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Some magneto-optical studies of paramagnetic salts at low temperatures Rieckhoff, Klaus Ekkehard 1959

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-SOME MAGNETO- O P T I C A L -OF PARAMAGNETIC •AT LOW  STUDIES SALTS  TEMPERATURES*! hy  .KLAUS E K K E H A R D  ,-B.So., U n i v e r s i t y  A  RIECKHOFF  of British  Columbia,  T H E S I S SUBMITTED I N P A R T I A L THE R E Q U I R E M E N T S FOR -MASTER  in  OF  1958  FULFILMENT  THE DEGREE  OF  SCIENCE  the Department of Physics  -We  accept t,o  this  thesis  the required  THE U N I V E R S I T Y  as  standard  OF B R I T I S H  September,  conforming  1959  COLUMBIA  OF  SOME M A G N E T O - O P T I C A L S T U D I E S OP PARAMAGNETIC  S A L T S A T LOA  7  TEMPERATURES  ABSTRACT  Short  of the theories  waves i n an  magnetic and  resumes  anisotropic  of the influence are  effect  polarized  given.  o f propagation o f  electro-  medium, o f t h e F a r a d a y  o f paramagnetic  resonance  The P o i n c a r e s p h e r e  effect,  on the Faraday  i s introduced  t o describe  light.  A paramagnetic  resonance  spectrometer i s described,  which  was m o d i f i e d s o a s t o a l l o w t h e s t u d y o f m a g n e t o - o p t i c a l phenomena  under  meter were 101  the influence  operated located  mode,  with  i n t h e 2-band u s i n g  i n a transmission  and immersed  holes allowing  direction system  o f paramagnetic  parallel  the passage  on t h e sample.  The  Glan-Thompson p r i s m  the  passing the analyzer  light  circuit  and could  be  magnetic  emerging  analyzer.  An  light  i n a  optical  ( ^ = 5461  t h e sample  The r e l a t i v e  could be measured  displayed  was p r o v i d e d  t h e sample  field.  from  samples  i n t h e TE  The c a v i t y through  spectro-  The  operating  monochromatic  light  through a  plier  of light  provided plane-polarized  incident  cavity  helium.  to the external  The  a 2K39 K l y s t r o n .  type  i n liquid  resonance.  A)  passed  intensity  of  by a p h o t o m u l t i -  as a f u n c t i o n  o f time  on an  oscilloscope. Experiments lattice and  relaxation  are described time  external magnetic  temperature  was m e a s u r e d  field.  and magnetic  i n detail  as a f u n c t i o n  In these  field,  i n which  the spinof  temperature  experiments, f o r a  the Faraday  r o t a t i o n was  given reduced  by  pulses The  o f microwave  r e t u r n o f t h e Faraday  equilibrium inferred  tensity  value after  from  transmitted  versus  time  Oerstedt The  seconds.  to  the resonance  the  between  time  except field  experiment  showed t h a t  forthis  laxation  could  Mention particular ation  f o rthis  i s strongly  o f . 0 0 1 t o .1 proportion-  and showed o n l y  small  corresponding  time  relaxation  salt.  of the length  could  ethylsulfate  No  o f neodymium  q u e s t i o n was t w i n n e d .  are given.  t o be inversely  no e f f e c t  tion  time  axis  of the c r y s t a l with  i s d e s c r i b e d , which  case  respect  be smaller  than  case. effect"  ethylsulfate.  dependent  of the  observed.  time must  i s g i v e n , by assuming In this  be  Within  accurate determination o f the r e -  b e made i n t h i s  effect  780 a n d 2540  dip at a field  i s made o f a n " o v e r s h o o t  crystal  relaxation  f o r t h e microwave f r e q u e n c i e s used.  on c e r i u m  1 millisecond  between  o f the order  f o ra large  the spin-lattice  time  records o f this i n -  1.38°K and 4.22°K  appeared  p u l s e s on t h e r e l a x a t i o n  An  be  of the light  o f the spin-lattice  f o rfields  accuracy o f the experiments  microwave  relationship  power o f t h e t e m p e r a t u r e  dependence,  to i t s  were o b t a i n e d and t h e r e l a x a t i o n  measured were  The r e l a x a t i o n  to the third  field  ethylsulfate  times  o f time  these records.  and temperatures  relaxation  t o the cavity.  power was c u t o f f c o u l d  Photographic  o f t h e measurement  o f neodymium  applied  as a function  v e r s u s time  relationship  from  length  t h e microwave  by t h e a n a l y z e r .  Results time  rotation  the intensity  time was deduced  al  power o f v a r y i n g  one may  observed A possible  that  explan-  the crystal i n  infer  that  on the orientation t o the external  i n one  the relaxa-  o f the o p t i c a l  magnetic  field.  iv, Tne r e s u l t s were found to disagree with present-day theories of paramagnetic relaxation.  Assumptions of doubtful v a l i d -  i t y i n the theory are discussed as possible reasons for such d i s agreement .  In p r e s e n t i n g the  this  thesis in partial  r e q u i r e m e n t s f o r an  advanced degree at the  of B r i t i s h Columbia, I agree that it  freely  agree that for  available  the  f o r r e f e r e n c e and  permission for extensive  s c h o l a r l y p u r p o s e s may  D e p a r t m e n t o r by  be  c o p y i n g or p u b l i c a t i o n of t h i s  gain  shall  p  Department of  a l l o w e d w i t h o u t my  h  y  £ i C £  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a . Date  Sep  I  ,  Columbia, 19  s  9  shall .I  the  make  further  copying of t h i s  his representatives.  be  study.  of  University  Library  g r a n t e d by  that  not  fulfilment  Head o f  thesis my  It i s understood  thesis for written  financial  permission.  V  ACKN OlYLEDGSMENTS The research described i n t h i s t h e s i s was supported by the National Research Council of Canada through research grants to Dr. J.M. Daniels and the award of a Bursary Studentship  (1958-59( and  (1959-60) to the author.  I am g r e a t l y indebted to my research supervisor, Dr. J.M. Daniels, who suggested of the work.  the research and helped me i n a l l phases  His help, advice, and contagious enthusiasm made  t h i s thesis p o s s i b l e .  I owe.-.' him a special word of thanks for  providing me with an assistant, Miss Id, Jay, i n the f i n a l phases of the work. Grateful acknowledgement i s made to Miss M. Jay, who not only drafted a l l the diagrams for this thesis, but also did most of the work involved i n reducing the hundreds o f photographs containing the experimental r e s u l t s to numerical data.  Her help was  Invaluable, To the members of the f a c u l t y o f the University o f B r i t i s h Columbia, who - be i t i n the classroom or i n private discussions provided me with knowledge and enthusiasm, I also owe a great debt of gratitude. I am very g r a t e f u l to the technicians o f the Department of Physics for t h e i r help, e s p e c i a l l y to Mr. J . Lees, Mr. W. Maier, and Mr, H. Zerbst. I also wish to express my appreciation to a l l my fellow students, who helped me i n many ways, and - l a s t not l e a s t - to my wife Marianne, who contributed to t h i s thesis by her encouragement and many personal s a c r i f i c e s .  vi  TABLE  OF  CONTENTS page  Chapter 1)  I:  Introduction  Propagation  of Electromagnetic  Anisotropic  Medium  2)  Faraday  3)  Poineare's  4)  Faraday  Rotation  5)  Faraday  Effect,  6)  2.  Rotation  Waves  i n an 1  and B i r e f r i n g e n c e  Representation  (1stpart)  of P o l a r i z e d  and Birefringence Paramagnetic  Light  (2nd P a r t )  Resonance,  5 10 13  and  Spin-Lattice  Relaxation  16  Spin-Lattice  Coupling  21  Chapter  II:  Experimental  Arrangement  23  1)  Description  o f the  apparatus  23  2)  Description  o f an Experiment  29  3)  Preparation  o f Samples  39  Chapter 1)  III:  Measurements  Linearity,  43  Reproducibility,  e t c . ,o f the  Apparatus 2)  Some  3)  Effect  4)  Measurements  43  Typical Results  Fields  Time  at Various  and Temperatures  6)  " O v e r s h o o t " Phenomenon  Bibliography  54  of Relaxation  Experiment;* w i t h  IV:  52  o f Pulse-Length  5)  Chapter  ..  Discussion  Cerium  55 Ethylsulfate  63 64 65  73  T  vii.'.  LIST  OF  ILLUSTRATIONS following  Figure  I  :  Views o f of  the  the  motion  electric  of  vector  the  tip  for  and Figure  i l  :  11  Decomposition of rized light into components  elliptically polacircularly polarized 11  Figure  III  :  Poincare  Figure  IV  :  Properties of  Figure  V  :  E f f e c t o f combined and b i r e f r i n g e n c e  Figure  VI  :  sphere  View o f  IS  the  apparatus  Poincare  sphere  Faraday  ....  effect 15  with  dewars  removed Figure  VII  :  View  23  of  apparatus  in  operating  condition  23  Figure  VIII  :  View  of  optical  system,  source  Figure  IX  :  View o f  optical  system,  detector  side  side X  :  Microwave  Figure  XI  :  Pulse  Figure  XII  :  Optical  Figure  XIII  :  Circuits  system  23  modulator  23 circuit  24  system  27  associated with  optical  system  27  Figure  XIV  :  C r y s t a l growing  apparatus  Figure  XV  :  Verdet  vs. -p-!  XVI t o : XXVII :  ..  23  Figure  Figures  12  Typical process  constant views o f  the '  013*  40 4  6  relaxation 53  page  v-ii-i  following page Figure XXVIII •: Figures XXIX to XXX  Rotation vs. time on semi-log scale  : Typical views of effect of pulse: Length ;'  Figures XXXI : Typical views of "overshoot t o XXXII : phenomenon"  -53 53 53  Figures XXXIII: Relaxation time vs. magnetic f i e l d ... to XXXIV : • •  61  Figures XXXV : Relaxation time vs. temperature to XXXVI :  62  Figure XXXVII : Poincare' representation o f overshoot effect  69  Chapter  of  I:  Introduction  In  the experiments  paramagnetic  resonance  measure t h e s p i n - l a t t i c e single  crystals  sections insofar  is  relaxation helium  described  time  In Chapter  with  account  of the results  finally  gives a  i s utilized  o f paramagnetic  temperatures.  f o r a proper  The  I I the apparatus used  the experiments.  Chapter  obtained  ions i n  following  f o r the  I I I gives a  knowledge.  1)  P r o p a g a t i o n o f E l e c t r o m a g n e t i c Waves  waves  equations  describing  i n general are  involved,  experiments i n con-  detailed  i n the experiments.  discussion o f the results  basic  to  understanding of the  day  magnetic  effect  the influence  a n d so a r e t h e e x p e r i m e n t a l t e c h n i q u e s u s e d  nection  The  thesis  c h a p t e r g i v e t h e t h e o r y o f t h e phenomena  as i t i s required  experiments.  i n this  on t h e Faraday  at l i q u i d  of this  described  Chapter  i n the light  o f present  i n an A n i s o t r o p i c  the behaviour  ( i n Gaussian  IV  of  Medium  electro-  units)  (l.la)  c  (1.1b)  div  (l.ld)  The  the  D.  symbols  field,JB  electric  density,  <}B  = 0  (1.1c)  magnetic  L  E  url  and  have  their  the magnetic  displacement, c  the velocity  u s u a l meaning, i n d u c t i o n , JE1 the current of light  i . e . H_  i s the  the e l e c t r i c  density,  i n vacuo•  g  the  f i e l d , J^. charge  2 JTor a l i g h t we  wave p r o p a g a t e d  have t h e s u b s i d i a r y  ±  (1.2a)  equations  g = 0  (1.2c)  B  =  H.  (1.2d)  Dj. =  €j  (l„2d)  dices,  Ej  L  i swritten  standing An  i n tensor  appearing  tensor  equations  (1.2),  x,y,2  u j  a r e dummy i n -  .  i n d i c a t e s summation tensor  over  H\  (curl  (l.3b)  ( c ^ L E ^  o f t h e medium.  theequations  (1.1) c a n be w r i t t e n  = ±  e  L]  § |  j  ^  divH = 0  (i.3c)  Considering Cartesian  a plane  coordinate  (1.4a)  E  (1.4b)  Ey  x  wave  system,  propagated  i n t h e2  direction of  l e tus p u t  t c o t -^e  =» T  e  *  e  i.«t-/Z  For  a l l i t s  notation)  (1.3a)  a  twice  i sthe permittivity Using  notation,  f o r the coordinates  index  values,  (in  dielectric  0  -  (l.Bb)  where  i nan anisotropic  such a plane  r[  wave, whatever  the form  of  E  2  , we  have  (1.4c)  =0  =  Substituting  (1.4)  into  (1.3b)  and  solving  Mc  f o r the  we  obtain  H = o A  Equation (1.4)  (1.3c)  and  i s satisfied  (1.5)  into  (1.6a)  -r-gp ^  (i.6b)  ctr^i  ( 1 . 3 a ) , we  °  e  c.^  the  ~  =~  ^  f o r EL  Substituting  *  +  €xy^+€xz5,y  \*Y*Z* yY'V Y*5' e  9  \ £ ? x ^ 6  2  s  ^ t 6 2  y  2  €  (1.6c) i s the  values  (1.5).  U*xf  6  0 = — e  where  by  get  ~~C"  e  (1.6c)  Equation  identically  same  as i s o b t a i n e d when  a n d »H  into  into  (1.6a)  equation  substituting  (1.3d)  Hence  (1,7)  Substituting  (1.7)  and  ( 1 . 6 b ) we  finally  get .a. , A  (1.8a)  Equations  (  SxK-^^i^)^ (1.8  polarizations change o f  +  ) represent of those  polarization,  and  (£xy- TTr )'ri §J  an  X  eigenvalue  waves w h i c h also  =• problem  and  give  are propagated  their propagation  the  without  constants  ^  4 Note  that H  propagation,  b u t EL  If so  the  lies  components  of the  coefficients.  polarized,  but  £ -  tensor  of the eigenvalue  eigenvectors are linear  real  to the d i r e c t i o n  of  does n o t .  are the c o e f f i c i e n t s  the  perpendicular  Thus  combinations  the stable  travel with  equations of  (1.8),  ^  propagated  different  are a l lr e a l ,  and  and  waves  velocities.  then  are  This  with  plane  i s known  as b i r e f r i n g e n c e . If, have  however,  a Faraday  attenuation, must  e-  the  effect.  tensor  imaginary  components,  F o r the waves t o be propagated  the eigenvalues  be H e r m i t i a n .  has  must  be  real,  Writing the eigenvalue  and  hence  equation  we  without  the (1.8)  tensor as  (1.9a)  (1.9b)  We  -loc^  +  -  eigenvectors are ^  polarized is  waves.  decomposed  different tween on  —  ^~4s  have  (1.10)  The  -  S  A  into  two  6  toe  , which represent  polarized  hence  a phase  a plane  circularly  wave e n t e r i n g s u c h  circularly polarized  circularly polarized  emerging to give  polarization.  £ iy^  plane  velocities,  t h e two  =  waves t r a v e l l i n g  difference components,  polarized  a medium  wave w i t h  i s introduced and a  these new  with be-  re combine  plane  of  5 2)  Faraday  Rotation  As of  light  shown  in a  a  an  s  ~  be  calculated  outline  of  how  Consider zero, with From  a  set  tensor  3 c  0  e  tCO"t  For  of the  this  be  may  an  ion i n a  of  excited  .  the  hy  propagation  the  substance.  £ -  This  the p r o p e r t i e s of  done  i s now  state  tensor.  a  tensor ions  and  , whose e n e r g y  I'YK^'  to  the  theory  given:  !'%>>  states  subjected  of  the  i s  E-k .  energy  Let  state-function  harmonic  of  perturbation  ^  = M  <  the -  by  Y o - i - i ;  of  chapter,  perturbation theory,  .  interaction  this  introduced i n electromagnetic  from  i s given  TOW  (2.1)  as  property  , when  TO'  of  directly  timedependent  ion  1)  part)  i s i n general governed  phenomenologieal  can  B i r e f r i n g e n c e (1st  in section  crystal  The is  and  <k?u 4- c o — l  radiation with  ~  e  matter,  as  e  )\  i s w e l l known,  we  q A can  replace  the  potential  ic  charge,  magnetic  for  an  momentum  V  and  by A  V  operator  +• oj y  and  _|0  hy_£  , w h e r e cj are  the  potentials  e  can  choose  for the  a gauge  3  ,  algebraic of the  and electron-  electro-  numerical  a t o m , w h i c h was  » olivA 0 . 3  i n which  electronic  originally  of  charge,  the  becomes  (2. )  i s the  r^r-  field. We  Writing  the  Be = ae -fa 0  A*  form  the  Hamiltonian  To  the f i r s t A ^  -  order,  For  d i r e c t i o n may  E  he  .  wave p r o p a g a t e d  i n the  expressed  i s the strength of the electric  0  vector  need this  Since  i n J\.  the term  E_ = & E o siv>{co(t--§•)}  Where  we  can drop  plane-polarized electromagnetic  (2.4)  unit  we  i n i t s direction  two  such  field,  ( i f t h e wave  and  i s not plane  i s a polarized,  terms).  wave  the wavelength  much g r e a t e r  than  an atomic  e~  (2.6)  of any l i g h t  which i s l i k e l y  dimension,  we  t o he used i s  can approximate  =• I  c  Hence  (2.7)  A = Using  perturbation for  i)  this  t h i s we  value  Hamiltonian  the perturbed  The In  §: ( e ofA  i n the expression  and s u b s t i t u t i n g  s t a t e - f u n c t i o n we  electric  )  +e  moment  M  (2.3) f o rt h e  i n the expression  get  i s given  by  •^^lexl'M'^  neglect:  A l l terms  o f degree  >|  i,n  El  0  £ ccot ii)  a l l terms w i t h  a time  dependence  other  than  e  (2.1)  7 and  we  write  <^O1^1YK>  (2.9) as  i s also well  = -  tY >  I r*«>«<%\r  k  known.  