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Crystal chemistry of the Olivines at elevated temperatures Lager, George Adolphe 1976

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CRYSTAL CHEMISTRY OF THE OLIVINES AT ELEVATED TEMPERATURES George Adolphe Lager B.S., St. Joseph's College (ind.), 1970 M.S., V i r g i n i a Polytechnic I n s t i t u t e and State University, 1972 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of Geological Sciences We accept t h i s thesis as conforming to the required standard ' THE UNIVERSITY OF BRITISH COLUMBIA June, 1976 0 George Adolphe Lager, 1976 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d that c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f ^^tJiliJ ^uinciS The U n i v e r s i t y o f B r i t i s h Co lumbia 2 0 7 5 W e s b r o o k P l a c e V a n c o u v e r , C a n a d a V 6 T 1W5 D a t e Jtc*-* rfji i i ABSTRACT Single c r y s t a l x-ray intensity data have been collected with a manual diffractometer u t i l i z i n g flat-cone geometry f o r N i -o l i v i n e , Ni 2SiO^, (25°, 300°, 600° and 900°C), m o n t i c e l l i t e , G ai.OO M s0.93 F e0.07 S 1 0it-' ^ 2 5 ° ' 3 3 5 ° ' 6 1 5 ° a n d 7 9 5 ° G ^ a n d giaucochroite, G a0.98 M n0.87 M g0.10 Z n0.05 S i%' ( 2 5 ° ' 3 0 0 ° ' 6 0 0 ° a n d 8 0 0 ° G ) -Weighted anisotropic refinements resulted i n residual (R) factors ranging from 0.026 to 0.0^5. Examination of these data together with data from high temperature refinements of f o r s t e r i t e (Smyth and Hazen, 1973)> hortonolite (Brown and Prewitt, 1973) and f a y a l i t e (Smyth, 1975) indicates that the o l i v i n e structure expands primarily as a r e s u l t of "bond length expansions i n the non-tetrahedral s i t e s . Three-dimensional s t r a i n theory has "been used to calculate the magnitude and directions of the p r i n c i p a l axes of the thermal expansion e l l i p s o i d s f o r the polyhedra i n the above o l i v i n e s . The shapes and orientations of the thermal expansion e l l i p s o i d s f o r M(l) and M(2) octahedra are related to the topology and chemistry of the octahedra. The basic shape of the e l l i p s o i d i s a function of the s t r u c t u r a l con-figuration of the octahedron while i t s orientation i s a function of the occupancy of the s i t e . The M(2) cation position i n a l l s i x o l i v i n e s i s temperature dependent. With increasing temperature, the cation i s displaced in a dir e c t i o n away from the triangle of shared edges. The magnitude of th i s displacement, measured r e l a t i v e to the centre-id of the octahedron, i i i i s the greatest for the non-transition metal cations (Mg , Ca ). The smaller displacements exhibited by the transition cations *i"2 +2 (Ni , Fe"^) may be related to metal-metal interactions between adjacent M(l) and M(2) octahedra. The magnitude of the displacements in the room temperature structures varies almost linearly with i n -creasing cation radius for the non-Ca olivines. The thermal ex-pansion data for M(l) and M(2) octahedra suggest that Fe behaves differently than the other octahedral cations as a function of temperature. Unit c e l l expansions are interpreted in terms of the polyhedral expansions by considering different paths through the crystal structure. For the non-Ca olivines, the thermal expansion of b c e l l edge i s governed primarily by expansions in the M(2) octahedra. The unit c e l l expansions in the (Mg,Fe) olivines are consistent with a decrease M2 , Ml in the strain component T e^- a-^ v e to £, ^  with increasing Fe content. Polyhedral distortion parameters have been defined in terms of the principal strain components. The parameter, £ s = (£^ - £,-,) + 2 2 (£ ^ £y) + ( £ y £- ^ ) » which measures the shape change or distortion associated with thermal expansion, is greater for M(l) octahedra than for M(2) octahedra for the temperature range of a l l olivine refinements. A comparison of strain ellipsoids as a function of temperature and octahedral site chemistry suggests that there i s a similarity "between the structural changes due to temperature and chemical substitution in the olivines. TABLE OF CONTENTS iv Page ABSTRACT i i TABLE OF CONTENTS iv LIST OF TABLES v LIST OF FIGURES v i i ACKNOWLEDGEMENTS x I. INTRODUCTION 1 II. EXPERIMENTAL DETAILS 2 Specimen Description and Data Collection 2 Refinements ^ III. RESULTS AND DISCUSSION. lA Olivine Structure 1^ Polyhedral Expansions 17 Thermal Parameters 29 Strain Analysis of Polyhedral Expansions 3^ Octahedral Distortions ^9 Unit Cell Expansions 5+ A Comparison of Thermal and Chemical Expansions 60 IV. SUMMARY OF RESULTS 68 SELECTED REFERENCES 71 APPENDIX Is Observed and Calculated Structure Factors 75 APPENDIX II: Application of Three-Dimensional Strain Theory to an Analysis of Polyhedral Distortions in Olivines 82 V L I S T OF TABLES T a b l e Page 1 C h e m i c a l A n a l y s e s f o r M o n t i c e l l i t e and G l a u c o c h r o i t e t^-2 R - f a c t o r s and Number o f O b s e r v a t i o n s f o r R e f i n e m e n t s o f N i - O l i v i n e , M o n t i c e l l i t e and G l a u c o c h r o i t e 8 3 P o s i t i o n a l P a r a m e t e r s , I s o t r o p i c T e m p e r a t u r e F a c t o r s and C o r r e s p o n d i n g R o o t Mean S q u a r e D i s p l a c e m e n t s f o r t h e Atoms i n N i - O l i v i n e , M o n t i c e l l i t e and G l a u c o c h r o i t e 9 k A n i s o t r o p i c T e m p e r a t u r e F a c t o r C o e f f i c i e n t s (x 10**) f o r N i - O l i v i n e , M o n t i c e l l i t e and G l a u c o c h r o i t e 10 5 M a g n i t u d e s and O r i e n t a t i o n s o f t h e P r i n c i p a l A x e s o f t h e T h e r m a l V i b r a t i o n E l l i p s o i d s f o r t h e A toms i n N i - O l i v i n e , M o n t i c e l l i t e and G l a u c o c h r o i t e 11 6 I n t e r a t o m i c D i s t a n c e s (X) i n N i - O l i v i n e , M o n t i c e l l i t e and G l a u c o c h r o i t e . . 12 7 I n t e r a t o m i c A n g l e s ( ° ) i n N i - O l i v i n e , M o n t i c e l l i t e and G l a u c o c h r o i t e 13 8 L i n e a r T h e r m a l E x p a n s i o n C o e f f i c i e n t s c . ( 0 C x 10^), S l o p e s o f R e g r e s s i o n L i n e s (fi/°G x l ( r ) and C o r r e l a t i o n C o e f f i c i e n t s f o r Mean M-0 D i s t a n c e s and U n i t C e l l P a r a m e t e r s i n S i x O l i v i n e s 20 9 The M a g n i t u d e ( £ ) and R a t e o f I n c r e a s e (X/°C x 10^) o f t h e D i s p l a c e m e n t o f t h e M(2) C a t i o n w i t h I n c r e a s i n g T e m p e r a t u r e f o r S i x O l i v i n e s 26 10 M a g n i t u d e s and O r i e n t a t i o n s o f t h e P r i n c i p a l A x e s o f t h e T h e r m a l E x p a n s i o n E l l i p s o i d s f o r t h e P o l y h e d r a i n S i x O l i v i n e s 37 11 L o n g i t u d i n a l S t r a i n s C h a r a c t e r i z i n g t h e P o l y h e d r a l T h e r m a l E x p a n s i o n s i n S i x O l i v i n e s 38 v i LIST OF TABLES (Cont.) Table Page 12 Distortion Parameters Characterizing the Volume ( £ ) and Shape Changes ( £ ) Associated with the Thermal Expansion of the Octahedra in Six Olivines.. 52 13 Unit Cell Parameters ($) for Ni-Olivine, Monticellite and Glaucochroite 55 lh Magnitudes and Orientations of the Chemical Expansion Ellipsoids for the Octahedra in Seven Room Temperature Refinements of Olivines 63 15 Longitudinal Strains Characterizing the Chemical Expansion of the Octahedra in Seven Room Temperature Refinements of Olivines 16 Observed and Calculated Structure Factors for Ni-Olivine 77 17 Observed and Calculated Structure Factors for Monticellite 79 18 Observed and Calculated Structure Factors for Glaucochroite 81 v i i LIST OF FIGURES Figure Page 1 A portion of the olivine structure projected on (lOO). Bold face lines indicate oxygen-oxygen edges which are shared between adjacent polyhedra. The M(l) cation labelled i s located at x - 0.50, y = 0.50 and z = 0.50. The labelling of the atoms i s consistent , with Tables 5 and 6 2 Plot of the mean M(l)-0 distances versus temperature for Ni-olivine (Ni), forsterite (Fo), hortonolite (Ho), fayalite (Fa), monticellite (Mo) and glaucochroite (Gl). Error bars refer to one estimated standard deviation 19 3 Plot of the mean M(2)-0 distances versus temperature for six olivines. Unless otherwise stated, the abbreviations in this and subsequent figures are the same as those in Figure 2 22 k Plot of the displacement of the M(2) cation from the centroid of the octahedron versus temperature for six olivines , 25 5 Plot of the displacement of the M(2) cation from the centroid of the octahedron versus M(2) cation radius for seven room termperature olivine structures. The abbreviation Ga refers to ^-Ca 2 Si02 4 , (Brown, 1970). Cation r a d i i are from Shannon and Prewitt (196?) 27 6 Plot of the equivalent isotropic temperature factors for the M(l) cations versus temperature for six olivines. Temperature factors for forsterite are from Hazen (personal communication). 30 7 Plot of the equivalent isotropic temperature factors for the M(2) cations versus temperature for six olivines. Temperature factors for forsterite are from Hazen (personal communication) 31 v i i i LIST OF FIGURES (Cont.) Figure Page 8 Plot of the equivalent isotropic temperature factors for sil i c o n versus temperature for six olivines. Temperature factors for forsterite are from Hazen (personal communication) -^2 9 Cyclographic projection of the principal axes of the M(l) thermal expansion ellipsoids for six olivines. Ellipsoids were calculated for the temperature range of each olivine refinement. Solid, open and dotted symbols refer respectively to the longest, intermediate and shortest axes of the ellipsoids ^1 10 Cyclographic projection of the principal axes of the M(2) thermal expansion ellipsoids for six olivines. Ellipsoids were calculated for the temperature range of each olivine refinement. Solid, open and dotted symbols refer respectively to the longest, intermediate and shortest axes of the ellipsoids 1 + 2 11 Bar diagram comparing the longitudinal strains i n M(l) and M(2) octahedra to the linear thermal expansion coefficients for the c e l l parameters i n six olivines. Strains are given as "strain/l°C" in order to make comparisons between olivines refined at different temperatures. The values for the longitudinal strains are (x 10 6) ^5 12 Plot showing the deformation of M(l) and M(2) octahedra as a function of temperature for the non-Ca olivines. Open and solid symbols refer respectively to M(l) and M(2) octahedra. The differences between strains were calculated from Table 11 for each temperature interval of a given refinement. See text for further explanation 4 7 13 Plot showing the deformation of M(l) and M(2) octahedra as a function of temperature for the Ca-olivines. Open and solid symbols refer respectively to M(l) and M(2) octahedra. The differences between strains were calculated from Table 11 for each temperature interval of a given refinement. See text for further explanation 48 ix LIST OF FIGURES (Cont.) Figure Page 14 Bar diagram comparing the distortion parameters £ and £ for M(l) and M(2) octahedra in six olivines. Parameters characterize the volume and shape changes, respectively, associated with polyhedral thermal expansions 53 15 Plot of the a c e l l edge versus temperature for , six olivines 16 Plot of the b c e l l edge versus temperature for six olivines 57 17 Plot of the c c e l l edge versus temperature for six olivines 59 18 Gyclographic projection of the principal axes of the M(l) chemical expansion ellipsoids for seven room temperature olivine structures. Ellipsoids were calculated for the chemical substitutions in Table 14. Solid, open and dotted symbols refer respectively to the longest, intermediate and shortest axes of the ellipsoids ( O ^ i —* Fao» • F a 0 - F a n n , A Fa — Fa. , 0 Fa — Fa , [=• F a 5 ° 0 - W1Q0, O ^ O - G J ? V Gl 3 -°* G a ) . ? ? . . . . 6 5 19 Plot showing the deformation of M(l ) and M(2) octahedra due to chemical substitutions i n the non-Ca (19a) and Ca-olivines (l9"b). The differences between strains were calculated for the chemical substitutions in Table 15. Open and solid symbols refer respectively to M(l) and M(2) octahedra.,.. „ 67 20 The undistorted and distorted cubes refer to an infinitesimal volume in a crystal before and after deformation. Material points in the undeformed crystal (for example, 0,0,X^) are displaced u = u-jj. + U2J + uok as a result of the deformation (modified after Bloss, 197l) 84 X ACKNOWLEDGEMENTS The writer gratefully acknowledges his major advisor, Dr. E.P. Meagher, for his guidance, constructive criticism and patience throughout this study. Thanks are also extended to Dr. G.E. Brown of Stanford University for providing samples of Ni-olivine and glaucochroite and to Drs. H.J. Greenwood, T.H. Brown, K.C. McTaggart, J.V. Ross, J. Trotter of the Department of Chemistry and G.V. Gibbs of the Department of Geological Sciences, Virginia Polytechnic Institute and State University who c r i t i c a l l y reviewed the thesis. The writer wishes to thank in particular Dr. G.V. Gibbs for thoughtful reading of the manuscript and helpful suggestions for i t s improvement. The section on strain theory benefitted from discussions with Drs. J.V. Ross, J. Ramsey of the Department of Mechanical Engineering and John Schlenker of the Department of Geological Sciences, Virginia Polytechnic Institute and State University. 1 I. INTRODUCTION Since the determination of the structure of olivine by Bragg and Brown in 1926, a number of precise crystal structure refinements have been undertaken. A review of room temperature refinements prior to 1971 is given by Brown (1970), who examined a number of olivine-type minerals ranging in composition from Ni^SiO^ to Ca^SiO^. More recent room temperature investigations include those of Wenk and Raymond (1973) ( F a 0 1> F & 1 0 > F a ^ , F a ^ ) , Wan and Ghose (1973) (Co 1 > 1 0Mg 0SiO^), Brown and Prewitt (1973) ( F a ^ , Fa 2 ?) and Rajamani, Brown and Prewitt (1974) (Ni^ Q^SQ ^ ^ i O ^ ) . Crystal structure refinements at high temperatures have been limited to compositions ranging from forsterite to fayalite (Brown and Prewitt, 1973: F a 3 Q ; Smyth and Hazen, 1973: Fa Q and Mg^^Fe^QMn^^SiO^; Smyth, 1975: ^ 1 0 Q ) • In the present investigation, the crystal structures of Ni-olivine (NigSiO^), monticellite (Ca^ QQ^SQ y/^'eo Q ^ ^ L ^ AN<^ glaucochroite (Ca^ n-Q^ o Q^SQ io^NO 05^ have been refined with data collected at temperatures ranging from 25° to 900°C. The com-plicated polymorphic relations of Ca-olivine (Smith et a l . , 196l, 19&5) -1 In this study, the term olivine-type refers not only to the forsterite (Mg^SiO^) - fayalite (Fe^SiO^) group but to a l l silicates isostructural with forsterite/fayalite. Ni^SiO^ represent end-members in this group of minerals with respect fo the size of the octahedral cations. N i + 2 and C a + 2 are, respectively, the smallest and largest cations that have been found to date to f u l l y occupy the octahedral sites. 