Solving modulated structures by X-ray and electron crystallography

X-ray diffraction can be used for accurately determining not only classical, ordinary structures, but also modulated ones. For structures with weak modulations, the modulation induced satellite reflections are often hard to be observed by X-ray diffraction, but they appear clearly in electron diffraction. In these cases, X-ray diffraction will give only average structures whereas electron diffraction will yield information about the modulations. Sr(1.4)Ta(0.6)O(2.9) is a complex modulated compound with weak modulation and small modulated domains. Here we demonstrate the power of combining X-ray and electron crystallography for studying modulated structures on powders. The modulations of Sr(1.4)Ta(0.6)O(2.9) were determined from electron diffraction (SAED) and high resolution electron microscopy (HREM) images. With specially developed image processing techniques, the weak modulations were enhanced, facilitating the interpretation of HREM images in terms of atomic structure.     

Modulated crystals and almost periodic measures

Modulated crystals and quasicrystals can simultaneously be described as modulated quasicrystals, a class of point sets introduced by de Bruijn in 1987. With appropriate modulation functions, modulated quasicrystals themselves constitute a substantial subclass of strongly almost periodic point measures. We re-analyze these structures using methods from modern mathematical diffraction theory, thereby providing a coherent view over that class. Similar to de Bruijn’s analysis, we find stability with respect to almost periodic modulations.

     

Determining the structure of phosphorus in phase IV

We explore the unknown structure of phosphorus in phase IV (P-IV phase) based on first-principles calculations using the metadynamics simulation method. Starting from the simple cubic structure, we find a new modulated structure of the monoclinic lattice. The modulation is crucial to the stability of the structure. Through refining the structure further by changing the modulation period, we find the structure whose x-ray powder diffraction pattern is in best agreement with the experimental pattern. We expect that the modulation period of the structure in the P-IV phase is very close to that found in this study and probably incommensurate.     

Diffraction management and sub-diffractive solitons in periodically driven Bose-Einstein condensates

We theoretically investigate the diffraction management in Bose-Einstein condensates (BECs) in one- (1D), two- (2D) and three-dimensional (3D) geometries. The management technique is based on the superposition of harmonic lattices’ potentials moving at a common speed but in different directions, leading to a harmonic spatio-temporal modulation of the potential. In this way a reduction in, and eventually the disappearance of usual diffraction and emergence of fourth-order diffraction are achieved. We show sub-diffractive solitons in such a diffraction managed system and demonstrate their stability in 1D, 2D and 3D. In 2D and 3D cases we investigate diffraction management by lattices of different symmetry, and study their influence on the isotropy of solitons.     

Incommensurability in crystals

The aim of this review is to present aperiodic crystals from a unifying point of view, showing why it is justified to call them crystals, despite the lack of a three-dimensional lattice periodicity, and to discuss in what sense they differ from periodic crystals in structure, symmetry and other physical properties. The extension of the concept of crystal has been based, during the last two decades, on investigation of incommensurate crystal phases. Among these, the most important ones are the modulated crystals. Their crystallographic nature, already apparent in the diffraction pattern, could be made explicit on the symmetry level by embedding in a higher-dimensional Euclidean space. The recent discovery of quasicrystal phases (representing a fairly different class of aperiodic crystals than the modulated ones) can also be approached in a similar way. Furthermore, it now appears that another class of incommensurate crystals, the so-called composite structures, represents a kind of intermediate case between the other two.

In § 1 basic concepts are presented, together with a number of compounds given as illustration for typical incommensurate crystal phases. In § 2 we deal with the general formalism allowing a crystallographic symmetry characterization. It is intended as a first approach to crystal-structure determination, which justifies the emphasis on the diffraction pattern and on the modulated-crystal case. The appropriate generalization to the quasicrystal case is considered in § 3. The crystallographic nature of the incommensurate phases is apparent in their growth forms. It has been known for centuries that crystal morphology is essentially based on lattice periodicity. It is therefore fascinating to discover how nature solves the problem in the incommensurate crystal case; we discuss this in § 4. The origin of incommensurability is the subject of § 5 (on a phenomenological level) and of § 6 (on a microscopic level). In § 7 the basic concepts needed in crystal-structure determination are discussed in more detail than was appropriate in § 2. In § 8 we discuss those physical properties which are more closely related to incommensurability, fitting nevertheless into the general frame of crystal physics. In § 9 we deal with defects, again in particular with the additional ones due to the more complex structure of aperiodic crystals, making clear at the same time why defects play an even more important role than in periodic crystals.     

