 In this lecture, we will learn about reciprocal lattice, as we see it is a reciprocal of the lattice. So, how does it evolve and what why is it essential we will learn about in this particular lecture. To learn about diffraction it is highly essential that we learn in terms of reciprocal lattice and how does it evolve we will talk about that in this particular lecture. We might be familiar already with the Bragg's equation and we know that Bragg's equation is given by 2 d sin theta is equal to n lambda. So, taking that particular equivalence we can directly see that if we keep our lambda as constant our sin theta is inversely proportional to the interplanar spacing which means that if I have a large atomic spacing it will produce a smaller diffraction angle. So, that is what we can see that sin theta is inversely proportional to 1 by d. So, as my lattice spacing increases I get diffraction which is at much smaller angles and also that the smaller atomic spacing will produce a larger diffraction angle. So, that is the inverse relationship of the interplanar spacing with the diffraction angle. Secondly, we can also see that if we have for a particular lattice spacing for interplanar spacing I can also see that my sin theta is inversely proportional to my energy or in other words if I start increasing my lambda or I start decreasing my energy term then obviously my theta also will start getting if I start increasing my lambda then my theta will basically decrease. So, that is what is the equivalence what we can see here because as my energy is higher my lambda is lower. So, eventually my theta also has to decrease. So, for higher energy terms I can see diffraction at much closer point or theta is very very low when I have very high energy wavelength. So, I can see that the diffraction pattern will vary with the incident wave particle energy and the diffraction angle they become smaller when once I increase my incident energy or once I reduce my lambda. So, my theta increases my theta decreases as I decrease my lambda. So, that is the direct dependence I can see with theta with in terms of energy or the wavelength at the same time if I keep my n as either plus 1 or minus 1 I can see that this law will be again valid and it shows that there is some sort of symmetry which is basically which has to be incorporated for different values of n. So, automatically diffraction is it will have the same probability whether my n is equal to plus 1 or n is equal to minus 1. So, my diffraction pattern automatically inculcates some symmetry into itself. So, just summarizing it if I keep my lambda as constant my sin theta and d have inverse proportionality or in other words if I have my d value as constant I have direct proportionality of theta and lambda. It means if I reduce my lambda or if I increase my energy my theta value will be very very low. So, that tells whether my diffraction parts will finally appear in terms of Bragg's equation. Secondly, the Bragg's law it is not a positive law it is a negative law which means that if I am not satisfying Bragg's law no diffraction will occur, but once my Bragg's equation is satisfied I may see those diffraction and diffraction is well defined as reinforced coherent scattering. So, it is basically reinforcement of the coherent beams which are scattered. So, that defines my diffraction as such and again Bragg's law or 2 d sin theta equal to n lambda this particular equation is a negative law and that tells me clearly that if I am not satisfying this particular equation my reflection will not occur, but if this condition is satisfied it may occur that is what is being defined by the Bragg's law and again there are certain factors of scattering by various electrons because initially we will have a scattering by individual electron that brings out the polarization factor. I can have the whole atom because atom is consisting of so many electrons it will have something called atomic scattering factor and all those atoms will be now arranged or organized in a particular unit cell in certain fashion. So, it will result out my unit cell structure factor. So, I have polarization factor which evences from the electron or I have whole atom which has many electrons. So, this particular factor is dependent on my atomic number then I have the structure factor which is dependent on so many atoms which constitute or define a particular unit cell. So, this is how my intensity basically varies. And coming on to that coming on to the next part of it I see that my overall structure factor is not dependent only on the positions of my atoms which are there in the unit cell. So, I can see that structure factor is independent of the shape and size of the unit cell it just depends on the location or the positions of atoms and what their atomic scattering factor is. So, that part we can clearly see from this particular part the overall structure factor is not dependent only on the my miller indices and the location of my atoms which are located out there in a unit cell. So, it is independent of the shape and the size of unit cell and again it can have various intensities the overall atomic scattering factor can be different for if I have two kinds of atoms atom A and atom B and what I will get finally, is kind of a resultant amplitude which is arising from the scattering of different kinds of atoms. So, overall I can see that my overall scattering factor is dependent on only on the location of atoms in a particular unit cell and