2 prevented a high temperature structure determination of this mineral. The above refinements of high temperature data w i l l he examined to-gether with those of forsterite (Smyth and Hazen, 1973), hortonolite (Brown and Prewitt, 1973) and fayalite (Smyth, 1975). Only those olivines which do not show a change in the state of order upon heatin, are considered. In addition to characterizing the response of the olivine-type structures to elevated temperatures, strain analyses of the polyhedra within these structures are compared as a function of "both chemical occupancy and temperature and related to the unit c e l l thermal ex-pansion. It i s believed that studies of this type contribute not only to our understanding of the olivine-type minerals but provide some insight into the high temperature crystal chemistry and the mechanisms of thermal expansion in other s i l i c a t e groups as well. II. EXPERIMENTAL DETAILS Specimen Description and Data Collection With the exception of monticellite, the crystals used in this study were the same as those examined by Brown (1970). The monticellite crystal, which was picked from a specimen from Crestmore, California, was kindly donated for this study by the Department of Geological Sciences at the University of Br i t i s h Columbia. The chemical analysis of the synthetic crystal of Ni-olivine has been reported as pure NigSiO^ (Brown, 1970). The natural crystals of monticellite and glaucochroite were chemically analyzed on a ARL-3 SEMQ electron microprobe by averaging approximately 15 compositions over each crystal. The data were reduced using the empirical correction factors determined by Bence and Albee (1968). The mean chemical analyses and the estimated standard deviations of the mean values are presented in Table 1. Equidimensional crystals approximately 0.05 - 0.10 mm in diameter were oriented parallel to either c or a in tapered fused-s i l i c a capillaries which were then cemented to a platinum wire attached to the goniometer. In the case of glaucochroite, heating experiments +2 prior to data collection indicated probable oxidation of Mn . As a result, the crystal was mounted directly on a tapered s i l i c a capillary using a high temperature cement composed of Sauereisen binder no. 29 2 and powdered kaowool f i l l e r . The crystal mounted on the capillary was then inserted into a larger capillary of approximately 0.3 mm in diameter which was flushed with hydrogen gas, evacuated with a vacuum pump and sealed under vacuum using an oxy-acetylene mini-torch. The furnace used in this study was modified from the design of Foit and Peacor (1967). The furnace, which requires flat-cone geometry, has been adapted to a Weissenberg camera for space group and c e l l dimension work and to a manually-operated Supper-Weissenberg diffractometer for three-dimensional single crystal intensity data collection. The diffractometer i s used with Zr-filtered Mo-K<t radiation and a pulse height discriminator. The furnace temperature Babcock and Wilcox Refractories, Burlington, Ontario. 4 Table 1. Chemical Analyses for Monticellite and Glaucochroite. Oxide Weight % Oxides Monticellite Glaucochroite MgO 23.61(20) 2.27(20) SiO 37.28(26) 33.40(20) CaO 35.37(24) 29.99(22) MnO - 33.29C24) ZnO - 2.35(35) FeO 3-22(18) -Total 99.48 101.30 Number of Atoms Per 4 Oxygens Mg 0.936 0.103 S i 0.992 1.000 Ca 1.008 0.979 Mn - 0.866 Zn - 0.052 Fe 0.071 -5 was calibrated with a Pt/Pt-13% Rh thermocouple which was centered in the furnace in place of the sample "before each high temperature run. The estimated accuracy of the temperature, based on the thermocouple measurements and the melting points of Zn(4l9°C) and KBr (730°C), is t 15°G. The unit c e l l parameters for each olivine at high temperature were determined from a least-squares refinement of film data using a cassette modified for the Weissenberg camera (Meagher, 1975)- In addition to the O-level Weissenberg photographs used for c e l l dimension work, upper level flat-cone photographs were also recorded at each temperature. These photographs were checked for systematic absences and found to be consistent with space group Pbnm in a l l cases. With the exception of glaucochroite, the same crystal was used for c e l l parameters and intensity collection. Approximately 250 symmetry non-equivalent reflections of sin < 0.5 were collected with a manual Weissenberg s c i n t i l l a t i o n counter diffractometer for Ni-olivine (25°, 300°, 600° and 900°), monticellite (25°, 335°, 615° and 795°) and glaucochroite (25°, 300°,600° and 800°). For each crystal, data were recollected at room temperature after heating to determine i f any permanent changes had occurred as a result of the heating cycle. The small number of observable re-flections indicates both the small size of the crystals and the geometric restrictions imposed by the design of the furnace and diffractometer. The latter factor imposed an upper limit on the value of the /( angle which resulted in the loss of reflections of 6 the type X= 6 for Ni-olivine and monticellite and h = 5 for glaucochroite. Each reflection was scanned and traced on a str i p chart recorder. Relative intensities were determined using an integra-tion planimeter and reduced to J F o ^ s | using a program which corrects for Lorentz and polarization effects. Transmission factors were calculated for each crystal and indicated no appreciable absorption (Burnham, 1966). Refinements F u l l matrix least-squares refinements were calculated for each data set using a modified version of ORFLS (Busing, Martin and Levy, 1962) and the neutral atomic scattering curves of Doyle and Turner (1968). Starting positional and thermal parameters were taken from Brown (1970). The i n i t i a l refinements on atomic positional parameters and isotropic temperature factors were calculated u t i l i z i n g unit weights. Upon convergence, the isotropic temperature factors were converted to the anisotropic form and the refinement was continued with a modified version of the Gruickshank weighting scheme (Cruickshank, 1965). Anisotropic refinements of each olivine at room temperature resulted in a number of non-positive temperature factors. In the case of Ni-olivine, a strong correlation was noted between one of the scale factors and the negative temperature factor coefficient, ^>jy of M(l). When this scale factor was held constant during the f i n a l refinement, the correlation was reduced and B^^ refined to a positive value. For 7 m o n t i c e l l i t e and g l a u c o c h r o i t e , i s o t r o p i c t e m p e r a t u r e f a c t o r s were c o n v e r t e d t o t h e a n i s o t r o p i c f o r m h u t t h e v a l u e s o f f ° r M ( l ) i n m o n t i c e l l i t e and B ^ f o r 0 ( l ) i n g l a u c o c h r o i t e were n o t r e f i n e d . The p o s i t i o n a l p a r a m e t e r s o b t a i n e d i n t h e r e f i n e m e n t s w h i c h r e s u l t e d i n n o n - p o s i t i v e B ' s were i d e n t i c a l to t h o s e o b t a i n e d i n t h e r e f i n e m e n t s i n w h i c h s c a l e f a c t o r s o r t e m p e r a t u r e f a c t o r s had b e e n l o c k e d . I n t h e r e f i n e m e n t o f N i - o l i v i n e , l a r g e n e g a t i v e d i f f e r e n c e s f o r t h e s t r o n g l o w a n g l e r e f l e c t i o n s were n o t e d . A s a r e s u l t , a s e c o n d a r y e x t i n c t i o n p a r a m e t e r o f t h e f o r m F - F obs c a l c / F \2 E x t = sumw ( F , - c a l c ; O D S 1 + E I was i n c l u d e d i n t h e r e f i n e m e n t , where EXT i s t h e f u n c t i o n b e i n g m i n i m i z e d , w i s t h e w e i g h t a s s i g n e d t o e a c h r e f l e c t i o n , E i s t h e e x -t i n c t i o n c o e f f i c i e n t and I i s t h e u n c o r r e c t e d i n t e n s i t y . The f i n a l w e i g h t e d (R ) and u n w e i g h t e d (R) r e s i d u a l s f o r t h e w a n i s o t r o p i c r e f i n e m e n t s and t h e number o f o b s e r v a t i o n s u s e d i n e a c h r e f i n e m e n t a r e p r e s e n t e d i n T a b l e 2. The f i n a l p o s i t i o n a l p a r a m e t e r s , t h e i s o t r o p i c e q u i v a l e n t s o f t h e a n i s o t r o p i c t e m p e r a t u r e f a c t o r s and t h e i r c o r r e s p o n d i n g r o o t - m e a n - s q u a r e d i s p l a c e m e n t s a r e g i v e n i n T a b l e 3. The a n i s o t r o p i c t e m p e r a t u r e f a c t o r c o e f f i c i e n t s a r e p r e s e n t e d i n T a b l e h and t h e d a t a f o r t h e t h e r m a l v i b r a t i o n e l l i p s o i d s a r e g i v e n i n T a b l e 5» The o b s e r v e d and c a l c u l a t e d s t r u c t u r e a m p l i t u d e s a r e compared i n A p p e n d i x I ( T a b l e s 16, 17, 18). P o s i t i o n a l p a r a m e t e r s o b t a i n e d f r o m r e f i n e m e n t s o f d a t a c o l l e c t e d b e f o r e a n d a f t e r h e a t i n g a r e s t a t i s t i c a l l y i d e n t i c a l and a r e w i t h i n 2 <T o f t h o s e r e p o r t e d by 8 Table 2. R-factors and Number of Observations for Refinements of Ni-Olivine, Monticellite and Glaucochroite. Ni-olivine Number of Observations R* R ** w K 228 • 037 .045 300 226 .032 .048 600° 233 .027 .036 900° 230 .027 .036 Monticellite 25° 223 .030 .039 3 3 5 o 228 .025 .032 625° 226 .028 .026 795 220 .029 .032 Glaucochroite 233 .024 .026 300° 230 .024 .026 600° n 213 .030 .033 800° 201 .028 .029 * R= £ obs F R = w [£»(| obs calc F calc F obs wF obs Table J. Positional Parameters, Isotropic Temperature Factors and Corresponding Root Mean Square Displacements for Ni-Olivine, Monticellite and Glaucochroite. Atoa Parameter 25°C Ni-Olivine 300°C 600°C 900°C 25°C Monticellite 335°C 615°C 795°C 25°C Glaucochroite 300°C 600°C 800°C M(l) X y z B " 0.0 0.0 0.0 0.34(2)* 0.066 0.0 0.0 0.0 0.64(2) 0.090 0.0 0.0 0.0 0.97(2) 0.111 0.0 0.0 0.0 1.43(2) 0.135 0 0 0 0 0 0 0 0 29 (1) 061 0 0 0 0 0 0 0 0 74(3) 097 0.0 0.0 0.0 1.42(4) 0.134 0 0 0 1 0 0 0 0 92(4) 156 0 0 0 0 0 0 0 0 54(3) 083 0 0 0 1 0 0 0 0 16(3) 121 0.0 0.0 0.0 1.95(4) 0.157 0 0 0 2 0 0 0 0 51(4) 178 M(2> X y z B <„> 0.9924(2) 0.2738(1) 0.25 0.34(2) 0.066 0.9925(3) 0.2745(1) 0.25 0.66(2) 0.091 0.9922(2) 0.2746(1) 0.25 1.01(2) 0.113 0.9919(2) 0.2750(1) 0.25 1.47(2) 0.136 0 0 0 0 0 9771(3) 2766(1) 25 38(2) 069 0 0 0 0 0 9764(2) 2774(1) 25 88(2) 106 0.9759(3) 0.2781(1) 0.25 1.44(3) 0.135 0 0 0 1 0 9753(3) 2789(1) 25 93(3) 156 0 0 0 0 0 9803(5) 2780(1) 25 41(4) 072 0 0 0 0 0 9797(5) 2789(1) 25 85(4) 104 0.9805(6) 0.2797(2) 0.25 1.48(5) 0.137 0 0 0 1 0 9806(6) 2802(2) 25 90(5) 155 Si X y z B <y> 0.4276(4) 0.0944(2) 0.25 0.29(3) 0.061 0.4272 0.0947 0.25 0.52(4) 0.081 0.4276(4) 0.0941(2) 0.25 0.70(3) 0.094 0.4277(4) 0.0943(2) 0.25 1.00(3) 0.113 0 0 0 0 0 4110(3) 0815(1) 25 28(3) 060 0 0 0 0 0 4091(3) 0817(1) 25 62(3) 089 0.4087(3) 0.0824(2) 0.25 1.07(3) 0.116 0 0 0 1 0 4076(4) 0829(2) 25 42(3) 134 0 0 0 0 0 4161(5) 0868(2) 25 39(5) 070 0 0 0 0 0 4150(5) 0875(2) 25 80(5) 101 0.4156(7) 0.0880(3) 0.25 1.30(7) 0.128 0 0 0 1 0 4153(7) 0895(3) 25 72(7) 148 0(1) y z B <y> 0.7703(9) 0.0935(6) 0.25 0.53(8) 0.082 0.7697(13) 0.0923(7) 0.25 1.02(11) 0.114 0.7693 0.0930 0.25 1.18(10) 0.122 0.7682(10) 0.0934(6) 0.25 1.67(9) 0.145 0 0 0 0 0 7460(9) 0768(4) 25 45(8) 075 0 0 0 1 0 7434(8) 0776(4) 25 24(8) 125 0.7428(10) 0.0785(5) 0.25 2.00(10) 0.159 0 0 0 2 0 7398(10) 0785(5) 25 66(11) 184 0 0 0 0 0 7466(14) 0843(5) 25 61(12) 088 0 0 0 1 0 7452(14) 0841(5) 25 27(14) 127 0.7429(18) 0.0840(6) 0.25 1.83(18) 0.152 0 0 0 2 0 7378(19) 0840(7) 25 33(20) 172 0(2) X y z B <li> 0.2197(11) 0.4455(6) 0.25 0.36(8) 0.068 0.2187(14) 0.4454(7) 0.25 0.97(11) 0.111 0.2163(13) 0.4459(6) 0.25 1.07(10) 0.116 0.2166(11) 0.4464(5) 0.25 1.56(8) 0.141 0 0 0 0 0 2466(8) 4491(4) 25 54(7) 083 0 0 0 1 0 2476(8) 4501(3) 25 01(5) 113 0.2463(9) 0.4509(4) 0.25 1.47(8) 0.136 0 0 0 1 0 2463(9) 4521(4) 25 86(9) 153 0 0 0 0 0 2305(14) 4528(5) 25 60(6) 087 0 0 0 0 0 2277(14) 4542(5) 25 92(12) 108 0.2312(20) 0.4555(6) 0.25 1.80(18) 0.151 0 0 0 2 0 2311(19) 4549(6) 25 20(18) 167 0(3) X y B 0.2754(8) 0.1633(4) 0.0310(10) 0.50(6) 0.080 0.2770(10) 0.1632(5) 0.0324(12) 1.00(8) 0.113 0.2778(9) 0.1630(4) 0.0328(11) 1.36(7) 0.131 0.2787(8) 0.1631(4) 0.0333(9) 1.74(6) 0.148 0 0 0 0 0 2725(6) 1466(3) 0451(7) 55(5) 083 0 0 0 1 0 2738(6) 1473(3) 0472(6) 02(5) 114 0.2731(6) 0.1476(3) 0.0472(7) 1.63(6) 0.144 0 0 0 2 0 2729(7) 1479(3) 0477(8) 12(6) 164 0 0 0 0 0 2818(9) 1527(3) 0493(6) 58(9) 086 0 0 0 1 0 2827(9) 1530(3) 0500(6) 06(9) 116 0.2819(11) 0.1539(4) 0.0519(8) 1.74(12) 0.148 0 0 0 2 0 2831(12) 1539(4) 0526(8) 22(13) 168 * The numbers in parentheses in this and subsequent tables are estimated standard deviations and refer to the last decimal place cited. ** Isotropic equivalents of the anisotropic temperature factors (Hamilton, 1959). Table 4 . Anisotropic Temperature Factor Coefficients (x 10 ) for Ni-Olivine, Monticellite and Glaucochroite. 25 C 5 5 ( 5 ) 6 ( 1 ) 2 0 ( 9 ) - 0 ( 1 1 ) - 2 ( 2 ) - 3 ( 1 ) 6 0 ( 5 ) 1 ( 1 ) 3 2 ( 5 ) - K l ) 5 1 ( 8 ) 2 ( 1 ) 2 5 ( 8 ) 1 ( 3 ) 2 6 ( 1 7 ) 1 1 ( 5 ) 6 5 ( 2 3 ) - 3 ( 8 ) 8 5 ( 2 0 ) 3 ( 4 ) 1 4 ( 2 4 ) 7 ( 7 ) 7 3 ( 1 4 ) 1 2 ( 3 ) 2 5 ( 1 6 ) - 2 ( 5 ) 1 0 ( 1 1 ) 1 (6) 7 6 ( 6 ) 1 0 4 ( 5 ) 1 3 3 ( 4 ) 2 4 ( 7 ) 1 9 ( 1 ) 2 9 ( 1 ) 3 9 ( 1 ) 3 4 ( 9 ) 5 6 ( 1 1 ) 1 0 0 ( 6 ) 22 - 2 ( 2 ) - 5 ( 2 ) - 4 ( 2 ) - 3 ( 2 ) - 5 ( 3 ) - 7 ( 3 ) - 1 2 ( 3 ) - 7 ( 5 ) - 5 ( 2 ) - 8 ( 1 ) - 9 ( 1 ) - ! ( 2 ) 8 4 ( 6 ) 1 2 5 ( 5 ) 1 6 2 ( 5 ) 58(6) 1 5 ( 1 ) 2 2 ( 1 ) 2 9 ( 1 ) 9 ( 1 ) 4 3 ( 6 ) 7 1 ( 5 ) 1 2 0 ( 5 ) 1 1 ( 5 ) - 1 ( 2 ) 1 ( 2 ) 1 ( 1 ) 1 ( 2 ) 3 6 ( 1 0 ) 5 4 ( 9 ) 7 6 (7 ) - 4 ( 7 ) 1 2 ( 2 ) 1 6 ( 2 ) 2 4 ( 1 ) 7(1) 5 1 ( 1 1 ) 6 6 ( 1 0 ) 9 1 ( 8 ) 18 (7 ) 1 ( 3 ) 8 ( 3 ) - 0 ( 2 ) - 2 ( 2 ) 7 9 ( 2 5 ) 2 2 ( 6 ) 1 0 4 ( 3 2 ) 1 ( 1 2 ) 9 3 ( 2 6 ) 1 0 ( 5 ) 1 1 8 ( 3 6 ) 1 7 ( 9 ) 9 8 ( 1 8 ) 3 5 ( 4 ) 4 7 ( 2 1 ) - 8 ( 7 ) - 4 ( 1 4 ) 1 3 ( 7 ) 1 0 8 ( 2 2 ) 3 5 ( 5 ) 7 9 ( 2 8 ) - 1 ( 1 0 ) 1 7 1 ( 2 4 ) 1 8 ( 5 ) 6 5 ( 2 6 ) 2 5 ( 9 ) 1 4 8 ( 1 6 ) 3 5 ( 4 ) 9 1 ( 1 9 ) 5 ( 6 ) - 9 ( 1 3 ) 5 ( 6 ) 1 0 5 ( 1 9 ) 4 8 ( 5 ) 1 4 4 ( 2 4 ) - 5 ( 8 ) 2 0 4 ( 2 1 ) 2 2 ( 4 ) 1 3 3 ( 2 5 ) 2 1 ( 7 ) 1 8 1 ( 1 5 ) 4 8 ( 3 ) 1 1 0 ( 1 6 ) - 1 ( 6 ) - 1 8 ( 1 1 ) 1 1 ( 6 ) 9 ( 1 9 ) - 5 ( 6 ) ' 7 ( 1 7 ) 4 ( 3 ) 7 1 ( 2 0 ) 6 ( 6 ) 4 4 ( 1 2 ) 1 6 ( 2 ) 2 9 ( 1 1 ) 8 ( 4 ) - 8 ( 9 ) 2 ( 4 ) 9 2 ( 7 ) 1 7 ( 1 ) 1 1 ( 9 ) 4 9 ( 6 ) S(2> 2(21 32(15) 12(41 112(19) -8(6 1 1 1 7 ( 1 8 ) 1 4 ( 3 ) 7 5 ( 1 8 ) 1 0 ( 6 ) 9 5 ( 1 1 ) 2 4 ( 2 ) 6 0 ( 1 0 ) - 0 ( 4 ) 1 4 ( 8 ) 1 0 ( 4 ) elliti. 