THE STRUCTURESOF BINARY CRYSTALLINE FLUORIDES Thai: for tho Dean. of Ph. D. MICHIGAN STATE COLLEGE Wiliiam Gordon Thomas 1954 may: I ___-._.l-L ’1 W LffiRARY A 1- (C; Q I v I r \ ' ' -‘ I / I Z ' .Jih";1‘.n by» ' , I Univt’fi‘“ I)" . 1 ”'4'" '0" I THE STRUCTURES OF BINARY CRYSTALLINE FLUORILES BY William Gordon Thomas A THESIS Submitted to the School of Graduate Studiee of Michigan State College of Agriculture and Applied Science in pertinl fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemistry Year 1954 .'/ J chq ;1 ‘11! .51 $1 X ' 3 Acknowledgements The author wishes to eXpress his sincere thanks to Dr. M. T. Rogers, under whose constant supervision and unfailing interest this investigation was undertaken and to whom the results are herewith dedicated. He is also greatly indebted to Dr. A. J. Smith for the use of the X-ray diffraction equipment of the Metallurgical Engineering Department. Grateful acknowledgement is also due to Dr. J. L. Spiers for hie valuable aid in the construction of equipment and guidance in experimental procedures, and to Drs. L. L. QUill, De Te E'ing, Fe Ba Dutton, Ks Gs Stone, Jr., Be He Dickinson, P. L. Lressel, and others for their helpful suggestions and assistance in one way or another. The investigator extends his sincere thanks to Mrs. Frances McDermott for the preparation of many of the charts and diagrams included in the dissertation. William Gordon Thomas candidate for the degree of Loctor of PhilosOphy Dissertation: The Structures of Binary Crystalline Fluorides mtline of Studies ' Major subject: Physical Chemistry minor subjects: Inorganic Chmmdstry, Physics Biographical Items Born, July ll, 1917, Ipswich, South Dakota Undergraduate Studies, South nakota State College, 1935-1939 Graduate Studies, South Dakota State College, 1946, University of Eichigan, 1946-49, Michigan State College 1950-1954 Experience: member United States Army, 1939-46, In- structor in Chemistry, South Dakota State College, 1946, Teaching Fellow in Chemis- try, University of Michigan, 1946-49, As- sistant Professor, Board of Examiners, Michigan State College, 1949-53, Assistant Professor in Chemistry, Central Michigan College of Education, 1955-54 Member of American Chemical Society 11 Table of Contents List of Tables....................................... iv List of Figures...................................... vi I. Introduction................................... 1 II. Historical Background.......................... 2 A. Elements of Crystal Structure.............. 2 B. Structure of Binary Crystalline Fluorides.. 10 III. Experimental................................... 26 'A. Preparation of Materials................... 26 B. Preparation of Samples..................... 32 C. X-ray Technique............................ 38 IV.C Theoretical.................................... 47 A. Determination of Unit Cells................ 47 B. Determination of Space-groups.............. 57 0. Determination of Atomic Positions.......... 59 V. Results and Liscuseion......................... 76 A. Manganic Fluoride.......................... 76 B. Silver Difluoride.......................... 112 0. Chronic Fluoride Trihydrate. . . . . . . . . . . . . . . . 122 D. Chromium.Tetrafluoride..................... 161 E. Chronic Fluoride........................... 151 F. Stannic Fluoride........................... 159 VI. Summary........................................ 145 List of Symbols and Abbreviations.................... 147 utor‘turo CitedOOIOOOOOOOOOO0.0.0.000...00.0.0000... 150 iii I. II. III. IV. V. VI. VII. VIII. X. XI. XII. XIII. XIV. XV. XVI. xv11.' List of Tables Characteristics of crystal systems........... Structures of binary monofluorides........... Structures of binary difluorides............. Structures of binary trifluorides............ Structures of higher binary fluorides........ X-ray exposure data.......................... Diffraction pattern of manganio fluoride with iron K1.radiation....................... Diffraction pattern of manganic fluoride with molybdenum xx radiation. . . . . . . . . . . . . . . . . Diffraction patterns of manganic fluoride and cobalt trifluoride....................... Proposed unimolecular cells.............f.... Interplanar spacings for unimolecular cells.. Proposed bimolecular cells................... Interplanar spacings for bimclecular cells... Comparative interplanar spacings............. Structure factor data for calculations Vith iron K“ radiation....................... Relative intensities for proposed - structures 1 - 6 with iron K“ radiation...... Relative intensities for proposed structures '7 - 12 with iron Ki radiation..... iv 11 13 16 19 42 77 78 84 8'7 88 89' 9‘0 ' 94 100 101 102 XVIII. XIX. XXI. XXII. XXIII. XXIV. XXV. XXVI. xxvn. XXVIII. List of Tables (continued) Relative intensities for proposed structures 13 - 17 with iron K‘ radiation.... Structure factor data for calculations with molybdenum K4 radiation................. Relative intensities for preposed structures with molybdenum K4 radiation...... Diffraction pattern of silver difluoride with molybdenum Kdradiation................. Diffraction pattern of silver difluoride with copper Kiradiaticn..................... Diffraction pattern of chromic fluoride trihydrate with copper Kiradiation.......... Diffraction pattern of chromic fluoride trihydrate with molybdenum K“ radiation...... Diffraction pattern of chromic fluoride prepared from chromium tetrafluoride......... Diffraction pattern of chromic fluoride with copper Ki radiation..................... Diffraction pattern of chronic fluoride with molybdenum K4 radiation................. Diffraction pattern of stannic fluoride with molybdenum Kat radiation. . . . . . . . . . . . . . .. . 105 104 104 113 115 152 134 155 140 1. 2. 3. 4. 5. 6. 7. 10. 11. 12. List of Figures The fourteen space lattices..................... Intersection of a plane on the crystal axes indicating the calculation of Miller indices.... Diagram of the unit cell of ferric fluoride..... BJurstrom chart for the tetragonal system....... BJurstrom chart for the hexagonal system........ Bjurntrom chart for the rhombohedral system..... Diffracting power of manganese atoms and . fluoride ions as a function of scattering angle........................................... Total angle factor for small or large reflection angles as a function of scattering angle........................................... Total angle factor for intermediate reflection angles as a function of scattering angle........................................... Absorption factor for manganic fluoride (iron fix radiation) as a function of scattering angle................................ Absorption factor for manganic fluoride (molybdenum Ki radiation) as a function of scattering angle................................ X-ray powder diffraction photographs of manganic fluoride (mnFa)........................ vi 62 65 66 69 70 80 13. 14. 15. 16. 17. 18. 19. List of Figures (continued) Relative intensities, observed and calculated for proposed structures (2) and (3)............. Relative intensities, observed and calculated for proposed structures (8) and (ll)............ Liagram of the pronosed unit cell of manganic fluoride (MnFa)........................ X-ray powder diffraction photographs of silver difluoride (A599)........................ X-ray powder diffraction photographs of chronic fluoride trihydrate (CrF5°5ngo)......... X-ray powder diffraction photographs of chronic fluoride (CrFa)......................... X-ray powder diffraction photograph of Btannlc fluoride (Sn?.)......................... vii 107 108 111 117 123 156 145 I. NTIKHL'CCTIOI-I The structures of a large number of the binary crystal- line fluorides have been reported, but there is almost as large a number for which no structural data are available. The structures of the higher fluorides which have been re- ported are, in general, not well established. It is the purpose of this investigation to provide information concern- ing the structures of several fluorides of unknown structure. This dissertation undertakes a discussion, based on X- ray powder diffraction photographs, of the structures of the following fluorides: manganic fluoride (MnFa), silver di- fluoride (AgFg), chromic fluoride (CrFa), chronic fluoride trihydrate (CrF3-5H.O), and stannic fluoride (SnF,), and in addition, a short reference to the structure of chronic tot-y rafluoride (CrF.). Single crystals of most of these com- pounds, suitable for single crystal X-ray investigation, have to date not been prepared. The structures of the fluorides are of particular inter- est since they are the most ionic compounds (and sometimes the only ionic compOLnds)of the metals. A knowledge of the crystal structures is essential to the interpretation of the properties of the metallic fluorides, many of which are be- coming of industrial importance. II. HISTURICAL EACKGROUED A. Elements of Crystal Structure 1. 'Unit coll:types or crystal systems:5 A crystal con- sists of a large number of repetitions of a basic pattern of atoms. Just as in many textile materials and wall-papers a pattern is repeated over and over again on a surface, so in a crystal a particular grouping of atoms is repeated many times in space. The reason for the formation of regular patterns is that atoms, ions or molecules tend to settle down in positions of minimum energy; for each atom, ion, or molecule a particular environment of neighbors has a lower energy content than any other, and there is therefore a tend- ency for this arrangement to be taken up everywhere. The only patterns of exactly repeated environments capa- ble of indefinite extension are those in which successions of pattern-units lie on straight lines. The choice of origin does not matter, and the whole structure can be divided into units of volume by joining similarly situated points by straight lines. The unit of pattern in a crystal can be con- sidered as a box bounded by three pairs of parallel sides. The shape and dimensions of the box, that is, the lengths of its three different edges (so, b0, co) and the angles (4,’ , i) between them, are characteristic for each different crystal -2- species; in some crystals the box is a cube, in others it is rectangular with unequal edges, in others the angles are not right angles, and so on. ihe boxes that have been disou uses- are the simple unit cells. As the choice of oririn does not matter, it is al- nays possible to select a unit call so that id entiCsl stars are situated at the corners of the unit cell. As these stems are shared by the 81 ht cs 11. that meet at the orners, there is one atom, one patternnunit, to each cell. It is as of ”53:22 euce, C 54 (l L? sometimes more appropriate to use, for par: 0 a J. fiat» h) a *3 :3 O t.) *3 a p H L: a cell containing more than or e unit 0 tso reasons far this. The first and sore i=”\““a t roasen is that the symmetries of the complete arrange;m nt are tee same as those of arran"ements in which the saa,r e of the true pat- tez'n is the aha; e of the large; cell. The second reason is to obtain a more convenient frame of reference. The types 0. cempound unit 091 la are: facewcentcrsd, with identical pat- tern-units at the centers of all three pairs of 03,031te faces in addition to those at the corners; body~ccntercd, with iden- tical pattern~units in the ct~anters of the cells as well as at the corners; and end-cen;ered, with id entical pat: <2tn—lnlts at tee centers of one pair of oppos its fee as in addition to those at the corners. The arran .enent of the pattern-units, the asaem C13: :6 cf? in 3 each cf w;ich re“r93Jr*3 one pattern- unit, is called the space- ~lat tics. The goints of the space- lattics - the lattice points - are thus corners of the true unit of pattern; the conventionally accepted unit cell may be simple or compound) it may contain two or more space-lattice points. There are seven crystal systems or unit cell types, each with a simple unit cell. In addition, there are seven compound unit cells, two in the cubic system, two in the te- tragonal system, and three in the orthorhombic system. There is, then, a total of fourteen space lattices in all. The unit cell types for the different crystal systems are summarized in Table l (p. 5). The unit cells, simple and compound, are often designated by the letters P, P, I, and C. A simple unit cell is desig- nated by P, a face-centered unit cell by.F, a body-centered unit cell by I. and an end-centered unit cell by C. Diagrams1 of the fourteen space lattices appear in Figure l (p. 6). It should be noted that the hexagonal unit cell is designated as end-centered, whereas it was stated above that the only unit cell in the hexagonal system was a simple unit cell. The end-centered unit cell in the diagram is not a new or different unit cell, merely a different method of represent- ing the simple unit cell. 2. £9int:grogps and spaceggroup§,5 In the previous section the various possible arrangements (both simple and compound) of lattice points have been mentioned. Now the arrangements of atoms around each lattice point must be con- sidered. In speaking of the symmetries of the arrangements around a lattice point it is customary to use the term point- -4- TnBLE I CHARACTERISTICS OF CRYSTAL SYSTEHS System Unit Cell Triclinic Anglesfi ,5 , and (unequal and not 90“. ‘ Edges a0, b0, and co unequal. Monoclinic sad-=90". (not 90°. 