Phason Space Research Articles

Higher-order topological sound transport in synthetic spaces

We design a four-dimensional system with straight waveguides composed of coordinate-dependent coupled waveguides. By controlling the phase in synthetic space and the excitation of the corresponding sound source mode, we show that the pumping incorporating adiabatic modulation of discretely varied coupling waveguides along a given orbit in the phason space occurs only in specific space directions. In addition, robust corner-to-corner topological pumping is achieved in finite-sized two-dimensional modulated acoustic structures. Our results provide a general strategy to achieve topological pumping through discrete property modulations and pave new paths for studying higher-dimensional topological phases in discrete systems.

Phason space analysis and structure modelling of 100 Å-scale dodecagonal quasicrystal in Mn-based alloy

The dodecagonal quasicrystal classified into the five-dimensional space group P126/mmc, recently discovered in a Mn–Cr–Ni–Si alloy, has been analysed using atomic-resolution spherical aberration-corrected electron microscopy, i.e. high-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM) and conventional transmission electron microscopy. By observing along the 12-fold axis, non-periodic tiling consisting of an equilateral triangle and a square has been revealed, of which common edge length is a = 4.560 Å. These tiles tend to form a network of dodecagons of which size is a ≈ 17 Å in diameter. The tiling was interpreted as an aggregate of 100 Å-scale oriented domains of high- and low-quality quasicrystals with small crystallites appearing at their boundaries. The quasicrystal domains exhibited a densely filled circular acceptance region in the phason space. This is the first observation of the acceptance region in an actual dodecagonal quasicrystal. Atomic structure model consistent with the electron microscopy images is a standard Frank-Kasper decoration of the triangle and square tiles that can be inferred from the crystal structures of Zr4Al3 and Cr3Si. Four kinds of layers located at z = 0, ±1/4 and 1/2 are stacked periodically along the 12-fold axis, and the atoms at z = 0 and 1/2 form hexagonal anti-prisms consistently with the 126-screw axis. The validity of this structure model was examined by means of powder X-ray diffraction.

Spatial decay of the phason field in quasicrystal linear elasticity

We present numerical simulations showing that a conservative phason self-action, i.e. the stationary part of the phason diffusion, promotes in quasicrystal linear elasticity the phason spatial decay. The proof of the possible existence of such a phason self-action is based on the invariance requirements of the external power of all actions over a quasicrystal, with respect to isometry-based changes in observers in the physical and phason spaces. The numerical result confirms a 2011 theoretical prevision by Colli and Mariano.

Ising spin orders and magnetic interactions analyzed in phason space in a 2-dimensional Penrose lattice

Simulated annealing calculations were performed on a Penrose lattice with the aim of clarifying the relation between the interaction connections and the magnetic orders in the phason space. Alternating signs were assumed in the magnetic interactions up to the second shortest distances between spins. Two different types of long-range antiferromagnetic orders were found for two cases with reversed sign of the interactions. The two types of order were most adequately interpreted as the orders of subdivisions in the occupation domain in the phason space. Reasonable interpretations of the mechanism of the subdivisions are given on the basis of the physical interactions projected to the phason space.

Simulation study on antiferromagnetic order of Ising spins in a Zn–Mg–Ho model structure

A long-range antiferromagnetic order was studied on Ising spins in a Zn–Mg–Ho model structure for Ruderman–Kittel–Kasuya–Yosida (RKKY)-like alternating magnetic interaction in simulated annealing calculations. The order showed sharp magnetic diffraction peaks, with similar widths as ordinary lattice diffraction, but diminished intensity of the central peak. This indicated a long-range antiferromagnetic order. The order was most suitably interpreted as a separation of the occupation domain in the phason space. The energetic mechanism of the order was qualitatively analysed in magnetic interaction connections in the phason space.

Ising Spin Orders Conforming to Local Matching Rule and Long-Range Regularity of Interactions in Penrose Lattice

A simple arithmetic rule is proposed to correlate the matching rule in Penrose lattice with long-range Ising spin orders with the aim of clarifying the energetic mechanism of the orders under Ruderman–Kittel–Kasuya–Yoshida-like (RKKY) alternating interactions. Three different types of antiferromagnetic orders were found to be compatible with the rule. The orders showed sharp magnetic diffraction spots with the same peak widths as the ordinary lattice diffraction, which were consistent with the long-range orders enforced by the matching rule. The three types of the orders were most adequately interpreted as the orders in the occupation domain in the phason space. The RKKY interactions resulted in two of the orders in simulated annealing calculations. The results indicate the importance in the energetic mechanism of the consistency of the connectivity of the interaction signs inherent in the quasiperiodic structure.

