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UBC Theses and Dissertations

Magnetic feedback and quantum oscillations in metals Van Schyndel, André John 1980

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el  MAGNETIC FEEDBACK AND QUANTUM OSCILLATIONS  IN METALS  by  ANDRE JOHN/VAN SCHYNDEL B.Sc,  McMaster  University,  1978  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE  in THE FACULTY OF GRADUATE STUDIES Department o f  We accept t h i s  thesis  to the r e q u i r e d  Physics  as  conforming  standard  THE UNIVERSITY OF BRITISH COLUMBIA October 0  1980  Andre John Van Schyndel ,  1980  )  In p r e s e n t i n g t h i s  thesis  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  an advanced degree at the U n i v e r s i t y of B r i t i s h the L i b r a r y s h a l l I  f u r t h e r agree  make i t  freely available  that permission  for  Columbia,  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 o f  this  that  study. thesis  f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s of  this  representatives. thesis  It  is understood that copying or p u b l i c a t i o n  f o r f i n a n c i a l gain s h a l l  written permission.  Department of The U n i v e r s i t y o f B r i t i s h  Columbia  2075 Wesbrook Place Vancouver, Canada V6T 1W5  Date  Oct.  )Q  ) q % 0  not be allowed without my  i i  A B S T R A C T  A feedback technique is presented f o r the r e d u c t i o n of the Shoenberg magnetic  interaction  a l l o w s the s p i n s p l i t t i n g  parameter g  in m e t a l s . c  The method  f o r extremal o r b i t s on  the Fermi s u r f a c e to be o b t a i n e d from de Haas-van Alphen measurements,  now e s s e n t i a l l y  severe d i s t o r t i o n s  resulting  f r e e from the o f t - t i m e s from magnetic  feedback technique a l s o o f f e r s effects,  the most  several  interaction.  advantageous  important one being a simple and  The  side reliable  method f o r d e t e r m i n i n g a b s o l u t e amplitudes o f de Haas-van Alphen o s c i l l a t i o n s .  E x p l i c i t formulae are d e r i v e d  the dependence of s e v e r a l  showing  key o b s e r v a b l e q u a n t i t i e s  on the  amount of magnetic feedback, and these formulae are found be in good agreement w i t h experiment. a p p l i e d to the d e t e r m i n a t i o n o f g in Pb.  c  The t e c h n i q u e  f o r the [110]  y  to  is  oscillations  TABLE OF CONTENTS page Abstract  ii  Tab 1 e o f Contents  ....  L i s t of Tables  iii v  L i s t of Figures  vi  Acknowledgements  viii  CHAPTER I  -  INTRODUCTION  1  CHAPTER II  -  SPIN SPLITTING OF LANDAU LEVELS  5  CHAPTER III  - THE SHOENBERG MAGNETIC  INTERACTION  EFFECT CHAPTER  CHAPTER  13  IV - REDUCTION OF MAGNETIC INTERACTION USING A FEEDBACK TECHNIQUE  20  V  27  - EXPERIMENTAL  DETAILS  5.1  Sample P r e p a r a t i o n  27  5.2  D e t e c t i o n Apparatus  31  5.3  Modulation C o i l and Superconducting Magnet  5.4  C r y o g e n i c Apparatus  37  5.5  Signal  39  CHAPTER VI  6.1  Processing  -  EXPERIMENTAL TEST OF THE FEEDBACK TECHNIQUE  Preliminary Considerations  ....  36  50 50  iv  page  6.2  M i n i m i z a t i o n of Sidebands  6.3  The Mass P l o t s  6.4  The Beat P a t t e r n  69  6.5  Phase Information  86  6.6  The L i n e a r i t y of A J / A J v s .  6.7  Conclusions  CHAPTER VI I  -  CHAPTER VIII -  52 62  (A /A ) ]  91  2  2  ••••  EXTRACTION OF THE g' A j / A ^ v s » (A ^ / A^)  c  FACTOR FROM •••• ••••  A SEARCH FOR THE 4 MG OSCILLATIONS..  8.1  P r e l i m i n a r y Remarks  8:2  Review o f the Standard Weak Modulation  91  9^  99 99  Solution  100  8.3  Large Modulation  103  8.4  M o d i f i c a t i o n to the Apparatus and A n a l y s i s  f o r the F ^ 4MG Search  112  APPENDIX A:  F l e x i b l e Gear R o t a t o r  114  APPENDIX B:  The D i s c r e t e F o u r i e r T r a n s f o r m  119  APPENDIX C:  Computer Programs  122  BIBLIOGRAPHY  134  V  LIST OF TABLES Page I II  I II  F o u r i e r C o e f f i c i e n t s p, q E f f e c t i v e Mass f o r Observed in Pb:H||/[110l Ranking  the Terms  16 Oscillations 64 109  vi  LIST OF FIGURES page FIGURE 1.  The S p i n - s p l i t  M a g n e t i z a t i o n at  Absolute  Zero  9  2.  The dHvA M a g n e t i z a t i o n at F i n i t e Temperature  3.  Crystal  Diameter as a F u n c t i o n o f  ....  10  Melt  Heater V o l t a g e  28  *t.  D e t e c t i o n Arrangement  ....  3^  5.  General Schematic C r y o g e n i c Assembly  38  6.  F i n e Tuning C i r c u i t  kO  7.  E q u i v a l e n t C i r c u i t f o r Tuning Arm  8.  B l o c k Diagram o f Apparatus  9.  C i r c u i t Diagram of  ....  kO ....  I n t e g r a t o r and Adder  10.  S y n c h r o n i z a t i o n of the Time Window  11.  dHvA Osci 1 lations;! i.nLead Along  12.  F o u r i e r Transforms Around 1 6 0 MG  13.  F o u r i e r Amplitudes as a F u n c t i o n of Feedback  [110]  kk kS A8 51 53  Gain  5k  \k.  Mass P l o t s with No Feedback  65  15.  Mass P l o t s w i t h Near-Optimal  16.  Mass P l o t s  17.  Beat Envelopes with No Feedback  18.  Beat Envelope w i t h Near Optimum Feedback  19.  C a l c u l a t i o n o f Apparent Beat P e r i o d s  Feedback  at Optimum Feedback  66 70 73 Ik 79  vi i  page 20.  Calculated Solutions  21.  The Beat Envelope w i t h Optimum Feedback . . .  84  22.  Ideal  85  23.  Individual O s c i l l a t i o n s at "Magic F i e l d "  the  Individual Figure 22  l/H- of  2k.  25.  26.  L.K.  for  Beat Envelope  O s c i l l a t i o n s at  88  89  Phase D i f f e r e n c e and Amplitude o f A2 at a "Magic F i e l d " A . / A , vs. back^  (A./Aj  w i t h and without  90 Feed92  1  27.  Fundamental y  28.  Sample R o t a t o r Assembly  29.  80  Beat Envelope  Sample R o t a t o r Assembly w i t h Gear and C o i l  Former  97 ....  115  Driving ....  . .1  117  ACKNOWLEDGEMENTS  It for his  is a s i n c e r e p l e a s u r e to thank Dr. A.V.  Gold  support and d i r e c t i o n in the s u p e r v i s i o n of  work, and h i s c l o s e personal  interest  this  in p r o v i d i n g a d v i c e  and encouragement. I am g r a t e f u l  to the N a t i o n a l  S c i e n c e s and  E n g i n e e r i n g Research C o u n c i l f o r t h e i r f i n a n c i a l in the form of a Postgraduate  Scholarship.  support  1  CHAPTER ONE  INTRODUCTION  In  1930, de Haas and van Alphen n o t i c e d that  the  m a g n e t i z a t i o n of bismuth o s c i l l a t e d as a f u n c t i o n of an e x t e r n a l l y a p p l i e d magnetic f i e l d at  low temperatures.  This  f o r almost  remained a l a b o r a t o r y c u r i o s i t y  until  i t was  effect  r e a l i z e d that  this  de Haas-van Alphen  could be used as a powerful  the Fermi s u r f a c e of m e t a l s .  20 years  tool  Valuable  (dHvA)  in the study  of  i n f o r m a t i o n on the  d e t a i l e d shape of the Fermi s u r f a c e can be o b t a i n e d from the  frequency o f : t h e magnetizatton o s c M l a t i o n s i . i n  i n v e r s e f i e l d domain, and oscillations  i t was soon  found that  are e x h i b i t e d by most metals  the  -  the  in the p e r i o d i c  table. There  is a l s o a wealth o f  information contained  the harmonic content of the o s c i l l a t i o n s ,  in p a r t i c u l a r  about the s p i n p r o p e r t i e s of c o n d u c t i o n e l e c t r o n s . measurements of usually  the fundamental  straightforward  in  Amplitude  frequency component are  (although a b s o l u t e  determinations  2  of  the amplitude r e q u i r e great  difficulties  care).  However,  are encountered when s t u d y i n g  the higher  harmonics and these d i f f i c u l t i e s o f t e n make i t i m p o s s i b l e to o b t a i n meaningful  various  well-nigh  i n t e r p r e t a t i o n s of the d a t a .  The most s e r i o u s of these d i f f i c u l t i e s  is  the  harmonic d i s t o r t i o n caused by the Shoenberg  significant  magnetic  interaction effect. In t h i s  t h e s i s , we p r e s e n t an o r i g i n a l  the m i n i m i z a t i o n of  technique f o r  the Shoenberg e f f e c t , thereby a l l o w i n g  the s p i n parameters to be determined r e l i a b l y , and without the use of c o r r e c t i o n f a c t o r s . Lande s p i n s p l i t t i n g orbits  factors g  c  a p p r o p r i a t e to c y c l o t r o n  in the m e t a l , and the r e l a t i o n of  the harmonic amplitudes Chapter  These s p i n parameters are  the f a c t o r s g  in the dHvA e f f e c t  is  to  c  reviewed  in  I I.  In Chapter how i t a r i s e s ,  I I I we d i s c u s s  the magnetic i n t e r a c t i o n ,  and how i t has been d e a l t with  (to a very  l i m i t e d e x t e n t ) by e x t r a o r d i n a r i l y tedious d e c o n v o l u t i o n of the experimental d a t a . In the p a s t , the magnetic  s e v e r a l attempts have been made to reduce  i n t e r a c t i o n e x p e r i m e n t a l l y , but these have met  with o n l y modest s u c c e s s . approaches  A f t e r summarizing  to the problem, we present  these experimental  in Chapter  IV  the  p r i n c i p l e s of the feedback technique which is c e n t r a l to thes i s.  this  3 Putting is  the idea o f feedback to work  the s u b j e c t of Chapter VI.  observable q u a n t i t i e s  in the  laboratory  The dependences of v a r i o u s  key  on the amount of feedback a r e c a l c u l a t e d ,  and compared w i t h experiment. The experimental apparatus measurements section  is d e s c r i b e d  includes  in Chapter  used f o r  in d e t a i l  the feedback  in Chapter V.  This  the c i r c u i t r y which the concepts developed  IV d i c t a t e , along with s p e c i a l  design  considerations  to make the technique s i m p l e , p r a c t i c a l , and r e l i a b l e . Having developed the procedure f o r o b t a i n i n g data which are e s s e n t i a l l y Chapter VII  f r e e o f magnetic  the f i r s t  a p p l i c a t i o n of  to the d e t e r m i n a t i o n of g along  i n t e r a c t i o n , we present  c  in  the feedback technique  f o r the y o s c i l l a t i o n s  in  lead  [110]. Recent o b s e r v a t i o n of o s c i l l a t i o n s  lead u s i n g sound a t t e n u a t i o n (Shubnikov-de Haas e f f e c t ) oscillations  analytically  by e x i s t i n g  d e r i v e an exact s o l u t i o n  similar  The d e t e c t i o n of such  benefitsgreatly  f i e l d s , of an amplitude  long p e r i o d in  magneto-resistance  prompted a search f o r  in the dHvA e f f e c t .  period o s c i l l a t i o n s modulation  and the  of'very  by the use of very larger  formulations.  long large  than can be t r e a t e d  In Chapter VIII we  f o r the response of dHvA  to a modulation f i e l d of a r b i t r a r y a m p l i t u d e .  oscillations  This  is  followed  by the d e t a i l s o f an experiment in which a c o n c e r t e d but u n s u c c e s s f u l attempt was made to find" the l o n g - p e r i o d o s c i l l a t i o n s .  h  Had they been found, the feedback technique would  have  shown them e i t h e r to be genuine dHvA o s c i l l a t i o n s ,  or  oscillations  generated by magnetic  interaction.  The best we  could do was p l a c e an upper l i m i t on t h e i r amplitude three major symmetry d i r e c t i o n s  [100],  [110], and  in the  [ill].  We conclude w i t h some s u g g e s t i o n s f o r f u r t h e r work, both  in the technique i t s e l f ,  and  its  application.  CHAPTER TWO  SPIN SPLITTING OF LANDAU LEVELS  IN METALS  In the same year as de Haas and van A l p h e n ' s d i s c o v e r y Landau  (1930) independently remarked that the  m a g n e t i z a t i o n of a metal would be expected to show oscillations  because o f the q u a n t i z a t i o n of the h e l i c a l  o r b i t s of the c o n d u c t i o n e l e c t r o n s . Onsager  (1952) p r e d i c t e d on the b a s i s o f general  semi-classical  arguments  that the p e r i o d i c i t y was  simply  r e l a t e d to extremal areas of the Fermi s u r f a c e normal to the magnetic f i e l d . Kosevich  Shortly  t h e r e a f t e r L i f s h i t z and  (1955) confirmed O n s a g e r ' s p r e d i c t i o n and p r o -  ceeded to work out express ions f o r the amplitudes of oscillations.  The r e s u l t of t h i s  with some m o d i f i c a t i o n s by D i n g l e , general  the  r a t h e r b e a u t i f u l work,  (1952) is equation [I],  form of the de Haas-van Alphen m a g n e t i z a t i o n :  6  [la]  M = I orbits  I A r  [lb]  A=  D ( B ) ( r X / s i n h rX)  [lc]  X = 2TT  r  r"  3 / 2  S = g  c  sin  [ 2 i r r ( f - - Y ) ± 'n/k]  exp (-rXTp/T)  D  m*/2m  magnetization,  i n c l u d e the steady m a g n e t i z a t i o n a r i s i n g  orbital  q u a n t i z a t i o n and s p i n .  in 1/B,  and each o r b i t has  The o s c i l l a t i o n s  areas and n e g a t i v e f o r maxima.  rX/sinh  from  are p e r i o d i c  i t s own c h a r a c t e r i s t i c frequency F.  The phase f a c t o r TT/A is p o s i t i v e f o r minimal  magnetic  (rirS)  D  The symbol M r e f e r s to the o s c i l l a t i n g and does not  cos  m* k T/efiB  2  C  [Id]  r  D(B)  c r o s s e c t ional  is a f u n c t i o n o f  the  i n d u c t i o n , and a l s o of Fermi s u r f a c e parameters. rX is a measure of the thermal broadening of  quantized o r b i t s . in a s i m i l a r  The i m p e r f e c t i o n s of the c r y s t a l  the result  b r o a d e n i n g , and a r e c h a r a c t e r i z e d by the D i n g l e  temperature T^ of the c r y s t a l . The f a c t o r cos  (TTTS) = cos  major concern in t h i s  thesis  (rirg m"/2m) c c  is the one o f  s i n c e from i t g  c  itself  is  determi ned. The e l e c t r o n s p i n w i l l  i n t e r a c t with  magnetic f i e l d s y m m e t r i c a l l y s p l i t t i n g  the a p p l i e d  the Landau l e v e l s by  7 the amount g e 1i B/2m, where electron,  g  is a s p l i t t i n g  free e l e c t r o n value Each Landau l e v e l  is  m  is  the mass of the f r e e  f a c t o r which may d i f f e r from  (2.0023) because of thus  split,  amount 2TT (g m " / 2 m ) .  spin-orbit-coupling.  resulting  l e v e l s each separated by e fi B/m",  in two s e t s  but s h i f t e d  Each set w i l l  give  In  1/B,  the amplitude of the 2 s e t s w i l l  amount  gm"/2mF,  ]r Z  [M(-+ Q  that  oscillations  frequency  in the absence o f s p i n  + M(-  splitting.  -  gm* AmF)  become  ] .  :B  •The c o s i n e spin< f a c t o r " i n the amplitude e x p r e s s i o n follows  F,  be d i s p l a c e d from one another by an  so the r e s u l t i n g m a g n e t i z a t i o n w i l l  gm"/4mF)  of  in phase by the  similar  in the m a g n e t i z a t i o n with the same fundamental and h a l f  immediately when t h i s average  form of the kind given  in  [lb]  ''  is a p p l i e d to a. wave -  the e f f e c t of the s p i n  [ 1 a ] - [ 1d] at a b s o l u t e zero (T=0)  perfect crystal  In  r  = D(B)  n  r"  3 / 2  cos  this  case,  iA  discontinuous  [lb]  reduces  splitting, and  to  nrS  which are the F o u r i e r c o e f f i c i e n t s of a S = 0.  '  1  l e t us examine equations (T =0).  i ^  [la] .  To g i v e a c l e a r e r p i c t u r e of  A  its  change  when the uppermost Landau l e v e l  c u s p l i k e waveform  in the m a g n e t i z a t i o n becomes d e p l e t e d as  it  for  occurs crosses  in a  8 the Fermi energy. is  The e f f e c t of cos  TTTS as we have j u s t  to sum the c o n t r i b u t i o n s from the two s e t s of Landau  spin s p l i t  about the v a l u e at S = 0.  that shown in F i g u r e 1,  matter to determine the phase s h i f t waves, and thereby determine S. us  levels  The two sawtooth wave-  forms, along w i t h t h e i r sum is shown in F i g u r e waveform such as  seen  1.  With a  i t would be a s i m p l e  between the two sawtooth  At temperatures a v a i l a b l e  to  in the l a b o r a t o r y , the thermal damping f a c t o r s p r e f e r e n t i a l l y  reduce the higher harmonics, that shown in F i g u r e 2, which  resulting  in a waveform more  is an experimental  with t y p i c a l experimental parameters. sinusoidal, directly  but  i t ;is  c  r e c o r d i n g taken,  The t r a c e is not p u r e l y  q u i t e e v i d e n t that S cannot be measured  from the waveshape.  To e x p l o i t the cos the g  like  nrS dependence in the hopes of e x t r a c t i n g  f a c t o r from dHvA amplitude d a t a ,  i t would be convenient to  o b t a i n a method in which the o t h e r amplitude f a c t o r s played o r no r o l e . pulation of  Gold and Schmor [lb]  (1976) showed that with some mani-  an a l g o r i t h m c o u l d be o b t a i n e d using  t h r e e harmonic amplitudes  to determine the v a l u e  Forming the dimens ion 1 ess q u a n t i t y  a  a  o  Ilk  [1 + 1/3 t a n h a  2  a = A^/AjA^,  X]  [1 + 1/3 t a n h  2  X]  where [2b]  a  CO  (/3/2)(l  - tan  2  ^S) /(l 2  - 3 tan  2  the  first  S.  g i ves  [2a]  little  ^S)  [lb]  F i g u r e 1.  The Spin S p l i t M a g n e t i z a t i o n at A b s o l u t e Z e r o a , b : c o n t r i b u t i o n s from each of the 2 s p i n d i r e c t i o n s c: r e s u l t a n t m a g n e t i z a t i o n  F i g u r e 2.  The dHvA M a g n e t i z a t i o n at Temperature  Finite  11  and the s u b s c r i p t s Using  r e f e r to the l i m i t i n g  the harmonic content  itself  cases  X -»• 0 and X  °°.  as an i m p l i c i t gauge of  the bath temperature, the h y p e r b o l i c f u n c t i o n s o f X can be el Imki a ted between [ l b ]  for  r = 1, 2 and  [2a]  to g i v e  the simple  relation  [3a]  A /A ]  3  = a.  [(Aj/A^  -  2  1/4  ( A ^ ) * ]  where [3b]  ( V V f j  =  2 / 2  " exp  is  independent of  be o b t a i n e d as  (XTp/T)  the temperature T.  cos  TTS/COS 2TTS  The v a l u e a  the s l o p e of a s t r a i g h t  can t h e r e f o r e  M  l i n e p l o t of A^/A^ v s .  2 (Aj/A£)  as the temperature is v a r i e d and the f i e l d held  constant. From [2b]  it  is c l e a r that the s o l u t i o n 2  o b t a i n e d from a q u a d r a t i c equation is^ t h e r e f o r e m u l t i v a l u e d .  arises  difficulty  is  TTS and the s o l u t i o n solution  the harmonic  the fundamental  from the f a c t  amplitudes.  from the p e r i o d i c nature of  Equivalent solutions  be  and  the D i n g l e temperature T^ which can be  T h i s m u l t i v a l u e d nature a r i s e s  plicity  will  r e l a t i v e phase measurements  o b t a i n e d from the f i e l d dependence of  the a b s o l u t e s i g n of  S  The p h y s i c a l l y meaningful  can be s e l e c t e d with the a i d of a rough e s t i m a t e of  in tan  for  A  amplitude.  that we do not know further multi-  tan irS in  [2b].  are ± S ± p where p is an i n t e g e r .  inherent  This  in the use of quantum o s c i l l a t i o n s  to  determine g [lb].  c  and a r i s e s  from the p e r i o d i c i t y of  the c o s i n e  in  One can use a band s t r u c t u r e c a l c u l a t i o n to h o p e f u l l y  r e s o l v e the  ambiguity.  