11. More on electron-specimen interaction

Added: 2026-05-07

[00:00:16]In the previous lectures, we've seen what happens when an electron beam interacts with a specimen. the concepts of scattering, the classifications of scattering and various signals that are generated. Now we look at how you can quantify these and how can we build towards contrast mechanisms and how do we understand the various forms of data that are generated in a transmission electron microscope.

[00:00:44]In short, you can get either images or defraction patterns in a conventional transmission electron microscope, which is a very basic microscope without any specific special detectors. Most modern day microscopes have a stem detector, a stem bright field or a dark field detector and so on. But to start with, if you look at the old school microscopes, you could simply perform just imaging and defraction. In the initial part of the uh course following this lecture, we'll look at how the contrast mechanisms for conventional bright field and dark field imaging are.

[00:01:23]What are the different mechanisms in which the images are formed? And we'll look at a few examples. Following this we'll look at some of the demonstration sessions on how to acquire this data on a TEN. But now let us just quickly recollect what are the different concepts of scattering that we've seen.

[00:01:42]If you look at scattering in principle for any electromagnetic radiation uh owing to the dual nature of electromagnetic radiation you can quantify it either as behaving as a particle or as a wave. Now in reality you cannot separate them. Inherently you would have both the wave nature and the particle nature or the dual nature together. But for our simplicity and understanding and to quantify the mechanisms we look at the electromagnetic wave either as a particle or as a wave. If you look at the initial ideas from Rutherford's back scattering experiment because the concepts were all developed at a time period when the ideas of quantum mechanics were not uh really existent or in the very formative stages. most of the fundamental known classical physics principles were applied and the interaction of electromagnetic radiation with specimens were treated as problems of collision. So going by this if you look at the particle scattering concepts most of the interactions initially are looked upon as particle particle collisions they are governed by cross-sections and mean free path and let me refresh your mind on some of the concepts covered earlier the expression we had was something based on the angle of scattering which is again uh having its foundations in the rutherford back scattering experiment. And uh you can in principle relate that the probability of scattering or the scattering cross-section is in fact not a real uh physical quantity like an area on which you have a successful collision.

[00:03:40]It is somewhat like a probability function which gives you that when a successful scattering event has happened there is a deviation in the path of the electron beam and this can be estimated based on the angles of scatter with some expressions and uh this strongly depends on the atomic number also we had something on the coherency of the electron waves. We introduced a few concepts. In fact, if you remember, we touched this concept called coherency when we saw the lectures on electron sources. But the coherency of electron waves is also again related to the scattering angle. Initially the Rutherford experiment was considered as giving you an expression for a successful scattering event and most of the collisions were considered elastic and largely within a transmission electron microscope which is operated at 200 or 300 kileron volt. This angle of scattering is typically lower than 10 to 20 m radians or at the most 2 to 3°.

[00:04:51]So if you apply to real specimens in practice the forward scattered electrons are elastically scattered through angles which are relatively small.

