11.8.3: Lines and Directions in Crystals

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11.62.png — Figure 11.62: Vector directions

The absolute location of a line in a crystal is not often significant, but directions (vectors) have significance because they describe the orientations of symmetry axes, zones, and other linear features. Crystallographers designate directions with three indices in square brackets, [uvw]. As with point locations, we give numbers describing directions in terms of unit cell dimensions. In Figure 11.62a, direction V has indices [132] – it goes from the origin to a point at 1a, 3b, 2c.

The drawings in 11.62b and 11.62c show other vectors pointing the same direction as [132]. We clear fractions and divide by common denominators when giving the indices of a direction. So directions [½ 1½ 1] and [264] would be simplified and described as [132]. Although no commas are included in [132], reading sequentially, as we do for point locations, we articulate it “one-three-two.” By convention, commas separate indices only if they have more than one digit.

11.63.png — Figure 11.63: Examples of line indices

Figure 11.63 shows several lines on a two-a dimensional lattice. Parallel lines have identical indices, and choosing an origin is unimportant. The direction [240] is the same as [120], and when we clear the common denominator [240] becomes [120]. For line [120], the bar over the 1 is equivalent to a negative sign, indicating that the line goes in the negative a-direction. We articulate the indices as “bar-one-two-zero.” Because all directions in Figure 11.63 lie in the a-b plane, they have indices [uv0]. This notation indicates that the first two indices may vary, but the third is 0.