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

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EPR OP SUBSTITUTIONAL FE 5* IN A NATURAL CRYSTAL OP BROOKITE (TiO 2) by J u a n A. Rostworowski B.Sc. Universidad Nacional de Ingenieria • Lima (1969) A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of Physics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1972 In present ing th is thes is in pa r t i a l f u l f i lmen t o f the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, 1 agree that the L ib ra ry sha l l make it f r ee l y ava i l ab le for reference and study. I fu r ther agree that permission for extensive copying o f th i s thes i s for s cho la r l y purposes may be granted by the Head of my Department or by h is representat ives . It is understood that copying or pub l i c a t i on o f th i s thes i s fo r f inanc ia l gain sha l l not be allowed without my wr i t ten permiss ion. J . A. R o s t w o r o w s k i Department of ^ftySiOf? The Un ive rs i t y of B r i t i s h Columbia Vancouver 8, Canada Date A p r i l 21, 1972 ABSTRACT EPR spectra of Pe^+ in a natural crystal of brookite have "been investigated at X- and Q-band frequencies at room temperature and 573°K. Part of the paramagnetic resonance spectrum observed has been interpreted on the assumption that Pe-5* occupies eight equivalent ? i ^ + sites in brookite, with four inequivalent orientations.. The spectra show an "intermediate" zero-field s p l i t t i n g at X-band and a "normal" zero f i e l d s p l i t t i n g at Q-band frequencies. 'Hie spin He.miltonian parameters which f i t the spectra are the following:: v g r 2.002 ± 0.005 . D= (1170 i 30) x 10-4cm"1 E r (330 ± 20) x 10~4cm"! [ p a * ( l / l ^ o i O = ( 1 5 ± 1 0> x 10~4cm~1 [pa+(l/12>i?] 1 0 0 = (-1315) x lO^cm- 1 [ga+{L/12)i?]QQ1 = (-66+4) x lO^cm" 1 i i i 'TABLE OF CONTENTS Page Abstract i i Li s t of Tables iv Li s t of Figures v'. Acknowledgements v i i . . 1. Introduction 1 2. Results of Previous Investigations on Brookite 3 Structure of Brookite 3 Phase Transformation 10 Other Properties 12 3. The Spin Hamiltonian 13 -4. Samples and Experimental. Techniques 17 Natural Crystals of Brookite 17 . X-Band and Q-Band Spectrometers 17 Angular Dependence of the Spectra 19 High Temperature Measurements ' 21 5. Experimental Results 23 Preliminary Observations 23 Determination of the Spin Hamiltonian Parameters 29 Adjustment of the Hamiltonian Parameters 33 6. Discussion of Results and Conclusions 38 Bibliography 41 .Appendix A 46 Appendix B 48 Appendix C 50 i v LIST OF TABLES Table I L a t t i c e Parameters 7 Table II Ion Coordinates f o r a Sample TiOg Octahedron 7 Table III Ion Distances for a Sample TiOg Octahedron 8 Table IV T i - T i Distances 9 Table V Summary of the Atomic Movement i n Topotaxy i n the Ti02 System 11 Table VI Other Properties 12 Table VII Transition P r o b a b i l i t i e s 34 Table VIII •' Hamiltonian Parameters 35 LIST OP FIGURES is* Figure 1 The p o s i t i o n of the ions i n the projections of a u n i t c e l l of brookite 4 Figure 2 Projections i n the (100) plane of some TiOg octahedrons. The common edges of the octahe-drons are drawn heavier 5 Figure 3 Double hexagonal closed packed (DHCP) arrangement 5 Figure 4 Block diagram of the EPR spectrometer arrangement f o r X- and Q-band frequencies 20 Figure 5 EPR spectra at Q-band of a natural s i n g l e brookite c r y s t a l with K p a r a l l e l to [loo] at room temperature and approximately 200°C 24 Figure 6 EPR spectra at Q-band of a. natural single brookite c r y s t a l with H p a r a l l e l to', [bid] at room temperature and approximately.200°C 25 Figure 7 EPR spectra at Q-band of a natural single brookite c r y s t a l with II p a r a l l e l to [ooij at room temperature and approximately 200°C 26 Figure 8 EPR spectrum at Q-band of a natural single brookite c r y s t a l with H s l i g h t l y o f f jfooi] i n approximately the (110) plane at room temperature 27 Figure 9 Angular dependence of the EPR l i n e s of the spectrum observed at high temperature i n brookite at Q-band with H rotated i n the three p r i n c i p a l c r y s t a l planes 28 Figure 10 Energy l e v e l s of s u b s t i t u t i o n a l Fe^ i n . . . . "brookite v.'ith H p a r a l l e l to each of the p r i n c i p a l crystallographic axes 37 Figure 11 Positions of the EPR t r a n s i t i o n s within the lower and upper Kramer doublets of Fe-?-1" f o r E/D = 0.25 49 v i i ACKHOV/LSSC-BMEI-JTS I would l i k e to thank Br. C. P. Schwerdtfeger f o r h i s he l p f u l guidance i n the preparation of this t h e s i s . I an also g r a t e f u l to Br. K. Horn f o r the many d i s -cussions on the in t e r p r e t a t i o n of the observed spectra. I am also indebted to A. Harnik of the C r i s t a l l o g r a p h i c I n s t i t u t e of E.T.H., Zurich, f o r the supply of many specimens of Brookite. The research o f thi s thesis was supported f i n a n c i a l l y by the National Research Council, grants number A-2228 and A-7121. Additional f i n a n c i a l aid was obtained from the University of B r i t i s h Columbia's 1970-71 President's Committee on Research. 1 1. IKTROBUGTION It was Klaproth who f i r s t recognised a polymorphism i n 1798, that i s , that the same chemical compound may c r y s t a l l i z e i n d i f f e r e n t forms. Klaproth observed the polymorphic forms of calcium carbonate ( c a l c i t e , aragonite)."*" That titanium dioxide has several natural polymorphic forms ( r u t i l e , anatase a,nd brookite) has been known aiready fo r many years. Recently, a fourth polymorphic form was s y n t h e t i c a l l y produced ( T i 0 2 H ) 2 . R u t i l e , anatase and brookite are often found i n nature as good single c r y s t a l s . S y n t h e t i c a l l y one can obtain single 3 c r y s t a l s of r u t i l e and p o l y c r y s t a l l i n e anatase . To date, a l l attempts to produce brookite a r t i f i c i a l l y , even i n poly-c r y s t a l l i n e form, have been unsuccessful. As a r e s u l t most EPR studies have been made with r u t i l e 4 _ 7 and some with natural c r y s t a l s of anatase8-13. No EPR studies of brookite have been reported. An EPR comparative study of d i f f e r e n t polymorphic forms can be i n t e r e s t i n g from a mineralogical point of view. Only recently, EPR has been acknowledged to be a to o l i n mineralogy and geology. Reviews by W. Low1* and S. Ghose^ have shown that EPR can be h e l p f u l i n c l a r i f i n g some mineralogical problems such as the r e l a t i o n between the paramagnetic impurity and the mineral host ( s i t e preference). Information about coordination and l o c a l symmetry (non-equivalent s i t e s , o r i e n t a t i o n and values of the c r y s t a l f i e l d parameters, charge compensation), nature of chemical bonding, color centers and other l a t t i c e defects, order-disorder, and r e l a t i v e abundance of d i f f e r e n t para-magnetic impurities or d i f f e r e n t valence states of a given impurity may be obtained from analysis of EPR measurements. Since impurities are a factor that determine the habit formation of the minerals i t i s also remotely possible that EPR can be h e l p f u l i n this geological problem'. Remotely since non paramagnetic impurities are impossible to observe or large concentrations of paramagnetic impurities w i l l not be seen or . make analysis extremely.difficu.lt v/ith EPR. In the present thesis the r e s u l t s of an EPR study of s u b s t i t u t i o n a l i r o n impurities i n a natural sin g l e c r y s t a l of brookite are described. Ho attempt to analyse other EPR t r a n s i t i o n l i n e s has been made to date although some specu-l a t i o n has been attempted. 