80, b0, and co unequal. Orthorhombic “=’*7'90°. a0, be: and co unequal. Hexagonal de’=90°. 13120°. a°=b°. 00 different from a0 and b0. Rhombohedral *808i, not 90°. .a°=bo=co. Tetragonal " e-i-th90°. aonbo. 00 different from a0 and b0. Cubic '4‘9‘7‘9ooo ‘o‘bo°°o° -5- 8 I) 7’ no P P Triclinic Monoclinic I Q’ Q t» if ’ 4 I C Orthorhombic Q. 0. 0o 0 o C R Hexagonal Rhombohedral Q 00 P I F Cubic Figure l. The fourteen Space lattices. -5- group, and for the symmetries of the couplets arrangerent in the crystal the term space-group. There are four types of symmetry elements that make up the point-group symmetries. These are the center of symmetry, the plane of symmetry, the axes of symmetry, and the inversion axes. A center of sym-: metry exists if, along any direction from the selected lat- tice point, the arrangement encountered is exactly repeated in the diametrically opposite direction. A plane of symmetry. exists if any plans divides the crystal into halves, each the 'mirror image of the other. An axis of symmetry exists if the atomic arrangement is exactly repeated two, three, four, or six times when the crystal is rotated around that axis through one complete revolution. An inversion axis exists if the atomic arrangement is exactly reproduced by combined rota- tion through sixty, ninety, or one hundred and twenty degrees and inversion through a point. All the possible point-group symmetries are different combinations of the symmetry ele- ments described, a total of thirty two point-groups. The possible ways of placing the various types of atomic arrangements in the various types of space-lattices must now betconsidered. The molecule can only be placed in a lattice having the appropriate symmetry if both molecule and lattice are to retain their original symmetries. Thus, each of the tillirty two point-groups must be placed, correctly oriented, 111 a lattice having appropriate symmetry. Bearing in mind tale existence of compound lattices having the same symmetries «7-. as sinnl ones, it can he realized that the number of ar- ’1 anaenents possible on this basis is considerably :roator than thirty two. Eat this does not include all the pos- sible arrangements. Treviously only those symmetry Opera» tions which lead from one atom in the crystal to another associated with the same lattice point have been considered. These are tne point-group symmetries. New in many space patterns two adaitional types of ynnetry operations can be discerned - types which involve translation and therefore do not occur in point-groups. These elements are glide :lanss, which involve simultaneous reflection and translation, and screw axes, which involve simultaneous rotation and trans- ‘3 lstion. Continued repetition 01 these symmetry Operations does not lead to another atom associated with tn: same lat- tice point, but to a corresponding atom associated with the next lattice point, and then the next, and so on throughout the crystal. the total number of possible stemic arrange- ments, that is, the number of possible combinations of all these symmetry elements, is two hundred and thirty. Lheso are the two hundred and thirty space-groups which are com- pletely described in the Internationale Tabellen zur Bestimmung von aristallstraxturen.7 5. figmenclature of crystal planes. $hcrs is an ex- tremaly larbe number of ways in whlca a crystal Can be a1- vided into layers by sets of parallel planes passing through lattice yolnts. Each of these sets of parallel planes is -7}- described by three numbers such as 210 or 132, the meaning of which can probably be best understood by an example. It must be remembered that, in general, the unit lengths (unit cell edges) along the three crystallographic axes are differ- ent. If the 312 crystal planes are considered as an example, the numbers mean that these planes cut the g,axis at inter- vals of ao/s (so being the unit length in this direction), the Q’sxis at intervals of bo/l, and g.sxis at intervals of 00/2. The index numbers are celled Miller indices and are defined as the reciprocals of the intercepts of the crystal planes on the crystallographic axes. t‘rom the definition, it can be seen that planes parallel to one of the crystallo- graphic axes will have zero as one of the index numbers. As an example, crystal planes 210 out the g’axis at intervals of su/Q, the y’axis at intervals of bo/l, and are parallel to the g_axis. These indices are indicated in a general way by the letters hki. The meaning of the Miller indices may be better under- stood by reference to the diagram (p. 10).4 The diagram rep- resents an orthorhombic crystal with threemutually perpen- dicular crystallographic axes. A crystal plane is shown cut- ting the ;,axis at so, the y’axis at b0, and the g_axis at 00/2. The intercepts arel, l, and is their reciprocals are l, l, and 2. The Miller indices for this crystal plane would be written 112. ' -9- Figure 2. Intersection of a plane on the crystal axes indicating the calculation of Miller inuices. B. Structure of Binary Crystalline Fluorides l. ggown structures. In Tables II - V are listed the binary fluorides for which structural data are available to- gether with the various structural characteristics reported for them. Unless otherwise noted, all of these structures are described by Wyckoff.14 The unit cells are of importance, but perhaps even more so are the structure-types. The sodium chloride (NaCl) structure-type is very well known and is quite characteris- tic for the uni-valent metal fluorides. In this structure- type, each metallic ion is surrounded by six equidistant non-metallic ions in an octahedral arrangement, and each non-metallic ion is surrounded by six equidistant metallic ions in an octahedral arrangement. It should be noted that -10- TABLE II STRUCTURES OF BINARY EONOFLUORIFES Compound Structure-type Unit Cell Dimensions Ref. La? Sodius.chloride Cubic, a°-4.0l73 2. 14 Ha? Sediun chloride Cubic, 10'4o520 Z. KP Sodium chloride Cubic, a°-5.34 K. Rb? Sodium chloride Cubic, a,-5.64 2. Ce? Sodium chloride Cubic, aot6.008 X. AgF Sodium chloride Cubic, a,-4.92 I. TlF Sodium chloride Orthorhemgic, a085.180.x., b.-5.495 4.. c.'6.080 A. On? Zinc blende Cubic, ao-4.255 X. 58.? Wurtsite Hexagona , s°.4.39 K.. 00.7.02 . 33.F. Unknown Reported Tetragonal A3,? Cadmiun.iodide Hexagonal. aoe2.989 3., 00.5 .710 A. -11- thallium fluoride (TlF) is listed as having the sodium chlo- ride structure-type, but the structure is slightly distorted. The zinc-blends structure-type has cubic symmetry, as has the sodium chloride structure-type, and the eurtzite structure- type has hexagonal symmetry, but the atomic arrangements are such that, in both the wurtzite and zinc-blends structure- types, each metallic ion is surrounded tetrahedrally by four non-metallic ions and each non-metallic ion is surrounded in the same manner by four metallic ions. Ammonium fluoride is included because the ammonium salts are so often classed as binary salts. Its structure, in addition to the ordinary bonds between ammonium ions and fluoride ions, contains N-H-F hydrogen bonds. Eisilver fluoride (£32?) is not a univalent metal fluoride, but is included in Table II for lack of a better place to classify it. It has the cadmium iodide structure, a layer structure with hexagonal symmetry, in , which each metallic ion has three nonumetallic ions as neigh- bore, and each non-metallic ion has six metallic ions as neighbors. The difluoridos, in general, may be divided into three 13 If the groups depending on the size of the metallic ions. radius ratio, that is, the ratio of the crystal radius of the metallic ion to the crystal radius of the fluoride ion, is less than 0.414, the difluorides have one of the quartz structures. In these structures, each metallic ion is sur- rounded tetrahedrally by four non-metallic ions, but each -12- 9""T"T‘-‘_.‘(‘ 7:“ "7f '3 7"".7'7 ""7 “7“? ' ilqk« L‘54‘.AI~' OJ B}..‘él‘.|. 6.1L; .‘JJ‘JltIJI..-..‘ Compcund R‘ Type Unit Cell Dimensions Ref. 36?. 0.25 (bquarts Hexagonal6 aot4.72 3., ' 14 mar. -O.48_ Rutile Totrngona , a,-4.66 3.. 00’3008 O . Co?“ 0.53 Rutile Tetragona‘, 3084.69 3., ' 00‘3019 X. FeF. 0.55 Rutlle Tetrsgonal s°=4.670 3.. 0033.297 :0 nnr, 0.59 Rutile Tetragonal a,=4.865 K., KiF. 0.51 Rutile Tetragonal, ao=e.710 §., 00330118 A. Par,' 0.68b Rutile Tetragonsl,a°=4.931 3., 00:30:36? . Zur. 0.54 Rutile . Tetrsgone , a°=4.715 3.. 3033013 O 01?. 0.75 Fluorite Cubic, o°=5.451 3. arr. 0.85 Fluorite Cubic, 3035.784 K. Bar. 0.99 Fluorite Cubic, eo-6.187 3. RIP. 1.01b Fluorite Cubic, 3086.568 K. CdF. 0.71 Fluorite Cubic, eo¢5.40 X. EuF, 0.85b Fluorite Cubic, s°-5.796 i. TABLE II I (continued) w Compound 5“ Type Unit Cell Eimcnsions Ref. 83?, 0.81 Fluorite Cubic, c°I5.54 i. fl-Pbr, 0.89 Fluorite Cubic, oou5.935 E. A-er, 0.89 Lead Di- Orthoruomgio, ao-7.6103., chloride b°=6.41 A., c°=5.80 A. CuF. 0.55b Unknown Monoclinic . 6 SnF. 0.77b Unknown Tetragonal,o&o=7.78 3., 8 00:7.52 A. ‘Radius ratio. b Estimated value. °c.lculatod by the author from the published X-ray powder diffraction pattern. -14- y .1 \ lea s. q ‘1. .J I) \ up Q. .3. ...J -lJ J‘.‘ 38 2 fl .LJ .‘5 o. .10 c1 7. -..a th .1 ‘5 \. 9.. L113 33-1323 ll .ol cl. ..ro v.58 Ll ‘p' i . A. in ‘a—t...-I-J loo ‘J P. u c .5 ad +9 r 0 1 r. .l 1J3." :. o‘l \ 'JZ-A Kc H- I 3 1'... -L- :17 1; l I“ a no A. .o a i Q: I "1 L 1'3 , - 1 a '4 c v 4' . h hvfls§h¢ L-.. \’ 0 l .u l a; o r . A 3 ... a ..o. .A ‘va .. I. Iv. L. .u .o .4 2. . i .ll 1“ .u l I ‘- 4|.“ « u ._...u J 3 n: L.“ O A e l \A J J C . i). i e . n. ...u cu l 3 .o C i . A. 3 1. s... . l A 1* .ru 1.. av Q l .u 1,. J to m. n9 .d 3 t .. .1. e u .. l” 1... . l .t W. no I. u.’ .4 Lb .r» 1.x. .1 d .L.. V .-.. 8 o I O .1. .- .— V C) 1‘ 170 I'd - ‘5 \ .0 .§'-“~l‘ Lu 0 "\ n ‘1 . ,‘ J. t .a .LJ...‘ 011.”. 3 .. ._ .ru 3 WV J _.i re a. in. 3 n. all . u.. ’ - l & As}- A.viJ :13 ‘r U be Fouls 3' 1134 O I\ iuc, Q 0 ‘fl - \ 'n “Aafi -‘J’\JL .2. 0' _‘ ‘ zrlctlrc. 4;: U alllscs in .1 U '1 '1', ‘1‘ J \4 1 lit) .1.” 3 Li 0 '& .- :- Ur .10 ‘J 3 13-. .Ak' ori guy..- -9 .30,- 2‘1“, \‘1 W L the a‘ -, fi. - ‘7 Qt L. {1’3 3.. L.‘ U 0'4 3 33.3 s .vnfi “a: Liv...J.L 2.11.13 0 ”a mi a. i very ;‘ ,, '- llucrldes t3 "‘3 R! A. 'L t 11.-{'>Ll uni D 3 ‘ -.- x v ‘1 «(3.5.4.94 '3 .1) 0‘] VA *. -h 3-“? 3 1-- - n .1 U1. dutls C 183 rl L- I I r" ‘o‘v } I“ '7 a”. '- n. -~. 4'" oiuhl \‘\¢ MIVOO 40‘! "u U all N ’33:," ' ‘? .‘Jm‘ ' fl 4‘. 'r\ ".‘,.'.‘:l l , '3; ,_ Ud-‘ I, V have not, in try .""" .\ QuILL4U '1 .L “ Q ohm one .....‘:l ._ . a n}- old.) i..l a. ., . l. -1 . ‘ .a «A AC ULAJ. 'u ids (F835) s ‘13 flue: Paw» b \Is A ‘ T}. 1"; L33 IV r' , ' 1“ 1*"?331',‘ (“*1 it"“‘.‘_r *'~rj o, pl. turf 1111' “own Ill... Compound Type Unit Cell Dimensions Ref 0 Alp, AlF. nhouoohodrol, .,-s.029 fi.. 14 “58’31'o For, For, Rhombchedral, a°=5.39 2., 4.58.0 . e CoF, For, Rhombehedrsl, .,=5.so 3.. 4-57°0.0 Par. For, Rhombohedra , a°n5.56 1., 4.5‘.0.o th. For, Rhombohocrol, a085.34 3.. “54°20'e Sor, 3.0. Rhomtohoerol, a,=4.025 X.. 4-89'84'. Yr, YF. Ortberhemb c, a,-e.ces I., 16 b..6e850 e. 0"‘03’3 ‘0. SmFa YF. Orthorhonbio, a.'6.669 ., 16 b.-7.059 A.. o,-e.4oe ., Eur, ‘YF. Orthorhonb c, 10-6.622 ., 16 h.‘70°19 e. 00'4.896 0. car, Yr, Orthorhomb c, .,-6.570 I , 16 . baflo93‘ e. 0.. ‘0393 ‘0’ TbFo YF. orthorhm 0 ' .0“ 0513 e g 16 bo'flo949 e, Co"038‘ e. DyF. YF. Orthorhcnb c, 80.6.460 .. 15 b6.5‘906 e. 00"0576 e. 80F. YF. Ortherhefib c, 10-6.404 ., 16 bo-6.875 ., o.-4.sve ., -16- Compound Type Unit Cell Dimensions Ref. ErF, YF. Orthorhomb c, ao=6.354 §., 16 b°=o.846 ., co=4.seo .. Tmr, yr, Orthorhorhic, a°=6.285 §., 16 b°.60811 go. 00:40408 O YbFa YF. Orthorhonb c, a°=6.216~§., 16 b0260786 0. 30340454 to m5 YFQ Orthor’hor‘tb C. 8.026.151 §eg 16 boas. 5:3 1., 00:34.46? 0 p-Bir, yr. Ortuorhomgio, a°a6.56 3.. 15 bo=7.03 i., co=4.86X. Ce?a CeFa Henugonal, ao=7.llé X., 14 o°=7.273 i. La?a CeF’a Hexagonal, s087.163 3., c°=7.329 . NdFa CeFa Hexagonal, 3037.021 K., c087.196 X. Prr, Cor, Hexagona1,°ao=7.061 2., 00370218 he Ac?a CcPa Hexagonal 3027.40 3., 0037.54 . AmFa CeF. Hexagonal,oa087.068 K., HpF° Cor, Bexsgonal,oa087.129 §., 60370287 A. PuFa 00F, Hexagonal,9ao=7.092 K., c°=7.254 A. UP. CeFa Hexagona1,6eo=7.lal 2., 00370348 A0 -17— TABLE IV (continued) Compound Typo Unit Cell Dimensions Ref. 9 SmF. 06F, Hexagona1,0a686.956 A., 16 00.70120 A. EuF. CeFa Haxagonal,olofl6. 916 A., 16 c°=/.09l A. Hop, Co?a - Hexagonal,oa°86.833 3., 16 0086.9o4 A. TmF. 00F, Hoxaaonal,°1°=6. 763 A., 16 SbPa Unique Orthorhomgic, a°~7.25° A., 14 b 0037 *9 A0, 0084.65 A. (FBlF. Fluorite Cubic, 3035.853 A. (modified) TABLE V STRUCTURES OF HIGHER BIIARY FLUORIDES compound TIP. Un1t 0011 Dinonaiono Rot. cor. arr. lpP, zrr. pbp. UiP. «#0:. 