How to Read Hexagonal Crystal Directions and Planes (Miller-Bravais Indices)

If you’re anything like me, your introductory materials science class was going great until the prof switched from cubic Miller Indices to hexagonal Miller-Bravais indices.

Miller-Bravais indices are a 4-axis coordinate system for 3-dimensional crystals, based on the unit cell. This coordinate system is based on the 3-axis Miller index, but with an extra axis which is used for hexagonal crystals. The system can indicate directions or planes, and are often written as (hkil). Some common examples of Miller Indices on a hexagonal prism include $[11\bar{2}3]$ , the body diagonal; $[11\bar{2}0]$ , the face diagonal, and $(10\bar{1}0)$ , the face plane.

How can you have 4 axes in only 3 dimensions? Why would you cram 4 axes into 3 dimensions? How do you switch between the “easy” [hkl] index and the “hard” [hkil] notation? Keep reading for the answers to these questions!

Outline

Review of Miller Indices
Hexagonal Miller-Bravais Coordinate System for Directions
Hexagonal Miller-Bravais Coordinate System for Planes
Converting Vectors between Hexagonal Miller-Bravais and Regular Miller Indices
Converting Planes between Hexagonal Miller-Bravais and Regular Miller Indices
Why use Hexagonal Miller-Bravais instead of Regular Miller Indices?
Example Problems
Summary
References and Further Reading

Review of Miller Indices

I’ll put this in collapsible text for the upperclassmen who just need a quick refresher. If you need a full explanation of Miller indices, check out that article. Be sure you understand it, because Miller-Bravais indices build right on top of this foundation.

Click here to find out more.

Miller indices plot a direction or plane along 3 axes that correspond to the 3 lattice parameters of the crystal. A value of 1 means you have traveled the full distance of that lattice parameter. In cubic systems this is exactly like cartesian coordinates, but in other systems the axes may be different lengths and may not be perpendicular to each other.

The notation for Miller indices is

Square brackets $[hkl]$ for a specific direction. For example, in a cubic system $[100]$ and $[010]$ are perpendicular directions.
Angle brackets $\langle hkl \rangle$ indicate a family of directions. For example, in a cubic system $\langle 100 \rangle$ includes $[100]$ , $[\bar{1}00]$ , $[010]$ , $[0\bar{1}0]$ , $[001]$ , and $[00\bar{1}]$ .
Parenthesis $(hkl)$ indicate a specific plane. For example, in a cubic system $(100)$ and $(010)$ are perpendicular planes.
Curly brackets $\{hkl\}$ indicate a family of planes. For example, in a cubic system $\{100\}$ includes $(100)$ , $(\bar{1}00)$ , $(010)$ , $(0\bar{1}0)$ , $(001)$ , and $00\bar{1})$ .

“h,” “k,” and “l” are the distance along each lattice vector. For example, $[121]$ has h=1, k=2, and l=1. This means that the vector moves a distance “a” along the a vector, “2b” along the b vector, and “c” along the c vector. Alternatively, you could imagine this as ½ a along a, 1b along b, and ½ c along c.

For planes, you use reciprocal space. The $(hkl)$ of the plane is the inverse of the value where the plane intersects the lattice vectors. For example, $(102)$ intersects the a vector at a distance of “1a,” never intersects the b vector, and intersects the c vector at a distance of “½ c.”

Hexagonal Miller-Bravais Coordinate System for Directions

When we looked at hexagonal crystals in the regular Miller system, I showed that it was easier to imagine the primitive, hexahedral version of the hexagonal cell. For Miller-Bravais indices, I’ll go back to the conventional unit cell.

For Miller-Bravais indices, we need to label 4 axes in the hexagonal crystal. In the basal plane, we have 3 axes of equal length each separated by 120º, which we call a₁, a₂, and a₃ (they are each the same as the lattice parameter “a”). Then there is the c axis, perpendicular to those three.

Now, the question is, how do we go from a 3-axis system to a 4-axis system?

Let me put the math in collapsable text–those that aren’t interested can skip straight to the final formula.

Click here to find out more.

Let’s use “u,” ‘v,” and “w,” to represent the distance along the a₁, a₂, and c vectors in the 3-axis system. When we add a₃, we can use capital “U,” “V,” “T,” and “W” to represent the distance along the a₁, a₂, a₃, and c vectors.

