Photoelectron Wave Function Research Articles

Inner-shell photoelectron angular distributions from fixed-in-space OCS molecules: comparison between experiment and theory

Photoelectron angular distributions (PADs) for O 1s, C 1s and S 2p1/2, 2p3/2 ionization of OCS molecules have been measured in shape resonance regions. These PAD results are compared with the results for O 1s and C 1s ionization of CO molecules, and multi-scattering Xα (MSXα) calculations. The mechanism of the PAD formation both for parallel and perpendicular transitions differs very significantly in these molecules and a step from a two-centre potential (CO) to a three-centre potential (OCS) plays a principal role in electron scattering and the formation of the resulting PAD. For parallel transitions, it is found that for the S 2p and O 1s ionization the photoelectrons are emitted preferentially in a hemisphere directed to the ionized S and O atom, respectively. In OCS O 1s ionization, the S–C fragment plays the role of a strong ‘scatterer’ for photoelectrons, and in the shape resonance region most intensities of the PADs are concentrated on the region directed to the O atom. The MSXα calculations for perpendicular transitions reproduce the experimental data, but not so well as in the case of parallel transitions. The results of PAD, calculated with different ℓmax on different atomic centres, reveal the important role of the d (ℓ = 2) partial wave for the S atom in the partial wave decompositions of photoelectron wavefunctions.

Theory of CaL2,3-edge XAS using a novel multichannel multiple-scattering method

A new method for calculating X-ray absorption spectroscopy (XAS) at the L2,3 edges of Ca and transition metals is presented. It is based on the multichannel multiple-scattering theory by Natoli et al. [Phys. Rev. B, (1990), 42, 1944-1968] combined with the eigen-channel R-matrix formalism. Atomic multiplet-like effects, owing to the Coulomb interaction of photoelectrons and the 2p hole, are taken into account through a configuration interaction ansatz for the final-state wavefunction. The various multiplet states lead to a set of channels for the photoelectron wavefunction, which is calculated in multiple-scattering theory. The method is applied to Ca, an important element for biological applications of XAS. An L3:L2 branching ratio of 3:4 is found, in good agreement with experiment but in contrast to the statistical value 2:1 obtained in all one-electron approaches. By using a linear mixture between statically screened (approximately 90%) and unscreened (approximately 10%) core-hole potential, the line shape, too, agrees well with the experimental one.

Variationally stable calculations for molecular systems: polarizabilities and two-photon ionization cross section for the hydrogen molecule.

The variationally stable method of Gao and Starace [B. Gao and A. F. Starace, Phys. Rev. Lett. 61, 404 (1988); Phys. Rev. A 39, 4550 (1989)] has been applied for the first time to the study of multiphoton processes in molecular systems. The generalization in theory is presented, as well as the calculation of properties such as the static and dynamic polarizabilities of the hydrogen molecule and the generalized two-photon ionization cross section. The Schwinger variational iterative method [R. R. Lucchese and V. McKoy, Phys. Rev. A 21, 112 (1980)] has been applied in the achievement of the photoelectron wave function, while a Hartree-Fock representation has been used for the target. This research has been motivated by the scarceness of ab initio calculations of molecular multiphoton ionization cross sections in the literature.

Density functional study on the circular dichroism of photoelectron angular distribution from chiral derivatives of oxirane.

The linear combination of atomic orbitals B-spline density functional method has been successfully applied to a series of four chiral derivatives of oxirane, to calculate the photoionization dynamical parameters, the circular dichroism in the angular distribution effect, and to identify trends along the series. The computational algorithm has proven numerically stable and computationally competitive. The photoionization cross section, asymmetry, and dichroic parameter profiles relative to valence orbitals have been systematically studied for the states which retain their nature along the series: the identified trends have been ascribed to the different electronic properties of the substituents. A rather unexpected sensitivity of the dichroic parameter to changes in the electronic structure has been found in many instances, making this dynamical property suitable to investigate the electronic structure of chiral compounds. The magnitude of the circular dichroism in the angular distribution effect does not seem to be associated with the initial state chirality, but rather to be governed by the ability of the delocalized photoelectron wave function to probe the asymmetry of the molecular effective potential.

