 Hello everyone, I am Dr. Nilesh Prakash Gaurav, an assistant professor with the Department of Material Science and Engineering at Indian Institute of Technology, Kanpur. I will be covering a small module in the course, Advanced Characterization Techniques. So, the different things that we are going to cover in this module are as follows. First and foremost, we are going to have a brief introduction to x-rays, which I am sure you would have already had in preliminary courses on characterization of materials. Having done that, we are going to have a basic diffraction theory review in a couple of slides. After that, however, we will look at advanced techniques involving scattering and diffraction like small angle x-ray scattering and grazing incidence small angle x-ray scattering. After understanding these advanced techniques, we are then going to move to electron diffraction and study low energy electron diffraction as well as reflection high energy electron diffraction. After this, we are going to cover extended x-ray absorption fine structures and surface extended near-edge x-ray absorption fine structures. After covering x-ray and electron diffraction, we will go and have an understanding about neutron diffraction and essentially study small angle neutron scattering. I will like to bring to your notice that what all techniques that we have enumerated over here are pretty advanced techniques and are techniques which we do not use on a day-to-day basis in our labs. Having said that, the basic purpose of this course is to acquaint you with this state of the art scattering and diffraction techniques which you will be using for specific research related problems. So, let us first go back and have a look at the basics of x-ray diffraction. So all of us are aware that x-rays were discovered by W. E. Ronchin in 1895. X-rays as we are aware are electromagnetic radiation with wavelength of the order of 0.01 to 100 nanometer and frequency of 3 into 10 power 16 to 3 into 10 power 19 hertz. All of us are aware that x-ray are generated using x-ray tubes and which produce beam standing as well as characteristic x-ray peaks. We use optics to separate the beam standing x-rays from the characteristic x-rays and do conventional x-ray diffraction studies. Having said that another important tool I am going to talk about different sources of x-rays and how they are generated as well as the optics in the due course of this lecture. But before that we are going to study how exactly diffraction or for that matter how do x-rays interact with matter to give us sufficiently important information. So, let us first try to understand the interaction of radiation with matter. We know that whenever a radiation I mean an electromagnetic radiation interacts with a matter it can lead to absorption as well as emission of photons. At the same time it can lead to what is known as scattering. Scattering can be classified as Rayleigh scattering where the energy is conserved. Resonance scattering which occurs for liquids as well as remaining scattering. Part of these scattering events which I have mentioned here will be covered in the course of this module as most of the characterization techniques to be introduced in the later part of this course are dependent on various scattering events. We are also aware that scattering can lead to reflection, refraction and diffraction. The most important of it is diffraction which is essentially defined as superposition of scattered waves such that the diffraction pattern inherits the periodicity of the entity. Having said that I am going to spend a couple of slides later focusing on what exactly is diffraction. But before we do that let us try to understand how do x-ray interact with matter. We know that the propagation of x-rays through matter is characterized by below mentioned three basic processes namely coherent scattering wherein the x-ray photons do not change their energy while there is also what is known as incoherent scattering in which the photons deflect from their initial direction and part of the photon energy is transferred to the recoil electron and is classically known as the Compton effect due to elastic scattering. Another important phenomena which occurs with x-rays interaction with matter is x-ray absorption where the transmitted intensity of the x-ray radiation is less than the incident intensity. Having said that our major interest lies in diffraction. Diffraction I would like to mention refers to spreading and interference of waves passing by an object or aperture that disrupts the wave. It is pertinent for sound water as well as electromagnetic waves. If you want to study diffraction the best way is to go out any reservoir of water where the water is stable and try to throw two pebbles and see how the two waves are formed and how do they interact with each other. Another important point that is to be mentioned regarding diffraction is that the diffraction effect is most valid if the wavelength is of the order of the size of the diffracting objects. These diffracting objects may be apertures or may be other objects which lead essentially to spreading or for that matter scattering of waves. It is to be mentioned that a diffracted beam may be defined as a beam composed of a large number of scattered rays which are reinforcing one another. The most important point to be noted in the above discussion is that the scattered rays which mutually reinforce one another that means their amplitudes add up leads to diffraction. So before we go ahead with diffraction let us spend some time on understanding the wave particle duality of electromagnetic radiation. We all know that light or any electromagnetic radiation can be considered as a particle with energy h nu which is a photon if you talk about quantum mechanics while in terms of classical physics we can always consider it as a