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 Symmetry and Correlations with the Bravais Lattices

Let us start a new lecture, which is on symmetry and correlations with the Bravais lattices. So, before we get into it, First read these two lectures 1. Bravais Lattices Symmetry in Crystals and 2. Symmetry in Crystals

We learned there about symmetry based on well-defined criteria. And we defined that there are four types of symmetry elements, first is translational, which is a given for every system. So, translational is something which is generally not talked when we define the class because translational symmetry has to be there for a crystal for a periodic system. So, we have translational symmetry, reflection symmetry, rotation, and inversion. So, these four symmetries operations-basically completed. There are some other symmetry operations which are glide and screw. However, these are the four primary symmetry operations to define the crystal systems and Bravais lattices. And then there are finer distinctions between the same class or Bravais lattices; there are different materials with different motifs and different symmetry elements come into the picture.

However, these are four basic symmetry operations (1. Translational, 2. Reflection, 3. Rotation and 4. Inversion) that define Bravais lattices and crystal systems. And we also saw that what is the defining symmetry for various crystal systems? 

The marvel of one of the man-made symmetries. The famous architecture has a perfect fourfold of symmetry about the axis vertical through the center of the center dome. Source : Wikipedia_Taj Mahal

Now, that is mostly governed by rotation. So, for example, for the cubic system, you need to have four 3-folds. for tetragonal, you need to have one 4-fold, and for orthorhombic, we need to have four 2-folds, and so on. So, we had defining symmetry for 7 classes of crystal systems, and then we looked on to Bravais lattices, what is the correlation of these Bravais lattices with the symmetry? So, for example, we looked at 7 crystal systems and we defined them in categories of P, I, F, C. We saw that in the case of the cubic, we have primitive, body-centered and face-centered, in case of the tetragonal, you had only primitive and body-centered, and in case of the orthorhombic you had only you had all four of them and so on. So, the question was, why are some of these missing?.

So, for example, why is a C - centered cubic missing? Why is face-centered tetragonal missing? Why is C - centered tetragonal missing? And then, hexagonal which again had the only primitive system, rhombohedral also had only primitive and then monoclinic again had only primitive, and monoclinic also had C - centered, triclinic had only primitive.

Why C - centered cubic is not there, the reason is that the cubic can be defined as bodycentered tetragonal and C - centered cubic does not fulfill the criteria of four 3-folds which must be present in a cube. So, although it may look like a cube, it is not a cube, it has a smaller unit cell, and it fulfills symmetry criteria of tetragonal unit cells. So, Ccentered becomes body-centered tetragonal.

Similarly, why we do not have a face-centered tetragonal? So, we will not go into all of them, but I will give you some examples as to why some of them are not present. So, let us say a face-centered tetragonal here. So, let me draw a unit cell here tetragonal unit cell.

So, we will draw two unit cells, and you might have guessed by now that they are not there because either they do not make a valid lattice or they convert into something else which has either higher symmetry or smaller size. So, this does not look like the two are slightly different in sizes, but nevertheless. So, let us put the atoms here, these are two tetragonal cells adjacent ones, so, we are saying that why is face-centered tetragonal not there. So, we draw atoms at the center of the faces, so, we have drawn this here you can see that you can construct a smaller tetragonal cell in this fashion, which is a bodycentered tetragonal. So, it has the same tetragonal symmetry but a smaller cell. So, basically, we prefer a smaller cell; as per our earlier discussion, there are two criteria one is a smaller size, second is the symmetry. So, in the case of a cube, you saw it does not follow the symmetry. In this case, we can see that there is a smaller cell size, which is preferred. As a result, it converts as a body-centered tetragonal. That is why facecentered tetragonal is not present in the Bravais lattices, because it can be represented by a smaller body-centered tetragonal unit cell. So, this is why FCT is not there and why is FCT not a Bravais lattice.

Why is C - centered tetragonal cell is not a Bravais lattice? Let me draw a C - centered tetragon again, and I will have to make two unit cells. Because you can always make a simple tetragonal cell with the same symmetry. So, the answer is C - centered tetragonal is nothing but simple tetragonal so, that is why this does not exist.

 Similarly, for hexagonal, you can see there is no FCH, BCH, or CCH. The reason for that is, the moment you put body-centered and face-centered, you lose the 6-fold rotation symmetry, it no longer remains as a hexagonal. So, if you try putting an atom at the center of the unit cell and try to operate, the 6-fold will lost. Similarly, you try to do that in facecentered tetragonal C - centered tetragonal you can see that you will lose the 6-fold symmetry.

For example, in cubic FCC or BCC unit cell over their primitive counterparts? You saw that one FCC is made of four primitive lattices, what is the shape of that lattice? It is a parallelopiped, and it is not a regular shape like a cube shape or something like that. So, the reason why you choose FCC over the primitive counterpart is that FCC has higher symmetry in the cube and has higher symmetry elements; it has four 3-folds, 2-folds and 4-folds. Whereas, if you choose only the primitive unit cell, you will lose some of the symmetry elements. So, that is why FCC, although it is a bigger unit cell than the primitive unit cell. So, higher symmetry despite the larger size, the same is true of BCC, the same is true of any other non-primitive structure which is chosen in comparison to the primitive structure.

