I‘ Burl-V k 37:- \i-Uhd - . ~10 F" J ' , ' 0.9:: V lmigfiv' ‘ ' ‘0 fl'luf'a‘ I 3 ' "l . 5" - ‘2 < a' < :‘J _.._.y—‘ 1 '15} M'l'. ' ' .! ,f'br'u' .1'2 .. in ' Til/{*L‘Wi: 4-; ' - _\_ , H; .m I, ”641‘" ' "I -.. . w my n I ‘ 9§.(h Iii-c " I)- 71: 1 *4 .17).: . . #3:)”; THESiS r --—' ”W3”! :1?" "-r T.’J’._‘-..'\! { i . 4 a r FifiCLZ 1 I téfite i Umveraswy x.— J This is to certify that the thesis entitled I. CRYSTAL STRUCTURE OF A 2:1 COMPLEX OF 12-Crown—4 WITH SODIUM PERCHLORATE and II. PREPARATION AND CRYSTAL STRUCTURE OF (Sr/Y)Cl AND (Sr/Yb)C1 p%é8()6%ted by 2+x Eileen Mason has been accepted towards fulfillment of the requirements for Ph . D . degree in Chemistry \Zffi 2;] a £61 ‘ ajor professor Date VZ/[m [637 l 0-7639 W {16’ \~-1“‘ III” ‘ 'II’ OVERDUE FINES: 25¢ per day per item RETURNING LIQRARY MATERIALS: Place in book return to move charge from ctrcuiet‘lon records I. CRYSTAL STRUCTURE OF A 2:1 COMPLEX OF lZ-Crown-U WITH SODIUM PERCHLORATE and II. PREPARATION AND CRYSTAL STRUCTURE OF (Sr/Y)C12.05 AND (Sr/Yb)Cl2+X By Eileen Mason A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemistry 1981 ABSTRACT I. CRYSTAL STRUCTURE OF A 2:1 COMPLEX OF l2-Cr0wn—U WITH SODIUM PERCHLORATE By Eileen Mason The crystal structure of Na(C8OUH16)2ClOu has been determined by a single crystal X-ray diffraction study. The crystal belongs to space group P21/a with lattice parameters a = 15.1120 (8) K, g = 15.2145 (10) K, _c_ = 9.650 (5) A, §’= 92.13 (N)°. The intensities of “027 reflections were measured on a Picker-FACS-I automatic diffractometer (Mo Ka radiation) and refined after initial positions had been indicated by MULTAN by alternating least squares cal- culations with subsequent difference maps. Full matrix least squares refinement of non-hydrogen atom positions and anisotropic thermal parameters, and isotropic hydrogen thermal parameters converged at B1 = .071 for lUOl reflec- tions with |F2| greater than 30 (F2). The sodium ion forms a sandwich structure with two 12C“ rings. The eight-fold coordination at the metal is approximately square anti— prismatic. The perchlorate is severely disordered. Cation- anion interaction appears to be minimal. Eileen Mason II. PREPARATION AND CRYSTAL STRUCTURE OF (Sr/Y)Cl2.05 and (Sr/Yb)Cl2+X Recent developments in solid state chemistry have focused on the detection of long range order in compounds of complex stoichiometry. A series of anion excess species of composition Mnx2n+l which exhibit the Vernier struc- ture has been identified for certain rare earth ions and combinations of rare earth ions with strontium or yttrium. Attempts to prepare a similar Vernier-type species con- taining only strontium and yttrium have resulted in a dif- ferent type of anion-excess structure. An apparently iso— morphous strontium-ytterbium analog has also been prepared. The compounds form colorless crystals which maintain the SrCl2 unit cell with Z = u. The crystal structure was refined only for the yttrium compound which has cell parameters a = 9_= g,= 6.967 (1)3. The intensities of 26u5 reflections were measured on a Picker FACS-l automatic dif- fractometer (Mo Ka radiation) and initial positions indi- cated by a Patterson synthesis were refined by least squares calculations with subsequent difference maps. Two struc- tural solutions are suggested. In the proposed vacancy model, the extra Y3+ charge is balanced by the simultaneous removal of a Sr2+-Cl- ion pair from the SrCl2 structure. The individual ion sites are only partially occupied and are not equivalent. The Eileen Mason inequivalence of the sites strongly suggests that some ordering of the vacancies occurs, but the exact nature of the ordering cannot be determined. Refinement in space group P23, which specifies that the anion sites will form two distinguishable interpenetrating tetrahedra, yields a residual value of R = .1013 for 218 reflections with [F]2 greater than 0 (F2). Refinement in space group P1, which allows all sites to refine independently, leads to a residual value R = .0873 for the same data set. The second model describes the structure in terms of a Willis cluster of defects and includes both anion vacancies and interstitial anions. Full-matrix least squares refine- ment in space group Fm3m, with positions analogous to those in U02.13 and (Ca/Y)F2.10, converged at R = .0633 for 11“ reflections with |F|2 greater than 0 (F2). Refinement of all data, including reflections inconsistent with face centered symmetry converged at R = .1126 for 11h reflections with IFI2 greater than 0 (F2). It is believed that these additional mixed index reflections indicate ordering of the defect clusters. But and and the love science — they are both gambles; if you try to win sun’s light, you must be prepared also to lose every day. Diane Wakoski, "In This Galaxy" ACKNOWLEDGMENTS Above all, I wish to thank Dr. Harry Eick for his unflagging confidence and optimism. It is largely due to his influence that I returned to graduate school at all, and, without his encouragement my hopes would surely have succumbed to the pitter patter of a myriad little defeats. I want to extend my appreciation to Dr. A. I. Popov for providing the crystal of Na(12—crown-H)2010u and for helpful discussions. I am grateful to Dr. W. E. Braselton and Steven Ton- sager for chemical analyses by induction coupled plasma spectrometry. The results of the analyses aided me im- measurably in this study. I would like to thank Dr. Donald Ward for the data collections and his assistance with the crystallographic calculations. I acknowledge the Department of Chemistry of Michigan State University and the National Science Foundation for financial support throughout my graduate career. Lastly, I want to thank the large number of Friends and Relations who have extracted me, like Pooh, from many a Place of Great Tightness. ii TABLE OF CONTENTS Chapter LIST OF TABLES. . . . . . . . . . . . . . . LIST OF FIGURES CHAPTER I - BACKGROUND MATERIAL IN SYMMETRY AND CRYSTALLOGRAPHY . . . . . . Reciprocal Space. . . . . . . . . . . Methods of Data Collection. The Precession Method Data Collection with an Automated Diffractometer. . . . . . . . . . Transformation of Unit Cells. Data Reduction. Obtaining a Trial Structure from a Patterson Map . . . . . . . . . Obtaining a Trial Structure by Direct Methods. . Refinement of the Trial Structure CHAPTER II - CRYSTAL STRUCTURE OF Na(CBOqu6)2ClOu . . . . Introduction. Experimental. Structure Solution and Refinement. . . . Results and Discussion... 111 Page vii ll 16 23 3O 32 38 HO HO U2 U2 149 Chapter Page CHAPTER III - PREPARATION AND CRYSTAL STRUCTURE OF (Sr/Y)C12. 05 and (Sr/Yb)C12+x. . . . . . . . . 59 Introduction. . . . . . . . . . . . . . . . . . 59 Experimental. . . . . . . . . . . . . . . . . . 69 Preparation of (Sr/Y)Cl2+X and (Sr/Yb)012+. . . . . . . . . . . 69 Data Collection and Refinement of (Sr/Y)Cl2+x. . . . . . . . . . . . . . . 73 Preliminaryx Investigation of (Sr/Yb)012+. . . . . . . . . . . . . 99 Results and Discussion. . . . . . . . . . . . . 105 REFERENCES. . . . . . . . . . . . . . . . . . . . . 116 ANNOTATED BIBLIOGRAPHY. . . . . . . . . . . . . . . 123 APPENDIX. . . . . . . . . . . . . . . . . . .'. . . 126 iv Table 10 LIST OF TABLES Crystal Data. . . . . . . . . . . Final-cycle Refinement Indicators Positional and Thermal Parameters with Associated ESD's Bond Distances and Angles Crystal Data for (Sr/Y)C12.05, First Data Collection . Results of Rhombohedral Refinement Positional and Thermal Parameters with Associated ESD's Crystal Data - Second Data Col- lection . Comparison of Refinement in Space Group Pm3m Using Both Data Col— lections. . . . . . . . . . . . . Discrepancy Indices for Refine- ments with Different Cation:Anion Occupancy Ratios. . . . . . . . . Results of Chemical Analysis by Induction Coupled Plasma Spectrometry. Page “3 A7 A8 52 75 77 81 85 91 9A Table 11 12 13 1A The Vacancy Model Results of Refinement. . Structural Parameters of Average Cells The Willis Cluster Model Results of Refinement Percent Occupancy of Zr Sites in Zr0.77S- vi Page 95 98 100 109 Figure LIST OF FIGURES Page Relationship between the direct and reciprocal lattice axes . . . . . . . . A Above: The reciprocal lattice and the sphere of reflection. Below: The direct plane. Figure redrawn from Stout and Jensen, see Bibliography. . . . . . . . . . . . . . . . 6 (a) A zero—level plane of the reciprocal lattice whose normal, t, is precessing about the direct X-ray beam with precession angle u. Redrawn from The Precession Method, Buerger. See Bibliography. (b) Reciprocal lattice plane, tangent to sphere of reflection, and intersecting the sphere of reflection. Redrawn from Stout and Jensen. (c) Film parallel to reciprocal lattice plane showing circular trace of circular intersec- tion with sphere of reflection. Re- drawn from Stout and Jensen . . . . . . . . 10 V11 Figure 10 Page A four-circle diffractometer. . . . . . . . 12 Scan mode geometry. Redrawn from Stout and Jensen. . . . . . . . . . . . . . 12 The sodium-dicrown unit illustrating 50% probability ellipsoids and the atom labeling scheme. H-atoms are designated by the number of the carbon atom to which they are attached . . . . . . 50 The packing in Na(120A)2ClO¢ Sodium ions are located at (0.5 i 0.0076); chlorine atoms at (i0.00B3).. . . . . . . . 55 The severe distortion of the per- chlorate moiety is illustrated by the presence of seven partially-occupied oxygen atom positions surrounding the chlorine atom . . . . . . . . . . . . . . . 56 Clusters in (Ca/Y)F2+x. Above: 2:2:2 cluster, two F' ions, two F" ions and two normal ion vacancies. Below: an extended 3:A:2 cluster with two F" ions and A F' ions. Y3+ positions assumed in calculating diffuse scattering. Redrawn from Reference A8. . . 62 The 5/11, 6/13 and ll/2A Vernier structures. . . . . . . . . . . . . . . . . 65 viii Figure 11 12 13 1A 15 16 17 Page (a) The cubic fluorite unit cell. (b) Tetragonal basis cell derived from the fluorite cell. (c) Rhombohedral basis cell derived from fluorite unit cell. (d) The rhombohedral basis cell and suggested superstructures. (Figure redrawn from Reference 57.). . . . . . . . . . . . . . . 66 Crystal mounting apparatus. . . . . . . . . 72 Zero layer precession photographs of (Sr/Y)Clz+x. Above: An apparent six-fold axis. Below: A pattern of triplets. . . . . . . . . . . . . . . . . . 7A Relationship between the axes of the rhombohedral “ and cubic — unit cells. . . . . . . . . . . . . . . . . 80 Imax/I vs ¢ for two representative azimuthal scans. No absorption cor- rection has been applied. . . . . . . . . . 83 A zero layer precession photograph showing two of the cubic axes . . . . . . . 10A Above: 2:2:2 Willis cluster con- taining two normal anion vacancies ( D) two X' interstitials ( 0), two X" interstitials (. ), and cubic (%,%,8) sites (+). ix Figure 17 18 Page Below: A 3:A:2 Willis cluster. Figure redrawn from Reference A7. . . . . . 107 Illustration of the atomic positions in the monoclinic superstructure, space group C2/m. The monoclinic unit cell is shown by the heavy lines. The NaCl-type ZrS cells are shown as thin—lined cubes. Re— drawn from Reference 57 . . . . . . . . . . 109 CHAPTER I BACKGROUND MATERIAL IN SYMMETRY AND CRYSTALLOGRAPHY The first portion of this dissertation is designed to stand apart from the remainder of the work. It is intended to provide a brief and qualitative overview of crystallo- graphy and symmetry. For the sake of brevity, the list of topics is eclectic; for the sake of simplicity, mathemati- cal derivations have been excluded whenever possible. Because of the generalized approach, references have been deliberately omitted. The reader is directed to the appended bibliography for sources of more detailed analysis. Although it is not complete, the bibliography is annotated to provide a quick guide to the type and complexity of information contained in each work. RECIPROCAL SPACE The experimental data required for X—ray analysis of a crystal structure consist of measurements of the direction of scattering of the X-ray beam by the crystal and the intensities of these diffracted beams. The first value reveals the size and shape of the unit cell. The second value may be analysed to give the positions of atoms within the unit cell. X-ray diffraction from the planes of a crystal occurs when n1 = 2dsin6, where A is the wavelength of the incident X-ray beam,ciis the interplanar spacing, and 29 is the angle between the incident beam and the diffracted beam. Bragg's Law which describes the conditions for diffraction is more easily visualized in reciprocal space than in direct space. For each cell in direct space, a corresponding cell in reciprocal space can be constructed. The relationship between the direct and reciprocal lattices requires that for a, b, c, vectors in real space, and a*, b*, c*, vectors in reciprocal space, a* is perpendicular to b and c, etc., while a is perpendicular to b* and c*, etc. The reciprocal lattice can be generated by designating one direct lattice point as an origin and defining the normals from this point to all possible (hhfi) direct lattice planes. The reciprocal lattice points will be positioned along these vectors at a distance l/dhh£ from the origin if dhh£ is the perpendicular distance between the direct lattice planes of the set (th). The indices hhl will define positions in reciprocal space in the same way that the coordinates xyz define positions in direct space. For orthogonal systems, a=8=Y=a*=B*=y*=90° and the reciprocal space axes are coincident with the axes in direct space. For radiation of wave length A, the translation in direct space, 1 is related to a translation in reciprocal uvw’ space, by the relationship a dhhz’ * . UUW The axes of the unit cell in direct space are simply special cases of translation: - _. * a - £100 ' A(]/d 100) b = ‘010 = M1”‘0101 c = 1001 ‘ “(7/d*001’ For non-orthogonal systems, the situation is more complicated because the reciprocal lattice axes will not be coincident with the direct cell axes. The interplanar distance measured along the normal may be different from the interplanar distance measured along the cell axis. For example, if the angle between a and c is 8, then leO = a sin 8 = a sin 8* (B*=l80°—B) * = *=—_—— and leO a a sinB 001 100 Figure 1. Relationship between the direct and reciprocal lattice axes. Note that for 90° angles, sin 8 = l and a* = %. Consider a reciprocal lattice plane in a crystal which is bathed in a beam of X-rays of wavelength A. The beam is parallel to this plane, and is indicated by a line (X0) passing through the origin (0) of the reciprocal lattice. A unit circle in direct space corresponds to a circle of radius 1/1 in reciprocal space. Such a circle, with radius l/l passes through 0. Its center (C) falls on line XO. P is any reciprocal lattice point located on the cir- cumference of the circle. (Figure 2) Angle OPB will be equal to 90° because it is inscribed in a semi-circle, and sin OBP = sin e = 8% = 925x A. Rem- ember that P is a reciprocal lattice point. Therefore, OP equals l/dhhi’ and from the Bragg equation (l/n°d )X sin e = 2"“ = [(1/2 dhizflfll. Through this derivation, we see that Bragg reflections occur whenever reciprocal lattice points fall on a circle of radius l/A which is tangent to the origin of the recipro— cal lattice. Diffracted Beam The direct plane. Figure redrawn from Stout and Jensen, see Bibliography. The reciprocal lattice and the sphere of Below: Above: reflection. Figure 2. METHODS OF DATA COLLECTION Measurements of the direction of scattering and the intensity of the scattered beam may be made photographically or electronically. The methods are complementary, for, al- though a diffractometer with a scintillation counter provides numerical, quantified information, film techniques are fre- quently more reliable guides to the symmetry of the cell, since the scintillation counter records only one reflection at a time and must be specifically positioned for each reflec- tion. Photographic techniques record an entire plane of the reciprocal lattice on a single film and minimize the chance of overlooking reflections. Furthermore, the film is tan- gible and easily visualized, while the diffractometer pro- duces pages of numbers. In this study, photographic data collection by the precession method preceeded electronic data collection with an automated diffractometer. Measurement of the films gave approximate values for the cell parameters, but the major benefit of the technique was in the identification of the cor- rect space group. The relative intensities were considered only in relation to the symmetry properties. The Precession Method In the precession method, the incident X—ray beam is parallel to a direct lattice axis and perpendicular to a plane in the reciprocal lattice which contains 2 axes. This layer is tangent to the sphere of reflection at the origin. Arc settings on the goniometer head are used to bring the crystal into proper alignment. The crystal is rotated at an angle, u, about an axis perpendicular to the beam. Movement of the crystal is such that the direct axis vector now revolves about the beam, maintaining constant angular separation of u°~ The recipro- cal lattice zero layer now cuts the sphere and the inter- cepted circle precesses about the origin as the direct lattice axis precesses about the incident beam (Figure 3). Additional reflections from other layers are intercepted by a metal screen. The film also precesses and always remains parallel to the reciprocal lattice plane. Reflections are recorded when a reciprocal lattice point falls on the intersection of the lattice layer and the sphere of reflection. Upper layers of the reciprocal lattice are recorded by moving the film closer to the crystal by a distance corresponding to the measured spacing of the rows. Each precession layer photo- graph gives an undistorted image of all reflections in a particular plane of reciprocal space. Measurements of the precession film, when scaled, give the cell parameters of the reciprocal lattice. Since the symmetry of the reciprocal lattice will be the same as the symmetry of the direct lattice, examination Figure 3. a) b) c) A zero-level plane of the reciprocal lattice whose normal, t, is precessing about the direct X—ray beam with precession angle u. Redrawn from The Precession Method, Buerger. See Bibliography?