3 2. RESULTS OF PREVIOUS INVESTIGATIONS ON BROOKITE STRUCTURE OF BROOKITE Brookite has an orthorhombic symmetry and belongs to the Pbca ( B ^ ) space group. R. Vfeyl 1^ remeasured the l a t t i c e constants of brookite, previously determined by L. Pauling and 1 7 J.H. Sturdivant , and found t h e i r r e s u l t s to l i e within tne pre c i s i o n l i m i t s of 0.1$ attained by his measurements. These values are: o a - 9.164 A b = 5.447 X c - 5.145 A . However he found some discrepancies f o r the values of the l a t t i c e parameters and these are l i s t e d i n Table I. Using the values of Table I and the coordinates of equivalent positions f o r the Pbca space group"^'"^ one i s able to f i n d the projections of the u n i t c e l l . Figure 1 shows these projections. Each titanium ion i s surrounded by s i x oxygen ions at the v e r t i c e s of an octahedron ana i t also shares three edges with other octahedrons. This i s shown i n Figure 2. These . common edges are shorter i n comparison with the other edged of 20 tne octahedron which i s i n accordance witn Pauling's Rule (Rutile shares two edges and anatase shares four"^' ^ ) . Tables II and I I I give the distances between the titanium and the oxygens as well as the distances between the oxygens. Clearly, t h i s octahedron i s d i s t o r t e d . Furthermore, o • 0 o o o o o o o ' 0 o o o o o o 0 ; o • o 0 o o o 0 Figure 1.- The position of the ions i n the pro lections of. a unit c e l l of brookite o o2-.0 v4' Figure 2.- Projections i n the (100) plane of some TiOg octahedrons. The common edges o f the octahedrons are drawn heavier. • J\ o o o o o o o o Q O O O O O O O O ^ o o o o o o o o ( 7 0 0 0 0 . o o o o /\ 0 0 0 0 o o o o B o o 0 0 0 0 o o j\ • • o o 0 0 b o o o Q O ' 0 0 0 0 o o o Figure 3.- Double hexagonal closed packed (DHCP) arrangement 3. Weyl has calculated that the T i i n the octahedron i s . displaced by 0.2i0.lA from the center toward Ojy and. away OJ-J-( i n r u t i l e as well as anatase the titanium ion i s i n the middle of the TiOg octahedron) and that t h i s ion, T i Q , i s further away by 0.12t0.06 i from the neighbouring ions, T i 1 and T i 2 than from the t h i r d nearest neighbouring ion T i ^ , as can be calculated from Table IV. (Y.'here the subscripts on T i r e f e r to the ions l a b e l l e d i n Figure 2). In the brookite structure each octahedron i s bound through two common.edges to'two other octahedrons forming a chain i n the [001J d i r e c t i o n and by the t h i r d common edge to another' such chain forming a net, p a r a l l e l to the (100) plane. The u n i t c e l l has t..'o such nets, one over the other which are bound by common corners. Tiie (001) plane i n Figure 1 shows d i s t i n c t i v e l y that the u n i t c e l l has four empty columns, hof.ever, i t i s not cl e a r that each such column contains t-'-o i n t e r s t i t i a l s i t e s sur-rounded by six oxygens at the corners of a larger d i s t o r t e d octahedron. Another way of viewing the structure of brookite i s th following: the oxygen ions are i n approximately double hexa-gonal closed packed (DROP) arrangement, i . e . (A3ACABAC..•) as shown i n the Figure 3. The closed packed plane i s the (100) plane, one h a l f of the octahedrons are f i l l e d with t i t a n i u m . • 7 T A B L E I L A T T I C E ? A R A : 3 T 5 R S (117 B R A C K E T S T H E V A L U E S O P ? A U L I : : G A~:B S T U R D I V A N T ) 0 1 0 2 Ti x 0.00 8 (0.010 ) 0 . 229 (0 . 230 ) 0.128 (0.127) y 0.147 (0.155) 0.110 (0.105) 0.098 (0.113) z 0.182 (0.160) 0.530 (0.535) 0.863 (0.873) T A B L E II 10IT COORBJ [ N A T E S F O R A S A M P L E TiO* O C T A K E I E C K ( G I V E N I N j f'R A C T I O N S O F T E E L A T T I C E C O N S T A N T S ) X y z 0,128 0/402 0,363 Ol 0,008 0,147 0,182 ° I I 0,229 0,110 0,530 ° I I I 0,271 0,610 0,530 ° I V -0,008 0,647 0,318 °v 0,229 0,330 0,030 ° V I 0,008 0,353 0,682 I O N DISTANCES FOR A SAMPLE TiO£ OCTAHEDR02? T io-°I ' T io-°H T i o - O l I I T io-°VI ° I I - ° V I ° V I - ° I I I ° I I I - ° I V °-I7-°V O v-0 T ° I I I " ° V o v-o_ V 11 0T-0 ° V I - ° I V •-URE 2 ) 2,0010,05) i 2,0 5+0,05) % 1,9410,05 1,84+0,05 1,9510,05 1.99*0,05 2,7110,08 2,55£0,08 2,90±0,08 2,7910,08 2,98±0,08 2,5510,08 2,7510,08 2,8610,08 2, 99+0,08 2,8110,08 2,4710,08 2,62±0,08 TABLE I V T i - T i DISTANCES (SEE PIGURB 2) T i 0 - T l 1 = T i - T i . (3,00 ^ ,0 3) A * - T i Q = T i - T l n 0 8- o 9 o T i Q - T i 3 (2,94*0,0 3) A Ti - T i = Ti - T i c (3,78+0-0 3) A O A O J T i Q - T i f i = T i -Ti„ (3,57^0,0 3) A Ti c — T i Q (3,53^0,0 3) A 10 ' PHASE TRANS FORMA TION The topotactic mechanisms f o r the Ti02 system are "based on a p r i n c i p l e of favoring as much as possible the maintenance of the oxygen close packing during the polymorphic t r a n s i t i o n (Anatase i s pseudo-cubic close packed (CG?) v/ith the close packed plane (112); R u t i l e , rough approximation of an hexagonal close packed (HOP), (100) or (010) planes; T i 0 2 H , HOP, (100) plane). Table V summarizes the observed and proposed topotaxy i n the titanium dioxide system . At t h i s point, i t should be mentioned that the k i n e t i c s and mechanism of the b r o o k i t e - r u t i l e transformation are very s i m i l a r to those of the anatase-rutile transformation^-* -^. Tlie rate of transformation and i t s a c t i v a t i o n energy are governed by the surface size and by the amount of impurities, e.g. the concentration of oxygen vacancies or i n t e r s t i t i a l s . But the a c t i v a t i o n energy, which i s mainly f o r the production of nucleation s i t e s , i s higher f o r the anatase-rutile case and the entropy of a c t i v a t i o n i s large and negative f o r the br o o k i t e - r u t i l e transformation compared with the small and po s i t i v e value f o r the anatase-rutile transformation. This may be understood i n terms of the change of symmetry (lower to higher) of the f i r s t case and the absence of such a change i n the second^. The e f f e c t of impurities i n general i s that oxygen vacancies accelerate v. here as i n t e r s t i t i a l ions i n h i b i t , the transformation 2^.' TABLE V SUGARY OF THE ATOMIC MOVEMENT IN TDPOTAXY IN THE TiOp SYSTEM Reaction Number of Ovypen Redistribution of the Layers Rearranged Titanium Atoms* RemarVs 21 - futile None One half in each layer Only reaction