Difference between revisions of "Weber indices"
From Online Dictionary of Crystallography
(Created page with "<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="pu...") |
BrianMcMahon (talk | contribs) (Tidied translations and added German (U. Mueller)) |
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| − | <font color="blue">Indices de Weber</font> (''Fr''). <font color=" | + | <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> ('' | ||
| − | |||
| − | Let '''A'''<sub>1</sub>,'''A'''<sub>2</sub>,'''A'''<sub>3</sub>,'''C''' be the four hexagonal axes, written as capital | + | 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 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'''. | ''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'''. | ||
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Now, because in the two-dimensional (001) plane only two of the three axes are linearly independent, the following identity can be established: | 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>1</sub> + '''A'''<sub>2</sub> + '''A'''<sub>3</sub> = 0 → '''A'''<sub>3</sub> = -('''A'''<sub>1</sub> + '''A'''<sub>2</sub>). |
A similar relation holds for the Weber indices: | A similar relation holds for the Weber indices: | ||
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''u'' = ''U''-''T''; ''v'' = ''V''-''T''; ''w'' = ''W'' | ''u'' = ''U''-''T''; ''v'' = ''V''-''T''; ''w'' = ''W'' | ||
| − | ''U'' + ''V'' + ''T'' = 0 | + | |
| + | ''U'' + ''V'' + ''T'' = 0 → ''T'' = -(''U''+''V'') | ||
so that: | so that: | ||
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To find the opposite relations, one has simply to subtract the second equation from the first multiplied by two and vice versa: | 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'' | + | 2''u''-''v'' = 3''U'' → ''U'' = (2''u''-''v'')/3<br> |
| − | -''u''+2''v'' = 3''V'' | + | -''u''+2''v'' = 3''V'' → ''V'' = (2''v''-''u'')/3<br> |
''T'' = -(''U''+''V'') = -(''u''+''v'')/3. | ''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. |
| + | |||
| + | <table border cellspacing=0 cellpadding=5 align=center> | ||
| + | <tr align=center> | ||
| + | <th=3> Miller indices</th> | ||
| + | <th>Bravais-Miller indices</th> | ||
| + | <th>Indices of the perpendicular direction</th> | ||
| + | <th>Weber indices of the perpendicular direction</th> | ||
| + | </tr> | ||
| + | <tr align=center> | ||
| + | <td=2>(001)</td> | ||
| + | <td>(0001)</td> | ||
| + | <td>[001]</td> | ||
| + | <td>[0001]</td> | ||
| + | </tr> | ||
| + | <tr align=center> | ||
| + | <td>(''hk''0)</td> | ||
| + | <td>(''hki''0)</td> | ||
| + | <td>[2''h''+''k'',''k''+2''k'',0]</td> | ||
| + | <td>[''hki''0]</td> | ||
| + | </tr> | ||
| + | <tr align=center> | ||
| + | <td>(100)</td> | ||
| + | <td><math>(10{\bar 1}0)</math></td> | ||
| + | <td>[210]</td> | ||
| + | <td><math>[10{\bar 1}0]</math></td> | ||
| + | </tr> | ||
| + | <tr align=center> | ||
| + | <td><math>(2{\bar 1}0)</math></td> | ||
| + | <td><math>(2{\bar 1}{\bar 1}0)</math></td> | ||
| + | <td>[100]</td> | ||
| + | <td><math>[2{\bar 1}{\bar 1}0]</math></td> | ||
| + | </tr> | ||
| + | </table> | ||
| + | |||
| + | Despite the advantage of getting the same numerical indices for a plane ([[Bravais-Miller indices]]) and for the direction perpendicular to it (Weber indices), 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. | ||
[[Category: Fundamental crystallography]]<br> | [[Category: Fundamental crystallography]]<br> | ||
[[Category: Morphological crystallography]] | [[Category: Morphological crystallography]] | ||
Latest revision as of 14:46, 20 November 2017
Indices de Weber (Fr). Weber-Indizes (Ge). Indici di Weber (It). ウェーバー指数 (Ja). Índices de Weber (Sp).
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 A1, A2, A3, C be the four hexagonal axes, written as capital letters to avoid any possible confusion with the rhombohedral axes a1, a2, a3, and let be uvw and UVTW the indices of a direction with respect to A1, A2, C or A1, A2,A3, C respectively. For a given direction the following identity must hold:
uA1 + vA2 + wC = UA1 + VA2 + TA3 + WC.
Now, because in the two-dimensional (001) plane only two of the three axes are linearly independent, the following identity can be established:
A1 + A2 + A3 = 0 → A3 = -(A1 + A2).
A similar relation holds for the Weber indices:
U + V + T = 0.
Substituting the above identities, one immediately gets:
uA1 + vA2 + wC = UA1 + VA2 - T(A1 + A2 ) + WC
uA1 + vA2 + wC = (U-T)A1 + (V-T)A2 + WC
u = U-T; v = V-T; w = W
U + V + T = 0 → T = -(U+V)
so that:
u = 2U+V; v = U+2V; w = W
To find the opposite relations, one has simply to subtract the second equation from the first multiplied by two and vice versa:
2u-v = 3U → U = (2u-v)/3
-u+2v = 3V → V = (2v-u)/3
T = -(U+V) = -(u+v)/3.
The Weber indices of the direction perpendicular to a lattice plane are the same as the Bravais-Miller indices of that plane.