Let's get going. Welcome back.
This is the 11th lecture of the condensed matter, of course, where we left off last time we were studying reciprocal space.
Reciprocal space. And we came to this understanding that the reciprocal lattice is somehow the Fourier transform of the direct lattice,
and that the reason we want to study reciprocal space is because it's going to make our life easier when we come to studying waves.
So it's a rather important thing to to understand well, and sometimes it's hard to understand for a transfer as well,
because they're not something that we think about in everyday life a little bit complicated.
So it turns out that there's a much nicer geometric way to understand a lot of reciprocal space, which is what we're going to discuss today.
So I need some definitions. As usual, let's start with a simple definition.
A lattice plane equals a plane containing containing three three, non linear, non linear, linear,
and therefore linear and therefore an infinite number and therefore are an infinity of lattice points.
Okay, so a lattice plane is a plane containing three non linear and therefore an infinite number of lattice points.
So here are some pictures of lattice planes here. On the left we have a BK lattice.
There's a plane cutting through three points. Therefore, if you continued out to infinity, it will cut to an infinity number of points.
That's a lattice plane. There's a on the right is a simple cubic lattice.
There's a plane that cuts the three last points. If you continue out to infinity, it cuts through an infinite number of lattice points.
Another definition, a family of lattice planes.
Of lattice planes, mama lattice plane.
Baby lattice plane. No, I'm joking. That's not what no one ever laughs in this class.
And again, it is families.
Large planes equals an infinite, infinite, infinite and finite set of equally spaced equals spaced lattice planes.
Such that. Such that all lattice points, all lad points are included.
Are included in one of the planes in one of the planes.
Let me know if my riding starts getting to be too messy to read. So here's a simple cubic lattice and a couple of families of lattice planes.
Each one of these is a lattice planes that cut through an infinite number of lattice points,
and they're equally spaced, and they include every single point in the lattice when taken all together.
This is a BK lattice and there's a family of lattice plans for the BK lattice cut through all the lattice points in the BC lattice.
This, however, is not a family of lattice planes for the BK lattice, even though they're lattice planes.
What's drawn? They are equally spaced, but they do not include every lattice point they miss.
These guys here in the middle of the conventional media cell. Therefore, it is not a family of lattice planes.
Okay. So the connections, the reciprocal lattice claim.
A family of lattice planes. A family of lattice planes.
It is perpendicular to our reciprocal lattice vector back, which I called G and further and the distance,
the spacing between between neighbour planes, between neighbour planes planes is d equals to pi over absolute d men.
We're g in the g men equals shortest.
Recep. Lattice vector.
Lat vec. Indirection.
Indirection. Of G OC.
This is important statement.
Every time we have a family of lattice planes orthogonal to this family of lattice planes, there is a reciprocal lattice vector.
And if you want to know the spacing between the lattice plane in the family of lattice planes,
you find the shortest reciprocal lattice vector pointing in that direction to pi over the
absolute value of that shortest reciprocal lattice factor gives you the spacing between planes.
Okay. This is a big statement. We haven't shown it.
It's useful to actually prove this statement because actually in the proof,
there's a lot of important geometry that we're going to need in coming up in a moment.
So let's go through the proof of that. This is true. So, first of all, so this is just a proof by geometry.
So a little bit of geometry that you probably remember.
First of all, it's define a plane pick of our vector g, g points X equals a constant C defines a plane defines a plane of points of points,
points X, perpendicular to G, perpendicular to g, where.
Okay, let me actually draw this over here. So here's the point zero.
Here's the vector g, here's the plane perpendicular to G, which I'm calling X is the plane of points X and the distance from zero to the plane x.
Let's call that distance X-Men here where X-Men the minimum distance to the plane equals C over absolute G.
Does that geometry look familiar from before it look reasonable?
Yeah. So the X-Men here is C over absolute G for defining the plane by G that x equals equals C.
Okay, so that's how are you going to define a plane now?
We're going to define a set of planes. Set of planes, of planes we define by each the I got x equals one.
So why does that define a set of planes? Well, of the i g that x equals one is the same thing as saying g dot x equals 2 p.m. where m is an integer.
There's an element of integers, right? So.
So that means we have a whole bunch of planes whose minimum distance to the origin is given by 2 p.m.
So X-Men equals two pi am over absolute g.
So space in between these planes. Spacing between planes.
Plains D is two pi over absolute g.
Okay, let's actually I have a better picture of that than I can draw.
So the geometry we're interested in is if the object x equals one.
It defines a set of parallel planes. One of the planes will be g that x equals 2 p.m. another plane will be God.
X equals two pi and plus one. The distance to the first plane is two pi am over absolute g.
The second distance the second plane is two pi and plus one over absolute g.
So the distance between the two is two pi over G. Does that geometry look pretty good?
Yeah. Okay, good. All right. So now recall that if G is a reciprocal lattice vector, so g is reciprocal added factor, reciprocal lat.
Back then e to the i g r equals one for all four.
All are indirect. Lattice indirect.
That's a definition of the reciprocal lattice.
So that what that tells us is that every lattice point is going to be included inside this set of planes
because NPR will have to satisfy the IG that our equals one if these are simply reciprocal last factor.
So all of our lattice points in the direct lattice are going to be included in the set of planes that we just defined.
However, it does not mean that every plane cuts through lattice points.
Okay. Why not? Okay, here's a here's a picture to explain that.
So here we have a set of planes defined by some reciprocal last vector, which I'll happen to call G-men.
And here we have a set of planes divided by defined by twice that reciprocal lattice vector.
The spacing between the lattice planes is half as big because the spacing is always two pi over g.
You'll notice that in the second picture we have a plane which is defined by this reciprocal aspect of G, which doesn't intersect any lattice points.
What we're guaranteed by this law is that all direct lattice points must be included in one of the planes.
But we're not guaranteed that every plane has to cut through a lattice point.
So if we want to get a set of planes in which everyone every plane cuts through the lattice points, we have to choose.
So to get family of lattice planes, of lattice planes, and to be a lattice plane, it has to has to cut through a lattice point.
So if we want a set of planes where every plane cuts through a lattice point, we must choose.
Must choose. Game in the shortest reciprocal lattice vector in that given direction.
If it was not the minimum reciprocal last vector in that direction, you could make it smaller,
cut it in half, for example, and get something which has fewer planes.
And yet all of the lattice points should still be included. So you should take the minimum reciprocal lattice vectors so that you get enough planes.
There's guaranteed that all all your lattice points in the direct lattice will be included in those planes, but you don't get any extra OC.
Good. All right. So that's the the proof of that of that rather important theorem.
Now, it's it's useful to actually come up with some some way to describe these reciprocal lattice planes.
Are these are these family of lattice planes or these reciprocal lattice vectors.
And this is a type of notation that we will use very frequently known as Miller Indices.
For describing. For describing.
Describe families of planes, planes, planes or vectors in respect of space vectors and risk space.
And the idea of a mirror index is that you write in parentheses.
HK Well, to mean actually, maybe, maybe before I do that.
All right, we describe families of planes or vectors of reciprocal space in terms of terms of reference vectors,
reference vex b, sub i these bits of i's very important.
They may be primitive, primitive, reciprocal lattice vectors or they may not be.
Generally what we do is we write as we're starting to write a second go HK in
parentheses to mean h times b one plus k times b two plus l times b three.
And this notation should look a little bit familiar. We can compare it to the notation we use in real space, indirect space, direct space.
We use the notation u v w to mean u times A1 plus v times A2 plus w times A3.
So the important thing to take away here is that if we use curved parentheses here, we mean reciprocal space.
If we use square brackets, we mean direct space.
So don't confuse those with each other. Now, in both cases, we may or may not mean these A's of these B's to be primitive.
Okay, that's very important, because a lot of the questions that come up on exams.
Exactly reference this subtlety that sometimes you work with primitive lattice
reciprocal lattice vectors are primitive lattice vectors and sometimes you don't.
Why don't you? Sometimes because sometimes the primitive lattice vectors are primitive, reciprocal.
Large vectors are ugly angles. And if you choose to use a conventional unit cell rather than primitive unit cell,
you can use orthogonal vectors rather than using things that weird angles.
So if you can use orthogonal, you usually do, even if it means you're not using primitive lattice vectors.
Okay, so let's be be careful here being careful, being careful about our definitions.
The rules kind of go like this. First, choose direct real space vectors, direct space vectors, direct space vectors,
a sub I that's going to be these things which may or may not be primitive.
I'll even write that down. May or may not, may not be p elves.
P elves. Now for us,
you'll remember we had this rule that we're not going to be responsible for lattices in three dimensions where the axes are not orthogonal.
Okay. For us, for us in three D, we always choose.
Choose orthogonal or orthogonal A's.
Okay? Sometimes in two D, they'll ask a question where they're not orthogonal,
but in three D we always choose to choose orthogonal A's That means for simple cubic,
simple cubic cubic, these can be these are primitive lattice vectors players.
But for FCC, FCC and BC, these are not not pelvises, but they have the advantage of being orthogonal.
So they make your life a lot easier because it's a lot easier to work with things that are orthogonal and things that are not orthogonal.
Okay. So given that we've defined these A's, we then construct B's.
So can so call that to construct construct B's by v i talked a j equals to pi delta i j and we've seen this equation before.
If, if, and only if is our pelvis, then B's will be reciprocal.
Are reciprocal. PVS are reciprocal. That reciprocal lat plays primitive lattice vectors.
Okay. Again for us.
For us, since we are using in 3D, in 3D, since we are always using orthogonal axes in 3D,
it will always turn out to be the case that the absolute value of BI equals to pi over the absolute value of A.I. because axes are orthogonal.
Axes orthogonal. Okay.
So that makes life a lot easier.
So generally when we write down R, g equals L, we mean, whereas this thing over here, h, b, one plus k and we already again we mean hb1 plus k,
b two plus lb3 where the B's have been constructed in this way,
but always g defines a set of plain defines a set of plains, a set of plains plains via the idea that x equals one.
Either the i g dot x equals one.
So we can always define a set of plains for any vector g by either the x equals one,
and the spacing of these plains spacing is d h l equals to pi over absolute g.
Or I guess in assuming our orthogonal axes, assume orthogonal B's, orthogonal A's orthogonal to A's and B's are orthogonal or orthogonal.
Orthogonal. Then we have we can write that as two pi over our absolute value of h squared,
absolute b one squared plus k squared, absolute b two squared plus l squared, absolute b three squared.
Okay. And for cubic lattice for cubic for cubic where A1, A2 and a3 are the same length and orthogonal to each other,
therefore b1 b2 b three are the same length and are orthogonal to each other.
The spacing d h kcl is a the length of the primitive,
the length of the conventional used cell in indirect space over a squared plus k squared plus elsewhere.
Okay. And maybe I'll note this one more time.
If the A's our pelvis is our pelvis, then the bees are pelvis.
Then these pelvis, the right that above B's pelvis. And that means that hk l is a reciprocal lattice factor receptor laterally.
But it's not always the case if the A's are not pelvis to start with.
Okay. All right. So supposing someone hands you a set of planes and asks you for the major indices.
That sounds like it might be hard to figure out, but in fact,
there's an easy way to figure out geometrically what the Miller Indices are for a set of planes.
And this is known as the method of intercepts. Method of intercepts.
It goes kind of like kind of like this.
So suppose we have, uh, so here's three axes X axis, the y axis, and the z axis.
Let's assume orthodontics are the primitive lattice vectors in the three directions aren't necessarily the same.
So call this a and two A2 and this one will be, I guess, a one to a131 for a one here and then a three to a three and three.
A three. So suppose that we have a plane going through the origin.
So imagine there's a plane cutting through the origin here and then also cuts through the axes like this.
Okay. So you take the first two parallel planes. One of them cuts to the origin.
It's parallel to this plane, which cuts through the axes like this.
The rule is that hk l right this way.
H is to k is to l as a proportionality is the same as a one over the x intercept is to a two over the y intercept is to a three over the Z intercept.
So in this picture here, this would be a one over the x intercept is one over 4h2 is over,
the y intercept is one over two and a three over the z intercept is one over three.
So it should have the same ratios as this. If we multiply by the least common multiple, least common multiple,
we get three comma six come a four as Miller Indices that would be appropriate for this this plane and generally write it as 364 without any commas,
at least as long as there's no confusion as to where you should put the commas.
If you go over ten, then you have to indicate where the commas are.
Otherwise it's not clear where it would be. All right.
Now, if and this and what we mean by three, six, four here is that 364 is three B, one plus six, B two plus four, B three.
And that's what we mean by the Miller Indices.
And that that set of Miller Indices is gives a reciprocal last vector, which is normal to this to this plane.
There's a some cases which are a little more complicated.
If a plane if a plane does not intersect, an axis plane does not not intersect and axis not intersect, intersect an axis.
Then you get a miller index of zero, you get miller index index equals zero.
Let's see if we can explain what that is. So here again, three axes.
This is the A1 axis out here. Maybe this would be a two, this would be two, a two.
And maybe here we have three, a three.
And suppose we have an axis. Axis, always on a plane, always cutting through zero.
And then the next plane cuts here.
But it's parallel to the A1 axis here to the x axis, so it never intercepts it.
So what do we do with the Miller Indices? Well, we say H is two, K is two, L as well.
Okay. So the first question is, where do we intercept the X axis?
Well, if it doesn't intercept the x axis at all, you can say that is really like intersecting the x axis at infinity.
So we write one over infinity here and then k while intersects the y axis at two.
So one over two and we intersect the z axis of three.
So we write one over three and multiply by the least common multiple and we get one over infinity 0332 So the Miller indices here would be 032.
Okay. So let me show you a couple examples of this.
Here's a set of Miller Indices that are 100 because it intersects X at one and doesn't intersect Z or Y.
Here's one. That's one one, two, because it intersects exec one Y at one and Z at one half.
The reciprocal of one half is two.
So it's one, one, two. So you can have a situation.
R can have you can have Miller index less than zero index negative like one minus one zero, which you typically write as one one with a bar over it.
Zero. Okay. So the. R bar means negative.
Means negative. That's just the notation that's conventionally used now.
Okay, so here is a great big apology for the UN coordination of our community.
So depending on which side of the ocean you come from, the way you say this out loud is different.
If you're from the United States, you say one one, bar zero.
If you're from England, you say one bar one zero.
I don't know why it makes things incredibly confusing. When it's written, it's totally unambiguous.
The bar on top means negative, but when you say it out loud, it's totally confusing and no one knows what you mean.
So I recommend just writing it down, and then everyone knows what you're talking about.
Okay, let me show you an example where we have a negative index.
So this plane here actually is maybe it's not so clear, but but it's supposed to you supposed to sort of be able to extrapolate down with
your with your eyes and see that this plane is sort of starts up here and goes down.
It would intersect the z-axis at minus one half. And so its intercepts are one one minus one half.
So it's more indices or 112 with the bar over it. Negative two.
Okay, so here's a whole bunch of different Miller Indices for a bunch of different planes.
He's 001100010101, so forth and so on.
And you see that they actually they actually kind of come in families.
So if you look at the family across the top row, across the top row, you'll see that 001 and 100.
They kind of look similar and 010 looks kind of similar. And if you imagined actually having a sample,
it's a simple cubic lattice and you picked up the simple cubic lattice and you looked at it from one side,
100 would look exactly like 010, depending on if you picked it up and turned it sideways.
It's exactly the same. It just depends on which side you picked up and looked at first.
Similarly, with 101 or 110, if you picked it up and you looked at it sideways, it would look exactly the same.
The spacing between planes in those families would be exactly the same.
The way it cuts diagonally through the through the unit cell would look the same.
It just matters which way you picked it up in the first place. So we have a notation for equivalent sets of indices, equivalent sets.
Of Miller Indices, indices which we notate with curly brackets.