it is independent of the size of the unit cell or its shape. And this thing also we might have already learnt about it in the x-ray diffraction, but all those reflections which may be present in a simple cubic I have all the reflections which may be present for a body centered I have only h plus k plus l miller indices which the combination when it is even I get my reflection for face centered I have I should have h k and l which are unmixed it means all h k and I have to be all odd for everything has to be even and then for n centered I need to have h and k unmixed for the c centered one and for necessary absence reflections are totally absent for simple cubic I should have everything is present. So, I do not have any reflection which are absent actually absent when I have h plus k plus l is equal to odd for b c c and for f c c I have h plus k plus l they are all mixed they are all mixed. So, then I would not get any reflection. So, these are all the criteria for a diamond for a d c f c c b c c or s c cells now coming back to which forms the basis of my of the reciprocal lattices like if I start seeing a particular a particular crystal and I start defining each and every plane by their normals. So, I can see the points begin at a fractional coordinate of one. So, this is what is defining my particular unit cell. So, this is the end limits of a particular unit cell it will have a dimensions of one on each side, but as soon as start coming into the particular crystal I will see that I have the different planes plane two zero zero plane four zero zero they all start falling from the outside to inside. Similarly, I will have all those points available here as well I have one one zero. So, I will have my one bar one zero somewhere here along the same side I will have my two two bar one zero somewhere here two bar one zero and similarly, I will have more intensity of points along this side which is near the origin point. So, I will have more intensity of points which will start crowding my central location. So, this problem is that we have increase in the intensity as we start approaching the origin. So, this is my origin as soon as I start coming as soon as I start creating all the planes or labeling all the planes I will have more intensity in the central part because my one zero zero is at this particular part this particular point and two zero zero is at the half the distance. So, I will have more agglomeration of points in this particular regime. So, to basically come out from this particular problem we define something called a reciprocal lattice and the problem arises is say if you want to see a particular one zero zero plane I orient my crystal in approximately with one zero zero and which has a equal angle with the beam as well as the detector. So, I I can define that my theta is basically obeyed and I can obey the Bragg's law for a particular alignment. So, this becomes very easy for one zero zero plane or lower index planes, but what happens if I want to go for a higher or complicated higher index plane such as two four six. So, basically it creates some problem or complexity once I want to visualize all those things in two dimensions. So, actually what I have is 2 D planes which have intersection of a 3 D unit cell. So, this creates much more complexity. So, if you want to visualize this 2 4 6 plane what I can do I can just remove the one dimension of it. So, once I can remove one dimension I can see everything in two dimension and that is what I am more comfortable with. So, I can remove those one dimension from my unit cell and what I see is everything in the reciprocal lattice spacing I can see I can form the same crystal in a reciprocal lattice I remove one dimension now I much more simpler view for analyzing. So, I have everything representation of crystal itself in a reciprocal lattice and I can I am removing my one dimension now I have the same plane with much more simplicity and how do we go about that it is more like this that we have each set of parallel plane and those parallel plane have length which is equal to the reciprocal of the interplanar spacing. So, if say if I had reciprocal lattice for a cubic unit cell say if I d 1 1 1 is equal to a by my under root of h square plus k square plus l square for a cubic for a cubic cell. So, what I can do in my reciprocal space my new new spacing will become 1 by d 1 1 1. So, my d 1 1 1 star will become 1 by d 1 1 1. So, I have this set of parallel planes of 1 1 1 which have their length equal to the reciprocal of the interplanar spacing and the normal direction and the normal direction to this particular plane is the orientation of the corresponding set of parallel planes. So, I can also define its orientation I can also define its magnitude how what is the overall spacing between all those particular planes. So, each point which which is being observed in the reciprocal lattice that is nothing but a parallel set of planes. So, all the planes like 1 0 0 all the set of parallel planes which are shifted by a particular unit cell distance all those I can represent by a single point and I now I know its direction because I will have the normal direction which is orientation of a corresponding set of parallel planes I know its magnitude that is nothing but the 1 by inverse of the inverse of the interplanar spacing between those particular set of parallel planes. So, I can define each point in the reciprocal space as a parallel set of planes. So, what I was seeing here that