High School Notation for indexing a crystal direction

[etotheipi]

Given a crystal basis $\{\vec{a}, \vec{b}, \vec{c} \}$, the two lattice vectors $\vec{r}_1 = u_1 \vec{a} + u_2 \vec{b} + u_3 \vec{c}$ and $\vec{r}_2 = 2u_1\vec{a} + 2u_2 \vec{b} + 2u_3 \vec{c}$ both obviously point in the same direction whilst $\vec{r}_2$ is twice as long as $\vec{r}_1$. However, some people drop the common factors and index both as $[111]$, whilst others seem to keep the common factors and index them as $[111]$ and $2[111]$ respectively.

So, regarding this [...] notation, does it represent a direction or a vector?

 

[Twigg]

Those are (probably?) Miller indexes. If that's the case, [111] represents a plane perpendicular to your $\vec{r}_1$. Using the indexes to represent vectors and writing something like 2[111] seems like abuse of notation but I'm definitely no crystallographer.

https://en.wikipedia.org/wiki/Miller_index

 

[etotheipi]

Those are (probably?) Miller indexes. If that's the case, [111] represents a plane perpendicular to your $\vec{r}_1$. Using the indexes to represent vectors and writing something like 2[111] seems like abuse of notation but I'm definitely no crystallographer.

https://en.wikipedia.org/wiki/Miller_index

As far as I'm aware, planes are indexed by round brackets, e.g. $(hkl)$, whilst directions are indexed by square brackets, e.g. $[uvw]$. Usually it is the $hkl$ that are referred to as Miller indices.

But the $(hkl)$ plane is orthogonal to the reciprocal vector $h \vec{a}^* + k \vec{b}^* + l\vec{c}^*$. In a simple cubic system, the reciprocal basis equals the direct basis so it turns out that the $[hkl]$ direction in real space is orthogonal to the $(hkl)$ plane for a cubic system only.

 

[Twigg]

You're right, I mistakenly thought that [hkl] referred to a plane in the direct lattice, when it really refers to a direction in the direct lattice. Per your question, the wikipedia article seems insistent on [hkl] being a direction, not a vector. Personally, I can see why you might want to abuse notation and talk about [222] = 2[111] for something like, next-nearest neighbors effects. I think context is the key here.

 

[etotheipi]

I think I've found an answer! Essentially, a 'lattice vector' and a 'lattice direction (or zone axis)' are two slightly different concepts. The lattice vector is a true vector in the sense of linear algebra and is simply indexed with its components in the usual way, whilst a lattice direction is indexed by dividing through by common factors and given in the simplest whole number ratio:

The term zone axis, more specifically, refers to only the direction of a direct-space lattice vector. For example, since the [120] and [240] lattice vectors share a common direction, their orientations both correspond the [120] zone of the crystal. Just as a set of lattice-planes in direct-space corresponds to a reciprocal-lattice vector in the complementary-space of spatial-frequencies and momenta, a "zone" is defined as a set of reciprocal-lattice planes in frequency-space that corresponds to a lattice-vector in direct-space.

(https://en.wikipedia.org/wiki/Zone_axis)

Lattice directions are written the same way as lattice vectors, in the form [UVW]. The direction in which the lattice vector is pointing is the lattice direction. The difference between lattice directions and lattice vectors is that a lattice vector has a magnitude which can be shown by prefixing the lattice vector with a constant. By convention U, V and W are integers.

(https://www.doitpoms.ac.uk/tlplib/crystallography3/parameters.php)

Also an official reference here:
https://journals.aps.org/pre/authors/crystallographic-notation-h1