Plotting the trace of the zone [UVW] on a stereographic projection
In many analyses involving crystallographic features and/or diffraction, it is important to be able to determin ewhich planes lie in a given zone and what the trace of that zone is on a stereographic projection.
A Zone [uvw] is simply a collection of planes whose normals are perpendicular to the vector [uvw]. If those normals are given the indices [hkl], expressed in crystal co-ordinates then the Weiss Zone Law states that [uvw].[hkl] = 0.
To plot the zone [uvw] on a stereographic projection we need to recognize that the zone traces a great circle along the surface of the bounding sphere of the stereographic projection traversing a full 360° of rotation, that the trace of that great circle on the equatorial plane of the sterographic projection always touches the edge of the stereographic projection and thus contains the normal to the plane (hk0). We can plot the trace of the zone as follows.
(1). Take a standard stereographic projection centred on [001] and determine the angle between the x-axis, the vector [100], and the normal to the plane (hk0). First find h and k that statisfys the Weiss Zone Law.
if u0 then h=k.v/u otherwise k = h.u/v, l=0
So the angle required is
(2) Rotate the x-y axis so that the x-axis is parallel to [hk0]. The rotation about the x-axis, z-axis and y-axis is given by a simple set of matrices.
(3) Now find the angle between the vector [001] and the vector [uvw] so that the z-axis can be rotated around the new x-axis so that is aligned parallel to the vector [uvw].
(4) If the new x-axis vector is rotated around the new z-axis through one complete rotation (360°) then the x-axis vector will progressively trace out every possible vector that is perpendicular to the vector [uvw]. It then remains to only translate the vector onto the stereographic projection using the methodology outlined in the previous class.
The total rotation is given by multiplying the 3 matrices together in the correct order. The result of the multiplication of the matrices is a 3x3 matrix, the first row of which is the vector [hkl].
As q is varied from 0 to 360° then resultant vector is translated to its pole on the stereographic projection. In the expression below [uvw] is the resultant vector not the zone!
The poles [100],[010],[110],[-110],[101] and [011] are plotted above to give a standard stereographic projection for a cubic crystal.