my 1 0 0 was here in the normal in the normal nomenclature 2 0 0 out here 4 0 0 out here, but in terms of reciprocal lattice my 1 0 0 is out here my 2 0 0 is double the spacing my 4 0 0 is 4 times the spacing of 1 0 0. So, now I can see all this part I am basically opening it up to go on the other side and I can see my lattice much more clearly. So, my 1 0 0 is here 2 0 0 is out here 4 0 0 out is here. So, what I can see I can see uniform or periodic distribution of all those points which I can define by either by the vector by its vector quantity. So, now I have a star which is nothing but a reciprocal lattice and on this star I have b star and now I can see everything is now in a 2 D I have removed 1 1 dimension to it which is nothing but the 0 0 1 part of it I have already removed it that becomes a zone axis and I can present all my points all my planes so nicely by a single point. Now, I know my direction as well I know my direction this thing is my 1 0 0 direction and that is what I am going here with and this of my direction is 0 1 0. So, that is where I am going and it is perpendicular to the 0 0 1 1 plane. So, that is what I can see here that becomes the zone axis. So, now it becomes much more simpler for me to represent all the planes I know their magnitude what is the interplanar spacing that becomes 1 by D of the interplanar spacing in the reciprocal lattice spacing I know its direction. So, I can represent now set of parallel planes by a single point in a reciprocal lattice and the concept of reciprocal lattice comes more like this that I have incoming x-ray beam it can again be a electron beam depending on a kind of diffraction which is occurring. Then I have the Ewald sphere this is nothing but the Ewald sphere which is a which is a radius of 1 by lambda and upon interaction with the crystal some beam gets a transmitted but those which are following the which are obeying the Bragg's law they get diffracted out here and what I get is a diffraction spot on the Ewald sphere. So, this is my first spot which will come out here this is my transmitted beam and this particular distance is nothing but 1 by D or the inverse of the interplanar spacing. So, I have 2 D sin theta is equal to n lambda now I can define this sin theta is equal to 1 by 2 times D divided by 1 by lambda. So, I can see that if I take a particular thing like this I can define that if my incident beam interacts with the material crystal goes away then everything is falling on the Ewald sphere these 2 particular points I take A and B I can take them A and B and the half of it is my theta what I am seeing is 1 by 2 D this is my 1 by lambda. So, that is what is now being shown from this particular equation. So, I have particular crystal planes which are oriented in certain direction they give out my theta and 1 by lambda and I can now see that everything is now falling on the Ewald sphere this particular part. So, I can see this is my Ewald sphere and then I can what I get is spacing of 1 by D out here. So, that is what is being represented by the reciprocal lattice. So, it means that all the points which are falling on the Ewald sphere all those points on the reciprocal lattice which are falling on the Ewald sphere only they are producing my diffraction spot because they are the ones which are obeying the Bragg's law because we can see sin theta is equal to 1 by 2 D whole divided by 1 by lambda. So, I have my Ewald sphere which is a radius of 1 by lambda and also I am getting the theta value of 1 by 2 D or the 2 theta becomes 1 by D. So, that is what I am seeing out here in the Ewald sphere construction that I have my 1 by D I have my 2 theta out here and that is what is giving me the diffraction spot. And again this is what is being stated that diffraction will occur only when a reciprocal lattice point such as the sphere that is what is obeying and then that will create a diffraction spot. So, let this be the construction of a reciprocal space. So, I have my reciprocal lattice I have my reciprocal lattice out here R L this is a this is this can be individual plane which can be represented by all this intersection of all those points. So, I have my points it is a 2 D reciprocal lattice or it is 2 D to 1 0 0 1.0 2.0 3.0 and so on. So, and this is my Ewald sphere and this has a radius of 1 by lambda radius of 1 by lambda. So, what I can see this particular distance it is stretching the Ewald sphere this particular point is stretching the Ewald sphere. So, I have 1 by D of 4 1 this is again in a 2 D only 2 dimension. So, I am seeing that my Ewald sphere it has a radius of 1 by lambda and I am getting 1 by D which is of a particular plane which is being satisfied by 4 1 plane. So, when the point is stretching 4 1 I am getting a diffraction spot. So, my reciprocal lattice once it coincides with the Ewald sphere construction I get a diffraction spot. So, it is more like this I am sending a beam of a crystal plane which is oriented at certain theta value and once it obeys the theta value I have theta 1 1 1 and again in this particular case I have theta 2 2 2. So, once it is stretching the Ewald sphere say my plane is exactly oriented. So, I will get a diffraction spot out here for plane d 2 2 2 I will get a diffraction spot at some other point because now my theta 2 2 is different than theta 1 1 1 and for a particular material once I utilize a copper key alpha I fix my lambda value or my inverse of lambda it comes out to be 0.65 per angstrom. In this case I have