- I've had a lot of fun on Matt's channel
making a video about close-packing,
but it's very closely linked
to crystal formation, crystal defects, and stuff like that.
And I've actually built something
that demonstrates how crystals form and how defects form.
So, I'm gonna make a video on my channel about that.
And Matt, you're gonna stick around,
because there might be some maths in it.
- I'm all about the maths.
- So, occasionally Matt might say some numbers
or something like that.
- Six. - That's a good one.
- But one is a different number.
Sorry, carry on.
- So, yes, now there used to be a toy that you could get
in the 60s and 70s called Atomics.
And you can't get it anymore, which is a shame,
because it meant that I had to spend hundreds
of pounds making my own.
- You recreated it? - I've recreated the toy.
- Oh my goodness.
- So here it is.
It's thousands of
ball bearings. - Wow,
I can see where all the money went.
- Yeah, mostly in the etching of-
(Matt laughs) - youtube.com/stevemould.
- Well, I just thought if someone turns this into a GIF
and puts on Facebook, then at least I've got some branding.
- I reckon they could blow that out.
Pretty crazy. (Matt laughs)
Don't do that.
I mean, I probably overkill the fact
that you etched it into every single ball.
(laughs) - That was the expensive part.
Anyway, if you let the balls just drop like this,
they self arrange into a lattice,
which is related to the video that we did on Matt's channel,
about close-packing.
Definitely check that one out, by the way.
And if you haven't subscribed to Matt's channel,
do that also for it's all maths all the time.
To be honest, though, my subscribers are probably
a strict subset of Matt's subscribers, but anyway,
if for some reason
that's not the case. (laughs) - You used a strict subset.
That's nice.
- I know I'm good at maths,
I just don't do it on my channel.
I do sometimes when I'm talking about tau, mostly.
So anyway, so these naturally fall into a regular lattice.
What's brilliant about it is you can see
it's not perfect, right?
And the defects in this arrangement
of spheres closely matches the defects you get
in real crystals in the real world.
So this might be a good analogy for how, say,
aluminum atoms arrange themselves in a metal.
- So why you got these holes down?
- Yeah, so that's an example of a defect you might get.
That's called a point defect,
because it's happening at a point in the lattice,
a point where an atom should be.
And that's called a vacancy.
- Right, so there should be one there but there's not.
- And you see it's nice and stable, right?
Actually interestingly, in a real crystal,
you could have neighboring atoms
just kind of jumping
into that hole. - Oh, they pop into that gap?
- Yeah. - But that's just moving
the problem to the side.
- So brilliantly, yeah, you can have a hole that--
- So a hole moves through the lattice?
- Yeah. So let's have a look
at another defect.
- 'Cause I can a lot
of lines going all over. - Yeah! Okay,
so here's a region of lattice
and here's another region of lattice.
- [Matt] And these are all hexagonal packing?
- It's all hexagonal. - [Matt] Very efficient.
- Yeah, like a honeycomb.
Except that this region
and this region are at slightly different angles.
They don't line up.
So, you know, you've got this one like that
and this one's kind of--
- And when they meet it doesn't mesh.
- Yeah, and that happens in the real world as well.
It's called a grain boundary.
So if you've got a crystal growing over here
and a crystal growing over here, they collide.
That's what it's like, where they meet.
But here's another one that's pretty interesting.
(pearls jiggling)
Let's just re-jiggle that.
Okay. So here, let's see this one here.
- Oh yeah, there is a line there!
- It's a straight line, but it's a bigger gap than the rest.
So this crystal and this crystal they're oriented the same
but for some reason they don't quite match up.
- They're still not lining up. - Yeah.
This isn't really a perfect analogy
for anything in the real world, quite.
But it's close enough to a certain type of defect
that's really, really interesting.
It's interesting enough that I bought a whole load
of ball pit balls, and we're gonna actually try
and build a crystal.
Because there's something really really surprising
about close-packing that I just, yeah.
I couldn't believe when I saw it.
- Now, I'm interested.
I mean, the rest of it was great too, but close-packing.
- That was a bit clickbait, wasn't it?
When I just said, "You're not gonna believe--""
- "You're not gonna believe
where these things are centered."
- "Ah, this guy, this guy stacks--"
You won't believe what happens next.
Yeah.
Okay. The third layer of balls will make you--
(Matt laughing) - Blow your mind.
- Yeah. That's the only way, anyway.