wave and define it by a wave vector which is a complex vector which can be given by A e i power phi where which can be become which has a particular amplitude which is given by the product of the complex equation of the wave and its complex conjugate and is directly proportional to A square. We also know that electromagnetic radiation including light rays as well as x rays comprise of electric field and magnetic field which are mutually orthogonal to each other and that the direction of travel of light is normal to the electric field as well as the magnetic field. We will be using these concepts in understanding how does diffraction occur in the course of this module. So coming back to diffraction as I had already mentioned diffraction is bending of radiation from an object or spreading of radiation from an aperture it is also defined as directional scattering and is very important for revealing the geometry of the entity involved because the diffraction pattern essentially inherits the geometry of the entity that leads to diffraction. Having said that there is another important interesting phenomena which we have studied during the course of our high school physics which is interference from apertures and there is a very nice similarity between diffraction and interference and therefore in this module we are going to understand diffraction through interference and I would like to quote a very nice saying by another than Finman himself who said that no one has ever been able to define the difference between interference and diffraction satisfactorily. It is just a question of usage and there is no specific important physical difference between them. Therefore, we are going to follow an approach wherein we try to understand diffraction through interference. But before we do that let me just throw in a bit of history pertaining to x-ray diffraction as I had already mentioned it was in 1896 that x-rays were discovered by Ronshin. However, it was not until 1912 when von Lowe, Friedrich and Nipping passed x-rays through a crystal of zinc sulfide they obtained a periodic pattern that made them conclude that crystals are composed of periodic arrays of atoms and crystals cause distinct x-ray diffraction pattern due to atoms. Bragg and Lorenz determined relative positions of atoms within a single crystal in 1914 using x-ray diffraction for this the Bragg, Fathers and Duyo also won a Nobel Prize. It was much later when the diffraction was used to decipher the structure of DNA and the well known very famous double helix structure of DNA was deciphered first firstly using x-ray diffraction. Thus we see that x-ray diffraction has come a long way over the last century and the development of state of the art characterization techniques involving x-rays has led to fundamental path breaking discoveries over the last century. So before we go ahead and talk about diffraction let me just talk about interaction of light of waves for that matter any electromagnetic wave with matter or for that matter with an aperture so for this we are going to look at interaction of light waves in the visible spectrum with an aperture. So we know that whenever a light wave interacts with an aperture each and every point in the aperture acts as a source of spherical wave front according to Huygens wave principle. This is what has been shown in the first part wherein we can see how a spherical wave front is from the aperture however as we move away far from this the distance increases from the aperture we see that the wave front becomes parabola. This is the regime of what is known as Fresnel diffraction the condition for which is given right over there wherein the f is equal to a square by L lambda that is greater than or equal to 1 where a is the size of the aperture and L is the distance between the aperture and the image plane. Having said that what we are going to mostly deal with is the Fraunhofer diffraction in which the wave front which started as spherical then became parabolic at very large L it essentially becomes planar and this is what we are going to deal with mostly during the course of x-ray diffraction. Having said that as I had already mentioned we are going to try and understand diffraction in terms of interference. So whenever we have a planar wave front of light interacting with an aperture we see that instead of getting one high intensity we do know we get a spread of intensity as shown in the figure below. Just to give you an idea of how we can understand the mechanics of wave I have just noted down the wave equation and integrated it to show again that the intensity is directed proportional to square of the psi and that the wave function itself that you get on an interaction with the aperture is a function of sin of pi A sin theta by lambda over pi A sin theta by lambda which is nothing but a sin c function of pi A sin theta by lambda. Let us kind of neglect the mathematics for the time being and see what is the physical significance of it and we all know that whenever optical or visible light passes through a small aperture we do get a pattern like this wherein there is one single maxima at the same time there are some secondary maxima and the condition for maxima as well as minima is given below. So we can see that whenever sin theta by lambda is equal to n over A we do get a minima and there is no intensity. However whenever sin theta by lambda is 2n plus 1 over 2A where A is the length of the aperture we do get a maxima. Now this is for only one aperture what if we are having two apertures which is the classical Young's double slit experiment. This experiment and the results of it we have covered in our high school days and we all know that Young's double slit experiment leads to an interference pattern which can be explained on the basis of constructive interference and destructive interference. It is