 Let us begin our lecture. So, now I am back to you with a different topic altogether which is about crystallography. So, we have actually four lectures in what traditionally is called solid state physics or these days called condensed matter physics. The two lectures are more on the crystal aspect. So, these lectures I will be covering and two lectures which are more on the band theory and free electron theory aspect that will be called by Professor Suresh. What I am covering here is first lecture would be on just pure crystal aspects and the second lecture will be on the X-ray diffraction. So, this is the way I am planning the two lectures. So, let us begin with our basic crystallography. Now, when we start discussing crystallography, we start always first with certain definitions. So, let us start with the most important and most useful definition which is the definition of what we call as a Bravais lattice. So, let us go to the concept of Bravais lattice. When we talk of Bravais lattice, many times it is also referred as lattice, but one has to be clear because there can be also lattices which are not Bravais. I will give you example. But let us first be sure that when I am talking of a Bravais lattice in solid state physics, it is not the crystal. It is just a lattice which is a set of points. It has something to do with crystal. What it has to do with crystal and how do we generate this crystal that we will discuss later. But as such the lattice is not a crystal. It is also always infinite. You know you remember that crystal is always finite. On the other hand lattice, a Bravais lattice is always meant to be infinite. You can always ask the question that if I am going to provide a link with actual crystals which are always finite, why do we at all talk about an infinite lattice. That is a interesting question. The thing is that as I have been telling in physics, we always start with the most ideal situation and the most ideal situation is when you have perfectly periodic or what we call as infinite lattice. Really deep inside the crystal, then it does not matter because we do not see the surfaces. Only problem will come if you are looking at some specific properties at the crystal surface. That is where one has to be little more careful. In fact, surface physics has become a different branch of physics. So long I am talking about the properties which is essentially in the bulk of the material. Probably the surfaces will not be seen and we can treat I mean we can sort of start with an idealized concept of an infinite lattice. So let us come back. We say Bravais lattice is not a crystal, is always infinite. Of course it can be one dimensional, two dimensional or three dimensional. When we are talking about crystals, generally we will mean three dimensional. But often it is sort of to give examples to explain the concept, we may use the idea of one dimensional and two dimensional lattices. Now as I said, Bravais lattice is actually a pattern of points. It is just points which are filling the entire space because it has to be infinite. So they fill the entire space and not every set of points which is periodic will be or a regular arrangement will not be called Bravais lattice. In order that it is called a Bravais lattice, this set of points must have some specific conditions. So this is what I have written is a pattern of points filling the entire space obeying certain conditions. What are these conditions? This is what I am going to tell in the next transparency. So the condition is that Bravais lattice defined that is what I have said and infinite array of discrete points. They are not continuous, they are discrete points with an arrangement that appears exactly identical from whichever point the array is viewed. This is what is most important. As I said I will give you examples. The basic idea is that see let us suppose you are sitting on one of the Bravais lattice points. Now you have the idea of direction x, y, z. You know which is x direction which is y direction which is z direction. If I close your eye and move you to any other point of the Bravais lattice, then when you open your eye you should not be able to tell whether you are at the original point or you have moved from there. The environment would look exactly identical. Remember x, y directions are known to you. So looking at the x direction wherever you found points in the original condition you will also find the points exactly in the new condition also when you have been displaced. So it should look exactly identical as before from any point. You could move from one point to any other point of the lattice and it should appear to be perfectly identical. You should not be able to understand whether you are in the same point or you have moved to a different point. If you have such a regular pattern of points this will be called a Bravais lattice. We will also give you a methodical definition but let us first understand this particular aspect. Now for example let us take a two-dimensional simple square lattice. Here you have points. These are points of course I have drawn you know somewhat bigger circle but they are supposed to be points. Now and of course you assume that they are infinite it means they keep on moving exactly this pattern is repeated in this direction, in this direction, in this direction and in this direction because this is two-dimensional. So we are talking of only two-dimensional space. So you have exactly similar type of pattern filling entire space in two dimension, this direction. So I have just shown only a few points but let us imagine that these points are sort of infinite. You have points everywhere in the two-dimensions. Now as if you are supposed sitting on this particular point you will find out that let us suppose we call this direction as x direction and this direction as a y direction. So in the x direction and let us suppose this distance is a you will find out that distance a you find another point. If you move along y direction you will find another point at a distance of a. If you move in i plus j direction where i and j are unit vectors along the x and y direction then at a distance of root 2a you will find the point. Now if I move you from this particular point to this particular point nothing will change. Here again at a distance of a along the x direction you will find a point. Along the y direction at a distance of a you will find a point. Along i plus j direction you will find at a point at a distance of root 2a because this is supposed to be infinite. So if I have moved myself from here to here I will not find any difference to me everything would look exactly identical as before. So I will not be able to tell whether this is the original point or whether I have moved from that particular point to a different point. So this set of points is what we call as a Bravais lattice. Let us take another example what we call as a two-dimensional centered square lattice. So earlier we had only these points now we have another point here in between here. It may appear to you initially that this point is differently located from these two points because this point is at the center of a square while these points are at the corner of a square but it is not correct because here this particular point can also be looked as a corner point of this square. This what I have plotted as a red square and this point which was earlier appear to be at corner of this black square will now appear to be at the center of this particular red square. So depends on how you look any point of this lattice can also can be thought as a point on the center or on the point on the corner. Again I repeat if I move from this particular point I will find a point at a distance of 2a here at a point of this at a distance of root 2a here. If I move from this particular point let us say this particular point exactly identical will happen at a distance of 2a in this particular direction I will find a point at a distance of root 2a along this particular direction I will find a point. So if I moved from this particular point to this particular point or for that matter from this point to this particular point or this particular point to this particular point or this point you will find the environment to be exactly identical. This also sometimes we call as a translational symmetry because if you translate from one particular point to any other point, you find everything is 100 percent symmetrical is exactly like before. It does not appear to you that you have changed your position. So, this is what we call remember you have to move from one particular point to another point. So, this two dimensional centered square lattice also forms a Bravais lattice in of course, two dimension. As I said a lattice a Bravais lattice can be in one dimension, two dimension, three dimension the way I mean depends on your problem. Now, let us take an example of a non Bravais lattice. See here we have set of points which are like this which are also regular and let us assume that they are also infinite it means you have exactly similar pattern which is being followed everywhere right from minus infinity to plus infinity along the x direction right from minus infinity to plus infinity in the y direction. So, this is also infinite set of pattern this is also a regular pattern because you have exactly the same pattern which is being now sort of translated every time, but however there is a difference. I could call this is a lattice but I will not call it as a Bravais lattice because if I move from this particular point to this particular point then definitely my environment has changed because if I am sitting on this particular point if I move along this particular direction let us call this small distance as x at a distance of x I will find a point. But if I am moving from this particular point to this particular point at a distance of x here I will not find a point means definitely I will be able to tell because if I am sitting on this point I will find it out that along this direction at a distance of x there is a point if I am sitting on this particular point I do not see a point at a distance of x. So, I will definitely be able to tell that I am not in the original point but I have moved my position. So, these set of points do not form a Bravais lattice. Let us take one more example of a slightly more regular structure which also does not form a Bravais lattice and that is actually very very interesting type of structure which we call as a honeycomb