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Epr of substitutional fe3 in a natural crystal of brookite (tio2) Rostworowski, Juan Adalberto 1972

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EPR OP SUBSTITUTIONAL F E * IN A NATURAL 5  CRYSTAL OP BROOKITE ( T i O ) 2  by J u a n A. Rostworowski B.Sc. U n i v e r s i d a d N a c i o n a l de I n g e n i e r i a •• Lima (1969)  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE  i n the Department of Physics  We a c c e p t t h i s t h e s i s as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA April,  1972  In p r e s e n t i n g t h i s  thesis  an advanced degree at the L i b r a r y I  in p a r t i a l  the U n i v e r s i t y  s h a l l make i t  freely  fulfilment of of B r i t i s h  available  for  the  requirements  Columbia, 1 agree  that  r e f e r e n c e and s t u d y .  f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s  for  for  thesis  s c h o l a r l y purposes may be granted by the Head o f my Department o r  by  his  representatives.  of  this  It  thesis for financial  i s understood that c o p y i n g o r p u b l i c a t i o n gain s h a l l  not be allowed without my  written permission.  J.  Department o f  ^ftySiOf?  The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada  Date  April  21,  1972  A.  Rostworowski  ABSTRACT  EPR spectra o f Pe^ i n a natural c r y s t a l o f brookite +  have "been i n v e s t i g a t e d a t X- and Q-band frequencies at room temperature and 573°K.  Part o f the paramagnetic resonance  spectrum observed has been interpreted on the assumption that Pe- * occupies 5  eight equivalent ? i ^ s i t e s i n brookite, with +  four inequivalent orientations.. The spectra show an "intermediate" z e r o - f i e l d  splitting  at X-band and a "normal" zero f i e l d s p l i t t i n g at Q-band frequencies. 'Hie spin He.miltonian  parameters which f i t the spectra  are the following:: v  g r 2.002  ± 0.005  .  D=  (1170 i 30) x 10-4cm"  Er  (330  1  ± 20) x 10~4cm"!  [pa*(l/l^ oiO  =  [pa (l/12>i?]  = (-1315) x l O ^ c m -  +  [ga+{L/12)i?]  1  QQ1  0  0  (  1 5 ± 1 0  >  x  10~ cm~ 4  1  1  = (-66+4) x l O ^ c m "  1  iii 'TABLE OF CONTENTS Page Abstract  i i  L i s t of Tables  iv  L i s t of Figures  v'.  Acknowledgements  vii..  1.  Introduction  1  2.  Results o f Previous Investigations on Brookite  3  3. -4.  Structure of Brookite  3  Phase Transformation  10  Other Properties  12  The Spin Hamiltonian  13  Samples and Experimental. Techniques  17  Natural Crystals of Brookite  17  . X-Band and Q-Band Spectrometers  17  Angular Dependence of the Spectra High Temperature Measurements 5.  6.  19 '  Experimental Results  21 23  Preliminary Observations  23  Determination of the Spin Hamiltonian Parameters  29  Adjustment of the Hamiltonian Parameters  33  Discussion of Results and Conclusions  Bibliography  38 41  .Appendix A  46  Appendix B  48  Appendix C  50  iv  LIST OF TABLES  Table I  L a t t i c e Parameters  7  Table I I  Ion C o o r d i n a t e s f o r a Sample TiOg Octahedron  7  Table I I I  Ion D i s t a n c e s f o r a Sample TiOg Octahedron  8  Table IV  T i - T i Distances  9  Table V  Summary o f the Atomic Movement i n Topotaxy i n the Ti02 System  11  Table VI  Other P r o p e r t i e s  12  Table V I I  Transition  34  Probabilities  Table V I I I •' H a m i l t o n i a n Parameters  35  LIST OP FIGURES is* Figure 1  The p o s i t i o n o f the i o n s i n the p r o j e c t i o n s of a u n i t c e l l o f brookite  Figure  2  4  P r o j e c t i o n s i n the (100) plane o f some TiOg octahedrons.  The common edges o f the octahe-  drons are drawn h e a v i e r Figure  3  Double hexagonal  5  c l o s e d packed  (DHCP)  arrangement Figure 4  5  B l o c k diagram o f the EPR  spectrometer  arrangement f o r X- and Q-band f r e q u e n c i e s Figure 5  EPR  s p e c t r a a t Q-band o f a n a t u r a l s i n g l e  brookite  c r y s t a l with K p a r a l l e l to  room temperature Figure  6  EPR  and approximately.200°C  c r y s t a l w i t h II p a r a l l e l to  room temperature EPR  [ooij a t  and approximately 200°C  26  c r y s t a l w i t h H s l i g h t l y o f f jfooi] i n  a p p r o x i m a t e l y the  (110) plane a t room  temperature 9  25  spectrum a t Q-band o f a n a t u r a l s i n g l e  brookite  Figure  24  s p e c t r a a t Q-band o f a n a t u r a l s i n g l e  brookite  Figure 8  and approximately 200°C  c r y s t a l w i t h H p a r a l l e l to', [bid] a t  room temperature EPR  [loo] a t  s p e c t r a a t Q-band o f a. n a t u r a l s i n g l e  brookite  Figure 7  20  Angular dependence o f the EPR spectrum  27 l i n e s o f the  observed a t h i g h temperature  in  b r o o k i t e a t Q-band with H r o t a t e d i n the three principal  c r y s t a l planes  28  Figure 10  Energy l e v e l s o f s u b s t i t u t i o n a l "brookite v.'ith H p a r a l l e l  Positions  o f the EPR  in  ... .  to each o f the  principal crystallographic F i g u r e 11  Fe^  axes  37  t r a n s i t i o n s w i t h i n the  lower and upper Kramer d o u b l e t s o f Fe- " f o r ?-1  E/D  = 0.25  49  vii ACKHOV/LSSC-BMEI-JTS I would l i k e  to thank B r . C. P. Schwerdtfeger f o r h i s  h e l p f u l guidance i n the p r e p a r a t i o n o f t h i s  thesis.  I an a l s o g r a t e f u l to B r . K. Horn f o r the many d i s c u s s i o n s on the i n t e r p r e t a t i o n o f the observed s p e c t r a . a l s o indebted  I am  to A. H a r n i k o f the C r i s t a l l o g r a p h i c I n s t i t u t e o f  E.T.H., Z u r i c h , f o r the s u p p l y o f many specimens o f B r o o k i t e . The  research o f this  t h e s i s was supported  financially  by the N a t i o n a l Research C o u n c i l , g r a n t s number A-2228 and A-7121.  A d d i t i o n a l f i n a n c i a l a i d was o b t a i n e d  from the  U n i v e r s i t y o f B r i t i s h Columbia's 1970-71 P r e s i d e n t ' s on Research.  Committee  1 1.  IKTROBUGTION I t was K l a p r o t h who f i r s t r e c o g n i s e d  1798,  t h a t i s , t h a t the same chemical  a polymorphism i n  compound may  crystallize  i n d i f f e r e n t forms.  K l a p r o t h observed the polymorphic forms  o f calcium  (calcite,  carbonate  aragonite)."*"  That t i t a n i u m d i o x i d e has s e v e r a l n a t u r a l polymorphic forms ( r u t i l e , anatase a,nd b r o o k i t e ) has been known a i r e a d y for  many y e a r s .  Recently,  s y n t h e t i c a l l y produced  a f o u r t h polymorphic form was  (Ti0 H) . 2  2  R u t i l e , anatase and b r o o k i t e are o f t e n found i n nature as good s i n g l e c r y s t a l s .  S y n t h e t i c a l l y one can o b t a i n s i n g l e 3  c r y s t a l s o f r u t i l e and p o l y c r y s t a l l i n e anatase . a l l attempts to produce b r o o k i t e a r t i f i c i a l l y , c r y s t a l l i n e form, have been u n s u c c e s s f u l .  even i n p o l y -  As a r e s u l t most EPR  s t u d i e s have been made with r u t i l e 4 7 n d some with _  a  c r y s t a l s o f anatase8-13.  To date,  natural  No EPR s t u d i e s o f b r o o k i t e have been  reported. An EPR comparative study o f d i f f e r e n t polymorphic forms can be i n t e r e s t i n g from a m i n e r a l o g i c a l p o i n t o f view.  Only  r e c e n t l y , EPR has been acknowledged to be a t o o l i n mineralogy and  geology.  Reviews by W. Low * and S. G h o s e ^ have shown 1  t h a t EPR can be h e l p f u l i n c l a r i f i n g some m i n e r a l o g i c a l problems such as the r e l a t i o n between the paramagnetic i m p u r i t y and the mineral host and  (site preference).  l o c a l symmetry  (non-equivalent  Information sites,  about  coordination  o r i e n t a t i o n and v a l u e s  o f the c r y s t a l f i e l d parameters, charge compensation), n a t u r e o f chemical bonding, c o l o r c e n t e r s and o t h e r l a t t i c e  defects,  o r d e r - d i s o r d e r , and r e l a t i v e abundance o f d i f f e r e n t p a r a magnetic i m p u r i t i e s o r d i f f e r e n t v a l e n c e s t a t e s o f a g i v e n i m p u r i t y may be o b t a i n e d from a n a l y s i s o f EPR measurements. S i n c e i m p u r i t i e s are a f a c t o r t h a t determine the h a b i t f o r m a t i o n o f the m i n e r a l s i t i s a l s o remotely p o s s i b l e EPR  can be h e l p f u l i n t h i s g e o l o g i c a l problem'.  