the resonance frequency of the cavity vo, measured at the output of the lock-in amplifier, with the old cell was of the order of 0.2 V; with the new cell, it was of order of 2.5 mV. The smaller vo required that all offsets in the spectrometer had to be re-examined, since an offset that is negligible when v v  0  0  = 0.2 V, can be a major source of systematic error when  = 2.5 mV. It was found that spurious signals, which were reaching the detector through the high  Q confocal resonator and through the isolator between the klystron and the sideband  er 3. Electron Spin Resonance  1.480 G H z  Spectrometer  •  t  2-Phase L o c k i n t  Detector  i  :  2-Channel A / D  + +  Computer  Figure 3.3: The present 115 GHz spectrometer.  Chapter  3.  Electron  Spin  32  Resonance  generator, each contributed about 1 mV to v . 0  Previously , the klystron had been  operating at 112.5 GHz. Its output was mixed with the local oscillator output at the sideband generator to produce the 115 GHz working signal. However, due to decreasing power output at the lower frequency range of the aging klystron  7  we were forced to  operate it at 116.5 GHz, using the lower sideband as the working signal. The changes in the frequencies of the carrier and the unwanted sideband undoubtedly aggravated the leakage problems. A modified out-of-band transmission cavity filter, with a loaded Q of 380, was inserted into the waveguide going down to the cell; it suppressed the level of the unwanted signals incident on the cell, and hence, reduced the power reflected back up through the confocal resonator and to the detector mixer. The addition of a second isolator between the sideband generator and the injection cavity sufficiently isolated the detector from the sideband generator. With these changes and some minor optimization of all the components in the system, the offset in the spectrometer output was brought to less than 1 % of v and was no longer a significant source of uncertainty. 0  3.3.4  Data Acquisition  The experiment is a very computer intensive one, and has required a strenuous software writing effort throughout its various stages of development. A small computer controls the experiment via an IEEE interface. It also collects and stores the raw signals from the spectrometer. All the computer controlled devices used in the present version of the the spectrometer were designed by R. Cline and built by the U.B.C. Physics Electronic shop. During a recombination rate measurement run, the computer collects the data by periodically ramping the magnetic field sequentially through the ESR conditions for the different states of the atoms in the cell. The computer stores the digitized output We have twice replaced the klystron source.  7  Chapter  3.  Electron  Spin  Resonance  33  of the spectrometer on a hard disk. At the end of the run, the raw data is downloaded to a micro-computer for processing. The quantity of data gathered (usually 2 kbytes of data per scan, with each decay containing about 100 sets of scans) requires a computer for efficient analysis. Furthermore, the resonance cavity is severely perturbed by the resonant H | atoms, and the inphase and quadrature parts of the reflected signal are non-linear functions of the complex magnetic susceptibility. The signal requires fairly intensive processing in order for the absorption and dispersion curves to be retrieved.  ZA  Signal Analysis  To determine the absolute density of Hj atoms in the cell from a measurement of the susceptibility, the electrical characteristics of the microwave cavity must be known. By normal techniques it is extremely difficult to measure accurately the loaded quality factor Q L and the coupling constant 3 of a 115 GHz microwave cavity attached to the bottom of dilution refrigerator. Furthermore, since the H atoms strongly perturb both the Q and the resonant frequency of the cavity, the electrical characteristics of the cavity and waveguide combination are required in order to process accurately the signal and determine the complex susceptibility. The usual method for determining the susceptibility, when the measured signal is a convolution of the real and imaginary parts of the susceptibility, is to perform some trial deconvolution of the signal and then check whether the obtained curves satisfy the Kramers-Kronig relation. Until the curves satisfy the relation, a series of iterative transforms are performed. We describe a different method for finding the susceptibility. This method assumes that the absorption is localized and that the true absorption curve is found from the reflected signals, when the computed absorption curve is non-zero only in a permitted range. The method is applicable when the absorption peak is localized and the reflected signal extends beyond the range of the absorption  Chapter  3.  Electron  Spin  34  Resonance  peak. Not only can the method determine the magnetic susceptibility from the reflected signal, but it also measures the important cavity parameters. The ratio T of the reflected voltage to the incident voltage for a cavity filled with a magnetic material with susceptibility x = x' ~ x" is given by [42, 41] l  r =—  ^  <*)  + 7^  1  + *QL(£  (3.2)  -  where 7  u is the 0  ?2Sonance  4TTQ ( X " + ix'),  =  (3.3)  L  frequency of the empty cavity, u> is the frequency of the incident  microwave, and To is the T ratio of the empty cavity at the resonant frequency. It is shown in reference [42, 41] that T =  =  0  with Voo the incident voltage. Inverting  equation 3.2 for 7, we find that  r r-i 0  T] -  *QL 1 I.  1 - w.  (3.4)  When the frequency tuning error is zero, u — u> , equation 3.4 reduces to 0  £-5f  7 -  The measured voltage ratio T* is related to the T in equation 3.5 by T* =  (3.5) Texp(-iO),  where 6 is the phase error in the measured voltages, and this error is assumed to be constant throughout the sweep. Substituting the measured ratios into equation 3.5, and writing the ratios in terms of voltages, one finds that v  0  -  v*e  w  ,  x  To  The transform indicated by equation 3.6 is non-linear and is sensitive to the phase error, To, and v . The most stable method for recovering the 7 (and hence the susceptibilities) 0  is to note that the expected absorption peak is localized and zero everywhere else, and  Chapter  3.  Electron  Spin  35  Resonance  then to perform the transformation indicated in equation 3.6 while minimizing x"  2  i n  appropriate regions of the sweep by varying v , r , and 9. 0  0  The analysis of the highest density sweeps (where the method converges well) determines the value of IV Given that To depends only on the cavity coupling constant which remains constant as long as the apparatus is not warmed up to liquid nitrogen temperature, this value of To can be used to analyze the rest of the sweeps. With the reduction of the number of minimization variables to two, the method can be successfully applied to all but the weakest signals, when v is then also held fixed at a value determined by 0  a previous higher density sweep in the same run. The transformation of the raw signal sweeps by equation 3.6 produces 7 as a function of the sweep field. If the Q L of the cavity is known, then the susceptibility curves can be calculated from the 7 curve using equation 3.1. For simplicity, assume that the phase error in the measured voltages is adjusted to zero, so that the simpler transformation indicated by equation 3.2 is applicable for finding the 7 curve. Note that all the terms and/or ratios on the righthand side of equation 3.2 are of the order of 1, except for the Q L factor, which is approximately 5000. Analyzing sweeps of raw data taken with small deviations Aw in the excitation frequency from a>, allows the determination of 0  Q L , since, to a very good approximation, the effect of the tuning error is to shift the base line of the imaginary part of 7 (which is proportional to x') by — 2 Q L ^ . Moreover, because the x'  0 1 a  S  as  0 1  non-interacting spin particles intrinsically has zero mean, the  resonance frequency of the cavity corresponds to the frequency with zero baseline offset in the determined x' curve. As both Q L and 8 are known, the absolute density of the HJ. in the cell can be determined from either the total area of the absorption curve or a fit to the wings of the dispersion curve.  Chapter  3.  Electron  Spin  36  Resonance  The absorption and dispersion are related by the Kramers-Kronig relation  'H - x'(oo) = - 11/  X  ,  (3.7)  where the integral takes only the principal value. Out in the wings of the dispersion, in the region where x"^)  =  > the dispersion can be expanded as  u  x'M = E H ^ T . ,•=0 K - W) where A\ = £ /f^ a^x"^)? is  7 r _ 1  ( - ) 3  8  times the area under the absorption curve. Even in  samples with moderate densities, the reflected signal from the cavity tends to approach the saturated level, and consequently, the density determined by the area of the absorption curve is less accurate than the density found by a fit to the wing of the dispersion curve. Furthermore, the densities found using the dispersion curves are not as sensitive to the parameters used to reconstruct the susceptibility curves.  Chapter 4  Recombination Rate Studies in Spin-Polarized Hydrogen  4.1  Introduction  This chapter presents the two-body recombination rates measurements in HJ, between 0.2-0.8 K and discusses the observation of resonant recombination in HJ,. The two-body recombination rate in HJ. can be accurately determined only in a controlled environment, because we must be able to separate out the contributions of the two-body recombination processes to the total rate of decay, which has contributions from other two-body and one-body processes; so not only must the absolute density be measured, but also the conditions that affect the rates must be uniform throughout the sample space, so that the analysis of the data does not involve convolving temperature and magnetic field dependences into the effective rates of the system. We obtain this situation by trapping the HJ, in the cell with a fountain-effect valve that isolates the gas in the cell from the r.f. source and prevents the atoms from experiencing fluctuating conditions. The conditions of constant volume and temperature, and uniform magnetic field greatly simplifies the data analysis and increases the certainty of the recombination rates that are obtained. The observation of the surface bound atoms will be discussed in a later chapter. All the results we will discuss in this chapter have been previously published as references [44], [47], and [48]. An indepth study of low temperature resonant recombination processes in atomic hydrogen systems can be found in the thesis of Reynolds[49].  37  Chapter 4. Recombination Rate Studies in Spin-Polarized Hydrogen  4.2  38  Recombination Rate Measurements in Spin-Polarized Hydrogen  The early ESR experiments were concerned with measurement of the temperature dependence of the two-body recombination rates in Hj at temperatures below 0.5 K [41]. Since the work described in this chapter is a continuation of the measurements begun by B.Statt, we will only briefly describe the procedures for taking and analyzing the data. Further details can be found in the references [41] and [44]. The results of the present work suggest that, for T > 0.5K and the conditions of this experiment, the simple scaling rule for the two-body recombination rates in Hj as a function of magnetic field (as discussed in Chapter 2) is not generally valid. We will show that the breakdown of the scaling rule is due to the activation of a resonant recombination process. . A decay data set is obtained as follows: first, the cell is filled with Hj atoms created in the pulsed r.f. source with the "fountain valve" open. When the desired density is reached, the power to the "foutain valve" is removed and one starts to collect raw data sweeps, as previously described, while waiting for the valve to close. The sweeps are collected until almost all the atoms have recombined. Using the method described in chapter 3 the | a) and | b) atom densities can then be extracted from each sweep. The initial densities of atoms in the |a)- and |b)-states are determined from the sweeps taken immediately before the valve closed. A set of model rate equations are then fit to the density decay curves using numerical integration methods. 4.3  Rate Equations  The evolution of the densities of atoms in each of the four hyperfine states can be described by a set of four coupled nonlinear differential equations. When the total bulk density is less than 2 x 10  1S  cm  - 3  and temperatures greater than 0.2 K, only two-body  4.  Chapter  Recombination  Rate Studies  in Spin-Polarized  39  Hydrogen  recombination processes need be included in the rate equations . Since the active rel  laxation processes are also at most two-body, the appropriate rate equations are second order in the densities and can be written as ni =  ^Kijninj(l+6ij)-^(Gij,kininj-Gu,ijn^ i  i,k,l  j  (4.1) where the indices denote the hyperfine states of the participating H atoms, Kij is the two-body recombination rate involving | i)- and | j)-state atoms, Gjj,ki is the rate constant for the two-body relaxation reaction i + j —> k -f 1, and gy is the rate constant for one-body relaxation i —• j.  These rate equations can be simplified by assuming that  processes involving two upper-state atoms are negligible, and that the number of recombination constants can be reduced. The small upper-state atom densities and the larger recombination rates between upper- and lower-state atoms justify the use of the first approximation. In the situation where the Van der Waals process is the only active recombination mechanism and when the "sudden approximation" is applicable, each of the rates KJ; can be written as a linear combination of the zero-field recombination rates in hydrogen, K ho and K ort  para  (as shown in equations 2.26 - 2.29). We have used values  from the literature for the spin exchange [50], electronic [51], and nuclear spin relaxation [42]. Also, to speed up the fitting, only the rate equations for the |a) and |b) densities are integrated; the | c) and | d) densities are assumed to change more slowly than the | a)- and | b)-atom densities and they can be calculated from the rate equations by assuming that hd and n are negligibly small. Numerical integration of the four coupled c  equations have confirmed the validity of the adiabatic approximation for the conditions of our experiment. The effect of the microwave induced relaxation of the atoms during a ESR sweep of a resonance is also included in the numericalfittingprocedures. ^hree-body recombination processes will become important at T < 0.15 K. for the conditions in these experiments.  Chapter  4.4  4.  Recombination  Rate  Studies  in Spin-Polarized  Hydrogen  40  Results  The values obtained for  and K b from the fits to the decay curves are shown in a  Figure 4.1. Qualitatively, the rates have the expected temperature dependence. If the surface and bulk atoms are in thermal equilibrium, then the effective bulk two-body recombination rates, which have contributions from the bulk and the surface processes, can be written as  where K-j is the bulk process rate and K-j is the intrinsic surface rate, and  = 30cm  -1  is the area-to-volume ratio of the cell. The bulk rate can be expressed as Kl  = o v,  (4.3)  i3  where <r,j is the cross section for the reaction and v is the average relative speed of the atoms involved. Similarly, Kfj =  AJJV , 2  where Ajj is the crosslength of the surface process  and v is the 2-D relative average speed of the atoms involved in the reaction. As the 2  temperature is lowered, the surface contributions to the rates dominate and one expects that the plot of ln(KijT ) versus 1 / T will approach a straight line with a slope of 2Ee/kB. 1/2  Below 0.5 K, the recombination rates determined by the present experiment coincides with the results of previous measurements of the rates [44, 37], and above 0.5 K, the upturn in the K ^ values would at first sight appear to be in accord with the recombination rates at 40 kG that are predicted by the the low field measurements[53]. However, the measured rate of K b, which should follow the K a  u  rates and be also increasing as the  temperature increases, are not. We believe that the high temperature values of the K b, a  as determined from the fits to the decay curves, are distorted and pulled by the much larger K  M  rates, and should not be considered as reliable as the Kaa measurements. As  such they have not been included in the plot of the recombination rates. We expect that  Chapter 4. Recombination Rate Studies in Spin-Polarized Hydrogen  41  1/T (K) Figure 4.1: Temperature dependence of the measured two-body recombination rate constants Kaa (crosses) and K b (circles) in a magnetic field of 41 kG. The curve is the best fit to the sum of formation limited resonant recombination in the bulk and constant cross length Van der Waals recombination on the /- He surface. The dotted line is determined by appropriately scaling the zero-field results of Morrow et al. [52] assuming that there are no active resonant recombination processes. a  4  Chapter  4. Recombination  Rate  Studies  in Spin-Polarized  42  Hydrogen  if we could accurately measure the rates of the K b above 0.5 K, then we would find a  that the rates would continue to decrease at a rate set by the K b values below 0.5 K. a  Previous measurements of the rate constants at higher magnetic fields [54, 38, 46] and with zero-field [55] measurements have shown that the magnetic field dependence of the rates constants is given by e#. The zero-field recombination rate K in an unpolarized hydrogen gas can be derived from the 41 kG rates of  a  Kaa "t" K<xb 84 »  ,  T  =  K  and K b as / A A\ -  ( 4  4 )  if one assumes that the only active recombination process is the Van der Waals process. As shown in Figure 4.2, when the derived zero-field values are plotted with the measured zero-field results of Jochemsen et al. [55] (with the surface contributions scaled from A/V  = 4cm  -1  to our A/V), a large discrepancy between the two results at T > 0.5 K  is apparent. Where the derived values deviate from the zero-field measurements, the derived Ko values are larger than the zero-field results, but they increase more slowly with the temperature than the measured zero-field results. The upturn in the zero-field rate is due to the increase in the bulk reaction rate of H + H -f He —* H + He, which 2  follows the rapid increase of the He vapour pressure. Moreover, the results of Greben 4  et al. [10] indicate that, at temperatures above 0.7 K, the Van der Waals recombination process in HJ. results in K b > K a  M  (K tho > K or  para  ) . As the measured  rates are  much larger than the the corresponding K b values, some other process must be found a  to explain this discrepancy. The temperature dependence of the upturn in the K  M  at T > 0.5 K is suggestive of  a process with an activation energy of 5 K, which is approximately the Zeeman energy splitting in the 41 kG field. This temperature dependence of the K  M  is suggestive of a  recombination process initiated by rapid electronic relaxation of the HJ, atoms: hydrogen atoms in the upper states rapidly recombine with other HJ. atoms, since the rate of  Chapter  4.  Recombination  Rate  Studies  in Spin-Polarized  Hydrogen  43  Figure 4.2: The temperature dependence of. the zero-field values calculated from the measured high field values (open diamonds) and the values measured by Jochemsen et al.[55](squares), which were suitably modified as explained in the text.  • Chapter 4. Recombination Rate Studies in Spin-Polarized Hydrogen  44  these recombination processes are not suppressed by the t\ factor. We have considered and rejected this electronic relaxation enhanced process as a possible explanation. The measured two-body electronic spin-relaxation rate is consistent with theoretical [43] and previously measured values[51, 9]; this measured rate is too small to account for the observed upturn. The resonant recombination processes in HJ. at temperatures below 1 K have been previously considered[43, 1], but the rates of the processes in HJ, at these temperatures were considered to be negligible compared to the rate of the Van der Waals processes. The present results indicate that the resonant recombination process activated by the predissociation of the topmost bound state of H can be a very efficient channel for destroying 2  HJ,. Resonant recombination has already been discussed briefly in section 2.5; here we fill-in some of the details and consider the results of the calculations by Reynolds[49]. Basically, the resonant recombination processes in HJ. is not inhibited by the low temperature of the system, since in the appropriate magnetic field, two unbound HJ, atoms can have the same energy as a bound hydrogen molecule; consequently, as the two physical situations have the same energy, two free HJ. atoms can easily tunnel to the bound molecular state. The conservation of the symmetry of the wavefunction of the two hydrogen atoms involved in a resonant recombination event determines the allowed spin states of the initially free atoms. In a magnetic field B, the H2(14,4) state has a binding energy D, while a pair of well separated HJ. atoms with m = — 1 have a total potential energy A « —^ hB B  e  (as shown  in Figure 4.3) plus some kinetic energy. If the appropriate B field is applied, then the two configurations may have the same energy. The overall molecular state of H must be symmetric under the exchange of atoms. 2  The spin state of the H (14,4) state is | 0000), where | Smslmi) is a spin ket, with S 2  the total electron spin, I the total nuclear spin, and mi the spin projection along the  Chapter  4.  