Miller Indices Weiss Zone Law

So, in this is lecture, we will continue on hexagonal lattices a little bit followed by Weiss zone law, and we will see that although in a seemingly cubic lattice, the directions may look similar, symmetry may change that outcome completely. So, with that, we will finish the miller indices part.

This is the recap of the last lecture that we did. So, we have this hexagonal lattice, and within this hexagonal lattice, you have this green colored smaller unit cell which is the primitive unit cell of the hexagonal unit cell and then for the sake of representation you can represent it as a hexagon which consists of 3 of these unit cells.

So, in the normal unit cell, you need 3-axis, a1, a2, and c. The third axis, the one which is perpendicular to the basal plane. So, a 3-axis system is fine for a smaller unit cell, but when it comes to hexagonal representation, you see the problem arises because there are similar planes that are depicted differently if you continue with the 3-axis system.

So, for example, as I showed you last time (hkl) becomes (hkil) with the condition h + k + l = -i, and the direction [UVW] in 3-axis system becomes [uvtw] with the condition,

So, if we do the matching of variables, a, b components along a1, a2, and c-axis, so you will get U is equal to u, and this will be minus of t, V will be equal to v, t = -(u+v). So U = 2u + v, V = 2v + u, and W will remain as w. So, this is a very simple equivalence that you can set and determine what is [UVW] relation.

For example, let us say ABCD plane and another plane EACF, these 2 are equivalent planes right. So, this plane is you can say it is (100), this plane is (010), but if you want to depict now this plane let us say GEFH, this plane is nothing but the one which is diagonal, and it is smaller unit cell. So, you will say this plane is some sort of (110) type plane. So, in a 3-axis system, it will become (11̅0), but in the 4-axis system, this will become (101̅0), similarly, this will become (011̅0), and this will become (11̅00).

So, you can say that all of these are family of (101̅0) planes, so this will imply (101̅0), (011̅0), (11̅00) and so on and so forth. So, you will have 6 planes, few of them are parallel to each other. So, you will change those negative signs and so on. So, these are family of planes, that is why this 4-axis representation is better. So, this is what I did show you.

Similarly, as I showed, you could draw, (112̅2) plane. So, this is equivalent to the (112) plane. Now, if you draw all such planes, they may end up with different species. So, if you just see (112̅2) this is first, the second one will become (2̅112), and the third one will become (12̅12).

Now, try converting this in a 3-axis system; this will be (112), this will be (2̅12) and this will be (12̅2). So, you can see that this is all right, but these two are different. So, it gives you an impression as if they are different planes because that is what we learned in the cubic system or tetragonal system, (112) is equivalent to (212) and, but here we see the planes which are identical in hexagonal system they look differently in 3 coordinate system. So, (121), (12̅2), (2̅12) and (112) they can be represented all by (112̅2), (2̅112), (12̅12) and so on and so forth.

Now, this is a projection of a basal plane of the hexagonal system. So, we are looking down the c-axis, and we are looking at AB plane. So, direction [120], there is no translation in c. So, this is the direction, and the 3-coordinate system will become a 4-coordinate system, and it will become (011̅0). So, if we draw this now, this will be 0, so there is nothing around a1, 1 along with a2, there is -1 along with a3, and if you connect the final point, it is the same direction. So, all the directions of the kind (101̅0), (011̅0), (11̅00), and so on. Is that clear how to draw the direction in the hexagonal 4-axis system? Now, a little difficult example to work on and that is, (112̅1). So, this has all three coordinates. So, you shift the origin first and how will you draw (112̅1)? 1 along a1, 1 in a2, then 1 in c-axis. So, you connect this point with that point that will give (112̅1).

The 4-axis system has more advantages in terms of representation of planes is easier. Since direction is just a vector you can draw them with the 3-axis system within the same system, but if you draw a bigger unit cell, the hexagonal unit cell then 4-axis system helps. If you look at the direction [110], is nothing but (112̅0) direction, [120] is (011̅0). So, you will have to make the conversions, you have to use the formulas which I provided you.

So, you will need these equations to make these conversions. So, if you now convert back,

So, this is how you can make the conversion between the directions, if you draw it along the third axis, it will become even complicated, but you can do these exercises at home.

So, now I will give you some more examples of hexagonal crystals you have. So, this is courtesy professor Anandh Subramaniam, and he made very nice slides on the crystallography. So, this is about as I said (112̅0) plane. So, (112̅0) plane is nothing but 1 along a1, 1 along with a2 and away, and by default, the third intercept, which is minus 2, which is half right, the intercept will be 1, 1, ½, ∞. So, it has a one intercept here, one intercept there, minus half intercept along a3 and nothing along it is parallel to 3-axis. So, you can see that this is the plane. So, in 3D, the plane is going to look like that. So, all the planes, this is one kind of plane, this is another kind of plane, and you will have which is the other one and the equivalence at the bottom this will be another plane. So, these are all identical planes (112̅0), (2̅110), (12̅10) and so on and so forth.

So, now this is the one I showed you (100) plane example. So, these planes are identical and look differently in 3-axis system, but they look the same in the 4-indices system, this is another example of the plane, which is (21̅1̅0). So, (21̅1̅0) looks something like remembers you should always ensure in all these the sum of the indices is equal to 0.

So, (112̅0) sum of these 3 is 0, that equivalence must be maintained if that equivalence is not maintained that then something is wrong somewhere, so all the sums must be equal to (hki) and [uvt], these sums must be equal to 0.

