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EPR of substitutional and of charge compensated Fe3+ in anatase (TiO2) and its temperature dependence Horn, Manfred 1971

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EPR OP SUBSTITUTIONAL AND OP CHARGE COMPENSATED F e 5 + I'M ANATASE ( T i 0 2 ) AND ITS TEMPERATURE DEPENDENCE by Manfred Horn A THESIS SUBMITTED IN PARTIAL FULFILMENT THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of P h y s i c s We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA June, 1971 In present ing t h i s thes is in p a r t i a l f u l f i l m e n t o f the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ib ra ry s h a l l make i t f r ee ly a v a i l a b l e for reference and study. I fu r ther agree that permission for extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h is representa t ives . It is understood that copying or p u b l i c a t i o n o f th is thes is f o r f i n a n c i a l gain sha l l not be allowed without my wr i t ten permiss ion . Department The Un ive rs i t y of B r i t i s h Columbia Vancouver 8, Canada ABSTRACT Paramagnetic resonances were observed i n n a t u r a l s i n g l e c r y s x a l s of anatase between 1 °K ana 1230 °K and are i n t e r p r e x e d as due to r e g u l a r s u b s t i t u x i o n a l P e ^ + ( l ) 3 + and to Fe combined w i t h an oxygen vacancy at a n e a r e s t neighbour s i t e ( I I ) . The s p i n Harniltonian parameter h^0 of ( I ) decreases from +457 x 1 0 - ^ cm - 1 at 1 °K almost l i n e a r l y to -225 x 1 0 ~ 4 cm - 1 a t 1230 °K. This u n u s u a l l y s t r o n g tem-perature dependence of b 2° and the observed temperature dependence of the o r i e n t a t i o n of the magnetic axes of spectrum ( I I ) are both e x p l a i n e d by assuming t h a t the p o s i -t i o n s of the oxygen ions w i t h i n the u n i t c e l l are temperature dependent. i i i TABLE OF CONTENTS Page A b s t r a c t i i L i s t of Tables v L i s t of F i g u r e s v i Acknowledgements x 1. I n t r o d u c t i o n • • 1 2. R e s u l t s of previous i n v e s t i g a t i o n s on anatase , 4 2.1. C r y s t a l s t r u c t u r e 4 2.2. Phase t r a n s f o r m a t i o n to. r u t i l e . 7 2.3. Other p r o p e r t i e s of anatase 9 2.4- EPR s t u d i e s of anatase 10 3. S p i n Hamiltonian and c r y s t a l f i e l d 14 3:1 C St)in Hamiltonian. f o r F e ^ i n an i n t e r m e d i a t e c r y s t a l f i e l d 14-3.2. Z e r o - f i e l d s p l i t t i n g of S-state ions and i t s temperature dependence 18 4. Samples and experimental technique 22 4.1. N a t u r a l c r y s t a l s of anatase 22 4.2. S y n t h e t i c anatase powders 22 4.3. X-band EPR spectrometer 23 4.4. Angular p l o t s of EPR s p e c t r a 25 4.5. V a r i a b l e temperature measurements 30 5. E x p e r i m e n t a l r e s u l t s 35 5.1. P r e l i m i n a r y o b s e r v a t i o n s 35 5.2. EPR spectrum I -38 5.3. Temperature dependence of spectrum I '43 5.4. EPR s p e c t r a I i ' 4 8 5.5. Temperature dependence of s p e c t r a I I 56 i v Page D i s c u s s i o n of r e s u l t s 59 3+ 6 . 1 . EPR spectrum I : s u b s t i t u t i o n a l Fe 59 3+ 6 . 2 . EPR s p e c t r a I I : charge compensated Fe 65 6 . 3 - Conclusions 69 B i b l i o g r a p h y 71 Appendix A M a t r i x elements of the Spi n H a m i l t o n i a n ( 8 ) f o r S = 5/2 and the magnetic f i e l d H a l o n g one of the c o o r d i n a t e axes 78 Appendix B. Approximate c a l c u l a t i o n of b 0 2 and b 2 of spectrum I I o 80 Appendix C C a l c u l a t i o n of s p i n H a m i l t o n i a n parameters w i t h a l i n e a r i s e d l e a s t mean square f i t 83 LIST OF TABLES Table I -Table I I . Table I I I . Table IV. Table V . Table VI. Previous EPR r e s u l t s w i t h anatase Spin H a m i l t o n i a n parameters of s p e c t r a I and I I i n anatase. at 78 °K and 300 °K Linewidths of spectrum I i n anatase at room temperature D i r e c t i o n s of the magnetic axes f o r spectrum I I . i n anatase, expressed axes of the other three s p e c t r a I I are obtained by s u c c e s s i v e r o t a t i o n s M a t r i x f o r the s p i n H a m i l t o n i a n (8) f o r S = 5/2 and H // z. Ex p r e s s i o n s to be i n s e r t e d i n place of fc>n i n Table V. f o r the cases when H i s along the x or y a x i s VI LIST OF FIGURES. Page F i g . 1 The p o s i t i o n s of the io n s i n a u n i t c e l l of anatase 5 Fig.2 B l o c k diagram of the f i e l d l o c k e d EPR spectrometer arrangement which w i l l a u t o m a t i c a l l y t r a c e the angular de-pendence of EPR t r a n s i t i o n f i e l d s 27 Fig.3 E x p e r i m e n t a l arrangement f o r measuring angular p l o t s or r e l a t i v e angular po-s i t i o n s of EPR l i n e s 28 Fig.4 G r o s s - s e c t i o n of the c a v i t y w i t h the high-temperature h e a t i n g elements i n place 32 F i g . 5 EPR spectrum at X-band of a n a t u r a l anatase c r y s t a l w i t h H p a r a l l e l to [OOf] at room temperature -^ 6 F i g . 6 EPR spectrum at X-band of a n a t u r a l anatase c r y s t a l w i t h H p a r a l l e l to [100] a t room temperature 37 Fig.7 Angular dependence of EPR l i n e s of spectrum I i n anatase at X-band with. H r o t a t e d i n three d i f f e r e n t c r y s t a l planes F i g . 8 Peak h e i g h t s of the outer f i n e s t r u c t u r e l i n e s of spectrum I i n anatase w i t h H r o t a t e d i n the (001)-plane, n o r m a l i z e d Fig.9 EPR s p e c t r a at X-band of a n a t u r a l anatase c r y s t a l w i t h H p a r a l l e l to [001] at 449 °C, 495 °c; and 754 °0 Fig.10 Temperature dependence of EPR l i n e s Fig.11 Temperature dependence of and b^° of spectrum I i n anatase Fig.12 EPR s p e c t r a of a n a t u r a l c r y s t a l a t X-band w i t h H p a r a l l e l to [110] a t 239 °0 and 861 °<J and w i t h H p a r a l l e l to [100] at 721 °c to 1 of spectrum I at X-band of anatase w i t h H p a r a l l e l to [001] as a f u n c t i o n of temperature v i i i Page Pig.13 S p e c t r a I I i n the M O O ) - c r y s t a l plane of anatase at room temperature and 9.19 GHz. 23 KGauss was the h i g h e s t o b t a i n a b l e f i e l d . The lower p a r t of the f i g u r e shows a spectrum obtained by r o t a t i n g the c r y s t a l i n the (100)-plane at a constant f i e l d H = 7200 Gauss 49 Pig.14 EPR spectrum at X-band and room tempe-r a t u r e of s y n t h e t i c anatase powder doped w i t h 1 mol-% of Fe 52 Pig.15 Energy l e v e l s of a paramagnetic centre I I _: ^ ^ _(- . . . j _ i„ 11 „ „ n i _ "1 - L ,- i -O . u . . , J _ l OlJ-J.^v> • J C V O O ' I I J. u i i j . i j ^ C i , j _ C ^ J L o i VJCOL^I± V> J_ 0 i i c three magnetic axes. The observed t r a n -s i t i o n s are i n d i c a t e d and numbered as i n Pig.13 54 Pig.16 Temperature dependence of the angle 2 as d e f i n e d i n the t e x t . Approximately y?' g i v e s the magnetic axes of the f o u r spectra. I I , d e f i n e d i n Table IV 58 0 f P i g . 17 L a t t i c e sums ]J9° and (B^ ) , c a l c u l a t e d w i t h a p o i n t charge model of anatase as a f u n c t i o n of the oxygen, parameter u. 62 Pig.18 E f f e c t i v e g va l u e s a t low f i e l d f o r t r a n s i t i o n s w i t h i n the two lower Kramer doublets of the ground s t a t e of P e 5 + (Ref.51J Pig.1 9 P o s i t i o n s of the EPR t r a n s i t i o n s w i t h i n the two lower Kramer doublets of P e 3 + f o r E/D = 0.25 lRef.52) X ACKNOWLEDGEMENTS I would l i k e to thank Dr. C P . Schwerdtfeger f o r h i s s u p e r v i s i o n and a s s i s t a n c e i n the p r e p a r a t i o n of t h i s t h e s i s . The r e s e a r c h of the t h e s i s was supported f i n a n c i a l l y by the N a t i o n a l Research C o u n c i l of Canada through r e s e a r c h g r a n t s to Dr. C.P. Schwerdtfeger. I n a d d i t i o n , I am g r a t e f u l to Dr. E. Meagher f o r h i s help i n the X-ray a n a l y s i s of anatase and r e l a t e d problems. To Dr. M.H.L. Pryce I am t h a n k f u l f o r d i s c u s s i o n s on theore-t i c a l aspects of the i n t e r p r e t a t i o n of the observed s p e c t r a and the s p i n H a m i l t o n i a n . S c h o l a r s h i p s from the Ford Foundation and the 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, Peru, which I got through the kindness of Dr. V. L a t o r r e , are g r a t e f u l l y acknowledged. L a s t l y , I must acknowledge the constant encouragement and help I r e c e i v e d from my wi f e R o s a r i o i n completing t h i s re search. 1. INTRODUCTION I t has long been known tha t the same chemical com-pound can c r y s t a l l i z e i n d i f f e r e n t forms. This phenomenon i s 1 c a l l e d polymorphism . For many years i t has "been r e c o g n i z e d t h a t the mi n e r a l s anatase , b r o o k i t e and r u t i i e are polymorphic forms 2 of t i t a n i u m d i o x i d e . More r e c e n t l y , a f o u r t h polymorph of •5 TiOg, the s y n t h e t i c h i g h pressure T i O g H , was produced . Good c r y s t a l s of the three n a t u r a l l y o c c u r i n g p o l y -morphs of HO2 have been found. S y n t h e t i c a l l y one can grow s i n g l e c r y s t a l s of r u t i i e w i t h the V e r n e u i l method and p o l y -c r y s t a l l i n e anatase. The techniques to o b t a i n s y n t h e t i c s i n g l e c r y s t a l s of anatase and b r o o k i t e , or even p o l y c r y s t a l l i n e b r o o k i t e , are- however s t i l l not w e l l developed^'^. As a r e s u l t , most s t u d i e s on the p r o p e r t i e s of TiO^ have been made w i t h r u t i i e . T h i s i s p a r t i c u l a r l y true f o r i n v e s t i g a t i o n s u t i l i z i n g E l e c t r o n Paramagnetic Resonance (EPR) techniques. Whereas numerous EPR s t u d i e s have been r e p o r t e d on paramagnetic imp-u-r i t i e s and d e f e c t s i n r u t i i e ' , only a few of anatase ( d i s -cussed i n s e c t i o n 2.4.) and none of b r o o k i t e have been p u b l i s h e d . The i n t e r e s t i n EPR s t u d i e s of r u t i i e i s based mainly on two f a c t s . F i r s t , the b a s i c s t r u c t u r a l b l o c k s of r u t i i e In o l d e r l i t e r a t u r e a l s o named o c t a n e d r i t e , i n a l l u s i o n to i t s common o c t a h e d r a l h a b i t . 2 are TiOg octahedra, the same as i n BaTiO^ and other f e r r o -e l e c t r i c s w i t h p e r o v s k i t e s t r u c t u r e . An understanding of the p r o p e r t i e s of the s t r u c t u r a l l y and c h e m i c a l l y s i m p l e r r u t i l e can t h e r e f o r e help to e x p l a i n the f e r r o e l e c t r i c phase t r a n s i -t i o n s i n p e r o v s k i t e s . Second, r u t i l e has very good d i e l e c t r i c p r o p e r t i e s ( h i g h d i e l e c t r i c constant and low l o s s f a c t o r ) and the e l e c t r o n i c ground s t a t e s p l i t t i n g s and r e l a x a t i o n times of i r o n and chromium i m p u r i t i e s are a p p r o p r i a t e f o r i t s use as a 7 maser m a t e r i a l . The same arguments should a l s o apply f o r anatase and b r o o k i t e where the s t r u c t u r a l d i f f e r e n c e from r u t i l e l i e s mainly i n the d i f f e r e n t s t a c k i n g of the TiOg octahedra. A thorough EPR study of these m i n e r a l s was t h e r e f o r e thought to be In a d d i t i o n to t h i s , there e x i s t s a l s o an i n t e r e s t i n a comparative EPR study of the d i f f e r e n t polymorphic.forms of TiO^ from a m i n e r a l o g i c a l p o i n t of view. P a r t i c u l a r m i n e r a l o -g i c a l problems are: Which f a c t o r s determine t h a t TiOg c r y s -t a l l i z e s i n one case as r u t i l e and i n other cases as anatase or b r o o k i t e ? Which i m p u r i t i e s are found i n the d i f f e r e n t p o l y -morphs, i n which q u a n t i t i e s and on which s i t e s w i t h i n the c r y s t a l ? Reviews by W. l o w 8 and S. Ghose^ have shown t h a t EPR can help to c l a r i f y m i n e r a l o g i c a l problems of t h i s k i n d . In the present t h e s i s the r e s u l t s of an EPR a n a l y s i s of i r o n i m p u r i t i e s i n n a t u r a l s i n g l e c r y s t a l s of anatase are 3 d e s c r i b e d and are supplemented w i t h the a n a l y s i s of s y n t h e t i c p o l y c r y s t a l l i n e anatase doped w i t h i r o n . I n view of the s m a l l number of s t u d i e s of the p r o p e r t i e s of anatase and the absence of an up to date c o m p i l a t i o n of these data, a s h o r t summary of previous works on anatase w i l l be g i v e n before d i s c u s s i n g the r e s u l t s of the present i n v e s t i g a t i o n . The main r e s u l t s of t h i s i n v e s t i g a t i o n have been 1 0 p u b l i s h e d elsewhere 4 2. RESULTS OF PREVIOUS INVESTIGATIONS ON ANATASE 2•1• 0 r ys ta1 s t rue t ure Anatase lias t e t r a g o n a l symmetry and belongs to the space group 14^/amd (u|-^) (which i n c l u d e s a centre of symmetry, e.g. halfway between the oxygen ions A and F, as l a b e l l e d i n 1 1 Fig.'l) . In Fig.1 i s shown a u n i t c e l l . A l l T i ions s i t on s p e c i a l s i t e s w i t h p o i n t symmetry 42m ( f ^ ) , whereas the p o s i t i o n s of the oxygen i o n s , which have p o i n t symmetry mm, i s completely determined e x p e r i m e n t a l l y by measuring the "oxygen parameter" u, as i n d i c a t e d i n F i g . 