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Longitudinal nuclear spin relaxation in ³HE gas at low temperatures Chapman, Ross 1975

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LONGITUDINAL NUCLEAR SPIN RELAXATION IN 3HE GAS AT LOW TEMPERATURES by ROSS CHAPMAN B.Sc, McMaster University, 1968 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of PHYSICS We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June, 1975 In presenting th i s thesis in par t i a l fu l f i lment of the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f ree ly ava i l ab le for reference and study. I further agree that permission for extensive copying of th i s thes i s for scho lar ly purposes may be granted by the Head of my Department or by his representat ives. It is understood that copying or pub l i ca t i on of th is thes is fo r f i nanc ia l gain sha l l not be allowed without my writ ten permiss ion. Department of The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 i i Abstract The first measurements of the temperature dependence of the intrinsic 3 dipolar relaxation time T^ due to binary collisions in dilute He gas are reported. Sufficiently pure experimental conditions to observe T^B were prepared by coating bare pyrex sample cells with clean neon gas. The experiments are performed at low temperatures (1.7 K to 19.0 K) where the colliding atoms have very low energy, so the effect of the long-range attractive forces is strongly felt and the measurements are sensitive to the depth and nature of the helium attractive well. Calculations of T using the best available helium potentials have been IB fitted to the experimental results to test the parameters which describe the potential. The data favours a potential of the Bruch-McGee form, but having a slightly deeper attractive well of 11.5 K. 3 The experiment also probes the adsorbed phase of He via wall relaxation. Both wall relaxation and bulk gas relaxation are present in a l l measurements, but they can be identified by their density dependence. Measurements of wall relaxation time T ^ have been made on strongly relaxing and weakly relaxing surfaces, and the field dependence has been studied between 0.5 kG and 9.3 kG at temperatures of 2.6 K, 4.2 K and 8.0 K. The results of the field dependence are interpreted in terms 3 of a model which considers relaxation of He atoms diffusing on a plane and interacting via the dipolar coupling. i i i Table of Contents Page Abstract Lis t of Tables v L i s t of Figures ' y i Acknowledgements ~. F i i i Chapter 1 Introduction 1.1 Introduction to Intermolecular Forces 1 1.2 General Ideas of NMR in Monatomic Gases 5 1.3 NMR in 3He gas 7 3 2 Theory of Longitudinal Spin Relaxation in Dilute He Gas 11 2.1 Calculation of o(E) for dipolar coupling 12 2.2 Other Relaxation Mechanisms 19 2.3 Time Correlation Function Approach 20 2.4 Relaxation at the walls 23 3 Experimental techniques 3.1 Measurement of Longitudinal Relaxation Times 27 3.2 Experimental Apparatus 30 3.2.1 General Description 30 3.3 Radiofrequency Bridge Spectrometer 32 3.4 Coil and Resonant Circuit 40 3.5 Field Sweep Unit 43 3.6 Cryostat 43 3.7 Temperature Measurement 45 3.8 Gas Handling System 49 3.9 Methods of Suppressing the effects of Wall Relaxation 51 i v Page 4 Presentation of Results and Discussion 4 . 1 Introduction 5 5 4 . 2 Possible Sources of Error 5 6 4 . 3 Temperature Dependence of T-^ g 6 3 4 . 4 Density Dependence of T ^ B 7 0 4 . 5 An estimate of the Effects of 3 Body Collisions 7 2 4 . 6 Results of Wall Relaxation Measurements at 1 . 0 K Gauss 7 4 5 Field Dependence of the Wall Relaxation 5 . 1 Introduction 7 9 5 . 2 Presentation of Results 8 3 5 . 3 Measurements of Field Dependence by Optical Pumping Techniques 8 9 6 Suggestions for Future Experiments 9 1 Appendix A. Calculation of Relaxation Time in an Adsorbed Layer. 93 Appendix B. Effects of Diffusion on AFP 104 Bibliography 1 0 9 V List of Tables Table Page 4.1 Results of fitting temperature dependence by calculations of T-^ g using available helium potentials. 68 4.2 Wall Relaxation times T l w at 4.2 K, 1.0 K Gauss and p = 1.0 x 10~2 gm/cc. 77 5.1 Correlation time T and area per adsorbed atom a from wall relaxation measurements 87 List of Figures Figure Page 2.1 MS and Bruch-McGee helium Potentials 15 2.2 Dipolar cross-section vs. relative energy for MS and BM helium potentials 16 2.3 T1t> vs. T, calculations using MS and BM potentials 17 IB 2.4 Dipolar and Spin rotation cross-sections 19 3.1 AFP methods of measuring relaxation times 28 3.2 Block diagram of RF Spectrometer 31 3.3 Twin-T or Anderson Bridge 33 3.4 Possible coil-coax configurations 36 3.5 3.5 MHz Crystal Oscillator 37 3.6 Cascode Amplifier and detector 38 3.7 Field Sweep Unit 41 3.8 Variable temperature Cryostat 42 3.9 Thermometry Circuit 46 3.10 Gas handling System 48 4.1 Relaxation curve for low f i e l d measurement of T^  58 4.2 P/T 1 vs. p 2 60 4.3 Relaxation curve for high f i e l d measurement of T^  61 4.4 PT 1 B vs. T 64 4.5 PT1T) vs. T for best values of BM helium potential 66 IB 4.6 Log vs. log p 71 4.7 Estimate of 3 body effects 73 4.8 T,„ vs. p 75 Figure Page 5.1 1^  vs. H 80 5.2 Log T l w _ 1 vs. log H 83 5.3 Plots of T1IT "'"vs. UT for selected values of x 85 1W Al Plots, of 8^(t) f° r dipolar He-He surface interactions 96 A2 Plots of j^(wt) for dipolar He-He surface interactions 98 A3 Plots of gx(t) for dipolar He-paramagnetic site interactions 100 A4 Plots of j^(wt) for dipolar He-paramagnetic site interactions 101 Bl Dependence of relaxation on applied RF voltage 105 B2 Recovery of Magnetization at short times 108 : vi i i Ac knowled g ement s I am deeply grateful to Professor Michael Richards of the University of Sussex who not only provided the i n i t i a l stimulus for this project and worked with me during the f i r s t few months, but also continued to give encouragement and help throughout the course of the work. I also thank my research supervisor Myer Bloom for giving me the chance to work on a project of my own choice and continuing to have f a i t h and lend support during the extended search for the very long relaxation times. I have benefited greatly from the many discussions we have had. Professor Bernie Shizgal of the chemistry department kindly gave me the use of his programs for the calculation of T and has also contri-buted useful suggestions. Finally, I wish to thank Mr. John Lees for his willing assistance and sound technical advice in the use and construction of the glassware required in the experiment. I also appreciate the work of Mr. Ernie Williams, his assistant, who performed much of the glassblowing. 1 CHAPTER 1 INTRODUCTION 1.1 I n t r o d u c t i o n t o I n t e r m o l e c u l a r f o r c e s N u c l e a r s p i n r e l a x a t i o n i n d i l u t e ga ses has l o n g been r e c o g n i z e d as a s e n s i t i v e means o f s t u d y i n g the dependence o f i n t e r m o l e c u l a r f o r c e s on m o l e c u l a r o r i e n t a t i o n ^ . Because the n u c l e a r s p i n r e l a x a t i o n t imes a r e m e d i a t e d by i n t e r m o l e c u l a r t o r q u e s w h i c h change t h e m o l e c u l a r a n g u l a r momenta d u r i n g a c o l l i s i o n , r e l a x a t i o n t ime measurements have p r o v i d e d an e f f e c t i v e p r o b e o f t h e a n i s o t r o p i c p a r t o f t he i n t e r m o l e c u l a r (2) (3) p o t e n t i a l . R i e h l e t a l . and L a l i t a and B loom have s u c c e s s f u l l y o b t a i n e d i n f o r m a t i o n on the Harare gas p o t e n t i a l by n u c l e a r m a g n e t i c r e s o n a n c e (NMR) e x p e r i m e n t s on gaseous m i x t u r e s . The NMR r e l a x a t i o n r a t e o f m o l e c u l a r sys tems i s r e l a t e d by a w e l l e s t a b l i s h e d t h e o r y ^ ^ t o the c o r r e l a t i o n t ime f o r m o l e c u l a r r e o r i e n t a t i o n . The c r o s s s e c t i o n f o r m o l e c u l a r r e o r i e n t a t i o n i s c a l c u l a t e d i n terms o f the i n t e r m o l e c u l a r p o t e n t i a l , b u t the i n t e r p r e t a t i o n o f e x p e r i m e n t a l r e s u l t s i s d i f f i c u l t s i n c e even i n t h e s i m p l e s t m o l e c u l a r sy s tems n e i t h e r the i s o t r o p i c n o r the a n i s o t r o p i c p a r t s o f the p o t e n t i a l a r e w e l l known, and o n l y v e r y c r u d e f u n c t i o n a l forms o f t h e p o t e n t i a l a r e u sed i n c a l c u l a t i o n s . 3 By c o n t r a s t a sample o f He gas i s a s y s t e m o f v e r y l i g h t w e a k l y c o u p l e d s p i n % p a r t i c l e s i n w h i c h the w e l l known a n i s o t r o p i c d i p o l a r c o u p l i n g between s p i n s i s r e s p o n s i b l e f o r r e l a x a t i o n . The r e l a x a t i o n r a t e i s i m m e d i a t e l y r e l a t e d t o the c r o s s s e c t i o n f o r s p i n t r a n s i t i o n s 2 caused by the dipolar interaction and the cross section can be calculated in terms of adjustable parameters in the functional form of the isotropic 3 part of the helium potential. Although He gas is the most fundamental system in which to apply NMR techniques in order to investigate inter-particle potentials, difficulties associated with the preparation of sufficiently pure experimental conditions remained as a barrier to the experiments. Improved methods of sample preparation and cleaning techniques developed in this work have removed the experimental problems and permitted for the first time observation of the dipole-dipole relaxation 3 time T, in bulk He gas. XD The precise measurements of the relaxation time can be used to test the theory of NMR relaxation in a fundamental system, and also provide an independent experimental check of the parameters in the helium potential. Since very good agreement with the theory of relaxation is expected, we hope to stress the nuclear spin relaxation time measurements as a unique probe of the potential. The subject of intermolecular forces is enormous and only a brief discussion the scope of which is restricted to non-polar systems, will be presented. A discussion in greater depth can be found in Hirshfelder e t a l ( 5 ) . The simplest picture of the central part of the interatomic potential is a sum of two contributions, a long range electromagnetic attraction and a short range repulsion. Much of the empirical information about the central potential comes from conventional techniques such as experiments to measure v i r i a l coefficients, transport properties (viscosity, diffusion and thermal conductivity), and also beam 3 scattering experiments. Exact theoretical calculations, particularly of the short range part of the interaction, are extremely difficult multi-electron problems and have been attempted only for simple systems such as H-H, He-He and H^ -He. The helium potential has received much experimental and theoretical attention and several potentials exist in the literature. The potentials are obtained by fitting experimental data or calculations of the He-He interaction to a function of interatomic separation r in order to obtain values for adjustable parameters. The form usually chosen for the repulsive part is an exp(-r) and for the long range attraction an induced dipole-induced dipole (r ^ ) plus an induced dipole-induced —8 quadrupole (r ). The potentials give a range of depths of the attractive well from 10.3 K to 12.0 K at a separation of r = 2.9 A. Experimental accuracy is not sufficient to determine the well depth e to better than 1.0 K. Empirical information about the helium potential has been provided ( fi ) by measurements of several different physical properties ° including second v i r i a l coefficients in the range 1.5 K to 1500 K, coefficient of viscosity in the range 1.5 K to 2000 K, diffusion and spin diffusion between 1.2 K and 700 K, and thermal conductivity in the range 100 K to 800 K. Much of the data, however, covers the high temperature regime, with relatively sparse information in the region below 10 K where experiments would be more sensitive to the attractive well. Scattering experiments^ ^  ^  have been done at energies only as low as 5 times the well depth. A recent calculation of the helium-helium interaction was performed by Shafer and McLaughlin who obtained a well depth of 12.0 K, slightly 4 deeper than the empirically determined values. Although the normal isotope of helium is not observable by NMR, 3 3 i t is possible to detect He. A dilute gas of He atoms has long been recognized as a fundamental and simple system, from the point of view of NMR,in which the longitudinal nuclear spin relaxation is dominated by the anisotropic dipolar interaction,which is exactly known. It is possible to perform an accurate calculation of the relaxation rate from first principles in terms of the interatomic helium potential which determines (9) the cross section for nuclear spin reorientation in binary collisions. The relaxation rate is proportional to the thermal average of the product of scattering cross section and relative velocity of the atoms, — = n<av> , (1.1) T l where n is the number density of helium atoms with velocity v and a is the cross section. Results of relaxation time measurements at varying temperatures can be used as an independent test of the parameters in the potential. The NMR measurements reported in this work have comparable accuracy to the other transport property measurements and have been made at low temperatures including the range 1.7 K to 19.0 K. Since the experiments are performed in conditions in which most of the atoms have very low energies, information should be obtained on the nature and depth of the helium attractive well. In addition, i t should be possible to determine, the upper limit of other contributions to the relaxation, such as the contribution from a spin rotation interaction which dominates T^ in the 129v (10) heavier inert gas Xe. 5 It was with these intentions that a study of nuclear spin relaxation 3 in He gas was begun, but there was also the possibility that relaxation time measurements could provide information about surface effects i f relaxation at the sample chamber walls was the strongest relaxation in the system. Since i t was expected that bulk gas relaxation could be observed in only sufficiently pure systems, i t was apparent from the outset that both the interatomic potential and the surface interactions could be studied,depending on the preparation of experimental conditions. 