Recombination  Rate  Studies  in Spin-Polarized  Hydrogen  45  Figure 4.3: The singlet and the m = —1 triplet, plus the Zeeman energy, potentials for the case A = 2p B = 5.5K and D = 0.7K. s  e  Chapter  4. Recombination  Rate Studies  in Spin-Polarized  46  Hydrogen  polarizing magnetic field. Although the total electron spin operator does not commute with intra-atomic hyperfine interactions, which are the terms responsible for the predissociation of the H2(14,4) state, the total spin J and mj = mi + m do commute s  with them. As a consequence, the angular dependence of the wavefunction is conserved when a molecule in the H2(14,4) state dissociates via the intra-hyperfine interaction. The H2(14,4) molecule can only dissociate into two well separated | a)-atoms; therefore, the reverse resonant recombination process can only take place between two | a)-atoms and it will only contribute to the K rate. M  If the resonant recombination process is limited by the formation rate of the H2(14,4) molecules H2, then by equating the chemical potential of the  to that of the two free  j a)-atoms, and using the principle of detailed balance, the rate  can be written as (4.5)  where T is the lifetime of the predissociated state. Note that the main temperature dependence of equation 4.2, with  is the exp (—^p')- We originally fit the data to equation 4.1 using given by equation 4.5 and  The fit, which is shown in Figure 4.1, gives D = 2A /e aa  2 4  = 1.5(0.1)A, and E  B  independent of the temperature. 0.7(0.1)/f, r =  63(9)ps, X  para  =  = 1.15(0.02)A:[47]. The so derived value of D would  have improved considerably the spectroscopic value of 0.7(0.9)if , but recent calculations 2  indicate that the value of D is not well determined by the experiment. Reynolds and Hardy [49, 47] show that not only is the assumption of formation limited recombination invalid for the conditions of this experiment, but also that the spin-exchange corrections to the recombination rate must be included for an accurate description of the decay kinetics. They find that  must be modified from equation 4.5 to (4.6)  See footnote 9 in reference [47].  Chapter  4.  Recombination  Rate  Studies  in Spin-Polarized  47  Hydrogen  where E = A — D, and A is the thermal deBroglie wavelength for a particle with reduced u  mass u = mn/2, and r is the stabilization time due to collisions of the HJ with He atoms. 4  c  The rate r  - 1  = v 5nH , where v = (8kBT/7ru)2 is the average relative speed between a u  u  e  H2(14,4) molecule and He, n# is the He density, and ct is the cross section for relaxation 4  4  e  of HJ by collisions with He. 4  The coupled channel calculations show that the stabilization rate, due to collisions with the saturated He vapour, is much larger than (more than 10 times) the calculated 4  predissociation rate in a field for 41 kG, if the temperature is above 0.6 K . Therefore, the assumption of formation limited recombination is only valid at temperatures above 0.6 K. We recognized this, but given that at 0.6 K no sudden change in the Kaa rate was observed, we made the assumption that at the lower temperatures the collisions of HJ with H atoms are sufficiently frequent enough to stabilize the HJ molecules. It is not an unreasonable assumption to make. At 0.6 K , the Hj density in the cell (usually «  10 cm~ ) is comparable to the He vapour density and the HJ stabilization cross 15  3  4  section in collisions with HJ, atoms is probably as large as the He cross section. One 4  might expect the resonant recombination process to be always in the formation limited regime for the experimental temperature range. A more complete theory for the stablization of HJ in collisions with HJ.-atoms shows that the above assumption neglects the possibility of spin-exchange collision reactions during the stablization of the molecule. The stabilization reactions involving HJ, atoms are HJ + a  H  2  +a  (4.7)  HJ + b  H  2  +b  (4.8)  HJ + a  H  2  +b  (4.9)  HJ + b  H  2  +a  (4.10)  Chapter  4.  Recombination  Rate Studies in Spin-Polarized  48  Hydrogen  Previously, we had considered only the reactions 4.7 and 4.8, but all four reactions make significant contributions to the stabilization rate for the conditions of this experiment. Additional terms to the rate equations 4.1 are required when the exchange mechanisms are included in the resonant recombination process: n  =  h  =  b  where K  M  - ^ K ^ n i - n )n  (4.11)  2  a  a  1c  a  — K ( n - n )n a a  6  a  -  2 a  2K n tta  2 1  1c  is defined in equation 4.6, and T  c  = {K2 + K )(n 3  + n) + lf ,  (4.12)  e  h  a  in which T^ is the He contribution to the stabilization rate, K is the rate for reactions 4  e  2  4.7 and 4.8, and K3 is the rate for reactions 4.9 and 4.10. In the limit when T^ >e  K3(n -f rib), the effect of the spin-exchange stabilization is negligible and equations 4.11 a  reduce to ri  a  =  — 2K n . 2  aa  l  We have re-analysed the data with the modifications to  the rate equations given by equations 4.11, and find that over the entire experimental temperature the fits are better than the original fits, which assume no spin-exchange stabilization. Unfortunately, when both the direct and spin-exchange stabilization processes are included into the analysis, the accuracy of the determination of D is reduced. Since the present data does not indicate how dominant exchange stabilization is, some estimate of 3  the rates K and K 3 are required in order to determine D precisely. The original fits, 2  which assumed K  2  »  K , find that D = 0.7 ± 0.1K, and fits assuming K » K find 3  3  that D = 0.2 ± 0.1K. 3  T h e rates are difficult to calculate as the processes are essentially three-body in nature.  2  49  Chapter 4. Recombination Rate Studies in Spin-Polarized Hydrogen  4.5  Conclusions  The temperature dependence of the surface Van der Waals recombination rates are unknown; as yet there exists no microscopic theory for the surface processes comparable to the work of Greben et al. on the bulk processes[10]. Their results indicate that at very low temperatures the bulk rate for ortho-hydrogen formation K o  r t h o  should be zero,  whereas the rate for para-hydrogen K p ^ is finite. A simple heuristic argument suggests that the surface processes should have the same behaviour at very low temperature. In the "sudden approximation" theory of recombination of hydrogen, only pairs of | a) atoms that collide with even angular momentum / can recombine and they will form only para-hydrogen. On the other hand, only pairs of | a) and | b) atoms which collide with odd / can recombine and they will form only ortho-hydrogen. The selection rules for the recombination processes are due to the conservation of the angular symmetry of the spatial wavefunction during the collision. The collisions at very low temperatures are predominantly s-wave. Therefore, at low temperatures, mostly pairs of | a) atoms will recombine, which will result in K  p a r a  ^> K ho- This should be true whether surface or ort  bulk processes are considered. In Figure 4.4, 7 = ^f- = ffi"* are shown for the tempera4  ab  ortho  ture range where the surface contributions dominate. The plot shows that 7 increases as the temperature decreases. Wefindthe same general behaviour in 7 with temperature as Statt[44], but our values are generally smaller than those presented there. The present 7 values are more in line with the values found from the re-analysed data of Yurke et al. [46]. The temperature dependence of the ratio of the recombination rates suggests that an accurate measurement of E B cannot be found by taking the slope of a plot of In K T / ver1  2  sus 1/T as is commonly done. Most the measurements of E B suffer from the defect that they all must assume that some crosslength is independent of the temperature. To date,  Chapter 4. Recombination Rate Studies in Spin-Polarized Hydrogen  4.0  — —r 1  3.0 • XL  • • •  0  •  i  I  i  1  i  i  i —  I  50  r  •  •  2.0  as  1.0  0.0  i  2.0  i  i  j  i  2.5  3.0  1/T  i i_  3.5  4.0  (K" ) 1  Figure 4.4: Temperature dependence of 7 = j ^ - in a 41 kG field. The increase in gamma at the higher temperatures is due to the contribution from the resonant recombination process, which affect only Kaa.  Chapter 4. Recombination Rate Studies in Spin-Polarized Hydrogen  51  the most accurate measurement of the binding energy finds E B = 1.011(0.010)K[26] . The 4  discrepancies between the values of the binding energy measured by the pervious experiments are due to the temperature dependence of the cross-lengths, which are assumed to be temperature independent. In summary, we have observed resonant recombination in a gas of Hj in field of 41 kG and at temperatures below 1 K, and have identified the H.2(14,4) state HJj as the active resonant state. Resonant recombination in hydrogen is a two step process: a "resonant" state molecule is formed; then through a collision with another particle, the molecule makes a transition to a more deeply bound state of the hydrogen molecule. Initially, it was thought that these low temperature measurements of the resonant recombination rates would determine the energy of the H2(14,4) state more accurately than present spectroscopic methods. The resonant recombination data has been re-analysed using the recent theoretical results of Reynolds and Hardy[47]. Those results indicate that the energy of the H2(14,4) state is not as accurately determined as once thought. The initial simplifying assumption that, at the lower temperatures, only the direct stabilization collisions between the "excited" molecules in the H (14,4) state and the HJ. atoms are 2  important, is invalid for the actual experimental conditions: contributions from spinexchange stabilization collisions must be included when analyzing the data. During a spin-exchange relaxation collision, the HJ. atom not only stabilizes the H^ molecule, but it also exchanges its proton with one of the molecule's. The additional process channels introduced by the spin-exchange process drastically complicates the rate equations that describe the evolution of the HJ. densities. The data at present is unable to identify the contributions from the different relaxation processes. If the direct process dominates, then the dissociation energy D of the H (14,4) is 2  This value is found from surface frequency shift measurements in a cryogenic hydrogen maser. The size the shift depends on the lifetime of the HJ. adsorbed on the He surface. 4  4  Chapter 4. Recombination Rate Studies in Spin-Polarized Hydrogen  52  0 . 7 ( 0 . I f the exchange process dominates then D = 0.2(0.1) K. A more satisfactory analysis of the data awaits the calculation of the direct and spin-exchange stabilization rates, which will involve solving three-body problems. Perhaps, with a carefully applied study of the resonant recombination process by varying the magnetic field, temperature, and area-to-volume ratio of the cell, one might be able to separate out the contributions from the two relaxation processes and accurately determine the dissociation energy of the H2(14,4) state. Since the uncertainty in the spectroscopically measured dissociation energy Do,o of the ground state of the hydrogen molecule is the dominant source of uncertainty in determining the dissociation energy Di4 of the H (14,4) state, an accurate i4  measurement of the D i  4|4  2  would give an improved value of Do,o-  Chapter 5  Studies on Spin-Polarized Deuterium  5.1  Introduction  A natural extension of our work on recombination mechanisms and rates in HJ, is the study of similar processes in spin-polarized deuterium D|; experimentally it is much more difficult to study DJ. than it is to study HJ,. This is reflected in the lower quality and paucity of the DJ. recombination data; the surface two-body Van der Waals recombination rates in DJ. are measured over a very limited temperature range. In some of the samples, the preferential recombination of the | a)- and | /3)-atoms in the DJ. gas produced a gas of DJ.J., a gas in which both the nuclear and electron spins in the atoms are aligned antiparallel to the polarizing magnetic field. Some of the samples contained both DJ.J. and HJ.J.; a rather unusual mixture of two completely spin-polarized atoms. Simplified rate equations were fit to the recombination decays in the mixture. The results indicate that the zero-field surface crosslengths for recombination of hydrogen with deuterium atoms is about six times larger than the already very large cross lengths for recombination of two deuterium atoms. Three different measurements on DJ. are discussed in this chapter. Beginning with a quantitative study of the two-body recombination rates in DJ,, the measurements that are discussed will become increasingly more qualitative as the number of samples observed decreases. The decrease in the number of samples of DJ.J or mixtures of DJ.J; and HJ.J. studied is due to the difficulty in readily creating the samples and studying them without  53  54  Chapter 5. Studies on Spin-Polarized Deuterium  a major catastrophe interrupting the measurements . All of the experiments discussed 1  in this chapter have been previously published [56, 57].  5.2  Rate Equations for Gas of DJ.  Analogous to the evolution in a gas of HJ., the evolution of the densities of atoms in each of the Zeeman states of deuterium can be described by a set of six coupled second order differential equations. In principle, we could try to fit these equations to the measured decay curves. But, the quality of the data is not high enough to warrant such an effort . 2  Instead, we have fitted much simpler rate equations to the data. If only the Van der Waals recombination and one-body extrinsic nuclear relaxation processes determine the decay rate of the sample, then assuming that the densities in the upper energy Zeeman states are negligible, the rate equations are h  =  — 2K n  hp  =  — K pn np  n  =  — K rc n  a  7  2  a a  a  — K pn np a  a  — K ~ n 'y — Gi(ra — np) Q  t  (5-1)  a  a  — 2Kppnp — Kp^npn^ — G\(np — n^) + G\(n — np)(b.2)  a  a7  Q  a  7  — Kp^npn^ 4- G\(np — n ).  (5-3)  7  Within the context of the "sudden" approximation for recombination, K  7 7  = 0, since  the electronic spin wavefunction of two 17)-atoms has no projection onto the electronic singlet molecular state of deuterium. The relative sizes of the various rate constants in DJ. are not the same as in HJ.. In the case of HJ., the spin exchange rate of the only active channel, Gac—bd (or Gbd—ac) = 2.9 x 10~ cm" sec [50], is clearly the largest rate in the problem. The presence of 13  3  -1  the spin-exchange process strongly effects the decay process and must be included in the Due to the particular nature of the procedure used to generate the high densities of DJ. required to create DJ , there was always a chance that the valve might open or the dilution refrigerator might cease functioning and "crash", i.e. suddenly warm up. The plot in figure 5.1 is one of the best DJ recombination data. 1  1  2  Chapter  5.  Studies  on Spin-Polarized  55  Deuterium  T  1  r  TIME (s)  Figure 5.1: Representative decays of | a), | 8), and | 7) deuterium atoms, taken after the cell walls have been coated with H2. The decays are taken separately, using initial conditions as identical as possible. The solid lines are the computer fits generated using equations 5.1-5.3.  Chapter  5.  Studies  on Spin-Polarized  56  Deuterium  rate equations . In the case of DJ., however, the rates for the three exchange channels, 3  which are not inhibited by the large magnetic field, are not the largest rate constants in the system. The three channels, which are listed in sections 2.6, all involve collisions between a lower and upper energy Zeeman state atom. The fits to the data, which are described in the next section, demonstrate that the recombination rates Kuc between two atoms each in a lower energy Zeeman state are approximately 1 x 10~ cm sec 15  3  -1  throughout the temperature range of the measurement.  If Van der Waals recombination is the dominant decay mechanism and the use of the "sudden" approximation in calculating the rate constants is valid, then the recombination rate Kui between an atom in a lower energy Zeeman state and an atom in a upper energy state is related to Kik by (5.4)  Ku, where  is defined in section 2.3.2 and in a 41kG field e-, = 1.8 x 10" . Approximately 2  6  10 times larger than the exchange rate, K^ ~ 5 x 10 cm sec _lo  _3  -1  is the largest rate in  the problem; collisions between upper and lower energy state atoms will usually result in recombination and not in spin-exchange reactions. Hence, to first order, the spin exchange terms are not required in the rate equations. Moreover, the Kui rates are large enough that any atoms in the upper energy Zeeman states will immediately recombine with a DJ. atom and be removed from the gas. This chemical cleaning action will tend to ensure that the | 6)-, | e)- and | ^-densities are very small, and one can neglect terms involving these densities in the rate equations. This major assumption reduces the required number of equations to three. If the DJ, is not too highly polarized into one Zeeman energy state, then the intrinsic spin-relaxation terms in the rate equations act like correction terms on the recombination rate. Thus, those relaxation rates can be The rate is large enough that the results of the HJ fits are independent of the exact value of the exchange rate used in equations 4.1. 3  Chapter  5. Studies  on Spin-Polarized  57  Deuterium  also ignored. The electronic spin relaxation of the DJ. atoms to upper energy Zeeman states is inhibited by the electron spin-flip energy (5.5K), as the temperature of the gas is usually < 0.7K. Because under the conditions of these experiments the one-body nuclear relaxation rate G i is suppressed exponentially as a function of the Lamor frequency of the transition[58], the transition rate of | a)  | 7) is much smaller than the other G i  rates in DJ., and is ignored. To first order, the frequency splitting between the | a) and I 0), and I 8) and | 7) are the same. We make the approximation that these one-body nuclear relaxation rates are equal. With all these approximations, the rate equations reduce to the form in equation 5.1 -5.3. In the "sudden" approximation theory of Van der Waals recombination, the two-body recombination rates in equations 5.1-5.3 can be written as a linear combination of two fundamental rate constants K tho, the zero-field formation rate of o—D (I = 0,2), and or  2  Kpara, the zero-field formation rate of p—D (I = 1). Following the calculations in HJ., we 2  find that the DJ. rates can be written as Kao,  —  ie K 2  2 £> para —1\) [3K tho + Kpara] c  K0 a  =  Ko,-y  xv  or  — CJTJ  [Kortho + Kpara]  Kw  =  2^1) Kpara  /37  =  — t£)J\ rtho  K  0  With these relationships, the number of free parameters in equations 5.1 - 5.3 is reduced from 6 to 3.  58  Chapter 5. Studies on Spin-Polarized Deuterium  5.3  Results of the Experiments on DJ.  The two-body recombination rates in HJ. have been measured using the present ESR apparatus to the 5% level.; the measured corresponding rates in DJ. are rather less accurate. This is due to major differences between the behaviour of DJ. and HJ.. First, the maximum DJ. densities obtained are much lower than the maximum HJ, observed in the experiments: a DJ. density of 10 cm 14  -3  compared to 2 x 10  15  cm  - 3  in Hj. Second, the  lifetime of a sample of DJ., ~ 60 sees at T = 0.7K, is shorter than the lifetime of several hours for a typical sample of HJ.. The much shorter decay times of the DJ. samples require that we use a different method from the one used to measure the density decay curves in HJ.. It is impossible to accumulate enough density measurements on a sample of DJ. by repetitively and sequentially sweeping the magnetic field through the ESR resonances of all three of the DJ. states. For onething, rapid slewing of the magnetic field induces a significant amount of eddy-current heating. At best, the induced heating causesfluctuationsin the amount of /- He in the cell. The small changes in the amount of dielectric material in the mi4  crowave cavity modulates the phase of the reflected microwaves, which in turn distorts the measured voltages. At worst, the induced heating warms up the fountain pump and causes the valve to open. The modified procedure to collect DJ. data is simple, but only moderately satisfactory: the decay curve of each species is measured in separate fillings of the cell. During each filling, we check that the initial densities are the same as the previous ones by measuring all three steady state densities. Then before the valve closes, we start to repeatedly measure the density of one species. The three different curves are then synchronized by shifting their time axes so that the closing of the valve coincide. The rate constants are found by numerallyfittingthe rate equations to the constructed decay curve. Thefittingprocedure takes into account the perturbation effect due to the  Chapter  5. Studies  on Spin-Polarized  59  Deuterium  microwave excitation of the atoms during the sweep of the line and the fact that the measurements are taken on three separate samples. The initial set of DJ. recombination (high G) data was taken before the cell walls were coated with molecular hydrogen. A second set of data (low G) was taken after the cell walls were coated with a layer of H of the order of 1000 A thick . The layer of 4  2  molecular hydrogen reduces the extrinsic one-body nuclear relaxation rate, which is due to the magnetic impurities in the walls of the cavity . 5  The behaviour of the DJ. decays depends on the relative importance of the one-body relaxation process to the two-body recombination process. In samples with strong relax-. ation (Gi > Kn), the atoms in all three states recombine at a common rate ( | ) ( K  aa  +  K ^ - f Ko/j + Kc-y -r-K^-y). In the weak relaxation case (Gi <C Kn), the \ o>}- and |/?)-atoms decay at a faster rate than the | 7)-atoms, since K  7 7  = 0. Eventually, the DJ. sample  becomes electron and nuclear spin-polarized, and decays at a rate 2Gi. The degree of nuclear polarization obtained in the sample depends on the size of G i . Substantial polarization of the DJ, gas is observed only after the cell walls are coated. An example of the polarization of the gas sample is shown in figure 5.1. Only in the case of the low G recombination data could the numerical fitting routine determine separately K tho and K or  we find that K ho ^ K ort  p a r a  p a r a  . Over the narrow experimental temperature range  to within experimental uncertainty. The fits to the high G  data could not distinguish between the separate contributions of K ho and K ort  we assumed in the fitting routines that K ho = K ort  p a r a  p a r a  , so  , as found by the low G i fits. In  figure 5.2, we have plotted the rate constant, K = -^-(Kortho + Kpara), for the total density of DJ, under the condition that no nuclear polarization exits. Also included infigure5.2 T h e cell walls are coated with molecular hydrogen instead of deuterium since the low temperature source is much less efficient at producing D l and it would take an unreasonable amount of time to deposit deuterium. These measurements were taken with the "old" cell, which had more magnetic contamination in the wall. 4  5  Chapter 5. Studies on Spin-Polarized Deuterium  \2.  