okay um this will just remind me to start recording so don't forget some announcements yeah so quiz one is available and again this will only be uh for this time will be full credit if you attempt it it is a timed quiz so there's five questions um you have seven minutes to complete it it's very similar to the homework questions and this is what the quizzes will be like for the remaining quizzes as well and it's only available today so you have until midnight tonight to start and complete this quiz in seven minutes um homework two i'll post that today it will be due next wednesday at noon it has 20 questions i think i might actually i still have to edit one of the questions but that will be available uh this afternoon this evening next week we have a new ta in charge of quiz section that will be sid and homework 2 is due next week and again there will be another quiz on the same day in the similar format and then the week after that is your first midterm so i will not be doing a quiz section but um during quiz section i believe this is the case you'll be taking your midterm and we still don't know the format it's either going to be on canvas or through i mean it's definitely going to be on canvas but it might be similar to the quiz where you have like an hour to complete it and whatnot you also have homework due that that um that day as well okay and then make sure you're on top of the reading like i said the summer quarter things are going by a bit quickly so you know read ahead for next week this is this is the reading assignments for next week um also i've established my office hours on monday after class on monday every now and then i might offer some additional office hours and i'll send an announcement out okay uh so dr brush today talked about crystals and uh i wanted to talk more about that crystallography of course the study of the atomic arrangement in solids so you have a crystalline solid where it has a long range order and then opposed to an amorphous solid known as a glass material where it still has short range order right so you have silicon silicon still tetrahedrally coordinated to oxygen but there's no long range order in an amorphous solid all right so this is an example of quartz versus silica and quartz and silica they have the same chemistry the same components silicon and oxygen but one is crystalline one is amorphous and of course as you know silica makes up uh most glass is a major component of like window glass and other glasses like drinking glasses so here's another example of crystals these three crystals look very similar right they have the same sort of uh shape to them and uh although they're different colors so this is corundum ruby and sapphire and does anyone know the atomic excuse me the chemical formula for these three crystals or one of these three crystals okay the the chemical formula for corundum ruby and sapphire is aluminum oxide let me oops i move my mouse aluminum oxide for all three of these are made out of the same base material which is aluminum oxide a crystal crystalline aluminum oxide now for ruby and sapphire is interesting to make the color it gets doped doping which means we're adding some kind of impurity to the crystal in this case for ruby we're adding about one to three percent by weight i think it's weight not atomic percent chromium three plus and we're substituting it with aluminum inside the crystal structure so the crystal structure is still the same just some of the atoms have been replaced with chromium and that makes the crystal red and sapphire do the same thing with titanium and iron you have to have both in order to make it blue color so that's a little bit of interesting and but the point is that they they have the same crystal structure and that's what gives it this macroscopic shape and as dr brush had mentioned there's poly crystals a polycrystalline material have many smaller crystals and the boundaries of these crystals are called grain boundaries all right so each one of these crystal lights has its uh atomic arrangement this crystal structure but they might be in different orientations next to each other here's an example of a polycrystalline material not only is it polycrystalline but it's a poly composite material that's a you could consider it a composite right it has multiple minerals multiple different uh chemical formula excuse me chemical compounds that make up this granite so quartz mica feldspar uh here's a little quiz i i like to give this to um i do a lot of outreach for k through 12. so i'm going to give you guys the same kind of quiz you know which one of these materials contains crystals and uh there's there's one of these materials that does not contain crystals so in the in the chat where you can shout it out which one of these do you think does not contain crystals we have crystal wear copper metal graphite polyethylene all right it looks like most of you guys are saying uh the polyethylene jug does not contain crystals and that is incorrect yeah it's a kind of yeah it's a trick question the crystal wear is the only material on this uh list that does not contain crystals crystal wear is actually glass all right it's just called crystal wear because you know it has facets it's cut and it looks like crystals where you know this is something i learned i didn't know until like high school i took a material science camp here at the uw that metals contained crystals and that kind of blew my mind that metals and everything almost made up of crystals graphite of course has a crystal structure we talked about that on first week the polymer and this is where most people say is it doesn't have containing crystals polymers do will uh be kind of semi-crystalline where their polymer chains will arrange in a periodic fashion with long-range order and so that the polymers do contain crystals a crystal structure crystal wear is a specific type of glass does anyone know no the k-12 students also all said the polymer some of them