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This Week's Finds in Mathematical Physics (Week 262)

  1. Mar 30, 2008 #1
    Also available at http://math.ucr.edu/home/baez/week262.html

    March 29, 2008
    This Week's Finds in Mathematical Physics (Week 262)
    John Baez

    I'm done with teaching until fall, and now I'll be travelling
    a lot. I just got back from Singapore. It's an incredibly
    diverse place. I actually had to buy a book to understand
    all the foods! I'm now acquainted with the charms of appam,
    kaya toast, and babi buah keluak. But I didn't get around to
    trying a chendol, a bandung, or a Milo dinosaur, even though
    they're all available in every hawker center.

    Today I'll talk about quantum technology in Singapore, atom
    chips, nitrogen-vacancy pairs in diamonds, graphene transistors,
    a new construction of E8, and a categorification of sl(2).

    But first - the astronomy pictures of the week!

    First another planetary nebula - the "Southern Ring Nebula":

    1) Hubble Heritage Project, Planetary Nebula NGC 3132,
    http://heritage.stsci.edu/1998/39/index.html

    This bubble of hot gas is .4 light years in diameter. You
    can see *two* stars near its center. The faint one is the
    white dwarf remnant of the star that actually threw off the
    gas forming this nebula. The gas is expanding outwards at
    about 20 kilometers per second. The intense ultraviolet
    radiation from the white dwarf is ionizing this gas and making
    it glow.

    The Southern Ring Nebula is 2000 light years from us. Much
    closer to home, here's a new shot of the frosty dunes of Mars:

    2) HiRISE (High Resolution Imaging Science Experiment),
    Defrosting polar sand dunes, http://hirise.lpl.arizona.edu/PSP_007043_2650

    These horn-shaped dunes are called "barchans"; you can read
    more about them at "week228". The frost is carbon dioxide,
    evaporating as the springtime sun warms the north polar
    region. Here's another photo, taken in February:

    3) HiRISE (High Resolution Imaging Science Experiment),
    Defrosting northern dunes, http://hirise.lpl.arizona.edu/PSP_007193_2640

    The dark stuff pouring down the steep slopes reminds me of
    water, but they say it's dust!

    Meanwhile, down here on Earth, I had some good conversations with
    mathematicians and physicists at the National University of Singapore
    (NUS), and also with Artur Ekert and Valerio Scarani, who work here:

    4) Centre for Quantum Technologies, http://www.quantumlah.org/

    I like the name "quantumlah". "Lah" is perhaps the most famous
    word in Singlish: you put it at the end of a sentence for
    emphasis, to convey "acceptance, understanding, lightness,
    jest, and a medley of other positive feelings". Unfortunately
    I didn't get to hear much Singlish during my visit.

    The Centre for Quantum Technologies is hosted by NUS but
    is somewhat independent. It reminds me a bit of the Institute
    for Quantum Computing - see "week235" - but it's smaller, and
    still getting started. They hope to take advantage of the
    nearby semiconductor fabrication plants, or "fabs", to build
    stuff.

    They've got theorists and experimentalists. Being overly
    theoretical myself, I asked: what are the most interesting
    real-life working devices we're likely to see soon? Ekert
    mentioned "quantum repeaters" - gadgets that boost the power
    of a beam of entangled photons while still maintaining
    quantum coherence, as needed for long-distance quantum
    cryptography. He also mentioned "atom chips", which use tiny
    wires embedded in a silicon chip to trap and manipulate cold
    atoms on the chip's surface:

    5) Atomchip Group, http://www.atomchip.org/

    6) Atom Optics Group, Laboratoire Charles Fabry, Atom-chip
    experiment,
    http://atomoptic.iota.u-psud.fr/research/chip/chip.html

    There's also a nanotech group at NUS:

    7) Nanoscience and Nanotechnology Initiative, National
    University of Singapore, http://www.nusnni.nus.edu.sg/

    who are doing cool stuff with "graphene" - hexagonal sheets
    of carbon atoms, like individual layers of a graphite crystal.
    Graphene is closely related to buckyballs (see "week79") and
    polycyclic aromatic hydrocarbons (see "week258").