1.2 General ideas of NMR in Monatomic Gases ( 4 The technique of NMR is so well established in the literature that only a brief operational description of a relaxation time experiment will be presented. A nucleus with a net spin angular momentum I has a magnetic moment u proportional to the spin y = tiyl where y is the gyromagnetic ratio of the nucleus. In an applied magnetic field H,the spin precesses, classic-ally, at its Larmor frequency a)„proportional to H, u ) „ = yH. One relates this description to the quantum mechanical picture via the relationship of the Larmor frequency to the energy separation of nuclear Zeeman levels in H,AE = tm)o. Relaxation time measurements are made on a macroscopic sample of nuclear spins which has been brought to a net equilibrium magnetization M^  in H. The system can then be disturbed from equilibrium by irradiation with a weak radio frequency field H^ , perpendicular to H, which oscillates at the Larmor frequency. After removing H^  the spin system will return to equilibrium, dissipating its absorbed rf energy through a 6 spin-lattice coupling to the available degrees of freedom. The approach to equilibrium is given by phenomenological equations first derived by Bloch^ 1 1^ dt T 2 T 2 T x where the first term describes the motion of the spins in the applied field and the remaining terms describe relaxation to equilibrium. The time constants and describe respectively the relaxation parallel to the applied field and the relaxation in a plane perpendicular to H. Termed the longitudinal and transverse relaxation times by Bloch, T^ describes processes in which the spin system exchanges energy with its surroundings and T^ refers to those processes which do not change the total spin energy. It is important to note that the Bloch equations give an accurate description of relaxation in liquids and gases only, where the spin-spin interactions are weak and the duration of collisions during which the spin gives up its energy is very short. A spin on a particular atom experiences a fluctuating magnetic field from the spin on its collision partner which can be analyzed into spectralcomponents; the fourier components at W q and also 2 U) q are effective in causing spin transitions and the relaxation rate depends on the magnitude of these spectral components. A very important case known as the short correlation time limit occurs when the correlation time x of the interaction is much smaller c than OJ \ for then T, = T„ . The physical basis of this effect can be seen o 1 2 J 7 by transforming to a reference frame rotating at the Larmor frequency. In such a coordinate system, the external field is zero and only the fluctuating local field will be seen by a spin. If the surroundings are isotropic the transformation does not change the appearance of the local field since its fluctuations are much more rapid than to . The relaxation o in the x, y and z directions should proceed at the same rate. 3 1.3 NMR in He gas 3 The He atom in the ground state is spherically symmetric and has a nuclear spin I = ^ with a large y, comparable in magnitude to that of 3 protons. A dilute gas of He atoms is a weakly interacting spin system in which the interatomic dipolar coupling is the dominant relaxation (12) mechanism . That i s , the spin system dissipates energy absorbed from the rf field via collisional modulation of the dipole-dipole interaction 3 between pairs of atoms. In the framework of this model, a He atom experiences a changing magnetic field for a period of time of the order of the duration of a collision, but is unaffected between collisions. The local field seen by the nuclear spin is a series of spikes of width T ~ ^/v where d is the distance of closest approach and v is the c -12 thermal velocity. This time (10 sec) is sufficiently short that x << to ^ for fields attainable with laboratory electromagnets, c o In this limit T 2 Q = T2 3 1 1 ( 1 w e c a n e s t i m a t e T2 by considering the effects of binary collisions which occur incoherently at an average (13) frequency u ,, . I n each collision the transverse spin magnetization coll dephases by A<j> = Atox << 1 and after N collisions the mean squared value of the accumulated phase is given by a random walk argument to be 8 2 N<A<j» . An expression for is obtained by setting the accumulated phase angle equal to one radian: V1 = < A < f > 2 > °coll ' <A<{>2> can be estimated from the strength of the dipolar coupling»(y2ti/d3) and the duration of collisions, <A<j>2> = (f^yVd 6 ) (d/v)2. Finally, T~ l = T 1 B _ 1 = n(m/kT)% h 2 Y 4/d 2 - P/T3/2 (1.3) where n is the number density of atoms (mass m) in the gas. This is a crude treatment which ignores the precise nature of the interatomic potential and does not describe how i t affects the duration of a collision. There will also be a contribution from the transient spin rotation interaction associated with effects of electric polarization during a collision. Evaluation of 1.3 at 1 atm pressure shows T^  to be about 10^ sec 4 at room temperature and 10 sec at 4.2 K; the very long times indicate how weak the relaxation mechanism is and make i t apparent that the NMR experiment has hope of success only at low temperature. Previously (15-17) 3 reported measurements of T^ in He gas are a l l much shorter than the expected values,undoubtedly because of the presence of paramagnetic impurities or of wall relaxation which can easily short circuit the pure gas relaxation. Oxygen molecules are expected to make a significant contribution to ^ when their fractional concentration reaches 3 2 6 [y( He)/y(02)] ~ 10 . When making measurements at or ab ove 77 K i t is difficult to avoid concentrations of this magnitude, but at 4.2 K where the present data was recorded, a l l the impurities except 4He will be condensed out on the walls. Wall effects are more difficult to estimate because the probability 9 of relaxation at the wall depends on the nature of the surface and on the 3 bulk gas pressure which affects the total time a He atom resides near the wall. For sufficiently dirty surfaces a, the spin f l i p probability 3 per wall c o l l i s i o n i s large enough that a He atom is relaxed during the time i t spends near the surface and T,TT, the wall relaxation time, is 1W determined by the time necessary to reach the walls. In this limit can be estimated from a random walk argument, T1 = R2/ D « n (1.4) for a c e l l of radius R. is proportional to density via the diffusion coefficient D and is independent of a. In the clean surface regime an average particle diffuses to the wall but a i s so small that the atom makes too few wall collisions to f l i p i t s spin before returning to the bulk gas. The average wall c o l l i s i o n frequency in a sphere of radius R i s 3v/4R, and a spin requires on the average 1/a wall collisions to relax so, T l - ^ . (1.5) 3av Collisions resulting in adsorption w i l l be most effective in relaxing 3 He spins owing to much slower motion in the adsorbed phase and perhaps the presence of surface impurities. The spin f l i p probability a can then be expressed as a = a., T . , / T 1 a , where a., is the adsorption probability Ad Ad lAd Ad and T . , and T,. , are the time spent on the surface and relaxation time of Ad lAd the adsorbed phase respectively. It has been assumed that i ^ << T ^ j . x^j is given by the surface density of adsorbed atoms divided by the flux of particles leaving the adsorbed phase which at equilibrium is equal to the inward flux. The latter is proportional to n. Hence T^ a n in both limits and in particular is proportional to n and ^ in the clean surface regime. Since the wall relaxation and bulk gas relaxation have different density dependences i t is possible to identify each contribution to the observed relaxation rate by studying the dependence of on p , where p = nm. The observed relaxation rate can be written (1.6) The basic task is to prepare experimental conditions in which is suppressed sufficiently that bulk gas relaxation can be observed. Under such conditions, a temperature dependence of T at low temperature Ln will yield information on the He-He potential. There is also in each experiment, information on wall relaxation 3 and the behaviour of the adsorbed two dimensional phase of He atoms. In chapter 2 the detailed theory of relaxation in the bulk gas is discussed f i r s t . This is followed by a discussion of wall relaxation. Experimental details and results are discussed in chapters 3 and 4, respectively. Preliminary measurements of the field dependence of wall relaxation were made and these results are interpreted in chapter 5 in terms of a model of relaxation in the adsorbed phase presented in appendix A. 11 CHAPTER 2 THEORY OF LONGITUDINAL SPIN RELAXATION  IN DILUTE 3HE GAS 3 The nuclear spins in He gas are relaxed by fluctuating magnetic fields associated with the nuclear dipolar coupling between pairs of colliding atoms. The calculation of T^fi using simple c o l l i s i o n a l models described previously is suitable only for an order of magnitude estimate and does not allow a detailed interpretation of T^ in terms of the atomic and kinetic properties of the gas. Because of the simplicity of the dipolar coupling, i t should be possible to perform an exact calculation of T^fi from f i r s t principles using the best available helium potentials to describe the scattering of He atoms. The temperatures at which the nuclear spin relaxation experi-ments were performed re s t r i c t the relative energy of a colliding pair of atoms to very low energies and this permits a computational simplification since i t is necessary to include only the lowest par t i a l waves in treating the scattering problem. A formal kinetic theory for the calculation of T.. in dilute monatomic I D (18) gases was developed by Chen and Snider . The essential quantity in "their theory is the cross section a(E) for spin transitions resulting from the collisions of pairs of atoms interacting via the spherical potential and an anisotropic dipolar coupling which i s responsible for the spin 12 f l i p s . The relaxation rate is then obtained from 00 n(-2h]xsh f -E/kT a(E)EdE (2.1) where E is the relative energy of a colliding pair of reduced mass u and n is the number density of the gas. The kinetic formulation has the advantage that the dynamics of the binary collisions are accurately taken into account. In addition, r e a l i s t i c forms of the He-He potential can be used to evaluate the cross section, so that a comparison of the calculated and measured values of T1 can provide a test of the form of the potential used. 2.1 Calculation of the Cross Section a(E) 3 The interaction between He atoms is taken to be the ground state electronic interaction Vo(r) and the anisotropic dipolar interaction between nuclei, where r i s the relative position vector of atoms with nuclear spins 1^  and I„. In the centre of mass system the Hamiltonian i s V(r) = V 0(r) + V(rtIltI2) v ( f , i l t i 2 ) = (2.2) H = -(n 2/2u)V 2 + V(r) (2.3) 13 The cross section can be obtained by performing a standard partial wave analysis of the scattering. Since we are considering the collision of identical spin ^  particles, the total wave function ty is antisymmetric. It is expanded as a product of space and spin variables, with the spin part expressed in terms of total spin I = 1^  + 1^ which can be one of two pure states, either the symmetric 1=1 state or the antisymmetric 1=0 state. Because the 1=0 state does not contribute to the relaxation, the space part of ty must be antisymmetric and the partial waves necessary in the calculation are restricted to odd values of i. Over the entire range of E, the number of partial waves required to achieve convergence is small, -4 and at very low energy (E < 5x10 eV) , o"(E) is effectively given by the % = 1 partial wave only. Since the dipolar coupling is many orders of magnitude smaller than the spherical potential, it is possible to calculate the partial wave scattering amplitudes within the distorted wave Born ( 9 ) approximation. The details of the calculation are given by Shizgal . The properly symmetrized cross section was computed numerically in the energy range 0 - 0.07 eV using the best available He-He potentials, in particular the Beck^19^, Bruch and McGee (BM)^6 ^ , and McLaughlin and Shafer (MS) ; the latter two potentials are plotted in fig. 2.1. The Beck potential is a f i t of second virial coefficient data in the temperature range 25 K - 1500 K to the functional form VQ(r) = A exp(-ar - Br ) -0.869 (r*+az) [ 1 + 2.709 + 3a2 rl + a'1 ] (2.4) o , / c where a = 0.675A; a = 4.390A- ; g = 3.746 x 10 A ; A = 398.7 eV. 14 Bruch and McGee have fitted transport property and second v i r i a l coefficient data in the range 1.5 K - 2000 K to the functional form V Q(r) = £[exp(2c(l - x)) - 2ex P(c(l - x))] r ^  r 2 X " r / rmin ( 2 ' 5 ) V (r) = -1.47 r ~ 6 - 14.2 r 8 r > r o v 2 o and obtained £ = 1.484 x 10~15erg; r = 3.0238A; r 0 = 3.6828A and ° m 2 c = 6.12777. The MS potential is a f i t of calculations of the He-He interaction to the functional form V Q(r) = 455.23 exp(-ar - br 3) - 0.9213 r " 6 - 2.623 r ~ 8 (2.6) with a = 4.33A-1 and b = 0.01717A-3. -3 In addition a Lennard-Jones (6-12) potential with E = 0.887 x 10 eV and a = 2.56A was tried. Calculated values of a using the MS and BM potentials are shown in -3 -1 Fig. 2.2. At high energies E > 10 eV, a l l potentials give an E - 3 dependence with slight differences in magnitude, whereas below 10 eV there is a greater difference in the numerical results, although a l l -4 potentials indicate a maximum near 10 eV. T^B is readily obtained from Eqn. 2.1. The plot shown in fig. 2.3, the calculated values of T in the range 0 - 20.0 K, suggests that the temperature dependence is sensitive to the form of the spherical potential, and in Fig. 2.1 V (r) vs. r for the MS and BM helium potentials. Fig. 2.2 C a l c u l a t i o n s of cross s e c t i o n a(E) f o r spin t r a n s i t i o n s caused by d i p o l a r coupling vs. energy, using the MS and BM helium p o t e n t i a l s . 17 particular the details of the attractive well. At -h temperatures T > 10 K the value of T 1 T ) T approaches a constant. The most interesting region is T < 2.0 K in which a minimum occurs. Measurements at higher temperatures are not unimportant, however, because the potentials can be tested by fitting calculations of T to the experimental results. I D 2.2 Other relaxation mechanisms There will also be a contribution to the relaxation from the transient spin rotation interaction associated with effects of electric polarization during a collision. In brief, there is a distortion of the charge clouds owing to the van der Waals interaction during a collision; the rotation of the distorted charge distribution creates a fluctuating field at the nucleus which can excite spin transitions. It is possible to calculate a cross section for spin transitions owing to the spin-rotation coupling and estimate the strength of its contribution to T^. The calculation is formulated in the same manner as in the previous case of dipolar coupling. Figure 2.4 shows a plot of 3 dipolar and spin-rotation cross sections for He. It is evident that at low temperature there should be only a negligible contribution from the spin-rotation coupling. A further contribution to the relaxation could arise from magnetic field gradients through which the spins diffuse. Relaxation by this 3 mechanism has been observed in optical pumping studies of He gas at very (21) low density . The relaxation time can be estimated by a random walk argument, in which the spin experiences a changing magnetic field of mean 2 square value <6H > at intervals of time T equal to the average time 3.0 0 ' . : r — , 1 20 30 , . 