Anomalous elastic scattering of X-ray photons by the molybdenum atom

The differential cross section for elastic scattering of an X-ray photon by the molybdenum atom is calculated in the region of the ionization threshold of the 1s shell. The calculation is performed in the nonrelativistic approximation taking into account the rearrangement of electron shells of the atomic core in the field of the virtual 1 s vacancy and processes of one-photon virtual excitation (ionization) of the ground state. Good agreement with experimental results is obtained     

Resonance inelastic scattering of an x-ray photon by the xenon atom

The effect of many-particle and relativistic effects on the absolute values and the shape of the doubly differential cross section of resonance inelastic scattering of a linearly polarized x-ray photon by the free xenon atom in the energy region of the K ionization threshold is studied theoretically. The evolution of the spatially extended structure of the scattering cross section in the Kα,β structure of the x-ray emission spectrum of the xenon atom is demonstrated. The calculations were performed in the dipole approximation for the anomalous dispersion part of the total amplitude of probability of inelastic scattering and in the impulse approximation for the contact part of this amplitude. The radial relaxation of electronic shells, the spin-orbit splitting, the double excitation/ionization of the ground atomic state, as well as the Auger and radiative decays of the atomic core vacancies being formed were taken into account. In constructing the probability amplitude of the process, the relativistic effects were taken into account as a passage from the nonrelativistic Hartree-Fock wave functions to the relativistic Dirac-Hartree-Fock wave functions of the one-particle scattering states and as the passage (for the radiative transition amplitudes) to the relativistic form of the operator of the photon-atom interaction. The calculation results are predictive in character and, at the incident photon energy 34.42 keV, agree well with the results of the synchrotron experiment.     

Elastic scattering of a photon by an atom with an open shell

In the nonrelativistic approximation, the absolute values and the forms of the differential cross sections of the anomalous elastic scattering of a linearly polarized x-ray photon by manganese and copper atoms in the energy region of the ionization threshold of the 1s shell are calculated for the wave functions of one-electron states. We take into account the multiparticle effects of relaxation of atomic shells in the field of core vacancies, multiplet splitting, the configuration interaction, and Auger and radiative decays of vacancies, as well as the processes of double excitation/ionization of the ground state of the atom. The calculation results are in good agreement with the synchrotron experiment.     

Many-particle effects in resonant inelastic scattering of an X-ray photon by an atom

The influence of many-particle effects on the shape and values of the double differential cross section for the resonant inelastic scattering of a linearly polarized X-ray photon by a free atom near the K and KL₂₃ ionization thresholds has been theoretically analyzed for the neon atom. The calculations have been performed using the nonrelativistic Hartree-Fock approximation for single-electron wavefunctions and the dipole approximation for the anomalous dispersion component of the cross section. The analytical structure of the contact part of the scattering cross section has been obtained beyond the dipole approximation. The effects of the radial relaxation of electron shells, spin-orbit and multiplet splitting, and configuration interaction in the doubly excited atomic states, as well as the Auger and radiative decays of the produced vacancies, are taken into account. The nature and role of the effect of correlation amplitudes, which is responsible for the appearance of the nonzero amplitudes of nonradiative transitions between intermediate and final single-electron states of the same symmetry that are obtained in different Hartree-Fock fields, have been analyzed also. The calculations are predictive and, for an incident-photon energy of 5.41 keV, agree well with experimental results for the Kα X-ray emission spectrum of the neon atom.     

Fine structure of the cross section of contact inelastic scattering of an X-ray photon by an atom

The analytical structure of the double differential cross section of contact inelastic scattering of an X-ray photon by a free atom is established by the methods of the theory of irreducible tensor operators beyond impulse approximation. For the example of an Ne atom in the energy region of the ionization threshold of the 1 s shell, the existence of a fine structure of the cross section caused by transitions of electrons of the atomic core to excited states of the discrete spectrum is predicted theoretically. The results of the calculation with the effects of radial relaxation, inelastic scattering via the formation of intermediate states, and elastic Rayleigh scattering taken into account are predictive in character and, at the incident photon energy of 22 keV, are in good agreement with the results of the synchrotron experiment. Original Russian Text ? A.N. Khoperskiĭ, A.M. Nadolinskiĭ, A.S. Kasprzhitskiĭ, 2008, published in Optika i Spektroskopiya, 2008, Vol. 105, No. 1, pp. 5–10.     

Nonresonance Compton scattering of an X-ray photon by an atom with the core of the <Emphasis Type="Italic">d</Emphasis> symmetry

Using the Zn atom as an example, beyond the scopes of the impulse approximation and the incoherent scattering-function approximation, we study the absolute value and the shape of the double differential nonresonance Compton scattering cross section of an X-ray photon by an atom with the core of the d symmetry. We take into account the effects of radial relaxation of shells in the field of core vacancies and of elastic (Thomson and Rayleigh) scattering. Calculation results have a predictive character and, for the incident photon energies of 14.93 and 22.10 keV and the scattering angles of 141° and 133°, agree well with experimentally determined values of the differential Compton scattering cross section.     