The t h r e e harmonic method o f f e r s f o c u s i n g on the S dependence of require further scattering  rates  the major advantage of  the a m p l i t u d e .  i n f o r m a t i o n about  Other methods  Fermi s u r f a c e parameters  s i n c e complete c a n c e l l a t i o n of  amplitude f a c t o r s is not a c c o m p l i s h e d .  the o t h e r  or  13  CHAPTER THREE  THE SHOENBERG MAGNET IC-INTERACTI ON EFFECT  The d i s c u s s i o n oscillations  in the p r e v i o u s  chapter assumed that  were measured as a f u n c t i o n of  d u c t i o n B.  In p r a c t i c e , the o s c i l l a t i o n s  f u n c t i o n of  the a p p l i e d f i e l d  [k]  in-  are measured as a  H, r e l a t e d to  B  by  B = H + 4TT ( 1 - 6 ) M  f o r a second degree s u r f a c e w i t h axis.  the magnetic  the  <5 is  situation,  the demagnetizing 4TT(1 -6)M/H i  of a r a p i d l y o s c i l l a t i n g often constitutes  H  factor.  The s u b s t i t u t i o n of  amplitudes A  the normal  sinusoid  (see  B  [la]),  is  laboratory in the argument  the c o r r e c t i o n term  a l a r g e p a r t o f one c y c l e .  p o i n t e d out by Shoenberg  o f t e n hopeless  In  10 ^ , however s i n c e  The n e c e s s i t y of d i s t i n g u i s h i n g  equation for M  p a r a l l e l ' t o a principal,  between B and H was  first  (1962). [k]  into  [la]  results  as a f u n c t i o n o f H',, c o n v o l v i n g contortions,  in an  implicit  the harmonics  thereby s e v e r e l y m o d i f y i n g  and the phases of  the harmonics.  into  the i d e a l  Recovery of  the  ideal  amplitudes and phases  is  the c e n t r a l theme of  this  thes i s. There a r e v a r y i n g degrees o f the s e v e r i t y of magnetic  i n t e r a c t i o n (M.I.).  this  In the l i m i t of small  can be made, but o f t e n the d i s t o r t i o n s  M  corrections  are so severe that  it  is  i m p o s s i b l e to e x t r a c t the i d e a l amplitudes and phases from the data. To see the e f f e c t s of the term 4-rr(1 - 6 ) M , l e t us re-write  [1]  by s e t t i n g  8TT F — H 2  x = 2TT (jf "" Y )  K =  /. .\ (1-6)  2  C =  Z = KM  Since  | ku (1 - 6 ) M | « H ,  [5]  z = I C r  Thi:s  KA  r  R  sin  [la]  [r(x-z)  r  becomes  + -n/h]  impl i c i t ' e q u a t i o n f o r z can be s o l v e d by a s e r i e s  s u c c e s s i v e approximations  in a scheme developed by P h i l l i p s  Gold  approximation  (1969) where the n ^  z  and  z  ( n )  (  0  )  =  =  I r=l  0.  C  sin r  [r(x-z  ( n _ r )  )  is g i v e n by  +TTA]  of and  15 While the g a t h e r i n g  of  terms can become q u i t e  tedious (n)  a f t e r a few s t e p s , exact to 0(n) This  the procedure is  in the amplitude  / \ M  n I r=l  =  W  q  r >  [p  in T a b l e  r  sin  r  carried  cos  (rx + TT/4)  ]  out  to many more  (1976) w i t h the a i d of a computer  to perform the a l g e b r a i c  manipulations.  the amplitudes o f the r e s u l t i n g  in the i n t e r a c t i n g  of  1, where  (rx + TT/4) + q  been  o r d e r s by Perz and Shoenberg  To o b t a i n  and G o l d ,  r  i ' t e r a t i o n c " scheme-has  program designed  is  factors.  are most c o n v e n i e n t l y d i s p l a y e d as a t a b l e  Fourier c o e f f i c i e n t s p  The  in that z  scheme has been c a r r i e d out by P h i l l i p s  and the r e s u l t s  [6]  convenient  harmonics  t h e o r y , we merely f i n d the magnitude of  the  th r  term in the complex F o u r i e r expansion  A' = ( r r 2  KP  +  q M  2  r  )  For the f i r s t  [7a]  A, = A  1  ,  /  2  3 harmonics,  A„ = A,  [7c]  A  3  = A  3  8/2  the r e s u l t  is  0(3)  +  1 [7b]  i.e.,  /f  1  1/2  + 0(4)  3 /2  + 0(5)  Term  P  0(2)  0(1)  icA |  l  A  l  ic^ A | 8  2/2 KA  Q  0(h)  0(3)  A  +  1  Z  2/2  P  KA  A  2  KA  1  3  +  2  2  6/2  KA.A  2  1  2  ±  —  K A } 3  1  -  +  —  6/2  /2 3KA.A.  3  A  3"  - p ^ 2/2  3KA,A ± q  K A, A  2  2/2  P  -  1  /2  +  2  K A«  icA«A—  -  2/2  2  Q  2  l  3  2  + 2/2  3< A3 8  K A } 3  A.  L  ^  ±  1  Fourier Coefficients  p,q.  From P h i l l i p s  and Gold  (1969).  2KA.A.  1  ±  2  /2  ^A }  2  2  /2  Table  KA  __ i .  3/2 KA  %  2KJA.A. Ul  1  +  '  3/2  ±  ' /2  3  +  2< A A 2  2  1  0 2  We recover the amplitudes A^ , A^, A^ of the i d e a l these amplitudes are s u f f i c i e n t l y If,  as  small.  we o b t a i n Shoenberg's  fundamental" r e s u l t s .  These are found in  taking  and A^ approach z e r o ,  the l i m i t as  =  [8a]  A]  [8b]  A^ = - j  A  3  =  1  k  A  and  KA  i  +  0  (  5  oscillations consider just  by  the r e s u l t  is  )  dHvA f r e q u e n c i e s are p r e s e n t , M.I.  these should be  [7c]  0(4)  +  2  d i s t o r t i o n o f the harmonic c o n t e n t .  sidebands  "strong-  0(3)  A +  |  [7b]  There are o t h e r n o t i c e a b l e e f f e c t s of M.I.,  generates  if  i t o f t e n happens, the i d e a l amplitudes are swamped  by the terms generated by M.I.,  [8c]  theory  If acts  l i k e a mixer, and The s i m p l e s t  and d i f f e r e n c e f r e q u e n c i e s o f  have (assuming a long  rod 6  from two extremal  =0).  of  fundamental  from d i f f e r e n t o r b i t s on the Fermi s u r f a c e . the fundamentals  the  two or more fundamental  and combination t o n e s .  sum  besides  If we  s e c t i o n s , we  18 where the s u b s c r i p t s Since  —• [4ir(M 2  H  [10]  a  M =  A  r e f e r to the two s e c t i o n s .  + M,)]<<1, we can w r i t e b  s i n (x  a  a  -  a b x , = 2TT (—r* a ,b H  K  m) +  a  A,  b  [9] as  s i n (x, b  K,M)  b  F  where  8TT F  and  K  .  =  a,b  ) a ,b  +  L  .  2  1  v  a ,b  — —  ,2  From [7a] we see that f o r one of the fundamental  frequency alone the amplitude  remains unchanged to second o r d e r , so that  r e p l a c i n g M on the r i g h t  s i d e o f [10] by the i d e a l  Kosevich m a g n e t i z a t i o n should) be a reasonable  Li'fshitz-  approximation  f o r c a l c u l a t i n g the l o w e s t / o r d e r combination terms.  [11]  M =  +  Assuming terms  A  s i n [x - k A s i n (x ) - K A , s i n (x, )] a a a a a b b  a  s i n [x, - K , A b b a  A,  b  s i n (x ) - K . A a b a  s i n (x, ) ] • b  the q u a n t i t i e s K A to be s m a l l , we keep o n l y the l i n e a r  in such q u a n t i t i e s  K A  [12]  Vie then o b t a i n  M  giving  K A  2  V " 2  s  i  n  <  2 x  J a  a b - — ^ — [(K + K ,) l a b A  2  - - V ^ s i n 2  (2x.) b  A  s i n (x + x, ) - ( K - K , ) s i n ( x - x , ) ] a b a b a b  19 Thus, to lowest o r d e r , the amplitudes o f the sum and d i f f e r e n c e f r e q u e n c i e s a r e given by  [13a]  A  s  A A, a b 2  ( i e  a  +  K  b  )  and A A, a b 2  [13b]  The r e s u l t s o b t a i n e d so f a r apply when the a b s o l u t e amplitude of the dHvA o s c i l l a t i o n s AH - —  is  much s m a l l e r  than the f i e l d  or in the reduced n o t a t i o n , C < < l . r  where t h i s  is no longer t r u e .  spacing,  There are many cases  The dHvA m a g n e t i z a t i o n can approach  or even exceed the f i e l d s p a c i n g .  In such cases the m a g n e t i z a t i o n  f o r m a l l y becomes m u l t i v a l u e d , and the r e s u l t i n g m a g n e t i z a t i o n the one with the g r e a t e s t the m a g n e t i z a t i o n  thermodynamic s t a b i l i t y .  is  When C > l r  is no longer uniform i n s i d e the sample, and  Condon domains a r e formed (see Condon 1966, Condon and Walstedt 1968). The M.I.  results  discussed  in t h i s c h a p t e r c l e a r l y a l t e r  the temperature dependence from that g i v e n by the i d e a l Kosevich  (L.K.) amplitudes  [lb].  For example,  combination tones generated by M.I., of  Lifshitz-  in the case of  the temperature dependence  the amplitudes of the sum and d i f f e r e n c e f r e q u e n c i e s from X X  20  CHAPTER FOUR  REDUCTION OF MAGNETIC INTERACTION USING A FEEDBACK TECHNIQUE  From the d i s c u s s i o n s e v i d e n t that magnetic role  o f the p r e v i o u s c h a p t e r ,  it  is  i n t e r a c t i o n must play o n l y a very small  i f any i n f o r m a t i o n from the harmonic content  is  to be  obta i ned. It  is c l e a r t h a t the a b s o l u t e amplitude o f the magneti-  z a t i o n determines the r e l a t i v e s i z e s of the M.I. harmonics.  generated  One might c o n s i d e r reducing these troublesome  e f f e c t s by e x p l o i t i n g the temperature dependence.  M.I.  At a high  enough temperature, the a b s o l u t e amplitude can be made arbitrarily  small,  thereby reducing the M.I.  f o r t u n a t e l y the L.K. off  harmonics.  Un-  second and higher harmonic amplitudes  f a s t e r than t h e i r M.I.  c o u n t e r p a r t s with  drop  increasing  temperature thereby i n c r e a s i n g , not d e c r e a s i n g the waveform d i s t o r t ion. The dependence on the demagnetizing f a c t o r 6 in been used with some success of M.I.  to minimize or c o n t r o l  [k]  has  the e f f e c t s  E v e r e t t and G r e n i e r (1-978) have cut c r y s t a l s  into  21 ellipsoids  of v a r y i n g aspect  ratios  to study  the dependence  of the harmonic s t r u c t u r e on t h r e e d i f f e r e n t values of 6 . previous  g  c  f a c t o r measurements,  cut very t h i n  (0.5  fortunately, this  m m  )  d i s k s with 6 % 0 . 9  t h i n d i s k method is  undesirable side e f f e c t s . avoids  Gold and Schmor  to reduce M.I.  t e d i o u s , and has  field  a modulation c o i l  [15]  H  w i t h a small  so  If we s e p a r a t e h(t)  h(t)  feedback g a i n ,  [17]  i.e.,  the  method"  = h  produced by  m  + h  i n t o the  the seed f o r the feedback  f  if  4TT ( 1 - 6 )  enters  i n t o two components  3  is an e x p e r i m e n t a l l y  then the e q u a t i o n f o r  BM-=  quasi-static  M + h(t) .  that the modulation f i e l d h ( t )  and we l e t h^ = - 3R, where  of M,  the " d i s k  p e r t u r b a t i o n h(t)  e q u a t i o n f o r B in the same way as M is  [16]  some  that  B = H + 4TT ( 1 - 6 )  The f a c t  idea.  Un-  some advantages as w e l l .  E x p e r i m e n t a l l y , one u s u a l l y modulates background  have  The method about to be d e s c r i b e d  most o f the u n d e s i r a b l e f e a t u r e s of  and o f f e r s  (1969)  ln  M  B  adjustable  can be made independent  22 the M . i . w i l l necessary  be e f f e c t i v e l y s u p p r e s s e d .  to o b t a i n a s i g n a l  proportional  a c c o r d i n g l y , and a p p l y t h i s Suppression  o f M.I.  signal  their  sound v e l o c i t y measurements was wrapped t i g h t l y  therefore  to M, a d j u s t  by means of magnetic  a c h i e v e d by T e s t a r d i and Condon  in the c o i l  is  the g a i n  as a f i e l d to the sample.  first  a coil  It  feedback was  (1970)  in the c o u r s e o f  in b e r y l l i u m .  In t h e i r work,  around a c u b i c sample, and a c u r r e n t  approximated the e q u i v a l e n t s u r f a c e c u r r e n t s  in the  sample and thus c o u l d be made to c a n c e l the dHvA m a g n e t i z a t i o n . The a p p r o p r i a t e c u r r e n t was found by Imposing a n u l l c r i t e r i o n on an e x t e r n a l  pickup c o i l .  detection  In the T e s t a r d i - C o n d o n  arrangement,  the sample c o u l d not be r o t a t e d , and f o r a  c u b i c sample  , the m a g n e t i z a t i o n  is  .  inherently non-uniform.  We  have developed a d i f f e r e n t type o f t e c h n i q u e which a l l o w s  the  sample to be r o t a t e d , and  is  to e s t a b l i s h  the c o r r e c t amount o f feedback.  such c r i t e r i a , be d i s c u s s e d  1.  nFj  present  in p r a c t i c e w i l l  Combination Terms a c t s as a mixer  two genuine f r e q u e n c i e s F mF where 2  used  in t u r n .  M i n i m i z a t i o n of M.I.  ±  itself  There a r e s e v e r a l  and the ones which are e a s i e s t  As we have s e e n , M.I. if  in which the dHvA e f f e c t  n  in the i d e a l  and  1  m  and F  2  in the sense  a r e p r e s e n t , M.I.  are i n t e g e r s .  that generates  These terms a r e not  t h e o r y , of c o u r s e , and the c r i t e r i o n  becomes the m i n i m i z a t i o n combination f r e q u e n c i e s .  (ideally  the z e r o i n g )  of  these  2.  Mass  Plots  The temperature dependence o f the L.K.  harmonics  is  given  by A (T) a . * „ % X e" r s m h rX r  where  r X  X = Xm* T/H and X = 2ir C The M.I.  k /efi. D  2  D  terms have a d i f f e r e n t temperature dependence  f o r each harmonic, so t h a t optimum feedback is of  In A/T v s . T/H f o r the f i r s t ,  straight  3-  l i n e s w i t h the s l o p e s  found when p l o t s  second, and t h i r d harmonic a r e  p r e c i s e l y in the r a t i o  1:2:3.  The Beat Envelope When 2 s i g n a l s are c l o s e  beat.  The M.I.  of the f i r s t  in f r e q u e n c y , a l l  the harmonics  terms g e n e r a l l y beat at the d i f f e r e n c e frequency  harmonic because i t  is  the s t r o n g e s t  in a m p l i t u d e .  The feedback can be a d j u s t e d to make the harmonies beat at t h e i r proper f r e q u e n c i e s .  4.  Phase  Information  The r e l a t i v e phases of the harmonics of the i d e a l  L.K.  terms a r e e a s i l y c a l c u l a t e d , and s i n c e the M.I.  terms add in a  d i f f e r e n t phase, one need simply a d j u s t 8 u n t i l  the L.K.  relationships  phase  are e s t a b l i s h e d .  2 5.  L i n e a r i t y of A j / A ^ v s . As d i s c u s s e d  (Aj/A^  in Chapter Two, (see  [3a]),  a p l o t of A^/A^  2 vs.  (A^/A^)  amplitudes  should y i e l d a s t r a i g h t f o l l o w the i d e a l L.K.  in the presence of  M.I.  line  form.  if  the harmonic  Curved p l o t s a r e o b t a i n e d  As p r e s e n t e d , the feedback t e c h n i q u e seems  to  accomplish  the same d e s i r a b l e o b j e c t i v e s which have p r e v i o u s l y a t t a i n e d by e x p l o i t i n g Perhaps actual  the demagnetizing  the most t e d i o u s  preparation of  feature  the sample.  d i r e c t i o n can be s t u d i e d ,  The demagnetizing  orientation,  so that  the e x t e r n a l  field  special is  field.  in the disk-method  is  the  In each sample, o n l y one  the one p e r p e n d i c u l a r to the plane of  the sample.  H  been  field  is very s e n s i t i v e  to  c a r e must be taken to ensure  that  p r e c i s e l y p e r p e n d i c u l a r t o the p l a n e  4  of  the d i s k .  Disk-shaped  soft materials  such as  free of s t r a i n .  lead,  (1-6),  by the same  is d i f f i c u l t of  so that  ellipsoidal  shape  induction f i e l d ,  and when  using  to keep the  sample  the d e t e c t i o n apparatus  reducing M.I.  are a l l  r e l a t e d to the  the sample be a t h i n d i s k .  is no such c o n s t r a i n t ,  are f r a g i l e ,  reduces  is  the  factor.  The above drawbacks that  it  The s e n s i t i v i t y  a l s o dependent on sensitivity  samples  requirement  In the feedback method t h e r e  and any e l l i p s o i d  can be used.  The  i s n e c e s s a r y o n l y to a c h i e v e a u n i f o r m and  it  is  felt  that  f o r some a p p l i c a t i o n s  of  the feedback t e c h n i q u e , the sample need not even be e l l i p s o i d a l (discrimination below).  In  the c r y s t a l  between genuine dHvA terms and M.I.  the case of a s p h e r i c a l can be s t u d i e d  sample need be p r e p a r e d . subject  is  coils,  The s p h e r i c a l  and  sample  in  filling  is e v i d e n t l y  requirement of  the optimum shape f o r mechanical  the sphere has a l a r g e  directions  see  in the same e x p e r i m e n t , and o n l y one  to the p r e c i s e o r i e n t a t i o n  a sphere  sample a l l  terms;  the d i s k ,  stability.  factor for solenoidal  i t can be shaped q u i t e  precisely.  not and  Finally,  pick-up  There one  i s a d i s t i n c t advantage t o the feedback  i s f a c e d w i t h the problem o f d e c i d i n g whether o r not  observed  frequency  the feedback to  system when  fall  i s genuine,  g a i n from z e r o causes  in amplitude.  cases where M.I.  generates  the M.I.  Anderson, 1 9 7 3 ) .  generated  ( c f . van Weeren  the d i s k method, one would have to make a t l e a s t  a b s o l u t e amplitude measured.  in .  tens o f sidebands,many o f which  o f the dHvA o s c i l l a t i o n  In the p a s t , c a r e f u l  and  two d i s k s  i s t h a t the  can be v e r y  easily  measurement o f sample volume  g e o m e t r i c a l c o u p l i n g c o n s t a n t s between the sample and  detection coil  were r e q u i r e d , as w e l l as the net g a i n o f  a m p l i f i c a t i o n system, w i t h a l l i t s f i l t e r s . Using feedback,  i s the Gauss t o amp optimum feedback  magnetization  ratio  M  M  where 6 The  =  the  coil.  coil,  1975)  is required  simply measures the  c u r r e n t 1^ in the modulation  and  When amplitude the dHvA  i s g i v e n a b s o l u t e l y by  YI [ , 8 ]  y o f the modulation  i s a t t a i n e d , one  the  ( c f . Knecht  the o n l y c a l i b r a t i o n c o n s t a n t which  the feedback  may  ratios.  A v a l u a b l e s i d e e f f e c t o f u s i n g feedback  of  terms  To a c q u i r e the same i n f o r m a t i o n unambiguously  with d i f f e r e n t aspect  and  Increasing  T h i s can be v e r y h e l p f u l  l a r g e r than the genuine frequency  from  by M.I.  in amplitude, w h i l e the genuine dHvA f r e q u e n c i e s s t a y  the same o r r i s e  be  o r generated  an  3YI  F  Ml-fi)  =  i s the demagnetizing feedback  ~8tT  F  ( F O R  A  S  P  H  E  R  E  )  factor.'  p r i n c i p l e c o u l d be used w i t h advantage in  measurements o f the quantum o s c i l l a t i o n s  in o t h e r  electronic  p r o p e r t i e s o f metals attenuation e t c .  eg. Shubnikov-de Haas e f f e c t ,  One would s t i l l  magnetization o s c i l l a t i o n s feedback s i g n a l .  ultrasonic  need to measure the dHvA  in o r d e r to o b t a i n  the r e q u i r e d  For example, feedback c o u l d be used  determine whether a set o f quantum o s c i l l a t i o n s  to  i n , say,  the  u l t r a s o n i c a t t e n u a t i o n are genuine ones or generated by As we s h a l l  s e e , an  feedback l o o p , b r i n g i n g of  the feedback s i g n a l .  integrator  is  present  inevitable d r i f t s When t h i s o c c u r s ,  in the  in the zero  level  t h e r e is a  D.C.  c u r r e n t added to the modulation f i e l d which appears shift  in the e x t e r n a l f i e l d H.  insufficient  D.C. drift  the course o f  the  to m a t e r i a l l y a f f e c t  amplitude d a t a . However, these d r i f t s make the phase less  as a  In our experiment, t h i s  c o u l d e a s i l y be kept below 5 Gauss d u r i n g measurements,which was  M.I.  the  information  reliable. A slight  imbalance  in the pickup c o i l  to be a problem s i n c e t h i s would simply the modulation  range.  The c o i l s  does not t u r n out  i n c r e a s e or decrease  could e a s i l y be balanced  to  3 r e j e c t the homogeneous modulation f i e l d to 1 p a r t that a modulation most ± 1 Gauss.  