 Hello everyone, welcome to this material characterization course. So far we have seen a diffraction principle in TEM and then various techniques involved in a diffraction and how to extract information from the simply looking at the diffraction pattern of transmission electron microscope. And also we have seen that you know how to use this Kikuchi line and how to use the convergent beam electron diffraction and it is various advantages of using these techniques in much more briefly. Similarly, we will just go through the imaging techniques in transmission electron microscopy before you look at the laboratory demonstrations of the actual experiment in TEM. So, we will not get into the detailed analysis of TEM imaging in this course because it is only a part of a characterization course, but you will get the basic idea behind each imaging techniques in the conventional transmission electron microscope. So, if you look at the major classification of imaging, the first one goes like this amplitude contrast imaging that is formation of a bright field image by using only the transmitted beam. B, formation of a dark field image by using only one strong diffracted beam. So, the other category is a phase contrast imaging where the formation of image by more than one beam interfering each other. The contrast is therefore a composite of interference effects and amplitude changes produced by defects. So, we will first go through the amplitude contrast imaging or we can also say that it is a diffraction contrast. And the name itself says that once you say that it is a diffraction contrast or amplitude contrast, it directly related to the diffraction intensity. So, you have some idea about how the diffraction intensity is been perceived and which we have seen so far. But just for a clarification, we will go through that once again. And we in a X-ray diffraction we have seen that you know in a diffraction you have you know you have kinematical theory and assumptions. Similarly, in TEM also you have a couple of theories which arises out of this specimen. So, all the kinematic relation what we have seen in X-ray diffraction holds good here. And then we will say that in another theory called a dynamic theory and then we will see the difference the key difference between these two theory and how they are explaining the origin of the contrast in the image. So, these things we will see. So, if you look at the kinematical theory is applicable to only thin specimen and for conditions away from the exact Bragg condition which is S not equal to 0. You have to remember this point very clearly here. We will be using this parameter called deviation parameter called S which is which describes the diffraction takes place away from the Bragg position. So, as I mentioned in the previous lectures also this idea you have to keep in mind whether the diffraction takes place with exact Bragg condition where S is equal to 0 where the diffraction intensity where we have looked at where S is not equal to 0 that means away from the Bragg position. So, kinematical condition applicable to only in this situation for a thin foils. So, if you look at the diffracted intensity this is psi square D stands for diffraction which is equal to F by T whole square sin square pi T S by pi S square and the intensity for a transmitted beam psi square is equal to 1 minus F by T whole square sin square pi T S by pi S square. So, you can just simply write this psi square T plus psi square D is equal to 1. So, the kinematic theory predicts that the bright field and a dark field images are complementary because even in the x-ray diffraction we have mentioned it assumes that the only one diffracted beam and the one transmitted beam that is a two beam condition. So, that is why this equation predicts that you have a bright field image as well as a dark field image they are complementary to each other. So, the kinematical function predicts periodic variations in intensity with the thickness for for a constant S that is thickness fringes or variation in intensity with S for a constant thickness which lead to fringes about the Bragg contours. What we are now trying to say is this function whatever we have just said is the function is a kinematical function which predicts a periodic variation in the intensity with the thickness for a constant S suppose if you S is constant the so the how the periodically the intensity varies or the variation in the intensity with S for a constant thickness. So, these two will describe how you see a kind of fringes. So, the kinematical theory I try to explain this the fringes which we see in a bright field image T M image. The intensity maximum occurs in dark field whenever S is equal to N by T that is a constant thickness or when T is equal to N by S for a constant S. So, this we will see in an example in an example and before you go to the actual imaging let us look at this slide very interesting slide this is the most fundamental idea one should get before you get into the interpretation of image contrast. You see this is the assume that the schematic shows the a thin specimen of thickness T and then you have this incident beam psi 0 and which enters the specimen and then you have and from the diffraction phenomenon you know that the beam enters the specimen and then try to interact and some of the assume that if some of the beams are diffracting. So, the intensity of the transmitted