— Reciprocal lattice plane, tangent to sphere of reflec- tion, and intersecting the sphere of reflection. Redrawn from Stout and Jensen. Film parallel to reciprocal lattice plane showing circular trace of circular intersection with Sphere of reflection. Redrawn from Stout and Jensen. 10 sphere of reflection plane of zero layer plane of r.l. points Figure 3 11 of a properly indexed precession film will reveal the space group of the unit cell. It should be noted here that examination of only zero layer films may be misleading if the apparent symmetry of the zero layers is higher than that of the cell as a whole. In the zero layer, two—fold axes and mirror planes are in- distinguishable. Furthermore, centering or other symmetry conditions may lead to the systematic extinction of certain classes of reflections. For example, in a face-centered cell, h,k and 1 must be all odd or all even. Hence, the HKO layer will contain only even values of hand h. This will not be obvious, however, unless the HKO plane is com- pared to an HKL plane where L is odd and which contains only odd valued h's and h's. Failure to recognize systematic extinctions is a major cause of improper identification of the space group. It is not possible to solve a structure accurately, if at all, if an axis of the unit cell is erron- eously shortened to a fraction of its true length. Data Collection with an Automated Diffractometer The four circle diffractometer has four arcs, each with 360° of freedom. The crystal orienter which contains the o and x circles is mounted within the w circle and rotates with it. The 29 axis is colinear with w- When X = 0°, the axis of 9 is coincident with 20 and w. The crystal is Figure A. A four-circle diffractometer. 26 constant a) An w scan. /’ . [I <:::tounter b) A 6—29 scan Dcounter starting position final position Figure 5. Scan mode geometry. Redrawn from Stout and Jensen. l3 mounted on a goniometer head which has a height adjustment and may have are settings. The are settings are not re- quired (Figure A). The X-ray detector is mounted on the 20 circle. The source, which is fixed, the detector, and the center of the orientation assembly lie in a plane perpendicular to the m and 26 axes. Because of this restriction, it is neces- sary to reorient the crystal —- generally using the ¢ and X circles -- to bring the desired reciprocal lattice point into contact with the sphere of reflection in the source/ center/detector plane. After reorientation of the crystal, the detector is moved along the 29 circle until it is situated properly to record the reflection. The values of X: c, w and 20 must be determined and individually set for each reflection to be recorded. Both the cell constants and their orientation relative to the ¢ axis must be known; the angular settings are highly inter— dependent. A diffractometer is automated by being inter- faced directly with a computer which calculates settings as needed, drives the encoder motors to the preselected angle setting, and records the angular and intensity data. On the system used in this work a PDP8/I computer was inter- faced with a Picker diffractometer. An actual reflection is not a point in reciprocal space but has finite dimensions. To measure the reflections, the detector must scan through a small region of space which 1A encompasses the reflection. Two scanning modes are in com— mon usage: In the w scan, the crystal -- and hence the reciprocal lattice -- is rotated by the w circle so that lattice points pass from the outside to the inside of the sphere of reflection while the detector is stationary at the precalculated 2e setting. The detector observes a region of space along an are centered at the lattice origin and passing through the reciprocal lattice point. In the 6—20 mode, the counter moves at an angular rate twice that of the crystal, scanning along a portion of the straight line connecting the lattice point and the origin. This is approximately perpendicular to the arc traced in the w scan (Figure 5). The advantage of one scan mode over the other depends on the nature of background intensity as much as the shape of the reflections. A common technique for measuring back- ground is to obtain counts with the detector stationary at the beginning and end of the scan range, average the values obtained, and, after correcting the rate to the actual scan time, to subtract the background from the net counts of the reflection. If the background is uniform, either scan mode will be acceptable. An w scan however is perpendicular to the streaks caused by white radiation and will not reveal this increase in background intensity, while in the 6-29 scan mode, the detector moves along the radiation streak and will give a more reliable evaluation of the intensity of the reflection. 15 The use of crystal monochronometers has greatly reduced the problems of white radiation, reflecting almost pure Kal and KO!2 radiation. At small diffraction angles, the “1'02 divergence is small and a restricted scan range will encompass both peaks. At higher values of sine however, KO"1 and Ka2 broader and more diffuse. The scan range must therefore be gradually diverge, and reflections tend to be increased. If an extended range is used for all reflec- tions, then the intensities of the low—e reflections may be inflated by integration over too great a width. For this reason, the computer varies the scan width with 29 so that an optimum scan range will be employed for all reflections. Variations in background intensity due to the diffrac— tion of white radiation are systematic and predictable, but other factors may complicate the separation of background and reflection intensity. Scattering of X-rays by air, the mounting materials, protective capillaries and amor- phous materials present with the crystal may lead to random and unpredictable variations in background intensity. The crystal support and amorphous material may also absorb X- rays and lead to reduced intensity of the diffracted beam. Since it is difficult to correct for these influences, they should be minimized. A beam collimator will lessen air scattering and use of the thinnest possible mounting and capillaries will reduce both scattering and absorption. The quality of the crystal itself will also affect the l6 resolution of the reflections; a flat, diffuse reflection will not be easily separated from the background. Broad and split reflections and reflection peaks with shoulders are generally due to crystal flaws. A small fragment at- tached to the main crystal will lead to additional reflec tions. Depending on the intensity and location of these reflections, the main crystal may be unsuitable for analysis. Real crystals are rarely perfect but exhibit a mosaic structure composed of small, slightly misaligned crystal- lites. The degree of misalignment, the mosaic spread, af- fects the shape of the reflections: when it is too great, reflection peaks are excessively broad, but when misalign- ment of the crystallites is less than 0.5° it is actually beneficial, helping to maintain the same magnitude of extinc- tion for reflections of both high and low intensity. A discussion of extinction will be included in the data reduc- tion section. TRANSFORMATION OF UNIT CELLS It is possible that the space group initially identi- fied may be lower in symmetry than the true space group. In general, however, this will not interfere with solution of the structure, since a higher symmetry cell of the same volume can be found analytically, or the higher symmetry space group may become apparent as refinement proceeds. 17 When a different choice of unit cell is made, it is necessary to re-index the data in terms of the new unit cell. It is possible to use matrix multiplication to trans- form direct lattice vectors and the indices of planes or individual reflections from one cell to another. A related matrix will transform reciprocal lattice vectors, coordin- ates of positions in the unit cell, and zone axis symbols. The first step in interconverting two unit cells is to define the unit vectors of one in terms of the other. Per- haps the easiest way to do this is to draw a picture of the two cells and verify the relationship geometrically, and to write a series of equations of the type: a2 = sllal + S12b1 + Sigcl b2 = 82131 + s22bl + S23°1 C2 = S3131 * S32b1 * S33C1 that is: a1 811 S12 S13 a2 b1 S21 S22 S23 = b2 18 To convert the unit vectors of cell 2 to cell 1, the inverse matrix must be used, a2 811 S12 b2 821 S22 c2 S31 832 Similarly, the indices hihizi Liv: —1 S13 a2 f11 +12 S23 = b2 +21 +22 s33 c2 +31 +32 +13 al +23 = bl +33 c1 are transformed to h.h.£.. J J Transformation of reciprocal cell vectors requires not only exchange of the S and T matrices, but also interchange of the rows and columns within each matrix. ‘1' a1 +11 1' b1 1‘12 * c1 +13 and * 32 S11 * b2 812 * c2 S13 +21 +31 32* 1“.22 1‘32 = ”2* +23 +33 C2* S21 S31 al* S22 S32 = bl* S32 833 c1* Thus, 19 These matrices are used to convert the positions xigiz£ to xjyjzj. Again, the two conversion matrices share an inverse relationship. Interconversion is straightforward, provided that the Sij and +13 elements are all integral. Since 8 and T are inverses, this condition is satisfied only if the elements are equal to :E1 or 0. This will be true if the new cell has the same volume as the old cell. Transformation becomes more complex, however, when fractional matrix elements are required to transform the cell originally chosen into one with a different volume. Although the multiplication of matrices containing non—integral elements is well defined mathematically, physi- cal interpretation is ambiguous, since h,h and i are required to be integral. As an example, consider the interconversion of two of the possible hexagonal and cubic cells considered for (Sr/Y)C12+X. ah = -l/2 acu + 1/2 ccu bh = 1/2 aCu — 1/2 bcu ch = aCu + bcu + ccu This leads to the matrix relationship: 20 ah -2/3 2/3 1/3 acu bh -2/3 -u/3 1/3 = bcu ch u/3 2/3 1/3 ecu Consider now the transformation of the hexagonally- indexed reflections (001), (002), and (003) to their cubicly- indexed counterparts: 0 —2/3 2/3 1/3 1/3 0 -2/3 -A/3 1/3 = 1/3 1 u/3 2/3 1/3 1/3 0 -2/3 2/3 1/3 2/3 0 -2/3 —A/3 1/3 = 2/3 2 A/3 2/3 1/3 2/3 -2/3 2/3 1/3 1 -2/3 -A/3 1/3 = 1 A/3 2/3 1/3 1 21 If we clear the fractions by multiplying each set of indices by the lowest common denominator, we find that (001)h and (003)h, two parallel planes, are now both repre- sented by the same cubic indices, (lll)c. The multiplication has forced the planes to become equivalent. If, in order to avoid this pitfall, all the indices are multiplied by the same factor, we find that 1 -2/3 2/3 1/3 1/3 1 h cu _ E} - 1h -2/3 -A/3 1/3 - -5/3 5cu 1h A/3 2/3 1/3 7/3 7cu Since the original transform matrix was obtained geomet- rically, multiplication by the inverse of this matrix should restore the indices (lll)h. However, this is not the case: —1 1cu -2/3 2/3 1/3 1cu -1/2 0 1/2 3h scu -2/3 —A/3 1/3 = 5cu 1/2 —1/2 0 = 3h 7Cu u/3 2/3 1/3 7cu 1 1 1 3h Although multiplication of the entire set of indices main- tains the separate identities of parallel planes, it re- defines the unit cell. If the unit cell parameters have been determined cor- rectly, it is a straightforward procedure to transform 22 this unit cell to an equivalent cell of higher symmetry, provided such a cell exists. However, if certain classes of reflections have been overlooked, leading to an incor— rect unit cell determination, it may not be possible to reindex the reflections in terms of the correct cell. The outlook for successful transformation is especially bleak if the original cell is too small and both the length and direction of the axes are to be changed. Upper layer precession photographs are crucial to proper space group identification and may be required for the cor- rect determination of the unit cell axis lengths. If upper layer photographs are to be obtained, the crystal must be oriented with a principal axis parallel to the X-ray beam. Unfortunately, the arcs of the goniometer head are limited to adjustments of i25°, and it may be necessary to remoun: the crystal to achieve a suitable orientation. Remounting a crystal is frequently such a tedious task that it is im— possible in actual practice. This problem does not arise with a four circle dif- fractometer which has full freedom in all planes. With this instrument, difficulties in identifying the proper unit cell arise because reflections can be overlooked easily and this can lead to incomplete data collection. The correct directions of the axes may be identified, but the assumed lengths may be a fraction of their true values. Such a data set can be reindexed successfully, but it will contain "holes" 23 and a satisfactory structural solution may not be obtainable until the missing data have been collected. DATA REDUCTION Since X-rays and crystals are both periodic, the pattern of diffraction will also be periodic and hence can be des- cribed by a Fourier series in three dimensions. Fhkl the structure factor for the unit cell for the reflection (hh£) is given by = X f 2 h +k +1 + i X f i 2 h + + Fhkfi {J loos n( xJ yJ ZJ) j Js n w( x3 kyJ lxJ = X f ei2n(hxj+kyj+lzj) J J where f is the scattering factor of the 1th atom and 3 x3, yJ and 23 are its fractional coordinates. The structure factor can also be written: Fhkl = § chos ¢J + i stin p3, where ¢j is the value of the phase scattered by the jth atom. Unfortunately, the fractional coordinates cannot be cal— culated directly since the phase angles are not measurable 2A quantities and must be surmised from the intensity data. As in all wave phenomena, the intensity of the wave is proportional to the square of its amplitude and the in- tensity of each reflection is a measurable quantity. The result of a collection of X—ray diffraction data is a list of intensities and scattering angles identified by indices h, k and f. While the angular parameters are determined by the dimensions of the lattice, the intensities are governed by the nature and arrangement of the unit cell contents. After all background has been removed, the intensity of each reflection will be given by: I = K|F0|2(Tv)(Lp)(Ab) where K is a scale factor, Tv is related to the thermal motion of the atoms, Lp is a geometric effect and Ab includes absorption and extinction corrections. Lp, being proportional to l/cose, is totally independent of the cell contents. Calculation of the absorption correction may be aided if the number and types of atoms in the unit cell is known, but no knowledge of atomic positions is required. Absorption reduces the intensity of reflections. In any absorption phenomenon, the intensity, I, of a beam passing through an absorber of thickness T is given by 25 I = IOe-“T, where I0 is the intensity of the incident beam and u is the linear absorption coefficient. The absorption coefficient increases rapidly but not linearly with increas- ing atomic number, and is also dependent on the wavelength of the radiation used. Mass absorption coefficients, (u/pa where p is density) for most elements and most common radiations are listed in the International Tables for X-ray Crystallography, vol. III and absorption coefficients for compounds can be calculated. For a compound of density p, made up of Xn% of element En’ the linear absorption co- efficient for radiation of wavelength 1 is given by uA = p g (Xn/IOO)(u/p)A,En If uncorrected, absorption serves to reduce the thermal parameters to values less than the true values. Absorption effects are most noticeable for reflections collected at low values of sine. If the crystal is spherical, absorption varies exponentially with sin2e. For irregularly shaped crystals, the path length of the beam through the crystal, 1, will vary with the orientation of the crystal relative to the incident beam. In this case, it is possible to correct for absorption by calculating the path length through the crystal of the beam diffracted from every infinitesimal portion of the crystal and then inte- grating over the entire crystal volume. The crystal is 26 described in terms of the planes which comprise its faces and the distance of these planes from an arbitrary center. Calculation of a path length for each reflection is tedious and not exactly straightforward. Computer programs can apply such an analytic correction to every reflection. Unfortunately, difficulties arise in the cases where absorption effects are the greatest: in crystals containing heavy atoms. In these cases, where the greatest portion of the diffracted beam may be reflected by the outermost planes, the calculation of the pathlength will be extremely complex. An analytic correction may be not merely nonproductive but actually counterproductive if the calculated path length does not correspond closely to the actual path length. The attempted correction may increase the intensity differences of equivalent reflections rather than reduce the differ- ences. The best alternative may be to use an empirical absorption correction based on azimuthal scans of individual reflections. In such a correction, the reduced intensity of a reflection is multiplied by an appropriate factor to restore it to its original value. No knowledge of the unit cell contents is required. If the azimuthal scan is made for a reflection at X = 90°, the magnitude of the correction factor will depend on the values of ¢ and 20 and will simultaneously correct for absorption by mounting materials. An empirical correction adjustable for all four setting parameters, ¢, x, m and 29 would be even better. 