WEBVTT - autoGenerated
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The radiation motor. We will consider for the moment only its rays. Then we will give
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some more information about electrons or electrons. So the wave for x-ray may be described
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by a plane wave. EI, these are the countries, the electrical countries, is EI equals E0I,
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the exponential to pi v t minus x divided by c. The involved character E0I is the electrical
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field. As you know, this magnitude is oscillating like the wave. So the EI which is on the left
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side shows the oscillation. So the amplitude is E0I and the EI is the oscillating amplitude,
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time to time and point to point. So in the point x at the time t we will observe EI which is
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oscillating. Knee is the Greek letter Knee is the frequency of the wave. It is the classical
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wave of this field. So what we have to do now to model the interaction of radiation matter. Now we have
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the model of the radiation. And we have the model of the lattice of the crystal, the model of the radiation.
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And now we can do the fourth step, the interaction of radiation matter. Is it okay? We will use the results
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by Thompson. It is the Thompson scattering. Thompson is the man who characterized the electron,
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the charge of the electron, the mass of the electron and so on. So we see the number applied for that.
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So, Thompson made this observation. If we have an oscillating field E, if we have an oscillating field
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E on a charged particle with the charge E, for the moment the charged particle may be whichever,
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may be also a proton, may be an alpha particle, doesn't matter. E and the small e and m are the charge
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of the particle and the mass of the particle. Is it okay? Is it okay for all? Okay, I will repeat.
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Starting from the beginning. We want to have a radiation model. We will use the classical approach.
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That is, we describe the radiation like a plane wave. The amplitude of this wave is T0i, like a wave in the
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Humboldt hardware. So that is the amplitude of the wave. Then the amplitude at the time t and in the position x is Ei,
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which is E0i for this expression, which is the oscillating expression, because it is sine and cosine, right?
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This is the equation. N is the frequency and t is the time x, the side in which we measure Ei. Is it okay?
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Now, we have described the interaction between this plane wave and the matter. So, we will use the results by Thomson.
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What Thomson did? Thomson observed that if E is on a field, is an oscillating field, and we have a particle with a charge e, small e, and mass m,
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this particle will receive a strength, F, which is equal to the charge multiplied by the electric field.
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It is known by all of you. The strength is equal to e, small e, capital E. Is it okay?
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Now, but when a particle undergoes a strength, F, it will accelerate. So the acceleration will be equal to the strength divided by the mass, right?
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Is it okay? So, what will occur when a charged particle is accelerated by this strength?
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It will emit radiation. We are here in this campus here, so we know that. When a particle is accelerated, it must emit radiation.
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So, which kind of radiation? The radiation has to be spherical, no more plane, because it will be emitted in any direction.
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So, we will have an issue, but if we look as a function of the distance from the point in which the particle is to r, r is no more x, r is any distance from the position of the particle.
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It's okay. Now, we are doing with a spherical wave. So, if we look at r here or here or here, as I mentioned, at the distance r from the particle, what will we learn?
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We will have a spherical wave, ed, which is E0, the exponential to pi, i, mi, theta, r divided by c and so forth. Is it okay?
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So, what will be the electrical field in the position r, for example here, at the distance, in the position at the distance r from the particle?
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It will be ed equal to this one. Is it okay? So, this is the amplitude of this scattered wave. r is the distance. Of course, c is the speed of the light.
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This alpha may be different from E0, because it is not sure that the particle emits the wave at the same moment in which it receives the acceleration. Maybe a shift of time.
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And so, the calculation of Thompson really led to this formula here, the intensity emitted by a particle with charge e and mass m is this one.
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Is it okay? This is the formula by Thompson. We will explain now this formula.
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To explain the formula, we need some drawing. So, this in direction, it arrives a plane wave, this one.
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And the particle is here of charge e and mass m is in O. From O, spherical wave is emitted, which is this one. And we observe this wave in this point, q. Is it okay?
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x is the direction of the wave. So, it is easy to understand that the zeta and y direction may be rotating as we want. We are not too large to choose any direction for zeta and y.
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So, what we do is that zeta and y are made in such a way that observation point q is in the plane x, y. Is it okay? We can do that freely. So, we observe in the plane x, y.
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So, just to have in the hand the problem, suppose that we want to understand now better this. What says this formula?
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The formula says that the intensity of the spherical wave emitted by this charged particle is the charge according to Thompson.
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According to Thompson, the h is Thompson. Yes. Proportional to I, I. I, I, I is the intensity of an incident wave. Okay? I, I, it is normal. The larger intensity of this new wave, the larger intensity of the scattered wave.
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Then, we have here this ratio which is quite important. The charge of the particle power 4 and at the denominator, m square, r square, c4. And we will discuss this. But there is this coefficient which is interesting, c square, c square phi.
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What is phi? Phi is the angle between the acceleration direction of the electron, of the charged particle, if you want, and the observation direction. So, the observation direction is this because we choose q as point of observation. So, the direction of observation is this.
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And the acceleration, what is the acceleration? The direction. Suppose that the wave is playing, incident wave is playing, and is polarizing into the plane xz. It makes it this way, in this plane. Okay?
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Making this way in the plane. So, what will be the direction of acceleration? The same as phi. The same as phi. Remember that parallel to z would be parallel to z because the wave is displayed in this. So, it goes, the direction is x, but the wave makes it this way.
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So, when it is polarizing in this way, the o will oscillate along the z. If it is polarizing along y, the wave goes in this way, the electron will move along y. Okay? So, depending on the direction of the parallelization, we will have a different value of c square phi. Is it okay?
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So, in the first time, for the first case, for the first case, when the wave is polarizing in the plane xz, the acceleration will be along the z. And therefore, the angle with the observation direction will be phi.
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It will be pi by 2, sorry. It will be pi by 2. Is it okay? Because the acceleration is in this direction, and the observation is in the plane xy. Is it okay? Fine? If the wave is polarizing in direction y, then the electron will be accelerated in the y direction.
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So, the angle will be this one. Is it okay? So, the formula will take two different coefficients here, according to the direction of the parallelization. Is it okay?
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Remember that here, in the cyclotron here, the waves are polarizing. So, this factor there is very important to understand what you have to measure. Is it okay?
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If you are collecting data in your lab, the wave is slowly polarizing. So, it is very important. But here, it is very important. Is it okay?
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Now, look at this ratio here. If power 4 and square r square c4. When i, the intensity of the scattered radiation, or the radiation scattered by the electron is large, when?
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When? Please, let me know.
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I am not speaking about phi. I am speaking about this ratio. So, consider this ratio. When this ratio is...
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Because the mass is...
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Well, the mass must be as small as possible, and the charge must be as large as possible. Right? So, which particles have a mass as small as possible, and the charge as large as possible?
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So, electrons are the best candidate to scatter. Now, consider not the electrons, but the particles. How much lower will be the energy?
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This mass of a photon is more or less 1,800 more. There you see m2 square. So, the intensity of the photon's country will be 1 million or times less than the scattering of the electrons. Right?
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So, we can conclude that the protons are invisible. The each way.
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That is the reason why you, when you started with the crystallography, somebody who already started with the crystallography, always heard electron density. Electron density map. Why?
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Because the electrons scatter the radiation. And then we can speak of electron density. Is it okay? Fine?
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Newton can contribute to the discussion? No, because they do not hear charge. That is the reason why we speak about electron density. Is it okay? Fine.
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Now, the voice after two or three days has essentially started to be a little jarring, but after it moves.
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So, now, we should discuss this c2 phi. C2 phi depends on the polarization, we said. But for us, so for each polarization and for each percentage of polarization, we will have a different value here.
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So, the people working in the experiment perfectly know how to do that. So, you will receive the data already corrected for this polarization. Is it okay?
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But, to have an idea, what we can do is, okay, we can assume if you want a wave which is unpolite. In this case, it is easy to show you. If you want, I can demonstrate. You are not interested too.
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Okay, I will give you the formula. If you want, we can spend five minutes to demonstrate it. If you accept the formula as trivial, we can go. Is it okay?
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So, in case the wave is not polarized, the formula is this. So, that c2 phi is replaced by this. Okay?
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C2 phi is replaced by this, which is called polarization factor. Is it okay? So, if you are curious, we can demonstrate. If you are not curious, we can go on.
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Demonstrate.
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Demonstrate. So, what do we do? We assume that the wave, we assume it here. That way, it is polarized. And we can take, we assume first that it is polarized along the zeta. So, that the electron oscillates along the zeta.
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And we can also assume, as an example, that it is polarized along y, another wave. Is it okay? When the wave is not polarized, when the wave is not polarized, what we have?
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Right. It means that there are waves, single waves, which can assume any indirection. Okay? So, what we can do is to divide the incident beam into two beams. One polarized along zeta and one polarized along y.
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One half of the beam polarizes along the zeta. One half of the beam polarizes along the y. So, we have only some of the contribution of this two. Okay? Of this two. So, when this polarizes along the zeta, along the zeta, what do we have?
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Okay.
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Thank you. We are ready for any remains. Any face can occur via resolution. Fine. Fine. So, I started, I repeat. It's okay? So, suppose that the wave is unpolarized.
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So, we can subdivide this wave into part. One half of the beam polarizes along the zeta. And one half of the beam polarizes along y. Okay? So, remember, the intensity is one half of I E. One half of I E polarizes here.
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Okay. Now, let us consider first the half beam polarizes along zeta. We said that when the beam is polarized along the zeta,
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the acceleration direction is z itself. So, the angle between O cubed and all zeta is pi by two. Is it okay? So, this beam will be one over t. All that, all that here,
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I don't write it. I put it this way. It's okay? And then c squared phi is equal to one because we said phi is pi by two. It's okay? Now, we go to the half beam polarizes along y. In this case, which would be the angle between this direction and the zeta?
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In this direction? Would be an angle which now we define correctly. Suppose that two theta is the angle between x and O cubed. That is, be careful because we will follow always this. And this notation is followed by all the crystallography.
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Two theta, this is, is the angle between the incident radiation and the observed radiation. Two theta will be always the angle between the incident radiation and the observed radiation. Is it okay? So, two theta is this.
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So, phi, phi is pi by two minus two theta. Is it okay? What is the sign of pi by two minus theta? Cosine. Cosine of two theta. So, it will be cosine squared two theta. Is it okay? Fine for long?
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Is it okay? Fine? So, at the end, at the end, if we sum this intensity, if we sum this intensity, we will obtain one plus cos squared phi divided by two. Is it okay? So, this is the intensity of unproided beam.
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And this is called polarization factor. As you may observe, it is, absolutely to the contrary, it is the factor for a beam unprologizing. But it is called polarization factor because that is the effect. Yes?
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I have a question. What is divided by two? Because the beam, it is one half. Okay, so this is just one. Exactly. It's one half. One half of the beam, we should put one half and then one. Is it okay? Fine for long?
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So, now we know which is the intensity of an electron according to Thomson. It is this expression here. And two theta for us will be always, always the intensity of the angle between the incident radiation and the observed radiation. Is it okay?
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In our case, the diffracted intensity here, two theta, always. Fine?
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So, this is just when, for every direction there is the same probability? For each, for each, to theta. For each to theta, we have the same intensity. Changing to theta will change it intensity.
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Do you have an idea of polarization?
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Ah, yes, yes. The polarization is for each direction, we have the same probability of the wave. When the wave is practically polarized, we have one probability, one for that direction, zero for the other. Partially polarized means a different dispersion of the wave. Is it okay?
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Okay, now, we are now prepared to work with considerably only electrons in no other. That's important.
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And, but is it possible to work with other particles? Protons or neutrons? Neutrons, too. But not by working, that is, the scattering will be different, of course. And we will explain why.
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Let us start with electrons, because it is important that you have the idea of what it may be made for starting the matter. So, we can also use electrons. How to use electrons? A microscope. A microscope may be used. The microscope scatter the radiation.
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If the microscope is a quipid, well a quipid may do the refraction, of course. And the number, the wavelength used in microscopy are very, very short. Much, much shorter than the Israeli wavelength.
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And with which electrons should interact? Because we know that each way interacts with the electrons. And electrons will what they interact? Electron are charged particles. So, we will interact with the electrical field around the electrons.
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So, the electrical field is generated by the cloud of electrons and the nuclei. So, the field which you observe in the material is the results of the sum of the electrical fields generated by electrons and by the protons, which are in the nucleus.
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So, they will be scattered by the electron field. Is it okay? Since they use a very, it is necessary to use very short wavelength, why? Because the electrons as charged particles will be absorbed very soon.
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Unless the wavelength is very short. While the x-ray can cross up and provide the proof of our earth, of our disease, unfortunately, the electrons are immediately absorbed from the skin.
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So, they need to be a very small wavelength. Okay, that is the first point, immediately I will show it. But, there is also the importance of the electrons. Why? Because, maybe there is more particle may be started.
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Because, why there? Because x-ray are penetrating. That means that they cross a big amount of matter without interacting with it.
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And so, for observing x-ray, you need a minimum amount of matter. Is it okay? So, you need a crystal of dimension 0.1 micron, for example. Is it okay?
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But, if you want to study, if you want to study matter with a crystal very, very small, then electrons are better. Because, they strongly interact with the matter.
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So, nanocrystal may be studied by electrons. Is it okay? That is the most important tool of the electrons. Is it okay? The possibility of working with nanocrystals.
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But, on the same time, the theory we will see, valid for x-ray, is no more valid for electrons. So, the diffraction by electrons is very complicated.
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We will see, I will show you this point later on. So, the intensity we observe, the diffraction intensity we will observe, are not kinematic.
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Like x-ray, we will see what means kinematic. And the diffraction, really, is still an open problem for the electron diffraction. Is it okay? It is an unsolved problem.
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But, the idea is to push up, push this method to solve the problem of the multiple diffraction. We will see later on. It is okay?
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This is for the electrons. We can also use protons. But protons are destroyed, heavy atoms, heavy particles. So, very likely, destroy the matter.
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The second is the neutron. The last particle we could use are neutrons. Neutrons are not charged with particles and have a defect and also a good feature.
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The defect is that they are penetrating and very largely penetrating. And therefore, they interact weakly with the matter. We could not stay here if there is an experiment there with neutrons because neutrons can come here.
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So, that means that they need very large intensity to use for diffraction. Or, if you want, big amount of matter, big crystal, to generate diffraction.
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So, the neutrons are used only for a specific case. Is it okay? We will see a little bit what that means.
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Only for a specific case. But diffraction may be made. It is quite useful for powder diffraction because when you use crystal powders, you can use a larger amount of more crystals and then you can use it.
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So, neutron crystallography may be useful. But only a specific case and it is expensive. So, neutrons are used not really for crystallography, not mainly for crystallography, but for other reasons.
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For other types of material studies. The second point is, I will say that later on. To compare electron diffraction with the neutron diffraction and x-ray diffraction.
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These are the differences I will explain later on when I will speak about some scattering powers. I will give you this information later.
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So, practically, I will prepare you to work mostly with x-ray. Is it okay? You can use electrons for diffraction only in a specific case.
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Well, but of course, in electron crystallography there are other approaches which can be used, but not diffraction, but other things which are not of interest to the bond influence.
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Okay, now, what we are using is the classical theory of the scattering. It is not quantum mechanical theory.
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So, as soon as Compton made the job of calculating the classical scattering theory, Compton made the vice versa.
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He said that the theory is not good enough for the reason. And he tried to use the concept of focus, something which is practical and the way we see the terms of it.
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And then he observed that things are not really as Compton described, because when some free electron are submitted to radiation, this free electron receives some energy from the radiation.
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And therefore subtract the radiation to the scattering, subtract energy to the scattering.
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This subtraction of energy to the scattering means a difference in the wavelength.
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So, Compton observed that electrons produce scattering at wavelengths longer than the incident wavelength.
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And obtained with the formula which is there.
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Delta under equal 0.024, 1 minus cos 2 theta. 2 theta is always the angle between incident direction and observed direction. Is it okay?
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Is it fine?
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And he also observed that this phenomenon, this delta lambda feature is maximum when we look at 2 theta equal minus 1.
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It is when we are looking at the back scattering.
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Why? When we look just in direction, just in the same direction of the incident radiation, cos 2 theta is close to 1 and the lambda is 0. Is it okay?
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So, this effect, the radiation comes in this direction. If I look in this direction, the scattering in this direction, I see the lambda equal 0 because cos 2 theta is 1, 1 minus 1 is 0.
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But if I look, the radiation comes in this way, that I look, the back scattering on the opposite direction, cos 2 theta is equal minus 1 and then the delta lambda is maximum. Is it okay?
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That was Compton. And Compton received the number twice 2.
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That's an important thing. The modern physics started from that age, very important things.
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So, that in any case implies that there is a faster of energy from the radiation to the electrons because the delta lambda means a change of wavelength and therefore means a subtraction of energy.
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So, that implies a faster of energy from the incident radiation to be clear if this is the wave and here we have the particle.
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When the wave interacts with the particle, the particle goes in this direction and the wave comes in this direction with a longer wavelength. Is it okay?
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But luckily in crystallography, the transfer of the energy to the electrons to the orbital is not so easy because the electrons stay in stable orbitals.
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So, they should pass from one state to another state. So, it is like if the mass of the electrons increase because of the mass of the nucleus.
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It's okay because they are bonded to the nucleus. That is the response for free electrons. But they are connected to the nucleus and stay in orbitals, stable in orbitals.
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And therefore, sometime it occurs, we will see later on when it occurs, but often this phenomenon, in crystallography, may be neglected. Is it okay?
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So, we in crystallography will usually work with the Thomson for me. Is it okay? The Compton effect is prison. But this Compton effect, which kind of effect is really experimental, it is incoherent.
