Steps to Draw a Given Crystallographic Plane

Miller indices are one of the almost non-intuitive concepts most people meet in an introductory course. And, since a few notation differences can completely modify the meaning, advanced students also come back to review Miller Indices.

In this article, I'll explain what Miller indices are, why they're important, and how you can read and write them. Avant-garde students tin can skip straight to the Review if they're merely looking for a quick refresher.

Miller Indices are a 3-dimensional coordinate system for crystals, based on the unit cell. This coordinate organisation can signal directions or planes, and are often written as (hkl). Some common examples of Miller Indices on a cube include [111], the trunk diagonal; [110], the confront diagonal; and (100), the face plane.

Past the time you stop this article, yous'll know what those numbers and symbols mean!

Basic Note

The first assumption of Miller Indices is that you know the crystal family unit. If yous don't know what crystals are, this topic volition be super confusing–you might desire to cheque out this article first. If you're not certain nigh the different crystal families, yous can read an explanation in this article well-nigh Bravais Lattices, but as long as you know what a "cube" is, you can understand Miller Indices.

Every crystal can be depicted equally a hexahedron (that means it has vi faces, like a cube). There are some crystallographic coordinate systems which have "extra" dimensions, like the (hkil) Miller-Bravais system for hexagonal crystals, only you tin e'er reduce a conventional crystal cell into a primitive prison cell which is easily described past the (hkl) Miller Indices.

This commodity volition go on with the traditional Miller Indices.

Miller Indices are a coordinate organization (similar the cartesian or polar coordinate systems you learned in high schoolhouse), then the beginning thing you lot demand is an origin.

The origin is the indicate $(0,0,0)$ and y'all can define it anywhere in your crystal. In most cases, the dorsum left corner of the crystal is the most natural point to define the origin.

Y'all as well demand a sense of scale. For miller indices, the calibration is the size of the unit cell. In other words, the value "0" is the origin of one unit of measurement cell, and the value "1" is the origin for the next unit jail cell over.

Miller indices also have weird way of writing negatives. Allegedly this was developed to relieve space in one-time crystallography journals. Instead of writing negative i as "-one," we write it equally " $\bar{1}$ " and pronounce it as "bar one." If you saw $[0\bar{1}0]$ you would pronounce that like "the zero bar one zero direction."

It's also of import to remember that crystals are defined by their symmetry. That'due south why the choice of origin is arbitrary. In some cases, nosotros may want to distinguish between a specific direction, and all equivalent directions. Nosotros make this distinction with brackets.

Don't worry, these full general rules will make sense when we utilise them to the specific example of points, directions, and planes.

Crystallographic Coordinate Arrangement

Miller indices use a coordinate system which is very similar to the cartesian coordinate system. The cartesian system is the regular 2D or 3D coordinate system y'all used in loftier schoolhouse, which has three perpendicular axes 10, y, and z.

If you want a review of the cartesian system, click to expand.

Imagine you had a box with a length of 4, a width of two, and a top of 3. Put the bottom left corner of the box at the origin of your cartesian 3D system. What is the position of the top right corner?

If you followed this picture, you can see that the top right corner of the box is at the point $(4, 2, 3)$ . Its position is 4 spaces forth the 10-centrality, 2 spaces along the Y-axis, and 3 spaces forth the Z-axis.

By definition, the origin is at $(0, 0, 0)$ because it is 0 spaces along each axis.

The only thing that changes between the crystallographic cartesian arrangement and the version y'all learned in high school is the axes orientation.

In high school, you probably saw the X-centrality travel to the right, the Y-axis travel up, and the Z-axis wasn't shown, but travelled out of the page.

In crystallography, we employ the Z-axis much more than than in high schoolhouse math. A clearer way to depict these axes is to have the 10 axis travel towards you (downwardly and left), the Y axis travel to the right, and the Z-axis travel upwards.

So the betoken $(1, 2, 3)$ would exist one step toward yous, 2 steps to the right, and three steps up (NOT 1 step right, 2 steps up, and 3 steps out of the page).

The final thing to remember about crystallographic coordinates is that the X-, Y-, and Z-axes may not exist perpendicular to each other. In the cartesian system, they are always perpendicular. In a cubic crystallographic system, they are too perpendicular, because the cubic lattice parameters are perpendicular.

All the same, non all crystals have perpendicular lattice parameters. For example, a hexagonal lattice has 2 lattice parameters that are 120º to each other, which are both perpendicular to the third lattice parameter.

Every bit you can meet, the point $(1, 1, 1)$ looks a scrap dissimilar depending whether you have a cubic or hexagonal crystal structure. It's impossible to discuss Miller Indices without knowing the underlying crystal structure.

