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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 HE GAS AT LOW TEMPERATURES 3  by  ROSS CHAPMAN B.Sc,  McMaster U n i v e r s i t y ,  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 t h e s i s as conforming required  THE  standard  UNIVERSITY OF BRITISH COLUMBIA  June, 1975  to the  In p r e s e n t i n g t h i s  thesis  an advanced degree at the L i b r a r y s h a l l I  f u r t h e r agree  in p a r t i a l  f u l f i l m e n t o f the requirements f o r  the U n i v e r s i t y of B r i t i s h Columbia,  make it  freely available  that permission  for  I agree  r e f e r e n c e and  f o r e x t e n s i v e copying of t h i s  study. thesis  f o r s c h o l a r l y purposes may be granted by the Head o f my Department by h i s of  this  written  representatives. thesis  for  It  is understood that  f i n a n c i a l gain s h a l l  permission.  Department of The U n i v e r s i t y o f B r i t i s h  2075 Wesbrook Place Vancouver, Canada V6T 1W5  Columbia  that  or  copying or p u b l i c a t i o n  not be allowed without my  ii  Abstract The f i r s t measurements of the temperature dependence of the intrinsic 3 dipolar relaxation time T ^ due to binary collisions i n dilute are reported. T^  B  He gas  Sufficiently pure experimental conditions to observe  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 i s strongly f e l t 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. but  The data favours a potential of the Bruch-McGee  form,  having a slightly deeper attractive well of 11.5 K. 3 The experiment also probes the adsorbed phase of  relaxation.  He v i a wall  Both wall relaxation and bulk gas relaxation are present i n  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 f i e l d 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 f i e l d dependence are interpreted i n terms 3  of a model which considers relaxation of and interacting v i a the dipolar coupling.  He atoms diffusing on a plane  iii  Table of Contents Page Abstract L i s t of Tables  v  L i s t of Figures  ' yi  Acknowledgements  ~. F i i i  Chapter 1 Introduction 1.1 1.2 1.3  Introduction to Intermolecular Forces General Ideas of NMR i n Monatomic Gases NMR i n He gas  1 5 7  3  3 2 Theory of Longitudinal Spin Relaxation i n D i l u t e 2.1 2.2 2.3 2.4  C a l c u l a t i o n of o(E) f o r d i p o l a r coupling Other Relaxation Mechanisms Time C o r r e l a t i o n Function Approach Relaxation at the walls  3 Experimental 3.1 3.2 3.2.1 3.3 3.4 3.5 3.6 3.7 3.8 3.9  He Gas  11 12 19 20 23  techniques  Measurement of Longitudinal Relaxation Times Experimental Apparatus General Description Radiofrequency Bridge Spectrometer C o i l and Resonant C i r c u i t F i e l d Sweep Unit Cryostat Temperature Measurement Gas Handling System Methods of Suppressing the e f f e c t s of Wall Relaxation  27 30 30 32 40 43 43 45 49 51  iv  Page 4 Presentation of Results and Discussion 4.1 4.2 4.3 4.4 4.5 4.6  Introduction Possible Sources of Error Temperature Dependence of T-^g Density Dependence of T ^ B An estimate of the E f f e c t s of 3 Body C o l l i s i o n s Results of Wall Relaxation Measurements at 1 . 0 K Gauss  5 5 6 7 7  5 6 3 0 2  7 4  5 F i e l d Dependence of the Wall Relaxation 5.1 5.2 5.3  Introduction Presentation of Results Measurements of F i e l d Dependence by O p t i c a l Pumping Techniques  6 Suggestions f o r Future Experiments  Appendix A. Appendix B.  Bibliography  C a l c u l a t i o n of Relaxation Time i n an Adsorbed Layer. E f f e c t s of D i f f u s i o n on AFP  7 9 8 3 8 9  9 1  93 104  1 0 9  V  List of Tables Table 4.1  4.2  Page Results of fitting temperature dependence by calculations of T-^g using available helium potentials.  68  Wall Relaxation times T at 4.2 K, 1.0 K Gauss and p = 1.0 x 10~ gm/cc.  77  Correlation time T and area per adsorbed atom a from wall relaxation measurements  87  l w  2  5.1  L i s t 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  T  17  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  F i e l d Sweep Unit  41  3.8  Variable temperature Cryostat  42  3.9  Thermometry C i r c u i t  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  1t>  IB  vs. T, calculations using MS and BM potentials  / T vs. p  2  1  60  4.3  Relaxation curve for high f i e l d measurement of T^  61  4.4  PT  64  4.5  PT vs. T for best values of BM helium potential IB  66  4.6  Log  71  4.7  Estimate of 3 body effects T,„ vs. p  4.8  1B  vs. T  1T)  vs. log p  73 75  Figure  Page  5.1  1^ vs. H  5.2  Log T  5.3  Plots of T  _ 1 l w  80 vs. log H 1IT  "'"vs. UT for selected values of x  83 85  1W Al  Plots, of 8^(t) f° 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  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  A4  r  :  viii  Ac knowled g ement s  I am deeply g r a t e f u l to Professor Michael Richards of the University of Sussex who not only provided the i n i t i a l stimulus for t h i s 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 r e l a x a t i o n 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 h i s programs for the c a l c u l a t i o n of T  and has also c o n t r i -  buted u s e f u l suggestions. F i n a l l y , I wish to thank Mr. John Lees for h i s w i l l i n g assistance and sound technical advice i n the use and construction of the glassware required i n the experiment.  I also appreciate the work of Mr. Ernie  Williams, h i s a s s i s t a n t , who performed much of the glassblowing.  1  1  CHAPTER  INTRODUCTION  1.1  Introduction Nuclear  as  a sensitive  on m o l e c u l a r times  Intermolecular  spin  means  studying  orientation^.  momenta  provided  relaxation  of  are mediated by  angular  during  obtained  Riehl  et  resonance  (NMR)  T h e NMR  The c r o s s  (2)  t h e o r y ^ ^ to  the  is  difficult  the  isotropic  nor  and  only  crude  very  since  the  even i n  anisotropic  functional  part (3)  forces  relaxation  change  of  the  the  have  molecular have  intermolecular  successfully  by n u c l e a r  systems time  is  magnetic  r e l a t e d by  for molecular is  calculated  interpretation  the  simplest  parts  forms  recognized  mixtures.  reorientation the  spin  potential  correlation  for molecular  been  time measurements  and Bloom  gas  long  intermolecular  which  anisotropic  of molecular  intermolecular p o t e n t i a l , but  results  torques  on g a s e o u s  the  of  the n u c l e a r  relaxation  Harare  rate  gases has  the dependence  and L a l i t a  experiments  section  dilute  Because  of  on t h e  relaxation  established  in  a collision,  al.  information  forces  intermolecular  an e f f e c t i v e p r o b e  potential.  the  to  of  of  the p o t e n t i a l  the p o t e n t i a l  is  a system  reorientation. in  terms  systems are w e l l  are  used  neither known,  in  calculations.  3 By coupled coupling rate  is  contrast spin  a sample  % particles  between  spins  immediately  is  of  He  gas  i n which  t h e w e l l known  responsible  related  to  the  for  cross  of  very  section  light  anisotropic  relaxation. for  of  experimental  molecular  of  a well  The spin  weakly dipolar  relaxation transitions  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. system in which to apply NMR  Although  He gas is the most fundamental  techniques in order to investigate inter-  particle potentials, d i f f i c u l t i e s 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 f i r s t time observation time T,  in bulk  XD  3 He  of the dipole-dipole relaxation  gas.  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 i s 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, w i l l  be presented. A discussion in greater depth can be found in Hirshfelder etal  ( 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 d i f f i c u l t 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 f i t t i n g 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 i s 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 i s 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 i t is possible to detect  3 He.  A dilute gas of  NMR,  3 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 f i r s t 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, — T  =  n<av>  ,  (1.1)  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. NMR  The  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. are performed i n conditions i n which  Since the experiments  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 heavier inert gas  129  v  Xe.  (10)  5  It was with these intentions that a study of nuclear spin relaxation in  3 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 w i l l be presented. A nucleus with a net spin angular momentum I has a magnetic moment u proportional to the spin the nucleus.  y = tiyl  where y i s the gyromagnetic ratio of  In an applied magnetic f i e l d H,the spin precesses, classic-  ally, at i t s 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 f i e l d H^, perpendicular to H, which oscillates at the Larmor frequency.  After removing H^ the spin system w i l l  return to equilibrium, dissipating i t s absorbed rf energy through a  6  spin-lattice coupling to the available degrees of freedom. The approach to equilibrium is given by phenomenological equations f i r s t derived by Bloch^ ^ 11  dt  T  T  2  2  T  x  where the f i r s t term describes the motion of the spins in the applied f i e l d and the remaining terms describe relaxation to equilibrium. time constants  and  The  describe respectively the relaxation parallel to  the applied field and the relaxation i n 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 i t s 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 i n liquids and gases only, where the spin-spin interactions are weak and the duration of collisions during which the spin gives up i t s energy is very short.  A spin on a particular atom  experiences a fluctuating magnetic f i e l d from the spin on i t s collision partner which can be analyzed into spectralcomponents;  the fourier  components at W and also 2U) are effective in causing spin transitions q  q  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 i s 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 f i e l d w i l l be seen by a spin.  If the surroundings are  isotropic the transformation does not change the appearance of the local field since i t s fluctuations are much more rapid than to . The relaxation o in the x, y and z directions should proceed at the same rate.  3 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 1.3  NMR  in  in which the interatomic dipolar coupling is the dominant relaxation (12) mechanism  . That i s , the spin system dissipates energy absorbed from  the rf f i e l d 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 f i e l d 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  c  << to ^ for fields attainable with laboratory electromagnets, o In this limit  T  2Q  =  T  2  311(1  w e  c a n  e s t i m a t e  T  2 by considering  the  effects of binary collisions which occur incoherently at an average frequency u  ,, coll  (13)  .In  each collision the transverse spin magnetization  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  i s obtained by setting the accumulated phase  angle equal to one radian: V  1  =  < A < f > 2 >  °coll '  <A<{>> can be estimated from the strength of the dipolar coupling»(y ti/d ) 2  2  3  and the duration of collisions, <A<j>> = (f^yVd ) (d/v) . Finally, 6  2  T~ l  = T  _ 1 1 B  =  n(m/kT) h %  2  4 Y  /d  2  -  2  P/T 2  (1.3)  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 w i l l 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. (15-17)  Previously  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 d i f f i c u l t to avoid concentrations of this magnitude, but at 4.2 K where the present data was recorded, a l l the impurities except He w i l l 4  be condensed out on the walls. Wall effects are more d i f f i c u l t to estimate because the probability  9  of relaxation at the wall depends on the nature of the surface and on bulk gas pressure which affects the t o t a l time a the w a l l .  3  the  He atom resides near  For s u f f i c i e n t l y d i r t y surfaces a, the spin f l i p p r o b a b i l i t y 3  per wall c o l l i s i o n i s large enough that a time i t spends near the surface and T, , TT  He atom i s relaxed during  the  the w a l l relaxation time, i s  1W  determined by the time necessary to reach the walls. In this l i m i t =  T  1  for  can be estimated from a random walk argument, R/ 2  D  a c e l l of radius R.  «  (1.4)  n  i s proportional to density v i a the d i f f u s i o n  c o e f f i c i e n t D and i s independent of a. In the clean surface regime an average p a r t i c l e diffuses to the w a l l but a i s so small that the atom makes too few w a l l c o l l i s i o n s to f l i p i t s spin before returning to the bulk gas.  The average w a l l c o l l i s i o n  frequency i n a sphere of radius R i s 3v/4R, and a spin requires on the average 1/a w a l l c o l l i s i o n s to relax so, T l  -  ^  .  (1.5)  3av  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 most e f f e c t i v e i n relaxing 3  He spins owing to much slower motion i n the adsorbed phase and perhaps the presence of surface impurities. be expressed as  The spin f l i p p r o b a b i l i t y a  can then  a = a., T . , / T , where a., i s the adsorption p r o b a b i l i t y Ad Ad lAd Ad and T . , and T,. , are the time spent on the surface and relaxation time of Ad lAd 1  a  the adsorbed phase respectively.  It has been assumed that i ^ << T ^ j .  x^j i s given by the surface density of adsorbed atoms divided by the f l u x of p a r t i c l e s leaving the adsorbed phase which at equilibrium i s equal to the inward f l u x .  The l a t t e r i s proportional to n.  Hence T^  a  n i n both l i m i t s  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 p = nm.  on p , where  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 w i l l yield information on the He-He potential.  at low temperature  Ln  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 i s discussed f i r s t .  This i s followed by a discussion of wall relaxation.  Experimental details and results are discussed in chapters 3 and 4 , respectively.  Preliminary measurements of the f i e l d dependence of wall  relaxation were made and these results are interpreted in chapter 5 i n terms of a model of relaxation in the adsorbed phase presented in appendix A.  11  CHAPTER 2  THEORY OF LONGITUDINAL SPIN RELAXATION IN DILUTE HE  GAS  3  3  The  nuclear spins i n  He gas are relaxed by f l u c t u a t i n g magnetic  f i e l d s associated  with the nuclear dipolar coupling between pairs of  c o l l i d i n g atoms.  The  described previously  c a l c u l a t i o n of T^  fi  using simple c o l l i s i o n a l models  i s suitable only for an order of magnitude estimate  and does not allow a detailed i n t e r p r e t a t i o n of T^ i n terms of the atomic and k i n e t i c properties  of the  gas.  Because of the s i m p l i c i t y of the dipolar coupling, i t should be possible to perform an exact c a l c u l a t i o n of T^ using the best available helium potentials He atoms.  The  fi  from f i r s t p r i n c i p l e s  to describe the scattering  temperatures at which the nuclear spin relaxation  of  experi-  ments were performed r e s t r i c t the r e l a t i v e energy of a c o l l i d i n g p a i r of atoms to very low  energies and  t h i s permits a computational s i m p l i f i c a t i o n  since i t i s necessary to include only the lowest p a r t i a l waves i n t r e a t i n g the scattering problem. A formal k i n e t i c theory for the c a l c u l a t i o n of T..  (18) gases was  developed by Chen and  Snider  "their theory i s the cross section a(E)  .  i n d i l u t e monatomic  ID  The  e s s e n t i a l quantity i n  for spin t r a n s i t i o n s r e s u l t i n g from  the c o l l i s i o n s of pairs of atoms interacting v i a the spherical  potential  and an anisotropic dipolar coupling which i s responsible for the  spin  12  flips.  The r e l a x a t i o n rate i s then obtained from 00  2h]x h n(-  f  s  -E/kT  a(E)EdE  (2.1)  where E i s the r e l a t i v e energy of a c o l l i d i n g p a i r of reduced mass u and n i s the number density of the The  k i n e t i c formulation  has  binary c o l l i s i o n s are accurately  gas.  the advantage that the dynamics of taken into account.  the  In addition, r e a l i s t i c  forms of the He-He p o t e n t i a l can be used to evaluate the cross section, so that a comparison of the calculated and measured values of T  1  can provide  a test of the form of the p o t e n t i a l used.  2.1  C a l c u l a t i o n of the Cross Section  a(E)  3 The  i n t e r a c t i o n between  e l e c t r o n i c i n t e r a c t i o n Vo(r) and  He atoms i s taken to be the ground state the anisotropic d i p o l a r i n t e r a c t i o n between  nuclei, V(r)  v(f,i  =  l t  V (r) + 0  i ) 2  V(r I I ) t  lt  2  (2.2)  =  where r i s the r e l a t i v e p o s i t i o n vector of atoms with nuclear I„.  spins 1^  and  In the centre of mass system the Hamiltonian i s  H  = -(n /2u)V 2  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 oftymust 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 % = 1 partial wave only.  eV) , o"(E) is effectively given by the  Since the dipolar coupling is many orders of  magnitude smaller than the spherical potential, i t is possible to calculate the partial wave scattering amplitudes within the distorted wave Born (  approximation.  9  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^ ^, Bruch and McGee (BM)^ ^, and McLaughlin and 19  Shafer (MS)  6  ; the latter two potentials are plotted in fig. 2.1.  The Beck potential is a f i t of second v i r i a l coefficient data in the temperature range 25 K - 1500 K to the functional form 0.869 V (r) = A exp(-ar - Br ) Q  (r*+a ) z  o  where a = 0.675A;  ,  a = 4.390A ; -  2.709 + 3a [ 1 + r  l  + a'  1  / c  g = 3.746 x 10 A  2  ]  ; A = 398.7 eV.  (2.4)  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 (r)  = £[exp(2c(l - x)) - 2ex (c(l - x))]  Q  X  "  V (r) = o v  r ^ r  P  2  min  r/r  -1.47 r ~  ( 2  6  -  14.2 r  r >r  8  and obtained £ = 1.484 x 10~ erg; ° c = 6.12777.  r = 3.0238A; m  15  '  5 )  2  o  r = 3.6828A and 2 0  The MS potential i s a f i t of calculations of the He-He interaction to the functional form  V (r) = 455.23 exp(-ar - br ) - 0.9213 r " - 2.623 r ~ 3  6  (2.6)  8  Q  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 i s a greater difference in the numerical results, although a l l -4 potentials indicate a maximum near 10  eV.  T^ i s readily obtained from Eqn. 2.1. The plot shown i n f i g . 2.3, the B  calculated values of T  i n the range 0 - 20.0 K, suggests that the temperature  dependence i s sensitive to the form of the spherical potential, and i n  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 c r o s s s e c t i o n a(E) f o r s p i n t r a n s i t i o n s caused by d i p o l a r c o u p l i n g v s . energy, u s i n g the MS and BM h e l i u m 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  1T)  T  approaches a constant.  interesting region is T < 2.0 K in which a minimum occurs.  The most  Measurements at  higher temperatures are not unimportant, however, because the potentials can be tested by f i t t i n g calculations of T  to the experimental results. ID  2.2  Other relaxation mechanisms There w i l l 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 contribution to T^.  and estimate the strength of i t s  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 f i e l d 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 square value <6H > at intervals of time T equal to the average time 2  3.0  0 '  .  20  Fig. 2.4  :  ,  r —  30  ,  .  40  Cross sections for spin transitions via the dipolar and rotation interactions, k is relative wave number and d the classical turning point. . ... At kd - 31, where the section magnitudes are equal, the relative energy of the pair is 0.2 eV.  1 50  spin is crosscolliding  |^  20  necessary to d i f f u s e through 6H.  Schearer and W a l t e r s ^ ^ have estimated 22  this relaxation time to be  -1 T, = 1  2 2 G <u >x H 2~  (2.7)  o  where G i s the magnetic f i e l d gradient  and u the thermal v e l o c i t y of atoms  which make c o l l i s i o n s on the average at i n t e r v a l s T . experienced i n t h i s work (G ^  10  In the f i e l d  gradients  Gauss/cm i n l.OkG ) the time constant  i s of order 10"^ seconds.  2.3  The Time Correlation Function Approach By contrast Abragam^ ^ has developed a formal theory of 4  relaxation i n monatomic gases i n which the r e l a x a t i o n rate i s calculated i n terms of f o u r i e r transforms of the time c o r r e l a t i o n functions of the dipolar coupling.  