 Hello everyone, welcome to this material characterization course. In the last class, we just started looking at some of the diffraction principles in a transmission electron microscopy. I started talking about a conventional selected area diffraction as well as the convergent beam electron diffraction and then I was saying that a SAT pattern or SAID pattern is obtained in a TEM using a parallel beam and a small probe or a micro probe or nano probe is obtained using the selected I mean convergent beam electron diffraction. So we will continue that discussion and if you use an incident beam as convergent, the spots becomes discs and their diameters depending upon the convergent angle 2 theta or 2 alpha something like that. You can form a convergent beam electron diffraction pattern in a TEM mode in any TEM that can produce a small probe which is less than 1 micron with convergence angle greater than 20 milli radians. So I will just play some of the rate diagram using schematic which will distinguish between a conventional selected area diffraction pattern and a convergent beam electron diffraction pattern. What you are seeing in this schematic A is the a conventional electron diffraction where you have the transmission transmitted beam that is also called a zero order spot and then you have the diffracted spot which is there. This is a conventional parallel beam elimination. Otherwise you have a convergent beam operation where you see that the zero order spot and you have the diffracted spot which appears as a disc as compared to the spot in a conventional SAD pattern. So this is an idea. We will talk about this and its application much more detail in the coming slides and this is one of the actual electron diffraction pattern taken from this textbook and what it shows is 4 different kind of patterns you get depending upon the angle your angle. So the first one is 1.4 into 10 to the power minus 3 and second one is obtained with 2.3 and third one is 4.9 and fourth one is 8.1 milli radians. So you can obtain a micro diffraction or Kossel molestant diffraction pattern depending upon the different converging angles. So this is first information about the diffraction using small probes. What is important here is always we talk about a probe size and probe diameter. What is the physical meaning of this? So in order to explain that I brought this slide. So the probe normally what we see in an electron microscope is nothing but a full width half maximum. So you see that you know the corresponding you see that corresponding profile intensity profiles are shown in this here and then you have the actual image in the microscope they will appear on the fluorescence state like this. So you have the different probe for example if you take a spot size of 6 which will have a probe diameter of this order that is 14 nanometers, 4 nanometers and 2 nanometers and so on. You have a nano probe or a stem mode or a normal probe you will have this kind of an intensity profile and if you go to the HR stem high resolution mode you see that the probe diameter is as small as possible nearly 10 nanometers. So which belongs to the spot size 4. So if you go to the microscope they will give the in a dial with the numbers which one each number will belong to particular probe diameter. The physical meaning of the probe diameter is a full width of maximum. So just you have to appreciate that fact that is why I brought this slide. What is the meaning of a probe diameter? The most important thing is camera length. Just now in the last class we have seen one of the operations I mentioned that we need to calibrate the camera length we will look at the details how do we get this relation with the camera length and the de-spacing of the specimen. So the similar ray diagram what we have gone through incident beam this is a sample and then you have the back focal plane transmitted beam a diffracted beam the distance between the specimen and the back focal plane is L and then you have the 2 theta diffraction angle the distance between the transmitted beam and the one of the diffracted beam is R. So the angle that the diffracted beam makes with the incident beam is 2 theta but from the Bragg's law we know that sin theta is equal to theta since theta is too small which is equal to lambda by 2D each set of diffracting plane spacing DHKL produces a spot in the diffraction pattern at the distance R from the center and the diffraction sorry and the direction perpendicular to the planes R by L is equal to tan 2 theta which is approximately equal to 2 theta because theta is small so you can write lambda is equal to 2 D sin theta is equal to 2 D theta therefore R D can be written as L lambda. So this relation comes from this approximation and the camera length what we are calibrating or using is a projected length and not a physical distance in TM. So you have to be very careful about this aspect it is always a projected length and not the physical distance in the TM. So you have to know this basic idea about this camera length. If lambda L is known the camera constant and relates the distance of a spot from the origin of the diffraction pattern to the D spacing of the set of planes from which it comes. The camera constant can be determined using a standard and hence the D spacing corresponding to a reflection can be found once R has been measured. So this is the experimentally