Structure of Metals Packing Co-ordination Interstices

 Structure of Metals Packing Co-ordination Interstices

In this lecture, we will again continue talking about the structure of metals and alloys with a focus on the packing of atoms in these structures, coordination, and interstices.

So, let me recap the last lecture i.e structure of metals and alloys . So, in the last lecture, we started about the structure of metals. So, here we talked about how atoms packed closely in metallic structures, they packed themselves in face-centered cubic or cubic closed packed structure and hexagonal closed packed structure the by virtue of the arrangement of layers in AB. So, if you have an array of atoms like this and each layer underneath is like this. So, this is how and you can keep extending it in two directions.

So, this is how your first layer, which is A layer, and when you put that B layer on top, the B layer can go on this type of interstices or this kind of interstices. So, let us say these interstices will give rise to the B layer, and these interstices will give rise to the C layer. So, if you follow the stacking sequence of ABC ABC…, which can be described by FCC lattice with a motif at 0,0,0. If it is AB AB stacking kind of sequence, where you put the next layer at B, then the next layer again comes back on A, you do not have to necessarily go to C which is again the alteration of ABC ABC type of packing, but both are closed packed structure this is called as hexagonal closed packed structure. However, in this case, it is not primitive hexagonal with a motive 000 and 1/3 2/3 1/2. So, it is a two-atom motif, here, in this case, it is a single atom motif by which you can define the lattice single atom motif with FCC lattice with one atom at 000, 1/2 1/2 0 , 1/2 0 1/2, and 0 1/2 1/2.

So, this class, we will again look at some other structures so, not only in metals, atoms arrange themselves in ABC ABC type of packing, there are other structures as well, for example, if you have atomic arrangement like this, and you can put the next layer, a different color used, so this will be B layer, and the first layer will be A. So, in this case, you can see that the A layer is not closely packed, the number of atoms which surround each atom is different. So, numbers of nearest neighbors are in the closed packed case, what was the situation, it was 6. This means it is less closed packed so, you can find out the atomic density in this layer by just calculating in the number of atoms per unit area, and stacking gives rise to a structure that looks like that try to remake it. So, for the sake of illustration, the atoms are not touching here, and one atom will be at the center. So, these will be at 000 positions, and this will be at 1/2 1/2 1/2 position. So, this kind of lattice called as body-centered cubic.

So, here you can see that this structure is less closely packed as compared to a hexagonal closed packed or face-centered cubic structure. So, it has two atoms at the center, it has two atoms in the unit cell, one at the corner at 000 so, two atoms per unit cell, so, this lattice type is BCC which is I, when you define a lattice-type as non-primitive you do not need to define the coordinates of all the atoms in motif you just need to define only one atom which is at 000 it will. So, I will ensure that the number of atoms in the unit cell one is at 000, another is at 1/2 1/2 1/2, and they are identical lattice points. So, it has two lattice points per unit cell, and they are identical. So, I ensure that you have two lattice points, but to define the motif, you can just choose the single atom motif at 000. Similarly, in case of FCC, single-atom motif will be at 000, but it ensures that you have four identical lattice points one at 000, another at 1/2 1/2 0, 1/2 0 1/2, and 0 1/2 1/2.

So, you have atoms in this fashion, and this is layer 1, the second layer goes directly on top of it. So, this is one atom, this is another atom, they have kept apart for the sake of illustration, but they do touch each other, this is simple cubic. So, here, each atom has how many nearest neighbors? Six nearest neighbors, but within the plane, it has only four nearest neighbors. Likewise, in the case of BCC, the number of nearest neighbors was 8 in 3D.

So, this atom is surrounded by eight of the atoms. Similarly, this atom is surrounded by eight atoms. So, this is called as a simple cubic lattice, the lattice type is defined as P, which is primitive and the motif is at 000.

So, these are the Four types of lattices we typically find the examples, for example, HCP based metals like beryllium, cadmium, magnesium, one form of titanium are all metals, some alloys, brasses, titanium alloys they are also hexagonal. Then you have FCC based metals like platinum, gold, copper, nickel. And BCC based metals like iron, tungsten, molybdenum, and then simple cubic is polonium, and this is perhaps the only metal in the periodic table, which is a simple cubic structure. So, in this case of FCC structured materials, we define a closed packed plane the plane, which is closed packed.