say the copper metal too so it's all right the crystal where does anyone know what a type of glass crystal where is or what the chemi what the major uh what what's the chemical formula that makes crystal wear unique and doesn't look like anyone knows so crystal where boron silicate uh nope not quite uh nah but it's not a borosilicate glass like pyrex is a borosilicate glass crystal wear is a lead glass so you know if you have any fine crystal in your house it contains a bit of lead oxide and the reason why does anyone know why well i guess you might not know because you didn't know the first question the reason why crystal wear uses lead oxide is that the lead oxide component in glass adding lead oxides to this glass increases the index of refraction okay the refractive index so it makes it so the light that bounces in bounces out at a higher angle and so it makes it optically attractive right like diamond has a very high index of refraction cubic zirconia has a very high index of refraction that's why we use it as a substitute for diamond and jewelry crystal lead glass also has a fairly high index of refraction the other reason why lead glass is used for crystal wear is that it makes it easy to machine it makes it softer it's easier to cut these facets into the glass to make it as we call it a crystal but in fact it's actually amorphous um does it make it melt into molds yes but related to the manufacturability it makes it easier to cut to make these facets and there's actually been some studies that look at you know different types of beverages acidic beverages like wine and how much lead is actually leached out of the wine so you gotta be a little bit careful actually my family has a bunch of crystal glass that we use almost every day so i should probably check into how safe it actually is anyways just uh and here's another example of crystals in real life so you might have seen these guard rails or like light post lamp posts uh handrails that are zinc galvanized right so they're made out of steel but they're dip coated in zinc metal and uh to act as a corrosion protection for the the steel or iron and the zinc makes these very beautiful crystals and the the shape and size of these crystals this is called a spangle all right you can have different types of spangle and depending on the different additives you put into the zinc alloy you can make them smaller or bigger for different optical properties but it's mainly for corrosion prevention so next time you're at the street corner and you take a look at if uh the lamp post is uh zinc galvanized you can see spot out these giant crystals um some metals the element pure elements have a variety of different crystal structures and we're going to look at some of the more common crystal structures that duck brush has already covered but you can see how the crystal structures follow you know these different chemical groups so obviously you know how the atoms arrange depends on the the electron configuration uh in in the chemistry of the atoms um some so the common ones are the face centered cubic also known as cubic close packing all right so nickel copper silver gold aluminum those are the the very common uh face centered cubic metals and then bcc is another common iron at room temperature is bcc um and the alkali metals are also bcc and then hexagonal closed pack is also the common element or crystal structure we talk about titanium specifically is uh when when i think of hcp i always think of titanium metal um yes some some of the crystal structures will change at different temperatures uh iron is is the obvious example of that at a higher temperature i forget which what temp is like 700 and something degrees celsius 724 something like that it changes from bcc into fcc and then at even higher temperature it changes back to bcc another interesting thing about the crystal structure of iron is um when we start talking about alloys so we're substituting some of the iron atoms for some other atoms uh specifically stainless steel is made of a nickel iron alloy there's a large percentage of nickels so if you put a large percentage of nickel substitute with the iron what happens is your crystal structure changes uh this crystal structure of stainless steel is uh fcc not bcc all right let's uh move on uh so this we're coming to our first uh question first problem uh is unit cell calculations i i'm gonna have you guys calcul do these number of problems for each unit cell the first is the number of atoms in the unit cell so in this example is just the simple cubic lattice and as dr brush had said polonium is the only single element that forms a simple cubic lattice all right the crystal structure is simple cubic and we'll talk about this a bit more the crystal structure is made up of a lattice points and a basis in this case it's just a single atom all right and so the number of atoms in each unit cell you need to calculate for a simple cubic is rather easy each corner there's eight corners and each corner has one eighth of an atom so there's one atom per unit cell okay and then you want to find out the coordination number so this is the number of nearest neighbors uh for us any atom they're all they're all they're non-unique right they're all the same so if you take one atom here and find its nearest neighbors and so you have to kind of imagine what does the unit cell above and to the right and in front of it look like so if we take this corner atom there's these three here one two three and there's also going to be three there's one above one to the right and one in front so the coordination number is six which makes a octahedral coordination okay then last parameter a so that's the dimension of the unit cell okay and in terms of radius so a simple cubic is very easy uh it's just the two radii so a equals 2r and then also the atomic packing factor it's going to be the volume of atoms so four thirds pi r cubed times the number of atoms in the unit cell divided by the