615UC 157(8) 13(2) 67(1!) -12(3) -28(6) 1(11 175(61 24(1 1 8H (*!) 22 (11 X 1(81 - 2 ( 2 ) 88(181 49(5) 163(23) -6 (7) 149(20) 23(4) 1 1 2 ( 2 0 ) 1 1 ( 7 ) 1 7 5 ( 1 3 ) 3 3 ( 2 ) 9 6 ( 1 2 ) 2 ( 5 ) - 1 ( 1 0 ) 1 1 ( 5 ) 166(8) 4 1 ( 2 ) 130(17) - 1 3 ( 3 ) -42(7) - i ( ) I 3 2 ( 1 ) 1 3 0 ( 1 0 ) 107(7) 79(1 1 1 08( 11 ) - H i ) 137(21) 61 ( 5 ) 2 1 8 ( 2 8 ) - 0 ( 8 ) 1 4 7 ( 2 0 ) 3 0 ( 4 ) 1 6 2 ( 2 3 ) 1 ( 7 ) 2 3 2 ( 1 4 ) 3 8 ( 3 ) 1 3 5 ( 1 4 ) 9 ( 5 ) - 5 ( 1 1 ) 1 6 ( 5 ) 2 5 C 3 7 ( 8 ) 1 5 ( 1 ) 3 1 ( 2 ) - 3 ( 2 ) - 7 ( 4 ) - i ( l ) 3 9 ( 1 1 ) 1 0 ( 1 ) 2 0 ( 3 ) - 1 ( 2 ) 2 3 ( 1 4 ) 9 ( 1 ) S 0 ( 5 ) - 5 ( 3 ) 3 6 ( 3 3 ) 1 6 ( 4 ) 4 1 ( 1 2 ) - 7 ( 1 1 ) 52 8 ( 3 ) 5 3 ( 1 1 ) 1 0 ( 9 ) 5 6 ( 2 3 ) 1 4 ( 3 ) 2 9 ( 7 ) 2 ( 5 ) 1 ( 9 ) 3 ( 4 ) G l a u c o c h r o i t e 3 0 0 ° C 6 0 0 ° C 1 1 7 ( 7 ) 2 7 ( 1 ) 5 9 ( 2 ) - 5 ( 2 ) - 2 5 ( 4 ) - 5 ( 1 ) 1 1 1 ( 9 ) 1 6 ( 1 ) 4 0 ( 3 ) - 2 ( 2 ) 9 3 ( 1 3 ) 1 7 ( 2 ) 3 7 ( 4 ) - 3 ( 3 ) 1 2 8 ( 3 5 ) 2 8 ( 5 ) 7 1 ( 1 3 ) 4 ( 1 1 ) 5 7 ( 3 1 ) 2 2 ( 4 ) 6 6 ( 1 1 ) 4 ( 8 ) 1 0 6 ( 2 3 ) 2 5 ( 3 ) 5 3 ( 8 ) 6 ( 6 ) 5 ( 1 0 ) 0 ( 4 ) 0 f o r M(2), S I , 0 ( 1 ) a n d 0 ( 2 ) . 1 9 9 ( 9 ) 4 6 ( 1 ) 9 4 ( 3 ) - 1 2 ( 3 ) - 2 9 ( 5 ) - 1 0 ( 2 ) 1 7 5 ( 1 2 ) 2 7 ( 1 ) 8 1 ( 4 ) 6 ( 3 ) 1 4 8 ( 1 8 ) 2 7 ( 2 ) 6 4 ( 6 ) - 6 ( 4 ) 1 4 6 ( 4 4 ) 3 9 ( 7 ) 1 2 3 ( 1 8 ) 1 2 ( 1 5 ) 2 2 3 ( 4 6 ) 3 1 ( 6 ) 9 9 ( 1 5 ) 2 ( 1 2 ) 1 7 4 ( 2 9 ) 3 1 ( 4 ) 1 1 4 ( 1 1 ) - 1 ( 7 ) 4 ( 1 4 ) 1 2 ( 5 ) 800 c 246(10) 60(1 ) 121(4) -14(4) - 4 6 ( 6 ) -11(21 239(14) 34(1 ) 9 6 ( 4 ) - 0 ( 4 ) 1 6 6 ( ] 9 ) 35(2 J 1 0 1 ( 7 ) -12(51 239(49) 5 6 ( 7 ) 106(18) 7 1(17) 231(46) 3 4 ( 6 ) 1 54 ( 1 8 ) -17(1 1) 250(32) 5 2 ( 5 ) 9 3 ( 1 0 ) 9(8) 0 ( 1 5 ) 1 8 ( 5 ) Table 5 . Magnitudes and Orientations of the Principal Axes of the Thermal Vibration Ellipsoid; for the Atoms i n Ni-Olivine, Monticellite and Glaucochroite. Angle (°) of T± with Angle (<•) of 1^  with Angle (°) of T± with Angle (°) of r± with Angle (") of T± with Angle (°) of i± with Olivine I°C i 0 + * +b +c r.(5t) +a +b + a •H> +b rt(i) +b *c r.(i-) -H, Ni-Olivine 25 .048(7) 88(4) 42(30) 48(21) .024(11) 87(3) 3(3) 90 .030(12) 93(8) 3(8) 90 .053(19) 14 OB) 76(33) 90 .033(34) 101(11) 11(11) 90 .061(21) 108(19) 99(36) 20(25) .064(7) 86(9) 132(30) 42(29) .074(6) 90 90 180 .064(11) 90 90 180 .075(16) 76(38) 166(38) 90 .048(44) 90 90 180 .080(10) 96(31) 168(30) 101(34) .079(4) 5<8) 89(6) 95(8) .083(4) 3(3) 93(3) 90 .076(6) 3(8) 87(8) 90 .106(19) 90 90 0 .100(11) 11(11) 79(11) 90 .093(8) 19(19) 99(28) 73(20) 300 .074(9) 75(11) 68(8) 27(12) .087(6) 90 90 0 .062(9) 6(18) 96(18) 90 .095(15) 10(73) 100(73) 90 .052(25) 118(12) 28(12) 90 .083(21) 90(27) 109(12) 19(12) .093(4) 162(11) 94(16) 72(11) .089(4) 99(14) 171(14) 90 .080(6) 96(18) 174(1B) 90 .106(15) 100(73) 170(73) 90 152(12) 118(12) 90 .104(10) 167(11) 103(15) 94(26) .102(3) 100(14) 22(7) 110(10) .098(4) 9(14) 99(14) 90 .096(10) 90 90 0 .135(21) 90 90 0 .145(22) 90 90 0 .142(9) 103(11) 23(10) 71(10) 600 .092(8) 69<11> 68(5) 32(11) .108(2) 96(9) 6(9) 90 .072(7) 150(B) 60(B) 90 .111(12) 3(24) 87(24) 90 .081(15) 116(7) 26(7) 90 .119(10) 123(27) 63(20) 135(31) .111(3) 157{9) 91(10) 67(10) .113(4) 90 90 180 .098(5) 120(8) 150(8) 90 .119(21) 90 90 180 .108(22) 90 90 180 .134 (9) 39(39) 92(68) 129(37) .127(2) 100(6) 22(5) 110(7) .120(2) 6(9) 84(9) 90 .109(8) 90 90 0 .135(11) 93(24) 3(24) 90 .150(10) 26(7) 64(7) 90 .139(8) 72(57) 27(19) 70(54) 900 .117(2) 38(7) 71(3) 5B(6) .124(2) 96(6) 6(6) 90 .093(5) 0(8) 90(8) 90 .110(10) 5(9) 85(9) 90 .100(10) 111(7) 21(6) 90 .130(9) 60(18) 108(11) 36(14) .134(3) 52<7) 116(8) 131(4) .137(2) 174(6) 96(6) 90 .113(3) 90(B) 180(8) 90 .159(8) 85(9) 175(9) 90 .155(14) 90 90 180 .148(6) 146(19) 115(18) 69(17) .151(2) H9(4) 33(6) 123(6) .147(3) 90 90 0 .128(5) 90 90 0 .161(14) 90 90 0 .160(8) 21(6) 69(6) 90 .165 (6) 103(14) 32(15) 62(12) Honticellite 25 .043(10) 37(13) 63(14) 67(11) .047(12) 90 90 0 .050(8) 21(19) 69(19) 90 .044(44) 90 90 0 .038(26) 131(28) 41(28) 90 .059(13) 36(20) 110(10) 62(25) .066(7) 64(39) 153(17) 81(70) .073(4) 76(16) 166(16) 90 .061(12) 90 180 .053(18) 172(9) 98(9) 90 .067(15) 139(28) 131(28) 90 .082(12) 62(25) 96(20) 151(25) .070(4) 115(37) 93(69) 25(28) .083(4) 14<16) 76(16) 90 .066(6) 111(19) 21(19) 90 .112(10) 98(9) 8(9) 90 .121(17) 90 90 0 .104(7) 69(12) 21(10) 86(19) 335 .066(11) 60(7) 81(6) 31(8) .096(3) 100(5) 10(5) 90 .070(6) 7(7) 97(7) 90 .060(15) 7(5) 83(5) 90 .088(11) 114(12) 24(12) 90 .093(9) 54(19) 66(12) 134(13) .102(4) 107(13) 153(12) 70(8) .101(6) 90 90 180 97(7) 173(7) .142(8) 83(5) 173(5) 90 .123(9) 156(12) 114(12) 90 .111(7) 38(20) 68(15) .116(4) 145(7) 65(13) 66(9) .118(3) 10(5) 80(5) 90 .098(8) 90 90 0 .153(13) 90 90 0 .125(15) 90 90 0 .133(6) 79(11) 40(12) 52(10) 615 .106(7) 57(8) 81(6) 35(9) .122(2) 107(4) 17(4) 90 -099(4) 8(9) 82(9) 90 .102(11) 4(5) 86(5) 90 .114(10) 122(15) 32(15) 90 .12B(8) 90(19) 48(11) 137(12) .136(4) 121(10) 135(B) 61(8) .136(5) 90 90 180 .117(3) 82(9) 172(9) 90 .177(8) 86(5) 176(5) 90 .140(9) 148(15) 122(15) 90 .144(5) 171(19) 10(23! 81(19) .156(4) 131(6) 46(B) 72(7) .146(3) 17(4) 73(4) 90 .132(6) 90 90 0 .185(13) 90 90 0 .153(14) 90 90 0 .156(7) 84(23) 42(11) 49(12) 795 .120(5) 34(4) 70(4) 63(6) .140(2) 108(5) 18(5) 90 .113(4) 7(7) 83(7) 90 .128(10) 0(5) 90(5) 90 •133(9) 10(77) 100(77) 90 .140(7) 106(10) 37(8) 123(9) .164(4) 81(12) 155(9) 68(15) .162(2) 162(5) 108(5) 90 .135(3) 83(7) 173(7) 90 .197(9) 90(5) 180(5) 90 .138(9) 100(77) 170(77) 90 .168(5) 161(14) 98(18) 73(20) 3 .178(7) 123(7) 76(16) 36(14) .165(1) 90 90 0 .151(8) 90 90 0 .214(14) 90 90 0 .1B5(13) 90 90 0 .180(7) 81(24) 54(9) 38(14) G ochroite 25 1 .061(8) 29(10) 78(6) 64(9) .066(4) 90 90 0 .047(17) 22(13) 68(13) 90 .062(32) 17(25) 73(25) 90 .053(25) 128(13) 38(13) 90 .077(10) 93(91) 107(29) 17(21) 2 .083(4) 61(10) 107(9) 146(10) .068(9) 172(20) 98(20) 90 .079(6) 68(13) 158(13) 90 .094(14) 90 90 180 .092(14) 142(13) 128(13) 90 .082(17) 167(43) 77(44) B9(90) 3 -099O) 93(7) 21(7) 111(7) .081(4) 98(20) 8(20) 90 .080(6) 90 90 0 .102(14) 107(25) 17(25) 90 .106(11) 90 90 0 .097(9) 77(36) 22(28) 73(21) 300 1 .094(4) 53(3) 75(3) 41(3) .093(3) 90 90 0 .090(5) 90 90 0 .124(11) 90 90 0 .082(23) 8(19) 98(19) 90 .104(8) 118(49) 82(24) 29(53) 2 .132(3) 52(44) 139(62) 103(43) .100(3) 98(10) 172(10) 90 127(32) 143(32) 90 .124(16) 157(68) 67(68) 90 .119(10) 90 180 .113(10) 39(40) 114(29) 61(54) 3 .134(3) 59(49) 53(64) 128(19) •117(5) 8(10) 98(10) 90 .110(6) 143(32) 53(32) 90 .134(12) 67(68) 23(68) 90 .119(11) 82(19) 8(19) 90 .129(7) 65(26) 26(27) 85(16) 600 1 .125(4) 58(3) 70(3) 39(3) .129(4) 108(10) 18(10) 90 .118(5) 90 90 0 .129(20) 22(26) 112(26) 90 .140(12) 94(25) 4(25) 90 .131(9) 84(27) 32(12) 121(10) 39(8) 79(13) 127(4) -133(3) 90 90 1B0 .126(6) 127(24) 143(24) 90 112(26) 158(26) 90 .146(11) 90 90 180 .1*7(12) 173(26) 83(26) 89(23) 3 .177(3) 110(11) 23(7) 99(8) .149(5) 18(10) 72(10) 90 .140(6) 143(24) 53(24) 90 .163(12) 90 90 0 .166(17) 4(25) 86(25) 90 .165(8) 86(23) 59(11) 31(10) 800 1 .137(4) 54(3) 74(2) 41(3) .144(3) 90 90 0 •135(7) 37(13) 53(13) 90 .151(13) 90 90 0 .138(14) 63(17) 27(17) 90 .132(9) 86(10) 113(6) 24(7) 2 .190(3) 41(6) 81(12) 130(3) .148(3) 91(8) 179(8) .148(5) 90 90 180 .160(17) 32(19) 122(19) 90 .177(15) 27(17) 117(17) 90 .174(10) 158(21) 72(21) 78(11) 3 .201(3) 108(10) 19(7) 97(8) •172(5) KB) 91(8) 90 .158(5) 127(13) 37(13) 90 .200(14) 58(19) 32(19) 90 .183(11) 90 90 0 .193(8) 68 (22) 30(15) 70(7) Table 6. Interatomic Distances (£) in Ni-Olivine, Monticellite and Glaucochroi te. N i - O l i v i n e 300°c Tetrahedron M Si-O(l) 11] Si-0(2) [2] Si-0(3) <Si-0> 0(1)-0(2) 0(l)-0(3) O(2)-0(3) a 0(3)-0(3) a <0-0> M(l) Octahedron [2] M(l)-0(1) [2j M(l)-0(2) [2] M(l)-0(3) M o n t i c e l l i t e 335°C 615°C '2 0(1) -0(3)" 2 0(1) -0(3') 2 0(1) -0(2) b '2 0(1) -0(2') '2' 0(2) -0(3') X. 0(2) 0 ( 3 ) a <0-0> M(2) Octahedron [2: [2: M(2)-0(l) M(2)-0(2) »(2)-0(3) M(2)-0(3") <M(2)-0> 2] 0 ( l ) - 0 ( 3 " ) " 0 ( l ) - 0 ( 3 ) " 0(2)-0(3) 0(2)-0(3"') 0(3)-0(3')e 0(3)-0(3") 0 ( 3 " ) - 0 ( 3 " ' ) 1.620(5) 1.660(6) 1.637(5) 1.639 2.758(7) 2.765(6) 2.556(6) 2.590(11) 2.665 2.064(4) 2.060(4) 2.111(4) 2.078 2.806(5) 3.092(6) 2.838(7) 2.992(6) 3.297(7) 2.556(6) 2.930 2.105(5) 2.043(6) 2.172(5) 2.054(5) 2.100 2.969(7) 2.806(5) 3.146(6) 2.895(7) 2.590(11) 2.966(5) 3.323(11) 2.956 1.625(6) 1.666(7) 1.630(7) 1.638 2.755(9) 2.766(8) 2.560(8) 2.582(15) 2.666 2.067(5) 2.071(5) 2.123(5) 2.087 2.824(8) 3.096(8) 2.849(10) 3.001(8) 3.322(8) 2.560(8) 2.942 2.130(7) 2.040(7) 2.183(6) 2.062(6) 2.110 2.995(8) 2.824(8) 3.154(8) 2.901(9) 2.582(15) 2.980(6) 3.351(15) 2.970 1.627(6) 1.657(6) 1.634(6) 1.638 2.754(8) 2.766(7) 2.560(7) 2.585(13) 2.665 2.077(4) 2.083(4) 2.131(4) 2.097 2.835(7) 3.110(7) 2.870(9) 3.012(7) 3.347(7) 2.560(7) 2.956 2.131(6) 2.044(6) 2.193(5) 2.068(6) 2.116 3.000(7) 2.835(7) 3.170(6) 2.901(8) 2.585(13) 2.992(5) 3.366(13) 2.980 0-0 edge shared between two octahedra. 1.626(5) 1.661(6) 1.635(5) 1.639 2.759(7) 2.765(6) 2.564(6) 2.588(11) 2.668 2.089(4) 2.088(4) 2.142(4) 2.106 2.850(6) 3.127(6) 2.882(7) 3.023(6) 3.364(6) 2.564(6) 2.968 2.141(6) 2.054(5) 2.204(5) 2.073(5) 2.125 3.009(6) 2.850(6) 3.184(6) 2.913(7) 2.588(11) 3.002(5) 3.383(11) 2.991 Glaucochroite 300°C 6C 1.617(5) 1.656(4) 1.637(4) 1.617(5) 1.651(4) 1.629(4) 1.621(5) 1.651(5) 1.633(4) 1.614(6) 1.645(5) 1.632(5) 1.624(8) 1.659(6) 1.634(4) 1.637 1.632 1.635 1.630 1.640 2.768(6) 2.744(5) 2.556(6) 2.616(9) 2.767(6) 2.730(5) 2.556(5) 2.599(8) 2.768(6) 2.736(5) 2.559(6) 2.610(10) 2.753(7) 2.730(6) 2.554(6) 2.611(10) 2.764(9) 2.737(7) 2.582(5) 2.604(7) 2.664 2.656 2.661 2.655 2.664 2.186(3) 2.089(3) 2.113(3) 2.204(3) 2.090(3) 2.131(3) 2.217(4) 2.098(3) 2.137(3) 2.230(4) 2.100(3) 2.144(4) 2.251(5) 2.160(4) 2.218(4) 2.129 2.142 2.151 2.158 2.210 2.961(5) 3.117(5) 2.830(6) 3.204(8) 3.334(5) 2.556(6) 2.979(5) 3.150(5) 2.842(6) 3.219(5) 3.359(5) 2.556(5) 2.985(5) 3.170(6) 2.859(7) 3.235(5) 3.376(5) 2.559(6) 3.002(6) 3.182(6) 2.869(7) 3.245(6) 3.389(6) 2.554(6) 3.032(8) 3.282(6) 2.958(9) 3.271(6) 3.535(7) 2.582(5) 3.000 3.018 3.031 3.040 3.110 2.484(5) 2.316(5) 2.414(4) 2.291(4) 2.496(4) 2.330(4) 2.421 (4) 2.301(4) 2.501(5) 2.335(5) 2.431(4) 2.306(5) 2.519(5) 2.344(5) 2.439(4) 2.309(5) 2.447(6) 2.305(6) 2.417(4) 2.306(4) 2.368 2.378 2.385 2.393 2.366 3.607(6) 2.961(5) 3.609(6) 3.148(5) 2.616(9) 3.381(5) 3.767(9) 3.613(5) 2.979(5) 3.620(2) 3.171(5) 2.599(8) 3.385(4) 3.810(8) 3.613(6) 2.985(5) 3.635(6) 3.184(5) 2.610(10) 3.389(5) 3.826(10) 3.620(6) 3.002(6) 3.651(6) 3.199(6) 2.611(10) 3.392(5) 3.843(11) 3.522(6) 3.032(8) 3.600(6) 3.165(7) 2.604(7) 3.340(5) 3.884(7) 3.316 3.329 3.337 3.349 3.317 1.626(8) 1.617(10) 1 .599(11) 1.647(6) 1.653(8) 1 .677(8) 1.629(4) 1.630(6) 1 .619(6) 1.633 1.633 1 629 2.744(9) 2.749(12) 2 738(13) 2.733(7) 2.733(9) 2 712(10) 2.575(6) 2.574(7) 2 583(7) 2.602(8) 2.588(11) 2 583(11) 2.660 2.659 2 652 2.259(5) 2.273(6) 2 292(6) 2.169(5) 2.163(6) 2 169(6) 2.229(4) 2.243(5) 2 253(5) 2.219 2.226 2 238 3.048(8) 3.061(10) 3 095(11) 3.294(6) 3.319(8) 3 328 (8) 2.974(9) 2.969(12) 3 004(12) 3.281(3) 3.296(4) 3 301(5) 3.565(7) 3.576(10) 3 590(10) 2.575(6) 2.574(7) 2 583(7) 3.123 3.133 3. 150 2.465(6) 2.309(6) 2.429(4) 2.308(4) 2.488(8) 2.327(8) 2.424(6) 2.325(6) 2.511(9) 2.322(8) 2.435(6) 2.329(6) 2.375 2.386 2.394 3.532(6) 3.048(8) 3.619(6) 3.169(7) 2.602(8) 3.345(5) 3.902(8) 3.544(8) 3.061(10) 3.629(8) 3.212(9) 2.588(11) 3.346(6) 3.944(11) 3.556(8) 3.095(11) 3.630(8) 3.215(10) 2.583(11) 3.357(6) 3.960(12) 3.328 3.343 3.354 Table ?• Interatomic Angles (°) in Ni-Olivine, Monticellite and Glaucochroite. Monticellite Glaucochroite 300°C 600° C Tetrahedron 0(l)-Si-0(2) 0(l)-Si-0(3) 0(2)-Si-0(3) a 0(3)-Si-0(3') a M(l) Octahedron 0(1)-M(1)-0(3) U OCl)-M(l)-0(3'3 0(1)-M(1)-0(2) D 0(1)-M(1)-0(2') 0(2)-M(l)-0(3'i 0(2)-M(l)-0(3) a M(2) Octahedron '2] O(l)-M(2)-0(3'': 2] 0(l)-M<2)-0(3) >] 0(2)-M(2)-0(3) 2] 0(2)-M(2)-0(3" l ] 0(3)-M(2)-0(3')' 2] 0(3)-M(2)-0(3"; [ l ] 0(3'*)-M(2)-0(3 114 48(34) 113 69(43) 114 03(36) 114 19(33) 116 22(16) 116 32(25) 116 06(22) 115 94(19) 101 67(19) 101 91(24) 102 14(22) 102 15(19) 104 55(35) 104 72(47) 104 54(40) 104 60(36) 115.45(24) 114.96(16) 101.38(17) 106.04(29) 84.45(13) 84 75(24) 84 70(21) 84 69(18) 67.05(15) 95.55(18) 95 25(24) 95 30(21) 95 31(15) 92.95(15) 86.96(15) 87 02(20) 87 23(17) 87 27(15) 82.88(12) 93.04(15) 92 98(20) 92 77(17) 92 73(15) 97.12(12) 104.43(21) 104 77(24) 105 18(21) 105 37(19) 105.05(16) 75.57(21) 75 23(24) 74 82(21) 74 63(19) 74.95(16) 91.12(13) 91 16(17) 91 19(15) 91 11(13) 97.98(12) 82.00(13) 81 80(18) 81 91(16) 81 98(14) 74.36(11) 96.54(14) 96 60(18) 96 80(16) 96 76(14) 99.42(13) 89.95(14) 90 00(18) 89 74(15) 89 80(14) 86.20(11) 73.21 (27) 72 52(34) 72 23(31) 71 91(26) 65.60(20) 89.12( 9) 89 12(12) 89 16(11) 89 10( 9) 91.82( 9) 108.01(27) 108 68(33) 108 93(29) 109 37(25) 110.53(19) 115 70(23) 115 54(27) 115 29(29) 114 47(14) 114 49(17) 114 43(18) 102 39(15) 102 36(17) 102 42(18) 105 89(26) 106 11(31) 106 29(33) 86 80(14) 86 55(16) 86 67(18) 93 20(14) 93 45(16) 93 33(18) 82 87(12) 82 92(14) 82 95(15) 97 13(12) 97 08(14) 97 05(15) 105 46(14) 105 68(16) 106 01(16) 74 54(14) 74 32(16) 73 99(16) 97 67(11) 97 39(12) 97 07(12) 74 55(11) 74 49(13) 74 52(13) 99 25(11) 99 41(13) 99 50(13) 86 45(10) 86 65(11) 86 86(12) 64 94(18) 64 96(21) 64 75(22) 91 55 ( 7) 91 35(10) 91 17(11) 111 77(17) 112 12(21) 112 66(22) 114.71(35) 114.30(21) 103.28(22) 105.66(30) 85.46(18) 94.54(18) 84.23(18) 95.77(18) 107.72(17) 72.28(17) 95.59(14) 77.13(16) 99.33(18) 86.68(13) 65.19(18) 89.96(8) 114.71(21) 113.94(36) 114.21(22) 103.65(22) 106.01(31) 85.56(19) 94.44(19) 84.37(19) 95.63(19) 108.32(17) 71.66(17) 95.44(14) 77.05(16) 99.61(18) 86.70(13) 64.75(19) 89.80(10) 115.42(22) 114.42(48) 114.67(27) 103.29(29) 105.12(40) 85.36(25) 94.64(25) 84.02(24) 95.98(24) 108.51(23) 71.49(23) 94.79(17) 77.07(20) 99.57(23) 87.31(17) 64.51(25) 89.57(12) 116.05(28) 113.40(49) 114.93(29) 103.19(29) 105.85(43) 85.86(27) 94.14(27) 84.60(24) 95.40(24) 108.54(23) 71.46(23) 94.47(18) 77.46(21) 99.44(24) 87.46(18) 64.08(25) 89.59(13) 116.41(30) The numbers in brackets refer to the m u l t i p l i c i t y of the angle. Angle opposite 0-0 edge shared between an octahedron and tetrahedron. Angle opposite 0-0 edge shared between two octahedra. 