# Miller indices: How does one translates the orientation to the origin of the unit cell?

In a section discussing Miller indices, my textbook says the following:

The rules for determining the Miller indices of a direction or an orientation in a crystal are as follows: translate the orientation to the origin of the unit cell, and take the normalized coordinates of its other vertex.

What I'm confused about is how one translates the orientation to the origin of the unit cell?

Since the textbook just mentions this without elaboration or illustration, it is not clear precisely what this means or how one would go about it. I would appreciate it if people could please take the time to explain and demonstrate how this is (mathematically) done. Sources would be appreciated, if you have them.

• When you reach a confusing point like this, consult another book or source. I usually found it beneficial to read the same material from 2-3 books. Multiple sources gave me a better understanding of the topic being studied. – MaxW Jul 17 '19 at 15:53
• @MaxW I attempted to find clarification using a Google search before I posted this question. See the slide beginning Determine the Miller indices for the plane: www3.nd.edu/~amoukasi/CBE30361/Lecture__crystallography_B.pdf It's the same problem: The author mentions the process, but I don't see any explanation or illustration of how it is actually done. – The Pointer Jul 17 '19 at 16:01
• try reading this en.wikipedia.org/wiki/Miller_index – MaxW Jul 17 '19 at 16:23
• @MaxW I did, but there isn't even any mention of translations or "origin". – The Pointer Jul 17 '19 at 16:24
• @ThePointer Note that you may train yourself a bit, e.g. on doitpoms.ac.uk/tlplib/miller_indices/index.php – Buttonwood Jul 17 '19 at 20:53

Let's say your structure contains a short piece of helical DNA, and it (the helix axis) starts at point P1 and goes to point P2. You want to know the Miller index associated with that direction. So you move the entire helix (without rotating) so that P1 is now at the origin. The new position of P2, lets call it P2(prime), will be at:

$$P_2' = P_2 - P_1$$

Just checking that is is a translation (I have to subtract the same vector from the entire structure, otherwise I'm rotating or warping it):

$$P_1' = P_1 - P_1 = (0,0,0)$$

Again, the pair of two points still points in the same direction, but now $$P_1$$ is at the origin. Taking the fractional coordinates (i.e. along the unit cell axes) of $$P_2'$$, you can turn those into Miller indices $$h$$, $$k$$, and $$l$$ by multiplying with a large number and rounding to the next integer. Generally, the direction will be off because of the rounding. However, in special cases, e.g. if your DNA is on some crystallographic symmetry axis, the Miller indices will match your direction exactly.

• Could you clarify "and you have a Miller index. h, k, and l will not typically be integers unless your DNA is on some crystallographic symmetry axis.". Miller indices, by definition, must be integers. The direction doesn't need to be a symmetry axis to have (integer) Miller indices. – marcin Jul 18 '19 at 11:40
• @marcin I edited the answer. Here is an example: Let's say your fractional coordinates are (0.3147, 0.5324, 0.4143). Multiplying by 10000 tells me that the direction with Miller indices (3147, 5324, 4143) will point in just about the same direction (within sigfigs). You are unlikely to measure that reflection though. Multiplying by 32 gives (10.0704, 17.0368, 13.2576) so the Miller indices (10,17,13) have a similar direction to our direction of interest, and you might actually measure that reflection. – Karsten Theis Jul 18 '19 at 13:52