measured quantity from the film or from the diffraction image you measure the distance R and if you use nanometer unit for D spacing then the unit of a camera constant is nanometer mm which is not a standard unit but it is convenient for determining the D values in the nanometer scale. So we will appreciate this a simple concept when we actually try to index some of the diffraction pattern using a camera length as well as the corresponding distance R from a given pattern. So this is the typical calibration again I am repeating this the D spacing in nanometer of the first 9 diffraction rings from a gold and aluminum both are you know face centered cubic lattice. So you can have this values for the reference and this is a calibration chart we will come back to this when we do an exercises then that will make more sense rather than I go through this values. Now I will come back to the reciprocal lattice concept which I talked about during the fundamentals of electron optics as well as the I mean fundamental of this course itself we discussed about this this is the second time you are seeing this image and as such the reciprocal lattice concept has been discussed again in X-ray diffraction here also it is the same just to have much more confidence let us go through this what you are seeing is an optical diffraction pattern taken from the grid. Grid is nothing but kind of mesh it could be a metallic mesh or it could be of any material made of any material it is like something related to something similar to grating it is a mesh. So the pattern A, B, C, D and E they are all optical a diffraction pattern obtained from the grating the pattern A from a grating with the spacing of 0.126 mm pattern B from a 100 mesh grid with the spacing of 0.25 mm pattern C from a 200 mesh grid with the spacing of 0.125 mm pattern D from a 400 mesh grid with the spacing of 0.0625 mm pattern E pattern from the 2D crystal shown in F. So what you have to appreciate here is the there is a relationship between the distance between the spot in a diffraction pattern to the grid spacing there is a relationship. So you can appreciate that the as the spacing in the grid decreases the distance between the spot increases. So this is a reciprocal relation here. So we can make some statement about based on this observation there is a definite relationship between a periodicity and orientation of the object and the spacing and orientation of the spots in the diffraction pattern. So the diffraction pattern is known as a reciprocal lattice because of the inverse relationship it bears to the direct lattice that is the object. So we have seen this already I will just play some animation. So what you are seeing in this animation is this is a undeviated beam from the specimen and this is a diffracted beam here and what is shown in this right hand head image is the intersection of 2 planes plane 1 and plane 2 and this is the electron beam and this is the screen on which you see the diffraction spots appearing here and I want you to look at the schematic much more carefully because you see that a plane 1 is drawn in one color and plane 2 is drawn in another green color and the diffracted spot corresponding to plane 1 and plane 2 are also indicated with the same color of the plane. So you see that a plane 2 you see the diffracted spot appearing in the this line which is 90 degree to the plane orientation you can see that it is 90 degree to the plane orientation either this side or that side. Similarly you look at the plane 1 the diffraction diffracted spot appears exactly on that space which is 90 degree to the plane 1 orientation. So this is very important and for a clarity only 2 planes are shown here in principle you can have n number of planes or it could be more planes also where a diffraction can occur and then you will see the corresponding diffracted spot in the screen like this. Now we will take you to another concept which is very important in analyzing the diffraction pattern called a stereographic projection. So like you have an atlas where all the you know countries regions are drawn in one scale they are all called you know area true projections here you have an angle true projections. So stereographic projections are called angle true projections similar to atlas where you have area true projections. So the idea of the stereographic projection is what you are seeing on the screen is like you assume that you assume that you are placing a small cubic crystal inside the centre of the sphere and you try to draw a normal to this cube and then which comes and intersects the surface of the sphere and each one is each one plane normal which comes out and hits the surface of the sphere from the centre is called a poles each one poles. So I will just show one very nice animation where you will appreciate what I am trying to say you imagine that I have kept one a cubicle crystal inside the sphere now the assume that the sphere is transparent and I have just put that into a 001 orientation from this from the I mean the plane of the screen the plane normal of the screen itself 001. So in that projection I have put the crystal inside the sphere now I will try to rotate this sphere you will appreciate what I am trying to say you see that the crystal each plane normal which comes and intersect the surface of the sphere you can see these are all poles individual