 Hello everyone, welcome to this material characterization course. From today's class onwards, we will discuss about transmission electron microscopy. And we have found so far the characterization in terms of x-ray diffraction and also we used electron microscopy that is scanning electron microscopy and so on. And then we have enough background to take up this another advanced variant of microscopy in terms of electron optics and also in terms of diffraction physics. So with all that background, we will be able to appreciate this microscopic technique also without any problem I believe. And nevertheless, I will reemphasize some of the concepts then and there whether it is a diffraction physics or an electron optical system wherever we need to emphasize with respect to a transmission electron microscopy. So I will just begin the introduction of the transmission electron microscopy course. What I will do is I will first introduce a very general introduction I give about this technique what is that people expect or what is people want to do with this microscope, what are the information they get and then I will touch upon or I would say that we will refresh whatever the basic diffraction and then basic physics behind or optics electron optics once again we will touch upon. And then I will take you to the instrumentation details how what are the various parts and then functions especially and then how they facilitate the imaging system and so on. Then in the later what I will do is I will take to the diffraction in TEM in much more detail what are all the possible experiments one can perform in TEM which exploits a diffraction phenomena and what kind of information you get and then I will focus on imaging part and in imaging we will also discuss about what kind of contrast mechanisms very briefly because it is a part of a characterization course which I would like to finish in our 10 to 12 lectures the fundamentals of all this transmission electron microscope. So if you look at this in the fundamental of electron optics I would like to look at this expressions much more carefully and this kind of expression can be derived from the de Broglie's hypothesis and he showed that in an energy of the electron can be related to lambda or you can say that you know if the acceleration voltage increases you can bring down the lambda to the very small value. Of course this expression is valued I mean valued or you can use this if you ignore the relativistic effects and this is the basic relation but if you do not ignore this relativistic effect then you need to go to this kind of an expression because at the very high voltage the electron travels about one and half times the speed of light. So we cannot ignore this relativistic effect and if you ignore this relativistic effect then you can use this relation and then you can approximately derive lambda is equal to 1.22 divided by e to the power half where e is the electron volts please remember electron volt is the energy of the electron in acceleration and lambda is in nanometer for a 100 kilo electron volt electron we find that lambda is approximately equal to 4 p cometer which is much smaller than the diameter of an atom and we represent the acceleration accelerating voltage of the microscope and e we represent the energy of the electrons in the microscope. And then I would like to give you some very brief you know introduction about this images or the results which you can obtain from an electron microscope normally what we are interested is in a microstructure microstructure containing features in terms of you know if it is a metallurgical sample people are interested in defects and if it is you know physics and then you are interested in defect density and atomic positions and so on and so but what you have to keep in mind in totality I mean the TM image gives an average image in terms of the depth it does not have the depth sensitivity. So what you are seeing in the slide is the image of a dislocation in a gallium arsenide a band of dislocation threats through the thin specimen from the top to bottom but remain in focus through the foil thickness so what I try to say is from an TM image you will not be able to see these features are in the top surface or the middle of the surface or the bottom of the surface. So it gives what you are seeing is in a actually a projection which you will see what I mean by projection and what you are seeing is in an average you are not able to distinguish whether these features are lying in the top or whether these features are lying in the middle of the foil or bottom of the foil and so the TM micrograph will not have the depth sensitivity that is one information and what are the information you will get from this TM results in terms of crystallography you get the basic idea of the crystal structure lattice repeat distance specimen shape crystallographic symmetry analysis of miniscule crystals you can derive point group and space group and so on. So this is the typical electron diffraction pattern one can get from a TM and this is a TM diffraction pattern from a thin foil of aluminum lithium copper and these are the wide range of information one can derive from the transmission electron diffraction pattern. So another very important information from this you should as I mentioned that what you are seeing that there is a photograph where you see that two animals are standing behind one another but it is appearing as if the head of the animal is appearing in the same both sides but which is not the true. So the depth sensitivity is not available in the TM micrograph this is a point I would like to emphasis here you have to be very you have to keep that in my information in mind all TM information that we get images diffraction patterns spectra is average through the thickness of the specimen the most important information one have to keep in mind whether it is an image whether it is a diffraction pattern or a spectra is average through the thickness of the specimen what is the thickness of the thickness of the specimen how it varies we will see when we prepare the sample preparation and a typical thickness we will arrive at and how that is affecting the imaging condition that we