explained that whenever two waves meet in phase their amplitudes get added up and leads to constructive interference. However if the two waves meet out of phase then their amplitudes get cancelled and if they are having the same amplitude you get complete or total destructive interference and there is no intensity. The condition for constructive interference is that d sin theta is equal to m lambda which essentially means that the path difference between the two waves is an integral multiple of lambda. Therefore the two waves are meeting in phase and their amplitudes get added up. However if the phase difference is not an integral multiple of lambda and in fact if it is m plus half lambda the two phases the two waves rather are completely out of phase and therefore their amplitudes cancel each other and we get total destructive interference. Well having understood the Young's double slit experiment completely double slit experiment I like to show you how the Young's double slit experiment can be extended to 2D. So in Young's double slit experiment we had two slits which I just do not know it just does not work out. Just to give an understanding for interference and diffraction I have shown how we can get a interference pattern from two slits as observed in the figure above. At however if we are having an array of slits we do get a pattern comprising of dark spots which are shown here in yellow rather grey and bright spots which are shown here in terms of yellow colour. Similar thing we get when there is an interaction of electrons with matter and it leads to what is classically known as electron diffraction pattern which is quite routinely seen in a transmission electron microscope and this is because of diffraction occurring not from occurring from atoms. Herein we can clearly see the similarity between diffraction of electrons from atoms and interference of visible light with apertures. We all know from the basic course on characterization that we have undertaken that diffraction follows what is known as Bragg's law. In order to understand what exactly I will just repeat what is Bragg's law and we know that whenever there are two waves which are coming over a crystalline material the wave say 1 and 2 the part difference between these two waves is essentially d sin theta plus d sin theta which is nothing but ab plus vc. We know that if the overall part difference between waves 1 and 2 is an integral multiple of lambda that is if n lambda is equal to 2 d sin theta the two waves 1 and 2 are going to meet in phase and their amplitudes are going to get added and hence we will say that diffraction has occurred. So this is the generalized definition of Bragg's law where n is known as the order of diffraction lambda is the wavelength d is the interplanar spacing and theta is essentially the angle of incidence. Having said that I like to mention that many a times diffraction and reflection are used intermittently though these two are two different phenomena. Now let us try to understand the geometry of Bragg's law before we proceed the most important thing that is very essential for Bragg's condition to be followed is that the incident beam and the normal of the reflection plane as well as the diffracted beam how to be coplanar the angle between the diffracted beam and the transmitted beam is always 2 theta and this is what we measure. If you recollect a normal X-ray diffraction pattern essentially comprises of i versus 2 theta and another important thing regarding the choice of radiation for the kind of study that you are doing is determined by the wavelength wherein you know that sin theta cannot be more than unity and therefore it requires that lambda should be less than twice the d spacing we want to study. There are other conditions which are laid down on the selection of wavelength but will cover them as in the due course of the module. Another important thing which we would like to locate in Bragg's law is that is what is known as the order of reflection. So if you look at the Bragg's law which is nothing but n lambda over 2 d sin theta and rearrange the terms we do get lambda is equal to 2 sin theta into d over n and therefore we can say that a reflection of any order as a first order reflection from planes real or fictitious spaced at a distance of 1 by n of the previous spacing. Therefore an n order reflection can always be replaced with a first order reflection from real or fictitious planes spaced at a distance of 1 by n of the previous spacing. Therefore if we set d prime as d over n we do get lambda is equal to 2 d sin theta put in other words it means that an nth order reflection from hkl planes of spacing d may be considered as a first order reflection from the nth nk nl plane of spacing d prime wherein the spacing d prime is equal to d over n. The reason I am visiting Bragg's law again and again because is to emphasize the importance of what is going to be what is essentially known as reciprocal space. So again here I rearrange the Bragg's law and get d on one side and then I essentially separate out sin theta on the other side wherein I can write sin theta is equal to 1 over d or 1 over d prime divided by 2 over lambda. What is the basic advantage of writing the Bragg's law in this way will be enumerated in the next couple of slides. The easiest way of approximating it is to look at the value of n. So if we look at 1 0 0 set of planes we get the first order reflection wherein lambda is equal to 2 d sin theta. However if we look at the second order reflection we go back and we see that the plane is supposed to lie right in between the two planes which we have shown already similarly we can imagine that the 3 0 0 or the third order reflection is going to lie at one third the distance between the two planes. Now if you go back to the last slide we see that if you go in between two set of planes