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Crystal structure of spessartine and andradite at elevated temperatures Rakai, Robert Joseph 1975

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CRYSTAL STRUCTURE OF SPESSARTINE AND ANDRADITE AT ELEVATED TEMPERATURES by Robert Joseph Rakai B. Sc. (Honors), University of Manitoba, 1972 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the Department of Geological Sciences We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1975 In presenting th is thes is in pa r t i a l fu l f i lment o f the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f ree ly ava i l ab le for reference and study. I fur ther agree that permission for extensive copying of th is thes is for scho la r ly purposes may be granted by the Head of my Department or by his representat ives . It is understood that copying or pub l i ca t ion of th is thes is fo r f inanc ia l gain sha l l not be allowed without my wr i t ten permission. Depa rtment The Univers i ty of B r i t i s h Columbia Vancouver 8, Canada i i ABSTRACT Single crystal x-ray intensity data have been collected with a manual diffractometer u t i l i z i n g flat-cone geometry for a spessartine from Minas Gervais, Brazil at 25°, 350°, 575° and 850°C and for an andradite from Valmalen, Italy at 25°, 350°, 575° and 850°C. Weighted anisotropic least-squares refinements in space group Ia3d resulted in residual (R) factors ranging from 0.019 to 0.029. The refinements indicate that a l l of the positional parameters of the oxygen atom in andradite vary during thermal expansion. The x and y positional parameters of oxygen in spessartine likewise vary during thermal expansion while the z positional parameter remains essentially constant. Linear thermal expansion coefficients of the c e l l edges are 0.632(53) x 10~5 ° C _ 1 and 0.764(23) x 10~5 V 1 for spessartine and andradite, respectively. A high linear dependence is shown to exist for the c e l l edges of members of the s i l i c a t e garnet solid solution as a function of temperature and composition collectively. Si-0 interatomic distances in both spessartine and andradite (uncorrected for thermal displacement) show a zero or slightly negative expansion while the Mn-0 and Al-0 interatomic distances of spessartine and the Ca-0 and Fe-0 interatomic distances of andradite show significant positive expansions as a function of increasing temperature. The rates of expansion of the Al-0, Mn(l)-0(4), i i i and Mn(2)-0(4) interatomic distances in spessartine are 1.61(30) x 10 ** A °C _ 1, 2.12(28) x IO" 5 A °C~ 1, and 3.28(31) x 10~5 A °C""1, respectively. The rates of expansion of the Fe-0, Ca(l)-0(4) and Ca(2)-0(4) inter-atomic distances in andradite are 2.94(44) x 10 ^ X °C \ 2.31(20) x 10~5 A °C _ 1, and 3.28(13) x 10~5 A °C~ 1, respectively. Relations between these values and the values reported for pyrope and grossularite (Meagher, 1975) as well as the concomitant polyhedral adjustments are considered in detail. In both spessartine and andradite, the shared octahedral edge increases at a greater rate than does the unshared octahedral edge and the X(2)-0(4) distance increases at a greater rate than does the X(l)-0(4) interatomic distance during thermal expansion. The spessartine garnet structure f a c i l i t a t e s these variations mainly by rotation of the essentially ri g i d SiO^ tetrahedron while the andradite garnet structure f a c i l i t a t e s the same variations mainly by a distortion of the SiO. tetrahedron. The SiO. tetrahedron of andradite 4 4 becomes more ideal as a function of increasing temperature. TABLE OF CONTENTS iv Page ABSTRACT i i TABLE OF CONTENTS iv LIST OF TABLES v i LIST OF FIGURES v i i i ACKNOWLEDGEMENTS x i CHAPTER I. INTRODUCTION 1 General 1 Definition of Garnet Structure 2 Previous Silicate Garnet Refinements at Room Temperature 6 Previous Silicate Garnet Refinements at Elevated Temperatures 8 Purpose of Study 13 II . EXPERIMENTAL DETAILS 15 Elevated Temperature Method 15 Specimen Descriptions, Chemistry, Space Group, and Cell Dimensions 17 Intensity Collection and Reduction 21 I I I . REFINEMENTS 23 Continued.... Page IV. RESULTS AND DISCUSSION 37 Thermal Expansion of Cell Parameters 38 Oxygen Fractional Coordinates 44 Polyhedra Variations with Temperature . . . . 47 Silicate Tetrahedron 47 The YOg Octahedron 53 The XOg Triangular Dodecahedron 66 Isotropic Temperature Factors and Thermal Ellipsoids 73 Structural Adjustment during Thermal Expansion in Silicate Garnets . . . . 80 V. SUMMARY . 86 SELECTED REFERENCES 88 APPENDIX I: Observed and Calculated Structure Factors 91 v i LIST OF TABLES Table 1 Description of the Structure Sites of Garnet (modified after Geller, 1967) 2 The Number and Type of Shared Polyhedral Edges in the Garnet Structure (from Novak and Gibbs, 1971) 3 Chemical Compositions, Cell Parameters a (&), Cell Volumes V (& 3), and Calculated Densities p (g/cc) of the Spessartine and Andradite 4 Refinement Parameters: Number of Observations (NO), Number of Symmetry Nonequivalent Reflections (NNR), Number of Variables (NV), and Conventional and Weighted R-Factors (R and R ) . . . w 5 Final Weighting Parameters (F* and G*) and Error mean square (Ems) of the Weighted Anisotropic Refinements 26 6 Oxygen Positional Parameters, Isotropic Temperature Factors (B), and the corresponding R.M.S. Amplitude of Vibration (u) of the Atoms of Spessartine and Andradite 28 7 Anisotropic Temperature Factor Coefficients, 3 / . . v (x IO-*), of Spessartine and Andradite . . 29 8 R.M.S. Magnitudes (u) and Orientations (9) of Thermal Vibration Ellipsoids of the Atoms of Spessartine with Respect to the Crystallo-graphic Axes 31 9 R.M.S. Magnitudes (u) and Orientations (6) of Thermal Vibration Ellipsoids of the Atoms of Andradite with Respect to the Crystallo-graphic Axes 32 10 Interatomic Distances (A*) and Angles (°) of the Si04 Tetrahedron, AlOg Octahedron, and MnOg Triangular Dodecahedron of Spessartine 33 Page 20 25 v i i LIST OF TABLES (Cont.) Table Page 11 Interatomic Distances (A) and Angles (°) of the Si04 Tetrahedron, FeOg Octahedron, and CaOg Triangular Dodecahedron of Andradite 34 12 Metal - Metal Interatomic Distances (A) for Spessartine and Andradite . 35 13 Metal - Oxygen - Metal Angles (°) for Spessartine and Andradite . . . . . 36 14 Predicted and Observed Cell Parameters for pure End-member Garnets at Various Temperatures . 43 15 Bond Angle Strain in Pyrope, Spessartine, Grossularite and Andradite . 50 16 Ionic Radii, Rates of increase of Mean M-0 distances (A ° C - 1 x 10 + 5) and Mean Thermal Expansion Coefficients (°C _ 1 x 10 + 5) of Pyrope, Spessartine, Grossularite and Andradite 59 17 Equivalent Isotropic Temperature Factors, B250, at Room Temperature and Linear Rates of change of Isotropic Temperature Factors with Temperature 74 18 Predicted Cell Edges (a) and Positional Parameters (x, y, and z) for Almandine and Manganese-Grossularite at Elevated Temperatures 85 19 Observed and Calculated Structure Factors of the Spessartine 92 20 Observed and Calculated Structure Factors of the Andradite 93 v i i i LIST OF FIGURES Figure Page 1 Atomic asymmetric unit of garnet c e l l (modified after Menzer, 1928) 3 Portion of the garnet structure projected down the z-axis. Large circles represent oxygens, solid circles the Si cations, hatched circles the Y cations and quadrant ruled ones the X cations (modified after Gibbs and Smith, 1965) Cation and oxygen atoms of a portion of the garnet structure defining the components of the polyhedrons in accordance with the nomenclature uti l i z e d in the text (after Novak and Gibbs, 1971) Tetrahedral position angle (y) versus the mean radius of the X-cation, <r{X}>, for the aluminum s i l i c a t e garnets: pyrope (Py), almandine (Al), spessartine (Sp), Mn-grossularite (Mn-Gr) and grossularite (Gr). Data from Novak and Gibbs (1971, as in Meagher, 1975) 11 Tetrahedral position angle (y) versus temperature for pyrope (Meagher, 1975) 11 A portion of the garnet structure showing the effect of tetrahedral rotation to smaller y angles. The {X}-0(4) distance (equivalent to X(2)-0(4) of Figure 3) increases at a greater rate than the {X}-0(2) distance (equivalent to X(l)-0(4) of Figure 3) for a specified tetrahedral rotation rate (modified after Meagher, 1975) 12 Spessartine c e l l parameter versus temperature. Error bars represent ± one estimated standard deviation 39 Andradite c e l l parameter versus temperature. Error bars represent ± one estimated standard deviation 40 Fractional coordinates (x, y, z) of oxygen in spessartine as a function of temperature. Error bars represent ± one estimated standard deviation . . . . . . . . . . . . 45 ix LIST OF FIGURES (Cont.) Figure Page 10 Fractional coordinates (x, y, z) of oxygen in andradite as a function of temperature. Error bars represent ± one estimated standard deviation . . . . . . . 46 11 Variation of the unshared tetrahedral edge, 0(1)-0(3), of andradite with increasing temperature. Error bars represent ± one estimated standard deviation . . 49 12 Variation of the tetrahedral position angle (y) with increasing temperature in pyrope (Meagher, 1975) and spessartine. Error bars represent ± one estimated standard deviation 52 13 General position of s i l i c a t e tetrahedron in the garnet structure. Rotation of tetrahedron about 4" axis to smaller y angles results in a decrease in the y parameter (Ay) which is larger than the variation in the z parameter (Az). The 4 axis passes through the s i l i c o n atom and is perpendicular to the page 54 14 Idealized variation in the y and z positional parameters of oxygen in the garnet structure restricted such than no simultaneous change occurs in the y angle of the s i l i c a t e tetra-hedron. A 4 axis passes through the s i l i c o n atom and i s perpendicular to the page 55 15 Variation of the Al-0 interatomic distance with increasing temperature in pyrope (Meagher, 1975), spessartine and grossularite (Meagher, 1975). Error bars represent ± one estimated standard deviation 57 16 Variation of the Y-0 interatomic distance with increasing temperature in grossularite (Meagher, 1975) and andradite. Error bars represent ± one estimated standard deviation 58 17 Rate of change of the Y-0 interatomic distance with increasing temperature, d(Y-0)/dT, versus the mean size of the non-tetrahedral cations, <r>, for pyrope (Meagher, 1975), spessartine, grossularite (Meagher, 1975) and andradite. Error bars represent ± one estimated standard deviation . . 61 X LIST OF FIGURES (Cont.) Figure Page 18 Variation of the shared and unshared octahedral edges in spessartine with temperature 63 19 Variation of the unshared octahedral edge, 0(l)-0(5), with temperature in grossularite (Meagher, 1975) and andradite. Error bars represent ± one estimated standard deviation 64 20 Variation of the shared octahedral edge, 0(l)-0(4), with temperature in grossularite (Meagher, 1975) and andradite. Error bars represent ± one estimated standard deviation 65 21 X(l)-0(4), X(2)-0(4) and <X-0> distances of spessartine versus temperature. Error bars represent ± one estimated standard deviation . . . . . 68 22 X(l)-0(4), X(2)-0(4) and <X-0> distances of andradite versus temperature. Error bars represent ± one estimated standard deviation 69 23 <X-0> versus temperature in pyrope, spessartine, grossularite and andradite 71 24 Equivalent isotropic temperature factors, By, of the Y-site cations in pyrope (Py), spessartine (Sp), grossularite (Gr) and andradite (An) versus temperature . 76 25 Equivalent isotropic temperature factors, B x, of the X-site cations in pyrope (Py), spessartine (Sp), grossularite (Gr), and andradite (An) versus temperature 77 26 Linear rates of change of mean metal-oxygen distances with temperature, d<M-0>/dT, versus the linear rates of change of the cation isotropic temperature factors with temperature, dB/dT 78 ACKNOWLEDGEMENTS The writer gratefully acknowledges Dr. E.P. Meagher for suggesting the research topic and for .his guidance, constructive criticism, and patience throughout the preparation of this thesis. The s i l i c a t e garnet specimens used in this study were obtained from Dr. G.V. Gibbs of the Department of Geological Sciences, Virginia Polytechnic Institute and State University, Blacksburg, Virginia. During the course of this study, the author was supported by a National Research Council Postgraduate Scholarship. 1 CRYSTAL STRUCTURE OF SPESSARTINE AND ANDRADITE AT ELEVATED TEMPERATURES I. INTRODUCTION General The general s i l i c a t e garnet structural formula may be conveniently written as {X^ } [Y,,](Si 3)0 1 2 (Geller, Miller and Treuting, 1960). The bracket type defines the oxygen coordination pblyhedra formed around each cation: { } refers to a triangular dodecahedron (distorted cube) with a coordination number of eight, [ ] refers to an octahedron with a coordination number of six, and ( ) refers to a tetrahedron with a coordination number of four (Geller, 1967). Both Geller (1967) and Rickwood (1968) have reviewed the assortment of cations which are known to occur in the {X} and [Y] sites of the chemically diverse nesosilicate garnets. In the common end-member I | j | | | | | s i l i c a t e garnets {X} refers to Ca , Mn , Fe , Mg and [Y] refers to Te***, Cr4"4"1", Al*** (Rickwood, 1968). On the basis of physical properties and an apparent natural miscibility gap the s i l i c a t e garnets were divided into two groups by I | Winchell (1933): (1) the ugrandites, {X} = Ca with compositions between the end-members grossularite {Ca^}[Al,,] (Si^O^» uvarovite {Ca 3>[Cr 2](Si 3)0 1 2, and andradite {Ca3>[Fe2 ] ( S i 3 ) 0 1 2 ; and (2) the pyralspites, {X} * Ca vith compositions between the end-members pyrope {Mg 3}[Al 2](Si 3)0 1 2, almandine {Fe 3>[Al 2](Si 3)0 1 2, and spessartine {Mn 3>[Al 2](Si 3)0 1 2. More recently, Novak and Gibbs (1971) have concluded the phase relation studies of the last twenty-five years demonstrate complete miscibility exists between a l l the geologically important garnets even i f only under special conditions. Definition of Garnet Structure The f i r s t garnet crystal structure was solved by Menzer (1926). By 1928, Menzer established that grossularite, andradite, almandine and spessartine were isostructural with space group symmetry Ia3d^" (Menzer, 1928). Figure 1 shows the atomic arrange-ment of an asymmetric unit of the garnet c e l l as determined by Menzer. The cations are in special positions with no positional degrees of freedom while the oxygen atoms are in general equipoint positions with x = 0.04, y * 0.05 and z = 0.65 (Menzer, 1928). An account of the garnet structure sites i s given in Table 1. There are eight formula units per garnet c e l l . Nomenclature of the International Tables for X-ray Crystallography (1952). Figure 1. Atomic asymmetric unit of garnet c e l l (modified after Menzer, 1928). 4 TABLE 1. Description of the Structure Sites of Garnet (modified after Geller, 1967) Structure formula {X3> [Y 2] (Si-j) 0 1 2 Point symmetry 222 3 4 1 Equipoint* 24c 16a 24d 96h Coordination to oxygen 8 6 4 Polyhedra triangular octahedron tetrahedron dodecahedron Nomenclature of the International Tables for X-ray Crystallography (1952). In terms of coordination polyhedra, the garnet structure consists of independent SiO^ tetrahedra and YOg octahedra which share corners to form a continuous three dimensional framework within which each {X} atom is surrounded by eight oxygen atoms (Figure 2). Each oxygen is coordinated by one (Si), one [Y] and two {X} cations. The number and type of shared polyhedral edges in the garnet structure are documented in Table 2. Gibbs and Smith (1965) have observed that the coordination polyhedra of the garnet structure-have a large percentage of shared edges which leads to a tightly packed arrangement. Figure 2. Portion of the garnet structure projected down the z-axis. Large circles represent oxygens, solid circles the Si cations, hatched circles the Y cations and quadrant ruled ones the X cations (modified after Gibbs and Smith, 1965). 