40 50 Fig. 2.4 Cross sections for spin transitions via the dipolar and spin rotation interactions, k is relative wave number and d is the classical turning point. . ... At kd - 31, where the cross-section magnitudes are equal, the relative energy of the colliding |-pair is 0.2 eV. ^ 20 necessary to diffuse through 6H. Schearer and Walters^ 2 2^ have estimated this relaxation time to be -1 2 2 T, = G <u >x 1 H o 2~ (2.7) where G is the magnetic f i e l d gradient and u the thermal velocity of atoms which make collisions on the average at intervals T . In the f i e l d gradients experienced in this work (G ^  10 Gauss/cm in l.OkG ) the time constant is of order 10"^ seconds. 2.3 The Time Correlation Function Approach By contrast Abragam^4^ has developed a formal theory of relaxation in monatomic gases in which the relaxation rate i s calculated in terms of fourier transforms of the time correlation functions of the dipolar coupling. The theory demands that the dipolar interaction which couples the spins to an external reservoir be weak, so that the probability for spin f l i p s between nuclear Zeeman states can be calculated by perturbation theory. The relaxation rate i s T1B = 2 ^ 1 ( 1 + 1 } ( J l ( c o o ) + J2 ( 2 a )o ) : ) ( 2 - 8 ) where J (io ) = m o e ° gm(t)dt (2.9) a n d / Y 2 (GO Y* (0 (t) , q}(t)) 21 is the ensemble average of the product of matrix elements of the dipolar interaction at times separated by t. It is possible to obtain information about the interparticle potential by calculating the correlation function i n terms of the detailed atomic and kinetic properties of the gas. A possible method for calculating the correlation functions for a dilute monatomic gas i s provided by the theory (23) of Oppenheim and Bloom . These authors restrict consideration to binary collisions in which a given pair interacts v i a a potential V (r) and i s coupled to the reservoir by a general spin-lattice interaction F(r). They consider in detail the classical l i m i t in which for a pair of atoms with i n i t i a l separation r and momentum p, the separation at time t, r ( t ) , i s completely determined by solution of the equations of motion for the pair. Thus p(t) and r(t) are given by P(t) = p V V Q(r(t') dt' 1 f t r(t) - r + i P(t') dt' (2.11) o The ensemble average in Eqn.2.10 i s calculated in terms of a time dependent pair distribution function (TDPDF) g(r,r',t) which gives the probability that a pair of atoms i n i t i a l l y separated by r is separated by r(t) at time t :• g m(t) = N dr dr F(r) F(r') g(r,r',t) (2.12) The relationship of a measured relaxation rate governed by F(r) to the 22 microscopic behaviour of the gas is that it probes the TDPDF through F(r). Oppenheim and Bloom did not provide an exact solution of g(r,r',t) for any particular system. This would have involved numerical calculations of classical trajectories for specified interatomic potentials. Instead they developed a constant acceleration approximation (CAA) to the TDPDF to estimate the role of atomic or molecular scattering on T^  for a general potential. In their treatment of g(r,r',t) the acceleration between r and r(t) is essentially replaced by a constant equal to VQ(r) - VQ(r(t)) y|r - r(t) | With the CAA -5 . .. 5« g(r,r*,t) = g ( r r g(r') 2 P(r,r',t) (2.13) where g(r) is the radial distribution function and P(r,r',t) the free particle TDPDF. For a monatomic gas with dipolar interactions the relaxation rate has been determined by Bloom et a l . ^ 2 ^ 4, 2 „ 1 T i b-1 4nT-h- I(I+1) ( 2 ^ ( N / V ) Q ) ( 2 > 1 4 ) a where a is the atomic diameter and 1^ (2,0) is an integral over the radial distribution function and has been evaluated for a hard sphere and a Lennard-Jones potential. This approximate calculation should be compared with the exact calculation^^' 2* 3^ only at high temperatures where classical approximations are good, but it should not be valid at lower 23 temperatures in the quantum regime. 2.4 Relaxation at the Walls Contributions from wall relaxation were present in varying strengths in a l l measurements taken in this study and i t was not always possible to neglect their effect. Many of the previous studies of 3 (15-17) relaxation in He gas have been limited by wall relaxation, but only phenomenological theories have been used to explain the interactions at the surface. Wall effects are d i f f i c u l t to estimate without precise knowledge of the nature of the surface, in particular, what impurities are present. It is possible, however, to develop a relationship between 3 the wall relaxation time T 1 r T and the relaxation time T 1 4, of He atoms 1W lAd (25) adsorbed on the sample chamber surface . This section establishes the relation between T ^ and T^ ^ and i n Appendix A, a model of relaxation in the adsorbed layer is proposed. 3 We consider a spherical bulb containing a gas of N He atoms. The time rate of change of the magnetization M(r,t) is governed by diffusion in the gas and relaxation by collisions in the bulk gas and at the walls. The equation of motion for <M> , the expectation value of M(f,t) is therefore |- <M(r,t)> = DV2<M(r,t)> - - i - <M(r,t)> (2.15) d t T1B 3 where D is the diffusion coefficient of He gas. We w i l l neglect relaxation in the bulk gas and solve the diffusion equation subject to the boundary condition 24 — <M(r,t)> | = -u <M(r,t)> (2.16) r=R which accounts for a f i n i t e probability of relaxation at the surface via the parameter u. The boundary condition is obtained in the following way. Denoting a the probability of a spin transition in a single wall c o l l i s i o n , and J_^(r,t), J (r,t) respectively the flux of magnetization incident on and emerging from the surface, atoms leaving the surface have a definite probability of. being relaxed, and the incident and emerging fluxes are related by J_(r,t) = (1 - a) J + ( r , t ) (2.17) Since J ± ( r , t ) = v[h <M(Rt)> ? | |- <M(x;t)>] where A is the mean free path, substitution into equation 2.17 yields the boundary condition with u = 3a/2A(2 -a). The inverse of u has the dimensions of length (of order A/a) and i t is interpreted as the minimum distance a spin must travel near the surface before being relaxed. The system of equations 2.15 and 2.16 has been solved with the result that M(r,t) i s a summation over the modes of the diffusion equation. If the i n i t i a l magnetization is spatially isotropic, the solution i s <M(r,t)> = I A V Sin(<A) e~th 1 = D(co^) 2 25 where coV i s a root of the r a d i a l part of equation 2.15 subject to the boundary condition. The quantity observed i n experiment <M(t)> i s obtained by i n t e g r a t i n g <M(r,t)> over the volume of the sample. There are two l i m i t i n g cases of i n t e r e s t yR>>l and yR<<l. The condition yR>>l i s s a t i s f i e d for highly r e l a x i n g surfaces and or l a r g e pressures; i n t h i s l i m i t the so l u t i o n for i o V i s c o V = V T T / R with the r e s u l t 1/ V = DvV/R2 . (2.19) T The c o n d i t i o n yR<<l i s s a t i s f i e d i n the l i m i t of weakly r e l a x i n g walls and/or low pressures. An atom near the surface may c o l l i d e many times with the walls but return to the bulk gas before being relaxed. In t h i s l i m i t only til the f i r s t mode contributes since the amplitude of the v mode i s attenuated 2 2 1 2 by (yR) /v . The s o l u t i o n i s ( c o ) = 3yR which gives T ~ 1 = 3a v/4R . (2.20) lw It i s important to note that i n a bulk gas sample the con d i t i o n X<<R i s usu a l l y s a t i s f i e d so the stronger condition i n d e f i n i n g the l i m i t s i s the nature of the surface. It i s c l e a r i n the case of highly d i s o r i e n t i n g surfaces that i s proportional to density (the spins d i f f u s e to the w a l l and are relax e d ) , however the density dependence i n the opposite l i m i t i s not obvious. 3 At low temperatures i t i s known that one or two layers of He w i l l (27) be adsorbed on the walls and therefore i f yR<<l only c o l l i s i o n s r e s u l t i n g i n adsorption w i l l be e f f e c t i v e for r e l a x a t i o n . In t h i s 26 s i t u a t i o n the spin t r a n s i t i o n p r o b a b i l i t y i s a = a \ x /T Ad Ad' lAd (2.21) where a 'Ad TAd and T lAd are r e s p e c t i v e l y the p r o b a b i l i t y that a p a r t i c l e w i l l be adsorbed on c o l l i s i o n , the average time spent i n the adsorbed phase, and the r e l a x a t i o n time i n the adsorbed phase. I t has been assumed that T , , << T. . ,. T . , i s given by (surface density of adsorbed atoms)/ Ad lAd Ad ° J J (flux of atoms leaving the adsorbed phase). In equilibrium the f l u x leaving the surface equals the incident f l u x and the l a t t e r i s given by a , riv/4. Hence ad = 4/nv a, jO" Ad (2.22) where cr i s the surface area per adsorbed atom, and T RnaT lAd 1W (2.23) 3 showing that i n the weakly r e l a x i n g surface l i m i t as w e l l , i s prop o r t i o n a l to density. 27 CHAPTER 3 EXPERIMENTAL TECHNIQUE 3.1 Measurement of Longitudinal Relaxation Times Although many techniques exist for measuring nuclear magnetic relaxation times, the method of adiabatic fast passage (AFP) is most easily adaptable for making measurements on systems with very long relaxation times. The technique is well established in the literature so only an operational description of an AFP experiment will be given. Initially the spin system is polarized along the effective field in the rotating frame Hgfj ~ i + (H - IO0/Y) k > f a r enough off resonance that H is almost parallel to the applied field H. The field eff v r H (or frequency oo) can be varied through resonance in such a way that the magnetization M follows H isentropically and adiabatically in the sense that the angle between M and stays constant during the sweep. The adiabatic conditions are succinctly described by eqn. 3.1 1 dH 1. Y H1  > \ A £ > fT± O . D and are easily satisfied in a sample of pure H^e gas using sweep rates of 0.1 Gauss/sec and rf fields of about 60 mGauss; It is important to note that the above inequalities apply only to a gas in which the short correlation time limit is valid. At resonance a signal which is proportional to the i n i t i a l magnetization is induced in a coil surrounding the sample and at the end of the sweep M and H are antiparallel to their i n i t i a l orientations, (fig. 3.1) o UJ o Ul O < s H 0 + A H H 0 - A H TIME 2 O r -< N I -LU e o < 2 TIME o) PAIRWISE METHOD FOR MEASURING T, TIME 2 g < N 1-Ui H O < Ji TIME b ) METHOD FOR VERY LONG T, Fig. 3.1 29 The spin magnetization then returns to its equilibrium value according to the equation M(t) = M(») + (M(o) - MO)) e " t / T l (3.2) T^ is usually measured by making pairs of sweeps through resonance in which the return sweep monitors the recovery to equilibrium after a time t. The relaxation curve can be followed by varying the time t of the return sweep. This method is unsuitable for long relaxation times since one must wait several T^'s between pairs of sweeps for reestablish-ment of equilibrium. The very long relaxation times were measured by sweeping the field once through resonance to a resting field in order to invert the magnetization and at later times sampling the magnetization non-destructively by cycling the field to resonance and then immediately returning to the resting field. The magnetization continues on its original relaxation curve and the entire measurement can be made in a period of about 3T^. As long as the adiabatic conditions are satisfied there is negligible loss of (28) magnetization each time the signal is monitored It is important to ensure that is greater than the local field H and also the field gradient 6H over the sample. Otherwise a l l the spins are not inverted at the same time during the sweep and the signal strength is reduced by the factor H /H or H /<5H respectively. A weak H also creates X Lt X. I nonuniform magnetization in the sample which will persist until smoothed by diffusion. (Appendix B) At the sample chamber surface local fields H which are responsible for wall relaxation may not be small compared to and i t may be necessary 30 to consider their effect on the adiabatic passage. On resonance the magnetization is polarized along the effective field in the rotating 2 2 ^ frame and has energy levels proportional to (H.. + H ) 2. Transitions JL o between these levels can easily be excited by the local surface fields (but not the local fields in the bulk), and the relaxation rate may be very strong. The time of the passage through resonance must, therefore, be short compared to (walls) which is the wall relaxation time in the rotating frame. Measurement of (walls) could be made by stopping the sweep exactly on resonance and observing the decay. The results would be extremely useful since samples the low field spectrum of the wall relaxation rate. It was very difficult with the existing method of sweeping to stop exactly on resonance, so was not investigated in detail. A rough estimate was available, however, by comparing the signal strength of the i n i t i a l and return passages in each sweep to resonance. This gave ~ 4 minutes in a field of 1 KGauss at 4.2 K, and high density, p > 4.0 x 10 gm.cm . 3.2 Experimental Apparatus The success of the experiment depended upon the attainment of sufficiently clean conditions to allow observation of relaxation characteristic of the bulk gas. A l l other tasks were relatively minor. A discussion follows of the major components of the apparatus, describing the design of the electronics, cryostat and gas handling system. The final section deals with the preparation of clean surfaces. 3.2.1 General description Design of the detection system was greatly simplified by the 3.5 M H z R F Tuned Oscillator Bridge Amplifier Liquid He R F Coil Varian E l e c t r o m a g n e t B L O C K DIAGRAM O F R F S P E C T R O M E T E R Detector C h a r t Recorder 32 3 large signals expected, owing to the large y-factor of the He nucleus and the low temperatures and high gas densities characteristic of the (4) experiment. Signal strength vs^g> c a n be estimated from the relation v . - -ZT NAiu^ XH,-, where Q is the quality factor of a coil of N turns and sig 2 ° u n J 3 3 area A, and X is the spin susceptibility. For a 1 cm sample of He gas at 4.2 K and 1 atm. pressure, v . is several mV. sig A simple rf bridge spectrometer operating at 3.5 MHz was used to moni-3 tor the AFP signal from a sample of He gas contained in a pyrex bulb. The signal was amplified by a tuned rf amplifier, detected and fed directly to a chart recorder (Fig. 3.2). Al l measurements were made in a 12" Varian magnet with a 2.25" pole gap. The magnet had the attractive feature of reasonably good field homogeneity. At the optimum place between the pole faces the field was 6 3 constant to 7 parts in 10 per cm . For the purpose of high field relaxation experiments the magnetic field was calibrated to an accuracy of 3% using a standard Hall effect probe. 4 A variable temperature cryostat was constructed in which He exchange gas surrounding the sample stabilized temperature in the range 1.2 K to 20 K. The temperature was monitored by a carbon resistor. Glass dewars were built in the physics department glass blowing shop. 3.3 Radiofrequency Bridge Spectrometer The choice of NMR spectrometer was dictated by the nature of the signal detection: an AFP spectrometer must be sensitive to the in-phase component of the rf signal. And, in addition, long term stability is desirable since i t takes on the average several hours to complete a T^ measurement. 