Music

Yesterday we started...

Yesterday we discussed the basics of the kinetic crystals and started to talk about acoustic waves with periodic structures.

Actually we only discussed guided waves in media, which I'm not very good at.

So let's finish this today and then let's discuss a little bit more some actual devices.

Starting with discussing the alchomechanical band structures, so actual designs that have both bandgap in the atomic and atomic domain.

Then to introduce finger and point defects, talk a little bit about the nanotech device.

It's just been used to do ground-set cooling and the snowflake device, which is a one-dimensional alchomechanical crystal,

which can be used to do rather full-on and full-on stuff.

Then we'll discuss the alchomechanical coupling.

Finally we're starting to get into more experimental things, fabrication and testing. Next time we'll discuss experiments.

So we have this light, which is very soft. Remember, in the electric mechanics we have two transverse waves

and in mechanics we have these two waves, typically two transverse, also with different polarization.

Then we also have this pressure wave. We're going to discuss this two-dimensional slab and identify this as a kind of a retro-mode,

which is the in-plane section, which is a very slow dispersion mode here.

And we have the in-plane shear mode and we have the pressure mode here.

If you look at what happens in a nanotech device, you introduce another boundary.

That is, now you also have, typically, the x-axis, you have another set of boundaries.

What you'll find is that down here, the load of wave vectors looks rather similar, however it's more modes.

Particularly you have now two flexural modes, because also in the in-plane direction you can have flexural displacement.

This is the old one, the one we had before, it's the same mode as the red one.

It's basically in-plane flexural. The green one is this one.

This is in-plane flexural, this is out-plane flexural. And you have again the pressure wave.

And finally you have one additional wave, which you don't have in the mode, which you don't have in bulk material, and that's the torsional mode.

And actually people have used the fact that there is an impedance mismatch between such a beam and the bulk, so the bulk cannot support the mode,

to actually reduce clamping loss by coupling a mechanical object with this degree of freedom to bulk.

This is all about guided waves, and now let's introduce the periodicity in the mechanical domain.

For that we started with the point model.

So this is a diatomic chain, it's a one-dimensional chain with different masses, two objects in different masses,

they are periodic, and they are always mass one and mass two, and you have a lattice constant A, and I'm sure you have seen this model.

What you will find is two different types of normal modes.

There is the optical mode, where two neighbors move differentially, and the acoustic mode, where they usually move more in pump.

So one other way you can see is this here, the low, the high masses that move in here is more the loader masses that move.

So another way you can see is that the part of the lattice, because M1 is part of the lattice,

so it has two moves towards each other in the optical phase.

In any case, you can derive a dispersion relation for such a system.

And it's given here, and here is the spring mass that connects any two masses, and two and one are the masses, and this lattice is the wave vector.

And you get this plus sign for the optical, we will call it the optical mode, and the minus sign here will be called the acoustic mode.

So when you evaluate this, you get this on the back.

And what you see is that you have these acoustic waves, which are very linear dispersion at small k values,

and that basically tells you why wave packets of very long wavelength sound waves can propagate pretty nicely through solid materials.

At large k vectors, where that's not the case anymore, you have nonlinear dispersion.

So what is the span gap scale? It's pretty big.

So if you look at this term, there is obviously a proportionality with the lattice, sorry, with the spring concept,

which will be dependent on the elastic properties of your material.

But there is also like, you could be interested at this point, where k is pi over 8, and at this point, pi over 8 and you have sine pi half, which is 1,

so you have 4 over m squared for two identical masses, for example.

And this is also 1 over, 4 over m squared for two identical masses.

So if there is only one type of mass, then you don't have a span gap.

And in other words, if you want to have a big span gap, you need to make the masses very different from each other.