that  Remotely s i n c e  non paramagnetic i m p u r i t i e s a r e i m p o s s i b l e to observe o r l a r g e c o n c e n t r a t i o n s o f paramagnetic i m p u r i t i e s w i l l not be seen o r . make a n a l y s i s e x t r e m e l y . d i f f i c u . l t v/ith EPR. In the p r e s e n t t h e s i s the r e s u l t s o f an EPR study o f substitutional iron impurities i n a natural single crystal o f b r o o k i t e are d e s c r i b e d .  Ho attempt to a n a l y s e o t h e r EPR  t r a n s i t i o n l i n e s has been made to date a l t h o u g h some specul a t i o n has been attempted.  3 2.  RESULTS OF PREVIOUS INVESTIGATIONS ON BROOKITE STRUCTURE OF BROOKITE B r o o k i t e has an orthorhombic symmetry and belongs to  the Pbca  (B^)  space group.  R. V f e y l ^ remeasured 1  the  lattice  constants o f b r o o k i t e , p r e v i o u s l y determined by L. P a u l i n g and  1 7 J.H. S t u r d i v a n t  , and found t h e i r r e s u l t s  p r e c i s i o n l i m i t s o f 0.1$  to l i e w i t h i n tne  a t t a i n e d by h i s measurements.  values are:  These  o a - 9.164  A  b = 5.447 X c - 5.145  A .  However he found some d i s c r e p a n c i e s f o r the v a l u e s o f the l a t t i c e parameters  and these are l i s t e d i n Table I .  Using the v a l u e s o f Table I and the c o o r d i n a t e s o f e q u i v a l e n t p o s i t i o n s f o r the Pbca space group"^'"^ one i s able to f i n d the p r o j e c t i o n s o f the u n i t c e l l .  Figure 1  shows these p r o j e c t i o n s . Each t i t a n i u m i o n i s surrounded by s i x oxygen i o n s a t the v e r t i c e s o f an octahedron ana i t a l s o shares three edges with other octahedrons.  This i s shown i n F i g u r e 2.  These .  common edges a r e s h o r t e r i n comparison w i t h the o t h e r edged o f 20 tne octahedron which i s i n accordance w i t n P a u l i n g ' s Rule ( R u t i l e shares two edges and anatase shares f o u r " ^ ' ^ ) . Tables I I and I I I g i v e the d i s t a n c e s between the t i t a n i u m and the oxygens as w e l l as the d i s t a n c e s between the oxygens.  Clearly,  t h i s octahedron i s d i s t o r t e d .  Furthermore,  o  • 0  o  o  o o  o o  o o  o • o  0  o 0  o ' o 0  o o  o ;  o o 0  Figure 1.- The p o s i t i o n o f the ions i n the pro lections of. a u n i t c e l l of brookite  o  .0  o2  v' 4  F i g u r e 2.- P r o j e c t i o n s i n the (100) plane o f some TiOg octahedrons.  The common edges o f the o c t a h e d r o n s  heavier.  are drawn  •  J\  Q ^  ( /\  o O  7  0  B o j\ • • Q O '  o  0  o O  0 o  o  0  o  0  o  o O  0  0 o  0  F i g u r e 3.- Double hexagonal  0  o  . 0  O  o  0  0 0 0 0 0  o O  O  o  o 0  c l o s e d packed  o  o  o  o 0  b  o  o  O  o  o o  o o  o O  o  o  o o  o  o o  o  (DHCP) arrangement  o  3. Weyl has c a l c u l a t e d t h a t the T i i n the o c t a h e d r o n i s . d i s p l a c e d by 0 . 2 i 0 . l A from the c e n t e r toward O j y and. away OJ-J(in  rutile  as w e l l as anatase the t i t a n i u m  i o n i s i n the  middle o f the TiOg octahedron) and t h a t t h i s i o n , T i , Q  f u r t h e r away by 0.12t0.06 i and T i  2  is  from the n e i g h b o u r i n g i o n s , T i  1  than from the t h i r d n e a r e s t n e i g h b o u r i n g i o n T i ^ , as  can be c a l c u l a t e d from Table IV.  (Y.'here the s u b s c r i p t s on T i  r e f e r to the i o n s l a b e l l e d i n F i g u r e 2 ) . In  the b r o o k i t e s t r u c t u r e each octahedron i s bound  through two common.edges to'two  o t h e r octahedrons f o r m i n g a  c h a i n i n the [001J d i r e c t i o n and by the t h i r d common edge to another' such c h a i n f o r m i n g a net, p a r a l l e l  to the (100) p l a n e .  The u n i t c e l l has t..'o such n e t s , one over the o t h e r which a r e bound by common c o r n e r s . Tiie (001) plane i n F i g u r e 1 shows d i s t i n c t i v e l y the  u n i t c e l l has f o u r empty columns,  hof.ever, i t i s not c l e a r  t h a t each such column c o n t a i n s t-'-o i n t e r s t i t i a l rounded  that  sites  by s i x oxygens a t the c o r n e r s o f a l a r g e r  sur-  distorted  octahedron. Another way o f v i e w i n g the s t r u c t u r e o f b r o o k i t e i s t h following:  the oxygen i o n s a r e i n a p p r o x i m a t e l y double  g o n a l c l o s e d packed  (DROP) arrangement,  shown i n the F i g u r e 3.  hexa-  i . e . (A3ACABAC..•) as  The c l o s e d packed  plane i s the (100)  plane, one h a l f o f the octahedrons a r e f i l l e d  with  titanium.•  7 TABLE  LATTICE  I  ?ARA:3T5RS  (117 B R A C K E T S  OP  A~:B S T U R D I V A N T )  0  ?AULI::G  0  1  THE VALUES  Ti  2  x  0.00 8 (0.010 )  0 . 229 (0 . 230 )  y  0.147 (0.155)  0.110 (0.105)  0.098 (0.113)  z  0.182 (0.160)  0.530 (0.535)  0.863 (0.873)  TABLE  10IT  C O O R B J[ N A T E S  (GIVEN  Ol °II ° I I I ° I V  °v ° V I  IN  FOR  jf'R A C T I O N S  II  A SAMPLE OF TEE  0.128 (0.127)  TiO*  OCTAKEIECK  L A T T I C E CONSTANTS)  X  y  z  0,128  0/402  0,363  0,008  0,147  0,182  0,229  0,110  0,530  0,271  0,610  0,530  -0,008  0,647  0,318  0,229  0,330  0,030  0,008  0,353  0,682  I O N D I S T A N C E S FOR A S A M P L E TiO£ OCTAHEDR02?  •-URE 2 )  o-°I  2,0010,05) i  ' o-°H  2,0 5+0,05) %  Ti  Ti  o-OlII  T i  1,9410,05 1,84+0,05 1,9510,05  Ti  o-°VI  1.99*0,05 2,7110,08  ° I I - ° V I ° V I - ° I I I ° I I I - ° I V  2,55£0,08 2,90±0,08 2,7910,08  °-I7-°V  2,98±0,08  Ov-0T  2,5510,08  2,7510,08 ° I I I " ° V  o -o_ V 11 v  0T-0 ° V I - ° I V  2,8610,08 2, 99+0,08 2,8110,08  2,4710,08 2,62±0,08  TABLE I V T i - T i DISTANCES (SEE PIGURB 2)  Ti -Tl  1  Ti -Ti  3  0  Q  = Ti-Ti  .  o (2,94*0,0 3) A  Ti - T i = T i - T i c O  Ti -Ti Q  A  0  O  (3,78+0-0 3) A  J  = T i -Ti„  (3,57^0,0 3) A  = T i —- TT il 8o 9  (3,53^0,0 3) A  f i  T i* - T i  (3,00 ^ , 0 3) A  cQ  Qn  10 ' PHASE TRANS FORMA TION The  t o p o t a c t i c mechanisms f o r the Ti02 system a r e "based  on a p r i n c i p l e o f f a v o r i n g as much as p o s s i b l e the maintenance o f the oxygen c l o s e p a c k i n g d u r i n g the polymorphic (Anatase  i s pseudo-cubic  c l o s e packed  transition  (CG?) v/ith the c l o s e  packed plane  (112); R u t i l e , rough approximation  o f an hexagonal  c l o s e packed  (HOP), (100) o r (010) p l a n e s ; T i 0 H , 2  HOP, (100)  plane). Table V summarizes the observed i n the t i t a n i u m d i o x i d e system  and proposed  topotaxy  .  At t h i s p o i n t , i t s h o u l d be mentioned t h a t the k i n e t i c s and mechanism o f the b r o o k i t e - r u t i l e t r a n s f o r m a t i o n a r e v e r y s i m i l a r to those o f the a n a t a s e - r u t i l e t r a n s f o r m a t i o n ^ - * ^ . -  Tlie r a t e o f t r a n s f o r m a t i o n and i t s a c t i v a t i o n energy a r e governed by the s u r f a c e s i z e and by the amount o f i m p u r i t i e s , e.g.  the c o n c e n t r a t i o n o f oxygen v a c a n c i e s o r i n t e r s t i t i a l s .  But the a c t i v a t i o n energy, of nucleation sites, and  which i s mainly  f o r the p r o d u c t i o n  i s h i g h e r f o r the a n a t a s e - r u t i l e case  the entropy o f a c t i v a t i o n i s l a r g e and n e g a t i v e f o r the  brookite-rutile  t r a n s f o r m a t i o n compared w i t h the s m a l l and  p o s i t i v e v a l u e f o r the a n a t a s e - r u t i l e t r a n s f o r m a t i o n . may be understood  i n terms o f the change o f symmetry  h i g h e r ) o f the f i r s t the s e c o n d ^ .  