# May 9, 2006 {#week231}

[Enceladus](http://en.wikipedia.org/wiki/Enceladus_(moon)) is a moon of
Saturn with a cracked icy surface, twisted and buckled by tidal forces,
hinting at mysteries beneath:
$$\href{http://www.nasa.gov/mission_pages/cassini/multimedia/pia07800.html}{\includegraphics[max width=0.65\linewidth]{../images/enceladus.jpg}}$$

1) NASA, "Enceladus the storyteller", `http://www.nasa.gov/mission_pages/cassini/multimedia/pia07800.html`

Recently the NASA space probe Cassini has been getting a good look at
Enceladus. In March, Cassini discovered that it has geysers among the
cracks near its south pole --- geysers that spray water right out into
space!
$$\href{http://saturn.jpl.nasa.gov/news/press-release-details.cfm?newsID=639}{\includegraphics[max width=0.65\linewidth]{../images/enceladus_geyser.jpg}}$$

2) "NASA's Cassini discovers potential liquid water on Enceladus", `http://saturn.jpl.nasa.gov/news/press-release-details.cfm?newsID=639`

3) Special issue on Enceladus, _Science_ **311** (March 10th 2006).

The water freezes in microscopic crystals, which replenish Saturn's [E
ring](http://saturn.jpl.nasa.gov/multimedia/images/image-details.cfm?imageID=2091)
--- a diffuse bluish ring that was previously a mystery.

The currently popular theory for the geysers looks like this:
$$\href{http://www.nasa.gov/mission_pages/cassini/multimedia/pia07799.html}{\includegraphics[max width=0.65\linewidth]{../images/enceladus_geyser_model.jpg}}$$

4) NASA, 'Enceladus "cold geyser" model', `http://www.nasa.gov/mission_pages/cassini/multimedia/pia07799.html`

Enceladus is now the the only place besides Earth where liquid water has
been seen --- though people believe Jupiter's moon
[Europa](http://en.wikipedia.org/wiki/Europa_(moon)) has oceans under a
layer of ice, and maybe
[Ganymede](http://en.wikipedia.org/wiki/Ganymede_(moon)) and
[Callisto](http://en.wikipedia.org/wiki/Callisto_(moon)) do too.

While we tend to take it for granted, water is a very strange chemical:

5) Martin Chaplin, "Forty-one anomalies of water", `http://www.lsbu.ac.uk/water/anmlies.html`

As you probably know, the specific heat of water is unusually high,
which stabilizes the Earth's temperature. And no other simple compound
exhibits so many different forms. There at least 18 forms of ice! You
can tour them here:

6) Martin Chaplin, "The phase diagram of water", `http://www.lsbu.ac.uk/water/phase.html`

The hexagonal form of ice we find here on earth is called
[ice Ih](http://www.lsbu.ac.uk/water/ice1h.html). There's also a slightly
denser cubic phase, [ice Ic](http://www.lsbu.ac.uk/water/ice1c.html),
which forms when water vapor is condensed on a cold substrate. Below
$-130$ Celsius, a low-density amorphous solid form called
[LDA](http://www.lsbu.ac.uk/water/amorph.html#lda) is possible. By
compressing ordinary [ice Ih](http://www.lsbu.ac.uk/water/ice1h.html) to
high pressures, you get a different higher-density amorphous form,
called [HDA](http://www.lsbu.ac.uk/water/amorph.html#hda). And there's
an even denser amorphous form called
[VHDA](http://www.lsbu.ac.uk/water/amorph.html#vhda).

(It's unusual for a crystal to become amorphous when you compress it or
cool it, but ordinary ice is unusually light: it floats on liquid water!
That's because the powerful hydrogen bonds of water allow it to
maintain a very sparse hexagonal crystal structure --- so sparse you could
even fit extra water molecules in the gaps. When you crush this, it
becomes amorphous.)

There are also crystal forms called [ice II](http://www.lsbu.ac.uk/water/ice_ii.html)
through [ice XIV](http://www.lsbu.ac.uk/water/ice_xii.html#icexiv), in order of
discovery. It would take a few weeks to discuss all these, but luckily
Chaplin's website has a separate page on each kind, with nice
explanations and pictures of the crystal structures.