poles. So the top one is 100 the bottom one is the opposite in fact it should be bar 1 00 here it is 001 and the opposite side is you can see that 00 bar 1 you can see that. So like that you have all opposite poles here and there the idea is if you represent your I mean a plane and directions with this we will be able to analyze the crystal crystallographic orientation much more easily. So I have just shown one simple thing you can look at these kind of a projection system and as I just mentioned all these poles and which will have a perpendicular I mean particular angle for example 100 and 010 will have a 90 degree and 110 and 010 will have about 45 degree. So like that you can you can just visualize the angle between two plane normals so that means you will be able to identify the angle between the planes and then you will be able to represent the crystal crystallography orientation much more easily and then we will use this concept to I mean explain some of the reciprocal lattice concept as well as some evolved sphere concepts we will use it and this is just for your introduction just you have to just imagine how this stereographic projection can be viewed in all these four five directions so you can see all the important poles are indexed and then how they rotate. So now after going through this I believe that you have some idea about this what is stereographic projection. So now what you are seeing is all a 2D projection what I have just shown is an animation is a 3D so this is the same thing what I have shown 0 0 1 projection I have shown it is represented in the 2D like this. So this is the stereographic projections for a cube crystal with 0 0 1 parallel to the south north direction the sphere of the figure stereographic projection is 0 1 0 is parallel to the east west so this is what it is shown here. So here how you can just look at the some of the poles are written here this is the some of the nomenclature which I am not going into the detail and the basic idea I want to show it here is using the stereogram you will be able to represent the crystallographic direction that is all I would like to say and if time permits we will solve some small problems using the stereographic relation and also try to plot some of the poles on a graph like this a wolf net divided into 2 degree divisions you can actually plot some of the poles and then try to do some kind of an exercise of rotating each pole what will happen what kind of stereographic projections you will be able to visualize that we can do it and I am introducing this technique as a tool which can be with this tool we can represent the orientation of the crystal much more clearly that is that is the information I want to give at this point of time and when we actually go through some of the exercises are solving the tutorial problems we will be able to appreciate how this tool can be used so this is just for one references where some of the poles are being rotated here and you can appreciate that how the angle true projections is very effectively represented in a 2d fashion that is the basic idea and this is a standard stereographic projection for a cubic structures 1 is 0 0 1 and 1 1 0 C 1 1 1 and D 1 1 2 so here is this there is only one projection is shown here but other projections I have not shown and what you are seeing here is this is for a 0 0 0 1 projection for cubic system where all this spot shows the parallel orientations of another system like this and suppose if you look at the stereo projection of both BCC and FCC their poles are indexed like that so you will be able to measure the angle between the 2 poles of the different crystal system here and for example some of the poles are very closely parallel that means 0 1 1 of BCC is parallel to 1 1 1 of FCC or 1 1 1 of BCC is parallel to 0 sorry 1 0 1 of FCC and 2 1 1 of BCC is parallel to 1 2 1 of FCC something like that you can you can see all these things are if they are exactly parallel both of them both the points will coincide and if it is not parallel then you can see that how far they are away from each plane that means the angle will tell you the orientation of each plane that means you can relate the plane which is suppose if you have a 2 phase system one is BCC and an FCC you will be able to relate the parallel planes of the 2 systems so when they are in parallel to each other then you will form you will obtain a diffraction pattern straight so that means because diffraction patterns are obtained only when the planes are parallel to each other so by looking at the stereogram you will be able to identify the poles which are parallel that means the plane normals are parallel then that in actual physical meaning it is a planes are parallel and then you will get a corresponding diffraction pattern in a back focal plane you will be able to analyze both the patterns and then you will be able to correlate the relationship between or the orientation relationship between a 2 crystals using this stereographic projection so that is the basic idea about this stereographic projection I do not want to get into the detail due to the lack of time but you should know why this is being used when a number of planes are parallel to a single direction they are set to constitute a zone and the common direction is the zone axis or a zone direction for example if you go to this particular zone for example 001 