This (lower to  case and the absence o f such a change i n  The e f f e c t o f i m p u r i t i e s i n g e n e r a l i s t h a t  oxygen v a c a n c i e s a c c e l e r a t e v. here as i n t e r s t i t i a l i o n s i n h i b i t , the t r a n s f o r m a t i o n ^ . ' 2  TABLE V SUGARY OF THE ATOMIC MOVEMENT IN TDPOTAXY IN THE TiOp SYSTEM  Reaction  Number of Ovypen Layers Rearranged  21 - futile  None  Brookite-II  1 in 4  Redistribution of the Titanium Atoms*  RemarVs  One half in each layer  Only reaction that has been reversed. Experimental evidence inferred from work ol 1/2 1 /7 1/2 1/2 1/2 1/2 1 B/ e2 ndeiiany et al. (1966) Every Ti in two consecutive layers Reaction without SRO phase altering with two layers in which there formation. Hrookite has aigis no Ti motion. za£ TiO» octahedral chains as A B A C A B A C is found in II. A  B  A  B  A  B A B  1 1  This reaction dotsEver)' cth^r Ti in each layer in ordeErxpe tr oimentally observed Brookite - rutil*n ot involve an oxyf -orm straight Ti ociahcdral chains. topotaxy. gen shearing mech-A B C A B C A B C anism, but there is * shifting of all 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 oxygen. A n a 12 s-c—rulUe  This reaction dotsEvery other Ti in each layer in oru'eTroptootaxy experimentally obnot involve an oxyform straight Ti octahedral chains. served. A large distortion ocgen shearing mech-A B A C A B A C curs in the formation of anism but alt ojtystraight octahedral chains gen atoms shift 1/2 1/2 1/2 1/2 1/2 1/2 1 f/ r2 om zigzag chains. posit k>as.  Anatase—II  A in 6  A repeating sequence of all Ti andRean co tion always accompanied Ti motion beginrrlns with the Ti bwei-th SRO phase formation. tween the A ar.d C layers. A  B  1 Anitas*—brooLite  9 in 12  C  A  B  C  A  B  1 1 I Very complex  C  Reaction reported only one time.  • In most cases the mover-.er.t of the li:ai:ium ions is to adjacent tctrahcdral sites vrhich arc in the process of becoming octahedral M'IC; bcc.vjie of tlie accompanying oxygen nation. The legend for the tit.tnium motion is 1/2 every other titaniu:r. charters site 1 every titanium chan^rs ?ite 0 no titanium chan;;o ji'.e . The layering sthenic i> that vi the rcactanl.  OTHER PROPERTIES A l i s t o f these p r o p e r t i e s i s g i v e n i n Te.ble V I . t i e s o f r u t i l e and  anatase  Proper-  have been a l s o i n c l u d e d f o r  compari-  son? ' . 0  8  TABLE VI PROPERTIES Property  Anatase  Rutile  Density •  3.87-3.95g/ca  Eardness  5.5  Specific  n  n  4 .21-4 .25g/cn  3  - 6.0  7.0  4.13g/cm-  3  - 7.25  5.5-0.0  heat  a t R.T.  13.22cal/mol°C  Optical  Brookite  13.16cal/mol°C  Constants  5693& o  2.56  5893A  2  A  2.61  9  2  >  g  n 2.5831 * n, 2.5843  o  n 2.7004 f  Static  dielectric  constant  48  114  78  N o t i c e t h a t the v a l u e s o f these p h y s i c a l c o n s t a n t s f o r b r o o k i t e are between those measured f o r anatase EPR  rutile..  STUDIES  Hamiltonian and  and  parameters f o r s u b s t i t u t i o n a l P e ^ ,  P are g i v e n i n 10~^cm~\  Anatase  (3O0°K)  where D, E,  a  are:  g  D  E  a  p  2.005  308.7  -  10 2.8  6  2.00  6780  Rutile  (4.2-300 )°K  690  No EPR  s t u d i e s i n b r o o k i t e have been r e p o r t e d .  -280  230  3.  THE SPIN HAMILTONIAN 'The e l e c t r o n i c  c o n f i g u r a t i o n o f the Pe^* i o n i s 3d^  the f r e e i o n e l e c t r o n i c ground s t a t e i s ^S^yg*  magnetic  resonance spectrum s h o u l d be remarkably s i m p l e .  The  sextet  would have no s p l i t t i n g o t h e r than the Zeeman i n t e r a c t i o n , a s i n g l e l i n e would be observed, ho»ever,  experimentally  i s f a r from the t r u t h . ' Both f i n e and h y p e r f i n e have  been observed f o r i o n s with h a l f f i l l e d  shells.  and  and  this  structures  3d, 4f and  5f  S e v e r a l mechanisms have been proposed to e x p l a i n  31 tnis.  S p i n - s p i n and s p i n - o r b i t i n t e r a c t i o n s  can produce  seccnd o r h i g h e r o r d e r terms c o u p l i n g the ground s t a t e through higher' o r b i t a l s t a t e s to the c r y s t a l  field.  The m u l t i p l e f i n e s t r u c t u r e can c o n v e n i e n t l y be  repre-  sented by adding "bo the Zeeman term i n the s p i n H a m i l t o n i a n , terms o f h i g h e r powers i n S ., S y  and S , z  g r o u p i n g them i n t o  combinations o f s p i n o p e r a t o r s , each such o p e r a t o r b e i n g the e q u i v a l e n t to a combination of. s p h e r i c a l harmonics.  Such  Til  e q u i v a l e n t s p i n o p e r a t o r s denoted as 0' appear e x t e n s i v e l y i n  32. l i t era "o"ure. Che advantage o f u s i n g e q u i v a l e n t s p i n o p e r a t o r s i s tha one can choose immediately which terms w i l l appear ..in the s p i n Hamiltonian.  One  excludes the e q u i v a l e n t s p i n o p e r a t o r s  are o f odd degree i n S, s i n c e they are not i n v a r i a n t time r e v e r s a l .  Prom the o p e r a t o r s o f even degree one  o n l y those which r e f l e c t  the symmetry o f the c r y s t a l  which  under chooses field.  Furthermore, the number o f such o p e r a t o r s i s l i m i t e d by the  14 f a c t t h a t o p e r a t o r s o f higher.degree  than 23 may be o m i t t e d  s i n c e they have zero.matrix  elements between the s t a t e s under  consideration.  3+  Hence f o r Pe " one can w r i t e i n g e n e r a l  ? f = /JH-g.S * 1/3 ZZ-  m—— 2  b l o\  • 1/60 E Z  U s i n g the known t r a n s f o r m a t i o n  o| 0*  (1)  properties of  under  m—t;-  r o t a t i o n o f the c o o r d i n a t e system one can e v a l u a t e  the t r a n s -  f o r m a t i o n p r o p e r t i e s o f the b ^ under these r o t a t i o n s . pendix  i n Baker and W i l l i a m s  i s v e r y h e l p f u l f o r these c a l c u  l a t i o n s and has been used i n Appendix A to c a l c u l a t e Hamiltonian  f o r an a r b i t r a r y  The form.  spin Hamiltonian  The f i r s t  An ap-  the s p i n  orientation. may be expressed  i n an a l t e r n a t i v e  summation i n (1) can be v . r i t t e n a s : 3-D-S  where B i s a t e n s o r q u a n t i t y .  R e f e r r e d to the p r i n c i p a l axes,  t h i s becomes  where i t i s c o n v e n i e n t  to s e t the sum o f the three  coefficients  to zero •by s u b s t r a c t i n g the q u a n t i t y 1/3 (B\^B^B )(S,.. t3 2  z  2  *3 ) = 1/3 (B„+ B; + D„ )S (3*1) 2  which i s j u s t a c o n s t a n t  t h a t moves a l l energy l e v e l s up o r  down by the same amount. The  fact  t h a t one can s e t the t r a c e o f t e n s o r B to be  zero, means t h a t there a r e o n l y two independent  coefficients  x(s|_S|) * I)as| • D y s | + D„S|=  D S^ X  v.= i t h  (2)  D { s f - l / 3 3(S+1)} *  1/2  E(sf*sf)  D - 5/2 D,_ 2 = 1/2  (D  x  - IL,}  Thus the second term o f e q u a t i o n ( i ) i s e q u a l to. e q u a t i o n (2) if D.  the axes are chosen to he the p r i n c i p a l axes o f the t e n s o r These axes are commonly known i n S?R l i t e r a t u r e as the  magnetic axes o f a paramagnetic c e n t e r , and are determined e x p e r i m e n t a l l y by a b s o l u t e extreme p o s i t i o n s of. the EPR lines.  spectral  By e v e n t u a l permutation o f the axes one can a d d i t i o n a l l y  l i m i t the v a l u e o f E/D the p u r e l y a x i a l  to the range o^E/D*1/3, where zero i s  case^.  S i n c e there i s no argument  to suppose t h a t the p r i n c i p a l  axes o f the t e n s o r b^ c o i n c i d e w i t h the p r i n c i p a l axes o f bg, the l a t t e r h a v e been taken as d e f i n i n g  the magnetic  Most work p u b l i s h e d i n t h i s - f i e l d h a v e  i n c l u d e d o n l y the b°  and b' terms o f the second summation i n e q u a t i o n 4  sizes.  .  Ex-.  c e p t i o n s are made when d e a l i n g v/ith c u b i c o r pseudo-cubic crystals  (when one can c o n s i d e r the f i r s t summation as a s m a l l  p e r t u r b a t i o n o f the s e c o n d ) .  In which case the magnetic axes  are taken to be the p r i n c i p a l axes o f the b^ t e n s o r . i n c l u d i n g the "two terms d e s c r i b e d , u s u a l l y w r i t t e n as:  By  the H a m i l t o n i a n i s a l s o  16 ?< = / K H . g . S ) * D ( S  2 z  - l / > 3 ( 3 + 1 ) ] y 1/2 3 ( 3 , o f ) 2  --H 1/6 a ( s | + 3^ -A 3 | - 1/5 S ( 3 ^ 1 ) ( 5 3 ^  53-1 ) J  2  1/180 P { 3 5 3 a - 3 0 3 ( 3 + 1 ) S'| + 2 5 3 ^ \7iie.rc  J • *{ i £ r.; t n e c o o r d i n a t pie rr-r stem  T~ '3 Ii"' ° Cl 3_ CU  ••^ic-ld.  i n g e n e r a l , they a r e not the  p r i n c i p a l axes o f t h e t e n s o r Hamiltonian :° b  ( l ) are r e l a t e d -"  2  2  -  D  D.  63(3v-l)+33 (3+l) J 2  2  re-Ten  .cur-zio-LG  .cn  v>  (5)  'hie  t i - i o '  as  O  o f the c r y s t a l  C:.0  -C,  y ,  o:" - o a r a m e t e r s om j.1 "  to D, 3, a. and P by = I + P/3 --  2  «.  