range o f  in 10 , so  1 kG might have been a l t e r e d by at  27  CHAPTER FIVE  EXPERIMENTAL DETAILS  5.1  Sample  Preparation  The experiments were performed on s i n g l e Previous  use of the C z o c h r a l s k i  Phillips  and Gold  single  crystals  apparatus  1969)  showed  method f o r c r y s t a l its  great  A single  crystal  the meniscus  the seed  (cf.  in producing  possible.  In  (6NT grade)  in a vacuum o f  the process of  rates are 0 . 5 - 1 . 0  The diameter of  c y l i n d e r was found to be r a t h e r  10 ^ to 10  is  slowly  the r e s u l t i n g  insensitive  the  single-crystal  to p u l l i n g  to heater v o l t a g e ,  the need f o r a r a t h e r high" degree of  p u l l e d from the  c m / h r . , 0.5 being  speed,  F i g u r e 3 shows the  diameter on heater v o l t a g e .  is very s e n s i t i v e  but the  reaching e q u i l i b r i u m ,  but c r i t i c a l l y dependent on melt temperature. dependence o f c r y s t a l  The  is enough to keep a l l  turns upward, and the seed  T y p i c a l growth  diameter  growth  seed is dipped i n t o the m e l t , and the  submerged p o r t i o n s o l i d .  slowest  lead.  c e n t e r s around a melt of zone r e f i n e d lead  heat conduction through  melt.  success  of  o f extremely low D i n g l e temperatures.  from Cominco L t d . , e l e c t r i c a l l y heated Torr.  crystals  The c r y s t a l  and t h i s  dictates  long term s t a b i l i t y  and  measurement a c c u r a c y .  The more d i r e c t measurement of  with the use of a thermocouple had i r r e p r o d u c i b i 1 i t y drawbacks,  inherent thermal  the v o l t a g e  determined d i a m e t e r . absolute voltage adjust  lag and  r e q u i r i n g the o p e r a t o r ' s  a t t e n t i o n d u r i n g the growth p r o c e s s . one need o n l y set  temperature  In the v o l t a g e  to o b t a i n a c r y s t a l  Four f i g u r e accuracy was  constant measurement,  o f any p r e -  r e q u i r e d in the  (A.C.) v o l t a g e measurement and a S o l a 5008  constant  t r a n s f o r m e r p r o v i d e d the r e q u i r e d s t a b i l i t y . the v o l t a g e ,  a v a r i a c was used  To  in c o n j u n c t i o n with  two 3  rheostats. absolute  The f i n e c o n t r o l had a range of ± 1 p a r t  voltage.  Crystals  ranging  were p u l l e d r e p r o d u c i b l y using required a spherical  crystal  of  in 10  o f the  in diameter from 1 to 10 mm  t h i s method. roughly  The experiment  7 mm diameter, so  that  crystals  p u l l e d f o r the feedback experiment had a diameter  slightly  l a r g e r than  this.  After pulling a crystal  roughly  5 cm l o n g ,  was separated from the melt by r a i s i n g Once  the c y l i n d e r was  removed  the v o l t a g e on the h e a t e r .  from the  growing  i t was c a r e f u l l y mounted in a r o t a t i n g chuck. c i r c u l a r c y l i n d e r was used as a spark c u t t i n g kept below 0.010  i n , and the t o o l was  t o o l was  rotating crystal axes  intersected.  lowered with  its  apparatus,  A hollow copper tool.  rotated during  procedure s i n c e the t o o l erodes as w e l l as rotating  it  The w a l l  the c u t t i n g  the sample.  As  the  a x i s p e r p e n d i c u l a r to the  cylinder, a spherical  was  sample r e s u l t e d  if  the  30  The  i n t e r s e c t i o n was c l o s e r than  be c a r e f u l l y a d j u s t e d d u r i n g sure t h a t  0.002.  the f i n a l  the tool w a s c u t t i n g  on a l l  i n , as  this  could  c u t t i n g stages by making  of  the c i r c l e  inscribed  in the c r y s t a l . When the sphere was near c o m p l e t i o n , t h e r e remained two p o i n t s or " e a r s " on which the t o o l was c u t t i n g . the sphere to the unused p a r t of holds  the c r y s t a l  the endpiece to the sphere.  l a t t e r to cut through f i r s t ,  c y l i n d e r , the o t h e r  It was d e s i r a b l e f o r  so the t o o l  a x i s was  5° away from being p e r p e n d i c u l a r to the c r y s t a l both axes c o p l a n a r .  Using  this  forming b e t t e r than l%spheres Since of  lead  the c r y s t a l  is a strong  cutting  in s t r o n g e t c h a n t  keeping  crystals  obtained.  absorber of X - r a y s ,  the s u r f a c e adequate  A s u i t a b l e e t c h i n g procedure  was needed to remove the p i t t e d s u r f a c e  etch  axis,  must be very good in o r d e r to o b t a i n  spark e r o s i o n p r o c e s s .  the  positioned  procedure,  were u s u a l l y  Laue b a c k - r e f l e c t i o n photographs.  One held  l a y e r generated by the  T h i s procedure c o n s i s t e d of a 45 minute (250 cc g l a c i a l  d i s t i l l e d h^O, and 62.5 cc 30%  acetic acid,  187.5 cc  immediately f o l l o w e d by a  wash in e t h a n o l . A f t e r c a r e f u l l y mounting 5 minute P o l a r o i d X-ray The X-ray  process  photographs  included r o t a t i n g  to ensure both a s i n g l e orientation.  the c r y s t a l  on a goniometer,  could be taken f o r o r i e n t a t i o n . to major symmetry  c r y s t a l , and an .unambiguous  directions  final  31  5.2  D e t e c t i o n Apparatus In the i n d u c t i v e method f o r measuring magnetic  the sample  is placed in a balanced pickup c o i l  modulation c o i l .  and a l s o a separate  The l a t t e r p r o v i d e s a time dependent  sinusoidal)  deviation  pickup c o i l  is balanced to be i n s e n s i t i v e to t h i s  net m a g n e t i z a t i o n  susceptibility  in the steady background f i e l d ,  i n s i d e the balanced pickup c o i l  (often and the  change.  Any  induces a  dM voltage  in i t p r o p o r t i o n a l  consists  of  induction  to  .  The balanced pickup  two c o i l s , one f o r the d e t e c t i o n of the t o t a l  (the pickup c o i l )  and the o t h e r to buck out the  c o n t r i b u t i o n from the modulation f i e l d Since  (the bucking  coil).  l a r g e modulation was e n v i s a g e d , mechanical  was of primary concern s i n c e major source of n o i s e .  in t h i s  To t h i s  regime  end, the modulation c o i l  c e n t r i c solenoids  field.  and the counter-wound bucking c o i l ,  to a c h i e v e maximum mechanical  r i g i d i t y , were wound as  d i r e c t l y on top o f one a n o t h e r .  It  is  t r u e that some s e n s i t i v i t y  is  The c e n t r e  lost  in  arrangement because the f l u x due to the m a g n e t i z a t i o n of sample threads both the pickup and the bucking c o i l s . case of a long  rod sample, t h i s  than a f a c t o r of 2, but the gain it.  again  two c o n -  tap was made a v a i l a b l e to f i n e tune the balance when the were c o o l e d .  took  i n s i d e the bore  of the main magnet p r o v i d i n g the steady background The pickup c o i l  rigidity  v i b r a t i o n s a r e the  the form of a long s o l e n o i d , m e c h a n i c a l l y f i x e d  worst  coil  this the  For the  l o s s f o r our c o i l s  in s i g n a l - t o - n o i s e  coils  is  less  is w e l l worth  32 Some thought was g i v e n to the s i z e o f the w i r e which should be used.  A simple c a l c u l a t i o n t a k i n g  Johnson n o i s e given by the Nyquist duced s i g n a l  S gives  the s i g n a l  of the r a d i u s o f the w i r e  [19]  S/N £  where  be made as  to n o i s e  r.  [4TrM](£)(4k TAf)"  1 / 2  B  &  (1)  h is  the h e i g h t o f the c o i l  w is  the width of the c o i l  T is  the temperature  is  in-  r a t i o S/N as a f u n c t i o n  the m a g n e t i z a t i o n of the sample  p is [19]  e q u a t i o n and the t o t a l  M is  Af and  i n t o account the  the frequency bandwidth the r e s i s t i v i t y of the w i r e .  has no maximum as a f u n c t i o n o f small  r,  so that  as p o s s i b l e , w i t h o n l y the mechanical  of the w i r e to be c o n s i d e r e d .  r should strength  The r e s i s t i v i t y dependence suggests  pure copper w i r e or s u p e r c o n d u c t i n g w i r e . When t h i s  s o l e n o i d - o n - s o l e n o i d arrangement  must c a r e f u l l y c a l c u l a t e the r a t i o of turns that  in the bucking c o i l ,  in the pickup to  as the combined width o f the c o i l s  l i m i t e d by the a v a i l a b l e space. overwind the bucking c o i l  is used, one  One must a l s o be c a r e f u l  so t h a t  in the process of  is  to  balancing,  turns need be removed, not added. In the  final  model,  the  pickup-bucking c o i l  pair  had an i n s i d e diameter o f 0.300" and an o u t s i d e diameter o f 0 . 5 0 0 " .  33  The pickup c o i l  had 9500 t u r n s , and the bucking c o i l  turns o f #46 copper w i r e .  (0.0017 inches  had 6016 *  in d i a m e t e r ,  3  insulation  included). The b a l a n c i n g was done by c o n s t r u c t i n g a modulation similar  t o that  coil  in the c r y o s t a t , and p l a c i n g the sample-bucking  coil  arrangement  inside.  Turns were removed from the bucking  coil  u n t i l z e r o pickup r e s u l t e d .  T h i s c o u l d be done to an  accuracy o f h turn ( ^ l i n l O ^ at room temperature but worsening to ^  1 in 10  upon c o o l i n g to 4.2K) .  For best n o i s e  immunity, the  connection to the outermost windings was put at ground p o t e n t i a l . The s p h e r i c a l sample  (radius  a) and balanced pickup  form" a very convenient d e t e c t i o n arrangement. o f the sample ensures a uniform magnetizing c o n c u r r e n t l y forms a s p a t i a l l y  coil  The s p h e r i c i t y  field  i n s i d e , and  inhomogeneous d i p o l e magnetizing  f i e l d o u t s i d e the sphere as shown in F i g u r e k. The c o i l is given  former f o r the sample c o i l  (a f u l l  description  in Appendix A) formed the housing o f an i n t r i c a t e  r o t a t i o n system designed to r o t a t e the sample about an a x i s 90° away from the o n l y d i r e c t i o n o f a c c e s s . Sadler  (1969)  Pippard and  d e s c r i b e a system which uses very l i t t l e space in  the sample region by employing a Mylar gear w h e e l . modifications spherical  Several  to t h i s d e s i g n were made to accommodate our  sample, and compactness  requirements.  of the c o n s t r u c t i o n a r e presented in Appendix A.  The d e t a i l s  34  H_  +  h  m  -  /3  M  s<  balanced  N\\l///,  pick - up *  coil  i ' ' (~l in I0 ) 4  B (r<a):  add uniform field j  B (r>a):  add M ^  (-f)  3  {2? cos0 + 0 sintf}  (dipole field) F i g u r e 4.  Detection  4?M  Arrangement  Although to n o i s e total  [19] g i v e s a 1/p dependence to the s i g n a l  r a t i o , we were working  in the regime where the  n o i s e was predominantly determined by the  n o i s e of  the f r o n t end d i f f e r e n t i a l a m p l i f i e r .  quency response c o n s i d e r a t i o n s , a superconducting it  input  turned out that  measurement  sample c o i l  however,  was wound.  For  (see s e c t i o n  rendered the c o i l  d h of the modulation c y c l e where —=dt  5-5)  Unfortunately,  the l a r g e modulation employed  techniques  fre-  normal  in our in  parts  2  derivative  is  c a p a c i t a n c e of currents.  important s i n c e the c o i l  was  large.  i t , along with the  determines the i n t e r n a l  A c a l c u l a t i o n performed a f t e r  wound and used showed that these  self induced  the c o i l  was  induced c u r r e n t s exceeded  the c r i t i c a l c u r r e n t f o r the w i r e at we were working.  The second  the f i e l d s  in which  36  5.3  Modulation  Coil  and Superconducting Magnet  Both the use of the feedback t e c h n i q u e , and the d e t e c t i o n of  long p e r i o d dHvA o s c i l l a t i o n s  of  large modulation.  benefit greatly  Homogeneous  modulation of >1 kG peak  peak amplitude could be achieved w i t h the f i n a l Wrapped by Richard C h r i s t i e ,  the c o i l  superconducting wire cluded). of  (0.0065  a  niobium  inches  to  apparatus.  took the form of an  long s o l e n o i d w i t h an O.OO986 m inner r a d i u s . 316 turns each were wound using  from the use  0.0602m  Four l a y e r s  h$% t i t a n i u m  in d i a m e t e r ,  of  alloy  insulation  in-  The f i n i t e s o l e n o i d equation g i v e s a Gauss-to-amp r a t i o  y=251 f o r t h i s  geometry.  o f £ 1 kG, s e v e r a l the t r a n s m i s s i o n  In o r d e r to modulate w i t h  amplitudes  amps a r e r e q u i r e d to power the c o i l .  of this  '  Since  c u r r e n t would be a major heat l e a k to  the helium b a t h , a simple c a l c u l a t i o n was done to o p t i m i z e the diameter o f the leads sistive  to the modulation c o i l .  h e a t i n g and c o n d u c t i o n  w i r e gave roughly  Taking the r e -  i n t o a c c o u n t , copper and b r a s s  the same heat  leak f o r t y p i c a l  currents  optimum d i a m e t e r s were, of c o u r s e , very d i f f e r e n t ) . chosen because of For t y p i c a l  its  smaller  c u r r e n t s , the heat  level  Brass was  temperature c o e f f i c i e n t o f  resistance.  l e a k was estimated to be 0.1  f o r the 1.4 mm o p t i m i z e d diameter b r a s s w i r e . w i r e was c o n t i n u o u s l y  (the  Watt  Superconducting  s o l d e r e d to the brass up to the maximum  of the helium b a t h . In a d d i t i o n  to screw mounts,  grease was used  as  a  low temperature g l u e to ensure the r i g i d i t y of the mount.  The  main magnet was b u i l t by American Magnetics  (A.M.I.  #10066) q  and was  rated a t 80 kG w i t h a homogeneity of  over a 1 cm diameter sphere at current  its  centre.  1 part  Vapour c o o l e d  leads were used to m i n i m i z e t h e heat l o s s ;  at peak f i e l d was 65 Amperes.  in 10  the c u r r e n t  The flow r a t e o f helium through  these leads could be c o n s t a n t l y monitored w h i l e r u n n i n g . Gauss to amp r a t i o o f the main magnet was  1229.  The  The assembly  i n c l u d e d a p e r s i s t e n t c u r r e n t s w i t c h , which allowed the magnet to run w i t h o u t an e x t e r n a l power supply once i t was e n e r g i z e d .  5.4  The C r y o g e n i c Housing  the main magnet i s an Oxford l i q u i d helium dewar  w i t h the usual c o n t a i n super  Apparatus  l i q u i d nitrogen jacket.  The vacuum spaces  i n s u l a t i o n f o r maximum thermal  isolation.  t h i s o u t e r dewar is an inner dewar w i t h a t a i l the main magnet c o r e ,  i n s i d e the modulation  B u i l t by P e t e r Haas in the P h y s i c s  extending  machine shop at  v a l v e which when opened allowed a t r a n s f e r o f inner dewar.  into  coil.  the inner dewar f e a t u r e d an e x t e r n a l l y c o n t r o l l e d vacuum  the o u t e r dewar to t h i s  Inside  U.B.C., tight  l i q u i d helium from  Not o n l y does t h i s make  the t r a n s f e r process more c o n v e n i e n t , but the helium t r a n s f e r r e d to the inner dewar (and sample) air  particles..  c o u l d be f i l t e r e d to remove s o l i d  The vacuum j a c k e t o f the inner dewar allowed  the pumping of the helium i n s i d e to a t t a i n temperatures of 1.2K.  For best n o i s e  immunity, the inner dewar was  about  isolated  m e c h a n i c a l l y from the main magnet, and modulation c o i l .  This  38  TRANSFER VALVE  LEVEL DETECTORS Inner Dewar Outer Dewar LIQUID HELIUM (T-4-.2 K) LIQUID HELIUM ( l . 2 K < T M . 2 K)  INNER DEWARj HELIUM FILTER  INNER DEWAR  HELIUM RESERVOIR OF OUTER DEWAR  SUPERCONDUCTING MODULATION COIL MAIN MAGNET COIL  Figure 5-  General  Schematic C r y o g e n i c Assembly  39 isolation  c o u l d be checked e x t e r n a l l y with a c o n t a c t  resistance test.  Both the o u t e r and inner dewars  l i q u i d helium l e v e l  contain  d e t e c t o r s manufactured by American  F i g u r e 5 shows a composite drawing of the c r y o g e n i c  5.5  Signal  Magnetics.  apparatus.  Processing  As p r e v i o u s l y d i s c u s s e d  (Section  c o n s i s t e d of 2 counter-wound s o l e n o i d s to be i n s e n s i t i v e  to uniform f i e l d s .  5.2),  the pickup  balanced at  coil  room temperature  A lead from the c e n t r e tap  o f these c o i l s was made a v a i l a b l e a t the top o f the c r y o s t a t f i n e tune the balance as  the c o i l s were c o o l e d .  The major c o n s i d e r a t i o n c i r c u i t r y was ponents  to  in the d e s i g n o f the d e t e c t i o n  the e l i m i n a t i o n of any frequency dependent com-  in the feedback network.  S i n c e the pickup c o i l s ,  modulation c o i l s would have c o n s i d e r a b l e  i n d u c t a n c e , the  procedures were used to e l i m i n a t e p o s s i b l e by t h e i r r e a c t a n c e .  Care was taken  t u n i n g c i r c u i t to ensure t h a t p o i n t where phase s h i f t s  phase s h i f t s  and following  caused  in the d e s i g n of the f i n e  the c o i l s were not  loaded to the  might be important at the maximum  frequency. The pickup c o i l ' s  inductance was c a l c u l a t e d to be 18 mH as  an upper l i m i t , and 500 Hz was chosen as the maximum frequency to be handled by the feedback network. The b a l a n c i n g arrangement  i s shown in F i g u r e 6.  v a l e n t c i r c u i t f o r the b a l a n c i n g  circuit  The e q u i -  is shown in F i g u r e 7.  ho  i  ± LOW NOISE DIFFERENTIAL AMPLIFIER P.A.R. 113  iqure 6.  F i ne Tun i ng C i rcu i t  •Wr  R  R*  Figure  1.  ^1  Rj INPUT IMPEDANCE OF PAR 113  2  X  E q u i v a l e n t C i r c u i t f o r Tuning Arm  41  The e x t e r n a l If  R  load on the c e n t r e tap (C.T.)  is the p a r a l l e l  is a t most R^  combination o f R„ + R_ and R., 2 3 i  R (R R =  Then  V = o  + R ) — R. + R„ + R, i 2 3 V.R 1  Rg+jiol_2+Rj+R  V.R  (R +R  +R  [21] (Rg+R^R)  The phase s h i f t  [22]  2  -  M  2  + (ojL )  )  2  2  is  < | > = tan  R^+R^+R  The D.C. r e s i s t a n c e o f the counterwound c o i l was R^ = h2ti at 4.2K hence, f o r a phase s h i f t  o f <1° a t 500 Hz,  Rj + R > 450 ti  As  i t turns o u t , the s e l f  determining f a c t o r .  The s e l f  c a p a c i t a n c e o f the c o i l capacitance of a c o i l  good a p p r o x i m a t i o n , be r e p r e s e n t e d by a p a r a l l e l  was the  can t o a  c a p a c i t o r shown  as C<. in F i g u r e 6. The f a c t t h a t  this  c a p a c i t a n c e was the dominating  influence  on the frequency c h a r a c t e r i s t i c s was a s c e r t a i n e d by unbalancing the f i n e t u n i n g s l i g h t l y ,  and o b s e r v i n g  the frequency  response  of the pickup c o i l s  to the modulation f i e l d .  When performing  t h i s measurement, w i t h R j » 450 fi, the frequency response of pickup c o i l  rolled off  When the s e l f voltage  the  to -3db at 90 Hz.  capacitance  is  taken i n t o a c c o u n t ,  the  in F i g u r e 6 is given by  [23]  V  = - -JL coCc (  o  J  s  where  L  [24]  ^o V  j  (  a  , -J_) L  = Lj + l  Hence  l-a) LC -  jRcoC  2  =  i  (l-  2 U  LC)  2  + (wRC)  If we assume L is n e g l i g i b l y  v  ) i V  + R s  2  small,  2  1 1+ UR C 2 )  o V.  