27 Extinction is another process which results in an at- tenuation of intensity of a beam passing through a crystal. It arises from the geometry of diffraction. Each ray re- flected at a Bragg angle can also undergo a second reflec- tion back into the primary beam. Since a phase shift of n/2 occurs with each reflection, the twice—reflected beam differs in phase from the primary beam by n, and the triply- reflected beam traveling in the direction of the Bragg reflection differs in phase from the once-reflected beam by n. Because of the phase shift, addition of rays re- flected n times and rays reflected (n-2) times leads to a reduction of intensity. In a perfect crystal, the intensity of the diffracted beam would be proportional to IFOI and not |FO|2. Correction factors can be calculated for situations in which primary correction is significant, but mosaic struc— ture prevents the extension of perfect planes over appre— ciable regions of space and primary extinction is usually negligible. Secondary extinction, however, is frequently encountered and often appreciable. It arises for intense reflections when such a large percentage of the incident beam is re- flected by the very first planes encountered that the deeper planes are subjected to a lower intensity and hence produce reflections of lower intensity than expected. A large crystal will suffer more from secondary extinction as will 28 a crystal containing heavy atoms, or one in which the mosaic blocks are very nearly parallel. Reflections at low values of sine/l are also subject to greater secondary extinction effects, as they are generally stronger reflec- tions. Since secondary extinction frequently mimics absorption, the correction is customarily added to the linear absorption coefficient. The thermal parameters and scale factor can be esti- mated from knowledge of the nature of the cell contents without knowledge of the position of those contents. A very good approximation of these values can be derived by Wilson's method, noting that for a random distribution of N atoms, the local expectation value of 2 fJ (hhi) L1. IIMZ -——- 2 F = l heel 1 Consider now the relationship between Ith£|2 and IF I2 3 obs which is known only on an arbitrary scale. 2 _ 2 lFobAI ‘ K 'thzl and '———— 2 _ *——-2 _ IFobAI ' K thhLI ‘ K c... lle r 2(hhz) 1 J 29 Let us now assume that the effects of thermal motion can be described by including an additional factor so that 2 2e-(2B sin2e)/A2 f = of where °f is the scattering factor at 0°K as listed in the International Tables for Crystallography. Then 2 -M. 2 Z °f j J 2 2 Ke-(2B sin 6)/l .Q I and _ 2 2 An q -£n K? (28 sin 0)/A The best fitting straight line drawn through the plot of Anq vs. sin20/l2 will have a slope of -2B and an inter- cept of in K. The values of B and K can be adjusted from these approximate values during refinement. After appropriate manipulations, 2 2 2 Icorr = K |F| exp ( 2B sin e/A ) The structure factor of the cell is the sum of the scattering 30 factors of all atoms within the cell. If positions are chosen for all atoms in the asymmetric unit, amplitudes and phases of the structure factor can be calculated, and the calculated value compared to the observed value. Good agreement of these values as reflected in the discrepancy index, R = (F064 - FcaficVFobA ’ usually indicates that the structure proposed is correct. Until phase angles can be measured directly, crystal- lographic strategy will concentrate on methods of deriving suitable trial structures. OBTAINING A TRIAL STRUCTURE FROM A PATTERSON MAP One method of obtaining a trial structure is by analysis of a Patterson map. The Patterson function is given by the equation P(U'VW) = 71-- z 2: z IFI2cos 2n(hU+kV+£W) c all h,h,£ It requires knowledge only of the indices and [FW2 value for each diffracted beam. Because no phase information is required, only the relative positions of atoms will be 31 calculated. P(UVW) is evaluated at every point in a region identi- cal to the asymmetric unit. The value of P at each point corresponds to the sum of the appearances of the structure when the origin of the Patterson cell has been superimposed on every possible position in the structure. In other words, the structure is viewed from each atom in turn. A vector is drawn from this atom to every other atom in the structure. The height of the peak at the terminus of this vector is equal to the product of the atomic numbers of the atoms joined. The vectors are then superimposed at the origin of the Patterson map and their heights are summed. The map is always centrosymmetric because interatomic vector B-A will always have the same magnitude and the 0p- posite direction as vector A-B. If N atoms are contained in the unit cell, there are N2 interatomic vectors and (Nz-N)/2 vectors in the Patterson map. Unless the number of atoms in the unit cell is very small, the number of Patterson vectors is so large that the complexity of the map limits its usefulness. Nonetheless, the Patterson function is a powerful tool for analyzing structures containing heavy atoms. In these cases, the vector between the two heavy atoms produces a very intense peak which is readily recognizable. The co- ordinates of this peak can be related to the coordinates of this atom in the structure by examination of the symmetry 32 requirements of the space group of the unit cell. The heavy atom will have a much larger atomic scattering factor than the lighter atoms and will probably dominate the phase angles for the entire structure. Hence, it may be pos- sible to begin refinement of a trial structure even if no other atomic position is known. OBTAINING A TRIAL STRUCTURE BY DIRECT METHODS If the species under study contains no heavy atom, interpretation of the Patterson map may be extremely dif- ficult. In such cases, direct methods may be used to extract phase information from intensity data. Direct method cal- culations are simplified by using a normalized form of the structure factor, Ehh£' Since atoms have finite size, the values of the scattering factors decrease with increasing sine. The E-values, on the other hand, are not dependent on sine because the atoms are treated as zero-dimensional points. For this reason, the 5's are a good indication of the intensity of a reflection relative to other reflections of the same class rather than relative to all other reflec- tions. A relatively intense high order reflection will have a large |E|, even though a rather small |FO| may have been observed. The value of E is given by: 2 _ 2 lEhhil ‘ (K |F0,hh£' )/€ 33 where 9] is the spherically averaged scattering factor of th atom. If the molecule is known to contain well- the j characterized fragments, the molecular scattering factors of these groups can be used as g-values in the calculation of the E's. The a factor in the denominator depends only on symmetry, and compensates for the mutual cancellation of the expansion of Ith£|2 for certain classes of reflections. The phases of a few reflections must be known if direct phasing methods are to be applied to other reflections. In a centrosymmetric space group, the phase angle must be either 0 or n, and the phase angle will determine only the sign of (Fo,hk£)/|Fo,hk£|’ In a centrosymmetric space group there are eight po- tential origin sites, (000), (03s), (50%), (ago), (s00), (030), (00%) and (8%h), and the origin for a particular structure will be specified by the phase assignment of three reflections. The reflections chosen should have large values of [El and specific parity properties. Each index, A, h, or L, will be either odd or even. The entire set of in- dices can be grouped into the parity classifications: ccc, cco, coc, occ, coo, 00c, oco, or 000. The origin-defin- ing reflections must be chosen from different parity groups and must be linearly independent -- $LQL -- for any three reflections, the parity of (h+h', k+h', £+£') must be dif— ferent from the parity of (th), (h'h'i') and (h", k", K") 3A and the numerical values of all indices must be linearly independent. Knowledge of these phases can be used to derive the phases of other reflections by the use of the relation- ship [8(hh£)J[s(h'k'£')J[s(h—h',k—h',£-£')J z +1. where "3" means "sign of" and "=" means "is probably equal to", and the signs are multiplied such that (+)(+) = +, (+)(-) = (-)(+) = - and (-)(-) = +. A more general form of this equation, the X2 relation— ship, is given by 81:5,”sz 2 SEh'E'z'(Ehrk!£1)(Eh_hv,h_kt ,£_£1)] The summation is carried out over all vector pairs of known sign which form a closed triangle, a vector triplet, with hkz, egg., nu = 81—5;h'h'£' = 93'5';h-hv,!z-h',z—z' = 1710. This relationship would be of little value if the prob- ability of the correctness of the statement could not be evaluated. Fortunately, the probability of a positive phase is expressed by the formula: 35 P+hh£ = % + ktanh[(o3/og/2)a'] where On = ZZ n (Z = the atomic number of the jth atom) J v _ X and a ‘ IEhhz' h'h'z'Ehthh-h',h-h',z-z" The initial application of this method should be with reflec- tions with large IEI, since these relationships will have the greatest probability and will participate in the great- est number of vector triplets. Eventually, no more phases will be able to be assigned with confidence. In this case, new series can be begun by applying the Z2 formula to reflections and assigning symbolic signs., 112;: A, B, C, etc. At the end of the assignments, these symbolic signs must be evaluated as + or -. Another limitation of the £2 formula arises in space groups lacking glide plane or screw axis symmetry. Unless some other means of generating negative signs exists, the £2 formula will produce only positive Ehhl's' Current com- puting methods use the inventory of phases determined by high-probability £2 relationships as a basis for deriving other phases with the tangent formula: g EbEa—b Sin (¢b + ¢a-b) g EbEa-b COS (¢B + ¢a-b) tan ¢a = The tangent formula can also be used in phase assignment for 36 non-centrosymmetric systems, but its application is limited by the complexity of the calculations. An additional feature is the possibility of weighting the E's, so that those which are established with greater certainty have a greater ef- fect in establishing ¢. Remember, however, that the original origin-defining reflections were chosen semi-randomly, and phases were assigned arbitrarily. While it is possible to choose a "good" starting set of reflections, it is not always pos- sible to identify the best starting set until the phase assignment is complete. Since different starting sets can be chosen, different phase sets will be derived. The best phase set can be determined by calculating figures of merit for each set. One indicator of the plausibility of a phase set is the absolute figure of merit, given by M = (Z'Zrand)/(Zexp' Z The Z's are related to the probability weightings rand)‘ of the tangent formula. "Z" is derived from the weighted tangent formula and the absolute values of E's. All these are measurable quantities. Zrand is calculated for a random distribution of atoms, and Zex is derived from the phase p set assignment. For a correct phase set determination, Z will equal Zex and the absolute figure of merit should equal 9 one. In practice, correct sets usually have values in the range of 1.2 i -0.2, but on occasion, a correct phase set has had an absolute figure of merit as low as 0.7. 37 The $0 figure of merit is completely independent of the tangent formula and is based solely on the values of E: 1”0 = XI: EbEa—bI a b The summation over b includes all terms available from the set of phases under consideration, while the summation over a includes reflections for which Ea is approximately equal to zero. The expected value of "0 for a reliable phase set determination is small. The $0 value is sensitive to molecular position. The third figure of merit is the well known crystal- lographic "R index" calculated for E's rather than F's: liEalobA ’ IEa'cach ZIE a R = Z a alobA It is frequently found that a combined figure of merit is more reliable than any one index. The combined figure of merit may be given by M-Mmin + w (ADO )max—wO (I) l Mmax-Mmin 2 (wo)max-(wojmin C: + w max-R 3 Rmax-Rmin 38 The mJ's are weighting factors which are generally assumed to be unity. The maximum value of C is wl + w2 + w3 and the phase set that has a figure of merit closest to this value deserves the greatest consideration. Once phases have been reliably determined, the E's can be combined in a three dimensional Fourier synthesis in the same manner as the original F's, and a map of the synthesis can be used to establish initial positions for a trial structure. REFINEMENT OF THE TRIAL STRUCTURE Once a trial structure has been identified, difference maps rather than electron density maps can be prepared. In a difference map, IFCI is calculated using the phase angle determined for the trial structure. [Fcl is then subtracted from IFOI. A positive region indicates that more electron density is needed in that area, while negative areas are oversupplied with electron density by the trial structure. At early stages of the refinement, large peaks can indicate the proper positions for additional atoms, while a deep hole indicates that the atom at that position should probably be removed. A trench about an atomic position or a residual peak at an atomic position suggests that the temperature factor should be expanded or contracted. If the atomic site has a negative value but an adjacent area is positive, the input atom should probably be moved toward the peak. 39 Refinement of a structure may be carried out by alter- nating difference maps with least squares fitting sequences. The difference map indicates the region of the asymmetric unit where the trial structure is weak and suggests the type of adjustment needed. The least squares fit adjusts all variables to bring the calculated structure into the best possible agreement with the observed data. Although a low discrepancy index from the least squares calculation is one measure of the goodness of fit, a flat difference map is one of the best criteria of a good structure determination. CHAPTER II CRYSTAL STRUCTURE OF Na 30(F), no. F(000) P - see Reference 9 Q - see Reference 9 scan range \ 2267.0 1.391 .08 mm x .20 mm x .59 mm colorless parallelipiped mounted approximately along the normal to (110). Mo Ka, graphite mono— chromatized 1.972 0-20 0‘: 29 g 50 20 2.0 A027 lAOl 1008 0.02 0 .65° below al to 0.65° above 02 AA disordered. Initial positions for the atoms were as indicated by MULTAN. Two groups were input: the Na+-dicrown unit, and the perchlorate moiety. Thirty-two number sets were cal- culated, with Set 30 equivalent to Set 17. Set 18 was chosen as a basis for initial positioning of the atoms because it had the highest combined figure of merit. COMBINED ABS FOM PSI ZERO RESID FOM Rel. wts for combined FOM l l l ‘ Max. value 1.015A 1.752 AA.27 2.881 Min. value .A705 .867 26.6A .8798 Set 18 1.015A .972 26.6A 2.8811 The 23 atoms positioned by MULTAN at the most—likely distances and angles were input to LESQ and the positions of the remaining atoms were identified from difference maps. These were found to be close to the positions indicated by MULTAN. Refinement of the crown rings proceeded smoothly through a series of least squares calculations and dif— ference maps. Hydrogen positions were postulated by HFINDR and were not refined, although thermal parameters were re— fined isotropically. The structure was refined in sections: each ring with its hydrogens was treated separately and the perchlorate was refined either independently or with one ring at a time. A5 Refinement of the perchlorate moiety was complicated by the disorder of this anion. MULTAN initially indicated seven oxygen atoms surrounding the chlorine atom. "Bond lengths" from the central peak to these seven peaks varied from 1.1A A to 1.83 A, and "bond angles" centered on the "chlorine" position varied from 3A° to 16A°. After isotropic refinement of the Cl atom, the Na atom and the carbon and oxygen atoms of both rings, the four highest peak positions were chosen as potential sites for perchlorate oxygen atoms. These were all located at a reasonable distance from the chlorine atom (Cl-O bond distance in ClOu‘ should be about 1.A5 A), but after refine- ment, the angles at the chlorine atom varied from 80.92° to 136.8A°. Shifting the origin and dividing the chlorine into first two and then three positions, and fixing the chlorine in a special position resulted in substantial increases in the residual. Pairs of peaks with approximately correct bond distances and angles relative to the chlorine position were chosen from the difference map and additional positions for oxygen atoms were calculated by HFINDR. Several tetrahedra were con- structed. Invariably, least squares refinements would in- dicate that geometric perfection was not compatible with reality. Attempts were made to identify two tetrahedral units, A6 centered on either a single chlorine atom or distinct chlorine atoms. The occupation of each site was refined as a variable in ORFLS. Again, these efforts were not successful. As one peak was removed from the difference map, another would appear. When three oxygen atom positions suggested that another oxygen atom should be located at a fourth position, the ballooning thermal parameters and holes in the dif- ference map revealed that the fourth position was not oc— cupied as expected. During these calculations, the atoms were refined iso- tropically in order to facilitate the separation of the distinct positions for each configuration. When anisotropic thermal parameters were used, the painstakingly separated locations merged. Finally, seven positions were left sur- rounding the chlorine atom, and these positions were close to those postulated by MULTAN. A new series of refinements was begun by using the posi- tions originally indicated by MULTAN. The final results of refinement, including the final shift to error ratios are listed in Table~2. Positional parameters with associated errors are listed in Table 3, Atomic scattering factors were taken from the $2222? 