Crystals and quasicrystals: A continuum model

Radin, Charles

doi:10.1007/bf01205933

Cited by 19 publications

(18 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(ii) The ground states for n ≤ 29 can also be characterized, but due to the smallness of the structures, more degeneracies can occur. In particular, for n = 8, 9,12,15,18,21,29 there might be one octagon at the boundary. (We refer to Remark 7.11 for more details.)…”

Section: Resultsmentioning

confidence: 99%

“…For n = 10, 11, the presence of an octagon is excluded by Proposition 7.9 and the fact that each hexagon can share at most 2 atoms with an octagon (see Lemma 7.5 and Lemma 7.6). Consequently, for n ≤ 29, ground states may contain a boundary octagon only for n = 8,9,12,15,18,21,29. This is indeed possible as shown in Fig.…”

Section: Claimmentioning

confidence: 91%

“…To the best of our knowledge, the present paper provides a first rigorous mathematical crystallization result for two-dimensional molecular compounds. Let us mention that in case of different types of particles, simulations are abundant, but rigorous results seem to be limited to [3,6,29]. We also refer to [4] for a recent study about optimal charge distributions on Bravais lattices.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Crystallization in the hexagonal lattice for ionic dimers

Friedrich

Kreutz

2019

Math. Models Methods Appl. Sci.

View full text Add to dashboard Cite

We consider finite discrete systems consisting of two different atomic types and investigate ground-state configurations for configurational energies featuring two-body shortranged particle interactions. The atomic potentials favor some reference distance between different atomic types and include repulsive terms for atoms of the same type, which are typical assumptions in models for ionic dimers. Our goal is to show a two-dimensional crystallization result. More precisely, we give conditions in order to prove that energy minimizers are connected subsets of the hexagonal lattice where the two atomic types are alternately arranged in the crystal lattice. We also provide explicit formulas for the ground-state energy. Finally, we characterize the net charge, i.e., the difference of the number of the two atomic types. Analyzing the deviation of configurations from the hexagonal Wulff shape, we prove that for ground states consisting of n particles the net charge is at most of order O(n 1/4 ) where the scaling is sharp.2010 Mathematics Subject Classification. 82D25.

show abstract

Section: Resultsmentioning

confidence: 99%

Section: Claimmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Crystallization in the hexagonal lattice for ionic dimers

Friedrich

Kreutz

2019

Math. Models Methods Appl. Sci.

View full text Add to dashboard Cite

show abstract

“…A related extension for statistical models on quasiperiodic lattices seems even more realistic; however, the statistical mechanics on such lattices is quite different and is actually under investigation. The case of periodic hamiltonians having no periodic ground states (see [16]) is treated by using Peierls argument in [13]; this case is not considered here. It would be interesting to extend the theory in the directions mentioned above.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Extension of the Pirogov-Sinai theory to a class of quasiperiodic interactions

1988

View full text Add to dashboard Cite

Abstract. We extend the Pirogov-Sinai theory in such a manner that it applies to a large class of models with small quasiperiodic interactions as perturbations of periodic ones. We find general diophantine conditions on the frequency module of the quasiperiodic interactions and derivability conditions on the interaction potentials. These conditions allow to prove that the low temperature phase diagram is a homeomorphic deformation of the phase diagram at zero temperature.

show abstract

“…For any potential, the set of ground-state configurations is nonempty but it may not contain any periodic configurations [16,17,18,19,20,21,22,23]. We restrict ourselves to systems in which, although all ground-state configurations are nonperiodic, there is a unique translation-invariant probability measure supported by them.…”

Section: Classical Lattice Gas Models and Nonperiodic -Ground Statesmentioning

confidence: 99%

Stable Quasicrystalline Ground States

Miękisz¹

1997

Journal of Statistical Physics

View full text Add to dashboard Cite

Abstract. We give a strong evidence that noncrystalline materials such as quasicrystals or incommensurate solids are not exceptions but rather are generic in some regions of a phase space. We show this by constructing classical lattice gas models with translationinvariant, finite-range interactions and with a unique quasiperiodic ground state which is stable against small perturbations of two-body potentials. More generally, we provide a criterion for stability of nonperiodic ground states.

show abstract

Crystals and quasicrystals: A continuum model

Abstract: We construct the first model of particles in the plane with completely symmetric, short range, two body interactions which has quasiperiodic, but no periodic, ground states.

Cited by 19 publications

References 22 publications

Crystallization in the hexagonal lattice for ionic dimers

Crystallization in the hexagonal lattice for ionic dimers

Extension of the Pirogov-Sinai theory to a class of quasiperiodic interactions

Stable Quasicrystalline Ground States

Contact Info

Product

Resources

About