Symmetry-breaking bifurcations on cubic lattices

Callahan, T. K.; Knobloch, Edgar

doi:10.1088/0951-7715/10/5/009

Cited by 37 publications

(31 citation statements)

References 19 publications

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“…The point group of Γ y0 contains the 2π/3 rotations around the vertical axis, (α − ) 2 and (α − ) 4 , and the reflections on vertical planes (β − )(α − ), (β − )(α − ) 3 , (β − )(α − ) 5 . The point group of Σ y0 is α 2 , βα .…”

Section: Restrictionmentioning

confidence: 99%

On the projection of functions invariant under the action of a crystallographic group

Pinho

Labouriau

2014

Journal of Pure and Applied Algebra

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Abstract. We study functions defined in (n + 1)-dimensional domains that are invariant under the action of a crystallographic group. We give a complete description of the symmetries that remain after projection into an ndimensional subspace and compare it to similar results for the restriction to a subspace. We use the Fourier expansion of invariant functions and the action of the crystallographic group on the space of Fourier coefficients. Intermediate results relate symmetry groups to the dual of the lattice of periods.

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Section: Restrictionmentioning

confidence: 99%

On the projection of functions invariant under the action of a crystallographic group

Pinho

Labouriau

2014

Journal of Pure and Applied Algebra

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“…Since we found body-centered cubic crystals in Fig. 2(b), and since these cannot be represented in terms of the icosahedral basis vectors, we compute their free energy as a separate calculation, choosing a different set of basis vectors [34]. Figure 4 shows regions in the ðμ; νÞ plane, identifying the globally stable solution in each region.…”

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confidence: 99%

Three-Dimensional Icosahedral Phase Field Quasicrystal

et al. 2016

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We investigate the formation and stability of icosahedral quasicrystalline structures using a dynamic phase field crystal model. Nonlinear interactions between density waves at two length scales stabilize threedimensional quasicrystals. We determine the phase diagram and parameter values required for the quasicrystal to be the global minimum free energy state. We demonstrate that traits that promote the formation of two-dimensional quasicrystals are extant in three dimensions, and highlight the characteristics required for three-dimensional soft matter quasicrystal formation. DOI: 10.1103/PhysRevLett.117.075501 Periodic crystals form ordered arrangements of atoms or molecules with rotation and translation symmetries, and possess discrete x-ray diffraction patterns, or equivalently, discrete spatial Fourier spectra. In contrast, quasicrystals (QCs) lack the translational symmetries of periodic crystals, yet also display discrete spatial Fourier spectra. QCs made from metal alloys were discovered in 1982 [1] and attracted the Nobel prize for chemistry in 2011. QCs can be quasiperiodic in all three dimensions (e.g., with icosahedral symmetry), or can be quasiperiodic in two (or one) directions while being periodic in one (or two). The vast majority of the QCs discovered so far are metallic alloys (e.g., Al/Mn or Cd/Ca). However, QCs have recently been found in nanoparticles [2], mesoporous silica [3], and soft matter [4] systems. The latter include micellar melts [5,6] formed, e.g., from linear, dendrimer or star block copolymers. Recently, three-dimensional (3D) icosahedral QCs have been found in molecular dynamics simulations of particles interacting via a three-well pair potential [7].In recent years, model systems in two dimensions (2D) have been studied in order to understand soft matter QC formation and stability [8][9][10][11][12]. Phase field crystal models have been employed to simulate the growth of 2D QCs [13] and the adsorption properties on a quasicrystalline substrate [14]. The ingredients for 2D quasipattern formation are, first, a propensity towards periodic density modulations with two characteristic wave numbers k 1 and k 2 [15-18]. The ratio k 2 =k 1 must be close to certain special values; e.g., for dodecagonal QCs the value is 2 cosðπ=12Þ. Second, strong reinforcing (i.e., resonant) nonlinear interactions between these two characteristic density waves are required [17,19,20]. Earlier work on quasipatterns observed in Faraday wave experiments reveals similar requirements [19,[21][22][23]. We demonstrate here, following Mermin and Troian [24], that these same requirements suffice to stabilize icosahedral QCs in 3D. In contrast, nonlinear resonant interactions between density waves at a single wavelength are important in stabilizing simple crystal structures, such as body-centered cubic (bcc) crystals [25] although, with the right coupling, QCs can also be stabilized [26].We consider a 3D phase field crystal (PFC) model, appropriate for soft matter systems, that generates modulations with two l...

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“…In Golubitsky and Stewart [8, chapter 5] there is a complete description of this method used, for example, in Dionne and Golubitsky [6], Dionne [5], Bosch Vivancos et al [19], Callahan and Knobloch [4] and Dionne et al [7], where the spatially periodic patterns are sometimes called planforms.…”

Section: Introductionmentioning

confidence: 99%

Hexagonal projected symmetries

2015

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Abstract. In the study of pattern formation in symmetric physical systems a 3-dimensional structure in thin domains is often modelled as 2-dimensional one. We are concerned with functions in R 3 that are invariant under the action of a crystallographic group and the symmetries of their projections into a function defined on a plane. We obtain a list of the crystallographic groups for which the projected functions have a hexagonal lattice of periods. The proof is constructive and the result may be used in the study of observed patterns in thin domains, whose symmetries are not expected in 2-dimensional models, like the black-eye pattern.

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Symmetry-breaking bifurcations on cubic lattices

Cited by 37 publications

References 19 publications

On the projection of functions invariant under the action of a crystallographic group

On the projection of functions invariant under the action of a crystallographic group

Three-Dimensional Icosahedral Phase Field Quasicrystal

Hexagonal projected symmetries

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