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Numerical studies of the nuclear hyperfine interactions in manganese dichloride tetrahydrate McLeod, Beverly Ann 1983

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NUMERICAL  STUDIES OF THE NUCLEAR HYPERFINE INTERACTIONS IN MANGANESE DICHLORIDE TETRAHYDRATE by BEVERLY ANN MCLEOD B.ScUniversity  A THESIS SUBMITTED  Of B r i t i s h  Columbia,1979  IN P A R T I A L FULFILMENT OF  THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES D e p a r t m e n t Of  We a c c e p t to  this  Physics  t h e s i s as c o n f o r m i n g  the required  standard  THE UNIVERSITY OF B R I T I S H April  ©  COLUMBIA  1983  B e v e r l y Ann M c L e o d , 1983  In p r e s e n t i n g  this  thesis  in  partial  requirements  for  an a d v a n c e d  B r i t i s h Columbia, I agree that freely that  available  or  understood that financial  gain  the L i b r a r y  f o r extensive  s c h o l a r l y p u r p o s e s may by  be  his  her  not  Physics  The U n i v e r s i t y o f B r i t i s h 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5  Date:  April  27,1983  the  make  I further  the  Columbia  agree  Head  of  representatives. of t h i s  i t  thesis for my  Itis  thesis  be a l l o w e d w i t h o u t my  permission.  Department of  by  copying or p u b l i c a t i o n shall  shall  c o p y i n g of t h i s  granted  or  of  degree a t t h e U n i v e r s i t y of  f o r r e f e r e n c e and s t u d y .  permission  Department  fulfilment  for  written  Abstract The  nuclear  (NMRON)  is a  solids,  magnetic useful  systems.  experimental  The  NMRON  work  for  in this of  group a t  5  oriented  nuclei  t h e study of magnetic  important  studies  nuclear oreintation  of  technique for  and i s e s p e c i a l l y  dilute  resonace  investigating thesis  very  complements  "Mn-MnCl^.4H 0  by t h e  2  the University  of  British  Columbia. The  manganese  interaction  but which a l s o  to the i n t e r a c t i o n of  and  The  latter interaction  the  Zeeman  The  due  moment  g r a d i e n t eq  i o n i c charges i n the l a t t i c e .  and  results  e n e r g y l e v e l s , so t h a t  frequencies for  (Zeeman)  has a s m a l l c o n t r i b u t i o n  c a n be t r e a t e d  interaction  spin  hyperfine  t o a magnetic  the e l e c t r i c f i e l d  from t h e s u r r o u n d i n g  nuclear  a  between t h e e l e c t r i c q u a d r u p o l e  the nucleus  arising  experiences  w h i c h i s due p r i m a r i l y  interaction,  eQ  nucleus  as a perturbation in shifts  on  i n the  t h e r e a r e 21 r e s o n a n t  spin I .  electric field  gradient  i scalculated  by t r e a t i n g  these  i o n s a s p o i n t c h a r g e s , a n d a c o m p u t e r p r o g r a m t o do  this  calculaton  sum,  i sdeveloped.  e q , h a s t o be m u l t i p l i e d  shielding effect  factor  The r e s u l t by  on  lattice  the Sternheimer  anti-  which takes i n t o account the amplifying  due t o t h e d i s t o r t i o n by t h e f i e l d  electrons  of t h i s  gradient  t h e atom o f i n t e r e s t : t h i s f a c t o r  for  t h e manganese i o n .  MnF,  i s investigated  As a check, t h e case  of  of the i s known 5 5  Mn  in  b e c a u s e t h e v a l u e o f |e qQ| h a s been 2  derived The  results  f o r e qQ/h a r e : 2  1 1 .7±0.03 MHz  previous  8.8 MHz  For  to  calculation[1].  exper iment  our  In  independent  f r o m e x p e r i m e n t a n d an  calculation  calculation  t h e system  the f i r s t ( I ) ,  5  1  8.5±0.4 MHz  " M n - M n C l . 4 H i O two m o d e l s a r e 2  ionic  c h a r g e s o f -e a n d .5e a r e a s s i g n e d  t h e 0 and H i o n s i n t h e water m o l e c u l e s ,  by E l S a f a r [ 2 ] . electrically because the  as  principal  of  neutral.  This  case  i s more  magnetization  does  a x i s of t h e e l e c t r i c  P/12 = e q Q / [ 8 h I (21 + 1 ) ] x 2  in  complicated and  because  not. coincide  field  with  gradient tensor.  q u a n t i t y m e a s u r e d by e x p e r i m e n t a n d c a l c u l a t e d  Here  suggested  I n t h e s e c o n d ( I I ) , t h e w a t e r s a r e assumed  o f t h e l o w e r symmetry o f t h e c r y s t a l  axis  used.  principal  electric  an a s y m m e t r y p a r a m e t e r .  The  by us i s  { 3 c o s 6 - 1 + r\ cos2<P s i n 9 } 2  2  ( 6 , <P ) a r e t h e p o l a r a n g l e s o f t h e m a g n e t i z a t i o n the  a  field  axis  g r a d i e n t f r a m e a n d f\ i s  The r e s u l t s  f o r P/12 a r e :  experiment  :-  P/12 = +0.52±0.07 MHz  o u r c a l c u l a t e d model I  :-  P/12 = +0.015±0.002 MHz  o u r c a l c u l a t e d m o d e l I I :-  P/12 = +0.21±0.02 MHz .  Both models g i v e the c o r r e c t s i g n  .  f o r P,  but  model  o b v i o u s l y g i v e s much b e t t e r q u a n t i t a t i v e a g r e e m e n t .  m o d i f i e d by u s i n g a more r e c e n t a n d p r e c i s e v a l u e o f Q  II  We  are also  able  t o c a l c u l a t e t h e magnetic  interaction  between manganese  moment  and o b t a i n  easy  atoms  for a field  of  applied parallel  dipole to the  axis:  Antiferromagnetic  regime:-  yH'B S p i n f l o p regime the easy  d i p /  V  = -0.0256 A"  3  ( w i t h moments c a n t e d a t  d i p  ^  = [0.01 l 9 c o s c x - 0 . 0 4 9 2 c o s c x  2  2  Ferromagnetic  3  regime:A'*<x\?W  = +0.0119 A -  For  a field  MnCl  .4H O c r y s t a l B-C p l a n e we o b t a i n  applied perpendicular  (with  /^'"Bcl.p^  t o t h e easy a x i s i n t h e  2  A second  2  p r o g r a m h a s been  specrtrum  the  recovery  because  spectrum  has  a  resonance  due i n p a r t  5  *Mn  the  has  distinctive  5  are by  s p i n 1=3.  t o competing  the z  system  profile  generate  "Mn-MnC^.4H O.  i t could  techniques of  to  for  modifications  Numerical  3  written  specifically  However, w i t h minor generally.  to thed i r e c t i o n  = [ 0.0137cos /3 -0 . 0492cos/? s i n / ? 2  relaxation  for the  field):  - 0 . 0 2 5 6 s i n ^ ] A"  calculate  3  moments c a n t e d a t a n g l e &  the applied  NMRON  c<~ t o  sincx  2  -0 . 0290 s i n < X ] A"  regime  angle  axis):-  /?• B  of  magnetic  dipole  be  necessary  to  spin-lattice  The t h e o r e t i c a l  f o r the  second  applied  m=2/m=l  and f o u r t h  order  iv  multipole part in  c o n t r i b u t i o n s t o t h e gamma-ray i n t e n s i t y a n d i n  to spin-lattice relaxation.  the experiment[3].  experimental relaxation  spectra time  temperature at  temperature  Rev.  Fitting  of  yields T  i  This p r o f i l e i s observed the  values  theoretical  f o r the s p i n - l a t t i c e  = 3.0±0.5 x 10"  seconds  at  a  o f T = 0.081 K a n d T«. = 2.3±0.3 x 10" s e c o n d s T = 0.064 K.  [ 1 ] H. Y a s u o k a , T i n Ngwe, a n d V. 177, 667 ( 1 9 6 9 ) .  [ 2 ] Z. (1969) .  t o the  M.  E l Saffar,  J.  Chem.  [ 3 ] A. K o t l i c k i , B. M c L e o d , M. T u r r e l l , t o be p u b l i s h e d i n Hyp. I n t .  J a c c a r i n o , Phys. Phys.  52,  S h o t t , a n d B.  4097 G.  V  Table  of Contents  Abstract i i L i s t of Tables v i i L i s t of Figures viii Acknowledgements ix I. INTRODUCTION 1 1.1 The N u c l e a r Q u a d r u p o l e I n t e r a c t i o n 3 1.1.1 The I n t e r a c t i o n H a m i l t o n i a n 3 1.1.2 Q u a d r u p o l e E n e r g y L e v e l S p l i t t i n g 6 II. POINT SOURCE CALCULATIONS OF NUCLEAR QUADRUPOLE AND MAGNETIC DIPOLE ENERGIES 9 2.1 I n t r o d u c t i o n 9 2.2 C a l c u l a t i o n Of V : F o r A L a t t i c e Of P o i n t Charges 10 A. THE GENERAL EXPRESSION 10 B. REMOVING THE CONTRIBUTION FROM THE ORIGIN 13 C. THE F I N A L EXPRESSION 14 2.3 F i n d i n g The P r i n c i p a l V ^ 15 2.4 A P r o g r a m To F i n d V,j 16 2.5 The S t e r n h e i m e r A n t i s h i e l d i n g F a c t o r 20 2.6 S t r u c t u r e Of The L a t t i c e s 20 2.6.1 MnF 20 2.6.2 M n C l .4H 0 21 2.7 P o i n t C h a r g e R e s u l t s A n d C o m p a r i s o n W i t h Experiment 24 2.7.1 55Mn-MnF 24 2.7.2 54Mn-MnClj. .4H,.0 25 a. M n C l - ^ H ^ O W i t h W a t e r M o l e c u l e s Assumed E l e c t r i c a l l y Neutral 25 b. M n C l ^ H j O W i t h C h a r g e s Q(H) = .5e And Q(0)=-e 26 c. The E x p e r i m e n t a l V a l u e Of P F o r 54MnMnCl .4H 0 27 2.8 The M a g n e t i c D i p o l e I n t e r a c t i o n E n e r g y F o r M n C l j ^ H ^ O I n D i f f e r e n t Magnetic Phases ..28 III. DESCRIPTION OF AN NMRON SPECTRUM 31 3.1 The N u c l e a r R e s o n a n c e C o n d i t i o n 31 3.2 Gamma-ray E m i s s i o n 33 3.3 P o p u l a t i o n Of The S p i n L e v e l s 34 3.3.1 The E q u i l i b r i u m P o p u l a t i o n 34 3.3.2 The P o p u l a t i o n s A t R e s o n a n c e 35 3.3.3 The P o p u l a t i o n s D u r i n g R e l a x a t i o n 35 IV. PROGRAM TO GENERATE AN NMRON SPECTRUM A P P L I E D TO 54MNMANGANESE CHLORIDE TETRAHYDRATE 37 4.1 Gamma-ray E m i s s i o n F o r 54-Mn 37 4.2 A Two T e m p e r a t u r e M o d e l 38 4.3 The R e s o n a n c e C o n d i t i o n s F o r 54-Mn I n M n C l i ^ H ^ O ..38 4.4 P o p u l a t i o n Of The N u c l e a r S p i n L e v e l s 40 4.4.1 Equilibrium 40 4.4.2 R e s o n a n c e 41 4.4.3 R e l a x a t i o n 41 4.5 W i d t h Of The R e s o s n a n c e P e a k s 42 ;  A  2  2  z  2  2  vi  4.6 The P r o g r a m 4.7 O u t p u t Of The P r o g r a m F o r 54Mn-MnCl V. SUMMARY  .4H 0  BIBLIOGRAPHY APPENDIX A - CHECK OF EQUATION 2.19 APPENDIX B - PROOF THAT THE EIGENVALUES AND EIGENVECTORS OF V.j ARE P R I N C I P A L F I E L D GRADIENTS AND P R I N C I P A L AXES APPENDIX C - PROGRAM 1 APPENDIX D - OUTPUTS OF PROGRAM 1 FOR MANGANESE FLOURIDE AND MANGANESE CHLORIDE TETRAHYDRATE APPENDIX E - PROGARM 2 APPENDIX F - SAMPLE OUTPUTS OF PROGRAM 2  42 47 54 57 58 60 61 65 69 75  vi i  List  I. II.  of T a b l e s  P4/nmn u n i t c e l l p o s i t i o n p a r a m e t e r s f o r MnCl .41-^0 2  20 21  vi i i  List  1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.  of Figures  n u c l e u s a n d an e x t e r n a l c h a r g e 3 d e f i n i t i o n of the E u l e r angles 7 c o o r d i n a t e a x e s f o r V;j o f t h e p r o g r a m 17 S t r u c t u r e of the l a t t i c e s d i s p l a y i n g the o r d e r i n g of Mn a t o m i c m a g n e t i c moments b e l o w t h e N e e l temperatures 23 the a x i s of magnetization 25 phases o f an a n t i f e r r o m a g n e t 29 d e c a y scheme f o r 54-Mn 37 m o d i f i c a t i o n o f t h e Zeeman n u c l e a r s p i n e n e r g y l e v e l s by t h e q u a d r u p o l e i n t e r a c t i o n 39 o n s e t o f n u m e r i c a l i n s t a b i l i t y w i t h d e c r e a s e i n T ...45 c o m p u t e r g e n e r a t e d s p e c t r u m f o r Mn i n M n C l . 4 H O ...48 e x p e r i m e n t a l s p e c t r u m f o r "Mn i n M n C l . 4 H O w i t h a t h e o r e t i c a l f i t drawn i n 49 t h e o r i g i n a l m=2/m=1 t r a n s i t i o n l i n e s p e c t r u m 51 t h e a d j u s t e d m=2/m=1 t r a n s i t i o n l i n e s p e c t r u m , w i t h a t h e o r e t i c a l f i t drawn i n 52 t  5<t  z  5  1  i  z  ix  Acknowledgement  I  would  like  Professor  Brian  throughout  the  particular for  to  Turrell, progress  I would l i k e  the extra  thank  for  of  this  time and a s s i s t a n c e  final  s t a g e s of w r i t i n g ,  not  h a v e met t h e d e a d l i n e . I am g r a t e f u l t o as  many  various  help  to  Mark  supplied  Shegelski  support  work.  In  appreciation  given  me  during  without which I could  fellow as  s a n d w i c h e s o r good humour a t odd thanks  supervisor, his  t o e x p r e s s my  the  contributions  my  aid  students with  hours.  for  FMT, t o Special  f o r encouragement and  generously at c r i t i c a l  moments.  1  I. This  thesis  programs t o study the system  5  INTRODUCTION  develops  and  the nuclear  applies  hyperfine  two  computer  interactions  in  "Mn-MnCl^.4H O. z  i) Chapter field  II describes the calculation  g r a d i e n t and t h e magnetic d i p o l e f i e l d  solids.  Because  these  lattice  techniques, implement The Mn s i t e point  of  sums  and  tackled  electric  charge  field  gradient tensor  a n d f o r t h e «-form  z  models  with  obtained  obtained  quadrupole  f o r the Sternheimer compared  i n an  NMR  with  and t h e  on MnF  on " M n - M n C l ^ 4 H 0 r e c e n t l y  University  of B r i t i s h Columbia.  5  Z  magnetic  values  result  at the serves as  calculations.  dipole  fields  antiferromagnetic, the spin phases of ordered  2  The  a n d i n an NMRON  performed  The MnF  i n the  factor.  the experimental  experiment  f o r the numerical  from  interaction  case  anti-shielding  experiment  The  z  i n t h e l a t t e r , assuming a value given  r e s u l t s a r e then  using  MnCl .4H2,0  i n t h e former  5 5  literature  of  studies.  f o r the Mn nucleus  nucleus  a check  numerical  i scalculated at a  ion positions  and X-ray d i f f r a c t i o n  found  "Mn  by  FORTRAN p r o g r a m h a s been d e v e l o p e d t o  From t h e s e v a l u e s t h e e l e c t r i c  5  magnetic  t h e l a r g e number o f atoms i n v o l v e d , a r e best  a  in  them.  f o r MnF  neutron  is  of the e l e c t r i c  MNCl^^H^O.  flop  arecalculated and  the  for the  ferromagnetic  2  ii) Chapter intensity  I I I reviews the c a l c u l a t i o n  e m i t t e d f r o m an e n s e m b l e o f  o r e i n t e d by a h y p e r f i n e  radioactive  i n t e r a c t i o n which  magnetic  when t h e s y s t e m h a s been f i r s t  allowed  to  case  recover  i n which  interaction a  small  analytic change  by s p i n - l a t t i c e  t h e system  which  i s predominately  r e s o n a t e d and then  relaxation.  i s ordered  quadrupole  solution exists  emitted  nuclei  by  a  for the hyperfine  i s predominately magnetic, but which has  electric  of  o f t h e gamma-ray  the level  intensity)  contribution.  f o r I>3/2, populations  during  Since  no  rate  of  t h e time (which  relaxation  determine the  must  be  solved  numer i c a l l y . In  chapter  spectrum chapter 5  IV a program i s developed t o generate a  according  t o the theoretical  I I I . The p r o g r a m i s w r i t t e n  description  specifically  f o r the  "Mn-MnCl . 4H2_0 s y s t e m b u t w i t h s i m p l e m o d i f i c a t i o n s z  be a p p l i e d  profile  i s found  l i n e due t o c o m p e t i n g  fourth order intensity  could  generally.  