Triply Differential Cross Section Research Articles

Compton scattering revisited

We review the standard theory of Compton scattering from bound electrons, and we describe recent findings that require modification of the usual understanding, noting the nature of consequences for experiment. The subject began with Compton and scattering from free electrons. Experiment actually involved bound electrons, and this was accommodated with the use of impulse approximation (IA), which described inelastic scattering from bound electrons in terms of scattering from free electrons. This was good for the Compton peak but failed for soft final photons. The standard theory was formalized by Eisenberger and Platzman (EP) [1970. Phys. Rev. A 2, 415], whose work also suggested why impulse approximation was better than one would expect, for doubly differential cross sections (DDCS), but not for triply differential cross sections (TDCS). A relativistic version of IA (RIA) was worked out by Ribberfors [1975. Phys. Rev. B 12, 2067]. And Surić et al. [1991. Phys. Rev. Lett. 67, 189] and Bergstrom et al. [1993. Phys. Rev. A 48, 1134] developed a full relativistic second order S-matrix treatment, not making impulse approximation, but within independent particle approximation (IPA). Newer developments in the theory of Compton scattering include: (1) Demonstration that the EP estimates of the validity of IA are incorrect, although the qualitative conclusion remains unchanged; IA is not to be understood as the first term in a standard series expansion. (2) The greater validity of IA for DDCS than for the TDCS, which when integrated give DDCS, is related to the existence of a sum rule, only valid for DDCS. (3) The so-called “asymmetry” of a Compton profile is primarily to be understood as simply the shift of the peak position in the profile; symmetric and anti-symmetric deviations from a shifted Compton profile are very small, except for high Z inner shells where further p ⇒ · A ⇒ effects come into play. (4) Most relativistic effects, except at low energies, are to be understood in terms of simple kinematic modifications of nonrelativistic IA, plus using a relativistic charge density for high Z inner shell states; these shift the peak and change its height. However, for high Z, corrections to RIA persist in the peak region, even at extreme relativistic energies (correction of about 15% for Z = 92 ).

(e,2e) ionization of helium and the hydrogen molecule: signature of two-centre interference effects

Relative (e,2e) triply differential cross sections (TDCS) are measured for the ionization of the helium atom and the hydrogen molecule in coplanar asymmetric geometry at a scattered electron energy of 500 eV and ejected electron energies of 205, 74 and 37 eV. The He experimental results are found to be in very good agreement with convergent close-coupling calculations (CCC). The H2 experimental results are compared with two state-of-the-art available theoretical models for treating differential electron impact ionization of molecules. Both models yield an overall good agreement with experiments, except for some intensity deviations in the recoil region. Similar (e,2e) works were recently published on H2 with contrasted conclusions to the hypothesis that the two H nuclei could give rise to an interference pattern in the TDCS structure. Murray (2005 J. Phys. B: At. Mol. Opt. Phys. 38 1999) found no evidence for such an effect, whereas Milne-Brownlie et al (2006 Phys. Rev. Lett. 96 233201) reported its indirect observation. In this work, based on a direct comparison between experimental results for He and H2, we observe an oscillatory pattern due to these interference effects, and for the first time the destructive or constructive character of the interference is observed, depending on the de Broglie wavelength of the ejected electron wave. The experimental finding is in good agreement with the theoretical prediction by Stia et al (2003 J. Phys. B: At. Mol. Opt. Phys. 36 L257).

Electron impact single ionization of helium with large energy transfer

We report triply differential cross section (TDCS) for electron impact single ionization of helium at the incident energy of 729.6eV for a variety of scattering angles and ejection energies. The calculations are based on two different methods, namely, the Glauber approximation (GA) and the 3C approximation, also called the BBK method. In the GA the effects of post collision interaction have been taken into account by a classical method, whereas the BBK method has been studied with and without exchange. A comparison is made of the present TDCS with the results of other calculations and the experimental data from two different laboratories. For large energy transfer we find a significant contribution from the exchange effect. Both the GA–PCI and the BBK methods show some shifts in the position of binary maxima which are in reasonably good agreement with experiment.