[00:05:03]It is considered that the inelastics scattering phenomenon which is also there approximately 5 to 7% depending on the accelerating voltage you are at say at 300 KV approximately you have 5% of in elastic scatter and around uh 200 KV around 7% of inelastic scatter these are not absolute numbers but approximate ranges and they are deviated at slightly larger angles so the intensity of the scattered electrons is strongly influenced by the arrangement of atoms in the specimen as well. And this brings to a concept called as defraction. When you look at how electrons are scattered through a path, specifically when they interact with crystalline samples, like the way you have the Young's double slit experiment, when you have a couple of slits and two point sources are kind of forming an interference pattern at a screen collected at a distance, you see fringes of bright and dark uh variations in the intensity. And you can relate this the fringe spacing to the spacing between the slits and the path difference is related. And based on these sort of concepts, you might have seen in some of your earlier courses on X-ray crystalallography that the Braggs law is derived where you're assuming the planes of atoms within a crystalline sample act as centers where the X-rays that are incident are reflected at the same angle. In reality, this picture is extremely different in a transmission electron microscope. So the angle of scatter also strongly is influenced by the scattering phenomenon. So if I look at this defraction can be considered as a special case of scattering. In fact when I talk about defraction of light you consider that light has undergone defraction when it bends from edges of a sample. Whereas in defraction of electrons or in X-ray defraction you consider defraction as a case when the Braggs law is satisfied and you have a constructive interference. Likewise even in electron defraction when you consider planes of atoms or columns of atoms as scattering centers on undergoing interaction with the electron beam they are deviated at specific paths. And when you look at all of these collective planes or columns of atoms as scattering centers and you look at the constructive interference of them, the pattern that you obtain is called as defraction pattern. So the collective scattering of atoms produces defraction and this is explained by the wave nature of electrons. Once again bear in mind you cannot really separate the wave nature and the particle nature but the signature of the signal that you get the way you get a interference pattern in a Youngs double set experiment you get a constructive interference from the atomic planes and the columns and you produce a defraction pattern. So defraction is mainly controlled by the angle of incidence of the electron beam relative to the atomic planes. So let me give you an example and draw the general picture we have in mind when we think of uh defraction. So if you remember to understand X-ray defraction from planes, let us say there are two planes of atoms separated by a distance d and you had x-rays incident like this and at an angle theta. You see that they're also scattered at an angle theta and you derive uh an expression for path difference versus the angle of scatter. And then you say that the additional path that is traveled here which is this little number here is twice d * when this is theta and this is d it is 2D sin theta and if you equate it to an integral multiple of wavelength you get the expression for the bra defraction condition. Now what is important here in electron defraction is the angles of scatter are extremely small. That is because the wavelength of x-rays is typically around 1.54 angstroms and let us say my unit cell size is of the order of 10 angstroms in the maximum case. It ranges from 2 and a half angstros to 10 anstroms for most inorganic samples. This angle theta you see here is of the order of a few degrees. It can range from 10 to let us say 100° for crystalline samples when you're using X-ray as a source. But when this lambda goes to the order of 0.038 038 angstroms for a 100 KV electron beam the theta also becomes extremely small and it is typically less than or equal to 1°.

[00:10:21]So in practice you should treat the case of electron defraction as a case where when I have plane separated by a distance d the electron beam that is incident at a very small angle is also scattered at a very small angle and therefore the elastic scattering angles are extremely small as described a short while ago. So this again brings back to the point that defraction is mainly controlled by the angle of incidence and because the wavelengths are also so small the bra diffraction is a special case where more or less the electron beams incident parallel to atomic planes also satisfy the bragg.

[00:11:07]So the separation of the atomic planes also governs how far the uh electron beam is defracted and the interatomic spacing between the planes. So from this what you can see is the defraction is a kind of a special case of scattering or interaction of the electron beam with the specimen which you can treat as an angular distribution of intensity and image as we've seen some of the earlier lectures can be considered as a spatial distribution of intensity.

[00:11:49]So with all this background let us see what happens when an electron beam is incident on a specimen. Here is a sketch that I've taken from uh the Williams and Carter textbook. If I think that I have a near planar incidence on a thin specimen in a TM once the interaction happens I collect the forward scattered electrons and you can say that the image is sort of a distribution of intensity whereas the defraction pattern is actually an angular distribution of intensity. We'll look at this in a little more detail slightly later.

[00:12:32]But inside a TEM when a specimen is illuminated with a B beam of electrons you typically assume that you have a near planar incidence and you have coherent electrons interacting with the specimen. In practice in reality you really do not have coherent electrons.

[00:12:50]Uh there is also some incoherency and you need typically a monochromator or a chromatic aberration corrector to have a real coherent source of electrons.

[00:13:00]These beams are defined to uh or the these beams are restricted to well- definfined paths within the electron microscope and based on the principles of ruford back scattering and also a few concepts that you'll see when you look at the image formation in high resolution uh imaging you can define the trajectory you can also uh model the intensity variations and it is all reasonably well understood. There are a few important terms. If you look at the electron beam that is incident on the specimen, you call it as the incident beam.

[00:13:38]Scattered beam is also called as the electron beam which is scattered by the uh the specimen. And these could be elastic or inelastically scattered electrons. And the scattered beam can be the forward scattered, the defracted or the elastically scattered beam. or it can also be the direct beam or the undeviated beam which is along the principal axis of the microscope.

[00:14:04]So in short, this was also covered in the previous slide. Spatial distribution of the scattered beam is an image and an angular distribution of the scattered beam is a diffraction pattern. And bear in mind the angles of scatter are so small that when you are satisfying the bra defraction as well, you have a parallel beam of electrons propagating through a specimen. And when a parallel beam of electrons goes through a lens, naturally they would all converge at the back focal plane of the objective lens.