Kurt Vonnegut wrote a novel called "Cat's Cradle" starring a
substance called ice IX, which was supposedly more stable than liquid
water at ordinary temperatures and pressures. When it got loose, it
destroyed the world. Luckily the actual
[ice IX](http://www.lsbu.ac.uk/water/ice_ix.html) isn't like that,
and it couldn't be: the most stable form of water already prevails.

But enough about ice IX. I want to talk about
[ice X](http://www.lsbu.ac.uk/water/ice_vii.html#iceten)!

This is one of the most extreme forms of ice known. It's only stable at
pressures of about 50 gigapascals --- in other words, roughly 500,000
atmospheres.

Hmm. Do those quantities mean as little to you as they do to me? A
"pascal" is a unit of pressure, or force per area, equal to one newton
per square meter. An "atmosphere" is another unit of pressure,
basically the average air pressure at sea level here on Earth. This has
the annoying value of 101,325 pascals. Personally I have some trouble
getting a feel for how much pressure this is, since a newton per square
meter isn't much, but 101,325 of them sounds like a lot. So for me,
being an American, it's helpful to know that an atmosphere equals 2116
pounds per square foot. If you're a metric sort of person, that's
about the weight of 1 kilogram pushing down on each square centimeter.
That's a lot of pressure we're under! No wonder we feel stressed
sometimes.

(Yes, I know a kilogram is not a unit of weight. I mean the weight
corresponding to a mass of a kilogram in the Earth's gravitational
field at sea level. Sheesh!)

But I digress. I was saying that [ice X](http://www.lsbu.ac.uk/water/ice_vii.html#iceten) only forms at a
pressure of about 50 gigapascals. But I've actually read figures
ranging from 44 to 80 gigapascals. This raises the question: how do
people know these things? Do they actually know, or just guess?

Well, some overgrown kids get paid to study these issues by actually
squashing water to enormous pressures using "diamond anvil cells". Not
many substances can withstand such huge pressures, but diamonds can: as
you know, they're really hard! They're also transparent, so you can
see what's going on while you're squashing something. You basically
just stick something between two carefully carved diamonds, surrounded
by a metal foil gasket, and squash the heck out of it:

7) "Diamond anvil cell", Wikipedia, `http://en.wikipedia.org/wiki/Diamond_Anvil_Cell`

Apparently they can get pressures of up to 360 gigapascals this way,
which is the pressure at the center of the Earth.

Another method, which sounds even more fun, is to use a "light gas
gun". Here you explode a few kilograms of gunpowder to shoot a piston
down a tube. As it shoots forwards, the piston pushes some gas down the
tube. The tube narrows to a tiny tip at the end, so the gas is going
really fast by the time it shoots out. It shoots out into a much
narrower tube, where it pushes a projectile. You can then fire the
projectile into something, to generate very high pressures for a very
short time.

8) "Light gas gun", Wikipedia, `http://en.wikipedia.org/wiki/Light_Gas_Gun`

It's not called a "light" gas gun because it's wimpy --- in fact
they're huge, and everyone evacuates the lab when they run the one at
NASA! It's called that because the speed of the projectile is limited
only by the speed of sound in the gas, which is higher for a light gas
like helium --- or even better, hydrogen. Even better, that is, you don't
mind exploding gunpowder near highly flammable hydrogen! But, as you can
imagine, people who do this stuff are precisely the sort who don't
mind. You may enjoy reading how folks at Lawrence Livermore National
Laboratory used a light gas gun to compress hydrogen to pressures of up
to 200 gigapascals, enough to convert it into a metal:

9) Robert C. Cauble, "Putting more pressure on hydrogen", `http://www.llnl.gov/str/Cauble.html`

This supports the theory that the hydrogen at Jupiter's core is in
metallic form, which would explain its enormous magnetic field. They
know their hydrogen became a metal because they fired a laser at it and
saw it was shiny! In fact, they fired three lasers at it simultaneously,
just for kicks.

(By the way, this article erroneously says a "megabar" is 100 pascals.
It's a million atmospheres, or 100 gigapascals.)