1A national Tables for X-Ray Crystallography as were the real and imaginary dispersion corrections.15 An anomalous scattering factor of zero was assumed for hydrogen. The final difference Fourier map, while reflective of A7 Table 2. Final—cycle Refinement Indicators. :1 i—’ ll (ZlFo-Fc|)/2FO) = .0712 for 1A01 data with F > 30(F) 2}1/2 = {£(wtg x IFO—FCI2)/2wtg x F0 .0756 for A027 data :11 M II (including zero-weighted data) Standard deviation of an observation of unit weight = {£(wtg x IFO-Fcl2)/N }1/2 = 2.A09. ref‘Npar Shift-to-error ratios: maximum (non-H) .28377 average (non-H) .03A75 Final difference map: 1.A8A -l.672. max. positive density (eA_l) max. negative density (eA-l) A8 Table 3. Positional and Thermal Parameters with Associated ESD's. ' (A) Sodium chroun Non-Hydroren Atoms Atom x y 2 811 822 B33 Bl? 813 8;, Na .0127(3) .2L79(5\ .u92u(5) n.2(6) 5.0(6) 3.2(7) .0(5) - .6(5) - .3(6) 011 .0369(3) .1020(u) .3812(6) u.u(3) 5.3(3) 5.u(3) - .u(3) -1.3(3) -1 3(3) cl2 -.038(1) .062(3) . 26(9) h.8(5) u.5(6) 10.1(8) - .u(u) - .3(5) -2.1(5) 013 -.115(1) .081(1) .u10(2> 6.0(6) 5.2(6) 7.7(7) - .6(5) -1.8(5) .5(5) 01, -.129L(u) 1717(9) .8251(6) 6.3(3) 6.5(A) b.5(3) .0(3) .0(2) .0(3) :15 -.1755(9) 212(1) .310(2) 3.u(u) 7.7(6) 5.6(5) - .1(u) —1.2(u) 1.8(5) C16 -.160(1) 309(1) .321(2) n.6(5) 6.5(6) 6.1(6) .7(5) - .5(A) .5(5) 0 , -.0727(3) .3328(A) .3160(6) 5.2(3) n.8(3) u.5(3) .3(3) -1.1(2) .1(3) c1, -.0A2(1) .335(1) .182(1) 5.0(5) 6.3(6) 3.7(u> .7(u) - .6 .3(u> c19 .056(1) .332(1) .195(2) 8.1(6) 5.5(6) 3.7(8) - .8(5) .1(U) 1.2(u) o110 .0905(31 .2591(u) .272A(6) 5.2(3) 6.8(u) n.0(3) 1.3(3) - .2(2) - .5(3) c111 .093(1) .183tl) .193(2) n.6(5) 11.2(9) n.3(5) .7(5) .6(u) -1.3(6) C112 .107(1) .108(1) .295(2) u.2(5) 8.5(7) 5.u<5) - .2(5) .5(u) -1.7(5) o21 .1637(u) .2A06(A) .5922(6) 5.u(3) 6.0(A) n.8(3) 1.7(3) - .5(2) - .3(3) c22 .2006(7) .3208(8) .eue(1) 5.1(5) 6.9(6) 6.0(6) - .3(5) -2.5(u) - .9(5) c23 .1659t7) .395A(9) .561(1) n.8(55 7.3(7) 8.0(7) -3.0(5) 1.3(5) -1.2(5) 02“ .0689(A> .3965(u) .5589(6) 6.0(3) 5.A(3) 5.1(3) 1.3(3) - .5(3) -1.A(3) c25 .o322(9) .h3h9(9) .669(1) 10.0(7) n.8(6) n.7(5) 1.7(5) 1.8(5) - .9(u) 026 -.0626(8) .u131(9) .67u(1) 6.1(6) 5.3(6) 8.2(7) 2.0(5) -1.0(5) -1.0(5) o27 -.O7A2(3) .3190(A) .675€(5) 6.0(3) 6.1(A) 3.9(3) 1.0(3) .2(2) .0(3) 028 -.o638(8) .287(1) .811(1) 6.6(6) 9.3(7) 3.9(5) - .1(5) .2(b) - .8(5) C29 -.0533(8) .189(1) .79L(1) 5.9(61 11.2(8) 5.1(6) -2.8(6) 2.0(5) - .9(6) 0210 .0220(u) .1680(u) .715"(51 5.7(3) 6.2(3) 3.9(3) .6(3) -1.6(2) - .2(3) c211 .1018(7) .1719(9) .790(1) 5.8(5) 6 5(6) 6.8(5) 1.2(5) -2.9(A) .A(5) c212 .1703(7) .170(1) .686(1) 5.2(5) 6.0(6) 5.8(5) 2.2(A) .9(A) .3(5) (8) Perchlorate Atom: Atom x y z Mult. 811 822 833 812 813 8,, c1 .2u77(u) .A705(2) .0033(8) 1 7.8(3) 9.8(2) 5.u(3) - 0.2(2) - .1(3) 1.1(2) 031 .3396(8) .u71(1) .007(2) .716(u) 7.6(7) 15.n(1.5) 11.3(1.2) .u(8) 2.3(8) .9(9) 032 .2u(2) .u9(2) -.1u(3) .A86(3) 17.3(2.2) 10.1(1.8) 8.0(1.5) - 5.7(1.6) -6.3(1.5) 1 2(1 3) 033 .179(5) .neu(u) .0u9(8) .uou(3) A1.0(8.9) u7.2(10.2) 15.6(A.1) -25.7(8.0) 18 9(5.0) .1(5.h) 030 .236(1) .A86(1) .1u9(2) .791(A) 1A.5(1.1) 13.3(1.3) 8.A(1.0) 2.2(9) -1.3(8) -2.8(9) o35 .2h3(2) .557(2) -.ouz(3) .5uz(u) 23.9(2.6) 21.2(2.3) 22.2(3.1) - 8.1(2.6) -8.u(2.5) 19.6(2.5) 036 .237(1) .3779(7) -.012(2) .783(5) 18.1(1.1) 9.7(6) 9.0(8) - 6.3(1.0) 1.5(7) -A.1(9) 037 .16o(3) .h67(3) -.092(5) .309(2) 13.0(2.9) 9.1(2.3) 8.8(2.3) - 2.9(2.0) -9.5(2.5) 1.5(1.7) (C) Hydrogen Atom: Atom X Y Z 8180 Atom X Y Z 3130 H12, -.0268 -.0088 .325u 7.8(2.3) H22A .2711 .3187 .6AAA 8.2(2.u) H128 .0506 .08A6 .2201 9.2(7.8) H228 .1818 .3290 .7550 “.3(1.7) 13A -.1050 .0519 .5123 9.0(2.8) H23A .1892 .3887 .A566 12.1(3.n) H138 -.1723 .0518 .3589 3.5(1.6) 523B .189A .AS69 .6063 6.2(2.1) H15A -.2AA6 .1986 .3153 A.5(1.7) H25A .0397 .5059 .6633 9.7(3.1) ”153 -.1515 .1879 .2130 u.1(1.6) H258 .0655 .u106 .7632 u.8<1.8) H16A -.185A .3308 .8186 1h.6(u.0) H26A -.o968 .huos .583u 17.2(5.1) H16B -.1960 .3398 .23U9 5.2(1.8) H265 -.0888 .AAIO .7679 6.1(2.1) H18A -.0618 .395" .1305 5.A(1.9) H28A -.1208 .3017 .8706 7.6(2.h) H188 -.0653 .2788 .1232 6.8(2.2) H288 -.006u .3153 .8633 13.0(3 6) ngA .078A .3920 .2u59 11.5(3.7) H29A -.1110 .1635 .7397 9.2(2.8) 198 .0807 .3290 .0909 6.8(2.1) H298 -.ou65 .1593 .8963 6.7(2.1) 8111A .1869 .1863 .122u 18.5(A.8) H211A .1086 .1160 .860A 10.0(2.9) H1118 .0326 .17u2 .1337 2.2(1.2) 8,113 .1061 .2325 .8501 7.5(2.5) 8112A .1131 .0868 .2385 6.0(2.1) 8,12, .2335 .1721 .7u09 10.6(3.1) H1128 .1667 .1198 .327“ 3.0(1.A) H2128 .16u5 .1085 .6283 1A.0(h.1) A9 the disorder in the perchlorate region, does not indicate either the absence or the gross mispositioning of atoms. The relatively high residual electron density in the area of the anion will be discussed later. RESULTS AND DISCUSSION The sodium ion has a coordination number of eight and is coordinated in a sandwich configuration between the two 12C“ rings, being too large to fit into the cavity de- fined by the ring oxygen atoms (see Figure 6). The co- ordination about the cation may be described as a slightly distorted square antiprism. A few representative angles are tabulated below to indicate the extent of distortion. ANGLE OF IDEAL ATOM 1 ATOM 2 ATOM 3 ANGLE SQ: ANTIPRISM 019 011 017 H6.11° H5.00° 01“ 011 0110 91.77° 90.00° 017 011 0110 H5.66° “5.00° 014 011 02“ 71.79° 70.U6° 01“ 011 027 U5.U7° H2.81° 0110 011 021 62.00° 63.91° 0110 011 0210 98.61° 100.u9° The square antiprismatic configuration allows the closest approach of the rings to reduce cavity size to fit the metal ion. Sodium-oxygen distances average 2.A9(5) A, in ex— cellent agreement with the calculated distance of 2.5M A 16 (CNNa = 8; CNO = 3). Each set of four crown ether oxygen atoms coordinated 5O Figure 6. The sodium-dicrown unit illustrating 50% prob- ability ellipsoids and the atom labeling scheme. H-atoms are designated by the number of the carbon atom to which they are attached. 51 to the cation is essentially planar. Displacements from planarity for both Ring 1 and Ring 2 are 0.01 A. The Na+ ion is centered between the planes at a distance of 1.53 A from each. Within the ring, C—0 and C-C distances average 1.90 K and 1.50 K, respectively, with C-O-C and 0-0—0 angles averaging ll3.6° and lO9.9°. Complete bond and angle in- formation for the non-hydrogen atoms is listed in Table A. All values are in good agreement with published l2CU structures.7’8 The Na-O distances are slightly larger for 12Cu-Na+ complexes (2.“9 A) than for 15C5 complexes (2.39 A)17 but smaller than for 18C6 complexes (2.9M A).18 Because of the differences in coordination number and geometry, it would not be meaningful to correlate these bond distances to physical properties. Hydrogen atoms were refined isotropically, but all other atoms were refined anisotropically. The degree of anisotropy among atoms of the same species varied but the variations do not appear to be related to proximity to the disordered anion. The perchlorate groups form a layer with all chloride ions restricted to the region z = 1.0033“. The sodium-di— crown units form another layer, with the sodium ions fall- ing at z = .5 i .00756u. Thus, each ion is approximately centered in a rectangular parallelepiped of counter ions. (The x- and y-dimensions of this coordination polyhedron are approximately equal. This follows directly from the fact 52 Table A. Bond Distances and Angles. Crown Rings - Distances (3) 011-012 1.908 (39) 021-022 1.999 (6) 012-013 1.990 (59) 022—023 1.501 (13) 013-01“ 1.909 (16) 023-02“ 1.995 (15) 019-015 1.935 (18) 029-025 1.386 (12) 015-016 1.986 (22) 025-026 1.502 (15) 016-017 1.901 (16) 026.027 1.996 (19) 017-018 1.399 (12) 027‘028 1.398 (11) 018-019 1.512 (22) 028—029 1.512 (16) 019-0110 1.931 (17) 029—0210 1.999 (17) 0110-0111 1.392 (18) 0210—0211 1.902 (13) 0111-0112 1.519 (29) 0211-0212 1.989 (12) 0112-011 1.389 (18) 0212-021 1.908 (12) Crown Rings - Angles (Deg) 011-012—013 111.9 (50) 021—022—023 107.8 (57) 012“C13"019 112.2 (21) 022-023-029 111.5 (89) 013—0117415 119.6 (11) 023-0214425 119.2 (85) 014—015-016 107.0 (13) 029-025-026 111.0 (79) C16-Cl6-Ol7 119.6 (12) 025-026-027 109.9 (103) C16‘017'018 113.9 (11) 026—027-028 110.2 (87) 017-018—019 107.1 (11) 027-028-029 109.6 (73) 018-019-0110 119.7 (12) 028-029-0210 111.5 (107) 019-01104111 112.2 (11) 029-02104211 115.5 (89) Ono-0111.0112 106.1 (19) 0210—0211—0212 106.6 (66) C111'0112'011 109-0 (1”) C211-0212'021 112-5 (8”) 0112-011-012 116.0 (32) 0212-021—022 112.9 (52) Table 9. Continued. Perchlorate Distances (A) 01-031 1.92 (1) 01-035 1.39 (3) 01-032 1.92 (29) 01—036 1.93 (1) 01-033 1.18 (8) 01-037 1.91 (5) 01-03“ 1.99 (2) Perchlorate Angles (Deg) 031-01-032 93.7 (125) 032-01-037 70.1 (126) 031-01-033 153.9 (37) 033-01'039 57.9 (38) 031-01-0314 97.5 (11) 033-01-035 85.1 (33) 031-01-035 90.7 (19) 033-01-036 96.3 (31) 031-01-036 98.7 (9) 033-01-037 92.0 (92) 031-01-037 163.9 (22) 039-01-035 98.3 (19) 032—01-033 106.5 (13) O39‘C1’O36 109.2 (10) 032--01-03,4 115.5 (129) 035-01-036 159.2 (15) 032-01-035 59.5 (129) 035-01—037 89.2 (23) O32'C1'O36 95.8 (129) 036—01-037 79.9 (20) BU that the x5 and y-dimensions of the cell are approximately equal.) (See Figure 7.) There does not appear to be any significant interaction either between the perchlorate and the sodium ions, or between the perchlorate and the di- crown unit as a whole. Further evidence for this lack of interaction may be found in the severe disorder of the per- chlorate moiety. Perchlorate groups are frequently found to be disordered,19-2u but are often well-behaved when directly coordinated or hydrogen-bonded to another group.25 In this case, seven partially-occupied oxygen positions surround the chlorine atom. (See Figure 8.) Major devia- tions from the expected tetrahedral angles and 1.95 A bond lengths are noted. The actual values are given in Table14. Neither the angular distribution, nor the superposition of the unit cell symmetry with the tetrahedral symmetry of the free ion, nor the pattern of occupancy supports the hypothesis that there are two separate but distinctly tetrahedral anion configurations. Thermal parameters for the perchlorate oxygen atoms are extremely high and most are strongly anisotropic. The chlorine atom also deviated markedly from spherical electron distribution; however, attempts to resolve the positions with the largest thermal parameters into two or more posi- tions failed. Similarly, attempts to position atoms in a less-distorted tetrahedral configuration always resulted in sharp disagreement with the observed distribution of 55 mAesoo.o + .Ammoo.o+v pm mEOpw ocHLOHno m. ov pm oopmoofi ohm mcofi ESHoom .20Homazomflvmz Ca mcfixoma one .5 opsmfim 56 Figure 8. The severe distortion of the perchlorate moiety is illustrated by the presence of seven partially- occupied oxygen atom positions surrounding the chlorine atom. 57 electron density. Another unusual feature of this analysis is the pres- ence of the largest peak in the final difference Fourier map (1.98 eA-3) within an Angstrom of the chlorine position. Attempts to refine this residual peak as a partially-occupied chlorine center were unsuccessful. Since this "bonding distance" would place another atom within the radius of the Cl-O bonding Sphere, it appears that this and other similar peaks are artifacts and do not represent real atomic posi— tions. The ellipsoidal representation of electron density cannot reflect the directional bonding in the predominately covalent perchlorate. These peaks may correspond to the area enclosed in the ellipsoid, but outside the tetrahedron enclosing the bonding electrons of chlorine. Perhaps the most meaningful conclusion is that the anion is nearly spherical and oriented semirandomly. In such a case, the excessively large thermal parameters and resultant unexpected bond lengths and angles may reflect an unsuccessful attempt to describe a curved surface of electron density as an ellipsoid. One may also speculate that the perchlorate anions exhibit long-range structural modulation, but that the intensity differences due to this ordering are too subtle to be detected by the diffractometer. It is possible that certain anomalously large tempera- ture factors of the ring atoms and the relatively high back- ground noise on the difference map are artifacts of the 58 unresolved disorder about the perchlorate. Sandwich structures of transition metals have been noted in which the cation with its aqueous (or anionic) coordination sphere was nestled between two large crown rings.26’27 Alkali metal crown complexes can also show water-cation coordination if the metal is complexed to only 18 one crown ether. In dicomplexed species, aqueous layers may form infinite, two-dimensional networks between crown layers.7’8 An unusual aspect of this structure is the ab- sence of solvent molecules, a condition which permits the disorder of the perchlorate: the anion cannot be held in place by interaction with a solvent network, and it is pre- vented by the crown ligands from interacting with the cation. CHAPTER III PREPARATION AND CRYSTAL STRUCTURE OF (Sr/Y)Cl2. and (Sr/Yb)Cl2+X 05 INTRODUCTION In the past 10 to 15 years, sustained interest in the atypical low valent and mixed valent metal halides has been evidenced by inorganic chemists and materials scien- tists alike. Since any metal with more than one oxidation state (or any mixture of metals with different oxidation states) has the potential to form a mixed valent compound, it is not surprising that a bewildering array of phases has been identified. The Ln-LnX3 systems have proven especially rich in this type of chemistry.28"36 Although phase diagrams are available for a number of these halide systems, they are subject to repeated revis- ions as new phases are discovered.37’ul Often, it is learned that the phases were hindered by kinetic rather than by thermodynamic barriers. Many phases exhibit extremely complex stoichiometry and their structural char- acterization has lagged far behind their identification. Complex combining ratios are characteristic of the defect structures. These nonstoichiometric compounds 59 60 may maintain very similar physical properties even though their range of stable existence may span several atomic percent of one or more components. Current thinking in solid state chemistry now views these species not in terms of point defects and random, local distortions, but in terms of defect ordering, extending even to long range structural modulation. Short range ordering can also lead to a clustering of defects over a limited microdo- main, while long range ordering results in a lattice super— structure which can be detected by the presence of weak X-ray reflections occurring in an orderly fashion. Neutron diffraction and electron microscopy are even more powerful methods for investigation of ordering. The concept of long range ordering has led to the re— evaluation of many structures with the specific intent of searching for defect ordering. Many systems, such as alloys, certain oxides, and many minerals, are now thought to be well-ordered compounds rather than solid solutions.”2 Among the species subjected to reexamination are a number of fluorite-related structures, both anion-deficient and anion-excess, including oxides, oxide fluorides and other halides. Indeed, cubic solutions of the mineral yttro- fluorite and related phases of the general type (Ca,Y, RE)F2+x’ for 0 i x i 0.5, can serve as representative examples of nonstoichiometry. In the late 1950's, Brauer and Miller143 determined 61 that at 500°C the solubility limit of LaCl3 in Sr012 was about 22.5 mole percent. At this concentration, the fluorite structure of SrC12 was retained and cell param- eters increased less than 0.03A. They hypothesized that extra anions were accommodated interstitially at 8&8 and the translations of this position. This model, however, does not address the problem of accommodating additional anions in a structure which is already closely packed. In 1963, B. T. M. Willis published the results of a 99,95 2+x’ another anion-excess derivative of the fluorite structure. very precise neutron diffraction study of U0 For the composition ”02.13: he found that interstitial anions did not occupy the large cubic holes at gag, 500, 030 and 00%, but that the oxygen atoms located at 8%% and the symmetric equivalents of this position were shifted along towards the center of the cell, and the "interstitial" oxygen atoms were shifted about 1 A away from 88% along . A model consistent with the observed data requires that the defects be clustered if extremely short anion-anion distances are to be avoided. Neutron diffraction studies of (Ca/'){)F2.10 by Fender and co— 96-98 workers were also consistent with this model. (See Figure 9.) The structure of YbCl2 was once classified as a fluo- rite. Later, more precise measurements by Fishel"I9 sug— gested that it would be described more exactly as an 62 (m) (110) , l , o l (110) : (111)§ F. : ' .. _ FFIF{ Y F' C] F' (110>/ '--~--- 5” 1 Figure 9- Clusters in (Ca/Y)F2+x. Above: 2:2:2 cluster, two F' ions, two F" ions and two normal ion vacancies. Below: an extended 3:9:2 cluster with two F" ions and 9 F' ions. OY3+ positions assumed in calculating diffuse scattering. Redrawn from Reference 98. 