9-w- zrr. zrp. sz. er, zrr. ZrF, zrr. Unknown Un1quo Un1quo unique lonoolin1 , o,¢18.6 i., bo-1005 0, 00.813 ('126°. Ion0011n1, a..11.70 I, h 0.9.“ :' 00' 706' ’312695' 0 lonoclin1o 1,-12.69 1. , b. -1o.a4 1..° o,-o.sn Q3126°10'. h lonocl1n1o I .I12:O1.‘, b -1o.av 1.. no.0.“ 61.. (-126010'. [0110011111, ..-1s.1 {3 bo-ll . (3126-? ' 0°” Ionool1n1c A ..12.81 1 , b-10.74 t..' o.-o.41 9-126010'. lonocl1n1, o 0'11'71 b 0.908. 0' 80.7.6. [:126‘9'. Ropertod totrngonnl. 11 Tetragonnl a “.528 K. 0,-4.4?! 1. . ' Totrngonnl o,t11.473 1., 0,-5.208 1. .19. o. in- V with two molecules per unit cell in which each metallic ion is surrounded by six equidistant non-metallic ions in an ectahedral arrangement, but each non-metallic ion has only two metallic ions as close neighbors. The aluminum fluoride structure is nearly the same except that the arrangement of the fluoride ions is distorted and does not have true hex- agonal symmetry. The cerous fluoride (CeFa) structure has hexagonal symmetry with six molecules per unit cell. The yttrium trifluoride (YF.) structure has orthorhonbic symme- try with four molecules per unit cell. Scandium fluoride has been assigned to the cubic rhenium oxide (Reoa) structure- type, but it is actually a rhombohedral distortion of that structure-type. A molecular lattice has been described for antimony triflucride (SbF,). The structure of bismuth tri- fluoride (d-BiF.)”has been described as a fluorite structure with additional fluoride ions added in the interstices of the lattice. The zirconium tetrafluoride (zrr.) structure has monoclinic symmetry with twelve molecules per unit cell. A diagram of the unit cell reported for ferric fluoride (FeF.) appears in Figure 2, (p. 21). In the diagram, the ferric ions are designated by small circles, the fluoride ions by large circles. The fluoride ions belonging to this particular unit cell are designated by placing the coordi- nates of the atomic positions within the circles represent- ing the ions. The environment of one ferric ion is shown. Each ferric ion is surrounded by six fluoride ions in an -20- Figure 5. Diagram of the unit cell of ferric fluoride. -21- octahedral arrangement, three of the fluoride ions are in the same unit cell as the ferric ion, three are in adjacent unit cells. 2. Crystal structurefiproblena. Previously, the only factors that have been mentioned as having an effect on the crystal structure-type of a binary fluoride have been the size and oxidation state of the retallic ion. It should be made clear that many other factors are involved also. Ri- nary crystalline fluorides are considered as being essential- 1y ionic crystals, and as such the polarization of the bonds in the crystal has an important effect in the formation of many crystal structures. The particular electronic con- .figuration of the metallic ion may contribute to this bond lhalarisation, so that it is also an important factor. Eval- uation of the effect that these factors have on crystal Itznucture is an extremely difficult task, and a method giv- iné; completely consistent results has not been developed.10 The factors mentioned above are of particular importance because it appears that the point has been reached in crystal strtrcture determinations where it is necessary to establish some new methods or to make major modifications in the meth- ods 'that have been characteristically used for such deter- minations. It is realized that X-ray powder diffraction tectuaiques are exceptionally good only for rather simple "Natures, and that for more involved structures X-ray sin- Slfi-crystal diffraction techniques should be used. It -22- sheild be noted that at the preach time it appears that inplo structures (mono- and di-fluorides) have been quite well determined, that is, 'he structures to be determined arc the more involved structures, so that the powder techniques for structure determinations should be at, or near, an end. It should also be noted that the reason that many structures have not been determined in the past has been the failure to obtain single crystals with which to work. In the field of the binary fluorides there are a number of reasons for his. methods of preparation are of such a nature that usually only micro-crystalline materials are obtained. Sometimes the :materials are obtained only in an amorphous condition. To ‘prepare single crystals from such materials, they must be (lissolved in a suitable solvent and allowed to crystallize 'verqrslowly. The common solvent for aich preparatiols is nutter, but many fluorides are quite hygroscopic, form hy- drwites easily, hydrolyze easily, or are extremely reactive witfla water. Also, in many cases, it is impossible to pre- pazfie an anhydrous salt from a hydrated salt, because of de- conqoosition. Sublimation and melt techniques are troubled by Ireactivity and decomposition at high temperatures. In Partticular cases these difficulties may be overcome and a suithable method devised for obtaining single crystals. EVOII then, however, the problems may not be at an end, be- cause, the crystals may have to be completely protected from the atmosphere during the X-ray exposure, It appear-3 highly probable that some means must be devised to obtain sufficient inforxstion from powder data to assign suitable crystal struc- tures to many of these salts whose structures have not been determined. There is no doubt that the interpretation of such data will be extremely difficult and that hasty decisions may often be wrong. However, by careful individual interpreta- tion of such data by a number of investigators, it should be possible to arrive at satisfactory structures for several substances which are now relegated to that group for which one does not try to derive the crystal structure. If this field is to progress and an understanding of the manner in which substances crystallize is to increase, bringing with it an increase in the understanding of the forces acting be- tween ions and perhaps withi ions, it appears that such an endeavor must be made. One method of increasing our know- ledge of crystal structures has been illustrated in a recent investigation which interprets the X-ray powder diffraction xmtterns of ten different binary fluorides in terms of an IK-ray single crystal determination of a Very closely related fluoride.16 This group of eleven fluorides is listed in {Pablo I? as having the yttrium fluoride (YFBJ structure-type. ltn.X-ray single crystal determination was made of the struc- tnare of yttrium fluoride. but the K-ray powder diffraction Ikltterns of all eleven fluorides are so similar that all of tflnem have been assigned the same structure-type. It is heped -2 4-. that this dissertation will provide additional information for future investigators as well as provide crystal struc- ‘ taro data for selecuud fluoriuea. III. EX?T“IK?§Tir A; Preparation of Eaterialas 1‘ Fcnrun‘c fluoride (Vh?ll and silver dlfluoriéo Q A”? . Che mangcnic fluoride and silver difluorice used in this investigation were comrerclal yrodmcts with a guar- anteed minimum assay 99% HnFa and 99% AgFg, reapectlvely..' 2. C.romlo fluoride trihyfirate {Cryayzgtgi. Hydroua chromic oxide (Cr.036{fiso}x) was precipitated by audition of dilute amaonium hydroxlce solution slowly to a flletiOD of chronic chloride hcxohycruto (CrCla'fiflgO). When precipito- tlon was c mglcte, tho solution woo filtered can the residuo rexfiinlng on tho filter paper was transferred t; a platinum crucible. The reaicue was dissolved in Lyérofluoric acid (46% HF) and ovarorctnc to drynoaa on a atocm bath. The so- .lution 1n hydrofluoric acid an& subsequent evagoration was rcpcated twice. The product won dried at a temperature of ‘100 - 110°C. for twenty four hours. This rroduct, when con- ‘verted to chronic oxide (Crgoa) by ignition in a platinum crucible, yielded I conversion factor of 0.438, compared to a theoretical corverslon factor of c.468 for the conversion -___._- A__‘ +Ponnay1vunia Salt Company. or chronic fluoride trihydrata to chronic oxide. The chronic oxide was positively identified by its X-ray diffraction pat- tern. This was considered sufficisnt to identify the product as chronic fluoride trihydrats (CrF,°5H.O). It has been reported12 that chromic fluoride trihydrato hydrolyzes in moist air to chronic oxide at temperatures of 200’0. and above, although this hydrate is stable at lower temperatures. The salt showed no change in composition when dried ovsr phosphorus pentoxide for one month. The salt showsd no change in composition when dried at loo-110°C. for ens week. then heated in an open platinum crucible, ths salt appears almost to erupt from within as the temperature is in- cresssd. An experiment was designed to dstermins whether or not moist air was necessary for this reaction. A pyrex com- bustion tube was fitted with rubber stoppers and connected by means of glass and rubber tubing to s series of absorption towers. The combustion tube was placed in a combustion rur- nsss equipped with s direct-reading Hoskins thermocouple for tonpsraturs nsssurcment. A combustion boat containing the salt was plscsd in the combustion tube, after which the sp- paratus was closed by connecting it to s second series of ab- sorption towers. A stream of air, dried by successive pass sags through calcium chloride, magnesium perchlorate, and phosphorus pentoxide. was passed through the apparatus. The outlet for this air stream was protected by absorption towers containing magnesium perchlorate and calcium chloride. nith .27- the dry air stream continuing, the temperature was slowly increased. As the temperature approached 230°C., visible evidence of a reaction appeared - the combustion tube, near the air outlet was being etched by hydrofluoric acid. Exam- ination of the residue showed that it was a mixture of the hydrated salt and chromic oxide. At higher temperatures, the salt is completely converted to the oxide. Extremely small crystals of chronic fluoride trihydrate were obtained by a very slow evaporation of a water solution of this salt. hieroscopic examination showed that the crys- tall were in the form of six—sided hexagonal prisms. 3. Qhromium tetrafluoride (CrF‘l. Chromium tetrafluo- ride was prepared in an apparatus consisting of a two foot section of Eonel tubing (diameter onechalf inch) with Monel coupling flanges at each end. honel couplings could be bolted to these flanges. COLper tubing (diameter one-quarter inch) was passed through each coupling, the end of the tubing flanged and soldered to the coupling. These cepper tubes served as inlet and outlet tubes for supply of gas to the honel reaction tube. The reaction tube was placed in a com- bustion furnace equipped with a Hoskins direct-reading ther- mocouple for temperature measurement. Shavings of metallic chromium.were placed in a nickel boat, which was then placed in the reaction tube. The couplings with attached copper tubing were bolted to the reaction tube, and thefree end of the outlet tube was inserted in a Teflon test tube containing -28- Fluorolube oil which served as a flow meter. The inlet tube see attached to a cylinder of chlorine trifluoride.+ The ap- paratus was swept out with a stream of chlorine trifluoride to remove oxygen and moisture and to insure that the product would be a fluoride. bhile maintaining a slow stream of chlorine-trifluoride, the reaction tube-was heated to a tem- perature of 500°C., maintained at that temperature for one hour, and then allowed to cool. When the reaction tube had cooled sufficiently, the flow of chlorine trifluoride was stopped, the reaction tube opened, and the boat containing ‘ the product of the reaction removed and placed in a desicca- tor over barium oxide.’ The product, a brown powder, when converted to chronic oxide (Cr.0,) by ignition in a platinum crucible yielded a conversion factor of 0.652, compared to a theoretical conversion factor of 0.594 for the conversion of chromium tetrafluoride to chromic oxide. The high conversion factor might indicate that some lower fluoride was present in the product, or that some chromic chromate was present in the ignition residue giving high results, or that some hydrolysis of chromiun.tetrafluoride occurred while it was being weighed. The ignition product was identified as chronic oxide by its X-ray diffraction pattern. It was concluded that the reac- tion product was predominantly chromium tetrafluoride, but that lower fluorides might be present in small amounts. *Harshaw Chemical Company 4.; Chronic fluoride {Cr£.L. Chronic fluoride was prod psred by two different methods. In the first method, chro- mium tetrafluoride from the previous preparation was placed in a nickel boat, which was then placed in the Monel reaction tube previously used. The reaction tube was heated gradually to a.tenperature of 500.0. to bring about thermal decomposi- tion of the chromium tetrafluoride. The reaction tube was allowed to cool, and the boat containing the product was rusoved and placed in a desiccator over barium oxide. The product, a green powder, when converted to chronic oxide (0r.0.) by ignition in s platinum.crucible, yielded a con- version factor of 0.729, compared to a theoretical conver- sion factor of 0.697for the conversion of chronic fluoride to chronic oxide. Slightly high results are expected due to small amounts of chronic chronate in the residue. How- ever, partial hydrolysis to the nonchydrate of chronic oxy- fluoride (GNP-m0) may have occurred (improbable). Partial hydrolysis to chronic oxyfluoride (arc?) is probable, so that the reaction product was not definitely identified. The second method of preparation of chronic fluoride consisted of a dehydration by means of chlorine trifluoride of chronic fluoride trihydrate that had been previously prepared. Chronic fluoride trihydrate in a nickel boat was placed in the Monel reaction tube described previously. While maintaining a temperature of approximately 100°C., substantially below the decomposition temperature of chronic ~50- ,‘1 to I‘ ’a L‘- M ii fluoride trihydrate, chlorine trifluoride was passed through the apparatus for two hours. The reaction tube was allowed to cool, and the boat containing the product use removed and placed in a desiccator over barium oxide. The product, a green powder, when converted to chronic oxide (Craoa) by ig- :nition in a platinum crucible, yielded a conversion factor of 0.716, compared to a theoretical conversion factor of Ou697 for the conversion of chronic fluoride to chronic ox- ide. As the presence of chlorine trifluoride would prevent any hydrolysis, this is deemed sufficient to identify the product of the reaction as chromic fluoride. The ignition product was identified as chronic oxide by its X-ray dif- fraction pattern. 5. giennigzgluoridegLSng‘l, The same general method was used for the preparation of stunnic fluoride as for the preparation of chromium tetrafluoride. Granular tin ("O Inesh) was placed in a nickel boat and the boat then inserted in the honel reaction tube. The reaction tube was swept out with chlorine trifluoride to remove oxygen and moisture and to insure that the product obtained would be a fluoride. “1th the stream of chlorine triflucride continuing, the reac- tion tube was heated gradually to a temperature of 2GO°C., maintained at that temperature for two hours, and allowed to cool. then the reaction tube had cooled sufficiently, the stream of chlorine trifluorido was stopped, he reaction tube opened, and the boat containing the product removed and placed in a desiccator over barium oxide. The product, a white powder, when converted to stannic oxide (Sn0.) by is- nition in a platinum crucible, yielded a conversion factor of 0.745, compared to a theoretical conversion factor of 0.774 for the conversion of etannic fluoride to stannic ox- ide. As oxygen and moisture were eliminated, this is deemed sufficient to identify the product as stannic fluoride. B. Preparation of Samples 1. pr; box. Prior to this investigation it was known that silver difluoride is extremely reactive with even a trace of moisture, and that msngsnic fluoride is quite reac- tive with moisture. It was realized, that for proper manip- ‘ulation of these salts, it would be necessary to eliminate any possibility of water contamination. It was deemed essen- tial that a dry box be used for the manipulation of these salts. Not too much is known concerning the nature of chro- mium tetratluoride, but it was deemed best to prevent water tron coming in contact with this material also. Stannic fluoride is known to be hygroscopic, so that possible con- tact with moisture should be eliminated. Chromium trifluo- ride and chronic fluoride trihydrate are not affected ap- DP001£Vly by moisture. All of these salts are more easily nanipulatsd when very dry than when any appreciable amount of moisture is present. It was decided, therefore, to develOp a uniform method of procedure that could be used for the manipulation of all salts used in this investigation, but which could be modified, if necessary, for the individual salt concerned.* The dry box used was one that had been prepared previ- ously in the machine chap of the Chemistry Department, lichigan State College. It had been prepared from a fifty- five gallon oil drum, and was provided with an access port which could be sealed by means of a steel plate equipped with a rubber gasket. It had also been equipped with a three-eighths inch thick plastic viewing window, a pressure gauge, numerous stopcocks for evacuating or pressurising the dry box, electrical connections including a light, and glove ports. Heavy rubber gloves had been attached to the glove ports and sealed with a metal band. Steel plates equipped with rubber gaskets had been provided for the glove ports so that the ports could be sealed when the dry box was to be sub- Jected to pressure or evacuation. This was done to prevent damage to the rubber gloves and to maintain the glove seals. It was deemed advisable to thoroughly check this unit for vapor tightness before use. With the glove ports sealed with the steel plates provided, the dry box was placed under a slight positive pressure with nitrogen gas. Extensive tests and sealing operations were performed. When all leaks that could be found had been effectively sealed, the dry box was considered to be ready for use. .53- Later, it was found that all traces of moisture could not be kept out of the dry box. It was concluded that the most likely source of moisture was diffusion through the gloves themselves or that flexing of the gloves during the various manipulations allowed moisture to diffuse past the point of attachment of the gloves to the dry box. These glove attachments were periodically rescaled to eliminate as far as possible this possible source of moisture. It was concluded that the best possible conditions would be obtain- ed by maintaining a large supply of desiccant in the dry box, and by providing for a good circulation of air. This air circulation was provided for by installing a small electric fan, with a powerstat for control, in the dry box. This fan was kept in operation almost continually except during the actual manipulation of the salts. 2.‘ Qapillarytubgg, When making Z-ray powder diffrac- tion photographs, it would be ideal if the only material in the X-ray beams were the sample material itself. This is possible only if the sample is a coherent piece of material such as a metal wire. Powder samples must be held together in some manner, using as little extraneous material as pos- sible. Sometimes, powders are mixed with a trace of adhes- ive and stuck to a hair, or mixed with adhesive and extruded as a rod. For the samples in this investigation, however, the only method available is to pack the samples in capil- lary tubes, and to seal those capillaries to prevent mois- -34- ture from coming in contact with the salts. The capillaries used were silica capillaries.+ They were of uniform bore, 0.25 - 0.30 mm. inside diameter, and with a wall thickness of 0.035 - 0.050 mm. hose capillaries were cut into three- fourthe inch lengths and sealed at one end in an oxygen flame before use. 3. manipulation of materials. Since the dry box had not been used for some time, it was considered advisable to subject it to a period of preliminary drying. This prelimi- nary drying period was also utilized to dry all of the equip- ment necessary for the manipulation of the first material, manganic fluoride. For this purpose, the following equip- ment was placed in the dry box: a large supply of barium oxide to be used as desiccant, an agate mortar and pestle for grinding the sample, a supply of silica capillary tubes previously sealed at one end, forceps for manipulating the capillary tubes, Kalk Kord weather strip (a plastic, flexio ble, non-hardening putty) for sealing the open ends of the capillary tubes, the container of manganic fluoride, screw- driver for opening the container, a small desiccator, a num- ber of small vials, and a micro-spatula. The last three items were included so that a sample of manganic fluoride could be loaded into the vials and placed in the desiccator, thus obviating the opening of the sealed container of mane *Thermal Syndicate, New Yerk, N.Y. genie fluoride if more sample might be needed at a later date. As the investigation proceeded, additional capillary samples were desired, so that this precaution was justified. Barium oxide was considered to be a sufficiently good desiccating agent for the manipulation of manganic fluoride, was cheap, and readily available in large amounts. The desiccator was also charged with barium oxide desiccant. The small elec- tric fan and powerstat had been installed earlier. The dry box was then sealed, a supply of desiccant ex- posed, the fan placed in operation at medium speed, and the box dried for one month with periodic replenishment of des- iccant. At the end of this time, it was assumed that the interior of the dry box was sufficiently dry for exposure and manipulation of manganic fluoride. The container of manganie fluoride was Opened, a suit- able sample removed and placed in the mortar, and the con- tainer rescaled. Even though the manganie fluoride was in powder form, it was ground further to insure that the parti- cles were of sufficiently small size for the preparation of good diffraction photographs. After grinding, the powder was loaded into the capillary tubes. Two capillary samples were prepared at this time, each in the same manner. The capil- lary tube was grasped with the forceps, the open end of the capillary tube guided through the powder, and the forceps, still retaining the capillary tube, tapped on the edge of the mortar to cause the powder to fall to the bottom or sealed end of the capillary tube. This process was repeated until the capillary tube was filled and the powder packed in the capillary tube. As these manipulations were performed while using the heavy rubber gloves, in a rather confined position, and with such a small capillary tube, the grinding of the sample, and especially the loading of the capillary tube, was quite difficult and extremely tedious. After the capillary tube was loaded, the open end was sealed with the plastic putty, the same putty to be used later in attaching the capillary tube to the X-ray camera. The manganic fluo- ride not used in filling capillary tubes was loaded into the vials and placed in the desiccator for future use. The mor- tar and pestle, forceps, and spatula were removed from the dry box, cleaned, and made ready for another investigation. They were then replaced in the dry box. Barium.oxide was not considered to be a sufficiently good desiccating agent for use with silver difluoride. For this reason, the dry box was dried for one month using phos- phorus pentoxide as the deeiccating agent before the silver difluoride container was orened. However, the manipulation of the material and the loading of capillary tubes were ex- actly the same as for manganic fluoride. For the remainder of the materials, barium oxide was used as the desiccating agent. The dry box was kept sealed as much as possible in order to prevent large amounts of moisture from entering. It should be added that the X-ray powder diffraction photo- -57- 521vhe were pregered as raoiilg as possible after the prep- aretion of tIIG capillar3 aidplns to minimize possible de- on of the -31plesc teriorat‘ I C. X-Iay .ecnui‘ee '“W iffr action equififient. inc diffraction units used ‘71 l. l were Philips X-ray Iiffracticn Lnite, Kodel 30. 5001. The ap;aratus consists esser t1ally of a stable, shockproof, full- wave rectified generator prov1ded with suitable coanrols and mounted in a substantial metal enclosure. $1113 unit has an equipment table and a centrally mounted heavy leaded bronze I-ray tube housin', oeari as four gravity Operate; been shut- ters one for each beam port.° Phe tub e hoa31n5 accomoiates a series of quickly iufOTClah‘CabLO diffraction tune es, avail- able with a variety of tar5et materials. The hi5h-voltage generator delivers up to fil ymilliamperee at volta5ce up to sixty kilovolts. X-raye are produced when rapidly moving electrons col- lide wi h at me. The l-ray tubes available for this unit are of glass. evacuated and sealed. Electrons are emitted in these tubes by an electrically heated tungsten filament, and are accelerated to the target by means of the h 5h potential maintained between the fi ement and target. The X-rays pro- duced are characteristic of the target material. The fila- ment of the tube is surrounded by a shield which brings th M‘ ('7‘ I . P- ‘X-rays to an effective focus. The tubes are provided four suitably placed mica windows, as that with a six de5ree ‘ viewin5 cn5lo, i‘cur focal Spots are available, two oi which are es pecially 500d for cryst31105rspnic purposes. Each of the four Beam ports is set in a machined face of the tube housing at a six degree sn5le from the vertical, and normal to the X-ray been path. Four rotatable, inden- in5, filter selectors are incorporated, each with ive filo ters and one unobstructed p rt. Four integral, sutonstic, gravity opersted beam safety shutters are proV1'ded to close ports when not in use. The camera rsczet support so rm LStS o; a heavy casting, precision machined, upon which is mounZCd the Camera track. Vertical adjustnent is obtained by means of en adjustirg screw. the track and camera cs be angilat ed in the six de- gree plane with respect to the tube housing. The csncrcs used were Philips powder cameras, 11.159 cm. diameter, self-calibrating (Streamer is technique), equipped with high intensity resolution slit and pinhole svstem, and rotation mechanism. .These are cylindrical cameras in which 35 mm. film strips are used. 1 ems l synchlonous motor is mounted centrally wi him the camera bracket to Operate the rotation mechanism. "14‘“ 2. Preparation of clifrscti on :hoto rsphs. A ponds r J y... p. t C) .4.