14 Brown ( l9?0). The i n t e r a t o m i c d i s t a n c e s and a n g l e s and t h e i r e s t i m a t e d s t a n d a r d d e v i a t i o n s were c a l c u l a t e d w i t h t he p r o g r a m ORFFE ( B u s i n g , M a r t i n and L e v y , 1964) and a r e t a b u l a t e d i n T a b l e s 6 and 7> The i n t e r a t o m i c d i s t a n c e s a r e u n c o r r e c t e d f o r t h e r m a l m o t i o n a n d r e -p r e s e n t d i s t a n c e s be tween mean a t o m i c p o s i t i o n s . The t h e r m a l e x p a n s i o n o f s e l e c t e d p a r a m e t e r s i n t h i s s t u d y has b e e n r e p r e s e n t e d b y a l i n e a r t h e r m a l e x p a n s i o n c o e f f i c i e n t d e f i n e d a s = J ( / T " 4>5o) (1) i 2 5 0 CT - 25°) 9 Q o where X ^ and A ^ a r e t h e l e n g t h s o f s e l e c t e d p a r a m e t e r s a t 25 0 and a t some h i g h e r t e m p e r a t u r e T . I n t h i s s t u d y , t h e q u a n t i t y , (I - I ) _ T 25 , i s t h e s l o p e d e t e r m i n e d f r o m a s i m p l e l i n e a r (T - 25°) r e g r e s s i o n o f 1 v e r s u s T . I I I . RESULTS AND DISCUSSION O l i v i n e S t r u c t u r e A d r a w i n g o f t h e o l i v i n e s t r u c t u r e p r o j e c t e d on (100) i s g i v e n i n F i g u r e 1 . The b o l d - f a c e l i n e s i n t h e d r a w i n g r e p r e s e n t s h a r e d o x y g e n - o x y g e n e d g e s . The ma in s t r u c t u r a l u n i t i s t h e z i g z a g c h a i n o f o c t a h e d r a p a r a l l e l t o c . The M ( l ) o c t a h e d r a , w h i c h h a v e C^ symmet ry , s h a r e s i x o x y g e n - o x y g e n edges w i t h a d j a c e n t p o l y h e d r a : two edges a r e s h a r e d w i t h M ( l ) o c t a h e d r a , two w i t h M(2) o c t a h e d r a and two w i t h 15 Figure 1. A drawing of a portion of the olivine structure projected on (lOO). Bold-face lines indicate oxygen-oxygen edges which are shared between adjacent polyhedra. The M(l) cation labelled i s located at x = 0.50, y = 0.50, z = 0.50. The labelling of the atoms i s consistent with Tables 5 and 6. 16 17 si l i c a t e tetrahedra. The M(2) octahedra have G v symmetry and share a triangle of edges: two edges are shared with M(l) octahedra and one i s shared with a s i l i c a t e tetrahedron. The s i l i c a t e tetrahedra, one of which i s shown in Figure 1, link octahedral chains in adjacent layers along a. For the olivines considered in this study which are not pure end-members, M(l) and M(2) octahedra have the following cation occupancies respectively: hortonolite, Mg^ ^^Fe^ ^ ^Mg^ 705^e0 285' ^ ' ^ i 0 6 ^ ^ 6 » M S0.93 F e0.07 '• G a l . 00 ; gl^cochroite, MnQ_ Q^<±QZnQ_ ^: G a Q # ^ _ Q 2 . The reader i s referred to Birle s i a l . (19^8) and Brown (1970) for a more detailed discussion of the crystal structures and phase relations in the olivine group of minerals. Polyhedral Expansions Sili c a t e Tetrahedra The mean Si - 0 distances in the three olivines refined in this study show a slight contraction with heating. The largest change with temperature i s observed in glaucochroite where the mean distance de-creases by slightly more than 1 <r over the temperature range 2 5 u - » 800UG. The mean distances in forsterite, hortonolite and fayalite show a similar response with temperature. The decrease in the Si - 0 distances may not be r e a l i s t i c , however, since corrections have not been made for the increased thermal motion at high temperatures. Mean oxygen-oxygen distances in the tetrahedron remain es-sentially constant with temperature in Ni-olivine while in monticellite and glaucochroite they decrease slightly more than lo" (Table 6). The contraction in monticellite and glaucochroite primarily 18 reflects the decrease in length of the unshared edges o(l ) -0 (2 ) and 0(l ) - 0 (3 ) . The 0-0 edges in the tetrahedron which are shared with octahedra show relatively l i t t l e variation except for 0(3)-0(3) a in glaucochroite. The variations in the mean and individual 0-0 distances in forsterite and hortonolite are comparable to those discussed above. In fayalite the shared edge, 0 (3 ) -0 (3 ) a f decreases in length by 1.83^ (2.577A—yZ.530%) from 20°—»900°C. In view of the large errors associated with the O-Si-O angles, the only significant angular variation with temperature i s with respect to 0 (2)-Si -0 (3 ) a , the angle opposite the edge shared between the tetrahedron and the M(l) octahedron (Table 7)- With heating, this angle widens in Ni-olivine and monticellite. A similar variation i s observed in forsterite and hortonolite. In glaucochroite and fayalite, this angle shows no significant change. M(l) Octahedron The mean interatomic distances in the M(l) octahedron (Table 6) for the six olivines are plotted versus temperature in Figure 2. Ranking of the thermal expansion coefficients of the mean distances (Table 8) results in the following sequence: Mg-0 (forsterite) *~ Mg-0 (monticellite) -~ Mn-0~Ni-0 > Fe-0 ~ (Mg,Fe)-0. Thermal expansions of individual M(l)-0 distances decrease in.the order M(l)-0(3) > M(l)-0(2) > M(l ) - 0(l) for Ni-olivine and forsterite and M(l ) - 0(l)> M(l)-0(3) > M(l)-0(2) for hortonolite, fayalite, monticellite and 19 T 1 1 1 1 I I I I 1 1 1 1 1 1 1 200 400 600 800 1000 T°C Figure 2. Plot of the mean M(l)-0 distances versus temperature for Ni-olivine (Ni), forsterite (Fo), hortonolite (Ho), fayalite (Fa), monticellite (Mo) and glaucochroite (Gl). Error bars refer to one estimated standard deviation. 20 Table 8. Linear Thermal Expansion Coefficients (°C x 10^), Slopes of Regression Lines (£/°C x 105) and Correlation Coefficients for Mean M-0 Distances and Unit Cell Parameters in Six Olivines. Nias'-goo") Fo(25°-1000°) Ho(24°-710°) oC dl/dT r cx dl/dT r ex. dl/dT r <M(l)-0> 1.54(2) 3.21(5) 0.999 1.93(24) 4.05(50) 0.984 1.17(2) 2.48(5) 0.999 <M(2)-0> 1.31(10) 2.76(21) 0.994 1.72(8) 3.69(17) 0.998 1.22(7) 2.63(14) 0.999 a 1.18(5) 5.57(24) 0.998 0.87(11) 4.06(50) 0.985 0.61(1) 2.92(4) 0.999 b 1.09(4) 11.01(41) 0.999 1.54(2) 15.20(16) 0.990 0.96(7) 9.94(72) 0.997 c 1.11(3) 6.56(16) 0.999 1.33(2) 7.94(13) 0.974 0.97(10) 5.83(58) 0.995 Fa(20°-900°) Gl(25°-800°) OC dl/dT r dl/dT r dl/dT r <M(l)-0> 1.26(13) 2.73(29) 0.989 1.74(5) 3.71(14) 0.999 1.54(20) 3.42(45) 0.983 < M(2)-0> 1.50(9) 3.27(19) 0.997 1.33(9) 3.14(21) 0.996 1.52(4) 3.61(10) 0.999 a 0.99(8) 4.88(40) 0.987 1.01(8) 4.20(38) 0.992 0.87(11) 5.01(56) 0.988 b 0.95(9) 10.14(95) 0.983 0.99(2) 11.29(26) 0.999 1.02(3) 11.09(38) 0.999 c 1.19(9) 7.57(55) 0.990 1.13(4) 9.23(26) 0.999 1.45(8) 7.39(53) 0.995 21 glaucochroite. In a l l olivines except hortonolite and fayalite, the thermal expansion of a M(l)-0 distance i s generally proportional to i t s length at room temperature (i.e., the largest thermal expansions are associated with the longest bond lengths). The oxygen-oxygen edges in the M(l) octahedron are of three types: (l) those shared with tetrahedra, (2) those shared with octahedra and (3) those that are unshared. In general, thermal ex-pansions can be ranked as follows: unshared ]> shared with octahedra >> shared with tetrahedra. The 0-M(l)-0 angles generally change less than 1 ° over the temperature range of the six olivine refinements. The most s i g n i f i -cant changes are in 0(2)-M(l)-0(3) , which i s opposite the edge shared between the M(l) octahedron and the s i l i c a t e tetrahedron, and 0(2)-M(l)-0(3 ')- In a l l olivines studies, the former decreases by while the latter increases by~l°. M(2) Octahedron The mean M(2)-0 distances are plotted as a function of temperature in Figure 3- °^M (2) o > ^ a ^ e ^ decreases in the order Mg-0 > Ga-0 (glaucochroite)~Fe -0 > Ga-0 (monticellite)~ Ni -0 > (Mg.Fe)-O. Table 3 indicates a significant displacement in the position of the M(2) cation with temperature for a l l olivines studied. The dis-placement i s in a direction away from the triangle of shared edges 22 Figure J. P l o t of the mean M(2) distances versus temperature f o r six o l i v i n e s . Unless otherwise stated, the abbreviations i n t h i s and subsequent f i g u r e s are the same as those i n F i g . 2. 23 (see Figure l ) in the plane containing the M(2)-0(l) and M(2)-0(2) bonds (i.e., roughly toward 0(2) and away from 0 ( l ) ) . Cation dis-placements with temperature have also been noted in the M(2) poly-hedron in pyroxenes and the M(4) polyhedron in amphiboles (Cameron, Sueno, Prewitt and Papike, 1973. Sueno, Cameron, Papike and Prewitt, 1973)• The magnitude of the displacement observed in these investi-gations can be correlated with the size of the cation. In this study, the displacement i s measured relative to the centroid of the octa-hedron and i t s rate of change as a function of temperature decreases in the order Mo~Fo~ Gl > Ni > 'Fa~Ho (Figure 4 and Table 9). It i s interesting to compare Figure 4 with Figure 5> where the displacement i s plotted versus cation size for seven room temperature refinements of olivines. It is evident from Figure 5 that the dis-placements in the room temperature structures are a function of the size of the M(2) cation, i.e., the magnitude of displacement increases with increasing cation radius. Room temperature investigations of pyroxenes and amphiboles (Papike et a l . , 1969? Takeda, 1972) have also indicated a similar relationship between displacement and cation radius. The displacement at room temperature as a function of the M(2) cation radius may be explained qualitatively as resulting from the asymmetric distribution of shared edges about the M(2) octahedron. For example, increasing the effective radius of the M(2) cation from N i + 2 (0.70 Shannon and Prewitt, 1969) in NigSiO^ to Ca + 2 ( 1.00 A5) in CagSiO^ results in an insignificant increase in the shared edge 0(3) - 0(3'), whereas the opposite unshared edge 0(3' ') - 0(3' ' ') Figure 4. Plot of the displacement of the M(2) cation from the centroid of the octahedron versus temperature for six olivines. 25 T 1 1 1 1 1 1 1 1 1 Mo i 1 1 • i l 1— 400 600 800 1000 T ° C 26 Table 9. The Magnitude (£) and Rate of Increase (S/°G x 10^) of the Displacement of the M(2) Cation with Increasing Temperature for Six Olivines.* Olivine Displacement Olivine Displacement Ni-Olivine Hortonolite 25° 300° 600° 900° 0.206 0.212 0.217 0.222 24° 375° 710° 0.264 0.267 0.269 d l/dT 1.79(13) 0.81(18) Forsterite Fayalite 25° 350° 675° 1000 0.244 0.251 0.258 0.26© 20° 300° 700° 900° 0.293 0.295 0.299 0.301 d 1 /dT J 2.44(19) 0.95(19) Monticellite Glaucochroite 25° 335°, 795 0.304 0.313 0.317 0.323 25° 300° 600° 800° 0.304 0.312 0.316 0.321 d l /dT 2.35(23) 2.14(31) * The displacement of the M(2) cation in ^ -GaQSi0,. (Brown, 1970) at 25°C i s 0.311 &. * 4 27 0.28 0.32 C E <D u _D a «/> 25 c O 0.24 O U 3 - l r - i Ca • Mo Fa Si • • Ho • -Fo • • -N i -• J 1 1 1 _j i 1 0.70 0.80 0.90 M2 Cation Radius (A) 1.00 Figure 5- Plot of the displacement of the M(2) cation from the centroid of the octahedron versus M(2) cation radius for seven room temperature olivine structures. The abbreviation Ga refers to X -GagSiO^ (Brown, 1970). Cation r a d i i are from Shannon and Prewitt (1969)• 28 increases from 3.3C4 - 4.1J.6 X (Brown, 1970). This, coupled with an increased M(2)-0 distance, results i n a sh i f t of the M(2) cation away from the shared edge and the centroid of coordinating oxygens. The relatively constant 0(3) - 0(3') shared edge attests to the presence of antibonded repulsive forces "between the cations whether these forces "be largely ionic or a combination of ionic and covalent in character. The plot in Figure 5 indicates that the correlation between the radius of the M(2) cation and the displacement i s essentially linear from Ni-olivine to fayalite. However, the Ca-olivines, regardless of the size of the M(l) cation, cluster about one point and exhibit dis-placements only slightly greater than fayalite. Unfortunately, i t i s d i f f i c u l t to determine whether the non-Ca and Ca-olivines represent a continuous trend since there are no refinements',of end-member com-+2 +2 positions with a cations radius intermediate between Fe and Ca The cation displacements at high temperatures may be related to the asymmetry of the potential energy function for the M(2) cation. Due to the contribution of repulsive terms to the total potential, the "potential well" for the M(2) cation may have a steeper gradient in the direction of the triangle iof shared edges. Increased anharmonic vibration at high temperatures should cause a shift in the position of the cation away from this gradient. Bums (l9?0b) has suggested charge transfer between adjacent octahedral cations in olivine to explain the observed pleochroism in fayalite. It i s suggested that the smaller displacements in the Fe-containing olivines may be related to metal-metal interactions between I 29 +2 Fe cations in adjacent M(l) and M(2) octahedra. Tn this regard, i t is interesting that the M(l) - M(2) distances show a smaller thermal expansion in fayalite and hortonolite than in the non-Fe olivines. +2 The magnetic spin alignments on adjacent Ni cations in Ni-olivine are also thought to he related to metal-metal interactions (Newnham et a l . , 1965)• The thermal expansions of the 0-0 distances in the M(2) octa-hedron can he grouped as indicated for M(l). In regard to angular variations, the 0(3)-M(2)-0(3') a and 0(3 ' ') - -M(2)-0(3* ' ') angles show the greatest changes with heating. 0(3)-M(2)-0(3*)^ which i s opposite the edge shared between the M(2) octahedron and the s i l i c a t e tetrahedron, decreases by ~ 1 ° while 0(3 ' ')~M(2)-0(3' ' ') increases by approximately the same amount. In both M(l) and M(2), the angles opposite the edges shared with s i l i c a t e tetrahedra show significant decreases with temperature. Thermal Parameters The isotropic equivalents of the anisotropic temperature factors (B) are plotted versus temperature for M(l), M(2) and sili c o n cations in Figures 6, 7 and 8, respectively. For a given structure, the rate of increase in B i s greater for the M cations than the silic o n cations. The M(2), S i , 0(l) and 0(2) atoms in olivines occupy equipoints with point symmetrym . As a result, the orientation of the thermal vibration ellipsoids (and the strain ellipsoids as we w i l l see later) 30 Figure 6 . Plot of the equivalent isotropic temperature factors for the M(l) cations versus temperature for six olivines. Temperature factors for forsterite are from Hazen (personal communication). 31 B(A2) 2.50 2.00 •SO 1.00 0.50 1 — 1 1 — — 1 1 1 1 1 1 1 — — P Fa / fi 01 / / / Mo / Fo 1 1 1 —« 1 J -X 1 1 A 200 400 600 800 1000 T°C Figure 7 . Plot of the equivalent isotropic temperature factors for the M(2) cations versus temperature for six olivines. Temperature factors for forsterite are from Hazen (personal communication). 