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00:00Welcome everyone. Today we are going to immerse ourselves in the atomic arrangement of crystalline
00:14solids and see how these arrangements govern the properties we care about as engineers and
00:22scientists. Our focus will be based on the textbook on this subject and then I will mainly guide you
00:37based on the textbooks. This focus will be both systematic and then also very practical.
00:49We will start from the simplest the metal model, the atoms as hard spears. That's what we see in the
00:59textbook and also almost everywhere. And build up to real structure types in metals and ceramics.
01:12Along the way, we will connect these structures to density and ductility and other macroscopic
01:23behaviors in a later part. Here are our three guiding questions.
01:30And the first, what are crystal structures are common in metals and ceramics. We will map out
01:40like a simple cubic or the body centered cubic or face centered cubic and finally like a hexagonal
01:47closed packing for metals. Then turn to ceramic prototypes such as rock salt,
01:55asium chloride, zinc blend, fluoride, fluoride, and perovskite. You know, like these days,
02:03perovskite solar cells are quite a very strong area for that kind of research. And also zinc blendy
02:12is very the same structure for like a diamond like carbon link.
02:22So those structures are based on this the the solid arrangement and how depending on how they are
02:34arranged, then that covers the the properties and also the the slip mechanism which will be discussed
02:43in the later part. And we will discuss over that too. And second one, what features at the atomic level
02:52determine density? Density is very important too, right? Which is sometimes it's very heavy and
03:01and sometimes it's not. You know, have you like the try to hold a gold there with the same volume,
03:12which is much heavier than regular plastic the same with the same volume or regular like ice ice
03:22or the water ice. So all these properties can be determined by the atomic structure arrangement.
03:36Now we will see how the number of atoms per unit cell and the cell volume and atomic weights feed into
03:44theoretical density formula. So although we we go through the theoretical density, that can be compared
03:55to just the real the density of the metals or any material. And then we will discuss why two materials
04:04with similar structures can have very different densities. As I imagine here, like the same volume
04:12for ice and the gold or platinum, even like what is the lead, which can be used for blocking all the radiation,
04:24x-ray something, right? So they have a different density. And the third, how do how do ceramic structures differ from
04:36metallic one? So I, uh, I don't know why they have like, uh, uh, they are in the same chapter, like structure of crystal solids,
04:48like metal and ceramic. Ceramic can be treated in a different chapter, but it did before, but somehow
04:57they merged into one chapter, chapter three. I think they will split them again later,
05:04maybe in chapter under the version of 15th edition or some later. They have to change something to
05:13make the edition change, right?
05:18Um, by the end of this session, we should be comfortable describing the geometry of the major metallic structures,
05:28explaining clothes packed stacking, like packing, right? How do we pack? Um, you know, like, uh, that
05:36mentioned, I mentioned that like, Adam can be considered as a, like a very strong sphere, right? Just like,
05:44like, um, regular sphere, very round shaped one, right? Ball.
05:54And, uh, the, the, this, and, uh, how they can be, like, sequenced.
06:01And, uh, the computing theoretical deep densities for metals and ceramics.
06:09And, uh, predicting likely coordination numbers in ionic solids based on radius ratio.
06:15So, I don't know, we can do that easily or it doesn't matter if you, it takes some time, that's okay.
06:24But we can imagine how that those, uh, atoms are arranged in the space
06:31to make the solid structure or ceramic structures, whatever, right?
06:42Before we begin, think of a property you expect to be sensitive to crystal structure.
06:49What, like strength, ductility, electrical conductivity, thermal expansion?
06:56What, what's the major importance in your research area?
07:03For example, for me, um, I would like to have very like a stable structure.
07:11Uh, I don't need like a change the temperature a lot.
07:15I don't, my research do not treat the temperature up to like 400 and 500 degrees Celsius for the,
07:23for running. But some people may need it for like a space industry or space scientist or something.
07:33So, uh, what, what's your, um, focus for the material?
07:39That's what we are going to consider for the, for understanding this whole thing.
07:46Okay. Simply, we don't go, uh, that much further. Uh, we will go through this again in the, uh, in the, uh, next chapter.