A characteristic resonance  of  multipole  and t o  definitively  identifies  e f f e c t s of the second and  contributions  spin-lattice  f o r t h e m=2/m=1  t o t h e gamma-ray  relaxation.  This p r o f i l e  t h e e x p e r i m e n t a l m=2/m=1  resonance  line. Fitting allow  of the t h e o r e t i c a l and e x p e r i m e n t a l  the determination  c o n t r i b u t i o n s t o the  of  hyperfine  t h e Zeeman interaction  spectra  and quadrupole and of t h e  3  spin-lattice  The  quadrupole  the problems 1.1 1.1.1  The  relaxation  time.  i n t e r a c t i o n energy  tabled  and  i s discussed  Nuclear Quadrupole  The  due  i n the next  section.  Interaction  Interaction Hamiltonian  Consider a nucleus with p(x*)  i s common t o b o t h  centered to external  Figure The  total  due  to e l e c t r i c  about charges  a  charge  t h e o r i g i n i n an e l e c t r i c as p i c t u r e d  1 - n u c l e u s and  electric  localized  in figure  an e x t e r n a l  p o t e n t i a l at a point  charges q ( x ) k  external  density potential  1.  charge  x in  the  nucleus  t o the n u c l e u s  will  be  (1.1) k  Let  the e x t e r n a l  charges  be due  t o a l l the ions  in a  4  lattice nucleus  e x c l u d i n g t h e e l e c t r o n s o f t h e atom t o belongs.  Noting  much g r e a t e r t h a n (1.1)  the  which  the  that interatomic distances are  diameter  of  a  nucleus,  equation  c a n be e x p a n d e d i n a T a y l o r s e r i e s a b o u t t h e o r i g i n . 7=3  (1.2)  This  permits a m u l t i p o l e expansion  energy  f o r the c h a r a c t e r i s t i c  of t h e system.  £ ~J V o n p m dV - V(si  p(ir  ) d'x  Nuclear  (1.3)  Let due  drw  5. g  by V j .  be d e n o t e d  c  ;  t o e x t r a - n u c l e a r charges  Since  the  s a t i s f i e s Laplace's  potential equation  at the o r i g i n  Z V V- - o J  &  ( 1.4) the  third  quadrupole  term energy, E ,  in  the  expansion  (1.5),  called  the  E , c a n be w r i t t e n Q  - fc E  V  :>  Q  l  i  (1.5)  where  (1.6) is called  the quadrupole  moment  tensor  5  In  quantum  mechanics  E  Q  and Q  li  e x p e c t a t i o n v a l u e s o f quantum m e c h a n i c a l  correspond  t o the  operators.  These  e x p e c t a t i o n v a l u e s d e p e n d upon t h e q u a n t u m nucleus  which  projection  For a given s p i n ,  of the  by i t s s p i n , T,  i s characterized  of the spin along  state  some a x i s , I  and t h e  .  2  t h e expectation value of the  component o f t h e q u a d r u p o l e  { i K l Q ' M l I z ) = <Hzl  ij-th  tensor i s  " S^lSfD/otf) d'x III*')  JfsVxi  (1.7) Using  the Wigner-Eckart  t h e o r e m , t h i s c a n be w r i t t e n [ 1 0 ]  as  <ii iQ Mii;> f  2  <nj(3ri *i i)-s " r|tii> i  * f e r = y  ,  i  i  (1.8) where  Q  • < lI | e L (  - r * ) S(*x\) | T I> k  (1.9)  k  Thus t h e H a m i l t o n i a n  u  i ^ i  By  choosing  x  1 =  f o rthe quadrupole  interaction i s  d.io)  t.j-,.  X , x =Y, x =Z 2  3  t o be  the principal  axes  d e f i n e d by V.. and  using  quadrupole  Laplace's Hamiltonian  H.'SftiVi) where  = 0 fori * j  L  ^  equation  (1.4),  (1.11) the e l e c t r o s t a t i c  for a spin I nucleus  "  ^ -  reduces t o  nd/-^)]  (  ,.  12)  6  eQ  =  V  r\  and  2 z  =  (V  -  x x  V  )/\  yy  ( 1 . 1 3 )  .  x  By c o n v e n t i o n  t h e p r i n c i p a l a x e s a r e l a b e l l e d X, Y  in accordance  with  and  l xx I - ' vy I - l z x I v  The H a m i l t o n i a n  H  where and  i s o f t e n expressed  i n the convenient  14  form  a  a n d I_ a r e t h e s t a n d a r d  lowering  1  * i. )]  1  I  < - >  v  \ [3i;-I -  -i  q  v  Z  (1.15)  a n g u l a r momentum r a i s i n g  operators I =I +iI +  x  y  ,  I_=I -iI y  (1.16)  y  and  ^ 1.1.2  21(21 + 0  Quadrupole Energy L e v e l To d e t e r m i n e  it  axis  Splitting  t h e e n e r g y o f t h e m a g n e t i c s u b s t a t e s , m,  f o r which these  of m a g n e t i z a t i o n ,  azimuthal  angle  interaction axes  can  be  respect  rotation  to  0 and  quadrupole  a x e s X, Y, a n d Z, t h e m a g n e t i z a t i o n  o b t a i n e d ' from  the  principal  axes  [R(06<P)]~ ,  where  = R («)  R (^) R (y)  through  the Euler angles  z  the  of  F o r an  z, which l i e s a t p o l a r a n g l e  of the i n v e r s e r o t a t i o n  R(«pV)  a  states are eigenvectors.  with  principal  application  2.  ( 1 > 1 7 )  i s d e s i r a b l e t o w r i t e the H a m i l t o n i a n , H , i n terms  operators  is  21 ( 2 1 - l)  u  1  by  (1.18)  z  (a,^,^)  of f i g u r e  7  Figure  Applying  2 - definition  of t h e E u l e r  angles  the i d e n t i t i e s [7]  (1.19) and  (1.20) to e x p r e s s i o n  (1.1 ) f o r t h e q u a d r u p o l a r  Hamiltonian  gives  H i? {i(3c« e-/)(3i -i ) + IA A~ b - i B ^ e ^ e B  x  0  l  Q  x  [we ( 3 1 / - r )  + intend  < 1 , 2 1 )  i  l  + A  -B^ec^ejj  where  A - 1/ - I-  2  B •  and I , I T  operators  * d**  = I +il> , I with  (1  = I -il„ , a r e now a n g u l a r  respect  to  an  orthogonal  .  22)  momentum Cartesian  8  c o o r d i n a t e system  i n which  the z - a x i s  lies  a l o n g the  axis  of m a g n e t i z a t i o n . The  quadrupole  in the s t a t e I  E (ml Q  ( 1  '  = mh  i s now  • (Im|rl = 4?  2 3 )  Q  [ 3  readily  found:  llm>  Q  j w  e  -  I  - rj c«  * [ 3 ^ - 1(1 + Defining ?  of a s p i n T n u c l e u s  i n t e r a c t i o n energy  0]  2<P .  P by  *  ( 3 cos  [1  1  6  - 1)  +  |  cos  sin* e  ] (1.24)  the  quadrupole  levels  energy  difference  between a d j a c e n t  spin  i s obtained:  E (ml Q  - E  Q  (ir»-1) = P Cm- ± ) , (  1.25)  9  II.  POINT SOURCE CALCULATIONS OF NUCLEAR QUADRUPOLE  AND  MAGNETIC DIPOLE ENERGIES 2.1  Introduct ion To  levels  find of  the s p l i t t i n g equation  of  the  (1.25),  the  d e r i v a t i v e s of the e l e c t r o s t a t i c known.  A  nuclear second  charges.  solid The  calculations positions  to  be  a  positions are  of  taken  determined  the  to  by  spatial  ,  must  be  i s to consider  the  (  lattice  energy  order  p o t e n t i a l , V.  s i m p l e and o b v i o u s a p p r o a c h  crystalline  spin  of  classical  point  point  charges  i n the  be  the  time-averaged  X-ray  and  neutron  ion  diffraction  experiments. A t t h e t i m e t h i s work was the  magnetic  interaction only  one  nuclear III. P  Q  ,  en-ergy  levels  undertaken, due  to  the  had been m e a s u r e d i n MnF , b u t resonance  line  had  been  which  t e c h n i q u e t o be d i s c u s s e d  determines  interaction,  was  the  i n MnF , i t was 2  2  z  using in  f o r the  magnitude  seen t o a g r e e  MnCl .4H 0  observed  S i n c e the p o i n t charge c a l c u l a t i o n  of  quadrupolar  in  2  orientation  experiment  splitting  the  chapter parameter  of t h e q u a d r u p o l a r  i n o r d e r of m a g n i t u d e  with  hoped t h a t t h i s a p p r o a c h  would  a l s o g i v e a good e s t i m a t e o f P f o r M n C l . 4 H 0 . 2  It obtain  t u r n s out t h a t  the  quantities  a  V-  required  P c a n a l s o be u s e d t o e s t i m a t e t h e m a g n e t i c  i n t e r a c t i o n due different  t o t h e s u b l a t t i c e s of m a g n e t i c  p h a s e s of a c r y s t a l .  The  dipolar  to  dipole  ions i n the  field  acting  10  at  the  nuclear  hyperfine effect  field  site and  of a p p l i e d  is  a  contribution  needs t o be  f i e l d s on  known  to  in  the  total  analyzing  the n u c l e a r magnetic  the  resonance  f requency. We  will  due  i n t r o d u c e a p r a c t i c a l method f o r c a l c u l a t i n g  t o a l a t t i c e of  reader  of  how  the  values  of  V-  thus  implement  the  above  presented  and  applied  MnClj^H^O. are used  The  to  find  V-  obtained.  the  The  A  to  the  found  FORTRAN  two  spatial derivatives  the  from  the  program  to  i n A p p e n d i x C,  lattices  is  MnF^  and  calculations  F i n a l l y , the program  interaction  a  energies  is  for  the  2  F o r A L a t t i c e Of  r  Point  Charges  Expression  the quadrupole  the H a m i l t o n i a n  needed.  be  remind  MnCl .4H 0.  General  evaluate  may  and  , which i s l i s t e d  dipole  Of V  Calculation  To  are  principal  then compared t o e x p e r i m e n t .  A.  to  charges,  r e s u l t s of t h e p o i n t c h a r g e  m a g n e t i c p h a s e s of 2.2  point  interaction  energies  ( 1 . 1 5 ) , the v a l u e s of the second of t h e e l e c t r o s t a t i c  For a l a t t i c e  due order  potential,  of p o i n t c h a r g e s ,  V-- ,  i s given  by  (2.1 ) A lattice by  translation  can of  l a t t i c e v e c t o r s 1.  be  genetrated the  Let  from the  primitive u  be  or  primitive unit  the p o s i t i o n  cell  cell by  coordinates  the of  11  the  point  charges q(if) i n the l a t t i c e  may  be w r i t t e n w  .  unit c e l l .  Then V/j  y  (2.2) This  sum c o n v e r g e s s l o w l y  best  marginally  alternate  practicable  expression  convergence integral  d i r e c t summation  due  can  (2.2) w i t h  be  and F o u r i e r  is  to roundoff e r r o r .  f o r equation  properties  identity  The  so t h a t  obtained  much by  use  at An  better of an  analysis.  identity i s  iyi  " ~  nTtt nTtT ]  e  dp (2.3)  Differentiation  of t h i s  yields  (2.4) Notice  that  decreases values  right  rapidly with  look  Substitution  w  the  y  of  increasing values  equation  (2.4)  o f |5f| f o r l a r g e  in  Fourier  space.  of (2.4) i n t o (2.2) g i v e s  0  L  periodicity  II  * where  side  a t t h e expansion of V,--  expanded i n a F o u r i e r 2  hand  of p .  Now  By  the  ^  J  of  the l a t t i c e  the integrand  (2.5) c a n be  series  - l ^ j r t l V  _ y  cU.u  *  *  (2.6)  12  €  F  "  G  =  J [ 3x'^  L  Vl  (2.7)  1  e  Je  - i  k •u  ._  du  (2.8) Here V  i s t h e volume  L  Making t h e change (2.7)  of  the  lattice,  of v a r i a b l e s  V  =  L  8L'L L . 2  u -> 1+u a n d u s i n g  3  equation  t h i s becomes dx 1  - LtJ  i75 (2.9)  Using the following (r  -t.  U) ;k)  nl  p r o p e r t y of F o u r i e r =  (-,•*)" K I  transforms  ( £(x) J  k)  (2.10) gives  7^ e  (2.11)  <« -IT? Since  this  limits  o f i n t e g r a t i o n by ±oo  r-  ~  integral  . •  Ml  H  converges  rapidly,  replacing  the  produces  - |Je*l (2.12)  where N volume equation  i s t h e number o f u n i t c e l l s V . L  Substitution  of  lattice  (2.12) back  of into  (2.6) y i e l d s e  k  where v  the  equation  \frr  the  in  = V,./N  right  L  hand  i s the unit c e l l side  of  (2.13)  je  p» P 3  volume. falls  (2.13)  Noticing  that  off rapidly  with  13  increasing  k  equation  small  (2.4)  intermediate integral 'real'  for  into  limit  in  for  Together  splits parts  integration,  large  ^  one  two  space  for  values  of  equations  c~  J  of  Fourier  space  p,  (2.2),  e  k If; ;  the  by  G,  the  and  small  integral use  of  of  an  evaluates  values  of  the  p and  in  p.  (2.4),  and  (2.13)  give  dp  J  (2.14) 9v'3v  V Evaluation  v  ,-  * £  of  ^  u  )  the  (  J  J e  ImT  integrals  ^  V  yields  -  ^  e  e  /X()?r) (?r) t  -4where  dp J  ;  y  + PlVrn  D er~fc ( & I 7 x l )  (  2  *  1  5  )  ]J  N  (2.16)  B.  The is  still  Removing The C o n t r i b u t i o n From The  divergent contained  remedied  by  equation  (2.2)  integration in  contribution  a Taylor  in  subtracting where G. series  the  Writing about  due  equation |xP  to  =  (2.16).  from t h e  integral  T  right  now has  one  gets  1+0  = 0  This  may  hand s i d e  lower  b=rp and e x p a n d i n g b=0,  Origin  the  limit  be of of  integrand  1 4  Doing t h e i n t e g r a l s  Now  apply  partial  f o r each term i n t h e e x p a n s i o n  d e r i v a t i v e s with respect  to x  L  gives  and x t o J  obtain a^T^  l^nrj  dp  v 3  r ;  (2.17)  In t h e l i m i t  as r goes t o z e r o t h i s  r —» o  Ca  C.  Expression  the f i n a l  partial origin  ( 2 . 1 8 )  The F i n a l  Thus  expression  f o r t h e second  d e r i v a t i v e s of t h e e l e c t r o s t a t i c due t o a l a t t i c e  charge a t t h eo r i g i n V;,  becomes  -II  V ^ j  4=0 {  I  where e r f c  6*  W  charges  excluding  any  itself, i s  ^  iff*  t\  *  e  e  d,  /• _ ,  •i \  r  ~ll  (2.19)  C 'J  i s t h e complementary  erf C  This  point  at the  L  -Tv>3  If  of  potential  order  2  ( \j )  P°°  = jjp^j  expression,  -x e  e r r o r f u n c t i o n g i v e n by 1  dx  •  b u t w i t h t h e k=0 t e r m m i s s i n g , was  15  first  derived in  (2.19)  for  a  1924  [5].  simple  cubic  term must  be  presented  in Appendix A .  2.3  Finding  of  the  values  second  principal  respect  to  axes  eigenvalues,  I  equation  confirms that  algebraic  the  eigenvalues  ^  t h e "k*=0  verification  is  -  straight of  the  V••  the  be  solves  to  electric  the the  to  one  find  the  potential  coordinates  derivatives  to  respect  forward  define  principal  = O  ) unit  into x  known w i t h  are  with  taken.  components  secular  and  of  equation  a for  ,  k  3x3  back ( v  are  is  V , one  ( V- X I  is  it  the  matrix,  def  V-  which  which the  symmetric  (here  of  derivatives  Considering  its  This  Cartesian axes,  principal the  present.  lattice  checking  The P r i n c i p a l V i i  Once set  However,  i  matrix)  the  )  k-i.i,  £  then  original  k  -  S  (2.20)  )  substitutes  eigenvalue  these  equation  0  (2.21) and  solves  each  7i ' •.  for If  k  procedure  three  text  is  to  eigenvalues  slighty  of  the  the  three  different,  potential,  k  corresponding not  can be  V  eigenvectors,  x x  , V  y y  E , (  f o u n d i n any  shown t h a t  the  z  axes X, Y,and  the  principal  , and V E,  the  independent  but can be  are  to  distinct,  linearly  It  \  principal  E  are  on l i n e a r a l g e b r a .  corresponding  parallel  eigenvectors  finding  eigenvalues  derivatives the  the  for  eigenvectors standard  the  Z 2  and Z.  , and E  3  that  ,  A proof  are is  16  presented  i n A p p e n d i x B.  2.4 A P r o g r a m To F i n d V j :  Writing lattice  the  lattice  vectors,  I,  v e c t o r s , F, i n t e r m s o f t h e i r  J " f  = n,A + n j  K  IT - k n  and  basis  reciprocal  vectors,  •+ n C  (2.22)  3  « n.7, - n i  t  - n,K  (2.23)  3  where  Tf. - w ?i j<  = ^ ( ' = UL ( J , ? ) C  5  v  the  ( 3 - ct)  summations  V-  i n equation  is  (2.24)  )  cEa  e x p r e s s e d a s summations Since  y  real,  (2.19) o v e r 1 a n d "k* c a n be r e -  over t h e e'  integers  c a n be r e p l a c e d  Making these s u b s t i t u t i o n s i n equation final  expression  to evaluate w  _  n, ,  n , 2  by c o s ( k . u ) .  (2.19)  gives  the  V j: ;  dj:  v  L  j~?u+o  114  "  (2.25)  1  -f-  D  (iff  •+ u  )  The p r o g r a m was w r i t t e n t o accommodate l a t t i c e s most  3  u s e d i n t h e FORTRAN p r o g r a m o f A p p e n d i x C  n,,«i.,rx*  at  n .  one  of  the  angles  between  the l a t t i c e  where basis  17  v e c t o r s A, *B, ~C, i s not to  (specifically,  n i n e t y degrees.  permit  general  appropriate replacing  angles  The  lattices,  by  between the  lattice  b e t w e e n A and  be e a s i l y  reading  modified  in  three  basis vectors  vector expressions  by  and their  r e c t a n g u l a r C a r t e s i a n axes w i t h r e s p e c t t o which are obtained are  those  such  that  a l o n g t h e A * - a x i s , y l i e s a l o n g t h e B - a x i s and  along  C)  forms.  t h e d e r i v a t i v e s v.lies  However, i t can  t h e c o o r d i n a t e and  most g e n e r a l  the angle  the  C-axis.  The  A*-axis  is  z  d i r e c t e d along  x  lies the  v e c t o r BxC*. z t  Figure An  3 - c o o r d i n a t e axes f o r V j ;  o u t l i n e of the program  i) First summation f r o m M-1  (2.25).  program  follows.  N i s r e a d , where N-1  in equation  of the  i s the upper  N o t e t h a t an  limit  of  increase in N  t o M r e q u i r e s t h a t a f u r t h u r (12(m-1)+2) x C(N=1)  c o m p u t a t u i o n s be p e r f o r m e d , where C ( 1 ) computations  required  for  N=1.  i s the  number  In a l l l a t t i c e s  of  so f a r  18  t e s t e d , N=4 ii)  given convergence to the  Next the l a t t i c e  positions ions  has  in lattice  in  unit  position  The  six  to  lattice  the  charges  through  unit  cell  A total  of  coordinates  rectangular translated  35 are  Cartesian so t h a t of  the the  partial  are then c a l c u l a t e d  equation  by  this  d e r i v a t i v e s of  the  at t h i s o r i g i n unit  cell  ( 2 . 2 5 ) , t o an a c c u r a c y  due  of p o i n t determined  N. the p r i n c i p a l  t h e method o f s e c t i o n To  finite  minimize  error,  i n c r e a s e s w i t h N,  accuracy  t o which  the  with  u  the summation o v e r  present  found  2.3.  roundoff  over  are  ;  the computer, double  summation  inside  derivatives V j  number o f b i n a r y d i g i t s  r e p r e s e n t e d by  is  indpendent  generated  iv) F i n a l l y  the  and  c o o r d i n a t e r e a d becomes t h e o r i g i n  potential  the  the  d e f i n e d a b o v e and  electric  by  of  The  figure.  cell. iii)  by  i s permitted.  terms  c o o r d i n a t e system first  b a s i s parameters  coordinates are read.  i n the u n i t c e l l  re-expressed  fourth  e r r o r due  which  the  number  is  i s used,  and  a  precision  in equation  to  ( 2 . 2 5 ) i s done  n.  The  amount o f r o u n d o f f  and  may  be d e t e c t e d by  error  checking  Laplace's equation, equation  (1.4),  satisfied. Note  that  intermediate integral evaluated  G  in  limit  i d e n t i t y of in  Fourier  equation of  (2.25)  integration  equation space  (2.4) and  is which  just  the  splits  the  into  parts  in real  space.  to  be  Thus G  19  s h o u l d be c h o s e n or  Fourier  same  t h e two sums, one  s p a c e a n d one i n l a t t i c e  rate.  convergence  so t h a t  The  value  of  for lattices  in  space, converge  the  MnF  of c u b i c  was done by d i r e c t  2  program r e s u l t .  for  lattices  symmetry.  was  found  Laplace's  to  give  equation  expected round-off If point  location  will  ensure  while the  be  results.  i t  charge  will  Finally,  a  non-lattice  the  unit  of z e r o .  cell,  The z e r o  of  the  and  charge  to  be t a k e n a s t h e o r i g i n  a r e t o be  V- , ;  at which  determined.  proton  charge.  however, the c h o i c e of u n i t s  a c h a n g e of f o r m a t c o d e may user.  of G  be f o u n d by r e a d i n g i n t h i s  coordinate in a  at  c o n v e n i e n c e , o u t p u t f o r m a t s have been  in multiples  format,  the  of  chosen  c o m p a t i b l e w i t h l e n g t h s g i v e n i n angstroms and  given  zero  Thirdly, variation  gradients  t h e y may  i t s position  For  t o be  t h a t t h e p o i n t makes no c o n t r i b u t i o n  d e r i v a t i v e s V-  V..  f o u n d t o be o b e y e d t o w i t h i n t h e  field  desired,  with  First  error.  as t h e f i r s t  associating  change  was  the e l e c t r i c are  was c h e c k e d  symmetry.  no  .  summation and f o u n d t o m a t c h  S e c o n d l y , V,-:  of c u b i c  at the  G h a s been s e t t o g i v e e q u a l  The p r o g r a m was t e s t e d by f o u r methods for  reciprocal  be l e f t  to  charges  Aside  from  i s a r b i t r a r y , and by  to the  discretion  of  20  2.5 The S t e r n h e i m e r The  value  point charge which  of  V-  a t a n u c l e u s due t o a l a t t i c e o f  ;  nucleus r e s i d e s .  interactions  electrons  about  can  important  Inter-ionic distort  t h e n u c l e u s from  s i n c e t h e y a r e so c l o s e , an  Factor  " i o n s " n e g l e c t s t h e e l e c t r o n s o f t h e atom  the  electron  Antishieldinq  contribution  to  and i n t r a - a t o m i c  the  closed  spherical  these c l o s e d  shell  V- .  The  {  in  shell  symmetry, and e l e c t r o n s make actual  field  g r a d i e n t a t t h e n u c l e u s i s g i v e n by [ 1 2 ] ( v 7  .  I . , = <  22 Actual  where ^ It  i s called  can  be  1 _  *~  ) v  77  (2.26)  11  the Sternheimer  calculated  through  anti-shielding  perturbation  factor.  t h e o r y , and  t u r n s o u t i n g e n e r a l t o be a n e g a t i v e number. The  16 „  v a l u e of  c a l c u l a t i o n s which  used  follow-is  f o r the taken  Mn * 2  ion  i n the  from r e f e r e n c e [ 1 5 ] .  2.6 S t r u c t u r e Of The L a t t i c e s 2.6.1  MnFj_ MnF  space  group  lengths cell  i s an  z  A  orthorhombic  P4/nmn =  parameter,  B  .  The  crystal  lattice  = 4.865 A,  C  d e s c r i b e d by t h e  basis  vectors  = 3.284 A, a n d t h e u n i t  U , o f T a b l e I h a s t h e v a l u e U = .31 X/A  Y/B  have  Z/C  Mn ( 0 0 0 Mn ( .5 .5 .5 F ±( U U 0 F ±(U+.5 .5-U .5 T a b l e I - P4/nmn u n i t  ) ) ) ) cell  [13].  21  2.6.2 M n C l  .4H 0  The  f o r m o f manganous c h l o r i d e  monoclinic  c r y s t a l w i t h space group  has b a s i s v e c t o r l e n g t h s and  C  =  99.74°. located  A  tetrahydrate P2,/n.  = 11.186 A,  is  The u n i t  B  cell  9.523  A,  6.186 A, and t h e a n g l e b e t w e e n A and B i s  There  are four u n i t s of M n C l . 4 H 0 per u n i t 2  i n terms  of l a t t i c e  2  Z/C)  = (x,y,z;  .5+x,.5-y,.5+z) (See F i g u r e 4 ) .  positons  f o r t h e Mn, C I , a n d 0 atoms a r e t a k e n f r o m an X-  ray d i f f r a c t i o n  study [ 1 6 ] , w h i l e the  come f r o m n e u t r o n d i f f r a c t i o n  Mn 0. 2329 C M 1 ) 0. 061 0 Cl(2) 0. 3817 0. 3010 0(1 ) 0(2) 0. 1 568 0(3) 0. 1323 0(4) 0. 3695 H( 1 ) 1 0. 3831 H( 1 )2 0. 3005 H ( 2 ) 1 0. 0728 H(2)2 0. 1989 H ( 3 ) 10. 1 105 H(3)2 0. 0996 H ( 4 ) 1 0. 4356 H(4)2 0. 361 6  y/B  z/C  0. 1714 0. 3076 0. 3662 0. 1 1 27 0. 2280 0. 9736 0. 0381 0. 1418 0. 01 54 0. 2000 0. 1 967 0. 9277 0. 9292 0. 0716 0. 9389  0. 9865 0. 0938 0. 0355 0. 3334 0. 6446 0. 9590 0. 8764 0. 3882 0. 3657 0. 5993 0. 5288 0. 8199 0. 0681 0. 8061 0. 8540  Table I I - p o s i t i o n parameters The  experiment  interaction  was c a r r i e d  less  1  than  time-averaged  K, room  z  positions  f o r M n C l .4H 0  to  investigate  out  at  whereas  y,  analysis [2].  x/A  ATOM  hydrogen  x,  =  cell,  c o o r d i n a t e s a t (X/A, Y/B, The  a  crystal  the  quadrupole  temperatures  of  the data given i n Table I I are  temperature  positions.  However,  an  22  order-of-magnitude  analysis  expansion  coefficients  positions  due  changes i n the the  to  indicates  the  that  the  t h i s temperature d i f f e r e n c e  calculated  experimental  using  errors.  P v a l u e s by  appropriate shift will  amounts s m a l l e r  in cause than  23  Figure 4 - S t r u c t u r e of t h e l a t t i c e s d i s p l a y i n g t h e o r d e r i n g of Mn a t o m i c m a g n e t i c moments b e l o w t h e N e e l temperatures  24  2.7 P o i n t C h a r g e R e s u l t s And C o m p a r i s o n W i t h 2.7.1  Experiment  SSMn-MnF^ The  f o r MnF  2  principal  Using  output  of  i slisted  i n A p p e n d i x D, a n d g i v e s v a l u e s o f t h e  electric  these  t h e p r o g r a m d e s c r i b e d i n s e c t i o n 2.4  field V  /e = - 0 . 179 A '  3  x x  V  /e = -0.499 A '  3  y y  V  7 I  /e = +0.678 A"  (2.27)  3  values  axis directions  d e r i v a t i v e s of  i n equation  (see f i g u r e  (2.21) y i e l d s  principal  4) .  (2.28).  Z  H e r e , a n d i n t h e r e s t o f s e c t i o n 2.7, A, B, C, w i l l unit  vectors along  consideration.  5 5  t h e b a s i s v e c t o r s of t h e l a t t i c e Mn  has  spin  moment Q = 0.4±0 . 0 2 x 1 0 * c m 2  ant i s h i e l d i n g  factor  enhancement o f V  JZ  which the nucleus  2  I = 5/2 a n d a  [3].  ( 1 - Xo. )  =  Taking 9  Mn  i n MnF  Sternheimer  ion in  resides, gives (2.29)  Yasuoka e t a l .  [ 1 5 ] measure  | e q Q / h | = 11.7±0.03 MHz 2  and  quadrupole  due t o t h e e l e c t r o n s o f t h e Mn  2  5 5  under  [15] t o account f o r  e q Q / h = 8.5±0.5 MHz For  a  denote  (2.30)  calculate e q Q / h = 8.8 MHz 2  (where t h e c a l c u l a t i o n h a s b e e n m o d i f i e d by u s i n g t h e more recent value of Q given above),  i n good a g r e e m e n t w i t h o u r  25  est imate. 2.7.2  54Mn-MnCl,,4H 0 a  The polar  axis  angles  of  magnetization  6 = 7.6°  f o r MnCl^^H^O  l i e s at  and <t> = 3° i n t h e A*-B-C f r a m e [ 1 ] .  c  Figure  5 - t h e a x i s of m a g n e t i z a t i o n  In terms of u n i t v e c t o r s a l o n g crystal,  the  unit  magnetization  vector  c a n be  t h e A*, B a n d C a x e s o f t h e  lying  along  the  axis  written  m = 0.13 A* + 0.0069 B + 0.99 C a.  a  If  the water molecules  the e l e c t r i c  and  Assumed  Neutral  i n t h e p o i n t c h a r g e sum, t h e n  listing  (2.31)  MnCl .4H,0 W i t h Water M o l e c u l e s  Electrically  of  i n MnCl^^H^O  are disregarded  the p r i n c i p a l  p o t e n t i a l obtained  from  the  d e r i v a t i v e s of program  output  i n Appendix D are  from  V  xy  V  yy  V  z z  equation  /e = -0.0162 A " /e = -0.112 A " /e = +0.128 (2.21)  are  3  (2.32)  3  A"  3  found  to  correspond  to  26  principal  axis  directions  X = - 0 . 1 5 2 A + 0.882 B + 0.445 C Y = - 0 . 9 3 8 A * - 0.271 Z = Using  1= 3, a n d t h e e l e c t r i c  Q = 0.4±0.04x10 Q  cm  2 U  2  = 0.53±0.12  Substitution (1 . 2 4 )  of  for  5  "Mn [ 9 ] ,  MHz, and  (2.31),  Yo»  n  been  s  finds (2.34)  and (2.34)  into  is  (2.35)  due t o Q, and u n c e r t a i n t y due  It  is  not  ignored charge  respect  clear,  however,  that  electrostatically. electrostatic  good  agreement  assigned  the  respectively. and  Taking  oxygen  derivatives  t o t h e oxygen  atoms  t h e above  in M n C l . 4 H O a  z  of  the  cell  with  found  [2]  i n the u n i t has been H  atom p o s i t i o n s  d e t e r m i n e d by n e u t r o n d i f f r a c t i o n when c h a r g e s were  molecules  Minimizaton  energy  with  And Q(Q)=-e  the water  t o t h e h y d r o g e n atom p o s i t i o n s  give  to  neglected.  M n C l , . 4 H , 0 W i t h C h a r g e s Q(H)=.5e  point  .5e  equation  MHz .  b.  may be  to  one  moment,  yields  the u n c e r t a i n t y a  quadrupole  fl = 0.746  (2.33)  P = 2.3±0.2 where  (2.33)  0.312 A * - 0. 385 B + 0 . 8 6 9 C .  the s p i n ,  P  B + 0.217 C  atoms  charges  of  -e  and h y d r o g e n for  as and  atoms  the  hydrogen  gives p r i n c i p a l  potential  ( s e e A p p e n d i x D) V  and p r i n c i p a l  axes  ™ /e  = +0.003 A "  3  V  y y  /e  = +0.225 A "  V  z z  /e  = -0.258  k-  3  3  (2.36)  27  X =  0.056 A* + 0.681 B - 0.730 C  Y =' -0.242 A* + 0.718 B + 0.652 C Z =  The  equations  (1.13) and (1.17) and t h e v a l u e s  yield  P  q  From  0 . 9 6 9 A* + 0.140 B + 0.205 C .  definitons,  (2.35)  (2.37)  i\ = 0 . 9 8  = -0.1110.01 MHz, a n d  (2.31),  (2.37),  (2.38)  ( 2 . 3 8 ) , and ( 1 . 2 4 ) we  find  P = +0.18±0.02 MHz . c.  The E x p e r i m e n t a l  The  experimental  i s a b o u t 35 t i m e s the and  Of P F o r 54Mn-MnCl .4H 0  v a l u e o f P i s 6.2±0.8 MHz [ 6 ] .  the p o i n t charge  result  This  obtained  h y d r o g e n a n d o x y g e n atoms a r e a s s i g n e d  charges  when o f .5e  -e r e s p e c t i v e l y . The  that  discrepancy  the  Indeed  the  implicit  point  water m o l e c u l e s  of e r r o r  gave  experimentally  unknown.  in  charge  values  value.  may  indicate  t o n o n - i o n i c bonding  this result  model  i s too  f o r the case  Note t h a t  t h e c o r r e c t s i g n f o r P.  of t h e great.  where t h e  i s reasonably both  point  close charge  An u n c e r t a i n amount  h a s been i n t r o d u c e d i n c o n v e r t i n g t h e r e s u l t s f o r  principal  accuracy  these  are neglected e n t i r e l y  to the experimental models  between  p r o p o r t i o n of i o n i c  water m o l e c u l e s  the  Value  (2.39)  of  potential  derivatives,  determined the  theoretical  parameter,  V-- , P,  antishielding  to  the  because  the  factor  is  28  2.8 The M a g n e t i c D i p o l e In D i f f e r e n t M a g n e t i c Although calculation magnetic  the  I n t e r a c t i o n E n e r g y F o r MnCl,.4H^0  Phases  prime  objective  of t h e e l e c t r i c  dipole  quadrupole  interaction  the program d e s c r i b e d  o f t h i s work was t h e interaction,  c a n a l s o be e s t i m a t e d  energy  due  two m a g n e t i c s u b l a t t i c e s w i t h m a g n e t i c moments /a, the  site  using  i n s e c t i o n 2.4.  The m a g n e t i c d i p o l e i n t e r a c t i o n  at  the  o f an atom b e l o n g i n g  t o the f i r s t  to  a n d /T  2  sublattice  i s g i v e n by  °'P'  x  Sublattice  1  subl*tli«Z  Writing  *  J  k  the d i p o l a r magnetic  field,  t o a m a g n e t i c d i p l e moment /<  B  '  (2  40)  a t t h e o r i g i n due  d i p  a d i s t a n c e r from t h e o r i g i n  i n t e r m s o f t h e d i p o l e v e c t o r p o t e n t i a l ^^.p., ^d,  P  ~  V  * X  d i p  (2.41 )  .  