IJ6 r '  2J0  60  2.4  (K* ) 1  Figure 5.2: Plotted are the average recombination rate K in Dj calculated from the measured rates as explained in the text. The triangles are values obtained from the low Gi decays. The open circles are values calculated from high Gi decays. The solid circles were found from fits to the decay |7)-atoms, under the assumption of no nuclear polarization (see discussion at the end of the chapter). Also shown as crosses are the results of SW[11] where we have suitably scaled the rates for the A / V ratio and the magnetic field dependences as described in the text. The solid line is a fit to their data with A / V = 23 c m assuming a binding energy of 2.6 K. The dashed line is for A / V = 8.09 c m , the smaller of the limits given by SW, and the value later thought to be most likely [59] - 1  -1  61  Chapter 5. Studies on Spin-Polarized Deuterium  are results of SW[11] appropriately scaled for our values of the field and A / V ratio . We 6  have chosen A / V = 23cm" for their experimental cell so that their data extrapolates 1  smoothly to our data, assuming that a binding energy of 2.6 K. Recently SW[1] have argued that 8.09cm  -1  is the mostly likely valve for their A / V ; however, then the two  sets of data do not agree very well. But, given the quality of the present data, this lack of agreement is not so disturbing. This disagreement may be due to the presence of an active resonant recombination channel in the deuterium gas[49]. Unfortunately, the temperature range of the Dj measurements is too small to determine the binding energy of Dj to the /- He surface. The increasing slope of the plot of 4  KT» versus 1/T in figure 5.2 suggests that we are measuring the rate of a surface process, and in principle, we should be able to determine E B using equation 4.2 . 7  If we use E = 2.6(0.4)K[59] with A / V = 30 cm" and B = 40.7 kG in equation 4.2 1  B  and equation 5.5, we find that the zero-field surface recombination cross lengths = 5600A, or 1600 times larger than  Apara  A  p a r a  X rtho  w  0  in Hj.  Although, in principle the results of the Gi measurements on Dj could be used to determine E B , the complicated temperature dependence of the surface process (beyond the usual T^ exp(EB/kbT) dependence) does not allow the determination of E B of Dj to the /- He surface. 4  5.4  Discussion of the Dj Results  Spin-polarized deuterium, at low temperatures, is much more difficult to study than Hj due to a combination of a larger binding energy of Dj to the /- He surface and a very 4  large cross length for surface two-body recombination. The increase in the binding energy This assumes that we are measuring the surface two-body recombination rate. The temperature dependence of the recombination rate measured SW is indicative of a surface process. Although we know that this is not an accurate method for determining E B , a repeat of this measurement under better conditions would be useful. 6  7  Chapter  5. Studies  on Spin-Polarized  62  Deuterium  of DJ. to the /- He surface is due to a mass effect. Since deuterium and hydrogen are 4  iso-electronic, the Van der Waals attractive potential between the helium surface and the deuterium atoms is, to a good approximation, identical to the one between hydrogen and the surface. The more massive deuterium atom is more strongly bound by the potential well, since its zero-point motion is less than that of hydrogen. It has been suggested by Papoular[60] that the measured binding energy of deuterium on the /- He surface is too large and that the experiment of SW has measured an effective 4  binding energy. He suggests that DJ. adsorbed on the /- He surface form dimers and that 4  these dimers act as a very efficient path for recombination. The dimer enchanced surface recombination rate would have a temperature dependence similar to the usual Wan der Waals surface recombination process, but the usual 2EB coefficient in the exponential temperature dependence of the effective bulk rate of the process would be replaced by 2EB + | e<i \i where t is the binding energy of the dimer. The binding energy inferred a  from the two-body recombination rate measurement (as done by SW) would be E B + which is  larger than the actual binding energy of DJ. to the surface. Given the large  uncertainties in both the present estimate of the DJ, binding energy[61] and the measured value, the discrepancy between them does not appear to be very significant. The dimer theory would conveniently explain the much larger recombination cross lengths in DJ,, since one would expect the recombination process in a gas of bound dimers to be larger than between free DJ, atoms. However, any process involving dimers that we have been able to think of would result in a large difference of the two intrinsic rates, whereas we find that K ho = K ort  p a r a  . For example, the decay mechanisms of two atoms in  a dimer state could be similar to the resonant recombination mechanisms and would then contribute to the recombination rate in a similar manner. The intra-hyperfine interaction can induce transitions from the electronic triplet-state dimers to bound molecules, and since the potential interaction between two atoms is isotropic, the angular momentum /  Chapter  5.  Studies  on Spin-Polarized  63  Deuterium  is conserved in the recombination reactions. If the initial nuclear spin state of the dimer is I = 1 (as is the case of dimers with / = 0), then the process would yield only I = 0,2 molecules and would result in K ho ^ K ort  p a r a  . Recently, Kagan et al. [62] have calculated  the low energy scattering phase shifts in collisions of quasi-2 dimensional hydrogen atoms and have found that there is a bound state for tritiums atoms, but no bound states exist for HJ.-HJ. or DJ.-DJ, pairs. So, until conclusive evidence for the existence of dimers in DJ. adsorbed on a /- He surface is found, the simpler model of unbound atoms on the surface 4  will be used. Calculations show that a second bound state of DJ. adsorbed onto the /- He 4  surface exists, but the estimated binding energy is ~ lmK[61]. Since the second state is not appreciably occupied when T = 0.5K, its existence should not noticely affect the recombination rates that occur between atoms adsorbed on the surface.  Chapter 6  Doubly Spin-Polarized Systems  6.1  Introduction  During the decay of a sample of DJ. in the cell with low Gi, the preferential recombination of the | a)- and | /3)-atoms can create a sample containing mostly | 7)-atoms, a gas of DJ.J.. A measure of the polarization of the sample is the spin purity P, which we define as P =  ^ «7 +  .  np + n  (6.1)  a  Purities as high as 0.8 were measured near the end of standard decays, where the total density of the sample was usually about 5 x 10 cm . The highest initial densities of 12  -3  DJ. were created by pulsing He gas into the discharging source, just before the fountain 4  effect valve was about to close. The pulse of gas not only compressed DJ. into the cell, but also closed the fountain pump valve and trapped DJ. in the cell. The trapped sample density was typically 5 x 10  14  c m . With this pulsed technique, we obtained samples -3  with Oy « 3 x 10 c m , along with n^ and n 13  -3  < 3 x 10 cm~ , which is the sensitivity n  a  3  limit of the spectrometer. The corresponding spin purities of the samples were greater than 0.98. For the present experimental conditions, the observable decay processes in DJ.J. in a cell with weakly relaxing walls are rather uninteresting. The decay rate of such a sample is limited by the small Gi; therefore, only the rate Gi can be measured. We have measured the Gi rate in DJ.J. for the temperature range 0.3 to 0.7 K. These results are 64  Chapter 6. Doubly Spin-Polarized Systems  65  useful primarily as a consistency check of the accuracy of the fitting routines used to analyze the DJ. recombination data. At the temperatures where the recombination data for DJ. exist, the Gi found by the fits to the decay of DJ. agree (to within experimental uncertainty) with the values found from the measurements on DJ.J.. It is not possible to extract a reliable value of E B from the Gi data, since the Gi has an intrinsic temperature dependence, in addition to the temperature dependence associated with the fraction of time an atom spends on the surface. The size of the intrinsic temperature dependent factor cannot be easily predicted or measured, so any attempt to obtain a value for E B is futile. Consistent with the microscopic theory of Berlinsky et al.[58], our measurements demonstrate that the Gi rate in Dj is much larger than the corresponding rate in HJ.. At T = 0.69 K in the molecular hydrogen coated cell, the lifetime of a sample of DJ. J. is about 27 minutes; whereas, the lifetime of a sample of HJ,I in the cell at identical conditions is of the order of days. High density samples of DJ.J sometimes contained an appreciable density of HJ,J> The contaminating HJ.J. is a legacy of the method used to coat the cell walls with a nonrelaxing layer. When the source is loaded with deuterium, the output flux from the r.f. source is too low to coat the walls with molecular deuterium in a reasonable amount of time. So instead, we initially load the source with a very small amount of molecular hydrogen ( < 1% of the deuterium that is eventually added) into the source, and coat the cell walls with molecular hydrogen. Next, we bury the molecular hydrogen in the source by depositing a thick layer of deuterium, and then start the DJ. experiments. But, since the low temperature source is a more efficient atomic hydrogen source than a deuterium source, it was possible to obtain HJ. in the cell, especially when the discharging source was flushed with helium gas.  66  Chapter 6. Doubly Spin-Polarized Systems  6.2  Recombination Rate Measurements  A gaseous mixture of HJ.J. and DJ. j is an unusual physical system since both species are 100% electron and nuclear spin polarized, in the present experiments we have measured some of the two-body recombination rates between hydrogen and deuterium atoms. In the "sudden" approximation of the theory of two-body Van der Waals recombination in a gas of either HJ. or DJ,, one finds that there are two intrinsic rates, K tho and K or  PARA  ;  however, in the theory of recombination between DJ. and HJ, atoms, there is only one intrinsic rate KHD- There is only one rate constant, since the HD molecule does not require a definite parity in its wavefunction under the exchange of the two atoms. We have performed two different types of measurements on the mixtures. In both of the experiments, we have measured the recombination rates between DJ. and HJ. atoms, and the rates between H and D atoms where one of the pair is in an upper energy Zeeman state. In the first experiment, we observed the densities of the 17)- and | b)-atoms decay through the one-body nuclear relaxation of the | 7)-atoms (a "natural" decay). In the second experiment, we actively excited the | b)-atoms, in the mixture, into the | c)-state, while monitoring the|b) density ( a "burn" experiment). The decay, via natural relaxation, of the densities of the | ) , |b), and |/?)-atoms in a 7  sample which contains a mixture of DJ. J. and HJ.J, can be described by the set of equations n  =  —Gi(n — np) — Kp^npn  hp  =  Gi(n — np) — Kp npn — Kbpn^np  hb  =  — Kbpnbnp  7  7  7  y  y  y  (6-2)  In writing this set of rate equations, we have assumed that the Gi process in hydrogen can be neglected, since its rate is much smaller than the same process rate in deuterium. Assuming that only the | 7), | b), and | /?)-states are appreciably occupied, there are  Chapter 6. Doubly Spin-Polarized Systems  67  no active spin-exchange processes. The effect of including the | a)-atoms, which can be generated by the nuclear relaxation of the | /3)-atoms, was examined. Including the fourth state merely modifies the |/?)-state density by 1% and does not change the overall behaviour of the equations. As the | /?)-atom density is quite small in the highly polarized mixture, we assume that hp = 0. Solving for n in equation 6.2, we find that 7  n  i  —  TP  (-)  b  6  3  where we have neglected the Gi term in the denominator, since the recombination terms are much larger. Substituting equation 6.3 into 6.2 and parameterizing n with respect 7  to nj, we find that —^ = 1 + 2 r ^ , arii)  rib  (6.4)  where r = Kp /Kbp. The behaviour of the solutions to this parametric equation depend y  on the ratio of the recombination rate between two deuterium atoms in the lower energy Zeeman states to the rate between a hydrogen and deuterium atoms both in a lower energy Zeeman state. When the data, such as the decay shown infigure6.1, is fit using 6.4, we find that r = 0.68 at T = 0.68K. In a second experiment, the two-body recombination process between an atom in the an upper energy Zeeman state and an HJ. J. or DJ.J. atom is observed by using microwave radiation to rapidly flip the electron spins of the HJ. j-atoms, this is an induced electronic relaxation mechanism. If the induced one-body electronic relaxation rate is larger than any other relaxation rate, then the dominant decay process will be that of electronic relaxation of the H|j-atoms and subsequent recombination with either an HJ.J or DJ.jatom. The electronic relaxation of the | b)-atoms is induced by adjusting the magneticfieldto the center of the resonant absorption peak. With the assumption that the | c) population is always much less than the | b) population, the height of the absorption is proportional  Chapter  6. Doubly  Spin-Polarized  Systems  68  Figure 6.1: Natural decay of a sample of doubly spin-polarized D and H. The b-atom densities are shown as squares and the 7-atom densities are shown as triangles.  Chapter  6.  Doubly  Spin-Polarized  Systems  69  to the | b)-atom density of the sample. The | b) density during the experiment did not behave as expected. Infigure6.2, the plot of the absorption as a function of time shows a very different type of behaviour from what was expected: the expected rate of decay is proportional to the square of the density, that is its rate is faster than the rate of an exponential decay. The expected behaviour of the | b)-atom density which would be indicative of a recombination process that is bottle-necked by the microwave spinrelaxation rate. Exponential behaviour occurs near the end of the decay. In a similar sample, we have measured the | 7)- and | b)-densities at the end of a series of short irradiations and find that the discontinuity in the slope of the absorption curve roughly coincides with 17)-density decaying to zero. The result of the "burn" experiment can be explained if one assumes that spinexchange cross section between electron spin relaxed HJ.J and | 7)-atoms is larger than the recombination cross section. Then, the most likely result of the collision is the transfer of the depolarization to the I7), rather than the recombination of the pair. This process will continue to regenerate HJ. j. as long as the density of 17)-atoms is large enough. Since the initial D\U density is larger than the HJ.J. density, the relaxed DJ.J, atoms will tend to recombine with DJ.J. atoms rather than interact with a H|J. atom. The effect of the regeneration process is to slow down the decay rate of the HJ.J. density. Once the DJ.J. reservoir is sufficiently depleted, then the dominant reaction in the system will be the recombination of the spin-relaxed HJ,| into molecular hydrogen. This process would be characterized by a rate proportional to the density squared. From simple phase shift calculations of the C7 «-» b6 spin-exchange bulk cross section at low temperatures, we find thatCTHD~ 7.5 A . This is to be compared to the low temA 2 in D-D collisions. Since the recombination rates between D-D atoms are larger than the spin-exchange rates in D-D collisions, the simplest quantum mechanical calculations would suggest that the 07 «-* b8 spin-exchange  Chapter  6. Doubly  Spin-Polarized  Systems  Time (arb.)  Figure 6.2: Microwave induced relaxation decay of doubly spin-polarized H and D  70  Chapter  6.  Doubly  Spin-Polarized  71  Systems  reaction rate is smaller than the recombination rates in the system. For the calculations the of spin-exchange collisions in D-D and H-H, the simple phase shift calculation results for the rates are within 10% of the results of coupled channel calculations[50, 63]. Therefore, it is unlikely that the value of calculated exchange rate in HD is incorrect by more than one order of magnitude. A resonant exchange process would result in a much larger rate, but at low temperatures a mechanism similar to the one responsible for the resonant recombination reaction in HJ. cannot be active, since the relevant exchange collisions involve pairs of atoms with m = 0. s  Having established the relative sizes of the various rates, we ignore, for the moment, these results and analyse the burn experiment assuming that the spin-exchange process is the most dominant one, since we cannot otherwise analyse the data. Initially, the sample contains only I7) and |b)-atoms, but once the microwave power is applied, some | b)-atoms make a transition up into the | c)-state. The spin-exchange process, c+7 —» b-(-<5, will create | c5)-atoms in the sample. Because a "burn" measurement takes about two minutes, the one-body nuclear relaxation process in DJ.J. and HJ.J. may be neglected. For the same reasons as in the DJ. case, we neglect all the two-body relaxation terms and two-body recombination terms between atoms which are in the upper energy Zeeman states. With all these approximation, we find that the rate equations for the system are n  =  — K n n — K^n^n^ — c(n n — ni,n«)  h$  =  —Kybfi-fTii,  rib  =  -K6 Ti{,n — Ksbnsnb + e ( n n  h  =  —Kb n n  7  c  7 C  7  c  c  c  c  —  Kysn^ns  c  b  c  + e(n n — ribUs) c  c  — Ks nsn c  c  (6.5)  7  (6.7)  — ribn$) — W(rib — n )  7  c  — e(n n - n n ) c  (6-6)  7  7  b  s  + W(nb — n ),  (6.8)  c  where c is the spin-exchange rate and W is the microwave induced transition rate. If we assume that, for the time scales of interest, hs and h are small compared to n and nj c  7  Chapter  6.  Doubly  Spin-Polarized  72  Systems  terms, and that the spin-exchange rate is large enough to ensure n n = c  7  ribns,  then we  can write drib  = 1+  (6.9)  Tib  where R = K ,+K ' Integrating equation 6.9, we find that <t  7e  n  =  7  Cnf R  _  - n 1  , ,  ,  (6.10)  h  fiin  where C is a constant of integration. We have re-analysed our previously published data[57] using 6.10 and find that at T = 0.68K,  = 1.85 ± 0.03. We have attempted to  analyze the data without making the infinite spin-exchange rate approximation, but the quality of the present data is such that no useful results were obtained. If there are no active resonant recombination processes, then we can use the simplifying results from the calculations of the Van der Waals recombination rates in HH, DD and HD systems. The recombination rates between H and D atoms in a 41 kGfieldare related to the zero-field recombination rate KHD?  1 K K  sb  =  -K D  pb  =  yK  (6.11)  H  H D  where KHD is the zero-field formation rate of ortho-deuterium. Using the relationships in equation 6.11 and equations 5.5, we can write r as r =  = ^ ^ and R can be I  n  written R = ^ ^ . Although the results of the recombination rates calculated using the I  B  "sudden" approximation indicate that r should be equal to R, we find that our results are inconsistent with r = R, but given the quality of the "burn" data and the assumptions used to arrived at equation 6.10, the discrepancy should not be taken too seriously.  Chapter 7  Surface Atoms  7.1  Introduction  The existence of a bound state of HJ. on the /- He surface is detected in the recombination 4  rate studies by the temperature dependence of the rates at the lower temperatures (see equation 4.2). At even lower temperatures, the action of the preferential recombination processes in the gas sample chemically induces 100 % polarization of the electron and proton spins on the atoms, and results in a gas containing only | b)-atoms. If samples with bulk atoms densities greater than 3 x 10  13  cm  - 3  are brought to temperatures of  « lOOmK, then the atoms adsorbed on the surface can be directly detected . Whereas, at temperatures above 200 mK, the measured absorption spectrum of the b-state atoms has only one peak, which corresponds to the absorption of atoms in the bulk, in samples at a temperature around 100 mK, the absorption spectrum  1  of the | b)-line changes  dramatically: a peak appears on either side of the absorption peak of the bulk atoms. In consecutive scans on the decaying sample, the two side peaks move towards the central line, and eventually, the two peaks merge with the central one. The dipole-dipole magnetic interaction between the b-atoms adsorbed on the surfaces of the cell is the mechanism responsible for the shift in the resonant field of the two side peaks. In the bulk, the local field due to the other dipolar atoms averages to zero, but in 2-D, there can be a non-zero average local magneticfielddue to the other atoms adsorbed on the surface. The localfielddepends on the orientation of the surface to the polarizing 1  Recall that the spectrometer is afixedfrequency one and that the magneticfieldis swept. 73  Chapter  7. Surface  74  Atoms  magnetic field. Some of the properties of the local magnetic field can be qualitatively predicted from a simple classical model. We will eventually consider more accurate models for describing the dipole interaction between the atoms in the following chapters, but for now, we consider only a very simple classical model. This model ignores any pair correlation, and assumes that the atoms are uniformly distributed on the surface with density o. If the surface normal n makes an angle 6 with the polarizing magnetic field B , then for a classical dipole m = hj /2k, e  which is located at the origin, the magnetic  field at a position r is given by  ()  B  =  r  >  ~Z  t - ) 7  1  where f is the unit vector of r. The extra magnetic field in the z-direction B at the origin t  due to all the atoms, which are uniformly distributed with a surface density cr, is found by performing the integral  B, =  f* Jr  c  /V"»'(f)-1, Jo  (7.2)  r  z  where r is the distance of closest approach between the atoms and 6 is the angle between c  the polarizing field and r. As the axis of the cylindrical microwave cavity is oriented parallel to B , we only need to consider local fields due to dipoles confined in a plane either parallel to or perpendicular to B . If the atoms are adsorbed on a surface that is normal to B , then the average local field due to the other atoms is given by Bend =  -7rfc  -k  (7.3)  B ide = ^7e—k. r  (7.4)  7 e  rc  and if the surface is parallel to B  8  c  Chapter  7.  