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That is, there is no relation between the scattered radiation and the incident radiation. Therefore, it cannot enter into the diffraction.
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So, when all the wavelengths are strengthened into the diffraction, and this Compton scatter may be everywhere because it cannot participate in diffraction.
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And therefore, it is a background. So, this background is not too noisy, usually. And therefore, we will neglect it. Is it okay? Unless you are doing some study specifically on the background.
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But usually, it can be neglected. Is it okay? So, we can say, ah, good, because clearly, it will be simple, okay? Fine? Is it clear up to now? So, we will continue with Thomson. Fine.
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Now, now we know, we know that we have scattered, which does not imply any change in the wavelength. We will see later on, that we can produce,
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we want to produce some transfer of energy. We want to produce some transfer of energy, and specifically to exploit it. But that would be seen later.
00:47:20.000 --> 00:47:36.000
For the moment, we will work in the condition that the wavelength of the incident radiation is equal to the wavelength of the scattered radiation, and no transfer of energy is there. No contact. All Thomson. Is it okay?
00:47:36.000 --> 00:48:00.000
Now, we have now, when a wavelength enters in contact with a crystal, it will, let me say, interact not with only one wavelength, but with a million, a million, a million of electrons, right?
00:48:00.000 --> 00:48:25.000
So, we have to start to study what occurs when two electrons simultaneously scatter. We are not speaking about crystal for the moment. We are speaking about the minimum interactions, two electrons which scatter simultaneously, okay? Fine?
00:48:25.000 --> 00:48:46.000
And then, we will try to complicate things a little bit, from two electrons to many, from many electrons to an atom, from an atom to a molecule, from a molecule to a unit cell, from a unit cell to a crystal, and so on. Is it okay?
00:48:47.000 --> 00:49:06.000
Now, we will investigate the interference mechanism which govern the diffusion from an extended body. Is it okay? And we will start with two charged particles.
00:49:06.000 --> 00:49:26.000
One is in O and the other is in O prime. Is it okay? I will show you which O and O prime are. O is here and will serve as our origin of our system. Is it okay?
00:49:27.000 --> 00:49:33.000
And O prime is here, so we have two electrons, one here and one here.
00:49:34.000 --> 00:50:02.000
The direction of incident x-ray beam is S0. S0 means unitary vector, okay? In Italian, we have the name of bevso.
00:50:02.000 --> 00:50:20.000
But it is unknown in other countries. I don't know in Germany. Is bevso or something like that in Germany? No. It is an Italian word which is known. Bevso is unitary vector.
00:50:20.000 --> 00:50:47.000
And S0 is this unitary vector. So the beam, the plane wave is this along S0 and we observe the radiation along the unitary vector S.
00:50:50.000 --> 00:51:19.000
If we follow carefully that, we will see the first application of the mathematics we did yesterday.
00:51:19.000 --> 00:51:29.000
So we will understand why we started the Fourier transform. Okay? So follow carefully.
00:51:29.000 --> 00:51:52.000
Now, to study the interference between two waves, two scattered waves, we have to understand what is the difference in the wavelength in the wave path, okay?
00:51:52.000 --> 00:52:14.000
So, characterize first the geometry, which is the difference in the path between this wavelength and this, sorry, this wave and this wave.
00:52:14.000 --> 00:52:41.000
The difference is AO plus rho B, that is the wave which interact with the O run a path which is longer than O prime, right? This wave. This path and this path.
00:52:41.000 --> 00:53:01.000
If you compare them, you will see that this wave has a longer path than this because AO plus OB is in addition. Is it okay? It's fine for all?
00:53:01.000 --> 00:53:24.000
So, AO is the projection of r on the direction S0. AO is the projection, the distance, O prime with this r. And on this direction here, on this.
00:53:24.000 --> 00:53:52.000
So AO is minus r dot S0. Minus because this is in the opposite direction. Is it okay? And OB, it is the projection of r on the direction S. So it will be BO r dot S.
00:53:52.000 --> 00:54:17.000
Is it okay? So, the difference in the pathway between this wave and this wave is AO plus BO. Is it okay?
00:54:17.000 --> 00:54:43.000
Did I lose someone? Is it okay for all? So, but when you consider waves, what is important is the difference in the phase. So it is not important the difference in the millimeter or in angstrom.
00:54:43.000 --> 00:55:08.000
What is different is that the difference in pathway divided lambda, which is called the optical pathway difference. Because it is divided by lambda. So, AO plus BO divided lambda will be what? rS minus S0 divided lambda. Is it okay?
00:55:08.000 --> 00:55:33.000
So, AO plus BO is r dot S minus S0. So, AO plus BO divided lambda is there. Is it okay? So, this is the optical difference in the pathway.
00:55:33.000 --> 00:55:58.000
So, if we avoid to break with us S minus S0 divided lambda, which is a boring expression, we can replace it by r star. It doesn't mean nothing for the moment, okay? For us, r star is any variable.
00:55:58.000 --> 00:56:21.000
So, we will learn r dot r star. Is it okay? And r star is this one. Okay, now, what is the difference in the phase of the two wave? The difference in the phase.
00:56:21.000 --> 00:56:49.000
For knowing the difference in the phase, we have to multiply this r star by r, pi to pi. And then we will know which is the difference between the phase of this wave and the phase of this wave. Is it okay?
00:56:49.000 --> 00:57:17.000
So, the difference between two phases, between these two phases, is this one. Okay? Let us, to be clear, for a less accurate way, when I sum two vectors, I take into consideration the angle between the two vectors and I sum it. I must take into consideration this.
00:57:17.000 --> 00:57:36.000
If these two vectors are exactly with the same direction, the sum, the vectorial sum is different from the normal sum. But when I say vector in different direction, I have to take this into consideration.
00:57:36.000 --> 00:57:58.000
So, when I sum two waves, I have to take into consideration the phase difference between them two waves. If that wave have the same phase, they can be summed without any problem. But if they don't have the same phase, I have to consider the difference between the two waves. Is it okay?
00:57:58.000 --> 00:58:24.000
So, the fact that the second wave involve more parts means that the difference in the phase is created from this wave to this wave. And when I try to calculate the sum of the two waves, I have to take care of this. Is it okay?
00:58:24.000 --> 00:58:53.000
So, now, let us calculate first the models of this R star. Because for us, it's very important that this model is here. R star, as you see, is S minus S0 divided by lambda. S0 is this. Here, I put in this screen S0 divided by lambda.
00:58:53.000 --> 00:59:14.000
S0 divided by lambda. S0 divided by lambda. And here, I put S divided by lambda. Okay? Both have the same magnitude because S0 and S are unit vectors. So, this triangle is isosome.
00:59:14.000 --> 00:59:43.000
So, S minus S0 divided by lambda is just this vector, as you know. Is the vector connecting the tip of this vector with the tip of this vector? So, S star is this. And now, how to calculate this? This is easy because this is perpendicular to this.
00:59:43.000 --> 01:00:01.000
So, it will be 2c theta of lambda. C theta of lambda, C theta of lambda, 2c theta of lambda. So, the model of R star is 2c theta of lambda. Please take care of this notation because this notation will be used always.
01:00:01.000 --> 01:00:23.000
So, R star vector is the difference between S minus S0 divided by lambda. S0 is the radiation. S is the radiation of the observed radiation. And lambda is the wavelength. Okay?
01:00:23.000 --> 01:00:49.000
S modulus is 2c theta of lambda. Okay? Now, we want to make the sum of the intensity of these two waves. What we have to do? We have to take the sum of these two waves, taking account of the difference in the wavelength.
01:00:49.000 --> 01:01:10.000
So, without problem, we can assume that the phase of this wave is 0. It doesn't matter. We can choose the origin where we want. It's 0. But the second has to be a shift of the phase, which will be a0 prime, a0 prime multiplied exponential to pi I R star over R. Okay?
01:01:10.000 --> 01:01:35.000
Which is the shift of the phase due to the larger pathway. Okay? Now, so this is for two. But for n, it is easy. We can generalize this by making F R star the sum over all the scatters.
01:01:35.000 --> 01:02:04.000
I will take this point because some phase says that I should repeat. Okay? So, up to here, is it clear?
01:02:04.000 --> 01:02:25.000
We have no. What is the a0? A0 is the amplitude of this wave here with the phase 0 because the zero phase can be, what is important is the difference between one wave and the other wave. So, I can use 0 for the first wave.
01:02:25.000 --> 01:02:48.000
But the second must be the phase, the shift of phase, must be to pi I R star R because this is the shift of the phase to pi R star R. So, in this way, F R star will be the amplitude of the wave scattered by both.
01:02:48.000 --> 01:03:16.000
Okay? F R star is the amplitude of the wave scattered by both. And if I have not only two electrons scattered, but a lot of electrons, I just have to introduce a sum over all the electrons and then
01:03:16.000 --> 01:03:40.000
that is to dissipate the phase shift. Okay? Be careful. R j gives the position of the direction with respect to our polygon system. Okay? So, the amplitude we observe is depend on the structure of the electrons.
01:03:40.000 --> 01:03:58.000
Depend on the position of the electrons. Is it okay? Is it okay? It's fine?
01:03:59.000 --> 01:04:21.000
The modulus of R star should only come out as some wavelength. The modulus of R star, this one, the two centigrade of an electron comes from here. So, is it okay? It's this one. It is two centigrade of an electron. It's okay?
01:04:22.000 --> 01:04:39.000
So, what we have is that when we consider the interference of many scattered waves, the effect is that the interference depends on the structure, on the position of the electrons.
01:04:40.000 --> 01:04:51.000
And vice versa, the server scattering provides us information about the scattered position. Is it okay?
01:04:51.000 --> 01:05:13.000
Okay, so this is our formula. Now, R j is the amplitude of the scattered wave. R j is defined by Thomson. The Thomson formula, remember?
01:05:13.000 --> 01:05:42.000
The Thomson formula? What's this one? This one, okay? R j was defined in terms of Thomson formula. Which depend? Why is it dependent on j? Why? Thomson formula? Thomson?
01:05:42.000 --> 01:05:54.000
Because we are considering one electron. It can be here very well. Thomson. Now, we are considering the interference of electrons. So, we need the knowledge and the positions.
01:05:54.000 --> 01:06:06.000
Even for the same electron, A j is the same. Sorry? For every electron, A j is not the same? For every electron, A j may be the same for every electron.
01:06:06.000 --> 01:06:14.000
But later on, we will consider scattered waves, which are not the same directions for every electron.
01:06:14.000 --> 01:06:28.000
Maybe, atoms. Maybe, your electrons may be at a low temperature, and then the electron is at a larger temperature, and it takes difference.
01:06:28.000 --> 01:06:40.000
So, for the moment, we start in a general way, and then we will see that it's better to take this general approach. Is it okay?
01:06:40.000 --> 01:07:07.000
So, we have, then, this expression here. All of you are convinced? Is it okay? And what is important is that we know that the scattering gives information about the structure, about the body, the scattering body. Is it okay?
01:07:08.000 --> 01:07:13.000
Now, be careful.
01:07:13.000 --> 01:07:41.000
Now, we are not speaking about diffraction. We are speaking about a set of electrons.
01:07:43.000 --> 01:07:59.000
Now, A j is defined in terms of what? Of mass of the electron, charge of the electron, photos, and so on, and so on. Two complicated things. Okay?
01:07:59.000 --> 01:08:20.000
We will try to introduce, here, we place A j by F j, which is the scattering factor in terms of Thomson electrons. Something like, one electron is maybe one, for example, in some cases.
01:08:20.000 --> 01:08:42.000
So, F j is the capacity of the charged particle to scatter. So, what we will do is to take F, like the ratio between E, F squared, like E divided by A in Thomson.
01:08:42.000 --> 01:09:07.000
Or, if you want, F is one amplitude divided by amplitude according to Thomson. What is that? If my scattering is an electron, I go here to calculate for this electron what is Thomson intensity.
01:09:07.000 --> 01:09:33.000
And, I go back, and I see that it is this one, for example. Is it okay? Then, I have my scattering, my scatter there, which is not an electron, but any other thing. And, I make the ratio between the amplitude scattering by my scatter with respect to Thomson.
01:09:33.000 --> 01:09:54.000
So, in this way, with respect to intensity scattered by one electron according to Thomson, in this way, the ratio is a number. Three times the scattering of one electron according to Thomson. Three points, six times the scattering according to Thomson, and so on.
01:09:54.000 --> 01:10:15.000
So, in this way, we cancel a lot of problems, intensity of the scattering, power of radiation, mass charge, all these things, because all is referred to the intensity scattered by one electron. Is it okay?
01:10:15.000 --> 01:10:38.000
That is important, because when you will receive the data, the factor data, what you will really read is a ratio between the amplitude you observe and the amplitude of the electron according to Thomson.
01:10:46.000 --> 01:10:48.000
There.
01:10:48.000 --> 01:11:13.000
This is our formula.
01:11:13.000 --> 01:11:36.000
So, how to, in a negative way, see a faster, how you have to put one detector in direction as, and look how many photons you receive, right? Is it okay or not? That is what you have to do.
01:11:36.000 --> 01:11:55.000
So, you have to put two electrons in an empty space, and interact, you will send a wavelength, and then in direction, you measure the number of photons you have.
01:11:55.000 --> 01:12:23.000
On what parameter, on which parameter will depend this intensity? That is, the number of photon, will depend on the intensity of the incident field, right? And therefore, you go back, depend on the intensity of the beam.
01:12:23.000 --> 01:12:45.000
Of L, E, electron charge squared, or the M squared, or the R squared, how far you are, because if you are far from the two electrons, you will receive a small intensity. And, on cost to theta, all these parameters, it is terrible to take into account every time all that.
01:12:45.000 --> 01:13:05.000
I received two million of electrons. What means for you, if someone gives you, the reflection, this reflection has two million of, one million of, doesn't mean nothing, right? So, it is not the best way of giving the information.
01:13:06.000 --> 01:13:30.000
So, what we do, really, is not to calculate the number of photon arriving to the detector, but to replace this, to divide the amplitude you observe by the amplitude which should measure if one electron scatter in the same condition, at the same to theta.
01:13:31.000 --> 01:13:56.000
And then, you will obtain, sorry, what is called F. This F is the amplitude you observe divided by the amplitude which you should observe by an electron scattering in the same direction, under the same condition. It's okay?
01:13:57.000 --> 01:14:16.000
So, that will be a number. A more easy number, because it is a ratio between the scattering of your system and the scattering of one electron in the same condition, according to something. It's okay? It's fine?
01:14:16.000 --> 01:14:40.000
Okay. So, I wrote F squared like I divided I, but you can replace it by F equal A divided A times one. It's okay? The amplitude divided by the amplitude. It's okay? Follow?
01:14:41.000 --> 01:15:01.000
Fine? We're going. Now, I ask you, our system, for the moment, is a discrete system, but a body, usually, is not a discrete. It's a continuous system of scattles.
01:15:02.000 --> 01:15:20.000
What we have to do if our system is not discrete, but a continuous system of scattering? We should replace this by an integral, right? The sum should be replaced by an integral.
01:15:21.000 --> 01:15:38.000
And Rj should be substituted by R, genetic R, on which we integrate, because the summation of all Rj will be replaced by R, on which we integrate the integral. It's okay?
01:15:39.000 --> 01:16:04.000
And so, in this condition, what will be there? We will have this condition. Why? Because, and this is, this is the fluid transform of the electron density. Why of the electron density?
01:16:04.000 --> 01:16:24.000
Because, in this system, in this system, we have Fj, which is the ratio between the amplitude scattered by our system and the electrons. So it's, how many electrons do we have there? Is it clear?
01:16:24.000 --> 01:16:53.000
So, we will have, we will have the fluid transform. So we see, now, that Fr star, the amplitude of the phase scattered by anybody, any scattered body, scattering body, is the fluid transform of the electrons.
01:16:54.000 --> 01:17:08.000
Now, all of you understand, because we used a full one-day for displaying a little bit of mathematics. Is it okay?
01:17:24.000 --> 01:17:50.000
We will call Fr like the structure factor, because we are going on the structure of this electron, right? The structure factor.
01:17:50.000 --> 01:18:08.000
And is the fluid transform of the electron. That is what we have shown this morning, right?
01:18:20.000 --> 01:18:48.000
I see our beautiful art notes of the ladies, and our awful art.
01:18:50.000 --> 01:19:18.000
So, take always a lady for comparing, because she is very useful.
01:19:20.000 --> 01:19:24.000
What is the question?
01:19:24.000 --> 01:19:26.000
This is a row.
01:19:26.000 --> 01:19:48.000
Row. Row is the electron density. For us, from now on, it will be always, you know, the electron density. It's okay? So, please, row R is the electron density. Is a function of R, the position of the space, in the space.
01:19:48.000 --> 01:20:07.000
And the exponential 2 pi R, R star R, is the typical fluid transform term. Let us try to understand better things.
01:20:08.000 --> 01:20:25.000
F R star. What means? Because otherwise, you will assume 6 without a little bit of reflection on there. F R star is a function of R star. What is R star?
01:20:26.000 --> 01:20:33.000
But as we defined, right? As we defined.
01:20:33.000 --> 01:20:54.000
F R star defines the direction we are looking for. Is it okay? So, F R star means that F depends on the direction of the observation. Because it depends on S minus S0 divided by lambda.
01:20:54.000 --> 01:21:09.000
S0 is fixed, because R, the beam, and the beam cannot be moved, otherwise you should move all the secrets from there. So, the beam is fixed, but the observation may change.
01:21:09.000 --> 01:21:28.000
Of course, you can move your observation. So, R star is a function of the observation. So, please remember, row R is defined in terms of R, the real space in which there are the electrons.