    Some researchers believe that graphene transistors could
    operate in the terahertz range, about 1000 times faster than
    conventional silicon ones. The reason is that electrons move
    much faster through graphene. Unfortunately the difference
    in conductivity between the "on" and "off" states is less for
    graphene. This makes it harder to work with. People think
    they can solve this problem, though:

    8) Kevin Bullis, Graphene transistors, Technology Review,
    January 28, 2008, http://www.technologyreview.com/Nanotech/20119/

    Duncan Graham-Rowe, Better graphene transistors, Technology
    Review, March 17, 2008, http://www.technologyreview.com/Nanotech/20424/

    Ekert also told me about another idea for carbon-based computers:
    "nitrogen-vacancy centers". These are very elegant entities.
    To understand them, it helps to know a bit about diamonds.
    You really just need to know that diamonds are crystals made
    of carbon. But I can't resist saying more, because the geometry
    of these crystals is fascinating.

    A diamond is made of carbon atoms arranged in tetrahedra, which
    then form a cubical structure, like this:

    9) Steve Sque, Structure of diamond,
    http://newton.ex.ac.uk/research/qsystems/people/sque/diamond/structure/

    Here you see 4 tetrahedra of carbon atoms inside a cube.
    Note that there's one carbon at each corner of the cube, and
    also one in the middle of each face. If that was all, we'd
    have a "face-centered cubic". But there are also 4 more
    carbons inside the cube - one at the center of each tetrahedron!

    If you look really carefully, you can see that the full
    pattern consists of two interpenetrating face-centered
    cubic lattices, one offset relative to the other along the
    cube's main diagonal!

    While the math of the diamond crystal is perfectly beautiful,
    nature doesn't always get it quite right. Sometimes a carbon
    atom will be missing. In fact, sometimes a cosmic ray will
    knock a carbon out of the lattice! You can also do it yourself
    with a beam of neutrons or electrons. The resulting hole is
    called a "vacancy". If you heat a diamond to about 900
    kelvin, these vacancies start to move around like particles.

    Diamonds also have impurities. The most common is nitrogen,
    which can form up 1% of a diamond. Nitrogen atoms can take
    the place of carbon atoms in the crystal. Sometimes these
    nitrogen atoms are isolated, sometimes they come in pairs.

    When a lone nitrogen encounters a vacancy, they stick together!
    We then have a "nitrogen-vacancy center". It's also common for
    4 nitrogens to surround a vacancy. Many other combinations are
    also possible - and when we get enough of these nitrogen-vacancy
    combinations around, they form larger structures called
    "platelets".

    10) R. Jones and J. P. Goss, Theory of aggregation of nitrogen
    in diamond, in Properties, Growth and Application of Diamond,
    eds. Maria Helena Nazare and A. J. Neves, EMIS Datareviews
    Series, 2001, 127-130.

    A nice thing about nitrogen-vacancy centers is that they act
    like spin-1 particles. In fact, these spins interact very
    little with their environment, thanks to the remarkable properties
    of diamond. So, they might be a good way to store quantum
    information: they can last 50 microseconds before losing
    coherence, even at room temperature. If we could couple them
    to each other in interesting ways, maybe we could do some
    "spintronics", or even quantum computation:

    11) Sankar das Sarma, Spintronics, American Scientist
    89 (2001), 516-523. Also available at
    http://www.physics.umd.edu/cmtc/earlier_papers/AmSci.pdf

    Lone nitrogens are even more robust carriers of quantum
    information: their time to decoherence can be as much as a
    millisecond! The reason is that, unlike nitrogen-vacancy
    centers, lone nitrogens have "dark spins" - their spin
    doesn't interact much with light. But this can also makes
    them harder to manipulate. So, it may be easier to use
    nitrogen-vacancy centers. People are busy studying the options:

    12) R. J. Epstein, F. M. Mendoza, Y. K. Kato and D. D.
    Awschalom, Anisotropic interactions of a single spin and
    dark-spin spectroscopy in diamond, Nature Physics 1 (2005),
    94-98. Also available as arXiv:cond-mat/0507706.

    13) Ph. Tamarat et al, The excited state structure of the
    nitrogen-vacancy center in diamond, available as
    arXiv:cond-mat/0610357.

    14) R. Hanson, O. Gywat and D. D. Awschalom, Room-temperature
    manipulation and decoherence of a single spin in diamond,
    Phys. Rev. B74 (2006) 161203. Also available as
    arXiv:quant-ph/0608233

    But regardless of whether anyone can coax them into quantum
    computation, I like diamonds. Not to own - just to contemplate!
    I told you about the diamond rain on Neptune back in "week162".
    And in "week193", I explained how diamonds are the closest thing
    to the E8 lattice you're likely to see in this 3-dimensional world.