volume of the unit cell so it'll be in this case uh a cubed which is two r cubed okay so simple cubic is not very dense it's a 50 uh atomic packing factor all right i wanted to do this uh we're going to do it for body center cubic and face center cubic but not hexagonal it's a little bit more difficult i wanted to do this uh break you guys into breakout rooms um i haven't really prepared it and then i'll give you about five or so minutes to do it we're going to start with the face centered cubic and then i after five or so minutes i'll i'll call out on a group for the answer but uh so when i break you out into breakout rooms you're i think you're gonna lose this page so i what i suggest you doing is if you're on your computer uh if you open up like powerpoint or word you can quickly copy this uh this slide if you're on windows the shortcut for a screenshot is just windows shift s and then allow you to make a screenshot i don't know if you can see what i see because my friend my screen that gets frozen and then when you select the area you want to screenshot it copies it to the clipboard so for example i like to do everything in powerpoint so yeah there it is so that way you have it when i break you out in the breakout room so i'll go back here real quick um but this is going to be a little bit of an experiment for me because i haven't really done this before let's see we're going to have let's try to get four participants per room about three to four participants per room all right looks like everyone's back now so we're going to go over the answers i'm going to pick on people from different groups to give their answer now if you don't have an answer that's okay just say you didn't come up with an answer or whatever excuse you want to come up with but it's totally fine um so let's start with uh from let's see room one if you're there can you answer the first question how many atoms fit in the in one fcc unit cell count the number of nearest neighbors or neighbors are touching so below it directly below it we should have four right we have one and two we can see two here there's also two in the back that we can't see then along the equator here we have these four that we can see here and then directly above it there's gonna be another four all right so four times three is twelve and this makes this uh symmetry called a cubo octahedron all right so if you have your central atom and then this is the the nearest neighbors around it makes 12 nearest neighbors all right how about the next question last parameter a in terms of radius r so in this diagram i drew here is just of the face of the fcc unit cell and we're looking for a in terms of r the radius of the atom so here we can draw the shape we have a triangle um where the what is this the hypotenuse and it's been a long time since trigonometry is equal to 4r all right and then the legs of the triangle are a and we can use um what is that pythagoreans theorem a squared plus a b squared equals c squared and then you can solve for a in terms of r so a equals two r times the square root of two so if you know the radius of any atom okay you can calculate and you know it's fcc crystal structure then you can calculate its its lattice parameter a okay and then from that you can use it to calculate the density of the material if you know its molar mass 0.74 is correct yes so you just take the volume of the atoms in the unit cell so we have four atoms times the volume of the sphere divided by the volume of the unit cell and if you've you've calculated the parameter last parameter just a squared or q excuse me a cubed so it's that value cubed 0.74 so that's an important number to memorize 0.74 for fcc 0.74 also 12 is an important number and 4. so these those are the important numbers okay let's move on um and we're kind of it takes a bit of time to go through all this so i think if it's okay with you guys we're going to just not break out into breakout rooms for this next one so i can move on to the the next content um but the same thing for bcc so i'll go through it with you or we can just uh answer out loud uh so how many atoms fit in one oh this is a typo here it should say one bcc unit cell how many atoms are inside one bcc unit cell anyone want to shout out okay i think that should be two uh yes that is two so how about the coordination number eight eight is correct right so it's uh just take this one atom in the center and it's got eight corners surrounding it so it makes a cubic geometry how about the lattice parameter this one's a bit more time consuming so i'll just go through it so here i've drawn a shape i've drawn a plane that cuts from edge to edge right down the center of this cube okay and i've drawn out the two the two-dimensional projection of that plane of the atoms you'll see that there's a continuous three atoms down the center here so that's equal to four r four times the radius and then the edge is the last parameter a so that's what we're looking for in terms of r and then i've also drawn this triangle which is the base of the square and so i've kind of unfolded it out into this two-dimensional projection kind of like origami here and it has two last parameters as well these two edges here so again through pyrthagren's theorem a squared plus a squared equals c squared a squared b squared equals c squared so that makes this edge length here along the the diagonal a times the square root of two so now we have our our two legs a and b and that equals c so again pythagorean's theorem and you get a in terms of r so a equals 4r over square root of 3. how about atomic packing factor does anyone know the answer on the top of their head yeah 0.68 resets that's correct 0.68 so it's the same calculation you have the information you need but uh like i said for for cubic for fcc and bcc and hcp you should memorize these numbers because they might come up in a test or something and you don't have to calculate it yeah two atoms per unit cell eight nearest