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Papers by Shelomo I Ben-Abraham
Physical Review A, 1976
The effects of nonlinearity in nematic liquid crystals subject to electric and magnetic fields ar... more The effects of nonlinearity in nematic liquid crystals subject to electric and magnetic fields are emphasized. For the Freedericks transition, the distortion profile is calculated exactly. For the domain modes, the torque equation can be approximated to a high degree by the nonlinear sine-Gordon equation. An exact twodimensional periodic solution is found whose consequences are in good quantitative agreement with all experimental observations. The possibility of a first-order transition to a vortex pattern is predicted.
Acta Physica Polonica A, 2014
We recapitulate our previously developed recursive algorithm for creating paperfolding structures... more We recapitulate our previously developed recursive algorithm for creating paperfolding structures in arbitrary dimensions. Here we explain and apply it specically to three and four dimensions. We visualize the results by suitable projections. We also explicitly enumerate the number of folds in the rst four generations of the recursion. We conjecture (without proof) that the Fourier spectrum of the structures is pure point (the Bragg peaks).
Abstract.- The curvature $ of an electronematodynamic flow pattern obeys, in the steady state, th... more Abstract.- The curvature $ of an electronematodynamic flow pattern obeys, in the steady state, the nonlinear equation $ " + qZ −t,h3+b- t,h3 + b−t,h3+b$ ' = 0. The term $ $ I, called squint, is studied. Squint originates in the distortion torque and the distortion stress. Squint brakes the axial inversion symmetry of a simple flow roll but preserves the reflection symmetry of a roll pair. Flow patterns are predicted to show an observable pairing of domain stripes. Terms unjustifiably discarded in the one-constant approximation are explicitly introduced. The mean angle approximation which replaces the variable coefficients qZ and b by constant mean values is also discussed and justified. Applied electric, magnetic and thermal fields induce in nematic liquid crystals a variety of interesting flow patterns [l-41. The well known Williams domains may be taken as a typical representative. Recently, the importance of nonlinear phenomena in these flow patterns has been increasingly recognized...
Acta Crystallographica Section A Foundations and Advances, 2015
Physics Letters A, 1976
It is shown that both the classical and the quantum sine-Gordon theory depend on a single scaling... more It is shown that both the classical and the quantum sine-Gordon theory depend on a single scaling parameter and therefore the coupling constant cannot be freely chosen. To introduce a meaningful coupling constant it is proposed to include higher Fourier terms in the sine-Gordon ...
Physical Review Letters, 1970
The Frenkel-Kontorova model of a dislocation might be useful for the study of tachyons. The model... more The Frenkel-Kontorova model of a dislocation might be useful for the study of tachyons. The model has tachyonlike solutions which correspond to unstable states. This may explain why tachyons are unlikely to be observed. ... (Some reference links may require a separate subscription.)
Journal of Non-Crystalline Solids, 1993
Imperfect icosahedral quasicrystals are assumed to grow obeying a local rule forcing the twin ori... more Imperfect icosahedral quasicrystals are assumed to grow obeying a local rule forcing the twin orientation of like building blocks. The resultant vertex configurations are classified by their energies determined by calculating partial overlaps in their dual images. A previously established ...
Journal of Non-Crystalline Solids, 1993
We have constructed all 5450 combinatorially possible vertex configurations formed by the Ammann ... more We have constructed all 5450 combinatorially possible vertex configurations formed by the Ammann rhombohedra. The defectiveness of the vertex configurations is quantified by a penalty functional defined as the minimal sum of squared distances between the dual acceptance domains ...
I briefly summarize the discussion in the Zeitschrift für Kristallographie on „What is a crystal?... more I briefly summarize the discussion in the Zeitschrift für Kristallographie on „What is a crystal?“ [1]. The best upto-date answer is that it is a solid whose Fourier spectrum contains a pure point part, in other words, it shows Bragg peaks. That rises the more general question about „What diffracts and how?“ Hence, there is growing interest in quasiregular heterostructures [2]. These are layer structures artificially fabricated according to certain algorithms, mainly substitution sequences. Some outstanding examples are Fibonacci, Thue-Morse, Rudin-Shapiro, paperfolding and period doubling chains (finite parts of sequences) [3]. Their spectral properties are of great interest.