1 . D. Cromer and K. 12 H a r r i n g t o n determined u at room temperature, as w e l l as the c e l l dimensions a and c from X-ray powder data. T h e i r r e s u l t s are a = 3-785 - 0.001 A . • C = 9.514 - 0.006 1 (1 ) u = .0.413 - 0.002 A The u n i t c e i l as shown i n Fig.1 c o n t a i n s f o u r "mole-c u l e s " of TiOg' However, some authors r e f e r to a l a r g e r u n i t c e l l w i t h e i g h t molecules, w i t h the same c as given i n (1) but an a which i s f2* times l a r g e r . This u n i t c e l l i s r o t a t e d 45° around the c a x i s , [OOl] , w i t h r e s p e c t to the s m a l l e r one shown i n F i g . ' l . While d e f i n i n g c r y s t a l l o g r a p h i c axes one has t h e r e -f o r e to be c a r e f u l which i s the u n d e r l y i n g choice of the u n i t c e l l . As can be seen from. Fig.1 each t i t a n i u m i o n i s 6 surrounded Toy s i x oxygen ions at the corners of a d i s t o r t e d octahedron and every oxygen i o n by three t i t a n i u m i o n s . Each TiOg octahedron shares 4 edges w i t h other TiOg octahedra and 1 3 the shared edges ar e , i n accordance w i t h P a u l i n g s r u l e , shortened (thus i n c r e a s i n g the d i s t a n c e between ne i g h b o u r i n g h i g h l y charged T i ^ + i o n s ) . Prom the v a l u e s f o r a, c and u, as g i v e n i n ( 1 ) , one can c a l c u l a t e the n e a r e s t neighbour d i s t a n c e s to : 4 T i - 0 1 .937 A SA, SB, SC , SD 2 T i - 0 1 • 964 A SE, SP 4 0-0 2 .802 A AB, BC , CD, DA 4 0-0 2 .446 A BE , DE, AP, CP 4 0-0 3 .040 A AE, CE, DP, BP The l a s t column of (2) r e f e r s to the i d e n t i f i c a t i o n of i n d i -v i d u a l ions i n P i g . 1 . The t i t a n i u m i o n a t S has as n e a r e s t neighbours the f o u r oxygen ions at A-D. Fig.1 shows a l s o t h a t each u n i t c e l l c o n t a i n s f o u r i n t e r s t i t i a l s i t e s , one i s marked w i t h I, w i t h the same symmetry elements as the Ti^"'~ s i t e . The only d i f f e r e n c e from the T i ^ 4 " . s i t e i s t h a t the surrounding oxygen octahedra are longer along the [001] a x i s w i t h an I-O^" d i s t a n c e of 2.792 A. Another way of v i e w i n g the s t r u c t u r e of anatase i s the f o l l o w i n g : the oxygen i o n s , which have an i o n i c r a d i u s of 14 1.32 A , are approximately i n a cubic c l o s e packed arrange-ment with l a y e r s p a r a l l e l to the ( 1 1 2 ) - c r y s t a l planes. A l l the 7 t e t r a h e d r a l v o i d s of t h i s cubic c l o s e packed s t r u c t u r e are emp-ty and h a l f of the o c t a h e d r a l v o i d s are f i l l e d w i t h the s m a l l T i 4 + i o n s , w i t h an i o n i c r a d i u s of 0.69 I 1 4 , which l i e i n z i g - z a g l i n e s p a r a l l e l to l 2 2 l ] . I t i s i n t e r e s t i n g to compare one p a r t i c u l a r f e a t u r e of t h i s s t r u c t u r e w i t h t h a t of r u t i l e . In r u t i l e , where the i n d i v i d u a l TiOg octahedra have an orthorhombic d i s t o r t i o n and share only two edges w i t h other octahedra, the d i s t a n c e "between nei g h b o u r i n g T i ^ + and 0 2~ ions i s 1.946 1 and 1.984 A 1 2 . These d i s t a n c e s are g r e a t e r than the e q u i v a l e n t ones i n anatase and r e s u l t i n a volume f o r a TiOg octahedron of 9-9 A ^ , as compared to 9.4 1 f o r anatase. I n s p i t e of t h i s , the d i f f e r -ent s t a c k i n g r e s u l t s i n the d e n s i t y of r u t i l e b eing 7% h i g h e r 2.2. Phase t r a n s f o r m a t i o n to r u t i l e 1 5 A. Navrotsky and 0. Kleppa J have shown t h a t anatase i s , under a l l temperature and pressure c o n d i t i o n s , thermody-n a m i c a l l y metastable. Under h e a t i n g , anatase transforms i r r e v e r s i b l e to the s t a b l e r u t i l e . Of i n t e r e s t i n t h i s con-n e c t i o n i s the work of W. Beard and W. P o s t e r 1 6 who found anatase i n a quenched melt of T i 0 o and SiO,,, and T i O n and B o0^ ^ ^ 2 2 3 from 1660 °G and above, thus i n d i c a t i n g a h i g h temperature formation of anatase, whereas no r m a l l y i n the l a b o r a t o r y ana-tase i s formed by dehydration of p r e c i p i t a t e d t i t a n i u m 'hydrox-1 7 ide at lower temperatures . 8 S e v e r a l authors have s t u d i e d the k i n e t i c s of the t r a n s f o r m a t i o n of anatase to r u t i i e 1 8 " ~ 2 1 . The r e s u l t s can be summarized as follows.. The r a t e of 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 are governed by the surface s i z e and by the nature and amount of i m p u r i t i e s which determine the d e f e c t s t r u c t u r e of anatase, e.g. the c o n c e n t r a t i o n of oxygen va c a n c i e s or i n t e r s t i t i a l s . S p e c t r o s c o p i c a l l y pure anata.se powder transforms to r u t i i e o 19 a f t e r h e a t i n g above a t h r e s h o l d temperature of 610 'C . The a c t i v a t i o n energy i s 80 k c a l / m o l w h i c h i s mainly the a c t i v a -t i o n energy f o r the p r o d u c t i o n of the n u c l e a t i o n s i t e s . The r a t e of t r a n s f o r m a t i o n i s t h e r e f o r e governed by the r a t e of n u c l e a t i o n . The e f f e c t of 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 vacancies a c c e l e r a t e , i n t e r s t i t i a l i o n s i n h i b i t the 21 ° + + + t r a n s f o r m a t i o n . The former are produced by Cu11" , L i , Wa , C r ^ + , F e ^ + , which s u b s t i t u t e f o r T i ^ " + , whereas Ca^ +, S r ^ + , • 2+ - 5 + 6 + Zn , C l , P , S i n c o r p o r a t e as i n t e r s t i t i a l i ons or gene-3+ r a t e i n t e r s t i t i a l T i . A c c o r d i n g to the i m p u r i t i e s present, the t h r e s h o l d temperature f o r the t r a n s f o r m a t i o n can vary from 400 °C to 1200 °C. . 2 2 D a c h i l l e , Simons and Roy determined the i n f l u e n c e of pressure on the t r a n s f o r m a t i o n , e s t a b l i s h i n g apparent s t a b i l i t y ranges of anatase, b r o o k i t e , r u t i i e and T i 0 2 H i n a P,T diagram. 23 ' Shan on and Pask'"" proposed a mechanism f o r the t r a n s -formation of anatase to r u t i i e : s t a r t i n g w i t h the e x p e r i m e n t a l 9 evidence t h a i r u t i i e c rysta.ls formed i n transformed s i n g l e c r y s t a l s of anatase have p a r t i c u l a r p r e f e r r e d o r i e n t a t i o n s , they assumed r e t e n t i o n of the {112} pseudo c l o s e packed planes of oxygen i n anatase as the {100} pseudo c l o s e packed planes i n r u t i i e , and rearrangement of the t i t a n i u m and oxygen i o n s w i t h i n these planes. 2.3. Other p r o p e r t i e s of anatase Other p h y s i c a l p r o p e r t i e s of anatase t h a t have been i n v e s t i g a t e d are the b a s i c mechanical, thermal and o p t i c a l ones. I f not otherwise i n d i c a t e d , the f o l l o w i n g v a l u e s are taken from r e f e r e n c e s 17 and 24, where a l s o more, and u s u a l l y o l d e r , data can be found. Values i n parentheses r e f e r to r u t i i e . Anatase c r y s t a l s show p e r f e c t cleavage p a r a l l e l to (001) and {011}. The d e n s i t y i s 3.87 - 3-95 gm/cra5 (4.21 -. 4.25) and the hardness i s 5 . 5 - . 6 . 0 ( 7 . 0 - 7 . 5 ) . 25 Rao • measured w i t h an X-ray h i g h temperature powder camera the l a t t i c e constants a and c up to 712 °0. Over t h i s temperature range he obtained a continuous i n c r e a s e of c and a, w i t h c i n c r e a s i n g more r a p i d l y than a. At room temperature he obtained the thermal expansion c o e f f i c i e n t s 7.8 x 10~ 6 °<J~1, « x = 3.8 x 10" 6 V 1 and at 712 °0 * l ( = 19.5 x 10~ 6 V ' 1 and <^i = 9 . 5 x 1 0 CJ . Over a temperature range of 500 °U c i n c r e a s e d 0.6$ and a, 0.3$. S c h r o e d e r 2 6 measured o p t i c a l l y the' change of the q u o t i e n t < X , , / « X by measuring the angle of a 10 prism cut c o n v e n i e n t l y out of a y e l l o w anatase c r y s t a l from B i n n e n t a l , S w i t z e r l a n d , <rtu/ cKJL i n c r e a s e d l i n e a r l y up to 642 - 3 °'J and beyond t h i s temperature a.gain l i n e a r l y , but w i t h a d i f f e r e n t slope,up to 900 °G where anatase s t a r t e d to t r a n s -form i n t o r u t i l e . T h i s behaviour below 900 C was completely r e v e r s i b l e . Schroeder concluded t h a t at 642 °(J anatase undergoes a r e v e r s i b l e phase change to a h i g h temperature s t r u c t u r e , named - anatase, w i t h probably a l s o t e t r a g o n a l symmetry. The s p e c i f i c heat at room temperature i s 13.22 cal/mol°u (13.16; . S y n t h e t i c powders of anatase are w h i t e . N a t u r a l c r y s t a l s have v a r i o u s c o l o r s , from brown, y e l l o w , g r e e n i s h , p a s s i n g i n t o blue and b l a c k but r a r e l y c o l o r l e s s . The o p t i c a l c onstants vary markedly w i t h wavelength and temperature. The i n d i c e s of r e f r a c t i o n n and n decrease c o n t i n u o u s l y w i t h o e J t e m p e r a t u r e 2 6 up to 900 °G, w i t h n Q d e c r e a s i n g f a s t e r than n , so t h a t the b i r e f r i g e n c e decreases a l s o . At room temperature n5893A = 2 . 5 6 ( 2 . 6 1 ) , n 5 8 9 3 l = 2 . 4 9 ( 2 . 9 0 ) . Anatase i s an i n s u l a t o r at room temperature. The s t a t i c d i e l e c t r i c constant of the s y n t h e t i c powder i s i = 48(114) 2 1 . 2.4. EPR s t u d i e s of anatase The previous EPR r e s u l t s obtained w i t h anatase are summarized i n Table I . The parameters quoted there have the TABLE I. Previous EPR Results with Anatase Values i n parentheses r e f e r to r u t i l e Paramagn. species Location T °K Freq. band Seff g ( j g j. or S x Sv g z D 10 cm Remarks -4 -1 Energies i n 10 cm . • Ref. F e 3 + Substit. f o r T i 300 (4.2-300) X 5/2 2.009 2.002 (2,00) + 309 (+6780) natural single c r y s t a l F = + 308(+230) a =•+ 99(-280)(E = + 690) 28 C r 3 + Substit. f o r T i 4 + 300 (4.2-300) X 3/2 1.973 (197) 373 (-6800) Synthetic powder A53Cr = 1 7 ( 1 6 # 7 ) ( E = _ 2 2 7 0 ) 29 3+ T i Substit. f o r 83 -173 X.Q 1/2 1.990 1.959 (1.975) (1.953) Synthetic powder, doped 5+ with Sb 30, I n t e r s t i t . (4.2) 1/2 1.987 1.966 (1.972) (1.940) ( A4 7 » 4 9 T i - 1.9 - 2.2) 31 3+ ' ( T i O ) J + near s u r f a -ce, i n the bulk 300 X 1/2 2.023 2.052 1.983 Synthetic powder, heated under oxygen atm.at 450 C A47,49Ti = K , 32 on surface 83-300 X 1/2 three g values between 2.023 and 1.98 Synthetic powder, heat and oxygen treated. Also other l i n e s are observed, which are assoc.with Ti3+ and F-center on surface. 33 1 V. 1 I on surface 1.6-300 X,Q 1/2 1.977 1.399 2.022 Synthetic powder, heat and oxygen treated. 34 12 u s u a l meaning ^ f o r t h e i r e x p l i c i t d e f i n i t i o n see chapter 3 of t h i s t h e s i s ; . The only p u b l i s h e d s i n g l e c r y s t a l study was made by 2 8 D. Gainon and R. L a c r o i x . They r e p o r t e d a s i n g l e magnetic spectrum i n a n a t u r a l c r y s t a l of anatase, which they a s s o c i a -ted w i t h P e ^ + i m p u r i t i e s s u b s t i t u t i n g f o r T i ^ 4 . T h e i r two main c o n c l u s i o n s were t h a t : 3 + ( i ) "D i s much s m a l l e r than f o r the Re i o n i n the r u t i i e form of T i 0 2 " . ( i i ) "The ratio-D/F i s very near u n i t y , w h i l e i t i s at l e a s t more than ten times l a r g e r i n a l l the other cases we know". ( L i t e r a l q u o t a t i o n s j . As Gainon and L a c r o i x made only room . temperature measurements they c o u l d determine only the r e l a -t i v e s i g n s of the s p i n H a m i l t o n i a n parameters, however, they conclude t h a t the upper s i g n , as g i v e n i n Table I , " i s more l i k e l y c o r r e c t , because f o r a l l experiments yet known, a has a p o s i t i v e s i g n " . T. ±>arry2^ analysed ( J r ^ + doped anatase powder a t room temperature. He observed f i v e powder l i n e s which he cou l d d e s c r i b e by the s p i n H a m i l t o n i a n parameters g i v e n i n Table I . The c r y s t a l f i e l d term D (on l y the a b s o l u t e value was measured) i s s i m i l a r to that f o r Fe , as e s t a b l i s h e d by Gainon and L a c r o i x , and the s p e c t r o s c o p i c s p l i t t i n g f a c t o r i s 3 + t y p i c a l f o r Cr- i n weak o c t a h e d r a l c r y s t a l f i e l d . The hyper-'s 3 f i n e s p l i t t i n g constant A f o r -^Cr i s the same as i n r u t i i e . 13 In a d d i t i o n to these two s t u d i e s on paramagnetic i m p u r i t i e s i n anatase some EPR analyses on d e f e c t s i n anatase 30 have been r e p o r t e d . iVi. Che et a l . r e p o r t two EPR s p e c t r a 5+ ' i n Sb doped anatase pov/der, which they d e s c r i b e d as due to 3+ T i i ons i n d i f f e r e n t c r y s t a l l o c a t i o n s w i t h a x i a l symmetry, probably s u b s t i t u t i o n a l and i n t e r s t i t i a l . (J. Hauser and P. 32 Cornaz r e p o r t two EPR powder s p e c t r a w i t h orthorhombic symmetry of heat t r e a t e d anatase powder and i n t e r p r e t them as due to (TiO) complexes near the s u r f a c e . The l a s t two s t u d i e s , as g i v e n i n Table I , d e s c r i b e EPR l i n e s which are a s s o c i a t e d w i t h 0^ and 0^ r a d i c a l s a ttached to the surface of anatase powder. 14 5- SPIN HAMILTONIAN AND CRYSTAL FIELD In t h i s chapter the s p i n H a m i l t o n i a n and the theo-r e t i c a l i n t e r p r e t a t i o n of i t s parameters v / i l l be d i s c u s s e d i n 3+ as much as i t i s r e l e v a n t to the observed EPR s p e c t r a of Fe i n anatase. 