 If you draw an FCC unit cell, the question that I want to ask you is that, can this FCC be represented as body-centered tetragonal? For example, if I draw a neighbor to it, this is a neighbor, this is body-centered tetragonal. So, the question is, why can FCC not be represented as a BCT lattice? So, you can see the symmetry FCC has four 3-folds, it has 4-folds. So, three 4-folds and it has six faces so, three 4 -folds and it has six 2 -folds. In the case of tetragonal, you have one 4-fold and two 2-folds. So, although BCT has a smaller size than the FCC unit cell, the symmetry of FCC is higher. So, since the symmetry of FCC is higher, we choose a higher symmetry.

So, when you have this conflict of symmetry, then symmetry prevails when the symmetry is similar, then you choose the smaller size.

The two defining criteria are symmetry and size. The symmetry prevails over the size. Why do not we have 28 Bravais lattices? Why do you have only 14 Bravais lattices? And the reason lies in symmetry that some of them can be represented either by higher symmetry structures or by smaller size unit cells, or in some cases, they do not represent the symmetry of the crystal system at all. For example, in the hexagonal system if you try to draw C - centered or F - centered or I - centered unit cells, you tend to lose the defining symmetry of the crystal system itself.

So, these are certain considerations that we take into account when we talk about the crystal systems and symmetry. So, I hope now there is some clarity on why do we have 7 crystal systems? And which are defined based on symmetry, and each of these has a defining symmetry, and it is the combination of symmetry operations which defines in which class a particular shape will belong to. And the choice also, as we said, in the beginning, you have multiple choices of unit cells, you still choose a smaller unit cell, you used to choose a highly symmetric unit cell.

So, if you look at this, for example, as an example, this 1D, 2D lattice. So, here you can see now that we choose this unit cell in preference to either 1 or 2. So, 1 is preferred over 2 because of higher symmetry, and this is nothing but the combination of so; here, rotational symmetry plays an important role.

So, let me now summarize the whole crystallography in a few minutes so, what we did was we started with point lattice is nothing but a regular array of points in a space with each point having an identical neighborhood. So, a regular array of points with an identical neighborhood. Then we defined a unit cell, and the unit cell is defined as the smallest repeatable unit, which can be translated into the lattice without creating any gaps..

So, you will leave these gaps in between so, and there should be no gap at all so, that is why if you look at the rotation symmetry, there are certain operations which are so, you can see that 2-fold rotation, you have filled space. So, if you have rectangles lined together, these are rectangles that have 2-fold symmetry. So, all these rectangles will fill the space; there are no empty spaces. 3-fold rotation again, you will fill the spaces now, of course, this will make a hexagonal symmetry, but you can see in the case of a cube, for example. So, all these will fill the space is 3-fold and 6-fold again space-filling. So, space-filling is an important criterion.

The squares all of them will fill that space. So, when we talk about 3-fold you talk in the context of, for example, cube 3-fold, you cannot talk in cubic because triangles do not fill their space regularly. So, as a result, you do not talk about the triangular lattice. So, these will fill that space 4-fold will fill the space-filling.

However, when you look at the pentagon now, we look at the pentagon you try making a pentagon now in a bit around it. So, you will see that if you try making a pentagon like this, regular pentagons would not be able to fill the gaps. Now, these angles around a point, you need to have 3600completed, and since each of these angles is how much? This is 720; another pentagon will give you 720, but you cannot have five pentagons sitting around a corner. So, if you try building now, this is one if you try building around it, this will go something like that. So, you leave a gap here similarly, and if you try to do the same exercise on other points, you will try leaving gaps. So, pentagons do not fill the space. So, there are gaps so, there are gaps in the structure with pentagon filling.

Moreover, hence, 5- fold is not found in crystals in periodic crystals because there is no space-filling, you leave gaps in the structures. So, this was another thing which has to do with the. Then we looked at the concept of lattice parameters, which is a, b, c, α, β, nad γ, and correlations between these will are dependent upon the crystal systems and Bravais lattices.
        Then we looked at what is primitive and verses non-primitive lattice. And then, we looked at what is the concept of the motif because it will eventually determine how bigger the unit cell be, what kind of symmetry will it follow, and what kind of space and point group will it have. So, this is a very important concept, and then we moved on to the concept of crystal systems and Bravais lattices.
 
So, you have 7 crystal systems and 14 Bravais lattices. We have seen why do not we have 24?. There are possibilities are P, I, F, C. So, we saw the symmetry operations, and we saw that depending upon the defining symmetry of the crystal system you choose a unit cell which is either smaller in size it has higher symmetry and based on those considerations we come up with only 14 Bravais lattices, we do not have 28 Bravais lattices.
        So, this is sort of a short primer on crystal systems, Bravais lattices, symmetry, and crystallography. There is something which leads further from this is a space group endpoint group, but we will not consider that in this class, it is beyond the scope of this, but if somebody is interested he can he or she can look at the books which can give you more information.
 
Book on Crystallography will give you knowledge of point groups and space groups. So, point group and space groups are further classifications of crystals. So, within the cubic crystal class, you will have various other connotations depending upon how atoms and molecules I mean most of these are not single atoms, most of the compounds. So, in compounds, how motif arranges itself at various sites will be determined by their arrangement, and this will be this will give rise to us. They can be classified based on the point in space groups.
 
So, we will leave this out of the discussion post for this lectures; we will now move on to the next topic/update, which is on miller indices, which is a way to classify crystals and understand their various properties.
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