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: ]

 

Theory...

Hexagonal Miller plane indices to Miller-Bravais indices and back:
(hkl)⇔(hkil)

( h: k: l: )

( h: k: i: l: )

 

Theory...

Angle between two cubic directions:
[u1 v1 w1] ∠ [u2 v2 w2]

[ u1: v1: w1: ] [ u2: v2: w2: ]

cos(θ):  θ(°):  θ(rad): 

 

Theory...

Angle between two hexagonal directions:
[U1 V1 T1 W1] ∠ [U2 V2 T1 W2]

a: c:

[ U1: V1: T1: W1: ] [ U2: V2: T2: W2: ]

cos(θ): θ(°): θ(rad):

 

Theory...

Cubic lattice plane on which a given crystallographic direction is orthogonal:
(hkl) ⊥ [uvw]

( h: k: l: )

[ u: v: w: ]

 

Theory...

Approximate hexagonal lattice plane on which a given crystallographic direction is orthogonal:
(hkl) ⊥ [uvw]

a: c:

( h: k: l: )

[ u: v: w: ]

 

Theory...

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.