Analytical solutions for some defect problems in 1D hexagonal and 2D octagonal quasicrystals

We study some typical defect problems in one-dimensional (1D) hexagonal and two-dimensional (2D) octagonal quasicrystals. The first part of this investigation addresses in detail a uniformly moving screw dislocation in a 1D hexagonal piezoelectric quasicrystal with point group 6mm. A general solution is derived in terms of two functions φ ₁, φ ₂, which satisfy wave equations, and another harmonic function φ ₃. Elementary expressions for the phonon and phason displacements, strains, stresses, electric potential, electric fields and electric displacements induced by the moving screw dislocation are then arrived at by employing the obtained general solution. The derived solution is verified by comparison with existing solutions. Also obtained in this part of the investigation is the total energy of the moving screw dislocation. The second part of this investigation is devoted to the study of the interaction of a straight dislocation with a semi-infinite crack in an octagonal quasicrystal. Here the crack penetrates through the solid along the period direction and the dislocation line is parallel to the period direction. We first derive a general solution in terms of four analytic functions for plane strain problem in octagonal quasicrystals by means of differential operator theory and the complex variable method. All the phonon and phason displacements and stresses can be expressed in terms of the four analytic functions. Then we derive the exact solution for a straight dislocation near a semi-infinite crack in an octagonal quasicrystal, and also present the phonon and phason stress intensity factors induced by the straight dislocation and remote loads.      

Twelve-fold quasicrystal and its approximant of Ta62Te38 interpreted as modulated crystals

A transmission electron microscopy study reveals that the twelve-fold quasicrystal and its approximant in Ta62Te38 are crystals subjected to the structure modulation. It is composed of two modulated layers rotated by 30 degrees (or 90 degrees) to each other about their normal. Structures of the twelve-fold quasicrystal and its approximant can be related by modulation waves with the same directions but with slightly different wavelengths. The modulation is considered to be due to the rearrangement of atomic vacancies as a response to the occurrence of charge density waves.     

X-ray study of the anomalous thermal hystereses of the modulation wavevectors in Cs2HgCl4

A rich sequence of structural modulations in Cs₂HgCl₄ as a function of temperature was studied by means of X-ray diffraction. Accurate satellite-position measurements on the cooling and heating paths of the crystal revealed abnormal thermal hystereses for incommensurate phases and coexistences of neighboring commensurate phases. A well-defined X-ray picture of the a-axis modulated phases in the range of 221–184 K were observed on the heating path, while the c-axis modulated phases existing below 184 K were definitely detected on the cooling path. The proper conditions for a precise phase diagram of Cs₂HgCl₄ can be correlated with relatively defect-free transformations of a-axis modulations at heating and of c-axis modulations at cooling. The peculiarity of Cs₂HgCl₄ to switch modulation direction among the a- and c-axes at 184 K allows us deliberately accumulate and thus control a majority of mobile defects on the mutually perpendicular (100) or (001) planes by possessing crystal within temperature domain of a- or c-axes modulations, respectively.     

Periodic,quasi‐periodic,and random quadratic nonlinear photonic crystals

Quadratic nonlinear photonic crystals are materials in which the second order susceptibility χ⁽²⁾ is spatially modulated while the linear susceptibility remains constant. These structures are significantly different than the more common photonic crystals, in which the linear susceptibility is modulated. Nonlinear processes in nonlinear photonic crystals are governed by the phase matching requirements, which are determined by the reciprocal lattice of these crystals. Therefore, the modulation of the nonlinear susceptibility enables to engineer the spatial and spectral response in various three‐wave mixing processes. It enables to support the efficient generation of new optical frequencies at multiple directions. We analyze three wave mixing processes in nonlinear photonic crystals in which the modulation is either periodic, quasi‐periodic, radially symmetric or even random. We discuss both one‐dimensional and two‐dimensional modulations. In addition to harmonic generations, we outline several new possibilities for all‐optical control of the spatial and polarization properties of optical beams in specially designed nonlinear photonic crystals.     

