Weber indices

From Online Dictionary of Crystallography

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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 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.