Weber indices - Revision history

BrianMcMahon: Tidied translations and added German (U. Mueller)

2017-11-20T14:46:24Z

Tidied translations and added German (U. Mueller)

BrianMcMahon: Style edits to align with printed edition

2017-05-20T11:24:43Z

Style edits to align with printed edition

MassimoNespolo: LF missing

2017-04-02T15:54:31Z

LF missing

MassimoNespolo: extension

2016-06-12T15:25:40Z

extension

MassimoNespolo: Created page with "Indices de Weber (''Fr''). Índices de Weber (''Sp''). Indici di Weber (''It'').
2015-02-26T07:14:08Z

Created page with "Indices de Weber (''Fr''). Índices de Weber (''Sp''). Indici di Weber (''It''). Indices de Weber (''Fr''). Índices de Weber
 (''Sp''). Indici di Weber (''It''). ウェーバー指数　(''Ja'')

For trigonal and hexagonal crystals, the [[Miller indices]] are conveniently replaced by the [[Bravais-Miller indices]] which make reference to a four-axes setting. For lattice directions, a similar extension to a four-axes axial setting exists, known as the '''Weber indices'''.

Let '''A'''1,'''A'''2,'''A'''3,'''C''' be the four hexagonal axes, written as capital letter to avoid any possible confusion with the rhombohedral axes '''a'''1,'''a'''2,'''a'''3, and let be ''uvw'' and ''UVTW'' the indices of a direction with respect to '''A'''1,'''A'''2,'''C''' or '''A'''1,'''A'''2,'''A'''3,'''C''' respectively. For a given direction the following identity must hold:

''u'''''A'''1 + ''v'''''A'''2 + ''w'''''C''' = ''U'''''A'''1 + ''V'''''A'''2 + ''T'''''A'''3 + ''W'''''C'''.

Now, because in the two-dimensional (001) plane only two of the three axes are linearly independent, the following identity can be established:

'''A'''1 + '''A'''2 + '''A'''3 = 0 --> '''A'''3 = -('''A'''1 + '''A'''2).

A similar relation holds for the Weber indices:

''U'' + ''V'' + ''T'' = 0.

Substituting the above identities, one immediately gets:

''u'''''A'''1 + ''v'''''A'''2 + ''w'''''C''' = ''U'''''A'''1 + ''V'''''A'''2 - T('''A'''1 + '''A'''2 ) + ''W'''''C'''

''u'''''A'''1 + ''v'''''A'''2 + ''w'''''C''' = (''U''-''T'')'''A'''1 + (''V''-''T'')'''A'''2 + ''W'''''C'''

''u'' = ''U''-''T''; ''v'' = ''V''-''T''; ''w'' = ''W''
''U'' + ''V'' + ''T'' = 0 --> ''T'' = -(''U''+''V'')

so that:

''u'' = 2''U''+''V'';
''v'' = ''U''+2''V'';
''w'' = ''W''

To find the opposite relations, one has simply to subtract the second equation from the first multiplied by two and vice versa:

2''u''-''v'' = 3''U'' --> ''U'' = (2''u''-''v'')/3 
-''u''+2''v'' = 3''V'' --> ''V'' = (2''v''-''u'')/3 
''T'' = -(''U''+''V'') = -(''u''+''v'')/3.

The addition of the '''A'''3 axis modifies the indices ''u'' and ''v'', which become ''U'' and ''V'', and the relation ''T'' = -''U''-''V'' holds for ''U'' and ''V'' but not for ''u'' and ''v'', whereas for the [[Bravais-Miller indices]] the addition of the third axis does not modify ''h'' and ''k'' so that the relation ''i'' = -''h''-''k'' is applied directly. For this reason, the [[Bravais-Miller indices]] are widely used in crystallography, whereas the Weber indices are more used in fields like electron microscopy and metallurgy but seldom in crystallography.

