Group theory in solid-state physics is not dead yet alias some recent developments in the use of group theory in solid-state physics

In this article a review is given of the principal applications of group theory in solid-state physics.

Some of these applications are well established, such as the simplification of the forms of tensors representing physical properties of crystals, the labelling of electronic energy band structures, and the study of the splitting of atomic or ionic energy levels in crystals. The general principles involved in these applications are discussed. However, no attempt is made to give a comprehensive review of all the work which has been done in these areas; for further details references are given to the existing literature.

The main intention of the article is to show that apart from the well-established applications, which are adequately described in the existing literature, there have been many new developments in recent years. Group theory has come to be applied to many other types of problems in solid-state physics and these applications have not been discussed extensively in the existing review and textbook literature on the subject. These applications include: the study of the symmetry, in k space, of constant energy surfaces and in particular the symmetry of the Fermi surface; the labelling and the degeneracies of dispersion relations for phonons, magnons, and other kinds of quasiparticles; selection rules for processes involving various particle or quasiparticle states in crystals; structure determination and phase transitions; the use of two-dimensional space groups for surfaces and thin films; and the problem of the symmetry of a (non-magnetic) crystal situated in a uniform external magnetic field. The treatment given in the article is not restricted to the use of the classical point groups and space groups but, where magnetic ordering is important, the appropriate generalized symmetry groups are considered.     

Galaxies: A Very Short Introduction,by J. Gribbin

A brief outline of conventional symmetry studies of crystalline solids using Bravais lattices, point groups and space groups is presented, leading up to the recent development by Shubnikov of the idea of antisymmetry and the subsequent derivation of the black and white Bravais lattices and of the black and white point groups and space groups. These groups are also called magnetic groups or Shubnikov groups.

The results of neutron-diffraction experiments have made it clear that there are many crystals whose structures must be described by one of these Shubnikov groups rather than by one of the ordinary point groups or space groups. The relationship between a structure with the symmetry of one of the Shubnikov point groups and the magnetic properties, paramagnetism, diamagnetism, ferromagnetism, ferrimagnetism or anti-ferromagnetism, which it is able to exhibit is also considered.     

Effects of magnetic field and spin-orbit interaction on energy levels in 1D quantum wire: analytical solution

In this paper, we study the effects of a perpendicular magnetic field and spin-orbit interaction (SOI) on energy levels of a quasi-one-dimensional quantum wire. Here, we solve analytically the Schr?dinger equation to obtain energy levels and wave functions. The results show that the splitting at k?=?0 is not constant and it depends on the subband. This variable splitting is due to a symmetry breaking effect when magnetic field and SOI are simultaneously present.     

Nuclear ground states dressed with rf photons in Mössbauer–Zeeman 57Fe spectroscopy

We have applied the “dressed” state concept, developed in quantum electronics, to the situation in which spin 1/2 ground-state nuclear levels are coupled by rf photons. In particular, we have studied Mössbauer spectroscopy when there is Zeeman splitting of the nuclear levels and a further interaction due to an applied rf-radiation field when the rf frequency is in the neighborhood of the ground-state splitting. The dressed-state approach treats the coupling of the ground nuclear Zeeman levels, due to a radio frequency field, by considering the total system made up of: nucleus, static magnetic field, and rf field as one global quantum system. The energy levels and corresponding eigenstates of the system are calculated as a function of the rf frequency and the magnitude of the rf magnetic flux density. Mössbauer spectra are calculated for the ⁵⁷Fe case in which the source is subjected to both the static and radiation fields while the absorber nuclear levels are unsplit.

     

On the magnetic space group and Raman tensors of antiferromagnetic RbCoF3

The magnetic unit cell of the antiferromagnetic RbCoF₃ is identified theoretically, and the magnetic space group is found to be of type IIIb Shubnikov. The study of the splitting of the fundamental energy level of the magnetic ion Co²⁺(⁴F) at the center of BZ is presented, using the theory of crystalline field adapted for magnetic crystals. The symmetries of the electronic Raman activities are determined by group theory, and the respective Raman tensors are presented. Comparison between the theory and the existing experimental data is discussed.     

L'Étude de la Structure Électronique des Métaux des Terres Rares

In rare earth metals, one can neglect interactions between 4f shells centred on neighbouring sites. The conduction band is occupied by three sd electrons (eventually two in europium and ytterbium). These sd electrons are coupled to the f electrons through an interaction of the form where s _e is the spin of a conduction electron and Sf _i the spin of the ith f electron of a given ion. It is therefore possible to consider two groups of properties:

1. The ones, related to the nature of the conduction electrons, change very little through the series: this is the case of the crystalline structure, of the atomic volume.

