 Welcome to this material characterization course. In the last class, we started discussing the fundamentals of X-ray diffraction and then we just emphasize the basic physics of X-rays and how it is generated and so on. And in this class, we will continue to look at the properties of X-rays and since we have some basic understanding of this X-rays as an electromagnetic radiation which we have discussed in the fundamentals of the optical as well as scanning electron microscopy, the electromagnetic characteristic characteristics also will exactly fit with this X-rays but since we are going to talk about only the X-ray diffraction, we will recollect some of the concepts and the basic physics behind this and we will also look at the properties and then move on to the concept of diffraction in much more detail. So today what I am going to do is just X-ray waves. So we are looking at the X-rays as a wave. So we will look at some of the fundamental aspects or parameters which describes the X-rays as a wave and what are the things we have to look at. This is what I just introduced then we will discuss the properties much more detail. So the schematic which I have drawn to describe X-rays as the transverse waves or the waves where you have the oscillation in one plane and you are the direction, propagation direction is this. So there are oscillation direction as well as the propagation directions are mutually perpendicular. They are all called the transverse waves and then you have the amplitude A and you have the wavelength lambda and what I have marked here is a phase angle for a complete one particle of the wavelength. The phi is about 360 degree that is 2 pi. So we will just define these things so that when we use these parameters for explaining the wave properties this will be much more handy and this is a reference from where I have taken this introduction to diffraction in material science and engineering by Aaron D. Kravitz. So now we will see that we will write few remarks. So the wavelength which we have marked is the length unit of periodicity of the wave that is one full cycle that is the periodicity and then we also talk about the frequency of this waves. The frequency is the number of periodic wave cycles that pass through a fixed position in the path of the wave per second. So we use these terms quite frequently so it is better always to put it very clearly the basic meaning of this parameters that is why we are going through this. Now we will write an expression for the since the electromagnetic radiation travels with the speed of light so we can write this well known expression C by mu is the frequency and the wavelength of the x-rays they are related like this and C is the speed of the light that means that means one angstrom x-ray has a frequency mu of 3 into 10 to the power 8 per second. So few more points about this wave during one full cycle the wave amplitude oscillates through 360 degree or 2 pi radians of the phase angle 5. The phase angle for a wave travelling along the y axis is given by 5 is equal to 2 pi y by lambda. So this particular information we have already discussed in the phase contrast microscopy when we looked at the light optical system the similar thing we are doing here just for reinforcing the understanding and because we will be using this all the concept related to diffraction and imaging and so on not only here in electron diffraction as well everything is a common here. So the as the distance along y varies from 0 to lambda or from y to y plus 2 pi the phase angle varies from 0 to 2 pi radians or 0 to 360 degree. So you have to keep this in mind the wavelength we are measuring which is going to be related to the phase difference as well as the path difference like we discussed in the light optical microscope which is going to be discussed and quantified when we talk about diffraction and so on. So that is why we are introducing this again though we have already gone through it but it is better to have a clear idea about these parameters and the another thing is I want to draw a plane polarized light or plane polarized wave. What I have drawn the schematic is two polarized waves one is hatched the other one is which is there in the I mean the hatched wave is in the yz plane and the other wave which is perpendicular to this oscillation oscillation plane perpendicular to this plane is in xy plane and this is unpolarized light I mean like we have already discussed this or in this case we are talking about x ray waves not light. The amplitude vectors for a series of unpolarized waves travelling down the y axis so this is how it is going to look like. So for the that means what we are trying to show here is for an unpolarized light the amplitude vectors of the wave will be in the all over the all the directions compared to the plane polarized light like this what we are seeing here. Now few points we have to remember x ray waves can be represented in both trigonometric and a complex exponential notation the waves have sinusoidal periodicity so that their trigonometric representation is a sin 2 pi mu t or a cos 2 pi mu t we can keep an origin of for the wave can be expressed in terms of time that is t equal to 0 or a reference plane at y equal to 0 at a point y1 from a reference position there will be a phase shift given by 2 pi y by lambda that is the wave at y1 is described by a sin 2 pi mu t minus y1 divided by lambda. So in order to understand the phase relations which is very important in the some of the concepts like diffraction imaging so these fundamental parameters and their notations and how they are described is very important and normally people have lot of difficulty in getting these concepts that is why we are going little slowly and also you should remember this parameters like phase shift and then how they are represented for a given wave property. So now you will look at another property what I have written is for a diffraction it is convenient to use complex exponential notation to represent a wave in the form a e to the power 2 pi i into mu t minus y1 by lambda so this is what same thing. So what we are now trying to do here is look at the other notations which we will be using in the concept of diffraction something like an exponential notation like this and this is related to trigonometric representation by e to the power minus i y which is equal to cos y minus i sin y however it is the intensity i that is recorded in a diffraction pattern namely the product of the wave and its complex conjugate. So to appreciate that part let us consider a small volume of materials scatters in an incident x-ray wave if the volume is divided up into n point sources a wave from the point k is given by e k is equal to a k into e to the power 2 pi i into mu t minus y1 by lambda. So this is the wave from the point k from the n sources so that means if you want to sum up all the waves coming from n sources we will modify this expression accordingly. We can write so what we have done here is we have represented the wave from the point k from the material which has got n point source and we are now summing up all the scattered n scattered waves in the volume of the material that is e equal to summation over n e k e to the power 2 pi i into mu t minus y1 divided by lambda and you can express this exponential in terms of the trigonometric function like what we have just said here and it takes a form like this and then you can write the complex conjugate for this that is e star equal to e to the power minus 2 pi i mu t and then the trigonometric function. So the intensity is the product of the e star which is equal to sigma over n into e k cos 2 pi yk by lambda whole square plus sigma over n e k sin 2 pi yk by lambda whole square. So it is just to give you some idea how the waves are represented and how the each expression is looking like when you consider the wave properties. So we will be using some of this basic functions when we talk about diffraction as well as the some of the interference of these waves of x-rays. So after this I will start discussing about the diffraction and my first attempt is going to talk about diffraction in terms of phase relations. Since we talked about a phase and then I hope you have some idea about this phase and phase shift and we also have already seen the phase difference and path difference they are all measured in terms of wavelength. So we will relate this phase relations with diffraction and then we will try to give a complete explanation of how do we appreciate a diffraction in the I mean a diffraction of x-rays in the crystal lattice. So that we will see in the next class. Thank you.