277 research outputs found
The hard-core model on and Kepler's conjecture
We study the hard-core model of statistical mechanics on a unit cubic lattice
, which is intrinsically related to the sphere-packing problem
for spheres with centers in . The model is defined by the sphere
diameter which is interpreted as a Euclidean exclusion distance between
point particles located at spheres centers. The second parameter of the
underlying model is the particle fugacity . For the ground states of
the model are given by the dense-packings of the spheres. The identification of
such dense-packings is a considerable challenge, and we solve it for as well as for , where
. For the former family of values of our proofs are
self-contained. For our results are based on the proof of
Kepler's conjecture. Depending on the value of , we encounter three
physically distinct situations: (i) finitely many periodic ground states, (ii)
countably many layered periodic ground states and (iii) countably many not
necessarily layered periodic ground states. For the first two cases we use the
Pirogov-Sinai theory and identify the corresponding periodic Gibbs
distributions for and , ,
in a high-density regime , where the system is ordered and tends to
fluctuate around some ground states. In particular, for only a finite
number out of countably many layered periodic ground states generate pure
phases
Minimal Area of a Voronoi Cell in a Packing of Unit Circles
We present a new self-contained proof of the well-known fact that the minimal
area of a Voronoi cell in a unit circle packing is equal to , and
the minimum is achieved only on a perfect hexagon. The proof is short and, in
our opinion, instructive
High-density hard-core model on and norm equations in ring
We study the Gibbs statistics of high-density hard-core configurations on a
square lattice , for a general Euclidean exclusion distance .
The key point is an analysis of solutions to norm equations in the ring
. We describe the ground states in terms of
M-triangles, i.e., non-obtuse -triangles of a minimal area with
the side-lengths . Further, we identify -triangles as
elements of .
First, there is a finite class (Class S) formed by values generating
sliding, a phenomenon leading to countable families of periodic ground states.
We identify all with sliding. Each of the remaining classes is proven to
be infinite; they are characterized by uniqueness or non-uniqueness of a
minimal triangle for a given , up to -congruencies. For
values of with uniqueness (Class A) we describe the periodic ground
states as admissible triangular sub-lattices of
maximum density. By using the Pirogov-Sinai theory, it allows us to identify
the extreme Gibbs measures (pure phases) for large values of fugacity and
describe symmetries between them.
Next, we analyze the values with non-uniqueness. For some all
M-triangles are -congruent but not -congruent
(Class B0). For other values of there exist
non--congruent M-triangles, with different collections of
side-lengths (Class B1). Moreover, there are values for which both cases
occur (Class B2). The phase diagram for Classes B0, B1, B2 is determined by
dominant ground states. Classes A, B0-B2 are described in terms of cosets in
by the group of units.Comment: Supplemen
Observation of Anomalous Internal Pair Creation in Be: A Possible Signature of a Light, Neutral Boson
Electron-positron angular correlations were measured for the isovector
magnetic dipole 17.6 MeV state (, ) ground state
(, ) and the isoscalar magnetic dipole 18.15 MeV (,
) state ground state transitions in Be. Significant
deviation from the internal pair creation was observed at large angles in the
angular correlation for the isoscalar transition with a confidence level of . This observation might indicate that, in an intermediate step, a
neutral isoscalar particle with a mass of 16.70 (stat)
(sys) MeV and was created.Comment: 5 pages, 5 figure
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