A comparative study of the modulated structures of some A2BX4 compounds

The structures of the modulated phases of K₂SeO₄, Rb₂ZnCl₄, Rb₂ZnCl₄ and Cs₂CdBr₄ are compared. Both commensurate and incommensurate modulations are considered. Clear common features can be derived between the polarization vectors of their primary harmonic modulations. Even distortions with very different modulation wavevectors present a clear correlation.     

Theoretical study of the magnetism in the incommensurate phase of TiOCl

Going beyond a recently proposed microscopic model [D. Mastrogiuseppe, A. Dobry, arXiv:0810.3018v1] for the incommensurate transition in the spin-Peierls TiOX (X=Cl, Br) compounds, in the present work we start by studying the thermodynamics of the model with XY spins and adiabatic phonons. We find that the system enters an incommensurate phase by a first order transition at a low temperature T_c1. At a higher temperature T_c2 a continuous transition to a uniform phase is found. Furthermore, we study the magnetism in the incommensurate phase by density matrix renormalization group (DMRG) calculations on a one-dimensional Heisenberg model where the exchange is modulated by the incommensurate atomic position pattern. When the wave vector q of the modulation is near π, we find local magnetized zones (LMZ) in which spins abandon their singlets as a result of the domain walls induced by the modulated distortion. When q moves far away enough from π, the LMZ disappear and the system develops incommensurate magnetic correlations induced by the structure. We discuss the relevance of this result regarding previous and future experiments in TiOCl.     

Dynamical Casimir effect in superradiant light scattering by Bose–Einstein condensate in an optomechanical cavity

We investigate the effects of dynamical Casimir effect in superradiant light scattering by Bose-Einstein condensate in an optomechanical cavity. The system is studied using both classical and quantized mirror motions. The cavity frequency is harmonically modulated in time for both the cases. The main quantity of interest is the number of intracavity scattered photons. The system has been investigated under the weak and strong modulations. It has been observed that the amplitude of the scattered photons is more for the classical mirror motion than the quantized mirror motion. Also, initially, the amplitude of scattered photons is high for lower modulation amplitude than higher modulation amplitude. We also found that the behavior of the plots are similar under strong and weak modulations for the quantized mirror motion.     

Statistical approach to multiple-q modulated structures: average Patterson analysis

Using a statistical approach, the average unit cells have been constructed for modulated structures with different types of modulation, from simple sinusoidal to square-wave functions. The obtained unit cells fully describe diffraction intensities of the main reflections and their satellites. A universal distribution, valid for different lengths of the modulation vector, has been found. Average Patterson functions have been constructed and used to distinguish between different types of modulation.     

Conductivity of a generalized hopping system with internal dynamics

We derive explicit exact expressions for the frequency dependent conductivity of a system whose dynamics is described by a Master Equation in which the transition rates _ab can be functions of other dynamical variables. We apply this method to study the hopping of particles in a lattice with the hopping rate modulated by an additional harmonic variable. The results of this model are related to the observed microwave conductivity of superionic conductors. In particular we show that the modulation of the hopping rate can produce a structure similar to the one observed in AgI type systems.     

Imaging ripples on phononic crystals reveals acoustic band structure and Bloch harmonics

Broadband surface phonon wave packets on a phononic crystal made up of a microstructured line pattern are tracked in two dimensions and in real time with an ultrafast optical technique. The eigenmode distribution and the 2D acoustic band structure are obtained from spatiotemporal Fourier transforms of the data up to 1 GHz. We find stop bands at the zone boundaries for both leaky-longitudinal and Rayleigh waves, and show how the structure of individual acoustic eigenmodes in k space depends on Bloch harmonics and on mode coupling.     