where  (2.42) and a p p l y i n g t h e v e c t o r V  * (F * 7$)  -  identity  (G-V)  - (f-v) along  F  -  e  -  with Laplace's equation,  d,p  "  Substituting  £  (V - F F  (  )  v-<5)  (equation  (2.43  1.4) g i v e s  (2.44)  7  this expression  forB  d i p  back  into  equation  ( 2 . 4 0 ) , one o b t a i n s t h e m a g n e t i c d i p o l e i n t e r a c t i o n  energy  29  at  a n atom o f moment  E . a  - t  -  [  )  i n terms of the q u a n t i t i e s V . ;  ( / * . ) ' V.j  1  t  -  C  A  ^  ^  ^  ^  -  J  ^'  >P  where V  (2.45)  i sdefined,  ;i  f o r t h e pupose of t h i s  section  only,  without t h e charge f a c t o r q ( r ) ,  w  V a  and  the subscript  3* 11  7  —J —  ,  (2.46)  1 or 2 denotes the magnetic  sublattice  o v e r w h i c h t h e sum i s c a r r i e d o u t . When an i n c r e a s i n g m a g n e t i c to  the  easy  axis  antiferromagnetic  of  field  an  i sapplied  ordered  exchange i n t e r a c t i o n ,  parallel  magnet  with  t h e system  passes  through three phases: the a n t i f e r r o m a g n e t i c , t h e s p i n - f l o p and t h e f e r r o m a g n e t i c r e g i m e s  EASY AXIS  (see figure 6 ) .  EASY AXIS  EASY AXIS  B.  B.  B.  A'  A  ANTIF FERROMAGNETIC  SPIN  Figure For  a  system vectors  i n the spin and  the  FIELD  a field  flop  regime, the  l i e i nthe plane defined  by t h e e a s i e s t a n d n e x t - e a s i e s t a x i s For  TRANSVERSE  6 - phases of an a n t i f e r r o m a g n e t  biaxial  magnetization  FERROMAGNETIC  FLOP  (the yz-plane).  a p p l i e d p e r p e n d i c u l a r t o t h e easy a x i s i n  y z - p l a n e t h e m a g n e t i c moments a r e c a n t e d by  angle  30  to  the f i e l d  again  up t o a c r i t i c a l  i s ferromagnetically  field  v a l u e when t h e s y s t e m  ordered (see f i g u r e 6 ) .  From e q u a t i o n ( 2 . 4 5 ) a n d v a l u e s o f V a  Mn  atom f o r t h e two m a g n e t i c  shown i n f i g u r e flop  and  parallel E  = f\  calculated  sublattices  of M n C l . 4 H 0 4  6, t h e e x p r e s s i o n s f o r E  ferromagnetic  regimes  for a  t o t h e e a s y a x i s c a n be s u m m a r i s e d  at  in  the  field  2  spin  applied  by  ( .01 1 9 c o s « -.04 92cosoc s i n a -. 0 2 9 0 s i n « ) A " , 2  2  3  2  (2.48) where <x =0 c o r r e s p o n d s t o t h e f e r r o m a g n e t i c p h a s e . For  the case of a p e r p e n d i c u l a r  magnetic  dipole  figure  t u r n s o u t t o be  E  =/4  2 Mn  interaction  (.01 3 7 c o s 0 2  energy  applied  field,  f o r t h e phase  the  shown i n  -.0492cos^3 sin(3 - . 0 2 5 6 s i n ( S ) % ' . 2  3  (2.49) Note  t h a t /S =0 c o r r e s p o n d s t o t h e a n t i f e r r o m a g n e t i c  (field  applied parallel  t o t h e easy  axis).  phase  31  III. 3.1  The  DESCRIPTION OF  Nuclear  The  Resonance  hamiltonian  AN  NMRON SPECTRUM  Condition  of  a  spin I nucleus  magnet w i t h t h e e f f e c t i v e m a g n e t i c parallel  t o the z - a x i s i s g i v e n H  =  n  Here  H^  H h  •+  K  i s the  H n  average  isotropic  ordered  the  nucleus  at  by  H  (3.1)  " S N  Q  Zeeman m a g n e t i c  magnetic h y p e r f i n e thermal  +  field  i n an  interaction of  interaction  the H  interaction tensor  electron can  be  where A i s t h e  and  spin  <S>  is  moment.  the  For  an  written (3.2)  where g  N  i s the n u c l e a r  n u c l e a r m a g n e t o n , and H  g  is  gyromagnetic  B*  N  r a t i o , /i  i s the magnetic h y p e r f i n e  the quadrupole i n t e r a c t i o n h a m i l t o n i a n  ( 1 . 1 5 ) , and  H  $N  is  u  of  i s the Suhl-Nakamura h a m i l t o n i a n  the  field. equation  [  8 ]  where  (3.4) are c o u p l i n g constants. sublattice exchange  H e r e Y„  nearest  neighbors  interaction  J, A  the h y p e r f i n e c o n s t a n t ,  and  i s t h e number of with  which  there  opposite is  an  i s t h e t r a n s v e r s e component of f(r )  is  the  range  of  the  32  indirect the  notation  ~  Ur)  Notice of  spin-spin  that  interaction  of  reference  r  e  8)  r  goes a s y m p t o t i c a l l y  ,  f(r) falls  which for l a r g e  ~  oc  (using  as  1  ( 3 i  off exponentially  for large  5)  values  r.  Let dilute  us  our  attention  radioactive nuclei in a  Nakamura nuclear the  confine  interaction s p i n s , the  distance  case.  The  spins  solid.  effective  term i n H spins  is  only is  S N  SN  can  energies  equations  type, be  < I ™  ~  the  | H  +  M  H  | I  like  negligible  when  to  to f i r s t  (3.2)  p  Suhl-  i t i s in  the  this  number  of  suppressed.  radiactive  ( 1 . 2 4 ) and  the  very  between  l a r g e , as  is also  neglected  of  (1.23),  and  c a s e of  Since  second term i s p r o p o r t i o n a l  Thus H  E^  first  between  of a g i v e n  state  is  to the  are  order  i n the  spin-  which  from  nuclei, found to  be  m> (3.6)  giving  am  the  E„-  i  +  [?m  - I ( I + I ))  z  energy d i f f e r e n c e between a d j a c e n t s p i n  E _  -  a  -  i P  states  [ 2 - - i] (3.7)  (a  From t h e  expression  Plank's  constant  electromagnetic  -  -  V/ ) 2  Pm  f o r p h o t o n e n e r g y , E = hv, and  p  is  r a d i a t i o n , (3.7)  the yields  where h  frequency the  21  of  is the  first  33  order  resonance  h v„  =  ( a  conditions  -  p  /2 )  -+  ?  m  (3.8) 3.2 Gamma-ray The an  angle  oriented  Emission  normalised  gamma r a d i a t i o n  6 t o the quantization  axis  i n t e n s i t y emitted at  of a system of  by t h e h y p e r f i n e i n t e r a c t i o n o f e q u a t i o n  nuclei (3.2)  is  [11] '  W(Q)  A ( m, 9 ) P l m )  E  (3.9) where  P(m) i s t h e n o r m a l i s e d  p o p u l a t i o n of t h e s p i n  state  m a n d A(m,9) i s d e f i n e d by A (m,e)  U  = E  F^P  2 J  Z J !  (cos e)  f ^ n r  * I -m Here  F  and  corresponding  U  m  a r e angular  O J  respectively  vfiiTT o /  momentum  i t , P  polynomial  the  expression integers is  The oriented this  21,  i s a 3 j symbol  and  [7].  The  from z e r o t o t h e l e s s e r  the lowest  observed  order  spin  value  gamma t r a n s i t o n technique nuclei  of  coefficients  t o the observed  t o a n y d e c a y s w h i c h may p r e c e d e of  2<  t r a n s i t i o n and is a  last  Legendre  factor  summation  o f 2L o r 2 I  i n the  runs M  , where  over I  M l N  i n t h e cascade which precedes the o f m u l t i p o l a r i t y L. nuclear  magnetic  resonance  (NMRON) r e l i e s on m o n i t o r i n g  anisotropic  1 0 )  radiation  when  of  t h e change of  the  population  34  distribution  of the spin  conventional  NMR e x p e r i m e n t s ,  alternating  f l u x due t o p r e c e s s i n g  detected. spin  i s altered,  whereas  of  approximately  "ft - r a d i a t i o n  detectable, required  as  to  s t a n d a r d NMR 3.3 P o p u l a t i o n The  so  induce  an  nuclear  spins  Of The S p i n  nuclei  evolution  equilibrium,  radiation  of  are  depending  thermal  10  1 1  which i s  i s that  a  p a r t i c l e s can produce  changes  in  intensity  t o the approximately  10  observable  change  voltage  2 0  are  nuclei in  experiments.  poulations  oriented  that  opposed  in  i t i s t h e emf i n d u c e d by t h e  An a d v a n t a g e o f t h e NMRON t e c h n i q u e ,  system  enough  levels  Levels the  spin  governed upon is  by  whether being  states  of a system of  three  equations  the spin excited  of  system i s i n by  incoming  a t a resonance frequency, or i s r e l a x i n g  toward  equilibrium after excitation. 3.3.1  The E q u i l i b r i u m The  spin at  populations  Population of t h e s p i n  s t a t e s , m, f o r a  system i n thermal e q u i l i b r i u m with  a thermal  nuclear resevoir  t e m p e r a t u r e T, obey t h e B o l t z m a n n d i s t r i b u t i o n  _ E en) /|<t ?  where  b<^  E(m)  Boltzmann  =  £  -e<~>/  i s the energy of s p i n  constant.  { 3  kT  substate  *  1 1 )  m and k i s t h e  35  3.3.2  The  P o p u l a t i o n s At  Electromagnetic condition t h e m-1  of  and  radiation  equation  m nuclear  has  enough i n t e n s i t y ,  will  be  equalized  the v a l u e  of  Resonance satisfying  (3.8)  induces  spin levels.  the  resonance  transitions  If  the  RF  radiation  t h e p o p u l a t i o n s of t h e s e  two  ( s a t u r a t i o n c o n d i t i o n ) , each  t h e a v e r a g e of t h e i r  previous  between  taking  the  time  (3.12)  i n t e r v a l d u r i n g which the  is applied i s short populations  of  relative the  to fhe  other  on  populations.  1  If  levels  resonance  relaxation  levels  are  frequency time,  the  comparatively  una f f e c t e d . 3.3.3  The  Populations During  Once RF cease,  Relaxation  e x c i t a t i o n s between the n u c l e a r  i n t e r a c t i o n s w i t h the  populations  to  distributions. populations  relax  lattice  toward  The  rate  w i t h time  during  of  spin  cause the their  change  spin  levels level  equilibrium of  the  r e l a x a t i o n i s given  level  by  dPc*0 di  (3.13)  where W7  and  VH  are defined  i n t e r m s of  the  spin-lattice  36  relaxtion  time  T,  (3.14)  <Irrm|lt I I Here, the  s q u a r e m a g n i t u d e s of t h e m a t r i x = i<I  ,>r  m+.i  i  (3.15)  = I ( I+ I ) In that the  equations  (3.13)  spin level  energies  I )  ( 3 . 1 4 ) i t has  level  i s a f u n c t i o n o f t h e p o p u l a t i o n of  (3.13)  adjacent  is  a  derivative  level.  nondiagonal  differential  equations.  where  1^2,  this  since  solution  requires  finding  system  For  been a s s u m e d  increase with increasing  that  one  time  (M-+  Note  least  the  and  are  ^>r  ii  +  elements  |*  m.  of t h e p o p u l a t i o n of  one  its  own  and  F o r m = - I , -1+1,  ...  ,1-1,  system  first  general  can  of T  o n l y be  of  the  corresponding  the  r o o t s of a 21+1  21+1  a n d / o r T, solved  I  order  in  cases  numerically,  secular  order  at  equation  polynomial.  37  IV.  PROGRAM TO GENERATE AN NMRON SPECTRUM A P P L I E D TO 54MN-MANGANESE CHLORIDE TETRAHYDRATE  4.1 Gamma-ray E m i s s i o n The capture state  5 u  F o r 54-Mn  M n n u c l e u s has s p i n  and then  by an E2  of the daughter,  54  1=3.  5  I t d e c a y s by e l e c t r o n  gamma-ray  t o t h e ground  *Cr [ 4 ] .  Mn  3+  2+ E2 5  For  this  4  C r _  0  Figure  7 - d e c a y scheme f o r 54-Mn  decay,  the values  c o e f f i c i e n t s U , and F  lt  Jt  of the angular  of equation  momentum  (3.10) a r e f o u n d  t o be  U = 0 . 8 2 8 , U = 0 . 4 1 8 , F =-0.598, a n d F =-1.069. z  4  a  Substituting (3.30) y i e l d s  these  the  emitted  a t an  oriented  system  4  values  normalized  angle  9  gamma  radiation  t o the quantization  o f 54-Mn n u c l e i , W ( 6 ) .  W(0) = 1 + . 3 3 3 [ P ( 1 ) + P ( - 1 ) ] -  into equations  [P(3)+P(-3)]  +  (3.9) and intensity axis  o f an  F o r 6=0 t h i s i s .667[P(2)+P(-2)] (4.1)  38  4.2 A Two T e m p e r a t u r e If a fraction are  cooled  to  Model  of the r a d i o a c t i v e n u c l e i  a  temperature  appreciable orientation rest  remain  warm,  T  low  -  {(i  m  +  where WA and  f  {  WA  +  ( WCo)  i s the f r a c t i o n  the  measured  be  - IJ f  T  ( 4  ]  i s t h e c o u n t m e a s u r e d when t h e  T  while  t  N  I  solid  yco)}  £t  f  =  spins,  t h e n t h e gamma-ray i n t e n s i t y  -px)  -  a  enough t o p r o d u c e  of the n u c l e a r  along the spin quantization axis w i l l y  in  sample  of r a d i o a c t i v e  is  nuclei  -  2 )  warm,  which a r e  a t t e m p e r a t u r e T. 4.3 The R e s o n a n c e C o n d i t i o n s F o r 54-Mn I n MnCl,.4H,0 The (3.2)  Zeeman  interaction  parameter,  a,  1  in  constant,  magnitude  P, c a l c u l a t e d  than  i  the  quadrupole  i n chapter I I .  predicts  the  the  exaggerated labelled  spin  level  quadrupole  equation  (3.6)  e n e r g y d i a g r a m o f f i g u r e 8, i n  interaction  for clarity,  interacton  Since the r e s u l t s  of c h a p t e r I I i n d i c a t e t h a t P i s p o s i t i v e ,  label  equation  f o r a Mn n u c l e u s i n M n C l . 4 H 0 i s n e g a t i v e a n d much  greater  which  of  and  the  energy spin  has  been  s t a t e s m a r e be  i n o r d e r o f a s c e n d i n g e n e r g i e s by a  new  integer  n d e f i n e d by n = 3 - m  (Motivation arrays  for this  labelling  (4.3) comes f r o m t h e f a c t  that  i n t h e p r o g r a m m i n g l a n g u a g e FORTRAN must be i n d e x e d  39  by  positive integers.)  n  gives  Rewritng equation  (3.7) i n terms of  (4.4)  \  | \  m « -3  n•b  m-2  n-5  m=-l  n«4  l I  \ M  \  A  ,  E N E R G Y  •0  n-3  1  n-2  m --  m-2  \  \  Figure  8 - modification  n-0  i  F  •M  r3  r  n =l  +F  of t h e Zeeman n u c l e a r  spin  e n e r g y l e v e l s by t h e q u a d r u p o l e i n t e r a c t i o n The  minimum  energy  difference  o c c u r s b e t w e e n t h e n=0 a n d n=1, o r m=3  AE,  al  -  ! ?  between s p i n a n d m=2  levels  states.  (4.5)  40  From e q u a t i o n s between  (4.4)  adjacent  and  (4.5)  spin  the  l e v e l s can  t h i s minimum e n e r g y d i f f e r e n c e a s  A E „  Dividing  A E ^  =  by  •+  P  frequencies are  energies  be w r i t t e n i n t e r m s of  2  (n - I )  Planck's constant  transition  n»  the  I, 2, ••• ,  first  (  order  4 > 6  )  resonance  obtained  (4.7) n « i,  i,  where  4.4 4.4.1  P o p u l a t i o n Of  The  Nuclear  Spin  Levels  Equilibrium Since  equation  |P|  <<  |a|, the energy of the s p i n l e v e l s  ( 3 . 1 1 ) may  m = -3 ,-2,...,2,3  be a p p r o x i m a t e d  3  ( n l  m  E(m)=-ma f o r  giving -  ?  by  in  ^  n  a  /kT w  ^  (4.8)  n  The  experimental  v a l u e o f a/h  be w r i t t e n i n t h e r m a l  2  i s -508.2 MHz  [6]  which  can  u n i t s as a/k=-0.0244 K e l v i n s .  