Then  + complex The  conjugate  M  ratio  "^r-  defines  a component  tensor  OC^j a n d t h e p e r m i t t i v i t y t e n s o r  to  by  this  6cj  (2.11) To the  find,  x y  parallel  component to y  As  -  y  itions  and 1C >  absorption.  IYo> harmonies,  and and  w  consider to x  parallel  t  +  o f how  an i o n whose  OL  , we  k  , and whose  1a >  odtj  X<XlylY >|^(z^ ^^)<%lx|Yo>^  an example  consider  and  x  i s related  6 -  o c  and  of  , and  Then  us  £ ^  <&j •+* VtcoCtj  5 5  e.g. the  of M  component  o f the p o l a r i z a b i l i t y  .  a n d V, Take  I cc> X+  the Faraday effect  ground excited Assume c  as a x i s  state i s a Kramers state  i s a  about,l e t  are permitted of quantization  1Vo>  doublet,  i s another Kramers  f o r s i m p l i c i t y , that  c a n be w r i t t e n Ly  comes  doublet,  only  f o r dipole  transradiation  the a-axis.  Both  as a s e r i e s o f s p h e r i c a l  spherical  harmonic p r o p o r t i o n a l  to  .  8 e ^  and x-ty  1  e  to  .  components  and  differ  by  <'Volx-eylot> < Yo I x-ty  <Yo| X+tylcO*  Then  'Yo  of  m  numbers  i s a spherical  1.  Let  1 o c ^ s .  ct  vanishes  f o rwhich  It i seasily  unless  ^  a n d a r e  of the expansion  o f Y©  efficients  i n turn  because  arereal  both  a n d CL  there a r e  t h e magnetic seen  real  quantum  <Ylx+tylo<.>  that  <YoU+d/U> = ^ ,  efficients  imaginary  harmonic p r o p o r t i o n a l  0  ,  because  arereal.  the Hamiltonian  the co-  Thec o c o n t a i n s no  terms.  Then  Hence  < Y© l y I ot>  the product  added,  t h e time  90°  phase  the  oC  dependence o f of  (or€  reversing  other  component  the signs  f o r this  cot  (2.12) a r e , whereas t h e  (seeequation by an imaginary  (2.4).  This  element of  o f the Kramers  - X  this  ,  •s o f t h e o r i g i n a l  <V,  we r e p l a c e  component  doublet  Ix t  1  i s obtained  by  component.  c > yu,  by -  has a Faraday  I  i n the expression effect  o ft h e  sign. In  Faraday  w t  o f t h e v*\  component  and hence  opposite  x  imaginary  ) tensor.  <Y, \x-Ly\c>  Hence  i s a pure  i s a s cos  o6 ^  was a s  >  i n expression  s h i f t i s t o be represented  The  (2.13)  K. &> IX I Yo  Thus when t h e e x p o n e n t i a l s  dependence  Thus  /  0  number.  time  < Y „ l y l o t > = I *ir  <Y UU> =  2.13)  general  effect  components,  both  components  are occupied  and t h e t o t a l  i s a s t a t i s t i c a l average o f t h e e f f e c t s  o f t h e two  and i s therefore p r o p o r t i o n a l t o the differencesi n  9 population  (and t o  Now part  there  of the  modes o f  other  i s i n general  6 - tensor  besides). a set of  i s diagonal.  axes  i n which the  T h e n we  have  f o r the  real stable  propagation:  (e  (2.14a)  COCY^=  +  -i*^  (2.14b)  where  things  p  i n general  ^  }  «  g  Hence  (2.15) and  ^  =  therefore  or  The  phase  is proportional  difference  6  introduced  t o the d i f f e r e n c e between  per unit path  t h e two  length  propagation  constants, i . e .  (2.18) where In  S~ /A.  this  ponsible  i s an  average r e f r a c t i v e  representation  ft  gives  responsible  f o r the  Faraday  index.  the  f o r birefringence, while effect.  .  oc  contribution to gives  the  £  res-  contribution  10* 3)  Poinoare's  (a)  Right  Representation  and l e f t  Consider • +• 2  circularly  a right  Suppose  plane.  in  the direction  on  a  circle  polarized  handed  light  vector  Let the light o f —"2 ;  p o l a r i z e d " (RHCP),  said  "left  In the  (3.1)  by an observer  looking  i t i s said  plays  I t i s convenient  polarized of angular  t o take  of light,  momentum.  They a r e  i_p  + by  e  c  ^  vector  t o be  "right  direction,  moves hand i t i s  (LHCP).  the role  components  l+-> =  (Both m u l t i p l i e d be  i n the  polarized"  ,A.  of  t s 0  i f i n the opposite  circularly  vector potential  state-function.  functions  time  the quantum-mechanical d e s c r i p t i o n o f a  magnetic  circularly  or  i n the d i r e c t i o n  a t some  observed  direction,  circularly  hand  of axes  propagated lies  be  light.  i f the t i p of the e l e c t r i c  i n clockwise  to be  system  i s being  , and t h e e l e c t r i c  (y,2)  of Polarized Light  light  analogous  as b a s i c  -  i f t h e time  to a  states the  since these  l-> = ^  wave  are eigen-  t_p  dependence  i s to  stressed).  Since  JS^curlA.  (3.2) we  have  It  i s seen  LHCP  light.  that  carl £=- ^  represents  RHCP l i g h t  =  and  ^ c u r l A_  I—>  represents  11 Let I r>  us  and  normalize Il >  the  with  e x p r e s s i o n f o r EL.  phases  consistent  and  with  define  those  states I  of  *>  I ->  and  (3.4)  I r> = ~^=i  (see  diagram  Note  that  (or ± (h)  Figure  i s plane  lr-l>  ) and  circularly the  ponents  determine  diagram  "Y  the  is  then  such  (CP)  L -  t  the  axis  i)  axis.  that the  The  .  of  (EP)  (See B  i f there i s a ellipse  I r>  of  phase  i s inclined  , then of  G is a  the com-  ellipse.  difference at  II)  t h e two  an  In of  a n g l e -sj-  expression f o r this where  into  Figure  B  the  x  ).  diagram  phases  axes  to  light  a n d A—  The  the  normalized  cos QlC> + sln Se  (or_j_  polarized  +- B \\r&  parallel  y  to  b e A+  ellipse  direction  of  (PP)  components.  A1l>  of  i t i s seen  , the major  to  up  parallel  elliptieally  h a l f - a x e s of the i s made  the  of  polarized  PP  i s  polarized  ellipse  = ^  \l>  I)  lr*l>  Decomposition  Let  (j_-Ljj.)  ellipse  parameter  that  A - cos  (3.5)  0  B = sln  0e  Therefore  IAl - + I B ^  (3.6)  3  = l  ensuring normality. The the  angle  ellipticity <c>  of  instead  the  light  o f by  A  can  also  and  ft  be  or'by  described 0.  by  Views vector  of the motion for  I I > and  of t h e Ir >.  tip  of the  electric  FIGURE  Decomposition of  H  ellipticolly  polarized  light into  polarized  components.  Following  page  11  circularly  Then  .  and  therefore  (c)  The P o i n c a r e  (See  diagram  a point  resent  polarized The  P P l i g h t light on ing  state  of polarization  just  significance  respectively.  and only  light  polarized  by a p o i n t  by a p o i n t  only  phase  and R  rep-  by the point  Note  of a polarized here,  L  described i s represented  hemisphere.  the phase  o f t h e sphere  elliptically  i s represented  the lower  (state-function) i s described  The  represents  i s represented  ance,  The p o l e s  a n d RHCP l i g h t  light  point X  sen©  sphere*  on a sphere.  LHCP  ~~  Figure III)  Each by  cosO-st^ 0  I A | - 161  t<x* vj> = | A U I B \  (3.7)  that  there beam  i s no  way  LHEP RHEP  of represent-  has no  have p h y s i c a l  are represented  physical signific-  on the P o i n c a r e  sphere. (d) i) is  P r o p e r t i e s of the Poincare A  state  represented  coordinates ii)  of polarization  Two  sense)  as a point  a r e ^0,°Y  states which  are represented  sphere.  l * > = CosO\l>  on t h e s p h e r e  whose  +stinQ«  \r>  spherical  polar  . are orthogonal by opposite  ( i nthe  ends  of a  .  plane.  hemisphere,  - phase  differences  differences  ( o r ^)  on t h e e q u a t o r .  on the upper  light  phase  i n the x  E.  quantum-mechanical diameter:  F I G U R E  HI  L  X  R  P O I N C A R E  Following  page  12  S P H E R E  F I G U R E EZ"  Properties  Following  page  12  of  D  the  Poincare'  Sphere  If  l u >= cos@lL>  then  the orthogonal  lv> is  -  t h e same  + sirtQe  state  sthOe  state  | r>  1 ,  ll>-cos©lr>  as  I v> = - s t n O l l > + cos 0 e  •®)ll> +sin(*+0)  = cos lu>  Thus Tf  and I  , i . e .they  the  Poincare  iii)  e.g.  are located  where  state, then  of a state  iv)  ( I ot>  a beam  incident where  ends  by  as a combination states  states  o f two chosen:  and l w > i s  any  circle  i v>  ©e  angle  between  the points  rep-  lw> ; only  light  say) can be r e p r e s e n t e d represented  on t h e a n a l y z e r , i sthe great  Faraday Rotation We  's d i f f e r  o f a diameter o f  o f the basic  + slr\  which passes  of light  2-^  l r >  can write  cos 0 I u t >  and  An a n a l y z e r  state  we  i s t h e great  lut>  resenting  Y  , hut their  on opposite  states i s independent  2. 0 '  e '  sphere.  I w> =  4)  t h e same *Y  i f lu.> a n d l v > a r e t w o o r t h o g o n a l  other  If  have  The r e p r e s e n t a t i o n  orthogonal  !r>  t Y  by a point  by a point w  the amplitude  circle  angle  the effect  (2nd  polarization  ct o n t h e s p h e r e .  on t h e sphere i s  transmitted  between  and B i r e f r i n g e n c e  can discuss  o f a given  ot a n d w  i s cos  ^  ,  .  part)  o f a mixture  o f Faraday  effect  and  birefringence rather Continuing  The  eigenvectors  (/S  (4.1)  from  simply  using  the Poincare  sphere,  t h e end o f s e c t i o n 2 ) :  are given  +T/^ o6*  by  )^  +  + iocy^ =  0  or  (4.2)  §  where D  - - ~  ^  i s a normalizing  (  V  the eigenvectors  l x > and l y > r e p r e s e n t  and  axis respectively, the principal  the  light  are  where y  ^  ?  )  ^  plane  l x > •+• ^ l y ^ polarized along the axes  of the real  € - tensor. If  we  write  |u> = cos 0 U > + s L n 0 e  (4.4)  ly>  then (4.5) Taking  t h e upper  (4.6)  s i g n and  "^T = stnh^  - X  and  hence  tan Q  letting  +1 = cosh^  then  (4.7)  ^  /SVcFT^)  3  Ioc> a n d l v >  ^  denominator  D - = Xiot^^T  (4.3) and  =  = - e  part  15 14.8)  The  c  o  s  ratio  a  =  Q  -TP-  T  great  distant  fringence  can  Figure  i)  Find  now  V.  The  the the  great  lu.>  and  angle  I u.>  ,  I f plane  will  be  Let  this  to  easily  lx>  lu.^  , ly>  r  = ^ = ^  of  ot  ( a n d I V> I  , and  and ) lies  , and  on  is  .  Faraday  -  ly>  effect  by  i s as  and  bire-  referring  to  the  dia-  follows:  which represent  c i r c l e through this  polarized light  point  by  the  stable  lx>  ,  |y>  c i r c l e and  and  the  \  great  circle  , where  i s i n c i d e n t on  a point  on  the  and  Faraday  the  equator  of  crystal, the  this  sphere.  be  birefringence phase  + ^  the  effect  d i f f e r e n c e between the .  separately  and  ^ =  j  independent  visualized  i s e q u a l 2, 0  represented  a  , I l>  combined  l i e on  lx^  iv)  introduce  =  X  birefringence  Draw t h e  The  Ix^  construction  points  iii)  v)  be  l  representing  angle2,$  an  of  « l  t  imaginary,  point  by  effect  gram  ii)  pure  the  lx.>  from  of  = t o  c i r c l e through  The  waves  ^  T  is a  Therefore the  T  Hence  i s known, t h e  i f the  phase  t o t a l phase  two  combined waves  proportional  d i f f e r e n c e from  d i f f e r e n c e cf  will  is  each easily  calculated. vi)  The  emergent  construct I t>  .  a Go  small round  light  i s represented  circle with this  lu.>  c i r c l e an  as  the  angle  lo>  by  pole, cT  .  .  To  find  passing  lc\>  through  F I G U R E  Y  l> X  --------_\ly> /  \  o>  lu>  20  > '  0'  /  /  )  N  v  >  i a > :  /  /  lx>  |  li >  lr >  E f f e c t of combined Faraday E f f e c t and  F o l l o w i n g page 15  Birefringence  16  i vii) the  I f the  intensity passed I o>  through and  analyzer passes  I0>  O'.  ¥/hen vary,  lu.>  will  case  then,  and  form  by  6  lated  the  + cos  somewhat  small  J  time  a path  by  the  the  For  dependence  of  chapter o f /3  deviation  may  the  , to  find circle \ci>  between  analyzer i s  be  Effect,  but  the  o f oc larger  to  trace  the  are  of  circle  from the  from  angle since  simply have  a path  for  the  with  of  of  effect  oc  sufficiently i n the  ex-  ^O+Cos©')  neglected. Paramagnetic  Resonance,  and  of paramagnetic  resonance  on  Spin-Lattice  the  the  bearing  Relaxation The  re-  the  dependence that  oc  the  position  encountered  intensity  as  intensity  time  will  out  time  I I I i t i s shown (such as  of  , we  since  the  of  longer  deviations  will  The  where  latter  , however,,  i s no  a n d ^  the  manner  great  ^j=0  dependence, time  In  equator.  i s the  case  also  birefringence),  a function  Q  where  complex r e l a t i o n s h i p  relationship Faraday  along  i n a manner w h i c h  values  of  identical  vary  . I n  only  (no  .  a function  initial  0  i n an  then be  progressively  as  -  not  I a>  time  )  periments)  5)  angle  r e g a r d l e s s o f <X, .  with  out  /3  case  (6> Wet))}  and  the  lo>  (and/3  circle  passed  i n time,  intensity  a  great  intensity  analyzer w i l l  increasing /2  point  the  with  vary  trace  and  to  Let  f o r the  will  the  b o t h I u.>  l a >  a n a l y z e r , c o n s t r u c t the g r e a t  vary  coincide  1i>  between  and  except  { I  £  .  Then  oc  lo>will  passed  the  r e p r e s e n t e d by  0') .  cos  will  I <x>  and  he  by  light  Faraday  17 >  effect  was  first  mechanical effect  by  the  (V,  t h e o r y was  i t s e l f was IV,  (III,  latter  two  Kastler  by  Gpechowski  observed  by  macroscopic  was  given  ( I , 1951).  by  quantum-  ( I I , 1955),  Daniels  theory  A  for  and the  and  the  Wesemeyer case  S k r o t s k i i , Zyrianov  investigated and  Iziumov  1958).  effect, ing  of  for  the  the  the  of  the  to  Then the  a  the  system  state  will  2yU„ H  .  within  ature  I f moreover  the  .  s  The  Boltzmann  where  we  populations  N  be  and  N,  are  theory for  Faraday as  first  a  i n an  excited  the  spin  i t may  be  the  to  the  was  difference  dipole  moment  system said  levels will  kT  the  It  a. m a g n e t i c  of  in fact  a  state.  i s  to  in  a  levels  spin  temper-  equilibrium lattice  t h e n be  i s used  energy  thermodynamic  have  the  L  fieldH  magnetic  doublet w i t h  T  state  Now  T  the  =  arises  state.  have  s  understand-  ground  external  split  this  effect  ground  is proportional  having  the  of  chapter.  i t i s i n thermodynamic  =» N, e  Z  the  levels of  d i s t r i b u t i o n (which  (5,i)  the  ions  If  itself,  crystal lattice,  ature.  of  the  later  ion having  situated  equilibrium T  in a  for  two  of  necessary  s h o w n how  rotation  S =• x  spin  by  discussed  likewise  of  resume  points  i t was  Faraday  ground  separated  short  those  2)  and  consider  due  a  a paramagnetic  populations  us  only  section  doublet  that  is  experiments  case  Kramers seen  following  stressing  In  let  A  by  given  first  1958).  The  in  predicted  to  temper-  given  define  with  by T  s  the ).  *  number  of  ions  in  the  upper  and  lower  ,  18 state  respectively. The  tion  i n t e n s i t y o f paramagnetic  o f a quantum  of electromagnetic  depends o n t h e d i f f e r e n c e tween w h i c h population  take  number  T$  consequently  to  i n the lower  inducing  looses  between  system  The  Wo-V*  to  system Tj  rate  -  Tt.  .  spin  o f c h a n g e o f VI  and also  dependent  2 t  When m i c r o w a v e term  Stoa P  o f f , Yi  increases  value  and thus  i t will  temperature,  due  Y^© ( i . e . "cools  a t which point  i s supposed  equili-  Since the  always be a provided  to be  on t h e c o u p l i n g  The c o u p l i n g  o f change  3 )  t h e microwave  the spin-  i n equilibrium.  and l a t t i c e .  the rate  occur and  i srestored).  t o Y\  when  When  to the l a t t i c e  i sproportional  <{<r> ( 5  concerned  i s expressed  , called the s p i n - l a t t i c e relaxation  for  difference i n  i s reduced  i s switched  and l a t t i c e  of the instantaneous i s itself  than  o f the latter,  spin-system  Faraday r o t a t i o n  y\  to i t s equilibrium  energy  down" t o t h e t e m p e r a t u r e  measure  Let this  the levels  the transitions  spin-system  brium  "fceuaJl^H  o f t h e two s t a t e s b e -  place.  