“ a J" : 1.3” '3; uClJ uO 0131 camera consists essentially of an aperture 3 the X-ra* beam aholucr for the sumrlc a framework for hold- J 1 r) a -53- ing the photographic film and a beam trap to prevent fogging of the film by the primary beam. The cameras used contained c mechanism for rotation of the sample holder. The sample holder contains 1 small centrally loccted hole for the in- sertion of the specimen mount, 1 small, cylindrical, par- tially hollow, brace fitting. A oopillcry tube oemplo woe Iocured to the specimen mount with [elk Kord, the putty-ceclod end or the ocpillary being in contoot with the Ipocimcn mount. The specimen mount woo. then placed in the sample holder. The beam trcp woo removed trom.tho camera. .The capillary woo exactly centered in the X-rcy been by adjusting the rotation mochcniom with the add jucting screw provided on the camera. During this adjustment, the position of the oopillcry was observed by looking into the tip or the aperture system through the hole resulting from tho removcl or the boom trap. This adjustment was con- tinued until the capillary remained continually in the center of the x-rcy beam when the oomplo holder woe rotated manually. The aperture system woo then removed from.thc camera. The inner circumference of the comer: 1. 36 cm., but the upring locking device to hold the film tight cgainot the wall or the ocmorc permits use of only 35.5 on. film stripo.‘ Holc- for insertion of the cperturo system and the boom trcp :muot be punched in the film before use. A Philips Punch and Pilm.0uttor was used for this oporction as it is designed to -40- cut the correct length of film and to punch holes in the proper places. The following operations were performed under a safe- light. A roll of film (Kodak No-Screen‘X-ray Safet§HFilm) was removed from its Lox and unwrapped, after which it was inserted into the punch and film cutter while the two holes were punched and the correct length of film was cut. The film strip was removed from the punch and film cutter, in- serted in the camera, and locked in place with the locking device. A piece of black paper, the same dimensions as the film strip, was placed against the inner surface of the film strip, inside the framework for holding the film, to prevent any visible fluorescent radiation from reaching the film. The last two Operations were performed in an especially careful manner so as not to alter the position of the capil- lary in any way. The-aperture system and the beam trap were replaced in the camera. The cover was then placed on the '- camera. The wave-length distribution in the Xeray beam depends on.the.material of the target and the accelerating voltage 11sed. For powder photographs, it is essential to have a monochromatic beam, that is, a beamconsisting of one wave- length only. This-can be accomplished by placing a suitable. filter in the path of the X-ray beam. After the film had been loaded into the camera, the camera was mounted on the camera track and locked in place -41- TAELE VI X-RAY EXPOSURE DAlfi Sample Radiation Filter K7 Me Time MnF. Fe K4 Mn 55 15 4 hours KnF. Mo K4 Zr 55 15 2.1 hours 113?. Cu :4 Ni 55 15 4 hours AgF. Mo K5 Zr 55 10 4.4 hours CrF.-5H.O Cu K4 Ni 55 16 4 hours CrFG¢5H90 Ea K¢i Zr 55 15 2.1 hours CrF. Cu K“ Ni 55 15 4 hours m." Cu 1:4 Ni 35 16 4 hours Crl’.b Cu x4 Ni 55 16 4 hours our." Mo K4 Zr so 20 2 hours 811?. Mo K4 Zr 50 20 2 hours ‘CrF, prepared by decomposition of CrF.. bCrF. prepared by dehydration of CrF503H.O. -42- with the aperture system as near to the beam port as possi- ble. The selected X-ray tube was placed in the diffraction apparatus, the correct filter placed over the beam port, the sample holder set into rotation, and the sample exposed to X-radiaticn for a selected period of time. All the diffraction photographs were prepared in this manner. The exposure data for all photographs are listed in Table VI. In this table, Kv is the voltage across the tube in kilovclts, and Ma is the tube current in milliamperes. After exposure the film was removed from the camera under a safe-light, develOped for three minutes (Kodak Liq- uid.X9ray Developer), rinsed, fixed for twenty to thirty minutes in an acid-hardening (Kodak F-S) fixing bath, washed for two hours, and dried. All films were developed, fixed, and dried in the same manner. 3. Analysis eggphotogrgphs. When a narrow monochroé matie‘bemm of X-rays passes through a small specimen of a powdered crystalline solid, numerous cones of diffracted beams emerge from the specimen and, in this experimental ar- rangement, are recorded as arcs on a strip of film encircl- ing the specimen. Each cone consists of a large number of small diffracted beams, each from a small crystal. All of the diffracted beams in any one cone can be considered as reflections by one particular type of crystal plane. Any particular crystal plane can reflect monochromatic x-rays _cnly when it is at a particular angle 9 to the primary beam; -43- all the little crystals which happen to lie with this plane at this angle to the primary beam give a reflection. The angle of reflection is equal to the angle of incidence; hence the reflected beam makes an angle 20 with the primary beam. The reflected beams from all the little crystals which happen to be suitably oriented form a cone of semi- vcrtical angle 26 having the primary beam as its axis. Each different type of crystal plane requires a different angle of incidence, and therefore gives a reflected beam at a dif- ferent angle to the primary beam; thus, numerous cones of reflected beams are produced at specific angles, each cone coming from a different crystal plane. Diffraction by a three-dimensional array of atoms can be represented by the fundamental equation known as Bragg's law, n1 82d sin 9, Equation 1 where A is the X-ray wave-length, d the distance between suc- cesive identical planes of atoms in the crystal, 9 the angle between the xeray beam and these atomic planes, and n any whole number. For the method of indexing crystal planes given above (p. 8), n is always equal to one. It should be particularly noted that the position of the arcs on the dif- fraction photograph are dependent only on the interplanar spacings, d, and the x-ray wave-length. ' From the measured position of each are on a powder pho- tograph, 6 can be calculated, and thence d by the Bragg -44- equation. Since on powder photographs the position of the undsviated primary beam is usually not precisely defined, it is necessary to measure from an are on one side of the photo- graph to the corresponding are on the other side. In the experimental method used here, this is the only method for measuring the arc positions that can be used. The distance between corresponding area on the photographs were measured with a Philips Illuminator and Measuring Device, and cor- rected for film shrinkage assuming uniform shrinkage. The length of the film was determined bcth before and after ex- posure and a correction factor applied to all measured arc distances. This was done for each photograph, as the cor- rection factor varies slightly with different photographs. The corrected measurement between corresponding arcs was then halved yielding an arc distance, 8, measured in cen- timeters, which represents the distance between the primary X-ray beam and the diffracted beam. As the circumference of the camera used is 36 cm., the maximum S distance is 18 cm. An 3 distance of 18 cm. corresponds to a Bragg angle, 9, of 90°. Therefore, 9, expressed in degrees, can be ob- tained by multiplying the 8 distance by five. The x-rays used are not strictly monochromatic; copper K1 radiation, which passes through the nickel filter on the beam port, consists not of one wave-length but of two slightly different save-lengths, 1.5405 K. and 1.5444 K. ’These produce reflected beams from any particular crystal ~40- plane at slightly different angles, but in ordinary powder cameras the two reflections are not resolved except at an- gle: near 90°. Therefore, for the arcs at small angles a weighted average value of the wave-length must.be used for calculations. As the 31; wave-length is twice as strong as the K1. wave-length, it receives twice the weight, so that the weighted average value of the wave-length can be calcu- lated as follows: 2K + K 0 7L”. =- 3 . 1.5418 A. For iron radiation, K1; is 1.9360 5. and Ed. in 1.9419 3.. giving a weighted average value for the wave-length of 1.9379 2. For molybdenum radiation, Kd; is 0.7095 R. and TXd; in 0.7136 3., giving a weighted average value for the ‘wave~length of 0.7107 R. The procedure, then, in as follow-z the S distances, 11n.oentimetere, are obtained from film measurements; the ‘valuee of O, in degreee, are obtained by multiplying the S dietancee by five; and the values of d, the interplanar Ipacinga, are obtained by calculation using Bragg'e law, Equation 1 (p. 44). In addition to the film meaeuremente, the relative in- teneitiee of the diffraction arcs on each photograph were OI- timated vieually by comparing the area with each other. The intensities of the diffraction area are related to the posi- ‘tione of the atoms and will be diecueeed in Chapter IV, Part 00 IV. THEORETICAL A. Determination of Vnit Cells 1. Graphical indexing.3 It has been assumed above that the positions of diffracted beams depend only on the unit cell dimensions, while the intensities of the diffracted beams depend on the positions of the atoms in the unit cell. From a powder photograph all that can be obtained (apart from the intensities of the arcs, which are irrelevant to the present problem) is a set of values of d, the interplanar spacings. Each arc represents a reflection from a partic- xalar set of parallel crystal planes, but there is nothing to :1ndicate which set of crystal planes produces which arc; no- thing, that is, except the magnitude and ratios of the spac- ings themselves. The unit cell cannot be deduced directly; the only course is the indirectione - thinking what arrange- :nent of pattern-units has spacings of the observed magni- tudes. This can only be done for the more symmetrical ar- rangements; for those of low symmetry, the number of varia- bles defining the unit cell is too great for such a method to be possible, but for arrangements whose unit cells are Clarined by not more than two variables - that is to say, for crystals with cubic, tetragonal, and hexagonal (including bhombohedral) symmetry - it is readily accomplished. -47- Fdr the more symmetrical arrangements mentioned above (and sometimes for others), it is possible to assign indlces to the crystal planes, reflections from which formed the diffraction area, by trial and error calculations. Eut this would be a laborious process, and there is no need to pro- ceed in this way since tie problem can be solved graphi- cally. In a unit cell with totragonal symmetry, the relative spacing of the different planes are determined by the axial ratio cO/ao, if two crystals happened to have the same axial ratio but different actual cell dimensions, their patterns would show the same relative spacings, though one pattern would be more spread out than the other if the same x-ray wave-length were used. Graphs connecting the relative val- ues of d and co/ao can be constructed, and the whole set of arcs in a powder pattern identified by finding where their relative spacings fit the chart. In order to deal only with relative spacings so that only the shape (not the actual size) of the cell enters into the problem, the chart is made IOgarithmio with respect to d. The first such charts were published by Hull and Lavey (lfidl), who plotted 10g d for each crystal plane against co/ao. These charts are rather small, and for small values of co/ao do not extend far enough for some purposes. Bjurstrom (1951) developed a different type of chart which used a rectangular framework, plotting h'+k' on the 4r: . U- left vertical axis, l. on the right vert’c a1 axis and cg/ao on the horizontal axis. This tyre of 013 rt connect.s the re- lative val~ec of lfl’d and 00/33: but, as the chart is not ,4 ya agar thmic with respect to l/d‘, the scale to be used for rletting; these valu:s is not known. This difficulty can be removed by rakinj the chart lo35ri thmi c al on" tro vertical area. Thus, values of lo:(h'+k') arc rlotted along the left vertical axis, and values of log X. along the ‘.