S  S  i  From the frequency c h a r a c t e r i s t i c s , be 25.8 y F .  Using  the f u l l  equation  was determined t o  [24] w i t h L.=  18 m.H d i d  change the r o l l o f f f r e q u e n c y , p r o v i n g our assumption negligible. limiting  that L  The frequency c h a r a c t e r i s t i c o f the c o i l  factor  in the feedback l o o p , and  is  not is  the  in f a c t s e t s the upper  bound to the usable feedback band. The modulation c o i l amplifier.  The c o i l  was e n e r g i z e d with a Crown M600  was placed  5 fi n o n - i n d u c t i v e monitor  D.C.  in a s e r i e s combination with a  resistor.  The inductance o f the c o i l  h3 produced a l a r g e phase s h i f t 50 Hz.  This s i t u a t i o n  feedback.  is e a s i l y  remedied by using  A tap was taken between the 5  modulation c o i l voltage  even f o r f r e q u e n c i e s as  (see F i g u r e 9 ) ,  f o r the o p e r a t i o n a l  The Crown a m p l i f i e r gain  is  low as  current  resistor,  and the  and used to p r o v i d e a feedback  a m p l i f i e r f e e d i n g the Crown a m p l i f i e r . then immaterial  so  long as  it  is  large  enough and the c u r r e n t gain of the op-amp-Crown c o n f i g u r a t i o n determined by o n l y 2 r e s i s t o r s . shifts  this  i d e a , the phase  were e l i m i n a t e d up to 1 kHz, above which the g a i n s needed  would s e t  the system  There this  Using  into  accidental  its  gain, while s t i l l  being  sufficient  l i n e a r i t y , must be kept below the p o i n t where d i s c o n n e c t i o n o f the feedback r e s i s t o r  catastrophe.  The Crown gain s e t t i n g  without harming  the o v e r a l l  results  can be used as a  open-loop g a i n ,  the c u r r e n t needed (see F i g u r e 9)•  in  limit,  s i n c e the op-amp  p r o v i d e s most of the gain and merely s a t u r a t e s  set  at  S i n c e the Crown a m p l i f i e r can d e l i v e r enough c u r r e n t  to damage the a p p a r a t u s , to a s s u r e  oscillations.  is a s a f e t y p r e c a u t i o n which must be emphasized  point.  deliver  is  if  the Crown cannot  The Crown gain was  to d e l i v e r the maximum a l l o w a b l e output c u r r e n t f o r a 15V  input s i g n a l . The v o l t a g e  appearing on the w e l l - b a l a n c e d pickup c o i l s  is  dM proportional integration.  to  -pp  M can be r e t r i e v e d e a s i l y by  The s i g n a l  modulation s i g n a l  proportional  then added to the  and fed to the modulation c o i l  A b l o c k diagram o f the apparatus diagram appears  to M was  in F i g u r e 9.  appears  analogue  as a c u r r e n t .  in F i g u r e 8.  The c i r c u i t  kk  balanced pick up coil  + Differential amplifier  LA. h  VW  1  °  dM dt  Spectrum Analyzer  Voltage Gain ~500  balance trim  X-Y Recorder  Superconducting Modulation Coil h « 1 kG m  Integrator  oscillator  P-P  h -A/Wm  - / 3 M ~ 1G P-P Adder  ' h -/3M m  Power amplifier  Figure  8.  Block Diagram of  Apparatus  INTEGRATOR  INVERTER  ADDER AND AMPLIFIER  CLOSED FOR NO FEEDBACK  FEEDBACK GAIN ADJUST  >  FROM PAR 113  -WAr  10 K  10 K  F i g u r e 9-  Circuit  Diagram of  Integrator  and Adder  A simple arrangement was used f o r t e s t i n g response o f the e n t i r e c i r c u i t r y . balancing after  the pickup c o i l  slightly,  the f r e q u e n c y  The procedure i n v o l v e d unand o b s e r v i n g  the s i g n a l  i n t e g r a t i o n on one channel of a d u a l - t r a c e o s c i l l o s c o p e .  The feedback path was broken at the adder, so that o n l y modulation s i g n a l was fed to the modulation c o i l . signal the  the  The modulation  from the o s c i l l a t o r was monitored on the second t r a c e o f  'scope.  The two t r a c e s  were c a r e f u l l y superimposed at  some  low f r e q u e n c y , and then the frequency o f the modulation was increased u n t i l noticeable. was 0.2  a s e p a r a t i o n between the 2 t r a c e s became  The  u s a b l e feedback band determined in t h i s way  to 79 Hz.  bandwidth  is  It  must be p o i n t e d out t h a t  not very r e s t r i c t i v e .  this  limited  The low frequency l i m i t was  imposed by a 10 sec time c o n s t a n t on the i n t e g r a t o r which c o u l d be i n c r e a s e d  in the f u t u r e .  lowest frequency is  In the procedure used h e r e ,  t h a t of the f i e l d modulation and dHvA terms  can always be made to appear at h i g h e r Initially direction.  the  time f r e q u e n c i e s .  the lead sphere was o r i e n t e d c l o s e to the  [110]  So o r i e n t e d , lead e x h i b i t s a p a i r o f s t r o n g y  o s c i l l a t i o n s with F^'v 17.9, and  a s i n g l e a o s c i l l a t i o n with  F^d&OMG.  We wene most f o r t u n a t e to bb.ta i n^the use of a d i g i t a l ^ spectrum analyzer  (Hp3582A). wh-ich is  e s s e n t i a 11 y' an on-1 i ne, F o u r i e r t r a n s -  former. • When a p p r o p r i a t e l y set up, t h i s the harmonics of at F  cx  ± nF  described.  y  computer c o u l d  the y d o u b l e t , as w e l l as the sidebands  due to M.I.  resolve appearing  T h i s " a p p r o p r i a t e " setup wiell now be  47  A t r i a n g u l a r wave was used to p r o v i d e a ramp in the modulation f i e l d ,  and i t s amplitude was chosen to sweep  enough dHvA o s c i l l a t i o n s exceed ^ B)  = 18 MG.  h oscillations  resolve a l l  f o r the r e s o l u t i o n of the a n a l y z e r to  When a Hanning window  is used,  easily  In p r a c t i c e i t was not d i f f i c u l t  sweep through about 5 fundamental y o s c i l l a t i o n s number of c y c l e s could be i n c r e a s e d by working at  2  (see Appendix  of the fundamental were enough to  the harmonics.  because o f the H  through  at 60 kG; lower  to  this  fields  H dependence of the f i e l d s p a c i n g AH % y . 2  At 60 kG, the f i e l d at which most of the work was done, AH^ * 200  G, and a peak to peak ramp o f  1 kG sweeps through 5  y osci11 at ions. The spectrum a n a l y z e r o b t a i n e d  its  the f r o n t - e n d d i f f e r e n t i a l a m p l i f i e r  input d i r e c t l y from  (see F i g u r e  8).  The modulation frequency was chosen to be 1 Hz, hence the time window on the a n a l y z e r should be a l i t t l e l e s s 0.5  s e c , t r i g g e r e d at the b e g i n n i n g of the r i s i n g  the t r i a n g u l a r m o d u l a t i o n . of the t r i g g e r If  F i g u r e 10 shows the s y n c h r o n i z a t i o n  a l i m i t e d number of c y c l e s  is c o n s i d e r e d , the o s c i l l a -  p e r i o d i c in H, and a l i n e a r f i e l d ramp  t r a n s f o r m the dHvA o s c i l l a t i o n s  t h e i r time frequency is given by  [25]  ramp of  and the time window.  tions are e s s e n t i a l l y will  than  h F f , f = , ° m  H I  d  to the time domain where  48  TIME WINDOW  OPEN  LO  1  CLOSED  \-  OPEN  1.5  0.5 TIME (SEC)  F i g u r e 10.  S y n c h r o n i z a t i o n o f the Time Window  49  where  and  H  is  the steady background  h  is  the P-P modulation ramp amplitude  ^mod  '  F  is  the dHvA frequency  D  is  the duty c y c l e o f the modulation waveform.  s  t  '  i e  m  °d l tion u  a  field  frequency  For a t r i a n g u l a r wave, the duty c y c l e is  ^ _  2h F f H  For f  m o d  1/2  so  that  , mod  2  = 1 Hz, H = 60 kG,  h = 1 kG, the y  appear at a frequency of ^ 10 Hz, the a ' s  oscillations  at ^ 89 Hz.  50  CHAPTER  SIX  EXPERIMENTAL TEST OF THE FEEDBACK TECHNIQUE  6.1  Preliminary Initial  Considerations  experiments were performed on a s i n g l e  lead sphere o r i e n t e d with In  this  sisting  the t o t a l  d i r e c t i o n : , lead e x h i b i t s  3  of A  % A  a :  Y  applied f i e l d along  b  £  1,.  Gauss at  amplitudes  in a f i e l d of 60 kG.  There  Y  a r e a l s o somewhat weaker a o s c i l l a t i o n s  at a frequency of  and under the above c o n d i t i o n s , A^ £ 0.03 oscillations It 0.4  F^ ^ 18 MG.  have approximately equal 1.2K  [110].  s t r o n g y o s c i 11 at i ons c o n -  of two f r e q u e n c i e s separated by 0.42 MG at  These 2 f r e q u e n c i e s y ,  crystal  are shown in F i g u r e  Gauss.  These  11 (H ^ 60 kG, T ^ 1.2  turned out that the expansion parameter KA^ was  in the c o n d i t i o n s  to be q u i t e  o b s e r v a b l e q u a n t i t i e s on feedback g a i n . to agree w e l l with the expected r e s u l t s optimum feedback gain using  under non-optimal  about  high.  c h a p t e r , we i n t e r p r e t the dependence of  Much of what f o l l o w s ,  k)  under which we u s u a l l y worked, so we  expected the harmonic d i s t o r t i o n In t h i s  160 MG,  The technique appears f o r no M.I.  the c r i t e r i a presented  t h e r e f o r e is  several  at  in Chapter  the a n a l y s i s of the  feedback c o n d i t i o n s ,  to ensure an  the IV.  quantities  understanding  51  Pb  M || [110]  or , y Figure  plus 11.  M.I.  y  oscillations  generated  crimy in Pb:H|J [110]  de Haas-van Alphen O s c i l l a t i o n s . . Above: y o s c i l l a t i o n s , below: ^ 4 y oscillations which were e x p e r i m e n t a l l y suppressed t o b r i n g out the a o s c i l l a t i o n s and M.I. sidebands  52 of the mechanisms  i n v o l v e d , and at  the same time,  shows the very t e d i o u s mathematics one avoids feedback t e c h n i q u e .  While  must remember that at correction factors  it  is  important  it  by usinq  For each q u a n t i t y d i s c u s s e d , we s h a l l ,  the  to check t h i s  the optimum feedback p o s i t i o n ,  in any of the r e s u l t s  clearlv  once, one  the  are no longer needed.  in t u r n ,  look at  the  cases of no feedback, non-optimum feedback and optimum feedback. In some c a s e s ,  6.2  a v a r i e t y of feedback s e t t i n g s were used.  M i n i m i z a t i o n of  Sidebands  In o r d e r to o b t a i n an s e t t i n g was, first. Using  It  the m i n i m i z a t i o n - o f - s i d e b a n d s  is  the s i m p l e s t  the t r i a n g u l a r  oscillations  idea where the optimum feedback  and perhaps a l s o the most d r a m a t i c .  m o d u l a t i o n , about 5 fundamental  at F  the feedback gain u n t i l  nF  i  the m i n i m i z a t i o n of sidebands  v  were e a s i l y  criterion,  these sidebands  the sidebands  amplitude spectrum f o r s e v e r a l 12.  Figure  Using  the o b j e c t was to  adjust  reached a minimum.  Examples of  a + y,  feedback s e t t i n g s appear  and a-y  feedback,  the F o u r i e r  13 is a q u a n t i t a t i v e p r e s e n t a t i o n of  dependence of the a , gain.  resolved.  decreased f o r n e g a t i v e  increased f o r p o s i t i v e feedback.  Figure  y  the gain from zero in e i t h e r d i r e c t i o n produced the  expected r e s u l t s , and  used  were swept o u t , and analyzed by the spectrum a n a l y z e r .  About seven sidebands  Changing  c r i t e r i o n was  in the  amplitudes on the feedback  53  positive feedback  _  without feedback  negative ^ feedback  ar  or  a:  J  ~ optimum negative feedback Figure  12.  F o u r i e r Transforms Around 160 MG As A F u n c t i o n Of Feedback G a i n . (The transforms do not s t a r t at F=0) H = 6 1 . 7 9 kG T = 1-.2K  5k  .(M.I.) Ajty  * if f * \ -A  f f  J  (f ^A )±A J (f £A )  x  6 = 1 -  a  r  r  1  /  a  4TT(1-S)  H2  0.20r  —>  0  Sign of Feedback  0.2  0.4  0.6 TT  Figure  13-  (1-8)  0.8 -  F o u r i e r Amplitudes Feedback Gain  I  1.2  i - £  As A F u n c t i o n Of  1.4  55 To compare the observed a m p l i t u d e w i t h t h e o r y , use the convenient  let  us  notation:  2TT F f  =  e  = 1m (1-6) M  A"  1 -  =•  M = 4TT(1-6)M Y,<* Y,«  7 — 7 ^ — r -  4IT(1-6) We expect the s t r o n g e s t the Y term on a .  M.I.  The fundamental  c o n t r i b u t i o n from the e f f e c t of dHvA m a g n e t i z a t i o n  for  a  is g i v e n by  [26]  M  a  = A"  a  where h is a small  sin  F [2ir(-=^-> 1/2) D+n  change  -  TT/4]  in H.  The phase f a c t o r s  need not concern us f o r the p r e s e n t , and F h we can expand the denominator f o r << 1 to o b t a i n a f i r s t 2 o r d e r approximation H  M  = A  cos  a  assuming that in B is A'^ cos harmon i cs  [f  (h+eA a  Y  cos  f  h)] Y  the dominant c o n t r i b u t i o n to the t o t a l f h.  T h i s can  magnetization  immediately be decomposed  into  56  [27]  M  =  ( l )  A  J (f 0 a  I  eA*)[sin(f + f ) h + sin(f Y a y  a  e A * ) [ s i n ( f + 2f y a y  0  2  a  a  - A* J ( f a  f h  y  - A* J , ( f a  cos  eA")  n  a  -  f )h] a y  ) h + cos(f  a  + A* J , ( f e A * ) [ s i n ( f + 3f ) h + s i n ( f a 3 a Y a Y  where J  (x)  is  the Bessel  f u n c t i o n of  -  the f i r s t  2f)h] y  - 3f ) h j a y  kind of o r d e r  v  v and argument This  x.  first.step  generates  sidebands  next s t e p  in the c a l c u l a t i o n  f  to  itself  is  around f  to a l l o w  , and the  these sidebands  i n t e r a c t with the y o s c i l l a t i o n s .  and  The equations  a  are  identical  referring  to those  in the f i r s t  to the c o r r e s p o n d i n g  needs c a r e f u l bookkeeping. the  contributions.  we qet terms  in f  a+Y  step w i t h the  reversed r o l e s .  This  step  is  f  a-y  '.  One o n l y  c a r r i e d out o n l y  For example, when f and  subscripts  to  i n t e r a c t s with  The r e s u l t  i s as  y,  follows:  57  [28]  M  ( 2 )  = A* J Y  f  + A" J Y  f  A" J Y  f  + A" J Y  f  - A'"' J Y  f  + A J Y  f  + A" J Y  f  + A" J Y  f  + A" J Y  f  + A" J Y  f  x  eA*  )cos[f -(f  a+Y  Y  Y  a  + f •)•]  terms at f r e quency f  Y  Y  eA" ) [-cos{f +(f - f ')}] a-y y a y  Y  eA* ) s i n a  (f + f ) y  eA* ) sin Y a+zy  sin  a  (f - f ) Y a+zy  Y  eA*) a  Y  eA* . ) s i n a-zy  Y  eA*^ ) [ - c o s ( f + f ^ )] a+Y Y c Y  (f + f , ) Y a-2Y  Y  )[-cos(f  a+3y  - f  Y  a + y  terms at f r e quency f a-Y  terms at f r e quency f  +  eA*  Y  (f - f ) y a  terms at f r e quency f  a+2y  )]  a+3y  eA* ) c o s ( f . ~ f ) a-y Y OL-y  terms at q  eA* . ) c o s ( f + f Y a~3Y Y  U  e  n  C  y  . )  fref  a-2  Y  a~3Y  (2) In t h i s  second step  from those generated T h i s expansion  M  , the amplitudes A are to be taken  in the f i r s t  should  be  a  step  M ^ .  reasonable approximation to the  gain dependence in our p l o t s of the sideband amplitude The expansion  reduces, to the s i m p l e r form, given  (Figure  earlier  ([13a], [13b]) i f we c o n s i d e r o n l y the f i r s t 2 combination f r e q u e n c i e s , and a l l o w o n l y y to i n t e r a c t w i t h a.  We then o b t a i n  13)  58  A" = A J . ( f eA ) + A" J . ( f E A ) a+y a 1 a Y y \ y a t  and using  Ji(x)  the small  argument  - TT , x<< 1,  A* a+Y  this  e A" A" « — f J t 2  Similarly,  oi-y  ~-  reduces  (  f  +  to  )  f  a  Y  , , eA A' w  A"  approximation  c  -fJt ( . ) f  2  A'^ remains unchanged  f  a  y  in t h i s  approximation.  This  is  the  r e s u l t o b t a i n e d e a r l i e r with e = l f o r no feedback (see We immediately see that  the s i m p l e approximation  to d e s c r i b e the l a r g e e f f e c t M.I.  is  [ 13a] , [ 13b]) .  insufficient  has on our d a t a , s i n c e A* '  3  indeed change, and the sidebands  simplified  do not  rise  •  does  a  l i n e a r l y as  this  pred i c t s . In o r d e r to use the more d e t a i l e d formulae find  the a b s o l u t e m a g n e t i z a t i o n of at  terms. is  This  can be done in a few ways.  to measure A'^ by o b s e r v i n g  The r e l a t i v e magnitudes Fourier of  least  transform.  one o f  we must  the fundamental  The most s a t i s f y i n g  way  the feedback f i e l d at the optimum 3 .  o f A ^ and A'^ canbe read d i r e c t l y o f f 5  An a l t e r n a t e method  the a o s c i l l a t i o n s  [28],  to J (f eA" ) .  o a e = 1, and the m a g n e t i z a t i o n at '  Y  this  is  to f i t  When t h e r e point  the  the g a i n dependence is  no feedback,  is M , = A" J (f A" ) • a,e = l a o a Y  59 At optimum feedback, e = 0, and M = A , hence a,e=0 a n  r  T T ^ = J (f A ) M „ o a y e=0  which y i e l d s Using 6.2  A".  the feedback technique and [18],  Gauss in a f i e l d of 61.82  included  in t h i s  notation).  kG.  (Keep  A'  was measure to be  in mind t h a t 4TT(1-<$)  An independent f i t  to the J  is  curve o  y i e l d e d 6.0 Gauss.  This  these e a r l y s t a g e s ,  the feedback was not s e t p a r t i c u l a r l y c a r e -  fully.  The amplitude o f  is good agreement e s p e c i a l l y s i n c e  the a o s c i l l a t i o n s  by t h e i r r e l a t i v e s t r e n g t h  was  in  then o b t a i n e d  in the F o u r i e r t r a n s f o r m .  T h i s gave A* to 0.189 Gauss. Using the d e r i v e d formulae f o r A" , A" ,  •  a'  a+Y  , and A"  , we o b t a i n  a-Y  the feedback dependence:  [28]  A*(e)  [29]  A*  a  = A*(0)  a  - (e)  a±y  J (f eA* ) o a y  = A*(0)  a  = A*(0)  a  J . ( f eA" )  l a  y  J . ( f eA")  l  a  y  ± A * J . ( f eA" ) ± A * J , ( f e A*  y 1 y  a  y 1 a a+2y  ± A * J . [ f A * J ( f eA" )] y 1 Y a 0 a y n  + A" J . [ f A" J _ ( f eA" )] . y 1 y a 2 a y  The r e s u l t s o f t h i s the measured data  c a l c u l a t i o n are shown as s o l i d curves along w i t h  in F i g u r e 13.  are no a d j u s t a b l e parameters  It  should be mentioned t h a t  in the sense t h a t a l l  there  the q u a n t i t i e s  (e  60  needed for the calculations were measured by Independent means. The agreement is exceptionally good for A , and the general  trends  q  for the sidebands seem correct. An independent check on the A^(e) dependence is conveniently afforded by Aoki and Ogawa (1978) who used rod shaped samples. In their data the a o s c i l l a t i o n s  have a very small amplitude, and  only the sidebands appear in the transform.  We expect A*  a  ^  Q  ^  a  ^ '  We measure A^ = 6.0 Gauss, for a sphere, (6=1/3), so that for a rod (6=0), A ^ w i l l have increased by the r a t i o argument of the Bessel f  A  a  * Y  =  2* 0 60  MG)  (61.82 kG)  (].j/3)  =  3/2.  The  function becomes (  3  /  2  )  {  6  Q  )  ^  2  >  3  6  >  2  This is very close to the f i r s t zero of J  q  , which is 2.405.  Extrapolating our data, which was possible at a later date from an amplitude vs. f i e l d measurement, (see Figure 27) we can obtain a more r e a l i s t i c estimate at the f i e l d used by Aoki and Ogawa. At their f i e l d , 50.5 kG, we measure A* = 4.10.  f A* = a  Y  " (50.5)  2 7 7 ( 1 6 0  G )  For their f i e l d ,  (3/2) (4.10)'= 2.42  2  which again is in good agreement with their observation of small fundamental a amplitude. We notice from Figures 12 and 13 that the sidebands never completely disappear.  There are many reasons why this might occur,  the most obvious being a frequency dependence of the feedback network. There is a clue to the o r i g i n of this  imperfection in the  61  amplitudes of the sidebands r  The d a t a show a change must e x p l a i n t h i s  f  ,  and f  a+Y  a-y  r e l a t i v e to each o t h e r .  