If we want to translate from [uvw] to [UVTW], we need this equation to be true:

$ua_{1}+va_{2}+wc=Ua_{1}+Va_{2}+Ta_{3}+Wc$

We also know that $a_{1}+a_{2}+a_{3}$ will return to the starting point, so summing those vectors gives us zero. So

$a_{1}+a_{2}+a_{3}=0$ , or $a_{1}+a_{2}=-a_{3}$

We can then substitute this into the equation to get rid of a₃:

$Ua_{1}+Va_{2}+T(-a_{1}-a_{2})+Wc=(U-T)a_{1}+(V-T)a_{2}+Wc$

Thus we find:

$u= U-T$
$v = V-T$
$w = W$

However, this is still a problem, because we have too many unknown variables! There is an infinite set of solutions for this problem.

To reduce this to one “correct” solution, we need to introduce one more constraint. Just like $a_{1}+a_{2}+a_{3}=0$ , we can require $U + V + T = 0$ . If we do this, we get a unique solution which preserves index symmetry (see that section to find out why this is important).

We can rewrite u, v, and w without “T” because $T = -(U+V)$

$u=2U+V$
$v=2V+U$
$w=W$

This converts $[hkl]$ to $[hkil]$ , but what about the reverse? Rearranging a few terms gives us the answer.

$u=2U+V => U=\frac{u-V}{2}$
$v=2V+U => U=v-2V$

Therefore,

$\frac{u-V}{2}=v-2V$
$u-V=2v-4V$
$V=\frac{2v-u}{3}$

And don’t forget, we defined T as $T = -(U+V)$

Therefore, the final equation to convert from $[uvw]$ to $[UVTW]$ is

$U= \frac{2u-v}{3}$
$V= \frac{2v-u}{3}$
$T = -(U+V)$
$W= w$

So to write the hexagonal Miller-Bravais 4-axis indices, you just need to find the simple 3-axis Miller indices and convert using this formula.

If that seems like a lot of unnecessary work–don’t worry, there’s a big advantage to using the 4-axis system in hexagonal crystals. Read the section below about “Why use Hexagonal Miller-Bravais instead of Regular Miller Indices?”

Hexagonal Miller-Bravais Coordinate System for Planes

Thankfully, planes in the 4-axis system $(hkil)$ system are actually very similar to the 3-axis $(hkl)$ system–that’s because “h,” “k,” and “l” are the same in both systems. “i” is simply defined by the formula h + k = -i.

Otherwise, you can use the regular procedure for finding planes in Miller indices. You don’t even need an image to see that $(111)$ becomes $(11\bar{2}1)$ , $(100)$ becomes $(10\bar{1}0)$ , or $(\bar{1}\bar{1}0)$ becomes $(\bar{1}\bar{1}20)$ .

Another way to think about it, is to treat the a₃ axis just like the other axes. For example, the $(11\bar{2}0)$ intersects the a₁ axis at 1, the a₂ axis at 1, the a₃ axis at -½, and the c axis at infinity (never, because it’s parallel).

It will mathematically work out that the intersection at the a₃ axis will be the negative sum of the intersections at the a₁ and a₂ axes, so I personally never bother with a₃ and just think about a₁ and a₂, like regular Miller indices.

Converting Vectors between Hexagonal Miller-Bravais and Regular Miller Indices

We already covered the conversion from Miller indices to Miller-Bravais indices, but what about the reverse?

$u= \frac{3U+v}{2}$
$v= \frac{3V+u}{2}$
$w= W$

Converting Planes between Hexagonal Miller-Bravais and Regular Miller Indices

Converting planes from Miller-Bravais indices to Miller indices is trivial–just drop the “i” value in $[hkil]$ to get $[hkl]$ !

So $(11\bar{2}0)$ becomes $(110)$ , $(10\bar{1}0)$ becomes $(100)$ , $(7\bar{8}13)$ becomes $(7\bar{8}3)$ , etc.

Why use Hexagonal Miller-Bravais instead of Regular Miller Indices?

One of the most useful features of miller indices is allowing us to talk about things which happen symmetrically in crystals. For example, you may be familiar with the fact that the FCC crystal has slip planes in the close-packed direction, $[110]$ .

However, by symmetry there is no difference between $[110]$ , $[\bar{1}\bar{1}0]$ , or $[10\bar{1}]$ , or any of these permutations–these all belong to the same $\langle 110 \rangle$ family.

Visually, it’s obvious which indices belong in the $\langle 110 \rangle$ family, so you can instantly identify whether any given direction belongs to the close-packed direction.

