Phonon Green Function Research Articles

Electron—Phonon Interaction in Disordered Transition Metal Alloys

AbstractThe Hamiltonian of the electron‐phonon interaction is derived and applied to the description of disordered transition metal alloys. The coupled set of equations for the electron and phonon Green functions is derived with the help of the equation‐of‐motion technique. Neglecting the vertex corrections in the self‐energy operators the closed self‐consistent system of equations is obtained. The relevant configurational averaging is performed in the framework of the CPA. The electronic specific heat in the low‐temperature limit is calculated.

Deformation of quasiharmonic crystals

In the quasiharmonic approximation, the phonon Green's function of crystal deformation is determined and the dependence of the frequency on the deformation is found. The results obtained are applied to the description of isothermal and adiabatic elastic moduli.

Damping of low-lying collective modes in TbVO4near the structural phase transition

The damping of collective modes in TbVO4 near the structural phase transition is calculated with the use of a perturbation method applied to a pseudospin-phonon model. A pseudofermion representation is utilised for the pseudospin operator in order to take into account an intrinsic effect of fluctuation field caused by surrounding pseudospins on a given pseudospin. It is shown that the spectrum of singlet pair excitations splits into a pair of spin waves by the fluctuation field and at the same time it gives the damping of the collective modes of the order of the experimental values. The damping of both the acoustic phonon and the central doublet excitation are also derived by using relationships which hold exactly between pseudospin and phonon Green functions.

The coupled magnon—Phonon modes in magnetic crystals with shubnikov symmetry

The method of calculation of hybrid magnon—phonon states in an arbitrary magnetic crystal with the Shubnikov magnetic symmetry is worked out. The method is founded on the system of accurate equations for the phonon, magnon and hybrid Green functions. With the help of the corepresentations theory, this system is reduced to the independent susbsystems containing a small number of equations. The secular equation determining the frequencies of hybrid modes is obtained. The power of this equation is determined by the dimension of irreducible representation corresponding to phonon and magnon modes participating in the formation of the hybrid state, and also by the containance of this representation in the vibrational and spin-wave representations. The handy working formulas is derived. The common theory is illustrated with the example of antiferromagnetic FeF 2.

Strong-coupled electron—acoustic-phonon system in one dimension

A systematic strong-coupling perturbation theory is applied to the one-dimensional model of an electron interacting with acoustic phonons through the deformation potential. The Hamiltonian is diagonalized to the next order in the inverse coupling constant beyond the strong-coupling limit. The energy to this order involves a shift in the density of phonon modes due to the electron-phonon interaction. This shift is analyzed in terms of a phonon Green's function and is found to contain a resonance at phonon frequencies $\ensuremath{\sim}\frac{s}{R}$ where $s$ is the speed of sound and $R$ is the size of the polaron. There is also a depletion in the density of phonon modes due to the polaron, which iteself appears as a translational mode in the theory. This depletion is related to a form of Levinson's theorem.

Diagrammatic Perturbation Theory for a Spin–Phonon System

AbstractIn a diagrammatic expansion of the phonon Green function for a coupled spin–phonon system an essential quantity that enters the calculation is the polarisation matrix associated with the generalised spin susceptibility. Averages of time‐ordered products of spin operators that occur have been conveniently expressed in terms of semi‐invariants and are evaluated using a diagrammatic technique. Hitherto, only the zeroth‐ and second‐order contributions to the polarisation matrix are derived for a system of paramagnetic ions coupled to the phonons of the host lattice through an interaction that is linear in the strain. Near a spin resonance frequency, higherorder terms in the polarisation matrix become important. The complete expression for the fourthorder contribution to the polarisation matrix is reported. The result is valid for all temperatures and the concentration dependence of the paramagnetic ions is accounted for.

Theory of resonant Brillouin scattering via exciton-polaritons

The backscattering of normally incident light by a medium in which there are two polariton branches is considered. The exciton boundary condition is written as u+ mu du/dz=0, and explicit results are given for this form. The phonon Green function which governs the scattering includes the reflection term which is of importance in opacity broadening. If the wavevector of the scattered light does not vary significantly over the linewidth, then the lineshapes are of the form derived by Dervisch and Loudon (1976); in general numerical evaluation is required. The opacity broadening means that the positions of the Brillouin peaks are shifted from those given by earlier theories. The equivalence between the differential formulation used here and an integrodifferential formulation is established and discussed.

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 2 v 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.

Mode damping by electrostrictive interaction in a uniaxial ferroelectric

AbstractAn electrostrictive‐type coupling model is used to study mode damping in the paraelectric phase of a uniaxial ferroelectric with an anisotropic soft‐mode dispersion. Mode linewidths are obtained by solving the phonon Green's function equation of motion in the self‐consistent phonon approximation (SPA). The dominant contribution to the damping comes from the scattering of a soft optic phonon with the absorption or emission of an acoustic phonon. The soft‐mode damping γs shows a (T – T0)1/2 behaviour while, in the isotropic limit, it becomes linearly dependent on temperature. For large anisotropy, the longitudinal acoustic attenuation α1 ∝ ω2(T – T0)−1/2X XIn (T – T0)−1/2 or ω2(T – T0)−1/2 according as the soft‐mode is under or over‐damped in the opposite limit, the corresponding dependences are ω2(T – T0)−1/2 and ω2(T – T0)−3/2, respectively. If soft‐mode interaction with other modes (e.g. phonon density fluctuations) are considered, γs becomes nearly temperature‐independent in this case α1 behaves like ω2 (T – T0)−1 which compares favourably with experimental observations on lead germanate where the soft‐mode is overdamped.

