The Structure of the Ionic Crystals Brookite and Columbite Citation Sturdivant, James Holmes (1930) The Structure of the Ionic Crystals Brookite and Columbite. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/ANRX-9P07. https://resolver.caltech.edu/CaltechETD:etd-12012003-084446 Abstract No abstract. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: Chemistry Degree Grantor: California Institute of Technology Division: Chemistry and Chemical Engineering Major Option: Chemistry Thesis Availability: Public (worldwide access) Research Advisor(s): Pauling, Linus Thesis Committee: Unknown, Unknown Defense Date: 1 January 1930 Other Numbering System: Other Numbering System Name Other Numbering System ID Gates Chemical Laboratory 177 Gates Chemical Laboratory 195 Record Number: CaltechETD:etd-12012003-084446 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-12012003-084446 DOI: 10.7907/ANRX-9P07 Related URLs: URL URL Type Description https://doi.org/10.1524/zkri.1928.68.1.239 DOI Article adapted for ch. [1] https://doi.org/10.1524/zkri.1929.69.1.556 DOI Article adapted for ch. [2] Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 4702 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 01 Dec 2003 Last Modified: 03 Oct 2019 23:37 Thesis Files Preview PDF - Final Version See Usage Policy. 2MB Repository Staff Only: item control page

THE STRUCTURES OF THE IONIC CRYS'r.us BROOKITE Al\JD COLU.MBlTE Thesis by James Holmes Sturdivant In partial fulfillment of the requirements for the degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1930 For encouragement, e.cl.vice. e.nd direction in the course of the studies here reported, the author is grateful to Professor Linus Pauling. 239 · XV. The crystal structure of brookite. By Linus Pauling and J. H. Sturdivant in Pasadena. (With 6 figures.: I. Introduction. In the study of the structure of a crystal with X-rays the effort has been made by .many workers, especially Wyckoff and Dickinson in America, to eliminate rigorously all but one of the possible atomic arrangements consistent with the smallest unit of structure permitted by the experimental data, without reference to whether or not the arrangements were chemically reasonable or were in accord with assumed interatomic distances. The importance of this procedure arises from the certainty with which its results can be accepted. For although structure determinations by less rigorous methods have been found to be false, no important error has yet been detected in any investigation which interpreted photographic data solely with space-group theory aided by . merely qualitative assumptions regarding the factors affecting the intensity of X-ray reflection. It would accordingly be desirable to conduct all structure determinations by this method; but unfortunately the labor involved in its application to complex crystals, involving more than a very few parameters, makes this impossible. J.<'urthermore, if several different atoms are present in the structure, it is often necessary to make quantitative assumptions regarding their relative reflecting powers, so that for this reason too the rigorous method cannot be used. But complex crystals are of great interest, and it is desirable that structure determinations be carried out for them even at the sacrifice of rigor. The method which has been applied in these cases is this: one atomic arrangement among all of tl::ie possible ones is chosen, and its agreement with the experimental data is then examined. · If the agreement is complete or extensive, it is assumed that the structure is the correct one. Striking regularities· in the intensities of reflection from simple planes may suggest an approximate structure. This occurred in 16* 240 Linus Pauling and ,J. H. Sturdivant Dickinson's ,study of tin tetraiodide1); he then considered values of the five parameters involved which differed only slightly from those suggested by the intensities of simple reflections, and found a set giving complete agreement with the Laue photographic data. On account of the large amount of experimental information accounted for by it, this structure can be confidently accepted as correct. In other cases it is necessary to make use quantitatively of relative reflecting powers or F-curves, and often agreement is found between the assumed structure and data from rotation photographs or ionization chamber measurements only; then the struclure can usually be accepted as approximately correct, the uncertainty depending upon the paucity of the experimental data or the lack. of agreement with them. As a result of the recent increase in knowledge of the effective radii of various ions in crystals, Professor W. L. Bragg has suggested and applied 2) a simple and useful theory leading to the selection of possible structures. His fundamental hypothesis is this: if a crystal is composed of large ions and small ions, its structure will approximate a closepacked arrangement of the large ions alone, with the small ions tucked away in the interstices. In case all of the close-packed positions are not occupied by large ions an open structure results. To apply this theory one determines the unit of structure in the usual way, and finds by trial some close-packed arrangement of the large ions of known radius (usually oxygen ions with a crystal radius of about 1,35 A) compatible with this unit. The other ions are then introduced into the possible positions in such a way as to give agreement with the observed intensities, and if necessary the large ions are shifted somewhat from the close-packed positions. With the aid of this close-packing method Bragg and his co-workers have made a promising attack on the important problem of the structure of the silicate minerals. During the investigation of the structure of brookite, the orthorhombic form of titanium dioxide, another method of predicting a possible structure for ionic compounds was developed. This method, which is described in detail in Section III of this paper, depends on the assumption of a coordination structure. It leads to a number of possible simple structures, for each of which the size of the unit of structure, the space-group symmetry, and the positions of all ions are fixed. In some cases, but not all, these structures correspond to closepacking of the large ions; when they do, the method further indicates l) R. G. Dickinson, J. Am. Chem. Soc. 45, 958. 1923. 2) W.L.BraggandG.B.Brown, Pr.Roy. Soc., A, 110, 34. l01!6; \V.L.Bragg and .r. West, ibid., A, 114, 450. 1927. Tho cdato.1 structure: of brookite. 241 the. amount and nature of the distortion from the close-packed arrangement. The general method of investigation which we used is the following. Spectral photographs from the three pinacoids provided accurate dimensions of the possible units of structure. Data from Laue photographs then led to the determination of the smallest allowable unit, and the pl'esence and absence of reflections on Laue photographs served as criteria in finding the space-group symmetry. (Laue photographs rather than rotation photographs were used because the chance of error in assigning indices to Laue spots is very small1).) On applying the coordination theory we found that one of the predicted structures had the same space-group symmetry and unit of structure as the crystal itself. The predicted parameter values agreed reasonably well with the intens1t1es of reflections on rotation photographs, and better agreement was obtained by changing one of the nine parameters slightly. On account of lack of knowledge of F-curves the intensities of Laue spots were not accounted for. II. The unit of structure and space-group. Three forms of titanium dioxide, Ti0 2, are known. Of these the ' crystal !!truclures of the two tetragonal forms, rutile and anatase, have been thoroughly investigated 2); in each case only one parameter is involved, and the atomic arrangement has been accurately determined. The third form, brookite, is orthorhombic, with axial ratios o,sA.• 6 : 1 : o,9Ut. Data obtained from photographs of the K-radiation of molybdenum reflected from the ground pinacoids of a hrookite crystal (variety arkansite from .Magnet L:ove, Ark.)~) are given m Table L These lead to a unit with d10 0/n1 2,291 A, do10/n2 = 2,7181, and do 0ifn3 = 1,284- A, in which n 1 , n 2 , and n 3 are the orders of reflection of the first lines on the photographs. Laue photographs were taken with the incident beam of X-rays at various small angles with the normal to (100) of a thin plate of brookite tabular on {I 00} (from Iliedertobel, Uri Canton, l) As an example of an incorrect space-group determination result.ing from e~ror in assigning indices to reflections on rotation photographs mention may be made of the assignment of olivine to VR by Rinne, Leonba.rdt, and Hentschel (Z. Krist. 59, 548, 1924). Bragg and Brown (Z. Krist. 63, 538. 1926) determined the space· group of olivine to be V),0 and pointed out the source of lhe previous error. 2) L. Vegard, Phil. Mag. 1, 1151. 1926; G. Greenwood, ibid. 4:8, 654. HIM; M. L.Huggins, l:'hys. Rev. 27, 638. 1926. 