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When dealing with crystalline materials, sometimes it is necessary to refer to specific directions in crystal lattices (or) it may be of interest to know the crystallographic orientation of a plane or group of planes in a crystal lattice.
The basis for determining index values in the unit cell, with a coordinate system consisting of three (x, y and z) axes situated at one of the corners and coinciding with the unit cell edges.
To identify crystal planes and directions in cubic crystal structures, the Miller Notation is used
The Miller Indices of directions are specified by drawing a vector from the origin through a point in the desired direction and giving the coordinates of the point in terms of the lowest integers.
The orientation of a plane in space can be designated by a system of notation called the Miller indices of Planes.
The Miller Indices of a crystal plane are defined as the reciprocals of the fractional intercepts (with fractions cleared) which the plane makes with the crystallographic x, y and z axes of the three non parallel edges of the cubic unit cell.
This is especially important for metals and alloys that have properties which vary with crystallographic orientation.
Procedure to Determine the Miller Indices of Directions for a cubic crystal
Using a right hand Coordinate system, determine the coordinates of two points that lie on the direction.
Subtract the coordinates of the TAIL point from the coordinates of the HEAD point to obtain the number of Lattice Parameters traveled in the direction of each axis of the coordinate system.
Clear the fractions and, or reduce the results obtained from the subtraction to lowest integers.
Enclose the numbers in Square Brackets [ ]. If a negative sign is obtained represent the ve sign with a bar over the number.
Miller Indices of Directions for Hexagonal Crystals
The three a1, a2 and a3 axes are all contained within a single plane (called the basal plane), and at 1200 angles to one another. The z-axis is perpendicular to the basal plane.
Find the Coordinates of Head and Tail.
Subtract Head Tail.
Reduce to the lowest integers. These numbers are called as U, V and W
Find U, V and W by the following equations.
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
Write in proper form [U,V, t, W]
Crystallographic planes:
In all but the hexagonal crystal system, crystallographic planes are specified by three Miller Indices as (hkl)
Any two planes parallel to each other are equivalent and have identical indices.
Procedure for determining Miller Indices for Cubic Planes
If plane passes through the selected origin, either another parallel plane must be constructed within the unit cell by an appropriate translation or a new origin must be established at the corner of another unit cell.
At this point the crystallographic plane either intersects or parallels each of the three axes; the length of the planar intercept for each axis is determined in terms of the lattice parameters a, b, c.
The reciprocals of these numbers are taken. A Plane that parallels an axis may be considered to have an infinite intercept and therefore a zero index.
If necessary these three numbers are changed to the set of smallest integers by multiplication or division by a common factor.
Miller Indices of Planes for Hexagonal Crystals
Crystal Plane in HCP unit cells is commonly identified by using four indices instead of three.
The HCP crystal plane indices called Miller-Bravis indices are denoted by the letters h, k, I and l are enclosed in parentheses as (hkil)
These four digit hexagonal indices are based on a coordinate system with four axes.
The three a1, a2 and a3 axes are all contained within a single plane (called the basal plane), and at 1200 angles to one another. The z-axis is perpendicular to the basal plane.
The unit of measurement along the a1, a2 and a3 axes is the distance between the atoms along these axes.
The unit of measurement along the z- axis is the height of the unit cell.
The reciprocals of the intercepts that a crystal plane makes with the a1, a2 and a3 axes give the h, k and I indices while the reciprocal of the intercept with the z-axis gives the index l.
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