Angular distributions and dichroism of photoelectrons ejected from fixed-in-space molecules of definite symmetry: Application to the C2v symmetry group

A theoretical investigation of the angular distributions of photoelectrons (ADP) ejected from fixed-in-space molecules or clusters having C2v symmetry is presented. The consideration is based on an expansion of the photoelectron wave functions in symmetry-adapted spherical harmonics. The ADP is derived for linearly polarized, circularly polarized, and unpolarized light in the case when the light beam is parallel to the C2 axis of the molecule. The twofold symmetry of the target reveals itself most clearly for circularly polarized or unpolarized light if the dependence on the azimuthal angle is studied for a fixed polar angle. The circular dichroism in the angular distribution (CDAD) is investigated for the general case of an arbitrary direction of the photon beam. It is shown on model examples that the CDAD has characteristic zeros in the mirror planes of the molecule that may be used to determine the symmetry of the initial state. For linearly polarized light the photoelectron angular distributions have a more complicated dependence on the azimuthal angle, though it also may be used to distinguish between different initial states. The examples presented in this paper demonstrate the general properties of the angular distributions as well as properties of the circular and linear dichroism in the angular distribution.

Quantum theory of the effect of post-collision interaction on the absorption of X-ray photons by a deep atomic shell

A fundamental problem for modern atomic physics, the theoretical description of the many-particle effects of post-collision interactions in the photoionization of deep atomic shells outside the framework of quantum-mechanical perturbation theory is considered. The analytical solution of the stationary many-particle Schrödinger equation for the non-relativistic wavefunction of the system of interacting continuous spectra of atomic states is obtained using a class of weight-functions of Cauchy–Lorentz type. States considered are those resulting from: (i) direct photoionization of a deep shell of a free atom and (ii) Auger-decay of a deep vacancy with two (photo- and Auger-) electrons in the continuous spectrum. The solution takes into account: (a) the rearrangement of the continuum photoelectron wavefunction in the changing field of the atomic residue, and (b) the electrostatic interaction of photo- and Auger-electrons resulting from the Auger-decay of the vacancy. The corresponding analytical structure of the photoionization cross section of the deep atomic shell is derived. A generalization is proposed for: (a) two or more channels of Auger-decay of the deep vacancy, and (b) direct photoexcitation of the deep atomic shell.

MULTICENTERED THEORY OF MOLECULAR PHOTOIONIZATION

A new theory for near-threshold photoionization of inner electrons of atoms confined in multicentered atomic formations, e.g. molecules or clusters, is developed. The formulas for fixed-in-space molecules have been derived. The interaction of the photoelectron in the continuum with atoms that surround the atom being ionized, is replaced by the suitable boundary conditions imposed on the photoelectron wave function at the location of nuclei of these atoms in the molecule. The general formulas derived are used to calculate photoelectron angular distributions of diatomic molecules. The calculated data are in qualitative agreement with experimental data and results of other calculations.

Many-body theory of core-level photoemission by Keldysh Green’s functions

The many-body theory of X-ray photoelectron diffraction (XPD) spectra is developed by using Keldysh’s non-equilibrium Green’s functions, which was basically applied by Caroli et al. We have applied skeleton diagram expansion in terms of dressed one-electron Green’s function G and the screened Coulomb interaction W. The damping photoelectron wave functions under the influence of the optical potential play a central role in the present XPD theory. On the basis of the present theoretical framework, we provide detailed discussion on Debye–Waller and Franck–Condon factors in XPD spectra at finite temperature, resonance effects in particular multiatom resonant photoemission (MARPE) effects, and also on the loss effects. The XPD spectra are simply described by a product of the Franck–Condon factor and the XPD intensity including the Debye–Waller factor. This result is in contrast to the X-ray absorption fine structure (XAFS) where the Franck–Condon factor is smeared out by a convolution product. We have discussed the conditions which give rise to prominent MARPE spectra. In particular, symmetry lowering plays an important role giving rise to prominent MARPE spectra. The angular dependence provides us with useful information on near neighbor atomic arrangement. Some model calculations clarify the basic physical processes of MARPE spectra. To discuss the loss spectra, we stress the importance of the interference effects between the intrinsic and extrinsic losses. The explicit formulas, which include both inelastic and elastic scatterings, are newly derived in the present theoretical framework, and enable us to calculate an XPD pattern associated with losses.

