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Now we can start. So we learned that the reciprocal of the crystal system belong to the same crystal system, right?
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We didn't do the example of tricraiding because for that it is important to manage some spherical geometry, okay?
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So we skipped. But in any test book or histography you will find how to locate the angle and so on.
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The angle and the magnitude of the established crystal system. Is it okay?
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Now, this strange dual reference system is interesting because as some property, because if as no useful property, no people should consider it, right?
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So as very interesting property.
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So one of the property may be this one. Suppose you have into this dual reference system a star, b star, c star.
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You have a point with the coordinate x star, y star, b star is a point in this reference system, okay?
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And you have a point into the direct space system defined by x, y, z. So in the reference a, b, c, you have a point x, y, z.
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And in the reference system, a star, b star, c star, you have a point x star, y star, z star. Is it okay?
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So the vector which start from an origin and reach x, y, z, we know it. Is the vector.
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Here is the vector x1a plus its y1b plus zeta1c, right? This is the vector which connect the origin with the point x1y1 zeta1 into the unit cell.
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Is it okay? And look at the reciprocal space. And look at the point x star, b star, c star, sorry, x2 star, y2 star, z2 star in reciprocal space.
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That is, and the vector which connect the origin of the reciprocal space with this point x star 2, y star 2, and zeta star 2 is this vector here.
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x2 star, a star, plus y star, 2 star, b star, plus zeta2 star, c star. So we are interested to calculate the scalar product of r1 dot r2 star.
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Is a vector defined in that space the first and the second in reciprocal space? Can you do this dot multiplication?
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So all the people did the job. So all the people found this interesting relation.
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So when we have a vector defined into reciprocal space and multiply in a scalar way with a vector defined in direct space,
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what we have to do only is to multiply the corresponding coordinates, x6, yy, zeta, z. That's enough.
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So we spend a lot of time when we do this kind of multiplication just by using one vector in the reciprocal space and one vector in direct space.
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But it's a nice property. You can't set a lot of terms there, right? This is the first.
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The second important relation is this one. Sorry for the Italian, but I will translate it in English.
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r star h, the vector r star h with the h star plus kv star plus nc star, means hkl integer, hkl integer, is normal to the family of lattice plane hkl.
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So suppose you have in that space a family of planes, lattice plane hkl. Is it okay?
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hkl. So infinite planes hkl, you know how to draw it. Is it okay? Now, take hkl and go into the reciprocal reference and write the vector h star plus kv star plus nc star.
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Use this three hkl and take the vector in reciprocal space. This vector is perpendicular to the plane, to the family of plane hkl.
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So, for example, you are using hkl like 2 4 6, hkl 2 4 6. Well, the vector r star h, 2 h plus, I forgot what I said, 4 b star plus 6 c star is perpendicular to the family of plane 2 4 6 in direct space. Is it okay?
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This finder? This is the thesis. Now we have to demonstrate that. Because it is a very interesting relation, okay? I repeat.
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If we have a family of plane 2 4 6 in direct space and we know how to manage reciprocal space, of course, we go into reciprocal space with the reference r star b star c star.
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We draw the vector r star h equal 2 h star plus 4 b star plus 6 c star. This vector is perpendicular to the family 2 4 6 in direct space. Is it okay? Fine.
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Now, how to demonstrate it? You will do the job, eh? You will do this. So follow me. You will do the job.
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We have here a, b, c. And in this direct space, a, b, c, I write the first plane of the family hkl not passing through the origin. Right?
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Which will be a over h, b over k, c over n will be the intersection of this plane. Do you remember that? So a over h, b over k, c over n. That is the first plane of the family not crossing through the origin.
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Okay. So, in order to be sure that a vector is perpendicular to a plane, what do we have, what we need? To demonstrate that that vector is perpendicular to two line of the planes. If it is perpendicular to two line of that plane, it is perpendicular to the plane. Is it okay? It's fine?
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So, which two line? We can choose what we want. For example, choose this line here. This one. The line a, b. A, b, what is? Is a vector which is the difference between b, k, b over k, minus a over h. Do you remember the paragon?
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No? The difference between two vectors is the vector which is? This is a, this is b. The vector a plus b is this one.
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That's right? That's clear to all because I do. A, which is this, and this is b. So, a plus b, okay? But if I want a minus b, what I have to do? I take b in this direction, for example, and this is minus b, right?
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And I sum a plus minus b, and a plus minus b is this. Which, if you want, is this. Because this is a. So, this is a minus b, and this is a plus b. Is it okay?
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So, this would be b over k minus a over h. So, what we have to do is to demonstrate the r star h. h r star plus c d star plus l c star is perpendicular to this vector here. To demonstrate that it is perpendicular, what you have to do?
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The scalar part has to be zero. So, please do the scalar part between r star h and b over k. So, you have to calculate the star part between this, which is this.
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This scalar dot this one. Is it okay? That should be zero. To be sure that really is.
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All of you find the solution? All of you find the solution. So, we check it with this, but we can also check it with this. Because this would be c over l minus b over k. So, if you do the same, you will learn this.
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So, what you have is that r star h, capital h means h k l. r star h is perpendicular to the family of plane h k l. So, if you have everything in that space, a family of lattice planes h k l, and you want to know which is the orientation of this family of plane.
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It is easy. Go into this space and you calculate the vector h r star plus k b star plus l c star. And then you will see that it is perpendicular to the family h k l. Is it okay?
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Fine. Now, there is another property, which I will mention, and is the modulus of this vector of r star h, the modulus is one over d.
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The modulus is one over d. d is the interpolant distance of the family of plane h k l, distance between two planes, the closed planes. Is it okay? So, here is another important result.
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So, let us first see if it is a real insult. And then we will try to guess what is the importance of this. So, we want to demonstrate this. I show you here a, b, c. And then I again show you the first plane of the family.
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It is okay? Not crossing the origin. So, it will be a of h, b of k, c of l. Is it okay? And this is the first plane of this family. Now, okay, what is the distance from o to this plane? The distance d.
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The distance d is the perpendicular, which is constructed starting from o to the plane, the point m. So, this is this segment here. If this segment is perpendicular, from o we construct the perpendicular to the plane.
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And d is nothing else than o m. Is it okay? It is fine? The interpolant distance of this family is the segment of the line perpendicular to this plane conducted from o. Is it okay? All are convinced from that.
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So, now, this is d. The problem is to check how long is d. So, d will be the projection of a h, a over h, a over h on this direction, right? So, all m is the projection of a h on this direction.
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So, we can write e of h multiplied, multiply what? What is this direction? It is the r star h divided r star h in modules. Because r star h we know is perpendicular to this plane. We demonstrate just now.
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So, if we do r star h divide the modules, this is the unit vector along this direction. So, what we are doing is to calculate d h like the projection of e a divided h, which is this, by this unitary vector on the unitary vector here, r star h divided its modules.
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Are you able to do a multiplied r star h? We are only able to do a multiplied r star h. Do it.
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r star h is this. So, you have to do only a multiplied r star h.
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What you find is that really r star h is equal to 1 over d.
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Now, take care of this, because you have this information. Which information breaks you in a cyclical lattice family or plane? Which information?
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To define this family, hkl, you need the orientation and the interponal distance. This teaches you this family. When you know the orientation and d, you have calculate the family. Right?
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So, that means that r star h, this vector in the reciprocal space, represent all this family, because it defines the orientation of the family or plane and the interponal distance.
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So, only one vector, by means of this orientation and by means of these modules, fully calculate the family of the direct lattice plane, hkl.
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So, when for some reason you have to move this family of infinite planes, hkl, you rotate it, for example, you don't need to think about this family of planes, how they rotate.