The theory demands that the dipolar i n t e r a c t i o n which couples the  spins to an external reservoir  be weak, so that the p r o b a b i l i t y f o r spin  f l i p s between nuclear Zeeman states can be calculated by perturbation The  relaxation rate i s  T  where  a n d  theory.  1B  =  2  ^  1  J (io ) = m o  (  e  /Y  2  1  +  1 }  ( J  l  ( c o  o  )  +  J  2  ( 2 a )  o  ) : )  ° g (t)dt  -  8 )  (2.9)  m  (GO  ( 2  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 i s possible to obtain information about the i n t e r p a r t i c l e 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 d i l u t e monatomic gas i s provided by the theory (23) of Oppenheim and Bloom  . These authors r e s t r i c t consideration  to binary  c o l l i s i o n s i n which a given pair interacts v i a a potential V (r) and i s coupled to the reservoir by a general s p i n - l a t t i c e interaction F ( r ) . They consider i n d e t a i l the c l a s s i c a l l i m i t i n 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  r(t)  -  r  V V ( r ( t ' ) dt' Q  +  1 i  f t  P(t') dt'  (2.11)  o  The ensemble average i n Eqn.2.10 i s calculated i n terms of a time dependent pair d i s t r i b u t i o n 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 i s separated by r(t) at time t :•  g (t) m  =  N  dr  dr  F(r) F(r') g(r,r',t)  The relationship of a measured relaxation rate governed by F(r) to the  (2.12)  22  microscopic behaviour of the gas is that i t 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  V (r) - V (r(t)) Q  Q  y|r - r(t) | With the CAA  g(r,r*,t)  =  .  -5  .. 5«  g ( r r g(r') P(r,r',t)  (2.13)  2  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  Tib  -1  4, 2 4n -h- I(I 1) T  +  (  „ 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 c a l c u l a t i o n ^ ^ ' * ^ only at high temperatures where 2  3  classical approximations are good, but i t should not be valid at lower  )  23  temperatures i n the quantum regime.  2.4  Relaxation at the Walls Contributions from wall relaxation were present  i n varying  strengths i n a l l measurements taken i n this study and i t was possible to neglect t h e i r e f f e c t . (15-17)  Many of the previous  not always  studies of  3  relaxation i n  He gas  have been l i m i t e d by wall r e l a x a t i o n , but  only  phenomenological theories have been used to explain the interactions at the surface.  Wall e f f e c t s are d i f f i c u l t to estimate without precise  knowledge of the nature of the surface, i n p a r t i c u l a r , what impurities are present.  It i s possible, however, to develop a r e l a t i o n s h i p between 3 the wall r e l a x a t i o n time T and the r e l a x a t i o n time T , of He atoms 1W lAd (25) 1rT  1 4  adsorbed on the sample chamber surface r e l a t i o n between T ^  .  This section establishes the  and T^ ^ and i n Appendix A, a model of relaxation  i n the adsorbed layer i s 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) i s governed by d i f f u s i o n i n the gas and r e l a x a t i o n by c o l l i s i o n s i n the bulk gas and at the walls.  The  equation of motion f o r <M> , the expectation value of M(f,t) i s therefore | - <M(r,t)> = DV <M(r,t)> - - i - <M(r,t)> (2.15) 1B 2  d t  T  3 where D i s the d i f f u s i o n c o e f f i c i e n t of  He  gas.  We w i l l neglect relaxation i n the bulk gas and solve the d i f f u s i o n equation subject to the boundary condition  24  —  <M(r,t)> | r=R  = -u <M(r,t)>  (2.16)  which accounts f o r a f i n i t e p r o b a b i l i t y of r e l a x a t i o n at the surface v i a the parameter u.  The boundary condition i s obtained i n the following  Denoting a the p r o b a b i l i t y of a spin t r a n s i t i o n i n a single wall and J_^(r,t),  way.  collision,  J ( r , t ) respectively the f l u x of magnetization incident on  and emerging from the surface, atoms leaving the surface have a d e f i n i t e p r o b a b i l i t y of. being relaxed, and  the incident and emerging fluxes are  related by  Since  J_(r,t)  =  (1 - a)  J (r,t)  =  v[h  ±  J (r,t)  (2.17)  +  <M(Rt)> ?  |  |-  <M(x;t)>]  where A i s the mean free path, s u b s t i t u t i o n into equation 2.17 y i e l d s the boundary condition  with  dimensions of length  u = 3a/2A(2 -a).  (of order A/a)  and  The  inverse of u has  the  i t i s interpreted as the minimum  distance a spin must t r a v e l near the surface before being relaxed. The  system of equations 2.15  r e s u l t that equation.  M(r,t)  and  2.16  has been solved  with the  i s a summation over the modes of the d i f f u s i o n  If the i n i t i a l magnetization i s s p a t i a l l y i s o t r o p i c , the  solution i s  <M(r,t)>  1  =  = D(co^) 2  I  A  V  Sin(<A)  e~  th  25  where co i s a r o o t  o f the r a d i a l p a r t o f e q u a t i o n 2.15  V  boundary c o n d i t i o n . obtained  The q u a n t i t y  subject  observed i n experiment <M(t)>  to the is  by i n t e g r a t i n g <M(r,t)> over t h e volume o f t h e sample.  There a r e two l i m i t i n g  cases o f i n t e r e s t yR>>l and yR<<l.  c o n d i t i o n yR>>l i s s a t i s f i e d pressures;  i n this  limit  for highly relaxing surfaces  the s o l u t i o n f o r  io  V  is  co  V  =  VTT/R  The  and o r l a r g e with the  result  = DvV/R  V  1/  2  .  (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  low  pressures.  i n t h e l i m i t o f weakly r e l a x i n g w a l l s  and/or  An atom near t h e s u r f a c e may c o l l i d e many times w i t h t h e  w a l l s b u t r e t u r n t o t h e b u l k gas b e f o r e  being  relaxed.  In this limit  only  til the f i r s t mode c o n t r i b u t e s s i n c e the amplitude o f t h e v 2 2 12 by (yR) /v . The s o l u t i o n i s ( c o ) = 3yR w h i c h g i v e s  T  It  lw  ~  1  i s important  usually satisfied  =  mode i s a t t e n u a t e d  3a v/4R .  (2.20)  t o n o t e t h a t i n a b u l k gas sample t h e c o n d i t i o n so t h e s t r o n g e r  X<<R i s  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 t h e case o f h i g h l y d i s o r i e n t i n g s u r f a c e s  proportional to density however t h e d e n s i t y At  that  (the s p i n s d i f f u s e t o t h e w a l l and a r e r e l a x e d ) ,  dependence i n t h e o p p o s i t e  limit  i s not obvious.  low t e m p e r a t u r e s i t i s known t h a t one o r two l a y e r s o f  3  He w i l l  (27) be  adsorbed on t h e w a l l s  resulting  i n adsorption  is  and t h e r e f o r e  i f yR<<l o n l y  w i l l be e f f e c t i v e f o r r e l a x a t i o n .  collisions In this  26  s i t u a t i o n the s p i n 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  where a 'Ad  T  Ad  Ad  x  and T  a r e r e s p e c t i v e l y the p r o b a b i l i t y t h a t a p a r t i c l e  lAd  w i l l be adsorbed on c o l l i s i o n , phase, T  T  . , i s g i v e n by Ad ° J  ( f l u x of atoms l e a v i n g  a ad,  the average time spent i n the adsorbed  and the r e l a x a t i o n time i n the adsorbed phase.  t h a t , , << T. . ,. Ad lAd  leaving  (2.21)  /T  Ad' lAd  the s u r f a c e  riv/4.  (surface density  the adsorbed p h a s e ) .  J  I t has been assumed of adsorbed  In equilibrium  the f l u x  e q u a l s the i n c i d e n t f l u x and the l a t t e r i s g i v e n by  Hence  = 4/nv a, jO"  (2.22)  Ad  where cr i s t h e s u r f a c e  T  showing  atoms)/  RnaT 1W  that  proportional  a r e a per adsorbed atom, and  lAd  (2.23)  3  i n the weakly to d e n s i t y .  relaxing surface  l i m i t as w e l l ,  is  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 i s well established in the literature so  only an operational description of an AFP experiment w i l l be given. I n i t i a l l y the spin system is polarized along the effective f i e l d in the rotating frame resonance that H  eff  Hgfj  ~  i  +  (H -  IO /Y) 0  k > f  a r  enough off  is almost parallel to the applied f i e l d H. v  r  The f i e l d  H (or frequency oo) can be varied through resonance i n 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 Y H  1  >  dH  \ A £  >  1. T±  f  O.D  and are easily satisfied in a sample of pure ^He gas using sweep rates of 0.1 Gauss/sec and r f fields of about 60 mGauss; It i s 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 c o i l 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  H + AH 0  o Ul O <  H -AH 0  s  TIME  2  O r-  < N  TIME  I-  LU  e o < 2  o) PAIRWISE METHOD FOR MEASURING T,  TIME  2  g <  Ji  N  1-  TIME  Ui H O <  b)  METHOD FOR VERY LONG  Fig. 3.1  T,  29  The spin magnetization then returns to i t s equilibrium value according to the equation  M(t)  =  M(»)  +  (M(o) - MO))  e"  (3.2)  t / T l  T^ i s 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 i s unsuitable for long relaxation times since one must wait several T^'s between pairs of sweeps for reestablishment of equilibrium. The very long relaxation times were measured by sweeping the f i e l d once through resonance to a resting f i e l d 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 f i e l d .  The magnetization continues on i t s 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 i s negligible loss of (28) magnetization each time the signal i s monitored It i s important to ensure that  i s 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 X  or H /<5H respectively. Lt  X.  A weak H  also creates I  nonuniform magnetization in the sample which w i l l persist until smoothed by diffusion.  (Appendix B)  At the sample chamber surface local fields H for wall relaxation may not be small compared to  which are responsible 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 frame and has energy levels proportional to (H..  2  JL  + H  2 ^ ). 2  o  Transitions  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 very strong.  be  The time of the passage through resonance must, therefore,  be short compared to  (walls) which i s 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. useful since  The results would be extremely  samples the low f i e l d spectrum of the wall relaxation rate.  It was very d i f f i c u l t 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  a f i e l d of 1 KGauss at 4.2 K, and high density, p > 4.0 x 10  3.2  ~  4 minutes in gm.cm .  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  f i n a l 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  MHz  Oscillator  Liquid  RF  Tuned  Bridge  Amplifier  Detector  He Chart Recorder  RF  Coil Varian Electromagnet  BLOCK  DIAGRAM  OF  RF  SPECTROMETER  32  large signals expected, owing to the large y-factor  of the  3 He nucleus  and the low temperatures and high gas densities characteristic of the experiment.  