The stereographic projection

The orientation of any plane in a crystal can be just as well represented by the inclination of the normal to that plane relative to some reference plane as by the inclination of the plane itself. All the planes in a crystal can thus be represented by a set of plane normals radiating from some one point within the crystal. If a reference sphere is now described about this point, the plane normals will intersect the surface of the sphere in a set of points called poles. This procedure is illustrated in Fig. 2-25, which is restricted to the 100 planes of a cubic crystal. The pole of a plane represents, by its position on the sphere, the orientation of that plane. 

A plane may also be represented by the trace the extended plane makes in the surface of the sphere, as illustrated in Fig. 2-26, where the trace A BCD A represents the plane whose pole is Pj. This trace is a great circle, i.e., a circle of maximum diameter, if the plane passes through the center of the sphere. A plane not passing through the center will intersect the sphere in a small circle. On a ruled globe, for example, the longitude lines (meridians) are great circles, while the latitude lines, except the equator, are small circles. 

A lattice plane in a crystal is several steps removed from its stereographic projection, and it may be worth-while at this stage to summarize these steps  

The normal CP is represented by its pole P, which is its intersection with the reference sphere. 

After gaining some familiarity with the stereographic projection, the student will be able mentally to omit these intermediate steps and he will then refer to the projected point P as the pole of the plane C or, even more directly, as the plane C itself. 

The angular relationships among crystal faces (or atomic planes) cannot be accurately displayed by perspective drawings but if they are projected in a stereographic way they can be precisely recorded and then clearly understood. 

Let us assume a very small crystal is located at the center of a reference sphere (atomic planes are assumed to pass through the center of the sphere). Each crystal plane within the crystal can be represented by erecting its normal, at the center of the sphere, which pierces the spherical surface at a point known as the pole of the 

As it is inconvenient to use a spherical projection to determine angles among crystal faces or angular distances of planes on a zone, a map of the sphere is made, so that all work can be done on flat sheets of paper. 

The projection of the net of latitude and longitude lines of the reference sphere upon a plane forms a stereographic net—the Wulff net (Fig. 4). The angles between any two points can be measured with this net by bringing the points on the same great circle and counting their difference in latitude keeping the center of the projection at the central point of the Wulff net. 

Stereographic projections of low-index planes in a cubic crystal and in a hep crystal are given in Fig. 5. Only one side of the projection is visible thus it must not be forgotten that below  

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The N2(G) adducts are more stable than the N6(a) and 06(G) and NU(c) adducts. Because cis addition products are present, minor amounts of the other adducts are found. If only cis addition occurred, then the i(-) and Il(-) isomers would be stereoselected by N2(G), and the l(+) and Il(+) isomers would be stereoselected by N6(A) and 06(G). Although we did not perform calculations on the cis adducts, it can be seen from the stereographic projections that the change accompanied by a reflection of only the BPDE atoms through the plane of the pyrene changes the (+) into (-) isomers. Thus, the rules of stereoselectivity are reversed. However, the small amount of cis adduct yields these minor components the l(-)-N2(G) adduct is most prevalent (38) for reactions of BPDE i(-) with DNA and we assume that this arises from the cis addition. If both trans and cis addition occurred equally, we predict that stereoselectivity would not be observed. 

At this point, it is advantageous to switch to the use of stereographic projections to visualize fully the three-dimensional geometry.2 The stereographic projection in Fig. 24.6 shows the geometry of the shear illustrated in Fig. 24.5 when the shear... 

Other projections are possible, but crystallographers use the stereographic projection because the angles between planes in the projection are the true angles between the planes. 

Assuming that the a, phase is a kind of (3 phase, the indicatrix orientation should be compared to the structure of 0-(BEDT-TTF)2I3 (Fig. 2). However, analysis of the stereographic projections shows that a transformation of a phase into this transformed 0 phase requires large movements of molecules. In a first step the BEDT-TTF molecules of the a phase should be rotated around the L axis by 78°, 42°, or 30°, respectively, and then all L axes should be further inclined by 15°. On the other hand, the transformation from the a phase into the a, phase needs a rotation by only 34°, 3°, or 16° around L, and then the same inclination by 15°. This led to the assumption that the af-(BEDT-TTF)2I3 is a newer phase than the 0-(BEDT-TTF)2I3 [29]. 

