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Okay. Let's get started. Welcome back.

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Now that this is the 12th lecture of the condensed matter course.

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Hello. Hello. Is it Monday still? So when we left off, we were talking about scattering the standard scattering experiment.

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If you have some sample, you send in some way.

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That way vector k and something comes out at wave vector.

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K prime. And by measuring what comes out at various wave vectors.

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K prime, we're supposed to be able to deduce the structure, the microscopic structure of the sample.

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I want to emphasise again that this type of experiment is the way to understand the

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microscopic structures of things on a scale smaller than you can actually see with your eye.

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The number of Nobel Prizes that have been awarded for people doing this type of experiment is enormous.

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There's like 20, 25, 30 IG Nobel Prizes, just a couple of famous ones.

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You know, Oxford's own Dorothy Hodgkin, Somerville's favourite daughter, won a Nobel Prize for discovering the structure of penicillin and insulin.

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Using X-ray scattering, Watson and Crick very famously figured out the DNA double helix structure,

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which they figured out by looking at X-ray scattering data taken by Rosalind Franklin.

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Rosalind Franklin sadly died extremely young and was therefore denied a Nobel Prize as well as the rest of her life.

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It is very sad, but everyone should know her name nonetheless.

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Anyway, this is an extremely important experiment and that's why we're spending so much time on it.

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At least this lecture, next lecture, and maybe some of the lecture thereafter.

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One of the things we derived in the last lecture is that in order to get scattering from K to prime,

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we must satisfy in a crystal the Laue condition that k minus K prime must be a reciprocal lattice vector g.

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Now, at the very end of of the last lecture, I claimed that this condition,

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this Laue condition is very similar to a condition that you have already thought about when you considered just simple diffraction off of planes.

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If you have light or x rays or something, or neutrons coming into a set of parallel planes, a family of lattice planes and you have are different.

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We understand how you can get diffraction of this sort at certain angles off of parallel family of planes.

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And I'm going to claim that this this condition here is actually equivalent to that.

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So first, what we're going to do is we're going to review some things about diffraction off of planes.

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Then we're going to show that the LAUE condition and the diffraction condition the same.

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So let's abstract this picture a little bit to just a set of planes like this.

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And we'll imagine we have some wave coming in like this.

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It reflects off like this at an angle theta, but also another parallel wave coming in like this.

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Exactly the same wave just goes down to one layer further.

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The fracture off like this, the same angle theta like this.

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And I'm just going, I'm going to actually emphasise this and maybe I'll even write it down because it's an extremely common source of error.

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Total deflection angle. Deflection angle equals to theta.

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Why is that while there's theta there and another theta there.

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So the difference, the difference in angle between the incoming wave and the outcome wave is actually two theta.

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This this causes all sorts of grief to a lot of people.

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So make sure, you know, when you're talking about theta and when you're talking about two theta.

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So as you did this this type of calculation in prior years,

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you know that the what you need to do is you need to calculate the additional distance travelled by the second wave.

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So if I drop a perpendicular here and a perpendicular here, the additional distance here is d sine theta if d is the distance between the the layers.

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So in order to get constructive interference, constructive, maybe, maybe I'll write this first.

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The additional distance. Additional distance distance is then two d sine theta equals two d sine theta.

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So de sine theta on this side and then another d sine theta on that side.

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So the additional distance taken by the by the wave that reflected off the

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second plane is two d sine theta to get constructive interference constructive.

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Interference into Ference.

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I think as the term goes on, my writing gets worse. So we get is end lambda equals two sine theta.

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So you have to have an integer a number of wavelengths contained in that extra distance.

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Does that all look familiar from previous years? Yeah. Okay, so I'll put a box around it because it's an important equation.

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This is known as the Bragg condition, Bragg condition, after William Henry and William Lawrence Bragg,

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the father and son team who pioneered X-ray scattering.

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William Lawrence Bragg winning a Nobel Prize at the age of 25 for his work on X-ray, scattering for a chocolate bar.

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Does anyone know who the second youngest Nobel laureate was? It's a name you all know.

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So those are all really good guesses. But those are all wrong.

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Really close to all of them. Same era. Heisenberg.

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He said it up there. Oh, my gosh. Close enough. Yes, Heisenberg.

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Exactly. Dirac was just a few months older than then, then Heisenberg.

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All right, so now what I claim to you was that this condition, this Bragg condition, is actually forgetting constructive interference.

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You only get constructive interference coming off on certain angles from your crystal

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that that condition is actually the same condition as this lousy condition over here,

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which is the conservation of crystal momentum. So let's see if we can prove that you write this down, because this is important proof equivalence.

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Of Bragg, Bragg and Laue.

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Okay, so same diagram. We have two parallel planes in a family of lattice plane.

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You can actually you can kind of understand why it is that these two are going to be related.

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Remember from last lecture that reciprocal lattice vectors are associated with families of lattice

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planes so that gee up there is going to be telling you something about this family of lattice planes.

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So the fact that we have parallel planes is because we have a reciprocal lattice vector.

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Those are sort of equivalence statements. So here we have our family of lattice planes separated by some distance.

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D We have an incoming wave at wave vector.

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K And outgoing at wave vector. K Prime that.

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And then since we have a family of lattice planes, there is a orthogonal vector g which is a reciprocal lattice vector.

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That was one of the things we derive last time that every time we have a family of last planes,

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there's a reciprocal lattice vector perpendicular to it.

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Since we have since we have elastic scattering, the magnitude of K and the magnitude of K prime are the same.

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And let's actually define some unit vectors.

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Unit vectors, vacc k hat, k prime hat and g hat to be pointing in the the appropriate directions and then with a little bit of geometry.

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K hat dot jihad equals sine theta.

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Just take a second. Make sure you're convinced of that. If K hat and jihad were pointing in the same direction.

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Then K had a g hat would be one sine theta would also be one in that case.

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Whereas K prime hat which is going in the opposite direction, jihad is minus sine theta.

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Okay. And we can also use we're going to need that, that k vector.

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We can write as the magnitude of K which is two pi over lambda with lambda the way vec with the wavelength times k hat.

