Miller indices - Revision history

BrianMcMahon: Tidied translations.

2017-12-11T09:23:55Z

Tidied translations.

BrianMcMahon: Tidied translations.

2017-11-16T12:56:07Z

Tidied translations.

MassimoNespolo: /* See also */

2017-09-01T10:17:18Z

‎See also

BrianMcMahon: Style edits to align with printed edition

2017-05-22T09:57:55Z

Style edits to align with printed edition

BrianMcMahon: Style edits to align with printed edition

2017-05-15T16:54:50Z

Style edits to align with printed edition

MassimoNespolo: /* Reciprocal space */ co-prime to relatively prime

2016-08-17T09:17:30Z

‎Reciprocal space: co-prime to relatively prime

MassimoNespolo: /* Direct space */ co-prime to relatively prime

2016-08-17T09:16:42Z

‎Direct space: co-prime to relatively prime

MassimoNespolo: /* Direct space */ overemphasis ("instead") removed

2016-06-27T21:52:52Z

‎Direct space: overemphasis ("instead") removed

MassimoNespolo: /* Direct space */ typo

2016-05-18T10:05:37Z

‎Direct space: typo

MassimoNespolo: /* See also */ Laue indices (internal link)

2016-04-14T16:41:17Z

‎See also: Laue indices (internal link)

← Older revision		Revision as of 09:23, 11 December 2017
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−	<font color = "blue">Indices de Miller</font> (''Fr''). <font color = "red">Millersche Indizes</font> (''Ge''). <font color="black">Indici di Miller</font> (''It''). <font color="~~brown~~">~~Индексы Миллера~~</font> (''Ru''). <font color="~~purple~~">~~ミラー指数~~</font> (''Ja''). <font color="green">Indices de Miller</font> (''Sp'').	+	<font color = "blue">Indices de Miller</font> (''Fr''). <font color = "red">Millersche Indizes</font> (''Ge''). <font color="black">Indici di Miller</font> (''It''). <font color="purple">ミラー指数</font> (''Ja''). <font color="brown">Индексы Миллера</font> (''Ru''). <font color="green">Indices de Miller</font> (''Sp'').

	== Definition ==		== Definition ==

← Older revision		Revision as of 12:56, 16 November 2017
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−	<font color = "blue">Indices de Miller</font> (''Fr''). <font color = "red">Millersche Indizes</font> (''Ge''). <font color="~~green~~">~~Indices de~~ Miller </font>(''Sp''). <font color="~~black~~">~~Indici di Miller~~</font> (''It''). <font color="~~brown~~">~~Индексы Миллера~~</font> (''Ru''). <font color="~~purple~~"> ~~ミラー指数~~ </font>(''Ja'').	+	<font color = "blue">Indices de Miller</font> (''Fr''). <font color = "red">Millersche Indizes</font> (''Ge''). <font color="black">Indici di Miller</font> (''It''). <font color="brown">Индексы Миллера</font> (''Ru''). <font color="purple">ミラー指数</font> (''Ja''). <font color="green">Indices de Miller</font> (''Sp'').

	== Definition ==		== Definition ==

@@ Line 72: / Line 72: @@
 == See also ==
 *[[Laue indices]]
 *[[Law of rational indices]]

@@ Line 72: / Line 72: @@
 == See also ==
+*[[Laue indices]]
 *[[Law of rational indices]]
 *[[Reciprocal lattice]]
 *[[Weiss parameters]]
-*[http://journals.iucr.org/j/issues/2015/04/00/gj5135/ Discussion about non-coprime Miller indices]
+*Miller, W. H. (1839). ''A treatise on crystallography''. Cambridge: Pitt Press. [https://archive.org/details/ATreatiseOnCrystallography Online copy at archive.org]
-*[https://archive.org/details/ATreatiseOnCrystallography Miller's book at archive.org]
+*Nespolo, M. (2015). [http://journals.iucr.org/j/issues/2015/04/00/gj5135/ ''J. Appl. Cryst.'' '''48''', 1290&ndash;1298.] ''The ash heap of crystallography: restoring forgotten basic knowledge''. Discussion about non-coprime Miller   indices
 [[Category:Fundamental crystallography]]<br>
 [[Category:Morphological crystallography]]