    The reason is that in any dimension you can define a checkerboard
    lattice called Dn, consisting of all n-tuples of integers that
    sum to an even integer. Then you can define a set called Dn+ by
    taking two copies of the Dn lattice: the original and another
    shifted by the vector (1/2,...,1/2). D8+ is the E8 lattice,
    but D3 is the face-centered cubic, and D3+ is the pattern formed
    by carbons in a diamond!

    In case you're wondering: in math, a "lattice" is technically
    a discrete subgroup of R^n that's isomorphic to Z^n. Dn+ is
    only a lattice when n is even. So, the carbons in a diamond
    don't form a lattice in the strict mathematical sense. On the
    other hand, the D3 lattice is secretly the same as the A3
    lattice, familiar from stacking oranges. It's one of the
    densest ways to pack spheres, with a density of

    pi/3 sqrt(2) ~ .74

    The D3+ pattern, on the other hand, has a density of just

    pi sqrt(3)/16 ~ .34

    This is why ice becomes denser when it melts: like diamond,
    it's arranged in a D3+ pattern.

    (Do diamonds become denser when they melt? Or do they always
    turn into graphite when they get hot enough, regardless of
    the pressure? Inquiring minds want to know. These days
    inquiring minds use search engines to answer questions like
    this... but right now I'd rather talk about E8.)

    As you probably noticed, Garrett Lisi stirred up quite a media
    sensation with his attempt to pack all known forces and particles
    into a theory based on the exceptional Lie group E8:

    15) Garrett Lisi, An exceptionally simple theory of everything,
    available as arXiv:0711.0770

    Part of his idea was to use Kostant's triality-based description
    of E8 to explain the three generations of leptons - see "week253"
    for more. Unfortunately this part of the idea doesn't work, for
    purely group-theoretical reasons:

    16) Jacques Distler, A little group theory,
    http://golem.ph.utexas.edu/~distler/blog/archives/001505.html

    A little more group theory,
    http://golem.ph.utexas.edu/~distler/blog/archives/001532.html

    There would also be vast problems trying get all the dimensionless
    constants in the Standard Model to pop out of such a scheme - or
    to stick them in somehow.

    Meanwhile, Kostant has been doing new things with E8. He's mainly
    been using the complex form of E8, while Lisi needs a noncompact
    real form to get gravity into the game. So, the connection between
    their work is somewhat limited. Nonetheless, Kostant enjoys the
    idea of a theory of everything based on E8.

    He recently gave a talk here at UCR:

    17) Bertram Kostant, On some mathematics in Garrett Lisi's
    "E8 theory of everything", February 12, 2008, UCR. Video and
    lecture notes at http://math.ucr.edu/home/baez/kostant/

    He did some amazing things, like chop the 248-dimensional Lie
    algebra of E8 into 31 Cartan subalgebras in a nice way, thus
    categorifying the factorization

    248 = 8 x 31

    To do this, he used a copy of the 32-element group (Z/2)^5
    sitting in E8, and the 31 nontrivial characters of this group.

    Even more remarkably, this copy of (Z/2)^5 sits inside a copy
    of SL(2,F_{32}) inside E8, and the centralizer of a certain
    element of SL(2,F_{32}) is a product of two copies of the gauge
    group of the Standard Model! What this means - if anything -
    remains a mystery.

    Indeed, pretty much everything about E8 seems mysterious to me,
    since nobody has exhibited it as the symmetry group of anything
    more comprehensible than E8 itself. This paper sheds some
    new light this puzzle:

    17) Jose Miguel Figueroa-O'Farrill, A geometric construction
    of the exceptional Lie algebras F4 and E8, available as
    arXiv:0706.2829.

    The idea here is to build the Lie algebra of E8 using Killing
    spinors on the unit sphere in 16 dimensions.

    Okay - what's a Killing spinor?

    Well, first I need to remind you about Killing vectors. Given
    a Riemannian manifold, a "Killing vector" is a vector field that
    generates a flow that preserves the metric! A transformation
    that preserves the metric is called an "isometry", and these
    form a Lie group. Killing vector fields form a Lie algebra
    if we use the ordinary Lie bracket of vector fields, and this
    is the Lie algebra of the group of isometries.