Angular distributions of photoelectrons ejected from fixed-in-space molecules of C3v symmetry group

The angular distributions of photoelectrons from fixed-in-space molecules of C3v symmetry group are studied for the particular case when the light beam is incoming parallel to the C3 rotational axis of the molecule. The consideration is based on the expansion of photoelectron wave functions in symmetry-adapted spherical harmonics. The analytical equations for the angular distributions of photoelectrons ejected by linearly polarized, circularly polarized, and unpolarized light have been derived. The threefold symmetry of the target reveals itself most clearly for circularly polarized and unpolarized light when the dependence on the azimuthal angle is studied for a fixed polar angle. For linearly polarized light the photoelectron angular distributions have a more complicated dependence on the azimuthal angle. On the other hand, with linearly polarized light one can easily distinguish between the a1 and a2 initial states by observing the zeroes of photoelectron intensities in the symmetry plane ejected by light linearly polarized either in the symmetry plane or perpendicular to the symmetry plane. The numerical examples presented in the paper demonstrate the general properties of the angular distributions as well as properties of the circular and linear dichroism in the angular distribution.

Circular dichroism in core-level emission from O/W(110): Experiment and theory

Circular dichroism has been observed in the photoelectron diffraction of chemically shifted W $4f$ photoelectrons from the $\mathrm{W}(110)1\ifmmode\times\else\texttimes\fi{}1\ensuremath{-}\mathrm{O}$ surface. The diffraction peaks from first-layer W atoms bonded most directly to oxygen showed an azimuthal rotation similar to those reported in a prior experiment on Si(001). The maximum rotation angles are also in good agreement with those calculated using the formula $\ensuremath{\Delta}\ensuremath{\varphi}=m/kR{\mathrm{sin}}^{2}\ensuremath{\theta}$ derived previously by Daimon et al. [Jpn. J. Appl. Phys., Part 2 32, L1480 (1993)] and assuming $m=4,$ which is the maximum magnetic quantum number of the photoelectron wave function.

Cross sections and angular distributions of the photoelectron correlation satellites of the Xe atom

A theoretical and experimental study of the behaviour of the correlation satellites arising during 5s photoionization of xenon is presented. Many-body perturbation theory and configuration-interaction techniques have been applied to calculate the wavefunctions of the Xe II ionic states. An expression for the angular distribution parameter of the photoelectrons taking into account final-ionic-state configuration interaction is derived. Photoionization cross sections and angular distribution parameters were calculated for the 5s main line and the majority of the satellite lines and compared with our high-resolution measurements and earlier lower resolution measurements. The differences in the angular distribution parameter dependence on the photon energy for the 5s main line and satellites were analysed in terms of their origin. The most important mechanisms are: interference of several photoionization channels characterizing different orbital momenta of the photoelectron, the mixture of terms with different total orbital momenta in the final ionic state, and the dependence of the photoelectron wavefunctions on the total momentum of the photoelectrons.

Circular dichroism from non-chiral and non-magnetic materials observed with display-type spherical mirror analyzer

Strong circular dichroism in 2-dimensional photoelectron diffraction patterns was observed for the photoelectron from the Si 2p core on the Si(001) surface using a display-type spherical mirror analyzer, although this Si(001) surface has no chirality and magnetism. The forward focusing peaks in the pattern rotate in the same direction as the incident light, i.e., clockwise or counterclockwise when the helicity of the incident circularly polarized light is reversed. These rotations of the pattern are explained by the phenomenon that the wave-front of the photoelectron wavefunction rotates as it propagates outward. This is the first direct observation of the rotation of the wavefront of the electronic wavefunction and clarifies the correspondence between the classical and the quantum mechanical ideas of angular momentum. This phenomenon offers a new possibility to measure the three dimensional surface structure.

Off-center displacement of the Nb ions below and above the ferroelectric phase transition of KTa0.91Nb0.09O3.

