How to Draw Hexagonal Crystal Planes

If y'all'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-centrality coordinate organisation for 3-dimensional crystals, based on the unit of measurement cell. This coordinate organization is based on the 3-axis Miller index, simply with an extra axis which is used for hexagonal crystals. The system tin can indicate directions or planes, and are often written equally (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 tin y'all have iv 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? Proceed reading for the answers to these questions!

Review of Miller Indices

I'll put this in collapsible text for the upperclassmen who simply need a quick refresher. If you lot demand a full caption of Miller indices, check out that article. Be certain you understand it, because Miller-Bravais indices build right on top of this foundation.

Click hither to detect out more than.

Miller indices plot a direction or plane along iii axes that correspond to the 3 lattice parameters of the crystal. A value of one means you lot have traveled the full distance of that lattice parameter. In cubic systems this is exactly like cartesian coordinates, just 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 $(hkl)$ point a specific plane. For case, in a cubic system $(100)$ and $(010)$ are perpendicular planes.
Curly brackets $\{hkl\}$ signal a family of planes. For instance, in a cubic system $\{100\}$ includes $(100)$ , $(\bar{1}00)$ , $(010)$ , $(0\bar{1}0)$ , $(001)$ , and $00\bar{1})$ .

"h," "thousand," and "50" are the distance forth each lattice vector. For example, $[121]$ has h=1, k=2, and l=one. 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, y'all could imagine this equally ½ a forth a, 1b forth b, and ½ c along c.

For planes, you apply reciprocal infinite. The $(hkl)$ of the aeroplane is the inverse of the value where the plane intersects the lattice vectors. For instance, $(102)$ 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 organisation, I showed that it was easier to imagine the archaic, hexahedral version of the hexagonal cell. For Miller-Bravais indices, I'll go dorsum to the conventional unit prison cell.

For Miller-Bravais indices, we need to characterization four axes in the hexagonal crystal. In the basal plane, nosotros take 3 axes of equal length each separated past 120º, which we call a₁, a₂, and a_three (they are each the same every bit the lattice parameter "a"). Then at that place is the c centrality, perpendicular to those three.

Now, the question is, how practise we get from a iii-axis organisation to a four-axis system?

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

Click here to detect out more than.

Allow's employ "u," 'v," and "west," to correspond the distance along the a_one, a₂, and c vectors in the three-axis system. When we add a₃, nosotros can utilize capital "U," "V," "T," and "W" to represent the altitude along the a_ane, a₂, a₃, and c vectors.

If nosotros 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 besides know that $a_{1}+a_{2}+a_{3}$ will return to the starting bespeak, so summing those vectors gives us cipher. Then

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

We can and so 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 nosotros find:

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

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

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

We can rewrite u, v, and westward without "T" considering $T = -(U+V)$

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

This converts $[hkl]$ to $[hkil]$ , but what nearly the contrary? 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, yous just need to find the unproblematic 3-axis Miller indices and convert using this formula.

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

Hexagonal Miller-Bravais Coordinate Organisation for Planes

Thankfully, planes in the four-axis system $(hkil)$ system are actually very similar to the 3-axis $(hkl)$ organisation–that'due south considering "h," "chiliad," and "l" are the same in both systems. "i" is simply divers by the formula h + thou = -i.

Otherwise, you can use the regular procedure for finding planes in Miller indices. You don't even need an paradigm to run into 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 call up about information technology, is to treat the a₃ axis simply like the other axes. For example, the $(11\bar{2}0)$ intersects the a_one axis at 1, the a₂ centrality at 1, the a_three axis at -½, and the c axis at infinity (never, considering it's parallel).

Information technology volition mathematically work out that the intersection at the a_three axis volition exist the negative sum of the intersections at the a_ane and a_two axes, so I personally never carp 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 nigh 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–but drib 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?

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

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

Visually, it'due south 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 arrangement in hexagonal crystals, all the same, the indices don't align with the crystal symmetry. For instance, yous may realize that $[100]$ and $[010]$ are both shut packed directions and HCP, and thus belong to the same family unit of directions. Even so, $[001]$ does non 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 $[0\bar{1}0]$ . (retrieve that the c-axis has a dissimilar length than the a-axes, so can never change as well positive/negative within ane 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 too works for planes. Consider one of the side face planes in a hexagonal jail 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 four-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 certain that you lot don't have whatever problems with regular Miller indices. Instance 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 hither to check out the solution!

Call back that the difference betwixt 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 take, so nosotros can but write:

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

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

Click here to check out the solution!

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

$h+k=-i$

So in club to get $(hkl)$ , we simply have to skip "i"

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

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

Practice three. Notice Miller-Bravais indices of the aeroplane and management presented below.

Click here to check out the solution!

Let's focus on the aeroplane first. We tin 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 footling bit trickier to determine Miller-Bravais indices $[UVTW]$ for the management. Outset, permit'south attempt to notice out Miller indices $[uvw]$ . It should exist obvious that the answer is $[011]$ . (If you lot don't know how to practice information technology, yous should get familiar with Miller indices first). Once nosotros have a $[uvw]$ value, we can utilize 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 westward=1.

The obtained values are: U=-⅓, V=⅔, T=-⅓, and W=i. Now we demand to multiply everything by 3 (it's customary to write down directions every bit integers).

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

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

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

Click here to cheque 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 every bit 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): $[2\bar{1}\bar{1}0]$ , $[10\bar{1}0]$ , $[\bar{1}\bar{1}20]$

Exercise 6. Rewrite these iii 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 bear witness the inherent 6-fold symmetry, which is non captured in traditional 3-centrality Miller indices.

Converting planes betwixt 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 airplane indices $(hkl) \rightarrow (HKIL)$ :

$I=-(h+k)$

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

Simply 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 than about regular Miller indices.

If you want to check your work, you can detect a "Miller Index plane estimator" for cubic lattice from the University of Cambridge Dissemination of Information technology for the Promotion of Materials Scientific discipline.

If you're reading this commodity every bit an introductory student in materials science, welcome! I promise 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, y'all may be interested in my other crystallography manufactures. Here is this list, in recommended reading lodge:

Introduction to Bravais Lattices
What is the Difference Between "Crystal Construction" and "Bravais Lattice"
Atomic Packing Gene
How to Read Miller Indices
How to Read Hexagonal Miller-Bravais Indices
Shut-Packed Crystals and Stacking Social club
Interstitial Sites
Archaic Cells
How to Read Crystallography Notation
What are Point Groups
Listing of Indicate Groups

If you are interested in more details near 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
ii. Face-Centered Cubic
2a. Diamond Cubic
three. Trunk-Centered Cubic
four. Simple Hexagonal
4a. Hexagonal Shut-Packed
4b. Double Hexagonal Close-Packed (La-type)
five. Rhombohedral
5a. Rhombohedral Close-Packed (Sm-type)
6. Simple Tetragonal
seven. Body-Centered Tetragonal
7a. Diamond Tetragonal (White Can)
8. Simple Orthorhombic
ix. Base-Centered Orthorhombic
10. Face-Centered Orthorhombic
11. Body-Centered Orthorhombic
12. Elementary Monoclinic
xiii. Base of operations-Centered Monoclinic
fourteen. Triclinic

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Source: https://msestudent.com/hexagonal-miller-bravais-indices/

How to Draw Hexagonal Crystal Planes

Review of Miller Indices

Hexagonal Miller-Bravais Coordinate System for Directions

Hexagonal Miller-Bravais Coordinate Organisation 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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