will see it in the in the coming classes. A single TM image has no depth sensitivity so this aspect has been illustrated in the previous couple of slides and another important thing is you may get very different kind of features like this you may consider that you know one may think that this is a kind of a second phase particle and this is another coarsening effect of the precipitate something like that but you have to be very careful before we use these TM results this could be simply not at all related to the material at all it may be due to the some of the you know irradiation damage by the electromagnetic radiation itself. So unless you have a combination of all the you know the typical requirement to interpret the TM results I will just mention it when we come to the interpretation in an actual TM results you have to be very careful you cannot just under your well experienced in this field it is very difficult to interpret these results and then looking at this kind of image you can be very easily misled because you do not have enough supporting evidence to prove that these are all second phase particle or something like that. So one has to be very careful about presenting just a one bright field image or a dark field image something like that and then talking about a second phase particle and so on which will be highly you know misleading or maybe completely wrong also. So you need to have an appropriate results in combinations of crystallography data through diffraction and we have to prove the second phase particle through the a simple dark field imaging and then you have to correlate with that a bright field imaging and so on. So the what I am talking about all these are different techniques which I will be explaining it in due course of time. So what I the information I want to derive from this slide is you have to be very careful about talking about the features in a TM micrograph just putting at the one slide you have to have additional information to talk about the features of what you are seeing in the micrograph. Now the depth of focus in electron microscope is very high which we have already seen it. I will play a small animation what you have to look at it as this is the simple lens and then you see that in alpha and beta are the angle correspond to the object side and image side and this cross over is projected here and you can simply simple geometry you can derive that alpha image is equal to tan alpha image if you incorporate I mean if you consider this triangle and this distance is d objective and this distance is a small d objective and if you see this is in a beta objective and for example then if you consider this geometry triangle then you can derive an expression like this for alpha image similarly you can do the beta this is distance b d image and this is an alpha image angle and this is the small d image then you can write considering this triangle beta objective which is approximately equal to tan beta objective is equal to d objective by 2 that is this distance half distance divided by d objective by 2 this is this distance this half distance you can write alpha image by alpha and also alpha sorry alpha image as well as a beta objective angles. So what we do with this we can use this relation to calculate the angular magnification in the microscope that is m a which is equal to alpha image divided by beta objective this is angular magnification and then we can also calculate the transverse magnification which is m t is equal to d image by small d objective and which is nothing but m t is equal to 1 by m a that is inverse of angular magnification is your transverse magnification these are small relationship. So you should appreciate this you have enough background to appreciate this now and we also know that the depth of focus d image is equal to d objective by beta objective times m t square and the depth of field is d objective is small d objective divided by beta objective. So this is the relation between the depth of field and depth of focus with respect to that ray diagram what we have seen. So you have this background to appreciate this we have already discussed enough what is depth of focus in depth of field it is just to give you a recap and now I will just show a schematic where you see that in the electron optical systems they are mostly characterized by small aperture angles leads to a decisive advantage where the image focus is concerned. So you can see that the rays which are converging here and then converging there that is above and below the objective you have a finite distance where the image can be sharply focused that is in general the schematics in general displays the very small aperture angle effect and then the concerned depth of focus where alpha is the aperture angle and df that is this is a df is the most effective electron beam spot size for a collection semi angle of 10 milli radians and dob of 2 angstrom equation 2 tells us the depth of field will be 20 nanometers. So the d objective small d objective of 2 angstrom if you substitute that into the equation 2 what we have seen you will get about 20 nanometers this means a specimen of this thickness can all be in focus at the same time if you want to see a detail at the 2 angstrom level we need to use a magnification of about 50,000 x equation 1 tells us that under these conditions the depth of focus will be about 5 kilometers if we only need to see 2 nanometers we can use a magnification of 50,000 x and still the depth of focus is 5 meters. So all this ray diagram and the small small mathematical expressions illustrates a point that the depth of focus in an electron microscope is very very high and then you will see that the this aspect has been exploited in the in a transmission electron microscope hardware itself where you though you will see that image formation is occurring in the fluorescent screen in the on the table but your recording system will be much much below where you may have a plate camera or a film camera or it is a CCD which is much below but still whatever you are focusing that image on the fluorescent screen will be nicely recorded in a CCD camera of the same focus. So that is one evidence that the electron microscopes have a