beam as well as the diffracted beam will oscillate like this why it oscillate like this we have the reason we will see in a few minutes. But now look at this suppose if you assume that the beam enters the sample surface and as it go inside the deeper into the crystal then the transmitted intensity follow this oscillation and the diffracted intensity follows this oscillation. So, that point you remember and this distance that means if you look at this transmitted intensity this portion is characterized as psi g called extinction distance we will in a minute we will see what is it and so much of intensity is lost from the transmitted beam. So, this distance is a characteristic distance for a material and depends upon various parameters that also we will see and you see that depending upon the number of this psi g n psi g in the sample which is equal to thickness of the sample I mean thickness of the thin foil this also we will see how it is valid. But before you for time being ignore about this but what I am trying to show here is you please start thinking before you interpret the image contrast in a TEM the electron beam enters this sample and your transmitted and a diffracted beam have the intensity oscillation in this manner that is that point you remember to start with then we will see how we can manipulate this idea to understand the some of the image contrast in the TEM operation. So, when you what is written here the intensity oscillation in a perfect crystal section of a thickness n psi g. So, we are talking about a perfect crystal that means I do not have any fault within the top to a bottom surface of the crystal it is a perfect crystal I do not have any you know boundaries I do not have any vacancies I do not have any dislocations I do not have any interface and so on. So, you assume that it is an ideal crystal and then your electron beam transmit through the sample and then that time the intensity oscillation is of this kind we will just work on this idea as we move on move along the periodic variation of psi square that is intensity with the T leads to a primary extinction T naught is equal to 1 by s forget about this T naught is equal to 1 by s which is a relation which how the extinction distance has got we will we will talk about it in a in a while, but the periodicity of the potential energy that originates with the periodicity of the atom arrangements. So, now we will try to answer the the question why the periodicity why the intensity oscillates like that. So, we will try to have some kind of a qualitative understanding the periodic potential causes the amplitude of the high energy electron to be transferred back and forth dynamically between forward scattered and diffracted wave functions at precise law a condition for a for a strong diffraction the physical distance over which the wave amplitude is transferred back and forth once is called the extinction distance. So, now you see that you have the an accountability for why we see that intensity oscillation in the a perfect crystal. So, it is because of the periodic potential that causes the amplitude of the high energy electron to be transferred back and forth. So, we have the lattice lattice have the atoms and then the lattice has got a periodicity and then the periodicity of the potential energy of the lattice which affects the amplitude of the high energy electron. So, that is the basic idea. So, the interaction of these two will cause the beam to oscillate back and forth and at a precise law a condition for a strong diffraction the physical distance over which the wave amplitude is transferred back and forth once is called extinction distance. Now, you go back and see this it will make a sense. So, now you see that it goes forward and backward once and this distance is called a extinction distance very important in understanding the diffraction contrast or any fringe contrast in the right field image. So, this is explained again by a dynamical theory the usage of extinction distance and so on. In dynamical theory that means dynamic diffraction conditions the extinction distance psi g is defined as pi v divided by lambda fg where v is the volume of the unit cell and f the structure factor for the particular deflection. So, you have the idea about structure factor how it originates what it contains. So, you know at least you can now connect what why psi g is a characteristic of a material. At every integral number of extinction distances all the electrons end up in the forward direction that is transmitted direction and at every odd of multiple of extinction distances all the electrons end up in the diffracted directions. So, it clearly describes what we have seen in the the previous slide how the oscillation takes place and so on. And now having said that this is the the kinematical approximation and then how the dynamic dynamic theory explains this extinction distance. In kinematic theory we say that only one diffracted beam we are looking at and one transmitted beam we are looking at and there is no interaction between these two that is assumption of the kinematic theory. In a dynamic theory we say that these two beam try to interfere and then produce some effect. So, the equation which we are seeing in the slide describes this phenomenon interaction of the transmitted beam and a diffracted beam and then what it commits let us see d phi naught by d z is equal to i pi by psi naught