N o t e t h a t i f t h e s i g n o f P i s n e g a t i v e , t h e minimum e n e r g y d i f f e r e n c e o c c u r s b e t w e e n t h e two h i g h e s t e n e r g y levels. If t h e l e v e l s a r e l a b e l l e d by i n t e g e r s , n, w h i c h i n c r e a s e w i t h l e v e l e n e r g y , then i n eq. 4.6, P ( n - 1 ) — > 3P(2I-n)  41  4.4.2  Resonance  Assuming frequency the the  that  radiation is  spin-lattice nuclear  (4.7)  is  time  i n t e r v a l during  applied,  relaxation  spin levels  satisfied  for  t,  small  t i m e T, ,  during n=n  is  the  which  the  resonance  relative populations  time  when  •for  n - n ^  as  n=n -i R  (4.9)  2  4.4.3  Relaxat ion One  numerical  differential  a p p r o x i m a t i o n to  equations  (3.13)  - Wf  -  where  the  obtained (3.14) with  of  equation  can be w r i t t e n e x p l i c i t l y  K  to  expressions  m  element  k i - < i  as  that  of  linked  n  (4.10)  by  in ,1^+1) . r  for  W|(n+1,n) definition  level  energy  p r e v i o u s l y assumed.  terms  given  system  WKn.n-iJlPt.rtW-Fgim]  by s u b s t i t u t i n g  and n o t i n g  is  the  and (4.1)  Wf(n,n+1) into  increases  Doing  this  (3.13)  with n,  for  the  are and not  matrix  yields  i - i i n > r =  \ < i «  u . i i  n*,>r (4.11)  42  4.5 W i d t h Of The R e s o s n a n c e I n t e r a c t i o n s of t h e  Peaks  nuclear  spins  with  f i e l d s a b o u t them i n a s o l i d  produce a f i n i t e  resonance  gamma-emission  lines.  interaction  a n d an i n h o m o g e n e o u s s p r e a d  quadrupole  ineractions  contributions  to the l i n e width.  the  net  then  equation  will  the  local  width t o the  The  Suhl-Nakamura  i n t h e Zeeman  make  near  gaussian  I f i t i s assumed  e f f e c t may be a p p r o x i m a t e d  by a g a u s s i a n  and  that  profile  ( 4 . 2 ) i s r e p l a c e d by  (4.12) 4.6 The P r o g r a m The  FORTRAN p r o g r a m l i s t e d  spectrum to  of n u c l e a r magnetic resonance  t r a n s i t i o n s between s p i n l e v e l s  interaction previous W(0), 5  "Mn  for  i n appendix  equation and  the  t h e "Mn 5  lines  split  according t o the governing  sections  of  this  E generates  corresponding  by t h e h y p e r f i n e  equations  chapter.  of the  The e x p r e s s i o n f o r  ( 4 . 1 ) , used i n t h e program i s upper l i m i t s  specific  be c h a n g e d t o  accommodate any m a g n e t i c a l l y o r i e n t e d n u c l e a r s p i n i s governed  which  H  strongly  by  the hamiltonian equation  dominates.  parameters  a r e read  particular  experimental spectrum.  different  The  other  i n s o t h a t t h e y may be v a r i e d  input parameters  to  on t h e D O - l o o p s h a v e been s e t  s p i n o f 3, b u t t h e s e c a n e a s i l y  which  a  system (3.1)  in  spectrum to f i ta  Some s a m p l e o u t p u t s f o r  are l i s t e d  i n A p p e n d i x F.  43  The  spectra  change of  are  frequency  generated with  according  time;  both  decrease w i t h time are p e r m i t t e d . not  an  may  s p e c t r u m may versus  emission plot  be  be  in  presented  frequency,  population  one  of  increase  spin levels  intensity  of number of  forms; the  as a t a b l e of number  accompanied  of t h e  two  with  'counts'  a  and is  and  the or  i t may  be  versus  frequency.  computer c e n t e r s , p l o t s u b r o u t i n e s  included  in  subroutines  listing  of  13 were w r i t t e n by Mary Ann  in these  spectra are  The  equations  range  relaxation of  T, of  becomes  The  as  the  unphysical  counts which i s j u s t beginning and  s p e c t r a of  The  gamma a  different at  have not The  been  FORTRAN  figures 9  vertical  bars  (4.10) w h i c h i s used of  differential during  unstable  spinfor  w h i c h d e p e n d s upon the  frequency  oscillation t o appear =  a the  stepsize.  instability.  i s q u i t e n o t i c a b l e a t T, f i g u r e 9.  the  bars.  numerically  t e m p e r a t u r e and  of  available  level populations  below a c e r t a i n v a l u e the  As  E.  system  9 d i s p l a y s t h e o n s e t of t h i s  manifested  seconds  Potts. deviation  the  which govern the  input value Figure  "solve"  counts  o u t p u t as  s p e c t r a p l o t s of  s e t of d i f f e r e n c e e q u a t i o n s  numerically  lattice  standard  are  Appendix  used to produce the  of  normalized  time,  routines  generated  tabulation  different  the  and  by  facilities  the  linear  Frequency modulation  plotting  to  a  included. Output  to  to  It  is  i n t h e number o f at  T,  =  11000  10000 s e c o n d s f o r  s p e c t r a of t h i s  figure  have  44  the  same  input  parameters  For  even  lower  values  oscillation  of  except T,  f o r t h e v a l u e s o f T, .  the  amplitude  of  the  becomes s o g r e a t t h a t n e g a t i v e v a l u e s o f " t h e  number o f c o u n t s " a r e  produced.  45  MNCL2—T1=11000-SECONDS 240000 239200 238400 237600  to r—  5  g 236800 u_  o 236000 235200 234400 233600 232600 |232000  ax.i  FREQUENCY (MHZ)  MNCL2--Tl=1000u-SECQNDS 240000  239200 238400 237600 \ 236900 5  • 236000 c  235200 234400 233600 232800 232000 5M.1  S12.I  FREQUENCY IMHZ)  Figure  9 -  o n s e t of n u m e r i c a l d e c r e a s e i n T,  instability  with  46  Users  of  the  program  difference equations the o r i g i n a l  that  the  ( 4 . 1 0 ) become w o r s e a p p r o x i a m t i o n s  to  error.  of  the  numbers  of  at  permit  improve a c c u r a c y  difference  s h o u l d not  A change t o a h i g h e r  REAL*16) w i l l  formula  also  d i f f e r e n t i a l s y s t e m as A t  too s m a l l a v a l u e of A t off  should  is  machine  increases. c h o s e n due  precision  level  (e.g.  s m a l l e r v a l u e s of A t . to  equation formula  be  replace  note  the  However to  representaion f r o m REAL*4 t o A n o t h e r way  2-point  This  approximate  equation  (3.1).  may  alleviate  stability  p r o b l e m t o some d e g r e e .  the  to  difference  (4.10) w i t h a t h r e e or h i g h e r to  round-  point  derivatives  of  the above m e n t i o n e d  47  4.7  Output The  Of The P r o g r a m F o r 54Mn-MnCl, .4H,0  p r o g r a m was u s e d  estimated profiles  values  of  i n two ways.  the parameters,  were g e n e r a t e d .  r e s o n a n c e s , s o t h a t more d e t a i l e d  to  the experimental  referred  (e.g.  Figure  10  spectra,  T, )  to i nthis  inserting  initial  resonance  T h e s e were t h e n u s e d t o i d e n t i f y  the  parameters  First,  r u n s c o u l d be f i t t e d  allowing  other  t o be o b t a i n e d .  spectra  The e x p e r i m e n t s  s e c t i o n are those of r e f e r e n c e  shows*  the numerically  produced  gamma  e m i s s i o n spectrum along the magnetic q u a n t i z a t i o n a x i s f o r 5  "MnCl  .4^0.  a  From  diagram of f i g u r e may  be  hence  the  The  frequency  transition,  m=2/m=1  (3.7) and t h e energy  8, t h e r e s o n a n c e  identified.  lowest  m=3/m=2  equation  lines  smallest  of  t h e spectrum  energu d i f f e r e c e and  resonance  corresponds  transition  and so  on  line.  up  t o the highest  Only the f i r s t  r e s o n a n c e s show up i n t h e e x p e r i m e n t a l s p e c t r a , due  t o t h e s m a l l d i f f e r e n c e s between t h e  populations of the higher resolving  capability  to the  the next lowest frequency belongs t o  f r e q u e n c y m=-2/m=-3 t r a n s i t i o n  11,  level  energy  spin  of the apparatus.  levels  three  see f i g u r e equilibrium and t h e  48  Figure  10 -  computer  generated  spectrum  for  5 a  Mn  in  MnCl .4H2.0 1  I,  '1  ! / I  — ' 460.9  1  1  1  S00.1 501.3 502.5  1 503.7  1 504.B  1— 50B.0  1 507.2  1 50B.4  1 509.B  I  I  I  511.8 512.1 5 1 3 . 2  FREQUENCY (MHZ)  I  I  5 1 4 . 4 515 8  I 518.6  49  Figure  11 -  experimental  spectrum  M n C l . 4 H O with a t h e o r e t i c a l 2  i  fit  for  5ft  Mn  drawn  FREQUENCY (MHZ)  in  in  50  Note drop  that  i n t h e m=2/m=1 t r a n s i t i o n  i n W(0) f o l l o w e d by a r i s e a b o v e a n d t h e n r e c o v e r y t o  the e q u i l i b r i u m v a l u e .  Such a p o s t - r e s o n a n c e  been p r e v i o u s l y a n t i c i p a t e d . this  an e a r l y  run i n which a suspected Refined  modulation The  measurements  c o n t i n u o u s l i n e drawn  corresponds  through  toa spin-lattice  resonance  i n f i g u r e 12. the experimental  generated  relaxation  temperature  t h e program d e s c r i b e d i n t h e l a s t  T  spectrum.  t i m e o f T, = =  0.081 K.  s e c t i o n does n o t  f o r f r e q u e n c y m o d u l a t i o n , d i s c r e p a n c i e s between t h e  g e n e r a t e d a n d e x p e r i m e n t a l s p e c t r a c a n be a c c o u n t e d the  0.3  used  i nthe experiment.  the  time  MHz  function  modualtion  interval  totalled,  some  was  a s m a l l frequency  seen  11 i s t h e b e s t - f i t  3.0±0.5 x 10* s e c o n d s a n d a  allow  with  of t h e  F i g u r e 11 shows  m=2/m=1  then r e v e a l e d the p r o f i l e  data bars i n f i g u r e  Since  identification  i n the experimental spectrum.  observed.  bump h a d n o t  I tserves as a signature f o r  l i n e and a l l o w e d the p o s i t i v e  resonance  It  there i s a sharp  then  I f frequency  during  because  which  W(0) o f  modulation. difference  average  over  In particular  equation  of  t h e m=3/m=2  count  (4.1) i s a v a l u e w i l l be  could explain  resonance  over  counts a r e  t h e frequency  this  stepsize  i s modulated  gamma-ray  of frequency, the accumulated  weighted  spectra.  a n d 0.14 MHz f r e q u e n c y  f o r by  lines  range  of t h e  the height o f t h e two  51  Figure  12 - the o r i g i n a l m=2/m=1 t r a n s i t i o n  line  spectrum  NNCL2 V H=0 100QS * 5 4  243260 242478 241696 240914 5240132  ; 239350 |238568 237786 237004 236222 235440 502.1  JL  502.7  303.9  _i_  503. B  FREQUENCY (MHZ)  504.8  52  Figure  13 - t h e a d j u s t e d  m=2/m=1 t r a n s i t i o n  spectrum, w i t h a t h e o r e t i c a l  line  f i t drawn i n  240000 239200 238400  CO  237600  g 236800 ; 236000 \235200 234400 233600 232800 232000 _1_ 502.1  303.1  _1_  304.1 FREQUENCY  SOS.l  (MHZ)  53  More d i f f i c u l t y the  spectrum  of  was  encountered i n t r y i n g  figure  12.  to  analyze  In the b e s t o b t a i n a b l e  fit,  t h e h e i g h t o f t h e peak a b o v e t h e e q u i l i b r i u m  level  f o r the  theoretical  that  of  spectrum  experimental  was  value.  However,  shows t h a t t h e r e i s an counts  less  upward  than  half  inspection slope  in  i n c o u n t s due  t o the  suspected  the  c a n be s e e n , t h e f i t smooth  12,  number  When t h e  temperature  s u b t r a c t e d o f f , the spectrum of f i g u r e  adjusted  of f i g u r e  before the resonance d i p , which i s c o n s i s t e n t  a w a r m i n g of t h e s a m p l e d u r i n g t h e r u n .  the  the  of with  increase  increase  13 i s o b t a i n e d .  is As  w i t h the numerical spectrum, which i s  curve  superimposed  spectrum,  is excellent.  on  the d a t a b a r s of the  I t corresponds to  T,  =  2.3±0.3 x 10" s e c o n d s and a t e m p e r a t u t e T = 0.064 K e l v i n s .  54  V. Two  computer  programs  study of h y p e r f i n e first of  program  SUMMARY a r e developed as a i d s  interactions  i n magnetic  solids.  f i n d s t h e p r i n c i p a l second order  the e l e c t r i c potential  a t any p o s i t i o n  i n a l a t t i c e of  I t i s u s e d t o p r o v i d e an e s t i m a t e  magnitude  sign  of  the nuclear-lattice  i n t e r a c t i o n c o n s t a n t s P and P . q  T h i s program  used t o d e t e r m i n e t h e magnetic d i p o l e for  lines  spin-lattice the  relaxation, 5  i  levels  of  be  deduced.  relaxation For  subsequent  because  the  the nuclear  spin  The n u m e r i c a l c o m p u t a t i o n a l l o w s t h e  Comparison  of  parameters  with such  relaxations  experiment as  allows  the s p i n - l a t t i c e  time.  MnCl .4H 0 the point 2  =  2  0. 1 277  A"  3  = 0.746 when t h e w a t e r  electrically f\  I t was w r i t t e n of  resonance  i sapplied t o  of t h e resonance l i n e s and subsequent  determination  f\  The s e c o n d  ( w h i c h d e t e r m i n e s t h e gamma r a y e m i s s i o n ) c a n n o t be  profiles  V _/e  with  t h i s program  the populations  solved a n a l y t i c a l l y .  22  and  be  interaction energies  interaction  c a s e o f "Mn-MnCl^ . 4 H 0 .  redistribution  to  can a l s o  generates a spectrum of n u c l e a r magnetic  s h i f t e d by q u a d r u p o l e  of the  quadrupole  t h e p h a s e s o f an o r d e r e d a n t i f e r r o m a g n e t .  program  The  derivatives  point charges. and  i n the  neutral  and  an  calculations  asymmetry  molecules  units,  = 0 . 9 8 when t h e o x y g e n  charge  were  gave  parameter considered  of as  a n d V L ^ / e = 0. 02551 A " a n d  and hydrogen  c h a r g e s o f -e a n d +e/2 r e s p e c t i v e l y .  3  atoms were a s s i g n e d  55  Comparison experimentally theoretical difficult V  of  the  obtained  numerical quantity,  antishielding to  factor  determine.  , t h e above v a l u e s  result  with  the  P, i s d e p e n d e n t upon a whose  Incorporating  accuracy this  is  factor  into  c o r r e s p o n d t o P = 2.3±0.2 MHz a n d  = 0.18±0.02 MHz r e s p e c t i v e l y .  P  E x p e r i m e n t gave P = 6.2±0.8  MHz. Two  expressions  interaction of  summarize  energies  t h e Mn  magnetic  dipole  calculated f o r the d i f f e r e n t  phases  MnCl .4H O: i  z  \ 0. 0 111 eos ^. - 0. 0«-<?2cos ot s.« oc - 0 . 0 2 9 0 sm *"] A  , = At*  1  1  is  t h e d i p o l e energy  the  e a s y a x i s f o r t h e s p i n f l o p p h a s e where  moments  i n the case of a f i e l d  are canted  at  angles of ±«  In t h e case of a f i e l d i n the c r y s t a l  the  dipole  Mn  dipole  respect ^-d.pote.  /Vn  j6 =0 g i v e s  applied  B-C p l a n e ,  i n t e r a c t i o n energy  field  transverse  phase. t o t h e easy obtained f o r the  i n d i r e c t i o n s o f ±/5  with  directioni s  the magnetic d i p o l e phase  magnetic  f o r t h e phase i n which  [ 0.0 13? COS*j3 - 0 . 0 4 92 COS /S  antiferromagnetic  along  t o t h e easy a x i s .  the expression  moments a r e o r i e n t e d  to the applied =  the  o(=0 c o r r e s p o n d s t o t h e f e r r o m a g n e t i c  Note t h a t  axis  applied  -  0.0 25<p s m ^ ] A 1  i n t e r a c t i o n energy f o r t h e  (field  applied  along  t h e easy  axis). The  numerically  g e n e r a t e d NMRON s p e c t r u m f o r M n 5 4  in  56  MnCl .4H 0 z  produced an unexpected p r o f i l e  JL  resonance l i n e : intensity and  a f t e r an i n i t i a l  then  rose  subsequently  above t h e i n i t i a l  i nthe experimental  frequency  was r e d u c e d  experimental  t h e gamma-ray  e q u i l i b r i u m value  This behavior  t o the resonance  observed  and  decrease,  r e l a x e d back t o i t .  