Signal strength ^g> v  c a n  s  be estimated from the relation  (4)  v . - -ZT NAiu^XH,-, where Q is the quality factor of a c o i l of N turns and sig 2 ° 3 3 area A , and X is the spin susceptibility. For a 1 cm sample of He gas u  n  J  at 4.2 K and 1 atm. pressure, v . i s several mV.  sig A simple r f bridge spectrometer operating at 3.5 MHz was used to moni3 tor the AFP signal from a sample of He gas contained i n a pyrex bulb. The signal was amplified by a tuned r f amplifier, detected and fed directly to a chart recorder (Fig. 3.2). A l l measurements were made in a 12" Varian magnet with a 2.25" pole gap.  The magnet had the attractive feature of reasonably good f i e l d  homogeneity.  At the optimum place between the pole faces the f i e l d was 6 3  constant to 7 parts in 10  per cm . For the purpose of high f i e l d relaxation  experiments the magnetic f i e l d 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 r f signal.  And, in addition, long term stability i s  desirable since i t takes on the average several hours to complete a T^ measurement.  33  R  AAAr  c  2  Output  Input ©  TWIN-T  OR ANDERSON  At  4.2  BRIDGE  K:  c, = 2 7 pf C = C' = 4 . 7 pf CQ  = 194  c  =  10  =  1.25  R Fig. 3.3  2  pf  pf  Twin - T bridge circuit, with parameters for operation at 3.5 MHz and 4.2 K.  34  An rf bridge i s 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 i s 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 or Q  respectively) was used in the spectrometer. The  balance conditions are u) L(C + C + C 2  ( 1 + C/C^))  =  1  R R a) CC ( 1 + C./CJ p / l 2  where R i s 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 + cyc^)  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. the bottom, the coax opened into the exchange gas chambers; was therefore prevented from entering the tube.  At  liquid He  Such a long line is an  undesirable feature because i t s resistance lowers the circuit Q, but nevertheless one which could not be avoided since i t s 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 c o i l , 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 c o i l , 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  where  eff  =  C  I> L+V C  2  *  i s the part at low temperature and  temperature,  the remainder at higher  (Figure 3.4) shows three possible configurations with 3.4(b)  the actual arrangement.  36  b)  a) L C L  300 K  300 K  T =300 K c  Q=  r (300 L  300 K  K)  Q  r (4K)+ r x L  d)  a—  Li  =C )  4 K<Tc<300  K  T„ = 4 K  r ( 4 K) + r ( C / ( C + C ) ) 2 Q - = L  Fig.  3.4  x  a  a  b  OJL  r  ( 4  -  K  )  2.2K 820  2.4 K 0.01 fJL 3.5 MHz  3.3  —I Id—tf-  .OOl/i  .ooi^:  1.2 K:  3.3 K:  .  2N5087  •2.2K i L-WAr  :8.2K |24K  IN 617  3.5 MHz CRYSTAL  0.1  2N5087 fJL  ZZ O.OI/i 24 K  O.l/i v/.l /1~T"  50 pf .01 p- H2K 1  3.3K*  OSCILLATOR  Fig. 3.5  —i  • + 18 V 1 5 0 / t h  :47K  2N5486 Output  i —  Input  :33K 47 K 2 56  Fig.  D02d  3.6  CASCODE  AMPLIFIER  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 i t s 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. term d r i f t .  The major component of noise at the chart recorder was a long It rarely exceeded 100 mV/hr and was tolerable since the  baseline could be interpolated reliably i n the region of a resonance. The major source of d r i f t 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 d r i f t 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. when they occurred near a signal.  The spikes were a serious problem only In that situation, the resonance was  ignored and subsequently retaken. High frequency noise was filtered by the chart recorder i t s e l f 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 i s 1.7 x 10  volts.  This voltage is amplified by  the tuned amplifier with gain G and then fed to a peak detector whose 2h 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 chart recorder,  2 h , r 2 h <v > = 1//2 G<v > - 3 yV , for a gain of 250.  Owing to the strength of the microphonics and the d r i f t i t was not likely that the thermal noise would be observed. The spectrometer was capable of detecting resonances over a large -4 range of temperature and density, and a lower limit of p = 7 x 10  -3 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). -7 P ^ - 7 x 10 t  This gives  -3 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 i t s design was not a c r i t i c a l problem.  The c o i l consisted of 40  C: 4.7/t - 0 . 0 4 7 / 1  S +• |8V ° - 18V  Double poleDouble throw a) « Sweep b) = Hold  3  C  4f  39 K 100 •—vw2.0K (10 turn)  -wv  470 K  a)A ^ 3 MCI74I  binr  IK •AAAr  -w\— 10 K  1  :470K  IK  Output 400 K 100 K  FIELD  SWEEP UNIT  Fig. 3.7  Rubber "0" Ring Innerchamber pumping line  To vacuum Pumps  Feedthrough Seal Support  tube  Outerchamber pumping line  ._,  Cryostat  Copper  head  baffle  Brass flange X  Indium 0" Ring  Outer vacuum chamber  Coax  0.5mm glass capillary  Woods' Metal seal  Carbon resistor  Inner vacuum chamber  Heater coil  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). length of wire. 4.2 K.  Connection to the coax was made by a short  The resonant circuit Q was measured to be about 80 at  Although a large fraction of the tuning capacitance C  q  was  soldered directly across the c o i l , 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 i s 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 O.D.  tube.  A 5/8" O.D.  pumping tube surrounded the central 3/8" tube  3/8"  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. maintained above the flange.  The level of liquid helium was always 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 i t s 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 c o i l , 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 i n 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 l e f t 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 d r i f t 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 i t s 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 c o i l 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 the vapour pressure of liquid  against  4 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  R  2  Decade  Box  R  T  Carbon  Resistor  :  = MM  M A X  R  6  1.5 V  3.9  THERMOMETRY  CIRCUIT  2  47  Jackson and K u r t i  log  lfJ  V J < i /  R + K/log  lfJ  R  2b + a/T  where  R i s resistance i n 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 i s 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 i n 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 i s shown i n (Fig. 3.9); and needs l i t t l e elaboration.  The switch  i t i s fairly simple  controls the current to the  resistor and could be changed to allow lower current at lower temperatures.  a  To Vacuum Pump  Neon Cell  G l a s s - m e t a l Seal to .125" St. Steel 2 mm Glass Capillary  VV  -J-I-T-  ,  Leak Valve  Pressure Gauge  Reducing  T  7  <2H  I  .062" St. Steel  Glass-metal 1 * - Seal — i to .25"copper  77 K  Misch-Metal 0 getter  r*7  2  Charcoal Trap  0.5 mm Capillary  Thermocouple Pressure Gauge  Researchgrade 4  Sample Chamber  He He Storage  Fig. 3.10 G A S  Cell  HANDLING S Y S T E M oo  49  3.8  Gas Handling  System 3  Samples of  He gas were obtained from the Monsanto Corp. with  a quoted i s o t o p i c p u r i t y 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 for s t o r i n g the gas at room temperature and conveying chamber during experiments.  constructed  i t to the sample  The metal valves and tubing were necessary 3  to withstand  the required pressures of  He gas and the design featured  miniature s t a i n l e s s s t e e l Hoke valves joined to the tubes by fittings.  The glassware was  gyrolok  protected from the high pressures by a  leak valve and the two parts were joined at a standard Kovar glass-metal seal.  P u r i f i c a t i o n of the gas was  performed i n the glassware.  Before an experiment the e n t i r e gas handling system was 4  many times with research grade  He and then pumped to < 1 0  flushed  -3  torr.  The  3  He gas shipped from Monsanto was so i t was  not s u f f i c i e n t l y pure to use  directly  cleaned by s t o r i n g i t i n a 4 - l i t r e pyrex bulb l i n e d with a  permanent getter.  The getter was  made i n the glass blowing shop by (33)  slowly evaporating a t h i n surface of misch-metal bulb i n an argon atmosphere. with oxygen and was admitted  on the i n s i d e of the  The misch-metal surface i s highly r e a c t i v e  an e f f e c t i v e trap.  Gas from the storage tank was  to a d d i t i o n a l p u r i f i c a t i o n stages i n the gas l i n e as shown i n  F i g . 3 . 1 1 (A charcoal adsorption trap cooled to 7 7 K was  used i n  conjunction with the getter to take out nitrogen and r e s i d u a l heavier contaminants.) The clean gas was  then passed through the leak valve at  a slow rate i n t o the metal tubes and c o l l e c t e d i n the sample chamber, which had been cooled previously to 1 . 2 K.  Enough l i q u i d 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  through the leak valve and returning i t to the storage tank;  He gas 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 l i t r e 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 n i t r i c acid, acetone and d i s t i l l e d 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 f i n a l  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 al^^  to be a weakly relaxing surface for  He spins.  Timsit et  have conducted an extensive study of wall relaxation for ^He 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  probability of 3 He diffusing into the glass is very low (activation energy  51  for diffusion is ^ 6400 cal/mole whereas adsorption energy i s 230 cal/mole) and the relaxation rate becomes negligible.  3.9  Methods of Suppressing the Wall Relaxation By far the most d i f f i c u l t 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 T 1  The gas sample i t s e l f 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 i n  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 i t s e l f 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  created, whose surface was extremely weak in relaxing  3  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 also be suitable.  4 Below 4.2 K i t may be possible to coat with He sxnce  i t is preferentially.adsorbed owing to i t s 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 again became a problem.  