Inflation Symmetry Research Articles

The Rat Thoracic Ultrasound protocol: scanning technique and normal findings.

Respiratory diseases (especially pneumonia) are very common disorders in pet rats. The suspected diagnosis is mostly based on the clinical signs, thoracic auscultation, and thoracic radiography. However, auscultation is insensitive in determining the severity of the disease, and radiographs are often unremarkable. Non-cardiac thoracic ultrasonography is increasingly used in veterinary medicine; however, it has not been described in detail in rats. Thoracic ultrasonic examination was conducted on 400 client-owned conscious pet rats. The rats were examined in the period from June 2023 to August 2023 in two veterinary clinics. Due to the small size of the animal, different anatomical considerations, and different evaluation protocols, as well as to meet the optimal outcome of detailed thoracic ultrasound, a standard methodological protocol was developed, and the name RATTUS (Rat Thoracic Ultrasound) was proposed. Typical signs of normal RATTUS were described (bat sign, lung sliding, A-lines, abdominal curtain sign, ski jump sign, lung pulse, seashore sign in M-mode, and bamboo sign). The new evaluation of lung inflation symmetry by substernal access was also described. The methodical approach presented and the normal findings description are proposed to be used for a standard/routine thoracic ultrasound examination in pet rats.

Alloy Quasicrystals: Perspectives and Some Open Questions at Forty Years

Four decades have elapsed since the first quasiperiodic crystal was discovered in the Al–Mn alloy system, and much progress has been made during this time on the science of quasicrystals (QCs). Notwithstanding this, a significant number of open questions still remain regarding both fundamental and technological aspects. For instance: What are QCs good for? How can we improve the current provisional QC definition? What is the role of the underlying quasiperiodic order and the characteristic inflation symmetry of these compounds in the emergence of their unusual physicochemical properties? What is the nature of chemical bonding in QCs formed in different sorts of materials such as alloys, oxides, or organic polymers? Herein these and other closely related issues are discussed from an interdisciplinary perspective as well as prospective future work in the field in the years to come.

Spectral properties of substitutions on compact alphabets

AbstractWe consider substitutions on compact alphabets and provide sufficient conditions for the diffraction to be pure point, absolutely continuous and singular continuous. This allows one to construct examples for which the Koopman operator on the associated function space has specific spectral components. For abelian bijective substitutions, we provide a dichotomy result regarding the spectral type of the diffraction. We also provide the first example of a substitution that has countably infinite Lebesgue spectral components and countably infinite singular continuous components. Lastly, we give a non‐constant length substitution on a countably infinite alphabet that gives rise to substitutive Delone sets of infinite type. This extends the spectral theory of substitutions on finite alphabets and Delone sets of finite type with inflation symmetry.

Inflation versus projection sets in aperiodic systems: the role of the window in averaging and diffraction.

Tilings based on the cut-and-project method are key model systems for the description of aperiodic solids. Typically, quantities of interest in crystallography involve averaging over large patches, and are well defined only in the infinite-volume limit. In particular, this is the case for autocorrelation and diffraction measures. For cut-and-project systems, the averaging can conveniently be transferred to internal space, which means dealing with the corresponding windows. In this topical review, this is illustrated by the example of averaged shelling numbers for the Fibonacci tiling, and the standard approach to the diffraction for this example is recapitulated. Further, recent developments are discussed for cut-and-project structures with an inflation symmetry, which are based on an internal counterpart of the renormalization cocycle. Finally, a brief review is given of the notion of hyperuniformity, which has recently gained popularity, and its application to aperiodic structures.

Super-coloured tilings: a novel class of two-dimensional limit-periodic structures

A two-dimensional (2D) periodic coloured tiling has a colour symmetry if it is invariant against a symmetry operation involving the change of colours. The idea of colour symmetry may also be adopted for another type of symmetry, namely the inflation symmetry for self-similar structures. A general method for constructing substitution rules that generate a class of 2D aperiodic coloured tilings is presented, where we assume that there are only two colours and a single shape for the tiles. These structures are limit-periodic, and we will call them super-coloured tilings (SCTs). SCTs form an important class of aperiodic ordered structures with a perfect long-range order. Since their aperiodicity relies on the colours of the tiles, an SCT reduces to a periodic tiling if the colours of the tiles are disregarded.