lattice. In fact, I am using the word lattice let us also be little clear when we use the word lattice sometimes we do not use very very appropriately in talking I mean many times in the crystal we said what is the lattice type which always mean Bravais lattice type. A thing which is non Bravais in principle can also be called lattice but generally we sort of avoid that particular word though I mean in principle if you go strictly by definition these lattices can still be called lattices but they are not Bravais lattices. But often when we are talking of solid statistics because we will normally be dealing only with Bravais lattices. So, we just most of the time remove the word Bravais and we say what is the lattice type of a particular material. So, I mean depending upon the reference to context one should be able to judge what I mean whether I mean a Bravais lattice or non Bravais lattice. So, this is a honeycomb structure you can see these are hexagons these are points which are at the corner of hexagons. So, there is one point here there is one point here there is one point here like this then this is also a very regular arrangement this is filled at the entire space like this. This I insist that this does not form a Bravais lattice. If for example, if you are looking at this particular if you are sitting at this particular point here you find let us call at the distance of a one particular point and then you find whatever is this is also distance a but whatever is this angle in this particular direction a point at a distance a from here another point in a distance of a. Now, if I move from this particular point to this particular point here I do not see at a distance of a another point here I do not see a distance of another point. So, clearly the environment has changed in fact it is very very interesting in this particular thing because you will realize that as I move from this particular point to this particular point actually my environment has turned by 180 degrees because what was x axis I will observe exactly the same thing in minus x direction because in the x direction there was a pointed distance of a here now in the distance of minus x there will be a pointed distance of a while there was no point here but now you will find a point here. So, it appears as if the environment has rotated when I have moved from this particular point to this particular point but remember the condition of a Bravais lattice is that the environment should look exactly identical even if it is rotated that is not acceptable as a Bravais lattice. So, this does not form a Bravais lattice. Let me just tell you for your interest you can satisfy yourself is there as a point here at the centers of all these hexagons then that will form actually a Bravais lattice in the absence of that particular point at the center this does not form a Bravais lattice this we call as a honeycomb structure or honeycomb lattice which is not the Bravais lattice but if I put the points at the centers of this hexagon this will form is what we call as a hexagonal Bravais lattice. Now let us look at the more mathematical definition and more useful definition as the entire solid state physics is concerned and this is what I am calling as alternate definition. This definition has been given for a three dimensional lattice this can easily be extended to two dimension and one dimension if we are interested in. How we will see but let us first just take a proper mathematical definition of a three dimensional Bravais lattice. Now I will not show but it can be seen that the two definitions which I have given the first definition about the environment and the second definition which is more mathematical definition they are identical. So, let us assume that we have three vectors a, b and c and because we are talking of three dimension it should not be in the same plane. So, vector a, b and c are not there are three vectors which are not in the same plane. Now take a linear combination of these vectors which we call as a translation vector t as n1a plus n2b plus n3c where n1n2n3 are integers which could be positive negative or 0. Now keep on giving various values of n1n2n3 right from minus infinity to plus infinity but they have to be integral values you will get a set of infinite translational vectors infinite position vectors. Now add the position at the tip of these vectors you put a point these particular points will form what we call as a Bravais lattice. So, a Bravais lattice can be generated with the help of three vectors which we call as primitive vectors these are called primitive vectors which generate a given Bravais lattice of course these three primitive vectors should not be in the same plane. So, let me read a three dimensional Bravais lattice can also be defined as a set of all points the position vectors of which are given by t is equal to n1a plus n2b plus n3c here n1n2n3 are integers positive negative or 0. The vectors a b and c are not in the same plane and are called primitive vectors of the lattice they generate infinite set of points forming the Bravais lattice. So, in principle I can even look at the other way and for every Bravais lattice I could find what are the primitive vectors. Let us take some example of three dimensional lattices this is probably one of the most common and simple three dimensional Bravais lattice which we call as a simple cubic lattice. In order to imagine this we always assume that there is a cube of course there are no points here the points are only at the corners of these cubes these lines have been drawn just for making our I mean thought process little simpler. So, you have let us assume this cube and this cube is let us suppose is translated everywhere in three dimension from minus. So, you have similar type of cube know there is another cube which is just touching this particular cube exactly identical cube in this direction in this direction and of course in the third direction. So, in all the three directions you know you have exactly similar type of cube which has been repeated from minus infinity to plus infinity. Now, all the points which are points are only at the corners of the cube these points that I will generate will form a Bravais lattice and the primitive vectors of this particular Bravais lattice is written as a times a vector a will be a times i plus and b is a times j and c is a times k i a i of course a is same here because this is cubic lattice. So, this length this length and this length are all these three lengths are same. So, we have only a only these vectors their directions are different one is let us say in this particular direction another is particular direction third in this particular direction. So, these are the primitive vectors which generate this simple cubic lattice. Now, let us look at another cubic lattice which we call as a body centered cubic lattice it is like in two dimension we could consider a centered square lattice we can consider a centered cubic lattice in which in addition to these points which are on the corner there is also one particular point at the center of this particular cube all right. Now, these are the primitive vectors of this particular body centered cubic lattice. Let me just tell you that for the same lattice there can be a multiple ways of writing the primitive vectors there is no unique way of writing primitive vectors unfortunately because of symmetry in crystallography some of these things are not I mean are somewhat ambiguous. So, in fact what you could show that if I take primitive vectors is a i plus a j plus a by 2 i plus j plus k this will also would have served as a proper primitive vector for body centered cubic lattice. But on the other hand we normally like to take a better symmetrical type of primitive vectors we do not want specifically to be preferential or biased towards x y directions. So, we generally write the primitive vectors of this particular type of bravais lattice as a is equal to a by 2. See it is easier to remember when you are talking of the first lattice it is the first one which is negative sign second and third as positive sign when I am writing the second primitive vectors it is the second one which has negative sign first and third a positive sign when I am writing the third vector it is the third one which has negative sign and first and second are having positive sign. Now unfortunately because there is always a three dimensional thing involved which is not always easy to write on transparencies or on chalkboard I always tell students that imagine that you are sitting in a room whatever is your room is assumed to be a cube then you start imagining that particular cube. So, what you will say that this particular point which is saying minus i plus j plus cube if let us say this is x direction this is y direction. So, you are going in negative direction. So, you are going to a neighboring room like that and because j and k is positive this is supposed to be j and this is supposed to be k. So, this particular point starts from this particular origin and goes to the center of the neighboring room. So, similarly these a b c can be defined as a position vectors of similar type of cubes which are in the neighboring cubes. It is much simpler to think that you are sitting in a cube cubical room and you have a sort of infinite array of rooms in all the three directions and then try to visualize these things and imagine these things and then draw these vectors. The next lattice that we are going to talk about or let next, Bravais lattice that I am going to talk about is what you call as a face centered cubic Bravais lattice which is the one which I am giving here. Here there is no point at the body center you have points of course at the corners of the cube. But then you have remember in this particular room if you look you will have six faces four faces there are four walls one is ground another is roof. So, there are six things 1 2 3 4 and this is roof and this is your ground. So, there are six faces inside a cube and each of these faces is centered. So, here if you look at the roof is also centered if you look at your ground the ground is also centered then you look at all the four walls all these four walls are centered. So, there are six faces and all these six faces are centered. This is what we call as a face centered cubic lattice. In this case of course you can write the primitive vectors as this and in this case when you are writing the first one is missing i is not there when I am