Triply differential (e,2e) cross sections for ionization of the nitrogen molecule at large energy transfer

Measurements of the (e,2e) triply differential cross sections (TDCS) are presented for the ionization of the nitrogen molecule in coplanar asymmetric geometry at an incident energy of about 600 eV and a large energy transfer to the target. The experimental results are compared with state-of-the-art available theoretical models for treating differential electron impact ionization of molecules. The experimental TDCS are characterized by a shift towards larger angles of the angular distribution with respect to the momentum transfer direction, and by a large intensity in the recoil region, especially for ionization of the ‘inner’ 2σg molecular orbital. Such shifts and intensity enhancement are not predicted by the model calculations which rather yield a TDCS symmetrically distributed around the momentum transfer direction.

Nondipole effects in the triply differential cross section for double photoionization of He

Lowest-order nondipole effects are studied systematically in double photoionization (DPI) of the He atom. Ab initio parametrizations of the quadrupole transition amplitude for DPI from the {sup 1}S{sub 0} state are presented in terms of the exact two-electron radial matrix elements. Analytic expressions for these matrix elements within lowest-order perturbation theory (LOPT) in the interelectron interaction are also given. The corresponding parametrizations for the dipole-quadrupole triply differential cross section (TDCS) are presented for the case of an elliptically polarized photon. A general analysis of retardation-induced asymmetries of the TDCS including the circular dichroism effect at equal energy sharing is presented. Numerical LOPT estimates of nondipole asymmetries in photoelectron angular distributions for the cases of linear and circular polarization and of the circular dichroism effect at equal energy sharing are presented. We find that experimental observation of nondipole effects at excess energies of the order of tens to hundreds of eV should be feasible in TDCS measurements. Our numerical results exhibit a nondipole forward-backward asymmetry in the TDCS for DPI of He at an excess energy of 450 eV that is in qualitative agreement with existing experimental data.

Charge and energy-dependence of the Gaussian description of the triply differential cross sections for equal-energy sharing photo-double-ionization of two-electrons ions

We evaluate triply differential cross sections (TDCSs) for the photo-double-ionization (PDI) of He-like ions, and equal electron energy sharing, by using the SC3 model for the three-body final state. These cross sections are fitted with the usual dipolar Gaussian form which is found able to describe the theoretical TDCS, and could be applied for the interpretation of experimental data even at intermediate photon energies. We determine the dependence of the correlation factor on the excess energy $({E}_{f})$ and target nuclear charge $(Z)$. We find that its width has an ${E}_{f}^{1∕4}$ dependence near the atomic double-ionization threshold but departs from this law and attains a plateau as the excess energy increases. We compare our results with the predictions of classical and semiclassical Wannier approaches.

A new C3-like model for the photodouble ionization of He

We have recently proposed a new model to deal with the final two electron continuum state in photodouble ionization (PDI) processes. This final wave function introduces a dilatation parameter into the interelectronic effective coordinate to account for the nuclear dynamical screening. In this work we extend the analysis and evaluate the triply differential cross-section (TDCS), the simply differential cross-section (SDCS) in energy and the total cross-section for PDI of He. Comparison with the standard C3 model shows that the TDCS is enhanced in the threshold region by effect of the correlation introduced by the present model. A more accurate description of the intermediate energy region is also obtained. Comparison with recent experimental data shows good overall agreement of the angular and energetic distributions.

Angular distributions for photo-double-ionization of the He isoelectronic sequence

We describe the three-body final state resulting from the photo-double-ionization (PDI) of two-electron ionic atoms by a modified C3 wave, denoted SC3. This continuum wave function accounts for the nuclear dynamical screening in the inter-electronic motion. We analyze the scaling properties of the triply differential cross-sections (TDCS). For an easier comparison with possible experimental results we analyze the correlation factor in the Gaussian parametrization of the TDCS, for different nuclear charges. We determine the dependence of the half width at half maximum of the Gaussian with the nuclear charge and discuss possible physical mechanisms.