that has been A B A B A B A B reversed. Experimental evi-dence inferred from work ol 1/2 1/7 1/2 1/2 1/2 1/2 1/2 Bendeiiany et al. (1966) Brookite-II 1 in 4 Every Ti in two consecutive layers Reaction without SRO phase altering with two layers in which there formation. Hrookite has aig-is no Ti motion. za£ TiO» octahedral chains as A B A C A B A C is found in II. 1 1 Brookite - rutil* This reaction dots not involve an oxy-gen shearing mech-anism, but there is * shifting of all oxygen. Ever)' cth^r Ti in each layer in order to form straight Ti ociahcdral chains. A B C A B C A B C 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 Experimentally observed topotaxy. A n a 12 s-c—rulUe This reaction dots not involve an oxy-gen shearing mech-anism but alt ojty-gen atoms shift posit k>as. Every other Ti in each layer in oru'er to form straight Ti octahedral chains. A B A C A B A C 1/2 1/2 1/2 1/2 1/2 1/2 1/2 Toptaxy experimentally ob-served. A large distortion oc-curs in the formation of straight octahedral chains from zigzag chains. Anatase—II A in 6 A repeating sequence of all Ti and no Ti motion beginrrlns with the Ti be-tween the A ar.d C layers. A B C A B C A B C Reaction always accompanied with SRO phase formation. 1 1 I 1 A n i t a s * — b r o o L i t e 9 in 12 Very complex Reaction reported only one time. • In most cases the mover-.er.t of the li:ai:ium ions is to adjacent tctrahcdral sites vrhich arc in the process of becoming octahedral M'IC; bcc.vjie of tlie accompanying oxygen nation. The legend for the tit.tnium motion is 1/2 every other titaniu:r. charters site 1 every titanium chan^rs ?ite 0 no titanium chan;;o ji'.e . The layering sthenic i> that vi the rcactanl. OTHER PROPERTIES A l i s t of these properties i s given i n Te.ble VI. Proper-t i e s of r u t i l e and anatase have been also included for compari-s o n ? 0 ' 8 . TABLE VI PROPERTIES Property Anatase Ruti l e Brookite Density • 3.87-3.95g/ca3 4 .21-4 .25g/cn 3 4.13g/cm-Eardness 5.5 - 6.0 7.0 - 7.25 5.5-0.0 S p e c i f i c heat at R.T. 13.22cal/mol°C 13.16cal/mol°C O p t i c a l Constants n5693& 2.56 2.61 n 2.5831 o * n5893A 2 A 9 2 > g o n, 2.5843 n 2.7004 f S t a t i c d i e l e c t r i c constant 48 114 78 Notice that the values of these physical constants f o r brookite are between those measured for anatase and r u t i l e . . EPR STUDIES Hamiltonian parameters f o r s u b s t i t u t i o n a l Pe^, where D, E, a and P are given i n 10~^cm~\ are: g D E a p Anatase (3O0°K) 2.005 308.7 - 10 2.8 6 Rutile (4.2-300 )°K 2.00 6780 690 -280 230 No EPR studies i n brookite have been reported. 3. THE SPIN HAMILTONIAN 'The e l e c t r o n i c configuration of the Pe^* ion i s 3d^ and the free ion e l e c t r o n i c ground state i s ^S^yg* magnetic resonance spectrum should be remarkably simple. The sextet would have no s p l i t t i n g other than the Zeeman i n t e r a c t i o n , and a single l i n e would be observed, ho»ever, experimentally this i s f a r from the truth. ' Both fine and hyperfine structures have been observed for ions with h a l f f i l l e d 3d, 4f and 5f s h e l l s . Several mechanisms have been proposed to explain 31 t n i s . Spin-spin and s p i n - o r b i t interactions can produce seccnd or higher order terms coupling the ground state through higher' o r b i t a l states to the c r y s t a l f i e l d . The multiple fine structure can conveniently be repre-sented by adding "bo the Zeeman term i n the spin Hamiltonian, terms of higher powers i n Sy., S and S z, grouping them into combinations of spin operators, each such operator being the equivalent to a combination of. sp h e r i c a l harmonics. Such Til equivalent spin operators denoted as 0' appear extensively i n 32. l i t era "o"ure. Che advantage of using equivalent spin operators i s tha one can choose immediately which terms w i l l appear ..in the spin Hamiltonian. One excludes the equivalent spin operators which are of odd degree i n S, since they are not invariant under time r e v e r s a l . Prom the operators of even degree one chooses only those which r e f l e c t the symmetry of the c r y s t a l f i e l d . Furthermore, the number of such operators i s l i m i t e d by the 14 f a c t that operators of higher.degree than 23 may be omitted since they have zero.matrix elements between the states under consideration. Hence f o r Pe 3 +" one can write i n general ? f = /JH-g.S * 1/3 ZZ- b l o\ • 1/60 E Z o| 0* (1) m—— 2 m—t;-Using the known transformation properties of under r o t a t i o n of the coordinate system one can evaluate the trans-formation properties o f the b^ under these r o t a t i o n s . An ap-pendix i n Baker and Williams i s very h e l p f u l f o r these calcu l a t i o n s and has been used i n Appendix A to calcu l a t e the spin Hamiltonian f o r an a r b i t r a r y o r i e n t a t i o n . The spin Hamiltonian may be expressed i n an a l t e r n a t i v e form. The f i r s t summation i n (1) can be v.ritten as: 3-D-S where B i s a tensor quantity. Referred to the p r i n c i p a l axes, thi s becomes where i t i s convenient to set the sum o f the three c o e f f i c i e n t s to zero •by substracting the quantity 1/3 (B\^B^B z)(S,.. 2t3 2*3 2 ) = 1/3 (B„+ B; + D„ )S (3*1) which i s just a constant that moves a l l energy l e v e l s up or down by the same amount. The f a c t that one can set the trace of tensor B to be zero, means that there are only two independent c o e f f i c i e n t s x(s|_S|) * I) as| (2) D XS^ • D y s | + D„S|= D{sf-l/3 3(S+1)} * 1/2 E ( s f * s f ) v.= i th D - 5/2 D,_ 2 = 1/2 (D x - IL,} Thus the second term of equation ( i ) i s equal to. equation (2) i f the axes are chosen to he the p r i n c i p a l axes of the tensor D. These axes are commonly known i n S?R l i t e r a t u r e as the magnetic axes of a paramagnetic center, and are determined experimentally by absolute extreme positions of. the EPR s p e c t r a l l i n e s . By eventual 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 E/D to the range o^E/D*1/3, where zero i s the purely a x i a l c a s e ^ . Since there i s no argument to suppose that the p r i n c i p a l axes of the tensor b^ coincide with the p r i n c i p a l axes of bg, the l a t t e r have been taken as defining the magnetic sizes. Most work published i n t h i s - f i e l d h a v e included only the b° and b' terms of the second summation i n equation . Ex-. 4 ceptions are made when dealing v/ith cubic or pseudo-cubic cr y s t a l s (when one can consider the f i r s t summation as a small perturbation of the second). In which case the magnetic axes are taken to be the p r i n c i p a l axes of the b^ tensor. By including the "two terms described, the Hamiltonian i s also usually written as: 16 ?< = /KH.g.S)*D ( S z 2 - l / > 3(3+1)] y 1/2 3 ( 3 2 , o f ) --H 1/6 a ( s | + 3^ -A 3| - 1/5 S ( 3 ^ 1 ) ( 5 3 2 ^ 53-1 )J (5) 1/180 P { 353a - 303(3 + 1) S'| + 253^ - 6 3 ( 3 v - l ) + 3 3 2 ( 3 + l ) 2 J \7iie.rc tne c o o r d i n a t e T~ '3 Ii"'v> ° Cl 3_ CU pi rr-r r.; stem J • *{ i £ r e - T e n .cn .cur-zio-LG t i - i o ' O o f the c r y s t a l ••^ic-ld. i n g e n e r a l , they are not the a s C:.0 -C, y , Z p r i n c i p a l axes o f the tensor D. H a m i l t o n i a n ( l ) are r e l a t e d to D, 3, a. and P by 'hie o:" - o a r a m e t e r s om xne 3'om j.1 " •"-: ° 2 -" D b 2 -= I + P/3 -- 2« . 