writing the second one second one is missing that is j is not there you have k plus i when I am writing the third one the third one that is k is not there. So, when I am writing a is a by 2 j plus k i does not appear when I am writing b is a by 2 k plus i j does not appear if I am writing c is a by 2 i plus a. Of course you can again visualize if this is your origin if I am writing this is x y direction this is y direction i plus j divided by 2. So, this is the vector joining from this particular point to this particular face center which will be one of the primitive vectors. Similarly, when I am writing j plus k this is j plus k. So, a vector joining from this particular point to this particular point here at this center this is what will be called the second primitive vectors. This I leave is an exercise that you take a linear combination and find let us say position vector of this particular point as a linear combinations of i a b c. So, you can satisfy yourself that by taking appropriate values of n 1 and 2 and n 3 from these primitive vectors you can reach this particular point this point this point or any other point of the lattice when this is your origin. So, these are the primitive vectors of what we call as a face centered cubic lattice. Now, next important concept is what we call as the concept of basis. So, we started from a set of points. Now, what do I do with these set of points? These set of points I associate certain thing with these points this whatever we associate with these points that is what we call as a basis. Now, the concept of basis is little more general there are many ways in which basis can be used and is the concept of basis is useful. But thing is that you start with a bravais lattice and then each of the lattice point you try to associate something you associate either different set of points or eventually you associate atoms or a set of atoms to generate crystal structure. So, basis is something which is being used which is being associated with each point of the lattice. I will give you the example, but let us just see there are 3 ways in which are 3 usefulness or 3 concepts where we use the idea of the bravais lattice. First often we come across lattices which are not bravais. For example, I have given you the example of honeycomb structures and some other when there is another structure I give you example which are non bravais lattice. Now, what you can do you can start from a bravais lattice and by associating 2 or more points at each of these you may be able to generate a non bravais lattice or in other words if I started with a non bravais lattice I can convert this to a bravais lattice by taking more number 2 or more number of points as me associated with each of the bravais lattice I will give you an example. The second thing where second place where we use the concept of basis is when I want to convert a basis lattice a bravais lattice to a simpler bravais lattice. See we start with a lattice which is also a bravais lattice but this I want to convert into a simpler lattice because I want to emphasize certain type of symmetries and then I convert one bravais lattice to another simpler bravais lattice. So, one way is that convert a non bravais lattice to a bravais lattice. Other thing is that convert a bravais lattice also to a simpler bravais lattice by using the concept of basis means you start with a simple bravais lattice at each of the points you associate certain number of points exactly in identical way what is important here is that whatever is your associating basis the exactly the same basis must be associated at each of these lattice points at each of this bravais lattice points. Now, the third thing which is where we are we come into picture is that you put an atom or set of atoms atoms ions or whatever it is set an atom or set of atoms exactly in identical fashion at each of these points of the bravais lattice to generate an ideal crystal. I am using the word ideal crystal because as we know that crystals I mean a realistic crystal is never 100% ideal if nothing else at least it will have surfaces you cannot make it infinite but even you will always find certain defects in crystals a normal crystal will never be 100% perfect. So, this is the reason I am using the word that this is to generate an ideal crystal and as I have said in solid state physics you start always with an ideal thing and in solid state physics we always start with an ideal bravais lattice and then only we discuss the physics you know it produces certain amount of simplicity in the way we understand the condensed matter physics or solid state physics if we start with this concept of an ideal bravais lattice. Let us give examples of this particular type of all the three things. So, this is one non bravais lattice that we have talked now consider these two points as one set. Now, instead of this particular point take any point does not matter for example, take this particular point take this particular point. Now, what you say that you forget about another second point you just take only one point this as you know is a square lattice now at each of the bravais lattice points you associate two points one here another at this particular distance let us say minus i minus j at a distance of x the starting from a bravais lattice at each of the lattice points I started putting two points one at the original and let us say first we take this point as the origin I put one point itself there another point here at a distance of x in direction minus i minus j then these two points form the basis starting from a square bravais lattice by putting two points one at the origin another at this particular point I will be able to generate this type of non bravais lattice structure. Similarly, you go to this particular point now again you take your origin take this point as origin put one point at origin another point you put at a distance of x at minus i minus j direction you will get this particular point you take this particular point as origin put one point itself there another point at a distance of minus a by 2 i plus j. So, this non bravais lattice can be looked as a bravais lattice with two points basis. Now, let us look at the honeycomb structure honeycomb structure is exactly identical situation you take these two points these set of two points consider them together you can see that this generates is what we call as a simple hexagon lattice. If I had told you that if you have a hexagon and you have one particular point at the center then that does form a bravais lattice. So, for example you can take whatever you want to take this particular point you take this particular point you take this particular point you take this particular point look only one these points and put exactly identical you point this particular point thing and this particular thing then you go this you put this particular point here you go to this you put a exactly in a similar fashion this particular point you will be able to generate what we call as overall hexagonal bravais lattice. So, you start from hexagonal bravais lattice and at each point of this lattice you start putting two points one here another here one here another here starting from a hexagonal bravais lattice you will be able to generate these this honeycomb structure. So, honeycomb structure can be looked as a bravais lattice which is hexagonal simple simple hexagonal bravais lattice with two points basis. Now, conversion of a lattice this is something which I leave this is an exercise to you but you you can look into it that let us look at body centered cubic lattice I have already defined what is a body centered cubic lattice you start with a simple cubic lattice then in simple cubic lattice you start putting one point at the origin and another point at a distance of a by 2 i plus j plus k let me just write this thing to make it clear. Remember i plus i j k are unit vectors along x direction y direction z direction. So, you start with a simple cubic lattice then you take a by 2 i plus j plus k this is the point which will lead you to body centered. So, this was my simple cubic lattice where you have points at each corners. Now, let us suppose we take a particular point here put one particular point here another point at a by 2 i plus j plus k if I move half the way here half the way here and half the way here you I land up at the center of this particular cube. So, you put one point here one point here then you go here you put one point point here another point there you put the one point here another point here. So, from this simple cubic lattice by taking two points basis I will be able to generate a body centered cubic lattice. So, a body centered cubic lattice can also be thought as a simple cubic lattice with two points basis. So, remember body centered cubic in its own self is a Bravais lattice simple cubic lattice also is a Bravais lattice but simple cubic lattice is comparatively simpler. So, many times I would rather like to call body centered cubic lattice as a simple cubic lattice with two points basis this is often done in crystallography because if we start with a simple cubic lattice things become somewhat easier. So, often in the case of crystallography when I am looking at the crystal structure using x-rays I rather prefer to represent a body centered cubic or a face centered cubic lattice with a simple cubic lattice with a number of points basis. Let us go to the next one. Similarly, a face centered cubic lattice can be thought of a simple cubic lattice with four points basis. So, if you take a simple cubic lattice and start putting four points at each of the point one will be at origin let me just write this also. If you put one particular point start with a simple cubic lattice put one particular point at the origin another point you put at a by 2 i plus j third point you put at a by 2 j plus k and fourth point you put at a by 2. If you start from a simple cubic lattice then at each point you start putting four points one at that point itself and another three points at the position vectors which are given here you can assure yourself that what you will get will be exactly a face centered cubic lattice. So, a face centered cubic lattice can be thought of as a simple cubic lattice with four points basis. Now, create a crystal structure that is a third use of the basis. See often we say for example iron at room temperature has a BCC structure. When I say I mean this is again you know a simple way of putting it when I say