in t h e i r r e l a t i v e a m p l i t u d e , so the mechan i'sm  change.  The frequency response of the feedback network might be such a mechanism i f the time frequency o f the y term is c o r r e c t l y fed back, but that of the h i g h e r a term is n o t ; however, t h i s the c a s e , as was shown by a s i m p l e experiment. modulation frequency by a f a c t o r of components by the same amount. response  is  Decreasing  10 decreases a l l  not  the  the feedback  There is no doubt that the s y s t e m ' s  is very f l a t below 10 Hz, y e t even when a was made to  appear at o n l y 7 Hz, the sidebands  at optimum feedback s t i l l  appeared w i t h the same a m p l i t u d e . It was thought  that a s p a t i a l l y  f i e l d might a l s o cause the r e s i d u a l back f i e l d  itself  inhomogeneous sidebands.  is o n l y of the o r d e r of  feedback  While the f e e d -  1 Gauss, the s u p e r -  imposed t r i a n g u l a r modulation was about 500 Gauss at value.  This  peak  inhomogeneity would not however, cause the r e l a t i v e  sideband amplitudes inhomogeneous, on o r b i t  its  to change.  Even i f the f i e l d  is  spatially  t h e r e would be no dependence o f the feedback f i e l d  i.e. F Y  and F  would both be fed back the same.  This  a  inhomogeneity may impose a l i m i t on the minimum a m p l i t u d e of s i d e b a n d s , but would not change t h e i r r e l a t i v e a m p l i t u d e s . currents  induced in the sample would cause a s i m i l a r  The m a g n e t o - r e s i s t a n c e of  lead g i v e s  then should not be a problem, but  i f eddy c u r r e n t  Eddy  inhomogeneity.  i t a s k i n depth o f  1 meter f o r our t y p i c a l f i e l d s and f r e q u e n c i e s .  the  about  The s k i n d e p t h , inhomogeneities  62  were p r e s e n t , they would not account f o r the n o n - v a n i s h i n g s idebands. The most l i k e l y e x p l a n a t i o n sample.  is  phase smearing  in the  T h i s sample had been t h e r m a l l y c y c l e d between room  temperature and 4.2K t h r e e times b e f o r e these data were t a k e n . If  dislocations  and s t r a i n had b u i l t  up, the change  d e n s i t y would change the frequency o f the o r b i t s immediate a r e a , making oscillations  6.3  The Mass  in e l e c t r o n  in the  the optimum feedback a f u n c t i o n o f  the  under c o n s i d e r a t i o n .  Plots  The temperature dependence o f one dHvA harmonic amplitude is given by (see  \  ( T ) a  [lb])  7fnF7x  X  =  2  *  2  < k  B  T / ( e * B ) .  63 In the high temperature l i m i t  (X ~ 3),  the h y p e r b o l i c s i n e f u n c t i o n 1  can be r e p l a c e d w i t h Then,  its  exponential approximation: sinh X £ 2  letting  \ = 2 r m'k /(e'R ) = 146.9 ^ ,  we have  2  1  [31]  B  A (T)  a T  r  e  '  r  X  y  T  /  y =  H  ^  m  1  hence, a p l o t of £n ^ vs. T/H should y i e l d a s t r a i g h t slope -rAy.  T h i s a p p r o x i m a t i o n u s u a l l y h o l d s , but  of the fundamental  y osci11 at ions at  1°K and a f i e l d o f 60 kG is about exponential  1.4,  and the e r r o r in the A simple c o r r e c t i o n  from an i t e r a t i v e scheme to  determine y from data which arenot s a t i s f a c t o r i l y high thermal smearing  limit  (X ^ 3)•  =  r  rX s i n h rX  r X  + £n ( l - e  2 v r  A  )  + constant = - r X  or )A(l-e" Jin < r _  2 r X y  A  U  6  scheme,  2rXe" . -2rX 1-e  =  f i e l d we have A (y)  i n t o the  ,  where C is a temperature independent c o n s t a n t .  £n  far  To develop t h i s  we s t a r t w i t h the b a s i c h y p e r b o l i c form  CTA  in the case  [110], the v a l u e o f X at  approximation becomes about 6%.  term can be used, and i t f o l l o w s  l i n e of  T / H  ) ' ? =  -  rXy T/H  Then at  constant  X  If  n is  the o r d e r o f  i t e r a t i o n , the v a l u e y  is  obtained  from  [32]  4  h^(l-B-  , (0) where y ->- °°.  2  r  X  w  (  n  '  l  _ Convergence  to the d e s i r e d degree o f  harmonic. optimal  This  /  M  \}-rXy  15.  n  )  T/H  = y  in curves f o r a l l  shown in F i g u r e  Table  measured with near optimal  14.  but the  first  Application of  near-  out these curves  as  I I shows the e f f e c t i v e masses feedback  in the [110]  direction,  along w i t h t h e i r c o u n t e r p a r t s d e r i v e d from data given by and Gold  (n)  a r e observed w i t h o u t f e e d b a c k , a  feedback promptly s t r a i g h t e n s  shown in F i g u r e  (  accuracy.  T/H r e s u l t s is  T  , . . , (n-1) is a c h i e v e d when y  When the y o s c i 1 l a t i o n s p l o t o f Jin j vs.  )  Phillips  (1969).  Table  II  E f f e c t i v e Mass f o r Observed O s c i l l a t i o n s  Osc i11 at ion  y(Measured)  in Pb:HJj [110]  y(Derived)  a  1.08  1.1010.01  a+y , a-y  1.74  1.66±0.02  y fundamental  0.565  0.56±0.01  Y 2nd  Harmonic  1.00  1 .12±0.02  Y 3rd  Harmonic  1.35  1.68±0.03  [Values of y (measured) are d e r i v e d from the s l o p e s f o r non-optimal feedback]  of  F i g u r e 15  Figure  14.  Mass P l o t s With No Feedback  66  Near-Optimal Feedback  F i g u r e 15-  Mass P l o t s With Near-Optimal  Feedback  67  The agreement for the fundamental o s c i l l a t i o n s is  to be expected because they a r e e s s e n t i a l l y  The combination in the s i m p l e  ot+y, '  a-y,  a r e generated by M.I.,  M.I. and  A A  « J C (f  a± y  2  ± f a  ) y  The temperature dependence  is  -A(y +y [33]  u n a f f e c t e d by  approximation  -  A  terms  i s good, which  A  a ± y  (T)  = T  )T/H  e  Our measured v a l u e comes from the s l o p e o f a p l o t  of  A In  '"TY T the  VS.  T / H , and the d e r i v e d v a l u e  is  the sum of y  2  In  values  ideal  theory,  in T a b l e II  w i t h minimal  i n t e r f e r e n c e from M.I.  the presence o f residual  M.I.,  residual  [34]  1  the c o n v e n t i o n a l is  is  1:2:3.  The d e r i v e d  Upon measurement o f  in a c c o r d w i t h t h i s M.I.  ratio  these  suggesting  In o r d e r to take account o f  the  p r e s e n t because the feedback g a i n was not set  optimum, the c o n v e n t i o n a l  and A  the r a t i o y ^ r y ^ y ^  are y^, 2y^ and 3yj> s i n c e y^ can be measured  v a l u e s , we found them not  In  and y . . Y  theory o f M.I. theory,  2  ]l  applicable.  i f A r e p r e s e n t s the L.K.  the i n t e r a c t i n g a m p l i t u d e ,  A^ = A  is  to  amplitude,  68  To find the amount of M . I . A^ from [34] adjusting  e  s t i l l present, we c a l c u l a t e  u n t i l the temperature dependence plot  y i e l d s the Ideal e f f e c t i v e mass. Writing  A  C, X e  -X  . . and A  2  _ = ^  2  .. -2X Xe  to extract the temperature dependence, we obtain  [35]  A  2  = K  X e " { l - ? X + 1/2 £ 2X  2  X }  2  2  1 / 2  where and  [36]  In  X[1-SX+1/2S X ] 2  Plotting  the  2  1/2  l e f t side of this equation against  T/H  should  y i e l d an ideal slope of -2Xy^ - 165. A numerical c a l c u l a t i o n gave the value 5 = 0.62 as the best f i t line with the desired slope. With this value of ?, the correction factor [1-?X + V 2 t ; X ] 2  attains  the values  2  1 / 2  1.47 to 0.71 between 3.4K and 1.2K, respectively  which is appreciable, but not d r a s t i c enough to warrant further terms in the expansion. The same treatment can be applied to the t h i r d harmonic. The conventional  interacting theory in the presence of feedback  gives (see [7c]).  [37]  3K E AjA  3< e AjA  2  + 1/2  ft  2  A3  3< e A; " 2  8A„  2  V  /  2  69 -  As b e f o r e  = ^  n =  [38]  X e  ,  F  3X  introducing  the equation  becomes  A^ = A { l - 3 ? n X + 1/2 ( 3 ? n X ) [ l - l / 2 a] + ( 3 / 4 ? n X ) } 2  2  2  2  1 / 2  3  The c u r l y brackets A  SLn  , and  give  the c o r r e c t i o n f a c t o r , so that  plotting  3 r-  x{  y  /9  v s . T/H should g i v e a s l o p e of ~3Xy  u  c h o i c e o f r\. n = 1.1  f o r the a p p r o p r i a t e Y  yields  Using  the v a l u e o f t, c a l c u l a t e d b e f o r e , a v a l u e o f  the c o r r e c t s l o p e f o r the t h i r d harmonic  temperature.  dependence. The agreement o f the c a l c u l a t i o n w i t h the expected suggests we c o r r e c t l y understand feedback s e t t i n g ,  the mechanics o f the non-optimal  and hence we can use the slopes  as a c r i t e r i o n f o r r e a l i z i n g optimum feedback. at optimum feedback a r e shown  in F i g u r e  16.  of s c a t t e r was the temperature measurement; Figure  16 by a s y s t e m a t i c  shift  w i t h i n h%.  The mass p l o t s  The l a r g e s t this  is  made  source  indicated  The r a t i o o f the slopes  in  are now far  l i m i t , and the c o r r e c t i o n term  developed e a r l i e r was employed in t h e i r The Beat  plots  The y^ data were known to be i n s u f f i c i e n t l y  i n t o the high thermal smearing  6.4  o f the mass  in the p o i n t s o f each l i n e which  were taken at the same temperature. 1:2:3  results  analysis.  Pattern  The dominant c o n t r i b u t i o n to magnetic comes from the fundamental  interaction  dHvA frequency s i n c e  it  usually  is u s u a l l y  the  Figure  .020  .025  16.  Mass P l o t s  .030 T/H  at Optimum Feedback  .035 (K/  k G  )  .040  .045  strongest.  If  and b e a t i n g  occurs,  frequency w i l l mental beats  t h e r e are two n e i g h b o u r i n g  frequencies  present,  at  the second  harmonic  beat w i t h a frequency equal  to t h a t of  the f u n d a -  the terms  r a t h e r than at  from M.I.  twice the fundamental  as would normally be e x p e c t e d . harmonics.  As a r e s u l t ,  portional  to the harmonic  In  lead along  frequencies of  (see  10 s t r o n g e r  b  (r)  Figure  11).  t h e r e is no M . i .  The fundamental  is  roughly  0 . 4 2 MG.  M.I.  present. the y  is about a f a c t o r  In  the  ideal  theory we can  due t o the two y f r e q u e n c i e s  by  M  a  M  b  =  I A r  sin  a  [27rr(£ip - y)  . and  I A* s i n [ 2 T r r ( ^ p - r -.total ..a , ..b M = M + M  where  6F  The s i g n  if  the  o n l y be p r o -  [ 1 1 0 ] , we have such a s i t u a t i o n w i t h  r e p r e s e n t the m a g n e t i z a t i o n a  true f o r a l l  the beat frequency w i l l index  frequency  than the second harmonic, and the fundamental  beat frequency  (•Y ,y )  The same is  beat  =  = F  reversal  area o f one o f  -  a  of  b  and  F^F  y) + TT/4]  1 3  the phase f a c t o r IT/4  the c o r r e s p o n d i n g  w h i l e the o t h e r  is  required since  Fermi s u r f a c e areas  is a minimum.(see Ogawa and Aoki A  a  A  a  If we l e t n =  manipulations  F  -  lead  to  -  A 1 , then elementary + A  is a maximum,  (1978)).  b  b  the  trigonometric  72  [39]  M  t 0 t a 1  =  I (A;|+A ) f { ( l + n ) + ( l - n ) s i n ( 2 T : r 6 F / B ) } r b  V  2  where  = tan '{n tan ( u  r  F =  Unfortunately, to o r i e n t a t i o n .  1 / 2  - v) + i f ^ l  x s i n [2irr  and  2  r  i r r  D  S F  - ir A ) }  B  r —  the phase o f the beats  This  i s very  sensitive  is c l e a r l y shown by the l a r g e spread  f i e l d values o f the minima reported in the l i t e r a t u r e . sensitivity  results  oscillations  along  from the r e l a t i v e l y [110].  Beats  low symmetry  in o s c i l l a t i o n s  in  This  in the y  corresponding  to o r b i t s o f h i g h e r symmetry such as 3 a t [100] do not appear t o be so s e n s i t i v e . The beat envelopes f o r t h e f i r s t feedback a r e shown in F i g u r e harmonic appears  17.  t h r e e y harmonics  without  The envelope o f the f i r s t  as we e x p e c t , however that of the second  harmonic c l e a r l y has the p e r i o d i c i t y o f the f i r s t . harmonic beat envelope has a p a r t which  The t h i r d  i s b e a t i n g at t h r i c e  the fundamental beat frequency however, the p o s i t i o n a t the f i e l d c o r r e s p o n d i n g to a f i r s t amplitude,  indicating  harmonic maximum has h i g h e r  that a t l e a s t  some o f the amplitude  is  due to terms generated from M.I. With n e a r - o p t i m a l harmonic  appears  the fundamental a f f e c t e d by M.I.  feedback, shown in F i g u r e 18, the second  t o be approaching a b e a t i n g p a t t e r n at twice  frequency. effects.  The t h i r d harmonic  is  still  73  Pb  y  oscillation envelopes  0.015  0.016  0.017 '/H ( k G " ) 1  Figure  17.  H|| C MO]  0.018 —*>  Beat Envelopes Without (A., A_, A in Gauss)  Feedback  Ik  I  L  .015  .016  I  L_  .017  .018  l/H (kG" ) 1  Figure  18.  Beat Envelopes With Near-Optimum Feedback (A,, A „ , A_ in Gauss)  Concentrating  75  on- the second harmonic, we observe a  sequence of alternating large and small beat maxima.  The  apparent beat period also alternates. To explain these results we can c a l l upon a result derived e a r l i e r (see [7b]) presence of M.I.,  for the second harmonic amplitude in the and feedback, namely,  ,  .  1  i c e  A  2  K E  A? 2 1/2  8TT F 2  where  K = — = — H  If we take the limit as A  approaches zero, we obtain  2  Shoenberg's  "strong fundamental" result  [Al]  Upon  A  =  2  ± A  2  K  e  substitution of the beating amplitude of. the fundamental  Aj into [ 4 l ] , we can find the contribution of Aj at the second harmonic. We see from Figure 17 that i t is a good approximation to take the amplitudes of the individual A ^ = A ^ . ( a  y  o s c i l l a t i o n s to be equal,  The magnetization due to the fundamentals  (  f - 6F M  ( l )  = A^  {sin  +  sin  £2TT(  -  2 H  p" - — , [2TT( -  y)-  2  h  Y  )  TT/4]  + TT/4]}  or [42]  M^)  =  2A {sin[2^(^- - y) ] COS[2TT(JI) - TT/4]} 1  is then  76  The envelope is 2A, cos  given by .. [  ^  -  IT  A]  so t h a t the m a g n e t i z a t i o n a m p l i t u d e a p p e a r i n g at harmonic due t o the f i r s t  ~ (2) M  "JLL  =  { / ( A  2  the second  is  C O S  2  [2^^)  _  or [43]  M  =  ( 2 )  -  K  A  2  e{l  + cos  [2u(~)  - TT/2]}  The genui.ne second harmonic g i v e s  [2^( I iI - 2y) 2  M = A (sin 2  +  sin  +  [2ii  TTA]  + if A ] }  - 2y)  or [44]  M = 2A  Adding  2  sin  [27T(^f -  cos  2y)]  the 2 c o n t r i b u t i o n s g i v e s  -  the t o t a l  irA)'  magnetization  amplitude at the second harmonic  [45]  M = 2A  2  cos  - *A)  The c o n t r i b u t i o n from the f i r s t using  an a l t e r n a t e approach.  -  is  A  2  [1 + c o s ( ^ F  - ir/2) ]  harmonic can be v e r i f i e d by  From [12]  c o n t r i b u t i o n at the sum frequency  KB  f o r two f r e q u e n c i e s , the  77 A2  [  4  6  ]  A  SUM  =  s  a b - — f — e A  Substitution  O  i  n  a  ( 2 x  " -V-  }  S  i  n  ( 2  V  A  [(K  sin  K )  +  A  B  of the two y fundamental  (x  a  +  x )] b  frequencies  gives:  ^2 A  S  U M  =  {  +  [2TT(  2 F  S l n  sin  [2TT(  2 F  +  6  "  F  6  F  -  2 )  -  -  2 )  +  Y  Y  TT/4]  }  n  A -  2 £  4"  "  2  k  s  i  n  f ' 2  7 7  Elementary t r i g o n o m e t r i c m a n i p u l a t i o n s  [47]  M = -K E A  which reproduces  [cos  (^L  the amplitude  The i n t e r a c t i n g data.  2  resu,lt  in  [45]  In o r d e r to o b t a i n values  (  2  7 f -  lead  2Y)1-  to:  - Tr/2) + 1Isinl2Tr  [43]. should f i t  the n e a r - o p t i m a l  f o r Aj and  to f i t  one can f i n d the z e r o s , and match the p e r i o d r a t i o ,  feedback  the c u r v e ,  that  that every o t h e r beat p e r i o d be s h o r t e r by the observed  is,  insist  amount.  The zeros o f the c a l c u l a t e d second harmonic amplitude a r e determined from  2A  z 0  cos  (2TT~  F _  n  -  TT/4)  -  K  e A? { l + c o s C ^ I n 2  TT/2)  } = 0  or 2A  -J-  [48] K  A*  cos  - Tr/4) H  e  = 1 .+ cos  - TT/2) H  78  148] Is a transcendental equation which can be solved numerically, 2A and i t e r a t i o n leads to a value of — - r for which the r a t i o of the tcAjC 2  beat periods agree with the observed response.  The d e t a i l s are  simply c l e r i c a l and w i l l not be included, however, a plot of the r a t i o of the two apparent second harmonic beat periods as a function KA E 2  of  —  appears in Figure 19.  saying there is no M.I.,  At e = 0 which is equivalent to  the corresponding ordinate is 1,  indicating the equivalence of a l l beat 0.737  on  the  ordinate  corresponds  periods. to  that  near-optimal feedback setting of Figure 18.  The value of observed in the  This ordinate corres-  ponds to 2A — | KAJ  =4.91  e  A similar c a l c u l a t i o n was done assuming A <0 which yields the 2  result: 2A„ — |  = - 3.41.  KAJE  Using these solutions, the calculated interacting second harmonic was plotted along with the calculated f i r s t harmonic beat envelope in Figure 20.  The biggest difference between the positive and 2A negative solutions for — j - ' r e l a t i v e phase of the second 2  s  t  n  e  KAJ e  harmonic minima with respect to the f i r s t harmonic minima.  Upon  comparison with the measured data (see Figure 18), i t becomes obvious that A  2  is indeed negative.  With the negative  solution  F i g u r e 19.  C a l c u l a t i o n of the Ratio of the Apparent Periods of vs. KA E 2  (0.204,0.737)  KAf € 2Ao  Beat  BEAT  08  ENVELOPE  AMPLITUDE (Arbitrary units)  the agreement  is  r e a l l y q u i t e good, e s p e c i a l l y  in the  relative  phase o f the f i r s t  harmonic minimum w i t h r e s p e c t to the second  harmonic minimum.  The c a l c u l a t i o n p r e d i c t s  second harmonic to be d i s p l a c e d A (^ f  LL H  ) = 0.748 rad and 4.33  F i g u r e 16 is  A (-^fp)  0.75  =  similar  much g r e a t e r , e s p e c i a l l y determining  •  the  from a fundamental minimum by  rad .  The observed r e s u l t  rad and 4.5  The c a l c u l a t i o n of the i n t e r a c t i n g harmonic f o l l o w s  the 2 minima o f  calculations,  from  rad.  result  f o r the t h i r d  but the c o m p l e x i t y  in the t r a n s c e n d e n t a l  is  equations  S i n c e the procedure worked w e l l w i t h the second  harmonic, the t h i r d harmonic e q u a t i o n s were s o l v e d by computer. An o p t i o n  in the program enabled us to take the F o u r i e r  at each f i e l d s e t t i n g  requested which  is more in keeping with  way the data was o b t a i n e d e x p e r i m e n t a l l y . cluded  transform  T h i s program  is  the  in-  in Appendix C.  One c o u l d c o r r e c t the temperature dependence of the n e a r optimal  feedback data by e x t r a c t i n g  calculated  interacting  result.  t h i s dependence from the  T h i s was done at a f i e l d  corresponding  to a maximum in the i d e a l second harmonic beat e n v e l o p e .  At t h i s f i x e d  f i e l d H , the arguments of the c o s i n e s q  are c o n s t a n t , A.j and A^.  H  in the i n t e r a c t i n g  and the temperature dependence is e x t r a c t e d from q  is  independent o f feedback gain s i n c e  f i e l d at a maximum of the i d e a l beat e n v e l o p e . dependence  results  is:  it  is  the  The temperature  82  2A„ A (j)  a  2  COS  2  {~T^~~.  