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Everyone happy with this so far. Yeah. Okay, good.

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So we're going to assume the Laue condition. So assume Laue that is k minus K prime equals g.

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So assume that satisfied and I can rewrite that then as two pi over lambda times k hat minus k prime hat equals g

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then dot with g hat to get to pi over lambda g hat okay hat minus g hat okay prime hat equals g hat to g vector.

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Okay then we'll substitute in. We just had the jihad k hat is sine theta so we'll substitute that in so we get to pi over lambda.

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So this is sine theta here. The second term is minus sine theta and then g hat g vector is just the absolute value of g vector and one.

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Still happy. Okay. No, not happy.

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Yeah. Still happy. Okay, so just a little bit of rearrangement.

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This becomes two pi over absolute g vector times two sine theta equals lambda.

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And now remember what we said, what we derived last time about reciprocal lattice vectors.

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The spacing between lattice planes D is two pi over the shortest reciprocal lattice vector in a given direction.

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G min. So any reciprocal lattice vector g vector has to be some integer times the minimum lattice vector in that given direction.

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So PN is some integer integer.

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So substituting this equation in this equation into the equation just above gives us two d sine theta equals and lambda the bragg condition.

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So what we've shown is that the Laue condition,

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this conservation of crystal momentum is the same thing as requiring constructive interference or having the Bragg condition satisfied.

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Okay. Good. All right. So supposing that we have the bad condition or the lousy condition satisfied?

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One of the things we derived last time was that the intensity of scattering.

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Intensity of scattering is proportional to the structure factor.

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At G came out as K prime squared structure factor s equals the four eight transform of the scattering potential v of.

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Ah. Does that sound familiar from last lecture? I hope.

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Okay. But that still leaves up in the question what is V of ah and what v of are is what the potential your wave feels as it goes into the crystal.

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Depends on well, depends on the crystal, but it also depends on what kind of wave you're using.

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They the wave interacts with the crystal very differently depending on the type of wave that you're using.

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So the easier case to think about is neutrons.

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So case one, neutrons, neutrons interact.

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Mainly mainly are via nuclear forces by a short range short range nuclear forces and nuclear forces.

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So what does that mean? It means the neutrons go into the sample and they see very short range potentials associated with the nuclei.

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And we can write that we of our equals or is proportional to, I guess, some of our atoms,

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all atoms alpha in the whole system, some coefficient B alpha times a delta function potential at r minus R alpha.

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So basically saying that every nucleus.

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So our alpha is position of atom alpha. Position of atom alpha.

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And B Alpha is in the in the magnitude of the interaction with that nucleus.

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It's known as the nuclear scattering length. Scatter length, length.

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And so the essence of this equation is basically saying that the the neutrons coming in,

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they see very sharp peaks in potential at the positions of the nuclei.

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And and that's all they don't see anything else. Now these coefficient B, they have to do with nuclear physics.

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And nuclear physics can be extremely complicated.

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They can be positive or they can be negative depending on the particular nucleus we're talking about.

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Positive means you have a repulsive interaction with the nucleus. Negative means you have an attractive interaction with the nucleus,

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and you can have the either and the size of these B's can vary by lots, depending on the particular nucleus you're thinking about.

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And really the only way to know what B is for a particular nucleus is to just look it up on a table or measure it.

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You know, you could have two atoms on the periodic table that are right next to each other, like carbon, nitrogen and the nucleus.

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Scattering length is completely different. Some can be positive, some can be negative.

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It just goes all over the place. So it's actually something that we can't actually predict very easily.

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Even nuclear physicist, even good nuclear physicists have a hard time predicting these things.

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So we just have to resort to a table and assume that these things are known or they somehow we're going to be able to measure them.

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But given that we have this sort of functional form of the interaction with the nuclei,

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we can then calculate the structure factor, structure factor SMG,

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which remember the the definition is that it's actually deeper and therefore you transform in the unit

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cell the for you transform in the unit cell of e to the i g r times v of r something we derive last time.

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And if we actually plug in the, the form that we just wrote down above,

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we have some of our alpha b alpha delta, three dimensional delta function R minus hours of alpha.

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Now we'll let the Delta Function Act and we get the fairly simple outcome that the sum over Adams alpha Adams Alpha in unit

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cell of B sub alpha e to the i g dot our sub alpha where r sub alpha is the position of position of atom alpha and unit cell.

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Adam Alpha. So there we have it.

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So that's the structure factor for scattering, scattering of neutrons against nuclei.

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Now that's versus for neutrons. Neutrons being fairly simple.

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But there's a second case that we need to worry about, which is x rays and x rays and more complicated x rays.

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X-rays. Well, they're electromagnetism X-rays, you know, like light.

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They mainly scatter. Mainly scatter from electrons.

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Scatter from electrons. From from electrons.

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Electrons. Via Thompson scattering in space.

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Thompson scattering last year. So. Sound familiar? Yeah. So it may be scattering from electrons via Thompson scattering.

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There's a little bit of Thompson scattering off the nuclei, but really very, very,

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very small amount because the nuclei are very heavy compared to the electrons.

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And so they're very hard to scatter off of. It's hard to push them around. It's easy to push the electrons around because they're light.

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So mainly the of our four x rays, the effective interaction is proportional to the density of electrons.

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The density of electrons. So what's the density of electrons?

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Well, we can write sort of an approximate expression for the density of electrons,

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which will rate as some sum over atoms alpha Z, sub alpha Z here being the atomic number, atomic number of atom, alpha,

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atomic number being the number of electrons in the in the atom,

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also the number of protons in the atom times some function which I'll call g sub alpha of R minus R alpha,

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which I guess we can call something like a shape function. Which is normalised.

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So it's integral as one and we can compare that to what we have over here.

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Looks kind of similar, except here it's a delta function and here it's not a delta function.

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So the difference being that when you if you're a neutron, you you scatter off of only the nucleus.

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Whereas if you're an X-ray, you scatter off of the whole atom.

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So you can kind of think of this G as being a very, very fat delta function.

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It's spread out, spreading out the delta function into some functionality or just takes into account the entire size of all the electrons in the atom.

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So think of it as a sort of a fat helper function.