[[Category: Fundamental crystallography]] 
[[Category: Morphological crystallography]]

@@ Line 1: / Line 1: @@
-<font color="blue">Indices de Weber</font> (''Fr''). <font color="green">Índices de Weber
+<font color="blue">Indices de Weber</font> (''Fr''). <font color="red">Weber-Indizes</font> (''Ge''). <font color="black">Indici di Weber</font> (''It''). <Font color="purple">ウェーバー指数</font> (''Ja''). <font color="green">Índices de Weber</font> (''Sp'').
-</font> (''Sp''). <font color="black"> Indici di Weber</font> (''It''). <Font color="purple"> ウェーバー指数</font> (''Ja'').
 For trigonal and hexagonal crystals, the [[Miller indices]] are conveniently replaced by the [[Bravais-Miller indices]] which make reference to a four-axes setting. For lattice directions, a similar extension to a four-axes axial setting exists, known as the '''Weber indices''', ''UVTW''.

@@ Line 1: / Line 1: @@
 <font color="blue">Indices de Weber</font> (''Fr''). <font color="green">Índices de Weber
-</font> (''Sp''). <font color="black"> Indici di Weber</font> (''It''). <Font color="purple"> ウェーバー指数</font>　(''Ja'')
+</font> (''Sp''). <font color="black"> Indici di Weber</font> (''It''). <Font color="purple"> ウェーバー指数</font> (''Ja'').
 For trigonal and hexagonal crystals, the [[Miller indices]] are conveniently replaced by the [[Bravais-Miller indices]] which make reference to a four-axes setting. For lattice directions, a similar extension to a four-axes axial setting exists, known as the '''Weber indices''', ''UVTW''.
-Let '''A'''<sub>1</sub>,'''A'''<sub>2</sub>,'''A'''<sub>3</sub>,'''C''' be the four hexagonal axes, written as capital letter to avoid any possible confusion with the rhombohedral axes '''a'''<sub>1</sub>,'''a'''<sub>2</sub>,'''a'''<sub>3</sub>, and let be ''uvw'' and ''UVTW'' the indices of a direction with respect to '''A'''<sub>1</sub>,'''A'''<sub>2</sub>,'''C''' or '''A'''<sub>1</sub>,'''A'''<sub>2</sub>,'''A'''<sub>3</sub>,'''C''' respectively. For a given direction the following identity must hold:
+Let '''A'''<sub>1</sub>, '''A'''<sub>2</sub>, '''A'''<sub>3</sub>, '''C''' be the four hexagonal axes, written as capital letters to avoid any possible confusion with the rhombohedral axes '''a'''<sub>1</sub>, '''a'''<sub>2</sub>, '''a'''<sub>3</sub>, and let be ''uvw'' and ''UVTW'' the indices of a direction with respect to '''A'''<sub>1</sub>, '''A'''<sub>2</sub>, '''C''' or '''A'''<sub>1</sub>, '''A'''<sub>2</sub>,'''A'''<sub>3</sub>, '''C''' respectively. For a given direction the following identity must hold:
 ''u'''''A'''<sub>1</sub> + ''v'''''A'''<sub>2</sub> + ''w'''''C''' = ''U'''''A'''<sub>1</sub> + ''V'''''A'''<sub>2</sub> + ''T'''''A'''<sub>3</sub> + ''W'''''C'''.
@@ Line 10: / Line 10: @@
 Now,  because in the two-dimensional (001) plane only two of the three axes are linearly independent, the following identity can be established:
-'''A'''<sub>1</sub> + '''A'''<sub>2</sub> + '''A'''<sub>3</sub> = 0 --> '''A'''<sub>3</sub> = -('''A'''<sub>1</sub> + '''A'''<sub>2</sub>).
+'''A'''<sub>1</sub> + '''A'''<sub>2</sub> + '''A'''<sub>3</sub> = 0 &rarr; '''A'''<sub>3</sub> = -('''A'''<sub>1</sub> + '''A'''<sub>2</sub>).
 A similar relation holds for the Weber indices:
@@ Line 24: / Line 24: @@
 ''u'' = ''U''-''T''; ''v'' = ''V''-''T''; ''w'' = ''W''
-''U'' + ''V'' + ''T'' = 0 --> ''T'' = -(''U''+''V'')
+''U'' + ''V'' + ''T'' = 0 &rarr; ''T'' = -(''U''+''V'')
 so that:
@@ Line 34: / Line 34: @@
 To find the opposite relations, one has simply to subtract the second equation from the first multiplied by two and vice versa:
-''u''-''v'' = 3''U'' --> ''U'' = (2''u''-''v'')/3<br>
+''u''-''v'' = 3''U'' &rarr; ''U'' = (2''u''-''v'')/3<br>
--''u''+2''v'' = 3''V'' --> ''V'' = (2''v''-''u'')/3<br>
+-''u''+2''v'' = 3''V'' &rarr; ''V'' = (2''v''-''u'')/3<br>
 ''T'' = -(''U''+''V'') = -(''u''+''v'')/3.