If you use the same 3-axis system in hexagonal crystals, however, the indices don’t align with the crystal symmetry. For instance, you may realize that $[100]$ and $[010]$ are both close packed directions and HCP, and thus belong to the same family of directions. However, $[001]$ does not share symmetry with that family, but $[110]$ does.

In 3-axes Miller indices, the $\langle 100 \rangle$ family includes $[100]$ , $[110]$ , $[010]$ , $[\bar{1}00]$ , $[\bar{1}\bar{1}0]$ , and $[0\bar{1}0]$ . (remember that the c-axis has a different length than the a-axes, so can never change besides positive/negative within one family).

With the 4-axis system, the close-packed direction is $\langle 2\bar{1}\bar{1}0 \rangle$ family and its permutations: $[2\bar{1}\bar{1}0]$ , $[\bar{1}2\bar{1}0]$ , $[\bar{1}\bar{1}20]$ , $[\bar{2}110]$ , $[1\bar{2}10]$ , and $[11\bar{2}0]$ .

The symmetry also works for planes. Consider one of the side face planes in a hexagonal cell, such as $(100)$ .

The $\{100\}$ family would include $(100)$ , $(010)$ , $(\bar{1}10)$ , $(\bar{1}00)$ , $(0\bar{1}0)$ , and $(1\bar{1}0)$ .

With the 4-axis system, this $\{10\bar{1}0\}$ family includes $(10\bar{1}0)$ , $(01\bar{1}0)$ , $(\bar{1}100)$ , $(0\bar{1}10)$ , $(\bar{1}010)$ , and $(1\bar{1}00)$ .

Example Problems

Before attempting to solve our hexagonal Miller-Bravais Indices tasks, make sure that you don’t have any problems with regular Miller indices. Example problems for regular Miller indices are in this article.

Practice 1. Rewrite these 3 planes as hexagonal Miller-Bravais indices: $(111)$ , $(203)$ , and $(1\bar{1}0)$ .

Click here to check out the solution!

Remember that the difference between planes in Miller Indices $(hkl)$ and Miller-Bravais indices $(hkil)$ is the i, which is just:

$h+k=-i$

This can be calculated from the values we already have, so we can simply write:

$(11\bar{2}1)$ , $(20\bar{2}3)$ , $(1\bar{1}0)$

Practice 2. Rewrite these 3 planes as Miller indices: $(0001)$ , $(3\bar{1}\bar{2}1)$ , and $(10\bar{1}0)$ .

Click here to check out the solution!

Remember that the difference between planes in Miller Indices $(hkl)$ and Miller-Bravais indices $(hkil)$ is the i, which is just:

$h+k=-i$

So in order to get $(hkl)$ , we simply have to skip “i”

This can be calculated from the values we already have, so we can simply write:

$(001)$ , $(3\bar{1}1)$ , $(100)$

Practice 3. Find Miller-Bravais indices of the plane and direction presented below.

Click here to check out the solution!

Let’s focus on the plane first. We can see that it is parallel to a₁, a₂, and a₃. The plane intersects the c axis at 1. Thus, the Miller-Bravais indices for that plane are $(0001)$ .

It’s a little bit trickier to determine Miller-Bravais indices $[UVTW]$ for the direction. First, let’s try to find out Miller indices $[uvw]$ . It should be obvious that the answer is $[011]$ . (If you don’t know how to do it, you should get familiar with Miller indices first). Once we have a $[uvw]$ value, we can use the formula below to obtain Bravais-Miller indices.

$U= \frac{2u-v}{3}$
$V= \frac{2v-u}{3}$
$T = -(U+V)$
$W= w$

We know that in this case u=0, v=1, and w=1.

The obtained values are: U=-⅓, V=⅔, T=-⅓, and W=1. Now we need to multiply everything by 3 (it’s customary to write down directions as integers).

As a result, the direction is: $[\bar{1}2\bar{1}3]$

Plane: $(0001)$
Direction: $[\bar{1}2\bar{1}3]$

Practice 4. Draw the $(11\bar{2}0)$ plane and $[21\bar{3}3]$ direction. Convert Miller-Bravais indices $(hkil)$ and $[hkil]$ to Miller indices $(hkl)$ and $[hkl]$ .

Click here to check out the solution!

$(hkl)=(110)$
$[hkl]=[101]$

Practice 5. Rewrite these 3 directions as hexagonal Miller-Bravais indices: $[100]$ , $[210]$ , $[\bar{1}\bar{1}0]$ . Rewrite them as hexagonal Miller-Bravais indices.