what are all the parallel directions you will be able to identify with this stereogram so when you say a zone axis so you will be able to identify number of planes that are parallel to the single direction which can be easily identified by using stereographic projection the indices hkl of all the spots in the pattern are related to the indices of the zone axis uvw to which the beam is parallel by the v's zone law so you have the very powerful relationship which is given by v's zone law you can identify the hkl and hkl of the spots in a pattern and it is zone axis they will have the relations like this hu plus kv plus lw is equal to 0 so using this relation you will be able to identify some of the planes which are parallel to this zone so this is very powerful law and we will be able to we will be using this relation when we index some of the diffraction pattern and finding out the parallel planes now what I will do is I will again come back to this reciprocal lattice what you are now seeing is a schematic animation where you have the real lattice and it is relation with the corresponding reciprocal lattice here you will be able to draw this physically because all that you need to know the information about for example it is 1 by d of 1 0 0 plane which is equal to 1 by a or 1 by d 2 0 0 type of plane is equal to 2 by a and then the angle between them so you will be able to draw physically when you have the the basic information from the real lattice for example this is a and this is b and this is d spacing you will be able to generate this kind of a reciprocal lattice of your own similarly you can do it in a 3d because we know the relation so this is just to give you an idea please remember in an electron diffraction experiment even in a TEM what you generate in a microscope is a 3 dimensional reciprocal lattice but you project it on a 2d screen so you get only the 2d information so because in a please understand in a transmission electron microscope it is that your crystal is transparent to the electron beam so you get a 3d information it is a 3 dimensional information but of course as I mentioned in the beginning it is an averaged information but you get the projection on the 3d information projected on a 2d screen and which is seen as a spot we will get into the details why it appearing like a spot what is the physical meaning of the spot and so on in the due course so this is again a reciprocal lattice construction for a monoculating crystal so you can try one of this systems because you know the basic relationship between the real crystal and reciprocal lattice system so this can be a good exercise if you do as a an exercise for few systems then your ability to analyze this reciprocal lattice or a diffraction pattern will be a manifold you will be more comfortable if you theoretically solve and then get into the selection rule and then see that which are the spots will be allowed and which are the spots will be not allowed you can mark them and then if you if you can have the a basic idea of calculating this theoretically then you will have much more confidence in analyzing the electron diffraction pattern which I will show with some few examples on a on a on a blackboard or a tutorial class then you will appreciate this so this is just for your information this is for a hexagonal crystal lattice you have the reciprocal lattice system and another important thing is when you index the reciprocal lattice then you have the kind of 180 degree ambiguity so you have what I what what is the schematic showing is suppose if you have a set of spots where which you have indexed with one type of an indices and if you rotate this like this then the indices should be different you can see that indices are opposite so you have to confirm this it is not that once you rotate this the indices are having different sign it is not completely wrong but then there is an ambiguity you have to fix this which particular indices is given we will talk about it when we do an indexing exercise so this is coming because of the symmetry and this is some of the single crystal electron diffraction pattern for a primitive cubic system these are all available in the literature in most of the books electron diffraction book you will find it you will be able to suppose if you you are able to find out the ratio from your electron diffraction pattern you can directly directly see this kind of an indexed pattern here then you will be able to transfer these indices to your system if it is if it is a cubic system and if you are able to match the angle between the each I mean the orientation and the spacing and the ratio everything matches then you will be able to use this kind of an indexing system nevertheless it is better to calculate of your own and index I will demonstrate in couple of exercises it is always better to do it do a calculation and do it but in if it is a well-known system like the cubic system there is nothing wrong in looking at this pattern and then look at the solutions I will skip this these are all basic information the the another important point I want to emphasize here is whatever we have just seen so far is a single crystal electron diffraction pattern what you are now seeing is a diffraction pattern from a polycrystalline material so what you are seeing instead of a spots you are you are able to see rings here so if a area of