This app was built to translate between Miller and Miller-Bravais indices, to calculate the angle between given directions and calculate the normal to a given plane for both cubic and hexagonal crystal structures.

The hexagonal system is more conveniently described by 4 basis vectors, 3 of which are co-planar and therefore, not linearly independent. Hence, the tendency to describe planes in hexagonal crystal using the Miller-Bravais system* (hkil) *where \( i=-(h+k)\) and can be omitted in writting *(hk.l)* .

However, when attempting crystallographic computations this system proves cumbersome and transforming to Miller independent indices becomes more convenient. For instance, a crystallographic direction described in Miller-Bravais indices as *[UVTW]* where \( T \equiv -(U+V)\)* *is a vector \(U\mathbf{a_1} + V\mathbf{a_2} + T\mathbf{a_3} + W\mathbf{c}\) in a system in which \(\mathbf{a_3} = -(\mathbf{a_1} + \mathbf{a_2})\). The same vector can be described in a frame defined by independent basis vectors \(\mathbf{a_1}, \mathbf{a_2}, \mathbf{c} \) as *[uvw]*:

\(U\mathbf{a_1} + V\mathbf{a_2} + T\mathbf{a_3} + W\mathbf{c} = u\mathbf{a_1} + v\mathbf{a_2} + w\mathbf{c} \)

## Hexagonal Miller direction indices to Miller-Bravais indices and back:

[uvw]⇔[UVTW]

[ **u:** ** v:** ** w:** ]

[ **U**: ** V:** ** T:** ** W:** ]

## Hexagonal Miller plane indices to Miller-Bravais indices and back:

(hkl)⇔(hkil)

( ** h: **** k: **** l: ** )

( ** h: **** k: **** i: **** l: ** )

## Angle between two cubic directions:

[u_{1} v_{1} w_{1}] ∠ [u_{2} v_{2} w_{2}]

[ **u _{1}:**

**v**

_{1}:**w**] [

_{1}:**u**

_{2}:**v**

_{2}:**w**]

_{2}:

**cos(θ):** ** θ(°):** ** θ(rad):**

## Angle between two hexagonal directions:

[U_{1} V_{1} T_{1} W_{1}] ∠ [U_{2} V_{2} T_{1} W_{2}]

**a**: **c**:

[ **U _{1}:**

**V**

_{1}:**T**

_{1}:**W**] [

_{1}:**U**

_{2}:**V**

_{2}:**T**

_{2}:**W**]

_{2}: ** **

**cos(θ):** **θ(°):** **θ(rad):**

## Cubic lattice plane on which a given crystallographic direction is orthogonal:

(hkl) ⊥ [uvw]

( **h:** **k: ** **l: ** )

[ **u: ** **v: ** **w: ** ]

## Approximate hexagonal lattice plane on which a given crystallographic direction is orthogonal:

(hkl) ⊥ [uvw]

**a**: **c**:

( **h:** ** k:** ** l:** )

[ ** u:** ** v:** ** w:** ]

This tool was written during an undergraduate summer project by Albes Koxhaj under the supervision of Dr Carol Trager-Cowan and with some help from Elena Pascal.