6 In this text, oxygen-oxygen interatomic distances of poly-hedra and cation-anion interatomic distances are identified in accordance with the nomenclature of Novak and Gibbs (1971) as defined in Figure 3. In addition, the mean radius of the X-cation and the mean radius of the Y-cation are specified as <r{X}> and <r[Y]>, respectively. TABLE 2. The Number and Type of Shared Polyhedral Edges in the Garnet Structure (from Novak and Gibbs, 1971) Polyhedron Shared Edges Tetrahedron 2 Octahedron 6 Triangular dodecahedron 2 4 4 with triangular dodecahedra with triangular dodecahedra with tetrahedra with octahedra with other triangular dodecahedra Previous Silicate Garnet Refinements at Room Temperature Pyrope and grossularite represent end-members with respect to the size of the X-cation in the natural aluminum s i l i c a t e garnet series (Zemann, 1962). Grossularite was f i r s t refined by Abrahams and Geller (1958) and pyrope was f i r s t refined by Zemann and Zemann (1961). On the basis of the above two refinements Zemann (1962) considered the Figure 3. Cation and oxygen atoms of a portion of the garnet structure defining the components of the polyhedrons in accordance with the nomen-clature u t i l i z e d in the text (after Novak and Gibbs, 1971). 8 crystal chemistry of s i l i c a t e garnets. Euler and Bruce (1965) refined an iron bearing pyrope based on the Bragg reflections to which only the oxygen ions contribute while Gibbs and Smith (1965) determined the crystal structure of a pure end-member pyrope which was synthesized by Boyd (Boyd and England, 1959). In the following year, Prandl (1966) refined a grossularite by X-ray and neutron diffraction techniques while an andradite was refined by Quareni and de P i e r i (1966). Geller (1967) considered the results of the above crystal structure determinations in a lengthy review of the crystal chemistry of garnets. In 1971, the crystal structure of a pyrope, chromium-pyrope, almandine, spessartine, manganese-grossularite, grossularite, uvarovite, goldmanite, and andradite were reported by Novak and Gibbs (1971). In their a r t i c l e , Novak and Gibbs established that the crystal structure of any s i l i c a t e garnet can be closely predicted at room temperature from i t s chemical composition. Previous crystal structure refinements at room temperature, especially the comprehensive work of Novak and Gibbs, form a basis from which the structural details of the s i l i c a t e garnet solid solution can be extended to elevated temperatures. Previous Silicate Garnet Refinements at Elevated Temperatures The mechanics of atomic structural adjustment in a pyrope and a grossularite as a result of thermal expansion have been investigated by Meagher (1973). The pyrope structure was refined at 25°, 350°, 550° and 750°C and the grossularite structure was elucidated at 25°, 365° and 675°C. Meagher's (1973) results indicate the positional parameters of oxygen vary in pyrope but remain approximately constant in grossularite. The Si-0 interatomic distance of both garnets does not change significantly upon heating (Meagher, 1973). Ca-0 and Al-0 interatomic distances in grossularite and the Mg-0 and Al-0 interatomic distances in pyrope increase linearly with increasing temperature (Meagher, 1975). Meagher (1975) found that the essentially r i g i d SiO^ tetrahedron in pyrope rotates about i t s 4 axis with increasing tempera-ture while no detectable rotation occurs in grossularite. Concomitantly, in pyrope the shared octahedral edge lengthens at a greater rate than the unshared octahedral edge resulting in a linear decrease in the octahedral bond angle strain with increasing temperature. Meagher (1975) compares the above variations in the pyrope structure as a function of heating to changes in the aluminum s i l i c a t e garnet structure as a function of chemical substitution in the triangular dodecahedral site. Major structure response similarities are examined in detail in the following paragraphs. In order to characterize the orientation of the tetrahedron in s i l i c a t e garnets Born and Zemann (1964) defined a tetrahedral position 2 angle, y, as the smaller of the two angles formed by the tetrahedral Born and Zemann.(1964) called this angle a; Meagher (1975) redefined the angle as y. 10 edge, 0(l)-0(2), and the two crystallographic axis directions normal to the 4 axis. The 0(1)-0(2) tetrahedral edge is restricted to the plane normal to the 4 axis by symmetry requirements. For a r i g i d tetrahedron, variation of the y angle implies rotation of the tetrahedron about the 4 axis which passes through the s i l i c o n atom. As predicted by Born and Zemann (1964), the tetrahedral position angle in aluminum s i l i c a t e garnets decreases with increase in the size of the X-cation (Novak and Gibbs, 1971) as illustrated in Figure 4. Meagher (1975) demonstrates that in an analogous manner the tetrahedral position angle in pyrope decreases linearly with increasing temperature (Figure 5). This rotation of the essentially r i g i d tetrahedron about the 4 axis to smaller y angles upon heating in pyrope results in the X(2)-0(4) distance lengthening at a greater rate than the X(l)-0(4) distance (Figure 6). Octahedral distortion decreases with increasing temperature in pyrope because the shorter shared octahedral edge increases in length at a greater rate with increasing temperature than does the longer unshared edge (Meagher, 1975). In a similar fashion, octahedral distortion in aluminum s i l i c a t e garnets decreases as <r{X}> increases from 0.89 A xn pyrope to 1.01 A in a manganese-grossularite (Novak and Gibbs, 1971). Meagher (1975) reasons the pyrope structure's response to heating i s probably due at least in part to the fact the rate of change with temperature of the isotropic temperature factor, dB/dT, for Mg is greater than dB/dT of the remaining cations. Mg , therefore, has a greater effective radius at higher temperatures. Such is not the case 11 8 •g «»••»-to ? IU<-< Mn-Cr Gr e <r{X)> (A) Figure A . Tetrahedral position angle (y) versus the mean radius of the X-cation, <r{X}>, for the aluminum s i l i c a t e garnets: pyrope (Py), almandine (Al), spessartine (Sp), Mn-grossularite (Mn-Gr) and grossularite (Gr). Data from Novak and Gibbs (1971, as in Meagher, 1975). -1 1 I I 1 | T 100 200 3 0 0 4 0 0 300 6 0 0 7 0 0 Temperature °C Figure 5. Tetrahedral position angle (y) versus temperature for pyrope (Meagher, 1975). Figure 6. A portion of the garnet structure showing the effect of tetrahedral rotation to smaller y angles. The {X}-0(4) distance (equivalent to X(2)-0(4) of Figure 3) increases at a greater rate than the (X}-0(2) distance (equivalent to X(l)-0(4) of Figure 3) for a specified tetrahedral rotation rate (modified after Meagher, 1975). 13 in grossularite where dB/dT of Ca is approximately equal to dB/dT of Si"1"1" and Al' ' ' . Based upon the variations with temperature of the pyrope and grossularite structure, Meagher (1975) speculates that in an aluminum s i l i c a t e garnet with a mean X-catlon size intermediate to that of Mg and Ca one would expect the mean thermal expansion 3 coefficient of the Al-0 interatomic distance to be intermediate to that in pyrope and grossularite. Secondly, Meagher predicts the tetrahedron in such intermediate garnets should rotate to smaller values of y as a function of temperature with successively lower dy/dT values as <r{X}> increases. ' Purpose of Study This research was conducted for the purpose of determining the temperature dependence of the crystal structure of spessartine and andradite. Spessartine was chosen for this study because i t is an aluminum s i l i c a t e garnet with a X-cation which is intermediate in size relative to the X-cations of the grossularite and pyrope end-members. As such, the crystal structure of spessartine represents an excellent means of further clarifying the structural response with increasing temperature of the aluminum s i l i c a t e garnet 3 Thermal expansion coefficient i s defined on page 37. solid solution. Andradite was selected for study since i t represents the natural end-member calcium s i l i c a t e garnet with the largest known octahedral site cation (Geller, 1967). Variations in the response of the andradite and grossularite crystal structures upon heating are to be considered in evaluating the effect which the chemistry of the Y-cation has upon the thermal response character of the calcium s i l i c a t e garnet structure. 15 II. EXPERIMENTAL DETAILS Elevated Temperature Method Single crystals were maintained at elevated temperatures in an electrical resistance, platinum wire-wound, single crystal 4 furnace. The cylindrical heater, which was constructed by E.P. Meagher , is a near replica of the furnace which was designed by Foit and Peacor (1967). Electrical current was furnished by a voltage regulated, compensating current, power supply. Furnace design necessitated the u t i l i z a t i o n of flat-cone Weissenberg diffraction geometry (Buerger, 1942) wherein diffractions from each reciprocal lattice level occur in a direction normal to the rotation axis of the crystal. Such diffraction geometry was accommodated by the furnace through a 2.5 millimeter-wide planar s l i t which permitted unobstructed passage of the incident and flat-cone diffracted X-ray beams. Each reciprocal l a t t i c e level was diffracted through the s l i t by setting the angle u of the incident beam (Buerger, 1942). Other diffraction methods would have required broader openings in the heating unit and hence would have decreased overall furnace insulation and temperature s t a b i l i t y (Foit and Peacor, 1967). Department of Geological Sciences, University of British Columbia, Vancouver 8, British Columbia. 'Technal Model 3510/S; Electronic Services, 981 Pinewall Crescent, Richmond, British Columbia. 16 The heating unit was firmly clamped, via fixed support assemblies, to either a counter diffractometer base or the equivalent part of a film Weissenberg unit which rotates for the various y angles. The apparatus allowed three degrees of both translational and rotational freedom for accurate alignment of the furnace. Furnace temperature versus power supply was graphed to 1000°C in the preliminary stages of this study in order to f a c i l i t a t e a means of temperature selection. A Platinum - 13% Rhodium/Platinum thermo-couple was mounted on a goniometer head in place of a crystal and centered in the X-ray beam. The furnace was aligned about the thermo-couple. Temperature was monitored by a Leeds and Northrup volt potentiometer (model 8687) for uniform increments of voltage. Approximately 30 minutes was required for the furnace temperature to equilibrate after a power supply change. Accuracy of the curve was checked against the melting points of NBS standard zinc (melting point = 419.6°C) and a spectroscopic standard potassium chloride 7 (melting point = 776.0°C). Melting points of the two polycrystalline materials were sensed by X-radiation. In both cases, the difference between the graph and the observed melting was no greater than 10°C. This result compares well with Meagher's (1975) estimate that furnace temperatures are accurate to ±15°C. United States National Bureau of Standards. SPEX Industries, Inc., Metuchen, New Jersey, U.S.A. 17 At one atmosphere air pressure, spessartine melts congruently at about 1200° ±5°C (Snow, 1943) while andradite i s stable up to 1137° ±5°C where i t dissociates to pseudowollastonite and hematite (Huckenholz and Yoder, 1971). In order to remain well'within the thermal s t a b i l i t y fields of both spessartine and andradite, structure determinations were planned for 25°, 350°, 575° and 850°C. A preliminary study was initiated to assess whether or not divalent manganese in spessartine would oxidize in air at elevated temperatures. Small crystal fragments were heated at 850°G for approximately 48 hours and then were cooled to room temperature. The heating resulted in no visually detectable alteration of the homogeneous crystals. Immersion o i l s indicated the heating provoked no change in the room temperature refractive index. It was therefore assumed an oxygen evacuated environment would not be required for high temperature investigations of the spessartine. Later results confirm the v a l i d i t y of this assumption. Specimen Descriptions, Chemistry, Space Group and Cell Dimensions The spessartine garnet (U.S.N.M. # 107286) originated from Minas Gervais, Brazil. The andradite (U.S.N.M. # 116725) came from Valmalen, Italy. The crystal structures of both garnets have been refined previously at room temperature by Novak and Gibbs (1971). Chemical compositions (Table 3) were determined by Novak and Gibbs (1971) with an ARL-EMX electron microprobe analyser. Raw intensity 18 data were reduced with the Rucklidge and Gasparrine (1968) EMPADR V program. Elemental proportions were normalized to 12 oxygen atoms per formula unit and the relative cation portions were then adjusted on the basis of perfect stoichiometry. Composition of an andradite from Valmalen by Quarenl and de P i e r i (1966) compares very well with that specified by Novak and Gibbs (1971). For the high temperature X-ray analyses, microscopically homogeneous single crystals were wedged in tapered fused-silica glass capillaries and crystallographically oriented by oscillation and rotation methods. S i l i c a glass softens at approximately 1200°C (Smyth, 1970). The capillaries, with aligned crystals, were cemented (Sauereisen 8 9 binder and powdered kaowool f i l l e r ) to platinum wire and mounted on a goniometer head. Final acceptance for detailed X-ray examination was based upon the cl a r i t y and sharpness of diffraction spots on zero and upper level Weissenberg photographs. To a close approximation, dimensions of the selected spessartine crystal were 0.08 by 0.09 by 0.11 millimeters. The chosen andradite crystal was essentially equidimensional with a 0.07 millimeter diameter. Zero, f i r s t , second and third level Weissenberg photographs were obtained by the equi-inclination method at room temperature using Sauereisen binder no. 29, Sauereisen Cements Company, Pittsburgh, Pennsylvania, U.S.A. i • Babcock and Wilcox Refractories, Ltd., 1185 Walker Line, Burlington, Ontario. 