neighbors the coordination and point six eight sixty eight percent um hexagonal close fact uh not going to do any calculations really but it's just to visualize uh how the atoms are packed right um so here's a again a two-dimensional projection of just this first layer here and then i move up to this the layer above it okay so this is that j atom here on the layer above it and then the next layer on top of that okay so you get uh the parameter last parameter a is just 2r and there's two atoms per this unit cell in the smaller unit cell but the expanded unit cell this hexagonal unit cell there's six atoms but what's what is important like dr brush had mentioned in class is the similarity between hcp and fcc um they're they're both close-packed lattices and they they both have the same atomic packing factor right uh which was 0.74 i get that one right i always forget i haven't memorized it myself yeah 0.74 just double checking all right so they have the same uh packing factor and like dr bruce had mentioned it the only difference is the you know they have these close packed planes of atoms and the layering sequence is different where hcp is aba and fcc is abc and i've tried to visualize this uh further for the fcc so you can try to see where the cubic structure comes from in this orientation right so you can see where this is the fcc cube i've taken just this face and this the two-dimensional projection of that face and there's the face-centered face of it okay and then the coordination is the same for both of them but what is different is the the symmetry of that the geometry all right so for hcp which is aba it has this this weird shape triangular ortho by cupola you don't need to know the names of anything but you do need to know that coordination is 12. and so this is that visualization of the 12 and then fcc because it's abc it has a different geometry cuboctahedron and this is important and i i think the mechanical engineers in the group will find this interesting you know even though they have the same coordination the same atomic packing factor but because the difference in the stacking between hcp and fcc their mechanical properties are much different in general for metals that are fcc crystal structure that's aluminum copper nickel gold they are very malleable metals but metals that are hcp primarily titanium are not very malleable okay and that's because of the crystal structure the arrangement of these layers to make a metal malleable means we're we're plastically deforming the material it means we're physically having the atoms slide past each other when we pull a metal apart all right and so the question is how easy is it for atoms to slide past each other between these two different uh crystal structures and it depends on the number of directions that the plane that we're talking about a plane of atomic a plane b plane that can slide okay so i i try to visualize this this is a little bit beyond the scope of the class although i think we might talk a little bit more about it when we discuss mechanical properties but at this point you don't really need to know but so i've drawn out a plane of atoms all right this is a closed packed plane for fcc and hcp they both have these same close-packed planes but fcc has more than one of the same close-packed plane so i'm going to draw another one here here's another plane of closed-packed atoms okay the same shape the same arrangement of atoms where hcp does not have that arrangement you can't make another one of these hexagons in this geometry okay and then actually fcc has two more so it has a total of four closed plaque planes okay and so uh later on you'll talk about slip systems this is these are the systems of how atoms can slide past each other all right and so fcc has many more slip systems than hcp does acp only has three slip systems fcc has 12 that's why scc metals are in general are more malleable than hcp okay all right let's move on uh so yeah the equations to know apf atomic pack factor the volume of the atoms divided by unit cell this will also come up the density i think you've already probably seen it in last homework is uh of course mass over volume but when you're calculating using the atoms and the the crystal structure you can use this equation here n is the number of atoms in the unit cell a is the molar mass of the atom or atomic mass of the atom n is avogadro's number vc is the volume of the unit cell okay so as long as you know those parameters like the a and r then you you should be able to calculate density if you're given a molar mass and you should also know the number of atoms in the molar unit cell n is the number of atoms in the unit cell so we we calculated that for fcc it was four atoms bcc it was two atoms and so on a is the molar mass so it's going to be in a grams per mole okay n sub a is just avogadro's number was that 6.022 times 10 to 23 what is that atoms per molecule or just atoms right per mole atoms per mole excuse me so then that will cancel out the moles and you're left with grams per atom times the number of atoms divided by vc is the volume of the unit cell so you're left with you know the mass of the atoms in the immune cell divided by the volume of the unit cell and that's density okay so we're going to move on and talk about uh some symmetry of crystal structures and uh dr brush had briefly mentioned this the brave lattices so like he said all crystals can be arranged or categorized into one of these 14 brave lattices okay and so the ones that we typically deal with is are the cubic system because the symmetry is is easy mathematically it's easy to work with but again there's different lattice types and these are just lattices okay these are not crystal structures lattices are just a set of points in space it doesn't doesn't mean where the i mean atoms can be put onto these lattice points or a group of atoms can be put onto these lattice points and then there's a bit of complexity you know especially in the