A point group approach to selection rules in crystals

The problem of generation of the selection rules for a transition between Bloch states at any point of the Brillouin zone in crystals is equivalent to the problem of the decomposition of Kronecker products of two representations (reps) of a space group into irreducible components (the full group method). This problem can be solved also by the subgroup method where small reps of little groups are used. In this article, we propose a third method of generation of the selection rules, which is formulated in terms of projective reps of crystal point groups. It is based on a well-known relation between small irreducible reps (irreps) of little space groups and projective irreps of the corresponding little cogroups. The proposed procedure is illustrated by calculations of the Kronecker products for different irreps at the W point of the Brillouin zone for the nonsymmorphic space group O _h ⁷ which is one of the most complicated space groups for the generation of selection rules. As an example, the general procedure suggested is applied to obtain the selection rules for direct and phonon-assisted electrical dipole transitions between certain states in crystals with the space group O _h ⁷ .     

Multiphonon selection rules for cuprous oxide type crystalline systems

Experimental data on the IR and Raman spectra along with the slow coherent neutron scattering data for Cu₂O were examined for assignments of frequencies to phonon modes not only at the center but those at the boundary of the Brillouin zone. The inconsistency in assignments for the zone center modes and the lack of not only assignments but identifications of the zone boundary phonon required the use of second order Raman and IR spectra as the useful probes. The basis for the interpretation of these spectral data is given in terms of reduction coefficients for ordinary products and symmetrized squares and cubes of representations in the “cuprite” structure O_h⁴-Pn3m. They are presented for the R, X, M and Γ points of the Brillouin zone. They will serve for obtaining selection rules, invariants, and covariants for this structure. Additional tables are given that enable one to identify what pair of phonons correspond to electric dipole, magnetic dipole, electric quadrupole, or Raman transitions, respectively.     

Inversion relations,phase transitions and transfer matrix excitations for special spin models in two dimensions

The infrared and Raman active phonons (k=0) of orthorhombic TbF₃ and of isomorphous HoF₃ have been observed and attributed to the different irreducible representations of the factor groupD _2h. Moreover we investigated crystal-field excitations of the 4f-configuration of the Rare-Earth-ions and effects of their interaction with optical phonons, especially as a function of an external magnetic field. The infrared reflection spectra of these materials not only display reststrahlen bands of electric dipolar but also of magnetic dipolar type with a characteristic shape of the latter bands.On entering the ferromagnetic ordered state pertinent symmetry reductions (D _{2hC 2h}) become evident in the system of electronic transitions and optical phonons in TbF₃ by a change of selection rules and a magnetization-induced transfer of transition intensities to new components of the transition dipole or of the Raman polarizability tensor. These observations are in accordance with general symmetry considerations.The delocalization of the crystal-field excitations results in a Davydov splitting of the single-ion transitions.     

Symmetry analysis in neutron diffraction studies of magnetic structures: 3. An example: the magnetic structure of spinels

Using the symmetry analysis technique developed in the two earlier papers of this series a study is made of the magnetic structures in spinels (space group O⁷_h). The magnetic structures with the wave vector K = 0 and those with K ≠ 0 are considered in detail. As an example, an analysis is given of the magnetic ordering in MgV₂O₄ which is characterized by the three-arm star {bdK₁₀} (in Kovalev's notation) and of that in HgCr₂S₄ where a helical structure corresponding to the star {bdK₆} has been found. For each of the three stars we have determined the composition of the magnetic representation and calculated the basis functions of the irreducible representations. For the magnetic structures determined experimentally we have specified the irreducible representations by which these structures should be described. The examples furnished illustrate the typical situations liable to occur when performing symmetry analysis of magnetic structures of crystals.     

Structural distortions in families of perovskite-like crystals

The crystallographic and group theoretical analysis of the structural phase transitions in perovskite and perovskite-like crystals is reviewed. We include ABX₃ perovskites and their relative crystals of ReO₃ type (G₀ = O¹ _h), elpasolites, cryolites and their relatives (G₀ = O⁵ _h), layered crystals of T1AlF₄ series (G₀ = D¹ _4h), Aurivillius and Ruddlesden-Popper series (Go = D¹⁷ _4h). The structures in their initial phase G₀ often contain n layers (n = 1,2,3) of vertex linked octahedra. The distorted phases produced by one kind of tilt and by superposition of tilts in the slabs are enumerated. Most of the tilts correspond to symmetry changes, which can be associated to definite librational lattice modes irreducible representations of the G₀ group. The softening of modes associated to the PT has been found experimentally in many perovskites, elpasolites and layered crystals with n = 1. In contrast, no such soft modes have been found yet for even-layered (n = 2) crystals. Examples of successive phase transitions due to the superposition of tilts in these types of crystals have been collected.     