07:58So, uh, we will see. Let's warm up with, uh, like a thought experiment, like imagine things.
08:04Imagine atoms as identical hard sphere. So they are all hard sphere. So they do not call it the
08:11textbook. Do not call it like a ball because ball is very soft. And if you press it, it, it goes,
08:18goes in a little bit. So in this case, just spear, hard sphere.
08:26If you, uh, I had to arrange them on a tabletop to minimize the, the, the space. So how many balls,
08:36or no, how many spheres can be placed in the same area. So if we consider it this way,
08:43see, we have multi, many, many, like rooms on the right side, but left side, we don't see that like
08:54much room compared to the right side. So left one has like a higher packing and the right side is lower
09:02packing, right? That's what we see. How, so, uh, if you had to like arrange them with high density,
09:13how would you do that? Like a left side or right side? The best two dimensional arrangement is, uh,
09:21like hexagonal packing, hexagonal packing, right?
09:24Each sphere is surrounded by six neighbors. So let me choose one thing. One, two, three, four,
09:35five, six, six neighboring atoms are surrounded for just one atom in the center.
09:43Now extend this idea into three dimensions by stacking these two dimensional arrays.
09:50And then the key question becomes, how do successive layers sit relative to the voice in the,
10:03the layer beneath? So sometimes we, we have to make this shape, this structure in a layer by layer,
10:14or this structure can be placed layer by layer. There are two, uh, efficient ways to stack, uh,
10:24for this close hexagonal, like a hexagonal class packing for the left one, because, uh, this is a higher
10:32packing and, uh, this is a kind of lowest and, uh, it has a highest we will see in, in the next slides very soon.
10:42And then like the sequence, this can be placed as A and then the next one is somehow different, uh,
10:52arrangement can be B considered as B. So, um, once taking away is A, B, A, B. We will see that a little bit later.
11:02So we just imagine it in the, in your brain. The other thing is A, B, C and A, B, C that's later will be, uh, turned out,
11:14turned out to be like FCC face centered cubic, which is like a face centered cubic structure.
11:22And then both the, uh, achieve the maximum possible packing density for equal, uh, spheres.
11:30That's around like a 74%. And this is quite low, um, around like 50%. We will see them very soon.
11:40Uh, a third, very, uh, the important structure for metals is the body centered cubic.
11:48BCC. So HCP is the most important thing. And the other one is face centered one and later body centered cubic.
12:03So it's not clothes packed. So, uh, I mentioned that like a 50% and that's around like a 68% or something.
12:12And then 74% is the highest density packing.
12:19As we imagine, uh, examine each structure, I want you to visualize three things.
12:25So keep it in mind, keep them in mind, where the atoms sit in the unit cell, where the atoms sit in the unit cell,
12:36along which directions atoms actually touch and how many nearest neighbors each atom has.
12:44For example, this, this one has like a six neighboring atom.
12:48So, uh, let's guess if you have multiple friends around you, then you may have like a multiple
12:58bonding with them and that may strengthen you for emotional connection and like a physical connection too.
13:11So, um, the higher number of neighboring atoms may reflect some sort of the higher interaction.
13:23And normally they are connected through electrons and positive, negative, positive, so like a strong bond,
13:32which means it may have the higher strength or higher density and something like that, right?
13:43We can, uh, connect the, the characteristic between them and the, the atomic density and its property as like a ductility or strength or something.
13:55So metals in general, they favor dense, highly symmetric structures, right?
14:10Compared to non-metal one, metals normally like, uh, this, or they are discovered as a solid form.
14:19If they are not metal, you know, like oxygen or fluorine, they are all gas, right? Helium.
14:28So they don't want to, uh, what is that? Like, they hold each other.
14:35But in the case of metal, they are surrounded by neighboring atoms, friends,
14:42and then they have a strong connection between them.
14:45So, um, you know, like, uh, they tends to be densely packed, right?
14:54So solid form, they want to pack, to be packed, densely packed.
15:04So, uh, the comp, the, let's see the composition.
15:08Pure metals often consist of one element.
15:13For iron, just iron.