“1t vertf.ral axis, and all of these points are joined by lo 3: rithmic cirves. ?owever, the chart has one indecirsble fentzrc, at the two sides (that is, for very small or very larrc values of co/ao) sons of the lines are nearly parallel, and appear (2 .v ‘ .‘- 1- credit 0-.4'.’ tie very crowded. It is hotter, her efore, to 2 {ml ‘. I r) -7 ‘LJ ‘art in these regions br plotting a3tinst is considered to be the bee t type of chart, and was selected for preparation for tYis investigation. Charts for the hex- a3onal and rh01boi dral system a war e prepared in addition to the chart for the tetragonal system. Photostatic copies of traces of these charts were prepared commercially and appear in Figures 4, 5, and 6.31s construction of these charts was somewhat complicated and a brief outline of the method used appears below. The methznd can be reac ily understood by reference to the copies of the charts. On the chart for the tetragonal system, values or hk appear on the left vertical boundar y. At these points were plotted the values of log(h‘+k’) which correSpond to these -d9- l TETRAGONAL SYSTEM HEXAGONAL SYSTEM u..- M‘fl RHOMBOHEDRAL SYSTEM 33 J J #”'—+~—'—'i 5 J ‘ 1 4| 4I3 I J :2 J J . J ‘ os " 40 F 4 I J ‘ gtal . J m a m 220 30 J J J r \ J 3 m4 J ‘ J .2 J I J 2 Bjur om cLartJf J J J \ '13 l32 LOG .87 .7! co/.. -52- .57 .43 .29 hk values. On the right vertical boundary, values of 1 appear, at which points were plotted the corresponding val- ues of log 1'. On the horizontal axis appear values of oo/ao, but values of log co/ao were plotted. It is essen- tial to construct a framework of vertical lines as an aid in the construction of this chart. The vertical framework lines are plotted at values of log co/ao given by 1 ‘. I esss 0.1, 002. 003, g 0.90 The values of co/ao for these points are 0.53, 0.50, 0.65, 0.82, 1300, 1.23, 1.53, 2.00, and 3.00. Vertical lines are constructed at corresponding values of log co/ao. The left and right boundaries of the chart are theoretically at an infinite distance, so that they are placed at any convenient distance. It is best to begin construction of the framework lines at the center of the chart where co/ao equals one. ‘The construction of the curves can best be illustrated by an example. Consider the plotting of the 213 curve, which joins log :5 (1..., log (1.519)) an the left to log 9 (1.1... legj‘) on the right. logarithms of the nine intervening equally spaced numerical values, that is, log 5.4, log 5.8, log 6.2, log 6.6, log 7.0, log 7.4, log 7.8, log 8.2, and log 8.6 are plotted on succeeding framework lines. These plotted points, including the two boundary points, are then connected by a smooth curve. To the left or right of the framework, the chart is not completely dependable as the boundaries should be at an infinite distance as mentioned above. The construction of the chart for ‘he hexagonal system is identical with the construction of the chart for the tet- ragonal system with two exceptions: (1) values of log (h'+hk+k”) are plotted along the left vertical axis, and (2) the framework lines are plotted at values of log co/ao given by 4/5a’ = .1 .2 . ---- .9. Wfiawlco 0.0.03, ,0 The values of co/ao for these points are 0.29,.0.43, 0.57, 0.71, 0.87, 1.06, 1.52, 1.75, and 2.60. The framework lines are constructed at corresponding values of log co/so. The chart for the rhombohedral system is identical to the chart for the hexagonal system except that curves repre- senting diffraction arcs that would be given by a simple laexagonal crystal but not by a rhombohedral crystal were omitted. Only those hexagonal crystal planes for which 11 - k +,l - 3n, h + 2k +,3 a 3n, and o2h - k +,I 8 3n, where n may be any integer, give diffraction arcs if the true unit cell is rhombohedral. On the chart for the rhombonedral system, hexagonal indexing was used because calculations based on the hexagonal unit cell are much simpler than those based on the rhombohedral cell. The parameters of the true rhombohedral unit cell can be easily calculated from the parameters of the hexagonal cell. -34- These charts are used to give the indices of the crystal planes, reflections from which give the diffraction arcs on the diffraction photographs. The values of log l/da for all the arcs on the photograph were plotted on a strip of paper using the same scale as was used in the vertical plot on the charts. This strip was then moved about on one of the charts (or more if necessary}, keeping it always perpendicular to the log co/ao axis, until a good match between strip points and chart curves was found. (Some reflections may be ab- sent; this feature may be ignored.) more than one matching position may be found; the position giving the simplest in- dices refers to the simplest unit cell. When the correct position was found, the indices of all diffraction ares were read from the chart, and recorded. These charts are used to index X-ray powder diffraction jpatterns produced by crystals with cubic, tetragonal, hex- ‘egonal, and rhombohedral symmetry. It should be noted that e.tetragonal cell with an axial ratio equal to one is cubic, so that the chart of the tetragonal system at this position can be used for cubic crystals. Naturally, if a match posi- tion between strip points and chart curves, for which all diffraction arcs were satisfactorily indexed, was found, this method served to identify the shape of the unit cell. 2- Unit 10” thi-a After indexing has been completed, the length ao of the unit cell edge of a cubic crystal can be calculated from the interplanar spacing corresponding to -55- _any diffraction are from the expression agsd“(h'+k'*"). The results from area at large angles are more accurate than those from the first few arcs for two reasons: first, the errors due to the thickness and absorption of the specimen diminish with increasing diffraction angle, and second, the resolving power increases with 9, so that calculations can be made for each of the Hi components in the X-ray beam. For crystals with tetragonal symmetry, ao can be obtain- ed from the spacing of any hko arc, and 00 from any of the 001 area, the most accurate results being obtained from the arcs at the largest reflection angles. Arcs representing two or more different crystal planes with about the same spacing should be avoided. If unambiguous hko and OQI arcs are not available, both a0 and 00 can be calculated from the spacings of any two arcs having different hk and X values .from the equations h'+k’ ' '+k‘ ' dinT—L 4, 30.3] 3 d: [Ear—E. 9 .16.:- a 1. Equation 2e 0 0 ° ° 'The most accurate results are OthLHOd from a pair of area, tone of which comes from a plane with high hk and low I, and the other from a plane with low hk and'highwlg they should 'be fairly near together on the photograph so that absorption and other errors are about the same for both. Calculations should be made from several pairs of arcs, and the results averaged. For crystals with hexagonal symmetry, the method of calculation is the same as for tetragonal crystals except that the calculations must be based on the equation - \ dsE(Ltl%f_£_1) + 31...} :1, Equation 3. so 00 B. Determination of Space-Groups3 The number of possible space-groups for a crystal under investigation is limited by the knowledge of the crystal system to which it belongs. From this point it is often possible to identify the space-group unequivocally from the x-ray diffraction pattern. In examining a list of X-ray reflections for this pur- pose, it is best to look for evidence of the lattice type - 'whether it is simple or compound: systematic absences throughout the whole range of reflections indicate a com- ;pound lattice, and the types of absences show whether the cell is body-centered, end-centered, or face-centered. If all his! reflections with h+k+x odd are absent, the lattice is body-centered. If all hkR reflections with h+k odd are absent, the lattice is end-centered. If all hk! reflections ‘litheh+k or k+X or 1+h odd are absent, the lattice is face- centered. When the lattice type has been determined, a search is made for further absences. Systematic absences throughout a zone of reflections indicate a glide plane normal to the zone axis, while systematic absences of reflections from a single principal plane indicate a screw axis mortal to the plane. The result of such a survey, followed by an enxni- nation of the list of absences for all space-groups? may be to settle the space-group unequivocally. X-ray diffraction patterns do not always lead to the selection of a space-group. It may be possible to obtain additional information from other sources, such as an exam- ination of morphological features, or tests for piezo-elec- trio and pyro-electric effects, or by an optical examina- tion for evidence of rotation of the plane of polarized light. ' If the space-group is in doubt at the conclusion of the enaminstions and tests listed above, there is no other course than to proceed with the next stage in the interpre- tation of the X-ray diffraction patterns. There may be stereochemical reasons for supposing that one arrangement is Inore likely than others, and this arrangement will be tried first. Such possibilities cannot be discussed in general . terms; they are specific for each crystal. Familiarity with the general background of crystal chemistry and molecular stereochemistry is desirable. C. The Leteenination of Atomic Fositione 1. [utter of molecules Err unit cell. The fiPBt step in determining atomic rositions ie the celculation of the number of molecules per unit cell. If the density of the material is known or Can be determined, the number of mole- cules per unit cell, m, can be conveniently calculated from the equation Equation 4. E II vac ask: where d is equal to the density (grams per cubic centime- ter), V is equal to the volume of the unit cell (cubic cen- timeters), m is equal to the molecular weight, and f is equal to the mass of an atom of unit atomic weight (1‘6604 x 10-0; gram). The number of molecules per unit cell, in addition to being a fundamental criterion for determining atomic positions, also serves as a check on the space-group determination, as the selected space-group must be able to accommodate the calculated number of molecules per unit C611. 2. leculetion of intensities.3 Each are on an X-ray powder diffraction photo;reph may be regarded as the reflec— tion of X-rays by a particular set of parallel crystal planes. The intensity of this reflection is controlled by several factors - the diffracting powers of the atOms, the arrangement of the atoms with regard to the crystal planes, the bragg angle at which reflection occurs, the number of crystallographically equivalent sets of planes contributing towarcs the total intensity of the arc, tLe amplitude of LLB thermal vibrations of the atoms, and absorption of the X- ray beam by the powder Bangle. In any powder photeraph, two features are apparent; one, there is a general dininu~ tion of intensities with increasing reflection angle, and two, the intensities vary from one are to the next in an apparently irregular manner. Ghe general diminution of in- tensities with increasing reflection angle is due to a de- crecce in the diffracting powers of atoms with increasing reflection angle, to the polarization of the X-rays on re- flection (dependent on the reflection angle), to a geomet- rical factor, and to the thermal vibrations of the atoms. The apparently irregular variation of intensity from one are to the next is due to the effect of the relative posi- tions of the atoms in space - the structure factor, and to the variation in the number of equivalent sets of planes contributing to the arc - a number which depenes on the type of plane. 