Miller Indices for Points

Technically, Miller Indices don't exist for points–just materials scientists and crystallographers represent points with an unnamed annotation system that is very like to Miller Indices, so I volition explain that here. The main notation is that y'all use parentheses () and commas. The way y'all write them is exactly the fashion you lot write cartesian coordinates.

I've really used this notation system before in the article, and I'm sure y'all understood, considering it'south very intuitive. You simply need to follow the bones rules–0 is at the origin, and one is the altitude of 1 unit jail cell. We as well use the letter "h," k," and "fifty" to designate the 3 different lattice parameters.

For example, in the cubic organisation, the three lattice parameters have the aforementioned length and are all perpendicular to each other (this is the definition of cubic). And so, for all cubic crystals, "h" is the length of the cube's border in the x-direction. "k" is the length of the cube's edge in the y-direction. "l" is the length of the cube in the z-management.

The same rules apply fifty-fifty in noncubic cases, but since the vector's aren't perpendicular to each other, the terms "x-centrality, y-axis, and z-centrality" don't really brand sense.

The point $(1, 1, 1)$ will e'er exist the summit right corner, opposite the origin. The point $(\frac{1}{2}, \frac{1}{2}, \frac{1}{2})$ volition ever exist the heart of the crystal.

Recollect, when describing points you write the bespeak as (h, k, l). You as well apply negative signs, rather than the "bar" notation. In other words, you write $(-h, -k, -l)$ instead of $(\bar{h}, \bar{k}, \bar{l})$ .

Miller Indices for Directions

To describe a direction, all you need to know is the betoken you want to travel to, relative to the origin. For example, if y'all desire to travel to the right, the point directly to the correct of the origin $(0, 0, 0)$ is $(0, 1, 0)$ . All y'all need to do is have this point and properly format it.

If you think, the format for directions is a foursquare bracket $[hkl]$ . If you wanted to talk nearly the family unit of directions, utilize angle brackets $\langle hkl \rangle$ .

So, to indicate the direction "right" in a cubic crystal you would write $[010]$ . The direction "left" would be $[0\bar{1}0]$ .

The $[010]$ management looks a bit different in the hexagonal system, but it'due south nonetheless just the length and management of the 2d lattice parameter. If you lot wanted to show a line to the "right" in a hexagonal system (depending on where we ascertain the original axes), y'all would need to use a linear combination of 2 lattice parameters. In other words, $[110]$ .

Finally, information technology'south customary to reduce fractions. The "length" of the management doesn't matter. If you wanted to indicate a direction that travels ¼ upwards the ten-axis while going all the way across the y-axis, it'due south traditional to write that as $[140]$ instead of $[\frac{1}{4}10]$ , by multiplying that latter version by 4 until everything is a whole number.

Miller Indices for Planes

Reading Miller indices for planes is a scrap different, because we have to enter "reciprocal space."

Reciprocal space means you accept the inverse of whatever point you were thinking of. The inverse of ane is still 1, the inverse of two is ½, and the changed of 0 is infinity.

Here is the iii-stride procedure to discover the miller indices for planes.

Find the betoken where the plane intersects each centrality. If the airplane never intersects an axis because it is parallel to that axis, the intersection bespeak is ∞.
Have the inverse of each intersection indicate.
Put those 3 values in the proper (hkl) format. Remember that negatives are expressed with a bar, parenthesis () indicate a specific plane, and curly brackets {} signal the family of planes. Don't use any commas or spaces!

In a cubic organisation, information technology turns out that the direction $[hlk]$ will always be perpendicular to the plane $(hkl)$ . For example the $[110]$ management is perpendicular to the $(110)$ plane.

This is not necessarily truthful in not-cubic systems.

Directional Families

Directional families are the set of identical directions or planes. These families are identical because of symmetry.

Imagine that I handed yous a cube and asked you to draw the $[100]$ . Past now, I hope y'all could practice this easily! However, if I gave the same cube to someone else, they would probably depict a different $[100]$ , considering they chose a different origin or a unlike initial rotation.

The line you originally drew may expect like $[001]$ compared to the other person's version of $[100]$ .

In this way, we can say that $[001]$ and $[100]$ belong to the aforementioned directional family. The only manner to distinguish betwixt the two is to define a consistent rotational frame of reference. This means that any fabric holding which is truthful forth $[100]$ will also be true of $[001]$ or whatsoever other management in the $\langle 100 \rangle$ family.

To observe the unlike directional families, find all the permutations that tin supersede $[hkl]$ with a negative version, such as $[\bar{h}kl]$ or $[h\bar{k}\bar{l}]$ . If the lattice vectors are the same length and have the aforementioned angle between them, y'all can as well change the guild, such every bit $[klh]$ or $[hlk]$ .