units of 1 by length because it is a inverse construction of a particular length scale and that eventually brings out my 1 by D value of around 0.43 per angstrom. And if you realize if I calculate 1 by d 2 2 2 it comes out to be approximately 0.86 per angstrom. So, we will see that it is much more periodic in nature and my this length is a exactly half of my d 2 2. So, that is what brings the periodicity in this particular construction. So, if you want to see it again as a merger of these two I can see that I have plane 1 1 1 I have plane 2 2 2 and how do they can fall on the same Ewald sphere because my 1 by lambda term is constant for a particular radiation. And now I can see the plane 1 1 1 1 has to be aligned differently as compared to the plane d 2 2 2 to give me a diffraction spot. So, I need to have kind of a poly crystal material with a proper orientation with my plane 1 1 1 and plane 2 2 2 to produce a diffraction spot at particular location in the Ewald sphere. So, my reciprocal lattice has to be such that it basically comes out or touches the Ewald sphere. So, that I need to have a polycrystalline material. So, my reciprocal lattice will be accordingly to a particular plane and my Ewald sphere is constant because my lambda value is fixed for that. So, I can get diffraction spot at some location. So, coming back to it the direction and length of the reciprocal space will be more like this that my d h k l plane is pointing from origin to the point. So, this is origin and it is to a point. So, this is a plane 1 1 0 and now this particular point is now perpendicular to the first family of h k l. So, now I have plane 1 1 0. So, the overall point towards it or the d h k l pointing from origin to this particular point will be my particular plane in the reciprocal lattice with its direction of d 1 1 0. I have to construct a reciprocal lattice my directions say this is this with the normal lattice construction. So, I have a 1 vector and a 2 vector which are in the real space which is a length along a 1 direction is a 1. So, in reciprocal space I have this particular length which is a 1 that becomes now 1 by a 1 1 by a 1 in my reciprocal lattice. So, I had a length of unit 1 which becomes 1 by a 1. So, I had length of a 1 which becomes 1 by a 1 in my reciprocal space. So, this particular point which was a 1 by 2 now that becomes 2 by a 1. So, I had this particular length this becomes now 2 by a 1 and so on. Similarly, a 2 that becomes 1 by a 2 something like this becomes 1 by a 2 any point out here in the center part this particular length now becomes 2 by a 2. So, that is how the reciprocal lattice gets constructed. So, I had this 1 1 this particular point this now becomes this particular point. So, if I had some 2 2 comma 2 that becomes basically out. So, any point 2 comma 2 out here will now be at this particular location. So, this particular point now has converted to this point this point has now gone to this point. So, that is how I construct my reciprocal lattice like this one more condition to be know to be notice this that my new b 1 vector is perpendicular to is perpendicular to my a 2 vector. So, my reciprocal lattice vector of b 1 is perpendicular to my a 2 vector and my b 2 vector is perpendicular to a 1 vector. So, my reciprocal lattice it has a unit vector which are perpendicular to the my real lattice real space lattice vectors. So, seeing it properly again. So, we can have again this a 1 and a 2 which are the real space coordinates real space vectors and now my b 1 will be now perpendicular to a 2. So, my b 1 or the reciprocal lattice vector will be perpendicular to a 2. So, my perpendicular vector will come out approximately like this and that is nothing but this particular plane. So, that is what I am seeing that 1 0 plane is now falling here 2 0 is falling out here for a 1 it was it has become b 1 and now b 1 is perpendicular to a 2 and now my b 2 will be now like this in the reciprocal space that is what I am seeing out here that may now b 2 is perpendicular to a 1. So, that shows the perpendicularity of the various vector in the reciprocal lattice. So, it is very essential that it is not only diminishing the dimension by 1 by it is also becoming perpendicular to the original other axis. So, I had a 1 so it is now helping the b 2 to form which is perpendicular to a 1 and my b 1 is perpendicular to a 2 that is how it becomes the how it becomes in the reciprocal lattice spacing or the basis vectors in three dimension can also be defined by the cross product of the other two. So, my b 1 star becomes the a 2 cross a 3 divided by v where where is the volume or b 2 becomes the a 3 cross a 1 by v and b 3 becomes a 1 cross a 2 by v and again particular thing is again b 3 is again the reciprocal interplanar spacing and again b 3 will be perpendicular to both a 1 and a 2 that is how I can see that how the overall lattice reciprocal lattice spacing is being constructed. So, I can see my b 3 it will remain perpendicular to my a 1 as well as a 2 and this particular distance is nothing but 1 by the interplanar spacing of particular plane which can be 0 1 1 or it can be something else as well this will be 0 0 1 approximately here. So, that is what is being shown out here how construct my reciprocal lattice. So, what I can get finally, as the interplanar spacing is nothing but the miller indices plus the reciprocal lattice vectors. So, that is how I will define my 1 by d 1 by d of h k l that is given by the lattice reciprocal