Modes of vibrations due to (U-center + additive cation impurities) in alkali halides

Local and resonant modes due to hydride ions in various alkali halides containing additive cation impurities have been computed by the Green function technique. Local vibrations due to U-centers in alkali halides have O_h symmetry. When one of the six nearest neighbour cations is replaced by an additive impurity, the site symmetry of the system is lowered from O_h to C_4v. The phonon Green functions matríx is analysed according to the irreducible representation of the point group symmetry pertaining to the substitutional impurity. We have considered the vibrations of the hydride ion and all its six nearest neighbours. Analytical expressions have been derived for various modes of vibrations. Using group theory the 21 × 21 matrix has been block diagonalized into various irreducible representations. The effect of mass changes and the changes in short-range force constants have been taken into account. The computed results of the localized modes have been compared with the available experimental results. Good agreement has been found. Theoretical results on resonant modes are also displayed, which will be of use in future experiments on these systems.     

Polarization selection rules for the allowed optical transitions in europium-terbium double activated calcium tungstate phosphor

The paper is devoted to the problem of the optical anisotropy of the rare-earth ions occupying low-symmetry positions in crystals. The crystal field multiplets arising from LSJ terms of Eu³⁺ and Tb³⁺ ions in the crystal field of calcium tungstate scheelite (CaWO₄) are analyzed (S₄ point symmetry). The selection rules, in particular, polarization rules for the allowed electric dipole optical transitions in the electronic shells of the Eu³⁺ and Tb³⁺ in CaWO₄ host lattice are discussed. Special attention is paid to the study of the angular (polarization) dependence of the two-photon absorption that seems to be an effective tool for the understanding of the complicated optical pattern. The peculiarities of the anisotropy of the two-photon absorption prove to be specific for each allowed dipole transition in S₄ symmetry center.     

Incommensurate phases induced by an electric field

All magnetic symmetry classes for which the two-dimensional representations are physically irreducible are enumerated. These representations admit of the Lifshitz invariant induced by an external electric field aligned parallel to the principal axis of the crystal. In the majority of the cases, this field does not split two-dimensional representations into one-dimensional representations and, therefore, an incommensurate phase arises in an arbitrarily weak field. The representations of the magnetic classes C _4h and C′_4h are considered in close detail. The components of the toroidal momentum are chosen as the components of the order parameter.     

On the two-vector invariants of point groups

The paper is devoted to group-theoretical analysis of a function of two vectors, invariant under some point group (special attention is paid to the O_h group). In particular, the invariants in the direct product of spaces transforming according to the lth and l'th irreducible representations of the rotation group are studied. A compact formula determining the number of such invariants for the group O_h is found. It is shown that all possible invariants in the considered product space can be constructed from all possible scalar products of vector functions of both vectors transforming according to complex conjugate irreducible representations. The addition theorem for these functions is proved.     

Magnetic circular and linear dichroism in cubic high spin d5 complex

Magneto-optical parameters useful for the interpretation of magnetic circular dichroism (MCD) and magnetic linear dichroism (MLD) spectra of high spin d⁵ complexes in tetrahedral or octahedral crystalline fields have been calculated. Two possible intensity borrowing mechanisms—one involving spin-orbit (s.o.) interaction of the ground sextet with the excited quartet states and the other involving the s.o. interaction of an excited charge transfer sextet with the excited quartet states—have been considered for electric dipole transitions in T_d symmetry and for magnetic dipole transitions in O_h symmetry. Of the numerous other mechanisms which may induce electric dipole oscillator strength in O_h symmetry complexes, eight (those in involving odd parity T _1u or T _2u states vibronically induced via t _1u or t _2u vibrational modes) have been investigated in detail. In each of these mechanisms, parameters have been calculated for transitions to all individual s.o. components of all excited quartet states. It is shown how the MCD and MLD techniques, if used in combination, may provide specific information on the mechanism of intensity borrowing and on the spectroscopic assignments of individual s.o. components.     

