# 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 **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 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.