63 orthorhombic CeSi structure: two orthorhombically distorted fluorite cells lined up in the 0 direction. Fishel also prepared a new ytterbium chloride and characterized it by X-ray powder diffraction. Chemical analysis established 50 the formula as YbCl2.26. Subsequently, Lfike initiated a project to grow a single crystal of this phase and to determine its structure. When he was unable to prepare a single crystal suitable for X- ray analysis, he substituted a trivalent heteroion for Yb(III). The structure solved was for the compound YbSErCll351 and proved to be isostructural with Fishel's "YbCl2.26". The difference in reported composition is as- cribed to inaccurate chemical analysis. Yb6C1l3 can be described as another ordered and regular variant of the fluorite structure. Six fluorite units lined up in the b direction are required to accommodate the additional anions in Yb6Cll3. This compound may be regarded as a Vernier structure with overlaid hexagonal and rec- tangular networks. It is one member of a homologous series of structures of composition MnX2n+l formed by the rare earth halides and certain mixtures of the rare earth and alkaline earth halides. Lfike's work was beautifully complemented by that of Barnighausen who simultaneously solved the structure of 52 Dy5C1ll and the isostructural SruDyClll. The ordering of this last compound appears to be in direct conflict 69 with the Brauer—Mfiller results. Vernier structures have now been identified for other 53 59 5C111, SmSBrll, 55 Cf9GdC111 and Cf9GdBr11’ while the M6Xl3 series is repre- 56 elements. In the M5Xll series we find Ho 52 59 13, Sm6Brl3, and possibly SmSYBr13. In addition, the phase SmllBr2u appears to be an inter— sented by Dy6Cl growth of the 5/11 and 6/13 phases. (See Figure 10.) Greis and co-workers have employed a systematic index- ing of superstructure lines of powder diffraction patterns to examine the crystal chemistry of yttrofluorite species in the systems LnF2-LnF3 (Ln=Sm,Eu,Tm,Yb) and MF2-REF3 (M=Ca,Sr,Ba; RE=Y,La,Sm,Eu,Tm,Yb).57'63 They have identi- fied two different series of homologous compounds, MmF2m+5 and MmF2m+6' For the composition region where m = l3, 19, 15 or 19, both tetragonal and rhombohedral structures were found (See Figure 11) and cationic order- ing was present. Anionic ordering is known to exist in the fluorine- rich rhombohedral phases which Greis designates as B and y . In these species, anionic clustering occurs as a result of cation ordering. At intermediate fluorine concentrations, anion clusters appear to be built up from both interstitial and shifted fluorite-structure anions. At fluorine con- centrations where cubic symmetry prevails the microdomains are too small to be revealed by X-ray diffraction, but Greis believes that partial ordering occurs. 65 .nmsaeesaen aefiese> :N\HH can ma\e .HH\m 0:9 .OH mtswaa A .m k 66 I» a) b) xi lg c) ,AAAAA ‘ 3’? A 51591619919 '7 “ .‘_wlII“uI.Ll( «9 {%‘1§Av ‘ V‘smn " AXALAWWV 99$¢$wyngMM1 « new Figure 11. a) The cubic fluorite unit cell. b) Tetragonal basis cell derived from the fluorite cell. c) Rhombohedral basis cell derived from fluorite unit cell. d) The rhombohedral basis cell and suggested superstructures. (Figure redrawn from Reference 57.) 67 The Russian literature reports europium6u’65 and Samarium66 chlorides as having the Vernier—type Mnx2n+l stoichiometry and also reports mixed metal species con- taining SP012 and Ba012 with trivalent rare earth ions.66 Unfortunately, no structural details are available. 68 have identified fluorite- Fedorov and co—workers derived structures in the CaF2—LnF3 systems for Ln=Y, Ho-Lu. These phases have an idealized composition Ca8Ln5F31, the same MmF2m+5 composition as that described by Greis, but have been indexed on the basis of pseudo— cubic unit cell parameters with each axis approximately 13 times that of CaF . More recently, phase diagrams have 2 been investigated for the systems SrF -LnF3 for Ln=Y, 2 La-Nd, Sm, Gd-Lu. The phases identified have had either fluorite, fluorite-derived, or tysonite (LaF3) structures. To date, only Lfike and Baringhausen have studied the rare earth Vernier phases through the analysis of single crystals: all structures except YbSErCll3 and DySClll have been solved by refinement of powder diffraction data. Because of the similarity of electron density of Yb(II) and Er(III), not to mention the similarity of Dy(II) and Dy(III), the site symmetry of the +2 and +3 ions has never been specified crystallographically. In addition, no studies of either single crystal or powder diffraction data have indicated whether these structures are stable in the complete absence of f—block elements. It is also of 68 interest to investigate the limits of cell size: what are the limits of "n" in Mnx2n+l when the Vernier structure is maintained? As initially planned, the objectives of this research were to grow single crystals of Vernier compounds and to solve their structures to determine: 1) whether f—block metals are necessary to stabilize the Vernier structure; ii) what, if any, is the site symmetry of the metal coordination? What are the specific locations of the +2 and +3 ions? Many combinations of divalent and trivalent ions were proposed on the basis of radius ratios in known Vernier structures. However, in view of the overwhelming dif- ficulty in identifying crystallographically suitable crystals, investigation was eventually limited to combina- tions containing SrCl2, a compound noted for its ability to accommodate high concentrations of chemical defects, and form high quality crystals even in the presence of hetero- ion dopants. Furthermore, SruDyCl1 exists as a prece- 1 dent for the incorporation of Sr(II) into the Vernier struc- ture. The similarity of size of Y(III) to that of Dy(III) = 1.0272070 sug- (at CN=VIII, R = 1.0191; R Y(III) Dy(III) gested YCl3 as a component worth investigating. A mixture of SrCl2 and Y013 would address the first objective. It would be reasonable to anticipate the formation of 69 Sr YCl2n+l’ analogous to SruDyClll. A mixture of SrCl2 n-1 and YbCl3 was chosen to elucidate the site symmetry and ordering of the cations. It was hoped that substitution of Sr(II) for Yb(II) would aid in the formation of a crystallographically suitable crystal in the same manner as the substitution of Er(III) for Yb(III). Although a high quality crystal of (Sr/Y)Cl2+x was identified, it proved to have a different structure. The research project completed is not the project originally envisioned. The question of the necessity of f—block elements in the Vernier structures has not yet been ad- dressed: absence of proof is not proof of absence. The structure of (Sr/Y)Cl2.05 appears to be related to the cluster model of U02.13 proposed by Willis. This is the first Willis-cluster chloride described. EXPERIMENTAL Preparation of (Sr/Y)tl2+X and (Sr/Yb)Cl2+X The rare earth halides and, to a lesser extent, SrCl2 are hygroscopic. Heating the hydrated species leads to the formation of oxide halides rather than to simple de- hydration.7l’72 To avoid oxide halide formation, a modi- fication of the method of Taylor and Carter,73 dehydration of hydrated chlorides in the presence of excess NHuCl, was used to prepare the starting materials. 70 Yb203 and NHuCl were dissolved in 6 N hydrochloric acid. The resultant hydrated YbCl3 and coprecipitated NHuCl were placed in a glass bulb attached to a high vacuum line, and the water and NHuCl were eliminated by heating to about 320°C overnight under dynamic vacuum. After de- hydration was complete, the glass bulb was cut free with a torch and opened only in an Argon atmosphere glove box. Anhydrous Sr012 was prepared similarly from Sr(OH)2. The overall reactions are: 6N HC1 Yb203 + 6 NHuCl 2 YbCl3 + 3 H20(g) + 6 NH3(g) 6N HCl Sr(0H)2 + 2 NHuCl --' SrCl2 + 2 H20(g) + 2 NH3(g) The YCl3 was used as purchased from Research Inorganic Chem- icals. Since the major interfering impurities were expected to be hydrolysis products, X-ray powder diffraction was used to check for purity. Powder patterns include a charac— teristic grouping of additional lines when hydrolysis products are present even in low concentrations. The mixed valent chlorides were prepared by high tem- perature reactions. Samples containing 5:1 molar ratios Of SPC12 and YCl3 or YbCl3 were sealed in evacuated out— gassed quartz ampoules which were placed in a tube furnace and heated to about 900°C for two hours to insure complete melting. Thereafter, crystals grew from the melt as the 71 temperature was automatically lowered at an average rate of l.5° per hour. After the samples were annealed at 900°C for several hours or overnight, the furnace was shut off and allowed to cool to room temperature. The ampoules were Opened in the glove box. In all cases, the sample had formed a hard, compact mass which adhered to the glass. As great a portion of the contents as possible was chipped free and lightly ground in a mortar. The crushed material was transferred to a Schlenk tube fitted with an optical window and removed to the outside of the glove box where the sample was protected by flowing Argon and examined under a microscope. Individual crystals were selected and sealed into Lindemann capillaries. (See Figure 12.) The selected crystals were subjected to microscopic examination under polarized light. Those which did not exhibit birefringence were examined by the precession method to determine the approximate cell parameters and the space group. Particular attention was directed to searching for weak reflections indicative of superstructure. It was necessary to examine many crystals before one of suitable quality was identified. Eventually, a rather ex- tensive collection of photographs of (Sr/Y)Cl2+X was ac- quired. The habit of these crystals was such that the preferred orientation in the capillary characteristically led to a zero-layer photograph showing an apparent six-fold 72 .mdpmgmoam wcfipcsoe Hmpmzpo .mH opzwfim ET-«(DZL'H 09:9 mafimoao mamaaaomo .m Eopm poacH goom> ooze coauoopopm mpwaaaqwo mopmpmoa< wgapcsoz [fl xoooQOpm Endom> AEOppom pmamv ooze Lommcwne BOUCHZ Hmoapao com mmmau <: m C) D T 73 axis, and a perpendicular zero—layer showing an unusual pattern of triplets separated by a row of reflections of greatly reduced intensity. (See Figure 13.) Camera geometry prevented me from obtaining a complete set of upper layer photographs. After calculation of cell parameters the crystal was referred to Dr. Donald Ward for data collection. Data Collection and Refinement of (Sr/Y)Cl2+X Crystal data are listed in Table 5. Volume calculations based on SrCl2 and YCl3 indicate that the chloride anion when associated with cation oc- cupies about 93 or 99 A3. The calculated cell volume of 2521333 should hold about 5.7 anions. Possible formulas would be SrYClS, or, if the cell were doubled, SquC1ll. Unfortunately, there was no basis for enlarging the unit cell: a search had been made along the 00K and 0&0 axes for evidence of superstructure, but none was found. The data were reduced and standard deviations were cal— culated as a function of counting statistics.9 Refinement was attempted using the entire system of Zalkin's programs (MAGPIK, ABSOR, INCOR, ORDER2, FORDAP, LESQ, DISMAT, DISTAN) as modified for local use.11 Eventually refinement was accomplished using ORFLS.l3 Work was begun using an unmerged data set because ORDER was not operational at the time. An analytic absorption 79 o-nql‘f‘Kv' 3 Figure 13. Zero layer precession photographs of (Sr/Y)— C12+x. Above: an apparent six-fold axis. Below: a pattern of triplets. 75 Table 5. Crystal Data for (Sr/Y)Cl2 05, First Data Col— lection. ' formula (Sr/Y)Cl2+x, x unknown space group R3 or R3 systematic absences unit cell, hexagonal axes: a,b,A O c,A Y,deg -h+k+£#3n (hexagonal axes) 9.923(1) l2.096(5) 120.0 Unit cell parameters were calculated from a least-square fit of 18 strong reflections in the range 35° 3 28 3 95° V,A3 crystal description radiation u, cm scan type limit Bragg angle, deg background count time scan speed, deg in 20/min unique data, no. unique data with F2 > 20(F), no. P - See Reference 9 Q - see Reference 9 scan range 252.8 colorless, approximately 0.5 x 0.77 x 0.89 mm Mo Kc, graphite mono- chromatized 157 8-28 0 g 28 i 60 20 sec. 2.0 503 977 0.02 O l.0° below 01 to l.0° above 02 76 correction was applied to the raw data. The path length on which this correction was based was calculated from measure- ments of the crystal through the protective capillary. Preliminary ion positions were determined from a Patter- son synthesis and were reasonable in the context of the space group chosen. Refinement was begun with a succes- sion of least squares calculations and difference maps. With a metal ion located at approximately 0 0 3/8 and a chloride at approximately 0 0 7/8, the structure refined quickly to R = 0.1716 and Rw = 0.1951. (See Table 6.) Unfortunately, attempts to add more anions to the structure resulted in higher residuals and unacceptable bond distances. Without additional anions, the cation:anion‘ratio was 1:1 with both species located in special positions of the type 002. Placing a chloride ion in a general position, however, would result in a cationzanion ratio of 1:9. Stoichiometry could be satisfied by placing anions at selected special positions, but these were not consistent with either Patterson synthesis or bond length require- ments. This impasse suggested that either the wrong unit cell or the wrong space group had been chosen. Since ORDER was operational again, the data set was merged in rhombohedral symmetry. A comparison of supposedly equivalent reflec- tions revealed that their intensities were drastically dif- ferent. If the correct space group were chosen and if 77 Table 6. Results of Rhombohedral Refinement Positional and Thermal Parameters with Associated ESD's. Atom x y z B Sr .00 .00 .3729(2) .9l(7) Cl .00 .00 .830 (3) 10.1(7) R1 = (ZIFO-FCI)/XFO) = .1716 for 977 data with F2 20 (F)2 2 1/2 = 2 R2 = {2(wtg x IFOxFCl )/2wtg x FO } .1951. Shift-to-error ratios: maximum .355 average .129 Final difference map: 53.3 -3.0 max. positive density: (ex-3) max. negative density: (eA-3) 78 the data had been adequately corrected for absorption, the intersities of these reflections should be similar. Because an analytic absorption correction had already been applied to the data, it appeared that the wrong unit cell and hence the wrong space group had been chosen. Precession photographs of (Sr/Y)C12+x were reexamined for evidence of a different unit cell. Both orthorhombic and monoclinic cells were tentatively identified and the indices of the reflections were transformed by matrix multiplication. Neither transformation was pursued, since merger of the data set in neither of the new symmetry classes reduced the discrepancy index of ORDER significantly. The failure of this value to respond to changes of space group suggested that the problem lay in the absorption cor- rection and not in the symmetry. A stereogram confirmed that there were distinct areas where the reflections were either more or less intense than average. Corresponding discrepancies were noted in a series of four azimuthal scans, and remained unaffected when the physical description of the crystal was changed in the absorption correction program. Comparison of the intensities of the azimuthal scans before and after the analytic correction was applied showed that this method was ineffective in eliminating the absorption problem. The necessary data for an empirical correction could not be obtained because the crystal had deteriorated. 79 I had been unable to solve the structure of (Sr/Y)Cl2+X in the rhombohedral space group and had been unable to identify evidence of a more promising space group. There— fore, because of the strong precedent for the retention of the fluorite subcell in related anion-excess species, I set about to locate a cubic unit cell with edges of about 7A. After finding this SrCl2 subcell, I hoped to find ad- ditional reflections which would reveal the presence of superstructure. A new crystal was selected and a new series of preces- sion photographs was examined. Because of the orientation of the crystal in its protective capillary and the geometry of the goniometer head, it was not possible to obtain photo- graphs showing a cubic unit cell. Examination of a scale model however revealed the relationship between the desired cubic cell and the rhombohedral cell. (See Figure 19.) The second crystal was moved to the diffractometer. Once the cubic cell had been identified all three axes were scanned for superlattice reflections. No evidence of superstructure was found. Because of the difficulties encountered with the first crystal extraordinary efforts were directed towards an- ticipating additional complications with the second crystal. A triclinic data set was collected over the region 9 ik, iz with the 20 range extending to 65°. This crystal 3 was found to have a primitive cubic cell, even though SrCl2 8O Figure 19. Relationship between the axes of the rhombo- hedral In. and cubic— unit cells. Table 7. Crystal Data - Second Data Collection. formula space group a,b,c,A a,B,Y,deg (Sr/Y)C12.05 apparently cubic. data set was collected. tions in space groups P1, P23, P93m and Fm3m are proposed. 