Non-relativistic Green Function Research Articles

Relativistic many-body XMCD theory including core degenerate effects

A many-body relativistic theory to analyze X-ray Magnetic Circular Dichroism (XMCD) spectra has been developed on the basis of relativistic quantum electrodynamic (QED) Keldysh Green's function approach. This theoretical framework enables us to handle relativistic many-body effects in terms of correlated nonrelativistic Green's function and relativistic correction operator Q, which naturally incorporates radiation field screening and other optical field effects in addition to electron-electron interactions. The former can describe the intensity ratio of L2/L3 which deviates from the statistical weight (branching ratio) 1/2.In addition to these effects, we consider the degenerate or nearly degenerate effects of core levels from which photoelectrons are excited. In XPS spectra, for example in Rh 3d sub level excitations, their peak shapes are quite different: This interesting behavior is explained by core-hole moving after the core excitation. We discuss similar problems in X-ray absorption spectra in particular excitation from deep 2p sub levels which are degenerate in each sub levels and nearly degenerate to each other in light elements: The hole left behind is not frozen there. We derive practical multiple scattering formulas which incorporate all those effects.

S-wave Quarkonium Spectral Functions Above Tc

The temperature-dependence of charmonium and bottomonium spectral functions is discussed in a nonrelativistic Green's function approach using different potentials. The results are compared to lattice data.

Ground State Quarkonium Spectral Functions above Deconfinement

We discuss the temperature-dependence of S-wave quarkonium spectral functions in a nonrelativistic Green's function approach and compare these to lattice QCD results.

Nonrelativistic Green's Function for Systems with Position-Dependent Mass

Given a spatially dependent mass we obtain the two-point Green's function for exactly solvable nonrelativistic problems. This is accomplished by mapping the wave equation for these systems into well-known exactly solvable Schrodinger equations with constant mass using point canonical transformation. The one-dimensional oscillator class is considered and examples are given for several mass distributions.

Relativistic corrections in magnetic systems

We present a weak-relativistic limit comparison between the Kohn-Sham-Dirac equation and its approximate form containing the exchange coupling, which is used in almost all relativistic codes of density-functional theory. For these two descriptions, an exact expression of the Dirac Green's function in terms of the non-relativistic Green's function is first derived and then used to calculate the effective Hamiltonian, i.e., Pauli Hamiltonian, and effective velocity operator in the weak-relativistic limit. We point out that, besides neglecting orbital magnetism effects, the approximate Kohn-Sham-Dirac equation also gives relativistic corrections which differ from those of the exact Kohn-Sham-Dirac equation. These differences have quite serious consequences: in particular, the magnetocrystalline anisotropy of an uniaxial ferromagnet and the anisotropic magnetoresistance of a cubic ferromagnet are found from the approximate Kohn-Sham-Dirac equation to be of order $1/c^2$, whereas the correct results obtained from the exact Kohn-Sham-Dirac equation are of order $1/c^4$ . We give a qualitative estimate of the order of magnitude of these spurious terms.

Kinetic equation for strongly interacting dense Fermi systems

We review the non-relativistic Green's-function approach to the kinetic equations for Fermi liquids far from equilibrium. The emphasis is on the consistent treatment of the off-shell motion between collisions and on the non-instant and non-local picture of binary collisions. The resulting kinetic equation is of the Boltzmann type, and it represents an interpolation between the theory of transport in metals and the theory of moderately dense gases. The free motion of particles is renormalised by various mean field and mass corrections in the spirit of Landau's quasiparticles in metals. The collisions are non-local in the spirit of Enskog's theory of non-ideal gases. The collisions are moreover non-instant, a feature which is absent in the theory of gases, but which is shown to be important for dense Fermi systems. In spite of its formal complexity, the presented theory has a simple implementation within the Monte-Carlo simulation schemes. Applications in nuclear physics are given for heavy-ion reactions and the results are compared with the former theory and recent experimental data. The effect of the off-shell motion and the non-local and non-instant collisions on the dynamics of the system can be characterised in terms of thermodynamic functions such as the energy density or the pressure tensor. Non-equilibrium counterparts of these functions and the corresponding balance equations are derived and discussed from two points of view. Firstly, they are used to prove the conservation laws. Secondly, the role of individual microscopic mechanisms in fluxes of particles and momenta and in transformations of the energy is clarified.

Nonrelativistic dynamics of heavy particles near the production threshold

A solution to the Schrödinger equation for the nonrelativistic Green function which is used for describing the heavy quark–antiquark pair production near the threshold in e +e − annihilation is presented. A quick comparison with existing results is given. A choice of the effective mass scale for the nonrelativistic system with Coulomb interaction is discussed.