taken as a s i g n a t u r e  f o r t h e m=2/m=1  and indeed  was  s p e c t r u m when t h e m o d u l a t i o n  t o a small value.  spectra  c a n be  The  c a n be f i t t e d  theoretical  reasonably  well,  a l l o w i n g h y p e r f i n e i n t e r a c t i o n p a r a m e t e r s t o be e v a l u a t e d . Of  the s i x resonance  existence within the  of  a  quadrupole  small  equilibrium  Allowing modulation  for  frequency,  lowest  transition  energy line.  predicted  interaction,  the resolving c a p a b i l i t y  two h a d d e f i n i t i v e  the  effects  a reasonable  of  was  experimental spectrum. a suspected run, and this  value  the  f i t was  of  obtained f o r  and strongest resonance,  t h e m=3/m=2  From t h i s v a l u e s o f T, = 3.0±0.5  not possible  the  experimental  x 10"  obtained.  t o get a  f i t within the  Adjustment of the spectrum t o take warming o f t h e sample o v e r  an i n i t i a l  owing t o  profiles.  u n c e r t a i n t i e s f o r t h e raw m=2/m=1  permitted  to the  o n l y t h r e e were  differences  s e c o n d s a n d T = 0.081 K e l v i n s were It  due  of the experiment  population  l e v e l s , and only the f i r s t  the  lines  transition into  account  the d u r a t i o n of t h e  a g o o d f i t f o r T, = 2.3±0.3 x 10* s e c o n d s temperature  for T  o f 0.064  Kelvins.  Note  that  does not i n c l u d e any u n c e r t a i n t y f o r  adjustment o f the spectrum.  57  BIBLIOGRAPHY 1.  A l t m a n , R. F.; S p o o n e r , S.; L a n d a u , D. P.; a n d R i v e s , J . E.; P h y s . R e v . B, 11, 606 ( 1 9 5 8 ) .  2.  E l Saffar,  3.  H a n d r i c h , E.; S t e u d e l , A.; a n d W a l t h e r , Lett. 29A, 486 ( 1 9 6 9 ) .  4.  K a r l s s o n , E.; W a p p l i n g , R.; e d s . H y p e r f i n e I n t e r a c t i o n s : S t u d i e s i n N u c l e a r R e a c t i o n s and Decays, ( A l m q u i s t and W i k s e l l I n t e r n a t i o n a l ; S w e d e n ) .  5.  K o r n f e l d , H.; Z e i t s .  6.  K o t l i c k i , A.; M c L e o d , B.; S h o t t , M. ; a n d T u r r e l l , G.; t o be p u b l i s h e d , Hyp. I n t .  7.  M e s s i a h , A.; Quantum M e c h a n i c s V o l . 1 1 , H o l l a n d ) , A p p e n d i x C.  8.  N a k a m u r a , T.; P r o g r . (1958).  9.  N i s s e n , L.; a n d H i u s k a m p , W. ( 1970).  Z.  M.; J .  Chem P h y s .  Physik.  5 2 , 4097  22  Theoret.  (1970).  H.; P h y s .  (1940). B.  (North-  P h y s . ( K y o t o ) 20, 542 J . ; P h y s i c a 50, 259  10.  S l i c h t e r , C h a r l e s P.; ' P r i n c i p l e s o f M a g n e t i c R e s o n a n c e ' , pp 164-170, ( H a r p e r & Row; 1 9 6 3 ) .  11.  S t e f f e n , R. M.; a n d A l d e r K.; i n 'The Electromagnetic I n t e r a c t i o n i n Nuclear S p e c t r o s c o p y ' , e d . W. D. H a m i l t o n ( N o r t h 1975) C h . 12.  Holland;  12.  Sternheimer, Phys. Rev. (1954) .  R. M.; P h y s . R e v . 84, 244 ( 1 9 5 1 ) ; 8 6 , 316 ( 1 9 5 2 ) ; P h y s . R e v . 9 5 , 736  13.  S t o u t , J . W.; G r i f f e l , M.; P h y s . R e v . 7 6 , 144 ( 1 9 4 9 ) ; J . Chem. P h y s . 18, 1455 ( 1 9 5 0 ) .  14.  S u h l , H.; P h y s .  15.  Y a s u o k a , H.; Ngwe, T i n ; J a c c a r i n o , V., P h y s . 177, 667 ( 1 9 6 9 ) .  16.  Z a l k i n , A.; F o r r e s t e r , J . Chem. 3, 529 ( 1 9 6 4 ) .  Rev.  109, 606  (1958).  D.; T e m p l e t o n , D.;  Rev. Inorg.  58  APPENDIX A - CHECK OF EQUATION 2 . 1 9 F i r s t we show d i r e c t l y t h a t V vanishes at the o r i g i n f o r a s i m p l e c u b i c l a t t i c e i n a c u b i c volume c e n t e r e d on t h e o r i g i n . V i s d e f i n e d by v  Evaluating  2 2  Z  -  (  fcj)  the d e r i v a t i v e ,  lattice  this i s  h*  =  The are  &  s  (A.  vectors  f o r a simple cubic  r = a( n , i + n j + n k 2  where n, , n , n (A.1) gives 2  3  are integers.  5  lattice  1 )  of s i d e 'a'  )  (A.2)  Substituting this , ,  into  x  "  3n»* - ( to," + ru* + in» ) 2  a  =- N  If it  we d i v i d e t h e d e n o m i n a t o r i n t o t e r m s i n n , n i s c l e a r t h a t t h i s sum i s z e r o .  "  N  _ v  ~  N  and n  "I  If equation ( 2 . 1 9 ) f o r V i s c o r r e c t , then the r i g h t hand s i d e o f t h i s e q u a t i o n must a l s o v a n i s h f o r a s i m p l e cubic l a t t i c e . F o r such a l a t t i c e t h e r e c i p r o c a l l a t t i c e vectors are rj  g = 2 /a(n,i+n 5 *k)  (A.3)  + n  x  and  t h e volume of t h e u n i t c e l l i s v  ££Lt  = a  (A.4)  3  Using the f a c t t h a t the e x p r e s s i o n i s independent of the v a l u e o f G, a s l o n g a s G i s p o s i t i v e , we c h o o s e p a r a m e t e r G s o t h a t t h e e x p o n e n t i a l t e r m s i n t h e two sums o f e q u a t i o n (2.19) a r e e q u a l . This requires G  =v/Tf/a  (A. 5)  The g=0 t e r m i s f o u n d by s e t t i n g n = n = n = n i n e q u a t i o n (A.3) a n d t a k i n g t h e l i m i t a s n g o e s t o z e r o . 1  Substituting  (A.2),  (A.3),  2  3  ( A . 4 ) , and (A.5) i n t o  59  ( 2 . 1 9 ) and u s i n g  the f a c t  0  a s was  shown a b o v e  -+ V  that  f o r a simple  cubic  lattice  o  f o r t h e s p e c i a l c a s e m=5,  ;  =r 2.  II )  gives  =  0  Thus t h e r i g h t hand s i d e o f e q u a t i o n ( 2 . 1 9 ) d o e s i n d e e d vanish f o r a simple cubic l a t t i c e . T h i s w o u l d n o t be t r u e i f t h e g*=0 c o n t r i b u t i o n o f 1/3 were n o t p r e s e n t .  60  APPENDIX B - PROOF THAT THE EIGENVALUES AND EIGENVECTORS OF V ARE P R I N C I P A L F I E L D GRADIENTS AND P R I N C I P A L AXES Let the u n i t e i g e n v e c t o r s E\ of t h e m a t r i x which has c o m p o n e n t s V;j be t h e b a s i s v e c t o r s o f t h e x c o o r d i n a t e s y s t e m , a n d l e t e be u n i t b a s i s v e c t o r s o f t h e x c o o r d i n a t e system, such t h a t K  >x"" 2*  (B.I)  J  then  U . - ^ U x - E , - )  -  By w r i t n g t h e e i g e n v e c t o r s the x - c o o r d i n a t e system K  the  right  -  (EwY  t  -  V.i  (  i n terms of t h e i r  (£*Y&*  B  -  2  )  components i n  - ( £ „ 1 * 6 ,  (B.3)  hand s i d e o f ( B . 3 ) becomes  v , . - fcYW y A  k l  b u t s i n c e Ej i s an e i g e n v e c t o r  (B  .  4)  this i s  V„ - (£<)"{ii)' ^  .>  (B  5  By t h e c o m m u t a t i v e p r o p e r t y o f p a r t i a l d i f f e r e n t i a t i o n a n d the r e a l n e s s of the e l e c t r i c f i e l d , t h e m a t r i x w i t h components V i s symmetric. I t i s t r i v i a l t o show t h a t the e i g e n v e c t o r s of a symmetric m a t r i x f o r d i s t i n c t e i g e n v a l u e s a r e orthogonal (and i f the e i g e n v a l u e s a r e not d i s t i n t , c a n be c h o s e n o r t h o g o n a l ) . Hence i t i s shown V 'j  '  $ <j ^ j  (B.6)  The d e r i v a t i v e s o f t h e e l e c t r i c p o t e n t i a l V w i t h r e s p e c t t o t h e c o o r d i n a t e system which has t h e ( o r t h o g o n a l ) e i g e n v e c t o r s of t h e m a t r i x w i t h components V j as i t s basis, are the p r i n c i p a l d e r i v a t i v e s . C  61  APPENDIX C -  C C C C  PROGRAM 1  PROGRAM TO SUM BY EWALDS METHOD, THE SECOND ORDER P A R T I A L D E R I V A T I V E S OF THE E L E C T R I C POTENTIAL AT THE FIRST INPUT COORDINATE POSITION DUE TO POINT CHARGES AT THE L A T T I C E P O I N T S , FOR L A T T I C E S IN WHICH AT MOST ONE OF THE ANGLES BETWEEN L A T T I C E B A S I S VECTORS IS NOT NINETY DEGREES  * * * *  IMPLICIT REAL*8(A-H,0-Z) REAL U(6,35,3),CHARGE(10),G,GS0,X(3),RL(3),S1ZER0(6),S1(6).S2(6) REAL 0 1 ( 6 ) . 0 2 ( 6 ) , V I J ( 6 , 1 0 ) , S U M 1 ( 6 ) , S U M 2 ( 6 ) , O R I G I N ( 3 ) INTEGER N E A C H ( 1 0 ) PI=3. 141592653 ROOTPI=DSQRT(PI) C THE SUMS FOR VIO WILL BE COMPUTED UP TO 1 NX 1=1 NY 1 = 1N21=N-1 READ(5,30) N C INPUT L A T T I C E B A S I S VECTOR LENGTHS IN ANGSTROMS C AND THE ANGLE IN RADIANS BETWEEN VECTORS A AND C READ(5,20) A,B,C,BETA C G IS THE INTERMEDIATE L I M I T OF INTEGRATION WHICH S P L I T S THE INTEGRAL C INTO A FOURIER OR RECIPROCAL SPACE PART (SUM 1) AND A REAL SPACE PART (SUM2) C THIS CHOICE OF G GIVES GOOD CONVERGENCE RATES FOR BOTH PARTS C FOR L A T T I C E S WITH NEAR CUBIC SYMMETRY. G= ROOTPI/C C PREPARE CONSTANTS TO BE USED IN THE SUMMATION LOOPS GSO=G»G SBETA=DSIN(BETA) CBET A = DCOS(BET A) AX = A * SB ETA AZ=A*CBETA WRITE ( 6 , 7 7 ) A . B . C . S B E T A C0NST1=PI/(AX*B*C)*(-4.) C INPUT AND OUTPUT CHARGE AND POSITIONS OF THE IONS IN THE C PRIMITVE C E L L WHICH ARE USED TO GENERATE THE L A T T I C E R E A D ( 5 , 2 0 ) NTYPES READ(5,20) (CHARGE(K),K=1.NTYPES) R E A D ( 5 , 2 0 ) (NEACH(K), K =1,NTYPES) WRITE ( 6 , 2 6 ) WRITE(6,28) DO 11 1=1.NTYPES NIONS = NEACH(I ) WRITE(6,27) CHARGE(I) DO 11 U=1.NI0NS READ(5,22) XCOORD.Y.Z WRITE(6,29) XCOORD.Y.Z C RE-EXPRESS L A T T I C E COORDINATES IN ORTHOGONAL CARTESIAN COORDINATES U(I,J,3)=C*Z+AZ*XC00RD U ( I , J . 1 )=AX*XC00RD 11 U ( I ,<J,2)=B*Y C THE D E R I V A T I V E S V I J OF THE E L E C T R I C POTENTIAL ARE DETERMINED AT THE O R I G I N . C TRANSLATE THE P R I M I T I V E C E L L SO THAT THE FIRST ION READ I N , C THE ION L A B E L L E D 0 = 1 , OF CHARGE L A B E L L E D 1=1. L I E S AT THE O R I G I N . DO 133 K=1 .3 133 ORIGIN(K)=U(1,1,K) DO 12 1=1,NTYPES NIONS=NEACH(I) DO 12 0=1,NIONS DO 12 K = 1 , 3 12 U(I,U,K)=U(I,U,K)-ORIGIN(K) C RECIPROCAL L A T T I C E B A S I S FOR A MONOCLINIC L A T T I C E R1=2.*PI/AX R2=2.*PI/B  62  R3Z = 2 . * P I / C R3X=-2.*PI*CBETA/(C*SBETA) RX=R1+R3X RYS0=R2*R2 RZS0=R3Z*R3Z RLSQ=RX*RX+RYSQ+RZSQ C F I R S T TERM IN THE RECIPROCAL L A T T I C E SUM S1ZERO(1 )=RX*RX/RLSQ S1ZERO(2)=RYSQ/RLSQ S1ZER0(3)=RZSQ/RLSQ S1ZER0(4)=RX*R2/RLSQ S1ZERO(5)=RX*R3Z/RLSO S1ZERO(6)=R2*R3Z/RLSQ C WRITE TABLE HEADINGS WRITE(6,31 ) WRITE(6,32) WRITE(6,33) WRITE(6,23) C INT LABELS THE TERM IN THE SUMMATION WHICH IS STORED IN VIJ C 0 1 , 0 2 , CORRESPOND TO THE CURRENT TOTALS OF Q ( I ) S U M 1 ( I ) , 0 ( I ) S U M 2 ( I ) INT=0 DO 615 1 = 1 , 6 02(I)=0. 615 01(I)=0. DO 116 1 = 1 , 3 116 Q 2 ( I ) = 4 . * G S Q * G / ( 3 . * R O O T P I )*CHARGE(1) C LOOK AT UNIT C E L L S UP TO N-1 C E L L S DISTANT DO 5 NX= 1 ,N N1=NX-1 55 CONTINUE DO 4 NY =1,N N2=NY-1 54 CONTINUE DO 3 N Z = 1 , N N3=NZ-1 53 CONTINUE C FIND CONTRIBUTIONS FROM EACH TYPE OF ION DO 2 1=1,NTYPES DO 91 K=1 ,6 SUM1(K)=0. 91 SUM2(K)=0. C SUM CONTRIBUTIONS FROM ALL IONS OF ONE TYPE NIONS=NEACH(I) DO 1 J = 1 , N I 0 N S C ION C A R T E S I A N COORDINATES FROM THE P R I M I T I V E C E L L ORIGIN U 1 . U 2 . U 3 U1=U(I,U, 1 ) U2=U(I,d,2) U3=U(I,d,3) C DO RECIPROCAL L A T T I C E SUM, SUM 1 RL(1)=N1*R1+N3*R3X RL(2)=N2*R2 RL(3)=N3*R3Z RLSO=RL(1)*RL(1)+RL(2)*RL(2)+RL(3)*RL(3) IF ( R L S O . E O . O ) GO TO 14 REXP=RLSQ/(4.*GSO) IF ( R E X P . G T . 1 7 4 ) GO TO 15 S=DCOS(RL(1)*U1+RL(2)*U2+RL(3)*U3)/(RLSQ*DEXP(REXP)) DO 17 I S = 1 , 3 17 S1(IS)=RL(IS)*RL(IS)*S S1(4)=RL(1)*RL(2)*S  63  S 1 ( 5 ) = RL(1 ) * R L ( 3 ) * S S1(6)=RL(2)*RL(3)*S GO TO 16 14 DO 140 I S = 1 , 6 140 S1(IS)=S1ZER0(IS) 16 DO 160 I S = 1 , 6 160 SUM1(IS)=SUM1(IS)+S1(IS) 15 CONTINUE C DO REAL SPACE L A T T I C E SUM, SUM2 C ION COORDINATES WRT ORIGIN ABOUT WHICH MOMENT IS MEASURED ARE DENOTED X(I) X(1 )=N1*AX+U1 X(2)=N2*B+U2 X(3)=N1*AZ+N3*C+U3 RR=X(1)*X(1)+X(2)*X(2)+X(3)*X(3) IF ( R R . E O . O . ) GO TO 10 R=DSQRT(RR) IF ( ( G S O * R R ) . G T . 1 7 4 . ) GO TO 6 GREXP=1./DEXP(GSQ*RR)*2./R00TPI*G GO TO 66 6 GREXP=0. 66 EF = DERFC(G*R ) DO 4 0 0 I S = 1 , 3 DSO=X(IS)*X(IS) 0UADRU=(3.*DSQ-RR)/(RR*RR) 400 S2(IS)=2.*DSQ/RR*GSQ*GREXP+QUADRU*(GREXP+EF/R) RRR=RR*R TERM=(GREXP*(2.*GSQ*G+3./RRR)+3./RRR*EF)/R S2(4)=X( 1 ) * X ( 2 ) » T E R M S2(5)=X( 1)*X(3)*TERM S2(6)=X(2)*X(3)*TERM DO 120 I S = 1 , 6 120 SUM2(IS)=SUM2(IS)+S2(IS) 10 CONTINUE 1 CONTINUE 0=CHARGE(I) DO 123 K=1 ,6 0 1 ( K ) = 0* SUM1(K) + Q 1 ( K ) 123 02(K)=0*SUM2(K)+02(K) 2 CONTINUE IF ( ( N 1 . N E . N 2 ) . 0 R . ( N 2 . N E . N 3 ) ) GO TO 44 INT=INT+1 DO 126 K=1 ,6 126 VIJ(K,INT)=C0NST1*Q1(K)+Q2(K) 44 CONTINUE IF ( N 3 . L E . 0 ) GO TO 3 N3=-1*N3 GO TO 53 3 CONTINUE IF ( N 2 . L E . 0 ) GO TO 4 N2=-1*N2 GO TO 54 4 CONTINUE IF ( N 1 . L E . 0 ) GO TO 5 N1=-1*N1 GO TO 55 5 CONTINUE C OUTPUT THE 6 INDEPENDENT PARTIAL D E R I V A T I V E S OF THE E L E C T R O S T A T I C POTENTIAL LAST=2*N-1 WRITE(6,60) ((VIJ(L,K),L=1,6).K=1.LAST) CALL PRINC(VIU(1,LAST).VIJ(2,LAST),VId(3,LAST),VId(4.LAST),VIJ(5,LAST).VIJ(6,LAST))  64  60 20 22 23 24 25 26 27 28 29 30 31 32 33 77 C c  FORMAT(6F13.6) FORMAT(6G7.4) F0RMAT(3E13.4) F O R M A T ( / 7 X , ' V X X ' , 10X, ' V Y Y ' , 10X, ' V Z Z ' , 10X, ' V X Y ' , 1 0 X , ' V X Z ' , 10X, ' V Y Z ' , / , 7 8 ( ' * ' )) FORMAT(13,3E17.4) FORMAT(G5.2 ) FORMAT ( / / . ' I O N POSITIONS IN THE UNIT CELL IN L A T T I C E C O O R D I N A T E S ' ) F O R M A T ( / , ' I O N S OF CHARGE ' , F 6 . 2 ) FORMAT( ' X/A Y/B Z/C ' , / , 5 3 ( ' * ' ) ) FORMAT(3F10.4) FORMAT(15) F O R M A T ( / / ' 2 N D ORDER PARTIAL D E R I V A T I V E S OF THE E L E C T R O S T A T I C POTENTIAL AT THE P O S I T I O N ' ) F O R M A T ( ' O F THE F I R S T ION, IN UNITS OF ANGSTROMS, WHERE X L I E S ALONG THE A * - A X I S , ' ) F O R M A T ( ' Y L I E S ALONG THE B-AXIS, AND Z L I E S ALONG THE C-AXIS OF THE L A T T I C E ' ) F O R M A T ( / / , ' L A T T I C E PARAMETERS: A , B, C (ANGSTROMS)',3F9.3/,20X,'SIN(BETA)',F9.4) END  SUBROUTINE PR I N C ( V X X , V Y Y , V Z Z , V X Y , V X Z , V Y Z ) *****************************************************************************  C FINDS AND OUTPUTS THE P R I N C I P A L E L E C T R I C F I E L D GRADIENTS VII FOR GIVEN V I J * ******»**»»**»*,,**,»***»»»»*»*,*****»*****»»**»**.****»*******,»,,„*»»****»* c  15 16  2  COMPLEX ROOT,S1 , S 2 , V ( 3 ) , I M , R E A2=VXY*VXY B2=VXZ*VXZ C2=VYZ*VYZ X1=VXX*VYY+VXX*VZZ+VYY*VZZ-A2-B2-C2 XO=A2*VZZ+B2*VYY+C2*VXX-2*VXY*VXZ*VYZ-VXX*VYY*VZZ 0=X1/3. P=X0/(-2.) ROOT=CSQRT(CMPLX(Q*Q*Q+P*P.O.)) S1=CEXP((CLOG(P+ROOT))/3.) S2 = C E X P ( ( C L O G ( P - R O O T ) ) / 3 . ) V( 1 )=S1+S2 IM=(0. , 1 . ) * S Q R T ( 3 . ) / 2 . * ( S 1 - S 2 ) RE = - . 5 * V ( 1 ) V(2)=RE+IM V(3)=RE-IM DO 15 1=1,3 IF ( A I M A G ( V ( I ) ) . N E . O . ) W R I T E ( 6 , 1 6 ) FORMAT( ' A P R I N C I P A L VII IS COMPLEX: INPUT V I J ARE NOT PHYSICALLY VALID') V1=REAL(V( 1 ) ) V2 = R E A L ( V ( 2 ) ) V3 = R E A L ( V ( 3 ) ) WRITE(6,2) V1.V2.V3 F O R M A T ( / / ' T H E P R I N C I P A L D E R I V A T I V E S A R E : ' , / , ' V 1 1 =' ,E 1 2 . 