15:15For gold, and gold.
15:18Right? So the atomic spheres have nearest the same radius, right?
15:25The, this uniform formality makes a periodic and symmetric packing straightforward.
15:32That's quite obvious, isn't it?
15:40And, uh, like, uh, the bonding, the second one, metallic bonding is not directional.
15:47Oh, that's quite interesting.
15:49I thought they could have some sort of bonding directions, but it says it doesn't.
15:56Metallic bonding is non-directional.
15:59The valence electrons are delocalized.
16:04Like, uh, we already learned about this.
16:06Like, the electron C,
16:09전자 바다, right?
16:11That screens the, the positively, uh, charged ion, the cores.
16:17Because, uh, the bond isn't, uh, tied to a specific orientation, unlike covalent bonds,
16:28atoms can arrange to maximize overall cohesion without the, the directional constraints and
16:37covalent networks impose.
16:39And, uh, the third, energetics.
16:47The systems tend to lower their energy by bringing, uh, the oppositely charged species closer,
16:56balanced against core to core repulsion.
17:00Right?
17:00So if they get too close, then positive, positive, they have, we will have repulsion.
17:07Allowing smaller nearest neighbor, uh, neighbor distance,
17:12and a higher coordination of neighbors, both of which lowers the overall energy.
17:24So nearest neighbor distance tend to be small in order to lower bond energy.
17:31And fourth, electron cloud shells cores from each other.
17:38Otherwise, uh, the, the charge cannot be, uh, what is that?
17:43Like, uh, canceled out, right?
17:47So electron cloud shells cores from each other.
17:51That's because the, the high-symmetric crystals often
17:56represent the robust energy minima for a wide range of temperatures and, uh, the compositions.
18:10And, uh, the, so these are the, why we have, like, uh, the dense packing.
18:18Metals have the simplest, the crystal structures.
18:28Metals have the simplest crystal structures.
18:32Then what's others that we will examine three such structures.
18:37So, uh, we are, like, uh, saying some metal structure.
18:42Then what, what else do I have here?
18:45We said, like, a ceramic structures in the later part.
18:49That's quite different.
18:50And then metals have, like, a simplest crystal structure.
18:55We will see what are those are simplest, the crystal structure.
19:02Three such structures.
19:04We will see three.
19:05What are three?
19:06We will see.
19:08I think it could be, like, a four.
19:12Okay, let's start, uh, with the simplest, uh, possibility.
19:20The simple cubic structure.
19:22Simple cubic.
19:23Just a simple cubic.
19:26Cubic.
19:28And, uh, the picture, the,
19:34of this, the cubic.
19:36Imagine this one.
19:37So, every corner, we will have, uh, atoms.
19:42And, uh, like, uh, it's, like, uh, eight corners.
19:47And no other atoms in the cell.
19:50Here, atoms touch along the, the cube edges.
19:54If we model them as hand spheres, uh, heart spheres, like, with, uh, just, uh, the right radius.
20:04Right?
20:11The coordination number, which is the number of neighbor, uh, the neighbors,
20:17the nearest neighbors, each atom has, is a six in simple cubic.
20:23Right?
20:24Let's see why it's a six.
20:26So, let me choose this one.
20:29One.
20:30Two.
20:32Three.
20:33This is not.
20:34This is not.
20:35Right?
20:35So, this is longer than this one.
20:38So, it's, it's the,
20:43what is that?
20:44Like, a closest one.
20:46And then, we will see the upper one.
20:50The symmetrical.
20:51This one and the symmetric the other way.
20:54And this one has the symmetric the other way.
20:56This one has the symmetric the other way.
20:58So, one, two, three, one, two, three, four, five, six.
21:04Six coordination.
21:07Coordinated atoms.
21:11So, can you guess?
21:13Oh, that's the, the, the drawing of the right side.
21:17This one.
21:25Right?
21:26Despite its conceptual simplicity,
21:30simple curie is rare among metallic the elements.
21:34The packing is not very efficient.
21:37We will see that in the later, um, slide.
21:41And the resulting lower number of neighbors
21:44generally makes its arrangement less favorable for a metallic bonding.
