# Weber indices

### From Online Dictionary of Crystallography

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.

Miller indices | Bravais-Miller indices | Indices of the perpendicular direction | Weber indices of the perpendicular direction |
---|---|---|---|

(001) | (0001) | [001] | [0001] |

(hk0) |
(hki0) |
[2h+k,k+2k,0] |
[hki0] |

(100) | [210] | ||

[100] |

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.