The (112̅1) will look like the 1 on top and the 1 on the bottom. (101̅1) which is parallel to the a2-axis. So, this plane is nothing but (101) plane, but in 4-axis system, it looks (101̅1). So, (101̅1), (1̅011), (01̅11) are all similar planes.

So, this is about the planes and directions in various crystal systems, you have seen how we can draw the plane, how we can draw a direction, how we can identify a plane, how we can identify a direction and we have also seen a cubic system that direction lies perpendicular to the plane direction. So, [hkl] is perpendicular to (hkl), but that is only true of a cubic system.

So, it is only true for certain directions in these systems depending upon the indices in the orthogonal system, and you still have some cases where perpendicularity is maintained, but in non-orthogonal systems is not maintained at all. So, in a hexagonal system, for example, [001] is perpendicular to (001) plane, but that is the only direction which is perpendicular to (001) plane. So, this relation is true only for the cubic system for hexagonal tetragonal and orthorhombic it depends upon which direction in which plane you were talking about.

And another crystallographic thing that you need to know is that Weiss zone law, Weiss zone law is a very useful law especially for plaster deformation, any phenomena that happen within the crystallographic planes where you need to know the anisotropy and you need to find out which direction lies common to certain planes.

So, this law says that for a direction [UVW] lying in a plane (hkl), dot product h.U+k.V+l.W=0, this is called as Weiss zone law. So, what it means is that if you have multiple planes, and all these multiple planes, intersect each other, and there is a common direction to these planes. This common direction is called a zone axis. This is of very much importance especially in transmission electron microscopy studies when you make that diffraction patterns.

In diffraction patterns, what you will do is that when the electron hits the material along a certain axis, that may be a certain crystallographic direction. So, all the planes which are containing that direction may be represented on the diffraction pattern. So, sometimes they may be diffraction spots from a few planes, and you need to identify, but using the Weiss zone law, you can identify other spots from one particular plane because they must have a common zone axis.

So, this is of importance in transmission electron microscopy quite a bit. So, if you have a direction [UVW] which is common to two planes (h1k1l1) and (h2k2l2) this is called as a zone axis, and it can be found as,

So, you can write these two equations, and these two equations can be solved in this fashion. So, that is how you can find the common axis between the two planes. You can see here in this case you have let us say two planes, one plane is (010), and another plane is (11̅0).

You have a plane (11̅0) and you have a plane (100). So, for the first case, it will become u - v = 0. For the second case, it will become u = 0. So, this plane is (111), what is this plane, if I choose this as an origin, then this is an intercept of 1 along x, intercept of -1 along y. So, this becomes (11̅0). So, now you can see that these two intersect each other. So, if they intersect each other, in that case, u + v + w = 0, and u – v = 0. So, in this case, you can see that u = 0 and v = 0, which means your w is undefined. So, the direct direction which is parallel to these two planes contain is [00w] means [001] direction. So, both the planes will have [001] direction as common. But the second case, (111) so, u + v + w = 0, and u - v = 0, and w = -2u. So, the direction would be -2u. So, this is [112̅]. So, this is how you can find common directions.

Now, let me finally come to another point, which is based on symmetry, that the set of directions is always related by symmetry.

So, this is again courtesy professor Anandh Subramaniam so, and here you have a grid of points, and in this case, if there is a 4-fold rotation within the plane, then [1 0] = [0 1], [1 1] and [1̅1] directions are equivalent.

So, if you rotate it by 4-fold, you will get back to the same configuration. So, this can be rotated by 900 because there is a 4-fold, similarly [0 1] is equivalent to [01̅] because there is a 2-fold axis that is present. Now, this is about when the motif is a single atom.

If we change the motif, you have a 4-fold, but again there are no mirrors. If there are no mirrors, then the two directions, which could be equivalent because of a mirror they will not be the same. So, for example, [12] and [1̅2] do not belong to the same because there is no mirror. The symmetry changes the classification of direction quite dramatically.

So, this is again an example where which directions are equivalent and which directions are not equivalent, it depends upon the type of motif. So, in a square motif, if you have a different motif, then also it changes, but in a non-square lattice with a simple motif again, things change.

So, it depends upon the symmetry whether you have a 4-fold present, whether you have a 2-fold present, whether you have a mirror present. So, the family of directions, the concept changes from system to system which is determined by symmetry.

To summarize you, we have learned about the direction [UVW], which are Miller indices of a direction which is nothing but a vector and <UVW> is Miller indices of a family of symmetry-related directions. So, since cubic is the most symmetric system, all of them happens to be of the same type, but in the tetragonal system, they become different. Similarly, in (hkl) is to represent a single plane, {hkl} is to represent the family of symmetry-related planes, and we do not typically allow separators in the miller indices of directions and planes they are written together, you write them (123) unless, of course, the magnitude is in double-digit. So, that is where we will finish this particular part, and we now go to the next part which is the structure of metals.

For Next Lecture Click below

Structure of Materials

Structure of Materials : Bonding in Materials

Structure of Materials : Correlation between bond and physical properties

Crystal Structure: Lattice and Basis

Primitive and Non-primitive Lattices

Crystal Systems and Bravais Lattice

Bravais Lattices Symmetry in Crystals

Symmetry in Crystals

Symmetry and Correlations with the Bravais Lattices

Miller Indices (Planes and Directions)

Miller Indices (Planes and Directions) contd....