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...") |
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</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'''. | + | 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 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: | ||
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''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]] | ||
Revision as of 15:25, 12 June 2016
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, UVTW.
Let A1,A2,A3,C be the four hexagonal axes, written as capital letter 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.