the specimen selected by the diffraction aperture contains a crystal in several orientations the diffraction pattern will consist of some of individual patterns in the case where the specimen consists of very many crystals of random orientation the spots are so close together that they fall on a series of continuous rings so what does it mean you you have in a single crystal pattern of different orientation whatever you have just seen some of the I just showed as for a cubic systems FCC, BCC and HCP systems is that they are all single crystal that means a diffraction is occurring from a single crystal or a single grain when you when you are aperture focusing a region which that region contains a lot more a crystals single crystals or the crystals oriented in much more random positions or orientations then your diffraction will be not a spot but a ring because the pattern will consist of a sum of all individual patterns suppose if you put all the patterns together then that will form fall in the the ring so all the individual or independent orientations that can be put are superimposed that is why it is called sum of individual patterns then you will find a ring pattern that means the physical meaning is all the you know the many crystals of random orientations will contribute to the a diffraction conditions that is all it means suppose if you look at the basic diffraction conditions that means you have all the orientations that means many many crystals are obeying the Bragg conditions and then contributing to the diffraction intensity that is all it means so now this is one simple example of how to use this camera constant to identify the single electron I mean single crystal electron diffraction pattern so this is a diffraction pattern from a fels bar which is a C phase centered triclinic crystal structure so the electron diffraction pattern appears like this the camera constant for this particular fels bar is lambda L is equal to 3.6 nanometer millimeter this is this how you have to do the measurement from the center that is a transmitted beam to the diffracted beam is r1 this is an r2 so r1 is about 9.5 mm and r2 is 6.35 mm using the relations you can lambda L is equal to rd d1 is equal to 0.379 nanometers and d2 is equal to 0.567 nanometers so like that we can readily index this electron diffraction pattern provided the camera constant is calibrated and well known otherwise this kind of indexing procedure is not valid I will show some of the other procedures of indexing the diffraction pattern in some of the tutorial class. So now we will look at the how the Ewald sphere is related to a reciprocal lattice so the Ewald sphere it links the reciprocal lattice to the Bragg law this we have already seen in an x-ray diffraction phenomenon just to recall you see that this is a Ewald sphere and here you have this sample incoming ray this is a transmitted beam and this is a diffracted beam for this geometry the radius is 1 by lambda and then you can write sin theta for this geometry OP by OX which is equal to 1 by D divided by 2 by lambda or 2D sin theta is equal to lambda this is simply a Bragg law derivation from this schematic so what it signifies the signification is so wherever the Ewald sphere cut through the lattice the where the reciprocal lattice point exactly intersects with the surface of the sphere then you will though only those spots will appear in the diffraction pattern what is the physical meaning only those spots will satisfy the Bragg law so that is that is why only those spots are appearing in the diffraction pattern not the other pattern. Now we can also look at this Bragg law using this vector equation so you have the Ewald sphere and this is an incident beam that is a k0 and this is a diffracted beam with vector k and this is the G vector it is a diffraction vector and G is equal to k-k0 so this is a vector equation alternative to the Bragg's law is shown like this this geometry is more appropriate for x-ray diffraction where the wavelength is of the same order of the magnitude as an atomic dimension that is the only difference and what you are now going to see an animation is very interesting animation you see that as I mentioned that in the beginning of the today's lecture or when I talked about the difference between an x-ray diffraction or electron diffraction the only difference is the wavelength one of the important aspect of the diffraction between x-ray and electron so here you see that I will play this animation again what you see you are now the Ewald sphere is so big that you know most of your diffracted spot are falling exactly on the periphery of this Ewald sphere so you are able to see the many number of spots in an electron diffraction pattern as compared to x-ray diffraction pattern in x-ray diffraction pattern you see only few peaks not so many number of peaks as you see in an electron diffraction pattern you have n number of spots around a transmitted beam so that is because the lambda is so small so that 1 by lambda is you know so big so your Ewald sphere become very big and then all this reciprocal lattice point are exactly intersecting the cutting through the or I would say that Ewald sphere is cutting through all the reciprocal lattice point so you are able to see so many spots around the transmitted beam so in electron diffraction the wavelength is 2 orders of magnitude smaller