gmcm  at which wall relaxation  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 i s slow (compared to the motion i n the bulk), the spectral density of the fluctuating surface fields w i l l have a cut off frequency 1/T which may be comparable to the frequencies a) c  o  used i n the experiment.  o  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 i s weaker.  The spectrum of fluctuating  fields i n the bulk gas cuts off at a much larger frequency than CO as q  discussed i n the introduction so the bulk gas relaxation i s effectively independent of frequency over the range accessible i n this work. In this way relaxation curves of the very long relaxation times characteristic of the low density gas could be measured, although i n 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 T  1  IB  and T_ . 1W rT  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 f i e l d (1.0  kG  ) and then the f i e l d was increased to a higher f i e l d (a 9.3 kG  f i e l d 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) where MJJ and respectively.  =  MJJ + ( ^ - i y e "  t / T  l  ,  are the equilibrium magnetizations at high and low fields,  54  Then the magnet was returned to the measuring f i e l d , the signal monitored and finally cycled back to the relaxing f i e l d .  The entire sequence was  repeated with different resting times at 9.3 kG .  Since the  measurement i s made in a time much less than T^, there i s 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 JLB  below 4.2 K .  measurements  in the interesting range of temperature  55  CHAPTER 4  PRESENTATION OF RESULTS AM) DISCUSSION  4,1  Introduction In t h i s section a systematic study of T^ i n the gas at  temperatures i n the range 1.7  K to 19.0 K and at varying density i s reported.  Each observation of T^ contains information on the bulk gas r e l a x a t i o n and on wall r e l a x a t i o n and the two components can be separated t h e i r density dependence. equation 1.6  T  l  The constants  on the basis of  Denoting the observed 'rate by T^~\  then  gives  1  =  C  l  p  +  C  2  / p  C^ and C^ describing bulk gas and w a l l r e l a x a t i o n , r e s p e c t i v e l y , -1  are obtained by f i n d i n g the slope and y-intercept of a p l o t of pT^  2 vs p .  The phenomenological expression above i s v a l i d only under c e r t a i n conditions and can c l e a r l y lead to systematic errors i f other mechanisms dependent on density are present.  V a l i d i t y i s ensured for the bulk  gas  r e l a x a t i o n i n the d i l u t e gas l i m i t of binary c o l l i s i o n s , but there may l i m i t a t i o n s from w a l l r e l a x a t i o n . the surface area may  be  For instance, at extremely low d e n s i t i e s  be f r a c t i o n a l l y covered, with the coverage increasing  with density. There i s empirical j u s t i f i c a t i o n for using  eqn. 1.6  i n the density  range observed and f o r the kinds of surfaces prepared i n 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  .  seconds to 10  Comparing these values with the estimate for relaxation in  the magnetic field gradients of the magnet, i t i s 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 T  1  and the f i e l d dependence of T..  were taken  Lvl  Lo  only on neon coated surfaces. This section f i r s t discusses the general sources of error i n 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 d i f f i c u l t  to estimate a baseline over the slow d r i f t ,  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 a f f e c t r e s u l t s 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 f a r away from resonance. As previously mentioned, the adiabatic conditions, equation 3.1 were e a s i l y s a t i s f i e d .  An r f f i e l d strength of  = 60 mG was  sufficient  to invert the magnetization, but create no serious heating at the sample. The s i g n a l could, i n p r i n c i p l e , be observed with a wide range of sweep rates which s a t i s f i e d 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 , the. response time of the chart pen ( x ^0.3 s e c ) . p  p  The spins were brought into equilibrium at a f i e l d H s l i g h t l y above resonance and then swept to a resting f i e l d approximately the same distance below resonance. complete.  Since H-H  Q  - 25 Gauss, inversion of the magnetization was  There was no saturation of the l i n e at this distance from 2  resonance since AM /M - (H /AH) . z o 1 n  A small correction was necessary,  however, to account f o r the fact that the spins were relaxing i n a r e s t i n g f i e l d d i f f e r e n t from the i n i t i a l p o l a r i z i n g  field.  Measurements at temperatures greater than 4.2 K were made by the nondestructive sampling technique at low f i e l d s . the  Although the p u r i t y of  system was very high, there was a small component of w a l l relaxation  which could not be ignored, p a r t i c u l a r l y at the high temperatures where the wall and bulk components were nearly equal i n strength.  From equation 4.1  i t i s obvious that the c r i t e r i o n f o r judging whether bulk gas r e l a x a t i o n was observed during experiments was  that T^ ^ scaled with density. -1  rigorous check was also made by p l o t t i n g pT  1  2  vs p .)  (A more  1.0  8  tn i o tn  r  i  1  T  120  140  0.3-  1> W  0.1  20  Fig. 4.1  40  60  80  time (min )  Relaxation curve for typical low field T _2 T = 4.2 K  100  p = 1.7 x 10  gm/cc.  1  measurement  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 i n 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 magnetization 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 r f f i e l d was weak as described i n appendix B .  Values of T^, however, were not  obtained from the graph but by f i t t i n g 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/a where a i s the error determined from 2  the  experimental trace.  If the estimated error was reasonable, then o~  was in close agreement with the curve f i t t i n g 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 f i e l d , o 8.5 K,  A  6.5 K.  i  £  0.3 + I  I/)  0.1 30  Fig. 4.3  60  90  time (min)  <-Trr.-fr.al hi eh f i e l d T, measurement. Relaxation curve for typical nign n e x u ± ±  T = 2.1 K  p = 0.4 x 10~ gm/cc.  120  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 + CT and then the results 2  at a number of temperatures could be analyzed according to Equation 1.6 in order to obtain Values of  and  and C^.  Plots of pT^ * vs p 'are shown in Fig. 4.2. 2  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 i s limited to very low values and T^ i s so long that pure gas relaxation could not be observed unless measurements were made at high fields.  With this  technique, only the i n i t i a l slope of the relaxation curve was measured. Fig. 4.3 shows a plot of (S(t) - S )/(S U  T  H  L  - S ) vs. t where S„ and S. are u  ri  rl  L  the high f i e l d and low field equilibrium signals respectively, and S(t) i s the signal amplitude at the end of the time spent at high f i e l d .  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 f i e l d signal had to be made from the i n i t i a l signal at low f i e l d .  The signal after long times at high f i e l d  was checked for f u 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 f i e l d . -2 during one of the runs, at 4.2 K and 1.0 x 10 between the two methods.  This was done  -2 gm cm  , with good agreement  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 a l l 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 d i f f i c u l t i e s in the analysis caused by wall relaxation and weak signals.  The f i r s t 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. known that  4.3  (It i s  3 (36 ") He vapour satisfies the Curie law to temperatures of 2.0 K) '  Temperature Dependence of T^g Experimentally determined values are plotted against temperature  in  figure  4.4  for comparison with calculations of T  using the poten-  1 D  1  D  t i a l s 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, f i t s 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 —  •  I"  1  1  1  1  l  1  1  1  >  -  150 >  7.  e  100  -  9 I—  ^  4 +  50  (a)  •  <—,  i  1  • • 5  i  1  •  i  1  •  i  1  10  •  |  i  1  1  I  •  1  15  .  i  1  ,  •  1  i 20  T(K) r—— —i— 1  150  1  1  1  1  1  I  "1  -T  I  4-  A' E U  +  "100 V)  50 -I  (O Fig. 4.4  T(K)  10  Temperature dependence of T^. of T  11]B}  (a) Beck potential using (a)  (c) BM potential.  1  r ~  15  20  Broken curve is a calculation (b) MS potential and  65  temperature.  The  least value of the sum  of squares was  obtained with the  BM p o t e n t i a l . It i s clear that relaxation v i a the dipolar coupling alone i s s u f f i c i e n t to explain the relaxation.  An upper l i m i t can be placed on the r e l a x a t i o n  rate for r e l a x a t i o n v i a the spin r o t a t i o n i n t e r a c t i o n by considering  the  agreement with experiment at high temperature where' the spin r o t a t i o n cross section i s largest.  The empirically determined relaxation rate f o r bulk  gas relaxation can be expressed as a sum (T  1 S R  of dipolar (T  )and spin r o t a t i o n  ) components  T  At 19.0  1B  _ 1  K and 10  =  -2  T  1 D "  -3 gmcm  1  +  T  1 S R  T,_  -1  IB  _  < - >  1  = 6.17  4  ± 0.2  x 10  -5  sec  -1  2  , so the upper  l i m i t of the spin r o t a t i o n relaxation rate i s roughly the experimental uncertainty  0.2  x 10  sec  There are no simple  \ p r e s c r i p t i o n s f o r d i r e c t inversion of  property data to obtain parameters of the interatomic p o t e n t i a l .  transport Having  chosen a p a r t i c u l a r form which s a t i s f i e s the data as w e l l as the potentials used, i t i s possible to test the s e n s i t i v i t y of T._  to the p o t e n t i a l 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  f i t s with a l l three potentials are remarkably good. to see how  s e n s i t i v e T,  the  But i t i s useful  i s to the p o t e n t i a l and i f possible determine  IB  which p o t e n t i a l and what value of the well depth.gives the best f i t . r e s u l t s of the f i t s with adjusted BM and MS  The  potentials are shown i n table  4.1. Variations i n the potentials were achieved by first.determining  the  I  1  2  4  1  1  6  1  I  8  10  I  1  I  1  I  1  I  12  14  16  18  T ( K )  Fig. 4.5  Temperature dependence of T._.. ' Broken curve i s a calculation of T^  IB  B  using best parameters of BM potential.  23  s e n s i t i v i t y of the w e l l depth to each parameter and then varying the most s e n s i t i v e parameter to obtain a desired depth.  