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You just spare your cleverness, just looking at the r star h, how move r star h, because you know that as r star h moves, the family plane is always perpendicular.
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So, you don't control directly the family hkl. You only control r star h. It's much more simple. And the modules gives d. The direction is perpendicular to the family.
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So, you see, like this dual space reference system, which we call the cyclical lattice, is able to make simpler the calculation and to make also simpler our understanding of crystallography.
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So, when we will manage family of planes in direct space, please forget the direct space and think that there is one vector which represents this family of planes.
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One reciprocal lattice point, one vector. The vector is hkl, integer into the reciprocal space. So, that means it's a point in the reciprocal space.
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So, each point of the reciprocal space represents a family of direct space lattice planes. Is it okay? Is it okay?
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So, at my question, what represent a direct lattice point? What do you answer? Direct lattice point.
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What represent a direct lattice point?
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Direct lattice point should be a family of planes in reciprocal space.
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Yes, but we didn't make this jump. Please remain to what we did before. To what we did, I teach you before. What represent a direct lattice point?
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We made this statement at the beginning. You did this job during your preparation at home, at work, because it was a question there.
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What represent a direct lattice? What represent a direct lattice point? And then all we agreed about it.
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Then, yesterday, in the last half an hour, I asked you, what represent a direct lattice point? What represent a direct lattice point?
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So, please, try to remember.
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Okay. So, a direct lattice point represents the content of the unit cell. Is it okay?
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What represent a direct lattice? The direct lattice?
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The translation symmetry. The translation of the repetition symmetry of our question. Is it okay?
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Is it fine there? Okay. What represent a reciprocal lattice point?
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You forgot? You didn't say what you said. A friend of direct lattice planes. Is it okay?
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Can one of you, good willing, can repeat all these three definitions, please?
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The lattice point is an object of symmetry. A lattice point represent the content of the unit cell.
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Okay. What represent the lattice, direct lattice, symmetry or a crystal? What represent a reciprocal lattice point?
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A family of dark space, dark space. In the dark space. Is it okay? Is it okay? Why it represent a reciprocal dark space, crystallographic space? Why?
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Because its modules, our star modules, is equal to one over d. And this direction is a very good to the family of planes. Is it okay?
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In my experience, to long experience, I know that that will be vaporized in three two days. So I ask another good willing people to repeat all this, okay? So please you.
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The direct lattice point represents the content of the unit cell. The direct lattice is the repetition or the symmetry of the crystal.
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The repetition symmetry? And the reciprocal lattice point represents the family of planes.
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Dark space planes, crystallographic planes. Why that? Because this direction, its direction is a family of the planes in dark space and because its magnitude is one over the internal space.
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Our star is one over the internal spacing of the family of the planes. Is it okay? The H is the internal spacing of the family of planes. Is it okay?
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I want them all to be able to repeat this.
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Okay, so a lattice point represents the content of the unit cell, which is the elliptic. Therefore, the direct lattice represents the symmetry of the repetition.
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And the reciprocal lattice points correspond to a family of crystallographic planes in the direct space. And this is because the direction of the vector in reciprocal space is perpendicular to the direct plane.
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And the distance is the reciprocal of the distance between each other. Is it okay for you now? Sure? Okay, so that now is perfectly clear.
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We will check later on. Because I know that it is not always so.
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I will not start before she comes. Because otherwise we don't understand nothing. So if someone has to go there, otherwise we don't understand nothing.
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I have a question. So we have the angles between the C and C star and then A and A star and we define them by 90 minus beta.
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90 minus beta?
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It means in monochromatic. In monochromatic.
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In monochromatic. Okay, we have some time because some people...
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Okay, so we are in the monochromatic system. B is perpendicular to A and C. And beta may be richer, may be heavy, right?
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So B star is perpendicular to A and C. Because we are in monochromatic, right? B star is perpendicular to A and C.
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So B is perpendicular to A and C. Because I have a monochromatic. B star has been assumed to be perpendicular to A and C. Definition one.
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Therefore, B star will be parallel to B. Okay? Fine. A star has to be, by definition, perpendicular to B and C. To B perpendicular has to be in the plane AC. Because the plane AC is perpendicular to B.
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And to B star, of course. So A star has to be in this plane, the plane AC. Okay, now A star has to be at 90 degrees from C. Because we know that A star point dot C is zero.
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By definition. Definition one. A star dot C zero. So A star has to be at 90 degrees from C. Therefore, if all this angle here, all this angle is here is beta.
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And this has to be 90. This one has to be beta minus 90. Because this has to be 90, right? So this has to be beta minus 90. The same we can do for C star.
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It has to stay in the plane AC. And it has to be at 90 degrees from A. So this has to be beta minus 90. And the rest is clear. We are all here. Some colleague of yours is out. No. We can start.
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Now, what we are to do now? Biologists will not be happy for that. But they need to do the job. So I will try to make things as simple as possible.
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But we need to have a main mathematical background. And my students are, let me say, resistant. We are resistant in the first day. But at the end we are very happy.
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Because they learn a lot. By means of this mathematics. Because it will be applied to an armada. To an armada. It will be applied to anything you can imagine.
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So we have to do this mathematical background. And then the diffraction will be very simple. Okay? So please do not lose any step of this. Because otherwise you will not receive anything. Okay? Fine.
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Now, what we have really to do for going into the diffraction. We need to modelize the things we are in our hand. Which we have to modelize to model the crystal.
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We have to model the wave. The waves which interact with the crystal. And we have to model the interaction between the wave and the crystal. Is it okay? So we have to model at least three objects.
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The crystal. When I say model, I mean mathematically model. So we need to be able to model all this object. The crystal. The wave. The interaction wave crystals. Is it okay?
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To model the crystal. We have not to forget that the crystal has a lattice and the content of the unit cell. So in some way we have to take care of both these elements.
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The simplest way for obtaining this rule is to try to model the lattice before. Appoints. Regular points. So we can find easily a good model. Is it okay? Then we try to model the crystal.
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Because the crystal is more than its lattice. Okay? And then we will model the wave or the radiation. It is the same wave. And then we have to model the interaction, the radiation, the crystal. Is it okay?
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Now. We go through.
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Sorry, one moment.
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Okay. So. We need to introduce the Dirac delta function. Okay?
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So we have to introduce this Dirac delta function because it is very, very useful for us. Dirac delta function. The Dirac delta function centric into the point r0, r0. And we will write it like delta r minus r0.
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Here is a function highly discontinuous. It is zero everywhere. It is zero when r is different from r0. It is infinite in r0. Is it okay?
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It is fine for the two definitions. Delta r minus r0 is a function because it is a function of r, as you see. Of the space. And r is the variable into the space.
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So this function is highly discontinuous because it is zero everywhere except in the point r equal r0. Well, it is infinite. Is it okay? And the second definition is that the integral of this delta function over all the space.
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So you see below the integral the s, which means all the space, is zero. Is one, sorry. Is one. Is it okay? So the delta function will be defined by two relations.
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One, it is everywhere zero except in r0 where it takes infinite value. And the integral of this is one. This function is one over all the space. Is it okay? Now, which kind of function may, how this function may appear to us?
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In green you have the reference system. And I wrote a very sharp function, a very infinitely sharp function. Of course, I could not draw an infinitely sharp function. An infinitely sharp function which is infinite in r equal r0 and zero everywhere. Is it okay?
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Is it fine now? All for all is clear then? Fine. Now, how can we approximate this function just to have something which may be manageable? How can we approximate this extremely sharp function?