3+ 3.1. S p i n Hamiltonian f o r Fe i n an i n t e r m e d i a t e c r y s t a l f i e l d In most cases a c r y s t a l f i e l d i s not st r o n g enough to break down the LS-coupIing of the f r e e i o n s of the 3 d -t r a n s i t i o n elements, but i s st r o n g e r than the s p i n - o r b i t coupl-i n g . As a r e s u l t , the t o t a l angular momentum J l o s e s i t s meaning but L ( o r b i t a l angular momentumj and S ( s p i n angular momsntuis) r C i L i a . n i '• go'«j& Quaii i/urn i i u i i i i j e i B ' • .LULLS i a osii6& o i i y i n t e r m e d i a t e c r y s t a l or l i g a n d f i e l d case. The e l e c t r o n i c c o n f i g u r a t i o n of the i o n Fe i s d and 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^/g. ^ n a n i n t e r m e d i a t e c r y s t a l f i e l d the ground s t a t e i s an o r b i t a l s i n g l e t , Owing to the c r y s t a l f i e l d and/or an externa], mag-n e t i c f i e l d H I t s 6 - f o l d spin-degeneracy i s l i f t e d . N e g l e c t -i n g a p o s s i b l e i n t e r a c t i o n w i t h surrounding n u c l e a r s p i n s or the n u c l e a r s p i n of the 2°/o abundant ^'Fe, the f i n e s t r u c t u r e of the ground s t a t e i s c o n v e n i e n t l y d e s c r i b e d by the s p i n H a m i l t o n i a n (Abragam and i j l e a n e j r ^ ; t h i s i s the main r e f e -rence source f o r the present c h a p t e r j 15 « = a S . g . 1 + 1 £ = 2 b 2 m o 2» + ^ l - 4 b 4» o 4» (3) The f i r s t term i n (3) r e p r e s e n t s the u s u a l magnetic f i e l d de-pendence, 6 i s the Bohr magneton and g the s p e c t r o s c o p i c s p l i t t i n g t e n s o r . E x p e r i m e n t a l l y (and a l s o t h e o r e t i c a l l y ) g i s found to he n e a r l y i s o t r o p i c and to d i f f e r at most a 3+ f r a c t i o n of ^cfo f o r Fe i n a v a r i e t y of host c r y s t a l s . • The b n m i n (3) can be c o n s i d e r e d as experimental constants and' the 0 n m are operator e q u i v a l e n t s , formed from polynomials of the s p i n operator S, so t h a t they have the same t r a n s f o r m a t i o n p r o p e r t i e s as the corresponding s p h e r i c a l harmonics X ^ . In the present case the important 0 n m a r e ^ 6 0 2° = 3 S z 2 - S(S + 1) ; 0 2 2 = 1/2 ( S + 2 + S_ 2) -= - S y 2 • 0 4° = 3 5 S z 4 -•30S(S+1)S Z 2 + 2 5 S z 2 . - 6S(S+1) + 3 S 2 ( S + 1 ) 2 (4). 0 4 4 = 1/2 ( S + 4 + S_ 4) 0 42 = l / 4 { [ 7 S z 2 - S ( S + 1 ) - 5 ] ( S + 2 + S _ 2 ) + ( S + 2 + S _ 2 ) [ 7 S z 2 - S ( S + 1 ) - 5 J \ A p p r o p r i a t e b a s i s f u n c t i o n s f o r (3) are the e i g e n f u n c t i o n s 1M>, M = - 5/2, - 3/2, . + 5/2 of the operator S . The z operators 0 n m connect then a s t a t e |M.,> only w i t h |Mp>, where | I\(L - M ? | = m. • 16 Spin operators of higher degree than f i v e need not to be i n c l u d e d i n (3) because a l l of t h e i r m a t r i x elements are zero w i t h i n the ma n i f o l d of S = 5/2. S p i n operators of odd degree have a l s o zero m a t r i x elements, because they do not f u l f i l l time r e v e r s a l i n v a r i a n c e (the components of S are odd operators under time r e v e r s a l ) . The constants b m are r e l a t e d to D, E, a and P used i n many EPR s t u d i e s "by the r e l a t i o n s b 2° = D b 4° = a/2 + P/3 (5) b 2 2 = 3E b 4 4 ='.5a/2 • Using the known t r a n s f o r m a t i o n p r o p e r t i e s of Y n m under r o t a t i o n of the coo r d i n a t e system, one can evaluate 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 of the b n m under these r o t a t i o n s . E x p l i c i t formulae are given by V. Vinokurov et a l . ' ^ ' ^ 8 and 39 J . Thyer et a l . . The number of necessary c o e f f i c i e n t s b v i f f i can be reduced by choosing a s p e c i a l c o ordinate system and i s f u r t h e r reduced a c c o r d i n g to the symmetry of the c r y s t a l f i e l d . In p a r t i c u l a r , a cubic c r y s t a l f i e l d V c ( x 4 + y 4 -i- z 4 - 3/5 r 4') • (6) 17 g i v e s r i s e to an operator e q u i v a l e n t 0^° + 5 0^ , so t h a t i t can he d e s c r i b e d by a s i n g l e parameter a = 2/5 b^.4 = 2 b ^ 0 p r o v i d e d one chooses the three f o u r f o l d axes as the r e f e r e n c e system. A l l other b n m are then zero. I f one uses, however, a f o u r f o l d a x i s as z - a x i s and the two p e r p e n d i c u l a r t w o f o l d axes as x- and y'-axes, one o b t a i n s t ^ 4 , and t h e r e f o r e a, of opposite s i g n and an a d d i t i o n a l apparent a x i a l d i s t o r t i o n 2 8 F = -3a. I t i s now apparent t h a t Gainon and L a c r o i x must have used t h i s l a t t e r c o o r d i n a t e system to d e s c r i b e the EPR 3+ spectrum of s u b s t i t u t i o n a l Fe i n anatase, r e s u l t i n g i n an unusual n e g a t i v e value f o r a and a very l a r g e F (Gainon and L a c r o i x e s t a b l i s h e d only t h a t a has an opposite s i g n to D, but D i s p o s i t i v e , as w i l l be d i s c u s s e d below). * The observed E P K s p e c t r a of Fe'' i n anatase i n d i c a t e 3+ t h a t the Fe i o n s are l o c a t e d i n two d i f f e r e n t c r y s t a l f i e l d s The f i r s t one i s mainly cubic w i t h an a d d i t i o n a l d i s t o r t i o n a long a t e t r a g o n a l a x i s . The r e s u l t i n g a x i a l c r y s t a l f i e l d i s r e p r e s e n t e d by the s p i n H a m i l t o n i a n = g„ 13 H z S z + 1 g ± 13 H x ( S + + S J + 3 b2 °2 + 6*0 b4 °4 + £0 V °4 (7) The o n l y symmetry of the c r y s t a l f i e l d at the second s i t e s i s g i v e n by a r e f l e c t i o n plane through the paramagnetic '6 A r e i n t e r p r e t a t i o n • of the r e s u l t s of Gainon and L a c r o i x was g i v e n by M. Horn and O.F. Schwerdtfeger^O. 18 i o n s . The a p p r o p r i a t e c o o r d i n a t e system i s then g i v e n by the x-and z-axes i n t h i s plane and the y - a x i s p e r p e n d i c u l a r to the r e f l e c t i o n plane ( r e f . 3 5 , p . 6 6 7 j . This makes b n m = 0 f o r m<0. The d i r e c t i o n s of the two axes i n the plane can be chosen so t h a t b 2 = 0. This i m p l i e s the choice of the p r i n c i p a l axes . of the second order tensor b,-,m as coo r d i n a t e axes and i s s p e c i a l l y a p p r o p r i a t e i f the terms b 2 m are much l a r g e r than the f o u r t h order terms b^m, because then the e x p e r i m e n t a l l y determined "magnetic axes", d e f i n e d by extreme p o s i t i o n s of the EPR s p e c t r a l l i n e s , c o i n c i d e w i t h the c o o r d i n a t e axes. As w i l l be d i s c u s s e d i n chapter 5 , only b^111 w i t h m even are considered i n the d e s c r i p t i o n of the observed EPR s p e c t r a of anatase, so t h a t the r e s u l t i n g s p i n H a m i l t o n i a n has the form "K = B rf • g • S + i - b 2° 0 2° + 1 b 2 2 + 1- 5 b 4 m 0 4 m (8) 3 3 6 0 m=0,2,4 By e v e n t u a l permutation of the axes one can a d d i t i o n -a l l y l i m i t the value of b 2 2 / b 2 ° to the range 0 < b 2 2/b 2° L i ( r e f . 4 1 J . 3 - 2• Z e r o - f i e l d s p l i t t i n g of S-state i o n s and i t s •feraperature dependence 3 + I f the ground s t a t e of Fe i n an i n t e r m e d i a t e 6 c r y s t a l f i e l d were a pure i t s s i x f o l d s p i n degeneracy would not be l i f t e d and a l l b n m of the s p i n H a m i l t o n i a n ( 3 ) would be zero. E x p e r i m e n t a l l y however, one observes a f i n e 1 9 s t r u c t u r e of the ground s t a t e and s e v e r a l mechanisms have been proposed to e x p l a i n i t . (An up to date l i s t of r e f e r e n c e s i s 42 g i v e n by A. Serway ).. 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 produce second or higher order 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 f i e l d . The r e s u l t s f o r b,-,0 {=!)) can be summarized i n b 2° = b Q + c, B 2 ° + c 2 ( B 4 0 ) ' + c 3 (B 2°j 2 • . ( 9 ) The ti^ are c o e f f i c i e n t s of a development of the e l e c t r i c c r y s t a l f i e l d V at the paramagnetic i o n s i t e i n s p h e r i c a l harmonics Y m n 1 / 2 V = (JULY B./ 1 r n Y „ m . ( 1 0 ) n,m >2n+'!> — (. B^ 0 j 1 i s the remaining p a r t of a f t e r subs t r a d i n g the p a r t corresponding to a p e r f e c t cubic c r y s t a l f i e l d . b accounts f o r cov a l e n t bonding and o v e r l a p of the e l e c t r o n s of the paramagnetic i o n w i t h the surrounding l i g a n d s and the' c_j are constants determined by p r o p e r t i e s of the paramagnetic i o n , such as s p i n - o r b i t c o u p l i n g a n d . o r b i t a l l e v e l s p l i t t i n g i n a cubic c r y s t a l f i e l d . R. Sharma, T. Las and R. Or b a c h 4 3 have c a l c u l a t e d b 9° f o r Mn 2 + i n o c t a h e d r a l c o o r d i n a t i o n of ZnP 2. They f i n d b 2° = - 0.07 B 2° + 0.36 ( B 2 0 ) 2 + 4.34 ' (11) 20 i f b^ 0 i s measured i n cm 1 and ti^® i n e 2 / 2 a o n + 1 ; e i s the elementary charge, a , the atomic u n i t of. l e n g t h . In some cases , even the use of a p o i n t charge model to c a l c u l a t e B^ 0 and ( B ^ 0 / , whi l e n e g l e c t i n g b Q , has r e s u l t e d i n a c a l c u l a t e d value of b^ 0 i n reasonable agreement 43 w i t h experiment. A p o i n t charge model c a l c u l a t i o n y i e l d s B 2° = £ ~ ci. 13 cos^O. - 1) / R.^ (12j ( B . 0 ) ' = y " q . ( - (35 c o s 4 0. - 30 c o s 2 0. + 3J Z c.-in 4 Q ono/i rK 1 / T) 5 4 ~ " d ' f J J ' " J U 3 ) where the e x t e r n a l p o i n t charges q.. |e| are s i t u a t e d , a t {R . , 0., dp . j w i t h r e s p e c t to an o r i g i n taken at the s i t e the paramagnetic i o n . of. The above b r i e f l y o u t l i n e d theory e x p l a i n s a l s o to . some extent the temperature dependence of the a e r o - f i e l d s p l i t t i n g of S-state i o n s , because w i t h changing temperature the i n t e r i o n i c d i s t a n c e s and t h e r e f o r e the components jj 111 of the c r y s t a l f i e l d w i l l change. Thus R. Sharma 4^ q u a l i t a t i v e l y e x p l a i n e d the temperature dependence of D f o r Mn i n UdOlg, by c a l c u l a t i n g i ^ 0 and (.B^0/ w i t h a p o i n t charge model of t h i s c r y s t a l f o r d i f f e r e n t temperatures, u s i n g the known t h e r -mal expansion of Odulg. 21 if) W. Walsh et a l . d i s t i n g u i s h between t h i s " i m p l i c i t " temperature dependence caused "by thermal expansion and an " e x p l i c i t " temperature dependence of the s p i n H a m i l t o n i a n parameters caused by l a t t i c e v i b r a t i o n s . F o r m a l l y , these i m p l i c i t and e x p l i c i t e f f e c t s can always be separated f o r any v a r i a b l e which i s completely spec-i f i e d by the temperature T and the volume V of the c r y s t a l , e.g. f o r D one can w r i t e 3T / P \ "5 T / V K \ i> p / T where the f i r s t term on the r i g h t - h a n d side i s the e x p l i c i t c o n t r i b u t i o n to D'^J and the second term on the r i g h t — h a n d s i d e i s the i m p l i c i t term. 6 =v(~T"T"^P ^ i e " t n e r m a l volume expansion c o e f f i c i e n t and K=--^-(-^4)m , the c o m p r e s s i b i l i t y . The i m p l i c i t temperature dependence of 1) may t h e r e f o r e be e v a l u a t e d i f 13, K and the i s o t h e r m a l pressure dependence of D are known. On.the basis of a simple " e f f e c t i v e p o i n t char-ge model" W. Walsh et a l . 4 6 c a l c u l a t e t h a t the i m p l i c i t temperature dependence of D and a should depend cn the i n t e r -i o n i c d i s t a n c e r as 7 —91 D <* r 1 ; a r where an i s o t r o p i c thermal expansion i s supposed. 22 4. SAMPLES AND EXPERIMENTAL TECHNIQUE 4.1. N a t u r a l c r y s t a l s of anatase The f i v e n a t u r a l c r y s t a l s of anatase s t u d i e d i n t h i s work o r i g i n a t e d i n Tavetch, Graublinden, S w i t z e r l a n d and B i n n t a l , W a l l i s , S w i t z e r l a n d . They were k i n d l y s u p p l i e d by A. Harnik of the C r y s t a l l o g r a p h i c I n s t i t u t e of ETH, Z u r i c h , and E. Meagher of the Geology Department of U.B.C., Vancouver. T y p i c a l diameters of the c r y s t a l s were 1-3 mm. The two Tavetch c r y s t a l s had pyramidal h a b i t , opaque gray c o l o r and t h e i r EPR s p e c t r a , although q u a l i t a t i v e l y s i m i l a r to those of the B i n n t a l c r y s t a l s , showed broad and p o o r l y r e s o l v e d l i n e s . The three c r y s t a l s from B i n n t a l were y e l l o w and t r a n s p a r e n t . They possessed d i f f e r e n t h a b i t s but always had c l e a r r e c o g -n i z a b l e {110}-planes, which helped to o r i e n t a t e them. One c r y s t a l was s p l i t i n three p a r t s , u s i n g the (001)-cleavage• planes. The o r i e n t a t i o n of the c r y s t a l axes was checked w i t h b a c k - r e f l e c t i o n Laue X-ray photographs. A l l the B i n n t a l c r y s t a l s showed the same EPR s p e c t r a and the s i n g l e c r y s t a l r e s u l t s r e p o r t e d here were obtained w i t h them. 4.2. S y n t h e t i c anatase powders The s i n g l e . c r y s t a l s t u d i e s were supplemented w i t h s y n t h e t i c powder s t u d i e s . The powders were prepared by 23 h y d r o l y s i s of T i C l ^ w i t h NIL. OH ( r e f . 17, p.229). The p r e c i p i -t a t e d T i 0 2 powders were c r y s t a l l i z e d by h e a t i n g f o r 24 hours a t 400 °C i n open p o r c e l a n v e s s e l s . The powders co u l d be doped by adding 1 mol - f o or 0.01 mol - f o of aqueous s o l u t i o n s of P e C l ^ or lvmClg to the i n i t i a l T i C l ^ . Debye-Scherrer X-ray a n a l y s i s showed f o r a l l powders only anatase l i n e s . 