hexagon so i just screen clipped this from wikipedia if you look up like crystal structure or something uh you know between hexagonal the crystal family system and the lattice system and and trigonal hexagonal it's not you know you don't need to know it's not too important but we'll talk a little bit when we start looking at hexagonal clothes packed and which system it's in but what this is showing is the brave lattice and it's just a way to categorize the symmetries of these crystals in a graduate level courses we talk about space group symmetries which are just further ways of defining the symmetry of the the different crystals and so on um and so what i want to emphasize and a lot of students miss this and even i've even had some professors that have have incorrectly said this um but i want to emphasize that the crystal structure is made up of the lattice plus a basis and so this often times this is also called the motif so i use these interchangeably motif and basis so the lattice like i said is just a group of points in space it doesn't represent uh you know the crystal structure physically it's just a group of points so fcc is a lattice it's a brave lattice and for example gold the metal has the fcc crystal structure so in this case the lattice and crystal structure have the same name okay but it's not the same it's not necessary for other crystals uh here's another example bcc of iron has a bcc crystal structure and then inert gases this is methane and other noble gases if you cool it down to solidify them generally form in the fcc crystal structure so in this case the basis is a group of atoms it's a it's a molecule and if you cool it down enough it forms the fcc crystal structure but then it can get more complex so here's an example this is the cesium chloride crystal structure so what do you think the base excuse me what do you think the lattice is of this crystal structure what is the the lattice you know specifically what is the brave lattice all right so it's going to be one of these cubic thing bcc because it's incorrect and this is the point i'm trying to make the lattice in for this crystal structure is not bcc um it's actually simple cubic okay so i have drawn this out the motif is cesium chloride it's a group of two atoms okay so that's the motif there there's the simple cubic lattice points okay so if you were to take this motif you know let's take this chlorine ion with the cesium attached you move you translate it to each one of these lattice points that's how you expand to this cesium chloride crystal structure so the crystal structure is not bcc that's why i've heard i've heard some professors say oh this is bcc no it's not bcc bcc would be if if the atom in the center is the same as the atoms at these lattice points but the atom in the center is not the same okay the lattice structure is simple cubic the bravais lattice is simple cubic it has a motif of two atoms so if you translate this motif to all these different lattice points that's how you build up the crystal and the crystal structure is called cesium chloride it's the other uh like cesium bromide caesium iodide i think they all make the cesium chloride crystal structure so does that make sense why does the central atom not count so in bcc i i should have included this i had a picture of what it would look like if it was bcc in bcc all the atoms are the same okay like uh iron is bcc okay but in this case this the atom in the center is not the same as the atom on the corner yeah it's simple cubic of cesium chloride like of the caesium chloride molecule i guess you could say all right so this is what i wanted to emphasize that this is not the bcc structure and i have some more examples that should make it clear um so here are a set of different crystal structures and they're their respective names so sodium chloride is called the rock salt crystal structure it's also called the halite crystal structure the same thing so what is the lattice what is the bravais lattice for sodium chloride what do you guys think all right it's not bcc this time this time it's fcc yeah and the motif what do you think the motif is for fcc and you can see it here you know just look at these blue atoms right blue which which atom is the blue atom it is uh smaller so it's going to be the cation typically cations are smaller right we're giving away like electrons so um so you see the sodium makes this fcc lattice here so what is the basis yeah so sodium chloride is right two atoms so sodium chloride is the basis so if you took this sodium attached to this chlorine even if you just put took the middle or center of mass and you put it at each one of these lattice points you're going to construct the sodium chloride crystal structure okay how about calcium fluoride or fluorite crystal structure what's the lattice yeah this one's also fcc um and how about the basis for this one yeah so this one was you see you know here's the fcc last right here the was this the calcium makes that fcc structure all right i'll move on the motif for this is just a group of three atoms calcium and two fluorines so if you were to take calcium and two fluorines and translated it to every lattice point in fcc you're going to make the fluorite structure right so if you took say these three atoms right here calcium and two fluorines you translate this over to the next one you're gonna you're gonna get you're gonna build up the the symmetry right okay and there shouldn't be any overlapping when you translate it there's no like duplicates that wouldn't work they're all unique okay now we get to titanium the hcp yeah so i want to emphasize this as well because a lot of students get this incorrect hcp is a crystal structure it is not a bravais lattice okay if we go if you go back to the bravais lattices all right and this will give away the next answer you know hexagonal does not have any uh it doesn't have face center doesn't have body center all right there's only simple hexagonal