International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shumeli © International Union of Crystallography 2010 |
International Tables for Crystallography (2010). Vol. B, ch. 3.1, p. 406
Section 3.1.9. Planes^{a}Department of Chemistry, University of Kentucky, Chemistry–Physics Building, Lexington, Kentucky 40506–0055, USA |
Among several ways of characterizing a plane in a general rectilinear coordinate system is a description in terms of the coordinates of three non-collinear points that lie in the plane. If the points are U, V and W, lying at the ends of vectors u, v and w, the vectors , and are in the plane. The vectoris normal to the plane. Expansion of (3.1.9.1) yieldsMaking use of (3.1.7.1),If now x is any vector from the origin to the plane, is in the plane, andFrom (3.1.9.2),Rearrangement of (3.1.9.4) with on the left and on the right, and using (3.1.9.3) for z on the left leads toIf, in particular, the points are on the coordinate axes, their designations are , and , and (3.1.9.6) becomeswhich may be writtenorin which the vector h has coordinatesThat is, the covariant components of h are given by the reciprocals of the intercepts of the plane on the axes. The vector h is normal to the plane it describes (Sands, 1982a) and the length of h is the reciprocal of the distance d of the plane from the origin; i.e.,
If the indices are relatively prime integers, the theory of numbers tells us that the Diophantine equation (3.1.9.8) has solutions that are integers. Points whose contravariant components are integers are lattice points, and such a plane passes through an infinite number of lattice points and is called a lattice plane. Thus, the for lattice planes are the familiar Miller indices of crystallography.
Calculations involving planes become quite manageable when the normal vector h is introduced. Thus, the distance l from a point P with coordinates to a plane characterized by h iswhere a negative sign indicates that the point is on the opposite side of the plane from the origin.
The dihedral angle between planes with normals h and isA variation of (3.1.9.13) expresses in terms of vector u in the first plane, vector w in the second plane, and vector v, the intersection of the planes, as (Shmueli, 1974)
A similar calculation gives angles of torsion. Let and be, respectively, the projections of vectors t and u onto the plane with normal h.The angle between and represents a torsion about h (Sands, 1982b). Another approach to the torsion angle, which gives equivalent results (Shmueli, 1974), is to compute the angle between and using (3.1.8.3).
References
Sands, D. E. (1982a). Vectors and Tensors in Crystallography. Reading: Addison Wesley. Reprinted (1995) Dover Publications.Sands, D. E. (1982b). Molecular geometry. In Computational Crystallography, edited by D. Sayre, pp. 421–429. Oxford: Clarendon Press.
Shmueli, U. (1974). On the standard deviation of a dihedral angle. Acta Cryst. A30, 848–849.