Hither is a list of the private directions in the directional families $\langle 100 \rangle$ , $\langle 110 \rangle$ , $\langle 111 \rangle$ . If two directions belong to the same directional family, their corresponding planes will as well belong to the same planar family.

Since the cubic lattice has the well-nigh symmetry, there are the most number of identical directions in each directional family unit. Imagine, notwithstanding, that yous had a tetragonal crystal that was longer in the $[100]$ management than the $[001]$ direction. In this instance, they would NOT belong to the aforementioned family. $[100]$ and $[010]$ would vest to the aforementioned family unit, which yous could telephone call $\langle 100 \rangle$ or $\langle 010 \rangle$ . All the same, the $\langle 001 \rangle$ family would just include $[001]$ and $[00\bar{1}]$ .

Identifying directional families becomes especially confusing if the lattice parameters h, m, and 50 are non perpendicular to each other. This was the main motivation for creating Miller-Bravais indices, which only apply to hexagonal crystals and convert the 3-term (hkl) values into 4-term (hkil) values. This conversion is a chip circuitous, but allows you to identify hexagonal directional families merely based on the numerical value of the index.

Alternative Notations

Advanced topic, click to expand.

This is going into collapsable text because it's an advanced topic, most different letters that may exist used to designate different axes or positions along the axes.

In this commodity, I've tried to use h, g, and fifty, for the values inside the Miller Index, and x-axis, y-centrality, and z-axis for the directions.

It'due south besides common to use "U," "V," and "W" to designate directions, as in [UVW] vs (hkl).

Additionally, the way I used x-, y- and z-axes is technically incorrect. We're technically supposed to use the lattice vectors, rather than cartesian axes. In the cubic system, they are the aforementioned, but they are not the aforementioned in other crystal systems.

Lattice vectors are often described using the letters " $a$ ," " $b$ ," and " $c$ ." Sometimes you lot might see the latters " $a_1$ ," " $a_2$ ," and " $a_3$ ," although this notation is typically used but with primitive cells.

I have tried to write this article in a way that is most understandable for people trying to learn Miller Indices, but I think it's important to know that you'll see minor notational differences in existent scientific journals or textbooks that hash out some theory of the indices. In most applied cases, you will just need to understand the meaning of basic indices such every bit $[100]$ , $\langle 111 \rangle$ , $(220)$ , and $\{110\}$ .

Review

Now you know how to read and write Miller indices! For a quick review of annotation:

(h, k, l) is for points. Remember to use the negative sign (-h) instead of bar sign ( $\bar{h}$ ) and don't reduce fractions–these rules apply to directions and planes.
[hkl] is for a specific direction.
<hkl> is for a family of directions.
(hkl) is for a specific plane. Remember almost reciprocal (inverse) space in planes!
{hkl} is for a family unit of planes.

Before you go, yous may be interested in practicing a few instance problems.

Example Problems

Practice ane. Describe the $[100]$ , $[111]$ , and $[010]$ directions in a cubic crystal.

Click here to check out the solution!

Ascertain an origin. I'll choose the back left corner to ascertain as my $(0, 0, 0)$ betoken
Find the corresponding $(1, 0, 0)$ , $(1, 1, 1)$ , and $(0, 0, 1)$ points considering they take the same $(h, k, l)$ values as the directions you want.
The line from the origin to these points, extending infinitely, is your management.

Practise ii. Write the Miller Indices of the indicated direction.

Click hither to check out the solution!

First, define an origin. It's e'er okay to move the origin afterward, since crystals-by definition–repeat between unit of measurement cells. In this instance, you lot demand the origin to intersect forth the indicated management, and so y'all tin motion the origin the back left corner. Alternatively, yous could simply translate that vector and then it intersects with the dorsum-left corner.

Either fashion, yous'll see that it takes a movement of 0 unit cells in the 10-direction, 0 unit cells in the y-direction, and one unit cell in the z-direction to move along that vector. Thus, the direction is $[001]$ .

Practice 3. Write the position of the indicate, and the Miller Index for the direction from the origin to the point. Assume the origin is at the back left corner.

Click here to check out the solution!

Hopefully it's straightforward to find this point, especially since I labelled the position for yous. You need to translate 1 unit of measurement cell in the x-direction, ⅔ unit jail cell in the y-direction, and ½ unit of measurement cell in the z-direction. Thus, the bespeak is at location $( \frac{1}{2} ,\frac{2}{3}, \frac{1}{2})$ .