4.3. X-band EPR spectrometer A l l EPR s p e c t r a were taken at X-band. The X-band EPR spectrometer was of c o n v e n t i o n a l "balanced "bridge design. As the analysed anatase c r y s t a l s showed s t r o n g EPR s i g n a l s , no s p e c i a l a t t e n t i o n had to be g i v e n to operate the EPR spectrometer w i t h maximum s e n s i t i v i t y . The microwave bridge was formed by a magic T. A r e f l e x - K l y s t r o n ( V a r i a n V-153/6315; max. output 70 mW), a. one-way f e r r i t e i s o l a t o r and a f l a p - a t t e n u a t o r were connected to the input arm, a c r y s t a l d e t e c t o r (Microwave 1N 23B) to the output. One side arm ended i n a T E ^ Q 2 resonance c a v i t y (mostly used V a r i a n multi-purpose c a v i t y V-453'l) , coupled through an a d j u s t a b l e i r i s . The other r e f e r e n c e arm con-t a i n e d a s l i d e screw tuner and a matched l o a d . The c r y s t a l diode was biased (100-300 ^ A) by a d j u s t i n g e i t h e r the s l i d e screw tuner or the c o u p l i n g of the resonance c a v i t y . The K l y s t r o n was frequency l o c k e d to the resonance 24 c a v i t y "by modulating the K l y s t r o n r e f l e c t o r v o l t a g e w i t h 10 kHz and u s i n g the corresponding phase s e n s i t i v e detected output from the c r y s t a l d e t e c t o r as an e r r o r feedback s i g n a l . The magnetic f i e l d was modulated w i t h 100 kHz through s m a l l modulation c o i l s a t t a c h e d d i r e c t l y to the r e s o -nance c a v i t y . The modulation amplitxide used was 0.2-12 Gauss a c c o r d i n g to the l i n e w idth of the analysed EPR t r a n s i t i o n . 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 detected at 100 kHz (PAR L o c k - i n a m p l i f i e r , model 121). The output of the L o c k - i n was u s u a l l y connected, d i r e c t l y to a s t r i p c h a r t r e c o r d e r . In t h i s way one obtained a r e c o r d i n g of the f i r s t d e r i v a t i v e of the EPR a b s o r p t i o n l i n e s . T h e frequency of the microwaves was measured w i t h a d i g i t a l frequency meter. (Hewlett-Packard Frequency Con-v e r t e r 5255A and E l e c t r o n i c Counter 5245L). For most experiments a 9-5" magnet (iVlagnion) wets used, w i t h a r o t a t i n g c o i l f i e l d sensor iMagnion FFC-4 power s u p p l y ) . The maximum o b t a i n a b l e f i e l d was 23 kG. The d i r e c t r e a d i n g f i e l d sensor d i a l s were c a l i b r a t e d a g a i n s t a N1V1R-gaussrneter, The permanent magnetic f i e l d was always perpen-d i c u l a r to the microwave magnetic f i e l d at the sample i n the c a v i t y . 25 4.4. Angular p l o t s of EPR s p e c t r a In order to i d e n t i f y and to analyse the paramagnetic cen t r e s i n a s i n g l e c r y s t a l one has to measure 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 of the o r i e n t a t i o n of the c r y s t a l i n the s t a t i c magnetic f i e l d H. U s u a l l y one records an EPR spectrum f o r a given o r i e n t a t i o n as a f u n c t i o n of the f i e l d s t r e n g t h H, then r o t a t e s e i t h e r the sample or the mag-net through a c e r t a i n angle i n a p a r t i c u l a r c r y s t a l plane, records a new spectrum and so on. To o b t a i n complete graphs of H^. versus angle 9 one r e q u i r e s many s p e c t r a i m p l y i n g a great amount of time and energy. This i s p a r t i c u l a r l y true i f the EPR l i n e s cross one another and i f t h e i r i n t e n s i t i e s are s t r o n g l y angular dependent.' I n i t i a l l y , t h i s method a l s o was a p p l i e d i n the pre-sent work to analyse the EPR s p e c t r a of the anatase c r y s t a l s . L a t e r , however, a new technique was used. I t was found t h a t one can o b t a i n these graphs more e a s i l y by l o c k i n g the mag-n e t i c f i e l d to a p a r t i c u l a r EPR l i n e and then r o t a t i n g the c r y s t a l s l o w l y w i t h a s m a l l motor. In t h i s way one can r e c o r d d i r e c t l y on an X-Y p l o t t e r the d e s i r e d graphs by s u p p l y i n g to the Y-input a v o l t a g e p r o p o r t i o n a l to the instantaneous mag-n e t i c f i e l d , deduced from a f i e l d sensor, and to the X-input of the p l o t t e r a vo l t a g e i n d i c a t i n g the o r i e n t a t i o n of the sample. A d e s c r i p t i o n of t h i s technique was gi v e n bv M, Horn and C P . Schwerdtf e g e r 4 ? . 26 Pig.2 g i v e s the block diagramme of t h i s method. E s s e n t i a l l y i t i s the c o n v e n t i o n a l EPR spectrometer as des-c r i b e d i n the previous s e c t i o n 4.3. But whereas u s u a l l y the EPR s i g n a l output of the b o c k - i n a m p l i f i e r goes d i r e c t l y to a r e c o r d e r , i t i s now used as an e r r o r s i g n a l f o r the con-t r o l u n i t of the magnet power supply. This technique i s the same as t h a t used to r e g u l a t e the magnetic f i e l d w i t h an RMR s e n s o r 4 8 . As i n every feedback technique one has to ensure t h a t the e r r o r s i g n a l has the c o r r e c t p o l a r i t y . The g a i n and time constant, of the L o c k - i n a m p l i f i e r have to be a d j u s t e d to give s u f f i c i e n t output v o l t a g e to d r i v e the c o n t r o l u n i t . of the magnet power supply but to a v o i d o s c i l l a t i o n s ( o ver-s h o o t i n g ; of the magnetic f i e l d . However, these adjustments are not c r i t i c a l . In order to monitor the angular p o s i t i o n of the specimen, the c r y s t a l was mounted d i r e c t l y on the elongated s h a f t of a 10 t u r n r e a r - s h a f t type H e l i p o t potentiometer l,Pig.3), which was connected across a b a t t e r y as shown i n P i g . 2 . The u p p e r " l i m i t of angular r e s o l u t i o n obtained i n t h i s way was ca. 0.3° , t h i s corresponded to the jumps of the movable contact of the potentiometer from one wire t u r n to the next. On the other end of the H e l i p o t s h a f t was mounted a s m a l l DO motor w i t h a gearbox a l l o w i n g r o t a t i o n s i n both d i r e c t i o n s w i t h 0.1 - 4 RPH. To search f o r an EPR l i n e , one can e i t h e r sweep the magnetic f i e l d f o r a p a r t i c u l a r c r y s t a l o r i e n t a t i o n or r o t a t e !00 KHz O S C KLYSTRON F R F n LOCKED TP CAVITY; CRYSTAL DET r SAMPLE IN CAVITY SAMPLE HOLDER J / s LOCK- IN <— MAGNET POWER SUPPLY MOTOR 0.1-4 RPH f I T HELIPOT f—* PVTr*Pf \ i A j FIELD REGULATOR FIELD OUTPUT F i g .2 Block diagram of the f i e l d locked EPR spectrometer arrangement which w i l l automatically trace the angular dependence of EPR t r a n s i t i o n f i e l d s 'ro Revolution Counter Gear Box 36000= 1 Protractor Potentiometer-Crystal Holder— f b ^ - D C Motor Cavity-o oj i/f!.5 •Pointer —r v ^Wav$ Guide llKsl' E x p e r i m e n t a l arrangement f o r measuring angular p l o t s or r e l a t i v e angular p o s i t i o n s of EPR l i n e s 2 9 the c r y s t a l at a p a r t i c u l a r f i x e d magnetic f i e l d . I f one f i n d s an EPR l i n e i n the l a t t e r case the magnetic f i e l d w i l l a u t o m a t i c a l l y l o c k i n on t h i s l i n e and f o l l o w i t as the c r y s -t a l i s f u r t h e r r o t a t e d . I f two EPR l i n e s c r o s s , i t can happen th a t the system does not- f o l l o w the o r i g i n a l l i n e . T h i s i s immediately r e c o g n i z e d by a sudden change i n the slope of the curve. To t r a c e the c o n t i n u a t i o n of the o r i g i n a l l i n e , one must unlock the f i e l d and sweep i t to an approximate good p o s i t i o n i n the H,9 plane and l o c k i n a g a i n . The Magnion PFO-4 magnet power supply could f o l l o w an e x t e r n a l c o n t r o l s i g n a l w i t h a maximum change of the magnetic f i e l d of 1 k(Jauss/15 sec, thus l i m i t i n g the v e l o c i t y of c r y s t a l r o t a -t i o n i f the EPR f i e l d H.^ changes r a p i d l y w i t h angle. A m o d i f i c a t i o n of the above d e s c r i b e d technique was used to measure a c c u r a t e l y r e l a t i v e angular p o s i t i o n s of EPR l i n e s . Por t h i s purpose, the magnetic f i e l d was set to a constant value and the EPR s i g n a l output of the L o c k - i n a m p l i f i e r was r e g i s t e r e d on a s t r i p c h a r t r e c o r d e r w h i l e r o -t a t i n g s l o w l y the c r y s t a l w i t h the motor. With a r e v o l u t i o n counter connected to the in p u t of the gearbox (see Pi g . 3 ) the r o t a t i o n angle could be measured: 1 t u r n corresponded to 0.01 degree of r o t a t i o n of the c r y s t a l . B y ' s e l e c t i n g appro-p r i a t e EPR l i n e s the angle between the magnetic axes of d i f f e r e n t paramagnetic cen t r e s c o u l d be measured i n t h i s way to an accuracy of - 0.01 degree. 4.5. V a r i a b l e temperature measurements In the e a r l y stages of the present work i t was re c o g n i z e d t h a t the EPR s p e c t r a of the anatase c r y s t a l s are very temperature dependent. As the analysed temperature range was i n c r e a s e d d i f f e r e n t methods were t r i e d to achieve these temperatures. I n i t i a l l y , the temperature of the sample was e s t a b l i s h e d by a temperature c o n t r o l l e d Ng-gas f l o w . For t h i s purpose a quartz dewar was passed through the sample • hole s of the c a v i t y . The sample was f i x e d w i t h r e f r a c t o r y cement ( S a u r e i s e n No.29 Zirconium Base Cement) to a s m a l l brass rod (heat s i n k ) which i t s e l f , was attached to a t h i n w a l l e d s t a i n l e s s s t e e l tube. .The sample was i n s e r t e d from the top i n t o the dewar and co u l d be r o t a t e d about the v e r t i c a l a x i s . The temperature was measured w i t h a c a l i b r a t e d Chromel-Alumel thermocouple f i x e d to the brass rod. The lower p a r t of the dewar, out s i d e the microwave c a v i t y , c o ntained a Kanthai wire heater and another thermocouple as monitor. Through a lower hole of the dewar ir^-gas (p r e c o o l e d f o r tem-peratures between -190 °C and room temperature) was i n t r o -duced. T h i s method i s s i m i l a r to t h a t a p p l i e d by V a r i a n w i t h i t s v a r i a b l e temperature accessory V-4557, which permits sample temperatures between -190 °c and +300 ° 0 . Using the d i f f e r e n c e between the volt a g e of the monitor thermocouple and a standard v o l t a g e .source as an e r r o r feedback s i g n a l f o r the power supply of the hea t e r , the temperature of the sample 31 could be c o n t r o l l e d very a c c u r a t e l y . But, t h i s method imposed an upper l i m i t on the sample temperature of. approximately 6 0 0 °C. The l a r g e s p a c i a l s e p a r a t i o n between heater and sample (approx. 7 cm) caused a l a r g e temperature g r a d i e n t between them. In a d d i t i o n , i t was d i f f i c u l t to expose the h e a t i n g wire u n i f o r m l y to the Ng-gas f l o w , so t h a t the tempe-r a t u r e of some part of the h e a t i n g wire r e g u l a r l y exceeded the m e l t i n g p o i n t w h i l e t r y i n g to i n c r e a s e the sample tempe-r a t u r e above 600 °C. l a t e r on a design was used which i s a m o d i f i c a t i o n 49 of t h a t used by D. G i a r d i n o and L. P e t r a k i s , who themselves 50 m o d i f i e d a design proposed by 1. Singer et a l . . The main d i f f e r e n c e of the new desig n , shown i n P i g . 4 , i s t h a t i t . permits the use of a Y a r i a n standard c a v i t y . The r e s i s t i v e h e a t i n g element c o n s i s t s of two s t r i p s of platinum about 1 mm wide and bonded to opposite s i d e s of a 6 mm o.d. t h i n w a l l e d quartz tube. The l e n g t h s of the plat i n u m s t r i p s were about 24 mm, matching the c a v i t y h e i g h t . The ends of the tube were completely coated w i t h p l a t i n u m . To o b t a i n these bonded s t r i p s , p l a t i n u m paste (Platinum Paste No.6 0 8 2 , manufactured by the Hanovia L i q u i d Gold D i v i s i o n of Engelhard I n d u s t r i e s Inc.) was a p p l i e d w i t h a f i n e brush and f i r e d i n a furnace at about 7 0 0 ° c . S e v e r a l platinum l a y e r s were a p p l i e d i n order to o b t a i n a room tem-perature r e s i s t a n c e of the heater of about 2 ohms. 32 Ceramic Post with Thermocouple Platinum Coated Quartz Tube Copper Wire Slotted Cavity Wall Fig-4 C r o s s - s e c t i o n of- the c a v i t y w i t h the high-temperature h e a t i n g elements i n place 33 This heater tube was placed c o a x i a l l y i n a 10 mm o.d. quartz tube w i t h 1 mm w a l l t h i c k n e s s . The two tubes were separated w i t h asbestos between them i n the upper and lower p a r t s where the heater tube was completely coated w i t h p l a -tinum. Using s m a l l t e f l o n spacers (not shown i n Pig.4) the combined quartz tubes were then placed i n the c a v i t y through the 11.4 mm i . d . sample h o l e s , so th a t the heater s t r i p s c o i n c i d e d w i t h the c a v i t y . Copper wires were bonded near the ends of the heater tube and connected to a r e g u l a t e d DC-power supply (Kepco PR-38-15M). Cooled Ng-gas was i n t r o d u c e d i n the c a v i t y through the s l o t t e d c a v i t y w a l l . This gas escaped through the s m a l l gaps between the 10 mm quartz tube and the in n e r w a l l s of the sample holes of the c a v i t y . The sample was glued witn. a . r e f r a c t o r y cement d i r e c t l y to a Chromei-Alumel thermocouple p a s s i n g t h r o i i g h a ceramic r o d . This design proved to be t r o u b l e f r e e at o p e r a t i o n temperatures of up to 1000 °C. With a moderate f l o w of cooled Ug-gas the outer c a v i t y w a l l s remained below room temperature. In order to o b t a i n 900 °C at the sample, an e l e c t r i c power i n p u t of about 100 W was needed. To determine the abso l u t e signs of the s p i n Hamil-t.onian parameters of the analysed Pe^ i m p u r i t i e s i n anatase, EPR measurements at 1•°K and 4.2•°K were made. Por t h i s pur-pose a g o l d - p l a t e d brass c a v i t y , o p e r a t i n g i n TE 1 A.