Why does regularity bring stability?

[pillow47]

Hi PF,

I've been wondering why lattice structures form in metals and in salts. Why do fcc or bcc structures reduce the energy of a system so that regular lattices are favorable over those whose atoms are randomly placed?

Thanks,
Pillow

 

[daveyrocket]

Well one thing to keep in mind is there are many materials with amorphous phases, where they don't have a regular structure.

But the simpler situations will give you a lattice. It's because atoms attract each other from a distance, but if they get too close they repel each other. So there's a happy place right in the middle where the energy is as low as possible. Because of this, atoms want to get close together at this distance. You can get more atoms packed in a regular structure than you can in a random structure.

Example: Get yourself a handful of pennies. Place one on the desk. Next, place pennies to get as many touching the first one as you can (it's 6). They will form a regular structure. You can repeat it on the others. You will see that in order to maximize the number of pennies that are close together you will need a regular structure. Crystals are just the 3D analog of that.

 

[snorkack]

You can get more atoms packed in a regular structure than you can in a random structure.

Example: Get yourself a handful of pennies. Place one on the desk. Next, place pennies to get as many touching the first one as you can (it's 6). They will form a regular structure. You can repeat it on the others. You will see that in order to maximize the number of pennies that are close together you will need a regular structure. Crystals are just the 3D analog of that.

Not quite.

Yes, in 2D, there is one close packing, and it is regular.
This is not true in 3D.

Imagine that you are packing cannonballs. First layer is hexagonal close packing. Unique, and regular.

But where will you put the second layer?

Obviously, put the second layer balls in holes between three balls in first layer.

But which ones?

Each ball in first layer is surrounded by six holes. Each hole is surrounded by only three balls.

There are twice as many holes between balls as there are balls.

Once you choose a hole for first ball of second layer, you can put the whole second layer in place. Filling half the holes, leaving the second half in place.

But where does the third layer go?

You have to choose between two sets of holes. Again. And now they are no longer equal!

One half of the holes is directly above the balls in first layer. The other half... is not.

If you choose placement ABABAB..., what you get is hexagonal close packing. If you choose placement ABCABC..., what you get is cubic close packing.

Both are regular. But both are equally closely packed - because the packing of A and B is not affected by what comes next.

The only difference is the feeble long distance interactions through a layer of other atoms in between.

But why should the packing be regular at all?

Why should ABC be followed by another ABC, and not by BCB...? So long as there are no AA, BB or CC, how about a packing where BAB, BAC, CAB and CAC are interspersed in completely random manner with no long term regularity? It is still just as dense as either of the regular packings... why then are the regular packings so common?

 

[daveyrocket]

Yes, the 2D situation was meant to be illustrative because it didn't seem to me that the OP was ready for or asking for details at this kind of depth. The question you're asking is a bit more complicated, and has a lot more to do with the quantum physics of the situation. Also a similar question is why is bcc so common, since it isn't actually close packed.

The second neighbor interactions are not quite so feeble as to be irrelevant. A part of it is the crystal field that is generated for a central atom. in ABA stacking this arrangement is a trigonal prism, and in ABC stacking it's octohedral. These cause different splittings of the degenerate energy levels of a single atom, and depending on the filling of the energy levels and how strong the crystal field is this can have an effect on how the atoms arrange. The hexagonal arrangement also allows an extra degree of freedom, since the c/a ratio is not fixed and few elements are actually right at the ideal value. This has to do with the tendency some materials have for stronger bonds to form in a plane and weaker bonds to form perpendicular to that plane. Carbon is an example of this, in the ground state of carbon (graphite), it's hexagonal with a very large c/a ratio.

In a realistic situation, hcp and fcc may be close in energy. Many such materials will have a phase diagram that has both phases somewhere on there. Also, if a material is quenched (cooled very rapidly through the solid-liquid transition), disorder will get frozen in. It may take a long process of annealing to really get the idealized order to show itself in the crystal structure.

This sort of question is difficult to answer with much generality because when one starts to look closely at different elements, the reasons for them having one ground state phase vs. another can be different for each element. Si has an fcc Bravais lattice for very different reasons than Cu does.