 Hello, this is lecture 4 in the series. Here I will give a recap at the beginning. Until the last lecture, I have derived the scattering amplitude for X-rays and neutrons. For neutrons, I used Fermi-Golden rule and went ahead and derived the scattering amplitude and the intensity also. The square of the scattering amplitude will give us intensity. I did the same for X-ray to compare the two scattering amplitudes. I mentioned to you in the previous lecture that the scattering amplitude remains same except we have a summation over Bj e to the power iq dot rj when we have a potential at the site rj. This is for a delta function potential in case of thermal neutrons. In case of X-rays, it is almost similar only instead of Bj over some j I have something called Fj and then e to the power iq dot rj part remains same. So this Fj is a form factor which I introduced for X-ray which is a Fourier transform of the charge cloud at a point and which falls in space. Whereas in case of neutrons I had Bj which actually if I plot Fj against q it falls but Bj it remains constant. This is because it is a delta function in real space for nuclear potential and hence the Fourier transform of the delta function remains same all over q. It is a constant value in q. This is one advantage of neutron diffraction but all the derivations done so far were for a rigid lattice or a lattice at zero Kelvin. Today I will introduce you to the temperature effect for the same diffraction and then I will go over to bring in dynamics in our formalism that is the proposal for today. Now I will continue with the lecture. So as I said derived scattering amplitude for neutrons, derived scattering amplitude for X-rays. So far we dealt with a crystallographic structure at zero Kelvin no thermal effect. So what does temperature do? That is the question. So temperature causes thermal motion we know. So scientists was apprehensive about the fact that when we have a finite temperature then even if we have an underlying crystal lattice they may look somewhat like what I am showing here in this picture. Because these are the mean positions it is a square lattice so mean positions are indicated from the grid but at any particular temperature because of thermal motion the atoms will move and if I look at any instant of time it might look something like this. So the structural experiment I will come to it later also basically it is an average of such instantaneous pictures either over time and also over the entire ensemble of atoms or the whole crystal. So that means at any instant of time scientists felt before the first diffraction experiment was done that there will be no regularity in the lattice. So that was the apprehension and I would like to share with you a bit of history just a little bit so that we know that this apprehension was not by anybody but some of the major scientists. So once Laue had the idea that materials or crystallographic material can act like a three dimensional grating in front of x-rays and he wanted to look at the diffraction pattern from this grating. Grating we the word is familiar to you we use in case of optical rays. So I will just share this part of the history that he shared this proposal in on a holiday the Easter vacation he was in Munich University and they were in they used to go to Alps for skiing. And he discussed this problem with Sommerfeld and other we all big experiment place and others and what was the result. The result was they said encountering a strong disbelief in a significant outcome of any diffraction experiment based on the regularity of the internal structure of the crystals because they said the regularity will be disturbed. It was argued that inevitable temperature motion of the atoms would impair the regularity of the grating to such an extent that no pronoun diffraction maxima could be expected. This was a view of the the most established experimental at that day. Dubai will later attack this issue and I will show you what happens. But the fact is that it was also decided by some if not all that since x-ray machines are available at University of Munich. Why don't you do the experiment and I'm showing you historically the very fast x-ray diffraction that was done on copper sulphate. The fast x-ray diffraction was done by two experimentalists and you will be surprised to know that these experiments were pushed in the busy schedule of the experiment in University of Munich during off hours. But this experimental results were first time you can see the Laue spots or the diffraction spots on the photographic plate as expected by Laue. So this is a Laue pattern today known as Laue pattern at that time it was not called Laue pattern that diffraction spots could be seen. And very quickly they improved the experimental setup and they took Laue pattern of zinc plane and zinc blend is ZNS and you can clearly see it is a single crystal. The four fold symmetry over here and the three fold symmetry over here. So it was established that even at a finite temperature at that time it whatever was the room temperature in the experimental hall Laue pattern or the diffraction pattern do appear. They don't disappear because the thermal effect will make the place or position of the atoms uncertain and kill the periodicity. So Debye was one to solve it. Again after an animated argument at a coffee table in a cafe called Cafe Lutz and he writes I came to conclusion to the conclusion that the sharpness of the interference lines would not suffer. But that their intensity should diminish with increasing angle of scattering the more so the higher the temperature Peter Debye. So what he said is this if if you see a lower temperature if you see a lower temperature pattern then you can see the atoms are located. Or say 0 degree Kelvin the atoms are located at the rigid lattice size in a crystallographic structure. Now when I raise the temperature it is not that the atoms they run away from that place rather their vibration around the mean position increases. So as if the atom is expanding in size with temperature and not running away. So from smaller size the atoms become larger and if you remember I told you the larger the size of the atom but this is a dynamical effect the atom is really not becoming larger. What we see is an average picture in time that because of the extended vibrational capability of the atom it takes the as if it occupies larger space around the mean position. And then because of this you have something called a Debye Waller factor which is I0 the intensity at 0 degree Kelvin it to the power minus one third u square which is the mean value of the displacement and q square where q is the q will be g for drag diffraction. So it does not get destroyed rather it gets diminished in intensity and in general the displacement is not spherical but an ellipsoid called thermal ellipsoid. Because in every direction the bondings are not same and the solid will not allow equal amplitude of vibration in all directions. And then it will rather allow an ellipsoid at this ellipsoid will be oriented inside the crystallographic lattice I am talking about it is