significant depth. So few more remarks on the depth of focus the depth of focus is related to the depth of field through the magnification M where capital D is equal to D M square by alpha compared to the object plane the extra factor of M square for the depth of focus arises because the image is larger by a factor M. So the ray intersections defining the image plane move M times more rapidly than those on the object plane rays of different angles that converge at the same point on the image have mutual angles M times smaller than what they left the object plane. So now we will try to demonstrate whatever we have just read through with through a schematic you just observe that this is a glass lens and then this is an object for example a solid line D 1 and then see that there is a small correction it has to be you know that is the these two lines are supposed to intersect and then diverge and it is drawn as a parallel line it is not true it has to be an intersection but I will assume that it is intersecting and then diverging like this and then diverging in this direction like this. So you see that suppose if you assume that the solid red arrow is an original object and it is being imaged and what you are seeing here is suppose if you see that you know the distance D 2 is the it is a limit of the blurring image because the you know which when you use this when you move this D 1 slightly to the distance D 1 with the dotted line then these two rays will trace like this you can follow this suppose if you assume that yes this is my solid line original object if I move that into slightly a position D 1 then the divergence happens and then the green ray will trace like this because of that the distance D 2 the blurring will occur and your D 1 is because of the mispositioning of the your the image plane where you have one here and one here intersection so it is a misposition of the image plane so that causes a D 2 so what we have just seen as a if you go back you can you can just verify this that is the depth of focus is related to the depth of field through it is a m square times the I mean depth of field that is that we can prove here suppose if you have this this distance is approximately you know I mean the the object here is magnified here approximately 2.5 times this is to the scale and you can see that if you square of this that is you can see that 6.3 D 1 times the D 2 that means the depth of focus is equal to 6.3 times the the distance of depth of field depth of focus is m square times the D 1 so that you can geometrically prove this this illustration clearly shows that the geometrical demonstration for the factor m square it is not m 2 it is m square so that clearly shows about depth of focus now we will quickly review the resolution of the electron lens we have already seen this in a fundamental of electron optical system and just to recap you would like to go through this a resolution is defined as minimum solvable distance and then if you consider the theoretical resolution if there is no aberration at all the resolution of any lens that is glass or electromagnetic is customarily defined in terms of Rayleigh criterion which is also a practical definition the criterion gives as a merit in terms of the eyes ability to distinguish images of two self luminous incoherent point sources a single point source will not be imaged as a point even if no aberrations or astigmatism are present I will play this animation for these two self luminous point sources which are trying to converge and then these two point sources will be recognized as a independent source only with the distance of 0.61 times the lambda that we have already seen it so you just recall that earlier discussion the finite size of the lens results in diffraction of the rays at the outermost collection angle of the lens usually by limiting aperture this diffraction results in a point being imaged as a disc called the array disc so you remember this is an array disc which we have already seen it so I will not discuss that further which has a cross section intensity profile and Rayleigh stated that if the maximum from one source lies over the first to minimum from the other source then the overall intensity profile exhibits a dip in a middle at about 80% of I max so this also we have seen so the minimum of the next source will be matching with the maximum of the first source so and this dip will occur at about 80% of the I max so this also we have seen previously the eye can discern this dip as two overlapping images thus indicating the presence of two separable sorry two separate objects under this circumstances the distance apart of two incoherent point sources is defined as theoretical resolution of the lens and rth and it is given by the radius of the array disc rth that is theoretical resolution is equal to 0.61 times lambda by beta and we have also seen about the spherical aberration where CS is the constant for a particular lens called the spherical aberration constant and B is a semi angle of collection in the objective lens the resolution of the object is given by some combination of Rayleigh criterion and aberration error so we will now look at the some treatment by Hawke's gives the particularly a clear description of how this combination leads to a value for a resolution in a microscope. So suppose if you include the spherical aberration coefficient how it is going to be so this is a Hawke's treatment suppose if you assume that we are taking a spherical aberration into Rayleigh criterion and take the combination of Rayleigh and spherical aberration discs in the quadrature rth is equal to rth square plus r's spherical aberration because of spherical aberration r due to spherical aberration is called rxph square to the power half we can now thus find how r varies with beta using this relation r as a function of beta is equal to 0.61 times lambda by beta whole square plus CS into beta cube to the power I mean to the power 2 whole to the power 1 by half that means the square root of all this expression the two terms vary differently with the aperture collection semi angle beta a compromise value exist when dr by d beta is equal to 0 is if you can differentiate that expression you will