32 T°C Figure 8. Plot of the equivalent isotropic temperature factors for silicon versus temperature for six olivines. Temperature factors for forsterite are from Hazen (personal communication). 33 are constrained by symmetry to certain directions. Two axes of the ellipsoid must l i e in the mirror plane whereas the third axis l i e s in a direction perpendicular to this plane. The M(l) and 0(3) atoms, on the other hand, are in general positions and there are no symmetry constraints on the orientations of their vibration ellipsoids. The thermal ellipsoids for the M(l), M(2) and s i l i c o n atoms in N x-olivine, monticellite and glaucochroite are generally t r i a x i a l . With heating, they increase in size but maintain approximately the same shape. Brown and Prewitt (1973) suggested that the shape of the thermal ellipsoid for M(2) in hortonolite at high temperatures was, in part, related to the effects of positional disorder. As discussed in the previous section, the temperature dependence of the cation position in the M(2) octahedron varies from structure to structure. For example, the Mg cation (forsterite) exhibits a much larger dis-placement than the Fe cation (fayalite). Thus, at elevated tem-+2 +2 peratures, the Mg and Fe cations in hortonolite w i l l occupy slightly different positions along b relative to their positions at room temperature. This w i l l tend to increase the magnitude of the ellipsoid parallel to b. Brown and Prewitt (1973) noted a decrease in the anisotropy of the thermal ellipsoid for M(2) as a function of temperature which was primarily due to an increase in the magnitude of the ellipsoid axis parallel to b. The effects of positional disorder on the shapes of the thermal ellipsoids in pyroxenes and amphiboles has been discussed by Cameron et a l . (1973) and Sueno et a l . (1973)' 34 The thermal ellipsoids for the oxygen atoms in a l l structures are t r i a x i a l and do not vary appreciably in orientation with increasing temperature. In general, the shortest axes of the ellipsoids are oriented parallel to the associated S i - 0 bond. Strain Analysis of Polyhedral Expansions One of the d i f f i c u l t i e s in describing the thermal expansion of a polyhedron i s to decide specifically what parameters most quantitatively define the expansion. In the past, workers have used the variation in individual distances or angles or, more commonly, the variation in 'mean distances as a measure of polyhedral thermal expansion. This method has obvious disadvantages since i t i s impossible to evaluate the true nature of the thermal expansion which i s the resultant of simultaneous changes in the numerous distances and angles within a polyhedron. In addition, this method provides no information as to the directional character of the thermal expansion. Three dimensional strain theory provides a quantitative means for characterizing the thermal expansion of a polyhedron. Mathe-matically this involves determining the thermal expansion tensor (a second rank tensor) and the corresponding magnitude ellipsoid (Nye, 1957) for a polyhedron. It turns out that this can be easily cal-culated from the positional parameters obtained from the least-squares refinement. The calculation, which i s discussed in detail in Appendix II, compares atom positions of a polyhedron at two different temperatures and then, using an iterative least-squares method, deter-35 mines a 'best-fit' ellipsoid to describe the shape and volume changes associated with the thermal expansion. This ellipsoid i s the familiar strain ellipsoid evaluated over a particular temperature interval. If so desired, the rotations of the polyhedra due to heating can also be calculated. The magnitudes and directions of the principal axes of the thermal expansion ellipsoids for M(l), M(2) and Si polyhedra have been calculated for the six olivines and are presented in Table 10. The longitudinal strains ( £ ^ , ^ £ 2 ' ^33^' wn^-c'1 a r e "^ne diagonal components of the tensor (Table 11), are defined as the strains parallel to the unit c e l l edges. In Table 10, the strain components are given as "strain/l°G" in order to make comparisons between olivines at different temperatures. The results of the above calculations afford a quick means of calculating the % volume expansion of a polyhedron. Consider a unit sphere. In general, due to thermal deformation, the lengths of i t s axes w i l l become ( l + £ ^ ) , ( l + £g) a n ^ +^"}) a n <^ "^ne ^ volume ex-pansion (or dilation) with respect to the i n i t i a l , undeformed—'state. w i l l be given by = Volume ellipsoid - Volume sphere Volume Sphere - d + e±) (i -+e 2 ) (i + £ 3 ) -1 (2) If the strains are small, the crossproduct terms can be neglected and (2) reduces to Magnitudes and Orientations of the Principal Axes of the Thermal Expansion Ellipsoids for the Polyhedra in Six Olivines. 37 H2K 8 = 8 8 = 8 SSS 8=8 88= SSS 8=8 8=8 SSS 8=8 ==8 ass S. £ S 5 S c § § § c S 5 £ o o o S?iu-1 § ° 5 ooo SSS S3g SS"" SKS 33£ SS" ""SIC 3 3 . ? 3 S • = S S = 5 SKI 8=8 8=8 S2K =88 8=8 8*5 =88 88= JES 8=8 88= KSS S ° = 8g = SS3 S8S S8S KSS 83= S S S K3K S S S K S 8 S C S KSK S S S S S S S S S S § 8 SSS S8R S83 SSS 835 S83 KSS 8 S 5 S S S £ 3 K K8K S S S S S S S S S S S S s s ; sss s s ; " = ~ ~ = « « = 2 s^s sss sss sss ass sss s a : sss sss 8ss ess sss S S J sss tgg d d -dd doc ~~= ~ " = <? = ° ""= ~ ~ ° =<?? - = ? " = ~ 7 ? ~ - » ~ ~ = ~ » © " « 7 ~ - ° d d « « -J d ~ - d - o « ««o « -3 d d -i •» i i s * s j s • a j = s " s 8 * * " a * " * i x M a * * « s ZSK S S S 388 S 5 S 3 S S S S S S S S S S S J 8 S SUSI 382 £ g s " 3 § s 2 G S S 2 S 3 s s s 5 S I a 8 3 S 3 £ 5 S 1 * 2 S 5 S S s 8 ; SSS 528 E 3 S S S S S 8 S SSJ S K E S 8 2 2 8 S £ S S 38X S 8 " S5S CS! S S S S S S S S S S S i E S S S S S 8 8 3 S S S S8" S S " 235 33! 352 3JS 3SK S S S SSS SS8 SSS SSS SS32 SSS ES8 5~S S3g S S S SS8 SSS SS2 3S5 S S S S R S 5S2 x » " g * J = I * o r K Table 11. Longitudinal Strains Characterizing the Polyhedral Thermal Expansions in Six Olivines Temperature Range Ni-Olivine 25 -*• 300 Longitudinal Strains Temperature Range, Longitudinal Strains Temperature Range, e33 Longitudinal Strains e22 Mtt) M(2) M(l) M(2) M(l) M(2) Forsterite 25 -* 350 M(l) M(2) M(l) M(2) 675 -> 1000 M<1) M(2) 0.0078 0 0008 0 0028 0.0041 0 0058 0 0041 -0.0001 0 0034 -0 0030 0.0079 0 0020 0 0029 0.0023 0 0030 0 0032 0.0045 0 0034 0 0036 0.0054 0 0036 0 0031 0.0045 0 0034 0 0036 -0.0002 0 0017 0 0011 0.0096 -0 0057 0 0024 -0.0001 0 0101 0 0040 -0.0064 -0 0093 -0 0042 0.0077 0 0028 0 0028 0.0053 0 0066 0.0033 -0.0032 0 0025 0 0007 0.0068 0.0122 0 0058 0.0048 0 0014 0 0075 -0.0010 0 0108 0 0039 Hortonolite 24 - 375 Fayalite 20 -<- 300 M(l) K(2) M(l) M(2) Monticellite 25 -* 335 0.0038 0.0039 -0.0028 0.0032 0.0043 0.0027 -0.0001 0.0031 -0.0013 0.0031 0.0025 0.0029 0.0017 0.0030 -0.0034 0.0038 0.0048 0.0004 M(l) 0 0058 0 0020 0 0018 M(2) 0 0043 0 0045 0 0064 Si 0 0017 -0 0003 -0 0152 M(l) 0 0063 0 0054 0 0036 M(2) 0 0054 0 0013 0 0055 Si -0 0040 0 0017 -0 0062 Monticellite 335 •* 615 M(l) 0 0041 0 0024 0 0025 M(l) 0.0043 0 0035 0.0042 M<2) 0 0025 0 0049 0 0032 M(2) 0.0018 0 0013 0.0042 Si -0 0003 0 0019 -0 0009 Si 0.0015 -0 0001 0.0042 375 -* 710 615 -»795 M(l) 0 0051 0 0012 0 0034 M(l) 0.0057 -0 0003 0.0027 M(2) 0 0024 0 0036 0 0043 11(2) 0.0036 0 0028 0.0032 Si -0 0015 -0 0025 -0 0004 Si -0.0039 -0 0026 0.0003 Glaucochroite 25 » 300 M(l) M(2) M(l) M(2) MU) M(2) 0.0080 0.0017 -0.0038 0.0008 0.0063 0.0023 0.0120 0.0073 -0.0090 -0.0007 0.0042 -0.0031 0.0032 0.0025 0.0010 0.0040 0.0012 0.0053 0.0021 0.0028 -0.0012 0.0036 0.0058 -0.0052 0.0014 0.0022 -0.0019 U 0 CO 39 The % volume expansion calculated from (3) w i l l be approximate (- 5% of true % expansion) because the ellipsoids are only "best-f i t s " . Table 10 indicates that the anisotropy of the thermal ex-pansion, as measured by the difference in the longest and shortest axis of the ellipsoid, i s greater for M(l) than for M(2) octahedra in a l l olivines except monticellite and glaucochroite. This i s primarily due to the topology of the octahedron. The M(l) octahedron shares two 0-0 edges with s i l i c a t e tetrahedra (Figure l ) . Since these edges exhibit a zero to negative thermal expansion in a l l olivines, the strain or expansion in this direction i s always small relative to that observed in any other direction. In the M(2) octahedron, there i s only one edge shared with a tetrahedron. This shared edge, 0 ( 3 ) - 0 ( 3')i i s oriented perpendicular to the mirror plane at 0.25£ and parallel to the unshared edge 0 ( 3 ' ' ) - 0 ( 3 ' ' ' ) i which shows the largest expansion of any 0-0 distance in the octahedron. Two axes of the thermal ex-pansion ellipsoid for M(2) are constrained by symmetry to l i e in the mirror plane. The third axis l i e s perpendicular to the mirror plane and reflects the thermal expansions of both the shared and unshared edges. As a result, the anisotropy of the thermal expansion for M(2) i s generally less than for M(l). As indicated in Table 10, there i s a significant difference between the non-Ca and Ca-olivines with respect to the nature of the thermal expansion for M(2) octahedra. For Ca-olivines, the anisotropy 40 of thermal expansion for M(2) i s approximately equal to that for M(l) and the M(2) and M ( l ) ellipsoids are similar in shape. The cyclographic projections of the principal axes of the thermal expansion ellipsoids for M ( l ) and M(2) octahedra are plotted in Figures 9 and 10 for the temperature range of each olivine refine-ment . For M(l) octahedra, the direction of the shortest axis of the ellipsoid remains essentially invariant for a l l six olivines. This direction i s approximately parallel to the 0-0 edge shared between M ( l ) octahedra and s i l i c a t e tetrahedra. The close grouping reflects the near constancy of this edge with heating in a l l olivines. With increasing mean cation radius, ^ ( ^ ( i ) ^ +^ < - ^ M(2)' >^ ' ^ e l 0 1 1 ^ 3 ^ axes of the ellipsoids rotate in a plane approximately perpendicular to the direction of the shared edge. The magnitudes of the ellipsoids, as given hy the dilation in equation (4), are consistent with the thermal expansions of the mean M(l)-0 distances. In contrast to M ( l ) octahedra, the orientations of the thermal expansion ellipsoids for M(2) do not appear to he related directly to cation size. It i s interesting, however, that i f only the non-Fe olivines are considered i n Figure 10, a correlation i s noted between strain direction and the size of the M(2) cation. As for M ( l ) octa-hedra, the size of the ellipsoids r e f l e c t the thermal expansions of mean M-0 distances. The thermal expansion ellipsoids for s i l i c a t e tetrahedra have approximately the same shape for the temperature range of each refine-ment. In general, the tetrahedra in a l l structures contract in the directions approximately parallel to a and _c. 4 1 Figure 9. Cycle-graphic projection of the principal axes of the M(l) thermal expansion ellipsoids for six olivines. Ellipsoids were calculated for the temperature range of each olivine  refinement. Solid, open and dotted symbols refer respective-l y to the longest, intermediate and shortest axes of the ellipsoids. A l l axes plotted are at an angle less than 90 with respect to c. 42 Figure 1 0 . Cycle-graphic projection of the principal axes of the M(2) thermal expansion ellipsoids for six olivines. Ellipsoids were calculated for the temperature range of each olivine  refinement. Solid and open symbols refer respectively to the longest and shortest axes of the ellipsoids. The intermediate axes are constrained by symmetry para l l e l to c. 43 The thermal expansions of the octahedra can also he represented in terms of the longitudinal strains (Table 11). These are plotted for M(l) and M(2) octahedra in Figure 11 for' the temperature range of  each refinement. The linear thermal expansion coefficients for the unit cell parameters, which are also represented in the plot, will he discussed in a later section. Inspection of Figure 11 indicates that there is a significant difference in the nature of the thermal deforma-tion in M(l) versus M(2) octahedra in a l l olivines except glaucochroite. For the M(2) octahedra, the deformation is characterized hy the following relationships between strains: (l) £ 2 2 > ^ 33^ "^11 Fo and Ho), ( 2 ) £ ^ > £ 2 2 (Fa, Mo) and ( 3 ) ^ 1 1 > £33 7 £ 2 2 (Gl). In general, the relationship £ ^ > ^33 7 ^ 2 2 holds for the M(l) octahedra in a l l six olivines. The above relationships for Ni-olivine, forsterite and hortonolite reflect, in large part, the difference in the magnitude of strain parallel to b ( 6 2g) in M(l) and M(2). A S discussed pre-viously, 0-0 edges in M(l) and M(2) which are shared with tetrahedra exhibit a zero to negative thermal expansion. In M(l), these edges have their largest vector component parallel to b and consequently the smallest strain in M(l) parallels this direction. The magnitude of strain parallel to b in M(2), evaluated over the temperature range of each refinement, is greater by a factor of 2 than that observed in M(l) for the non-Ca olivines. The relative difference in the magnitude of the £ 2 2 component suggests that the thermal expansion of the b cell edge in these olivines is primarily controlled by expansions in the M(2) octahedra. / Figure 1 1 . Bar diagram comparing the longitudinal strains in M(l) and M(2) .octahedra to the linear thermal expansion co-efficients for the c e l l parameters in six olivines. Strains are given as "strains/ 1 G" in order to f a c i l i t a t e comparisons between olivines refined at different temper-atures. The values for the longitudinal strains are (x 1 0 ° ) . 