Okay. We need to go into 3D for this one to work
because this one is a stacking defect or a stacking fault.
So.
- [Matt] I'm not genuinely excited.
- If you bring up the box.
- All right, so here's your box of green balls.
- So I've already gotten
the first layer - They're packed
very efficiently.
Good job. - Yeah, so hexagonal packing,
on the first layer of green balls.
So to distinguish the next layer, we're gonna use red balls.
- Okay. So what we're doing is like the next layer
as if these were atoms of some kind of crystal lattice.
- So yeah, these one's nestled into the little dips created
by three green balls that are touching.
- This is kind of fun. - Yeah.
- I can see why kids like playing with these.
- I don't think they do this in ball pools.
- What! I do. - Okay.
- All right, so we've got two layers.
- Yeah. So far so good.
Let's add the third layer. - Third layer, easy.
- But we're on different teams now.
- Oh, okay.
Yeah. I'll go yellow ones
and I'm gonna start stacking everything.
To be honest, I don't see what the challenge is yet.
This is all very straight forward.
- You will see.
Okay.
- Okay.
(both laughing)
I was about to say that if you put that down,
I can't put mine in my lattice,
but then I realized that's kind
of the point of this, isn't it?
(indistinct) checkmate is that?
- Isn't it interesting? - [Matt] I can put one there,
- So it turns out that when you put the balls down,
there's actually two choices. - [Matt] I got two options.
- Yeah, and they're incompatible with each other.
So you started putting yours there.
I started putting mine here and they come together.
And you can't finish the lattice,
that we've got this gap now running through.
- So we were two bits of the crystal growing,
we've hit here and we get, what did you call this thing?
- So this is a stacking fault.
What that means is, where I've placed my balls relative
to the previous two layers is different to where you--
- I lined mine up with the green ones.
- [Steve] Yeah. - So mine is the same
as the bottom third layer?
- [Steve] Yeah. If you look down
and I look through the holes,
between the red balls, I either see a green ball immediately
below it--
- [Matt] Or you don't.
- [Steve] Yeah, you don't. You see a gap.
- And you went for the gaps?
- I went for the gaps.
Whereas you went for--
- The deeply pleasing alignment with the layer (indistinct).
- So yeah.
So your layer is exactly in align with the green layer.
Mine is not.
- I mean, to make it efficient
every layer has to be different
to the one directly below it, but then you've got a choice.
It can be the same or different to the one below that.
- Yes.
- And then that's it?
Like the next layer has to be one
of those three again, right?
There's no fourth option?
- Yeah.
So, the way we stack the blue balls here,
the first, second, and third layer are all different.
So we can label those A, B and C.
- Correct. - And if you carry on,
thank you very much, confirmation from a mathematician,
I mean that's all I ever want on my channel.
- That's why I'm here.
(Steve laughs)
- So a lattice that goes ABC, ABC, ABC, ABC ,
is a face centered cubic. - Correct.
(both laughing)
- Lattice.
Whereas what you've done with the yellow balls,
(Steve laughs) - Carry on.
- It goes, AB AB AB, which is
a hexagonal closed, no.
- I was almost gonna go, "Incorrect!"
- Hexagonal close-packing lattice.
- Yes.
Can I show you a diagram?
- Yes. - Okay.
Okay. So you have correctly labeled
that you've got three options.
You've got A, B, this is not tidy writing,
and C for your different layers.
And I think of it as a triangle of options.
And so if I switch colors,
what I was doing over here,
is I was just bouncing between A to B to A to B.
So I'm going backwards and forwards on one edge.
If you pick anything going backwards and forwards
on a given edge, you get the hexagonal close-packing.
And then if you do it over here, what you're doing,
is you're doing laps around the triangle.
If you do regular laps around the triangle
that gives you your cubic face packing.
And then-- - [Steve] Cubic face packing.
- [Matt] Yes.
(both laughing)
Same thing. Face centered (indistinct).
Right, so what you can do obviously,
is there any other combination of those?
Right? So if we did another one, right?
So this one will be a bit more consistent, and we did ABC.
There's no reason we can't then go back to B
and then to C, and then to A and then to B,
and then, and so.
- You can just travel around the--
- Yeah. You can skip around wherever you want, right?
And so if you get the two cycles,
you get the two regular ones. - [Steve] Yeah.
- For the rest of them you just get,
all sorts of around the things happening.
And because there's no ultimately, there's no limit,
it means there are infinitely many possible ways
to create your crystal lattice by stacking the spheres.