Zeitschrift für Kristallographie, 2007
In April 1991 the Commission on Aperiodic Crystals [1] adopted the following definition of a crys... more In April 1991 the Commission on Aperiodic Crystals [1] adopted the following definition of a crystal:“[In the following] by ‘crystal’ we mean any solid having an essentially discrete diffraction diagram, [and by ‘aperiodic crystal’ we mean any crystal in which three-dimensional lattice periodicity can be considered to be absent.]”This definition has, in my opinion, two flaws. …
Journal de Physique, 1982
2014 Nous étudions l'instabilité d'ondulation dans un smectique A sous cisaillement parallèle aux... more 2014 Nous étudions l'instabilité d'ondulation dans un smectique A sous cisaillement parallèle aux couches. Nous cherchons la dilatation au seuil d'apparition d'une ondulation dont le vecteur d'onde fait un angle 03B8 avec la direction de l'écoulement. Nous trouvons une relation intégrale qui montre que la dilatation est nécessairement positive. Nous développons une méthode de perturbations pour calculer la dilatation explicitement en fonction de l'angle 03B8 et du taux de cisaillement. Nous montrons qu'une ondulation de vecteur d'onde perpendiculaire à la direction de cisaillement n'est pas affectée par l'écoulement et apparait pour un seuil de dilatation minimum et égal à la valeur statique. Nous concluons qu'une texture rectangulaire se développe pour des dilatations plus grandes en accord avec le réseau de domaines focaux observé expérimentalement. Abstract. 2014 We study the undulation instability in a smectic A in the presence of shear flow parallel to the layers. We look for the dilation at the threshold where an undulation appears with a wave vector inclined at an angle 03B8 to the direction of flow. We find an integral relation showing that the dilation is necessarily positive. We develop a perturbation method to calculate the dilation explicitly as a function of the angle 03B8 and the shear rate. We show that an undulation with wave vector perpendicular to the shear direction is not affected by the flow and has a minimum threshold dilation equal to the static value. We conclude that a rectangular texture develops at higher dilations in agreement with the rectangular focal domain pattern observed experimentally.
Acta Crystallographica Section A Foundations and Advances, 2021
Acta Crystallographica Section A Foundations and Advances, 2019
I present an overview of multidimensional deterministic aperiodic structures which I constructed ... more I present an overview of multidimensional deterministic aperiodic structures which I constructed together with my collaborators, mainly my own students. First I present the octagonal two-color Ben-Abraham-Gähler structure constructed by maximal covering of the plane by two octagonal patches. It serves to interpret the structure of the octagonal MnSiAl quasicrystal found by Kuo et al. and investigated by electron crystallography by Hovmöller et al. It is an infinite layer structure with layers alternating in color reflecting the physical structure where alternating layers are antiphases of each other. Next I show our multidimensional generalizations of the Paperfolding and Period doubling sequences. In the sequel I present the Color coding of the chair tiling in arbitrary dimension and the multidimensional Brick tiling (my generalization of the table tiling) as well as its color coding. The latter has a singular continuous part in its Fourier spectrum.Finally I show my two newest structures: The Crossed cube structures-generalizations of the Crossed square tiling whose singular continuous Fourier spectrum is equivalent to that of the non-Pisot squiral tiling.At last I present the multidimensional generalization of the Golay-Rudin-Shapiro sequence whose Fourier spectrum is essentially absolutely continuous.
Acta Crystallographica Section A Foundations of Crystallography, 2000
conference abstracts s16 ECM Nancy Acta Cryst. (2000). A56 (Supplement), s16 s1.m1.o5 Cluster cov... more conference abstracts s16 ECM Nancy Acta Cryst. (2000). A56 (Supplement), s16 s1.m1.o5 Cluster covering of dodecagonal and related structures. S.