2. The others, such as the magnetic properties, are related to the internal shells and vary with the filling of the 4f shell. Experiment shows a correlation between those two groups of properties. De Gennes formalism, essentially valid in the hypothesis of tightly bound 4f electrons, gives a satisfactory picture of the properties of the metals in the second half of the series, but it does not give as good a picture for the first rare earth metals, especially for cerium. In the cerium free atom, the 4f, 5d, 6s states have comparable energies and one might think that, in the trivalent metal, the 4f states are broadened in energy by resonances with the extended sd states, but still do not overlap from one atom to the other. They would then occupy virtual bound states analogous to the virtual bound states described by Blandin and Friedel for the transition impurities in noble metals.

An identical situation seems to occur in ytterbium under pressure: one observes a huge increase of the electrical resistivity which goes back to low values at very high pressures. This might also be the case of the actinide metals, especially of Plutonium, in which the 5f states begin to stabilize. So we have to consider two cases:

1. The 4f electrons occupy bound states.

2. The 4f electrons occupy virtual bound states.

In the first part (§ 2), we use de Gennes formalism for 4f bound states. The energy related to magnetic interactions is computed making the assumption of a spherical Fermi surface. A correlation between the crystalline structure and the magnetic properties shows up. In the second half of the series, one can neglect the crystalline field effects and the total energy is the sum of the magnetic term and of the elastic term due to the contribution of the conduction electrons. For every state of magnetic order, the crystalline structure is well defined, corresponding to the minimum of the total energy, and conversely. It is possible to explain in this manner:

1. The b.c.c. structure of europium, which is unusual for a divalent transition metal.

2. The variation of the c/a ratio of the h.c.p. structure both through the series and with temperature.

3. The anomalies in the thermal expansion coefficient observed below the magnetic order-disorder transitions.

4. The helix pitch of the magnetic configurations of this type.

The anomalies of the thermoelectric power observed at the transition points are related to the different dependences of the spin correlations above and below the transition temperatures. The agreement between theory and experiment is satisfactory. Some discrepancy can be attributed to the rather crude approximation of a spherical Fermi surface.

In the second part (§ 3), we deal with a situation where the 4f electrons occupy virtual bound states. These levels are very narrow, about 10^?2 ev wide, and separated in energy by the correlations between electrons. Using Blandin's formalism we calculate the electrical and magnetic properties associated with such a situation. Calculations lead to very strong magnetic coupling; the indirect interaction between magnetic ions is antiferromagnetic for first nearest neighbours, whereas in the case of 4f bound states it is ferromagnetic. Finally, it is possible to explain the properties of cerium and ytterbium.

1. In Cerium, the two first levels overlap at the Fermi level, in such a way that the f electron be almost entirely distributed in the first level.

2. In ytterbium, under pressure, the fourteenth level comes across and above the Fermi level. The maximum resistivity is obtained for a half filling of this level.

In the third part (§ 4), we attempt to apply this model of virtual bound states to plutonium, although in this metal, the 5f shells have a larger spatial extension than the 4f orbitals in rare earths. Anomalies in several physical properties of plutonium seem to indicate a magnetic transition at about 65° K, but no anomaly shows up in the magnetic susceptibility. Using a virtual bound state model associated with a very small polarization of the 5f states, it is possible to explain all the physical properties of plutonium. This model leads to a very small magnetic moment, that cannot be detected by experiment.     

Some crystalline field effects in metals

We discuss different physical effects which are caused by the crystalline electric field splitting of rare earth ions in metals. Thus, the rare earth ions may be impurities dissolved in a metallic matrix or they may form a regular lattice. In the former case we distinguish between the cases of a normal conducting and a superconducting matrix. The influence of the crystalline field splitting on the properties of the conduction electrons is calculated. In the case of a normal matrix anomalous behaviour of the thermoelectric power is found due to the impurity levels. If the matrix is superconducting large deviations result from the theory of Abrikosov and Gorkov which describes the influence of non-split magnetic impurities on superconductivity. A comparison of the theory with available experiments is presented.

For the case that the rare earth ions form a regular lattice we discuss various aspects of the collective excitations in the paramagnetic state. Special attention is paid to the soft mode problem of exchange induced ferromagnets. Furthermore we discuss the influence of impurities on the excitation spectrum (localized modes, resonant modes etc.) and on the magnetic ordering temperature.     