5. Liffractinggpcwcrs of atoms.3 X-rays are diffract- ed, not by the nucleus of an atom, but by the cloud of elec- trons forming the outer parts of the atom. The diffract- ing power of an atom is determined, in the first place, by the number of electrons surrounding the nucleus, that is, ‘by the atomic number or the element. Atoms in crystals can- -6’1 y- 0 wt not be regarded as scattering points. -ne diameter of the electron cloud of an atom is of the same order of size as 3 . I 4- .9 i .. ,. t... 4 an . s... . ohe (1-8 ts..ce between tin) 00;.11301‘6 Oi {inurciLUl’it 470...»). .oAU be rederded es sohores of defi- etoms in many crystals may ‘ nite radius in Contact with each other, which means that the electron clouds of adjacent s one new be regarded as just touching each othnr. The consequences of this are that waves of x-rsys diffracted by outer electrons of the atom may not be in phase with waves diffracted by inner electrons of the atom. This results in a reduction in the intensity the diffracted been. In this circumsténce, the reduction 0 '$ in intensity is not large, because the electron density in the outer regions of the atom is low compared with the den- sity near the center. For the higher order diffraction ores (reflections from closely spaced planes), when the phase difference for waves from the inner regions of adjacent atoms is several wave-lengths, waves from regions of similar electron densities interfere with each other, and the inten- sity of the diffracted beam is much reduced. The apparent diffracting power of an atom is dependent on the spacing of ‘the reflecting plenes. It is usually given as a function of sin 9A1, and the diffracting powers of all atoms (symbol- ized to) for a wide range of values of sin 9AL,are tabu- lated.7 Plot: of the values of to as a function of sin GAL for the manganese atom and for the fluoride ion were made for this investigation, copies of which appear in Figure 7. -51- 25 15- 10 Mn 5— F- 4 1 1 I l O O 0.2 0, Figure 7. Liffracting power of manganese atoms and fluoride ions as a functfon of scatter*ng angle. 4. ang;g_£gggggg.3 In addition to the effect that the reflection angle haa on the diffracting powere of atoms, there are three angle factore that muat be considered. The polarization of x-raye which occurs on reflection caueee a diminution of intensitiee in the diffracted beams with in- creaeing angle of reflection. Thia effect reduces the in- tensity of any diffraction are by the factor (l+coe'29)/2. An additional angle factor ie known ae the Lorentz fac- tor. Moat eryetale are not perfect, and during rotation re- flect over a range of aeveral minutes of arc. Thie effect ie cauaed by different portiona of the lattice not being quite parallel to each other. The Lorenta factor ia unique for each type of photograph, and for the powder diffraction photograph hae a value of two. The third factor ie geometrical in nature. All the re- flected beame from all the little cryatala are apread over a cone which ie narrow for reflectiona at enall anglea but Inch wider for reflectiona at larger anglea, when 29 1a near 90'. The fraction of inteneity_per unit length of are (which decidee the degree of blackening of the film) 1a less at the larger anglee than at the analler ones. The conea are amaller again for back reflectione at Bragg anglee ap- proaching 90', ac that here again there ia a greater fraco tion of intenaity per unit length of arc. Thie effect changee the intensity of a diffraction are by the factor l/(ein'e cos 9). -63- These three angle factors are combined into one ex- pression for the calculation of the intensities of the dif- fraction arcs, equal to l + cos'ZQ . ein‘e cos 9 Values of this total angle factor for values of 6 from 0 - 90‘ are tabulated.7 Plots of the values of the total angle factor as a function of 6 were prepared for this in- vestigation, copies of which appear in Figures 8 and 9. 5. Thermal vibrations.3 Atoms in crystals vibrate at ordinary temperatures with frequencies very much lower than those of x-rays. At any one instant, some atoms are dis- placed from their mean positions in one direction while those in another part of the crystal are displaced in anoth- er direction. Diffracted x-rays which would be exactly in phase if the atoms were at rest, are actually not exactly in phase, and the intensity of the diffracted beam is lower than it would be if all the atoms were at rest. For crystal planes of large spacing (those giving reflections at small angles), the thermal displacements of the atoms are small fractions of the plane-spacing, and do not affect the in- tensities appreciably. For the more closely spaced planes, the atomic displacements may be comparable with the plane- spacing, and the intensities of these reflections may be greatly reduced. The effect is greater the larger the angle of reflection. -64- 246' (a (2032 ze)/sl n26 cos 6 0° 10° 20° 3 , 80° 90° 6 Figure 8. Total angle factor for small or large reflection angles as a function of scattering angle. -55- H .b (D m _ O 0 , «2‘3 '63 > 0 N8_ 0 0 :2; 4—- 1 I J l I 20° 40° 60° 80° Figure 9. Total angle factor for intermediate reflection angles as a function of scattering angle. -66— This effect reduces the intensity of a diffracted beam by the factor -3 as)" e 7\ where B is a constant for a particular crystal. B is re- lated to the amplitude of vibration of the atoms. It can usually be estimated only approximately.7 An inaccurate estimation of B means that the intensities of the diffrac- tion arcs decrease more slowly or more rapidly than they should with increasing angle of reflection. The use of the above sapression implies that all the atoms vibrate with equal amplitudes, which is not strictly true. In general, thermal vibrations are different for every crystallograph- ieally different atom in a unit cell. An additional assump- tion is implied in the use of the expression given above: that the thermal vibrations of the atoms have the same mag- nitude in all directions in the crystal. This is not strict- ly true, but is a sufficient approximation to the truth for nany crys tals . It was not found possible to estimate an acceptable ‘ value of B for the calculations made in this investigation, therefore the correction factor for the thermal vibrations of the atoms was not included in the calculation of the in- tensities of the diffraction arcs. However, in Chapter V consideration is given to the effect that thermal vibrations of the atoms would be expected to have on the intensities of the diffraction arcs. —67- 6. Absorption. The effect of absorption of X-rays in a powder specimen is to diminish the intensities of diffrac- tion arcs at small angles much more than those at higher angles. Corrections can be calculated for cylindrical spec- imens of known diameter.2’ 7 The absorption factor is de- pendent on 9, but is also dependent on the productfir, where ,0 is the linear absorption coefficient of the crystal, and r is the radius of the crystal specimen. ya can be obtained from the equation x“ = '32“ where n is the number of molecules in the unit cell, V is the volume of the unit cell in cubic centimeters, and [4“ is the atomic absorption coefficient (obtained from tables7). Values of the absorption factor are available in tables7 for a range of values of/nr and for reflection angles from O - 90'. The procedure for obtaining the absorption factor is as follows: obtain the values of/q‘ from the tables, the number of molecules per unit cell from Equation 4 (p. 59), the volume of the unit cell from the parameters, and calcu- late/a and/4r: for this value offlr, obtain from the tables the values of the absorption factor for given values of G, and plot the absorption factor as a function of 6. The ab- sorption factors used in this investigation were those de- veloped by Classen.7 It is recognized that a different set of absorption factors has been developed,2 but, for the/qr values encountered in this investigation, there is no appre- n68- (3 IO Figure 10. S 0'; < 1) ,5 Cl— 0 C: O... O Q '1 O 9 for manganic fluoride (iron K4 not?on of scattering angle. 1? - r . .— §A4;-/ e ”*9.- v 0.26 0.24 OO Absorption factor for manganic fluoride (molybde- num Kq radiation) as a function of scattering angle. -70- oiable difference between the two sets of values. Copies of the plots of absorption factors versus 9 appear in Figures 10 and 11. 7. Number of eguivalent geflections. The number of equivalent reflections or multiplicity can probably be under- stood best by an example. A crystal with cubic symmetry has three planes of type 200, that is, 200, 020, and 002. How- i ever, a crystal turned to all possible orientations would ' give reflections from this type of plane for six different 3* orientations with respect to the X-ray beam (002 and 0C5 ‘J being reflections in opposite directions from the same plane). Therefore, the multiplicity for this type or crys- tal plane is six. There are twelve different planes of type 211, which would give a multiplicity of twenty four. The intensity of any diffraction arc is directly dependent on the number of equivalent reflections from the type of crys- tal plane producing the arc, and that number (designated by p) must be included when calculating the intensity of the diffraction arcs. The multiplicities for all types of crystal planes have been tabulated.7 8. Structure amplitude. The structure amplitude is the amplitude of the radiation scattered by one unit cell. In order to calculate the structure amplitude, it is neces- sary to compound the eaves scattered by all the atoms in the unit cell in different directions of diffraction, the directions of diffraction being determined by the inter- -71- planar spacings in the crystal.1 In most crystals, the co- ordinates of some or all the atoms are not simple fractions of the unit cell edges, and the phase relationships between waves from different atoms are not simple. It is possible to compound the waves graphically or vectorially, but it is extremely difficult, especially for complex structures, and is not necessary. In practice the compounding of waves from the different atoms is done by calculation:5 The expression for compound- ing waves from different atoms (diffracting power f0) situ- ated at different points in the unit cell (coordinates x, y, z in fractions of the unit cell edges) is, in general, for any reflecting plane hkl, Fa where A =3 [facesQflhx+ky+Xs), B a Z:f°sin2u1hx+ky+!z), and F is the structure amplitude. This is valid for all a A. + B”, Equation 5. crystals, whatever their symmetry, but when a center of sym- metry exists at the origin of the unit cell, the sine terms add up to zero. Equivalent atoms (those related to each other by symmetry elements) have coordinates which are re- lated to each other in a simple way, and on this account the cosine terms for the whole group may be combined to form a single expression, the evaluation of which is usually more rapid and convenient than the process of dealing with each atom separately. The sine terms may be combined in a sim- ilar way. If the combined expressions are used, it is only necessary to consider each reference atom in turn. The co- sine term for each independent group is evaluated and then all the cosine terms are added together. Sine terms for all the independent groups are likewise added together. For an ideally imperfect crystal the value of F8 obtained from Equation 5 (p. 72) is proportional to the intensity the bk! reflections would have if the atoms really Were in the postulated positions. The combined expression for the contribution of a set of equivalent atoms in the general positions have been calculated for each space group.7 It should be noted that these expressions do not include the diffracting power to. 9. gomglete expression for intensit If relative intensities are being calculated, it is suffi- cient to multiply the square of the structure amplitude by all the correction factors mentioned above. For an K-ray powder diffraction photograph, the intensity of each arc is preportional to a . Fap l_:u£2§_§§_.T A, Equation 6. sin 9 cos 9 where T is the factor for the thermal vibrations of the atoms, and A is the absorption factor. The other symbols are given above. The variable parameters (coordinates of atomic posi- tions) are determined by calculation of the relative inten- slties for selected atomic positions. It is nonceeary to postulate likely positions for the atoms, to calculate the intensities which these oosltlone would give, and to com- pare these calculated intensities with those observed. The prospects of success depend on whether the postulated posi- L) ions are anywhere near the correct position—, let“; some measure of agreement with observed intensities. If the postulated positions are near the correct ones, the correct positions can be found by judicious small displacements of some or all the atoms from the positions first chosen. 10. Background of crystal cficuletry.3 Ideally, crys- tal structures should be deduced from the X~rej diffraction patterns of crystals (together with such persical proper- ties as are rigorously determined by internal symmetry) without making any stereococmicsl assumptions. Most of the simple structures, and some of the more comrlex ones, have been determined in this way. In the early days of the use of x-ray methode for the determinatl n of crystal struc- tures, it was necessary that structures should be deduced by rigorous reasoning from physical data, so_that the foun- dations of crystal chemistry should be well and truly laid. In some of the more complex structures, however, it would be difficult to determine all the atomic positions by such methods alone. In these circumstances the obvious course 18 to make use of the wealth of information contained in the large number of crystal structures already established, -74- as well as stereochemical information obtained by other methods, and those physical preperties which have been shown by experience to give reliable structural informa- tion. There is no reason why the fullest possible use should not be made of the generalizations resulting from previous studies, providing one retains an open mind with regard to the possibilities of deviations from or excep- tions to these generalizations. After all, such consider- ations are only used to indicate approximate atomic posi- tions which, it is hoped, will give approximately correct X-ray intensities. The atoms are then moved about indea pendently until the best possible agreement between the calculated and observed intensities of X-ray reflections from a wide range of planes is obtained. The proof of the correctness of the structure is this agreement, and it does not matter how it is attained, whether by rigid deduction from the X-ray diffraction pattern alone or by reasonable induction from general principles arising from a survey of previously determined structures. :75. f .u- _._.._ . .fiq .'