is the only bravais lattice for hexagonal so the crystal structure is hcp what is the lattice i already gave away it's hexagonal the motif is actually a two of titanium atoms okay so that's the point i want to make that lattice points and crystal structure are different okay they're not the same lattice is just a set of points in space and depends on what you put onto those points in space and that can build up and that's how we get so many different types of crystal structures not just 14 there's way more than 14 crystal structures there's but there's only 14 brave lattices okay yeah simple hexagonal let's move on uh so here's a question and i i don't i i don't remember if brush got to this in his lecture but it's definitely part of the reading i believe now here's an example of a ionic solid magnesium oxide so magnesium is a cation two plus oxygen is the anion two minus it the ionic solid so oxygen arranges itself into an fcc anion lattice so here are the lattice points for the oxygen so you can assume that oxygen is going to be on each one of these points okay the question is where will the magnesium go and this deals with what we call interstitials interstitials are the spaces in between so in fcc there are two different types of interstitials there is this red one here is the octahedral interstitial interstitial right that's made up from the faces and then on the corners is the tetrahedral interstitial so for each unit cell well there should be eight there should be eight uh octahedral excuse me there should be eight tetrahedral interstitial sites and i believe there's also let's see there's one on each face seven uh no it's each edge so there's what one two three four that's eight there's twelve uh thirteen and oh yeah i forgot to divide by two anyways the question is uh where will magnesium go will magnesium go into the octahedral sites or we'll go into the tetrahedral sites does anyone know how to go about solving the problem what do we have to look at and i i believe this was part of the reading for this week yeah the radii so we look at the atomic radii and what do we look about you know what what do we what do we need to do make sure they don't overlap um i think there's a it's a more general rule to determine whether it's octahedrally coordinated or tetrahedrally coordinate so i'll i'll go into that so this is from callister it might be chapter 14 actually which is part of your reading assignment and you what you do is take the ratio of the cation to the anion radius okay and it's a general rule of thumb if they fall in one of these ranges that will be the coordination of the ani or the cation to the anion and remember usually the cation is the smaller anion excuse me i'm getting my words messed up cation is usually the smaller ion because you're giving away electrons your electron shell is getting smaller the anion is usually the bigger ion because it's receiving electrons it's uh less it's more loosely bounded to the atom and so in this case you just take the ratio between the two okay and so the ratio between magnesium and oxygen 0.51 so that falls into here coordination number six which is an octahedral all right so magnesium is going to fit into the octahedrals not into the tetrahedrals it's it's more energetically favorable and that's why these these numbers are determined it's it's which coordination makes it more energetically favorable if you have a too small of an atom inside the octahedral is not very energetically stable you can think of like a small atom rattling around in a cage it rather be too big than it than too small so this is what that looks like if we take the magnesium put it at each of these edges and inside the center and then if we expand this octahedral this is the next door neighbor octahedral interstitial site i think there was another question i had yeah what's the crystal structure of magnesium oxide what do you guys think and you should be able to just look at this and know right away what the crystal structure it's one of the crystals that we already talked about what is the crystal structure and what i'm looking for is the name of the crystal structure right so not bcc bcc is only a crystal structure if the entire material is the same uh composition like like iron it's all made of iron that can that's the only time bcc can be a crystal structure otherwise it's a lattice and uh [Music] yeah you're right halite is correct uh sodium chloride crystal structure or rock salt this is the halite rock salt or sodium chloride crystal structure all right it's the same as what we saw before i won't go all the way back okay let's go into miller indices okay and crystallographic directions so you know if we want to describe uh the range or the the plane of atoms inside a unit cell or perhaps what the atoms look like at the surface of different single crystals we use what's called miller indices all right here's an example of the spinel crystal structure this is iron uh what we call this iron 304 it's also iron two three oxide because it has mixed of iron two plus and iron three plus it's called magnetite it makes the spinel crystal structure and these nanoparticles of that spinel and macro particles crystals having the same shape and the shape is determined of course by the crystal structure but the question is you know how do you how can you describe the surface of these crystals and we use miller indices miller indices have this abbreviation hkl and these are integer values so here's another example of some kind of nano crystal i forget but it describes the different surfaces of this crystal with the miller indices and then there's also you should just distinguish it from the direction where are the crystal structures such as halite found um i don't quite don't quite understand the question where crystal structures such as halite found you mean like on earth oh like a list of crystal structures um i i'm pretty sure chapter three if not chapter three it's chapter 14 i