45 1.60| 1.20^  .40h r 24 16 24 -3 3 •22 -3 3 Ni Fo Ho Fa Mo Gl Ni Fo Ho Fa Mo Gl Ni Fo Ho Fa Mo Gl 46 The above differences between M(l) and M(2) octahedra become more apparent when the "deformation paths" for the octahedra are plotted. In Figures 12 and 13, the difference between the longitudinal strains ( £ ^ - ^22) Plotfed- versus the difference ( € ^  - £33) f ° r the non-Ca and Ca-olivines, respectively. The interior of the plot i s contoured in ( £ - ^22)' ^ e s a ^ e °^ c l a r i f y i only one of these controur lines i s shown - the one for which £ = The difference between strains was evaluated for each temperature interval of a given refinement. For example, in the refinement of Ni-olivine, the differences were calculated from Table 11 for the temperature intervals 25°-»-300° , 300°-=»600° and 600°—»900°. When these differences are plotted, as in Figures 12 and 13, they trace out de-formation paths for M(l) and M(2) octahedra as a function of tempera-ture. It i s important to note that a l l paths originate from the point (0, 0, 0) which corresponds to the i n i t i a l "thermally undeformed" state. Therefore, these figures imply nothing about the octahedral strains in the room temperature structures. Strain differences are additive, i.e., the terminating point on each path represents the sum of the differences for each temperature interval. In Figure 12, i t i s interesting to compare the M(2) deformation paths for forsterite, hortonolite and fayalite. The M(2) path for forsterite primarily reflects the large strain parallel to b. In hortonolite, the strains parallel to b and c are approximately equal and the deformation path calculated l i e s near the line for which £ ^= £ • The deformation path__f or fayalite indicates that the greatest hi Figure 12. Plot showing the deformation of M(l) and M(2) octahedra as a function of temperature for the non-Ca olivines. Open and solid symbols refer respectively to M(l) and M(2) octahedra. The differences between strains were calculated from Table 11 for each temperature interval of a given refinement. See text for futher explanation. 48 Figure 13. Plot showing the deformation of M(l) and M(2) octahedra as a function of temperature for the Ca-olivines. Open and solid symbols refer respectively to M(l) and M(2) octahedra. The differences between strains were calculated from Table 11 for each temperature interval of a given refinement. See text for further explanation. 49 component of strain i s parallel to c rather than b. Thus, i t appears that with increasing Fe content in the (Mg,Fe) olivines, there i s a gradual shift in the direction of greatest strain from b to c. This trend may reflect the differences in the thermal expansions of M(2)-0 distances noted for these olivines. The strain components plotted in the preceding figures are dependent on (l) the unit c e l l parameters determined at high temperatures, since these are used to calculate the atom positions of the polyhedra at each temperature and (2) the relative changes in the positions of the atoms with temperature. Unfortunately, i t i s d i f f i c u l t to assess the relative contributions of each factor to the observed trends discussed in this section. It cannot be argued, however, that the trends reflect solely the unit c e l l parameters because similar trends were obtained even when the c e l l parameters at 25°G were used to calculate the strain components at higher temperatures. Octahedral Distortions A number of different parameters have been proposed in recent years to characterize the distortion in coordination polyhedra. Parameters commonly used are the angle variance and the quadratic elongation (Robinson, Gibbs and Ribbe, 1971) which are essentially measures of the mean deviation of interatomic angles and distances from their ideal values. Recently, Dollase (1974) has suggested a more quantitative approach which employs an iterative least-squares method. Distorted polyhedra can be compared with least-squares-fitted polyhedra 50 of any desired symmetry. The degree of distortion can then he evaluated from the degree of f i t . Unfortunately, i t i s not always possible to find a 'best-fit' polyhedron for a distorted polyhedron (i.e., the M(2) octahedron in olivine, Dollase, p. 517). In this study, the methods of strain analysis have been used to define the distortions of M(l) and M(2) octahedra at high temperatures. The total elastic strain energy stored in an isotropic body equals the total work done during deformation, £ <£Q<£> which consists of two parts. One part i s that used in changing the volume (£ ) and the other part i s that used i n distortion (£ _ ) (change of shape s without a change in volume). This may be expressed as follows £ T 0 T = £ V + £ S (5) or in terms of the strain ' E TOT = 3 E , £ 2 + -2- G ^ o 2 ( 6 ) T 0 T 2 (l-2v) + 4 ° where E and G are the moduli of ela s t i c i t y and r i g i d i t y , respectively, v i s Poisson's ratio, £ i s the mean strain defined by £ = (^u + ^ 2 2 + 3^3^  — and % i s the octahedral unit shear (Mase, 1970) where ' „ - - ! - . [ ( ' ! - £ 2 ) 2 + C £ 2 - S ) 2 + ( £ 3 - £ l ) 2 J * <7) Substituting for 8 = ( £ ^  + £22+ ^ 33^  = ^ 1 + 2^ + 3^^  3-11(1 3 3 i equation (6) becomes £ T 0 T = K ( £ 1 + £ 2+ £- 3) 2 + K [ C £ I - 6 2 ) 2 -+ .( £ 2- + ( &3 " ^ l ^ (8) where E, G and v are a l l assumed constant. 51 The quantities £ and £ , which are proportional to the strain energies of dilation and distortion, respectively, have been calculated for M(l) and M(2) octahedra for the six olivines considered in this study. These parameters are presented in Table 12 and com-pared graphically in Figure lk. I t i s important to'-.note that £ and £ were evaluated for the temperature range of each refinement and therefore do not reflect the relative distortions of the octahedra at room temperature. When interpreting the significance of these parameters i t i s also important to bear in mind that (l) the thermal expansions of the octahedra are not isotropic and (z) the values of E, G and v are not constant from one polyhedron to another nor are they generally constant as a function of temperature. In spite of these assumptions, the method i s of use since relative trends and not absolute magnitudes are considered. In Figure 14, the £ values are simply reflecting the volume expansions of the octahedra since equals the dilation given by equation (3). It is interesting, however, that for the Fe-containing M2 Ml olivines £ v > . In view of the previous discussions on M(2) cation displacements and the thermal expansions in the M(l) and M(2) +2 octahedra, this suggests that Fe may behave differently than the other cations as a function of temperature. There has been considerable interest in the octahedral distortions in olivines, particularly in regard to the problem of Mg-Fe ordering. In hortonolite, Brown and Prewitt (1973) noted that the distortion of 52 Table 12. D i s t o r t i o n Parameters Characterizing the Volume ( £ ) and Shape Changes ( £ ) Associated with the Thermal Expansion of the Octahedra i n Six Olivines. O l i v i n e Oc tahedron £ v (x 10 V £ s (x I O 4 ) * * N i - O l i v i n e M ( l ) 13 .24 5 .35 ( 2 5 ° - 9 0 0 ° ) M(2) 11 .57 0 .10 F o r s t e r i t e M ( l ) 21 .03 9 .99 (25 ° -1000° ) M(2) 18 .46 1.05 H o r t o n o l i t e M ( l ) 3.49 2 . 6 0 ( 2 4 ° - 7 1 0 ° ) M(2) 4 .37 0 .22 F a y a l i t e M ( l ) 7 .33 3 .50 ( 2 0 ° - 9 0 0 ° ) M(2) 12 .04 1.05 M o n t i c e l l i t e M ( l ) 12 .77 7.54 ( 2 5 ° - 7 9 5 ° ) M(2) 8 .48 4 .16 G l a u c o c h r o i t e M ( l ) 11 .94 9 .08 ( 2 5 ° - 8 0 0 ° ) M(2) 11 .84 5 .86 * £ v = ( £ x +€ ** 2 + £ 3 ) 2 2 ) 2 + <e2 - £ 3 ) 2 + ( £ 3 - £ { ) 2 ; £ i > £ 2 > £ 3 F i g u r e B a r d i a g r a m c o m p a r i n g t h e d i s t o r t i o n p a r a m e t e r s £ a n d £ f o r M ( l ) a n d M(2) o c t a h e d r a i n s i x o l i v i n e s . P a r a m e t e r s c h a r a c t e r i z e t f i e v o l u m e a n d s h a p e c h a n g e s , r e s p e c t i v e l y , a s s o c i a t e d w i t h p o l y h e d r a l t h e r m a l e x p a n s i o n s . Un UO 54 M(l) octahedra, as measured hy the angle variance (Robinson, Gibbs and Ribbe, 1971), increased at a greater rate with temperature than the distortion of M(2) octahedra. Similar analyses of octahedral distortions in other olivines using the angle variance parameter (Lager and Meagher, 1974; Smyth, 1974), however, failed to indicate any di f f e r e n t i a l increase in distortions. Recently, on the basis of a high temperature Mossbauer study of six natural (Mg,Pe) olivines, Shinno, Hayashi and Kuroda (1974) conclude that the difference in distortion between M(l) and M(2) octahedra i s apparently greater at high temperatures, confirming the findings of Brown and Prewitt (1973)' In this study, the parameter £ indicates that the shape change or distortion associated with thermal expansion is greater for M(l) than M(2) octahedra in a l l olivines studied. The fact that the angle variance did not predict this result i s not surprising since the cal-culation does not account for changes in distances. Unit Gell Expansions With the possible exception of forsterite, unit c e l l parameters exhibit an essentially linear relationship with temperature (Figures 15, 16 and 17; Table 13). A comparison of linear thermal expansion coefficients (Table 8) indicates the following relationships: (l) ex. a ' v o t y ^ o C ^ for Ni-olivine, (2) oC increases relative to o 2 90 ) / l l where 9. i s the octahedral angle. i=l 55 Table 13. Unit Cell Parameters (X) for Ni-Olivine, Monticellite and Glaucochroite. Olivine T G Ni-Olivine 25 300 600 900 4.726(1; 4.744(1 4.760(3) 4.775(1) 10.118(2 10.151(2 10.179(4) 10.216(2) 5.913(1 5.933(2 5.951(3) 5-971(3) Monticellite 25 335 615 795 4.825(1 4.834(1 4.848(2) 4.857(3) 11.111(1) 11.147(2) 11.176(6) 11.199(6) 6.383(2 6.409(2 6.436(5) 6.454(6) Glaucochroite 25 300 600 800 4.913(1 4.924(1 4.937(2) 4.953(2) 11.151(2 11.178(2 11.212(4) 11.237(5) 6.488(1 6.504(6 6.532(8) 6.5^3(10) 56 57 Figure 17. Plot of the c c e l l edge versus temperature for six olivines. 59 200 400 600 800 1000 T °C 60 b in the (Mg,Fe) olivines with increasing Fe content and (3) *. ^ , °<- for the Ca-olivines. c b a The above relationships can be qualitatively interpreted in terms of the thermal expansions of the polyhedra by considering the paths through the crystal structure which define each cell edge ex-pansion. For example, the a cell edge expansion can be represented in terms of the strains in the M(l) octahedron and the silicate tetrahedron which are parallel to a. In strain notation, this can be expressed as Ml Si the sum £ + 6 n • In a similar manner, one*-can define the ex-pansions parallel to b (2 £ ^ + 2 £ 22p a n d " — ^ 2 ^ 33^' In Ni-olivine, the strains along the three paths are approximately equal and <*• ~ «c , ~ <K . The large thermal expansion along b in forsterite is due primarily to the magnitude of the strain parallel to b in the M(2) octahedron. In hortonolite and fayalite, the magnitude M2 Ml of 8 o n is decreased relative to £ and <*. shows an increase relative 22 33 c to ^ ^ (refer to discussion of M(2) deformation paths on page 46). The thermal expansions in the Ca-olivines reflect the magnitude of the component £ ^ i . The linear thermal expansion coefficients are compared with the octahedral strains in Figure 11. A Comparison of Thermal and Chemical Expansions It is of interest to compare the thermal expansions in olivines to the structure changes or chemical expansions which result from octahedral cation substitutions in the room temperature structures. Such a comparison has some validity since thermal expansions reflect increased thermal vibrations of the octahedral cations in much the 61 same way as chemical expansions reflect an increase in octahedral cation radius. Although the effective radius of "both oxygens and cations increase with temperature, the rate of increase in B with heating i s generally greater for the octahedral cations than for the oxygens in a l l olivines studied. In order to compare thermal and chemical expansions, strain ellipsoids were calculated for the cation substitutions l i s t e d in Table 14 following the procedures outlined in Appendix II for f i n i t e strain analyses. The f i n i t e strain calculation was used because, in some cases (i.e., Ga-olivines) ; the magnitude of strain exceeded 5%. Infinitesimal strain theory begins to break down when the strains are approximately greater than 2%, The magnitude and direction of the principal axes of these ellipsoids are presented i n Table 14. The longitudinal strains are given in Table 15-The cyclographic projections of the principal axes of the M(l) strain ellipsoids are plotted in Figure 18. I t i s informative to compare these ellipsoids with the thermal expansion ellipsoids in Figure 9- For M(l) octahedra, there i s a general correlation between the strains due to the effects of chemical substitution and temperature. In both Figures 9 and 18, the axes of the ellipsoids l i e approximately in the same area of the stereonet. . The orientations of the ellipsoids for chemical strain, however, are relatively insensitive to the chemistry of the octahedral sites, which i s in contrast to that noted previously for the thermal expansion ellipsoids. This result was rather unexpected since the orientations of the M(l) thermal expansion Table 14. Magnitudes and Orientations of the Chemical Expansion Ellipsoids for the Octahedra in Seven Room Temperature Refinements of Olivines. Compositional Range Principal Strain Component E± (xl02) Angle +a C) of +b E t with +c Ni 2-»Mg 2 M(l) 1.70 150 103 64 0.98 70 69 29 -0.55 69 155 77 M(2) 2.04 65 25 90 1.24 90 90 0 0.97 25 115 90 Mg 2-*Mg 1 > 8Fe 0. 2 M(l) 0.59 66 45 54 0.27 48 131 70 -0.05 129 105 43 M(2) 0.27 90 90 0 -0.01 148 122 90 -0.05 58 148 90 M8l.8 F e0.2"* M8l. 4 F e0.6 M(l) 1.40 31 64 75 0.71 128 62 41 0.15 78 140 53 M(2) 0.86 75 151 90 0.58 90 90 0 0.48 151 105 90 M8l.4 F e0.6"'' M(l) 1.06 35 69 64 0.08 111 105 27 -0.26 116 26 85 M(2) 0.87 76 166 90 0.26 90 90 0 -0.23 166 104 90 MgFe - Fe 2 M(l) 2.63 47 62 57 1.26 133 47 75 -0.64 105 124 38 M(2) 1.45 76 166 90 0.93 90 90 0 0.21 14 76 90 Mo-* Gl M(l) 6.65 37 69 62 4.32 123 39 71 -0.44 104 121 35 M(2) 1.92 90 90 0 1.04 16 74 90 -2.42 74 16 90 Gl-»Ca M(l) 11.92 43 62 60 4.28 133 56 62 0.87 93 133 43 M(2) 5.31 90 90 0 3.42 154 116 90 -0.69 64 154 90 *The E^'s were calculated according to the methods outlined for fi n i t e strain analyses. 64 Table 15. Longitudinal Strains Characterizing the Chemical Expansion of the Octahedra i n Seven Room Temperature Refinements of Olivines. Compositional Range E l l Unit Extensions* E 2 2 E33 Ni 2-*Mg 2 M(l) M(2) 0.0135 0.0117 -0.0024 0.0187 0.0105 0.0125 M82"* M g1.8 F e0.2 MU) M(2) 0.0020 -0.0002 0.0041 -0.0004 0.0021 0.0027 M S l . 8 F e 0 . 2 " * M 8 l . 4 F e 0 . 6 MU) M(2) 0.0119 0.0050 0.0051 0.0084 0.0055 0.0058 M S l . 4 F e 0 . 6 " > M 8 F e M(l) M(2) 0.0068 -0.0017 -0.0007 0.0081 0.0027 0.0026 MgFe ~»Fe 2 M(l) M(2) 0.0179 0.0029 0.0098 0.0138 0.0048 0.0093 Mo-»Gl M(l) M(2) 0.0536 0.0077 0.0329 -0.0217 0.0169 0.0191 Gl-»Ca M(l) M(2) 0.0802 0.0268 0.0420 0.0004 0.0435 0.0517 * E i i = + 2E^^)l/2 _ i ] where the Ej^'s are the diagonal components of the f i n i t e s t r a i n tensor. 