This Week's Finds in Mathematical Physics (Week 231)

In summary, This week John Baez discusses Enceladus, a moon of Saturn, and the recent discoveries made by the NASA space probe Cassini. The moon is believed to have geysers that spray water into space and replenish Saturn's E ring. Water is a very strange chemical, with 18 known forms of ice. One of the most extreme forms, ice X, is only stable at pressures of about 50 gigapascals. Scientists use methods such as diamond anvil cells and light gas guns to study these extreme pressures.
  • #1
John Baez
Also available at http://math.ucr.edu/home/baez/week231.html

May 9, 2006
This Week's Finds in Mathematical Physics (Week 230)
John Baez

Enceladus is a moon of Saturn with a cracked icy surface, twisted
and buckled by tidal forces, hinting at mysteries beneath:

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!

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 - a diffuse bluish ring that was previously a mystery.

The currently popular theory for the geysers looks like this:

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 has oceans
under a layer of ice, and maybe Ganymede and Callisto 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.
There's also a slightly denser cubic phase, ice Ic, which forms when
water vapor is condensed on a cold substrate. Below -130 Celsius,
a low-density amorphous solid form called LDA is possible. By
compressing ordinary ice Ih to high pressures, you get a different
higher-density amorphous form, called HDA. And there's an even denser
amorphous form called 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 through ice XIV, 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" staring 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 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!

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 50,000
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 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 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:

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. They can form a right-side-up tetrahedron, or an
upside-down tetrahedron.

Each oxygen has 4 hydrogens next to it. If you compress water a bit
less than enough to make ice X, you get ice VII. 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, you get ice VIII, 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 "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
Phase diagram at http://math.ucr.edu/home/baez/cavazzoni_ice_phases.jpg

It seems that when you heat up ice X, 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,
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 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.

For more on the representation theory of quivers, see the references
in the "Addenda" to "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: C -> Vect

where 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 "week230"), the tame quivers
(which have a countable 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 V-enriched if not 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 Australian tricks. But next Monday I'm off to the
Perimeter Institute, to indulge the physics side of my personality...

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

Quote of the Week:

"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

-----------------------------------------------------------------------
Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at

http://math.ucr.edu/home/baez/

For a table of contents of all the issues of This Week's Finds, try

http://math.ucr.edu/home/baez/twfcontents.html

A simple jumping-off point to the old issues is available at

http://math.ucr.edu/home/baez/twfshort.html

If you just want the latest issue, go to

http://math.ucr.edu/home/baez/this.week.html
 
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  • #2
John Baez wrote:
>
> Also available at http://math.ucr.edu/home/baez/week231.html
>
> May 9, 2006
> This Week's Finds in Mathematical Physics (Week 230)
> John Baez

[snip]

> 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.


Illustrating the deep sourcing of Intelligent Design, the IDiots and
we note that the average weight of the average apple is one Newton.

> (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!)


[snip]

> 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


To my mind, a most interesting seriously big squishie would be the
zeta-pinch facility at Sandia. Adding the delight of a huge aligned
magnetic field to the study of compressed light magnetic nuclei is the
right thing to do.

http://zpinch.sandia.gov/
Not the gizmo, just its power supply.
http://physicsweb.org/articles/news/7/4/7/1/z-machine
another view

> 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.


Consider the planned popping of 700 tons of ANFO to model a
nuclear-based underground bunker buster the US is not Officially
developing. Given the physics of long rod penetration, the depth of
the presumed targets, and the (radiative) political inconvenience of a
megatonne shallow depth detonation, one extrapolates the merry
weaponeers are aiming for a nuclear (or nukular, all things
considered) Monroe effect.

<http://www.globalsecurity.org/military/systems/munitions/bullets2-shaped-charge.htm>

[snip]

> -----------------------------------------------------------------------
> Previous issues of "This Week's Finds" and other expository articles on
> mathematics and physics, as well as some of my research papers, can be
> obtained at
>
> http://math.ucr.edu/home/baez/
>
> For a table of contents of all the issues of This Week's Finds, try
>
> http://math.ucr.edu/home/baez/twfcontents.html
>
> A simple jumping-off point to the old issues is available at
>
> http://math.ucr.edu/home/baez/twfshort.html
>
> If you just want the latest issue, go to
>
> http://math.ucr.edu/home/baez/this.week.html