01:21:29.000 --> 01:21:45.000
F R star. It is F R star, not F R. R star is the space of the direction, in some way. It is because it is S minus S0 divided by lambda.
01:21:46.000 --> 01:22:07.000
So, F R star depends on the direction in which you are measuring. Is it okay? This, the modulus of this, pay attention, the modulus of this R star is 2 C theta by lambda. And what is C theta? Or if you want, 2 theta?
01:22:08.000 --> 01:22:27.000
2 theta is the angle between the observation and the direction of the incident wave. So, that means that F R star depends on 2 theta. If you are looking at the direction, this is the direction of Israel.
01:22:28.000 --> 01:22:41.000
If I look in the same direction, F R star will take a value. But if I change my direction and go up, it will change the F R star. Is it okay?
01:22:42.000 --> 01:23:01.000
And at the moment, please, don't forget that F R star does not change in a new form way. F R star is a vector. R star is a vector. So, maybe the same thing will observe here something and here something. That's a metaphor. Is it okay?
01:23:02.000 --> 01:23:13.000
Fine? Can one of you, I will ask you questions and they will answer you to check the police.
01:23:14.000 --> 01:23:18.000
Which is the name of F R star?
01:23:19.000 --> 01:23:24.000
Structure factor. Why it is called a structure factor?
01:23:25.000 --> 01:23:27.000
Because of the structure of the star.
01:23:28.000 --> 01:23:33.000
F R star. What is the mathematical definition of F R star?
01:23:33.000 --> 01:23:52.000
We are speaking now on crystal or any. Any for the moment. Crystal is not entered in our discussion, right? Is it okay? So, anybody.
01:23:53.000 --> 01:24:00.000
F R star is a function of?
01:24:01.000 --> 01:24:14.000
R star. R star, which is? S minus zero divided by R star. And what is that? F R star change with?
01:24:14.000 --> 01:24:26.000
The position of observation with respect to this. Okay? Changing that will change us. Is it okay? Fine?
01:24:27.000 --> 01:24:47.000
Are you involved in this phenomenon at the moment? Yes, always. Because I see you, because you are the scatters at this moment. You are the scatters.
01:24:48.000 --> 01:25:04.000
And your light, the light is scatters by your body and comes in my eye. So, what there is here, here is F R star. One angle with respect to you.
01:25:05.000 --> 01:25:13.000
Another angle with respect to her. Another angle with respect to her. So, all that makes the complex image I see. Is it okay?
01:25:13.000 --> 01:25:22.000
So, at this moment, we have to say F R star. Every one of us scatters. We can also say I am scattering.
01:25:22.000 --> 01:25:45.000
Okay? So, question the product. We are not involved with the crystal now, right? We are absolutely general. F R star is continuous or not?
01:25:46.000 --> 01:26:01.000
Yes. So, that is the reason why I see you continuously. Because I move. I move the angle. I see you because your scattering is continuous. Don't be too proud of that.
01:26:01.000 --> 01:26:28.000
Okay. So, now, we are in this condition and we learn about this. Now, why that is important? We are talking about it because I have access to the F R star.
01:26:28.000 --> 01:26:49.000
F R star, my eye may transform this fully transformed by inverse fully transformed by my crystalline in the eye and I can have rho. So, rho may be the inverse fully transformed of F R star. That is what we do continually.
01:26:49.000 --> 01:27:11.000
So, in this case, I have rho. Luckily, the light may be forged by our eyes and I can calculate the fully transformed. With each ray, that is not possible. Because each ray has a short wavelength.
01:27:11.000 --> 01:27:32.000
That means that each ray penetrates our crystalline and which cannot focus the radiation. Therefore, our process and no, and it is impossible to make an inverse fully transformed.
01:27:32.000 --> 01:27:46.000
That is the reason why we have problems with crystals. Because we are using, we are obliged to use crystals, but we use each ray and therefore we cannot make the inverse fully transformed.
01:27:46.000 --> 01:27:53.000
It's okay? It's fine for? Yes? Okay.
01:28:03.000 --> 01:28:28.000
So, now, we can consider better, in a better way, the scattering amplitude of an electron or one electron. What we have to do to consider that? Okay, we can make an incursion into the quantum mechanics.
01:28:29.000 --> 01:28:45.000
Which is the electron density of one electron? Remember that we are speaking of electron bonded to nuclei. Because we are in crystals, they have all the crystals.
01:28:45.000 --> 01:29:11.000
The electron density will be, as you know, the square of the wave, of the state wave function. So, will be C R squared. Can't perfectly know this C, so we perfectly know what is rho of the electrons.
01:29:11.000 --> 01:29:34.000
It's okay? So, if we want to calculate the scattering factor, the scattering factor of the electron, f is called, the small f will always call scattering factor of the electron.
01:29:34.000 --> 01:29:52.000
It will be the integral over cold space of the electron density, over the electron, by the exponential typical of the Fourier transform. It's okay?
01:30:05.000 --> 01:30:06.000
Is it fine?
01:30:07.000 --> 01:30:18.000
I didn't say the first rule, but this one.
01:30:18.000 --> 01:30:44.000
One, do you remember the kind of fundamental theory? He says that every electron around the nucleus stay in a stable state, and this state will be described by orbital. So, you have R, S, P, D, R, E as well.
01:30:44.000 --> 01:31:00.000
So, one electron may be described, therefore, as in this state. And the probability, when the electron moves in its orbital, it will have a distribution of intensity in the space.
01:31:00.000 --> 01:31:14.000
And the density, electron density, along this orbital, is C R squared, which is the state function for that orbital, for that time of the electrons.
01:31:14.000 --> 01:31:33.000
And then, the scattering factor of the electrons will be always functional over R star, of course, like for all, and will be the Fourier transform of this electron density. It's okay?
01:31:33.000 --> 01:31:52.000
Now, to make an example, suppose that you are in the carbon state, the carbon atom, and we have two electron of type 1S, two electron of type 2S, two electron of type P. Is it okay?
01:31:52.000 --> 01:32:14.000
If we use an asperical approximation, the state 1S has this distribution. This is, let me say, the center of the nucleus. At zero, this is the center of the nucleus.
01:32:14.000 --> 01:32:31.000
So, this electron, the 1S, is close to the nucleus, and we learn this distribution. So, it is very close to the nucleus. 2S is more of this person, around the nucleus.
01:32:31.000 --> 01:32:45.000
And this will be 4 pi R squared, where it will become distribution function in terms of the radium. So, these are the two distribution.
01:32:46.000 --> 01:33:06.000
When we calculate the, this distribution may be represented by this Slater formula, which is a formula which, more or less, gives this distribution.
01:33:06.000 --> 01:33:30.000
And the low E, 2S, by this Slater formula, for the 2S distribution. Is it okay? So, if we calculate the Fourier transform of this, it's not too complicated, because our symbol of exponential is more or less what we did yesterday.
01:33:31.000 --> 01:33:57.000
We have this, 1S and 2S. We could expect that, right? We could expect that. Why? Suppose these are close to the bell form, to a bell form.
01:33:58.000 --> 01:34:07.000
This is a Gaussian, this is a Gaussian. What do you expect when you calculate the Fourier transform? The reverse, yes?
01:34:07.000 --> 01:34:11.000
Yeah, that a narrow Gaussian becomes a very wide one, and that a...
01:34:11.000 --> 01:34:12.000
That's right, it's lower.
01:34:12.000 --> 01:34:18.000
That a narrow Gaussian becomes a wide one Fourier transform, and vice versa.
01:34:18.000 --> 01:34:28.000
Yes? You have the sharp distribution becomes larger, and the larger distribution becomes sharp. Does that do to you, sir?
01:34:28.000 --> 01:34:29.000
Yeah.
01:34:29.000 --> 01:34:47.000
So, what you see is that 1S will become, which is sharper here, will become flat here, and 2S, which is flat here, becomes sharp here. Is it okay?
01:34:47.000 --> 01:34:55.000
But in this f, e, there is a complex function right now.
01:34:55.000 --> 01:35:11.000
In this f, f, yes, we are speaking about this function of centrosymmetric. So, when you consider this, this appear.
01:35:12.000 --> 01:35:18.000
So, at the end, you have this expression. Is it okay?
01:35:21.000 --> 01:35:27.000
Now, we start from 1. Why we start from 1?
01:35:28.000 --> 01:35:45.000
Why this scattering angle is... because, sorry, here, I forgot to encourage you to answer.
01:35:45.000 --> 01:35:52.000
I observe that this is distribution of the electrons in the real space, so we have R.
01:35:52.000 --> 01:36:03.000
This is the distribution in function of R star. Do you remember R star? What is 2C theta or lambda as a model?
01:36:03.000 --> 01:36:16.000
So, this is in function of 2C theta or lambda. So, that means that when C theta is close to 0, it is when theta is close to 0, we have the maximum.
01:36:17.000 --> 01:36:25.000
And when we increase theta, the intensity decreases. Is it okay?
01:36:26.000 --> 01:36:47.000
So, physically, what means that if I have a beam comes in this direction, and I observe in the same direction of the incident beam, the radiation, I have the maximum.
01:36:48.000 --> 01:36:56.000
But, when I move from this in the radiation, in this way, the intensity will decrease. Is it okay?
01:36:56.000 --> 01:37:01.000
In this way or in this way or in this way? Is it okay?
01:37:01.000 --> 01:37:03.000
What is the negative pressure?
01:37:03.000 --> 01:37:10.000
Oh, it is just the free transform of an approximation. Don't worry about it.
01:37:11.000 --> 01:37:14.000
This is an approximation of the force.
01:37:15.000 --> 01:37:21.000
Okay? It is okay then? So, be careful.
01:37:21.000 --> 01:37:40.000
When this one says a very important thing, you will suffer a lot from this when you take the...
01:37:41.000 --> 01:37:53.000
And the region stays here. So, be careful about this in the south because that will link your theta.
01:37:53.000 --> 01:38:01.000
And quite often, your boss will be unsatisfied for your force because of this.
01:38:01.000 --> 01:38:13.000
So, try to understand what this means. These are the distributions of the electron in real space.
01:38:13.000 --> 01:38:24.000
One s is close to the energy. Two s is dispersed. Is it okay?
01:38:24.000 --> 01:38:31.000
Now, we look at the scattering of these two electrons.
01:38:31.000 --> 01:38:39.000
I still have a question. So, we have a structure as scattering factors for all the electrons of one atom?
01:38:39.000 --> 01:38:44.000
This is for one electron, third electron or third electron?
01:38:44.000 --> 01:38:53.000
Okay, because before we said that this scattering factor is the amplitude we observed divided by the amplitude.
01:38:53.000 --> 01:38:56.000
Right, right. Thank you.
01:38:56.000 --> 01:39:13.000
I didn't put in any of this. When we are speaking about electrons, I said that rho e is exactly the square amplitude of the state function, right?
01:39:13.000 --> 01:39:23.000
The state function defines an orbital. This orbital, each point of this orbital, scatters.
01:39:23.000 --> 01:39:31.000
So, the electron density is the electron density for the fluid orbital. Is it okay?
01:39:31.000 --> 01:39:37.000
And each point of this orbital scatters.
01:39:37.000 --> 01:39:41.000
So, the orbital from one electron and at each point of this orbital?
01:39:41.000 --> 01:39:48.000
Yeah, because you can imagine that there are millions and millions of electrons.
01:39:48.000 --> 01:40:01.000
So, each electron, it is the scattering may occur from each point of the orbital.
01:40:01.000 --> 01:40:10.000
So, you have to consider this effect when you consider the scattering of one electron. Is it okay?
01:40:10.000 --> 01:40:18.000
Because each point has an electron density rho different from the zero of the orbital. Is it okay?
01:40:18.000 --> 01:40:28.000
I have another question. I didn't understand why the scattering factor is maximum equal to one.
01:40:28.000 --> 01:40:31.000
I will.
01:40:31.000 --> 01:40:34.000
Now, is it okay then?
01:40:34.000 --> 01:40:42.000
So, we are applying the electron fluid transform to the distribution c of the orbital.
01:40:42.000 --> 01:40:55.000
And then we will take this term. Remember, f is f. This f is f in terms of Thomson scattering.
01:40:55.000 --> 01:41:00.000
Is it okay? Of Thomson scattering.
01:41:00.000 --> 01:41:06.000
And therefore, what we will see are not number of photon which arrive and settle.
01:41:06.000 --> 01:41:18.000
But the ratio between the amplitude of the radiation scattered by the orbital with respect to the Thomson electron scattering.
01:41:18.000 --> 01:41:20.000
Is it okay? It was the definition.
01:41:20.000 --> 01:41:46.000
So, and therefore, we will start from one and then decrease. Now, look at this.
01:41:46.000 --> 01:42:00.000
This is the orbital. It doesn't matter if it's true or not. And then the radiation comes here.
01:42:00.000 --> 01:42:07.000
And we start to consider the intensity into the incident radiation.
01:42:07.000 --> 01:42:24.000
So, we look at. I want to make a beautiful picture. We look at this.
01:42:24.000 --> 01:42:34.000
Do you think there is some difference in the pathway from one angle from one way and another way?
01:42:34.000 --> 01:42:51.000
No. So, it is the max. So, every wave entering in contact with the electron density has no change in the phase with respect to the other.
01:42:51.000 --> 01:43:06.000
So, there is no decrease of intensity in this case. And therefore, the intensity scattered is exactly the same as the electron was there and scattered like a point.
01:43:06.000 --> 01:43:17.000
Is it okay? Because the electron is concentrated here or it is distributed here. It's the same. There is no difference. So, we have one.
01:43:17.000 --> 01:43:41.000
Okay? When we go in this direction, it is obvious that between this and this start to be difference in the pathway between one wave and another wave.
01:43:41.000 --> 01:43:57.000
And therefore, the intensity will decrease. The larger is 2 theta. Remember, this is 2 theta. The larger is 2 theta. The larger will be the path, the difference in the path.
01:43:57.000 --> 01:44:26.000
And therefore, the decrease. Is it okay? Is it fine? So, we now understand that here there is one because we always consider the ratio between the scattering amplitude of the system and the electron amplitude according to toxic.
01:44:26.000 --> 01:44:48.000
Right? Now, we are considering one electron and when we consider this radiation here, the diffuse radiation here, we have one. When we go out with the 2 theta angle, then it will decrease.
01:44:48.000 --> 01:44:56.000
It's okay? That is universal. So, you always will face this problem. Is it clear?
01:44:56.000 --> 01:45:02.000
That's because our star is 0 when 2 theta is 0.
01:45:02.000 --> 01:45:15.000
Our star is 0 when 2 theta is 0? Yes? Yes. It's okay? It's fine? Okay. Now, come to the conclusion.
01:45:16.000 --> 01:45:36.000
Which is your preferred scatter? Is 1s or 2s? Which is your preferred scatter? 1s or 2s? 2s? 1s.
01:45:37.000 --> 01:45:47.000
What do you prefer? 1s or 2s like a scatter? 1s.
01:45:47.000 --> 01:45:59.000
Look at the other. An opinion. 1s or 2s? Which scatter do you prefer?
01:46:07.000 --> 01:46:18.000
Think about it. The best scatter is that one providing information at any angle. Right?
01:46:18.000 --> 01:46:35.000
So, the best scatter is 1s because it provides information at more or less any angle of scattering. Okay?
01:46:35.000 --> 01:46:45.000
On the contrary, 2s stops with respect to 1s. So, that's providing information. Okay?
01:46:45.000 --> 01:46:57.000
So, that means that when your incident wave comes in this direction, with 2s you can collect only the linear set of data.
01:46:58.000 --> 01:47:18.000
After this, 2 theta here, not later. 1s slightly gives information more. Is it okay? 5-4? 5-6? Is it clear or bad? No? No.
01:47:18.000 --> 01:47:38.000
Any questions? No question. So, we know now, just to repeat the idea, that when an electron scatter, we will call each electron's form.
01:47:38.000 --> 01:47:48.000
The electron's form for the electron density line scattering factor. Okay? Electron scattering factor. Small f.
01:47:48.000 --> 01:48:05.000
The electron scattering factor is the
01:48:06.000 --> 01:48:24.000
2 curves there, 2 state function, and we have 2 different scattering factors. All the scattering function we will see are a function of r star.
01:48:25.000 --> 01:48:42.000
That is, 2c theta 1, always. Even in literature, when you will go to the national table to check the scattering factors, as always, you will see a function of r star, 2c theta or lambda.
01:48:43.000 --> 01:48:58.000
It's clear, so be careful taking your memory. r star is equal to the lambda. So, just to check, what is r star? Between?
01:48:58.000 --> 01:49:10.000
r star, plus, minus, minus, minus, zero, divided by. What is this modulus? Or lambda? The scattering factor is given here as a function of? r star.
01:49:10.000 --> 01:49:21.000
r star. Okay? And usually, it's r star like modulus because the system is considered spherical, but it's region wide.
01:49:21.000 --> 01:49:34.000
Okay, so the best scatters are those close to the nucleus that we receive for one vector,
01:49:34.000 --> 01:49:39.000
but of course you can see for a larger system.
01:49:39.000 --> 01:49:42.000
So the best scatters are close to the nucleus.
01:49:42.000 --> 01:49:47.000
The worst scatters are called the valence vectors.
01:49:47.000 --> 01:49:48.000
Is it okay?
01:49:48.000 --> 01:49:50.000
Is it fine for that?
01:49:53.000 --> 01:49:56.000
So, see you enough now.
01:50:18.000 --> 01:50:20.000
No, it's okay.
01:50:20.000 --> 01:50:21.000
It's okay.
01:50:21.000 --> 01:50:22.000
We have one more.
01:50:22.000 --> 01:50:23.000
One more.
01:50:23.000 --> 01:50:24.000
One more.