21:50And, uh, the poluminum or the, is a most, the, the famous exceptions.
21:59It adopts the simple cubic lattice.
22:02We don't know about like a polonium.
22:04Have you heard about it?
22:06No, I, I don't think we will have a chance to see that
22:11polonium for your research.
22:15Maybe just one or two for the whole graduate students in our, uh, graduate school.
22:25But that one is, uh, is favoring this simple cubic structure.
22:31They don't know much about it, why they choose this kind of,
22:36but there are different kinds of, uh, they, they will have their own situation.
22:41For most metals, more neighbors and, uh, higher packing efficiencies
22:49proved, uh, energetically superior.
22:52So we will see what's next.
22:54So in this case, they, uh, calculated the simple cubic atomic packing factor.
23:04Uh, uh, atomic packing factor is, uh, quite important to see
23:09the, the, to calculate the density of each metal.
23:13As we mentioned before, like, uh, density of the metal is quite important to
23:20see how stable that metal could be, right?
23:24So, uh, here, if we, um, put a for the, the, the length of the edge,
23:34then this is simple cubic.
23:36So this is a, and, uh, this is a, a, a, a, and everything is a.
23:41And then, um, so if we have like an atom that is considered as R,
23:50they are facing at the edge.
23:53So this R and this R, two R becomes just a, right?
23:59So R should be half a.
24:04So, um, atomic packing factor can be simply calculated by
24:10the, the, the weight.
24:13This is weight.
24:15And this is volume of the, uh, unicell.
24:20So, uh, how many atoms are existing here?
24:24Just, this is the one eighth of the atom.
24:27One, two, three, four, five, six, seven, eight.
24:29We watched that video before, right?
24:32So that's eight, uh, eight times one of eight.
24:39So that's one atom.
24:41So one atom.
24:43So, uh, the, the weight of one atom is this one.
24:50And, uh, they calculate this way.
24:52And then finally discovered like a 52%.
24:57Oh, I'm sorry.
24:57I mentioned like a 54% before.
25:01So that's 52.
25:02I'm sorry.
25:03Let me correct that 0.52.
25:06That's just half of the, uh, the volume.
25:11Right?
25:12So I don't think this is the, uh, um,
25:16the high packing for the metal.
25:19Now we move on to the, um, the next slide.
25:27Let's move to the, uh, the body centered cubic structure,
25:31which is one of the, the workhorses of the metallurgy.
25:37That means like a chromium, tungsten, iron.
25:40We heard about that a lot, right?
25:42So titanium, molybdenum, they are having like a body centered cubic.
25:48Okay.
25:49In BCC, uh, atoms occupy the eight corners of the cube,
25:56like the simple cubic before.
25:58So we remember that this eight, uh, cornered one makes just one atom.
26:05If we collect them, put into the, put into, uh, just the whole thing,
26:12that becomes one.
26:13And then it says like a body centered cubic, right?
26:18So it's, it's cubic system, but in the center of body,
26:23we have another atom there.
26:26So that's why body centered cubic in the body centered cubic,
26:33still cubic.
26:34So, uh, if we consider only this etched atom, that's 52%, but we are not,
26:43have another atom inside there.
26:45So due to that atom, this doesn't meet each other any longer.
26:50So, uh, so they are, have some gap between them.
26:53So, uh, in this case, we, we still put a on, on the etched, etched distance.
27:01Then a is not, uh, 2r, not any longer, right?
27:11So if we draw this way, that can easily show us where the atoms are located.
27:18But actually they are faced, uh, they are facing each other.
27:22So the actual density drawing is looking like left side.
27:27So, uh, we consider the contacts here, actual touching between spheres.
27:44The key contact direction is along the body diagonal, right?
27:48Body diagonal.
27:50So we will see the next slide.
27:52that's, uh, the, if this is virtual materials, some sort of like, um,
28:01you can go to the textbook website and then they are providing,
28:07they provide some, the three dimensional working system for this metal arrangement.
28:19But somehow it doesn't work that much.
28:23So I tried to use it before, but, uh, I couldn't, but just try some other website to
28:35see like a three dimensional one.
