A bstract The Kähler potentials of modular symmetry models receive unsuppressed contributions which may be controlled by a flavor symmetry, where the combination of the two symmetry types is referred to as eclectic flavor symmetry. After briefly reviewing the consistency conditions of eclectic flavor symmetry models, including those with generalised (g)CP, we perform a comprehensive bottom-up study of eclectic flavor symmetry models based on Ω(1) ≅ ∆(27) ⋊ T ′, consisting of the flavor symmetry ∆(27) in a semi-direct product with the modular symmetry T ′. The modular transformations of different ∆(27) multiplets are given by solving the consistency condition. The eight nontrivial singlets of ∆(27) are related by T ′ modular symmetry, and they have to be present or absent simultaneously in any Ω(1) model. The most general forms of the superpotential and Kähler potential invariant under Ω(1) are discussed, and the corresponding fermion mass matrices are presented. Based on the eclectic flavor group Ω(1), two concrete lepton models which can successfully describe the experimental data of lepton masses and mixing parameters are constructed. For the two models without gCP, all six mixing parameters vary in small regions. A nearly maximal atmospheric mixing angle θ 23 and Dirac CP phase δ CP are obtained in the first model. After considering the compatible gCP symmetry and the assumption of $$ \mathfrak{R}\tau $$ R τ = 0 in the first model, the μ − τ reflection symmetry is preserved in the charged lepton diagonal basis. As a consequence, the atmospheric mixing angle and Dirac CP phase are predicted to be maximal, and two Majorana CP phases are predicted to be π .
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Symmetries in hard polygon systems determine plastic colloidal crystal mesophases in two dimensions
Orientational ordering is a necessary step in the crystallization of molecules and anisotropic colloids. Plastic crystals, which are possible mesophases between the fluid and fully ordered crystal, are translationally ordered but exhibit no long range orientational order. Here, we study the two-dimensional phase behavior of hard regular polygons with edge number n = 3–12. This family of particles provides a model system to isolate the effect of shape and symmetry on the existence of plastic crystal phases. We show that the symmetry group of the particle, G , and the symmetry group of the local environment in the crystal, H , together determine plastic colloidal crystal phase behavior in two dimensions. If G contains completely the symmetry elements of H , then a plastic crystal phase is absent. If G and H share some but not all nontrivial symmetry elements, then a plastic crystal phase exists with preferred particle orientations that recover the absent symmetry elements of the crystal; we call this phase the discrete plastic crystal phase. If G and H share no nontrivial symmetry elements, then a plastic crystal phase exists without preferred orientations, which we call an indiscrete plastic crystal.
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- Award ID(s):
- 1409620
- NSF-PAR ID:
- 10090824
- Date Published:
- Journal Name:
- Soft Matter
- Volume:
- 15
- Issue:
- 12
- ISSN:
- 1744-683X
- Page Range / eLocation ID:
- 2571 to 2579
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract The deformation of crystalline materials by dislocation motion takes place in discrete amounts determined by the Burgers vector. Dislocations may move individually or in bundles, potentially giving rise to intermittent slip. This confers plastic deformation with a certain degree of variability that can be interpreted as being caused by stochastic fluctuations in dislocation behavior. However, crystal plasticity (CP) models are almost always formulated in a continuum sense, assuming that fluctuations average out over large material volumes and/or cancel out due to multi-slip contributions. Nevertheless, plastic fluctuations are known to be important in confined volumes at or below the micron scale, at high temperatures, and under low strain rate/stress deformation conditions. Here, we develop a stochastic solver for CP models based on the residence-time algorithm that naturally captures plastic fluctuations by sampling among the set of active slip systems in the crystal. The method solves the evolution equations of explicit CP formulations, which are recast as stochastic ordinary differential equations and integrated discretely in time. The stochastic CP model is numerically stable by design and naturally breaks the symmetry of plastic slip by sampling among the active plastic shear rates with the correct probability. This can lead to phenomena such as intermittent slip or plastic localization without adding external symmetry-breaking operations to the model. The method is applied to body-centered cubic tungsten single crystals under a variety of temperatures, loading orientations, and imposed strain rates.
- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1039/C9SM00016J