[00:14:41]And also if you remember the function of a lens, we know that when a specimen is placed very close to the focal plane of an objective lens, the information that is propagated through the lens is first a FIA transform which is at the back focal plane and the image that is formed at the image plane is again an inverse FIA transform of the information you have at the back focal plane. So this brings me to the question, how do we switch between images and defraction patterns in a transmission electron microscope?

[00:15:20]This is an important question which would give you an idea of how the microscope is constructed and why a few apertures are placed at specific locations along the axis of the microscope. So let us look at the image of a microscope and also the ray diagram. Let me bring back to the question that we started this discussion with. How do you switch between an image and a diffraction pattern? So in a transmission electron microscope, it is known and established in the past few lectures that the specimen is brought more or less into the same plane as the objective lens and it is brought very close to the position of the lens and within the uh focal length more or less at the same as the focal plane of the objective lens. So if I say the specimen is having a potential of XY here at the back focal plane I have the FIA transform of sin not XY from the specimen and at the back focal plane I have the FIA transform of sin not XY and at the image plane I have the inverse this fia transform of.

[00:16:48]So switching between an emerging plane and the back focal plane is simply by pressing a small switch called as the defraction uh button where you're making the back focal plane as the object plane for the projection lenses in a TEM. when I want to record a diffraction pattern.

[00:17:10]But when I want to record an image, I simply look at the image plane as the object plane for the projection lenses and information is recorded on the screen.

[00:17:20]Now an important concept that has been introduced is the idea of uh scattering and the idea of defraction. And if I want to understand more on this on how the intensity or the amplitude of the scattered electron beam changes, we look at something called as atomic scattering factor. This is a very important concept for the foundation of uh not just imaging but also defraction in electron microscopes. So let us say I have an isolated atom and if I have an incident beam and if I represent this as say uh s is sin 2 pi * k dot r this incident beam now under goes scattering and let us say if s i is the incident beam with s not sin 2 pi k dot r when it under goes scattering if I say s is the scattered beam I have s not sin 2 pi if I write this k dot r as some zed in the z direction of the electron beam I can introduce a phase shift of say a small angle five. And if I do the expansion of sin of a + b, I can write it as s not time cos 5 plus cos 2 pi z sin Okay. Now we've established that in a transmission electron microscope the angles of scatter are typically very small. It is almost 0 to 1° and for such small angles sin phi is nearly five and uh cos 5 becomes 1.

[00:19:56]So what you then have is the total intensity of the scattered beam is c is sin sin 2 pi kz plus 5 sin 2 pi kz plus<unk> by 2. So what I've done is the cosine here I have written as sine of<unk> by2 + the same value that I have here. So this expression has a form s total or the total forward scattered intensity is some s + i scattered.

[00:20:56]So in principle I can write the scattered intensity.

[00:21:02]So this is the expression or this is a kind of a schematic that we saw a little while earlier towards the end of my previous lecture that if you have an isolated atom that also causes a phase shift and the same phase shift has been introduced here and the final expression you get for a scattered wave is size scattered is sin not e power 2 pi k1 r plus there is some value F theta e power 2 pi kr / r. This value of f theta what you have is called as the atomic scattering factor or the change in the amplitude caused to the forward scattered electron beam as a consequence of interaction with an an isolated atom.

[00:21:56]And if this is resulting in a deviation in the angle of scattering by theta or phi, it causes a phase shift. And you can always relate this amplitude change to the angle of scatter through this sort of an expression.

[00:22:12]Now in practice in reality you do not have an isolated atom in most of the specimens. You would have atoms arranged in a very periodic fashion in a crystalline sample. So let us see how does this evolve and change when I expand this idea to a real crystal.

[00:22:34]So this concept of expanding the amplitude that I get from a crystalline sample as a consequence of scattering from atoms which are arranged in a periodic manner is called as structure factor. And uh let me just refresh or recollect the expression that we had earlier for the intensity of a scattered uh wave. Uh remember I have actually arrived at this expression by taking the concept of superposition of waves. That means if I have two waves, let us say uh this is incident and it is propagating forward and it is interacting with this wave which is deviated through an angle say theta or phi here. The total scattered wave can be written as sum of two complex numbers which is s incident plus scattered. And from here we've done a small simplification where we made the angle of scatter extremely small and we arrived at this expression. So if I now extend it to a crystalline lattice and let us say I have a periodic arrangement of atoms like this.

[00:23:51]For now for simplicity let us look at the propagation of electron beam horizontal not vertical. And let us say a beam is incident at an angle like this and it is scattered through a small angle like this.