^ mode, was submerged i n l i q u i d helium. The frequency of the f i e l d mo-d u l a t i o n was 4-00 Hz i n these measurements (modulation c o i l s a t t a c h e d to the magnet p o l e s ) . 35 5 . EXPERIMENTAL RESULT'S 5.1. P r e l i m i n a r y o b s e r v a t i o n s Since anatase has t e t r a g o n a l symmetry one would expect t h a t i t s EPR s p e c t r a would a l s o show t h i s symmetry. This was e x p e r i m e n t a l l y v e r i f i e d . . The o v e r a l l EPR s p e c t r a of the analysed a.natase c r y s t a l s were the same w i t h the d i r e c t i o n of the magnetic f i e l d H, expressed i n p o l a r c o o r d i n a t e s , along (9, (j> ) and along (0, - + n IT/2) , where n i s an i n t e g e r . The EPR s p e c t r a at 9-36 GHz and room temperature are shown i n Pig.5 w i t h H p a r a l l e l to the c r y s t a l l o g r a p h i c [001]-axis and i n Pig.6 w i t h H p a r a l l e l to 1100] ( i n the l a t t e r case more l i n e s e x i s t at hi g h e r magnetic f i e l d ) . As a r e s u l t of the study of the angular dependence, of these EPR t r a n s i t i o n s , one c o u l d separate them i n t o two groups, which w i l l be c a l l e d s p e c t r a I and I I and which w i l l be d i s c u s s e d s e p a r a t e l y . Some weaker EPR l i n e s were observed. However, t h e i r i n t e n s i t i e s and the f a c t t h a t they overlapped w i t h other l i n e s d i d not permit even a q u a l i t a t i v e i n t e r p r e t a t i o n . One of these l i n e s i s i d e n t i f i e d w i t h a question-mark i n P i g . 6 . 2 4 6 KGAUSS Fig.6 EPR spectrum at X-band. of a na t u r a l anatase c r y s t a l with H p a r a l l e l to [lOO] at room temperature 38 5.2. EPR spectrum I The f i v e , l i n e s marked w i t h s i n g l e arrows i n P i g . 5 are c a l l e d spectrum I.. T h e i r angular dependence i s shown i n P i g . 7 f o r three c r y s t a l planes. This angular dependence, the o b s e r v a b i l i t y of the spectrum at room temperature, the r e l a t i v e i n t e n s i t i e s of the f i v e l i n e s and the absence of a 3 + h y p e r f i n e s t r u c t u r e i n d i c a t e t h a t these l i n e s belong to Pe i m p u r i t i e s i n a c r y s t a l f i e l d of t e t r a g o n a l symmetry w i t h r e s p e c t to the [ 0 0 1 ] c r y s t a l a x i s . Using the c r y s t a l l o g r a p h i c [ 1 0 0 ] , [010 ] and [001] axes as x, y and z axes r e s p e c t i v e l y , the spectrum c o u l d be d e s c r i b e d by the s p i n H a m i l t o n i a n ( 7 ) w i t h parameters g i v e n i n Table I I . These values were obtained by f i t t i n g the t r a n -s i t i o n f i e l d s w i t h H p a r a l l e l to [001] , [100] and [11 o ] to ( 7 ) , evaluated to second order. The r e s u l t s were checked by d i r e c t d i a g o n a l i z a t i o n of the energy m a t r i x w i t h a computer programme. The absolute s i g n s of the s p i n H a m i l t o n i a n para-meters were deduced from the r e l a t i v e i n t e n s i t i e s of the EPR l i n e s at l i q u i d helium temperature. At low temperatures the h i g h f i e l d l i n e s were more i n t e n s e , l e a d i n g to the i d e n t i f i c a -t i o n of the i n d i v i d u a l f i n e s t r u c t u r e l i n e s as g i v e n i n P i g . 7 . Whereas the s p i n H a m i l t o n i a n ( 7 ) w i t h the parameters g i v e n i n Table I I , d e s c r i b e s completely the angular dependence of the p o s i t i o n s of the l i n e s of spectrum I , i t does not F i g . 7 Angular dependence of EPR l i n e s of spectrum I i n anatase at X-band with H rotated i n three d i f f e r e n t c r y s t a l planes Vx1 40 Table I I Spin H a m i l t o n i a n Parameters of Sp e c t r a I and I I i n Anatase at 76 °K and 300 °K _4 -1 A l l e nergies are i n 10 cm b b, Spectrum I 3 + ( S u b s t i t u t i o n a l - Fe ) Spectrum I I 3+ (Charge compensated Fe ) 78 °K 300 °K 78 °K 300 °K 2.004 - 0.001 2.005 - 0.001 2.004 i-0.002 2.005 - 0.002 + + 4 4 7 - 1 + 55.9 - 1 + 267 - 3 + + 308.7 - 1 + + 53-5 - 1 257 - 3 + 2.002 - 0.005 5110 t 30 - 4970 - 30 3540 - 30 - 3720 ±- 30 + - 26 - 10 + 60 - 50 - • 14 - 10 + 87 - 50 + + 1 0 0 - 1 0 + 7 0 - 1 0 e x p l a i n the angular dependence of the l i n e w i d t h . In Table I I I the l i n e w i d t h s f o r some o r i e n t a t i o n s of H are g i v e n . Table I I I l i n e w i d t h s (between Peaks" of D e r i v a t i v e Curves) of Sp^cJirum I i n Anatase at Room Temperature, i n Gauss T r a n s i t i o n H // [1 00] H // [110] H // [001] |l/2>«-»l- 1/2 > 12 ± 2 12 t 2 12 - 2 \± 1/2>**I± 3/2 > 70 ± 20 12 ± 2 14 ± 2 |±.3/ 2>Hi 5/2 > unmeasurable 1 9 - 3 19 - 3 41 The outer f i n e s t r u c t u r e l i n e s are p a r t i c u l a r l y broad when H i s p a r a l l e l to the magnetic x or y a x i s . T h i s behaviour i s i n d i c a t e d i n F i g . 8 . There the peak to peak h e i g h t s of the outer EPR f i n e s t r u c t u r e l i n e s , n o r m a l i z e d to 1 f o r H //[110], are p l o t t e d as a f u n c t i o n of the o r i e n t a t i o n of H i n the (OOl)-plane. The peak h e i g h t s drop very f a s t when H i s r o t a t e d away from the [110] d i r e c t i o n . Assuming a constant l i n e shape,, these peak h e i g h t s are a s e n s i t i v e measure of the l i n e w i d t h s . • . T h i s angular dependence of the l i n e w i d t h s cannot be e x p l a i n e d w i t h a mosaic s t r u c t u r e of the c r y s t a l s i n c e t h i s would cause a broadening of the l i n e s f o r i n t e r m e d i a t e o r i e n t a t i o n s and not f o r d i r e c t i o n s along the magnetic axes where the l i n e s have extremum p o s i t i o n s . F o r m a l l y , t h i s l i n e broadening along the [100] and [010] d i r e c t i o n s can be d e s c r i b e d by i n t r o d u c i n g an o r t h o -rhombic term ( S x 2 - 3 y 2 ) ' (16) i n the s p i n H a m i l t o n i a n ( 7 ) , w i t h 5" v a r y i n g randomly between - 1<(f<+ 1 from one paramagnetic centre to the next. This term w i l l inhomogeneously spread out the I - 1 / 2 / W j l 3/2 > t r a n s i t i o n s along the x and y axes by 18b 2 /gB ( c a l c u l a t e d to f i r s t order) and the \- 3/2> ±- 5/2 > t r a n s i t i o n s by 36b 0 2/g(3. 42 Fig.8 Peak h e i g h t s of the outer, f i n e s t r u c t u r e l i n e s of • spectrum I i n anatase w i t h H r o t a t e d i n the ( 0 0 1 j -plane , normalized to 1 at K jj [110] 43 but i t w i l l not a f f e c t the t r a n s i t i o n f i e l d s i n the [001"] and L110] d i r e c t i o n s . The l i n e w i d t h of 70 Gauss of the |± 1/2}*"M— 3/2> t r a n s i t i o n s along the x and y axes leads then a t room temperature to bg^/gS = 3 0 - 6 Gauss. 5.3. Temperature dependence of spectrum I The p o s i t i o n s of the i n d i v i d u a l l i n e s of spectrum I , w i t h e x c e p t i o n of the centre l i n e , depend very s t r o n g l y on tem-p e r a t u r e . Pig;. 9 and P i g . 5 show t h i s spectrum f o r H //[001] f o r f o u r d i f f e r e n t temperatures and i n Pig.10 the l i n e p o s i -t i o n s f o r the same d i r e c t i o n of the magnetic f i e l d are p l o t t e d as a 1 iiiic i j i o i i ox iiue otsiup&ra. ourfc oeovveen i iv ancx 1 u f 0 The s p i n H a m i l t o n i a n constants at 78 °K are g i v e n i n Table I I . At t h i s temperature the spectrum shows some s a t u r a t i o n . Pig.11 shows the temperature dependence of b p 0 0 o and b^ . Whereas b^ decreases only s l o w l y w i t h i n c r e a s i n g , temperature, bv,0 decreases from + 457 x 1 c m - 1 at 1 °K almost l i n e a r l y w i t h temperature over most of the analysed temperature range, passes through zero at 800 - 10 °K and reaches - 225 x 10""4 cm - 1 at 1230 °K. The f i n e s t r u c t u r e of spectrum I i s a l s o c l e a r l y v i s i b l e w i t h H //[110] up to h i g h temperatures as can be seen i n P i g . 12, whereas the broad \- 1/'2>«-H± 3/2 > t r a n s i -45 500 46 IlKiAl • Temperature dependence of b 9° and b 0 of spectrum I i n anatase • "~ 4 48 t i o n s w i t h H//[100] broaden w i t h i n c r e a s i n g temperature s t i l l more, so t h a t at h i g h e r temperatures only the centre t r a n s i t i o n i s v i s i b l e . One r e c o r d i n g f o r t h i s d i r e c t i o n i s . 2 i n c l u d e d i n Pig.12. This broadening i n d i c a t e s t h a t b^ m the random orthorhombic term (16J i n c r e a s e s w i t h temperature. In a d d i t i o n , the spectrum has no cubic symmetry at 800 °K, where b^° = 0, s i n c e the s p e c t r a l o r H||z and H|/x (or y) are d i f f e r e n t . At the h i g h e s t analysed temperature, 1230 °K, the anatase c r y s t a l a l r e a d y began to t r a n s f o r m to r u t i l e . W i t h i n 30 min. the i n t e n s i t y of spectrum I decreased to about one h a l f of i t s i n i t i a l v a l u e . The t r a n s f o r m a t i o n to r u t i l e s t a r t e d on the surface of the c r y s t a l and the transformed p a r t was m i l k y - y e l l o w , opaque and under the microscope appeared p o l y c r y s t a l l l n e . 5.4- EPR s p e c t r a I I In a d d i t i o n to the EPR l i n e s of spectrum I , one observes i n the anatase c r y s t a l s other r a t h e r i n t e n s e EPR l i n e s . They are marked, i n Pig.5 and 6 w i t h double arrows. These, l i n e s are c h a r a c t e r i z e d by a l a r g e a n i s o t r o p y and s t r o n g angular dependent i n t e n s i t i e s . P i g . 1 3 shows i n the upper part the angular dependence of these l i n e s i n the ( 1 0 0 ) - c r y s t a l plane at room temperature and 9.19 GHz. The i n t e n s e l i n e s between 1383 Gauss and 1783 Gauss have a l i n e -w i dth of 11 - 14 Gauss. 49 H = 7 2 0 0 G ».6 IlE^Jl Spectra I I i n the ( 1 0 0 ) - c r y s t a l ulane of anatase at room temperature and 9.19 GHz. 23 KGauss wa,s the h i g h e s t o b t a i n a b l e f i e l d . The lower p a r t of the f i g u r e shows a spectrum obtained by r o t a t i n g -the c r y s t a l i n the (100)--plane at a constant f i e l d H - 7200 Gauss 50 These l i n e s can be d i v i d e d i n t o f o u r separate s p e c t r a which, only d i f f e r from one another i n 90° r o t a t i o n s of t h e i r magnetic axes about the [001]~ c r y s t a l a x i s . These f o u r s p e c t r a are designated s p e c t r a H A _ T Q - The magnetic axes of these spectra,expressed i n p o l a r c o o r d i n a t e s , are g i v e n i n Table IV. The absolute accuracy of these d i r e c t i o n s i s ± 1 ° . Table IV D i r e c t i o n s of the Magnetic Axes f o r Spectrum 11^, Expressed i n P o l a r Coordinates (Q, <^>) . The Axes  of the Other Three S p e c t r a I I are Obtained by Suc-c e s s i v e R o t a t i o n s of 90° about the [00l] C r y s t a l A x i s . Magnetic a x i s D i r e c t i o n i n c r y s t a l ( 9 ^ ) X (y>, 180°) y (90°, 90°) = [010] z (90° -y, 0°) 300 WK: 6.32 - 0.01 78 °K: y= 6.15 i 0.01 At the top of Pig.13 are i n d i c a t e d • t h e s e magnetic axes which l i e i n the (100)-plane, y means the y - a x i s of spectrum 11^, e t c . The lower p a r t of Pig.13 shows a spectrum obtained by r o t a t i n g the c r y s t a l at a constant f i e l d H = 7200 Gauss i n the (100)-plane. Prom t h i s spectrum one deduces d i r e c t l v an accurate value f o r y>, the angle which i s i n c l u d e d i n the s p e c i f i c a t i o n of the magnetic axes as g i v e n i n Table IV. Spectra II,- as spectrum I , are caused by i r o n impu-t i t i e s i n the anatase c r y s t a l . This i s confirmed by the EPR a n a l y s i s of the s y n t h e t i c powders. Pig.14 shows the EPR. spectrum of anatase powder doped w i t h 1 mol -^o of Pe. The three EPR powder l i n e s marked w i t h arrows correspond to the s t r o n g l i n e s i n the s i n g l e c r y s t a l at 1617, 1783 and 1383 Gau w i t h H a l o n g the x, y and z axes, r e s p e c t i v e l y . These EPR l i n e s are not found i n pure anatase powder, nor anatase doped w i t h Cr or to. The i n t e n s i t y , the l i n e w i d t h s and the absence of a h y p e r f i n e s t r u c t u r e , together w i t h the powder a n a l y s i s , i n d i c a t e t h a t s p e c t r a I I are a l s o produced by f e r r i c i r o n i n a h i g h s p i n s t a t e (S = 5/2). Prom the s t r o n g a n i s o t r o p y of the l i n e s and the absence of a l i n e near g = 2, one can however i n f e r t h a t the zero s p l i t t i n g i s l a r g e . compared to the Zeeman energy at X-band. The quadrupolar terms bg™ i n the s p i n H a m i l t o n i a n (3) t h e r e f o r e have to be dominant and t h e i r p r i n c i p a l axes w i l l determine the o r i e n t a t i o n s of the magnetic axes. Hence, an i s o t r o p i c g = 2 .0023 was assumed and i n i t i a l l y a l l b ^ m were set equal to zero. A f i r s t guess f o r the remaining b ?° and b o 2 was . 2 obtained by u s i n g graphs g i v e n by Troup et a l . 5 1 and A a s a 5 2 , as d i s c u s s e d i n Appendix B. Troup et a l . 5 1 c a l c u l a t e the 5 3 e f f e c t i v e g value f o r the EPR t r a n s i t i o n s w i t h i n the three ground s t a t e doublets of Fe^ 4" i n a s t r o n g c r y s t a l f i e l d (D/hv»l)as a f u n c t i o n of E/D = b ^ / ^ 2 ° . . With the observed t r a n s i t i o n s i n anatase one obt a i n s E/D = 0 . 