19 Ni-filtered Cu-radiation. Systematic absences of a l l reflections except: (hkl) with h + k + 1 = 2n, (hhl) with 1 = 2n and 2h + 1 = 4n, and (Okl) with k = 2n and 1 = 2n confirmed the space group of both garnets as Ia3d. Although the andradite i s slightly optically birefringent, precession photographs exposed up to 200 hours by Novak and Gibbs (1971) failed to detect any departure from Ia3d space group symmetry. Back-reflection Weissenberg photographs were collected at room temperature for both garnets using unfiltered Cu-radiation. Cell edges were calculated by a least-squares refinement of the data. Both c e l l parameters (Table 3) are s t a t i s t i c a l l y identical to the edges of Novak and Gibbs (1971). Diffractions were recorded on Weissenberg photographs at elevated temperatures in a specially designed cassette using unfiltered 4 Cu-radiation. The dismantleable camera was constructed by E.P. Meagher to f a c i l i t a t e crystal orientation by the standard oscillation technique without disturbing the furnace. Crystals were heated to the desired temperature, given 24 hours to equilibrate, and then were reoriented before measurements commenced. The elevated temperature c e l l parameters of spessartine and andradite were obtained by a least-squares refinement of the film Weissenberg data corrected for absorption by the Nelson-Riley method (Nelson and Riley, 1945). Room temperature c e l l parameters measured after the heating were s t a t i s t i c a l l y identical to the pre-heating c e l l edges. Cell parameters, c e l l volumes and calculated densities are given in Table 3. TABLE 3. Chemical Compositions, Cell Parameters a (A), Cell Volumes V (X3), and Calculated Densities p (g/cc) of the Spessartine and Andradite Garnet Composition Spessartine { M n2.58 F e0.34 C a0.08 } [ A 11.99 F e0.01 ] ( s i 3 ) o 1 2 * Andradite { C a2.97 M g0.02 M n0.01 }t F e1.99 M0.01 ] ( s i 3 ) o 1 2 * Garnet a (A) v ( f t P (g/cc) Spessartine 25°C* 11.612(1)** 1565 4.21 Spessartine 25°C 350°C 575°C 850°C 11.613(1) 11.634(2) 11.653(3) 11.673(5) 1566 1574 1582 1591 4.19 4.17 4.15 4.13 Andradite 25°C* 12.058(1) 1753 3.85 Andradite 25°C 350°C 575°C 850°C 12.059(1) 12.088(2) 12.109(2) 12.135(2) 1754 1766 1776 1787 3.84 3.82 3.80 3.77 * Novak and Gibbs (1971). Number in parentheses refers to one estimated standard deviation. Intensity Collection and Reduction Three dimensional single-crystal intensity data were obtained for both garnets at 25°, 350°, 575° and 850°C. Intensity data were collected on a manual flat-cone Weissenberg diffractometer employing a s c i n t i l l a t i o n counter and a pulse height discriminator set to accept approximately 90 percent of the diffracted Zr-filtered MoKa radiation. t|), $ and y settings were calculated with an IBM 370/168 using Foreman's^ settings program. Magnitude of diff e r e n t i a l absorption of the Zr-filtered MoK^  radiation by the tapered s i l i c a glass capillaries was calculated to be insignificant over the investigated interval of y. Diffractions from the f i r s t nine levels of the reciprocal l a t t i c e in the range sin9 < 0.5 of the type h > k were traversed and monitored on a strip chart recorder. Standard reflections were recorded at one hour intervals as a check of electronic fluctuations and crystal alignment. Relative intensities were obtained from the strip charts with an integration planimeter. Andradite intensities were corrected for Lorentz and polarization factors. Groups of symmetry equivalent reflections demonstrated that d i f f e r e n t i a l absorption was negligible and hence absorption corrections were not applied. N. Foreman, Department of Geology, University of Michigan, Ann Arbor, Michigan, U.S.A. 22 Spessartine intensity data sets for 25 ° , 350° and 850°C were collected from the original spessartine crystal. Symmetry equivalent reflections displayed considerable differential absorption after Lorentz and polarization factor corrections. The intensities were corrected for absorption u t i l i z i n g a modified version of the program written by Burnham (1966) on an IBM 370/168. The 575°C spessartine intensity data were obtained from a second spessartine crystal after the f i r s t crystal was jarred from i t s capillary. The second spessartine crystal was essentially equidimen-sional with a diameter of 0.08 millimeters. Relative intensities were corrected for Lorentz and polarization factors. Sets of symmetry equivalent reflections showed negligible differential absorption and hence absorption corrections were not applied. 23 III. REFINEMENTS Each set of observed structure amplitudes, as defined by the reduced intensity data, were submitted to f u l l matrix least-squares refinements in space group Ia3d using a modified version of ORFLS (Busing, Martin and Levy, 1962) on an IBM 370/168. Neutral atom r e l a t i v i s t i c Hartree-Fock scattering factors (Doyle and Turner, 1968) were employed without correction for anomalous dispersion. Starting positional and thermal parameters were taken from Novak and Gibbs (1971). Oxygen positional parameters, isotropic temperature factors and scale factors were varied in the i n i t i a l unit weighted cycles of refinement. The i n i t i a l refinements of the four andradite data sets revealed systematic positive variances (|Fo|-|Fc|) between observed (|Fo|) and calculated (|Fc|) structure amplitudes for those structure amplitudes which possessed large magnitudes. This trend was indicative of secondary extinction and, therefore, an extinction coefficient was added as a refinement variable (Zachariasen, 1963). Linear hypothesis testing (Hamilton, 1965) of the weighted R-factor ratio of the non-extinction to extinction corrected refinements at every study temperature indicated that the extinction corrected refinements were s t a t i s t i c a l l y more significant at the seventy-five to ninety-five percent confidence levels. Extinction corrections were carried through a l l subsequent cycles of the 25°, 350°, 575° and 850°C andradite refinements. After three cycles of isotropic refinement, the symmetry equivalent observed structure amplitudes of each of the four 24 spessartine refinements were normalized to one scale, i f necessary, and averaged. A l l additional cycles of refinement were calculated with the average structure amplitude of each symmetry nonequivalent reflection. Average symmetry nonequivalent structure amplitudes were not used in the andradite refinements due to the extinction correction. Refinements were continued by converting from unit weights to weights (w) assigned to each observed structure amplitude following Cruickshank (1965): w = [(1 + ((.Fo -•E*)lQ*)2)~hYi where F* and G* are adjustable parameters. F* and G* were adjusted to furnish essentially constant values of (w(|Fo|-|Fc[ • scale factor) ) -for-essentially equally populated, groups of increasing |Fo|. The resultant parameters from each of the eight structures were submitted to f i n a l weighted least-squares refinements by converting to anisotropic temperature factors. Table 4 gives the number of structure amplitudes sampled (NO), the number of symmetry nonequivalent reflections (NNR), the number of parameters varied (NV), and the conventional and weighted R-factors (R and R ) of the unit weighted isotropic and f i n a l weighted w . anisotropic refinements. The weighting parameters (F* and G*) and the error mean square (Ems) of the anisotropic spessartine and andradite refinements are contained in Table 5. The magnitudes of the f i n a l observed (|Fo)) and calculated (|Fc|) structure factors are tabulated in Appendix 1. The f i n a l oxygen positional parameters are given in Table 6 together with the isotropic equivalents of the anisotropic temperature TABLE 4. Refinement Parameters: Number of Observations (NO), Number of Symmetry Nonequivalent Reflections (NNR), Number of Variables (NV), and Conventional and Weighted R-Factors (R and R ) Isotropic Refinement Anisotropic Refinement Unit Weight Weighted Garnet NO NNR * NV R+ ** R w NV R R w Spessartine 25°C 140 76 9 0.027 0.028 16 0.021 0.029 350°C 132 72 8 0.028 0.027 16 0.022 0.029 575°C ' 129 68 8 0.038 0.038 16 0.025 0.032 850°C 121 67 9 0.034 0.032 16 0.019 0.028 Andradite 25°C 154 85 10 0.026 0.026 18 0.022 0.029 350°C 110 66 11 0.032 0.034 19 0.029 0.039 575°C 105 63 10 0.026 0.028 18 0.024 0.028 850°C 98 58 9 0.034 0.035 17 0.022 0.027 * Number of parameters varied. t Conventional R-Factor = I(|Fo| - |Fc|) /IlFo| . ** Weighted R-Factor = (Zw( |Fo| - |Fc|) 2/IwFo 2). 26 TABLE 5. Final Weighting Parameters (F* and G*) and Error mean square (Ems) of the Weighted Anisotropic Refinements t Garnet G* Ems Spessartine 25°C 25.0 35.0 1.129 350°C 30.0 40.0 1.246 575°C 40.0 50.0 1.843 850°C 30.0 35.0 1.110 Andradite 25°C 20.0 60.0 1.487 350°C 20.0 65.0 2.452 575°C 25.0 135.0 2.511 850°C 35.0 55.0 1.731 t Error mean square = ((w||Fo| - |Fc|| )/(N0 - NV)) . 27 factors (B) (Hamilton, 1959) and their root mean square (R.M.S.) - - 2 equivalents (y) where y = (B/8TT ) 2 (Buerger, 1960). The number contained In parentheses adjacent to reported values in a l l tables refers, unless stated otherwise, to one estimated standard deviation. The estimated standard deviations of the R.M.S. displacements (c-) 2-were calculated following Novak and Gibbs (1971): a- = a„/K>Ti y where y B a_ is the estimated standard deviation of the isotropic temperature factor. The room temperature (25°C) results compare well with the results of Novak and Gibbs (1971) which are also given in Table 6. Both the spessartine and andradite crystals were subjected to long intervals at elevated temperatures (while obtaining the c e l l edges) before the 25°C intensity measurements were conducted. The reasonable correspondence of the two 25°C spessartine structure determinations reinforce the earlier hypothesis that no detectable oxidation of Mn would occur at the elevated temperatures. The anisotropic temperature factor coefficients (3^j) of the f i n a l refinements are tabulated in Table 7 for spessartine and andradite. The » ^ 1 3 a n (* ^23 a n i s o t r o P l c thermal parameters of the [Y] atom were locked at a small positive value in the f i n a l cycles of a l l refinements. The root mean square (R.M.S.) displacement magnitudes (y) and orientations of the thermal vibration ellipsoids (8) with respect to the crystallographic axes were calculated with Busing, Martin and Levy's (1964) ORFFE program. The estimated standard deviations were also calculated by ORFFE. Table 8 gives the magnitudes and orientations of the spessartine atoms at the four study temperatures. Table 9 contains the same information for the andradite atoms. 28 TABLE 6. Oxygen Positional Parameters, Isotropic Temperature Factors (B), and the corresponding R.M.S. Amplitude of Vibration (u) of the Atoms of Spessartine and Andradite ATOM 25°* 25° 350° 575° 850°C SPESSARTINE Mn Al Si 0.03510(8)+ 0.0355(3) 0.0357(4) 0.0359(4) 0.0367(3) 7 0.04766(11) 0.0476(3) 0.0470(3) 0.0470(4) 0.0465(3) z 0.65261(10) 0.6526(3) 0.6526(3) 0.6528(4) 0.6530(3) 2 + t 0.378(14) 0.369(61) 0.783(70) 0.907(86) 1.301(68) u 0.069(1) 0.068(6) 0.100(4) 0.107(5) 0.128(3) B 0.477(10) 0.439(35) 0.938(46) 1.342(70) 1.646(53) u 0.078(1) 0.075(3) 0.109(3) 0.130(3) 0.144(2) B 0.430(13) 0.221(66) 0.547(66) 0.646(77) 0.948(72) u 0.074(1) 0.053(8) 0.083(5) 0.090(5) 0.110(4) B 0.350(14) 0.343(52) 0.587(64) 0.726(100) 1.110(69) u 0.067(1) 0.066(5) 0.086(5) 0.096(7) 0.119(4) ANDRADITE Oxygen Ca Fe X 0.03986(15) 0.0396(2) 0.0400(5) 0.0404(5) 0.0407(5) 7 0.04885(13) 0.0488(2) 0.0486(5) 0.0484(5) 0.0482(4) X 0.65555(15) 0.6554(2) 0.6552(6) 0.6559(5) 0.6562(5) B 0.577(21) 0.671(56) 0.768(105) 1.089(89) 1.351(81) u 0.084(2) 0.092(4) 0.099(7) 0.117(5) 0.131(4) B 0.383(9) 0.432(45) 0.565(142) 0.770(137) 1.455(185! u 0.069(1) 0.074(4) 0.085(11) 0.099(9) . 0.136(9) B 0.494(17) 0.337(36) 0.514(63) 0.809(55) 1.107(50) u 0.079(1) 6.065(3) 0.081(5) 0.101(3) 0.118(3) SI 0.477(25) 0.077(2) 0.494(62) 0.079(5) 0.846(226) 0.104(14) 1.269(245) 0.127(12) 1.190(229) 0.123(12) t tt Novak and Gibbs (1971). Number ln parentheses refers to one estimated standard deviation. Isotropic equivalents of the anisotropic temperature factors, Hamilton (1959). 29 TABLE 7. Anisotropic Temperature Factor Coefficients, 0 / t ,v (x 10 5), of Spessartine and Andradite • Atom i j of 8(iJ> 25 350" 575° 85CTC SPESSARTINE 0 11 75(7)*v 102(24) 174(28) 215(36) 222(26) 22 88(6) 57(24) 174(27) 196(31) 290(27) 33 56(6) 46(23) 86(26) 90(29) 204(26) 12 13(6) -11(18) 15(21) -32(32) -4(23) 13 -15(7) 9(19) -29(23) -47(29) -40(24) 23 -2(6)- -43(18) -23(22) 0(29) 0(24) Mn** 11 71(3) 34(9) 120(13) 131(22) 188(13) 22 100(2) 105(17) 200(22) 305(32) 359(26) 23 14(3) 22(9) 28(14) 36(23) 26(15) A l ' 11 76(2) 41(26) 101(26) 119(30) 174(28) 12 -3(4) 1 1 1 1 S i " 11 68(5) 55(16) 115(20) 153(36) 211(20) 33 ^46(9) 68(24) 105(29) 124(42) 200(32) ANDRADITE 0 11 102(8) 105(19) 170(37) 189(30) 263(27) 22 115(8) 116(22) 117(39) 226(34) 259(31) 33 71(9) 125(21) 107(39) 142(33) 166(28) 12 16(9) -11(13) 20(27) -11(25) -10(29) 13 -18(8) 14(17) -28(37) -59(29) -39(27) 23 2(6) -1(16) 20(34) 23(31) -22(27) Ca** 11 69(7) 55(13) 18(49) 38(46) 179(63) 22 92(4) 84(19) 145(54) 178(59) 281(70) 23 5(7) 7(10) 17(34) 31(40) 63(34) Fe' 11 67(2) 58(13) 88(23) 138(20) 188(18) 12 -6(4) 1 1 1. 1 S i " 11 69(11) 77(19) 176(89) 265(91) 150(81) 33 107(18) 89(26) 129(74) 192(86) 228(84) * Novak and Gibbs (1971). ** Symmetry constraints require 0(22) - 0(33) and 0(12) - 0(13) - 0.0. V 0(11) = 0(22) - 0(33), and 0(12) - 0(13) = 0(23) - locked at 0.00001 for refinement. vv 0(11) - 0(22) and 0(12) - 0(13) -0(23) - 0.0. *' Number in parentheses refers to one estimated standard deviation. 30 Interatomic distances and angles and their estimated standard errors were calculated with the ORFFE program (Busing, Martin and Levy, 1964). Bond lengths and angles of the SiO^ tetrahedron, AlOg octahedron and MnOg triangular dodecahedron of spessartine are given in Table 10. Bond lengths and angles of the SiO^ tetrahedron, FeOg octahedron and CaOg triangular dodecahedron of andradite are given in Table 11. The relative frequency of occurrence of the different bond lengths and angles are also tabulated in Tables 10 and 11. Metal-metal interatomic distances of the spessartine and andradite are tabulated in Table 12. The metal-oxygen-metal angles about the oxygen atom are given for the spessartine and andradite in Table 13. The interatomic distances presented are the distances between mean atomic positions uncorrected for thermal displacements. Novak and Gibbs's (1971) corresponding room temperature results are also given in Tables 10 through 13 for comparison. 31 TABLE 8. R.M.S. Magnitudes (u) and Orientations (9) of Thermal Vibration Ellipsoids of the Atoms of Spessartine with Respect to the Crystallographic Axes Atom 0 or u r* Oxygen u(r) X 1 0.0226(367)" 0.0712(123) 0.0713(137) 6(r,x) 1 90(12) 75(14) 71(11) 51(15) e(r,y)° 1 49(11) 80(12) 85(13) 89(10) 6(r.z) 1 41(11) 18(9) 20(12) 39(15) u(r) X 2 0.0758(116) 0.1046(85) 0.1111(113) 0.1090(84) 0.1324(77) e(r,x)° 2 128(34) 47(29) 117(20) 141(16) e(r,y)° 2 126(22) 137(29) 149(22) 95(43) e(r.z) 2 58(20) 92(17) 76(15) 51(15) u(r) X 3 0.0880(92) 0.1173(89) 0.1307(104) 0.1415(65) 8(r,x)° 3 38(34) 46(28) 34(19) 95(33) 8(r,y)° 3 118(23) 49(29) 121(22) 5(42) e(r,z) 3 66(21) 107(9) 104(11) 87(29) Mn u(r) X 1 0.0479(67) 0.0908(50) 0.0950(79) 0.1140(40) 6(r,x)° 1 0 0 0 0 6(r,y) 1 90 90 90 90 e(r,2) 1 90 90 90 90 u(r) X 2 0.0751(52) 0.1087(45) 0.1362(59) 0.1516(34) e(r,x)° 2 90 90 90 90 e(r,y)° 2 45 45 45 45 e(r,z)° 2 135 135 135 135 u(r) X 3 0.0931(52) 0.1249(55) 0.1531(72) 0.1632(50) 6(r,x)° 3 90 90 90 90 G(r,y) 3 45 45 45 45 e(r,z)° 3 45 45 45 45 Al** u(r) X 1 0.0525(110) 0.0827(71) 0.0902(73) 0.1093(57) 2 0.0525(110) 0.0827(71) 0.0902(73) 0.1093(57) 3 0.0545(58) 0.0839(38) 0.0913(40) 0.1103(31) S i v u(r) X 1 0.0615(90) 0.0848(116) 0.0923(157) 0.1176(95) 2 0.0681(118) 0.0843(116) 0.0923(157) 0.1176(95) 3 0.0681(118) 0.0887(76) 0.1028(120) 0.1206(57) * r ( l ) , r(2), and r(3) are ellipsoid axes and x, y, and z are crystallographic axes. ** . Vibration ellipsoid of Al is statistically isotropic. V Vibration ellipsoid of Si is statistically isotropic. vv Number in parentheses refers to one estimated standard deviation. 32 TABLE 9. R.M.S. Magnitudes (u) and Orientations (8) of Thermal Vibration Ellipsoids o f the Atoms of Andradite with Respect to the Crystallographic Axes Atom 0 or u r* 25" 350" 575" 850 C Oxygen u(r) X 1 e(r,*)° 1 8(r,y)° 1 8(r.z)° 1 u(r) 8 2 e(r.x)° 2 8(r,y)° 2 ©(r.z)° 2 u(r) X 3 S<r,x)° 3 S(r.y)° 3 e(r,z)° 3 Ca u(r) X 1 6(r,x)° 1 e(r.,y)° 1 e(r,z)° 1 u(r) X 2 0(r.x)° 2 e(r.y)° 2 e(r,z)° 2 u(r) X 3 6(r.x)° 3 6(r,y)° 3 e(r.z)° 3 0.0831 (83) v v 32(23) 67(36) 111(28) 0.0932(88) 79(45) 148(40) 119(46) 0.0996(75) 60(24) 111(47) 37(43) 0.0637(73) 0 90 90 0.0753(59) 90 45 135 0.0816(68) 90 45 45 0.0763(205) 69(16) 125(30) 42(27) 0.0989(164) 92(36) 143(31) 127(32) 0.1165(128) 21(17) 79(31) 107(28) 0.0366(491) 0 90 90 0.0974(136) 90 45 135 0.1098(167) 90 45 45 0.0867(147) 57(12) 96(13) 33(11) . 0.1240(102) 129(27) 135(36) 72(22) 0.1362(113) 124(28) 46(36) 63(16) 0.0528(323) 0 90 90 0.1044(163) 90 45 135 0.1247(159) 90 45 45 0.1048(109) 71(11) 78(12) 23(11) 0.1404(86) 93(113) 165(63) 76(40) 0.1438(83) 19(23) 98(117) 108(30) 0.1155(202) 0 90 90 0.1277(149) 90 45 135 0.1603(110) 90 45 45 Fe** u(r) X 1 2 3 S i ' u(r) X 1 2 3 0.0651(49) 0.0651(49) 0.0668(25) 0.0752(95) 0.0809(118) 0.0809(118) 0.0802(70) 0.0802(70) 0.0816(37) 0.0976(281) 0.0976(281) 0.1143(288) 0.1010(47) 0.1010(47) 0.1021(25) 0.1196(267) 0.1196(267) 0.1402(242) 0.1182(37) 0.1182(37) 0.1191(20) 0.1057(284) 0.1304(240) 0.1304(240) * r(l), r(2), and r(3) are ellipsoid axes and x, y, and z are crystallographic axes. ** Vibration ellipsoid of Fe is statistically Isotropic. V Vibration ellipsoid of Si is statistically isotropic. vv Number in parentheses refers to one estimated standard deviation. 33 TABLE 10. Interatomic Distances (X) and Angles (°) of the S10^ Tetrahedron, AlOg Octahedron, and MnOg Triangular Dodecahedron of Spessartine Distance or angle f* 25°** 25° SI - 0 ( l ) v 4 1.637(1)" 1.633(3) 0(1) - 0(2) 2 2.520(2) 2.518(6) 0(1) - 0(3) 4 2.746(2) 2.737(6) Mean 0 - 0 2.671 2.664 0(1) - S i - 0(2) 2 100.67(7) 100.90(22) 0(1) - S i - 0(3) 4 114.04(4) 113.92(12) 350" 1.631(4) 1.632(5) 1.624(4) 2.516(7) 2.517(10) 2.511(8) 2.735(7) 2.736(9) 2.721(7) 2.662 2.663 2.651 100.92(27) 100.93(33) 101.23(28) 113.91(15) 113.90(18) 113.74(15) A 1 " 0(1) -6 1.902(1) 1.902(3) 1.904(3) 1.909(4) 1.915(3) 0(1) - 0(4) 6 2.678(2) 2.682(6) 2.691(6) 2.699(8) 2.719(6) 0(1) " 0(5) 6 2.702(2) 2.698(5) 2.695(6) 2.701(8) 2.698(7) Mean 0 - 0 2.690 2.690 2.693 2.700 2.709 0(1) - AI - 0(4) 6 89.48(5) 89.66(13) 89.91(18) 89.95(23) 90.45(18) 0(1) - Al - 0(5) 6 90.52(5) 90.34(13) 90.09(18) 90.05(23) 89.55(18) Mn(l) - 0(4)*' 4 2.247(1) 2 Mn(2) - 0(4)v* 4 2.408(2) 2 Mean Mn - 0 2.328 2 0(1) - 0(2) 2 2.520(2) 2 0(1) - 0(4) 4 2.678(2) 2 0(4) - 0(6) 4 2.824(3) 2 0(4) - 0(7) 2 2.822(2) 2 0(1) - 0(7) 4 3.367(1) 3 0(8) - 0(7) 2 3.953(2) 3 Mean 0 - 0 3.004 3 0(1) - Mn(2) - 0(2) 2 68.21(5) 68 0(1) - Mn(2) - 0(4) 4 70.14(5) 70 0(4) - Mn(2) - 0(6) 4 74.59(5) 74 0(4) - Mn(2) - 0(7) 2 71.75(4) 71 0(1) - Mn(2) - 0(7) 4 92.58(4) 92 0(8) - Mn(2) - 0(7) 2 110.40(5) 110 .249(3) 2.254(4) 2.258(5) 2.267(4) .408(3) 2.419(4) 2.423(5) 2.436(4) .329 2.337 2.341 2.352 .518(6) 2.516(7) 2.517(10) 2.511(8) .682(6) 2.691(6) 2.699(8) 2.719(6) .827(6) 2.844(8) 2.848(10) 2.869(8) .815(6) 2.826(8) 2.829(9) 2.830(8) .365(2) 3.372(2) 3.377(4) 3.383(3) .959(6) 3.980(8) 3.989(11) 4.019(9) .004 3.015 3.020 3.034 .07(16) 67.86(18) 67.76(26) 67.26(19) .22(15) 70.20(15) 70.30(21) 70.54(16) .68(11) 74.89(15) 74.86(20) 75.09(15) .55(15) 71.46(19) 71.42(23) 71.03(19) .49(9) 92.31(12) 92.27(16) 91.94(12) .61(15) 110.68(19) 110.75(22) 111.16(19) * Numbers in this row refer to the frequency of occurrences within the specified polyhedron. ** Novak and Gibbs (1971). v Oxygen atom designation as in Figure 3. w Number ln parentheses refers to one estimated standard deviation. Mn(l) is equivalent to X(l) of Figure 3. »* Mn(2) i s equivalent to X(2) of Figure 3. 34 TABLE 11. Interatomic Distances (X) and Angles (°) of the SiO^ Tetrahedron, FeO, Octahedron, and CaO. Triangular Dodecahedron of Andradite Distance or angle f* 25°** 25° 350° 575° 850°C Si 0(1)' 4 1.643(2)" 1.646(3) 1.647(6) 1.640(6) 1.638(6) 0(1) - 0(2) 2 2.564(4) 2.568(6) 2.576(13) 2.562(12) 2.558(11) 0(1) - 0(3) 4 2.739(3) 2.746(5) 2.746(11) 2.735(10) 2.731(10) Mean 0 - 0 2.681 2.686 2.689 2.677 2.673 0(1) - Si - 0(2) 2 102.64(7) 102.53(18) 102.84(45) 102.69(40) 102.70(41) 0(1) Si - 0(3) 4 112.99(8) 113.05(10) 112.88(24) 112.97(21) 112.96(22) Fe 0(1) "6 2.024(2) 2.021(3) 2.025(7) 2.037(6) 2.045(5) 0(1) - 0(4) 6 2.890(3) 2.884(5) 2.894(11) 2.916(9) 2.932(10) 0(1) - 0(5) 6 2.834(4) 2.832(5) 2.832(13) 2.844(10) 2.851(9) Mean 0 - 0 2.862 2.858 2.863 . 2.880 2.891 0(1) _ Fe - 0(4) 6 91.12(7) 91.03(12) 91.24(28) 91.44(23) 91.60(23) 0(1) Fe - 0(5) 6 88.88(7) 88.97(12) 88.76(28) 88.56(23) 88.40(23) Ca(l) 0(4)*' 4 2.366(2) • 2.364(3) 2.374(7) 2.377(6) 2.383(6) Ca(2) - 0(4)'* 4 2.500(2) 2.500(3) 2.509(7) 2.518(6) 2.527(5) Mean Ca - 0 2.433 2.432 2.441 2.448 2.455 0(1) - 0(2) 2 2.564(4) 2.563(6) 2.576(13) 2.562(12) 2.553(11) 0(1) - 0(4) 4 2.890(3) 2.884(5) 2.894(11) 2.916(9) 2.932(10) 0(4) 0(6) 4 2.936(4) 2.936(6) 2.953(15) 2.957(12) 2.967(11) QC4) 0(7) 2 2.847(4) 2.852(5) 2.854(12) 2.861(11) 2.867(11) 0(1) - 0(7) 4 3.485(2) 3.486(2) 3.497(5) 3.498(4) 3.504(4) 0(8) - 0(7) 2 4.175(3) 4.172(6) 4.191(14) 4.212(12) 4.231(11) Mean 0 - 0 3.134 3.134 3.145 3.153 3.162 0(1) - Ca(2) - 0(2) 2 65.64(10) • 65.79(14) 65.70(31) 65.20(27) 64.92(27) 0(1) - Ca(2) - 0(4) 4 72.82(10) 72.66(13) 72.64(30) 73.06(25) 73.26(25) 0(4) - Ca(2) - 0(6) 4 74.17(7) 74.20(12) 74.37(28) 74.27(23) 74.27(21) 0(4) - Ca(2) - 0(7) 2 69.43(9) 69.55(12)' 69.33(29) 69.23(25) 69.13(27) 0(1) - Ca(2) - 0(7) 4 91.44(5) 91.51(8) 91.43(18) 91.17(16) 91.02(15) 0(8) - Ca(2) - 0(7) 2 113.25(9) 113.09(12) 113.29(28) 113.50(24) 113.66(26) * Numbers in this row refer to the frequency of occurrences within the specified polyhedron. ** Novak and Gibbs (1971). V Oxygen atom designation as in Figure 3. vv Number in parentheses refers to one estimated standard deviation. *' Ca(l) i s equivalent to X(l) of Figure 3. '* Ca(2) i s equivalent to X(2) of Figure 3. TABLE 12. Metal - Metal interatomic Distances (£) for Spessartine and Andradite Metal - Metal 25°* 25° '350° 575° 850°C Spessartine Mn - A l 3.248 3.2459(2) v v 3.2513(5) 3.2571(7) 3.2627(12) Mn(2) - S i . 3.558 3.5557(2) 3.5622(4) 3.5680(6) 3.5741(11) Mn(l) - S i 2.905 2.9032(2) 2.9085(5) 2.9132(7) 2.9182(12) A l - S i 3.248 3.2459(2) 3.2518(5) 3.2571(7) 3.2627(12) Mn(l) - Mn(2)** 3.558 3.5557(2) 3.5622(4) 3.5680(6) 3.5741(11) S i - S i 3.558 3.5557(2) 3.5622(4) 3.5680(6) 3.5741(11) A l A l 5.032 5.0286(3) 5.0377(5) 5.0459(8) 5.0546(13) Andradite Ca _ Fe 3.370 3.3706(2) 3.3787(5) 3.3846(5) 3.3918(5) Ca(2) - S i 3.692 3.6923(2) 3.7012(4) 3.7076(4) 3.7156(4) Ca(l) - S i 3.015 3.0147(2) 3.0220(5) 3.0272(5) 3.0337(5) Fe - S i 3.370 3.3706(2) 3.3787(5) 3.3846(5) 3.3918(5) Ca(l) - Ca(2)? 3.692 3.6923(2) 3.7012(4) 3.7076(4) 3.7156(4) S i - S i 3.692 3.6923(2) 3.7012(4) 3.7076(4) 3.7156(4) Fe - Fe 5.221 5.2217(3) 5.2343(5) 5.2434(5) 5.2546(5) Novak and Gibbs (1971). Mn(l) and Mn(2) are equivalent to X(l) and X(2), respectively, of Figure 3, Ca(l) and Ca(2) are equivalent to X( l ) and X(2), respectively, of Figure 3. Number i n parentheses refera to one estimated standard deviation. TABLE 13. Metal - Oxygen - Metal Angles (°) for Spessartine and Andradite Metal - 0 - Metal 25°* 25° 350° 575° 850°C Spessartine A l - 0 - S i 133.08(7)* v 133.25(17) 133.66(24) 133.66(32) 134.21(25) A l - 0 - Mn(2) 102.72(4) 102.56(14) 102.58(16) 102.51(18) 102.22(16) A l - 0 - Mn(2) 97.11(5) 97.04(13) 96.83(15) 96.78(21) 96.42(16) Mn(l) - 0 - S i 95.55(5) 95.52(14) 95.61(16) 95.65(22) 95.76(17) Mn(2) - 0 - S i 122.05(6) 122.13(15) 121.94(17) 122.02(20) 122.07(18) Mn(l) - 0 -' Mn(2)** 99.65(4) 99.50(11) 99.27(16) 99.25(20) 98.86(16) Andradite Fe - 0 - S i i 133.37(10) 133.37(18) 133.62(42) 133.73(34) 133.86(32) Fe - 0 - Ca(2) 100.03(8) 100.18(11) 100.06(23) 99.85(21) 99.71(22) Fe - 0 - Ca(2) 95.75(8) 95.83(11) 95.79(25) 95.41(22) 95.21(21) Ca(l) - 0 - S i 95.86(10) 95.84(12) 95.73(29) 96.06(26) 96.19(25) Ca(2) - 0 - S i 124.77(10) 124.60(14) 124.59(32) 124.81(27) 124.91(26) Ca(l) - 0 - C a ( 2 ) v 98.68(6) 98.73(11) 98.54(26) 98.43(21) 98.31(21) * Novak and Gibbs (1971). ** Mn(l) and Mn(2) are equivalent to X(l) and X(2), re s p e c t i v e l y , of Figure 3. v Ca(l) and Ca(2) are equivalent to X(l) and X(2), re s p e c t i v e l y , of Figure 3. vv Number i n parentheses refers to one estimated standard deviation. ON 37 IV. RESULTS AND DISCUSSION Variations of the crystal structure of spessartine and andradite during thermal expansion are described and discussed in this chapter. The format of this chapter is influenced by the fact that the garnet crystal structure i s completely defined by specifying the unit c e l l size and the fractional coordinates of the oxygen atom (Menzer, 1926). The cations, as was indicated in an earlier chapter, are in special positions. Unit c e l l expansion, oxygen fractional coordinate variations, polyhedral variations and atomic thermal vibration are considered in detail. Variations of structural parameters with temperature are characterized in this text by linear rates of change and conventional linear thermal expansion coefficients. The linear thermal expansion coefficient i s defined as: 1 A g - A 2 5p  aA " A 2 5o T - 25° l C ' where A 2^ Q is the value of a parameter at 25 C, Aj, is the value of the same parameter at temperature T (°C) and is the linear thermal expansion coefficient. The quantity represents the variation of parameter A per degree temperature increase divided by the room temperature value of parameter A. In this study, the magnitude: \ ." A25° T - 25° 38 i s taken as the slope obtained by linear regression of parameter A versus temperature. The linear relations shown in the figures of this chapter represent the results of such simple linear regression analyses. Unless stated otherwise, these relations are linear within the errors of the experimental determinations. Estimated standard deviations of the regression slopes were combined with the estimated standard deviations of the A^o terms by propagation of error techniques (Hamilton, 1964) to arrive at estimated standard deviations for the thermal expansion coefficients. Error terms obtained in this manner are relatively large in a l l cases as a result of the compounding of the A^o and regression slope error terms. Errors of the regression slopes are comparatively smaller. For this reason, rates of change are generally favored in this text for comparison purposes. The results presented in this chapter indicate, however, that a l l thermal expansion coefficients display the same trends as the rates of change. Thermal Expansion of Cell Parameters Available elevated temperature c e l l edge data reveal each s i l i c a t e garnet c e l l parameter can be closely modeled as a linear function of temperature. The c e l l edge of the grossularite measured by Meagher (1975) increased linearly with increasing temperature within the errors of the c e l l parameter determinations. C e l l edges of the spessartine and andradite measured in this study vary linearly with temperature i n an analogous manner (Figures 7 and 8; linear correlation coefficient for spessartine = 0.9987 and for andradite = 0.9999). 39 12.06 12.05 200 400 600 800 Temperature (*C) Figure 7. Spessartine c e l l parameter versus temperature. Error bars represent ± one estimated standard deviation. 40 11.69 11.68 11.60 1 i 1 r • , 1 — i . 0 200 400 600 800 Temperature (°C) Figure 8. Andradite c e l l parameter versus temperature. Erro bars represent ± one estimated standard deviation. 41 Skinner (1956a) measured the cel l edges of apparently pure end-member grossularite, andradite, almandine, spessartine, and pyrope to approximately 750°C. The cel l edge of each sample was determined at approximately 15 different temperatures. Skinner listed the estimated standard deviation of his powder camera technique (Skinner, 1956b) as 0.001 £ . Based on this accuracy, the variation of the cell edge of each garnet with temperature cannot be described as truly linear. However, the correlation coefficients of first order linear regression analyses of Skinner's cel l edges versus temperature vary between 0.9971 and 0.9993. Therefore, Skinner's (1956a) measurements demonstrate that the cell edges of most end-member silicate garnets can, to a first approximation, be modeled as a linear function of temperature. It has been recognized for some time that the cel l edges of silicate garnets vary linearly with the sizes of the six- and eight-coordinated cations. McConnell (1966) presented a regression formula relating garnet cel l edge to Ahren's cation radii . Novak and Gibbs (1971) updated the same formula using data from 56 chemically analyzed silicate garnets with well determined cel l edges. Their equation relates the cel l parameter to the mean radius of the X-cation and the mean radius of the Y-cation using Shannon and Prewitt's (1969) effective radii . The multiple linear correlation coefficient of Novak and Gibbs's (1971) analysis (0.996) manifests the strong linear dependence of the silicate garnet cel l edge upon composition. Silicate garnet cel l edges therefore might be expected to possess a near-linear dependence upon both temperature and composition. 42 A multiple linear regression analysis was calculated to test this hypothesis with the c e l l edges of: 1) this study, 2) the c e l l edges of pyrope and grossularite as reported by Meagher (1975), and 3) the nine edges determined by Novak and Gibbs (1971) at room temperature. The sample population was comprised of the data from twenty-four c e l l edge measurements. The resulting equation (multiple linear correlation coefficient = 0.9993) relates the c e l l parameter, a, to the equilibrium temperature, T°C, the mean radius of the X-cation, <r{X}>, and the mean radius of the Y-cation, <r[Y]>, using Shannon and Prewitt's (1969) effective r a d i i as follows: a = 8.969(17) + 1.872(37)<r[Y]> + 1.684(18)<r{X}> + 8.73(46)xl0 - 5T where the number in parentheses are the estimated standard errors of the regression coefficients. The reported regression coefficients are s t a t i s t i c a l l y significant contributors to the equation at the 99% confidence level based upon the par t i a l F-ratios of the coefficients at the appropriate number of degrees of freedom. The effective radius of 8-coordinated {Fe+^} was taken as 0.91 X and a value of 0.98 X was +2 employed as the radius of {Mn } in compliance with Novak and Gibbs (1971). Other than the three pyrope c e l l edges used by Meagher (1975), Skinner's (1956a) c e l l parameter data were not included in the regression analysis. The chemical compositions of the garnets studied by Skinner were never determined. Skinner's (1956a) andradite, grossularite, and pyrope data, however, provide a means of checking the presented regression formula. Table 12 shows a good correspondence exists between Skinner's observed 43 TABLE 14. Predicted and Observed Cell Parameters for pure End-member Garnets at Various Temperatures Garnet Temperature (°C) Cell Parameters observed* calculated** Pyrope 25.0 11.459 X 11.462 X 122.3 11.467 11.471 247.1 11.478 11.482 328.1 11.486 11.489 441.0 11.497 11.499 597.1 11.514 11.512 698.1 11.524 11.521 758.0 11.531 11.526 Grossularite 25.0 11.851 11.846 108.8 11.857 11.854 185.5 11.863 "11.860 277.7 11.871 11.868 387.9 11.881 11.878 480.1 11.889 11.886 589.3 11.900 11.896 707.7 11.912 11.906 Andradite 25.0 12.048 12.062 149.3 12.059 12.072 349.0 12.078 12.090 501.8 12.094 12.103 589.2 12.103 12.111 629.9 12.108 12.114 690.0 12.114 12.120 * As reported by Skinner (1956a). ** Calculated with the regression equation of this study using the effective r a d i i of Shannon and Prewitt (1969). 