state.  becomes l a r g e r  spin-lattice relaxation  the  o f energy  = N, -  microwave t r a n s i t i o n s between  field  radiation  due t o a b s o r p -  be  y\ excess  resonance  i n population  the transitions  (5.2) the  J  proportional  between  spin-  by a parameter  time.  We  have  then  YI  of  Y>O - v\ =  radiation  "  T  T  o f appropriate  h a s t o be s u b t r a c t e d  frequency  on the r i g h t  i s present,  hand  side,  a  where  19 ' P  i s the t r a n s i t i o n p r o b a b i l i t y per unit  2- r\ by  r e s u l t s from  St,  .  T h u s we  the fact  -  equilibrium  we  (5,5)  each  and t h e f a c t o r  t r a n s i t i o n changes  r\  have  (5.4)  At  that  time  ~  have  ^  =  a  0  and hence  I +a,PT, P  Now  assuming Po  where spin  a resonance i s a  relaxation  line  constant. time  7^  of width  & f  we  can write  I t i s customars?- t o d e f i n e  b y "J^ = T  a  a n d h e n c e we  a  may  P  a  j|-  spinalso  write P Po "Hi Then a t e q u i l i b r i u m 3  At  low temperatures  quite T,  »  and  large ,  liquid is  =  ft  (5.6)  i-t-ap T;T; 0  and f o r c o n c e n t r a t e d s a l t s  (due t o s p i n - s p i n  interactions)  e.g. i n the s a l t s discussed  Helium  temperatures  of the order o f i n this  case  10 ' s e c  equation  T  .  tends  and hence  i n this  of the order  a  cff  of  thesis 10  sec  At high temperatures  ( 5 . 6 ) may b e w r i t t e n  t o be  we  have  we  have a t  whereas  T  4  TV,"** T ,  as  n0  (5.7)  ua.PoT,* For  the s a l t s  investigated  (5.7) i s a p p l i c a b l e  at liquid  nitrogen  temperatures. Since  P  0  i s proportional  t o the square  o f t h e amplitude  of  t h emagnetic  field  cerned w i t h magnetic for  sufficiently  able  change  hence  strong v\  transitions  microwave  from  0  t h e microxvave , we w i l l  proportional  only),  radiation  i t s equilibrium  (we a r e c o n -  i t i s seen  that  there i san appreci'Ylo  value  i n the Faraday  radiation  then observe  to  rotation  i s suddenly  a change  ,and due t o t h e  cut off at a  i n Faraday  as given i n equation  time  rotation  (5.3), which  may  be w r i t t e n  ot (n»-y\Ck))  nop - TO(t)  (5.8)  =  I f we w r i t e  K  i sa constant o f proportionality,  K { V  =  (5.10)  ^ (.i)  Hence b y o b s e r v i n g one may f i n d Note temperature lattice valid  rotation  §Ct) a K v O O  where  off,  T>  ~  f o r the instantaneous Faraday  (5.9)  ing  radiation  radiation.  If  also  dipole  a c o r r e s p o n d i n g change  microwave  t a  of  o f t h e microwave  that T  u  T  (  0  -  after  =  remains  liquid  implicitly  relaxation  assumed  constant throughout,  i f the coupling helium bath  T  e  t h e microwave power  r e p r e s e n t s an i n f i n i t e heat  only  between  i s such  find  K{y, ^^C0)} e '  , the spin-lattice  we h a v e  we  that  sink.  that  time.  the lattice  l i e . that the  This  the l a t t i c e therate  has been c u t  assumption i s and the surround-  o f cooling  of the  21  l a t t i c e by the b a t h exceeds the r a t e of h e a t i n g by the r e l a x i n g spin-system.  This  assumption  was j u s t i f i e d w i t h i n the o r d e r s o f  magnitude observable i n t h e experiments 6)  described.  Spin-Lattice Coupling  T h e mechanism by which energy i s t r a n s f e r r e d from the spin-system t o the l a t t i c e i n an i o n i c c r y s t a l and v i c e - v e r s a , i s agreed t o be the f o l l o w i n g : Lattice  v i b r a t i o n s (phonons) modulate the c r y s t a l l i n e  e l e c t r i c f i e l d and the spin-system f e e l s the e f f e c t o f t h i s modu l a t i o n v i a t h e s p i n - o r b i t i n t e r a c t i o n , s i n c e the s p l i t t i n g o f the o r b i t a l l e v e l s i s i n g e n e r a l determined electric field periments  by the c r y s t a l l i n e  (at l e a s t i n the c r y s t a l s i n v e s t i g a t e d i n the ex-  described i n t h i s thesis}.  Van  Vleck  ( V I , 1940)  made some d e t a i l e d c a l c u l a t i o n s o f  the s p i n - l a t t i c e r e l a x a t i o n time f o r v ^ v the complex Cs Tt  and  OGH^O  Ti • <o  , and  ( S O O ^ / 12. H^O  and T t  H^O  ions i n  as found  inKCrCSO^I^O  on t h e b a s i s o f t h i s model.  d i s t i n g u i s h e d between two p r o c e s s e s :  He  A " d i r e c t p r o c e s s " i n which  spin-system absorbs o r emits a quantum o f l a t t i c e energy and a p r o c e s s " i n which a quantum o f l a t t i c e energy  "Ramann  ically  s c a t t e r e d , i . e . the spin-system absorbs a quantum of one  energy and emits one w i t h a d i f f e r e n t energy While  is Inelast-  the " d i r e c t p r o c e s s " i s the important  simultaneously. one at v e r y low tem-  p e r a t u r e s ( t h e r e l a x a t i o n time b e i n g p r o p o r t i o n a l t o  T  in  g e n e r a l ) , the " R a m a n n p r o c e s s " i s dominant at h i g h e r  temperatures  (where the r e l a x a t i o n time i s p r o p o r t i o n a l t o a much h i g h e r power of,  ~Y  Van  Vleck  » e«S»  I  f o the case o f r  C T  and T t  a l s o obtained a dependence o f t h e r e l a x a t i o n time on  )  22 the  external  tions  magnetic  are o f  particular  great  salt  „ His  laxation  on  time  and  ions  the  field  were  not  with  increase  at the  on  i)  ii)  a  can  higher  lattice  larger  ever,  this  relaxation netic  priori  dependence  time  the  agreed ion.  of the  re-  in  decreases very  time  This  i s  the  directly  For  the  decrease  of  gen-  rapidly  relaxation  "direct  process"  very rapidly  as  because: of  the  lattice  i s greater,  and  lattice  energy  vibrations modes  of  numerous.  transition  the  i s s t i l l  probability  since  effects,  expected  an  heat  compared  This  f o r higher  the wavelength  specific  i s small  increases.  field. should  t o t h e s e two  effect  the dependence  time  field,  increases  field  Vleck's result  temperature  magnetic  a r e more  Contrary field  extent on  y  discusses  modes i s g r e a t e r ,  netic  Van  the r e l a x a t i o n  also  interact  a  and  a large  calcula-  e x p e r i m e n t a l l y confirmed, though  relaxation  energy  The  and  i s increased.  With  which  Vleck  the  field  depend t o  detailed  i n temperature.  the e x t e r n a l  least,  used,  the  _  i t i s known t h a t  time  However  w  eral  Van  H.  c o m p l e x i t y and  .  results  field  •  to  shorter.  increase  of t h e  i ) and  i s  energy  o f the  spin  i i ) , and  system.  i s not  born  How-  the  decrease r a p i d l y  prediction  mag-  as  out  the  mag-  by  experiment. There of its  which  will  be  validity.  Vleck  (VII,  are many  assumptions  discussed  Some  1941).  of  i n Van  i n Chapter  these  are  Vieek's theory,  IV, which  discussed  give  i n two  some  doubt  papers  by  for Van  23 The no  fact  detailed  it  ions  theoretical  a firm  ered  Chapter  II:  1)  Description  sisted  relaxation  h a v e b e e n made,  inherent  topics  i n the problem  relaxation  f o r research  e.g. paramagnetic  Experimental  time  of  emphasises of obtaining  has not been  have  been  discov-  resonance.  Arrangement  of the  apparatus  essentially  state  paramagnetic  easier  i n t h e meantime,  The  of the s p i n - l a t t i c e  understanding o f the relaxation-processes, that  since  i n 1940 a n d 1941  1  difficulties  emphasises  fashionable,  Van V l e c k s papers  i n the solid  quantititive  also  since  calculations  paramagnetic the  that  used  Apparatus (see plates  Figure  V I t o IX)  o f a c o n v e n t i o n a l paramagnetic  con-  resonance  spectrometer, which  had been m o d i f i e d t o a l l o w m a g n e t o - o p t i c a l  studies.  constituents  The m a j o r  o f t h e s p e c t r o m e t e r were t h e  following: a)  A microwave  microwave power or  continuously,  investigation.  system  capable  i n the  X  t o a resonant  cavity  reflex-klystron  microwave  power.  the  s u p p l y v o l t a g e s were  stabilized  DC  I t s filament  power pack  former  from  passed  v i awaveguide  attenuator  ( 3 CM  -band  (See b l o c k diagram  A  other  of delivering  t h e 1 1 5 v AC m a i n s .  to a  through  directional  t o 140mw o f  wavelength), containing  either  pulsed  t h e sample  under  Figure X). provided the source o f the  was  fed via  up  supplied  by a 6v  obtained from a Sola  electronically  constant voltage  The o u t p u t  a variable  an  car. b a t t e r y ,  from  trans-  the k l y s t r o n  susceptance and an  coupler-; w h e r e  some  o f t h e power  was  FIGURE  VI  A P P A R A T U S W I T H DEWARS  REMOVED  FIGURE V I I  APPARATUS I N OPERATING  To  f o l l o w page  23  CONDITION  FIGURE  VIII  OPTICAL SYSTEM, SOURCE SIDE  jrlGURE EC  OPTICAL SYSTEbi, uETECTuR  To follow page 23  SIDE  FIGURE STABILIZED POWER SUPPLY  WAVE METER  X  DUMMY LOAD  KLYSTRON PULSE MODULATOR OSCILLOSCOPE  MICROAMMETER ATTENUATOR  Willi  |  I ©  VARIABLE CRYSTAL TUNABLE  M  GALVANOMETER  SUSCEPTANCE DIODE END CAVITY  SAMPLE  MICROWAVE  SYSTEM  24'  deviated meter  to  and  a branch  containing  a  PRD  -B  crystal  rectifier  directional  coupler  was  another  attenuator,  the  m i c r o w a v e power was  coupled  from waveguide  The  coaxial cable  resonant  cavity operating  cavity  was  achieve  a  cavity to  with of  liquid  holes  between which ment.  a  helium  depending  light  by  of  2mm  through input  the  fed  ing  a  noise  the  silicon  low  detector  this  monitoring  to  of The  an  to  oscilloscope. output  was  Coupling could  the  mounted  at  the  axis to  the  5000  passage  top  during  the  an  input  face,  The  experi-  coupling,  section  galvanometer  In  the  choke-plunger  detector. a  of  to  provided  allow  a  bent  of  c a v i t y was  on  into  be  order  cable. type  Q*  loaded  a waveguide  diode  fed  transmission  the  this  coaxial  to  loops,  to  to  faces  identical  silicon be  was  following  which  The  a vertical  output  could  modulation (see  the  of  reflector  normally  the  t h e s i s the  multivibrator  amplitude  loop,  Following  contain-  output or  from  via  a  i n v e s t i g a t i o n s des-  used only  for tuning  and  purposes.  Pulse  variable  coupling  c o u p l i n g was  coaxial cable  in  101  coupling.  around  and  cribed  of  open  sample which  rotated  video-amplifier  mode.  TE  sample u s e d .  output  output  tunable  i n the  t e m p e r a t u r e s was  the  and  the  diameter i n i t s broad  and  c o u l d be The  on  ammeter.  microwaves to  small  s u i t a b l e degree  at  9000  provided  micro  p r e c i s i o n wave-  a  fed the  with  529  diagram  the  i n length  the  k l y s t r o n was  Figure  XI)  i n  at  steps  between  between a rate  0 of  and  provided  capable  klystron essentially  continuously repetitive  of  2 msec 45  to  of  by  delivering  rectangular and  124  85v.  approximately  to  pulses  msec  The 1  a  and  pulses  pulse  every  i n were 2  B  220 V  P U L S E O U T P U T  TRIGGER O U T P U T A /  ' M A N U A L C O N T R O L  1  SINGLE  A.  M  l w  S _i  4  7  K  IW —  P U L S E S  FIGURE PULSE  MODULATOR  XI CIRCUIT  25 seconds.  However,  the repetition  factor  o f about  switch  was p r o v i d e d ,  Synchronizing cope .were  also  the leading  of  these  and t r a i l i n g  trigger  pulses  triggering,  and t h e o s c i l l o s c o p e .  6)  A water-cooled  a magnetic leaving  field  rheostats centre in  a  from  a llOv  inner  filled  with  liquid  helium  the  resonant  volume side the  helium  allowing  over  amplitude control  small f o r  the trigger  capable  of  out-  producing  the pole-pieces  air..  o f water  An a x i a l  was p r o v i d e d  hole  cooled  at the  f o r the passage  of  light  field.  a l l m e a s u r e m e n t s t o b e made a t t e m p e r a -  dewar f l a s k  1 . 4 ° K. topped  This  v i a a syphon  This  thermometer,  dewar, w h i c h  connections  the r i m o f the brass  as a  t o pressure  v a c u u m pump.  could  be f i l l e d  manner,  as well  lubricated with  consisted o f  cup which  could  i n a conventional  speed r e t a r y sleeve  cryostat  by a brass  c a v i t y of the spectrometer  t o p by a rubber  between  v i a a network  DC g e n e r a t o r .  pieces  a i r , and a high  slipped  i t was r a t h e r  oerstedt with  o f the magnetic  liquid  The  inches.  down t o a p p r o x i m a t e l y  "soft"  pulses.  of the amplitude  electromagnet,  o f up t o 6000  of the pole  A cryostat  tures  but since  core  and a manual  differentiation  edges o f t h e p u l s e s .  magnet was s u p p l i e d  the direction  c)  iron  a g a p o f 2.15 The  the m u l t i v i b r a t o r by  a n a m p l i f i e r was u s e d  put  length),  by a  t h e sweep o f t h e o s c i l l o s -  was i n d e p e n d e n t  pulses,  be i n c r e a s e d  the application of single  triggering from  could  on t h e p u l s e  allowing  obtained  the modulating  stable  (depending  pulses,for  of  for  20  rate  with  contained constant  gauges,  I t was  glycerine,  be  out-  sealed at which  cup a n d t h e e n d o f t h e dewar.  To  allow  holding  free  of light,  t h e c a v i t y , was  dewar was as  passage  a hard  precoolant.  allow  passage  dewar  cannot  vered  tail  troduce  left  vacuum  of light. be  end.  noise  dewar  e n d was  Yet bubbles  i n the signal  light  nearly path  the  inner  the  tail  dewar,  mension were and  To e l i m i n a t e  dewar.  o f about  of light.  4 b y 4 mm  dewar, u n t i l  tight  f i t o f the copper obtained  foil  a tunnel  was  b e t w e e n t h e two  does  most measurements were  inner  dewar  could  unsil-  i n  -point  this  i n the  wrapped  into  around  positions  cut i n the f o i l  recto  o f t h e same i n n e r d i -  No.  33  electrical  and the outer  the thickness  dewar w a l l s  strength,  r a d i a t i o n o f heat  were  was  path i n -  and  described  to the inner  of bubbles.  below t h e X  By means  i n the  put  walls  tape  walls  was  of the  such  of the  that  of  which remained  Bubbling  a  outer  into position.  i n the d i r e c t i o n o f propagation  and hence f r e e  not occur  foil  the l a t t e r  t h i s manner  coolant  brand  a t t h e windows  when  outer  such bubbles  Square windows  cut out of Scotch  inner  created  reduce  to  i n the  light  At t h e a p p r o p r i a t e  i n s e r t e d between the copper  dewar was  the actual  of the kind  inner  unsilvered  particularly  a c y l i n d e r o f c o p p e r - f o i l was  openings  passage  left  the  liquid a i r  investigated o f a type  a n d a t t h e same t i m e  of the inner  tangular "allow  Impossible.  to hold  of the coolant  passing  w h i c h w o u l d make m e a s u r e m e n t s thesis  flask  avoided,  dewar,  Surrounding  likewise  Bubbling  completely  end o f the  unsilvered.  outer  Its tail  the t a i l  In  light free  i n the helium  (2.19°K ) , t h e r e g i o n  of bath  i n which  made.  of the high-speed be r e d u c e d  over  r o t a r y pump the l i q u i d  pressure  helium  bath  i n the to  reach  27 as low as 1.38°K.  temperatures be  measured by a mercury-and  which  indicated  dewar. an  The c o n s t a n t  indication  precooling d)  polarized  prior  diagram  through  source.  ripple  to  lens  and t o s t a b i l i z e  source  emerging used  giving  from  complete  o f i n c i d e n c e up t o 3 0 ° .  light  passed line  through  plane-  o f t h e mag-  analyzed.  (See  as  a DC g e n e r a t o r  giving  ap-  designed  the collimator  1 converging the axial  lens hole  served  converg-  p a i n s were  truly  taken  parallel,  contained a  polarization  f o rlight  Glanhaving  an  Following the collimator the  ) and then  the polarizer  genera-  i n the arc (see dia-  No p a r t i c u l a r  filter  through  o f 1 b y 1 cm c r o s s - s e c t i o n  through  t o reduce  was f o l l o w e d b y a  a coloured glass  ( X=5461 A  Thompson p r i s m  through  during  served  i n t h e experiment  angle  and  to obtain  H 100-4A  the current flow  s e r v i n g as a c o l l i m a t o r .  