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Pole Figures in Crystallography

Pole figures are applied in crystallography and materials science for visualizing the three-dimensional orientation of crystals and the orientation distribution (texture) of grains in polycrystalline materials. A pole figure maps the directions of a set of symmetrically equivalent "poles," that is, normal directions of lattice planes, onto the - plane of a Cartesian -- coordinate system. In order to work with pole figures, it is important to acquire experience in correlating these figures with the corresponding three-dimensional crystal lattice orientation. This Demonstration shows how pole figures are constructed for the example of a cubic crystal lattice. You can choose three different sets of symmetrically equivalent poles. Moreover, to illustrate how pole figures correlate with the three-dimensional orientation of the crystal lattice, you can rotate the lattice about an arbitrary axis by an arbitrary angle.

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Snapshot 1: This shows the principle of constructing pole figures. Think of the crystal as positioned in the center of a sphere. The normal directions of sets of crystallographically equivalent planes, in this example the planes (space-diagonal planes of the cube), are extended to the points where they intersect with the sphere (blue arrows). The intersection points on the southern hemisphere are discarded. The intersection points ("poles") on the northern hemisphere are connected to the south pole of the sphere (blue dashed lines). The actual "pole figure" consists of the points (blue discs) where the connection lines (dashed) intersect with the - plane.
Snapshot 2: This shows the pole figure of the nonrotated cubic crystal, viewed in plan view, that is, in the direction of the axis. The planes correspond to the faces of the cube. There are six such faces: , , , , , . However, only five of them appear in the pole figure because the sixth one, , is pointing into the southern hemisphere (in the direction) and therefore is discarded.
Snapshot 3: This shows the pole figure of the nonrotated cubic crystal, viewed in plan view, that is, in the direction of the axis. The planes correspond to the space-diagonal planes of the cube. There are eight such planes, but only four of them appear in the pole figure because the other ones are pointing into the southern hemisphere and therefore are discarded.
Snapshot 4: This shows the pole figure of the nonrotated cubic crystal, viewed in plan view, that is, in the direction of the axis. The planes correspond to the face-diagonal planes of the cube. There are 12 such planes, but only eight of them appear because the other ones are pointing into the southern hemisphere and therefore are discarded.
Snapshot 5: To observe how pole figures indicate the orientation of the crystal, you can rotate the crystal about an arbitrary axis by an arbitrary angle. You can adjust the orientation of the rotation axes by choosing appropriate integer components from the corresponding , , popup menus. The rotation axis is shown in red, and its pole appears as a red disk in the pole figure. This snapshot shows a pole figure after rotating the cube that represents the cubic crystal lattice by an arbitrary angle (150°) about an axis in the direction (red). The effect of rotations becomes particularly clear by automatically incrementing the rotation angle ("Play" in the animation controls of "rotation angle"). If all three components of the rotation vector are set equal to zero, the rotation is undefined, and the rotation angle cannot be set different from zero.
Snapshot 6: This is a plan view showing the pole figure after the cube has been rotated by 54.7° about the direction. This rotation turns a pole (plane normal) into the direction of the axis. Correspondingly, the resulting pole figure shows the three-fold symmetry of the cube about the normal of a plane, that is, the direction, and one of the poles coincident with the axis (center).
Snapshot 7: This is a plan view showing the pole figure after the cube has been rotated by 45.0° about the direction. This rotation turns a pole (plane normal) into the direction of the axis. Correspondingly, the resulting pole figure shows the two-fold symmetry of the cube about the normal of a plane, that is, the direction, and one of the poles coincident with the axis (center).
Reference
[1] Wikipedia. "Pole Figure." (Aug 14, 2014) en.wikipedia.org/wiki/Pole_figure.
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