2 5 - 0 . 0 2 . The axes i n Table IV are a l r e a d y chosen to give a f i t w i t h 0 i b p 2 / b o 0 H . The next step was to use t h i s value of E/D 5 ? 3 + and the graphs of Aasa^ , who c a l c u l a t e d f o r Fe^ the t r a n s i -t i o n f i e l d s H versus D, u s i n g E/D as a parameter. The r e s u l t , was D = 1 4 . 2 — 1 GHz. An adjustment of these parameters was then made w i t h a computer programme g i v e n by J . Hebden et 5 3 ' a l . , u s i n g a t r i a l and e r r o r method. This programme c a l c u -l a t e s , f o r g i v e n b p 0 and bp 2) the EPR t r a n s i t i o n f i e l d s and p l o t s the energy l e v e l s . F i g . 1 5 shows the r e s u l t s f o r the cases w i t h H p a r a l l e l to the three magnetic axes. The observed u r an s J. t i Oils are iituj-oa. u u u. c x x i u . niai'xCc u i c e r i uxcaxlv as i n i i g . i 'j . A f i n a l l e a s t mean square computer c a l c u l a t i o n of bp 0 , b p 2 and i n c l u d i n g b^ 0, b^ 2 and b ^ 4 i n the s p i n Hamil-t o n i a n , was made to f i t a l l 1 3 EPR l i n e s i n F i g . 1 3 . This method i s d i s c u s s e d i n Appendix 0 . The r e s u l t s are i n c l u d e d i n Table I I . The absolute signs of the b m were again n ° e s t a b l i s h e d w i t h an EPR measurement at l i q u i d helium tempera-ture . I t should be po i n t e d out tha t the b^111 of Table I I are only an order of magnitude e s t i m a t e , s i n c e there e x i s t s no reason to suppose t h a t the p r i n c i p a l axes of the tensor c o i n c i d e w i t h the p r i n c i p a l axes of b 0 m , the l a t t e r being taken as ENERGY IN CM"' 15 Energy l e v e l s of a paramagnetic centre II i n anatase with H p a r a l l e l to each of the three magnetic axes. The observed transitions- are indicated and numbered as i n Fig.13 d e f i n i n g the magnetic axes. But by i n c l u d i n g b^ , b^ and b^ the 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 agree w i t h the measured ones to w i t h i n the experimental e r r o r , which i s ca. 0.1$, and an 1 3 a d d i t i o n a l i n c l u s i o n of b^ and b^ i n the s p i n H a m i l t o n i a n d i d not seem warranted. In order to o b t a i n an estimate of the r e l a t i v e s p i n c o n c e n t r a t i o n of c e n t r e s of type I and I I , the s p e c t r a at room temperature w i t h H p a r a l l e l to the [001] - a x i s were used ( P i g . 5 ) . Along t h i s d i r e c t i o n the f o u r s p e c t r a I I c o i n c i d e and the l i n e i n t e n s i t i e s A^ of the centre l i n e 1+ 1/2>*»|- l/2> of spectrum I and A-[I of the |+ 3/2><^|- 3/2> l i n e of s p e c t r a I I ( l i n e No.14- of Pig.13) were compared. The h i g h f i e l d nomenclature |- 3/2 > i s used here only to i d e n t i f y these two s t a t e s as b e l o n g i n g to the i n t e r m e d i a t e Kramers doublet of centre I I . The r e l a t i v e s p i n c o n c e n t r a t i o n i s g i v e n by N H ^ I I , N I ~ N I , i where ^ T i s the t o t a l number of spins I , I I i n the sample, N I , I I , i i s t h e n u m b e r o f s p i n s I , I I i n the s t a t e | i > , and \ < i | s | ^ ~ ( f > j 2 i s the induced t r a n s i t i o n p r o b a b i l i t y between s t a t e s \ i > and If> f o r the microwave magnetic f i e l d p a r a l l e l t o [ l 0 0 ] . A I I \<± I S1Q0 I ° l A. (17) 56 1/2 > p - 9/4 3/2 > | 2 5 * 4.4 ( r e f . 5 4 ) a value N^/N-j- = 0.3 - 0.1 i s obtained. This means t h a t i n the analysed c r y s t a l s f o r each paramagnetic centre of type I I there e x i s t some 3-5 c e n t r e s of Type I . 5-5. Temperature dependence of s p e c t r a I I The s p i n H a m i l t o n i a n parameters which d e s c r i b e o s p e c t r a I I were i n a d d i t i o n determined f o r 78 K u s i n g the same procedure as d i s c u s s e d i n the previous s e c t i o n . The r e s u l t s c*r6 aj_oG x i i o l u d c d ±11 l E D i e .LA.. a ' l oh.i.s Lower tempe-r a t u r e the orthorhombic term bp 2 i s s m a l l e r , whereas b g 0 i s l a r g e r than a t room temperature. S p e c t r a I I show only a s m a l l s a t u r a t i o n e f f e c t even at 1 °K. On the other hand, i f the temperature i s r a i s e d above room temperature the l i n e s broaden and 'for temperatures h i g h e r than + 300 °C the s p e c t r a become unobservable. S p e c i a l a t t e n t i o n has been g i v e n to an accurate measurement of the angle y> , determining the o r i e n t a t i o n of the magnetic axes of s p e c t r a I I . The weak l i n e s between Q - - 20° i n the spectrum shown i n the lower p a r t of Pig.13 were used to measure the angle y. These l i n e s belong p a i r w i s e to t r a n s i t i o n s U sing | < V 2 | S : { 0 0 | -I < 5/2 | < 0 0 ! ~ 57 i n the lowest Kramer's doublets of s p e c t r a Il-g and 11-^ (near l i n e No.13 of F i g . 1 5 j and are p a r t i c u l a r l y a p p r o p r i a t e f o r an accurate measurement of y. F i r s t , they are near G = 0(H//[001]), an o r i e n t a t i o n which can be achieved a c c u r a t e l y because there the f o u r s p e c t r a of type I I c o i n c i d e , thus g i v i n g a check f o r the c o r r e c t o r i e n t a t i o n of the c r y s t a l i n the magnetic f i e l d H. Second, the s e p a r a t i o n betv/een these l i n e s does not change measurably f o r s m a l l r o t a t i o n s of the c r y s t a l out of the C'100)-plane. The r e s u l t s at 300 °K and 78 °K are g i v e n i n Table IV. At the lower temperature y i s s l i g h t l y s m a l l e r . The weak l i n e s used to determine y> at 300 °K and 78 °K c o u l d not be f o l l o w e d to higher temperatures. ... T h e r e f o r e , the s t r o n g t r a n s i t i o n s i n the h i g h e s t Kramers doublets of II^ and II-Q, marked w i t h arrows i n the spectrum of F i g . 13, were used above room temperature. These t r a n s i t i o n s occur, however, a t i n t e r m e d i a t e angles 9 where no r e f e r e n c e o r i e n t a - • t i o n e x i s t s to a l i g n e x a c t l y the c r y s t a l and, i n a d d i t i o n , the s e p a r a t i o n between these l i n e s changes markedly i f the magnetic f i e l d i s r o t a t e d s l i g h t l y out of the (/I 00 j-plane . Consequently, the s e p a r a t i o n between these l i n e s , now d e s i g -nated 2 jo 1 , i s shown i n Fig.1b. I t i n d i c a t e s only q u a l i t a t i -v e l y the temperature dependence of the d i r e c t i o n s of the mag-n e t i c axes. These angles d i f f e r by about 1°.from 2y as g i v e n i n Table IV. F i g . 16 Temperature dependence of the angle 2y>', as defined i n the text. Approximately y' gives the magnetic axes of the four spectra I I , defined i n Table IV 5 9 6. DISCUSSION OF RESULTS 3+ 6.1. EPR spectrum I : s u b s t i t u t i o n a l Fe As d i s c u s s e d i n see.tion 2.1., a l l T i 4 f " s i t e s i n — 3+ anatase have 42 p o i n t symmetry. I f Fe s u b s t i t u t e s f o r T i 4 + then a s i n g l e paramagnetic spectrum w i t h t e t r a g o n a l sym-metry along the [0011-axis i s expected. Spectrum I i s i n t e r -preted as r e s u l t i n g from such a c e n t r e . From the symmetry of the spectrum i t i s a l s o p o s s i b l e 3+ that the Fe ions are at i n t e r s t i t i a l s i t e s such as I i n F i g . 1 . However, t h i s would generate a s t r o n g l o c a l charge i n e q u a l i t y . In a d d i t i o n , i n r u t i l e one observes paramagnetic 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 only i f the i m p u r i t i e s are too l a r g e to enter the s u b s t i t u t i o n a l s i t e 6 . I n r u t i l e these s i t e s are d i s t i n g u i s h a b l e because they have other symmetry axes than the s u b s t i t u t i o n a l s i t e s . In anatase, however, the f o u r n e a r e s t neighbour d i s t a n c e s are the same f o r the i n t e r s t i t i a l as f o r the s u b s t i t u t i o n a l s i t e s and, i n o c t a h e d r a l c o o r d i n a -3+ t i o n , the i o n i c r a d i u s of Fe , 0.73 A, i s a l s o n e a r l y the same as t h a t of T i (0.69 A ) Thus one can conclude t h a t 3 + the Fe ions enter anatase s u b s t i t u t i o n a l l y . There e x i s t two main d i f f e r e n c e s between the EPR s p e c t r a of s u b s t i t u t i o n a l F e 3 + In anatase and i n r u t i l e 3 3 ' ^ 6 : the zero f i e l d s p l i t t i n g i s s m a l l e r by a f a c t o r of 20 i n ana-tase and, c o n t r a r y to the case of r u t i l e , i s s t r o n g l y s e n s i -60 t i v e to temperature. These d i f f e r e n c e s are s u r p r i s i n g because the s t r u c t u r e and p h y s i c a l p r o p e r t i e s of the two c r y s t a l s are very s i m i l a r . An attempt was made to c a l c u l a t e the s p i n H a m i l t o -o 43 n i a n parameter bp f o l l o w i n g the ideas of Sharma et a l . , as d i s c u s s e d i n s e c t i o n 3.2. For t h i s purpose the l a t t i c e sums Bp 0 and ( B ^ 0 ) ' , d e f i n e d i n (12) and (13), were c a l c u l a t e d w i t h a p o i n t charge model of anatase, u s i n g the c r y s t a l l o -g raphic data of (1). The sums were c a r r i e d out w i t h a compu-t e r programme over a l l i o n s -within a sphere of 20 A, 30 1 and 40 & around a T i 4 + i o n s i t e . The r e s u l t s f o r the two l a r g e r spheres d i f f e r l e s s than 1$, i n d i c a t i n g a good convergence of the l a t t i c e sums (a sphere w i t h a r a d i u s of 40 K i n c l u d e s some 25?000 i o n s ) . The r e s u l t s are B ° = + 154 x 10 4 e 2/2a 3 ^ o ( B A 0 ) ' = + 4.2 x 10" 4 e 2/2a 5 4 ' o I n t r o d u c i n g these values i n (11) r e s u l t s i n b 9° -+ 8.2 x 10 4 cm 1 which compares w i t h the experimental value b2° = + 308.7 x 1 0 - 4 cm" 1. Since the n u m e r i c a l f a c t o r s i n (11) were c a l c u l a t e d f o r . M n 2 + i n ZnFp, they w i l l not apply e x a c t l y i n our case, but the same order of magnitude would be expected 3 + i o r Fe . One p o s s i b l e e x p l a n a t i o n f o r t h i s discrepancy i s the h i g h p o l a r i s i b i l i t y of anatase. This would r e s u l t i n a deformation of the neighbouring oxygen ions and a change of 61 t h e i r p o s i t i o n s when r e p l a c i n g T i ' " by the s m a l l e r charge 3+ of Fe . This discrepancy between c a l c u l a t e d and measured b 2° e x i s t s a l s o i n r u t i l e where a p o i n t charge c a l c u l a t i o n g i v e s B 2° = - 45 x 10" 4 e 2 / 2 a Q 3 , (B 4°)' = - 251 x 10~ 4 e 2 / 2 a Q 5 , w i t h the z a x i s p a r a l l e l to [110]. No e x p l a n a t i o n f o r the l a r g e 3+ d i f f e r e n c e s of the observed zero f i e l d s p l i t t i n g of Fe i n anatase and r u t i l e can t h e r e f o r e be deduced from these c a l -c u l a t i o n s • • . Ne v e r t h e l e s s , the p o i n t charge model seems adequate to e x p l a i n q u a l i t a t i v e l y the temperature dependence of the s p i n H a m i l t o n i a n parameters i n anatase. In the c a l c u l a t i o n of the l a t t i c e sums B 2° and ( B ^ 0 ) ' i t was found t h a t they are very s e n s i t i v e to s m a l l changes i n the oxygen parameter u (see F i g . 1 ) . The r e s u l t s are gi v e n i n F i g . 17. An i n c r e a s e of u from 0.413 A*, i t s value at room temperature, to 0.452 A decreases B 2° to zero and ( B ^ 0 ) ' to - 7-3 x 1 0 - 4 e 2 / 2 a Q 5 . A t f i r s t g l ance, t h i s r e s u l t i s s u r p r i s i n g i n as much as an in c r e a s e of u deforms s t i l l more the oxygen octahedron surrounding T i 4 + . But the displacements of the oxygen ions. A-D (see Fig.1) have an opposite e f f e c t on B 2° than the displacements of the ions E and F. I t was concluded t h a t the temperature dependence . of the s p i n Hamiltonian parameter b 2° i s produced by an in c r e a s e of the oxygen parameter u w i t h temperature. A l s o 62 •0.39 0.41 0.43 045 OXYGEN PARAMETER U IN A F i g • 17 L a t t i c e sums lip and (B, )' , c a l c u l a t e d w i t h a p o i n t charge model of anatase as a f u n c t i o n of the oxygen parameter u i n support of t h i s argument i s the temperature dependence of the o r i e n t a t i o n s of the magnetic axes of s p e c t r a I I as w i l l he d i s c u s s e d below i n s e c t i o n 6.2. This s h i f t of the oxygen ions does not change the symmetry of the c r y s t a l . I t changes e s s e n t i a l l y only the bond angles but not the bond l e n g t h s between a T i 4 + i o n and i t s f o u r n e a r e s t oxygen i o n s . I n r u t i i e t h i s i s not p o s s i b l e . There, a change of the oxygen parameter a f f e c t s d i r e c t l y the 4 + 2 -n e a r e s t neighbour T i - 0 d i s t a n c e and hence r e s u l t s i n a l a r g e r change of the c r y s t a l energy. This would e x p l a i n the absence of the st r o n g temperature dependence of the EPR s p e c t r a i n r u t i i e . U n f o r t u n a t e l y , no h i g h temperature c r y s t a l l o g r a p h i c X-ray data of anatase are a v a i l a b l e i n order to check t h i s proposed temperature dependence of the oxygen parameter u. An attempt was made to determine at h i g h temperature the c r y s t a l s t r u c t u r e of anatase w i t h an X-ray s i n g l e c r y s t a l study u s i n g a h i g h temperature Weissenberg camera and - counter, c o n s t r u c t e d by E. Meagher of the Geology Department of U.B.C. I t appeared t h a t the i n v e s t i g a t i o n would be too le n g t h y , mainly because of the i n v o l v e d experimental d i f f i c u l t i e s , to be i n c l u d e d i n t h i s t h e s i s . More i n s i g h t i n t o the o r i g i n of the temperature dependence of b p 0 c o u l d be gained i f the i s o t h e r m a l pressure • dependence of b ° were known. This would permit the c a l c u l a -64 t i o n of the i m p l i c i t p a r t of the temperature dependence of h p 0 w i t h (.1.4). E v i d e n t l y , the proposed temperature dependence of the oxygen parameter u does not correspond to an i s o t r o p i c thermal expansion, which, a c c o r d i n g to (15), would r e s u l t i n an i n c r e a s e of b p 0 w i t h temperature. The i m p l i c i t temperature dependence of b p 0 as gi v e n by Walsh et a l . 