4 / , ' V22 = ' ,E 1 2 . 4 / , ' V 3 3 RETURN END  =',E12.4)  APPENDIX D - OUTPUTS OF PROGRAM 1 FOR MANGANESE FLOURIDE AND MANGANESE CHLORIDE TETRAHYDRATE  1 2 3 4 5 6 7 8 g 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38  MANGANESE LATTICE  FLOURIDE  PARAMETERS:  ION P O S I T I O N S X/A  A , B, C (ANGSTROMS) SIN(BETA) 1.0000  IN THE UNIT C E L L Y/B Z/C  IN L A T T I C E  4.870  4.870  3.310  COORDINATES  ***************************************************** IONS OF CHARGE 2.00 0.0 0.0 0.5000 0.5000  0.0 0.5000  IONS OF CHARGE - 1 . 0 0 0.3100 0.3100 -0.3100 -0.3100 O.8100 0.1900 -0.8100 -0.1900  0.0 0.0 O.5000 -0.5000  2ND ORDER P A R T I A L D E R I V A T I V E S OF THE ELECTROSTATIC POTENTIAL AT THE POSITION OF THE F I R S T I O N , IN UNITS OF ANGSTROMS, WHERE X L I E S ALONG THE A * - A X I S , Y L I E S ALONG THE B - A X I S , AND Z L I E S ALONG THE C-AXIS OF THE L A T T I C E VXX VY Y VZZ VXY VXZ VYZ ****************************************************************************** 0.092355 0.095193 0.325637 -0.275065 0.004786 0.006808 0.066398 0.050975 0.158975 -0.175824 -0.000489 -0.000492 0.031702 0.024977 -0.049819 -0.043569 -0.000020 -0.000023 0.028311 0.024937 -0.049842 -0.043093 0.000001 -0.000005 0.024924 0.024924 -0.049859 -0.042854 0.000001 -0.000005 THE V11 V22 V33  PRINCIPAL DERIVATIVES ARE: = 0.6778E-01 = -0.4985E-01 = -0.1793E-01  66  1 • 2 3 4 5 6 7 8 9 10 1 1  MnC12*4H20 n e g l e c t i n g LATTICE  PARAMETERS:  12 13 14 15 16 17  18 19 20 21 22 23 24 25 26 27 28  29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46  POSITIONS X/A  molecules  A , B, C (ANGSTROMS) SIN(BETA) 0.9856  1  ION  the water  IN THE UNIT C E L L Y/B Z/C  IONS OF CHARGE 2 0 . 0 0 0.2329 0 ,.1714 0.7329 0 . 3286 0.7671 0 . 8286 0.2671 0 . 6714  0 . 9865 0 . 4865 0 . 0135 0 . 5135  IONS OF CHARGE 0.0 0.0610 0 . 3076 0.5610 0 . 1924 0.9390 0 . 6924 0.4390 0 . 8076 0.3817 0 . 3662 0.8817 0 . 1338 0.6183 0 . 6338 0.1183 0 . 8662  0 . 0938 0 . 5938 0 . 9062 0 . 4062 0 . 0355 0 . 5355 0 . 9645 0 . 4645  IN  11.860  9 .513  6 . 186  L A T T I C E COORDINATES  2ND ORDER P A R T I A L D E R I V A T I V E S OF THE E L E C T R O S T A T I C POTENTIAL AT THE POSITION OF THE F I R S T I O N , IN UNITS OF ANGSTROMS, WHERE X L I E S ALONG THE A * - A X I S , Y L I E S ALONG THE B - A X I S , AND Z L I E S ALONG THE C-AXIS OF THE L A T T I C E VXX -0.027778 -0.029913 0.048122 0.005525 -0.036709 -0.036745 -0.036781 THE V11 V22 V33  VYY 0 . 105880 0. 292075 0.278755 0.274889 0.274417 0.274413 0.274408  P R I N C I P A L D E R I V A T I V E S ,ARE : = 0.5804E+00 = -0.9479E+00 = 0.3675E+OQ  VZZ  VXY  -0.386016 -0.211707 -0.237241 -0.237235 -0.237536 -0.237549 -0.237562  -0.329176 -0.335763 -0.312874 -0.313642 -0.314513 -0.314495 -0.314475  -0 -0 -0 -0 -0 -0 -0  VXZ VYZ ******************** .5596 18 - 0 . 596682 .528808 - 0 . 493323 .518889 - 0 . 500053 .519308 - 0 . 500403 .519076 - 0 . 500361 .519050 - 0 . 500358 .519024 - 0 . 500355  67  1 2 3 4 5 6 7 8 g 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 . 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60  MANGANESE LATTICE  CHLORIDE  TETRAHYDRATE  PARAMETERS:  A , B, C (ANGSTROMS) SIN(BETA) 0.9856  11.860  9.513  I ON POSITIONS IN THE UNIT CELL IN L A T T I C E COORDINATES X/A Y/B Z/C ***************************************************** IONS OF CHARGE 2.00 O.2329 0 . 1 7 14 0.7329 0.3286 0.7671 0.8286 0.2671 0.6714  0.9865 0.4865 0.0135 0.5135  IONS OF CHARGE - 1 . 0 0 0.0610 0.3076 0.5610 0.1924 0.9390 0.6924 0.4390 0.8076 0.3817 0.3662 0.8817 0.1338 0.6183 0.6338 0.1183 0.8662  0.0938 0.5938 0.9062 0.4062 0.0355 0.5355 0.9645 0.4645  IONS OF CHARGE - 1 . 0 0 0.3010 0.1127 0.8010 0.3873 0.6990 0.8873 O.1990 0.6127 0.1568 0.2280 0.6568 0.2720 0.8432 0.7720 0.3432 0.7280 0.1323 0.9736 0.6323 0.5264 0.8677 0.0264 0.3677 0.4736 0.3695 0.0381 0.8695 0.4619 0.6305 0.9619 0.1305 0.5381  0.3334 0.8334 0.6666 0.1666 0.6446 0.1446 0.3554 0.8554 0.9590 0.4590 0.0410 0.5410 0.8764 0.3764 0.1236 0.6236  IONS OF CHARGE 0.50 0.3831 0.1418 0.8831 0.3582 0.6169 0.8582 O.1169 0.6418 0.3005 0.0154 0.8005 0.4846 0.6995 0.9846 0.1995 0.5154 0.0728 0.2000 0.5728 0.3000 0.9272 0.8000 0.4272 0.7000 0.1989 0.1967 0.6989 0.3033 0.8011 0.8033  0.3882 0.8882 0.6118 0.1118 0.3657 0.8657 0.G343 0.1343 0.5993 0.0993 0.4007 0.9007 0.5288 0.0288 0.4712  L  6.186  6 8  61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98  0.3011 0.1105 0.6105 0.8895 0.3895 0.0996 0.5996 0.9004 0.4004 0.4356 0.9356 0.5644 0.0644 0.3616 0.8616 0.6384 0.1384  0.6967 0.9277 0.5723 0.0723 0.4277 0.9292 0.5708 0.0708 0.4292 0.0716 0.4284 0.9284 0.5716 0.9389 0.5611 0.0611 0.4389  0.9712 0.8199 0.3199 -0.1801 0.6801 0.0681 0.5681 0.9319 0.4319 0.8061 0.3061 0.1939 0.6939 0.8540 0.3540 0.1460 0.6460  2ND ORDER P A R T I A L D E R I V A T I V E S OF THE ELECTROSTATIC POTENTIAL AT THE POSITION OF THE F I R S T ION. IN UNITS OF ANGSTROMS, WHERE X L I E S ALONG THE A * - A X I S . Y L I E S ALONG THE B - A X I S . AND Z L I E S ALONG THE C-AXIS OF THE L A T T I C E VXX VY Y VZZ VXY VXZ *********************************************************** 0.025639 0.063927 -0.013792 0.151370 0.007280 -0.019269 0.015972 0.013264 -0.001291 -0.000324 -0.016840 0.013157 0.009904 -0.007849 -0.009113 -0.019769 0.012807 0.009922 -0.007880 -0.009166 -0.022666 0.012773 0.009911 -0.007921 -0.009157 -0.022672 0.012772 0.009908 -0.007921 -0.009153 -0.022677 O.012772 0.009905 -0.007922 -0.009149 THE V11 V22 V33  PRINCIPAL DERIVATIVES ARE: = 0.2551E-01 = -0.2576E-01 = 0.2438E-03  VYZ -0.020507 0.010725 0.011133 0.011092 0.011094 0.011094 0.011094  69  APPENDIX E - PROGARM 2  C PROGRAM TO GENERATE THE NMRON SPECTRUM FOR 54-MN IN M N C L 2 * ( 4 H 2 0 ) * C WITH QUADRUPOLE S P L I T RESONANCES AND S P I N - L A T T I C E RELAXATION * ***».*.»,**»**«.*»»******»***,*»»,**,».***. ,**•****,****».,*«,*•***** c  C  c***.***********************************************»**********»********,***** C MAIN PROGRAM * C READS SPECTRUM PARAMETERS, FINDS AND OUTPUTS NUMBER OF COUNTS VS FREQUENCY * »*****.************.*************************«»**********»**********.»**»**** COMMON P ( 8 ) , P 1 ( 8 ) , W D ( 8 , 8 ) , P T 1 ( 8 ) , W U ( 7 , 8 ) ,INT,LEX, *LSEP,KEND,WA,A,B,LINE,COLD CDMMON/BC/SIGMA COMMON/CC/STEP C C V A R I A B L E TYPES c  LOGICAL*1 L A B E L ( 2 5 ) R E A L M X(4096) , Y(4096) INTEGER A . B . C  C C INPUT AND ECHO SPECTRUM PARAMETERS £****«»***************************** 101  201 1001 102  104 1004 202 1002 103 203 303 106 1003  WRITE(6,101) FORMAT( ' E N T E R ( R E A L ) L E V E L SEPARATION ( K E L V I N S ) & " C O L D " L A T T I C E TEMPERATURE ( K E L V I N S ) ' ) READ(5 , 1001 ) E K , T WRITE(G,600) EK,T WRITE(G,201) F O R M A T ( ' E N T E R ( R E A L ) RELAXATION TIME & WARM C O U N T ' ) R E AD(5 . 1001) T1,WA W R I T E ( 6 , 6 0 0 ) T1.WA FORMAT(4G13.5) WRITE(6,102) F O R M A T ( ' E N T E R ( I N T E G E R ) DWELL T I M E , TIME & NUMBER OF FIRST E X C I T A T I O N ' ) READ(5,1002) INT,LEX,LINE WRITE(G,601) INT,LEX,LINE W R I T E ( 6 , 104) F O R M A T ( ' E N T E R ( R E A L ) FREQUENCY S T E P S I Z E , FREQUENCY OF FIRST E X C I T A T I O N IN THE SPECTRUM') READ(5,1004) STEP,EXFREQ WRITE(6,600) STEP,EXFREQ FORMAT(2G13.5) WRITE(6,202) F O R M A T ( ' E N T E R ( I N T E G E R ) E X C I T A T I O N SEPARATION IN T I M E , END OF RUN TIME ( S E C O N D S ) ' ) R E A D ( 5 , 1 0 0 2 ) LSEP.KEND WRITE(6,601) LSEP.KEND F0RMAT(4I8) W R I T E ( 6 , 103) F O R M A T ( ' E N T E R 1 IF FORWARD RUN, 0 IF BACKWARD RUN ' ) READ(5,1003) A WRITE(G,203) F O R M A T ( ' E N T E R 1 FOR WRITEOUT 0 O T H E R W I S E ' ) READ(5,1003) B WRITE(6,303) F O R M A T ( ' E N T E R 1 IF PLOT IS DESIRED OTHERWISE 0') READ(5,1003) C WRITE(G,106) F O R M A T ( ' E N T E R 1 IF THE E X C I T . FREQ. DISCONTINUITY IS TO BE SMOOTHED, 0 OTHERWISE') R E A D ( 5 , 1 0 0 3 ) NOJUMP WRITE(6,602) A,B,C.NOOUMP F0RMAT(3I3) WRITE(6,105)  3WI11-X=31 I N I ' X ' t=3 w i n L oa (\+lNI/(X31-ON3X))*d31S-03adX3=iSN00 6 O i 0 3 ( r '03'V) d l U ****************** 0 ( o = v ) N n a aavwxova o (909'9)3iiart  (909'9)31ia«  u o i oo ( o o a a j d i **************************************************************0 U = a) a s i s a n o s a s i i n d i n o ivoiaswnN v J I SONIOVSH s i a v i s n a w o t+iNI/0N3>l = lSVT i +aN3X = >( #***************************************************************************0 N o u o a a i a  Nna iviN3wia3dX3  S A (A)SiNnoo/*' l a s o 3 o=(8'i)n« •o=(i'8)a« znoa»(w'uw)a«=(t+w'w)n« s A ' t =w G o a ( ( i - w ) * ( t - i - w ) - 1 * ( + i ) )*OM=(W' t +w)a« P L 'i=w f o a e=i avaHG s i s n s i o n N S H I d o W H I N S W O W a v i n o N v i v i o i a (znoa+i. )/n/i=o« ( i / X 3 )dX3/l- = z i ~ i o a  OSIVOIONI  t+w <- w a o d ( t + w ' w ) n « W <- l + W aOd (W'l+W)0M  3 H i aod ( x ) A 0 N 3 n 0 3 d d  :S13A3~I  l+W S W N 3 3 M 1 3 3 xvi3a S N i i n o a a n s  N O I i l S N V d i dO 3 i V d N I 3sn aod aavdsad  wns/(w)d=(w)d z.' i = w e o a ( w ) d + wns=wns L ' t =w s o a •o=wns (1/»3*W*'t-)dX3=(W)d i't=w i oa  o 0 3 3 e z  t  S13A31 N i d s 3 H i a o d ( w ) d N O i i n a i a i s i a N v w z i i o a 3 x v i s a ONV SiNnoo s s N i i n o a a n s N I s s n aod aavaaad 3 3 0=(8)lid 0=(8)Id 0=(8)d *********************************************************D N l d S H i 8 , 3 H i dO N O I i V l R d O d ONIHSINVA V S O n a O d i N I 3 x v T 3 a 3 N i i n o a a n s N I S N O i s s 3 a a x 3 A d i i a w i s 01 o 3 VW9IS*'I-=VW9IS ( 0 ' 0 3 ' V ) d l ***************************+*******************************3 s s n v o 3 N i i n o a a n s a o d v w o i s d o N O I S a s a a A a a Nna a a v w x o v a d i o 3 anoD (009'9)3iiaM a n o 3 (voov ' s ) a v a a i v ) a i o o 3av HOIH/A S N i a s do N 0 i i 3 v a d ( n v s a ) a 3 i N 3 - ) i v w a o d sot (801'9)3iia« VWOIS (009'9)3ild« vwsis (toot's)av3a  w A 3 a3~mavn  (-(3Aoav  i 3aniva3dW3i  (/ ( S 0 N 0 3 3 S ) N O i i n a i a i S i a  Nvissnvo  3Hi do N O i i v i A a a  swa ( i v 3 a ) a 3 i N 3 , U v w a o d  OL  got  71  CALL C O U N T S ( L E . Y Y ) INDEX=(LTIME-1)/INT+1 X(INDEX)=CONST+INDEX*STEP Y(INDEX)=YY 7 CONTINUE GO TO 19 C FORWARD RUN (A=1) £•****•*******,»,»*** 9  CONST=EXFREO-STEP*(1+LEX/INT) DO 8 L T 1 M E = 1 , K . I N T LE = L T I M E - 1 NL=LE/INT+1 CALL C O U N T S ( L E , Y Y ) X(NL)=CONST+STEP*NL Y(NL)=YY CONTINUE  8 C C SMOOTH THE  EXCITATION  DISCONTINUITIES  IN  ^COUNTS IF  REQUESTED  ( ; , * » « » * * * » * » » » * » « * * * * * * * * , » » » * * « * » » « » » » * , * * • • » , » * . * , * » * * » » » * * « »  19  IF ( N O J U M P . E O . O ) GO TO 500 DO 50 L= 1 , L A S T YL=Y(L) Y(L ) = SMOOTH(L,YL)  50 C C PLOT NUMBER OF COUNTS VS FREQUENCY IF REQUESTED (C=1) C MOVEC, FREAD, PICT ARE SUBROUTINES WRITTEN BY MARY ANN POTTS  (;;*****»*******»*».»»***»*•***»*.*,»*,*••*******»**«,»»*****»*, 500  IF ( C . E O . O ) GO TO 89 CALL M 0 V E C ( 2 5 , ' ',LABEL(1)) CALL FREAD(6,'S:'.LABEL,25) DX = X(2 )-X( 1 ) CALL PICT(LAST,X,Y,DX,LABEL)  C C OUTPUT NUMBER TABLE OF COUNTS VS FREQUENCY IF REQUESTED (B=1) ;;»,»****«»****•****«*»***,»*,»*««,,*,,*»,,*»*•*»»»*•,,*•*.*»,*** 89 99 10 20 30 60 70 80 600 601 602 605 606 C  IF ( B . E Q . O ) GO TO 99 WRITE(6,70) W R I T E ( 6 , 8 0 ) ( X ( I ) , Y ( I ) ,1 = 1 , L A S T ) RETURN F O R M A T ( 2 ( F 7 . 5 . 2 X ) , F 7 . 1 , 2 X , 4 ( 1 5 , 2 X ) , F 9 . 1 , 3 ( 2 X , I 1)) FORMAT( ' INPUT DATA ' , / 2 E 1 3 . 5 . / 2 E 1 3 . 5 , / 2 1 8 , / 2 1 8 , / 3 1 3 , / 2 E 1 3 . 5 ) F O R M A T ( ' PT1 =',7E13.5,/) F0RMAT(25A1,//) FORMAT ( / / 2 X , ' FREQUENCY ^COUNTS' , / 2 X , ' * * * * * * * * * * * * * * * * * * * * * * * » * * ' ) F0RMAT(E13.5,2X,E13.5) F0RMAT(2E13.5) F0RMAT(3I8) F0RMAT(4I3) F0RMAT(//2X,'TIME',4X,'NORMALIZED INTENSITY',6X,'POP PROB OF THE SPIN L E V E L S ' ) F O R M A T ( 3 0 X , 6 X , ' P 1 ' , 1 1X, ' P 2 ' . 1 1X, ' P 3 ' , 11X, ' P 4 ' , 11X, ' P 5 ' , 11X, ' P 6 ' , 11X, ' P 7 ' ) END  SUBROUTINE COUNTS(L.Y) »»»*„*«******»»»*»***»*»»**»««»»*»*»,»**.,*,*»**»»«,*«»**,•*•**••*.*** C C A L L E D FROM MAIN c  C FINDS THE  EXPECTED NUMBER  OF COUNTS FOR THE FREQUENCY  COMMON P ( 8 ) , P 1 ( 8 ) , W D ( 8 , 8 ) , P T 1 ( 8 ) , W U ( 7 , 8 ) . I N T , L E X , *LSEP,KEND,WA.A,B,LINE,COLD C0MM0N/AC/YMAX(6),YMIN(6)  LABELLED  BY  L *  72  COMMON/BC/SIGMA INTEGER A , B IF ( A . E O . 1 ) GO TO 11 LP=LEX+5*LSEP C FOR BACKWARD RUN IF FREQUENCY > HIGHEST  ************  FREQUENCY OF E X C I T A T I O N  C POPULATION OF THE S P I N L E V E L S IS GOVERNED BY THE BOLTZMAN DISTRIBUTION * £************************************************************************* IF ( L . G T . L P ) GO TO 12 GO TO 14 Q*********************************************************************** C FOR FORWARD RUN, I F FREQUENCY < LOWEST FREQUENCY OF E X C I T A T I O N C POPULATION OF THE S P I N L E V E L S IS GOVERNED BY THE BOLTZMAN DISTRIBUTION ^************************************************************************* IF ( L . L T . L E X ) GO TO 12 11 GO TO 14 IF ( ( L . N E . O ) . A N D . ( L ,, N E . K E N D ) ) GO TO•33 12 DO 13 M=1 ,7 P1(M)=P(M) 13 CALL EMISSN(Y) GO TO 33 C  *  L CORRESPONDS TO THETRANSITION ENERGY BETWEEN 2 ADJACENT C C IF I .E. CORRESPONDS TO ONE OF THE 6 RESONANCE FREQUENCIES C AVERAGE AND EQUALIZE THE POPULATIONS OF THESE TWO L E V E L S Q******************************************************************** 14 N=1 15 LEX2 = LEX+(N-1 ) * L S E P IF ( ( L . G T . L E X 2 - 1 ) . A N D . ( L . LT . LEX2+1 ) ) GO TO 16 GO TO 18 16 DO 38 M= 1 ,7 38 PT1(M)=P1(M) LEVEL=LINE+N AVRAGE = ( P 1 ( L E V E L - 1 ) + P 1 ( L E V E L ) )/2 PT1(LEVEL)=AVRAGE PT1(LEVEL-1)=AVRAGE DO 39 M=1,7 39 P1(M)=PT1(M) Q********************************************************** C STORE ^COUNTS IMMEDIATELY BEFORE AND AFTER RESONANCE(N) * C IN Y M I N ( N ) , YMAX(N) FOR USE IN SUBROUTINE SMOOTH * Q********************************************************** YMIN(N)=Y CALL EMISSN(Y) YMAX(N) = Y GO TO 33 18 N=N+1 IF ( N . L T . 8 - L I N E ) GO TO 15 C C IF NOT A RESONANCE FREQUENCY THEN LEVEL POPULATIONS RELAX TOWARD EQUILIBRIUM Q*********************************************************************** CALL RELAX CALL EMISSN(Y) C C PRINT A NUMERICAL TABLE OF POPULATION OF L E V E L S VS TIME I F REQUESTED Q********************************************************************* 33 C  IF ( B . E Q . 1 ) RETURN END  CALL  WRIT(L.Y)  73  SUBROUTINE  RELAX  C****.****».*...«********.**»,**.***,**»*.