Erratum: X-ray cross-correlation analysis and local symmetries of disordered systems: General theory [Phys. Rev. B<b>82</b>, 104207 (2010)]

Altarelli, M.; Kurta, Ruslan P.; Vartanyants, Ivan A.

doi:10.1103/physrevb.86.179904

Cited by 50 publications

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“…Local order and local symmetries are ubiquitous also in larger molecules or even disordered matter [7]. One way to detect local order on the atomic scale uses X-ray cross correlation analysis [7][8][9][10]. In view of the existing powerful methodology to tackle global symmetries the question arises whether the presence of local symmetries also allows for the development of a predictive formalism and corresponding mathematical framework.…”

Section: Introductionmentioning

confidence: 99%

Dynamics of local symmetry correlators for interacting many-particle systems

Schmelcher

Krönke

Diakonos

2017

The Journal of Chemical Physics

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Recently (PRL 113, 050403 (2014)) the concept of local symmetries in one-dimensional stationary wave propagation has been shown to lead to a class of invariant two-point currents that allow to generalize the parity and Bloch theorem. In the present work we establish the theoretical framework of local symmetries for higher-dimensional interacting many-body systems. Based on the Bogoliubov-Born-Green-KirkwoodYvon (BBGKY) hierarchy we derive the equations of motion of local symmetry correlators which are offdiagonal elements of the reduced one-body density matrix at symmetry related positions. The natural orbital representation yields equations of motion for the convex sum of the local symmetry correlators of the natural orbitals as well as for the local symmetry correlators of the individual orbitals themselves. An alternative integral representation with a unique interpretation is provided. We discuss special cases, such as the bosonic and fermionic mean field theory and show in particular that the invariant two-point currents are recovered in the case of the non-interacting one-dimensional stationary wave propagation. Finally we derive the equations of motion for anomalous local symmetry correlators which indicate the breaking of a global into a local symmetry in the stationary non-interacting case. * pschmelc@physnet.uni-hamburg.de

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Section: Introductionmentioning

confidence: 99%

Dynamics of local symmetry correlators for interacting many-particle systems

Schmelcher

Krönke

Diakonos

2017

The Journal of Chemical Physics

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“…25,35,45 However, a persisting problem of these protocols is that they are prone to introduce defects in the superlattice structure with some degree of granularity and significantly smaller grain sizes, which poses difficulties in determining the angular correlation with a meaningful statistical distribution. 11,17,24,41,46,47 Using a conductive MC on the basis of PbS NCs, 35 we show how X-ray cross correlation analysis (XCCA) [48][49][50][51] (Fig. 1c) with the corresponding real-space image (Fig.…”

Section: Introductionmentioning

confidence: 99%

“…Note that the real shape and binding mode of the ligand sphere may be more complex than that displayed in In order to determine the angular correlations and unit cell parameters within each domain quantitatively, we applied an XCCA approach (see Methods for XCCA details). [48][49][50][51][59][60][61][62] This method is based on the evaluation of a two-point angular cross-correlation function (CCF) that can be calculated for each diffraction profile as 48,49 (…”

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Quantifying Angular Correlations between the Atomic Lattice and the Superlattice of Nanocrystals Assembled with Directional Linking

et al. 2017

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We show that the combination of X-ray scattering with a nanofocused beam and X-ray cross correlation analysis is an efficient means for the full structural characterization of mesocrystalline nanoparticle assemblies with a single experiment. We analyze several hundred diffraction patterns of individual sample locations, i.e. individual grains, to obtain a meaningful statistical distribution of the superlattice and atomic lattice ordering. Simultaneous small-and wide-angle X-ray scattering of the same sample location allows us to determine the structure and orientation of the superlattice as well as the angular correlation of the first two Bragg peaks of the atomic lattices, their orientation with respect to the superlattice, and the average orientational misfit due to local structural disorder. This experiment is particularly advantageous for synthetic mesocrystals made by the simultaneous self-assembly of colloidal nanocrystals and surfacefunctionalization with conductive ligands. While the structural characterization of such materials has been challenging so far, the present method now allows correlating mesocrystalline structure with optoelectronic properties. Mesocrystals (MC) are three-dimensional arrays of iso-oriented single-crystalline particles with an individual size between 1 -1000 nm. [1][2][3][4][5] Their physical properties are largely determined by structural coherence, for which the angular correlation between their individual atomic lattices and the underlying superlattice of nanocrystals (NC) is a key ingredient. 1,2 Colloidal NCs stabilized by organic surfactants have been shown to pose excellent building blocks for the design of synthetic MCs with tailored structural properties which are conveniently obtained by self-assembly of NCs from solution on a solid or liquid substrate by exploiting ligand-ligand 3 interactions. [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25] Typically, the utilized ligands consist of wide-gap, bulky hydrocarbons which render the MCs insulating. [26][27][28][29][30][31][32][33] MCs obtained in this way exhibit average grain sizes of ~150 µm 2 , which enables a detailed characterization by electron and/or X-ray microscopy. 34 Since the optoelectronic properties of PbS NC ensembles bear many opportunities for applications in solar cells or photodetectors, a number of ligand exchange procedures with small organic or inorganic molecules as well as single atom passivation strategies have been developed, all of which greatly increase the carrier mobilities within the SL of NCs. 28,33,[35][36][37][38][39][40][41][42][43][44] Due to the short interparticle spacing imposed by these ligands, structural coherence is mostly lost in such superlattices, but in rare cases it has been demonstrated that significant long-range order and even mesocrystallinity can be preserved. 25,35,45 However, a persisting problem of these protocols is that they are prone to introduce defects in the superlattice structure with some degree of granularity and signifi...