@@ Line 2: / Line 2: @@
 </font> (''Sp''). <font color="black"> Indici di Weber</font> (''It''). <Font color="purple"> ウェーバー指数</font>　(''Ja'')
-For trigonal and hexagonal crystals, the [[Miller indices]] are conveniently replaced by the [[Bravais-Miller indices]] which make reference to a four-axes setting. For lattice directions, a similar extension to a four-axes axial setting exists, known as the '''Weber indices'''.
+For trigonal and hexagonal crystals, the [[Miller indices]] are conveniently replaced by the [[Bravais-Miller indices]] which make reference to a four-axes setting. For lattice directions, a similar extension to a four-axes axial setting exists, known as the '''Weber indices''', ''UVTW''.
 Let '''A'''<sub>1</sub>,'''A'''<sub>2</sub>,'''A'''<sub>3</sub>,'''C''' be the four hexagonal axes, written as capital letter to avoid any possible confusion with the rhombohedral axes '''a'''<sub>1</sub>,'''a'''<sub>2</sub>,'''a'''<sub>3</sub>, and let be ''uvw'' and ''UVTW'' the indices of a direction with respect to '''A'''<sub>1</sub>,'''A'''<sub>2</sub>,'''C''' or '''A'''<sub>1</sub>,'''A'''<sub>2</sub>,'''A'''<sub>3</sub>,'''C''' respectively. For a given direction the following identity must hold:
@@ Line 37: / Line 37: @@
 ''T'' = -(''U''+''V'') = -(''u''+''v'')/3.
-The addition of the '''A'''<sub>3</sub> axis modifies the indices ''u'' and ''v'', which become ''U'' and ''V'', and the relation ''T'' = -''U''-''V'' holds for ''U'' and ''V'' but not for ''u'' and ''v'', whereas for the [[Bravais-Miller indices]] the addition of the third axis does not modify ''h'' and ''k'' so that the relation ''i'' = -''h''-''k'' is applied directly. For this reason, the [[Bravais-Miller indices]] are widely used in crystallography, whereas the Weber indices are more used in fields like electron microscopy and metallurgy but seldom in crystallography.
+The Weber indices of the direction perpendicular to a lattice plane are the same as the Bravais-Miller indices of that plane.
 [[Category: Fundamental crystallography]]<br>
 [[Category: Morphological crystallography]]

← Older revision		Revision as of 15:54, 2 April 2017
Line 23:		Line 23:

	''u'' = ''U''-''T''; ''v'' = ''V''-''T''; ''w'' = ''W''		''u'' = ''U''-''T''; ''v'' = ''V''-''T''; ''w'' = ''W''
		+
	''U'' + ''V'' + ''T'' = 0 --> ''T'' = -(''U''+''V'')		''U'' + ''V'' + ''T'' = 0 --> ''T'' = -(''U''+''V'')