MILLER INDICES GROUP ACTIVITY 1 ABIY KIFLE JUSTINE YOO FALL 2009 HISTORY AND DISCRIPTION y Developed by William Hallowes Miller y Describes the orientation of planes in a lattice y Helps to investigate events in material science Continued y The position of the planes or the direction of a vector in the lattice are usually described by three integers MAIN GOALS y To avoid infinite intercepts y Miller indices correspond to the xyz coordinate y Since crystal lattice structures are periodic structures the whole structure can be constructed as the multiple of the unit cell structure HOW TO FIND MILLER INDICES y Determine the intercepts of the plane intersecting the xyz coordinate a1 x direction a2 y direction and a3 z direction y Take the reciprocals of the intercepts h 1 a1 l 1 a2 and m 1 a3 y Reduce the result to the smallest integer y hkl for single plane and hkl for a group of planes CONTINUED y Negative Miller values are denoted by a bar on top Example kl y Planes without an intersecting intercept are taken as infinite planes DIRECTION y The direction of a lattice plane is usually normal to the plane but not necessarily always y Directions are represented by vector components that are integer multiples of the basis vector y Directions are denoted by def and def for a family of directions PROCEDURE y Determine the coordinated of the vector y Reduce the coordinate in to the smallest integer values y Present the resulting integers as def y Family of directions as def y Example 100 010 and 001 As a family 100 EXAMPLES y 1 For the intercepts x y and z with values of 3 1 and 2 respectively find the Miller indices SOLUTION y Intercepts x 3 y 1 and z 2 y Take the reciprocal of the intercepts h 1 x k 1 y and l 1 z h 1 3 k 1 1 l Multiply h k and l by 6 to find the smallest integer hkl 263 EXAMPLE SOLUTION Notice that the plane does not intersect the y axis Therefore take the intercept as y Assume the intercepts of the x and z are a and b y Take the reciprocals of the intercepts h 1 a k 1 and l 1 b Multiply by a b to reduce to an integer value hkl b0a In cubic structures where the intercepts x z a Then h 1 a k 1 and z 1 a a hkl 101 y EXAMPLE 4 z y x In what direction is the line segment shown above oriented SOLUTION y In cubic lattice the length of each plane is the same Therefore in our problem we have 100 planes y Find the distance the line traveled from each plane From the x axis 1 unit y axis 1 unit z axis 1 unit So the line intersects and travels in the direction of 111 PROBLEMS y 1 Answer the following multiple choice problems Which statement below is false about 121 and 121 A They are in the same set of planes B They are in the same family of planes C They are parallel with each other D They are perpendicular with each other PROBLEM 2 What miller index plane is shown below z y x A 0 2 1 C 0 4 1 B 0 1 2 D 0 0 0 PROBLEM 3 What six miller indices does the set 100 include Can this set be written another way 4 Find the Miller indices for the plane below Assume a 1by 1 by cube z y x
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