Bloch waves and non-commutative tori of magnetic translations

Since Head first demonstrated the advantages of computer displayed theoretical intensities from defective crystals, computer display techniques have become important in image analysis. However the computational methods employed resort largely to numerical integration of the dynamical equations of electron diffraction. As a consequence, the interpretation of the results in terms of the defect displacement field and diffracting variables is difficult to follow in detail. In contrast to this type of computational approach which is based on a plane-wave expansion of the excited waves within the crystal (i.e. Darwin representation ), Wilkens assumed scattering of modified Bloch waves by an imperfect crystal. For localized defects, the wave amplitudes can be described analytically and this formulation has been used successfully to predict the black-white symmetry of images arising from small dislocation loops.

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Guided and standing Bloch waves in periodic elastic strips

Waves in Random and Complex Media ◽

10.1080/17455030802541566 ◽

2009 ◽

Vol 19 (2) ◽

pp. 321-346 ◽

Cited By ~ 10

Author(s):

Samuel D.M. Adams ◽

Richard V. Craster ◽

Sebastien Guenneau

Keyword(s):

Bloch Waves

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On the derivation of ℏ ⋅ d k/dt=F, the k‐space form of Newton’s law for Bloch waves

American Journal of Physics ◽

10.1119/1.14685 ◽

1986 ◽

Vol 54 (2) ◽

pp. 177-178 ◽

Cited By ~ 8

Author(s):

Herbert Kroemer

Keyword(s):

Space Form ◽

Bloch Waves ◽

Newton's Law ◽

Newton’S Law

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Rayleigh–Bloch waves along elastic diffraction gratings

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2014.0465 ◽

2015 ◽

Vol 471 (2173) ◽

pp. 20140465 ◽

Cited By ~ 18

Author(s):

D. J. Colquitt ◽

R. V. Craster ◽

T. Antonakakis ◽

S. Guenneau

Keyword(s):

Boundary Conditions ◽

Numerical Simulations ◽

Elastic System ◽

Diffraction Gratings ◽

Effective Medium ◽

Scalar Wave ◽

Trapped Modes ◽

Bloch Waves

Rayleigh–Bloch (RB) waves in elasticity, in contrast to those in scalar wave systems, appear to have had little attention. Despite the importance of RB waves in applications, their connections to trapped modes and the ubiquitous nature of diffraction gratings, there has been no investigation of whether such waves occur within elastic diffraction gratings for the in-plane vector elastic system. We identify boundary conditions that support such waves and numerical simulations confirm their presence. An asymptotic technique is also developed to generate effective medium homogenized equations for the grating that allows us to replace the detailed microstructure by a continuum representation. Further numerical simulations confirm that the asymptotic scheme captures the essential features of these waves.

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Evanescent Bloch waves and the complex band structure of phononic crystals

Physical Review B ◽

10.1103/physrevb.80.092301 ◽

2009 ◽

Vol 80 (9) ◽

Cited By ~ 111

Author(s):

Vincent Laude ◽

Younes Achaoui ◽

Sarah Benchabane ◽

Abdelkrim Khelif

Keyword(s):

Band Structure ◽

Phononic Crystals ◽

Bloch Waves

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Modeling of electron transport in nanoribbon devices using Bloch waves

2018 76th Device Research Conference (DRC) ◽

10.1109/drc.2018.8442279 ◽

2018 ◽

Author(s):

Akash A. Laturia ◽

Maarten L. Van de Put ◽

Massimo V. Fischetti ◽

William G. Vandenberghe

Keyword(s):

Electron Transport ◽

Bloch Waves

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Bloch wave homogenisation of quasiperiodic media

European Journal of Applied Mathematics ◽

10.1017/s0956792520000352 ◽

2020 ◽

pp. 1-21

Author(s):

SISTA SIVAJI GANESH ◽

VIVEK TEWARY

Keyword(s):

Bloch Wave ◽

Periodic Operator ◽

Almost Periodic ◽

Periodic Media ◽

Direct Integral ◽

Bloch Waves ◽

Direct Integral Decomposition ◽

Quasiperiodic Media ◽

Integral Decomposition ◽

Periodic Operators

Quasiperiodic media is a class of almost periodic media which is generated from periodic media through a ‘cut and project’ procedure. Quasiperiodic media displays some extraordinary optical, electronic and conductivity properties which call for the development of methods to analyse their microstructures and effective behaviour. In this paper, we develop the method of Bloch wave homogenisation for quasiperiodic media. Bloch waves are typically defined through a direct integral decomposition of periodic operators. A suitable direct integral decomposition is not available for almost periodic operators. To remedy this, we lift a quasiperiodic operator to a degenerate periodic operator in higher dimensions. Approximate Bloch waves are obtained for a regularised version of the degenerate operator. Homogenised coefficients for quasiperiodic media are obtained from the first Bloch eigenvalue of the regularised operator in the limit of regularisation parameter going to zero. A notion of quasiperiodic Bloch transform is defined and employed to obtain homogenisation limit for an equation with highly oscillating quasiperiodic coefficients.

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