Stereographic projection

1.8 Stereographic projection

Projective geometry makes it possible to represent the relative orientation of crystal planes and directions in three-dimensional space in a more convenient two-dimensional form. The standard stereographic projection is frequently used in the analysis of crystal behavior; X-ray diffraction anal- yses usually provide the experimental data. Typical applications of the method are the interpretation of strain markings on crystal surfaces, portrayal of symmetrical relationships, determination of the axial orientations in a single crystal and the plotting of property values for anisotropic single crystals. (The basic method can also be adapted to produce a pole figure diagram, which can show preferred orientation effects in polycrystalline aggregates.)

A very small crystal of cubic symmetry is assumed to be located at the center of a reference sphere, as shown in Figure 1.11a, so that the orientation of a crystal plane, such as the (1 1 1) plane marked, may be represented on the surface of the sphere by the point of intersection, or pole, of its normal P.

The angle φ between the two poles (0 0 1) and (1 1 1), shown in Figure 1.11b, can then be measured in degrees along the arc of the great circle between the poles P and P ′ . To represent all the planes in a crystal in this three-dimensional way is rather cumbersome; in the stereographic projection,

Atoms and atomic arrangements 21

Pole P to N

(1 1 1) plane

Pole P⬘ to (0 0 1) plane

P⬘

Equatorial plane

P f (0 0 1) (1 1 1)

f Primitive circle

Q Angle between (0 0 1)

S and (1 1 1) planes (a)

(c) Figure 1.11 Principles of stereographic projection, illustrating: (a) the pole P to a (1 1 1) plane;

(b)

(b) the angle between two poles, P, P ′ ; and (c) stereographic projection of P and P ′ poles to the (1 1 1) and (0 0 1) planes, respectively.

Diad Triad 111 axis

Figure 1.12 Projections of planes in cubic crystals: (a) standard (0 0 1) stereographic projection and (b) spherical projection.

the array of poles which represents the various planes in the crystal is projected from the reference sphere onto the equatorial plane. The pattern of poles projected on the equatorial, or primitive, plane then represents the stereographic projection of the crystal. As shown in Figure 1.11c, poles in the northern half of the reference sphere are projected onto the equatorial plane by joining the pole P to the south pole S, while those in the southern half of the reference sphere, such as Q, are projected in the same way in the direction of the north pole N. Figure 1.12a shows the stereographic projection of some simple cubic planes, {1 0 0}, {1 1 0} and {1 1 1}, from which it can be seen that those crystallographic planes which have poles in the southern half of the reference sphere are represented by circles in the stereogram, while those which have poles in the northern half are represented by dots.

22 Physical Metallurgy and Advanced Materials As shown in Figure 1.11b, the angle between two poles on the reference sphere is the number of

degrees separating them on the great circle passing through them. The angle between P and P ′ can

be determined by means of a hemispherical transparent cap graduated and marked with meridian circles and latitude circles, as in geographical work. With a stereographic representation of poles, the equivalent operation can be performed in the plane of the primitive circle by using a transparent planar net, known as a Wulff net. This net is graduated in intervals of 2 ◦ , with meridians in the

projection extending from top to bottom and latitude lines from side to side. 8 Thus, to measure the angular distance between any two poles in the stereogram, the net is rotated about the center until the two poles lie upon the same meridian, which then corresponds to one of the great circles of the reference sphere. The angle between the two poles is then measured as the difference in latitude along the meridian. Some useful crystallographic rules may be summarized:

1. The Weiss Zone Law: the plane (h k l) is a member of the zone [u v w] if hu + kv + lw = 0. A set of planes which all contain a common direction [u v w] is known as a zone; [u v w] is the zone axis (rather like the spine of an open book relative to the flat leaves). For example, the three planes (1 ¯1 0), (0 ¯1 1) and (¯1 0 1) form a zone about the [1 1 1] direction (Figure 1.12a). The pole of each plane containing [u v w] must lie at 90 ◦ to [u v w]; therefore, these three poles all lie in the same plane and upon the same great circle trace. The latter is known as the zone circle or zone trace. A plane trace is to a plane as a zone circle is to a zone. Uniquely, in the cubic system alone, zone circles and plane traces with the same indices lie on top of one another.

2. If a zone contains (h 1 k 1 l 1 ) and (h 2 k 2 l 2 ) it also contains any linear combination of them,

e.g. m(h 1 k 1 l 1 ) + n(h 2 k 2 l 2 ). For example, the zone [1 1 1] contains (1 ¯1 0) and (0 1 ¯1) and it must therefore contain (1 ¯1 0) + (0 1 ¯1) = (1 0 ¯1), (1 ¯1 0) + 2(0 1 ¯1) = (1 1 ¯2), etc. The same is true for different directions in a zone, provided that the crystal is cubic.

