If not all closely packed positions are occupied by large ions, an open structure is created. Using this close-packing method, Bragg and his colleagues have made a promising attack on the important problem of the structure of silicate minerals. Tho cdato.1 structure: from brookite. amount and nature of deformation due to the close-packed arrangement.
Spectral photographs of the three pinacoids provided accurate dimensions of the possible structural units. Applying coordination theory, we found that one of the predicted structures had the same space group symmetry and structural unity as the crystal itself. Part of the brookite structure, showing the three edges of each octahedron shared with adjacent octahedrons.
Testing the structure with the intensities of X-ray reflections
These reflections were not taken into account because of the likelihood that their intensities would be largely influenced by absorption in the crystal.). The structure factors were calculated using predicted parameter values and assuming that the relative repulsive strengths of titanium and oxygen atoms are proportional to their atomic numbers. These structural factors reproduced the significant features of the observed intensities for the fifty forms; this general agreement is strong evidence that the predicted structure closely resembles the actual one.
Therefore, the parameter values varied little and a change of - 0.004 in X was found to be sufficient to remove most of the discrepant points.
Discussion of' the structure and the coordination method
01newhal different fl~curve;:; (which corresponds to no11-Jineary extrapolation) .. vould ghe better Rutile provides an example of a structure that can be easily deduced using coordination theory, but in which the oxygen ions do not approach any kind of close packing. The basic unit of the arrangement is an octahedron of oxygen ions around a titanium ion; the octahedra are combined as shown in fig. In our space group determination, only the presence or absence of reflections on Laue photographs was used. The presence or absence of the residual reflection, (O 3 3 ), cannot be determined from J, au e photographs because of its superimposition on (022). The appearance in the Schroder photograph of the echo from (15l) and not that from (251) arises from its small angle of oscillation of the crystal (20°). The unit cell must therefore contain three or four molecules of (Fe,llfn)M 2 o 6 • With. the first alternative must be a sample consisting of 73.5%. Of the remaining seventeen space groups, only one requires absences of the first order, which have been regularly observed.;. These absences are exactly what is needed for any Vh-based structure, so this is considered a proper 14 space group. The minimum number of: parameters to be evaluated to determine the atomic positions of 4 Fe, 8 Cb, and 24 O is seven, and a larger number is necessary if the atoms are to be distributed homogeneously throughout the unit cell. The joining of octahedra to form the finished structure will be accomplished without the separation of faces (three oxygens) between the columbian octahedra or. Finally, it is convenient to assume that the oxygen framework is derivable from the distortion of a tight stack with the layer normals parallel to the b -axis of columbite. Closest-packing arrangements of ions are described lO in terms of the order of superposition of. The tn. of triple hexagonal close stacking lacks a diagonal slip plane normal to the layers and containing a twofold rotation axis parallel to them. The normal for layers of closed bicubic order has only one distinct set of vvi sliding planes with a diagonal translation, and the .normal for these planes and for layers only one distinct set of planes of symmetry with. But for the equivalent value of the parameter u there is no other value of ~ that places the iron atoms inside reasonably. regular or spaced octahedra of order 2. Small parameter variations, with d~ on the order of 1/8, must obviously be taken into account when testing the structure with the observed reflection intensities. The examination of the hexagonal, close-packed arrangement (with the hexagonal axis parallel to b) is. A model representing half of the structural unit of oolumbite, combining the strings of octahedrons by sharing only corners. The iron octe.hedra share two eclges each to form rows parallel to the ~ axis, and the columbian octahedra do likewise; each string of stacked octahedra joins two adjacent strings only by sharing the corners. A choice between Structures C and E is easily made. from com:parison bf observed reflection intensities from. 200) and (400) with intensities calculated from each of the two structures, which differ only in distribution. As an additional precaution against interchange of prisms and pyramids, each photograph of the oscillation was repeated with the crystal rotated a few degrees about the normal to the reflecting surface; when ~· ss,y. The relative intensity of useful lines in different photographs was assessed by visual comparison with a scale of eighteen darkening lines corresponding to different intensity values of the !-ray product. The factor l/~ is a correction for the different specific exposure times of the planes included in the axis. Although the K-radiation of molybdenum (v/R = 1287) is softer than and near the K-edge of chrombium (v/R = 1401), no correction is necessary for the change in amplitude ratio in a dispersion region abnormal. .16 The effect of the presence of tantalum on the observed ratio of Cb atoms: Ta atoms. 89:11 was estimated from the F curve for tantalum according to the Thomas model to be equivalent to an increase of. Since this effect is opposite to that of anomalous dispersion, and., in function of the inaccuracy of the intensity estimation as well as of. The greatest possible contribution of the oxygen atoms to anything was taken into account; this maximum for higher orders. Satisfactory agreement was reached with Ycb. Values of high-order reflections of the form {hkO} (Table V, Photo C) were treated similarly as functions of the single:parameter xCb' with -0.25 ~ x ~ 0.25; only. The Discussion of the Structure The close-packing distortion that followed the insertion of the large coordinated ions resulted in a broadening of all the octahedra, with an average increase of the columbium octahedra edges from the originally assumed length or 2.76 A by 4.% and the iron octahedra by 8%. Although the principle that the edges shared by octahedra should be shortened as much as possible was not applied, this rule is followed by the Columbia octahedra; since it appears to be invalid for iron octahedra, the Fe?.+ - Fe'H recoil momentum is negligible compared to other interionic forces. Although the structure of C conforms to most of the principles that have been proposed for ionic structures, the deviation of the electrostatic valences of oxygen from the value of two is large. Qualitatively, the bond strength as defined by Pauling7 (ie, the positive potential at the anion site originating from the adjacent oa). e should increase with decreasing anion-cation distance, while the columbium-oxygen distances should be. consistent with this prediction, the most satisfactory agreement between the calculated and observed intensity appears to be when the Cb-Oz distance is shorter than . 18 Now it can be seen that the structure of columbite is also the same as that of brookite; the two differ in the manner in which the consecutive cords are connected, and consequently in the number of edges per octahedron. The finding that by replacing tetravalent cations with. by changing from a Ti30s molecule to a FeCb205 molecule, ~he. the number of shared edges per octahedron is reduced from three to two in otherwise similar structures. Structure A constructed in the study of brookite, 19 with the exception of the tripling of the unit caused by the presence of the two types of metal atoms. They led early in the present investigation to the prediction that the columbite unit would prove to be a triple of the unit of structure A, and to its rapid derivation by coordination. theory alone, without assumption of close packing, of both structures C and E. Only when it was proven that no structure with proper symmetry containing sets of octahedra could exactly satisfy the electrostatic principle was the hypothesis of olooo psi.eking replaced . for the l:cy'potheaia of strings of octahedra. Using Laue and oscillation photographs, Fe,Mn)(Cb,Ta)2o6, which mainly contains oolumbi te, was found to have an orthorhombic unit of structure and dimension. It may be noted that the axial ratio given by Schrauf corresponds to the X-ray unit, excluding the inter-radiation. For convenience in calculating structure factors, the origin of the coordination system used above has been transferred to a center of symmetry, (f,0,-.f), referred to the Wyckoff origin.Summary
The Unit Cell and Space Group
The Application of the Coerdination Principles, with Closest Packing?
Testing the Structure with the Est.imated Intensities of Reflection
ABLE V
