Spectral geometry of crystal lattices

2003 ◽  
pp. 271-305 ◽  
Author(s):  
Motoko Kotani ◽  
Toshikazu Sunada
Keyword(s):  
Spectral Geometry ◽  
Crystal Lattices

The Interpretation of Crystal Lattice Images

1972 ◽  
Vol 30 ◽  
pp. 550-551
Author(s):  
J. M. Cowley ◽  
Sumio Iijima
Keyword(s):  
Crystal Orientation ◽  
Diffraction Theory ◽  
Phase Changes ◽  
Electron Wave ◽  
Crystal Lattices ◽  
Heavy Atoms ◽  
Lattice Images ◽  
The Right ◽  
Intuitive Interpretation ◽  
Suitable Approximation

The imaging of detailed structures of crystal lattices with 3 to 4Å resolution, given the correct conditions of microscope defocus and crystal orientation and thickness, has been used by Iijima (this conference) for the study of new types of crystal structures and the defects in known structures associated with fluctuations of stoichiometry. The image intensities may be computed using n-beam dynamical diffraction theory involving several hundred beams (Fejes, this conference). However it is still important to have a suitable approximation to provide an immediate rough estimate of contrast and an evaluation of the intuitive interpretation in terms of an amplitude object.For crystals 100 to 150Å thick containing moderately heavy atoms the phase changes of the electron wave vary by about 10 radians suggesting that the “optimum defocus” theory of amplitude contrast for thin phase objects due to Scherzer and others can not apply, although it does predict the right defocus for optimum imaging.

Atomic resolution Z-contrast imaging of bulk crystal surfaces in Scanning Reflection Electron Microscopy

1990 ◽  
Vol 48 (4) ◽  
pp. 414-415
Author(s):  
Z. L. Wang ◽  
J. Bentley
Keyword(s):  
Electron Microscopy ◽  
Elastic Scattering ◽  
Dark Field ◽  
Contrast Imaging ◽  
Crystal Lattices ◽  
Small Probe ◽  
Scanning Transmission ◽  
Annular Dark Field ◽  
Image Position ◽  
Z Contrast

The success of obtaining atomic-number-sensitive (Z-contrast) images in scanning transmission electron microscopy (STEM) has shown the feasibility of imaging composition changes at the atomic level. This type of image is formed by collecting the electrons scattered through large angles when a small probe scans across the specimen. The image contrast is determined by two scattering processes. One is the high angle elastic scattering from the nuclear sites,where ϕNe is the electron probe function centered at bp = (Xp, yp) after penetrating through the crystal; F denotes a Fourier transform operation; D is the detection function of the annular-dark-field (ADF) detector in reciprocal space u. The other process is thermal diffuse scattering (TDS), which is more important than the elastic contribution for specimens thicker than about 10 nm, and thus dominates the Z-contrast image. The TDS is an average “elastic” scattering of the electrons from crystal lattices of different thermal vibrational configurations,

Synthesis, Characterisation and Structural Studies of [Zn(phen)3][Fe(CN)5(NO)]·2H2O·0.25MeOH and [(bipy)2(H2O)Zn(μ-NC)Fe(CN)4(NO)]·0.5H2O

2003 ◽  
Vol 58 (9) ◽  
pp. 916-921 ◽  
Author(s):  
Amitabha Datta ◽  
Samiran Mitra ◽  
Georgina Rosair
Keyword(s):  
Infrared Spectroscopy ◽  
Thermogravimetric Analysis ◽  
Octahedral Geometry ◽  
Distorted Octahedral Geometry ◽  
Structural Studies ◽  
Crystal Lattices ◽  
X Ray ◽  
X Ray Crystallography ◽  
Distorted Octahedral ◽  
Solvent Molecules

Two new bimetallic complexes [Zn(phen)3][Fe(CN)5(NO)] · 2 H2O · 0.25 MeOH, (1) and [(bipy)2(H2O)Zn(μ-NC)Fe(CN)4(NO)] · 0.5 H2O, (2), have been isolated (where phen = 1,10-phenanthroline and bipy = bipyridyl) and characterised by X-ray crystallography [as the 2 H2O · 0.25 CH3OH solvate for (1) and hemihydrate for (2)] infrared spectroscopy and thermogravimetric analysis. Substitution of phenanthroline for bipyridyl resulted in a cyano-bridged bimetallic species rather than two discrete mononuclear metal complexes. The bond angles of Fe-N-O were shown to be practically linear for both 1 [179.2(7)°] and 2 [178.3(3)°], and the Zn atoms have distorted octahedral geometry. The solvent molecules in both crystal lattices take part in forming hydrogen-bonded networks.

scholarly journals Spectral geometry of κ-Minkowski space

2006 ◽  
Vol 47 (6) ◽  
pp. 062105 ◽  
Author(s):  
Francesco D’Andrea
Keyword(s):  
Minkowski Space ◽  
Spectral Geometry

On the stability of crystal lattices IX. Covariant theory of lattice deformations and the stability of some hexagonal lattices

1942 ◽  
Vol 38 (1) ◽  
pp. 82-99 ◽  
Author(s):  
M. Born
Keyword(s):  
Elastic Constants ◽  
Hexagonal Lattice ◽  
Covariant Theory ◽  
Tensor Calculus ◽  
Crystal Lattices ◽  
New Form ◽  
The Stability ◽  
Central Forces

The theory of lattice deformations is presented in a new form, using the tensor calculus. The case of central forces is worked out in detail, and the results are applied to some simple hexagonal lattices. It is shown that the Bravais hexagonal lattice is unstable but the close-packed hexagonal lattice stable. The elastic constants of this lattice are calculated.

Solid-state dynamics and single-crystal to single-crystal structural transformations in octakis(3-chloropropyl)octasilsesquioxane and octavinyloctasilsesquioxane

2017 ◽  
Vol 19 (40) ◽  
pp. 27516-27529 ◽  
Author(s):  
A. Kowalewska ◽  
M. Nowacka ◽  
M. Włodarska ◽  
B. Zgardzińska ◽  
R. Zaleski ◽  
...  
Keyword(s):  
Solid State ◽  
Single Crystal ◽  
Structural Transformations ◽  
Crystal Lattices ◽  
Thermally Induced ◽  
Crystal Structural

Thermally induced formation of symmetric crystal lattices in functional POSS proceeds via different mechanisms and results in unique reversible phenomena.

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