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The computation of chemical equilibrium and the distribution of Fe, Mn and Mg among sites and phases.. 1987

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THE COMPUTATION OF OF FE, MN AND MG CHEMICAL EQUILIBRIUM AND THE DISTRIBUTION AMONG SITES AND PHASES IN OLIVINES AND GARNETS by CHRISTIAN DE CAPITANI D i p l . Mineralog-Petrograph, U n i v e r s i t y of Berne, 1983 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of G e o l o g i c a l Sciences We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1987 © CHRISTIAN DE CAPITANI, 1987 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the The U n i v e r s i t y of B r i t i s h Columbia, I agree 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 study. I f u r t h e r agree that permission f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s or her r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of G e o l o g i c a l Sciences The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: September, 1987 A b s t r a c t A g e n e r a l a l g o r i t h m f o r the computation of chemical e q u i l i b r i a i n complex systems c o n t a i n i n g n o n - i d e a l s o l u t i o n s has been developed. The method i s a G-minimization based on repeated l i n e a r and n o n l i n e a r programming s t e p s . A computer program (THERIAK) based on t h i s a l g o r i t h m has been w r i t t e n and was used to sol v e a great v a r i e t y of problems, ranging from a simple b l a s t furnace c a l c u l a t i o n to l i q u i d - l i q u i d unmixing i n a four component s i l i c a t e melt. The computing times are i n the magnitude of 1/2 to 2 seconds f o r each c a l c u l a t i o n . The method can a l s o be used to t e s t the consequences of thermodynamic models and data i n systems of i n t e r e s t to many f i e l d s , i n c l u d i n g chemistry, geochemistry and m e t a l l u r g y . I n t e g r a t e d powder d i f f r a c t i o n i n t e n s i t i e s can be used to measure Mn-Mg and Fe-Mg s i t e occupancies i n o l i v i n e s because of the d i f f e r e n c e i n s c a t t e r i n g f a c t o r s between Mg and Mn or Fe. T h e o r e t i c a l l y c a l c u l a t e d i n t e n s i t y r a t i o s are s u b j e c t to u n c e r t a i n t i e s from p o s i t i o n a l parameters ( l e s s than 3.5 % for peaks with a r e l a t i v e i n t e n s i t y g r e a t e r than 25 %) and unknown charge d i s t r i b u t i o n (up to 30 % ) . S e v e r a l peak r a t i o s are l e s s s u b j e c t to t h i s l a s t u n c e r t a i n t y and may be used to measure the s i t e occupancies i n o l i v i n e s . 27 s y n t h e t i c Fe-Mn-Mg o l i v i n e s (800 °C, vacuum, with g r a p h i t e ) were i n v e s t i g a t e d with Mossbauer spectroscopy and XRD i n t e n s i t y e v a l u a t i o n , producing occupancies a c c u r a t e to approximately 0.03 per s i t e . A thermodynamic s p e c i a t i o n i i model r e p r e s e n t s the data very w e l l . P r e l i m i n a r y Fe-Mn-Mg exchange exper iments i n v o l v i n g o l i v i n e and garnet p l a c e some l i m i t s on element d i s t r i b u t i o n s between these two m i n e r a l s . Table of Contents A b s t r a c t i i Table of Contents i v L i s t of Tables v i i i L i s t of F i g u r e s x Acknowledgements x i i INTRODUCTION 1 1. THE COMPUTATION OF CHEMICAL EQUILIBRIUM IN COMPLEX SYSTEMS CONTAINING NON-IDEAL SOLUTIONS 3 1.1 The fo r m u l a t i o n of a gen e r a l a l g o r i t h m 3 1.1.1 I n t r o d u c t i o n 3 1.1.2 T h e o r e t i c a l fundamentals 4 1.1.2.1 Non-unique s o l u t i o n s to the e q u i l i b r i u m problem 5 1.1.2.2 Formulation of the chemical e q u i l i b r i u m c o n d i t i o n 10 1.1.3 Phases with f i x e d compositions 11 1.1.4 Adding s o l u t i o n phases to the system 15 1.1.5 Summary of the a l g o r i t h m 19 1.2 D e t a i l s of the computation of chemical e q u i l i b r i u m 21 1.2.1 The i n i t i a l assemblage 21 1.2.2 The n o n l i n e a r programming ( s t e p 1 of algorithm) 21 1.2.2.1 I d e a l s o l u t i o n s 23 1.2.2.2 Nonideal s o l u t i o n s 24 1.2.3 The l i n e a r programming ( s t e p 2 of algorithm) 31 1.2.4 The change of base o p e r a t i o n ( s t e p 3 of algorithm) 34 1.3 A p p l i c a t i o n s of the computation of chemical e q u i l i b r i u m 34 i v 1.3.1 I n t r o d u c t i o n 34 1.3.2 Example 1: B l a s t furnace problem 35 1.3.3 Example 2: S i l i c a t e melts 35 1.3.4 Example 3: mica - f e l d s p a r e q u i l i b r i a 45 1.3.4.1 Thermodynamic models ..45 1.3.4.2 Excess phases 47 1.3.4.3 A d d i t i o n of c a l c i t e 48 1.3.4.4 A d d i t i o n of a b u f f e r 49 1.3.5 D i s c u s s i o n 54 2. ORDER-DISORDER IN TERNARY FE-MN-MG OLIVINES 56 2.1 I n t r o d u c t i o n 56 2.2 The s y n t h e s i s of o l i v i n e s 57 2.3 Some methods f o r s i t e occupancy d e t e r m i n a t i o n c i t e d i n the l i t e r a t u r e 61 2.3.1 Mossbauer spectroscopy 62 2.3.2 S i n g l e c r y s t a l s t r u c t u r e refinements 62 2.3.3 The R i e t v e l d method 63 2.3.4 The a-b p l o t 65 2.3.5 V i b r a t i o n a l s p e c t r a 66 2.3.6 CHEXE spectroscopy 66 2.4 Choice of methods 66 2.5 The determination of F e - s i t e d i s t r i b u t i o n s by Mossbauer spectroscopy .68 2.5.1 I n t r o d u c t i o n 68 2.5.2 Experimental set-up 72 2.5.3 E v a l u a t i o n of the Mossbauer spectroscopy measurements 75 2.5.4 D i s c u s s i o n of the r e s u l t s 82 2.6 C a l c u l a t i o n of i n t e g r a t e d X-Ray i n t e n s i t i e s 83 v 2.6.1 The data needed f o r the c a l c u l a t i o n s 83 2.6.2 Equations 86 2.7 The de t e r m i n a t i o n of Mn-Mg or Fe-Mg s i t e occupancies by XRD a n a l y s i s 90 2.7.1 I n t r o d u c t i o n 90 2.7.2 The measurement of l a t t i c e parameters 91 2.7.3 U n c e r t a i n t i e s i n c a l c u l a t e d i n t e n s i t i e s due to atom c o o r d i n a t e s and temperature c o r r e c t i o n f a c t o r s 98 2.7.4 U n c e r t a i n t i e s i n c a l c u l a t e d i n t e n s i t i e s due to charge d i s t r i b u t i o n 102 2.7.5 The i n t e n s i t y measurements and the occupancy i n t e r p r e t a t i o n 104 2.7.6 Example of a s i t e occupancy d e t e r m i n a t i o n 109 2.7.7 The S i t e occupancies of the s y n t h e t i c Fe-Mn-Mg o l i v i n e s 120 2.8 Thermodynamic model f o r the t e r n a r y Fe-Mn-Mg o l i v i n e s o l i d s o l u t i o n s at 850 °C 121 2.8.1 I n t r o d u c t i o n 121 2.8.2 Model assuming independent s i t e mixing ...122 2.8.3 A s l i g h t l y more general model 124 2.8.4 F i t t i n g the models to the data 125 2.9 Co n c l u s i o n s 126 BIBLIOGRAPHY 128 APPENDIX A: DESCRIPTION OF INPUT FOR THE PROGRAM THERIAK 141 A.1 Problem input 141 A. 2 The data-base 145 A. 2.1 S e c t i o n *** ... MINERAL DATA 147 A.2.2 USE and CODE ( s e l e c t i n g phases) 151 A. 2.3 S e c t i o n *** ... SOLUTION DATA 153 A. 2.4 S e c t i o n *** ... MARGULES 157 v i A.3 C a l c u l a t i o n of A^G i n an e x t e r n a l s ubroutine ...158 A.4 C a l c u l a t i o n of a c t i v i t i e s i n an e x t e r n a l subroutine 159 APPENDIX B: LISTING OF PROGRAM THERIAK 161 APPENDIX C: THE MOESSBAUER FURNACE 213 APPENDIX D: LISTING OF PROGRAM LATEX 217 APPENDIX E: THE POLYNOMIAL FUNCTION FOR THE LATTICE PARAMETERS 228 APPENDIX F: TABLES USED FOR THE APPROXIMATION OF THE POSITIONAL PRARMETERS AND THE RESULTING UNCERTAINTIES 229 APPENDIX G: THE PW1710 DIFFRACTOMETER 242 APPENDIX H: SOME EXPERIMENTAL EQUILIBRIUA BETWEEN OLIVINE AND GARNET IN THE SYSTEM FE-MN-MG-AL-SI"0 ...245 H.1 The s y n t h e s i s of the garnets 245 H.2 The exchange experiments 245 H.3 The q u a n i t a t i v e a n a l y s i s of the run products. ..246 v i i L i s t of T a b l e s Table Page 1. E q u i l i b r i u m c a l c u l a t i o n s f o r the b l a s t furnace example 36 2. I n t e r a c t i o n parameters f o r the quaternary system MgO - CaO - A 1 2 0 3 - S i 0 2 , a c c o r d i n g to Berman ( 1983) 42 3. C a l c u l a t e d c o e x i s t i n g quaternary l i q u i d s at 1400 °C, 1 bar 44 4. A fG [J/mol] f o r a l l c o n s i d e r e d phases i n example 3. P = 4 Kbars 50 5. C a l c u l a t e d mica - f e l d s p a r e q u i l i b r i a 52 6. Computing times f o r the examples on a 48 megabyte Amdahl 5850 computer 55 7. Isomer s h i f t ( 5 ), Quadrupole s p l i t t i n g (AE n) and the f u l l width at h a l f h e i g t h ( D f o r the w s y n t h e t i c Fe-Mn-Mg o l i v i n e s 79 8. Mossbauer area r a t i o s and Fe-occupancies f o r the s y n t h e t i c Fe-Mn-Mg o l i v i n e s 80 9. The l a t t i c e parameters f o r the s n t h e t i c Fe-Mn-Mg o l i v i n e s 92 10. The range of values f o r the p o s i t i o n a l parameters and the temperature c o r r e c t i o n f a c t o r s 99 11. The r e s u l t of the m u l t i p l e l i n e a r r e g r e s s i o n . The co n s t a n t s are used to c a l c u l a t e the p o s i t i o n a l parameters from known s i t e occupancies 101 12. The d i f f e r e n c e s between c a l c u l a t e d and observed i n t e g r a t e d i n t e n s i t y r a t i o s f o r f a y a l i t e , t e p h r o i t e and f o r s t e r i t e 107 13. Example: The s i t e occupancy f o r MnMgSiO a 111 14. normalized i n t e g r a t e d i n t e n s i t i e s 114 15. S i t e occupancies f o r the s y n t h e t i c Fe-Mn-Mg o l i v i n e s 117 16. The m a t e r i a l s used i n the vacuum furnace 216 17. Fe-Mn-Mg O l i v i n e s from the l i t e r a t u r e w i t h r e p o r t e d s i t e occupancies and/or s t r u c t u r e data ....230 18. P o s i t i o n a l parameters f o r Fe-Mn-Mg O l i v i n e s from the l i t e r a t u r e 232 v i i i 19. Temperature c o r r e c t i o n f a c t o r s f o r Fe-Mn-Mg O l i v i n e s from the l i t e r a t u r e 234 20. Estimated e r r o r s i n the c a l c u l a t i o n of i n t e g r a t e d i n t e n s i t i e s : Example f o r T e p h r o i t e 236 21. Estimated e r r o r s i n the c a l c u l a t i o n of i n t e g r a t e d i n t e n s i t i e s : Example f o r f a y a l i t e 238 22. Estimated e r r o r s i n the c a l c u l a t i o n of i n t e g r a t e d i n t e n s i t i e s : Example f o r f o r t s e r i t e 240 23. L i s t of s y n t h e s i z e d garnets 249 24. L i s t of usable exchange experiments 250 25. Summary of microprobe a n a l y s e s 253 ix F i g u r e L i s t of F i g u r e s Page 1. Examples of non-unique e q u i l i b r i a 8 2. Example of computing the assemblage having minimum A fG 13 3. I l l u s t r a t i o n of the a l g o r i t h m i n c l u d i n g s o l u t i o n phases 18 4. I l l u s t r a t i o n of the use of the Gibbs-Duhem equation 27 5. M i s c i b i l i t y gaps i n the CaO - A l 2 0 3 - S i 0 2 melts ....38 6. Phase diagram f o r the system CaO - A l 2 0 3 - S i 0 2 a t 1400 °C 41 7. Experimental set-up f o r the o l i v i n e s y n t h e s i s 58 8. F u g a c i t i e s of 0 2, CO and C0 2 i n the s y n t h e s i s 59 9. Schematic set-up of the Mossbauer apparatus 73 10. Mossbauer spectrum f o r fay 6 teph 4 fo 2 77 11. Mossbauer spectrum f o r fay 2 teph 10 78 12. The l a t t i c e parameter a of the Fe-Mn-Mg o l i v i n e s ....94 13. The l a t t i c e parameter b of the Fe-Mn-Mg o l i v i n e s ....95 14. The l a t t i c e parameter c of the Fe-Mn-Mg o l i v i n e s ....96 15. The volumes of the Fe-Mn-Mg o l i v i n e s 97 16. U n c e r t a i n t y of i n t e g r a t e d i n t e n s i t i e s due to u n c e r t a i n t i e s i n atom c o o r d i n a t e s and temperature c o r r e c t i o n f a c t o r s 103 17. Comparison of c a l c u l a t e d and measured i n t e g r a t e d i n t e n s i t i e s f o r t e p h r o i t e , f a y a l i t e and f o r s t e r i t e .110 18. I n t e n s i t y r a t i o s versus s i t e occupancies f o r MnMgSiOft. (values < 1) 112 19. I n t e n s i t y r a t i o s versus s i t e occupancies f o r MnMgSi0 4. (values > 1) 113 20. The observed l n ( k _ ) values i n f o r the s y n t h e t i c Fe-Mn-Mg o l i v i n e s 123 21. The Mossbauer furnace 214 22. The sample holder f o r the Mossbauer furnace 215 x Roozeboom diagrams f o r some exchange experiments i n the b i n a r y Fe-Mn and Mn-Mg o l i v i n e - garnet systems 252 x i Acknowledgements I am g r a t e f u l to my committee members, H.J. Greenwood, T.H. Brown and E.P. Meagher, who supported t h i s work and c a r e f u l l y e d i t e d and reviewed many e a r l i e r v e r s i o n s of my t h e s i s . I a l s o wish to thank Dr. C. McCammon f o r her d e d i c a t e d and e f f i c i e n t h e l p with the Mossbauer apparatus and Mr. Doug Poison who improved and b u i l t the vacuum furnace. I a l s o thank P r o f . S.E. C a l v e r t from the oceanography department for h i s f r i e n d l y h e l p with the PW1710 d i f f T a c t o m e t e r and Ms. Maggie P i r a n i a n f o r the i n t r o d u c t i o n to the CAMECA SX50 microprobe. For s t i m u l a t i n g d i s c u s s i o n s and suggestions my thanks are due to Dr. R.G. Berman, Dr. M. E n g i , P r o f . J . Rice and a l l my c o l l e g u e s and f r i e n d s i n the Department of G e o l o g i c a l S c i e n c e s . The r e s e a r c h r e p o r t e d here was made p o s s i b l e by funding through NSERC grant A4222 to H.J. Greenwood and the award of a UBC U n i v e r s i t y Graduate F e l l o w s h i p and a K i l l a m P r e - d o c t o r a l f e l l o w s h i p . x i i INTRODUCTION The p a r t i t i o n i n g of elements between c o e x i s t i n g phases and between d i f f e r e n t s t r u c t u r a l s i t e s of a s i n g l e phase i s of c e n t r a l importance f o r geothermometry and geobarometry. In a system with more than two exchangeable elements, as i n most n a t u r a l assemblages, the e q u i l i b r i u m i n v o l v i n g d i f f e r e n t phases can become a very complex problem f o r experimental and t h e o r e t i c a l work. The aim of the present t h e s i s was to i n v e s t i g a t e problems r e l a t e d to heterogeneous e q u i l i b r i a i n multicomponent and multiphase systems. N a t u r a l l y , only a few of these problems can be pursued i n more d e t a i l . The sample system chosen c o n s i s t s of the t e r n a r y Fe-Mn-Mg o l i v i n e s and gar n e t s . The r e a c t i o n s beween Fe-Mg o l i v i n e s and garnets have been proposed as a geothermometer by Kawasaki and Matsui (1977) and O ' N e i l l and Wood (1979). These authors a l s o p o i n t out that i n order to i n v e s t i g a t e the thermodynamic p r o p e r t i e s of the garnet s o l i d s o l u t i o n s , experimental work with o l i v i n e as c o e x i s t i n g phase may prove more u s e f u l than with the more complex pyroxene s o l i d s o l u t i o n s . A te r n a r y system was chosen to emphasize the t y p i c a l c o m p l i c a t i o n s encountered i n higher than b i n a r y order systems. Although Ca may be a more important element than Mn, i t was not c o n s i d e r e d because i t s mixing p r o p e r t i e s are expected to be more complicated (Dempsey (1980)) and because of my s u b j e c t i v e sympathy f o r Mn m i n e r a l s . At a very e a r l y stage of the i n v e s t i g a t i o n i t became obvious that no f i n a l model f o r o l i v i n e - garnet e q u i l i b r i a would be 1 2 p o s s i b l e . Because the experimental p a r t of the problem demands e x t e n s i v e a n a l y t i c a l work and the department microprobe was not a v a i l a b l e f o r two y e a r s , only few, p r e l i m i n a r y exchange experiments were planned and c a r r i e d out. T h e i r r e s u l t s are presented i n Appendix H. One of the most important fundamental problems when d e a l i n g with complex multiphase systems i s the computation of the e q u i l i b r i u m assemblage at given c o n d i t i o n s . Although the l i t e r a t u r e on the subject i s huge and s t i l l growing, a completely g e n e r a l method f o r n o n - i d e a l systems has to my knowledge not been p u b l i s h e d . One s p e c i f i c approach, the " l i n e a r programming method" (White et a l . (1958)), seemed very promising and i s developed to a g e n e r a l a l g o r i t h m i n the f i r s t p a r t of the present t h e s i s . T h i s p a r t was a l s o submitted f o r p u b l i c a t i o n (de C a p i t a n i and Brown (1987)). The second problem i n v e s t i g a t e d i n more d e t a i l i s the d i s t r i b u t i o n of Fe, Mn and Mg i n t e r n a r y o l i v i n e s . The n o n - l i n e a r behaviour of the l a t t i c e parameters f o r the Fe-Mn and Mn-Mg o l i v i n e s (Annersten et a l . (1984)) suggest that these s o l u t i o n s are n o n - i d e a l . For a thermodynamic mixing model i t i s important to chose a r e a l i s t i c entropy of mixing term based on our knowledge of o r d e r - d i s o r d e r of that phase. The s i t e d i s t r i b u t i o n of the t e r n a r y o l i v i n e s c o u l d be measured with a combination of Mossbauer spectroscopy and XRD i n t e n s i t y e v a l u a t i o n . The p r i n c i p l e s of the l a t t e r method were a l s o presented at a AGU meeting, (de C a p i t a n i and Greenwood (1985)). 1. THE COMPUTATION OF CHEMICAL EQUILIBRIUM IN COMPLEX SYSTEMS CONTAINING NON-IDEAL SOLUTIONS 1.1 THE FORMULATION OF A GENERAL ALGORITHM 1.1.1 INTRODUCTION One of the important problems i n t h e o r e t i c a l p e t r o l o g y i s the de t e r m i n a t i o n of the e q u i l i b r i u m phase assemblage f o r any s p e c i f i e d set of c o n d i t i o n s . Computations of e q u i l i b r i a permit comparison with n a t u r a l or experimental assemblages and thus t e s t the v a l i d i t y of t h e o r e t i c a l models and thermodynamic data. The computed r e s u l t s can be used to design f u t u r e experiments or to c a l c u l a t e assemblages at c o n d i t i o n s beyond the range of c u r r e n t experimental techniques. In many r e s p e c t s the computational approach to e q u i l i b r i u m resembles the experimental approach, because both operate on the p r i n c i p l e of p r o g r e s s i v e l y r e d u c i n g the f r e e energy u n t i l a minimum i s reached. We cannot guarantee, however, that the ab s o l u t e minimum has been reached i n e i t h e r case. The experiment may terminate with metastable phases due to d i f f i c u l t y i n n u c l e a t i n g the s t a b l e phases. S i m i l a r l y , a computation may converge to a l o c a l minimum or lack the presence of a s t a b l e phase due to i t s omission from the data base. I t i s c l e a r t h at very c a r e f u l i n t e r p r e t a t i o n i s r e q u i r e d f o r both computed and e x p e r i m e n t a l l y determined e q u i l i b r i a . 3 4 The aim of the present i n v e s t i g a t i o n i s to present a technique f o r minimizing the Gibbs f r e e energy which i s an ex t e n s i o n of an a l g o r i t h m p u b l i s h e d by Brown and Skinner (1974). The new method shows some s i m i l a r i t i e s with the " l i n e a r programming method" o r i g i n a l l y proposed by White, Johnson and Dantzig (1958). In the d i s c u s s i o n below, I attempt to give enough d e t a i l to allow the i n t e r e s t e d reader to program or improve the a l g o r i t h m h i m s e l f . A program ( c a l l e d THERIAK), w r i t t e n i n FORTRAN 77 i s d e s c r i b e d and l i s t e d i n appendix A and B. 1.1.2 THEORETICAL FUNDAMENTALS We w i l l r e s t r i c t our d i s c u s s i o n here to some of the assumptions and i m p l i c a t i o n s that are important f o r the computation of chemical e q u i l i b r i a . For a more d e t a i l e d treatment of the axioms and laws of thermodynamics the reader i s r e f e r r e d to any one of a number of good textbooks (e.g. P r i g o g i n e and Defay (1954), Denbigh (1971) or Becker (1978)). The systems we c o n s i d e r are a l l c h e m i c a l l y c l o s e d ; that i s , the bulk composition i s f i x e d . We assume that there are only two p o s s i b l e types of energy, namely heat and energy r e l a t e d t o pressure-volume work. T h i s assumption f i x e s the number of independent v a r i a b l e s f o r any f u n c t i o n of s t a t e to two, which i s a l s o the maximum degree of freedom i n the phase r u l e sense. The most general f o r m u l a t i o n of the chemical e q u i l i b r i u m problem f o l l o w s from the second law of thermodynamics, and p r o v i d e s that the 5 e q u i l i b r i u m s t a t e i s the s t a t e i n which the t o t a l entropy of the c l o s e d system and i t s surroundings has reached the maximum v a l u e . I f we take the independent v a r i a b l e s to be pressure and temperature, t h i s s t a t e i s a l s o c h a r a c t e r i z e d by a minimum i n the Gibbs f r e e energy of the c l o s e d system. Thus a chemical experiment can be simulated by s o l v i n g a mathematical o p t i m i z a t i o n problem which, f o r independent P and T, corresponds to the m i n i m i z a t i o n of the Gibbs f r e e energy, G (G=U-TS+PV). Although d i f f e r e n t c h o i c e s of independent v a r i a b l e s are p o s s i b l e , I have r e s t r i c t e d the present i n v e s t i g a t i o n to the c a l c u l a t i o n of e q u i l i b r i u m assemblages f o r constant P and T. In the past twenty years many a l g o r i t h m s have been developed f o r s o l v i n g the chemical e q u i l i b r i u m problem. Comprehensive reviews can be found i n papers by Smith and Missen (1982) or Smith (1980a,1980b). E a r l i e r work was summarized by Z e l e z n i k and Gordon (1968), Van Zeggeren and Storey (1970) and Holub and Vonka (1976). Most popular a l g o r i t h m s are based on B r i n k l e y (1947), Huff et a l . (1951) (NASA-algorithm), White et a l . (1958) (RAND-algorithm), N a p h t a l i (1959,1961), V i l l a r s (1958), C r u i s e (1964) and Smith and Missen (1968). A l l these a l g o r i t h m s assume that the chemical e q u i l i b r i u m problem has a unique s o l u t i o n . 1.1.2.1 Non-unique s o l u t i o n s to the e q u i l i b r i u m problem A s o l u t i o n to the chemical e q u i l i b r i u m problem i s non-unique i f more than one assemblage f u l f i l l s the c o n d i t i o n s of e q u i l i b r i u m . S t r i c t l y speaking these 6 c o n d i t i o n s are only f u l f i l l e d f o r the g l o b a l minimum G assemblage and f o r most cases t h i s assemblage i s the unique s o l u t i o n to the chemical e q u i l i b r i u m problem. However, most mathematical f o r m u l a t i o n s of the e q u i l i b r i u m c o n d i t i o n s may allow an a l g o r i t h m to converge to a s o l u t i o n which i s not the true e q u i l i b r i u m assemblage. For a computation these mathematically non-unique s o l u t i o n s are very important and we w i l l t h e r e f o r e use the term "unique s o l u t i o n " only f o r e q u i l i b r i u m problems that have a s i n g l e s o l u t i o n i n the mathematical sense. For these problems the only assemblage to which an a l g o r i t h m can converge to i s the e q u i l i b r i u m assemblage. The types of non-unique s o l u t i o n s that may e x i s t depend on the computational method. In f i g . 1 three s i t u a t i o n s with non-unique e q u i l i b r i a are shown. An i n d i f f e r e n t s t a t e ( a l s o c a l l e d degenerate non-uniqueness) r e s u l t s i f s e v e r a l phase assemblages are s t a b l e and have the same Gibbs f r e e energy. ( F i g . 1a). Although t h i s i s a case with m u l t i p l e g l o b a l minima i t i s g e n e r a l l y not c o n s i d e r e d a s e r i o u s problem, because t h i s s i t u a t i o n i s u s u a l l y only encountered on u n i v a r i a n t curves i n the P-T-plane. L o c a l minima ( F i g . 1b) can be very d i f f i c u l t to a v o i d and f i n d i n g them can be of i n t e r e s t , because they may correspond to metastable s t a t e s found i n experimental and n a t u r a l assemblages. However, i t i s important and u n f o r t u n a t e l y d i f f i c u l t to recognize the d i f f e r e n c e between a l o c a l and the g l o b a l s o l u t i o n i n a G-minimization. The t h i r d example i s a c o n s t r a i n e d minimum 7 F i g . 1: (page 8) Examples of non-uniqe e q u i l i b r i a . The bulk c o m p o s i t i o n i s g i v e n by the arrow. a) : The phase assemblages S^-P and S 2 - P have the same Gibbs f r e e energy , so t h a t both s a t i s f y the e q u i l i b r i u m c r i t e r i a of minimum A^G f o r a g i v e n c o m p o s i t i o n . b) : The mathemat ica l f o r m u l a t i o n of the e q u i l i b r i u m c o n d i t i o n s may be s a t i s f i e d by two d i f f e r e n t assemblages . The assemblage S , - P r e p r e s e n t s a l o c a l minimum. S 2 ~ P i s the g l o b a l minimum. c) : S 2 - P i s a p o s s i b l e " s t a b l e " assemblage i f we do not e x p l i c i t l y a l l o w f o r the p o s s i b i l i t y of two c o e x i s t i n g phases from the same s o l u t i o n . The t r u e e q u i l i b r i u m assemblage i s S , - S 3 . 8 Fig. 1: Examples of non-unique equilibria A,G A,G A,G 0 X a) Indifferent State b) Local Minimum t l l l l l l l l l c) Constrained Minimum 1 9 (Fig 1c), meaning not only that the ranges of certain variables are r e s t r i c t e d , but also the number of variables we consider. If an algorithm i s constrained to assume that no more than one solution phase can be part of the stable assemblage, then the global minimum for t h i s problem i s the assemblage S 2 _P. The true equilibrium 8 ,-83 can only be found i f we allow two co-existing phases from the same solution. Whereas systems with only ideal solutions have unique e q u i l i b r i a (Shapiro and Shapley (1965)), t h i s i s generally not the case i f nonideal solutions are involved. This fact and some implications for computational algorithms were pointed out by Caram and Scriven (1976), Othmer (1976), Heidemann (1978) and Gautam and Seider (1979). Many algorithms used today in connection with nonideal solutions were o r i g i n a l l y developed for use with ideal systems and assume unique solutions, (e.g. Fic k e t t (1963), Eriksson (1971,1974), and Harvie and Weare (1980)). If deviation from i d e a l i t y i s large and the p o s s i b l i t y of phase separation is not considered e x p l i c i t l y , e.g. by phase-splitting (Gautam and Seider (1979)), such algorithms may converge to an improperly constrained minimum. The published algorithms that were s p e c i a l l y designed for nonideal solutions are usually r e s t r i c t e d to one phase separation or assume that a maximum of two phases from the same solution can co-exist (e.g. Kaufmann and Bernstein (1970), Sanderson and Chien (1973), George et a l . (1976) and Prausnitz et a l . (1980)). 10 As w i l l be d i s c u s s e d below, the reason why most a l g o r i t h m s are not c o m p l e t e l y g e n e r a l f or n o n i d e a l systems l i e s i n the f o r m u l a t i o n of the e q u i l i b r i u m c o n d i t i o n . 1.1.2.2 F o r m u l a t i o n of the c h e m i c a l e q u i l i b r i u m c o n d i t i o n For any system of f i x e d c o m p o s i t i o n the g e n e r a l e q u i l i b r i u m c o n d i t i o n i s : N min imize G = Z n , « g , k=1 K K s u b j e c t to the c o n s t r a i n t s tha t a l l n ^ O and that the mass ba lance e q u a t i o n s are s a t i s f i e d . where n^ i s the number of moles and g^ i s the molar G i b b s f r e e energy of the phase k. At the s t a r t of any c o m p u t a t i o n , the t o t a l number of c o e x i s t i n g phases (N) i s not known, and because the s o l u t i o n phases have a range of p o s s i b l e c o m p o s i t i o n s , t h e i r Gibbs f r e e e n e r g i e s at s t a b i l i t y a l s o are not known. T h i s seemingly c h a o t i c s i t u a t i o n u s u a l l y l eads to an a l t e r n a t e f o r m u l a t i o n of the e q u i l i b r i u m c o n d i t i o n , where g^ i s r e p l a c e d by the c h e m i c a l p o t e n t i a l of species j . The t o t a l number of s p e c i e s i s known and each can be a s s i g n e d to one phase . U n f o r t u n a t e l y t h i s f o r m u l a t i o n assumes a unique s o l u t i o n to the e q u i l i b r i u m p r o b l e m , because each s p e c i e s can be presen t in on ly one amount n j . A g e n e r a l i z a t i o n i n c l u d i n g p o s s i b l e phase s e p a r a t i o n s becomes extremely d i f f i c u l t . 11 A p o s s i b l e way to use the g e n e r a l e q u i l i b r i u m c o n d i t i o n with no f u r t h e r assumptions was proposed by White, Johnson and Dantzig (1958). They c a l l e d i t the " l i n e a r programming method". The procedure as o r i g i n a l l y d e s c r i b e d assumes that the Gibbs energy f u n c t i o n s of the s o l u t i o n s are convex, but the b a s i c idea of v a r y i n g the number of phases c o n s i d e r e d d u r i n g computation can e a s i l y be used f o r n o n i d e a l systems in which the G f u n c t i o n s l o c a l l y may be concave. Although s e v e r a l authors have used phases and not s p e c i e s i n t h e i r computations, (e.g. Brown and Skinner (1974), Connoly and K e r r i c k (1984,1986)) no completely g e n e r a l a l g o r i t h m seems to have been p u b l i s h e d . The f o r m u l a t i o n proposed here i s s u r p r i s i n g l y simple and much more e f f e c t i v e than i n i t i a l l y a n t i c i p a t e d . 1.1.3 PHASES WITH FIXED COMPOSITIONS Because G as a f u n c t i o n i s only c a l c u l a b l e r e l a t i v e to a r e f e r e n c e s t a t e , the f o l l o w i n g d i s c u s s i o n r e f e r s to the Gibbs f r e e energy f u n c t i o n as A^G, the AG of formation from a r e f e r e n c e s t a t e . Let us f i r s t c o n s i d e r a system of f i x e d T, P and bulk composition f o r which each phase has an i n v a r i a b l e composition and a c a l c u l a b l e A^G. In t h i s case, the problem of f i n d i n g the minimum A^G assemblage can be sol v e d by any l i n e a r programming method. How such a method may work i s o u t l i n e d below and i l l u s t r a t e d i n F i g . 2. With two independent v a r i a b l e s the maximum number of s t a b l e phases i s equal to the number of components (C) i n the 12 F i g . 2: (page 13) Example of computing the assemblage hav ing minimum A^G for a s e l e c t i o n of phases w i th f i x e d c o m p o s i t i o n s . The bu lk c o m p o s i t i o n i s g i v e n by the arrow. S t a r t i n g w i t h an i n i t i a l guess (here A - B ) , one phase at a t ime i s r e p l a c e d by another to a c h i e v e a lower t o t a l A ^ G . In t h i s example A i s r e p l a c e d by C , then B by D, and so on u n t i l the s t a b l e assemblage , of lowest A ^ G , i s found . The sequence of c a l c u l a t i o n s w i l l depend on the order of c o n s i d e r i n g new phases . 13 Fig. 2: Example of computing the equil ibrium assemblage for phases with fixed compositions A t A f G i r 0.0 t i i I i i i i 0.5 X 1.0 1 4 system. We w i l l s t a r t i n i t i a l l y w i t h any assemblage of e x a c t l y C l i n e a r l y independent phases . For a s p e c i f i e d bu lk c o m p o s i t i o n some of these phases may have zero amounts. A l l o t h e r phases c o n s i d e r e d are r e l a t e d to t h i s i n i t i a l assemblage by one r e a c t i o n e a c h . F i g . 2 shows an example of a s imple b i n a r y system where we have chosen the i n i t i a l assemblage to be A+B. As we go through the l i s t of phases c o n s i d e r e d ( i n t h i s case in a l p h a b e t i c a l o r d e r ) we n o t i c e tha t the r e a c t i o n C = A+B has a A r G > 0. T h i s means tha t C i s more s t a b l e than A+B and for the chosen bu lk c o m p o s i t i o n , C+B becomes the new s t a b l e assemblage . For the next phase i n the l i s t (D) , a s i m i l a r procedure ( A r G of D=C+B i s > 0) l e a d s to the lower A^G assemblage C+D and so on , u n t i l the s t a b l e assemblage H+E i s l o c a t e d . Each time a phase i s found t h a t l i e s below the A^G t i e - l i n e of the c u r r e n t l y c o n s i d e r e d assemblage , t h i s phase r e p l a c e s one phase from t h a t assemblage . Which phase i s r e p l a c e d depends on the bulk c o m p o s i t i o n . We must go through the l i s t of c o n s i d e r e d phases s e v e r a l t i m e s , u n t i l we f i n d the t r u e s t a b l e assemblage where a l l o t h e r phases l i e above (or on) the A^G t i e - l i n e . For mult icomponent systems the components and A^G d e f i n e a hyperspace and the A^G t i e - l i n e s become A^G h y p e r p l a n e s , but the procedure remains the same. I f A r G between two or more assemblages happens to be e x a c t l y zero then t h i s i s c a l l e d an " i n d i f f e r e n t s t a t e " (see P r i g o g i n e and Defay (1954) ) . An e q u i l i b r i u m c a l c u l a t i o n w i l l f i n d o n l y one assemblage , but because the phases of the o t h e r 1 5 assemblages have the same c h e m i c a l p o t e n t i a l s , t h i s s i t u a t i o n i s e a s i l y r e c o g n i z e d . 1.1.4 ADDING SOLUTION PHASES TO THE SYSTEM I f we now add s o l u t i o n phases to the system the mathemat i ca l treatment becomes more complex. As a f i r s t a p p r o x i m a t i o n we c o n s i d e r a l l the phases of f i x e d c o m p o s i t i o n p l u s a l l endmembers of the s o l u t i o n s and c a l c u l a t e f o r t h i s s i m p l i f i e d system the minimum A^G assemblage by l i n e a r programming as d e s c r i b e d above . The essence of the new a l g o r i t h m i s tha t we now s i m p l i f y f u r t h e r c a l c u l a t i o n s by t r a n s f o r m i n g the r e f e r e n c e s t a t e of the complete system so t h a t the phases of the " c u r r e n t l y s t a b l e assemblage" a l l have A f G = 0, but t h a t the A r G of a l l p o s s i b l e r e a c t i o n s i n the system remain unchanged. I f o r i g i n a l l y A^G meant " format ion from the elements" and the s t a b l e assemblage c o n s i s t s of the phases A , B and C , then a f t e r t h i s t r a n s f o r m a t i o n the v a l u e s of A^G have the meaning " format ion from A , B and C " , but a l l s t a b i l i t y r e l a t i o n s remain the same. T h i s change of base o p e r a t i o n i s e a s i l y a c c o m p l i s h e d by l i n e a r a l g e b r a . For each s o l u t i o n we now s e a r c h f o r the c o m p o s i t i o n wi th the s m a l l e s t A^G. ( T h i s i s a n o n l i n e a r programming problem and i s d i s c u s s e d i n more d e t a i l be low. ) The c o m p o s i t i o n s so de termined are now t r e a t e d as i f they were phases of f i x e d c o m p o s i t i o n and a new minimum A^G assemblage i s computed by l i n e a r programming. Aga in we t r a n s f o r m the system, f i n d for each 1 6 s o l u t i o n the minimum A^G composition and c a l c u l a t e the s t a b l e assemblage by l i n e a r programming. These three steps are repeated u n t i l some convergence c r i t e r i a i s s a t i s f i e d . F i g . 3 shows an example with two n o n i d e a l b i n a r y s o l u t i o n s . In F i g . 3a only the four endmembers A 1 f A 2, B, and B 2 are c o n s i d e r e d and the s t a b l e assemblage i s c a l c u l a t e d to be pure B, + pure B 2. The step from F i g . 3a to F i g . 3b i n v o l v e s the change of base o p e r a t i o n such t h a t the A^G t i e - l i n e B,-B 2 becomes h o r i z o n t a l and A f G B 1 = A fG f i 2 = 0. ^ f G A 1 ~ ^ f G B 1 ^ a n c ^ ^ f G A 2 ~ ^ f G B 2 ^ r e m a i - n unchanged. Now we search f o r the minima of both s o l u t i o n s , add these p o i n t s as phases to those a l r e a d y c o n s i d e r e d and compute the minimum ACG assemblage again with l i n e a r programming, f i n d i n g the assemblage B 3-A 3. The step to F i g . 3c i s by the same procedure with the A^G t i e - l i n e of the c u r r e n t l y most s t a b l e assemblage B 3-A 3 being transformed to a new zero l e v e l , whereupon a new s t a b l e assemblage i s computed. The convergence i s r e l a t i v e l y f a s t (averaging about 6 i t e r a t i o n s ) i n b i n a r y s o l u t i o n s but, of course, slows down with a l a r g e r number of endmembers. T h i s procedure always computes an assemblage with C (=number of components) phases, although some of these phases may be present i n a zero or, owing to round o f f e r r o r s , near-zero amount. I f two or more s t a b l e phases from the same s o l u t i o n have almost i d e n t i c a l compositions, (e.g. a l l mole f r a c t i o n s w i t h i n 10~ 5) these have to be i n t e r p r e t e d as corresponding to one phase o n l y . The a l g o r i t h m makes no assumptions about the 17 F i g . 3: (page 18) I l l u s t r a t i o n of the a l g o r i t h m i n c l u d i n g s o l u t i o n phases. The system has two components with two n o n - i d e a l s o l u t i o n s A and B. The endmembers of the two s o l u t i o n s have i d e n t i c a l c ompositions. The temperature i s 1 K and A^G f o r A1, A2, B1 and B2 are -3, -8, -6 and -9 (J/mol) r e s p e c t i v e l y . The excess f u n c t i o n s are 2 1 - x j ^ - x ^ f ° r s o l u t i o n A and 15«x^ 1'Xg2 + 2 3 * x B I " X B 2 L = o r S O x U t i ° n B « T ^ e bulk composition i s 0.6 moles of the f i r s t component and 0.4 moles of the second. The s t a b l e phases a r e : A 10.28658 A 20.71342 a n d B 10.83061 B 20.16939 Fig. 3: I l lustrat ion of the a lgor i thm inc luding solution phases 19 number of phases i n a s t a b l e assemblage and some unexpected r e s u l t s stemming from t h i s approach are d i s c u s s e d i n the examples l a t e r . The success of the a l g o r i t h m n a t u r a l l y depends on how f a s t and how a c c u r a t e l y we can compute the minimum A^G composition of a s o l u t i o n . 1.1.5 SUMMARY OF THE ALGORITHM The f o l l o w i n g i s a general summary of the a l g o r i t h m . Each step i s d i s c u s s e d i n more d e t a i l i n the next c h a p t e r . We assume t h a t we are given a temperature T, a pr e s s u r e P and a bulk composition. The l i s t of c o n s i d e r e d phases c o n t a i n s i n i t i a l l y the names, the compositions and A^G f o r a l l p o s s i b l e phases of f i x e d c omposition. F u r t h e r , A^G of each s o l u t i o n i s a known f u n c t i o n of the composition. We s t a r t with a f e a s i b l e assemblage of C (=number of components) l i n e a r l y independent phases, each having A^G = 0. F e a s i b l e means that a l l amounts are non-negative and that the mass balance equations are s a t i s f i e d . The f i r s t i t e r a t i o n i s used to generate a b e t t e r i n i t i a l assemblage and t h e r e f o r e step 1 i s skipped. 1. Search each s o l u t i o n phase f o r the composition with the sm a l l e s t A^G. ( n o n l i n e a r programming problem). Add these compositions as "new phases" to the l i s t of co n s i d e r e d phases. 20 2. F i n d from a l l c o n s i d e r e d phases the assemblage wi th minimum A ^ G . ( l i n e a r programming problem) The phases computed i n s t ep 1 which t u r n out to have z e r o amount a f t e r t h i s s t e p can be d i s c a r d e d from f u r t h e r c o n s i d e r a t i o n . 3. The assemblage wi th minimum A^G (from s t ep 2) i s c o n s i d e r e d to be the new r e f e r e n c e s t a t e . A^G of a l l phases are computed r e l a t i v e to t h i s assemblage . T h i s change of base o p e r a t i o n l e a v e s us i n a s i t u a t i o n i d e n t i c a l to that b e f o r e s t ep 1. Cont inue a g a i n w i t h s t ep 1. A f t e r the l i n e a r programming ( s t e p ,2), the most s t a b l e assemblage d e f i n e s a A^G r e f e r e n c e h y p e r p l a n e , and no c o n s i d e r e d phase can have a A^G below i t . In the next i t e r a t i o n we can on ly c a l c u l a t e a more s t a b l e assemblage i f new s o l u t i o n phases w i th a A^G below that r e f e r e n c e hyperp lane are c o n s i d e r e d . The s earch for phases wi th A^G below the the r e f e r e n c e h y p e r p l a n e i s g r e a t l y f a c i l i t a t e d by the change of base o p e r a t i o n ( s t ep 3), because now a l l phases w i th A^G below have A^G < 0 and a l l phases above have AjG > 0. I f a s o l u t i o n phase has any p o i n t s w i th A^G < 0 a f t e r s t ep 3, then the g l o b a l minimum ( s t ep 1) i s guaranteed to have a A^G l e s s than z e r o and t h e r e f o r e the f o l l o w i n g s t e p 2 w i l l compute a more s t a b l e assemblage . The a l g o r i t h m c o n t i n u e s to c a l c u l a t e new assemblages as long as any phases 21 with a AjG below the c u r r e n t r e f e r e n c e hyperplane e x i s t . The i t e r a t i o n s are terminated when the sum of a l l n o n - p o s i t i v e A^G before step 3 exeeds a t h r e s h o l d v a l u e , (e.g. - 1 0 ~ 8 ) . 1.2 DETAILS OF THE COMPUTATION OF CHEMICAL EQUILIBRIUM 1.2.1 THE INITIAL ASSEMBLAGE We have s t a t e d above that i n i t i a l l y we s t a r t w i t h a f e a s i b l e assemblage f o r which a l l phases have A^G = 0. Because the bulk composition i s o f t e n given i n terms of components, (elements, oxides) i t i s convenient to d e f i n e imaginary "phases" having the same composition as these components and a A^G of zero. These "phases" are l i n e a r l y independent and are used as i n i t i a l assemblage. To a v o i d problems with r e a l phases which may have p o s i t i v e A^G, the "phases" of the i n i t i a l assemblage are assign e d an abs u r d l y h i g h A^G (e.g. 10 2°) j u s t before the f i r s t l i n e a r programming ste p . A f t e r t h i s , t h e i r values are r e s t o r e d to zero, but i n l a t e r loops they are not c o n s i d e r e d a g a i n . 