[HN Gopher] Wigner Crystals (2017)
___________________________________________________________________
Wigner Crystals (2017)
Author : leephillips
Score : 36 points
Date : 2021-08-15 16:48 UTC (6 hours ago)
(HTM) web link (johncarlosbaez.wordpress.com)
(TXT) w3m dump (johncarlosbaez.wordpress.com)
| contravariant wrote:
| The author seems to get quite close to a proof that there must be
| 6 'defects' but then gives up for some reason.
|
| To get the result all you really need is the additional
| requirement that the vertices on the edge each have 4 neighbours.
| Once you have that then you can just glue it to its mirror image.
| This now functions as a triangulation of the sphere so it must
| have 12 more vertices with 5 neighbours than vertices with 7
| neighbours, since it's composed of 2 equal halves each half must
| have exactly 6 more vertices with 5 neighbours.
|
| You can probably generalize this by letting vertices on the edge
| have one neighbour more or fewer as well, but things get a bit
| messy.
| kmill wrote:
| The answer to the puzzle is in the comments on that page (Prof.
| Baez gives your argument in one of them, too).
|
| It would be interesting knowing why there are roughly six
| clusters of defects. The figure seems to technically have seven
| clusters, but it seems plausible that low-energy configurations
| tend to have at most six.
|
| One point of view is that in a triangulation, the degree d of a
| vertex corresponds to something like the curvature there, where
| the curvature is concentrated into a cone angle of 60 * d. The
| total cone angle defect (the sum of 360 - 60 * d over all
| vertices) is 720 degrees for a sphere, either by Euler
| characteristic or Gauss-Bonnet, so if the only vertex degrees
| you see are 5 and 6, there are exactly 12 vertices of degree 5.
| So, fancifully, it seems like the potential function makes the
| space "want" to be flat (i.e., have vertices of degree 6 for
| cone angle defect of 0), so it seems plausible that these
| defects, some kind of fundamental particle of curvature, would
| "want" to be concentrated in packets of total-
| defect-60-degrees, and that these would "want" to be relatively
| far apart from one another.
| jhoechtl wrote:
| > I'd like to explain a conjecture about Wigner crystals, which
| we came up with in a discussion on Google+.
|
| There is a link to the Google+ discussion. No, you can't read
| what has been discussed there, another abandoned and folded
| Google service.
| [deleted]
| c1ccccc1 wrote:
| So the 2d crystals make sense to me, but how do the 3d crystals
| form? Charge is usually concentrates on the boundaries of a
| conductor, right? Shouldn't the same apply to an empty region of
| space?
| contravariant wrote:
| Your objection applies equally to the 2D crystal.
|
| As far as I can tell the fact that charges reside on the
| boundary only really applies to induced charges. When the
| conductor itself has a net charge then you'd normally expect
| these to configure themselves so they're as far apart from each
| other as possible. If the conductor doesn't have any kind of
| weird shape this should normally result in a roughly uniform
| distribution of charge (if it has spikes then most charges will
| move there to be away from the bulk).
| c1ccccc1 wrote:
| I think for my objection to apply in the 2D case, you'd need
| for the force law to be 1/r instead of 1/r^2. That would
| happen in a truly 2D world, but not in this case where we
| have charges confined to a 2D plane, but with 3D electric
| fields extending above and below that plane.
|
| I think charge still goes to the surface, even if not
| induced. A spherical conductor with a net charge would have a
| uniform distribution of charge on its surface, and 0 charge
| density inside the bulk. Any net charge density inside the
| conductor would create a diverging electric field around
| itself, which would cause a current to flow, dissipating that
| net charge in the process. The same argument should apply to
| a bunch of free electrons, shouldn't it?
| contravariant wrote:
| Ah you're right by definition a conductor _does_ have a
| current provided there 's any electrical field inside it
| (divergent or not). So indeed they can't have charges
| inside them in equilibrium.
|
| However if you induce an electrical field in a vacuum then
| no charges at all will flow because a vacuum is the perfect
| insulator. And even if there is nonzero electric field in
| and around an individual electron they won't move as long
| as things cancel out at their exact position.
|
| Really the theory of conductors and charge densities seems
| to break down somewhat once you get to the point of
| individual electrons near absolute zero in a vacuum. One
| way to put this to the test would be to charge a conductor
| to its absolute limit, basically removing all free charges
| from it and seeing if the rest will crystalize. This might
| require an impractical amount of energy.
___________________________________________________________________
(page generated 2021-08-15 23:01 UTC)