<p>This is not a four- digit number but four separare numbers called <strong>Bravais-Miller indices</strong>.  Bravais-Miller indices descrive the orientation of a crystal plane relative to the symmetry axes of a crystal, as described in <a href="https://en.m.wikipedia.org/wiki/Miller_index" rel="nofollow noreferrer">Wikipedia</a>.  Put very briefly, a zero index means the plane is parallel to that axis, while nonzero indices encode the relative values of the intercepts on axes to which the plane is not parallel.  These pieces of information about parallel and nonparallel axes, together, specify the orientation of any plane that passes through the lattice points of a crystal.</p>
<p>In cubic crystals there are three principal, fourfold axes and thus you render three (not two) numbers for the Bravais-Miller indices of a plane; this is adapted to most other crystal systems.  But, exceptionally for hexagonal and rhombohedral systems, there are four symmetry axes; three of them in the hexagonally packed basal plane and the fourth perpendicular to them along the sixfold or theefold rotational axis.  Thus four indices are commonly used for that case.  The illustration below (<a href="https://www.researchgate.net/figure/a-Single-crystal-X-ray-diffractometer-results-of-Al-2-O-3-surfaces-with-miller-indices_fig6_303392726" rel="nofollow noreferrer">Source</a>) gives some commonly encountered planes in hexagonal crystals.</p>
<p><img src="https://i.sstatic.net/eWmyP.png" alt="enter image description here" /></p>
<p>Putting the first three axes into the same plane means the intercept designations on these axes must be dependent on each other; in fact those three numbers should add up to zero.  Thus <span class="math-container">$[1010]$</span> is not correct; it should be <span class="math-container">$[10(-1)0]$</span> or using the more common notation for a negative index, <span class="math-container">$[10\overline10]$</span>.  As shown in the above illustration, physically this corresponds to the lateral face of a hexagonal prism.</p>

CC BY-SA 4.0
2022-10-08T00:24:26.737Z
My understanding is that even in this crystal system three indices is sufficient mathematically but the fourth is added for convenience, e.g. "Thus four indices are commonly used for that case." Have I got that right? Also, an outstanding (as in *currently unanswered*) question in Matter Modeling SE: [Straightforward formalism to get four sets of 2D hexagonal lattice vectors of fcc(111) planes that I can also cite?](https://mattermodeling.stackexchange.com/q/6364/201)
uhoh
CC BY-SA 4.0
2022-10-08T00:32:21.970Z
Yes. Four indices are used for hexagonal crystals for convenience. Three would have been enough mathematically. In contrast, with the advent of quasicrystals we now have some crystals that mathematically require as many as six indices!
Oscar Lanzi
CC BY-SA 4.0
2022-10-08T00:41:44.363Z
in that case you may also find interesting [Proposing a 2D quasicrystal; what are the necessary and sufficient conditions? (If it looks like a duck and quacks like a duck, or...?)](https://mattermodeling.stackexchange.com/q/6993/201)
uhoh
CC BY-SA 4.0
2021-01-10T11:11:01.123Z
2021-01-16T19:39:50.647Z
2021-01-16T19:39:50.647Z
Oscar Lanzi
<p>On Wikipedia's page for apatite, the crystal cleavage is listed as [0001] or [0101], which I assume are Miller indices....</p>
<p>But how can there be four numbers/digits instead of two, like there usually are?</p>

<geology><mineralogy><minerals><crystallography><crystals>
How can the 'crystal cleavage' of apatite have a four-digit number?
Kurt Hikes
# How can the 'crystal cleavage' of apatite have a four-digit number?

(Asked by: username_0 on 2021-01-08)

On Wikipedia's page for apatite, the crystal cleavage is listed as [0001] or [0101], which I assume are Miller indices....

But how can there be four numbers/digits instead of two, like there usually are?

**Tags:** geology, mineralogy, minerals, crystallography, crystals


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## 1 answers

### Answer by username_1 on 2021-01-10. Score: 7

This is not a four- digit number but four separare numbers called **Bravais-Miller indices**. Bravais-Miller indices descrive the orientation of a crystal plane relative to the symmetry axes of a crystal, as described in [Wikipedia](https://en.m.wikipedia.org/wiki/Miller_index). Put very briefly, a zero index means the plane is parallel to that axis, while nonzero indices encode the relative values of the intercepts on axes to which the plane is not parallel. These pieces of information about parallel and nonparallel axes, together, specify the orientation of any plane that passes through the lattice points of a crystal.

In cubic crystals there are three principal, fourfold axes and thus you render three (not two) numbers for the Bravais-Miller indices of a plane; this is adapted to most other crystal systems. But, exceptionally for hexagonal and rhombohedral systems, there are four symmetry axes; three of them in the hexagonally packed basal plane and the fourth perpendicular to them along the sixfold or theefold rotational axis. Thus four indices are commonly used for that case. The illustration below ([Source](https://www.researchgate.net/figure/a-Single-crystal-X-ray-diffractometer-results-of-Al-2-O-3-surfaces-with-miller-indices_fig6_303392726)) gives some commonly encountered planes in hexagonal crystals.

![enter image description here](https://i.sstatic.net/eWmyP.png)

Putting the first three axes into the same plane means the intercept designations on these axes must be dependent on each other; in fact those three numbers should add up to zero. Thus $[1010]$ is not correct; it should be $[10(-1)0]$ or using the more common notation for a negative index, $[10\overline10]$. As shown in the above illustration, physically this corresponds to the lateral face of a hexagonal prism.

> **Comments:**
> **username_2**: My understanding is that even in this crystal system three indices is sufficient mathematically but the fourth is added for convenience, e.g. "Thus four indices are commonly used for that case." Have I got that right? Also, an outstanding (as in *currently unanswered*) question in Matter Modeling SE: [Straightforward formalism to get four sets of 2D hexagonal lattice vectors of fcc(111) planes that I can also cite?](https://mattermodeling.stackexchange.com/q/6364/201)  
> **username_1**: Yes. Four indices are used for hexagonal crystals for convenience. Three would have been enough mathematically. In contrast, with the advent of quasicrystals we now have some crystals that mathematically require as many as six indices!  
> **username_2**: in that case you may also find interesting [Proposing a 2D quasicrystal; what are the necessary and sufficient conditions? (If it looks like a duck and quacks like a duck, or...?)](https://mattermodeling.stackexchange.com/q/6993/201)  

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