But I'm digressing again. I was saying [ice X](http://www.lsbu.ac.uk/water/ice_vii.html#iceten) forms at a pressure
of around 50 gigapascals. It's pretty far-out stuff. It's a cubic
crystal with density 2.5 times that of ordinary liquid water. It's so
compressed that separate water molecules no longer exist! Instead, the
oxygen atoms form a body-centered cubic. This means they lie at the
corners of a lattice of cubes, but with one at the center of each cube
too, like the red dots in this picture by Cavazzoni:
$$\includegraphics[max width=0.65\linewidth]{../images/cavazzoni_ice_X.jpg}$$

10) Carlo Cavazzoni, _Large scale first-principles simulations of water and ammonia at high pressure and temperature_, Ph.D. thesis, Scuola Internazionale Superiore di Studi Avanzati, October 1998. Figure 4.10: symmetric and super-ionic ice X structures, p. 57. Available at `http://sirio.cineca.it/~acv0/thesis.html`

Hydrogen ions --- in other words, protons --- sit at the midpoints of half
the edges connecting cube corners to cube centers. There are two ways
they can do this, illustrated by the yellow and gray dots shown above.
They can form a right-side-up tetrahedron, or an upside-down
tetrahedron.

A body-centered cubic can also be visualized as two interpenetrating
cubic lattices, labelled A and B here:
$$\includegraphics[max width=0.65\linewidth]{../images/cavazzoni_ice_X_2.jpg}$$

Each oxygen has 4 hydrogens next to it. If you compress water a bit less
than enough to make [ice X](http://www.lsbu.ac.uk/water/ice_vii.html#iceten), you get
[ice VII](http://www.lsbu.ac.uk/water/ice_vii.html). This is almost the same,
but two of those hydrogens are closer to the oxygen than the other two,
so there are still separate water molecules! It's completely random
which two hydrogens are closer than the other two. But if you cool down
[ice VII](http://www.lsbu.ac.uk/water/ice_vii.html), you get
[ice VIII](http://www.lsbu.ac.uk/water/ice_viii.html), where it's *not*
random.

So, Nature explores all the options.

Recently people have gotten interested in ice at even higher pressures ---
and also higher temperatures, to understand the interiors of planets
like Neptune and Uranus. Here pressures range from 20 to 800
gigapascals, and temperatures from 2000 to 8000 kelvin. In
["Week 160"](#week160) I mentioned that on Neptune it may rain
diamonds, formed by methane in the atmosphere. But what happens to the
water, and the ammonia? If they became good electrical conductors, that
might explain the magnetic fields of these planets.

People have done computer simulations to study this:

12) C. Cavazzoni, G. L. Chiarotti, S. Scandolo, E. Tosatti, M. Bernasconi and M. Parrinello, "Superionic and metallic states of water and ammonia at giant planet conditions", _Science_ **283** (January 1999), 44--46. Also available at `http://www.sciencemag.org/cgi/content/full/283/5398/44`

$$\includegraphics[max width=0.65\linewidth]{../images/cavazzoni_ice_phases.png}$$

It seems that when you heat up [ice X](http://www.lsbu.ac.uk/water/ice_vii.html#iceten), it goes into a
"superionic" state where the little tetrahedra of hydrogen ions in
each cube are constantly randomizing themselves, instead of remaining
fixed. It's a curious hybrid of a solid and a liquid, since the
hydrogens are moving around, while the oxygens stay in their
body-centered cubic crystal.

But if you heat it even more, the oxygen melts too! As you can see from
the phase diagram above, it then becomes an ionic fluid.

As you heat it even more, you enter the region labelled "gap closure",
where the water starts to act like a metallic plasma. Then it's a
really good conductor of electricity.

The curve labelled "Neptune isentrope" describes the pressures and
temperatures you'd experience if you unwisely jumped into Neptune!

As you fell in, it would keep getting hotter and the pressure would keep
rising until you entered this chart, at a temperature of about 2000
kelvin. At this point you'd see molecular fluid water --- I say this
because at temperatures above 650 kelvin (the critical point for water),
there's no sharp difference between liquid and gas. Then the fluid
would become ionic... and then you'd start drifting towards gap
closure and the metallic plasma phase. Down deep, metallic plasmas of
water and ammonia might explain the magnetic field of this planet.