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A limit. A limit of another function. Right? Which function can you think about? Gaussian. The Gaussian function, as you remember, has a sigma. A sigma which says us how sharp is the function. So we can write in one dimension just in one dimension.
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The limit for sigma going to zero over one over sigma root two pi exponential x minus x zero squared divided two sigma squared. This is the Gaussian function. All of you are accustomed to this. Right?
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This Gaussian is like a, what is the name of a bear? It's like a bear? It's a bear. And so it's like a bear.
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And if you take sigma going to zero, it becomes sharp and sharp and sharp and sharp up to where the sigma is infinite, of course. This becomes infinite and the functions go to infinite. Is it okay?
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Fine, follow. Please, you all arrive here for this function because some exercise will make more familiar what we are saying. Okay?
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This one? Yeah. One over sigma root squared two pi. Here's two sigma. It's okay, follow. Now, if that is clear, unfortunately, that will not be useful for us.
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So, not forget that because for some exercise we will be useful, so we will recall it. So, took your notes carefully because sometime I will say look at that, look at that and you will check it. Okay. So, at the moment, the best way to represent this function is this one, like an integral.
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So, delta x minus x zero is the integral between minus infinity to infinity exponential two pi i x star x minus x zero plus x star. So, this is a delta function. We will see why, of course.
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Is it okay? So, all of you wrote that. Now, we have to give some explanation why this strange function. This strange function.
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Okay. Let us follow me in this. This is the function approximating our, the function, the delta function. Right? The exponential in one dimension. What you wrote.
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So, instead of writing integral between minus infinity to infinity, we can write like limit for g going to infinity, integral between minus g and g. Right? To this trivial.
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Okay. Now, I remember you what means, what means a i x. All of you remember that exponential i x is cosine x plus plus i.
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So, this exponential here may be divided in two parts. One is the cosine and the other is the sine. Right? Plus i sine. Is it okay?
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Fine for long? Trigger position. Really. So, you have no possibility of confusing you. So, now we have to make the integral. Let us do the integral of the sine.
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Or the sine. The sine, do you remember, the integral of a sine is? The integral of a sine is? Cosine. The integral of a sine is a cosine. Right?
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It is a cosine. It is what I would do if I'm not. It is a cosine. Okay? So, we have to do now, if we iterate this sine, we will have a cosine. And the cosine, when this cosine is calculated in minus g and g is the same. Cosine is minus g and cosine g is the same.
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When we subtract this, these two limits is zero. Right? The integral of a sine is a cosine. So, we should calculate cosine in g and minus g. So, when we do cosine minus g and cosine g is the same. Because the function cosine is even function. So, that disappears.
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And what we have is only the sine. So, the integral of a cosine is a sine. And we will have sine divided by the integral of this between minus g and g. But this disappears because the integral of a sine is a cosine and therefore disappears.
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So, what do we have to do? I wrote here. I wrote here. So, plus i for this. But cosine in minus g and g is the same. So, we can cancel it. And what remains is this. So, that integral really is the limit for g going to infinity of this function. Which is a real function.
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Is it okay? Why not?
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I'm lost at the step when you said the integral of sinus is cosine. And then, no that's fine, but then the next step comes.
00:48:06.000 --> 00:48:18.000
The integral of a cosine is a sine. So, when we calculate the integral function of this, it is this one. Or this, it is this one. And of this, it is this one.
00:48:18.000 --> 00:48:40.000
So, we have to calculate this function in minus g and g. And this function in plus g minus g. So, when we go here, cosine, when we replace x star by minus g, we have cosine 2 pi of minus g something. And minus cosine plus g something.
00:48:41.000 --> 00:48:54.000
x minus x is z. And therefore, what remains is only this. So, what we have is this function here.
00:48:54.000 --> 00:49:19.000
So, at the end, this strange formula, the delta function is this limit here. So, we have to understand how sinus this function. So, this function is made in this way.
00:49:25.000 --> 00:49:28.000
It is made in this way.
00:49:28.000 --> 00:49:34.000
Can you explain the last step again, please?
00:49:34.000 --> 00:49:40.000
This one? What do you want to explain?
00:49:40.000 --> 00:49:48.000
Why this is zero? Why this is zero?
00:49:49.000 --> 00:50:13.000
Cosine y, between minus g and g, means calculate cosine y, y minus g. So, this cosine g minus g. Minus cosine, calculate in g.
00:50:14.000 --> 00:50:22.000
Cosine minus g is equal to cosine g. So, we have cosine g minus cosine g, which is z.
00:50:24.000 --> 00:50:25.000
Is it okay?
00:50:32.000 --> 00:50:33.000
So...
00:50:43.000 --> 00:50:46.000
Why this?
00:50:53.000 --> 00:51:03.000
Because when you calculate this function, you are calculating this function for x star minus g. And x star is equal to g.
00:51:03.000 --> 00:51:17.000
At the very last time, it says, under the division P, it writes as x minus x0.
00:51:17.000 --> 00:51:21.000
Should it be 2P, or how did the two disappear there?
00:51:21.000 --> 00:51:22.000
Should it be?
00:51:22.000 --> 00:51:28.000
Like in the upper equation, it's 2P, it writes as x minus x0.
00:51:28.000 --> 00:51:31.000
It's not zero, because this is the variable.
00:51:31.000 --> 00:51:33.000
One lower division?
00:51:33.000 --> 00:51:34.000
No, no, here.
00:51:34.000 --> 00:51:35.000
No, in the next time.
00:51:35.000 --> 00:51:36.000
Oh, in the next time.
00:51:36.000 --> 00:51:39.000
She's asking, where is the two?
00:51:39.000 --> 00:51:41.000
Well, this one is zero.
00:51:41.000 --> 00:51:43.000
Because you took the sine of g.
00:51:43.000 --> 00:51:46.000
The sine is two times, you know?
00:51:46.000 --> 00:51:48.000
Two times.
00:51:48.000 --> 00:51:54.000
So, I take with this, minus this, times two times.
00:51:54.000 --> 00:51:57.000
So, you have two at the numerator.
00:51:57.000 --> 00:51:59.000
That is the reason.
00:51:59.000 --> 00:52:10.000
So, now, we show this function.
00:52:10.000 --> 00:52:19.000
Not just taking x0 equal to zero, just to make simple things, okay?
00:52:19.000 --> 00:52:20.000
For x0, y0.
00:52:20.000 --> 00:52:22.000
You see, the form of this function.
00:52:22.000 --> 00:52:26.000
The form of this function is this one.
00:52:27.000 --> 00:52:31.000
For g equal one, g does it this way.
00:52:31.000 --> 00:52:34.000
For g equal three, does it this way.
00:52:34.000 --> 00:52:40.000
So, it is a function which decreases with x.
00:52:40.000 --> 00:52:44.000
And then, when g increases, goes up.
00:52:44.000 --> 00:52:45.000
It's okay?
00:52:45.000 --> 00:52:53.000
So, when g goes to infinite, like this function should be to approximate the delta function,
00:52:53.000 --> 00:52:55.000
this goes to infinite.
00:52:55.000 --> 00:52:57.000
This disappears.
00:52:57.000 --> 00:53:02.000
And the remaining function, infinite three, shall.
00:53:02.000 --> 00:53:03.000
Is it okay?
00:53:03.000 --> 00:53:10.000
When g goes to infinite, see, g equal one, this is two.
00:53:10.000 --> 00:53:13.000
g equal three, this is six.
00:53:13.000 --> 00:53:16.000
g equal 10, that is 20.
00:53:16.000 --> 00:53:18.000
And so on, okay?
00:53:18.000 --> 00:53:26.000
So, when g becomes infinite, like in the representation, the function is enormously sharp.