iron has a BCC structure or a BCC lattice what it means that you start with a body centered cubic lattice and put one atom of iron at all these points remember the atom is supposed to be much bigger. So, often we show this structure in like points it is not really point the atom is not like a point no atom has much bigger if it as we will be discussing it is much more simpler to consider an atom as a sphere or a much bigger object. So, sometimes you know people always get confused students get confused that these points are atom and they are essentially hanging with one particular line in between that is not really the correct picture. See what we are representing when we are representing BCC structure is only those points the points and you know where the center of these particular atoms could lie. So, when we say iron has a BCC structure it means if you start with a BCC lattice and put one atom at each of these points you will generate an ideal iron structure. Similarly, a sodium chloride can be thought of as a face centered cubic lattice with two atoms basis this I have found that a very large number of students and many times teachers are sort of confused because of some funny reason because they always feel that sodium chloride is a simple cubic structure. Remember if we want to talk of a Bravais lattice everything is to be identical in sodium chloride. Yes if sodium and chlorine would have been exactly same atoms this would have formed a simple cubic structure. Sodium and chlorine are different atoms or rather different ions they cannot be treated at one point. If you look if you remove your chlorine things and look only at the sodium ions then you will probably you will get in fact possible I will bring a structure and probably show you that in that particular case if you look only at sodium ions you will see that sodium ions are placed in a face centered cubic positions. So you must start with a face centered cubic lattice each point of this particular lattice let us say at origin you put sodium ion and then you go you start with iron atom and then you put at a point AA by 2 I plus J plus K a chlorine and ion then you will be getting a sodium chloride structure probably if I remember I will bring the next time a structure and try to show you how the sodium chloride structure can be thought of as a face centered cubic structure. So it must start from face centered cubic lattice and it must contain two ions basis at each of the point you put one particular atom at origin and another atom at a distance of A by 2 I plus J plus K and you generate a sodium chloride structure. Similarly cesium chloride structure if cesium and chlorine would have been identical this would have formed a body centered cubic structure but cesium and chlorine are not the same atoms so I cannot call it a body centered cubic structure I actually call it a simple cubic lattice with two atoms basis or rather two ions basis one ion again being put at origin another at ion being put at A by 2 I plus J plus K. See in sodium chloride we start with a face centered cubic lattice while in cesium chloride I start with a simple cubic lattice. Now diamond is another structure where you are having exactly two type of atoms basis but both of them are exactly identical unlike sodium chloride or cesium chloride where sodium is different from chlorine in diamond you have diamond is carbon you have only carbon atoms but these carbon atoms if you just replace these carbon atoms by points that will not form a bravais lattice. So if I have to look at diamond if I have to describe the structure of diamond I must describe a lattice and a basis if at every structure is defined in terms of a bravais lattice and a basis when I take the bravais lattice of diamond it will be face centered cubic lattice and in the case of diamond you start with a face centered cubic lattice and put two carbon ions one at origin another a distance of A by 4 I plus J plus K. If a diamond structure are one atom at the origin another at A by 4 remember A by 4 I plus J plus K which is one fourth the length of the body diagonal this will give you a diamond structure. So as we say we start from a bravais lattice then we start talking about basis which consists of atoms or ions and generate a crystal structure. So a crystal structure in principle is defined as you have to define two things one is a bravais lattice another is the basis. Now let us talk about the planes because when we are going to talk about the extra crystallography one thing which you always talk as all of you know we talk of the planes because Bragg condition talks about the planes and the distance between the planes. So let us first see what we mean by planes inside a lattice because this is again an aspect in which I find students are somewhat confused. The thing is that a plane is something which you draw within a lattice a plane is a plane the standard plane there is nothing special about that thing. But when I am talking of a plane inside a lattice this must contain some lattice points this must contain lattice points in fact every plane will consist infinite lattice points. It must pass through lattice points I cannot think of a plane I mean when I am talking of a plane inside a crystal I am not talking of a plane which does not contain any point. In principle it is always possible to draw a imaginary plane which is let us say between two points for example you know let us take a two dimensional structure. So you have here you have point here you have point here you have point here okay. I can imagine a plane which is something like this of course in two dimensions it will be a line okay this does not contain any point. Now this according to me is not a valid line or a valid plane okay but a line like this which is passing through these lattice points does represent a proper line or a proper plane. So a particular line or plane in three-dimensional line in two-dimension must contain lattice points then only we call it a normal plane whenever we are talking of the planes okay it always must contain points. We say many set of planes can be drawn through lattice points planes must contain lattice point. Then by symmetry because remember we our starting point of Bravais lattice is that the lattice is a perfect translational symmetry. So if a plane has to pass through a point okay then through every point a parallel plane can always be drawn okay. So every lattice point through every lattice plane a parallel plane has to pass. So I am giving an example here. For example this is just taking a two-dimensional thing it is easier to draw two-dimensional lattice. If you are working on three-dimensional thing you can imagine that this structure is also moving in the third direction out of the plane of the paper okay and these lines are not longer but a plane like this okay. For example I have drawn one set of planes or one set of line which are like this. Now these are parallel lines okay similarly you can draw a parallel line which is passing through these planes. Similarly through every plane you can draw a parallel line okay which will pass through some of the plane. Remember this line or this plane will not pass through all the points. It will pass through some of the points but nevertheless because this lattice is infinite. So this will contain infinite points okay but a parallel line or parallel plane must pass through every point. So through this particular point also exactly parallel plane must pass. Now there is another set of planes which I have drawn okay joining this particular point to this particular point. Now there is another set of planes which is parallel to this which is also passing through these lines. Similarly you can imagine of another plane which is passing through this particular points okay. So there will be point somewhere here to which it could join something like this okay. So basic question is that a particular plane must pass must contain a point and through every point a parallel plane must pass. Then we will call these as the planes inside a lattice. Now in order to identify a set of parallel planes what is important word is a parallel planes. Remember we are always when I am drawing Miller indices of a plane we are always characterizing the set of parallel planes. It is not just one plane. One plane generally does not have a meaning because of symmetry because of translation symmetry whatever happens to this particular plane has to happen for any other parallel plane through any other lattice points because all points are equivalent there. I mean there is absolutely no difference between these points. Their environments are exactly identical. So the way a particular point is going to behave exactly the same way another point is also going to behave. That is what is the starting point of a Bravais lattice okay. So this Miller indices is used to characterize a set of parallel planes because many times especially in X-ray diffraction we have to talk from which particular plane this diffraction has come okay. We have to identify these planes okay and therefore when I say a plane it always means a set of parallel planes. So this is used to characterize a set of parallel planes and this is the way we do it probably all of you know about it because this generally I am sure you must be teaching at some of the things and even otherwise at high schools people give the idea of Miller indices okay. But just to complete the story I am giving this concept of Miller indices. Take intercepts on one of the planes of the set of along suitable set of axis I think this language is little odd. Take intercepts of one of the planes okay. What I am saying that out of the parallel planes you choose one of the plane okay. Take that particular plane and find out intercept on a set of suitable axes okay. These suitable axes for example could be your primitive vectors themselves ABC. Sometimes we may not use these primitive vectors as I will be discussing later okay. To emphasize symmetry I may use a slightly different type of vectors. For example when I am talking of let us say body centered cubic or face centered cubic I often draw in terms of x, y, z okay or in other words I convert these lattices into simple cubic lattices as I said just now said how to convert we have just discussed this aspect and using that particular way express these you know set of axes okay these intercepts with those set of axes okay. Generally from the reference of context it is clear whenever I am talking of a