01:50:24.000 --> 01:50:25.000
One more.
01:50:25.000 --> 01:50:26.000
One more.
01:50:26.000 --> 01:50:27.000
One more.
01:50:27.000 --> 01:50:28.000
One more.
01:50:28.000 --> 01:50:29.000
One more.
01:50:29.000 --> 01:50:30.000
One more.
01:50:30.000 --> 01:50:31.000
One more.
01:50:31.000 --> 01:50:32.000
One more.
01:50:32.000 --> 01:50:33.000
One more.
01:50:33.000 --> 01:50:34.000
One more.
01:50:34.000 --> 01:50:35.000
One more.
01:50:35.000 --> 01:50:36.000
One more.
01:50:36.000 --> 01:50:37.000
One more.
01:50:37.000 --> 01:50:38.000
One more.
01:50:38.000 --> 01:50:39.000
One more.
01:50:39.000 --> 01:50:40.000
One more.
01:50:40.000 --> 01:50:41.000
One more.
01:50:41.000 --> 01:50:42.000
One more.
01:50:42.000 --> 01:50:43.000
One more.
01:50:43.000 --> 01:50:44.000
One more.
01:50:44.000 --> 01:50:45.000
One more.
01:50:45.000 --> 01:50:46.000
One more.
01:50:46.000 --> 01:50:47.000
One more.
01:50:48.000 --> 01:50:49.000
One more.
01:50:49.000 --> 01:50:50.000
One more.
01:50:50.000 --> 01:50:58.000
One more.
01:50:58.000 --> 01:51:09.000
One more.
01:51:09.000 --> 01:51:16.000
One more.
01:51:16.000 --> 01:51:28.000
This is completely different than the scale of planes for each vehicle.
01:51:35.000 --> 01:51:39.000
Like the front of the crystal, the back of the crystal, they're urging on the spot.
01:51:39.000 --> 01:51:42.000
And we've got some small size of these and size of the front of the crystal.
01:51:42.000 --> 01:51:44.000
The size of the might be increased.
01:51:44.000 --> 01:51:46.000
Everything is the same.
01:51:46.000 --> 01:51:48.000
Everything is the same.
01:51:48.000 --> 01:51:50.000
Everything is the same.
01:51:50.000 --> 01:51:52.000
Everything is the same.
01:51:52.000 --> 01:51:54.000
Everything is the same.
01:51:54.000 --> 01:51:56.000
Everything is the same.
01:51:56.000 --> 01:51:58.000
Everything is the same.
01:51:58.000 --> 01:52:00.000
Sure, sure.
01:52:00.000 --> 01:52:02.000
So I've got these planes, right?
01:52:02.000 --> 01:52:04.000
My light comes in here.
01:52:04.000 --> 01:52:08.000
And it's towards the front of the crystal.
01:52:09.000 --> 01:52:16.000
If the back of the crystal comes up here, it comes up here, right?
01:52:16.000 --> 01:52:18.000
Okay, can we start?
01:52:21.000 --> 01:52:30.000
One other question you can ask me is, okay, we understand that there are some,
01:52:30.000 --> 01:52:39.000
that we have to use the classical theory of the scattering concept.
01:52:39.000 --> 01:52:47.000
But we see also that this scattering of the electron decreases.
01:52:47.000 --> 01:52:51.000
So there are some loops of energy.
01:52:51.000 --> 01:52:59.000
And this loops of energy is, let me say, is incoherent energy,
01:52:59.000 --> 01:53:02.000
according to the compound.
01:53:02.000 --> 01:53:04.000
So it is present.
01:53:04.000 --> 01:53:09.000
But we don't care of it usually because it's not too big.
01:53:09.000 --> 01:53:10.000
It's okay?
01:53:10.000 --> 01:53:16.000
So now, so that was here, but it doesn't matter.
01:53:16.000 --> 01:53:23.000
Now, we go to a little bit more complicated system, right?
01:53:23.000 --> 01:53:26.000
Which is the atomic scattering factor.
01:53:26.000 --> 01:53:30.000
So we consider a more complicated system, the atom.
01:53:30.000 --> 01:53:32.000
Is it okay?
01:53:32.000 --> 01:53:35.000
Fine?
01:53:35.000 --> 01:53:40.000
Now, what is, the first question is always the same.
01:53:40.000 --> 01:53:46.000
What is the electron density of the atom, like we did before?
01:53:46.000 --> 01:53:49.000
What is the electron density of the electron?
01:53:49.000 --> 01:53:52.000
And now, what is the electron density of the atom?
01:53:52.000 --> 01:54:00.000
Of course, it's the sum of the electron density of the single electrons.
01:54:00.000 --> 01:54:08.000
And therefore, it's the sum of the Cj squared, which are the state function for each electron.
01:54:08.000 --> 01:54:12.000
Is it okay?
01:54:12.000 --> 01:54:17.000
We are lucky because the chemistry didn't work for us.
01:54:17.000 --> 01:54:24.000
So we know, we know in a perfect way, in a really perfect way,
01:54:24.000 --> 01:54:28.000
which are the state function of each electron.
01:54:28.000 --> 01:54:39.000
And therefore, we can have a clear vision of what is the electron density of the atom.
01:54:39.000 --> 01:54:41.000
Is it okay?
01:54:41.000 --> 01:54:45.000
Fine?
01:54:45.000 --> 01:54:50.000
What would be the scattering, the atomic scattering factor,
01:54:50.000 --> 01:54:56.000
would be the Fourier transform of the electron density.
01:54:56.000 --> 01:54:59.000
Is it okay?
01:54:59.000 --> 01:55:04.000
Well, the Fourier transform of the electron density
01:55:04.000 --> 01:55:12.000
would be the sum of the Fourier transform of the electron density.
01:55:12.000 --> 01:55:15.000
So that is clear.
01:55:15.000 --> 01:55:21.000
We have only to make the Fourier transform of this sum.
01:55:21.000 --> 01:55:25.000
So it is known, and that is done.
01:55:25.000 --> 01:55:30.000
And usually, you will see into the international table,
01:55:30.000 --> 01:55:37.000
you will see the scattering, atomic scattering factor for each atom.
01:55:37.000 --> 01:55:39.000
Is it okay?
01:55:39.000 --> 01:55:41.000
Fine?
01:55:41.000 --> 01:55:44.000
Calculated for each of them.
01:55:44.000 --> 01:55:55.000
Now, let us see the picture describing the atomic scattering factor.
01:55:55.000 --> 01:56:03.000
We have always to put this atomic scattering factor in function of C theta lambda.
01:56:03.000 --> 01:56:07.000
Always, C theta lambda, C theta lambda doesn't matter.
01:56:07.000 --> 01:56:08.000
Is it okay?
01:56:08.000 --> 01:56:14.000
Because that is the parameter on which we should calculate.
01:56:14.000 --> 01:56:16.000
Fine.
01:56:16.000 --> 01:56:20.000
For something, we start from 16.
01:56:20.000 --> 01:56:24.000
All of you understand because we start from 16?
01:56:24.000 --> 01:56:26.000
Why?
01:56:26.000 --> 01:56:32.000
Because in that direction, there is no difference in the pathway.
01:56:32.000 --> 01:56:37.000
So at zero C theta lambda, that would be 16.
01:56:37.000 --> 01:56:43.000
And it would be for Na plus, it would be 10.
01:56:43.000 --> 01:56:47.000
For Na would be 11.
01:56:47.000 --> 01:56:49.000
For oxygen would be 8.
01:56:49.000 --> 01:56:52.000
For carbon would be 6.
01:56:52.000 --> 01:56:53.000
Is it okay?
01:56:53.000 --> 01:56:55.000
We will start from 6.
01:56:55.000 --> 01:57:02.000
And then, they will go down, unfortunately.
01:57:02.000 --> 01:57:03.000
It will be for now.
01:57:03.000 --> 01:57:05.000
Is it okay?
01:57:05.000 --> 01:57:09.000
The form, more or less, is similar.
01:57:09.000 --> 01:57:11.000
The form, more or less, is similar.
01:57:11.000 --> 01:57:15.000
All goes down, more or less, in the same way.
01:57:15.000 --> 01:57:20.000
Is it okay?
01:57:20.000 --> 01:57:26.000
It is evident that if we are some idiot,
01:57:26.000 --> 01:57:29.000
this scattering would be larger.
01:57:29.000 --> 01:57:35.000
Both at small theta and at larger theta.
01:57:35.000 --> 01:57:42.000
So let me say, the heavy atom are more visible than the light atom.
01:57:42.000 --> 01:57:44.000
Is it okay?
01:57:44.000 --> 01:57:52.000
Fine for now.
01:57:52.000 --> 01:57:57.000
And this may be a problem.
01:57:57.000 --> 01:58:00.000
The first problem was already discussed,
01:58:00.000 --> 01:58:06.000
because it was the decay of the scattering factor with theta.
01:58:06.000 --> 01:58:09.000
If the decay is very strong,
01:58:09.000 --> 01:58:14.000
in this position, you will see very small intensity.
01:58:14.000 --> 01:58:18.000
They will contribute in a very small way.
01:58:18.000 --> 01:58:22.000
So that theta, the contribution of that theta
01:58:22.000 --> 01:58:26.000
will be not visible after a given c theta.
01:58:26.000 --> 01:58:28.000
Is it okay?
01:58:28.000 --> 01:58:30.000
Or lambda?
01:58:30.000 --> 01:58:32.000
No?
01:58:32.000 --> 01:58:38.000
So if we have the, this is the direction of the incident radiation,
01:58:38.000 --> 01:58:42.000
we can go up with c theta or lambda in this way.
01:58:42.000 --> 01:58:46.000
But at a certain moment, over to that decay,
01:58:46.000 --> 01:58:50.000
we will not measure nothing.
01:58:50.000 --> 01:58:55.000
Because any atom does not contribute anymore.
01:58:55.000 --> 01:58:56.000
Is it okay?
01:58:56.000 --> 01:59:06.000
So we have a limited c theta lambda in which we can measure theta.
01:59:06.000 --> 01:59:09.000
Is it clear?
01:59:09.000 --> 01:59:12.000
Fine?
01:59:12.000 --> 01:59:13.000
Okay.
01:59:13.000 --> 01:59:24.000
The second point is we will more easily skip the light atom.
01:59:24.000 --> 01:59:28.000
The thing is, the light atom, because of that curve,
01:59:28.000 --> 01:59:38.000
will contribute to the total scattering in a very weak way.
01:59:38.000 --> 01:59:43.000
So maybe we will not see them into electron density,
01:59:43.000 --> 01:59:46.000
because they don't contribute to the f.
01:59:46.000 --> 01:59:48.000
Is it okay?
01:59:48.000 --> 01:59:57.000
If a scattering atom does not contribute to the f in a substantial way,
01:59:57.000 --> 02:00:01.000
you easily will escape it, of course.
02:00:01.000 --> 02:00:04.000
If you have some light inside,
02:00:05.000 --> 02:00:12.000
and this lamp is, and there are some hydrogens close to the lamp,
02:00:12.000 --> 02:00:19.000
it is lost time to try to see the hydrogen there.
02:00:19.000 --> 02:00:20.000
Is it okay?
02:00:20.000 --> 02:00:22.000
Is it fine?
02:00:22.000 --> 02:00:29.000
So let me say, in some way,
02:00:29.000 --> 02:00:35.000
the scattering power of the atoms
02:00:35.000 --> 02:00:43.000
is proportional to the atomic number.
02:00:43.000 --> 02:00:52.000
At any theta, because the H curve has all the curve has similar models.
02:00:52.000 --> 02:01:05.000
So the scattering power of one atom is proportional to its atomic number.
02:01:05.000 --> 02:01:09.000
That is the view for each ray.
02:01:09.000 --> 02:01:12.000
But to enlarge your point of view,
02:01:12.000 --> 02:01:18.000
I remember that it is not so for electrons.
02:01:19.000 --> 02:01:30.000
Electrons are more sensible to the, less sensible to the atomic number,
02:01:30.000 --> 02:01:35.000
because they depend on the electric field,
02:01:35.000 --> 02:01:41.000
and the electric field has a positive and negative charge inside,
02:01:41.000 --> 02:01:47.000
which in some way destroys each other.
02:01:47.000 --> 02:01:54.000
So less sensitive to the atomic number.
02:01:54.000 --> 02:02:01.000
So it is much more easy to find hydrogen atoms and lithium atoms
02:02:01.000 --> 02:02:05.000
by electrons that by each ray.
02:02:05.000 --> 02:02:06.000
Is it okay?
02:02:09.000 --> 02:02:12.000
Neutrons, now come to the neutrons.
02:02:12.000 --> 02:02:18.000
Neutrons interact not with the electrical field,
02:02:18.000 --> 02:02:21.000
but with the neutral particle.
02:02:21.000 --> 02:02:26.000
So they interact with the nucleus.
02:02:26.000 --> 02:02:29.000
A neutron interacts only with the nucleus,
02:02:29.000 --> 02:02:33.000
does not see electrons.
02:02:33.000 --> 02:02:36.000
They do not exist for a neutron.
02:02:36.000 --> 02:02:38.000
They interact with the nucleus.
02:02:39.000 --> 02:02:42.000
When they interact with the nucleus,
02:02:42.000 --> 02:02:47.000
for a micro, micro, micro, micro, micro, microsecond,
02:02:47.000 --> 02:02:53.000
they make a new nucleus, and then it is emitted again.
02:02:53.000 --> 02:02:56.000
So they interact with the nucleus.
02:02:56.000 --> 02:02:59.000
So that means that we see the nucleus.
02:02:59.000 --> 02:03:01.000
We do not see the electrons.
02:03:01.000 --> 02:03:05.000
And the scattering power of the neutron
02:03:05.000 --> 02:03:10.000
depends on the nuclear force,
02:03:10.000 --> 02:03:15.000
and it may be negative.
02:03:15.000 --> 02:03:19.000
So we can have also negative scattering factor there,
02:03:19.000 --> 02:03:25.000
because they are, let me say, emitted after pi,
02:03:25.000 --> 02:03:33.000
after some time, as a phase shift of pi.
02:03:33.000 --> 02:03:36.000
So in some way, you see, for example,
02:03:36.000 --> 02:03:45.000
a very hydrogen is a big scattering factor,
02:03:45.000 --> 02:03:52.000
but negative, and also is not coherent quite often.
02:03:52.000 --> 02:03:58.000
So what they do is to replace the hydrogen with the eutarium,
02:03:58.000 --> 02:04:02.000
and the eutarium is a very big scattering.
02:04:02.000 --> 02:04:06.000
And therefore, you can find the hydrogen by a neutron
02:04:06.000 --> 02:04:10.000
in a careful way, because in one of the biggest categories.
02:04:10.000 --> 02:04:15.000
So when you are entrusted to hydrogen positions,
02:04:15.000 --> 02:04:21.000
you have not to use its ray, but you have to use neutrons.
02:04:21.000 --> 02:04:24.000
And the neutrons define the position of the nucleus,
02:04:24.000 --> 02:04:26.000
not of the electrons.
02:04:26.000 --> 02:04:30.000
So a little bit shifted each other.
02:04:31.000 --> 02:04:36.000
So any radiation has its own advantage
02:04:36.000 --> 02:04:40.000
and its own disadvantage, is it okay?
02:04:40.000 --> 02:04:47.000
So when you want to study some material,
02:04:47.000 --> 02:04:49.000
you have to choose the correct radiation
02:04:49.000 --> 02:04:52.000
according to what you want to do.
02:04:52.000 --> 02:04:55.000
Is it okay?
02:04:55.000 --> 02:04:59.000
Now, yes?
02:04:59.000 --> 02:05:02.000
Yes, you said that nuclear neutron scattering
02:05:02.000 --> 02:05:04.000
is based on nuclear forces?
02:05:04.000 --> 02:05:06.000
It's based on that.
02:05:06.000 --> 02:05:11.000
The nucleus, the neutron interacts with the nucleus.
02:05:11.000 --> 02:05:16.000
It constructs, let me say,
02:05:16.000 --> 02:05:24.000
a very short-time leading modified nucleus.
02:05:24.000 --> 02:05:27.000
So my question is because the nucleus
02:05:27.000 --> 02:05:29.000
is a very small comparison of all atoms.
02:05:29.000 --> 02:05:33.000
So that means that scattering would be like almost known.
02:05:33.000 --> 02:05:36.000
Yeah, that is. Thank you for this.
02:05:36.000 --> 02:05:40.000
I forgot the one thing to say. I forgot this.
02:05:40.000 --> 02:05:48.000
Your colleague said, really, that the nucleus is very small.
02:05:48.000 --> 02:05:50.000
What means that?
02:05:50.000 --> 02:05:55.000
When I described this interaction,
02:05:55.000 --> 02:05:58.000
when I described this interaction for the electrons,
02:05:58.000 --> 02:06:03.000
which has an orbital,
02:06:03.000 --> 02:06:07.000
the orbital is much bigger than the nucleus, of course.
02:06:07.000 --> 02:06:16.000
So what we observe is the decay of the scattering intensity
02:06:16.000 --> 02:06:19.000
with the computer, right?
02:06:19.000 --> 02:06:22.000
But the nucleus is more.
02:06:22.000 --> 02:06:26.000
So this effect is not present.
02:06:26.000 --> 02:06:32.000
It is like one very small electron there.
02:06:32.000 --> 02:06:34.000
So we are not decay.
02:06:34.000 --> 02:06:36.000
And therefore, for neutrons,
02:06:36.000 --> 02:06:40.000
the decay is constant with the theta.
02:06:40.000 --> 02:06:48.000
That is the most important feature of the neutrons.
02:06:48.000 --> 02:06:50.000
The decay is constant with theta.