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So now I should say, however, that this this form where we're summing up over the shapes of all these different atoms,

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times the number of electrons in each atom is a little bit approximate.

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And the reason it's a little bit approximate is because there's an assumption,

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a hidden assumption here, that the shape of the atom is independent of its environment.

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If an atom happens to be bonded to one particular type of atom, it will actually distort the electron cloud of the atom a little bit.

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So the shape will change a little bit, and we've ignored that.

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And if it's bonded to a different type of atom, its shape will will change a little bit and maybe another direction.

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It will depend on which direction it's bonded and so forth and so on.

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But to a very good first approximation, this is a fairly good way to describe the the shape of the density of electrons in any crystal,

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just you only have a shape of for a particular type of atom and you add it up over

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all atoms and you ignore the fact that neighbouring atoms will distort the shape.

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Okay, good. All right.

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So we can take this form of the, of the density of electrons and again construct the structure factor as a g which is now sum over all atoms.

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Atoms alpha integral D three are very similar to what we did just over here, over a unit.

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So then we have the I got R, we have Z Sub Alpha and we have g sub alpha.

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Of our ways are alpha. If we do that for you, transform.

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What we get is extremely similar to what we get over here, very similar to this equation here.

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What we get is of the following form sum over atoms, alpha and unit cell of each of the.

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I got our efforts of alpha use different letter as a function of g.

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This is known as the atomic form factor. Form factor.

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It depends on the type of atom we're talking about.

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And if you want to know what the atomic form factor actually is, so actually maybe just compare it to over here.

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The atomic form factor takes the place of the nuclear scattering length over here of this takes the place of the B alpha.

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But the overall structure of that of the of the structure factor here out of this equation stays pretty much the same.

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The only difference is the atomic form factor is also a function of the, of the the reciprocal lattice factor g.

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Whereas over here it's not in the reason it's not over here is because we have a nice simple

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delta function interaction over here and we have a more complicated interaction over here.

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If you want to know the actual form of F of G, it's actually an integral d3r, it's just the 48 transform g r r g sub alpha.

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I guess maybe this about for the z sub alpha first, then g of alpha of R.

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So it's just the for the transform of the shape function g.

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The surprise maybe is this integral is actually the overall space, not just over the moon itself.

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And that I'm not going to go through the whole derivation.

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I think there's a revision homework assignment that asks you to try to work through it maybe.

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Or there was last year or maybe it's not there this year. But it's worth trying to do.

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It's in the book. But to try to convince you that the integrals should be over all space, let me make the following argument.

200
00:21:50,690 --> 00:21:55,100
So these the shape functions g have long tails.

201
00:21:55,340 --> 00:21:59,240
I mean, this tails are small, but they are. But they go out a long way.

202
00:21:59,720 --> 00:22:03,770
If you're only adding up over atoms within the unit cell,

203
00:22:03,950 --> 00:22:10,910
you would never feel the tail of one of those those shape functions because you're only adding up over alphas within the unit cell.

204
00:22:11,330 --> 00:22:14,500
What you should be including the tails of all your neighbours.

205
00:22:14,510 --> 00:22:20,090
So there's atoms way out there and other unit cells. Then you should be feeling the tails of those of those shape functions.

206
00:22:20,390 --> 00:22:26,150
And so the fact that you're integrating all overall space is keeping track of both the

207
00:22:26,150 --> 00:22:31,780
short range part of G and the tails of the of the GS from other atoms farther away.

208
00:22:31,790 --> 00:22:35,510
So they all sort of get re some and come into the the form factor here.

209
00:22:35,880 --> 00:22:39,230
Okay. So I'm not going to do the derivation. It's probably not worth doing.

210
00:22:39,560 --> 00:22:43,020
But. Might be worth doing for fun.

211
00:22:43,170 --> 00:22:48,809
Anyway, the the form of this form factor more or less looks like this.

212
00:22:48,810 --> 00:22:50,640
So we'll draw alpha here.

213
00:22:51,030 --> 00:23:04,680
We'll draw G over here for g equal to zero for g equals zero equals zero f sub alpha is just z sub alpha just basically counting.

214
00:23:04,800 --> 00:23:09,840
You can kind of see if you insert zero into that integral all over space.

215
00:23:09,840 --> 00:23:13,350
G is a normalised shape function, so you just get Z back out.

216
00:23:13,950 --> 00:23:19,200
So, so if g zero you get z sub alpha up here,

217
00:23:19,470 --> 00:23:28,710
but then it sort of decays slowly down to something much smaller and the decay length is more or less one over the radius of the atom.

218
00:23:30,090 --> 00:23:38,170
Radius of atom. Very frequently, people just make the approximation that F of alpha is a constant,

219
00:23:38,500 --> 00:23:44,530
and the constant is just the number of electrons because you know, if you have more electrons, you scatter more off of it, off of the atom.

220
00:23:44,740 --> 00:23:52,090
So just let's assume that the form factor is independent of G and it's basically just proportional to the number number of electrons in G.

221
00:23:52,330 --> 00:23:58,569
It's not a horrible approximation. It isn't completely right, but it's not too bad and makes your life a lot simpler.

222
00:23:58,570 --> 00:24:05,620
It makes the the x ray problem look a lot more like the the neutron problem where there's this form factor.

223
00:24:05,620 --> 00:24:09,340
This B doesn't depend on G anymore. It doesn't depend on the reciprocal lattice factor.

224
00:24:09,490 --> 00:24:13,570
But strictly speaking, F should decay as a function of g as well.

225
00:24:14,620 --> 00:24:21,069
So generally, whether we're talking about X-rays or we're talking about neutrons,

226
00:24:21,070 --> 00:24:27,820
we will always have s of g of the following form some over atoms, alpha atoms, alpha in units.

227
00:24:27,820 --> 00:24:41,049
So each of the i g are alpha and then well I'll write it as f of g but or or b sub alpha.

228
00:24:41,050 --> 00:24:48,700
If we're talking about neutrons, it's convenient frequently to rewrite this in terms of Miller Indices.

229
00:24:49,030 --> 00:24:56,140
So if we rewrite these statements in terms of Miller Indices, we would have some of our alpha and units all.

