Transformation from Miller-Bravais to Orthonormal Coordinate System

Miller-Bravias Coordinate System

In HCP material systems, it is convenient to employ the 4-index Miller-Bravais (M-B) coordinate system to describe slip directions and slip plane normals. The basal plane contains three directions $(a_1, a_2, a_3)$ that form angles of 120 $^\circ$ with each other and one direction $(c)$ that is perpendicular to the basal plane. With this particular system, the symmetry of the directions is apparent through the shuffling of the 3 non-zero indices. For example, the basal slip directions in the 4-index M-B coordinate system are $[\bar{1}2\bar{1}0]$ , $[\bar{2}110]$ , and $[\bar{1}\bar{1}20]$ .

However, determining direction cosines directly from the 4-index M-B coordinate system can be somewhat confusing because it is not orthonormal. Therefore, we will step through the procedure for moving from a 4-index M-B coordinate system to an orthonormal system.

The first step is to convert the 4-index system to a 3-index system. This is done by eliminating the $a_3$ direction. Let’s assume the 4-index indices are $[ u v t w]$ and the 3-index indices are $[U V W]$ . The conversion follows:

$U = u - t; V = v - t; W = w$

Now that we have a 3-direction coordinate sysem, A simple transformation is performed to convert the 3-index M-B $[U V W]$ to the orthonormal system $[x, y, z]$ . The conversion follows:

$\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 & -\frac{1}{2} & 0 \\ 0 & \frac{\sqrt{3}}{2} & 0 \\ 0 & 0 & \frac{c}{a} \end{bmatrix} \begin{bmatrix} U \\ V \\ W \end{bmatrix}$

Note that this transformation aligns two directions, $a_1 \| x$ and $c \| z$ .

References

M. Niewczas (2010). “Lattice correspondence during twinning in hexagonal close-packed crystal,” Acta Mat 58, 5848-5857.