Crystallographic Planes
are written parenthesis (hkl), where h, k and l represent integers that are called the Miller Indices for that crystal plane.
The following methodology is used to label crystallographic planes, and thus get the values of the integer Miller indices h, k and l.
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Draw an orthogonal coordinate system somewhere on the crystal.
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Determine the x,y,z coordinates where the plane intersects the axes. These
will be integer values and will represent a number of lattice points. -
Take the reciprocals of each of these points of intersection and then determine
the smallest set of integers that give the same ratios between the 1/x, 1/y and
1/z and these will be your h,k and l indices and thus the (hkl) crystal plan.
Example 1.6:
Find the label for Plane I in Figure 1.12.
- Intercepts: 4~a, ∞~b
- Taken plane is the reciprocals: 1/4 , 1/∞
- Reduce to smallest set of integer that has same ratio 1/4 , 0 → (1, 0) Plane
Example 1.7:
Find the label for Plane II in Figure 1.12.
- Intercepts: (x=2) and (y=2)
- Reciprocal 1/2 , 1/2
- Plane (1,1) (Figure 1.12 Plane II)
Example 1.9:
Find the label for a 3D Crystal Plane with indices (2,1,4).
- Intercepts at (x=2, y=4, z=1)
- Reciprocals: 1/2 , 1/4 , 1/1
- Plane (2,1,4) (See Figure 1.14)
