Wulff net

Small circles on the sphere also project as circles, but their projected center does not coincide with their center on the projection. For example, the circle AJEK whose center P lies on AEBW projects as jection is at C, located at equal distances from A at P located an equal number of degrees (45° in The device most useful in solving proble projection is the Wulff net shown in Fig. 2-29. It... 

We o]ten wish to rotate poles around various axes. We have already seen that rotation about an axis normal to the projection is accomplished simply by rotation of the projection around the center of the Wulff net. Rotation about an axis lying in the plane of the projection is performed by, first, rotating the axis about the center of the Wulff net until it coincides with the north-south axis if it does not already do so, and, second, moving the poles involved along their respective latitude circles the required number of degrees. Suppose it is required to rotate the poles and Bi shown in Fig. 2-34 by 60° about the NS axis, the direction of motion being from IF to F on the projection. Then Ax moves to A2 along its... 

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. 

Construct a Wulff net, 18 cm in diameter and graduated at 30° intervals, by the use of compass, dividers, and straightedge only. Show all construction lines. 

An alternative method of indexing plotted poles depends on having available a set of detailed standard projections in a number of orientations, such as 1001, HOI, and 1111 for cubic crystals. It is also a trial and error method and may be illustrated with reference to Fig. 8-6. First, a prominent zone is selected and an assumption is made as to its indices for example, we might assume that zone fl is a <100> zone. This assumption is then tested by (a) rotating the projection about its center until Pg lies on the equator of the Wulff net and the ends of the zone circle coincide with the N and S poles of the net, and (b) rotating all the important points on the projection about the NS-a is of the net until Pb lies at the center and the zone circle at the circumference. The new projection is then superimposed on a 100 standard projection and rotated about the center until all points on the projection coincide with those on the standard. If no such coincidence is obtained, another standard projection is tried. For the particular case of Fig. 8-6, a coincidence would be obtained only on a 1101 standard, since Fg is actually a HOI pole. Once a match has been found, the indices of the unknown poles are given simply by the indices of the poles on the standard with which they coincide. 

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. 

Figure 3.21 Wulff net that shows the stereographic projection of lines of equal longitude and equal latitude at 10° intervals.

A net of meridians (great circles) and parallels (small circles) allows us to define the coordinates of the points s on the sphere in analogy with terrestrial geography. If we project this net stereographically, we obtain the Wulff net. This allows us to easily determine the ratios a b c as well as the indices of the faces and zones starting from the angles between the faces of a crystal measured with a goniometer. 

Fig. 1.7. Wulff net with the poles situated in the stereographic plane...

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