[00:24:10]What happens is you would have consequently several scattering uh phenomenon possible which is also called as plural scattering or multiple scattering which is a case of uh dynamic defraction or you could have a single scattering event and when I have a broad beam of electrons incident on a specimen the electron beams are all incident and interact interacting with these atoms that are arranged in a periodic fashion and the output that I get the scattered wave is a resultant of all these uh waves that are scattered due to these scattering centers which are atoms arranged in a periodic fashion. And I actually can relate this with some kind of a simple superposition again. And uh if I now look at this as a simple vector summation of the incident beams which are deviated through small angles. And let us say if an electron beam which is incident under goes a phase change through a small angle 51 as a result of one scattering event.

[00:25:30]Overall if I say the resultant scattered wave and the amplitude of a resultant scattered wave is represented by an expression as capital F hkl and if small f_s f_sub_1 f_sub_2 are consequences or deviation in the path of electron beam due to numerous successive scattering events or due to numerous scattering instances happening within the crystalline lattice. I can look at this resultant total phase shift as an expression of this form where capital FHKL is simply a summation of all of these atomic scattering factors with a periodic arrangement of atoms within a crystalline lattice. Now if you've done any courses on crystalallography you know that you can represent the crystal structure as a motive plus a latis and the motive is represented with the coordinates x y and z and you have an occupancy of atoms within a specific crystal structure. So what this expression tells you is as a consequence of periodic arrangement of atoms, the resultant amplitude of the scattered electron beam is a function of the arrangement of atoms within a crystalline lice. And this is actually called as the uh structure factor. And within crystalline samples, the structure factor strongly depends on multiple factors. One is the atomic number whether it is a single uh element or a crystal structure with only one element or whether you have multiple elements acting as scattering centers and you have a resultant intensity.

[00:27:30]There is something also called as a primitive crystal structure or an ordered crystal structure and there are many many variations which give you a change in the intensity distributions.

[00:27:42]So when small fn is the atomic scattering uh factor and if the atoms are arranged in periodic fashion with coordinates at positions x y and z you can write this expression as fhkl is an integral of f theta * e R 2 pi I K dot R where R is a position vector which includes the positions of the atoms which can be given by the Miller's indices of the planes within a crystalline sample. So if I have a different uh crystal structures, the resultant amplitude from a cell can be given by an expression of this form. I've given you this in an integral form earlier but you can write the resultant amplitude from a cell as an expression like this where the position vector gives you the positions of the uh atoms that are occupied within a crystal structure and this is my reciprocal vector and you can see that I can easily relate from the real space positions of the atoms to the uh reciprocal uh latice or the image that I record or the defraction pattern that I record which is again information in the reciprocal space. And for a crystalline solid you have the structure factor giving you constructive interference when this expression becomes unity or an integral number. And based on this for different crystal structures and for different crystal systems you can actually get how the intensity changes as a function of the atomic scattering factor with the number of lattice points and this gives you varying diffraction patterns for different crystal structures.

[00:30:12]We will look at this in much more detail when we look at the demonstration sessions on how to obtain defraction patterns and how to analyze defraction pattern. But the whole objective here is to give you a flavor that defraction is one kind of a special case of scattering. And scattering can be viewed as a phenomenon where electron beam interacts with the specimen and there's a change in the angle of scattering and you can have elastic scattering, you can have inastic scattering, you can have single scattering event, you can have plural scattering events and uh you can also have defraction which is a special case of scattering and the intensity variation that happens as a consequence sequence of scattering can be quantified in terms of the rutherford scattering expression and when I extend it to real specimens. You can see that as the atoms are arranged in a periodic fashion within crystalline samples you also end up getting variations in the amplitudes.

[00:31:20]Now if I look at this variations in the amplitude or the atomic scattering factor as a consequence of the angle of scattering within amorphous and crystalline materials for an amorphous sample you would see that the scattered amplitude as a function of the angle of scatter would be rather diffuse of this form. Whereas for a crystalline material at some specific periodic locations you would get variations in the intensities and the height of these peaks or the value of this intensity would actually be related to the way the atoms are arranged within a crystalline sample and that is what we saw in the previous slide.

[00:32:11]Also as you can see if I go very far away and for larger angles of scattering the intensity also dies down and this is what is reflected in an electron defraction pattern.

[00:32:26]So with this I hope you got some clarity that as a consequence of scattering and the interaction there are variations in the amplitude and this can be strongly related with the kind of information you get from a sample.