00:53:26.000 --> 00:53:32.000
And this disappears, so it becomes a delta function.
00:53:32.000 --> 00:53:34.000
Is it okay?
00:53:34.000 --> 00:53:35.000
Fine?
00:53:35.000 --> 00:54:04.000
So, what I would like to transmit to you is the idea, believe me, when I say that the delta x minus x0 may be represented by this limit.
00:54:04.000 --> 00:54:05.000
Is it okay?
00:54:05.000 --> 00:54:08.000
Seems strange for us for the moment.
00:54:08.000 --> 00:54:20.000
But I can permit that this is the Fourier transform.
00:54:20.000 --> 00:54:24.000
Is it a Fourier transform?
00:54:24.000 --> 00:54:26.000
Is the expression a Fourier transform?
00:54:26.000 --> 00:54:29.000
We will study later on, okay?
00:54:30.000 --> 00:54:32.000
Is it a Fourier transform of one?
00:54:32.000 --> 00:54:35.000
A flat function equal to one.
00:54:35.000 --> 00:54:36.000
Okay?
00:54:36.000 --> 00:54:41.000
So, it is important for you to understand what is the Fourier transform.
00:54:41.000 --> 00:54:47.000
Because the Fourier transform will be used in crystallography when you calculate the electron density.
00:54:48.000 --> 00:54:50.000
It will be used in NMR.
00:54:50.000 --> 00:55:04.000
It will be used in any, any, any, even when you do a control of your hand and you do your picture.
00:55:04.000 --> 00:55:11.000
That picture in advanced cases are treated by Fourier transform to increase the clearness.
00:55:11.000 --> 00:55:13.000
So, it is everywhere.
00:55:13.000 --> 00:55:15.000
The Fourier transform is everywhere.
00:55:15.000 --> 00:55:24.000
And I can also say to you that you, you do from your, from the first moment in which you go, you are doing Fourier transform.
00:55:27.000 --> 00:55:29.000
You don't know that problem.
00:55:29.000 --> 00:55:33.000
But you, you do that continuously.
00:55:33.000 --> 00:55:38.000
So, you have to know the Fourier transform, okay?
00:55:38.000 --> 00:55:41.000
And this is the first application.
00:55:41.000 --> 00:55:48.000
The delta function may be expressed by this function, which we will understand later on.
00:55:48.000 --> 00:55:54.000
Is it a Fourier transform of a flat function with a value equal to one?
00:55:54.000 --> 00:55:55.000
Is it okay?
00:55:55.000 --> 00:55:57.000
Fine for now.
00:55:57.000 --> 00:55:58.000
Okay.
00:55:58.000 --> 00:56:00.000
It's a little bit creepy, this.
00:56:00.000 --> 00:56:01.000
But it doesn't matter.
00:56:01.000 --> 00:56:04.000
We will show it.
00:56:04.000 --> 00:56:14.000
Now, we want to expand to a three-dimensional space, this definition, which was in one dimension.
00:56:14.000 --> 00:56:29.000
So, we say that the delta r minus r0 is the integral over all the space, all the space,
00:56:29.000 --> 00:56:37.000
r star, exponential 2 pi i, r star, r minus r0, the r star.
00:56:37.000 --> 00:56:48.000
Okay?
00:56:48.000 --> 00:56:57.000
I write this function because we are using it.
00:56:57.000 --> 00:57:02.000
Fine for now.
00:57:02.000 --> 00:57:10.000
Look at what we did because it is probably something which you don't do usually.
00:57:10.000 --> 00:57:27.000
You use an auxiliary function, auxiliary function, x star to define a function into the x space.
00:57:27.000 --> 00:57:31.000
So, x star is another variable.
00:57:31.000 --> 00:57:38.000
Another space, it doesn't matter, is an auxiliary variable to define a function x.
00:57:38.000 --> 00:57:49.000
So, you introduce this strange x star, which has nothing to do with the x, and you integrate it over x star to obtain x.
00:57:49.000 --> 00:57:50.000
Is it okay?
00:57:50.000 --> 00:57:59.000
So, in some way, you are using function x star, variable x star, to define a delta function in x.
00:57:59.000 --> 00:58:04.000
And this x star disappears because it is integrated.
00:58:04.000 --> 00:58:05.000
Is it okay?
00:58:05.000 --> 00:58:12.000
So, for the moment, x star does not mean nothing for us.
00:58:12.000 --> 00:58:18.000
Has no relation with the reciprocal space points, right?
00:58:18.000 --> 00:58:20.000
Has no relation with that.
00:58:20.000 --> 00:58:22.000
It is x star.
00:58:22.000 --> 00:58:28.000
Later on, we will see that has a relation because of that star.
00:58:28.000 --> 00:58:31.000
Has something to do with the reciprocal space.
00:58:31.000 --> 00:58:32.000
Is it okay?
00:58:32.000 --> 00:58:45.000
For the moment, it is only occasional that we use x star because we could use y, we could use t, we could use delta, doesn't matter, any variable.
00:58:45.000 --> 00:58:58.000
But I use x star because later on, we will see that this auxiliary variable is related to the reciprocal space.
00:58:58.000 --> 00:58:59.000
Is it okay?
00:58:59.000 --> 00:59:00.000
For all?
00:59:00.000 --> 00:59:01.000
Fine?
00:59:01.000 --> 00:59:02.000
Good.
00:59:02.000 --> 00:59:21.000
So, delta r minus our zero, the three-dimensional delta function, is this integral on the space, x star, the space in which is defining this auxiliary variable.
00:59:21.000 --> 00:59:22.000
Is it okay?
00:59:22.000 --> 00:59:24.000
Is it okay?
00:59:24.000 --> 00:59:34.000
S is the space in which x is defined, while s star is the space in which x star is defined, or r star.
00:59:34.000 --> 00:59:36.000
Is it okay?
00:59:36.000 --> 00:59:39.000
Fine for?
00:59:39.000 --> 00:59:45.000
Can I go on, or you are questioning?
00:59:45.000 --> 00:59:46.000
Fine.
00:59:46.000 --> 00:59:47.000
We go.
00:59:47.000 --> 00:59:55.000
Now, some properties, some properties are very useful.
00:59:55.000 --> 01:00:00.000
Writing r minus r zero, or r zero minus r, is the same.
01:00:00.000 --> 01:00:01.000
Why?
01:00:01.000 --> 01:00:09.000
Because the function is an infinitely sharp function, so if you look on this way, on this way is the same.
01:00:09.000 --> 01:00:12.000
So, it is exactly equal.
01:00:12.000 --> 01:00:39.000
The other point is a strange, strange relation, because if you look, f r delta r minus r zero, with r variable, with r variable, is equal to f r zero delta r minus r zero.
01:00:39.000 --> 01:00:43.000
It is strange, right? Because f r varies.
01:00:43.000 --> 01:00:47.000
On this side, we have only r zero.
01:00:47.000 --> 01:00:49.000
So, it seems strange.
01:00:49.000 --> 01:00:57.000
When you work with the infinity, really, there are some strange results.
01:00:57.000 --> 01:01:00.000
So, why that?
01:01:00.000 --> 01:01:05.000
Look at this result.
01:01:05.000 --> 01:01:15.000
Let us calculate the first side when r is different from r zero.
01:01:15.000 --> 01:01:23.000
What would be the first side of that equation, when r is different from r zero?
01:01:23.000 --> 01:01:27.000
It would be zero, because delta is zero everywhere, except in r zero.
01:01:27.000 --> 01:01:29.000
So, that would be zero.
01:01:30.000 --> 01:01:36.000
And the second member of the equation?
01:01:36.000 --> 01:01:38.000
Zero.