get this kind of value from this equation the optimum value of beta can be obtained like this b opt is equal to 0.77 times lambda to the power 1 by half divided by CS to the power 1 by half so this is called a spherical aberration limited resolution and for a 100 kv electrons a lambda is 0.0037 nanometers for an instrument with a CS is equal to 3 mm gives a beta opt value of 4.5 milli radians so you have r minimum is equal to 0.91 times CS lambda to the power cube whole to the power 1 by 4 this expression that gives the practical resolution of the microscope typically the value for the r minimum is 0.25 to 0.3 nanometers but for high resolution instruments have they are minimum which is approximately equal to 0.15 nanometers so what we have now trying to say is we have already stated that spherical aberration is very important operation which is very difficult to eliminate from the lens so if you keep that spherical aberration into a system and then how the resolution is getting modified that is the bottom line and these are all the small I mean steps or derivations which demonstrates to you what is the how the resolution expression get modified as well as the how the beta angle getting optimized so that is the basic information nothing to get confused here so now we will again go back to some of the basics look at this animation what you are seeing is suppose if you assume that this is a thin specimen which being irradiated by the electron beam and then your transmitted beam will have or if you the image of the specimen will have an oscillation in the intensity that is scattered electrons with varying intensity you will see okay and also you have the incident beam you have a diffraction pattern as well as the forward scattered beam of electrons so these two you are going to get in the transmission electron microscopy where you have a thin specimen is placed so what you are seeing here scattering within the specimen changes both the spatial and angular distribution of emerging electrons so that is the idea you have to appreciate this the scattering within this thin specimen changes both spatial angular distribution of emerging electrons that is it and you can see other schematic so the schematic is self-explanatory so you have a background to understand this so a coherent incident beam is falling on the thin specimen then you have back scattered electron second electron and coherent elastic scattered electrons and then you have direct beam incoherent in a in elastic scattered electrons and so on so if it is a bulk specimen there is nothing like you see the forward scattered I mean signals forward scattered signals only you get the backward scattered signals only a thin specimens permits electrons to be scattered in both the forward and backward directions while the bulk specimen only back scatters the incident beam electrons so very fundamental idea you know it but you have to why we are saying this because in transmission electron microscopy we use only the forward scattered electrons we do not look at the backwards scattered electrons and will quickly rush through this basic idea again once again elastic scattering is usually coherent if the specimen is thin and crystalline elastic scattering usually occurs at relatively low angles 1 to 10 degrees that is in the forward direction at higher angles for example greater than 10 degrees elastic scattering becomes more incoherent inelastic scattering is almost always incoherent and relatively low angle that is less than 1 degree as the specimen gets thicker less electrons are forwarded forward forwarded forward scattered and the more are backwards scattered until the primary incoherent back scattering is detectable in bulk non-trans specimen specimens for this point you have to remember forward scattering causes most of the signals used in the TEM so the convenient definition of small angle is about 10 milli radians in TEM we can control the angle of incidence of electrons on the specimen and we will define the semi angle of incidence as alpha in the TEM we use apertures and detectors to collect the collect a certain fraction of scattered electrons and we will define any semi angle of collections as beta we will define all this scattering semi angles controlled by the specimen as theta and this may be a specific angle such as twice the Bragg angle where theta is equal to 2 theta B or a general scattering semi angle theta so again you can look at the schematic how the electron beam comes and this is an alpha whatever we have just stated in the previous slide you can just look at them as a schematic this is an alpha beam convergent semi angle and this is a specimen and then you have a general scattering angle theta and the collection semi angle is beta and this is your aperture and this is an optic axis so I can play it again so that takes care of all their definitions in a TEM and these are all the typical diffraction pattern one get in a TEM and you should know as a beginner what is the difference between all four of them what you are seeing is a diffused ring which typically comes from an amorphous material and this is a single crystal electron diffraction pattern and this is a polycrystalline single I mean polycrystalline electron diffraction pattern as a ring sharp rings and this is a convergent beam electron diffraction so we will explain I will explain all these things when we discuss the diffraction in a TEM and what is the reason you see this kind of a pattern that also will be discussed in detail and I am just trying to give you an introductory feel that what kind of diffraction you will be able to get from these transmission electron microscopy so you have these four typical types of electron diffraction is possible and then they are very powerful I mean information it gives you can derive little more significant microstructural aspects from this diffraction pattern so we will go through them when we come to that section and little more fundamentals again let us again a recap the atomic scattering factor f theta which is elastic f theta is a measure of amplitude of an electron wave scattered from an isolated atom is proportional to the scattered intensity f theta depends in lambda theta and z