Incommensurability and recursivity: lattice dynamics of modulated crystals

Lange, C. de; Janssen, Τ.

doi:10.1088/0022-3719/14/34/009

Cited by 45 publications

(14 citation statements)

References 19 publications

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“…For each value of the modulation wavevector the spectrum is projected on a vertical energy axis. Large-gap openings are observed from de Lange & Janssen (1981). mensurately modulated phases.…”

Section: Figurementioning

confidence: 92%

“…If the modulation is one third, two gaps will open and so on. In the limit of the irrational modulation and infinite cell, one thus expects an infinite number of gaps (de Lange & Janssen, 1981). The analysis carried out on these 1D modulated chains shows the fractal character of the excitation spectrum in the limiting aperiodic case on one hand, but also the rapidly vanishing size of the gaps on the other hand (de Lange & Janssen, 1981).…”

Section: Excitation Spectra Phonons and Phasons In Aperiodic Crystalsmentioning

confidence: 94%

“…The effect of the shape of the modulation was also shown in this model, with the occurrence of a phason gap for nonsinusoidal modulation. The second 1D system of interest is the Frankel-Kontorova model for which the phason branch only goes to zero in the regime of weak coupling (de Lange & Janssen, 1981). This model can also be viewed as a model of a composite with two different sublattices, but Ted Janssen and his co-worker, with the double-chain model, developed a much better composite approximation.…”

Section: Figurementioning

confidence: 99%

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Ted Janssen and aperiodic crystals

Boissieu

2019

Acta Cryst Sect A

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This article reviews some of Ted Janssen's (1936–2017) major contributions to the field of aperiodic crystals. Aperiodic crystals are long-range ordered structures without 3D lattice translations and encompass incommensurately modulated phases, incommensurate composites and quasicrystals. Together with Pim de Wolff and Aloysio Janner, Ted Janssen invented the very elegant theory of superspace crystallography that, by adding a supplementary dimension to the usual 3D space, allows for a deeper understanding of the atomic structure of aperiodic crystals. He also made important contributions to the understanding of the stability and dynamics of aperiodic crystals, exploring their fascinating physical properties. He constantly interacted and collaborated with experimentalists, always ready to share and explain his detailed understanding of aperiodic crystals.

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Section: Figurementioning

confidence: 92%

Section: Excitation Spectra Phonons and Phasons In Aperiodic Crystalsmentioning

confidence: 94%

Section: Figurementioning

confidence: 99%

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Ted Janssen and aperiodic crystals

Boissieu

2019

Acta Cryst Sect A

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“…The fact that one may consider a Bloch electron in an external magnetic field as a problem in an aperiodic crystal explains why the spectra of the Hofstadter problem and those of the problem of phonons in an aperiodic crystal are very similar [11,12].…”

Section: Generalized Magnetic Space-time Groupsmentioning

confidence: 99%

Development of Symmetry Concepts for Aperiodic Crystals

Janssen

2014

Symmetry

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An overview is given of the use of symmetry considerations for aperiodic crystals. Superspace groups were introduced in the seventies for the description of incommensurate modulated phases with one modulation vector. Later, these groups were also used for quasi-periodic crystals of arbitrary rank. Further extensions use time reversal and time translation operations on magnetic and electrodynamic systems. An alternative description of magnetic structures to that with symmetry groups, the Shubnikov groups, is using representations of space groups. The same can be done for aperiodic crystals. A discussion of the relation between the two approaches is given. Representations of space groups and superspace groups play a role in the study of physical properties. These, and generalizations of them, are discussed for aperiodic crystals. They are used, in particular, for the characterization of phase transitions between aperiodic crystal phases.

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“…This is perhaps the simplest non-trivial model available which describes the effects of phonons onto electronic conduction and is sometimes referred to as the "modulated crystal" [1,9,25,26,32]. Since the potential in (1) isn't of the form cos(x − 1 2 at 2 ), the unitary transform propsed in [35] and leading to Bloch oscillations doesn't apply here (see also [33] for an AC-type modulation).…”

Section: Preprint Submitted To Elsevier Sciencementioning

confidence: 99%

Impurity bands and quasi-Bloch waves for a one-dimensional model of modulated crystal

Gosse¹

2008

Nonlinear Analysis: Real World Applications

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This paper investigates a simple one-dimensional model of incommensurate "harmonic crystal" in terms of the spectrum of the corresponding Schrödinger equation. Two angles of attack are studied: the first exploits techniques borrowed from the theory of quasi-periodic functions while the second relies on periodicity properties in a higher-dimensional space. It is shown that both approaches lead to essentially the same results; that is, the lower spectrum is splitted between "Cantor-like zones" and "impurity bands" to which correspond critical and extended eigenstates, respectively. These "new bands" seem to emerge inside the band gaps of the unperturbed problem when certain conditions are met and display a parabolic nature. Numerical tests are extensively performed on both steady and time-dependent problems.Key words: Quasi-periodic potential, impurity band, quasi-Bloch wave, quantum chaos, spectral algorithm, deformed crystal, Charge-Density Wave (CDW). 1991 MSC: 81Q05, 81Q20 IntroductionThis article is a step further in the numerical study of electronic motion in a one-dimensional model of "harmonic crystal", that is to say, a modeling in solid-state physics going beyond the rigid periodic lattice picture. Indeed, it consists in assuming that, within the BornOppenheimer approximation (the motion of nucleons is much slower than the electron's ones and thus decouples adiabatically), ionic cores are linked together by springs and vibrate like classical coupled oscillators; consult the book [1] and the recent paper [14] to which we are about to borrow notation.In [14], the following rescaled problem describing the motion of a free electron inside a 1-D lattice of atoms which display single-mode vibrations has been investigated within the * Corresponding Author Email address: l.gosse@ba.iac.cnr.it (Laurent Gosse). Preprint submitted to Elsevier ScienceWKB semi-classical framework [5,19] for the following Schrödinger equation,and under the fundamental assumption that k and Ω(k) = | sin(kπ)| are rational and commensurate. This is perhaps the simplest non-trivial model available which describes the effects of phonons onto electronic conduction and is sometimes referred to as the "modulated crystal" [1,9,25,26,32]. Since the potential in (1) isn't of the form cos(x − 1 2 at 2 ), the unitary transform propsed in [35] and leading to Bloch oscillations doesn't apply here (see also [33] for an AC-type modulation). A particular feature pointed out in [39] and called lattice tracking which refers to the dragging effect of heavy nucleons onto the much lighter electron was easily observed in the numerical results.The commensurability hypothesis is of critical importance in this global picture in order to keep on applying the classic Bloch decomposition for (1); we assume the reader familiar with this theory and refer only to [1,5,15,19] for details. The first step in this line of thinking is the derivation of "energy bands", which are dispersion relations for electrons around certain levels of excitation. Solids are clas...

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Incommensurability and recursivity: lattice dynamics of modulated crystals

Cited by 45 publications

References 19 publications

Ted Janssen and aperiodic crystals

Ted Janssen and aperiodic crystals

Development of Symmetry Concepts for Aperiodic Crystals

Impurity bands and quasi-Bloch waves for a one-dimensional model of modulated crystal

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