particular plane what are the axes with respect to which I am defining these planes. Sometimes there could be ambiguity that is important to tell you when I am talking defining these planes which are the set of axes that I am taking choosing which I am defining these planes. So take intercepts of one of the planes of this particular set along a suitable set of axes and express in terms of lattice constant. So I am giving you this example later. Take reciprocal of these numbers multiply by suitable number to make the numbers integer with no common factors okay. This is important often in X-ray crystallography we also give a planes which are common factors we also talk of 2, 2, 2 planes or 3, 3, 3 planes okay which means slightly different okay a few of time we will discuss in the when we are discussing the X-ray crystallography. So this is a simple example you know let us suppose we are talking of this particular plane which is here alright let us suppose these are three axes these need not be even mutually perpendicular to each okay it could be just some of the axes of our choice. So let us suppose this is one axis and the intercept here is twice lattice constant if I intercept may not be even at one particular lattice constant depends on what type of plane you are talking but as I am saying taking this is simple example. So you have intercept here as 2A here then along the B direction which is I am calling as a second direction here you have intercept as 4B 1, 2, 3, 4 okay this distance and this distance need not be same okay because this lattice need not be really cubic lattice then along the third direction of just C direction the intercept is this 3 times C. So this is intercept is 2A this intercept is 4B this intercept is 3C. Now I remove ABC because I have to express this in terms of lattice constant so I will just call this as 2, 4, 3 then take reciprocal okay reciprocal will be 1 upon 2, 1 upon 4, 1 upon 3 then you multiply by suitable number to convert this as three numbers okay in fact and they should not be a common factor and there should be no fraction. So you will find that the number which I need to multiply it by is 12 if I multiply it by 12 this becomes 6, this becomes 3, this becomes 4. So Miller index of this particular plane will be called 634. So this is a 634 plane and remember infinite set of parallel planes which all will have Miller indices 634. Just to give an example if you are talking of simple cubic lattice this is 1, 0, 0 plane in fact this is the plane to which X direction is normal. So you can see the intercept along the X direction is lattice constant along the Y direction and Z direction the intercepts will be infinite if I take invert of infinite I will get 0. So 1 upon 1, 1 upon infinity, 1 upon infinity so this becomes 1, 0, 0. So this is the 1, 0, 0 plane in simple cubic lattice. This is what will be 1, 1, 0 plane which divides the cube into the half diagonally okay. You can see intercept along X direction is A, intercept along Y direction is A, intercept along C direction is infinite because C this C direction will never intersect this particular plane. So if you take the reciprocal it will become 1, 1, 0 so this is 1, 1, 0 plane. You take 1, 1, 1 plane this is 1, 1 plane you take the corner you take this corner this corner join these three this is 1, 1, 1 plane okay. You can see the intercept along the X direction is A, Y direction is A, Z direction is also A. So this is actually a 1, 1, 1 plane of a simple cubic lattice. Sometimes we have to give the directions also an electron moves in this particular direction. So we have to say in which direction it moves with the crystal structure. So we have to many times define directions in fact even the band structures and many properties of the solid depend in the which direction I am trying to look for them alright. So there is other indices here these are much more simpler than the Miller indices because here I do not take any reciprocal. If I have a direction I mean a direction can always be represented by a vector quantity. So if I have a direction of UA plus VB plus WC where A, B, C are any three vectors appropriately chosen for example this could be primitive vectors or this could be what we call as often a conventional vectors. Then in this case UVW written in a square bracket remember in the case of Miller indices for plane I had written that in smaller bracket. If you write this in square bracket this UVW will be the direction or will be the index for the direction. There are some other conventions also in fact many times we have to represent a set of equivalent planes. For example let us talk of a cubic structure. Now what I am calling is exactly x direction in a cube I could have called as a y direction or I could have called as a z direction because all these three directions have to be equivalent purely by symmetry whatever happens along the x direction would also happen along the y direction would also happen along the z direction because these are 100 percent symmetrical directions. There is nothing which is different I mean what I as I said whatever I am calling is x direction I could have called as a y direction my lattice constant would not have changed nothing would have changed. So whatever property I expect along in the 1 0 0 plane I will also expect that same property in 0 1 0 plane or 0 0 1 plane. So these planes for example could be all equivalent planes planes like 1 0 0 0 1 0 or 0 0 but let us talk of planes. Now whatever property happens along 1 0 0 plane will also happen along 0 1 0 and will also happen along 0 0 1 direction. So if I have to talk for all three of them together I use a slightly different decks and then I write this as 1 0 0 plane. See if I write this with the curly bracket essentially it means that 1 0 0 or all its equivalent directions. This equivalent direction has to be referred from the point of view of symmetry. I have given you example of a cubic lattice which has the highest symmetry if the symmetry goes down then accordingly appropriately chosen equivalent planes will be referred by 1 0 0 putting in curly bracket. Similarly if I am looking at equivalent directions see whatever happens along 1 0 0 direction should also happen along 0 1 0 direction then I write this into angular bracket. So this will be 1 0 0 directions. So this is what we said a set of equivalent planes equivalent world has to be looked from the point of view of reference of context can be written as curly bracket hkl. Similarly a set of equivalent direction again equivalent being defined depending upon what type of crystal structure you are talking will be written into the angular bracket uvw. So uvw represent a set of direction uvw not only uvw but also equivalent of uvw. Then I come to the concept of unit cell. Unit cell is a very very important thing. See advantage of defining a Bravais lattice is that most of the time you define a property within a particular volume of a space and you just translate this particular thing over the entire space. So a particular volume of the space if it is translated through all the primitive vectors of the lattice all the translation vectors of the lattice if it fills the space completely without leaving any void and without any overlap then this particular unit cell is called a primitive unit cell. So as I said a volume in three dimension when translated through all possible translation vectors fills the entire space without any overlap and without any voids is called a primitive unit cell. Unfortunately there are many ways of representing a unit cell there are multiple possibilities. You have to normally appropriately choose some of these things. But also it must contain on the average of one particular point. Let me just give you some example of a two dimensional structure. So then just to for example if you are having a set of points let us say square lattice that is what we had considered earlier. For example this could be one of the unit cell. If you translate this way, this way, this way or this way it completely fills the space. You translate through any of the translation vectors. Of course translation vectors I have defined only for three dimension. If I want to choose two directions then I have to take only two vectors in order to be only there will be two primitive vectors a and b not in the same line and then you will have only two integers n1 and n2. If you are talking about one dimension then there will be only one vector. So translate through any of the primitive vectors here and then you will be able to fill this space completely. Now this could be one of the primitive unit cell. For example this could be another possibility of primitive unit cell which will have exactly the same condition. So there are multiple ways of having a primitive unit cell which will represent the same lattice. Now depends on your convention depends on your choice which you would like to use as your unit cell. Of course as I said it must contain one point on the average. Let me just come back to this particular figure. See I would like to mention here that it contains only one point on the average. See here if you look at this thing it is very clear that it contains only one point on the average. Here if you look at this particular corner there are four corners all right in this unit cell. But each one of them is being shared by four squares one square here another square here third square as fourth square here. So therefore when I am taking on the average this particular point only one fourth has to be accounted. When I count one fourth and there are four such points so on the average I will get one particular point per unit cell. Now often to emphasize symmetry I do not use a primitive unit cell but I choose a unit cell which is of dimension n times primitive unit cell or volume n times primitive unit cell where n is an integer. This is what we call as a conventional unit cell. So a conventional unit cell is a volume in three dimension when translated through a subset not all possible vectors but through a subset of all possible translation vectors fills the entire space without any overlap, lab and without any void. This is often used to emphasize symmetry. For example whenever we are talking about body centered cubicle lattice and face centered cubicle lattice we always draw a simple cubic unit cell because that is little more convenient. It is again putting back the same words that I represent a body centered and a face centered cubicle lattice as a simple