4. -SAMPLES AMP EXPERIMENTAL TECHNIQUES  NATURAL CRYSTALS OP BROOKITE Several natural c r y s t a l s of brookite have been used i n t h i s work. They originated i n Maderanerthai, U r i , Switzerland; Valser Tal, Graubunden, Switzerland; and Magnet Cove, Arkansas, USA. She Maderanerthai and Valser Tal samples studied i n t h i s wcrk had the same habit,, they '-.ere translucent flakes p a r a l l e l to the (010) plane, and were e i t h e r l i g h t brown or yellowish i n color. The Magnet Cove sample also c a l l e d Arkansit was almost a perfect octahedron and black i n color. The EPR spectra of the Maderanerthai and Valser Tal samples are undistinguishable, a l l observed EPR t r a n s i t i o n l i n e s have the same r e l a t i v e i n t e n s i t i e s . Any comparison with Arkansit was not possible since these samples had a very high loss f a c t o r and hence no EPR spectrum was obtainable. X-3ANB A'-MO Q-3ANB SPECTROMETERS The EPR spectrometer was a conventional balanced bridge design. -The microwave bridge u t i l i s e d , a magic T at X-band and a c i r c u l a t o r at Q-band. A r e f l e x k l y stron (X-band: Varian V-153/6515max. output 70 mv, Q-band: OKI 35V10;: max. actual output 40mw), a one way f e r r i t e i s o l a t o r , and a flap, attenuator were connected to the input arm, a c r y s t a l detector, to the output. In X-band one arm ended i n a TE 102 resonance cavity (Varian multi-purpose cavity V-4531) coupled through an 18 adjustable i r i s , the other reference arm contained a s l i d e -screw tuner and a matched load. In Q-band the t h i r d arm of the c i r c u l a t o r contained a slide-screw tuner and a c i r c u l a r TE 012 resonance cavity (Ventron sample cavity SG-10-ka) and v.as coupled through adjustable t e f l o n s p a c e r ^ , The c r y s t a l diode was biased (100-300 MA) by adjusting the s l i d e screw tuner. The klystron was frequency locked to the resonant cavity.by modulating the r e f l e c t o r voltage with a 10KHz s i g n a l and using the corresponding phase se n s i t i v e detected output from the c r y s t a l detector as an error s i g n a l . The magnetic f i e l d was modulated at 100KHz. through small modulation c o i l s attached to the resonant cavity at room temperature. At l i q u i d Helium temperature a 400Hz modulation arrangement was used through la r g e r modulation c o i l s attached to the faces of the magnet. The modulation amplitude used- was 10-15 gauss. This high modulation was safel y used since a l l l i n e s were over 30 gauss wide and hence no modulation broadening e f f e c t s would be observed. The pre-amplified output from the c r y s t a l detector was phase se n s i t i v e detected at lOOKEz (400Hs) (PAR Lock i n Amplifier/phase detector, model 121). The output was then connected to a s t r i p chart recorder. The frequency of the microwaves at X-band was measured with a d i g i t a l frequency meter (Hewlett-Packard Frequency counter 5255A with plug-in. adapter HP 5245 ), at Q-band a marker !9 w&s introduced (DPPE powder) to "the cavity. The experiments were performed on a 9.5" Magnion Magnet with a r o t a t i n g c o i l f i e l d sensor (Magnion FPC-4 power supply) the d i r e c t f i e l d readings by the sensor '..ere c a l i b r a t e d through a NMR gaussmeter. A l l measurements were made 7vith the . magnetic f i e l d perpendicular to the microwave magnetic f i e l d at the sample. Figure 4 gives a block diagram of the- experimental set-up. ANGULAR DEPENDENCE 0? TKE SPECTRA I t i s possible, i n general, to determine experimentally the p r i n c i p a l axes of the tensor D by f i n d i n g the extremum positions of the EPR s p e c t r a l l i n e s T o achieve t h i s , one usually measures the EPR t r a n s i t i o n f i e l d s as a function of the o r i e n t a t i o n of the c r y s t a l f i e l d with respect tc the s t a t i c magnetic f i e l d . In the p a r t i c u l a r case of brookite, i t was found that i t was operationally very d i f f i c u l t to f i n d the extreme t r a n s i t i o n f i e l d s since, what was one l i n e at an o r i e n t a t i o n p a r a l l e l to a p r i n c i p a l c r y s t a l l o g r a p h i c axis became four l i n e s at a random or i e n t a t i o n . To t r y to follow a l i n e by r o t a t i n g the c r y s t a l or magnet i n such a random plane proved to be an impossible task. The other p o s s i b i l i t y was to take the spectra at pre amplifier cryst. cletec. variable coupler / Q-BAND ' MICROWAVE Set-up L \ klystron isolator attenator magic T slide-srew tuner crystal detector / Al a / V amplifier r. lock-in 1/ t w wave generator chart recorder 19-magnet pow. sup. rot. coil field sensor modulation coils j Figure 4.- Block diagram of the EPR spectrometer arrangement f o r X- and Q-band frequencies. field regulator d i f f e r e n t orientations i n the three c r y s t a l l o g r a p h i c (001, 010, 100) planes where each t r a n s i t i o n l i n e s p l i t into two and thus hopefully receive some information on the projections of the magnetic axes i n these planes, then to reconstruct where the r e a l magnetic axes are, and f i n a l l y obtain enough information to i d e n t i f y and analyse the paramagnetic centers involved. The new method described by K. Horn and C. Schwerdt-feger°> ^ ° f or making angular plots was used only on a li m i t e d basis, since i t proved almost inoperative owing to the large number of l i n e s and t h e i r respective cross-overs. However i t did. prove useful i n this case for the more accurate alignment of the c r y s t a l which was previously oriented with. 37 a back r e f l e c t i o n Laue method ( C u l l i t y ) . HIGH TEMPERATURE MEASUREMENTS In the angular plots made i n the (00ij, (010) and (100) planes there are so many t r a n s i t i o n l i n e s that i d e n t i f i c a t i o n of a paramagnetic center from symmetry considerations alone was not possible. Transition l i n e s disappeared at some orientations, they crossed v/ith other l i n e s and no consistent trends could be found. A discriminating mechanism was necessary. The temperature dependence spectra o f anatase ' J showed that spectra due to substitutional, i r o n could s t i l l be seen e a s i l y at temperatures over S00°C while spectra of sub s t i -t u t i o n a l i r o n complexes with an oxygen vacancy disappeared at 22 much lower temperatures. I t was a discr i m i n a t i n g device. Naturally there was no reason to "believe that a charge compen-sation ^center was present, hut s p i n - l a t t i c e r e l a x a t i o n times for d i f f e r e n t impurity ions are i n general always d i f f e r e n t , so that t h i s discriminating t o o l could very well work. High temperature measurements at X-hand were made with a flow of heated nitrogen gas (Varian V4557), at Q-band a hot brass finger was used. The discriminating t o o l proved suc-c e s s f u l . 