 Hello everyone, welcome to this material characterization course. In the last class we looked at the x-ray diffraction conditions through Laue equations and we compared that Laue equations with the Bragg's law and then we found that there is a difference in these two laws where the Bragg's law explains the diffraction conditions with the in terms of parallel row of atoms in the planes whereas in the Laue conditions we were able to obtain the diffraction conditions for the rows and net as well as the three dimensional lattices. And then we also try to relate the Bragg's law with the reciprocal lattice and we briefly went through the concepts of reciprocal lattice and we also showed or demonstrated how this a wall sphere links the Bragg's law with the reciprocal lattice. And then we started discussing about the intensity of x-ray diffraction which is which is what primarily we are interested in when we use x-rays as a probing tool to analyze the crystal systems. And then in that discussion we were we were also saying that in order to arrive at the the intensity expressions you need to understand how the x-ray scattered by the single electron is understood. So, in that direction we will continue our discussion today. So, look at this first slide where some of the important remarks about scattering by an electron x-ray scattering by an electron. Since an electric field exerts a force on a charged particle such as an electron the oscillating electric field of an x-ray beam will set any electron it encounters into oscillatory motion about its mean position. An electron which has been set into oscillation by x-ray beam is continuously accelerating and decelerating during its motion and therefore emits electromagnetic wave electron scatters x-rays. The scattered beam has the same wavelength and the frequency as the incident beam and is said to be coherent with it. Since there is a definite relationship between the phase of scattered beam and that of the incident beam which produced it. So, we have the definition of what is a coherent beam that means it should have the same wavelength and the frequency as the incident beam. So, we we also know that what you mean by the the phase relations by now x-rays are scattered in all directions by an electron the intensity of the scattered beam depends on the angle of scattering. J. J. Thomson demonstrated that the intensity i of the beam scattered by a single electron of charge e column and the mass m in kg at a distance r meters from the electron is given by i equal to i naught into mu naught by 4 pi square into e to the power 4 divided by m square r square sin square alpha which is equal to i naught k by r square sin square alpha where i naught is the intensity of the incident beam mu naught is equal to 4 pi into 10 to the power minus 7 meter kilogram per c square k is equal to constant and alpha angle between the scattering direction and the direction of the acceleration of the electron. Suppose if you look at this kind of coherent scattering of x-ray by a single electron assuming that this is the coordinate where you have all this electric vectors in the mutual perpendicular direction x, y, z and then we can try to account for the intensity of the x-rays scattered by an electron. So, look at this remarks an unpolarized incident beam such as that issuing from x-ray tube as its electric vector e in a random direction in the y, z plane e square is equal to e y square plus e z square on the average e y will be equal to e x since the direction of e is perfectly random therefore we can assume e y square is equal to e z square is equal to half e square. The intensity of these two components of the incident beam is proportional to the square of their electric vectors since e measures the amplitude of the wave and the intensity of a wave is proportional to the square of its amplitude. So, we can write the intensity expression like this I o y is equal to I o z is equal to half I naught with respect to this schematic. The y component of the incident beam accelerate the electron in the direction o y it therefore gives rise to a scattered beam whose intensity at p is found from the above equation to be I p y is equal to I o y times k by r square. Since alpha is equal to y o p which is equal to pi by 2 similarly the intensity of the scattered z component is given by I p z is equal to I o z times k by r square cos square to theta and we can look at the other expressions since alpha is equal to pi by 2 minus 2 theta the total scattered intensity at p is obtained by summing the intensity of these two scattered components I p is equal to I p y plus I p z which we can substitute in this form k by r square into I o y plus I o z cos square to theta and then we can rearrange them into I naught into k by r square into 1 plus cos square to theta by 2. This is the Thompson equation for the scattering of an x ray beam by a single electron where 1 by 2 into 1 plus cos square to theta is called a polarization factor which we will be incorporating in all the intensity equation whenever we are going to write. If monochromator is used with the diffractorometer then this expression is modified into this form that is 1 by half into 1 plus cos square to theta cos square to theta m. Now we will look at another form of scattering which is called a Compton effect which describes the elastic collision of photon and electron. It occurs when the stream of x ray quanta are photons encounters loosely bound or free electrons. So there is a slightly different from what scattering which we talked about previously and here it is the x ray quanta. We are talking about a quanta which encounters a loosely bound or free electrons like two billiard balls which are colliding with each other something like that. So you have the h v 1 is colliding with an electron before impact like this then after impact it goes like this into two different directions. The wavelength lambda 2 of the scattered radiation is thus slightly greater than the wavelength lambda 1 of the incident beam. The magnitude of the change being given by the equation delta lambda which is nangstrom unit is equal