1.2.2 THE NONLINEAR PROGRAMMING (STEP 1 OF ALGORITHM) The c e n t r a l part of the a l g o r i t h m i s the procedure that searches the minimum A^G composition f o r every s o l u t i o n . The d e t a i l s of how t h i s n o n l i n e a r programming r o u t i n e i s set-up depend on the fo r m u l a t i o n of the s o l u t i o n models. I t i s r e q u i r e d that each s o l u t i o n be d e f i n e d over i t s e n t i r e c o m p o s i t i o n a l and s t r u c t u r a l range. I have chosen t o use the 22 concept of endmembers. That i s , each s o l u t i o n i s c o n s i d e r e d to be made up of any number of s p e c i e s , where each of these s p e c i e s i s a s t r u c t u r a l l y v a l i d endmember. The only c o n s t r a i n t f o r a s o l u t i o n i s that the mole f r a c t i o n s of the s p e c i e s be non-negative. Most s o l u t i o n models used i n geochemistry f o l l o w t h i s concept. Let us take as an example a b i n a r y orthopyroxene ( F e , M g ) 2 S i 2 0 6 . The obvious endmembers are F e 2 S i 2 0 6 and M g 2 S i 2 0 6 . If the model should i n c l u d e i n f o r m a t i o n on s i t e occupancies f o r the two s i t e s M, and M 2, then the endmembers ar e : F e F e S i 2 0 6 , F e M g S i 2 0 6 , M g F e S i 2 0 6 and MgMgSi 20 6. Every p o s s i b l e and s t r u c t u r a l l y d i f f e r e n t c o n f i g u r a t i o n of the s m a l l e s t u n i t c o n s i d e r e d becomes an endmember i t s e l f . Such s p e c i a t i o n models f o r complex s o l u t i o n s are f r e q u e n t l y used f o r gaseous systems and were a l s o proposed f o r s o l i d s by Brown and Skinner (1974), Brown and Greenwood ( i n p r e p a r a t i o n ) , and was used by Engi (1983) i n m o d e l l i n g the t e r n a r y s p i n e l s o l i d s o l u t i o n . A completely d i f f e r e n t approach would be to d e f i n e the s o l u t i o n as a mixture of F e M 1 S i 0 3 , F e M 2 S i 0 3 , M g M 1 S i 0 3 and M g M 2 S i 0 3 , with the c o n s t r a i n t that the amount of M1-species i s equal to the amount of M2-species. In that case the s t r u c t u r a l homogenity i s c o n t r o l l e d by an a d d i t i o n a l e x p l i c i t c o n t r a i n t , and would r e q u i r e a d i f f e r e n t n o n l i n e a r programming r o u t i n e than the one proposed. The problem i s to f i n d the minimum A^G composition of a given s o l u t i o n . 23 ne : Number of endmembers ( s p e c i e s ) i n the s o l u t i o n . x : 1 x ne row v e c t o r : Xj = c o n c e n t r a t i o n of endmember j i n s o l u t i o n phase. Where s e v e r a l phases from the same s o l u t i o n are d i s c u s s e d t h e i r c o n c e n t r a t i o n v e c t o r s w i l l be x „ , x" e t c . n m M° : 1 x ne row v e c t o r : M°J = chemical p o t e n t i a l of pure endmember j . M : 1 x ne row v e c t o r : Mj = chemical p o t e n t i a l of endmember j i n s o l u t i o n phase. Mj = Mj + a-R«T«ln(aj) (where a = s i t e occupancy i n t e g e r ) I f s e v e r a l phases from the same s o l u t i o n are d i s c u s s e d t h e i r chemical p o t e n t i a l v e c t o r s w i l l be H , M_ e t c . A^G of a s o l u t i o n phase wi th the c o n c e n t r a t i o n v e c t o r x i s : ne A f G ( x ) = Z x.«M- = x « 7 i T (Gibbs-Duhem equat ion ) j = 1 -1 -1 where M t = t r a n s p o s e of M 1.2 .2 .1 I d e a l s o l u t i o n s Mj = Mj + a « R - T - l n ( X j ) The A j G - f u n c t i o n of an i d e a l s o l u t i o n i s everywhere convex and there i s a unique minimum where Mi = M 2 = ••• = U N E The c o n c e n t r a t i o n v e c t o r x for t h i s minimum can be c a l c u l a t e d i n one s t e p . F i r s t the endmember J w i t h the s m a l l e s t i s found . Then a l l x - ' s are c a l c u l a t e d as x . = e-(»]-»°j)/("-*'?) j = i , 2 , . . . n e The exponent i s always ^ 0, but i f i t i s smaller than -150 Xj i s set to ^ 1 0 " 6 6 . ( = e " 1 5 0 , t h i s i s about 10 magnitudes bigger than the s m a l l e s t number our computer can h a n d l e ) . The sum of a l l w i l l be between 1 and ne. Thus we can s a f e l y d i v i d e a l l Xj by t h i s sum and get the c o n c e n t r a t i o n vector x f o r minimum A^G. 1.2.2.2 Nonideal s o l u t i o n s There are of course many ways t o . f i n d the minimum of a nonideal s o l u t i o n . Good i n t r o d u c t i o n s i n t o the t o p i c are e.g. Dantzig (1963), McCormick (1980) and Dixon (1980). For our s t r a t e g y the f o l l o w i n g key p o i n t s were c o n s i d e r e d : • I t i s r a t h e r i n e f f e c t i v e . t o use a h i g h l y s o p h i s t i c a t e d (and computer i n t e n s i v e ) method, because the r e f e r e n c e hyperplane i s moved a f t e r each i t e r a t i o n . T h i s r u l e s out any method that uses d i r e c t l y or i n d i r e c t l y s e c o n d - d e r i v a t i v e i n f o r m a t i o n ( i . e . Newton's method). On the other hand, to guarantee l o c a t i n g the minimum A^G assemblage the a l g o r i t h m r e q u i r e s the exact and g l o b a l minimum. Our compromise i s that we use a s t r a t e g y that can r e f i n e a minimum to any d e s i r e d p r e c i s i o n , i f r e q u i r e d , but may f o r standard use converge to a more approximate v a l u e . •Although we are p r i m a r i l y s e a r c h i n g f o r the g l o b a l minimum, the l o c a l minima prove u s e f u l as w e l l , 25 e s p e c i a l l y i f two (or more) phases of the same s o l u t i o n are found to c o e x i s t . A l l l o c a l minima found are added to the l i s t of co n s i d e r e d phases i n the hope t h a t at l e a s t one of them i s c l o s e to the g l o b a l minimum. (Adding more phases than j u s t those d e f i n i n g the l o c a l minima may a c c e l e r a t e the a l g o r i t h m i n some c a s e s ) . The s t r a t e g y that we have adopted i s a g r a d i e n t method. S t a r t i n g from an i n i t i a l guess, one step c o n s i s t s of sea r c h i n g a new s o l u t i o n phase with sm a l l e r A^G along a descent d i r e c t i o n v. The steps are repeated u n t i l a minimum i s reached or some other stopping c r i t e r i a i s s a t i s f i e d . The phase found i n the l a s t step i s added to the l i s t of co n s i d e r e d phases. For each st e p f i r s t a descent d i r e c t i o n v and an approximate s t e p s i z e A i s c a l c u l a t e d , then the a c t u a l s t e p s i z e i s determined i n a s o - c a l l e d s t e p s i z e procedure. Because A^G of n o n - i d e a l s o l u t i o n s may have l o c a l minima, s e v e r a l i n i t i a l guesses are used, to in c r e a s e the p r o b a b i l i t y of f i n d i n g the g l o b a l minimum. During the mi n i m i z a t i o n we w i l l make use of some s p e c i a l p r o p e r t i e s of the A^G f u n c t i o n : a) Af (3(x n) = x n'ju^ : T h i s a l l o w s us t o c a l c u l a t e e a s i l y a f u n c t i o n value on the tangent hyperplane to x n at any p o s i t i o n x . T h i s w i l l be: x •MJ'. (see F i g . 4 ) . i f c m m n 3 x m«Mp < A^G(x n) then the v e c t o r v = ( x m _ x n ) s t a r t i n g at x n and p o i n t i n g towards x*m i s a descent d i r e c t i o n . F i g . 4: (page 27) I l l u s t r a t i o n of how the r e l a t i o n A^G(x) = x«7iT i s used d e t e c t descent d i r e c t i o n s and p o s s i b l e l o c a t i o n s of o n e - d i m e n s i o n a l minima. (See t ex t for d e t a i l s ) . Fig. 4: I l lustrat ion of the use of the Gibbs-Duhem equation 28 F u r t h e r : i f (x -x„) i s a descent d i r e c t i o n or m n A j G ( x m ) < A^G(x n) and Xp'M* < ^ f G ^ m ) then we know that there i s a minimum along a l i n e between x m and x\ . (The 3 m n absence of a minimum cannot be proved). b) I f x has a component x^ = 0 then Mj = -°°.(in p r a c t i c e Mj may be set to a l a r g e negative number, e.g. - 1 0 2 0 ) and any v e c t o r v s t a r t i n g from x with p o s i t i v e v^ i s a descent d i r e c t i o n . Thus any descent d i r e c t i o n w, s t a r t i n g from anywhere i n the s o l u t i o n , with negative Wj w i l l c o n t a i n a minimum w i t h i n the f e a s i b l e r e g i o n , ( a l l x^ ^ 0). T h i s i s a necessary c o n d i t i o n to use a simple s t e p s i z e procedure, and i n f a c t allows us to t r e a t each A^G m i n i m i z a t i o n as an unconstrained o p t i m i z a t i o n problem. The i n i t i a l guesses: If one or more s o l u t i o n phases are pa r t of the " c u r r e n t l y s t a b l e assemblage", these w i l l be used as i n i t i a l guesses. In a d d i t i o n , i f one endmember i s s t a b l e or the number of s t a b l e phases from t h a t s o l u t i o n i s zero or changed a f t e r the previous i t e r a t i o n , we scan the s o l u t i o n f o r a reasonable s t a r t i n g p o i n t . I t has u s u a l l y proved s u f f i c i e n t to choose a g r i d width of 1, that i s to s e l e c t the endmember with the lowest A^G, but very i l l - b e h a v e d s o l u t i o n s may r e q u i r e a f i n e r g r i d or more i n i t i a l guesses, (e.g. a l l 29 endmembers). The c a l c u l a t i o n of v and A : The re f inement of each i n i t i a l guess i s done in s teps each d e f i n e d by a d i r e c t i o n v and a s t e p s i z e A . In the f o l l o w i n g x „ i s the c o n c e n t r a t i o n v e c t o r a f t e r the n t l 1 s t e p . v i s always d e f i n e d in a way tha t the sum of a l l the v^ e q u a l s zero and the l e n g t h of v i s one. T h i s i s not a b s o l u t l y neces sary but g i v e s a u n i f o r m meaning to the s t e p s i z e A . F o r the f i r s t two s teps v i s always the d i r e c t i o n of the s t e e p e s t d e s c e n t : (v i s then n o r m a l i z e d so t h a t | v | = 1) and A i s e i t h e r an i n i t i a l v a l u e A- •. ( e . g . 0.02) i f the m i t 3 s t a r t i n g p o i n t i s a new i n i t i a l guess , or a v a l u e computed i n a p r e v i o u s m i n i m i z a t i o n for the same s o l u t i o n phase i f the s t a r t i n g p o i n t i s a p r e v i o u s minimum. To prevent z i g z a g g i n g , f or f u r t h e r s t e p s , v and A depend on the p o s i t i o n of a supposed minimum in the d i r e c t i o n of n / ne (x -x n n-2 ) : i f x n-2 < x (note : x" = A f G ( x ) n n r n then v s t eepes t descent A I / 2 e l s e v n-2 ( N o r m a l i z e d : | v | = 1) 30 The refinement i s terminated when e i t h e r the maximum number of steps allowed i s reached or A becomes smal l e r than the s m a l l e s t s t e p s i z e A m ^ n c o n s i d e r e d . The number of steps should be kept minimal (e.g. =3). The only s i t u a t i o n that r e q u i r e s more steps i s i f the minimum i s c l o s e to a hyperplane d e f i n e d by other p o s s i b l y s t a b l e phases. The s t e p s i z e procedure: Now that we have e s t a b l i s h e d v and A, we search f o r a p o i n t c l o s e to the minimum along the descent d i r e c t i o n . Each time a new composition v e c t o r x n + 1 i s c a l c u l a t e d , i t has to be d e f i n e d so that the sum of the x^ = 1 and a l l x^ > 0. T h i s i s achieved by s e t t i n g a l l x. < A . „ to zero and n o r m a l i z i n g J 3 l min 3 x . . to a sum of 1. n+1 The new composition v e c t o r x n + 1 i s c a l c u l a t e d as f o l l o w s : x n + 1 ( m ) = x n + m«A«v (m=0,1,2...) where m i s i n c r e a s e d u n t i l A f G ( x n + 1 ( m ) ) < A f G ( x n + 1 ( m + 1 ) ) or m > allowed maximum (e.g. 100) If m a f t e r t h i s step i s zero then x + 1 ( m ) = x n + V'A/(2^m) (m=1,2,...) where m i s i n c r e a s e d u n t i l A f G ( x n + 1 ( m ) ) < A^G(x n) or m > allowed maximum or A/(2«m) < A m ^ n ( i n the l a s t two cases x n + 1 = x n ) . 31 If A i s i n the magnitude of the optimal s t e p s i z e , m w i l l be a small number. A f t e r the f i r s t few i t e r a t i o n s the d e s c r i b e d procedure to c a l c u l a t e A y i e l d s nv*3. (with some notable exeptions where m==50.) Lik e a l l g r a d i e n t methods the convergence can become very slow. T h i s may occur i f the a l g o r i t h m i s "caught i n a narrow trough". T h e r e f o r e , i f A becomes smaller than A m. we should 3 ' min t e s t whether we are c l o s e to a minimum or whether the above s i t u a t i o n i s encountered. F o r t u n a t e l y , s o l u t i o n s show these narrow troughs u s u a l l y along t i e - l i n e s of s p e c i e s with low A^G. We can thus escape by d e f i n i n g a d d i t i o n a l steps i n the d i r e c t i o n towards each endmember (e.g. with A = A^ n^ t) If the computed minimum becomes a s t a b l e phase, i n the next i t e r a t i o n i t w i l l be used as an i n i t i a l guess with a s t e p s i z e A = |x n - X Q | or A m ^ n whichever i s g r e a t e r . 1.2.3 THE LINEAR PROGRAMMING (STEP 2 OF ALGORITHM) nc: Number of components i n the system np: Number of c u r r e n t l y c o n s i d e r e d phases n: 1 x np row v e c t o r : n^ = number of moles of phase k g: 1 x np row v e c t o r : g^ = A^G of one mol of phase k B: nc x 1 column v e c t o r : b^ = amount of component i i n system ||X||: nc x np matrix of phase compositions. X., = amount of component i i n phase k 32 A f e a s i b l e assemblage i s c h a r a c t e r i z e d by: ||X||«n T = B and a l l n k > 0 The problem to f i n d the e q u i l i b r i u m assemblage from a set of f i x e d composition phases can be formulated as: minimize G = n«g T subject to the c o n s t r a i n t s : ||X||«n T = 5 and a l l n^ > 0 T h i s can be sol v e d with a s p e c i a l case of the simplex method (see e.g. Dantzig (1963)). The f o l l o w i n g procedure assumes that at the beginning of step 2 of the a l g o r i t h m the f i r s t nc columns i n ||X||, n and g d e f i n e a f e a s i b l e assemblage of l i n e a r l y independent phases and t h a t the rank of ||X|| i s nc. These c o n d i t i o n s are s a t i s f i e d by the i n i t i a l assemblage as d e f i n e d e a r l i e r and remain so throughout a l l i t e r a t i o n s . F i r s t the matrix ||X|| i s reduced, so that the f i r s t nc columns are the i d e n t i t y - m a t r i x . (To a v o i d accumulations of rounding e r r o r s , the X ^ are f i r s t expressed i n terms of the system-components and then reduced). Each column j=1,2...nc rep r e s e n t s a " c u r r e n t l y s t a b l e phase". Each column k=nc+1,nc+2,...np can be i n t e r p r e t e d as a r e a c t i o n r e l a t i n g the phase k to the " c u r r e n t l y s t a b l e phases".: nc phase, = E X., phase-i=1 1 K 1 with a A rG(k) = Each r e a c t i o n i s t e s t e d f o r A G(k) > 0. I f one i s found then I ( 9 i - X i k ) q k 33 that phase k i s more s t a b l e than a l i n e a r combination of the " c u r r e n t l y s t a b l e " phases. At t h i s p o i n t a 5-step procedure i s s t a r t e d : -A- Locate the " c u r r e n t l y s t a b l e phase" to be re p l a c e d by the newly-found phase k: The s m a l l e s t p o s i t i v e n./X.. 1 1K (i=1,2,..nc) i s c a l l e d F and the corresponding i i s A c a l l e d i . Phase i w i l l be d i s p l a c e d by phase k. A A -B- Force agreement with the c o n s t r a i n t s ||X||•n T = B and n. ^ 0 by d e f i n i n g new phase-amount v e c t o r s n- = n i - F x - X i k (i=1 ,2, ...nc) , n k = F x ' (Note that the number of moles of phase i x becomes zero.) -C- Switch columns k and i i n the matrix ||X|| , g, n and other book-keeping a r r a y s . T h i s a c h i e v e s replacement of phase i by phase k. A -D- Reduce ||X|| with the p i v o t i n g element now occupying p o s i t i o n i x , i x . T h i s normalizes the r e a c t i o n s to r e f e r to the phases now occupying the f i r s t nc columns of ||X||. -E- Search f o r the next phase wi t h A rG(k) > 0. If none i s found, terminate the l i n e a r programming, otherwise r e t u r n to step -A-. 34 1.2.4 THE CHANGE OF BASE OPERATION (STEP 3 OF ALGORITHM) T h e o r e t i c a l l y i t would be p o s s i b l e to hide t h i s step i n the n o n l i n e a r programming r o u t i n e (minimizing relative to the " c u r r e n t l y s t a b l e assemblage"). The advantage of performing the change of base o p e r a t i o n a f t e r each l i n e a r programming step i s that the a l g o r i t h m becomes more tra n s p a r e n t and thus e a s i e r to program. I t transforms the gene r a l chemical e q u i l i b r i u m problem i n t o a s e r i e s of i n d i v i d u a l f u n c t i o n m i n i m i z a t i o n s . For each phase g^ i s r e p l a c e d by: nc Q k ' - q k x i k ' q i i = 1 For k<nc the above equation y i e l d s g^' = 0. 1.3 APPLICATIONS OF THE COMPUTATION OF CHEMICAL EQUILIBRIUM 1.3.1 INTRODUCTION The a l g o r i t h m presented i n t h i s chapter i s e s p e c i a l l y designed f o r e q u i l i b r i u m c a l c u l a t i o n s i n complex systems i n v o l v i n g phase s e p a r a t i o n i n more than one phase. It would be r a t h e r i n e f f i c i e n t to use i t f o r an i d e a l one-phase system. The examples are chosen p r i m a r i l y to demonstrate the v a r i e t y of problems that can be s o l v e d with the d e s c r i b e d method. Although some of the data or models may be of q u e s t i o n a b l e u t i l i t y , the c a l c u l a t e d r e s u l t s show that the a l g o r i t h m a l l o w s a quick i n s i g h t i n t o complex s t a b i l i t y 35 r e l a t i o n s . (Or at l e a s t i n t o the models we have of them.) For most of the f o l l o w i n g a p p l i c a t i o n s the models are taken d i r e c t l y from the l i t e r a t u r e or are given i n a t a b l e , so that a c r i t i c a l re-examination of the r e s u l t s i s p o s s i b l e . 1.3.2 EXAMPLE 1: BLAST FURNACE PROBLEM Th i s example i s taken from Smith and Missen (1982, pp. 208-211) who use i t to demonstrate the c a p a c i t i e s and l i m i t a t i o n s of d i f f e r e n t computation methods. The problem was o r i g i n a l l y formulated and s o l v e d with s l i g h t l y l e s s CH„ in the bulk composition by Madeley and Toguri (1973). A l l data and the r e s u l t are l i s t e d i n t a b l e 1. As we would expect from a b l a s t furnace the s t a b l e phases are i r o n , a gas phase and i m p u r i t i e s (CaO). T h i s problem has a unique s o l u t i o n and i n a d d i t i o n n i t r o g e n i s an i n e r t s p e c i e s i n the gas phase. Our method cannot take advantage of any of these s p e c i a l cases but s t i l l converges at very a c c e p t a b l e speed and the c a l c u l a t e d c o n c e n t r a t i o n s are i d e n t i c a l to those p u b l i s h e d by Smith and Missen (1982). 1.3.3 EXAMPLE 2: SILICATE MELTS The system c o n s i d e r e d i s CaO - A l 2 0 3 - S i 0 2 . A l l data for the s o l i d phases and the t e r n a r y n o n - i d e a l melt model are taken from Berman and Brown (1984). Let us f i r s t c o n c e n t r a t e on the melt a l o n e . The thermodynamic model of the melt phase i n v o l v e s a 4t*1 degree Margules-type excess f u n c t i o n with 12 i n t e r a c t i o n parameters. F i g . 5 shows the 36 T a b l e 1.: E q u i l i b r i u m c a l c u l a t i o n s f o r the b l a s t furnace example. (1050 K, 1 atm) Spec i e s A f G [ J / m o l ] Bulk S t a b l e m. f . S o l i d phases C ( G r a p h i t e ) 0 85.5901 F e ( I r o n ) 0 3.5270 42. 8270 CaO(Lime) -529190 0.6063 0. 7562 F e 3 O f l (Magnet i te ) -762660 13.1000 C a C 0 3 ( C a l c i t e ) -942450 0.1499 < FeO(Wust i t e ) -193930 i d e a l qas 282. 2639 0 2 0 20.4600 6. 1303- 10" 2 2 H 2 0 2. 2766- 10" 2 N 2 0 187.1000 6. 6286< 10" 1 C H 2 0 -86110 8. 3 5 4 3 « 10" 9 CHO -62550 3. 7258- 10" 9 OH 22590 2. 8093. 10" 1 3 CO 2 -395970 2. 3608- 10" 2 H 2 0 -189870 1.7750 1 . 5723- 10- 3 CH« 24850 2.5540 2. 3201 - 10" 5 CO -204640 2. 8918- 10" 1 B u l k : i n i t a l m i x t u r e of phases i n moles , m . f . : mole f r a c t i o n i n s t a b l e gas phase . 37 F i g . 5: (page 38) M i s c i b i l i t y gaps i n the CaO - A l 2 0 3 - S i 0 2 me l t s a c c o r d i n g to the data from Berman and Brown (1984) . The c o n c e n t r a t i o n s are i n mol %. The s t r a i g h t l i n e s are c a l c u l a t e d t i e - l i n e s between c o e x i s t i n g phases . The d o t t e d area i s a f i e l d where three d i f f e r e n t t e r n a r y mel ts are i n c o e x i s t e n c e at 1400 ° C . T h e i r c o m p o s i t i o n s are i n terms of ( C a O / A l 2 0 3 / S i 0 2 ) : ( 0 .02135 /0 .13489 /0 .84376) , (0 .09798/0 .77994/0 .12208) and (0 .08535/0 .20656/0 .70809) . Fig. 5: Miscibi l i ty gaps in the CaO - A 1 2 0 3 - SiO g melts S i 0 8 39 r e s u l t s of e q u i l i b r i u m computations f o r v a r y i n g compositions at 1400° and 1500° C. For the two diagrams some 70 c a l c l a t i o n s were performed. There has been some d i s c u s s i o n about the temperature extensions of that model (Barron (1985), Berman and Brown (1985b), Connolly and K e r r i c k (1986)) to which we can now add a new s p i c e : below ca. 1478 °C there i s a f i e l d with three c o e x i s t i n g melts. T h i s a r i s e s from the i n t e r f e r e n c e of two two-phase f i e l d s . Although i t i s metastable r e l a t i v e t o p o s s i b l e s o l i d phases and has no f u r t h e r i m p l i c a t i o n s f o r t h i s system we should once more be reminded that multicomponent s o l u t i o n s may exsolve to more than only two phases. S i m i l a r r e s u l t s have been rep o r t e d f o r the same system by C o n n o l l y and K e r r i c k (1986) at temperatures higher than those f o r which the model was c a l i b r a t e d . F i g . 6. i s the phase diagram f o r t h i s system at 1400 °C i n c l u d i n g the s o l i d phases. M u l l i t e i s not c o n s i d e r e d because of inadequate thermodynamic data. I f m u l l i t e were i n c l u d e d the f i e l d with two c o e x i s t i n g melts i n the S i 0 2 ~ r i c h corner might prove to be metastable. For the quaternary system MgO - CaO - A l 2 0 3 - S i 0 2 the mixing p r o p e r t i e s of the melt were i n v e s t i g a t e d by Berman (1983). The model i s s i m i l a r to the t e r n a r y system but now has 31 Margules parameters. They have a v a l i d i t y range d e f i n e d by the l o c a t i o n of the l i q u i d u s s u r f a c e and e x p e r i m e n t a l l y determined t w o - l i q u i d f i e l d s and are l i s t e d i n t a b l e 2 together with a c a l c u l a t e d t i e - l i n e i n t a b l e 3. 40 F i g . 6: (page 41) 0 Phase d iagram for the system CaO - A 1 2 0 3 - S i 0 2 at 1400 ° C . A l l thermodynamic data are from Berman and Brown (1984) . The l i q u i d u s f i e l d i n c l u d e s a two-phase r e g i o n in the S i 0 2 - r i c h c o r n e r . The s t i p p l e d areas are the s t a b l e melt f i e l d s . The t i e - l i n e s between the melt and the s o l i d phases are o m i t t e d . Fig. 6: Phase diagram for the system CaO - A1 2 0 3 - S i 0 2 at 1400 °C S i 0 8 42 Table 2.'. I n t e r a c t i o n parameters f o r the quaternary system MgO - CaO - A 1 2 0 3 - S i 0 2 , a c c o r d i n g to Berman (1983). The index i s as d e f i n e d by Berman and Brown (1984). W = WH - T-Ws Binary S i 0 2 - A L 2 0 3 index W. H SSSA -161039.81 SSAA 1803871.61 SAAA 258911.07 W, -60.41 844.79 110.47 Binary S i 0 2 - CAO SSSC SSCC SCCC -25525.64 •341962.81 •960867.88 34. 19 56.62 •247. 1 1 Binary S i 0 2 - MgO SSSM SSMM SMMM 94145.91 •270581 .70 •610906.87 52.77 -59.98 •195.01 Binary A 1 2 0 3 - CaO AAAC AACC ACCC •1 97743.47 •734020. 10 •617537.61 -1.11 •251 .94 -76.83 Binary A 1 2 0 3 - MgO AAAM AAMM AMMM -641890.87 727706.30 -691193.17 -224.47 290.88 -227.94 43 Table 2. (continued) W = WH - T-Wg Binary CaO - MgO index CCCM CCMM CMMM W H 318144.87 590616.72 114076.93 154.79 322.37 58.66 Ternary S i 0 2 - A 1 2 0 3 - CaO SSAC -2685775.05 -917.87 SAAC -2833471.13 -976.80 SACC 580678.70 526.17 Ternary S i 0 2 - CaO - MgO SSCM -1143506.91 -279.90 SCCM -2464803.70 -669.00 SCMM -2026666.90 -555.03 Ternary A 1 2 0 3 - CaO - MgO AACM 343546.79 160.77 ACCM -2440837.52 -526.78 ACMM -3334297.25 -1148.10 Ternary S i 0 2 - A 1 2 0 3 - MgO SSAM -1828080.99 -693.44 SAAM -3201173.35 -1382.29 SAMM -1828080.99 -693.44 Quaternary SACM S i 0 2 - A 1 2 0 3 - CaO - MgO 2179011.74 1328.50 44 T a b l e 3. : C a l c u l a t e d c o e x i s t i n g q u a t e r n a r y l i q u i d s at 1400 ° C , 1 bar . The bulk c o m p o s i t i o n i s 30% S i 0 2 , 40% A 1 2 0 3 , 10% CaO and 20% MgO. A l l c o n c e n t r a t i o n s are i n mol%. 12.023 % 87.977 % l i q u i d 1 l i q u i d 2 S i 0 2 32.625 29.641 A 1 2 0 3 38.721 40.175 CaO 22.914 8.235 MgO 5.740 21.949 A l t h o u g h the a l g o r i t h m was deve loped hav ing main ly b i n a r y and t e r n a r y systems i n mind , i t a l s o works w e l l f o r c a l c u l a t i n g q u a t e r n a r y s o l v i . The computing time seems to i n c r e a s e rough ly in p r o p o r t i o n to the square of the number of endmembers. I f the method were used to s tudy phase s e p a r a t i o n models in h i g h e r o r d e r mult icomponent m e l t s , i t would be u s e f u l to a c c e l e r a t e the convergence for t h i s s p e c i a l c a s e . In p r i n c i p l e , t h i s can be a c h i e v e d by u s i n g a p r e v i o u s l y c a l c u l a t e d e q u i l i b r i u m to generate the i n i t i a l guess and by add ing more than j u s t the minimum A^G c o m p o s i t i o n phases at each i t e r a t i o n ( e . g . some p o i n t s i n the v i c i n i t y of these minima and around the bulk 45 composition.) 1.3.4 EXAMPLE 3: MICA - FELDSPAR EQUILIBRIA 1.3.4.1 Thermodynamic models Let us assume we were i n t e r e s t e d i n doing an experimental study on the r e a c t i o n s between white mica (paragonite - muscovite) and f e l d s p a r . T h i s system i s r e l a t i v e l y well-known and we can p r e d i c t most of the r e s u l t s . The data-base used was compiled by Berman et a l . (1985) and i s c o n s i s t e n t with most experiments performed i n t h i s system. A l b i t e and K - f e l d s p a r have l o w - f e l d s p a r standard s t a t e p r o p e r t i e s . A d i s o r d e r i n g f u n c t i o n i s used, so that at high temperatures t h e i r A^G's are those of the h i g h - f e l d s p a r s . The a - 0 t r a n s i t i o n i n quartz i s modelled a c c o r d i n g to Berman and Brown (1985a). A ^ G of C0 2 i s c a l c u l a t e d u sing a m o d i f i e d Redlich-Kwong equation of s t a t e ( K e r r i c k and Jacobs (1981)) and A ^ G of H 20 a c c o r d i n g to Haar et a l . (1984). The apparent A ^ G f o r a l l c o n s i d e r e d phases are l i s t e d i n Table 4. The most important r e a c t i o n s f o r which the data have to be c o n s i s t e n t are paragonite + quartz = a l b i t e + a l u m o s i l i c a t e + water ( C h a t t e r j e e (1972)). and muscovite + quartz = K - f e l d s p a r + a l u m o s i l i c a t e + water. ( C h a t t e r j e e and Johannes (1974)). The mixing model f o r muscovite - paragonite i s taken from C h a t t e r j e e and Froese (1975) which they d e r i v e d from t h e i r own molar volume measurements and the solvus data of Eugster 46 et a l . (1972) f o r w h i t e m i c a s . The n o t a t i o n f o r the Margules parameters i s the same as i n t r o d u c e d by Berman and Brown (1984). A f G ( M i c a ) = x p a - A f G ( P a ) + x M u«A fG(Mu) + R . T . x p a . l n ( x p a ) + R . T . x M u . l n ( x M u ) + W 1 2 2 . x p a . x M u + W 1 1 2 . x p a . x M u [ J / m o l ] . Where : W 1 2 2 = 12230.2504 + 0.7104432-T + 0.665256-P W 1 1 2 = 19456.0184 + 1.6543536-T - 0.456056«P For t he f e l d s p a r s t he model of G h i o r s o (1984) was chosen, which on rearrangement, y i e l d s : A f G ( f s p ) = x A b - A f G ( A b ) + x 0 r - A f G ( O r ) + x A n«A fG(An) + K - ^ A b - ^ A b ^ A n ^ + R ' T - x O r - l n ( x O r ( l - x A n ) ) + R-T-x, - l n ( x a (1+x. ) 2 / 4 ) + W 1 1 2 . x j An An An Ab Or + W ' ^ - X A b * X A n + W ' 3 3 ' x A b ' x A n + W ^ - x 0 r . x A n + w i 2 3 - x A b . x 0 r . x A n [J/mol] Where : W 1 1 2 = 17062.352 + 0.3606608-P W 1 2 2 = 30978.336 - 21.5476-T + 0.3606608-P (Thompson and H o v i s (1979)) W, , 3 = 8471.3448 W 1 3 3 = 28225.6824 (Newton e t a l . (1980)) 47 W 2 2 3 = 27983.0104 + 20.21022624-T W 2 3 3 = 67469.0920 - 11.062496-T W 1 2 3 = 76225.45088 - 8.43465112«T (Ghiorso (1984)) The mixing p r o p e r t i e s f o r H 20 and C0 2 are a c c o r d i n g to K e r r i c k and Jacobs (1981). The f i r s t column i n t a b l e 5 shows the c a l c u l a t e d assemblages at v a r i o u s temperatures and 4 Kbars f o r a bulk composition equal to 1/2 p a r a g o n i t e , 1/2 muscovite p l u s water. 1.3.4.2 Excess phases C h a t t e r j e e and Froese (1975) have c a l c u l a t e d the e q u i l i b r i u m assemblages i n the same system, assuming that i n a d d i t i o n to H 20, q u a r t z i s a l s o always pre s e n t . Whether quartz i s s t a b l e i n the chosen P-T-range depends on the bulk composition. We can always f o r c e quartz i n t o the s t a b l e assemblage by adding enough S i 0 2 to the system. A l t e r n a t i v e l y we can c a l c u l a t e the e q u i l i b r i u m assemblages with excess q u a r t z by assuming the chemical p o t e n t i a l of S i 0 2 i n the s t a b l e assemblage to be equal to A^G of q u a r t z . In that case we have to allow t h a t quartz may be present i n a negative amount. Because the l i n e a r programming r o u t i n e i s based on the c o n s t r a i n t that a l l amounts are p o s i t i v e , I propose a s l i g h t l y d i f f e r e n t approach: we d e f i n e a phase c a l l e d -quartz with the composition - S i 0 2 and a A^G which i s the negative of that of q u a r t z . The phases quartz and -quartz have the same chemical p o t e n t i a l s and i f quartz alone i s s t a b l e ( r e l a t i v e to t r i d y m i t e , c r i s t o b a l i t e , S i + 48 0 2 e t c . ) then one of the two w i l l always be p a r t of the s t a b l e assemblage: q u a r t z , i f enough S i 0 2 i s p r e s e n t and - q u a r t z i f n o t . The number of moles of s t a b l e - q u a r t z i s the amount of S i 0 2 we would have t o add t o the system t o s a t u r a t e i t w i t h q u a r t z . In column 2 of t a b l e 5 the r e s u l t s w i t h e x c e s s q u a r t z a r e l i s t e d . As l o n g as two w h i t e micas a r e s t a b l e t h e i r c o m p o s i t i o n s a r e the same as i n column 1. For 600 °C C h a t t e r j e e and Fr o e s e (1975) c a l c u l a t e d x Q r = 0.0416 and xMu = u'8"738. The s m a l l d i f f e r e n c e i n our c o m p u t a t i o n a l r e s u l t s i s e x p e c t e d from the s l i g h t l y d i f f e r e n t endmember p r o p e r t i e s used. 1.3.4.3 A d d i t i o n of c a l c i t e A dding c a l c i t e t o the system changes the s i t u a t i o n d r a s t i c a l l y . The f e l d s p a r s have an a d d i t i o n a l endmember, the f l u i d becomes a m i x t u r e of H 20 and C 0 2 and more phases have t o be c o n s i d e r e d . These changes a r e r e s p o n s i b l e f o r q u i t e d i f f e r e n t e q u i l i b r i u m c o m p o s i t i o n s (column 3, t a b l e 5 ) . The t h r e e c o e x i s t i n g f e l d s p a r s a t 550 0 a r e a consequence of G h i o r s o ' s (1984) model ^which was f i t t e d t o d a t a between 600 0 and 900 °C w i t h the i n t e n t i o n t o e x t r a p o l a t e t o higher t e m p e r a t u r e s . The c o n f i g u r a t i o n a l e n t r o p y assumes d i s o r d e r e d a l b i t e / K - f e l d s p a r and f u l l y o r d e r e d a n o r t h i t e ( a c c o r d i n g t o Newton e t a l . ( 1 9 8 0 ) ) . Our e x t r a p o l a t i o n t o lower t e m p e r a t u r e s i s i n t e n d e d o n l y t o demonstrate the a b i l i t y of the a l g o r i t h m t o handle such complex s t a b i l i t y r e l a t i o n s . 49 1.3.4.4 A d d i t i o n of a b u f f e r In many e x p e r i m e n t a l t e c h n i q u e s b u f f e r s a re used t o c o n t r o l the a c t i v i t i e s of one or more s p e c i e s i n the f l u i d phase. S i m i l a r (and p r o b a b l y more complex) b u f f e r s p l a y an i m p o r t a n t r o l e i n g e o l o g i c a l p r o c e s s e s . I f the c o n t r o l l i n g r e a c t i o n s a r e s i m p l e , the con c e p t of b u f f e r i n g can e a s i l y be s i m u l a t e d i n a A ^ G - m i n i m i z a t i o n . We w i l l h y p o t h e t i c a l l y assume t h a t the r e a c t i o n b r u c i t e = p e r i c l a s e + H 20 i s b u f f e r i n g the a c t i v i t y of H 20. T h i s r e a c t i o n was chosen because the a c t i v i t y of H 20 w i l l v a r y c o n s i d e r a b l y over the c o n s i d e r e d temperature range. We d e f i n e a new phase which we c a l l B P - b u f f e r . I t s c o m p o s i t i o n i s H 20 and A^ G ( B P - b u f f e r ) = A ^ G ( b r u c i t e ) - A ^ G ( p e r i c l a s e ) . We now have t o add enough H 20 t o the system, so t h a t the B P - b u f f e r becomes s t a b l e . I f too much i s added t h i s does not change the assemblage, because a l l the e x c e s s H 20 w i l l go i n t o the b u f f e r phase. For the c a l c u l a t i o n of column 4, t a b l e 5 I d i d not add H 20 but have used the same t e c h n i q u e as f o r the ex c e s s q u a r t z and have d e f i n e d a n e g a t i v e b u f f e r phase. The b i g g e s t i n f l u e n c e of the b u f f e r on the s t a b l e assemblage i s a t 400 and 450 °C, where the H 2 0 - a c t i v i t y i s s m a l l . 50 Table 4.: A fG [J/mol] for a l l considered phases in example 3. P = 4 Kbars 400 C 450 C 500 C 550 C eoo C 650 C 700 ' C ALBITE ANDALUSITE ANORTHITE BOEHMITE CALCITE CORUNDUM CRISTOBALITE DIASPORE GEHLENITE GIBBSITE GROSSULAR dADEITE KAOLINITE KYANITE LAUMONTITE LAWSONITE LIME MARGARITE MEIONITE MUSCOVITE -4075566.88 -2654752.33 -4365243.60 -1026914.47 -1270609.53 -1714917.47 -935769.21 -1026442.54 -4120291.55 -1347493.52 -6820716.11 -3122061.28 -4265285.94 -2655718.71 -7581274.72 -5035238.32 -661949.37 -6428119.55 -14346560.88 -6185113.43 -4096864.29 -2665986.52 -4386277.47 -1032301.70 -1279423.62 -1721749.27 -940463.02 -1031171.02 -4141299.37 -1356203.25 -6850766.32 -3137126.10 -4288420.54 -2666485.69 -7627497.71 -5061677.06 -665876.09 -6458716.13 •14420543.81 -6216798.67 -4119235.37 -2677855.52 -4408347.79 -1037984.64 -1288639. 10 -1728988.88 -945383.80 -1036195.21 -4163277.56 -1365454.08 -6882433.84 -3152980.36 -4312781.45 -2677890.85 -7676025.56 -5089582.82 -669982.32 -6490976.10 -14498043.78 -6250141.51 -4142625.94 -2690326.99 -4431402.77 -1043950.40 -1298236.69 -1736615.54 -950519.82 -104 1502.22 -4186175.27 -1375224.21 -6915633. 14 -3169584.01 -4338308.43 -2689902.21 -7726741.68 -5118891.96 -674257.68 -6524818.08 -14578886.74 -6285060.13 -4166984.24 -2703371.50 -4455395.50 -1050187.11 -1308198.85 -1744610.38 -955860.42 - 1047080. 18 -4209946.62 -1385493.37 -6950286.68 -3186900.74 -4364946.79 -2702490.61 -7779540.35 -5149546.07 -678692.92 -6560168.00 -14662915.02 -6321479.82 -4192261.32 -2716962. 17 -4480283.36 -1056683.82 -1318509.58 -1752956. 19 -961395.88 -1052918.14 -4234550.00 -1396242.70 -6986324.01 -3204897.48 -4392646.66 -2715629.44 -7834325.31 -5181491.30 -683279.72 -6596958.26 -14749985.14 -6359332.25 -4218411.29 -2731074.42 -4506027.43 -1O63430.4O -1329154.25 -1761637.21 -967117.33 -1O59005.97 -4259947.55 -1407454.63 -7023680.91 -3223544.00 -442 1362.44 -2729294.30 -7891008.64 -5214677.94 -688010.56 -6635126.99 -14839966.08 -6398554.76 Ln O 51 Table 4. (continued) 400 C 450 ' C 500 * C 550 ' C GOO ' C 650 ' C 700 ' C PARAGONITE POTASSIUM FELDSPAR PREHNITE PYROPHYLLITE QUARTZ SILLIMANITE STELLERITE TRIDYMITE CA-AL PYROXENE WAIRAKITE WOLLASTONITE PSEUDOWOLLASTONITE YUGAWAIRALITE ZOISITE CLINOZOISITE CARBON DIOXIDE STEAM BP-Buffer (=BRUCITE-PERICLASE) -6141210.88 -4114965.26 -6401100.31 -5808596.81 -938516.34 -2654516.62 -11378078.94 -936240.56 -3400457. 16 -6915351.05 -1687570.02 -1685346.75 -9453495.64 -7105190.98 -7102840.97 -489581.23 -337079.75 -351616.51 -6172241.75 -4136665.82 -6432810.31 .-5836291.37 -942969.67 -2665892.71 -11449229.40 -940873.57 -3416215.49 -6953799.26 -1696055.53 -1693992.11 -9508560.64 -7138292.66 -7135417.64 -498243.82 -343366.37 -355636.63 -6204913.85 -4159424.00 -6466161.13 -5865503.69 -947658.46 -2677898.25 -11523978.12 -945736.62 -3432792.79 -6994141.16 -1704948.08 -1703041.89 -9566381 .59 -7173133.63 -7169733.61 -507116.90 -349892.75 -359867.93 -6239143.93 -4183184.04 -6501073.20 -5896161.71 -952574.78 -2690502.49 -11602143.94 -950817.94 -3450146.98 -7036280.80 -1714226. 16 -1712474.65 -9626819.52 -7209628.74 -7205703.72 -516186.73 -356643.87 -364299.76 -6274855.89 -4207894.55 -6537474.24 -5928199.70 -957712.45 -2703677.47 -11683562.49 -956106.76 -3468239.87 -7080131.10 -1723870.29 -1722270.96 -9689748.21 -7247700.69 -7243250.66 -525441.37 -363605.91 -368922.36 -6311980.05 -4233508.34 -6575298.31 -5961557.49 -963066.98 -2717397.67 -11768084.10 -961593.20 -3487036.71 -7125612.71 -1733862.84 -1732413. 18 -9755052.74 -7287279.04 -7282304.00 -534870.33 -370765.97 -373726.80 -6350452.50 -4259982.05 -6614485.08 -5996179.88 -968653.89 -2731639.75 -11855572.00 -967268 . 19 -3506505.75 -7172653. 15 -1744187.72 -1742885.21 -9822628.04 -7328299.35 -7322799.30 -544464.30 -378111.94 -378704.91 I—1 52 Table 5.: Calculated mica - feldspar e q u i l i b r i a . Bulk 0.5 Parag + 0.! 5 Muse 0.5 Parag + 0 5 Muse 0.5 Parag + 0.5 Muse + 1 H20 0.5 Parag + 0.5 Muse + 1 H*0 comp.: + 1 HiO + 1 H;0 + excess Qz + 2 Qz + 2 Cc + 2 Oz + 2 Cc (+ BP-Buffer) 400 ' C Pa(0.09O7)Mu(0 .9093) Pa(0.0907)Mu(0.9093) Pa(0.0907)Mu(0.9093) Pa(0.0370)Mu(0.9630) 4 Kbar Pa(0.9537)Mu(0 .0463) Pa(0.9537)Mu(0.0463) Pa(0.9537)Mu(0.0463) Ab(0.7911)0r(0.007 1)An(0. . 2018) Ab(0.8877)Or(0.0261)An(0 .0862) Steam Steam Hi 0(0.9928)C0*(0.0072) H,0(0.0279)C0*(0.9721) Quartz Quartz, Ca l c i t e , Z o i s i t e Quartz, C a l c i t e , Z o i s i t e 450 ' C Pa(0.1158)Mu(0 .8842) Pa(0.1158)Mu(0.8842) Pa(0.1158)Mu(0.8842) Pa(0.0326)Mu(0.9674) 4 Kbar Pa(0.9387)Mu(0 .0613) Pa(0.9387)Mu(0.0613) Pa(0.9387)Mu(0.0613) Ab(0.5782)0r(0.OO4O)An(O. 4178) Ab(0.7549)0r(0.0271)An(0 .2180) Ab(0.1187)0r(O.OOO3)An(O. 8810) Steam Steam HJ0(0.9369)C0*(0.0631) H;0(0.0600)CO «(0.9400) Quartz Quartz, Calcite Quartz, C a l c i t e , Kyanite 500 ' C Pa(0.1448)Mu(0 .8552) Pa(0.1448)Mu(0.8552) Pa(0.0308)Mu(0.9692) Ab(0.0568)0r(0.9372)An(0. .0060) 4 Kbar Pa(0.9208)Mu(0 .0792) Pa(0.9208)Mu(0.0792) Ab(O.5157)0r(0.O159)An(O.4684) Ab(0.5175)0r(0.0215)An(0. .4610) Ab(0.1535)0r(O.OO23)An(O. 8442) Ab(0.15O8)0r(O.OO30)An(O. 8462) Steam Steam H;0(0.7502)C0z(0.2498) H*0(0. 1 152)C0z(0.8848) Quartz Quartz, Calcite Quartz, C a l c i t e Ln to Table 5 . (continued) Bulk 0 . 5 Parag + 0 . ! 5 Muse 0 . 5 Parag + 0 . 5 Muse 0 . 5 Parag + 0 . 5 Muse + 1 H * 0 0 . 5 Parag + 0 . 5 Muse + 1 HJO comp.: + 1 HJO + 1 H z O + excess Oz + 2 Qz + 2 Cc + 2 Qz + 2 Cc (+ BP-Buffer) 5 5 0 ' C P a ( 0 . 1 7 8 3 ) M u ( 0 . 8 2 1 7 ) P a ( 0 . 1 7 8 3 ) M u ( 0 . 8 2 1 7 ) A b ( 0 . 0 7 3 0 ) 0 r ( 0 . 9 1 8 5 ) A n ( 0 . 0 0 8 5 ) A b ( 0 . 0 7 3 0 ) 0 r ( 0 . 9 1 8 5 ) A n ( 0 . 0 0 8 5 ) 4 Kbar P a ( 0 . 8 9 9 3 ) M u ( 0 . 1 0 0 7 ) P a ( 0 . 8 9 9 3 ) M u ( 0 . 1 0 0 7 ) A b ( 0 . 4 4 1 2 ) 0 r ( O . O 2 2 3 ) A n ( O . 5 3 6 5 ) Ab(O . 4 4 1 2)0r(O . O 2 2 3)An(O . 5 3 6 5 ) A b ( 0 . 2 0 4 0 ) 0 r ( 0 . 0 0 6 5 ) A n ( 0 . 7 8 9 5 ) A b ( 0 . 2 0 4 0 ) 0 r ( 0 . 0 0 6 5 ) A n ( 0 . 7 8 9 5 ) Steam Steam H * 0 ( 0 . 6 6 6 7 ) C 0 * ( O . 3 3 3 3 ) H,0(0.2049)COz(0.7951) Quartz Quartz, C a l c i t e Quartz, C a l c i t e 6 0 0 * C P a ( 0 . 2 1 6 9 ) M u ( 0 . 7 8 3 1 ) P a ( 0 . 1 4 0 4 ) M u ( 0 . 8 5 9 6 ) A b ( 0 . 0 8 9 2 ) 0 r ( 0 . 8 9 9 3 ) A n ( 0 . . 0 1 1 5 ) A b ( 0 . 0 8 9 2 ) 0 r ( 0 . 8 9 9 3 ) A n ( 0 . , 0 1 1 5 ) 4 Kbar P a ( 0 . 8 7 3 4 ) M u ( 0 . 1 2 6 6 ) A b ( 0 . 9 4 3 2 ) 0 r ( 0 . 0 5 6 8 ) A b ( 0 . 3 0 7 8 ) 0 r ( 0 . 0 1 6 8 ) A n ( 0 . . 6 7 5 4 ) A b ( 0 . 3 0 7 8 ) 0 r ( 0 . 0 1 6 8 ) A n ( 0 . . 6 7 5 4 ) Steam Steam H * 0 ( 0 . 6 6 6 7 ) C O * ( 0 . 3 3 3 3 ) H * 0 ( 0 . 3 4 6 O ) C 0 * ( 0 . 6 5 4 0 ) Corundum Quartz, S i l l i m a n i t e Quartz, C a l c i t e Quartz, C a l c i t e 6 5 0 ' C P a ( 0 . 0 9 6 3 ) M u ( 0 . 9 0 3 7 ) A b ( 0 . 8 1 0 5 ) 0 r ( 0 . 1 8 9 5 ) A b ( 0 . 1 0 4 4 ) 0 r ( 0 . 8 8 0 4 ) A n ( 0 . 0 1 5 2 ) A b ( 0 . 1 O 4 4 ) 0 r ( O . 8 8 O 4 ) A n ( O . . 0 1 5 2 ) 4 Kbar Ab(O . 8 6 6 6)0r(O . 1 3 3 4 ) A b ( 0 . 3 9 5 2 ) 0 r ( 0 . 6 0 4 8 ) A b ( 0 . 3 0 2 5 ) 0 r ( 0 . 0 2 2 9 ) A n ( 0 . 6 7 4 6 ) A b ( 0 . 3 0 2 5 ) 0 r ( 0 . 0 2 2 9 ) A n ( 0 . 6 7 4 6 ) Steam Steam H * 0 ( 0 . 6 6 6 7 ) C 0 z ( 0 . 3 3 3 3 ) H j 0 ( 0 . 5 6 9 1 ) C 0 2 ( 0 . 4 3 0 9 ) Corundum Quartz, Si l l i m a n i t e Quartz, C a l c i t e Quartz, C a l c i t e 7 0 0 ' C A b ( 0 . 5 0 0 0 ) 0 r ( 0 , . 5 0 0 0 ) A b ( 0 . 5 0 0 0 ) 0 r ( 0 . 5 0 0 0 ) A b ( 0 . 1 1 8 3 ) O r ( 0 . 8 6 2 2 ) A n ( 0 . 0 1 9 5 ) A b ( 0 . 1 1 8 3 ) O r ( 0 . 8 6 2 2 ) A n ( 0 . 0 1 9 5 ) 4 Kbar A b ( 0 . 2 9 7 2 ) 0 r ( 0 . 0 3 0 4 ) A n ( 0 . 6 7 2 4 ) A b ( 0 . 2 9 7 2 ) 0 r ( 0 . 0 3 0 4 ) A n ( 0 . 6 7 2 4 ) Steam Steam H ; 0 ( 0 . 6 6 6 7 ) C 0 ; ( 0 . 3 3 3 3 ) H ! 0 ( O . 9 1 7 8 ) C 0 ! ( 0 . 0 8 2 2 ) Corundum Quartz, S i l l i m a n i t e Quartz, C a l c i t e Wollastoni te 54 1.3.5 DISCUSSION The method presented f o r computing phase e q u i l i b r i a can be used to s o l v e a great v a r i e t y of problems and to t e s t the consequences of thermodynamic models and data i n a reasonable amount of computer time. The t i m i n g s presented i n t a b l e 6 show that even f o r very complex problems the computing times are in the order of one to two seconds. For e x t e n s i v e c a l c u l a t i o n s i n a s p e c i f i c system i t might be more e f f i c i e n t to use a more s p e c i a l i z e d a l g o r i t h m , and to use the general a l g o r i t h m only i f d i f f i c u l t i e s are encountered or to check some of the assemblages. For example, the p o s i t i o n of the l i q u i d u s i n F i g . 6 can be c a l c u l a t e d much f a s t e r with an a l g o r i t h m used by Berman and Brown (1984), but t h e i r procedure w i l l not r e c o g n i z e the t w o - l i q u i d f i e l d i n the S i 0 2 - r i c h c o r n e r . E x p l o r i n g the P-T-plane fo r s e v e r a l systems with v a r y i n g bulk compositions i t was found t h a t the a l g o r i t h m always converged, but as with a l l other methods, we cannot completely exclude the p o s s i b i l i t y that some of our r e s u l t s are l o c a l minima. In f u t u r e a p p l i c a t i o n s I w i l l i n c l u d e f u r t h e r s o l u t i o n models (e.g. e l e c t r o l y t e s ) and a d i f f e r e n t c h o i c e of independent v a r i a b l e s . 55 T a b l e 6: Computing t imes for the examples on a 48 megabyte Amdahl 5850 computer 1. B l a s t furnace example ( t a b l e 1) 0.43 sec 2. S i l i c a t e me l t s ( f i g s . 5 & 6, t a b l e 2) two b i n a r y l i q u i d s c a . 0.10 sec One t e r n a r y l i q u i d 0.40 - 0.60 sec two t e r n a r y l i q u i d s 0.30 - 0.40 sec three t e r n a r y l i q u i d s c a . 0.50 sec one t e r n a r y l i q u i d + one s o l i d phase 0.25 - 0.35 sec one t e r n a r y l i q u i d + two s o l i d phases c a . 0.20 sec t h r e e s o l i d phases c a . 