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 4 v . 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.

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

Abstract 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 4 v . The phonon Green functions matrix 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.

Defect modes due to substitutional impurities in FCC lattice: Green's function approach

Lattice dynamical study of monoatomic fcc crystals containing substitutional impurities has been made by the Green's function technique, using group theory. The impurity and its 12 nearest neighbors constitute an XY 12 impurity space having O h symmetry. The phonon Green function matrix is analyzed according to the irreducible representations of the point group pertaining to the substitutional impurity in the fcc lattice. The effects due to change in mass at the impurity site and the change in nearest neighbor force constants for the impurity-host atom interactions are taken into account. Analytical expressions for the various modes of vibrations pertaining to the defect space have been obtained. Local mode frequencies due to various substitutional impurities, corresponding to F 1u mode (defect atom moving) have been computed. A special model is chosen for the defecthost force and it is assumed that there are no distortions of the lattice structure due to the defect. A Kihara hardcore potential with parameters fitted to neutron data has been used to compute lattice dynamics and Green functions of the host lattices. Our theoretical results have been compared with available experimental and theoretical results. Our results show reasonably good agreement with experimental results.

Phonon scattering by paramagnetic ions with S>1/2. I. Zeeman levels in the absence of a zero-field splitting

In an interacting spin-phonon system, coherences in the phonon scattering from different ions lead to coupled spin-phonon modes. The lifetime of these excitations in the phonon regime is calculated for a crystal containing ions with S>1/2. The poles of the phonon Green function are related directly to those of a generalized cumulant spin Green function for which a diagrammatic expansion in terms of a polarization matrix is presented. Averages of time-ordered products of spin operators occur and these are expressed in terms of semi-invariants. To evaluate the latter, a generalized Wick's theorem is developed that deals directly with operators that are of higher order than linear in the spin components. A detailed analysis of the attenuation is made in the case when paramagnetic ions are in sites of octahedral symmetry in a cubic crystal. An explicit expression is given in a situation of experimental interest, namely, for transverse wave propagation along a cubic axis with the propagation and polarization vectors respectively perpendicular and parallel to the axis of quantization.

Lattice dynamics of rare‐gas solids by Kihara's potential III. Green functions

AbstractThe phonon Green functions are presented for solid argon, krypton, and xenon. The Green functions are computed by studying the lattice dynamics of these solids, using the Kihara hard core potential. A set of potential parameters fitted to neutron data is used in the computation. The localized mode due to neon in argon is studied as an example of the use of these Green functions in defect problems.

Phonon scattering by paramagnetic ions in a concentrated paramagnetic crystal

In an interacting spin-phonon system, coherences in the phonon scattering from different ions lead to coupled spin-phonon modes. The lifetime of the excitations is calculated for a fully concentrated crystal containing ions with S=1/2. The poles of the phonon Green function are related directly to those of the generalized spin susceptibility for which a diagrammatic expansion in terms of a polarization matrix M is presented. Averages of time-ordered products of spin operators that occur are expressed in terms of semi-invariants and evaluated using the Vaks diagrammatic technique. Inclusion of the diagrams in the expansion of M to second order lead away from resonance, to a phonon lifetime consistent with that obtained in previous treatments using the drone-fermion representation. A new feature of the present method however is that a partial summation of M to all orders, at least at low temperatures, is possible that enables the lifetime of the coupled excitations to be found over the entire frequency range. Results for the lifetime of the excitations close to the spin resonance frequency are presented.

A continued fraction representation of the mass operator

We explore some further possibilities of application of the projection operator method of Zwanzig to the theory of Green's functions of quantum statistical mechanics, initiated by lchiyanagi, and present a continued fraction representation of the mass operator involving a hierarchy of the random forces. As an application of the theory, we calculate the polarization operator of the phonon Green's function of the Frohlich Hamiltonian in the first approximation which corresponds to the assumption that the electron momenta are orthogonal to the phonon momentum.

Analytical results for torsional vibrations of polymers: dynamical matrix for random conformations and phonon Green function for Pitzer's model

The matrix elements of the lattice Green function for Pitzer's model (describing the out-of-plane skeletal vibrations of an extended polyethylene chain) have been calculated analytically and discussed as to singularities, asymptotic behaviour, density of states, incoherent neutron scattering cross section and localized modes due to an isotropic mass defect. In addition, analytical formulae are presented for the torsional part of the dynamical matrix for a chain in arbitrary conformation.

Comparison of the molecular cluster model with the phonon model for Jahn—Teller active impurities in crystals

For comparing the molecular cluster model with the phonon model in describing the Jahn-Teller interaction for magnetic impurities in crystals, the Jahn-Teller energy and the Ham reduction factors are expressed in terms of the phonon Green's function of the host crystal for an orbital triplet coupled to E g modes of vibrations. Numerical results for the case of MgO lattice have been obtained using the phonon Green's function derived from the breathing shell model.

Self-Consistent Theory of Peierls Transition in One-Dimensional Electron-Phonon Systems

The effect of fluctuations on the Peierls transition of one-dimensional systems of the half-filled band is investigated. The spectrum of the phonon Green's function becomes completely softened at the Brillouin zone boundary as T2|log T|-3/2 when T tends to zero.

Calculations on two phonon bound states in diamond and germanium

A careful numerical calculation is reported searching for two phonon bound states in diamond and germanium. Using the experimental phonon dispersion in germanium and an improved force model for diamond we examined the two phonon Green function for a bound state pole, working in a formulation in which Rayleigh's theorems can be used. Our results are consistent with the possibility of a bound state in diamond and germanium.