Thompson p r i s m  passage  dewar  and subsequently  network  The l i g h t  the polarizer  mercury  only  the axis  mercury a r c type  190v v i a a f i l t e r  make t h e l i g h t  since  i n the inner  monochromatic  along  T h i s was s u p p l i e d f r o m  gram F i g u r e X I I I ) . ing  t h e sample  o i l B),  helium,  of producing  c o u l d be p a s s e d  General-Electric  proximately tor  i n the inner  could  Figure XII), A  light  capable  which  o f the helium  of liquid  4.2° K  (Apiezon  t h e r m o m e t e r was u s e d  t o the t r a n s f e r  system  light  field  pressure  volume  below  an oil-manometer  o f the temperature  An o p t i c a l  netic  the vapour  Temperatures  a system t o focus  i n t h e magnet  selecting  the polarizer a n d 2 cm l e n g t h .  consisting the plane  into  the green a  GlanAfter  of 1 diverging  of the polarizer  t h e sample  i n the reso-  2 H  FIGURE  CRYSTAL Hg-ARC  POLARIZER ANALYZER  FILTER  PHOTOCAVITY MAGNET  OPTICAL  CORE  SYSTEM  MULTIPLIER  FIGURE  H E  5 Hy  1500  A  h V W H 4 0 UF  8 0 UF  AMMETER CURRENT REGULATOR  GENERATOR  Hg-ARC  5819  'SHUNT GALVANI O M E T E R 10 K  rWVH .01 U F  —  OSCILLOSCOPE BATTERIES  CIRCUITS  Following  • 90 V  EACH  ASSOCIATED  page  27  UNIT PULSER  WITH  OPTICAL  SYSTEM  28 nant the  cavity. axial  From  hole  i n the  telescopic  system  analyzer  again a  mounted the  -  i n a  lier use  of  a  dynode  dynodes  and  the  through  capacitor be  and  eloped  by  across  the  l o a d was  with  of the  of  type  Figure  multiplier  through  through  were  the  shunted  by  o f 381  was  box  1049  oscilloscope Ilford  BP-3  of  the  without  0  film.  The  was  photo-  plate a  and  supmica-  which  could  99 JCL  .  the  beam  V  devDC-  4  oscilloscope  1428.  connected on  light-detecting  The  signal  the  output  markers  was  .OlyuF  to  camera model  the  following  cable to  double  t o p r o v i d e time  diagram  of  grounded,  fed v i a coaxial model  photomultip-  resistance  a resistance  resistor  of the The  onto  tube.  between  plate.  polarizer  finally  of the  applied  the  the  and  directly  between each and  -,  to  a  The  from  a  to the the  i n -  second  circuit  see  XIII).  .01 yuF  capacitor  p r o v i d e d the  milliseconds,  scale  response  1217-A u n i t - p u l s e r  oscilloscope  then  photomultiplier  resistor  Cossor  a Cossor  (For a f u l l  The  a  loaded with  General-Radio  90v  a galvanometer  input  passed  identical  B-Batteries  180v  load  shunted  light  t h e magnet,  prism  of the  dynode  load  c a m e r a was  diagram  final  1,  the  equipped  trace.  no.  of  (5819)  radio  of  amplifier  put  by  a 1 0 KSX  shorted, or  plate-side  RCA  network.  and  half  with vernier  linearity  cathode  plied  an  emerging  c o n v e r g i n g l e n s e s f o l l o w e d by  ring  supplied  divider  second  the  Glan-Thompson  of  assure  i t was  sample  o f two  divided  photocathode To  the  thus  across the p l a t e - l o a d  output w i t h  serving  t o reduce  a  time-constant the  of of  the  photo-,i:  .1  noise bandwidth  of  the  29 '  -system, the  effectively  photomultiplier  ducing thus  a l l signal  allowing  constants scope  2)  triggered  H.  from  Description  pared,  the  account  of  on  of  of  some  cavity,  edges  vacuum  a  suitable  p l u n g e r , and  the  actual  measurements were  was  s w i t c h e d on  to  current  of the  sible  fatigue  before they the  fall  do  any not  expected  effects  during  50  at  ^  made.  to produce  f o r the  once  closed  i t could by  be  a rubber  of  been  to  pre-  has  inserted  to  come t o  been an  maxiwas  are  slow,  sensitivity  experiment. to  the  the  pos-  equilibrium  effects  from  photo-  allow  established  from  Then  the  done  attached  removed  photomultiplier to  i n the  i n  before  producing a  Since these changes  hours  experiment  T h i s was  dynodes  stop.  10  the  (usually  h e l i u m t r a n s f e r was  through which  -  features  had been  equivalent  used d u r i n g  sample  to  the a c t u a l  equilibrium  of i n t e n s i t y  choke-plunger w i t h the  a p p a r a t u s was  sudden  sample  to begin,  amp.).  measurements were  syphon  )',  i t s proper orientation  light  final  level  opening  of  the  tend  cn,  e t c . has  Some 8  the photocathode  photomultiplier,  used  planned  amount  order of  approximate  the  an  onto  microwave  under  system,  oscillo-  1958),  approximately established.  allowed  of the  and  time-  sweep o f t h e  of the mechanical  cavity  intensity  second  decay-processes with internal  the  mum-light  cycles per  repro-  Experiment  a choke  and  1000  as mentioned  (VIII,  en  to  The  to each experiment  mounted  0  leading  cavity  Wesemeyer  Prior  from  from  source, while f a i t h f u l l y  reproduction  to the  detailed  by  light  components  s p e c t r o m e t e r , e.g.  given  the  high-frequency shot noise  1 millisecond.  applied A  and  correct  ^  was  pulses  the  eliminating  the  the  inner  at  the  The  apparatus  cavity, top  of  of  and  the  the  dewar  and  30 the to  inner 3  cm  walls  dewar w a l l s w e r e mercury-pressure  f o r the purpose  during precooling. was  transferred,  f o r several  o f a i r were  admitted  o f conducting heat  This  thus  evacuated  a i r was  creating  frozen  hours.  About  t o the inner  away f r o m  dewar  the inside  o u t when l i q u i d  the necessary  1  helium  h a r d vacuum  when  needed, From  a cylinder  dewar t h r o u g h pressure pheric  a charcoal  reached  At this  through which  s e r t e d was r e m o v e d was  quickly  c e s s was never The  choke  position, of  Since  the rubber  felt  p l u n g e r was  the helium  and the o u t e r dewar w e r e  dewar filled  c o n t e n t s were  phere. the  an i n s i d e This  crystals  temperatures  continually  pressure  slightly  elaborate precooling used  inside  had been  took  tended  pro-  predetermined  cup on t o p  a i r and k e p t  that was  vacuum  t o crack when  under  experiments.  the  admitted into  procedure  attached  place, i t  a i rtemperature.  above  be i n -  during this  a s the. b r a s s  d i d n o t s t a n d up u n d e r  and a l s o  t h e sample  into  atmos-  could  with  t h e system  the  closing the  W i t h i n 1 t o 1§ hours  cooled to l i q u i d  h e l i u m g a s was  above  sample  up w i t h l i q u i d  dewar  maintain  with  oriented  as w e l l  the end o f t h e experiment.  time  stop  the inner  a i r , until  d u r i n g any o f t h e numerous immediately  into  mercury  and i f i t a c t u a l l y  until  this  cm  2 to 3  of a i r into  passed  i n liquid  and the choke-plunger  diffusion  itself  the inner  immersed  point  k e p t to a minimum,  made  then  the choke-plunger  inserted.  overpressure,  trap  approximately  pressure.  channel  h e l i u m g a s was  full  the  inner  During  t h e system  to  o f t h e atmosnecessary at  because  elevated  cooled too  suddenly.  1,  31 Once air  temperature,  line,  leading  ferred it  the inner  from  through  were to  taken  K  t o a recovery  the container  evaporated .  As soon  helium  the return  opened  to a i r .  and  ( i ncase  time).  Then  opened ture tem  was  the  helium  below ing  a second  4.2° K  t h e system the A  valves  cooled filled  t h e dewar  was  done  pump.  could  f o ra  at a  valves  r e q u i r e d when  although  changes  d i s s i p a t e d i n the c a v i t y took  later  and the sys-  When p u m p i n g  temperature  i t was  place.  by  vary-  two  not found  within  some m a n i p u l a t i o n  i n t h e amount  away  evaporated  by adjusting  constant  line  tempera-  be a d j u s t e d  Usually  the temperature  t h e measurements  t h e syphon,  or i t s free end  closed again  achieved  i n t h e pumping l i n e .  d i f f i c u l t to maintain  liquid  t o t h e a i r and t h e r e t u r n  The t e m p e r a t u r e was  .5  t h e system  from  .4 t o .5 l i t r e s o f h e l i u m  This  helium  down t o  came t o e q u i l i b r i u m a t t h e d e s i r e d  speed.  pressing  with  contemplated  speed r o t a r y  throughout  er  from  line  trans-  Of t h e s e  was removed  For the experiments  about  -point.  t h e pumping  was  container  the return  was  dewar b y  and simultaneously  t r a n s f e r was  to the high  vapour,  dewar  dewar was  closed  liqtiid  return  helium  a transfer.  the inner  e i t h e r removed  to  to 3 l i t r e s o f l i q u i d  during  was c l o s e d  t h e s y s t e m was  opened  parallel too  line  and l i q u i d  2.5  cooled  t o the helium  into the inner  as the inner  simultaneously.  below  before  while  The s t o r a g e  t h e s y p h o n was  closed  Usually  had been  opened  system,  container  t h e syphon. from  contents  t h e system was  a storage  .8 l i t r e  4,2°  dewar  ft  .02° K  of the  o f microwave  pow-  32, While  the  switched  on  with  a r c and  the  as  approximately ignition up,  the  ately was  the  cryostat  follows: ' 190V  the arc  A DC  Amp.  opened  potential drew 4  the  and  resistance  a  to  Amp  meter  to  to  of  shunting  the  the  regulator  was  approximately  i f necessary  time  to  The  adjusted until  type  of  time  arc.  the  across i t .  then  from  As  This  6  to  tube. and  arc  heated  approxim-  to maintain the proper  tube  variable  the  voltage  7V.  This i s  Subsequently  the  was  Upon  current regulator  was  checked  the  was  series  i t reached  circuit  operating region for this  arc  circuit.  ignite  When  mercury  placed i n  current.  connected  best  justed  applied  5 Amp  across  was  m e t e r was  required  drop  voltage  down, t h e  decreased.  switch  10V  i n the the  10  pumped  were  current gradually  1.1  was  resistance  the  the  read-  operating point  of  the  tube. As alignment maximum  at  of  the  had  was  room  as  to  not  and  to  direction  of  the  sible place.  of be  hence  done  parallel  cavity  to  as the  done  t o have  photomultiplier  the  to  cavity since  at l i q u i d on  of  assure  the  the  a  final  passage sample.  position  visual of  a  This of  temperature  i n n e r dewar  the as  shifted  the  in a direction perpendicular  light. system  i t s optic  axis  the as  placed  final  nearly  of propagation of  following was  and  helium  optical  direction i n the  made  slightly  of the  stabilized,  stage  A l s o pumping  of propagation  so  T h i s was The  the  this same  had  was  through  at  the  alignment  crystal,  system  light  temperature.  After  temperature  exactly  latter the  the  optical  intensity  alignment cavity  soon  alignment as  light,  postook  manner: in position  behind  the  33 analyzer by  using  was  used  that  and a  i t s output  fast,  as  a  part of  d i s p l a y e d on  untriggered, repetitive  reference the  optic  line.  and  except  a  incandescent  atory,  which provided  occupied sure  by  the  t h a t no  measurable  t i l  photocurrent  of  multiplier position  line  the  In  falling at  ever  way  the  the  intensity  and  flat  discrimination  the  mm  indirect'light  to  the  region  precautions  the  from  i t was  unwanted  line  made.  &  difficult.  Having  set  the  crystal  carefully  ing  the  sources  a n a l y z e r was  rotated  as  by  evidenced  the  r e p r e s e n t i n g the  top  edge  the  of  i n the  unposi-  photo-  oscilloscope to  the  reference  intensity  detected  of  light  rapidly  and  galvanometer  This procedure  was  followed  intensity  or  Particularly setting  near  curve  relationship,  the  are  not  analyzer  rotated  of propagation  in a of  for  comparison intensity  direct  identically  horizontal  light,  until  i n -  broad  extrem-  intensity  plane, no  of  visual  zero  f o r minimum  when-  minima  i s extremely  making  to  the  the  analjrzer cos  made  observation of  o f minima which  direction  switched o f f , labor-  small changes  by  the  the  light  above t h e  ely  was  to  corresponding  the  beam  magnet  edge  of  second  around  made o n  f o r minimum  versus to  far  line  draped  Adjustments were  very  were  the  a  of  a minimum  photocurrent.  levels  from  room were  oscilloscope,  a c c u r a t e l y as  the  due  at  The  as  corner  these  Now  sweep.  c l o t h was  in a  of  oscilloscope  p h o t o m u l t i p l i e r c o u l d be  adjustments  tensity  was  .5  about  this  as  measuring  By  amount  the  bottom  onto  least  on  output.  photocurrent line.  bulb  photomultiplier.  the  i n the  subdued  apparatus.  the  tion  very  black  extending  a l l lights  reached the  A  system  photomultiplier single  the  contain-  further  improvement or  the  stal as  of  the  crystal  had  c o u l d be  t o be  such  counterclockwise  wise, its  by  sides covered also  With tical  system  ments were was  the  with  and  the  begun,  no  with  the  klystron  with  small  The  Faraday  way  from  0  Amp,  each  field  tensity  of  to  of  the  ship,  back  Amp,  various  stage  to  the  of  the  cry-  clockwise  thus  light,  linearity  making  on  as  well  (other-  presenting  an  excellent  0  that  the  as  resonance  a  amp. 15  A  cooling-water  i n the or  analyzer  a  the  mode cavity  attenuation. intensity  recorded,  amp.  typical 10  of  was  i.e.  operating at  frequency  35  field  cavity,  f o r minimum  Amp.,  measure-  function of  f o r maximum  at  op-  klystron.  t h e m a g n e t was  the  and curve  then  with  working  down  may  be  Amp.,5 Amp.,  setting  with  0  the  taken Amp.  At  f o r minimum i n -  averaged.  calibration  usually the  stabilized,  sure  ally  analyzer  between  of  at  starting  and a  cavity  power p r e s e n t  Amp, of  angles  a n a l y z e r was  the  25  obtained  this  o f f a x i s and  the  set  the  to  5 readings  were At  tem  35  adjustment  analyzer  increase i n intensity  rotation  not  attenuators  curve  the  properly aligned, actual  o f f the  currents, arrays  hysteresis  either  rotation  r h e o s t a t s , and  either  position  of  first  current f l o w i n g through  various  90°  crystal  microwave  output  both  i n an  plasticine  after  the  with  with  final  slightest  crystal  of  obtainable'.} .  measured  and  the  temperature  first  The  resulted  s u p p l i e d t o magnet, At  adjustment  made.  that  t u r n i n g the  minimum was  no  minimum b y  of  incoming made,.to  the  response  polarization check  on  p h o t o m u l t i p l i e r and  of  the  and  sys-  setting  t h e co ^ r e l a t i o n oscilloscope  35 response, ic ity was  and a l s o  o f the l i g h t  done  as  above set  intensity,  t o the range  exposure  rotated sion) was  At  w a s made  each  shunted Next  tuned  the r e f l e c t o r  to  tune  pulses  was made  half  frame  t h e range  an hour  s t i l l  supplied  of the oscilloscope o b t a i n e d i n a.  The a n a l y z e r  (depending  o n t h e same  a one was  then  on t h e o c c a frame;  this  covered, whereupon t h e set of  exposures  0° t o 90° counterclockwise. on t h e  recorded.  switched  on a n d a l l o w e d t o warm  b e f o r e making  and the whole  further  microwave  measure-  system  was  then  frequency of the cavity,  itself  by l o w e r i n g t h e  by a p p r o x i m a t e l y 40V. as f a r as p o s s i b l e  being able  s e tf o r  slightly  t h e p h o t o c u r r e n t , as m e a s u r e d was  positioned  the analyzer  and another  from  to the resonance  the klystron  was  of the traces,  90° had been  by d e t u n i n g t h e k l y s t r o n  of  while  ellipt-  calibration  positioned  the gain  degrees  galvanometer  The k l y s t r o n  carefully lowed  a full  also  trace  brightness  t h e k l y s t r o n was  f o r at least  ments.  