3) We are indebted to Mr. Rene Engel for the crystals used in this research. Linus Pauling and J. H. Sturdivant 242 Table I. Spectral data. ·hkl) ' I Oril~r of I 1Hellecfi'on Line. . I Angle of I (MoK; , Reflection ~~~ , - ,~ = (010) (001) i.,,.1 a 3n1 3nt 4n1 4 n1 4n1 a1 a2 ;' fJ u1 l d/n· ~:54,~C-~ 1 586 ,~ 5 t13,t1 4,tis~ 13 24 13 29 15 4 t 15 59 17 59 18 6 4,580 4,581 4 ,585 4,583 4,584 6 1,0 2,748 2,716 7 30,5 13 H 2,718 1a 25 2,720 15 5 2,721 15 H 2,718 13 58 1,2835 14 14 16 0 1,':!833 1,':!841 1,2840 H 6 Switzerland). The short wave-length limit of X-radiation present in the incident beam was 0,24 A; values of nA. calculated for Laue spots on the basis of the unit obtained by putting n 1 = n 2 = n 3 = 1 were found often to be much smaller than 0,24 A, so that this unit cannot be the correct one. The smallest unit which will account for the Laue data is that with n 1 = &., n 2 = 2, n~ = &.; i. e. 7 with 1) dioo 9,·16(} A, d010 = iJ,436 A, d001 ts,1::rn A. The presence of all observed Laue spots, produced by planes belonging to over two hundred different forms, is accounted for by this unit. {Representative Laue data are recorded in Table II.) :i'o evidence was found for the existence of a larger unit; hence this unit may be accepted as the true one. All indices used in this paper (including Table II) are referred to the axes of this unit of structure. The unit contains 8 Ti02 ; the density calculated from the X~ray data is 4- 112, within the range 4.,03-4,22 of the experimental determinations reported in Groth 1 1) During the preparation of this manuscript, ~. note ha~ appeared (A.Schroder, Z. Krist. QG, 403. 1028) giving the dimenaions of the unit of structure for brookito as a= 9,136 + 0,02, b 5,'t39 ± 0,01, r.i 5,153 ± o,03 A. No space-group determination or discussion of the atomic arrangement was reported. 243 The cristal structure of brookite. Table II. Laue data for n'A between o,35 and o,~5 A. (hlcl; du1 ---~·--- 2 4 1 . 2 1 4 1 4 2 1 2 4 3 4 2 1,25A. 1,20 nA ---------------- 1,18 1,l5 1,H 0,42.A ,36 ,44 ,44 ,37 Estimated inlonsity --~ 0,02 ,5 ,8 ,1 ,5 a 4 3 1,03 ,40 2 3 7i 3 5 1 l 5 1,02 1,00 1,00 ,37 ,41 ,44 0,3 ,01 I ti 2 4 7i 1i 3 4 5 1 2 5 0,99 ,36 ,1 ,96 ,95 ,2 ,2 2 5 3 ,89 25 3 5 3 ,ss ,42 ,36 ,35 ,42 ,35 ,87 ,40 2 li I\ ,R7 ,H ,0:1. 3 3 5 1 l 6 1 1 6 4 6 1 ,85 ,84 ,84 ,38 ,05 1" 2 5 5 3 5 6 1 ,95 ,83 ,s2 ,so ,so ,4 ,8 ,5 ,35 ,43 ,39 ,15 ,H ,2 ,43 ,37 ,43 ,1 ,02 ,03 ,2 ,15 3 5 4 3 5 4 1 6 3 ,79 '45 5 6 2 ,77 ,36 ,42 ,so ,1 '" ,2 1 3 G ,77 2 7 1 ,75 5 5 7 2 ,74 ,74 ,72 ,40 ,2 ,2 '71 ,70 ,45 ,05 2 7 ,36 ,1 2 7 ,70 ,35 ,01 2 2 7 ,70 ,69 ,45 ,44 ,40 ,37 ,43 ,45 ,44 ,05 ,l5 0,2 ,H ,05 1 4 6 5 3 6 2 3 2 7 3 7 1l 5 7 2 7 3 6 7 7 1 2 7 4 3 8 1 ,68 ,68 ,66 ,66 ,65 ,65 ,43 ,42 ,39 ,2 ,05 ,02 ,05 ,o5 ,1 a Linus Pauling and J. H. Sturdivant 244 Table II (cont.). (hkl1 dhkl nl.. Estimated intensity 4 G 5· O,li5 o,ss 772 7 T7 ,Gt. ,41 ,63 ,.,.1 ,1 7 2 7 ,G2 6 7 4 5 r; 7 4 s lf ,61 ,03 ,l ,61 ,37 ,4 1 ,4;) ,60 '41 ,o3 8 ,oo ,37 ,1 2 ~ 0,1 ,15 ,or. 5 6 6 ,59 ,3!\ ,02 7 8 3 3 7 6 ,57 ,51) ,04 7 5 7 9 4- ,55 ,54 ,39 ,43 ,41 ,38 4 2 9 ,54 ,43 ,02 ,o 3 4 4 !l ,~I .H ,os 5·H·2 5·10. 2 10·1·9 10 .1. 9 ,51 ,51 ,47 ,45 ,39 ,36 ,38 ,47 ,05 ,05 ,02 ,02 ,02 ,o 3 and in good agreement with the experimental value i,125 obtained by Schroder. The presence of first- order reflections from all types of pyramidal planes (Table II) eliminates from consideration all space-groups based on any but the simple orthorhombic lattice T 0 • Of these the following are further definitely eliminated 1) by the occurrence of first-order reflections from the prism planes given in Table III: C:j_v, Cfv• CJv, qv, q'v• q~, v~, v~, v~, v~, v~, v~, v~0 , 1~:.2 1 v~;\ v~.\ v~6. If the structure is based on one of the space-groups a1 ~v' 2 04 as v1 v2 V3 vc1 vi vi y;, or v21, 02v' 2v' "1!' l l ' ' h' J1l hl " it is very difficult to account for the fact that no first-order reflections were observed from planes with h 0 and k odd, k 0 and l odd 1 or l = 0 and h odd; these space-groups may accordingly be removed from consideration. But just these absences are required for any arrangement derived from V~5, which may hence be accepted as the correct space-group. 1) Space-group criteria are given by Bozorth and Pauling, J. Am. Chem. Soc. 47, 1568. 1925. Cf. also references in ootnote 13, ibid. 245 The crystal structure of brookite. Table IH. Prism rellections. a) Incident beam normal to (100). (hkl) n.< dhkl Estimated intensity 1,33 1,26 1,26 1,00 0,43 a ':rn 0,15 ,37 ,43 ,20 1,00 o,7G '4;) a a ,2s 0,03 ,73 ,73 ,49 ,H a a 0 ·7 ,70 ,12 a 5 0 11 ,60 ,41 0,02 l 0 1 0 4 2 5 2 0 r) 2 7 7 4 7 b) Crystal tipped 14° 0 3 1, 70 I a 1, 44 0,47 ,44 ,42 3,5 l, 31 ,a:; ',2 1,07 1,06 1,04 ,48 a ,32 1,8 1, 01 ,31 ,43 1 0 6 ,Ra ,33 0 5 4 ,s2 ,24 a a v, 3 a 0 5 ,81 ,21, a 2 0 ,76 ,41 o,s 6 -7 0 ,69 ,63 ,48 ,38 ,2 3 () 3 2 2 3 1 5 0 3 0 3 1 0 ;; 8 I 1,47 a ,1 c; Crystal tipped 20°. 0 2 T 0 4 1 0 T 2,39 1,31 1,31 0,29 I ,47 ,27 2 l ,9 4 1, 'l4 ,40. 0,5 a 3 1,06 1,06 1,06 1,04 ,44 0,3 ,36 ,24 a a ,43 l ,02 ,a9 1 5 3 0 5 0 2 !) 4 0 5 0 6 1,00 ,97 ,34 a a a a ,96 ,85 ,36 ,49 ,40 u (i 4 ,82 ,82 ,34 ,33 5 1 5 T 0 3 3 5 0 0 5 ,93 ,30. 0,2 a o,; a 0,1 246 Linus Pauling and J. H. Sturdivant Table III (cont.). c) Crystal tipped 20° (cont.). Estimated intensity (hkl) dhkl 5 6 0 o,s• A o,47 A a 5 0 6 ,77 ,42 0,2 2 7 0 4 0 7 '76 ,73 ,72 ,69 ,63 ,60 nl. I I -~- 2 0 7 6 ·7 0 3 0 8 7 8 0 2 9 0 3 8 0 ,59 ,50 I ,47 ,6 ,32 ,43 ,42 a O,l 7 ,4 3 ,1 a ,39 a ,B2 ,35 0,02 a The coordinate positions for equivalent atoms provided by V~ 5 are1) 4a: 0 0 0; t t 0; 0 { t; t 0 l; 4b:itt; 00-~; -~00; 01;0; 8c:x,y,x; x,y,z; ·t+x,i--y,z; x, t+y, t-x; x, l-Y, t .t + x, y, l - ;v. -~· x, t -~--x, y, x; y, t+x; Six parameters must be evaluated to determine the positions of 8 Ti and 16 0 in the unit in case that both the no-parameter positions 4 a and 4 b are occupied, and nine parameters in case that aU atoms are in positions 8 c. A structure dependent on such a large number of parameters can not be determined by. rigorous methods. III. The prediction of a structure with the coordination theory. The structures of rutilc and ano.to.se, represented in Figs. 1 and 2 7 seem at first sight to have very little in common beyond the fact that each is a coordination structure, with six oxygen atoms about each titanium atom and three titanium atoms about each oxygen. On closer examination, however, it is found that from a certain point of view the structures are closely similar. Let us consider as the basis of the structures an octahedron with a titanium atom at its center and an oxygen atom at each corner. The rutile and anatase structures are both made of such octahedra sharing edges and corners with each other; in rutile two eclgeis uf each octahedrun are shared and in anatase four. In both crystals the titanium-oxygen distance is a constant, with 1) R. W. G. Wyckoff, i~The Analytical Prf!8fmtation of lhA flA.•nlfs nft.hP. Tlrnory of Space-Groups", Puhl. Carnegie Institution No. 34 8, 1922. 247 The crystal structure of brookite. the value i 195-i ,96 A. The basic octabedra are only approximately regular; they are in both rutile and anatase deformed in such a way as to cause each shared edge to be shortened from 2, 76 A (the value for regular oc_tahedra) to 2,50 A, other edges being correspondingly lengthened. (The distorted octahedra occurring in rutile and in anatase are indicated in Figs. 1 and 2.) Just such a distortion is to be expected, for the Coulo.mb repulsion of the two quadrivalent metal ions brought near each other when an edge is shared will cause the titanium-titanium distance to increase until the repulsion of the two oxygen ions defining Fig. 1. Fig. 1. The structure of rutile. Large circles represent the centers of titanium a.toms, small circles those of oxygen atoms. One octahedron with oxygen atoms at its cornex: and a titanium atom at its center is shown; two of its edges, those indicated by arrows, are shared with adjoining octahedra. Fig, 'I. Tho structure of ana.tase. Four odgcs of the octahedron shown, indicated by arrows, are shared with adjoining octahedra. Fig. 2. the shared edge becomes large enough to counteract the effect. The theoretical discussion 1) of the rutile and anatase structures with the help of the known properties of ions has led to results which are in each case in quantitative agreement with the observed contraction of shared edges. As a result of these considerations the following assumptions were made: 1. Brookite is composed of octahedra, each with a titanium atom at its centP.r and oxygen atoms at its corners. 1) Linus Pauling, Z. Krist. 68, 377. 1928. 248 Linus Pauling and J. H. Sturdivant ~. The octahedra share edges and corners with each other to such an extent as to 3ive the crystal the correct chemical composition. 