TTA) {1+C0S(  ~  +  2 T  ^ - TT/2)}  K A j£  2  a 2A  {cosC ^ - TT/4) + o 2  2  but  K A^E  K£ eX 2  2A,  K A£ -^A2  K 5 -|E  2  2? X  2E  2  2  a l l the temperature dependence,  2  KAj£  (-^—)  2A,  ( = ~3.4l in our case)  Q  then  K  A e 2  X X  2A„  '2A > K  2  A £j 2  letting  a = cos (^rr— ~ TT/4)  and  b = {1 + cos  H  O  - TT/2)}  f2A, 2 KA^ e  we o b t a i n a temperature dependence o f  A (T) 2  hence,  plotting  = 2A  2  [a + bXj  - TT/2)]}  2  l e t t i n g X be the value of X where • o  K A,£  o  i  The q u a n t i t y X = Ay T/H c o n t a i n s so,  []+cos(^-  83  vs.  In  should y i e l d a s l o p e o f In our d a t a ,  T/H  -2Xy. a = 0.642,  and we obtain, a s l o p e of  167  b = 0.352,  kG/K which corresponds  to y = 0.568.  This  r e s u l t , c a l c u l a t e d from the second harmonic near-optimum feedback  data  is  in e x c e l l e n t agreement w i t h t h a t d e r i v e d from the fundamental  amplitude  (see T a b l e  ll).  The above agreement again demonstrates d e t a i l e d mechanisms  involved  the understanding of  in n e a r - o p t i m a l  feedback, and we now  move on to the optimum feedback p o s i t i o n , the r e s u l t s F i g u r e 21.  appearing  With optimum feedback, the beat f r e q u e n c i e s of the  t h r e e harmonics a r e in the r a t i o  the  in first  1:2:3.  From the i d e a l n o n - i n t e r a c t i n g beat envelopes which are now a v a i l a b l e to us  thanks  to the use of optimum feedback,  t h a t t h e r e is a favoured  it  appears  f i e l d s e t t i n g w i t h i n each fundamental  beat  c y c l e where the t h r e e beat envelopes a r e s i m u l t a n e o u s l y c l o s e  to  their maximum v a l u e s , and the s l o p e s a r e not very l a r g e .  occurs  at 1/H  roughly  1/3 of the way  (shown as  1/H^  i n t o . t h e beat envelope p l o t t e d a g a i n s t  in F i g u r e 2 2 ) .  These p o s i t i o n s ,  c a l l e d "magic f i e l d s " a r e the optimum f i e l d s harmonic measurement.  This  affectionately  to perform a t h r e e  The amplitudes o f the harmonics a r e c l o s e  to but not at t h e i r beat maxima, so that a f i e l d dependence measurement  is needed to determine the a c t u a l  from those measured at the "magic f i e l d " . appears  r e l a t i v e amplitudes  S i m u l a t i o n o f such a p l o t  in Figure 22, wi th the c o r r e c t ion = .for the fundamental shown  84  0.015  0.016  0.017  '/ ( k G - ) 1  H  F i g u r e 21.  0.018  —  The Beat Envelope With Optimum Feedback  ENVELOPE AMPLITUDE  58  86 The c o r r e s p o n d i n g e m p i r i c a l  r e s u l t f o r the fundamental  appears  in F i g u r e 2 7 .  6.5  Phase  Information  The v a l u e o f the argument dHvA e f f e c t thus  is q u i t e  require great  large  the s i n u s o i d  (^10^).  precision  to be c o n s i d e r e d r e l i a b l e .  of  describing  A b s o l u t e phase  measurements  in f i e l d and o r i e n t a t i o n F o r t u n a t e l y , the phase  between harmonics can be measured r e l i a b l y . considering d i f f e r e n t frequencies  (i.e.,  the  if  they a r e  relationships  However, s i n c e we a r e  F^, F^=2F^ ,  F^=3F^),  the  r e l a t i v e phase must be d e f i n e d w i t h some c a r e . The standard of the form s i n  d e f i n i t i o n of the phase s h i f t between a v a r i a t i o n  (u)t+ij>j) and  its  r**  1  harmonic sin(roit+ij^)  involves  the c o n s t r u c t i o n o f a r e f e r e n c e s i n u s o i d w i t h frequency u> c r o s s i n g z e r o w i t h a p o s i t i v e s l o p e a t some a r b i t r a r y  t = 0 (any c o n v e n i e n t  t = 2mir/cj where  m = 1, 2, 3 . . .  this  r e f e r e n c e is another at a frequency no c r o s s i n g  fundamental  would a l s o d o ) .  z e r o w i t h a p o s i t i v e s l o p e a t the same t = 0. d i f f e r e n c e between the fundamental and the c o r r e s p o n d i n g q u a n t i t y quantity  r^  -  ^  signal  f o r the r  Associated  If  with  the phase  and the r e f e r e n c e is <f» j t h  is a c o n s t a n t and s e r v e s  harmonic as  is  , then the  the d e f i n i t i o n o f  the  phase d i f f e r e n c e . When d e a l i n g w i t h a s i g n a l frequencies,  which  the sum o f two c l o s e  the s h o r t - r a n g e modulation may not be enough to  the i n d i v i d u a l  frequencies  in the F o u r i e r t r a n s f o r m .  one must c a l c u l a t e the r e s u l t a n t phase experiment.  is  In  this  resolve case,  in o r d e r to compare w i t h the  87 We have a l r e a d y p r e s e n t e d the r e s u l t s o f a numerical substitution  in the envelope e q u a t i o n f o r a p a i r of b e a t i n g  oscillations  (see [33]  fields  plotted in F i g u r e 2 2 ) .  The two  inverse  1/Hj and l / h ^ correspond to F i g u r e s 23 and 2k where the  individual  oscillations  are p l o t t e d out to determine the r e l a t i v e phase.  With r e f e r e n c e to F i g u r e 2 2 , both the second and t h i r d harmonics have undergone one z e r o c r o s s i n g has n o t .  At  between 1/Hjand 1 / H  compared to t h a t at  1/H^.  This  harmonic when  indeed shown in F i g u r e s 23 and (see  section  F i g u r e 23 shows t h a t a t the f i r s t o f these f i e l d s  one w i t h the lowest v a l u e o f first  is  1/Hj corresponds to one o f the "Magic f i e l d s "  6.4).  t h r e e harmonics  phases i|>  150]  r  first  l/h^ we t h e r e f o r e expect the second and t h i r d harmonics  to have the o p p o s i t e phase r e l a t i o n s h i p to the f i r s t  2k.  but the  2  i.e.  the  1/H, the phase d i f f e r e n c e between the  is z e r o .  G e n e r a l i z i n g [ 6 ] to a l l o w f o r the  in the presence o f b e a t s , and f e e d b a c k , we o b t a i n  A  2  = A  2  sin  (2x+^ ) 2  -  1/2  KC  A  2  sin(2x+2^ ). 1  In a d d i t i o n , from [ 3 9 ] Tor small n we see that  and i|> can a t t a i n  o n l y 2 values,^-0 and TT.  in [50]  T h i s makes  the two terms  2  phase o r TT out o f phase depending on the s i g n o f A , 2  T(> . 2  If  the two terms compete  the second term in [50]  2  we expect  i f the magnitude of  exceeds t h a t o f the f i r s t .  In any case the  measured phase d i f f e r e n c e should f o r small n (narrow beat w a i s t s ) 0 o r TT.  in  and the v a l u e of  in the presence of M.I.,  a phase r e v e r s a l o f the measured phase o f A  either  be  The measured phase d i f f e r e n c e as a f u n c t i o n o f feedback g a i n  F i g u r e 23.  I n d i v i d u a l Osci11 at ions Near the "Mag i c F i e l d " l/H, of F i g u r e 22  F i g u r e 2k.  Individual O s c i l l a t i o n s 1/H of F i g u r e 22 2  Near  Figure  25.  Measured Phase D i f f e r e n c e and Amplitude of y at a Magic F i e l d (61.739 kG)  91 at a magic f i e l d appears is  in F i g u r e 25.  Included  the second harmonic amplitude dependence.  that  From 150]  should be a l i n e a r f u n c t i o n o f e f o r small Phase measurements  at o t h e r p a r t s  not r e l i a b l e s i n c e the amplitudes of changing  6.6  in t h i s  n  figure we see  in  [39].  o f the beat c y c l e were  the o s c i l l a t i o n s  were  q u i c k l y , and the l a r g e modulation smears the phase.  The L i n e a r i t y of In Chapter  I I,  Aj/A^  (Aj/A^  vs.  we found that  information  leading  to the  g  c  2 f a c t o r comes from the s t r a i g h t This  line  is  straight  only  F i g u r e 26 shows p l o t s o f with optimum feedback. with feedback  is  the i d e a l L.K.  this  behaviour  kind f o r data without  i n f o r m a t i o n , and how t h i s  is  realised.  in the graph  good to one who has made these  to deal w i t h M.I.  the d r a s t i c way  (h^/A^) .  feedback and  The l i n e a r i t y and low s c a t t e r  surprisingly  using o t h e r techniques dramatically  if  l i n e p l o t of A ^ / A ^ v s .  in which M.I.  This  plots  f i g u r e shows very  i n t e r f e r e s with amplitude  i n t e r f e r e n c e has been  successfully  removed by the feedback t e c h n i q u e .  6.7  Conclusions B e f o r e using  such as the g working  c  the feedback t e c h n i q u e to measure  quantities  f a c t o r , we must be c o n f i d e n t that the t e c h n i q u e  p r o p e r l y , and know the l i m i t s w i t h i n which we can work.  T h i s c h a p t e r demonstrates which feedback has  the c o n s i s t e n c y and the e f f e c t i v e n e s s  in reducing  M.I.  is  I50r92  Pb  H|| CIIOJ  y -oscillations  XlO'  100  500  Figure  1000  26.  Aj/A^  vs  1500  (A,/A )' 2  2000  With and Without Feedback.  H is constant at 61.13k kG, and the bath temperature T is v a r i e d .  93  The c a l c u l a t i o n s  i n v o l v i n g non-optimum and near-optimum  feedback g i v e c o n s i s t e n t agreement w i t h the experimental demonstrating experiment.  results,  the understanding of the r o l e of feedback in the Only f o r the a + y sideband amplitude  (Figure  13)  do we f i n d a s y s t e m a t i c d e v i a t i o n from t h e o r e t i c a l e x p e c t a t i o n . This  is presumably  and a -y  sidebands  understood.  r e l a t e d to the n o n - v a n i s h i n g  of the a + y  at optimum feedback, and is not yet  fully  In every o t h e r case o f optimum feedback, the data  conform to the r e s u l t s must be s t r e s s e d  that  expected f o r  ideal  in each s e c t i o n of  L.K. this  b e h a v i o u r , and  it  c h a p t e r , the  phrase "optimum feedback" r e f e r s to the same feedback gain  i.e.,  the optimum s e t t i n g  the  others.  f o r one experiment is  the same f o r a l l  T h i s c o n s i s t e n c y g i v e s one c o n f i d e n c e that the same  optimum feedback s e t t i n g w i l l  give  reliable g  c  f a c t o r measurements.  While most o f the chapter d e a l s w i t h non-optimum feedback, i t c l e a r l y demonstrates c o r r e c t i o n s f o r M.I.  that at the optimum feedback g a i n no  need be a p p l i e d .  94  CHAPTER SEVEN  EXTRACTION OF g A^/A^ v s .  FACTOR FROM  {A /A )  PLOTS  2  }  2  To apply the a l g o r i t h m presented in Chapter II recall  l e t us  a few r e s u l t s :  2 [3a]  A,/A  [3b]  ( A  [2b]  a  3  = am  1 2^0 / A  =  [ ( A ^ r  2  ^  6  X  2  -  P  ( X T  D  1/4  / T  (A,/A )  ^  ]  2  C  O  S  i  r  S  /  c  o  s  2 i t S  A?  -  (/3/2)(l-tan  nS)  /(1-3  tan  »S) -  OO  Um ( — ) X + » AjA X »  3  (A^/A2)Q  From [3a]  is  independent of the temperature T  we see t h a t the s l o p e o f the graph of  |A^/A | 3  vs.  2 (Aj/A,,) T)  [51]  is  (holding .  [2b]  tan  the f i e l d  H  c o n s t a n t , and v a r y i n g the temperature  can e a s i l y be i n v e r t e d to g i v e  2  rrS = 1 - /3  <v  V  ±  ^~-W  95 The v a l u e  |a } from a l e a s t  F i g u r e 2 6 i s '|'a ] = 0.392. imaginary This S =  0.330  or  The square  < 0.  root  to the p o i n t s  in  in [ 5 1 ] y i e l d s  an  The r e a l s o l u t i o n s These values  0.197.  fit  = + 0.392 so we must conclude that a =-0.392,  result for a  implies A ^ / A ^  squares  of  [51]  are  are modulo 1 because of  the  2  p e r i o d i c i t y of the f u n c t i o n tan principal  values  T O d e c i d e between these two  f o r S we measure the a b s c i s s a  F i g u r e 2 6 to o b t a i n ,  1/4 (A / A ) 1 2  [3b]  TTS.  intercept  . as can be seen from  c o r r e s p o n d i n g to the two s o l u t i o n s A./A )  T  [52]  (  = - f - £n [ D  X  »  n  (  1  2  2/2  From the l e a s t squares  n  [3a].  0  can e a s i l y be i n v e r t e d to g i v e D i n g l e  u  in  temperatures  for S .  cos 2TTS ° ] cos TTS  fit  to the p o i n t s  in F i g u r e  26,  2  1/4 used  ( A J / A ) Q = 91.0.  Using  2  [ 5 2 ] and the experimental  parameters  in the experiment, along with the e f f e c t i v e mass p found  Chapter VI,  our p r e v i o u s  following Dingle  solutions  f o r S correspond to the  temperatures  S =  0.330,  T  D  = 1.42K,  A /A  S =  0.197,  T  D  = 0.749K,  A. /A  The D i n g l e  T  T  2  <  0  >  0  temperature can a l s o be o b t a i n e d from the f i e l d  dependence o f the fundamental amplitude estimate  in  is n e c e s s a r y .  and  [ l b ] ; only a  rough  In the a p p r o x i m a t i o n X * 3, the complete  f i e l d and temperature dependence is given by  96  Aj cx(T//B)  so  exp {-  Xy(T/B)(l  + Tp/T)}  that  [53]  Jln(A /B)  = -Xy (T/B) (1 + Tp/T)  1  we see that a p l o t o f In  From  [53]  give  the D i n g l e  not change t h i s ,  temperature. as  long as  vs.  (AJ/B)  1/B  can  The f a c t t h a t A ^ is b e a t i n g the p o i n t s  used on the graph  at  the same p o s i t i o n  in the beat c y c l e .  The obvious  is  to use the f i e l d and the amplitude at  does are  choice  the maxima o f  the  beat p a t t e r n . F i g u r e 2 7 shows a p l o t o f A ^ v s . fit  squares  to the maxima f o r T = 1 . 2 5 K g i v e s  -Xy(T/B)(1+T /T)  =  D  It  1 / H , and a l e a s t  - 1 6 4 . 5 or  is q u i t e e v i d e n t t h a t S = 0.197  The phase measurements g i v e A ^ / A We a r e thus according  2  J  =  Q  is  0.750K  the proper p r i n c i p a l  > 0 c o n s i s t e n t with t h i s c h o i c e .  l e f t w i t h o n l y the t r i g o n o m e t r i c m u l t i p l i c i t y  to which p o s s i b l e  each o f which g i v e s  solutions  are S = ± 0 . 1 9 7  i d e n t i c a l experimental  For s i m p l e metals  such as  restricts  ± P,  P e 1  results.  l e a d , where the band s t r u c t u r e  can be d e r i v e d from a weak p s e u d o p o t e n t i a l spin-orbit  value.  i n t e r a c t i o n , a physical  together with  the  argument g i v e n by P i p p a r d (1969)  the range o f p o s s i b l e v a l u e s o f S a c c o r d i n g to the  98 inequality  S  1 (rn* /m)  where, s  is  electron  (s/2)  +  the number of Bragg r e f l e c t i o n s undergone by an  in one c y c l o t r o n o r b i t .  t, o r b i t normal  to [110]  so that with rrT/m =  This value  which g i v e s  is 3 f o r the  r i s e to the y o s c i l l a t i o n ,  0.560, we have  0 < S < 2.06  Of the  4 values of S which f a l l  into t h i s  1.197, 0.803, 1.803) two g i v e A /A ]  2  <  interval  0 which is  with the phase and D i n g l e temperature c r i t e r i a . l e f t with the two p o s s i b l e  S = g  c  (0.197,  inconsistent We are  thus  solutions.  m"/2m = 0.197 and 1.803 c  corresponding to g  c  = 0.704 and 6.44  respectively.  The u l t i m a t e c h o i c e between these 2 r e l i e s on a band c a l c u l a t i o n which  includes  the s p i n - o r b i t  interaction.  CHAPTER EIGHT  A SEARCH FOR THE 4MG OSCILLATIONS  8.1  P r e l i m i n a r y Remarks Quantum o s c i l l a t i o n s  of u n u s u a l l y  long p e r i o d  have been observed r e c e n t l y in lead using de Haas e f f e c t (Tobin e t . a l . , (ivowi  and Mackinnon,  It  1969)  B r i l l o u i n zone.  Shubnikov-  and sound a t t e n u a t i o n  1976).  has been suggested that these long  might a r i s e from small  pockets of e l e c t r o n s  While pockets o f t h i s  empty l a t t i c e band s t r u c t u r e , a l l fitted  the  oscillations in the 4th  kind appear  r e a l i s t i c band  in the  calculations  to the Fermi s u r f a c e data show the 4th zone to be  empty, ( c f .  Anderson and G o l d , 1965).  wonder whether the long o s c i l l a t i o n s generated by  We are thus  led to  might be an a r t i f a c t  M.I.  A c o n c e r t e d e f f o r t was made.to d e t e c t s i m i l a r tions  4MG)  oscilla-  in the dHvA e f f e c t w i t h the hope t h a t they c o u l d then be  s t u d i e d w i t h the feedback t e c h n i q u e . e v i d e n c e f o r these long o s c i l l a t i o n s  U n f o r t u n a t e l y , no c o u l d be found, so t h a t on  an upper l i m i t on t h e i r amplitude r e s u l t e d .  In the process  of  100 the s e a r c h , some u s e f u l exact s o l u t i o n  8.2  ideas were developed i n c l u d i n g the  to the problem of  large m o d u l a t i o n .  Review of the Standard Weak-Modulation  Solution  F i e l d m o d u l a t i o n , f o l l o w e d by p h a s e - s e n s i t i v e d e t e c t i o n , is  the most w i d e l y used technique f o r o b s e r v a t i o n o f the  Haas van Alphen e f f e c t .  The problem of c a l c u l a t i n g the e.m.f.  induced in a p i c k - u p c o i l in d e t a i l  de  surrounding  f o r weak modulation f i e l d s .  c i r c u m s t a n c e s which warrant  the sample has been s o l v e d There a r e , however,  r a t h e r l a r g e modulation  fields,  l a r g e enough so that some o f the approximations made in the weak-modulation  treatment may no longer be v a l i d .  circumstance  the d e t e c t i o n of  as  is  long-period o s c i l l a t i o n s  those r e p o r t e d b y Tob i ri e t . a 1 . (rl 969)  F ^ li MG.  It  ~ 1 kG which  One such  havfing f r e q u e n c i e s  is- then d e s i r a b l e to modulate with an amplitude is a s i z e a b l e f r a c t i o n of the q u a s i - s t a t i c  ground f i e l d .  We f i r s t  f o r a r b i t r a r y s t r e n g t h of modulation its  back-  review the standard f o r m u l a t i o n f o r  weak m o d u l a t i o n , and then develop an e x a c t , e x p l i c i t  In  solution  field.  present w i d e l y - u s e d form, the modulation f i e l d  is s i n u s o i d a l ,  small w i t h r e s p e c t to the l a r g e background  H, and is p a r a l l e l  to  f i e l d H + h s i n ait. sweep s l o w l y ,  such  so that  it.  h field  Thus, the sample e x p e r i e n c e s a net  The l a r g e background f i e l d in t r e a t i n g  is made to  the modulation f i e l d , we can  regard the background f i e l d as e s s e n t i a l l y c o n s t a n t .  The  101  c r i t e r i o n for this  (dh.) , dt  R  M  < <  assumption  is  dJi dt  S  The treatment f o r weak modulation just  is w e l l  known, and w i l l  be o u t l i n e d here. Without  constant  loss of g e n e r a l i t y , we can  •  M =  S  m  where h(t) quency.  various  phase f a c t o r s , and w r i t e the o s c i l l a t o r y  the m a g n e t i z a t i o n  M  ignore  [  r H  simply  2 T r F  +  h(t)  = h s i n cot  The equation  the amplitude f a c t o r s  as:  i ]  and u> is is  the modulation angular  f o r a .reduced m a g n e t i z a t i o n ,  incorporated  into  it.  approximation,  panded, and only the l i n e a r term in 77 is rl  [*f  (1 -  S i n c e the approximation  careful  so  that  the denominator r e t a i n e d , so  is  ex-  that  ^ ) ]  is made in the argument of a  (—rr~' Xj 10^ t y p i c a l l y ) n to s t a t e the j u s t i f i c a t i o n c o r r e c t l y .  oscillating  with  inhomogeneity due to eddy c u r r e n t s .  In the c o n v e n t i o n a l  M^sin  fre-  We might a l s o  add that we must work at a low enough frequency to we have no f i e l d  p a r t of  sine function,  rapidly  one must be  102  The c r i t e r i o n to be s a t i s f i e d argument only  is  if  at most f i r s t  order  must a s s u r e  in  •p- .  