230
00:24:58,680 --> 00:25:13,499
E to the two pi i h times you alpha plus k times v alpha plus l times w alpha f sub alpha hk l so here hk

231
00:25:13,500 --> 00:25:31,650
l hk l is the Miller Indices Miller Indices of g g vector and you alpha v alpha w alpha equals position.

232
00:25:33,690 --> 00:25:38,780
Position of atom alpha alpha in unit cell.

233
00:25:42,480 --> 00:25:47,050
Okay. That make people happy. Good so far.

234
00:25:47,920 --> 00:25:51,550
All right. Let's actually do an example to make make some of this more clear.

235
00:25:52,180 --> 00:25:58,510
So example, a material we talked about before, caesium chloride.

236
00:25:58,990 --> 00:26:04,090
So remember, caesium chloride is simple cubic with basis.

237
00:26:04,870 --> 00:26:17,769
With basis. And the basis has a caesium at 000000 and a chlorine at one half.

238
00:26:17,770 --> 00:26:23,200
One half. One half. And I think I may have a picture of that.

239
00:26:23,730 --> 00:26:26,590
Yeah, I hope this looks familiar. We spent some time discussing it.

240
00:26:27,970 --> 00:26:38,060
So using that expression for the structure factor, we can we can write s at a scale equals.

241
00:26:38,560 --> 00:26:41,890
So it's going to be a sum over the two atoms in the unit cell.

242
00:26:42,280 --> 00:26:47,680
So the first atoms at 000. So each of the two pi, then we have a bunch of zeros upstairs.

243
00:26:47,980 --> 00:26:55,630
So the first time will just be f caesium because either the two pi is zero is just going to give you one good.

244
00:26:55,900 --> 00:27:01,040
And the second term will be f chlorine times e to the two pi.

245
00:27:01,390 --> 00:27:09,430
And then we have h times one half plus k times one half, plus L times one half.

246
00:27:11,230 --> 00:27:22,450
Plugging in the position of chlorine, UVA and W both being one half for chlorine and I can rewrite that as F strictly speaking.

247
00:27:23,170 --> 00:27:32,640
Strictly speaking, I should have given f it caesium an f chlorine h can l dependencies as I just ranted about up up there.

248
00:27:32,650 --> 00:27:39,730
So they're strictly speaking, they're also functions of h can't l weak functions of h candle as well.

249
00:27:39,760 --> 00:27:46,450
They depend slightly on on the magnitude of G two, but I typically won't write that right, that dependence.

250
00:27:46,450 --> 00:27:51,040
But you have to remember that it's actually there so we can rewrite that as f caesium in here.

251
00:27:51,040 --> 00:27:57,759
I'm going to drop the dependence on h can't l for convenience times f plus f chlorine and this factor of each of the

252
00:27:57,760 --> 00:28:04,450
two pi i h times one half plus k times one half was l times one half is actually minus one in the H plus plus l.

253
00:28:06,390 --> 00:28:09,870
Good. So this is the structure factor. And you square this thing.

254
00:28:10,050 --> 00:28:18,240
This will give you the equity squared. It will give you the amplitude of scattering to O with k minus k prime given by Miller Indices.

255
00:28:18,240 --> 00:28:22,170
H can l. Okay. All right. Let's do another example.

256
00:28:23,400 --> 00:28:27,060
Example 2x2. Pure caesium.

257
00:28:28,020 --> 00:28:30,430
Pure caesium. Now you remember that that's BC,

258
00:28:31,200 --> 00:28:49,110
but PCC can be thought of as simple cubic with a basis with basis where the basis is caesium at 000 and caesium at one half, one half, one half.

259
00:28:49,800 --> 00:28:53,340
So in other words, we're just replacing the chlorine by another caesium.

260
00:28:53,670 --> 00:28:59,420
We discussed this before. And so there's a picture of it.

261
00:28:59,450 --> 00:29:06,000
So you just replace the chlorine with the caesium. So we get exactly the same result, except now chlorine.

262
00:29:06,020 --> 00:29:09,050
This is the chlorine from the centre of the unit, the conventional unit cell.

263
00:29:09,350 --> 00:29:21,830
It's going to be the same as if caesium. So now we'll get s h cl equals F caesium plus F caesium times minus one to the H plus k plus l.

264
00:29:23,090 --> 00:29:31,760
And we want to we can factor all the F caesium and F caesium times one plus minus one, the eight plus plus L.

265
00:29:32,600 --> 00:29:38,270
Now you'll notice that this vanishes unless write this down vanishes.

266
00:29:40,940 --> 00:29:47,800
Unless. H plus k plus l is even age plus k plus l is even.

267
00:29:49,930 --> 00:29:54,420
This is what's known as a selection rule or a systematic absence.

268
00:29:54,430 --> 00:30:02,020
Maybe I'll write those down. Selection rule. Rule or systematic absence.

269
00:30:06,750 --> 00:30:10,630
Systematic absence. Absence. Absence.

270
00:30:10,920 --> 00:30:14,610
Sense. Systematic absence.

271
00:30:15,360 --> 00:30:20,760
The idea being that you will get no scattering unless h plus k plus l is even.

272
00:30:20,940 --> 00:30:27,120
So all of the Miller indices with H plus people to L being odd have no scattering at all.

273
00:30:27,120 --> 00:30:31,109
They are systematically absent from this, from the scattering or another way words.

274
00:30:31,110 --> 00:30:36,089
In order to get scattering, you must satisfy this selection rule.

275
00:30:36,090 --> 00:30:41,520
And this is generally true for BC lattices, for BC lattices, for BC lattices.

276
00:30:41,610 --> 00:30:46,530
These are the selection rules now.

277
00:30:48,390 --> 00:30:56,820
So whenever you do a scattering experiment, if you discover that all the Miller indices scattering by these wave vectors with H plus K plus L,

278
00:30:57,690 --> 00:31:05,549
any time if you discover that H was odd that there's no scattering at those Miller indices, but you do find scattering, for instance.

279
00:31:05,550 --> 00:31:10,560
Kate Purcell even. It's a very, very good bet that what you have is a BC lattice.

280
00:31:11,280 --> 00:31:15,960
However, you can be fooled. How can you be fooled? You can be fooled.

281
00:31:16,110 --> 00:31:18,660
For example, let's consider the case of caesium chloride.

282
00:31:19,020 --> 00:31:26,580
Suppose just coincidentally, F caesium and F chlorine were equal to each other by some coincidence,

283
00:31:26,850 --> 00:31:30,210
in which case you'd get the same vanishing over here. You'd have the same selection wrong.

284
00:31:30,750 --> 00:31:37,139
But if they're different, maybe by a little bit, then the selection rule is almost satisfied.

285
00:31:37,140 --> 00:31:42,990
Meaning that you have to look really, really carefully to see that there was some amplitude because these would almost exactly cancel.

286
00:31:43,770 --> 00:31:48,090
So caesium chloride does not have this property of caesium and chlorine are not very similar,

287
00:31:48,180 --> 00:31:52,650
but there's a very, very similar material for which these two are extremely close.

288
00:31:52,860 --> 00:31:57,330
And you might be mistaken, you might if you looked at the data, you might make a mistake.

289
00:31:57,540 --> 00:32:00,990
And unless you looked at it really carefully, you would think that you would have the same selection role.

290
00:32:01,110 --> 00:32:09,180
Anyone want to guess what that material is? Well reduction potassium chloride so that we have there.

291
00:32:09,780 --> 00:32:14,250
I'll give this to you anyway, but it's not the right answer. Potassium chloride is a very good gas.

292
00:32:14,520 --> 00:32:20,849
And the reason the reason thallium chloride is a good gas is because remember that these aff these form factors are

293
00:32:20,850 --> 00:32:29,370
basically proportional to the atomic number and the atomic number of caesium and chlorine are extremely different.

294
00:32:30,180 --> 00:32:36,090
Chlorine is what is chlorine? Chlorine is number 17 and caesium is number number 55.

295
00:32:36,330 --> 00:32:43,310
So these two numbers will be extremely different. But if you replaced the chlorine, the caesium with potassium, potassium is right above it.

296
00:32:43,320 --> 00:32:51,959
On the periodic table it's chemically very similar. Then potassium has as atomic number 19 and chlorine has atomic number 17.

297
00:32:51,960 --> 00:32:55,440
So these would be pretty close to each other and you'd have a near cancellation.

298
00:32:56,480 --> 00:32:58,520
Potassium chloride is not the right answer because in fact,

299
00:32:58,520 --> 00:33:02,299
potassium chloride does not take this crystal structure and it takes a different crystal structure.

300
00:33:02,300 --> 00:33:05,780
It's FCC. It's like it's like sodium chloride. It's FCC with a basis.

301
00:33:06,350 --> 00:33:09,379
There's another material, but that's a hint. What's the other material?

302
00:33:09,380 --> 00:33:13,300
That would be more like caesium chloride. No.

303
00:33:13,350 --> 00:33:17,150
No, no. Magnesium. No, no, no.

304
00:33:17,230 --> 00:33:20,500
All right. I'm going to eat this one myself. It's a caesium iodide.

305
00:33:21,220 --> 00:33:27,400
If you replace chlorine with iodine, which is which is to two spaces under it on the periodic table,

306
00:33:28,570 --> 00:33:33,820
that iodine has has atomic number 53 and caesium has atomic number 55.

307
00:33:34,030 --> 00:33:37,929
So the cancellation between these two would be almost exact.

308
00:33:37,930 --> 00:33:42,820
I mean, only within a few percent you would, you would have a slight non cancellation.

309
00:33:42,970 --> 00:33:49,570
So you're very easy to make the mistake and look at the data and think that you have a BSC lattice because you think there's a systematic absence.

310
00:33:49,750 --> 00:33:53,650
But if you look at it much, much more closely, you discover that in fact, it's not perfectly cancelling.

311
00:33:54,040 --> 00:33:57,810
Okay. And we're happy. All right. So.

312
00:34:00,200 --> 00:34:04,220
Right. We can do the same thing with FCC lattices.

313
00:34:05,420 --> 00:34:08,990
Consider the case of FCC. FCC lattices.

314
00:34:12,230 --> 00:34:19,700
So for FCC we can think of FCC equals equals a basic, simple cubic times a basis simple cubic.

315
00:34:22,690 --> 00:34:27,760
Time's a basis. Incidentally, I should I should have mentioned this.

316
00:34:27,940 --> 00:34:30,010
If we were looking at nuclear scattering lengths,

317
00:34:30,340 --> 00:34:35,350
then there's absolutely no reason you should ever expect any two would be the same independent of their atomic number.

318
00:34:35,350 --> 00:34:40,419
Because, I mean, you could have a strange coincidence that the scattering length of hydrogen happens

319
00:34:40,420 --> 00:34:43,960
to happen to match the scattering length of iron or something crazy like that.

320
00:34:44,110 --> 00:34:50,740
But just because they are close to each other in the periodic table doesn't mean that they have close nuclear scattering links completely unrelated.

321
00:34:51,160 --> 00:34:55,810
So it's sometimes good if you think there's a near cancellation to switch from x rays to neutrons,

322
00:34:56,020 --> 00:34:59,890
because a near cancellation with x rays won't be a near cancellation with neutrons generally.

323
00:35:00,550 --> 00:35:13,720
All right, back to FCC. The FCC. If we can think of it as simple cubic with a basis where the basis equals basis equals 0000 and one half,

324
00:35:13,720 --> 00:35:22,630
one half zero and one half, zero one half and zero one half one half.

325
00:35:24,730 --> 00:35:30,639
Actually, I think I have a picture of that as well. Remember this from from a couple of lectures ago.

326
00:35:30,640 --> 00:35:35,160
The FCC lattice is for inter penetrating simple qubits.

327
00:35:36,530 --> 00:35:39,099
Okay, so we can write the structure factor.

328
00:35:39,100 --> 00:35:49,180
The structure factor is going to be the sum over all these four four lattice points within the conventional unit cell of E to the two pi,

329
00:35:49,300 --> 00:36:06,490
and I'll abbreviate this hk l dotted into you the w point alpha, which we can then write as one plus e to the i pi h plus.

330
00:36:07,060 --> 00:36:11,230
So one is from 000. Then we'll have h plus k plus.

331
00:36:11,230 --> 00:36:17,230
Either the i pi h plus l plus the i pi k plus l,

332
00:36:19,150 --> 00:36:30,550
which is then equal to one plus minus one to the h plus k plus minus one to the plus L plus minus one to the K plus l.

333
00:36:31,960 --> 00:36:47,680
And I claim that this vanishes unless h cancel h can't l all even or all odd.

334
00:36:50,420 --> 00:36:53,510
And to convince you of that, if at all.

335
00:36:53,510 --> 00:36:57,080
Even then, all those sets of exponents are even so it's one plus one plus one plus one.

336
00:36:57,530 --> 00:37:02,659
If h can't l are all odd. Then again two odds added together always give you an even.

337
00:37:02,660 --> 00:37:09,140
So it's again one plus one plus one plus one. But if you have, for example, one odd and two evens, let's think about that case.

338
00:37:09,410 --> 00:37:15,889
Let's suppose H is odd and canal or even then h plus k is odd, h plus l is odd, but k plus l is.

339
00:37:15,890 --> 00:37:19,520
Even so we'll have one minus one, minus one plus one and vanishes.

340
00:37:19,850 --> 00:37:27,499
And you can convince yourself that unless h can either all even or all odd, you'll get that will vanish.

341
00:37:27,500 --> 00:37:33,950
So this is a selection role rule for for FCC.

342
00:37:37,250 --> 00:37:40,610
So where do these where do these selection rules actually come from?

343
00:37:41,570 --> 00:37:47,240
If you remember back to the last lecture or the lecture before we discussed families of lattice planes and how how

344
00:37:47,240 --> 00:37:53,840
it is that certain sets of Miller indices do not correspond to families of lattice planes for BCS and FCC lattices.

345
00:37:54,170 --> 00:38:03,080
So remember, for example, with 100 family of lattice planes are reciprocal, lattice vector corresponds to this these sets of planes for simple cubic.

346
00:38:03,500 --> 00:38:12,709
If I tried to use that 1004 BC, it would not be a family of last planes because you would miss the the guy in the centre of the conventional use.

347
00:38:12,710 --> 00:38:18,710
So you have to go up to 200 in order to get that second the extra plane in the middle,

348
00:38:18,890 --> 00:38:25,879
a smaller plane spacing in order to catch that additional additional last point in the middle of the conventional unit.

349
00:38:25,880 --> 00:38:29,270
So OC two, does this sound familiar? Vaguely familiar.

350
00:38:29,420 --> 00:38:38,870
I hope so. Now, this is why 200 satisfies the basic selection rule,

351
00:38:38,870 --> 00:38:46,189
whereas 100 dozen for BC the selection rule is that h plus k plus l should be even that is true for 200.

352
00:38:46,190 --> 00:38:49,310
It is not true for 100. That's the origin of the selection rule.

353
00:38:49,550 --> 00:38:56,330
The selection rule is asking when is this set of no indices actually a reciprocal lattice vector for this particular lattice?

354
00:38:56,690 --> 00:38:57,589
That's where it's coming from.

355
00:38:57,590 --> 00:39:06,709
You can see another case with one one, one, one, one one makes these diagonal planes here cutting across the unit cell here.

356
00:39:06,710 --> 00:39:09,860
But it would miss the back point if it were thinking about PCC.

357
00:39:09,860 --> 00:39:12,530
We'd be missing the point in the centre of the unit cell.

358
00:39:13,100 --> 00:39:19,730
If you went to 2 to 2, you have twice the density of of lattice plane of planes here and you would catch the,

359
00:39:20,270 --> 00:39:21,679
the lattice point in the centre of the unit.

360
00:39:21,680 --> 00:39:28,280
So 111 does not satisfy the BC selection rule that h plus capable cell should be even whereas 2 to 2 does.

361
00:39:28,550 --> 00:39:31,970
Okay, so that's how you tell when you have a real reset.

362
00:39:31,980 --> 00:39:36,170
When the indices actually represent a reciprocal lattice vector,

363
00:39:36,170 --> 00:39:40,100
they represent a reciprocal lattice vector when they satisfy the selection role and

364
00:39:40,100 --> 00:39:44,350
they do not represent a reciprocal lattice vector when they do not represent it,

365
00:39:44,360 --> 00:39:47,690
satisfy the selection rule. Okay. Makes sense. Yeah. Good.

366
00:39:48,200 --> 00:39:58,370
All right. One thing that's really cool about these about these selection rules is that, in fact, when you can we can describe any crystal,

367
00:39:58,370 --> 00:40:03,019
we can describe a crystal as a lattice, any one of these 14 lattice types times a basis.

368
00:40:03,020 --> 00:40:07,879
This is one of the things we went over when we did crystal structure, for example, sodium chloride is an FCC lattice,

369
00:40:07,880 --> 00:40:13,160
but then times a basis which tells you there's a sodium is 000 and a chlorine at one half, one half, one half.

370
00:40:13,490 --> 00:40:18,680
The selection rules remain the same independent of the basis selection rules.

371
00:40:20,450 --> 00:40:25,400
Rules are independent, independent in depth of basis.

372
00:40:29,270 --> 00:40:32,479
So there's a mnemonic for how it is that people remember.

373
00:40:32,480 --> 00:40:38,720
This is frequently said that S is S for the lattice times s for the basis.

374
00:40:40,310 --> 00:40:43,640
So if s for the lattice vanishes, then the whole thing vanishes.

375
00:40:44,030 --> 00:40:49,999
Okay, so let's see if we can put some, you know, some equations behind this.

376
00:40:50,000 --> 00:40:53,270
What I mean by s for the lattice times s for the basis. All right.

377
00:40:53,270 --> 00:41:07,340
So let's recall that S is going to be a sum over atoms, alpha and units l of f sub alpha in the i g r alpha.

378
00:41:07,940 --> 00:41:15,860
And then we can write any atom alpha in terms of a lattice point and a basis vector x.

379
00:41:15,860 --> 00:41:19,010
So this is lattice point, they are flat point.

380
00:41:19,340 --> 00:41:21,500
An X is the basis factor basis.

381
00:41:23,030 --> 00:41:31,250
So that its position are alpha is the position of the lattice vector R for that atom and the basis vector X for that atom.

382
00:41:31,520 --> 00:41:35,210
So for example, x for sodium here is zero.

383
00:41:35,570 --> 00:41:39,049
It's sitting right on the lattice where X for chlorine is one half, one half,

384
00:41:39,050 --> 00:41:43,100
one half, which tells you that you take to find the position of a chlorine,

385
00:41:43,160 --> 00:41:48,319
you go to a lattice point and then you just place it by a vector X for chlorine, which is one half, one half, one half.

386
00:41:48,320 --> 00:41:53,520
So I'm just writing the. Coordinates of the atom in terms of its lattice coordinate and the basis displacement.

387
00:41:53,810 --> 00:42:05,970
Okay. All right. So then we can write s in terms of this as some over R in lattice, some over X in basis.

388
00:42:08,380 --> 00:42:16,630
Of F the form factor is only going to depend on X because it's only cares what type of atom you're talking about, whether it's a sodium or a chlorine.

389
00:42:16,900 --> 00:42:26,890
So F only depends on X here times each the i g and then the position are smaller is bigger plus bigger plus b x.

390
00:42:29,450 --> 00:42:31,240
Okay, then we can factor this.

391
00:42:31,870 --> 00:42:54,190
Write it as equal to sum over r in lattice times e to the i g dot r times sum over x and basis in basis f sub x in the i g dot x.

392
00:42:54,820 --> 00:42:58,300
Okay, so this is what we just calculated above.

393
00:42:59,140 --> 00:43:05,050
This is ask for the lattice and this here is s for the bases.

394
00:43:06,730 --> 00:43:14,049
So it's really pretty much strictly true that the structure factor for any crystal is a structure factor for the lattice alone times,

395
00:43:14,050 --> 00:43:17,260
the structure factor this factor here for the basis.

396
00:43:17,530 --> 00:43:20,679
So if the structure factor for the lattice vanishes, in other words,

397
00:43:20,680 --> 00:43:25,960
if you're not satisfying the selection rule for that, for the lattice type, then the whole thing is going to vanish.

398
00:43:26,230 --> 00:43:28,480
So it doesn't matter how complicated your basis is.

399
00:43:28,630 --> 00:43:35,320
If it's FCC, in order to get scattering, you have to satisfy the FCC scattering rules that h can all have to be all even or all odd.

400
00:43:35,770 --> 00:43:42,159
Okay. Now there's a way to interpret this statement that that you have to satisfy the selection

401
00:43:42,160 --> 00:43:47,710
rule for the lattice alone in order to get scattering from a complicated crystal.

402
00:43:49,450 --> 00:43:55,670
So let's think about this picture here with sodium chloride for a second. Imagine for a second that we ignore the chlorine.

403
00:43:55,690 --> 00:44:00,040
Just get rid of the chlorine. And then what we have left is sodium is on an FCC lattice.

404
00:44:00,430 --> 00:44:06,520
In order to get X-ray scattering or neutron scattering from those sodium, you have to satisfy the selection rule.

405
00:44:06,520 --> 00:44:10,360
Otherwise you get no scattering from it. Now let's get rid of the sodium and put the chlorine.

406
00:44:10,360 --> 00:44:18,100
Then the chlorians are also forming a complete lattice and in order to get scattering from the chlorians, you have to satisfy the selection rule.

407
00:44:18,310 --> 00:44:23,230
Now, if you put them back, both back in by superposition in order, it's just going to be the sum of the two.

408
00:44:23,260 --> 00:44:26,680
Scattering is a scattering from the sodium and the salt and the scattering from the chlorians.

409
00:44:26,860 --> 00:44:32,020
So if you're not satisfying the selection rule for one of them, you're not you're not going to get any scattering at all.

410
00:44:32,200 --> 00:44:35,470
So it's basically just saying that you add up the scattering from each of the pieces separately.

411
00:44:35,920 --> 00:44:47,440
Okay. All right. So in the next lecture, what we're going to do is we're going to yeah, we're going to actually look at some real X-ray data,

412
00:44:47,680 --> 00:44:53,260
and we're going to analyse the X-ray data and deduce what is in some real materials.

413
00:44:54,850 --> 00:45:05,350
But one of the things that we're expected to know that sort of has to be discussed is how you make X-rays in the first place, how to make make X-rays.

414
00:45:09,340 --> 00:45:16,630
So there's two ways to make X-rays, which we may only be able to get to the simplest of them today.

415
00:45:16,900 --> 00:45:23,650
But the simple way, the cheap way, simple and cheap is to use an X-ray tube.

416
00:45:24,340 --> 00:45:35,620
X-ray tube. This is a method discovered by Röntgen way, back in around 1919, 1900, 1901.

417
00:45:35,980 --> 00:45:42,940
The idea is you take a great big voltage here and you put it across cathode on this side,

418
00:45:43,090 --> 00:45:46,780
just a piece of metal on this side and a big target on this side.

419
00:45:47,410 --> 00:45:58,750
Target on this side. And by having this huge voltage, you spit electrons off the cathode and the electrons hit the target and off comes X-rays.

420
00:45:59,050 --> 00:46:06,580
So what's really happening here is that the electrons are fast electrons.

421
00:46:07,780 --> 00:46:15,690
Fast electrons kick out electrons from core orbitals kick out electrons from core.

422
00:46:15,700 --> 00:46:24,629
Orbitals from core. And the x rays you see,

423
00:46:24,630 --> 00:46:29,700
the x ray lines you actually see are mainly from other electrons falling down

424
00:46:29,880 --> 00:46:33,750
from the high orbitals down to the look to the core orbitals to refill them.

425
00:46:34,080 --> 00:46:42,240
So see emission from transitions, from electron transitions.