Atomic scale coexistence of periodic and quasiperiodic order in a 2-fold Al-Ni-Co decagonal quasicrystal surface

Decagonal quasicrystals are made of pairs of atomic planes with pentagonal symmetry periodically stacked along a 10-fold axis. We have investigated the atomic structure of the 2-fold surface of a decagonal Al-Ni-Co quasicrystal using scanning tunneling microscopy. The surface consists of terraces separated by steps of heights 1.9, 4.7, 7.8, and $12.6\phantom{\rule{0.3em}{0ex}}\mathrm{\AA{}}$ containing rows of atoms parallel to the 10-fold direction with an internal periodicity of $4\phantom{\rule{0.3em}{0ex}}\mathrm{\AA{}}$. The rows are arranged aperiodically, with separations that follow a Fibonacci sequence and inflation symmetry. The results indicate that the surfaces are preferentially Al-terminated and in general agreement with bulk models.

Inflation and wavelets for the icosahedral Danzer tiling

The distribution of atoms in quasi-crystals lacks periodicity and displays point symmetry associated with non-crystallographic modules. Often it can be described by quasi-periodic tilings on built from a finite number of prototiles. The modules and the canonical tilings of five-fold and icosahedral point symmetry admit inflation symmetry. In the simplest case of stone inflation, any prototile when scaled by the golden section number ? can be packed from unscaled prototiles. Observables supported on for quasi-crystals require symmetry-adapted function spaces. We construct wavelet bases on for the icosahedral Danzer tiling. The stone inflation of the four Danzer prototiles is given explicitly in terms of Euclidean group operations acting on . By acting with the unitary representations inverse to these operations on the characteristic functions of the prototiles, we recursively provide a full orthogonal wavelet basis of . It incorporates the icosahedral and inflation symmetry.

Quasiperiodicity in decagonal phases forced by inclined net planes?

It is generally assumed that decagonal quasicrystals show periodically arranged atomic layers only on net planes perpendicular to the tenfold axis and quasiperiodically arranged ones parallel to it. However, there also do exist only slightly puckered atomic layers that are periodically arranged and inclined to the tenfold axis. They coincide with the net planes of the periodic average structures of the decagonal phase and are related to the strongest Bragg reflections. Since they link quasiperiodic and periodic directions, inclined net planes may play a crucial role for growth and stabilization of decagonal quasicrystals. In fact, it is shown how ideal quasiperiodic long-range order and inflation symmetry allow for the existence of inclined net planes with small corrugation and reinforce the relation with the periodic average structures.

Vertex frequencies of the hierarchical dodecagonal tiling

The frequency of occurrence of a configuration, such as a vertex, is an important characteristic of a tiling. Conventionally one deduces this frequency from the volume of the corresponding acceptance domain (alias window) in perpendicular space. The method fails in the case of the Stampfli tiling which has a fractal window. We have overcome this difficulty by using the inflation symmetry of the square-triangle tiling.

The ‘random’ square–triangle tiling: simulation of growth

Some alloy systems, such as Ni–Cr, V–Ni–Si and Ta–Te, have quasicrystalline phases with 12-fold symmetry. These structures may be described in terms of dodecagonal tilings by equilateral triangles and squares. The formation of quasicrystals still poses a problem, since local information is insufficient for the construction of a perfect quasiperiodic structure. The growth of real quasicrystals may be due to several mechanisms. We have simulated the growth of a quasicrystal from a melt, consisting of squares and equilateral triangles of equal edge length. We are interested in the abundancies of the vertex configurations formed, both regular and defective. Unrestricted random growth tends to result in segregation of triangles from squares. Favoring triangles to attract squares and vice versa brings about nearly perfect patterns with nearly perfect vertex abundancies, as well as realistic defect concentrations. We have also calculated the exact vertex frequencies of the ideal square–triangle tiling by relying on inflation symmetry.

Modulated states in Mg–Al alloys and classical Frank–Kasper phases: a high resolution X-ray diffraction study

The Mg–Al system shows for composition in the 50–60at.% Al range several states, either periodic or quasiperiodic, related to tetrahedral and icosahedral motif structures. The Mg–Al (61at.%) rapidly solidified alloy shows a cubic quasiperiodic state with inflation symmetry characterized by the algebric number 2−√3. The main question is to know if this state is an example of a new kind of quasicrystal or if this inflation property is only due to a continuous variation of the wavenumber. We report high resolution X-ray diffraction studies of this Mg–Al (61at.%) metastable state and the related stable Frank–Kasper phase (β-phase). These experiments clearly show fluctuations of the wavenumber. The quasiperiodic Mg–Al (61at.%) state appears thus to have typical features of incommensurately modulated phases, but with a persisting connection with Frank–Kasper phases.