Anomalous elastic scattering of X-ray emission by an atom with open shells

In the nonrelativistic approximation for one-electron wave functions, the differential cross section for anomalous scattering of linearly polarized X-ray emission by a sodium atom near the ionization threshold of the 1s ² shell is studied. The analysis takes into account the many-body effects of relaxation of atomic shells in the field of a deep vacancy, multiplet splitting, and the double excitation/ionization processes. The theoretical results obtained for the anomalous-dispersion scattering region are predictive.     

Resonant scattering of an X-ray photon by a heavy atom

The influence of many-body and relativistic effects on the absolute values and shape of the double differential cross section for the resonant scattering of a linearly polarized X-ray photon by a free xenon atom near the K-shell ionization threshold has been theoretically analyzed. The evolution of the spatially extended structure of the scattering cross section to the K _{α, β} structure of the X-ray spectrum of the xenon atom emission has been demonstrated. The calculations have been performed in the dipole approximation for the anomalous dispersion component of the total inelastic scattering amplitude and in the impulse approximation for the contact component of this amplitude. The contribution of the Rayleigh (elastic) scattering component is taken into account using the methods developed in Hopersky et al., J. Phys. B 30, 5131 (1997). The effects of the radial relaxation of the electron shells, spin-orbit splitting, double excitation/ionization of the atomic ground state, as well as the Auger and radiative decays of the produced main vacancies, are considered. Using the results obtained by Tulkki, Phys. Rev. A 32, 3153 (1985) and Biggs et al., At. Data Nucl. Data Tables 16, 201 (1975), the nonrelativistic Hartree-Fock wavefunctions are changed to the relativistic Dirac-Hartree-Fock wavefunctions of the single-particle scattering states when constructing the process probability amplitude. The calculations are predicting and are in good agreement with the synchrotron experiment on the measurement of the absolute values and shape of the double differential cross section for the resonant scattering of an X-ray photon by a free xenon atom reported by Czerwinski et al., Z. Phys. A 322, 183 (1985).     

Resonance inelastic scattering of an x-ray photon by a Ne-like atomic ion

The absolute values and the shape of the double differential cross sections of the resonance inelastic scattering of a linearly polarized x-ray photon in the energy region of the ionization threshold of the deep 1s shell of the Ne atom and the Si⁴⁺ and Ar⁸⁺ neon-like ions are calculated in the nonrelativistic approximation for the wave functions of one-electron states and in the dipole approximation for the scattering probability amplitude. The multiparticle effects of radial relaxation of electronic shells in the field of a deep vacancy and of stabilization of the deep vacancy as a neutral atom transforms to a multiply charged positive ion, as well as the effect of the spin-orbit splitting of the 2p valence shell, are taken into account. The calculation results are predictive in character.     

Inelastic scattering of an X-ray photon by a manganese atom

The influence of multiplet and many-particle effects on the shape and magnitude of the double differential cross section of resonant inelastic scattering of a linearly polarized X-ray photon by a free atom with an open shell in the ground state within the energy range of the K ionization threshold is theoretically investigated using a manganese atom as an example. The radial relaxation of electron shells, spin-orbit and multiplet splittings, configuration mixing in intermediate scattering states, and Auger and radiative decays of newly formed vacancies are taken into account. The results of the calculation are predictive.     

Theoretical study of the X-ray Kβ emission spectrum of an argon atom

The influence of many-particle effects on absolute values and the shape of the doubly differential cross section of resonant inelastic scattering of a linearly polarized X-ray photon by a free Ar atom close to the K- and KM ₂₃-ionization thresholds is studied theoretically. The evolution of the spatially extended profile of the scattering cross section to the principal Kβ_1,3 and satellite Kβ₅ structures of the Ar X-ray Kβ emission spectrum is demonstrated. The calculations are performed in the nonrelativistic Hartree-Fock approximation for the wave functions of one-electron states and in the dipole approximation for the anomalous-dispersion amplitude of the scattering probability. The effects of the radial relaxation of electron shells, the correlation loosening, vacuum correlations, the spin-orbit and multiplet splitting, as well as the Auger and radiative decays of the vacancies produced are taken into account. The calculation results have a predictive character and, for the case of the incident photon energies of 3199.2 and 3245.9 eV, they are in good agreement with the results of the synchrotron experiment on measuring the X-ray Kβ emission spectrum of a free Ar atom.     

Resonant inelastic contact scattering of X-ray photons on atoms and ions

The existence of an extended resonance structure outside the X-ray emission regions is theoretically predicted in the total double differential cross section for the scattering of linearly polarized photons on free atoms (ions). This structure is almost entirely determined by inelastic photon scattering of the contact type. The amplitude of the inelastic contact scattering probability is described using an analytical expression for a non-relativistic transition operator, which was previously obtained by the author outside the dipole and momentum approximations. The resonant inelastic contact scattering of X-ray photons on a neon atom and neonlike ions of argon, titanium, and iron has been studied. Calculations were performed in a nonrelativistic approximation for the wave functions of the scattering states, with allowance for many-body effects of the radial relaxation of one-electron orbitals in the Hartree-Fock field of a deep 1s vacancy and (for neon atom) the double excitation/ionization of the ground atomic state.     