Surface  75  Atoms  For electrons, the energy difference between the state with rru = —|and| is just ^7 BLocaie  If the bulk atoms resonate at a field B, then the atoms on surfaces parallel to B will resonate at a lower field B—|Bgide|, and atoms on a surface perpendicular to B will resonate at a higher field B +|B d|- Moreover, the model predicts that the ratio | g*^- |= 2, and 1  en  that average field shift of the peaks from the central line is proportional to the surface density. If we take, as the distance of nearest approach, a hard core radius of 4 x 10  -8  cm,  the theoretical results agree quite well with the experimental data. But the close agreement is largely a coincidence . Since the thermal deBroglie wavelength of the hydrogen 2  atom at T = O.IK is 55 A, and average spacing between the surface atoms (for a typical density of 5 x 10 cm ) is 141 A, the quantum mechanical corrections to the classi11  -2  cal values can be non-negligible. However, the physical mechanism responsible for the presence of the wing peaks is understood. The original interest in the satellite peaks stemmed, among other things, from the possibility of directly measuring the surface binding energy. With a simultaneous measurement of the surface and bulk densities, and the assumption that the two systems are in thermodynamic equilibrium, the binding energy can be found using equation 2.25. However, the temperature of the system must be accurately known. The preliminary measurements of the surface binding energy have been presented elsewhere[15]. A major goal of this present experiment was to measure E B more accurately. Unfortunately, a careful analysis of the recent data taken with the new cell indicates that the temperature gradient due to the large Kapitza resistance between the liquid helium and the copper walls is too large to permit an accurate determination of E B . On the other hand, the ESR resonance lineshape of the atoms adsorbed on the /- He 4  surface is very intriguing and very interesting to study. The measured absorption peaks of the surface bound atoms are very asymmetric. This is due, in part, to the saturation of the See Chapter 9.  2  Chapter  7. Surface  76  Atoms  line by the exciting fields. The measurements with the lowest power levels indicate that the intrinsic lineshape is somewhat less assymmetric, but lineshape is still not lorentzian or gaussian. The exact kinetics of the mechanism which is responsible for the unusual shape is at this point not understood even after many attempts to do so. In the next section, we will discuss the relevant physical properties of the surface system. In the chapter 9, we will present the results of some simple model calculations. Here we will discuss the kinetic properties of the b-atoms adsorbed on the /- He film, 4  and the possibility of detecting the Kosterlitz and Thouless superfuild transistion in the 2-D system. The results of the measurements of the binding energy, the lineshift and the lineshape are presented in the next chapter.  7.2  Kinetic Properties of Hydrogen Atoms on a /- He Surface 4  The kinetic behaviour of the adsorbed Hj is usually treated as ideal gas-like. For surface densities less than the saturated density of 1.8 x 10 cm , the ideal gas approximation 3  14  -2  is usually appropriate[64]. This approximation can be used, since there is very small penetration of the hydrogen wavefunction into the liquid helium and there is a freezing out of the thermal ripplons at these low temperatures. Figure 7.1, which is taken from Edwards and Mantz[65], clearly demonstrates the weakness of the interaction of the atom with surface (aside from the Van der Waals interaction). The weakness of the ripplon interaction has been confirmed by the calculations of Zimmerman and Berlinsky[29], who found that the mean free path of Hj atoms due to ripplon interactions is greater than 30 thermal deBroglie wavelengths for T < 0.3K (a mean-free time between collisions of 2.4 x 10~ sec for T = 0.1K). They also found that the effective mass of the adsorbed 8  atom is within 1% of the bare mass. 3  This value has been corrected for the experimentally measured binding energy.  Chapter  7.  Surface  Atoms  77  Figure 7.1: The square of the hydrogen wavefunction above the liquid helium surface p as calculated by Edwards and Mantz. [65].  Chapter  7.  Surface  78  Atoms  The behaviour of the two-body wavefunction, for separations less than 10 A, can strongly affect the average dipolar interaction between the atoms. The kinetic 2-D scattering crosslength for adsorbed Hj is 0.9 A[62] and is not too different from the triplet s-wave scattering length in 3-D of 0.72 A[66]. Since the 2-D crosslength is much less than the ~ 10 A extent of the adsorbed Hj atom wavefunction, the motion perpendicular to the surface is not strictly two-dimensional. Most of the recent calculations use a 2|-D model for the surface state, which amounts to performing an average over the motion perpendicular to the surface before solving Schrodinger's equation for the system. A detailed study of the atomic motion during a collision was conducted by J. P. H. W. Van der Eijnde[67], who examined both 2|- and 3-D models. The implications of his results for the lineshape problem will be discussed in Chapter 9.  7.3  Superfiuid Transition in 2-D  The driving force behind most of the research in Hj is the quest to observe Bose-Einstein condensation (BEC) in this weakly interacting bose gas. The critical temperature T for c  the BEC transition is given by 4  ft  / \ /  2  T  < =  3  -  3  1  2  ^  ( i )  3  <-> 7  5  where m is the mass of the atom, n is the bulk density, and g is the spin degeneracy, 2/+1 . For a bulk density of 1.6 x 10 c m , T = 0.1 K. The highest densities obtained 5  19  -3  c  to date are densites in the range of 10 cm [68, 69, 70], but with temperatures limited 18  _3  to about 550 mK. It has been observed by Edwards and Mantz[65] that the superfiuid transition in the 2-D Hj gas should be much easier to observe than the BEC transition in the 3-D gas. Although derived for a non-interacting gas, this formula remains valid for a weakly interacting gas. Instead of the chemical potential approaching zero at T , it approaches the interaction energy. In hydrogen, this is the number of occupied hyperfine states of the gas. 4  c  5  Chapter  7.  Surface  79  Atoms  They note that, due to the compression effect of the surface, a 2-D density of 10 c m 14  is equivalent to a volume density  -2  (which is much larger than the bulk densities that  6  have been observed) of 10 c m . They, then, conjecture that the adsorbed HJ. may be 21  -3  a superfluid at densities and temperatures which are all ready experimentally accesible. The 2-D transition temperature should follow the universal Kosterlitz-Thouless equation: i k  T  c  ^ / sc -2m^ 7r  rp2D B  =  / «\ -  2 ?  7  ( ?  where p is the superfluid mass per unit area just below the transition at T sc  to predict T  2 D  accurately, p  sc  2 D  6 )  . In order  must be evaluated from a microscopic theory for the HJ.  film. Berlinsky[71] takes ^ = a/10 and finds that T  2 D  Bc  with the most optimistic estimate for p  sc  (= cr), T  2 D  = 76mK if a = 10 cm~ . Even 14  2  = 7.6mK for a surface density of  10 cm . Since, in the new cell, the heating of the Z- He surface due to the recombination 12  -2  4  processes in the HJ,j gas (at surface densities near 10 cm ) prevents the surface from 12  -2  cooling below 90 mK, even reaching the Kosterlitz-Thouless transition temperature for a reasonable density of HJ. j, will be a very challenging experiment.  Effectively tr/lOA  6  Chapter 8  Results of Experiments on Surface Atoms  8.1  Data  The ESR apparatus was originally designed to study two-body recombination rates in Hj, where the typical lifetime of the sample is of the order of many minutes or several hours. Given this limitation, we have completed some measurements on samples of HJ,| at temperatures around 0.1 K, where a sample with a density of 1 x 10 c m 14  - 3  has a  lifetime of the order of 5 seconds. The minimum period of 2 seconds between scans of the ESR absorption peak of the b-atoms, is due to the inherent rate limitations of the present data acquisition system and the eddy current heating introduced by the sweeping of the magnetic fields. At most, three useful sweeps were usually taken on a given sample. It is not possible to study recombination processes at these low temperatures with the present apparatus. We can, however, measure the binding energy of the | b)-atoms to the /- He film and study the lineshape and lineshift of the ESR absorption lines of surface 4  |b)-atoms. For convenience in describing the different sets of data, we will use the notation that a "b-line" is a scan of the j b)-atom ESR absorption peak taken while the applied magnetic field is increased; whereas, a "d-line" is a scan of the same peak taken while decreasing the applied field. An example of a b-line scan is shown in Figure 8.1; a d-line scan taken under the same experimental conditions is shown in Figure 8.2. In a b-line spectrum, the wing peak on the left of the central peak corresponds to the absorption due to atoms  80  Chapter  ?  8.  Results  of Experiments  0.8  F  0.6  r  on Surface  81  Atoms  0.4  0.2 h  -0.0 0.0  1.0  2.0  3.0  4.0  Magnetic field (G) Figure 8.1: Absorption spectrum of doubly spin-polarized hydrogen taken while increasing the magnetic field.  Chapter  8. Results  of Experiments  82  on Surface Atoms  0.4  1  or 0.2  -0.0 0.0  1.0  2.0  3.0  4.0  Magnetic field (G) Figure 8.2: A spectrum taken while decreasing the applied magnetic field.  Chapter  8. Results  of Experiments  on Surface  Atoms  83  adsorbed on the sidewalls of the cell cavity. The peak to the right of the central peak corresponds to the absorption due to atoms adsorbed on both ends of the cell. The wing peaks have been observed with cell temperatures in the range of 80 to 125 mk, but the binding energy results indicate that the temperature of the gas never dropped below 90 mK, even when the temperature of the cell was much colder. An unfortunate aspect of the surface atom work was that the data was not analysed until after the experiment was shutdown because the necessary analysis software was not ready. An upgrade of the laboratory computer to a SUN 3/60 workstation was one of the circumstances that required that new programs be written. Although previous working analysis programs existed, they were written to run in a micro VAX II environment. In addition, the old programs were meant to analyse the absorption curves taken at higher temperatures and not the new spectra with the side wings, which are more difficult to analyse; the programs for the new spectra had to be very interactive in order to properly transform the voltage spectra to absorption spectra. It proved impossible for one person to simultaneously run a HJ. experiment of this type and write the required software. The question of whether the transition was saturated by the applied microwave field or not was answered by visually inspecting the voltage spectra taken with decreasing excitation levels. Initially, the unusual lineshape of the wing peaks were thought to be the intrinsic lineshape of HJ. atoms adsorbed on MHe. The sweep direction independence of the observed lineshapes seemed to support the fact that the intrinsic lineshape was being measured. But, a more thorough analysis of the lineshape data indicates that the microwave power used to probe the transition was, at least partially, saturating the line. Since the applied microwave field was 0.1 milligauss B in the rotating frame, and the observed width of the peaks was typically one gauss, saturation of the absorption peak by the field applied in these experiments was not expected. Fortunately, some routine measurements of the effect of the microwave power on the lineshape had been taken, and  Chapter  8. Results  of Experiments  on Surface  Atoms  84  this set of data indicates that, although some of the lineshape characteristic are distorted by the saturation of the line, several of the quantities of interest are not affected. Since some of the characteristic properties of the absorption spectrum axe insensitive to the level of saturation present in this experiment, most of the following discussions neglect this effect, and only in the last section of this chapter do we consider the effects of saturation.  8.2  The Binding Energy  In principle, it should be straightforward to deduce the binding energy of the Hj absorbed on the MHe surface from a simultaneous measurement of the bulk and surface densities, and the temperature. When in thermodynamic equilibrium with each other, the densities of an ideal gas populating a 3-D and a 2-D system are related, as given by equation 2.25. The surface densities of the atoms on the end plates and the sides of the cylindrical cavity are simply related to the area of the corresponding absorption peaks [72]. The temperature of the cell is measured with a carbon resistor that was calibrated in a 40 kG field using a He melting curve thermometer[45]. 3  But, the rapid decay rate of the sample and the destructive nature of the ESR technique complicates the measurement of the binding energy. The procedure for determining the absorption curve from the measured signal requires that each scan be wide enough to include the baseline microwave signal far enough away from the resonance peaks. This requirement of a minimum width, coupled with a maximum usable sweeprate of 7.236 Gs"  1  for the magnetic field, results in a typical time of 0.5 sees to complete a scan  of the absorption peak. The instantaneous decay rate of | b)-atoms with a density of 8 x 10 c m 13  - 3  at T = 0.090 K is 5.2 x 10 cm sec . Since the positions of the wing 13  _3  _1  peaks in the absorption spectrum depend on the surface density, not only axe the areas  Chapter  8. Results  of Experiments  on Surface  85  Atoms  of the peaks decreasing in time, but the peaks are also moving quite rapidly towards the central peak. Both effects require a correction to be applied, as does the destruction of the HJ. atoms due to the microwave excitation. Figure 8.3 shows a plot of the apparent binding energies, deduced from various measurements of the surface density of atoms adsorbed on the ends of cell, plotted against the temperature of the cell. The temperature dependence of the calculated E B and the weak density dependence of E B (see Figure 8.4) are indicative of the existence of a thermal gradient in the cell that is more strongly dependent on the temperature of the system than on the level of recombination heating. Since it appears that the thermal gradients are too large to permit an accurate determination of the binding energy, we shall instead determine the temperature of the HJ.J. gas in the experiments. If we take E B = 1.010(10)[26], and infer the temperature Tp of the /- He film from the measured densities, we find that Tp is almost always 4  1  greater than the measured temperature of the cell (as shown in Figure 8.5). Unresolved irregularities in the He melting curve temperature calibration of the carbon resistance 3  thermometer on the.cell have complicated the analysis of the data, but the uncertainty in the temperature for T < O.IK is still less than 2 mK. At temperatures below 0.1 K, the thermal gradients in the cell can be as large as 25 mK, so it is difficult to accurately determine the binding energy from these measurements. By averaging the E B found at the highest temperatures (~ 130mK), where the thermal gradients should be negligible, we find that E B = 1.03(1) K to be compared to 1.00(5) K we reported in our earlier experiment [15], and to the value of 1.011(10) K which was obtained by Hardy et al. using a method that is inherently more accurate, but less direct. The temperatures of the gases in the cell are determined by the temperature of the /-Hefilmas shown in the next section. 1  4  Chapter  8. Results  of Experiments  on Surface  86  Atoms  1.10  1.00 \-  «  0.90 h  0.80 h  0.70 60.0  80.0  100.0  120.0  140.0  Temperature of Cell (mK) Figure 8.3: The binding energies derived from measured surface densities on the ends of the cell.  Chapter  8. Results  of Experiments  on Surface  1.10  1.00  87  Atoms  ->  r  •  '  r ••  0.90  0.80  0.70  1  2.0  •  ' 4.0  •  ' 6.0  CJendCxlO  •  10.0  8.0 11  1  Cm" ) 2  Figure 8.4: The figures shows the density dependence of the binding energies inferred from the all the data taken during the experiment.  Chapter  8. Results  of Experiments  on Surface  88  Atoms  0.140  0.06  0.08  0.10  0.12  0.14  Tcell ( K )  Figure 8.5: The Temperature of the /- He film plotted against the the measured temperature of the cell. The open squares were calculated from surface atoms density measurements of the ends walls and the open triangles were calculated from the density measurements of the atoms on the sidewalls. 4  Chapter 8.  8.3  89  Results of Experiments on Surface Atoms  Thermodynamic Equilibrium in the System  In all experimental situations, a gas of Hj is only metastable; but, since the rates of processes that ensure thermodynamic equilibrium in the system are usually larger than the inverse lifetime r of the sample, the temperature of the system is a well defined quantity. In situations where r is very short, thermodynamic equilibrium may not exist in the system. At low enough temperatures, where the effective bulk recombination rates are increasing very rapidly, it is possible for the system of 2-D Hj j to be not in equilibrium with the bulk gas. Many authors have considered[28, 73, 74] or measured[75, 76, 77] the kinetic mechanisms responsible for producing thermodynamic equilibrium between the Hj and /- He surface. Many of schemes for achieving Bose-Einstein condensation depend 4  crucially on the behaviour of these mechanisms[78, 7, 8]. In the cell, there are four separate systems in contact with each other: the bulk Hj, the 2-D Hj, the /- He film, and the copper walls of the cavity. The kinetic process 4  responsible for the thermalization of the bulk, 2-D gases, and /- He film is the sticking 4  of the bulk Hj onto the /- He surface . The thermalization of the bulk gas proceeds via 4  2  the sticking process, since the bulk density is so low that the particle-wall interaction is much more effective in thermalizing the atoms in the bulk than the 3-D particle-particle interactions. In the case of the atoms adsorbed on the surface, the particle-particle and particle-ripplon interactions rates are approximately a thousand times faster than the inverse lifetime of the particle on the surface; therefore, the adsorbed particle is quickly thermalized to the surface temperature, and the ejected particle is at the surface temperature. Since, at 0.1 K, the thermal heat capacity of the Hj gas is about 10 times that of the /- He film covering the walls of the cavity and the thermal resistance between 4  the bulk gas and the surface is largest in the system, except for the Kapitza resistance It has been shown that below 500mK the thermal accommodation rate of the bulk gas to the /- He surface proceeds through the sticking process[74, 79]. 2  4  Chapter  8. Results  of Experiments  on Surface  90  Atoms  between the /- He and the wall of the container, the thermalization time of the bulk gas 4  to the surface temperature is the longest time in the system. The kinetic energy of the newly formed H2 is much larger than 1 K, and it is most probably that this fast moving H2 deposits its energy directly into the liquid helium film, since the mean free path due to collisions with HJ. atoms is much larger than the typical dimension of the cell (~ 0.2cm). This approximate picture is sufficiently accurate for the work here. If we use the results of Goldman [28] and calculate the Kapitza temperature gradient due to energy flow between the gas and the /- He film for the heat flow due to 4  recombination of the atoms, we find that it is completely negligible. Since the heat is deposited into the /- He film and the thermal equilibrium of the 4  system is mediated by the /- He film, the thermal time constant of the system consisting 4  of the 2-D and 3-D gas, and the film is determined by the slowest thermal time constant in the system, which is that of the 3-D gas and at T = 0.1K this is ~ 2ms. If the bulk density is 8 x 10 c m , then the initial decay rate of the sample at 13  -3  T = 0.1K is 5.2 x 10 cm s , which corresponds to a lifetime of 1.5 sees. Since the 13  3  _1  sample lifetime is much longer than the thermal relaxation time, the total system is at the same temperature. 8.4  Kapitza Resistivity  It is possible to use the ESR results to measure the degree of the heating of the /- He film 4  above the temperature of the copper walls. Given that the recombination heat input to the film is known, the heating of the surface is a direct measure of the Kapitza resistivity between the /- He and the walls. 4  The Kapitza resistivity Rk at an interface can be measured by using Rk =  AAT Q  (8.1)  91  Chapter 8. Results of Experiments on Surface Atoms  where A is the area of the interface, A T is the temperature gradient across the interface, with A T <IC T, and Q is the thermal current across the interface. At very low temperatures, the measured Kapitza resistivity between /- He and a clean copper surface can be 4  fitted by [80]. u x i r i K  _  T  W  erg  J  Since the measured temperature gradients are not small (see Figures 8.5), one must use the integrated form of equation 8.1 _  6  A(Tf - T )  erg  4  ^  4(1.1 x lO" ) K cm sec 5  4  2  K  '  }  where A = 0.306 cm in the present cell, Ti is the temperature of the gas and surface, 2  and T is the temperature of the copper walls. Ti is inferred from the surface and bulk c  densities measured in a scan. We use a = nA exp(1.01/Ti), taking the E B measured tri  by Hardy et al.