Lecture note 4 (09-15-2011)

Lecture note 4 (09-15-2011) - MAE 581 Advanced Materials...

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1 MAE 581 Advanced Materials Science Lecture note 4 We first give more information and examples about Miller index notation. Then we will discuss stereographic projection
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The orienta t on of a surface or a crystal plane may be de f ned by considering how the plane (or indeed any parallel plane) intersects the main crystallographic axes of the solid. The applica t on of a set of rules leads to the assignment of the Miller Indices , ( hkl ) ; a set of numbers which quan t fy the intercepts and thus may be used to uniquely iden t fy the plane or surface. The following treatment of the procedure used to assign the Miller Indices is a simpli f ed one (it may be best if you simply regard it as a "recipe") and only a cubic crystal system (one having a cubic unit cell with dimensions a x a x a ) will be considered. The procedure is most easily illustrated using an example so we will f rst consider the following surface/plane: Miller Index Notation http://www.chem.qmul.ac.uk/surfaces/scc/scat1_1b.htm 1 of 5 9/16/2010 7:38 AM
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Step 1 : Iden t fy the intercepts on the x , y and z axes. In this case the intercept on the x axis is at x = a ( at the point ( a ,0,0) ), but the surface is parallel to the y and z axes strictly therefore there is no intercept on these two axes but we shall consider the intercept to be at in f nity ( ) for the special case where the plane is parallel to an axis. The intercepts on the x , y and z axes are thus Intercepts : a , , Step 2 : Specify the intercepts in frac t onal co ordinates Co ordinates are converted to frac t onal co ordinates by dividing by the respec t ve cell dimension for example, a point ( x , y , z ) in a unit cell of dimensions a x b x c has frac t onal co ordinates of ( x / a , y / b , z / c ). In the case of a cubic unit cell each co ordinate will simply be divided by the cubic cell constant , a . This gives Frac t onal Intercepts : a / a , / a , / a i.e. 1 , , Step 3 : Take the reciprocals of the frac t onal intercepts This f nal manipula t on generates the Miller Indices which (by conven
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Lecture note 4 (09-15-2011) - MAE 581 Advanced Materials...

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