GROUP ACTIVITY 1ABIY KIFLEJUSTINE YOOFALL 2009MILLER INDICESHISTORY AND DISCRIPTIONyDeveloped by William Hallowes MilleryDescribes the orientation of planes in a latticeyHelps to investigate events in material scienceContinued y The position of the planes or the direction of a vector in the lattice are usually described by three integers.MAIN GOALSy To avoid infinite intercepts.y Miller indices correspond to the xyz coordinatey 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 INDICESy 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.CONTINUEDy Negative Miller values are denoted by a bar on top. Example - (ħkl).y Planes without an intersecting intercept are taken as infinite planes (∞).DIRECTIONy 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.PROCEDUREy 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>EXAMPLESy1 – For the intercepts x, y, and, z with values of 3,1, and 2 respectively, find the Miller indices.SOLUTIONy Intercepts: x=3, y= 1, and z=2y Take the reciprocal of the interceptsh = 1/x, k=1/y, and l=1/zh=1/3k = 1/1l = ½.Multiply h, k, and l by 6 to find the smallest integer.(hkl) = (263)EXAMPLESOLUTIONy 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 interceptsh = 1/a, k = 1/∞, and l= 1/bMultiply by a*b to reduce to an integer value(hkl) = (b0a)In cubic structures, where the intercepts x =z = aThen, h=1/a, k=1/∞, and z=1/aa*(hkl) = (101)EXAMPLE4.z?yxIn what direction is the line segment shown above oriented?SOLUTIONy 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 unity – axis - 1 unitz – axis - 1 unitSo, the line intersects and travels in the direction of <111>PROBLEMSy 1. Answer the following multiple choice problems:Which statement below is false about (121) and (121)?A. They are in the same set of planesB. They are in the same family of planesC. They are parallel with each otherD. They are perpendicular with each otherPROBLEM2. What miller index plane is shown below?z½yxA. (0 2 1) B. (0 1 2)C. (0 4 1) D. (0 0 0)PROBLEM3. 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.
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