Top quark production near threshold: NNLO QCD correction

We calculate the cross section of the process e +e −→t t ̄ near threshold by resumming Coulomb-like terms with next-to-next-to-leading (NNLO) accuracy. The nonrelativistic Green function formalism and the method of “direct matching” are used. The NNLO correction turns out to be large, of the same size as the NLO correction. It changes the position and the normalization of the 1 S-peak. The obtained results are compared with results existing in the literature.

Generalized Sturmian expansions of Coulomb Green's functions and two-photon Gordon formula

A useful representation for the radial part of the Coulomb Green's function is derived in the form of a double series in Laguerre polynomials with two free (arbitrary) parameters α and α′. An appropriate choice of α and α′ leads to a cardinal simplification in calculations of matrix elements involving Green's functions. The results are valid both in the nonrelativistic and in the relativistic case of the second-order Dirac equation. The momentum form of the nonrelativistic Green's function with free parameters is presented and a simple example of a two-photon extension of the well-known Gordon formula is given.

2D Saddle Potential Schrödinger Green's Function in the Presence of an Arbitrary Time-Dependent Electric Field

We derive the retarded nonrelativistic Green's function for a Schrödinger electron confined to a plane and subject to a parabolic saddle potential and an electric field having arbitrary time dependence and orientation on the plane. This derivation is carried out using Heisenberg equations of motion for position and momentum operators, following an earlier analysis of Schwinger, to obtain the retarded Green's function in closed form, as an explicit function of position and time variables.

A unified treatment of the non-relativistic and relativistic hydrogen atom II: the Green functions

For pt.I see ibid p.79-94. This is the second in a series of three papers in which it is shown how the radial part of non-relativistic and relativistic hydrogenic bound-state calculations involving the Green functions can be presented in a unified manner. In this paper the nonrelativistic Green function is examined in detail; new functional forms are presented and a clear mathematical progression is shown to link these and most other known forms. A linear transformation of the four radial parts of the relativistic Green function is given which allows for the presentation of this function as a simple generalization of the non-relativistic Green function. Thus, many properties of the non-relativistic Green function are shown to have simple relativistic generalizations. In particular, new recursion relations of the radial parts of both the non-relativistic and relativistic Green functions are presented, along with new expressions for the double Laplace transforms and recursion relations between the radial matrix elements. A novel proof of the Sturmian form of the radial Green functions is given in an appendix.

Theoretical analysis of spin-resolved photoemission data from ferromagnetic Fe(001)

To interpret recent spin-, energy- and angle-resolved experimental photoemission spectra at photon energies ranging from 20 to 70 eV, one-step calculations on the basis of a non-relativistic Green function formalism have been performed together with a calculation of the corresponding bulk band structure and momentum-resolved layer-by-layer quasi-particle density of states. The theoretical spectra are in good agreement with the data. Individual features are explained in terms of bulk interband transitions and emission from a majority-spin surface resonance. Self-energy corrections are found to be important and in qualitative disagreement with recent microscopic theory.

Perturbation theory for QED bound states

A new method is presented for deriving a systematic perturbative expansion for QED bound states, which does not rely upon solving any new or old equation. The starting point is a given nonperturbative zeroth order Green's function, obtained by a suitable “relativistic dressing” of the nonrelativistic Green's function for the Schrödinger equation with Coulomb potential, which embodies the Coulombic bound states and is known. The comparison with the complete Green's function as given by standard perturbative QED gives a perturbative kernel which is then used for the expansion of the QED Green's function in terms of the given non-perturbative zeroth order Green's function.

GREEN'S FUNCTION METHOD FOR ENERGY BAND CALCULATIONS IN THE SCALAR RELATIVISTIC APPROXIMATION (SRA-KKR)

In this paper, a Scalar Eelativiatic Green's Function method ( SRA-KKR) neglecting spin-orbit coupling effect has been proposed for the energy band calculation. With this method, only a few modifications to the Non-Relativistic Green's Function method would be needed in order to obtain the relativiatic effect on the electronic band structure of solid.

Effects of barrier penetration on energy spectrum and wave functions in the path integral formalism

We present a systematic approximation scheme for the non-relativistic Green function in the path integral representation which takes into account barrier penetration effects on the energy spectrum and wave functions for potentials having several minima. The method is applied to the cases of a symmetric potential with two minima and to a periodic potential. The energy spectrum is shown to agree with the usual WKB results. For the first mentioned case we also determine the wave functions in the classically allowed and forbidden regions from the residues of the Green function. They agree with those obtained in the standard WKB approximation.

Self-Consistent Energy Bands of Calcium by the Green's-Function Method

The nonrelativistic Green's-function method is used to compute self-consistent energy bands of calcium. The results show the state $U$ to be occupied in the metal and the state $W$ to be unoccupied. Corrections due to relativistic effects and to unfreezing of the core are discussed.

Coulomb Green's Function in Closed Form

A simple derivation of the nonrelativistic Green's function is given. Other representations of the function and their agreement with this and various other representations are discussed. (D.C.W.)