[26] and calculate Ti using the measured densities. The appropriate recombination rates are used to calculate the flux of heat in the system. We then plot 'Q > as a function of the surface density. C  The Kapitza resistivity could only be measured in a very narrow temperature region around 0.1 K. At T = 0.1K, ' T T  approaches 2 x 10~ K s/erg in the limit of small  T c  5  4  surface densities. The limiting value corresponds to a coefficient value of 1.5 x 10 instead -6  of 1.1 x 10  -5  in Equation 8.2. The smaller than expected Kapitza resistivity (about 7  times smaller) is probably due to the presence of a thick layer of molecular hydrogen on 3  the walls which improves the acoustic impedance match between the phonons in the /- He 4  and the copper wall. It is worth noting that in the previous cell, in which the surfaces were not polished as carefully as in the present one, no temperature gradient due to the heating of the surface was observed in case of surface densities less than 5 x 10 cm . 4  11  2  T h e layer was deposited on the copper walls to reduce the one-body nuclear relaxation rate due to magnetic impurities in the walls. T h e importance of the surface characteristics i n determining the Kapitza resistivity is well known, 3  4  Chapter  8.5  8. Results  of Experiments  on Surface  92  Atoms  Absorption Line Data  The most striking feature of the absorption line spectra at very low temperatures is the very unusual lineshape of the wing peaks. In spite of extensive efforts to understand these data, the lineshape feature is still not well understood. First, we consider the other properties of the spectra. The observed absorption spectra are corrected for the systematic distortion introduced by the the decay of the sample during the scan and the motion of the peak during the scan. The fractional correction to the measured average field shift of a wing peak from the central peak is one-half of the fractional correction to the measured density of the peak. Since, for 90mK < T < lOOkmK and o < 8 x 10 cm , the fractional correction to 11  -3  the average magnetic field shift of the peak is 2-3 % and the scatter in the average field data is much larger, we do not apply any correction to the field value. The correction to the measured surface density due to microwave induced recombination is 3 % for samples at 100 mK. Some of the interesting characteristics of the absorption spectra are the the average field shift of the wing peak from the average field of the bulk peak, the RMS widths of the wing peaks, and the separation between the sharp edge of a wing peak and the nearest onset edge of the bulk line. We have examined the density dependence of all three quantities. As the sidewalls and endplates of the cavity cell have a different orientation with respect to the polarizing magnetic field, and the measured spectra may depend on the direction of the field sweep, these subsets of the data are analyzed separately. The data sets were combined if no difference was detected in the results. Figure 8.6 and 8.7 show the average field shifts of a wing peak from the average field of the central line. There is a systematic shift in the average separation associated with but the requirement of veryflatsurfaces was deemed to be more important in these experiments.  Chapter  8.  Results  of Experiments  on Surface  Atoms  93  the direction of the sweep. The reason for the difference in the measured shifts is not understood at this moment, but an explanation of it may involve the kinetics of how the resonance peaks are excited. A similar effect is seen in the plot of the onset fields. The plots of the RMS linewidths versus the surface density do not have a noticeable directional dependence. The RMS width of the bulk peak is independent of the density, but it does have a small dependence on the sweep direction (b-line RMS width of 0.087 G and d-line RMS width of 0.067 G), which is much smaller than the scatter in the surface RMS width data. Table 8.1 summarizes the results of least square fits to the various sets of data.  Chapter  '8.  Results  of Experiments  on Surface  94  Atoms  1.25 •iH  2  1.00 0.75  (D  0.50  CD  S  0.25  > ^  0.00 0.0  2.0  4.0  (Jend  6.0  8.0  10.0  (xlO cm" ) 11  2  Figure 8.6: The open triangles correspond to average field shift measured for the peak due to atoms on the endplates, when swept by a b-line scan, and the squares correspond to the shift measured, when a d-line scan were used.  Chapter  8. Results  of Experiments  on Surface  Atoms  95  Figure 8.7: The open triangles correspond to average field shift measured for the peak due to atoms on the sidewalls, when swept by a b-line scan, and the squares correspond to the shift measured, when a d-line scan were used.  Chapter  8. Results  of Experiments  Quantity AB d e n  AB  s i d e  r> onset end D  D onset aide D  on Surface  Subset Fit to b and d b d b and d b d b and d b d b and d b d  96  Atoms  Least Square Fit 0.204 + 1.12 X 10~ < T 0.230 + 1.15 X lO" ' C 7 12  end  1 2  e n d  0.001 + 1.31 x 10"  <7  12  0.007 + 7.43 x 10" -0.014 + 7.37 x 10~ 0.027 + 8.09 x 10" -0.087 + 5.91 x 10" -0.095 + 5.50 x lO" -0.095 + 5.50 X 10" 0.103 + 7.39 x 10~ 0.118 + 8.15 x 10" -0.123 + 9.64 x 10" 13  cr de 8i  13  13  end  cr a cr a„  8ide  8ide  13  end  13  e  13  13  13  cr cr  13  d  (Tend 8ide 8ide  cr e 8id  Table 8.1: The table lists the results of the least squares fit (a + b a)to the line parameters for the b and d sweeps.  Chapter  8.  Results  of Experiments  on Surface  97  Atoms  The average shift of a wing peak from the central line can be calculated for a 2jD quantum mechanical model, which takes into account the finite spatial extent of the hydrogen wavefunction perpendicular to the surface. The results of Van der Eijnde et a/.[81] or Statt[82] can be used to infer the average dipole-field Had in a 2|-D H | | gas. Verhaar[83] has calculated that the average shift of the wing peak is given by Hdd  = rP2(cOS0)8cr7l7 (]r -^p-) thermal average  ^  e  TO  (8.4)  *  where k is the wavevector of the atom, 0 is the angle between the surface normal and the polarizing magnetic field, and the r  mtn  's are quantum mechanical quantities defined  by Ahn et al. [84]. Although Eijnde et a/.[81] do not give the value of the thermally averaged quantity of interest, they note[67] that the result of Statt[82] should be 10 % larger. With their suggested correction to the averaged sum that was calculated by Statt, we find that H  d d  = 1.17 x I O  -12  P (cos0) <7cm- G.  (8.5)  2  2  The position of the corresponding wing peak in the absorption spectrum is, therefore, displaced by Hdd from the bulk atom absorption line. In the case of H | atoms adsorbed on the ends of the cell (8 = 0), equation 8.5 implies a shift of 1.17 x 10~ <r cm~ G, 12  2  which agrees well with result in Table 8.1. The agreement is not so good for the average shift of the absorption peak of atoms adsorbed on the side walls of the cavity, but there the experimental uncertainty is larger.  8.6  Effect of the Microwave Power Level on the Line  The effects of saturation of the ESR line by the applied microwave power was not noticed until the voltage signals were analysed. The initial conclusion that there is no saturation of the line was based on a visual inspection of the voltage signals taken with varying  Chapter  8. Results  of Experiments  on Surface  Atoms  98  power levels. The dependence of the lineshape on the power level is mostly obscured by-the decrease in.the signal to noise level S/N of the measured spectra. Even with no attenuation of the incident microwave power, the S/N of the voltage spectra is marginal. Unfortunately, since it was thought that there was no saturation of the line, most of the data was taken without attenuation of the power level. When the microwave power is decreased by a factor of 4, the S/N of the voltage spectra becomes unacceptably 5  low. A systematic study of the degree of saturation of the line with the microwave power level would be useful, although it is not clear whether, with the present microwave spectrometer, it could be done properly. The saturation of the resonance line with increasing power levels is clearly visible in the analysed absorption spectra. As a routine measurement, the absorption spectra of the wing peaks were measured at three different power levels, with identical experimental conditions. The spectra were taken at full, one half, and one quarter power levels. These are shown in Figures 8.8-8.11. As the power level is decreased, there is a definite change in the shape of the wing peaks: the peaks become narrower and more symmetric. There is also a density dependence of the wing peaks which is more apparent in the full power spectra. In the preceding sections, we have studied some of the characteristic properties of the absorption spectra as functions of the surface density. The same properties have been plotted for the saturation effect spectra. Some of these plots are show in Figures 8.12-8.14. Since there are so few data points, only qualitative conclusions can be made. The results of the plots of the saturation data indicates that the data taken with no attenuation of the incident microwave power is useful. The RMS width of the wing peaks are I t is very difficult to measure the power incident on the cavity. The attenuation levels are inferred from the heating of the cell due to the incident power. s  Chapter  8.  Results  of Experiments  on  Surface'Atoms  99  sensitive to the power level, but the average shift of a wing peak from the central peak seems to be insensitive to the power level. The results of the average field shift analysis of the preceding section is, therefore, meaningful; the other results are probably not.  Chapter  8.  Results  of Experiments  on Surface  100  Atoms  0.6 F  0.4 h  or 0.2 h  -0.0 0.0  1.0  2.0  3.0  4.0  Magnetic field (G) Figure 8.8: A b-line absorption spectrum taken with the maximum incident microwave power.  Chapter  8. Results  of Experiments  on Surface  101  Atoms  0.4 -  Or 0.2  0.0 0.0  1.0  2.0  3.0  4.0  Magnetic field (G) Figure 8.9: B-line absorption spectra taken using one half the maximum available microwave power. The spectra were taken with the same sample but at a different point of the decay. The top spectrum has been offset by 0.1 so that the details of both lines can be seen.  Chapter  8.  Results  of Experiments  1  on Surface Atoms  |  102  -:  T-  0.4  or ^  0.2  0.0  1.0  2.0  3.0  4.0  Magnetic field (G) Figure 8.10: B-line absorption spectra taken with one quarter of the maximum microwave power. The top spectra was taken with a higher density sample then the bottom one.  Chapter  8.  Results  of Experiments  on Surface  103  Atoms  0.8  0.6  1 9 " 0.4 0.2  0.0  1.0  2.0  3.0  4.0  Magnetic field (G) Figure 8.11: A composite of b-line absorption specta taken with various power levels, but with similar atom densities. The bottom spectrum was taken with one quarter of the maximum power, the middle one was taken with one half of the maximum power, and the top one was taken with full power.  Chapter  8. Results  of Experiments  on Surface  104  Atoms  1.0  0.8 -  2 5  o  O  0.6 -  0.4  0.2 2.0  4.0  CJ  6.0  (xlO cm"- 2) 11  end  8.0 2  Figure 8.12: The measured local field shifts for atoms on the ends of the cell are shown for various strengths of the microwave power used to probe the system. Measurements with maximum power are shown as open squares, those with half power are shown as open triangles, and those with one quarter power are shown as open diamonds.  Chapter  8. Results  of Experiments  0.40  '  1  on Surface  1  1  i  105  Atoms  |  i  |  |  i  i  •  O  0.35  --  A A  •  ri  0.30 0  •  171  0.25 0  0.20  i  2.0  3.0  .  i  I  4.0  5.0  CJend  (xlO  .  1 1  I  .  I  6.0  .  7.0  8.0  cm" ) 8  Figure 8.13: The measured RMS width of the absorption peaks of atoms on the ends of the cell are shown for various strengths of the microwave power used to probe the system. Measurements with maximum power are shown as open squares, those with half power are shown as open triangles, and those with one quarter power are shown as open diamonds.  Chapter  8. Results  of Experiments  0.70  3  on Surface  1  1  -  106  Atoms  i  •  j  i  0.60 0.50  •r-l  CD O  0.40  •  -  -  0.30  • 0.20  A A  0.10 2.0 a  e  i  i  4.0  6.0  n  d  (xlO  1 1  8.0  cm" ) 2  Figure 8.14: The measured onset separation between the absorption peak of the surface atoms on the end of the cell and the central peak for various strengths of the microwave power used to probe the system. Measurements with maximum power are shown as open squares, those with half power are shown as open triangles, and those with one quarter power are shown as open diamonds.  Chapter 9  The ESR Lineshape of Hj j Adsorbed on a MHe Surface  9.1  Introduction  The ESR absorption lineshape of Hj j adsorbed on the /- He surface with the two density 4  dependent side peaks is unusual. Although an explanation for all the characteristics of the line has yet to be found, the mechanisms responsible for some of the characteristic properties are understood. For example, two side peaks are due to the non-zero average of the dipole-dipole interaction in a 2-D system. The side peaks were positively identified by sweeping the line while a linear magnetic gradient was applied along the B direction and noting the new position of the peaks. For example, the side peak corresponding to atoms on the ends of the cylindrical cavity split into two smaller replicas of itself. The average separation of the two new peaks was exactly equal to the difference in the polarizing field between the two ends of the cavity. In chapter 7, a model assuming classical dipole-dipole interactions was used to calculate the size of the local fields for atoms adsorbed on a surface parallel or perpendicular to the polarizing field B and the direction of the shifts. The good agreement of those calculations with the measured quantities is partly accidental: taking the hardcore radius of the triplet H-H potential as the distance of nearest approach just compensates for the neglect of the quantum mechanical correction to the classical dipole-dipole interaction. When the local fields are calculated with a more realistic distance of nearest approach and include the quantum mechanical effects, the new values of the splitting are almost  107  Chapter  9.  The ESR Lineshape  of HJ.J. Adsorbed  on a 7- He Surface 4  108  the same as the old values. There are several properties of the line that are incompletely understood. The two most intriguing of these are the following: a non-linear response of the spin system is observed with the application of a microwave field that is very much smaller than the measured linewidth of the peak; and the lineshape of peaks is not describable with a lorentzian or a gaussian function and, moreover, they are definitely not symmetric. The observed asymmetry of the line is partially due to the saturation of the line during the sweep, but even for the smallest applied microwave field, the lineshapes are not lorentzian or gaussian. It is not uncommon to observe absorption lines that are neither lorentzian nor gaussian (for example, the response of spin systems with quadrapolar interactions or in systems where there is only partial motional narrowing of the line). However, the sharp ramp lineshape is very unusual. The  measured linewidth of the HJ.J. atoms on the surface require a factor of one  hundred reduction in the linewidth from a static distribution of neighboring HJ.J. atoms. In the case of such large motional line narrowing, the lineshape would be expected to approach a lorentzian, contrary to what is observed. In ferromagnetic resonance experiments on solids, where the spins are strongly coupled, a non-linear response can be produced by applying a r.f. field that is much smaller than the linewidth of the resonance line [85]. In the next chapter, we will consider the spin wave instabilities that are responsible for the early onset of non-linearities in the ferromagnetic systems. But we will find that for the H | | adsorbed on the surface the onset, as predicted by this model, this onset occurs at much larger fields than what we have used in these experiments. Most of this chapter is concerned with describing the various models used to calculate the intrinsic linewidth of the ESR response of the spin system. Using some "back-ofthe envelope" calculations, we will discuss, in the next section, some of the basic line  Chapter 9. The ESR Lineshape of HJ.J. Adsorbed on a f- He Surface 4  109  characteristics of .the bulk and surface HJ, j. These calculations are important since the results help elucidate the mechanisms that should be included in a more complete model. Several different methods for determining the ESR response of the spin system are then considered. All are numerical calculations that rely on stochastic techniques to calculate the absorption line.  9.2  Preliminary Calculations  Since a significant number of the properties of the absorption lines of HJ. in the bulk or adsorbed on a surface, as well as the effects that applying a magnetic gradient has on the lineshape, can be explained with simple models, we start with these calculations. We consider the ESR absorption line of bulk | b) atoms under the conditions of these low temperature experiments (T«0.1 K). In chapter 10, we show that there are no effective Ti processes in the gas. The effective Ti of the spin system is the lifetime of the sample. There are several active processes that contribute to the intrinsic linewidth of the bulk atoms, but we will find that the intrinsic width is very narrow when compared to typical ESR linewidths (which can be of the order of several gauss or more). Since the chemical reactions in the gas effectively removes any hydrogen atoms which are not in the | 6) state, the contribution to the linewidth from the spin-exchange collisions is negligible. The dipole-dipole interaction appears to be the major mechanism for broadening the line. The finite lifetime of the bulk atom also contributes to the linewidth, since, during the sweep, the bulk atoms can repeatedly adsorb onto the surface and acquire some extra phase there. In the bulk, the average field due to the other spins is zero, since the dipole-dipole interaction averages to zero in an uniformly distributed gas in 3-D; however, there are fluctuations in the localfielddue to collisions with each other. The effect of the collisions  Chapter  9.''The ESR Lineshape  of HJ.J Adsorbed  On a /- He 4  110  Surface  on the linewidth can be estimated with the following model: we consider the situation of classical dipoles colliding in a 3-D gas. By assuming that every collision contributes the maximum phase shift (that is for headon collisions) to the spins and calculating the time required for an ensemble of similar dipoles to dephase by 1 radian, we can estimate a value for T2. As the maximum strength of the dipole-dipole interaction is given by 7  *, the phase (f> that can be acquired by a spin in a headon collision is approximately  u  m  av 2  where each spin has an electron moment, a is some distance of nearest approach which we take to be the hardcore radius (3.7 A), and v is the average thermal speed of the hydrogen atom, {^~Y^ - As the mean free path of the atoms in the gas is much larger 2  than the dimensions of the cell, we assume that walls of the cavity are magnetically inert, and the atoms just reflect off the walls and are not dephased by the collision. We find that the mean free flight time T in a sample with density 1 x 10 c m 14  - 2  is 1.5 x 10 sec. _4  The number of collisions an atom encounters in a time T is ^, and the variation in the number of collisions for an ensemble of spins is ( J ) / . Thus, we take T to be the time 1  2  2  T such that <f,  m  « 1.  (9.2)  Solving equation 9.1 for T , we find that 2  T  2  = -L  (9.3)  and for the experimental conditions, we find that T = 0.07 sec, or a corresponding 2  linewidth of 0.8 nanoG. If the linewidth were due only to dipolar interactions, then the linewidth of the bulk atoms would be very narrow. Also within this model, we can determine whether the intrinsic linewidth is narrowed by the rate of the fluctuations. The maximumfieldfluctuationduring a collision is  «  340 G, in this particular  Chapter  9.  The  ESR  Lineshape  of  HJ.J  Adsorbed  on a  Z- He 4  111  Surface  system. If there is motional narrowing of the line, then the deviations in the angular frequency Su> during the time for the collision process r must satisfy c  6UT  <  C  (9.4)  1.  The quantity on the left hand side of equation 9.4 is equal to the phase accumulated per collision, so thus we are in the regime for motional narrowing of the line. The process of bulk atoms adsorbing onto the /- He surface provides a stronger mech4  anism for line broadening. At 0.1 K, the time that a bulk atom spends between sticking events is 1.4 msec, and the residency time of a surface atom is ~ 0.33 msec. For an order of magnitude calculation, we take the average time between transitions between the surface and the bulk to be 1 msec, and with this we find an increase in the linewidth due to the finite lifetime effective is 60 pG. While the atom is on the surface, it is subject to an extra 1 Gfielddue to the local dipolarfieldson the surface. Since SUJT  =  2 x 10 ^> 1, 4  we should observe the intrinsic bulk line and not detect any linepulling effect due to the surface states, aside from the finite lifetime affect on the bulk atoms[86] lineshape. Using a similar argument, we can show that increasing the homogeneity of the polarizing field will appreciably affect the linewidth: there is no motional narrowing for the inhomogeneously broadened linewidth. With A B as the difference in the applied field across the cell, if 7 e  AB-  > 1,  (9.5)  v  where / is a length across the cell along which the gradient is applied and v is average speed of the spin, then the shape of the observed line will be sensitive to the field profile and not to just the average field. For a typical direction in the cell, condition 9.5 implies that for A B > lmG the gradient will not be averaged. This result is consistent with the observed property of the bulk absorption line. By carefully shimming the polarizing magneticfield,with superconducting trim coils, we have narrowed the observed linewidth  Chapter  9.  