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HOME > PAST ISSUE > May-June 2009 > Article Detail

MARGINALIA

The Squeeze Is On

How do molecules behave at extremely high pressure?

Roald Hoffmann

Close-packed Is Not Close Enough

Stacked%20orangesClick to Enlarge ImageAs people discovered millennia before scientists began to think about it, the densest way to pack spheres (such as fruits) together is to make a hexagonal honeycomb layer, in which each sphere has six nearest neighbors, and then to stack such layers on top of each other. There are infinite ways to keep on doing this; in each, a sphere has 12 nearest neighbors.

The volume of space left empty in these closest packings turns out to be 29.6 percent of the overall volume, a fact long known, but proved mathematically only a decade ago.

There are many non-close-packed ways to arrange atoms or molecules. The gut feeling until a few years ago was that under pressure everything in the world would go into one of the two most regular close-packed structures, so-called hcp (hexagonal closest packing) or fcc (face-centered cubic, sometimes also called ccp).

Well, here’s what happens to elemental barium (Ba), as determined by the research of Richard J. Nelmes, Malcolm I. McMahon and their colleagues at the University of Edinburgh: At ambient pressure, Ba has a bcc structure (body-centered cubic, with 8 nearest neighbors for each Ba). At 5.5 GPa it goes fcc, and everyone is happy. But as the pressure is increased even further, Ba leaves the fcc formation for a seemingly less close-packed (yet denser) structure. This new arrangement is comprised of a “host” and a “guest” lattice (both made of Ba), which are incommensurate with each other. At still higher pressures, over 13 GPa, Ba falls into an incredible structure with nearly 300 atoms per unit cell.

In the past decade, nearly every alkali metal and alkaline earth metal structure has been found to go out of close-packing formations under an increase of pressure. Density rules. So how can you leave close packing? Easy. The following are some ways to think about it.

First of all, our prejudices are that atoms are spherical. And indeed spheres are limited to 70.4 percent packing efficiency. But who says atoms have to remain spherical as you push on them? Think about the extreme of cubes. They pack with 100 percent filling of space, right? Not that electron density in atoms will go cubical. But it can deform in that direction, or toward that of a number of other space-filling polyhedra.

Here is what Stephen Hales wrote in 1727, in his Vegetable Staticks (the quote is from H. S. M. Coxeter):

I compressed several fresh parcels of Pease in the same Pot, with a force equal to 1600, 800, and 400 pounds; in which Experiments, tho’ the Pease dilated, yet they did not raise the lever, because what they increased in bulk was, by the great incumbent weight, pressed into the interstices of the Pease, which they adequately filled up, being thereby formed into pretty regular Dodecahedrons.

Secondly, equal-diameter spheres do pack maximally to occupy 70.4 percent of space. But if you can have spheres of different sizes, you could put small spheres into the holes “between” the large ones, and so on. The problem of the efficiency of packing spheres of two or three unequal diameters has not been solved, to my knowledge. I bet that for a number of radius ratios one will get more dense packing than 70.4 percent.

So if you have an arrangement of equal atoms, and a limit to their packing if they remain equal, but a denser packing is available if they become unequal in size, perhaps they’ll do it. How can they become unequal? For instance, by shifting some electron density from one sublattice in the solid to another, a kind of electronic disproportionation, symbolized by ([A]n → [A+]n/2[A]n/2).


A third factor: In a remarkable theoretical prediction, a few years ago my colleagues Neil W. Ashcroft and Jeffrey B. Neaton of Cornell University postulated that metallic lithium (Li) would move away from closest packing (it’s bcc at 1 atmosphere, fcc under higher pressure). Li atoms should pair up, and electron density moves into the crevices between Li pairs. With differences in detail, this risky prediction of a new mechanism for compaction was confirmed in 2000.

Atoms and molecules will do what they have to do to get denser, not what our simple minds tell them to do.




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