    Now, if our manifold has a spin structure, a "Killing spinor" is
    a spinor field psi such that

    D_v psi = k v psi

    for some constant k for every vector field v. Here D_v psi
    is the covariant derivative of psi in the v direction, while
    v psi is defined using the action of vectors on spinors.
    Only the sign of the constant k really matters, since rescaling
    the metric rescales this constant.

    It's a cute equation, but what's the point of it? Part
    of the point is this: the action of vectors on spinors

    V tensor S -> S

    has a kind of adjoint

    S tensor S -> V

    This lets us take a pair of spinor fields and form a vector
    field. This is what people mean when they say spinors are
    like the "square root" of vectors. And, if we do this to
    two *Killing* spinors, we get a *Killing* vector! You can
    prove this using that cute equation - and that's the main point
    of that equation, as far as I'm concerned.

    Under good conditions, this fact lets us define a "Killing
    superalgebra" which has the Lie algebra of Killing vectors
    as its even part, and the Killing spinors as its odd part.

    In this superalgebra, the bracket of two Killing vectors
    is just their ordinary Lie bracket. The bracket of a Killing
    vector and a Killing spinor is defined using a fairly obvious
    notion of the "Lie derivative of a spinor field". And, the
    bracket of two Killing spinors is defined using the map

    S tensor S -> V

    which, as explained, gives a Killing vector.

    Now, you might think our "Killing superalgebra" should be a
    Lie superalgebra. But in some dimensions, the map

    S tensor S -> V

    is skew-symmetric. Then our Killing superalgebra has a chance
    at being a plain old Lie algebra! We still need to check
    the Jacobi identity. And this only works in certain special cases:

    If you take S^7 with its usual round metric, the isometry group is
    SO(8), so the Lie algebra of Killing vectors is so(8). There's
    an 8-dimensional space of Killing spinors, and the action of so(8)
    on this gives the real left-handed spinor representation S_8^+.
    The Jacobi identity holds, and you get a Lie algebra

    so(8) + S_8^+

    But then, thanks to triality, you knock ourself on the head and
    say "I could have had a V_8!" After all, up to an outer
    automorphism of so(8), the spinor representation S_8^+ is the
    same as the 8-dimensional vector representation V_8. So, your
    Lie algebra is just the same as

    so(8) + V_8

    with its obvious Lie algebra structure. This is just so(9).
    So, it's nothing exceptional, though you arrived at it by a
    devious route.

    If you take S^8 with its usual round metric, the Lie algebra of
    Killing vector fields is so(9). Now there's a 16-dimensional
    space of Killing spinor fields, and the action of so(9) on this
    gives the real (non-chiral) spinor representation S_9. The Jacobi
    identity holds, and you get a Lie algebra structure on

    so(9) + S_9

    This gives the exceptional Lie algebra f4!

    Finally, if you take S^{15} with its usual round metric, the Lie
    algebra of Killing vector fields is so(16). Now there's a
    128-dimensional space of Killing spinor fields, and the action of
    so(16) on this gives the left-handed real spinor representation
    S_{16}^+. The Jacobi identity holds, and you get a Lie algebra

    so(16) + S_{16}^+

    This gives the exceptional Lie algebra e8!

    In short, what Figueroa-O'Farrill has done is found a nice
    geometrical interpretation for some previously known algebraic
    constructions of f4 and e8. Unfortunately, he still needs to
    verify the Jacobi identity in the same brute-force way. It
    would be nice to find a slicker proof. But his new interpretation
    is suggestive: it raises a lot of new questions. He lists some
    of these at the end of the paper, and mentions a really big one
    at the beginning. Namely: the spheres S^7, S^8 and S^{15} all
    show up in the Hopf fibration associated to the octonionic projective
    line:

    S^7 -> S^{15} -> S^8

    Does this give a nice relation between so(9), F4 and E8? Can
    someone guess what this relation should be? Maybe E8 is built
    from so(9) and F4 somehow.

    I also wonder if there's a Killing superalgebra interpretation
    of the Lie algebra constructions

    e6 = so(10) + S_{10} + u(1)

    and

    e7 = so(12) + S_{12}^+ + su(2)

    These would need to be trickier, with the u(1) showing up from
    the fact that S_{10} is a complex representation, and the su(2)
    showing up from the fact that S_{12}^+ is a quaternionic
    representation. The algebra is explained here:

    18) John Baez, The octonions, section 4.3: the magic square,
    available at http://math.ucr.edu/home/baez/octonions/node16.html

    A geometrical interepretation would be nice!