Acta Crystallographica Section A Foundations and Advances
Aperiodic 2018 ("9th Conference on Aperiodic Crystals"), 2018
The three-dimensional brick tiling and its color code were introduced at ICQ 13 [1, 2] as a gener... more The three-dimensional brick tiling and its color code were introduced at ICQ 13 [1, 2] as a generalization of the table tiling and its color code [3]. Here we present their generalizations to arbitrary natural dimension. We also show how to reduce the color code.
Physical Review B, 1999
A likely mechanism for the formation of quasicrystals is by maximally covering space with overlap... more A likely mechanism for the formation of quasicrystals is by maximally covering space with overlapping, stable atomic clusters. This notion is here applied to the experimentally determined layered structure of octagonal MnSiAl quasicrystals, which can be described in terms of a decoration of the octagonal Ammann-Beenker tiling. This decoration is abstractly represented by a two-color version of the tiling. The covering cluster of the quasicrystal corresponds to an octagonal covering patch of the colored tiling. This covering patch appears in two variants with complementary colors. The three-dimensional quasicrystal has a centered octagonal translation module, and its space group is I8 4 /mcm. ͓S0163-1829͑99͒15525-5͔
Le Journal de Physique Colloques, 1979
Squint originates in the distortion torque and the distortion stress. Squint brakes the axial inv... more Squint originates in the distortion torque and the distortion stress. Squint brakes the axial inversion symmetry of a simple flow roll but preserves the reflection symmetry of a roll pair. Flow patterns are predicted to show an observable pairing of domain stripes. Terms unjustifiably discarded in the one-constant approximation are explicitly introduced. The mean angle approximation which replaces the variable coefficients qZ and b by constant mean values is also discussed and justified.
Acta Crystallographica Section A Foundations of Crystallography, 1996
Acta Crystallographica Section A Foundations of Crystallography, 2006
Physical Review A, 1976
The effects of nonlinearity in nematic liquid crystals subject to electric and magnetic fields ar... more The effects of nonlinearity in nematic liquid crystals subject to electric and magnetic fields are emphasized. For the Freedericks transition, the distortion profile is calculated exactly. For the domain modes, the torque equation can be approximated to a high degree by the nonlinear sine-Gordon equation. An exact twodimensional periodic solution is found whose consequences are in good quantitative agreement with all experimental observations. The possibility of a first-order transition to a vortex pattern is predicted.
Acta Physica Polonica A, 2014
We recapitulate our previously developed recursive algorithm for creating paperfolding structures... more We recapitulate our previously developed recursive algorithm for creating paperfolding structures in arbitrary dimensions. Here we explain and apply it specically to three and four dimensions. We visualize the results by suitable projections. We also explicitly enumerate the number of folds in the rst four generations of the recursion. We conjecture (without proof) that the Fourier spectrum of the structures is pure point (the Bragg peaks).
Abstract.- The curvature $ of an electronematodynamic flow pattern obeys, in the steady state, th... more Abstract.- The curvature $ of an electronematodynamic flow pattern obeys, in the steady state, the nonlinear equation $ " + qZ −t,h3+b- t,h3 + b−t,h3+b$ ' = 0. The term $ $ I, called squint, is studied. Squint originates in the distortion torque and the distortion stress. Squint brakes the axial inversion symmetry of a simple flow roll but preserves the reflection symmetry of a roll pair. Flow patterns are predicted to show an observable pairing of domain stripes. Terms unjustifiably discarded in the one-constant approximation are explicitly introduced. The mean angle approximation which replaces the variable coefficients qZ and b by constant mean values is also discussed and justified. Applied electric, magnetic and thermal fields induce in nematic liquid crystals a variety of interesting flow patterns [l-41. The well known Williams domains may be taken as a typical representative. Recently, the importance of nonlinear phenomena in these flow patterns has been increasingly recognized...
Acta Crystallographica Section A Foundations and Advances, 2015
Physics Letters A, 1976
It is shown that both the classical and the quantum sine-Gordon theory depend on a single scaling... more It is shown that both the classical and the quantum sine-Gordon theory depend on a single scaling parameter and therefore the coupling constant cannot be freely chosen. To introduce a meaningful coupling constant it is proposed to include higher Fourier terms in the sine-Gordon ...
Physical Review Letters, 1970
The Frenkel-Kontorova model of a dislocation might be useful for the study of tachyons. The model... more The Frenkel-Kontorova model of a dislocation might be useful for the study of tachyons. The model has tachyonlike solutions which correspond to unstable states. This may explain why tachyons are unlikely to be observed. ... (Some reference links may require a separate subscription.)
Journal of Non-Crystalline Solids, 1993
Imperfect icosahedral quasicrystals are assumed to grow obeying a local rule forcing the twin ori... more Imperfect icosahedral quasicrystals are assumed to grow obeying a local rule forcing the twin orientation of like building blocks. The resultant vertex configurations are classified by their energies determined by calculating partial overlaps in their dual images. A previously established ...
Journal of Non-Crystalline Solids, 1993
We have constructed all 5450 combinatorially possible vertex configurations formed by the Ammann ... more We have constructed all 5450 combinatorially possible vertex configurations formed by the Ammann rhombohedra. The defectiveness of the vertex configurations is quantified by a penalty functional defined as the minimal sum of squared distances between the dual acceptance domains ...
I briefly summarize the discussion in the Zeitschrift für Kristallographie on „What is a crystal?... more I briefly summarize the discussion in the Zeitschrift für Kristallographie on „What is a crystal?“ [1]. The best upto-date answer is that it is a solid whose Fourier spectrum contains a pure point part, in other words, it shows Bragg peaks. That rises the more general question about „What diffracts and how?“ Hence, there is growing interest in quasiregular heterostructures [2]. These are layer structures artificially fabricated according to certain algorithms, mainly substitution sequences. Some outstanding examples are Fibonacci, Thue-Morse, Rudin-Shapiro, paperfolding and period doubling chains (finite parts of sequences) [3]. Their spectral properties are of great interest.
Zeitschrift für Kristallographie, 2007
In April 1991 the Commission on Aperiodic Crystals [1] adopted the following definition of a crys... more In April 1991 the Commission on Aperiodic Crystals [1] adopted the following definition of a crystal:“[In the following] by ‘crystal’ we mean any solid having an essentially discrete diffraction diagram, [and by ‘aperiodic crystal’ we mean any crystal in which three-dimensional lattice periodicity can be considered to be absent.]”This definition has, in my opinion, two flaws. …
Journal de Physique, 1982
2014 Nous étudions l'instabilité d'ondulation dans un smectique A sous cisaillement parallèle aux... more 2014 Nous étudions l'instabilité d'ondulation dans un smectique A sous cisaillement parallèle aux couches. Nous cherchons la dilatation au seuil d'apparition d'une ondulation dont le vecteur d'onde fait un angle 03B8 avec la direction de l'écoulement. Nous trouvons une relation intégrale qui montre que la dilatation est nécessairement positive. Nous développons une méthode de perturbations pour calculer la dilatation explicitement en fonction de l'angle 03B8 et du taux de cisaillement. Nous montrons qu'une ondulation de vecteur d'onde perpendiculaire à la direction de cisaillement n'est pas affectée par l'écoulement et apparait pour un seuil de dilatation minimum et égal à la valeur statique. Nous concluons qu'une texture rectangulaire se développe pour des dilatations plus grandes en accord avec le réseau de domaines focaux observé expérimentalement. Abstract. 2014 We study the undulation instability in a smectic A in the presence of shear flow parallel to the layers. We look for the dilation at the threshold where an undulation appears with a wave vector inclined at an angle 03B8 to the direction of flow. We find an integral relation showing that the dilation is necessarily positive. We develop a perturbation method to calculate the dilation explicitly as a function of the angle 03B8 and the shear rate. We show that an undulation with wave vector perpendicular to the shear direction is not affected by the flow and has a minimum threshold dilation equal to the static value. We conclude that a rectangular texture develops at higher dilations in agreement with the rectangular focal domain pattern observed experimentally.