The  to the i n t e n s i t i e s  o r 15  exposure  covering  exposure  r a y tube,  on t h e camera.  moved t o t h e n e x t  suitably  up  suited  a n d medium  and another  taken,  f o r the  on the o s c i l l o s c o p e  trace. With  c l o c k w i s e b y 10  was  limit  the crystal.  and t h e s i g n a l  repeated u n t i l  film  trace  best  experiment  second  from  of the cathode  the reference  given  was  reference  the bottom  minimum  emerging  an upper  follows:  The near  to establish  t o p u l s e i t back  by the m u l t i v i b r a t o r .  I t was  always  fol-  potential attempted  away  from  into  resonance w i t h the  Usually  resonance,  the klystron  was  36  then no  about  indication  tained was or  20,megacycles of a  signal  at the output  adjusted  pulse.  to the desired  delivered  An  the swing  the  output  reduced  pulse-length  to the cavity of this  condition due  magnet  measurements were  current  was  now  corresponding  t o be made.  5 r e a d i n g s were correspond  the s l i g h t e s t  under  "no p u l s e "  larly  near  position  at a  15  (forfaster  to  the second If  change to  suitable  sweep,  63 of  of the by  observ-  o f power  from  was  averaged). Faraday  rotation  observed,  This  going  entering  then  1500  the  pulses  per second,  cavity  again triggered  o r 500  were  msec even applied  lOmsecwide) .  i n i n t e n s i t y due t o  application  the approximate  since  particu-  down t o 1 5 0 , 5 0 , o r  The t i m i n g  as be-  minimum  rotation,  covering  i . e .change upon  were  rotation,  s e t t o i t se x t e r n a l l y  30 p u l s e s  signal,  pulses  then  at which  f o r minimum  the normal  (usually  and  strength  The p u l s e s w e r e  times).  (usually  i n the Faraday  the cavity,  sweep  but a t times  a reasonable  adjusted  and  reduced  speed  relaxation beam  taken  field.  the oscilloscope  single  ob-  a maximum  t o 35Amp.  o f microwave power  condition  applied,  a  to the bursts  to the normal  amount  the resonance  in  17, 35,  obtained  The m u l t i v i b r a t o r  fore  even  8,  until  to the f i e l d  c u t o f f and the analyzer  not always  was  be  or  multivibrator  the duration  increased  temporarily  did  could  the  (usually  little  cavity.  to the value  (again  Next  increased  during  on t h e g a l v a n o m e t e r  o f the  very  through the cavity  and i t soutput  indication  ing  The  coming  and e i t h e r  on t h e g a l v a n o m e t e r .  124 m i l l i s e c o n d s ) ,  power was  o f f resonance;,  of pulsed  change  power  i n rotation  37, was to  determined by p o s i t i o n i n g coincide  analyzer  until  previously which  t h e base  frequently  those  giving  signal the  Q  often  characteristic, Exposures  t h e n made  during  outside  light  laboratory (readings with  analyzer or the  to  to f a l l  region  analyzer  checked  essary.  t h e more  could e a s i l y  used.  then This  from  since  then the  linear  region of  accuracy o f the pulses  setting  any d i s t u r b i n g  since  also  made  of the  manually,  i n near  amount o f  region  of the  darkness  and o t h e r a d j u s t m e n t s were made  made  at varying  a t minimum, 4 5 ° o f f minimum sweep  t o sweep  cos^G  t h e whole  Exposures were  of signal  be done  Observation o f the o s c i l -  d i d n o t cause  and w i t h d i f f e r e n t  speeds.  entirely  Whenever  possible  or mainly over the  characteristic  (approximately 30°  4 t o 6 exposures were u s u a l l y  made a t  setting.  Throughout was  rate  of a flashlight).  6 0 ° o f f t h e minimum).  each  This  on the f i l m ,  settings(base  s i g n a l was  linear  exposures  on t h e a n a l y z e r  otherwise)  through  was  different  due t o s i n g l e  o c c u p i e d b y t h e a p p a r a t u s was  the help  The a n g l e  i n greater  o f the signals  the position  on t h e camera, w h i l e t h e s c r e e n o f t h e o s c i l l o -  t o t h e low r e p e t i t i o n  loscope  with  of the signal,  resulting  so as  by the pulses.  settings  t o sweep o v e r  scope was o b s e r v e d v i s u a l l y . due  produced  f o rt h e base  b e made  coincided  t o achieve this,  repeated f o ranalyzer  measurements. where  i n rotation  trace  and then r o t a t i n g the  of the signal.  had t o be r o t a t e d  a minimum  could  cos?  of the signal  o f t h e change  of the timing  o f the signal,  o c c u p i e d by the peak  the analyzer  a measure was  w i t h t h e peak  the base  a l l measurements t h e h e l i u m vapour  and adjustments  o f t h e pumping  The v a r i o u s p a r a m e t e r s  (magnet  pressure  s p e e d made when  current,  helium  nec-  vapour  pressure, scope  pulse  gain) At  length,  associated times  at intervals  pulses  o f a sequence  ^  1 every  noticeable enough meters, netic in  i n the r e l a x a t i o n  field,  and/or  a l s o made minutes  of single  oscillo-  at t h e normal  process.  different length,  pulses  and o f t h e f i r s t repetition  long Once  time  conditions  and/or  5 to'12 rate  effects  satisfied  obtained f o ra particular  pulse  speed,  were  that  seto f para(different  temperature)  mag-  were  made  fashion.  typical  exposures,  essentially  sweep  exposure were r e c o r d e d .  t o see whether  measurements under  A  25  of pulses  2 seconds)  an i d e n t i c a l  more  of several  e x p o s u r e s were  settings,  with each  exposures were  spaced  (  analyzer  successful  experiment  yielded  representing  relaxation  processes under  different  sets  of conditions  2 0 0 t o 300 o r  (field,  10 t o  temperature,  pulse-length). Subsequent oped  f o r 10 m i n u t e s  Kodak A c i d washed,  Fixer  dried  t o an e x p e r i m e n t D  i n Kodak  w i t h Hardener  the exposed  f i l m s were  -76 d e v e l o p e r a n d f i x e d i n  f o r 10 t o 20 m i n u t e s ,  a t room t e m p e r a t u r e  devel-  and f i n a l l y  extensively  evaluated  as  fol-  lows: The millimeter and  between a g l a s s - p l a t e  a binocular  and  microscope,  relationship  was  By means o f t h e e x p e r i m e n t a l d a t a and  versus time  i n the linear  low-power  v e r s u s time  curves the intensity-time  to rotation  entirely  under  flat  the intensity  and t a b u l a t e d .  calibration  verted  were p l a c e d  graph-paper  f o r each decay  measured the  frames  relationship  data, except  intensity-rotation  was  when t h e s i g n a l  range.  The  conwas  rotation  39, versus  time  sulting  p o i n t s were  straight fined  line  The  t o l i e to a f i r s t  parameters.  samples  used  i  s  prepared  amounts  X  Barium  sulfate  stands  t h e bottom,  sulfate orate  from  they  a t room  as a  results are  were  crystals  0  s  of molecular  and barium  equival-  ethylsulfate,  respectively).  of the rare-earth t o eliminate  i n a desiccator  washed w i t h  small  in a refrigerator slowly  allowed t o  a l ltraces  under  i n detail used  was  of barium  the solution reduced  amounts o f  until  settle  ethylsulfate  further  at room-temperature).  ethylsulfate  i n the laboratory  and was  obtained by l e t t i n g  has been d e s c r i b e d barium  (C^ H  sulfate  o r Mol  filtered  were  single  according to the reaction;  temperature  kept  Nd  o u t of the mixture  t o decompose  The prepared  and  the solution  salts  and then  paration  fell  and the s a l t s  tend  typical  was de-  respectively,i . e .  aqueous s o l u t i o n s  for Ce  carefully  The water  Some  ethylsulfate  of the particular  (virtiere  decanted,  line  then tabulated  i n t h e experiments were  a  BctfCjfcHsSO^a/ 2,Ha,0  at  of this  o f Samples  - a n d neodymium  x  ent  was  The r e -  approximation on a  of the slope  time, which  Ce(C H SO>) -?H ,0 They were  semilog graph-paper.  i n Chapter I I I .  Preparation  cerium  onto  and t h e i n v e r s e  of the various  discussed  of  found  as the r e l a x a t i o n  function  3)  d a t a was p l o t t e d  b y l" aeger ( !  pressure.  distilled use  (since  The p r e IX,  1914).  i n t h e r e a c t i o n was  according to B e i l s t e i n  evap-  also  (X, 1 9 1 8 ) .  40 Commercially from  the  cerium was  available purified  Fisher  salt.  The  prepared  supplied  by  Scientific  praseodymium, Single a  variety  the  salts:  a)  Simply  evaporate pressure  this  to  Chemical  .4%  to  in a  99%  pure The  following:  samarium,  .5%  other  from  the  desired  clear  dust  fairly  large  dish,  placed  at  desiccator,  yielded,  essential to  rare  oxide in  the  to  earth  .4% oxides.  e t h y l s u l f a t e s were  free  saturated  (^  amount  room  .1  30 cm  temperature besides  leave  solution  the  ) of  unusable  large  obtained  solutions  and  developed  the  t  neodymium  impurities  of  H 0  NOIJL ( ^ O i j . ^ * S  Co.  a  i t was  preparation  the  c r y s t a l s , some n i c e l y  case  i n the  be  c r y s t a l s of  letting  from  ( SO*)* S H ^ O  Ce^  sulfate  used  sulfate,  claimed  o f ways  in a  twinned  are  .1  was  laboratory  Lindsay  neodymium o x i d e  in  neodymium  i n the the  Co.  cerous  of  solution  atmospheric small  and  specimens.  In  completely  un-  disturbed. b)  Filling  letting the  a  the  seed  surface  access of  small  crystal  of  the  a i r yet  solutions,  ature.  30%  ly  large  in  larger  c)  Using  to  single  preventing  50%  of  the  by  This  joined  at  and  resistance wire  H.  5cm)  rest  with the  bottom  covering  the  beakers  dust p a r t i c l e s f r o m  tended  to  This  method  had  Wesemeyer  (VII,  growing machine"  bottom  take  seeds  machine by  saturated  at  evaporation  crystals.  XIV).  of  either  letting  Figure  to  solution,  "crystal  top  (1  and  beakers a  beakers  consisted .  were wound  of of  place  develop  or  solution, float  so  as  to  falling at  on  into  room  into  previously  admit  temper-  sufficientbeen  used  1958). own two  .5 c m  diameter  below  the  design small  (See  glass  diagram beakers  glass-tubing.  beaker  having  a  10  larger  n  F I G U R E  2L¥  Heater wi re  ^—7i ^  Saturated  \\\\\\|  Mercury  CRYSTAL  page  pool  Ground  glass  Crystal  seed  cover  C ylinder of f ilte r paper containing crystals  o o o o  Following  solution  40  GROWING  APPARATUS  diameter. meter the  a  At  pool  the  so  covered glass made  filled  that by  a  small  of perforated  glued,  so  in  large  the  order  of  that  .1  started to  erature  gradient  the  about  long  some o f  1  small the  supersaturated stal  produced ulties  one  this  m a n n e r was  methods, was  crystal  possible.  taken, other  since  planes  crystals  almost  completely  beaker.  When a  through  i n the  degrees  large  place.  into The  because  no  the  to  crystals  The  had  on  solution  and  became allowing  cry-  i n principle  clean  and  diffic-  and  re-  obtained the  a i r i n the  in  other  crystals  method were  been o b t a i n e d  in  dissolved  grown by  this  temp-  contents  the  crystal  those  a  However,  solution  was  convection  beaker  crystal.  long)  the  same t i m e  cup  the  solution  a  solution  paper  deposition of  further refinements  enough  the  cooler beaker,  seed.  superior  of  wire,  machine worked  i n keeping the  At  warmed  ethylsulfate  growth t o  heater  Below  i n the  warmer t h a n  filter the  raw  current  diameter  centigrade  i n the  cerium  small  the  The  with  immersed  system.  the  evaporation.  were  already  by  underthe  methods. For  which  No  beakers  filled  optically  probably  the  intro-  and  were e x p e r i e n c e d  stricting  and  was  1.5c*y»  paper  avoid  seed  1.7e»v> d i a m e t e r ,  when p a s s i n g  good  pool  a  operation  cup  passed  take  solution,  In  dia-  cylindrical  crystals  growth t o  located.  mercury  diameter beaker.  raw  smaller  to  flow  2  beaker w i t h  was  3  the  developed,  to  lctv\ )  saturated  on  filter  was  longer  glass plate  diameter  current  being  with  i t was  Amp.  the  ( *^  i t floated  a ground  plate  of  o f mercury-  s y s t e m was  duced  bottom  are had  use  i n the  hexagonal to  be  cut  experiments  in and  the  optic  axis of  s t r u c t u r e , had  t o be  identified,  polished perpendicular  to  the  i t .  crystals, and The  42 appearance Jaeger was of  o f these  c r y s t a l s has been  (IX, 1914).  difficult,  described  I d e n t i f i c a t i o n of the planes  particularly  i n small  samples.  e s t a b l i s h i n g the i d e n t i t y o f the planes  p r i s m s was  found  found  that  perfect  only,  took place  pressed  gently The  leather out  t o be b y t h e i r cleavage  when  cleavage  of these  a razor  b l a d e was  planes were  of progressively  and rouge.  A brass  ground  smaller  grain,  having in a  part  of i t s axis  and s l i d i n g  cylinder,  held  the crystal  w h i c h was t i e d  in  so t h a t  the sector,  the  flat  placed the  on a f l a t  cylinder  protruding  edge  and  was p r e s s e d  er.  By g e n t l e  the to  glass  inserted  (along  into  rested  stainless  edges  or  leather  ground  and s t e e l  rested above was  so t h a t t h e  on the g r i n d i n g  down u n t i l  steel  s e t on top, and  cylinder,  by t h e weight  with  by  o f fprotruded  surface  o f the brass  r o t a t i o n a l and t r a n s l a t i o n a l motion was  fine  a 120° sector cut  cylinder  the steel  and  and p o l i s h e d  angle  paper  the steel  down s l i g h t l y  top o f the brass  The  plate,  of the c r y s t a l  the crystal  the o p t i c  Corundum  with  and p o s i t i o n e d  t o be ground  t o p o f t h ec y l i n d e r .  brass  block,  the part  planes  oriented  heavy  one o f i t s 1 2 0 ° o u t s i d e  while  I t was  axis.  flat  cylinder,  way  t h e hexagonal  and these  properly  along  plasticine,  The s u r e s t  properties.  planes,  by  o f the c r y s t a l s  forming  i n the d i r e c t i o n of the optic  appropriate  corundum p a p e r  i n detail  of the  i t s f a c e was  c y l i n d e r s , and thus  cylindsteel  flush  with  perpendicular  axis. samples  the optic  used, i n  axis)  the experiments  b e t w e e n 3 a n d 5 mm,  varied  i n length  and i n c r o s s - s e c t i o n a l  st area  between 4  a n d 1 0 mm  left  much t o be d e s i r e d .  .  The o p t i c a l  Even though  quality o f the crystals  the best,  i . e . the clearest,  43 p a r t s were  cut out of the single  they  h a d some  still  regions  of imperfect  imperfections .produced sulting ing  tended  some  growth  (mainly  diffuse  crystals  are not only  also  decompose  Chapter  I I I : The  samples,  and s m a l l  the latter). the light  of the incident  o f randomly p o l a r i z e d  These  path,  and  light, re-  light  i n t h e emerg-  had t o be h a n d l e d  very  soft,  rapidly  results  performed•with  ethylsulfate  Reproducibility,  a)  Microwave  system:  The  frequency  o f t h e microxvaves  t h e wavemeter  drift  from  a well  i n frequency  during smaller  was  immersed  * .215 M c after  noticed  drifts  i n 10 ) 5  cavity.  care,  since  soluble, but  35°C.  i n this  i n a water  The u p p e r  warm  as  .1  as f l u c t u a t i o n s  . 5  the help Mc  o i l bath  limit  no f r e q u e n c y  up, even Mc  though  (that  i n the output  and  i t d i d n o t show a n y for drift  can c o n s e r v a t i v e l y be g i v e n  In practice  as s m a l l  £  cooled  after  the i n i t i a l  Apparatus  than  supply  .  except  be m e a s u r e d w i t h  of better  warm-up.  refer  mentioned.  e t c . , o f the  could  chapter  ethylsulfate  r e g u l a t e d power  a p e r i o d o f 15 m i n u t e s  evident  part  t o an accuracy  the klystron  quoted  i s explicitly  Linearity,  driven  above  neodymium  1)  Since  extreme  and e a s i l y  at temperatures  experimental  cerium  brittle  with  Measurements  experiments  where  ant  a i rbubbles  associated with  scattering  for  beam.  they  of  and used  to obstruct partially  i n a component  The  to  inclusions  crystals  d r i f t was  as ever  one s h o u l d  i s approximately o f t h e tuned  have 1  reson-  The  frequency o f the klystron  when p u l s e - m o d u l a t e d , accuracy, and  ing  i n this  pulses.  