Z  xne  3'om •"-  4. -SAMPLES AMP  EXPERIMENTAL TECHNIQUES  NATURAL CRYSTALS OP BROOKITE Several natural c r y s t a l s o f brookite t h i s work.  They o r i g i n a t e d i n Maderanerthai, U r i , S w i t z e r l a n d ;  V a l s e r T a l , Graubunden, S w i t z e r l a n d ; USA.  have been used i n  and Magnet Cove, Arkansas,  She M a d e r a n e r t h a i and V a l s e r T a l samples s t u d i e d i n t h i s  wcrk had the same habit,, they '-.ere t r a n s l u c e n t f l a k e s p a r a l l e l to the (010) plane, in color.  and were e i t h e r l i g h t brown o r y e l l o w i s h  The Magnet Cove sample a l s o c a l l e d A r k a n s i t  was  almost a p e r f e c t octahedron and black i n c o l o r . The EPR s p e c t r a o f the Maderanerthai and V a l s e r T a l samples are u n d i s t i n g u i s h a b l e , l i n e s have the same r e l a t i v e  a l l observed EPR  intensities.  transition  Any comparison w i t h  A r k a n s i t was not p o s s i b l e s i n c e these samples had a v e r y l o s s f a c t o r and hence no EPR spectrum was  high  obtainable.  X-3ANB A'-MO Q-3ANB SPECTROMETERS The EPR spectrometer was a c o n v e n t i o n a l  balanced bridge  design.  The microwave bridge u t i l i s e d , a magic T a t X-band and a  c i r c u l a t o r a t Q-band.  A r e f l e x k l y s t r o n (X-band: V a r i a n  V - 1 5 3 / 6 5 1 5 m a x . output 70 mv,  Q-band: OKI 35V10; max.  output 40mw), a one way f e r r i t e  :  isolator,  and a f l a p , a t t e n u a t o r  were connected to the i n p u t arm, a c r y s t a l d e t e c t o r , output.  actual  to the  In X-band one arm ended i n a TE 102 resonance c a v i t y  ( V a r i a n multi-purpose c a v i t y V-4531) coupled  through an  18 adjustable i r i s ,  the o t h e r r e f e r e n c e arm  screw tuner and a matched l o a d .  contained a s l i d e -  In Q-band the t h i r d arm  of  the c i r c u l a t o r c o n t a i n e d a s l i d e - s c r e w tuner and a c i r c u l a r TE 012 resonance  cavity  (Ventron sample c a v i t y SG-10-ka) and  v.as coupled through a d j u s t a b l e t e f l o n s p a c e r ^ , diode was  b i a s e d (100-300 MA)  The  crystal  by a d j u s t i n g the s l i d e screw  tuner. The k l y s t r o n  was  c a v i t y . b y modulating  frequency l o c k e d to the resonant  the r e f l e c t o r v o l t a g e with a 10KHz s i g n a l  and u s i n g the c o r r e s p o n d i n g phase s e n s i t i v e from the c r y s t a l d e t e c t o r as an e r r o r f i e l d was  temperature  through l a r g e r magnet.  signal.  The  magnetic  modulated a t 100KHz. through s m a l l modulation  a t t a c h e d to the resonant c a v i t y Helium  d e t e c t e d output  a t room temperature.  a 400Hz modulation arrangement was  modulation  At l i q u i d used  c o i l s a t t a c h e d to the f a c e s o f the  The modulation amplitude used- was  h i g h modulation was  coils  10-15  gauss.  This  s a f e l y used s i n c e a l l l i n e s were over  gauss wide and hence no modulation  broadening  effects  30  would be  observed. The p r e - a m p l i f i e d output from the c r y s t a l d e t e c t o r was phase s e n s i t i v e  d e t e c t e d a t lOOKEz (400Hs) (PAR  A m p l i f i e r / p h a s e d e t e c t o r , model 121). connected The  Lock i n  The output was  then  to a s t r i p c h a r t r e c o r d e r . frequency o f the microwaves a t X-band was  with a d i g i t a l frequency meter (Hewlett-Packard  measured  Frequency  counter 5255A with plug-in. adapter HP 5245 ), a t Q-band a marker  !9 w&s  introduced The  (DPPE powder) to "the c a v i t y .  experiments were performed on a 9.5" Magnion Magnet  with a r o t a t i n g c o i l f i e l d the d i r e c t f i e l d  sensor (Magnion FPC-4 power s u p p l y )  r e a d i n g s b y the sensor '..ere c a l i b r a t e d through  a NMR gaussmeter. A l l measurements were made 7 v i t h the . magnetic perpendicular Figure  field  to the microwave magnetic f i e l d a t the sample. 4 gives  a b l o c k diagram o f the- e x p e r i m e n t a l s e t -  up.  ANGULAR DEPENDENCE 0 ? TKE SPECTRA It i s possible,  i n general,  to determine e x p e r i m e n t a l l y  the p r i n c i p a l axes o f the t e n s o r D by f i n d i n g the extremum p o s i t i o n s o f the EPR s p e c t r a l l i n e s T o  a c h i e v e t h i s , one  u s u a l l y measures the EPR t r a n s i t i o n f i e l d s as a f u n c t i o n o f the o r i e n t a t i o n o f the c r y s t a l f i e l d w i t h r e s p e c t magnetic  field.  In the p a r t i c u l a r case o f b r o o k i t e , was o p e r a t i o n a l l y v e r y d i f f i c u l t fields  t c the s t a t i c  i t was found t h a t i t  to f i n d the extreme t r a n s i t i o n  s i n c e , what was one l i n e a t an o r i e n t a t i o n p a r a l l e l to a  p r i n c i p a l c r y s t a l l o g r a p h i c a x i s became f o u r orientation.  l i n e s a t a random  To t r y to f o l l o w a l i n e by r o t a t i n g the c r y s t a l  o r magnet i n such a random plane proved to be an i m p o s s i b l e task. The  o t h e r p o s s i b i l i t y was to take the s p e c t r a a t  pre amplifier  isolator  klystron  magic  attenator  detector  tuner  amplifier cryst.  variable coupler  /  /  modulation  L  j  B l o c k diagram o f the EPR  Q-band f r e q u e n c i e s .  19-  V  / coils  w  wave  chart  generator  recorder  magnet  a  MICROWAVE  F i g u r e 4.-  Al  \  Q-BAND  Set-up  lock-in  r.  1/  t  cletec.  '  crystal  slide-srew  T  pow.  sup.  rot. coil field sensor  spectrometer arrangement f o r X-  field regulator and  d i f f e r e n t o r i e n t a t i o n s i n the three 100)  c r y s t a l l o g r a p h i c (001, 010,  planes where each t r a n s i t i o n l i n e s p l i t i n t o two and thus  h o p e f u l l y r e c e i v e some i n f o r m a t i o n magnetic axes i n these planes, r e a l magnetic axes are,  on the p r o j e c t i o n s o f the  then to r e c o n s t r u c t where the  and f i n a l l y o b t a i n  enough  to i d e n t i f y and a n a l y s e the paramagnetic centers The  new method d e s c r i b e d  information involved.  by K. Horn and C. Schwerdt-  feger°> ^° f o r making angular p l o t s was used o n l y on a l i m i t e d b a s i s , s i n c e i t proved almost i n o p e r a t i v e owing to the l a r g e number o f l i n e s and t h e i r r e s p e c t i v e c r o s s - o v e r s . it  did. prove u s e f u l i n t h i s case f o r the more  However  accurate  alignment o f the c r y s t a l which was p r e v i o u s l y o r i e n t e d  with.  37 a back r e f l e c t i o n Laue method ( C u l l i t y )  .  HIGH TEMPERATURE MEASUREMENTS In the a n g u l a r p l o t s made i n the (00ij, (010) and (100) p l a n e s there are so many t r a n s i t i o n l i n e s t h a t i d e n t i f i c a t i o n o f a paramagnetic c e n t e r possible.  The  alone was n o t  T r a n s i t i o n l i n e s d i s a p p e a r e d a t some o r i e n t a t i o n s ,  they c r o s s e d be found.  from symmetry c o n s i d e r a t i o n s  v/ith o t h e r  l i n e s and no c o n s i s t e n t trends  could  A d i s c r i m i n a t i n g mechanism was n e c e s s a r y . temperature dependence s p e c t r a o f anatase  t h a t s p e c t r a due to s u b s t i t u t i o n a l , i r o n c o u l d s t i l l  '  J  showed  be seen  e a s i l y a t temperatures over S00°C w h i l e s p e c t r a o f s u b s t i t u t i o n a l i r o n complexes w i t h an oxygen vacancy disappeared a t  22  much lower temperatures. Naturally  I t was a d i s c r i m i n a t i n g  device.  there was no reason to "believe t h a t a charge  s a t i o n ^center  compen-  was p r e s e n t , hut s p i n - l a t t i c e r e l a x a t i o n times  for d i f f e r e n t impurity  ions are i n g e n e r a l  always d i f f e r e n t ,  so t h a t t h i s d i s c r i m i n a t i n g t o o l could v e r y w e l l work. High temperature measurements a t X-hand were made w i t h a flow o f heated n i t r o g e n brass f i n g e r was used. cessful.  