28:38You can just simply go to the, uh, the, the YouTube to find out the, the visualized,
28:47the cue, the body centered cubic.
28:52So here, as I mentioned before, like, uh, this one is like a, the, um, what is that?
29:01root three, a, and then root two, a, and a, right?
29:09So, uh, they are facing each other.
29:12That's, so we have to figure out which is, which direction is, uh, uh,
29:20like a making contact.
29:23It doesn't have any like empty space.
29:26See here, this, this line, it has empty space.
29:30So we cannot use like a two R equal a, it's not working here.
29:36But this diagonal one, like a three dimensional diagonal one,
29:41they are facing each other.
29:43How do we know if they are facing?
29:47Because textbook says that way, then we have to keep that in mind.
29:53If we draw that, they have to face it, right?
29:58So they calculate it this way.
29:59And then they figure out the, uh, the packing density is 0.68.
30:06Do you remember the packing density for simple cubic?
30:100.52.
30:13I was wrong.
30:13I got 0.54 before, but 0.52.
30:17And, uh, that's a simple cubic one.
30:21And then now like a BCC 0.68.
30:27So, uh, I, uh, I hope you remember how you can calculate this,
30:34like a packing density.
30:35If you see this kind of situation for your midterm exam,
30:39then you have to figure out how you can calculate it, right?
30:44So, I, uh, I know you, you can do that.
30:51So the coordination number,
30:55what's the coordination number for this biocentered cubic?
30:59So, uh, let's see, um, then to figure out how many atoms are touching this, uh, the one atom.
31:12In the case, it's easier because, uh, if we consider this body centered one, right?
31:18That is contacting one, two, three, four, five, six, seven, eight.
31:23Eight, uh, spheres are making contact with this centered one.
31:29Correct?
31:33Along the body diagonal direction.
31:38Each, like a corner atoms, contacts the center atoms,
31:42plus the equivalent neighbors in, uh, adjacent cells,
31:47maintaining the same, uh, count per atom.
31:54Hmm.
31:54So, uh, the,
32:01we already observed the, like a samples of this body centered cubic.
32:05What was it?
32:05Like a titanium and, uh, chromium and molybdenum,
32:11and, uh, titanium or titanium?
32:14Or, um, the...
32:19Yeah, that's, that's the one we would like to see.
32:23I, uh, I, uh, I hope you remember all this, like a packing factor.
32:31Now, let's move to, like, uh, what is that?
32:37Face centered cubic one.
32:38Um, I think face centered cubic is very important.
32:48So, you know, when we consider something,
32:52important thing comes a little bit behind, right?
32:55So, it's a little bit behind from like a simple cubic,
33:00and then the body centered cubic, and face centered cubic.
33:06The face centered cubic is the, uh, the arguably,
33:10the most important metallic structure.
33:15Based on the textbook, that's what I'm saying.
33:17Here, atoms occupy the cube, uh, corners, right?
33:23Like a cube corners.
33:28And the centers of all six faces.
33:32You know, what's the meaning of face?
33:35See, it looks like dice.
33:39Dice.
33:40You know what dice is?
33:44Dice.
33:44You draw a dice to see number.
33:50Right?
33:54So, uh, here.
33:56So, if we do not have this,
34:00then what?
34:01That's simple cubic.
34:03If we have something in the center, that's body centered cubic.
34:08Now, we still are considering like a cubic,
34:11but the atoms are located in the faces.
34:16How many faces do we have here?
34:18Six faces.
34:21So, in six faces, we have, uh, atom.
34:26If we cut it, then half of the atom is in that, that cube inside, right?
34:36So, how many of them?
34:37Like six.
34:39Six times a half sphere is three.
34:44And then outside, that becomes what?
34:46One.
34:49Total four.
34:51Four atoms in a cube.
34:54Simple cubic has just one.
34:57And then body centered cubic has only two.
35:01Now we have four.
35:03That's quite interesting.
35:06You know, one, one atom just for a simple cubic.
35:10But now we have like a four atoms for the simple cubic tool.
35:14And then body centered cubic is zero point six, eight.