02:06:50.000 --> 02:06:54.000
Indeed, if you go into the international table,
02:06:54.000 --> 02:06:59.000
you will see, for example,
02:06:59.000 --> 02:07:02.000
the eutarium 2.5.
02:07:02.000 --> 02:07:05.000
Enough. No more.
02:07:05.000 --> 02:07:09.000
And because it is constant with theta,
02:07:09.000 --> 02:07:13.000
while if you go into the international table
02:07:13.000 --> 02:07:17.000
to check the scattering factor of the x-ray,
02:07:17.000 --> 02:07:20.000
you will see curves.
02:07:20.000 --> 02:07:22.000
Is it okay?
02:07:22.000 --> 02:07:24.000
Is it fine?
02:07:24.000 --> 02:07:30.000
Other observations?
02:07:30.000 --> 02:07:31.000
Okay.
02:07:31.000 --> 02:07:41.000
So...
02:07:41.000 --> 02:07:50.000
Now, we have to discuss another interesting point.
02:07:50.000 --> 02:07:56.000
Canists found the electron density,
02:07:56.000 --> 02:08:02.000
like the sum of the electron density of the electrons,
02:08:02.000 --> 02:08:05.000
which comes into the atom.
02:08:05.000 --> 02:08:12.000
By considering that the atom is stable,
02:08:12.000 --> 02:08:16.000
the atom is stable.
02:08:16.000 --> 02:08:22.000
And they gave the state function for each electron
02:08:22.000 --> 02:08:28.000
belonging to the atom for a steady atom.
02:08:28.000 --> 02:08:33.000
Let me say at the time of zero curve,
02:08:33.000 --> 02:08:40.000
it was the 10th day.
02:08:40.000 --> 02:08:46.000
Unfortunately, when we work with the crystals,
02:08:46.000 --> 02:08:51.000
we are unable to work at zero curve.
02:08:51.000 --> 02:08:57.000
And we are obliged to work at room temperature
02:08:57.000 --> 02:09:00.000
or more frequently at a temperature
02:09:00.000 --> 02:09:03.000
which can be reached in some way.
02:09:03.000 --> 02:09:05.000
Is it okay?
02:09:05.000 --> 02:09:12.000
So, that means a big problem, we will see,
02:09:12.000 --> 02:09:14.000
for the scattering.
02:09:14.000 --> 02:09:23.000
So, what I will show you is how big is this problem
02:09:23.000 --> 02:09:51.000
for the scattering of history.
02:09:51.000 --> 02:09:53.000
Sometimes this is our reference system,
02:09:53.000 --> 02:09:55.000
Cartesian reference system,
02:09:55.000 --> 02:09:58.000
because we are not speaking about crystal so far, right?
02:09:58.000 --> 02:10:01.000
About the Cartesian reference system.
02:10:01.000 --> 02:10:09.000
And we have here one electron,
02:10:09.000 --> 02:10:14.000
where the electron is going around.
02:10:14.000 --> 02:10:19.000
Is it okay?
02:10:19.000 --> 02:10:29.000
This is rho R,
02:10:29.000 --> 02:10:37.000
rho R because the nucleus is on the origin of our system
02:10:37.000 --> 02:10:44.000
and rho R describing the state of the electrons.
02:10:44.000 --> 02:10:47.000
Is it okay?
02:10:47.000 --> 02:10:58.000
So, suppose now that this atom receives some inputs
02:10:58.000 --> 02:11:00.000
from the temperature.
02:11:00.000 --> 02:11:08.000
So, some photon goes interact with him.
02:11:08.000 --> 02:11:16.000
And so, what they will do for how the atoms can answer this.
02:11:16.000 --> 02:11:24.000
Remember that the atoms find the equilibrium position in a material,
02:11:24.000 --> 02:11:29.000
because they stay, they are bonded with atoms
02:11:29.000 --> 02:11:34.000
and their position corresponds to a minimum of energy, always.
02:11:34.000 --> 02:11:37.000
Otherwise, they should move that position.
02:11:37.000 --> 02:11:42.000
So, any atom is in a stable position.
02:11:43.000 --> 02:11:52.000
And when it receives strong inputs from photons,
02:11:52.000 --> 02:11:54.000
because the temperature is high,
02:11:54.000 --> 02:11:58.000
and therefore they receive something, what it has to do?
02:11:58.000 --> 02:12:02.000
It is, energetically, it is a minimum.
02:12:02.000 --> 02:12:06.000
It will oscillate around this minimum, okay?
02:12:06.000 --> 02:12:08.000
Is it fine?
02:12:08.000 --> 02:12:12.000
It will become an oscillator.
02:12:12.000 --> 02:12:16.000
And according to the inputs they arrive,
02:12:16.000 --> 02:12:21.000
it will oscillate in all the directions, like in a true.
02:12:21.000 --> 02:12:24.000
It is true, the English word?
02:12:24.000 --> 02:12:28.000
The truth.
02:12:28.000 --> 02:12:31.000
The truth.
02:12:31.000 --> 02:12:34.000
The truth.
02:12:34.000 --> 02:12:36.000
The truth.
02:12:36.000 --> 02:12:39.000
The truth, in a truth.
02:12:39.000 --> 02:12:45.000
So, they will oscillate in a truth, okay?
02:12:45.000 --> 02:12:54.000
Therefore, we should consider it like an oscillator, okay?
02:12:54.000 --> 02:12:56.000
Yes, okay?
02:12:56.000 --> 02:13:02.000
So, in this sense, when our material is not a zero cable,
02:13:02.000 --> 02:13:05.000
any atom is an oscillator.
02:13:05.000 --> 02:13:07.000
It's okay?
02:13:07.000 --> 02:13:12.000
Because it moves around this minimum equilibrium position.
02:13:12.000 --> 02:13:14.000
It's okay.
02:13:14.000 --> 02:13:22.000
Now, suppose that now it goes in this position.
02:13:22.000 --> 02:13:29.000
In our system, it goes here.
02:13:29.000 --> 02:13:34.000
And all the electrons have a movement in the same direction.
02:13:34.000 --> 02:13:36.000
Is it okay?
02:13:36.000 --> 02:13:42.000
So, this is the position of our prime.
02:13:42.000 --> 02:13:44.000
Is it okay?
02:13:44.000 --> 02:13:49.000
The position at which the instant thing is there.
02:13:49.000 --> 02:13:51.000
And then it will continue to move, of course.
02:13:51.000 --> 02:13:53.000
It will continue to move.
02:13:53.000 --> 02:13:57.000
But at the instant thing is in a plane.
02:13:57.000 --> 02:14:03.000
And then it will bring itself all the electrons.
02:14:03.000 --> 02:14:05.000
Is it okay?
02:14:05.000 --> 02:14:09.000
What will be this, the distribution of the electron density now?
02:14:09.000 --> 02:14:11.000
Will be?
02:14:11.000 --> 02:14:18.000
R minus rA, right?
02:14:18.000 --> 02:14:23.000
At the instant thing.
02:14:23.000 --> 02:14:25.000
Half right, sorry.
02:14:25.000 --> 02:14:27.000
Half right.
02:14:27.000 --> 02:14:29.000
Half minus half right.
02:14:29.000 --> 02:14:34.000
Is it okay?
02:14:34.000 --> 02:14:39.000
So, at the instant thing,
02:14:39.000 --> 02:14:47.000
but how much this position is frequented by the atom,
02:14:47.000 --> 02:14:52.000
there will be a particular probability for each position, r prime.
02:14:52.000 --> 02:14:55.000
Because maybe that in this direction,
02:14:55.000 --> 02:15:02.000
the atom oscillates more in this direction and less in this direction.
02:15:02.000 --> 02:15:06.000
So, there will be a probability P, r prime,
02:15:06.000 --> 02:15:10.000
where the atom not from this is, okay?
02:15:10.000 --> 02:15:16.000
So, we will have a probability P, r prime,
02:15:16.000 --> 02:15:22.000
that this position is occupied, okay?
02:15:22.000 --> 02:15:25.000
And therefore, if we change our prime here,
02:15:25.000 --> 02:15:27.000
there will be another, another, another.
02:15:27.000 --> 02:15:35.000
At the end, it's electron density will cover a bigger dominion
02:15:35.000 --> 02:15:41.000
than the original one because of this.
02:15:41.000 --> 02:15:45.000
Okay, so, what will be there?
02:15:45.000 --> 02:15:51.000
So, if the atom was original atom in a steady position with this,
02:15:51.000 --> 02:15:58.000
after that, it will occupy a much larger position because of this movement.
02:15:58.000 --> 02:16:05.000
So, what will be the Fourier transform of an atom which is 10th of February,
02:16:05.000 --> 02:16:13.000
agitated? What will be the Fourier transform of
02:16:13.000 --> 02:16:20.000
over rho over the atom, technically agitated,
02:16:20.000 --> 02:16:27.000
moved in r prime for the probability that r prime is occupied,
02:16:27.000 --> 02:16:30.000
integrated over all the positions.
02:16:30.000 --> 02:16:32.000
Is it okay?
02:16:32.000 --> 02:16:45.000
So, it is rho A t that will be the electron density over the atom,
02:16:45.000 --> 02:16:50.000
technically agitated, will be the electron density
02:16:50.000 --> 02:16:55.000
moved in r prime, the electron density at 0 kV,
02:16:55.000 --> 02:17:01.000
moved in r prime, multiplied the probability to be in r prime,
02:17:01.000 --> 02:17:07.000
integrated over all the r prime positions.
02:17:07.000 --> 02:17:12.000
That is, is it okay?
02:17:12.000 --> 02:17:15.000
What is that?
02:17:15.000 --> 02:17:17.000
A convolution.
02:17:17.000 --> 02:17:22.000
So, because what we do is take r prime,
02:17:22.000 --> 02:17:26.000
the second has to be moved in r prime
02:17:26.000 --> 02:17:30.000
and change it to minus r in minus r.
02:17:30.000 --> 02:17:34.000
Is it okay? So, this is the convolution.
02:17:34.000 --> 02:17:43.000
So, we write r prime convolution with r prime, okay?
02:17:43.000 --> 02:17:50.000
And now, we have to calculate the Fourier transform of a convolution,
02:17:50.000 --> 02:17:54.000
but we know how to calculate the Fourier transform of a convolution.
02:17:54.000 --> 02:18:00.000
So, what we have to do?
02:18:00.000 --> 02:18:03.000
The product of both.
02:18:03.000 --> 02:18:06.000
So, because of the convolution theorem,
02:18:06.000 --> 02:18:17.000
so we need F A t r star is equal to the Fourier transform of
02:18:17.000 --> 02:18:22.000
L A prime, which is this, F A,
02:18:22.000 --> 02:18:25.000
multiplied the Fourier transform of P r prime.
02:18:25.000 --> 02:18:28.000
Is it okay?
02:18:28.000 --> 02:18:31.000
Yes?
02:18:31.000 --> 02:18:35.000
Is the product of this Fourier transform
02:18:35.000 --> 02:18:39.000
by means of the convolution theorem?
02:18:39.000 --> 02:18:45.000
Is it okay for all or not?
02:18:45.000 --> 02:18:47.000
Yes.
02:18:47.000 --> 02:18:51.000
Which is Q r star?
02:18:51.000 --> 02:18:54.000
I was not at the Q for the moment.
02:18:54.000 --> 02:18:56.000
I was here.
02:18:56.000 --> 02:19:01.000
Here is the Fourier transform of P r prime.
02:19:01.000 --> 02:19:05.000
Q is the Fourier transform of P r prime.
02:19:05.000 --> 02:19:08.000
Is it okay?
02:19:08.000 --> 02:19:12.000
So, I will read.
02:19:12.000 --> 02:19:18.000
When the atom is steady, zero Kelvin,
02:19:18.000 --> 02:19:25.000
the Fourier transform, the electron is Rho A, the atom is steady,
02:19:25.000 --> 02:19:30.000
and the Fourier transform is there to be seen before.
02:19:30.000 --> 02:19:36.000
But if the atom, if our material is not at the zero Kelvin,
02:19:36.000 --> 02:19:39.000
the atoms become oscillators,
02:19:39.000 --> 02:19:44.000
and they will oscillate around the equilibrium position
02:19:44.000 --> 02:19:51.000
according to the domain which is around it, okay?
02:19:51.000 --> 02:19:57.000
So, once we want to consider it, we know that the electron density,
02:19:57.000 --> 02:20:01.000
Rho A, which is considered here,
02:20:01.000 --> 02:20:05.000
will be moved at the time t in the position R prime,
02:20:05.000 --> 02:20:11.000
and the electron density will become Rho R minus R prime, this one.
02:20:11.000 --> 02:20:16.000
And therefore, the electron density over an atom,
02:20:16.000 --> 02:20:22.000
thermically agitated, will be the electron density over an atom
02:20:22.000 --> 02:20:28.000
moved in R prime, multiply the probability that that position is occupied,
02:20:28.000 --> 02:20:36.000
indelated over all the positions, which is the convolution of Rho A by P.
02:20:36.000 --> 02:20:45.000
And therefore, the Fourier transform of Rho A t will be F A t.
02:20:45.000 --> 02:20:48.000
F A t will be, by the convolution theorem,
02:20:48.000 --> 02:20:54.000
the product of this Fourier transform and this Fourier transform.
02:20:54.000 --> 02:20:59.000
So, F A R star, multiply the Fourier transform of P,
02:20:59.000 --> 02:21:03.000
the probability of occupying one position,
02:21:03.000 --> 02:21:10.000
and it will be F A R star, atom steady, multiply the Q, Q R star,
02:21:10.000 --> 02:21:12.000
the Fourier transform of P.
02:21:12.000 --> 02:21:14.000
Is it okay?
02:21:14.000 --> 02:21:21.000
So, see how things are easy because of the magnetism here we did yesterday.
02:21:21.000 --> 02:21:32.000
So, what we can do is, okay, suppose that the movement of our atom
02:21:32.000 --> 02:21:41.000
because of the thermal agitation is isotopic and is Gaussian.
02:21:41.000 --> 02:21:46.000
This is the simplest assumption we can do, right?
02:21:46.000 --> 02:21:54.000
So, it is Gaussian, so P R prime is this one, is a Gaussian,
02:21:54.000 --> 02:22:07.000
and U is the square displacement average, the average square displacement from the position.
02:22:07.000 --> 02:22:13.000
So, you know which is the Fourier transform of a Gaussian?
02:22:13.000 --> 02:22:17.000
Yes, because you did the exercise, right?
02:22:17.000 --> 02:22:24.000
It is again a Gaussian function, but this time the variance
02:22:24.000 --> 02:22:30.000
instead of stay at the numerator, stay at the numerator, right?
02:22:30.000 --> 02:22:33.000
Do you remember the exercise we made yesterday?
02:22:33.000 --> 02:22:34.000
Yeah.
02:22:34.000 --> 02:22:38.000
Fourier transform of a Gaussian.
02:22:38.000 --> 02:22:58.000
So, what we left is this.
02:22:58.000 --> 02:23:21.000
Questions?
02:23:21.000 --> 02:23:24.000
Up to this position, do you have questions?
02:23:24.000 --> 02:23:25.000
No.
02:23:25.000 --> 02:23:30.000
So, what is R star?
02:23:30.000 --> 02:23:35.000
R star by definition, the modulus of R star is 2 c theta over lambda.
02:23:35.000 --> 02:23:45.000
So, we replace R star here by 2 c theta over lambda and we obtain this.
02:23:45.000 --> 02:23:53.000
Yes, the Fourier transform of the probability is a function which depends on c theta square of lambda.
02:23:53.000 --> 02:23:58.000
Is it okay?
02:23:58.000 --> 02:24:06.000
This 8 P squared pi square U may be written like a b,
02:24:06.000 --> 02:24:12.000
and so is exponential minus b c squared theta over lambda.
02:24:12.000 --> 02:24:21.000
This is the Fourier transform defect of the thermal agitation.
02:24:21.000 --> 02:24:22.000
Is it okay?
02:24:22.000 --> 02:24:32.000
So, the effect of the thermal agitation modify the scattering factor, atomic scattering factor
02:24:32.000 --> 02:24:42.000
when the atom is steady by this Q factor which is exponential minus b c squared theta over lambda.
02:24:42.000 --> 02:24:43.000
Is it okay?
02:24:43.000 --> 02:25:01.000
What is the a square?
02:25:01.000 --> 02:25:31.000
A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A
02:25:31.000 --> 02:25:48.000
A. A. A. A.
02:25:48.000 --> 02:25:54.000
R star squared, which it wouldn't be, because then you're pulling out a constant.
02:25:54.000 --> 02:25:55.000
Sorry.
02:25:55.000 --> 02:25:56.000
Yeah.
02:25:56.000 --> 02:25:58.000
You're incorporating that into B.
02:25:58.000 --> 02:25:59.000
Incorporating B.
02:25:59.000 --> 02:26:00.000
Okay.
02:26:00.000 --> 02:26:03.000
So, B is this one.
02:26:03.000 --> 02:26:13.000
So, B is called thermal factor, thermal factor, or temperature factor, like it was.
02:26:13.000 --> 02:26:21.000
And B, as you see, is proportional to U.
02:26:21.000 --> 02:26:31.000
U is the square displacement of the oscillator.
02:26:31.000 --> 02:26:32.000
Okay?
02:26:32.000 --> 02:26:41.000
So, the larger is the temperature, the larger would be U, because it's the average square
02:26:41.000 --> 02:26:44.000
displacement of the oscillator.
02:26:44.000 --> 02:26:47.000
And the larger would be B.
02:26:47.000 --> 02:26:49.000
It's okay?
02:26:49.000 --> 02:26:54.000
It's fine for all.