33 Input © c 2 R AAAr Output T W I N - T OR ANDERSON BRIDGE At 4 .2 K: c, = 2 7 pf C = C' = 4 .7 pf CQ = 194 pf c 2 = 10 pf R = 1.25 Fig. 3.3 Twin - T bridge circuit, with parameters for operation at 3.5 MHz and 4.2 K. 34 An rf bridge is extremely simple and can easily be crystal controlled for better stability. Moreover i t can be made sensitive to either component of the signal; in practice the in-phase component was slightly imbalanced to provide a carrier and the quadrature component nulled^ Its main disadvantages are single frequency operation and high sensitivity to microphonics, however the former is not relevant in a crystal controlled circuit. (29) The twin-T or Anderson bridge (Fig. 3.3) which offered the convenience of independent balance of the in-phase and quadrature components (by adjustment of C Q or respectively) was used in the spectrometer. The balance conditions are u)2L(C + C + C ( 1 + C/C^)) = 1 R R a)2CC ( 1 + C./CJ p / l where R is the effective resistance of the tuned circuit. P The sensitivity of the Anderson bridge has been analysed in detail by Gheorghin and V a l e r i u ^ ^ . They compared the S/N ratio of the bridge to that of a constant current source spectrometer, with the result S/N (bridge) c S/N (constant current) = (i+£,(i + c y c ^ ) 2 The strategy in the design was to make this factor nearly unity i.e. to set < by increasing and R. Great care was taken in construction of the bridge to suppress sources of microphonics. The most sensitive part was the coaxial cable (coax) from 35 the cryostat head to the sample coil and several designs had to be abandoned because of noisy performance. The most serious problem was boiling of liquid He refrigerant which had leaked into the line. A highly rigid, low capacitance coax was ultimately constructed of concentric thin wall 316-stainless steel tubes, 1/4" and 1/16" O.D. of length 32.5", with star shaped teflon spacers at, 2" intervals along the length. The line was sealed at the top by a Kovar feed-through and connected to the bridge by a short length of standard RG-58 cable. At the bottom, the coax opened into the exchange gas chambers; liquid He was therefore prevented from entering the tube. Such a long line is an undesirable feature because its resistance lowers the circuit Q, but nevertheless one which could not be avoided since its capacitance forms part of the tuning capacitance. It could be eliminated, of course, by immersing the entire bridge in the liquid He bath near the coil, but this was impractical because of the small space available in the dewar. A reasonable compromise was reached by soldering most of the tuning capacitance directly across the coil, leaving only a small variable capacitor at room temperature to allow easy adjustment of balance. (31) Alderman has considered this configuration and has shown that the effective resistance of the coax is R e f f = C I > C L + V 2 * where is the part at low temperature and the remainder at higher temperature, (Figure 3.4) shows three possible configurations with 3.4(b) the actual arrangement. 36 a) L C L 3 0 0 K T c = 3 0 0 K Q = r L ( 3 0 0 K) b) 3 0 0 K Q 3 0 0 K r L ( 4 K ) + r x 4 K < T c < 3 0 0 K d) a— Li =C ) T„ = 4 K Q = O JL r L (4 K) + r x (C a /(C a +C b ) )2 - r ( 4 - K ) Fig. 3 . 4 2.4 K 3.5 MHz Id—tf-—I .OOl/i 0.01 fJL 2.2K 3.3 K: 3.3 . 2N5087 . o o i ^ : IN 617 :8.2K |24K 24 K 1.2 K: •2.2K i L-WAr 0.1 fJL ZZ O.OI/i 820 O.l/i v/.l /1~T" 2N5087 50 pf .01 p-1-3.3K* H2K 3.5 MHz CRYSTAL OSCILLATOR Fig. 3.5 —i 1 5 0 / t h :47K 2N5486 i — Input 47 K 2 56 :33K D02d Fig. 3.6 CASCODE AMPLIFIER • + 18 V Output AND DETECTOR OJ co 39 Rf was supplied to the bridge by the amplified output of a 3.5 MHz crystal oscillator; the circuit is shown in (Fig. 3.5). The output of the bridge was amplified by a tuned single stage cascode amplifier (Fig. 3.6) capable of a voltage gain of 400. The cascode configuration was chosen because of its low noise properties and the design featured low noise FET's. During any particular experiment the gain was set at about 250 and never altered. After detection, the voltage was D.C. coupled to a strip chart recorder. The spectrometer as designed was useful over a wide range of gas densities. The major component of noise at the chart recorder was a long term drift. It rarely exceeded 100 mV/hr and was tolerable since the baseline could be interpolated reliably in the region of a resonance. The major source of drift arose from the components of the bridge outside the cryostat which were susceptible to ambient temperature variations. This effect was greatly reduced by shielding the components with a secure l i d on the spectrometer. Superimposed on the drift was a microphonic chatter with occasional bursts of narrow, well defined spikes of amplitude up to a few millivolts. The source of this noise was not clear since the cryostat was extremely insensitive to microphonics at room temperature, but at low temperature the boiling of liquid helium could have caused sudden shaking of the assembly. The spikes were a serious problem only when they occurred near a signal. In that situation, the resonance was ignored and subsequently retaken. High frequency noise was filtered by the chart recorder itself which had a response time of 0.3 sec for f u l l scale deflection. The thermal noise voltage can be estimated from Nyquist's formula <v > = 4KTRAv where T is the temperature of the tuned circuit which has effective resistance R and Av is the bandwidth of the system. For this detection system Av is the bandwidth of the chart recorder and R can be estimated from the bridge balance conditions. Using Av = 1 Hz, 4 R = 5 x 10 and estimating T - 100 K, the rms noise voltage at the —8 output of the bridge is 1.7 x 10 volts. This voltage is amplified by the tuned amplifier with gain G and then fed to a peak detector whose 2 h output is G<v > . Since only the in phase component of noise is detected, a factor of 1//2 appears in the final expression for rms noise at the 2 h , r - 2 h chart recorder, <v > = 1//2 G<v > - 3 yV , for a gain of 250. Owing to the strength of the microphonics and the drift i t was not likely that the thermal noise would be observed. The spectrometer was capable of detecting resonances over a large -4 -3 range of temperature and density, and a lower limit of p = 7 x 10 gmcm at 1.4 K was reached at which the signal strength was 19 mV and S/N - 4. Values of S/N less than 4 are not practical for making T^ measurements by the techniques described here. A lower limit of the observable density can be estimated by taking the ratio of the signal voltages at the observed limit and at the theoretical limit of purely Johnson noise (with S/N = 4). This gives -7 -3 Pt^ - 7 x 10 gmcm for T = 1.4 K, a density which is several orders of magnitude below the density of saturated vapour at that temperature. 3.4 Coil and Resonant Circuit Since noise was not limited by thermal noise in the tuned circuit its design was not a critic a l problem. The coil consisted of 40 39 K 2.0K (10 turn) +• |8V ° - 18V C 4f 100 •—vw--wv a)A ^ 3 MCI74I 470 K binr -w\—1 10 K :470K C: 4.7/t -0 .047 /1 S 3 Double pole-Double throw a) « Sweep b) = Hold IK •AAAr I K Output 400 K 100 K FIELD SWEEP UNIT Fig. 3.7 To vacuum Pumps Support tube ._, Brass flange Outer vacuum chamber 0.5mm glass capillary Carbon resistor Heater coil Rubber "0" Ring Innerchamber pumping line Feedthrough Seal Outerchamber pumping line Cryostat head Copper baffle X Indium 0" Ring Coax Woods' Metal seal Inner vacuum chamber RF coil with 110 pf tuning capacitor Lead *0' Ring Fig. 3.8 Variable Temperature Cryostat turns of 0.018" copper wire epoxied to the end of a %" copper sleeve provided at the bottom of the cryostat and the inner vacuum can provided an rf shield (Fig. 3.S). Connection to the coax was made by a short length of wire. The resonant circuit Q was measured to be about 80 at 4.2 K. Although a large fraction of the tuning capacitance C q was soldered directly across the coil, the coax and variable air capacitor at room temperature accounted for about 40% of the total. 3.5 Field Sweep Unit The static magnetic field was varied by applying a changing voltage to the "NMR Sweep" input of the Varian magnet. A sweep unit capable of generating the desired trapezoidal voltage was built using a motorola MC1741 operational amplifier as an integrator (Fig. 3.7) . Sweep speeds of a wide range were available by adjusting the continuously variable resistor R or the discreetly variable capacitance C . The unit was always operated manually during an experiment and the output voltage monitored on the second pen of the chart recorder. 3.6 Cryostat A variable temperature cryostat was constructed to operate in the range 1.2 K to 25 K and is shown in Figure 3.8 . To achieve temperatures greater than 4.2 K a two chamber system was necessary with exchange gas providing thermal contact with the sample. The inner copper can was suspended by a triangular array of V thin wall stainless steel tubes (one of which was the coax) and a similar 3/8" O.D. tube. A 5/8" O.D. pumping tube surrounded the central 3/8" tube 44 and suspended a brass flange; a l l outer tubes were fixed at the flange and at the 3 baffles to form a very rigid unit. The outer brass can was sealed at the flange by an indium "0" ring and at the bottom of the chamber a brass plug made contact with the inner can at a lead "0" ring joint. The level of liquid helium was always maintained above the flange. Thermal contact between the inner chamber and the bath was achieved via the brass plug. It was designed to provide a slow heat leak so that the higher temperatures could be attained with a reasonable heat input and thus a minimum increase in boiling rate of the liquid helium bath. The inner chamber was machined from copper to achieve a small temperature gradient along its length. It was sealed at the top by Wood's metal; both this seal and the indium "0" ring were cycled to room temperature many times without destruction. Non-magnetic feed-throughs were required for electrical connections into the vacuum chambers. Two standard Oxford Instruments feed throughs were used at the flange and a lead-glass platinum wire feed through was used at the inner chamber. To keep heat losses low #40 copper wire was used to make connections between the heating element and thermometer near the sample and a 4 terminal plug at the cryostat head. Surrounding the inner chamber was a heater coil, 24" of cotton coated nichfome wire expoxied to the surface of the can. Temperatures below 4.2 K were achieved by pumping on the helium bath while exchange gas in both chambers at a pressure of 0.5 mm provided thermal contact with the sample. Temperatures higher than 4.2 K were achieved by pumping out the brass chamber, admitting 1 mm exchange gas to the inner chamber and applying current to the heater. A steady state was subsequently reached between the heat input and the leak to the bath via the brass plug. If experimental conditions were altered, the system was left for several hours during which time the temperature was monitored and measurements begun only after the temperature variation was less than 0.1 K in an hour. Temperature was monitored during a T^ measurement above 3 4.2 K by observing the He gas pressure as well as the carbon resistor circuit; the largest drift was 0.1K over the total time of a measurement 4 Below 4.2 K the sample temperature could be monitored by the He vapour pressure and there was no drift observed greater than 0.05 K during a measurement. It was necessary to keep the system cold for several days in order to collect sufficient data at each temperature to separate the bulk gas and wall components of the observed relaxation rate, so each run involved many liquid helium transfers. The i n i t i a l transfer required about 4 litres to cool the cryostat and f i l l the dewar to its capacity of 2.5 litres. Subsequent transfers to top up the dewar required far less liquid He for cooling. 3.7 Temperature Measurement A 47 ft, 1/8 watt carbon resistor was used to monitor temperature within the copper can. The resistor was fixed by teflon tape to the coil frame at the topmost winding in order to be as close to the sample as possible. The resistor was calibrated in situ in a field of 1 kG against 4 the vapour pressure of liquid He using the 3 parameter f i t given in Hoare R, • I - 2 K - 1.5 M 2-15 K - 150 K 15-40 K - 15 K R 2 : Decade Box R M A X = M M R T Carbon Resistor 6 R 2 1.5 V 3 . 9 THERMOMETRY CIRCUIT 47 Jackson and Ku r t i V J < i / log l f J R + K/log l f J R 2b + a/T where R is resistance in ohms and T the temperature in degrees Kelvin. The parameters K, b and a were calculated using calibration points in the range 1.2 K to 4.2 K and then used to calculate values of resistance for temperatures to 15 K. At higher temperatures this extrapolation becomes too inaccurate and the extrapolation was extended by a smooth curve to a reference point at 77 K. Below 4.2 K the calibration is good to ± 0.1 per cent but the accuracy decreases above that temperature and is estimated to no better than 5% in the range 10 K to 20 K. This uncertainty represents the difference in temperatures read from the calibration curve and temperatures obtained from measurements of the NMR signal strength using the signal at 4.2 K as a reference point and assuming the validity of the Curie law, i.e. signal strength proportional to 1/T. The resistor was cycled to room temperature many times in the course of experiments and no significant change was ever detected. During each run, the calibration was checked at 77 K, 4.2 K and 1.2 K while precooling and admitting the sample. The thermometer circuit is shown in (Fig. 3.9); i t is fairly simple and needs l i t t l e elaboration. The switch controls the current to the resistor and could be changed to allow lower current at lower temperatures. To V a c u u m P u m p G l a s s - m e t a l Seal to .125" St. Steel a Neon Cell G l a s s - m e t a l 1 * - Seal — i to .25"copper 2 mm Glass Capil lary V V T - J - I - T - , I Leak Reducing Valve Pressure Gauge 7 Researchgrade 4 H e 77 K Charcoal T r a p M i s c h - M e t a l 0 2 getter r*7 Thermocouple Pressure Gauge <2H . 0 6 2 " St. Steel 0.5 mm Capillary He Storage Cell Sample Chamber Fig. 3.10 G A S H A N D L I N G S Y S T E M oo 4 9 3 . 8 Gas Handling System 3 Samples of He gas were obtained from the Monsanto Corp. with a quoted isotopic purity of 9 9 . 8 mole % and less than 0 . 1 mole % of other gases such as nitrogen and oxygen. A glass and metal gas handling system (Fig. 3 . 1 0 ) was constructed for storing the gas at room temperature and conveying i t to the sample chamber during experiments. The metal valves and tubing were necessary 3 to withstand the required pressures of He gas and the design featured miniature stainless steel Hoke valves joined to the tubes by gyrolok f i t t i n g s . The glassware was protected from the high pressures by a leak valve and the two parts were joined at a standard Kovar glass-metal seal. Purification of the gas was performed in the glassware. Before an experiment the entire gas handling system was flushed 4 - 3 many times with research grade He and then pumped to < 1 0 torr. The 3 He gas shipped from Monsanto was not sufficiently pure to use directly so i t was cleaned by storing i t i n a 4 - l i t r e pyrex bulb lined with a permanent getter. The getter was made in the glass blowing shop by ( 3 3 ) slowly evaporating a thin surface of misch-metal on the inside of the bulb i n an argon atmosphere. The misch-metal surface i s highly reactive with oxygen and was an effective trap. Gas from the storage tank was admitted to additional purification stages in the gas line as shown in Fig. 3 . 