0.10 sec two q u a t e r n a r y l i q u i d s 1.40 sec 3. mica - f e l d s p a r example ( t a b l e 4) column 1 0.20 - 0.30 sec column 2 0.20 - 0.30 sec column 3 0.50 - 1 .20 sec column 4 0.30 - 1.10 sec 2. ORDER-DISORDER IN TERNARY FE-MN-MG OLIVINES 2.1 INTRODUCTION Thermodynamic models of s o l u t i o n s always i n c l u d e a c o n f i g u r a t i o n a l entropy term which i s based on our knowledge of o r d e r i n g . A very important type of o r d e r i n g i s the " s i t e p r e f e r e n c e " of c e r t a i n atoms i n c e r t a i n s t r u c t u r e s . Because the o l i v i n e s have two o c t a h e d r a l c a t i o n s i t e s of d i f f e r e n t s i z e and symmetry, i t can be expected that there w i l l be a n o t i c a b l e o r d e r i n g . According to the c r y s t a l chemical c o n s i d e r a t i o n s of Ganguli (1977), the l a r g e r atoms w i l l p r e f e r the l a r g e r M2 s i t e . T h i s may be demonstrated with d i f f e r e n t techniques (e.g. Shinno, 1981) f o r K i r s c h s t e i n i t e (CaFeSiO«), Fe-Mn o l i v i n e s (Annersten et a l . , 1984) Fe-Ni and Fe-Ni-Mg o l i v i n e s (Annersten et a l . , 1982, Nord et a l . , 1982). The Fe-Mg d i s t r i b u t i o n i s almost d i s o r d e r e d , (Brown and P r e w i t t (1973), Wenk and Raymund (1973), Basso et a l . (1979)) but the d i f f e r e n t degrees of o r d e r i n g given by d i f f e r e n t authors suggest that a dependency on some i n t e n s i v e v a r i a b l e s must e x i s t , (e.g. oxygen f u g a c i t y as proposed by W i l l and Nover (1979). The aim of the i n v e s t i g a t i o n presented i n t h i s chapter i s to determine the c a t i o n occupancies i n s y n t h e t i c t e r n a r y Fe-Mn-Mg o l i v i n e s . T h i s knowledge w i l l be used to propose an entropy of mixing model f o r these o l i v i n e s . 56 57 2.2 THE SYNTHESIS OF OLIVINES Over the l a s t h a l f century, many o l i v i n e s and o l i v i n e type s t r u c t u r e s have been s y n t h e s i z e d . They are u s u a l l y c r y s t a l l i z e d from a melt or g l a s s at 1100 to 1400 °C. The method d e s c r i b e d below takes i n t o account that the s y n t h e s i s temperature was intended to be very c l o s e to the temperature of the planned exchange experiments. The s t a r t i n g m a t e r i a l s were : Fe70i: F i s h e r C e r t i f i e d Reagent. LOT 725030. F i r e d at 600 °C f o r 13 hours. Powder s t o r e d 110 °C. A microprobe a n a l y s i s of s y n t h e t i c f a y a l i t e r e v e a l e d l a t e r t h a t t h i s substance was contaminated with c a . 0.6 % Mn. A l l compositions were then r e c a l c u l a t e d with t h i s c o r r e c t i o n . MnCQ3; F i s h e r C e r t i f i e d Reagent. LOT 734606. Lot a n a l y s i s of 48.4 % assay as Mn used. Powder s t o r e d i n d e s i c c a t o r . MgO; F i s h e r C e r t i f i e d Reagent. LOT 741694 Powder s t o r e d at 110 °C. S i Q 2 : From S i 0 2 « x H 2 0 , M a l l i n c k r o d t A n a l y t i c a l Reagent. LOT KMNG. The s i l i c i c a c i d was converted to C r i s t o b a l i t e by slowly h e a t i n g to 800 °C i n a platinum c r u c i b l e , and l e a v i n g the sample at t h i s temperature f o r 24 hours. Powder s t o r e d at 110 °C. For each d e s i r e d o l i v i n e composition an oxide/carbonate mixture i s prepared, then enough g r a p h i t e i s added so that the bulk composition c o n t a i n s at l e a s t one mole g r a p h i t e f o r each mole of F e 2 0 3 and MnC0 3. The mixture i s then p l a c e d i n t o a Ag-tube, (ca. 3 cm long, 1.0 cm diam. and c l o s e d at the bottom) i n s i d e a S i 0 2 g l a s s tube (1.2 cm inner diam.) 58 F i g . 7: E x p e r i m e n t a l s e t - u p f o r t h e O l i v i n e s y n t h e s i s o 6 8 10 12 a £ 16 a a 1 8 • f-i fl 20 o • rH £ 22 w f£ 24- 26 28 30 32 34 C 3 2 800 820 840 860 880 900 Temperature [°C] •a c 3 3 cr to vacuum pump and Nitrogen •Pt wire -Ag tube with sample 3 2 3 0 2 8 2 6 2 4 2 2 2 0 c6 18 o 14 1 2 1 0 8 6 0 - - - - - - U 2 - - F i g . 8: C a l c u l a t e d a c t i v i t i e s i n t h e s y s t e m C — 0, w i t h g r a p h i t e p r e s e n t - - A - \ \ \ Cv J \ - 100 Bar s U J l - 1 i 0.003N Bars 1 • \~~\ 1 - 1 — 1 1 0.003 Bars 1 Bar 10 Bars 100 Bars 0.003 Bars 1 Bar 10 Bars 100 Bars 0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0 8 0 0 9 0 0 T e m p e r a t u r e [ ° C ] 60 and c o n n e c t e d t o the vacuum pump, a f t e r b e i n g f l u s h e d t h r e e t o f o u r t i m e s w i t h n i t r o g e n . Then i t i s heated t o 850 °C f o r c a . 3 h o u r s . T h i s p r e l i m i n a r y h e a t i n g i s n e c e s s a r y t o l e t a l l MnC0 3 d e c a r b o n a t e . In the s m a l l e r d iameter tubes used f o r the s y n t h e s i s , t h e r e would be too much l o s s of m a t e r i a l . Most of the F e 2 0 3 w i l l a l s o have r e a c t e d t o Fe 3Oj, and FeO. Then the sample i s c o o l e d and t r a n s f e r r e d t o a s m a l l e r Ag-tube (0.6 cm diam., c a . 8 cm l o n g ) . A s m a l l g r a p h i t e c y l i n d e r i s i n s e r t e d on t o p and the whole tube i s p l a c e d i n a S i 0 2 tube (0.7 cm i n n e r d i am.). (See. F i g . 7) I t i s exposed t o 850 °C f o r 3 t o 4 hours w h i l e c o n n e c t e d t o the vacuum pump. At about t h a t time a g r a p h i t e d e p o s i t d e v e l o p s i n the S i 0 2 - t u b e a t a p o s i t i o n c o r r e s p o n d i n g t o a temperature of 600-700 °C. We may s p e c u l a t e t h a t t h i s r e f l e c t s t he r e a c t i o n : 2-CO = C 0 2 + C The tube i s then s e a l e d and l e f t a t 850 °C f o r one t o t h r e e weeks. F i g . 7 shows the run c o n d i t i o n s and the temperature d i s t r i b u t i o n i n the f u r n a c e . Because up t o t h r e e samples c o u l d be accomodated i n the f u r n a c e , which were not a l l i n s e r t e d or removed a t the same t i m e , the temp e r a t u r e v a r i e d up t o ±20 °C f o r s h o r t t i m e s d u r i n g the s y n t h e s i s . The p r e s s u r e c o u l d not be c o n t r o l l e d and v a r i e d around one b a r , the lower l i m i t b e i n g the l i m i t of the vacuum pump (0.003 Bars) and the upper l i m i t the s t r e n g t h of the s i l i c a g l a s s (around 100 B a r s ) . The c a l c u l a t e d a c t i v i t i e s f o r 0 2 , CO and C0 2 are shown i n F i g . 8. The presence of g r a p h i t e i n the 61 experiments guarantees a 0 2 - a c t i v i t y l e s s than 1 0 " 2 ° . The products were examined with a magnet, o p t i c a l l y and by X-ray d i f f r a c t i o n . A good product i s non-magnetic, monomineralic a c c o r d i n g to o p t i c a l and X-ray d i f f r a c t i o n and has d i f f r a c t i o n peak widths at h a l f maximum around 0.3 °20. A l l products not conforming with the above c r i t e r i a were r e j e c t e d and the s y n t h e s i s repeated e i t h e r with the same sample or with a new mixture. I t turned out that f o r the s y n t h e s i s c o n d i t i o n s chosen, o l i v i n e s with a composition of more than 75% f o r s t e r i t e c o u l d not be s y n t h e s i z e d even a f t e r four weeks without being contaminated with MgO, c r y s t o b a l i t e and Fe- and Mn-oxides. Pure f o r s t e r i t e was s y n t h e s i z e d at 1100 °C i n two weeks. 2.3 SOME METHODS FOR SITE OCCUPANCY DETERMINATION CITED IN THE LITERATURE There i s no simple, e a s i l y a v a i l a b l e method that c o u l d be used to measure a l l s i t e occupancies of a l l elements. However s e v e r a l methods are very a c c u r a t e f o r c e r t a i n types of atoms and a combination of methods may be used to e x t r a c t the d e s i r e d i n f o r m a t i o n . V a r i o u s methods used f o r o l i v i n e s are b r i e f l y reviewed below. The l i t e r a t u r e c i t e d i s mainly r e s t r i c t e d to a p p l i c a t i o n s i n the t e r n a r y Fe-Mn-Mg o l i v i n e system. 62 2.3.1 MOSSBAUER SPECTROSCOPY Within ten years of the d i s c o v e r y of the Mossbauer e f f e c t , the f i r s t o l i v i n e s were being i n v e s t i g a t e d with t h i s method. ( E i b s c h i i t z and G a n i e l (1967) ) . The Fe po p u l a t i o n on the two s i t e s M1 and M2 can be d i s t i n g u i s h e d i f the sample i s heated to c a . 300 °C. No other element has been s u c c e s s f u l l y measured i n o l i v i n e s by Mossbauer spectroscopy, and t h e r e f o r e the method a p p l i e s only to Fe bea r i n g s o l u t i o n s . Systematic i n v e s t i g a t i o n s i n v o l v i n g s y n t h e t i c o l i v i n e s were made by Annersten (1982), Nord et a l . (1982), Shinno (1981) and Annersten et a l . (1984) . i n a d d i t i o n the technique i s of t e n used as an a d d i t i o n a l source of inf o r m a t i o n to c h a r a c t e r i z e n a t u r a l or s y n t h e t i c phases. (Bush et a l . (1970) , V i r g o and Hafner (1972) , and o t h e r s ) . 2 .3 .2 SINGLE CRYSTAL STRUCTURE REFINEMENTS The c r y s t a l s t r u c t u r e refinement u s i n g X-ray d i f f r a c t i o n i s s t i l l the most important method to obta i n s t r u c t u r e d ata. I f a p a r t i c u l a r l a t t i c e p o s i t i o n i s occupied by more than one atom, and these d i f f e r only s l i g h t l y i n the s c a t t e r i n g f a c t o r s , (e.g. Fe and Mn) then an accurate s i t e occupancy f o r each atom cannot be obtained by d i f f r a c t i o n a lone. F u r t h e r , a minimum g r a i n s i z e of a s i n g l e c r y s t a l (=sl00jum) i s r e q u i r e d f o r the method. The c r y s t a l s t r u c t u r e of o l i v i n e ( f o r s t e r i t e ) was f i r s t determined by Bragg and Brown (1926) . L a t e r refinements were r e p o r t e d by Belov et a l . (1951), Hanke and Zeman (1963), B i r l e e t a l . (1968) and 63 o t h e r s . (See a l s o l i s t g i ven i n appendix F ) . S i t e occupancies of o l i v i n e s from s i n g l e c r y s t a l s t r u c t u r e refinements were r e p o r t e d by Wenk and Raymund (1973), Brown and P r e w i t t (1973), Basso et a l . (1979), F r a n c i s and Ribbe (1980) and Brown (1980). 2.3.3 THE RIETVELD METHOD A very promising method i s the XRD p r o f i l e refinement as proposed by R i e t v e l d (1967,1969). The method c o n s i s t s of measuring a l a r g e number of i n t e n s i t i e s (approximately 1000) along a p r o f i l e and f i t the l a t t i c e parameters, p o s i t i o n a l parameters, occupancies e t c . to these i n t e n s i t i e s . I n i t i a l l y the method was used f o r neutron d i f f r a c t i o n p a t t e r n s with gaussian peak shapes. The major problem of a p p l y i n g the method to XRD data i s that the peak shapes have to be known i n order to f i t the data. Although a V o i g t or pseudo-Voigt f u n c t i o n was found to be very a c c e p t a b l e , ( H i l l (1984), David (1986)) some problems are u n r e s o l v e d . Sakala and Cooper (1979) c o u l d show that the s t r u c t u r a l parameters obtained by a R i e t v e l d p r o f i l e refinement were not the same as those obtained from an i n t e g r a t e d i n t e n s i t y refinement of the same data and that the standard d e v i a t i o n s of the parameters are i n c o r r e c t . They argue that a systematic e r r o r i s i n t r o d u c e d due to c o r r e l a t i o n s between r e s i d u a l s a c r o s s a Bragg peak. A d d i t i o n a l d i f f i c u l t i e s were p o i n t e d out by Thompson and Wood (1983), who propose that much more i n f o r m a t i o n must be c o n s i d e r e d f o r an a c c u r a t e p r o f i l e 64 refinement, (e.g. f i n i t e specimen l e n g t h , e c c e n t r i c i t y e r r o r s , broadening due to s t r a i n or p a r t i c l e s i z e and s p e c t r a l d i s t r i b u t i o n of the i n c i d e n t X - r a y s ) . A l l these problems l e d s e v e r a l authors (Cooper et a l . (1981), W i l l et a l . (1983)) to c o n c e n t r a t e more on p a t t e r n decomposition methods, where the u s e f u l i n f o r m a t i o n such as i n t e g r a t e d i n t e n s i t y , the peak maximum p o s i t i o n and the peak width of i n d i v i d u a l r e f l e c t i o n s are obtained f i r s t and then the s t r u c t u r e parameters are r e f i n e d analogous to a s i n g l e c r y s t a l s t r u c t u r e refinement. The newest t r e n d among c r y s t a l l o g r a p h e r s was i n i t i a t e d by Prawley (1981) and l a t e r Toroya (1986). T h e i r methods are whole-powder-pattern f i t t i n g s without r e f e r e n c e to a s t r u c t u r a l model. In t h i s approach, the i n t e n s i t i e s and the p o s i t i o n s of a l l r e f l e c t i o n s can vary i n d i v i d u a l l y . We may summarize that f o r X-ray d i f f r a c t i o n , the R i e t v e l d method i s f a r from being an e s t a b l i s h e d r o u t i n e i n v e s t i g a t i o n and i s at present mainly r e s t r i c t e d to refinements of s o l v e d , simple s t r u c t u r e s . As f u r t h e r developments in d i f f r a c t i o n techniques w i l l produce data with a r e s o l u t i o n much s u p e r i o r to that p r e s e n t l y a v a i l a b l e , powder d i f f r a c t i o n p r o f i l e refinements may develop more and more importance f o r the whole range of s t r u c t u r e refinements. Few R i e t v e l d refinements have been p u b l i s h e d f o r the r e l a t i v e l y c o m p l icated o l i v i n e s t r u c t u r e . (Lager et a l . (1981), H i l l and Madsen (1984)). C a t i o n d i s t r i b u t i o n s i n 65 o l i v i n e - t y p e s t r u c t u r e s were i n v e s t i g a t e d by Nord (1984) and Nord et a l . (1985), who had v e r y e n c o u r a g i n g r e s u l t s , d e s p i t e numerous d i f f i c u l t i e s , ( e . g . n e g a t i v e t e m p e r a t u r e c o r r e c t i o n f a c t o r s ) . 2.3.4 THE A-B PLOT A more e m p i r i c a l method f o r the occupancy d e t e r m i n a t i o n i n o l i v i n e s i s mentioned by Lumpkin and Ribbe (1983) and d e v e l o p e d i n more d e t a i l s by Lumpkin e t a l . (1983), M i l l e r (1985) and M i l l e r and Ribbe (1985,1986). They found t h a t a l i n e a r r e g r e s s i o n c o u l d a c c u r a t e l y be f i t t e d f o r each u n i t c e l l parameter i n terms of t h e e f f e c t i v e r a d i i of the c a t i o n s o c c u p y i n g the M1 and M2 o c t a h e d r a l s i t e s , u s i n g the r a d i i of Shannon (1974). The a 0 _ c e l l edge was found t o be s t r o n g l y dependent on the r a d i u s of the c a t i o n s o c c u p y i n g M1, and the b 0 - c e l l edge on t h o s e o c c u p y i n g M2. T h i s l e d t o the c o n s t r u c t i o n of a 0 v e r s u s b 0 p l o t s f o r b i n a r y o l i v i n e s t h a t a l l o w p r e d i c t i o n of the b u l k c o m p o s i t i o n s t o g e t h e r w i t h the M1 and M2 s i t e o c c u p a n c i e s . The method r e q u i r e s f i r s t a c e r t a i n number of known o c c u p a n c i e s , and because i t i s not a d i r e c t measurement, i s s u b j e c t t o m i s i n t e r p r e t a t i o n due t o charge d i s t r i b u t i o n s or s t r a i n s i n the c r y s t a l s . The advantage i s t h a t i t a l l o w s an e s t i m a t i o n of o c c u p a n c i e s from p u b l i s h e d b u l k c o m p o s i t i o n and l a t t i c e p a r a m e t e r s . So f a r the method has o n l y been a p p l i e d t o b i n a r y systems. 66 2.3.5 VIBRATIONAL SPECTRA Huggins (1973) proposed a s i m p l i f i e d i n t e r p r e t a t i o n of v i b r a t i o n a l s p e c t r a which allows an e s t i m a t i o n of the s i t e o c cupancies. However h i s theory may only be used when a pronounced n o n - l i n e a r i t y has been demonstrated f o r a given frequency - composition r e l a t i o n s h i p . T h i s c o u l d not be done fo r the Fe-Mg o l i v i n e s and t h e r e f o r e the method i s r e s t r i c t e d to the Fe-Mn and Mn-Mg b i n a r i e s . 2.3.6 CHEXE SPECTROSCOPY A new a n a l y t i c a l t r a n s m i s s i o n e l e c t r o n m i c r o s c o p i c technique known as C/mnneling Enhanced X-ray Emission spectroscopy was used by Smyth and T a f t ^ (1982) to determine the s i t e occupancies of Fe, N i , Mn and Ca i n a n a t u r a l o l i v i n e . The s t r e n g t h of the method i s a very high accuracy at very low (0.001 %) c o n c e n t r a t i o n s . 2.4 CHOICE OF METHODS For the present i n v e s t i g a t i o n with Fe-Mn-Mg o l i v i n e s , the ch o i c e of the methods used f o r the determination of the s i t e occupancies was n a t u r a l l y i n f l u e n c e d by the small g r a i n s i z e of the samples ( u s u a l l y <5 um) and by the equipment a v a i l a b l e at UBC. Because the i n v e s t i g a t e d system i n c l u d e s F e - o l i v i n e s , the Mossbauer spectroscopy was chosen as an i d e a l t o o l f o r determining the F e - s i t e d i s t r i b u t i o n . At f i r s t the i n t e n t i o n was to use a R i e t v e l d p r o f i l e refinement to r e s o l v e the Mn-Mg d i s t r i b u t i o n . U n f o r t u n a t e l y some 67 p r e l i m i n a r y i n v e s t i g a t i o n s showed that the peak shapes i n the p r o f i l e s were not symmetric and that the widths were not a smooth f u n c t i o n of 20. T h i s may be due to s t r a i n s in the s y n t h e t i c c r y s t a l s . At present a standard R i e t v e l d p r o f i l e refinement cannot take i n t o account a l l these a d d i t i o n a l problems, and the development of a more s o p h i s t i c a t e d p r o f i l e refinement exceeds the aim of the present i n v e s t i g a t i o n . The c a l c u l a t i o n of i n t e g r a t e d i n t e n s i t i e s , however, showed that s e v e r a l peak i n t e n s i t i e s are s t r o n g l y dependent on the Mn-Mg s i t e d i s t r i b u t i o n and t h a t a simple method, using only few i n t e n s i t i e s c o u l d very w e l l be used to determine the s i t e occupancies. The i n t e n s i t y r a t i o s used must be only s l i g h t l y dependent on the p o s i t i o n a l parameters and the e f f e c t i v e charge d i s t r i b u t i o n . 68 2.5 THE DETERMINATION OF F E - S I T E DISTRIBUTIONS BY MOSSBAUER SPECTROSCOPY 2 .5 .1 INTRODUCTION The Mossbauer e f f e c t i s based on the o b s e r v a t i o n t h a t i n c r y s t a l l i n e subs tances the n u c l e a r resonance a b s o r p t i o n i s e s s e n t i a l l y r e c o i l - f r e e i f T << © D . ( © D = Debye t e m p e r a t u r e ) . (Mossbauer ( I 9 5 8 a , b ) ) . For a d e t a i l e d i n t r o d u c t i o n i n t o the p r i n c i p l e and the methodology see V e r t e s et a l (1979), Herber (1971), Gibb (1976) and Gonser (1975) . We w i l l r e s t r i c t our overview to 5 7 F e . ( 1 For a 5 7 F e n u c l e u s , the energy d i f f e r e n c e between an e x i t e d s t a t e and the ground s t a t e i s = 14.4 keV. I f a 7-quantum w i t h the energy E ^ i s e m i t t e d from a f r e e n u c l e u s , the nuc leus i s a c c e l e r a t e d in the o p p o s i t e d i r e c t i o n , because of the c o n s e r v a t i o n of momentum and energy . The v e l o c i t y of r e c o i l i s E ^ / M - c = 81 .5 m/s , r e s u l t i n g in a r e c o i l energy of ( E ^ 2 ) / ( 2 - M - c 2 ) 1 .9-10" 3 eV, thereby r e d u c i n g the energy of the e m i t t e d 7 - r a y . S i m i i a r i l y the energy of a 7-quantum w i l l be decreased by another =*1.9- 10" 3 eV when i t i s absorbed by a f ree n u c l e u s . As the n a t u r a l l i n e width of 5 7 F e i s on ly ^ 4 . 7 - 1 0 ' 9 eV, no resonance a b s o r p t i o n can occur under the 1 summary of c o n s t a n t s : c = 2 .997925-10 8 [m/s] k = 1 . 3 8 0 4 - 1 0 - 2 6 [ J - K - 1 ] h = 6 . 6 2 5 6 - l O " 3 4 [ J - s ] X = h/2rr = 1 . 0545 • 1 0~ 3" [ J - s ] N = 6 . 0 2 2 5 2 - 1 0 2 6 [ m o l e c u l e s / k g - m o l ] M = Mass of 5 7 F e = 57/N [kg] 1eV = Energy of wave w i t h v = 8068 c m - 1 where v = 1/X = v/c = h-v = h - 7 - c = 1 .602551 - 1 0 " 1 9 [J ] 69 above assumed c o n d i t i o n s . As a consequence of thermal motion, the atoms of the absorber and the source approach and move away from each other, and the frequency (energy) of the 7-quanta change as a r e s u l t of the nuclear D o p p l e r - e f f e c t . Let v x be the v e l o c i t y of the e m i t t i n g nucleus i n the d i r e c t i o n of the emission, E 0 the e x i t a t o n energy of the nucleus and the energy of the 7-ray. Then, because of c o n s e r v a t i o n of energy: E 0 + (M-v 2)/2 = E^ + (M* (v +v) 2 )/2 X j A and E 0 - E = AE = (M«v 2)/2 + M-v-v 7 x where the r e c o i l v e l o c i t y v = E /(M«c) 7 thus: AE = E 2/(2'M-c 2) - E^'Vx/c. Depending on v x , AE may become zero and i f such a 7-ray then h i t s an absorbing nucleus which i s moving such that AE a l s o equals zero then resonance a b s o r p t i o n may occu r . However the p r o b a b i l i t y i s very low, because f o r 5 7 F e , v x would have to be = 163 m/s to compensate f o r the t o t a l energy d i f f e r e n c e of 3.8*10~ 3 eV. The v e l o c i t y due to thermal v i b r a t i o n can be approximated by the v e l o c i t y v f c with the maximum p r o b a b i l i t y i n the i s o t r o p i c Maxwell v e l o c i t y d i s t r i b u t i o n : PT 1/2 v x " v t = 2-k«T M For 5 7 F e at room temperature t h i s i s =9.3 m/s. In a c r y s t a l l i n e l a t t i c e the r e c o i l energy i s not 70 absorbed by the e m i t t i n g nucleus but can be t r a n s m i t t e d as l a t t i c e . v i b r a t i o n to the whole c r y s t a l . In a c l a s s i c a l i n t e r p r e t a t i o n , the p r o p o r t i o n of 7-rays that are r e c o i l f r e e i s : (e.g. V e r t e s et a l . , 1979, p 19) f = e-<x 2>A' where <x> i s the mean amplitude of the l a t t i c e v i b r a t i o n . T h i s p r o p o r t i o n approaches u n i t y , i f the amplitude of the l a t t i c e v i b r a t i o n i s small r e l a t i v e to the wave l e n g t h 71 of the 7 r a d i a t i o n . The amplitude of the l a t t i c e v i b r a t i o n decreases with d e c r e a s i n g temperature and i n c r e a s i n g bond i n t e r a c t i o n s . The f a c t o r f i s c a l l e d the Debye-Waller f a c t o r and i t s exact c a l c u l a t i o n i s only p o s s i b l e u sing quantum mechanics, (see e.g. Gonser (1975)). A Mossbauer spectrum i s obtained by moving the source r e l a t i v e to the absorber at a speed up to some cm/s and measuring the a b s o r p t i o n as a f u n c t i o n of r e l a t i v e speed. The energy of the 7 rays w i l l change a c c o r d i n g to the Doppler r e l a t i o n E _ = E - E -w/c. For E =14.4 keV and D 7 7 7 assuming that we can c o r r e l a t e a b o r p t i o n events with speed i n t e r v a l s on the order of 0.05 mm/s we can d e t e c t energy d i f f e r e n c e s between d i f f e r e n t n u c l e i or between d i f f e r e n t energy l e v e l s of a nucleus on the order of 10" 9 eV. To f a c i l i t a t e the i n t e r p r e t a t i o n of the r e s u l t s , some f r e q u e n t l y used terms are e x p l a i n e d below: L i n e width r The n a t u r a l l i n e w i d t h T of a 7-ray with the energy E can 71 be c a l c u l a t e d on the b a s i s of the Heisenberg u n c e r t a i n t y p r i n c i p l e . (At«AE where At = r = l i f e t i m e of the e x i t e d s t a t e . ) Assuming that the l i n e w i d t h at low temperatures f o r emission and a b s o r p t i o n i s the n a t u r a l l i n e w i d t h , t h i s r e l a t i o n can be used to measure the l i f e t i m e of the e x i t e d s t a t e . (Mossbauer (1958b)). The measurable l i n e w i d t h at room temperature i s u s u a l l y c o n s i d e r a b l y g r e a t e r than r , due to thermal motion and f i n i t e sample t h i c k n e s s . In a d d i t i o n , any v i b r a t i o n i n the experimental set-up (vacuum pump, a i r c o n d i t i o n i n g e tc.) w i l l c o n t r i b u t e to l i n e broadening. T h e r e f o r e , to o b t a i n an optimal spectrum the source, the absorber and the d e t e c t o r must be suspended c a r e f u l l y . Isomer s h i f t 6 Accord i n g to the theory of quantum mechanics the e l e c t r o n s , e s p e c i a l l y from the s - o r b i t a l s , have a non-zero d e n s i t y i n s i d e the nucleus, and i n t e r a c t with the nu c l e a r charge. T h i s i n t e r a c t i o n i n f l u e n c e s the energy l e v e l s of the nucleus, and i f source and absorber have d i f f e r e n t chemical environments, t h e i r e x c i t a t i o n e n e r g i e s w i l l a l s o be d i f f e r e n t . T h i s energy d i f f e r e n c e (= isomer s h i f t ) i s of the order 10" 8 to 10" 6 eV and can be compensated using the Doppler e f f e c t by moving the source r e l a t i v e to the absorber at a speed of some mm/s. The isomer s h i f t i s not g e n e r a l l y given i n u n i t s of energy i n the l i t e r a t u r e , but as a v e l o c i t y [mm/s] and not as an absolute s h i f t , but r e l a t i v e to a standard, (e.g. i r o n f o i l at room temperature). 72 Inseparable from the isomer s h i f t i s the temperature s h i f t : When a 7-photon i s emitted, the mass of the nucleus i s decreased by Am = h-v/c2. At any temperature above 0 K t h i s mass decrease i n f l u e n c e s the l a t t i c e v i b r a t i o n and t h e r e f o r e the phonon energy. T h i s has e x a c t l y the same e f f e c t as i f the r e c o i l energy was not z e r o . Quadrupole s p l i t t i n g AE^ If the e l e c t r o n s around a molecule are not s p h e r i c a l l y symmetric, they g i v e r i s e t o an e l e c t r i c f i e l d g r a d i e n t at the s i t e of the nucleus. The i n t e r a c t i o n with the quadrupole moment of the nucleus p a r t i a l l y removes the degeneracy of the (21+1) p o s s i b l e s t a t e s , r e s u l t i n g i n a s p l i t t i n g of the energy l e v e l s . These d i f f e r from the mean energy by approximately 10" 8 eV and can be measured i n a Mossbauer spectrum. 2.5.2 EXPERIMENTAL SET-UP F i g u r e 9 shows a schematic diagram of the Mossbauer apparatus used i n the present i n v e s t i g a t i o n . The furnace i s d e s c r i b e d i n more d e t a i l i n Appendix C. The source i s 5 7 C o in a rhodium matrix. 5 7 C o decays to the e x i t e d s t a t e of 5 7 F e with T ^ 2 of 270 days. The sample was mixed with g r a p h i t e (which shows l i t t l e a b s o r p t i o n of the 7-rays) to o b t a i n an optimal c o n c e n t r a t i o n of approximately 7 mg iron/cm 2 and p l a c e d between two g r a p h i t e or boron n i t r i d e d i s c s i n the sample h o l d e r . I t was found by s e v e r a l authors ( E i b s c h u t z Fig. 9: Schematic Set—up of the Moessbauer Apparatus Absorber Source Proportiona Counter Velocity transduce \ EX Air Air high voltage Pulse Amplifier Discriminator Multichannel Analyzer Moessbauer driving unit Function generator Oscilloscope 7 Co — w F e Decay Scheme W C o = 270 d sir- Fe 1=5/2 > y 137 KeV '91% 1 4 - 4 K e V l £ l 0 - ' s •0.0 KeV 74 and G a n i e l (1967), Bush et a l . (1970)) that the M, and M 2 s i t e s i n o l i v i n e cannot be r e s o l v e d i n a Mossbauer spectrum recorded at room temperature, but that by h e a t i n g the absorber to 300-400 °C the s p e c t r a corresponding to the two s i t e s were w e l l separated, although the l i n e w i d t h T d i d i n c r e a s e . The observed e f f e c t at e l e v a t e d temperatures i s a g e n e r a l l y i n c r e a s e d d i f f e r e n c e of the quadrupole s p l i t t i n g between M, and M 2. E i b s c h u t z and G a n i e l (1967) e x p l a i n that t h i s i s because at low temperatures only the inner e l e c t r o n s c o n t r i b u t e s i g n i f i c a n t l y to the quadrupole s p l i t t i n g . These inner s h e l l s are n e a r l y i d e n t i c a l f o r the M, and M 2 s i t e s . The symmetry d i f f e r e n c e (1 f o r M, and m f o r M 2) becomes n o t i c a b l e i n a Mossbauer spectrum only above ^100 °C. The absorber temperature was kept constant at around 300±1 °C. The s p e c t r a were accumulated f o r two to three days, u n t i l two a b s o r p t i o n peaks at high v e l o c i t y c o u l d be e a s i l y r e s o l v e d by eye. The s p e c t r a were recorded with a multichannel a n a l y z e r with 512 channels in m u l t i - s c a n n i n g mode. The mechanical v i b r a t o r f o r the source was d r i v e n i n a constant a c c e l e r a t i o n mode with a t r i a n g u l a r v e l o c i t y wave form. (See F i g . 9). 75 2 . 5 . 3 EVALUATION OF THE MOSSBAUER SPECTROSCOPY MEASUREMENTS The da ta c o l l e c t e d in the m u l t i c h a n n e l a n a l y z e r were t r a n s f e r e d to a c o m p u t e r - r e a d a b l e d i s k for f u r t h e r a n a l y s i s . The c o n v e r s i o n from channe l number to v e l o c i t y was c a l i b r a t e d w i t h F e - f o i l w i t h known l i n e p o s i t i o n s . ( -5 .3089 , - 3 . 0 7 7 1 , - 0 . 8 4 0 4 , 0 .8403, 3.0772 and 5.3150 [mm/sec] ) . The s p e c t r a were then f i t t e d w i t h a l i n e a r combinat ion of L o r e n t z i a n c u r v e s . The number of counts c o r r e s p o n d i n g to the v e l o c i t y v can be expressed a s : where N(°°) i s the h e i g t h of the b a s e l i n e , and A ^ , v? and are the a m p l i t u d e , p o s i t i o n and h a l f width of the i f c ^ l i n e r e s p e c t i v e l y . A l l c a l c u l a t i o n s were performed wi th a computer program w r i t t e n by R . J . P o l l a r d u t i l i z i n g s u b r o u t i n e s from the CERN l i b r a r y . T h i s i s a g e n e r a l purpose f i t t i n g program u s i n g f i r s t a n o n l i n e a r s implex approach and then r e f i n i n g the r e s u l t w i t h a g r a d i e n t method. A l l s p e c t r a were f i t t e d w i t h t h r e e quadrupo le d o u b l e t s . In some cases a s u c c e s s f u l convergence c o u l d o n l y be a c h i e v e d by c o n s t r a i n i n g the p o s i t i o n or the l i n e w i d t h of the s m a l l F e + 3 d o u b l e t , r e s u l t i n g i n an i n c r e a s e d u n c e r t a i n t y for these measurements. The assignment of the two main d o u b l e t s to the M, and M 2 s i t e s was made a c c o r d i n g to Anners t en et a l . (1982), Shinno (1981) and F i n g e r and V i r g o (1971) . The A. I n N(v) = N ( » ) - L i=1 1 + 76 i n t e r p r e t a t i o n of Annersten et a l . (1984), where the low v e l o c i t y peaks are c r o s s e d over f o r the Fe-Mn o l i v i n e s gave a l e s s s a t i s f a c t o r y f i t f o r our data. I t should be noted however, that t h i s c r o s s over makes no s i g n i f i c a n t change i n the area r a t i o s f o r the M, and M 2 s i t e s . The F e + 3 doublet i s d i s c u s s e d by Shinno (1981). Heating s e v e r a l samples in a i r , he showed that the F e + 2 p o p u l a t i o n s of M 2 decreased with i n c r e a s i n g F e + 3 content, l e a v i n g unchanged w i t h i n the e r r o r l i m i t s . We may t h e r e f o r e a s s i g n as a good approximation a l l F e + 3 to M 2. T h i s assumption i s a l s o supported by the measurement of pure f a y a l i t e (Table 7), and the c o n s i s t e n t K Q values d e s p i t e the F e 3 + v a r i a t i o n (Table 16). F i g . 10 and 11 show two examples of Mossbauer s p e c t r a . The f i r s t ( F i g . 10) converged e a s i l y , r e s u l t i n g i n an u n c e r t a i n t y f o r the r e l a t i v e occupancies of -2%. The second ( F i g . 11), because of the low Fe content and the strong p r e f e r e n c e of Fe to occupy the M, s i t e , i s more d i f f i c u l t to i n t e r p r e t . The u n c e r t a i n t y i n t h i s case i s =5%. The r e s u l t s of the Mossbauer measurements are summarized i n Tables 7 and 8. F i g . 1 0 : M o e s s b a u e r s p e c t r u m f o r f a y 6 t e p h 4 f o 2 + C , a t 2 9 8 ° C 1 0 0 . 2 0 1 0 0 . 0 0 - ^ 9 9 . 6 0 - -l-> S 9 9 . 4 0 - O O 9 9 . 2 0 - 99 .00-4 9 8 . 8 0 - - - I - _ y — IIII 4 I I I I I I I I 3 U N llll 2 un llll 1 IIII ( I I I I ) I I I I I I I I I I I I e, llll > HIT] llll 3 nil] mr V e l o c i t y [ m m / s e c ] 0 . 2 0 o.oo - 0 . 2 0 - A J V TV y " \j MA* . — IIII 4 IIII 3 IIII llll 2 '"I llll 1 IIII ( mi ) llll llll im > llll 3 mi B 0 = 42247000 AE Q 6 r A , % Fe|*(M,) 2.0282 0.94914 0.39000 1.50140 61.30 Fe (M 8) 2.4836 1.00460 0.34887 1.02130 37.30 Fe 0.8607 0.19385 0.30793 0.04319 1.39 78 F i g . 1 1 : M o e s s b a u e r s p e c t r u m f o r f a y 2 t e p h 1 0 + C , a t 3 0 1 ° C 1 0 0 . 0 5 1 0 0 . 0 0 9 9 . 9 5 1 1 9 9 . 9 0 9 9 . 8 5 -t-> 9 9 . 8 0 9 9 . 7 5 o o 9 9 . 7 0 9 9 . 6 5 9 9 . 6 0 9 9 . 5 5 - +-H Pf?f -+HH t \ + - + - V - - - — 4 — IIII 3 llll llll 2 II 11 mi 1 IIII ( IIII ) IIII IIII mi > IIII nil] llll A n irj V e l o c i t y [ m m / s e c ] 0 . 0 5 0.00 - 0 . 0 5 - \ llll mi V 1111 •ŷ 'll Vu l™ ^ llll llll llll llll rm 1 y»< llll im 1111 r mr 1 y*—" rfA/v HIT mi F~ llll B 0 = 51755000 AE, (5 T A, % Fe*(M4) 2.1096 0.95486 0.36366 0.67922 74.56 Fe"+(M8) 2.4389 0.99917 0.32011 0.16369 15.82 Fe 0.8475 0.15366 0.36017 0.08846 9.62 79 Table 7.: Isomer s h i f t (6). ' Quadrupole spl i tt i ng ( AE o l i v i nes o ) and the f u l l width at half heigth Al l units are in mm/sec. (r> for the synthet i c Fe-Mn-Mg NR . 6Fe ! + A E Fe 2 + PFe ! + 6Fe' + AE Fe> + rFe* + 6Fe 3 + AE Fe' + TFe 5 + M1 0 M1 M1 M2 0 M2 M2 0 1:120000 0 .9402 2 .0096 0 .3125 1 .0006 2. 4753 0 . 2946 0 .0953 0 .8001 0 .2905 4:080400 0 .9508 1 .9880 0 .3702 1 .0010 2 . 4682 0 . 3333 0, , 1988 0 . 7697 0 . 2503 5:050700 0 .9360 2 .0201 0 . 3441 0 .9935 2 . 4138 0 .2944 0, , 1395 0 . 7501 0 .2702 6:040800 0 .9508 2 .0477 0 .3759 0 .9946 2. 4309 0 .3184 0. ,2000 0, .8605 0 .2503 7:030900 0 .9600 2 .0866 0 .4047 0 .9921 2. 4524 0 .3318 0, , 1087 0. . 7507 0 . 2519 8:021000 0 .9549 2 . 1096 0 .3637 0 .9992 2. 4389 0 . 3201 0. . 1537 0, .8475 0 . 3602 11:100002 0. .9407 2 .0189 0. .3148 0 .9942 2 . 4732 0 . 2855 0. .2067 0, .8041 0 . 201 1 12:080202 0. .9442 1 .9971 0, , 3498 1 .0051 2. 4648 0, .3180 0. 1001 0. ,7502 0 .3717 13:060402 0. .9491 2 .0282 0, ,3900 1 .0046 2. 4836 0, , 3489 0. 1939 0. 8607 0 . 3079 14:040602 0. .9465 2 .0304 0. 3800 0 .9994 2. 4626 0. ,3297 0. 1004 0. 7502 0 .4989 15:020802 0. .9302 2 .0827 0. ,3567 0 .9828 2 . 4283 0, , 2908 0. 1430 0. 7525 0 .3060 17:080004 0. ,9446 2 .0761 0. 3499 0 .9977 2. 4890 0. .3212 0. 1 197 0. 8644 0 .4082 18:060204 0. .9294 2 .0246 0. 3499 1, .0017 2. 4563 0. ,3177 0. 1237 0. 8968 0. .4218 19:040404 0. 9410 2 .0410 0. 3800 0. ,9985 2 . 4637 0. 3235 0. 1990 0. 7846 0. ,4742 20:020604 0. 9543 2 . 1039 0. 4060 1. 0313 2. 5357 0. 3552 0. 1111 0. 8509 0. ,4741 22:060006 0. 9279 2 .0998 0. 3190 0. 9853 2 . 5025 0. 2814 0. 2999 0. 8659 0. 7055 23:040206 0. 9368 2 .0809 0. 3500 0, ,9978 2. 4854 0. 3171 0. 1044 0. 7510 0. ,4974 24:020406 0. 9337 2 . 1347 0. 3599 0. 9916 2 . 5091 0. 3293 0. 1096 0. 7521 0. 3103 26:040006 0. 9206 2 . 1803 0. 3308 0. 9901 2. 5203 0. 2719 0. 2334 0. 8852 0. 7993 27:020208 O. 9422 2 . 1067 O. 3600 0. 9913 2. 4926 0. 3205 0. 1094 0. 7500 0. 3822 The uncertainties in the Mossbauer parameters for the F e + ! are: 6 = ±0.02 mm/s, AE = ±0.03 mm/s and r = ±0.04 mm/s. For The F e + 3 the errors are ±0.04, +0.06 and ±0.08 mm/s respectively. " 80 Table 8 . : Mossbauer area ratios and Fe-occupancies for the synthetic Fe-Mn-Mg o l i v i n e s . Area rat ios in % Reiat ive occupancies in % Absolute occupancies NR. Fe* + Fe ! + Fe 3 + Fe* + Fe 2 + Fe 3 + x ( F e ! t ) x ( F e ! + ) x ( F e 3 + ) M1 M2 M1 M2 M1 M2 1 :120000 49, . 19 48 .88 1 .92 50. .05 48 .24 1 .71 0 .9950 0 .9590 0 .0340 4 :080400 60 .77 33 .24 5 .99 61 . 85 32 .82 5 . 33 0 .8198 0 .4350 0 .0706 5 :050700 65, .84 28 .33 5 .83 66. .89 27 .92 5 . 20 0, .5541 o. .2312 0 .0431 6 :040800 68 , .41 26 .72 4 .87 69, , 38 26 . 29 4 . 34 0 .4593 0 . 1740 0 .0287 7 :030900 71 . 17 27 .20 1 .63 71 , ,90 26 .66 1 .44 0 .3573 0 . 1325 0 .0072 8 :021000 74 , .56 15 .82 9 .62 75, .80 15 .60 8 .61 0, .2512 0, .0517 0, .0285 1 1 :100002 51 , .54 46 .08 2 .38 52, ,42 45 .46 2 . 12 0. .8684 0, .7531 0, .0351 12 :080202 54 , .23 36 .66 9 . 1 1 55 . .47 36, .38 8 . 15 0, . 7351 0. .4821 0. . 1080 13 :060402 61 . .30 37 .30 1 . 39 62. , 1 1 36 .66 1 .23 0. .6174 0. . 3644 0, .0122 14 :040602 63 , .53 29 .83 6 .64 64. 64 29. .44 5, .92 0. ,4283 0. . 1951 0. .0392 15 :020802 70. .03 29 . 15 0. .82 70. 72 28 , 55 0, .73 0. .2344 0, ,0946 0, .0024 17 :080004 51 . .82 44 .78 3, .40 52. 76 44. .22 3 . 02 0. ,6993 0. ,5861 0. .0400 18 :060204 55 . .74 41 . 52 2, .74 56. 64 40. ,93 2 . ,43 0. 5630 0. ,4068 0. ,0242 19 :040404 60. 63 33. .21 6, . 17 61 . 71 32. 79 5. ,50 0. 4089 0 . 2173 0. 0364 20 :020604 66 . .99 24 . .83 8, , 18 68. 18 24 . 52 7. ,31 0. 2259 0. 0813 0. 0242 22 :060006 50. 56 41 , .02 8, .42 51 . 75 40. 72 7. 53 0. 5144 • 0 . 4048 0 . 0748 23 :040206 59 . 29 39 . .01 1 . .70 60 . 12 38 . 37 1 . 50 0. 3984 0 . 2542 0. 0100 24 :020406 65 . 37 33 . 07 1 . 56 66 . 15 32 . 46 1 . 38 0. 2192 0 . 1076 0. 0046 26 :040008 52. 89 35 . ,78 1 1 . .32 54. 25 35 . 60 10. 15 0. 3594 0 . 2359 0 . 0673 27 :020208 57 . 70 40. .25 2. .05 58. 56 39. 63 1 . 82 0. 1941 0 . 1313 0 . 0060 For the conversion from area ratios to r e l a t i v e occupancies the constants are C i = 0 .97 and C i = 1.13. oo o The r e l a t i o n between s i t e p o p u l a t i o n s and area r a t i o s can w r i t t e n as: xFe2+ , Fe2 + _ J _ . Fe2+ , Fe2+ _ XM1 / XM2 " C, M1 ' AM2 " * 1 Fe3 + / Fe2+ _ \_ Fe3+ / , Fe2 + _ x 7 x t o t a l _ C 2 A 7 A t o t a l " Y z where A i s an a b s o r p t i o n peak area and x the Fe s i t e p o p u l a t i o n on or M 2. (Bancroft and Brown (1975)). The constants and C 2 are r e l a t e d to the r a t i o s of the l i n e w i d t h s , the f r a c t i o n s of r e c o i l f r e e a b s o r p t i o n and a s a t u r a t i o n c o r r e c t i o n . The s i t e p o p u l a t i o n s are then c a l c u l a t e d as: x F e 3 + Fe2+ XM1 Fe2 + XM2 1 + Y 2 Y 1 1 + 1 1  + 1 L 1 + Y 2 J We w i l l use c, = 0.97. T h i s i s c o n s i s t e n t with the value obtained by Annersten et a l . ( l 9 8 4 ) (They used a s l i g h t l y d i f f e r e n t f o r m u l a t i o n e q u i v a l e n t to = 0.98) and the measurement of f a y a l i t e (Fa - 1) of Shinno (1981). The c o r r e c t i o n i s sma l l e r than the expected u n c e r t a i n t y , but i n c l u d e d i n order to a v o i d systematic e r r o r s as much as p o s s i b l e . C 2 i s set a c c o r d i n g to the chemical a n a l y s e s of 82 Shinno (1981) to 1.13. Because of the s m a l l F e 3 + c o n c e n t r a t i o n s and the r e l a t i v e l y l a r g e u n c e r t a i n t i e s , the v a l u e of C 2 i s i r r e l e v a n t . 2 . 5 . 4 DISCUSSION OF THE RESULTS Our s y n t h e t i c o l i v i n e s ' c o n t a i n between 1% and 10% of the t o t a l i r o n as F e 3 + . Compared w i t h n a t u r a l o l i v i n e s t h i s f a c t i s not s u r p r i s i n g , as many c h e m i c a l a n a l y s e s ( e . g . those reviewed by Deer , Howie and Zussmann (1972)) show a p p r e c i a b l e amounts of F e 3 + . In c o n e c t i o n w i t h Mossbauer d a t a , however, F e 3 + was not r e p o r t e d u n t i l 1981 (Sh inno , 1981). T h i s i s p r o b a b l y due to the f a c t tha t the samples for Mossbauer a n a l y s i s are o f t en c a r e f u l l y s e l e c t e d , ex tremely pure phases where F e 3 + was s y s t e m a t i c a l l y e x c l u d e d . In s y n t h e t i c o l i v i n e s F e 3 + has not been r e p o r t e d b e f o r e , p r o b a b l y because most syntheses are done at 1000 to 1400 °C w i t h extremely low 0 2 f u g a c i t i e s . (Shinno (1981), Anners ten et a l . (1984)) . 83 2.6 CALCULATION OF INTEGRATED X-RAY INTENSITIES Most equations and data to c a l c u l a t e i n t e g r a t e d i n t e n s i t i e s can be found i n the INTERNATIONAL TABLES FOR X-RAY CRYSTALLOGRAPHY, 1974 ( a b b r e v i a t e d I.T.) The f o l l o w i n g o u t l i n e i s mainly i n f l u e n c e d by Yvon et al.(1977) and Borg and Smith (1969). 2.6.1 THE DATA NEEDED FOR THE CALCULATIONS Space group s p e c i f i c data For a s i t e s = (x/a,y/b,z/c) = (x',y',z') the e q u i v a l e n t p o s i t i o n s p^ are d e f i n e d by the symmetry o p e r a t i o n s f o r the space group: p. = a. + |'|T.| | - s T i = 1 ,2,3. ..N e q Where N e q i s the number of symmetry o p e r a t i o n s . For Pbnm the e q u i v a l e n t p o s i t i o n s a r e : (I.T. 1/151) p, = ( 0 , 0 , 0 ) + 1 0 0 0 1 0 0 0 1 x' Y' z' - 1 , 0 , 0 x' p 2 = (1/2,1/2,1/2) + 0 , 1 , 0 • y' 0 , 0 , - 1 z' p 3 = ( 0 , 0 ,1/2) + •1 0 0 0 •1 0 0 0 1 X 1 • y' z' pfl = (1/2,1/2, 0 ) + 1 0 0 0 •1 0 0 0 •1 x' y' z' Because Pbnm has a symmetry c e n t r e at the o r i g i n there are 84 at most 8 e q u i v a l e n t p o s i t i o n s : P i , P 2 f P 3 f Par "Pi» _p2 , - p 3 and -p«. For c e r t a i n s i t e s these may not a l l be d i f f e r e n t i f taken modulo 1. We t h e r e f o r e d e f i n e a m u l t i p l i c i t y f a c t o r f o r each s i t e : number of n o n - i d e n t i c a l p. modulo 1 The c o n d i t i o n s f o r n o n - e x t i n c t i o n are a l s o given i n I.T. 1/151. For Pbnm they a r e : hOl : h+1 = 2n and Okl : k = 2n Element s p e c i f i c data The s c a t t e r i n g f a c t o r s f° f o r the atom (or ion) m i s approximated with the a n a l y t i c a l f u n c t i o n : , , s i n 20^ b- (—T Z — > - The c o e f f i c i e n t s a^, b^ and c f o r each element are l i s t e d i n I.T. IV/Table 2.2B. For O 2 - they were taken from Tokonami (1965). The c o r r e c t i o n f o r anomalous d i s p e r s i o n i s made by adding a r e a l and an imaginary d i s p e r s i o n c o r r e c t i o n term: f = f 0 + M ' + / • Af" m m m m (I.T. IV/Table 2.3.1.) They are dependent on the wavelength X but assumed to be independent of 0. 85 The L o r e n t z - P o l a r i s a t i o n f a c t o r (LP) For a d i f f T a c t o m e t e r powder p a t t e r n the L o r e n t z - P o l a r i s a t i o n f a c t o r i s (I.T. 11/314): 1 + c o s 2(20) (LP) = s i n © • s i n(20) Information about the i n d i v i d u a l c r y s t a l Besides the l a t t i c e parameters a, b, c, a, 0 and y we have to know f o r each s i t e the atom c o o r d i n a t e s , the occupancy f a c t o r s f o r each element and the temperature c o r r e c t i o n f a c t o r s . The numerical assumptions about these parameters i s d i s c u s s e d i n d e t a i l i n chapter 2.7. - The temperature c o r r e c t i o n f a c t o r (or Debye-Weller f a c t o r ) B f o r a given s i t e can e i t h e r be assumed constant (= B e <^ = i s o t r o p i c e q u i v a l e n t of the temperature c o r r e c t i o n f a c t o r ) or i s : B = b ^ - h 2 + b 2 2 « k 2 + b 3 3 « l 2 + b 1 2 « h « k + b 1 3 » h « l + b 2 3 « k . l (= a n i s o t r o p i c temperature c o r r e c t i o n ) . In a s t r u c t u r e refinement the b ^ j are u s u a l l y s o - c a l l e d f i t parameters which should not be s t r e s s e d f o r a strong p h y s i c a l i n t e r p r e t a t i o n . The d i f f e r e n c e i n the c a l c u l a t e d i n t e n s i t i e s using one or the other ( i f both are given) i s too s m a l l to be important f o r our c a l c u l a t i o n s . We w i l l t h e r e f o r use Be<^. 86 2 . 6 . 