cubic lattice. So we just generate we just write a simple cubicle unit cell or a cubic unit cell. Remember a conventional unit cell which is of the form of a cube for a body centered cubicle lattice will contain two points on the average while in the case of face centered cubicle lattice it will contain four points. Now in fact there are examples which I will not work it out which you can always see. In fact most of the text book this has been given so I will not go into the detail. In three dimensions there can be 14 different type of rival lattice. When we say and these are classified in seven crystal systems when we talk of crystal systems and lattices they are always defined by the symmetry. In principle you take the least symmetrical lattice and every lattice will turn out to be a special case of that particular lattice. We do not go by that. Whenever the symmetry becomes higher we try to talk of a different lattice. So for example out of all these 14 rival lattices cubic lattices have the highest symmetry. In fact generally in crystallography if you have a proper course of crystallography they will talk about what are these particular symmetries and using those symmetries they will define crystal systems and crystal lattice. It becomes more of a problem group theory. But at this particular point we will just say that we just take this result from that particular crystallography thing and we say that in three dimension there can be 14 rival lattices and seven crystal systems. There are three cubicle lattices which we have already defined body centered cubic, face centered cubic and simple cubic and this along with what we call later as hexagonal closed pack which is not really a Breville lattice. A very large number of solids crystallize in one of these lattices. So this I mean in a first course on crystallography most of the time we talk only of these lattices no other lattices. But of course there are many other lattices which are actually present many complicated solids or many complicated structures can be much more difficult than just simple cubic just cubic or hexagonal structures. Now we come to the last part of this particular topic which is actually very interesting thing which we call as a closed pack structures. These closed pack structures are of interest because of multiple reasons. These are the structures which actually form which waste least amount of space when we are forming these structures. See earlier I had mentioned in this particular lecture that is not a very good idea in fact that is generally misconception among students thinking that these are atoms which are just lying where there are two here and there is a small line which is joining these two. It is not a correct picture. A slightly better picture is to treat these atoms hard spheres which touch each other. Actually treating them as hard sphere is sphere is also not correct because then we cannot consider the vibrations. So they are strictly spring not hard they are somewhat elastic but as I said we start from the simplest thing and when we talk of crystallography we generally talk in terms of the assuming that these atoms are hard spheres and they touch each other. So we do not have that type of concept we are talking. So this picture is not a correct picture in that sense that you have atom and the most you can say there is a center of the atoms but you cannot imagine that your atom is so tiny and there is a big bond which is starting to form this thing. A much better picture would be something like this touching. Now when you need assume that these atoms are sphere there is always going to be certain amount of space here for example which is going to be wasted where there is not going to be any atom. So assume atom as hard sphere and then try to find out a structure where you waste least amount of space which is packed to the maximum extent and when we talk of closed structures remember we are talking of single atom of course you can get a better packing if you have multiple atoms and a tiny atoms fill this particular small space which is being created as a void here. But when we talk of a general closed packed structures we mean only those structures in which there is only one type of atom. So consider a solid with only one type of atom assume atoms to be hard spheres we want to create structures which are packed to a zest. Now let us try to do this in one dimension. So this is a hard sphere this is another hard sphere and if you have only one dimension if you have only one line there is only one possibility that you can always put these atoms such that they touch each other. So you will not like to create a gap between these two these atoms must touch each other so that they waste least amount of space in one dimension. Now let us assume that you have created one line of atoms like this infinite line you have another similar lines now you want to put it on the top here at the bottom and want to create it in two dimension. Now if I put another line and put this line on the top of it so that these are just above it. So you have another line where the atom is here another atom is here then you can see a amount of space that is going to be wasted here is going to be extremely large. So better would be take similar type of line and push it slightly on the right hand side or on the left hand side so that this particular bottom part of the sphere actually comes and fits here. If it fits here then the amount of space that you are going to waste is going to be exceedingly small. So if I have to pack in two dimension I take a similar line and put another line so that this line is displaced by half the distance here from here to here actually comes here. So you can see that particular thing coming here which is like this. So I started with a line you put another line on the top of it this has been displaced this way this is forming a triangle. Now you can see that amount of space this we call as dimples small dimples each atom this each sphere is surrounded by six dimples in fact you can see that this is forming a hexagonal type of structure a simple hexagonal type of structure there is also an atom here. So the amount of space that is wasted here is least if I am putting this atom here if I have displaced this atom here and put on the top of it that time this particular dimples would have been much larger. So just by pushing it slightly this side so this is in two dimension forming a cross pack structure of course you can assume the fourth line which is going to be put this is again going to be put so that this displaced like this so it will be a similar line which is going to be put here. So this line exactly identical is going to be put here then this line exactly identical line is going to be put here. This is the way you are going to create a cross packing in two dimension. So all I wanted to say that you look at this particular thing the center of these atoms these are actually forming hexagonal simple structure you not only have points here center but you also have one center here which you know forms a bravillatus as I have been mentioning earlier if this point was not there then this would have been honeycomb structure which does not form a bravillatus but here you have this particular point and therefore this does form a hexagonal bravillatus and remember there are six triangles within one hexagon. Now question is that if I want to now go into the third dimension then I take similar type of cross packed two dimensional structure put on the top of this. If I want to put on the top of it I will have two possibilities that I will show in the next picture. One possibility I mean it is very clear that if I have to put I cannot put just exactly on the top of it then again I am going to waste a lot of space. So the center of the atom which is on the top line if it is occupying if it is just placed above this particular dimple then in that particular case you will waste lesser amount of space. So next plane of atoms I am going to put so that the center lies here but if I put one atom on the top of here because this is going to be sort of spherical I will not be able to put another atom at here this is what I will also show in the next picture. So you can put atoms in such a fashion that it occupies only the alternate of dimples. So if you put one atom on the top here then the second atom will come here third atom will come here. So only alternates of the dimples or what I will call the atoms has to be put at the top of this particular triangle. So only on the alternate triangles you will be able to put this particular atoms you will not be able to put the atoms on each triangle because this particular sphere is going to occupy certain amount of volume. Now the two possibilities either I put in such a fashion that the center occupies this then I will not be able to put an atom on the top of here. Second possibility is that I put one atom on the top here but then I will not be able to put atom here only on the alternate dimples I will be able to put the other atoms. So there are two possibilities in which I could create this particular direction three dimensional structure which I am showing in the next transparency. So this is one of the ways which the first one I call as A second one I call as B. So you see this is yellow thing and I have put it slightly transparent so that you can see the dimple behind it. So I have put this particular atom which is exactly identical atom on the top of this dimple but then I cannot put here because if I have to put it here then again this is going to occupy the space which is not possible. So here I put one atom on the top of this dimple on the second one then I am putting alternate one then I could not put an atom on the top of this dimple then now I am putting atom on the top of this dimple. So alternate of the dimples are being left out this has been left out on this I have put this has been left out on this I have put this one has been left out on the top of this I have put. So only on the alternate of these dimples I have been able to put the points. This let us call as A B. So first one was A position second one was B position. Now this could have been another way I could not need not have chosen these dimples I would have chosen these dimples then situation will be slightly different this will be slightly displaced but that was also A exactly possible way. So let us look at the second possibility now this particular dimple has been left out on this dimple I have put