Resonant inelastic scattering of an X-ray photon by a linear molecule

The double differential cross section of resonant inelastic scattering of a linearly polarized X-ray photon by a spatially oriented HF molecule in a gas phase is theoretically described in the energy range of the ionization threshold of the deep molecular orbital 1σ. The effects of radial relaxation of the wave functions of the core and excited scattering states in fields of the formed core vacancies, the vibronic effects, and the effects of Auger and radiative decays of the vacancies are taken into account in the nonrelativistic Hartree-Fock approximation. The results of the calculations are predictive and agree well with the results of the experimental measurements of the Kα emission spectrum of the HF molecule at an incident photon energy of 2 keV, which considerably exceeds the energy of the 1σ ionization threshold.     

Effects of radial relaxation and intershell correlations during resonance inelastic photon scattering by an atom

The absolute value and shape of the double differential cross section of the resonance inelastic scattering of a linearly polarized photon in the ionization threshold energy range of the subvalence s shell of the free neon or argon atom are calculated in the nonrelativistic approximation for one-electron wavefunctions and in the dipole approximation for the anomalously dispersive scattering probability amplitude. The effects of radial relaxation, intershell correlations, bremsstrahlung, spin-orbit splitting, and a finite decay width of s vacancies are taken into account. The effects of radial relaxation and intershell correlations substantially affect the near-threshold scattering intensity: they decrease the contribution of the leading Compton anomalously dispersive component of the total cross section calculated in the one-electron approximation by several times. The calculation results have a predictive character.     

SCF theory for excited states

Equations of Hartree-Fock type are formulated for a many-electron system described by a wave function containing any number of open and closed shells. The solutions define ‘optimal orbitals’ which minimize the average energy of the states of a configuration in which each shell contains a specified number of electrons. Analogues of the Koopmans theorem are constructed.

The results obtained are of special interest in discussing highly excited states (e.g. in ESCA and Auger spectroscopy) where ‘holes’ may be present in both inner and outer shells.     

Nonresonance Compton scattering of an X-ray photon by a Ni-like atomic ion

The absolute values and the shape of the double differential cross section for nonresonance Compton scattering of an X-ray photon by an atomic ion with d symmetry in the core are studied theoretically beyond the impulse approximation for the example of the Ni-like atomic ions Zn²⁺, Kr⁸⁺, and Mo¹⁴⁺. It is established that, as the nuclear charge of an ion increases, (a) in all the scattering channels, the leading harmonics of transition to states of the continuous spectrum are concentrated and (b) the integral intensities of nonresonance Compton scattering are increasingly redistributed to the energy region of resonance Lands-berg-Mandelstam-Raman scattering between the threshold of termination of the Compton profile and the line of elastic (Thomson and Rayleigh) scattering. The calculated results are predictive in character.     

Elastic scattering of an electron by the negative lithium ion

In the framework of the many-body theory, the differential and total cross sections of elastic scattering of slow electrons by the negative lithium ion Li⁻ are obtained. Calculations are performed both in the Hartree-Fock single-particle approximation and with regard to many-electron correlations, which take into account the dynamic polarization of the core. Features observed in the behavior of the phases and cross sections for p and d partial waves are associated with resonance scattering of electron waves. Considering the dynamic polarization of the core by an incident electron heightens the diffraction character of the scattering. The real process is compared with particle scattering in models with a repulsive potential.     

Hartree-fock equations for atoms with two open shells

We have formulated the Hartree-Fock equations for multielectron systems with two open shells (the Huzinaga method) in terms of the density matrix in the LCAO approximation. In order to solve the Hartree-Fock equations, in the algebraic approximation we have obtained expressions for the derivatives of the energy with respect to the density matrix elements and the nonlinear atomic orbital parameters (the orbital exponents). We discuss the question of calculating the open shell parameters (the vector coupling coefficients) in the configurations s¹p^N and s¹d^N within the Huzinaga method. We have calculated the energy for a series of atoms with two open shells in these configurations. Using rather narrow basis sets of Slater-type atomic orbitals), we have obtained energy values close to the results of a numerical solution of the Hartree-Fock equations with sufficiently high accuracy of the virial ratio. __________ Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 74, No. 2, pp. 145–152, March–April, 2007     

Large scale multi-configuration Hartree-Fock calculation of the hyperfine structure of the ground state of vanadium

The hyperfine structure of the ground state of vanadium, ⁵¹VI, is calculated in the nonrelativistic framework of the multi-configuration Hartree-Fock approximation. A configuration state function limiting algorithm is used to make the calculations feasible and to study the influence of core, valence and core-valence correlations in detail. The obtained configuration state function space captures the most important orbital correlations within 2%. Further correlations are included through configuration interaction calculation. The atomic state functions are used to evaluate the magnetic dipole hyperfine factor A and the electric quadrupole factor B. It turns out that the ab initio calculation can not capture the core polarization of the 2s shell. It introduces an error that is higher than the Hartree-Fock approximation. However, the detailed correlations being observed suggest the introduction of a wrong correlation orbital due to the algorithm being used. Neglecting this orbital leads to good agreement with 2% deviation from the experimental values for the A factors.