called a thermal ellipsoid. Now I will give a simple derivation of this is important for us that in diffraction. Now I will so if you remember I wrote and for x-rays this was a form factor it I went to the most primitive part of it actually for drag diffraction q becomes equal to g and then it changes. And this fj is the Fourier transform over the charge cloud. Now let me just write that this rj was at 0 degree Kelvin what I wrote. Now at any time t at any temperature at any sorry at a temperature t it will be the position at 0 Kelvin plus an oscillation around it at time t. This is what I have written comes and this tab ut it comes from dynamics and it comes from due to thermal vibration atoms and to be more precise it is coming from the phonon vibrations. So now the g dot rj it becomes e to the power minus i g dot rj e to the power minus i g dot ut this part is a thermal part. And we have to find out an average of this thing so this remains fixed so I need an average of this so e to the power minus g dot sorry I need an average of this a thermal averaging over the entire ensemble. So now this one this term I can break open like this we can expand the exponential I am sorry I am missing an I everywhere this was an I my apologies I everywhere so I have missed here. So now equal to 1 minus i dot u because this is a g dot u plus other terms I under the assumption that g dot u is small I can neglect the other terms. Then it becomes equal to 1 minus averaging g dot u minus g square u square plus square theta. Now g dot u you can see this is linear term and u can be the displacement can be in this direction if it is symmetric it can be equally in this direction if it is in this direction for a sphere it can be equally in this direction. So g dot u the average of this dynamical average should be equal to 0 and in g square u square cos square theta average of that it becomes g square no averaging u square. And over a sphere I need to find out the average of cos square theta so some of you must have done it that average of cos square theta is equal to cos square theta theta sin theta d theta d phi over a sphere 0 to pi. And 0 to 2 pi and you may check that it is come out to be one third. So then this becomes equal to one third then this becomes equal to g square u square one third. I plug this back into the expression of expand where expanded the exponential term and it becomes 1 minus one third u square g square. And I may put it back under the assumption that other terms are much smaller and then right is equal to exponential minus one third sorry it was half g square u square. So half I missed this term one half one third in the expansion it becomes one sixth now it becomes one sixth u square the average value of u square and some leverage g square. For expansion of the term e to the power minus i g dot u it comes equal to this. So now my form factor has one sixth u square g square and the intensity square of that. So I will have i equal to i 0 square of that one third g square u square that means my intensity at 0 Kelvin if it is i 0 it gets diminished by a factor e to the power minus one third g square u square. So as we raise the temperature u square increases i increases e to the power minus one third g square u square decreases i decreases. Also if you go to higher and higher q values that means you go to higher angles of diffraction then the g value increases again i decreases. So this is most important effect that when we raise the temperature when we raise the temperature we don't lose the Bragg peaks but because of vibration of the atoms are on the mean position their size increases this u square the size apparent size increases to the e to the power minus one third g square. So this is the maximum of the x-rays or neutrons even and then the intensity falls with the g square e to the power minus one third g square u square factor. This is a backdoor entry for dynamics. Basically I am trying to evaluate the intensity or the structure factor for a lattice and the finite temperature brings in dynamics and also this factor. Just to give you a taste of it I had did I had done the experiment with silicon powder using copper k alpha radiation. So I am just showing you that simple data which I took at room temperature and at 200 degree centigrade of silicon powder. The red one is the data taken at room temperature and the black one is the peak same peak at 200 degree centigrade. So you notice that the intensity has reduced and that is the cause of thermal vibration. This is basically the Debye Waller factor for this specific peak that I was measuring. But you have also noticed I am sure that the peak position has shifted. We have not talked about peaks shifting in our derivation so far. This is for the simple reason that when I heated the silicon powder not only that thermal vibration caused the reduction in intensity but the lattice expanded. So that means if it is a 2D sin theta this is a Bragg's law. If the lattice expands for the same lambda the theta has to decrease to maintain the equality in Bragg's law. And this is the signature of that. So you have the Debye Waller factor reducing the intensity of the lattice and the expansion of the lattice caused shifting of the peak. You might have also noticed that there is a small hump here that is related to instrumental resolution. Actually copper has two alpha lines K alpha 1 K alpha 2 and also K beta. This is splitting of the two lines in the experiment. So this is an experimental result which really demonstrates that the Debye Waller factor takes care of the thermal effect. And actually in our diffraction experiment this is also one parameter which is fitted to find out the thermal parameter. We can say that the thermal parameter or the extension to which the atom is vibrating is given by the Debye Waller factor. And I have also fitted as a parameter and this is basically a powder. I will discuss this in more details later. This is a powder diffractometer using position sensitive detectors. I am throwing terms at you and which I will explain later in which the data taken looks somewhat like this. This is how the peak positions come in a neutron diffraction experiment. They are fitted using a specific package known as Reedfeld package extremely popular today among all the people who want to fit their neutron or excel diffraction data. And typically this is how it looks at a finite temperature. But how long this will continue? I must say that the scientists at the beginning were not wrong. When you keep heating your sample, once the sample starts melting, then the lattice is destroyed. And once the lattice is destroyed, the picture that I showed earlier, which was the apprehension of the scientists, that indeed happened. When the lattice structures start melting, you have the atoms moving from the mean positions. It goes to a liquid state and you lose the Bragg peaks. You do lose the Bragg peaks. But till then, till the lattice is intact, it is the vibration which goes by the thermal motion. And you don't lose the Bragg intensity, but you have a reduction in intensity. With that, I end this module.