@@ Line 1: / Line 1: @@
-<font color = "blue">Indices de Miller</font> (''Fr''); <font color = "red">Millersche Indizes</font> (''Ge''); <font color="green">Indices de Miller </font>(''Sp''); <font color="black">Indici di Miller</font> (''It''); <font color="brown">Индексы Миллера</font> (''Ru''); <font color="purple"> ミラー指数 </font>(''Ja'')
+<font color = "blue">Indices de Miller</font> (''Fr''). <font color = "red">Millersche Indizes</font> (''Ge''). <font color="green">Indices de Miller </font>(''Sp''). <font color="black">Indici di Miller</font> (''It''). <font color="brown">Индексы Миллера</font> (''Ru''). <font color="purple"> ミラー指数 </font>(''Ja'').
 == Definition ==
@@ Line 6: / Line 5: @@
 === Direct space ===
-The [[law of rational indices]] states that the intercepts, ''OP'', ''OQ'', ''OR'', of the natural faces of a crystal form with the basis vectors '''OA''' = '''a''', '''OB''' = '''b''', and '''OC''' = '''c''' are inversely proportional to prime integers, ''h'', ''k'', ''l'', called '''Miller indices''' of the face. This definition can be extended to any reticular plane whose intercepts with the basis vectors are ''OP = C a/h'', ''OQ = C b/k'', and ''OR = C c/l'' (see Figure 1). When the lattice is indexed with respect to a [[primitive basis]], ''h'', ''k'', ''l'' are relatively prime integers (''i''.''e''. not having a common factor other than +1 or -1); this restriction does not hold when a [[Centred_lattices|centred unit cell]] is chosen. ''C'' is an integer which, when made variable, identifies a specific lattice plane of the infinite set having the same orientation with respect to the basis vector. This infinite set of planes defines a ''family'' of lattice planes, denoted by the Miller indices in parentheses: (''hkl''). The Miller indices of the equivalent faces of a [[crystal form]] are denoted by {''hkl''}.
+The [[law of rational indices]] states that the intercepts, ''OP'', ''OQ'', ''OR'', of the natural faces of a crystal form with the basis vectors '''OA''' = '''a''', '''OB''' = '''b''', and '''OC''' = '''c''' are inversely proportional to prime integers, ''h'', ''k'', ''l'', called '''Miller indices''' of the face. This definition can be extended to any reticular plane whose intercepts with the basis vectors are ''OP = C a/h'', ''OQ = C b/k'', and ''OR = C c/l'' (see Fig. 1). When the lattice is indexed with respect to a [[primitive basis]], ''h'', ''k'', ''l'' are relatively prime integers (''i''.''e''. not having a common factor other than +1 or -1); this restriction does not hold when a [[Centred_lattices|centred unit cell]] is chosen. ''C'' is an integer which, when made variable, identifies a specific lattice plane of the infinite set having the same orientation with respect to the basis vector. This infinite set of planes defines a ''family'' of lattice planes, denoted by the Miller indices in parentheses: (''hkl''). The Miller indices of the equivalent faces of a [[crystal form]] are denoted by {''hkl''}.
-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:
+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''
+''hx'' + ''ky'' + ''lz'' = ''C''.
 </center>
@@ Line 17: / Line 16: @@
 === 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'' = ''1''/''d'' where ''d'' is the lattice spacing of the family. When a centred unit cell is used in direct space, [[integral reflection conditions]] are observed in the reciprocal space which correspond to the non relatively prime Miller indices of the family of lattice planes.
+The [[reciprocal lattice]] vector associated with 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'' = 1/''d'' where ''d'' is the lattice spacing of the family. When a centred unit cell is used in direct space, [[integral reflection conditions]] are observed in the reciprocal space which correspond to the non relatively prime Miller indices of the family of lattice planes.
 === [[Bravais-Miller indices]]  (hexagonal axes) ===
-In the case of an hexagonal lattice, one uses four axes, '''a<sub>1</sub>''', '''a<sub>2</sub>''', '''a<sub>3</sub>''', '''c''' and four indices, (''hkil''), called [[Bravais-Miller indices]], where ''h'', ''k'', ''i'', ''l'' are again inversely proportional to the intercepts of a plane of the family with the four axes. The indices ''h'', ''k'', ''i'' are cyclically permutable and are related by
+In the case of a hexagonal lattice, one uses four axes, '''a<sub>1</sub>''', '''a<sub>2</sub>''', '''a<sub>3</sub>''', '''c''' and four indices, (''hkil''), called [[Bravais-Miller indices]], where ''h'', ''k'', ''i'', ''l'' are again inversely proportional to the intercepts of a plane of the family with the four axes. The indices ''h'', ''k'', ''i'' are cyclically permutable and are related by
 <center>
-''h'' + ''k'' + ''i'' = 0
+''h'' + ''k'' + ''i'' = 0.
 </center>
@@ Line 33: / Line 32: @@
 === Rhombohedral crystals ===
-The Miller indices ''h<sub>R</sub>'', ''k<sub>R</sub>'', ''l<sub>R</sub>'' referred to rhombohedral axes are related to the corresponding indices, ''h<sub>H</sub>'', ''k<sub>H</sub>'', ''i<sub>H</sub>'',''l<sub>H</sub>'' reffered to hexagonal axes by:
+The Miller indices ''h<sub>R</sub>'', ''k<sub>R</sub>'', ''l<sub>R</sub>'' referred to rhombohedral axes are related to the corresponding indices, ''h<sub>H</sub>'', ''k<sub>H</sub>'', ''i<sub>H</sub>'', ''l<sub>H</sub>'' referred to hexagonal axes by:
 <table align=center>
@@ Line 52: / Line 51: @@
    </tr>
 <tr>
-      <td> ''l<sub>H</sub>'' = ''h<sub>R</sub>'' + ''k<sub>R</sub>'' + ''l<sub>R</sub>''</td>
+      <td> ''l<sub>H</sub>'' = ''h<sub>R</sub>'' + ''k<sub>R</sub>'' + ''l<sub>R</sub>''.</td>
       <td>  </td>
       <td> </td>
@@ Line 65: / Line 64: @@
 [[Image:MILLER-2.gif|center]]
-The Miller indices of the planes <font color="green">''ABC' '' </font>, <font color="red">''ABC''</font>, <font color="blue">''ABC" ''</font>, <font color="purple"> ''AA"BB" '' </font> are <font color="green">(112) </font>, <font color="red">(111)</font>, <font color="blue">(221)</font>, <font color="purple"> (110)</font>, respectively. These planes have <font color="green">''AB'' </font>, or <math> [1{\bar 1}0]</math>, as common zone axis.
+The Miller indices of the planes <font color="green">''ABC'''</font>, <font color="red">''ABC''</font>, <font color="blue">''ABC" ''</font>, <font color="purple"> ''AA"BB" ''</font> are <font color="green">(112)</font>, <font color="red">(111)</font>, <font color="blue">(221)</font>, <font color="purple"> (110)</font>, respectively. These planes have <font color="green">''AB'' </font>, or <math> [1{\bar 1}0]</math>, as common zone axis.
 == History ==
-The Miller indices were first introduced, among others, by W. Whewell in 1829 and developed by W.H. Miller, his successor at the Chair of Mineralogy at Cambridge University, in his book ''A treatise on Crystallography'' (1839) - see ''Historical Atlas of Crystallography'' (1990), edited by J. Lima de Faria, published for the ''International Union of Crystallography'' by Kluwer Academic Publishers, Dordrecht.
+The first to introduce indices to denote a crystal plane was C. S. Weiss. His notation was modified independently by his student F. E. Neumann and W. Whewell whose indices are the inverse of the [[Weiss_parameters|Weiss]] indices. These indices were systematically used by Whewell's
 == See also ==
-*[[law of rational indices]]
+*[[Law of rational indices]]
 *[[Laue indices]]
-*[[reciprocal lattice]]
+*[[Reciprocal lattice]]
 *[[Weiss parameters]]
 *[http://journals.iucr.org/j/issues/2015/04/00/gj5135/ Discussion about non-coprime Miller indices]