44 cell parameters and the calculated equivalents. Measured linear thermal expansion coefficients of the andradite and spessartine cel l edges are 0.764(23)xl0 C and 0.632(53)xl0 ^ °C \ respectively. Skinner's (1956a) data gives 0.819x10"""' °C 1 and 0.885x10"^ °C~* for the andradite and spessartine, respectively. Some difference in results would be expected since the garnets examined in the two different studies are not known to be of the same composition. Lack of correspondence, however, is very large for spessartine. Possibly, this should have been anticipated as Skinner (1956a) questioned the accuracy of his spessartine and almandine data. Oxygen Fractional Coordinates Meagher (1975) found the oxygen positional parameters in grossularite do not change up to 675°C while in pyrope the y fractional coordinate decreases during thermal expansion. Fractional coordinates of the oxygen in spessartine, as'determined in this study (Table 6), are plotted versus temperature in Figure 9. The lines shown in Figure 9 were obtained by first order linear regression analysis. The y fractional coordinate decreases significantly while the x coordinate increases with increasing temperature. The regression slope of the x parameter versus temperature is nonzero at the 90 percent confidence level. The z fractional coordinate remains statistically constant over the temperature interval 25° to 850°C. Collectively, these results indicate the z positional parameter of oxygen remains essentially constant during temperature variation in the three end-member aluminum 45 0 CO c TJ o o o (0 c o o CD c o CD X o 0 0. 0. 0, 0. 0. 0. 0. 0-0. 0. 0. 0. 0. 0 0 0 0 0 0 0 0 6544 6 5 4 0 6 536 6 53 2 6528 6524 6520 .0482 • 0478 ,0474 .04 70 0466 0462 0458 0372 0368 0364 0 36 0 0 3 5 6 0 3 5 2 -\ 034 8 0 3 4 4 i 1 — — i — — — - 1 2 0 0 400 600 800 T e m p e r a t u r e ( ° C ) Figure 9. Fractional coordinates (x, y, z) of oxygen in spessartine as a function of temperature. Error bars represent ± one estimated standard deviation. 46 0) +•> c T3 O o o CO c o o CO c >« X O X 0 0 0 0 0 0 0 0, o. 0. 0. 0. 0. 0. 0. 0. 0 0 0 0 0 0 .6570 .6566 .6562 .6558 .6554 .6550 .6546 .6542 0492 0488 0484 0480 0476 0472 ,0416 ,0412 ,0408 ,0404 ,0400 0396 ,0392 ,0388 y 200 400 600 800 T e m p e r a t u r e ( ° C ) Figure 10. Fractional coordinates (x, y, z) of oxygen in andradite as a function of temperature. Error bars represent ± one estimated standard deviation. 47 s i l i c a t e garnets. The y positional parameter exhibits the largest variation with temperature in both spessartine and pyrope. The fractional coordinates of oxygen in andradite (Table 6) are plotted as a function of temperature in Figure 10. Oxygen displays a s t a t i s t i c a l l y significant decrease in the y coordinate and a s t a t i s t i c a l l y significant increase in the x coordinate as a function of increasing temperature. This is consistent with the trend shown by spessartine. In contrast to the z fractional coordinate of oxygen in the aluminum s i l i c a t e garnets, the z coordinate in andradite displays an apparent increase of slightly more than two estimated standard deviations (Figure 10). Variations in the character of the coordination polyhedra of spessartine and andradite as a result of the observed oxygen positional parameter shifts are discussed in the following section. Polyhedra Variations with Temperature Silicate Tetrahedron Meagher (1975) observed that the s i l i c a t e tetrahedra of both grossularite and pyrope remain r i g i d during thermal expansion. The results of this study indicate: 1) the s i l i c a t e tetrahedron of spessartine may not remain r i g i d as a function of temperature and 2) the s i l i c a t e tetrahedron of andradite distorts during thermal expansion. The Si-0 interatomic distance, the 0(l)-0(2) shared tetra-48 hedral edge and the O-Si-0 angles of the tetrahedra in spessartine and andradite exhibit no s t a t i s t i c a l l y significant variations over the examined temperature interval based on the sample populations of this study (Tables 10 and 11). Both spessartine and andradite, however, exhibit quite well defined apparent decreases in the 0(l)-0(3) unshared tetrahedral edge as a function of increasing temperature. The 0(l)-0(3) unshared edge decreases slightly more than two estimated standard deviations over the examined temperature interval in both garnets. It i s reasoned that this decrease is significant at least in the case of andradite. The x fractional coordinate of oxygen in the garnet structure i s defined in the crystallographic direction which is parallel to the 4 axis of the garnet tetrahedron (Menzer, 1926). This 4 point, symmetry (Table 1) dictates that variation in solely the x fractional coordinate must result i n variation of the 0(l)-0(3) unshared tetrahedral edge. Since the x fractional coordinate of oxygen in andradite increases at a significant rate as a function of increasing temperature (Figure 10), one can argue the observed variation in the 0(1)-0(3) edge must be real. On a s t a t i s t i c a l basis, the same reasoning is not s t r i c t l y true for spessartine where the x parameter variation with temperature is nonzero only at the 90 percent confidence level. The observed variation of the 0(1)-0(3) unshared tetrahedral edge in andradite upon heating is plotted in Figure 11. This variation causes the SiO^ tetrahedron to become more regular as a function of increasing temperature as is evidenced by the decrease in the tetrahedral bond angle strain (Table 15). 49 2.770 2.76 0 °< 2.7 5 0 O 2.72 0 \ 2.710 1 • • • • • • 0 200 400 600 800 Temperature ('C) Figure 11. Variation of the unshared tetrahedral edge, 0(l)-0(3), of andradite with increasing temperature. Error bars represent ± one estimated standard deviation; TABLE 15. Bond Angle S t r a i n i n Pyrope, Spessartine, Grossularite and Andradite Pyrope * 25°C 350° 550° 750° Spessartine ** 25°C 350° 575° 850° Gr o s s u l a r i t e * 25°C 365° 675° Andradite ** 25°C 350° 575° 850° -9.97 +5.21 -2.19 -0.22 +1.03 +1.45 +1.18 -9.63 +5.02 -1.78 -0.55 +1.15 +1.70 +0.83 -10.12 +5.29 -1.81 -0.64 +0.97 +1.73 +1.11 -10.10 +5.28 -1.67 -0.72 +0.88 +1.90 +1.08 -8.57 +4.45 -0.34 -1.38 +0.77 +2.98 -0.15 -8.55 +4.44 -0.09 -1.59 +0.75 +3.19 -0.24 -8.54 +4.43 -0.05 -1.69 +0.85 +3.16 -0.28 -8.24 +4.27 +0.45 -2.19 +1.09 +3.39 -0.67 -6.77 +3.43 +1.60 -2.15 +0.05 +4.90 -1.50 -6.77 +3.53 +1.60 -2.25 +0.25 +4.70 -1.50 -6.97 +3.63 +1.50 -2.25 +0.45 +4.50 -1.40 -6.94 +3.58 +1.03 -3.66 +3.21 +2.50 -2.15 -6.63 +3.41 +1.24 -3.75 +3.19 +2.67 -2.37 -6.78 +3.50 +1.44 -4.25 +3.61 +2.57 -2.47 -6.77 +3.49 +1.60 -4.53 +3.81 +2.57 -2.57 * Meagher (1975). ** This study. 51 The refined values of the Si-0 distance display a slight apparent decrease with heating i n both spessartine and andradite (Tables 10 and 11). Smyth (1973) likewise noted the Si-0 distances in the tetrahedra of an orthopyroxene might decrease as a function of temperature. The tetrahedral bond angle strain (Table 15) in both spessartine and andradite also apparently decreases slightly as a function of increasing temperature. Fyfe (1954) suggested increasing 3 the regularity of a SiO^ tetrahedron should permit the sp -hybrid orbitals of oxygen to be in a position for a more favorable overlap with those of s i l i c o n thus causing a decrease in the Si-0 bond length. The apparent trends observed for the Si-0 distance and O-Si-O angles of both spessartine and andradite as a function of temperature are consistent with Fyfe's hypothesis. As in pyrope, the y angle of the spessartine tetrahedron exhibits a linear decrease with increasing temperature; the angle decreases from 26.1(1)° at 25°C to 25.6(2)° at 850°C. Both numbers in parentheses are one estimated standard deviation and refer to the last significant figure. As predicted by Meagher (1975), the rate of change of y with temperature in spessartine i s less than in pyrope (Figure 12). Earlier in this text i t was specified: 1) y angle variation of the s i l i c a t e tetrahedron describes r i g i d body rotation about the 4 axis and 2) the y angle i s measured in the plane perpendicular to the 4 axis of the s i l i c a t e tetrahedron. In a hypothetical case, such rotation could cause variation in only the y and z fractional coordinates of 52 2 8.0° CO O a 26.5 c 0 200 400 600 800 Temperature (°C) Figure 12. Variation of the tetrahedral position angle (y) with increasing temperature in pyrope (Meagher, 1975) and spessartine. Error bars represent ± one estimated standard deviation. 53 oxygen since the x fractional coordinate is measured in the direction which is parallel to the 4 axis (Menzer, 1926). The decrease of the y angle in spessartine and pyrope during thermal expansion can therefore be attributed to the observed decrease in the y positional parameter of oxygen (Figure 9). The position of the oxygen atom in the garnet structure i s such that rotation of the tetrahedron about i t s 4 axis to a slightly smaller y angle results in a change in the y positional parameter which is considerably larger than the change concurred simultaneously by the z positional parameter. This is explained graphically i n Figure 13. The y angle in both andradite and grossularite does not vary significantly during thermal expansion. The y angle in grossularite remains at 24.5° over the temperature interval 25° - 675°C (Meagher, 1975). Although the y parameter of oxygen in andradite does decrease significantly during thermal expansion, the apparent increase ,in the z parameter (Figure 10) maintains the y angle at an essentially constant value. This mechanism i s graphically explained for the ideal case i n Figure 14. The fact that the y angle in andradite is relatively large compared to the y angles of most s i l i c a t e garnets (Figure 4) actually enhances this effect. The y angle in andradite does not vary significantly from 27.3(2)° over the temperature interval 25 - 850°C. The Y0^ Octahedron The size of the X-cation is known to influence the length of the Al-0 bond in aluminum s i l i c a t e garnets at room temperature. The Al-0 bond length equals 1.895 ± 0.01 X for <r{X}> less than 1.10 X and 54 • z a x i s +x axis projection x,y,z = oxygen parameters before r i g i d body rotation x,y',z f = oxygen parameters after r i g i d body rotation Figure 13. General position of s i l i c a t e tetrahedron in the garnet structure. Rotation of tetrahedron about 4 axis to smaller y angles results in a decrease in the y parameter (Ay) which is larger than the variation in the z parameter (Az). The 4" axis passes through the s i l i c o n atom and is perpendicular to the page. 55 +x axis projection x,y,z = i n i t i a l oxygen parameters x,y',z' = positional parameters of oxygen after variation of y and z parameters accompanied by no simultaneous variation in the y angle of the s i l i c a t e tetrahedron Figure 14. Idealized variation in the y and z positional parameters of oxygen in the garnet structure restricted such that no simultaneous change occurs in the y angle of the s i l i c a t e tetrahedron. A 4 axis passes through the s i l i c o n atom and i s perpendicular to the page. 56 1.924 i for <r{X}> larger than 1.10 % (Novak and Gibbs, 1971). Zemann (1962) had predicted two Al-0 bond length populations for Y=A1 s i l i c a t e garnets based upon the crystal structures of pyrope and grossularite. Figure 15 indicates the Al-0 distance in pyrope, spessartine, and grossularite increases linearly with increasing temperature. The thermal expansion coefficient of the Al-0 distance in spessartine at 0.846 x 10 °C ^ (Table 16) is intermediate in magnitude relative to the thermal expansion coefficients of the Al-0 interatomic distance in pyrope (0.737 x 10~5 0 C - 1 ) and grossularite (1.281 x 10"5 ° C - 1 ) . Collectively, these values indicate the rate of thermal expansion of the Al-0 distance apparently increases regularly with the size of the X-cation in the aluminum s i l i c a t e garnets. Y-0 bond lengths in X=Ca s i l i c a t e garnets increase linearly with the mean radius of the Y-cation (Novak and Gibbs, 1971). Figure 16 shows the Y-0 distance in andradite and grossularite also varies linearly with temperature. The linear rate of increase of the Fe-0 bond in andradite is slightly larger than that observed for the Al-0 distance in grossularite. The numerical values are given in Table 16. A similar relation has been observed in sodium pyroxenes. The linear rate of increase of the Fe-0 distance during thermal 3+ expansion in the M(l) octahedron of acmite (NaFe Si20g) is apparently larger than the rate of increase of the analogous Al-0 distance of jadeite (NaAl 3 +Si 20 6) (Cameron et a l , 1973). The rates of change of the Y-0 distance with temperature in 57 1.880 1 > > • • • « •— 0 200 400 600 800 Temperature (°C) Figure 15. Variation of the Al-0 interatomic distance with increasing temperature in pyrope (Meagher,,1975), spessartine and grossularite (Meagher, 1975). Error bars represent ± one estimated standard deviation. 58 2.050 i 1.910 \ 1.900 1 . 1 . , , - r — 0 200 400 600 800 Temperature (°C) Figure 16. Variation of the Y-0 interatomic distance with increasing temperature in grossularite (Meagher, 1975) and andradite. Error bars represent ± one estimated standard deviation. TABLE 16. Ionic R a d i i , Rates of Increase of Mean M-0 distances (X °C x 10 ) and Mean Thermal Expansion C o e f f i c i e n t s (°C~^ x 10+ )^ of Pyrope, Spessartine, Grossularite, and Andradite Garnet <r{X}>* <r[Y]>** <r>*** Bond M-0 Coordination Number d(M-0)/dT (X V 1 x 10 + 5) aM-0 ( V 1 x 10 + 5) Pyrope t 0.890 X 0.530 X 0.746 X Sl-0 4. +0.O9O(275)*t +0.055 Al-0 6 +1.39(16) +0.737 Mg(l)-0 8 +1.61(64) +0.733 Mg(2)-0 8 +4.43(14) +1.892 <Mg-0> 8 +3.10(22) +1.366 Spessartine t t 0.977 X 0.530 X 0.798 X Si-0 4 -0.954(412) -0.584 Al-0 6 +1.61(30) +0.846 Mn(l)-0 8 +2.12(28) +0.943 Mn(2)-0 8 +3.28(31) +1.362 <Mn-0> 8 +2.70(28) +1.159 G r o s s u l a r i t e t 1.118 X 0.532 X 0.884 X Si-0 4 +1.06(29) +0.644 Al-0 6 +2.46(7) +1.281 Ca(l)-0 8 +2.21(67) +0.953 Ca(2)-0 8 +2.00(4) +0.803 <Ca-0> 8 +2.02(40) +0.840 Andradite t t 1.118 X 0.644 X 0.928 X Si-0 4 -1.01(39) -0.614 Fe-0 6 +2.94(44) +1.455 Ca(l)-0 8 +2.31(20) +0.977 Ca(2)-0 8 +3.28(13) +1.312 <Ca-0> 8 +2.81(6) +1.155 t Meagher (1975). t t This study. * Mean radius of X-cation. ** Mean radius of Y-cation. *** Mean radius of nontetrahedral cations. * t Estimated standard error by weighted l i n e a r regression. 