phi naught plus i phi by psi g into phi g into exponential 2 pi i s z this is for the transmitted beam and this is a diffracted beam d phi g by d z is equal to i pi by psi g times phi naught into exponential minus 2 pi i s z plus i pi by psi naught into phi g. These two equations are popularly known as how we well in equations describe the variation of the amplitudes phi of the undeflected and diffracted waves as a function of z the distance through a perfect crystal. The first term in each equation arises from the scattering from the undeflected beam and the second term arises from the scattering from the diffracted beam. They show that the amplitude of each wave changes as the wave progresses through the crystal due to a contribution from each other. So, these two equations explains the interaction of transmitted beam as well as the diffracted beam. So, that is the physical meaning of these two equation. Now, we will just come back to the other features of this dynamical theory the exchange of electron density between the transmitted and diffracted beams is exactly analogous to the motion of two coupled harmonic oscillators which periodically exchange all the vibration energy of the system. The subsidiary maxima occurs when s square plus psi g to the power minus 2 into t square is equal to integral. So, this is one you can see the subsidiary maxima in the image contrast. For thin crystals the value of t that is thickness can be determined from the measurement of s at subsidiary fringes either in the image or in the diffraction patterns provided that the values of psi g are known. You will be able to find out the thickness of the foil if you know the psi g of a given material. So, we will see that also. So, the contrast effects in a perfect crystal are expected to be due to changes in thickness that is wedge fringes fringes at inclined defects. Changes in s will result in a bragg contours changes in orientation changes in s as well as g. So, we will now demonstrate all this predictions by the both the theories with the simple examples how the contrast varies or what kind of contrast variation you will be seeing by change of thickness or change of s or change in orientation of the foil. So, the important microstructure features include the changes in orientation with or without change in the structures or composition such as grain, twins, precipitates, structure of boundaries, lattice defects, point defects, line defects, planar defects and volume defects. Multi-phase systems it changes in the composition, but not in the structure example spinodals changes in the composition and structure general precipitation changes in the structure, but not composition for example, modern site interface interfaces coherent partially coherent incoherent. So, all this features can be identified with the contrast mechanisms predicted by the both the theories like kinematic as well as dynamic theory. So, we will see one by one by one. So, this schematic shows the general very general diagram this is I just brought it just for the completion you know the basic idea behind it this is the object and then you have the incident beam coming through this object and you have the lens and this is the back focal plane where your diffraction pattern is recorded and you have the image plane. So, this schematic is illustrating condition for imaging periodic structures if D is large enough and alpha small enough the diffracted beam A or A dash recombines with the direct beam B to give a magnified image of the planes D. The diffraction pattern is formed in the back focal plane A A dash N represent the diffraction pattern and Q magnifies the image of this pattern for the beams A and B. So, a simple ray diagram you all familiar with all the components here and now we will now go to specific operation in a TEM the schematic what you are seeing is a contrast in the bright field image arises through the local variation of the intensities of the diffracted beams and the diffracted beams is stopped by objective aperture. So, what you are seeing here is this is the specimen you have the electron beam coming through this and then you have a direct beam and then this is a diffracted beam and this is an objective aperture in a bright field image you see that all the diffracted beams are stopped and only direct beam is allowed to pass through the objective aperture. On the other hand you have the direct beam is stopped by the objective aperture only diffracted beam is allowed to pass through the objective aperture and that is how the both the bright field and dark field images are allowed to form and this operation can be done through beam tilt and this again we have seen it when we discussed about the diffraction phenomenon this is just for a recall and now we will look at this typical bright field transmission electron micrograph and this micrograph belong to a thin foil of molybdenum showing light grains dark grains fringes and sub grains. So, now our idea is why do we see this a contrast why certain positions or certain grains are appearing darker and why certain positions appear in gray color and why particular grains are appearing in bright. So, that is that is where we have to look at the theories and how we understand this. We should also understand why we see this fringes in the bright field image and we will go back to this image and then first try to explain why this grain is appearing