Difference between revisions of "Miller indices"
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| − | + | (French: ''Indices de Miller''). Spanish: ''Indices de Miller''.) | |
| − | === | + | == Definition == |
| + | |||
| + | === Direct space === | ||
| + | |||
| + | Planes of a given family of lattice planes with Miller indices (''h'', ''k'', ''l'') make intercepts ''OP = C a/h'', ''OQ = C b/k'', and ''OR = C c/l'' with the unit-cell axes '''OA''' = '''a''', '''OB''' = '''b''', and '''OC''' = '''c''' (see Figure 1), here ''h'', ''k'', ''l'' are integers and ''C'' is a constant integer. This property results from the [[law of rational indices]]. The variation of the orientation of the planes with the ratios of the Miller indices is illustrated in the attached [["Miller_Examples| examples]]. The equation of the planes of the family is: | ||
| − | + | <center> | |
| + | ''hx'' + ''ky'' + ''lz'' = ''C'' | ||
| + | </center> | ||
| − | == | + | === Reciprocal space === |
| + | |||
| + | The [[reciprocal lattice]] vector associated to the family of lattice planes is '''OH''' = ''h'' '''a*''' + ''k'' '''b*''' + ''l'' '''c*''', where '''a*''', '''b*''', '''c*''' are the reciprocal lattice basis vectors. '''OH''' is perpendicular to the family of lattice planes and ''OH'' = ''C''/''d'' where ''d'' is the lattice spacing of the family. | ||
| + | |||
| + | == History == | ||
| + | |||
| + | The Miller indices were first introduced, among others, by W. Whevell in 1829 and developed by W.H. Miller, his successor at the Chair of Mineralogy at Cambridge University, in his book ''A treatise on Crystallogtaphy'' (1839). | ||
| − | |||
| + | == See also == | ||
| − | + | [[law of rational indices]]<br> | |
| + | [[reciprocal lattice]] | ||
| − | |||
Revision as of 06:41, 3 February 2006
(French: Indices de Miller). Spanish: Indices de Miller.)
Definition
Direct space
Planes of a given family of lattice planes with Miller indices (h, k, l) make intercepts OP = C a/h, OQ = C b/k, and OR = C c/l with the unit-cell axes OA = a, OB = b, and OC = c (see Figure 1), here h, k, l are integers and C is a constant integer. This property results from the law of rational indices. The variation of the orientation of the planes with the ratios of the Miller indices is illustrated in the attached examples. The equation of the planes of the family is:
hx + ky + lz = C
Reciprocal space
The reciprocal lattice vector associated to the family of lattice planes is OH = h a* + k b* + l c*, where a*, b*, c* are the reciprocal lattice basis vectors. OH is perpendicular to the family of lattice planes and OH = C/d where d is the lattice spacing of the family.
History
The Miller indices were first introduced, among others, by W. Whevell in 1829 and developed by W.H. Miller, his successor at the Chair of Mineralogy at Cambridge University, in his book A treatise on Crystallogtaphy (1839).
See also
law of rational indices
reciprocal lattice