02:26:54.000 --> 02:27:08.000
Can you now, by yourself, imagine the effect of this on the thermal, on the, upon the spectrum factor?
02:27:08.000 --> 02:27:10.000
I will.
02:27:10.000 --> 02:27:37.000
I will assign here...
02:27:37.000 --> 02:27:47.000
This is the scattering factor of what?
02:27:47.000 --> 02:27:50.000
Of the oscillator.
02:27:50.000 --> 02:27:51.000
Of the oscillator.
02:27:51.000 --> 02:27:53.000
Right?
02:27:53.000 --> 02:28:05.000
Now, this is the scattering factor of F A, as calculated by the therapist and faces.
02:28:05.000 --> 02:28:06.000
Okay?
02:28:06.000 --> 02:28:09.000
So, now, the effect of the channel.
02:28:09.000 --> 02:28:21.000
Can you qualitatively modify this because of the temperature factor exponential minus B C squared theta parameter?
02:28:21.000 --> 02:28:24.000
Let us suppose B is equal to 2.
02:28:24.000 --> 02:28:25.000
Okay?
02:28:25.000 --> 02:28:26.000
2.
02:28:26.000 --> 02:28:34.000
How this is modified?
02:28:35.000 --> 02:28:36.000
How is it modified?
02:28:36.000 --> 02:28:47.000
When the theta is equal to 0, when theta is 0, the correction exponential minus B is 1.
02:28:47.000 --> 02:28:48.000
There is no correction.
02:28:48.000 --> 02:28:49.000
Okay?
02:28:49.000 --> 02:28:54.000
So, at here, there is no correction.
02:28:54.000 --> 02:28:55.000
Here.
02:28:55.000 --> 02:29:00.000
But, after that, it goes here.
02:29:01.000 --> 02:29:02.000
You all right?
02:29:02.000 --> 02:29:03.000
Yes.
02:29:09.000 --> 02:29:17.000
When we are at C squared theta over lambda, whichever B is, that's better.
02:29:17.000 --> 02:29:28.000
This is 0 because when we are at 0 here, or C squared over lambda, the exponential minus 0.
02:29:28.000 --> 02:29:30.000
The exponential minus 0 is 1.
02:29:30.000 --> 02:29:35.000
Therefore, the correction is multiplied by 1.
02:29:35.000 --> 02:29:37.000
Therefore, no correction here.
02:29:37.000 --> 02:29:47.000
But, when C squared theta over lambda squared is not 0, it will decrease rapidly, depend on B.
02:29:47.000 --> 02:29:54.000
The larger B, this is B equal to 1.
02:29:54.000 --> 02:29:58.000
This will be B equal to 2.
02:29:58.000 --> 02:30:02.000
And it will be B equal to 6, for instance.
02:30:02.000 --> 02:30:05.000
Is it okay?
02:30:05.000 --> 02:30:10.000
So, which B do you prefer?
02:30:13.000 --> 02:30:14.000
0.
02:30:14.000 --> 02:30:15.000
0.
02:30:15.000 --> 02:30:21.000
There is more B, because you want to collect theta at highest integral lambda.
02:30:21.000 --> 02:30:29.000
Therefore, the most important point is to take B as small as possible.
02:30:29.000 --> 02:30:38.000
Because when B increases, the scattering factor will become the same factor with decrease.
02:30:38.000 --> 02:30:49.000
There is the reason why most of your data collection are made at low temperature by cryogenic devices.
02:30:49.000 --> 02:30:51.000
Is it okay?
02:30:51.000 --> 02:30:58.000
We could reduce as much as possible the effect over the term of the fire.
02:30:58.000 --> 02:31:01.000
Is it okay?
02:31:01.000 --> 02:31:11.000
Fine for all.
02:31:11.000 --> 02:31:22.000
Which atoms are less sensible to the scattering, to the thermal agitation?
02:31:22.000 --> 02:31:27.000
What do you see?
02:31:28.000 --> 02:31:30.000
The light atom.
02:31:30.000 --> 02:31:36.000
The light atom you think are more sensible or less sensible?
02:31:36.000 --> 02:31:37.000
Less sensible.
02:31:37.000 --> 02:31:39.000
Less sensible, the light atom.
02:31:39.000 --> 02:31:42.000
So, think about it.
02:31:42.000 --> 02:31:43.000
No more.
02:31:43.000 --> 02:31:44.000
No more.
02:31:44.000 --> 02:31:45.000
Right.
02:31:45.000 --> 02:31:50.000
Because if you have a light atom, the weight is very small.
02:31:50.000 --> 02:31:54.000
One photon, sufficiently energetic moving.
02:31:54.000 --> 02:32:02.000
If the atom is heavy, you need more strength to move it.
02:32:09.000 --> 02:32:17.000
We are assuming that there is no transfer of energy to the single electron, only to the atom itself.
02:32:17.000 --> 02:32:23.000
When the photon made the electron, he would not transfer the energy to the single electron.
02:32:23.000 --> 02:32:27.000
He may only move the electron.
02:32:27.000 --> 02:32:36.000
So, the atoms are less sensible to the thermal agitation, of course.
02:32:36.000 --> 02:32:41.000
Because the same energy moves the electron less than the light atoms.
02:32:41.000 --> 02:32:43.000
That is the first point.
02:32:43.000 --> 02:32:57.000
The second point, do not forget, that also depends on the stability of that electron.
02:32:57.000 --> 02:33:12.000
So, if my atom is very stable, because the form of my trough is very sharp,
02:33:12.000 --> 02:33:17.000
it will move really not so much.
02:33:17.000 --> 02:33:25.000
But if the trough is large, it will easily move.
02:33:25.000 --> 02:33:32.000
So, if one atom is bounded in a strong way, it is very stable,
02:33:32.000 --> 02:33:41.000
then it will not move or will move less than one atom, which is bounded in a weak way.
02:33:41.000 --> 02:33:50.000
So, we start to make some difference, for example, between the atom which are inside the molecule
02:33:50.000 --> 02:33:53.000
and the atom which are peripheral.
02:33:53.000 --> 02:33:59.000
Because the atom which are peripheral will move more than the atom which are inside.
02:33:59.000 --> 02:34:01.000
Is it okay?
02:34:01.000 --> 02:34:06.000
How that is represented in crystallography?
02:34:06.000 --> 02:34:12.000
In crystallography, when you want to show the thermal factor,
02:34:12.000 --> 02:34:22.000
you put there one ball which increases with the thermal factor agitation, okay?
02:34:22.000 --> 02:34:29.000
So, balls larger will denote bigger agitation of that atom.
02:34:29.000 --> 02:34:37.000
So, not all the atoms, each atom has its own agitation, okay?
02:34:37.000 --> 02:34:44.000
According to the species, but also according to the bonds, okay?
02:34:44.000 --> 02:34:47.000
Maybe more stable or stable.
02:34:47.000 --> 02:34:53.000
Central or peripheral, that is what may be done.
02:34:53.000 --> 02:35:06.000
But remember that we are really describing the phenomenon in a very simple way,
02:35:06.000 --> 02:35:14.000
because we assumed that the agitation is isotopic, but it is not suitable.
02:35:14.000 --> 02:35:18.000
The agitation is usually anisotopic.
02:35:18.000 --> 02:35:20.000
Is it okay?
02:35:20.000 --> 02:35:22.000
Is it okay for all?
02:35:22.000 --> 02:35:32.000
So, we want to face the problem of the anisotopic movement of the atoms.
02:35:32.000 --> 02:35:34.000
Is it okay?
02:35:34.000 --> 02:35:39.000
But before introducing this point in a short time,
02:35:39.000 --> 02:35:48.000
I want to say that people working with cropping are more sensitive to the thermal agitation
02:35:49.000 --> 02:35:52.000
than people working with this small molecule,
02:35:52.000 --> 02:36:05.000
because you know that protein molecules are flexible because they have to do some work,
02:36:05.000 --> 02:36:13.000
which is not allowed, not foreseen from organic, for example, molecules.
02:36:13.000 --> 02:36:21.000
So, they are flexible, that means that the bonds allow movement of these electrons,
02:36:21.000 --> 02:36:32.000
and also they are into a solvent, which also allow this movement.
02:36:32.000 --> 02:36:44.000
So, the protein atoms more, very, very different, have a thermal factor of 120,
02:36:44.000 --> 02:36:50.000
often they arrive to 45, B or B.
02:36:50.000 --> 02:37:00.000
While there is no molecule, for example, for minerals, B is equal to 1, B is equal to 0.5.
02:37:01.000 --> 02:37:09.000
So, there is a strong difference between the B of protein and the B of stable molecules,
02:37:09.000 --> 02:37:11.000
like all organic molecules.
02:37:11.000 --> 02:37:15.000
All organic molecules are in between, okay?
02:37:15.000 --> 02:37:27.000
So, we have B of 4, 5, 6, 4 organic molecules, and 40, 20, 30, 4 proteins.
02:37:27.000 --> 02:37:32.000
Okay, is it clear the distribution?
02:37:32.000 --> 02:37:43.000
Now, go to the anisotropic, movement of the atoms.
02:37:43.000 --> 02:37:55.000
This one is a description very simplified, because, of course, let us suppose a simple molecule,
02:37:55.000 --> 02:37:57.000
a multiple linear.
02:37:57.000 --> 02:38:06.000
So, we have inside C, and we have, what does it matter, O, AO, at 180 degrees, does it matter?
02:38:06.000 --> 02:38:08.000
A linear.
02:38:08.000 --> 02:38:15.000
What do you think the atom will move more easily, in this way, perpendicular to the bond,
02:38:15.000 --> 02:38:21.000
or in this way, along the bond?
02:38:21.000 --> 02:38:23.000
Perpendicular to the bond.
02:38:23.000 --> 02:38:34.000
Perpendicular to the bond, because, otherwise, it should contrast the strength of the bond.
02:38:34.000 --> 02:38:38.000
The bond is a very strong strength, and the minimum is in a fixed position.
02:38:38.000 --> 02:38:43.000
When you move it from this position, you have to contrast a lot.
02:38:43.000 --> 02:38:47.000
The angle is much more easy to move.
02:38:47.000 --> 02:38:52.000
So, the movement is, in general, anisotropic, okay?
02:38:52.000 --> 02:39:02.000
And if we rest for a moment, two days after the anisotropic movement,
02:39:02.000 --> 02:39:16.000
and you look at what we see in literature, we see some ellipsoid, three axis ellipsoids.
02:39:16.000 --> 02:39:24.000
And the sides of those ellipsoids is proportional to the movement of the atoms.
02:39:24.000 --> 02:39:36.000
So, if you see the ellipsoid elongates in this direction, you should understand that the movement is strongly in this direction, okay?
02:39:36.000 --> 02:39:45.000
So, see this picture, and you will see that the internal atoms have smaller ellipsoids.
02:39:45.000 --> 02:39:50.000
Is it right? The internal atoms have a smaller ellipsoid.
02:39:50.000 --> 02:39:55.000
And the pedophilic atoms are larger ellipsoids.
02:39:55.000 --> 02:40:01.000
Is it okay? So, that is what I said before.
02:40:01.000 --> 02:40:11.000
The pedophilic atom, you have a larger movement at central atoms, and the ellipsoid is described by the ellipsoid.
02:40:11.000 --> 02:40:13.000
Is it okay?
02:40:14.000 --> 02:40:22.000
Of course, when you want to define the structure, and some of your atom is on symmetry elements,
02:40:22.000 --> 02:40:31.000
you have to take into account that the ellipsoid has to take into account the symmetry, the side symmetry of the atom.
02:40:31.000 --> 02:40:39.000
So, if you have this atom on a threefold axis, the ellipsoid cannot be three axis episodes.
02:40:39.000 --> 02:40:44.000
It has to be two axis ellipsoids because of the symmetry, okay?
02:40:44.000 --> 02:40:55.000
So, there are, in crystallography, a lot of constraints on the ellipsoids because of the symmetry,
02:40:55.000 --> 02:41:07.000
and therefore, you have to know how to correct a threefold, a three-axis ellipsoid into a two or in a sphere,
02:41:07.000 --> 02:41:11.000
because it is possible and it is in a sphere.
02:41:11.000 --> 02:41:18.000
Is it okay? So, that is more or less what you will see in literature.
02:41:18.000 --> 02:41:27.000
When you see a picture or something, and there is described some anisotopic term of factor, you will see something like that.
02:41:27.000 --> 02:41:36.000
Is it okay? If you look critically at this, you will see that the internal atoms are smaller ellipsoids,
02:41:36.000 --> 02:41:43.000
the external atoms are larger because they move up, and the movement is bigger.
02:41:43.000 --> 02:41:48.000
Is it okay? For peripheral. Is it okay for?
02:41:48.000 --> 02:41:59.000
Now, science is complicated. So, I have also to say something more about it. Why?
02:41:59.000 --> 02:42:10.000
Because you can understand that the atoms are not, cannot move as they want, right?
02:42:10.000 --> 02:42:19.000
Because they have both, also you speak in your friend, but the lane later you say, stop, because I don't hear.
02:42:19.000 --> 02:42:29.000
So, you can prevent some of your movement, okay? So, all the atoms prevent the movement over there.
02:42:29.000 --> 02:42:42.000
So, if one tries to go strongly in this direction, the other misses because this bond cannot be shorter than it should be.
02:42:42.000 --> 02:42:52.000
So, what is the final way we're looking at? That the atoms move in a continental way, all the atoms of the molecule,
02:42:52.000 --> 02:43:00.000
over the atoms of your monomer, for example, in protein, move in a continental way.
02:43:00.000 --> 02:43:09.000
So, you have to distinguish the moment of all the molecule, which is organized and coherent in all,
02:43:09.000 --> 02:43:22.000
and the moment of each one inside that molecule. This kind of study is made from people who are interested in the movement of the molecule, okay?
02:43:22.000 --> 02:43:40.000
Not from usual protein histology, which provide an isotopic or in lucky things an isotopic motion of the atoms.
02:43:40.000 --> 02:43:55.000
We don't take care of the overall organization of this motion, but we have to think that always, it is in this way, very strongly for protein.
02:43:55.000 --> 02:44:07.000
Because you cannot imagine that a protein atom can move by half an inch in one direction, because the other atoms should push it out.
02:44:07.000 --> 02:44:18.000
They say no for it. So, it is impossible, because, and therefore, you should imagine that the molecule is itself a oscillator.
02:44:18.000 --> 02:44:29.000
And this moment is the lead vibration of the molecule, is it okay? So, this is the molecule that liberates into the space,
02:44:29.000 --> 02:44:55.000
and simultaneously, each atom has its own movement. So, you will see pathways in the peripheral region of the monomer, more oscillation, because the molecule moves and you know that when you are on the periphery of the motion, they will move a lot, right?
02:44:56.000 --> 02:45:03.000
It's okay? It's okay, you all?
02:45:03.000 --> 02:45:14.000
So, please have over the thermal motion an idea which is more complicated that you can see at the first time, okay?
02:45:14.000 --> 02:45:32.000
So, you have to correct the atomic scattering factor of the steady atoms, those you find in nature, you find into the international table, you have to correct for the thermal factor.
02:45:32.000 --> 02:45:45.000
You cannot solve any structure without the correction for the thermal factor, because otherwise, the data should be completely incoherent with the model.
02:45:45.000 --> 02:46:09.000
So, usually, at each atom is associated a specific thermal factor, larger for the protein, a little bit medium for the organic molecule, low for the inorganic molecule, in average.
02:46:09.000 --> 02:46:38.000
So, this model is a rough representation of the real movement, because when you associate to a protein atom, B over 50 means that the protein atom moves by more than 0.5 extra, which is in conflict with the bond energy.
02:46:38.000 --> 02:46:59.000
So, therefore, the only solution is that the movement is organized, and this organized movement is called the vibration movement of the molecule, and therefore, you have to see the molecule move in this way, and each atom has its own movement in agreeing with it.
02:46:59.000 --> 02:47:14.000
It's okay? It's fine for long? Questions? Okay? Fine.
02:47:14.000 --> 02:47:42.000
So, now you have the idea of what means you will need your B when you will finish your splatula, means more than epsilonation only.
02:47:42.000 --> 02:48:07.000
One. Now, we go to consider the scattering amplitude of a molecule. Remember, we are not here in the crystal. We are considering a molecule.
02:48:07.000 --> 02:48:33.000
And this is a good way, they wouldn't decide for considering a single molecule, because one of the goal of a ferr is just to have a scattering and a shoulder, and know the structure by only considering one molecule, of the next goal, I suppose.
02:48:34.000 --> 02:48:59.000
So, let us consider this scattering amplitude of a molecule. Consider the role J is the electron density for one atom, isolated, thermically agitated, and located at the origin. Okay?
02:48:59.000 --> 02:49:16.000
It's okay? So, the role J includes the agitation. The atom is moved, occupied, recognized, occupied in a larger region. But it's located in the origin.
02:49:16.000 --> 02:49:35.000
Now, if we want to provide to this atom any position, we should change role J by role R minus role J. It's okay? It's fine? Here we could put role J R minus role J.
02:49:35.000 --> 02:50:00.000
Well, it's moved in RJ. It's okay? Please add role J here, because concern the J is there. It's okay? It's fine? Okay. Now, let role N be the electron density of a molecule.
02:50:05.000 --> 02:50:32.000
Role N will be the sum of the electron density of the single atoms moved in their position. It's okay? We move role J in position RJ, and the molecule will be the sum of each atomic position,
02:50:32.000 --> 02:50:51.000
located at their correct side position. It's okay? Do you have some objection today? I refer mostly to chemists.
02:50:51.000 --> 02:51:16.000
Because when you consider a molecule, always the distribution of the electron are not exactly equivalent to the distribution of the sum of the single atoms, because the valence electron are involved in it. So, this is an approximation. It's okay? So, we will consider this.