ticular  power  particular  remained  were  made  t o have  frequency,  so as t o o b t a i n  saturation  of the energy  the  Faraday  b)  Magnet:  It  in  was  ments less field  1 1 Imp.,  was made  and  have  varied  throughout  t o measure  constancy.  great-  any p a r -  i t and no  However,  a t t h e resonance  sufficient  to get appreciable  levels  power  and hence  observable  drew a c u r r e n t  time  were  the accuracy were  a n d t h e ammeter calibration  done b y a f l i p - c o i l  - even  changes i n  that  o f 3 5 Amp.  The  no measurable  water-  change  over p r o l o n g e d p e r i o d s o f  d i dnot exceed  and i t s s t a b i l i t y  The  £ . 1° K  optimum m a t c h i n g  was o b s e r v e d  of the r e l a x a t i o n  calibration  within  and a l s o  t o be s o e f f i c i e n t  - i fthe current  than  any experiment,  i n the cavity.  to assure  t h e magnet  found  the current  time  stable  contain-  rotation,  maximum f i e l d  cooling  noticed  of the c a v i t y  t o t h e c a v i t y may  taken  h a d t h e same  o f t h e h e i g h t o f t h e mod-  constant throughout  No a t t e m p t  p a i n s were  t o have  of the pulses,  was o c c a s i o n a l l y  frequency  t o experiment,  experiment.  attempts  by adjustment  delivered  experiment  the duration  be s a i d  a drift  o f helium took place  The from  remained  the temperature  boiling  case  The r e s o n a n c e  the crystal  provided  ly  can safely  had t o be c o r r e c t e d  ulating  no  though  during  25 Amp.  done w i t h  Since a l l measurefields  requiring  i n the determination o f the  limited  only  b y the accuracy o f t h e  used.  o f magnet-current  and checked  versus  field  by proton resonances  had been  at various  45, points the  o f t h e upper  curve was u s e d  magnet then The  current  reduced ammeter  that  was  used  c)  Cryostat:  In  the cryostat  power the  varied, quid  error than  ±  input  during  occasional  kept w i t h i n from  —  at fixed  the vapour  ±  kept  at a  field. so  .001°K i f t h e constant.  fixed  of t h e pumping  temperatures.,  The a b s o l u t e v a l u e pressure  Since  temperature  speed  on thel i -  variations  constant. could  o f t h e temperature  i n the cryostat  be  was  on t h e b a s i s  of  (21, 1958).  tures  below  could  b e made b e t w e e n  that  .1 Amp.,  t o maintain the temperature  M e a s u r e m e n t s w e r e made  boiling  s y s t e m was  an experiment  required  "1958 s c a l e "  and up) and  b e t w e e n 1.38 a n d 4 . 2 ° K c o u l d be  adjustments  .5%.  i . e .the  the desired —  of  85 O e r s t e d t .  the microwave  a l l measurements  the  t o produce  part  i n the determination o f the magnetic  temperatures  from  This  ( 3 5 Amp.  had a reading accuracy of  helium were  derived  to saturation  f o r p e r i o d s up t o 12 h o u r s w i t h i n  input  power  increased  curve.  i n the experiments,  to the value required  smaller  maintained  o f the hysteresis  exclusively  was  t h e maximum  field  In  part  the X  of helium  the intensity  - point. the X  caused  No  a t 4.22°K relaxation  - point  bubbles  fluctuations  and at several time  measurements •  a n d 4.2°K s i n c e  i n the cavity i n the light  tempera-  violent  t o such  beam  a  degree  obscured any  signal. A  check  m e n t s w a s made  on t h e c o n s i s t e n c y by p l o t t i n g  o f the temperature  the Verdet  constant  measure-  (rotation  i n °  46 'per  cm  path length  per  various  temperatures  erature  i n  stant turn we  °K  t h i s was  used  between in  measuring  to  be  pect  that  The  slight  the c r y s t a l s  which i n  As  .013°,  the  o f t h e two  dia-  crystal  discrepancy i n slope  explained by  the optic  con-  l / T -  to  the origin.  f o r each  i s readily along  Since the Verdet  susceptibility,  through  temp-  axis  are very  the  exactly  difficulty ( i t i s  delicate with  res-  to h a n d l i n g ) , Optical  System:  a l l measurements were  rotation  to be  avoid  ii)  obtain  light  o f the plane  taken  i)  intensity reliable  passed  The  by  fluctuations  i n the sample,  ter  network and  contribution  not a r i s i n g  the analyzer as a f u n c t i o n  o f t h e DC  i ) was  satisfied  - generator  t o the s i g n a l  assured noise  of the mercury-arc. left  of  due  care  had  much  intensity  that  Although  of  relationship.  sufficient  in conjunction with  from  to be  to a  rotation,  time,  the i n t e n s i t y - r o t a t i o n  stabilizer  system  from  measurements o f t h e r e l a t i v e  requirement  stability  optical  on i n t e n s i t y  of p o l a r i z a t i o n  fluctuations  faithfully  The  output  based  to  iii) establish  the  crystals  i s the  i s proportional  i n d e e d t h e case  i t slength  remembered  Since to  plot  obtained at  , where T  t o the magnetic  line  field)  Figure ZV),  i n the experiments.  t h e two  QJ^®  ethylsulfate)  a straight  shows  parts  d)  (See d i a g r a m  ( f o r neodymium  gram  y- _  against  i s proportional  expect  Oerstedt magnetic  t h e r e was  no  fluctuations the mechanical  degree. the  f i l -  noticeable  i n the  light-  stability  desired, w i t h proper  care  of  FIGURE VE&DET  XZ  CONSTANT VS.  I/(T-.0I3°)  CRYSTAL I o CRYSTAL H  deg. cm" 0  .02  .2  Following  page  46  .3  , .4  7  (°Kj-  47  (that  i s avoiding  system w h i l e ing  from  air  was  shifts kept  path to  the  %  could  In  out of -  the  of  the  point.  not  be  for  the  the  scope  depended  deviation of  this  from  * the  10%  ing  to  on  multiplier  the  ratio  was  On  the  of  time  inherent  at  only  care  fatigued  large of  as by  remained  serving to  scale  gave  on 15%.  clear  the  light  considerably  a linear the  res-  oscillo-  screen,  Although  part  the r o t a t i o n - i n t e n s i t y an  inaccuracy  since  the  f o r measurements  the  cause  limitations.  the  position of  the  of  up  to  position  could  not  traces  of be  serv-  purposes.  to  the  hum  pick-up  to n o i s e - r a d i o  same  the  variability  of  of  up the  of  the  the  of  conditions  the of  of  traces,  The  the. s i g n a l t o  t o £. 5%  introduced  base  of  at circuit  limitation  o s c i l l o s c o p e , which  as  time  photo-  and  final  of magnitude were  the  photomultiplier  somewhat.  spot-size  order  errors  random n o i s e  a broadening  favourable  the  within  trace  taken  aris-  4.22°K.  the  as  in  contributed  p o s i t i o n of being  not  bubbles  of  exactly  r e s o l u t i o n was otherwise  and  satisfied  intensity levels  signal  under  - point,  response  contributed  the  %  the  liquid  optically  the  screen  intensity levels  reduced of  the  low  the  The  could  b a t h was  r e s u l t i n g measurements,  calibration At  helium  nevertheless  correspond  for  high  there  The  avoided.  thus  in  Intensity  However,  linearity  i n the  traces  made  the  be  and  when p r e v i o u s l y  n o n l i n e a r i t y was  calibration,  beam,  measurements  intensity. on  could  entirely  i i ) was  photomultiplier to  Above  displacements  fluctuations in  beam  light  avoided  Requirement  ponse  light  intensity.  signal noise  The  vibrations or  making measurements)  fluctuations below  mechanical  p  the  hum  and  noise  random  noise.  due  the  to  oscilloscope.  48 Any  distortion  to  the photographic  to  he c o m p l e t e l y  corded  traces  process  to numerical  get misreadings  greater,of sulting log  though  versus has  of  position  t o be  state  i n each  i i i ) was  also  experiment  o f the analyzer that  entering perfectly  polarized,  resulting  i n a pure  lationship  between r o t a t i o n  the  of analyzer  of  a corresponding  been the  said  i n Chapter  response  portional in  Chapter  analyzer of  to square  rotation. 3)  be  the l i g h t  case  light  incident  of a fixed  In the i d e a l  0  +  case  be 1 0 0 % p l a n e  C©s@)  identical  re-  f r o m what  diagram  light  Figure  (intensity  i n Figure  l<x>  by a point  discussion  and  to the e f f e c t  follows  i tpasses plane under  on t h e semi-  and t h e c o n s t r u c t i o n  i s represented (since  interval, re-  was p e r f o r m e d , i t  (See a l s o  to polarized  i t possible  and i n t e n s i t y ,  This  4) a n d i l l u s t r a t e d  and i n the i d e a l  representing  will  a  satisfied.  this will  s'  under  of the i n t e n s i t y  with  COS*"^ ~  o f amplitude)  the Poincare' sphere  only),  done  aligned  I, Section  I, Section  i n our case  calibration  b y an angle  o f an analyzer  time  of the points  relationship  found  10%,whichever i s  the analyzer.  rotation  Faraday  or t  2°  made  due  the r e -  by measurement  not completely  t h i s was  crystal  effect  a  was  reducing  i n a given  scatter time.  of polarization  a perfect  to t  versus  remembered  the traces  i n the signals  in rotation severe  of r o t a t i o n Requirement  Even  relationships  corresponding  the change  relationship  However, when  the noise  i n occasional  plot  of recording  negligible.  low-power m i c r o s c o p e , to  o f the intensity-time  has IV) on  i s pro-  described  V.  The  on t h e e q u a t o r  polarized  light  the point  I  on t h e a n a l y z e r  lies  also  o> on t h e  49 equator along  the equator,  sphere.  keeping  Now whether we  lo> f i x e d ,  l©t> r e m a i n s f i x e d , w e w i l l  while ion  of t h e Poincare  r e l a t i o n s h i p ( i f there  remains 4)).  always  movement  of  and  analyzer  r o t a t i o n by  lot> o n t h e P o i n c a r e  nor p e r f e c t l y aligned. diffuse  former tically ent.  ticity In  i tdoes  tation  I o>  from  itself  with  the pole  will  To o b t a i n possibly  an estimate  introduced  more  of actual t h e same since  birefringence  or less  with  ellip-  componcom-  intensity r o -  are inclined  on t h e a n a l y z e r , 6  measurement.  t h e two p o i n t s  t o t h e maximum  Moreover  now  t o each  ellipticity as the point  (See diagram  o f propagation)  o f t h e sphere.  vary  some  per-  correction forthe ellip-  d e p e n d now o n t h e F a r a d a y  also w i l l  optically  of the unpolarized  planes  varying  ( t h e s t a b l e mode  birefringence  error  care  whose  incident  i t s path  case  away  field.  takes  corresponding  of the light  follows  this  <p  to a  introducing an unpolarized  the conditions  circles  I, Section  i n the crystal, the  the analyzer  r o t a t i o n o f l«.>will n o t g i v e  by an angle  lu,>  place  r e l a t i o n s h i p as r o t a t i o n o f l o >  obtainable  ^  consequence  us t h e exact  under  o n two d i f f e r e n t  other  In  not give  of the light  general  move  the calibration  lo>  0'.  by  i s neither  takes  entering  polarized, the latter While  ponent,  the light  rotat-  corresponds  sphere  As a  scattering of light  making  rotates,  intensity  as discussed, i n Chapter  Npw i n p r a c t i c e t h e c r y s t a l fect  I o>  i s no b i r e f r i n g e n c e a n d hence  on t h e e q u a t o r  C l e a r l y an  rotate  o r whether  g e t t h e same  i  Figure V ) .  lies  an  angle  the position of  r o t a t i o n , and t h e  the a p p l i c a t i o n of the magnetic  o f the order  b y misalignment  o f magnitude  o f the  o f the crystal, the  50  maximum change An  ellipticity  i n r o t a t i o n were  upper  imum  limit  (i.e.  on t h e e l l i p t i c i t y  the l i g h t  neglecting  L e t <p b e  (3.8) Chapter  cident  passed  I.  passed  Let X  as given  and I  by the analyzer  maxand  polarized  as follows: i n equations  (3.7)  b e t h e m i n i m u m a n d maximum  respectively, and X  by the analyzer  ease.  by comparing  elliptically  component)  parameter  with  i n one p a r t i c u l a r  completely  the unpolarized  i n elliptieity  i s obtained,  of light  t o be  the e l l i p t i c i t y  intensity  as t h e change  investigated  and minimum i n t e n s i t y  considering  and  as w e l l  the  i n -  intensity.  Then  Hence  i-5/1 ¥ was  found  inclusions ing be  = « r c cos  t o have  an upper  i n the crystals  i t i s more r e a l i s t i c a l smaller  ing  10°.  t h e change  large  Faraday  (unless gross ions which  (a.14)  This  t o (2.18)  i s responsible  rapidly  with  caused  increase  as large  was  diffuse  the e l l i p t i c i t y  also  borne  as the f i e l d  introduced. prevails) that  and those  as 3 0 ° .  considerable  t o estimate  i n ellipticity  misalignment  T  limit  estimate  r o t a t i o n was  z/  { +  o u t when  &  <p  I t may  s a f e l y be  the  oC  to  consider-  <increased  i n Section  while  scatter-  was  and a * assumed  of the equat-  4 ) . o f Chapter  f o r the Faraday r o t a t i o n , w i l l In f i e l d ,  Since  (causing  I,  increase birefringence)  51 will  stay nearly  creased  oC  towards  the pole  it  was  becomes  found  i t remains  bute  the changes  fact  we  lw.>  point  circle  angle  increasingly  dominant  t o be  seen  what  i n rotation  of 10°.  We  and l t x > w i l l  be  of r o t a t i o n  i s Introduced  observed S  from  i s i n shifted of  changed b y a p p r o x i m a t e l y  error  , and hence  i s separated  as t h e f i e l d  F o r a change  the e l l i p t i c i t y  ^ 0  have  T h i s means t h a t  of the sphere.  that  Thus  The  constant.  =  entirely j/cj  the pole  10°. attri-  t o oc* , w h i l e i n oc  +  1  i f we  520°  o f t h e sphere  by a great  had p r e v i o u s l y  I  c o s 3,0 =•  A  3  where £ L @ was (see  diagram  the great Figure  circle  V).  angle  between  Then, i n terms  Ix >  and  luc> vj>  o f the parameter  , we  have  sC  ^  Therefore  <S -  and which  only  point to our  —  Y ^ F ^ p  ** V \  1  to an e r r o r  15% i s obtained. moves  on a  the e q u a t o r i a l relaxation  although  small  plane,  time  Thus  of~  no  relaxation  times  obtained  that  that  that  there  f o r different  and t h e l i g h t  even  an  error  though the  "wobbles" w i t h  respect  on t h e a c c u r a c y  are expected  possible small misalignment  to the crystal  calculation  serious effects  measurements  I-Olcc  ~  2%.  i t i s seen  circle  i t i s not surprising  A  ^-VWi  =  i s a s l a r g e a s 3 0 ° , b y t h e same  lo>  respect  =  i n  =  corresponds i f y  Even of  ^ \  " <f ~ TT yr&f  from  i s some analyzer  this  source,  disagreement  on  settings.  of the magnetic path would  of  have  field no  with  serious  52 effects of  on  the r e l a x a t i o n  rotation  parallel  would  to  the  then  optic  time  measurements.  correspond to axis, while  the  the  The  absolute values  component  of  the  relative  changes  that  results  field  would  be  unaffected. F om  the  r  liable tions This  at  least  within  from  the  averages  gives  us  overall  2)  accuracy  typical  decay  the  limit  f o r the  change a  root  ent  sweep  XVT  falls  into  from the  pulse the  two  field  fixed  Figure  XX  the  and  condition,  XVIII  on  of  -  averaged  re-  devia-  parameters.  20%.  Compari-  i t reasonable to  the  following  assume  relaxation  times.  of and  XXI  68.5,  109  and  IXX  a  analyzer the  corresponding to of  microwave  was  The and  relaxation  113  59.4,  to XXVII  of  decays  and show  times  times 62.0  to  2.056°K  a  frommin-  relation obtained  a  124msecpulse  a t two  the  o b t a i n e d from milliseconds  decays  differ-  respectively.  f o r no-pulse but w i t h  a  power  corresponding to  due  a t minimum  same  53.5°  milliseconds  show d e c a y s  temperature set  set at  a t two  intensity-rotation  relaxation  78.1,  Figures XXII  e-vtuecpulse  an  approximately 44°  show t h e The  to  pages r e p r e s e n t  experiments.  