@@ Line 6: / Line 6: @@
 === Direct space ===
-The [[law of rational indices]] states that the intercepts, ''OP'', ''OQ'', ''OR'', of the natural faces of a crystal form with the basis vectors '''OA''' = '''a''', '''OB''' = '''b''', and '''OC''' = '''c''' are inversely proportional to prime integers, ''h'', ''k'', ''l'', called '''Miller indices''' of the face. This definition can be extended to any reticular plane whose intercepts with the basis vectors are ''OP = C a/h'', ''OQ = C b/k'', and ''OR = C c/l'' (see Figure 1). When the lattice is indexed with respect to a [[primitive basis]], ''h'', ''k'', ''l'' are co-prime integers; this restriction does not hold instead when a [[Centred_lattices|centred unit cell]] is chosen. ''C'' is an integer which, when made variable, identifies a specific lattice plane of the infinite set having the same orientation with respect to the basis vector. This infinite set of planes defines a ''family'' of lattice planes, denoted by the Miller indices in parentheses: (''hkl''). The Miller indices of the equivalent faces of a [[crystal form]] are denoted by {''hkl''}.
+The [[law of rational indices]] states that the intercepts, ''OP'', ''OQ'', ''OR'', of the natural faces of a crystal form with the basis vectors '''OA''' = '''a''', '''OB''' = '''b''', and '''OC''' = '''c''' are inversely proportional to prime integers, ''h'', ''k'', ''l'', called '''Miller indices''' of the face. This definition can be extended to any reticular plane whose intercepts with the basis vectors are ''OP = C a/h'', ''OQ = C b/k'', and ''OR = C c/l'' (see Figure 1). When the lattice is indexed with respect to a [[primitive basis]], ''h'', ''k'', ''l'' are co-prime integers; this restriction does not hold when a [[Centred_lattices|centred unit cell]] is chosen. ''C'' is an integer which, when made variable, identifies a specific lattice plane of the infinite set having the same orientation with respect to the basis vector. This infinite set of planes defines a ''family'' of lattice planes, denoted by the Miller indices in parentheses: (''hkl''). The Miller indices of the equivalent faces of a [[crystal form]] are denoted by {''hkl''}.
 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: