See that gray stuff inside Jupiter? It’s metallic hydrogen—according to theory, anyway.

But how much do you need to squeeze hydrogen before the H₂ molecules break down, the individual atoms form a crystal, and electrons start roaming freely so the stuff starts conducting electricity and becomes shiny—in short, becomes a metal?

In 1935 the famous physicist Eugene Wigner and a coauthor predicted that 250,000 times normal Earth atmospheric pressure would do the job. But now they’ve squeezed it to 3.6 million atmospheres and it’s still not conducting electricity! Here’s a news report:

• Probing hydrogen under extreme conditions, PhysOrg, 13 April 2012.

and here’s the original article, which unfortunately ain’t free:

• Chang-Sheng Zha, Zhenxian Liu, and Russell J. Hemley, Synchrotron infrared measurements of dense hydrogen to 360 GPa, Phys. Rev. Lett. 108 (2012), 146402.

Three phases of highly compressed solid hydrogen are known, with phase I starting at 1 million atmospheres and phase III kicking in around 1.5 million. I would love to know more about these! Do you know where to find out? Some people also think there’s a liquid metallic phase, and a superconducting liquid metallic phase. In fact there are claims that liquid metallic hydrogen has already been seen:

• W.J. Nellis, S.T. Weir and A.C. Mitchell, Metallization of fluid hydrogen at 140 GPa (1.4 Mbar) by shock compression, Shock Waves 9 (1999), 301–305.

1.4 Mbar, or megabar, is about 1.4 million atmospheres of pressure. Here’s the abstract:

Abstract. Shock compression was used to produce the first observation of a metallic state of condensed hydrogen. The conditions of metallization are a pressure of 140 GPa (1.4 Mbar), 0.6 g/cm (ninefold compression of initial liquid-H density), and 3000 K. The relatively modest temperature generated by a reverberating shock wave produced the metallic state in a warm fluid at a lower pressure than expected previously for the crystallographically ordered solid at low temperatures. The relatively large sample diameter of 25 mm permitted measurement of electrical conductivity. The experimental technique and data analysis are described.

Apprently the electric resistivity of fluid metallic hydrogen is about the same as the fluid alkali metals cesium and rubidium at 2000 kelvin, right when they undergo the same sort of nonmetal-metal transition. Wow! So does that mean that at 2000 kelvin but at lower pressures, these elements don’t act like metals? I hadn’t known that!

Another reason this is interesting is that if you look at hydrogen on the periodic table, you’ll see it can’t make up its mind whether it’s an alkali metal—since its outer shell has just one electron in it—or a halogen—since its outer shell is just one electron short from being full! You could say compressing hydrogen until it becomes metallic is like trying to get it to break down and admit its an alkali metal.

Apparently the metal-nonmetal transition for for liquid cesium, rubidium and hydrogen all happen when the stuff gets squashed so much that the distance between atoms goes down to about 0.3 times the size of these atoms in vacuum… where by ‘size’ I mean the Bohr radius of the outermost shell.

How did Huntington and Wigner get their original calculation so wrong? I don’t know! Their original paper is here:

• Eugene Wigner and Hillard Bell Huntington, On the possibility of a metallic modification of hydrogen J. Chem. Phys. 3 (1935), 764-771.

It’s not free; I guess the American Institute of Physics is still trying to milk it for all it’s worth. One interesting thing is that they assumed the crystal stucture of metallic hydogen would be a ‘body-centered cubic’… it’s rather hard to compute these things from scratch without computers. But this more recent paper claims that a diamond cubic is energetically favored at 3 million atmospheres:

• Crystal structure of atomic hydrogen, Phys. Rev. Lett. 70 (1993), 1952–-1955.

I explained the diamond cubic crystal structure in my recent post about ice. Remember, it looks like this:

Since the body-centered cubic is one of the crystal lattices I didn’t talk about in that post, let me tell you about it now. It’s built of cells that look like this:

… which explains its name. In the same style of drawing, the face-centered cubic looks like this:

In my post about ice, I mentioned that if you pack equal-sized spheres with centers at points in the face-centered cubic lattice, you get the maximum density possible, namely about 74%. The body-centered cubic does slightly worse, about 68%.

So, I always thought of it as a kind of a second-best thing. But apparently it’s the best in some ‘sampling’ problems where you’re trying to estimate a function on space by measuring it only at points in your lattice! That’s because its dual is the face-centered cubic.

Eh? Well, the dual of a lattice in a vector space consists of all vectors in the dual vector space such that is an integer for all points in The dual of a lattice is again a lattice, and taking the dual twice gets you back where you started. Since the Fourier transform of a function on a vector space is a function on the dual vector space, I don’t find it surprising that this is related to sampling problems. I don’t understand the details, but I bet I could find them here:

• Alireza Entezari, Ramsay Dyer and Torsten Möller, From sphere packing to the theory of optimal lattice sampling.

So: from the core of Jupiter to Fourier transforms! Yet again, everything seems to be related.