than x-ray diffraction as you can see a large number of reciprocal lattice points are close to the surface of the reflecting sphere because the radius of the sphere is so large so that we have seen so the other information about this reflecting sphere or Ewald sphere is you have to appreciate you have to see why we see a spot in a in an electron diffraction so what I really mean here is the sphere the diffraction spots are not really spots but are elongated perpendicular to the surface of the specimen if the specimen is of the thickness t the reciprocal lattice point is straight out to a length of 1 by t producing a rail rod that is a reciprocal lattice rod so since we are talking about a reciprocal relationship in the diffraction intensity so it is suppose if your specimen is having a thickness in one direction that is important suppose in the suppose this is the specimen direction I mean and this is your electron beam so this is the thickness of the sample then your diffraction intensities are going to be produced perpendicular to the I mean the orientation of the specimen so that is what it is shown here so in the 3 dimension what I just mentioned before when I showed a theoretical calculation of the reciprocal lattice in 3 dimension I just mentioned about the each spot in an electron microscope I mean the reciprocal lattice produced in electron microscope is in 3 dimension and then each spot intensity profile depending upon the specimen shape suppose if your specimen is a foil or a thin sheet of a thickness t then your intensity profile will be perpendicular to that orientation so similarly you have a cubic specimen then your intensity profile will be uniform in all the 3 directions so if your sample is oriented in this direction and your intensity profile will be in the opposite direction so it is a reciprocal relationship so similarly if you in most of the conventional TM analysis your sample is a thin foil and your electron beam is passing through the thin foil perpendicular to this so you will see that your reciprocal actual point will have intensity profile like a rod that is what it is shown here so you actually you produce a rod like this in 3D since you are I mean a wall sphere cuts through all the rail rod and when you cut the rod in a cross section you will see a circle so that is what you see as a circular point of intersection that is what is being projected on the fluorescent screen that is why you see as a very small small circular spot as a diffraction spot in a actual diffraction pattern so this is one of the concepts you will be able to appreciate in fact the very effective usefulness of the wall sphere concept is this where you are able to appreciate the actual what you are seeing on a screen why it is appearing in a circular spot so this is one of the usefulness so you can see that clearly in this schematic A where the wall sphere cuts through the rail rod and then you see that particular in fact it is not cutting in a line it is a complete it is a sphere so it is a surface so the whole surface cuts through the reciprocal rod in a 3 dimensions so you get a 2D surface like this what is shown in this schematic here and you can also appreciate that if the orientation of the you know your reciprocal lattice is slightly different you can see that this is a 0 order a diffraction spot and then you see the first order and second order and so on so another important aspect is so whatever we conventionally get a diffraction pattern is the 0 order diffraction pattern and as we talked about in a I mean conversion beam electron diffraction you will see that a higher order law by zones will be identified and you can see that how this higher order zones are being identified using this a wall sphere concept you can see that this is a 0 order this is a first order suppose if I have a reciprocal lattice here then it will appear like this first order second order and so on so this is another important aspect you have to remember what you are seeing is a cross section of I mean reciprocal lattice rod in 3 dimension what you see in a 2 dimension as a diffraction pattern so you can see that some of the examples this is a 0 order zone this is a first order in a TEM you can see the second order I mean this first order and second order higher order zones are visible using this concept you can see that the electron beam is parallel to the zone axis the crystal has been tilted slightly with the consequence that the first order law by zone is visible so this is a first order for that we need to just do the tilting experiment you will be able to appreciate this remember using a convergent beam electron diffraction this experiment can be done and we will be able to identify higher order law by zones which will enable as to characterize some of the symmetry elements in a material so we will see it when in the convergent beam electron diffraction when we take it up. So I will stop here and then I will continue this diffraction concept in a TEM and in the next class I will discuss the importance of kikuchi lines and how they are generated what to do with the kikuchi lines and so on and similarly we will look at little more detail about a convergent beam electron diffraction and its use in general very briefly we will look at it if not much more detail and then we will move on to some of the imaging aspects of TEM thank you.