The depth could  be changed by varying parameters i n either the short range or the long range part;  only the short range part of the BM p o t e n t i a l was varied,  whereas both parts of the MS p o t e n t i a l were varied i n turn. No errors are quoted f o r the parameters i n the BM p o t e n t i a l ; the authors quote only the accuracy to which the helium transport property data i s f i t t e d by the p o t e n t i a l below 50K and i t i s not known how s e n s i t i v e the data i s to the parameters.  This p o t e n t i a l was  tested with s l i g h t l y deeper wells i n order to improve the f i t to the NMR T  1  data below 3K.  The best f i t ( F i g . 4.5) was obtained with a well  depth of 11.5K which was achieved by a 7% v a r i a t i o n of the parameter e (eqn. 2.5). The NMR data i s reproduced to within 3%, but i t i s not known how w e l l the new adjusted BM p o t e n t i a l f i t s the other data o r i g i n a l l y used by Bruch and McGee. Results are given i n Table 4.1 f o r calucations of T^g using the MS p o t e n t i a l and v a r i a t i o n s obtained by adjusting the a t t r a c t i v e part to achieve w e l l depths between 9.3 K and 14.5 K.  The best f i t i s obtained  for w e l l depths s l i g h t l y more shallow than the quoted value, but the high temperature r e s u l t s are consistently underestimated.  An attempt  was made to f i t the data by making the p o t e n t i a l less repulsive, but the high temperature r e s u l t s were again f a r too low. A two parameter Lennard-Jones model i s capable of f i t t i n g the r e s u l t s over a r e s t r i c t e d 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  115  above 4.2 K .  A better f i t to the high temperature data can be obtained  only with values of w e l l depth e or range a which are greatly out of l i n e with the accepted values.  68  Table 4.1 Potential  Adjusted Parameters  Well depth  Sum of Squared Deviations 2  Beck  none  10.3 K  MS  none  12.0 K  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  BM  £ = 1.563x10  BM BM  erg  42.0 '  94.0  198  11.2 K  11.7  £ = 1.063xl0~ erg  11.5 K  10.6  £ = 1.643xl0~ erg  11.8 K  11.5  15  15  (sec.gm/cc)  69  The  NMR d a t a f a v o u r s  the BM p o t e n t i a l w i t h a p o t e n t i a l o f t h a t  form h a v i n g a s l i g h t l y deeper a t t r a c t i v e w e l l p r o v i d i n g  Perhaps the most i n t e r e s t i n g f e a t u r e is  o f the temperature dependence  the minimum i n t h e neighbourhood o f 1.0 K.  obtained  the best r e s u l t s .  A l t h o u g h d a t a was n o t  a t low enough temperatures t o pass through t h e minimum, i t i s  i n t e r e s t i n g to d i s c u s s The  the e f f e c t .  s o u r c e i s i n p a r t due t o t h e c r o s s  calculation.  (A T^ minimum would n a t u r a l l y r e s u l t from the average o v e r  energy i n eqn.  2.1.)  A s i m p l e p h y s i c a l p i c t u r e o f the c r o s s described  s e c t i o n maximum i n S h i z g a l ' s  as f o l l o w s .  Shizgal's  p a r t i a l wave c o n t r i b u t e s  c a l c u l a t i o n s show t h a t o n l y  t o a ( E ) a t low e n e r g i e s .  the e f f e c t i v e p o t e n t i a l i s Y Q ( r ) L = 1 wave.  s e c t i o n maximum can be  plus  the L = 1  F o r p-wave s c a t t e r i n g  the c e n t r i f u g a l b a r r i e r f o r the  A l t h o u g h i t i s tempting t o d e s c r i b e  the maximum as a p-wave  resonance, S h i z g a l p o i n t s out t h a t t h e L = 1 phase s h i f t does n o t e q u a l 12 a t the energy where the maximum o c c u r s , must be f a r more a t t r a c t i v e b e f o r e  and i n f a c t  a resonance appears.  the p o t e n t i a l (The He-He  p o t e n t i a l s u p p o r t s n e i t h e r a bound s t a t e n o r a m e t a s t a b l e s t a t e c a p a b l e o f causing  a resonance.)  Q u a l i t a t i v e l y , the maximum o c c u r s because the wave  f u n c t i o n i s concentrated part of V ( r ) . Q  i n t h e r e g i o n o f t h e w e l l owing t o t h e a t t r a c t i v e  The c o l l i s i o n  time i s e f f e c t i v e l y i n c r e a s e d ,  a greater p r o b a b i l i t y of a nuclear Eventually fugal  at s u f f i c i e n t l y  low e n e r g i e s  b a r r i e r and the s p i n f l i p  the d i s t a n c e  spin f l i p  during  l e a d i n g to  the c o l l i s i o n .  the atom i s s c a t t e r e d o f f the c e n t r i -  t r a n s i t i o n p r o b a b i l i t y i s reduced  o f c l o s e s t approach o f the p a i r i s l a r g e r .  since  The T^ minimum  ( r e l a x a t i o n r a t e maximum) o c c u r s as the temperature i s v a r i e d through t h e  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,„ ID  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 i n part 2.4 , as shown i n 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 f i n a l runs i n 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 i s of order d, the atomic diameter, T^ varies approximately as p 3 (34) This would be the case i n liquid He and i s expected i n a regime where (37)  many body collisions are important -2 (p < 5.0 x 10  . At moderate densities  -3 gmcm  ) a region of bulk gas relaxation i s observed, the  extent of which depends on the purity of the system. with (density)  The results scale  i n 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 i s entered and the curve passes through a maximum. The values of pT^ for the uncoated pyrex and the neon surface are  1  I  0.4  I I 1t I 0.6  0.8  1.0  ^ Fig. 4 . 6  I  1 2.0  I 4.0  (10~ gm.cm.- ) 2  3  Density Dependence of  Curve 1:  ^ pyrex walls 1st run in cleaning series  Curve 2:  • pyrex walls f i n a l run in cleaning series  Curve 3:  o neon coated walls  l i l t  1 8.0  8.0  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 i n using the phenomenological theory to analyze the data.  The properties of the two surfaces are  distinctly different and i t i s not clear what effect the substrate w i l l have on surface relaxation.  It i s expected that the effect of a neon  surface w i l l be less drastic since i t has a weaker van der Waals attraction and i t i s 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 i n estimating wall relaxation on a pyrex surface w i l l be more strongly f e l t .  Since the neon value i s larger and was  obtained under conditions of greater purity pT.. (neon) was selected as the best experimental measure of T-,..  IB  IB  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. For annealed neon walls a 4.5  =  10  -9  , but for bare pyrex a ~ 10  -8  An Estimate of the Effects of 3 Body Collisions In the region of where bulk gas behaviour i s observed there may  be effects at higher densities owing to three body collisions.  The data  £o 1.7 8  V  1.6  5  I  \ 1.3 1.2  h  2  3 4 5 /> (10 gm.cm-5) -2  Fig. 4.7  Graphical determination of coefficient  74  plotted i n 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.  V'obs  We can write for the observed relaxation rate  =  C  l  p  +  C  3  p 2  +  C  2  / p  by adding to eqn. 1 the term C^p which accounts for three body collisions. 2  This equation can be rearranged to obtain  \ P  E  p / T  The constant  l obs " 2 C  "  ]  l  C  +  3  P  can be estimated by considering the gas to be dilute i n  f i r s t approximation.  In Fig. 4.7  a plot of the — E /T  is independent of p, suggesting that  4.6  C  P  0  p  z  1  ,  I obs  - C_H vs p Z  i s negligibly small.  Results of Wall Relaxation Measurements at 1.0 K Gauss Wall relaxation was studied  in sample chambers satisfying both  limits of cleanness described i n part 2.4.  The time evolution of the  magnetization of the sample was shown to be a sum of modes of the diffusion equation  M(t) °= Z y v  A.  -t/x e  v  (eqn. 2.18).  v  Strongly Relaxing Walls In the limit of strongly relaxing walls, UR>>1, the time constant  of the v*"* mode in a spherical geometry i s 1  T  =  R /_  o  2  75  76  The higher modes are very quickly damped with respect to the f i r s t mode. -4 In addition, since the ratio of the amplitudes i s v  , i t may be possible  to neglect a l l but the f i r s t 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 i s a calculation based on the theory with R = 0.4 cm and D taken from data of Luszczynski et a l ^ ^ . It i s evident from the magnitude of T ^ that the bulk gas contribution to relaxation can be neglected i n this case. B.  Weakly Disorienting Walls For weakly disorienting surfaces, pR«l, only the f i r s t mode  contributes to the relaxation and  T-__ IW  =  RanT... , —j- lAd  . Wall relaxation  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 i n 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). Using p = 1.0 x 10 T.. ^ 10 1 Ad of T^  —3  -2  gm cm  -3  , m = 5 x 10  -24  (38} sec -1.0 sec ^ % we obtain  -16 2 (38) a = 15 x 10 cm , T 6 IW 1 10 > ~ — > 10 . The values p 3 gm,  ^ 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  i s very large. The indication is that T,„ 1W 1W i s approximately constant over the entire range (Table 4.2). 1TT  Table 4.2  Temperature (°K)  T 1 W  _2 (p=1.0xl0 gm/cc) (sec.)  Field (kG-)  4.2  (1.5 ± 3) x 10  1.0  6.5  (1.2 ± .1) x 10  4  1.0  8.5  (1.3 ± .1) x 10  4  1.0  12.0  (1.0 ± .1) x 10  4  1.0  15.0  (1.5 ± .2) x 10  4  1.0  19.0  (1.7 ± .3) x 10  4  1.0  4  78  CHAPTER  5  F i e l d dependence of the wall r e l a x a t i o n 5.1  Introduction Measurements of T^ at varying f i e l d were made i n the l i m i t of  very clean walls to investigate the empirical observation relaxation was much weaker at high f i e l d s .  that w a l l  The e f f o r t i s only a  preliminary study and the r e s u l t s are interpreted i n the framework 3  of a theory which i d e n t i f i e s the r o l e of relaxation of He atoms adsorbed on the surface.  The r e l a t i o n s h i p between T ^ and T ^ ^ , the  relaxation i n the adsorbed phase, developed previously i n section 2, i s used to evaluate the wall r e l a x a t i o n time i n 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 c a r e f u l l y deposited on vycor glass, graphite and zeolite.  Nuclear spin relaxation times are s e n s i t i v e to the l o c a l  field  fluctuations on the surface which have c o r r e l a t i o n time T , and measurements of T^ and  furnish information about the motion of atoms i n the (39) adsorbed f i l m . Although some Russian authors have calculated the 3 NMR lineshape f o r He monolayers, no attempt has been made to formulate a theory of r e l a x a t i o n i n the adsorbed layer, other than to suggest that the relaxation rate i s governed by the i n t e r a c t i o n with surface paramagnetic centres and i s proportional to x / (l+u) x ) , where x i s the c o r r e l a t i o n 2  P  2  P  time f o r the i n t e r a c t i o n between adsorbed surface^ ^. 