Dual model for magnetic excitations and superconductivity in UPd 2 Al 3

The Heavy Fermion state in UPd₂Al₃ may be approximately described by a dual model where two of the three U-5 f electrons are in a localized state split by the crystalline electric field into two low lying singlets with a splitting energy Δ≃ 6 meV. The third 5 f electron has itinerant character and forms the Heavy Electron bands. Inelastic neutron scattering and tunneling experiments suggest that magnetic excitons, the collective propagating crystal field excitations of the localized 5 f electrons, mediate superconducting (sc) pairing in UPd₂Al₃. A theory for this novel mechanism is developed within a nonretarded approach. A model for the magnetic exciton bands is analyzed and compared with experiment. The sc pair potential which they mediate is derived and the gap equations are solved. It is shown that this mechanism favors an odd parity state which is nondegenerate due to the combined symmetry breaking by the crystalline electric field and the AF order parameter. A hybrid model including the spin fluctuation contribution to the pairing is also discussed. Received 22 October 2001 and Received in final form 28 February 2002     

Anisotropic p−ƒ mixing mechanism explaining anomalous magnetic properties in Ce monopnictides

Anomalously small crystalline field splitting in the paramagnetic region and extremely strong magnetic anisotropy in the ordered phases in CeSb and CeBi are explained based on the anisotropic mixing mechanism between the 4? states and the valence bands. In the paramagnetic region, the mixing gives the effective crystalline field splitting which is estimated to cancel the splitting of the point charge model in good agreement with experiment. The anisotropy energy of CeSb calculated by using the realistic valence bands is consistent with the strong anisotropy observed experimentally.     

Photon and axion splitting in an inhomogeneous magnetic field

The axion–photon system in an external magnetic field, when the direction of propagation of axions and photons is orthogonal to the direction of the external magnetic field, displays a continuous axion–photon duality symmetry in the limit the axion mass is neglected. The conservation law that follow in this effective (2+1)$(2+1)$-dimensional theory from this symmetry is obtained. The magnetic field interaction is seen to be equivalent to first order to the interaction of a complex charged field with an external electric potential, where this fictitious “electric potential” is proportional to the external magnetic field. This allows one to solve for the scattering amplitudes using already known scalar QED results. From the scalar QED analog the axion and the photon are symmetric and antisymmetric combinations of particle and antiparticle. If one considers therefore scattering experiments in which the two spatial dimensions of the effective theory are involved nontrivially, one observes that both particle and antiparticle components of photons and axions are preferentially scattered in different directions, thus producing the splitting or decomposition of the photon and axion into their particle and antiparticle components in an inhomogeneous magnetic field. This observable in principle effect is of first order in the axion–photon coupling, unlike the “light shining through a wall phenomena”, which is second order.     

Zeeman splitting factor of the Er3+ ion in a crystal field

Numerical computations are presented on the energy levels of the Er³⁺ ion in crystalline fields of cubic, trigonal, tetragonal and orthorhombic symmetry. Zeeman splitting factors were obtained from the level splitting in an additional magnetic field. For the quartet Γ₈ states in cubic symmetry the Zeeman effect is described by an effective Hamiltonian ℋ= gμ_BBJ+ uμ_BBJ³ with the parametersg andu calculated for mixed fourth- and sixth-order potentials. For the eight doublets in the lower symmetry of an axial trigonal or tetragonal crystal field the principalg tensor components g_∥ and g_⊥ were calculated. The results of such calculations for a ground-state doublet can exactly account for the experimental data obtained on around 70 erbium centers in various crystalline hosts. However, sometimes different sets of parameters give comparably good results. An empirical rule of constant trace g_∥ + 2g_⊥ is supported by the calculations. In contrast to analytical treatments the effect of the crystalline field can be followed over a continuous range of the crystal field parameters. This allows one to establish relations on the relative signs of tensor components. It is found that the measured trace of tensors |g_∥| + 2|g_⊥| is not always equal to their real trace g_∥ + 2g_⊥. In an exploratory calculation a nonaxial center was simulated in an orthorhombic field, with calculation of the three principal values g_x, g_y and g_z. A good agreement is obtained for the recently reportedg values of an erbium center in silicon.     

Spezifische Wärme der Nitride Seltener Erden

The specific heat of rare earth nitrides has been measured at temperatures between 1.7 ans 270°K by means of an adiabatic calorimeter. The various terms contributing to the total specific heat have been separated. The total splitting of the ground-state multiplet of the trivalent RE-ions in the octahedral crystal field has been determined. The experimental values agree reasonably well with those calculated with the point charge model. The spontaneous magnetization and the magnetic specific heat below the ordering temperature are calculated with the molecular field theory by taking into account all the 2J+1 energy levels of the ground-state multiplet and including exchange interaction and crystal field splitting. Comparision is made with the experimental results, and the values obtained for the exchange energies are listed.     

Gruppentheoretische Behandlung der Aufspaltung von Kristalltermen vermöge der Wechselwirkung äquivalenter Gitterteilchen

First an analysis of the Hamiltonians related toBethe's crystal levels and to the energy levels of the whole crystal is given. Then the splitting ofBethe's levels by the interaction between the electrons of equivalent lattice particles is described in the group ring (factor group splitting,Davydov splitting). By thisWinston's rule derived for this splitting is deduced from a new group theoretical point of view. FinallyBethe's levels degenerated according to time reversal symmetry are considered and spin is regarded.     

