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트랜스크립트
00:00오늘의 내용은 우리의 개념을 만들어볼까요?
00:07우리의 개념을 이해하는 것입니다.
00:10우리의 개념을 이해하는 것입니다.
00:13첫 번째는,
00:15무엇이 그 difference between
00:18크리스탈과 non-crystalline solids?
00:22In crystalline material, atoms are arranged in a repeating 3D pattern, and we call that long-range order, and just one of them could be like a unit cell, and we learned about them already, like a cubic system, or a body-centered one, or a face-centered one, right?
00:48So that could be like a simple cubic or any kind of the repeating unit in 3D one, but in non-crystalline or amorphous material, there is no long-range repeating pattern.
01:06The arrangement looks more random, like a frozen liquid, like a glass, right?
01:14That could be like an arguing sample.
01:18And second, how do we name crystallographic directions and planes, right?
01:26So this is the main point in this extended chapter, like crystalline direction and planes.
01:35That's quite important to discuss over today.
01:38We use indexing schemes, square bracket for directions, and parenthesis for planes.
01:48Like the square bracket is shaped like this, and then like a parenthesis is like this.
01:58And third, when and why do material properties depend on directions?
02:04That's what we are going to see through the model, like the properties like stiffness, electrical conductivity, and the optical index can change from one direction to another.
02:29In polycrystalline materials, with random grain orientations, properties tend to be isotropic.
02:36That's what the term we already learned in the previous slide, previous lecture, and then because the directional effects average out, that could be isotropic.
02:51And by the end of this lecture, you should be able to describe a crystal versus non-crystal structure.
03:00And just take a look at the textbook, read and assign directions.
03:07That's your mission for this whole chapter part.
03:12Okay, let's see the energy and packing part.
03:19So a key idea in crystallography is that atoms tend to arrange themselves to minimize the energy.
03:28Minimize the energy.
03:29Dance.
03:30Ordered packing lowers the energy.
03:32Like dense and ordered packing has lowered energy compared to, if we compare the value, this value, and this value, A and B.
03:45A, B value is B, the absolute value is greater.
03:52That's negative value, so B could be smaller, but the absolute value is obviously larger, right?
04:03So, dance ordered packing lowers the energy of a solid.
04:08You can think of atoms like spheres.
04:11And if you stack them efficiently, they occupy less space and maximize the shielding and bonding interactions, which could stabilize the structure.
04:24That's why many metals adopt close-packed crystals.
04:32See, like FCC or the HCP.
04:37In that case, we had like a atomic packing factor up to 0.74, right?
04:44That's the highest number for our calculation.
04:48In contrast, non-dense or random packing is usually higher in energy.
04:57So, a simple cubic and a body-centered cubic, they might have like the higher energy in terms of the energy part.
05:09Whether a material becomes crystalline or non-crystalline depends on both the chemistry and the cooling history that may change the density.
05:20If atoms have simple shapes and bonding, and they cool slowly enough to find their best position, they open crystallize with this high density.
05:33If the structure is complex or the cooling is very rapid, then they do not have time enough to find this dense packing.
05:43So, they might stop at certain point which is not completely dense yet, right?
05:52So, it explains why metals are open dense and crystalline.
05:58Why glasses are amorphous, right?
06:02And why processing can push a material into one structure or the other.
06:09We will learn about that a little bit later.
06:11We also learned about some iron or what was that, like titanium?
06:16Depending on the temperature, they change their structures, right?
06:21Okay, let's compare crystalline and non-crystalline one.
06:32Which one can you say, which one is crystalline?
06:36It says some here, non-crystalline.
06:39So, it's quite ordered one, but it's not clearly ordered, it's torted, right?
06:52So, let's compare these two crystalline and non-crystalline materials using silicon dioxide as an example.
07:00On the crystalline side, atom speck in a period 3D layer.
07:073D layer, it's just like the 2D layer, but 3D part, it also virtually can be periodic 3D array.
07:20Many metals, many ceramics, and some polymers do this under normal condition.
07:26In crystalline silica, each silicon bonds with surrounding oxygen.
07:32So, this could be silicon and surrounding oxygen, like this.
07:36In a regular, repeating network.
07:42On the non-crystalline cell side, atoms have no periodic packing.
07:48So, see, this could be like same as this one, but we don't see that kind of periodicity in this non-crystalline.
07:58So, some of them looks like some periodic ones, but actually they are not, right?
08:05This often happens in a complex structure, or when cooling is very rapid.
08:13That's what we mentioned like many times, right?
08:16For non-crystalline silica, commonly like glass, each silicon still bonds to oxygen atoms,
08:23but beyond the nearest neighbors, the arrangement lacks long-range order.
08:29The structure looks disordered.
08:34Arrangement lacks long-range order, right?
08:39This difference matters for properties.
08:43The crystalline ceramics are open, opaque, and translucent.
08:51I don't know if this is really true, but the textbook says it that way.
08:55You know, glass is a little bit non-crystalline, disordered one,
09:01but they still look very clear, and the crystalline silica, that's quartz.
09:09And that's also very clear, very transparent.
09:13So, depending on which direction we are looking at,
09:17so that's very important to determine that transparencies.
09:25Okay, so let's move on, move to unicell, crystal structure.
09:31To describe like a crystal structures, we use the idea of unicell, unicells.
09:38Imagine lacking the infinite crystal and finding the smallest 3D tile here.
09:47This is the tile.
09:49It can make, we can make just one tile like this,
09:53or one tile like this to make the repeating the same thing, right?
09:58Or, as I mentioned that before, like this one.
10:02Either one is fine, right?
10:04That is kind of unicell.
10:07But if we do this way, this one does not like tell us where the atoms are located,
10:19how far to each other in this unicell.
10:23But if we choose this one as a unicell, then we know where the atoms are located,
10:31and then how far they are away from each other.
10:35So that's easier to figure out.
10:38And the distance between the atoms, that's also like a, represented by a,
10:45that's also quite easier to see through the unicell, right?
10:51So, but your answer could be this one, but I prefer this one to give more information about the structure itself.
11:03So, in three dimensionally, that unicell could be like this way, or this way.
11:12See, like if we cut it, that actually the unicell should be like this, not like this.
11:20Right? Because if that repeating unit should make the whole three dimensional metal structures.
11:32So we cannot like build up this one onto another one of this one if we build this way, right?
11:40So this is more correct version of the unit cell for the 3D part.
11:53So remember that like a common metallic, the crystal structures include face-centered cubic, face-centered cubic.
12:02So this is the face-centered cubic, right?
12:07And body-centered cubic, body-centered cubic.
12:11And hexagonal closed packing that is not shown in this picture.
12:16We will see them a little bit later.
12:19So a few, like a few key terms, the coordination number.
12:25So here is like some coordination that occurs.
12:29And then we will talk about that coordination number.
12:32And that is the number of the nearest neighbors touching a given atom.
12:38So in this picture, we don't know actually they are touching or not.
12:45But actually they should because we cannot leave this space empty.
12:49It's like a packing.
12:51So although we drew this way, actually it's not like this one.
12:57It should be like this one.
13:01And based on that one, we already calculated like AFP before, right?
13:06APF, I'm sorry.
13:08That's quite confusing abbreviation.
13:13APF, Atomic Packing Vector.
13:18It tells us what a fraction of the unit cell.
13:21That's what we calculated before.
13:24So I have a few examples here.
13:28Can you find some repeating unit of this picture?
13:33You can stop this file.
13:39Press the stop button.
13:41And then find out what could be the repeating unit.
13:45For example, you know, it has three colors.
13:48So it should have three of them.
13:52Would you like to draw some sort of...
13:58I don't know.
14:03Do you think that could be a repeating unit?
14:06No, right?
14:07What if I drew...
14:10Include...
14:11Red one.
14:12I don't know.
14:13Actually, there is no repeating unit for this figure.
14:17So you figure out why.
14:19Then here, where do we see this?
14:29It looks like some Joseon Dynasty door design or something.
14:34Or some sort of any kind of design.
14:42Then I choose this way.
14:47You figure it out too.
14:49So maybe you could make some other way.
14:52But it's okay.
14:54And what if we choose just a simple this one?
14:58Do you think that's a repeating unit?
15:01If we put exactly this one onto this one,
15:06That doesn't make this one, right?
15:10That's like a mirror image.
15:12They are mirror images.
15:14So we cannot put this one into this one to make this structure.
15:24And here is another example you want to figure it out.
15:28This is the kind of a hexagonal one.
15:31Hexagonal.
15:34I'm sorry.
15:35It's a little bit...
15:36What is it?
15:37Round shape.
15:38It's turning to some other direction.
15:40But imagine that's like a flat one.
15:43And the left side,
15:45Maybe we can have some sort of...
15:49I don't know exactly.
15:52But somehow...
15:54But never...
15:55You can never have like a triangle shape for the repeating unit.
16:01Because if we make a triangle,
16:03Then triangle...
16:05This space is empty.
16:07So we cannot fill up the whole area with a triangle.
16:12We cannot rotate it.
16:15So this one is different from this one.
16:19Right?
16:20So that's very important.
16:23That's why I mentioned that here...
16:26This is not a repeating unit.
16:30Right?
16:31So that should be flipped.
16:33So practice.
16:34Yeah, it says this way.
16:35But you can figure it out.
16:36There are many.
16:37So let's keep working on our textbook.
16:51The base.
16:52And you know...
16:54Like a crystal structures can be grouped by unit cell geometry into seven crystal systems.
17:02That's cubic, tetragonal, hexagonal, orthorhombic, rhombohedral, monoclinic and triclinic.
17:13Depending on the ABC and also the angle of something.
17:20And then we describe a unit cell by three etching rays like ABC and three actual angles, alpha, beta and gamma.
17:32For example, in cubic crystal, A should be the same as B equals C and then all angles should be 90 degrees.
17:42In hexagonal crystals, you know, in hexagonal way, A should be the same as B but C, the long way is a little bit longer.
17:58That's ideal ratio was 1.6 something.
18:04That's like the textbook, chapter three, the practice question, which could be on your midterm exam.
18:14So you figure it out.
18:16And the two angles are 90 degree and the third is 120 degrees.
18:22So when we combine this geometries and specific atom position, we get the 14 brevet lattice, crystal lattice.
18:33We also call it like brevet.
18:37It's a French physics guy.
18:40So the right pronunciation is like brevet.
18:48But we just call it like brevet.
18:51Sometimes the, you know, like Paris.
18:58In United States, we, in English, we read it like Paris.
19:03But, you know, in France, they do not call it Paris.
19:07It's Paris, right?
19:09Paris.
19:10Paris.
19:11Paris.
19:12Like that.
19:13Here, they call his name like Brevet.
19:17Brevet.
19:18But a lot of the science book is written in English in the United States.
19:29So depending on the people, they also call like brevet.
19:35Brevet.
19:36But they are the same.
19:38So whatever you prefer, you just call it that way.
19:43Like brevet crystals.
19:46Sometimes I call it like brevets and sometimes like brevet.
19:51I don't have any preference.
19:57But I, if I have in mind, I normally say like brevet.
20:03Brevet.
20:04Brevet.
20:05Latice.
20:06So, the Unicell is our coordinate system for the everything that follows.
20:16We will use it to define points and some kind of directions and planes.
20:24Remember, like different symmetries lead to different equivalences.
20:31In cubic crystals, many directions and planes are like equivalent.
20:36When you have, let's say you have a dice.
20:42So this plane looks different from this plane.
20:48But actually they are the same when you rotate it.
20:51And then whatever you choose, they look different but actually they are the same.
20:59Right?
21:00So, sometimes they are, the planes are, or directions are all the same way.
21:07But sometimes they are different.
21:10So, for example, we, let's say we have like some body centered one.
21:16So, this plane is quite different from this plane, which is diagonal one with the atom in the center.
21:30So, that shape is different.
21:33The size is different.
21:34Their direction is different.
21:36Right?
21:37So, they are totally different.
21:39So, depending on the crystal system, we expect different properties and some, what is that, like a stiffness or whatever the properties are.
21:58So, from now on, like a point coordinates.
22:02So, if we choose some point, then how we name it?
22:08So, in this case, although, to reach this point, we should come to like a amount to x-axis, and then b amount to y-axis, and c amount to z-axis.
22:25Right?
22:26And then, if we, like, choose this kind of point, that we go half and half and half.
22:35So, that should be, if we have some, but this is not like a body centered cubic, because it doesn't look like a cubic, a and b and c.
22:44If they are the same way, the same distance, that's body centered cubic.
22:49But actually, they expressed as a, b, c. Currently, we don't know if they are the same or not.
22:56So, but still, we know, like, this is the center of the body.
23:03Body centered, it's not cubic, but what is that, like, the, some, some geometry shape.
23:16And then, if it's the same way, half, half, half, that's body centered cubic.
23:22Right?
23:23So, and then, the point coordinates for unicell corners are one, one, one.
23:31Right?
23:32And, if we consider, like, y and g, y and g plane, then we can, like, this is b and the center part.
23:46If we, like, move exactly the same, that could be translated into the number.
23:52Right?
23:53So, we could do that way.
23:56So, it's, uh, it's like a regular, what is that, mathematics class.
24:03So, uh, this, we can use this kind of thing to place atoms within the cell.
24:11Uh, corner atoms are shared among, uh, eight cells.
24:16Like, if a center, the atoms are shared by two cells.
24:20Right?
24:21And, uh, interior atoms belong fully to one cell.
24:24Like that.
24:26Okay?
24:27That's our, like, um, approach to see the, the point, like, atom in the unicell.
24:36So, crystallographic direction.
24:40So, remember that, so, when we have a direction, we have, like, a, uh, bracket.
24:46Right?
24:47This kind of thing, we are going to use.
24:49Then, parenthesis, that's for the plane.
24:52So, um, uh, let me see if I have some, some, like, uh, when we have direction, the, it's
25:06not plane direction.
25:07It's just, uh, the point to point direction.
25:11Right?
25:12So, uh, actually, the point one is, like, a zero point.
25:17And point two is some part X, X, the, uh, here it's shown.
25:23Like, A amount.
25:25Then Y, it didn't go that way.
25:28Right?
25:29It's, it's stick to X, G plane.
25:32So, G is, uh, half of C.
25:36In that case, we, the, we normalize the coordination differences in terms of lattice parameter here.
25:45It's shown here.
25:46So, X, this is delta X, right?
25:51This is delta Y, delta Z.
25:54Could be divided by A.
25:56And then, uh, adjust the smallest integer numbers.
26:01That's gonna be done this way.
26:10And then, like, enclose in a square bracket here.
26:14No commas.
26:15Like this.
26:17So, if we do that, that becomes this way.
26:23And then, another one.
26:28In this case, it's not starting from zero point.
26:32It's starting from some other points.
26:35We can, uh, put this point if we believe X, Y, Z.
26:40That could be A amount and half B and zero.
26:47Right?
26:48That's point one.
26:50And point two is what?
26:53Uh, X, that negative A.
26:56And Y is B.
27:00C is, G is C.
27:04Right?
27:05That's the crystallographic direction.
27:07And then, you can do the subtraction.
27:10See?
27:11Uh, the same thing.
27:13Right?
27:14And then, we do the same thing.
27:16Multiply two to make this one as an integer.
27:19You know, integer.
27:21Like, an integer is, uh, 정수.
27:23We cannot make, like, uh, 0.5 for this something.
27:28So, we multiply that to make it this way.
27:32And then, negative four becomes a four bar.
27:36That's the way we can express the direction.
27:46This is like a point direction.
27:48You know, when we have plane,
27:51Do you remember this plane direction?
27:55That's perpendicular to this plane.
27:57Right?
27:58So, this is the plane direction.
28:01And this is the point direction.
28:09And, uh, the HCP, uh, the crystallographic direction is more complicated.
28:16This is, uh, very important.
28:18Uh, you will see something in your midterm exam for this drawing HCP crystallographic directions.
28:27Obviously.
28:28Um, Miller-Brevet coordination.
28:33So, there are two ways.
28:36Here, there's one, one, one method.
28:40There will be another method, which has been explained, uh, in, in the textbook.
28:46Uh, two.
28:47We will, we will see that one here, too.
28:50So, um, like, here, again, hexagonal crystal, these, uh,
28:57small twist.
28:58It's kind of a twist.
29:00Some equivalent directions in hexagonal symmetry don't look equivalent in the three index system.
29:07So, with your four index scheme, like, um, the, it says, like, U, V, T, W, somehow.
29:18Um, called Miller-Brevet.
29:19So, Miller-Brevet.
29:20Brevet.
29:21Brevet.
29:22So, Miller-Brevet.
29:23Brevet.
29:24Brevet.
29:25Brevet.
29:26Whatever.
29:27So, um, the three, like, uh, A-axis lie 1 and 20 degree, 120 degree apart in the basal plane.
29:40A1, A2, here, A1, A2, A3.
29:47And G is perpendicular.
29:48That's obviously different.
29:50The direction indices related to actual vector components along A1, A2, A3, and G.
29:59A practical algorithm, here, like, remove brackets.
30:04But we don't know the brackets, yet.
30:08And then, um, so, here, they said, like, divide by largest integers so the values are less than
30:161.
30:17So, here, um, normally, like, 3 or 2 or something, uh, we will do, in most cases, we will divide
30:24into 3 pieces.
30:25So, here, A1, this is 0 point, 1, 2, 3, so, 3 point.
30:34So, uh, A2, here, 1, 2, 3.
30:38A3, there is, like, uh, here, 1, 2, 3.
30:43So, if we make this way, then, uh, whatever that point is pointing by or pointing through
30:52or pointing to, then, we can go through that.
30:57Let's see the example here.
30:59Uh, draw 1 bar, 2 bar, 1, 3 direction.
31:05It says, like, remove brackets.
31:07So, it goes this way.
31:10Right?
31:11So, bar becomes negative.
31:14And it's easy.
31:16A1 is, A1 is this way.
31:19And then, negative 1, 1, this way.
31:24And then, A2, negative 2.
31:26A2, A2, negative, this is the positive value.
31:30So, negative 2 value is this way.
31:321, 2, 2, 2 steps.
31:36And then, A3, 1.
31:39A3 is this way.
31:41So, A3 is this one, this way.
31:441, 1 part.
31:46And then, Z is 3.
31:481, 1, 2, 3.
31:51Up to the top.
31:52Right?
31:53It goes up to the top here.
31:55So, actually, it should be, like, direction.
31:59It goes this way.
32:01So, that's the direction.
32:16How we can go through.
32:18So, work on this.
32:21It's not that difficult.
32:24I will just change a little bit.
32:26It's not the whole thing.
32:27To let you find some of the crystallographic directions in HCP.
32:36There's another way.
32:37Which I prefer the previous one.
32:40The Braves Miller calculation.
32:44But this is a little bit more twist.
32:47Maybe if you practice a little bit, then you still like the first one.
32:54And let's draw specific HCP directions in a hexagonal unit.
32:59That's what we had a regular way before.
33:03And then, the same thing.
33:07It's complicated.
33:09So, it says, like, we have u prime, v prime, w prime.
33:15And then, once we have this value.
33:18Then, the calculation occurs through this one.
33:23We have to remember this calculation.
33:25If needed, I will give this equation on your mid-term exam.
33:31So, you don't have to memorize all this equation.
33:35But just take a look at this way.
33:37And then, let's see the example.
33:50So, tail location.
33:520, 0, 0.
33:53And then, head location.
33:55A, A.
33:57A, A.
33:58So, A1, A2.
34:00And we don't consider A3 any longer.
34:03So, this is the A1, A2.
34:06If it didn't go the other way, that's negative way.
34:11Right?
34:12So, negative, negative could be.
34:15So, bracket.
34:16It becomes 1, 1, 0.
34:18And then, this is u prime, v prime, w prime.
34:23And then, using this one.
34:27Remember that, u, v prime, the previous equation.
34:31u prime, v prime, w prime.
34:36And then, this number could be here.
34:39And v here.
34:40And w prime is here.
34:43So, that goes into that equation.
34:46And then, we find out the value.
34:52Whatever you prefer.
34:54One or two.
34:56I prefer the first one.
34:58That's why they explained the first way.
35:01And this is the second way.
35:03You know, the second way, we don't even know why we have to do all this conversion factor working.
35:10Right?
35:11So, the previous one might give you more precise way to understand the directions.
35:25Right?
35:26So, I hope you work with that way.
35:35And then, here, crystallographic planes.
35:39Let's apply the Miller procedure.
35:42Example number one here.
35:44And the plane that intercepts that x, the x and y axis at a and b, a and b.
35:55And it's parallel to z.
35:58So, they are like a 001 plane preference to the origin at point o.
36:07So, this one, 001.
36:12Actually, you should figure out why this is 001.
36:19And then, 110.
36:21110.
36:22And then, also, 111.
36:28So, when you look at the planes, then you figure it out why they treat as the same plane.
36:45This plane is the same as this one and this one.
36:48Right?
36:49Like some kind of shifting.
36:51I hope you remember all this kind.
36:56You are getting used to all these planes.
37:01And then, Miller indices, reciprocals of the three actual concepts of the plane.
37:07That way, we will go through that.
37:10For example, we just saw it, right?
37:13So, a, b, c.
37:15The point, it's like one.
37:19And b is a b.
37:21It's touching a and b.
37:23So, divided by a and b.
37:25So, it's one and one.
37:27But it never touches any z-axis.
37:30That's like infinity.
37:32So, and then, reciprocal.
37:36They divide by one.
37:42And this becomes zero.
37:44So, one, one, zero.
37:47This plane is called one, one, zero.
37:50Why one, one, infinity?
37:54What if we call like one, one, infinity?
37:57Isn't it explaining more detail for, I would imagine?
38:04So, I don't know why they have like a, they have to take these reciprocals to explain this plane.
38:11But, but that way, we can go.
38:16It's not that difficult to see.
38:20So, zero, zero, one.
38:26One, then if we have a body centered cubic.
38:29See, the atoms are here.
38:33Right?
38:34And then, this is one zero zero.
38:38And we do not have any atom here for body centered cubic.
38:43We, but we have one atom there.
38:46So, depending on what kind of the, the, the, the crystal system, the metal or the material has, then the atomic density could be changing.
39:02See, if we choose one zero zero plane, we see only one atom in, in square plane.
39:10But we, we see like one, two, three, four, five atoms on this plane.
39:15Let's say we are making some, the catalyst.
39:22That catalyst should have like a more atoms on the surface to, to react with some other materials on the surface.
39:34Then, we need some, some planes to be exposed to the, to outside of the material.
39:43Right?
39:44If that material is exposed to this one, one zero way, then we will have five atoms on the surface.
39:52But if we have one zero zero planes on the outside, we only have one atom to the surface.
40:01Then, which is more reactive to the material outside?
40:05This one, obviously.
40:06Right?
40:07So, um, uh, when we design some, the material way, if we don't want to make a contact with the material to make lower the energy or more,
40:21more inert one, then we will design that as a, this one.
40:27If we want that surface is very reactive to some other material, then we will go this way.