4  3  P  He and a paramagnetic s i t e on the  V- 40H  (a)  3  3  4 5 H (KGauss)  4 5 H (KGauss)  Fig. 5.1 Field dependence of T A  2.6 K,  top figure  •  4.2 K, o 8.0 K  p = 1.6 x 10~ gm/cc.  bottom figure  2  p = 0.6 x 10~ gm/ 2  80  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 w i l l be mentioned now. spent i n the adsorbed layer,  The time  i s independent of the magnetic f i e l d  but w i l l be characteristic of the substrate since the binding i s via the van der Waals interaction, atoms on the surface.  w i l l also be different for each layer of  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 i n the limit of small jumps or jumps very large compared to atomic dimensions.  This motion i s independent of the magnetic f i e l d  but i s dependent on temperature, at least at the high temperatures observed in this work.  Relaxation times of paramagnetic spins, T^ , are sensitive g  to the magnetic f i e l d and can influence relaxation since the time varying electronic moment creates a changing magnetic f i e l d at the nucleus of a 3  He atom. Measurements of the f i e l d 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 a l . and indicates that T J ^ J i s 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 i s d i f f i c u l t to control the nature of the adsorbed phase, and at best we can say that in equilibrium a multilayer of  3 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 f i r s t layer much more strongly bound than successive layers. (The binding energy i s -20  for the  (41) f i r s t 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 i s much more probable than exchange with other layers. In the f i r s t approximation we shall assume that the neon coat i s 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. 3  The  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 i s calculated in Appendix A.  The temperature dependence i s implicit in the  diffusion coefficient and no attempt i s 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 i s not  restricted to systems in which the surface to volume ratio i s very large in order to obtain strong signals. The background relaxation rate of the bulk gas in carefully prepared systems i s very small so the technique can be directly sensitive to  82  100  1  I I I I II  1  1  1 1 I III I  '  i  10 •o a* cn  LO i  o  J  ' i i i '  1 '  1 H  Fig. 5.2  2.6 K,  0  4.2 K,  5  t  i  10  (KGauss)  Field dependence of wall relaxation rate, A  i i i i  o  8.0  K,  p = 0.6 x 10  gm/cc.  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.. continued  lis  to increase with field and reached T  at the highest field  X  lo  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 s t a t i s t i c a l l y 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. T^  1  vs. H is shown in Fig. 5.2.  At 4.2K and 8.OK,  but not 2.6K,  data i s consistent with the predicted field dependence of H f i e l d (Eqn.  A12).  A plot of the  at large  Fig. 5.3  Plots showing calculated f i t s to wall relaxation rate at 8.0 K and 0.6 gm/cc. (a) T = 2 x 10" sec (b) x = 2.0 x 10" sec (c) T = 2.0 x 10 sec. 8  -6  7  85  Using the theory developed in the f i r s t 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 u where  AA  _ 1 l w  =  A (j (o)i) + 3J (OJT) + 4J (2OJT) + 12J (2OJX) o  2  o  2  T = —3TT -Y V I ( I + 1 )-— • 20 nRd^a  ( 5 , 1 )  1  2  It i s possible to estimate a value of the correlation time x and the 3 effective separation d of  He atoms in the f i r s t layer by f i t t i n g 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 i s just d . A number of f i t s 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 i s constant to within  86  experimental error and of magnitude such that t o r - 1 for a f i e l d of 1 kG. The smaller value of T predicted at the lowest temperature i s 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 i s 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 i n 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 i n eqn. A.18 by the following expression -1 T  L W  3ir Y Y h S(S+l)T = ^--^-2 (J'(COT) + 12J|(COT) + 5 nRZ a a„ o p He 2  I  2  2  s  3j'(coT))  4  where  (5.2)  and o"^ are the surface areas per adsorbed paramagnetic and helium e  atom respectively.  Since the spectral densities i n each model are of the  same order of magnitude, i t i s possible to compare the calculations and estimate the surface concentration of paramagnetic sites necessary to account for the observed relaxation.  The ratio of T, ^ for the two models i s IW TT  approximately T.„  - 1  (model 1)  -^-T  T  2  = _ i l w  (model 2)  y I(I+1) a Z * — £ g S(S+l) o^d"  2  -  (5.3)  2  Y  Using a reasonable estimate of Z  q  - 8 x 10  cm, eqn. 5.3 indicates that the  concentration of paramagnetic sites i s 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 (OK)  T (sec. )  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  a (cm ) 2  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 f i e l d  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 f i r s t layer in which mobility of the adsorbed atoms increases with temperature.  89  5.3  Measurement of F i e l d Dependence by Optical Pumping Techniques 3 F i e l d dependence of the r e l a x a t i o n i n very d i l u t e He gas was (21)  also studied by o p t i c a l pumping techniques by Barbe et a l .  . Those  authors obtained very long relaxation times at 4.2 K by coating t h e i r sample chambers at low temperatures as suggested i n section 3.9.  They used  a v a r i e t y of gases including Ne, A, Kr, Xe, and U^, but obtained s a t i s f a c t o r y r e s u l t s only with H^.  At the low pressures c h a r a c t e r i s t i c of  3  t h e i r experiments ( He gas at 1 t o r r at room temp.) the r e l a x a t i o n i n the 3  bulk gas was l i m i t e d by d i f f u s i o n of the f i e l d gradients over the sample.  He atoms through weak magnetic  The r e l a x a t i o n time i s proportional to  2 H  f o r t h e i r experimental conditions: wx  time of  3 He atoms across the c e l l .  >> 1 where x i s the d i f f u s i o n o o - 1 2 A plot of T^ vs. H was l i n e a r and  extrapolated at H = 0 to a w a l l r e l a x a t i o n time of about 60 hours. As suggested e a r l i e r , the bulk gas relaxation observed i n our NMR experiment i s not limited by relaxation i n magnetic f i e l d gradients, but i s a d i r e c t measure of the dipole-dipole relaxation.  I t i s i n t e r e s t i n g , however,  to compare the values of T ^ obtained i n our work and the experiment of reference 20  .  In both cases, the gas pressure i s high enough  that the d i f f u s i o n l i m i t i s s a t i s f i e d  f o r atoms i n the c e l l , so we  expect the relaxation time to scale with bulk gas density according to eqn. 2.23. T / n (NMR) lw  cc.  Comparing the r a t i o T ^ / n f o r the two experiments we obtain =  5 x 10~  19  sec/gm/cc and T ^ / n (Barbe)  =  6 x 10~  sec/gm/  12  Since the r e s u l t s do not scale, we must look f o r the source of  discrepancy. 3 A possible explanation i s that owing to the low pressures of  He gas  the  surface i n the o p t i c a l pumping experiment i s not f u l l y covered, and  the  relaxation time i n the p a r t i a l l y f i l l e d adsorbed layer i s longer than  90  i n a f u l l y covered surface.  A rough idea of the f r a c t i o n a l  coverage can be obtained from a simple model which considers s i t e s 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 s i t e can be calculated  and expressed i n terms of the bulk gas pressure.  (The r e l a t i o n 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 - -20 K , the f r a c t i o n a l coverage i s 5 x 10 . I f we now take the 15 w  r a t i o T^/flaR f o r each experiment, we obtain the r a t i o of adsorbed relaxation  layer  times:  ( o p t i c a l pumping)/T  T lAd  lAd  (NMR)  -  Although the model i s crude, the r e s u l t i s not unreasonable account f o r the observed  results.  and could  91  CHAPTER  6  SUGGESTIONS FOR FUTURE EXPERIMENTS  The r e s u l t s obtained i n this work for the temperature dependence of T^  B  (covering a range of 1.7 K to 19.0 K) i n d i c a t e that dipolar r e l a x a t i o n 3  i s the dominant i n t r i n s i c mechanism i n bulk  He gas.  The data can be f i t t e d  adequately by c a l c u l a t i o n s using the accepted helium p o t e n t i a l s , with the Bruch-McGee p o t e n t i a l providing the best f i t .  This p o t e n t i a l was i n fact  derived from analysis of low temperature data on other transport properties and gives a w e l l depth of e = 10.8 K.  The present r e s u l t s , however,  suggest a s l i g h t l y deeper a t t r a c t i v e w e l l , e = 11.5 ± 0.5 K.  The NMR  experiment should give r e l i a b l e information on the depth and nature of the w e l l since the measurements probe the He-He scattering at very low energies where the e f f e c t of the a t t r a c t i v e forces should be strong. This experiment was unable to probe the region below 1.7 K where a T^g minimum i s predicted. conditions to measure  I t should be possible to prepare s u f f i c i e n t l y pure by the AFP techniques described here at  temperatures as low as 1.0 K, with improved design of the spectrometer. S e n s i t i v i t y could be improved by immersing the e n t i r e bridge i n the l i q u i d helium bath, as suggested i n 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 r a t i o as w e l l since the most suspect source of microphonics, be eliminated.  namely the long coax leading from the c o i l , would  To make measurements  below 1.0 K i t i s l i 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 i t 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 a  P  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 i t would be necessary to use liquid the spin dynamics.  He in the study of  93  APPENDIX A Calculation of Relaxation Time i n an Adsorbed Layer. 3  Suppose that N i s at an angle  ®  He atoms d i f f u s e on a plane of area A whose normal to the f i e l d H, and i n t e r a c t with each other v i a the  dipolar coupling.  The vector r = (r,8,<J>) between a pair of atoms i s  made time dependent by the d i f f u s i v e motion i n two dimensions.  