 see hexagonal unit cell, okay so I will just draw it here, the face is a hexagon, hexagonal unit cell may, the face will be a hexagon, right, so face is a hexagon, okay and in this the extra symmetry will have along this place, you rotate like this every 60 degree because angle is what, angle is 60 degree, so every 60 degree you will get the same impression, okay that's only one such lines are possible, this is possible hexagonal unit cell but cubic unit cell may will have only 23 total, okay. Next write down cubic crystal system, cubic crystal system, the first one is simple cubic unit cell, simple cubic unit cell means what, simple cubic means primitives and in primitive unit cell write down all atoms are present only at the corners, all atoms are present only at the corners, okay, all atoms are present only at the corners, okay, see in all these unit cell now we are going to discuss simple cubic, VCC, FCC, three things we are going to discuss, okay and all these unit cell we have to find out three things mainly that is rank, we need to find out the coordination number, coordination number, we need to find out packing fraction, packing fraction, packing fraction, volume occupied, right, see atoms right, so 100% volume to that one occupied, you see be a atom if you arrange in a cubic crystal lattice, so there will be some void present, okay, so what is the volume occupied, fraction of volume occupied, that is packing fraction of that because we will find out that, but we need to find out the volume occupied by the atoms in the unit cell, and we know the volume of the unit cell, volume occupied by the atoms divided by the volume of what, that is the volume of the cube, right, divided by a cube, gives you the packing fraction, so what we need to find out, a cube that we have already, we need to find out the volume occupied by the atoms or molecules, and since the atoms are spherical in shape, so its volume is what, 4 by 3 pi r cube, but how many atoms are present, that number into 4 by 3 pi r cube gives you what, the volume occupied by the atoms in the unit cell, understood right, so we need to find out what the number of atoms present, which is the effective number of atoms, and that is the rank of the atoms, rank is what, the effective number of atoms, okay, so rank we need to find out, the coordination number is the number of atoms that surrounds one particular atom, this is one atom, it is one atom, right, how many atoms surrounds one particular atom, that is the coordination number of that particular unit cell, correct, so packing fraction of what I said, volume occupied by the atoms divided by volume of the unit cell, if you do not write this, volume of atoms, suppose rank, another rank here, okay, we can find out these, into what we can write, 4 by 3 pi r cube, because the volume of one atom is 4 by 3 pi r cube, r is the radius of the atom, right, divided by a cube, now this value, if you need to find out, you must have the relation of a and r, what is a edge length of the cube, what is r, r is the radius of the atom, you must have the relation of a and r, so for all these of unit cell, different unit cell, we must have the relation of a and r, a and r are the relation, okay, so first we will discuss a and r relation, then we will see rank, then we will see coordination number, and last we will see, packing fraction, packing efficiency, okay, a and r relation is why important, because to find out packing fraction, r and a are relation we must have, okay, so the simple cubic unit cell, if I draw the diagram here, it is this one, so the exact diagram is difficult to draw, because the three dimensional, yeah, simple cubic unit cell, now here the atoms are present at the corners, okay, atoms are corner represented like this, only corners, and in this chapter where radius is represented by r always, edge length is represented by a always, got it, a is the edge length always, r is the radius, now what you have to keep in mind here, that atoms at the corner are in touch with each other, yeah, simple cubic unit cell property, suppose this edge is a v, so a v curve where you have to draw, okay, this is a point and this is v, so actually here it is not appearing that actual structure, but when the atoms will have the radius like this, this is one atom, it occupies on the corner, a is the corner, b is the corner, and another atom occupies like this, okay, so atoms at the corner are in touch with each other, okay, so what is the distance here, this is what, 2 r, and 2 r is equals to, okay, the relation of a and r is what, a is equals to 2 r, so this you write down, in simple cubic unit cell, the atoms at corner are in contact or touch with each other, this is very important, like this you should know the a and r relation in other units also, okay, so once you know the position how it arranged in a red cell, you