02:51:16.000 --> 02:51:40.000
What we have to do, the fully transform of this electron, basically. So, it will be the sum of this electron, the sum of the electron, basically, over the valence electron. So, it will be integral OS, although J R minus role J exponential 2 pi R star R in the electron. It's okay?
02:51:46.000 --> 02:52:07.000
Okay?
02:52:07.000 --> 02:52:32.000
Fine? To calculate this, what we have to do? The integral on a sum is certainly the sum on the integral. That's right? So, certainly, we can write it this way. But before we integrate this, it is better to introduce a change of variable.
02:52:33.000 --> 02:52:59.000
The variable here is R, as you see here, no? And instead of putting R minus R J, we write R capital J. Can you do this job? Change the variable R by R J. We are changing the variable R by R J.
02:52:59.000 --> 02:53:06.000
So, please do this job and see what you obtain.
02:53:29.000 --> 02:53:32.000
Thank you.
02:53:59.000 --> 02:54:12.000
What is this thing?
02:54:12.000 --> 02:54:30.000
The thing here, R is changed by R J, right? So, you will learn here that R J, instead of R, R J plus R J, right?
02:54:43.000 --> 02:55:00.000
We are changing the variable R by R J, not the variable R J, because R J is fixed by capital R J.
02:55:01.000 --> 02:55:03.000
So, it's 1.
02:55:09.000 --> 02:55:14.000
And here, instead of this, you have two things.
02:55:19.000 --> 02:55:26.000
It's okay for all of you here? So, you obtain this. It's okay?
02:55:27.000 --> 02:55:42.000
So, now, look at this. This expression, may it be simplified, because the variable now is R J.
02:55:43.000 --> 02:55:57.000
What we can do is to put R, or does not depend on R J. So, what does not depend on R J? Exponential to pi R small R J. That should be out.
02:55:58.000 --> 02:56:06.000
Is it okay? So, we take this, and we obtain this, which is out of the integral.
02:56:06.000 --> 02:56:16.000
In the inside of the integral, what remains? It remains R J R J multiplied exponential to pi R star multiplied R J.
02:56:16.000 --> 02:56:22.000
Please write this, even if I didn't write, you can do it.
02:56:27.000 --> 02:56:29.000
It's okay.
02:56:57.000 --> 02:57:19.000
You obtained it? Yes? Did you obtain that? Okay.
02:57:20.000 --> 02:57:29.000
Capital R J is any variable. You can replace it with X, with R, with any of it.
02:57:29.000 --> 02:57:37.000
So, what is this? R J, exponential to pi R star.
02:57:38.000 --> 02:57:53.000
The way it works for? Of this. Of the, of the vector basically over the J theta. And this is F J. Right?
02:57:54.000 --> 02:58:10.000
So, we can write F M R star is summation from J to the number of atoms of F J R star exponential to pi R star R J.
02:58:10.000 --> 02:58:24.000
It's okay. Let's not think a little bit about this formula, because this formula is, let me say, your bada mecon.
02:58:24.000 --> 02:58:32.000
Always in your pot, it doesn't mean. Because in the vector, always in your pot.
02:58:33.000 --> 02:58:58.000
So, the structure factor of the molecule is equal to the sum of the contributions coming from each atom of the molecule.
02:58:58.000 --> 02:59:10.000
In the first position, we put here the scattering factor of the atom, of the atom J, of a genetic atom, calculating the position of star.
02:59:10.000 --> 02:59:23.000
Multiplied exponential to pi R star, not R J. So, that is the structure factor of the molecule.
02:59:23.000 --> 02:59:37.000
For calculating it, what you really need. You really need to calculate F J for that, and to know the position of the atoms.
02:59:37.000 --> 02:59:48.000
Is it okay? And then for which R star you want, you can calculate F M R star. Is it okay for all? Yes?
02:59:48.000 --> 03:00:03.000
Now, I should complicate this a little bit, because F M R star is a function, which depends on R star, as you suppose.
03:00:04.000 --> 03:00:18.000
So, first question, F M R star, is it continuous? Yes, because it is a molecule. We are a molecule in some way.
03:00:18.000 --> 03:00:28.000
So, you see me from any position, and therefore the scattering is continuous for a single molecule, or for an assembly of molecules.
03:00:28.000 --> 03:00:53.000
Not for the crystal. Your body is continuous. So, this F M is continuous. So, if you take a molecule, one molecule, you put on a beam, you will receive in any place, in any orientation, you will receive intensity.
03:00:53.000 --> 03:01:11.000
Is it okay? Is it fine? No. Now, we want to calculate F M R star in any generic position R star. That is, with any angle, it is okay.
03:01:11.000 --> 03:01:23.000
So, to calculate it, we need, of course, the position of R J. That is clear. But we need also F J R star.
03:01:23.000 --> 03:01:45.000
So, when I want to calculate in a given R star, I have to introduce, I have to see here, this is infinity of lambda, which is R star, and I have to go to check the R star in which I want to calculate the structure factor.
03:01:45.000 --> 03:02:14.000
Where is here? Is here. So, since my thermal factor is 1, I take this value as F J R. If my thermal factor is 6, I will take this value here, and the R star in which I want to calculate,
03:02:14.000 --> 03:02:38.000
when I change R star, I will move from this position to this position. Okay? So, F M R star implies that you change the scattering factor, and according to your thermal factor, you will go here, and here, and here. Is it okay? Is it fine?
03:02:45.000 --> 03:03:02.000
Since the F J decrease, decrease with R star, on the average, F M R star will decrease with R star. Right?
03:03:03.000 --> 03:03:18.000
So, the intensity of the I C T double lambda reflection will be smaller than the intensity at small angle reflection. Is it okay? Fine for?
03:03:18.000 --> 03:03:47.000
When we go at very high I C T double lambda, we will not observe nothing. Fine for? Right? Okay, so remember, when you calculate F M R star in a point R star of this space, you should put inside F J R star,
03:03:48.000 --> 03:03:59.000
at that R star, because F J is a function. Is it okay? You should go there, and take the correct value. Is it fine? Okay.
03:03:59.000 --> 03:04:19.000
Now, we come to the question. Scattering amplitude of an infinite question?
03:04:20.000 --> 03:04:33.000
Semi-infinite? So, semi-infinite? Infinite. Infinite, what? What is the question? If it's half infinite, semi-infinite, or like?
03:04:33.000 --> 03:04:55.000
Infinite crystal. Infinite crystal, because we will see that it has not been infinite crystal, polycrystal. You know that crystal has dimension of angle. So, when you have one medium of molecules in this direction, it's like an incident. So, let's consider an infinite crystal.
03:04:56.000 --> 03:05:24.000
And then we will see when the number of units decreases. It's okay? Now, the electron density of an infinite crystal, how can it be written? We know that. The electron density of a crystal is the convolution between the lattice function
03:05:24.000 --> 03:05:53.000
and the content of the unit cell. But surely, we did not the content of the unit cell. We spoke about one molecule only. There is some difference. Yes, in principle, but for us, we can consider the content of a unit cell like a big molecule, considered this way. So, there is no problem for that. We can consider the content of a unit cell like
03:05:53.000 --> 03:06:22.000
a big molecule. It's okay? So, we should put there the rho n, which is the content of the unit cell, multiplied by the lattice function. It's okay? That was we obtained yesterday. It seems too far yesterday, right?
03:06:23.000 --> 03:06:37.000
Because it is just yesterday. So, don't forget. So, rho n is the rho n convolution with the other. What is the fluidus form of a convolution?
03:06:37.000 --> 03:07:06.000
The product of the fluidus of a convolution. So, t rho infinite is t rho n, but t rho n, we know what it is because we have just time to work. It is f n r star, right? There is one of the few which is always appeared. You remember, I mentioned that, but you were neglectful.
03:07:07.000 --> 03:07:33.000
So, f n r star. And then, we have to multiply them for the fluidus form of the lattice function, the direct lattice function. What is the fluidus form of an infinite lattice function? The less you put a lattice function. So, we can write summation h k l to infinity delta r star minus r star h.
03:07:37.000 --> 03:07:58.000
And what are r star h? Are the position of the reciprocal lattice form. So, r star h is equal to h star plus k star plus l star, where h k l are integer numbers. Is it okay?
03:08:07.000 --> 03:08:30.000
So, I repeat the concept in order to be sure that I didn't lose this information in this critical part.
03:08:30.000 --> 03:08:59.000
The electron density of the infinite crystal, denoted here like rho infinite r, r because we are in direct space, is rho m r, which is the content of the human cell, the motif, which is repeated, convoluted with the lattice function in direct space.
03:08:59.000 --> 03:09:28.000
In direct space, it is function of r. Is it okay? Now, in direct space, all is function of r. When we calculate the fluidus form of this equation, we will have transform of rho infinite r will be the product of the two fluidus forms.
03:09:29.000 --> 03:09:51.000
The first fluidus form is f m r star. It is the fluidus form of the unit cell. Multiply the fluidus form of the direct lattice. But the fluidus form of the direct lattice is the reciprocal lattice.
03:09:51.000 --> 03:10:04.000
And therefore, we write some issue from minus infinity to infinity over r star r star h. r star h are the position of the reciprocal lattice. Is it okay?
03:10:05.000 --> 03:10:28.000
So that is our expression. The structure factor of a crystal is equal to the structure factor of the single unit cell.
03:10:28.000 --> 03:10:46.000
And let's allow us to make a calculation of only on a unit cell and not on a billion of unit cell, on a unit cell, multiply this summation here. Multiply the reciprocal lattice. Is it okay?
03:10:46.000 --> 03:10:48.000
Can you say it again?
03:10:48.000 --> 03:10:49.000
Sorry?
03:10:49.000 --> 03:10:52.000
Can you say the last sentence again?
03:10:52.000 --> 03:11:20.000
The structure factor of the crystal f h is, sorry, I have to say something before. Okay. This f m h multiplied the reciprocal lattice function, which is this one. Is it okay?
03:11:20.000 --> 03:11:31.000
So I have to specify one. I need to say something about it. And r star h is this. So I have to specify something. Is it okay?
03:11:31.000 --> 03:11:55.000
So be careful now. This is what we have obtained. This is. Now I will explain why we can pass this. Okay? For the moment, we remain here. Remain here. Up to here, is it clear of all, I suppose? Up to here. Right?
03:11:55.000 --> 03:12:11.000
It's just the application of the Fourier transform. Now, which is the practical consequence of this expression?
03:12:11.000 --> 03:12:17.000
I have a question to this. What is the volume?
03:12:17.000 --> 03:12:37.000
One of V? One of V. V is the volume. And comes out because we are working into a space which is not Cartesian. All the coordinates are not in x, y, z Cartesian, but in a percentage of the unit cell.
03:12:37.000 --> 03:12:53.000
So when you apply the mathematics in this not Cartesian space reference system, then you need to put one over V. Over V. But it's a constant. It doesn't matter. Don't worry about it.
03:12:53.000 --> 03:13:17.000
I think it's entered into this detail because I should do the transformation of the volume so it is not important. So it is a constant, a state constant. Okay? So now, all of us agree about that.
03:13:17.000 --> 03:13:35.000
But this is a strong difference with what you obtained before. Why? Do you see the difference? It's discrete. It's discrete. Indeed.
03:13:35.000 --> 03:14:01.000
The delta function has always zero M and Y. Except in R star H. Therefore, it means that F, the transformation of log A, which is F R star. Let us call it F R star. Sorry. I put here F R star.
03:14:06.000 --> 03:14:31.000
So F R star is this one, explicitly. F R star. F R star is zero everywhere. Because the delta function, which are here, are zero everywhere except in the reciprocal of its point.
03:14:32.000 --> 03:14:50.000
So this function is highly disconfused. Don't confuse this with the scattering from a model or from a single sensor. That is confused. But when you are in a crystal, that becomes disconfused.
03:14:51.000 --> 03:15:08.000
So the intensity you will observe, the intensity you will observe, which is F square, will be zero everywhere. You will only have intensity on the reciprocal of its point. It's okay?
03:15:08.000 --> 03:15:25.000
So the experiment, or the fractional experiment, what is real is just look at the reciprocal of its point and measure the intensity in those positions. Not in terms. It's okay?
03:15:25.000 --> 03:15:41.000
It's okay? Because it is absolutely unusual to measure between the reciprocal and let us call it. There is nothing there. It's okay? It's fine for all.
03:15:41.000 --> 03:16:06.000
Fine for all? So instead of using F R star here, we can use F H. Because the F H is different from zero, only on a reciprocal of its point, H V L. So we can put here F H.
03:16:06.000 --> 03:16:28.000
H here and H or R star H here. So we can write F H here. So H is the rest of the point? Like the percentage of R star H, right? So F H instead of writing F R star H. It's okay?
03:16:28.000 --> 03:16:47.000
So this is the reason why you collect data in which there are F H K L. You go from there to the H K L reciprocal of its point. It's okay? Fine for all?
03:16:48.000 --> 03:17:16.000
This is what I have to tell you. Now, for the moment we see that really this effect of diffraction, the small domain, the zero domain, because our point or the reciprocal space, but we will see that there are
03:17:16.000 --> 03:17:40.000
physical phenomena which suggest us that are not points, are domain to receive it. It's okay? Fine for all? So we know now that when we do this happen from a crystal, we should be prepared to have intensity
03:17:40.000 --> 03:18:05.000
only at the reciprocal lattice points, no other. So we will have F H no more F R star. Okay? But F H only. So in the reciprocal space we look only to the reciprocal lattice point. It's okay?
03:18:10.000 --> 03:18:32.000
And the good news is that for calculating F H, we don't need to sum on all the atomic positions in the crystal, but only on the unit cell, because this F M is the content of the unit cell. It's okay? This fine?
03:18:33.000 --> 03:18:52.000
This is the capital H, also like R H. I mean, could I write F R H? I could write F R star H. F R star H, yes. But doesn't matter. It is sufficient to say H K L is enough. It's okay?
03:18:53.000 --> 03:19:21.000
So when you were preparing the data, this data has been worked by the man within the experiment, because they have indexes in the data. They know where R B C R star D star C star R, they gave the correct indexing H K L to each point.
03:19:21.000 --> 03:19:42.000
And then you will receive the data. It's okay? So there are some preliminary work made by the experimentalists, which recognize where R star D star C star R and which each point of the reciprocal space we are measuring. Okay?
03:19:42.000 --> 03:20:00.000
And then you will receive this data, corrected by polarization effect. It's okay? Fine? Do you have any questions?
03:20:00.000 --> 03:20:26.000
Okay. Now, this is the picture we need to have in mind. You see the crystal, the electron density, which is raw, right?
03:20:27.000 --> 03:20:51.000
And each right beam goes up to the crystal. R point X bar goes up to the crystal. And the interaction produce the fluid transform, which is here, which is in this space here.
03:20:51.000 --> 03:21:19.000
So F is the fluid transform of R. Please note, now the fluid transform of R is no more continuous, because we are in the crystal. And the direction of the diffracted beams are specific, where comply with the reciprocal lattice.
03:21:19.000 --> 03:21:32.000
Okay? So that is the reason why here I put some regular points. It's okay? And the intensity is only there, not in another point.
03:21:32.000 --> 03:22:01.000
So now suppose that our methods, physical methods, are able to make the inverse fluid transform. So from this point here, if we measure,
03:22:02.000 --> 03:22:19.000
models and phase of this, measure that F, F is a complex number. So if we have this complex number, we can apply the inverse fluid transform here and obtain the crystal.
03:22:20.000 --> 03:22:38.000
If we simulate the optical situation, what we should do? We have here an object. What we should have here?
03:22:39.000 --> 03:23:01.000
A continuous distribution of intensity, which is the fluid transform. And what should be this for collapse to make the inverse fluid transform?
03:23:02.000 --> 03:23:26.000
The crystalline in our heart. So here should be the crystalline, which makes the inverse fluid transform, and then we should see the object. Is it okay? So in the usual life, this is a single object. This should be continuous.
03:23:26.000 --> 03:23:55.000
This is our crystalline, which performs the fluid transform. When we use each ray, we have the crystal, we have this continuous beams, and if we were able to have F in models and phase of this complex number, we can do it.
03:23:56.000 --> 03:24:14.000
Otherwise, we cannot do it. Usually, we measure only the modules on this F, because we measure intensity. So that passage cannot be made. Is it okay? At least it cannot be made immediately. Is it okay?
03:24:15.000 --> 03:24:30.000
That is the reason why I was paid for 40 years. Because for this problem, to solve this problem of recovering the phases from the model. Is it okay?
03:24:31.000 --> 03:24:51.000
I am not sure if I remember one thing. So even if we had modules and phases, would it be possible to recover? Because still, we have two dimensions of reciprocal space, but we need three dimensions of direct space. So how can we do from this?
03:24:52.000 --> 03:25:11.000
The direction, the directions are not the same. The theta are not the same. So when you make an experiment, you make an experiment in such a way to avoid super overlapping process.
03:25:12.000 --> 03:25:29.000
When you do all the experiments, you have superposition of the reference. Now, any superposition of the reference is not there, because you see every reciprocal expression is taken independently from the experiment.
03:25:30.000 --> 03:25:45.000
For example, one zero zero and two zero zero are in the same direction. But one zero zero has a theta, and two zero zero has another theta. Another aspect. So, a thing in different time.
03:25:45.000 --> 03:26:14.000
So, is it clear this picture? Okay, now. Now, suppose that, I should say something wrong. Oh, I should say before passing through this.
03:26:15.000 --> 03:26:41.000
Electrons. Electrons have a very nice feature. Electrons can preserve the field. Why? Because electrons are charged particles, right? They are deflected by the crystal. And by means of electromagnetic glass, I can't solve it.