gas ( V a r i a n V4557), a t Q-band a h o t  The d i s c r i m i n a t i n g t o o l proved s u c -  5.  • EXPERIMENTAL RESULTS PRELIMINARY OBSERVATIONS S i n c e b r o o k i t e has orthorhombic  expect t h a t i t s EPR was  symmetry one would  s p e c t r a would a l s o show t h i s symmetry,  experimentally v e r i f i e d . EPR  s p e c t r a a t a p p r o x i m a t e l y 36 GHz are shown i n  F i g u r e s 5, 6 and 7 w i t h the permanent magnetic to two  field  parallel  the p r i n c i p a l c r y s t a l l o g r a p h i c axes, each f i g u r e c o n t a i n s s p e c t r a , one a t room temperature,  mately 20Q°C.  the o t h e r a t a p p r o x i -  F i g u r e 8 d i s p l a y s a spectrum  [oioj a x i s i n approximately the s i t i o n l i n e s p l i t s into four  (101)  s l i g h t l y o f f the  plane, here each  response o f the observed i n t o a t l e a s t two  groups.  tran-  lines.  As a r e s u l t o f the a n g u l a r dependence and t r a n s i t i o n s one The  temperature  c o u l d s e p a r a t e them  f i v e l i n e s marked w i t h arrows i n  F i g u r e s 5, 6 and 7 form p a r t o f one o f the groups. is  this  concerned o n l y w i t h these l i n e s .  This t h e s i s  The a n g u l a r dependence  w i t h i n the c r y s t a l l o g r a p h i c p l a n e s (100, 010  and 001)  are  shown i n F i g u r e 9. At  f i r s t , o n l y s p e c t r a a t X-band were taken,  nary a n a l y s i s i n d i c a t e d t h a t the zero f i e l d  A prelimi-  s p l i t t i n g was  o f the  o r d e r o f the frequency used.  S i n c e i t i s e a s i e r to a n a l y z e the  case for- which the zero f i e l d  s p l i t t i n g i s less  than the a p p l i e d  frequency, d a t a were taken and a n a l y z e d a t Q-band. measurements, consequently, were o n l y used as a  The X-band  check.  Pi'jure 5.-  EPR s p e c t r a a t -Q-band o f a n a t u r a l s i n g l e b r o o k i t e c r y s t a l w i t h  p a r a l l e l to [lOOJ a t room temperature  and approximately 200°C  Figure  6.- EPR spectre, a t Q-band o f a n a t u r a l s i n g l e b r o o k i t e c r y s t a l with H  p a r a l l e l to [010] a t room temperature and approximately  200°C  8  9  F i g u r e 7.-  10  SPPL  11  12  s p e c t r a a t Q-baid o f a natural  13  U  15  single brookite c r y s t a l with H  p a r a l l e l to [OOl] a t room temperature and approximately 200°C  c%  11  MAGNETIC  FIELD  Figure 8 . -  12  13  (Kgauss)  EPR 'spectrum  slightly off  [CIO]  a t Q-baiid o f a n a t u r a l s i n g l e b r o o k i t e c r y s t a l w i t h H  i n a p p r o x i m a t e l y the ( 1 0 1 ) plane' a t room  temperature  F i g u r e 9.-  Angular dependence' o f the EPR l i n e s o f , t h e spectrum observed a t h i g h  temperature i n . b r o o k i t e a t Q-band with H r o t a t e d i n the three p r i n c i p a l planes  crystal  29  PSTER 1 •*II"AT 10N 0? THE S P I N HAI.'.ILTONIAM PARAMETERS The f i r s t step i s to determine for the D tensor.  a set of p r i n c i p a l axes  The angular dependence of the t r a n s i t i o n  l i n e s i n the three c r y s t a l l o g r a p h i c planes where measurements were taken v.ere c a r e f u l l y recorded. Prom Figure 9 i t can be seen that the angular dependence i n the [OlO] plane indicates that there i s a magnetic axis near the (lOQj plane, i . e . , between zero and 10 degrees from i t . The (lOCj plane angular dependences are presented i n the sane figure.  The t r a n s i t i o n l i n e s reach an extremum f o r two d i f f e r -  ent o r i e n t a t i o n s .  The extremum f o r the /OQlJ d i r e c t i o n i n the  010 plane i s most l i k e l y due to the magnetic axis whose projection i s -25° from the [OQlJ  d i r e c t i o n i n the (l.OOj plane  because i t i s closer to the (OlO) plane.  One l a b e l s t h i s axis  Y, noting that t h i s i s done only as an a i d i n t h i s ckLscussion, As yet. there are no grounds to assume that t h i s i s the Y magnetic a x i s .  In the f i n a l analysis, the Y axis s a t i s f i e s the  i n e q u a l i t y 0 - E ±1/3  13  Further the other projection i n the (lOC-J plane indicates that there i s another magnetic axis ~"12 degrees from the 001 plane.  Consider the (OOlj plane i n the f i g u r e .  Again i t i s  most l i k e l y that the magnetic axis responsible f o r the extremum in the (lOOJplane near the [OlO] c r y s t a l l o g r a p h i c axis i s also  30 r e s p o n s i b l e f o r the extremum a t ~30° the (001) plane,  from the  [oiO]  in  soils  t h i s i s labeled, the Z a x i s .  I f these two  axes are near the planes  can assume f o r the moment t h a t the  discussed,  one  s p e c t r a measured i n the  g i v e n p r o j e c t i o n s • a r e going to be near the s p e c t r a along true magnetic axes.  Furthermore, the  fine structure  i s g r e a t e r i n the d i r e c t i o n l a b e l l e d Z,  than i n the  the  splitting one  l a b e l l e d Y, hence the l a b e l l i n g becomes more meaningful, one  has  labelled  by doing  the Y a x i s c o r r e c t l y as Y and  so the s i g n o f E and  Next an estimate  D are  the v a l u e s  and  not X because  the same f o r t h i s  choice.  o f D and E must be made.  T h i s i s done by c o n s i d e r i n g the maximum f i e l d s f o r t h e - t r a n s i t i o n l i n e s c l o s e to the Y and  Z directions respectively.  The  d i f f e r e n c e i n magnetic f i e l d between t r a n s i t i o n s |3/2><-* |l/2> and  |-l/2>«-»|-3/2)  i n both d i r e c t i o n s i s n e a r l y the same, t h i s  i s an i n d i c a t i o n t h a t the  centre under o b s e r v a t i o n  i s nearer  to  a f u l l y orthorhombic c o n f i g u r a t i o n than i t i s to a p u r e l y axial configuration.  Eence E w i l l be  c l o s e r to 1/3  than to  5R  S  zero.  The  d a t a was  calculated for F e ^  then compared to the graphs o f Aasa^ , the t r a n s i t i o n f i e l d s H v e r s u s  measured i n u n i t s o f liV u s i n g E/D i s hV/D  »12  f o r Q-band w i t h an S/D  F i n a l l y , one  can see  as a parameter. o f 0.25.  with  o f power f o u r i n S i s not n e g l i g i b l e .  s p e c t r a c l o s e to the Z and  D both The' r e s u l t Appendix  by several-arguments t h a t  c o n t r i b u t i o n o f the term i n the K a m i l t o n i a n operators  (See  Y axes.  The  who  B).  the  equivalent Consider  the  d i f f e r e n c e i n magnetic  field  between the  are almost  | 3/2>«-» 11/2>  and  1-1/2—1-3/2>  e q u a l , but the d i f f e r e n c e f o r the  | _3/2>«—»|-5/2)  t r a n s i t i o n s i s no  l o n g e r be e x p l a i n e d i f one  transitions  |5/2>"-*|3/2>  l o n g e r the same.  does not  T h i s can  take i n t o account  directions,  say  c o n s i d e r s three m u t u a l l y  no  equiva-  l e n t o p e r a t o r s o f f o u r t h power i n 3 i n the H a m i l t o n i a n . same is. true i f one  and  The  perpendicular  the p r i n c i p a l c r y s t a l l o g r a p h i c axes.  o r d e r p e r t u r b a t i o n theory c a l c u l a t i o n s o f the  Second  transitions,  n e g l e c t i n g f o u r t h o r d e r terms show t h a t the algebraic sum  of  the d i f f e r e n c e s between  for  three mutually sum  |3/2>*-*|l/2> and  p e r p e n d i c u l a r d i r e c t i o n s s h o u l d be  o f the d i f f e r e n c e s i s 300  gauss.  c a l c u l a t i o n s the d i f f e r e n c e between 1-1/2)*—1-3/2>  zero.  In the  F u r t h e r , u s i n g the same |3/2>*—»|l/2>  should be h a l f o f t h a t f o r the  |-3/2)«—»|-5/2> t r a n s i t i o n s . LOlOl  |-l/2>«-*|-3/2>  and  |5/2>*->|3/2>  T h i s i s c l e a r l y not true (see  and  e.g.  direction). From the above d i s c u s s i o n , i t i s c l e a r a t t h i s  t h a t because o f the l a r g e u n c e r t a i n t y i n f i n d i n g  stage  experimentally  the o r i e n t a t i o n s o f the p r i n c i p a l axes o f the t e n s o r D,  and  s i n c e the g v a l u e f o r Fe^  crystals  is  i n the l a r g e v a r i e t y o f h o s t  t h a t o f the f r e e electron.to-vaVEhina f r a c t i o n o f 2 $ .  