*********************************************** C C A L L E D FROM SUBROUTINE COUNTS * C FINDS NORMALISED POPULATIONS P 1 ( M ) , OF THE SPIN L E V E L S DURING S P I N - L A T T I C E RELAXATION * C**************************«*************^ COMMON P ( 8 ) , P 1 ( 8 ) . W D ( 8 , 8 ) , P T 1 ( 8 ) , W U ( 7 , 8 ) , I N T , L E X , *LSEP,KEND,WA,A,B PT1(1)=P1(1)+INT*(WD(2,1)*(P1(2)-P(2))-WU(1,2)*(P1(1)-P(1))) DO 41 M=2,7 41 PT1(M)=P1(M)+INT*(WD(M+1,M)*(P1(M+1)-P(M+1))-WU(M,M+1)*(P1(M) * - P ( M ) ) + W U ( M - 1 , M ) * ( P 1 ( M - 1 )-P(M-1) ) - W D ( M , M - 1 ) * ( P 1 ( M ) - P ( M ) ) ) DO 42 M= 1 ,7 42 P1(M)=PT1(M) RETURN END C SUBROUTINE E M I S S N ( Y ) *******»**********************************,****************»*******.***»* c  C C A L L E D FROM SUBROUTINE COUNTS * C FINDS THE NUMBER OF COUNTS EXPECTED FOR THE POPULATION OF L E V E L S P1(M) * ******»*****»************************************************************ c  COMMON  P(8),P1(8),WD(8,8),PT1(8),WU(7,8),INT,LEX,  * LSEP,KEND,WA,A,B,LINE,COLD **************************************************************** C GAMMA+1 = W(O), IS THE NORMALISED GAMMA-RAY INTENSITY EMITTED * C ALONG THE AXIS OF MAGNETISATION FOR COLD=1, * C WHERE COLD IS THE FRACTION OF MN-54 NUCLEI AT TEMPERATURE T * Q***********************************,**************************** GAMMA= . 3 3 3 * ( P 1 ( 3 ) + P 1 ( 5 ) ) + . 667 *(P 1 (2 )+ P 1 (6 ) ) - ( P 1 ( 1 ) + P 1 ( 7 ) ) ******»««»,*«,»*».*.*»**»»*,«*****»*****,************************** c  c  C THE NORMALISED GAMMA-RAY INTENSITY EMITTED FOR COLD ARBITRARY C Y/WA = ( 1 - C O L D ) + COLD*W(0) c  IS * *  *«*****„*.**»*»»***,*****,*»*».*.»«*»*»»***»********,»******»,*».*,  C  Y = WA*( 1+COLD*GAMMA ) RETURN END  SUBROUTINE W R I T ( L T I M E , Y ) Q***************************************************************************** C CALLED FROM SUBROUTINE COUNTS  C PRINTS OUT A TABLE OF POPULATION OF L E V E L S P1(M) AND R E L A T I V E ^COUNTS VS TIME * Q***************************************************************************** COMMON P(8),P1(8),WD(8,8),PT1(8),WU(7,8),INT,LEX, * LS E P , K E N D , W A , A , B INTEGER A , B REL=Y/WA WRITE(6,40) LTIME.REL WRITE(6,30) (P1(M),M=1,7) 40 F0RMAT(I6,3X,E13.5) 30 F0RMAT(30X,7E13.5) RETURN END C FUNCTION G A U S S ( N . N E X ) Q*********************************************************** C C A L L E D FROM SUBROUTINE SMOOTH * C GENERATES GAUSSIAN D I S T R I B U T I O N WITH RMS D E V I A T I O N SIGMA * Q* ********************************************************* * COMMON P(8),P1(8),WD(8,8),PT1(8),WU(7,8),INT,LEX, *LSEP,KEND,WA,A,B  74  87 88 C  COMMON/BC/SIGMA COMMON/CC/STEP XPON=(NEX-N)/INT*STEP/SIGMA XPON=XPON*XPON/2. IF ( X P O N . L T . 1 7 4 ) GO TO 87 GAUSS=0. GO TO 88 GAUSS=1 . / E X P ( X P O N ) RETURN END FUNCTION S M O O T H ( L S , Y )  C C A L L E D FROM MAIN C SMOOTHS THE E X C I T A T I O N FREOUENCY DISCONTINUITY  116  114 115  117 118 111  * WITH A GAUSSIAN *  COMMON P(8),P1(8),WD(8,8),PT1(8),WU(7,8),INT,LEX, *LSEP,KEND,WA,A,B C0MM0N/AC/YMAX(6),YMIN(6) IF ( A . E O . O ) GO TO 114 L=(LS-1)*INT IF ( L . G E . L E X + 5 * L S E P ) GO TO 118 N= 1 LEXN=LEX+(N-1)*LSEP IF ( L . L T . L E X N ) GO TO 117 N = N+1 GO TO 116 L=KEND-INT*(LS-1) IF (L . LE . L E X ) GO TO 118 N=6 LEXN=LEX+(N-1)*LSEP IF ( L . G T . L E X N ) GO TO 117 N = N-1 GO TO 115 SMOOTH=GAUSS(L,LEXN)*(YMAX(N)-YMIN(N))+Y GO TO 111 SMOOTH=Y RETURN END  75  APPENDIX F - SAMPLE OUTPUTS OF PROGRAM 2  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19  E N T E R ( R E A L ) L E V E L SEPARATION ( K E L V I N S ) & " C O L D " L A T T I C E TEMPERATURE ( K E L V I N S ) 0.24000E-01 0.70000E-01 E N T E R ( R E A L ) RELAXATION TIME & WARM COUNT 0.40000E+03 0.2S422E+06 E N T E R ( I N T E G E R ) DWELL T I M E , TIME 8. NUMBER OF F I R S T E X C I T A T I O N 20 300 1 E N T E R ( R E A L ) FREQUENCY S T E P S I Z E , FREQUENCY OF FIRST E X C I T A T I O N IN THE SPECTRUM 0.15000E+00 0.50040E+03 E N T E R ( I N T E G E R ) E X C I T A T I O N SEPARATION IN T I M E , END OF RUN TIME (SECONDS) 400 2400 ENTER 1 IF FORWARD RUN, 0 IF BACKWARD RUN ENTER 1 FOR WRITEOUT 0 OTHERWISE ENTER 1 IF PLOT IS DESIRED OTHERWISE 0 ENTER 1 IF THE E X C I T . FREQ. DISCONTINUITY IS TO BE SMOOTHED, 0 OTHERWISE 1 O 1 O E N T E R ( R E A L ) RMS D E V I A T I O N OF THE GAUSSIAN D I S T R I B U T I O N (SECONDS) 0.20000E+00 E N T E R ( R E A L ) FRACTION OF SPINS WHICH ARE COLD (AT TEMPERATURE T ABOVE) 0.1O0O0E+01  FORWRRD—NO-SMOOTHING  261348 258859 256370 253881 S251392 ; 248903 246414 243925 -  ''I  241436 -  ^ffimH r BniHnin jiffliiHnrHnn'innBiBBinHiil,[  1  238947 236458 I3.1  439.8  500.5  SOI.7  502.9  5 0 4 . 1 505.3  _l_ 306.5  507.7  306. B  _1_ _l_ _1_ 51D.1 5 1 1 . 5 5 1 2 . 5 513.7 514.9  FREQUENCY (MHZ)  518.1  76  End  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19  of  f i l e  E N T E R ( R E A L ) L E V E L SEPARATION ( K E L V I N S ) & " C O L D " L A T T I C E TEMPERATURE ( K E L V I N S ) 0.24000E-01 0.70000E-01 E N T E R ( R E A L ) RELAXATION TIME & WARM COUNT 0.40000E+03 0.26422E+06 E N T E R ( I N T E G E R ) DWELL T I M E , TIME & NUMBER OF FIRST E X C I T A T I O N 20 200 1 E N T E R ( R E A L ) FREQUENCY S T E P S I Z E , FREQUENCY OF F I R S T E X C I T A T I O N IN THE SPECTRUM 0.15000E+00 0.5O040E+O3 E N T E R ( I N T E G E R ) E X C I T A T I O N SEPARATION IN T I M E , END OF RUN TIME (SECONDS) 400 2400 ENTER 1 IF FORWARD RUN, 0 IF BACKWARD RUN ENTER 1 FOR WRITEOUT 0 OTHERWISE ENTER 1 IF PLOT IS DESIRED OTHERWISE 0 ENTER 1 IF THE E X C I T . FREO. DISCONTINUITY IS TO BE SMOOTHED, 0 OTHERWISE 1 0 1 1 E N T E R ( R E A L ) RMS D E V I A T I O N OF THE GAUSSIAN D I S T R I B U T I O N (SECONDS) 0.20000E+00 E N T E R ( R E A L ) FRACTION OF SPINS WHICH ARE COLD (AT TEMPERATURE T ABOVE) 0.10000E+01  GRUSSIRN-SMQQTHING  261348 -  I  258859 256370 253881 g251392 |•248903 246414 |243925 241436 238947 236458 .9 500.1 581.3 S02.5 503.7 504.B 506.0 507.2 50B.4 509.B 511.fl 312.1 513 2 314.4 SI J • M  FREQUENCY (MHZ)  »  77  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1G 17 18 19  E N T E R ( R E A L ) L E V E L SEPARATION ( K E L V I N S ) & " C O L D " L A T T I C E TEMPERATURE ( K E L V I N S ) 0.24000E-01 0.70000E-01 E N T E R ( R E A L ) RELAXATION TIME & WARM COUNT 0.40000E+03 0.26422E+06 E N T E R ( I N T E G E R ) DWELL T I M E , TIME & NUMBER OF FIRST E X C I T A T I O N 20 300 1 E N T E R ( R E A L ) FREQUENCY S T E P S I Z E , FREQUENCY OF F I R S T E X C I T A T I O N IN THE SPECTRUM 0.15000E+00 0.50040E+03 E N T E R ( I N T E G E R ) E X C I T A T I O N SEPARATION IN T I M E , END OF RUN TIME (SECONDS) 400 2400 ENTER 1 IF FORWARD RUN, 0 IF BACKWARD RUN ENTER 1 FOR WRITEOUT 0 OTHERWISE ENTER 1 IF PLOT IS DESIRED OTHERWISE 0 ENTER 1 I F THE E X C I T . F R E Q . DISCONTINUITY IS TO BE SMOOTHED, 0 OTHERWISE 0 0 1 0 E N T E R ( R E A L ) RMS D E V I A T I O N OF THE GAUSSIAN D I S T R I B U T I O N (SECONDS) 0.20000E+00 E N T E R ( R E A L ) FRACTION OF SPINS WHICH ARE COLD (AT TEMPERATURE T ABOVE) 0.10000E+01  BRCKWRRD-RUN 261500 258977 256454  to  253931  • 251408 ; 248865 246362  \  243839 241316  inn ^uiiiHEiiiHH^EmiiiniiEiiinE iiiHiimiiimiii jiinnMniimii'  238793 236270 464.8  4B5.B  4D7.0  490.2  JL  409 4  JL  430.B  J_  431.8  JL  433.B  494.2  JL  435.4  JL  498.0  FREQUENCY (MHZ)  J_  437.8  JL 499.0  500.2  JL  J_  5 0 1 . 4 SDZ.8  78  1 2 3  ENTER(REAL)  LEVEL  0.24000E-01 ENTER(REAL)  SEPARATION  0.40000E+03  0.26422E+06  5  ENTER(INTEGER)  DWELL T I M E .  6  20  8  ENTER(REAL)  & "COLD"  LATTICE  TEMPERATURE  (KELVINS)  RELAXATION TIME & WARM COUNT  4  7  (KELVINS)  0.7OOOOE-01  300 FREQUENCY  0.15000E+00  TIME  & NUMBER OF FIRST  EXCITATION  1 STEPSIZE,  FREQUENCY  OF F I R S T  EXCITATION  IN  THE  SPECTRUM  0.50040E+03  9 10 11 12 13 14 15  E N T E R ( I N T E G E R ) E X C I T A T I O N SEPARATION IN T I M E , END OF RUN TIME (SECONDS) 400 2400 ENTER 1 IF FORWARD RUN, 0 IF BACKWARD RUN ENTER 1 FOR WRITEOUT O OTHERWISE ENTER 1 IF PLOT IS DESIRED OTHERWISE 0 ENTER 1 IF THE E X C I T . F R E Q . DISCONTINUITY IS TO BE SMOOTHED, 0 OTHERWISE 0 0 0 1  16  ENTER(REAL)  17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60  0.20000E+00 E N T E R ( R E A L ) FRACTION 0.10000E+01  RMS D E V I A T I O N OF THE GAUSSIAN D I S T R I B U T I O N OF SPINS WHICH ARE  FREQUENCY /fCOUNTS ************************** 0.48465E+03 0.24043E+06 0.48480E+03 0.24039E+06 0.48495E+03 0.24014E+06 0.48510E+03 0.23924E+06 O.48525E+03 0.23766E+06 0.48540E+03 0.23676E+06 0.48555E+03 0.23807E+06 0.48570E+03 0.23883E+06 0.48585E+03 0.23927E+06 0.48600E+03 0.23954E+06 0.48615E+03 0.23972E+06 0.48630E+03 0.23985E+06 0.48645E+03 0.23996E+06 0.48660E+03 0.24004E+06 0.48675E+03 0.24012E+06 0.48690E+03 O.24018E+06 0.48705E+03 0.24024E+06 0.48720E+03 O.24028E+06 0.48735E+03 0.24032E+06 0.48750E+03 0.24035E+06 0.48765E+03 0.24038E+06 0.48780E+03 0.24042E+06 0.48795E+03 0.24050E+06 0.48810E+03 0.24077E+06 0.48825E+03 0.24123E+06 0.48840E+03 0.24148E+06 0.48855E+03 0.24070E+06 0.48870E+03 0.24027E+06 0.48885E+03 0.24009E+06 0.48900E+03 0.24006E+06 0.48915E+03 0.24009E+06 0.48930E+03 0.24015E+06 0.48945E+03 0.24022E+06 0.48960E+03 0.24028E+06 0.48975E+03 0.24034E+06 0.48990E+03 0.24038E+06 0.49005E+03 0.24042E+06  . . . .  COLD (AT  (SECONDS)  TEMPERATURE  T ABOVE)  E N T E R ( R E A L ) L E V E L SEPARATION ( K E L V I N S ) 6 "COLD" L A T T I C E TEMPERATURE ( K E L V I N S ) 0.24000E-01 0.64000E-01 E N T E R ( R E A L ) RELAXATION TIME 8. WARM COUNT 0.23000E+05 0.26422E+06 E N T E R ( I N T E G E R ) DWELL TIME. TIME 8 NUMBER OF FIRST E X C I T A T I O N 1000 11000 2 E N T E R ( R E A L ) FREQUENCY S T E P S I 2 E , FREOUENCY OF FIRST E X C I T A T I O N IN THE SPECTRUM 012500E+00 0.50350E+03 E N T E R ( I N T E G E R ) E X C I T A T I O N SEPARATION IN TIME. END OF RUN TIME (SECONDS) 39000 31000 ENTER 1 I F FORWARD RUN, 0 IF BACKWARD RUN ENTER 1 FOR WRITEOUT O OTHERWISE ENTER 1 I F PLOT I S DESIRED OTHERWISE O ENTER 1 IF THE E X C I T . FREQ. DISCONTINUITY IS TO BE SMOOTHED, 0 OTHERWISE 1 1 0 0 E N T E R ( R E A L ) RMS D E V I A T I O N OF THE GAUSSIAN D I S T R I B U T I O N (SECONDS) 0.20000E-01 E N T E R ( R E A L ) FRACTION OF SPINS WHICH ARE COLD (AT TEMPERATURE T ABOVE) O.10000E+01  -J TIME  NORMALIZED INTENSITY  0  0 .89445E+00  1000  0 .89445E+00  2000  0 .89445E+00  3000  0 .89445E+00  4CO0  O .89445E+00  5000  0 89445E+00  6000  0. 8 9 4 4 5 E + 0 0  7000  0. 89445E+00  8000  0. 89445E+00  9000  0. 8 9 4 4 5 E + 0 0  lOOOO  0. 89445E+00  110O0  0. 8 8 2 3 5 E + 0 0  12000  0. 89438E+00  13000  0. 89964E+00  14000  0. 9 0 1 7 8 E + 0 0  POP P1  PROB OF THE P2  SPIN LEVELS P3  P4  P5  P6  P7  0 . 33713E+00  0 . 23171E+00  0 .15925E+00  0 .10945E+00  0 .75225E -01  O .51701E -01  0 .35534E -01  0 .33713E+00  0 .23171E+00  0 .15925E+00  0 .10945E+00  0 .75225E -01  0 .51701E -01  0 .35534E -01  0 .33713E+00  0 . 23171E+00  0..15925E+00  0..10945E+00  0 .75225E -01  0 .51701E -01  0 . 3 5 5 3 4 E-01  0..33713E+00  0 .23171E+00  0,.15925E+00  0. 10945E+00  0 .75225E -01  0 .51701E -01  0.. 3 5 5 3 4 E-01  0..33713E+00  0, 23171E+00  0. 15925E+00  0. 10945E+00  0 75225E -01  0 51701E -01  0..35534E -01  0. 33713E+00  0..23171E+00  0. 15925E+00  0. 10945E+00  0 .75225E -01  0. 51701E -01  0. 3 5 5 3 4 E -01  0. 33713E+00  0. 23171E+00  0. 15925E+00  0. 10945E+00  0. 75225E -01  0. 51701E -01  0. 3 5 5 3 4 E -01  0. 33713E+00  0. 23171E+00  0. 15925E+00  0. 10945E+00  O. 75225E -01  0. 51701E -01  0. 3 5 5 3 4 E -01  0. 33713E+00  0. 23171E+00  0. 15925E+00  0. 10945E+00  0. 75225E -01  0. 51701E -01  0. 35534E -01  0. 33713E+00  0. 23171E+00  0. 15925E+00  0. 10945E+00  0. 75225E -01  0. 5 1 7 0 I E -01  0. 3 5 5 3 4 E -01  0. 33713E+00  0. 23171E+00  0. 15925E+0O  0. 10945E+00  0. 75225E -01  0. 51701E -01  0. 3 5 5 3 4 E -01  0. 33713E+00  0. 19548E+00  0. 19548E+00  0. 10945E+00  0. 7 5 2 2 5 E -01  0. 51701E -01  0. 3 5 5 3 4 E -01  0. 33153E+00  0. 21683E+00  0. 17203E+00  0. 11715E+00  0. 75225E--01  0. 51701E--01  0. 35534E--01  0. 32983E+00  0. 2244GE+00  0. 16577E+00  0. 11585E+00  0. 76861E--01  0. 51701E -01  o.35534E--Ol  15000  0.90242E+0O  16000  0.90234E+00  170O0  0.90193E+00  18000  0.90137E+00  19000  0.90078E+00  20000  0.90020E+00  21000  0.89966E+00  22000  0.89916E+00  23000  0.89871E+00  24000  0.89830E+00  25000  0.89794E+00  26000  0.89762E+00  27000  0.89733E+00  28000  0.89708E+00  29000  0.89685E+00  30000  0.89664E+00  31000  0.89646E+00  FREQUENCY 0COUNTS ************************** 0.50213E+03 0.50225E+03 0.50238E+03 0.50250E+03 0.50263E+03 0.50275E+03 0.50288E+03 O.50300E+03 0.50313E+03 O.50325E+03 0.50338E+03 0.50350E+03 0.50363E+03 0.50375E+03  0.23633E+06 O. 23633E+06 0.23633E+06 0.23633E+06 0.23633E+06 0.23633E+06 0.23633E+06 0.23633E+06 0. 23633E+06 0. 23633E+06 O. 23633E+06 O.23314E+06 0.23632E+06 0.23770E+06  O.32948E+00  O.22777E+00  0. 16340E+00  0.11440E+00  O.77425E-01  0.51991E-01  O.35534E-01  0.32969E+00  0.22933E+00  0. 16228E+00  0.11338E+00  0.77482E-01  0.52275E-01  0.35564E-01  0.33011E+00  O.2301lE+OO  O. 16165E+00  O.11267E+00  0.77367E-01  0.52471E-01  0.35621E-01  0.33061E+00  0.23051E+00  O. 16123E+00  0.11216E+00  0.77208E-01  0. 52583E-01  0.35689E-01  0.33112E+00  0.23073E+00  0.16093E+00  0.11178E+00  0.77047E-01  0.52637E-01  0.35759E-01  0.33161E+00  0.23085E+00  0.16069E+00  0.11149E+00  0.76898E-01  0.52654E-01  0.35823E-01  0.33206E+00  0.23091E+00  0.16049E+00  0.11125E+00  0.76762E-01  0.52648E-01  O.35880E-01  0.33248E+00  0.23096E+00  0.16032E+00  0.11105E+00  0.76640E-01  O.52630E-01  O.35927E-01  O.33285E+00  0.23099E+00  0.16018E+00  0.11088E+00  O.76531E-01  0.52603E-01  O.35965E-01  0.33320E+00  0.23101E+00  O.16006E+00  0.11074E+00  0.76432E-01  0.52573E-01  O.35994E-01  0.33351E+00  0.23103E+00  0.15995E+00  0.1 1061E+00  0.76342E-01  0.52540E-01  0.36015E-01  0.33379E+00  0.23105E+00  0.15986E+00  0.11050E+00  0.76261E-01  0.52507E-01  O.36030E-01  0.33404E+00  0.23107E+00  0.15978E+00  O.11040E+00  0.76188E-01  0.52474E-01  O.36039E-01  0.33427E+00  0.23109E+00  0.15971E+00  0.11032E+00  0.76121E-01  0.52442E-01  O.36043E-01  0.33448E+00  0.23111E+00  0.15965E+00  0.11024E+00  0.76060E-01  0.52410E-01  O.36043E-01  O.334G7E+00  0.23113E+00  0.15960E+00  0.11017E+00  0.76004E-01  0.52378E-01  0.36040E-01  0.33484E+00  0. 23115E+00  0.15956E+00  0.11011E+00  0.75953E-01  0.52348E-01  0.36033E-01  0.33500E+00  0.23117E+00  0.15952E+00  0.11006E+00  0.75906E-01  0.52319E-01  0.36025E-01  

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