3. The titanium-oxygen distance throughout is about ,1,95 A. The octahedra are not regular, but are distorted in such a way as to maintain the titanium~oxygen distanr,es rrrnstnnt. nnrl to shorten each shared edge to the length 2,50 A. An infinite number of structures satisfy these conditions. The number of simple structures, that is, structures referable to a small unit, which do so is small. vVe made no attempt to consider exhaustively the possible simple structures, but instead investigated the two which presented themselves first. The second of these we have found to be the structure of brookite. In rutile there occur strings of octahedra in parallel orientation connected by shared edges; these strings, which extend in the direction of the c-axis, then are connected by sharing corners to form the complete crystal. This suggested that staggered strings such as are represented in Fig. 3, with two edges of each octahedron shared, might combine to form a reasonable structure. If the combination is effected by the sharing of corners alone the simple structure rep resented in Fig. 4 results. This structure, Structure .A, has orthorhombic symmetry, and in fact the spar.e-gronp RymmPt.ry VA4 lts unit of structure contains four octahedra A staggered string of (4 Ti0 2 ), and, if distortion is neglected, has Fig. 3. octahedra with shared edges. thA dimensions 4,79 A, ~,t:i'I 1, and 5,52 A. It is not the structure of brookite. The next simplest structure built from these staggered strings results if each octahedron shares one edge with au uclahedruu uf l:lll aujuiniug o. string. This structure is shown in Fig. It is orthorhombic, and has the space-group symmetry V},5. The unit of structure contains eight octahedra (8 Ti0 2 ). Its dimensions, if the octahedra are assumed regular with edges 2,76 A long, are d 100 = 9,02 A, d010 = 5,52 A, and d001 = 4, 79 A. The space-group permits cyclical but not non-cyclical permutations of these dimensions. The crystal structure of brookite. 249 Fig. 4. A model representing a possible structure for Ti0 2 (Structure A), composed of staggered strings of oclahedra (Fig. 3) combined hy sharing octahedron corners only. Fig. 5. A photograph of the model representing the structure of brookite. 250 Linus Pauling and J. H. Sturdivant The fact that brookite also was found experimentally to have the 5 and to have 8 Ti02 in the unit suggests strongly that symmetry of our second structure is to be attributed to this crystal. There is also approximate agreement (within 7%) in the dimensions of the unit. It · is worthy of mention that the /.. choice of axes for comparison is not completely arbitrary: assuming that 9,02 A (model) is to be compared with 9,166: A (crystal), then the space-gronp requires that 5152 Abe compared with 5,4-36 A and 4- 179 A with 5,135 .A. Since the analytical consideration of the effect of deformation appearedto be too complicated to permit its execution, the following method based upon V1 L. t.hP. l'.ondruction and measure- ment of a model was used. The basic distorted octahedron employed is shown in Fig. 6; in it all titanium-oxygen distances are equal to 1195 1; the three shared edges are 2,50 A long; and the other edges are chosen so as to be distorted as little as possible from their ori 0 inul lengths. Such octahedra were F'ig. 6. A portion of the brookite structure, constructed to scale (2,54 cm. showing the three edges of each octahedron = 1 1) from heavy paper, und which are shared with adjoining octahedra, were then glued together to reand the resulting deformation. present a section of the crystal including one unit of structure (Fig. 5). The carefully measured dimensions of the unit were found to be 23 165 cm., ·14,0 cm., and 13,2 cm.; if these are reduced in the ratio 2,57 cm. ·I A (the increase in scale being due to incomplete coirn;iueuce uf ei.1 0 e:-s autl corner:; uf Lhe uclC!heura) they give the values shown in Table IV, which agree within the limit of error of the measurement with the experimental values for brookite. The suggested structure places eight titanium at.oms and two groups of eight oxygen atoms each in the positions 8c, with three sets of 251 The crystal structure of brookite. Table IV. Dimensions of the unit of structure. From_ model,. with undistorted octahedra From model, with distorted octahedra Experimental, for brookite d100 do10 d-001 9,02.A 9,20 5,52A 5,45 5,436 4,79 A 5,13 5,i35 9,166 parameter values. In order to predict values of these parameters the coordinates of the oxygen atoms were measured on the model, and positions were assigned the titanium atoms in the octahedra such as to make the titanium-oxygen distances nearly constant. Representing by x, y, x; §, r;, ~; and X, Y, Z the parameters for oxygen of the first kind, oxygen of the second kind, and titanium, respectively, the values that were predicted are 02 Ti 0,230 r; = 0,105 ~=- 0,465 0,131 Y= 0,113 Z= - 0,127. 01 X = 0,010 y = 0,155 x = 0,180 §= X= These may be compared with those corresponding to undistorted octahedra, namely, 01 02 r;= 0,250 0,125 y y ~ o,417 z X = 0 1000 §= y 0,124. x = 0,250 Ti 0,125 0,125 - 0,083. It is seen that the parameter changes due to distortion are considerable, being equivalent to translations of atoms as large as 0,35 A. IV. Testing the structure with the intensities of X-ray reflections. A number of rotation photographs were made with molybdenum K-radiation filtered through a zirconium oxide filter to isolate the Ka line. The positions of useful reflections, the indices of the planes producing them, and their visually estimated intensities are given in Table V. The factor placed beside the estimated intensity is a correction for the varying time of reflection, namely Y 1 - (t('A/d sin 2 tJp, where 'A is the wave-length, and u and d represent respectively the index of the axis of rotation and the unit translation along it1). (A number of reflections l) Ott, Z. Phys. 22, 201. 1924. Linus Pauling and J. H. Sturdivant 252 not included in the tables occurred on each photograph; namely, those produced by planes whose traces in the developed or ground face of the crystal [the faces (100), (01 O), and (001) for Table VJ were not perpendicular to the axis of rotation of the crystal. These reflections were not considered because of the probability that their intensities would be I_argely affected by absorption in the crystal.) Table V. Spectral reflections 1). a) Planes of form (Okl). 1. Beam incident on (0 1 0). 4 6 2 k= I w l= o s w .Ji 4 2. Beam incident on (0 0 1). l= le= o I ll 4 6 a mw 9 vw 0,8 b) Planes of form (hOl). 1. Beam incident on (1 0 0) 6 7 2 4 8 5 h= l= 0 vvw ms w 0,2 3,5 1,7 l=2 s 0,8 12 9 vs 17 5 10 12 a mw w 0,5 8 5,5 w o,s mw0,9 vvw 0,9 mw0,9 a 3,5 4 3 12 3,5 14 2. Beam incident on (0 0 4} l= h=l h=2 2 4 m vw w /; ,~ " :lit:\ 12 mw0,9 a 0,6 2,5 6 h=3 h=4 6 mw 0,8 a 7 3 a mw0,9 19 1) The estimated intensities of the various reflections are indicated by the abbreviations vs, very strong; s, strong; ms, medium strong; m, medium; mw, medium weak; w, weak; vw, very weak; vvw, very very weak; a, absent. The number placed below the estimated intensity for each plane gives the computed structure-factor. 253 The crystal structure of brookite. c) Planes of form (hkO). l. Beam incident on (1 0 0) 2 h= k=1 k 6 8 10 s U,'/ VW U,!l m a m 8 2 4 3 10 w0,8 2 a vw 0,9 a 1,5 o,7 2,5 2 2. Beam incident on (0 1 0) k h=2 s 0,7 8 h=4 4 7 2 3 vw 0,9 2,5 s 0,9 a a a a 10 o,s 5 6,5 H w 0,6 a 0,2 vw 0,9 4 a 2,5 0,4 mw0,9 6 5 i; Structure factors were calculated with the use of the predicted parameter values and with the assumption that the relative reflecting powers of titanium and oxygen atoms are proportional to their atomic numbers. These structure factors reproduced the salient features of the observed intensities for fifty forms; this general agreement is strong evidence that the predicted structure closely approximates the actual one. There was, however, disagreement in a few cases, when one reflection was observed to be stronger than another from a plane with larger interplanar distance despite the fact that its calculated structure factor was the smaller of the two. Accordingly, the parameter values were varied slightly, and a change of - 0,004 in X was found sufficient to remove most of the points of discrepancy. Structure factors calculated for the new parameter values with the assumption of relative reflecting powers proportional to atomic numbers are included in Table V. It will be seen that when qualitative consideration is taken of the effect of other factors causing a decrease in intensity with decrease in interplanar distance, the agreemerit for all fifty forms is good. The changed parameter values given in Table VI may accordi,ngly be accepted as correct. Table VI. Final parameter values for brookite. x=0,010 ~= o,2ao X= y r;= 0,105 Y= 0,155 %=0,480 Zeitschr. f. Kristallographie. 68. Bd. !:= - o,465 z 0,127 O,H3 o,n7 Linus Pauling and J, H. Sturdivant 254 'I'll'' ftrw rie11111ining dhwrP.panf'i~l'l :1r.~ proh:1hly due lo P1Tnr in the> assumed relative rellecling powers. To test lhis, we made use of rn /r'-cu!'ve for o= obtained lry linear exfrapolalion from ,\",{' <111d or, and Ulli> for 'l'iY" frf,nJ (fl ;:ind Ii' 'l'h~;;;1• /r'-1•111·ve;; (·whieh HNl nnl 1•1,p1·n- duc·~d lL!rn ht\c.rnse uf uncertuilllJ in tl1tir dahaiion) J.ead to s!ruclurL falll•:..rs which are, for Hie 6311.1.:: final paraMeler 1 alue;,;, also JJJ i:,uud J,111 ni.1 r•fompl:'.lf• :1grf!f•lll·!llt \Vith il1R n! ••;erv..:d inl d1>iti·!,;. P.