lattice vector which is again nothing but the corresponding to the real lattice plane. So, I have this g vector corresponding to the 1 by d vector in the real space and again this reciprocal lattice construction becomes the alternate way of representing my real space real lattice. So, that is what is the so good about reciprocal lattices I am eliminating 1 dimension at the same time I am able to mimic my real lattice. So, that is how that is how it assess me visualizing things much more clearly and again coming back to the relations of the reciprocal lattices I can see that my a star b a star c or my reciprocal lattice spacing with the original b c it becomes 0 or it is perpendicular to the original axis is similarly, my b star the dot product of my b star with a and c it again brings 0 and again my c star with a and b it brings out 0 it means everything is all these things are perpendicular to the other two axis which have which have it form. And again the reciprocal axis is are the scalar products of the axis from the direct lattice that is how it basically forms the reciprocal lattice and this is nothing the v volume is nothing but the a dot b cross c or b dot c cross a or c dot a cross b that is how I get the volume of the direct lattice. And for a simple cubic lattice I can see that if I take the dimensions to form a cube again I will have to go again 1 in i direction 1 in j direction and 1 in k direction and that is how I get my reciprocal lattice parameters of a b and of a star b star and c star all the three axis to form a particular unit cell where small a this particular thing is nothing but my lattice parameter. So, I get i by a j by a and k by a ideally 2 pi comes this factor particular comes because to maintain the periodicity of a particular structure. So, it is converting it into more into the radiance part. So, that is how it maintains as well the periodicity of the particular crystal in terms of taking it to the reciprocal lattice. So, finally, I get a star is equal to 2 pi a multiplied by i b star is equal to 2 pi j by a and c star is equal to 2 pi k by a. And now coming to the b c c lattice this a is again my lattice constant for a cubic for a cubic lattice and again a b c are the new axis which will really form the new crystal. So, to define a particular volume I again take a dot b cross c and it eventually comes out that eventually it comes out that my a star is equal to 2 pi by a i plus j b star becomes 2 pi a j plus k c star becomes 2 pi by a k plus i. And this ideal construction comes from the from the equality of my point which will form a particular unit cells. So, I can see that from origin I have to go certain distances out here like this to 3 different cells to be able to form 3 axis of a of the new crystal or the new lattice which will form. So, I can see for constructing particular axis for a particular axis my a direction I have to go i minus j plus k whole by 2 for second thing for b I have to go minus i plus j plus k or for c I have to go i plus j minus k. So, eventually taking doing this all those vector algebra or all the cross product I can see that it eventually comes out i plus j j plus k and k plus i to form my final crystal for final lattice. Similarly, for a FCC lattice my 3 equivalent point basically are these particular point which are sitting on the 3 different faces. So, those correspond to i plus j j plus k and k plus i. So, I can see I have to traverse traverse 1 1 in each direction because this is my bottom phase this is my front phase and this is my side phase the center of that. So, I can see that I have to travel i plus j by 2 j plus k by 2 and k plus i by 2 and taking the periodicity of 2 pi I take my again the cross product of b cross c and I divided by volume. So, I can get some parameters which are more like this eventually turns out that the reciprocal lattice of BCC is equal to FCC. So, I can see that this particular part is similar to what I started with in a BCC. So, I can see that this particular part and this particular part is now similar to my i minus j plus k. So, it is basically i minus j plus k it is similar to this particular part. So, I can see that reciprocal lattice of FCC is equal to BCC and reciprocal lattice of BCC is equal to FCC. So, that shows the correspondence of correlation from the BCC to FCC lattice at simple cubic remains a simple cubic, but my FCC becomes BCC in reciprocal lattice spacing and then my FCC becomes BCC. So, what I can see out here is the reciprocal lattice is the reciprocal of the primitive lattice and it is purely geometrical. Since it is only geometrical it is not dealing any anyhow with the intensity point. So, what I am getting particular point it is just showing me that it is reciprocal lattice or it is forming a reciprocal of the primitive lattice without taking care of the intensity of any particular point. So, I can have a some primitive cells something with the non primitive cells, but again how do I decorate them it really depends on the location of my particular points. It might also happen that some reciprocal lattice points may go missing or they can be scaled up or they can be scaled down in the terms of intensity or it can have a decoration of some motive because depending on the scattering power and again the Ewald's square construction can further select those points which are actually absorbed in the diffraction experiment. It is again a probability that whether it will come or not it totally depends on the scattering