The ESR Lineshape  of HJ.J. Adsorbed  on a /- He 4  112  Surface  to 0.1 G FWHM. This width corresponds to an homogeneity of 9 x 10  -6  per cm over the  volume of the cell; whereas the specified homogeneity of the commercial superconducting magnet used in these experiments is 10~ per cm. 4  The mechanisms responsible for the properties of the bulk absorption line can be determined using simple order of magnitude calculations, but such is not the case with surface lines. A few example calculations will demonstrate this. As with the bulk atoms, the spin-exchange collision processes are not effective in broadening the line of the surface atoms, but the dipolar interactions are very effective. In order to simplify the discussion, we will consider the classical dipolar broadening effects for atoms adsorbed on a plane which is perpendicular to the polarizing field. Using the same assumptions as for the bulk atoms case, we assume that all the extra phase due to the dipolar interactions is acquired through headon collisions and each contribute the maximum phase perturbation. We find that the maximum phase shift is given by (f>  m  = 0.07. Since <f)  m  is small compared to 1, the line shape of the line should  be very close to a lorentzian. Using equation 9.3, we find that T  2  =  5.2 x 10 sec, _7  which corresponds to a linewidth of 110 mG. If the dipolar interactions were the only perturbing mechanism in the system, then the observed linewidth of the end side peak should be about 110 mG, quite contrary to what we see. The observed linewidth of the bulk atoms is much narrower than that of the surface atoms. Since the divergence of the magnetic field is zero, the fields near the walls of the cell must be very similar to the fields right at the surface . Since the kinetic motion of 1  the bulk atoms does not average the effect of large magnetic gradients on the lineshape, The cell walls were specially prepared to reduce the magnetic impurity contamination. The quality of the surface is evidenced by the fact that even without coating the walls with molecular hydrogen, the one-body nuclear relaxation in the cell was as small as the rate in the previous cell with 500 Alayer of H . The lineshape measurements were taken with about 1000 A layer of H on the surface, so that the effect of any magnetic impurities should be very small, and if it not small, then the bulk atoms would also be affected by the extra fields. 1  2  2  Chapter  9.  The ESR Lineshape  ofHH  Adsorbed  on a /- He 4  Surface  113  if the linewidths of the surface atoms were due to field inhomogeneities, then the bulk line width would also be approximately as wide. The average residency time (~0.33 msec) of the atoms in the surface state results in a minimum linewidth of 0.2 mG in the surface absorption line due to lifetime broadening. A more complete model for describing the effect on the absorption line due to the hopping of the atoms from one resonance site to another is the Anderson and Weiss hopping model[86]. We have evaluated this model in the case for 3 sites, where the atoms are either on an endplate or the sidewall of the cell or in the bulk. The model predicts that the linewidths of the peaks are ~0.8 mG. This value is quite close to the initial estimate of the lifetime broadening of the line. It is very difficult to accurately calculate the lineshape for this situation if quantum mechanical effects are included. We will not attempt such a sophisticated calculation, but instead will determine a better estimate of the linewidth using classical models.  9.3  Numerical Simulations of the Absorption line  Any model seriously attempting to simulate the ESR lineshape must involve a quantum mechanical calculation of the effects of the dipolar interactions between identical particles, include the appropriate kinetic motion of the adsorbed atoms, and the effects of the microwave field which is coherently and simultaneously tipping the spins of a large number of particles. It appears that all this must be done in a regime where the dipolar interactions cannot be treated as a perturbation. This point will be discussed in the next chapter. All the models discussed in this chapter use the classical description of the dipolar interaction, although the calculated fields are partially corrected for the quantum  Chapter  9.  The ESR Lineshape  of H j j Adsorbed  on a /- He 4  Surface  114  -mechanical nature of the spins. If the spins are only weakly interacting, then the effect of the quantum correction to scale the classically determined magnetic field by 3/2(Abragam[87]). The triplet interaction between the adsorbed hydrogen atoms is approximated by a hard core potential, whose radius is a parameter in the simulations. Each particle carries a magnetic moment of ^ -, which aligns itself with the polarizing £  field. We ignore the motion of the atoms perpendicular to the surface, and treat the motion the particles on the surface like that of an ideal gas. Three models are studied. One model calculates the static local field distribution of a collection of particles distributed on a surface, with an average density cr. The second model is an analytic model, which uses stochastic techniques to simulate the effect of the collision between the spins. The model calculates the phase accumulated by magnetic moment on a test particle under the assumption that the effects of the collisions with other spins can be be described by a modified Campbell process. The third model numerically simulates in a limited manner the kinetic motion of all the particles, but allows hardcore interactions only at the site of the test particle. Even for the absorption line of the bulk atoms, there was some concern that during the sweep of the resonance line the Bloch equations might not describe the behaviour of the rapidly moving spins, since the intrinsic linewidth is so narrow. The rapidly moving bulk atoms are subject to a series of slightly saturating passages as they move back and forth through the non-uniform polarizing magnetic fields. We have verified that for an applied field of 0.2 mG, the Bloch equations adequately describe the effect of the sweep, these simulations are discussed in Appendix A. 9.3.1  Static Distribution  The distribution function for the local fields at the site of each of spin is calculated by using a program to model a set of particles, each carrying a classical dipole of strength  Chapter  9.  The ESR Lineshape  of Hji, Adsorbed  on a /- He 4  Surface  115  | ^ 2 % randomly distributed on a surface. The dipoles are aligned by a large polarizing field, which is either parallel to the surface or perpendicular to it. This allows us to simulate the static field distribution for atoms adsorbed on either the endplates or the sidewall of the cavity. The particles, which are placed randomly on the surface with a uniform distribution, have an average density of o. A hardcore interaction, with radius r prevents particles from lying on top of one another. At every site, the local field due c  to all the other particles is calculated, and a histogram of the distribution is computed. Histograms for typical densities observed in the experiments were generated for different hardcore radii. An example of field distribution histogram, which was generated for a  =  5 x 10 cm n  -2  and r = 6.88A, is shown in Figure 9.1. By studying models of c  HJ, adsorbed on the /- He surface in which the motion of the adsorbed hydrogen atoms 4  was allowed to be 3-D and not artifically confined to 2-D, den Eijnde[67] found that the distance of nearest approach is 6.88 A. The result of least square fit to average local field as a function of the density for r = 6.88 A gives c  ABend = -1.20(4) x I O  o (G)  (9.6)  o (G).  (9.7)  -12  and AB  s i d e  = 0.49(2) x I O  -12  A comparison of these results with the experimental values in Table 8.1 shows that the model closely simulates the local field shifts, but the ratio ^g™  ri  = 2.4 for the results  of the simulation, whereas the ratio of the measured quantities is 1.5. 9.3.2  Kinetic Models  If there were no mechanism to motionally narrow the dipolar field broadened line, then the absorption lineshape would be described by the static local field distribution. We have attempted to simulate the possible effects of the kinetic motion in the H | system  Chapter  9.  The ESR Lineshape  0.15  I  -1.0  1  of HJ,j Adsorbed  1  -0.8  •  1  -0.6  on a l- Ue 4  1  1  1  -0.4  Local field  116  Surface  r  -0.2  0.0  B (G)  Figure 9.1: Static field distribution for particles on a surface perpendicular to the polarizing field.  Chapter  9.  The ESR Lineshape  of HJ.J Adsorbed  on a /- He 4  117  Surface  on the lineshape by two; models, both of which introduce stochastic processes into the system.  Both models calculate the resonance absorption line by simulating the time  evolution of the normalized transverse magnetization m+(t) = m + im after an | pulse x  y  rotates the z-magnetization, m = 1, of the test particle to the x-axis, and using the z  fact that the Fourier transform of the autocorrelation function Re(m.|_(r)m+(0)) is the absorption lineshape [87, 88]. In a frame rotating with frequency u = 7 B 0 , the transverse 0  magnetization at time t after the | pulse is given by m+m  =  exp(i f dt'u{t')\  Jo  (9.8)  where u>(t) = 7B(t), with B(r) the randomlyfluctuatinglocal field. Thus, the simulations calculate S(r) = ( M ) +  ensemble)  (9.9)  (since m+(0) = 1) where the () mbie is the ensemble average. The two simulations differ ense  by the methods used to compute the ensemble average and in the processes included in the averaging. The first model includes only the contributions to the local field from particles that collide with the test particle. The second model includes the local field contributions from all the spins on the surface, but does not completely implement the correct kinetic behaviour of the particles. Both models use the ergodic hypothesis and replace the average over the ensemble with a time average for the system. Thefirstmodel also serves as a check on the second model, in one particular limit. The first model takes the conditions of the static model discussed in the previous section and allows the particles to move on the surface with a fixed velocity. The only effect of the hardcore interaction of the particles is to set a distance of nearest approach of the spins to the test particle. We make the simplification that the local field has contributions only from spins whose ballistic motion cause their paths to intersect with that of the test particle. With the assumptions stated above, we can replace the integral  Chapter  9. "The ESR Lineshape  of HJ,| Adsorbed  on a /- He 4  118  Surface  in 9.8 with a sum  f  dt'u(t')  =  (9-10)  £ ( « M ' " T ) " <M-T )), B  n  where <j> (t — r ) — </> (—T„) is the phase contribution due to spin n at time t after the | n  n  n  tip and r is the time that spin n collides with the test particle. The sum is over all the n  collision that occur in the time interval —T < r  n  < T 4-1, with T choosen so that the  contributions to the total phase from collisions that occur before -T or after t + T are negligible. Substituting 9.10 into equation 9.8, we find that S(<) = ( e x p ( » ^ ( r - T ) - ^ ( - r „ ) ) ) > , B  n  (9.11)  n  n  where the average is over the distribution of the number of particles that interact with the test particle, and the type of each collision. Using standard techniques for stochastic processes (see for example van Kampen [89]), we can write S(<) = Q o £  n^ (r )/exp(e^ (r-T )-^n(-T )))\ (  g  n  n  n  ,  (9.12)  where Q = exp(— /f^, drg(r)), with q(r) the number of an events occurring in a time 0  interval around T . The average () is over the velocities of the particles that collide and p  the type of collisions. The assumptions we have made in order to reach equation 9.12 are as follows: the net effect of all the dipole interactions can be calculated by considering only two particle interactions; the motion of the particles are uncorrelated, since the extra correlation due to diffusion of particles in 2-D is ignored; and for the relevant densities, the frequency of collisions of the spins with the test particle can be described by an independent poisson distribution q(r). For independent processes we can write equation 9.12 as S(t) = exp (/_^ dTq(T)((e^-^-*^\  - 1)) ,  (9.13)  where we have used the definition of QQ. We assume that the most probable number of  Chapter 9. The ESR Lineshape of H j j Adsorbed on a /- He Surface  119  4  spins colliding with the test particle per unit time is a constant given by q = 2 x (impact parameter) x flux of atoms.  (9-14)  Although in principle, we could numerically evaluate the average and perform the integral, we make the further assumptions that all the particles have the average relative thermal speed v and that the effect of any collision is the same as a headon collision. We take a  the impact parameter to be some radius r , an effective hardcore radius. For the case c  where the surface is perpendicular to the polarizing field, and including the quantum correction to thefield,thefieldat the origin due to an hydrogen atom at r is  * -  -  f  <  «  «  >  We have evaluated equation 9.13 for various hardcore radii and surface densities. We find that for the surface density range 0.5 x 10 to 10 x 10 c m , and a r = n  11  -2  c  6.88A, the linewidth slowly increases with the surface density and has a minimum width of 13 mG. If we take r = 3.7A, the hard core radius of the triplet hydrogen potential, c  we find the linewidth behaves the same, but the minimum width is 30 mG. Since only a fraction of the number of dipolar interactions that occur between the spin and the rest of the system are included in this simulation, we expect that the intrinsic linewidth is wider than the simulated one. Since the average local field should not be affected by the motional narrowing of the line, evidence that only a small number of the total dipoledipole interactions are include in this model is demonstrated by the magnitude of the slope of the average local field as a function of the surface density, which it is only 19 % of the same quantity simulated by the static model. The effect of including the neglected interactions in the linewidth calculations should be to further broaden the line. The last model we will consider includes the local field contributions from all the spins in the system.  This model sums the contributions from all the spins within a  Chapter 9. The ESR Lineshape of Hj. j Adsorbed on a /- He Surface 4  120  certain radius and approximates the contributions from the spins outside the circle by an integral. Again, we assume that direction of the spins on the particles remain rigidly fixed. The model includes the kinetic dynamics of the mobile particles, but still only allows collision interactions between the test particle and the other particles. With these simplifying assumptions, implementing this model for atoms adsorbed on a surface that is perpendicular to the polarizing field is relatively uncomplicated, since on this surface the rotational symmetry of the dipole field allows us to separately rotate the position of each particle until all the particles have velocities that are in the same direction. The problem reduces to modelling the kinetics of a 2-D collimated flux of particles which are incident on a scattering site, and recording the phase, as a function of time, of the test spin at the site. The density of particles in the flux is the surface density ,and the speed distribution of the particles is given by the Maxwell Boltzman distribution for particles diffusing out through a hole in a container which is at 0.1 K. The width of the collimated beam was taken to be wide enough so that contributions to the local field from particles outside of a disk, with a diameter equal of the collimation width, can be approximated by an integral. Since the measured absorption curves for sweeps of high density samples extend out to 2 gauss, and since there was a possibility that the simulated line with might be as narrow as a few milligauss, there were strong restrictions on the time scales used in the simulation. The angular frequency resolution of the Fourier transform of a function, which is defined for a period T , is m  JJT^J  and the frequency range of transform is  ,  where A T is interval on which the function is known. In order for the Fourier transform of the simulated S(t) to have the required resolution and span the required width, S(t) was generated at intervals of 20 nsec for a period of 20 //sec. The average mean free time between collisions is 20 nsec, so in order to include all the collisions the simulation used time steps of 0.1 nsec, but only stored the change in the phase of the magnetization  Chapter  9.  The  ESR Lineshape  of  Hj j Adsorbed  ?- He Surface  121  4  oh a  every 20 nsec. The evolution of the phase of the transverse moment was simulated for 30 psec. The averaged quantity ( S(t)) was constructed by averaging the S(t) generated by starting at different initial times. Hardcore radii of 3.7 A and 6.88 A were used in the simulations. The calculated absorption line for r = 6.88 A and a density of 5 x 10 c m 11  c  - 2  is shown in Figure 9.2. A  fit to the average local field due to the dipolar interactionsfindsthat AB  e n d  = 1.29(17) x 10  -12  cr,  (9.16)  for r = 6.88 A . This result agrees with the result of the static model calculation which c  found a slope of —1.20(4) x 10"  12  Gem  -2  (see equation 9.6 ). The linewidth for those  conditions is 90 mG FWHM, independent of the density. The results of this model for the special case that includes contribution to the phase only from spins which collide with the test particle are consistent with the results of the first kinetic model. In summary, we have shown that the lifetime broadening of the absorption line of the adsorbed atoms is 0.8 mG, this is the minimum linewidth of the system.  For a  distance of nearest approach of 6.88 A , the hardcore particle simulations produce the same average shift in the position of the absorption peak of the atoms adsorbed on a surface perpendicular to the polarizing field as are measured in the experiments. The simulated lines are quite symmetric and are quite narrow, and no lineshape remotely similar to the observed ramp shapes was produced. For the case of r = 6.88 A and c  a surface density of 5 x 10 cm , the linewidth is 90 mG FWHM. This is very close to 11  2  the initial estimate of the linewidth and indicates that the large deviations of the field during the collision do not cause any unsual effects in the lineshape.  Chapter 9. The ESR Lineshape of HJ.! Adsorbed on a J- He Surface 4  122  Figure 9.2: Simulated Line of HJ.J. on a surface perpendicular to B . The squares are the results of the simulation and the curve is a spline fit to the data.  Chapter 10  Phenomenological Theory  10.1  Introduction  Since we have yet to discover the mechanism responsible for the unusual lineshape of HJ.J. adsorbed on the surface, we shall take the alternative approach for describing our results and attempt to find a set of equations that can simulate the observed results. The alternative to an ab initio theory is a phenomenological model. There are two separate, but very closely related observations to explain: the very early onset of saturation due to the applied r.f. field; and the unusual lineshape, even allowing for slight saturation of the line. The two effects are likely due to the same mechanism, but this need not be the case. We note that similar lineshapes have been observed in ferromagnetic resonance experiments on solid systems using high power levels[90]. The lines observed in the ferromagnetic resonance experiments have a ramplike shape, and non-linearity affects are seen with r.f. fields much smaller than those predicted from their measured Tj and T values. Guided by the similarity of the results of the ferromagnetic experiments to the 2  results we have obtained, we have attempted to model our results using Bloch equations that include demagnetization terms [86], which can be quite large since all the HJ.J atoms are nearly perfectly aligned. Since the ferromagnetic resonance theories are developed to describe the response of 3-D solid systems, we have adapted them to describe the response of a 2-D system[91] . 1  In the theories of high power ferromagnetic resonance of solids, the modified Bloch equations are used as an argument for the plausibility of the existence of instabilities in their systems, and eventually, 1  123  Chapter 10. Phenomenological Theory  124  We have numerically simulated the resonance lineshapes of a system whose response to the applied r.f. is described by the modified Bloch equations. These simulations are discussed in section 10.3. For the parameters appropriate to the HJ,| system, we find that if an active T i process existed in the HJ.| system at 0.1 K, then the response of the system described by the modified Bloch equations are similar to what we observe. Unfortunately, we cannot find an active Ti process for H | | adsorbed on the Z- He surface 4  at 0.1 K. In order to observe the non-linear behaviour in the modified Bloch equation, the applied field must be larger than a critical value which depends on the value of the Ti in the system. For an applied rotating field of 0.1 mG, we find that (in case where the linewidth is determined by the Ti) the T i must be less than 29 psecs in order to observe non-linear effects. We note that for the intrinsic linewidth found in the simulation discussed in chapter 9 the applied field must be larger than the critical field of 25 mG in the rotating frame as given by condition 10.11. In the next section, we consider the relevant T processes in HJ.]. for the experimental a  conditions, and show that none of those T i process would result in a rate that can be used in the modified Bloch equations to account for our results. Numerical integrations of the modified Bloch equations show that even with only a T  2  process in the system, the response of the system shows hysteresis with the sweep  direction with a 0.2 mG driving field. Since the presence of hysteresis in the system is an indication that instabilities may also exist in the system, we have studied the modified Bloch equations in some detail. This is described in section 10.3. microscopic spin wave theories are developed to explain the observed results. We will use a strictly phenomenological approach and determine whether these equations describe the observed results.  