35:16Then why it has packing density, uh, has some number, like the, what is that?
35:27We will see the next one here.
35:29So, we remind that like a simple cubic, zero point five, two.
35:38And then body centered cubic, zero point six, eight.
35:42Now, like a FCC and hexagonal close pack, they are both have zero point seven, four.
35:49So, this is the highest number for metal packing.
35:55But here we have coordinated just atom inside.
35:59We have only one and here two and have four.
36:03Then it's quite interesting.
36:06So, it's like a four times greater, but it doesn't go four times.
36:11Because the cubic size has been changing too.
36:20Although this, the cubic, the size is changing for the repeating unit.
36:26Still, it has the lower void volume inside.
36:33That's what we are talking about, right?
36:36So, uh, we also want to like do some, uh, counting atoms.
36:45As I mentioned before, like each face centered atom is shared by two cells.
36:51So, uh, contributes to half.
36:54And six faces, that's three atoms.
36:56And, uh, add three eight corners.
36:59Eight corners, uh, each contribution is, uh, eight and four, one.
37:04All together, four atoms.
37:06And how about the coordination number?
37:09Remember that?
37:11So, how many atoms are making contact, touch?
37:18This is quite, uh, difficult to understand, but you, you will see.
37:26Each atom is surrounded by a dozen neighbors at the same distance.
37:32A dozen, somehow, a dozen.
37:40So, let's see this one.
37:43They are saying this distance and this distance, because they are making contact, right?
37:49So, they are all the same.
37:51One, two, three, four, five, six, seven, eight.
37:55Then we will have another one on the top, the same way, right?
37:59This one goes another way here, too.
38:02So, four, four, four.
38:05So, total 12.
38:10This reflects the, uh, the highly symmetric, like, close-packed nature of the lattice.
38:16Many common and industrial, uh, crucial matters, uh, FCC, here, aluminum, copper, or gold, too.
38:26Like, red, nickel, platinum, and silver.
38:30All the expensive, uh, metals are, in this way, face-centered cubic.
38:37Uh, can you guess what are the characteristics for those?
38:46Aluminum, copper, gold, nickel, and then nickel, platinum, silver.
38:52Especially silver and gold.
38:57FCC metals generally exhibit excellent ductility.
39:02Remember that, like, ductility, ductility.
39:10What's the meaning of ductility?
39:14You have to understand that words, for many cases in this, uh, in this semester,
39:19because we will consider them a lot, consider the word a lot, like ductility.
39:25Yusunam, because they, uh, possess multiple close-packed sleep system,
39:32with low barriers to dislocation motion.
39:37Those words are quite complicated, right?
39:40We will learn about that in the later chapter, too.
39:43So, uh, at this moment, you don't have to understand many things,
39:47but I want you to keep it in mind, FCC
39:51has, uh, FCC metals are, like, copper, gold, silver, platinum, all that, like, ductile metal,
40:05compared to hexagonal close-packed one.
40:12So, uh, that FCC, FCC stacking sequence are, uh, various.
40:20We will see that.
40:25To really see FCC, imagine slicing the crystal along a close-packed plane,
40:37which in cubic crystals is the 1-1-1 family.
40:41That's, we will talk about later, too.
40:44Each of these planes show atoms arranged in a triangular lattice.
40:50See? Triangular lattice.
40:53But actually, triangle cannot be a kind of, uh, uh, the unit cell.
41:01Stack those, uh, triangular layers, triangular layers,
41:06are, uh, in the, uh, in the following sequence.
41:12The first layer is A, A, A, A, and then B can sit on here.
41:18If we put this way, then, um, the, this part is, like, still we can see through.
41:29They are, like, empty.
41:31If we put the, the next ball on top of this area,
41:36that, that blocks every, uh, the, every empty, uh, line through.
41:43We can, we don't see that through.
41:45So if we consider this way, if we place this, uh, the green and the blue,
41:56this one, this one is empty.
41:58Right? This one is empty, this one is empty.
42:08But the next layer, if it goes another way, this way,
42:12the green one, then still this lane is, uh, this, the, the hole is still empty.