2 EQUATIONS Summary of symbols a , b , c , a , 0 , 7 : L a t t i c e parameters X: Wavelength of p r i m a r y X - r a y 0: G l a n c i n g ang le of r e f l e c t i o n ( h k l ) F h k l : s t r u c t u r e f a c t o r f o r r e f l e c t i o n ( h k l ) M: Number of d i f f e r e n t e lements i n the s t r u c t u r e I (m): Number of d i f f e r e n t s i t e s an element m may occupy J ( i ) : Number of e q u i v a l e n t p o s i t i o n s i n u n i t c e l l f or s i t e i f^: S c a t t e r i n g f a c t o r f o r Element m. (a f u n c t i o n of s i n 0 / X ) A f ^ : r e a l c o r r e c t i o n term f o r anomalous d i s p e r s i o n A f m : imaginary c o r r e c t i o n term f o r anomalous d i s p e r s i o n M s : M u l t i p l i c i t y of s i t e i ° m i : 0 c c u P a n c Y f a c t o r for element m on s i t e i B^: Temperature c o r r e c t i o n f a c t o r for s i t e i x n , i 4 » y m ^ F z m ^ : Atom c o o r d i n a t e s i n A for element m on mi] •'mi] mi] J e q u i v a l e n t p o s i t i o n of s i t e i p M : M u l t i p l i c i t y of r e f l e c t i o n ( h k l ) L P : L o r e n t z - P o l a r i z a t i o n f a c t o r f o r angle © The c a l c u l a t i o n of © f o r a l l r e f l e c t i o n s ( h k l ) X 2 r s i n 2 © = ^ , v - i ( h » b ' C - s i n a ) 2 + ( k « a « c « s i n / 3 ) 2 + ( l « a « b « s i n 7 ) + 2 • k • 1 « a 2 • b « c • (cos/3 • cos7~cosa) + 2 « h « l « a - b 2 «c • ( c o s 7 « c o s a - c o s / 3 ) + 2 « h « l « a « b ' C 2 ' (cosa*cos/3-COS7)j 87 Where v i s the volume of the u n i t c e l l : v = 2• a• b• c / s in—^—'-•sin £ — L « s i n — = — L « s i n — £ — L V 2 2 2 z F o r an upper 20 l i m i t ( 2 0 m a x ) the maximal v a l u e s for | h | , | k | and | 1 | a r e : h f a x - b i g g e s t i n t e g e r < s i n ( 2 0 m a x ) - 2 - v i = r 3 X ' b « c « s i n a |kf» - biggest integer < " ^ ^ s i ^ " " U f - - biggest integer < l i ^ f C ^ We can now combine a l l p o s s i b l e h , k and 1 and check whether they f u l f i l l the c o n d i t i o n s f o r n o n - e x t i n c t i o n and whether t h e r e are i d e n t i c a l r e f l e c t i o n s due to the symmetry of the c r y s t a l system. By c o n v e n t i o n a powder l i n e i s l a b e l e d wi th the h i g h e s t index (ordered w i t h r e s p e c t to h , k and 1) and the number of i d e n t i c a l r e f l e c t i o n s i s c a l l e d the m u l t i p l i c i t y of t h a t p powder l i n e M . In the or thorhombic system e . g . : (0 2 0) = (0 2 0) M P = 2 and ( 1 3 1 ) = ( T 3 1 ) = ( 1 3 1 ) = ( 1 3 T ) = ( 1 3 T ) = ( T 3 1 ) = ( 1 3 1 ) = ( T 3 T ) M P = 8 88 The s t r u c t u r e f a c t o r and the i n t e g r a t e d i n t e n s i t y One p o s s i b l e way of c a l c u l a t i n g the s t r u c t u r e f a c t o r i s given below, m i s used as an index f o r a the elements, i f o r the s i t e s these elements can occupy and j f o r t h e i r e q u i v a l e n t p o s i t i o n s . h k l M = Z m=1 ( £ o + A f . ) + , Km) 1 J. \ • s " B i Mr»0 . « e 1 mi sin 2© J ( i ) • Z [cos(27r(h.x m. • + k«y . . + 1-z . .)) . , mil m i l m i l 3 = 1 + ' • s i n ( 2 7 r ( h . x m i j + k . y m i j + l . z m i j > > ] C o l l e c t i n g the r e a l and imaginary terms s e p a r a t e l y : R I ( m ) s = z m i=1 j I (m) s i = 2 j ( i ) r s [ M?-°mi- -B s i n 2 © i r~ . c o s ( 2 7 r ( h . x m i j + k . y m i j + l - z m . . ) ) ] J ( i ) j „ s i n 2 © ~ B i ~ T z _ f , [ M i , 0 m s i n ( 2 ^ ( h . x m i j + k . y m i j + l - z m i j > > ] f R = f° + A f m m m f 1 = Af" m m Then: h k l M r = L m=1 L m m m m J and i p hkii 2 M M , m m , m m m=1 m=1 M M 2 ( s l - f * ) + 2 ( S ? « f l ) m m . m m m= 1 m=1 The i n t e g r a t e d i n t e n s i t y of a powder l i n e i s d e f i n e d as: I h R 1 = M P . ( L P ) . | F h k l | 2 T h i s i s c a l l e d the ab s o l u t e i n t e n s i t y (Hubbard et a l . (1975)). A l l f a c t o r s that are constant f o r a given d iffT a c t o m e t e r or s t r u c t u r e are not i n c l u d e d . The example appendix D i s f o r a d i s o r d e r e d O l i v i n e of the composition F e 0 . 3 3 3 3 M n 0 . e e e y M g S i O , . 90 2.7 THE DETERMINATION OF MN-MG OR FE-MG SITE OCCUPANCIES BY XRD ANALYSIS 2.7.1 INTRODUCTION If we know the s t r u c t u r e , the l a t t i c e parameters, the p o s i t i o n a l parameters and the s c a t t e r i n g f a c t o r s f o r each s i t e , we can c a l c u l a t e a l l i n t e g r a t e d i n t e n s i t i e s as o u t l i n e d i n chapter 2.6. Because the s c a t t e r i n g f a c t o r f o r a s i t e occupied by Mn (or Fe) d i f f e r s s t r o n g l y from one occupied by Mg, the d i s t r i b u t i o n of Mn (or Fe) and Mg on M1 and M2 has a n o t i c a b l e i n f l u e n c e on the r e s u l t i n g i n t e n s i t i e s . In order to use t h i s dependency f o r the determination of s i t e occupancies, a l l other parameters should i d e a l l y be known. The method presented here i s based on the f a c t that the s t r u c t u r e and the bulk composition i s known, the l a t t i c e parameters can be measured by X-ray d i f f r a c t i o n and that the p o s i t i o n a l parameters can be approximated with a s u f f i c i e n t accuracy from p u b l i s h e d s i n g l e c r y s t a l s t r u c t u r e refinements. The main task i n developing the method i s to determine the u n c e r t a i n t y i n the i n t e r p r e t a t i o n due to the use of approximate p o s i t i o n a l parameters and an unknown e f f e c t i v e charge d i s t r i b u t i o n . I t was found that f o r each o l i v i n e at l e a s t four i n t e n s i t y r a t i o s c o u l d be used f o r the occupancy d e t e r m i n a t i o n . These r a t i o s show a reasonable dependency on o r d e r i n g , but remain p r a c t i c a l l y independent of small e r r o r s i n the p o s i t i o n a l parameters and of charge d i s t r i b u t i o n . The most important 91 r a t i o s are (112)/(130) and (112)/(131), which proved u s e f u l f o r almost a l l s y n t h e t i c o l i v i n e s . The estimated u n c e r t a i n t y of the method i s ±3% f o r the occupancy on each s i t e . 2.7.2 THE MEASUREMENT OF LATTICE PARAMETERS One set of data needed to c a l c u l a t e i n t e g r a t e d i n t e n s i t i e s are the l a t t i c e parameters a 0 b 0 and c 0 . These can e a s i l y be measured from an X-ray powder d i f f r a c t i o n p r o f i l e . The i n t e r n a l standard was n a t u r a l q u a r t z , which i n turn was s t a n d a r d i z e d a g a i n s t S i metal. The parameters obtained f o r quartz are a 0 = 4.9135 ±0.0002 c 0 = 5.4059 ±0.0003 which i s i n e x c e l l e n t agreement with the v a l u e s i n the A S T M - f i l e . The programs used f o r f i t t i n g were JOB9214 from the U.S. G e o l o g i c a l Survey (Evans et a l . (1963)) and the n o n - l i n e a r programming monitor (NLP) of the UBC computing c e n t r e . In Table 9 a l l measured l a t t i c e parameters are l i s t e d . F i g u r e s 12 to 15 are an attempt to v i s u a l i z e the t e r n a r y c o m p o s i t i o n a l v a r i a t i o n of a 0 , b 0 , c 0 and volume. Each diagram has one independent c o m p o s i t i o n a l v a r i a b l e and i s o l i n e s f o r the other two. The l i n e s were c a l c u l a t e d from a polynomial f u n c t i o n f i t t e d to the data. (See appendix F.) Although the l a t t i c e parameters f o r M n - o l i v i n e s show a d i s t i n c t non l i n e a r dependence on the composition, the volumes are almost a l i n e a r combination of the endmember Table 9.: The l a t t i c e parameters f o r the s n t h e t i c Fe-Mn-Mg o l i v i n e s Nr. X_ X M n X„ a 0 [ A ] b 0 [ A ] c 0 [ A ] v o l [ A 3 ] 1 0. 9940 0. 0060 0 .0000 4 .8198 10. 4839 6. 0919 307 .8255 4 0. 6627 0. 3373 0 .0000 4 .8420 10. 5574 6. 1 443 314 .0900 5 0. 41 42 0. 5858 0 .0000 4 .8640 10. 5883 6. 1799 318 .2741 6 0. 331 3 0. 6687 0 .0000 4 .8688 10. 5964 6. 1 944 319 .5800 7 0. 2485 0. 751 5 0 .0000 4 .8780 10. 6051 6. 2094 321 .2227 8 0. 1 657 0. 8343 0 .0000 4 .8890 10. 6085 6. 2285 323 .0409 9 0. 0828 0. 9172 0 .0000 4 .8959 10. 6071 6. 2460 324 .3629 9 0. 0828 0. 9172 0 .0000 4 .8982 10. 61 39 6. 2486 324 .8585 10 0. 0000 1 . 0000 0 .0000 4 .9033 10. 6030 6. 2579 325 .3463 1 0 0. 0000 1 . 0000 0 .0000 4 .9030 10. 6076 6. 2619 325 .6756 1 1 0. 8283 0. 0050 0 . 1 667 4 .81 06 10. 4409 6. 0752 305 . 1390 1 2 0. 6627 0. 1 707 0 . 1 667 4 .821 9 10. 471 1 6. 1 001 307 .9977 1 3 0. 4970 0. 3363 0 .1667 4 .8310 10. 5045 6. 1 263 310 .8928 1 4 0. 331 3 0. 5020 0 .1667 4 .8428 10. 5351 6. 1 606 314 .3100 1 4 0. 3313 0. 5020 0 . 1 667 4 .8468 10. 5400 6. 1 633 314 .8539 1 5 0. 1 657 0. 6676 0 . 1 667 4 .8560 10. 5577 6. 1875 317 .2219 1 6 0. 0000 0. 8333 0 .1667 4 .8770 10. 5683 6. 2283 321 .0165 1 7 0. 6627 0. 0040 0 .3333 4 .7998 10. 3815 6. 0539 301 .6605 1 7 0. 6627 0. 0040 0 .3333 4 .7983 10. 3830 6. 0554 301 .6846 Table cont inued on next page. 93 Table 9. (continued) Nr. X F e X M n X M g a 0 [ A ] b 0 [ A ] c 0 [ A ] v o l [ A 3 ] 18 0. 4970 0. 1697 0. 3333 4 .8108 10. 4218 6. 0829 304 .7894 19 0. 3313 0. 3353 0. 3334 4 .8171 10. 4451 6. 1 047 307 .1585 19 0. 3313 0. 3353 0. 3334 4 .8193 10. 4492 6. 1097 307 .671 2 20 0. 1657 0. 5010 0. 3333 4 .8292 10. 4914 6. 1473 31 1 .4534 21 0. 0000 0. 6667 0. 3333 4 .8417 10. 51 55 6. 1825 314 .7690 22 0. 4970 0. 0030 0. 5000 4 .7914 10. 3425 6. 0387 299 .2481 22 0. 4970 0. 0030 0. 5000 4 .7895 10. 3425 6. 0387 299 .1294 23 0. 3313 0. 1687 0. 5000 4 .7962 10. 3837 6. 071 0 302 .3498 24 0. 1657 0. 3343 0. 5000 4 .8060 10. 4184 6. 0989 305 .3770 25 0. 0000 0. 5000 0. 5000 4 .8133 10. 441 1 6. 1 255 307 .8440 25 0. 0000 0. 5000 0. 5000 4 .81 34 10. 4441 6. 1 276 308 .0444 26 0. 3313 0. 0020 0. 6667 4 .7759 10. 2986 6. 0235 296 .2664 27 0. 1657 0. 1677 0. 6666 4 .7806 10. 331 1 6. 0501 298 .8075 28 0. 0000 0. 3333 0. 6667 4 .7857 10. 3669 6. 0750 301 .3982 31 0. 0000 0. 0000 1 . 0000 4 .7548 10. 2004 5. 9823 290 .1467 Cell parameter a *>> ^ ^ ^. ^ ^ I Ul/ I I I g l̂ u L i a i . L i vL I I I I I — ' • 00 I—•* CD o CD 3 CD <r»- CD •1 P o O t - r » CD 3̂ CD I I CD 00 ^ ^ ^ g s ̂  ^ ^ ^ cn os CO CO O r-^ 76 F i g . 1 3 : T h e c e l l p a r a m e t e r b 0 o f t h e F e - M n - M g o l i v i n e s F i g . 1 4 : T h e c e l l p a r a m e t e r c 0 o f t h e F e - M n - M g o l i v i n e s F i g . 1 5 : T h e v o l u m e o f t h e F e - M n - M g o l i v i n e s 98 volumes. 2.7.3 UNCERTAINTIES IN CALCULATED INTENSITIES DUE TO ATOM COORDINATES AND TEMPERATURE CORRECTION FACTORS V a r i a t i o n of the p o s i t i o n a l parameters f o r Fe-Mn-Mg o l i v i n e s i s r e l a t i v e l y s m a l l , as may be seen i n t a b l e 10. (The l i t e r a t u r e data are t a b u l a t e d i n appendix F ) . The c a l c u l a t i o n of i n t e g r a t e d i n t e n s i t i e s using an average value f o r each c o o r d i n a t e with an u n c e r t a i n t y of 0.02 - 0.05 A, would y i e l d i n t e n s i t i e s with u n c e r t a i n t i e s i n the range of 15 - 70 %. To a v o i d as much of t h i s source of e r r o r as p o s s i b l e a m u l t i p l e l i n e a r r e g r e s s i o n was f i t to the p u b l i s h e d s t r u c t u r e refinement d a t a . The independent v a r i a b l e s are the s i t e occupancies x p e ( M l ) f x M n ( M l ) , x p e ( M 2 ) and x p e ( M 1 ) . The r e s u l t of t h i s r e g r e s s i o n i s t a b u l a t e d i n t a b l e 11. To estimate the i n f l u e n c e of each p o s i t i o n a l parameter on the r e s u l t i n g X-ray d i f f r a c t i o n p a t t e r n , a p r i m i t i v e s e n s i t i v i t y a n a l y s i s was performed. The estimated e r r o r s f o r the p o s i t i o n a l parameters were taken to be the maximal r e s i d u a l s from the m u l t i p l e l i n e a r r e g r e s s i o n . T h i s corresponds to ^3a. Because most r e p o r t e d s t r u c t u r e refinements are f o r compositions c l o s e to the b i n a r y j o i n s , e s p e c i a l l y f a y a l i t e - f o r s t e r i t e , i n t e r p o l a t i o n i n s i d e the te r n a r y r e g i o n i s l e s s c e r t a i n than along the b i n a r i e s . Changing one atom c o o r d i n a t e or one temperature c o r r e c t i o n f a c t o r at a time by the amount of the maximal r e s i d u a l and 99 Table 10.: The range of valu e s f o r the p o s i t i o n a l parameters and the temperature c o r r e c t i o n f a c t o r s . HCP Ranqe var. max.res. XM1 0.0 0.0 0.0 0.0 yM1 0.0 0.0 0.0 0.0 ZM1 0.0 0.0 0.0 0.0 XM2 0.0 0. 98510 - 0 .99169 0.00659 0.0011 yM2 0.25 0. 27739 - 0 .28030 0.00291 0.0002 ZM2 0.25 0.25 0.0 0.0 X S i 0.375 0. 42260 - 0 .43122 0.00522 0.0013 y S i 0.08333 0. 09100 - 0 .09765 0.00665 0.0002 Z S i 0.25 0.25 0.0 0.0 x01 0.75 0. 75776 - 0 .76870 0.01094 0.0003 y01 0.08333 0. 08670 - 0 .09363 0.00693 0.0012 Z01 0.25 0.25 0.0 0.0 X02 0.25 0. 20760 - 0 .23010 0.02250 0.0008 y02 0.41667 0. 44721 - 0 .45510 0.00789 0.0012 Z02 0.25 0.25 0.0 0.0 X03 0.25 0. 27723 - 0 .28897 0.01174 0.0004 y03 0.16667 0. 15900 - 0 .16563 0.00663 0.0011 Z03 0.0 0. 03304 - 0 .04140 0.00836 0.0020 Table continued on next page. 100 T a b l e 10. ( cont inued) Range v a r . m a x . r e s . BM1 0.26 -- 0.62 0.36 0.16 BM2 0.22 -- 0.56 0.34 0.22 B S i 0.08 -- 0.52 0.44 0.23 B 01 0.26 -- 0.70 0.44 0.22 B 02 0.24 -- 0.68 0.44 0.25 B 0 3 0.28 -- 0.70 0.42 0.23 HCP: " i d e a l " hexagonal c l o s e - p a c k e d o l i v i n e model . (Hazen, 1976) Range: s m a l l e s t and b i g g e s t v a l u e from the l i t e r a t u r e , v a r . : d i f f e r e n c e between b i g g e s t and s m a l l e s t v a l u e , m a x . r e s . : maximal r e s i d u a l in the m u l t i p l e l i n e a r r e g r e s s i o n . 101 Table 11.: The r e s u l t of the m u l t i p l e l i n e a r r e g r e s s i o n . The constants are used to c a l c u l a t e the p o s i t i o n a l parameters from known s i t e occupancies f = a 0 ' X Fe(M1) + a 1* xMn(MD + a 2 * X F e ( M 2 ) + a 3* xMn(M2) a 0 a i a 2 a 3 a„ XM2 = 0. 99065 + 0. 0079070 + 0. 0014940 - o . 0132500 -0. 004212 y M2 = 0. 27740 + 0. 0001013 + 0. 0013330 + 0. 0026840 + 0. 001676 X S i = 0. 42639 +0. 0045390 + 0. 0059540 -0. 0010360 -0. 004791 y S i = 0. 09403 +0. 0019040 + 0. 0063790 + 0. 0015420 - o . 003984 X01 = 0. 76575 + 0. 0007991 + 0. 0001765 + 0. 0017330 - o . 008172 y o i = 0. 091 65 + 0. 0007996 + 0. 0082890 -0. 0005220 -0. 006306 X02 = 0. 221 67 -0. 0115900 -0. 0222700 -0. 0019070 + 0. 011480 y02 = 0. 4471 2 -0. 0008320 + 0. 0050220 + 0. 0075800 + 0. 001551 X03 = 0. 27732 + 0. 0126300 +0. 0096130 - o . 0011740 + 0. 0001291 y03 = 0. 16306 + 0. 0017360 + 0. 0058690 + 0. 0000079 -0. 0050850 Z03 = 0. 0331 4 -0. 0005452 + 0. 0038170 + 0. 0057090 + 0. 0044400 BM1 = 0. 41954 -0. 4030000 + 0. 2199900 + 0. 5038800 - o . 0198600 BM2 = 0. 43665 -0. 3399100 -0. 0238300 + 0. 3529300 + 0. 1180900 B S i = 0. 31248 -0. 3063000 + 0. 0188800 + 0. 3729400 + 0. 0493900 B01 = 0. 47866 + 0. 1597800 + 0. 0390600 - o . 1653900 + 0. 0178700 B02 = 0. 48532 -0. 2139000 + 0. 0699900 + 0. 1610200 - o . 0145500 B03 = 0. 50677 + 0. 0164300 + 0. 0887800 + 0. 0331200 - o . 0176900 1 02 r e c a l c u l a t i n g the XRD i n t e n s i t i e s p r o v i d e s an estimate of the e r r o r i n t r o d u c e d i n the i n t e n s i t y c a l c u l a t i o n s by u n c e r t a i n t y i n the p o s i t i o n a l parameters. For the three endmembers f a y a l i t e , t e p h r o i t e and f o r s t e r i t e these e r r o r s t a b u l a t e d i n appendix F. The t o t a l u n c e r t a i n t i e s are a l s o shown i n f i g . 16. I t may be seen that a l l peaks with a r e l a t i v e i n t e n s i t y g r e a t e r than 25 % have a t o t a l u n c e r t a i n t y of l e s s than 3.5 %. T h i s i s true f o r a l l three endmembers and was a l s o confirmed i n s e v e r a l sample c a l c u l a t i o n s of s o l i d s o l u t i o n s . As a general r u l e , the u n c e r t a i n t i e s i n c r e a s e s l i g h t l y with i n c r e a s i n g Mg content. 2.7.4 UNCERTAINTIES IN CALCULATED INTENSITIES DUE TO CHARGE DISTRIBUTION. A f u r t h e r f a c t o r c o n t r i b u t i n g to u n c e r t a i n t y i n the c a l c u l a t i o n s i s the e f f e c t i v e charge d i s t r i b u t i o n . For f a y a l i t e , t e p h r o i t e and f o r s t e r i t e the charges on each s i t e have been r e f i n e d by F u j i n o et a l . (1981). T h e i r r e s u l t s show strong d e v i a t i o n s from i d e a l i o n i c charges, but at present they do not allow a p r e d i c t i o n of the charge d i s t r i b u t i o n of the s o l i d s o l u t i o n s . In a d d i t i o n , the presence of F e 3 + i n our samples does not allow a d i r e c t comparision with t h e i r d ata. The only simple way to e x t r a c t s i t e occupancies from i n t e n s i t y measurements i s to use i n t e n s i t y r a t i o s that a re, w i t h i n a c c e p t a b l e l i m i t s , independent of the e f f e c t i v e charge d i s t r i b u t i o n . In order to f i n d these r a t i o s we make 103 F i g . 1 6 : U n c e r t a i n t y o f I n t e g r a t e d I n t e n s i t i e s d u e t o u n c e r t a i n t i e s i n A t o m c o o r d i n a t e s a n d T e m p e r a t u r e c o r r e c t i o n f a c t o r s 10 9 - 8 7 - S 6 .S 5- cd ^ A O 4 fl fl 3 - 2 - 0 o °A o a A o • D tf A • D ' O A o DA 0m o fi1 O A • A • a A • o • © A Tephroite Mn 2 Si0 4 Fayalite Fe 2 Si0 4 Forsterite Mg 8 Si0 4 8 a a • o 1 I 1 I 1 I 1 I 1 I 1 I 1 I 1 I 1 I 1 0 10 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0 r e l a t i v e i n t e n s i t y 104 two c a l c u l a t i o n s f o r every sample: one assuming n e u t r a l atoms in the l a t t i c e and one with i d e a l l y charged i o n s . Only those r a t i o s t h at d i f f e r by l e s s than 5 % i n the two c a l c u l a t i o n s are used f o r i n t e n s i t y measurements. The two extreme assumptions do not span the whole range of p o s s i b l e charge d i s t r i b u t i o n s , but i t was found that even very i r r e g u l a r charges (e.g. only M1 and h a l f of the 03 charged) gi v e i n t e n s i t y r a t i o s between the two extreme v a l u e s . 2.7.5 THE INTENSITY MEASUREMENTS AND THE OCCUPANCY INTERPRETATION The X-ray d i f f r a c t i o n p r o f i l e s were measured on two types of d i f fT a c t o m e t e r . An o l d e r (1965) P h i l i p s PW1050 goniometer and a P h i l i p s PW1710 automatic d i f fT a c t o m e t e r . In the f i r s t case the peaks of i n t e r e s t were scanned at 2 cm/min and 0.125 deg. 2©/cm. The peak shapes and the background were drawn i n by hand. The r e s u l t i n g p r o f i l e s were d i g i t i z e d with a T a l o s CYBERGRAPH d i g i t i z e r and the areas c a l c u l a t e d with a po l y g o n a l approximation. The important peaks (130), (131) and (112) were u s u a l l y scanned twice and each d i g i t i z i n g was repeated three times. The d i f f e r e n c e s between the d i f f e r e n t measurements average approximately 0.5 %. I f the same peak i s scanned with d i f f e r e n t s c a l e s , the d i f f e r e n c e s are i n c r e a s e d t o 2 %. To each peak an e r r o r i s assign e d which i s one q u a r t e r of the background n o i s e over the measured l e n g t h p l u s 2 % of the measured area to account f o r p o s s i b l e systematic e r r o r s . The 105 t o t a l u n c e r t a i n t y thus assigned to the r a t i o s v a r i e s from 3 % to 7 %, which we c o n s i d e r a c c e p t a b l e . The measurements made with the PW1710 d i f f T a c t o m e t e r were ev a l u a t e d s i m i l a r l y , except that the d i g i t i z i n g i s done i n constant steps of 0.05° 20, and that the background was approximated with a polynomial f u n c t i o n . The measuring programs are l i s t e d i n appendix G. A f i n a l d i f f i c u l t y i s that s e v e r a l peaks cannot be measured i n d i v i d u a l l y i n a XRD p r o f i l e . Instead of simple r e f l e c t i o n s i t i s necessary to c o n s i d e r groups of two or three r e f l e c t i o n s . In these cases the t o t a l u n c e r t a i n t y due to approximate p o s i t i o n a l parameters and temperature c o r r e c t i o n f a c t o r s can be estimated from the t a b l e s i n appendix F. The measuring s t r a t e g y may be summarized as f o l l o w s : • The r e f l e c t i o n s or groups of r e f l e c t i o n s t h at may be used should have a t o t a l r e l a t i v e i n t e n s i t y g r e a t e r than 25 %. Thi s i s necessary to assure that the u n c e r t a i n t y r e s u l t i n g from the approximate p o s i t i o n a l parameters and temperature c o r r e c t i o n f a c t o r s i s l e s s than =3.5 %. • For these peaks we can c a l c u l a t e a l l r a t i o s f i r s t using uncharged atoms and then using i d e a l l y charged i o n s . Those r a t i o s t h at remain w i t h i n a range of =5 % can be used f o r the d e t e r m i n a t i o n of s i t e occupancies. Which r a t i o s we can use must be determined s e p a r a t e l y f o r each i n d i v i d u a l sample. 106 As we can measure i n t e n s i t y r a t i o s with an accuracy of =5 %, we may p r e d i c t that the maximal d i s c r e p a n c y between c a l c u l a t e d and measured i n t e n s i t y r a t i o s w i l l be i n the magnitude of 10 %. T h i s p r e d i c t i o n can be t e s t e d with the three endmembers f a y a l i t e , t e p h r o i t e and f o r s t e r i t e . In t a b l e 12 the c r u c i a l r a t i o s are l i s t e d together with the c a l c u l a t e d and observed v a l u e s . A l l d i f f e r e n c e s are l e s s than 10 %. The r a t i o s 112/130 and 112/131 are usable f o r a l l three endmembers. Not a l l peaks can always be measured. The reasons are that f o r some compositions they may not be separable from neighbouring peaks of the o l i v i n e or they o v e r l a p with g r a p h i t e or s i l v e r peaks which are always present i n v a r y i n g amounts i n the experimental products. U s u a l l y at l e a s t four usable and measurable i n t e n s i t y r a t i o s were found i n the s o l i d s o l u t i o n s . The s o l i d s o l u t i o n s were e v a l u a t e d by f i r s t measuring the i n t e n s i t i e s . Then f o r the whole range of p o s s i b l e o r d e r i n g s (the Fe d i s t r i b u t i o n i s taken from the Mossbauer measurements) the i n t e n s i t i e s were c a l c u l a t e d i n steps of xMg ( M 2 ) = n * uE> f o r uncharged atoms and f o r i d e a l l y charged i o n s . Those r a t i o s t h at remain w i t h i n 5 % f o r both c a l c u l a t i o n s were used. Comparing with the measurements and assuming a 10% t o l e r a n c e , each r a t i o y i e l d s a range of p o s s i b l e occupancies. T h i s range i s not a t r u e a b s o l u t e bracket and t h e r e f o r e the f i n a l i n t e r p r e t a t i o n i s taken as the weighted mean ±1a of the midpoints of the a l l ranges, the weight being the r e c i p r o c a l of the width of the range. 1 07 Table 12.: The d i f f e r e n c e s between c a l c u l a t e d and observed i n t e g r a t e d i n t e n s i t y r a t i o s f o r f a y a l i t e , f o r s t e r i t e . t e p h r o i t e and 1) c a l c . obs. d i f f [%] f a y a l i t e 122/111+120 0.37672 0.38649 2.59 222+240+123/111+120 222+.. not measured 022+041/130 0.45960 0.49389 7.46 112/130 1.08995 1.10898 1 .75 222+240+123/130 222+.. not measured 112/022+040 2.37152 2.24539 5.62 222+240+123/022+040 222+. . not measured 112/131 1.54171 1.66188 7.79 200+041/131 0.43868 0.44257 0.89 t e p h r o i t e 222+240+123/111+120 1.52454 1.55106 1 .74 200+041/121+022 2.35904 2.33459 1 .05 112/130 1.15534 1.17306 1 .70 222+240+123/130 1.07927 1.05567 2.24 131/022+040 1.65471 1.68087 1 .58 112/022+040 2.57255 2.45424 4.82 112/131 1.55468 1.46011 6.48 108 T a b l e 12. ( cont inued) f o r s t e r i t e 212+002/020 1. 200+041/020 210/111+120 0. 222+240+123/111+120 3. 131/121+002 3. 112/130 1. 210/130 0. 112/131 1. 222+240+123/210 8. 08861 1.09177 0.29 200+. . not measured 35267 0.34161 3.24 12378 2.95109 5.85 30228 3.03487 8.81 59151 1.62960 2.39 16209 0.17813 9.89 24503 1.26579 1.67 85752 8.63869 2.53 1): r a t i o s p r e d i c t e d to have a d i f f e r e n c e between c a l c u l a t e d and measured v a l u e s s m a l l e r than 10 %. The c a l c u l a t e d v a l u e s are for uncharged atoms. 109 As may be seen i n the f o l l o w i n g example, t h i s c o i n c i d e s roughly with the range where a l l u s e f u l measured r a t i o s are w i t h i n 10% of the c a l c u l a t e d ones. (Table 13.) A l l measured i n t e n s i t i e s are compared with the c a l c u l a t e d ones i n f i g . 17. 2.7.6 EXAMPLE OF A SITE OCCUPANCY DETERMINATION The measured i n t e n s i t i e s f o r MnMgSiO a are l i s t e d together with the other r e s u l t s i n t a b l e 14. The r e f l e c t i o n s with i n t e n s i t i e s g r e a t e r than 25% and not o v e r l a p p i n g with peaks from other phases are (130), (131), (112), (121), (122) and (222). In t a b l e 13 the r e s u l t of the c a l c u l a t i o n s i s summarized. The f i v e usable r a t i o s are a l s o p l o t t e d i n f i g u r e 18 and 19. The boxes on the curves represent the ranges where the observed values are w i t h i n 10% of the c a l c u l a t e d ones. The (110) r e f l e c t i o n c o u l d not be observed and i s assumed to have a r e l a t i v e i n t e n s i t y l e s s than =1%. I t i s i n t e r e s t i n g to note that t h i s i s a l s o c o n s i s t e n t with the i n t e r p r e t a t i o n of x M n ( M 1 ) = 0.336 because much smaller or l a r g e r values would r e s u l t i n a more in t e n s e (110) r e f l e c t i o n . 110 Fig. 17: Comparison of ca lcu lated and m e a s u r e d integrated intensit ies for tephroite , fayal i te and forsterite 100 9 0 - 80 • T H 70 w fl i5 60 fl X I CD > U CD W O 5 0 - 4 0 - 30 2 0 - 10- 0 -ffl • : Tephroite Mn 2 Si0 4 O : Fayalite Fe2SiO« A : Forsterite Mg 2 Si0 4 A / A A 43 / 6 * ' A / / / / I 1 I 1 I 1 I 1 I 1 I 1 I 1 I 1 I 1 0 10 20 30 40 50 60 70 80 90 100 calculated intensity 111 Table 13.: Example: The s i t e occupancy f o r MnMgSiO« r e f l e x uncharged charged d i f f comments xMn(M1) xMn(M1) 130/121 0 .354±0 . 036 0.31710. 035 0 .037 O.K. 131/121 0 .387±0 . 074 0.35410. 068 0 .033 O.K. 112/121 0 .356±0 . 034 0.33710. 030 0 .019 O.K. 122/121 0.42110. 042 0.29110. 042 0 .130 too b i g d i f f . 222/121 0.36110. 048 0.29810. 042 0 .063 too b i g d i f f . 131/130 0.326+0. 069 0.28510. 065 0 .041 O.K. 112/130 0.42010. 420 0.57210. 373 0 .152 too f l a t 122/130 0. 13110. 131 0.438+0. 184 0 .307 too b i g d i f f . 222/130 0.36010. 204 0.42810. 206 0 .068 too f l a t 112/131 0.32710. 059 0.32110. 054 0 .006 O.K. 122/131 0.46110. 1 03 0.19210. 1 06 0 .269 too b i g d i f f . 222/131 0.30410. 1 04 0.21210. 099 0 .092 too b i g d i f f . 122/112 0.17310. 1 24 0.47810. 122 0 .305 too b i g d i f f . 222/112 0.36710. 1 39 0.47410. 132 0 . 1 07 too b i g d i f f . 121 : 121+002 122 : 122+140 222 : 222+240+123 10 % bracket, weighted mean 0.32910.022 0.33610.024 Fig. 18: Intensity ratios versus site occupancy for Mn6Mg6Si6024 (ratios < 1) cz o TD OJ c n ro sz u 110/112 121+002/130 121+002/112 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 CD O -M ro ro Z D Q J 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 v \ \ \ v \ \ \ \ \ V s. \ 110/112 121+002/130 121+002/112 0.0 0.1 0,2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 XJM1) Fig. 19: Intensity ratios versus site occupancy for r1n6Mg6Si6024 (ratios > 1) 10.0-a in a o OJ c n ro sz u 0.0 -7- . .._ . f :z: / - 7 ' f 7 s 131/121+002 112/131 130/131 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1 14 Table 14.: normalized i n t e g r a t e d i n t e n s i t i e s The u n c e r t a i n t i e s are ±(2 + 0.02*1) 01 10 1 1 1 2 13 1 4 15 16 020 8.8 9.4 4.0 9.2 1 .4 2.1 2.3 5.7 110 8.4 5.9 1.3 1 .2 1 .4 0.8 021 11.9 22.3 101 6.0 111+120 62.6 58.0 37.2 57.6 36.7 39.0 39.3 51 .6 121+002 10.2 16.5 6.9 12.5 12.7 13.0 11.1 10.2 130 90.2 85.2 77.7 85.4 73.4 72.5 68.9 77.3 022+040 44.5 40.7 30.3 37.4 33.0 31 .3 28.8 24.2 131 60.2 68.5 60.4 62.6 67.9 67.8 67.6 55.3 1 12 100.0 100.0 100.0 100.0 100.0 1 00.0 100.0 100.0 200+041 26.6 25.2 26.3 26.2 24.2 24.2 22.6 17.9 210 5.2 6.4 24.2 5.0 23.5 17.7 122+140 24.2 27.3 31.6 26. 1 32.3 33.7 35.0 211+220 5.0 11.4 222+240+123 87.5 90.0 1 36.2 90.4 121.0 1 22.0 127.8 84.9 152 17.3 11.4 —'— 20.8 23.0 01:12.00.00 1 0:00.12.00 11: 10.00.02 12_:08.02.02 13:06.04.02 14:04.06.02 15:02.08.02 16:00.10.02 1 15 Table 14.: (continued) 17 18 19 20 21 22 23 24 020 5 .9 9 .2 4 .3 1 .2 4 . 1 4 . 1 3 .9 3 . 1 1 10 021 2 .3 2 .0 <1 <0 .5 <1 <0 .5 <0 .5 <0 .5 101 111+120 34 .9 50 .2 43 .5 39 .3 68 .4 25 .9 38 .8 38 .9 121+002 1 1 .0 1 1 .8 1 1 .0 14 .3 21 .3 9 .2 15 .3 15 . 1 130 84 .3 86 .7 78 .0 70 .4 85 .4 64 .4 73 .8 71 .3 022+040 25 .3 24 . 1 23 .8 21 .5 23 .5 1 4 .2 1 7 . 1 16 .4 131 68 .6 72 .4 72 .9 76 .2 83 .5 59 .4 71 .8 77 .5 1 12 1 00 .0 100 .0 100 .0 1 00 .0 1 00 .0 1 00 .0 100 .0 1 00 .0 200+041 21 .5 24 .5 24 .3 22 .6 25 .7 — — — — 20 .0 210 7 .7 6 . 1 30 .0 6 .2 — — — 10 . 1 122+140 211+220 222+240+123 39 .6 34 .9 39 . 9 40 .4 38 .2 — 49 .5 47 .3 1 1 5 .7 100 .2 121 .2 1 26 . 1 109 .7 1 1 5 . 9 129 .2 1 32 . 1 1 52 16 .2 1 6 . 1 — — 17:08.00.04 18:06.02.04 19:04.04.04 20:02.06.04 21:00.08.04 22:06.00.06 23:04.02.06 24:02.04.06 116 Table 14.: (continued) 25 26 27 28 31 020 2.7 7.8 4. 1 10.2 23.8 110 <0.5 <0.5 <0.5 <1 <0.5 021 68.7 101 26.4 111+120 44.0 27.7 33.4 56.9 32.0 121+002 23.6 12.9 18.4 27.0 26.0 1 30 79.6 68.4 66.6 80. 1 61 .4 022+040 14.5 10.1 4.3 12.1 131 95.2 68.2 78.9 90.5 79.0 1 12 100.0 100.0 100.0 100.0 100.0 200+041 26.0 16.4 44.5 18.8 11.4 210 24.8 9.6 11.5 9.2 10.9 122+140 49.8 55.3 58.5 50. 1 78.0 211+220 19.8 222+240+123 102.6 111.1 1 17.5 105.1 77.4 1 52 10.2 25:00.06.06 26:04.00.08 27:02.02.08 28:00.04.08 31:00.00.12 1 17 Table 15.: Site occupancies for the synthetic Fe-Mn-Mg o l i v i n e s NR . 1:120000 ca l c . : 4:080400 calc. : 5:050700 calc. : G:040800 ca1c. : 7:030900 ca l c . : 8:021000 ca l c . : 11:100002 11:100002 ca l c . : 12:080202 ca l c . : 13:060402 ca l c . : 14:040602 14:040602 ca l c . : Fe 0.9940 Mn 0.0060 Mg 0.0000 Fe(M1) Fe(M2) Mn(M1) Mn(M2) Mg(M1) Mg(M2) 0.6627 0.3373 0.0000 0.4142 0.5858 0.3310 0.6687 0.2485 0.7515 0.1657 0.8343 0.8283 0.8283 0.0050 0.0050 0.0000 0.0000 0.0000 0.0000 0.1667 0.1667 0.6626 0.1707 0.1667 0.4970 0.3363 0.1667 0.3313 0.3313 0.5020 0.5020 0.1667 0.1667 0.9950 0.9974 0.8198 0.8010 0.5541 0.5655 0.4593 0.4679 0.3573 0.3623 0.2512 0.2484 0.8684 0.8490 0.8435 0.7351 0.7407 0.6174 0.6079 0.4283 0.4283 0.4399 0.9930 0.9906 0.5056 0.5244 0.2743 0.2629 0.2027 0.1943 0.1397 0.1347 0.0802 0.0830 0.7882 0.8076 0.8131 0.5901 0.5845 0.3766 0.3861 0.2343 0.2343 0.2227 0.003 0.0026 0.1811 0.1990 0.4459 0.4345 0.5407 0.5320 0.6427 0.6377 0. 7488 0.7516 * 0.002 * 0.002 0.0022 0.0871 0.0870 0.21 10 0.2008 0.3400 0.3510 0.3496 0.009 0.0094 0.4944 0.4756 0.7257 0.7371 0.7973 0.8058 0.8603 0.8653 0.9198 0.9170 0.008 0.008 0.0078 0.2543 0.2544 0.4616 0.4718 0.6640 0.6530 0.6544 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 O.0000 o.0000 0.0000 O.0000 0.0000 0.0000 0.1276 0.1470 O.1544 0. 1778 O.1722 0.1716 0.1913 0.2317 0.2207 0.2105 O.0000 0.0000 0.0000 0.0000 O.OOOO 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.2058 O.1864 0.1790 0.1556 0.1612 0.1618 O.1421 0.1017 0.1127 0.1229 1n(KD)' 1.4876 1.1902 1.2063 1.3207 1 .3474 ln(K ) 2 ln(K ) 3 method D D 1 .2912 1.2772 1 .2726 1.2240 0.5749 M 0.2875 PW 0.0863 -1.2048 M,D 0.4355 -0.8416 M.PW -0.2202 -1.4928 M,PW -0.0689 -1.2929 M,D 118 Table 15. (continued) NR. 15:020802 ca1c.: 16:001002 16:001002 ca1c. : 17:080004 17:080004 ca l c . : 18:060204 cal c. 19:040404 c a l c . : 20:020604 c a l c . : 21:000804 c a l c . : 22:060006 22:060006 c a l c . : 23:040206 ca l c . : Fe 0. 1657 0.0000 O.0000 0.6627 0.6627 Mn 0.6676 0.8333 0.8333 0.0040 0.0040 0.4970 0.4970 0.0030 0.0030 Mg 0. 1667 0. 1667 0.1667 0.3333 0.3333 Fe(M1) Fe(M2) Mn(M1) Mn(M2) Mg(M1) Mg(M2) l n ( K D ) : 0.4970 0.1697 0.3333 0.3313 0.3353 0.3334 0.1657 0.5010 0.3333 O.0000 0.6667 0.3333 0.5000 O.5000 0.3313 0.1687 0.5000 O.2344 O.2361 O.0000 O.0000 O.0000 0.6993 0.6970 0.6847 0.5691 O.5630 0.4089 0.4100 0.2259 O.2222 O.0000 O.0000 0.5144 0.5015 0.5209 0.3984 0.3803 0.0970 0.0953 O.0000 0.0000 0.0000 0.6261 0.6284 0.6407 0.4294 0.4310 0.2537 0.2526 0.1055 0.1092 0.0000 O.OOOO 0.4796 0.4925 0.4731 0.2642 0.2823 0.5480 O.5354 0.7600 0.7770 0.7559 * 0.002 * 0.002 0.0018 0.0922 0.0877 O.2030 0.2025 0.3650 0.3524 0.5300 0.5393 * 0.001 * 0.001 0.0013 0.0900 0.0882 0.7872 O.7997 0.9066 0.8896 0.9107 * 0.006 * 0.006 0.0062 0.2472 0.2517 0.4676 0.4681 0.6370 0.6496 0.8034 0.7941 * 0.005 * 0.005 0.0047 0.2474 0.2492 0.2176 O.2284 0.2400 0. 2230 0.2441 0.2977 0.3000 0.3135 0.3403 0.3493 0.3881 0.3874 0.4091 0.4254 0.4700 0.4673 0.4831 0.4960 0.4778 0.5116 0.5315 O.1158 O.1050 0.0934 0.1104 0.0893 O.3689 0.3666 O.3531 0.3263 0.3173 0.2787 0.2794 0.2575 0.2412 0.1966 O.2059 0.5169 0.5040 0.5222 0.4884 0.4685 1.2445 1.2679 1.3117 1.3183 ln(K )* ln(K ) ] D D 0.2515 -0.9930 0.3250 0.3041 0.2396 -1.1201 -0.8384 1.0282 0.1462 -1.1655 0.2984 •1.0198 •1 .2875 1 .4319 0.1377 0.0341 0.3643 -1.0576 method M,PW PW D M PW M, D M,D M.PW M PW M,PW CO Table 15. (continued) NR. x x x Fe(M1) Fe(M2) Mn(M1) Mn(M2) Mg(M1) Mg(M2) ln(K )' ln(K ) ! ln(K )' method Fe Mn Mg D D D 24 020406 0 1657 0 3343 0 5000 0 2192 0 1 122 0 2140 0 4546 0 5668 0 4332 1 .4231 0 4009 -1 .0222 M,PW ca l c . : 0 2074 0 1240 0 2042 0 4644 0 5883 0 41 16 25 000606 O 0000 0 5000 0 5000 O 0000 0 OOOO 0 3360 0 6640 O 6640 0 3360 -1 3623 PW ca l c . : 0 0000 0 0000 0 3552 0 6448 0 6448 0 3552 26 040008 0 3313 0 0020 0 6667 0 3594 0 3032 0 * 001 0 * 003 0 6391 O 6943 0 2529 M 26 040008 0 3313 0 0020 0 6667 0 3625 0 3001 0 * 001 0 * 003 0 6360 0 6974 0 281 1 PW ca l c . : 0 3521 0 3105 0 0009 0 0031 0 6469 0 6865 27 020208 0 1657 0 1677 0 6667 0 1941 0 1373 0 0960 0 2394 0 7099 0 6233 1.2600 0 2161 -1 0439 M, PW ca1c. : 0 1927 0 1390 0 0882 0 2455 0 7190 O 6155 28 000408 0 0000 0 3333 0 6667 0 0000 0 0000 0 2000 0 4666 0 8000 0 5334 - 1 2525 D c a l c . : 0 0000 0 0000 0 2059 0 4607 0 7941 0 5393 *: For these samples the occupancies for Mn are estimated 1: (k ) i = (Fe(M1 ) Mn(M2))/(Fe(M2)-Mn(M1)) D 2: (kph = (Fe(M1 ) Mg(M2))/(Fe(M2) Mg(M1)) 3: ( k D > 3 = (Mn(M1)•Mg(M2))/(Mn(M2)•Mg(M1)) method: M: Mossbauer spectroscopy, D: XRD inten s i t i e s using old diffractometer, PW: XRD Intensities using P h i l i p s PW1710. The calculated values are according to 'model 2' described in chapter 2.8. 120 2.7.7 THE SITE OCCUPANCIES OF THE SYNTHETIC FE-MN-MG OLIVINES In Table 14 a l l measured i n t e g r a t e d i n t e n s i t i e s are l i s t e d . Knowing the s i t e d i s t r i b u t i o n of Fe from the Mossbauer spectroscopy, the Mn-Mg d i s t r i b u t i o n was determined as d e s c r i b e d above. In a d d i t i o n , the b i n a r y f a y a l i t e - f o r s t e r i t e s e r i e s was a l s o analyzed with t h i s method, i n order to compare the r e s u l t s with the Mossbauer data. The measured F e 3 + was ass i g n e d to M2 as d i s c u s s e d i n chapter 2.5.3. The f i n a l r e s u l t with the M1 and M2 occupancies of a l l s y n t h e t i c o l i v i n e s i s presented i n t a b l e 15. For the te r n a r y compositions the u n c e r t a i n t i e s i n the Fe and Mn s i t e d i s t r i b u t i o n are h i g h l y c o r r e l a t e d . Because the Fe and Mn have s i m i l a r s c a t t e r i n g f a c t o r s , the XRD method pr o v i d e s i n f o r m a t i o n on Fe+Mn versus Mg occupancies. An ov e r e s t i m a t i o n of the Fe occupancy i n the Mossbauer spectroscopy w i l l r e s u l t i n an underestimation of the Mn occupancy on the same s i t e i n the XRD method. 121 2.8 THERMODYNAMIC MODEL FOR THE TERNARY FE-MN-MG OLIVINE SOLID SOLUTIONS AT 850 °C 2.8.1 INTRODUCTION The general framework f o r the mixing model of the o l i v i n e s i s chosen to be the " s p e c i a t i o n model" of Brown and Greenwood ( i n p r e p a r a t i o n ) . (See a l s o chapter 1.2.2.) T h i s model i s a g e n e r a l i z a t i o n of a l l s i t e mixing models and t h e r e f o r e very a p p r o p r i a t e f o r our problem. Because three c a t i o n s (Fe, Mn and Mg) can be d i s t r i b u t e d on two s i t e s (M1 and M2) the complete s t r u c t u r a l v a r i a t i o n u sing t w o - s i t e s p e c i e s can be d e s c r i b e d with nine s p e c i e s . Among these there are 6 independent r e a c t i o n s : (FeFe) + (MnMn) = 2-(FeMn) (1) (FeFe) + (MnMn) = 2-(MnFe) (2) (FeFe) + (MgMg) = 2-(FeMg) (3) (FeFe) + (MgMg) = 2-(MgFe) (4) (MnMn) + (MgMg) = 2'(MnMg) (5) (MnMn) + (MgMg) = 2«(MgMn) (6) For an i d e a l s p e c i a t i o n model, the a c t i v i t y of each s p e c i e s i s equal to i t s c o n c e n t r a t i o n and the r e a c t i o n c o n s t a n t s become: 2 2 xFeMn _ xMnFe K 1 _ K 2 - xFeFe' xMnMn xFeFe' xMnMn 122 K 3 = XFeMg R = xMgFe xFeFe' xMgMg xFeFe* xMgMg 2 2 xMnMg „ xMgMn K 5 - K.6 = xMnMn'xMgMg xMnMn'xMgMg 2.8.2 MODEL ASSUMING INDEPENDENT SITE MIXING If the o r d e r i n g i n the o l i v i n e s i s assumed to be a pure s i t e p r e f e r e n c e then the p r o b a b i l i t y of f i n d i n g f o r example Fe on a s p e c i f i c M1-site i s equal to x F e ( M l ) , independent of what the nearest neighbours a r e . We may t h e r e f o r e w r i t e : x(FeMn) = x F e ( M l ) ' x M n ( M 2 ) T h i s assumption i s q u i t e f r e q u e n t l y used to approximate the a c t i v i t i e s of endmembers i n s o l u t i o n s with i d e a l s i t e mixing, (e.g. Wood and Fr a s e r (1972)). The i m p l i c a t i o n f o r a s p e c i a t i o n model i s that only two independent r e a c t i o n c o n s t a n t s remain: K, = K i 1 = K D(Fe-Mn) = K K 3 = K;1 = K D(Fe-Mg) 5 = K i 1 = (Mn-Mg) = x F e ( M D * x M n ( M 2 ) * F e(M2) • x M n ( M 1 ) * F e < M l ) ' x M g ( M 2 ) x p e(M2) * x M g ( M 1 ) x M n ( M l ) ' x M g ( M 2 ) X M n ( M 2 ) • x M g { M l ) 123 F i g . 20: T h e o b s e r v e d l n ( K D ) v a l u e s f o r t h e s y n t h e t i c F e - M n - M g o l i v i n e s 2.0 X ( M n ) 0.0 0.2 0.4 0.6 0.8 1.0 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1.5 — 1.0 — 0.5 — a o.o -0.5 — 1.0 -1.5 -2.0 K = Ee(Mjj^n(M_.) D Mn(M ,)Fe(M.) Mg(M ,)Fe(M,) Mn(M,)Mg(M_.) Mg(M ,)Mn(M,) 1 I 1 I 1 I 1 I 1 I 1 I 1 I 1 I 1 I 1 0.0 0.2 0.4 0.6 0.8 1.0 X ( M g ) 124 where k 5 = k 3/k, because K^(Mn-Mg) = K D(Fe-Mg) K D(Fe-Mn) We can t h e r e f o r e p r e d i c t the thermodynamic p r o p e r t i e s of the t e r n a r y s o l u t i o n , knowing only two b i n a r i e s . As an example we may assume we had only the ten s i t e occupancy measurements on the f a y a l i t e - f o r s t e r i t e and on the f a y a l i t e - t e p h r o i t e j o i n . A f i t to the data y i e l d s ln(K D(Fe-Mg)) = 0.237 and ln(R D(Fe-Mn)) = 1.326. We can now c a l c u l a t e ln(K^(Mn-Mg)) = -1.089. C o n s i d e r i n g how few data were used, t h i s i s an e x c e l l e n t p r e d i c t i o n . A f i t to a l l data, assuming k 5 = K 3/k, y i e l d s : l n ( k , ) = ln(k D(Fe-Mn) = 1.332 l n ( k 3 ) = ln(k D(Fe-Mg) = 0.206 model 1 l n ( k 5 ) = ln(k D(Mn-Mg) =-1.126 2.8.3 A SLIGHTLY MORE GENERAL MODEL The most general i d e a l model would assume as the only c o n s t r a i n t that a l l k's are c o n s t a n t . A f i r s t attempt to f i t the data under these c o n d i t i o n s f a i l e d to converge ( l n ( k 3 ) growing i n f i n i t e ) and was l a t e r not repeated because the u n c e r t a i n t i e s of the data do not r e a l l y j u s t i f y too complicated a mixing model. The only e x t e n s i o n we w i l l a l l o w i s t h a t the k D ' s are not equal to the k v a l u e s , but that the r e l a t i o n s k , = k 2 1 , 125 k 3 = k« 1 and k 5 = k i 1 s t i l l h o l d . T h i s assumption i s e q u i v a l e n t to the statement that any i n t e r n a l r e a c t i o n where the s i t e occupancies remain the same has a A rG of zero. ( k r = 1). For the b i n a r i e s , k D w i l l s t i l l be equal to k but the k Q values vary with i n c r e a s i n g c o n c e n t r a t i o n of the t h i r d c a t i o n . The improvement over the independent s i t e mixing model i s that the data of the three b i n a r i e s are much b e t t e r taken i n t o account. The k values obtained a r e : ln ( k , ) = - l n ( k 2 ) = 1.295 l n ( k 3 ) = - l n ( k 4 ) = 0.185 model 2 l n ( k 5 ) = - l n ( k 6 ) =-1.193 T h i s model i s represented by the h o r i z o n t a l l i n e s i n Fig.20. 2.8.4 FITTING THE MODELS TO THE DATA With the u n c e r t a i n t i e s given i n t a b l e 15, there i s no problem to f i t a model which i s c o n s i s t e n t with a l l data. For the "best f i t " the f o l l o w i n g f u n c t i o n was minimized: n Z SORT i=1 (x obs Mg (M1 ) The d i s t r i b u t i o n of s p e c i e s was c a l c u l a t e d with the program THERIAK d e s c r i b e d i n pa r t one of the present t h e s i s . The m i n i m i z a t i o n of the above f u n c t i o n was made with the same 1 26 g r a d i e n t method as used f o r the G-minimization of s o l u t i o n phases i n THERIAK, except that the d e r i v a t i v e s were c a l c u l a t e d by f i n i t e d i f f e r e n c e s . The c a l c u l a t e d occupancies are compared with the measured ones in Table 15. 2.9 CONCLUSIONS Using i n t e g r a t e d powder d i f f r a c t i o n i n t e n s i t i e s , we can measure the s i t e d i s t r i b u t i o n of two elements, p r o v i d e d t h e i r s c a t t e r i n g f a c t o r s are s u f f i c i e n t l y d i f f e r e n t . The method presented y i e l d s u n c e r t a i n t i e s f o r the occupancies i n the order of 0.03. Combined with Mossbauer data i t was p o s s i b l e to determine the s i t e d i s t r i b u t i o n s of Fe, Mn and Mg i n o l i v i n e s . T h e o r e t i c a l l y the method i s not r e s t r i c t e d to o l i v i n e s , and i t might provide u s e f u l i n f o r m a t i o n i f i t were a p p l i e d t o other systems, e.g. the orthopyroxenes. Because the p r e c i s i o n of the method i s l i m i t e d , i t may be important f o r experimental p e t r o l o g i s t s to c o n c e n t r a t e on the growth of l a r g e r c r y s t a l s , so that s i n g l e c r y s t a l s t r u c t u r e refinements are p o s s i b l e . Larger and homogeneous c r y s t a l s would a l s o be very d e s i r a b l e as microprobe standards. To o b t a i n s m a l l e r u n c e r t a i n t i e s d e a l i n g with f i n e g r a i n e d samples, i t may be necessary to i n v e s t i g a t e u sing newer methods, l i k e CHEXE (Smyth and T a f t j ^ ( 1982)) or the e f f e c t s of anomalous d i s p e r s i o n (Waseda (1984)). A f u r t h e r unresolved problem i s that the knowledge of s i t e occupancies d e s c r i b e s only long-range o r d e r i n g , and i t 1 27 should t h e r e f o r e be an aim f o r f u t u r e i n v e s t i g a t i o n s to recognize a l s o short-range o r d e r i n g . In order to i n c o r p o r a t e the knowledge of occupancies i n t o a thermodynamic model, we need a v e r s a t i l e s i t e mixing model and a robust method f o r c a l c u l a t i n g chemical e q u i l i b r i a . The model used to d e s c r i b e the o l i v i n e s o l i d s o l u t i o n s i n my t h e s i s i s the " s p e c i a t i o n " model a c c o r d i n g to Brown and Greenwood ( i n p r e p a r a t i o n ) . T h i s model c o n s i d e r s each c o n f i g u r a t i o n of a small s t r u c t u r a l u n i t as a s p e c i e s , s i m i l a r to the ones used i n gas mixing models. (Compare a l s o chapters 2.8.1. and 1.2.2.). The method f o r computing chemical e q u i l i b r i a presented i n the f i r s t p a r t of my t h e s i s ("THERIAK"), uses a n o n - l i n e a r programming technique (step 1 of the a l g o r i t h m ) , which i s s p e c i a l l y designed to be used with n o n - i d e a l " s p e c i a t i o n " models. The combination "THERIAK"/"speciation" pro v i d e s t h e r e f o r e a powerful t o o l f o r m o d e l l i n g even very complex mineral e q u i l i b r i a . BIBLIOGRAPHY Annersten H., Ad e t u n j i J . and F i l i p p i d i s A.