01:01:38.000 --> 01:01:41.000
Zero, again, because delta is zero everywhere.
01:01:41.000 --> 01:01:44.000
We are calculating in r different from zero.
01:01:44.000 --> 01:01:46.000
So, we will have zero and zero.
01:01:46.000 --> 01:01:48.000
That is okay.
01:01:48.000 --> 01:01:54.000
Now, we calculate the first member in r zero.
01:01:54.000 --> 01:01:57.000
And then we have f r zero.
01:01:57.000 --> 01:01:59.000
What f r zero?
01:01:59.000 --> 01:02:05.000
So, as you see, this equality is correct.
01:02:05.000 --> 01:02:15.000
So, for us, f r delta r minus r zero is exactly as we say f r zero delta r r zero.
01:02:15.000 --> 01:02:17.000
Constant, f r zero, constant.
01:02:17.000 --> 01:02:19.000
It is okay?
01:02:19.000 --> 01:02:35.000
Of course, this function f is not important when it is different from zero, because the product is zero everywhere.
01:02:35.000 --> 01:02:37.000
It is okay?
01:02:37.000 --> 01:02:39.000
That is interesting.
01:02:39.000 --> 01:02:45.000
Assign, please, this carefully, because we will make some exercise.
01:02:45.000 --> 01:02:49.000
And then looks at this.
01:02:49.000 --> 01:02:56.000
And can you explain this beautiful, simple expression?
01:02:56.000 --> 01:03:04.000
The integral of any function f r, doesn't matter how complicated it is,
01:03:04.000 --> 01:03:15.000
multiplied by delta r minus r zero function, which for us is a function very complicated like this.
01:03:15.000 --> 01:03:25.000
Well, this integral is only f r zero.
01:03:25.000 --> 01:03:28.000
Look, integral of f r delta.
01:03:28.000 --> 01:03:30.000
That itself is an integral.
01:03:30.000 --> 01:03:32.000
So, it is an integral of an integral.
01:03:32.000 --> 01:03:39.000
And all this complication is very easily reduced to f r zero.
01:03:39.000 --> 01:03:42.000
Can you guess about it?
01:03:42.000 --> 01:03:47.000
Well, you just have to apply the integral to the right side of the second line.
01:03:47.000 --> 01:03:51.000
Then you have a constant, which you can pull out of your integral.
01:03:52.000 --> 01:04:06.000
If we use this equation here, and we replace f r delta r minus r zero by f r zero delta r minus r zero,
01:04:06.000 --> 01:04:10.000
f r zero goes out of the integral because it is a constant.
01:04:10.000 --> 01:04:13.000
Multiply the integral of delta.
01:04:13.000 --> 01:04:16.000
The integral of delta is one.
01:04:16.000 --> 01:04:18.000
So, we have f r zero.
01:04:18.000 --> 01:04:20.000
Is it okay?
01:04:20.000 --> 01:04:31.000
So, as you see, this strange and complicated things sometimes give us some very simple results.
01:04:31.000 --> 01:04:40.000
So, remember that the integral of any function, any function, doesn't matter how complicated this function,
01:04:40.000 --> 01:04:47.000
multiply the delta function concentrated on r zero is just r zero.
01:04:47.000 --> 01:04:51.000
The integral over all the space, of course.
01:04:51.000 --> 01:04:53.000
Is it okay?
01:04:53.000 --> 01:04:57.000
Okay, that are the...
01:04:57.000 --> 01:05:06.000
So, if it is so, if it is so,
01:05:06.000 --> 01:05:10.000
how can we represent a lattice?
01:05:10.000 --> 01:05:17.000
A lattice are points, which are highly discontinuous.
01:05:17.000 --> 01:05:28.000
Let me say, the weight of each lattice point is exactly equal to another lattice point.
01:05:28.000 --> 01:05:36.000
So, if we consider the weight, like the integral of a delta function, which is zero everywhere except on the point,
01:05:36.000 --> 01:05:47.000
we can represent the points, the lattice point, like a delta function centered on the lattice point.
01:05:47.000 --> 01:05:55.000
So, we can write the lattice function l r.
01:05:55.000 --> 01:05:57.000
Sorry, can you repeat this?
01:05:57.000 --> 01:05:59.000
Yes, I will repeat.
01:05:59.000 --> 01:06:03.000
The direct lattice points are what?
01:06:03.000 --> 01:06:11.000
Are. All equal each other, because all represent the same thing, the content of the unit cell.
01:06:11.000 --> 01:06:19.000
Highly discontinuous, because it is a point.
01:06:19.000 --> 01:06:25.000
And the weight of each point is exactly equal to the weight of the other point.
01:06:25.000 --> 01:06:32.000
So, if we take as weight the integral of a delta function, which is one,
01:06:32.000 --> 01:06:45.000
we can represent each lattice point by a delta function centered on the coordinates of the lattice point.
01:06:45.000 --> 01:06:53.000
So, the general lattice point is in r u v w, u a plus v b plus w c.
01:06:53.000 --> 01:06:59.000
And these are the delta function centered.
01:06:59.000 --> 01:07:05.000
Can you read the question first?
01:07:05.000 --> 01:07:07.000
Yes, I can write.
01:07:07.000 --> 01:07:17.000
r u v w is u a plus v b plus w c.
01:07:17.000 --> 01:07:24.000
These are the coordinates of a generic lattice point with the coordinates u v w.
01:07:24.000 --> 01:07:26.000
Is it okay?
01:07:26.000 --> 01:07:41.000
And the delta, the lattice function l r is equal, a sum over u v w, a sum over the coordinates u v w,
01:07:41.000 --> 01:07:49.000
from minus infinity to infinity, over delta r minus r u v w.
01:07:49.000 --> 01:07:51.000
Is it okay?
01:07:54.000 --> 01:08:00.000
I have the consequence for answering your question.
01:08:00.000 --> 01:08:06.000
Okay?
01:08:06.000 --> 01:08:08.000
It's okay?
01:08:08.000 --> 01:08:17.000
So, it is clear now that we have already a lattice method.
01:08:17.000 --> 01:08:24.000
Because the lattice may be represented by this lattice function.
01:08:24.000 --> 01:08:27.000
This lattice function is different from zero.
01:08:27.000 --> 01:08:33.000
Only in the points, is different from zero.
01:08:33.000 --> 01:08:40.000
Only in the points r, which are the points?
01:08:40.000 --> 01:08:48.000
u v w, holding the points r u v w.
01:08:48.000 --> 01:08:57.000
It's like r zero, r u v w.
01:08:57.000 --> 01:09:16.000
This is the lattice point.
01:09:16.000 --> 01:09:26.000
I do the lattice point like material points to avoid the observation of your colleague.
01:09:26.000 --> 01:09:27.000
Okay?
01:09:27.000 --> 01:09:38.000
So, now, these lattice points are infinitely sharp.
01:09:38.000 --> 01:09:40.000
Why?
01:09:40.000 --> 01:09:50.000
And then, suppose we have here some delta function on every lattice point.
01:09:50.000 --> 01:09:53.000
Infinitely sharp, infinitely sharp.
01:09:53.000 --> 01:10:01.000
So, we are substituting any lattice point by a delta function.
01:10:01.000 --> 01:10:02.000
Is it okay?
01:10:02.000 --> 01:10:16.000
So, this lattice point, where there are in r u v w, which is u a plus v b plus w c.
01:10:16.000 --> 01:10:17.000
Is it okay?
01:10:17.000 --> 01:10:28.000
So, if this is the origin, this will be 2a plus b.
01:10:28.000 --> 01:10:29.000
Right?