This  that is  the second o r d e r term i s very much l e s s  The second o r d e r term  2TTF H  ,hjth H  V  the  true  if  and  than 2TT.  is:  2  ;  So the j u s t i f i c a t i o n  is  H  or  simply  Fh —  If  . 1.  2  «  this  inequality  is  not s a t i s f i e d ,  the argument  s i n e must be taken to be a t  least quadratic  "weak m o d u l a t i o n " c r i t e r i o n  is  laboratory series  ~  situation,  as  usually  in h.  of the  This  met in the normal  and M can be developed  in a F o u r i e r  follows  .  r TTF  h(t)\  2  M .= s i n [ —  1  -r-r Ve 2i  (1  i  1  —) ]  2TTF /, . v (hsinwt) H2-^'=»  2TTF .--rr—  e  -i  H  -i :  -e  2TTF,, . ^ —r-(hsinwt) H  2  i e  . 2TTF H  103 Us^ng the i d e n t i t y  -i  00  u sin y  I  J>>"  e  - i n y  we o b t a i n  e  The c o i l  -intot  H  -e  2TTF -i H  s u r r o u n d i n g the sample g i v e s a v o l t a g e p r o dM  portional  •J-  dt  to  % -  .  Taking t h i s d e r i v a t i v e ,  I no) J (—r—) n ,,Z n=-°° H  cos  (-rj H  nut)  We now s e p a r a t e the t and H dependences; , and f i n d a f t e r a l i t t l e man i p u l a t ion  dM „ v „ • /27TFhx . ,2TTF TI\ . , TU — % - V 2nco J S i n ( - r j - + n;d s i n (ntot + — ) at , n ,,c n L C n=l " H N  This  is  the c o n v e n t i o n a l  note in p a s s i n g  that the same  of the two phase f a c t o r s  8.3  result • for result '  s  weak m o d u l a t i o n . is o b t a i n e d  if  We  the s i g n  reversed.  Large Modulation By l a r g e modulation we mean that our  initial  assumption  about the l i n e a r i t y of the s i n e argument breaks down. particular,  f o r the long o s c i l l a t i o n s  In  having F ^ k MG in an  104 a p p l i e d f i e l d of 50 kG, f o r maximum response, we should  modulate over something makes h ^ 325 Gauss.  l i k e one c y c l e of the waveform which  Our c r i t e r i o n f o r n e g l e c t of the second,  and high o r d e r terms was  /P/F  h «  Any such  long o s c i l l a t i o n  in the dHvA e f f e c t would be swamped  by the s t r o n g y o s c i l l a t i o n s  /P7F  % 2.7 kG  For t h i s  with a frequency of  17 MG,  making  (H = 50 kG)  situation,  i t cannot be s a i d  much l e s s than / H / F , and i t was f e l t 3  t h a t h is  then very  that a deeper study  into  the e f f e c t s of the q u a d r a t i c , and h i g h e r o r d e r terms, was warranted. Bessel object,  A common p r a c t i c e is  functions in p a r t ,  z e r o s when using  to e x p l o i t the z e r o s of  to e l i m i n a t e the unwanted o s c i l l a t i o n s . is  to determine p o s s i b l e s h i f t s  l a r g e modulation  of  v a l i d f o r any s t r e n g t h  Our  these  fields.  We now present an exact F o u r i e r decomposition which  M = s i n [rjrr  the  of m o d u l a t i o n .  In our b a s i c  is  equation,  rj  H+hcoso3t  we can use the c o s i n e phase of the modulation without generality,  loss  s i n c e the r e s u l t cannot be dependent on the  of  origin  105  of time.  This  c h o i c e of phase makes M an even f u n c t i o n of  time t which s i m p l i f i e s sine  the planned F o u r i e r expansion of  the  argument. Because o n l y the even, c o s i n e terms can s u r v i v e , we  can w r i t e the expansion  2TTF u . L. T H + hcoscot  =  r Z  „ n=0  as:  a  n  cos ncot  where  2TT/CO a  =  2co_  2TTF H + hcoscot  TT  and  dt  2TT/CO CO a  n  =  2TTF H + hcoscot  7  These i n t e g r a l s obtain  a  o  cos ncot dt  may be reduced to standard form, and we r e a d i l y  (Gradshyeyn and Ryzhik,  1965)  2TTF =  ATTF h_ H where  P ="  H  h/H  nr.  -  2  v  n  <-'  n  106  and 0  < p < 1  We now wri t e f o r M  M = sin[2a (l/2 0  ( ia = Im, { e  Making  e  2 3 - p cos uit + p cos 2wt - p cos 3^t + . . . ) ]  -2iagpcosa)t e  Q  2  '  a Q  P  2 cos2cot  e  3 - 2 i a p cos3<*>t e ... Q  use o f the i d e n t i t y  -iycosy  J (.„"  =  J  n  (  v  )  -my  e  -co  n=  and i t s complex conjugate  e  .ycosy  =  l  n  (  l  )  n ^  e  -ny  =-»  We o b t a i n the f o l l o w i n g  M = In,  Je' ° a  (  I (-i)  n  J (2a p) n  n n . . . = -«> r  0  e ,mt  J (i)  J (2  n  n  k-i  L  2  [  i e  B  n  t  p )a 2  a o  2 i n u t  k ^ -  )  k  k  A g a i n , we need the time d e r i v a t i v e , which i s e a s i l y o b t a i n e d :  n  k ] }  107  dM = Im dt  r iwt | ( - ) H ] X I6  This  is  M  the  exact s o l u t i o n f o r  dM , given dt  that  2TTF = sin [• H + hcoswt •I  The solution  is v a l i d for a r b i t r a r y  In order to obtain a tractable  h h, provided only that j | l <  and useful formula for  dM , it  is necessary to find  suitable  approximations  i n f i n i t e sums and products in the exact s o l u t i o n . done to any desired accuracy.  We are usually  for the  This can be  interested in the  phase-sensitive detection at a p a r t i c u l a r harmonic of the modulation frequency OJ. required integers  For the nth time harmonic (nw), the  (±n) are related to the various  indices occurring in the exact result  integral  by:  n  where n^ can be any integer between determines a l l the sets {n > k  -«° and °°. This equation  for any desired time-harmonic nu>,  each set giving one term in the s o l u t i o n .  A procedure w i l l now  be given for ranking these sets in order of importance.  108  The p a r t o f the s o l u t i o n which determines the magnitude of a p a r t i c u l a r  f  J  k=l  n  time harmonic  is  (2a p ) K  K  0  In most c a s e s , 2agp' -«l <  f o r k i 2,.  In a l l  c a s e s , 2agP^<<l  a l a r g e enough v a l u e o f k, s i n c e 0<p<l. k, a l l J  (2a^p' ) <  n  relative  are of order  for  For s m a l l e r v a l u e s o f  1, and a l l must be c o n s i d e r e d .  k  In the normal  laboratory s i t u a t i o n ,  there  i s but one such t e r m .  k When 2agP « 1 , which i s u s u a l l y the c a s e f o r rank the s e t s by t h e i r estimate.  result  Given the s e t (n^}  > k—,2 , one may  in the f o l l o w i n g o r d e r o f  magnitude  , the c o r r e s p o n d i n g term i s  approximately:  » (a.p ) If — — . k k c n  !  where the product s t a r t s a t a v a l u e k teger s a t i s f y i n g 2aQp^«l  (typically  c >  k  c  which i s the lowest  in-  = 2).  n^O  In p r a c t i c e ,  o n l y f o r s m a l l k ( k ~ 3 ) , making t h i s a q u i c k method o f  ranking.  One can see t h a t the o r d e r depends somewhat on the v a l u e s of ag and  ,p.  T a b l e III  i s an example of t h i s  o r d e r o f magnitude of  the c o r r e s p o n d i n g t e r m s .  the second time h a r m o n i c , and t y p i c a l ag and p.  ranking, along w i t h It  the  i s done f o r  v a l u e s were chosen f o r  109  n=2,  a = 5 x l 0  2  ,  p = TO  TABLE III Ranking The Terms  Order of Mag. 1  n  l  +2  n  2  n  3  k  n  0  0  0  0  ±1  0  0  10'  3  10"  3  ±k  ±1  0  0  10-  6  ±2  ±2  0  0  io"  ;i  0  ±1  0  IO-  ±4  ±3  0  0  6  9  10"  9  ±1  ±1  ±1  0  10"  9  ;3  ±1  ±1  0  10"  9  +3  +i  ±1  0  The next term is of order 10  The most important term is invariably n^ = ± n, n^ = 0 for k ^ 1.  We shall now calculate it separately, and compare  it to the result for weak modulation.  110  lm|e' °(-i) a  + e  , a  n J ^ a / )  J (2a p)  n  0  n  K  6  0  Q  oo  k  = 2nw s ! n ( n u t )  J - ( 2 a P ) n J (2a p  = 2nu J ( 2 a p )  n J ( 2 a p ) sin(nuit)  Q  n  To compare t h i s  Q  n  Q  Q  U  (-i na>)  t  sin(a  to the e a r l i e r c o n v e n t i o n a l  ^  let  get  sin(nwt  - -2naj J ( 2 a p )  k\ n J (2a p )  . / ^ . nir \ . / s i n ( n w t + !f )s\n(a  Q  By c o n t r a s t ,  k  0  0  K  n  o  Q  the r e s u l t  _ . /2TrFhx . = -2nu J ( — — ) s i n  i  ^  . niTv  (nut + —)  F i r s t l y and perhaps most is d i f f e r e n t .  the e x a c t r e s u l t  gives  - ^  )sin(a  Q  f o r weak m o d u l a t i o n  In c o m p a r i s o n , t h e r e a r e two  frequency  which us sub-  n <l (2a p )  Q  n  a  e  result,  £ -2no> J ( 2 a p )  n  dM  to  ' nl  n  - y- )  Q  s t a r t e d w i t h the s i n e phase o f the m o d u l a t i o n , s t i t u t e t -> t - ^  )j  (ina,  ( ) lm.7i, I • ( - J )  Q  0  n  e 1™  °(i)"(-)"j (2a p)n J (2a p ) n  " '  e  0  -  !f)  . nir + >f )  is  . /2TTF , s i n (-q— + —)  differences.  importantly,  The c o n v e n t i o n a l  the measured dHvA result  i s F, whereas  T h i s means t h a t t h e r e measured f r e q u e n c y .  i s a second o r d e r c o r r e c t i o n  in  Under most c i r c u m s t a n c e s , t h i s  the  shift  is  s m a l l , but g i v e n the h i g h degree of a c c u r a c y which dHvA work boasts, this  in some cases may be i m p o r t a n t .  It  is  to note t h a t a l t h o u g h o n l y the f i r s t  term in the  was t a k e n , t h i s  is exact, that  frequency c o r r e c t i o n  o f the h i g h e r terms change t h i s The o t h e r d i f f e r e n c e  \ 0 J  = i  ( 2 a  -  (A  this  infinite  order  it  Q  i s the a m p l i t u d e  « 1 for  correction  k > 1.  p r o d u c t converges to S 1.  One can show t h a t In  i s s m a l l e r than the c o n v e n t i o n a l  does not s h i f t  i s , none  result.  the  : t e r m , we can say t h a t the a m p l i t u d e f o r  modulation that  solution  k  S i n c e 2a p' % 1, 2 a p Q  important  strong  result,  the z e r o s o f the B e s s e l  The o b v i o u s a l t e r n a t i v e  f i r s t ,  and a l s o  function.  to e x p a n s i o n in a F o u r i e r s e r i e s  i s an e x p a n s i o n in a T a y l o r s e r i e s , namely  costot  Gathering a l l t a s k , but  +  (TJ)  cos  tot  -  cos  (77)  tot  +  the terms f o r any harmonic nco i s a f o r m i d a b l e  i f we r e t a i n  terms o n l y to second o r d e r  in  (77), we  .  note  t h a t t h e dHvA f r e q u e n c y  which  are the leading  terms  becomes  in a Taylor series of our exact  result  8.4  Modifications F ^ 4 MG  t o the Apparatus  consideration  a p p a r a t u s was s e n s i t i v i t y  fications. remained  output  and s i g n a l  the frequency  While  in the design of the detection to noise.  response d i c t a t e d  the sample-detection c o i l  t h e same, i t s o u t p u t now d r o v e  transformer  The s h i f t i n several  the primary of a  lOOfi) o f t h e d e t e c t i o n c o i l s .  was d e t e c t e d o n t h e s e c o n d  harmonic  modi-  arrangement  (P.A.R. M o d e l A M - l ) t o t a k e a d v a n t a g e  impedence  modulation  f o r the  Search  The m a j o r  emphasis from  and A n a l y s i s  o f t h e low The s i g n a l  o f a 4 1 . 7 Hz s i n u s o i d a l  f i e l d w i t h a P.A.R. 124 p h a s e - s e n s i t i v e d e t e c t o r .  Its  notch f i l t e r  (Q=50) was u s e d  and  a Krohn-Hite  (model  c e n t e r e d on t h e s e c o n d  t o b l o c k t h e fundamental  3322R) b a n d p a s s f i l t e r  (Q=l) was  harmonic.  The a m p l i t u d e o f t h e m o d u l a t i o n was made t o v a r y a s H which  kept  this  amplitude spanning  t h e same number o f dHvA  2  113  c y c l e s a t any f i e l d H.  Using the z e r o s o f the Bessel  function  response of the observed m a g n e t i z a t i o n ori the m o d u l a t i o n amplitude  (see s e c t i o n 8 . 3 ) ,  the dominant o s c i l l a t i o n s i n any  d i r e c t i o n c o u l d be a t t e n u a t e d by about a f a c t o r o f 5 0 , a l l o w i n g an i n c r e a s e in s e n s i t i v i t y o f the same f a c t o r saturation.  without  The r e s u l t i n g s i g n a l was d i g i t i a l l y  20 b i t r e s o l u t i o n on magnetic t a p e .  recorded w i t h  These data were s u b -  s e q u e n t l y F o u r i e r transformed w i t h the use o f the main U . B . C . computer (Amdahl 4 7 0 ) and a program o u t l i n e d  in Appendix C.  With t h i s arrangement, a l l o f the o s c i l l a t i o n s p r e v i o u s l y were e a s i l y i d e n t i f i e d , o s c i l l a t i o n s appeared.  in lead seen  however, no s i g n o f the k MG  The s e a r c h i n c l u d e d e x a m i n a t i o n o f the  F o u r i e r t r a n s f o r m s a t the second harmonic (^ 8 MG) to a l l o w f o r the p o s s i b i l i t y o f a s p i n s p l i t t i n g  z e r o a t the f i r s t  harmonic. The r e s u l t o f t h i s n e g a t i v e experiment p l a c e s an upper bound on the a m p l i t u d e o f these long p e r i o d o s c i l l a t i o n s magnetization.  In each of the t h r e e major symmetry  [ill],  [100],[110],  directions  t h e i r a m p l i t u d e must be l e s s than  1  in 10 o f the m a g n e t i z a t i o n o f the dominant o s c i l l a t i o n s each d i r e c t i o n . \ M G  2  0  0  U  G  This l i m i t  in a b s o l u t e terms  1.(1969)  part in  i s about  '  The o s c i l l a t i o n s of Ivowi and Mackinnon et.a  in  thus remain an enigma.  It  (1976)  is f e l t  and T o b i n  that t h i s area  of s t u d y would b e n e f i t g r e a t l y by a c o l l a b o r a t i o n o f the feedback t e c h n i q u e w i t h the Shubnikov-de Haas e f f e c t o r sound a t t e n u a t i o n , where these o s c i l l a t i o n s appear v i v i d l y .  APPENDIX A FLEXIBLE GEAR ROTATOR  An apparatus was b u i l t  to r o t a t e  the sample about an  a x i s which was 90° away from the a x i s o f the magnet bore (the o n l y d i r e c t i o n o f  a c c e s s ) based on an idea g i v e n by  P i p p a r d and S a d l e r (1969). original  The m o d i f i c a t i o n s  made to the  d e s i g n were e x t e n s i v e enough to w a r r a n t  description restricted  in t h i s a p p e n d i x .  Our compactness  further requirement  t h e s i z e o f the apparatus to a degree where the  mechanisms would be s u b s t a n t i a l l y  s m a l l e r than any t h a t had  p r e v i o u s l y been b u i l t s u c c e s s f u l l y . The e n t i r e a p p a r a t u s i s c o n s t r u c t e d from nylon rod except f o r a M y l a r g e a r .  T h i s c i r c u l a r M y l a r gear was c u t  from a p i e c e o f 0.003 inches t h i c k made to c u t 32 t r i a n g u l a r  sheet.  t e e t h w i t h a r a z o r b l a d e i n roughly  circular starting material.  A square h o l e ( s i d e  0.075 inch) was c u t in the c e n t r e w i t h a square h o l e , a r e t a i n e r  A s p e c i a l j i g was  punch.  length Through the  f a s t e n e d a r i n g to the gear so t h a t  the a x i s o f the r i n g was p e r p e n d i c u l a r to the gear a x i s , and  Figure 28.  Sample Rotator Assembly  i n t e r s e c t i n g i t ( s e e F i g u r e 28).  The r e t a i n e r ,  so p l a c e d ,  was welded to the r i n g w i t h a s o l d e r i n g i r o n .  The square  h o l e ensured the absence o f s l i p p i n g when the gear was t u r n e d . The s p h e r i c a l sample was glued to the r i n g w i t h a s m a l l drop of G . E . v a r n i s h .  Care was taken to a p p l y a minimum amount  of v a r n i s h to the sample as d i f f e r e n t i a l  c o n t r a c t i o n would  cause s t r a i n upon c o o l i n g . A f t e r a l l o w i n g 2k hours f o r the v a r n i s h to d r y , assembly was p l a c e d i n s i d e a c y l i n d r i c a l  tube by bending the  M y l a r gear to conform to the shape o f the t u b e .  When in p l a c e ,  s m a l l a x l e p i n s h e l d the gear a x i s s t a t i o n a r y w h i l e allowing retain  it  to r o t a t e .  As the gear r o t a t e s ,  i t s c y l i n d r i c a l s h a p e , and r o t a t e s  the a x i s o f the g e a r .  the  it  still  flexes  to  the sample about  Only the t e e t h at the top of the gear  p r o t r u d e from the c y l i n d e r , where they mesh w i t h a d r i v i n g The 16 t o o t h d r i v i n g gear was made by pushing a hot  gear.  brass negative into a c y l i n d r i c a l  n y l o n b l a n k , and subsequent  machining p r o v i d e d a c o u p l i n g to the top of the c r y o s t a t . body o f the c o i l  The  former h e l d the d r i v i n g gear in the proper  p l a c e to mesh w i t h the M y l a r g e a r .  In o r d e r to keep the  t e e t h of the d r i v i n g gear i d e n t i c a l  to those of the M y l a r g e a r ,  one f i n d s t h a t the d r i v i n g gear must r o t a t e about an a x i s which is o f f coil  centre.  former  The d r i v i n g gear w i t h the r o t a t o r assembly and  i s shown in F i g u r e 29.  A beryllium-copper  was used to ensure i n t i m a t e c o n t a c t of the  spring  gears when c o o l e d  117  BeCu  Spring  Slots in both sides 2-56  Bolt to fit in slots  »ff centre driving gear (16 teeth) to mote Mylar gear  0.500"  Coil f o r m e r 0.281"  c  ^ 0.853  i  1 1  i/32 hole for positioning  Threaded ^16-20  Teeth of Mylar gear  inside NF  Spherical Sample in sample ring  Axle pins Slot in base for positioning  JIR-20  F i g u r e 29.  NF  Sample R o t a t o r With D r i v i n g and Coi1 Former  Rear,  to l i q u i d h e l i u m t e m p e r a t u r e s .  The c y l i n d e r h o l d i n g  the  sample was i n s e r t e d i n t o the bottom of the c o i l former and h e l d r i g i d l y w i t h a 7 / 1 6 - 2 0 (NF) n y l o n b o l t . i t s proper r o t a t i o n a l  The l o c a t i o n of  p o s i t i o n was found by pushing a  temporary w i r e through a s m a l l h o l e in the bottom of  the  coil  the  f o r m e r , and i n t o a s l o t m i l l e d  sample c y l i n d e r . the n y l o n  i n t o the base of  T h i s w i r e was removed a f t e r  tightening  bolt.  The e n t i r e assembly was i n s e r t e d i n t o the t a i l the inner dewar shown in F i g u r e 5cryostat,  At the top of  p r o v i s i o n was made to r o t a t e  by hand, o r by e l e c t r i c  motor.  of the  the c r y s t a l  either  119  APPENDIX B THE DISCRETE FOURIER TRANSFORM  S i n c e both the spectrum a n a l y z e r , and the programs use d i s c r e t e F o u r i e r t r a n s f o r m s , finitions  will  A(k)  the b a s i c d e -  be p r e s e n t e d in t h i s a p p e n d i x .  Fourier transform  = V j=0  where X j , j = 0 ,  The d i s c r e t e  i s d e f i n e d by  X. e "  2 i r I j  k  /  k = 0, 1, . . .  N  , N-l  J  1, . . .  , N-l  The i n v e r s e t r a n s f o r m  B(J) = V  computer  A(k) e  2 l V i j  i s a s e t of complex numbers .  is  k  /  N  j  = 0,  ...  N-l  k=0  where B ( j )  = NXj.  The f a s t F o u r i e r t r a n s f o r m programs s u p p l i e d by o f t e n r e q u i r e the input data to be e i t h e r symmetric.  symmetric o r  Any s e t of data can be s e p a r a t e d i n t o i t s  symmetric and symmetric components.  If  libraries antianti-  the s e t X. c o n t a i n s  120  the o r i g i n a l  data v a l u e s , then the a n t i s y m m e t r i c v a l u e s are  g i v e n by  X  x  N/2+l-j  3  =  I  =  N / 2  X  ( X  =  N/2-j+l  "  X  N/2+j+l  )  j  =  2  '  •••  N  /  "  2  1  °' g  A s i n e t r a n s f o r m can then be a p p l i e d t o X .  The symmetric  S  v a l u e s X.  are g i v e n by  S  1  N/2+1-j  X  S X  I  =  X  T  X  =  S N/2  2  (X  =  X  N/2+j+l  =  X  N/2-j+l  )  j  =  2  '  ••'  N  /  "  2  1  N/2. s  A c o s i n e t r a n s f o r m can then be a p p l i e d to X . A p p l y i n g a window to the o r i g i n a l results  in a t r a d e o f f o f  sidelobes  encountered  data values u s u a l l y  r e s o l u t i o n and s i d e f e b e s .  