Okay, that is a terrible crummy bracket like that h cal, which means equivalent sets of 12 sets of h cal and regular parentheses brackets.
So for example, in this picture, curly brackets, 100 would mean 100 or 010 or 001 or one bar zero zero or zero one bar zero or 001 bar.
And coincidentally, these planes look like the surface of a cube, and you can pick up the surface of a cube and look at it six different ways.
And it would look equivalent from all of those six different angles.
We also have what's known as the multiplicity of each cal, which is the number equals the number of parentheses.
H cal in bracket.
H cal. So in this case we would have multiplicity of 100 is six because it includes all those six different sets of indices that I wrote down.
Okay. So for the first candy bar of the day, what is the multiplicity of one, two, three?
Now. Well, thanks. Now, who was it?
Someone said something. 48 Yeah, that's who said that.
48 This one's yours. Very good. I can't throw it that far.
48 The answer is 48 So let's see why the multiplicity of this is 48 First of all, one, two and three can be in any order.
So that six back that's three factorial gives you six.
One, two, three, one, three, two, two, three. One, two, one, three, three, one, two, three, two, one, six.
Possible ways of ordering them. But then in addition, any one of those numbers can have a bar over it or not.
So this one has two possibilities. This has two possibilities. This two has two possibilities that makes eight possibilities.
So it's eight times six gives you 48. So multiplicity of one, two, three equals 48, which is kind of interesting.
It means if you have the family of lattice planes associated with a one, two, three Miller Indices,
you can pick up a simple cubic lattice and look at it from 48 different directions
and have lattice planes with exactly the same spacings looking at you.
Sort of surprising. But. But that's true. Okay, so now why is it we care so much about these these lattice planes besides the fact that
they're going to tell us a lot about reciprocal space and we're interested in waves and waves,
you know, reciprocal spaces, language waves. Well, the reason one reason we might be interested in these families of lattice planes in reciprocal
space is because they actually tell you a lot about the macroscopic structure of the crystal.
It's kind of interesting that all of this information about lattice planes and reciprocal lattices,
this was all worked out in the middle of the 1800s before people even knew that there was such thing as atoms.
And the way it was worked out is because people knew that minerals,
nice crystal and minerals broke and so they fractured only along certain axes, and that enabled them to deduce all of this information.
So, for example,
with a simple cubic lattice you'll discover or any type of cubic lattice that the lattice will will break the material will break along simple,
simple sets of Miller indices. So like the 100 face here or the 010 phase here is where the material will will break most easily.
This is a law it's not a hard and fast law, but it's a very, very good rule of thumb by Broadway,
known as Bravo's Law, the same guy who worked out all the lattices.
And it actually tells you that when you when you take a cubic crystal and you break it, you get only the 100 phases.
Occasionally you'll get the 110 faces and even less occasionally you'll get one one, too.
So here's a picture of sodium chloride, which is a cubic crystal.
It's actually FCC crystal, and it breaks almost exclusively along the 100 are lattice planes.
You can see that these these breaks are entirely at 90 degrees quartz.
Okay. We don't study quartz this year. It's just a very nice picture.
So I put it in there and we don't study it this year because the axes are not orthogonal to each other.
It's a tricky little crystal, but the axes very clearly, the faces are very clearly at very fixed angles to each other,
which are dictated by how the lattice planes are arranged.
Okay, so I'm going to emphasise this one fact one more time because it shows up on exam so frequently.
So this is important fact. If as our please then the of if and only if A's are PVS, then bees are PVS.
I think I've written it down three times today that bees are pelvic responses.
Reciprocal pelvis, reciprocal pelvis, for example.
This is true for simple cubic, which means any h kcl is hb1 plus kb2.
This is the definition of HK I'll be three.
So any of these is a receptor lattice vector receptor lat vec because integer sums are primitive lattice vectors give you an element of the lattice.
If we want a family of lattice planes, a family of lattice planes, of flat planes, we should have, should have, that should be shortest.
The shortest. In a given direction.
Shortest recap flat vec in a given direction.
So for example, if we're considering the direction to four six, the shortest reciprocal lattice for the simple cubic lattice,
the shortest reciprocal lattice vector in that direction is one, two, three.
So you divide by divide by greatest common divisor.
So if you want to describe a set of lattice planes, you should always choose the reciprocal lattice factor, which is shortest in that given direction.
However, however, if the A's are not pelvises, then and this is then these are not PVS, not reciprocal.
These. Which means that not all hk l r recep lattice factors are recep lattice factors, and this is the source of most most good exam problems.
Okay, so let me show you exactly what we mean by this.
So recall this picture and the lattice planes that the family of last planes have drawn in this picture.
Here is the 100 family of lattice planes and and the minimum reciprocal last vector in that direction is, in fact, 100.
This is simple cubic lattice. So this is definitely a are a family of lattice planes and a reciprocal lattice vector.
Now, twice that 200 is a reciprocal lattice factor, but it's not a family of lattice planes because you have extra planes,
you have a plane that doesn't cut to any lattice points, half the spacing of planes.
So it's not a family of last planes, but it is a reciprocal lattice vector.
Now, let's think about what happens when we go to either a BK or an FCC lattice with a boxy and FCC lattice,
we're still going to use the same three orthogonal A's.
We're going to choose to use the conventional unit cell to define our A's, just like we did with a simple cubic lattice.
And we're doing this. They're not primitive lattice vectors, and that's a problem,
but we're doing it because we want to keep everything orthogonal and life is just a lot easier if you keep everything orthogonal.
Okay, so here's a picture with a bunch of Miller Indices, a bunch of lattice planes drawn for the simple cubic lattice.
Here's the 100 set of last planes just like I drew before. Now, in the back case, if you added one more point in the centre of that unit cell,
in the centre of that convention, in the cell, it would not be intersected by the 100 set of planes.
It is intersected by the 200 set of planes. So 200 is a correct family of lattice planes for BK.
It is not for four. Simple cubic is 100 isn't even a reciprocal lattice vector for BK because it does not intersect every lattice point.
Okay, that's that's the subtlety. And we're going to come back to this subtlety over and over for a BK.
You have to choose Miller Indices to make sure that every one of your lattice points is included in one of those planes.
Same for FCC. And the reason this is a problem is because we're using lattice vectors a which are not the primitive lattice vectors.
We're using orthogonal vectors because they're orthogonal makes your life easier, believe it or not.
Same with FCC. If we consider the FCC lattice and you consider the 111 planes, these planes here, we would not intersect some of these lattice points.
These would be missing. So 100 is not a reciprocal lattice vector for FCC.
200 is okay. Similarly, while we can look at 111 here, so 111 is a perfectly good set of family of lattice planes for a simple cubic.
It would not be a family of lattice planes for BK because it would miss the point in the centre of the unit cell.
But actually it is a family of lattice planes for FCC because it just happens to catch all those points in the centre of the faces.
Okay, now later on, we're actually going to derive a condition which will tell us when we have Miller Indices,
which are family of lattice planes for a particular lattice. But it's important to realise not every set of not every set of Miller Indices will
give you a reciprocal lattice vector or a family of lattice planes for an FCC lattice.
And the origin of this problem is we're not using primitive lattice vectors to describe our convention itself.
We're using the edges of the convention yourself. Is that clear?
Have I warned you enough? All right, good.
All right, so that's everything we have to say about reciprocal space.
And now we can get on to some applications of reciprocal space.
And perhaps the most important application we're going to run into is scattering experiments.
Scattering experiments, which will be the subject of the rest of this lecture and probably much of the next two lectures.
So the general scattering experiment starts with a sample sample.
You send in some wave with wave vector K,
some of it goes through and scattered comes out at wave vector K and some of it comes out at wave vector K prime.
By observing what comes out at wave vector K prime, you're supposed to be able to deduce what is in the sample.
Now, it's hard to overstate the importance of this type of experiment.
You may have studied it in other courses, especially subatomic,
but it is not an over is not an overstatement to say that almost everything we know about the microscopic world,
everything the fact that we know that there are nuclei, the fact that we know the structure of atoms,
the fact that we know the structure of materials, the fact that we know the structure of the proton.
All of these things come from this kind of experiment, this type. A scattering experiment.
You know, we grew up in a world where even kids know what atoms look like and they know what DNA looks like,
this double stranded helix, they know what water molecule looks like. They may even know what salt looks like.
Sodium chloride, sodium chloride. All of this stuff that we know and we take for granted comes from doing this kind of experiment.
The only thing that's different when we do this experiment first is when the high
energy physicist doing this experiment is what wave we put in to start with.
Okay, now we're interested in seeing atoms. And atoms are on the scale of about an angstrom.
It turns out to be a really good idea to choose a wave whose wavelength is about the same scale of the thing you want to see.
So for us, they're going to be a couple of options. One is x rays, electromagnetic waves whose length scale is about an angstrom.
So that will enable us to see things roughly of the size of an angstrom.
But, you know, we live in this quantum mechanical world, and we know that waves and particles are the same thing.
So we're entitled to use not only things we usually think of as ways, but things we usually think of as particles.
So a particle which is very convenient to do scattering experiments with as neutrons,
neutrons, you know, I think of it in particles, but they're also waves.
It turns out to be fairly easy to get neutrons to have wavelengths which are on about the same scale of about an angstrom,
which makes it easy to see things which are on the scale of roughly an angstrom.
These are pretty much the two types of waves that we're going to be responsible for this year.
I should mention that people frequently use electron waves as well,
but we don't study them this year because they have some complications, which we don't want to delve into.
So whatever type of wave we send into our material,
the thing that we're supposed to imagine is that the atoms in sample atoms in the sample
provide a potential a potential V of our that scatters that scatters wave scatters the wave.
And our objective is actually a bit of an inverse problem where instead of trying to calculate
what is the potential due to the wave much more frequently we're told what the wave has done.
We've measured what the wave has done, and we're trying to deduce the potential.
The potential is going to tell us where the atoms are in the material.
And we're trying to deduce that from observing what physically came out of our sample.
Now, just a little bit of a caveat.
It's easy to think about a potential for neutrons and neutron moves around it, feel some potential as it moves around.
It's a little bit less natural to think about of potential for for x rays, which are electromagnetic waves.
You can if you want, you can kind of think of it as a potential for photons,
but much more appropriate is you really should be thinking of it as spatially dependent optical index or spatially dependent dielectric constant.
But you won't go too far wrong to think about it as just a potential for particles as a particle move through your material.
So the way we're going to actually scat, we're going to actually calculate how much get scattered out in some way.
Vector. K prime is to use. Fermi's Golden rule.
Fermi's Golden Rule rule condensed matter joke.
This is a symbol for gold anywhere near one. So I made that up on the spot.
It was because I was trying to save my arm from writing. All right, never mind.
I'm not going to be a comedian. That's why I'm a physicist. Okay. So the scattering rate,
the scattering rate brings rates from K to K prime is given by you probably probably seen from his golden rule to PI over h bar.
There's a matrix element K prime the K that gets squared and an energy preserving delta function vectors are energy preserving delta function.
Now to simplify your life, we are going to assume we assume elastic scattering, elastic scatter, which means that e equals e prime.
The energy coming in and the energy coming out are the same. That will also mean the absolute value of K equals the absolute value of K prime.
There's several reasons why we assume elastic scattering. First of all, it's easier.
Secondly, when you actually do these experiments, most of the scattering that you measure actually is elastic.
It's actually quite interesting to look at inelastic scattering as well.
I might mention that a couple of lectures further on with inelastic scattering,
you actually get you leave some energy behind in the sample and that actually allows you to figure
out what the energy of the possible excitations are that you can excite in the sample as well.
But that's a little bit more complicated. So we're not going to worry about that for now.
So we're just going to assume elastic scattering. And the only thing that's difficult in that expression for me is golden rule is the matrix element.
So let's write out the matrix element. So we have K prime v k equals what?
Well, we'll use the, the real space representation of these cats.