125  Chapter 10. Phenomenological Theory  10.2  Possible T i Processes in H j i at 0.1 K  HJ.J. & be created from a high density sample of HJ. by lowering the temperature of c a  the gas, until the recombination rates are much larger than the relaxation rates in the system. Since the rate of decay of | a) atoms is larger than that for | b) atoms, under those conditions the | a) atoms are quickly removed from the system. The favourable conditions for creating and maintaining a sample of HJ,j are also the conditions that result in the lack of an effective Ti process due to spin-exchange or dipole-dipole interactions in the system. This will be demonstrated by comparing the lifetimes associated with spin relaxation with those associated with recombination. The number of active processes in the gas is limited since the density of | a), | c), and | d) atoms is very small in a sample of HJ.J, at 0.1 K. The major mechanism for creating | c) atoms is the ESR sweep of the resonance line, which excites approximately 7.5 % of the 16) atoms on the surface into | c) atoms. Other active processes are nuclear spin relaxation during the collision of two | b) atoms or by the magnetic field due to the impurities in the walls. The electronic spin transition bb —* be is greatly inhibited by the 5.5 K energy required to flip a electron spin up, as are any other electronic relaxation process, which involve both atoms initially in the lower energy states. Furthermore, every spin exchange rate (except the bd ^ ae) is suppressed by the high polarizing magnetic field[50]. The spin exchange or spin relaxation processes which can contribute to a T\ process are be —* bb, ac — > ab, cc —• bb, and ba —• bb. The time constants associated with these processes for o — 5 x 10  11  cm"" are show in Table 10.1. The rate of the 2  one-body relaxation, which are due to magnetic impurities in the walls of the cell, b ^ a and b ^ c are the smallest rates in the system and can be ignored.  Chapter10.  Phenomenological  126  Theory  Time constant (sec) Process ba -» H 51 be H 3.2 x 10" be -+ bb 8.9 x l f r cc —*• bb 8.9 x 10" ac —> bd 103 2  7  2  2  1  Reference [15] [15] [50] [50] [50]  Table 10.1: The time constant for the active surface processes in HJ.J, at 0.1 K with a surface density of 5 x 10 cm . The small concentration of | a) in the sample are calculated from the measured magnitude of the one-body nuclear relaxation rate and the a-b recombination rate. 11  -2  Chapter 10. Phenomenological Theory  127  From Table 10.1, we see that for any pair of atoms, except for the two | c) atoms, _ that the rate of the recombination process is much larger than the rate of the possible relaxation process; thus, almost always the atoms recombine, instead of contributing to a T i process. The Ti due to the relaxation of the two | c)-atoms into | b)-atoms is very much larger than the 29 psec T i we require. There is the possibility that the recombination processes are contributing to a Ti rate; however, although we have not examined carefully how recombination processes might be included in the Bloch equations, we do not believe this is the case in for the Hjj system.  10.3  Modified Bloch equations  We have adapted the model used to describe ferromagnetic resonance in solids to the 2-D system of adsorbed Hjj[92]. Anderson and Suhl[93] used this model to demonstrate the effects that nonlinear mechanism can have on the spin system; they then proceded to develop a microscopic theory of spin waves to explain the occurrence of the observed effects in a larger class of systems. We will use the modified equations as a basis for phenomenologically describing the observed lineshapes. The failure of this model to describe the observed results is an indication that, as in the case for ferromagnetic systems, the mechanism responsible for the unusual response of the system are excitations which cannot be described by the demagnetization terms that were included in the Bloch equations. The model for describing the evolution of the magnetization of the sample is the usual set of Bloch equations with modifications: we include additional terms to the Bloch equations that incorporate the demagnetization effects due to the sample. Since the Hj j atoms are almost perfectly aligned, the net effect of a coherent tip of the spins (due to the applied microwave field) will be to modify the average local field. Assuming only classical dipole interactions in a system of uniformly distributed dipoles on a surface  128  Chapter 10. Phenomenological Theory  with a density o~, we find that, in the rotating frame, the local demagnetization field is given by (H ) - = Am.n,-, m  where A =  (10.1)  f  , m,- are the normalized magnetizations per unit area, and the n, are the  demagnetization factors, which depend on the orientation of the surface to the polarizing field H . If the surface is perpendicular to H , then if the z-axis is parallel to H , n,  =  1  (10.2)  n  =  1  (10.3)  =  -2  (10.4)  y  n  z  and if the surface is parallel to H , with the y- and z-axis in the plane, then n  x  n  y  n  z  =  -2  (10.5)  =  1  (10.6)  =  1  (10.7)  The Bloch equations for the normalized magnetization in a frame rotating with angular frequency u> are dm  —  (m,i - 6 ) - — -,  = -7e(m x H«) -  i3  .  (10.8)  where T , = T , T = T i , and x  y  2  z  H , = (H - - ) i + hi + H 0  m  ,  (10.9)  where h is the rotating component of the applied field h cos(wt)i that is polarized along the x-direction in the rotating frame. The resonant field H of the system described by r  equation 10.8 and 10.9, in the limit of not too large response, is given by H  r  ~ H + 0  (rij. — n )Am, %^ > 1 + Oside  K  z  z  (10.10  Chapter 10. Phenomenological Theory  where 8 id a  e  =  129  1 if the surface is perpendicular to H„ and 0 otherwise. The resonant  field of the bulk atoms is H . The H predicted by 10.10 agrees with the results of the r  0  theories discussed in chapter 9. For simplicity in the rest of this chapter, we shall only consider the case of H perpendicular to the surface. Since the resonant field condition depends on m , if the response of the system is not 2  small, then H changes as m evolves in time. In particular, if the applied bias field is r  z  slightly below that given by 10.10, then any response of the system decreases m and z  moves H closer to the the bias field, and so further increases the response. If a large r  enough microwavefieldis applied, then the response of the system can become unstable, and the observed absorption line will be dependent on the direction in which the line is swept. Following the theory of Anderson and Suhl[93], we find that the instability occurs in a magnetic disk, which is perpendicular to H , when (10.11) where h is the applied circularly polarized microwavefieldand A H is the FWHM of the T  absorption line for very small response . 2  We have integrated equations 10.10, while sweeping the applied polarizing field inorder to simulate a sweep of the resonance line. In these simulations, we assume that initially the spins are perfectly aligned along the z-axis. If we assume T  2  = 2Ti, take T i to correspond to a linewidth of 50 mG ( assuming  that the linewidth is due to the Ti[93]), take r  = 6.88 A and o = 5 x 10 c m , 11  c  -2  and satisfy condition 10.11 by using a microwave field of 25 mG, we observe the steep onset and the long tail in the simulated absorption curves when swept in either direction, although the position of the step is different in these cases. A plot of the simulated Physically, condition 10.11 is, to within factors of order unity, the requirement that the resonant field of the response shift more than the intrinsic linewidth during the sweep of the line. 2  130  Chapter 10. Phenomenological Theory  : absorption is shown in Figure 10.1. The behaviour of the absorption (m ) during the y  • sweep reproduces many of the major characteristic of the observed line, but the linewidth is too narrow to correspond to the observed line and we cannot be further increase it without increasing the rate of the Tj process.  Also, the microwave field required to  produce this type of absorption line is very much larger than the 0.2 mG applied in the experiments and the value of the Ti process is much shorter than any listed in Table 10.1. If we assume that 0.2 mG is large enough to be greater than the critical field required to produce these effects, then we find that T i < 2.8 x 10 sec, which is much smaller than -5  any T i rate listed in Table 10.1. If the simulation is run without a strong T! process, then the sample is completely destroyed before the sweep in completed. If the equations are integrated with h  =  O.lmG and T  2  =  1.13 x 10  -6  sec  (AH = 50 mG), then the lineshape still depends on the direction of the sweep, but the lineshapes are now quite symmetric, as seen in Figure 10.2. For T  2  =  2.27 x 10  -6  sec (AH = 25 mG, there is very strong distortion of the  line only when swept in one direction. This large effect is not seen in the measured lineshapes, although there does appear to be a some small sweep-direction dependence in the measured quantities. The modified Bloch equations to this point have not included the effect that a high-Q cavity can have on the result of a resonance line measurement. An oscillating magnetic moment induces a magnetic field in the cavity; if the loaded Q of the cavity is large, then the induced magnetic field can be appreciable and act back on the spin. For small perturbations of the magnetization, radiation damping in the system will introduce some terms into the Bloch equations that look vaguely similar to a T i process. Following the theory of Bloom[94], the effects of radiation damping are include in the equations by introducing an extra magnetic field in the xy-plane, which lags by | the projection of  Chapter 10. Phenomenological Theory  131  Figure 10.1: The absorption line simulated using the modified Bloch equation, but with a T i process in the system.  Chapter  10.  Phenomenological  -0.2  1  0.0  132  Theory  '  1  0.5  '  1  '  1.0  1  1.5  Applied field (G) Figure 10.2: A line produced from integrating the modified equations, but with no T i process in the system.  Chapter 10. Phenomenological Theory  133  the total magnetization onto xy-plane and whose magnitude is given by k(M + M ,) / 2  2  1  (10.12)  = kM (m + m ) ' ,  2  2  2  1  2  0  where k = 27rQif, with f the filling factor for the cavity, and M  = o^fljl.  0  When the  filling factor is calculated for the atoms absorbed onto the endplates of a cavity operating in the TMon mode, we find that A k = 4*r^Qi, s  (10.13)  where A is the total area of the endplates, and V is the volume of the cavity. The net 8  c  effect of adding the damping field to equation 10.9 is shown in the set of equations for the normalization magnetization given below Au OA ~, AUrny — 3Am m +i r/„ Km ™m  /  m  x  y  = -7  m  y  —AHm + 3Am m x  x  -hm where K then K  = kM , Ti = 0  =  z  z  x  + Km m y  - K(m  2  y  oo and A H =  z  \  z  /  + hm  + m)  z  H -  /  If <r =  T  2  T  2  \  K - i ) \ T  2  x  mjL  3  5  x  (10.14)  10 cm" , 11  2  2.33 mG. The results of the simulations are not noticeably modified  by including the radiation damping terms.  This is not an unexpected result, since  the size of the radiation damping term is very much smaller than the T ( 1.136 x 2  10" sec which corresponds to afieldA H = 50 mG) and the demagnetization factor (3A 6  = -0.623 Gauss), which all ready included in the equations.  10.4  Spin Waves  Nuclear spin waves due to identical particle effects have have been predicted[95, 96] to exist in Hj and have been observed[97]. The same theories predict electronic spin waves in Hj4[98], and so one might conjecture that the unusual lineshapes observed in these  Chapter  10.  Phenomenological  134  Theory  experiments are due to the presence of spin waves in the system. Since the instabilities observed in the high power ferromagnetic resonance experiments are due to interactions of the magnetostatic modes with the spin waves and observed lines are very similar to the lines observed in the HJ.J. experiments, electronic spin waves in Hj j might be the mechanisms responsible for the observed lineshapes in the present experiments. Although electronic spin waves, which are due to identical particle effects, may be detectable in some experiments on Hjl, for the conditions of these experiments it appears that they should not significantly alter the resonance response of the system. Bouchaud and Lhuiller[98] show that for a 3-D gas of only |b) atoms, the electronic spin waves in the limit of small perturbations are described by a diffusion equation with a complex diffusion coefficient:  & =r - ^ 2  dt  1 -  < >  v  1(U5  i p  where D* is the nuclear spin-dependent diffusion coefficient for the electronic spins, p is a measure of the efficiency of the identical spin rotations effects for coherently propagating the perturbation to the damping effect due to the random collisions. The z is the perturbation in the x-y magnetization due to the spin waves. Since the 2-D equivalent equation to equation 10.15 would be identical and the coefficients should be approximately the same to within factors of unity, we can use the results of Bouchaud and Lhuillier [98] for the bulk gas to estimate the surface spin wave properties. For the bulk gas, they calculate that p is -12 at 0.1 K. We can estimate the diffusion coefficient of the 2-D electronic spin waves by taking their value, 1 x 10 c m s , for 18  _1  _1  the product of the diffusion coefficient and the density in HJ, j , and using an effective bulk density derived from our surface density of 5 x 10 c m . We assume that the wavevector 11  -2  of the spinwave corresponds to a perturbation with a wavelegth of approximately the cell diameter. With the substitution of all the parameters into equation 10.15, the equation  Chapter  10.  Phenomenological  Theory  135  reduces to a form which allows it to be included in the modified Bloch equations 10.14. The terms due to the spin waves affect the transverse components of the magnetization appear exactly like the x and y terms due to the demagnetization effects of the whole disk. Since the effective field associated with the spin wave terms (13 pG) is much smaller than the associated with the demagnetization terms (0.6 G for the corresponding surface density), the spin waves terms have negligible effect on the behaviour of the system.  Chapter 11  Discussions and Conclusions  11.1  ESR at 115 G H z in H | and DJ.  Using the apparatus built by B. Statt as part of his doctoral work, we have studied the two-body recombination processes in HJ. and DJ. at temperatures below 1 K in a 41 kG field using ESR at 115 GHz. We have extended the temperature range over which the two-body recombination rates in HJ. had been measured and reduced the systematic uncertainties in the measured rates. With further improvements of the apparatus, we have studied the intrinsic ESR absorption line of HJ.J. adsorbed on the /- He film that 4  covers the microwave cavity cell. We will discuss the results of the surface lineshape studies in the next section. At the lower temperatures, the measured recombination rates were found to be in accord with the rates predicted by a "sudden" approximation model, which has been successful in unifying the results of two-body recombination rate measurements taken under different magnetic field conditions. In particular at temperatures below 0.5 K, the measured rates at 41 kG could be satisfactorily related to the zero-field decay rates given by Jochemsen et a/.[55]. Above 0.5 K, the behaviour of the measured rates was markedly different from that predicted by the appropriately scaled zero-field recombination measurements. We have shown[47] that the anomalous recombination rates are due to the resonant recombination of two |a) atoms into the (v,J) = (14,4) bound molecular state, a process which becomes possible in large enough magnetic fields. The resonant  136  Chapter  11.  Discussions  and  Conclusions  137  recombination process does not contribute to the zero-field rates. We were the first to directly observe adsorbed atomic hydrogen on the surface of the /- He which coats the cavity cell walls. The measurement of the bulk and surface densities 4  allowed us to directly determine the binding energy of the hydrogen to the surface. In experiments using the original cell, we found a binding energy of 1.00(5) K. With the second cell used in the surface lineshape studies, we found a value of 1.03(1) K; the most accurate measurement to date of the binding energy [26] gives E B = 1.011(10) K. We have also measured the two-body recombination rates in atomic deuterium in a 41 kGfieldat temperatures below 1 K. Under comparable conditions, atomic deuterium was found to be much more difficult to observe than atomic hydrogen, which is due to fact that the recombination rates in atomic deuterium are much larger than in hydrogen. During the experiments we were able, for the first time, to produce and observe doubly spin-polarized deuterium (deuterium with both electron and deuteron spins aligned). We also obtained samples which contained doubly spin-polarized deuterium and doubly spin-polarized hydrogen, a mixture of two fully spin aligned gases. Although it would have been interesting to study the spin transport properties of such a system, with the present apparatus we were only able to measure the recombination rates between atomic deuterium and hydrogen. At 0.7 K and in a 41 kG field, the effective bulk rate for DH recombination is approximately equal to the effective bulk DD rate, and it is 150 times larger than the effective bulk rate for HH recombination. The main goal of much of the research in HJ. has been to observe Bose-Einstein condensation in a weakly interacting gas which may or may not be achieved in the near future. In the mean time, there are many interesting physical mechanisms to be studied in the non-Bose-condensated atomic gases. One advantage of studying processes in the atomic hydrogen system is that many of necessary quantum mechanical calculations required to understand the details of a particular phenomenon can be done from first principles, and  Chapter 11. Discussions and Conclusions  138  the results of experiments directly compared with the predictions of theories. As an exvample, it would be useful to improve the accuracy of the resonant recombination rate in atomic hydrogen, to better determine the position of the (14,4) level of molecular hydrogen. The result can be used to to determine whether the non-adiabatic and relativistic corrections to the H-H potential have been properly carried out in the theories. The atomic deuterium system has not been as thoroughly studied as the hydrogen system, in part this is due to the fact that it is very difficult to observe spin-polarized deuterium. It is difficult to measure the recombination rates in deuterium with the present apparatus because the lifetime of a deuterium sample is of the order of a minute. However, there are some interesting measurements that can be made on the deuterium system. For instance, it should be possible to determine the binding energy of deuterium to the /- He surface by simultaneously measuring the surface and bulk atom densities of 4  deuterium, just as we have done for the hydrogen system. Since the binding energy of deuterium appears to be about 3 times larger than that of hydrogen, one should be able to detect deuterium surface atoms at temperatures below 0.3 K.  11.2  Lineshape of Hjj on the Surface  At temperatures of approximately 0.1 K, adsorbed HJ.J atoms can be directly detected by ESR. On a plane surface, the averagefielddue to the magnetic moment of the aligned dipoles is non-zero, and depends on the orientation of the polarizing magnetic field to the surface. Accordingly, the ESR absorption due to atoms on the different surfaces appears as separate peaks displaced from the bulk atom peak. For a given surface density, the average field shifts of the side peaks from the bulk line can be predicted from very simple models. This was first observed by us[15] using the original cell. Subsequently, further studies were made using a new cell having microscopically flat surfaces.  Chapter  11.  Discussions  and  Conclusions  139  After extensive attempts to model the detailed shape of the line associated with the surface atoms, we have yet to discover the mechanisms responsible for the unusual lineshape or for the very early onset of nonlinear effects in the spin system. The observed surface lines have a ramp-like shape, which depend only weakly on the direction in which the resonance is swept. The shape of observed surface lines is not due to the inhomogeneities in the polarizing field or to magnetic impurities in the walls of the cell, since these mechanism also affect the lineshape of the bulk atoms. The observed bulk line width is quite narrow (w 0.1 G) and determined by the inhomogeneity in the applied field; whereas, the typical surface atom linewidth is about one gauss. We have observed a non-linear response of the surface atoms to the applied microwave power. These effects are observed with applied r.f. fields of 0.2 mG. We have simulated the absorption line of systems described by a modified Bloch equations, which include the demagnetization effects of the aligned dipoles, the radiation damping effects, and the spin wave interactions; and we have seen interesting effects. However, we have not been able to obtain correspondence with the observed lineshape or find a mechanism for the early onset of non-linear effects. The elucidation of this mystery will require further work, perhaps both theoretical and experimental. The theoretical analysis of the lineshape is complicated since we seem to be observing some non-linear effects induced by the applied microwave magnetic fields which partly obscures the intrinsic lineshape of the absorption peak. Since the signal to noise level of the present spectrometer is marginal for the case of surface experiments, further experiments with the same apparatus would not be very useful, since one cannot substantially lower the driving field without seriously lowering the S/N ratio. The primary source of the noise in the detection system is the high frequency diode mixer detector, the noise temperature of which has been measured to be about 6500 K[41]. The room temperature diode mixer needs to be replaced with one that has a lower noise temperature, such as a  Chapter  11.  