believe it was part of the reading there's a small reading of simon chapter 14. i think that talks about ceramic crystals 14 or 12. oh did i mix that up it might be 12 then if i'm i don't have it right in front of me so i'm not sure it might be 12 not 14. uh the chapter is about ceramic crystals or ionic crystals i believe and it should give you a different list of crystal structures um and then the direction are just vectors but you need to make sure that you're using the correct notation if you're talking about a plane in space for miller indices it's using parentheses or sometimes these curly brackets we're talking about a family of planes and if it's a direction you use the square brackets or these uh little what do you call those i don't know carrots arrows whatever for a family of directions so i have a couple problems but i think we'll just go through them because we don't have much time and like i said don't feel obligated to stay i this recording will be posted uh this evening once it finishes compiling so for crystallographic directions all right using the square brackets is is very simple just vector math right so you take the the final coordinate point in real space and minus the initial coordinate point um or excuse me this should be lattice space because this this uh you know here i show x y and z and to think of three dimensions but that's only correct if your unit cell is orthogonal okay if it has these right angles but you as you know some unit cells don't have right angles at the origin so it's not really correct to say x y and z but we're talking about lattice space the the vector math will still be the same so you should use these unit vectors instead of x y and z but in these examples they're all cubic so it's it's all the same but like in this example you know just doing the vector math finding the n n coordinate the beginning coordinate and that will give you and you should multiply the fractions out so in this example the n coordinate is 1 and then three quarters zero because there's no there's no change in the z dimension um initial is the origin so you need to do is multiply by four to remove fractions okay so there's no fractions in the miller notation for directions or for planes uh so this direction will be called brackets four three zero okay i had some questions i was going to have you guys do yourself and but because there's not enough time and i'm pretty sure your homework deals with questions like this as well um but here's i'll just go through this so this direction to point a right the coordinates are one-half one-half one-half all right and it starts at the origin so then if we multiply out the fractions so we multiply everything by two this is going to be one one one all right in brackets one one one this is the one one one direction this one is not starting at the origin okay so you have to kind of do the math here uh so we start and you can always move this to the origin to make it easier and that's what i'll try to visualize so in the x direction we're moving over one third we're going from two thirds to one okay and then in the z direct y direction we're moving one and the z direction we're moving negative one uh for negatives in miller notation negatives are written with a bar over the number okay so we need to multiply out the fractions so this was what i say 1 3 for the x direction so we're going to multiply everything by 3 so it's going to be turned into three and then one in the y direction and then one excuse me i'm sorry i messed that up it's going to be one three three bar okay one three three bar okay three bar means negative and then the last one we're going from this corner here we're moving uh positive one in the x direction negative one half in the y direction and negative one in the z direction okay we remove the fraction so we're going to multiply everything by two so it's going to be two one bar one bar let's see oh excuse me i did that wrong two one bar two bar yeah i forgot okay uh now for miller indices talking about planes it's a bit more complex it's not as simple there's a uh we need to change it into what we call reciprocal space um but it's still not too complicated and a lot of your homework will be uh will have some of these problems in fact this might even be a problem from your homework let's start with uh b for example so we want to find out what the miller indices of plane b is b is easier than a so we'll start with b so what you need to do is from the origin find the intercepts of this plane with the different axes all right so this b-plane intercepts x-axis at one-half okay it intercepts the y-axis at one-half and intercepts the z-axis at one okay and then the next step is to take the reciprocal so that's why this this is how it's different than the regular vector notation we're going to take the reciprocal of this these intercepts so it becomes 2 2 1 reciprocal of 1 is one and then if there's any fractions you you multiply them out and then the final notation is two two one for this plane that's how we describe this point uh for for a it's a bit more complex because a intersects our origin so we need to move our origin we can't have it intersect the origin we need to move the origin in order to calculate so if we move the origin to this position here so we consider this 0 0 0. oh sorry in the x axis intercepts at 2 3 the y axis now we're going to negative intersects at 1 and the z axis intercepts at one-half uh let me see there's a question how do we know when to write two two one that we mean the plane and not the vector oh yeah so the vector is always going to be this is the square brackets whenever you see a square bracket you you know it's a vector a direction but when it's a plane we use the the parentheses so that's how you know um so moving going back let's see where was i the the the y direction is negative one and the z direction is one half right here's the excuse me the z intercept is one half then take the reciprocal so