EPITAXIAL RELATIONSHIPS OF CUPROIYS OXIDEatomic scale (111) facets which make up the structure of the (012) plane. The electron microscope pictures, in addition to confirming the orientations determined by electron diffraction, gave specific data on the shapes of the oxide growth and the particular faces present. 0nly the three most densely packed faces occurred on the oxide.The authors would like to thank Dr A. T. Gwathmey and Dr N. Cabrera for many helpful discussions. This work was sponsored by the Office of Naval Research.References BRADLEY, D. E. (1954). J. Inst. Met. 83, 35. DONY-HENAVLT, O. (1910). Bull. Soc. Chem. Belg. The fitting of a plane or a line to a set of points by least squares is discussed, and a convenient numerical method is given.In the description of a crystal structure, it is sometimes desired to fit a least-squares plane to the positions found for some approximately coplanar set of atoms.Because it seems that an incorrect method is often used for doing this, we would like to discuss the problem and recommend an alternative method that is both correct in principle and convenient in computation. It becomes evident that the problem of the plane is essentially equivalent to the problem of finding the principal plane of least inertia for a set of point masses and that the problem of the best line is similarly equivalent to the very closely related problem of the least axis of inertia. The discussion therefore naturally covers line as well as plane and essentially recapitulates parts of a classical mechanical theory in deriving what is special to the present application. We first formulate the problem of the plane and present the recommended alternative method of solution, including a detailed numerical example, then discuss the prevalent incorrect method as well as various special cases, and finally consider the problem of the line and give a convenient method for handling it. * Contribution No. 2287 from the Gates and Crellin Laboratories.We find it convenient to use both ordinary vector notation, as in equations (1), (2), and (3), and matrix notation, as in equation (11), sometimes side by side. We also use two summation conventions: the Gaussian bracket [ ], to express summation over a set of points (cf., e.g., Whittaker & Robinson, 1940), and the convention of dropping the operator Z whenever it applies to repeated alphabetic indices. Definitions we often express as identities. The least-squares planeWhat is desired is to find the plane that minimizes , the weighted sum of squares of distances D~ of points k from the plane sought. These points are defined by the vectors r -xlal +x2a2+xSas -= x'a,.(The plane is defined by its unit normal and by the origin-to-plane distance d, whereupon the distance from the plane to a point is

A method of least-squares refinement is described in which the subsidiary conditions are treated like observational equations. The advantages of the method are its generality, its adaptability to machine computing, and the possibility of relaxing the subsidiary conditions to any desired degree by appropriate changes in the weighting. In suitable cases the method extends the range for which least-squares refinements converge to the correct solution.It is useful at times to refine positional parameters of atoms in such a way that these atoms by necessity represent a molecule of known and specified geometry. In the special case that this molecule is rigid, one may define the positions of all atoms in a molecular coordinate system and consider the parameters describing the orientation and translation of the molecular coordinate system relative to the crystallographic system as parameters to be refined (e.g. Sparks, 1958).A molecule that consists of several linked rigid portions may be treated by an extension of the same method. However, with increasing flexibility of the molecules considered the procedure becomes cumbersome.A more general method of imposing geometrical conditions on the positional parameters of the atoms is to subject them to subsidiary conditions in the least-squares treatment by the classical and powerful method of Lagrange multipliers (e.g. Hughes, 1941).The equations that express the desired conditions (e.g. specifying that certain interatomic distances and bonding angles have preassigned values) are multiplied by undetermined factors --the Lagrange multipliers --and the results added to the weighted sum of the squared residuals to be minimized. Differentiation with respect to the parameters results in equations for the parameter shifts as functions of the Lagrange multipliers. These equations and the subsidiary conditions may then be solved for the parameter shifts and the Lagrange multipliers. While elegant, this method is often cumbersome in numerical applications and in particular proved to be unsuitable to machine computations for the particular application of interest here.The idea of treating the desired subsidiary conditions exactly like observational equations presented itself. The squares of residuals resulting from the subsidiary conditions suitably weighted are simply added to the sum of the weighted squares of the * Contribution No. 2934 from the Gates and Crellin Laboratories of Chemistry. residuals coming from the usual observational equations.This appears to be an entirely general method of introducing conditions into the least-squares procedure, and in particular when refining positional parameters. In the limit that the residuals originating from the subsidiary conditions are supplied with infinite weights these conditions are satisfied precisely. It is, however, possible to relax the conditions to any desired degree by suitably decreasing the weights of the corresponding residuals. In this respect the present method is more general than the method of Lagrange multipliers. It h...

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Results are derived which show that in the analysis of x-ray or neutron-scattering data from mixtures of amorphous solids or liquids, a meaningful scattering function, i(s), can always be computed without recourse to simplifying assumptions regarding the atomic scattering functions. Fourier inversion of i(s) yields a distribution function, H(r) = ΣΣxixjHij(r), where xi is the atomic fraction of species i, and Hij(r) is the convolution of the true net radial-distribution function, hij(r), with a particular function of the atomic scattering factors. The much-used assumption that all atomic-scattering factors are proportional to the same function yields an H(r) that is a weighted sum of the hij(r), but does not offer any means of ascertaining the individual hij(r) terms if only one scattering experiment is performed.