power of a particular plane or a particular element or particular atom which is being present there. Only the location and the scattering power itself the combination of that decides whether the particular diffraction spot will be present or not. So, the construction of Ewald's square plus the diffraction which is coming here they jointly say whether a particular spot will be present or absent. So, seeing it in a different manner, so I can see that in my diffraction once the diffraction is occurring my row spacing in the diffraction pattern has become 1 by d. So, initially when I had a interplanar spacing of d it becomes 1 by d in my reciprocal spacing and again I can see this surface diffraction either in terms of rows of atoms and draw a kind of a qualitative picture of the diffraction pattern just by considering what kind of planes are present. So, just by considering the space between those to those rows I can construct my reciprocal lattice. So, from that I can eventually bring out my diffraction pattern. So, if I had a real space like this where I have say a surface of FCC 1 1 0 plane then I can see that in my FCC 1 0 0 plane it becomes more like this I have atoms which are associated at this particular location as well on the phase centers. So, I have something on the phases as well. So, I can get something more like this and seeing their 1 0 0 particular plane this particular plane I can see that I have planes which are more like this. I will get some sort of a construction where these atoms are much closer as compared to these particular atoms. So, that is what we have seen here that in this particular case we have a is less than b it means I can even have the atoms touching one another or they can have spacing much smaller than that of a b. So, my spacing along plane b interplanar spacing along b is much more scarce or spreaded quite far apart in comparison to my lattice spacing along this particular a direction. But once I bring it to the reciprocal spacing I can see if I keep my b constant if I keep keep my b and represented in terms of b star it becomes 1 by my b. So, I have this particular thing as 1 by b and now I can see that in this particular case in the real in the real lattice I had a closer to one another. But in diffraction once I am getting a diffraction pattern this a has gone to a has gone much far apart. So, this particular 1 by a has become much more sparse or much spreaded apart. So, here I had a less than b. So, now what I will get in reciprocal lattice spacing is 1 by a is much greater than my 1 by b. So, that is what happens in my reciprocal lattice this happens in the real lattice this happens in the other diffraction pattern I get 1 by a much greater than 1 by b. So, that part is being retained in the diffraction pattern this part I can also see is that the spacing of the rows this was the spacing of rows in a this is now become much more further apart. My spacing of b was this much. So, let me use a different color pen. So, I can show it more clearly I had something b around this part and now it has decreased as compared to the a. So, in the real space I had a I had b in this case I had a less than b, but in this case in the reciprocal or the diffraction pattern my a dash has increased in comparison to the b dash. So, that is what we are able to see from the diffraction of those particular things once I consider only about the rows. Alternately seeing or following up only with this particular rows or the spacing between the rows it becomes much more difficult. So, sometimes it is necessary that instead of following my rows I can go back to the particular atom and take their diffraction pattern. So, now I can also follow the position of atoms or the position of planes and then I can come back to my diffraction pattern. So, just tracking the rows or the spacing between the rows it is sometimes not convenient. So, what I will try to go I will try to go with individually atom. So, let me come back to a 1 0 0 plane of a simple cubic and if I take the real space I have the vector a 1 as a vector a 2 and this is the lattice vector and they will form a primitive unit cell. If I take a particular box it will form my primitive unit cell. Now, if I come back to my reciprocal lattice what I can see is my b 1 is perpendicular to the a 2 or I can also give it by a 1 dash. Now, my a 1 dash is perpendicular to a 2. So, my a 1 was out here and my now a 2 dash or my a 2 dash is now perpendicular to a 1. So, I can see that my a 2 dash is now perpendicular to a 1 that is that part I am seeing out here and again my a 2 is now creating a lattice vector that is in the reciprocal space which is perpendicular to a 2. So, now my a 1 dash is perpendicular to a 2 and I have my a 2 dash which is perpendicular to my a 1. At the same time I had a 1 the distance a 1 which is equal to distance a 2. So, I have this particular unit length this 1 was a this 1 is also a and once I come to the reciprocal lattice I can see that the distances are now similar. So, this was forming a square in the in the real space it remains that 1 by a is similar out here 1 by a is also similar in the along the a 2 axis. So, my a 1 x a 1 dash axis is similar to the a 2 dash axis in terms of its magnitude. So, I have that it is still forming a particular square. So, this particular part I can see. So, I have my because in this case I had a is equal to b. So, 1 by a remains equal to 1 by b that is what I am seeing in the reciprocal space of particular