6.967(1) 90.0 A triclinc Solu- Unit cell parameters were calculated from a least squares fit of 13 strong reflections in the range 91° 1 28 i 95° crystal description radiation U: cm-l scan type limit Bragg angle, deg. background count time, sec. scan speed, deg in 28/min unique data, no. unique data with F > 0(F) P - see Ref. 9. Q - see Ref. 9 scan range Colorless, broad, flat wedge with larger dimensions approxi- mately 0.3-0.9 mm Mo Ka’ graphite monochromatized 157 0—28 8_<_28=65 20 2.0 925 221 — primitive 119 - face centered 0.02 O l.0° below a to l.0° above a l 2 82 has face-centered cubic symmetry. Crystal data are sum- marized in Table 7. The change from face centering to a primitive cell indicates that the interstitial anions, if present, cannot be restricted to the position 88%: addition of another ion at this site would not destroy face centered symmetry. The second crystal was shaped like a broad, flat wedge and the intensity variations due to absorption were extreme. A series of azimuthal scans indicated that the intensity of a given reflection might vary by an order of magnitude as ¢ varied over 180°. Seven azimuthal scans were col— lected at X = 90° and these were used to calculate an empirical absorption correction. Unfortunately, this cor- rection was less effective as X deviated from 90° and varia- tions as great as four-fold were noted for individual reflections in azimuthal scans collected at other values of x. Because of the limited success of the absorption cor- rection, first attempts at refinement were made using only a portion of the data -— i.e., that collected in the 0 region where absorption effects were most consistent. It was necessary to strike a balance between obtaining the greatest number of data points and the smallest variations in intensity. (See Figure 15.) The best compromise was identified by comparing the residual values when the data in each slice were averaged in space group Pm3 (#200) by ORDER. 83 mwom. mama. .ooflacam Coos mm: sewpoop (goo coHBQLOQO oz .mcmom Hmcpsewmm o>flpmpcowogqog 03p Low 9 m> H\meH .mH ogzmwm moopmo 1 com com U 9 ooa o _ _ _ _ / lo; / Io.m 17/ IO.M .llo.: Ilo.m \+ .0. + e .I . o m /0 fr log. GJ%/¥l+ . llo.m / ‘k\ loom mwm oamumwa mm: momlowfi cam oomlmma woma. IIHIIIII m mCOHpoonom a omcwg e 89 The data collected over the region 170° 1 0 1 205° seemed to offer the best balance between a reasonable number of data points and a low discrepancy index. Preliminary atom positions were deduced from a Pat- terson map. Major peaks were found at (000), (580), (kkk) and lesser peaks at (95%) and (0&0). All other peaks were insignificant. If one assumes that the origin is occupied by an ion, these Patterson peaks are consistent with the SrC12 structure. In SrC12, all atoms are located in special positions. Even when the symmetry is lowered from face-centered cubic to primitive cubic, the positions remain non-refinable. Furthermore, attempts to refine thermal parameters with all atoms in special positions resulted in massive, random fluctuations of the temperature factors and the scale factor. Invariably, LESQ failed unless only the scale factor was refined and it refined in one cycle. After that, no further change was noted. The data from the first crystal were then transformed from rhombohedral to cubic coordinates. When atoms were placed at the same sites the residual was comparable to that of the second data collection. (See Table 8.) Due to the symmetry chosen for data collection, many reflections were not counted during the first data col- lection. (See Chapter 1, section on Transformation of Unit Cells.) Since all reflections counted were compatible 85 Table 8. Comparison of Refinement in Space Group Pm3m Using Both Data Collections. % Atom X Y Z B Mult. Occupancy Srl 0 0 O .7 .0208 100 Sr2 0 .50 .50 .7 .0625 100 C1 .25 .25 .25 1.0 .1667 100 For the first data collection, transformed to cubic indices: R1 = (ZIFO-FCI/XFO) = .1832 for 186 data with |F|2 greater than 20(F2) _ 2 2 l/2 _ R2 — {Z wtg x IFO x Fcl )/Zwtg X F0} - .2999 For the second data collection: R1 = (ZIFO-FcI/ZFO) = .2379 for 97 data with |F|2 greater than 20(F2) _ 2 21/2- R2 — {X wtg x IFO X Fcl )/Z wtg. X F0} - .2666. 86 with face-centered symmetry, this may have biased the data in favor of the fluorite structure. The lower residual noted for the first data collection is perhaps due to the selective nature of the collection. 10 An attempt to use MULTAN to extract additional in- formation from the second data set was not successful. Because the reflections which violated the face-centered extinction conditions were relatively weak, assignment of phase relationships was difficult. Information obtained from MULTAN either reiterated that obtained from the Pat— terson synthesis or was of questionable validity. The cubic space group may place restrictions on refine- ment. As noted, an impasse was reached when all atoms were located in unrefinable special positions. Furthermore, the true symmetry of the unit cell is almost certainly not cubic: if the position of the trivalent heteroion is not random, this in itself would destroy the cubic symmetry. Because there was no evidence to suggest that the unit axes had been chosen incorrectly, the same lattice was considered in terms of other, less restrictive space groups. Data were merged using ORDER, and the discrepancy index of ORDER was assumed to be a reliable criterion of the correct- ness of the space group chosen. This strongly suggested that the true symmetry was triclinic, space group P1, however, since very few values were averaged in P1, the low discrepancy index may reflect only the small number of 87 equivalent positions. Subsequently, least squares fits were carried out with the entire data set merged according to P1 symmetry. Atomic positions were specified individually. It was hoped that this approach would (1) avoid biasing the data in favor of any particular structure and (2) eventually allow positional refinement. With all positional and thermal parameters fixed, the residual, R, was found to be 0.2028 for 665 data of inten- sity greater than 20 and Rw was found to be 0.2779. A dif- ference map revealed that all major remaining peaks were satellites of the atoms already positioned. Fixing the scale factor and positions while the thermal parameters were refined was not successful: all temperature factors became negative. The same sequence was tried with ORFLS with similarly discouraging results. It appeared that the high symmetry of the cell was limiting the efficiency of the programs. When the cubic unit cell was used, I was unable to refine either posi— tional or thermal parameters with LESQ, even when the system was described as triclinic. The discrepancy index values of ORDER had already suggested that describing the cubic cell in terms of lower symmetry would have little effect on the goodness of fit. For this reason, I attempted to lower the symmetry by actually changing the unit cell to tetragonal symmetry, using the relationships: 88 “tat = “cu + bcu “tot = 2“cu + Zbcu btet = ”acu + bcu and biz: = 'Zacu + Zbcu ate: 7 ccu tht = ccu The first is a face-diagonal of the cubic cell, while the second, supported by measurements of the precession films, equates btet and etet to twice the face diagonal of the cubic cell. Attempts to average the transformed data with ORDER resulted in residual discrepancy indices so large that it was obvious that the matrix multiplication cell transform was not operating correctly. The approach was abandoned. Until this time, two assumptions underlay all attempts at refinement: (1) that the residual discrepancy index of ORDER was a reliable guide to the true symmetry of the unit cell; and (2) that the additional positive charge of the tri- valent heteroion would be balanced by interstitial anions. Although the first assumption was reasonable in View of the quantified data collected by the diffractometer, the qualitative information from the precession films clearly indicated a high degree of symmetry. This information 89 casts doubt on the first assumption, as the continuing fail- ure to identify a different sized unit cell that would refine casts doubt on the second assumption. The SrC12 lattice is too closely packed to accommodate interstitial anions without displacing the parent structure anions. One would then expect to find reduced anion site occupancy and residual peaks in the difference synthesis at the locations of the interstitials. The difference syn- thesis had not revealed any obvious interstitial sites. Assumption (1) could be tested by merging the data in the lowest cubic symmetry and examining the results of further refinement. If Assumption (2) were to be discarded, another means of maintaining charge balance had to be postu- lated. This could be achieved by the deletion of ions: perhaps by the removal of three Sr2+ ions for every two Y3+ ions incorporated (a net loss of one cation) or by paired 2+ Schottky defects, with two Sr ions and a Cl' ion removed for every Y3+ ion included (net loss of one cation and one anion). Since there are twice as many anion sites as cation sites, if the second mechanism were in operation, a 2x% reduction in cation occupancy would accompany an x% reduction in anion occupancy. The entire set of reflection data was averaged in cubic symmetry, space group P23. Atomic positions were input according to the equivalent positions of space group Pm3m. Thermal parameters, positions, and 100% occupancy were all 90 fixed and the scale factor was refined. Then the scale factor was fixed and occupancy was refined. A sudden, dramatic decrease in the weighted residual was noted, and after two cycles of refinement, the oc- cupancies were: Atom, Occupancy Found Normalized Values 01' 199% 100% Sr2+(000) 102% 68% Sr2+(055) 112% 75% Clearly, these values indicated a differential in occupancy and suggested that partial occupancy of sites was a good model for the structure. To obtain a better value for the ratio of cationzanion occupancy, I selected several reasonable values and refined the scale factor for one cycle. Atoms were input in space group Pm3m. A minimum value of the residual was noted at 80% Sr:90% C1 and these values were chosen as a first ap— proximation of the relative occupancies. (See Table 9.) Atomic positions were assigned and refined individually. Cycles of refinement alternated between the occupancy, while the thermal parameters and scale factor were fixed, and the thermal parameters and scale factor, while the oc- cupancy was fixed. Variations in occupancy and thermal parameters appeared 91 Table 9. Discrepancy Indices for Refinements with Dif— ferent Cation:Anion Occupancy Ratios. INPUT POSITIONS Space Group Pm3m X Y Z Biso Srl 0 0 0 1.5 Sr2 0 .5 .5 1.5 Cl .25 .25 .25 3.0 R(F) Occupancy All Data 1 Sigma Data 100% .3997 .3027 Sr=90/Cl=95 .3999 .3029 Sr=80/Cl=90 .3937 .3016 Sr=70/Cl=85 .3965 .3096 92 among the twelve positions. It was clear that they were not equivalent. The results of refining the positions individually suggested that the anions were divided into two sets of different occupancy forming interpenetrating tetrahedra. Two refinements were begun at this point: one in space group P1 and one in space group P23. Refinement of positional parameters in space group P1 resulted in only insignificant shifts, and no further attempt was made to refine these values. The final dif- ference map on the Pl refinement did not indicate the ab- sence or gross mispositioning of any atoms, although the lowest valley of -5.7 eAIBin the difference map of the P23 refinement was suspect. Occupancy values gave a composi- tion of MCI2.17’ close to the expected stoichiometry. Only after this structure analysis had been completed did it become possible to analyze the crystal chemically. Quantitative metal determination had not been done earlier because of the small amount of compound available. The ampoule contents were frequently inhomogeneous and the sample for analysis had to be selected crystal by crystal using a microscope joined to a glove bag. About 30 crystals which were visually similar to the (Sr/Y)Cl2+x data crystal were submitted to Dr. W. E. Brasel- ton for analysis by induction coupled plasma spectrometry. The results of this investigation proved that phase 93 separation had occurred, for the SrzY ratio of the sample was 18:1, corresponding to a formula of MCl2 05 rather than the expected 5:1 ratio and MCl stoichiometry. 2.17 Subsequently, the (Sr/Y)C12+ data crystal and several x visually dissimilar crystals of (Sr/Yb)Cl2+x were analyzed. The results are listed in Table 10. The (Sr/Y)Cl2+X crystal on which data were collected was similar in composi— tion to the bulk sample analyzed previously. The cation ratio of the strontium-ytterbium mixtures was highly variable and visual appearance proved to be a poor indica- tion of composition. Hamilton's testfll provided no evidence that either solution was the more probable. To insure conformity with the analytic results, the P23 structure was constrained to the composition MC12.05. The final results of the Pl refinement and the con- strained P23 refinement are summarized in Table 11. The vacancy model solution of (Sr/Y)Cl been 2.05 had obtained empirically. Every attempt had been made to avoid biasing the refinement in favor of a predetermined solu- tion. However, an unprejudiced opinion is not necessarily true. The data were now approached in terms of the Willis 99-98 cluster model. Because the Willis cluster model was specifically developed for fluorite solutions in this com- position range, the investigation was undertaken even though the model required that certain constraints be placed on 99 Table 10. Results of Chemical Analysis by Induction Coupled Plasma Spectrometry. ppb Sample Yb Y Sr Ratio Sr/Yb-white 119.7(2.9) ---------- 1657.0(33.1) 27. Sr/Yb-clear 2 crystals 99.2(l.9) ---------- 827.6(16.6) l7. Sr/Yb-clear 1 crystal 35.9(7) ---------- 85.1(l.7) 9.7 Sr/Y-clear data crystal ---------- 83l.2(l6.6) 15756(315) 19. Sr/Y-clear bulk sample ---------- 210(9) 3750(80) 18.1 ICP operating conditions: 955 Plasma Atom Comp - forward power (kw) flame height, mm nebulizer pressure, psi argon coolant flow, 8pm auxiliary flow, £pm sample flow, Rpm Jarrell-Ash 1. 16. 19 18 0. 0 l Table 11. 95 The Vacancy Model Results of Refinement. Space Group Pl Positional and thermal parameters with associated esd's. % Atom X Y Z B Mult. Occupancy 011 .75 .25 .25 2.8(2) .95(3) 95.2 012 .75 .75 .75 1.3(2) .87(2) 87.9 013 .25 .75 .25 2.0(2) .89(3) 89.1 019 .25 .25 .75 1.8(2) .87(3) 87.9 015 .75 .25 .75 2.5(3) .72(3) 72.2 016 .75 .75 .25 2.8(3) .79(3) 79.1 Cl? .25 .75 .75 .2(1) .73(2) 73.6 018 .25 .25 .25 .8(2) .72(2) 72.9 Srl .00 .00 .00 1.09(6) .759(7) 75.9 Sr2 .00 .50 .50 1.02(6) .750(8) 75.0 Sr3 .50 .00 .50 .57(9) .767(7) 76.7 Sr9 .50 .50 .00 1.62(7) .753(8) 75.3 R1 R2 Shift-to-error ratios: Final difference map Final Cycle Refinement Indicators (ZIFO-Fcl)/XFO) = .0873 for 117 data with |F| than 20 (F2) {2(wtg x IFO—FC|2)/zwtg x 902}1/2 = .0839. maximum positive density (eA-B) maximum negative density (eA-3) average maximum .2836 .08815 2 greater .802 -1.976 96 Table 11. Continued. Space Group P23 Positional and Thermal parameters with associated esd's % Atom x Y z B Mult. Occupancy 019 .25 .25 .25 .80(6) .209(8) 62.8 018 .75 .75 .75 1 95(7) .310 93.0 Srl .00 .00 .00 .29(3) .0692(9) 77.0 Sr2 .00 .50 .50 1.31(2) .1892(9) 75.7 C1 set A includes positions .25 .25 .25, .25 .75 .75, .75 .25 .75, .75 .75 .25. Cl set B includes positions .75 .75 .75, .75 .25 .25, .25 .75 .25, .25 .25 .75. Final Cycle Refinement Indicators R1 = (ZIFO-FCI)/£FO) = .1013 for 218 data with |F|2 greater than 0 (F)2 _ 2 2 1/2 - R2 - {£(wtg x IFO—Fc| )/zwtg x F0 } — .0899. Shift-to—error ratios: maximum .3578 average .0597 Final difference map 1.958 —9.869 maximum positive density (e9-3) maximum negative density (eA-3) 97 the data. All reflections inconsistent with face centering were deleted from the data set because both U02.l3 and (Ca/y)— F2.10 had been solved in Fm3m symmetry. As indicated pre— viously, none of the primitive reflections deleted was very intense, and refinement of the face-centered data would provide a good indication of the validity of the model. To begin evaluation of the Willis cluster model for (Sr/Y)C12.05, coordinates and occupancy values were input exactly as given for (Ca/Y)F2.1O.u7 (See Table 12.) Positional, thermal and occupancy factors were fixed while the scale factor was refined. Then thermal parameters, anion multiplicity, the scale factor and the positional parameters of the displaced anions were refined separately with ORFLS in alternating cycles. Separate refinements were necessary because of the high correlation of many of the variables. Although fewer parameters were varied in this refine- ment than in that of the triclinic vacancy model, the residual of the Willis cluster model quickly fell below that obtained with the vacancy model. The total anion occupancy was constrained to equal 8.20 (9 x 2.05) for the entire unit cell. After refinement of the site occupancies with ORFLS had stabilized, refinement was continued with LESQ for evaluation of an empirically determined 98 * Table 12. Structural Parameters of Average Cells.“8 (Ca/Y)F2.10 Contribution to 2+x x Y z in (Ca/Y)F2+x Lattice F 0.25 0.25 0.25 l.88(.09) Interstitial F' .50 v v .l9(.03) Interstitial F" w w w .08(.03) v = 0.36(.01) w = 0.92(.01) Contribution to 2+x U02.12 in UO2+x Lattice O 0.25 0.25 0.25 l.87(.03) Interstitial 0' .50 v v .08(.09) Interstitial 0" w w w .l6(.06) v = 0.38(.01) w = 0.91(.01) * Data from Reference 98. 99 extinction factor. A final difference map did not indi- cate either the absence or gross mispositioing of any atom. Final cycle refinement indicators and the final values of all parameters are presented in Table 13. Refinement of the entire primitive data set was at— tempted in space group P93m, the symmetry of the composite Uu09 cell described by Williszus Refinement of thermal parameters was not possible: they fluctuated without restraint until the program failed. The result of the in- complete refinement is also summarized in Table 13. Direct, quantitative comparison by Hamilton's method?