5. • EXPERIMENTAL RESULTS PRELIMINARY OBSERVATIONS Since brookite has orthorhombic symmetry one would expect that i t s EPR spectra would also show t h i s symmetry, th i s was experimentally v e r i f i e d . EPR spectra at approximately 36 GHz are shown i n Figures 5, 6 and 7 with the permanent magnetic f i e l d p a r a l l e l to the p r i n c i p a l c r y s t a l l o g r a p h i c axes, each figure contains two spectra, one at room temperature, the other at approxi-mately 20Q°C. Figure 8 displays a spectrum s l i g h t l y o f f the [oioj axis i n approximately the (101) plane, here each tran-s i t i o n l i n e s p l i t s into four l i n e s . As a r e s u l t of the angular dependence and temperature response of the observed t r a n s i t i o n s one could separate them into at l e a s t two groups. The f i v e l i n e s marked with arrows i n Figures 5, 6 and 7 form part of one of the groups. This thesis i s concerned only with these l i n e s . The angular dependence within the c r y s t a l l o g r a p h i c planes (100, 010 and 001) are shown i n Figure 9. At f i r s t , only spectra at X-band were taken, A p r e l i m i -nary analysis indicated that the zero f i e l d s p l i t t i n g was of the order of the frequency used. Since i t i s easier to analyze the case for- which the zero f i e l d s p l i t t i n g i s l e s s than the applied frequency, data were taken and analyzed at Q-band. The X-band measurements, consequently, were only used as a check. Pi'jure 5.- EPR spectra a t -Q-band of a natural single brookite c r y s t a l with p a r a l l e l to [lOOJ at room temperature and approximately 200°C Figure 6.- EPR spectre, a t Q-band of a natural single brookite c r y s t a l with H p a r a l l e l to [010] at room temperature and approximately 200°C 8 9 10 11 12 13 U 15 Figure 7.- SPPL spectra at Q-baid of a natural single brookite c r y s t a l with H p a r a l l e l to [OOl] at room temperature and approximately 200°C c% 11 12 13 MAGNETIC FIELD (Kgauss) Figure 8 . - EPR 'spectrum at Q-baiid of a natural single brookite c r y s t a l with H s l i g h t l y o f f [CIO] i n approximately the (101) plane' at room temperature Figure 9.- Angular dependence' of the EPR l i n e s of,the spectrum observed at high temperature in.brookite at Q-band with H rotated i n the three p r i n c i p a l c r y s t a l planes 29 PSTER 1 •*II"AT 10N 0? THE SPIN HAI.'.ILTONIAM PARAMETERS The f i r s t step i s to determine a set of principal axes for the D tensor. The angular dependence of the transition lines in the three crystallographic planes where measurements were taken v.ere carefully recorded. Prom Figure 9 i t can be seen that the angular dependence in the [OlO] plane indicates that there i s a magnetic axis near the (lOQj plane, i.e., between zero and 10 degrees from i t . The (lOCj plane angular dependences are presented in the sane figure. The transition lines reach an extremum for two di f f e r -ent orientations. The extremum for the /OQlJ direction in the 010 plane i s most l i k e l y due to the magnetic axis whose pro-jection is -25° from the [OQlJ direction i n the (l.OOj plane because i t i s closer to the (OlO) plane. One labels this axis Y, noting that this i s done only as an aid in this ckLscussion, As yet. there are no grounds to assume that this i s the Y mag-netic axis. In the f i n a l analysis, the Y axis satis f i e s the inequality 0 - E ±1/3 13 Further the other projection in the (lOC-J plane indicates that there is another magnetic axis ~"12 degrees from the 001 plane. Consider the (OOlj plane in the figure. Again i t i s most l i k e l y that the magnetic axis responsible for the extremum in the (lOOJplane near the [OlO] crystallographic axis i s also 30 responsible f o r the extremum at ~30° from the [oiO] soils i n the (001) plane, t h i s i s labeled, the Z a x i s . I f these two axes are near the planes discussed, one can assume f o r the moment that the spectra measured i n the given projections•are going to be near the spectra along the true magnetic axes. Furthermore, the fine structure s p l i t t i n g i s greater i n the d i r e c t i o n l a b e l l e d Z, than i n the one l a b e l l e d Y, hence the l a b e l l i n g becomes more meaningful, and one has l a b e l l e d the Y axis c o r r e c t l y as Y and not X because by doing so the sign of E and D are the same f o r this choice. Next an estimate the values of D and E must be made. This i s done by considering the maximum f i e l d s for the-tran-s i t i o n l i n e s close to the Y and Z d i r e c t i o n s r e s p e c t i v e l y . The difference i n magnetic f i e l d between t r a n s i t i o n s |3/2><-* |l/2> and |-l/2>«-»|-3/2) i n both di r e c t i o n s i s nearly the same, thi s i s an i n d i c a t i o n that the centre under observation i s nearer to a f u l l y orthorhombic configuration than i t i s to a purely a x i a l configuration. Eence E w i l l be closer to 1/3 than to S 5R zero. The data was then compared to the graphs of Aasa^ , who calculated f o r F e ^ the t r a n s i t i o n f i e l d s H versus D both measured i n u n i t s of liV using E/D as a parameter. The' r e s u l t i s hV/D »12 f o r Q-band with an S/D of 0.25. (See Appendix B). F i n a l l y , one can see by several-arguments that the contribution of the term i n the Kamiltonian with equivalent operators of power four i n S i s not n e g l i g i b l e . Consider the spectra close to the Z and Y axes. The difference i n magnetic f i e l d between the | 3/2>«-» 11/2> and 1-1/2—1-3/2> tr a n s i t i o n s are almost equal, but the difference f o r the |5/2>"-*|3/2> and | _3/2>«—»|-5/2) t r a n s i t i o n s i s no longer the same. This can no longer be explained i f one does not take into account equiva-l e n t operators of fourth power i n 3 i n the Hamiltonian. The same is. true i f one considers three mutually perpendicular di r e c t i o n s , say the p r i n c i p a l c r y s t a l l o g r a p h i c axes. Second order perturbation theory ca l c u l a t i o n s of the t r a n s i t i o n s , neglecting fourth order terms show that the algebraic sum of the differences between |3/2>*-*|l/2> and |-l/2>«-*|-3/2> f o r three mutually perpendicular d i r e c t i o n s should be zero. In the sum of the differences i s 300 gauss. Further, using the same calculations the difference between |3/2>*—»|l/2> and 1-1/2)*—1-3/2> should be h a l f of that for the |5/2>*->|3/2> and |-3/2)«—»|-5/2> t r a n s i t i o n s . This i s c l e a r l y not true (see e.g. LOlOl d i r e c t i o n ) . From the above discussion, i t i s c l e a r at t h i s stage that because of the large uncertainty i n f i n d i n g experimentally the orientations of the p r i n c i p a l axes of the tensor D, and since the g value f o r Fe^ i n the large v a r i e t y of host c r y s t a l s i s that of the free electron. to-vaVEhin