4 ^ i n the second term on the r i g h t - h a n d s i d e of (14) should t h e r e f o r e be s p l i t i n t o two p a r t s to correspond to the t e t r a g o n a l symmetry of anatase. One p a r t w i t h the pressure a p p l i e d p a r a l l e l to the o p t i c a l a x i s and one w i t h the p r e s s u r e ^ p e r p e n d i c u l a r to i t . A c c o r d i n g to P i g . 1 1 , b^ 0 decreases 12.5$ by i n c r e a s -i n g the temperature 500 °C. N e g l e c t i n g the s m a l l c o r r e c t i o n F i n ( 5 ) , t h i s i m p l i e s t h a t the cubic constant a should a l s o decrease 12.5%- This can be e x p l a i n e d by the i m p l i c i t term g i v e n i n (15), i f an i n c r e a s e i n the i n t e r i o n i c d i s t a n c e s of 0.6/o i s used. This i s c o n s i s t e n t w i t h the i n c r e a s e of the 2 5 l a t t i c e parameter c as measured by Rao . One can conclude t h a t the temperature dependence of the s p i n H a m i l t o n i a n parameter a i s mainly g i v e n by the average i n c r e a s e of the i n t e r i o n i c d i s t a n c e s and i s not a f f e c t e d by minor d i s t o r t i o n s which, on the other hand, a f f e c t s t r o n g l y the second order term bp 0 . The l a t t i c e sums c a l c u l a t i o n s , u s i n g a p o i n t charge model of anatase, may a l s o e x p l a i n the s t r o n g broadening of the outer f i n e s t r u c t u r e l i n e s of the EPR spectrum I when the magnetic f i e l d i s p a r a l l e l to {100}. For t h i s purpose the l a t t i c e sums B 2 ,(33^ )' and B 2 2 = if £ h s i n 2 e . : 0 0 3 2 T3 7 R3 were c a l c u l a t e d at p o i n t s a l o n g the z a x i s . The r e s u l t was a g a i n s u r p r i s i n g : a displacement of 0.02 1 from the centre of the oxygen octahedron along the z a x i s changes B 2° l e s s ' 2 than 1 $ but the generated orthorhombic component B 2 a l r e a d y —A ° 3 reaches 12 x 10 e V 2 a . Random, s t r e s s induced d i s p l a -cements of the F e 3 + ions along the z a x i s by 0.02 A or l e s s c o u l d thus e x p l a i n the observed angular dependence of the f i n e s t r u c t u r e l i n e w i d t h s . These displacements are so s m a l l t h a t the n e a r e s t oxygen i o n along the z a x i s v/ould s t i l l be 3 + f a r t h e r away from the Fe i o n than the f o u r n e a r e s t n e i g h -bours at A-D of F i g . 1 . Whereas t h i s i s one p o s s i b i l i t y , other random deformations of the c r y s t a l s t r u c t u r e or displacements 3 + of the Fe ions could have the same e f f e c t on the l i n e w i d t h s . I t i s perhaps a p p r o p r i a t e f o r completeness to s t a t e t h a t the temperature dependence of spectrum I , measured up to 960 °C,, does not show any d i s c o n t i n u i t y at 642 °C, where 2 6 Schroeder " ( c f s e c t i o n 2.3) proposed a r e v e r s i b l e phase-change to a h i g h temperature form of anatase. 6.2. EPR s p e c t r a I I : charge compensated Fe- ) + Replacement of a T i 4 + i o n by a F e 3 + i o n causes a nega t i v e charge excess of one elementary charge. This excess 65 ( 1 8 ) 66 has to be compensated to 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 . The charge compensation can be accomplished i n d i f -f e r e n t ways, e.g. other 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 or oxygen vacancies - c e n t r e s ; . These may oe randomly d i s -t r i b u t e d over the c r y s t a l or i n a f i x e d g e o m e t r i c a l r e l a t i o n 3 + to the Fe i o n s . The most reasonable e x p l a n a t i o n f o r the s p e c t r a I I 3+ i s t h a t they o r i g i n a t e from Fe - V Q centre p a i r s , produced 3+ " 4 + by Fe ions at T i s i t e s w i t h charge compensating oxygen vaca n c i e s a t one of the four n e a r e s t neighbour s i t e s . This e x p l a i n s the f o u r s p e c t r a I I (vacancies at A-D of F i g . l J , t h e i r symmetries and the d i r e c t i o n s of t h e i r magnetic axes. S i m i l a r EPR s p e c t r a of charge compensated Fe^' have 57 been observed i n other c r y s t a l s , e.g. SrTiO^ , which has the same b a s i c s t r u c t u r a l blocks of 'I'iOg octahedra. In f a c t , there e x i s t s a remarkable s i m i l a r i t y oetween the EPR s p e c t r a 3 + of Fe i n anatase and i n the t e t r a g o n a l low temp of SrTiO^. b 2° of normal s u o s t i t u t i o n a l F e 3 + i n S r l i O ^ i s a l s o s t r o n g l y temperature dependent and,as w i l l De d i s c u s s e d below f o r - a n a t a s e , t h e temperature dependent s h i f t of the oxygen Ions i n SrTiO^ could De detected by a r o t a t i o n of the magnetic axes of the charge compensated F e 3 + c e n t r e s 5 8 . In ru-t i l e weak EPR s p e c t r a of ur' , charge compensated oy oxygen, va-c a n c i e s a t second and t h i r d n e a r e s t neighbour s i t e s 5 * 9 and of Co 2', charge compensated at n e a r e s t neighbour s i t e s 6 0 , have been observed, however, not of F e 3 + . Again, at present no 67 d e f i n i t e answer can be g i v e n to the q u e s t i o n why the charge compensation of F e y i n r u t i l e i s random and t h e r e f o r e not 3 + observed, w h i l e i n anatase a s t r o n g EPR s i g n a l from Pe - V Q p a i r s i s observed. An e x p l a n a t i o n may be t h a t the a t t r a c t i o n between an Fe i o n and a V centre i s s m a l l e r i n r u t i l e than i n anatase because of the higher d i e l e c t r i c constant of the' f i r s t (114 vs. 48). . From the symmetry of the s p e c t r a I I a l o n e , i t i s p o s s i b l e t h a t the oxygen v a c a n c i e s are l o c a t e d one l a t t i c e d i s t a n c e f a r t h e r i n the same d i r e c t i o n s , 5.68 A from the 3 + Fe s i t e s . In r u t i l e , the zero f i e l d s p l i t t i n g of pure 3+ 3+ s u b s t i t u t i o n a l Cr and of Cr charge compensated a t the t h i r d neighbour, which i s 4.09 A d i s t a n t , d i f f e r by 20$ . I t seems t h e r e f o r e u n l i k e l y t h a t an oxygen vacancy 5.68'A 3+ away from a Fe i o n could produce the l a r g e value of the 3+ c r y s t a l f i e l d at the Fe' ' s i t e necessary to give the observed zero f i e l d s p l i t t i n g i n anatase. 3 + I f the charge compensation of Fe occurs only 3 + through V c e n t r e s , then f o r every two Fe i o n s one oxygen vacancy i s r e q u i r e d to o b t a i n charge n e u t r a l i t y . I f , i n 3 + a d d i t i o n , the a f f i n i t y between Fe ions and V ce n t r e s i s o s u f f i c i e n t l y l a r g e to i n s u r e t h a t a l l V c e n t r e s are l o c a t e d a t nearest neighbour s i t e s of J?eJ + , the r a t i o of F e 3 + - V o p a i r s to pure s u b s t i t u t i o n a l F e 3 + would be 1 : 1. This com-pares to the experimental value N J T / N J = 0-3 --0.1 and im p l i e s , t h a t the charge compensation does not occur w h o l l y through F e 3 + - V • p a i r s . o 68 An i n t e r p r e t a t i o n of the temperature dependence of b ?° of spectrum I i s a s h i f t of the oxygen ions.. T h i s would imply t h a t the axes of the c r y s t a l f i e l d of the charge com-pensated centre should s h i f t w i t h temperature. An i n c r e a s e of u should i n c r e a s e y , the angle d e f i n i n g the magnetic axes . of s p e c t r a I I ( f o r u = 0 one expects y» = 0 ) . Lowering the temperature should t h e r e f o r e r e s u l t i n a decrease of y . This decrease of y w i t h temperature was observed as shown i n Table IV and Fig.1b. At room temperature one can e m p i r i c a l l j ' ' r e l a t e y> 4+ 2-w i t h the known value of cN , the angle between the T i - 0 bond and the v 1 00,)-plane ; Y> = 0.502o<. I f t h i s r e l a t i o n remains v a l i d a l s o f o r o t h e r s , one o b t a i n s a t 77 °K a value c(,^o = 12.26°. This means t h a t the oxygen parameter u should decrease from 0.413 A to 0.401 A by lower-i n g the temperature from 300 °K to 78°K.. There remains the q u e s t i o n as to why the charge com-pens a t i o n occurs only at the f o u r oxygen i o n s A-D and not a t E or F of Mg. 1. From symmetry one should expect t h a t the magnetic axes of these l a t t e r paramagnetic c e n t r e s c o i n c i d e w i t h the c r y s t a l axes and t h a t the rhombic term Dp2 be r e l a -t i v e l y s m a l l . I t i s p o s s i b l e t h a t the weaker l i n e s which c o u l d not be i d e n t i f i e d would account f o r these paramagnetic c e n t r e s , but another e x p l a n a t i o n might be t h a t a YQ centre at E or F i s u n stable and switches to one of the p o s i t i o n s A-D. 69 6.3- Conclusions The main r e s u l t s of the r e s e a r c h of t h i s t h e s i s can be summarized as f o l l o w s . EPR measurements w i t h n a t u r a l s i n g l e c r y s t a l s of 3-i-anatase show two types of s p e c t r a , generated by Fe ions i n c o r p o r a t e d i n the c r y s t a l s t r u c t u r e of anatase at two d i f f e r e n t s i t e s . At one s i t e , designated I , the c r y s t a l f i e l d 3+ has a x i a l symmetry and s p l i t s the ground s t a t e of Fe only, weakly, at the other s i t e , designated I I , the c r y s t a l f i e l d has orthorhombic or lower symmetry and s p l i t s the ground s t a t e of Fe about 16 times more than at s i t e I . For each Fe at s i t e I I there are 3-5 F e 3 + at s i t e I . The EPR s p e c t r a change markedly w i t h temperature over the whole analysed range from 1 °K to 1230 °K. With i n c r e a s i n g temperature the a x i a l component of the c r y s t a l f i e l d at s i t e I decreases ( i t i s the l a r g e s t temperature dependence of b p 0 of any EPR spectrum known to the author) and the c r y s t a l f i e l d at s i t e I I becomes more o r t h o -rhombic and the d i r e c t i o n s of i t s axes change s l i g h t l y . • These EPR s p e c t r a are i n t e r p r e t e d as due to P e 3 + ions r e g u l a r l y s u b s t i t u t i n g T i 4 + ( s i t e I) and due to F e 3 + s u b s t i t u t i n g T i 4 + w i t h an oxygen vacancy (V -centre) at a n e a r e s t neighbour s i t e ( s i t e I I ) . The temperature dependence of the two types of EPR s p e c t r a can be e x p l a i n e d c o n s i s t e n t l y by assuming t h a t the 70 p o s i t i o n s of the oxygen ions w i t h i n the u n i t c e l l change w i t h temperature. A d e f i n i t e c o n f i r m a t i o n of t h i s proposed oxygen s h i f t could be g i v e n w i t h an X-ray c r y s t a l s t r u c t u r e d e t e r m i -n a t i o n at h i g h temperature. The d e s c r i b e d EPR s p e c t r a of i r o n i m p u r i t i e s i n anatase are r a t h e r d i f f e r e n t from t h a t obtained i n r u t i i e , i n s p i t e of the s i m i l a r i t y of the s t r u c t u r e of the two c r y s t a l s . A d d i t i o n a l experimental i n v e s t i g a t i o n s w i l l be necessary to understand more f u l l y the EPR spectrum of Pe i n anatase and i t s d i f f e r e n c e from t h a t i n r u t i i e . Of p a r t i c u l a r i n t e r e s t would be to understand the d i f f e r e n t d e f e c t s t r u c t u r e s of anatase and r u t i i e 1 . EPR should be a u s e f u l t o o l to c l a r i f y these problems. The measurement of the pressure dependence of the EPR spectrum of Fe , the d e t e r m i n a t i o n of the i n f l u e n c e of r e d u c i n g or o x i d i z i n g atmospheres on the c o n c e n t r a t i o n of the P e 3 + - V o c e n t r e s and the a n a l y s i s of EPR s p e c t r a of other t r a n s i t i o n element i m p u r i t i e s , e s p e c i a l l y those of d i f f e r e n t v a l e n c y , would y i e l d u s e f u l i n f o r m a t i o n . However, to c a r r y out these experiments, the technique of growing s i n g l e c r y s t a l s of anatase w i l l have to be developed f i r s t . 71 B IB L10 G RA P HY 1. The h i s t o r i c a l development of The knowledge on p o l y -morphism i s d e s c r i b e d by M. Buerger and M. Bloom i n " C r y s t a l Polymorphism", Z. K r i s t . 96, 182 (1937). A c c o r d -i n g to these authors, K l a p r o t h W a s the f i r s t person to recognize a polymorphism i n 1798 ( c a l c i t e , a r a g o n i t e ] 2. A d e s c r i p t i o n of the e a r l y d i s c o v e r i e s on these m i n e r a l s i s g i v e n oy P. .aarolan i n "Untersuchungen zur K r i s t a l l -chemie von PepO^ und TiOp , sowie i h r e r A l k a i i v e r o i n d u n g e n " , Schweiz. min. und P e t r o . ivii t . 2_3, 293 ( 1 9 4 3 ) 3 . Simons, P. and B a c h i l l e , P., "The S t r u c t u r e of T i O g l l , a High. Pressure. Phase of TiOp" > A c t a u r y s t a l l o g r . 2_3, 334 (1967) 4. uzanderna, A., C l i f f o r d , A . and Honig, J . , " P r e p a r a t i o n of H i g h l y P u r i f i e d TiOg (anatase;", J . Am. Dhem. Soc. 79, 5407 (1957) ' 5. Bach, H. , "Zur l i i l d u n g von ±srookit i n dtinnen TiOp S c h i c h t e n " , Naturwiss. 