1 1 (A charcoal adsorption trap cooled to 7 7 K was used in conjunction with the getter to take out nitrogen and residual heavier contaminants.) The clean gas was then passed through the leak valve at a slow rate into the metal tubes and collected i n the sample chamber, which had been cooled previously to 1 . 2 K. Enough liquid was collected 50 to f i l l the chamber and then the leak valve was closed. The bath was allowed to warm up to 4.2 K, a temperature at which the relation between ^ He gas density and pressure was known Measurements of pressure were made with a calibrated Marsh gauge which 3 was accurate to 0.3 psig. Pressure was varied by bleeding off He gas through the leak valve and returning i t to the storage tank; in this way no new contamination was introduced during a run. The gas was therefore recycled from one experiment to the next and with very small losses since the dead volume in the capillary tubes was far less than the 4 litre storage tank capacity. Keeping in mind the high cost 3 of He ($130/litre at NTP), the recovery was a necessary practice. The sample chamber was an approximately spherical bulb of inner diameter .5cm blown from pyrex glass and fixed to a 35" length of 0.5 mm pyrex capillary which had been cleaned with solutions of nitric acid, acetone and distilled water. At the top of the cryostat the glass was joined to the 1/16" stainless steel line by a Kovar glass metal seal and gyrolok reducer. The length of pyrex capillary served as the final cleaning trap since the gas was admitted slowly enough that any residual impurities condensed on the walls of the tube. Pyrex was chosen for the sample chamber since i t appears from several 3 experiments to be a weakly relaxing surface for He spins. Timsit et a l ^ ^ have conducted an extensive study of wall relaxation for H^e on several kinds of glasses and concluded that only a Corning #1720 aluminosilicate glass provided a better surface than pyrex. However, their experiments were done at higher temperatures where diffusion into the glass was important in relaxing the spins. At temperatures below 20 K the 3 probability of He diffusing into the glass is very low (activation energy 51 for diffusion is ^  6400 cal/mole whereas adsorption energy is 230 cal/mole) and the relaxation rate becomes negligible. 3.9 Methods of Suppressing the Wall Relaxation By far the most difficult experimental problem was preparing a sufficiently clean system to observe bulk gas relaxation. There was no way of measuring beforehand how effective a particular cleaning operation would be; the only suitable test for our purposes was simply the value of T1-The gas sample itself was purified as much as possible by the methods already mentioned, so i t remains to discuss the cleaning of the chamber. Standard methods of purifying surfaces such as baking under a vacuum were not good enough to remove a l l adsorbed particles. Some success in preparing a pyrex surface clean enough to observe bulk liquid T^ was (34) achieved by Horvitz who used a complicated and lengthy procedure of baking, pumping and rf discharging to drive impurities off the walls. After several days of cleaning T^ would be measured at 4.2 K, and then the entire process repeated until there was no change observed in successive measurements. This method was attempted with moderate success and T^ was observed to saturate in each series of measurements, but i t was eventually abandoned because the process itself was extremely tedious, and reproducible cleanness could not be obtained in different series of experiments. Instead of trying to drive impurities off the walls, a totally different and far more simple approach was developed. Clean surfaces, which could be reproduced from run to run were achieved by coating the 52 sample chamber walls with enough clean neon gas to build up several monolayers and cover any impurities. In effect, a neon container was 3 created, whose surface was extremely weak in relaxing He spins. The neon was admitted to a pressure of 2 mm at room temperature just before beginning to precool the apparatus. (In this way the gas served also as a heat exchange in the capillary and sample chamber). As the system cooled during the liquid helium transfer, the neon condensed on the walls, forming a layer over the impurities and higher boiling point gases which were not removed by the traps. Further improvement was made by annealing the neon coating, i.e. heating the surface at about 18 K (35) for several hours . T^ values taken after the.annealing process were consistently 15% longer than those on unannealed surfaces at the same conditions of temperature and density. Wall coatings have been used previously in optical pumping experiments to create weakly disorientation (26) surfaces. Bouchiat coated cells with paraffin in her studies of alkalis atoms. Neon was chosen as the coating substance since i t is spinless, has a low dielectric constant and a low boiling point.other gases such as Argon 4 also be suitable. Below 4.2 K i t may be possible to coat with He sxnce i t is preferentially.adsorbed owing to its higher isotopic mass . Barbe et a l ^ ^ have successfully used a coating of J{ to reduce ^ He wall interactions in low density optical pumping experiments. These preparations were successful in allowing observation of pure gas -2 -3 relaxation at high densities (p < 5.0 x 10 gmcm ) but eventually a -2 -3 lower limit was reached around 1.0 x 10 gmcm at which wall relaxation again became a problem. It was s t i l l possible, however, to observe the 53 pure gas relaxation beyond this limit by making the measurements at higher fields where the wall relaxation was much weaker. This effect was observed empirically and can be made plausible by the following argument which applies to the regime of weak wall relaxation discussed previously. 3 If motion of He atoms at the walls is slow (compared to the motion in the bulk), the spectral density of the fluctuating surface fields will have a cut off frequency 1/Tc which may be comparable to the frequencies a) o o used in the experiment. For such a condition, in T„ - 1, wall relaxation O O is very effective, but at higher frequencies the intensity of the spectrum diminishes rapidly and relaxation is weaker. The spectrum of fluctuating fields in the bulk gas cuts off at a much larger frequency than COq as discussed in the introduction so the bulk gas relaxation is effectively independent of frequency over the range accessible in this work. In this way relaxation curves of the very long relaxation times characteristic of the low density gas could be measured, although in practice only the i n i t i a l slope of the curve was taken, in order to conserve time; otherwise the system could not be kept cold long enough to collect a sufficient amount of data to separate T1 and T_rT. IB 1W Since the spectrometer operated at one frequency only, the measurement of T^  at different fields was accomplished in the following way. The spin system was brought to equilibrium at the measuring field (1.0 kG ) and then the field was increased to a higher field (a 9.3 kG field was chosen as a safe upper limit) where the spins began to relax to a new equilibrium for an arbitary period of time, following the law M(t) = MJJ + ( ^ - i y e " t / T l , where MJJ and are the equilibrium magnetizations at high and low fields, respectively. 54 Then the magnet was returned to the measuring field, the signal monitored and finally cycled back to the relaxing field. The entire sequence was repeated with different resting times at 9.3 kG . Since the measurement is made in a time much less than T^, there is only a small correction to be made for the time spent at 1.0 kG . In this way, the i n i t i a l slope of the relaxation curve could be measured and the range -2 -3 of density was extended to about 0.2 x 10 gmcm (the lowest measured) to make possible T measurements in the interesting range of temperature JLB below 4.2 K . 55 CHAPTER 4 PRESENTATION OF RESULTS AM) DISCUSSION 4,1 Introduction In this section a systematic study of T^ in the gas at temperatures in the range 1.7 K to 19.0 K and at varying density is reported. Each observation of T^ contains information on the bulk gas relaxation and on wall relaxation and the two components can be separated on the basis of their density dependence. Denoting the observed 'rate by T^~\ then equation 1.6 gives T l 1 = C l p + C 2 / p The constants C^ and C^ describing bulk gas and wall relaxation, respectively, -1 2 are obtained by finding the slope and y-intercept of a plot of pT^ vs p . The phenomenological expression above i s valid only under certain conditions and can clearly lead to systematic errors i f other mechanisms dependent on density are present. Validity i s ensured for the bulk gas relaxation i n the dilute gas limit of binary c o l l i s i o n s , but there may be limitations from wall relaxation. For instance, at extremely low densities the surface area may be fractionally covered, with the coverage increasing with density. There is empirical j u s t i f i c a t i o n for using eqn. 1.6 in the density range observed and for the kinds of surfaces prepared in this work, since 56 plots of pT^ 1 vs p Z always yielded straight lines (Fig.4.2). Possible errors arising from three body effects in the bulk gas are discussed in a later section. Relaxation times were typically one to three hours depending on experimental conditions, and the values of bulk gas and wall relaxation times associated with the observations were of order 10 seconds to 10 seconds . Comparing these values with the estimate for relaxation in the magnetic field gradients of the magnet, i t is unlikely that bulk gas relaxation was influenced by the field gradients. Measurements were made in sample chambers prepared both by the method of Horvitz and by surface coating, and in conditions of purity lying in both limits of wall relaxation described in section 3.9. Results for the temperature dependence of T1 and the field dependence of T.. were taken Lo Lvl only on neon coated surfaces. This section first discusses the general sources of error in the techniques before presenting the results. 4.2 Possible Sources of Error Signals were recorded directly on chart paper during recovery to equilibrium. To determine T^ i t was necessary to read from the graph only the signal amplitudes and times. In most cases i t was not difficult to estimate a baseline over the slow drift, and the error in reading each amplitude was estimated from the noise in the trace. A possible source of error was the balance condition of the bridge. Although the system was quite stable, there was a slow drift off balance 57 which could affect results at long times from the i n i t i a l sweep. The balance condition was always monitored during a measurement and when necessary, the out-of-phase component was nulled while the spin system was resting far away from resonance. As previously mentioned, the adiabatic conditions, equation 3.1 were easily satisfied. An rf f i e l d strength of = 60 mG was sufficient to invert the magnetization, but create no serious heating at the sample. The signal could, i n principle, be observed with a wide range of sweep rates which satisfied the adiabatic conditions, but the actual sweep rate was set so that the time taken to sweep through the resonance was roughly ten times T p , the. response time of the chart pen ( x p ^ 0.3 s e c ) . The spins were brought into equilibrium at a f i e l d H slightly above resonance and then swept to a resting f i e l d approximately the same distance below resonance. Since H-H Q - 25 Gauss, inversion of the magnetization was complete. There was no saturation of the lin e at this distance from 2 resonance since AM /M - (Hn/AH) . A small correction was necessary, z o 1 however, to account for the fact that the spins were relaxing in a resting f i e l d different from the i n i t i a l polarizing f i e l d . Measurements at temperatures greater than 4.2 K were made by the nondestructive sampling technique at low f i e l d s . Although the purity of the system was very high, there was a small component of wall relaxation which could not be ignored, particularly at the high temperatures where the wall and bulk components were nearly equal in strength. From equation 4.1 i t i s obvious that the criterion for judging whether bulk gas relaxation was observed during experiments was that T^ ^ scaled with density. (A more -1 2 rigorous check was also made by plotting pT1 vs p .) 1.0 i r 1 T 8 0.3-tn i o tn 1> W 0.1 20 40 60 80 time ( m i n ) 100 120 140 Fig. 4.1 Relaxation curve for typical low field T1 measurement _2 T = 4.2 K p = 1.7 x 10 gm/cc. 59 It was necessary therefore to take data at three different densities, at least, for each temperature. In each determination a minimum of seven sweeps to resonance were recorded spanning a time of about 3T^. It was important to wait at least 5T^  before beginning each measurement in order to get an accurate value of the equilibrium magnetization. This value should perhaps be checked during each measurement by waiting many T^'s and observing the recovery at long times (M^), but because relaxation times were usually several hours, the test was not always feasible. The recovery at long times was, however, checked in a particularly short high density Tj measurement. Full recovery, within experimental error, indicated that the method was reliable and that there was essentially no loss of magnet-ization on each sweep to resonance, when the adiabatic conditions were satisfied. It should be noted also that the technique was compared to the usual method of measuring T^  by taking the signals in pairs, with a long wait for equilibrium between each pair. Agreement between the two methods was satisfactory. A rough check of exponential recovery was made each time by plotting y = (S(t) - S(»))/(S(o) - S ( « 0 ) vs. t on semi log paper, (Fig. 4.1). Non exponential recovery was observed only when the strength of the rf field was weak as described in appendix B . Values of T^, however, were not obtained from the graph but by fitting the data to the equation logy = mt + b by a UBC computer library least squares f i t routine (LQF). Each data point was weighted by 1/a2 where a is the error determined from the experimental trace. If the estimated error was reasonable, then o~ was in close agreement with the curve fitting error given by the program. In practice, data was taken at chosen settings of heater current for a given density and then the entire sequence (lasting several days) was Fig. 4.2 p/T^ vs. p . Measurements taken at low field, o 8.5 K, A 6.5 K. i £ 0.3 + I I / ) 0.1 30 60 90 time ( m i n ) 120 <-Trr.