a f r a c t i o n of 2$. I t w i l l be meaningless i n t h i s work to try to f i t a correct value f o r g and hence the value of g-2.002. was assumed. Furthermore, there i s no reason to believe that.the p r i n c i p a l axes for the term i n the Hamiltonian v/ith equivalent operators of degree four i n 5 has the same p r i n c i p a l axes as tensor D. I f one assumes that the values of a and ? f o r brookite are of the order c f those of anatase and r u t i l e , one can cal c u l a t e that the contribution to f i r s t order perturbation theory i s of the same order as the c o n t r i b u t i o n to second order perturbation theory f o r E. Hence i t seems only meaningful to try to f i t the t r a n s i t i o n f i e l d s using f i r s t order perturbation theory f o r the terms i n the Hamiltonian o f degree four i n S, The f i r s t order perturbation theory f o r t h i s term i s proportional to pa+1/12 q ? where p = 5/2(l4-m^->:r-3/5) and q -35003^6 30cos 2S -*-3. The p r o p o r t i o n a l i t y f a c t o r f o r t h i s ' polynomial i s 2, -2.5, 0, 2.5 and -2 depending on the tran-s i t i o n one i s a c t u a l l y looking a t 3 2 . Notice must be talc en that these p r o p o r t i o n a l i t y factors are true only i f the eigenstates of the unperturbed Hamiltonian are pure. Since one does net ha.ve an a p r i o r i idea c f the value of p, there i s no way one can give the values of "a" and "E". The most one can give i s the value of pa+(l/l2)qr(-H) f o r any d i r e c t i o n . A rough estimate o f t h i s value i s e a s i l y found by calcu-l a t i n g the magnetic t r a n s i t i o n f i e l d s from second order perturbation theory i n D and E and adding the term E with i t s corresponding' p r o p o r t i o n a l i t y factor, hv = g/JH 5/ 2 +2 X * 2 R +32Y - 1Z-h v - g/jH 3/ 2 +1 X - 5/2 R - 4 Y + 5/4 Z h ir = g 0 l l y 2 -fO X •» 0 R - 16 Y + 2 Z hvr = g/?H_ 1 / / 2- 1 X + 3/2R - 4 Y + 5/4 Z. h^= g/?K_^ ?-2 X - 2 R f 32 Y - 1 Z t h i s i s a system of f i v e equations and four unknowns and thus e a s i l y solvable f o r R. Tne unknowns X, Y and Z are functions of D, E, 6 and Q".41 ASJUSTr3NT OP THE HAMILTONIAN PARAMETERS An adjustment of the estimated parameters was then made with a computer program given by J . Hebden et al.^9 Using a. t r i a l and error method. This program calculates for a given . Gxx, Gyy, Gzz, D and E, the EPR t r a n s i t i o n f i e l d , the corre-sponding t r a n s i t i o n p r o b a b i l i t y , and plots the energy l e v e l s . A correction was made to the t r a n s i t i o n f i e l d s by considering the contribution of the term R. The p r o p o r t i o n a l i t y ; factors were taken as true f o r Q-band analysis since the t r a n s i t i o n p r o b a b i l i t i e s at t h i s frequency indicate that the eigenstates are f a i r l y , pure i n t h i s range (they are near the 5:6:9:6:5 of the pure case). The t r a n s i t i o n p r o b a b i l i t y c a l c u l a t i o n . f o r the X-band range, on the other hand, indicates a very strong mixture of states, hence no correction was made to the t r a n s i t i o n f i e l d s and consequently the calculated t r a n s i t i o n f i e l d s do not f i t the experimental data and- i n some cases, they are as much as 500 gauss o f f . Figure 10 shows the plots of the energy l e v e l s for. the cases with E p a r a l l e l to the three p r i n c i p a l c r y s t a l l o g r a p h i c axes, and Table VII gives the t r a n s i t i o n p r o b a b i l i t y calcu-l a t i o n s . •TABLE V I I 34 Q-Band '—"ran s TRADITION PROBABILITIES i t i o n "010 100 001 |5/2>,«— |3/2>' 5.33 5.45 -4.89 13/2 »|l/2> 8.25 8.37 7.75 |l/2>'*-» |-l/2> 8.73 8.74 8.85 |-l/2>" 1-3/2) 7.21 7.10 8.12 1-3/2 >*— |-5/2 ^  4.20 4.05 5.27 X-Band* |5/2>'^|3/2/ 0.27 |3/2> *-»|l/2>' 6.16 |l/2)'«—|-l/2>' 6.27 |_l/2><-* |-3/2>' 3.09 1-3/2;*-* |-5/2> 0.68 4.93 0.85 4.30 (0.74) 8.23 5.87 4.99 5.96 4.19 1.06 1.33 * S e v e r a l o t h e r transitions which are possible from energy differences alone have not been included since the probability calculations give values of less than 0.10. TABLE VIII  HAMILTONIAN PARAMETERS g = 2.002 ± 0.005 D = (1170 t 30) x l O ^ c m " 1 E = (330 ^  20) x lO-^-cnT1 ( p a + l / l 2 q P ) 0 1 0 = (13+10) x l O ^ c n " 1 (pa+-l/12qP) 1 0 0 = (-13±5) x l O ^ c n T 1 (pa-»l/i2qP) 0 0 1 = (-6614) x lO'^cm" 1 Hie following polar angles give the o r i e n t a t i o n of the magnetic axes of one of the s u b s t i t u t i o n a l s i t e s , the orientations of the other seven, are e a s i l y calculated by symmetry considerations, The error f o r the following l i s t of angles i s - 3 ° . e ' # z 81° 55° y 149° 231° x 60° 210° The d i r e c t i o n between the longest and shortest Ti-0 bond, which are nearly opposite, corresponds to the z magnetic a x i s . 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 o f D, S, B , T , H 1 0 Q , R Q 1 0 and R , was made to f i t a l l 15 B?R l i n e s , 5 per a x i s . ft and f are the Euler angles as given by G o l d s t e i n 4 0 ) . This method i s discussed i n Appendix G. The absolute signs of D and B were established.with an E?R measure-ment at l i q u i d helium temperatures. The r e s u l t s are included i n Table V I I I . I t should be pointed out that low temperature measure-ments indicate that the temperature dependence of the para-meters i n the Hamiltonian vary at most 1$ compared to those found at room temperature. Hence the temperature behavior of brookite i s comparable to that found i n r u t i l e ^ and not to that found i n anatase 0'. ENERGY IN cm each of the p r i n c i p a l c r y s t a l l o g r a p h i c axes Jo 6. DI5GUG3IGI? 0? PEGUITS A.'D C0:?CHJ3I0K5 Prom Table II in. Chapter two, one can cal c u l a t e the pro-j e c t i o n of the segments j o i n i n g the T i to the neighbouring oxygens and the neighbouring titaniums. Comparison with the' angular dependence of the spectra enables one to i d e n t i f y the magnetic exes. The Z axis corresponds to t h e ~ ( T i Q - 0 ) d i r e c t i o n and the X and' Y d i r e c t i o n s correspond to the • A.(Ti - 0 T) andv(Ti - 0T.) d i r e c t i o n s r e s p e c t i v e l y , o l o V -The p o s s i b i l i t y " of i r o n occupying an i n t e r s t i t i a l s i t e has also been explored, but no l o g i c a l correspondence has been found, and, on the other hand t h i s would generate a strong l o c a l charge i n e q u a l i t y . In addition, i n r u t i l e one observes paramagnetic impurities i n the la 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 impurities are too large to enter, the s u b s t i t u t i o n a l s i t e . 4 2 In octahedral coordination, the i o n i c radius of F e 3 + i s 0.73°, compared to that of T i 4 which i s 0 . 6 9 ° . ^ Thus, one can conclude s a f e l y that the Fe ions go into s u b s t i t u t i o n a l s i t e s . The spectra a d d i t i o n a l l y show that a l l eight s i t e s are equally ocuppied by Pe^*" i o n s . A comparison between the r e s u l t s of the EPR spectra o f 3+ s u b s t i t u t i o n a l Pe i n brookite, anatase and r u t i l e i s i n t e r e s t ing. The Hamiltonian parameters f o r brookite are found to be between those f o r anatase and r u t i l e , and contrary to the case of anatase, the spectra were found i n s e n s i t i v e 'to temperature. 