Energy band structure of multistream quantum electron system

In this paper, using the quantum multistream model, we develop a method to study the electronic band structure of plasmonic excitations in streaming electron gas with arbitrary degree of degeneracy. The multifluid quantum hydrodynamic model is used to obtain N-coupled pseudoforce differential equation system from which the energy band structure of plasmonic excitations is calculated. It is shown that inevitable appearance of energy bands separated by gaps can be due to discrete velocity filaments and their electrostatic mode coupling in the electron gas. Current model also provides an alternative description of collisionless damping and phase mixing, i.e., collective scattering phenomenon within the energy band gaps due to mode coupling between wave-like and particle-like oscillations. The quantum multistream model is further generalized to include virtual streams which is used to calculate the electronic band structure of one-dimensional plasmonic crystals. It is remarked that, unlike the empty lattice approximation in free electron model, energy band gaps exist in plasmon excitations due to the collective electrostatic interactions between electrons. It is also shown that the plasmonic band gap size at first Brillouin zone boundary maximizes at the reciprocal lattice vector, G, close to metallic densities. Furthermore, the electron-lattice binding and electron-phonon coupling strength effects on the electronic band structure are discussed. It is remarked that inevitable formation of energy band structure is a general characteristics of various electromagnetically and gravitationally coupled quantum multistream systems.

The wavevector star channel and symmetry group

The concepts of `wavevector star channel' and `wavevector star channel group' are newly defined, which allow the effective study of phase transitions considering directly the translational symmetry breaking in crystals. A method is suggested by which the wavevector star channels can be found using the image of the representation of the translational group. According to this method, the wavevector star channels are found for the 80 Lifschitz stars in the reciprocal lattice. The wavevector star channel group is defined as the set of symmetry elements of the parent phase which leave the star channel invariant, and the wavevector star channel groups with one, two, three and four arms are calculated. It is shown that the complicated symmetry changes in the pyroelectric crystal Pb1−x Ca x TiO3 (PCT) can be described using the new five-component reducible order parameter transformed according to the representation of the wavevector star channel group, rather than the nine-component one based on the theory of the full irreducible representation of the space group.

Dynamical effects in the integrated X-ray scattering intensity from imperfect crystals in Bragg diffraction geometry. II. Dynamical theory

The analytical expressions for coherent and diffuse components of the integrated reflection coefficient are considered in the case of Bragg diffraction geometry for single crystals containing randomly distributed microdefects. These expressions are analyzed numerically for the cases when the instrumental integration of the diffracted X-ray intensity is performed on one, two or three dimensions in the reciprocal-lattice space. The influence of dynamical effects, i.e. primary extinction and anomalously weak and strong absorption, on the integrated intensities of X-ray scattering is investigated in relation to the crystal structure imperfections.

Umklapp collisions and thermal conductivity

“Umklapp collisions and thermal conductivity” deals with heat conduction in a dielectric solid. Collisions of phonons are divided into Umklapp and normal according as a reciprocal lattice vector is or is not involved in the phonon momentum balance. A local temperature is defined by appeal to local thermodynamic equilibrium. An equilibrium phonon distribution can be off-centred, yet non-decaying, if the only collisions are “normal”, conserving the total phonon momentum. Then heat flow does not decay, even if a representative collision reverses the phonon group velocity. Conversely, in an Umklapp collision it is the non-conservation of phonon momentum that causes heat flow to decay.

Improvement of precision in refinements of structure factors using convergent-beam electron diffraction patterns taken at Bragg-excited conditions

A local structure analysis method based on convergent-beam electron diffraction (CBED) has been used for refining isotropic atomic displacement parameters and five low-order structure factors with sin θ/λ ≤ 0.28 Å−1 of potassium tantalate (KTaO3). Comparison between structure factors determined from CBED patterns taken at the zone-axis (ZA) and Bragg-excited conditions is made in order to discuss their precision and sensitivities. Bragg-excited CBED patterns showed higher precision in the refinement of structure factors than ZA patterns. Consistency between higher precision and sensitivity of the Bragg-excited CBED patterns has been found only for structure factors of the outer zeroth-order Laue-zone reflections with larger reciprocal-lattice vectors. Correlation coefficients among the refined structure factors in the refinement of Bragg-excited patterns are smaller than those of the ZA ones. Such smaller correlation coefficients lead to higher precision in the refinement of structure factors.

Crystallographic analysis of the lattice metric (CALM) from single electron backscatter diffraction or transmission Kikuchi diffraction patterns

A new software is presented for the determination of crystal lattice parameters from the positions and widths of Kikuchi bands in a diffraction pattern. Starting with a single wide-angle Kikuchi pattern of arbitrary resolution and unknown phase, the traces of all visibly diffracting lattice planes are manually derived from four initial Kikuchi band traces via an intuitive graphical user interface. A single Kikuchi bandwidth is then used as reference to scale all reciprocal lattice point distances. Kikuchi band detection, via a filtered Funk transformation, and simultaneous display of the band intensity profile helps users to select band positions and widths. Bandwidths are calculated using the first derivative of the band profiles as excess-deficiency effects have minimal influence. From the reciprocal lattice, the metrics of possible Bravais lattice types are derived for all crystal systems. The measured lattice parameters achieve a precision of <1%, even for good quality Kikuchi diffraction patterns of 400 × 300 pixels. This band-edge detection approach has been validated on several hundred experimental diffraction patterns from phases of different symmetries and random orientations. It produces a systematic lattice parameter offset of up to ±4%, which appears to scale with the mean atomic number or the backscatter coefficient.

On the use of the scattering amplitude in coherent X-ray Bragg diffraction imaging

Lens-less imaging of crystals with coherent X-ray diffraction offers some unique possibilities for strain-field characterization. It relies on numerically retrieving the phase of the scattering amplitude from a crystal illuminated with coherent X-rays. In practice, the algorithms encode this amplitude as a discrete Fourier transform of an effective or Bragg electron density. This short article suggests a detailed route from the classical expression of the (continuous) scattering amplitude to this discrete function. The case of a heterogeneous incident field is specifically detailed. Six assumptions are listed and quantitatively discussed when no such analysis was found in the literature. Details are provided for two of them: the fact that the structure factor varies in the vicinity of the probed reciprocal lattice vector, and the polarization factor, which is heterogeneous along the measured diffraction patterns. With progress in X-ray sources, data acquisition and analysis, it is believed that some approximations will prove inappropriate in the near future.

Crystallography and Diffraction

Crystalline materials have a periodic arrangement of atoms, exhibit long range order, and are described in terms of 14 Bravais lattices, 7 crystal systems, 32 point groups, and 230 space groups, as tabulated in the International Tables for Crystallography. We introduce the nomenclature to describe various features of crystalline materials, and the practically useful concepts of interplanar spacing and zonal equations for interpreting electron diffraction patterns. A crystal is also described as the sum of a lattice and a basis. Practical materials harbor point, line, and planar defects, and their identification and enumeration are important in characterization, for defects significantly affect materials properties. The reciprocal lattice, with a fixed and well-defined relationship to the real lattice from which it is derived, is the key to understanding diffraction. Diffraction is described by Bragg law in real space, and the equivalent Ewald sphere construction and the Laue condition in reciprocal space. Crystallography and diffraction are closely related, as diffraction provides the best methodology to reveal the structure of crystals. The observations of quasi-crystalline materials with five-fold rotational symmetry, inconsistent with lattice translations, has resulted in redefining a crystalline material as “any solid having an essentially discrete diffraction pattern”