Discussions  and  140  Conclusions  cooled diode mixer or even better an SIS mixer. In addition, if the mixer were moved from its present location at the top of the cryostat down into the /- He bath, then the losses in 4  the waveguides would be substantially reduced, further improving the S/N of the system. However, this would require the use of low temperature microwave circulators and high-Q tunable filter: at frequencies of 115 GHz,  this would be a non-trivial feat of cryogenic  microwave engineering. The increased S/N should easily allow the measurement of the absorption lines without the complications from non-linear effects.  Appendix A  Numerical Simulations of the ESR Response of the Bulk Atoms  The simplest model for describing the response of a spin system in a continuous wave ESR experiment assumes that only a small fraction of the atoms are excited by the applied power and that the number of atoms, which are excited, can be calculated using first order perturbation techniques. This model is valid in systems where thefluctuationsin the local fields prevent a coherent rotation of the spins by the applied field. In terms of the intrinsic  T of the spin system, this requirement is that 2  h < ± . ,  (A.1)  where 7 is the gyromagnetic ratio of the spin and h is the applied microwave field. In our experiments, the observed linewidth, 0.1 mG, of the bulk absorption peak is due to the inhomogeneity in the polarizing magnetic field. For the relatively low densities of the samples obtained in these experiments, the intrinsic linewidth of the bulk Hj-atoms is very narrow, since for a dilute Hj gas, the T due to the spin-exchange processes is of the 2  order of seconds, and the T due to the fluctuations in the local field of the dipole-dipole 2  interactions is also of the order of a second. As a result, the applied microwavefieldmust be smaller than 5 x 10 G in order to satisfy condition A . l . For the experiments on the -8  surface atoms, the appliedfieldin the rotating frame has been determined to be 0.1 mG (see Appendix B). For the experiments that measured the recombination rates, it was considerably smaller, approximately 5 //G. However in both cases, the applied microwave fields are much larger than 5 x 10~ G, so that it is not clear whether the perturbation 8  model is applicable for describing the system's response. 141  c  Appendix A. Numerical  Simulations  of the  ESR Response  of the Bulk  Atoms  142  The extremely long mean-free path in the low-density bulk H | gas adds a further complication to the ESR response of the system. While the observed resonance peak is being swept, the H|-atoms in the sample are moving back and forth across the cell and through the non-uniform polarizing magnetic field; consequently, they are subject to a large number of slightly saturating passages. During a slightly saturating passage, the magnetization of the spin is significantly rotated around an effective magnetic field, a major component of which is the microwave field. Because the spins can, on average, make several hundred passages across the cell and not lose their phase coherence in a collision with another atom, the effects of the slightly saturating tips can be greatly enhanced, and as a result, dramatically change the response characteristics of the system to the applied field. Thus, it is not apparent whether even the Bloch equations are adequate for describing the ESR response of this system. In order to determine if a more sophisticated description of the system is required, we have numerically simulated two different models and determined the predicted loss, as a function of the applied microwave field, of the total magnetization due to a sweep of the resonance peak. One simulation uses the Bloch equations to describe the ESR response of the system, taking T from the observed linewidth. The other simulation considers the 2  effects of the multiple, slightly saturating, passages on the total magnetization during the time that the resonance peak is swept. Treating the applied microwave field as a parameter, the Bloch equations were integrated as the polarizing magnetic field was swept through the resonance condition. We used the experimentally observed line width, 0.1 G, as the width of the line due to the T  processes, and ramped the polarizing field at 7.26 Gsec , the value used in all the -1  2  experiments. We are interested in determining the decrease in the magnetization of the sample due to a sweep of the resonance, since the fraction of atoms removed by a sweep is a quantity that can be measured accurately (see Appendix B) in our experiments. The  Appendix  A. Numerical  Simulations  of the ESR Response  of the Bulk  Atoms;  143  simulation results for the fractional loss in the longitudinal magnetization as a function of the applied microwave magnetic field are shown in Figure A . l . Also, drawn in the same Figure is a line, which is proportional to the microwave power applied to the atoms. The results of the simulation behave generally as expected. For very small power levels, the loss in the total magnetization is proportional to the applied power. The low field results of the simulation are not sensitive to the exact value of T . But, even at moderately 2  low amplitudes of the appliedfield(compared to the linewidth of 0.1 mG), the loss is no longer proportional to the applied power and the saturation effects due to the applied radiation are seen. We have used a simple model to simulate the effects of the slightly saturated passages on the total magnetization of the sample.  Although the model makes some drastic  simplifications, it should contain all of the important components of the physical system. The physical system of a cavity in a non-uniform polarizing field and filled with many spins, which are weakly interacting with each other and free to move around in the cell, is replaced with a model that considers one classical spin that is moving in afieldwith a linear gradient and trapped inside an one dimensional box. As the spin bounces back and forth between the walls, which are separated by a typical length of the cavity, it moves through a polarizingfieldthat is set to the resonant condition of the spin and upon which is superimposed a linear gradient, which is zero at the center of the box. The gradient is chosen so that the difference in the magnetic field across the cavity is equal to the measured linewidth of the absorption peak. The perturbing microwave field is assumed to be uniform across the box. Starting with the spin completely aligned anti-parallel to the polarizing field and setting a value for the microwave field, we integrate the Bloch equation describing the spin, while it bounces inside of the box with a speed that changes each time it collides with a wall. We introduce some thermal averaging into the model by  Appendix  A. Numerical  Simulations  of the ESR Response  of the Bulk  Atoms  144  using the Maxwell Boltzmann distribution, appropriate for a temperature of 0.23 K ,to 1  determine the new speed assigned to the spin at every bounce. It is assumed that the spin retains full memory of its phase, unless it collides with another magnetic particle. Since the mean free time between collisions with each other is larger than the time required to sweep the magnetic field through the resonance peak of the sample (and the wall are non-magnetic), the spin retains track of its phase for the duration of the sweep; thus, we integrate the Bloch equations of the spin for the time it takes to sweep the peak. The results of the simulations are also presented in Figure A . l , where we, again, have plotted the fractional decrease in the total magnetization, as a function of the applied microwave field, due to a sweep of the resonance peak. The results of this simulation are quite insensitive to the value of the intrinsic T of the spin, as long as the linewidth, 2  associated with the T , is much smaller than 0.1 G. The uncertainties in the values 2  returned by the simulation are quite large, this is due to the simple averaging method used in this simulation. The results of the slightly-saturating-passages (ssp) simulations agree with the Bloch equation simulations for the very low to moderate applied field, but differ when the applied fields exceed 0.2 mG. Since the applied linearly polarized microwave field was 0.2 mG, the two results agree, given the uncertainty in the ssp simulations; moreover, they predict the value of the fractional decrease in the magnetization quite well. The both simulations predict a loss of « 9%. The measured fractional decrease in the magnetization of a low density sample of HJ, is 5 %, but a correction for the distribution of the microwavefieldmust be applied in order to compare the results. As the simulations assume an uniform microwave field, whereas the microwave field in a cavity operating at the TMon mode is non-uniform, an effective volume correction of 2, as discussed in Appendix B, must be applied to the experimental value. See Appendix B for an explanation for choosing this value of the temperature.  1  Appendix  A. Numerical  Simulations  of the ESR Response  of the Bulk  Atoms  145  In conclusion, for the low microwave fields applied in the present ESR experiments, the magnetization loss due to partially saturating sweeps can be calculated from the Bloch equations taking T from the experimentally observed linewidth. 2  Appendix  A. Numerical  Simulations  of the ESR Response  of the Bulk  Atoms  146  1.00 N  o  0.80  m O  0.60 -  fl  0.40 -  o  •rH  O  Ctf rH rXH  0.20 -  0.00 0.00  0.10  0.20  0.30  0.40  Applied field (mG) Figure A . l : The fractional decrease in the total magnetization during a sweep, as found by the different simulations, are plotted as a function of the applied linearly polarized microwave field strength. The results of the slightly saturated passages simulation are plotted as open triangles. The Bloch equation results are shown as open squares, and the the dashed line is a spline fit to these results. The solid line is a curve that is proportional to the applied microwave power.  Appendix B  Exciting Atoms in a Cavity  In our studies of the recombination rates in HJ,, we have generally assumed that the simplest standard models for the ESR or NMR response of spins in a microwave cavity were sufficient in the case of ESR .in HJ.. Our later efforts to understand the exact mechanisms responsible for the unusual ESR response of HJ,J, atoms adsorbed on the /- He 4  surface have raised questions which require that we examine in detail the validity of these earlier assumptions. Appendix A is concerned with determining the appropriate models for describing the ESR response of the atomic hydrogen system.  The ESR response  of the surface atoms is reminiscent of the characteristic spectra found in high-power ferromagnetic resonance studies on solids[90], and it has led us to consider non-linear effects. In this appendix, we are primary interested in developing a reliable method for experimentally determining the amplitude of the applied microwave field, which is a critical parameter in the non-linear effects of a spin system, since a reliable measurement of the field is imperative. Given that it is a practical impossibility for us to directly measure the incident power on the cavity with any precision, alternative methods had to be devised. Two methods are considered. Thefirstmethod does not assume any model for the response of the spin system; it is only concerned with the total energy absorbed by the HJ,-atoms from the microwave field. The second method assumes that the magnetization of the system is tipped by a slightly saturating passage through the resonance. In fact, as we will see, good agreement between the results of the two methods is obtained for low to moderate  147  Appendix  B. Exciting  Atoms  in a  Cavity  148  microwaves fields; this lends support to the validity of describing the ESR response of the bulk atoms with Bloch equations of the total magnetization of the sample. The reason for questioning the validity of using the Bloch equations for the total magnetization to describe the system is that the intrinsic linewidth of the H|-atom is very narrow, and a typical atoms can move several hundred times back and forth across the cell during the time it takes to sweep the resonance peak. Since there is a gradient in the polarizing magnetic field, the rapidly moving atoms may be subject to slightly saturating passage each time it crosses the cell. For an arbitrary strength of the microwave field, the behaviour of such a system is certainly not well described by a single set of Bloch equations. Before describing the procedures, we first describe a method for determining the number of atoms removed during a sweep. This method requires only the properties of that particular sweep, and in addition, does no assume any model for how the atoms evolve during sweep. This method is distinct from, but uses the results from, the procedure of performing many sweeps in rapid succession through the resonance, and noting the decay of the amplitude between successive sweeps. The microwave power transmitted to a sample in a cavity cell depends on the power incident on the cavity cell, the coupling of the cavity to the transmission line, and the Q of the cavity. If the circuit parameters and the incident power are known, then the total energy that is absorbed by the Hj-atoms when a resonance peak is swept can be found by evaluating an integral depending on the reflected signal and on the absorption curve of the resonance. Normally, not all of the required parameters are known, in particular, the incident power is not known with any accuracy. For the experiments described in this thesis, the relevant integral can only be evaluated up to an unknown scaling factor. However, the total energy that was absorbed by the spin system can also be determined from the number of atoms that were destroyed when the resonance line was swept, as  Appendix  B. Exciting  Atoms  in a  149  Cavity  described in the preceding paragraph. This allows us to determine the scaling constant of the integral. In order to determine the scaling factor and, thereby, the strength of the field, a series of calibrations sweeps were taken on a sample of H j j at T « 0.23K. These conditions were chosen for the following reasons: the decay kinetics in the system are simple, when only | b)-atoms are present; there are no complications due to the presence of surface state atoms for temperatures above 0.1 K; and the intrinsic decay rate of the sample is quite small at these temperatures, so that the microwave induced recombination rate completely dominates the decay. An equivalent circuit [99] for a cavity that is filled with magnetic atoms and coupled to a transmission line is shown in Figure B . l . The imaginary part of the impedance (including the cavity and the effects of the magnetic atoms) is given by Z ; whereas, c  the loss mechanisms due to the cavity and to the atoms are represented respectively by fcavity and r  a t o m s  . The transmission line is taken to have a characteristic impedance Zo-  At plane S, the right going (incident) voltage signal is given by Re(y_e~' ) wt  going (reflected) voltage signal is given by Re(v  Plane S is chosen so that v is  e~ ). ,iJjt  +  and the left +  real. The net time averaged power crossing plane S, and being absorbed in the combined cavity and atom system is P =  5^(1  v | +  2  -  | v_ | ). 2  (B.l)  The signal reflected from the cavity is mixed in the heterodyne detection system, and finally, recorded as signal voltage v which is related to v_ by v = av.,  (B.2)  where a is some constant that depends on the parameters of the detection system. If the cavity reflection coefficient were 1, then v would have the value corresponding  150  Appendix B. Exciting Atoms in a Cavity  Figure B.l: The equivalent circuit of a cavityfilledwith magnetic atoms. Z contains all the imaginary part of the impedance due to the cavity and the effects of the atoms. c  Appendix  B. Exciting  151  in a Cavity  Atoms  to v , i.e. +  v =  VQO  = av .  (B.3)  +  Since v is chosen to be a real voltage, substituting equations B.3 and B.2 into equation +  B.l, we get  Since Z is pure imaginary, the fraction of power x that is absorbed by the atoms is given c  by x =  —  ,  (B.5)  fatoms "4" ^cavity  where r ms = 47ra>L jt x", and r i t cav  at0  y  c a v  = woL avity<3o  y  C  Thus, the power absorbed by  the atoms, P t o m s , is given by a  Patoms = P  m  (B.6)  X  ^oQo  We use that Qo = Q ( l +  , . ~ + 47ru>x"  where Qo is the unloaded Q of the cavity and Q is the m  loaded Q of the cavity connected into the system, but without atoms in the cell. Defining -y" = 4 7 r Q x " , we find that the total energy E absorbed by the atoms during a sweep m  is given by  In practice u>o = u> to within one part in a million. After the absorption x" and, hence, 7" are found from the signal voltages, the integral B.7 can be evaluated to within the constant Zo | a. | . 2  During a sweep, a number of atoms N  e x  absorb energy and make a transition into the  excited state. The energy absorbed from the microwave field is (B.8)  Nexfiwt).  The total number of atoms removed by a single sweep N is measured by observing r  the decrease in the sample density as a rapid series of sweeps are performed on the  Appendix  B. Exciting  Atoms  in a  152  Cavity  sample. The number of atoms excited by the microwave radiation is related to the number removed by the sweep as u.  (B.»)  = $.  where M is the number of atoms removed from the gas when one atom is put into an excited state, and is usually close to 2. Thus, assuming a value for M , the direct measurement of N gives us the N ^^o that can be equated to the result of the integral in r  ex  B.7, and thereby, determining the constant Z | a | . As long as the incident power is not 2  0  changed, the number of atoms lost in a particular sweep can be determined by evaluating integral B.7. Since this method treats identically the situation where the resonating atoms strongly perturbs the cavity and where the response is weak, this method is particularly useful in experiments where large variations in the total susceptibility of the sample are encountered. We now proceed to describe the two methods for determining the absolute microwave field in the cavity during the sweep. The first method uses some of the results derived in the proceeding paragraphs, and aside from a well justified assumption regarding the lineshape, makes no assumption of the exact process by which the atoms make their transitions to the excited state. The second method assumes that the atoms are subject to a slightly saturating passage when their resonance peak is swept. The first method for determining the absolute microwave field is only applicable in the limit of small magnetizations, because it assumes that the cavity is not appreciably perturbed by the resonating atoms.  In these experiments, at densities of 10 cm , 13  -3  the real part of the reflected signal is essentially identical to the absorption curve; this condition is the necessary criterion for small perturbations of the cavity parameters by the resonating atoms. The lineshape of a sweep of low density bulk atoms can be well approximated by a rectangle, whose width is the F.W.H.M. of the line and height is the  153  Appendix B. Exciting Atoms in a Cavity  maximum value of the line. In this approximation, near the center of the resonance the power into the atoms is given by P„ =  (B.10)  where T is the time it takes to sweep through the peak. The fraction of power absorbed by the atoms is given by B.5, thus the total power absorbed by the cavity P is given by c  P c = P i n ( J " 1).  (B.ll)  The power dissipated in the cavity is also given by P  = S5«  c  (B.12)  where U is the total stored electromagnetic energy in the cavity. The total stored electromagnetic in the T M n mode in the ESR cell is 0  U = ^M SV,  (B.13)  E  where V is the volume of the cavity and Eo is the amplitude of the electric field of the mode. For the TMon mode, the maximum microwave magnetic field in the cavity is numerically equal to 0.728 E (c.g.s. units). Combining equations B.5 and B.10 through 0  to B.13 together, we find that the maximum microwave field is / 2  feN  (MVT I ; (  If we assume that M =  \ W  X  . )  1 / 2  •  < - > B  I4  2, then, for our typical experimental conditions, we find  that the maximum amplitude of the linearly polarized microwave field in the cavity was 2.1 x 10  -4  G. This is equivalent to a field of 1.05 x 10~ G in the rotating frame. 4  The second method is based on a result due to Goldman et. al [100], who formulated a method for calibrating the strength of r.f. field used in nuclear magnetic resonance  154  Appendix B. Exciting Atoms in a Cavity  experiments on solids. Using the Provotorov equation , they considered the problem of 1  a linear passage through a resonance line under the condition that the sweep produces only a slight decrease in the value of the longitudinal magnetization. If the conditions that ^  «  ^  «  7e#, ,  (B.15)  2  with Hi = ^ r - , and (B.16)  are satisfied in the resonance experiment, then the decrease in the longitudinal magnetization 6M is given by Z  H  6M  Z  =  - M  z  7 r  7  e  2  (B.17)  - # , dt  independent of H;, and where H ./. is an applied uniform field in the rotating frame. Since r  the microwave field in the T M n is non-uniform, the R . j in B.17 is the average over the 2  0  cavity volume of the square of the applied field, and is given by E  2 rJ  = I V  For the T M n mode H , =  /  dv H .. 2  (B.18)  J cavity  After a series of slightly saturating passages, the  2  0  magnetization of the sample is H  M  2  t  2  = M exp(-n7T7e-j^-), 0  (B.19)  dt  if the duration of the experiment is much shorter than the T i in the system. The time taken for the entire series of sweeps is t and n is the number sweeps per second. Since the linewidth of the absorption peak is set by the inhomogeneity of the applied polarizing field and since the applied r.f. field is very small, conditions B.15 and B.16 are The same results can also be derived using the standard formula for the transition probability under the effect of a harmonic perturbation. 1  Appendix  B. Exciting  Atoms  in a  155  Cavity  , satisfied in the present experiments. 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