'Tlie strong temperature dependence of the anatase case has been explained i n terms of a s h i f t of the oxygen ions"'-'. This . 3 9 s h i f t does not change the symmetry c f the aiiatase c r y s t a l , whereas i t would i n r u t i l e . No a n a l y s i s has been as yet done on the second o r more groups o f s p e c t r a which have been a l s o seen a t room tempera- • •tares. Each one o f these l i n e s has, a l s o the c h a r a c t e r i s t i c o f s p l i t t i n g i n f o u r a t a random o r i e n t a t i o n s . Replacement o f a T i ^ * i o n by a F e - 5 * i o n causes a negative charge excess o f one elementary charge a t the s i t e . This excess has to be com-pensated 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 . This may be achieved by o t h e r i m p u r i t i e s , i n t e r s t i t i a l i o n s , o r an oxygen vacancy. The p o s s i b i l i t y o f the f i r s t case i s u n l i k e l y s i n c e t h i s i m p u r i t y i o n would have to be o f a. p o s i t i v e charge h i g h e r than f o u r . From symmetry c o n s i d e r a t i o n s o f the s p e c t r a the e t h e r two are p o s s i b l e , the l a t t e r case o n l y i f the oxygen vacancy i s not present i n the nearest neighbours. Such a case has been r e p o r t e d i n r u t i l e f o r a s u b s t i t u t i o n a l Cr3+ by Ikebe A3 e t a l . T h i s assumption, i f t r u e , i s i n t e r e s t i n g compared w i t h the other polymorphic forms o f Ti C ^ . Anatase has a nearest neighbour oxygen vacancy"*9 a s s o c i a t e d w i t h F e 5 * where as r u t i l e lias none. A second p o s s i b i l i t y to e x p l a i n these l i n e s 'would be the presence of o t h e r i m p u r i t i e s , on the o t h e r hand the EPR s p e c t r a o f the !<x.deranerthai and V a l s e r t a l samples are i n d i s t i n g u i s h a b l e j , a l l observed EPR t r a n s i t i o n l i n e s have the same r e l a t i v e i n t e n s i t i e s . I t i s not c l e a r a t t h i s stage i f t h i s i n d i c a t e s t h a t a l l t r a n s i t i o n l i n e s can be a t t r i b u t e d to Pe^ or that the r e l a t i v e concentration of the paramagnetic impurities i s the same f o r both samples, and that this i s d i r e c t l y responsible f o r the f a c t that the samples have the same habit. An attempt to i l l u c i d a t e t h i s point was made by inv e s t i g a t i n g a sample from Arkansas (Arkansit) but no spectra were obtainable because of a ..high l o s s f a c t o r . This leads to another unanswered question. Prom where does the high loss f a c t o r come? The p o s s i b i l i t y of a high concentration of some impurity i s somehow doubtful since when the c r y s t a l was trans-formed to r u t i l e by heating, a sharp spectrum was obtained. At l i q u i d nitrogen and lower temperatures a d d i t i o n a l s p e c t r a l l i n e s have been observed centered at g values near two. Any analysis of these spectra v / i l l p r o v e — d i f f i c u l t be-cause of strong overlaping of t r a n s i t i o n l i n e s i n thi s region. In conclusion, the high temperature EPR. spectrum i s explained by assuming that P e 3 t substitutes f o r T i 4 + , a l l eight equivalent T i 4 + " s i t e s being occupied by Pe 3 _ f with equal p r o b a b i l i t y . The scope of this thesis has l e f t many questions un-answered which farther EPR investigations can possibly lead to i n t e r e s t i n g r e s u l t s . 41 BIBLIOGRAPHY 1. Buerger M., Bloom M., "Crystal Polymorphism", Z. • K r i s t a l l o g r 96, 182 (1937) 2. Simmons P., Dachille ?., "The Structure of T i 0 2 I I , a High Pressure Phase o f T i 0 2 " , Acta C r y s t a l l o g r . 2J>, 334 (1967) 3. Czanderna A., C l i f f o r d A., Honig J., "Preparation of Highly P u r i f i e d Q?i02 (anatase)", J . Am. Chem. Soc. 79_, 5407 (1957) 4. Low W., Offenbacher E., "Electron Spin Resonance of Magnetic Ions i n Complex Oxides" i n 33P (Edited by P. S e i t z and D. Turnball), Vol. 17 p.135, Academic Press, Hew York (1965) 5. Iyengar R., Codell M., Karra J., Turkevitch J . , "E3H Studies of the Surface Chemistry of Ru t i l e " , J . Am. Chem. Soc. 88, 5055 (1966) 6. Carter D., Okaya A., "EPR of P e 5 + i n T i 0 2 (Rutile)", Phys. • H e v « ll£> 1^65 (I960) 7. Lichtenberger G., Addison J . R., "?- and X-band Spectros-copy on P e 5 + i n Ru t i l e " , Phys. Rev. 134, 331 (1969) 8. Horn M., "EPR of S u b s t i t u t i o n a l and of charge Compensated Pe^ 4 i n Anatase ( T i 0 2 ) and i t s Temperature Dependence", PhD. Thesis, U. of B r i t i s h Columbia (1971) 9. Horn M., Schwerdtfeger C. P., "EPR of S u b s t i t u t i o n a l and Charge Compensated Pe^* i n Anatase (Ti02)", J« p h y s « Chem. 42 of S o l i d s 22, 2529 (1971) 10. Gaiiion D., Lacroix R., "EPR of Fe^ "*" Ion i n Anatase", Proc. Phys. Soc. (London) 79, 658 (1962) 11. Barry "SSR of C r ^ + i n Anatase ( T i 0 2 ) , S o l i d State Comm. 4, 125 (1966) 12. Che I'., Grave l i e P., Meriandeau P., "Etude par Resonance Paramagnetique Blectronioue d' un Bioxyde de Titane (Anatase) Contenant des Ions Antimoine", C. R. Acad. Sc. (Paris) 263C, 763 (1969) 1 3 . Eauser C., Cornaz P., "Evidence by EPR o f a Complex TiO^ + i n the C r y s t a l of T i 0 2 Anatase", Chem. Phys. L e t t e r s 5, 226 (1970) 14. Low W., "Electron Spin Resonance - a Tool i n Mineralogy and. Geology", Adv. i n Electr... and E l e c t r o n Phys. 24, 51 (1968) 15. Ghoose S., "Application of SSR i n S i l i c a t e Minerals", i n Short Course Lectures Motes on Resonance Spectroscopy i n Mineralogy, Am. Geol. I n s t i t u d (1968) 16. 'Veyl R., "Brasis ionsbes timmung der K r i s t a l i s t ruktur des Brookites, T i 0 2 " , Z. K r i s t a l l o g r . 111.' 401 (1959) .17. Pauling L., Sturdivant J . H., "The Cry s t a l - S t r u c t u r e o f Brookite", Z. K r i s t a l l o g r . 68, 239 (1923) '• 18. International Tables f o r X-Ray Crystallography 19. Wyckoff R.77.C, "Crystal Structures Handbook*, V o l . I, Chap. IV, p.7, Interscience Publishers, Inc., New York (1953) 20. Pauling I., "The P r i n c i p a l Determining the Structure of Complex Ionic Crystals", J . Am. Chem. Soc. 51, 1010 (1929) 21. Bragg I., C l a r i n g b u l l G., "-The C r y s t a l l i n e State", V o l . TV, p.109, Cornell University Press, New York (1966) 22. Simmons P., Dachille P., "Possible Topotaxy i n the T i 0 2 System", Am. Kin. 55, 403*(1970) 25. Czanderna. A:., Rao C , Honig J., "The Anatase - Ruti l e Transition", Trans. Faraday Soc. 5£, 1069 (1958) 24. Rao C., "Kinetics and Thermodynamics of C r y s t a l Structure Transformation ofSpectroscopically Pure Anatase to R u t i l e " , Can. J . Chem. 39_, 498 (1961) 25. Yoganarasimhan S., Rao C , "mechanism of C r y s t a l Structure Transformations", Trans. Faraday Soc. 58, 1579 (1962) 26. Shannon R., Pask J., "Kinetics of Anatase R u t i l e Trans-formation", J . Am. Germ. Soc. £Q, 391 (1965) 27. Dachille P., Simmons P., Roy R., "Pressure-Temperature Studies of Anatase, Brookite, R u t i l e and T i 0 2 H A m . Min. 53, 1929 (1966) 28. Shannon R., Pask J.,. "Topotaxy i n the Anatase-Rutile Transformation", Am. Kin. 49, 1707 (1964) 2 9 . Eao G., Yoganarasemhan S., Faeth D., "Studies on the Brookite-Eutile Transfo ma t i o n " , 'Trans. Paraday Soc. 57, 504 (1961) 50. Pascal P., "Nouveau Traite de Chimie- Minerale", V o l . IV, p.96, Masson et Gie., Paris (1965) 31. Serway R., 'temperature Dependent Spin Hamiltonian Para-meters o f Mn i n Trigonal s i t e s of CaCO^", r'hys. Rev. j>, 603 (1971) 32. Abragam A., Bleaney 3., "Electron Paramagnetic Resonance of Transition Ions", Clarendon Press, Oxford (1970) 33. Baker <T. K.,. Williams P. I . 