It will  be meaningless i n t h i s work to t r y to f i t a c o r r e c t v a l u e f o r g and hence the value o f g-2.002. was Furthermore, there i s no reason  assumed. to b e l i e v e t h a t . t h e  p r i n c i p a l axes f o r the term i n the H a m i l t o n i a n o p e r a t o r s o f degree f o u r i n 5 has tensor  D.  v/ith e q u i v a l e n t  the same p r i n c i p a l axes as  I f one  assumes t h a t the v a l u e s o f a and  the o r d e r c f those o f anatase  and  t h a t the c o n t r i b u t i o n to f i r s t  ? f o r b r o o k i t e are o f  rutile,  one  calculate  o r d e r p e r t u r b a t i o n theory i s o f  the same o r d e r as the c o n t r i b u t i o n to second theory f o r E .  can  order perturbation  Hence i t seems o n l y m e a n i n g f u l  to t r y to f i t  the t r a n s i t i o n f i e l d s u s i n g f i r s t o r d e r p e r t u r b a t i o n t h e o r y f o r the terms i n the H a m i l t o n i a n o f degree f o u r i n S, The  f i r s t order p e r t u r b a t i o n theory f o r t h i s  term i s  p r o p o r t i o n a l to p a + 1 / 1 2 q ? where p = 5/2(l -m^->:r-3/5) and 4  q -35003^6  3 0 c o s S -*-3.  The  2  p o l y n o m i a l i s 2, -2.5, s i t i o n one  0,  proportionality factor for this'  2.5  and -2  i s actually looking a t  3 2  .  depending on  the  tran-  N o t i c e must be talc en t h a t  these p r o p o r t i o n a l i t y factors are t r u e o n l y i f the e i g e n s t a t e s of  the u n p e r t u r b e d S i n c e one  H a m i l t o n i a n are  does net ha.ve an a p r i o r i i d e a c f the v a l u e  p, there i s no way most one  pure.  one  can g i v e the v a l u e s o f "a" and  "E".  of The  can g i v e i s the v a l u e o f pa+(l/l2)qr(-H) f o r any  direction. A rough e s t i m a t e o f t h i s v a l u e i s e a s i l y  found by  l a t i n g the magnetic t r a n s i t i o n f i e l d s from second p e r t u r b a t i o n t h e o r y i n D and E and a d d i n g corresponding' p r o p o r t i o n a l i t y hv  =  g/JH /  hv  -  g/jH /  h ir  =  g 0 l l y  hvr =  5  3  g/?H_  2  2  2  1//2  order  the term E w i t h i t s  factor,  +2  X * 2 R  +1  X - 5/2  R - 4 Y + 5/4  -fO  X  -  - 1 X  •»  0  R  + 3/2R  calcu-  +32Y - 1Z-  16  Y  +  2  - 4 Y + 5/4  Z  Z  Z.  h^=  g/?K_^ -2 X - 2 R  t h i s i s a system o f f i v e e a s i l y solvable  f o r R.  o f D,  Q".  equations and unknowns X,  An adjustment o f the e s t i m a t e d parameters was  then made  41  THE  HAMILTONIAN PARAMETERS  w i t h a computer program g i v e n by t r i a l and  e r r o r method.  Gxx,  Gzz,  Gyy,  D and  E,  J . Hebden e t a l . ^ 9  A c o r r e c t i o n was  the EPR  made to the  c o n t r i b u t i o n o f the  transition field, p l o t s the  term R.  The  f a i r l y , pure i n t h i s range (they  the  pure c a s e ) .  X-band range, on  f i e l d s and fit as  the 500  considering ;  the t r a n s i t i o n  are  eigenstates  near the 5:6:9:6:5 o f  t r a n s i t i o n p r o b a b i l i t y c a l c u l a t i o n . f o r the  the o t h e r hand, i n d i c a t e s a v e r y s t r o n g  o f s t a t e s , hence no  corre-  proportionality factors  p r o b a b i l i t i e s a t t h i s frequency i n d i c a t e t h a t the are  the  energy l e v e l s .  t r a n s i t i o n f i e l d s by  true f o r Q-band a n a l y s i s s i n c e  The  U s i n g a.  This program c a l c u l a t e s f o r a g i v e n .  sponding t r a n s i t i o n p r o b a b i l i t y , and  were taken as  Y and  thus  functions  E, 6 and  Tne  f o u r unknowns and Z are  ASJUSTr3NT OP  the  f 32 Y - 1 Z  ?  c o r r e c t i o n was  consequently the  experimental data  mixture  made to the t r a n s i t i o n  c a l c u l a t e d t r a n s i t i o n f i e l d s do and-  i n some cases, they are  not  as much  gauss o f f . Figure  10  shows the p l o t s o f the energy l e v e l s for. the  cases w i t h E p a r a l l e l to the axes, and lations .  Table VII  gives  three p r i n c i p a l c r y s t a l l o g r a p h i c  the  transition probability calcu-  34 •TABLE V I I T R A D I T I O N PROBABILITIES Q-Band '—"ran si t i o n  "010  100  001  |5/2> «— |3/2>'  5.33  5.45  -4.89  13/2  8.25  8.37  7.75  |l/2>'*-» |-l/2>  8.73  8.74  8.85  |-l/2>"  1-3/2)  7.21  7.10  8.12  1-3/2 >*— |-5/2 ^  4.20  4.05  5.27  |5/2>'^|3/2/  0.27  4.93  0.85  |3/2> *-»|l/2>'  6.16  4.30 (0.74)  8.23  |l/2)'«—|-l/2>'  6.27  5.87  4.99  |_l/2><-* |-3/2>'  3.09  5.96  4.19  1.06  1.33  ,  »|l/2>  X-Band*  1-3/2;*-* |-5/2>  0.68  * S e v e r a l o t h e r t r a n s i t i o n s which are possible differences alone have not been included since c a l c u l a t i o n s give values o f less than 0.10.  f r o m energy the probability  TABLE V I I I HAMILTONIAN  g = 2.002  PARAMETERS  ± 0.005  D = (1170 t 30) x l O ^ c m " E = (330 ^ 20) x lO-^-cnT (pa+l/l2qP)  0 1 0  (pa+-l/12qP)  0 0 1  lO^cn"  1  = (-13±5) x l O ^ c n T  1  = (-6614) x  1  Hie f o l l o w i n g p o l a r angles g i v e  The  seven, a r e e a s i l y  the o r i e n t a t i o n s o f  c a l c u l a t e d by symmetry  e r r o r f o r the f o l l o w i n g l i s t  e  The  lO'^cm"  the o r i e n t a t i o n o f the magnetic  axes o f one o f the s u b s t i t u t i o n a l s i t e s , the o t h e r  1  = (13+10) x  100  (pa-»l/i2qP)  1  considerations,  o f angles i s - 3 ° .  '  #  z  81°  55°  y  149°  231°  x  60°  210°  d i r e c t i o n between the l o n g e s t and s h o r t e s t T i - 0 bond, which  are n e a r l y o p p o s i t e ,  corresponds to the z magnetic a x i s .  A f i n a l l e a s t mean square B , T , H  1 0 Q  , R  5 per a x i s . Goldstein ). 4 0  Q 1 0  ft  and R  computer c a l c u l a t i o n o f D, S,  , was made to f i t a l l 15 B?R l i n e s ,  and f are the E u l e r a n g l e s as g i v e n by  T h i s method i s d i s c u s s e d i n Appendix G.  The  a b s o l u t e s i g n s o f D and B were e s t a b l i s h e d . w i t h an E?R measurement a t l i q u i d helium temperatures.  The r e s u l t s are i n c l u d e d  i n Table V I I I . I t s h o u l d be p o i n t e d o u t t h a t low temperature ments i n d i c a t e  t h a t the temperature  measure-  dependence o f the p a r a -  meters i n the H a m i l t o n i a n v a r y a t most 1$ compared to those found a t room temperature.  Hence the temperature  behavior o f  b r o o k i t e i s comparable to that found i n r u t i l e ^ and n o t to t h a t found i n anatase '. 0  ENERGY  each o f the p r i n c i p a l c r y s t a l l o g r a p h i c  IN  axes  cm  Jo 6.  DI5GUG3IGI? 0? PEGUITS A.'D C0:?CHJ3I0K5 Prom Table I I i n . Chapter two,  one  j e c t i o n o f the segments j o i n i n g the T i  can c a l c u l a t e  to the n e i g h b o u r i n g  oxygens and the n e i g h b o u r i n g t i t a n i u m s .  Comparison  a n g u l a r dependence o f the s p e c t r a e n a b l e s one magnetic  exes.  d i r e c t i o n and A.(Ti  o  - 0 ) l T  The  the pro-  w i t h the'  to i d e n t i f y  Z a x i s corresponds to t h e ~ ( T i  Q  - 0  the  )  the X and' Y d i r e c t i o n s c o r r e s p o n d to the •  a n d v ( T i - 0 .) d i r e c t i o n s o V  respectively, -  T  The p o s s i b i l i t y " o f i r o n occupying an i n t e r s t i t i a l has a l s o been e x p l o r e d , but no l o g i c a l  site  correspondence has  been  found, and, on the o t h e r hand t h i s would generate a s t r o n g local  charge i n e q u a l i t y .  paramagnetic if  In a d d i t i o n ,  in rutile  In o c t a h e d r a l c o o r d i n a t i o n ,  4 2  i s 0.73°,  compared to t h a t o f T i  can conclude s a f e l y t h a t the Fe sites.  observes  i m p u r i t i e s i n the l a r g e r i n t e r s t i t i a l - s i t e s  the i m p u r i t i e s are too l a r g e to enter, the  site.  one  4  substitutional  the i o n i c r a d i u s o f  which  only  is 0.69°.^  i o n s go i n t o  Fe  Thus,  3 +  one  substitutional  The s p e c t r a a d d i t i o n a l l y show t h a t a l l e i g h t s i t e s are  e q u a l l y o c u p p i e d by Pe^*" A comparison  ions.  between the r e s u l t s o f the EPR  spectra of  3+  s u b s t i t u t i o n a l Pe ing.  i n b r o o k i t e , anatase and r u t i l e  The H a m i l t o n i a n parameters  i s interest  f o r b r o o k i t e a r e found to be  between those f o r anatase and r u t i l e ,  and c o n t r a r y to the case  o f anatase, the s p e c t r a were found i n s e n s i t i v e 'to temperature. 