So, this plane is (111) plane, and in this plane lies a closed packed direction, which is of [110]. So, of course, if you take this type of (111) plane, then the directions which will lie in this will be (1̅10), (101̅) and (011̅). The dot product of the direction which lies in the plane should be equal to 0. So, there are closed packed planes and directions in case of FCC, in the case of HCP, the closed packed plane is (0001), which is also called a basal plane.

So, if you look at the hexagonal system, this is the plane, the closed packed directions. So, you will have [101̅0] type of directions. So, this will ensure [101̅0], [011̅0] and [11̅00], these three directions will lie within this plane. So, there are only one plane and three directions in case of FCC, there are multiple choices (111) planes, there are 4 types of (111) planes finally, we have 8, but if you do not consider equivalent planes, and there are these four and then you have three types of directions in each of them. So, the total possibilities are 4*3 will be 12.

So, we will see this in mechanical behavior, called a slip system. In the case of mechanical properties, we consider this as a slip system because, on this plane, the mechanical deformation happens in FCC structure materials. So, there are 12 slip systems in FCC. In the case of HCP, there are only three slip systems, one multiplied by three.

Now in the case of BCC, there is no closed packed plane, the plane which has the highest density of atoms. So, there is no closed packed plane, but there is a closed packed direction.

This is the BCC unit cell so, a closed packed plane means plane having the highest atomic density. So, BCC has a closed packed direction, which is [111] and there are multiple types of this [111]. So, basically, since it is a cubic structure, you can write this as <111>. So, there are few possibilities, [111] direction may lie in a plane which is the (110) type of plane, it may lie in a plane which is (112) type of a plane, it may lie in a plane which is (123) type of plane, the combinations will be total of 48. So, this is again something which is used in the mechanical behavior of materials, in case of simple cubic, the closed packed direction is <100> type of direction. So, we generally do not consider the slip system in this case. So, now, once we have seen the four types of these lattices with their closed packed planes and directions, we will move on to what is called a coordination number.

The coordination number is the number of nearest neighbors. So, for HCP or FCC case, as we have seen in the closed packed structure. Next atom goes let us say this is the atom, which is the next layer atom, and then again, the next layer, if it is the next layer, will go either it will go on top of A or it will go on top of B so, either it will be ABC ABC, or it will be no matter what the case is this atom will be surrounded by. So, within the plane, how many neighbors will it have? One. So, within the, and let us say one layer, and within the layer, it has six neighbors.

So, each atom has 12 nearest neighbors in FCC or HCP structure. In the case of BCC, the coordination number is 8. In the case of a simple cubic, the coordination number was 6. We can see that the number of nearest neighbors is largest in the case of a closed packed structure. As a result, there is a quantity called as atomic packing factor, which is highest in the case of FCC.

What is the atomic packing factor? Which is APF, is basically the volume of all the atoms in the unit cell divided by the volume of the unit cell. So, let us say let us see in case of FCC, you have a situation like this. So, I will not draw the atoms touching with each other, but they do touch each other for the sake of illustration. Let us say the unit cell parameter is a, so unit cell parameter is a radius of an atom is r.

So, this is how you determine the atomic packing factor. As an exercise, you can determine the atomic packing factor of BCC, simple cubic, and HCP structures.

As an exercise, you can determine the atomic packing factor of BCC, simple cubic and HCP structures. So, this will be the atomic packing factor of these materials. The last thing I wanted to cover in this perhaps if you would not be able to go to interstices is in case of HCP lattices.

There is something called 𝑐/𝑎 ratio, now in case of a hexagonal closed packed, we know that there is A layer, and the next set of atoms goes to B layer, and another next layer will be slightly displacing it. So, what we need to determine is a and c when atoms are touching each other what is the ideal 𝑐/𝑎 in the FCC structure? So, we need to find out if you look at the structure now ideal 𝑐/𝑎 ratio, if you connect this atom with the atoms below it makes a regular tetrahedron. So, what you have here is, you have something like that a regular tetrahedron like this. So, if I draw so, this let us say it is the center of this tetrahedron. So, let us say this point is a, the distance between these two is h, which is equal to 𝑐/𝑎.

So, this is where we will end this lecture, in the next lecture, we will talk about interstices in these structures and the structure of some common metals and alloys.

For the Previous Lecture Click below

Structure of Metals and Alloys

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