It's integral D and d are of either the signs in the i k prime dot r v of our times into the i kkr have the signs, right?
Yeah. Okay. And if you were careful, we would normalise this by one over square root of volumes.
This is the the way vector. This is the wave function k prime. This is the wave.
Function K, there's v r in the middle.
And we realise that what that is is all right this way d three are either minus i k prime minus k dotted with r v of r that is just well we'll
write it as v twiddle of k minus k prime k minus k prime where v twiddle is means for a transform v twiddle equals 48 transform of v of r.
This is a very, very general rule that the scattering amplitude from a K prime is just the 40 transform of the scattering potential.
Now in the last lecture.
And so that actually always, almost always holds true for anything that you're going to do.
Scattering experiments on that, the scattering of using Fermi's golden rule,
the scattering amplitude from some K2 k prime is just the four transform whatever the scattering potential is now.
In the last lecture, we considered 48 transforms of periodic functions.
So we're going to assume we have a crystal. Let's assume a periodic periodic scattering potential the of R.
So if we have anything that's crystalline, the potential will be periodic.
And so we can use the the law that we derived in the last lecture that the 40th transform of periodic
potential came on as k prime equals equals a sum over our reciprocal lattice vectors g of a delta function,
three dimensional delta function of k minus k prime.
Right that again k minus k primaries, g times structure factor of g.
I guess there's some normalisation factor to pi cubed over the volume of unit cell volume.
And so. And the structure factor here as of g o coming out of space,
r s of g equals the integral is equal to the 40 transform within a unit cell integral d three are in the i i g r v r r right the rs the same way,
and it's integral over the unit cell. Okay, a couple of important.
So this is a law that we derived last time and a couple of important facts about this.
This law we mentioned last time. First of all, the free transfer of this periodic function is zero.
Unless we're looking at values of K, which are elements of the reciprocal lattice factor.
So point one scattering is zero is zero and less and less k minus K prime is a reciprocal lattice vector g.
This is important, but the box around it, this is known as the Laue condition.
After Max von Laue, who won a Nobel Prize for discovering this in X-ray scattering Nobel Prize was actually exactly 100 years ago now.
And then there's this complication about whether it should be called the Laue condition or the Von Laue condition because he was born Max Laue.
But then his father was elevated to Noble Hood much later, and he took the name Von Laue.
But when he discovered this, his name was still Laue.
So it's usually called the Laue condition, not the found life condition, but other people will disagree.
So anyway, it's either Laue condition of the Von Laue condition, and but we've seen this thing before.
What this is, is conservation of conservation of crystal momentum.
You only scatter if you can conserve crystal momentum, right?
F k minus k prime is a reciprocal lattice vector that conserves crystal momentum.
It doesn't change the crystal momentum of the wave.
The second important fact that we're going to use later is that the intensity of scattering, intensity of scattering,
scatter is the structure factor s of well, delta delta k squared where delta k is the reciprocal lattice factor G which is scattered by.
Okay. So the point is that if you measure the intensity of the various scattering, you can figure out the structure factor.
What you're measuring is the structure factor at given reciprocal lattice factors.
The structure factor of giving your reciprocal lattice factors is the for transform of the scattering potential.
So in principle, you're measuring the scattering potential.
That's the point. By measuring the what is scattered, you're actually measuring the scattering potential now.
So I'm going to go over about one minute here. It may seem a little bit surprising this condition,
one that you only get scattering under certain conditions only if crystal momentum is satisfied, only if you if satisfy this lousy condition.
Now, that might seem a little bit weird, but I'm going to convince you that this is something that you know very well.
So imagine you have a crystal like this.
Then you think of there's really this is a bunch of lattice planes, it's a family of lattice planes.
And you send some wave into a family of lattice planes and it bounces off.
But it can bounce off this layer, too. And you know very well that if you bounce off planes like this, you can get diffraction.
And diffraction only occurs at certain angles.
You only get constructive interference at certain angles.
And what we're going to see is that this Laue condition is actually exactly the same condition as having constructive interference.
And we'll do that next time I see you Wednesday.