dark. So, if you if you can go back and then look at this schematic what you are seeing here is here is a specimen and then this is an electron beam passing through and this is a direct beam and this is a diffracted beam. Just assume that this diffracted beam which is coming out of particular region or a set of planes or set of grains being stopped by the objective aperture which is not reaching in the image plane. So, you can now correlate by looking at this schematic and this image that means this is a bright field image I said. So, the region which is appearing very dark are diffracting very strongly and they are not getting entered into the objective aperture. For example, the complete set of planes in the grain and this region and this region they are strongly diffracting, but they have not entered into the objective aperture. So, that regions will appear dark and the region which are appearing very bright that means they are they may not diffracting strongly or the diffracted beam also entering into the objective aperture we can consider that way also. So, you can either consider it is not diffracting strongly or the diffracted beam also entering into the objective aperture. Basically, this contrast arises because of the orientation of the grains. So, you can see that a strong diffraction not diffraction are in between. Some diffracted beams are entered something is not entered into the objective aperture. So, you can apply all the knowledge which you got in the diffraction phenomenon including the deviation parameter and so on. We can put together to understand this to give some perspective why you get this contrast. So, you can also look at some of the other very intricate features like a fringes. I know that a grain which is very dark we can say that that particular grain is strongly diffracting, but when you have the fringes like this why I get this a very systematic absence of a dark and bright dark and bright systematic absence of intensity. Why do you see that? That we will look at with this example or a schematic illustration of the origin of fringe contrast in the images of the crystal. So, look at this schematic you have the thin foil which has got a hole inside. After electrolytic polishing typically you produce a hole like this in a metallic foil that means you create a nice edge by jet thinning. Assume that your hole is prepared I mean in this manner and as I just showed in the two slides before this is the you know the electron beam enters the sample in this direction and then you have the transmitted beam that are direct beam or a diffracted beam. They have the intensity oscillation like this and the thickness of the foil is T from top surface P to the bottom surface P prime and then please note that the diffracted beam is out of phase by pi by 2 with respect to their direct beam. So, your direct diffracted beam is out of phase by pi by 2 in this. So, now you just see that the bottom portion of the schematic is B which shows some kind of a bright dark bright dark bright lines they are called fringes here also it is the fringes are in circular shape here it is in a vertical lines they are all fringes. So, now the question is how to account for this intensity oscillation in this fringes. So, if you look at this a transmitted intensity oscillation it is a diffracted intensity oscillation this periodicity is T naught prime the T naught prime which is again equivalent to psi g or an extinction distance we can say that. So, you look at this suppose if you have a boundary like this in the material and then you can just look at the how the diffracted beam will travel through this a boundary then because of that how the intensity variation will appear in a bright field image that is what we are interested suppose if you have a boundary inclined like this AB you see that you have the the diffracted intensity maxima here it comes on the bright field it will appear like a dark line wherever the diffraction maxima is there you see in a bright field image it will appear darker because the diffraction maxima is corresponding to minima in the direct beam intensity oscillation. So, that is where the the dark lines are seen remember that intensity is not 0 there but it is minima. So, when you have this kind of a diffraction intensity which oscillation which come across any inclined boundary like this or this you will have this intensity oscillation that is called a fringes let us go through the a caption. So, this is the illustration of the origin of the fringe contrast in the image of crystal section through the crystal showing the kinematical intensity oscillation of the direct and diffracted beams the depth periodicity T0 prime is equal to 1 by S and AB the grain boundary or it could be a stacking fault CD is a wedge and E is a hole B is a section normal to the beam corresponding to the bright field image showing dark fringes F or a thickness extinction contours. So, this particular fringe contrast is related to S that is inversely related to S 1 by S T is equal to T0 prime is equal to 1 by S and then we will show some more examples for this. So, now this is accounts for this fringes what we are seeing any fringes we are seeing. So, not only that even if you look at the dislocations but what we are seeing here the sub boundaries sub boundaries or dislocation whatever we are seeing here it clearly related to the surface unevenness which will have ups and