.' - i‘ a!" V. RESULTS AND LISCUSBION A. Manganic Fluoride F l. L;ray;results. The X—ray data are shown in Tables 5 A..- 4 ‘1»..44 “-04-. ' a v VII and VIII, along with calculated values of the reflection angle, 9, and the interplanar spacing, d, for each diffrac- gp.-- 4.. ._- I tion are. The killer indices,”hk9, included in these tables, were derived from the proposed bimolecular unit cell number seven (Table XII) to be described later. Photostatic copies of the X-ray negatives are shown in Figure 12. It can be noted that the pattern obtained with iron Kd radiation is somewhat faint, but that the diffrac- tion arcs are distinct and seem to appear as doublets. On the pattern obtained with molybdenum K4 radiation, there ap- pears serious overlapping of the diffraction area, especially for the diffraction arcs at low angles of reflection. At the higher reflection angles, the area are, in general, resolved. 2. Determination of the unit cell. Trivalent manga- nese has nearly the same crystal radius (0.62 K.) as has trivalent iron (0.60 1?“).1:5 so that it might be expected that manganic fluoride would have the same or nearly the same crystal structure as ferric fluoride. It is known14 -75- TABLE VII BIFFRACTICH PA TLHN OF HANGANIC FLUORILE EITH 1303 K4 RADIATION 8(0130) 9(°) 6(3.) I(E3t.) hki 3.017 15.09 3.722 vs 101 3.067 15.44 3.640 s 110. 4.146 20.73 2.737 vs 112 4.314 21.57 2.635 vw 211 4.443 22.22 2.562 vvw 101 5.077 25.39 2.260 I 102 5.250 26.25 2.190 I 201.210 6.291 31.46 1.656 I 202 6.452 32.26 1.615 VI 220 6.744 33.72 1.745 vw 113 6.667 34.34 1.717 vw 213 7.069 35.35 1.675 vw I02,312 7.173 35.67 1.653 vvw 201.321 7.396 36.99 1.610 vvw i21 6.194 40.97 1.472 - 130 6.343 41.72 1.466 - 133 6.935 44.66 1.376 - 224 9.447 47.24 1.320 - 242 9.915 49.56 1.272 - 13i,i31 10.536 52.660 1.216 - 212,212 10.692 54.460 1.190 - i30,561 .77- .I‘ TABLE VIII EIFFRACTION PATTERN OF MANGANIC FLTYORILE WITH MOLYBDFN'Ui KR RADIATION 3(0m.) 9(°) d(z.) I(Est.) hk2° 1.107‘ 5.535 3.666 vvs 101.110 1.539‘ 7.695 2.654 s 112,121 1.591b 7.955 2-568 101 1.643‘ 9.215 2.219 6 102,201 021.120 1.969 9.645 2.076 vvw 200 2.212 11.060d 1.656 s 202 2.257 11.265 1.616 s 220 2.361 11.905 1.722 w 213 2.472‘ 12.360 1.661 s 102,201 312.321 2.517 ‘ 12.565 1.631 s 121 2.672 13.360 1.536 vw 121 2.785 13.925 1.476 s 130 3.006 15.030 1.370 vs 224 3.124 15.620 1.320 w 242 3.241 16.205 1.274 w 13'1','1'31 3.363 16.615 1.226 w 212,212 3.467 17.335 1.193 w 130.301 3.566 17.630 1.161 vw 213 3.660 16.400 1.126 vw 204 478. a. ‘i' mna. . :7) a ‘ -. 3v" TABLE VIII (continued) 8(om.) 0(°) d(A.) I(E8t.) hk2° 3.792 16.960 1.094 vw 240 4.006 20.040 1.037 vw 400 4.361 21.605 0.957 vw - - 4.647 23.235 0.901 vw - - 4.773 23.665 0.676 vw - - 4.693 24.465 0.656 vw - . 5.373 26.665 0.767 vw - - 5.521 27.605 0.767 vvw - - 5.666 26.430 0.746 vvw - - ‘Uareselved doublet. bit is doubtful that this are was resolved from the preceding doublet. cThe hk‘ values appear different in Table VIII than in Table VII only because of unresolved doublets. dThe fifth digit in this column is not significant. «'79 . Figure 1.2. X-ray powder diffraction photographs of manganic fluoride with iron Kiradiation (above) and mo- lybdenum Kai radiation (below). .80- that ferric fluoridc ha3 a crystal structure with rhombo- hodral symxetry, and can be considered as having either a uni.-nolecu lar or bimolecular unit cell. The same general considerations hold true for the other closely related tri- 1luorides, those 01 alu:ni.um, cobalt, nickel, rhodium, and palladiu“. First consideration was, therefore, given to a rhombohedral type unit coll. The method for use of Lne hjurstron charts for the de- termination of the cryst tal 6"stem to ahi~h a crystal be- longs, and for the determination of tiller indices for the crystal planes which produce the diffraction arcs, has been given above (p. 55). for this determination the values of the intorplanur spacings from the diffraction pattern ob- tained with iron lid, radiation were used (Table VII . The values of log l/d' for all the derived d values were plotted on a strip of paper using the same scale as used on the ver- tical axes of he Bjurm rem charts. This strip of paper was then moved aroundo the ohaTts, vith particular considera- tion being given to the chart of the rhombohedral system, until the best possible as to cliing position was found. It was found that the best possible matching positions were found on the chart for the rhombohedral system at 00/60 equal to 1.52 and 2.64. These were selected as the best matching positions, but it should be noted that, at these positions, the chart has single curves while the strip of paper has doublet lines. .8;- The doublet nature of the diffraction arcs could be ex- plained in several ways: (1) absorption doablete, (2) a mix- ture of two closely related uninoleeular unit cells, (5) a mixture of two closely related binoleculer unit cells, (4) a mixture of a uninoleciler unit cell and a binoleculer unit cell, the two not giving identical diffraction arcs, and (5) a distorted rnombohedral unit cell. The doublet separation on the photograph obtained with iron Ed radiation is consid- ered too large to be caused by absorption as only very nar- row doublets are known to be caused by absorption of K-rays.3 Furthermore, manganic fluoride does not absorb iron KQSX- rays to such an extent that absorption doublets would be ex- peeted. In addition, a few doublets can be seen on the photograph obtained with molybdenum Ki radietion, which is not highly absorbed by mundanic fluoride. It was concluded that the doublets were not caused by absorption or the x- rays. than the observed interplanur spacings were separated into two 5r ups in an attempt to establish two unit cells with closely related pareneters, it was found that one group could be obtained that would closely approximate the pattern expected from a rhombohedrel unit cell, but that the second group had no apparent relationehipe within itself. For this reason it was concluded that a mixture of very similar unit cells was not the cause of the observed doublet nature of the diffraction area. There remains to be investigated the . .- E. ”r... --7 - 4”,? possibility that a distorted rhombohedral unit cell pro- duced the observed diffraction pattern. It was found5 that the observed diffraction pattern of manganic fluoride corresponds more closely to that of cobalt .trifluoride than to that of any of the other trifluorides. A comparison of the diffraction patterns, both obtained with iron Kd.radiation, of these two trifluoridee can be found in Table IX. It is evident that there are six diffraction arcs in the pattern for cobalt trifluoride that appear as dou- blets in the pattern for manganio fluoride. In addition, there are six single area that appear on both patterns at about the same reflection angle. There are only four arcs in the observed pattern of manganic fluoride that are not related to the pattern of cobalt trifluoride. Furthermore, there are only two arcs in the cobalt trifluoride pattern for which there are no corresponding arcs on the pattern of manganio fluoride. The similarity in the diffraction pat- terns was the basis for concluding that the unit cell of manganic fluoride is very similar to that of cobalt tri- fluoride. The unit cell of cobalt trifluoride can be con- eidered as a unimolecular unit cell (rhombohedral) with ao I 3.664 K. and 0(- 87°20', or as a bimolecular unit cell with a0 '- 5.30 2. and e‘ .- 57“. It was concluded that the unit cell of manganic fluoride was to be considered as a distortion of either a unimoleoular or a bimolecular rhombo- ~83- TABLE IX DIFFRACTION PATTERNS OF MANGANIC F‘LUQRILE AND COBALT TRIFLUORIDE 6'(MnF.) 6°(CoF.) 9°(unr.) s°(Cor.) 15.085a f 55.545a 15.50 55.55 15.545 55.865 57.0 20.750 58.990 21.45 58.55 21.570 40.970 40.7 22.215 41.715 22.5 41.5 25.585 44.675 26.05 47.0 26e250 ’ 47e255 27.7 49.6 51.455 49.595 52.0 52.5 52.280 52.580 54.1 55.720 54.460 55.6 54.555 r aThe fifth digit in this column is not gnificant. hedral unit cell, and that the lattice parameters were prob- ably very similar to those of the unit cell of cobalt tri- fluoride. Possible diffraction arcs of manganous fluoride (rnPa), manganic oxide (Nngoa), and manganese dioxide (Kn0.) were calculated, and found to be absent from the observed diffraction pattern of manganic fluoride (ana). Based on the observed diffraction pattern, the nature of the distortion that Could be expected was considered. From the photograph, it can be seen that only doublets of the diffraction arcs were evident. to higher splitting of the arcs was observed. It is conceivable that the unit cell of mangsnic fluoride could be a uninoleculsr monoclinic unit cell, but it is much more likely that the unit cell contains no.right angles. Based on the doublet diffraction arcs, the following combinations of parameters are possible: (1) three equal sides, two equal angles, one angle different, (2) two equal sides, one side different, three equal angles, and (5) two equal sides, one side different, two equal an- gles, one angle different, and the different side opposite the different angle. All other combinations of lattice para- meters would cause further splitting of the diffraction arcs. If any calculations were to be made to establish which of these three combinations was the correct one, it was necessary that the unit cell be regarded as a triclinic cell, and that the equation relating lattice parameters to interplanar spacings be used. This equation is given below. a s a l— 3 X Lsin'ok + 53-3111“? + Lain”! + all"-3-(cosficosrwcosdd d‘3 I: be c: o°o + Erlé-(cosXcosq-cose) din}; (003460354085) 0 0 ° 0 __J, h l ' are x . 1 - coa‘d.- eos'? - cos”¥ + Ecosdcosecosx Some simplification of this equation can be made when some of the lattice parameters are identical, but no general sim- plification can be made. It is a formidable and tedious task to use this equation to calculate the lattice parameters of a unit cell that would give the observed diffraction pat- tern. Hosever, several hundred such calculations were made, based on possible unimolecular and bimolecular unit cells. The proposed unit cells that give interplanar spacings most nearly like those derived from the observed diffraction pat- tern are described in Table x (unimclecular cells) and in Table III (bimclecular cells}. The interplanar spacings of these proposed unit cells are compared with the interplanar spacings derived from the observed diffraction pattern in Table XI (uninolecular cells) and in Table XIII (bimolecular ‘cells). The indexing for these unit cells is in agreement with the indexing of the diffraction pattern of cobalt tri- fluoride. The indices given with the diffraction pattern of cobalt trifluoride are henagonal indices, but they can be converted to rhonbohedral indices by the transformations H.h-§-g' K=h+2§+8' msz-2n.-1§+X' -BU- f“ w??? '4 33.4.8413 . 1" 7" "'r, " 4"! '7' ' ‘ - ‘ ’ LA‘LJ OLA-wk}; .LA‘. 1 2 3 4 6 6 7 .(X.) 3.74 3.75 3.75 3.74 3.74 3.74 3.75 3(3.) 3.74 3.76 3.73 3.74 3.74 3.74 3.73 6(3.) 3.65 3.66 3.66 3.64 3.64 3.65 3.66 ‘1‘- 87 86 87 86 89 68.5 66.3 .0 87 66 87 88 89 '86.5 66.3 87 86 86 66 86 63.3 83.6 Cc O IETERPLAfiAR *3 :1 {7) t3 >< 14 SPACINGS FOR UNImOLECULAR CELLS hkx 1, 2 3 _4 5 6 7 Obs. 100 3.73 3.73 3.74 3.73 3.73 3.73 3.73 3.72 X. 001 3.64 3.64 3.65 3.64 3.64 3.65 3.66 3.64 X. 110 2.71 2.73 2.74 2.73 2.73 2.75 2.79 2.74 X. 101 2.67 2.70 2.68 2.65 2.63 2.64 2.64 -2.64 X. 10$ 2.63 2.56 _2.56 2.56 2.58 2.55 2.58 2.56 X. 110 2.57 2.53 2.55 2.55 2.55 2.54 2.50 111 2.25 2.29 2.26 2.24 2.21 2.23 2.26 2.26 Z. 111 2.17 2.10 ,2.12 2.14 2.16 2.16 2.18 2.19 X. 11” 2.17 2.09 2.10 2.09 2.09 2.06 2.06 2.08 K. ~88» ‘ 33 63 36.33 36.33 63 63 63 63 36.33 3.33 33.33 33 33 36.33 36.33 36.33 63 63.33 36.33 63 36.33 3.33 33.33 33 .3 33 36.33 36.33 36.33 63 63.33 36.33 63 36.33 3.33 33.33 33 61 33.3 33.3 33.3 33.3 33.3 33.3. 33.3 33.3 33.3 33.3 33.3 33.3 ..m.o 33.3 33.3 33.3 33.3 33.3 33.3 33.3 ”3.3 33.3 H3.3 33.3 33.3 «.«33 33.3 33.3 33.3 33.3 33.3 33.3 33.3 33.3 H3.3 33.3 H3.3 33.3 ..«.¢ 33 ”a o” o 3 6 3 3 3 3 m a mudmo mdQDomAOMHm Qmwoaomm HHN mqm