3. The Law of Vector Addition: the direction [u 1 v 1 w 1 ] + [u 2 v 2 w 2 ] lies between [u 1 v 1 w 1 ] and [u 2 v 2 w 2 ].

4. The angle between two directions is given by:

where u 1 v 1 w 1 and u 2 v 2 w 2 are the indices for the two directions. Provided that the crystal system is cubic, the angles between planes may be found by substituting the symbols h, k, l for u, v, w in this expression.

When constructing the standard stereogram of any crystal it is advantageous to examine the sym- metry elements of that structure. As an illustration, consider a cubic crystal, since this has the highest symmetry of any crystal class. Close scrutiny shows that the cube has 13 axes of symmetry; these axes comprise three fourfold (tetrad) axes, four threefold (triad) axes and six twofold (diad) axes, as indicated in Figure 1.13a. (This diagram shows the standard square, triangular and lens-shaped symbols for the three types of symmetry axis.) An n-fold axis of symmetry operates in such a way that after rotation through an angle 2π/n, the crystal comes into an identical or self-coincident position in space. Thus, a tetrad axis passes through the center of each face of the cube parallel to one of the edges, and a rotation of 90 ◦ in either direction about one of these axes turns the cube into a new position which is crystallographically indistinguishable from the old position. Similarly, the cube diagonals form a set of four threefold axes, and each of the lines passing through the center of opposite edges form a set

8 A less-used alternative to the Wulff net is the polar net, in which the N–S axis of the reference sphere is perpendicular to the equatorial plane of projection.

Atoms and atomic arrangements 23

Tetrad (1 of 3)

Triad

Plane of

(1 of 4)

symmetry (1 of 6)

Center of symmetry

Center of symmetry (1)

Plane of symmetry (1 of 3)

(a) (b) Figure 1.13 Some elements of symmetry for the cubic system; total number of elements = 23.

of six twofold symmetry axes. Some tetrad, triad and diad axes are marked on the spherical projection of the cubic crystal shown in Figure 1.12b. The cube also has nine planes of symmetry (Figure 1.13b) and one center of symmetry, giving, together with the axes, a total of 23 elements of symmetry.

In the stereographic projection of Figure 1.12a, planes of symmetry divide the stereogram into 24 equivalent spherical triangles, commonly called unit triangles, which correspond to the 48 (24 on the top and 24 on the bottom) seen in the spherical projection. The two-, three- and fourfold symmetry about the {1 1 0}, {1 1 1} and {1 0 0} poles, respectively, is apparent. It is frequently possible to analyze a problem in terms of a single unit triangle. Finally, reference to a stereogram (Figure 1.12a) confirms rule (2), which states that the indices of any plane can be found merely by adding simple multiples of other planes which lie in the same zone. For example, the (0 1 1) plane lies between the (0 0 1) and (0 1 0) planes and clearly 0 1 1 = 0 0 1 + 0 1 0. Owing to the action of the symmetry elements, it can be reasoned that there must be a total of 12{0 1 1} planes because of the respective three- and fourfold symmetry about the {1 1 1} and {1 0 0} axes. As a further example, it is clear that the (1 1 2) plane lies between the (0 0 1) plane and (1 1 1) plane, since 1 1 2 = 0 0 1 + 1 1 1 and the {1 1 2} form must contain 24 planes, i.e. an icositetrahedron. The plane (1 2 3), which is an example of the most general crystal plane in the cubic system because its h k l indices are all different, lies between the (1 1 2) and (0 1 1) planes; the 48 planes of the {1 2 3} form make up a hexak-isoctahedron.

The tetrahedral form, a direct derivative of the cubic form, is often encountered in materials science (Figure 1.14a). Its symmetry elements comprise four triad axes, three diad axes and six ‘mirror’planes, as shown in the stereogram of Figure 1.14b.

Concepts of symmetry, when developed systematically, provide invaluable help in modern struc- tural analysis. As already implied, there are three basic elements, or operations, of symmetry. These operations involve translation (movement along parameters a, b, c), rotation (about axes to give diads, triads, etc.) and reflection (across ‘mirror’ planes). Commencing with an atom (or group of atoms) at either a lattice point or at a small group of lattice points, a certain combination of symmetry operations will ultimately lead to the three-dimensional development of any type of crystal structure.

The procedure provides a unique identifying code for a structure and makes it possible to locate it among 32 point groups and 230 space groups of symmetry. This classification obviously embraces the seven crystal systems. Although many metallic structures can be defined relatively simply in terms of space lattice and one or more lattice constants, complex structures require the key of symmetry theory.

24 Physical Metallurgy and Advanced Materials

(b) Figure 1.14 Symmetry of the tetrahedral form.

(a)