Tables for
Volume B
Reciprocal space
Edited by U. Shumeli

International Tables for Crystallography (2006). Vol. B, ch. 3.1, pp. 349-350   | 1 | 2 |

Section 3.1.9. Planes

D. E. Sandsa*

aDepartment of Chemistry, University of Kentucky, Chemistry–Physics Building, Lexington, Kentucky 40506-0055, USA
Correspondence e-mail:

3.1.9. Planes

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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 [{\bf u} - {\bf v}], [{\bf v} - {\bf w}] and [{\bf w} - {\bf u}] are in the plane. The vector [{\bf z} = ({\bf u} - {\bf v}) \wedge ({\bf v} - {\bf w}) \eqno(] is normal to the plane. Expansion of ([link] yields [{\bf z} = ({\bf u} \wedge {\bf v}) + ({\bf v} \wedge {\bf w}) + ({\bf w} \wedge {\bf u}). \eqno(] Making use of ([link], [{\bf z} = \varepsilon_{ijk} (u\hskip 2pt^{j} v^{k} + v\hskip 2pt^{j} w^{k} + w\hskip 2pt^{j} u^{k}) {\bf a}^{i}. \eqno(] If now x is any vector from the origin to the plane, [{\bf x} -{\bf u}] is in the plane, and [({\bf x} - {\bf u}) \cdot {\bf z} = 0. \eqno(] From ([link], [{\bf u} \cdot {\bf z} = {\bf u} \cdot {\bf v} \wedge {\bf w}. \eqno(] Rearrangement of ([link] with [{\bf x} \cdot {\bf z}] on the left and [{\bf u} \cdot {\bf z}] on the right, and using ([link] for z on the left leads to [\varepsilon_{ijk} x^{i} (u\hskip 2pt^{j} v^{k} + v\hskip 2pt^{j} w^{k} + w\hskip 2pt^{j} u^{k}) = \varepsilon_{ijk} u^{i} v\hskip 2pt^{j} w^{k}. \eqno(] If, in particular, the points are on the coordinate axes, their designations are [[u^{1}, 0, 0]], [[0, v^{2}, 0]] and [[0, 0, w^{3}]], and ([link] becomes [x^{1}/u^{1} + x^{2}/v^{2} + x^{3}/w^{3} = 1, \eqno(] which may be written [x^{i} h_{i} = 1 \eqno(] or [{\bf x} \cdot {\bf h} = 1 \eqno(] in which the vector h has coordinates [{\bf h} = (1/u^{1}, 1/v^{2}, 1/w^{3}). \eqno(] That 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[link]) and the length of h is the reciprocal of the distance d of the plane from the origin; i.e., [h = 1/d. \eqno(]

If the indices [h_{i}] are relatively prime integers, the theory of numbers tells us that the Diophantine equation ([link] has solutions [x^{i}] 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 [h_{i}] 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 [p^{i}] to a plane characterized by h is [l = (1 - {\bf p} \cdot {\bf h})/h, \eqno(] where a negative sign indicates that the point is on the opposite side of the plane from the origin.

The dihedral angle [\tau] between planes with normals h and [{\bf h}'] is [\tau = \cos^{-1} [-{\bf h} \cdot {\bf h}'/(hh')]. \eqno(] A variation of ([link] expresses [\tau] 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[link]) [\tau = \cos^{-1} [({\bf u} \wedge {\bf v}) \cdot ({\bf v} \wedge {\bf w})/|{\bf u} \wedge {\bf v}| |{\bf v} \wedge {\bf w}|]. \eqno(]

A similar calculation gives angles of torsion. Let [{\bf t}_{h}] and [{\bf u}_{h}] be, respectively, the projections of vectors t and u onto the plane with normal h. [\eqalignno{ {\bf t}_h &= {\bf t} - ({\bf t} \cdot {\bf h}){\bf h}/h^{2} &(\cr {\bf u}_h &= {\bf u} - ({\bf u} \cdot {\bf h}){\bf h}/h^{2}. &(}%(] The angle between [{\bf t}_{h}] and [{\bf u}_{h}] represents a torsion about h (Sands, 1982b[link]). Another approach to the torsion angle, which gives equivalent results (Shmueli, 1974[link]), is to compute the angle between [{\bf t} \wedge {\bf h}] and [{\bf u} \wedge {\bf h}] using ([link].


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.

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