Quasiperiodic long-range ferrimagnetic order in Ising model simulation in an icosahedral quasicrystal model structure

A new type of ferrimagnetic order was found in a Monte Carlo simulation of quasiperiodic magnetic system at low temperatures, which includes Ising spins arranged on the F-type quasiperiodic lattice. The spin occupation domains (or acceptance regions) in the phason space revealed the new type of ferrimagnetism consisting of two subdomains corresponding to up- and down-spins, which definitely indicated the quasiperiodic long-range nature of this ferrimagnetic order. A structure model of the Zn–Mg–Ho icosahedral phase was adopted as a three-dimensional quasicrystal model for this simulation. Ruderman–Kittel–Kasuya–Yosida (RKKY)-like alternating exchange interaction was assumed to extend to the fourth shortest distance. The elastic magnetic diffraction patterns from spin-dependent scattering consisted of sharp spots indicating a long-range order of the magnetic structure. Each diffraction spot could be indexed by six integers of all odd or all even, and this fact indicated F-type symmetry of the ferrimagnetic order which reflects the F-type symmetry of the structure.

Elastic theory of icosahedral quasicrystals - application to straight dislocations

In quasicrystals, there are not only conventional, but also phason displacement fields and associated Burgers vectors. We have calculated approximate solutions for the elastic fields induced by two-, three- and fivefold straight screw- and edge-dislocations in infinite icosahedral quasicrystals by means of a generalized perturbation method. Starting from the solution for elastic isotropy in phonon and phason spaces, corrections of higher order reflect the two-, three- and fivefold symmetry of the elastic fields surrounding screw dislocations. The fields of special edge dislocations display characteristic symmetries also, which can be seen from the contributions of all orders.

Static fluctuation distributions and their influence on diffraction patterns

Probability distributions of fluctuations of atomic positions around scattering vector dependent reference lattice point have been calculated for various crystalline and quasicrystalline structures and their influence on the diffraction pattern has been discussed. For incommensurate scattering vectors flat fluctuation distributions are found, which change drastically when the scattering vector approaches the diffraction peak position. Moments of these distributions have been used to calculate the diffraction pattern. Limitations of the Debye-Waller approximation are discussed and a new more accurate method of peak intensity estimation is proposed and tested for real space and phason space fluctuations.

New approach to diffraction pattern calculation

Abstract A new method of calculating diffraction patterns is proposed and tested on quasicrystals. Using an appropriate defined reference lattice the structure factor can be well approximated by a rapidly convergent series of variable u, which describes the nearest distance between atomic positions and points of the reference lattice. Only the first few terms are significant for the diffraction pattern calculation. The possibility of using the Debye-Waller approximation is also discussed. In this case an appropriate shift of the reference structure is required. Calculations based on the Debye-Waller formula in real and phason spaces give similar results.

Why do quasicrystals have sharp diffraction peaks?

The diffraction patterns of various two-dimensional structures have been analyzed in both real and phason spaces. It is shown that existence of diffraction peaks is strictly related to fluctuations in the variable u describing the nearest distance between atoms of the real structure and a one-dimensional reference lattice. Only if the minimum value of the second moment of the distribution of variable u is independent of the number of atoms and is lower than π 2/3 k 2, is a Bragg peak observed for a particular scattering vector k.

Use of the Debye-Waller approximation in diffraction-pattern calculation

A new method of diffraction-pattern calculation is proposed and tested on quasicrystals. With use of an appropriately defined reference lattice, a structure factor can be well approximated by a rapidly convergent series expansion of a variable u that describes nearest distances between atomic positions and points of the reference lattice. Only the first few terms are significant for difffraction-pattern calculation. The possibility of using the Debye–Waller approximation is discussed. In this case an appropriate shift of the reference structure is required. Calculations based on the Debye–Waller formula in real and phason spaces give similar results.