01:10:29.000 --> 01:10:32.000
So, r in this point.
01:10:32.000 --> 01:10:45.000
So, to put the delta function in this position, we have to write delta r minus r u v w.
01:10:45.000 --> 01:10:46.000
Is it okay?
01:10:46.000 --> 01:10:52.000
And this is a delta function centered on this specific u w.
01:10:52.000 --> 01:11:05.000
If you want to write on those points, we have to do some of u v w from minus infinity to
01:11:05.000 --> 01:11:09.000
plus infinity of this one.
01:11:09.000 --> 01:11:12.000
Is it okay?
01:11:12.000 --> 01:11:20.000
So, our first goal has been reached.
01:11:20.000 --> 01:11:31.000
We have a function l r, which is the sum of a delta function centered on all the points
01:11:31.000 --> 01:11:34.000
of the lattice, of the direct lattice.
01:11:34.000 --> 01:11:41.000
And this delta function are infinitely sharp.
01:11:41.000 --> 01:11:52.000
And we can now, let me say, manage this delta function because we know the definition and
01:11:52.000 --> 01:11:55.000
the main results which are interesting for us.
01:11:55.000 --> 01:11:57.000
Is it okay?
01:11:57.000 --> 01:12:08.000
So, l r is therefore the mathematical model of the delta function, of the direct lattice.
01:12:08.000 --> 01:12:10.000
Is it okay?
01:12:10.000 --> 01:12:12.000
Fine for all?
01:12:12.000 --> 01:12:15.000
I didn't say l r.
01:12:15.000 --> 01:12:17.000
I didn't say l r correctly.
01:12:17.000 --> 01:12:19.000
The last time you just said.
01:12:19.000 --> 01:12:21.000
You have to repeat?
01:12:21.000 --> 01:12:22.000
Yes.
01:12:23.000 --> 01:12:36.000
So, l r, the so-called lattice function, is the mathematical model of the direct lattice.
01:12:36.000 --> 01:12:42.000
And it is the sum of a delta function centered on the lattice point.
01:12:42.000 --> 01:12:48.000
So, we replace every lattice point by a specific delta function.
01:12:48.000 --> 01:12:50.000
Is it okay?
01:12:53.000 --> 01:12:55.000
Fine?
01:12:59.000 --> 01:13:03.000
Sometime, this is the infinite crystal of what?
01:13:03.000 --> 01:13:04.000
Infinite lattice.
01:13:04.000 --> 01:13:08.000
Sometime we have a finite crystal and a finite lattice.
01:13:08.000 --> 01:13:16.000
So, we can also write for the finite lattice that the lattice is summation from minus p
01:13:17.000 --> 01:13:23.000
over delta x minus n a, because we have n a is the period.
01:13:28.000 --> 01:13:30.000
I don't remember.
01:13:31.000 --> 01:13:35.000
Suppose we have this finite lattice from minus 4 to 4.
01:13:35.000 --> 01:13:40.000
So, a lattice, origin lattice here, then this is the period a.
01:13:40.000 --> 01:13:43.000
A lattice point here, here, here, and here.
01:13:43.000 --> 01:13:47.000
What we do is to replace this lattice point by the delta function.
01:13:47.000 --> 01:13:48.000
Is it okay?
01:13:48.000 --> 01:13:58.000
And then, this should be summation from minus p to p, delta x minus n a.
01:13:58.000 --> 01:14:08.000
Z n a, 1 n a, 2 n a, 3 n a, 4 n a, minus 1 n a, minus 2 n a, minus 3 n a, minus 4 n a.
01:14:08.000 --> 01:14:09.000
Okay?
01:14:10.000 --> 01:14:11.000
Is it okay for?
01:14:12.000 --> 01:14:20.000
So, this is a finite lattice, and this is, and before we spoke about the infinite lattice.
01:14:21.000 --> 01:14:22.000
Is it okay?
01:14:22.000 --> 01:14:24.000
Fine for questions, please.
01:14:32.000 --> 01:14:36.000
Sorry, why did you say that it's minus 4 n a, and it's not?
01:14:36.000 --> 01:14:37.000
Minus 4 a.
01:14:37.000 --> 01:14:38.000
4 a, okay.
01:14:38.000 --> 01:14:39.000
Minus 4 a.
01:14:39.000 --> 01:14:40.000
Okay.
01:14:41.000 --> 01:14:42.000
Is it okay?
01:14:43.000 --> 01:14:44.000
Other questions?
01:14:45.000 --> 01:14:46.000
No.
01:14:47.000 --> 01:14:53.000
So, we have this morning the mathematical model of a lattice.
01:14:53.000 --> 01:15:00.000
Like a sum of delta function centered on the point of the lattice, okay?
01:15:00.000 --> 01:15:01.000
Now.
01:15:10.000 --> 01:15:13.000
Now, we have 15 minutes.
01:15:13.000 --> 01:15:18.000
Okay, we can introduce some other concept.
01:15:18.000 --> 01:15:22.000
We need now to make the model of the crystal.
01:15:24.000 --> 01:15:28.000
The crystal we know is the crystal is?
01:15:28.000 --> 01:15:40.000
If we put an object to all the lattice points, that's crystal.
01:15:41.000 --> 01:15:43.000
A little more concrete.
01:15:43.000 --> 01:15:47.000
Is the regular repetition or a motif?
01:15:48.000 --> 01:15:50.000
The regular repetition or the motif?
01:15:50.000 --> 01:15:52.000
That is the crystal, okay?
01:15:52.000 --> 01:16:01.000
So, but the regular repetition of a model is not mathematically clear, what it is.
01:16:01.000 --> 01:16:09.000
We understand what is the repetition or the model, but mathematically we cannot represent it.
01:16:09.000 --> 01:16:17.000
So, we now have to make a step going from the definition, repetition of a model,
01:16:17.000 --> 01:16:21.000
to a mathematical definition of the crystal, okay?
01:16:21.000 --> 01:16:23.000
So, that is what we have to do.
01:16:23.000 --> 01:16:28.000
So, for that, we need the concept of convolution.
01:16:29.000 --> 01:16:31.000
Concept of convolution.
01:16:37.000 --> 01:16:40.000
Don't be afraid of that, it is simple.
01:16:43.000 --> 01:16:46.000
So, speak about convolution.
01:16:51.000 --> 01:16:55.000
I said before that my students were very happy at the time.
01:16:55.000 --> 01:16:59.000
So, I'm sure you will be too.
01:17:00.000 --> 01:17:03.000
So, convolution.
01:17:04.000 --> 01:17:07.000
The convolution involves two functions.
01:17:08.000 --> 01:17:11.000
The function rho and the function g.
01:17:12.000 --> 01:17:13.000
Is it okay?
01:17:14.000 --> 01:17:19.000
And the convolution is c, c.
01:17:20.000 --> 01:17:26.000
C, u is the convolution, is another space, let me say, another reference system.
01:17:26.000 --> 01:17:27.000
Doesn't matter.
01:17:27.000 --> 01:17:33.000
C, u is rho, r star, g, r.
01:17:33.000 --> 01:17:36.000
Star is an operation which will be defined now.
01:17:36.000 --> 01:17:37.000
Is it okay?
01:17:44.000 --> 01:17:50.000
Otherwise, it will be impossible to count.
01:17:50.000 --> 01:17:54.000
If you have some questions on the moment before.
01:17:54.000 --> 01:18:19.000
I will after demonstrate that we are immersed in convolution.
01:18:19.000 --> 01:18:26.000
So, yes, we will see that this is that way.
01:18:26.000 --> 01:18:31.000
So, we already know the point that it is, okay?