We have measured the off-center displacements of the Nb ions in ${\mathrm{KTa}}_{0.91}$${\mathrm{Nb}}_{0.09}$${\mathrm{O}}_{3}$ as a function of temperature both below and above the ferroelectric transition temperature ${\mathit{T}}_{\mathit{c}}$=85.6 K. The purpose of these measurements was to elucidate the basic nature of the ferroelectric phase transitions in mixed perovskites. The off-center displacements were determined by x-ray-absorption fine-structure (XAFS) measurements. The data were analyzed by calculating the XAFS spectra theoretically as a function of structural and other parameters and by refining these parameters to obtain the best fit with the experimental results up to the third coordination shell. In these calculations we used curved photoelectron wave functions and took into account multiple scattering up to and including, the third-order collinear scattering. It is found that these scattering contributions are both necessary and sufficient in order to account for all the scattering contributions with a total scattering path length of 8 \AA{} or less. At 70 K the off-center displacement, relative to the oxygen octahedra, is found to be 0.145 \AA{} in the [111] direction. These displacements change by less than 20% from 70 K to room temperature, namely, from below to far above the phase transition. These results lead to the conclusion that the magnitude of the displacements hardly changes over a very wide temperature range. Thus, the ferroelectric transition is not displacive and the main change taking place at the phase transition is from a disordered to orientationally ordered arrangement of displacements.

Rotational branching ratios at low photoelectron energies in resonant enhanced multiphoton ionization of NO

We report calculated rotational branching ratios for very low energy (50 meV) photoelectrons resulting from (1+1′) resonant enhanced multiphoton ionization (REMPI) via the Ji =1/2, 3/2, 5/2, and 7/2 levels of the P11 branch of the A 2Σ+ (3sσ) state of NO. Even angular momentum transfer (ΔN≡N+−Ni) peaks are dominant in these rotational distributions, in agreement with the selection rule ΔN+l=odd. Angular momentum coupling in the photoelectron wave function arising from the molecular ion potential leads to smaller but appreciable ΔN=odd peaks. The calculated ΔN=0 to ΔN=+2 peak ratios show the same strong decrease when Ji increases from 1/2 to 3/2 as seen in the experimental zero-kinetic-energy (ZEKE) photoelectron spectra [Sander et al., Phys. Rev. A 36, 4543 (1987)], but do not show the rapid die-off of the ΔN≠0 peaks for higher Ji observed experimentally. The calculated trend in the ΔN=+2 vs ΔN=0 peaks could be understood on the basis of simple angular momentum transfer arguments. These same arguments indicate that this trend in the ΔN=0 and +2 peaks with increasing angular momentum is not generally expected in other branches. Spectra via the R21 ( J) branch are presented to support this assertion. We also present photoelectron angular distributions which show a strong dependence on ΔN reflecting the changing composition of the photoelectron wave function.

Studies of the photoionization cross sections of CH4

We present cross sections and asymmetry parameters for photoionization of the 1t2 orbital of CH4 using static-exchange continuum orbitals of CH+4 to represent the photoelectron wave function. The calculations are done in the fixed-nuclei approximation at a single internuclear geometry. To approximate the near-threshold behavior of these cross sections, we assumed that the photoelectron spectrum is a composite of three electronic bands associated with the Jahn–Teller components of the distorted ion. The resulting cross sections reproduce the sharp rise seen at threshold in the experimental data and are in good agreement with experiment at higher energy. The agreement between the calculated and measured photoelectron asymmetry parameters is, however, less satisfactory.

Photon-induced post-collisional Auger decay: a modified 'shake-down' model

A modified 'shake-down' model describing both Auger and photoelectron lineshapes is derived from the two-potential formulation of post-collision interaction following photon impact. The model gives the correct expression for the lineshapes at high photoelectron energies and finite Auger energies. The distortion of the lineshapes is shown to be given by the overlap between the photoelectron wavefunctions in the final and smeared intermediate states, for which a construction is explicitly given.