plane of 1 0 0 of simple cubic. So, in 1 0 0 plane of simple cubic I have the unit cell which has length which are similar in the 2 directions. So, I have my lattice vector along a side of the x axis it is equal to a along y axis is also again equal to b and a equal to b. So, once I take it to the reciprocal lattice spacing I get vectors which are magnitude of 1 by a and 1 by b and since a is equal to b I get a similar reciprocal space which is again 1 by a is equal to 1 by b. So, it forms a it forms a diffraction spot which are which are basically much more periodic much more similar and with a similar spacing between the diffraction spots. So, along this side my distance is same as along this side in the particular case of simple cubic of 1 0 0. Now, let me go back to a different way in which we had concepted the FCC lattice. So, in this particular case I take a particular FCC cell I take it 1 1 0 particular plane 1 1 0 and I can see I have particular atoms which are sitting along this side let me come back to the different color. So, I can see that my atoms are located along this particular points. So, what I can see is it is similar to the real space lattice like here. So, this is a 1 1 0 plane of a FCC and then what I can see along the a 2 I have much more larger distance as compared to the a 1 and this will be the real space unit cell for this particular case. But, once I come to reciprocal space I can track each and every atom like this or particular row of atoms for a particular plane what I can see is now my a 1 it has become 1 by a 1 and now my a 2 has become 1 by a 2 so now I can see that initially I had this particular magnitude in a 2 which is now decreased and I had now a 1 which is now increased. So, in this particular case I had a 1 which was lesser than a 2 in this case I can see my a 1 dash is now greater than a 2 dash that is how it is forming my reciprocal lattice and the distances are being defined by the original magnitude of the lattice vectors and again the directionality is also being retained. So, I can see the symmetry part is also being retained in the reciprocal space as well and that thing is again being given in the 1 1 1 plane of the FCC. So, if I take the 1 1 1 plane of the FCC that becomes more like this if I take 1 1 1 plane so that this becomes the 1 1 1 plane of the FCC and in there I can see that I have atoms which are located along the sides. So, those are basically located out here out here out here then again on the centre of the edges and if I if I keep expanding it I get something which is more like this. Here and I can see that a 1 and a 2 they are not perpendicular to one another and they have some lattice vector were given along certain directions certain angles. So, I have some value of alpha which is associated now with the real lattice spacing. So, but my a 1 dash will remain perpendicular to a 2 that is what I can see this is my a 2 and now my a 1 will now be perpendicular to my a 2 direction. So, this is my a 2 and the a 1 dash is now perpendicular to a 2 and I had my a 1. So, now my a 2 dash will be perpendicular to a 1. So, I can see that my a dash 2 is perpendicular to a 1 and my a 1 dash is perpendicular to a 2 that is what the requirement of a reciprocal lattice and again the spacing part the spacing is basically same between the between this particular point to this particular point. So, from here to here and from here to here my lattice spacing is the same. So, I can see that the space which has been traversed from this particular point to here or from here this point to here remains the same. That is what is being shown out here at the same time I can see that this particular symmetry is now being retained in the reciprocal lattice as well. So, this particular part I can see it is being I have my this particular point it has been retained as such. So, I can see that the symmetry part is being now retained in the reciprocal lattice though it has changed by a particular angle, but the symmetry part is now it is also 6 fold this also remains the 6 fold symmetry in the reciprocal lattice. So, I can see that my real and the reciprocal lattice is they have the same symmetry. So, that part is also retained in the reciprocal lattice out here. So, all these kind of similarities I can again show them that a 1 and a 2 are perpendicular a 1 and a 2 dash are perpendicular at the same time a 1 and a 1 dash are parallel in my 1 1 0 f c c, but that is not true in the second case where I have a 1 and a 2 they are not perpendicular, but a 1 and a 2 dash are perpendicular and a other way also a 2 and a 1 dash are perpendicular, but a 1 and a 1 dash are not parallel. So, there can be certain complexities which can generate, but how it retains the symmetry of a particular crystal that is being given in the reciprocal lattice that it retains the symmetry of the real lattice. So, in certain cases once we have some surface adsorption. So, I can have some matrix which has some adsorption of certain particular entity which is again ordered in certain manner. So, I can have this adsorbate which has a kind of symmetry in the symmetry of couple of monolayers on the top of a particular matrix and this is a very common phenomena because we can we do see there is some specific adsorption of couple of monolayers on a particular material. So, how do they give out a particular diffraction pattern is more like this the matrix the overall structure can be divided into the matrix and the adsorbate which is