60 pyrope, spessartine, grossularite and andradite are plotted versus the mean size of the nontetrahedral cations in Figure 17. The relation i s very closely modeled by second order linear regression (correlation coefficient = 0.9996) however only a f i r s t order linear regression model is s t a t i s t i c a l l y warranted based on the magnitudes of the determination errors. On the basis of the trend shown In Figure 17 one might speculate that: 1) the rate of thermal expansion of the Al-0 interatomic distance in almandine (mean radius of the nontetra-hedral cations = <r> = 0.762 X) should be intermediate to that of spessartine and pyrope and 2) the rate of thermal expansion of the Cr-0 interatomic distance in uvarovite (<r> = 0.915 R) should be intermediate to that of the Y-0 distance in grossularite and andradite. The garnet octahedron shares half of i t s edges with triangular dodecahedra. The remaining octahedral edges are unshared. In direct contradiction to Pauling's electrostatic bonding principles (1929) the octahedral shared edge, 0(1)-0(4) in Figure 3, i s longer than the unshared edge, 0(l)-0(5), in the X=Ca s i l i c a t e garnets (Abrahams and Geller, 1958; Prandl, 1966; Quareni and de P i e r i , 1966; Novak and Gibbs, 1971). The shared edge is shorter than the unshared edge in pyrope, almandine and spessartine (Zemann and Zemann, 1961; Gibbs and Smith, 1965; Euler and Bruce, 1965; Novak and Gibbs, 1971). The variable character of the s i l i c a t e garnet octahedron led Geller (1967) to predict that i t should be possible for a s i l i c a t e garnet to possess a completely regular octahedron at room temperature. The point symmetry of the garnet octahedral site i s such that an ideal octahedron would exist when the shared and unshared octahedral edges are equal in length. 61 0.70 0.75 0.80 0.85 0.90 0.95 < r > (A) Figure 17. Rate of change of the Y-0 interatomic distance with increasing temperature, d(Y-0)/dT, versus the mean size of the non-tetrahedral cations, <r>, for pyrope (Meagher, 1975), spessartine, grossularite (Meagher, 1975) and andradite. Error bars represent ± one estimated standard deviation. 62 Novak and Gibbs (1971) found this to be the case in a manganese-grossularite. In pyrope (Meagher, 1975) and spessartine (Table 10) the octahedral shared edges expand with increasing temperature while the unshared edges remain essentially constant. Concomitantly the 0(1)-Al-0(4) angle increases while the 0(l)-Al-0(5) angle decreases. This results in a decrease of the octahedral bond angle strain [(O(l)-Al-0(4)) -90°] with heating in pyrope (Table 15). In spessartine the octahedral bond angle strain decreases with heating u n t i l both the shared and unshared edges become equal as shown in Figure 18. This condition corresponds to the existence of an ideal octahedron in spessartine. Upon continued heating the shared edge in spessartine becomes larger than the unshared edge and the resultant octahedral bond angle strain i s analogous to that typical of X=Ca s i l i c a t e garnets at room temperature (Table 15). This trend i s analogous to that observed in the octahedra of aluminum s i l i c a t e garnets at room temperature with variation in the size of the X-cation (Novak and Gibbs, 1971). Unlike spessartine and pyrope, the unshared octahedral edge in both grossularite and andradite increases during thermal expansion. The unshared octahedral edge of andradite increases at an apparently lesser rate than the unshared octahedral edge of grossularite (Figure 19). The shared octahedral edge of andradite, on the other hand, increases at a greater rate than the octahedral shared edge of grossularite (Figure 20). The shared octahedral edge increases at a greater rate than the unshared octahedral edge in andradite (Table 11).. This i s also the 63 2.740 2.730 2.720 • • (A: 2.710 2.700 unshared ^ o 1 o 2.690 2.680 2.670 2.660 2.650 • 0 200 400 Temperature 600 800 CO Figure 18. Variation of the shared and unshared octahedral in spessartine with temperature. 64 2.862 .] t o O I O 2.7 00 1 2.690 2.68 0 i -2.670 d r a ^ e T 800 Temperature (°C) Figure 19. Variation of the unshared octahedral edge, 0(l)-0(5), with temperature in grossularite (Meagher, 1975) and andradite. Error bars represent ± one estimated standard deviation. 65 2.94 0 2.930 2.920 2.91 0 ^ - 2.900 o , 2.880 j -X 2.7 9 0 O 2.7 8 0 2.7 7 0 2.760 2.750 2.740 i T 1 — r 200 400 600 800 T e m p e r a t u r e C C ) Figure 20. Variation of the shared octahedral edge, 0(l)-0(4) with temperature in grossularite (Meagher, 1975) ' and andradite. Error bars represent ± one estimated standard deviation. 66 case in spessartine and pyrope. As in pyrope (Meagher, 1975), this can be attributed to the greater rate of increase of the X(2)-0(4) interatomic distance relative to the rate of increase of the X(l)-0(4) distance during thermal expansion in both spessartine and andradite (Table 16). In andradite the resultant octahedral bond angle strain increases as a function of increasing temperature (Table 15). The shared and unshared octahedral edges in grossularite, however, increase at the same rate during thermal expansion (Meagher, 1975). The XOg Triangular Dodecahedron The 8-fold coordination polyhedron about the X-cation has two symmetry non-equivalent X-0 distances, X(l)-0(4) and X(2)-0(4). The X(2)-0(4) distance i s longer than the X(l)-0(4) distance in a l l s i l i c a t e garnets analyzed to date. This is in direct contradiction to what one would be led to expect by considering cation-cation repulsion forces. X-Si repulsions across the edge shared with the s i l i c o n tetrahedron would be expected to cause X(l)-0(4) to be longer than the X(2)-0(4) distance (Zemann, 1962). Zemann (1962) and Gibbs and Smith (1965) have proposed the X(2)-0(4) distance i s longer in order to satisfy geometric require-ments of the garnet structure; unreasonable Y-0 and unshared octahedral edge distances would result i f X(2)-0(4) were shorter than X(l)-0(4). More recently, Meagher (1975) arrived at the same conclusion u t i l i z i n g the Distance Least Squares computer program of Meier and V i l l i g e r (1969). Novak and Gibbs (1971) found X(l)-0(4), X(2)-0(4) and the mean X-0 distance, <X-0>, increase linearly with <r{X}> in the aluminum si l i c a t e garnets and with <r[Y]> in the calcium s i l i c a t e garnets. In a 67 somewhat analogous manner, the X-0 bond lengths in pyrope, spessartine, grossularite and andradite increase essentially linearly during thermal expansion. As in pyrope (Meagher, 1975), the X(2)-0(4) distance in spessartine increases at a greater rate than the X(l)-0(4) distance (Figure 21). Figure 22 shows the X(2)-0(4) distance in andradite also increases at a greater rate than the X(l)-0(4) distance during thermal expansion. Linear thermal expansion coefficients and the rates of change of the X(l)-0(4), X(2)-0(4) and <X-0> distances are documented for spessartine and andradite in Table 16. Meagher's (1975) results for pyrope and grossularite are also given in Table 16. The differences in the rates of expansion of the two X-0 distances decrease as the mean radius of the X-cation increases i n the aluminum s i l i c a t e garnets. In grossularite the two symmetry non-equivalent Ca-0 distances vary at essentially the same rate during thermal expansion. The spessartine refinements confirm that the relative rates of increase of the two symmetry non-equivalent X-0 distances in the aluminum s i l i c a t e garnets during thermal expansion are directly related to the rate of rotation of the SiO^ tetrahedron as shown in Figure 6 (Meagher, 1975). The response in andradite is quite different. The greater rate of increase of the X(2)-0(4) distance relative to the X(l)-0(4) distance i s achieved primarily by a distortion of the SiO^ tetrahedron. The SiO^ tetrahedron of andradite, as was discussed earlier in this test, becomes more regular as a function of increasing temperature. 68 o I c 800 Temperature (°C) Figure 21. X(l)-0(4), X(2)-0(4) and <X-0> distances of spessartine versus temperature. Error bars represent ± one estimated standard deviation. 69 o I CO O 2.5 3 0 2.520 2.51 0 2.5 0 0 2.460 2.450 -I 2.44 0 2.4 3 0 2.390 i 2.3 8 0 2.370 2.360 2.350 200 400 600 •I-800 Temperature (°C) Figure 22. X(l)-0(4), X(2)-0(4) and <X-0> distances of andradite versus temperature. Error bars represent ± one estimated standard deviation. 70 Linear variations of <X-0> with temperature are illustrated for pyrope, spessartine, grossularite and andradite in Figure 23. The slopes of the correspondences, d<X-0>/dT, which are also tabulated in Table 16, indicate the rate of increase of the <X-0> distance i s apparently dependent on not only the size of the X-cation, but also the size of the Y-cation. Two interesting trends are discernible when one considers the [Y] = Al and {X} = Ca garnets as two individual groups. In the aluminum s i l i c a t e garnets d<X-0>/dT decreases with increase in the size of the X-cation. This trend was recognized by Meagher (1975). The value of d<X-0>/dT for spessartine further c l a r i f i e s this trend as being essentially linear. Secondly, the andradite and grossularite data indicate that d<X-0>/dT apparently increases with increase in the size of the Y-cation in the calcium s i l i c a t e garnets. The manner in which these two trends might apply in general to the s i l i c a t e garnet solid solution was tested by multiple linear regression analysis. The equation obtained from the multiple linear regression analysis relates d<X-0>/dT to <r{X}> and <r[Y]> using the same effective ionic r a d i i as Novak and Gibbs (1971) as follows: d<X-0>/dT = 0.3711(49) x IO - 4 + 0.6362(97) x 10~ 4 <r[Y]> - 0.4479(49) x IO - 4 <r{X}> where the numbers in parentheses are the estimated standard deviations of the regression coefficients. A multiple linear correlation coefficient of 0.9999 indicates the regression f i t i s extremely good; one must rec a l l however that only four data sets were available for 71 2.4 6 0 1 V 2.30 0 f 2.290 2.280 2.270 -2.260 1 1 - i 1 . . 1 . r — 0 200 400 600 800 Temperature (°C) Figure 23. <X-0> versus temperature in pyrope, spessartine, grossularite and andradite. 72 the analysis. The regression coefficients are s t a t i s t i c a l l y significant contributors to the equation at the 99.9% confidence level based upon the partial F-ratios at the appropriate number of degrees of freedom. On the assumption that the multiple linear regression model is at least qualitatively valid, one would expect d<X-0>/dT to: 1) increase with increase in <r[Y]> in a s i l i c a t e garnet solid solution with an isochemical X-site cation and 2) to decrease with increase in <r{X}> in a s i l i c a t e garnet solid solution with an isochemical Y-site cation. Oxygen-oxygen edges of the triangular dodecahedron of both spessartine and andradite (Tables 10 and 11) increase essentially linearly during thermal expansion with the exception of the 0(l)-0(2) distance. The 0(1)-0(2) distance is the edge shared with the tetra-hedron and i t does not vary significantly upon heating in either garnet. An ideal triangular dodecahedron has two unique 0-X-O angles -69.45° and 71.70° (Lippard and Russ, 1968). Table 15 l i s t s the bond angle strains for a selected number of the bond angles of the triangular dodecahedron in pyrope, spessartine, grossularite and andradite at the specified study temperatures. The 0-X-O bond angles in both spessartine and andradite generally either increase or decrease linearly with increasing temperature (Tables 10 and 11) within the errors of the determinations. The only exception is the angle in andradite which is opposite the edge shared between dodecahedra, 0(4)-X(2)-0(6). The 0(4)-X(2)-0(6) angle of 73 andradite shows no s t a t i s t i c a l l y significant variation with temperature while the same angle in spessartine increases with temperature. The angle opposite the edge shared with the octahedron, 0(l)-X(2)-0(4), and the angle opposite the unshared edge 0(8)-0(7), 0(8)-X(2)-0(7), increase with temperature in both spessartine and ^ andradite. The angle opposite the edge shared with the tetrahedron, 0(l)-X(2)-0(2), and the angles opposite the unshared edges 0(4)-0(7) and 0(l)-0(7), 0(4)-X(2)-0(7) and 0(l)-X(2)-0(7), decrease linearly with temperature in both spessartine and andradite. Isotropic Temperature Factors and Thermal Ellipsoids The Isotropic equivalents of the anisotropic temperature factors of the atoms in spessartine and andradite at room temperature (Table 17) show reasonable agreement with the corresponding values of Novak and Gibbs (1971). Novak and Gibbs's (1971) and Meagher's (1975) isotropic temperature factors for the atoms in pyrope and grossularite at room temperature are also given in Table 17 for comparison. Equivalent isotropic temperature factors of each atom in pyrope, spessartine, grossularite and andradite generally increase linearly with increasing temperature within the errors of the determina-tions. The only two exceptions are the isotropic temperature factors of the calcium and s i l i c o n atoms of andradite. For qualitative comparative purposes, however, the variations of the isotropic temperature factors of these two atoms with temperature w i l l be approximated by linear relations since they show l i t t l e deviation from linearity. This appears a sensible approach as i t is widely recognized that isotropic temperature TABLE 17. Equivalent Isotropic Temperature Factors, B^o, at Room Temperature and Linear Rates of change of Isotropic Temperature Factors with Temperature Atom Garnet B 2 5o * B 2 5o ** dB/dT Oxygen pyrope spessartine grossularite andradite 0.502(21)t 0.378(14) 0.764(21) 0.577(21) 0.47(11) 0.369(61) 0.39(9) 0.671(56) 0.00130 0.00109 0.00116 0.00086 X Mg Mn Ca Ca pyrope spessartine grossularite andradite 0.789(29) 0.477(10) 0.612(14) 0.383(9) 0.93(20) 0.439(35) 0.39(10) 0.432(45) 0.00195 0.00145 0.00120 0.00120 Y Al Al Al Fe pyrope . spessartine grossularite andradite 0.397(23) 0.430(13) 0.664(19) 0.494(17) 0.40(6) 0.221(66) 0.40(6) 0.337(36) 0.00092 0.00085 0.00083 0.00095 Silicon pyrope spessartine grossularite andradite 0.194(20) 0.350(14) 0.558(17) 0.477(25) 0.29(11) 0.343(52) 0.30(11) 0.494(62) 0.00063 0.00090 0.00108 0.00093 Novak and Gibbs (1971). B2^o of pyrope and grossularite from Meagher (1975); B25° °^ spessartine and andradite from this study. Number in parentheses refers to one estimated standard deviation. t 75 factors are sensitive to data collection and refinement procedures. The linear rate of change of the isotropic temperature factor of each atom with temperature, dB/dT, is tabulated in Table 17. Each dB/dT value was obtained by simple linear regression analysis. A number of interesting relations can be observed from the dB/dT values of Table 17. The rates of increase of the 8-coordinated cations are greater than that of the 6-coordinated and 4-coordinated cations. Secondly, the rates of increase of the oxygen isotropic temperature factors l i e in general between those of the 6- and 8-coordinated cations. And thirdly, the rates of increase of the X-site cations display the greatest range while those of the Y-site cations exhibit the least variation. Figure 24 illustrates the variations of the isotropic temperature factors of the Y-site cations with temperature. Figure 25 displays the analogous variations of the isotropic temperature factors of the X-site cations. The linear relationships depicted in both Figure 24 and Figure 25 were obtained by f i r s t order linear regression analyses. The rates of increase of the isotropic temperature factors of each cation are plotted versus the rates of thermal expansion of each corresponding metal-oxygen distance in Figure 26. This relationship is of interest since bond length expansions w i l l , in part, reflect increased amplitudes of anharmonic vibration of atoms. In a qualitative sense, Figure 26 does show that the cations with the largest dB/dT values also correspond to the metal ions with the greatest bond length expansions. In detail, however, a direct correlation does not appear to exist. For example, dB/dT of the Y-site cation remains relatively constant while the 76 1.6 Figure 24. Equivalent isotropic temperature factors, By, of the Y-site cations in pyrope (Py), spessartine (Sp), grossularite (Gr) and andradite (An) versus temperature. 77 CQ 0 . 