“ of the two refinements is not possible unless both are based on the same number of data. Since the P93m refinement was incomplete, the vacancy model was refined in space group F23, with final discrepancy indices of R1 = .0919 and R2 = .0711 for 119 data with F greater than 0(F). These values were compared to those for the Fm3m refinement of the Willis cluster model. Preliminary Investigation of (Sr/Yb)Cl2+x The bulk sample of (Sr/Yb)C12+x was visibly less uni- form than that of (Sr/Y)Cl2+x. At least three types of ytterbium—substituted species were seen: water clear crystals, white crystals which seemed to be of marginal quality, and a white powder. The clear crystals were chosen for X-ray investigation and precession photographs 100 mH~.I woo.a Amlmov mpfimcoo o>fiumwoc Esefixme “mummy zpfimcoo o>fipfimoo Esefixme Home oocogommflo Hmcflm sma. owmpo>m zmz. Esefixme "mofipmg Loopo o» amanm emeo. u mxagmom x meswxam_omxom_ x mesva u mm Ammv 6 echo aoemoam N_m_ seas some :HH hoe mmoo. u nomw\_omuom_wv u Hm muoa x mm.m u noeomo coaeoeaoxm (nun memo. Amvom. oo. oo. 00. pm mo. Aoaoaoo. Aa.mvm.m Amvfizm. Amvazm. Amvaam. mao am. onmaoo. Am.mvm.o Amvmoa. Amvmoz. om. mao ze.a memo. AmVHm. mm. mm. mm. HHo mo.mHoA»\amv ea Amo.mv .eHsz m N a x sop< or soapsoaaeeoo m.omo oopmfioommm spa: mgoposmomo Hmaoonp com Hmcofipfimom Emsm asopu oomam .eeoEoeaeom co noflsnom Hoooz toensfio naaaaz 0:9 .MH ofiome 101 .owao>coo po: op mcoficm Hmaufipmnopcfi one you whopoEmhma HmEhmne omma. Aflvmo. om. om. oo. mam maze. Amvma.m oo. oo. oo. + Above: 2:2:2 Willis cluster containing two normal anion vacancies (D ) two X' ( interstitials ), two X" interstitials (Q ), and cubic (5,8,8) sites (+). Below: A 3:9:2 Willis cluster. Figure redrawn from Reference 97. 108 from that of the atoms in the composite structure with partial occupancy of one or more sites. In this case, the symmetries of the substructure and the composite structure are very similar because the majority of the sites oc- cupied in the composite are identical to those of the fluorite subcell. This fact makes identification of the true cell extremely difficult. Because the vacancy model is also closely related to the fluorite cell it should not be rejected out of hand, even though the Willis cluster model appears preferable. The discrepancy index of the vacancy model is reason- able. Furthermore, it is possible to refine the entire data set, not merely the face centered portion, in terms of this model, and the difference map reveals no great peaks. Precedent exists for this type of structure. Other vacancy structures are well documented. ZrS is a non- stoichiometric compound with a range of homogeneity extend- ing from ZrO 67S to ZrSlO2.76 The ideal 1:1 compound has the NaCl-type structure, but X-ray examination of a single crystal 0f ZPO.77S revealed a superstructure of the NaCl cell with partial occupancy of certain sites, as summarized in Table 19 and Figure 18. The pattern of vacancies becomes apparent if ZrO 77S is described as a monoclinic superstruc- ture of NaCl. Titanium monoxide is another grossly nonstoichiometric species, with a composition range extending from T100 9 109 Table 19. Percent Occupancy of Zr Sites in ZrO 77S. Zr Site (Wycoff Notation) Position in Fm3m Occupancy Error a (000) 107 :8 b (55%) 27 :8 c (ktko,(3/9 3/9 3/9) 63 :8 d (0990. (20%). (093/9) (3/9 0 1/9) ((1/9 3/9/0) 79 18 e (£00), (0x0). 69 :7 (x00), (O—xO), 69 +7 f (xxx),(x-x-x),(—xx-x), (-x-xx),(-x-x-x),(-xxx), ((x-XX),(XX-X) 85 :7 O Zr positions a,b,c,d,f labels correspond to table 0 S positions. Figure 18. Illustration of the atomic positions in the monoclinic superstructure, space group C2/m. The monoclinic unit cell is shown by the heavy lines. The NaCl-type ZrS cells are shown as thin-lined cubes. Redrawn from Reference 57. 110 to TiO at temperatures below 990°C. Depending on com- 1.25 position, either the cation or anion lattice can be nearly perfect, with vacancies in the counterion lattice. Even in the 1:1 compound, TiO, about 15% of both cation and anion sites are vacant.77 The observed electron microscope data can be explained if half the titanium and half the oxygen atoms are alternately missing in every third (110) plane of the original cubic cell. The high percentage of vacancies in these species can be supported only by long range ordering. The equilibrium concentration of defects in a solid can be approximated, but this value is characteristically on the order of 10-7 even at 900°C. This calculation is based on short range electrostatic forces operating within an essentially un— changed lattice. Structural modulation arises from the interaction of long range forces, such as stress and strain, with the short range forces. The ordering of defects to produce a new structure permits an overall lowering of internal energy through a balancing of long and short range forces. In the vacancy model of (Sr/Y)012.05, occupancy values for both anions and cations appear to be less than 100%. Balance of charge appears to be achieved not by incorpora- tion of interstitial ions, but by the removal of ions. Anion occupancy as well as cation occupancy is reduced, and suggests that for each metal site occupied by Y3+, lll another metal site and a chloride site are vacated. The alternative scheme of removing three Sr2+ ions whenever two Y3+ ions are incorporated, seems less likely, as it cannot explain the partial occupancy of the anion sites. If the absences occurred on a purely random basis, all sites would be equivalent. If the procedure used in the data handling had any effect at all, it would tend to ob- scure this inequivalence rather than enhance it. Averaging the data to alleviate the absorption discrepancies would tend to make the sites more equivalent rather than less equivalent, and the persistence of the inequivalencies argues for their significance. The obvious inequivalence of both the cation and anion sites, especially in the P1 refinement, strongly suggests an ordered structure. As in the cases of ZrS and TiO, the high percentage of vacancies also supports the hypothesis of long range ordering: such a high concentration of disordered defects would be thermo- dynamically impossible. While the level of confidence is greater for the Willis cluster model than for the vacancy model, both hypotheses are consistent with a structure based on a fluorite-like subcell. There is reason to believe that long range ordering of these subcells exists, but unfortunately the refinements of these data do not reveal the extended period of the crystal. It is highly possible that superstructure reflections which would define a larger unit cell are present but are 112 too weak for easy detection. Other hypotheses can be ad- vanced to explain why the exact modulation of the struc- ture cannot be specified at this time: (1) Crystal imperfections, either specific to the crystal chosen for data collection, or characteristic of the bulk sample. A longer period of annealing would minimize crystal imperfections unless the composition of the crystal is at a boundary region between two phases of different structure. If the crystal represents such a metastable state, longer annealing will lead to continued separation of the phases. (2) Absorption problems obscuring the ordering. More powerful absorption correction techniques and grinding the crystal to a spheroidal shape before data collection would minimize this problem. Density measurements would unambiguously determine whether a vacancy model was reasonable for this structure. The calculated density of the vacancy structure is about 2.3 3, while the calculated density of a structure with g cm- interstitial ions is about 3.1 g cm'3, slightly greater than that of SrC12. Unfortunately, these measurementscould not be performed because of the difficulty of obtaining a sufficiently large, unhydrated sample of homogeneous com- pound. Each crystal should be selected individually and even in a glove bag the material is attacked by 113 atmospheric moisture. The experience of selecting crystals for ICP analysis indicated that 50 mg. of visually similar crystals would be a relatively large sample. On such small amounts of compound, inherent weighing errors and weight changes due to hydrolysis could easily exceed the dif- ference in weight calculated for the two models. The samples analyzed typically contained a much smaller percentage of trivalent ion than expected. Walker's work with SrCl278 suggests a solubility limit of 5-6% RECl3 in SPC12, and it appears that a separation of phases has occurred. Since analysis of visually dissimilar portions of the bulk sample did not identify a fraction rich in tri- valent ion, I conclude that the MCl3 component combined with the reaction container either physically or chemically and was discarded with the used ampoules. If precipitated YbCl3 were in contact with the glass, side reactions must be considered. Ayasse79 has charac- terized the compound Yb3(SiOu)2Cl, formed by a side reaction when YbOCl is heated under a chlorine atmosphere in a quartz container. He also noted, but did not characterize, another phase of "thin white sheets and slivers found adjacent to and on the tube wall". This description is consistent with the etching of the ampoule walls which occurred during the reaction of SrC12 and YbCl3 or YC13, Yttrium species analogous to the ytterbium products would not be unexpected. Chlorosilicate formation with SrCl2 would not be anticipated. 119 (Sr/Y)Cl2.05 is the first chloride formally described in terms of the Willis cluster and the first Willis cluster compound in its composition region to exhibit superstruc- ture reflections. The significance of the primitive re— flections must be emphasized. They appear to arise from the anion superlattice, and study of these reflections may be the key to understanding long range ordering in fluorite related phases of the composition MX2+x for o i x i 0.2, Neither the Vernier structures nor the ordered superstruc- tures discussed by Greis has been found in this composition region. It should be noted that the cubic fluoride phases may exhibit similar superstructures, but the chloride systems are more likely to yield to X-ray investigation than the fluorides because of the greater electron density of the chloride ion. The major significance of this work lies in the sc0pe of the future work it suggests: namely, an investigation of the phase diagrams for systems of the type SrClZ-MC13 with emphasis on the identification of ordered phases in the composition region of 0 to 25 mole percent MCl3. 67,80 A number of phase diagrams have specified a phase M3M'Cl9 where M=Sr or Ba, and M'=Ce,Nd,Pr,Sm or Pu. The identification of a Willis cluster structure for (Sr/Y)Cl2 05 now suggests that these M3M'019 phases may be analogous to the known structure of U909' It would also be interesting to look for a structural 115 basis for the eutectic found in these systems at about 90 mole percent M'Cl3. The evidence of an ordered structure for (Sr/Y)C12.05 suggests that the crystal chemistry of fluorite derivatives is even more complex than it has seemed. REFERENCES REFERENCES Hughes, B. B., Haltiwanger, R. C., Pierpont, C. 0., Hampton M. and Blackmer G. L. Inorg. Chem. 19, 1801. ’ ’ ’ AQQQ’ Popov, A. I., and Lehn, J.-M., "Physicochemical Studies of Crown and Cryptate Complexes" in Chemistry of Macro- cyclic Compounds, G. A. Melson, Ed., Plenum, N.Y., 1979. Cahen, Y. M., Handy, P. R., Roach E. T., and Popov, A. I., J. Phys. Chem., 1975, 79, 80. Cahen, Y. M., Dye, J. L., and Popov, A. I., J. Phys. Chem., 1975, 19, 1289. Cahen, Y. M., Dye, J. L., and Popov, A. I., J. Phys. Chem., 1275, 19, 1292. Smetana, A. J., and Popov, A. I., J. Solution Chem., 1299, 2, 183- van Remoortere, F. P., and Boer, F. P., Inorg. Chem., 1279, 2. 2071. Boer, F. P., Neuman, M. A., van Remoortere, F. P., and Steiner, E. C., Inorg. Chem., 1219, 13, 2826. £23,2768T.’ and Ward, D. L., Acta Cryst. Sec. B, 1976, RAWI COUNT - [TIME x (1/2)(Bl + B2)/CT] SIGI (COUNT + SIGBKG2)l/2 SIGBKG = SIGB12 x (l/2)(TIME/CT) _ 310312 is the greater of (Bl + 132)”2 and /2 (IBl - B2|) raw intensity 8101 standard deviation of RAWI B1 and B2 = background counts measured at the ends of a scan of CT seconds duration. COUNT = counts recorded while scanning for TIME seconds. After applying factors to RAWI to correct for Lorentz polarization, decay, and absorption (where necessary) the square of the structure factor (F80) and its standard deviation are obtained. 116 RAWI 117 Equivalent and duplicate data are combined to give averaged values for FSQ and SIGFSQ for unique reflec- tions: N FSQ = ( 2 av i FSQi)/N l N Z (SIGFSQ):]2/N i—l SIGFSQaV = [ When the scatter of data from the averaged value ex- ceeded SIGFSQav by a factor of four or more, SCATTER was used in place of SIGFSQav for this reflection. N SCATTER = {< ZIFSQi — FSQ (2)/[(N—1)N1}1/2 i=1 av The factors P and Q can also be used in the full- matrix least squares calculation to modify SIGFSQ: Q2Jl/2 SIGFSQmod = [SIGFSQ2 + (P X FSQ)2 + The structure factor, FOBS = (FSQ)l/2, can be cor— rected by using an extinction factor, EF: FOBScor = (1 + EF x RAWI) x FOBS The standard deviation of the structure factor (SIGMA) is given by the equations: /2 SIGMA SIGFSQl when RAWI : SIGI and, SIGMA FOBS - (FSQ - SIGFSQ)1/2 for FSQ Z SIGFSQ. In the least squares refinement, structure factors are weighted by l/(SIGMA)2. 10. ll. l2. l3. l9. l5- l6. 17. 18. 19. 20. 21. 22. 23. 29. 25. B, 1978, 39, 936. — mmmm '—— Sec. B, 1980, 36, 193. ———__ 'b’VL’b — 118 Main, P., Hull, 8. E. Lessinger, L., Germain, G., De Clercq, J. P., Woolfson, M. M. Multan 78, "A System of Computer Programmes for the Automatic Solu- tion of Crystal Structures from X-ray Diffraction Data", 1978, Univ. of York, England. These programs were acquired from Dr. Allan Zalkin by Dr. Donald Ward and modified for use on the MSU CDC 6500 and Cyber 750. Johnson, C. K., ORTEP, Report ORNL-3799, Oak Ridge National Laboratory, Oak Ridge, TN, 1965. Busing, W. R., Martin, K. O., and Lerry, H. A., ORFLS9 based on ORFLS, Oak Ridge National Laboratory, Oak Ridge, TN, June 1978. "International Tables for X—Ray Crystallography", Kynoch Press: Birmingham, England, 1979, Vol. IV, pp. 72 ff. Ibid, p. 199. Shannon, R. D., Acta Cryst. Sec. A., 1975, 3g, 751. Bush, M. A., and Truter, Mary R., J. Chem. Soc., Perkin, 1272. 391- Dobler,M ., Dunitz, J. D. , and Seller, P., Acta Cryst. Sec. B, 1979, 30, 2791. Raston, C. L., White, A. H., and Wilks, A. C., Aust. J. Chem., 1978, 31, 915. Griggs, C., Hasan, M., Henrick, K., F., Matthews, R. W., and Tasker, P. A., Inorganica Chimica Acta, 1277: 2.2, 129- Fair, C. K., and Schlemper, E. 0., Acta Cryst. Sec. Maluszynzka, H., and Okaya, Y., Acta Cryst., Sec. B, 197,7. 33. 3099. Enders, R., Keller, H. J., Martin, R., and Traeger, U., Acta Cryst., Sec. B, 1982, 35, 35. Drew, M. G. B., and Hollis, 8., Acta Cryst., Sec. B, 1980,36, 2629. 'VM'VL West, D. X., Pavhovic, S. F., Brown, J. N., Acta Cryst., 26. 27. 28. 29. 30. 31. 32. 33- 39. 35. 36. 37. 38. 39. 90. 91. 92. 119 Vance, T. B., Jr., and Holt, E. M., Acta Cryst., Sec. 5. 1,999. _3_§, 1950. Vance, T. B., Jr., Holt, E. M., and Holt, S. L., Acta Cryst., Sec. B, 9999, 36, 153. Mellors, G. W. and Sanderoff, S., J. Phys. Chem., 1222,.22, 110- Keneshea, F. J., Jr. and Cubicciotti, David, J. Chem. Engr. Data, 999%, 6, 507. Druding, Leonard F, Corbett, John D., and Ramsey, Bob N., Inorg. Chem. 9969, 3, 869. Mee, Jack E. and Corbett, John D., Inorg. Chem., 1965, u, 88. 'b'VL/b Corbett, John D., Pollard, David L. and Mee, Jack E., Inorg. Chem., 9969, 6, 761. Corbett, John D. and McCollom, Bill C., Inorg. Chem., 9288’ 19., 938- Caro, P. E. and Corbett, John D., J. Less-Common Met., 122%, 11. 1- dekner, Ulrick and Corbett, John D., Inorg. Chem., 1272. 11. 926- Druding, Leonard F. and Corbett, John D., J. Am. Chem. Soc., 999%,.63, 2962. Simon, Arndt, Mattausch, Hansjfirgen, and Holzer, Norbert, Angew Chem., 9919, 68, 685. Adolphson, Douglas G. and Corbett, John D., Inorg. Chem., 9999, 16, 1820. Daake, Richard L. and Corbett, John D., Inorg. Chem., 9999, 16, 2029. Poeppelmeier, Kenneth R. and Corbett, John D., Inorg. Chem., 9999, 16, 1107. Poeppelmeier, Kenneth R., Corbett, John D., McMullen, Tom P., Torgeson, David R. and Barnes, Richard G., Inorg. Chem., 9999, 19, 129. Bevan, D. J. M., "Non—Stoichiometric Compounds. An Introductory Essay", in Comprehensive Inorganic Chem— istr , Vol. IV ed, (Oxford: Pergammon Press, 1973) 953, ff. 93. 99. 95. 96. 97. 98. 99. 50. 51. 52. 53. 59. 55. 56. 57. 58. 59- 60. Brauer, G. 122%, 322’ Willis, B. Willis, B. Cheetham, Taylor, R. 222«, 1111, Cheetham, J. Phys. 120 , and Mfiller, 0., Z. Anorg. Allg. Chem., 218. T. M., Nature, 1969, 191, 755. T. M., Journal de Physique, 1969, 16, 931. A. K., Fender, B. E. F, and Steele, D. and F. and Willis, B. T. M., Solid State Com- Q. 171 A. K. Fender, B. E. F., 0.. 1211. 5. 3107. Fender, B. 293. Fishel, No Fishel, No and Cooper, M. J., E. F., "Theories of Non-Stoichiometry" in Solid State Chemistry, MTP International Review of Science,ilnorganic Chemistry, Series 1, vol. 10, 1972, rman A., Ph.D. Thesis, Michigan State Uni- versity, East Lansing, MI, 1970. rman A. and Eick, Harry Chem., 1911, 11: 1198. Luke, H. a Conf. 12th, nd Eick, Harr A. Proc. A., J. Inorg. Nucl. Rare Earth Res. 121% 1’“ Barnighaus 1219’ l, 9 en, R., Proc. Rare Earth Res. Conf., 12th., 09. L6ckner, U., Barnighausen, R., and Corbett, John D., Inorg. Chem., 1911, 16, 2139. Barnighausen, M. and Haschke, John M., 1111, 11, Haire, R. R. L., Pro Pezzoli, Paul A., 18. Inorg. Chem., G., Young, J. P., Peterson, J. R., Fellows, c. Rare Earth Res. Conf., versity, East Lansing, MI 1977. Greis, O. Greis, . 1111 L3" Greis, 0., Greis, 0., 13th, 1977. M. S. Thesis, Michigan State Uni- Z. Anorg. Allg. Chem., 1919, 936, 175. and Petzel, T. ,1. Anorg. Allg. Chem., J. Solid State Chem., Z. Anorg. Allg. Chem., 121 61. Gettman, W. and Greis, 0., J. Solid State Chem., 1978, 26, 255. mmmm 62. Greis, 0., Proc. Rare Earth Res. Conf., 19th, 1979, g, 167. m m 63. Kieser, M. and Greis, 0., Z. Anorg. Allg. Chem., 1980, 969, 169. ” ““ 69. Astakhova, S. S., Laptev, D. M., Kulagin, N. M., Zh. Neorg. Khim., 1911,,3g, 1702. 65. Astakhova, S. S., Laptev, D. M., Kulagen, N. M., Zh. Neorg. Khim., 1911, g3, 2003. 