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section%204.3 - 4.3 M iller I ndices Objectives Calculate...

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4.3 Miller Indices Objectives Calculate Miller indices for directions and planes. Given Miller indices, draw directions and planes. We are now going to ignore the atoms in the unit cells for a little while to develop a concept called Miller indices . Miller indices are sets of coordinates for directions and planes in crystals. Miller indices were developed by several British mineralogists, and the idea was published by William Hallowes Miller in 1839. They were created to provide a simpler method for calculating the positions and angles between the faces of crystals. In the introduction to his book, A Treatise on Crystallography, Miller writes, “The expressions which in this Treatise have thus been obtained, are remarkable for their symmetry and simplicity, and are all adapted to logarithmic computation.” In fact, Miller indices are so remarkable that they are still the standard way of describing crystals today, and are used for everything from describing the positions of atoms in a crystal to predicting a material’s strength. The first step in determining Miller indices is to define atomic positions in a unit cell. Figure 4.3.1 shows a unit cell with some axes. A point within this unit cell is defined by a coordinate system that is placed on the edges of the unit cell. There are three important rules to be followed when assigning the coordinates of a point: 1. The coordinate system must be right-handed. 2. The unit length is the lattice parameter, a. If the unit cell has unequal sides, the unit length for each direction is the length of the cell edge in that direction. This means that the unit lengths may be different in different directions. 3. When listing the coordinates of the point, no parentheses are used, and the unit vector is no included. So for example, point A in Figure 4.3.1 has the coordinates 1, ½, 0.
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Figure 4.3.1: Defining atomic positions in a unit cell. Point A has the coordinates 1, ½, 0.
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