-fr.al hi eh field T, measurement. Fig. 4.3 Relaxation curve for typical nign n e x u ± ± T = 2.1 K p = 0.4 x 10~ gm/cc. 62 repeated at a new density. Since the heater setting was a coarse adjust-ment, i t was not possible to reproduce the temperatures exactly, so an interpolation was necessary in some cases. The data for each density was fitted to the empirical expression T^  = A + BT + CT2 and then the results at a number of temperatures could be analyzed according to Equation 1.6 in order to obtain and C^. Plots of pT^ * vs p2'are shown in Fig. 4.2. Values of and were not obtained from the graphs but by the least square f i t routine LQF. At temperatures lower than 4.2K the density range for the gas is limited to very low values and T^  is so long that pure gas relaxation could not be observed unless measurements were made at high fields. With this technique, only the initial slope of the relaxation curve was measured. Fig. 4.3 shows a plot of (S(t) - S U)/(S T - Su) vs. t where S„ and S. are H L ri rl L the high field and low field equilibrium signals respectively, and S(t) is the signal amplitude at the end of the time spent at high field. The time spent in the low field during the measurement of the AFP signal was always much shorter than T^  in that field so that there was no serious error introduced. It was again essential to establish equilibrium before beginning the measurement since an estimate of the high field signal had to be made from the i n i t i a l signal at low field. The signal after long times at high field was checked for fu l l recovery in a suitable case when T^  was short. It was possible to compare the two methods only at a temperature and density at which bulk gas T^  was observed at low field. This was done -2 -2 during one of the runs, at 4.2 K and 1.0 x 10 gm cm , with good agreement between the two methods. 63 In one particularly good run the relaxation rate was observed to scale with density down to 2.4 K, but at lower temperatures, and in all other experiments, there was some evidence of wall relaxation and Equation 1.6 was used to analyze the results. Data was collected during several runs in an attempt to reach as low a temperature as possible. Measurements were made as low as 1.4 K but the results are trusted only to 1.7 K owing to difficulties in the analysis caused by wall relaxation and weak signals. The first measurement in a run was always made at 4.2 K so that the density could be measured on the Marsh gauge. At subsequent lower densities at lower temperatures where readings on the gauge were off scale, the NMR signal strength was used to measure the density. (It is 3 (36 ") known that He vapour satisfies the Curie law to temperatures of 2.0 K) ' 4.3 Temperature Dependence of T^g Experimentally determined values are plotted against temperature in figure 4.4 for comparison with calculations of T 1 D using the poten-1 D tials chosen in this work. The sum of squared deviation between theory and experimental data was calculated for each potential and is presented in Table 4.1. Agreement with the calculated values is remarkably good. The BM potential, with a well depth of 10.8 K, fits the data closely in the range 3 K to 19 K, but the calculated values are slightly higher at temperatures below this range. The Beck potential, with a more shallow well, deviates at each end of the. measured range, underestimating at high temperature and overestimating at low temperature. The MS potential which has the deepest well, gives the best f i t at low temperature but underestimates at high 64 1 5 0 7. e 1 0 0 9 I— ^ 4 5 0 + (a) 1 — I " • 1 1 1 1 1 l 1 1 > -> ---• i i i i i | I . 1 , 1 < — , 1 • • 1 • 1 • 1 • 1 5 1 0 T(K) r — — 1 — i — 1 1 1 1 1 1 • 1 15 "1 I i • i 2 0 - T I 1 5 0 4-E U " 1 0 0 + V ) 5 0 A' (O T(K) 10 - I 1 r ~ 15 20 Fig. 4.4 Temperature dependence of T^. Broken curve is a calculation of T 1 ] } using (a) (c) BM potential. 1 B Beck potential (b) MS potential and 65 temperature. The least value of the sum of squares was obtained with the BM potential. It i s clear that relaxation via the dipolar coupling alone i s sufficient to explain the relaxation. An upper limit can be placed on the relaxation rate for relaxation via the spin rotation interaction by considering the agreement with experiment at high temperature where' the spin rotation cross section i s largest. The empirically determined relaxation rate for bulk gas relaxation can be expressed as a sum of dipolar (T )and spin rotation ( T 1 S R ) components T 1 B _ 1 = T 1 D " 1 + T 1 S R _ 1 < 4- 2> -2 -3 -1 -5 -1 At 19.0 K and 10 gmcm T,_ = 6.17 ± 0.2 x 10 sec , so the upper IB limit of the spin rotation relaxation rate is roughly the experimental uncertainty 0.2 x 10 sec \ There are no simple prescriptions for direct inversion of transport property data to obtain parameters of the interatomic potential. Having chosen a particular form which satisfies the data as well as the potentials used, i t is possible to test the sensitivity of T._ to the potential by IB adjusting the parameters by t r i a l and error and comparing with experiment. It may be argued that adjusting the potentials i s unnecessary since the f i t s with a l l three potentials are remarkably good. But i t i s useful to see how sensitive T, is to the potential and i f possible determine IB which potential and what value of the well depth.gives the best f i t . The results of the f i t s with adjusted BM and MS potentials are shown in table 4.1. Variations in the potentials were achieved by first.determining the I 1 1 1 I 1 1 1 I 2 4 1 6 8 10 I 12 I 14 I 16 18 2 3 T ( K ) Fig. 4.5 Temperature dependence of T._.. ' Broken curve is a calculation IB of T^B using best parameters of BM potential. sensitivity of the well depth to each parameter and then varying the most sensitive parameter to obtain a desired depth. The depth could be changed by varying parameters in either the short range or the long range part; only the short range part of the BM potential was varied, whereas both parts of the MS potential were varied in turn. No errors are quoted for the parameters in the BM potential; the authors quote only the accuracy to which the helium transport property data i s f i t t e d by the potential below 50K and i t i s not known how sensitive the data i s to the parameters. This potential was tested with slightly deeper wells in order to improve the f i t to the NMR T 1 data below 3K. The best f i t (Fig. 4.5) was obtained with a well depth of 11.5K which was achieved by a 7% variation of the parameter e (eqn. 2.5). The NMR data i s reproduced to within 3%, but i t i s not known how well the new adjusted BM potential f i t s the other data originally used by Bruch and McGee. Results are given in Table 4.1 for calucations of T^g using the MS potential and variations obtained by adjusting the attractive part to achieve well depths between 9.3 K and 14.5 K. The best f i t i s obtained for well depths slightly more shallow than the quoted value, but the high temperature results are consistently underestimated. An attempt was made to f i t the data by making the potential less repulsive, but the high temperature results were again far too low. A two parameter Lennard-Jones model i s capable of f i t t i n g the results over a restricted range, and can perhaps be used as a rough guess of the potential. The accepted Lennard-Jones helium parameters (section 2.1) provide a good f i t of the low temperature data but seriously underestimate T 1 1 5 above 4.2 K. A better f i t to the high temperature data can be obtained only with values of well depth e or range a which are greatly out of line with the accepted values. 68 Table 4.1 Potential Adjusted Parameters Well depth Sum of Squared Deviations 2 Beck none 10.3 K 42.0 (sec.gm/cc) MS none 12.0 K ' 94.0 BM none 10.8 K 25.0 MS D = 1.040 14.5 K 128.0 MS D = 0.899 11.5 K 64. MS D = 0.861 10.9 K 39. MS D = 0.823 10.2 K 45. MS D = 0.766 9.3 K 198 BM £ = 1.563x10 erg 11.2 K 11.7 BM £ = 1.063xl0~15erg 11.5 K 10.6 BM £ = 1.643xl0~15erg 11.8 K 11.5 6 9 The NMR data favours the BM p o t e n t i a l with a p o t e n t i a l of that form having a s l i g h t l y deeper a t t r a c t i v e w e l l providing the best r e s u l t s . Perhaps the most i n t e r e s t i n g feature of the temperature dependence i s the minimum i n the neighbourhood of 1.0 K. Although data was not obtained at low enough temperatures to pass through the minimum, i t i s i n t e r e s t i n g to discuss the e f f e c t . The source i s i n part due to the cross s e c t i o n maximum i n Shizgal's c a l c u l a t i o n . (A T^ minimum would n a t u r a l l y r e s u l t from the average over energy i n eqn. 2.1.) A simple p h y s i c a l p i c t u r e of the cross s e c t i o n maximum can be described as follows. Shizgal's c a l c u l a t i o n s show that only the L = 1 p a r t i a l wave contributes to a(E) at low energies. For p-wave s c a t t e r i n g the e f f e c t i v e p o t e n t i a l i s YQ(r) plus the c e n t r i f u g a l b a r r i e r f o r the L = 1 wave. Although i t i s tempting to describe the maximum as a p-wave resonance, Shizgal points out that the L = 1 phase s h i f t does not equal 12 at the energy where the maximum occurs, and i n f a c t the p o t e n t i a l must be f a r more a t t r a c t i v e before a resonance appears. (The He-He p o t e n t i a l supports neither a bound state nor a metastable state capable of causing a resonance.) Q u a l i t a t i v e l y , the maximum occurs because the wave function i s concentrated i n the region of the w e l l owing to the a t t r a c t i v e part of V Q ( r ) . The c o l l i s i o n time i s e f f e c t i v e l y increased, leading to a greater p r o b a b i l i t y of a nuclear spin f l i p during the c o l l i s i o n . Eventually at s u f f i c i e n t l y low energies the atom i s scattered o f f the c e n t r i -f u g a l b a r r i e r and the spin f l i p t r a n s i t i o n p r o b a b i l i t y i s reduced since the distance of cl o s e s t approach of the p a i r i s l a r g e r . The T^ minimum (re l a x a t i o n rate maximum) occurs as the temperature i s v a r i e d through the 70 range in which most probable energy in the Maxwell-Boltzman energy distribution roughly matches the height of the centrifugal barrier. 4.4 Density Dependence of T,„ I D Theory predicts that the bulk relaxation rate T^g is proportional to density, but according to eqn. 1.6 the observed relaxation rate at constant temperature should pass through a minimum (or T^ pass through a maximum) at densities low enough that wall relaxation becomes significant. This behaviour was observed in samples prepared in both ways described in part 2.4 , as shown in Fig. 4.6, which plots log T^ vs log p at 4.2 K and 1.0 K Gauss for samples of varying purity. Curves 1 and 2 give the results for an uncoated pyrex surface cleaned by the method of Horvitz, (the i n i t i a l and final runs in the cleaning process) and curve 3 shows a typical curve for an annealed neon surface. There are three regions of relaxation which can be distinguished, as defined by the gas density. At high density where the mean free path - 2 X is of order d, the atomic diameter, T^ varies approximately as p 3 (34) This would be the case in liquid He and is expected in a regime where (37) many body collisions are important . At moderate densities -2 -3 (p < 5.0 x 10 gmcm ) a region of bulk gas relaxation is observed, the extent of which depends on the purity of the system. The results scale with (density) in this regime, as indicated by the line of slope -1 drawn through the data. Eventually at low density where A >> d, the region of wall relaxation is entered and the curve passes through a maximum. The values of pT^ for the uncoated pyrex and the neon surface are 1 I I I 1 t I 1 I I 1 l i l t 0.4 0.6 0.8 1.0 2.0 4.0 8.0 8.0 ^ (10~2gm.cm.-3) Fig. 4 . 6 Density Dependence of Curve 1: ^ pyrex walls 1st run in cleaning series Curve 2: • pyrex walls final run in cleaning series Curve 3: o neon coated walls 72 slightly different, with the pyrex surface giving a systematically smaller value. The values reported are the mean values of several experiments performed under each kind of conditions. pT^ (pyrex) 66 ± 2 sec.gm.cm (at 4.2 K) pT^ (neon) 74 ± 1 sec.gm.cm The discrepancy may arise from errors in using the phenomenological theory to analyze the data. The properties of the two surfaces are distinctly different and i t is not clear what effect the substrate will have on surface relaxation. It is expected that the effect of a neon surface will be less drastic since i t has a weaker van der Waals attraction and i t is likely free of complications such as paramagnetic impurities and dangling bonds. Empirically,pyrex was a less pure system with T ^ about an order of magnitude smaller than that observed in the neon system, so errors in estimating wall relaxation on a pyrex surface will be more strongly felt. Since the neon value is larger and was obtained under conditions of greater purity pT.. (neon) was selected as IB the best experimental measure of T-,.. I B An estimate of sample chamber purity can be made using the constant C2 and eqn.l.5to get a value of a the probability of relaxation at the wall. - 9 - 8 For annealed neon walls a = 10 , but for bare pyrex a ~ 10 4.5 An Estimate of the Effects of 3 Body Collisions In the region of where bulk gas behaviour is observed there may be effects at higher densities owing to three body collisions. The data £ 1.7 o 8 1.6 V 5 I \ 1.3 1.2 h 2 3 4 5 /> (10 - 2 gm.cm-5) Fig. 4.7 Graphical determination of coefficient 7 4 plotted in Fig. 4.6, curve 2, for densities less than 6.0 x 10 Z gmcm 3 where A > 5 atomic diameters was analyzed to determine the presence of any three body effects. We can write for the observed relaxation rate V'obs = C l p + C 3 p 2 + C 2 / p by adding to eqn. 1 the term C^p2 which accounts for three body collisions. This equation can be rearranged to obtain \ E p / T l obs " C 2 ] " C l + C3 P P The constant can be estimated by considering the gas to be dilute in first approximation. In Fig. 4.7 a plot of the —0 E P/T 1 , - C_H vs p p z I obs Z is independent of p, suggesting that is negligibly small. 