The management would be identical, except that we adopt to avert fractions. Remember that directions extend infinitely, so we can hands multiply the $( \frac{1}{2} ,\frac{2}{3}, \frac{1}{2})$ value by 6, which is a mutual denominator. Thus, the direction is actually $[343]$ .

Do four. Draw the directions $[\bar{1}00]$ and $[010]$ , and write the Miller Alphabetize for the aeroplane mutual to both directions.

Click hither to bank check out the solution!

Hmm…What should y'all practice well-nigh a negative value? Travel out of the unit cell? That'southward a perfectly valid procedure, because there is an identical unit cell behind the original version. If you detect, in that 2nd unit jail cell it looks similar the vector comes out of the front end-left corner.

Because in that location'south always translational symmetry between unit of measurement cells, you can freely define a user-friendly origin (such as the front end-left corner), alter your frame of reference to a different unit jail cell, or just interpret the direction to stay inside your current unit of measurement cell.

If we draw both directions like this, it's articulate that the plane between them is the "basal" plane, or the floor/roof (remember by translation, the floor of ane unit of measurement cell is the roof of another).

To find the plane, we need to decide where it intersects with the lattice parameters.

This plane never intersects with the x-centrality or y-axis, because it is parallel to them. Thus, the h value is ∞ and the k value is ∞. The aeroplane intersects with the z-axis at point 0. By translation, 0 is also 1, so the 50 value is i.
The reciprocal of ∞ and one is 0 and 1.
Thus, the (hkl) value of the airplane is (001).
As a sanity check, retrieve that in cubic systems, the $[hkl]$ direction volition be perpendicular to the $(hkl)$ plane. By at present I promise it's piece of cake to draw the $[001]$ direction, which you tin can see is perpendicular to the $(001)$ aeroplane.

Practice five. Describe the $(220)$ and $(111)$ planes in a cubic crystal.

Click here to bank check out the solution!

By the reciprocal rule, the $(220)$ plane intersects the ten-axis at ½, the y-centrality at ½, and never intersects the z-centrality. The $(110)$ plane intersects the 10-axis at 1, the y-axis at 1, and never intersects the z-centrality. We tin draw that like this.

Notice that the $(220)$ and $(110)$ are parallel, but not identical. If I had an atom at $(0, \frac{1}{2}, \frac{1}{2})$ it would intersect the $(220)$ airplane simply not $(110)$ plane. This distinction matters more than if you exercise diffraction experiments.

Practice half-dozen. Draw the $[110]$ direction and $(110)$ plane in the hexagonal lattice.

Click here to check out the solution!

Let's first identify the direction. Remember, the hexagonal lattice parameters are not perpendicular, merely I'll keep calling them the x-, y-, and z-axes considering that is more familiar for almost of my readers.

The $[110]$ is i pace in the x-management and one step in the y-direction, like so:

To find the plane, let's plot our intersection points. It will intersect the x- and y-axes at the cease of the unit jail cell (reciprocal of ane is 1), and will be parallel to the z-axis (reciprocal of 0 is infinity).

References and Further Reading

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

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

If y'all're reading this article considering you're taking a form 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 Departure Between "Crystal Structure" and "Bravais Lattice"
Atomic Packing Factor
How to Read Miller Indices
How to Read Hexagonal Miller-Bravais Indices
Shut-Packed Crystals and Stacking Order
Interstitial Sites
Primitive Cells
How to Read Crystallography Note
What are Point Groups
List of Betoken Groups

If you are interested in more details about whatsoever specific crystal construction, I have written individual manufactures about unproblematic 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. Elementary Hexagonal
4a. Hexagonal Shut-Packed
4b. Double Hexagonal Shut-Packed (La-type)
five. Rhombohedral
5a. Rhombohedral Close-Packed (Sm-blazon)
half dozen. Uncomplicated Tetragonal
7. Body-Centered Tetragonal
7a. Diamond Tetragonal (White Tin can)
8. Simple Orthorhombic
9. Base-Centered Orthorhombic
10. Face-Centered Orthorhombic
11. Body-Centered Orthorhombic
12. Simple Monoclinic
13. Base-Centered Monoclinic
14. Triclinic

wilsongremand.blogspot.com

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

Steps to Draw a Given Crystallographic Plane

Basic Note

Crystallographic Coordinate Arrangement

Miller Indices for Points

Miller Indices for Directions

Miller Indices for Planes

Directional Families

Alternative Notations

Review

Example Problems

References and Further Reading

0 Response to "Steps to Draw a Given Crystallographic Plane"

Post a Comment

Iklan Atas Artikel

Iklan Tengah Artikel 1

Iklan Tengah Artikel 2

Iklan Bawah Artikel