Kinetic correlation in the final-state wave function in photo-double-ionization of He

We evaluate the triply differential cross section (TDCS) for photo-double-ionization of helium. We use a final continuum wave function which correlates the motion of the three particles, through an expansion in products of two-body Coulomb functions. This function satisfies a set of appropriate physical conditions in the coalescence points, in addition to the correct asymptotic behavior condition. We analyze the effect of this correlation in the TDCS and compare our results with experimental data.

Photo double ionization of He: C3-like wave function for the two electron continuum

We evaluate the triply differential cross-section (TDCS) for photo double ionization of helium. We use a C3 final continuum wave function with an interelectronic effective coordinate to express the nuclear screening. Comparison with the standard C3 model shows that the TDCS is enhanced in the threshold region by effect of the reduced interelectronic repulsion introduced by the present model. A more accurate description of the intermediate energy region is also obtained. Comparison with recent experimental data shows a good overall agreement of the angular distributions. The theoretical PDI total cross-section shows a relevant improvement in the intermediate energy region relative to the C3 model, which converges to data for photon energies larger than 1 keV.

Triply differential cross-section measurements in the double photoionization ofD2and He with asymmetric kinematic conditions

The breakup of ${\mathrm{D}}_{2}$ following the absorption of a single photon, leading to four free charged particles interacting via the Coulomb force, has been investigated. Triply differential cross sections (TDCSs) of the two ejected electrons were measured for selected asymmetric kinematic conditions, namely, a 20 eV electron in coincidence with a 5 eV ``reference'' electron fixed at four different angles. The results are compared with helium TDCSs and, although the respective angular distributions have similar features, there are surprising differences that could have their origin in two-center interference effects.

Equal energy-sharing double photoionization of helium from near-threshold to high energies

For a given excess energy E above the helium double photoionization (DPI) threshold, just two parameters are necessary to generate any angular set of the fully resolved triply differential cross sections (TDCS) in the case of the two outgoing electrons sharing E equally. It is shown that these parameters, the width and magnitude of the Gaussian ansatz, may be estimated ab initio from the convergent close-coupling (CCC) theory. The CCC calculations have been performed for E in the range of 3--80 eV, allowing for the generation of equal energy-sharing TDCS at arbitrary orientation of the two electrons and polarization of the photons.

Triply differential cross section for ionization of multielectronic targets

The Continuum Distorted Wave-Eikonal Initial State (CDW-EIS) approach is used to calculate the Triply Differential Cross Section (TDCS) for ionization of multielectronic atomic targets in coincidence with the outgoing projectile direction and recoil momentum of the target. A generatrix state method is used to evaluate the transition amplitudes for a general initial atomic state. The structures presented in both TDCSs are compared standing out the double collision mechanisms. The results obtained in CDW-EIS approximation are compared with previous classical trajectories Monte Carlo (CTMC) calculations.

Helium Double Photoionisation: An Accurate Solution of a Three-body Coulomb Problem

We present here the solution of the helium double photoionisation problem by the convergent close-coupling (CCC) method. This method allows us to obtain the most detailed description of the double photoionisation process in the form of the fully resolved triply differential cross section (TDCS). The accuracy of our model is tested by calculating the TDCS in the three different forms of the electromagnetic operator which produces essentially identical results. We compare our calculation with the most accurate experimental and theoretical data available to date.