4.6 Results of Wall Relaxation Measurements at 1.0 K Gauss Wall relaxation was studied in sample chambers satisfying both limits of cleanness described in part 2.4. The time evolution of the magnetization of the sample was shown to be a sum of modes of the diffusion -t/x v equation M(t) °= Z y e (eqn. 2.18). v v A. Strongly Relaxing Walls In the limit of strongly relaxing walls, UR>>1, the time constant of the v*"*1 mode in a spherical geometry is T = R /_ o 2 75 76 The higher modes are very quickly damped with respect to the first mode. -4 In addition, since the ratio of the amplitudes is v , i t may be possible to neglect a l l but the first mode even at times that are not too long. This conjecture was upheld by experimental results since exponential behaviour was always observed for the recovery of magnetization. The results for sample chambers prepared with sufficiently dirty surfaces to completely short circuit the bulk gas relaxation are shown in Fig. 4.8 The straight line is a calculation based on the theory with R = 0.4 cm and D taken from data of Luszczynski et a l ^ ^ . It is evident from the magnitude of T ^ that the bulk gas contribution to relaxation can be neglected in this case. B. Weakly Disorienting Walls For weakly disorienting surfaces, pR«l, only the first mode contributes to the relaxation and T-__ = RanT... , . Wall relaxation IW —j- lAd makes only a small contribution to the rate observed in experiment and values of T ^ must be obtained from eqn. 1.6 by data analysis proposed in section 4.1 . At 4.2 K the results obtained for wall relaxation on pyrex T and neon surfaces were — — - 10 and 10 sec gm cm respectively, results which are easily shown to agree with predicted values (eqn. 2.23). -2 -3 -24 -16 2 (38) Using p = 1.0 x 10 gm cm , m = 5 x 10 gm, a = 15 x 10 cm , T —3 (38} 6 IW 1 T.. ^ 10 sec -1.0 sec ^ % we obtain 10 > ~ — > 10 . The values 1 Ad p 3 of T^ ^ have been obtained in experiments on monolayers of He adsorbed on vycor glass and graphite, and are sensitive to the coverage and nature of adsorbed phase. Results for the temperature dependence of the wall relaxation in a field of 1.0 KG were obtained over a range 4.2 K - 19.0 K in sample chambers 77 with annealed neon coatings. Owing to the nature of the measurements, the error in the value of T 1 T T is very large. The indication is that T,„ 1W 1W is approximately constant over the entire range (Table 4.2). Table 4.2 _2 Temperature T (p=1.0xl0 gm/cc) Field (°K) 1 W (sec.) (kG-) 4.2 (1.5 ± 3) x 104 1.0 6.5 (1.2 ± .1) x 104 1.0 8.5 (1.3 ± .1) x 104 1.0 12.0 (1.0 ± .1) x 104 1.0 15.0 (1.5 ± .2) x 104 1.0 19.0 (1.7 ± .3) x 104 1.0 78 CHAPTER 5 Field dependence of the wall relaxation 5.1 Introduction Measurements of T^ at varying f i e l d were made in the limit of very clean walls to investigate the empirical observation that wall relaxation was much weaker at high f i e l d s . The effort i s only a preliminary study and the results are interpreted in the framework 3 of a theory which identifies the role of relaxation of He atoms adsorbed on the surface. The relationship between T ^ and T^^, the relaxation in the adsorbed phase, developed previously i n section 2, is used to evaluate the wall relaxation time in terms of a simple model of surface relaxation. 3 The adsorbed phase of He atoms has been studied by NMR techniques ( 38) previously and T^ and have been measured i n monolayers (or 3 multilayers) of He carefully deposited on vycor glass, graphite and zeolite. Nuclear spin relaxation times are sensitive to the local f i e l d fluctuations on the surface which have correlation time T , and measurements of T^ and furnish information about the motion of atoms in the (39) adsorbed film. Although some Russian authors have calculated the 3 NMR lineshape for He monolayers, no attempt has been made to formulate a theory of relaxation i n the adsorbed layer, other than to suggest that the relaxation rate i s governed by the interaction with surface paramagnetic centres and i s proportional to x / (l+u)2x 2) , where x i s the correlation P P P 3 time for the interaction between adsorbed He and a paramagnetic site on the sur f a c e ^ 4 ^ . (a) V- 40H 3 4 5 H (KGauss) 3 4 5 H (KGauss) Fig. 5.1 Field dependence of T A 2.6 K, • 4.2 K, o 8.0 K top figure p = 1.6 x 10~2 gm/cc. bottom figure p = 0.6 x 10~2 gm/ 8 0 There are, however, several characteristic times which can influence the time evolution of a spin on an adsorbed atom, and the choice of a correlation time for an adequate model of surface relaxation is not easy. A number of possible correlation times will be mentioned now. The time spent in the adsorbed layer, is independent of the magnetic field but will be characteristic of the substrate since the binding is via the van der Waals interaction, will also be different for each layer of atoms on the surface. The diffusion time, T^, of He atoms moving in a layer or exchanging between layers may be sufficiently long to satisfy the condition IOT^ - 1 for efficient relaxation. Diffusion can be considered in the limit of small jumps or jumps very large compared to atomic dimensions. This motion is independent of the magnetic field but is dependent on temperature, at least at the high temperatures observed in this work. Relaxation times of paramagnetic spins, T^g, are sensitive to the magnetic field and can influence relaxation since the time varying electronic moment creates a changing magnetic field at the nucleus of a 3 He atom. Measurements of the field dependence of T^^ in the temperature range 0 . 2 K to 2 . 4 K indicate a linear relationship T^^ <* H^^'^^\ a result not in accord with the simple prediction given above. The temperature dependence has been measured by Brewer et al. and indicates that T J ^ J is roughly independent of temperatures below 1.5 K. The experiments performed in this thesis are sensitive to relaxation in the adsorbed layers via wall relaxation in a macroscopic gas sample. It is difficult to control the nature of the adsorbed phase, and at best 3 we can say that in equilibrium a multilayer of He exists on the walls. 81 The binding energy and the residence time of adsorbed atoms are, however, quite different for each layer, with the first layer much more strongly bound than successive layers. (The binding energy is -20 for the (41) first layer on neon and about-2.3 K for the second layer.) It may be necessary to consider the interaction between layers, but in the simple models proposed here we shall consider relaxation in a single layer in which diffusion in the plane is much more probable than exchange with other layers. In the first approximation we shall assume that the neon coat is sufficient to shield the adsorbed spins from any paramagnetic sites on the surface and calculate the correlation function for the dipolar interaction between adsorbed helium atoms diffusing on a plane. An obvious extension of this calculation accounts for the effect of a paramagnetic site at fixed distance below the adsorbed layer. The 3 He-paramagnetic spin coupling can be made time dependent either by diffusion of the adsorbed atoms or by the relaxation of the paramagnetic spin. The frequency dependence of T^ for each of these interactions is calculated in Appendix A. The temperature dependence is implicit in the diffusion coefficient and no attempt is made to account for the effect of the temperature dependence of other parameters such as the surface density. There are definite advantages to using gas phase T^ measurements as a surface probe. In contrast to experiments on monolayers one is not restricted to systems in which the surface to volume ratio is very large in order to obtain strong signals. The background relaxation rate of the bulk gas in carefully prepared systems is very small so the technique can be directly sensitive to 82 100 1 I I I I I I 1 1 1 1 I I I I I 10 •o a* cn LO i o J 1 ' ' i i i ' ' i i i i i t i 1 5 H (KGauss) 10 Fig. 5.2 Field dependence of wall relaxation rate, p = 0.6 x 10 gm/cc. A 2.6 K, 0 4.2 K, o 8.0 K, relaxation mechanisms in the substrate as well as in the adsorbed layer. Some groups have employed liquid helium-3^^ as the background medium instead of the gas but although the liquid is clearly more useful at very low temperatures, i t does not offer the same promise of control of surface density via the wide range of bulk gas density available at temperatures even as low as 2.0 K. The technique is limited by the fact that surface relaxation is sometimes only a small component of observed relaxation and the error which must be associated with T ^ is unavoidably large, as explained below. 5.2 Presentation of Results. Relaxation times were measured by the method outlined in section 3.9. Two densities were studied and the temperature was varied between 2.6 K and 8.0 K. At the higher density T^ approached a constant value equal to T._ at relatively low fields (Fig. 5.1) but at lower density, T.. lis X continued to increase with field and reached T at the highest field l o only for 8.0 K (Fig. 5.1). The behaviour in Fig. 5.1 was expected since at high density the contribution of wall relaxation is a small fraction of the observed relaxation, except at very low fields. Values of T^^ were obtained by subtracting the bulk gas relaxation determined in earlier measurements from the measured relaxation. This analysis gives statistically significant results only for the lower density, but the error is s t i l l unavoidably large especially at high field where the observed relaxation is largely relaxation in the bulk gas. A plot of T ^ 1 vs. H is shown in Fig. 5.2. At 4.2K and 8.OK, but not 2.6K, the data is consistent with the predicted field dependence of H at large field (Eqn. A12). Fig. 5.3 Plots showing calculated fits to wall relaxation rate at 8.0 K and 0.6 gm/cc. (a) T = 2 x 10"8 sec (b) x = 2.0 x 10" 7 sec (c) T = 2.0 x 10 - 6 sec. 85 Using the theory developed in the first model of appendix A and equation 2.23, the wall relaxation rate can be related to the 3 spectral density of He atoms diffusing on a plane and coupled by the dipolar interaction: T l w _ 1 = A (jo(o)i) + 3 J 2(OJT ) + 4J o(2OJT) + 12J2(2OJX) u A 3TT Y V I ( I + 1 ) T ( 5 , 1 ) where A = — -1 -— • 20 nRd^a2 It is possible to estimate a value of the correlation time x and the 3 effective separation d of He atoms in the first layer by fitting the experimental results by a least squares routine to calculations of the spectral density. For a chosen value of x the spectral density was fitted to the data and a value generated of the parameter A for best f i t . The sum of squares of the deviations of the data and the fitted curve was then calculated. This procedure was repeated for other judiciously selected values of x, with each f i t giving a best value of A. The value of x at which the minimum of the sum of squared deviations occurred was chosen as the best estimate of the correlation time. From the value of A generated by this x we can estimate the effective separation using eqn. 5.1 and noting 2 that a, the surface area per adsorbed atom is just d . A number of fits to the data at 8.0 K are shown in Fig. 5.3 which plots the fitted curves for the best estimate of x and values much greater and much smaller than the best value. In table 5.1 the values of x and d are given for the temperatures studied. At the higher temperatures, the effective separation is approximately constant and in good agreement with the expected separation for a completed monolayer. The correlation time is constant to within 86 experimental error and of magnitude such that t o r - 1 for a field of 1 kG. The smaller value of T predicted at the lowest temperature is not reasonable because weaker relaxation than that observed would be expected. It is possible however that the model breaks down at very low temperature where T is approximately equal to the binding energy of the second layer. At 2.6 K a second layer, much less strongly bound may be starting to f i l l up and we can expect qualitatively a shorter correlation time for motion in this layer. We also expect stronger relaxation, in accord with the experiments, since the number of particles on the surface has increased owing to the presence of the partially f i l l e d second layer. The second model in appendix A considers the effect of paramagnetic sites near the surface. The wall relaxation rate can be expressed in terms of the spectral densities given in eqn. A.18 by the following expression -1 3 i r Y I 2Y s 2h 2S(S+l ) T T L W = ^--^-2 ( J'(C O T ) + 12J|(COT) + 3 j ' ( c o T ) ) (5.2) 5 nRZ 4a a„ o p He where and o"^ e are the surface areas per adsorbed paramagnetic and helium atom respectively. Since the spectral densities in each model are of the same order of magnitude, i t is possible to compare the calculations and estimate the surface concentration of paramagnetic sites necessary to account for the observed relaxation. The ratio of T,TT ^ for the two models is IW approximately T . „ - 1 (model 1) y 2I(I+1) a Z * -^-T = — £ - 2 - (5.3) T l w _ i (model 2) Yg2S(S+l) o^d" Using a reasonable estimate of Z q - 8 x 10 cm, eqn. 5.3 indicates that the concentration of paramagnetic sites is roughly 3 orders of magnitude lower 3 than the He surface concentration. The experiments performed up to now cannot distinguish between Table 5.1 Temperature T a (OK) (sec. ) (cm 2) 8.0 2 x 10~7 3.6 x 10 4.2 1 x 10~7 3.4 x IQ' 2.6 3 x 10~8 2.6 x 10 88 relaxation via surface concentrations of paramagnetic centres and 3 3 relaxation via He- He dipolar coupling. The theory developed in the appendix does not predict a linear relationship of with field as has been observed in some experiments, and these results remain unexplained. The relaxation times we have reported were, however, measured at higher temperatures than the results mentioned above, and our results for the frequency and temperature dependence are in qualitative agreement with the simple picture of a completed tightly bound first layer in which mobility of the adsorbed atoms increases with temperature. 89 5.3 Measurement of Field Dependence by Optical Pumping Techniques 3 Field dependence of the relaxation in very dilute He gas was (21) also studied by optical pumping techniques by Barbe et a l . . Those authors obtained very long relaxation times at 4.2 K by coating their sample chambers at low temperatures as suggested in section 3.9. They used a variety of gases including Ne, A, Kr, Xe, and U^, but obtained satisfactory results only with H^. At the low pressures characteristic of 3 their experiments ( He gas at 1 torr at room temp.) the relaxation i n the 3 bulk gas was limited by diffusion of the He atoms through weak magnetic f i e l d gradients over the sample. The relaxation time i s proportional to 2 H for their experimental conditions: wx >> 1 where x is the diffusion o o 3 - 1 2 time of He atoms across the c e l l . A plot of T^ vs. H was linear and extrapolated at H = 0 to a wall relaxation time of about 60 hours. As suggested earlier, the bulk gas relaxation observed in our NMR experiment i s not limited by relaxation in magnetic f i e l d gradients, but i s a direct measure of the dipole-dipole relaxation. It i s interesting, however, to compare the values of T ^ obtained in our work and the experiment of reference 20 . In both cases, the gas pressure i s high enough that the diffusion limit i s satisfied for atoms in the c e l l , so we expect the relaxation time to scale with bulk gas density according to eqn. 2.23. Comparing the ratio T^/n for the two experiments we obtain T l w/n (NMR) = 5 x 10~ 1 9 sec/gm/cc and T^/n (Barbe) = 6 x 10~ 1 2 sec/gm/ cc. Since the results do not scale, we must look for the source of discrepancy. 3 A possible explanation i s that owing to the low pressures of He gas the surface i n the optical pumping experiment i s not f u l l y covered, and the relaxation time in the par t i a l l y f i l l e d adsorbed layer i s longer than 90 in a f u l l y covered surface. A rough idea of the fractional coverage can be obtained from a simple model which considers sites on the surface of area a which may be unoccupied or occupied by one atom. 3 Assuming a binding energy E for He on hydrogen, the average occupancy D per site can be calculated and expressed in terms of the bulk gas pressure. (The relation between surface coverage and pressure i s the well-known Langmuir adsorption isotherm ). For the experimental conditions of 2 Barbe et a l . , gas pressure p - 17.5 dynes/cm at 4.2 K, and assuming (5) -4 E w - -20 K , the fractional coverage i s 5 x 10 . If we now take the 15 ratio T^/flaR for each experiment, we obtain the ratio of adsorbed layer relaxation times: T lAd (optical pumping)/T lAd (NMR) -Although the model i s crude, the result i s not unreasonable and could account for the observed results. 