65 Figure 18. Cyclographic projection of the principal axes of the M(l) chemical expansion ellipsoids for seven room temperature olivine structures. Ellipsoids were calculated for the chemical substitutions in Table lk. Solid, open and dotted symbols refer respectively to the longest, inter-mediate and shortest axes of the ellipsoids. A l l axes plotted are at an angle less than 90° with respect to c. See text for further explanation. 66 ellipsoids appear to refle c t the structural differences between M(l) octahedra as a result of an increase in cation size. The lack of a detailed correlation seems to suggest that other factors, in addition to those accounted for by the chemical strain model, make important contributions to the M(l) strains at high temperatures. Precisely what these other factors are, however, i s not clear. Attempts to interpret the variation in terms of individual thermal vibration ellipsoids have been unsuccessful probably because the strains at high temperature reflect the interaction between the vibrations of a l l atoms in the polyhedron. The above correlation between the thermal and chemical expansion ellipsoids i s less evident for M(2) octahedra. The chemical deformation paths for both M(l) and M(2) octahedra are plotted in Figure 1 9 . A comparison of this figure with those for thermal expansion indicates, as did Figure 1 8 , a similarity between the effects due to temperature and chemical substitution. The correla-tion i s particularly well illustrated in the case of monticellite +2 which may refle c t the fact that the rate of increase in B for Mg is +2 greater than for Ga a. CN O 8->< CN x.. \ 2-l i i i \ / / / w / • yy -8 -6 4 1 / /* / -4-/ / / / 2 4 o 8 ( E ^ x l O 2 A Mo->GI OGI-»Ca A M I AM2 b . Figure 19. Plot showing the deformation of M(l) and M(2) octahedra due to chemical substitutions in the non-Ca (l9a) and Ca-olivines (l9b). The differences between strains were calculated for the chemical substitutions i n Table 15 Open and solid symbols refer respectively to M(l) a n d M ( 2 ) octahedra. O x ^ 3 68 IV. SUMMARY OF RESULTS The olivine structure expands with increasing temperature primarily as a result of the bond length expansions in the non-tetra-hedral sites. The shapes and orientations of the polyhedral thermal expansion ellipsoids indicate that the octahedral expansions are related to the topology and chemistry of the M(l) and M(2) sites. The basic shape of the ellipsoid i s due to the structural configuration of the octahedron while i t s orientation reflects the occupancy of the site. For M(l) octahedra, the orientation i s apparently related to the mean cation radius. The results are less evident for M(2) octahedra. The M(2) cation position in a l l six olivines i s temperature dependent. With increasing temperature, the cation i s displaced in a direction away from the triangle of shared edges. The magnitude of this displacement, measured relative to the centroid of the octahedron, 4* 2 "H 2 i s greatest for the non-transition metal cations (Mg , Ga ). The 2+ 2+ \ smaller displacements exhibited by the transition cations (Ni ,Fe ) may be related to metal-metal interactions between-adjacent M(l) and M(2) octahedra. The magnitude of the displacements i n the structures at room temperature varies almost linearly with increasing cation radius for the non-Ca olivines. This i s consistent with a simple hard sphere model which i s based on the observation that the length of the edge shared between the M(2) octahedron and the s i l i c a t e tetrahedron remains essentially constant with composition. The significance of the displacements in the Ca-olivine i s d i f f i c u l t to evaluate since no data are available for intermediate end-member compositions. On the basis of 6 ? the M(2) cation displacements at high temperatures and the thermal expansion data for M(l) and M(2) octahedra, i t is suggested that Fe behaves differently than the other octahedral cations with heating. The magnitudes of the expansion (A = £^ + €^ + £3) for the octahedra are proportional to the thermal expansions of mean M-0 distances. For the olivines considered in this study, mean Mg-0 distances ex-hibit the greatest thermal expansions while the mean distances in the Fe-containing olivines generally exhibit the smallest expansions. This i s in agreement with the results of the pyroxene high temperature studies (Cameron et a l . , 1973) ' The polyhedral distortion parameter £ , which defines the shape changes associated with thermal expansion, is greater for M(l) than M(2) octahedra for the temperature range of a l l olivine refinements. Brown and Prewitt (1973) a n ( i more recently Shinno et-al.(1974) have also indicated a di f f e r e n t i a l rate of increase in the octahedral distortions in (Mg,Fe) olivines. The variation in the unit c e l l parameters with temperature can be expressed in terms of the octahedral strains within the crystal structure. This study indicates that the b c e l l edge in the non-Ca olivines i s primarily controlled by expansions in the M(2) octahedra. The thermal expansion coefficients for the c e l l parameters in the (Mg,Fe) olivines are consistent with the decrease in the strain com-M2 Ml ponent £ ^ relative to £ ^  with increasing Fe content. A compari-son of thermal and chemical expansions indicates that a general correlation exists between the effects due' to temperature and chemical substitution. The lack of a detailed correlation suggests that room 70 temperature data f o r the o l i v i n e s cannot he used to p r e d i c t the magnitude and d i r e c t i o n of the s t r a i n s at high temperatures. 71 SELECTED REFERENCES Bence, A.E., and A.L. Albee (1968) Empirical correction factors for the electron microanalysis of s i l i c a t e s and oxides. J. Geol. 76, 382-403. Bird, G. (1972) Calculating a l l the eigenvalues and eigenvectors of a symmetric matrix. University of Br i t i s h Columbia Computing  Center. Birle, J.D., G.V. Gibbs, P.B. Moore, and J.V. Smith (1968) Crystal structures of natural olivines. Amer. Mineral. 53, 807-824. Bjerhammer, A. (1973) Theory of Errors and Generalized Matrix Inverses, Elsevier Scientific Publishing Co., New York. Bowdler, H., R.S. Martin, C. Reinsch, and J.H. Wilkinson (1968) The QR and QL algorithms for symmetric matrices. Numer. Math. 11, 293-306. Bragg, W.L., and G.B. Brown (1926a) Die struktur des olivins. Z. Kristallogr. 63, 538-556. Brown, G.E. (1970) Crystal Chemistry of the Olivines, Ph.D. Thesis, Virginia Polytechnic Institute and State University, Blacksburg, Virginia. Brown, G.E., and G.T. Prewitt (1973) High-temperature crystal chemistry of hortonolite. Amer. Mineral. 58, 577-587. Burnham, CW. (1966) Computation of absorption corrections and the significance of end effect. Amer. Mineral. 51, 159-167. Burns, R.G. (l970b) Mineralogical Applications of Crystal Field Theory, Cambridge University Press, Cambridge, England. Busing, W.R., K.O. Martin, and H.A. Levy (1962) ORFFE: A fortran crystallographic least-squares refinement program. U.S.  Clearinghouse Fed. Sci. Tech. Info. Doc..., ORNL-TM-305. Busing, W.R., K.O. Martin, and H.A. Levy (1964) ORFFE: A fortran crystallographic function and error program. U.S. Clearinghouse  Fed. Sci. Info. Doc., ORNL-TM-306. Cameron, M., S. Sueno, CT. Prewitt, and J.J. Papike (1973) High-temperature crystal chemistry of acmite, diopside, hedenbergite, jadeite, spodumene and ureyite. Amer. Mineral. 58, 594-618. 72 Cruickshank, D.W.J. (1965) Errors in least-squares methods. In J.S. Rollett, Ed. Computing Methods in Crystallography, Pergamon Press, New York. Dollase, W.A. (1974) A method of determining the distortion of co-ordination polyhedra. Acta. Cryst. A30> 513-517-Doyle, P.A., and T.S. Turner (1968) Relativistic-Hartree-Fock x-ray and electron scattering factors. .Acta. Crystallogr. A24, 390-397. Foit, F.F., and D.R. Peacor (19&7) A high temperature furnace for a single crystal x-ray diffractometer. J. Sci. Instrum. 44, 183-185. Ford, H. (1963) Advanced Mechanics of Materials, John Wiley and Sons, Inc., New York. Frederick, D., and S.T. Chang (1965) Continuum Mechanics. Allyn and Bacon, Boston. Hamilton, W.C. (1959) 0 n "the isotropic temperature factor equivalent to a given anisotropic temperature factor. Acta. Crystallogr. 12, 609-610. +2 Ghose, S., and C. Wan (1974) Strong site preference of Co in olivine, Co. . ~Mg~ __Si0i.. Contrih. Mineral. Petrol. 47, 131-140. 1 , 1 0 0 , 9 0 4 Goluh, G., and C. Riensch (1970) Singular value decomposition and least squares solutions. Numer. Math. 14, 403-420. Householder, A.S. (1965) The Theory of Matrices in Numerical Analysis } Blaisdell Publishing Co., New York. Lager, G.A., and E.P. Meagher (1974) The crystal structure of Ni-olivine (NigSiO^) at high temperatures. (Abstr.) EOS 5, 350. Lager, G.A., and E.P. Meagher (1975) Strain analysis of polyhedral distortions in olivines at high temperatures. (Abstr.) Geol. Soc. Amer. Abstr. Programs 7» 1158. Malvern, L.E. (1968) Introduction to the Mechanics of Continuous  Medium, Prentice-Hall, New Jersey. Martin, R.S., and J.H. Wilkinson (1968) Reduction of the symmetric eigenvalue problem Ax = ^ Bx and related problems to standard form.. Numer. Math. 11, 99-110. 73 Mase, G.E. (1970) Schaum's Outline of Theory and Problems of Continuum  Mechanics, McGraw-Hill Book Co., New York. Meagher, E.P. (1975) The crystal structures of pyrope and grossularite at elevated temperatures. Amer. Mineral. 60, 218-228. Nelson, J.E., and D.P. Riley (19^5) An experimental investigation of extrapolation methods in the derivation of accurate unit-cell dimensions of crystals. Proc. Phys. Soc. (London) 57> 160-177-Newham, R.E., R.P. Santoro, J.H. Fang, and S. Nomura (1965) Antiferromagnetism in nickel orthosilicate. Acta. Crystallogr. 19, 147-148. Noll, W. (1974) The Foundations of Mechanics and Thermodynamics: Selected Papers, Springer-Verlag, New York. Novoghilov, V.V. (1953) Foundations of the Nonlinear Theory of  Elasticity, Graylock Press, Rochester, New York. Nye, J.F. (1957) Physical Properties of Crystals, Oxford University Press, London. Papike, J.J., Malcolm Ross, and Joan R. Clark (1969) Crystal-chemical characterization of clinoamphiboles based on five new structure refinements. Mineral. Soc. Amer. Spec. Paper 2, 117-136. Rajamani, V., G.E. Brown, and G.T. Prewitt (1975) Cation ordering in Ni-Mg olivine. Amer. Mineral. 60, 292-299. Ramsay, J.G. (19^7) Folding and Fracturing of Rocks, McGraw-Hill Book Go., New York. Robinson, K., G.V. Gibbs, and P.H. Ribbe (1971) Quadratic elongation: A quantitative measure of distortion in coordination polyhedra. Science, 172, 5^7-570. Rust, B.W., and W.R. Burrus (1972) Mathematical Programming and Numerical  Solution of Linear Equations, Elsevier Scientific Publishing Go., New York. Saada, A.S.(1974) El a s t i c i t y : Theory and Applications, Pergamon Press, Inc., New York. Shannon, R.D., and G.T. Prewitt (1969) Effective ionic r a d i i in oxides and fluorides. Acta. Crystallogr. B25, 925-946. Shlnno, I., M. Hayashi, and Y. Kuroda (1974) Mossbauer studies of natural olivines. Mineral. J. Japan 7, 344-358. 74 Smith, D.K., A.J. Majumhar, and F. Ordway ( l 9 6 l ) Re-examination of the polymorphism of dicalcium s i l i c a t e . J. Amer. Geram. Soc. 44, 405-411. Smith, D.K., A. Majumhar, and F. Ordway (1965) The crystal structure of f -dicalcium s i l i c a t e . Acta. Crystallogr. 18, 787-795-Smyth, J.R., and R.M. Hazen (1973) The crystal structures of forsterite and hortonolite at several temperatures up to 900°C. Amer. Mineral. 5§, 588-593-Smyth, J.R. (1975) High temperature crystal chemistry of fayalite. Amer. Mineral. , impress. Streat, J. (1973) Singular value decomposition of a matrix. University  of B r i t i s h Columbia Computing Center. Sueno, S., M. Cameron, J.J. Papike, and C.T. Prewitt (1973) High-temperature crystal chemistry of tremolite. Amer. Mineral. 58, 649-664. Takeda, H. (1972) Grystallographic studies of coexisting aluminan orthopyroxene and augite of high pressure origin. J. Geophys. Res. 72, 5798-5811. Timoshenko, S., and J.N. Goodier (1970) Theory of Ela s t i c i t y , McGraw-H i l l Book Co., New York. Wan, C , and S. Ghose (1974) Strong site preference of Co in olivine, Co^ 1 0Mg Q 9 o S i 0 4 * G° n trih. Mineral. Petrol. 47, 131-140. Wayman, CM. (1964) Introduction to the Crystallography of Martensitic Transformations, Macmillan Co., New York. Wenk, H.R., and K.N. Raymond (1973) Four new structure refinements of olivine. Z. Kristallogr. 137, 86-105. 75 APPENDIX I Table 16. Observed and Calculated Structure Factors for Ni-Olivine. 7 7 C S S R F S a : s g S g S S S S 3 K S 3 S g ; E S S S r 2 S S = S S f S E K S 5 g 3 S 2 : E P S - K X K ! ; 5 R Table 17. Observed and Calculated Structure Factors for Monticellite. 79 £ i s s s ^ s 2 s s 5 *' s s s s ^  ^-o^^^^^n'^^^^v'^v^^ Table 18. Observed and Calculated Structure Factors for Glaucochroite. 81 | 82 i APPENDIX I I 83 APPLICATION OF THREE-DIMENSIONAL STRAIN THEORY TO AN ANALYSIS OF POLYHEDRAL DISTORTIONS IN OLIVINES Strain Theory Any general deformation can he characterized hy a set of deformation equations such as X i = f(x 1 , x 2 , x 3 , t ) ( l ) where x^,x^,Xj are the coordinates of the 'material point' in the reference state and X^X^X^ are i t s coordinates at time t. When the reference state i s the i n i t i a l undeformed state (at t=0), the deformation i s said to he described in terms of the Lagrangian description. This method of description w i l l be used throughout the discussion. Assume that the undistorted cube in Figure 2 0 represents an infinitesimal volume in a crystal which undergoes a deformation. If x = x ^ i + XgJ + Xjk defines the position of any point (for example, 0,0,X^) in this cube relative to the origin of the coordinate axes at 0, u = u^i + U g j + u^k i s the displacement of a point due to deformation and X = X^i + X^j + X^k defines i t s current position at time t, then the functional relationship between the vectors x^ and X^ can be expressed by the deformation-gradient tensor as dX. = < J- X i. dx, - F, .dx. ( 2 ) 1 ~d%7 3 1J 3 where the X^'s and the x / s are the components of the respective 1 1 Figure 20. The undistorted and distorted cubes refer to an infinitesimal volume in a crystal before and after deformation. Material points in the undeformed crystal (for example, 0 , 0 , X 3 ) are displaced u= U j _ i + u 2J + u^k as a result of the deformation (modified after Bloss, 1971) • 85 position vectors along the coordinate axes. Since X^ = x^ + u^, where u^ = f(x^, x^, t ) , partial differentiation of X^ with respect to x. yields the result. 