426
00:46:45,930 --> 00:46:51,719
So an electron in a high up orbital drops down to fill one of these empty spaces in the in

427
00:46:51,720 --> 00:46:58,990
the core orbitals and you will see a nice x ray line coming and nice x ray coming off.

428
00:46:59,010 --> 00:47:07,100
Now the energy of that of that transition is basically about the rig burg times the atomic number squared.

429
00:47:07,110 --> 00:47:17,010
Remember, when you have a hydrogen atom, the energy of the are the I can say to the hydrogen atom, I'll go as the nuclear charge squared.

430
00:47:18,170 --> 00:47:22,520
So, you know, the red berg is 13 electron volts, but the atomic number of your atom,

431
00:47:22,520 --> 00:47:27,049
if you're using something like copper, something like 40, so you can have a huge, huge energy.

432
00:47:27,050 --> 00:47:30,870
This can be tens of kilowatts of kbps.

433
00:47:31,310 --> 00:47:41,720
And that will correspond very nicely to what we want to correspond to what we want, which is lambda approximately one angstrom.

434
00:47:42,020 --> 00:47:44,240
Now, there's a couple of really nice things about this technique.

435
00:47:44,510 --> 00:47:50,150
One thing that's really nice about this technique is since it's an atomic transition, it's very, very sharp.

436
00:47:50,390 --> 00:47:57,320
In other words, there is a very sharp line, a very, very distinct transition between two distinct energy levels.

437
00:47:57,320 --> 00:48:00,380
You know, very clear energy level up here, a very clear energy level down there.

438
00:48:00,560 --> 00:48:09,740
And so you get a very, very well-defined, well-defined, defined lambda out of this type of technique.

439
00:48:09,740 --> 00:48:14,930
And this is basically what what Röntgen was doing, you know, 100, 115 years ago.

440
00:48:15,260 --> 00:48:19,639
Now, in modern in the modern era, I may or may not have have a picture of this.

441
00:48:19,640 --> 00:48:23,060
Let me see if I have it. Uh, yeah.

442
00:48:23,060 --> 00:48:32,390
Here in the modern era, there's another way to make X-rays, which is sort of favoured as a better technique, is only these X-ray tubes.

443
00:48:32,640 --> 00:48:34,220
They're they're incredibly cheap.

444
00:48:34,550 --> 00:48:40,850
You can you know, this is what you have in the the dentist's office when they X-ray your mouth is one of these things or,

445
00:48:40,850 --> 00:48:44,750
you know, X-ray broken bones. It's these things. And there's probably dozens them around this building.

446
00:48:44,990 --> 00:48:50,420
You can build one yourself if you wanted to, but frequently you need a much more high tech X-ray source.

447
00:48:50,690 --> 00:48:57,310
And the modern high tech X-ray source is known as the Synchrotron Prana.

448
00:48:57,770 --> 00:49:00,920
And it's in this picture. It's one of these big buildings. This is just down the road.

449
00:49:01,100 --> 00:49:09,050
This is the X-ray synchrotron source at Diamond Lab, just just up the road by about 10 minutes by car.

450
00:49:09,470 --> 00:49:13,340
These things are about ten to the ten times brighter. Ten to the ten.

451
00:49:13,580 --> 00:49:19,550
Yeah, 10 to 10 times brighter. More photons than X-ray tubes.

452
00:49:20,300 --> 00:49:31,400
Than X-ray tubes. And that's actually quite useful to have if for a lot of modern experiments which

453
00:49:31,790 --> 00:49:36,650
examine either extremely small samples or you need to collect data extremely quickly.

454
00:49:36,860 --> 00:49:42,890
I mean, at the end of the day, you're always counting photons. You're sort of counting how many photons are scattering in a particular direction.

455
00:49:43,040 --> 00:49:45,320
If you have more photons, you always get better data.

456
00:49:45,560 --> 00:49:53,060
If you're if it's particularly if you're looking at a very small sample where it's hard to focus those electrons in on the small sample.

457
00:49:53,060 --> 00:49:56,870
The general idea of the synchrotron is you take this great big ring,

458
00:49:57,110 --> 00:50:02,180
which you can see in this picture, this synchrotron source, you spin electrons around this ring.

459
00:50:02,510 --> 00:50:12,470
So the electrons are going around this ring at very high velocities a bit gives so giga electron volts and then in order to get X-rays out of it,

460
00:50:12,950 --> 00:50:18,529
you take your electrons and you send them through a bunch of magnets.

461
00:50:18,530 --> 00:50:29,930
So north, south, south, north, north, south, this is known as an undulating or a Wigler sometimes.

462
00:50:31,730 --> 00:50:33,950
And the idea is that the electrons going through this,

463
00:50:33,950 --> 00:50:42,620
they're going through very fast and you make them accelerate through and as you know, accelerate accelerating electrons emits radiation.

464
00:50:42,620 --> 00:50:52,429
And in this case, they will emit directed X-ray radiation, which is extremely colonnaded, extremely well defined frequency.

465
00:50:52,430 --> 00:50:56,990
And if you're not happy with the sort of a technique, this is sort of important to know.

466
00:50:57,110 --> 00:51:08,390
If you're not happy with how well-defined the frequency is, if you want to specify a frequency more precisely, specify a wavelength more precisely.

467
00:51:13,190 --> 00:51:22,100
What you do is you refract off a known crystal different tract off the known crystal,

468
00:51:23,250 --> 00:51:31,610
known crystal at a particular angle, and that will pick out a particular wavelength that you're interested in using.

469
00:51:31,820 --> 00:51:35,990
Now, whenever you're doing this, you're throwing away lots and lots and lots of the photons that you have.

470
00:51:36,170 --> 00:51:39,950
But if you have ten to the ten times the number of photons that you need to begin with,

471
00:51:40,100 --> 00:51:44,420
you can throw away a ton of them and you can refract multiple times to really specify

472
00:51:44,630 --> 00:51:48,050
that wavelength extremely precisely and do extremely high precision measurements.

473
00:51:48,470 --> 00:51:51,710
I think that's all I have to say about this. I'll see you tomorrow. Wow.

474
00:51:51,790 --> 00:51:52,900
Wow. Okay.