-1,•ild) ~01newhal different fl~curve;:; (correspoilding to no11-Jinear extrapolation) •vould ghe better <lgreement, but because of the arhitrarines;; of this [H'neMlurP. no :itlP.rnpl. wa;; m;iclr. ·to ulilizA ii It is worthy of mention that the rotation photographs yield information substantiating lhe choiee of 1T as the correct space-group. Thu:; no r<>tlf!r.liomi or.1'11rrr.d from {·100}, poo}, {iHlO}, {700}, {900}, {11.0.0}, 1a.o.o}, {ao1 }, {t>oq, {701 J, {901 J, {1 HJ·'1}, {401}, {601}, {so.q, 10.0 ..f}, {12·0·il}, etc. on one photograph recorded in Table V, although { { 1 1 p!HnP.;; of all the;;A form;;; were in pn;;it.ions faxorahle to rellf.etion. Similnr failures to reflect were observed on the uther photographs. On account of the lack of delinite knowledge of the F-cunes no attempt was made to account for the intensilies of Laue spots. V. Discussion of' the structure and the coordination method. The structure found for hrookile is that shown in Fig. 5. Each titanium atom is surrounded by six oxygen atoms and each oxygen by three titanium atoms. The various interatomic distances, given in Table VII, are similar to those in rutile and anatase; for example, the a\·erage titanium-oxygen dis lance is 1,95 .\.. The oxygen-oxygen distances and the distortion of the basic octahedra have been discussed in the previous sections in the course of derivation of the structure. Table VII. tnteratom1c distances in brookite. 'l'i-01 1,919 A 01-03 t.,767 A :J'i-0~ 1,94 2 02-0;i .t,8'1lS 1ri-Oa Ti-04 'l'i-0; '1'(-0.; i ,929 1 ,975 1,983 i ,953 02-01 Oz-06 Ui-0., Oi- 01 01-Ch :!,508 04-0;, 04-011 t,817 2,731 2,:;14 a,012 2,s3:i 01-a1 2,869 O,,-nd 2,H3 0 1-05 2,;HI :!,997 255 The crystal structure of brookite. We believe that our c.onclusions can be accepted with considerable certainty, for the agreement lielweeu Lhe predil.:ted structure and the experimental results in regard to space-group symmetry, size of the unit of structure, and intensities of reflections on rotation photographs is so striking as to remove nearly completely from consideration the possibility of its being accidental. The arrangement of the oxygen atoms approximates double hexagonal close-packing 1). It is evident, then, that Bragg's close-packing method of deriving a possible structure might have b"een used. But both hexagonal and double hexagonal close-packing of oxygen ions are roughly compatible with the observed unit (compare Rows 1 and 3 of Table IV). To decide between these possibilities intensity comparisons would be required, and in the absence of information regarding the nature of the deformation from regular close-packing this procedure might be very laborious. In anatase the oxygen ions approximate cubic close-packing; the axial ratio is, however, increased from 2,0 to 2,5 by deformation, and a knowledge of the size of the unit of structure would not suggest the close-packed arrangement with much certainty. Rutile provides an example of a structure which is easily derivable by means of the coordination theory, but in which the oxygen ions do not approach any type of close-packing whatever. The close-packing method is more generally applicable in the case of crystals containing cations smaller than the titanium ion and showing correspondingly less deformation. The method is very powerful and simple, and already in the hands of Professor Bragg has led to a number of interesting structure d.eterminations. The coordination method, though also simple in principle, is somewhat less easy to apply. The two methods are not .distinct and mutually exclusive, but instead supplement each other and should be used together. VI. Summary. Brookite, Ti0 2 , is found with the use of Laue and spectral photographs to have an orthorhombic unit of structure with d100 9,166 A, d010 5,436 A, and d001 5,135 A, containing 8 Ti0 2 , and to have the symmetry of space-group Vh5· A structure predicted by means of the 1) There are three close-packed layers. from which all of the infinite number of close-packed structures can be built; namely, A; with an atom at X = o, Y = O (X, Y, Z being hexagonal coordinates); B, with an atom at X = :J-, Y =}; and 0, with an atom at X = t, Y = f. For hexagonal close-packing these layers are superposed in the order AB.AB.AB ... , for cubic close-packing in the order .ABC.ABO ... , and for double hexagonal close-packing in the order AB.A CABA 0 ... 17* 256 Linus Pauling and J. H. Sturdivant, The cristal structure of brookite. coordination theory had the same unit and space-group symmetry, and gave an atomic arrangement accounting for the intensities observed on rotation photographs. In this structure the eight titanium atoms and the sixteen oxygen atoms (in two groups of eight each) are in positions 8 e witl:t the parameter values given in Table VI. The basic unit of the arrangement is an octahedron of oxygen ions about a titanium ion; the octahedra are combined as shown in Fig. 5 so that each shares three edges with adjoining ones. Interatomic distances are given in Table VIL Gates Chemical Laboratory, California Institute of Technology. Contribution No. 177. Received May ~9th, 1928. 557 Sonderdruck aus ,,Zeilschrift fi.ir Kristallographie". Bd, 69, Heft 5/6. Akademische Verlagsgesellschaft m. b, H. in Leipzig, rn2t1. J. H. Sturdivant and Linus Pauling (Pasadena): Note on the Paper of A. Schroder: Beitrlige zur Kenntnis des Feinbaues des Brookits usw. (With 2 figures.) We have recently found 4) the space-group symmetry of brookite, the orthorhombic form of titanium dioxide, to be that of VJ,'. In our spacegroup determination, use was made of the presence or absence of reflections on Laue photographs only. Reflections from all types of bipyramidal planes were observed, requiring that the structure be based on the simple orthorhombic lattice I'0 • No first- order reflections were observed from prism planes of the following classes, although many such planes were in positions favorable to reflection: k = 0 and l odd; l = (} and h odd; h = 0 and k odd. The space-group criteria for Vt,," are that no odd- order reflections shall occur from just these planes. These criteria were afterwards found to be satisfied by reflections observed on several rotation photographs. ~) Gemeint ist dabei die Deformation der farblosen geliisten lonen Hg++ und Jbei ihrer Vereinigung zum Salz. 2) H. Ley, Z. angew. Ch. H, 845. 1928. ~) G. Scheibe, Chem. Zentralbl. 1927, II. 2i5L Z. Elektrochem. 34:, 497. 1928. 4) Linus Pauling and J. H. Sturdivant, Z. Krist. 68, 239. ~928. 558 Kiirzere Originalmilteilungen und Nolizen. On page 241 of our paper we pointed out the desirability of basing a spacegr·oup determination on Laue photo13ra pli:; n1Ll1er tlrnu uu rulaliun photographs, on account of the small chance of erro1· in assigning indices Lo the planes producing Laue spots and the much larger uncertJinty in identifying planes producing reflections on rotation photographs. An example ol' an incorrect choice of space-gt'oup arising from the use of rotation photographs was cited. A. Sc hr oder 1) has published data on rota lion photographs of brookite which are incompatible with the space-group V),". He reports the observation of odd-orde1• reflections from the following claaacs of prian1 planes: k = 0 and l even, l = 0 and h even, h = o without restriction. Reflections reported by Schroder which do not satis(y the criteria for V);' are: (051), (032), (052), (0,13), (033), (012), (015), (031), (034). I 013 I OZJ I 01.1 I I I I I I O~J I 052 I I I ""' I I I I O•O I I I I I I I I I I I I OJl. Ofl.L I I I I 051 021 I 1.il ZSJ I 0.'i2 020 060 Fig. 1. OZ1 I I 0711 I I I I II O'l1 151251 l I II I I I 060 211 I I I I I I I II I 080 r Fig. 2. Fig. L Diagram of Sch1•oder's rolalion photograph No. 3, showing his assignment of indices. The X-ray beam was incident on (01 O), the crystal rotating through 20° about (001]. Fig, 2. Diagram of a photograph made by the writers wilh the crystal oriented as for Fig. 1, but rotating through 45°. The proper assignment of indices is shown for those lines which it is considered practical to identify. On our Laue photographs the following planes, listed in Table Ill of our paper, failed to reflect in the fast order, although in position to do so: (Oaf), (051), (032), (015), (031), (034). The following planes not listed in our paper also failed to produce Laue spots, allhough in a position to refle~t radiation of the wave-lengths indicated: (052) at n'A = 0,28 and 0138 A, (o 13) at n'A 0,59 A, (012) at n'A = 0,64 1. These observations show that eight of the nine reflections reported by Schroder to disagree with the criteria of V)," do not occur on Laue photographs. The presence or absence of the remaining reflection, ( O3 3 ), cannot be determined from J, au e photographs on account of its superposition on (022). There is additional evidence that Schroder's supposed observation or reflections which we find not to occur arose from error in his assignment = 1) A. Schroder, Z. Krist. 67, 485. 1928. Kii.rzere Originalmitteilungen und Notizen. of indices Lo reflections on rotation photographs. 