downs when a such a crystal will interact with the the diffracted beam under I mean a electron beam I would say which undergoes a intensity oscillation as we have seen in the schematic then this kind of contrast is possible is that is a very simple qualitative explanation one can get without getting into the any complication. You can see that the thickness extinction contours and the bend contours in the samples for a molybdenum crystal the grain boundaries lie normal to the plane of the foil here you can see that the boundaries and you can see this fringes or you can say that contours bright and dark line because of the it is orientation and the ups and downs or it could be ups and downs. So, if it is an inclined surface you are bound to have this intensity oscillation due to the the we are because it is intersecting with the the diffracted and transmitted intensity oscillation. So, that is what it is the this the micrograph shown here is foil of aluminum you have some features marked a b c d and so on you have a two low order strong reflections hkl and minus h minus k minus l the single contours b c d belongs to a higher orders of hkl. So, you have this particular band is because of the strong reflection from particular hkl plane and they are all belong to higher order hkl this dark lines comes from and please note that though you may consider the whole band as a strong a diffraction or whatever it is appearing bright and dark where the kinematical theory explains there is a variation within this band bright and dark which is explained only by the dynamical theory where the transmitted and the diffracted beam interacts and produces this oscillation. And if you apply this concept of kinematical theory and we can look at some of the pits and the holes which forms the specimen and they also produces a fringes. So, look at the schematic this is considered a thick specimen which has got a hole in it. So, this is the intensity oscillation of your diffracted beam and this is a transmitted beam and so on and this is the extinction distance and since the extinction distance is 3 times of the thickness so we can write it is equal to t is equal to 3 psi g and you can look at the fringes from the this is the thick full thickness after that you form a fringe pattern like this circular fringe and it clearly shows that the distance between these 2 fringes is related to extinction distance like this. So, if you simply count the the distance the number of fringes we will be able to calculate the thickness of the foil this is of course pertaining to that particular location where you are seeing a pit or a hole in the foil. So, you will be you are seeing this fringes then you can relate this fringes and its count to the the total thickness of the foil. So, this is a geometrical description of formation of image pattern at holes or a similar thickness irregularities in thin foils dark fringes correspond to the intensity minima in the bright field. So, whatever the dark fringes you are seeing as I mentioned the previous slide it is not 0, but it is an intensity minima in a bright field image. So, you can see that a typical example which is shown in this diagram this is a not sorry it is not a diagram this is a micrograph actual micrograph you see that hole which is formed in a foil has got the fringes like this and here again another schematic which illustrates the the bend extinction contours. So, you have the foil of this nature which is a bend and this is the electron beam entering into the foil and it will produce the the contours of this nature and this suppose if you assume that this is the bright field image and this is the intensity oscillations and if it is a dark field image this the dark will appear brighter and this will appear darker and so on. So, this is the geometrical origin of a bend contours for the idealized case in thin foil section observed in the bright field illumination and this is again another example of the fringe contrast and and etched pits in a steel. So, you will get a symmetrical contours like this because you can identify the symmetrical nature of the contours by the SAD pattern which is given as an insert here. So, the point which I want to emphasize here is if you understand the electron beam interaction with the perfect crystal and then if you assume that the intensity oscillation of the transmitted beam and a diffracted beam and any irregularities or a thickness variation or any of the microstructural features they are going to give rise to a intensity oscillation or called a fringes or bend contours and so on. So, here is another example where the where you have the sample which has got an inclined boundary and you have this intensity oscillation of the diffracted beam. So, the geometrical origin of the fin contrast at the phase shift or interface plane the number of fringes depends on the extinction distance for the operating reflection G. So, please understand one thing very important thing this extinction distance is a characteristic of an operating reflection G that is for a given Bragg condition the extinction distance is valid. So, the number of fringes as I just mentioned depends upon the extinction distance for the given operating reflection. So, here is a typical micrograph given for a grain boundary. So, you see that the fringes in the inclined boundaries intersecting the polycrystalline section is shown here. See the boundary