FIGURE 2.13. The stereographic projection for representing, in two dimensions, the arrangement and directions of faces in a three-dimensional crystal, (a) The crystal is surrounded by a sphere. The points at which normals (perpendiculars) to the faces hit this sphere are noted by points labeled 100, 110, etc., the same cis the crystal faces they represent, (b) Each point (1 to 5, for example) representing a crystal face is joined to the opposite pole ( south if the point is in the northern hemisphere). 

The stereographic projection or stereographic triangle shown in Fig. 4.4 serves as a visual framework for considering all the surfaces that can be cleaved from a face centered cubic lattice. All points in or on the stereographic triangle uniquely... 

The convention used to classify the handedness of the natnrally chiral surfaces is based on the relative sense of rotation among the (111), (100), and (110) microfacets. Because all surfaces within the portion of the stereographic projection shown in Fig. 4.4 have the same relative sense of counterclockwise rotational... 

Figure 1.18. The schematic of how to construct a stereographic projection. The location of the center or inversion is indicated using letter C in the middle of the stereographic projection.

Figure 1.20. Examples of the stereographic projections with tetragonal (left) and cubic (right) symmetry.

More information about the stereographic projection can be found on the World Wide Web in the International Union of Crystallography (lUCr) teaching pamphlets and in the International Tables for Crystallography, vol. A. 

E. J. W. Whittaker, The stereographic projection, http //www.us.iucr.org/iucr-top/comm/ cteach/pamphlets/11/index.html. 

The lattice reorientation caused by twinning can be clearly shown on the stereographic projection. In Fig. 2-40 the open symbols are the 100 poles of a cubic crystal projected on the (001) plane. If this crystal is FCC, then one of its possible twin planes is (TT1), represented on the projection both by its pole and its trace. The cube poles of the twin formed by reflection in this plane are shown as solid symbols these poles are located by rotating the projection on a Wulff net until the pole of the twin plane lies on the equator, after which the cube poles of the crystal can be moved along latitude circles of the net to their final position. 

The reader may detect an apparent error in nomenclature here. Pole 5 for example, is assumed to be a 100 pole and spot 5 on the diffraction pattern is assumed, tacitly, to be due to a 100 reflection. But aluminum is face-centered cubic and we know that there is no 100 reflection from such a lattice, since hkl must be unmixed for diffraction to occur. Actually, spot 5, if our assumption is correct, is due to overlapping reflections from the 200, 400, 600, etc., planes. But these planes are all parallel and are represented on the stereographic projection by one pole, which is conventionally referred to as 100). The corresponding diffraction spot is also called, conventionally but loosely, the 100 spot. 

Figure 8-7 shows the stereographic projection in a more complete form, with all poles of the type 100, 110, and 111 located and identified. Note that it was not necessary to index all the observed diffraction spots in order to determine the crystal orientation, which is specified completely, in fact, by the locations of any two 100 poles on the projection. The information given in Fig. 8-7 is therefore all that is commonly required. Occasionally, however, we may wish to know the Miller indices of a particular diffraction spot on the film, spot 11 for example. To find these indices, we note that pole 11 is located 35° from (001) on the great circle passing through (001) and (111). Reference to a standard projection and a table of interplanar angles shows that its indices are (112). 

As mentioned above, the stereographic projection of Fig. 8-7 is a complete description of the orientation of the crystal. Other methods of description are also possible. The crystal to which Fig. 8-7 refers had the form of a square plate and was mounted with its plane parallel to the plane of the film (and the projection) and its edges parallel to the film edges, which are in turn parallel to the NS and EW axes of the projection. Since the (001) pole is near the center of the projection, whjch corresponds to the specimen normal, and the (010) pole near the edge of the projection and approximately midway between the E and 5 poles, we may very roughly describe the crystal orientation as follows one set of cube planes is... 

Fig. 8-9 Use of the Greninger chart to plot the axis of a zone of planes on the stereographic projection. is the axis of zone A.

Parenthetically, it should be noted that the positioning of the crystal surface and the axis A A at equal angles to the incident and diffracted beams is done only for convenience in plotting the stereographic projection. There is no question of focusing when monochromatic radiation is reflected from an undeformed single crystal, and the ideal incident beam for the determination of crystal orientation is a parallel beam, not a divergent one. 

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