559 The distance between n:llcdiuus tluc Lu IJIH.ues uf furm (o kl) and of form (I kl) on a photograpll made with the film t O cm. from the crystal is 0,7 mm. Measurements of the position of the central point of a given line referred to the central point of the undeviated image can accordingly scarcely be rendered snf'ficiently trustworthy to decide between two possible sets of indices differing so slightly as ( t kl) and (okl). The distance between two reflections which lie close together, however, can afford a more decisive test. We have made a rotation photograph with the X-ray beam incidcul on the (010) plane of brookite, and with the crystal rotatiug through 45° about the a-axis; the crystal-tofilm distance was t 0 cm. and the JJfoKa radiation was isolated with a zirconia filter. This photograph corresponds to Schroder's photograph No. 3, reproduced with his assignment of indices in fig. f. On our photograph, the principal reflections of which are represented in fig. 2, there appears on the right of the line numbered (05f) by Schroder a slightly :.tronger reflection. The11e two reflection& are Llue lo twu uf the vI1.1ues (051 ), ( 151 ), and (2 5 t ). In both lines, the Ka doublet was resolved. We calculate the following theoretical separations for the MoKa 1 reflections from these planes: (051) to ( tlil) 0 7074 cm. (05t) to (251) = 0 7221 cm. ( t5 t ) to (2 5 t) O, U 7 cm. = The separation of the two Ka 1 reflections was measured on the film as o,143 + 0,003 cm. These two refleetions are accordingly due to the planes (t5t) and (25t), and Schroder's assignment of one of them to the prism plane (051) is incorrect 1). The same error was no doubt made in his assignment of indices to the other reflections. Summary. It is shown that Schroder's reported observation of X-ray reflections from hrookit.e not allowed for 11 strnr.tnre hlHPd nn thl? «pi:1ee-group Vf' is due to error in his assignment of indices to lines on rotation photographs. Gates Chemical Laboratory, California Institute of Technology. Contribution No. t 9 5. Received October f 3th, t 9 2 8. t) The occurrence on Schroder's photograph of the relleclion from (15l) and not that from (251) arises from his small angle of oscillation of the crystal (20°). TRE CRYSTAL ST:i.UCTURE OF COLUJ~:BITE Columbite and tantalite 1 form a series of mixed crystals of composition (Fe,W.n)(Cb,Ta) 2 o6 , with orthorhombic holohedral symmetry. The crystal used in the present investigation was a short columbite prism from Norway, 2 showing developed pinacoids, blackt with a density of 5.71 ~ C.l (determined on a powdered sample with a pycnometer). Laue and oscillation photographs pre:farCd with it served for determining the space group and unit cell, while i. to find possible atomic pos}ions use was made of the coBrdination theory and of close packing. I. The Unit Cell and Space Group Oscillation photographs for the determination of the lattice constants were made with molybdenum K-radia.tion ref~lpted from the pinacoidst a calcite spectrum being recorded simultaneously for a comparison. obtained. Table I shows the data The indices of the reflecting pinacoids are referred. to a.x~s which will later be seen Lu be indicetted by the spa.ce group with Wyckoff's distribution of symmetry 2 elements; the space group axes ~re related to the crystallographic axes (after Dana) by cyclic permutation: b ·-Dana ....,-... -c ' -Dana c . . . . . . . -a. a_ --JJana '-"'"' b, - The tabulated interferences lead to a unit with d100/n 1 = 2. 541 l, d 010/n 2 = 7 .119 A, d001 /n = 3 2.865 l; n1 , n 2 , and n 3 are the orders of reflection of the first lines on the respective photographs. Laue photographs were taken with the incident beam of X-rays inclined at various small angles to the normals of three thin slips ground ~1rallel to (100), (010), and (001), respectively. The short wave- length limit of the incident spectrum (from a tungsten target) was 0.24 1. The smallest unit which will give calculated values of n~ not less than 0.24 l for all Laue spots is that corresponding to n1 = n 2 = n 3 = 2; since this unit accounts for all reflections observed on the two Laue photographs which were completely analyzed, it is to be accepted as the correct one. (Table II lists a few typically useful Laue interf er- ences appearing on one photograph; on the two photographs mentioned, ·a total of approximately two hundred forms reflected.) The fundamental translations are: 0 0 0 dlOO:;;.:.. G.082 A, do10 ~ 14.238 A, doo1 ~ 5.730 A. These are conf.:ti?ed by the application of the rola.nyi formula. to the layer lines occurring on all oscillation photographs. All indices occurring in this paper are referred to the axes 3 Table I Meaeu.rementl:i of tht: axial lengths Line 9 (hkl) Order (100) n1· f3 70 B. n1 0( 8 2 n1 (3 14 24. 2.540 2 n1 ex" 16 11. 2.541 2 n1 O(~ 16 17. 2.541 2 n2 ~ 5 43.5 7.117 3 nl') ,.;, ~ 7 38.5 ?.120 3 n2 C(4 8 35. 7.119 n3 (3 6 19.5 2.865 °" °'" 7 6.5 2,.868 (010) (001) n3 2 n3 2 n3 (Mo K) C(?. 1.5 Average d/n ' 0 2.542 A 2.543 14 19. 2.863 14 24. 2.864 .. 2.541 A 7.119 2.865 4 Table II Selec.ted first-order pyramidal reflections from a ty~ical Laue photograph X-ray beam at small angle to (010); 170 mah., 51.5 kv. peak. Form n~ {hklJ .A 311 .40 • • 44 434 .30, .3b, .38, .43 335 • 30, .. 33, .39, .43 136 .30, .31, .. 41, .42 405 .24, .28, • 3.0, .31 337 .25, .27 731 .. 24, .25 762 .35, .37, .40, .41 499 .32, .34, .38, .40 895 .31, .34, .35, .38 5 of this unit of structure. The axial ratio calculated from the X-ray data is a:b:c = 0.3569 : 1 : 0.4024. This may be compared with the axial ratio given by Dana for a columbite crystal: 3 aD a.na. :bD a.na :cD· 3 x 0.3G80 : l ana ::::::. 0.82850 : 1 : 0.88976, or a;b;c -=- : 3 x 0.4023. If the crystal were pure columbite, with any ratio of iron to manganese, the observed unit would contain 4.24 molecules of (Fe,N.in)Cb 2 o6 • The com.position (Fe,1.il:n)Ta 2 o6 , corresponding to pure tantalite, would require 2.?9 molecules in the unit cell. The unit cell must therefore contain either three or four molecules of (Fe,llfn)M2 o6 • With the former alternative, the specimen must consist of 73.5% Ta 2 o5 , 11.5% Cbz05; with the latter, of 14.1% Ta 2 o 5 , 65.8% Cb 2 o5 • A decision between the two possibilities may be made by reference to l.!a.rignac's table showing the relation between density and composition or columbites an~ tantalites 4 (or to other summaries of analytical results): Density 5.36 5 .. 65 5.70 5.74 5.92 Percent Ta 2o5 3.3 15.8 13.8 13.4 27.l 6.05 30 .. 4 ... ... 7.03 65.6 The former composition is correlated with a density of 7; 6 the latter with a density of 5.7. The unit of columbite therefore contains four molecules. Since first-order Laue reflections are observed from all types of pyramidal planes (Table II), the space group is based on the simple orthorhombic lattice r;. Of the s:pa.ce groups derived from ~ the following are eliminated5 by observed first-order reflections on Laue photographs from the prism planes listed in Table III: 3 8 9 10 2 3 4 6 8 . 9 VlO ul2 Vl5 C2v' 0 2v• 0 2v 1 0 2v' Vh, Vh, Vh, Vh, Vh, Vh, h ' vh ' h • Of the remaining seventeen space groups, only one requires the first-order absences which were regularly observed.; na.~ely, forms having h =- 0, k + 1 odd, k = O, i odd, or i, ~ O, h odd. These absences are exactly those required for any structure 14 based on Vh , and therefore this is taken to be the correct space group. No observable reflections upon the oscillation photographs contradict the preceding conclusion; over one hundred absences upon them confirm it. The four groups of coerdin~te positions permitted 14 by Vh for equivalent atoms are: 6 4b: o, o' o·, i2•2"•2> i i. o•<ift:2J 1 i. i2• o' o• t.o.!; o.t.o; ~.t,o; o,o,i. 4c: !,O,u-t; ~.t.!-u; ~.o,t-u; f,t,u+1. 4a. •• 7 Table III Prism reflections on Laue photographs Intensity I > 0 unless otherwise noted. (hkl} d.11...kl n1' 011 - 5.31 l 0.34 A 021 4.46 .48 0 021 " .29 0 031 3.65 .29 031 .. .23 031 " .39 071 1.92 ,44 013 1.89 .38 0.10.1 1.39 .5? 0 0.11.2 1.18 .46 0 0.12.1 1.16 .49 0 035 1.11 .45 0.13.l 1.08 .38 0.13.l tt .42 0.13.2 1 .. 00 .28 0.15.1 .94 .29 0.15.1 " .32 0.15.2 .90 .55 0 0.15.2 " .41 0 - I 0 0 8 Table III, Continued (hkl} dhkl. nA ! O.l7.2 a.so 1 o.31 1 0 0.19.l • ?4 .33 0.19.2 .73 .35 0 038 • '71 .26 0 o.~i.2 .66 .29 0 0.23.2 .60 • 24 0 0.23.3 • 59 .37 o.21.5 • 58 .35 0.25.3 • 55 .41 0.9.11 .49 .34 0.9.11 •• ..28 105 1.12 .43 0 105 " .54 0 501 1.00 .37 0 501 tt .32 0 502 .96 .38 106 .94 .29 106 It .39 106 n .48 107 .73 .27 108 .71 • 30 ?02 .?O .32 702 .?O .36 - - 0 Table III, Continued nA (hkl) dhkl -702 0.70 A () I 0 0.33 A '702 It .38 703 .60 ,30 308 .66 .42 308 " .47 ?04 .65 .41 902 .55 .25 902 " .34 3.0.10 • 54 .32 3.0.10 ll .3? 904 .53 • 36 904 " .39 904 .37 904 " " 230 2.24 • 31 310 1.68 .46 0 1.10.0 1.3? .49 0 290 l.~4 .31 1.11.0 1.25 .46 430 1.23 .44 i.12.0 1.15 .43 0 1.12.0 It .48 0 0 0 .41 0 10 Table III, Continued nA I 1,07 A o.42 A. 0 1.13.0 " .40 0 i.14.o 1.00 .4~ 0 1.14.0 u .36 0 1.14.0 " .38 0 530 .99 .38 0 530 H .4? 0 1.15.0 .93 .3? 0 1.15.0 • 32 0 1.15.0 .. .. .36 0 1.16.0 .88 .32 0 1.17.0 .83 .32 0 1.18.0 • '78 .26 0 ?30 .72 .25 0 3.25.0 .54 .36 0 (hkl) dh.kl. 1.13.0 - .,.__ 0 11 Bd: .x-t,y,z-f; x+f, -~-Y.•~-z; x-f,!-y,z+f; x+i,y.i-z; The minimum number of :parameters which must be evaluated to determine the atomic positions of the 4 Fe, 8 Cb, and 24 O is seven, and a larger number is necessary if the atoms are to be homogeneously distributed throughout the unit cell. Recourse to the non-rigorous principles developed in recent years for the solution of complex ionic structures is accordingly indicated. II. The Application of the Coerdination Principles, with Closest Packing? 0 Since the crystal radius of Fe~+ is 0.75 A, of Cb5 t- is 0.10 1, and of o= is 1.40 1. 