- Nice. And it doesn't matter how you go on this diagram.
It's the most efficient way to pack spheres.
- It doesn't change the percentage volume you're filling
with the spheres. - Yes. Now, interestingly
there is another type of fault
in crystal lattice - Really?
- Yeah. So it's called an Anti phase boundary.
So I toasted a regular crystal that is face centered cubic.
We'll go ABC, ABC, ABC, ABC.
But maybe at some point
it's gonna go ABC, ABC, ABC, ABC CBA, CBA CBA.
- Whoa! - So they'll switch.
- So you'll be going ABC, ABC, ABC,
BAC, BAC, BAC?
- Yeah. - How nice!
- Both of these crystals are face centered cubic,
but they just kind of--
- Mash up against each other.
- And you take the mirror image essentially halfway through.
Yeah. So that is, what's really nice,
is that the face centered cubic has a symmetry
that the hexagonal cross packing doesn't have,
and you can see it here.
So you've got, if I take away these two red ones here,
see this one as well, can you see this square?
- Oh yeah! - Of balls?
So you can't make a square in this one,
but you can in this one.
So you've got three by three square here.
- [Matt] That's nice.
- [Steve] Isn't that great? - Oh yeah!
So actually, if you were to reorientate it
you get a square based pyramid structure, going that way
and you get a kind of tetrahedral
triangle-based pyramid structure going this way?
- Yeah. - That's very nice.
- Yeah it's very nice.
There's one final thing that my Atomics rip-off does
which is really, really nice.
I want to show you.
So,
you take your crystal lattice.
- These are like stainless steel or something, right?
- Yeah, they're just, I think--
- Actually I've got some new,
these are the spare ones that you had in the beaker.
I'll just pull these out,
- Pour it out on a beaker. - A beaker.
- And you can see that they're tiny little spheres.
- Yeah.
- They're pretty good as far as spheres go.
I'd take two mil, you know what I mean?
- Really? - I'm guessing,
it's (indistinct)
- I can't remember, I bought them.
- Or we can just read the package. (indistinct)
Okay. Right.
So you've put loads of-- - Oh, two millimeters,
It says on the packet.
- There you go.
I may have seen the packet earlier.
(both laughing)
So you just, two mil?
- Yeah, do you know what's amazing?
- They should put it in giant letters on the, oh, they have.
- Okay. Just some estimation time.
- Estimation time!
- All of these plus all of these,
- [Both] How many?
That's probably 10 to there,
So that's maybe a hundred, 3000.
- 10,000! - 10,000!
- 10,000, yeah I know. Surprise.
- I thought by an order of magnitude, embarrassing.
(Steve laughs)
- First time we showed it, somebody said,
"I don't know, a thousand?"
But it's just, yeah, it's surprising.
I should have gone a hundred by a hundred,
I would have hit 10,000.
- Nearly got it.
- Lacked my convictions.
- Anyway what you do is, for the final trick in the arsenal,
you get some wool, a hat here and just give it a rub.
And that charges up.
- [Matt] You're polishing your--
- [Steve] Yeah, plaque.
But now all of this. - [Matt] Oh! Oh my goodness!
- So they're all slightly charged,
which means they're trying to get away from each other.
- That's so nice. Look at that!
- And so what you end up with is a phase boundary.
So you've got this, imagine this is the liquid phase
and this is the gas phase and they are in equilibrium.
- And in transitions from one to the other.
- Actually, maybe that's a solid, maybe this is
an example of sublimation.
So you've got solid to a gas. - Solid to a gas.
And even these guys are kind of close packing.
- Yeah. But if you touch it you'll destroy everything.
- I stole your electrons. - Thanks.
- I do apologize.
- Or maybe you donated electrons, we don't know.
- Who knows?
Who's keeping count?
- I don't remember, I mean, someone will know 'cause it'll--
- It'll be in the comments.
It was great fun doing a video with Steve Mould
on your science, YouTube channel about lattices
and how you can get all these fantastic inclusions
and effects in beads.
But I thought it might be nice if we also did a video
on my channel about some of the mathematics behind just
how efficiently you can pack these spheres in.
So to do that, people,
if you want to learn more about--
- [Steve] Matt did actually make that video.
If you want to see it head over to his channel,
Standup maths, the link is on the screen
and in the description.
There's also loads of other brilliant maths videos
on that channel, so don't forget to subscribe.