3., "E3R i n t.v0 S a l t s Containing C-adolinium", Proc. Phys. Soc. (London) 7_6, 1340 (1961) 34. Troup C-., Hutton D., "Paramagnetic Resonance of Pe^ i n Kyanite", B r i t . J . Appl. Phys.. 15, 1495 (1964) 35. C-ordon J . P., "Variable Coupling R e f l e c t i o n Cavity f o r Microwave Spectroscopy", Rev. Sc. I n s t r . 32, 658 (1961) 35. Horn M., Schwerdtfeger C. P., "A, Method to obtain D i r e c t l y the Angular Dependence of the EPR Spectra i n Single C r y s t a l Studies", Rev. Scient. Inst. 42, 880 (1971) 37. C u l l i t y 3. 3., "Elements of X-Ray De f r a c t i o n " , Addison-Wesley, Mass. (1967) 38. Aasa R., "Powder Line Shapes i n the Electron paramagnetic 45 Resonance Spectra of High-Spin F e r r i c Complexes11, J . Chem. Phys..52, 5919 (1970) 39. Byfleet C.., Chong D., Hebden J., McDowell C., "Calculation of the EPR Transition F i e l d s and T r a n s i t i o n P r o b a b i l i t i e s f o r a General Spin Hamiltonian", J . Mag. Res. 2, 69 (1970) 40. Goldstein H., "Mecanica C l a s i c a " , Aguilar, Madrid (1965) 41. Low W., "Paramagnetic Resonance", SSP Supplement II Academic Press, Hew York (1961) 42. Gerritsen H., "Paramagnetic Resonance of Transition Metal Ions i n Rutile (TiC'2)", i n Proceed, of I Intern. Conf. on Paramag. Res. (Edited by W. Low), V o l . I, p.5, Jerusalem (1962) 45. Ikebe M., Miyako Y., Date M., "ESR of Or 5* Ions Coupled with Oxygen Vacancies i n . R u t i l e " , J . Phys. Soc. Japan 26, 45 (1969) 44. Tiiyer J . R., Quick S. M., Holuj P., "ESR Spectrum of Fe^ + i n Topaz", Can. J . Phys. 45, 5597 (1967) 45. Vinokurov V. M., Zapirov M. M., Stepanov V. G., Soviet Phys. S o l i d State 6, 566 (1964) and 6, 570 (1964) 46. S c i e n t i f i c Subroutine Package (Subroutine LL3Q) IBM Corp. Publ. D i v i s i o n (1965) 46 The matrix elements of the spin Hamiltonian of equation ( l ) f o r an a r b i t r a r y o r i e n t a t i o n of the magnetic f i e l d have been calculated by using the general formulae given by Baker and Williams->-* and using the matrix elements f o r the equiva-l e n t spin operator 0° given by Abragam and Bleaney 3 2. The c a l c u l a t i o n have been made by s e t t i n g tne b^, h£, b|, b£ para-meters i n equation (1) equal to zero, 'The r o t a t i o n a l trans-formation o f the operators 0^ are also given by Thyler et a l . 4 ' and Vinokurov et a l . 4 * " The r e s u l t of this c a l c u l a t i o n i s the following:-. 1 o CD Imio C M In k -< r n 1 + m l ° ! K ^ |]K vo 1 i n i CM ^ • H + m l C M t —"—-v. 1 co k n i o ^ I C M CXJlH o t VO m H 1 m LCM ; i n rH. + . CM i n 1 rH C M j c n 1 ^ , m iCM I Lm vo 1 i n I 1. IS VO m iH 1 m o CM OJ + rH co':n 1 e> rHICvJ 1 . ^ m rH -f-t n i ^ M l i n I IS cn m l§|m rH[CM + i n [ H m m rH 1 t n [ C M j l A CM M . OJ H c o i t n I o, H J C M o vo Si ! (CM U"\ rH 1 m ICM ( O (r-t t n 1 t n f O J HQ + OJ ^ • t n 1 H CMJ i n 1 e> tnjCM *>}" ,W ICM m rH 1 t n vo k >* (CVI m rH 1 i n U4 . LCM o CO CM|rH ^ \ m"OJ CM + rH ^ + m l C M VO 4-t n vo 1° CTi +• m Into t n co h n f o ^>|CM CM|rH O tnjCM m l C M m!c\i rHICM rH|C\J m l CM K 1-b£(3cos^-l)+ b o ( l - c o s^ A ) c o s 2 ^ 6.. . IT. Kr,z (35cos^ _30cos2^> + 3 )f ££sin 4^ cos4°^ 3- 8 o 2 K^-6b2Sin^cos^+2b2(sin|b cosf cos2^+6siii|& sinSt) K,- °° [6 sin^b coS j i?-sin^cos^ ( 3 +cos^i )]+^sin^&(cosy*cos4*.+ <-sin4«C ) Kqr 3bpsin^4-1Qj? [(l+cos^ )cos2^-f-2 looses i n 2-6] K.-= -25'D^ s i r A i - p i sin 2/j, |(l+3cos 2 /a )cos4°^+t- cos 2A (3+cos 2 4 )sin4«6l ° "2Ta ' 240 ' ' ' K _=• -35b°sin^ cosy4+-bASin^ [cosy& (3 +cos^6 )cos4& + <-(l+3cos2/2> )xsin4asj Co 60 Kg; 55b^sin^^.bf ^(l^occs^^-cos 4^ )cos4«c+4'cos/i(l+coSyiJ ).sin4*6j where 6^ ) are the a z l s u t h a l and polar angles of the axis of quantization with respect to the c r y s t a l f i e l d axes, and the following well known r e l a t i o n s hold: tan^= tan $ and tan/3 =• tan 9. r-BZ APPROXIMATE CALCULATED H OP D AMD E Plie main features of the EPR spectrum of F e ^ can he described by neglecting the fourth order .terms o f the Hamilto-nian i n equation ( 5 ) . 'Thus c ^ = /?(H.g.S) + D { 3 2 - 1/3 S (3 -v 1 ) ] + ' 1 / 2 E ( 3 2 + ) (4) vhibh implies that two parameters, D and E, together \. i t h the di r e c t i o n s of the magnetic axes, are s u f f i c i e n t to characterize the spectrum. These parameters can be estimated with the help of graphs given by Aasa-^ i f the t r a n s i t i o n f i e l d s along the magnetic axes are measured. Aasa ca l c u l a t e s f o r 3=5/2 the t r a n s i t i o n f i e l d s H as a function of D, using E/D as a parameter. D and H are measured i n units of hV. Figure II i s a reproduction of those parts of Aasa's figures 1 and 2 which correspond to E/D=0.25. The. t r a n s i t i o n s f o r H p a r a l l e l to the d i r e c t i o n s within the 1 p r i n c i p a l c r y s t a l l o g r a p h i c planes nearest to .the Y and Z mag-ne t i c axes are shown. .2 .5 I 2 5. 10. gOH/hv re 11.- Positions of the SPP t r a n s i t i o n s within the upper Kramer doublets of Pe^ *". f o r S/D = 0.25 A P P E N D I X C CALCULATIONS O P THE SPIN HAMILTONIAN PARAMETER'S "ITS A LINEARIZED LEAST MEAN SQUARE P I T As discussed i n Chapter 5, the observed EPR tran-s i t i o n f i e l d s I l f / ^ - , k * l , 15 of the spectrum i n brookite are f i t to the truncated Hamiltonian (equation 4 ) v/ith a t r i a l and error method using a computer program given by Hehden et 39 a l . Then a l l the parameters are i n d i v i d u a l l y changed by a small amount p 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 from H ° are c a l c u l a t e d by diagonalising with the Hebden c a l computer programme. H v i s now defined by H,°_a l = Hf * H D • ' ' This represents a system of k l i n e a r equations i n the unknown p . The best values of p are determined by minimizing H ( H f P - H g a 1 ) 2 Ic Notice must be taken that i t may happen that the new parameters w i l l not converge n e c e s s a r i l y to some better values because of untrue l i n e a r i z a t i o n which occurs i f the H? are not exp near tne values. I t ma;/ hence be necessary to repeat the procedure several times. 

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