8 it may be assumed that 0 the metal-to-oxygen distances in tantalite are 2.0 ± 0.2 A. From the univalent cryetal ra.diuB of Cb 5+, a. cuOrdinatiun number of six for the ion in oxides has been predicted 9 ; z+ and although the univalent crystal radius of Fe has not been calculated, its crystal radius suggests thatthis ion also will co~rdinate six oxygen· ions about it. The octahedra of oxygens will be distorted, but because of the deformability of the iron octahedron, resulting from the large size of 12 the central ion and from its low valence compared to t:t.at of the columbium, the distortion will be left for determination by the intensity data. Union of the octahedra to form the completed structure will be accomplished without the sharing of faces (three oxygens) between columbium octahedra or between columbium and iron octahedra, and the number of shared edges should be small. Lastly, it is expedient to assume that the oxygen framework is derivable by distortion of a close-packed assemblage with the normal to the layers parallel to the b-axis of columbite. The four assumptions of reasonable metal-to-oxygen distances, of fairly regular octahedra, of limited sharing of oxygens, and of close packing suffice to definet under the restrictions imposed by the space group and unit cellt a small class of structures from which one may be selected for detailed testing with the observed intensities of reflection. Arrangements of ions in closest packing have been described lO in terms of the order of superposition of the three fundamental layer5 from v;hich all of the inf'ini te number of close-packed structures can be built; namely, A, with an atom at X= 0, Y = 0 (X,Y,Z being hexagonal co6rdinates); B, with an aton1 a.t X = · 1/3, Y::::. 2/3; and C, with an atom at X = 2/3, Y==- 1/3. The effective thickness of a layer of oxygen ions is a:pproxima.tely 2.3 l. If distortion is assumed 13 to alter the height per layer by a maxirmlm of 15%, the lene;th oi' the b-axie of columbite then requires that the suecession of layers have a periodicity six. Accordingly, the I.ayers may be superposed in the order ABABAB ••• , glving hexagonal close packing; in the order .A.B.ABAC ••• • g:i.v:ing triple-hexagonal close packing; in the order ABCABC •..• givir1g cubic close packing; in the o!'der ABCACB ••. , giving double-cubic close J'.QCking; and in no other non-equive.lent sequence. Tn.e triple-hexagonal close-packed assemblage lacks a diagonal glide plane normal to the layers and containing a two-fold rotation axis parallel to them. The unit with space-group symmetry v~ 4 therefore cannot be derived from such an oxygen skeleton. 11 The cubic close-packed framework of oxygens ·with triad a.xis parallel to b is similarly eliminated. Normal to the layers of the double-cubic closepacked arrangement there is only one distinct set of glide planes vvith a diagonal translation, and .normal to these planes and to the layers only one distinct set of symmetry planes with a simple glide. The clooc-:pa.oked into a unit with spa.ce group aseembl~ge can be inserted vi:i4 in only one distinct way. The oxygens are in three sets of 8d :positions. with parameter valn es! 14 = 1/2 x, - 5/6 xz.:::. 5/6 xJ yI - 1/12 Yz. = 5/12 Y;, =- 3/4 z 2.. = 1/8 Z3 = 1/8 If the oxygen ion is assigned the diameter 2.76 x, the z, - 1/8 ... - 4. 78 1, b = 13. 5 A, c:::. 5.52 A; they 0 a.xi al lengt11s o,re (:!- - 0 deviate by 6, 5, and 4%, respectively, from thoo e of the actual structure. This is the arrangement shown in Figure 2. If four iron atoms are inserted into this oxygen :framework in the unique :positions 4a or 4b, there are no reasonably regular octahedra of oxygen ions about them, smd the shortest iron-to-oxygen distance is only 1.4 i; the unique :positions are therefore not available. If, however, they are placed in positions 4c, w1th u = ~/8, each is surrounded 0 by oxygen atoms at a distance of 1.95 A, arranged in a regular octahedron, and each octahedron shares two edges with the adjoining ones to form parallel to the~ axis strings 12 of the ty:pe which were observed in brookite (Fig.1). Save for the equivalent parameter value u ::;::: -1/8, there is no other value of~ which places the iron atoms within reasonably 0 regular octahedra or at distances of the order of 2 A from the neighboring oxygens. (Small parameter variations, with d~ of the order of 1/8, are of course left to be considered when the structure is tested with the observed intensities of reflection.) Similarly, there are bu l. l.wo 15 Fig. 1. A string of staggered octahedra. 16 5.52A~-- ~. r• ·r •-r---------\--• I \ > 3 ; 3 / 1 Ii - I )-.;. I U I • 3 :\ \ 5/ T Q\ 0 \\ 0 , 7 ' @ !A ~, ~®\ \J; U 5 I @ ,, 1I,\\ 3 • / 6 ' '6 !HA 1· I\ M4 /~\/B\ {/ 1\ 6 I \51' \ 1~ ~ ~ ~:r~ "6 ')31, 7 0 3 ff : !J r\ I • \ •\ : L • : q. U . )( ~ I \ \>< I I \ \ 9 l!f ~ I 3 I I 7 o ' I I? 7 0 i9 ~Ai6 Fe • Fig. 2. undistorted. I 9 • 5 0 Cb @ 0 0 The projection of (100) of Structure E, The edges of the unit cell are outlined, and the uppermost face of each octahedron is indicated. The strings lie normal to the plane of projection; edges shared to form them are marked by double lines. The numeral beside each atom gives its depth in twelfths of a. 17 sets of positions for the columbium atoms; both place the metals within strings of staggered octahedra, with metalto-o~ygen distances 1.95 l. If the columbiums are in one of these sets of positions, x 8d: = 2/3, y = 1/6, z = 3/8, each colu~bium octahedron shares two edges with columbium octahedra of the same string, two edges with an iron octahedron, and two corners with columbium octahedra of an adjacent string. This set of :parameters is improbable; it is discarded for this and for another more convincing reason which will appear later. 13 If the second set of available parameter values is chosen, 8d: x = 2/3, y = l/6, z = 7/8, each columbium string is joined to an adjacent iron string ~y the sharing of one additional edge per octahedron (the type of linking observed in brookite); and only corners are shared between successive columbium strings. The electro- static valences of the three groups of oxygens are l~, 2, and 2i, respectively. Fig. 2 shows this structure, hereafter called Structure E. The examination of the hexagonal close-packed arrangement (with its hexagonal axis parallel to b) is similar to that of the double-cubic close packing. Only one distinct unit can be derived from it which has the 14 space group Vh and the basal translations of which approach~ and£ within 40%. The oxygens are in three sets 18 9 ®· 3 ® 5 z • I ZI ?26 I JB I I I 3 I @ r ~6 I 5 0 Fig. 3. I I ,, 5 I 37 ),\ i 1 ? ® I~ 1: 6 6 The projection on (100) of Structure c. See Fig. 2 for explanation of symbols. The lower numeral beside an atom, when two are present, refers to the numbering in Table VI. 19 Fig. 4. A model representing half of the unit of structure of oolumbite, showing the strings of octahedra qombined by sharing corners only. 20 of 8d positions (see Fig. 3); their ' parameters are: x= 1/3 f X2 = 1/3 X3= l/3 Y1 == l/12 y 2 = 5/12 Y3=- 3/4 z.:::. 1/8 z 2.:::. 1/8 z :s 0::::: 1/8 The iron atoms must lie in positions 4c with u = 5/8. (Settfng u = 1/8 produces an equivalent arrangement.) The columbium atoms must be in one of two sets of 8d positions. If they have the parameter values x = 0. y :: 1/6. z = 5/8. each columbium octahedron shares a face with an iron octahedron; this structure disagrees with our original hypotheses. They must then have the positions corresponding to x = 0 ' y :::. 1/6. z = 1/8 • The resulting unit, hereafter called Structure C, is shown in Fig. 3, and e. model of a portion of it in Fig. 4. been found to be the structure of columbite. It hae All of the metal atoms lie within regular octahedra of oxygen atoms. Iron octe.hedra share two eclges each to form strings parallel to the ~ axis, and columbium octahedra do likewise; each string of staggered octahedra is joined to two adjacent strings by the sharing of corners only. The electrostatic valences of the three groups of oxygens are 2, li, a.nd 2i", respectively. In short, Structure C differs from Structure E only in the sharing of corners instead of edges between 21 iron and columbium octahedra.. The principles which have been adopted indicate that Structure C is slightly more probable than the other. A choice between Structures C and E is easily made by com:parison bf the observed intensities of reflection from (200) and (400) with intensities calculated from each of the two structures, which differ only in the distributions parallel to~· (It is assumed that the relative intensities of the lower orders will not be largely affected by distorticnJ Osc1llation photographs show a normal decline in intensity for even orders fpnn (100); i.e., they suggest approximate " equality of the structure factors fo~ these orders. The calculated. structure factors a.re: · Structure E: 8 200 = - 2 Fe+ 2 Cb - 4 o. 8 400 = Structure C: 2 Fe - 2 Cb. = - 2 Fe - 4 Cb+ 12 o. 2 Fe+ 4 Cb - 12 o. 8400: 8 200 Structure C is accordingly selected as suitable for detailed comparison with the intensity data. III. Testing the Structure with the Est.imated Intensities of Reflection Oscillation photographs were prepared