We wish  to c a l c u l a t e the c o r r e l a t i o n functions G (t)  =  H(  2m  _2M  oo r  X  \  r(t)  3 Q  3  A  /  assuming that a l l pairs are equivalent. We f i r s t make a coordinate transformation so that the new z'-axis i s perpendicular to the surface. expressed  The s p h e r i c a l harmonics can then be  i n terms of the new coordinates by  Y  "  2m *> ( 9  I  D  Xm  ( 2 >  (  °> >° e>  )  2\<%>**>  Y  A  '  2  A  and we note i n t h i s coordinate system  Y  2 Q  =  -  Y  D^^ ^ 2  ^7l6  2 ± 2  -  -  /T  ^15732  Y  e  ± 2 1  *'  2 1  =  =  /T2  0  e  (0,9,0) i s generalized s p h e r i c a l function of 2  2 i  C D  ^'  order  In order to perform the average i n A . l we solve the 2-dimensional d i f f u s i o n equation f o r the p r o b a b i l i t y PCr^.r.t) which gives the chance of f i n d i n g a pair of atoms separated by r at time t given that the  1  94  separation was i n i t i a l l y r .  P(r  -  ,  .  5 i t )  o  o  _ i _ e-(-r ) /8Dt 2  0  °  with P(f ,r,0)  P(r ,r,t) i s given by  A  >  3  8rrDt  <5 (r-r ) , where D i s the diffusion coefficient of He  =  atoms on the surface. The correlation function becomes  ,  f " I'" v 2  G (t) = ^ J rdr Jr d r j _ d * J d * I m ' ' A d d o o , Q  u  n  o  u  U  (2)  D  n  Am  A l u  (2)  D _ r  "  A  A A  Y  2A( o*o)  Y2X,<e*)  Q  n  r  m  3 Q  r  3  -(?-r ) /8Dt 2  Q  x  A.4  8TrDt  N  where  /A i s 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 i s uniform for r > d. -4  M  /—  To solve this integral we replace e dimensional fourier transform <f ( ? - r  2 o)  /8Dt  = M TT  t  2n  j  -  j .}  . j, kdk  H  O  /3T){*  °  .CO  d  7  e  2 -2Dtk  by i t s two-ikXr-r )  e  ° n  .  A. 5  _  O  ik. r and make use of the expansion for e  e  -  I i V * ^  J (kr). A  A.6  £ = -<*>  where J^(kr) i s a Bessel function of the f i r s t kind andty-tyi s the angle -ik.r between k and r. There i s an analogous expression for e * °.  95  Substituting A. 6 into A. 5 and integrating over ij; we obtain oo  -(r-r r/8Dt n  V  kdk  = 2Dt  J.  o  e  1=  2 -2Dtk e  -i£(d>-(b 0  )  ,  „  V^r)  -°°  .  J,  Jt  .(kO Ji  x  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  -2Dtk V 2  rdr  o  o  d  u  Am  A  kdk e  -A-m  k  r  A=0±2  >  r  V  k  r  r o  3  3  A.8  To do the integrals over k and r we make the transformation  x  m< >  G  where  C  = r/d  =  ~ A Ad  4  and  I  A=0±2  g ( /d ) Dt  y = kd . Then  D<  A x X  X  ydy e  2  x  2)  m  D J> (  A.9  ( /d2) Dt  - " A  g> m  2 ,,2 / jr» J (xy) -2Dty /d  (J  x  x  2  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 (* i»2(«i ..< /* )**2<i 2<«i i»2<-)i ) Ad )  2  Dt  D  2  )  2+  )  2  0  x  g ( /d )) 2  A. 10  o>  t  96  -2  Q (NJ  -3  cn CD O  -S  0  -6 -2  ft  -I  LOG  Dt/d  LOG  Dt/d  2  -i  rg "U O CD  -2  -3  CD  o  -5  -6 -2  -1 2  Fig. A l Numerical calculations of g A ( ^ y ) vs. between helium atoms on the surface.  ,Dt, for dipolar coupling  (-^y)  97  In dealing with a spherical sample as was used i n our experiments the function in eqn. 10 should be averaged over a l l angles;  Ojt) m  =  [ g( /d2) °  3 g ( /d2) ]  Dt  16Ad*  Dt  +  2  the result i s  A.11  Z  Dt o The Fourier transform of g, ( /d ) i s readily obtained and we define the dimensionless transform  JiCwt) A.12 3  oo  y  d  - J (xy)  v 2  x  y  i 2 2 o y +o) T  ~  1  1  T  -  d  x  )  X  where x = d /2D . Results for the numerical integrations in eqn. 12 The spectral densities J (u) follow from the  are plotted in Fig. A.2. •CO  definition J (OJ) = m  e  la)t  G(t) dt , and the spherically averaged functions  J _co  are  J  m  N  (to)  2~ 32Ad D  (3 (w' ) T  0  +  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  T  = lAd  — 5  yV  1(1+1) [J.(w) + 4 J (2u) ]  A. 14  98 0  -i  -2  -3  CD O -4  0  -I  ]  LOG  wt  CD O  -6 -2  -1  0  LOG F i g . A2  Numerical  I  wt  c a l c u l a t i o n s o f j A ( u , t ) v s . wt f o r d i p o l a r  between h e l i u m atoms on the  surface.  coupling  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 Z  o  below the surface.  /A at a fixed distance  In our experiment, Z may be taken as the thickness o  of the neon coat. The calculation proceeds in the same way as in the previous case except that r i s now the distance between the paramagnetic spin and a 3  He atom.  The correlation functions /  G (t) =  N/ P  V  Y  ? J  Q  2 m  r  3  *  J J  9m  ° °  Vt)<Kt) \  r(t)  o  ) '  3  A.15  become under a rotation of the coordinate system •CO  Oft) "  J  A  rdr o  |-2TT  (•  r dr d<j> d* I D. • o Am o o J o XX» o o  ( 2 )  L  Y  x  2X 2  DJ  2 )  2  X'm  _  2 ' ) o '  o  °  d  3 /  /  " " o> / ( f  e  T,  °  (r +Z *) o o  (9,<)>,)  (r +Z  X  ?  2  4 D t  A. 16  2  4TrDt  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) m  =  - ^ J 8AZ o  [ g l ( / Z ) + 12 . ( / Z ) + 3 g * ( / Z ) ] A. 17 D t  2  0  Dt  g  2  Q  Dt  2  o  100  -11  -i  -2  1  <N1  a  cn CD O -4  -4  -5  -5 1  -6 -2  .,,  -I  0  .. .  I  -t  LOG D t / d  a  LOG D t / d  2  -2  Q  cn CD O  -s  -2  3  LOG D t / d Fig. A3  2  Numerical calculations of g*A(Dt/Z ) vs. (Dt/Z ) for dipolar o o coupling between a helium atom and a paramagnetic surface site. 2  2  2  101  -I  -2  CSJ  -3  CD O  —i  -i  o  i  0  2  ft  -6  3  -  2  -  LOG wt  1  1  0  LOG wt  -i  -2  O  -3  CD O  -6 -2  F i g . A4  Numerical c a l c u l a t i o n s  -I  0  1  LOG  of j ' A ( u ) t )  WT  v s . cut f o r d i p o l a r  between a h e l i u m atom and a paramagnetic  surface  site.  coupling  -«  «•  102  g^C^Vz ^ ) plotted in Fig. A.3 are obtained numerically  where the functions  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 i n Fig. A  A. 4. We can at this point account for the influence of the paramagnetic spin —t /T relaxation time T  by multiplying G (t) by the factor e  l e  le which describes  the correlation between the spin of the ion at times separated by t, assuming that there i s no correlation between the diffusive motion of the atoms and the ion relaxation. 2 2 /Z + /T, ) for /Z i n o le o the exponent of the fourier transformed probability density and the net result i n the spectral density i s a shift of the lower limit of the Z This amounts to substituting (  2 D y  2  +  2 D y  t  2  2  integration over k from zero to / ^e with x = T  j'^(cox).  ° /2D i n the functions  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 . The spherically averaged spectral densities J (a)) are given below; m q  J  ( m  w  )  =  N  P  8AZ  9  o  D  (J  +  1  2  j/(wx) + 3  j ' f a T ) ) Z  A.18  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 i s a good approximation to assume that the paramagnetic spin system i s i n equilibrium with i t s surroundings, in which case eqn. 87 in Chapter 8 of Abragam gives the result  1 0 3  f ~  =  Y  iV  h 4 s(s+1)  C  l l  J  ( 0  »r s u  ) +  2W  +  !  J 2  K V  lAd A. 19 f o r the r e l a x a t i o n r a t e . J ,  and J  J  «  Q  »  satisfied  S i n c e a> >> u) c  T  i t follows  t h a t when a i « 0  1  a r e o f comparable  magnitude b u t t h a t when CO,T >> 1 then  J, and may be dropped  i n A.19. T h i s 1a t t e r a p p r o x i m a t i o n i s  2  f o r T > 10 ^ s e c i n a f i e l d  o f 1 KGauss.  104  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 c e l l s .  It  was d i f f i c u l t 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 c o i l .  Relaxation curves for several applied rf voltages are shown  in Fig.Bl. An explanation i s 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 c e l l of flipped and unflipped spins respectively.  Diffusion w i l l create  in time a uniform distribution, but i n i t i a l l y there w i l l 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  Fig. Bl  «  8  t  t  n  N  j  ,6  20  Relaxation curves at 1.0 KGauss, (a) (b)  2.3 volts p-p to rf c o i l 0.7 volts p-p to rf c o i l  24  28  32  4.2 K and 2.5 x 10  36  gm/cc.  106  where  W = e  and  K = YH  /dH  X  /  dt  W is a function having values between 0 and 1. place over a narrow range of H^.  Since  Most of the variation takes  is a function of position in  the c e l l 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 *  adiabatic region F = 1  a < x < a+e  F varies rapidly between 0 and 1  a+e  non-adiabatic region F = 0  * x - L  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 i n i t i a l l y in the adiabatic region which remain there after a time t. can be written  The general solution  107  n  + TT  2  = _co  p=0,±l,-2  exp-[(2riL+pa)  /4Dt)]  2 where <K°0  i s an error function.  In the l i m i t 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 r e l a x a t i o n curve to t = 0. Empirical r e s u l t s of measurements of the recovery at short times are plotted i n F i g . B2, along with the expression f o r P ( t ) . The agreement i s reasonable.  Values of D are taken from reference 15.  These considerations indicate a possible method of measuring the d i f f u s i o n c o e f f i c i e n t D i n systems with very long r e l a x a t i o n times.  By  constructing an appropriate c o i l and sample chamber configuration one could e a s i l y create d i s t i n c t regions of f l i p p e d and unflipped spins. r e s u l t i n g time evolution of the AFP s i g n a l could be used to measure D. E f f e c t s associated with d i f f u s i o n of the spins i n the combined inhomogeneous dc and r f magnetic f i e l d s during a single sweep through resonance have also been considered and found to be n e g l i g i b l e f o r the experimental conditions used here.  The  (sec*)  /f  F i g . B2.  Recovery of magnetization at short times. S o l i d l i n e i s a c a l c u l a t i o n of P(t) 2.0 v o l t s (p-p) applied to c o i l . p = 6.5 x 10 gmcm~ ; _2  a = 0.9 cm  3  T = 4.2 K;  H = l.OkG;  109  Bibliography 1.  M. Bloom and I. Oppenheim (1967). Advances i n Chemical Physics, 12_ 549. Edited by J.O. Hirshfelder (John Wiley & Sons, New York).  2.  J.W. R i e h l , J.L. Kinsey, J.S. Waugh and J.H. Rugheimer (1968). J . Chem. Phys., 49_, 5276.  3.  K. L a l i t a and M. Bloom (1971).  4.  A. Abragam (1961). P r i n c i p l e s of Nuclear Magnetism. University Press, London).  5.  J.O. Hirshfelder, C F . C u r t i s s , R.B. Bird (1954). Molecular Theory of Gases and Liquids. (John Wiley & Sons, Inc., New York).  6.  L.W. Bruch and I . J . McGee (1970).  7.  J.M. Farrar and Y.T. Lee (1972).  8.  D.R. McLaughlin and H.F. Schaefer (1971).  9.  B. Shizgal (1973).  Can. J . Phys., 49_, 1018.  12. N. Bloemberger New York).  J . Chem. Phys., 52, 5884. J . Chem. Phys., 56, 5801. Chem. Phys. L e t t . , 12, 244.  J . Chem. 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