'Tlie s t r o n g temperature  dependence o f the anatase  case has  e x p l a i n e d i n terms o f a s h i f t o f the oxygen ions"'-'.  This  been  . s h i f t does n o t change t h e symmetry c f t h e a i i a t a s e  3  9  crystal,  whereas i t w o u l d i n r u t i l e . No a n a l y s i s has been as y e t done on t h e second o r more groups o f s p e c t r a w h i c h have been a l s o s e e n a t room tempera- • •tares.  Each one o f t h e s e l i n e s  has, a l s o t h e c h a r a c t e r i s t i c  o f s p l i t t i n g i n f o u r a t a random o r i e n t a t i o n s . o f a T i ^ * i o n by a  Fe- * 5  Replacement  i o n causes a n e g a t i v e charge e x c e s s o f  one e l e m e n t a r y charge a t t h e s i t e .  T h i s e x c e s s h a s t o be com-  p e n s a t e d t o keep the c r y s t a l e l e c t r i c a l l y n e u t r a l .  T h i s may  be a c h i e v e d by o t h e r i m p u r i t i e s , i n t e r s t i t i a l i o n s , o r a n oxygen v a c a n c y .  The p o s s i b i l i t y o f t h e f i r s t case i s u n l i k e l y  s i n c e t h i s i m p u r i t y i o n w o u l d have t o be o f a. p o s i t i v e h i g h e r than f o u r .  charge  From symmetry c o n s i d e r a t i o n s o f t h e s p e c t r a  the e t h e r two a r e p o s s i b l e , the l a t t e r case o n l y i f t h e oxygen vacancy i s n o t p r e s e n t i n the n e a r e s t n e i g h b o u r s .  Such a case  has been r e p o r t e d i n r u t i l e f o r a s u b s t i t u t i o n a l Cr3  +  by Ikebe  A3  et a l . T h i s  a s s u m p t i o n , i f t r u e , i s i n t e r e s t i n g compared  w i t h t h e o t h e r p o l y m o r p h i c forms o f T i C ^ . n e i g h b o u r oxygen vacancy"*9 a s s o c i a t e d w i t h rutile  A n a t a s e has a n e a r e s t F e  5  *  where a s  lias none. A second p o s s i b i l i t y t o e x p l a i n t h e s e l i n e s 'would be  the p r e s e n c e o f o t h e r i m p u r i t i e s , on t h e o t h e r hand t h e EPR s p e c t r a o f t h e !<x.deranerthai and V a l s e r t a l samples a r e i n d i s t i n g u i s h a b l e j , a l l o b s e r v e d EPR t r a n s i t i o n l i n e s have t h e same r e l a t i v e  intensities.  this indicates that a l l  I t i s not c l e a r a t t h i s stage i f  t r a n s i t i o n l i n e s c a n be a t t r i b u t e d t o  Pe^  o r t h a t the r e l a t i v e  c o n c e n t r a t i o n o f the paramagnetic  i m p u r i t i e s i s the same f o r both samples, and t h a t t h i s i s d i r e c t l y r e s p o n s i b l e f o r the f a c t same h a b i t .  t h a t the samples have the  An attempt to i l l u c i d a t e  t h i s p o i n t was made by  i n v e s t i g a t i n g a sample from Arkansas ( A r k a n s i t ) but no s p e c t r a were o b t a i n a b l e  because o f a ..high l o s s f a c t o r .  another unanswered q u e s t i o n . f a c t o r come?  This l e a d s to  Prom where does the h i g h l o s s  The p o s s i b i l i t y o f a h i g h c o n c e n t r a t i o n o f some  i m p u r i t y i s somehow d o u b t f u l s i n c e when the c r y s t a l was t r a n s formed to r u t i l e  by h e a t i n g , a sharp spectrum was  obtained.  At l i q u i d n i t r o g e n and lower temperatures a d d i t i o n a l s p e c t r a l l i n e s have been observed centered two.  Any a n a l y s i s o f these  a t g values  near  s p e c t r a v / i l l p r o v e — d i f f i c u l t be-  cause o f s t r o n g o v e r l a p i n g o f t r a n s i t i o n l i n e s i n t h i s  region.  In c o n c l u s i o n , the h i g h temperature EPR. spectrum i s e x p l a i n e d by assuming t h a t P e equivalent T i  4 +  3 t  substitutes f o r T i  " s i t e s being occupied  by P e  3 _ f  with  4 +  ,  a l l eight  equal  probability. The  scope o f t h i s  t h e s i s has l e f t many q u e s t i o n s un-  answered which f a r t h e r EPR i n v e s t i g a t i o n s can p o s s i b l y l e a d to  interesting  results.  41 BIBLIOGRAPHY  1.  Buerger M., Bloom M., " C r y s t a l Polymorphism", Z. • K r i s t a l l o g r 96, 182 (1937)  2.  Simmons P., D a c h i l l e ?., "The S t r u c t u r e o f T i 0 I I , a High 2  Pressure Phase o f T i 0 " , A c t a C r y s t a l l o g r . 2J>, 334 (1967) 2  3.  Czanderna A., C l i f f o r d A., Honig J . , " P r e p a r a t i o n o f H i g h l y P u r i f i e d Q?i02 (anatase)", J . Am. Chem. S o c . 79_, 5407 (1957)  4.  Low W., Offenbacher E., " E l e c t r o n S p i n Resonance o f Magnetic Ions i n Complex O x i d e s " i n 33P ( E d i t e d by P. S e i t z and D. T u r n b a l l ) , V o l . 17 p.135, Academic P r e s s , Hew York (1965)  5.  Iyengar R., C o d e l l M., K a r r a J . , T u r k e v i t c h J . , "E3H S t u d i e s o f the S u r f a c e Chemistry o f R u t i l e " , J . Am. Chem. Soc.  6.  C a r t e r D., Okaya A., "EPR o f P e •  7.  88, 5055 (1966)  H e v  «  in Ti0  2  (Rutile)", Phys.  ll£> 1^65 (I960)  L i c h t e n b e r g e r G., Addison J . R., "?- and X-band S p e c t r o s copy on P e  8.  5 +  5 +  i n R u t i l e " , Phys. Rev. 134, 331 (1969)  Horn M., "EPR o f S u b s t i t u t i o n a l and o f charge Compensated Pe^  4  i n Anatase ( T i 0 ) and i t s Temperature Dependence", 2  PhD. T h e s i s , U. o f B r i t i s h Columbia (1971) 9.  Horn M., Schwerdtfeger C. P., "EPR o f S u b s t i t u t i o n a l and Charge Compensated Pe^* i n Anatase (Ti02)", « J  p h  y « s  Chem.  42 o f S o l i d s 22, 10.  11.  Gaiiion D.,  L a c r o i x R.,  Phys. S o c .  (London) 79, 658  Barry 4, 125  12.  2529 (1971)  "SSR  of Cr^  "EPR  o f Fe^"*" Ion i n Anatase", P r o c . (1962)  i n Anatase  +  ( T i 0 ) , S o l i d S t a t e Comm. 2  (1966)  Che I'., Grave l i e P., Paramagnetique  Meriandeau P.,  "Etude par  B l e c t r o n i o u e d' un Bioxyde de  Resonance  Titane  (Anatase) Contenant des Ions Antimoine", C. R. Acad. S c . ( P a r i s ) 263C, 763 13.  E a u s e r C.,  (1969)  Cornaz P.,  i n the C r y s t a l o f T i 0  "Evidence by EPR 2  o f a Complex T i O ^  Anatase", Chem. Phys. L e t t e r s  5,  226 (1970) 14.  Low W.,  " E l e c t r o n S p i n Resonance  - a Tool i n Mineralogy  and. Geology", Adv. i n Electr... and E l e c t r o n Phys. 24,  51  (1968) 15.  Ghoose S.,  " A p p l i c a t i o n o f SSR  in Silicate  S h o r t Course L e c t u r e s Motes on Resonance Mineralogy, Am. 16.  'Veyl R.,  2  (1968)  Z. K r i s t a l l o g r . 111.' 401  P a u l i n g L., S t u r d i v a n t J . H., Brookite",  18.  Spectroscopy i n  " B r a s i s i o n s b e s timmung der K r i s t a l i s t r u k t u r des  Brookites, T i 0 " , .17.  Geol. I n s t i t u d  Minerals", i n  (1959)  "The C r y s t a l - S t r u c t u r e o f  Z. K r i s t a l l o g r . 68, 239  (1923) '•  I n t e r n a t i o n a l T a b l e s f o r X-Ray C r y s t a l l o g r a p h y  +  19.  Wyckoff R.77.C, " C r y s t a l S t r u c t u r e s Chap. IV, p.7,  Interscience  Handbook*, V o l . I,  Publishers,  Inc., New  York  (1953) 20.  Pauling  I . , "The P r i n c i p a l D e t e r m i n i n g the S t r u c t u r e  Complex I o n i c C r y s t a l s " , J . Am. 21.  22.  "-The C r y s t a l l i n e S t a t e " ,  p.109, C o r n e l l U n i v e r s i t y  Press, New  System", 25.  D a c h i l l e P.,  Am.  Czanderna. A:., Rao  Rao  C.,  "Possible  C,  (1966)  Topotaxy i n the  Trans. Faraday Soc. 5£, 1069  Yoganarasimhan  S.,  Dachille  (1961)  Rao  C,  "mechanism o f C r y s t a l  Germ. Soc. £Q, 391  P., Simmons P.,  Roy R.,  S t u d i e s o f Anatase, B r o o k i t e , Min. 53, 1929 28.  Shannon R.,  Structure  Structure  (1962)  Pask J . , " K i n e t i c s o f Anatase R u t i l e Trans-  f o r m a t i o n " , J . Am. 27.  2  Pure Anatase to R u t i l e " ,  Transformations", Trans. Faraday Soc. 58, 1579 Shannon R.,  Ti0  (1958)  " K i n e t i c s and Thermodynamics o f C r y s t a l  Can. J . Chem. 39_, 498  26.  