From Online Dictionary of Crystallography
Revision as of 14:46, 20 November 2017 by BrianMcMahon (Tidied translations and added German (U. Mueller))
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)
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.