Click here to check out the solution!

We need to use this formula:

$U= \frac{2u-v}{3}$
$V= \frac{2v-u}{3}$
$T = -(U+V)$
$W= w$

The obtained values are (respectively): $[2\bar{1}\bar{1}0]$ , $[10\bar{1}0]$ , $[\bar{1}\bar{1}20]$

Practice 6. Rewrite these 3 directions as Miller indices: $[11\bar{2}0]$ , $[\bar{1}2\bar{1}3]$ , $[21\bar{3}3]$ . Rewrite them as hexagonal Miller-Bravais indices.

Click here to check out the solution!

We need to use this formula:

$u= \frac{3U+v}{2}$
$v= \frac{3V+u}{2}$
$w= W$

The obtained values are (respectively): $[110]$ , $[011]$ , $[101]$

Summary

The 4-axis Miller-Bravais indices are useful for hexagonal crystals because the index values show the inherent 6-fold symmetry, which is not captured in traditional 3-axis Miller indices.

Converting planes between Miller indices and Miller-Bravais indices is simple, and just requires adding or removing the “i” value in [hkil], remembering that “i” is always constant i=-(h+k).

Converting directions between Miller indices and Miller-Bravais indices is trickier, and requires a formula:

$U= \frac{2u-v}{3}$
$V= \frac{2v-u}{3}$
$T = -(U+V)$
$W= w$

Useful equations when converting Miller and Miller-Bravais indices:

Miller plane indices to Miller-Bravais plane indices $(hkl) \rightarrow (HKIL)$ :

$I=-(h+k)$

Miller-Bravais plane indices to Miller plane indices $(HKIL) \rightarrow (hkl)$ :

Just drop the $``I''$

Miller direction indices to Miller-Bravais direction indices $[hkl] \rightarrow [HKIL]$ :

$U = \frac{2u-v}{3}$
$V = \frac{2v-u}{3}$
$T = -(U+V)$
$W= w$

Miller-Bravais plane indices to Miller plane indices $[HKIL] \rightarrow [hkl]$ :

$u= \frac{3U+v}{2}$
$v= \frac{3V+u}{2}$
$w= W$

References and Further Reading

Check out our article, if you want to read more about regular Miller indices.

If you want to check your work, you can find a “Miller Index plane calculator” for cubic lattice from the University of Cambridge Dissemination of IT for the Promotion of Materials Science.

If you’re reading this article as an introductory student in materials science, welcome! I hope you can find many other useful articles on this website. You may be interested in a related article I’ve written about Atomic Packing Factor.

If you’re reading this article because you’re taking a class on structures, you may be interested in my other crystallography articles. Here is this list, in recommended reading order:

Introduction to Bravais Lattices
What is the Difference Between “Crystal Structure” and “Bravais Lattice”
Atomic Packing Factor
How to Read Miller Indices
How to Read Hexagonal Miller-Bravais Indices
Close-Packed Crystals and Stacking Order
Interstitial Sites
Primitive Cells
How to Read Crystallography Notation
What are Point Groups
List of Point Groups
What are Space Groups
List of Space Groups
The 7 Crystal Systems

If you are interested in more details about any specific crystal structure, I have written individual articles about simple crystal structures which correspond to each of the 14 Bravais lattices:

1. Simple Cubic
2. Face-Centered Cubic
2a. Diamond Cubic
3. Body-Centered Cubic
4. Simple Hexagonal
4a. Hexagonal Close-Packed
4b. Double Hexagonal Close-Packed (La-type)
5. Rhombohedral
5a. Rhombohedral Close-Packed (Sm-type)
6. Simple Tetragonal
7. Body-Centered Tetragonal
7a. Diamond Tetragonal (White Tin)
8. Simple Orthorhombic
9. Base-Centered Orthorhombic
10. Face-Centered Orthorhombic
11. Body-Centered Orthorhombic
12. Simple Monoclinic
13. Base-Centered Monoclinic
14. Triclinic

Outline

Review of Miller Indices

Hexagonal Miller-Bravais Coordinate System for Directions

Hexagonal Miller-Bravais Coordinate System for Planes

Converting Vectors between Hexagonal Miller-Bravais and Regular Miller Indices

Converting Planes between Hexagonal Miller-Bravais and Regular Miller Indices

Why use Hexagonal Miller-Bravais instead of Regular Miller Indices?

Example Problems

Summary

References and Further Reading

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