Modulation parameters in incommensurate modulated structures with inflation symmetry

Inflation symmetry is one of the peculiar features of the diffraction pattern of a quasicrystal. However, it is not an exclusive property of quasicrystalline structures and it may be present in incommensurately modulated structures, as shown recently in the Al-Mg system (Donnadieu P et al 1996 J. Physique I 6 1153-64). The conditions that a single modulation parameter of an incommensurate structure must fulfil in order to have inflation symmetry are determined. Although the number of possible distinct inflation-symmetric quasilattices is infinite, from physical/experimental arguments it can be concluded that, in practice, only a few of them can be experimentally observed, the reported phase of the Al-Mg system being one of these particular cases. A quantitative criterion to classify the modulation parameters that give rise to quasilattices with observable inflation symmetry is proposed. The generalization of the analysis of incommensurate structures with more than one single modulation parameter is also discussed. Finally, the inflation parameters of diffraction patterns with rotational point groups of finite order, CN, are compared with the parameters of the one-dimensional case.

Geometric Models for Quasicrystals I. Delone Sets of Finite Type

This paper studies three classes of discrete sets X in $\Bbb R$ n which have a weak translational order imposed by increasingly strong restrictions on their sets of interpoint vectors X-X . A finitely generated Delone set is one such that the abelian group [X-X] generated by X-X is finitely generated, so that [X-X] is a lattice or a quasilattice. For such sets the abelian group [X] is finitely generated, and by choosing a basis of [X] one obtains a homomorphism $\varphi : [X] \rightarrow {\Bbb Z}^s$ . A Delone set of finite type is a Delone set X such that X-X is a discrete closed set. A Meyer set is a Delone set X such that X-X is a Delone set. Delone sets of finite type form a natural class for modeling quasicrystalline structures, because the property of being a Delone set of finite type is determined by ``local rules.'' That is, a Delone set X is of finite type if and only if it has a finite number of neighborhoods of radius 2R , up to translation, where R is the relative denseness constant of X . Delone sets of finite type are also characterized as those finitely generated Delone sets such that the map ϕ satisfies the Lipschitz-type condition ||ϕ (x) - ϕ (x')|| < C ||x - x'|| for x, x' ∈X , where the norms || . . . || are Euclidean norms on $\Bbb R$ s and $\Bbb R$ n , respectively. Meyer sets are characterized as the subclass of Delone sets of finite type for which there is a linear map $\tilde{L} : {\Bbb R}^n \rightarrow {\Bbb R}^s$ and a constant C such that ||ϕ (x) - $\tilde{L}$ (x)|| $\leq C$ for all x∈X . Suppose that X is a Delone set with an inflation symmetry, which is a real number η > 1 such that $\eta X \subseteq X$ . If X is a finitely generated Delone set, then η must be an algebraic integer; if X is a Delone set of finite type, then in addition all algebraic conjugates | η ' | $\leq$ η; and if X is a Meyer set, then all algebraic conjugates | η ' | $\leq$ 1.

Atomic and electronic structure of quasiperiodic and crystalline Mg-61 at.% Al alloys

Mg-Al alloys with composition Mg-61 at.% Al display two types of structure: a stable crystalline state (the so-called β-Mg2Al3) and a metastable one showing both quasiperiodicity and inflation symmetry. The atomic as well as electronic structures of stable crystalline β-Mg2Al3 and a metastable Mg39Al61 phase that exhibits quasiperiodicity and inflation symmetry have been studied thanks to x-ray spectroscopy techniques. Extended x-ray absorption fine-structure experiments performed above the Al and Mg K absorption edges have probed the local atomic order around Mg as well as Al atoms. X-ray emission and absorption spectroscopies have investigated occupied and unoccupied electronic states around Mg and Al. The local order has been found to be the same around Al atoms in both alloys, whereas around Mg atoms differences are seen for higher shells than first neighbours. This suggests that the same clusters must be involved in both phases and that the quasiperiodicity is connected with modification of the Mg atom environment forming the linkage between large clusters. On the other hand, the electronic distributions show differences from β-Mg2Al3 to the quasiperiodic phase that are consistent with the local order studies. A pseudogap is observed in the Mg and Al electronic structures which is more marked in the quasiperiodic phase than the crystalline alloy. By analogy to quasicrystals, we suggest the enhancement observed is due to the occurrence of an inflation mechanism in the arrangement of the clusters in the quasiperiodic alloy with respect to the β-Mg2Al3 crystal.

Cubic superspace symmetry and inflation rules in metastable MgAl alloy

The diffraction properties of a quenched Al-Mg alloy which has been recently termed as a “cubic quasicrystal” are quantitatively reanalyzed. It is shown that the phase can be interpreted within the superspace formalism as an ordinary incommensurately modulated structure. The cubic six-dimensional superspace group that describes its symmetry properties has been determined. The additional inflation symmetry features exhibited by the diffraction diagram can be summed up by its invariance for the inflation factor \(\), but this property has its origin in the specific value of the modulus of the modulation wave vectors, which is composition dependent. Other particular values of this modulus can give rise to similar scaling properties. Further experiments are required to ellucidate if the mentioned inflation symmetry is a fortuitous situation in a composition dependent wave vector, or has indeed the physical significance which would allow to describe the system as a “cubic quasicrystal”.