Recently people have done some experiments with water at extremely high
pressures, checking what theorists like Cavazzoni and company predict.
For example, this paper says that using "extremely large lasers",
people have studied water at pressures near a terapascal --- 1000
gigapascals:

13) P. M. Celliers et al, "Electronic conduction in shock-compressed water", _Plasmas_ **11** (2004), L41--L48.

They also mention that "a single datum at 1.4 terapascals from an
underground nuclear experiment has never been repeated." Some people
just don't know when to stop in the quest for higher pressures.

While I'm at it, I should mention a few more interesting articles on
weird forms of ice. There's a lot of research on this subject! Here's
a quick overview:

14) Nancy McGuire, "The many phases of water", American Chemical Society, `https://web.archive.org/web/20051201104533/http://www.chemistry.org/portal/a/c/s/1/feature_pro.html?id=c373e9fbed0a01c78f6a4fd8fe800100`

Here's a webpage with some nice pictures and an interesting story:

15) J. L. Finney, "The phase diagram of water and a new metastable form of ice", `http://www.cmmp.ucl.ac.uk/people/finney/soi.html`

And finally, there's a paper that talks about how ordinary ice Ih but
also silica and ice XI become amorphous when you squeeze them enough:

16) Koichiro Umemoto, Renata M. Wentzcovitch, Stefano Baroni and Stefano de Cironcoli, "Anomalous pressure-induced transition(s) in ice XI", _Physical Review Letters_ **92** (2004), 105502-1. Also available at `http://www.cems.umn.edu/research/wentzcovitch/papers/Phys._Rev._Lett._92_105502_(2004).pdf`

There's some interesting math in here, because they do computer
simulations of the transition from a crystal to an amorphous substance,
which is interesting to study using Fourier analysis. The idea is that
certain vibrational modes of the crystal "go soft", so they get easily
excited. When a bunch of modes go soft that have wavelengths not equal
to the crystal lattice spacing, the crystal structure becomes unstable,
and there can be a transition to an amorphous state.

There's also interesting math lurking in Cavazzoni et al's models of
ice X! If you think particle physics is hard, just wait until you try
understanding something complicated, like water.

I've been sort of [obsessed with ice](math.ucr.edu/home/baez/diary/diary_2006.html) lately. If
you like it too, I recommend this book for general information:

16) Mariana Gosnell, _Ice: The Nature, the History, and the Uses of an Astonishing Substance_, Alfred A. Knopf, New York, 2005.

but I bought this one, because it tells an interesting history of the
science of climate change as seen from icy peaks:

17) Mark Bowen, _Thin Ice: Unlocking the Secrets of Climate in the World's Highest Mountains_, Henry Holt & Co., 2005.

Now for some math. Last week I said a bit about quivers, the McKay
correspondence, and string theory. I want to dig deeper into the
relation between these subjects, because Urs Schreiber has some
interesting ideas about them, which he's mentioned here:

18) Urs Schreiber, "A note on RCFT and quiver reps", `http://golem.ph.utexas.edu/string/archives/000794.html`

But, I'm not feeling sufficiently energetic to explain these ideas
right now, especially since he already has! For some more clues, try
this:

19) Paul Aspinwall, "D-branes on Calabi-Yau manifolds", section 7.3.1, The McKay correspondence, p. 101 and following. Available as [`hep-th/0403166`](https://arxiv.org/abs/hep-th/0403166)

For more on the the representation theory of quivers, see the references
in the "Addenda" to ["Week 230"](#week230), and also this
excellent book:

20) David J. Benson, _Representations and Cohomology I_, Cambridge U. Press, Cambridge 1991.

You'll see how the non-simply-laced Dynkin diagrams get into the act! A
more thorough treatment, fascinating but somewhat quirky, can be found
here:

21) P. Gabriel and A. V. Roiter, _Representations of Finite-Dimensional Algebras_, Enc. of Math. Sci. **73**, Algebra VIII, Springer, Berlin 1992.