with the 22 m~lybdenum K~ lines, isolated by a zirconia filter, reflected from the ground pinacoida of the crystal, a crystallographic axis being made the axis of rotation. Reflections employed in the limitation of the parameters were identified by measurement of'. latitude and azimuth on the film; £or a control, the spheres of reflection in the reciprocal lattice were construct¢ed. As an added precaution against the con- fusion of prisms and pyramids, each oscillation photograph was repeated with the crystal rotated a few degrees about the norma.l to the re:rlecting face; when ~· ss,y. was the axis of rotation and (010) the reflecting face, this displaced vertiaa.lly reflections from forms with t :f o. Because of the length of the b-axia, refleotiona from planea with k - 0 and with k = l were superposed, save when E was the axis of rotation; photographs from (100) with Q the axis of rota- - tion, and from (001) with a the axis of rotation were therefore not used in the testing of the structure. All reflections from planes whose traces in the reflecting face were not normal to the axis of rotation were likewise ignored, because of the probability that the intensities of such reflections are strongly influenced by absorption. The relative intenaitiee of useful lines on the various photographs were estimated by visual comparison with a scale bearing eighteen lines of darkenings corresponding to various values of the product !-ray intensity 23 t\mea ti•~ from 0.25 to 30. The origin of the unit cell of columbite has been chosen at a center of symmetry. In this case, neglecting extinction, ana. with the oscillating crystal method a.a described, a quantity proportional to the integrated intensity of reflection from a plane (hkl) is ~~e, where "'f A· cos(ht: "" .. +kn .. + tt'r .lJ. . )t A=- L-. i.j l 1' 1J {\. I lJ l/w = [1 - (uA./dein 2-S)2j'/'ll, The factor l/~ is a correction for the varying specific times of illu.mini;.tion of planes incl:in ed to the axis of rotation; ~ is the index (i.e., h, k, or 1) corresponding to ... the axis of rotation, and £! is the unit translation along that axis. 14 The triple ( ~ n are the - r iJ ' J ij • rr ij ) co6rdinates of the jth atom of the 1th species referred to the axes of the unit cell. and the summation is over a certain half of the atoms of each kind in the unit cell; namely, over that half from which the others are obtainable by inversion through the origin. !i denotes the usual ! factor for the ith kind of atom. The quantityw~z, and less accurately!• must approximately parallel the estimated relative intensities of reflection on any one 24 photograph. The l curves employed,for iron and oxygen are taken from the ts.bula.tion by Bragg and West. 15 The 'E curve for c,olumbium is calculated. from Thomas' atomic model according to the method given by the same authors, and is corrected for the heat motion by an arbitrarily chosen factor less than unity and decreasing with increase of-8; the resulting! values are given in Table IV. Al though molybdenum K~ radiation (v/R = 1287) is softer than the K edge of coldmbium (v/R = 1401) and near it, no correction for the change of the amplitude-ratio in a region of anomalous dispersion is necessary.16 The effect of the presence of tantalum in the observed ratio atoms Cb: atoms Ta 89:11 was estimated from the F curve for tantalum according to the Thomas model to be equivalent to an increase of about 10% in the ! curve employed for columbium. Since this effect is opposite to that of the anomalous dispersion, and., in view of the inexactness of the intensity estimA-tes as well as of the! curves themselves, the presence of tantalum was likewase neglected. The parameters of the metal atoms along each axis were fixed successively. ! values of high orders from (010) were first examined as functions of the single parameter YCh on the interval 0 ~ Ycb L 0.25. Allowance was made for the maximum possible contribution to any~ 25 Table IV F values for columbium sin .S )\ F o.o 41. .1 35.5 .2 30.0 .3 24.8 • '1 21.3 .5 18,7 .6 16.5 .8 12.9 1.0 10.4 1.2 8.6 26 from the oxygens; for the higher orders this maximum· (equal to 12 A0 } is conveniently small. Observed inequalities of various lines (Table V, Photographs a and .E) sufficed to fix 0.159 ~ Ycb 6 0.170, and the most satisfactory agreement was obtained with Ycb-= 0.163. The oxygen parameters y , y , y were then chosen to give 2 1 3 agreement for all orders from {010). especially for those for which A -+A is small. With_Ycb evaluated, the Cb Fe A values of high-order reflections of form {hkO} V, Photograph C) (Table were treated similarly as functions of the single :parameter xCb' with -0.25 ~ x ~ on the subinterval -0.04 :::::: x Cb ~ Cb ~ 0.25; only 0.04 did the A curves . duplicate the observed order of intensities. - To explain simply the facts that the intensities of high-order reflections The oxygen parameters x , x , (hkO) appear to be independent of h and to vanish when k ia odd, ~Cb was made zero. 17 x 3 1 were then chosen with regard to the intensities of lower orders. In like fashion, ~Cb and~ large-angle ref;lections of form (Oki) E). 2 were limited by (Table V, Photograph They were varied over the intervals 0 £ zCb ~ 0.25, o. 50 ~ u ~ o. 75. The use of inequalities alone requires 0 .. 12: zcb ~ 0.22, and u <.. 0.70; satisf~tctory agreement is obtained only within a much smaller region. The following set of parameters was found to give 2? the , best agreement with all of the 127 prisms and pinacoids :i_d ':mtifi ed on the oscillation photographs: 1 XCb := 0. 000 ~ = Ycb = 0.163 y ZOb = 0.075 Z 1 o. 333 x 2 = 0.410 y 0.333 x = 0.090 y 0.155 Z0 - 1 - u 2 "' = = 0.150 3 3 = 0 .. 320 ~ 0 .. 750 z3 = 0 .. 170 = 0.600 Comparison of t11ese parameter values with those for the undistorted Structure C shows that distortion has produced - only slight relative shifts along the a and. b axes, and a .... maximum change of 0.05 parallel to the £axis. Table V gives the values of <.UA2. calculated with the corresponding atomic positions. If any one of the four metal parameters is altered by as much as 0.005, the ~ualita­ tive agreement is markedly poorer. 'l'he oxygen parameters along b seem to be fixed within the same limits. along~ Those and £, however, are determined by the intensities at best to a few hundredths; they are fixed at the values given by the 1::u:ii:mmption that the dietance from a colurnbium atom to an oxygen is 2.0 + _ 0.1 A. 0 IV. The Discussion of the Structure Fig. 4 is a photograph of a model representing one-half of the unit of structure of columbite. Any one string contains only iron or only columbium ions at the TABLE Photograph A: ., V X-rays incident on (100), .£.-axis vertical; oscillation 0-45°~ 42 kv. peak; 220 miilia.mpere hours 3rd Layer Line 2nd Layer Line Equator 6th Layer Line 4th Layer Line (c)A2 .10·4 (hkl) I <iJA2 .10· 4 (hkl) l C4>A2 .l'o•4 .24* 230 8.6 2.9• 240 .5 .93* 3.3 430 .5 .19 440 .25 .13 2.0 1.93 630 .oo .oo .20 .53 wA2 .10· 4 (hkl) I 17. 13.9* 220 .01 400 7.5 600 800 (hkl) I 200 I (hkl) I ~A2 .10- 4 460 ., .o 4.0 660 .6 .91 860 .2 .59 N ()) 29 T.ABLE Photograph B: V X-ray beam ineident on (010), a-axis vertical; oscillation 0-30°; 40 kv. peak; 75 milliampere-hours 2nd Layer Line Equator wA2 .10- 4 (hk:l) I CtJAe .10.. 4 (hkl) I 040 ' 1.5 2.5* 240 .35 1.3"' 250 v vw .. 02• 260 10 060 5.5 7.4· .o4"" 280 .oo .oo 290 .3 .39* z.10.0 .6 .67• 2.11.0 .oo .01* 2.12.0 2.5 3.1* 2.13.0 .oo .Ol* 2.14.0 vw .03* 2.15.0 vvw .o4* 2.16.0 .2 .31* 2.11.0 .oo .oo z.1a.o • ti 1.00 270 080 0 .10.0 0 .12.0 0 .14.0 0 .16.0 0 .1a.o vw .35 vi5. .oo .2 .5 .03"' .45"' 4.1 .oo .30 .n 23.* .03"' 30 TABLE V Photograph C: X-ray beam incident on (010), !,-a.xis vertical; oscillation 20-50°; 40 kv. peak; 225 milliampere-hours Equator (hkl) 0.16.0 0.19.0 0.20.0 0.22.0 0.;24.0 0~26.0 I l.O 1.7 .2 .35 .a vi.1 2nd Layer Line WA2 .lo- 4 .30 • 72 .04 .Hi .37 .02 4th Layer Line (hkl) I wA2 .10- 4 z.15.o vi.1 ,.03 2.16.0 .6 .21 4.16.0 .35 .,14 2.11.0 .oo .oo 4.17.0 .oo .oo 2.1s.o 2 .. 5 1.00 4.18.0 1.6 .78 2.19.0 .oo .oo 4.19.0 .oo .oo 2.20.0 vvw .01 4.20.0 vvw .. 12 2.21.0 .oo .oo 4.21.0 .oo .. oo 2.22.0 .25 .13 4.22.0 2.23.0 .oo .oo 4.23.0 .oo .oo 2.24.0 .5 .31 4.24.0 .35 .30 2.25.0 .oo .oo 2.26.0 vi.1 .03 (hkl) I Vl.2 wA2 .lo-4 .11 TABLE V Photograph D: X-ray beam incident on (010); .,2_-axis vertical; oscillation C-30°; 40 kv. peak; 75 milliampere hours 1st Layer Line Equator <uA2 .10- 4 {hkl) I 020 1.4 1 2.4* 040 2.2 2.6* 060 e. '7 .4* 080 0.10.0 05 VI •. .5 .os* I c.uA2 .10- 4 031 12. 13.5* 051 .oo .oo* 071 .2 .10* 091 2. l.4 .oo .oo .oo 0.11.1 4.5 4.1 0.14.0 .oo .oo 0.16.0 .2 .30 0.13.l 0.15.1 .5 .5 3rd Layer Line 2nd Layer tine (hkl) .45* 0.12 .o o.1s.o • (hkl) I CA>A2 .lo- 4 062 5. 4.1* 082 .oo .oo* 0.10.2 .5 .45* 0.12.2 1.5 .43 0.14.2 .oo .01 .05 .11 .25 .31 .41 0.16.2 .oo .01 0.19.l vvw .02 0.18.2 .72 Q)A2 .lo- 4 2.5 6.6 0.11.3 .oo .oo 0.13.3 .2 .28 0.15.3 1.5 1.3'7 093 .01 0.11.1 I (hkl) ti) V\ ... (N ....... TABLE V Photograph E: X-rays incident on ( 010); .e_-axis vertical; osoillaticn 20-50°; 40 kv. peak; 250 milliampere hours 1st Layer Line Equator (hkl) I '4lA2 .10· 4 0.16.0 1.2 .so o.1s.o 2.3 .12 0.20.0 .25 .04 0.22.0 .5 .13 0.24.0 1.0 .37 .1 .02 .2 .os (hkl) o.1s.1 0.17.1 0.19.l 0.23 .. 