Theoretical and experimental vibrational property of 1,3,4,6-tetramethylpiperazine-2,5-dione

1,3,4,6-tetramethylpiperazine-2,5-dione was synthesized, and the crystal structure was characterized by single-crystal X-ray diffraction method, vibrational spectral measurements were obtained by Fourier transform infrared spectroscopy and Fourier transform Raman spectroscopy. The measurements agree well with the calculated geometrical structure and harmonic vibrational wavenumbers (usingab initio and density functional theory Becke’s three parameters hybrid method with the Lee, Yang, and Parr non-local functions methods with 6-311G and 6-311G** basis sets).     

Diffraction theory of quasicrystals

The diffraction problem of quasicrystals is solved with the aid of shift operators, s _q(x, y, z) (q = x, y, z). Starting from certain sets of latent lattices which have to be optimized according to chemical composition, density, and diffraction pictures, it is shown that periodic structures are determined by as many parameters as the number of independent positions. Furthermore, the Fourier coefficients (Fc's) are strictly periodic. Structure factors are given by sums of products of Bessel functions (Bfs). The amplitudes of Fc's enter the arguments of Bf's, while their phases determine phase factors. The diffraction pictures show periodic properties, as far as the orders of Bf's and Fc's are concerned. This periodicity is violated only by the arguments of Bf's. Hence direct calculation of phases of reflections with certain ambiguities is possible. Quasicrystals are characterized by incommensurate modulations with an infinite number of possible Fc's which can not be derived from diffraction patterns. This difficulty is overcome by the observation of extinction rules, limiting the number of reflections observed. The orders of Fc's, Bf's respectively, entering the sum mentioned above, are correlated by convolution properties in diffraction. Taking into account their periodicities and the existence of sets of latent lattices, regularities in the diffraction pattern result which are briefly discussed.     

Phase retrieval and coherent diffraction imaging by a linear scanning pinhole sampling array

We propose a new method for phase retrieval and coherent diffraction imaging by a specially designed pinhole sampling array (PSA) based on a liquid crystal spatial light modulation. We demonstrate that the phase and the amplitude of the wave front passing through a pinhole sampling array plate can be directly extracted from the inverse Fourier transform of the far-field diffraction intensity pattern. Scanning the whole surface of the wave front by such a series of the PSA plates, we can assemble the extracted complex amplitude to a two-dimensional discrete distribution of the sampled wave front covering the entire PSA plate plane in the scanning consequence. We called it linear scanning pinhole sampling array (LSPSA). Thus the wave front can be reconstructed which avoids any iterative algorithms. Furthermore, it provides a feasible approach for lensless coherent diffraction imaging in real-time.     

Observation of antiphase dynamics and harmonic resonance in a modulated dual-wavelength laser

We observe a nonlinear response of a dual-wavelength Nd:YAG laser when subjected to low-frequency periodic modulations of cavity losses. The modulation frequency is far from the relaxation oscillation frequency. The harmonic resonances of the two laser wavelengths associated with antiphase intensity oscillations are demonstrated and resonances up to the fourth order were observed. For relatively weak modulation, the intensity oscillation frequency of the laser is equal to the modulation frequency. Harmonic resonances occur under a stronger modulation. We find that more harmonic components appear when the modulation frequency is increased. Furthermore, with enhancing the modulation, the dominant frequency of the intensity oscillations of both wavelengths is shifted toward the higher-order harmonic frequency.     

Competition between local potentials and attractive particle-particle interactions in superlattices

Naturally occuring or man-made systems displaying periodic spatial modulations of their properties on a nanoscale constitute superlattices. Such modulated structures are important both as prototypes of simple nanotechnological devices and as particular examples of emerging spatial inhomogeneity in interacting many-electron systems. Here we investigate the effect different types of modulation of the system parameters have on the ground-state energy and the charge-density distribution of the system. The superlattices are described by the inhomogeneous attractive Hubbard model, and the calculations are performed by density-functional and density-matrix renormalization group techniques. We find that modulations in local electric potentials are much more effective in shaping the system’s properties than modulations in the attractive on-site interaction. This is the same conclusion we previously [M.F. Silva, N.A. Lima, A.L. Malvezzi, K. Capelle, Phys. Rev. B 71 (2005) 125130.] obtained for repulsive interactions, suggesting that it is not an artifact of a specific state, but a general property of modulated structures.