two thirds becomes three halves negative one is still negative one one half becomes two and then you need to multiply out the refractions so multiply everything out by two so this becomes three two bar four so that is the miller indices for this plane a okay uh there's another question i missed how do you know the right order of the numbers are we talking about daniel this was your question the right order of the numbers are we talking about the directions here so it's it's always going to be like x y z or for the planes the right order um it's again the when we're talking about intercepts it'll be x y z as always in that order and like i said x y z is a bit misleading especially if we if we're changing into uh this is supposed to be lattice space and not real space but for cubic the lattice space is the same as real space but if you're if your unit cell you know like trigonal or monoclinic is going to have different angles the procedure is all the same yeah um so here's some uh problems that we'll go over i think there's a typo here this one half is by itself it shouldn't be there uh so this we're going to find the miller indices of these different planes of the cubic unit cell so this one here is a bit different because we don't have uh it doesn't intersect right in fact you could expand this plane and intersect way up here but instead what we'll do is we'll move the origin we're going to move the origin to this top corner here and so in the x direction it's negative one in the y direction is negative one and the z direction is negative one so this this miller indices of this is going to be one bar one bar one bar let's see yeah that's right okay and this plane is equivalent to uh one one one all right one bar one bar one bar is equivalent to one one one uh it's also equivalent to two two two and so on all right but so one this you know the correct answer would be one by one bar one bar the next one let's see so this problem is a bit different than the other ones because there it never intersects the y-axis all right this plane never intersects the y-axis uh so that in that case for y it'll be zero let's start with x though x is going to be one y is zero because it never intersects y and then uh z is one half so remember we need to take the reciprocal so we start one zero one-half take the reciprocal is one zero two and then there's no fraction so that should be the answer one zero two is the miller indices for this plane this next one is a bit more difficult because again the the intersection with the x-axis is not in the view of this unit cell so you would have to extrapolate this plane down so if this is one-half here and this is one to one half and then continue down then it's going to end up intersecting x at two so x is two y is one and z is one and this was starting at the origin again okay so it was two one one oh i got that wrong what did it happen oh i forgot to take the reciprocal yeah yeah so it's two one one before the reciprocal but remember we need to take the reciprocal reciprocal so 2 turns into one-half so then it's one-half 1 1 then multiply out the fractions it becomes 2 1 1. excuse me one two two all right so i got that one wrong because i forgot to do the reciprocal so always remember for planes you need to take the reciprocal uh before the final answer okay and then here's another question find the 1 1 1 plane of this unit cell this fcc unit cell and then also draw the 2d atomic arrangement of that 1 1 1 plane now there's a question on your homework that's very similar to this and i i want to uh you know it's it's it the homework said it was a hard a difficult question but it could be very easy if you know this trick if you know this i guess so 1 1 1 of course the x intercept is one here the y is one here and z is one here so that makes this triangular plane okay and then if we were looking at the atoms that intersect this plane as you draw out those atoms you know you get the closed pack structure the closed plaque plane so for fcc unit cells one one one is the closed packed plane of atoms whenever you see this hexagonal symmetry right this hexagonal symmetry of atoms like this this close packed plane you should know right away that's either it's either the one zero zero plane of hcp one zero zero of hcp or a one one one plane of fcc all right so there's a question on your homework that might seem at first a bit difficult because it wants you like to calculate something in order to find it it gives you like an image of this and you should know right away when you see this oh that's the one one one plane of fcc or the one zero zero plane of hcp but the question made it obvious is fcc okay the same thing for bcc find the one one zero plane and then draw the atomic arrangement of that um so this one 1 1 0 the intercept on the x axis 1 here and the intercept on the y axis is 1 here and there's no intersect on the zero on z so this is going to be a slice along the diagonal here okay is the one one zero plane and if we were to draw out the atoms on this plane this is what it look looks like and remember we talked previously what the unit cell last parameters were and uh so this is the atomic arrangement of the one one zero plane for bcc so if you had a uh iron for example iron is a bcc metal if you had a single crystal of iron for example if you had a single crystal of iron and it was cut along this 1 1 0 plane so that the surface of the single crystal was 1 one zero at the surface the atoms would be arranged in this pattern all right so that's what that means there okay um i think next i'm going to talk about uh x-ray diffraction but we've gone a bit over time and dr brush has not yet talked about x-ray diffraction yet so i think i'm going to end it here and leave that for dr brush and sid to talk about next time so i'm going to stop the recording and if you guys have any last questions you can go ahead and ask and we can go back if you need to because i did go through it a bit quickly