Exact eigenstates of tight-binding Hamiltonians on the Penrose tiling

We investigate exact eigenstates of tight-binding models on the planar rhombic Penrose tiling. We consider a vertex model with hopping along the edges and the diagonals of the rhombi. For the wave functions, we employ an ansatz, first introduced by Sutherland, which is based on the arrow decoration that encodes the matching rules of the tiling. Exact eigenstates are constructed for particular values of the hopping parameters and the eigenenergy. By a generalized ansatz that exploits the inflation symmetry of the tiling, we show that the corresponding eigenenergies are infinitely degenerate. Generalizations and applications to other systems are outlined.

Thermal Conductivity of Quasicrystals and Associated Processes

Quasicrystals (QCs) exhibit unusual physical properties that are significantly different from those of crystalline materials and are not expected for alloys consisting of normal metallic elements. Opposite to conventional metallic alloys, their thermal conductivity and diffusivity are unusually low (with a positive temperature coefficient)—practically that of an insulator—which is atypical for materials containing about 70-at.% aluminum. Moreover the thermal conductivity decreases when the structural perfection is improved. One observes a low, if any, electronic contribution to the heat capacity and thus a vanishing density of electronic states at the Fermi level.The origin of this unexpected behavior was first attributed to the existence of a deep pseudogap at the Fermi level with a localization tendency of electrons near the Fermi level.However experimental evidence led to an alternative approach related to the structure of quasicrystals. In QCs, well-defined atomic clusters form self-similar subsets of the structure over which electronic and vibrational states are expected to extend. According to the inflation symmetry of the icosahedral structure, the so-called recurrent localization effects may then explain the conduction behavior and other striking features of quasicrystals (e.g., brittleductile transition at high temperature, corrosion resistance, low friction, high hardness).In the following, we first present the main thermal properties of quasicrystalline alloys compared to those of conventional materials with an emphasis on the variation of the thermal conductivity with temperature. The combination of such peculiar conduction, mechanical, and tribological properties gives the quasicrystalline alloys a technological interest for applications where superficial thermal and mechanical conditions are of prime importance. This is illustrated with two examples involving a QC coating on a base Al substrate: (1) thermal insulation for which a low conductivity is needed and (2) quenching heat-transfer modification due to a low-effusivity superficial effect. These processes are then explained in the third part of this article in terms of the cluster-modes delocalization mechanism responsible for the low conductivity of the quasicrystals.

Magnetic states induced by electron-electron interactions in a plane quasiperiodic tiling

We consider the Hubbard model for electrons in a two-dimensional quasiperiodic tiling using the Hartree--Fock approximation. Numerical solutions are obtained for the first three square approximants of the perfect octagonal tiling. At half-filling, the magnetic state is antiferromagnetic. We calculate the distributions of local magnetizations and their dependence on the local environments as U (Hubbard repulsion strength) is varied. The inflation symmetry of quasicrystals results in a corresponding inflation symmetry of the magnetic configurations when passing from one tiling to the next in the progression towards the infinite quasicrystal.

Representation of certain self-similar quasiperiodic tilings with perfect matching rules by discrete point sets

Simple quasiperiodic tilings with 8-fold and 12-fold symmetry are presented that possess local de-/inflation symmetry and perfect matching rules. The special feature of these tilings is that the full information is already derivable from the set of vertex sites alone. This means that the latter is a valid representative of the corresponding equivalence class of mutual local derivability

Absence of inflation symmetric commensurate states in inflation symmetric networks

In this paper we present measurements of the normal-to superconducting phase boundary T c (H) for three different networks possessing inflation symmetry. Fluxoid quantization constraints induce the formation of a lattice of fluxoid quanta for any non-zero perpendicular magnetic field, and at particular fields, T c (H) exhibits cusp-like structure indicating that the lattice is commensurate with the underlying network geometry. For inflation symmetric netrvorks studied in the past, all commensurate states have always been related to the inflation symmetry. In the three networks studied here, none of the commensurate phase boundary structure derives from the inflation symmetry. We propose that this structure can instead be understood by considering the Fourier transform of the network geometry and that the transform actually provides a more universal prescription for the identification of commensurate states. The relevance of the transform (as opposed to the inflation symmetry) in determining commensurate states in two dimensions is consistent with analytical work performed for one dimensional systems