we do see that there are no atoms. So there is going to be no interaction between the x-rays and the atom which is absolutely essential for diffraction to occur. So how do we understand diffraction if at all there are no atoms in this plane that is where the concept of reciprocal lattice comes for our help. If we look at the right the bottom right equation sin theta is equal to 1 over d by 2 over lambda we can hope that diffraction occurs when this 2 over lambda which is the wavelength of choice that we have chosen corresponds to any one of the point lying on 1 over d. We know that as I have mentioned or rather you must be aware that all the diffraction or interference that we studied can be essentially described in terms of Fourier analysis of electron or electron concentration in the diffraction diffracted pattern. Every crystal therefore has a crystal lattice at the same time a reciprocal lattice associated with it. A diffraction pattern of a crystal is nothing but the map of the reciprocal lattice of the crystal. In order to decipher what exactly is a reciprocal lattice let me first say let me first tell you that in a real space we do have crystals which has a particular crystal structure. Now the crystal structure essentially comprise of a lattice or a crystal lattice and a unit cell content while the diffraction pattern that we get using x-rays or for that matter electrons is essentially comprising of the reciprocal lattice and the structure factor which determine the intensity of diffraction. In order to decipher how do we construct reciprocal lattice let me not go into all the maps of it but just start with a simple 2D presentation wherein I have shown you a plain lattice which can be seen over here and you can see that this lattice which I have over here is characterized by basis vectors x and y. Now if I were to draw a normal to these 2 vectors which I have shown over here as y prime as well as x prime where y prime is parallel to y and x prime is parallel to x and plot 1 over the distance. So you can see over here that along y the distance between the 2 atoms is very large while then I go to reciprocal space I not only go 90 degrees to y but reduce the distance and that is where you see that my y prime over here is smaller than x prime while in real space my x is smaller than y. So not only have we gone and change the directions but also we have changed the length the magnitude of the directions. We can define y prime as being normal to y but the distance between any 2 reciprocal points along y prime is 1 over the distance along y in the real space. Similarly we can define x and x prime the whole concept that I am trying to convey is that if you have a real lattice which is shown over here it leads to a reciprocal lattice shown over here and there is a mapping function or we can go from the real lattice we can go to the reciprocal lattice which we see using x ray diffraction. So if we can understand how we go from real space to reciprocal space we are going to measure actually the reciprocal space by diffraction. So we can always come back and get some idea about the kind of real lattice that is existing. So just a few words about reciprocal lattice if you go from 2D to 3D as I had already mentioned that y prime here was normal to y. Similarly if I start with say a cubic or orthogonal lattice I can start with ABC as the lattice parameters I can always get A star in the reciprocal space which is nothing but B cross C divided by A dot B cross C where A dot B cross C essentially gives me the volume of the unit cell and B cross C gives me a vector which is normal to the 2 vectors B and C in case of cubic lattice my A star is parallel to A but it is not a necessary condition and for other lattices this condition may not be true. So this is true only for cubic however A star may not be parallel if the lattice is non-cubic however what are A rather AB and C if you are looking at FCC, BCC and HCP structure I would like to mention that the ABC that we have chosen here are not the general lattice parameters that we use like A equal to B equal to C and alpha equal to beta equal to gamma equal to 90 for any cubic lattices instead it is the primitive lattice or the basis vectors that we have to choose therefore for FCC the A, B and C that we choose are essentially the 3 different types of 101 where in A is 101, B is 0, 1, 1 and C will be 110. So these are the 3 basis vectors that we have to choose another interesting thing which you must be aware is that the reciprocal lattice of FCC is BCC while that of BCC is FCC. For HCP lattices we do have another HCP lattice but the C by A ratio is changed one when we go from real space to reciprocal space for simple cubic the real lattice and the reciprocal lattice are the same. Now in order to understand diffraction as I had mentioned in the earlier slide that whenever it satisfies this particular condition of sin theta equal to 1 by d over to over lambda to understand the geometry we can see how the actually diffraction occurs. So I can always draw a radius or rather a sphere with radius 2 over lambda which is shown over here which is the limiting sphere and the eval sphere is essentially the small sphere that have shown over here with radius 1 over lambda and diffraction as already mentioned occurs at particular point when the sphere or the circle as shown in this figure and any amount of the point the reciprocal lattice points which are shown over here these are my reciprocal lattice points interact. So therefore you see that for this particular point where in this point as well as say something like this point is lying on the eval sphere diffraction occurs. Having said that you can now imagine that by varying either the reciprocal lattice or the eval sphere we can get diffraction in a material or in the crystal of our interest. This is what has been