66. Polyachenok, 0. G., and MoroZov, G. 8., Russ. J. Inorg. Chem., 1969, 6, 1978. 67. T'en, Fam Ngok and Morozov, J. 8., Russ. J. Inorg. Chem., 1911, 16, 1215. 68, Fedorov, P. P., Izotova, 0. E., Alexandrov, V. B., - and Sobolev, B. P., J. Solid State Chem., 1919, 9, 368. 69. Sobolev, B. P. Seranian, K. B., Garashina, L. S. and Federov, P. P., J. Solid State Chem., 1999, 99, 51. 70. Shannon, R. D., Acta. Cryst., Sect. A., 1976, 32A, 751 . ’VVVL _""' 71. Wendlandt, Wesley W., J. Inorg. Nucl. Chem., 1997, 2, 118. m “ 72. Wendlandt, Wesley, J. Inorg. Nucl. Chem., 1999, 9, 136. “ 73. Taylor, M. P. and Carter, C. P., J. Inorg. Nucl. Chem. 1111s a, 38- 79. Hamilton, Walter C., Acta Cryst., 1999, 16, 502. 75. Belbeoch, B., Piekarski, C. and Perio, P., Acta Cryst., 1991, 19, 837. 76. Conrad, B. R. and Franzen, B. F., "The Crystal Struc- ture of ZrS and Zr0.1 S" in The Chemistry of Extended Defects in Non-Stoich ometric Solids, ed Eyring, LeRoy and O'Keeffe, Michael (Amsterdam: North-Holland Publishing, 1970), 207. 77. 78. 79. 80. 122 Watanabe, D. and Terasaki, 0., "Electronmicroscopic Study on the Structure of Low Temperature Modification of Titanium Monoxide Phase", Ibid, 238. Walker, P. J., J. Crystal Growth, 1378, fig, 187. Ayasse, Conrad, and Eick, Harry A., Inorg. Chem., 1213., .12, 11110- Leary, T. A., and Johnson, K. W. R., J. Metals, 1261, 13, 670. AN ANNOTATED BIBLIOGRAPHY OF CRYSTALLOGRAPHY This bibliography is intended as an introductory guide to some of the crystallography books which are readily available at Michigan State University. It is not intended to be comprehensive: many other volumes may be equally useful or perhaps of even greater value. The reader is encouraged to examine the longer bibliog- raphies included in most of these works. Ahmed, F. R., with Hall, S. R. and Huber, C. P., Crystal- lographic Computing, Copenhagen: Munksgaard, 1969. These articles were presented at an International summer school and hence address specific computing problems, detailed and specific discussion of computing methods used in crystallography. Ahmed, F. R., ed., Crystallographic Computinngechnigues, Copenhagen: Munksgaard, 1976. Highly technical and mathematical discussions of cur- rent computing practice. Articles written by the authors of the programs. Best detailed discussion of MULTAN, but the utility of the material is limited by its difficulty. Arndt, U. W. and Willis, B. T. M., Single Crystal Dif- fractometry. Cambridge: Cambridge University Press, 1966. Unintelligible due to the obscurity of excessive detail. Avoid it. Buerger, Martin J., Crystal Structure Analysis. New York: John Wiley and Sons, Inc., 1960. Fundamental crystallography written clearly and in detail. Contains both theoretical and "nuts and bolts" discussions. 123 l2“ . The Precession Method in X—Ray Crystal- lography. New York: John Wiley and Sons, Inc., 196H. Everything you always wanted to know about the preces- sion method but didn't have time to ask. Written by the man who invented it. The definitive discussion. . Vector Space and its Application in Crystal Structure Investigation. New York: John Wiley and Sons, Inc., 1959. Verbose. May tell you more than you really want to know. Excellent discussion of Patterson functions. Glusker, Jenny Pickworth and Trueblood, Kenneth N., Crystal Structure Analysis: a Primer. New York: Oxford University Press, 1972. Do not confuse simplicity with triviality. This is an excellent introductory and reference text. Good index, glossary and bibliography. "International Tables for X-Ray Crystallography". vol I-IV. Birmingham, England: Kynoch Press. Useful discussions of theory and methods as well as invaluable tabulated data. Index is at the end of Volume IV. Use it: it may save you hours of hunting for the original articles. Ladd, M. F. C. and Palmer, R. A., Structure Determination by X-Ray Crystallography. New York: Plenum Press, 1979. Similar to Stout and Jensen (below) but more up to date. Includes examples. Main, Peter, Hull, Se. E., Lessinger, L., Germain, G., DeClercq, J. P., Woolfson, M. M., MULTAN78: "A System of Computer Programmes for the Automatic Solution of Crystal Structures from X-Ray Diffraction Data". University of York, England, 1978. The manual maintained by Dr. Ward is well documented and includes specific references to the original publications. Understanding of the program is en- hanced by reading the manual. Rollett, J. 8., ed., Computing Methods in Crystallography. Oxford: Pergamon Press, 1965. Based on a lecture series in crystallography. Covers algebra, statistics, phase determination and programming. 125 The last section is out of date, but the rest is valuable and not overly obscure. Illuminating illus- trations. Stout, George H. and Jensen, Lyle, R., X-Ray Structure Determination. New York: Macmillan Co., 1968. Crystallographic theory as well as "how to" sections. Readable and well illustrated. APPENDIX 126 011-..r' (1.13192! VICI1DR. «Inngant 07.1n11"nx. Inc olivtltncts t! 1.2) ton In 1u- '11-1‘-rn3.n-n ~Inc~10nn'r Ice.o.91 = 2612 I. ~ <11~nv- !'n'"n=‘ "hIl'll\ 1! Ina. or; : ItonI-Ilcnl. . 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I n! 3 25 2 2 u 3. 1. .g. .3 g . .5. .1 12 . .. 1 1 19 9 11 n0 ~n 19 1o on 12 In a on 1 o 12 | o .3. n 21 2 -2 1 I. 9 2 2 19 g o 1 o 11 .3. .g 3 2 g g g 1; .1. 2 lo 10 1o 2n n. o 21 ~on 12 12 2 2n 2 n n 2 o 1 o 9 12 1 n0 2 2 1| ~2n 3 3. g .1 2 3 11 .1. .1 g 1. .1. 3 1| n 3.. 3 a... 1. 2 ~1 1. n on an o 1o ~30 1 9 1o .0 I 12 on o lo 2 o o o 0 non I In 9 on . 3. 2 01 3 1. n 9. o 1. . .. 2 n 11 ~2n . ~1o 2 12 ~20 ~2 In . ~on 19 11 ~.0 n..- 9. o n 9 19 ~19n 1 lo 1 2 o . n on o 2 lo 2' I 2. 2 2 n o 11 .10. 3 o n .1. 3 a 11 p. 9 ~19 I 11 I0 01 20 .u I. n I n .0 n u 1 2 I! n ... 1 n o ~90 o 29 n . 2. 1 2 9 1 II to 2 n 1 I. o n 12 .0 0.1: ~1. o 12 n0 o 12 19 |n~|o o n ~on 1 9 o .0 o o 12 -II0 2 99 1 o n 12 I on o o 11 ~n0 1 29 1 I o I. 1 ~90 3 o o ~90 9 n 11 .n 0n In ~12 o In ~2n 1 o 19 n~~1n 9 9 ~20 2 91 19 ~1nn 2 9 11 ~.0 9 9 9 I.Lo 1.. 9 V n 11 ~19 I 29 2 ~| 1 19 9 In n 12 I .. . 12 . Ion ~2 9 ~12 o 11 ~2n 2 19 . 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I on 12 . 12 ~2n lo 2 1 9n o 9 o ~2~~1| o 2 ~90 ~32 o o ~90 Int! 11. 2 1o 2 I 10 01 9 I3 ~10 9 o 0. 99 2 9 . n 12 ~|on ~1 9 2 9 9 2o 2 n2 Iotn 9. . 1 o 0 ~10 1 2o 2 2 ~1o 19 o [In .31 . 1| s..12 1. .n 11 1 . 1. .. . 12 2. n I: 0. o 9 ~90 9 n 11 n20 0. 1n 1 ~9 lo I o 9n~12 1 11 ~|n 12 o lo ~n0 2 1n . . ~9 o 1| ~10 .33 1. 1 5...; I 1. I. I... 1.. 3 ~3 I 1. on n.tx ~n 29 2 I|n 1o 11 9 on ~9 lo on 11 n 9 090-10 In 9 1.0 12 9 9 on 2 lo ~|n ~. 9 12 ~00 .. 23 2 ~| ~1o 12 1| 20~Io I 2 30 n. . In ~10 01 12 n! 19 9 ~90 Intn 1. 9 n. 11 1 ~| 12 1 12 .0011 n 11 ~10 19 lo 9 on n 9 ~20 ~1 11 12 ~on .. a 1. .3. .n I 11 .1. 0 1 11 20 ~2 In 11 on ~2 9| 9 n 09 In Io Ion ~| 29 | 2 1| 9 12 10 o .0 Intn lo. I Iotn 11. n. 12 11 on -1 o 9 ~ . .. I 11 ~3. .. 2 I In ~2 11 11 1n I..- nl 12 9 .0 ~o o In ~20 n2 1. 1 ~1 19 o 11 n 11 ~1|n~|n o o on o n .0 ~o n 11 ~on .. . a 1. .1 I 1) ~|n n1 . n .- ~1 o 11 ~20 o n o .1 1 n1 n1 9 1. In ~| 1n 2 . 19 12 9 o 11 .3..33 o o .0 1 o ~2n ~n o 19 ~10 .9 o o 0.. .. 12 19 s. .. 1a 1: .9. o |. 3 1.- 1 2 1 1n n ~9n 0. ll 1 .0 o no I Ionn 9. I ~2 o 11 ~2n~12 o n ~10 2 n 9 on ~2 o 19 on n. o 1 ~90 09 In 11 Ion ~9 1o 2 2 1 o 11 ~.0 2 n 2 n0 2 ~12 n9 1 1| 10 1 o . ~00~1I 12 1 on no 21 o n~~11 29 2 o I o n ~on 0. o 29 ~n. .3 3 3 3. o. 2. 1 0. . o I 0.0 2 n I. ~10 n..= o 19 n 20 n. o 11 ~20 2 19 2 9 ~19 1o In on ~n 9 1| ~2n~|o 1n 2 ~9 n 12 1o ~|n ~1 22 1 no .3 .3 3 .1 .3 o 12 on. 03 11 12 In 3 . 1. .0 ~1 . n 1. n n .3 o 11 ~20 2 92 1 1 ~12 11 II on ~. 11 n 1 .0 o In 19 non o o 11 non .3 . 1 2. .2 1. . on 02 1n 2 2 n I 12 ~10 o o 9 on 2 2 02 . 1| ~2n n 1o 2 9 ~12 2 11 ~1|n no 1 1o ~.0 no 19 2 ~1 . 2 9 on 1 22 n 1 . 2 1| ~90 n1 1 1| 09- 0| 1 n ~2n 9 1| 9 1n- 1 n o 2. 2 I ~| o 12 ~n0 9 1o 2 9 ~11 19 o ~on ~2 1| 2 In n! 22 1 ~| 1 12 9 ~lnn 2 n 19 on 1 21 o o I. 090 o o 1 ~10 o I 12 90 2 2o 2 9 n 11 9 0.0 . 22 2 no ~1o 12 ~n~ ~| lo 9 .0 ~o 29 | ~| o 21 n n 2 11 o 90 g 3. 3 .| 3 n 13 1. u 1. 2 ~| 1 9 I) 20 2 n9 . ~1 | . 1| ~.0 2 19 2 n ~9 1o 9 Ion o 12 o In ~9 29 1 1 9 22 I . n 1. 19 On 3 g .3 .3. 2 3o 12 9. 2 a n .3. I... 13. n n o 11 ~90 2 II I. 10 n n 9 o0 0. o lo ~20 1 .1 o n -n 9 2 n0 on o 12 ~90 n o ~00 . 1. 3 n 13 ... 2 1. 9 ~10 0. In 12 90 —. .—...—.. . JOOOHU’IUONdu. a-vonuanooonnnav-oouwao I. lbuO~b~uVflu¢luOuodu’o.‘u‘ul~o O I-'-.:J\I.I,‘ 128 OBSERVED STRUCTURE FACTORS9 STANDARD DEVIATIONS9 AND DIFFERENCES (X 1.0) FOR (SR/Y1CL2.059 VACANCY MODEL F08 AND FCA ARE THE OBSERVED AND CALCULATED STRUCTURE FACTORS. = IFOB/ - IFCA/o $6 = ESTIMATED STANDARD DEVIATION OF F08. * INDICATES ZERO UEIGHTED DATA. H FOB K9L= 1 0 K9L= 1 62 K9L= 1 1 K9L= 1 52 K9L= 2 18 K9L= 2 1 K9L= 2 1 K9L= 2 1 K9L= 2 1 K9L= 2 27 K9L= 2 1 K9L= 2 100 K9L= 2 1 K9L= 2 22 K9L= 3 1 K9L= 3 1 K9L= 3 1 K9L= 3 1 K9L= 3 70 K9L= 3 1 K9L= 3 67 K9L= 3 0 K9L= 3 1 K9L= 3 2 K9L= 3 2 K9L= 3 0 K9L= SG DEL 09 O 1 0* 19 '1 3 '33 19 0 0 0 19 1 25 -Q7 09 0 3 7 09 1 O 0 19 '1 1 0* 19 0 1 0* 19 1 1 0* 29 '2 1 11 29 '1 1 0* 29 0 8 '53 29 1 2 0* 29 2 8 6 09 0 1 -1* 09 1 2 -1* 09 2 1 0* 19 '2 1 0* 19 '1 7 -9 19 0 1 0* 19 1 7 -9 19 2 2 -1* 29 '2 2 -1* 29 ‘1 1 0 29 0 1 0* 29 1 2 -2* 29 2 H FOB SG DEL H FOB SG 0 3 1 2 -1* K9L= 39 K9L= 39 '3 Q 2 2 3 63 1 5 K9L= 39 K9L= 39 '2 Q 3 1 3 2 1 0 K9L= 39 K9L= 39 '1 Q 3 1 3 66 6 0 K9L= 39 K9L= 39 0 Q 2 3 3 2 1 0 K9L=. 39 K9L= 39 1 Q 3 2 3 61 7 -6 K9L= 39 K9L= 39 2 Q 3 1 3 2 2 0* K9L= Q9 K9L= 39 3 Q 88 5 3 56 11 '2 K9L= Q9 K9L= O9 0 Q 2 3 Q 98 13 '26 K9L= Q9 K9L= 09 1 Q 20 2 Q 2 1 0 K9L= Q9 K9L= 09 2 Q 2 3 Q 20 1 Q K9L= Q9 K9L= 09 3 Q 87 6 Q 2 2 -1* K9L= Q9 K9L= 19 '3 Q 2 3 Q 2 2 -1* K9L= Q9 K9L= 19 '2 Q 19 2 Q 2 2 “1* K9L= Q9 K9L= 19 -1 Q 2 3 Q 3 1 1 K9L= Q9 K9L= 19 0 Q 82 9 Q 1 2 -1* K9L= 09 K9L= 19 1 5 2 2 Q 2 1 0 K9L= 09 K9L= 19 2 5 1 2 Q 2 2 -1* K9L= 09 K9L= 19 3 5 2 2 Q 2 0* K9L= 09 K9L= 29 '3 5 3 2 Q 2 2 “1* K9L= 09 K9L= 29 ‘2 5 1 3 Q 98 9 '9 K9L= 19 K9L= 29 '1 5 3 3 Q 2 2 0* K9L= 19 K9L= 29 0 5 55 5 Q 19 2 2 K9L= 19 K9L= 29 1 5 2 2 Q 2 1 O K9L= 19 K9L= .29 2 5 58 6 Q 95 10 '12 K9L= 19 K9L= 29 3 5 2 2 Q 2 3 '1* K9L= 19 K9L= 39 '3 5 57 5 Q 3 2 0 K9L= 19 DEL EL -2 -1. '1 -1. -1. H FOB 5 1 K9L= K9L‘ U! "I m x at :x x C O O C r- r r- r 1| UI 0| 2 0 0| 2 o x x O O mt- r'ar- r lib-"tall~l00\"(nIINIIOIuluIIMIlh-I X . r K UI O U”- H 56 DEL 2 19 q 19 3 29 3 29 1 29 2 29 1 29 2 29 2 29 3 29 -1. 3 1 q -19 -9 0 -3 0 -2 0 -1 0 0 -1. 1 -1. 2 -1. 3 0 Q -1. -4 -1. -3 0 -2. -3 -2 -1 09 -1. 09 F(09090) = 22Q H FOB SG DEL K9L= Q9 2 5 Q 2 0 K9L= Q9 3 5 Q 1 O K9L= Q9 Q 5 3 3 -1* K9L= 59 '5 5 3Q 3 1 K9L= 59 ‘Q 5 2 Q -2* K9L= 59 '3 5 Q1 3 2 K9L= 59 '2 5 2 3 -2* K9L= 59 '1 5 QQ 3 2 K9L= 59 0 5 3 3 -1 K9L= 59 1 5 Q2 Q *1 K9L= 59 2 5 2 3 -1* K9L= 59 3 5 Q2 2 3 K9L= 59 Q 5 Q 5 0* K9L= 59 5 5 33 1 0 K9L= 09 0 6 19 2 5 K9L= 09 1 6 2 3 -1* K9L= 09 2 6 83 5 -1 K9L= 09 3 6 3 2 0 K9L= 09 Q 6 13 0 1 K9L= 09 5 6 3 3 -1* K9L= 19 '5 6 3 Q 0* K9L= 19 'Q 6 3 3 -1* K9L= 19 '3 6 2 3 *1* K9L= 19 ‘2 6 3 2 0 K9L= 19 '1 6 Q 1 1 K9L= 19 0 6 1 2 -2* K9L= 19 1 STRUCTURE FACTORS CONTINUED FOR (SR/YDCL29059 VACANCY MODEL H F08 SG DEL 3 -1* 19 2 3 0* 19 3 2 0 19 Q 2 0 19 5 2 1 29 '5 Q -2* 29 “Q 5 -1 29 '3 3 '1* 29 '2 2 1 29 '1 3 0* 29 0 7 10 29 1 2 0 29 2 0 1 29 3 3 -2* 29 Q Q *1 29 5 2 0 39 -5 Q -1* 39 'Q 2 0 39 '3 2 0 39 '2 Q -3* 39 '1 3 -2* 39 0 2 -2* 39 1 Q -2* 39 2 3 -1* 39 3 3 -1* 39 Q Q 0* 39 5 Q 0* Q9 '5 Q -2* Q9 -Q 1 2 H FOR 80 DEL K9L= 6 Q K9L= 6 69 K9L= 6 Q K'L: 6 13 K'L: 6 3 “'L: 6 67 K9L= 6 Q KQL: 6 13 K9L= 6 Q o m x x x O C O r r r H N I 0‘ X C P mllOllOIIOTIOIIOOIOlll-l x . P 0‘ X C r UHSIIU‘HDH 0‘ X C bl" 0" K . r- 1 Q9 4 Q9 4 Q9 4 Q9 0 Q9 4 Q9 6 Q9 4 Q9 1 Q9 5 59 1 59 4 59 4 59 4 59 4 59 3 59 2 59 3 59 2 59 5 59 6 69 6 69 5 69 3 69 5 69 1 69 Q 69 4 69 3 69 -3 09 -2 O -1 09 O 0 1 -l. 2 -3 3 09 4 2 5 09 -s 3 -9 -1. -3 09 -2 0 -1 a 0 ill m9-O\H£nCLah-u¢:hac9dcao 9 9 -1. -4 09 *2 *1 -1. NHHNO 129 H F09 6 11 K9L= 6 3 K9L= 6 53 K9L= 6 5 K9L= 6 13 K9L= 7 2 a q q x x x x. O O C O P r' l' r N N Q s] X X X X 0 O O O I'- l' f" LfllO-DIIUIIUDIUIIUHOHUH 'u w u x x z 0 C O P.9r’ rcur' "TOIIOIIUII SG DEL 1 1 69 3 Q -1* 69 Q 1 Q 69 5 5 1* 69 6 2 Q 09 0 3 -1* 09 1 3 °1* 09 2 1 1 09 3 2 0 09 Q 3 '1 09 5 Q *1* 09 6 Q -1* 19 '6 5 0* 19 '5 3 2 19 'Q Q -1* 19 '3 3 1 19 '2 2 0 19 ’1 3 *2 19 0 3 -1* 19 1 3 '3 19 2 3 ‘1* 19 3 3 '1 19 Q Q 0* 19 5 2 1 19 6 2 1 29 -6 5 1* 29 *5 Q -1* 29 -Q Q -1* 29 '3 3 '1* H FOB q ‘1 N s! ‘l ‘1 ‘1 N ‘1 X c r X o r X C r 2 o r x X x C . . X o r C C C O O r r' r r- r TaunmonMIIUIucuu.pnuannwuuuaucuu O P 0‘ O C P o o 0 PP?“ UIIIU‘NUIIUNONO "9‘ "MI! 86 DEL H FOB 29 '2 7 3 Q 0* K9L= 29 '1 7 Q 2 2 K9L= 29 0 7 5 2 1 K9L= 29 1 7 3 2 0 K9L= 29 2 7 Q 3 0 K9L= 29 3 7 7 Q -1* K9L= 29 Q 7 5 2 1 K9L= 29 5 7 5 Q '2* K9L= 29 6 7 26 Q 1 K9L= 39 '6 7 5 2 1 K9L= 39 '5 7 30 2 '1 K9L= 39 -Q 7 6 2 1 K9L= 39 *3 7 3Q 3 2 K9L= 39 '2 7 1 2 0 K9L= 39 '1 7 32 3 0 K9L= 39 0 7 3 3 '1‘ K9L= 39 1 7 32 Q '1 K9L= 39 2 7 1 Q -1* K9L= 39 3 7 30 2 2 K9L= 39 Q 7 Q Q -1* K9L= 39 5 7 5 1 1 K9L= 39 6 7 Q 5 -Q* K9L= Q9 '6 7 5 3 2 K9L= Q9 '5 7 2 Q -1* K9L= Q9 *Q 7 5 5 '1. K9L= Q9 '3 7 Q 2 1 K9L= Q9 '2 7 5 3 1 K9L= Q9 '1 7 6 Q -2* K9L= Q9 0 7 6 PAGE 2 SG DEL Q 09 Q9 1 2 0 Q9 2 2 1 Q9 3 5 -1* Q9 Q 5 0* Q9 5 2 3 Q9 6 6 1* 59 *6 6 1* 59 '5 1 0 59 -Q 6 1* 59 ’3 2 '1 59 '2 1 2 59 '1 3 1 59 0 3 -3* 59 1 3 '1 59 2 Q -1* 59 3 1 1 59 Q 5 -3* 59 5 1 3 59 6 5 0* 69 *5 5 0* 69 -Q 5 0* 69 ‘3 5 0* 69 *2 Q -2* 69 ‘1 2 1 69 0 Q 0* 69 1 6 1* 69 2 2 2 69 3 3 2 130 STRUCTURE FACTORS CONTINUED FOR (SR/YDCL29059 VACANCY MODEL H FOB SG DEL H F08 SG DEL H F08 SG DEL H F08 K9L= 69 Q 8 2 5 *3* K9L= Q9 *3 9 Q 7 1 6 *3* K9L= 29 *6 8 2 Q *2* K9L= K9L= 69 5 8 Q2 3 0 K9L= Q9 *2 9 5 7 7 3 2 K9L= 29 *5 8 10 1 0 K9L= K9L= 79 *3 8 5 Q 1 K9L= Q9 *1 9 Q 7 25 1 1 K9L= 29 *Q 8 3 Q *1* K9L= K9L= 79 *2 8 10 2 0 K9L= Q9 0 9 2Q 7 Q 5 0* K9L= 29 *3 8 Q9 3 *5 K9L= K9L= 79 *1 8 3 Q *1* K9L= Q9 1 9 5 7 26 1 0 K9L= 29 *2 8 5 5 1* K9L= K9L= 79 0 8 57 Q *1 K9L= Q9 2 9 27 7 3 5 *1* K9L= 29 *1 8 11 1 1 K9L= K9L= 79 1 8 5 2 1 K9L= Q9 3 9 5 7 2Q 1 *2 K9L= 29 0 8 3 59 *1* K9L= K9L= 79 2 8 11 1 0 K9L= Q9 Q 9 29 7 7 2 2 K9L= 29 1 8 Q6 2 1 K9L= K9L= 79 3 8 Q Q 0* K9L= Q9 5 9 3 7 27 1 2 K9L= 29 2 8 Q 7 0* K9L= K9L= 09 0 8 56 Q *3 K9L= 59 *Q 9 29 8 6Q Q 0 K9L= 29 3 8 2 6 *2* K9L= K9L= 09 1 8 3 Q *1* K9L= 59 *3 9 3 8 Q 3 0 K9L= 29 Q 8 Q 6 0* K9L= K9L= 09 2 8 9 1 *1 K9L= 59 *2 9 27 8 11 1 0 K9L= 29 5 8 Q 3 0 K9L= K9L= 09 3 8 5 Q 1 K9L= 59 *1 9 3 8 3 Q 0* K9L= 29 6 8 6 3 2 K9L= K9L= 09 Q 8 Q2 1 1 K9L= 59 0 9 23 8 53 3 *1 K9L= 39 *6 8 5 5 1* K9L= K9L= 09 5 8 6 5 2 K9L= 59 1 9 Q 8 5 3 1 K9L= 39 *5 8 5 5 0* K9L= K9L= 09 6 8 5 Q 1 K9L= 59 2 9 Q 8 11 1 2 K9L= 39 *Q 8 5 5 1* K9L= K9L= 19 *6 8 5 2 1 K9L= 59 3 9 3 8 3 5 *1* K9L= 39 *3 8 Q 5 *1* K9L= K9L= 19 *5 8 Q Q 0* K9L= 59 Q 9 6 8 3 Q *1* K9L= 39 *2 8 5 7 0* K9L= K9L= 19 *Q 8 Q Q 0* K9L= 69 *3 9 Q 8 3 Q *1* K9L= 39 *1 8 Q 5 0* K9L= K9L= 19 *3 8 Q 2 0* K9L= 69 *2 9 6 8 3 5 *1* K9L= 39 0 8 Q1 3 *1 K9L= K9L= 19 *2 8 5 2 1 K9L= 69 *1 9 Q 8 Q 3 1 K9L= 39 1 8 7 2 K9L= K9L= 19 -1 8 1 5 -39 K9L= 69 9 2 8 3 Q *1* K9L= 39 2 8 8 1 * K9L= K9L= 19 0 8 1 Q *3* K9L= 69 9 Q 8 2 Q *1* K9L= 39 3 8 5 3 K9L= K9L= 19 1 8 5 Q 1 K9L= 69 9 6 8 Q 2 1 K9L= 39 Q 8 Q2 2 K9L= 3 0 1 1 1 2 0 K9L= 19 2 8 Q 5 0* K9L= 69 3 8 ‘3 Q -1* K9L= 39 -5 8 3 5 -1* K9L= K9L= 19 3 8 3 5 -29 K9L= 09 o 8 Q 3 0 K9L= 39 6 9 2 Q -2 K9L= 19 Q 8 7 2 3 K9L= 09 1 8 2 Q *2* K9L= Q9 *5 9 5 2 1 K9L= 19 5 8 Q 5 09 K9L= 09 2 8 Q 5 09 K9L= Q9 -Q 9 Q 5 -1 K9L= 19 6 8 Q3 3 -2 K9L= 09 3 SG DEL 5 09 09 Q 1 5 09 -5 -1 -4 1 *3 *1 *2 0 *1 *2 0 -1. 1 *2 2 -1. 3 -2 4 -1. 5 -1 -s -19 -9 09 *3 -1. *2 2 -1 09 H F08 K9L= 9 Q K9L= 9 26 K9L= 9 3 K9L= 9 2Q ~n o \D o x a: x O O O P r r' Ulld‘"«DIIOIUUIHCJIIU|Hhafl X o r o a x x O O r r m X X o o P F PAGE 3 SG DEL 39 0 Q 0 39 1 2 *2 39 2 5 *1* 39 3 1 *2 39 Q 5 *2* Q9 *3 5 0* Q9 *2 5 1* Q9 *1 3 1 Q9 0 5 09 Q9 1 5 0* Q9 2 3 2 Q9 3 5 *1* 59 *2 7 *1* 59 *1 1 *2 59 0 Q 0* 59 1 1 *2 59 2 5 *1* 09 D 2 *2 09 1 5 0* 09 2 1 *1 09 3 3 2 19 *3 3 2 19 *2 6 1* 19 *1 3 1 19 0 5 *2* 19 1 2 2* 19 2 3 3 19 3 6 *2* *2 131 OBSERVED STPUCTUKL FLCTUNS9 (SP/712L2915 iTAthRL DEVIATICCS9 HILL!“ CLUSTER AND DIFFERENCES (X2397) FOR F(39090) = 17Q2 FOB AND FCA ARE THE OrSERVED AND CALCULATED STRUCTURE FACTORS9 $6 = ESTIMATED STANDARD DEVIATIOL 0F FOB. DEL = IFDE/ * IFCA/o * INDICATES ZERO HEIGHTED DATA9 L FOB SG DEL L FOB SG DEL l FOL SG DEL H9K= 19 1 *2 117 1? T ‘ 86 Q *2 *1 569 29*119 3 61C 57 *2? 2 Q5? 35 *1Q 1 Q5Q219-23Q 2 115 3 *2 h 72 11 *1 H9K= 29 C Q 566 34 21 t 338 5 *Q 0 1QQ 25 *Q2 H9K= 69 Q H9K= 89 Q H9K= 29 2 *Q 106 6 9 *Q 3QQ 26 *2Q *2 216 12 32 *2 568 37 2? * 79 8 5 0 987 78*151 C 13C 3 *1 39C 27 *38 2 178 62 *5 2 5Q7 Q9 2 2 5 7 12 H9K= 39 1 Q 10! E 1; Q 368 16 0 *1 610 63 5 H9K= 69 6 H9K= 89 6 1 58Q 62 *20 *6 63 Q5 3 *2 328 22 *1Q H9K= 39 3 *Q 3°Q 26 *2 " 62 11 0 *3 526 9 63 *2 9Q 5 5 2 336 18 *6 *1 552 Q8 39 '0 Q88 29 22 P9K= 99 1 1 51Q 6O 1 2 89 6 10 *5 191 5 0 3 Q58 93 *5 Q Q23 5 27 *3 210 15 *11, H9K= Q9 0 6 99 16 39 *1 230 12 *10 0 9DQ12Q *Q H9“: 79 1 I 228 11 *13 2 160 11 *Q *5 275 21 1Q 3 212 Q *9 ,H9K= Q9 2 *3 318 28 7 F 183 6 *9 *2 872 79 76 *1 326 26 *9 H9K= 99 3 0 1Q9 13 *15 1 319 23 *16 *3 193 5 *16 2 8Q0 89 QQ 3 307 2C *3 *1 209 1Q *12 H9K= Q9 Q 5 268 1Q 7 l 215 17 *11 *Q 7Q2 Q5 1Q8 H9K= 79 3 3 189 5 *19 *2 158 12 27 *5 238 17 *3 89K: 99 5 0 7QQ 51 31 *3 318 27 21 *1 182 5 *9 2 152 13 21 *1 313 26 2 1 176 5 *16 Q 690 75 96 1 305 28 *5 H9K: 109 D H9K= 59 1 3 299 19 20 5Q 18 *8 *3 QQ5 37 3Q 5 2Q6 5 5 2 323 5 *20 *1 Q78 50 I9 H9K= 79 5 H9K: 109 2 1 Q68 Q1 9 *5 208 Q 3 *2 5 22 *3 3 Q28 36 16 *3 239 15 *2 ‘ 31Q 5 *29 H9K= 59 3 *1 268 2C 7 2 7C 13 11 *3 372 32 7 1 256 23 *5 *1 Q50 33 38 3 250 5 9 1 Q17 Q7 5 5 235 6 3C 3 369 27 5 H9K= 79 7 H9K= 59 5 *3 200 5 8 *5 268 21 10 *1 209 5 Q *3 330 23 2Q 1 191 6 *1Q *1 356 27 19 3 211 5 19 1 337 28 *1 H9K= 89 3 3 337 16 31 O 517 3Q 1Q 5 261 6 3 2 89 5 f H9K= 69 0 Q Q25 25 *3 0 152 16 15 6 87 10 2Q 2 703 Q2 62 H9K= 89 2 Q 106 3 5 *6 33Q 23 *8 H9K= 69 2 *Q 80 12 6 *Q 563 Q2 18 *2 Q60 36 *Q