r e p r e s e n t decays  region.  analyzer  o f f minimum.  t r a c e s were  f o r the  square  set o f  temperature, o f 1.685°K  t r a c e s were  2040 0  sweeps w i t h  due  Since the  i n rotation of  XVII  and  linear  Figures change  and of 8°  speeds.  f o r no  to XXXII  2540 0  of  imum  10%  mean  are  Results.  in rotation  field  -  the  error  curves o b t a i n e d i n the  Figures  450  of  obtained f o r a  of  F i g u r e s XVI  a  the range  an u p p e r  Some T y p i c a l  at  i t appears  o f t h e r.mn.s.deviations o b t a i n e d m a k e  son an  foregoing  at  different condition.  analyzer  set  these respectively.  corresponding to  a  change  53  i n r o t a t i o n of approximately f i e l d of  2040 #  89°,  due to a  and a temperature of  124msec  1.381°K.  pulse at a  Figures XXII and  XXIII were pulsed from minimum i n t e n s i t y , Figures XXIV and XXV from 4 5 ° off minimum and Figures XXVI and XXVII from 8 0 ° off minimum.  The relaxation times obtained from Figures XXII to XXV  were 1 2 4 , 1 2 8 , 1 2 1 , 1 2 6 milliseconds r e s p e c t i v e l y . The graph on Figure XXVII shows a semilog plot of the i n t e n s i t y versus time r e l a t i o n s h i p obtained by measuring the traces of Figures XVI, XX, and XXII. Figures XXIX and XXX are pictures of decays due to a 6 3 millisecond and 8 millisecond pulse r e s p e c t i v e l y , with a l l other parameters i d e n t i c a l , i . e . 31°,  magnetic f i e l d 1 5 4 0 , 0 '  4 5 ° off minimum.  a change of r o t a t i o n approximately , temperature  1,685°  JC, pulsed from  The relaxation times obtained were 6 3 . 0 and  69.2 milliseconds, showing no s i g n i f i c a n t difference. The second trace on a l l these p i c t u r e s ( u s u a l l y at the bottom) i s a time-marker of  30  pulses per second  10  and  23w»secwide  respectively. Figures XXXI and XXXII are evidence of the "overshoot" phenomenon observed i n one p a r t i c u l a r c r y s t a l - see section of this Chapter - .  Figure XXXI shows the decay due to a  6  124wsec  pulse causing a change i n rotation of approximately 2 3 ° from 4 5 ° off minimum at a f i e l d of 1790 0 1.38°K.  and a temperature of  The pulse i s seen to shift the trace up from near the  centre of the p i c t u r e .  The decay then l e t s the trace swing be-  yond the equilibrium position downward, before i t return again to the s t a r t i n g l e v e l of i n t e n s i t y . Figure XXXII displays this e f f e c t even more s t r i k i n g l y .  FIGURE  See  To f o l l o w  page  53  XVI  Text  jj'IGUH3  See  To  follow  page  53  ZYIII  I'ezt  FIGURE XX  £TGURE  See  To  tollow  page  53  XXII  Text  S'lGUJOC  See  To f o l l o w  page  53  XXIV  Text  FIGURE  See  irIGURS  To  follow  page  53  XXVI  Text  XXVII  FIGURE ROTATION  X.XVIII  VS. TIME  ON S E M I L O G  SCALE  AS OBTAINED FROM  100°  •  FIGURE  O  FIGURE X X  3  FIGURE ~ W T  •  F  50  Following  53  100  150  msec.  T.Q f o l l o w  page  53  FIGURE  TTTT  54 ,  "A 124  msec p u l s e  imately  1.38°K  roughly  .5  tangular  to  field  of  1540  i s expected  to  cause  at  1°  pulse  a  from  the  at the  swings  the  point  before  returning  to  the  same of  pulse  was  direction  intensity  relaxation 3)  Effect If  energy  tect  such an  otherwise  The field  of  the  a  expect  since  the  of  The  the  decay the  i s  XXXI  this  the  and  rec-  following  in  XXXII the  the  starting  intensity  displayed  downward m o t i o n o f on  to  approx-  of  original  original  overshoot  of  rotation  corresponds  beyond  prior  the  increase  trace).  No  crystal.  a  using  these  and  results  and  a l l of  them  energy  levels:  of  may In  the  No  more  than  and  10%  from  in  was  the  to  be  de-  12*  have  a l -  been  noticeable.  foregoing  section,  obtained. shows a  the  r e l a x a t i o n times  temperature  sufficient  to  of  obtained  1.685°K f o r  cause  Msec  relaxation  observed,  should  phon-  in  made t o  8 msec  was  not  the  on  which  comparing the  10%  the  excitation of  attempt  would  described  length  lattice,  effect  than  saturate  pulse  pulse-lengths  pulses.  to  cause  An  conditions,  XXX,  flf  1540  effect  change o f  following table of  an  phonons"  Effects smaller  of the  sufficient  long pulse  "hot  by  from  XXIX  i s  r e l a x a t i o n time.  systematic  puise-lengths, tion  15°  (in Figures  to  identical  obtained  typical  a  may  effect  Figures  at  one  effect  detectable.  are  the  pulse  i . e . create  any  Since  microwave power  time,  may  though  to  temperature  change  trace.  as  i t .  a  pulse-length  levels,  times  the  times were"determined  turn  under  of  minimum, the  a  This  as much  corresponds  the  on-bands,  a as  of  •relaxation  trace  and  minimum.  left  pulse  #  complete  various satura-  Pulse  Relaxation  length  Time  124  74.2 68,9  63  63.0 84.0  35  78.2 67.5  17  68.7 82,0  8  73.4 69,2  The  results  at different  fields  4)  Measurements o f R e l a x a t i o n  were Time  similarly at Various  T  inconclusive. Fields  and  Temperatures The of  following  the relaxation-time  temperatures value  a n d magnet  and r.m.s.  Temp.  ( °K) 1.380 .322  -  780  i s a  summary  o f neodymium fields.  It also  )  Relax.Time (msec) 146 150  of the  measurements  e t h y l s u l f a t e at  d e v i a t i o n f o r each  Mag.FieId  (  table  gives  set of T  the  various average  parameters.  avg. (msec)  r.m.s.dev. (insefc)  162.5  11.1  178.1  21.8  165 176 165 173 1030  135 157 171 185 186 207 186 198  Temp.  ( °K) 1.380  1.542 .434  Mag.i'ield ( )  Relax.Time (msec)  1290  206 203 186 195 208 195  198.8  8.4  1540  101 100 90 103  98.5  5.1  1790  129 134 122 118 125 146  129.0  9.1  2040  124 128 108 105  116.2  10.5  2290  122 101 101 135 109 127  115.8  13.3  2540  104 133 125 115  119.2  11.4  780  110 119  114.5  4.5  1030  112 121 125  119.3  6.1  1290  128 136 142  135.3  6.5  1540  80 91  87.7  4.9  92  T  avg. (msec)  r.rn.s.de-v (msec)  Temp. (  °K)  1.542  1.685 .522  Mag.Field (  )  Relax.Time (msec)  T  avg. (msec)  r.m.s.dev, (msec)  17.90  95 99 102 95  97.8  2.9  2040  108 125 111 100 113 114  111.8  7.9  2290  95 98 100 137 121  110.2  16.2  2540  110 89 138 98  108.8  18.2  780  109 127 99 130 114 129 107 130 111 130 112 118 100 104  115.7  11.3  1030  113 138 138 144 141 114 129 149 120 143 138 149  133.8  12.0  Temp. ( o j K  1.685  Mag.Field ( )  Relax.Time (msec)  1030  119 126 137 140 117 139 136 147  1290  121 139 •147 155 124 120 147 168 115 153 144 160 131 156 156 164 135 166 156 163  T  avg. (msec)  146.0  r.m.s.dev. (msec)  16.2  1540  69 74 63 84 68 78 69 82 69 73  72.9  6.3  1790  112 120 128 114 120 126 140  122.9  8.2  Temp. ( OK)  Mag.Field ( )  Relax.Time (msec)  1.685  2040  135  .5 22  T  avg. (msec)  r.m.s.dev. (msec)  147.8  10.9  117.0  7.0  162 140 159 136 142 139 163 143 159 2290  2540  102 116 123 114 128 117 118 112 120 127 119 120 112 116 112 116 113 135 110 111 115 125 110 110 109 105 112 104 120 94 100 98 104 112 110 118 116 109 109 107  108.2  5.5  Temp. (  °)  Mag.Field (  )  Relax.Time (msec)  Tm  avg.  (msec)  r.m.s.dev. (msec)  1.685 .522  2540  107 103 106 113 109 113  108.2  5.5  1.875 .629  780  55 48  51.5  3.5  1030  63 66  64.5  1.5  1290  69 70  69.5  .5  1540  51 52  51.5  .5  1790  59 61  60.0  1.0  2040  61 64 74  66.3  5.9  2290  57 59  58.0  1.0  2540  52 66  59.0  7.0  1030  63 70 68 64  66.2  3.8  1290  65 61 62 69  64.2  4.1  1540  42 40 37 38  39.2  2.7  1790  60 51 55 51 51 50  53.0  3.5  2.060 .723  61  Temp. ( °K)  Mag.Field ( )  2.060 .723  Relax.Time (msec)  63.3  5.1  2290  60 67 59 59  61.2  4.2  2540  58 65 53 67  60.8  5.6  1290  7.3  7.3  : . •.  1540  4.7 4.7  4.7  1790  6.0 4.8 6.5  2040  2.8 2.1 3.2  second  number  relaxation the r e s u l t s increase  nounced the  separation  same  and XXXIV  at different  show  between 0  7800  of the microwave  .35  2.7  .46  and 1290  cavity.  0  i s an  common  The w i d t h  of  feature  approximately-  followed by  corresponds  o r d e r o f magnitude as t h e w i d t h  dependence A  the "resonance  levels  T)  the f i e l d  temperatures.  , i . e .near  of the energy  5.8  i s In  at the various temperatures  d i p a t 1540  frequency the  time  ,  i n t h e c o l u m n Temp,  • F i g u r e s XXXIII  linear  r.m.s.dev. (msec)  68 78 59 62 59 64 62 59 59 65 61  4.22  "tThe  of  avg. (msec)  2040  •  the  T  a  field" to the  of this  prowhere resonance  dip i s of  o f the resonance  line  FIGURE  x x x in  RELAXATION TIME VS. MAGNETIC FIELD msec.  •  T=I,685"K  3T=2  060 K , ,  OT=4.22°K  150 -  100  9  9  3  9  50  a  .9  3  O  O K0  Following  page 6 1  FIGURE  XXXIV  RELAXATION TIME VS. MAGNETIC FIELD • T = 1.380 °K O T= 1.542 °K  msec.  T=I.875°K  3  200  •  G  o  •  0  100 o 3 3  9  3  « o  «3  8• o  <• ^ 3  3  K0 Following  page  61  ( rvy 250j2f). the  field  various  At higher  fields,  i . e . between 2040 #  d e p e n d e n c e i s somewhat o b s c u r e d .  t e m p e r a t u r e s do n o t g i v e  a n d 2540gr  The r e s u l t s f r o m  identical patterns.  I t appears,  however, t h a t f i e l d  dependence i n t h i s r e g i o n i s s m a l l ,  ling  a b o u t midway b e t w e e n t h e h i g h e s t  o f f at a value  level-  and  lowest  point o f the preceding d i p . The v a l u e s high i n a b s o l u t e  at a temperature  values,  i s p l o t t e d versus  field  This i s p a r t i c u l a r l y  temperature  on a l o g - l o g s c a l e f o r  straight line  t h e m e a s u r e d p o i n t s on t h e l o g - l o g p l o t , obtained  f o r the various  Slope  the following  o f l o g T,  780 1030 1290 1540 1790 2040 2290 2540  -  From t h e s e  f i t through slopes  fields.  (0)  i-iagnetic f i e l d  fields  notice-  strengths.  ./hen m a k i n g a l e a s t - s q u a r e  are  with  jJigures TKA7 a n d 2XCvT, i n vvhich t h e r e l a x a -  a b l e when s t u d y i n g  various  somewhat  e x c e p t a t t h e d i p , when compared  those a t the other temperatures.  t i o n ti-ie  o f 1.685°K a p p e a r  Versus  logT  3.48 2.62 2.99 2.83 2.88 3.64 1.94 1.95  r e s u l t s i t c a n be c o n c l u d e d  t h a t a t low  and i n t h e r e g i o n o f t h e d i p t h e r e l a x a t i o n t i i a e i s p r o 1  p o r t i o n a l to approximately may be p r o p o r t i o n a l t o T high f i e l d s  there  data  I  , this  .  , whereas a t h i g h e r However,  i s a l a r g e amount  since p a r t i c u l a r l y at  o f s c a t t e r i n t h e measured  dependence i s n o t v e r y  reliable  p o i n t s a r e a v a i l a b l e f o r 4.2°K a t t h e s e 1.6850K,  fields i t  ( u n f o r t u n a t e l y no  fields).  The d a t a a t  2.06°K, 4.22°K and 1.380OK, 1.542°K, 1.875°K were  63  obtained  with  two d i f f e r e n t  (crystal  I and c r y s t a l a  be  approximation  5)  Experiment An  sulfate  dependence  with  experiment  resonance  times  as i n t h e case An  was  obtained  6000 0  with  increased of  the rotation  aneously  (within  microwave  o f only  microwave to  only  large time.  than  radiation  enough  signal  The w h o l e  while  pulsing  could  be o b t a i n e d The  radiation  with  o f the micro-  then  instant-  within  decreased  t o 1.42°  to the application  to the influence .  This  of  o f the  change  measurements  o f the  60000 was c a r e f u l l y  microwaves.  amounted  No  a  relaxation  investigated  s u f f i c i e n t l y large  signal  field.  failure to find  on t h e F a r a d a y  2.4%  a n d was i n s u f f i c i e n t t o o b t a i n  up t o  a t any  up t o  the temperature  almost  .6%;  a strong  rotation  )  a change o f F a r a d a y r o -  o f 5600 0  f o r pulse-type  region  by  that  per^  At f i e l d s  removal  increased  i t had p r i o r  used  relaxation  of approximately  sudden  be a t t r i b u t e d  3 ° i n t h e sample  ,01°K.  the temperature  at a f i e l d  ethyl-  ( , 2 7 7 ° p e r cno  to the cavity  .2 s e c o n d s )  concluded  .6% c o u l d  i  decrease  Upon  while  I t was  appears t o  o f paramag-  and measuring  rotation  the rotation  to the value  power.  these  ethylsulfate.  applied  observed.  less  the influence  o f 1.41  and a s l i g h t  2 minutes,  i t returned  tation  was  time  a c r y s t a l o f cerium  effect  Faraday  power  to the cavity  approximately K,  large  a t a temperature  t o 1.45°K  wave-input  w a s made w i t h  o f neodymium  microwave  calculating  Ethylsulfate  on t h e F a r a d a y  extremely  When  respectively  fields.  of establishing  netic  XV).  crystal  of the relaxation  at a l l  Cerium  i n t h e hope  o f t h e same  I I of figure  separately, the best  T  parts  influence  o f t h e microwave  may b e a t t r i b u t e d  to a  very  64 short  relaxation  saturation  of  time,  the  and  energy  insufficient  levels  microwave  power  to  cause  at  the  external  magnetic  fields  use  the  relation  (5.6)  Chapter  used. At I,  best  S e c t i o n 5)  relaxation  )  making  ( X I I , 1951)  tion  (less by  order  o f magnitude  suitable  quote  below  assumptions  Bogle,  "Overshoot" one  ethylsulfate  a very  broad  that  the  1 msec).  Cooke  and  This  particular  crystal  of  observed  as  illustrated  in figures  XXXI and  XXXII.  consistently  under  power  an  the  (usually  on  account  laxation  the  prior  to  application  below  the  pulses, Close  of  rotation  the  of  the  level)  0)  .02.  i s very  the  observa-  examination  before  •  the  saturation.  further  • : (though  duration  increasing  of the  edge  of  the  Upon  always  The  i t  effect than  re-  had  remaining  value  "no-pulse" pulse  place  following  value  (124msec)  increased with  case  r e d u c t i o n took  the  condition.  this  value.  i n the  beyond  and  microwave  In  i t s normal  an  appeared  "no-pulse"  i t returned to the  "no-pulse"  leading  ethylsulfate  This effect  saturation);  pulse  long  neodymium  below  increased f i r s t  previous  also  of  pulses a  complete  with pulses of  and  that  reduced  the microwave  no-power  pronounced  condition  r o t a t i o n was of  to  ±-  salt  with  700  ( f*j  line, 3.70  and  d e s c r i b e d i n S e c t i o n 2)  a p p r e c i a b l e amount  application  ponding  Cooke  .  Phenomenon.  was  Faraday  Bogle,  the  Whitley.  effect  the  value  this  i s consistent  "overshoot"  caused  of  on  for the  of  relaxation  of  estimate  resonance  flu^^O  2.5°K w i t h  show  than  In  may  o b t a i n an  measurements  short  one  concentrated cerium  appearing  Our  6)  to  time,  For Whitley,  then,  correswas  with  more shorter  saturation.  r e v e a l e d no  co-  65  rresponding  increase  When  the crystal  ated  after  on w h i c h  attempts  explanation present  results  given  with  erature  (direct)  poor. Cooke  later). Previous  Also  deterior-  unusable,  i n other  crystals.  satisfactory  can be given  at the  be as T  Thus  ment w i t h  says  that  H  which  i s t h a t o f Van  the temperature  T , should  i n this  , and that  and as  decrease  very rap-  f o r t h e l o w temp-  there  agreement w i t h  show a s i m i l a r  poor  only  and Weidner  Van V l e c k ' s  with  agreement w i t h a temperature  dependence  (XIV, 1953) found  Thus,  2.5  on H  although  T  will  i s very, and  Van Vleck's dependence  i s observed. .  Only  at low temperatures  of T , proportional to theory.  w h i c h we  theory  4  depend-  e.g. by Benzie  Y\ a p p r o x i m a t e l y  little  little  , (about  salts,  a t low t e m p e r a t u r e s  dependence  i s very  Van Vleck's  on o t h e r  show t h a t T  thesis  f o r t h e d i p a t 1540 0  measurements  u s u a l l y shows  temperature  theory with  a t low temperatures,  Further  proportional to T  Eschenfelder  theory  proportional to  l  except  In.general  T ,  I, the only  process.  (XIII, 1950),  theory. Tj  behaviour  phenomenon  measurements r e p o r t e d  on f i e l d ,  comment  of  being  proportional to  ence  i tbecame  I I I can b e compared  Van V l e c k ' s  temperatures.  field,  The is  observed  so f a r , a n d no e n t i r e l y  i n Chapter  T J should  of  at high  idly  failed  of the pulse.  Discussion  ( V I ,1940).  1  had been  so t h a t  was s t a t e d i n C h a p t e r  s  dependence T  have  application  effect  a similar  of the "overshoot"  IV: A  Vleck  to find  upon  time.  Chapter  the  this  s e v e r a l experiments,  a t t e m p t s w e r e made These  i n rotation  and thus  the results  a  i n agreereported  66 ih  this  thesis  do  the r e s u l t s It  very  because  should  salts.  be n o t e d  m e a s u r e m e n t s made  i n this  powders,  theory,  there  until  have  been,  relaxation  has been aroused  solid  state  3 years  neither  on s i n g l e  have  crystals  and r e l a x a t i o n  processes  i n parasubject,  Almost  reveal  a l lt h e  on powder  including  on s i n g l e  usually  -  are important i n  a g o w e r e made  b e e n made  recently,  i n this  Maser.  t h e r e c e n t measurements,  thesis,  time  lately  o f the relaxation process  up t o about  , whereas  Measurements  that  of the 3 level  functioning  reported  w i t h Van Vleck's  of spin-lattice  Interest  the details  specimens-  well  of others.  few measurements  magnetic  the  do n o t a g r e e  the ones  crystals.  details  a r e no e x c e p t i o n  n o t seen i n  to this  general  statement. Let  us review  the basic  assumptions  o f Van V l e c k ' s  i)  That  there  i s thermal  equilibrium  i n the spin-system.  ii)  That  there  i s thermal  equilibrium  i n the lattice.  iii)  That  there  i s good  That  the frequency  That  spin-spin  heat  contact o f the l a t t i c e  with  theory:  the helium  bath. iv)  spectrum  of the l a t t i c e  i s a Debye  spec-  trum. v)  It valid. netic  field  i s that  the line  spins.  has almost  Taking  and that  the value  o f the spin-spin  case,  the l i n e reduced  any v a l u e  i s homogeneously  broadened,  c a n be n e g l e c t e d  i n this  moment o f t h e c r y s t a l  ously  value  i s reasonable,  The r e a s o n  magnetic that  interaction  may b e  (i)  i s  s a t u r a t e d , a n d t h e mag^  below  i sfreely  2500 0  .  time  This  rather than exchanged  f o r the line-width,  relaxation  that  t o z e r o , when t h e e x t e r n a l  broadened,  energy 2500  t o assume  indicates  inhomogenebetween t h e  we  of the order  arrive of  10  at a  67  seconds. within  H e n c e we  the )  spin  than  expect  system  Van  Vleck  ( V I I , 1941)  b o t t l e n e c k i n the  the  helium bath and  not  ally  a l l failed  to  observe  wave  the  power.  would  upset  an  The the  range  populated  region  it  were  ther  at  effect  i n the  be  these  experiments,  pur© and  the  i ti s .  of  the  described  in this  thesis  in  salt,  system  lattice  effect  varying  that  the  a long  to  vibra-  ions  and  the  experiment-  system  have  been  o f the  two  decay  process.  over p o p u l a t i n g spectrum.  would  see  than  seen,  the  i s that  perfect of  I t cannot  be  power  one  This  small  over-  lattice  as  o f the  Tothe  i s only from  delay though Ano-  i f there were spin  of  Both  effect one  these  two  system  accuracy  exponential.  there said  micro-  expected to  a pulse length  that  tried  i t a c t u a l l y was.  exponentials.  absence  be  the r e l a x a t i o n  i s a  of  o f microwave  phonon" band) might  ionic  the  pulse  by  i t was  pulse length  frequency  e x p o n e n t i a l decay*,, i n d i c a t e  which  of  an  lattice  o b s e r v a t i o n s , the  relaxation  spin  such  equilibrium  might  compound  and  typical  the  between  order  .  question,  system  energy  T\  in a  from  (by an  to detect  processes i n series,  would  these  by  (or "hot  which  that,  the  a much h i g h e r t e m p e r a t u r e  relaxation a  effect  i d e a was  since  of  to  established  so f a r .  lattice  frequency  relaxation,  transfer  experiments  such  smaller  open  transmission  attempts  t o be  experimentally for  calculated  heat  i n the  However,  In  v e r y much  should occur w i t h i n  lattice. have  times  ( i i ) i s much more  the  tions,  In  those measured  Assumption fact  thermal equilibrium  and  the  predomin-  experiments  68  It appears reasonable to consider assumption ( i i i ) to be v a l i d , since the c r y s t a l i s small and immersed d i r e c t l y i n the helium bath.  The existence of a pronounced temperature gradient  i n the sample would moreover cause considerable deviation from the cos? O i n t e n s i t y versus r o t a t i o n r e l a t i o n s h i p , which was  not  observed. The  correctness of assumption (iv) may  be questioned.  However, even major deviations from a true Debye spectrum would not seriously affect the relaxation process,  since only a small f r e -  quency band of the l a t t i c e vibrations Is instrumental i n the relaxation. The most important c r i t i c i s m i s that assumption (v) i s c e r t a i n l y not v a l i d , two  i t has  been shown experimentally  that, when  energy l e v e l separations are equal, or one i s an i n t e g r a l  multiple of the other, the relaxation time i s markedly reduced. This i s because, f o r example, one  spin can turn over turning back  two other spins of h a l f the energy.  The energy of the f i r s t  i s then communicated to the l a t t i c e via these other two  spins.  This, being a process i n p a r a l l e l with the straightforward reduces the relaxation time.  spin  process,  Such processes are used'technically  to control maser operation, and such a state of a f f a i r s can be brought about by adjustment of the f i e l d d i r e c t i o n r e l a t i v e to the c r y s t a l axes, or by the i n c l u s i o n of impurities. isotopes, Nd 143 abundant X —  12.2% also.  abundant  X-\  Neodymium has  and Nd 145  two 8>3%  The ions containing these n u c l e i have an  extensive hyperfine structure, and the conditions for two  and  three spin processes are well s a t i s f i e d .  However, the energy-  l e v e l s are so "broadened by spin-spin i n t e r a c t i o n that no sharp dips i n T,  as a function of magnetic f i e l d could be expected.  The dip i n T, the cavity.  at 1540 0 l i e s at the resonant f i e l d of  Its occurence at t h i s f i e l d i s too much of a coincid-  ence to expect i t to be due to anything but some influence of the cavity and the microwave system.  However, we  have not been able  to think of a mechanism which would shorten the relaxation time to the extent observed. that may crease T,  A l l e f f e c t s which we have thought of  be the klystron was  (e.g.  not properly shut off) ought to i n -  rather than decrease i t . Further experiments on t h i s topic are  continuing.  The "overshoot e f f e c t " described i n Chapter III, Section 6) could not be investigated further, since the c r y s t a l on which i t was  observed had decomposed.  Hence a l l that can be said about  i t at the present time i s that i n p r i n c i p l e such an overshoot effect i s possible i f the following conditions are s a t i s f i e d : i)  The  sample c r y s t a l i s twinned so that i t consists of two  parts, the o p t i c a l axes of which are not ii)  parallel,  The relaxation time depends strongly on the orientation of the  optic axis with respect to the external magnetic f i e l d , (lihat t h i s can be so has been v e r i f i e d i n other experiments). How  an overshoot may  arise under these conditions can be v i s u a l -  ized by considerations on the Poincare' sphere (See Figure XXXVII) . Suppose the sample c r y s t a l i s twinned such that i t cons i s t s of a part I, whose optic axis i s p a r a l l e l to the external  mag-  FIGURE  0  XXXVII  (r> POINCARE OF  Following  page 69  REPRESENTATION  OVERSHOOT  EFFECT  70 netic  field,  ular  to  the  followed external  representing from  I,  the  and  duced  in  I  magnitude  H  of  Faraday  Chapter the  be  on  II.  be  used  point  I  itV  the  by  angle  see  we  modes o f p r o p a g a t i o n  birefringence  and  Faraday-rotation  pulse  of microwave  brought  to  as  incident  that  modes and  of  zero,  on  say.  a  now  represented  microwaves, crystal  will  the  shift  sufficiently  at  approximately a  great  in  large,  circle  .  Now  Faraday  only such  be  l o ' > to the  same  rate  path,  and  denoted  be  the  l u > and  light  emerging  and  II  the  Also  left  same  stable  light  l * >  II,  from  the  emerging  in the by  I  i f the  and  path 1.  traced A  from  microwave  II w i l l  be  out  type  the  in  emerging  a  is  application of  , and  l v >  combined  Thus upon light  in  Suppose  by  polarized  light  de-  birefringence  of  .  birefringence  l © >  rotation  I  the  described  the  lt>  plane  any  application of  II w i l l by  and/or  have  the  rotation in  be  that  , say.  that  intro-  light.  by  i f upon  on  now  representing  from  the  /  II w i l l  l o >  by  point  X  I  represented  such  i . e . represented  difference  is  nearly  I,  /  may  the  introduced  incident  There w i l l  ducing be  phase  the  light  propagation  l y ^  l o  by  are  be  of  combined  to  emerging  between l i >  construction  happens  difference  power  the  the  sphere  rotation  U'>  have  perpendic-  the  light  angle of  is  on  length  to  in II  phase  i s represented  the  U>  what  the  II  amount  will  Let  from  circle  the  therefore  to  ,  choose  I I , now,  r o t a t i o n and  I may  stable  the  axis  the  great  may  optic  l t > be on  The  we  that  part  I I , whose  Let  determined  so  In  part  incident  (in principle  siredvalue). and  light  will  a  field.  incident  l t / >  and  by  J  introII  will  the the power  destroyed will  suitably placed  be  very analy-  zer, represented by l o t > , would pass during t h i s time l i g h t of continuously decreasing i n t e n s i t y .  As soon as the microwave power  goes t o zero at the end o f the pulse, the rotation i n I end I I w i l l reappear at a rate governed by the relaxation times o f I and H  respectively. Now suppose the relaxation time i n I i s very short com-  pared with that i n I I .  Then r o t a t i o n w i l l return i n I while II  exhibits only birefringence. appear slowly.  The r o t a t i o n i n I I w i l l then r e -  In t h i s case the point representing the l i g h t  emerging from the c r y s t a l w i l l trace out path 2 when returning from  (o>  to \o'> .  When i t reaches the point le>  along t h i s  path the i n t e n s i t y o f the l i g h t passed by the analyzer w i l l be i d e n t i c a l to the i n t e n s i t y due to 1©'^ (the equilibrium p o s i t i o n ) . Then the i n t e n s i t y w i l l f i r s t increase above i t s equilibrium value u n t i l a maximum i s reached and then decrease again to the equilibrium l e v e l .  This i s exactly what i s observed i n the case of  the "overshoot" e f f e c t .  However, i f the microwave pulse i s short,  we may have only p a r t i a l saturation i n I, hence only p a r t i a l r e moval of r o t a t i o n i n t h i s part.  Then i t i s seen that as a r e s u l t  path 1 and path 2 would be closer together and the amount of overshoot would be small, whereas with increasing pulse-length the path-difference and hence the amount of overshoot would increase, as was actually observed i n the experiments. The extent and appearance  of such an "overshoot" e f f e c t ,  while possible i n p r i n c i p l e , i s seen to depend on many f a c t o r s  72 (comparative crystal,  difference  polarization twinning  with  T  y  of incident  effect,  depends  respect  light,  times,  t h e n we  strongly  on  must  parts  amount  position  i s accepted  t o the external  Further  i n t h e two  i n relaxation  of the c r y s t a l  "overshoot" time  amounts o f r o t a t i o n  conclude  cause that  the orientation  investigations  of  twinned  birefringence,  of analyzer  as t h e  magnetic  of the  etc.).  underlying  the  If the  relaxation  of the optic  axis  field.  in this  direction  are planned.  73 BIBLIOGRAPHY References they  first  appear  are  numbered  i n the  A.  Kastler,  II. )  W.  Opechowski,  III. )  J.M. D a n i e l s and (1958).  H,  IV. )  H. W e s e m e y e r (1958).  J.M.  V. )  G.V.  Compt.rend.,  and  Skrotskii,  Van  232,  i n the  in  which  36,  405  Z e i t s c h r . P h y s . , 152,  591  and  Ixptl.  953  Rev.Mod.- P h y s . , wesemeyer,  P.S.  Theoret.Phys.(USSR), J.H.  listed  order  text.  I. )  VI. )  and  Daniels,  Zyrianov, 35,  1471  V l e c k , Phys.Rev.,  (1951).  25,  264  (1953).  Can.Journ.Phys.,  T.G.  Iziumov,  J.  (1958).  57,  426,  (1941).,  VII. ) J . H . V a n V l e c k , P h y s . R e v . , 5 9 , 7 2 4 , 730 (1941). V I I I . ) H. W e s e m e y e r , U n i v . o f B r i t . V o l . , P h . D . - T h e s i s IX. )  M.P.M. J a e g e r , K e c u e i l  X. )  Beilstein, (Springer,  XI. )  H. V a n D i j k , M. D u r i e u x , 24, S129(1958).  XII. )  G.S. 931  B o g l e , A.H. (1951).  X I I I . ) R.J. Benzie (1950)-. XIV. )  Handb. der 1918).  and  A.H. E s c h e n f e l d e r a n d (1953).  travaux  Organischen  Cooke  A.H.  des  J.R.  and  Cooke,  R.T.  chimiques,  33,  Chemie, V o l . 1 ,  Clement  and  S. ¥ h i t l e y ,  Proc.  (1958).  Phys.  Weidner,  J.K.  342  (1914).  p«325,  Logan,  Physica  Proc.Phys.Soc.  Soc.  Phys.  A,  Rev.  63,  92, ~  A,  201  869  64,  Views vector  of the motion for  I L > and  of t h e I r >.  tip  of the  electric  F I G U R E  TI  Decomposition of elliptically polarized  light  into circularly  polarized components.  F I G U R E  /  T H  i  R  P O I N C A R E  S P H E R E  FIGURE  Properties  of  the  32T  Poincare'  Sphere  F I G U R E  E  E f f e c t of combined Faraday E f f e c t and  Birefringence  FIGURE STABILIZED POWER SUPPLY  WAVE METER  X  DUMMY LOAD  KLYSTRON  T PULSE MODULATOR MICRO A M M E T E R  mi |  ATTENUATOR |  VARIABLE  GALVANOMETER  SUSCEPTANCE  0  CRYSTAL  DIODE  ~^\-  TUNABLE  END  M  /  CAVITY  SAMPLE  MICROWAVE  SYSTEM  OSCILLOSCOPE  B 220 V 220 K J T ,47 K S IOOK<  V<-[1  ( MEG  ISK^f-J,  <?  1/2 6 S L 7  9  PULSE WIDTH  PULSE oWtOTH  IE  PULSE HEIGHT HI  ^0/2  , PULSE OUTPUT  6SL7  I  4.7  K<  01 UF  100 K W  V  10 K  <!>o M A N U A L | > / CONTROL S_ FOR SINGLE P U L S E S W  4  <> >47K  ** IW  K  1  FIGURE XE PULSE MODULATOR  CIRCUIT  TRIGGER OUTPUT  FIGURE  XE  CRYSTAL Hg-ARC  POLARIZER  ANALYZER  I  FILTER  PHOTOCAVITY MAGNET  CORE  OPTICAL S Y S T E M  MULTIPLIER  FIGURE  H E  5 Hy 1500  A  hAAAH 4_ 40  UF  80  UF  AMMETER CURRENT REGULATOR  GENERATOR  CIRCUITS  Hg-ARC  ASSOCIATED  WITH  OPTICAL  SYSTEM  F I G U R E  —/)  \\\\\\|  XI2  Saturated Mercury  solution pool  Ground  glass  Crystal  seed  o o o o  CRYSTAL  cover  C y l i n d e r of f i l t e r paper containing crystals  GROWING  APPARATUS  FIGURE  VERDET  X E  CONSTANT VS. l/(T-.OI3°) CRYSTAL I  deg. cm"' 0"'  O CRYSTAL H  .08  .06 -  .04  .02  J  L  FIGURE  ROTATION  XXVIII  VS. TIME  ON S E M I L O G  SCALE  AS OBTAINED FROM •  FIGURE  O  FIGURE  3  FIGURE  100  50  100  150  msec.  FIGURE  xxx m  RELAXATION TIME VS. MAGNETIC FIELD msec.  •T=I.685°K 3T=2.060°K OT=4.22°K  150 -  100  9  •  •  3  c  (I  50 3  O  o KO  FIGURE RELAXATION T I M E  xxxiv  V S . MAGNETIC  msec.  FIELD  •  T=I.380°K  O  T= 1.542 °K  3  T-1.875 °K  200  O o  O  u  •  100 h  O  ©8  °  o 3 3  3  3  « 3  » ~  3  3  K0  FIGURE  RELAXATION T I M E VS. ON  msec.  XXXV  TEMPERATURE  LOG L O G S C A L E  o H = 1290 0 , Slope =-2.99 € H = 1540 0 , Slope = - 2 . 8 3 ® H =1790 0 , Slope = - 2 . 8 8 • H = 2 0 4 0 0 , Slope = - 3 . 6 4  200 150 100 50 -  25 -  10 -  2.5  4  K  FIGURE RELAXATION TIME ON  LOG  VS. LOG  <• H = 7 8 0 0 , ,  1.2  o  H = 1030 0,  1.4  1.6  xxxvi TEMPERATURE SCALE SLOPE = -3.48 SLOPE = -2.62  1.8  2.0 2.2  °K  POINCARE OF  REPRESENTATION  OVERSHOOT  EFFECT  


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