one atom now this has been left out on this particular dimple I have put one atom this has been left out on this particular dimple I have put on the atom. So this is exactly similar type of thing but then the alternate dimples that I am using is become different. So earlier I had put here but not here now I have put it here but then I cannot put it here. So this situation I will call alternate I am using only alternate triangles. So I had used one of the set I have used now another set. So this I am calling as A C. Now let us imagine that I have chosen first as A B now I have to put third one when I put the third one again I have possibilities of two types. So let us suppose I have put this or let us say A C it does not matter. Now if I have to put on this particular thing I will have again two possibilities either I use these crosses as dimples if I put take another plane and put on the top of it this particular atom if it lies above here it cannot lie above here. So either I choose this crosses where I am going to put this third layer such that the center of the atom lie here then this will occupy we will get occupied this will get occupied this will get occupied I mean remember this is the third dimension on the top one then these will not get occupied or I put exactly like what I have said I put on these dimples which are also alternate dimples. So either I choose these blue ones with a cross within or I use this particular red one with a cross arm just a simple cross. So either I choose these particular things or I choose these particular things. Now just I wanted to point out certain things if I decide to put on these red crosses this was my original A thing this turns out to be exactly on the top of the first layer. So your first layer then second layer and the third layer which is exactly on the top of it. On the other hand if I decide to put on this particular thing this is not coming on the top of any of these original spheres on the layer A this is on the dimples which were left out. So remember in the earlier case here these dimples were left out. Now I can put another set of layers on which such that the atoms are on the top of this dimple which was the left out dimple or I can choose this one if I choose this one this would be exactly on the top of the A atoms. So you have two possibilities of putting the third layer remember this is the third layer I am putting. So I can either put here or I can put here. Now in this particular thing I mean there are both the possibilities existing which will give you rise to two different type of structures as we will be talking just now. Let us assume that I am putting on the dimples which have been left out. If I am not putting on the top of the A layers but this layer third layer is being put on the dimples which have been left out when I had put the second layer. In that particular case this is what the situation would look like. So this was your originally first A layer. This I had called it C but you can call it B does not matter because B and C is always relative. Now the glue the third one things have been put on the all these are not on the top of green you remember you can see. This has been put now on the top of this left out dimples. So this has been put on the left out dimples. This has been put out on the left out dimples. Remember this yellow was put here this was left out on the top of this this is the blue. This on this particular thing there was an atom. Now this particular thing as you see here this was the original blue yellow this was left out. So I had put a green thing like third one there was a yellow here. So I have not put a point here. Fourth one was left out here on this I have put it here. All right. So there are two different ways of putting this is what we call as A B C type of stacking. If I have put this particular layer which was exactly on the top of green this I would have called A B A type of because you have gone down the third layer which is exactly on the top of the first layer but remember all these will give me the closest packing. The packing fraction as we call it will be exactly identical in both these cases because the amount of space that you are going to waste is exactly identical. Now this is what we call as two type of closed packed structures. If you have a stacking of A B A B A B type this is what we call as a hexagonal closed packed structure which as I will just now be telling does not form a Bravais lattice. And if I have a structure which is called which is A B C A B C A B C type the one which I have shown you then we call this as a cubic closed packed structure and this is exactly identical to what we call as a face centered cubic lattice. How I will just now show. Actually it is very simple to see that this is hexagonal closed packed structure because you can see the hexagons here. The problem when we are talking of FCC lattice you do not see these hexagons because you have to reorient themselves to make them cubic. Now let us look at this. This was an FCC unit cell which I have drawn. Now if you look at this point which is at this particular face center you look at this particular point which is at this face center. You have this point you have point this and you point this. So what is important is to realize that in face centered cubic structure 1 1 1 plane this is the 1 1 1 plane. If you join here this will be the plane here. This is the 1 1 1 plane which is the closest pack. So why do not we see that cubic structure when I am describing this closed packed structure from that particular thing because I am not I have to reorient I have to change this particular direction to see this cubic symmetry. So it is easier to see the other way you start with a face centered cubic structure and you say that on this particular and you can see visualize this particular hexagon. If this atom will come on the top of this particular angle this angle will come on the top of this particular angle exactly on the top when I am looking at the 1 1 1 orientation. So you can see it is not unfortunately so easy to see in this picture if you have three dimensional picture it is comparatively easier to see these aspects. So you can see this particular hexagon and you can see visualize that face centered cubic is actually representing a closed packed structure. And the closed packed plane is 1 1 1 what I have drawn as the original plane is actually the 1 1 1 plane of this face centered cubic lattice. Now this is my last transparency the hexagonal closed packed structure does not form a bravais lattice actually it can be represented as a hexagonal bravais lattice. In hexagonal bravais lattice you have one hexagon and another hexagon exactly on the top of it that is what is hexagonal bravais lattice which actually is a bravais lattice there is no in between points. So this can be represented as a hexagonal bravais lattice with two points basis one atom one point being put at the origin and another point at A by 3 plus B by 3 plus C by 2 which happens to be and ABC has to be appropriately defined which I am sorry I have not defined it probably I will give you that or let me just mention so if you take in plane hexagonal A this is hexagon in the plane you put exactly another hexagon on the top of it you will be generating hexagonal bravais lattice. Now A there are two conventions of defining A either you take this as A the one which I have used here is to choose this as B and C is this particular vector on the top which joins this particular plane exactly on the top of it sometimes instead of taking this B people take this as B. Now if you go A by 3 you come here in this particular direction one third of the way then you move one third of the way in this particular direction B you come here at the center of the triangle. So if you are forming if you take this particular triangle by going A by 3 in this direction going A by 3 in this direction you come at the center of this particular triangle. Now you go half the way between these two layers then you center this particular point. So that is what is when this was or what was your B layer and this particular C will be distance between one A layer and the second nearest A layer. So this is what will be giving you a bravais lattice with two points basis one point here another point at the prism that you will and if you have this triangle extended on the top this will be generating a prism. So that particular prism alternate of these prism will be centered. So this is it presents a hexagonal bravais lattice with two points basis another bus first at origin another at A by 3 plus B by 3 plus C by 2. Think I will end we can probably entertain five a few questions. Good morning sir. Good morning sir. Seat the slide number four. Yeah. Yes. Just my question is can we draw the here you have drawn for the 2D am I right? Yes sir you are right. For the SSC. Yeah. But can we draw for any shape like as a cuboid rectangle rectangle or any other? Yeah you can also have a rectangle lattice for example this A and B need not be same. This can also be hexagonal lattice example of which I have given as two dimension. There are multiple ways I do not remember how many but you know there is a certain number of different type of bravais lattices you can have in two dimension. I do not remember that number but if you look at some of the books you know you can tell how many number of lattices are possible. But you are right this is only one of the examples. But you know for example I give you a centered thing there is also possibility of hexagon which I have given you today morning. Okay this can also be rectangular where A and B are not same. All these will form the bravais lattice. Symmetry can be observed from any point of yeah see the basic idea of a bravais lattice is that all lattice points are equivalent. So you look at any particular lattice point and start looking at the symmetry. Okay. Now second question sir already you have mentioned four particular structure the Fc, Bcc, NaCl, Fcc but my questions how we can find out the four Fe it is F Bcc for NaCl is Fcc. Can you elaborate more? Because the only way I can find I am not the only way there are many ways okay but one of the important ways by which I can find the crystal structure is by using x-ray diffraction okay. So I will show you using x-ray diffraction how I can distinguish between a face centered cubic structure or a body centered cubic structure or a simple cubic structure okay. So in fact if you take a sodium chloride