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“…[6,12]. Angular XCCA enables direct determination of the Fourier components I n (q) from the ensemble of diffraction patterns measured at different positions on the sample [14,15]. For each diffraction pattern a two-point cross-correlation function of the form…”

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confidence: 99%

“…Information about the temperature evolution of the BO order parameters is typically obtained by fitting the measured azimuthal intensity distribution. We propose a different approach based on angular x-ray cross-correlation analysis (XCCA) [13][14][15][16][17] that allows determination of the BO order parameters directly, without fitting. Angular XCCA has been developed to study the structure of noncrystalline systems like liquids, amorphous systems, disordered ensembles of particles, etc.…”

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confidence: 99%

X-ray cross-correlation analysis of liquid crystal membranes in the vicinity of the hexatic-smectic phase transition

et al. 2013

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We present an x-ray study of liquid crystal membranes in the vicinity of the hexatic-smectic phase transition by means of angular x-ray cross-correlation analysis. By applying two-point angular-intensity cross-correlation functions to the measured series of diffraction patterns the parameters of bond-orientational (BO) order in hexatic phase were directly determined. The temperature dependence of the positional correlation lengths was analyzed as well. The obtained correlation lengths show larger values for the higher-order Fourier components of BO order. These findings indicate a strong coupling between BO and positional order. About 30 years ago it was realized that the phase transition between a two-dimensional (2D) crystal and the liquid phase can proceed through an intermediate hexatic phase [1]. The corresponding mechanism involves dissociation of dislocation pairs. The 2D hexatic phase is characterized by a sixfold quasilong-range bond-orientational (BO) order, while the positional order is short range and the shear modulus is zero. Phases with hexatic order have been found in several systems, such as electrons at the surface of helium [2], charged polymer colloids [3,4], and smectic liquid crystals (LCs) [5][6][7]. In the last case a hexatic phase was experimentally observed quite unexpectedly in three-dimensional (3D) stacked molecular systems [8]. In the 3D hexatic phase the positional order is short range, while the BO order persists over a long range. However, the mere existence of a hexatic phase does not imply a specific melting mechanism, and the origin of hexatic order and its relation to the defect-mediated melting transition is still controversial. Smectic liquid crystals are particularly suitable to investigate these problems, as they can be suspended over an opening in a solid frame. Such smectic membranes are substrate-free and have a controlled thickness ranging from two to more than thousands of layers [9].Smectic-A membranes can be described as stacks of liquid layers. The in-plane structure is liquidlike with positional correlations between the molecules decaying exponentially with a correlation length ξ 0 . While cooling a hexatic phase may occur, which shows a long-range BO order. This leads to a sixfold rotational symmetry and the BO correlations are characterized by the local ordering field ψ(r) ∝ exp[i6θ (r)], where θ (r) is the angle between the "bonds" and some reference axis. Upon decreasing the temperature, the width of the radial-intensity peak decreases simultaneously with a further development of the BO order. This indicates a coupling * Present address: The University of California, San Diego, La Jolla, CA 92093, USA.† Corresponding author: ivan.vartaniants@desy.de between the positional correlations and the BO order [10]. Additionally, at even lower temperatures a 3D crystalline phase appears with a hexagonal in-plane lattice or a rectangular lattice with a so-called herringbone order [6,7]. The common way to study the BO order is to perform x-ray or electron diffraction me...

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Erratum: X-ray cross-correlation analysis and local symmetries of disordered systems: General theory [Phys. Rev. B82, 104207 (2010)]

Cited by 50 publications

References 0 publications

Dynamics of local symmetry correlators for interacting many-particle systems

Dynamics of local symmetry correlators for interacting many-particle systems

Quantifying Angular Correlations between the Atomic Lattice and the Superlattice of Nanocrystals Assembled with Directional Linking

X-ray cross-correlation analysis of liquid crystal membranes in the vicinity of the hexatic-smectic phase transition

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