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I've recently been doing a spot of self-learning (Crystallography), and some of the examples provided in the Wikipedia article for Miller Indices have stumped me.

I can't for the world figure out how they were obtained!


Quick introduction to Miller Indices (the basic stuff):

Including this here, because it was what I was referring to when learning about this stuff

Miller Indices (hkl)

The orientation of a surface or a crystal plane may be defined by considering how the plane (or indeed any parallel plane) intersects the main crystallographic axes of the solid. The application of a set of rules leads to the assignment of the Miller Indices , (hkl) ; a set of numbers which quantify the intercepts and thus may be used to uniquely identify the plane or surface.

The following treatment of the procedure used to assign the Miller Indices is a simplified one (it may be best if you simply regard it as a "recipe") and only a cubic crystal system (one having a cubic unit cell with dimensions $a$ x $a$ x $a$ ) will be considered.

enter image description here

The procedure is most easily illustrated using an example so we will first consider the following surface/plane:

enter image description here

  • Step 1 : Identify the intercepts on the x- , y- and z- axes.

In this case the intercept on the x-axis is at x = a ( at the point (a,0,0) ), but the surface is parallel to the y- and z-axes - strictly therefore there is no intercept on these two axes but we shall consider the intercept to be at infinity ( ∞ ) for the special case where the plane is parallel to an axis. The intercepts on the x- , y- and z-axes are thus: $a$ , $∞$ , $∞$ (in that order).

  • Step 2 : Specify the intercepts in fractional co-ordinates

Co-ordinates are converted to fractional co-ordinates by dividing by the respective cell-dimension - for example, a point (x,y,z) in a unit cell of dimensions $a$ x $b$ x $c$ has fractional co-ordinates of ( x/a , y/b , z/c ). In the case of a cubic unit cell each co-ordinate will simply be divided by the cubic cell constant , $a$.

This gives: $a/a$ , $∞/a$, $∞/a$ ; i.e- $1$ , $∞$ , $∞$.

  • Step 3 : Take the reciprocals of the fractional intercepts

This final manipulation generates the Miller Indices which (by convention) should then be specified without being separated by any commas or other symbols. The Miller Indices are also enclosed within standard brackets (…) when one is specifying a unique surface such as that being considered here.

The reciprocals of 1 and ∞ are 1 and 0 respectively, thus yielding the Miller Indices ($100$)

So the surface/plane illustrated is the ($100$) plane of the cubic crystal.


Further on, it is mentioned:

If any of the intercepts are at negative values on the axes then the negative sign will carry through into the Miller indices; in such cases the negative sign is actually denoted by overstriking (i.e- adding a little bar/dash over) the relevant number.


Now I was going to through the Wiki article on Miller Indices (linked earlier on in this post), and the following examples were provided:

enter image description here

The corresponding Miller Indices have been provided in the boxes right under each entry. Also, the little "T"s are actually "1"s with a bar on top... kinda small, so it's easy to see it wrong.

Now the first six entries are pretty straightforward (I count left-right), but it's the others that are confusing.


Take for example, the seventh entry:

enter image description here

Okay, the third axis ($a_3$) lies along orange-colored plane, so the plane has no intercept with the ($a_3$) axis (i.e- intercept here is taken to be $∞$, as the orange plane and the axis in question are parallel). Which would naturally result in the Miller Index $0$ (third digit) which is the reciprocal of $∞$.

Alright, no issues there.

But what's the deal with the intercepts on the the first two axes ( $a_1$ and $a_2$ )?

The "intercept at infinity" logic doesn't work here (since the orange plane isn't parallel to $a_1$ or $a_2$).

According to the (box under the) entry, the orange plane makes an intercept of $1$ (second digit) with the $a_2$ axis. Moreover, it also (apparently) makes an intercept of $-1$ (first digit) with the $a_3$ axis.

This is incredible...I am totally lost.

The eighth and ninth entries, too, are made in the same vein (well, obviously... because the seventh, eighth and ninth entries are essentially the same thing)

enter image description here


My question:

How exactly, are the Miller Indices for the seventh entry (the one with the orange plane), obtained?


I've just asked for the rationale behind the Miller Indices of the seventh entry because I can easily work out the Indices for the eighth and ninth entries after that.

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    $\begingroup$ I'm still relatively new to Miller indices and I can't say I have an answer for a general procedure, but the Miller indices define an infinite family of planes. So while the particular plane shown for the $7^\text{th}$ entry doesn't have those intercepts, you can picture shifting that plane up to a location where it doesn't intercept $a_3$, intercepts $a_1$ at $-1$, and intercepts $a_2$ at $1$. The same sort of shifting will give you the correct intercepts for 8 and 9. $\endgroup$ – Tyberius Sep 20 '17 at 17:52
  • $\begingroup$ a similar question from physics.SE: physics.stackexchange.com/questions/62955/… $\endgroup$ – marcin Sep 20 '17 at 17:55
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IFF I remember correctly...

Sometimes you have to "move" the plane, in order to see where the axis crosses.

Take example 7, if you look at the example plane, the one actually shown, it is (0 ∞ ∞) not very useful. The ones making the examples have "moved the planes" (not really, but it is a useful fiction) in order to get the plane into the unit cell so they can show it. If they hadn't moved it, it wouldn't be visible.

The plane (-110) crosses $a_2$ axis at $a_2$ = $1$ and you see, using your spatial imagination, that it then needs to cross $a_1$ at $-1$ as well. It will never cross the $a_3$ axis, so $0$. The 8th and 9th are much harder to visualize but it is basically the same thing over again. The planes won't fit into the unit cell unless you "move" them.

Disclaimer: the (hkl) represents the set of all planes of the same err... "orientation", so it is technically not moving anything at all, it is merely choosing a representation of that said plane. If two planes in a certain crystal are equivalent, they can be represented with curly brackets. The sets (101),(010) and (001) are equivalent in a cubic crystal and can be represented collectively as {100}. Source: Basic Solid State Chemistry, A.R West 2nd ed. (1999)

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