3 du. =°* ui dx.. (3) 5 u ^ The partial derivative i , defines nine tensor components which 2XJ characterize the variation of the displacement u with position x i n an infinitestimal volume element. The tensor ^ U i i s sometimes denoted hy e ^ (Nye, 1957)• In (3) since the displacements are linear functions of the i n i t i a l coordinates, the deformation can he considered to he homogeneous (i.e., the state of strain does not vary from point to point in the cube in Fig. 20). A characteristic feature of homogeneous deformations i s that lines in a body which were straight before deformation w i l l remain straight after deformation and parallel lines w i l l remain parallel. In addition, a spherical volume element inscribed in the undistorted cube in Figure 20 w i l l deform into an ellipsoid (the magnitude ellipsoid, Nye, 1957, p.' 4?). I f the magnitudes of the r i g i d body rotations are to be considered, i t i s necessary to characterize the deformation-gradient tensor F ^ i n terms of irrotational (pure strain) and rotational (rigid body rotation) components. For a f i n i t e deformation, F^^ can be expressed as a product of a symmetric tensor, G. ., and an orthogonal tensor, R. . : F. . = R.T J CT . (4) (Noll, 1974). In (4), R.^ i s the nonlinear rotation tensor given by 86 R. .= l ^ ( l - c o s « ) + c o s w ^1^2^ l ~ c o s u ' ) ~ 1 3 s i n w l^l^Cl-cosu^+lgSinu; 2 l g l ^ C l - c o s ^ + l ^ s i n w lgCl-coswH-cosu> lgl-^Cl-cosw^-l^sinu* Jl^l^ d - c o s u ^ - l g S i n a ' l^lgd-cosu^+l^sinu/ l2(l-cosw)+cosu> where »*» i s the angle of rotation ahout the unit vector | l ^ , lgjl-^J If we represent in condensed form hy the matrix i t can he shown that = cos and that 111 h3 hi 122 x23 1 1 1 [«in + ^2 + *33 > - l ] (5) 1 = 132" 123 , 1 2 = 113 ~ 131, and 1 = 121 " 112 2 sin u/ 2 sin w 2 sin u/ 4 G^j i s the positive square root of the symmetric deformation tensor which can he expressed i n terms of the deformation gradient as S = A/ G^J where S i s referred to as the right stretch tensor (Mase, 1970). The positive square root of a symmetric matrix G can-be calculated from the transformation A. i T G 2 = XC 2 X . In the above, X i s the orthogonal matrix which diagonalizes G (j.e., the matrjx of eigenvector|), X^ i s the transpose of X and CI" = diag [(^^)"2, ( X 2 ^ ' ^ ^o)"2"]where the A 's are the eigenvalues of C (Wayman, 19647. 8? G. . = r r. 1 J Dx± dx. C. . i s related to the nonlinear strain tensor, E. ., which is given by where E. . = K c .-$..) ^ . . = 1 for i = j . = 0 for i i t j (6) Equation (6) i s generally expressed in terms of the displacement as E. . = | du. , 2u. , Pu 2u x + j + r r 3x. Px^ Px. 3x. (7) The summation on r in (7) i s from r = 1 to 3- Eor example, the E and E^^ components have the form 1 1 a x 1 L\ax t •/ v^ / " ^ x t / E1 2 = 2 fc+ ^ 1+ I [ ^ 1 ^ 1 + ^ 2 ^ ^ ^ 3 u [ 9 x 2 J ^x 2 ^x^ ^x 2 ^ ,?x, 3 2 J (8a) (8b) If the tensor F.^ is known, then the symmetric deformation tensor G. . can he determined since G. . = F.T. F. . (9) T (see, for example, Noll, 197^) where . i s the tensor transpose of F. .. The rotation tensor R. . can then he calculated from (4). The components of E. . (i.e.-, longitudinal or normal strains) are related to the unit extensions, E^^, along x^, Xg, x^ hy 88 E i. = r 11 L 1 + 2 E . C . l 2 - 1. l l J (10) The off-diagonal terms (i.e., shear strains) depend on "both the extensions and the shear angle, <j> .. , and can he formulated as E. s i n ^ ( 1 + E. .) ( 1 + E. .) ( H ) where ^ „ i s the angle between line segments which were parallel to the coordinate axes before deformation. If the assumption i s made that the displacement gradients are small, 1 (infinitesimal strain), the equations (7v 8, 10, ax. 11) developed for nonlinear f i n i t e strain can be simplified. For example, the product terms in (7) can be neglected in comparison to the linear terms and the deformation due to pure strain i s given by the linear strain tensor 5u, I J 2 3x. 9x. (12) I t can also be shown that the relations in (lO) and ( l l ) for the longitudinal and shear strains reduce to and i i £. . E..,. i i If we now define the linear rotation tensor as 3u. I 3 u 3 x ± J (13) the deformation equation in (4) can be conveniently expressed as a sum of symmetric and asymmetric parts. From (12) and (13)» (3) can be written as 89 du. = £.. dx . + <*/.. dx . • (14) i i j j i j j where the products £ . .dx . and UJ. .dx. express the displacement due to pure strain and rotation, respectively. Assuming that the £^'s and w^'s are constant as a function of position (homogeneous deformation) and integrating (l4), we obtain in expanded form u l = £ l l x l + £12 x2 + 6 i 3 x 3 " "l2 x2 + U ' l 3 x 3 ( l 5 a ) U2 e 2 1 X l + 622 X2 + £ 2 3 X 3 + "21X1 ' ^ 3 . ( l 5 b ) u3 = S l x l + 6 3 2 x 2 + e33 x3 + U 3 2 x 2 - w 3 1 x l ( l 5 c ) It can be shown (see, for example, Frederick and Chang, 19&5) that the displacements due to the rotational part of (l4) are given by the vector cross product yj x Xj_ where x^ i s the position vector and u v i s an axial vector (also called a pseudovector) denoted by 32- 13 a 21- . (16) The components ^Jj2' w13' w21 c a n ^ e S e o m e " ' : r i c a l l y interpreted as the mean angles of rotation about x^, x^, and x^, respectively, of line segments which were perpendicular to these axes before deformation. For example, W ^  i s the rotation about x^ towards Xg of a line segment parallel to x^ (Nye, 1957)• Since small rotations can be added vectorially, the tensor W ^± describes a counterclockwise rotation of | <x<| =J(U,^2^ + (bJ^j)^' + ^21^ r a < ^ i a n s about the directionu / . From elementary vector analysis, the orientation of is specified by 90 cos( LKI A x 1) = w / |^| cos( uj A x 2) = 00^3/ cos(u/ A x-j) = U/ 2 1/|u;| (17a) (17b) (17c) If the i n i t i a l positions, x^, and the f i n a l positions, X^, are known for the coordinating atoms in a polyhedron, then the i r -rotational and rotational components of strain can be easily deter-mined from (4) and (6) and (l4) for the cases of f i n i t e and i n -finitesimal strain, respectively. Considering f i r s t infinitesimal strains and expressing the system of linear equations in (15) in matrix notation, we have M = A b + m mn n m (18) where M is a (m x l ) displacement vector (u^'s), b i s a (n x l ) solution vector of strain tensor components ( £/. .'s and -*J . ,'s), A i s a (m x n) position matrix (x^'s) and f> i s a (m x l ) column vector of observational errors. The elements of the vectors and matrices in (l8) are: ' u l " — x i X 2 X3 " x 2 x 3 0 0 0 0 u 2 0 X l 0 X l 0 x 2 x 3 x 3 0 u 3 • p 0 • X l • 0 • • " X l - 0 X2. • x 2 • x 3 • l • 4 • 4 • x j  x 3 • -X3  X 2 • j x 3 0 t 0 • 0 • 0 4 0 4 0 x3' X l 0 xj  X 2 x j x 3 -X3 x 3 0 4 0 0 X3' X l 0 -X3 X l 0 xj  2 x j  2 4 + >1 £12 £13 21 Si £22 £23 32 e 33 /V 91 where the superscript j refers to the number of coordinating atoms in the polyhedron. In the most general case, for a p-coordinated site with (or l ) symmetry, m = 3P' When the displacement gradients were approximately greater than 0.05, the S. .'s and E. .'s calculated from (6) and ( l4) differed ^~ 3 ^-3 hy 3-1070. In such cases, the equations developed for f i n i t e strain apply and ( l8) has the form 11 X 2 X 3 j X 2 L X3J x l X 2 x 3 0 0 0 0 0 0 x l X 2 x 3 0 0 0 0 0 0 0 0 0 0 0 0 x l x 2 x 3 ' 0 0 0 0 0 0 0 0 0 x j  X 2 x j  X 3 0 0 0 0 0 x j  X l X5 X 2 x J  x 3 V + F 12 F13 F 21 F 22 F 2 3 F 31 F 32 F . 33. where Amn and jo axe defined as in (l8), u i s a (m x l ) position vector (X.'s) and h i s a solution vector of F. .'s. The strain and 1 i j rotation tensors can he calculated from the F. .'s according to the methods outlined in the previous section. If m > n and rank (A) = n, then the solution to (l8) i s unique and i s given hy the classical linear regression model as (21) where G i s the generalized inverse of A and satisfies the following four conditions: * A h = G u n mn m 92 i ) AGA = A i i ) GAG = G i i i ) (AG) T = AG iv) (GA) T = GA. Now i t has been shown that any matrix A with m> n can be factored in the form A = U "£ vJL (22) mn mm*- n n v ' where U i s a ( m x m) orthogonal matrix consisting of the orthonormalized T eigenvectors AA and V i s a (n x n) orthogonal matrix consisting T of the orthonormalized eigenvectors of A A. ^ i s a diagonal matrix of the form 2 = ; D = diag ( % ^ , f 2 , . . . f n ) where the ^ n ' s a x e "the non-negative square roots of the eigenvalues T of A A and are called the singular values of the matrix A. The above scheme i s called the singular value decomposition (SVD) of A. Because of the limited number of operations required in the de-composition, this technique probably gives the highest accuracy when computing a generalized inverse (Bjerhammer, 1973)• In this study, || UTU-I || , [I vVl ,-5 oO 1 T 1 I and A-U V were o O ' the order of 10 v where 11 A A is given by o n = max a. . 1 3 i J (23) (Bjerhammer, 1973)- In (23) the sums of the absolute values of the elements in each row of matrix A are computed. 11 A 11 i s the largest value calculated. From (22) i t can be shown (Golub and Reinsch, 1970) that the generalized inverse of A, G, can be expressed as 93 ^ + T G = V £ U i ' where 1+ = diag (% t ) and ' l/< . f or % .. > 0 ' ? i 11 0 for | , = 0. (i A + T The best estimate for b, b = Uu, i s the vector which minimizes ||u - A b j J 2 ( j|x|| 2 = ( $ x ? ) 2 for any vector x). The strain and rotational tensor components were calculated using a iterative least squares method adapted from Golub and Reinsch (1970) by J. Streat (1973)- The eigenvalues and eigenvectors of the symmetric strain tensor £j_j» i-e., the magnitudes and directions of the principal axes of the strain ellipsoid, were calculated from a subroutine modified by C. Bird (1972). The strain tensor £ . . i s reduced to a ^-3 symmetric tridiagonal matrix using Householder transformations (Martin, Reinsch and Wilkinson, 19^8). QL transformations are then used to find the eigenvalues and eigenvectors of £ ^  (Bowdler, Martin, Reinsch and Wilkinson, 1968). Error Analysis The variance of the matrix-vector product A b is * mn n var (Ab) = A var (b) A T = AMAT T where A = V£ U and M is the moment matrix (variance-covariance matrix). The variance of b i s therefore given by A A T var (b) = var (Au) = A var (u)A . (24) In (2 4) 94 var (u) = var E [ (u - Ab) (u - Ab) T] * S where E is the expected value (Rust and Burrus, 1972). If a l l u. 2 are uncorrelated and have the same variance, <T , S reduces to a diagonal matrix of form S = <r2I . (25) Substituting var (u) = <T I and A = V £ U in (25), we have var (b) = (v i u^ff^rCv i u T = <r2 (v i + u T) (u / v T ) = T 2 (V 1 + ) (UTU) ( £ + V T) = <r2 ( v l + ) ( v £ + ) T . 2 If the unbiased estimator of 0" i s 0- 2 = E(f T ) = £ (u - Ab) (u - Ab) T = g(u - Ab) 2 m-n u m-n u m-n then var (b\)= var (K) = g (u - Ah) 2 (26) u m-n (V $ ) (V £. ) . For f i n i t e strains, (27) gives the variance of the F. .'s. Propagation of error techniques can then be used to obtain estimates of the errors in the E. .*s and R. .'s. The above error analysis takes into account the accidental or random errors in a set of linear equations. Another kind of error which can be c r i t i c a l in a calculation of this type i s the rounding error. Reconsider the system of linear equations in (18) •95 u = A b + P m mn n i m i f the rounding error i n u i s denoted hy £ then u + p = A(h +Ah) where Ab i s the absolute e r r o r i n b. From ( 2 1 ) we have b + A b = G u + G | » - G^ > where G i s the generalized inverse of A. Since b = Gu Ab = G^ - Gj> = G(f ~ f l • Expressing ( 2 8 ) i n matrix norms (see Householder, 1964 f o r a discussion of norms), i t can be shown that f o r a p a r t i c u l a r ( 2 7 ) ( 2 8 ) s o l u t i o n u < Ab m\<- I 5 -K(A) / I N I ) (29). where ||x|| = || x|| = maxJx^J(Bjerhammer, 1973) and K.(h) = ||G | IA j ' l i s the condition number of matrix A. The quotients | |Ab and ||/°||/ 11 are measures of the r e l a t i v e error i n b and u, r e s p e c t i v e l y . Therefore, the i n e q u a l i t y i n (29) gives an upper A bound f o r the r e l a t i v e error i n b i n terms of the r e l a t i v e error i n u. I f || i s known, then J A b | | /||b|| can be e a s i l y determined from the procedure SVD since K(A) (30) where and ^ are, r e s p e c t i v e l y , the maximum and minimum singular values calculated i n the decomposition of A. Inspection of (29) indicates that f o r large values of K(A), i . e . , the matrix A i s poorly conditioned, small r e l a t i v e changes i n u. w i l l produce large r e l a t i v e changes i n % . 96 Application to the Structure of Olivine Since the M(l) octahedron in olivine has (or l ) symmetry, A ^ reduces to a (9 x 9) matrix and ( l8) can he solved exactly. For M(2) and S i polyhedra, which have G v (or m) symmetry, A ^ has dimensions ( l8 x 9) and ( l 2 x 9 ) , respectively. The higher symmetry of M(2) and S i polyhedra introduces a simplification in the strain and rotational tensors. The 13 and 23 components vanish and the tensors can he written as: £ . . = hi £12 0 0 12 0 '21 £ 2 2 0 U J 21 0 0 0 0 £ 33. 0 0 0 E. . = " E l l E 12 0 ; R ± J - ' R l l R 12 0 E21 E 22 0 R 2 l R 22 0 0 0 E 33 0 0 R 3 3 The standard errors ( <T ) calculated hy the methods discussed in the previous section are comparable in magnitude to the strain and rotational tensor components. Rather than tabulate the (T 's in Tables 11 and 15, the moment matrix, M, for a typical error calculation i s presented below; the M given i s for the thermal strain components (25—«-900°C) for the M(2) octahedron in Ni-olivine (see Table l l ) . 97 M = 2.86 .08 0 -.08 0 0 0 0 0 .08 1.03 0 • 13 0 .08 0 0 0 0 0 1.30 0 -.08 0 .04 .04 0 -.08 .13 0 1.30 0 .08 0 0 0 0 0 -.08 0 1.35 0 -.04 -.04 0 0 .08 0 .08 0 2.35 0 0 0 0 0 .04 0 -.04 0 1.23 -.05 0 0 0 .04 0 -.04 0- -.05 1.23 0 0 0 0 0 0 0 0 0 2.55 xlO In order to obtain an estimate of the errors in the magnitudes and orientations- of the principal strain components, the eigenvalues and eigenvectors of j^E.^ + 0"j , ~ °"] a n d- + ' [ E i j ~ were determined for a number of different strain states. The range in values calculated i n this manner were used as estimates of the errors. These indicate that the errors in the £.'s and E.'s are I I comparable to those in the £. .'s and E. .'s whereas the errors in the eigenvectors are approximately "*_15°. An estimation of the rounding error in b, A b , can be made from (29) and can be expressed as A b = [<(A) . ( \\/\\ I ||u|| )] . ||b|| . In a l l cases, 1.5 < K(A) < 2.0, where K(A) = 1 for an orthogonal matrix. From (23). < .00005 and, in general, < .10 and u < .20. Choosing the maximum estimate in each case, Ab ~ .00005 • A . A Thus, b i s not sensitive to errors m u. 

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