6 -I 0 .3 200 400 600 800 Temperature (°C) Figure 25. Equivalent isotropic temperature factors, Bx, of the X-site cations in pyrope (Py), spessartine (Sp), grossularite (Gr), and andradite (An) versus temperature. 78 3.0 in + O X 2.0 < I— T3 1.0 \ A o V 0.0 H -1.0 « M » - P y • C a - A n Mn —Sp • A l - G r C a — G r A l - S p • A l —Py O S i - G r O Si —Py Si —Sp O O S i — A n _2.0 " 1 1 1 1 1 1 r-0.5 1.0 ~ i — i i — i i i — i — i i 1.5 2.0 d B / d T (A2'C"1 x 1 0 ° ) Figure 26. Linear rates of change of mean metal—oxygen distances with temperature, d<M-0>/dT, versus the linear rates of change of the cation isotropic temperature factors with temperature, dB/dT. 79 Y-0 bond lengths exhibit quite a wide spectrum of rates of bond length expansion. The X-cations of the aluminum s i l i c a t e garnets display a strong correlation between dB/dT and d<M-0>/dT. In the two calcium garnets, on the other hand, the Ca-0 interatomic distances experience quite different rates of bond length expansion but possess equivalent dB/dT values for the calcium atoms. The thermal vibration ellipsoids of the manganese atom in spessartine and the calcium atom in andradite are anisotropic t r i a x i a l ellipsoids at room temperature (Novak and Gibbs, 1971 and Tables 8 and 9). Since the point symmetry of these X-cations are 222 (Table 1), the orientations of their thermal ellipsoids are constrained to certain crystallographic directions as shown in Tables 8 and 9. With increasing temperature the ellipsoids maintain roughly the same shape while their root mean square (R.M.S.) amplitudes of vibration increase in a l l directions. The thermal vibration ellipsoids of aluminum in spessartine and iron in andradite are s t a t i s t i c a l l y isotropic (uniaxial) with indeterminate orientations (Tables 8 and 9). Their R.M.S. amplitudes of vibration increase in a l l directions during thermal expansion with the vibration ellipsoids remaining s t a t i s t i c a l l y isotropic. This i s also the case for the s i l i c o n atom in both spessartine and andradite (Tables 8 and 9). The oxygen atom, as was indicated earlier in this text, occupies a general position in the garnet structure and thus neither the orientations nor the shape of i t s vibration ellipsoid are constrained 80 by symmetry requirements. The oxygen atoms of both spessartine and andradite have t r i a x i a l vibration ellipsoids at room temperature (Novak and Gibbs, 1971). The vibration ellipsoids apparently display no significant variations in shape during thermal expansion. A conclusive interpretation, however, cannot be made due to the relatively large determination errors. The experimental technique did not result in a resolution of the ellipsoid axis orientations which warrants a c r i t i c a l analysis. Structural Adjustment During Thermal Expansion  in Silicate Garnets The main objectives of this section are: 1) to bri e f l y review how the pyrope, spessartine, grossularite and andradite end-members of the s i l i c a t e garnet solid solution accommodate their observed differential metal-oxygen distance variations during thermal expansion and 2) to consider how other members of the solid solution might be expected to respond during thermal expansion. Polyhedral adjustments described in previous sections of this chapter indicate that the aluminum s i l i c a t e garnets possess, i n general, a different thermal response character than the calcium s i l i c a t e garnets. Thermal expansion in aluminum s i l i c a t e garnets is accomplished mainly by rotation of the f a i r l y r i g i d SiO^ tetrahedra to smaller y angles. The rates of rotation of the tetrahedra during thermal expansion decrease as the <r{X}> increases u n t i l the natural upper limit for the <r{X}> i s attained in grossularite. No significant rotation of 81 the SiO^ tetrahedra i s observed i n grossularite (Meagher, 1975). This tetrahedral rotation to smaller y angles is the main mechanism of accommodating: 1) the greater increase of the X(2)-0(4) distance relative to the X(l)-0(4) distance and 2) the greater increase of the shared octahedral edge relative to the unshared octahedral edge during thermal expansion in both pyrope (Meagher, 1975) and spessartine. The spessartine refinements of this study (Table 10) further suggest that a minor component of tetrahedral distortion may also occur during thermal expansion. Based on these results, the thermal response character of almandine (<r{X}> = 0.91 &") would be expected to be intermediate to that possessed by spessartine and pyrope. In andradite, as i n grossularite, the y angle of the tetra-hedron does not vary significantly during thermal expansion. Rather, the tetrahedron of andradite distorts by becoming more ideal. It is via this mechanism that the andradite structure accommodates: 1) a greater increase of the X(2)-0(4) distance relative to the X(l)-0(4) distance and 2) a greater increase of the octahedral shared edge relative to the octahedral unshared edge during thermal expansion. Since grossularite and andradite represent the two natural extreme end-members of the calcium s i l i c a t e garnets with respect to the <r[Y]> (Rickwood, 1968) one might expect calcium s i l i c a t e garnets with intermediate <r[Y]> w i l l similarly display no significant variations of their tetrahedral position angles (y) during thermal expansion. These garnets, however, might be expected to display small distortions of their SiO^ tetrahedra during thermal expansion. 82 A very interesting and significant characteristic of the polyhedral variations i s that essentially a l l of the metal-oxygen bond lengths, bond angles and oxygen-oxygen edges vary linearly with increasing temperature in each garnet within the errors of the deter-minations. Zemann (1962) was the f i r s t to recognize that a similar relation probably existed in the s i l i c a t e garnet structure at room temperature as a function of <r{X}> and <r[Y]> when he predicted that the c e l l edges, bond lengths and bond angles in almandine and spessartine should be intermediate to those in pyrope and grossularite. Novak and Gibbs (1971) reasoned further that since the bond lengths and bond angles in the garnet structure are determined in part by the positional parameters of oxygen (the cations are in special positions), the positional parameters of the oxygen atom might be linearly dependent on <r{X}> and <r[Y]>. In their a r t i c l e , Novak and Gibbs (1971) present linear regression equations which establish that a high linear dependence exists between the oxygen positional parameters and chemical compositions i n the s i l i c a t e garnet solid solution at room temperature. The equations allow one to predict the positional parameters of oxygen solely from the mean radius of the X- and Y-cations. Based on the above observations, one might suspect that the oxygen positional parameters of the s i l i c a t e garnet solid solution w i l l display a high linear dependence on both composition and temperature collectively. The number of s i l i c a t e garnet structures refined at elevated temperatures, however, are few. Only four members of the s i l i c a t e garnet solid solution have now been refined at elevated temperatures. Altogether, eleven elevated temperature crystal structure refinements 83 have been conducted. This does not constitute a sufficient number of refinements to warrant a serious attempt at modeling the whole s i l i c a t e garnet solid solution as a function of temperature. Three of the four s i l i c a t e garnets analyzed at elevated temperatures, however, are aluminum s i l i c a t e garnets. It i s reasoned that these refinements may be employed to obtain some indication of the f e a s i b i l i t y of modeling the oxygen coordinates of at least the aluminum s i l i c a t e garnets as a linear function of both temperature and composition. This proposed model was tested by multiple linear regression analysis. The sample population was comprised of the data from the crystal structure refinements of Meagher (1975), the spessartine refinements of this study and the room temperature refinements of almandine and manganese-grossularite of Novak and Gibbs (1971). In total, thirteen refinements were represented. Each of the three resulting equations relates an oxygen positional parameter to the equilibrium temperature (T°C) and <r{X}> using the same effective ionic r a d i i as Novak and Gibbs (1971) as follows: x = 0.0215(17)<r{X}> + 8.29(4.91) x 10~7T + 0.0140(17) y = -0.0207(17)<r{X}> - 8.09(4.83) x 10~7T + 0.0680(17) z = -0.0098(5)<r{X}> + 4.26(1.46) x 10~7T + 0.6620(5) where the estimated standard errors are given in parentheses and refer to the last significant figures. The resulting multiple linear correlation 2 2 2 coefficients squared are: R , N = 0.942, R , N = 0.939, and R , N = 0.975. (x) (y) . (z) A l l <r{X}> regression coefficients are s t a t i s t i c a l l y significant contributors in the equations based upon their partial F-ratios. In 84 both the x and y coordinate equations, the temperature term is a significant contributor only at the 80% confidence level. The temperature term is a s t a t i s t i c a l l y significant contributor at the 95% confidence level in the z coordinate equation. As such, these results indicate a moderate linear dependence does exist. With the above equations, one can semi-quantitatively predict the nature and relative magnitudes of the oxygen positional parameter shifts which one may expect to observe in almandine and manganese-grossularite as a function of increasing temperature. These predictions are tabulated in Table 18 along with predicted c e l l edges. For both almandine and manganese-grossularite, the equations predict that the y parameter should decrease significantly while the x parameter should increase with increasing temperature. The z parameter displays a small increase in both cases. 85 TABLE 18. Predicted Cell Edges (a) and Positional Parameters (x,y, and z) for Almandine and Manganese-Grossularite at Elevated Temperatures Garnet Temperature a** X y z Almandine 25°C* 11.531 & 0.0343 0.0486 0.6533 25°C 11.51 0.0338 0.0490 0.6531 200° 11.53 0.0339 0.0488 0.6531 400° 11.55 0.0341 0.0486 ' 0.6532 600° 11.56 0.0343 0.0485 0.6533 800° 11.58 0.0344 0.0483 0.6534 Manganese ' 25°C* 11.690 A* 0.0358 0.0463 0.6518 -grossularite 25°C 11.69 0.0360 0.0468 0.6521 200° 11.70 0.0361 0.0467 0.6521 400° 11.72 0.0363 0.0465 0.6522 600° 11.74 0.0365 0.04.64 0.6523 800° 11.75 0.0366 0.0462 0.6524 * Novak and Gibbs (1971). ** Cell edges obtained from equation presented in text. 86 V. SUMMARY Eight crystal structure refinements have been presented in this text. The almost pure end-member spessartine and andradite were refined by the least-squares technique in space group Ia3d at 25°, 350°, 575°, and 850°C. Final residual (R) factors range from 0.019 to 0.029. The spessartine and andradite structures at 25°C correspond favorably with the results of Novak and Gibbs (1971). Unit c e l l expansion, oxygen fractional coordinate variations, polyhedral variations and atomic thermal vibration have been described in detail. A high linear dependence has been shown to exist for the c e l l edges of the members of the s i l i c a t e garnet solid solution as a function of temperature and composition collectively. Si-0 interatomic distances in both spessartine and andradite (uncorrected for thermal displacement) show a zero or slightly negative expansion, while the Mn-0 and Al-0 interatomic distances of spessartine and the Ca-0 and Fe-0 interatomic distances of andradite show significant positive expansions as a function of increasing temperature. In both spessartine and andradite, the shared octahedral edge increases at a greater rate than does the unshared octahedral edge and the X(2)-0(4) distance increases at a greater rate than does the X(l)-0(4) interatomic distance during thermal expansion. The spessartine garnet structure f a c i l i t a t e s these variations mainly by rotation of the essentially r i g i d SiO^ tetrahedron while the andradite garnet structure f a c i l i t a t e s the same variations mainly by a distortion of the SiO^ tetrahedron. The SiO^ tetrahedron of andradite becomes more ideal as a function of increasing 87 ~^  temperature. The response of spessartine during thermal expansion has been compared with the responses of pyrope and grossularite in an attempt to further c l a r i f y the structural thermal response character of the aluminum s i l i c a t e garnet solid solution. Variations in the response of the andradite and grossularite crystal structures during thermal expansion were considered in evaluating the probable struc-tural thermal response character of other members of the calcium s i l i c a t e garnet solid solution. 88 SELECTED REFERENCES Abrahams, S.C, and S. Geller (1958) Refinement of the structure of a grossularite garnet. Acta. Crystallogr. 11, 437-441. Born, L., and J. Zemann (1964) Abstandsberechnungen und gitterenerge-tische Berechnungen an Granaten. Contrib. Mineral. Petrology 10, 2-23. Boyd, F.R., and J.L. England (1959) Pyrope. Ann. Rept. Director Geophys. Lab. 1320, 83-87. Buerger, M.J. (1942) X-ray crystallography. John Wiley and Sons, Inc., New York. Buerger, M.J. (1960) Crystal structure analysis. John Wiley and Sons, Inc., New York. Burnham, C.W. (1966) Computation of absorption corrections and the significance of end effect. Amer. Mineral. 51, 159-167. Busing, W.R., K.O. Martin, and H.A. Levy (1962) ORFLS: A fortran crystallographic least-squares refinement program. U.S. Clearing-house Fed. Sci. Tech. Info. Doc, ORNL-TM-305. Busing, W.R., K.O. Martin, and H.A. Levy (1964) ORFFE: A fortran crystallographic function and error program. U.S. Clearinghouse Fed. Sci. Info. Doc, ORNL-TM-306. Cameron, M., S. Sueno, C.T. Prewitt, and J.J. Papike (1973) High temperature crystal chemistry of acmite, diopside, hedenbergite, jadeite, spodumene, and ureyite. Amer. Mineral. 58, 594-618. Cruickshank, D.W.J. (1965) Errors in least-squares methods. In J.S. Rollett, Ed. Computing Methods in Crystallography. Pergamon Press, New York, 112-116. Doyle, P.A., and P.S. Turner (1968) Relativistic Hartree-Fock x-ray and electron scattering factors. Acta. Crystallogr. A24, 390-397. Euler, F., and J.A. Bruce (1965) Oxygen coordinates of compounds with garnet structure. Acta. Crystallogr. 19, 971-978. Foit, F.F., and D.R. Peacor (1967) A high temperature furnace for a single crystal x-ray diffTactometer. J. Sci. Instrum. 44, 183-185. Fyfe, W.S. (1954) The problem of bond type. Amer. Mineral. 39, 991-1004. Geller, S., C.E. Miller; and R.G. Treuting (1960) New synthetic garnets. Acta. Crystallogr. 13, 179-186. 89 Geller, S. (1967) Crystal chemistry of the garnets. Z. Kristallogr. 125, 1-47. Gibbs, G.V., and J.V. Smith (1965) Refinement of the crystal structure of synthetic pyrope. Amer. Mineral. 50, 2023-2039. Hamilton, W.C. (1959) On the isotropic temperature factor equivalent to a given anisotropic temperature factor. Acta. Crystallogr. 12, 609-610. Hamilton, W.C. (1964) Statistics in physical science. The Ronald Press Company, New York. Hamilton, W.C. (1965) Significance tests on crystallographic R-factors. Acta. Crystallogr. 18, 502-510. Huckenholz, H.G., and H.S. Yoder, Jr. (1971) Andradite s t a b i l i t y relations in the CaSi0»-Fe90. join up to 30 kb. N. Jb. Mineral. 