to lambda 2 minus lambda 1 is equal to 0.0486 sin square theta. So when an x ray beam encounters an atom each electron in its scatters part of the radiation coherently in accordance with the Thomson equation. However the nucleus has extremely large mass relative to the Duffen electron and cannot be made to oscillate to any appropriate extent. In fact the Thomson equation shows that intensity of coherent scattering is inversely proportional to the square of the mass of the scattering particle. The net effect is that coherent scattering by an atom is due to only the electrons contained in that atom. So now we have to ask some questions is the wave scattered by an atom simply the sum of the wave scattered by its component electrons? More precisely does an atom of an atomic number z that is an atom containing z electrons scatter a wave whose amplitude is z times the amplitude of the wave scattered by the single electron? The answer is yes. If the scattering is in the forward direction because the wave scattered by all the electrons of this atom are in in phase and the amplitudes of all the scattered waves can be added directly. See you have to understand this point very important. When we talk about a phase relation also we mentioned this aspect. If the scattering is in the forward direction for example we will look at one schematic about it and since all the waves will be in the same phase they will contribute to the intensity but in reality it is not so. You will have at least in this case the electrons will be there in different different directions in an atom but the atoms which are supporting the or forward scattering phenomenon they will contribute more to the intensity. So that is the point we are going to now prove in the coming schematic. So look at this schematic this is a nucleus and you have the electrons in the orbits and then you look at this a green line where the forward scattering rays are shown and they are meeting this wave front xx prime and then you have the x ray scattered in the other direction which are meeting in the yy wave front and now you know how to relate this a path difference. We have looked at those details in much much more examples we have seen. So now let us with respect to this schematic let us look at the remarks. The waves scattered in the forward direction by electrons A and B are exactly in phase on a wave front such as xx prime here because each wave has travelled the same distance before and after scattering. The other scattered waves shown have a path difference equal to CB minus AD that is CB minus AD is a path difference and are thus somewhat out of phase along a wave front such as yy prime. The path difference being less than one wavelength so they are going to scatter slightly different only partial interference occurs between the waves scattered by A and B with the result that the net amplitude of the wave scattered in this direction is less than that of the waves scattered by the same electrons in the forward direction. So this is the fundamental point we have to capture. So the waves which are not in the forward direction they are not going to contribute equally to the interference. So only a partial interference will occur between this waves and then they will contribute to some extent to the net amplitude of the wave scattered in that all directions. Here we have shown only one direction of that nature. You have to imagine that these kind of a partial interference will occur in most of the other directions as well and then finally you get a net amplitude that is the point you have to understand which is going to contribute to the integrated intensity. We will just talk about this integrated intensity because that is what we are interested. So right now you just appreciate this point how the forward scattering waves how they contribute or how about other rays which are contributing partially to the intensity. So at quantity F the atomic scattering factor is used to describe the efficiency of the scattering of a given atom in a given direction where F is equal to amplitude of the wave scattered by an atom divided by amplitude of the wave scattered by one electron. Suppose if F is equal to Z for any atom scattering in the forward direction as theta increases the waves scattered by the individual electrons become more and more out of phase and F decreases. So it is not just that you have all the electrons will be doing only forward scattering and then you will get a maximum intensity but it also depends upon the theta. So that is the point we are now explaining here this is for the atomic scattering factor of copper where you see that as the theta increases in fact it is not theta it is sin theta by lambda which decreases as the theta increases you can see that atomic scattering factor also decreases. So this is very important point and now we will just look at the whole picture effects produced by the passage of X-rays through a matter in general you have the incident beam and this is your specimen absorbing some of the X-rays will be absorbed and some will release as a heat and then you have fluorescent X-rays coming out and then you have scattered X-rays and you have electrons coming out. You have Compton recoil electrons and photo electrons and OJ electrons of these three category all the possibilities when you talk about a scattered X-rays you have unmodified coherent scattering the other one is Compton modified incoherent scattering. These are the two things we have seen today one is coherent scattering other was incoherent scattering in one case we see that after the collision the wave length is changed you get that means the because of the collision the energy is lost in the form of kinetic energy. So the lambda is slightly increased that is what we have seen in that expression where it is called a Compton effect it is an incoherent scattering the previous one where Thomson equation explains the coherent scattering. So these are all the two kinds of scattering just for your clarification. Now we will look at the structure factors which is very important for the intensity calculations. So the expressions