01:18:31.000 --> 01:18:42.000
So, we need that not only for crystallography, but for any other size.
01:18:43.000 --> 01:18:53.000
So, the convolution is the tool for having this mathematical model of the system.
01:18:56.000 --> 01:19:00.000
It is defined like rho star.
01:19:00.000 --> 01:19:03.000
Star is the operation of convolution.
01:19:03.000 --> 01:19:20.000
And it is the integral over the space S of rhoa multiplied g.
01:19:20.000 --> 01:19:28.000
But g calculated in the u minus rhoa.
01:19:31.000 --> 01:19:32.000
There.
01:19:33.000 --> 01:19:38.000
And we integrate over all the space S.
01:19:38.000 --> 01:19:39.000
Is it okay?
01:19:39.000 --> 01:19:41.000
This is what we have to do.
01:19:41.000 --> 01:19:46.000
When we do the convolution, our two functions, we need.
01:19:46.000 --> 01:19:47.000
Right.
01:19:47.000 --> 01:19:50.000
The first function as it is, rhoa.
01:19:50.000 --> 01:19:56.000
But the second function has to be changed in u minus r.
01:19:56.000 --> 01:20:00.000
And then we have to integrate over r.
01:20:00.000 --> 01:20:04.000
The function we will obtain, what it is.
01:20:08.000 --> 01:20:11.000
Is function of u.
01:20:11.000 --> 01:20:15.000
Because r is integrated and disappear.
01:20:15.000 --> 01:20:18.000
And then remain a function of u.
01:20:18.000 --> 01:20:19.000
Is it okay?
01:20:19.000 --> 01:20:25.000
So, the convolution has this.
01:20:25.000 --> 01:20:28.000
Make the passage from r to u.
01:20:28.000 --> 01:20:30.000
Is it okay?
01:20:30.000 --> 01:20:32.000
Is another variable.
01:20:32.000 --> 01:20:36.000
So, this is the convolution.
01:20:36.000 --> 01:20:39.000
The convolution is between two functions.
01:20:39.000 --> 01:20:47.000
Is it an operation which I wrote by star.
01:20:47.000 --> 01:20:49.000
Represented by star.
01:20:49.000 --> 01:20:53.000
Must have nothing to do with our star in the zebra space.
01:20:53.000 --> 01:20:56.000
Is it a single operation?
01:20:58.000 --> 01:21:02.000
And it is the integral over all the space S.
01:21:02.000 --> 01:21:05.000
Rhoa, exactly as it is.
01:21:05.000 --> 01:21:11.000
But g has to be modified in u minus r in there.
01:21:11.000 --> 01:21:12.000
Is it okay?
01:21:12.000 --> 01:21:13.000
Now.
01:21:13.000 --> 01:21:19.000
And when we do the integration, r disappear and becomes a function u.
01:21:19.000 --> 01:21:26.000
To have an idea of what I am doing and saying, we can see that.
01:21:27.000 --> 01:21:30.000
This is rhoa.
01:21:30.000 --> 01:21:32.000
This is a function.
01:21:32.000 --> 01:21:33.000
Rhoa.
01:21:33.000 --> 01:21:34.000
Is it okay?
01:21:34.000 --> 01:21:36.000
This is gr.
01:21:36.000 --> 01:21:38.000
Is this.
01:21:38.000 --> 01:21:39.000
Okay?
01:21:39.000 --> 01:21:41.000
What do we have to do?
01:21:41.000 --> 01:21:45.000
Rhoa remains like before.
01:21:45.000 --> 01:21:48.000
Rhoa, as you see in your formula.
01:21:48.000 --> 01:21:51.000
But g has to be changed.
01:21:51.000 --> 01:21:54.000
Has to be changed in u minus r.
01:21:54.000 --> 01:21:57.000
What means u minus r?
01:21:57.000 --> 01:22:02.000
U means that has to be translated by u.
01:22:02.000 --> 01:22:03.000
Okay?
01:22:03.000 --> 01:22:09.000
So it is moved from this position here to this position here.
01:22:09.000 --> 01:22:10.000
But not only.
01:22:10.000 --> 01:22:13.000
But has only to be inversion.
01:22:13.000 --> 01:22:18.000
So the form of this function, which is in this way, becomes in this way.
01:22:18.000 --> 01:22:21.000
Because r is changed with minus r.
01:22:21.000 --> 01:22:23.000
So it is.
01:22:23.000 --> 01:22:29.000
The function is inverted because you replace r by minus r.
01:22:29.000 --> 01:22:32.000
And it is translated by u.
01:22:32.000 --> 01:22:34.000
Is it okay?
01:22:34.000 --> 01:22:41.000
So while in your integral here, in the definition of the convolution,
01:22:41.000 --> 01:22:45.000
the rhoa function remains as it is.
01:22:45.000 --> 01:22:57.000
The function u becomes u minus r, which implies that the function g is inverted.
01:22:57.000 --> 01:23:04.000
That is, from r goes to minus r and translated by u.
01:23:04.000 --> 01:23:06.000
Is it okay?
01:23:06.000 --> 01:23:11.000
And the u is where we calculate the function u.
01:23:11.000 --> 01:23:13.000
So this is u.
01:23:13.000 --> 01:23:18.000
The product rhoa is this.
01:23:18.000 --> 01:23:27.000
By g u minus r, which is this and is this, is the product of this function.
01:23:27.000 --> 01:23:29.000
The product of this function.
01:23:29.000 --> 01:23:38.000
And I used this colored region here to represent the product.
01:23:38.000 --> 01:23:41.000
Because it is zero here, it is zero here.
01:23:41.000 --> 01:23:43.000
Because the product is zero there.
01:23:43.000 --> 01:23:45.000
One of the two functions is zero.
01:23:45.000 --> 01:23:49.000
The function g is zero here and zero here.
01:23:49.000 --> 01:23:53.000
So when we do the product, we only take this region here.
01:23:53.000 --> 01:23:55.000
Is it okay?
01:23:55.000 --> 01:23:57.000
So.
01:23:57.000 --> 01:23:59.000
Can you repeat this?
01:23:59.000 --> 01:24:02.000
In the integral, look at the integral.
01:24:02.000 --> 01:24:08.000
In the integral, we have to multiply rhoa by g u minus r.
01:24:08.000 --> 01:24:11.000
So we have to understand what we are doing.
01:24:11.000 --> 01:24:14.000
So rhoa is this.
01:24:14.000 --> 01:24:22.000
g u minus r means translation by u, a vector u, and inversion.
01:24:22.000 --> 01:24:26.000
So this function will appear inverted.
01:24:26.000 --> 01:24:29.000
Is it okay?
01:24:29.000 --> 01:24:39.000
So now, according to the integral, we have to multiply rhoa by g u minus r.
01:24:39.000 --> 01:24:41.000
So we have to do that before.
01:24:41.000 --> 01:24:45.000
And then, so we have to multiply these two functions.
01:24:45.000 --> 01:24:48.000
One and two.
01:24:48.000 --> 01:24:58.000
And I made this colored region because to show where it is different from zero.
01:24:58.000 --> 01:25:04.000
In this region here, g u minus r is zero.
01:25:04.000 --> 01:25:06.000
So there is nothing here.
01:25:06.000 --> 01:25:08.000
And there is nothing here, okay?
01:25:08.000 --> 01:25:10.000
So all we are in this region.
01:25:10.000 --> 01:25:16.000
So, but it is not only, we do not have only to make this product.
01:25:16.000 --> 01:25:20.000
But we have to integrate over all the r for the same u.
01:25:20.000 --> 01:25:24.000
For the same u, we have to integrate all that.