42:21But if we block them with, uh, like, another ball,
42:25then that's A, B, C, A, B, C.
42:28This is A, B, C, A, B, C.
42:30This is A, B, A, B.
42:35They are different.
42:36That's quite difficult.
42:43If you look straight down one of these closed-packed planes,
42:47you will see a repeating three-layer pattern.
42:51This three-layer periodically define, uh, defines FGC's stacking character
42:57and separates it from HCP.
43:00So this is HCP.
43:06This is FCC.
43:14Okay.
43:22Now, hexagonal closed-packing.
43:24Hexagonal closed-packed shares the same two-dimensional closed-packed one,
43:31as I mentioned before, like A, B, A, B, A, B.
43:36If we build up this way,
43:38its atomic packing factor is still 0.74.
43:43And then coordination numbers are the same way, like 12.
43:47And, uh, but it's not like a simple cubic, right?
43:50So in this, in the previous one, simple cubic, face-centered cubic, or some simple,
43:59body-centered cubic, BS, I'm sorry, simple cubic, body-centered cubic, face-centered cubic,
44:09now hexagonal closed-packing.
44:12So in, in the previous one, like C means like a cubic, body-centered cubic, face-centered cubic.
44:19But here, it's like a, it has a C, but C doesn't mean like a cubic.
44:25Hexagonal closed-packed this structure.
44:31That's cadmium, and the magnesium, titanium, and, uh, zinc.
44:38Like FCC, HCP has a coordination number of 12, and the same APF.
44:49That's important.
44:51And, uh, the C2A ratio, C2A ratio is 1.633.
44:57You don't have to remember this one.
44:59So, uh, you also have to keep it in mind.
45:02And for the test, the, if you need some sort of the equation, I'll give you the equation.
45:11And, uh, for, for example, this kind of ratio, I don't think we have to keep it in mind.
45:18So if needed, I will give it, give this number to the question.
45:23So you don't have to worry about remembering all these numbers.
45:28But you can, you should be able to calculate the atomic packing factor or coordination number, right?
45:38That's not that the hard to remember.
45:45This is also like a virtual material, some sort of three-dimensional image.
45:51So you can go through the, the YouTube or some other, the web demand, the three-dimensional
46:02designing software to see what it looks like.
46:11Okay.
46:11Now the theoretical density, we will see theoretical density.
46:17Um, let's move on to this formalized density.
46:23The theoretical density raw of a crystalline material can be calculated if we know these four things.
46:35The number of atoms in the unit cell, the, that's what we already learned.
46:40Like it's four simple cubic.
46:41We had just one atom and the body centered cubic two and the face centered cubic four, right?
46:48That's what we already know about them.
46:51And then, uh, the atomic weight.
46:54We also know about it, but we don't know the each atom's weight.
46:59So we have to know like Avogadro's number to calculate the atomic weight.
47:04And then the volume of a unit cell.
47:08So if we know the volume of the unit cell and then weight of atom and how many, that's density, right?
47:22The four cubic lattice, you know what's the things, right?
47:29So that's what we can calculate.
47:34So here, here's the example, chromium.
47:38The 52 gram per mole and body centered cubic.
47:43In the body centered cubic, we know we have two atoms there.
47:47And that's atomic, the atomic mass.
47:51And then we have to divide by Avogadro's number to get the number of the, the, the mass ratio.
48:02So it's, it's atom's mass.
48:06And then this is the volume, right?
48:07A, A, A.
48:09And this A can be replaced by R related one because we actually need to have like a real number for the volume.
48:19So if we calculate it, theoretical, the density is 7.18 gram per square, the cubic centimeter.
48:32But actually that's 7.19.
48:35They are very close, right?
48:37So that our imagine or our the approaching directions are all correct, right?
48:49That's what, uh, how we can prove this method is, uh, correct.
48:57It's kind of the, uh, thought experiments, right?
49:02So let me, uh, wrap up here for the first class, the, for chapter three.
49:14And then I will continue for the, uh, next part in the, in the next recording file.
49:20And then I will continue for the first class, the, in the next recording file.
49:22Thank you.
49:22Thank you.
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