Acta Crystallographica Section A Foundations and Advances, 2021
Acta Crystallographica Section A Foundations and Advances, 2019
I present an overview of multidimensional deterministic aperiodic structures which I constructed ... more I present an overview of multidimensional deterministic aperiodic structures which I constructed together with my collaborators, mainly my own students. First I present the octagonal two-color Ben-Abraham-Gähler structure constructed by maximal covering of the plane by two octagonal patches. It serves to interpret the structure of the octagonal MnSiAl quasicrystal found by Kuo et al. and investigated by electron crystallography by Hovmöller et al. It is an infinite layer structure with layers alternating in color reflecting the physical structure where alternating layers are antiphases of each other. Next I show our multidimensional generalizations of the Paperfolding and Period doubling sequences. In the sequel I present the Color coding of the chair tiling in arbitrary dimension and the multidimensional Brick tiling (my generalization of the table tiling) as well as its color coding. The latter has a singular continuous part in its Fourier spectrum.Finally I show my two newest structures: The Crossed cube structures-generalizations of the Crossed square tiling whose singular continuous Fourier spectrum is equivalent to that of the non-Pisot squiral tiling.At last I present the multidimensional generalization of the Golay-Rudin-Shapiro sequence whose Fourier spectrum is essentially absolutely continuous.
Acta Crystallographica Section A Foundations of Crystallography, 2000
conference abstracts s16 ECM Nancy Acta Cryst. (2000). A56 (Supplement), s16 s1.m1.o5 Cluster cov... more conference abstracts s16 ECM Nancy Acta Cryst. (2000). A56 (Supplement), s16 s1.m1.o5 Cluster covering of dodecagonal and related structures. S.
Acta Crystallographica Section A Foundations and Advances
Aperiodic 2018 ("9th Conference on Aperiodic Crystals"), 2018
The three-dimensional brick tiling and its color code were introduced at ICQ 13 [1, 2] as a gener... more The three-dimensional brick tiling and its color code were introduced at ICQ 13 [1, 2] as a generalization of the table tiling and its color code [3]. Here we present their generalizations to arbitrary natural dimension. We also show how to reduce the color code.
Physical Review B, 1999
A likely mechanism for the formation of quasicrystals is by maximally covering space with overlap... more A likely mechanism for the formation of quasicrystals is by maximally covering space with overlapping, stable atomic clusters. This notion is here applied to the experimentally determined layered structure of octagonal MnSiAl quasicrystals, which can be described in terms of a decoration of the octagonal Ammann-Beenker tiling. This decoration is abstractly represented by a two-color version of the tiling. The covering cluster of the quasicrystal corresponds to an octagonal covering patch of the colored tiling. This covering patch appears in two variants with complementary colors. The three-dimensional quasicrystal has a centered octagonal translation module, and its space group is I8 4 /mcm. ͓S0163-1829͑99͒15525-5͔
Le Journal de Physique Colloques, 1979
Squint originates in the distortion torque and the distortion stress. Squint brakes the axial inv... more Squint originates in the distortion torque and the distortion stress. Squint brakes the axial inversion symmetry of a simple flow roll but preserves the reflection symmetry of a roll pair. Flow patterns are predicted to show an observable pairing of domain stripes. Terms unjustifiably discarded in the one-constant approximation are explicitly introduced. The mean angle approximation which replaces the variable coefficients qZ and b by constant mean values is also discussed and justified.
Acta Crystallographica Section A Foundations of Crystallography, 1996
Acta Crystallographica Section A Foundations of Crystallography, 2006