40:35So, uh, the linear density or face density or some square density, surface density for atoms,
40:43that's very important to understand, uh, the solid material.
40:49And, uh, this is the reduction the same way.
40:58You should be able to do this to, um, get the high score for midterm exam.
41:04And then, also this, the solid surface, uh, one.
41:09Uh, miller indices six, three, four.
41:13And then, um, that, but I, I mentioned that like, uh, some family can be used as this, this one.
41:24Um, I don't know the name of the, this bracket, some, some sort of family bracket.
41:33So one, zero, zero, one, zero, zero, zero, one, something, something, something.
41:39They are all the same thing.
41:41So when we have this indices, uh, just the dice, then this surface should be the same as this one, this one, and the other one, right?
41:52That's what it says, like a planes are the same way.
41:57Although they are expressed in different numbers, actually, they are the same.
42:02Uh, yeah, this is the important thing.
42:09The hexagonal planes, then the same way, a1, a2, a3, that's what we saw before.
42:18In the case of a1, it touched a1, right?
42:21But it never touched a2, right?
42:24It, it, it's parallel to a2, it never touches that way.
42:28So, uh, infinity, but a3, that's, that's the other way, right?
42:36So, negative one, and, uh, it, this plane is touching this, uh, c, uh, c height.
42:44So, one, infinity, negative one, one.
42:48And then reciprocal, this way, and then reduction, and, uh, Miller-Brava indices becomes this way.
42:57That's plane, right?
43:01So, um, yeah, this is also very important for your Metam exam.
43:08So, I, I think you understood most of the thing.
43:15So, let's examine the, uh, the atomic packing on planes using iron, as an example.
43:22Iron foil can be used as a catalyst.
43:27Oh, that's what I mentioned in the previous slides.
43:31The atomic packing of the exposed plane is important.
43:35Draw 100 and 111 crystallographic planes for iron, and calculate the plan density for each of these planes.
43:45We already have the answer for this in the next slide.
43:49We will see, uh, one zero zero this way, and body centered cubic.
43:56Um, then the repeating unit this way, radius something, and then calculate, and then square meter, we have, uh, 10 to the 19th atoms around.
44:10You calculate it based on our thing.
44:17And then 111 iron, 111.
44:21You know, like, uh, 100 is kind of just, uh, the square one, and, but 111 is this way, right?
44:32So, in that case, we calculated, uh, 10 to the 19th, the same order.
44:41You remember that?
44:4310 to the 19th, 1.2 or 0.7.
44:52Which one has a higher number of iron on the surface?
44:56So, this one, right?
45:01So, uh, when we make an iron catalyst, we should make that material, uh, surface could be exposed to one zero zero plane, rather than one one one plane.
45:21Um, that also changing some other properties and some characteristics such as like a slip one or, uh, ductility, some, some softness or a lot of things.
45:36So, that's quite important.
45:38Right?
45:39So, we summarized this one, the whole lecture.
45:42So, um, you remember, I mentioned that, uh, although I, I had a chapter three, four teaching, that, uh, the, the, I, I added like a chapter 12 into the same one to describe the same thing.
46:03So, uh, we can understand the crystal structure all together.
46:10So, um, you have to read like a chapter three and chapter 12 to, uh, to understand this whole, uh, the, the structure, crystal structure part.
46:22And crystal, uh, uh, crystallographic points and direction planes are explained to understand here.
46:31But there is can be like a single crystal and polycrystalline.
46:35That's what the combined with this lecture and then previous slide lecture, right?
46:42So, um, I, um, I, um, I hope you understand, uh, like, uh, all this crystal, not all the crystal system.
46:51We didn't learn about other things and monocleaning or something.
46:55We just learned about base material, uh, simple cubic and, uh, body centered one, the base centered one and hexagonal close packing.
47:05Right?
47:06And, uh, I hope you understand those things in easier way from now on.
47:12Okay.
47:13Uh, thank you for taking this, uh, class.
47:17And then I will come back with the, another chapter soon.
47:22Thank you.
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