Peculiarities of the SERS Spectra of 4,4′-Bipyridine Molecule in a Single Molecule Detection Regime

The paper describes briefly main statements of the theory of the SERS spectra with regards to the single molecule regime, when the enhancement achieves the values ~10¹⁴?10¹⁵. Analysis of the spectra of 4,4′-bypyridine, obtained on the dimer lattice of sharp nanoparticles points out that the observed enhancement is caused exclusively by a strong quadrupole light-molecule interaction, which manifests in the presence of lines, caused by vibrations with the unit irreducible representations of the D₂ and D_2h symmetry groups, which apparently describe the symmetry properties of the molecule. The study of the spectra, obtained by tip enhanced spectroscopy demonstrates that the strong quadrupole light-molecule interaction still has a governing role, however the strong dipole interaction still manifests in the existence of very weak forbidden lines. This result apparently is associated with another experimental geometry.     

Magnetic structure of the Nd5Ge3 compound

Neutron diffraction measurements of the Nd₅Ge₃ intermetallic compound with a hexagonal structure (space group P6₃/mcm) have been performed at temperatures of ～10 and 293 K. The basis functions of irreducible representations of the space group D _6h ³ (P6₃/mcm), which are calculated as a result of the symmetry analysis of possible magnetic structures with the wave vector k = μb ₁, are used to facilitate the search for a real model of the magnetic structure of the compound.     

Permutation-inversion symmetry of C70 fullerite in the high-temperature phase

The permutation-inversion symmetry group of C₇₀ fullerite in its high-temperature phase is constructed with allowance for the rotation of its constituent molecules, and the local symmetry group of a rotating molecule in the crystal is identified. Irreducible representations of these groups are constructed that are compatible with the principle of wave-function symmetry with respect to permutations of identical nuclei. A group-theoretic classification is made of the quantum states of a rotating molecule and of the crystal in the high-temperature phase of C₇₀ fullerite. Selection rules are derived for electronic, vibrational, and rotational spectra in terms of irreducible representations of the permutation-inversion symmetry group of the crystal. Fiz. Tverd. Tela (St. Petersburg) 39, 1895–1901 (October 1997)     

The symmetry properties of the normal vibrations of C-15 type superconductor HfV2

The symmetry species of the normal vibrations of HfV₂ are analysed. The symmetry vectors at Г, δ and X, and phonon dispersion ω(δ) are given. The irreducible representation Г'₂ is responsible for the structural transition O_h⁷ → C_2v²⁰.     

Modes of vibrations due to (U-center + additive anion impurities) in KC1—II

Defect modes due to U-centers in potassium chloride containing additive anion impurities have been computed by Green's functions techniques. Local vibrations due to U-centers in alkali halides have cubic symmetry, but when one of the twelve nearest neighbour anions is replaced by an additive impurity, the site symmetry of the system reduces from O_h to C_2v and gives rise to new vibrational modes. The phonon Green's functions have been analysed according to the irreducible representation of the point group symmetry, pertaining to the substitutional impurity. We have considered the vibrations of the U-center, additive anion impurity and their nearest neighbours i.e. 36-dimensional defect space. Using group theory, symmetry coordinates were constructed and the 36 x 36 dimensional matrix was block diagonalised into various irreducible representations. The computed local mode frequencies have been conpared with the experimental measurements. Reasonably good agreement is found between them. The computed results for resonant mode frequencies are also displayed.     

Direct optical absorption selection rules for the hexagonal close-packed lattice

The direct electric dipole optical selection rules between Bloch electronic states at the symmetry points of the Brillouin zone of the hexagonal close-packed lattice are calculated. Bloch wave functions transforming according to the irreducible representations of both the single and double groups are considered.     

Isometric groups,selection rules and irreducible tensors of semi-rigid molecules

A systematic treatment of multipole selection rules of non-rigid molecules is presented, based on the isomorphism of the isometric group to the symmetry group of the rotation-internal motion hamiltonian. A classification of isometric groups and relations among the representations of the isometric group on various substrates are discussed. A set of general transformation formulae for irreducible tensor operators of semi-rigid molecular models is derived. These formulae are used for the derivation of dipole and quadrupole selection rules of a considerable number of semi-rigid models. The relations among the representations of the isometric group give rise to some theorems which allow a compact presentation of selection rules of non-rigid molecules.     

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.