is not straight here it is an inclined plane for this orientation. Suppose if you change the orientation you will be able to see a such a slope in this boundary also. So, that is where the tilting exercise is very sensitive in TEM. So, you see that the kind of fringes it produces that clearly shows that kind of a taper section that boundary has and then we can identify this boundary and the fringes comes because of the what we have seen in the schematic. The reason for the obtaining a fringe contrast is explained qualitatively by the kinematical theory and dynamical theory and so on. And we can also explain this diffraction contrast through something called phase amplitude diagram. What you are seeing here is let me first start with the polar representation of a complex number which is being used to construct the phase amplitude diagram. This is for an unit circle on a unit circle exponential i theta. This is a polar representation of complex number and what you are seeing here in this schematic is the wavelets diffracted from a unit cells at increasing depth rg in the expression diffracted intensity psi s is equal to summation over rg into fg exponential plus i 2 pi s dot rg. Recall that s is lying the vertical z direction. So, this the diffracted wave which travels inside the crystals. So, that is how the unit cells are represented by this because the total diffracted intensity is always summation of the individual diffracted wavelets like this. So, we will just look at that which more detailed using this phase amplitude diagram. The diagram which is shown in the here is vectors representing the individual terms that is nothing but your complex exponentials how this each diffracted wave is considered like this. So, this animation clearly shows that suppose if you have if you try to sum up the individual diffracted terms like this, then the total intensity of diffraction is given by this summation of all this exponentials complex exponentials in this fashion. So, that is how the diffraction intensity is appreciated or I would say is understood by this phase amplitude diagram. We will now see that how this phase amplitude diagram is useful in understanding the contrast what we see in a bright field image. So, this is the again representing the intensity psi psi star where you have the real part and imaginary part of the total intensity and you have this phase amplitude diagram for two conditions one is for s is greater than 0 or s is much much greater than much larger than the 0 that means you have the phase amplitude diagram for two values of s phase amplitude diagrams of diffracted intensity for two deviation parameters s the eight short vectors each have the same length, but different orientations. So, the very important information which you obtain from this phase amplitude diagram is. So, you will be able to say what kind of a contrast we are going to see just by looking at this s values or how this the circle the diameter of the circle the phase amplitude diagram circle how it varies by just looking at this we will be able to comment on how the intensity is going to vary. So, we will see an example like that. So, here is an schematic where you have a wedge kind of a sample and you see the electron beam coming and then how the the diffracted wave vectors will form how exactly it get you know rotated the I mean I would say that all this simple diagram shows that how the phase shift occurs as the function of the specimen thickness and then what you see here is the intensity of the transmitted beam and intensity of the diffracted beam you can see that it is very close to the the edge of the edge of the specimen you see that you know the intensity is high in a transmitted beam, but as you go up higher the thickness you are able to see the the diffracted intensity goes to the maximum and then as it comes down further thicker side that means you have that destructive interference contributes and then it comes down and then again it increases. So, you can see that clearly the phase amplitude diagrams for increasing thickness across a wedge shaped crystal showing a origin of thickness fringes in the diffraction contrast. So, all the contrast whatever we have discussed in the whether it is of a grain or a fault or boundaries whatever we have just discussed in the fringes they are all coming under diffraction contrast or amplitude contrast imaging and this particular illustration clearly shows that the the thickness variation will have a significant influence on the the the fringe contrast what you see. So, here you see a similar example where the phase amplitude diagram shows the for a same wedge shaped crystals, but with this yes smaller by a factor of 2 that means you reduce the yes that means what kind of changes it will show. If you go back to the the introduction where I showed here that means my phase amplitude diagram circle is going to be bigger if I reduce the yes the phase amplitude diagram or the circle is going to be bigger. So, this is what I am seeing. So, the the phase shift which is represented by this phase amplitude diagram is bigger compared to the different in the S value. So, similar effect we can see it in one of the examples we can take some examples and see what what what we have gone through is true or not and we can look at actual fringes in a metallic samples and then we will continue this discussion in the next class. Thank you.