Morphic effects — III. Effects of an external magnetic field on the long wavelength optical phonons

The changes in the frequencies of the k ≈ 0 optical vibration modes on the application of a static, external magnetic field to a non-magnetic crystal are determined to first order in the field strength. Second order effects are equivalent to the effects of an electric field in second order and they are not considered here. It is shown that the frequency of a nondegenerate mode is not altered to first order in the magnetic field. In the case of the noncubic crystal structures it is found that the magnetic field must have a component along the axis of highest symmetry in order that the doubly degenerate modes at k ≈ 0 have their degeneracy lifted. In the case of the cubic structures a magnetic field applied in any direction can completely split the degeneracy of modes which are triply degenerate at k ≈ 0. Expressions are given for the field induced changes in the normal mode frequencies. The modes whose frequencies are shifted are found to be right or left circularly polarized. A brief discussion is given of spatial dispersion effects, that is, splitting of the mode degeneracy linear in the phonon wave-vector. Finally, a review of the symmetry aspects of acoustical activity and Faraday effects of acoustical phonons is presented.     

Optical properties of semiconducting carbon nanotubes under axial magnetic field

The linear polarizability absorption spectra of semiconducting carbon nanotubes under axial magnetic field (B) have been calculated by the π-orbital tight-binding model and sum-over-state method. We have found that the optical spectra are split by the B-induced symmetry breaking and the amount of splitting increases with increase of magnetic field. Although the results are obtained within the noninteracting tight-binding model, the amount of splitting is still consistent with the experimental observation, offering a fast estimation of the B-induced splitting. Our numerical results also indicate that the splitting amounts of the second and third absorption peaks are close to that of the first one, which may be observed by the future experiments.     

Optical phonons of Rare-Earth halides in a magnetic field

A theory is developed which explains the observed magnetic field splitting of doubly degenerateE-phonons in Rare Earth Chlorides. It is based on first and second order magnetoelastic interactions. Quantitative calculations are done for CeCl₃ for which the field dependence of the splitting is derived and estimates of the coupling constants are given.Furthermore we propose a mechanism for the observed reduction of the phononlinewidth in a magnetic field.     

Thermalization of the electron-hole-liquid in Ge between magnetic field split valleys

The electron-hole liquid luminescence intensity and luminescence lineshape in Ge in magnetic fields H | 〈111〉 up to 18 T are studied. The field positions of the Landau oscillations and the absence of any splitting of the luminescence band establish that rapid intervalley thermalization between magnetic field split valleys occurs. Electrons in the two types of valley from one equilibrium system with common Fermi energy. These findings are contrasted with recent uniaxial stress and magnetic field results.     

The symmetry group paradox for non-rigid molecules

In many situations, the energy levels for a quantum system, whose Hamiltonian is invariant under a specific symmetry group, are split when the Hamiltonian is replaced by a new one with lower symmetry. In non-rigid molecules (NRM), fast quantum tunnelling processes allow the molecule to change between different geometrical configurations related by permutations of identical nuclei (or with inversion as well), resulting in the splitting of the energy levels for the rigid molecule (RM) case where tunnelling is absent. However, for NRM, there is apparently a paradoxical situation where although the original RM energy levels are associated with a symmetry group isomorphic to the point group for the geometrical configuration, the split NRM energy levels are associated with a symmetry group consisting of all permutations and inversions related to the fast quantum tunnelling processes between configurations, and for which the point group is a subgroup. The resolution of this paradox, where energy level splitting is evidently accompanied by an enlargement of the symmetry group, is the subject of this article.     

Fundamental Solutions of the Axial Symmetric Goursat Problem

Abstract

Fundamental solutions (FS) with a given boundary condition on the characteristics of relativistic problems with axial symmetry are considered. This is so-called the Goursat problem (GP) or zero plane formalism in Dirac’s terminology, or modification of the proper time method in the Fock-Nambu-Schwinger formalism (FNS).

Closed FS for the Volkov problem from the point of view of GP can be found. This means that integration over proper time in a FNS integral transformation can be performed. Using the special chosen dynamic symmetry of the initial state, FS for a particle in constant magnetic or constant electric field may also be calculated.     

Perturbative approach to the hydrogen atom in a strong magnetic field

We consider states of the hydrogen atom with the principal quantum number n≤3 and zero magnetic quantum number in a constant homogeneous magnetic field ?. The perturbation theory series is summed using the Borel transformation and conformal mapping of the Borel variable. Convergence of the approximate energy eigenvalues and their agreement with the corresponding existing results are observed for external fields up to n³?/?₀～5, where ?₀ is the atomic magnetic field. The possibility of restoring the asymptotic behavior of energy levels using perturbation theory coefficients is also discussed.