(1984): C a t i o n o r d e r i n g i n Fe-Mn s i l i c a t e o l i v i n e s . Amer. Mineral. 69: 1110- 1115 Annersten H., E r i c s s o n T. and F i l i p p i d i s A.(1982): C a t i o n o r d e r i n g i n Ni-Fe o l i v i n e s . Amer. Mineral. 67: 1212-1217 Bancroft G.M. and Brown J.R.(1975): A Mossbauer Study of C o e x i s t i n g Hornblendes and B i o t i t e s : Q u a n t i t a t i v e F e 3 + / F e 2 + R a t i o s . Amer. 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Lett. 9/9:1113-1116 139 Sundaram V.S., Gupta V.P. and Subbarao E.C.(1971): A M i n i a t u r e Vacuum Furnace f o r Mossbauer Spectroscopy. Rev. Sci. Instr. 42: 1616-1618 T h i e r r y P., C h a t i l l o n - C o l i n e t C , Mathieu J.C., Regnard J.R. and Amosse J.O981): Thermodynamic P r o p e r t i e s of the F o r s t e r i t e - F a y a l i t e (Mg 2SiO,-Fe 2SiO„) S o l i d S o l u t i o n . Determination of Heat of Formation. Phys. Chem. Minerals 7:43-46 Thompson J.B, J r and Hovis G.L.Q 979): Entropy of mixing i n s a n i d i n e . Amer. Mineral. 64: 57-65 Thompson P. and Wood I.G.(1983): X-ray R i e t f e l d Refinement using Debye-Scherrer Geometry. J. Appl. Cryst. 16: 458-47 2 Tokonami M.Q965): Atomic s c a t t e r i n g f a c t o r s f o r O 2". Acta crystallogr. 19:486 Toroya H.(1986): Whole-Powder-Pattern-Fitting without Reference to a S t r u c t u r a l Model: A p p l i c a t i o n to X-ray Powder D i f f T a c t o m e t e r Data. /. Appl. 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Spri nger-Verl ag N.Y., 183 pp. Wenk H.-R. and Raymond K.N.(1973): Four new s t r u c t u r e refinements of o l i v i n e . Zeitschr. Kristallogr. Bd. 137': 86-105 White W.B., Johnson S.M. and Da n t z i q G.B . 0958) : Chemical E q u i l i b r i u m i n Complex M i x t u r e s . /. Chem. Phys. 28: 751-755 W i l l G. and Nover G.(1979): I n f l u e n c e of Oxygen P a r t i a l Pressure on the Mg/Fe D i s t r i b u t i o n i n O l i v i n e s . Phys. Chem. Minerals 4: 199-208 (6: 247-248) W i l l G., P a r r i s h W. and Huang T.C.Q983): C r y s t a l - S t r u c t u r e Refinement by P r o f i l e F i t t i n g and Least-Squares A n a l y s i s of Powder D i f f T a c t o m e t e r Data. /. Appl. Cryst. 16:611-622 Wood B.J, and Fr a s e r D.G.(1976): Elementary Thermodynamics f o r G e o l o g i s t s . Oxford University Press, 303 p. Yoder H.S.Jr. and Sahama Th.G . ( l957): O l i v i n e X-Ray d e t e r m i n a t i v e curve. Amer. Mineral. 42: 475-491 Yvon K., J e i t s c h k o W. , Parthe E . 0 977); LAZY PULVERIX, a computer program f o r c a l c u l a t i n g X-ray and neutron d i f f r a c t i o n powder p a t t e r n s . /. Appl. Cryst. 10:73-74 Z e l e z n i k F . J , and Gordon S.(1968): C a l c u l a t i o n of Complex Chemical E q u i l i b r i a . Ind. Eng. Chem. 60/6: 27-57 APPENDIX A: DESCRIPTION OF INPUT FOR THE PROGRAM THERIAK A.1 PROBLEM INPUT The minimum input c o n s i s t s of two l i n e s (the f i r s t and the l a s t ) . These provide i n f o r m a t i o n about the temperature, the pre s s u r e , the bulk composition, the d e s i r e d output and the phases to be c o n s i d e r e d . f i r s t l i n e : TC P TC: Temperature i n °C P : Pressure i n bars l a s t l i n e : PRTCODE FORMULA USE PRTCODE: Allows a c h o i c e betweeen d i f f e r e n t outputs: <-1: p r i n t i n f o r m a t i o n about s e l e c t e d or r e j e c t e d phases from the data-base. NO EQUILIBRIUM CALCULATED =-1: p r i n t composition, c o n s i d e r e d phases and s o l u t i o n models. NO EQUILIBRIUM CALCULATED = 0: short output ( s t a b l e paragenesis) = 1: long output (composition, c o n s i d e r e d phases, s o l u t i o n models, s t a b l e paragenesis, a c t i v i t i e s of a l l c o n s i d e r e d phases 141 142 FORMULA: Bulk composition i n the form: component(value) component(value)... the values can be any numbers, but should be p o s i t i v e to av o i d u n p r e d i c t a b l e r e s u l t s . The maximum le n g t h of FORMULA i s 170 c h a r a c t e r s , i t may c o n t a i n s i n g l e blanks and must be separated from PRTCODE and USE by at l e a s t two blanks. USE: Code f o r s e l e c t i n g data from the data-base. For more i n f o r m a t i o n see A.2.2. Two examples of minimum i n p u t : 700 3000 -2 SI(1.0)AL(2.0)0(5.0) N 550.00 2000.0 1 MN(1) SI(8) CA(2) 0(19) 143 o p t i o n a l input Between the f i r s t and the l a s t l i n e , o p t i o n a l l i n e s may be i n s e r t e d . They c o n t a i n one or more statements i n the form: parameter=value separated by at l e a s t two blanks. The f i r s t e i g h t of the f o l l o w i n g p o s s i b l e parameters d e f i n e the s i z e s of a r r a y s and s t r u c t u r e s . They can be changed at execution time only i n the PL/1 v e r s i o n . In FORTRAN these have to be changed before c o m p i l a t i o n of the program. parameter d e f a u l t COMAX 15 PHMAX SOMAX EMMAX SITEMAX MAMAX MPOLMAX 100 10 Maximum number of components. (Can be l e s s than the number of components i n the database) Maximum number of c o n s i d e r e d n o n - s o l u t i o n phases. Maximum number of c o n s i d e r e d s o l u t i o n phases. Maximum number of endmembers f o r each s o l u t i o n phase. Maximum number of d i f f e r e n t s i t e s f o r i d e a l s i t e mixing model. Maximum number of margules parameters f o r each s o l u t i o n phase, Maximum degree of polynom f o r margules eq u a t i o n . 144 CALMAX 150 L01 MAX 80 TEST 10" 8 EQUALX 10 - 5 PGAS 1 LPP 60 DELXMIN 10" 7 DELXSCAN 1 DELXSTAR 0.02 STEPSTAR 2 Maximum number of phases to be kept s i m u l t a n o u s l y i n matrix ||X||. Maximum number of i t e r a t i o n s f o r the c a l c u l a t i o n . If LO1MAX < 0 then the number of i t e r a t i o n s w i l l be e x a c t l y |L01MAX|. The c a l c u l a t i o n s w i l l stop, i f the ab s o l u t e sum of a l l n o n - p o s i t i v e g^ a f t e r the l i n e a r programming s e c t i o n i s s m a l l e r than |TEST|. If TEST < 0 then the program p r i n t s a short summary f o r each i t e r a t i o n . Two phases of the same s o l u t i o n are c o n s i d e r e d equal i f E|Ax^| < EQUALX. The pressure f o r f l u i d phases i s P*PGAS. L i n e s per page, (has an e f f e c t only i n the PL/1 v e r s i o n ) . ^min : s m a l l e s t p o s s i b l e s t e p s i z e . = p r e c i s i o n f o r c a l c u l a t i n g x^'s i n non - i d e a l s o l u t i o n s . G r i d width f o r scanning n o n - i d e a l s o l u t i o n phases f o r i n i t i a l guess. ^ i n i t : ^•n^t^a-'- s t e p s i z e . Maximum number of steps, i f i n i t i a l guess r e s u l t s from a scan. 1 45 STEPMAX 3 Maximum number of steps i f i n i t i a l guess i s a p r e v i o u s minimum. GCMAX 100 Maximum number of AG f u n c t i o n c a l l s per step. (= maximum m i n s t e p s i z e procedure) Examples f o r o p t i o n a l i n p u t : TEST=-1E-7 LO1MAX=200 GCMAX=15 MPOLMAX=4 MAMAX=32 EMAX=10 A.2 THE DATA-BASE The format of the data-base f o l l o w s c l o s e l y the one used by E.H. P e r k i n s , R.G. Berman and T.H. Brown f o r the c a l c u l a t i o n of phase diagrams. The f i r s t few l i n e s of the data-base c o n t a i n g e n e r a l i n f o r m a t i o n on the components to be used : NC R COMPN(l) COMPN(2) COMPN(8) • • • COMPN( ) COMPN( ) COMPN(NC) MOLWT(l) MOLWT(2) MOLWT(8) • • * MOLWT( ) MOLWT( ) MOLWT(NC) NC: Number of components used i n the data-base. R : Gas constant [J/mol] 1 46 COMPN(i) : Components. (Format = 8A8) MOLWT(i) : M o l e c u l a r weights. (Format = 8F8.2) The r e s t of the data-base i s o r g a n i z e d i n s e c t i o n s . Each s e c t i o n begins with a l i n e having the s t r i n g '***' as the f i r s t three non-blank c h a r a c t e r s . The t e x t of that l i n e i s scanned f o r a s t r i n g d e f i n i n g the s e c t i o n . The s e c t i o n s r e c o g n i z e d by the program a r e : *** ... MINERAL DATA ... *** ... GAS DATA ... These two are t r e a t e d i d e n t i c a l l y . They c o n t a i n the phase d e f i n i t i o n s and the i n f o r m a t i o n to c a l c u l a t e A^G f o r v a r i a b l e p r e s s u r e s and temperatures. *** ... SOLUTION DATA ... D e f i n i t i o n s of s o l u t i o n s and s o l u t i o n models. *** ... MARGULES ... T h i s s e c t i o n c o n t a i n s Margules parameters for n o n - i d e a l s o l u t i o n phases. Other s e c t i o n s (e.g. *** ... COMMENTS ...) w i l l be skipped by the program. P r i n c i p a l l y the s e c t i o n s can be i n any order and may appear more than once. But i t i s obvious that i f we d e f i n e a s o l u t i o n phase, i t s endmembers must have been read i n p r e v i o u s l y . S i m i l a r l y the Margules parameters f o r a s o l u t i o n phase can only be a s s i g n e d p r o p e r l y a f t e r the s o l u t i o n phase i s d e f i n e d . Any l i n e anywhere, which i s empty or begins with a '!' i s c o n s i d e r e d a comment l i n e and i s skipped. 147 A.2.1 SECTION *** ... MINERAL DATA ... Each phase i s d e f i n e d with one phase d e f i n i t i o n l i n e and any number of data l i n e s . phase d e f i n i t i o n l i n e NAME FORMULA ABBREV [ CODE ] The four s t r i n g s are separated by at l e a s t two blanks. They a l l may c o n t a i n s i n g l e blanks. NAME : Name of the phase, (maximum 16 c h a r a c t e r s ) FORMULA : Chemical formula of the phase i n the form: component(value) component(value) ... The components have to be the same as d e f i n e d at the beginning i f the data-base. The values can be n e g a t i v e . ABBREV : A b b r e v i a t i o n f o r phase (maximum 8 c h a r a c t e r s ) CODE : [ o p t i o n a l ] (maximum 15 c h a r a c t e r s ) Each c h a r a c t e r i n CODE d e f i n e s a group to which the phase belongs. If the f i r s t c h a r a c t e r i s a '*' then the phase i s c o n s i d e r e d " s p e c i a l " . For more d e t a i l s see A.2.2. A l i n e w i t h i n t h i s s e c t i o n i s recognized to be a phase d e f i n i t i o n l i n e i f i t c o n t a i n s at l e a s t one ' ( ' ( l e f t 1 48 p a r a n t h e s i s ) and i s not a comment l i n e . Examples of phase d e f i n i t i o n l i n e s : HIGH ALBITE NA(1)AL(1)SI(3)0(8) H-ALB N ALMANDINE FE(3) AL(2) SI(3) 0(12) aim Data l i n e s The thermodynamic data f o r a phase f o l l o w the phase d e f i n i t i o n l i n e i n any ord e r . The f i r s t three non-blank c h a r a c t e r s i d e n t i f y the content of the input l i n e . These three c h a r a c t e r s must be w i t h i n the f i r s t f i v e columns of the l i n e . Columns 6 to 80 c o n t a i n at most f i v e f i e l d s of 15 columns with numerical data. (FORMAT = A5,5F15.4) 1. Standard s t a t e i n f o r m a t i o n . (25 °C, 1 b a r ) . A^G 0 i s not used and i t s f i e l d can be l e f t blank. ST A fG°[J/mol] A fH°[J/mol] S°[J/K.mol] V°[cm 3/mol] 2. Heat c a p a c i t y i n f o r m a t i o n : cp = k1 + k2«T + k3/T 2 + k4//T + k5«T 2 + k6/T + k7-/T + k8/T 3 + k9-T 3 [j/mol] CP1 k1 k4 k3 k8 CP2 k6 k2 k5 k7 k9 CP3 k1 k2 k3 1 49 3. Equation of s t a t e f o r gases (used to c a l c u l a t e /Vdp). I f the A^G c a l c u l a t i o n shows that the phase i s l i q u i d the name of the phase w i l l be changed t o : name(LIQ). a = a 0 + a,-T [J.cm 3] b = b 0 + b,«T [cm 3] Van der Waals: P = (R-T)/(V-b) + a/V 2 Redlich-Kwong: P = (R«T)/(V-b) - a/(i/T»V' (V+b) ) VDW a 0 a, b 0 b, R-K a 0 a, b 0 b. 4. Lambda t r a n s i t i o n s . Between the l i m i t s T e and T. a ret t r C p - l i k e f u n c t i o n i s added: C p t r = (T-AT).[1, + 1 2 - ( T - A T ) ] 2 where T r e f = T ° e f + AT T t r - T J r + AT and AT = (P-P 0)«T In a d d i t i o n AH A v t r / (5V/5T) and (6V/6P) are c o n s i d e r e d . TR1 T ° r T ° e f 1, 1 2 AH f c r TR2 T g AV f c r (6V/5T) (6V/5P) 5. Di s o r d e r c o n t r i b u t i o n s to A^G can be approximated by adding a C p - l i k e f u n c t i o n between the l i m i t s T Q and T^. d i s = d1 + d2-T + d3/T 2 + d4/i/T + d5-T 2 + d6/T + d7«i/f + d8/T 3 + d9-T 3 [J/mol] The volume i s taken from the shape of the enthalpy f u n c t i o n and i s s c a l e d to the data by the v a r i a b l e V 150 D1 dl d4 d3 V a d d6 D2 d2 d5 T T, o d 6. S p e c i a l phases: A^G at p and T i s c a l c u l a t e d i n an e x t e r n a l s u b r o u t i n e . A l l other data l i n e s w i l l be ignored. (Used e.g. to c a l c u l a t e A^G f o r water and steam a c c o r d i n g to Haar et a l . ( 1 9 8 2 ) ) . The Keyword can be maximal 10 c h a r a c t e r s long. (See a l s o A.3) SPC Keyword 7. To the A^G d e f i n e d by the standard s t a t e , Cp's e t c . we can add any combination of p r e v i o u s l y c a l c u l a t e d A^G's. T h i s i s u s e f u l e.g. f o r polymorphs or b u f f e r s . COM name[value] name[value] ... 151 A . 2 . 2 USE AND CODE (SELECTING PHASES) The phases i n the d a t a - b a s e may be d i v i d e d i n t o d i f f e r e n t g r o u p s . Each group i s coded as one c h a r a c t e r ( e . g . : A , B , C , D ) . The v a r i a b l e CODE i n the d a t a - b a s e [ o p t i o n a l ] d e f i n e s f o r each phase to which groups i t b e l o n g s . When the program i s run we can s p e c i f y i n the v a r i a b l e USE a l l groups tha t s h o u l d be c o n s i d e r e d . If USE = ' * ' then a l l groups are c o n s i d e r e d , i n c l u d i n g those w i t h no CODE d e f i n e d . s p e c i a l phases : I f The f i r s t c h a r a c t e r of CODE i s a ' * ' then the phase i s " s p e c i a l " ( e . g . a b u f f e r ) and does not be long to a s p e c i f i c group . I t w i l l o n l y be c o n s i d e r e d i f mentioned i n USE a f t e r a comma. The f i r s t comma in USE i s a d e l i m i t e r between the group names and the s p e c i a l phases . I f the f i r s t c h a r a c t e r of CODE i s a '+' then A f G of t h i s phase w i l l be c a l c u l a t e d even i f i t s c o m p o s i t i o n i s o u t s i d e the c o m p o s i t i o n a l space . In t h a t case t h i s phase can s t i l l be used to d e f i n e A^G of o ther phases , but w i l l i t s e l f not be c o n s i d e r e d f o r the e q u i l i b r i u m assemblage . In g e n e r a l a phase w i l l be c o n s i d e r e d i f : 1 . The group code i s matching 2. no phase w i t h i d e n t i c a l name i s a l r e a d y c o n s i d e r e d 3. i t s components are a subsystem of the bu lk components 4. enough da ta to c a l c u l a t e A f G i s p r o v i d e d . 153 Examples f o r USE and CODE: CODE USE Considered phases Phase 1 A A 1,4 Phase 2 BC B 2 Phase 3 C C 2,3,4 Phase 4 AC AB 1 ,2,4 Phase 5 AC 1 ,2,3,4 Phase 6 *QFM BC 2,3,4 Phase 7 *CB ABC 1,2,3,4 * 1,2,3,4,5 AB,QFM 1,2,4,6 AB,QFMr CB 1,2,4,6,7 ABC ,CB 1,2,3,4,5,7 A,CB 1 ,4,7 A.2.3 SECTION *** . . . SOLUTION DATA ... A s o l u t i o n phase i s d e f i n e d by one d e f i n i t i o n l i n e f o l l o w e d by any number of endmember l i n e s . s o l u t i o n d e f i n i t i o n l i n e SOLNAME (MODELL)[/DIV] N M 1 M 2 . .. M A l i n e w i t h i n t h i s s e c t i o n i s recog n i z e d to be a s o l u t i o n d e f i n i t i o n l i n e i f i t c o n t a i n s the s t r i n g ' (' ( b l a n k , b l a n k , l e f t p a r e n t h e s i s ) and i s not a comment l i n e . 154 SOLNAME : Name f o r s o l u t i o n (maximum 16 c h a r a c t e r s ) MODELL : T h i s s t r i n g i s scanned f o r c e r t a i n keywords: SKIP : The s o l u t i o n phase w i l l not be c o n s i d e r e d . IDEAL : An i d e a l s o l u t i o n model i s assumed. (a^=x^). T h i s i s a l s o the d e f a u l t i f no keyword i s r e c o g n i z e d . SITE : An i d e a l s i t e mixing model i s used. N : Number of d i f f e r e n t s i t e s ( i n t e g e r , can be z e r o ) . M^ : M u l t i p l i c i t y of s i t e i ( i n t e g e r ) . It i s assumed th a t the endmembers have the s i t e i occupied by M^ i d e n t i c a l elements or element groups. EXT : The a c t i v i t i e s of the endmembers are d e f i n e d i n an e x t e r n a l s u b r o u t i n e . A l l other keywords (except SKIP and MARGULES) N and M have no i n f l u e n c e . (See a l s o A.4. ) MARGULES : T h i s keyword s t a t e s , t h at there may be Margules parameters d e f i n e d f o r t h i s phase. They w i l l only be added i f t h i s keyword appears i n MODELL. /DIV : I f a s l a s h f o l l o w s immediately the (MODELL), then the number f o l l o w i n g i t i s used to c a l c u l a t e f r a c t i o n a l m u l t i p l i c i t i e s . A l l m u l t i p l i c i t i e s are d i v i d e d by DIV. Comments: N, M^ and DIV are i n t e g e r s by d e f i n i t i o n , but read as r e a l . I f decimal f r a c t i o n s are entered the program w i l l c a l c u l a t e a c c o r d i n g l y , but p r i n t a wrong model. N and M̂  are not o p t i o n a l , i f not used at l e a s t N=0 has to be d e f i n e d . 1 55 Endmember l i n e s EMNAME E , E - . . . E EMNAME : Name of the endmember. (maximum 16 c h a r a c t e r s ) . The endmember w i l l be added to the s o l u t i o n i f a c o n s i d e r e d phase with the same name e x i s t s . E^ : Element on s i t e i . (Maximum 8 c h a r a c t e r s ) . The element names do not have to correspond to a co m p o s i t i o n a l element. They are only used t o f i n d out which endmembers have which s i t e s occupied by the same element or element group. Examples: GAS PHASE (IDEAL) 0 CARBON DIOXYDE STEAM OXYGEN CARBON MONOXYDE HIGH FELDSPARS (IDEAL,MARGULES) 0 HIGH ALBITE HIGH SANIDINE GARNETS (SITE) 2 3 2 PYROPE MG AL ALMANDINE FE AL SPESSARTINE MN AL GROSSULAR CA AL ANDRADITE CA FE For the above examples the program w i l l p r i n t the models as shown on the f o l l o w i n g page. 15G Examples of printed models SOLUTION PHASES: 1 GAS PHASE 1 CARBON DIOXYDE ( 29) 2 STEAM ( 96) 3 OXYGEN ( 73) 4 CARBON MONOXYDE ( 30) SOLUTION MODELL: IDEAL ONE SITE MIXING A(C02) = X(C02) A(H20) = X(H20) A(02) = X(02) A(CO) = X(CO) HIGH FELDSPARS 1 HIGH ALBITE 2 HIGH SANIOINE MARGULES PARAMETERS: SOLUTION MODELL: IDEAL ONE SITE MIXING + MARGULES TYPE EXCESS FUNCTION ( 14) A(H-ALB) = X(H-ALB) ( 88) A(H-SAN) = X(H-SAN) W(112) = 18948.06 W(122) = 10322.68 GARNET 1 PYROPE ( 81) 2 ALMANDINE ( 15) 3 SPESSARTINE ( 94) 4 GROSSULAR ( 50) 5 ANDRADITE ( 18) SOLUTION MODELL: "IDEAL" 2 SITE MIXING 3 2 A(PYROPE) = [ X(PYROPE) ] [ X(PYROPE) + X(ALM) + X(SPESS) + X(GROSS) ] 3 2 A(ALM) = [ X(ALM) ] [ X(PYROPE) + X(ALM) + X(SPESS) + X(GROSS) ] 3 2 A(SPESS) = t X(SPESS) ] [ X(PYROPE) + X(ALM) + X(SPESS) + X(GROSS) ] 3 2 A(GROSS) = [ X(GROSS) + X(ANDR) ] [ X(PYROPE) + X(ALM) + X(SPESS) + X(GROSS) ] 3 2 A(ANDR) = [ X(GROSS) + X(ANDR) ] [ X(ANDR) ] 157 A.2.4 SECTION *** ... MARGULES ... The Margules parameters are grouped i n subsystems ( b i n a r y , t e r n a r y e t c . ) . Each subsystem i s d e f i n e d by a d e f i n i t i o n l i n e , f o l l o w e d by any number of parameter l i n e s . Margules d e f i n i t i o n l i n e EMNAME1 - EMNAME2 - ... A l i n e w i t h i n t h i s s e c t i o n i s recog n i z e d to be a d e f i n i t i o n l i n e i f i t c o n t a i n s the s t r i n g ' - ' (blank,minus,blank) EMNAMEn : Name of n t n endmember i n subsystem. parameter l i n e s k , k 2 k 3 . . . WH WS WV WCP The number of k^'s d e f i n e s the degree of the polynomial i n the Margules eq u a t i o n . T h i s does not have to be the same f o r a l l parameters. WG = WH - T-WS + p-WV + ... G E X = WG • x ( k 1 ) . x ( k 2 ) - x ( k 3 ) ... where the k^'s r e f e r to the i n d i c e s i n the d e f i n i t i o n l i n e . Example: HIGH ALBITE - HIGH SANIDINE 112 320,98.8 16.1356 0.46903 0 122 26470.9 19.3807 0.38702 0 158 A.3 CALCULATION OF A £G IN AN EXTERNAL SUBROUTINE If A^G i s to be c a l c u l a t e d i n an e x t e r n a l subroutine, (data l i n e : SPC Keyword), then the program c a l l s a subroutine named GSPEC. The parameters t r a n s m i t t e d a r e : NAME(name of phase), P ( p r e s s u r e ) , PGAS(pressure of f l u i d phases), T(temperature), CASE(=Keyword) and G(A^G). The subroutine c a l c u l a t e s G (or may read i t from a t e r m i n a l ) and re t u r n s c o n t r o l to the main program. Example of GSPEC i n PL/1: GSPEC:PROCEDURE(NAME,P,PGAS,T,CASE, G) ; DCL (NAME) CHAR0 6) VARYING; DCL (CASE) CHAR(10) VARYING; DCL (P,PGAS,T,G) DECIMAL FL0AT(16); DCL (WHAAR) ENTRY; G=0.0; IF CASE='HAAR' THEN CALL WHAAR(NAME,PGAS,T,G); RETURN; END GSPEC; 159 A.4 CALCULATION OF ACTIVITIES IN AN EXTERNAL SUBROUTINE If the model of a s o l u t i o n phase c o n t a i n s the s t r i n g 'EXT' then three subroutines (or three e n t r y - p o i n t s ) w i l l be c a l l e d : 1. PSOLINI. In t h i s subroutine we d e f i n e f o r a given s o l u t i o n phase the number and the names of the endmembers. 2. PSOLMOD. T h i s subroutine r e t u r n s a s t r i n g to be p r i n t e d , and which d e s c r i b e s the a c t i v i t y a c c o r d i n g to s o l u t i o n name and endmember number, ( o p t i o n a l ) 3. PSOLCAL. For each endmember the a c t i v i t y i s c a l c u l a t e d . The s o l u t i o n has to be d e f i n e d i n the data-base, but may have fewer endmembers and i n a d i f f e r e n t order than the same s o l u t i o n i n the e x t e r n a l s u b r o u t i n e s . The c a l l i n g parameters are r e l a t i v e l y complex, but to add a new s o l u t i o n phase to the alre a d y e x i s t i n g s u b r o u t i n e s , only few l i n e s have to be i n s e r t e d : Example ( f o r PL/1) in PSOLINI: IF SOLNAME='name of s o l u t i o n phase' THEN DO; N=number of endmembers; NAME(1)='name of endmember 1'; NAME(2)='name of endmember 2'; NAME(n)='name of endmember n'; END; 160 i n PSOLMOD: ( D e f a u l t : MODELL='NOT EXPLICITLY DEFINED';) IF SOLNAME='name of s o l u t i o n phase' THEN DO; IF K=1 THEN MODELL='activity of endmember 1'; IF K=2 THEN MODELL='activity of endmember 2'; IF K=n THEN MODELL='activity of endmember n'; END; in PSOLCAL: The independent v a r i a b l e s i n the formulas can be: P,T,X(1),X(2),..,X(n) IF SOLNAME='name of s o l u t i o n phase' THEN DO; A(l)=formula f o r a c t i v i t y of endmember 1; A(2)=formula f o r a c t i v i t y of endmember 2; A(n)=formula f o r a c t i v i t y of endmember n; END; APPENDIX B: LISTING OF PROGRAM THERIAK The major C O M M O N blocks are only listed in the main routine. INTEGER*4 CALMAX,COMAX,EMAX,MAMAX,MPOMAX,PHMAX,SITMAX,SOMAX PARAMETER (COMAX = 22,PHMAX = 150,SOMAX = 20,EMAX = 10,SITMAX = 5, * MAMAX = 31 ,MPOMAX = 4,CALMAX = 200) LOGICAL*4 MORE INTEGERS I001,I002,1,I2,II,COMAY,LLL,LPP REAL*8 FAR001(11),FF,MOLGEW(COMAX) CHARACTER*250 CH001,CH002,SYREC CHARACTER*8 CAR001(11) C C O M M O N BLOCK FOR SAVING INPUT DATA INTEGER*4 INDATA( 12,PHMAX) C O M M O N /DAIN/ INDATA REAL*8 REDATA(75,PHMAX) C O M M O N /DARE/ REDATA CHARACTER*!6 CHDATA(PHMAX) C O M M O N /DACH/ CHDATA LOGICAL*4 LODATA(5,PHMAX) C O M M O N /DALO/ LODATA C-—GLOBAL C O M M O N BLOCK REAL * 8 P, PGAS, PRAT, PO, R, RT,T,TC,T0 C O M M O N /SURE/ P,PGAS,PRAT,P0,R,RT,T,TC,T0 C — - C O M M O N BLOCK FOR DATABASE 1NTEGER*4 CHMCOD(COMAX),DIM,LUSE,NC,NPAR C O M M O N /ININ/ CHMCOD,DIM,LUSE,NC,NPAR REAL*8 CHE(COMAX),CHEM(COMAX),GGK(PHMAX),BULK(COMAX) C O M M O N /INRE/ CHE,CHEM,GGK,BULK CHARACTER*250 REC CHARACTER* 170 FORMUL CHARACTER*60 M O D C O D CHARACTER*25 CH,USE CHARACTER* 16 MANAM(EMAX),SONAM CHARACTER*8 ALPDIV(SOMAX +1 ),FORM,OXYDE(COMAX) CHARACTER*5 SITEL(SOMAX + 1,EMAX,SITMAX) CHARACTER*3 RECODE C O M M O N /INCH/ MANAM,SONAM,ALPDIV,FORM,OXYDE, >SITEL,RECODE,REC,FORMUL,MODCOD,CH,USE LOGICAL*4 GENUG(8),MARCOD(SOMAX+1),NULL(PHMAX+1) C O M M O N /INLO/ GENUG,MARCOD,NULL C — - C O M M O N BLOCK FOR G-CALCULATION INTEGER*4 NLANDA,NCOM,ICOM(10) C O M M O N /GCIN/ NLANDA,NCOM,ICOM REAL*8 AAT,AA0,ASPK(4),BBT,BB0,BSPK(4),DHTR(4),DVDP(4), >DVDT(4),DVTR(4),D1,D2,D3,D4,D5,D6,D7,D8,D9,GR,G0R,H0R, > K1, K2,K3, K4, K5, K6,K7, K8, K9,SQT,SQT0,S0R,TDMAX,TD0,TEQ(4), >TQ1 B(4),TRE(4),TT,TT0,VAD),V0R,FFCOM(10) C O M M O N /CCRE/ AAT,AA0,ASPK,BBT,BB0,BSPK,DHTR,DVDP, >DVDT,DVTR,D1,D2,D3,D4,D5,D6,D7,D8,D9,GR,G0R,H0R, > K1, K2, K3, K4, K5, K6, K7, K8, K9, SQT, SQT0, SO R,TDMAX,TD0,TEQ, >TQ1 B,TRE,TT,TT0,VADJ,V0R,FFCOM 161 162 CHARACTERS 6 CASE,NAM C O M M O N /CCCH/ CASE,NAM LOGICAL*4 DIS,RDK,VDW,LIQ,SPC,COM C O M M O N /CCLO/ DIS,RDK,VDW,L!Q,SPC,COM C — - C O M M O N BLOCK FOR THERIAK INTECERM EM(SOMAX + 1,EMAX),EMBCOD(SOMAX+1,EMAX),EMCODE(0:CALMAX), >EMNR(PHMAX),EMSOL(PHMAX),EQEM(SOMAX + 1,EMAX,SlTMAX,EMAX),CCMAX, > I NDX(SOMAX +1 ,MAMAX,MPOMAX),L001, LOI MAX, N EMBAS(SOMAX +1), >NEND(SOMAX+1),NEQEM(SOMAX+1,EMAX,SITMAX),NMARG(SOMAX+1),NPHA, >NSITE(SOMAX+1),NSOL,NUMMER(0:CALMAX),NUN,NUN2, > POLY(SOMAX +1 ,MAMAX), >PRTCOD,QQ(SOMAX+1,MAMAX,EMAX),STPMAX,STPSTA,SUGC(0:CALMAX),SUCNR C O M M O N /THIN/ EM,EMBCOD,EMCODE,EMNR,EMSOL,EQEM,CCMAX,INDX,L001, >L01MAX,NEMBAS /NEND,NEQEM,NMARC,NPHA,NSITE,NS0L,NUMMER,NUN,NUN2, >POLY,PRTCOD,QQ,STPMAX,STPSTA,SUCC,SUCNR REAL*8 ALPHA(SOMAX + 1),DXMIN,DXSCAN,DXSTAR,EQUALX,G(0:CALMAX), >CC(PHMAX),NN(0:CALMAX),SITMUL(SOMAX + 1,SITMAX),TEST, > WG(SOMAX +1 ,MAMAX),WH(SOMAX +1 ,MAMAX),WS(SOMAX +1 ,MAMAX), > WV(SOMAX +1 ,MAMAX), WCP(SOMAX +1, MAM AX), >X(0:CALMAX,COMAX),XEM(0:CALMAX,EMAX),XX(PHMAX,COMAX) C O M M O N /THRE/ ALPHA,DXMIN,DXSCAN,DXSTAR,EQUALX,C,CC,NN,SITMUL, >TEST,WG,WH,WS,WV,WCP,X,XEM,XX CHARACTER* 16 NAME(PHMAX),SOLNAM(SOMAX+1) CHARACTER*8 ABK(PHMAX),CHNAME(COMAX) CHARACTERS MODELL(SOMAX+1) C O M M O N /THCH/ NAME,SOLNAM,ABK,CHNAME,MODELL LOGICAL*4 PRTLOG(8) C O M M O N /THLO/ PRTLOG C—-END OF C O M M O N VARIABLES C — CHARACTER*3 YESNO C — DATA CAR001/'PCAS'/LO1MAX','EQUALX'/TEST','DELXMIN', *'DELXSCAN'/DELXSTAR','STEPSTAR','STEPMAX','GCMAX','LPP7 DATA FAR001/1.0D0,80.0,1D-5,1D-8,1D-7, *1.0D0,2D-2,2.0,3.0,100.0,60.0/ COMAY = COMAX LLL = 0 C-—READ T AND P FROM UNIT 5 MORE = .TRUE. READ (UNIT=9,FMT = '(A250)') SYREC CALL CELI(SYREC,TC) CALL CELI(SYREC,P) C—--READ OOPTIONAL NEW DEFAULT VALUES FROM UNIT 9 C THE FOLLOWING RECORD IS STORED IN SYREC 1001 IF (MORE) THEN READ (UNIT=9,FMT = '(A250)') SYREC MORE = (!NDEX(SYREC,'(').EQ.0) IF (MORE) THEN DO 501,12=1,11 I002 = INDEX(CAR001(I2),' ') IF (I002.EQ.0) 1002 = 9 I001=INDEX(SYREC,CAR001(I2)(1:I002-1)) IF (1001.NE.0) THEN CH002 = SYREC(1001:) I001=INDEX(CH002,' = ') CH001=CH002(I001+1:) CALL FIBLA(CH001,I001) CH002 = CH001(1001:) I001=INDEX(CH002,' ') CH001=CH002(1:1001-1) READ (UNIT=CH001,FMT = '(BN,D16.0)') FAR001(I2) END IF 501 CONTINUE END IF ELSE GOTO 1 END IF GOTO 1001 1 PRAT=FAR001(1) L01MAX=1DINT(FAR001(2)) EQUALX=FAR001(3) TEST=FAR001(4) DXMIN = FAR001(5) DXSCAN = FAR001(6) DXSTAR=FAR001(7) STPSTA = IDINT(FAR001 (8)) STPMAX=IDINT(FAR001(9)) CCMAX = IDINT(FAR001(10)) LPP = IDINT(FAR001(11)) IF (DXMIN.LE.0.0) DXMIN = 1D-7 IF (DXSCAN.LT.0.001) DXSCAN = 1.0D-3 IF (DXSCAN.GT.1.0) DXSCAN = 1.0D0 IF (DXSTAR.LE.0.0) DXSTAR=DXSCAN/10.0D0 IF (LPP.LT.30) LPP=30 C C READ NC AND R FROM UNIT 8 READ (UNIT = 8,FMT = '(A250)') REC CALL GELI(REC,FF) NC = lDINT(FF) IF (NC.CT.COMAX) THEN WRITE (UNIT=6,FMT = '(" COMAX = ", l5/" NC = ",I5)') COMAX, STOP END IF CALL GELI(REC,R) T0 = 298.15D0 P0 = 1.0D0 TT0=T0*T0 SQTO = DSQRT(TO) C READ ELEMENTS AND MOLCEW FROM UNIT 8 DO 502,1 = 1,NC,8 l001=MIN0(NC,l + 7) 502 READ (UNIT=8,FMT='(8A8)') (OXYDE(II),II = I,I001) DO 503,1 = 1,NC,8 l001=MINO(NC,l + 7) 503 READ (UNIT=8,FMT = '(8F8.2)') (MOLGEW(II),II = I,I001) C—-- C—-READ PRTCOD, FORMUL AND USE FROM SYREC C-—SET UP FIRST NUN COLUMNS OF MATRIX CALL CELI(SYREC,FF) PRTCOD = IDINT(FF) DO 650,1 = 1,8 650 PRTLOG(i) = .FALSE. IF (PRTCOD.LE.-2) PRTLOC(1) = .TRUE. IF (PRTCOD.EQ.-1) THEN DO 652,1 = 2,5 652 PRTLOC(l) = .TRUE. END IF IF (PRTCOD.EQ.0) THEN DO 654,1 = 5,6 654 PRTLOC(I) = .TRUE. END IF IF (PRTCOD.GE.1) THEN DO 656,1 = 2,8 656 PRTLOG(l) = .TRUE. END IF CALL TAXI(SYREC,FORMUL) CALL TAXI(SYREC,USE) CALL CHEMIE(COMAY,NC,OXYDE,FORMUL,CHEM) LUSE = INDEX(USE,' ')-1 IF (PRTCOD.EQ.0) TEST = DABS(TEST) CALL DBREAD IF (PRTLOG(D) STOP CLOSE(UNIT=9) C WRITE (7,FMT = '(I4)') NUN C WRITE (7,FMT='(7A10)') (CHNAME(I),I = 1,NUN) C WRITE (7,FMT='(I4)') NPHA C WRITE (7,FMT='(5A20)') (NAME(I),I = 1,NPHA) C WRITE (7,FMT='(I4)') NSOL C DO 550,1 = 1,NSOL C WRITE (7,FMT='(A20,I4)') SOLNAM(l),NEND(l) C WRITE (7,FMT='(1015)') (EM(I,1I),II = 1,NEND(1)) C 550 CONTINUE C IF (PRTLOG(2).OR.PRTLOC(3).OR.PRTLOG(4)) CALL PRININ IF (PRTLOG(5)) THEN WRITE (UNIT=6,FMT=140) TEST,L01MAX,EQUALX,DXMIN 140 FORMAT ('0TEST =',1PE11.4,8X,'L01MAX =',I4,13X,'EQUALX =', *1PE11.4,6X,'DELXMIN =',1PE11.4) WRITE (UNIT=6,FMT=141) DXSCAN,DXSTAR,STPSTA,STPMAX,CCMAX 141 FORMAT (' DELXSCAN =',1PE11.4,4X,'DELXSTAR =',1PE11.4,4X, *'STEPSTAR =',14,11X,'STEPMAX =',I4,12X,'GCMAX =',I5) END IF CALL GRECAL C START LOOPING 2111 WRITE (6,2010) 2010 FORMAT (' PRINT PARAMETERS:') YESNO = 'NO ' IF (PRTLOG(2)) YESNO = 'YES' WRITE (6,2012) YESNO 2012 FORMAT (' BEFORE CALCULATION: PRINT BULK COMPOSITION: ',A3) YESNO = 'NO ' IF (PRTLOG(3)) YESNO ='YES' WRITE (6,2013) YESNO 2013 FORMAT (21X/PRINT CONSIDERED PHASES: ',A3) YESNO = 'NO ' IF (PRTLOC(4)) YESNO = 'YES' WRITE (6,2014) YESNO 2014 FORMAT (21X/PRINT SOLUTION MODELS: ',A3) YESNO = 'NO ' IF (PRTLOC(6)) YESNO = 'YES' WRITE (6,2016) YESNO 2016 FORMAT (' AFTER CALCULATION: PRINT STABLE ASSEMBLAGE: ',A3) YESNO = 'NO ' IF (PRTLOG(7)) YESNO = 'YES' WRITE (6,2017) YESNO 2017 FORMAT (20X/PRINT COMPOSITIONS OF STABLE PHASES: ',A3) YESNO = 'NO ' IF (PRTLOG(8)) YESNO ='YES' WRITE (6,2018) YESNO 2018 FORMAT (20X/PRINT ACTIVITIES OF ALL PHASES: ',A3) WRITE (UNIT=6,FMT=2020) P,PGAS,TC,T 2020 FORMAT ('OP =',F9.2,' bar P(Cas) =',F9.2,' bar T =', *F8.2,' C = ',F8.2,' K') WRITE (6,2022) CHNAME(1),CHEM(CHMCOD(1)) 2022 FORMAT ('0BULK COMPOSITION: ',A8,1X,F11.6) DO 605,I = 2,NUN WRITE (6,2023) CHNAME(I),CHEM(CHMCOD(I)) 2023 FORMAT (19X,A8,1X,F11.6) 605 CONTINUE WRITE (6,2000) 2000 FORMAT ('0TP:NEW T AND PV BULK:NEW BULK', >/' PARA: NEW PARAMETERS'/' THER: CALL THERIAK7) READ (UNIT=5,FMT = '(A170)',END = 2099) CH001 IF (CH001.EQ.TP') THEN WRITE (6,2001) 2001 FORMAT (' ENTER NEW T AND P') READ (5,*,END = 2099) TC,P CALL GRECAL END IF IF (CH001.EQ/BULK') THEN WRITE (6,2002) 2002 FORMAT (' ENTER NEW FORMULA.:') READ (UNIT=5,FMT = '(A170)',END = 2099) FORMUL CALL CHEMIE(COMAY,NC,OXYDE,FORMUL,CHE) MORE = .FALSE. DO 601,1 = 1,NC IF (CHE(D.EQ.O.ODO.NEQV.CHEM(I).EQ.O.ODO) MORE = .TRUE. CHEM(I) = CHE(I) 601 CONTINUE IF (MORE) THEN WRITE (6,FMT = '(" RE-READ DATABASE")') CALL DBREAD CALL GRECAL ELSE DO 602,1 = 1,NUN 602 BULK(l) = CHE(CHMCOD(l)) END IF END IF IF (CHOOI.EQ.'PARA') THEN WRITE (6,2030) 2030 FORMAT (' PRINT BULK COMPOSITION?') READ (UNIT=5,FMT='(A3)') YESNO IF (YESNO(1:1).EQ.'Y') THEN PRTLOC(2) = .TRUE. ELSE PRTLOC(2) = .FALSE. END IF WRITE (6,2031) 2031 FORMAT (' PRINT CONSIDERED PHASES ?') READ (UNIT=5,FMT = '(A3)') YESNO IF (YESNO(1:1).EQ.'Y') THEN PRTLOG(3) = .TRUE. ELSE PRTLOC(3) = . FALSE. END IF WRITE (6,2032) 2032 FORMAT (' PRINT SOLUTION MODELS ?') READ (UNIT=5,FMT='(A3)') YESNO IF (YESNO(1:1).EQ.'Y') THEN PRTLOG(4)=.TRUE. ELSE PRTLOC(4) = . FALSE. END IF WRITE (6,2033) 2033 FORMAT (' PRINT STABLE ASSEMBLAGE ?') READ (UNIT=5,FMT='(A3)') YESNO IF (YESNO(1:1).EQ.'Y') THEN PRTLOG(6) = .TRUE. ELSE PRTLOG(6) = .FALSE. END IF WRITE (6,2034) 2034 FORMAT (' PRINT COMPOSITIONS OF STABLE PHASES READ (UNIT = 5,FMT='(A3)') YESNO IF (YESNO(1:1).EQ.'Y') THEN PRTLOG(7) = .TRUE. ELSE PRTLOG(7) = . FALSE. END IF WRITE (6,2035) 2035 FORMAT (' PRINT ACTIVITIES OF ALL PHASES ?') READ (UNIT=5,FMT='(A3)') YESNO IF (YESNO(1:1).EQ.'Y') THEN PRTLOG(8) = .TRUE. ELSE PRTLOG(8) = . FALSE. END IF END IF IF (CH001.EQ/THER') THEN CALL CALSTR IF (PRTLOC(2).OR.PRTLOG(3).OR.PRTLOC(4)) CALL PRININ CALL THER1A END IF GOTO 2111 2099 CONTINUE END C SUBROUTINE DBREAD INTEGERS CALMAX,COMAX,EMAX,MAMAX,MPOMAX,PHMAX,SITMAX,SOMAX PARAMETER (COMAX = 22,PHMAX = 150,SOMAX = 20,EMAX = 10,SITMAX = 5, * MAMAX = 31 ,MPOMAX = 4,CALMAX = 200) LOGICAL*4 IDCODE,FILEND INTEGERM 1001,1,11,ll,IS CHARACTER*3 SECTIO C - — C O M M O N BLOCK FOR SAVING INPUT DATA C-—GLOBAL C O M M O N BLOCK C - — C O M M O N BLOCK FOR DATABASE C C O M M O N BLOCK FOR G-CALCULATION C C O M M O N BLOCK FOR THERIAK C~—END OF C O M M O N VARIABLES REWIND (UNIT=8) NUN = 0 DO 505,1 = 1,NC IF (CHEM(I).NE.O.ODO) THEN NUN=NUN+1 IF (NUN.GT.COMAX) THEN WRITE (UNIT=6,FMT='(" COMAX = ", l5/" NUN = ",I5)') COMAX,NUN STOP END IF CHMCOD(NUN) = l END IF 505 CONTINUE IF (NUN.LE.0) THEN WRITE (UNIT = 6,FMT = '(" NUN = ",I5)') NUN STOP END IF DO 507,1 = 1,NUN I001=INDEX(OXYDE(CHMCOD(I)),' ') NAME(I) = ""//OXYDE(CHMCOD(I))(1:I001-1)//"" ABK(l) = OXYDE(CHMCOD(l)) DO 506,11 = 1,COMAX 506 XX(l,ll) = 0.0D0 XX(I,I)=1.0DO EMSOL(I) = 0 EMNR(l) = 0 BULK(l) = CHEM(CHMCOD(l)) 507 CHNAME(l) = OXYDE(CHMCOD(l)) NPHA= NUN SUGNR=NUN C—-- C~~-START READING DATABASE C — - FILEND = . FALSE. 168 NSOL=0 GOTO 5011 C START END-FILE ACTION 5010 RECODE='***' FILEND = .TRUE. REC = '** ENDFILE **' GOTO 5012 C-—END OF END-FILE ACTION 5011 DO 508,1 = 1,8 508 GENUG(I) = .FALSE. SECTIO = 'BAH' 1002 RECODE = ' ' C SKIP BLANK AND COMMENT RECORDS 1003 IF (RECODE.NE.' '.AND.RECODE(1:1).NE.'!') GOTO 3 READ (UNIT=8,FMT = '(A250)',END = 5010) REC CALL FIBLA(REC,I1) IF (I1.NE.0) RECODE = REC(l1:l1+2) GOTO 1003 C - — 3 IDCODE = .FALSE. IF (SECTIO.EQ.'MIN') IDCODE = (INDEX(REC,'(').NE.0) IF (SECTIO.EQ.'SOL') IDCODE = (INDEX(REC,' (').NE.O) IF (SECTIO.EQ.'MAR') IDCODE = (INDEX(REC,' - ').NE.0) 5012 IF (RECODE.EQ.'***'.OR.IDCODE) THEN IF (CENUG(D) THEN IF (SECTIO.EQ.'MIN') THEN CALL MINFIN CALL DASAVE(NPHA) END IF IF (SECTIO.EQ.'SOL') CALL SOLFIN DO 518,1 = 1,8 518 GENUG(I) = .FALSE. END IF IF (RECODE.EQ.'***') THEN IF (PRTLOGO)) WRITE (UNIT=6,FMT='("0",A132)') REC IF (FILEND.OR.INDEX(REC,'END').NE.0.OR. *INDEX(REC,'F1NISH').NE.0) GOTO 999 SECTIO = 'BAH' IF (INDEX(REC,'MINERAL DATA').NE.0.OR. *INDEX(REC,'GAS DATA').NE.0) SECTIO = 'MIN' IF (INDEX(REC,'SOLUTION DATA').NE.0) SECTIO = 'SOL' IF (INDEX(REC,'MARGULES').NE.O) THEN DO 519,IS=1,NSOL 519 IF (MARCOD(IS)) GOTO 5 5 IF (IS.LE.NSOL) THEN SECTIO = 'MAR' ELSE IF (PRTLOC(D) WRITE (UNIT = 6,FMT=105) 105 FORMAT (1X,'NO SOLUTION PHASE IS EXPECTING MARCULES-PARAMETERS') END IF ELSE IF (SECTIO.EQ.'BAH'.AND.PRTLOG(D) WRITE (UNIT=6,FMT=106) 106 FORMAT (1X/SECTION NOT READ: DATA NOT RECOGNIZED BY THERIAK') END IF ELSE IF (SECTIO.EQ.'MIN') CALL MINNEW IF (SECTIO.EQ.'SOL') CALL SOLNEW IF (SECTIO.EQ.'MAR') CALL MARNEW END IF ELSE IF (GENUG(D) THEN IF (SECTIO.EQ.'MIN') CALL M1NDAT IF (SECTIO.EQ.'SOL') CALL SOLDAT IF (SECTIO.EQ.'MAR') CALL MARDAT END IF END IF GOTO 1002 C - — C-—END OF READING DATABASE 999 RETURN END C £****************************** SUBROUTINE GRECAL 1NTEGER*4 CALMAX,COMAX,EMAX,MAMAX,MPOMAX,PHMAX,SlTMAX,SOMAX PARAMETER (COMAX = 22,PHMAX = 150,SOMAX = 20,EMAX = 10,SITMAX = 5, *MAMAX=31,MPOMAX = 4,CALMAX = 200) INTEGERM 1001,1,11,IS C — - C O M M O N BLOCK FOR SAVING INPUT DATA C GLOBAL C O M M O N BLOCK C C O M M O N BLOCK FOR DATABASE C — - C O M M O N BLOCK FOR G-CALCULATION , C - — C O M M O N BLOCK FOR THERIAK C-—END OF C O M M O N VARIABLES T=TC+273.15D0 IF (T.LE.0.0) THEN WRITE (UNIT=6,FMT = '(" T =",F8.2)') T STOP END IF IF (P.LT.0.0) THEN WRITE (UNIT=6,FMT='(" P =",F9.2)') P STOP END IF PGAS = P*PRAT RT=R*T TT=T*T SQT=DSQRT(T) DO 601,11 = 1,NUN 601 GGK(ll) = 0.0D0 DO 600,II = (NUN + 1),NPHA 1001=11 CALL DAREST(I001) NAM = NAME(II) IF (SPC) THEN CALL GSPEC(NAM,P,PGAS,T,CASE,GR) ELSE LIQ = .FALSE. CALL GCALC Ll fr*H3031Nl 1SHD (*)*H31DVeIVhD LOOHD 0Sc-*y313VHVI-O 33a (*)*H31DVaVHD (iSH3'D3y)ixvi 3Ni inoaans ON3 Nanisa (aOOI)LOOH3 = H3 3d (,(0'9La'Na), = lW3'9L0HD = l lNn) <XV3a (L-L00I:L)L00HD = 9L0HD (, /LOOHD)X3aNI = LOOI (:LOOl)H3=LOOH3 31 C3N3 Nani3H 00 = 33 N3HI (0'O3-L00l) 31 (L00l'H3)Viai3 11V3 (HD)N33 = 3V1 3VTL00I KH303J.NI 33 8*TV3a 9L0HD 9L*a3i3Vavl-0 LOOHD 0SZ*y313VHVH3 H D (*)*a31DVaVH3 (33'HD)H30 3Ni inoaans 0N3 0 = 11 (l+3VT03'll) 31 1 = 11 L L O1O0 (, ,3N'(H)hD) 31 LOS 3Vl 'L = l'lOS OQ (HD)N31 = 3V1 3Vl'l'll fr*H3D3JLNI HD (*)*y313V>JVH3 (n'HD)viai3 3Ni inoyans CIN3 SniMLLNCO 0L9 d*(rSI)AM+l*((l'SI)dDM*(Oi/l)OOia + (rSI)SM)-< (01-l)*(l'Sl)dDM + (l'SI)HM=(|/SI)DM (SI)DHVWN'L = I'0L9 OQ lOSN'L = Sl'0L9 OO SflNLLNCO 009 HD = (II)>IDD 31 ON3 ((l)WO3l)>IDD*(l)WCO33 + H0=yD 0S9 VNCON'L = ['0S9 oa N3H1 (WCO) 31 31 QN3 CALL FIBLA(REC,I1) IF (I1.EQ.0) THEN CHST=' ' RETURN END IF CH001 = REC(I1:) I1=INDEX(CH001,' ') CHST=CH001(1:I1-1) REC = CH001(I1:) RETURN END C £****************************** SUBROUTINE CHEMIE(COMAY,NC,OXYDE,FORMUL,CHE) INTEGERM I,I1,I2,13,NC,C0MAY CHARACTER* 170 FORMUL,CH170 CHARACTER*8 ELE,OXYDE(COMAY) REAL*8 CHE(COMAY),FF DO 501(I = 1,COMAY 501 CHE(l) = 0.0 CALL FIBLA(FORMUL,l1) 1001 IF (I1.EQ.0) GOTO 2 12 = INDEX(FORMUL/(') 13 = INDEX(FORMUL/)') IF (I2.LE.I1) 12 = 11 + 1 IF (I3.LE.I2 + 1) 13 = 12 + 2 ELE = FORMUL(l1:l2-1) DO 502,1 = 1,NC 502 IF (ELE.EQ.OXYDE(I)) GOTO 1 1 IF (I.EQ.NC + 1) THEN WRITE (UNIT=6,FMT = 100) FORMUL,ELE 100 FORMAT (' *** ',A170/' *** ELE= ',A8) STOP END IF CH170 = FORMUL(I2+1:13-1) READ (UNIT=CH170,FMT='(BN,D16.0)') FF CHE(I) = CHE(I) + FF CH170 = FORMUL FORMUL = CH170(I3 + 1:) CALL FIBLA(FORMUL,l1) GOTO 1001 2 RETURN END C — - £***************+************** SUBROUTINE MINNEW INTEGER*4 CALMAX,COMAX,EMAX,MAMAX,MPOMAX,PHMAX,SITMAX,SOMAX PARAMETER (COMAX = 22,PHMAX = 150,SOMAX = 20,EMAX = 10,SITMAX = 5, *MAMAX = 31,MPOMAX = 4,CALMAX = 200) C-—GLOBAL C O M M O N BLOCK C - — C O M M O N BLOCK FOR DATABASE C C O M M O N BLOCK FOR G-CALCULATION C - — C O M M O N BLOCK FOR THERIAK C-—END OF C O M M O N VARIABLES INTECERM COMAY,I,II,I001 CHARACTER*60 REJECT CHARACTER*!5 USECOD C O M AY = COMAX CALL TAXl(REC,NAM) CALL TAXI(REC,FORMUL) CALL TAXl(REC,FORM) CALL TAXI(REC,USECOD) IF (USECOD.EQ.' ') USECOD = ' N ' CENUC(1) = .FALSE. NULL(NPHA+1) = . FALSE. CALL CHEMIE(COMAY,NC,OXYDE,FORMUL,CHE) DO 520,1 = 1,LUSE IF (GENUG(1).OR.USE(l:l).EQ.7) GO TO 9 520 GENUG(1) = (INDEX(USECOD,USE(l:l)).NE.0.OR.USE.EQ.'*') 9 IF (USECOD(1:1).EQ.'*') THEN USECOD(1:1) = ',' I001 = INDEX(USECOD,' ') GENUG(1) = (INDEX(USE,USECOD(1:I001-1)).NE.O) END IF IF (.NOT.GENUC(I)) *REJECT = 'CODE NOT MATCHING : "7/USECOD//"" DO 521,11 = 1,NPHA IF (.NOT.GENUGd)) GO TO 10 IF (NAM.EQ.NAME(II)) THEN GENUG(1) = .FALSE. REJECT='DEFINED MORE THAN ONCE IN DATABASE' END IF 521 CONTINUE 10 DO 522,11 = 1,NC IF (.NOT.CENUG(D) GO TO 11 GENUG(1) = (CHE(ll).EQ.0.0.OR.CHEM(ll).NE.0.0) IF (.NOT.GENUC(1).AND.USECOD(1:1).EQ.' + ') THEN DO 702,1 = 1,NC 702 CHE(l) = 0.0D0 GENUG(1) = .TRUE. NULL(NPHA+1) = .TRUE. GOTO 11 END IF IF (.NOT.GENUGd)) REJECT ='COMPOSITION OUTSIDE DEFINED SPACE' 522 CONTINUE 11 IF (GENUG(D) THEN G0R=0.0D0 HOR=O.0D0 SOR = O.0DO VOR=O.0DO AA0 = 0.0D0 AAT = O.0DO BB0 = 0.0D0 BBT=O.0DO K1=0.0D0 K2 = 0.0D0 K3 = 0.0D0 K4 = 0.0D0 K5 = 0.0D0 K6 = 0.0D0 K7 = 0.0D0 K8 = 0.0D0 K9 = 0.0D0 NLANDA = 0 NCOM=0 DO 523,1 = 1,10 1COM(I) = 0 523 FFCOM(I) = 0.0D0 DO 517,1 = 1,4 ASPK(l) = 0.0D0 BSPK(l) = 0.0D0 TQ1B(I) = 0.0D0 TEQ(l) = 0.0D0 DVDT(l) = 0.0D0 DVDP(l) = 0.0D0 TRE(l) = 0.0D0 DHTR(l) = 0.0D0 517 DVTR(l) = 0.0D0 TD0 = 0.0D0 TDMAX=O.OD0 VADj = 0.0D0 D1=0.0D0 D2 = 0.0D0 D3 = 0.0D0 D4 = 0.0D0 D5 = 0.0D0 D6 = 0.0D0 D7 = 0.0D0 D8 = 0.0D0 D9 = 0.0D0 CASE = ' ' RDK=.FALSE. VDW = . FALSE. SPC = .FALSE. C O M = . FALSE. DIS = .FALSE. ELSE IF (PRTLOC(D) WRITE (UNIT=6,FMT=107) NAM,REJECT 107 FORMAT (' >',A16,' : ',A60) END IF RETURN END C - — SUBROUTINE MINDAT INTEGER*4 CALMAX,COMAX,EMAX,MAMAX,MPOMAX,PHMAX,SITMAX,SOMAX PARAMETER (COMAX = 22,PHMAX = 150,SOMAX = 20,EMAX = 10,SITMAX = 5, *MAMAX=31,MPOMAX = 4,CALMAX=200) C—-GLOBAL C O M M O N BLOCK C - — C O M M O N BLOCK FOR DATABASE C C O M M O N BLOCK FOR G-CALCULATION C — - C O M M O N BLOCK FOR THERIAK END OF C O M M O N VARIABLES INTEGERS I IF (RECODE.EQ.'ST ') THEN READ (UNIT=REC,FMT='(5X,4D15.0)') G0R,H0R,S0R,V0R GENUC(2) = .TRUE. V0R = V0R/10.0D0 END IF IF (RECODE.EQ.'R-K') THEN READ (UN1T = REC,FMT = '(5X,4D15.8)') AA0,AAT,BB0,BBT RDK=.TRUE. VDW = . FALSE. END IF IF (RECODE.EQ.'VDW) THEN READ (UNIT=REC,FMT='(5X,4D15.8)') AA0,AAT,BB0,BBT VDW = .TRUE. RDK = .FALSE. END IF IF (RECODE.EQ/CP1') THEN READ (UNIT=REC,FMT = '(5X/4D15.0)') K1,K4,K3,K8 CENUC(3) = .TRUE. END IF IF (RECODE.EQ/CP3') THEN READ (UNIT=REC,FMT='(5X,3D15.0)') K1,K2,K3 CENUC(3) = .TRUE. END IF IF (RECODE.EQ/CP2') THEN READ (UNIT=REC,FMT = '(5X,5D15.0)') K6,K2,K5,K7,K9 END IF IF (RECODE.EQ.'DT) THEN READ (UNIT=REC,FMT='(5X,5D15.0)') D1,D4,D3,VADJ,D6 DIS = .TRUE. END IF IF (RECODE.EQ/D2') THEN READ (UNIT=REC,FMT = '(5X,4D15.0)') D2,D5,TD0,TDMAX END IF IF (RECODE.EQ.'SPC) THEN CALL TAXI(RECCH) CALL TAXI(REC,CASE) SPC = .TRUE. CENUG(2) = .TRUE. GENUC(3) = .TRUE. END IF IF (RECODE.EQ.'COM') THEN CALL TAXI(REC,CH) CALL TAXl(REC,FORMUL) C O M = .TRUE. END IF IF (RECODE.EQ.'TRT') THEN NLANDA = NLANDA+1 l = NLANDA READ (UNIT=REC,FMT='(5X,5D15.0)') TQ1B(I),TRE(I),ASPK(1), *BSPK(I),DHTR(I) END IF IF (RECODE.EQ.TR2') THEN IF (NLANDA.GT.O) THEN l = NLANDA READ (UNIT=REC,FMT='(5X,4D15.0)') TEQ(I),DVTR(I),DVDT(I), *DVDP(I) END IF END IF RETURN END C-—- Q* **************************** * SUBROUTINE MINFIN INTEGERS CALMAX,COMAX,EMAX,MAMAX,MPOMAX,PHMAX,SITMAX,SOMAX PARAMETER (COMAX = 22,PHMAX = 150,SOMAX = 20,EMAX=10,SITMAX = 5, *MAMAX= 31 ,MPOMAX= 4,CALMAX= 200) C-—GLOBAL C O M M O N BLOCK C C O M M O N BLOCK FOR DATABASE C C O M M O N BLOCK FOR G-CALCULATION C - — C O M M O N BLOCK FOR THERIAK C-—END OF C O M M O N VARIABLES INTEGERS I,IX,I001,I1,I2 REAL*8 FF CHARACTER* 170 MODEL1 CHARACTER* 16 EM NAM IF (GENUG(1).AND.GENUG(2).AND.GENUG(3)) THEN NPHA= NPHA+1 IF (NPHA.GT.PHMAX.OR.NPHA.GT.CALMAX) THEN WRITE (UNIT = 6,FMT=100) P H MAX, CALM AX, N P H A 100 FORMAT (' PHMAX = ',I5/' CALMAX = ',I5/' NPHA = ',I5) STOP END IF IF (COM) THEN NCOM=0 CALL FIBLA(FORMUL,l1) 2001 IF (I1.EQ.0) GOTO 71 l2=INDEX(FORMUL,'[') IX=INDEX(FORMUL,']') IF (12.LE.I1) 12 = 11 + 1 IF (IX.LE.12 + 1) IX=l2 + 2 EMN AM = FORMUL(l1:12-1) DO 701,I = 1,(NPHA-1) 701 IF (EMNAM.EQ.NAME(I)) GOTO 72 72 IF (I.EQ.NPHA) THEN WRITE (UNIT = 6,FMT=300) FORMUL,EMNAM 300 FORMAT (' *** ',A170/' *** NAME= ',A16) STOP END IF MODEL1=FORMUL(l2 + 1:IX-1) READ (UNIT=MODEL1,FMT='(BN,D16.0)') FF N C O M = NCOM + 1 ICOM(NCOM) = l FFCOM(NCOM) = FF MODEL1 =FORMUL FORMUL = MODEL1(IX+1:) CALL FlBLA(FORMUL,l1) 176 GOTO 2001 END IF 71 CONTINUE SUGNR=SUGNR+1 NAME(NPHA) = NAM ABK(NPHA) = FORM DO 705,1 = 1,NUN 705 XX(NPHA,l) = CHE(CHMCOD(l)) EMSOL(NPHA) = 0 EMNR(NPHA) = 0 IF (PRTLOG(D) WRITE (UNIT=6,FMT='(1X,A16," O.K.")') NAM ELSE IF (PRTLOG(D) WRITE(UN1T = 6,FMT = 101) NAM 101 FORMAT (' >',A16,' : NOT ENOUGH DATA') END IF RETURN END C — - Q* **************************** * SUBROUTINE SOLNEW INTEGERS CALMAX,COMAX,EMAX,MAMAX,MPOMAX,PHMAX,SITMAX,SOMAX PARAMETER (COMAX= 22,PHMAX = 150,SOMAX = 20,EMAX= 10,SITMAX = 5, *MAMAX=31,MPOMAX = 4,CALMAX = 200) C—-GLOBAL C O M M O N BLOCK C - — C O M M O N BLOCK FOR DATABASE C C O M M O N BLOCK FOR G-CALCULATION C C O M M O N BLOCK FOR THERIAK C END OF C O M M O N VARIABLES INTEGERS l,K REAL*8 FF K=NSOL+1 NEND(K) = 0 CALL TAXI(REC,SONAM) CALL TAXI(REC,MODCOD) IF (INDEX(MODCOD,'SKIP').NE.0) THEN IF (PRTLOG(D) WRITE (UNIT = 6,FMT=108) SONAM 108 FORMAT (' >',A16,' : EXPLICITLY EXCLUDED') GOTO 40 END IF CALL GELI(REC,FF) NSITE(K)= IDINT(FF) IF (NSITE(K).GT.SITMAX) THEN WRITE (UNIT=6,FMT=109) SITMAX,K,NSITE(K) 109 FORMAT (' SITMAX = ',I5/1X,15,' : NSITE = ',l5) STOP END IF DO 523,1 = 1,NSITE(K) 523 CALL GELI(REC,SITMUL(K,I)) GENUG(1) = .TRUE. 40 CONTINUE RETURN END C-—- **************************** * SUBROUTINE SOLDAT INTEGERS CALMAX,COMAX,EMAX,MAMAX,MPOMAX,PHMAX,SITMAX,SOMAX PARAMETER (COMAX = 22,PHMAX= 150,SOMAX= 20,EMAX= 10,SITMAX= 5, * MAMAX = 31 ,MPOMAX = 4,CALMAX = 200) C GLOBAL C O M M O N BLOCK C C O M M O N BLOCK FOR DATABASE C C O M M O N BLOCK FOR G-CALCULATION C C O M M O N BLOCK FOR THERIAK C-—END OF C O M M O N VARIABLES INTEGER*4 l,ll,lll,K CHARACTER*16 EMNAM K = NSOL+1 ll = NEND(K) + 1 CALL TAXI(REC,EMNAM) DO 526,1 = 1,NSITE(K) CALL TAXI(REC,CH) 526 SITEL(K,ll,l) = CH DO 527,111 = 1,NPHA 527 IF (NAME(III).EQ.EMNAM.AND.