can establish the relation of a and r, okay, now next time, rank, it is the effective number of atoms, it is the effective number of atoms present, present within a unit cell, effective number of atoms present within a unit cell, what is the rank for simple cubic unit cell, 1, rank is the effective number of atoms, I will just discuss it here, okay, atoms are only at the corner, right, so corner to the atom is distributed among 8 cubes, 1, 2, 3, 4 and 4 on the top, okay, so what we can write, here right on the first point, the atoms present at the corner, the atoms present at the corner are distributed among, distributed among 8 different cubes or unit cell, 8 different unit cell and hence its contribution and hence its contribution in 1 unit cell is 1 by 8, okay, this is the corner cover, okay, okay, corner, second one, phase centre being an atom over, so what is the contribution in 1 unit cell of a space centre if you put an atom here, so it is 1 atom is distributed in 2 unit cell, 1 and 2, right, that is why I write down the atoms second point, the atoms present at the phase centre are distributed between 2 unit cell or distributed between 2 unit cell and hence its contribution in 1 unit cell is 1 by 2, okay, phase centre may be used to say, okay, 1 unit cell is 1 by 2, what about the atoms at the centre, centre may be used to say complete contribution, right, so write down, the atoms present at the centre of the unit cell, the atoms present at the centre of the unit cell will have contribution 1, next one, what about the edge centre, edge centre 1 by 4, 1 edge centre is here, so with this edge centre how many cubes you can draw, 1, 2, 4, 3rd and 4, okay, write down the atoms at the edge centre has contribution 1 by 4 in each unit cell, okay, so for this one we need to find out the rank, okay, so what is the rank, number of effective atom, okay, there are 8 atoms present at the corner and the contribution of one atom is what, 1 by 8, so rank is 1, right, it means we will have only one, effectively one atom present in the units, okay, so volume, okay, 1 into 4 by 3 pi r cube of the atoms, okay, this is one thing, now what is the coordination number, write down the definition, all these questions they ask rank, coordination number, packing fraction, all these questions they ask, coordination number, write down, it is the number of nearest atoms, it is the number of nearest atom which are in contact with each other, which are in contact with each other, what is the coordination number for this, 3, 4, 4, sorry, simple cubic units, what is the coordination number, 2, 4, 2, 3, 4, what is the coordination number, 3, 4, 3, okay, see if you have this atom, okay, this is in touch with this atom, because the relation was age goes to 1, so for this also this relation is true, what about this one, if we extend this edge, because this cube is above this edge, so there is a corner atom there also, so for this corner atom where it is, if you extend it, it is behind, if you extend it, it is on the right side, okay, but we are talking about crystal, we are not talking about a unit cell, I said properties, if you want to find out, you can study a unit cell, okay, but if you look at the coordination number in a crystal, in a crystal, it is in contact with crystal, why we are taking one unit cell, because it has the same property of the entire crystal, so whatever result we get from one unit cell, that will be same for the crystal lattice, correct, so coordination number is what, 6, not 3, okay, let it out, coordination number is 6, packing fraction is represented by 5, and it is defined by the volume occupied by the atoms divided by the volume of what, volume of unit cell, okay, volume occupied by the atoms, what will happen, rank into volume of one atom divided by volume of cube is 3 cube, okay, so that is the fraction, okay, examin fraction, 0.524, okay, so packing efficiency, this you understood this question, this relation, rank into volume of one atom, rank, volume of one atom is 4 by 3, 5R cube, 5A cube, and A in R correlation we already have, A is equals to 2R, we will substitute it here, okay, this becomes 4 by 3, 5R cube by 8R cube, 5 by 6, isn't it, 5 by 6, and that will be 0.524, packing fraction is 524, which you can also write 52.4 percent, what is the void, percent is void, 100 minus this, so 100 minus 52.4, all these questions they ask, okay, so 47.6 I guess, 47.6 percent is the percent is void, means volume of one atom, okay, so that is why we try to arrange the atom in such a manner, so that efficiency is the maximum, okay, so we will try to maximize the efficiency of units, and for that we will do the best possible arrangement of atoms, okay, so that we will discuss on the arrangement of atoms, and we will see how to do the arrangement, so that we can get maximum efficiency of units, clear, right.