    Finally - my former student Aaron Lauda has been working with
    Khovanov on categorifying quantum groups, and their work is starting
    to really take off. I'm just beginning to read his new papers, but
    I can't resist bringing them to your attention:

    18) Aaron Lauda, A categorification of quantum sl(2), available as
    arXiv:0803.3848.

    Aaron Lauda, Categorified quantum sl(2) and equivariant cohomology
    of iterated flag varieties, available as arXiv:0803.3652.

    He's got a *2-category* that decategorifies to give the quantized
    universal enveloping algebra of sl(2)! And similarly for all the
    irreps of this algebra!

    There's more to come, too....

    -----------------------------------------------------------------------
    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
     
  2. jcsd
  3. Apr 2, 2008 #2
    John Baez wrote:
    >
    > Also available at http://math.ucr.edu/home/baez/week262.html
    >
    > March 29, 2008
    > This Week's Finds in Mathematical Physics (Week 262)
    > John Baez

    [snip]

    > 7) Nanoscience and Nanotechnology Initiative, National
    > University of Singapore, http://www.nusnni.nus.edu.sg/
    >
    > who are doing cool stuff with "graphene" - hexagonal sheets
    > of carbon atoms, like individual layers of a graphite crystal.
    > Graphene is closely related to buckyballs (see "week79") and
    > polycyclic aromatic hydrocarbons (see "week258").
    >
    > Some researchers believe that graphene transistors could
    > operate in the terahertz range, about 1000 times faster than
    > conventional silicon ones. The reason is that electrons move
    > much faster through graphene. Unfortunately the difference
    > in conductivity between the "on" and "off" states is less for
    > graphene. This makes it harder to work with. People think
    > they can solve this problem, though:
    >
    > 8) Kevin Bullis, Graphene transistors, Technology Review,
    > January 28, 2008, http://www.technologyreview.com/Nanotech/20119/
    >
    > Duncan Graham-Rowe, Better graphene transistors, Technology
    > Review, March 17, 2008, http://www.technologyreview.com/Nanotech/20424/

    [snip]

    Ribbons of chemically defined graphene are trivially synthesized by
    the quadrillions (R lends solublity),

    <http://www.coronene.com/blog/wp-content/uploads/2008/03/polymer.png>

    Gold trace nanocircuitry is trivially emplaced upon silicon.
    alpha,omega-Terminate said graphene ribbons with (revealable) thiols
    and a four billion valve quad-CPU assembles itself. Then fire your
    chemist and reward your managers.

    --
    Uncle Al
    http://www.mazepath.com/uncleal/
    (Toxic URL! Unsafe for children and most mammals)
    http://www.mazepath.com/uncleal/lajos.htm#a2
     
  4. Apr 7, 2008 #3
    In article <47F0069F.7630F508@hate.spam.net>,
    Uncle Al <UncleAl0@hate.spam.net> wrote:

    >Ribbons of chemically defined graphene are trivially synthesized by
    >the quadrillions (R lends solublity),
    >
    ><http://www.coronene.com/blog/wp-content/uploads/2008/03/polymer.png>


    Cool! Do you know a reference besides that picture?
     
  5. Apr 8, 2008 #4
    In article <ftbt36$lep$1@glue.ucr.edu>, John Baez <baez@math.removethis.ucr.andthis.edu> wrote:

    > In article <47F0069F.7630F508@hate.spam.net>,
    > Uncle Al <UncleAl0@hate.spam.net> wrote:
    >
    > >Ribbons of chemically defined graphene are trivially synthesized by
    > >the quadrillions (R lends solublity),
    > >
    > ><http://www.coronene.com/blog/wp-content/uploads/2008/03/polymer.png>

    >
    > Cool! Do you know a reference besides that picture?


    And does the '48% over two steps' mean that the yield for ribbons ~10 steps
    long is (1-0.48)^10ppp0.15%, or can you get around it by multiple processing steps?

