布里渊区内能带E(k)=E(-k)的对称性

说明了布里渊区内能带Ｅ（ｋ）＝Ｅ（－ｋ）的对称性是时间反演对称性的结果，而与晶体的空间对称性无关．并且证明了当晶体具有中心反演对称性时，布里渊区内能带是二重简并的．     

二维光子晶体点群对称对不可约布里渊区的影响

采用平面波展开法,研究了二维光子晶体点群对称对不可约布里渊区的影响。在单柱体二维光子晶体单胞中增加一个柱体,设计成不同点群对称操作数的对称结构,计算了最低5条能带的本征频率值在第一布里渊区的分布图。结果表明:在计算二维光子晶体的能带结构时,如果光子晶体的点群对称操作数为n,所取的不可约布里渊区应为整个第一布里渊区的1/n。     

Lattice-vibrational symmetry of ZnP2 and ZnAs2 in the monoclinic system

The vibrational representation is decomposed into irreducible representations in the symmetry points of the Brillouin zone. The selection rules are found for a one-phonon IR absorption and Raman scattering of light. In the IR absorption, 17 frequencies are active with polarization along the second-order axis and 16 frequencies are active with a perpendicular polarization. There are 36 frequencies active in the Raman scattering. The basis vectors of the normal vibrations are found for symmetry points of the Brillouin zone and the nature of the long-wave vibrations is investigated. The decomposition of the direct product of irreducible representations is found (selection rules).Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 75–78, May, 1976.The authors thank G. F. Karavaev and N. V. Kudryavtsev for consultations.     

Band diagram of spin waves in a two-dimensional magnonic crystal

The dispersion curves of collective spin-wave excitations in a magnonic crystal consisting of a square array of interacting saturated nanodisks have been measured by Brillouin light scattering along the four principal directions of the first Brillouin zone. The experimental data are successfully compared to calculations of the band diagram and of the Brillouin light scattering cross section, performed through the dynamical matrix method extended to include the dipolar interaction between the disks. We found that the fourfold symmetry of the geometrical lattice is reduced by the application of the external field and therefore equivalent directions of the first Brillouin zone are characterized by different dispersion relations of collective spin waves. The dispersion relations are explained through the introduction of a bidimensional effective wave vector that characterizes each mode in this magnonic metamaterial.     

布里渊区中的对称平均点

本文提出了一些布里渊区中的对称平均点,用它可以方便地计算布里渊区中的平均值,其精度可以和特殊点相比而计算量却要小得多。文内给出了立方和六角晶格的对称平均点,并用具体的数值计算例子说明了它的计算精度。 关键词：     

Tunable reducibility of Brillouin zone and bandgap width in elliptical nanowire arrays

Rotating elliptical nanowire arrays as two-dimensional photonic crystals has been proposed and studied in this Letter. The analysis of the four lowest energy bands and the first bandgap width of some examples illustrates that the rotation and configuration of the primitive cell can have effects on the reducibility of the Brillouin zone.As the central element's orientation changes, the irreducible Brillouin zone could be expanded to the whole first Brillouin zone. Special attention has been paid to the nanowire arrays with adjacent elements perpendicular to each other, and the irreducible Brillouin zone unexpectedly retracted back to the 1/8 of the first Brillouin zone though the symmetry of elements is lower than that of the square lattice. Meanwhile, the first bandgap width of the perpendicular array can be adjusted by the rotation of each primitive element.     

Breaking bands degeneracy in two-dimensional hybrid triangular lattice phononic crystal: A theoretical and experimental study

Using symmetry reduction concept, the band gap properties of a two-dimensional hybrid triangular lattice phononic crystal were investigated both theoretically and experimentally. The band structure was calculated with the plane wave expansion (PWE) method. The results revealed that the band degeneracy at high symmetry point in the first Brillouin zone was lifted due to reducing symmetry, and hence a new low-frequency absolute band gap between the first and the second bands appeared. The measured transmission spectra were found to be in good agreement with the numerical results.     

Anisotropic photonic crystals: Generalized plane wave method and dispersion symmetry properties

This article presents a generalized vector plane wave expansion method, applicable to isotropic and anisotropic periodic dielectric media of arbitrary geometry and dimension. The influence of anisotropic material orientation on the symmetry properties of photonic crystal dispersion surface is discussed. It is shown that the overall Brillouin zone symmetry is formed by the intersection of the photonic crystal lattice symmetry and the symmetry determined by the anisotropic material orientation. This work explains how to define the irreducible Brillouin zone of a two-dimensional anisotropic photonic crystal and demonstrates that doing it correctly allows one to avoid erroneous results, when calculating band gap diagrams of anisotropic photonic crystals. With the help of the methods presented, the possibility of controlling the band gaps of anisotropic photonic crystals by means of external electric field is shown.     

Symmetry features of the energy bands of SiP2 with the pyrite structure

Results on the structure of silicon diphosphide are examined. The space group T _h ⁶ (Pa3) is used to derive the matrices for the irreducible representations of the symmetry groups for all types of wave vectors in the Brillouin zone. It is found that time inversion results in additional degeneracy at most points on the surface of the zone.Translated from Izvestiya VUZ. Fizika, No. 12, pp. 92–95, December, 1973.     

D46h结构空间群C-G系数的计算

在晶体拉曼散射张量的计算中,晶体空间群的Ｃ－Ｇ系数具有重要作用。本文利用本征函数法,计算了六角密排结构Ｄ６ｈ＾４空间各的Ｃ－Ｇ系数,同时给出波矢选择规则和Ｃ－Ｇ序列。     

Surface bands of the (001) surface of molybdenum

The electronic structure of the ideal (001) surface of Mo is calculated using a parametrized linear combination of atomic orbitals. Surface states and resonances are identified along the high symmetry directions of the two dimensional Brillouin zone. The local density of states near the surface is presented and the main modifications due to the surface states are discussed. Significant surface features are found in the range 0–0.1 Ryd below the Fermi energy in agreement with the experimental findings.     