51, 10 (1964) 6. G e r r i t s e n , h., "Paramagnetic Resonance of T r a n s i t i o n metal Ions i n R u t i l e (TiOp)", i n Proceed, of 1st I n t e r n , conf. on Paramagnetic Res. ( E d i t e d by W. Low;, V o l . 1 , p.3 , Jerusalem (19b2; 7. Low, W. and Offenoacher, P., " E l e c t r o n Spin Resonance of Magnetic Ions i n complex Oxides", i n S o l i d State P h y s i c s ( E d i t e d by P. S e i t z and D. T u r n o a l l ; , Vol.17, p. 135, Academic P r e s s , New York (1965) 8. Low, W., " E l e c t r o n Spin Resonance - A Tool i n mineralogy and Geo1ogy", Adv, " i n E l e c t r . and E1ectron Pays. 24, 51 (1968; 72 9- (ihose, S., " A p p l i c a t i o n of EPR i n S i l i c a t e M i n e r a l s " , i n Short Lecture Notes on Resonance Spectroscopy i n Mineralogy, Am. Geol. I n s t i t u t e (1968) 10. Horn, M. and Schwerdtfeger, C P . , "EPR of S u b s t i t u t i o n a l and Charge Compensated F e ^ + i n Anatase (TiOp)", J . Phys. Chem. of S o l i d s (to be p u b l i s h e d ) . 11. Bragg, L. and C l a r i n g b u l l , G., "The C r y s t a l l i n e S t a t e " , V o l . I T , pp 107, C o r n e l l U n i v e r s i t y P r e s s , New York (1966) 12. Cromer, D. and H e r r i n g t o n , K., "The S t r u c t u r e s of Anatase and R u t i i e " , J . Am. Chem. Soc. 77, 4708 (1955) 13. P a u l i n g , L., "The P r i n c i p l e s Determining the S t r u c t u r e of Complex I o n i c C r y s t a l s " , J . Am. Chem. Soc. _5_1_, 1010 (1929) 14. Whittaker and Muntus. Geoebem. Cosmochem. A c t a 34, 945 (1970) 15- Navrotsky, A. and Kleppa, J . , "Enthalpy of the Anatase R u t i i e Transformation", J . Am. Ceram. Soc. 5_0, 626 (1967) 16. Beard, W. and P o s t e r , W., "High-Temperature Formation of Anatase", J . Am. Ceram. Soc. 50, 493 (1967) 17. Grnelins Handbuch der Anorganischen Chemie , Vol. 4 1 : T i t a n , V e r l a g Chemie, We inhe im " (1951) 18. Czanderna, A., Rao, C. and Honig, J . , "The Anatase-R u t i l e T r a n s i t i o n " , Trans. Paraday Soc. 5_4, 1069 ( 1958) 19- Rao, C , " K i n e t i c s and Thermodynamics of the C r y s t a l S t r u c t u r e Transformation of S p e c t r o s c o p i c a l l y pure Anatase to R u t i i e " , Can. J . Chem. 39, 498 (1961) 73 20. Yoganarasimhan, S. and Rao, C , "Mechanism of C r y s t a l S t r u c t u r e Transformations", Trans. Faraday Soc. 58, 1579 (1962) 21. Shannon, R. and Pask, J . , " K i n e t i c s - o f the Anatase-R u t i l e Transformations", J . Am. Ceram. Soc. 48, 391 (1965) 22. D a c h i l l e , F., Simons, P. and Roy, R., "Pressure -Temperature S t u d i e s of Anatase, B r o o k i t e , R u t i i e and T i 0 2 I I " , Am. Min. 53, 1929 (1968) 23- Shannon, R. and Pask, J . , "Topotaxy i n the Anatase-R u t i l e Transformation", Am. Min. 49, 1707 (1964) 24- P a s c a l , ., "Nouveau Traite" de Chimie M i n e r a l e " , V o l . IX, p.96,- Masson et C i e . , P a r i s (1963) 25- Rao, K., Naidu, S. and Iyengar, I . , "Thermal Expansion of R u t i i e and Anatase", J . Am. Ceram. Soc. 5_3, 1 2 4 (1970) 26. Schroeder, A., " B e i t r a g e zur Kenntnis des Feinbaues des B r o o k i t s und des p h y s i k a l i s c h e n V e r h a l t e n s sowie der Zustandsanderungen der d r e i n a t i i r l i c h e n Titandj.oxyde" , . Z. f . K r i s t . 6 7 , 485 (1928) 27- E r r e r a , J . and K e t e l a a r , HI, " R e l a t i o n s entre l e s P r o p r i e t y Optiques et B i ^ l e c t r i q u e s et l a P o l a r i s a t i o n Ionique dans l e s S o l i d s " , J . Phys. et Radium 3, 239 (1932) 28. Gainon, D. and L a c r o i x , R., "EPR of F e 5 + Ion i n Anatase", Proc. Phys. Soc. (London) 7 9 , 658 (1962) 29. B a r r y , T., "ESR of C r 3 + i n A n a t a s e } „ , 3 o l i a s t a t e Comm. 4 , 123. (1966) 74 30. Che, M., G r a v e l l e , P. and Meriaudeau, P., "Etude par Resonance Paramagne"tique E l e c t r o n i q u e d'un Bioxyde de Titane (Anatase) Contenant des Ions Antimoine", C.R. Acad. Sc. ( P a r i s ) 26_8G, 768 (1969) . 31. Meriaudeau, P., Che, M. and Jorgensen, C., "Angular Overlap Treatment and ESR of Titanium ( i l l ) i n Anatase", Chem. Phys. L e t t e r s 5, 131 (1970) 32. Hauser, C. and Cornaz, P., "Evidence by EPR of a Complex "5 + TiO i n the C r y s t a l of TiOp Anatase", Chem. Phys. L e t t e r s 5, 226 (1970) 33- Cornaz, P., van Hooff, J . , P l u i j m , P. and S c h u i t , G., "Surface C o o r d i n a t i o n of Oxygen on Oxygen-Deficient TiOp and MoO^ as Revealed by ESR Measurements", D i s . Faraday Soc. 4_1_, 290 (1966) "ESR S t u d i e s of the Surface Chemistry of R u t i i e " , J . Am. Chem. Soc. 88, 5055 (1966) 35- Abragam, A. and Bleaney, B., " E l e c t r o n paramagnetic Resonance of T r a n s i t i o n Ions", Clarendon P r e s s , Oxford (1970) 36. Orbach, R., " S p i n - L a t t i c e R e l a x a t i o n i n Rare-Earth S a l t s " , Proc. Roy. Soc. (iiondon) 264A, 458 (1961 ) 37. Vinokurov, V., Zarip o v , M. and Stepanov, V., "EPR of Mn 2 + i n A p a t i t e " , Sov. P h y s . - S o l i d State 6, 866 (1964.) 38. Vinokurov, V., Zaripov, M. and Stepanov, V., "EPR of Mn + i n L i o p s i d e C r y s t a l s " , Sov. Phys.- S o l i d S t a te 6, 870 (1964) 7 5 39. Thyer, ,T. , Quick, S. and H o l u j , P., "ESR Spectrum of P e 5 + i n Topaz", Can. J . of Phys. 45, 3597 (1967) 40. . Horn, M. and Schwerdtf eger, C P . , "Re i n t e r p r e t a t i o n and 3 + Temperature Dependence of EPR i n T i 0 2 : Fe^ (Anatase)", S o l i d State Comm. 8, 1741 (1970) 41. Bowers, K. and Owen, J . , "Paramagnetic Resonance I I " ' , Rep. Progr. Phys. 18, p.321 ( 1 9 5 5 ) 42.. Serway, R., "Temperature-Dependent Spin H a m i l t o n i a n 2 + Parameters of Mn i n T r i g o n a l S i t e s of CaCO^, Phys. Rev. 3, 608 (1971) 43. Sharma, R., Das, T. and Orbach, R., " Z e r o - P i e l d S p l i t t i n g of S-State Ions. I . P o i n t - M u l t i p o l e Model", Phys. Rev. 149, 257 (1966) 44. J a , Y., "EPR of Pe^ and Mn i n N a t u r a l S i n g l e C r y s t a l s of P e t a l i t e L i A l S i ^ O ^ " , Aust. J . Phys. 2_3, 299 (1970)_ 45. Sharma, R., "Temperature V a r i a t i o n of the Z e r o - P i e l d S p l i t t i n g i n C d C l 2 : M n 2 + " , Phys. Rev. 2, 3316 (1970) 46. Walsh, W., Jeener, J . and Bloembergen, N., "Temperature-Dependent C r y s t a l F i e l d and Hyperfine I n t e r a c t i o n s " , Phys. Rev. L5_9, A1338 (1965) 47. Horn, M. and Schwerdtfeger, C P . , "A Method to Obtain D i r e c t l y the Angular Dependence of EPR Spectra i n S i n g l e C r y s t a l S t u d i e s " , Rev. S c i e n t . I n s t , (to be p u b l i s h e d ) 48. See f o r example: Ch. Poole " E l e c t r o n S p i n Resonance", I n t e r - S c i e n c e P u b l i s h e r , pp.367 (196.7); or a more rece n t a r t i c l e : Goodrich, R. et a l . , Rev. S c i e n t . I n s t . 4_1_, 245 (1970) '6 49. G i a r d i n o , D. and P e t r a k i s , I . , "High Temperature ESR Probe", Rev. S c i e n t . I n s t . 38, 1180 ( 1 9 6 7 ) 50. S i n g e r , I . , Smith, V/. and V/agoner, G., "Microwave C a v i t y f o r High Temperature ESR Measurements", Rev. S c i e n t . I n s t . 32, 213 ( 1 9 6 1 ) 51. Troup, G. and Hutton, D., "Paramagnetic Resonance of P e 5 + i n K y a n i t e " , B r i t . J . A p p l . Phys. J J 5 , 1 4 9 3 ( 1 9 6 4 ) 52. Aasa, R., "Powder Line Shapes i n the EPR Spe c t r a of High Spin P e r r i c Complexes", J . Chem. Phys. 52, 3 9 1 9 ( 1 9 7 0 ) 53- B y f l e e t , C., Chong, D., Hebden, J . and McDowell, J . , J . of Magn. Res. 2, 69 ( 1 9 7 0 ) . 54. H o l u j , P., "The Spin H a m i l t o n i a n and I n t e n s i t i e s of the ESR Sp e c t r a O r i g i n a t i n g from l a r g e Z e r o - P i e l d E f f e c t s on 6S S t a t e s " , Can. J . Phys. 44, 503 ( 1 9 6 6 ) 55.' L i c h t e n b e r g e r , G. and Addison, J.R., "F-and X-band Spectrosco; 381 (1969) scopy on Pe^4" i n R u t i l e " , . phys. Rev.. 184, 56. C a r t e r , D. and Okaya, A., "EPR of P e 5 + i n T i 0 o ( R u t i l e ) " , Phys. Rev. JM8, 1485 (1960) 57. K i r k p a t r i c k , E., M u l l e r , K. and Rubins, R., "Strong A x i a l EPR Spectrum of ¥eJ+ i n SrTiO-, Due to Nearest-5 Neighbour Charge Compensation", Phys. Rev. 235, A86 (1964) 58. M u l l e r , K. and B e r l i n g e r , W., " S t a t i c C r i t i c a l Exponents at S t r u c t u r a l Phase T r a n s i t i o n s " , Phys. Rev. L e t t e r s 26, 13 (1971) and r e f e r e n c e s quoted t h e r e i n . 7 7 5 9 - Ikebe, M., Miyako, Y. and Date, M., "ESR of C r 5 + Ions Coupled w i t h Oxygen 'Vacancies i n R u t i l e " , J . Phys. Soc Japan 2 6 , 4 3 ( 1 9 6 9 ) 6 0 . Miyako, Y., "ESR of C o 2 + Ions Combined w i t h Defects i n R u t i l e " , Phys. L e t t e r s 24A, 6 3 5 ( 1 9 6 7 ) APPENDIX A Matrix of the Spin Hamiltonian (8) f o r S = 5/2 arid the Magnetic F i e l d H along one of the Coordinate Axes Table V Matrix of the Spin Hamiltonian (8) f o r S = 5/2 and H//z h = g (i H 1 + 5 / 2 > | + 1 / 2 > | - 3 / 2 \ | - 5 / 2 ) | - l / 2 ) | + 3 / 2 ) 5/2 j 5 . , 10, o, , o 2 ^ J*2 + b 4 >/io\ 2 , 3'{ia 2 3 " b 2 + 2 0 b 4 yrb44 < + 1/2 [ \Eo, 2. 3 VTo7, 2 T°2 20"' b 4 i h - ^ b °+2b ° 2 3 2 4 0 <- 3/2 | 1 b * vr 4 /_ \ 5 / 2 5 10 0,. 0 - T h+ y- b 2 +b 4 i io\ 2 , siTcv", 2 .3 b 2 •"" 20 b 4 i ? 4 / \ -m\ 0 (io\ 2,afio", 2 3 b 2 + 20 \ - | b - | b 2 ° + 2 b 4 ° ,P5V 2 ^ \ 2 ^ b 2 " 4 b 4 / a. N. ' 3/21 f ? 4 3 , 2 0 „, 0 2 h ~ 3 b 2 " 3 b 4 79 Table VI E x p r e s s i o n s to be I n s e r t e d i n Place of i n Table V f o r the uases when H. i s along the x and  y A x i s This Table i s given by Thyer et a l . J . Some numerical e r r o r s are c o r r e c t e d . H // x H//y b2 - (b2° - b22j / 2 - (> 2° + b22; / 2 o b2° + b22] / 2 (3 b2° - b/) / 2 \ ° (3 3 (3 b4° + b 4 2 + V) / 8 • - (5 b4° - b 4 2 - b 4 4 ) / 2 ~ (5 V )/ 2 (35 b4° + 7 \2 + b/) / 8 (35 b4° - )/ 8 80 APPENDIX B Approximate C a l c u l a t i o n of bp and bp of Spectrum I I The main f e a t u r e s of the EPR spectrum of Pe^ (S = 5/2) i n a st r o n g c r y s t a l f i e l d can be d e s c r i b e d w i t h the s p i n Hamiltonian _ - - -= 28 . H . S. -i- D S 2 + E (S 2 - S^ 2) (19) This i m p l i e s t h a t two parameters, D (.= bp 0) and E (=.bp 2/3), together w i t h the d i r e c t i o n s of the magnetic axes, are s u f f i c i e n t to c h a r a c t e r i z e the spectrum.. These parameters can be estimated w i t h the help of graphs, g i v e n by Troup et a l . v ' and A a s a 7 ^ , i f . t h e t r a n s i t i o n f i e l d s along the'mag-n e t i c axes are measured. Troup et a l . c a l c u l a t e the e f f e c t i v e g values of the t r a n s i t i o n s w i t h i n the three Kramer doublets of the 3 + ground s t a t e of Fe - as- a f u n c t i o n of E/'D f o r D » hv . 81 51 hv i s the microwave photon energy. Pig.3 of Troup et a l , 1 i s p a r t i a l l y g i v e n here as Pig.18. -- z means a t r a n s i t i o n w i t h i n the doublet | i 1 / 2 > and the magnetic f i e l d H//z, e t c . T r a n s i t i o n s w i t h i n the doublet |- 5/2 > are not shown i n Pig.18. The e x p e r i m e n t a l l y observed t r a n s i t i o n s at X-band of spectrum I I i n anatase are i n d i c a t e d and numbered as i n Pig.13. One deduces from the graph E/D = 0.2 5 - 0.02. 52 Aasa c a l c u l a t e s f o r S = 5/2 the t r a n s i t i o n f i e l d s B as a f u n c t i o n of D, u s i n g E/D as a parameter. D and B are measured i n u n i t s of hv . In P i g . 1 9 are reproduced those p a r t s of Aasas f i g u r e s 1 and 7 which correspond to E/D = 0.25. The t r a n s i t i o n s f o r B p a r a l l e l to the coo r d i n a t e axes are shown i n the iipper p a r t of P i g . 1 9 f o r the lowest Kramer-doublet, i n the lower p a r t of the f i g u r e f o r the in t e r m e d i a t e Kramer doublet (Aasa l a b e l s the energy le A ^ e l s a c c o r d i n g to t h e i r r e l a t i v e magnitude, 3<-H means t h e r e f o r e a t r a n s i t i o n w i t h i n the |+ 3/2 > d o u b l e t ) . The observed t r a n s i t i o n s of spectrum I I i n anatase are again i n d i c a t e d and l a b e l e d as i n Pig.13. Prom these graphes a value of h v /D = 0.65 - 0.05 i s deduced. Together w i t h v = 9 - 1 9 GHz , t h i s r e s u l t s i n JD j = 14.2 i 1 GHz. 82 83 APPENDIX C • C a l c u l a t i o n of Spin H a m i l t o n i a n Parameters w i t h a L i n e a r i z e d  Least Mean Square F i t As d i s c u s s e d i n s e c t i o n 5.4-? the observed EPR t r a n -s i t i o n f i e l d s H^ x p , Ic = 1, 1 3 ? of spectrum I I i n anatase are f i t to the t r u n c a t e d Hamiltonian ( 1 9 ) w i t h a t r i a l and e r r o r 53 method u s i n g a computer programme gi v e n by Hebden et a l . . The best f i t , a t room temperature, i s obtained w i t h b 2° = 0.501 cm 1 b 2 2 = 0.375 cm 1 and g i v e s 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 H° . The above values of b 2° and b g 2 , together w i t h ^4° = b.2 = b 4 = 0, are used as approximate va l u e s (b m ) i n n ' o a Newton-Raphson i n t e r p o l a t i o n c a l c u l a t i o n to determine the constants of the s p i n Hamiltonian ( 8 ) . -The parameters ( b n m ) 0 are i n d i v i d u a l l y changed "by a s m a l l amount ob m and the. new t r a n s i t i o n f i e l d s , which d i f f e r by o h k from H^0, are c a l c u l a t e d by d i a g o n a l ! z i n g wiuh a com-puter programme the energy m a t r i x g i v e n i n Table V. H^ a l i s now d e f i n e d by 84 iy H c a l = o _,ry~ A b m k_ This r e p r e s e n t s a system of k l i n e a r equations i n the unknowns A b n m . The best v a l u e s of A b n m are determined by m i n i m i z i n g Z / Hexp u c a l v 2 k ( H k - H k } (21) The r e s u l t s , b n m = ( k n m ) 0 + A b n i n a r e i n c l u d e d i n T a b l e I I and d i s c u s s e d i n s e c t i o n 5.4. 

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