    Here's the blog post about it:

    http://www.coronene.com/blog/?pppp297

    Which points to this (mind the wrap)
    <http://pubs.acs.org/cgi-bin/abstract.cgi/jacsat/2008/130/i13/abs/ja710234t.html>

    J. Am. Chem. Soc., 130 (13), 4216 -4217, 2008. 10.1021/ja710234t
    Web Release Date: March 7, 2008
    Copyright © 2008 American Chemical Society
    Two-Dimensional Graphene Nanoribbons
    Xiaoyin Yang, Xi Dou, Ali Rouhanipour, Linjie Zhi, Hans Joachim Räder, and Klaus Müllen*

    --
    David M. Palmer dmpalmer@email.com (formerly @clark.net, @ematic.com)
     
  6. Apr 8, 2008 #5
    John Baez wrote:
    >
    > In article <47F0069F.7630F508@hate.spam.net>,
    > Uncle Al <UncleAl0@hate.spam.net> wrote:
    >
    > >Ribbons of chemically defined graphene are trivially synthesized by
    > >the quadrillions (R lends solublity),
    > >
    > ><http://www.coronene.com/blog/wp-content/uploads/2008/03/polymer.png>

    >
    > Cool! Do you know a reference besides that picture?


    http://www.coronene.com/blog/?p=297
    <http://pubs.acs.org/cgi-bin/abstract.cgi/jacsat/2008/130/i13/abs/ja710234t.html>
    J. Am. Chem. Soc. 130(13) 4216 (2008)

    Chemists make stuff, engineers make things. An engineer with a roll
    of adhesive tape sent off to peel graphite is a silly thing. The
    chemist will never make an interesting device. An interdisciplinary
    effort offends professional management whose casebooks are strictly
    linear anecdotes.

    We could be making self-assembled large scale integrated graphene
    circuits at will within a couple or three months. Pair stuff with
    things and toss in money. Nah. More studies are needed. Stop trying
    to do it and do it!

    An unpleasant precedent is WA Little and excitonic superconductors.
    Bill in Stanford physics had a cute idea and went slumming down the
    road to Stanford chemistry to flesh it out. They give him a polymer
    structure that could never be built, not even in wild dreams, and he
    published,

    William A. Little, Phys. Rev. 134 A1416-A1424 (1964)
    Exciton-based ambient temperature superconductors: polyacetylenes
    substituted with polarizable chromophores, [-C(Ar)=(Ar)C-]n.

    1) A linear trans-polyacetylene core,
    2) With pendant polarizable chromphores,
    3) Said chromophores' pi-clouds being no more than one sigma bond
    from the core,
    4) The core being entirely surrounded and enveloped by a
    cylindrical pi-cloud.

    Replace Bardeen-Cooper-Shrieffer large mass phonons (quantized lattice
    vibrations characterized by Debye temperature) with small mass
    excitons (quantized electronic excitations) possessing characteristic
    energies around 2 eV or 23,000 K. That is warmer than liquid
    nitrogen. Polymers are easy to fabricate into things like wires and
    cables.

    In the 21st century we can synthesize Little exciton supercon polymers
    at will with a hammer

    Acc. Chem. Res. 38(9) 745-754 (2005)

    or with a feather via acyclic diene metathesis (ADMET), Grubbs and
    Schrock catalysts. Lots of researchers pursue this elegant
    chemistry. I contacted a dozen folks who diddle with polyacetylenes
    (synthetic paths are easy) or high temp supercons (applications are
    BIG money). The chemists do not give a sparrow's fart about high temp
    supercons. The engineers know the answer is ceramic supercons (used
    to be metallic supercons) not plastic.

    http://www.mazepath.com/uncleal/benzen1.png
    Model polymer (multiple bonds and hydrogens omitted)
    One day in an organic lab, off the shelf.
    http://www.mazepath.com/uncleal/pyrene1.png
    Pricey. Pyrene is violently fluorescent.
    http://www.mazepath.com/uncleal/pave1.png
    There ya go, Pilgrim.

    Solubility during synthesis is a problem, but one easily solved. The
    result will be liquid crystal phases that spin into automatically
    aligned polymer chains (e.g., lyotropic spinning of Kevlar).

    It won't even fly as an undergrad project - risk of failure. What
    about risk of success? Little was not stupid. If he was off by a fat
    parameterization and it only superconducts to 23 C, will we be
    disappointed? Tc = 23 C, room temp, will get the kid a BS/Chem (and a
    trip to Sweden in December, and a $billion/year in royalties). Isn't
    it worth a university supporting then stealing that? Nah. Discovery
    is a orphan.

    --
    Uncle Al
    http://www.mazepath.com/uncleal/
    (Toxic URL! Unsafe for children and most mammals)
    http://www.mazepath.com/uncleal/lajos.htm#a2
     
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