91 CHAPTER 6  SUGGESTIONS FOR FUTURE EXPERIMENTS The results obtained in this work for the temperature dependence of T^ B (covering a range of 1.7 K to 19.0 K) indicate that dipolar relaxation 3 is the dominant i n t r i n s i c mechanism in bulk He gas. The data can be fi t t e d adequately by calculations using the accepted helium potentials, with the Bruch-McGee potential providing the best f i t . This potential was in fact derived from analysis of low temperature data on other transport properties and gives a well depth of e = 10.8 K. The present results, however, suggest a slightly deeper attractive well, e = 11.5 ± 0.5 K. The NMR experiment should give reliable information on the depth and nature of the well since the measurements probe the He-He scattering at very low energies where the effect of the attractive forces should be strong. This experiment was unable to probe the region below 1.7 K where a T^g minimum is predicted. It should be possible to prepare sufficiently pure conditions to measure by the AFP techniques described here at temperatures as low as 1.0 K, with improved design of the spectrometer. Sensitivity could be improved by immersing the entire bridge i n the liquid helium bath, as suggested in section 3.3 , to reduce the temperature of the tuned c i r c u i t from i t s present high value. This change could greatly improve the signal-to-noise ratio as well since the most suspect source of microphonics, namely the long coax leading from the c o i l , would be eliminated. To make measurements below 1.0 K i t is li k e l y that an amplifier with greater gain would be required, since the vapour density i s 92 an order of magnitude smaller than the minimum density observed in this work at 1.4 K. The more fruitful extension of this work would be a continuation of the wall relaxation measurements which probe the two dimensional adsorbed phase of helium. Preliminary results have shown that the gas phase T^  is a very effective probe of the interactions at the surface. More experiments are needed to establish the frequency and temperature dependence of T^  on the neon surface, and it is obvious that the relaxation should be studied with varying thickness of the coat and also different coatings such as H.2> ^2' ^ 4 a n c* P e r n aP s a heavier inert gas. The technique is not restricted to probing only the behaviour of the 3 adsorbed phase. Interactions between the substrate and He atoms near the surface can have a dramatic effect on T^ , so the measurements can be useful in probing properties of the underlying material. Nuclear spin relaxation measurements which probe the spin-dependent part.of the He-wall interaction are of interest to those attempting to determine the role of spin 3 dependent interactions in the transfer of energy hetween liquid He and solid (43) solids . At temperatures above 1.0 K where the gas behaves like a classical fluid, the gas phase T^  would be a simple and useful probe, but at 3 lower temperatures it would be necessary to use liquid He in the study of the spin dynamics. 9 3 APPENDIX A Calculation of Relaxation Time in an Adsorbed Layer. 3 Suppose that N He atoms diffuse on a plane of area A whose normal is at an angle ® to the f i e l d H, and interact with each other via the dipolar coupling. The vector r = (r,8,<J>) between a pair of atoms i s made time dependent by the diffusive motion in two dimensions. We wish to calculate the correlation functions G (t) = H ( 2m o o _2M \ A 1 X r Q 3 r ( t ) 3 / assuming that a l l pairs are equivalent. We f i r s t make a coordinate transformation so that the new z'-axis is perpendicular to the surface. The spherical harmonics can then be expressed in terms of the new coordinates by Y2m ( 9*> " I DXm ( 2 > (°> e >>° )  Y2\<%>**>  A ' 2 A and we note in this coordinate system Y 2 Q = - ^7l6 - /T Y 2 1 = 0 Y 2 ± 2 - ^15732 e ± 2 1 * ' = /T2 e 2 i^' D^^2^ (0,9,0) is generalized spherical function of 2 C D order In order to perform the average in A . l we solve the 2-dimensional diffusion equation for the probability PCr^.r.t) which gives the chance of finding a pair of atoms separated by r at time t given that the 94 separation was init i a l l y r . P(r o,r,t) is given by P ( r - , 5 i t ) . _ i _ e - ( - r 0 ) 2 / 8 D t A > 3 ° 8rrDt with P(f o,r,0) = <5 (r-r ) , where D is the diffusion coefficient of He atoms on the surface. The correlation function becomes , f 2 " I ' " v n ( 2 ) n ( 2 ) Y 2 A( Qo*o) Y 2 X , < e * ) r Q d r o j_d* J d * I D A m D r_ n G (t) = ^  J rdr J m ' ' A d d u u o o U A A , A l u A " m r Q 3 r 3 -(?-r Q) 2/8Dt x A.4 8 T r D t N where /A is the surface density of atoms and d their distance of closest approach. (The primes have been dropped). It has been assumed here that the i n i t i a l probability distribution for the pair separation is uniform for r > d. -4 - 7 M / — j . } /3T){* To solve this integral we replace e ° by its two-dimensional fourier transform t 2n <f ( ? - r o ) 2/8Dt = M j d H .CO 2 . j , -2Dtk - i k X r - r n ) . _ kdk e e ° A. 5 TT O O ik. r and make use of the expansion for e e - I i V * ^ J A(kr). A . 6 £= -<*> where J^(kr) is a Bessel function of the first kind and ty-ty is the angle -ik.r between k and r. There is an analogous expression for e * °. 95 Substituting A. 6 into A. 5 and integrating over ij; we obtain -(r-rnr/8Dt = 2Dt oo 2 V -2Dtk - i £ ( d > - ( b ) , „ . , x kdk J. e e 0 V ^ r ) J . ( k O o 1= - ° ° Jt J i o A. 7 This can be substituted directly into (4) and the integrations performed over <f> and <t> ; i t follows that o TTN A rdr o o u A Am - A - m d A=0±2 kdk e -2Dtk2 V k r > V k r o > r 3 r 3 o A . 8 To do the integrals over k and r we make the transformation x = r/d and y = kd . Then Gm<C> = ~ A I A x D<2) D(J> g >( D t/d2) Ad4 A=0±2 X X m - A " m A.9 where g x( D t/d 2) ydy e -2Dty /d 2 ,,2 / jr» J x(xy) x2 (J dx The integrals were done numerically, using a Simpson's rule routine, for A = 0 and 2 and the results are presented in Fig. A.l. (2) Using the properties of the 's we can expand eqn .A9 to obtain the following expression for the correlation function, v<> • ?V (*0i»2)(«i2..<Dt/*2)**2<iD2)<«i2+i»2)<-)i2) Ad x g 2( /d )) A. 10 t 96 -2 Q (NJ cn CD O -3 -S -6 -2 -I 0 ft LOG Dt/d 2 - i rg -2 "U O -3 CD CD o - 5 -6 -2 -1 LOG Dt/d 2 ,Dt, Fig. Al Numerical calculations of gA(^y) vs. (-^y) for dipolar coupling between helium atoms on the surface. 97 In dealing with a spherical sample as was used in our experiments the function in eqn. 10 should be averaged over a l l angles; the result is Ojt) = [ g( D t/d2) + 3 g 2( D t/d2) ] 16Ad* ° Z m A.11 Dt o The Fourier transform of g, ( /d ) is readily obtained and we define the dimensionless transform JiCwt) A.12 oo 3 y d y i 2 2 o y +o) T - J x(xy) v 2 1 ~ T - d x ) 1 X where x = d /2D . Results for the numerical integrations in eqn. 12 are plotted in Fig. A.2. The spectral densities J (u) follow from the •CO definition J (OJ) = e l a ) t G(t) dt , and the spherically averaged functions m J _co are J (to) m N 32Ad D 2~ (30(w'T) + 3 J 2 ^ U ) T ^ A.13 Using eqn. 76 in Chapter 8 of Abragam we obtain for the relaxation 3 rate He spins in the adsorbed layer = — y V 1(1+1) [J.(w) + 4 J (2u) ] T 5 lAd A. 14 0 9 8 CD O - i - 2 - 3 -4 - I 0 ] L O G w t CD O -6 -2 -1 0 I L O G w t F i g . A2 Numerical c a l c u l a t i o n s of j A ( u , t ) vs. wt for d i p o l a r coupling between helium atoms on the surface. 99 The model can easily be extended to account for the influence of 3 paramagnetic sites on the surface. We consider a He atom diffusing on a plane whose normal makes an angle 8 with the plane as before, but Np coupled to paramagnetic spins of surface density /A at a fixed distance Z below the surface. In our experiment, Z may be taken as the thickness o o of the neon coat. The calculation proceeds in the same way as in the previous case except that r is now the distance between the paramagnetic spin and a 3 He atom. The correlation functions / Y ? J Q J J *9 mVt)<Kt) \ G (t) = N P/ 2 m ° ° - ) A.15 V r 3 r ( t ) 3 ' o become under a rotation of the coordinate system O f t ) -" A •CO |-2TT (• rdr r dr d<j> J o o J • o o o d* I D . ( 2 ) D J 2 ) 2 X ° ° 3 / o L Am X'm _ o d/ o XX» (r +Z *) o o Y2X ( 9 , < ) > , ) e" ( f" ?o> 2/ 4 D t x T, A. 16 2 2 ' 2 (r +Z ) 4TrDt o ' Integration over <J>' and 4 ' gives non zero contribution for X = 0, ±1, ±2. o We obtain for the correlation functions averaged over a spherical sample chamber G . ( t ) = - ^ J [gl( D t/Z 0 2) + 12 g. ( D t/Z Q 2) + 3 g*( D t/Z o 2)] A. 17 m 8AZ o 1 0 0 -11 - 2 - 4 - 5 -6 - 2 . , , - I 0 . . . I LOG Dt/d 2 -i 1 <N1 a cn CD O - 4 - 5 1 -t a LOG Dt/d 2 -2 Q cn CD O -s -2 LOG Dt/d 2 3 Fig. A3 Numerical calculations of g*A(Dt/Z 2) vs. (Dt/Z 2) for dipolar o o coupling between a helium atom and a paramagnetic surface site. 1 0 1 -i o i LOG wt —i 0 ft 2 3 - I - 2 CSJ - 3 CD O -6 - « «• - 2 - 1 0 1 LOG wt -i - 2 O -3 CD O -6 -2 - I 0 1 L O G WT F i g . A4 Numerical c a l c u l a t i o n s of j ' A ( u ) t ) vs. cut f o r d i p o l a r coupling between a helium atom and a paramagnetic surface s i t e . 102 where the functions g^C^Vz ^ ) plotted in Fig. A.3 are obtained numerically in the same fashion as the functions g.. in the previous case. The fourier A i transforms of these functions are denoted j . (ojt) and are plotted in Fig. A A. 4. We can at this point account for the influence of the paramagnetic spin —t /T relaxation time T l e by multiplying G (t) by the factor e le which describes the correlation between the spin of the ion at times separated by t, assuming that there is no correlation between the diffusive motion of the atoms and the ion relaxation. 2 2 This amounts to substituting ( 2 D y /Z 2 + /T, ) + for 2 D y t/Z 2 in o le o the exponent of the fourier transformed probability density and the net result in the spectral density is a shift of the lower limit of the Z 2 integration over k from zero to /T^e with x = ° /2D in the functions j'^(cox). There is an appreciable effect i f the spin fluctuations of the paramagnetic ion are rapid with respect to the diffusion of the atoms over a distance of order Z q . The spherically averaged spectral densities J (a)) are given below; m Jm( w ) = N P 9 (J + 1 2 j/(wx) + 3 j ' f a T ) ) A.18 8AZ D Z o The relaxation rate for unlike spins coupled by the dipolar interaction may be obtained from Abragam. Since paramagnetic spin relaxation times are invariably much shorter than nuclear relaxation times, i t is a good approximation to assume that the paramagnetic spin system is in equilibrium with its surroundings, in which case eqn. 87 in Chapter 8 of Abragam gives the result 1 0 3 f ~ = Y i V h 4 s ( s + 1 ) C l l J 0 ( » r u s ) + 2 W + ! J 2 K V lAd A. 19 f o r the r e l a x a t i o n rate. Since a>c >> u)T i t follows that when ai 0 « 1 J Q , and J 2 are of comparable magnitude but that when CO,T >> 1 then J » « J, and may be dropped i n A.19. This 1 a t t e r approximation i s s a t i s f i e d f o r T > 10 ^ s e c i n a f i e l d of 1 KGauss. 1 0 4 APPENDIX B  EFFECTS OF DIFFUSION ON AFP Non-exponential relaxation with a very sharp i n i t i a l decay was observed in some experiments performed with very clean pyrex cells. It was difficult to trust the analysis of such results, even though T^ was clearly quite long. Wall relaxation was discounted as the source of this problem, because the system was satisfying the conditions of weakly relaxing walls. Experimental studies were undertaken to determine what conditions produced the non-exponential behaviour and i t was eventually possible to associate the effect with abnormally weak rf fields applied to the coil. Relaxation curves for several applied rf voltages are shown in Fig.Bl. An explanation is provided in the following simple model. For weak rf fields spins in some parts of the sample may not be flipped during the passage through resonance so that we can define two distinct regions within the cell of flipped and unflipped spins respectively. Diffusion will create in time a uniform distribution, but ini t i a l l y there will be a non-exponential recovery of the magnetization which is associated with the exchange of spins between the two regions, and not with the relaxation mechanism of the spins. The spin f l i p probability F can be expressed in terms of the adiabatic conditions F = 1 - 2W 105 0 « 8 t t n N j , 6 2 0 2 4 2 8 3 2 3 6 Fig. Bl Relaxation curves at 1.0 KGauss, 4.2 K and 2.5 x 10 gm/cc. (a) 2.3 volts p-p to rf coil (b) 0.7 volts p-p to rf coil 106 where W = e and K = YHX /dH/ dt W is a function having values between 0 and 1. Most of the variation takes place over a narrow range of H^ . Since is a function of position in the cell i t is possible for large cells to separate the sample into regions of adiabatic and non-adiabatic behaviour. 4 O-O-a o XT XT •+ X 0 ^ x * a < x < a+e a+e * x - L adiabatic region F = 1 F varies rapidly between 0 and 1 non-adiabatic region F = 0 We note that the region where F varies rapidly contributes very l i t t l e to the AFP signal. Neglecting the intermediate region we solve the diffusion equation to obtain the fraction P(t) of spins initially in the adiabatic region which remain there after a time t. The general solution can be written 107 n = _co p=0,±l,-2 + TT 2 exp-[(2riL+pa) /4Dt)] 2 where < K ° 0 is an error function. In the limit of short times t << a D, % 2 ^ n = 0 and the solution becomes P(t) - 1 - (4Dt/a rr) 2, The value of a can be estimated by extrapolating the long time recovery of the relaxation curve to t = 0. Empirical results of measurements of the recovery at short times are plotted in Fig. B2, along with the expression for P(t). The agreement i s reasonable. Values of D are taken from reference 15. These considerations indicate a possible method of measuring the diffusion coefficient D in systems with very long relaxation times. By constructing an appropriate c o i l and sample chamber configuration one could easily create distinct regions of flipped and unflipped spins. The resulting time evolution of the AFP signal could be used to measure D. Effects associated with diffusion of the spins in the combined inhomogeneous dc and rf magnetic fields during a single sweep through resonance have also been considered and found to be negligible for the experimental conditions used here. /f (sec*) Fig. B2. Recovery of magnetization at short times. Solid line is a calculation of P(t) 2.0 volts (p-p) applied to c o i l . p = 6.5 x 10_2gmcm~3; T = 4.2 K; H = l.OkG; a = 0.9 cm 109 Bibliography 1. M. 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Kelly, thesis (1974) Cornell University (unpublished). 41. J.R. Eckardt, D.O. Edwards, P.D. Fatouros, F.M. Gasparini, S.Y. Shen (1974). Phys. Rev. Lett., 32, 706. 42. C. Kittel (1969). Thermal Physics. (John Wiley & Sons, New York). 43. D.L. Mills and M.T. Beal-Monod (1974). Phys. Rev. A, 10, 343. 44. M.E. Rose (1957). Elementary Theory of Angular Momentum. (John Wiley & Sons, New York). 


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