V o l . TV,  Honig J . , "The Anatase - R u t i l e  Transformation o f S p e c t r o s c o p i c a l l y  25.  York  (1929)  K i n . 55, 403*(1970)  Transition", 24.  Chem. Soc. 51, 1010  Bragg I . , C l a r i n g b u l l G.,  Simmons P.,  of  (1965)  "Pressure-Temperature  R u t i l e and  Ti0  2  HAm.  (1966)  Pask J.,. "Topotaxy i n the A n a t a s e - R u t i l e  Transformation", Am.  K i n . 49, 1707  (1964)  29.  Eao G.,  Yoganarasemhan S., Faeth D.,  B r o o k i t e - E u t i l e Transfo m a t i o n " , 504 50.  P a s c a l P.,  Serway R.,  "Nouveau T r a i t e de Chimie- M i n e r a l e " , V o l . IV,  32.  (1965)  'temperature Dependent S p i n H a m i l t o n i a n Para-  meters o f Mn 603  'Trans. Paraday Soc. 57,  (1961)  p.96, Masson e t G i e . , P a r i s 31.  "Studies on the  i n T r i g o n a l s i t e s o f CaCO^", r'hys. Rev. j>,  (1971)  Abragam  A., Bleaney 3.,  " E l e c t r o n Paramagnetic Resonance  o f T r a n s i t i o n Ions", Clarendon P r e s s , Oxford 33.  Baker <T. K.,. W i l l i a m s P. I . 3.,  (1970)  "E3R i n t.v S a l t s 0  C o n t a i n i n g C-adolinium", P r o c . Phys. Soc. (London) 7_6, 1340 (1961) 34.  Troup C-., Hutton D.,  "Paramagnetic Resonance o f Pe^  K y a n i t e " , B r i t . J . A p p l . Phys.. 15, 1495 35.  C-ordon J . P.,  (1964)  "Variable Coupling R e f l e c t i o n Cavity f o r  Microwave Spectroscopy", Rev. S c . I n s t r . 32, 658 35.  Horn M.,  Schwerdtfeger C. P.,  "A, Method to o b t a i n  (1961) Directly  the Angular Dependence o f the EPR S p e c t r a i n S i n g l e S t u d i e s " , Rev. S c i e n t . I n s t . 42, 880 37.  Cullity  3. 3.,  Wesley, Mass. 38.  Aasa R.,  in  Crystal  (1971)  "Elements o f X-Ray De f r a c t i o n " ,  Addison-  (1967)  "Powder L i n e Shapes i n the E l e c t r o n paramagnetic  45 Resonance S p e c t r a o f H i g h - S p i n F e r r i c Complexes , J . 11  Chem. Phys..52, 5919 39.  (1970)  B y f l e e t C.., Chong D., Hebden J . , McDowell C.,  "Calculation  o f the EPR T r a n s i t i o n F i e l d s and T r a n s i t i o n P r o b a b i l i t i e s f o r a G e n e r a l S p i n H a m i l t o n i a n " , J . Mag. Res. 2, 69 40.  G o l d s t e i n H.,  41.  Low W.,  "Mecanica C l a s i c a " , A g u i l a r , Madrid  (1965)  "Paramagnetic Resonance", SSP Supplement I I  Academic P r e s s , Hew York 42.  (1970)  G e r r i t s e n H.,  (1961)  "Paramagnetic Resonance o f T r a n s i t i o n Metal  Ions i n R u t i l e  (TiC'2)",  i  n  Proceed, o f I I n t e r n . Conf. on  Paramag. Res. ( E d i t e d by W. Low), V o l . I, p.5, Jerusalem (1962) 45.  Ikebe M.,  Miyako Y., Date M.,  "ESR o f O r * Ions Coupled 5  w i t h Oxygen V a c a n c i e s i n . R u t i l e " , J . Phys. Soc. Japan 26, 45 44.  (1969)  Tiiyer J . R.,  Quick S. M.,  H o l u j P.,  "ESR Spectrum o f Fe^ +  i n Topaz", Can. J . Phys. 45, 5597 (1967) 45.  Vinokurov V. M.,  Z a p i r o v M. M.,  Phys. S o l i d S t a t e 6, 566 46.  (1964) and 6, 570  S c i e n t i f i c S u b r o u t i n e Package Publ. D i v i s i o n  (1965)  Stepanov V. G.,  Soviet  (1964)  (Subroutine LL3Q) IBM Corp.  46  The m a t r i x elements o f the s p i n H a m i l t o n i a n o f  equation  ( l ) f o r an a r b i t r a r y o r i e n t a t i o n o f the magnetic f i e l d been c a l c u l a t e d by u s i n g the g e n e r a l formulae  g i v e n by Baker  and Williams- -* and u s i n g the matrix elements f o r the >  l e n t s p i n o p e r a t o r 0 ° g i v e n by Abragam and  (1) e q u a l to z e r o ,  equiva-  Bleaney .  c a l c u l a t i o n have been made by s e t t i n g tne b^, meters i n e q u a t i o n  have  3 2  The  h£, b|, b£  para-  'The r o t a t i o n a l t r a n s -  formation o f the o p e r a t o r s 0^ are a l s o g i v e n by T h y l e r e t a l . ' 4  and Vinokurov The  et a l . * " 4  r e s u l t o f t h i s c a l c u l a t i o n i s the following:-.  CM CD  o  1  Imio In CM  |]K  k -< +  m  rn  1  K t  co  1  k n i o  o  ^ I C M  CXJlH  m  + mlCM t  i . CM  ; in  H 1  1 in  ^  VO  m  • H  vo  l°!  —"—-v.  ^  rH. +  Lm vo  in  1 rH jcn  CM  1  1  ^ , miCM  LCM  in  I .  VO  OJ m  1.  IS  iH 1  o  + rH co':n 1  e>  m  rHICvJ 1  m  rH[CM  rH 1  +  tn  in  m  [ H  tnfOJ  [CMjlA  OJ ^t n •  HQ +  1 H CMJ i n  1  CM ^ \ m"OJ  I  o,  m rH 1  tn  VO 4-  tn  m l  CM  ml  CM  m!c\i  -f-  tn i^Mlin  I  m  l§|m  U"\  (O (r-t  m  tn  rH 1  1  vo  k> *  CO  (CVI  m  o CM|rH  rH 1  in  U4 .  LCM  vo co  CTi +•  h n f o  m  Into  +  cn  rH  ICM  HJCM  1°  + rH  ^  o  coitn  W*>}" ,ICM  IS  m  Si! (CM  OJ  H  I  vo  CM M .  e>  tnjCM  ^  CM  rHICM  O  ^>|CM tn  rH|C\J  CM|rH  tnjCM  ml  CM  K -b£(3cos^-l)+ b o ( l - c o s ^ A ) c o s 2 ^ 1  6..  . IT.  ( 3 5 c o s ^ _30cos ^> + 3 )f ££sin ^ cos4°^ 38 o 2 K ^ - 6 b 2 S i n ^ c o s ^ + 2 b 2 ( s i n | b cosf cos2^+6siii|& s i n S t ) Kr,z  2  4  K,- °° [6 sin^b c o S j i ? - s i n ^ c o s ^ ( 3 c o s ^ i )]+^sin^&(cosy*cos4*.+ <-sin4«C ) +  Kqr 3bpsin^4- Qj? [ ( l + c o s ^ )cos2^-f-2 1  looses i n  2-6]  K.-= -25'D^ s i r A i - p i sin 2 /j, |(l+3cos a )cos4°^+ - c o s 2 A (3+cos 4 )sin4«6l ° "2Ta ' 240 ' ' ' 2  t  2  /  K _=• - 3 5 b ° s i n ^ cosy4+-bASin^ [cosy& (3 cos^6 ) c o s 4 & <-(l+3cos 2> )xsin4asj Co 60 +  +  2  /  Kg; 5 5 b ^ s i n ^ ^ . b f ^ ( l ^ o c c s ^ ^ - c o s ^ )cos4«c 4'cos/i(l+coSyiJ ).sin4*6j 4  where  ^6 ) are the a z l s u t h a l and p o l a r a n g l e s o f the a x i s o f  quantization with respect  to the c r y s t a l f i e l d  f o l l o w i n g w e l l known r e l a t i o n s h o l d : t a n 9. r-  B  Z  +  tan^=  axes, and the t a n $ and tan/3 =•  APPROXIMATE CALCULATED H OP D AMD E Plie main f e a t u r e s o f the EPR spectrum o f F e ^ can he d e s c r i b e d by n e g l e c t i n g the f o u r t h o r d e r .terms o f the H a m i l t o nian i n equation ( 5 ) . c ^ = /?(H.g.S) + D { 3 vhibh i m p l i e s  'Thus - 1/3 S (3 -v 1 ) ] + ' 1 / 2 E  2  of  spectrum.  2  )  (4)  t h a t two parameters, D and E, t o g e t h e r \. i t h the  d i r e c t i o n s o f the magnetic axes, a r e s u f f i c i e n t the  (3 +  to c h a r a c t e r i z e  These parameters can be e s t i m a t e d w i t h the h e l p  graphs g i v e n by Aasa-^ i f the t r a n s i t i o n f i e l d s a l o n g the  magnetic axes a r e measured. Aasa c a l c u l a t e s f o r 3=5/2 the t r a n s i t i o n f i e l d s H as a f u n c t i o n o f D, u s i n g E/D as a parameter. i n u n i t s o f hV. of  D and H a r e measured  F i g u r e I I i s a r e p r o d u c t i o n o f those p a r t s  Aasa's f i g u r e s 1 and 2 which c o r r e s p o n d to E/D=0.25. The.  transitions for H parallel  to the d i r e c t i o n s w i t h i n the  1  p r i n c i p a l c r y s t a l l o g r a p h i c p l a n e s n e a r e s t to .the Y and Z magn e t i c axes a r e shown.  .2  .5  I  2  5. 10.  gOH/hv re 11.- P o s i t i o n s  o f the SPP t r a n s i t i o n s w i t h i n  upper Kramer d o u b l e t s o f Pe^*". f o r S/D  = 0.25  the  APPENDIX  CALCULATIONS O P THE  C  SPIN HAMILTONIAN PARAMETER'S "ITS  A  LINEARIZED LEAST MEAN SQUARE P I T As d i s c u s s e d i n Chapter sition fields are f i t to the and  15 o f the spectrum  k * l ,  Ilf/^-,  5, the o b s e r v e d EPR  truncated Hamiltonian  tran-  i n brookite  ( e q u a t i o n 4 ) v/ith a  trial  e r r o r method u s i n g a computer program g i v e n by Hehden e t 39  al.  Then a l l the parameters are i n d i v i d u a l l y  s m a l l amount p and by  the new  changed by  a  t r a n s i t i o n f i e l d s , which d i f f e r  from H °  are c a l c u l a t e d by d i a g o n a l i s i n g w i t h the Hebden cal computer programme. H i s now d e f i n e d by v  H,°_ a l =  Hf * H  D •  '  '  T h i s r e p r e s e n t s a system o f k l i n e a r e q u a t i o n s i n the unknown p .  The b e s t v a l u e s o f p H Ic  (Hf P -  are determined Hg  a 1  )  minimizing  2  N o t i c e must be taken t h a t i t may parameters w i l l not converge n e c e s s a r i l y because o f untrue exp  by  happen t h a t the  new  to some b e t t e r v a l u e s  l i n e a r i z a t i o n which o c c u r s i f the H? are  near tne  values.  I t ma;/ hence be n e c e s s a r y  procedure  s e v e r a l times.  to r e p e a t  not the  

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