03:26:42.000 --> 03:27:07.000
Right? So, in principle, electrons can refocus the data. Indeed, you can see in some applications, the crystal directly there on the image. The electron density on the image.
03:27:08.000 --> 03:27:35.000
Because they are charged particles, which can be refocused by. But this feature is available, but is not easy to obtain. Only for a very special case, you cannot obtain that. Because the aberration of the microscope is very high.
03:27:36.000 --> 03:27:51.000
And therefore, you have images which are so distorted that you cannot be sure they are really that unique. It's okay. But remember, in principle, that is possible. It's okay.
03:27:52.000 --> 03:28:19.000
Now, go to the scattering amplitude of a finite crystal. How to represent a finite crystal? It is a very nice and elegant way of representing a finite crystal.
03:28:19.000 --> 03:28:42.000
So, the real crystal, finite crystal, is the real infinite crystal multiplied the form of the crystal, which is phi. Is it okay? You have the infinite crystal, but your crystal is finite.
03:28:42.000 --> 03:29:02.000
So, the finite crystal may be the infinite crystal multiplied by the form of the finite crystal. It's okay? So, it is zero outside the form. It is phi equal to one inside the form. It's okay. It's okay?
03:29:02.000 --> 03:29:31.000
So, phi is zero outside the form. Sorry, waste time. Zero inside the crystal. Why use this, it seems, very stupid way of doing things. But we are not, because if I represent the raw crystal like the infinite, multiplied phi, when I make the Fourier transform, I will learn that the F star is infinite.
03:29:32.000 --> 03:29:45.000
The Fourier transform of a finite crystal is equal to transform of a raw infinite convolution of the transform of phi. Is it okay?
03:29:45.000 --> 03:30:10.000
I don't want to go into detail, but do you remember the fact that when I made the convolution between two functions, the convolution of these two functions was flatter then?
03:30:10.000 --> 03:30:39.000
Okay. So, the same effect comes here. The convolution of these two functions give rise to a flatter function. So, the FMH are no more infinitely sharp, but are multiplied by n domain, which is not, no more delta, is d, which is the domain corresponding to phi.
03:30:39.000 --> 03:31:09.000
The transform of phi. So, what we will see here is the Fourier transform of the phi. Suppose phi is a point, is really small, so it's no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no,
03:31:09.000 --> 03:31:39.000
no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no,
03:31:39.000 --> 03:31:44.000
The Fourier transform of the form of the factor.
03:31:44.000 --> 03:31:45.000
Is it okay?
03:31:47.000 --> 03:31:49.000
Is it all right?
03:31:49.000 --> 03:31:57.000
So, what you will see is really something like that.
03:31:57.000 --> 03:31:59.000
And I will explain why.
03:31:59.000 --> 03:32:00.000
Okay.
03:32:00.000 --> 03:32:02.000
I repeat.
03:32:03.000 --> 03:32:11.000
Suppose that our crystal is small, is limited.
03:32:11.000 --> 03:32:23.000
Then the electron density of the crystal may be represented by the electron density of the infinite crystal multiplied the form of the crystal.
03:32:23.000 --> 03:32:28.000
One inside, the crystal zero inside.
03:32:29.000 --> 03:32:40.000
So, I can take the Fourier transform of this limited crystal, the rock curve, and I will obtain the convolution of this Fourier transform.
03:32:40.000 --> 03:32:46.000
The convolution enlarge the single function.
03:32:46.000 --> 03:32:54.000
So, the delta function is enlarged into a d function, which is the Fourier transform of the form.
03:32:54.000 --> 03:32:56.000
Is it okay now?
03:32:58.000 --> 03:33:07.000
So, what you will obtain is something like that.
03:33:07.000 --> 03:33:13.000
This is a limited lattice, one dimension.
03:33:13.000 --> 03:33:16.000
The reciprocal lattice is the contour.
03:33:16.000 --> 03:33:19.000
It will be flat, completely flat.
03:33:19.000 --> 03:33:29.000
And since this is limited, you will have some thickness here, because this is limited, okay?
03:33:29.000 --> 03:33:42.000
You have a very flat, bidimensional lattice, okay?
03:33:42.000 --> 03:33:46.000
This is infinitely sharp in this direction.
03:33:46.000 --> 03:33:50.000
It will be flat in this direction.
03:33:50.000 --> 03:33:52.000
Is it okay?
03:33:52.000 --> 03:34:00.000
This is long in this direction, will be short in this direction, and longer in this direction.
03:34:00.000 --> 03:34:01.000
Is it okay?
03:34:01.000 --> 03:34:03.000
It's clear.
03:34:03.000 --> 03:34:05.000
It's clear for all.
03:34:05.000 --> 03:34:12.000
What I have here, I have here an ellipse, I will have here also an ellipse.
03:34:12.000 --> 03:34:18.000
Very sharp in the vertical direction, flat in vertical direction.
03:34:18.000 --> 03:34:23.000
This is a sphere, I will see a sphere.
03:34:23.000 --> 03:34:38.000
So, all will be, the form will be the form of the lattice will have the Fourier transform of the crystal,
03:34:38.000 --> 03:34:39.000
or the form of the crystal.
03:34:39.000 --> 03:34:40.000
Is it okay?
03:34:40.000 --> 03:34:49.000
So, in this case, when the crystal is cubic, it will have something like a cube.
03:34:49.000 --> 03:34:50.000
Is it okay?
03:34:50.000 --> 03:34:51.000
And so on.
03:34:51.000 --> 03:34:52.000
It's clear.
03:34:52.000 --> 03:34:54.000
Fine?
03:34:54.000 --> 03:34:58.000
Now, experimentally, this is what we will see.
03:34:58.000 --> 03:35:00.000
This is Ellie.
03:35:00.000 --> 03:35:03.000
Can you go just back to the slides?
03:35:03.000 --> 03:35:05.000
We didn't finish.
03:35:05.000 --> 03:35:07.000
Again.
03:35:07.000 --> 03:35:10.000
I'm one of the mummers, yeah.
03:35:17.000 --> 03:35:20.000
What's the significance of changing the crystal, Megan?
03:35:20.000 --> 03:35:21.000
What?
03:35:21.000 --> 03:35:24.000
Omega is the volume of the crystal.
03:35:24.000 --> 03:35:25.000
This is changing the notation?
03:35:25.000 --> 03:35:26.000
Yes.
03:35:26.000 --> 03:35:28.000
Omega is the volume of the crystal.
03:35:28.000 --> 03:35:38.000
When I make the transform, the transform of the, in general, is over all the respect of the space.
03:35:38.000 --> 03:35:41.000
So, the Fourier transform is over all.
03:35:41.000 --> 03:35:48.000
But, phi is zero everywhere except in the domain, and then omega is the domain.
03:35:49.000 --> 03:35:56.000
So, all this integration will be reduced to the volume of the crystal because outside is zero.
03:35:56.000 --> 03:35:58.000
Right there?
03:35:58.000 --> 03:36:00.000
Then we don't need the phi there, right?
03:36:00.000 --> 03:36:01.000
Because it's one.
03:36:01.000 --> 03:36:02.000
We don't?
03:36:02.000 --> 03:36:06.000
We don't need a phi in the integral because it's one.
03:36:06.000 --> 03:36:07.000
It is one, of course.
03:36:07.000 --> 03:36:08.000
Yes.
03:36:08.000 --> 03:36:09.000
We don't need that.
03:36:09.000 --> 03:36:10.000
The only move is over.
03:36:10.000 --> 03:36:12.000
No, you don't need that.
03:36:12.000 --> 03:36:13.000
Exactly.
03:36:13.000 --> 03:36:14.000
Exactly.
03:36:14.000 --> 03:36:15.000
Is it okay?
03:36:15.000 --> 03:36:16.000
It's it.
03:36:16.000 --> 03:36:17.000
It's it.
03:36:17.000 --> 03:36:21.000
So, phi is the shape of the crystal.
03:36:21.000 --> 03:36:23.000
Phi is the shape of the crystal.
03:36:23.000 --> 03:36:24.000
One.
03:36:24.000 --> 03:36:28.000
It is one inside the crystal, zero outside.
03:36:28.000 --> 03:36:31.000
And, but shouldn't it also be a lattice?
03:36:31.000 --> 03:36:39.000
Like, in the previous, you know, like, for the infinite crystal, you have lattice.
03:36:39.000 --> 03:36:43.000
In infinite crystal, you need infinite lattice.
03:36:43.000 --> 03:36:44.000
Okay?
03:36:44.000 --> 03:36:50.000
And then, for an infinite lattice, we have delta functions.
03:36:50.000 --> 03:36:56.000
Over all the reciprocal lattice points, you have delta functions.
03:36:56.000 --> 03:37:07.000
If your crystal is finite, you will not have a delta function, but some small domain.
03:37:07.000 --> 03:37:12.000
The largest the crystal, the smaller a domain.
03:37:12.000 --> 03:37:13.000
Yeah.
03:37:13.000 --> 03:37:16.000
Is it okay?
03:37:16.000 --> 03:37:20.000
Is it okay?
03:37:20.000 --> 03:37:33.000
And the crystal will take, the reciprocal lattice point will assume the form over the,
03:37:33.000 --> 03:37:37.000
through the transform over the form of the crystal.
03:37:37.000 --> 03:37:38.000
Okay?
03:37:38.000 --> 03:37:39.000
Which is fine.
03:37:39.000 --> 03:37:41.000
For the infinite crystal.
03:37:41.000 --> 03:37:44.000
Of course, I did not mention it here.
03:37:44.000 --> 03:37:47.000
I did not mention it here.
03:37:47.000 --> 03:37:56.000
The, the, the secondary maxima, which are in between, of course, we, we started there.
03:37:56.000 --> 03:38:04.000
You show respect secondary maxima in between the one last edge and one another edge.
03:38:04.000 --> 03:38:09.000
So, we suppose, for the moment, they, they say that, but for the moment, we need lattice.
03:38:09.000 --> 03:38:10.000
Okay?
03:38:10.000 --> 03:38:24.000
So, in the reality, when you take a very small crystal, say, few, few, few units, then you
03:38:24.000 --> 03:38:34.000
will see that every lattice point has the form of the free transform of the form of
03:38:34.000 --> 03:38:39.000
the crystal and in between there are some secondary maxima.
03:38:39.000 --> 03:38:42.000
Is it okay?
03:38:42.000 --> 03:38:43.000
Yes.
03:38:43.000 --> 03:38:44.000
Yes.
03:38:44.000 --> 03:38:49.000
The reason that we get the lower group and not the point group when we do our experiments
03:38:49.000 --> 03:38:53.000
because of this information that you talked about here.
03:38:53.000 --> 03:38:54.000
No.
03:38:54.000 --> 03:38:57.000
Why do we get the lower group and we don't talk about point?
03:38:57.000 --> 03:39:00.000
Why do we see the other group later in this?
03:39:00.000 --> 03:39:02.000
And this inversion, how is...
03:39:02.000 --> 03:39:03.000
Which inversion is this?
03:39:03.000 --> 03:39:05.000
No, this is the inversion.
03:39:05.000 --> 03:39:10.000
If you have a symmetry in, it will show the symmetry out now.
03:39:10.000 --> 03:39:12.000
I see.
03:39:12.000 --> 03:39:14.000
We will see the symmetry.
03:39:14.000 --> 03:39:20.000
I understand what you mean, but after we have the point in between.
03:39:20.000 --> 03:39:21.000
Okay.
03:39:22.000 --> 03:39:24.000
So, is it okay?
03:39:24.000 --> 03:39:30.000
So, what we see experimentally is this one.
03:39:37.000 --> 03:39:38.000
Is it okay?
03:39:38.000 --> 03:39:39.000
Yeah.
03:39:40.000 --> 03:39:41.000
Fine, folks?
03:39:41.000 --> 03:39:54.000
So, at every point, every lattice point, this is the one plane of the reciprocal lattice
03:39:54.000 --> 03:40:02.000
that has been collected by a precession method, which was used in other times.
03:40:02.000 --> 03:40:03.000
Yes.
03:40:03.000 --> 03:40:11.000
And this shows directly a reciprocal lattice.
03:40:11.000 --> 03:40:16.000
And you see, this is a propane.
03:40:16.000 --> 03:40:19.000
I don't know exactly, it doesn't matter.
03:40:19.000 --> 03:40:24.000
And you see clearly, the reciprocal lattice here, in the periods,
03:40:24.000 --> 03:40:33.000
very small in this direction and larger in this direction, right?
03:40:33.000 --> 03:40:36.000
What means small in this direction?
03:40:36.000 --> 03:40:45.000
Long A, long C, and smaller A, right?
03:40:45.000 --> 03:40:46.000
Okay.
03:40:46.000 --> 03:40:53.000
Now, and you see that these are the form of, more or less, all these have the same form.
03:40:53.000 --> 03:40:56.000
You see?
03:40:56.000 --> 03:41:00.000
Nearly a section, of course.
03:41:00.000 --> 03:41:12.000
So, I have to say also that, okay, I can say something more in five minutes.
03:41:12.000 --> 03:41:17.000
Don't forget that we are dealing with the perfect question.
03:41:17.000 --> 03:41:27.000
I never spoke about imperfection, but the crystals, like humans, are very imperfect.
03:41:27.000 --> 03:41:33.000
So, we have to remember that.
03:41:33.000 --> 03:41:44.000
So, any crystal is never the perfect reputation of a model.
03:41:44.000 --> 03:41:56.000
In some way, the nature, the chemistry, the chemical department, and so on,
03:41:56.000 --> 03:42:00.000
does not allow to trust the revolution.
03:42:00.000 --> 03:42:03.000
And you see, so this is the perfect question.
03:42:03.000 --> 03:42:10.000
Do you remember when I showed you those ones, the displacements,
03:42:10.000 --> 03:42:14.000
in which at the end, I showed you some defects in the letters,
03:42:14.000 --> 03:42:19.000
and then you see on the other side, some diffuse scatter.
03:42:19.000 --> 03:42:20.000
Do you remember?
03:42:20.000 --> 03:42:27.000
Well, it is very important that one of the defects of the letters are two-stroke one.
03:42:27.000 --> 03:42:35.000
You will never ever have some basic purpose, basic purpose.
03:42:35.000 --> 03:42:40.000
You will never have some diffusion around.
03:42:40.000 --> 03:42:48.000
So, what is the way in which this baby came?
03:42:48.000 --> 03:42:59.000
Usually, the experimentalists take care of the dimension of the spots.
03:42:59.000 --> 03:43:05.000
When the spot is too large, there are two reasons.
03:43:05.000 --> 03:43:17.000
One is the crystal is very small, and then you have large spots, because it is inwards.
03:43:17.000 --> 03:43:23.000
And the second is that the crystal is very imperfect.
03:43:23.000 --> 03:43:26.000
Can we understand why?
03:43:26.000 --> 03:43:51.000
Suppose that these are these.
03:43:51.000 --> 03:43:59.000
Because it is a holographic clay, H-kyle, H-kyle.
03:43:59.000 --> 03:44:04.000
I made this as perfect a second, okay?
03:44:04.000 --> 03:44:13.000
The direction is the same, D is the same, but suppose that it is not in this way.
03:44:13.000 --> 03:44:17.000
So, what we do is, first I draw this,
03:44:17.000 --> 03:44:22.000
and the reciprocal axis, corresponding to this H-kyle,
03:44:22.000 --> 03:44:29.000
is, for example, this one, the arch star of H perpendicular to this.
03:44:29.000 --> 03:44:31.000
We know that, okay?
03:44:31.000 --> 03:44:39.000
But now suppose that here we start some other clay,
03:44:39.000 --> 03:44:46.000
which have a different orientation, a slightly different orientation.
03:44:46.000 --> 03:44:50.000
Because of imperfection.
03:44:50.000 --> 03:44:54.000
What will be this? The arch star for this?
03:44:54.000 --> 03:44:58.000
It will be rotated a little bit.
03:44:58.000 --> 03:45:06.000
And if this moment is continuous, it will correspond an arch.
03:45:06.000 --> 03:45:13.000
The reciprocal lattice point will be no more a point, but will be an arch.
03:45:13.000 --> 03:45:19.000
It's okay? If that is continuous.
03:45:19.000 --> 03:45:29.000
Let us suppose now that this day here is not constant, but changed a little bit.
03:45:29.000 --> 03:45:35.000
But the plane are oriented correctly.
03:45:35.000 --> 03:45:42.000
What will be left? Our stars will change, so we will go here.
03:45:42.000 --> 03:45:49.000
And if both are present, we will learn a domain. It's okay?
03:45:49.000 --> 03:46:06.000
So, as you see, the depth of the crystal may contribute to make large the domain of the diffraction spot.
03:46:06.000 --> 03:46:12.000
This, by crystallography, is called the MOSAIS.
03:46:12.000 --> 03:46:21.000
So, maybe that your crystal has a too big MOSAIS for checking or making measurements.
03:46:21.000 --> 03:46:29.000
So, people will say, no, it is not sufficient to make measurements.
03:46:29.000 --> 03:46:37.000
So, please check again and find a better crystal with the MOSAIS.
03:46:37.000 --> 03:46:47.000
And then you say, okay, I will try. You change your condition, physical condition, and grow some new crystals.
03:46:47.000 --> 03:46:53.000
And then, at the end, maybe you are lucky and you have a crystal with a small MOSAIS.
03:46:53.000 --> 03:46:58.000
Then you can collect it. It's okay? Is it fine?
03:46:58.000 --> 03:47:09.000
So, that is the situation at this moment. Do you have some questions?
03:47:09.000 --> 03:47:13.000
So, see you in one moment.