Analytical approach to the quantum-phase transition in the one-dimensional spinless Holstein model

We study the one-dimensional Holstein model of spinless fermions interacting with dispersion-less phonons by using a recently developed projector-based renormalization method (PRM). At half-filling the system shows a metal-insulator transition to a Peierls distorted state at a critical electron-phonon coupling where both phases are described within the same theoretical framework. The transition is accompanied by a phonon softening at the Brillouin zone boundary and a gap in the electronic spectrum. For different filling, the phonon softening appears away from the Brillouin zone boundary and thus reflects a different type of broken symmetry state.     

Angle-resolved photoemission from Fe(110): Determination of E()

Using angle-resolved photoemission spectroscopy utilizing polarized synchrotron radiation with a clean Fe(110) sample, we identified each of the Σ₁-, Σ₃- and Σ₄-band peaks with certainty using symmetry selection rules. We also determined energy band dispersion along the γ-Σ-N direction in the bulk Brillouin zone assuming direct transitions. There are small discrepancies between the observed bands and recent ground-state band calculations possibly due to Coulomb correlation effects. A surface state with odd symmetry about the [001] crystal direction and a binding energy of 0.15 eV at γ in the surface Brillouin zone was observed. Our results show that the ground-state calculations of ferromagnetic iron based on the local exchange correlation potential are basically in agreement with experimental dispersion relationship measurements.     

An LCAO calculation of the band structure of cadmium oxide

An LCAO method has been used to calculate the energy band structure of cadmium oxide along certain symmetry lines in the Brillouin zone. A satisfactory account of the observed direct and indirect band gaps is obtained.     

First principles studies of the dependence of magnetism on the crystal phase in 4d and 5d late transition metals

We investigate the possibility of inducing ferromagnetic order in 4d and 5d late transition metals through crystal symmetry change. First principles, self-consistent density functional theory calculations, with spin-orbit coupling included, performed at 0 K show that ferromagnetism occurs in the bulk of Rh and Pd at the optimum lattice constant if Rh is in the bcc and Pd in the hcp/dhcp phase. The ferromagnetic order originates in the d-band occupancy of Rh or Pd which locates the Fermi energy at the top of the highest peak of the respective (paramagnetic) density of states induced by the bcc or hcp/dhcp structure. This peak in the density of states is caused by flat bands which lie at the surface of the respective Brillouin zone. For a bcc crystal these flat bands have the eg character and are positioned at the surface of the bcc Brillouin zone along the N-P line. The origin of the flatness of the bands was found to be the translation symmetry of the cubic lattice which causes the bands with the eg character to be narrow along the k-lines whose k-vector directions are furthest off the directions to which the orbitals of the eg symmetry point. Due to the d-band occupancy of Rh these flat bands lie in the paramagnetic state at the Fermi energy, whereas in the ferromagnetic state they exhibit the largest energetic split. This indicates that a smaller degree of orbital overlap narrows electronic bands enhancing the tendency of the system for ferromagnetic band split. For the hcp/dhcp structure the states contributing to the high density of para-magnetic states at the Fermi level of Pd lie in the vicinity of the M-L line of the hcp Brillouin zone boundary, which possesses a high number of symmetry (M and L) points. Moreover, the M-L line is aligned with the stacking sequence direction ([0001]) which is furthest off the densest-packed atomic chain direction of an hcp-crystal and, consequently, the weakest-bond direction in the crystal. This makes the narrow bands along the M-L line flat. The instability of the bcc and the meta-stability of the hcp crystal phase modifications for metals with native close-packed crystal structures is subsequently analysed in order to find whether they can be grown as films on suitable substrates.     

Symmetry of lattice vibrations of hexagonal Cu2Te

The vibrational representation is expanded in terms of irreducible representations at the principal symmetry points of the Brillouin zone. Selection rules are found for single-phonon IR absorption, Raman scattering, and Mandel'shtam-Brillouin scattering of light. In IR absorption, a single frequency is active for light polarized parallel to the sixth order axis and a single frequency for perpendicular polarization. Five frequencies are active in Raman scattering of light. Expansions of the direct product of irreducible representations (selection rules) are found.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 45–48, November, 1981.     

Peculiarities of Phonon Spectra of Crystals with Cuprite Lattice Due to Sublattice Structure

Russian Physics Journal - The different translational symmetry of the cuprite crystal sublattices is taken into account by unfolding the phonon spectrum from the Brillouin zone of the crystal into...     

On the specificity of the electron spectrum of two-dimensional lattices

The possible types and features of the electron spectrum at the symmetry points of the Brillouin zone of a two-dimensional lattice have been presented.     

Orientational states of C60 molecule in crystals

Local symmetry of orientational states of the C₆₀ molecule in crystals has been investigated. It was shown that the various orientational phase transitions in different crystals are related to different orientational orbits. The model of orientational phase transitions based on a sequence of orientational states with different symmetry properties has been suggested. We have found that both the local symmetry of C₆₀ molecule and the symmetry of its internal vibrations become higher after a reduction of crystal spatial symmetry at the phase transition. This effect is fairly common and can be observed in the orientational order-disorder phase transitions with wave vectors at the Brillouin zone boundary. Feasible manifestations of the predicted effect in various experiments are discussed. Zh. éksp. Teor. Fiz. 113, 1081–1093 (March 1998)