(.NOT.NULL(III))) GO TO 15 15 IF (III.LE.NPHA) THEN NEND(K) = NEND(K) + 1 IF (NEND(K).GT.EMAX) THEN WRITE (UNIT=6,FMT=110) EMAX,K,NEND(K) 110 FORMAT (' EMAX = ',I5/1X,I5,' : NEND = ',I5) STOP END IF EM(K,NEND(K)) = III IF (PRTLOG(D) WRITE (UNIT = 6,FMT=111) SON AM,EMNAM 111 FORMAT (1X,A16,': ',A16,' O.K.') ELSE IF (PRTLOG(D) WRITE (UNIT = 6,FMT=112) SON AM, EMNAM 112 FORMAT (1X,A16,' >',A16,' NOT A CONSIDERED PHASE') END IF RETURN END C - — £****************************** SUBROUTINE SOLFIN INTEGERS CALMAX,COMAX,EMAX,MAMAX,MPOMAX,PHMAX,SITMAX,SOMAX PARAMETER (COMAX = 22,PHMAX= 150,SOMAX = 20,EMAX= 10,SITMAX = 5, *MAMAX=31,MPOMAX = 4,CALMAX = 200) C GLOBAL C O M M O N BLOCK C C O M M O N BLOCK FOR DATABASE C — - C O M M O N BLOCK FOR G-CALCULATION C - — C O M M O N BLOCK FOR THERIAK C-—END OF C O M M O N VARIABLES INTEGERM I,II,IE,IR001(EMAX),I001,1002,1003,11,K REAL*8 FF K=NSOL + 1 IF (NEND(K).GT.D THEN NSOL=K IF (NSOL.GT.SOMAX) THEN WRITE (UNIT = 6,FMT = '(" SOMAX = ",I5/" NSOL = ",l6)') SOMAX,NSOL STOP END IF ALPHA(K) = 1.0D0 ALPDIV(K) = ' ' 11 =INDEX(MODCOD,7') IF (I1.NE.0) ALPDIV(K) = MODCOD(l1:) IF (NSITE(K).EQ.O.AND.lNDEX(MODCOD,'SITE').NE.O) M O D C O D = ' ' IF (INDEX(MODCOD/SITE').EQ.O.AND.INDEX(MODCOD,'EXT').EQ.O) * MODELL(K) = T IF (INDEX(MODCOD,'SITE').NE.O) THEN 1 = 1 DO 509,1001 = 1,EMAX DO 509,1002= 1,SITMAX NEQEM(K,I001,I002) = 0 DO 509,1003 = 1,EMAX 509 EQEM(K,I001,I002,I003) = 0 MODELL(K) = 'S' 1007 IF (I.CT.NSITE(K)) CO TO 7 DO 514,IE = 1,NEND(K) DO 512,11 = 1,NEND(K) IF (SITEL(K,IE,I).EQ.SITEL(K,II,I)) THEN NEQEM(K,IE,I) = NEQEM(K,IE,I) + 1 EQEM(K,IE,I,NEQEM(K,IE,I)) = II END IF 512 CONTINUE IF (NEQEM(K,lE,l).EQ.NEND(K)) THEN SITMUL(K,I) = SITMUL(K,NSITE(K)) DO 513,1001 = 1,EMAX 513 SITEL(K,I001,I) = SITEL(K,I001,NSITE(K)) NSITE(K) = NSITE(K)-1 1 = 1-1 C O TO 6 END IF 514 CONTINUE 6 1=1+1 GO TO 1007 7 IF (NSITE(K).EQ.D THEN MODELL(K) = T ALPHA(K) = SITMUL(K,1) END IF END IF IF (INDEX(MODCOD,'EXT').NE.0) THEN MODELL(K) = 'P DO 515,IE = 1,NEND(K) 515 MANAM(IE) = NAME(EM(K,IE)) CALL SOLINI(SONAM,NEND(K),MANAM,NEMBAS(K),IR001) DO 710,IE = 1,NEMBAS(K) 710 EMBCOD(K,1E) = IR001(IE) IF (NEMBAS(K).LT.O) THEN WRITE (UNIT=6,FMT=102) SONAM,NAME(EM(K,-NEMBAS(K))) 102 FORMAT (1X,A16,' :THERE IS NO DATA FOR ',A16, *' IN SUBROUTINE') STOP END IF END IF IF (MODELL(K).EQ.'I') SITMUL(K,1) = ALPHA(K) IF ((MODELL(K).EQ.'l'.OR.MODELL(K).EQ.'S').AND. *ALPDIV(K).NE.' ') THEN CALL CELI(ALPDIV(K)(2:),FF) ALPHA(K) = ALPHA(K)/FF END IF SOLNAM(NSOL) = SONAM DO 516,1 = 1,NEND(NSOL) EMSOL(EM(NSOL,l)) = NSOL 516 EMNR(EM(NSOL,l)) = l IF (PRTLOC(D) THEN WRITE (UNIT=6,FMT=103) SONAM,(NAME(EM(K,l)),l = 1,NEND(K)) 103 FORMAT (1X,A16,' O.K., ENDMEMBERS: ',100(5(2X,A16)/35X)) END IF MARCOD(NSOL) = (INDEX(MODCOD,'MARCULES').NE.0) NMARG(NSOL) = 0 ELSE IF (PRTLOC(1)) WRITE (UNIT=6,FMT=104) SONAM 104 FORMAT (' >',A16,' : LESS THAN TWO ENDMEMBERS') END IF IF (PRTLOC(D) WRITE (UNIT=6,FMT = '(" ")') RETURN END C - — ^-****************************** SUBROUTINE MARNEW INTECER*4 CALMAX,COMAX,EMAX,MAMAX,MPOMAX,PHMAX,SITMAX,SOMAX PARAMETER (COMAX= 22,PHMAX = 150,SOMAX = 20,EMAX = 10,SITMAX= 5, *MAMAX = 31,MPOMAX = 4,CALMAX=200) C GLOBAL C O M M O N BLOCK C C O M M O N BLOCK FOR DATABASE C — - C O M M O N BLOCK FOR G-CALCULATION C C O M M O N BLOCK FOR THERIAK C END OF C O M M O N VARIABLES INTEGER*4 I DO 524,1 = 2,254 524 IF (REC(I-1:I + 1).EQ.' - ') REC(I-1:I + 1) = ' ' DO 525,1 = 1,EMAX CALL TAXl(REC,MANAM(l)) 525 IF (MANAM(I).EQ.' ') GO TO 13 13 DIM = I-1 NPAR=0 IF (DIM.GT.1) GENUG(1) = .TRUE. RETURN END C - — SUBROUTINE MARDAT INTEGER*4 CALMAX,COMAX,EMAX,MAMAX,MPOMAX,PHMAX,SITMAX,SOMAX PARAMETER (COMAX = 22,PHMAX= 150,SOMAX = 20,EMAX = 10,SITMAX= 5, *MAMAX = 31,MPOMAX = 4,CALMAX=200) C-—GLOBAL C O M M O N BLOCK C C O M M O N BLOCK FOR DATABASE C - — C O M M O N BLOCK FOR C-CALCULATION 180 C - — C O M M O N BLOCK FOR THERIAK C-—END OF C O M M O N VARIABLES INTEGERS 1,IIND(MPOMAX),IND(MPOMAX),POL,IS,I001,K REAL*8 WWCP,WWH,WWS,WWV LOGICALM CODE NPAR=NPAR+1 CALL TAXl(REC,CH) POL = INDEX(CH,' ')-1 READ (UNIT=CH,FMT='(100I1)') (IIND(l),l = 1,POL) CALL GELI(REC,WWH) CALL GELI(REC,WWS) CALL GELI(REC,WWV) CALL GELI(REC,WWCP) DO 532,IS = 1,NSOL IF (MARCOD(IS)) THEN CODE = .TRUE. DO 529,1 = 1,DIM IF (.NOT.CODE) GO TO 17 DO 528,K = 1,NEND(IS) 528 IF (MANAM(I).EQ.NAME(EM(IS,K))) GO TO 16 16 IND(I) = K CODE = (K.NE.NEND(IS) + 1) 529 CONTINUE 17 IF (CODE) THEN NMARC(IS) = NMARG(IS) +1 IF (NMARG(IS).GT.MAMAX) THEN WRITE (UNIT=6,FMT=113) MAMAX,lS,NMARG(IS) 113 FORMAT (' MAMAX = ',I5/1X,I5/ : NMARG = ',I5) STOP END IF POLY(IS,NMARG(IS))= POL IF (POLGT.MPOMAX) THEN WRITE (UNIT=6,FMT=114) MPOMAX,IS,POL 114 FORMAT (' MPOMAX = ',l5/1X,I5,' : POL = ',l5) STOP END IF DO 530,1001 =1,EMAX 530 QQ(IS,NMARG(IS),I001) = 0 DO 531,1 = 1,POL INDX(IS,NMARG(IS),l) = lND(IIND(l)) 531 QQ(IS,NMARG(IS),IND(IIND(I))) = QQ(IS,NMARG(IS),IND(IIND(I))) + 1 WH(IS,NMARG(IS)) = WWH WS(IS,NMARG(IS))=WWS WV(IS,NMARG(IS)) = WWV WCP(IS,NMARG(IS)) = WWCP C WG(lS,NMARG(IS)) = WWH+WWCP*(T-T0)-(WWS + DLOG(T/T0)*WWCP)*T+WWV*P IF (PRTLOG(D) WRITE (UNIT= 6,FMT='(1X,A16,": MARGULES-", •"PARAMETER ",10011)') SOLNAM(IS),(IIND(I),I = 1,POL) END IF END IF 532 CONTINUE RETURN END SUBROUTINE CCALC INTEGERS CALMAX,COMAX,EMAX,MAMAX,MPOMAX,PHMAX,SITMAX,SOMAX PARAMETER (COMAX = 22,PHMAX= 150,SOMAX = 20,EMAX = 10,SITMAX = 5, * MAM AX = 31, MPOMAX = 4,CALMAX = 200) -—GLOBAL COMMON BLOCK —COMMON BLOCK FOR G-CALCULATION —END OF COMMON VARIABLES REAL*8 CPRDT,CPRTDT,TD,VD INTEGER*4 1,1001 CPRDT= K1 *(T-T0) + K2*(TT-TT0)/2.0D0 *-K3*(1.0D0AT-1.0D0/T0) + 2D0*K4*(SQT-SQT0) * + K5*(TT*T-TT0*T0)/3.0D0+ K6*DLOG(T7T0) * + K7*(T*SQT-T0*SQT0)*2.0D0/3.0D0 *-K8*(1.0D0/(TT)-1.0D0/(TT0))/2.0D0 * + K9*(TT*TT-TT0*TT0)/4.0D0 CPRTDT=K1*DLOG(T/T0)+K2*(T-T0) *-K3 *(1.0D0/(TT)-1 .ODO/(TTO))/2.0DO *-2D0*K4*(1.0D0/SQT-1.0D0/SQT0) * + K5*(TT-TT0)/2.0D0-K6*(1.0D0/r-1.0D0/T0) * + 2D0*K7*(SQT-SQT0) *-K8*(1.0D0/(TT*T)-1.0D0/(TT0*T0))/3.0D0 * + K9*(TT*T-TT0*T0)/3.0D0 GR=H0R+CPRDT-T*(S0R + CPRTDT) IF (RDK.OR.VDW) THEN CALL GAS ELSE GR = GR + V0R*(P-P0) END IF DO 510,1 = 1,NLANDA 1001=1 510 CALL LANDA(I001) IF (DIS.AND.TD0.NE.0.0D0.AND.TDMAX.NE.0.0D0.AND.T.GT.TD0) THEN TD = DMIN1(T,TDMAX) CPRDT=D1*(TD-TD0) + D2*(TD*TD-TD0*TD0)/2.0D0 *-D3*(1.0D0TD-1.0D0/TD0) + 2D0*D4*(DSQRT(TD)-DSQRT(TD0)) * + D5*(TD**3-TD0**3)/3.0D0 + D6*DLOG(TDATD0) * + D7*(TD*DSQRT(TD)-TDO*DSQRT(TDO))*2.OD0/3.0D0 *-D8*(1.0D0/(TD*TD)-1.0D0/(TD0*TD0))/2.0D0 * + D9*(TD**4-TD0**4)/4.0D0 CPRTDT=D1 *DLOG(TD/TD0) + D2*(TD-TD0) *-D3*(1.0D0/(TD*TD)-1.0D0/(TD0*TD0))/2.0D0 *-2D0*D4*(1.0D0/DSQRT(TD)-1.0D0/DSQRT(TD0)) * + D5*(TD*TD-TD0*TD0)/2.0D0-D6*(1.0D0/TD-1.0D0/TD0) * + 2D0*D7*(DSQRT(TD)-DSQRT(TD0)) *-D8*(1.0D0/(TD**3)-1.0D0/(TD0**3))/3.0D0 * + D9*(TD**3-TD0**3)/3.0D0 IF (DABS(VADJ).GT.10.0D0) THEN VD = CPRDT/(10.0D0*VADJ) ELSE VD = 0.0D0 END IF GR = GR + CPRDT-(T*CPRTDT) + VD*(P-1.0D0) END IF 182 RETURN END C SUBROUTINE GAS INTEGER*4 CALMAX,COMAX,EMAX,MAMAX,MPOMAX,PHMAX,SITMAX,SOMAX PARAMETER (COMAX = 22,PHMAX = 150,SOMAX = 20,EMAX= 10,SITMAX = 5, *MAMAX = 31,MPOMAX = 4,CALMAX = 200) C GLOBAL C O M M O N BLOCK C C O M M O N BLOCK FOR G-CALCULATION REAL*8 AA,BB,DGGAS C O M M O N /GORE/ AA,BB,DGGAS C—-END OF C O M M O N VARIABLES REAL*8 KUB,KUC,KUD,OFT,VREF,VOL /VGAS,VFL,X1,X2,X2I,X3,F001 AA = AAO+T*AAT BB = BBO+T*BBT IF (PGAS.LE.0.0D0) THEN WRITE (UNIT = 6,FMT = '(" PGAS = ",F8.2)') PGAS STOP END IF IF (RDK) THEN OFT=P0*SQT KUB = -10.0DO*RT/P0 KUC = AA/OFT-BB*BB + BB*KUB KUD = -AA*BB/OFT END IF IF (VDW) THEN KUB = -BB-10.0D0*RT/P0 KUC = AA/P0 KUD = -AA*BB/PO END IF CALL KUBIK(KUB,KUC,KUD,X1,X2,X2I,X3) IF (X2I.NE.O.OD0) THEN VREF = X1 ELSE VREF=DMAX1(X1,X2,X3) END IF IF (RDK) THEN OFT=PGAS*SQT KUB = -10.0D0*RT/PGAS KUC = AA/OFT-BB*BB + BB*KUB KUD = -AA*BB/OFT END IF IF (VDW) THEN KUB = -BB-10.0D0*RT/PGAS KUC = AA/PGAS KU D = -AA * B B/PG AS END IF CALL KUBIK(KUB,KUC,KUD,X1,X2,X2I,X3) IF (X2I.NE.0.0D0) THEN VOL = X1 ELSE VGAS = DMAX1 (X1 ,X2,X3) VFL=DMIN1(X1,X2,X3) 183 DGGAS = -1D0 F001 = PGAS IF (VFL.GT.BB) CALL DELCAS(VFL,VCAS,PCAS;F001) IF (DGGAS.GT.O.ODO) THEN LIQ = .TRUE. VOL = VFL ELSE VOL=VGAS END IF END IF CALL DELGAS(VREF,VOL,PO,PGAS) GR=GR+DGGAS RETURN END C - — * * * * * * * * * * * * * * * * * * * * * * * * * * * * * SUBROUTINE DELGAS(V1,V2,P1,P2) INTEGERS CALMAX,COMAX,EMAX,MAMAX,MPOMAX,PHMAX,SITMAX,SOMAX PARAMETER (COMAX = 22,PHMAX = 150,SOMAX = 20,EMAX= 10,SITMAX= 5, *MAMAX = 31,MPOMAX = 4,CALMAX = 200) C-—GLOBAL C O M M O N BLOCK C - — C O M M O N BLOCK FOR G-CALCULATION REAL*8 AA,BB,DGGAS C O M M O N /GORE/ AA,BB,DGGAS C-—END OF C O M M O N VARIABLES REAL*8 V1,V2,P1,P2 IF (RDK) DGGAS = V2*P2-V1*P1-10*RT*DLOG((V2-BB)/(V1-BB)) * + (AA/(SQT*BB))*DLOG(V2*(V1+BB)/(V1*(V2 + BB))) IF (VDW) DGCAS=V2*P2-V1*P1-10*RT*DLOG((V2-BB)/(V1-BB)) *-AA*(1.OA/2-1.OA/1) DGGAS = DGGAS/10.0 RETURN END C - — * * * * * * * * * * * * * * * * * * * * * * * * * * * * * SUBROUTINE KUBIK(B,C,D,X1,X2,X2I,X3) REAL*8 B,C,D,Q,P,R,PI,PHI3,FF,X1,X2,X2I,X3 PI = 3.14159263538979D0 IF (C.EQ.0.0D0.AND.D.EQ.0.0D0) THEN X1=-B X2 = 0.0D0 X2I = 0.0D0 X3 = 0.0D0 RETURN END IF Q = ((2.D0*B*B*B)/(27.D0)-(B*C)/(3.D0) + D)/2.D0 P = (3.D0*C-B*B)/(9.D0) FF=DABS(P) R = DSQRT(FF) FF=R*Q IF (FF.LT.0.0D0) R = -R FF = Q/(R*R*R) IF (P.GT.0.0D0) THEN PHI3 = DLOG(FF+DSQRT(FF*FF+1.D0))/3.D0 184 X1=-R*(DEXP(PHI3)-DEXP(-PHI3))-B/(3.D0) X2I = 1D0 ELSE IF (Q*Q + P*P*P.GT.0.0D0) THEN PHI3 = DLOG(FF + DSQRT(FF*FF-1.D0))/3.D0 X1=-R*(DEXP(PHI3)+DEXP(-PHI3))-B/(3.D0) X2I = 1D0 ELSE PHI3 = DATAN(DSQRT(1.D0-FF*FF)/FF)/3.D0 X1 = -2.D0*R*DCOS(PHI3)-B/(3.D0) X2 = 2.D0*R*DCOS(PI/3.D0-PHI3)-B/(3.D0) X2I = 0.0D0 X3 = 2.D0*R*DCOS(PI/3.D0 + PHI3)-B/(3.D0) END IF END IF RETURN END SUBROUTINE LANDA(K) INTEGER*4 CALMAX,COMAX,EMAX,MAMAX,MPOMAX,PHMAX,SITMAX,SOMAX PARAMETER (COMAX = 22,PHMAX= 150,SOMAX = 20,EMAX= 10,SITMAX= 5, *MAMAX= 31,MPOMAX=4,CALMAX= 200) C GLOBAL C O M M O N BLOCK C C O M M O N BLOCK FOR G-CALCULATION C END OF C O M M O N VARIABLES REAL*8 A I ^ B I ^ C I ^ D I ^ C T R j g j R g ^ H S P ^ D S S P ^ C S P K , *AASP,ABSP,CCTR,TEQK INTEGER*4 K TEQK = TEQ(K)*(P-1.0D0) + TQ1B(K) CTR=TQ1B(K)-TEQK TR9=TRE(K)-CTR IF (T.GT.TEQK) THEN T9=TEQK ELSE T9 = T END IF D11=BSPK(K)*BSPK(K) AASP = ASPK(K)*ASPK(K) ABSP = ASPK(K)*BSPK(K) CCTR = CTR*CTR A11=AASP*CTR + 2.0D0*ABSP*CCTR + D11*CCTR*CTR B11=AASP + 4.0D0*ABSP*CTR+3.0D0*D11*CCTR C11 = 2.0D0*ABSP+3.0D0*CTR*D11 DHSPK = A11 *(T9-TR9) + B11 *(T9*T9-TR9*TR9)/2.0D0 * + C11 *(T9* *3-TR9* *3)/3.0D0 + D11 *(T9* *4-TR9* *4)/4.0D0 DSSPK=A11*(DLOG(T9)-DLOG(TR9)) + B11*(T9-TR9) + +C11*(T9*T9-TR9*TR9)/2.0D0 + D11*(T9**3-TR9**3)/3.0D0 IF ((T9 + CTR).LT.298.15D0) THEN DHSPK = O.OD0 DSSPK = O.OD0 END IF GSPK= DHSPK-T9*DSSPK IF (T.GT.TEQK) GSPK=GSPK-(DHTR(K)/TQ1B(K) + DSSPK)*(T-TEQK) GSPK=GSPK+DVDT(K)*(P-1.0D0)*(T9-298.15D0) * + (DVDP(K)/2.0DO)*(P*P-1.0DO)-DVDP(K)*(P-1.0DO) GR=GR + GSPK RETURN END C—-- SUBROUTINE DASAVE(N) 1NTEGER*4 CALM AX, C O M AX, EMAX, MAMAX,MPOMAX, PHMAX,SITMAX, SOM AX PARAMETER (COMAX = 22,PHMAX = 150,SOMAX = 20,EMAX = 10,SITMAX= 5, *MAMAX=31,MPOMAX = 4,CALMAX = 200) C - — C O M M O N BLOCK FOR SAVING INPUT DATA C — - C O M M O N BLOCK FOR G-CALCULATION C END OF C O M M O N VARIABLES INTECERM N,I,J INDATA(1,N) = NLANDA INDATA(2,N)=NCOM DO 500,1 = 1,10 500 INDATA(l + 2,N) = ICOM(l) C - — REDATA(1,N) = C0R REDATA(2,N) = H0R REDATA(3,N) = S0R REDATA(4,N) = V0R REDATA(5,N) = AA0 REDATA(6,N) = AAT REDATA(7,N) = BB0 REDATA(8,N) = BBT REDATA(9,N) = K1 REDATA(10,N) = K2 REDATA(11,N)=K3 REDATA(12,N) = K4 REDATA(13,N) = K5 REDATA(14,N) = K6 REDATA(15,N) = K7 REDATA(16,N) = K8 REDATA(17,N) = K9 DO 600,1 = 1,4 J = 9*(l-1) REDATA(J + 18,N) = ASPK(I) REDATA(J +19,N) = BSPK(I) REDATA(J + 20,N) = TQ1 B(l) REDATA(J + 21,N) = TEQ(I) REDATA(J + 22,N) = DVDT(I) REDATA(J + 23,N) = DVDP(I) REDATA(J + 24,N)=TRE(I) REDATA(J + 25,N) = DHTR(I) 600 REDATA() + 26,N)=DVTR(I) REDATA(54,N) = TD0 REDATA(55,N) = TDMAX REDATA(56,N) = VADJ REDATA(57,N) = D1 REDATA(58,N) = D2 REDATA(59,N) = D3 REDATA(60,N) = D4 REDATA(61,N) = D5 REDATA(62,N) = D6 REDATA(63,N) = D7 REDATA(64,N) = D8 REDATA(65,N) = D9 DO 610,1 = 1,10 610 REDATA(l + 65,N) = FFCOM(l) C - — CHDATA(N) = CASE C LODATA(1,N)= RDK LODATA(2,N)=VDW LODATA(3,N) = SPC LODATA(4,N) = C O M LODATA(5,N) = DIS RETURN END C - — SUBROUTINE DAREST(N) INTEGER*4 CALMAX,COMAX,EMAX,MAMAX,MPOMAX,PHMAX,SITMAX,SOMAX PARAMETER (COMAX = 22,PHMAX = 150,SOMAX = 20,EMAX = 10,SITMAX = 5, *MAMAX = 31,MPOMAX = 4,CALMAX=200) C — - C O M M O N BLOCK FOR SAVING INPUT DATA C — C O M M O N BLOCK FOR G-CALCULATION C END OF C O M M O N VARIABLES INTEGERM N,I,J NLANDA=INDATA(1,N) N C O M = INDATA(2,N) DO 500,1 = 1,10 500 lCOM(l) = INDATA(I + 2,N) C G0R = REDATA(1,N) H0R = REDATA(2,N) S0R = REDATA(3,N) V0R=REDATA(4,N) AA0 = REDATA(5,N) AAT=REDATA(6,N) BB0 = REDATA(7,N) BBT=REDATA(8,N) K1=REDATA(9,N) K2 = REDATA(10,N) K3 = REDATA(11,N) K4 = REDATA(12,N) K5 = REDATA(13,N) K6 = REDATA(14,N) K7=REDATA(15,N) K8 = REDATA(16,N) K9 = REDATA(17,N) DO 600,1 = 1,4 ] = 9*(l-1) ASPK(I) = REDATA(J +18,N) BSPK(I) = REDATA(] +19,N) TQ1 B(l) = REDATAO + 20,N) TEQ(I) = REDATA(J + 21,N) DVDT(I)=REDATA(J + 22,N) DVDP(l) = REDATA(J + 23,N) TRE(I) = REDATA(J + 24,N) DHTR(I) = REDATAO + 25,N) 600 DVTR(I) = REDATA(J + 26,N) TD0 = REDATA(54,N) TDMAX = REDATA(55,N) VADJ = REDATA(56,N) D1=REDATA(57,N) D2 = REDATA(58,N) D3 = REDATA(59,N) D4 = REDATA(60,N) D5 = REDATA(61,N) D6 = REDATA(62,N) D7 = REDATA(63,N) D8 = REDATA(64,N) D9 = REDATA(65,N) DO 610,1 = 1,10 610 FFCOM(l) = REDATAO + 65, N) C - ~ - CASE = CHDATA(N) C RDK=LODATA(1,N) VDW=LODATA(2,N) SPC = LODATA(3,N) C O M = LODATA(4,N) DIS = LODATA(5,N) RETURN END C—-- SUBROUTINE CALSTR INTECER*4 CALMAX,COMAX,EMAX,MAMAX,MPOMAX,PHMAX,SITMAX,SOMAX PARAMETER (COMAX = 22,PHMAX= 150,SOMAX = 20,EMAX = 10,SITMAX = 5, *MAMAX = 31,MPOMAX = 4,CALMAX = 200) C - — C O M M O N BLOCK FOR DATABASE C - — C O M M O N BLOCK FOR THERIAK C-—END OF C O M M O N VARIABLES INTEGERS N,l DO 500,N = 1,NPHA DO 600,1 = 1,NUN 600 X(N,I) = XX(N,I) NN(N) = 0.0D0 CC(N) = CGK(N) G(N) = CGK(N) NUMMER(N) = N EMCODE(N) = 0 IF (NULL(N)) THEN SUCC(N) = -N ELSE SUCC(N) = N END IF DO 605,1 = 1,EMAX 605 XEM(N,l) = 0.0D0 500 CONTINUE DO 505,N = 1,NUN SUCG(N) = -N 505 NN(N) = BULK(N) SUGNR = NPHA RETURN END C-—- SUBROUTINE PRININ INTEGER*4 CALMAX,COMAX,EMAX,MAMAX,MPOMAX,PHMAX,SITMAX,SOMAX PARAMETER (COMAX= 22,PHMAX= 150,SOMAX = 20,EMAX = 10,SITMAX= 5, *MAMAX=31,MPOMAX = 4,CALMAX = 200) C GLOBAL C O M M O N BLOCK C C O M M O N BLOCK FOR DATABASE C - — C O M M O N BLOCK FOR G-CALCULATION C - — C O M M O N BLOCK FOR THERIAK C—-END OF C O M M O N VARIABLES INTEGER*4 l,ll,IE,IS,IL00,IL02,N REAL*8 NNPC,SUMME CHARACTER*250 CH001 CHARACTER* 170 MODEL1,MODEL2 CHARACTER*20 MODEL0 WRITE (UNIT=6,FMT=120) P,PGAS,TC,T WRITE (UNIT=7,FMT=120) P,PGAS,TC,T 120 FORMAT ('1P =',F9.2,' bar P(Gas) =',F9.2,' bar T =', *F8.2,' C = ',F8.2,' K') IF (PRTLOG(2)) THEN WRITE (UNIT=6,FMT=119) WRITE (UNIT=7,FMT=119) 119 FORMAT('0COMPOSITION:',8X,'N',10X,'MOL%7' ') SUMME = 0.0 DO 540,1 = 1,NUN 540 SUMME = SUMME + DABS(NN(l)) DO 541,1 = 1,NUN NNPC = (NN(I)/SUMME)*100. WRITE (UNIT=6,FMT=121) l,CHNAME(l),NN(l),NNPC WRITE (UNIT = 7,FMT=121) l,CHNAME(l),NN(l),NNPC 121 FORMAT (1X,I3,2X,A8,1X,F11.6,2X,F11.6) IF (CHNAME(I)(5:6).EQ.' ') THEN CH = CHNAME(I) CHNAME(I) = ' 7/CH END IF 541 CONTINUE END IF IF (PRTLOG(3)) THEN I001=MIN0(15,NUN) WRITE (UNIT=6,FMT=122) (CHNAME(I),I = 1,I001) WRITE (UNIT=7,FMT=122) (CHNAME(I),I = 1,I001) 122 FORMAT ('0CONSIDERED PHASES:',15X,'C',7X,15(A5,1X)) DO 542,1001 =16,NUN,15 WRITE (UNIT=7,FMT=123) (CHNAME(I),I = I001,MIN0(I001+14,NUN)) 542 WRITE (UNIT=6,FMT=123) (CHNAME(I),I = I001,MINO(I001+ 14,NUN)) 123 FORMAT (42X,15(A5,1X)) WRITE (UNIT = 6,FMT='(" ")') WRITE (UNIT=7,FMT='(" ")') DO 545,K=1,NPHA WRITE (UNIT = CH001,FMT=124) K,NAME(K),GG(K) 124 FORMAT (1X,I3,2X,A16,1X,':',2X,F13.2) DO 544,1001 =1,NUN,15 DO 543,I = I001,MIN0(I001+14,NUN) I002 = 42 + MOD(I-1,15)*6 IF (XX(K,l).EQ.0.0) THEN WRITE (UNIT=CH001(I002+3:),FMT='("-")') ELSE WRITE (UNIT=CH001(I002:),FMT='(F6.2)') XX(K,I) END IF 543 CONTINUE WRITE (UNIT = 6,FMT='(A133)') CH001 WRITE (UNIT=7,FMT='(A133)') CH001 544 CH001=' ' 545 CONTINUE END IF IF (PRTLOG(4)) THEN IF (NSOL.NE.0) THEN WRITE (UNIT=6,FMT='("0SOLUTION PHASES:")') WRITE (UNIT = 7,FMT = '("0SOLUTION PHASES:")') END IF DO 600,IS = 1,NSOL WRITE (UNIT = CH001,FMT=130) IS,SOLNAM(IS) 130 FORMAT ('0',I3,2X,A16,' :',12X,'SOLUTION MODELL: ') IF (MODELL(IS).EQ.'I') WRITE (UNIT = CH001(54:),FMT=131) 131 FORMAT ('IDEAL ONE SITE MIXING') IF (MODELL(IS).EQ.'S') WRITE (UNIT = CH001(54:),FMT=132) NSITE(IS) 132 FORMAT ("'IDEAL"',I3,' SITE MIXING') IF (MODELL(IS).EQ.'F') WRITE (UNIT = CH001(54:),FMT=133) 133 FORMAT ('FROM EXTERNAL SUBROUTINE') IF (NMARG(IS).NE.O) THEN DO 550,1001=250,1,-1 550 IF (CH001(I001:I001).NE.' ') GO TO 20 20 WRITE (UNIT = CH001(I001+2:),FMT=134) 134 FORMAT ('+ MARGULES TYPE EXCESS FUNCTION') END IF WRITE (UNIT = 6,FMT = '(A133)') CH001 WRITE (UNIT=7,FMT = '(A133)') CH001 CH = ' ' 1001 = lNDEX(SOLNAM(IS),' ') DO 551,1002 = 1,1001-1 551 CH(I002:I002) = '-' WRITE (UNIT=6,FMT='(6X,A16)') CH WRITE (UNIT=7,FMT = '(6X,A16)') CH DO 599,IE = 1,NEND(1S) MODEL1 =' ' MODEL2 = ' ' MODEL0 = 'A(7/ABK(EM(IS,IE)) I001 = INDEX(MODELO,' ') MODELO(I001:) = ') = ' lL00 = l001+2 IF (MODELL(IS).EQ.'I') THEN MODEL2 = 'X('//MODEL0(3:IL00-2) IL02 = IL00-2 WRITE (UNIT = CH,FMT = '(F15.0)') SITMUL(IS,1) CALL FIBLA(CH,1001) I002 = INDEX(CH/.') CH(I002:) = ALPDIV(IS) I002 = I002 + INDEX(ALPDIV(IS)/ ')-1 IF (CH(I001-1:I002).NE.' 1 ') THEN MODEL1(IL02 + 1:) = CH(I001:1002-1) IL02 = IL02 + (I002-I001) END IF END IF IF (MODELL(IS).EQ.'S') THEN IL02 = 0 DO 552,1 = 1,NSITE(IS) MODEL2(IL02 + 1:) = '[' 1L02 = IL02 + 1 DO 553,11 = 1,NEQEM(lS,IE,l) MODEL2(IL02 + 1:) = ' X(7/ABK(EM(IS,EQEM(IS,IE,I,II))) I001=INDEX(MODEL2(IL02 + 1:),' ') IL02 = IL02 + I001-1 MODEL2(IL02 + 1:) = ') +' IL02 = IL02 + 3 553 CONTINUE MODEL2(IL02:) = ']' WRITE (UNIT = CH,FMT='(F16.0)') SITMUL(IS,I) CALL FIBLA(CH,I001) I002 = INDEX(CH,'.') CH(I002:) = ALPDIV(IS) I002 = I002 + INDEX(ALPDIV(IS),' ')-1 IF (CH(I001-1:I002).NE.' 1 ') THEN MODEL1(IL02 + 1:) = CH(I001:I002-1) IL02 = IL02 + (I002-I001) + 1 END IF 552 CONTINUE END IF IF (MODELL(IS).EQ.'F') THEN DO 554,1 = 1,NEMBAS(IS) 1001=1 554 IF (EMBCOD(IS,l).EQ.IE) C O TO 21 21 CONTINUE CALL SOLMOD(SOLNAM(IS),I001 ,MODEL2) IL02 = INDEX(MODEL2,' ')-1 END IF DO 555,1 = 1,IL02,85 K = MIN0(85,IL02-I + 1) N = IL00 + 37 CH001=' ' WRITE (UNIT=CH001(N + 1:),FMT = '(A85)') MODEL1(l:) WRITE(UNIT = 6,FMT='(A133)') CH001 WRITE(UNIT=7,FMT='(A133)') CH001 191 CH001=' ' IF (I.EQ.1) THEN WRITE (UNIT=CH001,FMT=135) IE,NAME(EM(IS,IE)),EM(IS,IE), *MODEL0(l:) 135 FORMAT (6X,I3,2X,A16,1X,'(',I3,')',3X,A15) END IF WRITE (UNIT=CH001(N + 1:),FMT = '(A85)') MODEL2(l:) WRITE(UNIT=6,FMT = '(A133)') CH001 WRITE(UNIT=7,FMT = '(A133)') CH001 555 CONTINUE 599 CONTINUE IF (NMARC(IS).CT.O) WRITE (UNIT = CH001,FMT=136) 136 FORMAT ('0',5X,'MARGULES PARAMETERS:') DO 558,1001 = 1,NMARG(IS),3 DO 557,1 = 1001,MIN00001 +2,NMARG(IS)) I002 = 30 + MOD(I-1,3)*28 CH001(I002:I002 + 1) = 'W(' 1002 = 1002 + 2 DO 556,K=1,POLY(IS,l) WRITE (UN1T=CH001(I002:I002),FMT = '(I1)') INDX(IS,l,K) 556 1002 = 1002 + 1 WRITE (UNIT=CH001(I002:),FMT=137) WG(IS,I) 137 FORMAT (') =',F13.2) 557 CONTINUE WRITE (UNIT=6,FMT = '(A133)') CH001 WRITE (UNIT=7,FMT = '(A133)') CH001 558 CH001=' ' WRITE (UNIT=6,FMT = '("0")') WRITE (UNIT=7,FMT = '("0")') 600 CONTINUE END IF RETURN END C—-- Q^* **************************** * SUBROUTINE THERIA INTEGERM CALMAX,COMAX,EMAX,MAMAX,MPOMAX,PHMAX,SITMAX,SOMAX PARAMETER (COMAX = 22,PHMAX = 150,SOMAX= 20,EMAX = 10,SITMAX= 5, *MAMAX = 31 ,MPOMAX = 4,CALMAX = 200) C-—GLOBAL C O M M O N BLOCK C — - C O M M O N BLOCK FOR THERIAK C END OF C O M M O N VARIABLES INTEGER*4 GNOM,LOCMIN(SOMAX+1),LOMIN1(SOMAX+1),NMAX, > STEM(SOMAX +1 ),TRY REAL*8 DELTAX,DELXXX(0:CALMAX),DlSTA,CSOL,GTEST,GTOT, >MINCOM(SOMAX+ 1,EMAX,EMAX),MUE(EMAX),VEKTOR(EMAX),XXEM(EMAX), > DXEND, DXLAST(SOMAX +1, EMAX) C O M M O N /TOIN/ GNOM,LOCMIN,LOMIN1,NMAX,STEM,TRY C O M M O N /TORE/ D E LTAX, D E LXXX, DI ST A, G SO L, GTEST, GTOT, MIN C O M , >MUE,VEKTOR,XXEM,DXEND,DXLAST INTEGER*4 l,ll,IX,l001,K,LOO2 REAL*8 FF,FX,SUMME CHARACTER*20 CH LOGICAL*4 CODE NMAX=NPHA GTOT=100.0D0 GTEST= DABS(TEST) TRY=1 IF (LO1MAX.LT.0) THEN GTEST=-100.D0 L01MAX = -L01MAX END IF IF (NMAX.GT.CALMAX) THEN WRITE (UNIT=6,FMT=142) CALMAX,NMAX 142 FORMAT (' CALMAX = ',I5/' NMAX = ',I5) STOP END IF DO 500,1 = 1,NUN IF (NN(I).EQ.O.ODO) THEN DO 502,II = NUN + 1,NMAX IF (X(ll,l).NE.0.OD0) G(II) = 1D25 502 CONTINUE END IF 500 CONTINUE DO 650,LOOl=1,LO1MAX IF (GTOT.LE.GTEST) THEN TRY = TRY +1 IF (TRY.GT.2) C O TO 25 DO 410,1 = 1,NSOL 410 STEM(I) = 1 END IF IF (TEST.LT.0.0D0) THEN WRITE (UNIT=6,FMT='(" ")') WRITE (UNIT=7,FMT='(" ")') END IF IF (L001.CT.1) THEN CALL ADDPH ELSE DO 400,1 = 1,NUN 400 G(I) = 1D20 END IF DO 602,1 = 1,NUN IX=I FX=DABS(X(I,I)) DO 601,11 = 1 + 1,NUN IF (DABS(X(ILD).GT.FX) THEN IX = II FX = DABS(X(II,I)) END IF 601 CONTINUE 1001=1 IF (IX.NE.l) CALL COLCHC(1001,IX) 602 CALL REDUCE(I001) CODE = .TRUE. DO 640,LOO2 = 1,25 IF (.NOT.CODE) GO TO 24 CODE = . FALSE. DO 630,K = (NUN + 1),NMAX IF (SUCG(K).LT.O) GOTO 630 SUMME = OD0 DO 603,11 = 1,NUN 603 SUMME = SUMME + G(II)*X(K,II) IF (SUMME-G(K).CT.ODO) THEN DO 604,1 = 1,NUN 604 IF (X(K,I).GT.0D0) CO TO 23 23 IF (I.LT.NUN + 1) THEN FX = NN(I)/X(K,I) IX = I DO 605,I=(IX + 1),NUN IF (X(K,I).GT.0D0) THEN FF = NN(I)/X(K,I) IF (FF.LT.FX) THEN FX = FF IX=I END IF END IF 605 CONTINUE DO 606,1 = 1,NUN 606 NN(I) = NN(I)-FX*X(K,I) NN(K) = FX NN(IX) = OD0 1001 =K CALL COLCHG(K,IX) CALL REDUCE(IX) CODE = .TRUE. END IF END IF 630 CONTINUE 640 CONTINUE 24 GTOT = 0.0D0 DO 641,1 = 1,NUN 641 GTOT = GTOT+DABS(G(l)) CALL CLEAN IF (TEST.LT.0.0D0) THEN l001 = LOO2-1 CH = ' ' WRITE (UNIT=6,FMT=150) LOO1,l001,GTOT,CH WRITE (UNIT=7,FMT=150) LOO1,l001,GTOT,CH 150 FORMAT (' LOOT =',I4,4X,'L002 =',I4,4X,'G(-) =',1PE12.5,2X,A17) CALL PRTSTR(1,NUN) END IF 650 CONTINUE 25 CALL PRTCAL RETURN END C — - SUBROUTINE CLEAN INTEGER*4 CALMAX,COMAX,EMAX,MAMAX,MPOMAX,PHMAX,SITMAX,SOMAX PARAMETER (COMAX = 22, PH MAX = 150,SOMAX = 20,EMAX = 10,SITMAX= 5, * M A M AX = 31 ,MPOMAX = 4,CALMAX = 200) C GLOBAL C O M M O N BLOCK C - — C O M M O N BLOCK FOR THERIAK C-—END OF C O M M O N VARIABLES INTECERM CNOM / LOCMIN(SOMAX+1 ),LOMIN1(SOMAX +1 ),NMAX, > STEM(SOMAX +1 ),TRY REAL*8 DELTAX,DELXXX(0:CALMAX),DISTA /CSOL,CTEST,GTOT, >MINCOM(SOMAX + 1,EMAX,EMAX),MUE(EMAX),VEKTOR(EMAX),XXEM(EMAX), > DXEND, DXLAST(SOMAX +1 ,EMAX) C O M M O N /TOIN/ GNOM,LOCMlN,LOMlN1,NMAX,STEM,TRY C O M M O N /TORE/ DELTAX,DELXXX,DISTA,GSOL,GTEST,GTOT,MINCOM, >MUE,VEKTOR,XXEM,DXEND,DXLAST INTEGERS K,li IF (L001.EQ.1) THEN DO 500,K=1,NMAX 500 IF (SUGG(K).LT.O) C(K) = 0.0D0 GTOT=100.0D0 END IF DO 501,K = (NUN + 1),NMAX DO 501,11 = 1,NUN 501 G(K) = G(K)-X(K,II)*G(II) DO 502,K=1,NUN 502 G(K) = 0.0D0 DO 503,11 = 1,NSOL LOMIN1(ll) = LOCMIN(ll) LOCMIN(II) = 0 503 STEM(II) = 0 K = 1 1001 IF (K.CT.NMAX) C O TO 1 IF (NUMMER(K).CT.O) THEN IF (EMSOL(NUMMER(K)).CT.0.AND.K.LE.NUN) * STEM(EMSOL(NUMMER(K))) = STEM(EMSOL(NUMMER(K))) +1 DO 504,11 = 1,NUN 504 X(K,II) = XX(NUMMER(K),II) CC(NUMMER(K)) = C(K) ELSE IF (K.GT.NUN.AND.(GTOT.GT.GTEST.OR.TRY.EQ.D) THEN IF (NUMMER(K).EQ.O) THEN CALL COLCHC(K,NMAX) N MAX = N M AX-1 K=K-1 END IF ELSE CALL XSOL(K) IF (K.LE.NUN) THEN LOCMIN(EMCODE(K))= LOCMIN(EMCODE(K)) +1 DXLAST(EMCODE(K),LOCMIN(EMCODE(K))) = DELXXX(K) DO 505,11 = 1,NEND(EMCODE(K)) 505 MINCOM(EMCODE(K),LOCMIN(EMCODE(K)),ll) = XEM(K,ll) END IF END IF END IF K=K+1 GO TO 1001 1 RETURN END 195 C - — Q* **************************** * SUBROUTINE ADDPH INTEGERS CALMAX,COMAX,EMAX,MAMAX,MPOMAX,PHMAX,SITMAX,SOMAX PARAMETER (COMAX = 22, PHMAX= 150,SOMAX = 20,EMAX= 10,SITMAX = 5, t *MAMAX = 31,MPOMAX = 4,CALMAX = 200) C—-'-GLOBAL C O M M O N BLOCK C C O M M O N BLOCK FOR THERIAK C—-END OF C O M M O N VARIABLES INTEGERS GNOM,LOCMIN(SOMAX+1),LOMIN1(SOMAX+1),NMAX, > STEM(SOM AX +1 ),TRY REAL*8 DELTAX,DELXXX(0:CALMAX),DISTA,GSOL,GTEST,GTOT, >MINCOM(SOMAX+1,EMAX,EMAX),MUE(EMAX),VEKTOR(EMAX),XXEM(EMAX), > DXEND, DXLAST(SOMAX +1, EMAX) C O M M O N /TOIN/ GNOM,LOCMIN,LOMIN1,NMAX,STEM,TRY C O M M O N /TORE/ DELTAX,DELXXX,DISTA,GSOL,GTEST,GTOT,MINCOM, >MUE,VEKTOR,XXEM,DXEND,DXLAST INTEGER*4 IS,I1,START,I001,I002 DO 502,IS = 1,NSOL 1001 =IS IF (NMARG(IS).GT.0.OR.MODELL(IS).NE.'l') THEN IF (STEM(IS).EQ.0.AND.LOMIN1(IS).EQ.LOCMIN(IS)) THEN I1=M1NO(1,LOCMIN(IS)) ELSE I1=-NEND(IS) END IF DO 501,START=I1,LOCMIN(IS) GNOM=0 1002 = START CALL MARMIN(I001,I002) 501 CALL NEWPH(IOOI) ELSE CALL MINGCI001) CALL NEWPH(I001) END IF 502 CONTINUE RETURN END C - — (2* **************************** * SUBROUTINE NEWPH(IS) 1NTEGER*4 CALMAX,COMAX,EMAX,MAMAX,MPOMAX,PHMAX,SITMAX,SOMAX PARAMETER (COMAX = 22,PHMAX = 150,SOMAX = 20,EMAX=10,SITMAX=5, *MAMAX = 31,MPOMAX = 4,CALMAX = 200) C GLOBAL C O M M O N BLOCK C C O M M O N BLOCK FOR THERIAK C — E N D OF C O M M O N VARIABLES INTEGER*4 GNOM,LOCMIN(SOMAX+1),LOMIN1(SOMAX + 1),NMAX, > STEM(SOM AX +1 ),TRY REAL*8 DELTAX,DELXXX(0:CALMAX),DISTA,GSOL,GTEST,GTOT, > Ml NCOM(SOMAX +1, EMAX, EMAX),MU E(EMAX), VEKTOR(EMAX),XXEM(EMAX), > DXEND,DXLAST(SOMAX +1,EMAX) C O M M O N /TOIN/ GNOM,LOCMIN,LOMIN1,NMAX,STEM,TRY C O M M O N /TORE/ DELTAX,DELXXX,DISTA,GSOL,GTEST,GTOT,MINCOM, > MUE,VEKTOR,XXEM, DXEND, DXLAST INTEGERS IS,I NMAX = NMAX+1 SUCNR=SUGNR+1 IF (NMAX.GT.CALMAX) THEN WRITE (UNIT=6,FMT=100) CALMAX,NMAX 100 FORMAT (' CALMAX = ',I5/' NMAX = ',I5) STOP END IF NN(NMAX) = 0.0D0 G(NMAX) = GSOL NUMMER(NMAX) = 0 EMCODE(NMAX) = lS DELXXX(NMAX)= DXEND DO 501,I = 1,NEND(IS) 501 XEM(NMAX,I) = XXEM(I) SUGG(NMAX) = SUGNR CALL XSOL(NMAX) l = NMAX IF (TEST.LT.0.0D0) CALL PRTSTR(NMAX,I) RETURN END C - — Q-***************************** * SUBROUTINE XSOL(K) INTEGERS CALMAX,COMAX,EMAX,MAMAX,MPOMAX,PHMAX,SITMAX,SOMAX PARAMETER (COMAX = 22,PHMAX = 150,SOMAX = 20,EMAX=10,SITMAX=5, *MAMAX = 31,MPOMAX = 4,CALMAX=200) C-—GLOBAL C O M M O N BLOCK C C O M M O N BLOCK FOR THERIAK C-—END OF C O M M O N VARIABLES INTEGERS GNOM,LOCMIN(SOMAX + 1),LOMIN1(SOMAX+1),NMAX, > STEM(SOMAX +1 ),TRY REAL*8 DELTAX,DELXXX(0:CALMAX),DISTA,GSOL,GTEST,GTOT, >MINCOM(SOMAX + 1,EMAX,EMAX),MUE(EMAX),VEKTOR(EMAX),XXEM(EMAX)/ > DXEND,DXLAST(SOMAX+1 ,EMAX) C O M M O N /TOIN/ GNOM,LOCMIN,LOMIN1,NMAX,STEM,TRY C O M M O N /TORE/ DELTAX,DELXXX,DISTA,GSOL,CTEST,GTOT,MINCOM, >MUE,VEKTOR,XXEM,DXEND,DXLAST INTEGER*4 l,K,N DO 501,N = 1,NUN X(K,N) = 0.0D0 DO 501,I = 1,NEND(EMCODE(K)) X(K,N) = X(K,N) + XEM(K,l)*XX(EM(EMCODE(K),l),N) 501 CONTINUE RETURN END C f^* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * SUBROUTINE MING(IS) INTEG ER * 4 CALM AX, COMAX, EMAX, M AMAX, MPOM AX, PHM AX, SITM AX, SOMAX PARAMETER (COMAX = 22,PHMAX = 150,SOMAX = 20,EMAX = 10,SITMAX = 5, * MAM AX = 31 , M POMAX = 4, C ALMAX = 200) C—-GLOBAL C O M M O N BLOCK C C O M M O N BLOCK FOR THERIAK C END OF C O M M O N VARIABLES INTECERM GNOM,LOCMIN(SOMAX+1),LOMIN1(SOMAX+1),NMAX, > STEM(SOM AX +1 ),TRY REAL*8 DELTAX,DELXXX(0:CALMAX),D1STA,GSOL,GTEST,GTOT, >MINCOM(SOMAX+1,EMAX,EMAX),MUE(EMAX),VEKTOR(EMAX),XXEM(EMAX), > DXEND, DXLAST(SOM AX +1, EMAX) C O M M O N /TOIN/ GNOM,LOCMIN,LOMIN1,NMAX,STEM,TRY C O M M O N /TORE/ DELTAX,DELXXX,DlSTA,GSOL,CTEST,CTOT,MINCOM, >MUE,VEKTOR,XXEM,DXEND,DXLAST INTEGERS IS,I REAL*8 CMIN,SUMME,RTA RTA=RT*ALPHA(lS) CMIN = CC(EM(IS,1)) DO 501,I = 2,NEND(IS) 501 GMIN = DMIN1(GMIN,GG(EM(1S,I))) SUMME = 0.0D0 DO 502,1 = 1,NEND(IS) XXEM(I) = 1D0/DEXP(DMIN1(150D0,(CC(EM(IS,I))-GMIN)/(RTA))) 502 SUMME = SUMME + XXEM(I) DO 503,1 = 1,NEND(IS) 503 XXEM(I) = XXEM(I)/SUMME CSOL = 0.0D0 DO 504,1 = 1,NEND(IS) 504 GSOL = CSOL + XXEM(l)*CC(EM(!S,l)) + RTA*XXEM(l)*DLOC(XXEM(I)) RETURN END C **************************** * SUBROUTINE MARMIN(IS,START) INTEGER*4 CALMAX,COMAX,EMAX,MAMAX,MPOMAX,PHMAX,SITMAX,SOMAX PARAMETER (COMAX = 22,PHMAX = 150,SOMAX= 20,EMAX= 10,SITMAX= 5, *MAMAX= 31 ,MPOMAX = 4,CALMAX= 200) C-—GLOBAL C O M M O N BLOCK C C O M M O N BLOCK FOR THERIAK C—-END OF C O M M O N VARIABLES INTEGER*4 GNOM,LOCMIN(SOMAX+1 ),LOMIN1(SOMAX+1 ),NMAX, > STEM(SOMAX +1 ),TRY REAL* 8 D E LTAX, D ELXXX( 0: CALM AX), DI STA, G SO L, GTEST, GTOT, >MINCOM(SOMAX+ 1,EMAX,EMAX),MUE(EMAX),VEKTOR(EMAX),XXEM(EMAX), > DXEND, DXLAST(SOMAX +1, EMAX) C O M M O N /TOIN/ GNOM,LOCMIN,LOMIN1,NMAX,STEM,TRY C O M M O N /TORE/ DELTAX,DELXXX,DISTA,GSOL,CTEST,GTOT,MINCOM, >MUE,VEKTOR,XXEM,DXEND,DXLAST INTEGER*4 l,IS,START,STEP,STEP2,NSTEP,L3,L1,ll,1001,1002 REAL*8 XXSC(EMAX),GSC,XXX(3,EMAX),GGG(3),AR001(EMAX), *AR002(EMAX),SUMME,F001,FN,FF,FX CHARACTER* 133 CH001 IF (START.EQ.0) THEN DO 501,I = 1,(NEND(IS)-1) 501 XXSC(l) = 0.0D0 XXSC(NEND(IS)) = 1.0D0 GSOL = GG(EM(IS,NEND(IS))) DO 502,1001 =1,NEND(IS) 502 XXEM(I001) = XXSC(I001) SUMME = 0.0D0 1001 XXSC(NEND(IS)-1) = XXSC(NEND(IS)-1) + DXSCAN SUMME = SUMME + DXSCAN XXSC(NEND(IS)) = 1.0D0-SUMME DO 503,I = (NEND(IS)-1),2,-1 IF (XXSC(NEND(IS)).GE.0.0D0) CO TO 1 XXSC(I-1) = XXSC(I-1) + DXSCAN SUMME = SUMME + DXSCAN DO 504,II = I,(NEND(IS)-1) SUMME = SUMME-XXSC(I) 504 XXSC(l) = 0.0D0 503 XXSC(NEND(IS)) = 1.0D0-SUMME 1 IF (XXSC(NEND(IS)).LT.0.0D0) THEN GO TO 10 ELSE CALL CNONID(IS,XXSC,GSC) IF (GSC.LT.GSOL) THEN DO 505,1001 =1,NEND(IS) 505 XXEM(I001) = XXSC(I001) GSOL = GSC END IF END IF GO TO 1001 10 DELTAX= DXSTAR NSTEP = STPSTA END IF IF (START.LT.0) THEN II =-START DO 701,I = 1,NEND(IS) 701 XXEM(l)=0.0D0 XXEM(II)=1.0D0 GSOL = GG(EM(IS,ll)) DELTAX= DXSTAR NSTEP = STPSTA END IF IF (START.GT.0) THEN DO 506,1 = 1,NEND(IS) 506 XXEM(l) = MINCOM(IS,START,l) GSOL = 0.0D0 DELTAX = DXLAST(IS,START) NSTEP = STPMAX END IF IF (TEST.LT.0.ODO) THEN WRITE (UNIT=CH001,FMT=100) DELTAX 100 FORMAT (' START: DELTAX = ',1PE12.5) DO 512,1001 =1,NEND(IS),10 DO 511,I = I001,MIN0(I001 +9,NEND(IS)) I002 = 34 + MOD(I-1,10)*10 511 WRITE (UNIT = CH001(I002:),FMT = '(F8.6)') XXEM(l) WRITE (UNIT = 6,FMT = '(A133)') CH001 WRITE (UNIT=7,FMT='(A133)') CH001 512 CH001=' ' END IF DO 513,1 = 1,NEND(IS) 513 XXX(1,I) = XXEM(I) CCC(1) = CSOL LI =1 IF (TRY.GT.LAND.CTOT.LE.CTEST) NSTEP=500 DO 516,STEP=1,MIN0(2,NSTEP) DO 514,1001 = 1,NEND(IS) 514 AR002(I001)=XXX(STEP,I001) CALL MUECAL(IS,AR002) CALL STEEP(IS) L1=STEP+1 CALL ETC(IS,AR002,GGG(STEP),AR001,GGG(L1)) DO 515,1001 =1,NEND(IS) 515 XXX(L1,I001) = AR001(I001) 516 CONTINUE DO 525,STEP = 3,NSTEP IF. (DELTAX.LE.DXMIN) GO TO 15 L3 = MOD(STEP-1,3) + 1 L1=MOD(STEP,3)+1 DO 517,1001 =1,NEND(1S) AR001(I001) = XXX(L1,I001) 517 AR002(I001) = XXX(L3,I001) CALL MUECAL(IS,AR002) GSC = 0.0D0 DO 518,1 = 1,NEND(IS) 518 GSC = GSC + MUE(I)*XXX(L1,I) CALL DISTAN(IS,AR001,AR002) DELTAX = DISTA IF (DELTAX.GT.DXMIN) THEN IF (GSC.LT.GGG(L3)) THEN CALL STEEP(IS) DELTAX = DELTAX/2.0D0 ELSE SUMME = 0.0D0 DO 519,1 = 1,NEND(IS) VEKTOR(l) = XXX(L3,l)-XXX(L1,l) 519 SUMME = SUMME + VEKTOR(l)*VEKTOR(l) SUMME = DSQRT(SUMME) IF (SUMME.GT.0.0D0) THEN DO 520,1 = 1,NEND(IS) 520 VEKTOR(l)=VEKTOR(l)/SUMME END IF END IF CALL ETC(IS,AR002,GGG(L3),AR001,GGG(L1)) DO 521,1001 =1,NEND(1S) 521 XXX(L1,I001) = AR001(I001) END IF 525 CONTINUE 15 CALL D1STAN(IS,AR001,XXEM) STEP2 = 0 IF (DISTA.LT.DXMIN) THEN DELTAX = DXSTAR STEP = 2-STEP FN = NEND(IS) 200 FX=DSQRT((FN-1.0D0)/FN) FF=-DSQRT(1.0DO/(FN*(FN-1.0DO))) DO 526,1 = 1,NEND(IS) 526 VEKTOR(i) = FF DO 530,STEP2 = 1,NEND(IS) IF (STEP2.NE.1) VEKTOR(STEP2-1) = FF VEKTOR(STEP2) = FX F001=DXMIN CALL VECADD(IS,AR001,F001,XXSC) CALL CNONID(IS,XXSC,CSC) IF (GSC.LT.CCC(LI)) THEN CALL ETC(IS,XXSC,CSC,AR001,CCC(L1)) END IF 530 CONTINUE END IF CALL DISTAN(1S,AR001,XXEM) DXEND = DMAX1(DISTA,DXMIN) DO 531,1 = 1,NEND(1S) 531 XXEM(I) = AR001(I) GSOL = CGC(L1) IF (TEST.LT.0.0D0) THEN I001=STEP-1 WRITE (UNIT = CH001,FMT=105) SOLNAM(1S),1001 105 FORMAT (1X,A16,2X,'STEP =',I4) DO 533,1001 =1,NEND(IS),10 DO 532,1 = 1001,MIN0(I001 +9,NEND(IS)) I002 = 34 + MOD(I-1,10)*10 532 WRITE (UNIT = CH001(I002:),FMT = '(F8.6)') XXEM(I) WRITE (UNIT=6,FMT='(A133)') CH001 WRITE (UNIT=7,FMT='(A133)') CH001 533 CH001=' ' WRITE (UNIT=6,FMT=106) D E LTAX, DI ST A, C N O M WRITE (UNIT=7,FMT=106) DELTAX,DISTA,GNOM 106 FORMAT (8X/DELTAX = ',1PE12.5,4X,'DISTA = ',1PE12.5, *4X,'CCALC = ',14) END IF RETURN END C Q^* **************************** * SUBROUTINE STEEP(IS) INTEGERS CALMAX,COMAX,EMAX,MAMAX,MPOMAX,PHMAX,SITMAX,SOMAX PARAMETER (COMAX = 22,PHMAX = 150,SOMAX = 20,EMAX= 10,SITMAX= 5, * MAMAX = 31 ,MPOMAX = 4,CALMAX = 200) C-—GLOBAL C O M M O N BLOCK C - — C O M M O N BLOCK FOR THERIAK C—-END OF C O M M O N VARIABLES INTEGER*4 CNOM,LOCMIN(SOMAX+1),LOMIN1(SOMAX+1),NMAX, > STEM(SOM AX +1 ),TRY REAL*8 DELTAX,DELXXX(0:CALMAX),DISTA,CSOL,GTEST,GTOT, >MINCOM(SOMAX + 1,EMAX,EMAX),MUE(EMAX),VEKTOR(EMAX),XXEM(EMAX), > DXEND, DXLAST(SOM AX +1, EMAX) C O M M O N /TOIN/ GNOM,LOCMIN,LOMIN1,NMAX,STEM,TRY C O M M O N /TORE/ DELTAX,DELXXX,DISTA,CSOL,CTEST,GTOT,MINCOM, >MUE,VEKTOR,XXEM,DXEND,DXLAST INTECERM IS,1 REAL*8 DELMUE,SUMME DELMUE = O.ODO SUMME = O.ODO DO 501,1 = 1,NEND(IS) 501 DELMUE = DELMUE + MUE(!) DELMUE= DELMUE/NEND(IS) DO 502,1 = 1,NEND(IS) VEKTOR(l) = DELMUE-MUE(l) 502 SUMME = SUMME + VEKTOR(l)*VEKTOR(l) SUMME = DSQRT(SUMME) IF (SUMME.GT.0.0D0) THEN DO 504,1 = 1,NEND(IS) 504 VEKTOR(l) = VEKTOR(l)/SUMME END IF RETURN END C - — SUBROUTINE DISTAN(IS,X1,X2) INTEGERM CALMAX,COMAX,EMAX,MAMAX,MPOMAX,PHMAX,SITMAX,SOMAX PARAMETER (COMAX = 22,PHMAX=150,SOMAX=20,EMAX = 10,SITMAX = 5, *MAMAX=31,MPOMAX = 4,CALMAX = 200) C-—GLOBAL C O M M O N BLOCK C - — C O M M O N BLOCK FOR THERIAK C-—END OF C O M M O N VARIABLES INTEGERS GNOM,LOCMIN(SOMAX+1),LOMIN1(SOMAX + 1),NMAX, > STEM(SOMAX +1 ),TRY REAL*8 D E LTAX, D E LXXX( 0: CALM AX), DISTA, GSO L, GTEST, GTOT, >MINCOM(SOMAX + 1,EMAX,EMAX),MUE(EMAX),VEKTOR(EMAX),XXEM(EMAX), >DXEND,DXLAST(SOMAX+1,EMAX) C O M M O N /TOIN/ GNOM,LOCMIN,LOMIN1,NMAX,STEM,TRY C O M M O N /TORE/ DELTAX,DELXXX,DISTA,GSOL,GTEST,GTOT,MINCOM, >MUE,VEKTOR,XXEM,DXEND,DXLAST INTEGER*4 1S,I REAL*8 X1(EMAX),X2(EMAX),DIFF DISTA = 0.0D0 DO 501,I = 1,NEND(IS) DIFF = X2(I)-X1(1) 501 DISTA = DISTA + DIFF*DIFF DI STA = DSQ RT( DI STA) RETURN END C SUBROUTINE ETC(lS,XSTART,GSTART,XNEW,GNEW) INTEGERM CALMAX,COMAX,EMAX,MAMAX,MPOMAX,PHMAX,SITMAX,SOMAX PARAMETER (COMAX = 22,PHMAX= 150,SOMAX= 20,EMAX = 10,SITMAX = 5, * MAMAX = 31 ,MPOMAX = 4,CALMAX = 200) C GLOBAL C O M M O N BLOCK C C O M M O N BLOCK FOR THERIAK C—-END OF C O M M O N VARIABLES INTEGER*4 GNOM,LOCMIN(SOMAX + 1),LOMIN1(SOMAX+1),NMAX, > STEM(SOMAX +1 ),TRY REAL*8 DELTAX,DELXXX(0:CALMAX),DISTA,GSOL,GTEST,GTOT, >MINCOM(SOMAX+1,EMAX,EMAX),MUE(EMAX),VEKTOR(EMAX),XXEM(EMAX), > DXEND, DXLAST(SOMAX +1, EMAX) C O M M O N /TOIN/ CNOM,LOCMIN,LOMIN1,NMAX,STEM,TRY C O M M O N /TORE/ DELTAX,DELXXX,DISTA,CSOL,CTEST,GTOT,MINCOM, >MUE,VEKTOR,XXEM,DXEND,DXLAST INTEGERS IS,I,II,I001 REAL*8 XSTART(EMAX),GSTART,XOLD(EMAX),GOLD,XNEW(EMAX), *GNEW,DELX DELX= DELTAX GOLD = CSTART DO 500,1001 = 1,NEND(IS) 500 XOLD(1001) = XSTART(I001) CALL VECADD(IS,XSTART,DELX,XNEW) CALL GNONID(IS,XNEW,GNEW) DO 502,11 = 1,CCMAX IF (CNEW.CE.COLD) CO TO 1 DELX=DELX+DELTAX GOLD = GNEW DO 501,1001 = 1,NEND(IS) 501 XOLD(I001) = XNEW(I001) CALL VECADD(IS,XOLD,DELX,XNEW) 502 CALL CNONID(IS,XNEW,CNEW) 1 GNEW = GOLD DO 503,1001 =1,NEND(IS) 503 XNEW(I001) = XOLD(I001) DO 504,1 = 1,GCMAX IF (GNEW.LT.GSTART.OR.DELX.LE.DXMIN) GO TO 2 DELX=DELX/2.0D0 CALL VECADD(IS,XSTART,DELX,XNEW) 504 CALL GNONID(IS,XNEW,GNEW) 2 IF (GNEW.GT.GSTART) THEN GNEW = CSTART DO 505,1001 =1,NEND(IS) 505 XNEW(I001) = XSTART(I001) END IF RETURN END C — - SUBROUTINE VECADD(IS,X1,DELX,X2) INTEGER*4 CALMAX,COMAX,EMAX,MAMAX,MPOMAX,PHMAX,SITMAX,SOMAX PARAMETER (COMAX = 22,PHMAX = 150,SOMAX=20,EMAX=10,SITMAX=5, *MAMAX = 31,MPOMAX = 4,CALMAX = 200) C—-GLOBAL C O M M O N BLOCK C - — C O M M O N BLOCK FOR THERIAK C — E N D OF C O M M O N VARIABLES INTEGER*4 GNOM,LOCMIN(SOMAX+1),LOMIN1(SOMAX+1),NMAX, > STEM(SOMAX +1 ),TRY REAL*8 DELTAX,DELXXX(0:CALMAX),DISTA,GSOL,GTEST,GTOT, >MINCOM(SOMAX+1,EMAX,EMAX),MUE(EMAX),VEKTOR(EMAX),XXEM(EMAX), > DXEND, DXLAST(SOMAX +1, EMAX) C O M M O N /TOIN/ GNOM,LOCMIN,LOMIN1,NMAX,STEM,TRY C O M M O N /TORE/ DELTAX,DELXXX,D!STA,CSOL,GTEST,CTOT,MINCOM, > MU E, VEKTOR,XXEM, DXEND, DXLAST INTECERM IS,I REAL*8 X1(EMAX),X2(EMAX),DELX,SUMME SUMME = 0.0D0 DO 501,I = 1,NEND(IS) X2(l)=X1(l)+VEKTOR(l)*DELX IF (X2(I).LT.0.0D0) X2(I) = 0.0D0 501 SUMME = SUMME + X2(I) IF (SUMME.GT.0.0D0) THEN DO 502,1 = 1,NEND(IS) 502 X2(I) = X2(I)/SUMME END IF RETURN END C Q* **************************** * SUBROUTINE GNONID(IS,XXX,CGC) INTEGER*4 CALMAX,COMAX,EMAX,MAMAX,MPOMAX,PHMAX,SITMAX,SOMAX PARAMETER (COMAX = 22,PHMAX = 150,SOMAX = 20,EMAX= 10,SITMAX = 5, *MAMAX=31,MPOMAX = 4,CALMAX = 200) C- - -GLOBAL C O M M O N BLOCK C — - C O M M O N BLOCK FOR THERIAK C-—END OF C O M M O N VARIABLES INTECER*4 CNOM,LOCMIN(SOMAX + 1),LOMIN1(SOMAX+1),NMAX, > STEM(SOMAX +1 ),TRY REAL* 8 D E LT AX, D E LXXX( 0:CALMAX),DISTA,CSOL,GTEST,GTOT, >MINCOM(SOMAX+1,EMAX,EMAX),MUE(EMAX),VEKTOR(EMAX),XXEM(EMAX), >DXEND,DXLAST(SOMAX + 1,EMAX) C O M M O N /TOIN/ CNOM,LOCMIN,LOMIN1,NMAX,STEM,TRY C O M M O N /TORE/ DELTAX,DELXXX,DISTA,GSOL,CTEST,GTOT,MINCOM, > MUE,VEKTOR,XXEM,DXEND,DXLAST INTECER*4 IS,N,I,I001 REAL*8 XXX(EMAX),AAA(EMAX),XM,CCG,RTA C N O M = C N O M + 1 CCC = 0.0D0 RTA=RT*ALPHA(IS) IF (MODELL(IS).EQ.'I') THEN DO 501,1001 =1,NEND(IS) 501 AAA(I001) = XXX(1001) ELSE CALL ACTIVI(IS,XXX,AAA) END IF DO 502,1 = 1,NEND(IS) IF (AAA(I).CT.O.ODO) * C C C = GCC + XXX(l)*CC(EM(IS,l)) + RTA*XXX(I)*DLOC(AAA(i)) 502 CONTINUE DO 504,N = 1,NMARC(IS) XM=1.0D0 DO 503,1 = 1,POLY(IS,N) 503 XM = XM*XXX(lNDX(IS,N,l)) 504 CCG = CGC + WC(IS,N)*XM RETURN END 204 C—-- SUBROUTINE MUECAL(IS,XXX) INTECERM CALM AX, C O M AX, EM AX, MAM AX, M POM AX, PHMAX,SITM AX,SOMAX PARAMETER (COMAX = 22,PHMAX =150,SOMAX= 20,EMAX =10,SITMAX = 5, * MAMAX = 31 , M POMAX = 4, CALMAX = 200) C-—GLOBAL C O M M O N BLOCK C C O M M O N BLOCK FOR THERIAK C-—END OF C O M M O N VARIABLES INTEGER*4 GNOM,LOCMIN(SOMAX+1),LOMIN1(SOMAX+1),NMAX, > STEM(SOMAX +1 ),TRY REAL* 8 D E LTAX, D ELXXX( 0: CALMAX), DI STA, GSO L, GTEST, GTOT, >MINCOM(SOMAX+1,EMAX,EMAX),MUE(EMAX),VEKTOR(EMAX),XXEM(EMAX), >DXEND,DXLAST(SOMAX + 1,EMAX) C O M M O N /TOIN/ G N O M , LOCMIN, LOM l N1, NM AX, STEM,TRY C O M M O N /TORE/ DELTAX,DELXXX,DlSTA,GSOL,GTEST,GTOT,MINCOM, >MUE,VEKTOR,XXEM,DXEND,DXLAST INTEGER*4 IS,IE,N,I,I001 REAL * 8 XXX( EM AX), AAA( EM AX), FF(M AM AX), RTA RTA=RT*ALPHA(IS) DO 501,N = 1,NMARG(IS) FF(N) = WG(IS,N) DO 500,1 = 1,POLY(IS,N) IF (FF(N).EQ.O.ODO) GO TO 1 500 FF(N) = FF(N)*XXX(INDX(IS,N,I)) 1 CONTINUE 501 CONTINUE IF (MODELL(IS).EQ.T) THEN DO 502,1001 = 1,NEND(IS) 502 AAA(I001) = XXX(I001) ELSE CALL ACTIVI(IS,XXX,AAA) END IF DO 504,IE = 1,NEND(IS) IF (AAA(IE).LE.O.ODO) THEN MUE(IE) = -1D20 ELSE MUE(IE) = GG(EM(IS,IE)) + RTA*DLOG(AAA(IE)) DO 503,N = 1,NMARG(IS) 503 MUE(IE) = MUE(IE) + FF(N)*(QQ(IS,N,IE)/XXX(IE) + (1-POLY(IS,N))) END IF 504 CONTINUE RETURN END C—-- SUBROUTINE ACTIVI(IS,XXX,AAA) INTEGER*4 CALMAX,COMAX,EMAX,MAMAX,MPOMAX,PHMAX,SITMAX,SOMAX PARAMETER (pOMAX = 22,PHMAX=150,SOMAX = 20,EMAX=10,SITMAX = 5, *MAMAX = 31,MPOMAX = 4,CALMAX = 200) C-—GLOBAL C O M M O N BLOCK C - — C O M M O N BLOCK FOR THERIAK C-—END OF C O M M O N VARIABLES INTEGER*4 GNOM,LOCMIN(SOMAX + 1),LOMIN1(SOMAX+1),NMAX, > STEM(SOMAX +1 ),TRY REAL*8 DELTAX,DELXXX(0:CALMAX),DISTA,GSOL,GTEST,GTOT, >MINCOM(SOMAX+1,EMAX,EMAX),MUE(EMAX),VEKTOR(EMAX),XXEM(EMAX), > DXE ND,DXLAST(SOMAX +1, EMAX) C O M M O N /TOIN/ GNOM,LOCMIN,LOMIN1,NMAX,STEM,TRY C O M M O N /TORE/ DELTAX,DELXXX,DISTA,CSOL,CTEST,CTOT,MINCOM, >MUE,VEKTOR,XXEM,DXEND,DXLAST INTECERM IS,IE,I,II REAL*8 XXX(EMAX),AAA(EMAX),XBAS(10),ABAS(10),SUMM IF (MODELL(IS).EQ.'S') THEN DO 502,IE = 1/NEND(IS) AAA(IE)=1.0D0 DO 502,1 = 1,NSITE(IS) SUMM = O.0D0 DO 501,11 = 1,NEQEM(IS,IE,I) 501 SUMM = SUMM + XXX(EQEM(IS,IE,I,ll)) 502 AAA(IE) = AAA(IE)*(SUMM**SITMUL(IS,I)) RETURN ELSE DO 503,I = 1,NEMBAS(IS) IF (EMBCOD(IS,I).EQ.O) THEN XBAS(l) = 0.0D0 ELSE XBAS(l) = XXX(EMBCOD(IS,l)) END IF 503 CONTINUE CALL SOLCAL(SOLNAM(lS),P,T,NEMBAS(IS),XBAS,ABAS) DO 504,1 = 1,NEMBAS(IS) IF (EMBCOD(IS,l).NE.0) AAA(EMBCOD(IS,l)) = ABAS(I) 504 CONTINUE RETURN END IF RETURN END C C++**************************** SUBROUTINE REDUCE(K) INTECER*4 CALMAX,COMAX,EMAX,MAMAX,MPOMAX,PHMAX,SITMAX,SOMAX PARAMETER (COMAX = 22,PHMAX = 150,SOMAX = 20,EMAX= 10,SITMAX= 5, *MAMAX=31,MPOMAX = 4,CALMAX = 200) C-—GLOBAL C O M M O N BLOCK C - — C O M M O N BLOCK FOR THERIAK C-—END OF C O M M O N VARIABLES INTEGER*4 CNOM,LOCMIN(SOMAX+1),LOMIN1(SOMAX + 1),NMAX, > STEM(SOMAX +1 ),TRY REAL*8 DELTAX,DELXXX(0:CALMAX),DISTA,GSOL,GTEST,GTOT, >MINCOM(SOMAX + 1,EMAX,EMAX),MUE(EMAX),VEKTOR(EMAX),XXEM(EMAX), > DXEND, DXLAST(SOMAX+1,EMAX) C O M M O N /TOIN/ GNOM,LOCMIN,LOMIN1,NMAX,STEM,TRY C O M M O N /TORE/ D E LTAX, D E LXXX, DI STA, G SO L, GTEST, GTOT, MIN C O M , >MUE,VEKTOR,XXEM,DXEND,DXLAST INTEGER*4 K,l,ll REAL*8 F,AR(COMAX) F = X(K,K) IF (F.EQ.O.ODO) THEN DO 501,11 = 1,NMAX 501 X(lI,K) = 0.0D0 ELSE DO 502,1 = 1,NMAX IF (X(l,K).NE.0.0D0) X(I,K) = X(I,K)/F 502 CONTINUE DO 503,1 = 1,NUN 503 AR(I) = X(K,I) DO 505,1 = 1,NUN IF (I.NE.K.AND.AR(I).NE.O.ODO) THEN DO 504,11 = 1,NMAX X(II,I) = X(II,I)-X(II,K)*AR(I) IF (DABS(X(II,I)).LT.1D-12) X(ll,I) = 0.0D0 504 CONTINUE END IF 505 CONTINUE END IF DO 506,1 = 1,NUN 506 X(K,l) = 0.0D0 X(K,K) = 1.0D0 RETURN END C - — SUBROUTINE COLCHG(K,l) INTECER* 4 CALM AX, C O M AX, EMAX, M AMAX, M POM AX, PHM AX, SITMAX,SOM AX PARAMETER (COMAX = 22,PHMAX = 150,SOMAX = 20,EMAX=10,SITMAX=5, *MAMAX = 31 ,MPOMAX = 4,CALMAX = 200) C-—GLOBAL C O M M O N BLOCK C - — C O M M O N BLOCK FOR THERIAK C-—END OF C O M M O N VARIABLES INTEGERM GNOM,LOCMIN(SOMAX+1),LOMIN1(SOMAX+1),NMAX, > STEM(SOMAX +1 ),TRY REAL*8 DELTAX,DELXXX(0:CALMAX),DISTA,GSOL,GTEST,GTOT, >MINCOM(SOMAX+1,EMAX,EMAX),MUE(EMAX),VEKTOR(EMAX),XXEM(EMAX), >DXEND,DXLAST(SOMAX+1,EMAX) C O M M O N /TOIN/ GNOM,LOCMIN,LOMIN1,NMAX,STEM,TRY C O M M O N /TORE/ DELTAX,DELXXX,DISTA,GSOL,GTEST,GTOT,MINCOM, >MUE,VEKTOR,XXEM,DXEND,DXLAST INTEGERM K,I,I001 DO 501,1001=1,COMAX 501 X(0,I001) = X(K,I001) NN(0) = NN(K) G(0) = G(K) NUMMER(0) = NUMMER(K) EMCODE(O) = EMCODE(K) SUGG(0) = SUGG(K) DELXXX(0) = DELXXX(K) DO 502,1001=1,EMAX 502 XEM(0,I001) = XEM(K,I001) DO 503,1001 = 1,COMAX 503 X(K,I001)=X(I,I001) NN(K) = NN(I) G(K) = C(I) NUMMER(K) = NUMMER(I) EMCODE(K) = EMCODE(I) SUGG(K) = SUGG(I) DELXXX(K) = DELXXX(I) DO 504,1001=1,EMAX 504 XEM(K,I001) = XEM(I,1001) DO 505,1001=1,COMAX 505 X(1,I001) = X(0,I001) NN(l) = NN(0) G(I) = G(0) NUMMER(l) = NUMMER(0) EMCODE(I) = EMCODE(0) SUGG(l) = SUGG(0) DELXXX(l) = DELXXX(0) DO 506,1001 = 1,EMAX 506 XEM(I,I001)=XEM(0,I001) RETURN END C - — SUBROUTINE PRTCAL INTEGER*4 CALMAX,COMAX,EMAX,MAMAX,MPOMAX,PHMAX,SITMAX,SOMAX PARAMETER (COMAX = 22,PHMAX = 150,SOMAX = 20,EMAX=10,SITMAX=5, *MAMAX = 31,MPOMAX = 4,CALMAX = 200) C GLOBAL C O M M O N BLOCK C C O M M O N BLOCK FOR THERIAK C-—END OF C O M M O N VARIABLES INTEGERM GNOM,LOCMIN(SOMAX + 1),LOMIN1(SOMAX+1),NMAX, > STEM(SOMAX +1 ),TRY REAL* 8 D E LTAX, D E LXXX( 0: CALM AX), DI STA, G SO L, GTEST, GTOT, >MINCOM(SOMAX+1,EMAX,EMAX),MUE(EMAX),VEKTOR(EMAX),XXEM(EMAX), > DXEND,DXLAST(SOMAX+ 1,EMAX) C O M M O N /TOIN/ GNOM,LOCMlN,LOMIN1,NMAX,STEM,TRY C O M M O N /TORE/ DELTAX,DELXXX,DISTA,GSOL,GTEST,GTOT,MINCOM, >MUE,VEKTOR,XXEM,DXEND,DXLAST INTEGER*4 1,11,1001,i002 LOGICAL*4 SPACE CHARACTER*16 TEXT(COMAX) CHARACTER* 133 CH001 REAL*8 SUMM(COMAX),EXPO,ACT,XACT,SUMME,NNPC,AR001(EMAX),F001, *TOTAL 1 = 1 1001 IF (I.GE.NUN) GOTO 1 IF ((NUMMER(l + 1).LT.NUMMER(l)).OR.