1 0.25 .. 1 o.2s.o V\ 2.5 .oo .2 cuA2 .l0"" 4 .2 \A.l .oo I wA2 .10· 4 0.14.2 V\.05 .01 0.16.2 .5 .11 l. .31 .01 .oo 0.22.2 .2 .06 0.24.2 .. 25 .os ,1 .01 .41 .01 o.1a.2 .02 .2 V\ .13 .01 .01 o.2s.2 0.21.1 3rd Layer Line {hkl) 0.20.2 0.21.1 0.26.0 I 2nd Layer Line VI {hkl) I CV }.2 .10~4 0.13.3 1. .28 0 .. 15.3 6. 1.37 0 .. 17.3 VI t15 .01 0 .. 19.3 ,5 .15 0.21.3 .s .43 0.23.3 .2 .02 0.25.3 .2 .08 .06 CJJ ro TABLE V Photograph F: X-rays incident on (001); b-axis vertical; oscillation 0-30°; 40 kv. peak; 190 milliampere hours (The estimation of the intensity of 0.0.10 is obtained from photographs at larger angles.) (hkl) 002 004 006 ooa 0.0.10 4lA2.10-4 I 15-30 a. 5. (hkl) I wA2.10·4 011 ..... 1. • 49* 013 .5 .3T* 015 .35 .25 3.1* .oo 024 2.0 026 1.12 057 V\.2 .oo (',)A2,10-4 .s .oo vvw (hkl) I wA2.10-4 031 15-30 17 .... 033 15-30 13.* 035 6. .21"' .04 I (',)A2.10 ... 4 042 1.2 1.09* 044 .oo .oo .1 .10 .88 .04 046 037 vvw (hkl) vvw Vl .01 .12 .. 01 6th Layer Line 6• 9.7* v ""w .21 • 09* 064 055 I 4th Layer Line 3rd Layer Line .03 028 062 vi.2 vvw .25 5th Layer Line 053 (hkl) 022 017 1. 2nd Layer Line 1st Laye~ Line Equator .21* 066 .02 5. 0 2.3}1 A 7th Layer Line 8th Layer Line 073 1.5 .49 075 .6 .31 084 vvw 9th Layer Line .04 095 5. .69 v.i v.:i 34 center af its octahedra; one layer of iron strings follows two layers of columbium strings. each oxygen ion. Table VI. Three metal ions are about The interatomic distances are shown in Distortion from close packing subsequent to the insertion of the large co5rdinating ions has resulted in an expansion of all octahedra, the mean increase in the edges of the columbium octaheara from the originally assumed length or 2.76 Abeing 4.% and of the iron octahedra 8%•. Although no use was made of the principle that edges shared between octahedra should be most shortened, the rule is obeyed by the columbium octahedra; since it appears to be invalid for the iron octahedra, the Fe?.+ - Fe'H re:pulsion is unimportant compared to other interionic forces. While Structure C is in accord with most of the principles that have been suggested for ionic structures, the deviation of the electrostatic valences of the oxygens from the value two is large. Qualitatively, the strength of a bond as defined by Pauling 7 (i.e., the positive potential at the location of the anion ariaing from a neighboring oation) ehould be considered to increase with decreasing anion-cation distance, and the columbium-oxygen distances should be observed to decrease in the order Cb-0 3 , Cb-01 , Cb-0 2 • In agreement with this prediction, what seems to be the moat satisfactory agreement between calculated and observed intensities appears when the distance Cb-Oz is shorter by 35 Table VI Interatomic distances in columbite Di stances equi va.l ent by synm1etry are bracketed; shared edges are marked by asterisks. Iron octahedron Columbium octahed1·on Elec. valence Elec. ve,J.ence of o:x:ygen of oxvgen Cb3-024 1.86 i it Fe2-01a Cb3-013 2.04 2 Fe2-021J 2.10 Cb3-012 2.05 2 Cb3-036 2.09 2t Cb3-03a 2.11 2-! Fe2-025 Ob3-037 2.12 2t 024-02J 2.79 012- 0 3aJ 2.61* Fe2-0l.:'J 2.12 A 0 2 it Fe2-025 Fe2-024J 2.14 it 0 25- 0 2 0 13-03'7 CJi3-021J 2.89 037-03s 2.?0 i8- 0~o 0 024-025J 2.93~ 021-026 012· 013 2 .. ?6 012- 0 24 2.84 036-03? 2.86 013-01a 3.07 012-03? 2.88 0 21- 0 25 3.07 0 24-036 2.88 013-024} 3.08 0 24- 0 3a ~.02 "18- 0 26 0 Oi.3-0251 3.22 013-024 3.08 0 1a- 0 24 Average 3.00 is- 0 36 3.04 035-03s 3.20 Average 2.87 36 approximately 0.1 A than any other columbium-oxygen distance. That Cb-03 is larger than Cb-01 is uncertain. Br&gger and Prior have observed a close crystallographic relationship between eolumbite and brookite. 18 It is now seen that the structure of columbite also ls e1m11ar to the structure of brookite; the two are differentiated by the manner in which successive strings are connected, and consequently by the number of edges aha.red per octahedron. It is satisfactory, but of doubtful significance, to find that with the replacement of quadrivalent cations by bivalent and quinquevalent ones. that is. with the change from the molecule Ti30s to the molecule FeCb205, ~he number of shared edges per octahedron in otherwise similar structures decreases from three to two. The undistorted Structure C is exactly the Structure A which was built up in the study of brookite, 19 except for the tripling of the unit caused by the presence of the two kinds of metal atoms. The two oscillation photographs shown in Figs. 5 and 6 evidence strikingly the pseudo-unit, with ab axis one-third that of columbite, and with the same distribution of symmetry elements. They le.d 20 early in the present study to the prediction that the columbite unit would prove to be a triple of the unit of Structure A, and to the rapid derivation by the coerdina~ion 3? Fig, 5. Photograph D of Table V. 38 Fig. 6. Photograph F of Table V. 39 theory alone, with no assumption of close packing, of both Structures C and E. Only when it was proved that no struc- ture of the proper symmetry containing strings of octahedra could satisfy the electrostatic principle exactly was the hypothesia of olooe psi.eking subatituted for the l:cy'potheaia of strings of octahedra. It has been seen that it was at least impracticable to apply the later hypothesis in a form more lenient than the earlier one. Summary A crystal of columbite-tantalite, (Fe ,Mn)(Cb,Ta) 2 o6 , containing principally oolumbi te was found by the use of Laue and oscillation photographs to have an orthorhombic unit of structure of dimensions 0 Q a= 5.082 A, b = 14.238 A, c molecules. co~rdination = 5.?30 A, containing four The space group is v~4 • 0 By the use of the theory and of close packing, a number of possible structures were devised, and one was shown to have the observed unit and symmetry, and. to explain the intem;iities of reflections on oscillation photographs. Its structural element is an octahedron of oxygen ions about a metal ion; each octe..hedron shares two edges with adjoining ones to form 40 strings of either columbium octahedra alone or of iron octahedra alone, and the strings are united by of corners. the sharing A diagram of the structure is shewn in Fig. 3, a model in Fig. 4. 41 Ref er enc es l. The crystallography and chemistry of the system are discussed in Dana, Mineralogx, 731-?~n (6th ed., New York 1 1920); P. Niggli and K. Faesy, Z. Krist., 61, 349354 \1925); Mellor, Treatise on Inorg. and Theoret. Chem., IX, 906-910 (New York, 1929);etc. -2. This crystal was furnished through the kindness of Professor J.R.Oppenheimer. 3. E.S.Dana, Z. Krist., 12, 271 (1887). Dana•s specimen (from Standish. Me.1 contained 68.99% Cba05 , 9.22% Ta~o5 ; had a density of 5.65. Within our limits of error, the composition and density are the same as those of the crystal used here. The axial ratio is further not very sensitive to changes in composition. It may be noted that the axial ratio given by Schrauf corresponds to the X-ray unit, save for the interchange of a and o. 4. Mellor, 1•£•• p. 908. 5. Space group criteria are given by Wyckoff, Z. Krist., 61, 431-443 (1925); C. Hermann, ibid., 68, 265-270 (1928); etc. 6. R.W.G.Wyckoff, "The Analytical Expression of the Results of the Theory of S~ace Groups," Publ. Carnegie Institution, No. 318 (1922). For convenience in the calculation of structure factors, the origin of the coerdinate system used above has been transferred to a center of symmetry, (f,0,-.f) referred to Wyckoff's origin. 7. Linus Pauling, J.Am.Chem.Soc., 51, 1010-1026 (1929). Concerning close pa.eking, see W.L.Bragg and G.B.Brown., P,R.S., A 110, 34 (1926); W.L.Bragg and J, West, ibid., A 114, 450 (1927). 8. Linus Pauling, J .Am.Chem.Soc., 49, pp. 771 and-· ?BO (192?). 9. - - -- - Pauling, reference 7, p. 1016. reference 7, p. 1022. 10. Pau~ing, 11. A convenient description of the symmetry of the 42 symmetry of the columbite unit is contained in the symbol according to Hermann (l.c.); it is D2hPVfO(, with the axes of the present paper. 'forVh',. 12. Linus Pauling and J.H.Sturdivant, Z. Krist., 68, 239-256 (1928). • 13. Comparison of the columbium r~rameters just given with those for Str~cture E below shows that the intensities of reflection from the two structures are the same for planes with i even. Hence the argument from intensities used at the end of this section to eliminate Structure E also elirLinates the arrangement just described. It may also be mentioned that the electrostatic valences of the oxygens in the structure are very unsatisfactory, being 2~• 2 t , and 1 t • 14. H. Ott, Z. Physik, 22, 201 (1924). 15. Z. Krist., 69, 139 (1928). 16. From an approximate formula given by Waller (Z. Physik, 51, 224, 1928) the corrected amplitude factor FT is found to be F - 2.8. That the :magnitude of the effect is small is confirmed by recent measurements of Wyckoff (Phys. Rev., 35, 215, 1930). 17. Since Ao is sm.a.11 one has for these reflections: 1 A= 4 Acb cos [h(xcb +- t)J cos kYcb + (-l)h/?.. A =: - 4 Acb sin [h(xCb + t) J sin kyCb 2 AFe for k even; for k odd; and h is necessarily even. 18. Er~gger, 19. G.T.Prior, Mineral.Nag., 13, 217 (1903); w.c. 71 (1906). !!11· Sddnorw. Granitpeg., Pauling and Sturdivant, l·£..·• pp. 248-249. 20. This conclusion was first drawn by Professor Pauling.