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Difference between revisions of "Weber indices"

From Online Dictionary of Crystallography

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<font color="blue">Indices de Weber</font> (''Fr''). <font color="green">Índices de Weber
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<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'')
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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''.  
 
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:
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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>3</sub> = -('''A'''<sub>1</sub> + '''A'''<sub>2</sub>).
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'''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:
 
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 --> ''T'' = -(''U''+''V'')  
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''U'' + ''V'' + ''T'' = 0 &rarr; ''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'' --> ''U'' = (2''u''-''v'')/3<br>
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2''u''-''v'' = 3''U'' &rarr; ''U'' = (2''u''-''v'')/3<br>
-''u''+2''v'' = 3''V'' --> ''V'' = (2''v''-''u'')/3<br>
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-''u''+2''v'' = 3''V'' &rarr; ''V'' = (2''v''-''u'')/3<br>
 
''T'' = -(''U''+''V'') = -(''u''+''v'')/3.
 
''T'' = -(''U''+''V'') = -(''u''+''v'')/3.
  

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 = 3UU = (2u-v)/3
-u+2v = 3VV = (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.

<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 align=center> <td=2>(001)</td> <td>(0001)</td> <td>[001]</td> <td>[0001]</td> </tr> <tr align=center> <td>(hk0)</td> <td>(hki0)</td> <td>[2h+k,k+2k,0]</td> <td>[hki0]</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 A3 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.