01:25:24.000 --> 01:25:30.000
So when we did this, what we learned is a function like this.
01:25:30.000 --> 01:25:34.000
And this function is flatter than the original one.
01:25:34.000 --> 01:25:36.000
Is it okay?
01:25:36.000 --> 01:25:38.000
It is flatter than the original one.
01:25:38.000 --> 01:25:42.000
This is the convolution of those two functions.
01:25:42.000 --> 01:25:46.000
You can say, wow, boring.
01:25:46.000 --> 01:25:48.000
What has this strength in?
01:25:48.000 --> 01:25:58.000
I will show you that it is not in this way.
01:25:58.000 --> 01:26:04.000
Can we just do this like one second more?
01:26:04.000 --> 01:26:06.000
This one?
01:26:06.000 --> 01:26:24.000
It seems very complicated, but after that we will see that it is more easy.
01:26:36.000 --> 01:26:38.000
Okay?
01:26:38.000 --> 01:26:47.000
Now, let us do this simple experiment.
01:26:47.000 --> 01:26:50.000
This is a wing wave.
01:26:50.000 --> 01:26:53.000
This is a wing wave, right?
01:26:53.000 --> 01:26:56.000
And this is whole material.
01:26:56.000 --> 01:27:03.000
This is a radiation which comes to the window here.
01:27:03.000 --> 01:27:12.000
And we want to know which is the intensity I can measure here in this point,
01:27:12.000 --> 01:27:19.000
which is far by the vector u from the origin of our system,
01:27:19.000 --> 01:27:22.000
which may be here, you see?
01:27:22.000 --> 01:27:24.000
This is the origin.
01:27:24.000 --> 01:27:26.000
This is our reference system.
01:27:26.000 --> 01:27:28.000
Is it okay?
01:27:28.000 --> 01:27:35.000
We want to know which is the intensity in the point u.
01:27:35.000 --> 01:27:47.000
So the intensity of this radiation is I0r because I0r.
01:27:47.000 --> 01:27:50.000
R is this vector here.
01:27:50.000 --> 01:27:56.000
This direction is r, the value of r.
01:27:56.000 --> 01:28:01.000
So I0r means that the intensity is different from according to r,
01:28:01.000 --> 01:28:07.000
maybe E0, 1, the r1 here, E0r to be very different, okay?
01:28:07.000 --> 01:28:13.000
This I0r may be different here, from here, from here.
01:28:13.000 --> 01:28:15.000
It is a function of this function, of this.
01:28:15.000 --> 01:28:17.000
Is it okay?
01:28:17.000 --> 01:28:24.000
And then we want to know which light we will receive in this point,
01:28:24.000 --> 01:28:27.000
which is different from this u.
01:28:27.000 --> 01:28:29.000
It is in u, okay?
01:28:29.000 --> 01:28:31.000
So what do we have to suppose?
01:28:31.000 --> 01:28:37.000
That each point of this window contributes to this, right?
01:28:37.000 --> 01:28:40.000
All this principle, each point of this window,
01:28:40.000 --> 01:28:45.000
give a contribution to the full intensity in this point u.
01:28:45.000 --> 01:28:50.000
So what will be the intensity of this point r?
01:28:51.000 --> 01:28:56.000
The contribution of this point r, this point r, to this.
01:28:56.000 --> 01:29:03.000
Suppose that here we have some absorber medium, like air, like water, absorber.
01:29:03.000 --> 01:29:10.000
And g is a function which describes us how much light is absorber.
01:29:10.000 --> 01:29:11.000
Is it okay?
01:29:11.000 --> 01:29:17.000
So then it will be the transferring function of the intensity here to here
01:29:17.000 --> 01:29:21.000
will be a function of u minus r, because this is r.
01:29:21.000 --> 01:29:22.000
This is u.
01:29:22.000 --> 01:29:24.000
This is u minus r.
01:29:24.000 --> 01:29:25.000
Is it okay?
01:29:25.000 --> 01:29:31.000
So it will be a function of u minus r.
01:29:31.000 --> 01:29:41.000
So the intensity E0r has to be multiplied by the transferring function g u minus r for this.
01:29:41.000 --> 01:29:47.000
And then if we want to know what is the full intensity on this point,
01:29:47.000 --> 01:29:52.000
we have to calculate the sum of this contribution here.
01:29:52.000 --> 01:29:55.000
And the sum is the integral.
01:29:55.000 --> 01:30:08.000
So as you see, the intensity we receive in the point u is exactly the convolution of the E0.
01:30:08.000 --> 01:30:12.000
The function comes here with the transferring function there.
01:30:12.000 --> 01:30:18.000
I will repeat it to be sure that you are...
01:30:18.000 --> 01:30:23.000
So this is a window.
01:30:23.000 --> 01:30:30.000
And the light comes here with the intensity E0, which is not constant with the change.
01:30:30.000 --> 01:30:31.000
So it's...
01:30:31.000 --> 01:30:32.000
What's the matter?
01:30:32.000 --> 01:30:37.000
It's a really wonderful experiment, this general experiment.
01:30:37.000 --> 01:30:40.000
E0r is this.
01:30:40.000 --> 01:30:43.000
It defines the point of the window.
01:30:43.000 --> 01:30:44.000
Okay?
01:30:44.000 --> 01:30:50.000
Now, I want to know which intensity I will observe in u.
01:30:50.000 --> 01:30:57.000
Each point of this window will contribute to u, because it gives a spherical function
01:30:57.000 --> 01:31:00.000
and they will be integrated.
01:31:00.000 --> 01:31:02.000
So which point will be u?
01:31:02.000 --> 01:31:10.000
So the point r, which is here, will contribute according to a transferring function,
01:31:10.000 --> 01:31:15.000
which is here because the light will be absorbed more or less.
01:31:15.000 --> 01:31:21.000
So there will be a function g for the transferring the intensity from this point to this point.
01:31:21.000 --> 01:31:30.000
So this transfer function will be a function of u on the side of this distance here.
01:31:30.000 --> 01:31:35.000
So the intensity will be I0r, which is this intensity,
01:31:35.000 --> 01:31:40.000
multiplied with the transferring function g mu minus r.
01:31:40.000 --> 01:31:47.000
But the total intensity will be the sum of each contribution of each point.
01:31:47.000 --> 01:31:50.000
That is the integral over the r.
01:31:50.000 --> 01:31:54.000
At the end, we will learn the intensity in u.
01:31:54.000 --> 01:31:57.000
Is it okay?
01:31:57.000 --> 01:32:06.000
Now, here, we are immersed in some convolution.
01:32:06.000 --> 01:32:08.000
Look at that.
01:32:08.000 --> 01:32:10.000
The window there.
01:32:10.000 --> 01:32:12.000
The window there.
01:32:12.000 --> 01:32:14.000
Suddenly, do the same.
01:32:14.000 --> 01:32:22.000
You will see light from each point, each window, small window, or there.
01:32:22.000 --> 01:32:26.000
And each point gives light to you.
01:32:26.000 --> 01:32:29.000
And therefore makes the convolution.
01:32:29.000 --> 01:32:33.000
So we are immersed in convolution.
01:32:33.000 --> 01:32:35.000
Okay?
01:32:35.000 --> 01:32:39.000
So that's to clarify the point.
01:32:39.000 --> 01:32:41.000
That's to clarify the point.
01:32:41.000 --> 01:32:48.000
So be more interested in convolution because it's part of our life.
01:32:48.000 --> 01:32:49.000
Okay?
01:32:49.000 --> 01:32:56.000
So this afternoon, this afternoon, we will try to apply that to the crystals.
01:32:56.000 --> 01:32:58.000
Is it okay?