114, 246-280. J International tables for x-ray crystallography (1952) Vol. I. symmetry groups. N.F.M. Henry and K. Lonsdale, eds. The Kynoch Press, Birmingham, England. Lippard, S.J., and B.J. Russ (1968) Comment on the choice of an eight-coordination polyhedron. Inorg. Chem. 9, 1686-1688. McConnell, D. (1966) Proprietees physiques des grenates. Calcul de la dimension de l a maille unite a partir de la composition chimique. Bull. Soc. Franc. Mineral. Cristallogr. 89, 14-17. Meagher, E.P. (1973) The crystal structures of grossularite and pyrope garnet at high temperatures (abstr.). Geol. Soc. Am. Abstr. Progr. 5, 735. Meagher, E.P. (1975) The crystal structure of pyrope and grossularite at elevated temperatures. Amer. Mineral, in press. Meier, Van W.M., and H. V i l l i g e r (1969) Die Methode der Abstands-verfeinerung zur Bestimmung der Atomkoordinaten idealisierter Gerustsrukturen. Z. Kristallogr. 69, 300-396. Menzer, G. (1926) Die Kristallstrukture von Granat. Z. Kristallogr. 63, 157-158. Menzer, G. (1928) Die Kristallstruktur der Granate. Z. Kristallogr. 69, 300-396. Nelson, J.E., and D.P. Riley (1945) An experimental investigation of extrapolation methods in the derivation of accurate u n i t - c e l l dimensions of crystals. Proc. Phys. Soc. (London) 57, 160-177. 90 Novak, G.A., and G.V. Gibbs (1971) The crystal chemistry of the si l i c a t e garnets. Amer. Mineral. 56, 791-825. Pauling, L. (1929) The principles determining the structure of complex ionic crystals. J. Amer. Chem. Soc. 51, 1010-1026. Prandl, W. (1966) Verfeinerung der Kristallstruktur des Grossulars mit Neutronen und Rontgenstrahlbeugung. Z. Kristallogr. 123, 81-116. Quareni, S., and R. Depieri (1966) La struttura Dell'andradite. Mem. Accad. Patavina, Sci. Mat. Nat. 78, 153-164. Rickwood, P.C. (1968) On recasting analyses of garnet into end-member molecules. Contrib. Mineral. Petrology 18, 175-198. Rucklidge, J., and E.L. Gasparrini (1968) EMPADR V. Department of Geology, University of Toronto. Shannon, R.D., and C.T. Prewitt (1969) Effective ionic r a d i i in oxides and fluorides. Acta. Crystallogr. B25, 925-946. Skinner, B.J. (1956a) Physical properties of end-members of the garnet group. Amer. Mineral. 41, 428-436. Skinner, B.J. (1956b) The thermal expansions of thoria, periclase and diamond. Amer. Mineral. 41, 39-55. Smyth, J.R. (1970) High temperature single-crystal studies on low-calcium pyroxenes. Ph.D. thesis, University of Chicago, Chicago, I l l i n o i s , U.S.A. Smyth, J.R. (1973) An orthopyroxene structure up to 850°C. Amer. Mineral. 58, 636-648. Snow, R.B. (1943) Equilibrium relationships on the liquidus surface in part of the Mn0-Al 90_-Si0 9 system. J. Am. Ceram. Soc. 26, 11-20. J Winchell, A.N. (1933) Optical mineralogy II. 3rd. ed., John Wiley and Sons, Inc., New York. Zachariasen, W.H. (1963) The secondary extinction correction. Acta. Crystallogr. 16, 1139-1144. Zemann, A., and J. Zemann (1961) Verfeinerung der Kristallstruktur von Synthetischeim Pyrope, Mg„Al„(SiO,)_. Acta. Crystallogr. 14, 835-837. 4 Zemann, J. (1962) Zur Kristallchemie der Granate. Beitr. Mineral. Petrology 8, 180-188. 91 APPENDIX 1 92 TABLE 19. Observed and Calculated Structure Factors of the Spessartine. 25"C 350 C 575 C 850"C Fc Fo Fc Fo h k 1 Fo Fc Fo 8 0 0 2 6 5 . 1 1 273 .17 250.69 4 4 4 2 4 8 . 9 1 250 .48 236.08 8 8 0 2 2 7 . 3 1 234.72 216.24 4 0 0 2 1 2 . 3 8 217.67 209.39 4 2 0 2 1 0 . 7 5 221 .37 211.26 6 4 0 2 0 2 . 5 1 206.25 195.48 8 8 8 1 9 2 . 9 0 191.58 168.16 6 4 2 1 6 9 . 9 5 172.4e 163.85 8 4 2 1 4 6 . 7 8 147.74 140.40 10 4 0 1 4 2 . 7 7 140.52 127.75 8 4 0 1 4 2 . 5 5 142.40 136.99 12 2 2 1 2 9 . 1 0 132.27 117.32 6 6 4 1 2 7 . 3 0 128.18 119.81 8 6 4 1 2 6 . 2 5 126.09 117.62 10 4 2 1 2 2 . 0 8 121.24 109.30 12 2 0 1 1 7 . 5 3 117.44 107.70 16 0 0 1 1 6 . 5 1 116.81 95.63 10 8 4 1 1 1 . 8 3 114.92 102.38 6 1 1 1 1 0 . 6 9 106.75 105.73 4 2 2 1 0 5 . 6 0 109.89 104.09 14 4 2 1 0 0 . 4 4 101.73 86.38 12 8 2 1 0 0 . 1 3 98 .79 85.21 12 6 0 9 9 . 9 5 100.11 87.22 8 8 4 9 6 . 8 1 97.32 89 .99 12 4 4 9 6 . 0 1 100.07 82.34 12 8 6 9 5 . 0 3 93 .02 79 .57 10 6 4 8 9 . 0 8 89 .72 77.78 14 4 . 0 8 8 . 5 7 91 .38 76.27 10 10 4 8 7 . 6 3 90 .10 76 .79 12 10 0 8 3 . 8 4 81 .21 68.11 5 2 1 8 2 . 3 5 78 .56 78 .29 12 6 6 8 2 . 0 8 81 .58 70 .63 12 10 2 8 1 . 5 2 79 .19 70.24 12 8 0 7 9 . 6 6 80 .64 69.41 .4 4 0 • 5 3 2 7 5 . 9 5 74 .59 74 .19 12 6 2 7 5 . 0 8 73 .44 66 .30 1" 7 n 7 2 . < 0 7 1 . S3 64.82 14 6 4 6 5 . 0 6 63 .05 53 .69 4 3 1 6 0 . 8 2 56 .09 59 .79 12 0 0 5 7 . 3 5 55 .66 55 .19 5 5 2 5 3 . 3 3 52 .42 49.88 8 5 3 4 5 . 0 7 4 6 . 3 3 42 .32 9 4 1 4 4 . 3 8 43 .49 40 .70 11 6 3 4 3 . 9 9 4 6 . 0 5 37.44 10 6 0 4 3 . 5 0 41 .98 37 .07 10 10 0 3 3 2 7 6 5 3 8 . 3 2 38 .36 34 .12 8 6 2 3 7 . 5 3 3 6 . 1 5 32 .97 9 7 6 3 6 . 6 7 38 .87 33.71 6 3 3 3 6 . 3 2 36 .37 34.27 7 7 2 3 5 . 2 0 36 .86 31.62 7 5 4 3 4 . 3 4 33 .61 30 .37 7 4 1 3 4 . 0 9 3 3 . 5 0 32.75 9 3 2 3 3 . 2 6 32 .20 28.82 8 5 1 3 3 . 0 0 33 .66 31.81 6 5 1 3 2 . 9 1 33 .58 31.42 10 7 1 3 2 . 2 7 30 .98 25.22 6 5 5 3 1 . 0 0 3 1 . 7 8 25 .45 2 2 0 29.74 9 6 1 2 7 . 8 2 26 .11 24 .80 12 8 4 2 7 . 5 2 27 .56 21.81 14 6 0 2 7 . 4 8 27 .46 25.63 10 5 3 2 7 . 4 4 2 7 . 9 3 24.51 14 7 1 2 6 . 9 1 25 .35 14 3 1 2 6 . 4 7 26 .99 10 5 1 2 6 . 2 7 25 .74 23 .56 9 6 3 2 5 . 9 9 26 .18 3 2 1 2 5 . 4 9 25 .06 25 .69 8 7 3 2 3 . 2 9 22 .76 9 2 1 2 3 . 2 2 23 .23 19.63 8 6 6 2 3 . 1 9 2 3 . 3 0 18.89 11 4 1 2 1 . 9 6 ' 22 .28 21.25 9 4 3 2 1 . 4 9 2 0 . 1 0 19.35 11 2 1 2 1 . 4 5 19.42 10 3 3 2 1 . 0 3 22 .75 17.98 8 2 0 2 0 . 6 6 20 .68 19.60 7 2 1 2 0 . 5 3 20 .63 17.92 6 2 0 1 9 . 8 2 19.15 19.98 6 6 0 Fe 262.70 249.89 253.85 236 .58 242 .92 243.11 225.82 234.75 227.52 228 .92 2 U . 3 3 201.67 200.85 187.06 188 .38 215.66 209.64 212.59 206.75 211.54 218.32 212.16 215.85 210.64 2 1 4 . 2 7 196.89 186.51 190.51 182.29 182.25 169.79 157.80 155.85 139.99 138 .95 165.94 156.48 163.02 154.72 157 .30 140.49 136.31 135.76 128.93 130.17 127.77 117.49 118.26 109.14 108 .45 135.64 132.02 131.60 127.95 126.17 119.13 110.22 115.57 104.99 106 .17 121.10 114.47 115.85 111.82 110 .30 115.66 109.35 108.36 101.56 101 .93 110.78 102.50 105.49 96 .47 9 9 . 0 0 105.91 100.09 98.85 92 .28 9 2 . 4 8 94.92 86.15 82.98 73 .47 7 1 . 5 3 99.76 83.17 89.35 79 .80 7 9 . 2 9 103.45 106.64 102.77 103.49 101 .54 107.39 101.89 106.85 99 .47 105 .86 88.62 80.45 81.20 70 .23 7 1 . 5 6 85.33 80.66 78.35 70.25 68 .47 86.44 79.81 79.02 73 .43 7 0 . 3 0 87.78 86 .49 85.14 77 .73 7 8 . 1 3 86.87 71.07 76.45 ' 64 .55 68 .34 78 .56 69 .57 70.07 61 .74 6 0 . 6 1 80.58 74 .09 75.66 66 .34 6 8 . 5 8 78.13 63.57 67.58 57 .38 57 .94 77.52 70.89 71.90 60 .87 ' 61 .54 68.03 62.16 59.18 52 .71 5 2 . 3 9 75.73 76.83 73.90 75J13 7 2 . 0 0 70.16 63.28 65.22 60 .09 5 8 . 6 5 67 .84 64 .59 62 .81 69.95 62.44 61.76 55 .56 5 5 . 2 7 77.53 75.46 71.98 70.83 69.90 70.45 6 9 . 3 9 65.57 63 .16 S2.33 55.26 55 .31 66.79 64.43 62 .69 56 .10 5 6 . 9 1 54.15 48.35 48.17 42 .77 4 1 . 6 8 55.42 57 .99 55.64 58 .35 5 5 . 0 0 52.88 46 .25 45 .83 40 .20 39 .32 48.83 48 .78 46 .29 44 .07 4 3 . 7 0 43.31 41 .91 42.71 39 .04 39 .53 40.36 38.84 39.13 37 .43 3 7 . 0 3 39.33 32.78 35.13 29 .95 3 1 . 3 5 36 .49 32.02 32.55 28 .99 28 .84 42 .54 43 .04 39 .40 37 .21 33.63 31 .99 31.22 28 .48 28 .52 32.48 32 .13 30.15 28 .67 2 8 . 4 3 33.17 31.04 29.48 26 .57 27 .44 34.33 34.27 34.14 34 .62 34 .24 32.59 28.33 29.57 28 .30 28 .57 31 .70 30 .63 30.88 29 .51 2 8 . 3 6 31.86 32.73 31.56 30 .65 3 0 . 6 5 28.68 24.97 24.48 25 .56 2 4 . 9 3 32.24 32.06 31.41 31 .12 2 9 . 4 6 30.75 30.62 29.65 28 .87 28 .21 26.22 23 .60 24.98 21 .29 22 .87 26.32 22 .70 23.29 21 .68 2 1 . 9 9 29.46 23.65 18.44 21.25 21 .45 19 .83 22.54 23 .09 24.57 23.85 23.33 22 .29 2 2 . 0 7 21.37 18 .99 18 .65 24.92 26 .43 24.11 24 .94 2 4 . 1 3 19.90 18.80 16 .57 14 .38 20.35 17.44 18 .16 17.63 18.00 16.87 18.50 16 .42 19.95 18.63 18 .22 18.11 19.57 20 .36 16.58 17 .42 18.08 18.41 15.53 16.28 15 .40 20.18 25.10 22.11 21 .45 21 .37 21 .68 21.21 93 h k. 1 4 0 0 8 0 0 8 8 0 8 0 8 6 2 4 6 4 2 4 2 6 8 8 8 4 2 0 4 0 2 8 4 0 8 0 4 4 2 2 4 0 8 6 4 0 4 0 6 6 0 4 12 2 2 6 6 4 6 4 6 10 4 2 10 2 4 16 0 0 4 4 4 8 8 4 8 4 8 14 4 2 8 4 2 4 2 8 8 2 4 14 2 4 10 6 4 10 4 6 10 4 0 12 8 0 10 0 4 12 6 6 12 10 6 12 0 8 10 10 4 8 4 6 8 6 4 6 0 2 6 4 8 6 2 0 12 10 2 12 6 2 12 0 2 12 8 8 10 4 8 12 2 0 12 2 6 12 0 0 10 8 4 14 6 4 10 10 0 6 1 1 14 4 6 16 4 0 16 0 4 12 8 2 12 6 8 12 2 8 16 4 2 12 6 0 12 0 6 12 8 6 14 8 4 14 4 0 16 2 4 14 . 4 . 8 12 10 0 14 0 4 5 2 3 3 2 5 8 2 2 3 3 2 5 1 2 5 2 1 4 1 3 8 4 4 3 3 2 3 2 3 4 3 I TABLE 20. O b s e r v e d and C a l c u l a t e d S t r u c t u r s F a c t o r s o f Che A n d r a d i t e . 2 5 ° C 3 5 0 ° C 5 7 5 ° C 8 5 0 ° C F o ' Fc Fo • Fc Fo • Fc Fo • Fc 2 9 0 . 6 5 284.05 277.73 280.53 284.02 279.34 284.73 276.24 2 7 2 . 2 0 273.61 261.78 266.77 257.94 257.75 247.24 247 .06 2 6 0 . 5 4 263.52 246.60 251.52 241.95 236.66 226.58 220 .85 2 5 8 . 4 9 263.52 249.70 251.52 238.03 236.66 218.41 220 .85 2 3 0 . 0 7 226.22 221.83 222.61 210.48 217.59 204.74 211.84 2 2 6 . 2 2 226.22 221.61 222.61 216.73 217.59 207.12 211 .84 2 2 5 . 3 7 226.22 232.44 222.61 214.88 217.59 209.11 211.84 2 1 0 . 8 7 207.77 203.72 196.51 183.99 180.54 163.80 161.77 1 8 9 . 0 4 185.95 183.07 184.78 190.54 183.24 188.51 182 .20 1 8 8 . 7 3 185.95 173.67 184.78 178.75 183.24 180.25 182 .20 1 8 1 . 7 1 185.52 178.69 182.76 175.77 174.36 166.45 167.22 1 8 1 . 2 1 185.52 182.75 182.76 169.86 174.36 161.21 167.22 1 8 0 . 7 1 181.23 169.50 179.62 173.56 177.92 170.77 175 .67 1 7 7 . 3 5 185.52 189.23 182.76 175.83 174.36 166.01 167.22 1 6 9 . 9 7 171.71 166.23 166.68 164.38 161.37 161.28 155 .36 1 6 9 . 8 1 171.71 172.14 166.68 159.16 161.37 154.08 155.36 169 .12 171.71 164.95 166.68 157.61 161.37 152.93 155.36 168 .54 166.50 159.42 160.90 148.34 151.05 136.02 142.15 1 6 8 . 1 8 166.45 163.37 161.44 154.28 154.44 148.58 147.37 1 6 5 . 2 9 166.45 169.16 161.44 156.88 154.44 149.40 147.37 1 6 3 . 9 2 161.73 150.35 155.05 148.17 146.35 131.99 137.07 1 6 3 . 8 3 161.73 157.00 155.05 142.41 146.35 132.97 137.07 1 4 4 . 2 8 145.81 128.60 132.61 114.59 116.40 100.27 98 .71 1 3 3 . 5 1 128.31 131.04 126.49 124.18 122.26 121.69 117 .40 1 3 1 . 8 5 129.42 128.47 123.29 116.96 115.93 111.20 108.28 1 2 8 . 0 6 129.42 113.42 123.29 117.47 115.93 111.83 108.28 1 2 7 . 8 9 125.92 117.47 117.07 109.55 104.54 90 .90 9 1 . 9 6 1 2 6 . 3 5 126.79 124.09 124.32 118.14 121.22 114.33 114.69 1 2 5 . 5 6 126.79 118.84 124.32 120.10 121.22 115.50 114.69 1 2 5 . 5 0 126.79 126.17 124.32' 120.39 121.22 112.59 114.69 1 2 3 . 4 8 125.92 112.00 117.07 98 .89 104.54 89 .97 9 1 . 9 6 1 2 1 . 5 3 121.75 116.64 114.36 105.97 106.49 100.52 97 .59 1 2 0 . 3 2 121.75 119.27 114.36 107.65 106.49 99 .29 9 7 . 5 9 1 1 8 . 9 5 117.66 112.51 112.28 108.01 104.30 97 .60 9 6 . 6 0 116 .53 114.08 103.74 103.37 94.90 94.01 83 .57 8 2 . 9 8 1 1 6 . 2 5 117.66 114.88 112.28 101.37 104.30 95 .96 9 6 . 6 0 1 1 2 . 5 4 .114.92 109.96 107.73 97..07 37.39 86.61 87 .22 " 1 1 2 . 1 2 109.24 104.13 100.28 88.88 .17.39 79 .66 75 .39 1 1 1 . 3 9 114.08 103.40 103.37 93.86 94.01 83 .31 8 2 . 9 8 l i n . B l H i . 2 5 10.1.11 105.66 92.55 95.28 84 .07 84 .54 1 1 0 . 3 7 107.83 106.60 102.25 98.11 97.46 89 .71 9 1 . 7 3 1 1 0 . 1 6 107.83 108.32 102.25 96.30 97.46 94.27 9 1 . 7 3 1 0 8 . 0 6 107.97 106.44 108.32 107.31 106.72 104.58 106 .13 1 0 7 . 3 9 107.83 94.61 102.25 98.65 97.46 95 .13 9 1 . 7 3 1 0 5 . 7 5 107.97 109.71 108.32 108.51 106.72 108.59 106 .13 1 0 3 . 0 0 100.74 94.52 94.59 86.48 83 .48 72.61 7 2 . 3 9 1 0 1 . 7 9 99 .25 93.72 94.71 88.94 86 .87 79.86 7 9 . 3 6 1 0 1 . 7 2 104.51 95.08 97.67 89.98 93.32 82 .89 8 5 . 1 8 1 0 0 . 4 0 100.19 89.13 91.31 81.44 79.59 69 .45 68;01 1 0 0 . 0 3 99.13 86.71 90.82 84.92 82 .45 74.54 74 .16 9 9 . 6 8 104.51 99.45 97.67 95.00 93.32 87 .54 8 5 . 1 8 98.63 99 .25 99.45 94.71 84.74 86 .87 77 .09 7 9 . 3 6 9 7 . 0 3 95 .66 92.08 89.05 87.57 85.25 76.62 77 .37 9 5 . 9 7 99.13 85.92 90.82 76.22 82.45 69.74 7 4 . 1 6 9 3 . 1 8 93 .66 85.58 85.91 75.17 75.97 63 .87 6 5 . 3 0 9 2 . 4 8 89 .67 91.17 89.80 79.15 80.58 78 .87 75 .19 9 1 . 4 3 91 .50 91.47 92.40 92.09 92 .09 87 .77 8 7 . 1 0 9 1 . 2 2 93 .66 85.05 85.90 71.06 75.97 65 .49 6 5 . 3 0 8 9 . 8 5 91 .19 81.17 83.84 76.30 74.21 61 .40 6 3 . 6 3 8 8 . 9 2 91 .19 84.64 83.84 73.59 74.21 63.54 6 3 . 6 3 8 8 . 8 3 86 .91 85.96 81.65 78.23 74.61 64.32 65 .96 8 6 . 2 6 85.12 79.29 79.01 72.65 71.34 61.27 60 .51 8 5 . 1 5 86 .91 72.88 81.65 74.55 74.61 66 .27 6 5 . 9 6 8 4 . 3 7 79 .68 68.91 72.57 65.00 64 .20 56 .46 5 4 . 2 5 8 3 . 6 6 84 .68 81.36 79.19 76.26. 72.07 68.42 66 .41 8 2 . 6 7 84 .68 84.91 79.19 69.80 72.07 65.97 66 .41 8 2 . 2 9 85 .12 75.31 79.09 70.27 71.34 59.15 60 .51 8 1 . 8 3 75.11 72.91 66.48 65.14 59.18 47 .33 4 8 . 3 0 8 1 . 1 6 79.54 72.39 72.41 64.32 64.43 56.64 56 .11 8 0 . 0 7 79.68 76.74 72.57 65.67 64 .20 52.14 54 .25 7 8 . 5 1 75.11 69.05 66.48 61.47 59.18 50 .93 4 8 . 3 0 7 8 . 3 8 76 .58 71.38 67.73 66.17 62.50 55.26 5 3 . 2 3 7 5 . 6 9 79.54 70.88 72.41 62.57 64.43 54.72 56 .11 6 6 . 3 2 63.78 66.72 63.81 60.80 63.55 59 .89 61 .24 6 6 . 3 1 63 .78 64.45 63.80 62.72 63.54 58.94 61 .24 6 5 . 7 5 ' 67.52 64.50 • 65.97 64.34 62.12 57.14 59 .03 6 5 . 1 9 63 .78 64.00 63.80 65.08 63.54 62.42 61 .24 6 2 . 0 1 60.54 58.40 60.55 61.53 59.30 59 .09 5 7 . 1 3 6 2 . 0 0 60.54 61.83 60.55 63.62 59.30 60 .13 57 .13 6 1 . 5 8 59 .20 62.96 59.35 61.23 59.06 57 .39 57 .96 6 1 . 0 5 59.76 60.30 59.58 53.85 56.29 55 .09 55 .05 6 0 . 9 9 58.75 61.47 57.82 58.80 56.43 59 .99 57 .52 6 0 . 6 8 58.75 62.66 57.82 59.51 56.43 58 .69 57 .52 6 0 . 1 6 59 .20 62.60 59.35 64.66 59.06 61 .20 57 .96 94 TABLE 20. continued. 25 °C 350°C 575°C 850°C h k 1 Po Fc Fo Fc Fo Fc Fo Fc 4 4 8 58.00 59.76 54.61 56.29 53.52 55.05 12 4 0 45.68 45.99 44.42 43.86 43.29 40.99 38.38 37.68 12 0 4 44.47 45.99 40.81 43.86 "39.86 40.99 37.22 37.68 12 4 4 44.42 46.21 41.24 40.51 37.31 38.72 8 3 5 43.21 42.45 41.90 41.52 9 4 1 42.24 43.68 42.13 43.19 41.80 40.34 34.90 38.66 5 3 8 41.54 42.45 39.24 41.52 37.15 38.96 34.28 36.30 9 1 4 41.38 43.68 42.67 43.19 38.94 40,34 10 6 8 38.67 37.83 39.76 36.76 8 5 3 38.63 42.45 41.33 41.52 36.72 38.96 38.12 36.30 6 6 0 36.62 37.06 40.52 38.30 34.96 36.23 36.46 36.11 10 S 6 36.18 37.83 39.77 36.76 36.84 33. IS 7 4 1 35.62 34.54 33.92 33.60 33.64 33.29 32.29 32.77 6 6 8 35.03 34.44 33.43 35.53 30.45 32.10 10 2 8 34.57 36.96 32.07 36.43 30.00 33.93 32.77 32.61 6 8 6 34.50 34.44 37.29 35.53 33.56 32.10 9 7 6 33.73 32.98 35.06 32.57 32.45 31.84 11 3 6 33.33 32.30 7 1 4 33.15 34.54 4 1 7 33.00 34.54 8 1 5 32.83 32.04 5 4 7 32.54 31.63 5 1 8 32.53 32.04 29.40 31.93 27.28 30.98 29.21 30.76 7 4 5 31.82 31.63 11 6 3 31.75 32.30 32.31 32.09 8 6 6 31.60 34.44 6 3 3 31.24 29.90 31.83 32.06 14 0 2 30.78 28.65 30.78 30.06 27.97 26.21 27.43 25.37 7 5 4 30.63 31.63 8 5 1 30.50 32.04 30.81 31.93 28.77 30.98 29.31 30.76 10 8 2 37.71 36.43 9 6 7 29.65 32.98 3 3 6 29.23 29.90 7 7 2 29.17 28.49 24.97 26.89 14 2 0 28.98 28.65 6 3 5 28.57 25.57 11 4 7 .28.47 26.49 7 6 5 27.99 27.11 7 6 1 27.98 27.40 25.75 26.29 26.99 24.12 27.13 26.81 3 2 5 27.47 26.97 7 2 7 27.29 28.49 11 1 4 27.13 26.93 7 S 6 26.94 27.11 6 5 3 26.83 25.57 5 5 2 26.69 26.97 25.51 25.55 24.80 24.79 5 3 6 26.61 25.57 9 3 4 26.21 24.91 7 1 6 25.46 27.40 6 5 7 25.35 27.11 11 4 1 25.24 26.93 6 1 7 25.09 27.40 9 4 3 24.75 24.91 8 7 3 24.52 23.61 4 4 0 23.35 23.74 20.68 23.79 23.45 23.96 24.17 24.09 11 5 8 22.84 21.44 7 3 8 22.07 23.61 8 6 2 21.72 20.21 9 3 6 21.34 18.94 9 3 2 21.24 21.67 6 5 S 21.19 21.31 6 2 8 21.10 20.21 14 3 1 20.56 19.41 10 7 1 19.94 21.78 9 6 3 18.79 18.94 3 4 3 17.99 18.14 4 3 5 17.76 18.14 6 5 1 16.13 14.87 7 5 8 16.07 16.96 4 3 7 15.04 14.12 14 8 2 ' 14.62 13.72 5 1 6 14.52 14.87 

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