(NUMMER(l).EQ.0.AND. *NUMMER(I +1 ).EQ.0.AND.EMCODE(l +1 ).LT.EMCODE(l))) THEN 1001=1 + 1 CALL COLCHG(I,I001) 1 = 1-1 ELSE 1 = 1 + 1 END IF IF (I.EQ.0) l = 2 GOTO 1001 1 CALL MULCHK IF (PRTLOG(6).OR.PRTLOG(7).OR.PRTLOG(8)) THEN TOTAL = 0.0D0 DO 491,1 = 1,NUN SUMME = 0.0D0 DO 490,11 = 1,NUN2 490 SUMME = SUMME + NN(ll)*X(ll,l) 491 TOTAL=TOTAL + SUMME*GG(I) TOTAL =-TOTAL SPACE = .TRUE. DO 502,1 = 1,NUN2 IF (NUMMER(I).LE.O) THEN TEXT(l) = SOLNAM(EMCODE(l)) ELSE TEXT(I) = NAME(NUMMER(I)) END IF IF (NUMMER(I).LE.NUN.AND.NUMMER(D.NE.O) SPACE = .FALSE. 502 CONTINUE l001=LOO1-1 WRITE (UNIT=6,FMT=100) P,PGAS,TC,T,NUN2,I001,GTOT,TOTAL,R WRITE (UNIT=7,FMT=100) P,PCAS,TC,T,NUN2,I001,GTOT,TOTAL,R 100 FORMAT ('1P =',F9.2,' bar',7X,'P(Gas) =',F9.2,' bar T =', *F8.2,' C =',F8.2,' KV STABLE PHASES:',l4,5X,'LOOP =',I4, */' G(-) =',1PE12.5,0P,5X,'G(System) =',F15.2,5X,'R =',F14.7) IF (.NOT.SPACE) WRITE (UNIT = 6,FMT=101) IF (.NOT.SPACE) WRITE (UNIT = 7,FMT=101) 101 FORMAT ('0COMPOSITION MAY BE OUTSIDE SPACE DEFINED BY PHASES') IF (NUN2.LE.0) RETURN END IF IF (PRTLOG(6)) THEN WRITE (UNIT=6,FMT=102) WRITE (UNIT = 7,FMT=102) 102 FORMAT(//9X,'PHASE',19X,'N',9X,'MOL%',35X,'X',14X,'X', *9X,'ACTIVITY',7X,'ACT.(X)7 *9X,' ',19X,'-',9X,'—',35X,'-',14X,'-', *9X,' ',7X,' ') SUMME = 0.0D0 DO 503,1 = 1,NUN2 503 SUMME = SUMME + DABS(NN(I)) DO 510,1 = 1,NUN2 NNPC = (NN(l)/SUMME)*1OO.0D0 WRITE (UNIT = CH001,FMT=103) NUMMER(l),EMCODE(l),TEXT(l),NN(l), •NNPC 103 FORMAT ('0',I3,I3,2X,A16,3X,F11.6,1X,F11.6) IF (EMCODE(I).GT.O) THEN DO 504,1001=1,EMAX 504 AR001(I001) = XEM(I,I001) CALL MUECAL(EMCODE(I),AR001) DO 505,11 = 1,NEND(EMCODEd)) EXPO = -CG(EM(EMCODE(l),ll))/(RT) IF (EXPO.LT.-150) THEN ACT=0.0D0 ELSE ACT=DEXP(EXPO) END IF EXPO = (MUE(ll)-CC(EM(EMCODE(l),II)))/(RT) IF (EXPO.LT.-150) THEN XACT=O.ODO ELSE XACT=DEXP(EXPO) END IF WRITE (UNIT = CH001(61:),FMT=104) NAME(EM(EMCODE(l),ll)), *XEM(I,II),XEM(I,II),ACT,XACT 104 FORMAT (A16,2X,F9.6,2X,3(2X,1PE12.5)) WRITE (UNIT=6,FMT = '(A133)') CH001 WRITE (UNIT=7,FMT='(A133)') CH001 505 CH001=' ' ELSE WRITE (UNIT=6,FMT = '(A133)') CH001 WRITE (UNIT=7,FMT = '(A133)') CH001 CH001=' ' END IF 510 CONTINUE END IF IF (PRTLOG(7)) THEN WRITE (UNIT = 6,FMT=105) WRITE (UNIT=7,FMT=105) 105 FORMAT (//' COMPOSITIONS OF STABLE PHASES [ MOL% ] WRITE (UNIT=6,FMT = '(" ")') WRITE (UNIT=7,FMT='(" ")') DO 512,1001 = 1,NUN,10 DO 511,1 = 1001,MIN0(I001+9,NUN) 1002 = 23 + MOD(I-1,10)*11 WRITE (UNIT = CH001(I002:),FMT = '(A5)') CHNAME(I) 511 CONTINUE WRITE (UNIT=6,FMT = '(A133)') CH001 WRITE (UNIT=7,FMT = '(A133)') CH001 512 CH001=' ' DO 513,1001 = 1,COMAX 513 SUMM(I001) = 0.0D0 DO 520,1 = 1,NUN2 WRITE (UNIT=CH001,FMT='("0",A16)') TEXT(I) SUMME = 0.0D0 DO 514,11 = 1,NUN SUMME = SUMME + DABS(X(I,II)) 514 SUMM(ll) = SUMM(II) + X(l,ll)*NN(l) DO 516,1001=1,NUN,10 DO 515,II = I001,MIN0(I001+9,NUN) I002 = 19 + MOD(II-1,10)*11 F001 =X(I,II)/SUMME*100.0D0 WRITE (UNIT=CH001(I002:),FMT='(F11.6)') F001 515 CONTINUE WRITE (UNIT = 6,FMT = '(A133)') CH001 WRITE (UNIT=7,FMT='(A133)') CH001 516 CH001=' ' 520 CONTINUE WRITE (UNIT=CH001,FMT=106) 106 FORMAT ('0TOTAL:') SUMME = O.ODO DO 521,1 = 1,NUN 521 SUMME —SUMME+ DABS(SUMM(I)) DO 523,1001 = 1,NUN,10 DO 522,I = I001,MIN0(I001+9,NUN) I002 = 19 + MOD(I-1,10)*11 F001 = SUMM(I)/SUMME*100.0D0 WRITE (UNIT=CH001(I002:),FMT='(F11.6)') F001 522 CONTINUE WRITE (UNIT=6,FMT = '(/,A133)') CH001 WRITE (UNIT = 7,FMT = '(/,A133)') CH001 523 CH001=' ' END IF IF (PRTLOG(8)) THEN WRITE (UNIT=6,FMT=107) WRITE (UNIT=7,FMT=107) 107 FORMAT ('1',8X,'PHASE',19X,'N',13X,'C',9X,'ACTIVITY7) CALL PRTSTR(1,NMAX) END IF 999 RETURN END C — - SUBROUTINE MULCHK INTECER*4 CALMAX,COMAX,EMAX,MAMAX,MPOMAX,PHMAX,SITMAX,SOMAX PARAMETER (COMAX = 22,PHMAX = 150,SOMAX = 20,EMAX= 10,SITMAX= 5, *MAMAX = 31,MPOMAX = 4,CALMAX=200) C—-GLOBAL C O M M O N BLOCK C C O M M O N BLOCK FOR THERIAK C — E N D OF C O M M O N VARIABLES INTEGERM GNOM,LOCMIN(SOMAX+1),LOMIN1(SOMAX+1),NMAX, > STEM(SOMAX +1 ),TRY REAL*8 DELTAX,DELXXX(0:CALMAX),DISTA,GSOL,GTEST,GTOT, >MINCOM(SOMAX + 1,EMAX,EMAX),MUE(EMAX),VEKTOR(EMAX),XXEM(EMAX), > DXEN D, DXLAST(SOM AX +1, EMAX) C O M M O N /TOIN/ GNOM,LOCMIN,LOMIN1,NMAX,STEM,TRY C O M M O N /TORE/ D E LTAX, D E LXXX, DI STA, G SO L, GTEST, GTOT, MIN C O M , >MUE,VEKTOR,XXEM,DXEND,DXLAST INTEGERM I,II,I1I,K,I001,I002 REAL*8 SUMME NUN2=NUN 1 = 1 1001 IF (I.GT.NUN2) GO TO 11 IF (DABS(NN(I)).LT.1D-12) THEN DO 501,K=I,(NUN2-1) I001=K+1 501 CALL COLCHG(K,I001) NUN2 = NUN2-1 1 = 1-1 GO TO 999 END IF IF (EMCODE(I).GT.O) THEN DO 510,II = (I + 1),NUN2 IF (EMCODE(II).EQ.EMCODE(I)) THEN 211 SUMME = O.ODO DO 502,111 = 1,NEND(EMCODEd)) 502 SUMME = SUMME + DABS(XEM(I,III)-XEM(II,III)) IF (SUMME.LT.EQUALX) THEN DO 503,111 = 1,NEND(EMCODEd)) 503 XEM(l,lll) = (NN(l)*XEM(l,lll) + NN(ll)*XEM(ll,lll))/(NN(l) + NN(II)) NN(I) = NN(I) + NN(II) NN(II) = 0.0D0 1001 = 1 CALL XSOL(I001) DO 504,K=II,(NUN2-1) 1001 =K 1002 = K+1 504 CALL COLCHC(I001,I002) NUN2 = NUN2-1 1 = 1-1 CO TO 999 END IF END IF 510 CONTINUE END IF 999 1 = 1 + 1 GO TO 1001 11 RETURN END SUBROUTINE PRTSTR(N1,N2) INTEGER*4 CALMAX,COMAX,EMAX,MAMAX,MPOMAX,PHMAX,SITMAX,SOMAX PARAMETER (COMAX = 22,PHMAX = 150,SOMAX= 20,EMAX = 10,SITMAX= 5, *MAMAX = 31,MPOMAX=4,CALMAX = 200) C—-GLOBAL COMMON BLOCK C-—COMMON BLOCK FOR THERIAK C-—END OF COMMON VARIABLES INTEGER*4 CNOM,LOCMIN(SOMAX+1 ),LOMIN1 (SOMAX+ 1),NMAX, > STEM(SOM AX +1 ),TRY REAL*8 DELTAX,DELXXX(0:CALMAX),DISTA,GSOL,GTEST,GTOT, >MINCOM(SOMAX + 1,EMAX,EMAX),MUE(EMAX),VEKTOR(EMAX),XXEM(EMAX), > DXEND, DXLAST(SOM AX +1, EMAX) COMMON /TOIN/ GNOM,LOCMIN,LOMIN1,NMAX,STEM,TRY COMMON /TORE/ DELTAX,DELXXX,DISTA,GSOL,GTEST,GTOT,MINCOM, >MUE,VEKTOR,XXEM,DXEND,DXLAST INTEGER*4 I,1I,N1,N2,I001,I002 REAL*8 EXPO,ACT CHARACTERS6 TEXT CHARACTERS33 CH001 DO 510,I = N1,N2 IF (SUGG(I).LE.NPHA) THEN CH001=' P' ELSE CH001=' S' END IF WRITE (UNIT = CH001(3:),FMT='(I5)') SUGG(I) IF (NUMMER(I).LE.O) THEN TEXT = SOLNAM(EMCODE(l)) ELSE TEXT= NAME(NUMMERd)) END IF EXPO = -C(l)/(RT) IF (EXPO.LT.-150.0D0) THEN ACT=0.OD0 ELSE ACT=DEXP(DMIN1(150.0DO,EXPO)) END IF WRITE (UNIT=CH001(8:),FMT=101) TEXT,NN(l),G(l),ACT 101 FORMAT (2X,A16,2X,1PE12.5,2X,0P,F12.2,2X,1PE12.5) IF (EMCODE(I).CT.O) THEN DO 502,1001 = 1,NEND(EMCODE(l)),5 DO 501,II = I001,MIN0(I001 +4,NEND(EMCODE(I))) 1002-70 + MOD(II-1,5)*12 501 WRITE (UNIT=CH001(I002:),FMT='(1PE11.4)') XEM(1,II) WRITE (UNIT=6,FMT='(A133)') CH001 WRITE (UNIT=7,FMT = '(A133)') CH001 502 CH001=' ' ELSE WRITE (UNIT=6,FMT = '(A133)') CH001 WRITE (UNIT = 7,FMT = '(A133)') CH001 END IF 510 CONTINUE RETURN END APPENDIX C: THE MOESSBAUER FURNACE In order to separate the M1 and M2 a b s o r p t i o n peaks i n a Mossbauer spectrum f o r o l i v i n e , the sample has to be heated to approximately 300 °C. ( E i b s c h u t z and G a n i e l (1967)). I t i s a l s o important that the surrounding can be evacuated or f i l l e d with an i n e r t gas, to prevent o x y d a t i o n . The vacuum furnace used i n t h i s i n v e s t i g a t i o n was b u i l t by Mr. Doug Poison f o l l o w i n g the design of Sudaram et a l . (1971,) with some major m o d i f i c a t i o n s . The design of the furnace i s shown i n F i g . 22 and 23 and the m a t e r i a l s used are l i s t e d i n t a b l e 16. The furnace i s mounted on two r a i l s which makes the s u p p o r t i n g l e g s unnecessary. The i n l e t and o u t l e t f o r vacuum pumping are d r i l l e d through the outer f l a n g e s . Only one Chromel-Alumel thermocouple i s used f o r measuring the temperature. I t was found that a constant v o l t a g e of c a . 50 V p r o v i d e d by a v a r i a c , r e s u l t e d i n a temperature of ca. 300 °C with a maximum v a r i a t i o n of ±2°C, as long as the vacuum pump was o p e r a t i n g c o n s t a n t l y . If the v a l v e i s c l o s e d and the pumpturned o f f , the temperature dropped approximately 10 °C i n 8 hours. A l l measurements were done with the pump working. 213 214 F i g . 21: T h e M o e s s b a u e r f u r n a c e c c S i coo l ing coi l s n n o o n o o o —7 p y r o p h y l l i t e spacers r a d i a t i o n sh ie lds UL o n o o o o C c power lead x glass to m e t a l seal t e f lon window c e r a m i c tube 4z h o l d i n g r o d ^ t h e r m o c o u p l e 3 0 f tef lon window Sample ho lde r Tj hea t ing e lement TJ—u~ D — c r cement p y r o p h y l l i t e f langes • g o o o o q o o o o o 0 r i n g brass she l l i n n e r brass flanges outer brass flanges 0 0 to v a c u u m pump Fig. 22: The sample holder for the Moessbauer furnace h o l d i n g rod 216 Table 16.: The m a t e r i a l s used i n the vacuum furnace d e s c r i p t i o n material,remarks thermocouple ceramic tube needle v a l v e s sample holder h o l d i n g rod windows g l a s s to metal s e a l h e a t i n g element cement r a d i a t i o n s h i e l d s ceramic spacers ceramic f l a n g e s 0 r i n g s s h e l l outer f l a n g e s inner f l a n g e s c o o l i n g c o i l s Chromel - Alumel, i n 2-hole ceramic tube Alumina, f o r hea t i n g element Brass, to vacuum l i n e s t a i n l e s s s t e e l , type 316 s t a i n l e s s s t e e l , type 316 t e f l o n f o r the thermocouple and the power leads Nichrome Sauerei sen s t a i n l e s s s t e e l , type 304 p y r o p h y l l i t e , supporting the s h i e l d s p y r o p h y l l i t e , s u p p o r t i n g the furnace Neoprene, d i f f e r e n t s i z e s brass brass brass f l e x i b l e copper tube APPENDIX D: LISTING OF PROGRAM LATEX LATEX:PROCEDURE OPTIONS(MAIN); DCL (BLAH,BAN,IDA,ABIN,MN1) FILE; DCL (KODLP,ISYMCE,NS,NPOS,NCOMPO,NREF,NBED,BED(10,9),IDX(-3:3), l,ll,lll,K,N,M,J,JJ,CODE,CODE2,ANOCOD,HMAX,KMAX,LMAX) BIN FIXED(31); DCL (A,B,C,ALPHA,BETA,GAMMA,WL,WWL4,ANO,PI2,TL,TH,SL,SH,FNEUT(8), SINA,SINB,SINC,COSA,COSB,COSC,SINABC,SINBC,SINAC,SINAB, VOL,AS,BS,CS,COAS,COBS,COCS,SX2,MAXlNT,INT,THETA,DWERT, RELINT,QQ,SMA,SMB,XN,YN,ZN,SCAT,FBRA,DUM,ARG,INSUM(13), AE,BE,CE,AE2,BE2,CE2,BC2,AC2,AB2,HH2,KK2,LL2,FF) DECIMAL FLOAT(16); DCL (XX(50),YY(50),ZZ(50)) DECIMAL FLOAT(16); DCL 1 INP(0:1000), 2 (HH,KK,LL) BIN FIXED(31), 2 (MUL,Q) DECIMAL FLOAT(16), 1 OUP(IOOO), 2 (VINT,STRU,LOPO,SUMA,SUMB,SUMAX,SUMBX) DECIMAL FLOAT(16), 1 ATOM(8), 2 ELEMT CHAR(4), 2 NA BIN FIXED(31), 2 (DELFR,DELFI,FA(4),FB(4),FC) DECIMAL FLOAT(16), 2 POS(50), 3 IDE CHAR(2), 3 (X,Y,Z,MUP,OCC,BT,BII(6)) DECIMAL FLOAT(16), 3 BTCODE BIN FIXED(31), 1 EQUK24), 2 (TS(3),FS(3,3)) DECIMAL FLOAT(16); DCL (TITEL) CHAR(200); DCL (HKL(3),CH(3),AA) CHAR(1); DCL (HKLCOD) CHAR(15) VARYING; DCL STR(3) CHAR(20) VARYING; DCL (WLCOD,SPGR,IDENT) CHAR(IO); DCL (ELE) CHAR(4); OPEN FILE(SYSPRINT) LINESIZE(132); AA = ""; GET FILE(BLAH) LIST(TITEL,A,B,C,ALPHA,BETA,GAMMA, AE,BE,CE,WLCOD,SPGR,TL,TH,ANO,KODLP); PUT EDIT(TITEL,'SPACE GROUP : ',SPGR)(COL(1),A,SKIP(2),A,A); ON ENDFILE(BLAH) GOTO AAFA; NCOMPO = 0; DO WHILE('VB); GET FILE(BLAH) EDIT(ELE)(COL(1),A(4)); CODE = 0; DO 1 = 1 TO NCOMPO WHILE(CODE = 0); IF ELE = ELEMT(I) THEN DO; CODE = l ; NA(I) = NA(I) + 1; END; END; IF CODE = 0 THEN DO; NCOMPO = NCOMPO+ 1; l = NCOMPO; ELEMT(I) = ELE; NA(I) = 1; END; ELSE I = CODE; GET FILE(BLAH) EDlT(IDE(l,NA(l)))(A(2)); GET FILE(BLAH) LIST(X(I,NA(I)),Y(I,NA(I)),Z(I,NA(I)), OCC(l,NA(I)),BT(l,NA(l))); IF BT(l,NA(l)) = 0 217 THEN DO; DO 11 = 1 TO 6; GET FILE(BLAH) LIST(BII(I,NA(I),II)); END; BTCODE(l,NA(l)) = 1; END; ELSE BTCODE(I,NA(I)) = 0; END; AAFArON ENDFILE(BAN) GOTO NOSP; CODE = 0; DO WHILE(CODE = 0); GET FILE(BAN) EDIT(IDENT)(COL(1),A(10)); IF SPGR=IDENT THEN DO; CODE=1; GET FILE(BAN) LIST(NS,ISYMCE,FBRA,NBED); DO 11 = 1 TO NS; GET FILE(BAN) LIST(TS(II,*),FS(II,*,*)); END; PUT EDITCBravais multiplicity =',FBRA)(SKIP(2),A,F(4,1)); PUT EDIT(NBED,'conditions for non-extinction :') (SKIP(2),COL(1 ),F(3,0),X(1), A); CH(1) = ' H ' ; CH(2) = 'K'; CH(3) = 'L'; DO 1 = 1 TO NBED; GET FILE(BAN) LIST(HKLCOD); PUT EDIT(HKLCOD)(COL(38),A); BED(I,*) = 0; DO J = 1 TO 3; HKL(J) = SUBSTR(HKLCODJ,1); IF HKL(J) = '0' THEN BED(IJ) = J; END; IF HKL(1) = HKL(2) THEN DO; BED(I,4) = 1; BED(I,5) = 2; END; IF HKL(1) = HKL(3) THEN DO; BED(I,4) = 1; BED(I,5) = 3; END; IF HKL(2) = HKL(3) THEN DO; BED(I,4) = 2; BED(I,5) = 3; END; DO J = 1 TO 3; IF (INDEX(HKLCOD/ + '|JCH(J))~=0 | INDEX(HKLCOD,' '||CH()))~ =0) THEN BED(l,5+)) = ); IF (INDEX(HKLCOD/-'||CH(J))~=0) THEN BED(I,5 + J) = -J; END; BED(I,9) = FIXED(SUBSTR(HKLCOD,INDEX(HKLCOD,' = ') + 1,D,6,0); END; END; END; NOSP:IF CODE = 0 THEN DO; PUT DATA(SPGR); STOP; END; PUT EDIT(NS/equivalent point positions :') (SKIP(2)/F(3/0),X(1),A); IF 1SYMCE = 1 THEN PUT EDIT('(+ symmetry centre at the origin)') (X(5),A); PUT SKIP(2) EDITC ')(A); CH(1) = 'X'; CH(2) = 'Y'; CH(3) = 'Z'; DO 1 = 1 TO NS; STR = ' '; DO J = 1 TO 3; STR(J) = STR(J)||CHAR(FIXED(TS(I,J),8,5)); STR(J) = SUBSTR(STR(J),3); DO K=1 TO 3; IF FS(IJ,K) = 1 THEN STR(]) = STR(J)||'+ '||CH(K); IF FS(IJ,K) = -1 THEN STR(J) = STR(J)||'-'||CH(K); END; IF INDEX(STR(J),'0.00000 + ' )~=0 THEN SUBSTR(STR(J),1,11) = ' '; IF INDEX(STR(J),'0.00000')~=0 THEN SUBSTR(STR(J),1,10) = ' '; END; PUT EDIT(STR)(COL(3),3 A(15)); END; ON ENDFILE(BAN) GOTO NOWL; CODE = 0; DO WHILE(CODE = OX- GET FILE(BAN) EDIT(IDENT)(COL(1),A(5)); IF WLCOD = IDENT THEN DO; GET FILE(BAN) LIST(ANOCOD,WL); CODE = 1; END; END; N O W U F CODE = 0 THEN DO; PUT DATA(WLCOD); STOP; END; DELFR = 0; DELFI = 0; FA = 0; FB = 0; FC = 0; ON ENDFILE(BAN) GOTO NOSC; CODE = -NCOMPO; DO WHILE(CODE<0); GET FILE(BAN) EDIT(IDENT)(COL(1),A(4)); CODE2 = 0; DO 1 = 1 TO NCOMPO WHILE(CODE2 = 0); IF ELEMT(I) = IDENT THEN DO; CODE2 = 1; CODE = CODE + 1; DO J = 1 TO 4; GET FILE(BAN) LIST(FA(I,J),FB(I,J)); END; GET FILE(BAN) LIST(FC(l),FNEUT(l)); DO J = 1 TO ANOCOD; GET FILE(BAN) LIST(DELFR(I),DELFI(I)); END; END; END; END; NOSC:IF CODE<0 THEN DO; DO 1 = 1 TO NCOMPO; IF FA(I,1) = 0 THEN PUT SKIP DATA(ELEMT(I)); END; STOP; END; PUT EDITCscattering factor coefficients :','a1'/b1'/a2','b2', 'a3'/b3'/a4'/b4'/c'/sum(ai) + c') (SKIP(2),A,COL(12),(4) (A,X(8),A,X(10)),A,X(12),A); DO 1 = 1 TO NCOMPO; PUT EDIT(ELEMT(l))(COL(1),A(4)); DO J = 1 TO 4; PUT EDIT(FA(I,J)/FB(I,]))(F(12,5),F(10,5)); END; PUT EDIT(FC(l),SUM(FA(l,'t:)) + FC(l))(F(12,5)/X(5),F(12/5)); END; PUT EDITCradiation :',WLCOD/landa =',WL) (SKIP(2)/A,X(1),A,A,F(9/6)); PUT EDITCanomalous dispersion correction terms :')(SKIP(2),A); PUT EDIT('delfr'/deIfi')(X(9),A,X(5),A); IF ANO = 0 THEN PUT EDIT('(not used for calcu!ations)')(X(4),A); DO 1 = 1 TO NCOMPO; IF (DELFR(l)~=0)&(DELFl(l)~=0) THEN PUT EDIT(ELEMT(l),DELFR(l),DELFI(l))(COL(40),A(4),2 F(10,4)); IF ANO = 0 THEN DO; DELFR(l) = 0; DELFI(I) = 0; END; END; DO M = 1 TO NCOMPO; DO 1 = 1 TO NA(M); DO J = 1 TO NS; XX(J) = TS(J,1) + FS(J,1,1)*X(M,]) + FS(J,1,2)*Y(M/I) + FS(J,1,3)*Z(M,I); YY(J) = TS(J/2) + FS(J,2,1)*X(M,l) + FS(J,2/2)*Y(M,l) + FS(J,2/3)*Z(M,i); ZZ(J) = TS(J,3) + FS(J,3,1)*X(M,I) + FS(J,3,2)*Y(M,I) + FS(J,3,3)*Z(M,I); END; IF ISYMCE = 1 THEN DO; NPOS = 2*NS; DO ] = (NS + 1) TO (2*NS); XX(J) = -XX(J-NS); YY(J) = -YY(J-NS); ZZ(J) = -ZZ(J-NS); END; END; ELSE NPOS = NS; DO J = 1 TO NPOS; XX(J) = MOD(XX(J),1); YY(J) = MOD(YY(J),1); ZZ(J) = MOD(ZZ(J),1); END; MUP(M,l) = NPOS; DO J = 1 TO NPOS-1; CODE = 1; DO j] = (J + 1) TO NPOS WHILE(CODE = 1); IF (ABS(XX(J)-XX(JJ))< 1 E-9)&(ABS(YY(J)-YY(JJ))< 1E-9) &(ABS(ZZ(J)-Z2(JJ))<1E-9) THEN DO; MUP(M,I) = MUP(M(1)-1; CODE = 0; END; END; END; MUP(M,l) = MUP(M,l)/NPOS; END; END; PUT EDITCatom positions')(SKIP(2),A); PUT EDIT('x/a','y/b'/z/c','Bt'/mult'/occ','N', 'atoms / unit cell') (COL(12),A,COL(22),A,COL(32),A,COL(41 ),A,COL(48) /A, COL(57),A,COL(65)ACOL(73),A); DO 1 = 1 TO NCOMPO; DO ] = 1 TO NA(l); PUT EDlT(ELEMT(l),IDE(l,)),X(l,)),Y(IJ),Z(l,J),BT(lJ), MUP(l /J),OCC(l,J),MUP(IJ)*NPOS /MUP(l,J)*NPOS*OCC(l,J)) (COL(1),AA3 F(10,6),F(8,4),2 F(8,4),F(6,1),X(6),F(11,5)); END; END; IF ISYMCE = 0 THEN DUM = 1; ELSE DUM = 2; PI2 = 8*ATAN(1); MAXINT=1; SINA = SIND(ALPHA); SINB = SIND(BETA); SINC = SIND(CAMMA); COSA = COSD(ALPHA); COSB = COSD(BETA); COSC = COSD(GAMMA); SlNABC = SIND((ALPHA+BETA + GAMMA)/2.0); SINBC = SIND((-ALPHA + BETA + GAMMA)/2.0); SINAC = SIND((ALPHA-BETA + GAMMA)/2.0); SIN AB = SI ND((ALPH A + BETA-GAMMA)/2.0); VOL = 2*A*B*C*SQRT(SINABC*SINBC*SINAC*SINAB); AS = B*C*SINAA/OL; BS = A*C*SINBA/OL; CS = A*B*SINC/VOL; COAS = (COSB*COSC-COSA)/(SINB*SINC); COBS = (COSC*COSA-COSB)/(SINC*SINA); COCS = (COSA*COSB-COSC)/(SINA*SINB) ; PUT EDITCa =',A,'alpha =',ALPHA,'b =',B,'beta =',BETA, 'c =',C,'gamma =',GAMMA,'vol =',VOL)(SKIP(2), (3) (COL(1)AF(9/5)/COL(18)AF(10,5)),X(7),A,F(10,4)); AE2 = AE*AE; BE2 = BE*BE; CE2 = CE*CE; BC2 = (B*C*BE*CE)**2; AC2 = (A*C*AE*CE)**2; AB2 = (A*B*AE*BE)**2; PUT EDITCanisotropy : ,,AE,BE,CE)(X(5)/A,(3) (F(9,5))); O N ENDPAGE BEGIN; PUT EDIT(WLCOD,WL,TITEL)(PAGE,A(6)/F(8,5),X(4)/A); PUT EDIT('H','K'/L','INT','ABS INT','2 TH','D','|F(hkl)|', 'MULT'/LP.'/sR*fR'/sl +fR'/sR*fl ' /sl*fl ') (SKIP(2) /COL(8),A,COL(12),A,COL(l6),A,COL(22),A,COL(31),A, COL(43),A /COL(54),A,COL(61 ),A,COL(71 ),A,COL(79) /A,COL(90), A,COL(102) ,A,COL(114),A,COL(126),A); PUT EDITC ')(SKIP,A); END; SH = SIND(TH)**2; SL = SIND(TL)**2; NREF = 0; WWL4=WL*WL/4.0; HMAX = FLOOR(SQRT(SHA/VWL4)/AS); KMAX = FLOOR(SQRT(SHAA/WL4)/BS); LMAX = FLOOR(SQRT(SHAVWL4)/CS); DO I = HMAX TO -HMAX BY -1; DO II = KMAX TO -KMAX BY -1; DO III = LMAX TO -LMAX BY -1; QQ = (WWL4)*((I*AS)**2 + (II*BS)**2 + (III*CS)**2 + 2*ll*lll*BS , , ,CS*COAS + 2*l*lll*AS*CS*COBS + 2*l*li*AS*BS*COCS); IF QQ>SH THEN GOTO NEXT; IF QQ<SL THEN GOTO NEXT; CODE = 1; IDX(-3) = -l)i; IDX(-2) = - l l ; 1DX(-1) = -I; IDX(0) = 0; IDX(1) = I; IDX(2) = II; IDX(3) = III; DO J = 1 TO NBED WHILE(CODE = 1); IF (IDX(BED(],1)) = 0)&(IDX(BED(J,2)) = 0)&(IDX(BED(J,3)) = 0) &(IDX(BED(J,4)) = IDX(BED(J,5))) &(MOD(IDX(BED(),6)) + IDX(BED(J,7)) + IDX(BED(J,8)),BED(J,9))~=0) THEN CODE = 0; END; IF CODE = 0 THEN GOTO NEXT; DO N = 1 TO NREF WHILE(CODE = 1); IF ABS(Q(N)-QQ)<1E-9 THEN DO; MUL(N) = MUL(N) + 1; CODE = 0; END; END; IF CODE = 0 THEN GOTO NEXT; IF (l = 0)&(M = 0)&(lll = 0) THEN GOTO NEXT; NREF = NREF + 1; IF NREF>1000 THEN DO; NREF=1000; GOTO ORD; END; Q(NREF) = QQ; MUL(NREF) = 1; HH(NREF) = I; KK(NREF) = l l ; LL(NREF) = III; N EXT: END; END; END; ORD:N = 1; DO WHILE(N<NREF); IF Q(N)>Q(N + 1) THEN DO; INP(0)= INP(N); INP(N) = INP(N + 1); INP(N + 1) = INP(0); N = N-1; IF N = 0 THEN N = 2; END; 222 ELSE N = N + 1; END; DO N = 1 TO NREF; /* PUT EDIT(N/)',HH(N),KK(N),LL(N)) (SKIP(2),COL(1),F(3,0)A(3) (F(4,0))); */ THETA = ATAN(SQRT(Q(N)/(1-Q(N)))); LOPO(N) = (1+COS(2*THETA)**2)/(SIN(THETA)*SIN(2*THETA)); SX2 = Q(N)/(WL*WL); SUMA(N) = 0; SUMB(N) = 0; SUMAX(N) = 0; SUMBX(N) = 0; DO M = 1 TO NCOMPO; SMA=0; SMB = 0; DO 1 = 1 TO NA(M); IF BTCODE(M,l)=1 THEN BT(M,I) = BII(M,I,1)*HH(N)**2 + BII(M,I,2)*KK(N)**2 + BII(M,1,3)* LL(N)**2 + BII(M,l,4)*2*HH(N)*KK(N) + BII(M,l,5)*2*HH(N)* LL(N) + BII(M,I,6)*2*KK(N)*LL(N); DO J = 1 TO NS; XN=TS(J,1) + FS(J,1,1)*X(M,I)+FS(J,1,2)*Y(M,I) + FS(J,1,3)*Z(M,I); YN=TS(J,2) + FS(J,2/1)*X(M,l)+FS(j<2/2)*Y(M,l) + FS(J/2,3)*Z(M/l); ZN=TS(],3) + FS(),3,1)*X(M,1) + FS(J,3,2)*Y(M,I) + FS(J,3,3)*Z(M,1); ARG = PI2*(HH(N)*XN + KK(N)*YN + LL(N)*ZN); SMA = SMA + COS(ARG)*MUP(M,l)*OCC(M,l)*EXP(-BT(M,l)*SX2); IF 1SYMCE~=1 THEN SMB = SMB + SIN(ARG)*MUP(M,l)*OCC(M /l)*EXP(-BT(M /l)*SX2); END; END; SCAT=0; DO 1 = 1 TO 4; SCAT = SCAT+FA(M,I)*EXP(-FB(M,I)*SX2); END; SCAT=SCAT+DELFR(M) + FC(M); SUMA(N) = SUMA(N) + SMA*SCAT; SUMB(N) = SUMB(N) + SMB*SCAT; SUMAX(N) = SUMAX(N) + SMA*DELFl(M) + FBRA*DUM; SUMBX(N) = SUMBX(N) + SMB*DELFI(M)*FBRA; END; SUMA(N) = SUMA(N)*FBRA*DUM; SUMB(N) = SUMB(N)*FBRA; STRU(N) = (SUMA(N)-SUMBX(N))**2 + (SUMB(N) + SUMAX(N))**2; VINT(N) = LOPO(N)*MUL(N)*STRU(N); HH2 = HH(N)*HH(N); KK2 = KK(N)*KK(N); LL2 = LL(N)*LL(N); FF = (AE2*BC2*HH2 + BE2*AC2*KK2 + CE2*AB2*LL2)/(BC2*HH2 + AC2*KK2 + AB2*LL2); FF = SQRT(FF); VINT(N) = VINT(N)*FF; MAXI NT = M AX(VI NT(N),MAXI NT); END; WEITER:N = N-1; PUT FILE(ABIN) EDIT(AA,TITEL,AA,' ')(COL(1),A,A(80),A,A); PUT FILE(IDA) EDIT(AA,TITEL,AA,' ')(COL(1),A,A(80),A,A); INSUM = 0; DO 1 = 1 TO N; IF MOD(l,55) = 1 THEN SIGNAL ENDPACE; DWERT=WLV(2*SQRT(Q(I))); THETA = ATAND(SQRT(Q(I)/(1-Q(I)))); RELINT = VINT(I)*100/MAXINT; IF HH(l) = 0 & KK(l) = 2 & LL(l) = 0 THEN INSUM(1)=VINT(I); IF HH(I) = 1 & KK(1) = 1 & LL(l) = 0 THEN INSUM(2)=VINT(I); IF HH(I)=1 & KK(I) = 1 & LL(I) = 1 THEN INSUM(3) = INSUM(3) + VlNT(l); IF HH(I) = 1 & KK(I) = 2 & LL(I) = 0 THEN INSUM(3) = INSUM(3) + VINT(I); IF HH(I) = 1 & KK(I) = 2 & LL(I) = 1 THEN INSUM(4) = INSUM(4) + VINT(I); IF HH(I) = 0 & KK(I) = 0 & LL(I) = 2 THEN INSUM(4) = 1NSUM(4) + VINT(I); IF HH(I) = 1 & KK(I) = 3 & LL(I) = 0 THEN 1NSUM(5) = VINT(l); IF HH(I) = 0 & KK(I) = 2 & LL(I) = 2 THEN INSUM(6) = INSUM(6) + VINT(I); IF HH(I) = 0 & KK(!) = 4 & LL(I) = 0 THEN 1NSUM(6) = INSUM(6)+VINT(I); IF HH(I) = 1 & KK(I) = 3 & LL(I) = 1 THEN INSUM(7) = VINT(I); IF HH(I) = 1 & KK(I) = 1 & LL(I) = 2 THEN INSUM(8) = VINT(I); IF HH(I) = 2 & KK(I) = 0 & LL(I) = 0 THEN INSUM(9) = INSUM(9) + VINT(I); IF HH(I) = 0 & KK(I) = 4 & LL(I) = 1 THEN INSUM(9) = INSUM(9) + VINT(I); IF HH(I) = 2 & KK(I) = 1 & LL(l) = 0 THEN INSUM(10) = VINT(I); IF HH(I) = 1 & KK(I) = 2 & LL(I) = 2 THEN INSUM(11) = INSUM(11) + VINT(I) IF HH(I) = 1 & KK(I) = 4 & LL(I) = 0 THEN INSUM(H) = INSUM(11) + VINT(I) IF HH(I) = 2 & KK(I) = 2 & LL(I) = 2 THEN INSUM(12) = INSUM(12) + VINT(I) IF HH(I) = 2 & KK(I) = 4 & LL(I) = 0 THEN INSUM(12) = INSUM(12) + VINT(I) IF HH(I) = 1 & KK(I) = 2 & LL(I) = 3 THEN INSUM(12) = INSUM(12) + VINT(I) IF HH(I) = 1 & KK(I) = 5 & LL(l) = 2 THEN INSUM(13) =VINT(I); PUT EDITd/)\HHd),KKd),LLd),RELINT,VINTd), 2*THETA,DWERT,SQRT(STRU(l)),MUL(l),LOPO(l),SUMA(l),SUMB(l), SUMAX(I),SUMBX(I)) ( C O L d ^ F O ^ X A ^ F(4,0),F(9,3),F(12,2)/Fd0/4),F(10,5)/ HT\A),H5,0),H10A)A F(12,5»; PUT FILE(IDA) EDIT(HH(I),KK(I),LL(I),2*THETA,RELINT,' ') (COLd),3 F(4/0)/Xd),F(9,4),X(1),F(9/3),A); PUT FILE(ABIN) EDlT(HH(l),KK(l),LL(l),VINT(l)/ ') (COLd),3 F(4,0),X(3),E(12,5/6),A); END; PUT EDIT(TITEL)(PACE,SKIP(2),A); PUT FILE(MN1) EDIT(AA,TITEL,AA,' ')(COL(1),A,A,A,A); PUT EDIT('222')(SKIP(3),COL(115),A); PUT EDIT('111','121V022','200','122' /'240') (COL(34),A,COL(43),A,COL(61 ),A,COL(88),A,COLd 06),A,COLd 15),A); PUT EDIT('020'/110'/120'/002'/130'/040'/13r/112'/041', '210','140'/123'/152') (COL(16),d3) (A,X(6))); PUT SKIP(2) EDIT('020')(COLd),A); PUT FILE(MN1) EDIT('020')(COLd),A); DO 1 = 1 TO 13; PUT EDlT(INSUM(1)/INSUM(l))(COL(12 + (l-1)*9),F(9/5)); PUT FILECMN1) EDIT(lNSUM(1)/INSUM(l))(COL(12 + (l-1)*9),F(9,5)); END; PUT SKIP(2) EDIT('110')(COL(1),A); PUT FILE(MN1) EDIT('110')(COL(1),A); DO 1 = 1 TO 13; PUT EDIT(INSUM(2)/INSUM(I))(COL(12 + (l-1)*9),F(9,5)); PUT FILE(MN1) EDIT(INSUM(2)/INSUM(l))(COL(12 + (l-1)*9),F(9,5)); END; PUT SKIP(2) EDIT('111 + 120')(COL(1),A); PUT FILE(MN1) EDIT('111+120')(COL(1),A); DO 1 = 1 TO 13; PUT EDIT(INSUM(3)/INSUM(l))(COL(12 + (l-1)*9),F(9,5)); PUT FILE(MN1) EDIT(INSUM(3)/INSUM(l))(COL(12 + (l-1)*9)/F(9,5)); END; PUT SKIP(2) EDIT('121+002')(COL(1),A); PUT FILE(MN1) EDIT('121+002')(COL(1),A); DO 1 = 1 TO 13; PUT EDIT(lNSUM(4)/INSUM(l))(COL(12 + (l-1)*9),F(9,5)); PUT FILE(MN1) EDIT(INSUM(4)/lNSUM(l))(COL(12 + (l-1)*9),F(9,5)); END; PUT SK1P(2) EDIT('130')(COL(1),A); PUT F1LE(MN1) EDIT('130')(COL(1 ),A); DO 1 = 1 TO 13; PUT EDIT(INSUM(5)/INSUM(l))(COL(12 + (l-1)*9),F(9,5)); PUT FILE(MN1) EDIT(INSUM(5)/INSUM(l))(COL(12 + (I-1)*9),F(9,5)); END; PUT SKIP(2) EDIT('022 + 040')(COL(1),A); PUT FILE(MN1) EDIT('022 + 040')(COL(1),A); DO 1 = 1 TO 13; PUT EDIT(INSUM(6)/INSUM(l))(COL(12 + (l-1)*9),F(9,5)); PUT FILE(MN1) EDIT(INSUM(6)/INSUM(l))(COL(12 + (l-1)*9),F(9,5)); END; PUT SKIP(2) EDn"('131')(COL(1),A); PUT FILE(MN1) EDIT('131')(COL(1),A); DO 1 = 1 TO 13; PUT EDIT(INSUM(7)/INSUM(i))(COL(12 + (l-1)*9),F(9,5)); PUT FILE(MN1) EDIT(INSUM(7)/INSUM(l))(COL(12 + (l-1)*9),F(9,5)); END; PUT SKIP(2) EDIT('112')(COL(1),A); PUT FILE(MN1) EDlT('112')(COL(1),A); DO 1 = 1 TO 13; PUT EDIT(INSUM(8)/INSUM(l))(COL(12 + (i-1)*9),F(9,5)); PUT FILE(MN1) EDIT(INSUM(8)/INSUM(l))(COL(12 + (l-1)*9),F(9,5)); END; PUT SKIP(2) EDIT('200 + 041')(COL(1),A); PUT FILE(MN1) ED1T('200 + 041')(COL(1),A); DO 1 = 1 TO 13; PUT EDIT(INSUM(9)/INSUM(l))(COL(12 + (l-1)*9),F(9,5)); PUT FILE(MN1) EDIT(INSUM(9)/lNSUM(l))(COL(12 + (l-1)*9),F(9,5)); END; PUT SKIP(2) EDIT('210')(COL(1),A); PUT FILE(MN1) EDIT('210')(COL(1),A); DO 1 = 1 TO 13; PUT EDIT(INSUM(10)/INSUM(I))(COL(12 + (I-1)*9),F(9,5)); PUT FILE(MN1) EDIT(INSUM(10)/INSUM(I))(COL(12 + (I-1)*9),F(9,5)); END; PUT SKIP(2) EDIT('122+140')(COL(1),A); PUT FILE(MN1) EDIT('122 + 140')(COL(1),A); DO 1 = 1 TO 13; PUT EDIT(INSUM(11)/INSUM(l))(COL(12 + (i-1)*9),F(9,5)); PUT FILE(MN1) EDIT(INSUM(11)/INSUM(l))(COL(12 + (]-1)*9),F(9/5)); END; PUT SKIP(2) EDIT('222 + 140 + 123')(COL(1),A); PUT FILE(MN1) EDIT('222 + 140 + 123')(COL(1),A); DO 1 = 1 TO 13; PUT EDIT(INSUM(12)/INSUM(l))(COL(12 + (I-1)*9),F(9,5)); PUT FILE(MN1) EDIT(INSUM(12)/INSUM(l))(COL(12 + (l-1)*9),F(9,5)); END; PUT SKIP(2) EDIT('152')(COL(1)/A); PUT FILE(MN1) EDIT('152')(COL(1),A); DO 1 = 1 TO 13; PUT ED!T(!NSUM(13)/INSUM(l))(COL(12 + (l-1)*9),F(9,5)); PUT FILE(MN1) EDiT(INSUM(13)/INSUM(l))(COL(12 + (l-1)*9),F(9,5)); END; END LATEX; 225 Example o f o u t p u t OLIVINE : M ( 1 ) : F E ( 0 . 1 6 7 ) M N ( 0 . 3 3 3 ) M G ( 0 . 5 0 0 ) M(2 ) :FE(0 .167 )MN(0 .333 )MG(0 .500 ) SPACE GROUP : P B N M B r a v a i s m u l t i p l i c i t y = 1.0 2 c o n d i t i o n s f o r n o n - e x t i n c t i o n 4 e q u i v a l e n t p o i n t p o s i t i o n s HOL: H+L=2N OKL: K=2N (+ symmetry c e n t r e at the o r i g i n ) O .50000 -X -X 0.50000+X O.50000+Y -Y O.50000-Y 0 .50000 -Z O.50000+Z - Z s c a t t e r i n g f a c t o r c o e f f i c i e n t s a1 b1 FE 11.76950 4 .76110 MN 11.28190 5 .34090 MG 5 .42040 2 .82750 6 .29150 SI 0 2 .43860 3 .04850 13.27710 a2 . 35730 . 35730 . 17350 b2 0 .30720 0 .34320 79.261 10 3 .03530 32.33369 2 .28680 5.70110 a3 3.52220 3.01930 1.22690 1.98910 1.54630 b3 15.35350 17.86739 .38080 .67850 .32390 0 . O. O. a4 2 .30450 2 .24410 2 .30730 1.54100 0 .86700 b4 76.88049 83 .75429 7 .19370 81 .69370 32.90889 c .03690 08960 85840 14070 25080 sum(a i )+c 25 .99040 24 .99220 11.98650 13.99760 7 .99940 r a d i a t i o n : CUA1 anomalous d i s p e r s atom p o s i t i o n s landa = 1 .540560 i o n c o r r e c t i o n terms : FE MN MG SI 0 d e l f r -1 . 1790 -0 .5680 0 . 1650 O.2440 0 .0470 d e l f i 3 .2040 2 .8080 0 .1770 O.3300 0 .0320 x / a y / b z / c Bt mu 11 o c c N atoms / un i t FE 1 0, 000000 0. .000000 0 .000000 0. . 5031 0 .5000 0 . 1667 4 .0 0 .66668 FE 2 0. .988850 0. .278870 0. .250000 0. .4702 0 .5000 0 . 1667 4 .0 0 .66668 MN 1 0. .000000 0. .000000 0. .000000 0. .5031 0 .5000 0 . 3333 4 .0 1 . 33332 MN 2 0. 988850 0. .278870 0. .250000 0. .4702 0 .5000 0 . 3333 4 .0 1 . 33332 MG 1 0. 000000 0. .000000 0. .000000 0. .5031 0 .5000 0 .5000 4 .0 2 .00000 MG 2 0. ,988850 0. .278870 0. .250000 0. .4702 0 .5000 0. .5000 4 .0 2 .00000 SI 1 0. 425030 0. .095400 0. .250000 0. .3463 0. .5000 1. .0000 4 .0 4 .00000 0 1 0. 763510 0. .092360 0. .250000 0. .4967 0. .5000 1. .0000 4 .0 4 .00000 0 2 0. 215820 0. 450440 0. .250000 0. 4950 0. .5000 1. ,0000 4 .0 4, .00000 0 3 0. 282480 0. ,163610 0. .036750 0. 5387 1 . .0000 1. ,0000 8 .0 8, .00000 a = 4 .80630 b = 10.48500 c = 6 .09630 a l p h a = 90 .00000 b e t a = 90 .00000 gamma = 90 .00000 vo l 307.2173 K5 (S3 226 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o d o o o o o o o o o o o o o o o o o o o o o o o o o o t o ^ i ^ o ) 0 ) c > ) ^ ^ c N i o i o i n i ^ o ) n a ) t ^ u ) t D i n c ^ n c o c o o i n o ) c s i n o ^ c n ^ o o o ) c n < > i C D t ~ t ~ c n ' t ' * ^ t o c n m ^ o ) C D O c o o ) t ~ ^ o i ~ t o O T O i n M - i D ^ ^ o o o i n i o c o c ^ i D i ^ o o o c o i o i n o o c o r - o o c o c ) o o i n t o c v i o ^ ' i n h - o ) U ) ^ 0 ) c \ i O ) ^ ^ o ) i n ^ o 5 < c ^ ^ o i ^ m c D ^ ^ ^ o ) U ) o o O i n t D t n r j O T c o o o i n " i } - o o o i ^ r o o c N ^ - i n o o ^ i c r ^ ^ o n ^ n o c ^ i n c ^ i D i n ^ ^ c N i o ) 0 ) o o i n c \ i - ^ c o i ^ c o r ) o o O O c o o a ) r - - O i n o j n o c N i U ) i ~ ' - 0 ) - ^ 0 ) O c o c ^ ^ r ~ i D c o i D ^ i n ( T ) ^ c o ^ i n n c > i t ^ i D U ) i ^ i n n o j o o ^ c ^ O c o c o o i n i n O t ~ ' - i n O c j - ^ 0 ' r o ) O f O O i n o O ' - i S ' - o i B n i ^ M ' - o o o o O n ' - N O n ' - o o O ' - i n o n ' - o o n t i D O i s o o i ' i r I I I I I - - I t - | | | | T - f l l l l o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o c n ^ c D L O ^ c M i n T - c N T t L n c n T t i n c n ^ c n L n T f c n T r c D C D L n t ^ ^ T t c o i n c n O T f r o cn T t c M c o c M - ' - T f 0 5 T r c o t - - a 5 T t T f c o T r i n c o CM CM co I CO CM CO CM i n r~ •» - CD Tf CD I I I I I I t - I I I ' - ^ n ' - l n o u ) ^ I l a ) 0 ) U l l n < O l n ' - » ^ I ) M ' - l I J l n o w o n ^ ( , ) n o ^ I ) ^ ! 0 ) 0 ) ' - ^ ^ N ^ < f l n o ^ f f l ^ t c ^ l ^ c o ^ c o ^ O c N c n ^ T f c o c M c o o ^ c o c o ^ c o c o c o c N C D c o ^ c o i n c n c o L n c j i ^ c D C D L ^ i n ^ c o c o c N C M O ^ i ~ c N O ^ C D C N c o O L n c o c o c N c n T f C D t ~ c o c D C 5 5 T f c M i n c D c o T t c N O T O I ^ ' - l I l O ^ " f p l O n n ^ t - t f n o ) ^ S l n ^ ^ ^ l J l ^ ( , ) n c D ^ ^ ^ ^ l l n • I ^ 1 0 0 0 0 l l O l n f f n r M ' - T f O L n c N c n e n T t T f c N O O c n c n o o c o r ^ t ^ f - c D C D C D L n L n L O i n T f T f T f T f T f T f T f T f T f T r T f t n w p i ' - ' - ' - ' - ' - ' - ' - C M T f T t T f O 0 T r 0 0 C N T f T f c M C O C 0 T t C N T f O 3 T r c 0 T f c 0 C D C O T f T f T f T r T f 0 0 T f 0 0 C 0 C 0 r a 0 5 Tt CM CO O O oo t- 0) O in r- CO CD O 0) I - l » CM r- t~ CO r- co i- CO r» TT in CD in •>- in o CM CO r- CO O in O Tf co CO I D co co CO ^ co oo O CN CM i in CO CO CM O O CO CO 00 co co i— CO 0) cn O co in cn o cn co in T~ TT 0) CO T— 00 0) o t— r» cn m If) u t- t T - CM Tt t- CM 0) CO CM CO O CO CM CO co O CO in CM 0) co in co CO t - T - CO oo CO co CO in o O CD CM CO 10 Tr CO 0) o cn Tr CD TT CM CO Tf TT r~ t- co co ^ CM O CO in CO oo in CD in CM o 0) 00 CD O CM CD l~ in CO CO CO CD in CM O co i ~ - c o c D i n r ~ O c o c o L O T - C O O 5 C O C O O 5 C O C D O C D C M C D O 5 C D ,- i n c M O O - t t O c o c n ^ - c o O i n t - , - CD co CO co Tt CM CO CN ••- • * 0 " d - c o t ~ 0 ) ' a - ' ^ c o M - i n c o CM CM CO t- * - CO CO T- oo CM in t- CO t- Tr CD O C 0 0 5 C D t - - C D a 5 L O r - ' - L O C M C n c O i n * - C M l D C D r ~ t " - T r T - i r ) t - c o r 3 C M i ~ O t - i D » - c M T ) - c o O t - T i r c o c M i n i n c n c D i n c M c n - a - ^ j - t - c M c o o o c D i n ' - ' j - o i c o c o c M O ' - c o T C M C M c o T r i o t - - r - - L O T f c D T r c N c o c M C D O ) O O T r T - o c o c o i o r - - L O c M c o o ) r - m i n o O c o c D C D i n ^ ' » t j - c o c o c o t - t - ' - o o ^ r ' - c o t x ^ c o c o c o c M c o ^ o o ' - O c o i n c D o o c N O c N • t c M O i ^ ' - c o c D c o i n c D O C D C o i ^ i n O c o i n c n c o c o O t ~ O c N ' » t ~ c O ' - t ^ c M c o c o i n ' - c M t ~ - ^ o ) 0 0 ) c N i n c o o o c o c M O ) o o c o t ~ i n M - c o c o i ^ r ~ c O ' 5 i - O i ^ C D ' » ' » c M O 0 ) 0 ) 0 ) c o c o c o o o c o c o c o c o r ~ t - t ^ r ~ t - - c o c D C D C O C D c o L O T r c O C O C O C O C O C O C N C N C M C M C M C M C M C M C M C M C N C M C M C M C N t- r- co co LO CM CM ^ c n c D ^ c n c o c ^ c M C D O C D c n c D C M C o O c n c M C D ^ - c o O c D o o c D c M l n c D • ^ ' - oo in in Tf oo 0 0 t - ' - ' 3 - L n O ) U O C D C O O O C 7 ) C M O C 7 ) C D C D t - C O C O t - - C O C O t ~ C 0 C M T r O 5 C O C 0 C O C O C 0 L O 0 0 CO CO T - CD CO 05 T— O * - 0) o " » i n in CM r - CM 05 r - in 01 t- CO 0) • q - ' - L n c n c M c o c n o c o i n i - - r - c o o o c M c n c n ' - i n c M O i n c o t - L O T t - C D C N oo co co in o CM CD 05 •'- cn CO CO CO CO 0) -r- CM CM 05 CD 05 C D l - C N C n ' - C M C D O ' S - i n C D ' ! 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O CO fe tn CO ro ro to co tn Ocncooofe-j^itococooofefe O CO -j O tn O tn oo CO oo ro cn oo tn co feoooooooooooooofeoooofeoooofefeoooofeoorooocofeoooofefefeoo U M U i o u r o N J i o u u r o M r o i o M U M u r o i o r o i o M f O r o r o i o c o u u -̂ -̂ -̂ rorororococococofefefecncncnoi--ioooooooooooooocooO-* cntntocn^icotoOtocncnofe-J-'fefecn-JO-'Cofefecncnro-jtoo 0~Jtncn->.tna)0)co-'OocnroOfefe-'Ofe-J-'Cncnoo-i.->.a)tooo -̂ coco--]OoooococnroO'--i(ncncooofe00ocorococo-̂ ro-̂ rooocntn < i i i -». i i i i i cntnro^.-^i-^cnifeico icnro-^-^co-^tncococotoro-^ifei Otocnroco-JO-'to(no)-itnto-'io-4-»-k-»oOrootnfetooocnfe Otncocoootococn- '^icDoocooocnootorotnoto-'Oo-JOtntoo-J tnoofeooo-^rofe-jtocofe-jcncnoorofe^icototoofetncnoooofe 0-~4fe-^-^cotn-*-<.oro-^~jooocn-^~4COCoiooofecn~j-^focofero tnoofe^-^.cootn-~jcocoooco--icocotoioio-'io-'OferoOfecococo rofeOroo-'tncofe<n^icococo-'->io-~itooo-'cn-40to~4rotocn o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o b o b b b b b b o o o o o o o o o o o b b b b b b b b o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o 1 o I I i - J . -*cnocooO~Jrocnorotootoo i i —̂  i —̂ -* O O CO i ro i 1 1 tn o O O fe O CO to CO ~i fe tn o fe ro o oo o 00 CO cn co ->• cn fe cn cn tn -4 O tn oo - 4 . cn O -•• cn fe ro t  tn — to o o fe tn - O co o tn ro to 00 to to ~4 CO o ro co co 00 to tn CO cn tn fe to -•• oo - i tn fe fe cn o -J ro cn co o ro oo 10 co tn tn o co •~i O _ t ro to cn cn to oo cn o ro tn cn to cn ro fe ro 00 CO tn CO tn co to 00 CO —*• cn o O -1 ro to oo tn O tn co tn oo to -•• ~i to cn tn tn tn ro co tn o o O o o o o o o o O O O O O O o o o o o o o o o o O O O O OOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOO APPENDIX E: THE POLYNOMIAL FUNCTION FOR THE LATTICE PARAMETERS In F i g 12 to 1 5 , a polynomial f u n c t i o n was used to approximate the values of the l a t t i c e parameters as a f u n c t i o n of composition. Although t h i s f u n c t i o n i s not used f u r t h e r , i t i s given here f o r the sake of completeness. f = F 1 . x p e + F 2 - x M n + F 3 - x M g + F 4 . x | e - x M n + F 5 . x | e . x M g + F'- xMn* xFe + F' , xMn , xMg + F ° - x M g ' x F e + F'* xMg* xMn The v a l u e s f o r the F^ are given below: a 0 b 0 Co v o l . F i 4 . 8 1 9 6 7 7 1 0 . 4 8 7 2 9 2 6 . 0 9 3 5 6 2 3 0 7 . 9 9 4 1 7 F 2 4 . 9 0 4 7 5 5 1 0 . 6 0 5 8 7 6 6 . 2 6 2 0 2 7 3 2 5 . 7 4 9 0 3 F 3 4 . 7 5 4 4 0 1 1 0 . 2 0 1 5 0 5 5 . 9 8 2 8 8 4 2 9 0 . 1 7 1 9 6 F« - o . 0 3 0 2 0 3 7 1 0 . 1 2 6 9 1 4 - 0 . 0 1 6 6 0 5 4 4 0 . 3 9 1 2 0 3 5 6 F 5 0 . 0 1 5 4 8 3 0 9 - o . 0 6 4 2 2 5 8 6 - 0 . 0 3 2 7 6 8 6 0 - 2 . 8 5 7 9 0 7 1 F 6 - o . 0 2 6 2 2 4 0 5 0 . 1 7 2 3 1 9 7 - 0 . 0 5 1 5 0 5 6 7 - 2 . 8 5 7 9 0 7 1 F 7 - o . 0 2 4 8 2 6 5 7 0 . 2 6 5 1 9 2 9 0 . 1 0 9 0 7 0 1 1 0 . 4 5 9 8 0 2 Fa - o . 0 0 3 0 1 9 8 7 0 . 0 5 9 5 9 1 6 1 0 . 0 5 2 7 4 2 7 9 3 . 7 4 4 5 6 8 9 F 9 - o . 1 0 7 2 2 2 6 0 . 0 8 3 1 5 1 7 7 - o . 0 5 3 7 1 6 0 - 8 . 8 4 8 9 1 0 9 2 2 8 APPENDIX F: TABLES USED FOR THE APPROXIMATION OF THE POSITIONAL PRARMETERS AND THE RESULTING UNCERTAINTIES The f o l l o w i n g t a b l e s were used i n chapter 2.7. The m u l t i p l e l i n e a r r e g r e s s i o n of t a b l e 4, was f i t t e d with the data i n t a b l e 17 and 18. Table 19 to 21 are the estimated u n c e r t a i n t i e s i n the c a l c u l a t i o n of i n t e g r a t e d i n t e n s i t i e s due to approximate p o s i t i o n a l parameters and temperature c o r r e c t i o n f a c t o r s . 229 230 T a b l e 17: Fe-Mn-Mg O l i v i n e s f rom the l i t e r a t u r e w i t h r e p o r t e d s i t e o c c u p a n c i e s a n d / o r s t r u c t u r e d a t a Ref . Fe Mn Mg Fe+Mn+Mg a b c Fe(M1) Mn(M1) Mg(M1) Fe(M2) Mn(M2) Mg(M2) ( 0 1 ) F a 1 .0000 0 .0000 0 .0000 1 .0000 4 .81950 10 .47880 6 .08730 1 .0000 0 .0000 0 .0000 1 .0000 0 .0000 0 .0000 ( 0 2 ) F a 1 .0000 0 .0000 0 .0000 1 .0000 4 .81800 10 .47000 6 .08600 1 .0000 0 .0000 0 .0000 1 .0000 0 .0000 0 .0000 ( 0 3 ) F a 1 .0000 0 .0000 0 .0000 1 .0000 4 .81800 10. . 47000 6 .08600 1 .0000 0 .0000 0 .0000 1 .0000 0 .0000 0 .0000 (OO)Fa 1 .0000 0 .0000 0 .0000 1 .0000 4 .81980 10. .48390 6 .09190 1 .0000 0 .0000 0 .0000 1 .0000 0 .0000 0 .0000 ( 0 4 ) F a 1 .0000 0 .0000 0 .0000 1 .0000 4 .821 10 10. .47790 6 .08890 1 .0000 0 .0000 0 .0000 1 .0000 0 .0000 0, .0000 ( 0 5 ) F a 1 .0000 0 .0000 0 .0000 1 .0000 4 .81900 10. .47000 6 .08600 1 .0000 0 .0000 0 .0000 1 . .0000 0. .0000 0, .0000 ( 0 6 ) F a 1 .0000 0 .0000 0 .0000 1 .0000 4 .81860 10, ,48220 6 .11080 1 .0000 0 .0000 0, .0000 1 .0000 0 .0000 0. .0000 (14 )0L5 0 .8900 0 . 1100 0. .0000 1 .0000 4 .82600 10. .51400 6 .10500 0 .9760 0 .0240 0, .0000 0. .8050 0 . 1950 0, .0000 (14)0L4 0 .6900 0 .3100 0. .0000 1 .0000 4 .84000 10, ,55600 6 .13500 0 .8260 0, . 1740 0, .0000 0 .5560 0 .4440 0, .0000 (14 )0L3 0 .4700 0 .5300 0, .0000 1 .0000 4 .85600 10, .58500 6 .16800 0 .6110 0 .3890 0. .0000 0. .3310 0. .6690 0, .0000 (14)0L2 0 .3000 0 .7000 0. .0000 1 .0000 4 .87100 10. .59400 6 .20000 0, .4040 0 .5960 0. .0000 0. . 1970 0. .8030 0 .0000 ( 14)0L1 0 .0900 0 .9100 0. .0000 1 .0000 4 .89600 10. , 60300 6 .24100 0, . 1350 0, .8650 0. .0000 0. .0450 0. .9550 0, .0000 (01 )Teph 0 .0000 1 . .0000 0. .0000 1 .0000 4 .90230 10. 59640 6 .25670 0, .0000 1 , .0000 0. .0000 0. .0000 1 , .0000 0. .0000 (OO)Teph 0 .0000 1 , .0000 0. .0000 1 .0000 4 .90330 10. 60300 6, .25790 0. .0000 1 , OOOO 0. .0000 0. .0000 1 . OOOO 0. .0000 (12 )Teph 0 .0000 1 . .0000 0. OOOO 1 .0000 4. .90000 10. 60500 6 , .26500 0, OOOO 1 . OOOO 0. .0000 0. .0000 1 . .0000 0. OOOO (18) 0 .5200 0. .4600 0. .0200 1 .0000 4 . 84400 10. 57700 6 . 14600 0, ,6600 0. , 2900 0. .0400 0. . 3700 0. ,6300 0. .0000 ( 1 7 ) F a 0 .9220 0. .0370 0. .0390 0 .9980 4. .81600 10. 46900 6 , 09900 (16) 0 .5500 0. .0750 0. 3750 1 .0000 4 . ,79800 10. 38700 6 . ,05500 0. 3610 0. 3890 (15)Fo51 0 .0010 0. ,4820 0. 5140 0 .9970 4 . 79400 .10. 49100 6. ,12300 0. OOOO 0. 0800 0. 9200 0. OOOO 0. ,8900 0. 1 100 (17 )Hya l 0 .4560 0. .0060 0. 5350 0 .9970 4 . 78500 10. 32500 6. 03800 (08)0G2B 0 . 2900 0. OOOO 0. 7100 1, .0000 4. .77500 10. 28000 6. 01600 0. 2920 0. OOOO 0. 7080 0. 2880 0. OOOO 0. 7120 (09)3G17 0 . 1050 0. .0000 0. 8950 1. .0000 4. .76600 10. 22500 5. 97300 0. 1050 0 . OOOO 0 . 8950 0 . 1080 0. OOOO 0. 8920 ( 1 7 ) F o 0 . 1000 0. .0000 0. 9000 1. .0000 4. .76200 10. 22500 5 . 99400 0 . 1000 0 . OOOO 0. 9000 0 . 1000 0. OOOO 0. 9000 ho CO 231 T a b l e 17: ( c o n t i n u e d ) Ref . Fe Mn Mg Fe+Mn+Mg a b c Fe(M1) Mn(M1) Mg(M1) Fe(M2) Mn(M2) Mg(M2) (09)3G12 0 .0950 0 .0000 0 . 9050 1 .0000 4 .76400 10. . 22000 5 .99200 0 .0970 0 .0000 0 .9030 0 .0950 0 .0000 0 .9050 (09)3G15 O . 0950 0 .0000 0 . 9050 1 .0000 4 .76900 10 .22900 5 .99400 0 . 1000 0 .0000 0 . 9000 0 .0880 0 .OOOO 0 .9120 (09)3G19 0 .0900 0 .0000 0 .9100 1 .0000 4 .76400 10. .22100 5 .99200 0 .0930 0 .0000 0 .9070 0 .0930 0 .0000 0 .9070 (09)3G51 0 .090O 0 .0000 0 .9100 1 .0000 4 .76200 10. .22600 5 .99200 0 .0900 0. .0000 0 .9100 0 .0860 0 .0000 0 .9140 (09 )2500 0 .0900 0. .0000 0 .9100 1 .0000 4 .76500 10. .23800 6 .00000 0 .0900 0 .0000 0 .9100 0 .0900 0 .0000 0 .9100 (09)2501 0 .0900 0 .0000 0 .9100 1 .0000 4 .75400 10. .20700 5 .98700 0 .0960 0, .0000 0 .9040 0 .0860 0 .0000 0 .9140 (09)3G9 0 .0850 0 .0000 0 .9150 1 .0000 4 .76900 10. 22600 5 .99000 0 .0850 0, .0000 0 .9150 0 .0850 0 .OOOO 0 .9150 (09)3G18 0 .0750 0. .0000 0 .9250 1 .OOOO 4 .76500 10. 22300 5 .99200 0 .07 50 0. .0000 0 . 9250 0. .0740 0 .0000 0. .9260 (09)2488 0 .0650 0 .0000 0 .9350 1 .0000 4. .77200 10. 2 1700 5 .99800 0 .0700 0 .OOOO 0. .9300 0 .0630 0 .0000 0. .9370 (10 ) -481 0 .0120 0. .0030 0 .9850 1 .0000 4. .75330 10. 19720 5 .98210 0 .0120 0. OOOO 0 .9880 0. .0180 0 .0000 0. .9820 ( 0 1 ) F o 0 .0000 0. .0000 1 .0000 1 .0000 4. .75340 10. 19020 5 .97830 0. .0000 0. .0000 1. .0000 0. ,0000 0 .0000 1. .0000 ( 1 1 )Fo 0 .0000 0, .0000 1. .0000 1 .0000 4. .75350 10. 19430 5. .98070 0. .0000 0. OOOO 1. .0000 0. OOOO 0. OOOO 1. .0000 ( 0 0 ) F o 0 .0000 0. .0000 1. .0000 1 .0000 4. 75480 10. 20040 5. .98230 0. OOOO 0. OOOO 1. .0000 0. .0000 0. .0000 1. .0000 ( 0 4 ) F o 0. .OOOO 0. 0000 1. .0000 1 .0000 4 . 75400 10. 19710 5 . . 98060 0. OOOO 0. OOOO 1. .0000 0. OOOO 0. OOOO 1. OOOO ( 0 5 ) F o 0. .0000 0. 0000 1. .0000 1 .0000 4 . 75100 10. 19700 5 . .97900 0. .0000 0. OOOO 1. OOOO 0 . OOOO 0 . .0000 1. OOOO ( 0 5 ) F o 0. .0000 0. 0000 1. .0000 1 .0000 4 . 75300 10. 19600 5 . .97900 0. OOOO 0. OOOO 1. .0000 0 . OOOO 0. .0000 1. OOOO ( 0 6 ) F o 0 .0000 0. .0000 1. .0000 1 .0000 4 . 75450 10. 20000 5 . .98140 0. OOOO 0. OOOO 1. OOOO 0. OOOO 0. .0000 1. OOOO (07 )Fo 0. .0000 0. 0000 1. .0000 1 .0000 4. 75600 10. 19500 5. .98100 0. OOOO 0. OOOO 1. OOOO 0 . OOOO 0 . OOOO 1. OOOO (OO)Fo 0. .0000 0. 0000 1. .0000 1 .0000 4. 75480 10. 20040 5. .98230 0. OOOO 0. OOOO 1. OOOO 0 . OOOO 0 . OOOO 1. OOOO ( 0 0 ) : T h i s work ( 0 3 ) : S m y t h (1975) ( 0 6 ) : L o u i s n a t h a n and Smith (1968) ( 0 9 ) : B a s s o e t a l . (1979) ( 1 2 ) : P e t e r s e t a l . ( 1973) ( 1 5 ) : F r a n c i s and R i b b e (1980) ( 1 8 ) : B r o w n (1980) ( 0 1 ) : F u j 1 n o et a l . (1981) (04) :Schwab und K u e s t n e r (1977) ( 0 7 ) : Y o d e r and Sahama (1957) (10):Wenk and Raymund (1973) ( 1 3 ) : B r a d l e y e t a l . (1966) (16 ) :Smyth and Hazen (1973) ( 0 2 ) : H a z e n (1977) ( 0 5 ) : F i s h e r and M e d a r l s (1969) ( 08 ) :Brown and P r e w i t t (1973) ( 1 1 ) : H a z e n (1976) ( 1 4 ) : A n n e r s t e n e t a l . (1984) ( 1 7 ) : B i r l e e t a l . (1968) r-o 232 T a b l e 18: P o s i t i o n a l p a r a m e t e r s f o r Fe-Mn-Mg O l i v i n e s f rom the l i t e r a t u r e Ref . X(M2) Y(M2) X ( S i ) Y ( S i ) X(01 ) Y(01 ) X(02) Y(02) X(03) Y(03) Z (03) ( 0 1 ) F a 0 .98598 0 .28026 0 .43122 0 .09765 0 .76814 0 .09217 0 .20895 0 .45365 0 .28897 0. .16563 0 .03643 ( 0 2 ) F a O .98510 O . 