The l a r g e  in the use o f a square window can h i d e  f r e q u e n c i e s o f s m a l l e r a m p l i t u d e which a r e a c t u a l l y f a r away in the space of the v a r i a b l e  k.  The Hanning window i s a good compromise, s i n c e not much resolution quickly  is l o s t ,  in k  space.  but the If  sidelobe  the o r i g i n a l  a m p l i t u d e decays very d a t a i s in time  a p p l i c a t i o n o f a Hanning window s i m p l y  involves  t,  multiplication  2  of  the o r i g i n a l  data by s i n  [Tr(t-tg)/T]  where t  is  the  smallest value of record.  In e f f e c t ,  t  and  T  i s the d u r a t i o n of  the Hanning window rounds o f f  the the sharp  c o r n e r s on the edges of the d a t a where the window i s opened and c l o s e d .  122  APPENDIX C COMPUTER PROGRAMS  The computer programs used to g e n e r a t e the in the body of t h i s  t h e s i s are l i s t e d  in t h i s  results  appendix.  For the most p a r t , the programs a r e w r i t t e n to be s e l f explanatory  in regards to t h e i r  use.  The " D a t a Reading and A d j u s t i n g " program reads data from the Stevenson i n t e r f a c e to EBDIC from A S C I I . translation After  routine  after  it  the  has been c o n v e r t e d  The c o n v e r s i o n was done by a s t a n d a r d ("TRANS) in the U . B . C . Computer  library.  r e a d i n g the d a t a , the F o r t r a n program, by use of  function  s u b - p r o g r a m s , a l l o w s the user to c r e a t e the proper  x and y c o o r d i n a t e s from the a v a i l a b l e d a t a .  The program then  c o n v e r t s the data to a format c o m p a t i b l e w i t h a l l remaining  the  of  the  programs.  The " S y n t h e t i c Data G e n e r a t o r " program a l l o w s the user to c r e a t e any data he p l e a s e s and puts T h i s program was l a r g e l y used to t e s t  it  in the proper  the o t h e r  B u i l t by A. Stevenson p r e s e n t l y a t TRIUMF.  format.  programs, and  check the  resolution.  The "Window" program was used to cut down the s i d e lobes of the t r a n s f o r m .  Instead o f u s i n g the Hanning window,  the data were m u l t i p l i e d  by a s i m p l e s i n e f u n c t i o n  0—rr  over the window.  T h i s g i v e s more r e s o l u t i o n  Hanning window, and the  s i d e l o b e s are s t i l l  spanning  than  not too  the large.  T h i s s i n e window was used because the data at one end of window (the h i g h f i e l d end) were the most i m p o r t a n t , were not cut o f f  the  and they  so d r a s t i c a l l y as w i t h the Hanning window.  The a n a l y s i s program t a k e s the F o u r i e r t r a n s f o r m o f data prepared by the e a r l i e r r e s o l u t i o n and the window i n  routines. k  the  One can choose the  space f o r  The power spectrum f e a t u r e was most o f t e n  the  transform.  used.  The " P l o t t i n g " program a c c e p t s data from a l l of  the  p r e v i o u s r o u t i n e s so t h a t r e a l space and F o u r i e r space d a t a can be p l o t t e d .  P l o t t i n g can be done on the p r i n t e r ,  g r a p h i c s t e r m i n a l , or the hard copy Calcomp p l o t t e r . axis  is labelled The M . I .  for M.I.  in the p r i n t e r  the The 1/H  plots.  S i m u l a t i o n program uses the f o r m u l a e developed  in Chapter I I ,  and c a l c u l a t e s the r e s u l t of our  experiment d e s c r i b e d in Chapter V I .  }2h Data  Reading  1 2 3 "  17  21 22 23 24  ?5 26 27 26 29 30 31 32 33  3U  F 0 R * A T (' M J " 9 E R OF RECORDS? READ 315.NNN 3 t 5 F0R*4T(T4) vjPPa2S*N'gM - R l TE f 3. <91 )NPP 91 F3RMATCT5) 00 20 L T N F i l ,M*N RE40(3,l)(4(Ti»n(I).I«l.SO) t F0R««4T(50(41,lx;F8,0n M J Ul,50,2 :  C*25aNJ <*ER M  OF o » T 4  nam,  C  IF(4(I),NE,CHAR1.0R.*(Tt1.NE,Crl4R2) 3  FORMATC  A8MDR**I»ITY  Is  L.TNE',13)  omavFu>i(0(inlD(I)>  18 19 20  P R I N T  314  15 16  Adjusting  DIMENSION 4t5O)7Dt50) T4T4CH4R1/'*'/ 0 4 T 4 CHAR2/'0'/  4  5 b 7 8 9 10 11 12 13 14  and  2 21 20  OtIl)sXFUM(0(In,0(in criMTtMUE wRTTE(2.2i)f0fIi.0(T-ll.ts2,5(i,2) F0R"4T(2F.14.7) cnNTjvjJE 3T0P E^O FUNCTION VFUM(X.Y) vFUMsvolo'. 5 E T 'J R ^ ENO FilMCTTON XF'JM(K.V) X F U S a l , / ( l . 22<»*y ) PETJHS END  PRJ.NT3,LI*E  PT P M ? 3 )  ( 1 4 ) •)  125 Synthetic Data Generator 1 2 J 4 5 6 7 8 9 10 11 t2  200 201 205 202 205  DIMENSION P T C t o n O l PTs3.1«15926 PRINT 2 0 0 F 0 R * A T ( » ENTER N U B E R 1? OAT* POINT P A I R S C i a ) ' ) READ 201,NNN FTRM T(TU) W9IT£(2,2051MN\) FORMATCIS1 PRINT 202 FORMAT ( • ENTER H"IN,MM X TN MS, ( 2 F 6 , 0 ) > ) READ 203,HMTN,MMAX FORMATC2F6.0) M  A  4  H T N C a ( H « A X - H M l N ) / F t . O A T ( N M N )  1J  ia  DO 100 1=1,1000 PT(T)s0'.  15  16 17 IB 19 20 21 22 23 ?U 25  100  6 2  26  5  1  7  CONTINUE PRINT J FORMATC'TO CREATE THE J U M OF T * E * P ( - 3 * n «C13 t 2. 1' ENTER fl.A.3.'.'. C5F6.0l'.'.'.OR ZER3 TO STOP' ) READ ?,B,A,0 FORMATC5F6.0) IMA.EQ'.o'.) 30 TO 99 PRINT 7.3,A,3 F0R*AT(3E10.3) MSHMAX 00  3  T s l . N N N  27  HsH-HTNC  28  Tsl./M  2R 30 31  5  32  99  PT(I)sPT(I)+T*EXP(-B*T)*C0SC2.*PI*A»T-3) CONTINUE GO TO 6 M=h"AX  33  Tsl'./M  3a  no  101  I«I,NVJN  35  HsH-HINC  36 37 38 39  Tsl'./M *RlTEC2.'4U,'TtT) F O R M A T (2E1«.7) CfNTlMUE  U0 Ul  «  101  STOP E  NO  « B T * A * T « 0  } . . , Ts 1  / M '  /  126 Wi ndow  t 2 J U 5 j, 7 B <t 10 11 12 15 ltt  15 tfc 17 18  I'  1 6 ^ 2  5 0  DIMENSION X(100n),V(1000) PT3i,tUt5"26 READ(5,1)NNN F0RM»T(I51 W9ITE(2,6)NNN FTRMATdSl 00 2 T » 1 , N N N READn,5ixm.virn C O R M A T (2E10.7) CONTIMU6 FsPI/<X(NNNl-XM OO a T3l,\iNN y a ) s v ( T ) * S T N f F * f X ( T ) - x ( l ))1 *RITE(2.5)X(I),Y(I)  F0RMAT(2El'a'.7) CONTINUE STOP E NO  121  Analysis 1 2 3 a 5 b 7 8 9 10 11 12 1 3 1 4 IS tb 17 18 1 9 20 21 ?2 23 2a 25 2b 27 28 29 30 31 32 33 34 35 36 37 38 39  315  10 9 1 ? 99 3 70 A 437  12  Program REAL T(1000),°Tf1000) PI2»2'.*3, 1415^265 REA0(3,315)NNN P0R"ATO5) XNMN«FLOAT(NN)N) 00 "» t»t»NNN QEAO(3»tO)TfI),BT(I) FORMAT (?E1<»'.7) CONTINUE PRINT 1 F0R"AT('0ANAI>STS?») REAO ? , I N A L FORMAT(Il)  GO STOPT0r3.4.5,b,7. > >).mi (  FORHIT!'OEOURIEP  <  "EAL  TRANSFORM'/'  M  FT^CsfFMAX-FMlMi/FLOATfM.IMF) FsF^IN 00 I t H t . N U M F 9IJMS0'. 00 12 J * l . N M N SJMsSU^+PT(J)*CPS(PI2*F*T(J)) CONTINUE SlJMsSU /XNNN M  13 11 4 3tt  F.3UM  *RITE(2.13) F0RMAT(2Eia'.7) FsF*FINC CONTINUE STOP PRINT 34 FORHATCOFOURIER IMAGINARY READ 8 , F M I N , F M , N S J M r  TRANSFORM/'  FMIM,FHAx.NUMF?(2F6 . 0 ,14 )  41<  ao  *RTT£(2.4371\IIHF FnC»CFHAX-FMlsjl/FLOAT (MJMF) FsFMTN  41 U2 «3  SJ »0'. 00 32 Jsl.MMM  00 31 Iat.NUMF M  au US "6 47 48 U9 50 51 52 53 54 55 5b 57  F M I N. F« A X , N'JMF ? ( 2F b . 0 . t 4 ) ' )  READ 8» FMIN.FMAX.NIMF F0R«AT(2Fb,0,l4) *RITE(2.437)^UMF F0R AT(I5)  32  31 5 54  S.JM33!JH*PT(J)*CPS(PI2*F»T(J)) CONTINUE S;J13SL)M/XNMN WRITE(2I13)F,3JM F«F*PINC CONTINUE STOP PRINT 54 FORMAT ( ' OPOWER SPECTRLJW,'/' FM T N, F* A X , 9EA0 R,FMTN,FMAX,NUMF wRlTEf2.a37)NUMF FINCB(FMAX-FHIN)/FLOAT(NJMF) F»FMT.N 00 51 I s t . N U M F  NJMF ? ( 2 F 6 , 0 , I « ) ' )  128  i SB 59 60  61 62  65  6a 65 66 67 6B  69 70 71 72 75 7a 75 76 77 7B 79 «0 SI 82 «S Bit 85 86 87 88 89 90 91 92  93 9U 95 96 97 98 99 100 101 102 105  52  51 6 7a  3UM«(SU*1 /XNNN)**2*(3UM2/XNNN1**2 wRITE(2.15)F,3UM FaF+FINC CONTINUE STOP PRINT 7U FORMAT ( ' O L A P l A C C TRANSFORMI/I REAO 8,SMTN,9MAX,NUM3  3M IN,3MA X , N J * 3 ? ( 2 F 6 , 0 , I a ) • )  «RITEC2.U37)\|JMS STNC«(3 AX-3MINI/FLOAT(NjMS) M  SSS^IN  OD 71 Ixl.NUMS 3U aO'. 00 72 J•1,NNN M  72  SJM«3UM*PT(.n*EXP(-3*T(J)) CONTINUE SJ a3J /XNNN W9ITE(2.13)3,SUM Sa3*3INC CONTINUE 3T0P PRINT 9tt M  71 7  9a  M  FORMAT('OLAPL*CF- 0*ER B  READ  SPECTRUM'/'  8,SMtN,SMAX,NJMS  8MIM,3MA X,NJMS?C?F6.0,  wRITEf2.U37)NUMS  105 1 06  PRINT 105 FOR^AT(> THE ANGULAR READ 106,A FORMAT(F6'.01  *RE3'IENC Y? ' )  S T N C « f 3M X-3MlN1/ri.0AT(NjM3) A  SaS"IN 00 91 I a l . N U M S SUMlaO, SUM2oO, 00 92 J s i . N N N S'JMlsSUMi+PT(J)*EXP(-3*T(T)l*C03(A*T(J)) 92  S'JM2 = S U « 2 * P T ( J ) * E X P ( - S * T ( J ) ) « S I N ( A * T ( J ) 1 . CONTINUE SUMa(3UMl/XNNN)*«2+(3JM2/XNNN)**2 wRITF(2,15)3,SJM  loa  105 106 107  3'J^loO, SUM2aO, DO 52 J « t , N N S SJMlsSU^l*PT(J)*C0S(9T2*r*T (J) ) SUM2sSU"2*PT(J)«SIN(Pl2*MTU)) CONTINUE  91  SsS*3TNC CONTINUE STOP END  129 Plotting t  DIMENSION  xnoon),YUOo<n,CH4nm  2 J 4 5 6 7 8 9 10 tl 12  LOGICAlM QUE REAL l < 7 5 ) OAT A CHAR/• 1,1*1/ C REAO IN THE DAT* READ(3.2)N 2 FORMAT(151 X M 4 X « . 1 '.E*50 YMAXaXMAX XMIN«1,E*S0 YMJNsXMIN OO 3 T 31» N  is  REAO(3.«)x(n.yi n  14 15 16 17 1R 19 20 21 22 23 2tt 25 26 27 f 28 29 30 31  32  33 34 35 36 37 38 39 UO ai 42 U3 44 US «6 U7 48 49 50 51 52 53 5a 55 56 57  ,  . a  FORMAT(2El«'.71 IF(X(I)'.LT.XMlMlXMTNaXfT) IF(X(t)^GT.XMAxiXMAXaX(I) IF(V(T).LT.VMlN1YMtNiY(t) IF(Y(t)'.GT.YMAXlYMAX»Yf I) 3 CONTINUE C F J NO S C A L I N G FACTORS X3»8,/(XMAX-XMIN) YSa8./(YMAX-YMIN) XSRs50,*XS/8. YSP«50,*YS/8. C PRINTER PLOT PRINT 7 7 F 0 R M A T ( • 00 YO'J '*ISH A P R I N T E R P L 0 T ? , . , K I N 0 L Y ENTER Y OR N') REAO 9,.QUE A FORMATUil I F (LCOMC (1 , 0 J E , »V« T.NE'.OlGO TO 9 PRINT 10 10 FORMAT ( i HOW MANY PRINTER PAGES wOULO YOU LIKE?'. .,(121 ' 1 REAO 11,NP 11 F0RMAT(I2) TNC»N/(NP*60) 00 l a I • 1 . 7 5 L(I)=CHAR(11 ta CONTINUE IPNTst 00 12 1=1.N,INC L(IPNT)sCHAR(t1 I?NTaIFIX((Y(I)-YMIN)*v3P*t1 L(TFNT)aCHAR(2l wRITE(2.151xrl).Y(I1,(L(J1,Jat,751 15 FORMAT(I i , 2 ( E l U , 7 , I X ) , • (', 754 1 ) 12 CONTINUE 9 PRINT 16 16 F0RMAT ( * 00 Y O j WISH 4 PFN PLOT?,,,(Y,N)•) RE40 17.QUE 17 F0RM4T(4H I F ( L C O M C (t » Q J E , ' Y i ) ' . NE'.0)STOP C PLOT CALL PL0T(XS*(X(l)-XMlN),YS*(Y(1)-YMIN),3) 00 5 I82.N CALL PL0T(XS*(xm-XM!Nl,YS*(YCI)»YMT.N),21 5 CONTINUE C » L L PLOTNO  13Q  M.I.  Simulation  1 2 3 a 5 6 7 ft 9 10 11 12 13 ia 15  DIMENSION X»l ( 5 O 0 ) , X A 2 f 5 0 0 ) > X A 3 C 5 0 0 ) DIMENSION X M ( 5 l ? ) , X l ( 2 5 « > ) , X 2 C 2 5 6 ) S E A L M,NliD,K4PP4,NCYC INTEGER y . s w COMMON XA1M»K,XA2MAX,XA3MAX,HTM4X.MIMIN DATA Y/'Y'/ PSI(RDUM)aATiNfETA*TAN(Pt*BDUM*DELf*MT-PI/«.)) AfRDU,AADU)»(AADU*AADJ*(1.-ETA)/(l.*ETAnft(SrjPT(2.)/2.)* 133RTm.*ETA*ETA)tU.-ETA*ETA)»SIN(2,»Pi»KDU*DELF*HI)) M(NU0.ALPHAn)«Ai*8IN(X*P3t(l.)J . t+A2*CSINf2,*X*PSI (2. n.0.5*NUD*SINC2.*X*2>PSI f I . ) ) ) ?*A'3'*fSlNf 3 , » X » P 3 I ( 3 . ) ) . 1 . 5 * N U 0 » A L P H A ! 3 » 3 ( S I N ( 3 ' . * X * P 3 I ( 1 . )*P3I (2.5 ) - 0 . 2 5 * N U O * 3 l N ( 3 . * X * 3 * P S I ( l . 1 ) ) ) AT(AB.Bfl,AL.3L)sS3RT((A8*SIN(AL)+38*SlM(BL))**2 1 •(AR*CO3(AL)*«9*C0SrBL))**2)  16  PT»3,1«I59  17  18 19 20 21 22 23 2U 25 26 27 I 28 29 30 31 32 33 34 35 56 37 58 39 UO 01 U2 U5 ait U5 U6 US U9 50 51 52 53  SO  55 56 57  PRINT 1  1  2  3 a 5  FORMAT (' P L E A S E  ENTERlA  M B  Ll 'JOES T  OF TM£  HI3HFST  FREQUENCY'/  1 ' F I R S T , 3 E C 0 N D , AND THll?0 HAR*TNTC ( I N S A J 3 S ) , AND E T A . . . C U F U . 0 ) ' ) READ 2.AA1,AA2,AA3,ETA PORMAT(UFa.O) AAlsAAl/1000. AA23AA2/1OOn. AAJ3AA3/1000. PRINT J FOR«AT<» P L E A S E ENTER £P 8 T L ON . A N1 J E L A , . ( ? F a . 0 ) ' ) READ a . E P S l L . O E l TA F0RMATC2FU.0) PRINT 5 T  FORMAT (' P L E A S E  ENTER  T ME  MEAN  FREQUENCY F , ' /  1 ' AND THE D I F F E R E N C E IM FREQUENCY OELF ( I N M G ) ( 2 F U . 0 )') READ 6 , F , D E L F 6 F0RMAT(2F«,0) PB1000,*F DELFsi000'.*DELF H K A P P A « 8 . * P T * P t « ( t ,-OEi_'TA)*F PRINT 9 9 F 0 R M A T ( ' DO YOU " I S H A NON-SMEARED PL 3 T ?.'. . f Y , N 1 ' ) READ 10.NSM 10 FORM T(Al^ I F ( N S M , M E . Y ) S O TO 200 PRINT 7 7 F 0 R M A T ( ' P L E A S E ENTER ,^TN,HMAX < '< 31 , NUMH , ' , , ( 2 F a . 0 , I 3 ) ' ) READ 3,MMIN,HMAX,NUMH 8 F0RMATC2Fa,0,131 C C A L C U L A T E THE NQN.sMEAREO A x P L l T J O E HIMINsl'./HMAX HIMAX«l'./HMIS HriNC»(HIMAX-HIMlN)/FLOAT(NUMM) HlaMIMIN XA1MAXB0, XA2MAX80. XA3MAX30, A  I n  DO 100 Tal.NjMH  HTaHI*HTINC KAPPAaHKAPPA«MI*MI A1aA(l AAl) a (  131  58 50 60 61 62 63 6U  XAtmaAl IFCXAtm'.GT.XAT^AXmi'MiXsXAHI) A2=A(2,,AA2) PSI2aPSI(2.) P3IlsPSlM.) X A 2 ( I ) » A T (Aa,-'.s*KAPP4*Al *A1 * E P S I L » 3 T a . 2 . * P 3 1 1 1 IF(XA?CT)'.GT.XA?«AX)XA2H X = XA2(I)  65  A3»A(3,.AA3)  67  AlNTaATn'..-0.25*«APPA.A|*Al/A2*EPSIti sn*o i2,3,.pgTn  66  6S 69 70 71 72 73 7u 75 76 77 78 79 80 81 82 S3 8U 85 86 87 88 89  90 91 92 93 oii 95 96 97 9fl 99 100 101 102 103 10K 105 106 107 108 109 110 111 112 113 110 115 116 117  I  B  4  PSI3aPSIC3.) p  3  PSIINT3ATANC(C03(P3Il*oSia)»0',a5«i<APPA*Al*Al/Aa*E 3lL*C0SC3.*PStl 1 ))/CSlM(P3n*BSl2)-0.25*KAPPA*Al*Al/Aa*EPSIL*SlN(3.*P3Il 2))) XA3(I1=A3*AT(l,.-1.5*A?*A2/(Al*A3)«<APPA*Al*At/Aa*EPSII.»AlNT,P3I3 l.PSIINT) IF(XA3(!)'.GT.XA3MAX)XA3MAXsXA3(I) too CONTINUE C NORMALIZE 00 101 Isl,NJ*n XAt(I)sXAt(I)/XAl*AX*l0. XA?tI)aXA2(n/X«2MAX*lo, XA3U)»XA3(I)/XA3<»AX*lo. 101 CONTINUE C PLOT XSCALE»15'./FLOAT (NUM4) C»LL °ISTR(ETA,FP9IL) C A L L PL0Tf2'.,XA1 ( 1 ) , 3 ) 00 102 T = 2 , N J M 4 CALL PL0T(XSCA 'F»FL0AT(I)»2,,XAim.2) 102 CONTINUE C*LL P L 0 T ( 2 . , X A 2 ( 1 ) . 3 ) o  L  DO  103  T=2,NJMH  CALL PLOT(XSCALF*FLOAT(T)+2,,XA2tI)•2) 103 CONTINUE CALL L 0 T ( 2 ' . , X A 3 ( t ) . 3 ) no 10fl T a 2 N J » M CALL PL0TfXSCALF*FL0AT(I)+2.,XA3(I).2) 10a CONTINUE C SMEARED AMPLITUDE 200 IFCNSM.EO'.YICALL PLOT ( ? 2 . . 0 , , - 3 ) PRINT 11 11 F O R A T ( ' 00 VOiJ «*ISH A SMEARED PL 0 ? . . . ( Y , N 1 ' 1 PEAO 12.SM 12 FORMAT(Al) IF(SM'.NE.Y)GO TO 999 PRINT 15 15 FORMATC' ENTER THE P-P MODULATION IN <3 (Fd'.O)') READ 16. PPMOD 16 FORMATCFo'.O) PRINT 17 17 FORMAT( P L E A S E ENTER HMTN.HMAX ( I N <G),AND N U M H . , , 1 , 1 1 (2F«'.0.13) • ) READ I8.HMIN.HMAX.NUMH 18 F O R M A T ( 2 F a , 0 , I 3) HIMINB1),/HMAX HTMAXal./HMTN CALL PL9TR(ETA,FP3IL) PM00sPPM0P/2. HIINCa(HIMAX.HIHlN)/FLO»T(NUMH) HI a H I M I N 3  #  M  T  1  13?  KAl^AXaO.  US 119  120 121 122 123 12<l 125 126 127 128 129  130  XA2MAXX0.  XA3 AXaO. OO 300 Ts1 , N J M M HlsHI*MTINC HTMINC»(l'./f r./Hl-PHOOj-l *./(t'./Ht*PM00n/512. HAMF«PI/(HIMINC#512») HTMS«J,/(1,/MI*PMOD) MT^iHIMS GENERATE TME ST3NAI. FROM j)NE MOOULATTON 9*EE<> DO 301 T=1.512 Hl*3HTM+HIMINC H  C  KAPPAsHKAPPA*HT*Ml  131  A 1 aA ( 1 , « A A 1 ) A 2 » A ( ? , , A A 2 )  1 3 2  133 13U 135 136 137 138 139 mo 1a1 1U2 U3 1UU t"5 1«6 1«7 ll«8 1U9 150 151 152 153 150 155 156 157 158 159 160 161 162 163 16U 165 166 167 168 169 170 171 172 173 17U 175 176 177  A3aA(3,.AA3) Xs2.»ol*cF*HtM. .5) C THF SINE IN THE NpxT LINE T3 THE MANNING *TNDO* XM(J)sMfEP3rL*!<»P A*AUAl/A2,A2*A2/CAl*A3-n* 13INtHANF*tMIM-HIM3)) 301 CONTINUE C SEPARATE XM INTO ITS ANTI3V" METflTC ( X t ) , A N D SYMMETRIC c CO PONENTS 00 302 J a 2 , 2 5 5 XI ( 2 5 7 - J ) 3 ( X M ( 2 s 7 - J 1 - X - ( 2 5 7 + jn/2. X2(25 7 - J ) s ( X M ( 2 5 7-J) + XM(257*jn/2. 302 CONTINUE XKllao'. X1C256) = 0'. X2C1)sXM(i) X2t2S6)sXM(256) C TAKE THE FOURIER TRANSFORM CALL S 5 t 2 f X 1 , X n CALL C512(X2,X2i C CALCULATE THE FIE|.n SPACING OF THE FTRST HAR"ONTC FSPlal,/(HI*Ml»F) C THE NUMBER OF CYCLES PER "«30 SWEEP,'.. NCYCaPPM00/F3Pl C THIS IS rfHERE THE FIRST HAR«ONIC I S IN THE TRANSFORM C FORM WINDOWS I*L'SNCYC/2' * l ' . T*iL2aMCYC*l , 5 * l ' H L 3 = NCYC*2,5*1, lxR3a.NCYC*3.5*l . C FIND THE MAXIMA OF THE A*9L!TU0E SPEC T RU *,'., XMAXsO, DO 3 i n J a t W L l i t w L ? A«PaS3RT(Xl(J)**2*X2(J1 * » 2 ) IF (A-P'.GT.XMAXIXHAXSAMP 310 CONTINUE XA1 (I)aXMAX IF(XMAX'.GT,XA1MAX)XA1MAXSXMAX XMAXaO, DO 311 JsT^I. 2» rwLl AMP«S3RT(X1(J1**2*X2(J^**2) IFfAMP.GT*. X M A X ) X M A X 3 A M p 311 CONTINUF XA2(I13XMAX IF(XM4X'.GT,XA2MAX)XA2MAX«XMAX XMAXaO, ,  P  M  fX2)  133  1 78 179 180 181 182 183 18(1 IRS lSfc 187 188 189 190 191 t92 193 19« 195  \b a  197 199 199 200 201 202 203 200 205 20(, 207 |208  209 210 211 212 213 210 215 216 217 218 219 220 221 222 223 220 ??5 2?6 227  00 31? Jat"(L3.tw»3 A«PsS3RT(Xl(.n**2*X2(!)*«2) IF(AMP,GT'.XMAX)XMAXaAM8 312 CONTINUE X4J(I1»XM4X IF(XMAX'.GT,XA3MAX)XA3*AXaXMAX 300 CONTINUE C °LOT'..'. C FIRST HARMONIC IN SQU4RE9, C SECOND HARMONIC IN TRIANGLES. C THIRO HARMONIC IN X'S C C N0RM4LIZE 00 320 Ial,NUMH X41(I)SXA1(I)/XA1MAX*10. x42(I)aX42(T)/X42MAX»lo, X43msXA3(T)/XA3M4X*lo. 320 CONTINUE C "LOT XSCALEa15'./FL' 4T(NUMH) 00 003 Ia1,NJMH xRNTsX3C4LE*FL04T(I)*2'. CALL S T M B r ) L f X P N T , X A l { t ) , . l « . 0 . 0 ' . , - n CALL 8YMBOL(XPNT,XA2(n,.lO,2,0.,-l) CALL SYMROL(XPNT.XA3(h..lO,O.0.,-l) 003 CONTINUE 9 9 9 CALL PLOTNO STOP E NO SUBROUTINE PLSTR(ETA,E = STL ) 1  COMMON  XA1MAX,XA2MAX,XA3-AX,MIM4X,MIMIN  CALL 4XrS(l'.,0..'FIRST HARMONIC AMPLITJ0E'« 20. 10.,90,, 10'. ,XA1MAX*100.) CALL 4X13(1.5,0'. # 'SECOND H4RM0NIC 4MP ITJOE',25.1 0 , , 1 90. , O'. , X A 2 M A X M O 0 ' . ) CALL 4X13(2'., 0.."'THIRD HARMONIC 4 M P L I T J O E ' , 2 0 , 1 0 . , 190,,0'.,XA3MAX*100.) CALL A X I S ( 2 . , 0 . ' l / H ( < 5 * « - l ) i . - l l , 1 5 . , 0 . 0 . H l M l N . 1(HI«4X-HIMIN)/JS.) CALL P L O T ( 2 . , l o ' . , 3 ) CALL ° L 0 T M 7 . . 1 0 , , 2 ) C4LL "LOT(17,,o'.,2) CALL 9T«ROL(1 7.?,~10.,.?8,22,0'.,-n CALL S V * 8 O L ( l 7 , u 8 , 1 0 . , , 2 » . , = , , 0 , , l ) CALL NUMflFR(l7.76,10.,.29,ETA,0.,2) CALL SYMBOLd?.?."., .2R.20.0. .-1) CALL SY-BOLd 7.UR.9. , . R, ts ' , 0 . , 1 ) CALL N U M R E R ( l 7 . 7 6 , 9 . , . 2 8 , E P 3 l L , n . . 2 ) RETURN END L  ,  t  ,  ?  134  BIBLIOGRAPHY A n d e r s o n , J . R . and G o l d , A . V .  P h y s . Rev. 139, No. 5A, Al459 (1965).  A o k i , H. and Ogawa, K. J . Low Temp. P h y s . 32. "131 (1978). Condon, J . H . P h y s . Rev. 145, 526 (1966). Condon, J . H . , and W a l s t e d t , R . E . P h y s . Rev. L e t t . 2J_ 612 (1968). Dingle,  R . B . P r o c . Roy. S o c . (London) A211, 500 (1952).  Dingle,  R.B.  P r o c . Roy. S o c . (London) A211, 517 (1952).  E v e r e t t , P . M . and G r e n i e r ,  C . G . P h y s . Rev. B ]8_, 4477 (1978).  Gradshyeyn, I . S . and R y z h i k , I.M. T a b l e s o f I n t e g r a l s and P r o d u c t s , Academic P r e s s , N . Y . , London (1965). G o l d , A . V . and Schmor, P.W. Can. J . P h y s . 54_» 2445 (1976). I v o w i , U.M.O. and Mackinnon, Knecht,  B.  Lifsbi'tz,  L.  J . P h y s . F'6_, 329 (1976).  J . Low Temp. P h y s . 2]_, 619 (1975). I.M. and K o s e v i c h , A . M . S o v i e t  P h y s . - JETP 2_, 636 (1956).  Ogawa, K. and A o k i , H . , J . Phys. F. <8, 1169 (1978). Onsager, L. P h i l . Mag. 43, 1006 (1952). P e r z , J . M . and Shoenberg, Phillips,  D . J . Low Temp. P h y s . 25, 275 (1976).  R . A . and G o l d , A . V . P h y s . Rev. J_78, 932 (1969).  Pippard, A . B . Physics of Metals, V o l . 1 : E l e c t r o n s , Cambridge U n i v e r s i t y P r e s s , London, P.113 (1969)• P i p p a r d , A . B . and S a d l e r , Shoenberg, Testardi,  D. P h i l . T r a n s .  F.T.  e d . by J . M . Ziman,  J . S c i . Inst, series 2  2_, 101 (1969).  Roy. S o c . (London) A255, 85 (1962).  L . R . and Condon, J . H . P h y s . Rev. B j _ , 3928 (1970) .  Tobin, P . J . , Sellmyer, 28A, 723 (1969).  D . J . and A v e r b a c h ,  van W e e r e n , J . H . and A n d e r s o n , J . R .  B.L. Phys. Lett  J . P h y s . F 3_ p^2109 (1973).  

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