which I have written all here all much familiar to you the first one is we have seen from the structure factor expression F hkl is the atomic scattering factor from the all the atoms in the unit cell. We will quickly look at this in few minutes what is the details here and this expression is the diffraction vector expression which we have seen yesterday and this is a reciprocal lattice vector and here this is a real lattice vector here which is given and then this form is the final form you get for the structure factor calculations and we will now take up some few examples how to use this and how this the equation gives the intensity for a given crystal system then how they contribute to the total diffraction intensity. So I will go to the blackboard and then so what I have written is if a unit cell contains atoms 1, 2, 3 up to the n atoms with the fractional coordinates u 1, v 1, w 1, u 2, v 2, w 2 etc then the atomic all the atomic scattering factors from that unit cell should be added like this and then if you sum it up and then the final equation appears like this. So this is the summation of all the atomic scattering factors multiplied by e to the power 2 pi i into h u n plus k v n plus l w n so on. I believe that you now appreciate this expression this is atomic scattering factor and then this is this complex exponential function we referred a scattered electromagnetic radiation in this case it is x ray is expressed in terms of complex exponential function and this is coming from the phase difference that also we have seen before this particular component is coming from a phase difference and this is the structure factor from the unit cell expression. So now we will apply this expression for individual unit cell we will take it up as simple case and I have taken this from this book BD quality and SR stock and you can go through for the entire description and much more detailed information is given there and you can also note down certain important relationship mathematical relationship in order to understand some of the factors which will come in between. So first let us go through that you can write some of the useful expressions like this you can keep in mind before we look at all the derivation for structure factor for a given crystal system the e to the power pi i or e to the power 3 pi i which is equal to e to the power pi i is equal to minus 1 and if it is e to the power 2 pi i then even number e to the power 4 pi i or 6 pi i is equal to 1 in general this is the expression e to the power n pi i is equal to minus 1 to the power n where n is any integer similarly we have e to the power n pi i is equal to minus n pi i where n is any integer and you have this expression as well e to the power pi pi plus e to the power minus i pi is equal to 2 cos x. So now we will take up an individual expression I mean unit system unit cell system unit cell containing only one atom so if you take up this expression the fractional coordinates 0 0 0 and the structure factor is f is equal to f into e to the power 2 pi i which is equal to and f square is equal to f square f square is independent of so if you consider if you consider a unit cell containing only one atom then the fractional coordinates can be the origin 1 I mean 0 0 0 and then if you substitute that into this structure factor equation then you see that f is equal to f into e to the power 2 pi i 0 which is f because this component becomes 1 and then you get capital F is equal to f this is intensity which is independent of hkl and the same for all reflections. So now we will take up the one more example which is having a 2 atoms per unit cell something like this a base centered unit cell and how do we calculate the structure factor the as usual we will start with the fractional coordinates that means atoms located in the coordinates for this particular position is you have 0 0 0 and half half 0 these are the fractional coordinates you can see that 0 0 0 position this is half half 0 position and then you can write the expression as usual capital F is equal to so you get the expression like this substitute this fractional coordinate into the structure factor equation and then you simplify this you get f into 1 plus e to the power pi i into h plus k. So we will write something about this expression so this expression may not be multiplied by the complex conjugate because h plus k is always the integral and f is a real and not complex so this is the first step we have to understand and then what is the implication of this condition is if h and k both even or odd that is unmixed indices then their sum will always be an even. So the implication of this condition is if h and k are both even or odd that is unmixed then their sum is always even and your e power pi i into h plus k has the value of 1. Therefore, you have f is equal to 2f for h and k unmixed and f square is equal to 4f square. If h and k are one even and one odd that is mixed indices then their sum is odd and e to the power pi i into h plus k has the value of minus 1. So this is the condition and we can now see that some of the indices what kind of reflection it will give. So if you apply this rule we can note that in either case the value of l index has no effect on this structure factor. So that is what we are seeing here. In both cases the indices l do not have any effect. For example, if we can take the reflections 1 1 1 1 1 1 2 1 1 3 and 0 2 1 0 2 2 0 2 3 all have same value of f namely 2f and similarly if you take the reflections 0 1 1 0 1 2 0 1 3 and 1 0 1 1 0 2 1 0 3 all have 0 structure factor and are systematically absent. So this is how you realize how the atom position contribute to the intensity through the structure factor. So this is one example. Similarly you can apply the similar conditions for a simple cubic system where you have body centered cubic lattice as well as face centered cubic lattice you can apply this and then realize that how the structure factor varies. We can we can look at couple of examples and also some of the ordered crystals we will apply this rule and then calculate the structure factor in the next class. Thank you.