28030 O . 42920 0 .09730 0 .76800 0 .09070 0 .20790 0 .45510 0 .28900 0. .16500 O .04030 ( 0 3 ) F a 0 .98530 O . 28000 0 .42920 0 .09750 0 . 76870 0 .09280 0 .20760 O .45290 0 . 28840 0. 16370 0 .03830 (01 )Teph 0 .98792 0 .28041 0 .42755 O .09643 0 . 75776 O .09363 0 .21088 0 .45369 0 .28706 0. 16384 O .04140 (18) 0 .98880 0 . 27990 0 .42770 0 .09520 0 .76180 0 .09040 0 .21410 0 .45180 0 .28810 0. 16270 0 .03880 ( 17)Fa 0 .98608 0, .28004 0 .43070 0 .09723 0 .76683 0 .09197 0. .21027 0 .45308 0 .28806 0. 16532 0 .03626 (16) 0. .98670 0. . 27920 0. .42870 0. .09570 - 0 . 76610 0 .09180 0. .21270 0. ,45140 0. , 28440 0 . 16330 0. .03570 (15)Fo51 0. .98700 0, ,27900 0, .42260 0. .09100 0, .75850 0 .08670 0. .23010 0. .44890 0, .27820 0. 15900 0, .03740 (17 )Hya l 0 .98598 0. . 27880 0, .42843 0 .09587 0, .76566 0 .09430 0. .21642 0. ,45084 0. .28264 0 . 16370 0, .03435 (O8)0G2B 0. .98800 0. ,27820 0. .42750 0. .09500 0, , 76660 0 .09190 0. .21790 0. .44890 0, , 28060 0 . 16380 0. .03400 (09)3G17 0. .98980 0. 27780 0. .42670 0. ,09440 0. .76610 0 .09180 0. 22050 0. ,44790 0. .27880 0 . 16320 0, .03370 ( 1 7 ) F o 0 .98975 0. 27743 0. .42693 0. .09434 0. . 76580 0 .09186 0. 22012 0. .44779 0, ,27810 0 . 16346 0, .03431 (09)3G12 0. ,99010 0 . 27760 0. 42670 0. 09430 0. 76590 0. .09160 0 . 22060 0. 44760 0. 27820 0 . 16310 0. 03370 (09)3G15 0. 99000 0 . 27770 0. 42690 0. 09440 0. 76620 0. .09180 0 . 22010 0 . 44780 0. 27840 0 . 16330 0. 03370 (09)3G19 0. .99010 0 . 27770 0. 42670 0. 09440 0. 76590 0. ,09170 0 . 22050 0 . 44770 0. 27850 0 . 16320 0. .03360 (09)3G51 0. ,99010 0. 27760 0. .42660 0. 09430 0. 76600 0. .09180 0 . 22040 0 . 44780 0. 27830 0 . 16310 0. .03360 (09)2500 0, ,99010 0. 27770 0. .42680 0. 09440 0. 76570 0. .09180 0 . 22050 0 . 44750 0. , 27830 0 . 16320 0. .03350 (09)2501 0. ,99010 0. 27770 0. 42670 0. 09430 0. 76590 0. .09170 0 . 22030 0 . 44770 0. 27850 0 . 16310 0. .03360 (09)3G9 0. .99010 0. 27770 0. 42670 0. 09430 0. 76580 0. .09170 0 . 22050 0. 44760 0. 27840 0 . 16320 0. 03360 (09)3G18 0. 99010 0 . 27760 0. 42680 0. 09430 0. 76610 0. 09160 0 . 22040 0 . 44770 0 . 27840 0 . 16320 0 . 03350 (09)2488 0. 99010 0 . 27760 0. 42650 0. 09440 0. 76630 0. 09160 0 . 22090 0 . 44760 0 . 27780 0 . 16330 0 . 03350 T a b l e 18: ( c o n t i nued) Ref . X(M2) Y(M2) X(S1 ) Y(S1 ) X(01) Y(01) X(02) Y(02) X(03) Y(03) Z(03) (10) -481 0 .99119 0.27744 0.42625 0.09409 0.76557 0.09144 0 .22163 0.44721 0 .27723 0.16311 ( 0 1 ) F o ( H ) F o 0 .99169 0.27739 0 .42645 0 .99150 0 .27740 0 .42620 0 .09400 0 .76570 0 .09130 0 .22150 0 .44740 0 .27770 0 .16280 0 .03315 0 .09403 0.76594 0.09156 0 .22164 0 .44705 0.27751 0 .16310 0 .03304 0 .03310 ( 0 0 ) : T h i s work ( 0 3 ) : S m y t h (1975) ( 0 6 ) : L o u i s n a t h a n and Smith (1968) ( 0 9 ) : B a s s o e t a l . (1979) ( 1 2 ) : P e t e r s e t a l . ( 1973) ( 1 5 ) : F r a n c i s and R i b b e (1980) ( 1 8 ) : B r o w n (1980) (01) (04) (07) (10) (13) (16) Fuj ino e t a l . ( 1981) Schwab und K u e s t n e r (1977) Yoder and Sahama (1957) Wenk and Raymund (1973) B r a d l e y et a l . (1966) Smyth and Hazen (1973) (02) (05) (08) (11) (14) (17) Hazen (1977) F i s h e r and M e d a r i s (1969) Brown and P r e w i t t (1973) Hazen (1976) A n n e r s t e n e t a l . (1984) B i r l e e t a l . ( 1968) T a b l e 19: T e m p e r a t u r e c o r r e c t i o n f a c t o r s f o r Fe-Mn-Mg O l i v i n e Ref . Fe Mn Mg B(M1 ) ( 0 1 ) F a 1 .0000 0, .0000 0 .0000 0 .52000 ( 0 2 ) F a 1 .0000 0, .0000 0 .0000 0 .48000 ( 0 3 ) F a 1 .0000 0. .0000 0 .0000 0, .57000 (01 )Teph 0 .0000 1 . .0000 0 .0000 0, . 62000 (18) 0 . 5200 0. .4600 0 .0200 0, . 39000 ( 17)Fa o . 9220 O. .0370 0 .0390 0. ,41000 (16) 0 . 5500 0 .0750 0 . 3750 0, .44000 (15)Fo51 o .0010 0. .4820 0 .5140 0. .42000 (17 )Hya l 0 .4560 0. .0060 0 .5350 0. .32000 (O8)0G2B 0 . 2900 0. .0000 0 .7100 0. .37000 (09)3G17 0. . 1050 0. .0000 0 .8950 0. .45000 ( 17)Fo 0 . 1000 0. .OOOO 0 .9000 0, ,33000 (09)3G12 o .0950 0. .0000 0 .9050 0, .42000 (09)3G15• 0 .0950 0. .0000 0 .9050 0. .40000 (09)3G19 0 .0900 0. .0000 0 .9100 0. ,50000 (09)3G51 0 .0900 0. .0000 0. .9100 0. .37000 (09)2500 0 .0900 0, .0000 0. .9100 0. ,52000 (09)2501 0 .0900 0, .0000 0, ,9100 0. .45000 (09)3G9 0 .0850 0. .0000 0. .9150 0. 49000 (09)3G18 0 .0750 0. .0000 0. .9250 0. 40000 (09)2488 0 .0650 0, ,0000 0, ,9350 0. 57000 from the l i t e r a t u r e B(M2) 0 .47400 0 .37000 0 .50000 0 .53100 0 .41000 0 .36000 0 .33000 0 .54000 0 .37000 0 .38000 0 .49000 O.36000 O.46000 0 .41000 0 .51000 0 .37000 0 .54000 0 .45000 0 .52000 0 .41000 0 .56000 B ( S i ) 0 .37200 0 .35000 0 .41000 0 .38200 0 .28000 O.27000 0 .33000 0 .36000 0 .19000 0 .29000 0.33O00 0 .20000 0 .30000 0 .34000 0 .36000 0 .25000 0 .41000 0 .33000 0 .40000 0 .34000 0 .52000 B(01 ) 0 .51000 0 . 36000 0 .53000 O.53700 O.54000 O.43000 0 .49000 O.50000 O.40000 0 .48000 0 .48000 O.35000 0 .47000 0 .48000 0 .51000 0 .44000 0 .59000 O.50000 0 .56000 0 .50000 0 .70000 B(02) 0 .52600 0 .33000 0 .42000 O.54200 0.41OOO 0 .48000 O.50000 O.48000 O.56000 0 .46000 0 .48000 0 .42000 0 .47000 O.5O0OO O.53000 0 .42000 O.58000 0 .48000 0 .57000 0 .51000 0 .68000 B(03) 0 .58300 0 .58000 0 .49000 0 .57900 0 .54000 O.52000 0 .55000 0 .50000 0 .50000 0 .50000 0 .52000 0 .41000 O.50000 0 .53000 O.57000 O.45000 0 .62000 0 .52000 O.60000 0 .54000 0 .70000 T a b l e 19: ( c o n t i n u e d ) R e f . Fe Mn Mg B(M1) B(M2) B ( S i ) B(01 ) B ( 0 2 ) B ( 0 3 ) ( 0 1 ) F o ( 1 1 ) F o 0 .0000 0 .0000 0 .0000 0 .0000 1.0000 1.0000 0 .44700 0 .26000 0 .44200 0 .22000 0 .30400 0 .08000 0 .40100 0 .26000 0 .40300 0 .24000 0 .43500 0 .28000 (00) T h i s work (01) Fuj ino et a l . ( 1981) (02) Hazen (1977) (03) Smyth (1975) (04) Schwab und K u e s t n e r (1977) (05) F i s h e r and M e d a r i s (1969) (06) L o u i s n a t h a n and Smith (1968) (07) Yoder and Sahama (1957) (08) Brown and P r e w i t t (1973) (09) B a s s o e t a l . ( 1979) (10) Wenk and Raymund (1973) (11) Hazen (1976) (12) P e t e r s e t a l . ( 1973) (13) B r a d l e y e t a l . ( 1966) (14) A n n e r s t e n e t a l . (1984) (15) F r a n c i s and R i b b e (1980) (16) Smyth and Hazen (1973) (17) B i r l e e t a l . (1968) (18) Brown (1980) 23S T a b l e 2 0 . : E s t i m a t e d e r r o r s i n the c a l c u l a t i o n of i n t e g r a t e d i n t e n s i t i e s : Example f o r T e p h r o i t e ( M n * S i O « ) hk l xM2 yM2 x S i y S i x01 y01 x02 y02 x03 y03 z03 bM1 bM2 b S i b01 b02 b03 e r r . % INT. 020 0 .52 - 0 . 7 3 - 2 . 5 3 1.50 - 4 . 3 8 . - 0 . 7 0 1.01 - 0 . 2 2 - 0 . 1 4 - 0 . 3 4 0 .35 5 .52 9.21 110 - 0 . 0 6 - 0 . 8 6 - 1 . 2 0 0 .24 0 .35 - 0 . 0 5 1.04 - 0 . 1 2 - 0 . 5 6 0.61 -1 .41 0 .37 0 . 7 9 - 0 . 0 2 0 .16 0 .15 2 .64 7 .15 021 . - 1 . 0 9 0 .23 0 .89 1.98 - 0 . 5 3 3.37 0 . 0 0 0 .59 - 0 . 8 6 - 0 . 4 8 0 .33 - 0 . 2 5 4 .36 12.55 101 3 .68 - 2 . 3 0 0 .02 0 .23 - 0 . 0 6 4 .55 0 . 0 0 0 .16 - 0 . 5 9 0 . 7 0 - 0 . 7 7 - 0 . 3 7 6 .42 7 .44 111 0 . 1 3 - 0 . 0 7 - 0 . 3 4 - 0 . 1 4 0 .09 0 .03 - 0 . 1 3 - 0 . 1 6 - 0 . 1 0 - 0 . 0 4 - 0 . 4 3 0 . 0 0 - 1 . 1 5 0 . 3 3 -0 .01 - 0 . 0 3 0 .04 1.36 54 .56 120 - 1 . 3 0 0 . 1 0 - 2 . 0 5 0 .13 0 .02 - 1 . 0 2 - 0 . 1 2 2 .23 - 0 . 1 9 - 2 . 2 9 . 0 . 0 0 0 .07 -0 .61 0.71 0 . 4 6 1.37 4 .49 6 .82 002 . . . . . . . . . . 2 .60 1.29 - 1 . 7 7 -1 .01 - 0 . 5 7 - 0 . 6 4 1.03 3 .79 7 .17 121 - 5 . 1 6 - 0 . 0 6 - 1 . 2 3 - 0 . 5 5 0.01 4 .02 0 .29 2 .28 0 .04 - 1 . 7 3 - 3 . 1 3 O.OO - 0 . 3 6 - 0 . 4 8 0 .62 - 1 . 4 8 0 . 3 9 8.11 3 .56 130 0 . 0 3 0 .36 0 .06 0 . 2 0 - 0 . 0 1 - 0 . 0 4 0 . 1 1 - 0 . 1 5 0 .17 0.01 - 0 . 5 5 - 0 . 4 1 - 0 . 0 9 0 . 0 0 0 . 0 4 - 0 . 1 1 0 .86 8 6 . 7 0 022 - 0 . 1 7 0 .22 0 .75 - 0 . 4 4 - 1 . 1 2 0 .64 - 0 . 9 7 - 1 . 2 4 0 .26 0 . 1 6 0 .38 0 .34 2.31 29.91 040 - 0 . 7 8 - 0 . 3 9 - 1 . 4 0 1.85 . 3.11 - 1 . 2 6 - 1 . 2 5 0 .72 0 .37 - 0 . 2 3 0.61 4 .48 8 .96 131 0 . 0 6 - 0 . 3 3 0 . 3 3 - 0 . 0 7 - 0 . 0 9 0.01 0 .19 0 .18 0 . 0 0 - 0 . 1 2 0.01 0 . 0 0 - 1 . 1 0 - 0 . 6 3 0 .02 0 . 0 9 - 0 . 0 0 1.39 64 .32 112 0.01 0 .16 0 .22 - 0 . 0 4 - 0 . 0 6 0.01 - 0 . 1 8 0 .02 - 0 . 0 8 0 .09 0 .12 - 0 . 7 7 - 0 . 2 0 - 0 . 4 3 0.01 - 0 . 0 9 0 .07 0 .99 100.00 200 0.31 1.21 0 .02 - 0 . 2 3 . 0 .22 . -1 .11 -1 .51 - 0 . 4 5 0 . 4 5 0 . 4 5 0 .84 2 .55 12.53 041 0 .85 - 0 . 4 7 -1 .51 0 .87 . - 0 . 5 5 - 1 . 4 0 0 . 0 0 - 1 . 4 9 - 0 . 7 9 - 0 . 4 6 0 .67 0 . 2 9 3 .14 12.80 210 - 3 . 9 4 -0 .01 - 0 . 8 9 0 .15 0.21 0 .06 0 .25 0 .34 0 .76 0.31 . 0 . 0 0 - 0 . 4 6 - 0 . 8 4 - 0 . 0 5 0 . 1 5 - 0 . 7 6 4 .33 6 .12 122 0 . 7 5 - 0 . 0 6 1.18 - 0 . 0 7 -0 .01 0 .55 0 .06 - 1 . 1 9 - 0 . 0 9 - 1 . 0 8 - 2 . 1 2 0 . 0 0 0 .09 0 . 8 0 - 0 . 8 8 - 0 . 5 8 - 1 . 4 8 3 .65 11.98 140 - 1 . 0 2 0 . 0 6 0 .06 0 .23 - 0 . 0 0 - 1 . 5 2 0 .08 - 0 . 8 5 - 0 . 0 7 2 .10 . 0 . 0 0 - 0 . 1 2 0 .42 -0 .51 - 0 . 7 2 - 1 . 2 0 3.31 10.79 211 0 . 8 8 - 0 . 0 7 - 1 . 4 7 - 0 . 1 2 0 . 3 4 - 0 . 0 5 - 0 . 9 4 0.11 0 . 1 3 - 0 . 1 5 0 .62 0 . 0 0 0 . 1 1 - 1 . 5 9 - 0 . 1 0 - 0 . 6 2 - 0 . 1 5 2.71 7 .56 132 - 0 . 1 0 - 1 . 2 5 - 0 . 2 0 - 0 . 6 8 0 .04 0 .12 - 0 . 3 6 0 .48 0 .48 0 .04 - 0 . 6 7 - 3 . 5 0 2 .65 0 .58 -0 .01 - 0 . 2 5 - 0 . 5 8 4.81 5 .59 ho 221 0 . 4 0 1.37 - 4 . 0 0 - 0 . 1 5 - 0 . 0 6 0 .96 - 0 . 4 4 1.90 - 0 . 1 7 -0 .51 3.26 0 . 0 0 - 2 . 6 9 2 .09 - 1 . 9 3 1.16 - 0 . 8 9 7 .16 2 .47 cr> 237 T a b l e 20 . ( c o n t i n u e d ) hk l xM2 yM2 x S i y S i x01 y01 x02 y02 x03 y03 z03 bM1 bM2 b S i b01 b02 b03 e r r . % INT. 042 1.43 . 0 .73 2 .46 - 3 . 1 7 . 4 .70 1.66 -3 .91 3 .97 - 2 . 2 9 - 1 . 0 9 0 . 7 0 1.58 9 .17 2 .29 150 - 0 . 1 3 - 1 . 2 4 0 .73 0.11 - 0 . 1 8 - 0 . 0 3 0 .05 - 0 . 8 9 - 0 . 2 0 - 1 . 4 0 . - 3 . 5 8 4.11 - 2 . 4 6 0 .07 0 .04 0 .29 6 . 3 9 2 .77 113 0.11 - 0 . 0 6 - 0 . 2 9 - 0 . 1 2 0 .07 0 .02 - 0 . 0 9 - 0 . 1 2 0 . 2 0 0 .09 0 . 7 0 0 . 0 0 - 3 . 6 5 1.01 - 0 . 0 3 - 0 . 0 8 - 0 . 3 0 3 .89 9 .95 151 0 .08 - 1 . 6 4 - 0 . 0 8 0 .96 0 .03 - 0 . 1 6 - 0 . 4 3 -0 .11 0 . 1 0 - O . 1 6 0 .45 0 . 0 0 -2 .91 0 .28 -0 .01 - 0 . 3 8 - 0 . 1 6 3 .58 5 .47 222 0 . 1 5 - 0 . 0 8 - 0 . 2 3 0 . 0 6 - 0 . 0 0 - 0 . 3 3 0 .09 0 . 1 7 - 0 . 0 4 0 . 4 4 - 0 . 2 5 - 1 . 0 4 - 1 . 3 1 0 . 1 6 - 0 . 1 5 - 0 . 3 3 - 0 . 3 0 1.88 69 .26 123 - 2 . 5 2 - 0 . 0 3 -0 .61 - 0 . 2 7 0 . 0 0 1.69 0 .12 0 .96 - 0 . 0 5 2 .02 2.91 0 . 0 0 -0 .51 -0 .71 0 .78 - 1 . 8 5 - 1 . 3 5 5 .45 2 .64 240 0 . 1 4 - 0 . 3 7 - 0 . 6 0 -0 .11 -0 .01 0 .62 - 0 . 0 5 -0 .71 - 0 . 0 7 - 1 . 2 0 -1 .31 - 1 . 2 9 0 .42 - 0 . 3 5 0 . 2 0 -0 .51 2.61 21 .67 241 0 . 2 5 0.71 0 .98 - 0 . 2 2 0.01 1.15 0 .23 - 0 . 5 8 - 0 . 0 5 0 .37 0 .95 0 . 0 0 - 2 . 4 5 - 0 . 7 4 0 .69 - 0 . 8 9 - 0 . 3 8 3.51 11.10 061 0 . 4 0 0 .47 1.44 . 0 .24 - 0 . 7 3 0 .09 0 . 0 0 - 2 . 2 0 - 0 . 6 6 - 0 . 2 5 - 0 . 7 3 - 0 . 0 4 2 . 9 9 12.29 133 0 . 0 6 - 0 . 3 3 0 . 3 3 - 0 . 0 7 - 0 . 0 8 0.01 0 .16 0 .15 -0 .01 0 .29 - 0 . 0 3 0 . 0 0 - 2 . 5 7 -1 .51 0 .04 0 . 1 7 0.01 3 .05 15.24 152 0 . 0 6 0 . 5 5 - 0 . 3 2 - 0 . 0 5 0 .07 0.01 - 0 . 0 2 0 .36 - 0 . 0 7 - 0 . 4 9 0 . 1 0 - 2 . 2 0 - 2 . 4 7 1.52 - 0 . 0 4 - 0 . 0 3 0 .14 3 .75 15.62 043 0 . 5 5 - 0 . 3 2 - 0 . 8 7 0 . 5 0 0 .86 1.76 0 . 0 0 - 2 . 1 3 - 1 . 1 8 - 0 . 5 8 0 . 8 5 - 0 . 9 9 3 .63 7 .60 310 - 0 . 1 4 - 0 . 2 2 - 2 . 0 5 0.01 - 0 . 2 2 0 .03 - 0 . 5 2 0 .07 0 .29 - 0 . 3 7 - 2 . 7 7 0.71 0 .14 0 .12 - 0 . 7 4 - 0 . 6 9 3 .75 4 .92 004 . - 1 . 3 6 - 0 . 9 6 - 1 . 3 3 - 0 . 7 8 - 0 . 3 5 - 0 . 3 9 - 0 . 3 6 2 .36 20 .52 242 - 0 . 4 1 1.08 1.82 0.31 0 .02 - 1 . 7 0 0 .15 1.98 - 0 . 1 8 - 2 . 8 8 - 1 . 0 3 - 4 . 9 8 5 .04 - 1 . 6 7 1.28 - 0 . 7 2 - 1 . 6 3 8 .88 2 .99 062 - 0 . 5 2 - 0 . 1 5 - 0 . 3 5 . - 0 . 8 8 0 .12 - 0 . 5 0 - 1 . 2 6 - 0 . 7 2 -0 .91 - 0 . 4 2 - 0 . 0 9 -0 .81 2 .29 23 .85 330 0 . 3 3 0 . 4 7 0.51 0 .02 0 .04 0 .12 - 0 . 2 7 0 .43 - 0 . 4 3 - 0 . 0 4 - 2 . 6 7 - 1 . 9 6 - 0 . 0 4 - 0 . 0 3 - 0 . 4 5 1.24 3.71 5.27 170 0 .04 0 . 1 9 0 . 0 9 - 0 . 3 7 - 0 . 0 2 0 . 0 5 - 0 . 0 5 - 0 . 2 7 - 0 . 0 7 0 .42 - 1 . 7 4 - 2 . 3 4 - 0 . 5 7 0 . 0 2 - 0 . 0 7 0 .17 3 .05 12.43 331 0 . 6 7 - 0 . 3 9 2 .67 -0 .01 0 .27 - 0 . 0 3 - 0 . 4 2 - 0 . 4 8 -0 .01 0 .33 - 0 . 0 4 0 . 0 0 - 4 . 1 4 - 0 . 2 2 - 0 . 1 8 - 0 . 7 4 0 .02 5.11 5 .58 312 0 . 1 3 0 . 2 0 1.88 T0 .01 0 .19 - 0 . 0 2 0 .44 - 0 . 0 6 0.21 - 0 . 2 7 - 0 . 3 4 - 3 . 0 9 - 0 . 8 0 - 0 . 1 5 - 0 . 1 3 0 .78 - 0 . 6 4 3.91 7 .85 322 - 1 . 0 9 0 .08 - 0 . 1 8 0 .08 - 0 . 0 3 0 .23 0 .22 - 0 . 3 7 - 0 . 3 0 - 0 . 3 5 - 0 . 6 9 0 . 0 0 - 0 . 3 7 -2 .51 - 1 . 0 1 -0 .51 - 1 . 3 5 3 .42 7.31 134 0 . 0 3 0 .38 0 . 0 6 0 .22 -0 .01 - 0 . 0 3 0 .09 - 0 . 1 2 0 .07 0.01 - 0 . 5 9 - 2 . 4 7 - 1 . 8 5 - 0 . 4 4 0.01 0 . 1 5 - 0 . 2 0 3 .22 12.26 238 T a b l e 2 1 . : E s t i m a t e d e r r o r s i n the c a l c u l a t i o n o f i n t e g r a t e d i n t e n s i t i e s : Example f o r F a y a l i t e ( F e ; S i O « ) h k l xM2 yM2 x S i y S i x01 y01 x02 y02 x03 y03 z03 bM1 bM2 b S i b01 b02 b03 e r r . % INT. 020 . 0 . 54 - 0 . 7 5 - 2 . 5 5 . 1.52 - 4 . 4 3 - 0 . 8 3 1.07 - 0 . 2 2 - 0 . 1 5 - 0 . 3 5 0 . 3 7 5.61 8 .98 110 - 0 . 0 7 -0 .81 - 0 . 9 7 0 .23 0.31 - 0 . 1 0 0 .94 -0 .11 - 0 . 5 0 0 .58 - 1 . 3 7 0 .36 0 .77 - 0 . 0 6 0 .16 0 . 1 5 2 .43 8 .46 021 - 1 . 1 6 0 .23 0 .97 2 .07 - 0 . 5 2 3 .50 0 . 0 0 0 .64 - 0 . 9 3 - 0 . 5 2 0 . 3 5 - 0 . 2 5 4 . 5 7 11.41 101 4 . 5 6 - 2 . 8 6 0 .05 0 . 3 0 - 0 . 0 7 5 .52 0 . 0 0 0 . 2 5 - 0 . 6 7 0 .88 - 0 . 9 7 - 0 . 4 3 7 .87 4 .98 111 0 . 1 5 - 0 . 0 7 - 0 . 2 9 - 0 . 1 4 0 .09 0 .06 - 0 . 1 2 - 0 . 1 6 - 0 . 0 9 - 0 . 0 3 - 0 . 4 3 0 . 0 0 - 1 . 1 8 0 .35 - 0 . 0 2 - 0 . 0 3 0 .04 1.38 58 .12 120 - 1 . 3 9 0 . 1 3 - 2 . 2 3 0 .12 0 .04 - 1 . 1 3 - 0 . 1 4 2 .35 - 0 . 2 0 - 2 . 4 7 0 . 0 0 0 . 0 9 - 0 . 5 9 0 . 7 6 0 . 5 0 1.47 4 .82 6 . 1 0 121 - 5 . 2 3 - 0 . 0 8 - 1 . 2 0 - 0 . 4 8 0 .02 3.91 0.31 2 .24 0 .04 - 1 . 5 6 - 3 . 1 6 0 . 0 0 - 0 . 4 6 - 0 . 4 2 0 . 6 6 - 1 . 5 0 0 .38 8 .07 3.61 002 . . 2 .46 1.43 - 1 . 9 5 - 1 . 0 8 -0 .61 - 0 . 6 9 1.13 3 . 9 0 6 .64 130 0 . 0 3 0 . 3 6 0 . 0 5 0 .20 -0 .01 - 0 . 0 9 0.11 - 0 . 1 5 0 .17 0.01 - 0 . 5 7 - 0 . 4 2 - 0 . 1 0 0 . 0 0 0 . 0 5 - 0 . 1 2 0 .88 91 .75 022 - 0 . 1 6 0.21 0 . 7 0 -0 .41 - 1 . 0 8 0 .58 - 1 . 0 0 - 1 . 2 8 0 . 2 5 0 . 1 6 0 .38 0 . 3 6 2 . 2 9 31 .69 040 - 0 . 7 4 - 0 . 3 4 - 1 . 3 5 1.70 . 2 . 90 - 1 . 2 3 - 1 . 2 3 0 . 6 9 0 .33 - 0 . 2 2 0 . 5 5 4.21 10.48 131 0 . 0 8 - 0 . 3 4 0 . 2 9 - 0 . 0 8 - 0 . 0 9 0 .02 0 .18 0 .19 0 . 0 0 - 0 . 1 2 0.01 0 . 0 0 - 1 . 1 6 - 0 . 6 5 0 . 0 5 0 . 0 9 - 0 . 0 0 1.44 64 .86 112 0.01 0 .16 0 . 1 9 - 0 . 0 4 - 0 . 0 6 0 .02 - 0 . 1 8 0 .02 - 0 . 0 8 0 . 1 0 0.11 - 0 . 8 2 -0 .21 - 0 . 4 5 0 . 0 3 - 0 . 0 9 0 .08 1.03 100.00 200 0 . 3 5 1.00 0 .04 - 0 . 2 2 0.21 - 1 . 0 7 - 0 . 4 4 - 0 . 5 0 0.41 0.41 0 . 7 7 1.93 15.07 041 0 .86 . - 0 . 4 7 -1 .41 0 .86 - 0 . 4 7 - 1 . 3 9 0 . 0 0 - 0 . 5 2 - 0 . 7 6 - 0 . 4 8 0 .67 0 . 2 7 2 .73 13.38 210 - 3 . 7 5 -0 .01 - 0 . 9 8 0 .13 0 .18 0 . 1 3 0 .23 0 .33 0 .69 0 . 3 0 0 . 0 0 - 0 . 5 6 - 0 . 7 5 -0 .11 0 .15 - 0 . 7 6 4 . 1 6 6 .98 122 0 . 7 5 - 0 . 0 7 1.19 - 0 . 0 6 - 0 . 0 2 0 .56 0 .07 - 1 . 1 6 - 0 . 0 9 - 1 . 0 9 -1 .91 0 . 0 0 0.11 0 .73 - 0 . 8 8 - 0 . 5 8 - 1 . 5 2 3 .53 12.20 140 - 0 . 9 8 0 .08 0 . 5 7 0 .20 -0 .01 - 1 . 3 8 0 .08 -0 .81 - 0 . 0 7 1.91 0 . 0 0 - 0 . 1 5 0 . 3 5 -0 .51 - 0 . 7 0 - 1 . 1 8 3 .14 11.99 211 0 . 8 9 - 0 . 0 9 - 1 . 7 2 -0 .11 0 .34 - 0 . 1 0 - 0 . 9 2 0 .12 0 .12 - 0 . 1 5 0 .64 0 . 0 0 0 . 1 5 - 1 . 5 0 - 0 . 2 4 - 0 . 6 8 - 0 . 1 5 2 . 8 3 7.41 132 - 0 . 1 2 - 1 . 2 5 - 0 . 1 9 - 0 . 6 7 0 .03 0 .28 - 0 . 3 4 0 .49 0 .47 0 .02 - 0 . 6 3 - 3 . 6 2 2 .72 0 . 6 5 - 0 . 0 3 - 0 . 2 7 - 0 . 6 2 3 .38 5 .85 221 0.61 1.70 - 4 . 4 2 -0 .21 - 0 . 1 6 1.18 - 0 . 5 6 2.24 - 0 . 2 0 - 0 . 5 7 3 .85 0 . 0 0 - 3 . 4 3 3 .08 -2 .31 1.41 - 1 . 0 0 8 .62 1.65 239 T a b l e 21 . ( c o n t i n u e d ) hk l xM2 yM2 xS1 y S i xQ1 y01 x02 y02 x03 y03 z03 bM1 bM2 bS 1 b01 b02 b03 e r r . % INT. 042 . 1.43 0 .69 2.51 - 3 . 1 0 4 .75 1.46 - 4 . 0 9 4 .18 - 2 . 3 8 - 1 . 0 5 0.71 1.57 9 . 3 3 2 .33 150 - 0 . 1 6 - 1 . 2 8 0 .64 0 .08 - 0 . 1 8 - 0 . 1 0 0 .06 - 0 . 9 4 -0 .21 - 1 . 4 2 - 3 . 7 6 4 .29 - 2 . 5 9 0 .16 0 . 0 5 0 .32 6 .67 2 .79 113 0 .14 - 0 . 0 6 - 0 . 2 5 - 0 . 1 2 0 .07 0 . 0 5 - 0 . 0 9 - 0 . 1 2 0 .18 0 .07 0 .74 0 . 0 0 - 3 . 8 2 1.12 - 0 . 0 6 - 0 . 0 9 - 0 . 3 0 4 .08 . 10.11 151 0 . 1 0 - 1 . 5 7 - 0 . 0 4 0 .92 0 . 0 3 - 0 . 3 4 - 0 . 4 0 - 0 . 1 1 0 . 0 8 - 0 . 1 5 0 .43 0 . 0 0 - 2 . 8 9 0 . 2 0 - 0 . 0 4 - 0 . 3 9 - 0 . 1 4 3 .52 6 .25 222 0 .18 - 0 . 0 8 - 0 . 2 0 0 .07 -0 .01 -0 .31 0 . 1 0 0 .16 - 0 . 0 5 0 .43 - 0 . 2 3 - 1 . 0 9 - 1 . 3 7 0 .18 - 0 . 1 6 - 0 . 3 3 -0 .31 1.95 69.71 240 0 .18 - 0 . 3 8 - 0 . 5 7 - 0 . 1 2 - 0 . 0 2 0 . 6 3 - 0 . 0 6 - 0 . 7 0 - 0 . 0 7 -1 .21 -1 .41 - 1 . 3 8 0 .52 - 0 . 3 3 0 . 2 0 - 0 . 5 0 2 .73 2 1 . 0 0 123 - 2 . 6 4 - 0 . 0 4 - 0 . 6 2 - 0 . 2 5 0.01 1:69 0 .13 0 .97 - 0 . 0 5 1.90 3.18 0 . 0 0 - 0 . 6 9 - 0 . 6 5 0 .87 - 1 . 9 7 - 1 . 3 7 5 .69 2 .42 241 0.31 0.71 0 .85 - 0 . 2 6 0 .03 1.04 0 .24 - 0 . 5 6 - 0 . 0 5 0 .32 0 .93 0 . 0 0 -2 .51 - 0 . 8 4 0 . 7 0 - 0 . 8 8 - 0 . 3 6 3 .49 11.46 061 . 0.41 0 .45 1.45 0 .22 - 0 . 6 6 0 .06 0 . 0 0 - 2 . 2 8 - 0 . 7 2 -0 .21 - 0 . 7 3 - 0 . 0 2 3 .04 12.56 133 0 .08 - 0 . 3 3 0 .29 - 0 . 0 8 - 0 . 0 8 0 .02 0 .16 0 .16 - 0 . 0 0 0 .28 - 0 . 0 2 O.OO - 2 . 7 3 - 1 . 5 9 0 .09 0 . 1 9 0.01 3 .22 15.08 152 0 . 0 7 0 . 5 5 - 0 . 2 8 - 0 . 0 3 0 .07 0 .04 - 0 . 0 2 0 .37 - 0 . 0 7 - 0 . 4 9 0 . 1 0 - 2 . 2 5 - 2 . 5 2 1.56 - 0 . 0 9 - 0 . 0 3 0 .15 3 .83 16.49 043 0 . 5 7 - 0 . 3 3 - 0 . 8 3 0 . 5 0 0 .77 1.88 O.OO - 2 . 2 8 - 1 . 2 0 - 0 . 6 3 0 . 8 9 - 0 . 9 8 3 .78 7 .27 310 - 0 . 1 9 - 0 . 2 4 - 2 . 1 3 0 .02 - 0 . 2 3 0 .07 - 0 . 5 3 0 .08 0 .29 -0 .41 . - 3 . 1 8 0.81 0 . 5 0 0 .32 - 0 . 8 6 - 0 . 7 9 4.21 4 .13 242 - 0 . 4 7 1.00 1.56 0 .33 0 .04 - 1 . 5 7 0 .15 1.77 - 0 . 1 7 - 2 . 6 7 - 0 . 8 3 - 4 . 8 5 4 . 9 0 - 1 . 9 0 1.13 - 0 . 6 7 - 1 . 4 9 8 . 5 0 3 .49 004 . . . . - 1 . 2 5 - 1 . 0 2 - 1 . 4 1 - 0 . 8 0 - 0 . 3 5 - 0 . 4 0 - 0 . 4 2 2 .39 20 .32 062 - 0 . 5 3 . - 0 . 1 6 - 0 . 2 9 - 0 . 8 6 0 .07 - 0 . 4 5 -1 .31 - 0 . 7 6 - 0 . 9 0 - 0 . 4 3 - 0 . 0 9 - 0 . 8 3 2.31 24 .38 330 0 . 4 0 0 . 4 5 0 . 5 0 0 .07 0 .03 0 .27 - 0 . 2 5 0 .43 - 0 . 3 9 - 0 . 0 2 -2 .71 - 1 . 9 4 - 0 . 1 4 - 0 . 0 5 - 0 . 4 7 1.25 3 .74 5 .72 170 0 . 0 5 0 . 2 0 0 .07 - 0 . 3 8 - 0 . 0 3 0.11 - 0 . 0 5 - 0 . 2 8 - 0 . 0 6 0 .44 - 1 . 8 1 - 2 . 4 2 - 0 . 5 4 0 .04 - 0 . 0 8 0 .17 3 .15 12.82 331 0 .73 - 0 . 3 4 2.21 - 0 . 0 2 0 .22 - 0 . 0 6 - 0 . 3 4 - 0 . 4 4 - 0 . 0 0 0 .27 - 0 . 0 2 0 . 0 0 - 3 . 7 5 - 0 . 6 4 - 0 . 3 8 - 0 . 7 0 0.01 4 . 5 9 7 .40 312 0 .14 0 .18 1.61 -0 .01 0 .16 - 0 . 0 5 0 .37 - 0 . 0 5 0 .18 - 0 . 2 5 - 0 . 2 9 - 2 . 9 5 - 0 . 7 4 - 0 . 4 8 - 0 . 2 8 0 . 7 6 -0 .61 3 .67 9 . 6 0 322 -1 .11 0 . 1 1 - 0 . 5 6 0 . 0 8 - 0 . 0 7 0 . 2 3 0 . 2 4 - 0 . 3 6 - 0 . 3 1 - 0 . 3 5 - 0 . 6 2 0 . 0 0 - 0 . 4 8 - 2 . 5 8 - 1 . 0 0 - 0 . 5 1 - 1 . 3 9 3 .53 6 .89 Co 240 T a b l e 2 2 . : E s t i m a t e d e r r o r s in the c a l c u l a t i o n o f i n t e g r a t e d i n t e n s i t i e s : Example f o r F o r s t e r i t e ( M g z S i O Q hkl xM2 yM2 x S i y S i x01 y01 x02 y02 x03 y03 z03 bM1 bM2 b S i b01 b02 b03 e r r . % INT. 020 . 0 . 2 5 - 0 . 7 9 - 2 . 7 3 1.84 - 4 . 8 0 - 0 . 4 5 0 .58 - 0 . 2 8 - 0 . 1 7 - 0 . 3 8 0 . 4 0 5 .96 22 .34 110 - 0 . 0 6 1.17 3.31 00 .69 -1 .01 0 .26 - 3 . 0 0 0 .28 1.67 - 1 . 3 0 . 2 .04 - 0 . 4 8 - 2 . 5 0 0 .16 - 0 . 3 6 - 0 . 3 5 6 .19 2 .76 021 - 0 . 4 0 0 .19 0 . 7 0 1.38 -0 .31 2 .54 0 . 0 0 0.21 - 0 . 6 9 - 0 . 3 8 0 . 2 9 - 0 . 1 6 3 .14 65.51 101 1.74 -2 .31 0 .04 0 .16 - 0 . 0 3 4 .60 0 . 0 0 0 .06 - 0 . 5 9 0 .74 - 0 . 8 3 - 0 . 3 2 5 .59 2 2 . 0 0 111 0 . 1 5 - 0 . 1 0 - 1 . 0 3 - 0 . 4 8 0.31 0 .19 - 0 . 4 9 - 0 . 3 8 - 0 . 2 7 - 0 . 0 8 - 1 . 0 9 0 . 0 0 - 1 . 9 7 1.21 - 0 . 0 7 - 0 . 0 9 0 . 0 9 2.91 14 .10 120 - 0 . 6 5 0 .04 - 2 . 3 7 0 .15 0 .04 - 1 . 2 5 -0 .11 2.46 - 0 . 1 6 - 2 . 6 3 . 0 . 0 0 0 .03 - 0 . 6 9 0 .86 0 . 6 5 1.72 5 .02 14.78 121 - 3 . 1 8 - 0 . 0 3 - 1 . 7 0 - 0 . 6 4 0 . 0 3 5.02 0 .25 3 .29 0 .03 - 1 . 7 9 - 3 . 9 7 0 . 0 0 - 0 . 1 8 - 0 . 6 6 0 . 8 9 - 1 . 9 1 0 . 4 3 8 .58 6 . 3 5 002 . . . . . . . . . . 2 .23 0 .72 - 0 . 9 8 - 1 . 1 6 - 0 . 6 5 - 0 . 7 4 1.25 3.21 17.97 130 0 .02 0 . 3 5 0 .09 0 .40 - 0 . 0 2 - 0 . 1 6 0 .19 - 0 . 2 4 0 .35 0 .03 - 0 . 5 6 - 0 . 3 8 - 0 . 1 6 0.01 0 . 0 6 - 0 . 1 8 1.03 6 2 . 8 3 131 0 .04 - 0 . 2 2 0 .47 - 0 . 0 9 - 0 . 1 4 0 .03 0.31 0 .17 0.01 -0 .11 0 .02 0 . 0 0 - 0 . 8 9 - 1 . 0 5 0 .06 0.11 - 0 . 0 0 1.53 8 0 . 3 2 112 0.01 0 . 1 3 0 .34 - 0 . 0 7 - 0 . 1 0 0 .03 - 0 . 3 0 0 .03 - 0 . 1 5 0 .12 0 .12 - 0 . 6 7 - 0 . 1 6 -0 .81 0 . 0 5 -0 .11 0 . 1 0 1.20 100.OO 200 0 . 9 5 10.12 0 .32 . - 1 . 4 6 1.45 - 4 . 7 4 - 6 . 4 3 - 4 . 5 8 4 .08 4 . 4 3 8 .26 17.23 0 . 5 0 041 . 0 . 7 3 . - 0 . 7 5 - 2 . 3 8 0 . 9 0 - 0 . 7 5 - 2 . 3 2 0 . 0 0 - 1 . 1 8 - 1 . 5 2 - 0 . 8 6 1.28 0.41 4 . 4 5 12.98 210 - 2 . 5 3 -0 .01 - 1 . 2 8 0 .19 0 .26 0 .16 0 . 4 0 0 .32 1.04 0.31 0 . 0 0 - 0 . 2 4 - 1 . 1 0 - 0 . 1 4 0 .17 - 0 . 7 9 3 . 4 0 10.19 122 0 . 2 9 - 0 . 0 2 1 . 0 5 - 0 . 0 7 - 0 . 0 2 0.51 0 . 0 5 - 1 . 0 0 - 0 . 0 6 - 0 . 9 9 - 1 . 5 6 0 . 0 0 0 .03 0 . 7 1 - 0 . 8 3 - 0 . 6 3 - 1 . 5 2 3 .13 4 2 . 5 5 140 - 0 . 4 6 0 . 0 3 0 .67 0.21 -0 .01 - 1 . 4 5 0 .06 - 0 . 5 4 - 0 . 0 5 2.21 0 . 0 0 - 0 . 0 5 0 .46 - 0 . 5 7 - 0 . 8 4 - 1 . 3 0 3.31 29 .48 211 0 . 4 0 - 0 . 0 3 - 1 . 7 2 - 0 . 1 2 0 .35 - 0 . 0 9 -1 .01 0 . 1 0 0 .12 - 0 . 0 9 0 .49 O.OO 0 .04 - 1 . 7 0 - 0 . 2 2 - 0 . 5 0 - 0 . 1 0 2 .79 19.89 132 - 0 . 0 6 - 1 . 1 7 - 0 . 2 9 -1 .31 0 .06 0 .47 - 0 . 5 7 0.71 0 .96 0 .08 - 0 . 7 5 - 3 . 4 3 2 .35 0 .99 - 0 . 0 4 -0 .31 -0 .91 4 . 9 9 4 . 5 3 230 2 .96 0 . 0 6 -2 .91 - 0 . 1 8 0 .57 - 0 . 1 2 1.25 0 .68 0 . 1 0 - 2 . 1 7 0 . 0 0 0 . 4 0 -3 .51 - 0 . 4 4 0 . 7 5 -0 .11 6 .13 3 .46 042 0 .97 1.19 3.91 . - 5 . 0 4 7.42 2 .12 - 3 . 1 7 3.41 - 3 . 6 0 - 1 . 6 9 0 . 6 9 2.81 12.18 2 .74 150 - 0 . 0 5 - 0 . 7 4 0 .72 0.21 - 0 . 1 9 - 0 . 0 9 - 0 . 0 5 - 0 . 6 9 - 0 . 2 0 -1 .11 - 2 . 0 5 2 .15 - 2 . 8 7 0 . 1 5 - 0 . 0 3 0 . 2 3 4 .48 6 .77 N3 o 241 T a b l e 22. ( c o n t i n u e d ) hk l xM2 yM2 x S i y S i x01 y01 x02 y02 x03 y03 z03 bM1 bM2 bS 1 b01 b02 b03 e r r . % INT. 113 0 .12 - 0 . 0 7 - 0 . 7 7 - 0 . 3 6 0 . 2 0 0 .12 -0 .31 - 0 . 2 4 0 .47 0 .14 1.69 0 . 0 0 - 5 . 5 2 3 .34 - 0 . 1 7 -0 .21 - 0 . 5 7 6 .79 3 .39 151 0 . 0 7 - 1 . 5 5 - 0 . 2 5 2 .02 0 .09 - 0 . 6 6 - 0 . 9 0 0 .09 0 .17 - 0 . 1 8 0 . 7 0 0 . 0 0 - 3 . 5 8 1.09 - 0 . 0 8 - 0 . 6 2 -0 .21 4 .78 3 .58 222 0 . 1 0 - 0 . 0 6 - 0 . 4 0 0.11 -0 .01 - 0 . 5 4 0.11 0 .35 - 0 . 0 6 0 .85 - 0 . 3 6 - 0 . 9 5 -1 .21 0 . 3 5 - 0 . 2 9 - 0 . 6 0 - 0 . 5 9 2 .18 66 .18 240 0 . 1 0 - 0 . 3 0 - 0 . 9 7 - 0 . 2 2 - 0 . 0 3 1.13 - 0 . 0 4 -1 .41 - 0 . 1 0 -2 .21 - 1 . 2 4 - 1 . 3 0 0 . 8 5 -0 .61 0 .24 - 1 . 0 6 3 .84 19.41 123 - 0 . 5 9 -0 .01 - 0 . 8 4 -0 .31 0.01 2 .06 0 . 1 0 1.35 - 0 . 0 3 2 .10 4 . 1 0 0 . 0 0 - 0 . 2 7 - 0 . 9 7 1.10 - 2 . 3 7 - 1 . 4 9 6.21 4 .62 241 0 . 1 3 0 . 5 6 1.49 - 0 . 3 5 0 .04 1.59 0 .26 - 0 . 5 6 - 0 . 0 5 0 .48 1.49 0 . 0 0 - 1 . 7 8 -1 .41 1.12 - 1 . 5 8 -0 .51 4 . 1 5 14.10 061 0 . 3 9 0 .79 2 .32 0 .95 -0 .91 0 .18 0 . 0 0 - 1 . 7 5 - 0 . 9 5 - 0 . 3 4 - 1 . 1 3 - 0 . 0 6 3 .64 13 .60 232 3 . 1 0 0 . 0 6 - 3 . 0 9 - 0 . 1 9 0 .55 -0 .11 1.21 0 .66 - 0 . 0 9 1.94 0 .07 0 . 0 0 0 .59 - 5 . 2 7 - 0 . 6 0 1.03 0 .14 7 .38 3 .42 133 0 .04 - 0 . 2 2 0 .47 - 0 . 0 9 -0 .11 0 .02 0 .26 0 .14 -0 .01 0 .26 - 0 . 0 4 0 . 0 0 - 2 . 0 9 - 2 . 4 9 0 .13 0 .22 0 .02 3 .33 19.62 152 0 . 0 7 1.10 - 1 . 0 8 -0 .31 0 .26 0 . 1 3 0 .06 0 .95 - 0 . 2 6 - 1 . 4 0 0 . 2 0 -4 .21 - 4 . 3 6 6 .15 - 0 . 2 9 0 . 0 6 0 . 4 0 8 .96 3 . 4 0 043 0 . 4 3 - 0 . 4 5 . - 1 . 1 9 0 .44 1.06 2 .87 0 . 0 0 - 1 . 5 4 -2 .01 - 0 . 9 5 1.41 - 1 . 2 7 4 .72 9 .79 310 - 0 . 1 0 - 0 . 2 3 - 4 . 2 5 0 .03 - 0 . 4 5 0 .12 - 1 . 2 2 0 .12 0 .68 - 0 . 5 8 . - 3 . 0 5 0.71 0 .76 0 .54 - 1 . 2 0 - 1 . 1 8 5 .84 3 .26 242 -0 .21 0 . 6 3 2 .10 0 .48 0 .05 - 2 . 1 9 0 .08 2 .80 - 0 . 1 8 - 3 . 9 5 - 1 . 1 5 - 3 . 4 4 3.68 - 2 . 4 2 1.61 - 0 . 6 2 -2 .51 8 .67 5 .16 004 . - 1 . 6 0 - 0 . 7 3 - 1 . 0 0 - 1 . 1 9 - 0 . 5 1 - 0 . 5 8 - 0 . 7 2 2 .57 2 9 . 2 0 062 -0 .31 - 0 . 1 6 . - 0 . 3 6 -1 .01 . 0.21 - 0 . 5 0 - 0 . 8 7 -0 .61 - 1 . 3 0 - 0 . 5 8 - 0 . 2 8 - 1 . 1 4 2 .47 41 .39 330 0 .26 0 .54 0 .93 0 .13 0 .07 0 .55 - 0 . 5 9 0.81 - 1 . 0 9 - 0 . 1 0 . - 3 . 1 8 - 2 . 1 2 - 0 . 1 9 - 0 . 1 0 - 0 . 6 9 2 .28 4 . 9 0 3 .04 331 0 .38 - 0 . 2 6 3 . 9 0 - 0 . 0 2 0 .38 - 0 . 0 8 - 0 . 7 7 - 0 . 4 5 -0 .01 0 . 3 0 - 0 . 0 6 0 . 0 0 - 3 . 3 4 - 0 . 8 5 - 0 . 5 7 - 0 . 9 5 0 .03 5 .44 7 .57 170 0 . 0 3 0 . 2 6 0 . 1 9 - 0 . 6 3 - 0 . 0 5 0 .17 - 0 . 1 4 - 0 . 2 9 - 0 . 1 3 0 .54 - 1 . 6 8 - 2 . 1 4 - 1 . 3 6 0 .07 - 0 . 1 6 0 .26 3.21 11.01 312 0 .07 0 .17 3 .20 - 0 . 0 2 0.31 - 0 . 0 8 0 .84 - 0 . 0 8 0 .42 - 0 . 3 6 - 0 . 3 6 - 2 . 8 0 - 0 . 6 5 -0 .71 - 0 . 4 7 1.05 - 0 . 9 3 4 .74 7 .88 322 - 0 . 5 0 0 .03 -0 .41 0 .09 - 0 . 0 6 0 .23 0 .18 - 0 . 4 0 - 0 . 2 4 - 0 . 4 0 - 0 . 6 3 0 . 0 0 - 0 . 1 4 - 2 . 6 2 - 1 . 0 2 - 0 . 7 0 -1 .71 3 .55 20.81 340 1.10 - 0 . 0 6 - 0 . 3 6 - 0 . 3 8 - 0 . 0 6 - 0 . 8 9 0 .34 - 0 . 3 0 - 0 . 2 8 1.24 0 . 0 0 0.31 - 2 . 3 3 - 0 . 9 7 - 1 . 2 9 -2 .01 4 . 0 3 7 .57 APPENDIX G: THE PW1710 DIFFRACTOMETER Because the diffT a c t o m e t e r i n the g e o l o g i c a l department was not o p e r a t i o n a l f o r s e v e r a l months, about h a l f of the measurements were done with a P h i l l i p s PW1710 automatic diffT a c t o m e t e r i n the oceanography department. U n f o r t u n a t e l y the microprocessor was connected to a t e l e t y p e t e r m i n a l and so a l l data had to be typed i n t o a computer f o r the a n a l y s i s . The output of the PW1710 was formatted i n a way that a l l o w s an easy check f o r t y p i n g e r r o r s . The measurements were done f o r 2 seconds at an i n t e r v a l of 0.01 20. The counts of f i v e measurements, corre s p o n d i n g t o an i n t e r v a l of 0.05 20 were added together and p r i n t e d . Each l i n e c o n t a i n s f i v e accumulated counts and at the end the sum oa s a l l counts on that l i n e . Parameters and programs f o r the PW1710 Nr. of r e g i s t e r s = 3. lower l e v e l = 35%, upper l e v e l = 75 % spinner ON, p r i n t i n g recorder OFF Program 154 SCH a b repeat from here to NXS f o r samples a t o b. HED p r i n t heading. LDC 4 load constant 4 i n t o accumulator STR 1 s t o r e content of accumulator i n r e g i s t e r 1 . CLR 2 c l e a r r e g i s t e r 2 CLR 3 c l e a r r e g i s t e r 3 SAN 15.000 set angle 20 at 15.00 deg. GAN put c u r r e n t angle i n t o accumulator DEC 2 set number of decimals i n output to 2. 242 243 BFR 0 s t o r e accumulator i n output b u f f e r DEC 0 set number of decimals i n output to 0, REP 155 200 repeat program 155, 200 times. REP 155 200 repeat program 155, 200 times. REP 155 200 repeat program 155, REP 155 200 repeat program 155, REP 155 200 repeat program 155, REP 155 200 repeat program 155, NXS l o a d next sample. 200 times. 200 times. 200 times. 200 times. Program 155 RCL 1 t r a n s f e r r e g i s t e r 1 to accumulator CNE 156 i f va l u e i n accumulator < 0, c a l l program 1 56 REP 157 5 repeat program 157, 5 times BFR s t o r e r e g i s t e r 2 i n output b u f f e r . CLR 2 c l e a r r e g i s t e r 2 RCL 1 t r a n s f e r r e g i s t e r 1 to accumulator. ADC -1 add -1 to accumulator STR 1 s t o r e accumulator i n r e g i s t e r 1 Program 156 LDC 4 l o a d constant 4 i n t o accumulator STR 1 s t o r e accumulator i n r e g i s t e r 1 PTR 3 s t o r e r e g i s t e r 3 i n output b u f f e r and p r i n t b u f f e r CLR 3 c l e a r r e g i s t e r 3 GAN put c u r r e n t angle i n t o accumulator DEC 2 set number of decimals i n output to 2 BFR 0 s t o r e accumulator i n output b u f f e r . DEC 0 set number of decimals i n output to 0 Program 157 MES 2.00 RCT measure the c u r r e n t angle f o r 2.00 seconds, t r a n s f e r number of counts to accumulator 244 ADD 2 add r e g i s t e r 2 to accumulator STR 2 s t o r e accumulator in r e g i s t e r 2 RCT t r a n s f e r number of counts to accumulator ADD 3 add r e g i s t e r 3 to accumulator STR 3 s t o r e accumulator i n r e g i s t e r 3 IAN 0.01 increment the a n g l e w i t h 0.01 The output on the t e l e t y p e has the f o l l o w i n g format: C=1 54 CAPI 24 09-12-1986 10.08 PROGRAM 154 15.00 45 41 44 31 33 194 15.25 31 43 46 37 43 200 15.50 29 33 42 46 44 194 e t c . APPENDIX H: SOME EXPERIMENTAL EQUILIBRIUM BETWEEN OLIVINE AND GARNET IN THE SYSTEM FE-MN-MG-AL-SI-Q H.1 THE SYNTHESIS OF THE GARNETS The s t a r t i n g m e t e r i a l s f o r the garnets were the same as f o r the o l i v i n e s (see chapter 2.4) p l u s : A 1 2 0 3 : From A 1 C 1 3 « 6 H 2 0 Baker Analyzed Reagent. LOT 429332. F i r s t the aluminium c l o r i d e i s heated over a Bunsen burner (ca . 300 to 400 °C) under constant shaking i n a beaker g l a s s , u n t i l a l l C l 2 and most H 20 i s removed. The substance i s then t r a n s f e r e d i n t o a Pt c r u c i b l e and heated 1/2 hour at 800 °C and 12 hours at 700 °C. The r e s u l t i n g m a t e r i a l i s 7~A1 20 3, and compares w e l l with the d e s c r i p t i o n of Rooksby (1951). The garnet mixture was f i r s t t r e a t e d the same way as the o l i v i n e s , mixed with g r a p h i t e and heated at 850 °C f o r 3 hours i n vacuum. The garnets used i n the exchange experiments were a l l s y n t h e s i z e d dry i n a s a l t - c e l l press at 1 5 - 2 0 KBars and 800 °C. (Table 23.) The Mn-rich products are o p t i c a l l y not completely i s o t r o p i c which means that t h e i r s t r u c t u r e i s not s t r i c t l y c u b i c . T h i s can however not be d e t e c t e d i n the X-ray d i f f r a c t i o n p r o f i l e s . H.2 THE EXCHANGE EXPERIMENTS The exchange experiments were made hydrothermally i n c o l d s e a l bombs at 2 KBars methane pressure and 800 °C, with H 20 p r e s e n t . The s t a r t i n g mixtures c o n s i s t of approximately 245 246 70-90 % garnet and 10-30 % o l i v i n e . The exact amount of each phase cannot be determined because both have an unknown excess of g r a p h i t e . The products were examined f i r s t by X-ray d i f f r a c t i o n , i n order to determine wether a r e a c t i o n took p l a c e , by comparing the d i s t a n c e i n 20 between the peaks (112) of o l i v i n e and (420) of garnet. Because of the r e l a t i v e l y small amount of o l i v i n e i n some samples, t h i s c o u l d not always be r e c o g n i z e d . The minimum time f o r an exchange r e a c t i o n was found to be around 30 days. T h i s i s a l s o around the maximum time, as one experiment l e f t f o r 62 days showed a completely d i f f e r e n t and (and a l s o unstable) assemblage (garnet, q u a r t z , oxides e t c . ) . T h i s a l s o p o i n t s to an a d d i t i o n a l fundamental problem: even a f t e r 30 days an un n o t i c e a b l e amount of an undesired phase may be present and may i n f l u e n c e the r e s u l t s . I t i s suggested that l a t e r work be c o n c e n t r a t e d on hig h e r p r e s s u r e s (and maybe higher temperatures) to a v o i d unstable assemblages. H.3 THE QUANITATIVE ANALYSIS OF THE RUN PRODUCTS. The sample p r e p a r a t i o n f o r the microprobe a n a l y s i s was suggested by Jack M. Ric e (pers. comm.) and a l s o used by Wang (1986). F i r s t the sample i s p l a c e d i n a g l a s s v i a l with a l c o h o l and put i n the u l t r a s o n i c c l e a n e r f o r 10 minutes. T h i s i n s u r e s that the i n d i v i d u a l g r a i n s , as long as they are not intergrown, become separated. A drop with suspended m a t e r i a l i s then taken and p l a c e d on a p o l i s h e d g r a p h i t e or diamond 247 s u r f a c e . On a good probe mount, the g r a i n s (ca. 2 - 5 p diam.) are b a r e l y v i s i b l e by eye and w e l l s e p a r a t e d . The advantage of using a diamond-mount i s that i f the sample looks not s a t i s f a c t o r y under the b i n o c u l a r microscope, i t can be wiped o f f e a s i l y and a new drop with suspended m a t e r i a l can be used. The samples were coated with carbon. For the q u a n t i t a t i v e microprobe a n a l y s i s on a CAMECA SX50 microprobe, s y n t h e t i c and n a t u r a l p o l i s h e d o l i v i n e and garnet standards were used. The composition was c a l c u l a t e d without any c o r r e c t i o n s from ( I s a m p l e / l s t a n d a r d ) • S t a n d a r d and normalized to 100% (Fe+Mn+Mg). T h i s method was chosen because the mass a b s o r p t i o n c o r r e c t i o n produced by the CAMECA software c o u l d not c o r r e c t l y d e al with a n a l y s e s having t o t a l s of only 30 to 70 %. S e v e r a l s y n t h e t i c o l i v i n e s and garnets with known composition were measured. Because the s u r f a c e c o n s i s t s of v a r y i n g g r a i n shapes, the measured c o n c e n t r a t i o n s show a v a r i a t i o n of 2-10 %, averaging around the c o r r e c t composition. T h i s a l l o w s an i n t e r p r e t a t i o n with an e r r o r of approximately 2 mol% f o r the compositions of i n t e r e s t . A summary of the a n a l y t i c a l r e s u l t s i s given i n Table 25. Only those a n a l y s i s which had S i and A l w i t h i n 10% of the expected v a l u e s f o r o l i v i n e s and garnets were used. The r e l a t i v e l y l a r g e a n a l y t i c a l e r r o r l i m i t s any i n t e r p r e t a t i o n . Further improvement on the sample p r e p a r a t i o n may reduce t h i s u n c e r t a i n t y . The r e s u l t s are presented i n Table 24 and F i g . 23. The dashed d i s t r i b u t i o n curves i n F i g . 23 i l l u s t r a t e approximately the e q u i l i b r i u m 248 f r a c t i o n a t i o n s . The exper iments are c o n s i s t e n t but not s u f f i c i e n t l y c o n s t r a i n i n g to j u s t i f y making a d e t a i l e d thermodynamic model . T a b l e 23. L i s t o f s y n t h e s i z e d g a r n e t s . Run x ( F e ) x(Mn) x(Mg) T PC] p [KBars ] t i me a o comments RCC2 0 . 0 0 0 1 .000 0 .000 700 3 13d 1 1 .624 a n i s o t r o p i c , homogeneous, y e l l o w RCC19 0 . 0 0 0 1 .000 0 • OOO 550 4 20d 1 1 .637 a n i s o t r o p i c , wh i t e RCC13 O.OOO 0.500 0 .500 600 2 22d --- q u a r t z , p y r o x e n e s , u n u s a b l e RCC23 0 .083 0.917 0 OOO 750 2 16d 1 1 . 542 q u a r t z , p y r o x e n e s , u n u s a b l e RCC24 0 .083 0.917 0 .000 780 2 10d q u a r t z , p y r o x e n e s , u n u s a b l e PC4 0 .083 0.917 0 .000 800 15 24h 1 1 .621 a n i s o t r o p i c PC5 0 .498 0.502 0. .000 800 15 24h 1 1 .577 i s o t r o p i c PC6 0 .498 0.502 0. .000 800 15 24h 1 1 .569 1 s o t r o p i c PC7 0 .746 0 .000 0. . 254 800 15 23h 1 1 .51 1 i s o t r o p i c PCS 0 . 249 0. 751 0 000 800 25 25h 1 1 .591 a n i s o t r o p i c PC9 0 .497 0.003 0. .500 800 15 27h 1 1 .487 i s o t r o p i c , few i m p u r i t i e s . ( p y r o x e n e ? ) PC 12 0 .248 0. 752 0. 000 800 20 46h 1 1 .597 an i s o t r o p i c , homogeneous PC 13 0 . 166 0.668 0. . 166 800 20 47h 1 1 .570 an 1 s o t r o p i c , homogeneous PC 14 0 . 0 0 0 0.667 0. ,333 800 20 49h 1 1 .573 an i s o t r o p i c , homogeneous PC15 O.OOO 0 .500 0. 500 800 20 46h 1 1 .540 an 1 s o t r o p i c , homogeneous PC16 0 . 166 0.668 0. 166 800 20 42h 1 1 .577 an i s o t r o p i c . homogeneous PC18 0 .663 0.171 0. 166 800 20 71h 1 1 .529 a n i s o t r o p i c , homogeneous 250 T a b l e 2 4 . : L i s t of u s a b l e exchange e x p e r i m e n t s Run x (Fe ) - g a r n e t x(Mn) x(Mg) x ( F e ) o l i v i ne x(Mn) x(Mg) RCC29/1 8 0 1 ' C , 2 K b , 3 1 d s t a r t i n g c o m p o s i t i o n f i n a l c o m p o s i t i o n 0 .4970 0 .5030 0 . 5 2 9 ± 0 . 0 2 0 .47 1 ± 0 . 0 2 0 .9940 0 .698+0.02 0 .0060 0 .302+0.02 RCC29/2 801' C , 2 K b , 3 1 d s t a r t i n g c o m p o s i t i o n f i n a l c o m p o s i t i o n 0 .4970 0 .5030 0 . 4 3 7 ± 0 . 0 2 0 .56310 .02 0 .0828 0 . 3 7 2 ± 0 . 0 2 0 .9172 O.628+0.02 RCC31/1 8 0 5 ' C , 2 K b . 4 3 d s t a r t i n g c o m p o s i t i o n f i n a l c o m p o s i t i o n 0 .5000 0 .5000 0 . 5 5 3 ± 0 . 0 2 0 .44710 .02 0 .5000 0 .266+0.02 0 .5000 0 . 7 3 4 1 0 . 0 2 RCC31/2 805' C , 2 K b , 4 3 d s t a r t i n g c o m p o s i t i o n f i n a l c o m p o s i t i o n 0 .5000 0 .5000 0.557+0.04 0 .44310 .04 0 .6667 0 .315+0.02 0 .3333 0 . 6 8 5 1 0 . 0 2 RCC31 /3 8 0 5 ' C , 2 K b ,43d s t a r t i n g c o m p o s i t i o n f i n a l c o m p o s i t i o n 0 .4970 0 .5030 0 .396+0.05 0 .60410 .05 0 .4142 0 . 5 9 8 1 0 . 0 2 0 .5858 0 .402+0.02 ho o 251 T a b l e 24. ( c o n t i n u e d ) Run • g a r n e t x ( F e ) x(Mn) x(Mg) x ( F e ) x(Mn) x(Mg) RCC32/1 s t a r t i n g c o m p o s i t i o n 0. 2485 0. .7515 0, .0000 1 . .OOOO 7 9 7 ' C , 2 K b , 3 7 d f i n a l c o m p o s i t i o n 0. 1 9 1 ± 0 . .02 0. . 8 0 9 ± 0 . 0 2 0. .074+0.02 0, . 9 2 6 ± 0 . 0 2 RCC32/2 s t a r t i n g c o m p o s i t i o n 0. 2485 0. .7515 0. ,2485 0. .7515 7 9 7 ' C , 2 K b , 3 7 d f i n a l c o m p o s i t i o n 0. 160+0. .03 0. .84010.03 0. . 4 0 5 ± 0 . 0 2 0. . 5 9 5 ± 0 . 0 2 RCC32/3 s t a r t i n g c o m p o s i t i o n 0 . 2485 0. 7515 0 . 3313 0. 6687 7 9 7 ' C , 2 K b , 3 7 d f i n a l c o m p o s i t i o n 0 . 176+0. 04 0 . 8 2 4 ± 0 . 0 4 0 . 4 1 7 ± 0 . 0 2 0. 5 8 3 ± 0 . 0 2 F i g , 23: Roozeboom diagrams f o r some exchange experiments i n the b i n a r y Fe-Mn and Mn-Mg o l i v i n e - garnet systems 1.0 0.8 OJ CI —I > —I '—I o u u. X OJ cz o c X 0.6 0.4 0.2 0.0 1.0 0.8 0.6 0.4 0.2 r z : R31/3 R32/3 R32/2 3. R29/2 R32/1 I I ' I 1 I ' I I I I I • I 1 I I I ' 0.0 0.2 0.4 0.6 0.8 1.0 X F c garnet R31/2 R31/1 o.o r i i i i i i • i i i • i i i i i i i i 0.0 0.2 0,4 0.6 0.8 1.0 X n n garnet 253 Table 25.: Summary of microprobe a n a l y s e s (Fe+Mn+Mg = 100%) measured expected o l i v i n e Fe 33.2(1 .7) 33. 1 040800 Mn 66.8(1.7) 66.9 n=1 1 Mg 0.0(0.0) 0.0 S i 49.8(2.8) 50.0 A l 0.0(0.0) 0.0 garnet Fe 50.2(1.2) 50.0 PC05 Mn 49.8(1.2) 50.0 n= 1 5 Mg 0.0(0.0) 0.0 S i 99.2(9.1) 100.0 A l 65.9(4.8) 66.7 garnet Fe 50.5(3.4) 50.0 PC06 Mn 49.5(3.4) 50.0 n=1 4 Mg 0.0(0.0) 0.0 S i 99.9(7.0) 100.0 A l 68.1(4.6) 66.7 o l i v i n e Fe 41.2(2.3) 41 .4 050700 Mn 58.8(2.3) 58.6 n=13 Mg 0.0(0.0) 0.0 S i 50.7(3.8) 50.0 A l 00.0(0.0) 00.0 254 Table 25. (continued) measured expected o l i v i n e Fe 07.7(0.5) 08.3 011100 Mn 92.3(0.5) 91 .7 n=7 Mg 0.0(0.0) 0.0 S i 49.4(3.2) 50 .0 A l 00.0(0.0) 00.0 garnet Fe 24.9(1.2) 24 .8 PC 1 2 Mn 75.1(1 .2) 75.2 n=1 3 Mg 0.0(0.0) 0.0 S i 98.0(4.2) 100.0 A l 66 . 8(3.6) 66.7 garnet Fe 39.6(4.0) from R31/3 Mn 60.4(4.0) n=1 5 Mg 0.0(0.0) S i 102.4(6.4) A l 67.0(6.4) o l i v i n e Fe 59.8(1.3) from R31/3 Mn 40.2(1 .3) n=5 Mg 0.0(0.0) S i 49 . 8(3.4) A l 00.0(0.0) 255 Table 25. (continued) measured expected garnet Fe 52.9(1.7) from R29/1 Mn 47.1(1.7) n=1 1 Mg 0.0(0.0) S i 99.3(4.2) A l 67.3(5.9) o l i v i n e Fe 69.8(1.4) from R29/1 Mn 40.2(1.4) n=4 Mg 0.0(0.0) S i 46.9(5.6) A l 00.0(0.0) o l i v i n e Fe 00.0(0.0) 00.0 000606 Mn 48.6(4.4) 50.0 n=1 9 Mg 51.4(4.4) 50.0 S i 49.8(4.0) 50.0 A l 00.0(0.0) 00.0 o l i v i n e Fe 00.0(0.0) 00.0 000804 Mn 65.8(4.5) 66.7 n=1 0 Mg 34.2(4.5) 33.3 S i 47.6(3.5) 50.0 A l 00.0(0.0) 00.0 256 Table 25. (continued) measured expected garnet Fe 47.9(4.9) 50.0 PC 15 Mn 52. 1(4.9) 50.0 n=1 1 Mg 0.0(0.0) 00.0 S i 101.3(5.2) 100.0 A l 66.6(4.2) 66.7 o l i v i n e Fe 99.4(0.2) 120000 Mn 00.6(0.2) n=6 Mg 0.0(0.0) S i 48.0(5.5) A l 00.0(0.0) garnet Fe 00.0(0.0) from R31/1 Mn 55.3(1.5) n=1 2 Mg 44.7(1.5) S i 100.3(5.5) A l 66.6(3.3) o l i v i n e Fe 00.0(0.0) from R31/1 Mn 26.6(1.3) n=5 Mg 73.4(1.3) S i 51 .8(2.8) A l 00.0(0.0) 257 Table 25. (continued) measured expected garnet Fe 00.0(0.0) from R31/2 Mn 55.7(3.6) n=8 Mg 44.3(3.6) S i 101.9(9.6) A l 68.0(6.5) o l i v i n e Fe 00.0(0.0) from R31/2 Mn 31.5(1.8) n=8 Mg 68.5(1.8) S i 48.6(4.2) A l 00.0(0.0) o l i v i n e Fe 65.8(1.6) 66.3 080400 Mn 34.2(1.6) 33.7 n=1 5 Mg 00.0(0.0) 00.0 S i 50.2(3.6) 50.0 A l 00.0(0.0) 00.0 garnet Fe 43.7(1.7) from R29/2 Mn 56.3(1.7) n=1 3 Mg 00.0(0.0) S i 100.1(6.8) A l 67.3(3.3) 258 Table 25. (continued) measured expected o l i v i n e Fe 37.2(1.4) from R29/2 Mn 62.8(1.4) n=4 Mg 00.0(0.0) S i 49.8(4.7) A l 00.0(0.0) garnet Fe 19.1(1.7) from R32/1 Mn 80.9(1.7) n=9 Mg 00.0(0.0) S i 101.6(6.6) A l 66.4(3.5) o l i v i n e Fe 7.4(1.6) from R32/1 Mn 92.6(1.6) n=6 Mg 00.0(0.0) S i 51.5(5.8) A l 00.0(0.0) garnet Fe 16.0(2.5) from R32/2 Mn 88.0(2.5) n=12 Mg 00.0(0.0) S i 100.9(5.3) A l 69.5(4.3) 259 Table 25. (continued) measured expected o l i v i n e Fe 40.5(1.3) from R32/2 Mn 59.5(1.3) n=5 Mg 00.0(0.0) S i 47.9(1.2) A l 00.0(0.0) garnet Fe 17.6(3.5) from R32/3 Mn 82.4(3.5) n=1 1 Mg 00.0(0.0) S i 98.1(5.5) A l 66.4(5.3) o l i v i n e Fe 41.7(1.3) from R32/3 Mn 58.3(1.3) n = 5 Mg 00.0(0.0) S i 52.3(3.3) A l 00.0(0.0) PUBLICATIONS De C a p i t a n i C . 0 9 7 7 ) ; Der S o l v u s im System C a C 0 3 - M n C 0 3 . V o r t r a g an der 52. Hauptversammlung der Schwe iz . m i n e r a l , p e t r o g r . G e s e l l s c h a f t i n B e r n , 8. Oktober 1977. Schweiz. mineral, petrogr. Mitt. 57: 463 De C a p i t a n i C . ( 1 9 8 3 ) : P e t r o g r a p h i s c h e Untersuchungen i n der Gegend F u r t s c h e l l a s - G r i a l e t s c h (Oberengadin) unter besonderer B e r i i c k s i c h t i g u n g der Manganerz-Vorkommen. Diplomarbeit Geol. Inst. Univ. Bern De C a p i t a n i C . and Brown T . H . Q 987): The Computat ion of C h e m i c a l E q u i l i b r i u m i n Complex Systems c o n t a i n i n g n o n - i d e a l S o l u t i o n s . Geochim. Cosmochi m. Acta (in press) De C a p i t a n i C . and Greenwood H . J . ( 1 9 8 5 ) : S i t e o c c u p a n c i e s i n Mn-Mg and Fe-Mg o l i v i n e s by XRD i n t e n s i t i e s ( a b s t r . ) . EOS 66/46:1134 De C a p i t a n i C . and P e t e r s T J . ( 1 9 8 1 ) : The S o l v u s i n the Sys ten C a C 0 3 - M n C 0 3 . Contrib. Mineral. Petrol. 76: 394-400 De C a p i t a n i C . and P e t e r s T j . 0 982): C o r r e s p o n d i n g S t a t e s in B i n a r y S o l u t i o n s , and G r a p h i c a l D e t e r m i n a t i o n of Margu le s P a r a m e t e r s . Contrib. Mineral. Petrol. 81:48-58 F i n g e r W . , M e r c o l l i I . , Kvindig R. , S t a u b l i A . , -> De C a p i t a n i C . , N i e v e r g e l t P . , P e t e r s T j . and Trommsdorff V . Q 982): B e r i c h t iiber d i e gemeinsame E x k u r s i o n der SGG und SMPG i n s Oberengadin vom 21. b i s 24. September 1981. Eclogae geol. Helv. 75/1: 199-222 1

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