and look at the structure you will indeed find it to be possessing all the properties of Fcc structure and not of a simple cubic structure. Okay okay by some XRD as well as some experimental way only you can define yes this is the structure of that one likewise. That is right that is right you are right 100% right. Yes now sir one more one more question is very simple yeah please separate. Sir already you have 37 the slide number 37 sir okay. Here we have drawn the closed spec structure am I right? Yeah you are right. Okay and it is in a three layer form. Yeah that's right. Yes sir but my question is sir is there whatever the space between the two atoms still it is it will be completely vacant or it will be filled up filled up by some material or something. There is some space which will always remain vacant in fact you can calculate the packing fraction this is a problem which we ask students to do it okay. The packing fraction is about 74% if I remember right okay it means still you have 26% of the space which is wasted. Now in principle I mean it does happen for example if you take a diatomic material you know it is possible that in the if you have very tiny atom you can fill in that particular case and you can have a structure which are closer packed but not with a single type of atom. If you have multiple type of atoms it may be possible that that vacant state can be filled by a very very tiny atom. Okay thanks so much sir. Good afternoon sir I have a question basic question in the first slide you have explained that diamond has two bases with two FCC lattice I have a small dot yeah please. Whether you are a diamond is filled only with carbon atoms and how can you justify that it has two bases can you please explain that sir. Okay see what happens the diamond structures can be looked into various type of ways okay one of the ways which is normally given in the textbook is that you take two FCC lattices which are exactly identical okay put one on the top of them displace another lattice with respect to the first lattice along the body diagonal distance of a by one fourth of the body diagonal then you create a diamond structure okay which is given in the almost all the textbook the way I like to explain to the students which is somewhat little more sort of comfortable I will put it like okay let me just try to draw this particular thing so I am I am just drawing a cubic unit cell all right let us assume this is face centered so you have points everywhere you know you have points here here and all the face center which I am not drawing all of them to make picture simple now what you do you divide this particular unit cell into eight parts by taking let us say half of the system so you have half of this okay go a by 2 okay you go a by 2 here okay go a by 2 here generate a smaller type of so this cube has a length of a by 2 and there will be eight such cubes okay you have eight such cubes which is filling the entire space now if you start from this particular point and go a by 4 distance of body diagonal this small cube will be centered all right which is the cube of length a by 2 so a diamond structure can be looked as you start with the FCC lattice FCC unit cell cubic unit cell then divide this particular unit cell into eight smaller unit cell okay then alternate of these particular unit cell will be centered if this is centered next one will not be centered then this will be centered next one will not be centered okay so you can generate this particular structure from FCC where you are putting one point here another point at this particular point then you start from this particular point take one particular point put another point here you start from this particular point put one point here another point here so you take FCC lattice and at each point you put at this origin and you put one point here another point at the center of a smaller cube which has a dimension of a by 2 another which has an edge of 80 by 2. Yeah, please go ahead. Is it clear? Yes, it's clear sir. It's clear for me sir. Thank you sir. Vulture courage please. Sangly. How to calculate density of lattice point for a particular plane? What do you mean by density of lattice points? You want total number of points per unit you sell? That's very easy. Or you want to find out packing fraction? In x-ray diffraction pattern for a particular peak, we get higher intensity. That's right. The reason is for a particular plane, the density of atoms or lattice points are large. I understand what is your question. First of all, it is not correct that the intensity depends on the number of atoms per unit on that particular plane. This is not correct. In fact, this is what I am going to discuss tomorrow. There are two major factors. One which you call as a structure factor and this is what is called atomic form factor. The intensity of a particular x-ray beam, x-ray plane lies on both these factors. Not just on the atomic geometrical factor but also on how x-rays are getting diffracted from various type of electrons and their wave function. So it's a little more complex. The intensity is to explain in terms of the density of planes are not very easy. But if you at all want to find out the density of the planes, that's not all that difficult. For example, look at 111 plane. Now what you have to find out, if you want to really find out what is the density points on this particular plane, you look at this particular triangle. Take this particular triangle, find out what is the area of this particular triangle. How many points this particular thing contains? This contains this particular point, it contains this particular point, it contains this particular point. Find out on the average, how many, this angle is 60 degrees. So this particular thing is being actually shared by six similar planes. Now looking at that particular thing, similarly here, you can find how many number of points are on the average lying on this particular triangle. Remember you have to account for sharing of each point by equivalent planes. Then find out the area, you can always calculate the total number of points per unit area on any given plane. That's not a very difficult exercise. You can even talk about how much of the area is occupied and all these problems. We generally ask our students to do in the condensed matter physics course. But what I want to tell you is that the intensity of X-ray line does not depend on this factor alone. There are other factors also which are important for determining the intensity of the lines. Sir, in cubic structure, the atoms are at the corners of the cube is called as a primitive shape. But in the hexagonal packed structure, the atoms are at the corners of the cube is called as a primitive shape. But atoms per shape in cube and the hexagonal packed structure, they are different. So how it is called as primitive shape? Okay, see think that the primitive unit cell has to be defined for any given type of lattice. Now when I am talking of let's say hexagonal closed packed structure, unfortunately this does not form a lattice. So I cannot call off a unit cell of that lattice. Now what is actually the lattice type? See in that case for HCP structure, I have to consider a Bravais lattice and I have to consider a basis. Now what you can show is the unit cell of the lattice, not of the hexagonal structure. Now if you show that unit cell, that unit cell will be like a prism. This triangle you extend from A point to another A point. It will contain two atoms of let's say whatever material, for example cobalt crystallizes in HCP. But see if you just replace cobalt by each of these atoms, that unfortunately will not form a Bravais lattice. Now also for example for cubic when I am talking of face centered cubic, the cube that I am drawing is not a primitive unit cell. It is a conventional unit cell. Primitive unit cell must contain one point. While FTC structure that we draw face centered cubic will contain on the average four points. If you insist on drawing a primitive unit cell for face centered cubic you can draw it. But generally we do not draw it because that will turn out to be very asymmetric and odd shaped. While it is much more simpler to understand when we have drawn a cubic structure. So we do not mind ignoring the idea of primitive unit cell and talking only in terms of conventional unit cell to describe the structure. Alright, I hope I have answered the question. Don Bosco, good afternoon. Good afternoon sir, I have two questions. The first question is, you have shown that when you are finding the Miller indices, you are using the intercepts that the planes make with the x, y and z axis. It need not be x, y and z. Let me correct it. It need not be x, y and z direction. It could be any three directions. If the plane contains one of the axis, one of the axis lies in the plane itself. You take a parallel plane. It does not matter. You take one plane which is parallel to it. See, Miller indices always represent a set of parallel planes. Do not take that plane but take a plane which is parallel to it. For example, when I drew 1, 0, 0, I did not take really the x, y plane or rather y, z plane. Because y, z plane, the axis will be lying in that particular plane. So I took a plane which is parallel to it. Sir, and my second question is, this is actually just to satisfy an intellectual question. So if I have the discussion, we have used Cartesian coordinates. So can we use maybe some other coordinate system to describe a lattice like say spherical or cylindrical? See, the thing is that we do not normally use spherical coordinate system because you do not have a spherical symmetry inside the solid. But the thing is that we definitely use different type of coordinate system. For example, if I am describing monoclinic or triclinic structure, where the ABC primitive directions do not have 90 degree angle in between them. Or for example, typical example of hexagonal which I have given you. In hexagonal, the A and B that I have defined are really not mutually perpendicular to each other. So I can define them into any trivial system which I personally feel. But spherical polar and cylindrical coordinate systems, because you do not have those type of symmetries inside the material. See, remember a crystal is a set of discrete points. A sphere is a continuous thing. I do not have that type of spherical symmetry here. So generally those type of coordinate systems are not used here. Thank you, sir. You are welcome. Okay, I think we will close.