shown in this slide and here in I have shown a 3D representation instead of a normal 2D representation and here we can see how exactly the eval sphere looks. So this is my eval sphere and this is how this is my reciprocal lattice. So different x-ray diffraction techniques use different ways of either changing the eval sphere or the reciprocal lattice in terms of crystal rotation so that they come into coincidence and we can get diffraction. So this is how I have shown Bragg's law in the real space on the left hand side wherein you can clearly see the S0 being the wave which is incident on the crystal over here on the crystal which gets diffracted in the real space while in the reciprocal space the same thing instead of having talking about lens we talk about reciprocal lens and therefore we have this 1 over lambda term coming over here and here we see how the incident wave is nothing but S0 by lambda while the diffracted one is S over lambda and this leads to a path difference which is nothing but which is equal to the interplanar spacing which is shown over here and this leads to the diffraction vector g and therefore it can be defined as g is equal to d prime hkl which is given by S-S0 over lambda. So this is the necessary condition for diffraction to occur in terms of in three dimensions as I had already mentioned the entire exercise now evolves about how to orient the crystals so that at least one of the point lies one or more point lies on the eval sphere so that diffraction occurs. So here in you can see that the 1 0 0 point lies and therefore you do not get diffraction however once you tilt the sample you do have 2 0 0 point of the reciprocal space which is lying on the eval sphere and we do get diffraction similarly the crystal can be reoriented so that we do get diffraction from the 4 0 0 reflection. So here in I hope you can appreciate that by changing the orientation of the crystal we can change the orientation of the reciprocal lattice that is associated with it and get diffraction. I have shown a simple normal theta to theta geometry which is routinely used in our x-ray labs for analyzing polycrystalline samples which comprise of multiple grains but I will cover the basics of rather the instrumentation part of it at a later stage. So before we go into the instrumentation part I will just like to go back and talk about you know we talked about a lot about how do waves interact and all those things but that is in a very layman's language actually diffraction is also the way it builds up is essentially with interaction of the wave with the electron or essentially the scattering of the x-rays from an electron by what is known as Thomson effect what is known as the Thomson effect which essentially tells us how the intensity of the x-rays vary when it is when it interacts with an electron at an angle theta. So we can correlate that the intensity of the x-rays on interaction with electrons changes as a function of theta. If you talk about an atom we know that an atom is an ensemble of electrons and it is directly and the scattering by an atom is directly proportional to the atomic number. However this is true only when there is direct incidence and as the incidence angle changes there is a decrease in the scattering atomic scattering factor of x-rays. However we know that it can be expressed in terms of a wave vector which may be expressed analytically as a e power i phi which is equal to a cos phi plus a i sin phi and phi is nothing but the amplitude sorry the phi is nothing but the phase difference and can be defined as 2 pi into h u plus k v plus l w. For any atom a e i power phi is nothing but f of e 2 pi i h u plus k v plus l w but this is true as you must be aware as soon as I talk about these locations h u k v and l w this is true for an atom not single atom which is not isolated but is in a unit cell. Therefore if you look at the scattering by unit cell which comprises of more than one atom as in a FCC unit cell or a BCC unit cell the simplest example that I had mentioned earlier was the simple cubic case wherein there is only one atom per unit cell. However when we go to BCC or FCC unit cells with two atoms and four atoms per unit cell the scattering by unit cell is given by summation of all the atoms in the unit cell and can be given as f of h k l is equal to summation over summation 1 over n small f e raise to 2 pi i h u plus k v plus l w and f the capital F the amplitude is nothing but the amplitude of waves scattered by all the atoms in the unit cell divided by amplitude of waves scattered by one electron. And therefore we come to the most important part of diffraction as in what all are the different reflectors that are possible with x-ray diffraction for different crystal structures. So if we look at simple cubic crystal structure we see that all the reflections are possible so that h square plus k square plus l square is equal to 1, 2, 3 up to 6 and then 8, 9, 10 we do not get 7 because knows sum of 3 squares never add up to 7. However when we come to BCC we have two atoms per unit cell which are located at 0, 0, 0 and 0.5, 0.5, 0.5 right at the center of the cube therefore we do get a condition that diffraction occurs only when h square plus k square plus l square is even that is 2, 4, 6, 8, 10 or 12 and therefore put in other words it essentially means that diffraction occurs only for peaks where in h plus k plus l is even and we do get diffraction for 1, 0, 1, 2, 0, 0 and 2, 1, 1 peaks. However when we go to FCC which has got 4 atoms per unit cell at 0, 0, 0, 0, 0.5, 0.5, 0.5, 0.5 we do get diffraction only if h k l are all odd or all even. And therefore we get diffraction at peaks of 1, 1, 1, 2, 0, 0, 2, 2, 0 and 1, 1, 3. Therefore the condition for diffraction can be summarized as h square plus k square plus l square is 3, 4, 8, 11 and so on. I think I am done with it.