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Indexing Directions and Planes > Directions in a 2-D Lattice

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Consider this 2-D lattice. The unit cell is defined by the lattice unit vectors: a and b. Move OD so it passes through the origin of the unit cell. Select any point that OD passes through - say P. Express the vector OP in terms of its number of unit vectors of a and b:     OP = 1a + 1/2b. A direction is expressed in the form [uv] where u and v are integers. The direction OD is thus given as [21]. In vector notation: OP =  1a + 1/2b. Direction of OD is [21].

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It is necessary to express directions in crystals with ease. For example, to state the direction in a lattice along which an electron beam is passing or in which planes of atoms may slip during deformation.

Directions in lattices are determined relative to the crystal axes defined by the unit vectors of the lattice unit cell. A direction is expressed in terms of its ratio of unit vectors.
Important

It is very important to be clear about which unit vectors of the lattice are being used.

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Note the use of square brackets and no separating comma. The ratio is always reduced to its lowest terms.

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A direction index, often called a Miller Index, in a 2-D lattice is expressed in the form [uv] where u and v are integers. If negative integers are required the notation is [u v], read as 'bar u bar v'.

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