# 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 **A**_{1}, **A**_{2}, **A**_{3}, **C** be the four hexagonal axes, written as capital letters 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 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>(*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**

_{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.