it decreases as theta increases and it decreases as lambda decreases and it increases with z for any value of theta so we have discussed this aspects while we discussing the x-ray diffraction so you have enough background for this so I will skip this which are inelastic process occur in the TEM process that generate x-rays process that generate other electrons something like secondary electrons processes that result from collective interaction with the many atoms there is a typo here atoms so these are all the general inelastic process in a TEM and now you recall this animation I will introduce an instrument through this animation what you are seeing is an electron source first where you have the high voltage applied and then you have an anode there is an aperture a condenser lens and then you have a specimen you have an objective lens and follows by an aperture intermediate lenses projector lenses and then final screen and what you have seen is how the electron beam comes through various apertures and lenses and falls on the specimen and it produces some signals second signals and then it further transmits through some of the electromagnetic lenses and apertures and it falls on the so you have the you are now familiar with this kind of an electromagnetic lens we have already seen the functions of this and how they are exploited here and also we have seen that you can look at the the corresponding light optical system where you have the condenser condenser lens and you have the specimen you have objective lens and you have a projector lens and then screen so you have one is to one comparison with the light optical system so both have almost similar I would say the ray diagram except that they are all electromagnetic lenses here it is a so the convergence angles alpha are so small that the ray diagrams are drawn with highly exaggerated angles and while the beam in the figure is not exactly the parallel to the optic axis alpha under this condition is less than 1 I mean less than 10 to 4 radians that is 0.0057 which is effectively a parallel beam and then we will look at the electron sources we will start with electron sources this also we have seen it in the introduction just for the sake of completion I will just go through this TEM will use a thermionic source or a field emission source and the two cannot be interchanged field emission source gives monochromatic electrons thermionic source or less monochromatic in nature and this is the typical I mean electron source or a gun design you have this electron gun this is a bayonet cylinder this is an actual photograph and this is an optic axis and you have the filament here and then you have the anode so corresponding anode is shown here and you have the gun crossover and applied voltage is there and we have looked at the function of this lens I mean the electron source and it is designed earlier also a high voltage is placed between the filament and the anode modified by the potential on the vernet cylinder which acts to focus the electrons into a crossover with a diameter d naught and the divergent divergence angle alpha naught and then if you look at the the thermionic sources for example it is a junction harping the tip of a junction harping filament and the distribution of electrons when the filament is under saturated and misaligned and then saturated aligned. So you have the this is an under saturated and misaligned beam will look like on the screen and and you have the under saturated aligned beam will look like this and this is the saturated beam. So we will look at this when we you will evidence this action while we operate the microscope and this is another thermionic source LAB 6 crystal and the electron distribution when the source is under saturated and aligned C is a saturated beam which will appear like this. This is a field emission gun tip electron pass from the field emission source showing a how a fine crossover is formed by two anodes acting as an electromagnetic lens and at one provides their extraction voltage to pull the electrons out of the tip and at two accelerates the electrons to 100 kV or more whichever is designed. So again we are looking at a second time we have discussed this. So we use apertures in the lens lenses to control the beam current and the convergence of the beam hitting this specimen. All lenses are imperfect in so far as they cannot gather all the radiation emitted by an object and so we can never create a perfect image. The image formed after each lens is rotated by 180 degree with respect to the object. We will see how this aspect is taken care in the modern microscope when we discuss the image and so on image formation in the TEM and a typical electromagnetic lens is shown in the schematic. You can see that these are all the copper coils which is this is a cross section of a electromagnetic lens which I have shown in the fundamentals of electron optical system as well. So you have the soft iron pole pieces and this is the bore and this is the gap and then you have the water inlet and outlet for the cooling and this is an optic axis electron optic system. The pole pieces surround the coils and when viewed in a cross section the bore and the gap between the pole pieces are visible. The magnetic field is weakest on the axis and increases in strength towards the side of the pole piece. So the electrons are more strongly deflected as they travel off axis. So you can see that why the schematic is shown in this manner is because of this effect. The bore to gap ratio is another important characteristic of such lenses controlling the focusing action of the lens. When we pass a current through the coil a magnetic field is created in the bore. This field is inhomogeneous along the length of the lens but axially symmetric. The strength of the field in a magnetic lens controls the ray path. So though we have we are going through this we have already seen the basic function of an electromagnetic lens just for the sake of completion and the recollection I am doing this. So we will continue to look at the some of the instrumental details and then we will go to the diffraction in TEM in much more detailed manner. So we will continue our lecture in the next class. Thank you.