being adsorb on the surface. So, for a particular matrix I can see my length is a 1, a 2 I can take it back to a dash 1 and a dash 2 in the reciprocal lattice this is the real lattice. I can take it to the reciprocal lattice this is the reciprocal lattice I can see that this particular part gives me certain diffraction pattern and in the second case I have my b 1 and b 2 which are much greater than which are twice the a 1 and my b 2 is also twice the a 2. So, from construction I can see it from here. So, I can see that my b 1 is equal to 2 times a 1 b 2 is equal to 2 times a 2. So, what happens in the reciprocal lattice is my b 1 becomes half of that and my b 2 becomes half of that. So, what I get for this I get much more diffraction spots as compared to my first case. So, I will see that my diffraction spots are much more in number once I have adsorbate which is spread in much more scarcely as compared to my overall matrix, but the problem is I have so less number of adsorbate on the surface of a matrix that its intensity might not be that great, but they will be overlapped with over one another. So, if I combine this and this I get something which is this. So, what I am seeing here is I am combining this part the diffraction pattern which is coming out from the matrix I am combining it with the diffraction pattern which is coming out from the adsorbate and then what I get finally, is the combined diffraction pattern and I can see I have a dark spots which is coming basically from the matrix out here and then I have also some overlapping of these particular points which are coming from the adsorbate. This is from the adsorbate there is some overlapping of these points even at this particular points. So, I can see those things particular things that I have a kind of combination between the my matrix the diffraction spots which are coming from the matrix diffraction spots which are coming from only from the adsorbate, but at the same time it is not telling anything about the intensity of the diffraction spots. So, I do not know what will their intensity will be, but I can predict that this adsorbate spots will be weaker because they are curing number the diffraction spots which will the intensity with which they will come out will be much more feeble. So, I can see that I can overlap all these particular intensity points in one because I have a combination of these two and so I can combine them to find out what eventually the diffraction pattern will look like will look like. So, in summary we can say that to eliminate the complexity of representing a particular real crystal the diffraction occurs by when the evolved sphere touches my reciprocal lattice planes and reciprocal lattice can be given by a set of parallel planes with and all those planes can be represented by a single point. So, I know their magnitude is equal to 1 by the inverse of the reciprocal of the interplanar spacing. So, in reciprocal lattice what I am getting is a point which represents a plane and the magnitude of that particular point from the origin is equal to the 1 by the interplanar spacing between the planes which it represents. So, that is how I can I can spread it around and I can open it up because if I take a real crystal all the points all the planes tend to agglomerate at the origin. So, there is much more segregation of all the planes if I start representing each and every plane. So, in reciprocal lattice I am more or less I am like opening it up. So, I can see a much more periodic way of representing all the planes all the set of parallel planes and I can define the direction I can define the magnitude. The magnitude is equal to 1 by d where d is the interplanar spacing and so I can in reciprocal space it will become 1 by d and I can also find their zone axis I can also find the overall magnitude and direction of all those planes. So, that eventually tells me how the diffraction spot will appear once it is once the material is getting diffracted. So, it is a construction or the merger of the evolved sphere and how it basically coincides with the lattice point the reciprocal lattice point to give me to obey the Bragg's law and give me a diffraction spot. Again Bragg's law is a negative law. So, even when the Bragg's law is being satisfied it may not give a diffraction spot depending on what is the overall scattering factor and how it is basically getting merged into or what is the intensity of that particular point. So, I may not even get a diffraction spot and also we learnt about how different planes can have how different planes can basically provided a kind of a reciprocal lattice. Now, we can either go about following the row or the spacing between the rows or we can go with individual atoms which can get constructed back into the reciprocal lattice. We also saw how we can combine if you have more than one particular matrix and how can we combine them to form a set of reciprocal lattice reciprocal lattice and it also maintains the symmetry of the real crystal. So, that is the advantage with the reciprocal lattice and it can have it can accommodate wide variety whether we have axis A 1 or A 2 whether they are parallel or not whether they are whether they are perpendicular or not we can always grab it back into the reciprocal lattice and the axis are perpendicular to the other two or other one axis which is which basically defines a particular unit cell. So, that is what we learnt about the reciprocal lattice I end my lecture here. Thanks a lot.