The orientation of a surface or a crystal plane may be defined by considering how the plane (or indeed any parallel plane) intersects the main crystallographic axes of the solid. The application of a set of rules leads to the assignment of the Miller Indices , (hkl) , a set of numbers which quantify the intercepts and thus may be used to uniquely identify the plane or surface. The following treatment of the procedure used to assign the Miller Indices is a simplified 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 first consider the following surface/plane: In this case the intercept on the xaxis is at x = a ( at the point (a,0,0) ), but the surface is parallel to the y and zaxes  strictly therefore there is no intercept on these two axes but we shall consider the intercept to be at infinity ( ∞ ) for the special case where the plane is parallel to an axis. The intercepts on the x , y and zaxes are thus Coordinates are converted to fractional coordinates by dividing by the respective celldimension  for example, a point (x,y,z) in a unit cell of dimensions a x b x c has fractional coordinates of ( x/a , y/b , z/c ). In the case of a cubic unit cell each coordinate will simply be divided by the cubic cell constant , a . This gives This final manipulation generates the Miller Indices which (by convention) should then be specified without being separated by any commas or other symbols. The Miller Indices are also enclosed within standard brackets (….) when one is specifying a unique surface such as that being considered here. The (100), (110) and (111) surfaces considered above are the socalled low index surfaces of a cubic crystal system (the 'low' refers to the Miller indices being small numbers  0 or 1 in this case). These surfaces have a particular importance but there an infinite number of other planes that may be defined using Miller index notation. We shall just look at one more … (i) in some instances the Miller indices are best multiplied or divided through by a common number in order to simplify them by, for example, removing a common factor. This operation of multiplication simply generates a parallel plane which is at a different distance from the origin of the particular unit cell being considered. e.g. (200) is transformed to (100) by dividing through by 2 . (ii) if any of the intercepts are at negative values on the axes then the negative sign will carry through into the Miller indices, in such cases the negative sign is actually denoted by overstriking the relevant number. e.g. (00 1) is instead denoted by (iii) in the hcp crystal system there are four principal axes, this leads to four Miller Indices e.g. you may see articles referring to an hcp (0001) surface. It is worth noting, however, that the intercepts on the first three axes are necessarily related and not completely independent, consequently the values of the first three Miller indices are also linked by a simple mathematical relationship. In the following diagram the three highlighted surfaces are related by the symmetry elements of the cubic crystal  they are entirely equivalent. In fact there are a total of 6 faces related by the symmetry elements and equivalent to the (100) surface  any surface belonging to this set of symmetry related surfaces may be denoted by the more general notation {100} where the Miller indices of one of the surfaces is instead enclosed in curlybrackets. Final important note : in the cubic system the (hkl) plane and the vector [hkl], defined in the normal fashion with respect to the origin, are normal to one another but this characteristic is unique to the cubic crystal system and does not apply to crystal systems of lower symmetry. Source.
Miller Index Notation
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