--
Uncle Al
http://www.mazepath.com/uncleal/
(Toxic URL! Unsafe for children and most mammals)
http://www.mazepath.com/uncleal/qz3.pdf
 
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  • #3
Uncle Al wrote:
> John Baez wrote:
>
>
> [snip]
>
>
>>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

>
>
> To my mind, a most interesting seriously big squishie would be the
> zeta-pinch facility at Sandia. Adding the delight of a huge aligned
> magnetic field to the study of compressed light magnetic nuclei is the
> right thing to do.
>
> http://zpinch.sandia.gov/
> Not the gizmo, just its power supply.
> http://physicsweb.org/articles/news/7/4/7/1/z-machine
> another view
>
>
>>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.

>


"Insanity Insanity Insanity ..."
spoken by British Corpsman
In the context of
'Bridge over the River Kwai'
Colonel Saito (Sessue Hayakawa)
Colonel Nicholson (Alec Guinness)

Do a Google Search on

Critical optical volume energy (COVE)

A pittance spent
to test structural elegance
resulting in benign energy
rather than bruit force
with massive destructive implications.

Richard
 
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  • #4
John Baez comments
> 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

snip
>So, Nature explores all the options.


Very true - even exploring negative pressure!
Chaplin makes a very brief mention of "clathrate" structures, as
"negative pressure" stable ice forms. These exist as methane (and other
small molecule) hydrates. The water molecules can form (at least two)
unstable low density cage structures, which are stabilized by trapping
random small molecules that effectively "negate" the ambient pressure.
Computing the stability of these non-stoichometric structures is an
interesting thermodynamic study, and methane hydrate is of immense
economic and ecologic importance.
Roger Beresford.
 
  • #5
Hello John and everyone!

John Baez wrote:
> 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


I heard this claim many time by now and I begin to wonder about it.
Most of the explanations of water's "strange" properties (like high
melting point, boling point, dielectric coefficient, viscosity etc.) I
heard
are based on hydrogen bonds. However, why is this phenomenon so
unique to water? For instance, why wouldn't hydrofluoric acid (HF)
exhibit the same properties? Fluorine is even more electronegative than
oxygen, so the molecule is bound to be very polar. Nevertheless, the
boiling point is merely 20C and the melting point is as low as -83C
(according to Wikipedia). So, there must be more to water than polarity
and hydrogen. Multipole moments possibly??Squark
 
  • #6
Squark wrote:

> Hello John and everyone!
>
> John Baez wrote:
>
>>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
>>

>
> I heard this claim many time by now and I begin to wonder about it.
> Most of the explanations of water's "strange" properties (like high
> melting point, boling point, dielectric coefficient, viscosity etc.) I
> heard
> are based on hydrogen bonds. However, why is this phenomenon so
> unique to water? For instance, why wouldn't hydrofluoric acid (HF)
> exhibit the same properties? Fluorine is even more electronegative than
> oxygen, so the molecule is bound to be very polar. Nevertheless, the
> boiling point is merely 20C and the melting point is as low as -83C
> (according to Wikipedia). So, there must be more to water than polarity
> and hydrogen. Multipole moments possibly??


There is a nice explanation of this in the Wikipedia article on
'hydrogen bond'.

Around each atom, you draw 4 bars, which represents 4 pairs of
electrons, except for hydrogen, which only needs one bar. eg:

H
|
water: |O|
|
_ H
HF: |F|
|
H
H
|
Amonia: |N-H

|
H

Next, a hydrogen bond is a bond between an H and a 'lone pair', eg:

H
| _
|O|...H-O-H
| -
H

Now (I just learned this from Wikipedia), water has the highest amount
of possible H-bonds per molecule, that is, if it binds to itself. HF
would have too many lone pairs, amonia too many H.

For the rest, the Wikepedia is a lot better than this post, except
perhaps drawing out the molecular diagrams explicitly, like I did in
ASCII, is helpful, at least it was for me.

Gerard
 

1. What is "This Week's Finds in Mathematical Physics"?

"This Week's Finds in Mathematical Physics" is a weekly blog written by mathematician and physicist John Baez. The blog features topics in the intersection of mathematics and physics, including but not limited to quantum field theory, general relativity, and string theory.

2. Who is John Baez?

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Yes, Baez welcomes submissions from readers who have interesting research or ideas in the field of mathematical physics. However, he receives a large number of submissions and cannot guarantee that all will be featured on the blog.

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