If you like category theory you may enjoy this book, because it's all
about representations of categories, i.e. functors
$$F\colon \mathcal{C} \to \mathsf{Vect}$$
where $\mathcal{C}$ is a category. It's full of nontrivial theorems about these,
starting with Gabriel's classification of quivers into those of finite
representation type (see ["Week 230"](#week230)), the tame quivers
(which have an infinite but still manageable set of indecomposable
representations), and the wild ones. But, you may be puzzled when you
read about "svelte" categories, or functors that "preserve
heteromorphisms".

I might as well say what those are. A category is "svelte" if its
isomorphism classes of objects form a mere set instead of a proper
class, like the category of finite-dimensional vector spaces. Most
people would say such a category is "essentially small".

And, a functor "preserves heteromorphisms" if it maps heteromorphisms
to heteromorphisms. Well, duh! But what's a "heteromorphism"? It's
their term for a morphism that's not an isomorphism. Most people would
say such a functor "reflects isomorphisms".

You may also be interested in what a "locular" category is, or a
"spectroid"... but I won't tell you! Read the book.

Speaking of category theory, this is my last week in Chicago, which is
really sad, because Steve Lack is just starting to give us a crash
course on "Australian category theory". Australia, you see, is the
center of macho category theory, where they're heavy on the calculus of
mates, doctrinal adjunctions are a dime a dozen, and everything should
be $\mathcal{V}$-enriched if not $\mathcal{W}$-enriched. But Chicago is starting to get macho
too: tomorrow Nick Gurski defends his Ph.D. thesis on "Algebraic
Tricategories"! So, the Chicago gang wants to learn some tricks from
the Australians. But next Monday I'm off to the Perimeter Institute, to
indulge the physics side of my personality....

------------------------------------------------------------------------

**Addenda:** I thank Colin Rust for correcting a serious typo. Uncle Al
points out that a newton is, quite appropriately, about the weight of an
average apple. Aaron Bergman had this to say:

> John Baez wrote:
>
> > Now for some math.  Last week I said a bit about quivers, the McKay 
> > correspondence, and string theory. I want to dig deeper into the 
> > relation between these subjects,
>
> You want
>
> 22) Tom Bridgeland, Alaistair King and Miles Reid, "Mukai implies McKay: the McKay correspondence as an equivalence of derived categories", available as [`math.AG/9908027`](http://www.arxiv.org/abs/math.AG/9908027).
>
> Much coolness there.
>
> > For more on the representation theory of quivers, see the references 
> > in the "Addenda" to ["Week 230"](#week230), and also this excellent book:
> > 
> > 20) David J. Benson, _Representations and Cohomology I_,  Cambridge U. Press, Cambridge 1991.
>
> There's a whole lot of cool things to say about non-Dynkin quivers.
> If you put a D-brane at an ADE singularity, you get the affine Dynkin
> quivers, but there are plenty of other singularities out there.
> Another nice set is the canonical bundle over a del Pezzo with the
> zero section collapsed. Quivers can be associated to this via a work
> of Bridgeland:
>
> 23) Tom Bridgeland, "T-structures on some local Calabi-Yau varieties", available as [`math.AG/0502050`](http://www.arxiv.org/abs/math.AG/0502050).
>
> with many of the same results in some physics papers by Herzog,
> Aspinwall and Wijnholt.
>
> My modest contribution to this story is where myself and Nick
> Proudfoot show that the moduli space of representations of these
> quivers (for a subclass of examples) has as a component the original
> cone:
>
> 24) Aaron Bergman, "Undoing orbifold quivers", available as [`hep-th/0502105`](http://www.arxiv.org/abs/hep-th/0502105).
>
> Aaron Bergman, "Moduli spaces for Bondal quivers", available as [`math.AG/0512166`](http://www.arxiv.org/abs/math.AG/0512166).
>
> Aaron

------------------------------------------------------------------------

> *That the glass would melt in heat,\
> That the water would freeze in cold,\
> Shows that this object is merely a state,\
> One of many, between two poles.* 
> 
> --- Wallace Stevens