Theory of the Stark effect

Theoretical photoabsorption cross sections for alkali-metal atoms in a static electric field $F$ are derived for energies near the zero-field ionization threshold ($\ensuremath{\epsilon}=0$), extending a previous development for hydrogen. Spectra result from zero-field dipole matrix elements and a density-of-states matrix ${D}_{{l}^{\ensuremath{'}}l}^{F}$. The factors ${D}_{{l}^{\ensuremath{'}}l}^{F}={({〈{\ensuremath{\Psi}}^{\ensuremath{'}}|\ensuremath{\Psi}〉}^{\ensuremath{-}1})}_{{l}^{\ensuremath{'}}l}$ represent a renormalization of the photoelectron's wave function ${\ensuremath{\Psi}}_{l}$ necessitated by the long-range Coulomb plus Stark potential. The matrix ${D}^{F}$ contains all spectral information on quasidiscrete Stark levels and continuum resonances, expressed as an algebraic function of (1) quantum defects ${\ensuremath{\mu}}_{l}$ embodying core effects, (2) a frame transformation between spherical and parabolic coordinates in the pure Coulomb field outside the alkali-metal core, and (3) factors ${H}_{{l}^{\ensuremath{'}}l}^{F}$ and ${h}_{{l}^{\ensuremath{'}}l}^{F}$ calculated from asymptotic amplitudes and phases of hydrogen-Stark wave functions of the parabolic dissociation channels. Hydrogenic parameters are calculated semianalytically within the WKB approximation; ${\ensuremath{\mu}}_{l}$ and dipole matrix elements are known independently. Predicted cross sections for photoionization of the excited $3^{2}P_{\frac{3}{2}}$ state of Na agree with experiment. The theory reproduces (a) asymmetric resonances observed at $\ensuremath{\epsilon}l0$, parametrized as a Beutler-Fano profile for a simple case, and (b) the oscillations observed at $\ensuremath{\epsilon}g0$, attenuated by factors $cos2\ensuremath{\pi}{\ensuremath{\mu}}_{l}$ from their predicted depth in H. Dependences on light polarization are sorted out for two-photon excitation.

Electronic relaxation effects on x-ray spectra of titanium and transition-metal carbides and nitrides

High-resolution titanium $K$ x-ray absorption spectra from pure metal, carbide, and nitride are measured using synchrotron radiation. It is shown that the local partial density of states at the metal site is adequate to describe the near-edge features observed. The magnitude of the electron screening to a suddenly switched-on core-hole potential is investigated and found to be correlated to the density of occupied electron states at the Fermi surface. The dynamical aspects of the screening process show up in the extended x-ray absorption fine-structure region of the spectra and influence the phase and amplitude of the photoelectron wave function. Several structural properties of these compounds are also considered.

EXAFS studies of disordered solids

The availability of intense sources of continuous radiation in the X-ray region has stimulated the development of a technique for structure determination which is particularly suited to studies of local environment in complex systems. The absorption cross section for the photoexcitation of an electron from a deep core state to a continuum state exhibits oscillations as a function of photon energy known as the extended X-ray absorption fine structure, or EXAFS. These oscillations result from the interference between the outgoing photoelectron wavefunction and that portion of it scattered back from neighboring atoms, and therefore contain considerable information about the local environment of the excited atom species. Analysis of the EXAFS can yield not only the distance but also the type and number of the nearest neighbors of the excited atom. The purpose of this paper is to discuss briefly the theoretical background and the practical aspects of EXAFS data analysis. The discussion is then illustrated through application of these techniques to EXAFS studies of arsenic dopants in amorphous silicon and of PdGe metallic glass.

Theoretical Calculation of Relative Intensities in ESCA

Calculations of the photoionization cross-sections of atomic and molecular levels for the A1Kα and MgKα lines depending on the wave function of the final and the initial states are analysed. The ns, np and nd photoionization cross-sections of atomic shells for Z ⩽ 36 and the HF, NH3, H2O, CH4, CH3F, CH2F2, CHF3, CF4, N2, CO and C2H4 molecular levels for the YMζ line and the photoionization cross-sections of atoms for Z ⩽ 36 and the CH4, N2 and H2O molecular levels for the ZrMζ line are calculated. The theoretical cross-sections of atoms for Z ⩽ 10 for the FKα line are presented too. Photoionization cross-section calculations using Hartree-Fock-Slater potential are in better agreement with the experiment than those using orthogonalized plane waves for photo-electron wave functions.