Absolute Triply Differential (e,2e) Cross Section Measurements for H with Comparison to Theory

Absolute triply differential ( $e,2e$) cross section measurements are presented for H (for incident energies, ${E}_{0}$, of 15.6 and 17.6 eV) for equal energy sharing and the ${\ensuremath{\theta}}_{12}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\pi}$ configuration. Results of distorted partial wave calculations agree with the measurements; those of convergent close coupling calculations agree with the relative angular distributions, but are lower than experiment by factors of from 2 to 7. Relative experimental results for H for ${E}_{0}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}14.6\mathrm{eV}$ show a qualitative change in shape, which agrees with theory. Implications of the absolute experimental results for the range of validity of the Wannier threshold law are discussed.

Three-body models of electron-hydrogen ionization

Although the importance of electron-electron correlations in (e,2e) reactions has long been realized, untill recently only little theoretical progress had been made including these correlations. Theoretical studies using “three-body” models (models with explicit dependence on the electron-electron coordinate) are hampered by the difficult six-dimensional integral for the scattering amplitude. “Two-body” models, on the other hand, retain only electron-ion interactions and the standard techniques of partial-wave analysis greatly simplify the integral. In contrast, the wave functions used in three-body models give rise to two-center integrals, which are not efficiently evaluated using partial-wave methods. Therefore, a different method was sought for. We have found that direct six-dimensional numerical quadrature is efficient (about five minutes on a workstation to compute one scattering amplitude). As a result, we now have the capability of calculating (e,2e) triply-differential cross sections using three-body wave functions of arbitrary complexity. This allows for a severe test of three-body models. The utility of our method is demonstrated by using two recently reported analytical three-body wave functions in an (e,2e) calculation.

Triply differential cross sections for the photo-double ionization of neon

The triply differential cross section (TDCS) for the photo-double ionization of neon to the final , , ionic states are calculated and compared with recent experimental results. The origin of maxima and minima (lobes) in the polar plots of the TDCS and their variation with detection geometry are analysed with the help of selection rules.

Angular distributions for double photoionization of helium: Discrepancies between theory and experiment.

We apply two different methods to the calculation of the triply differential cross section (TDCS) for double ionization of helium. In one method, the 3C method, the final state is described by a product of 3 Coulomb continuum wave functions, while in the other method, the 2SC method, the final state is described by a product of 2 screened Coulomb wave functions employing effective charges. We present results at the energies and geometries covered in the recent experiment by Lablanquie et al. [Phys. Rev. Lett. 74, 2192 (1995)]. There are substantial discrepancies. We also comment on a recent measurement of the absolute TDCS by Schwarzkopf and Schmidt [J. Phys. B 28, 2847 (1995)]. The measured result, as originally reported, is a factor of two smaller than the value for the absolute TDCS calculated using the 2SC method. We discuss this discrepancy. \textcopyright{} 1996 The American Physical Society.

Oriented and aligned two-electron continua

The triply differential cross section (TDCS) for electron-impact ionization of atoms is analysed in terms of tensorial parameters representing orientation and alignment in electron - electron coincidences. Particular attention has been paid to excited alkali states because of experimental feasibility. Three dynamical models for the calculation of the TDCS have been considered. The plane-wave impulse approximation (PWIA), the first Born approximation (FBA), and the use of a correlated double continuum wavefunction as employed by Brauner et al, hereafter referred to as BBK. It is shown that the PWIA forbids any orientational effect, but allows for an alignment. The FBA generally shows an orientational dichroism, the structure of the TDCS within the FBA has been worked out for non-statistical M-state populations. Finally, final-state correlation is shown to modify drastically the FBA results even at intermediate to higher energies. Numerical calculations for H(2p) are presented for the purpose of illustration.

Absolute triply differential cross sections for double photoionization of helium at 10, 20, and 52.9 eV above threshold.

We have calculated the coplanar triply differential cross section (TDCS) for double ionization of helium by one photon at total energies of 10, 20, and 52.9 eV above threshold. Our results agree very well with recently measured and calculated angular distributions [O. Schwarzkopf, B. Kr\"assig, J. Elmiger, and V. Schmidt, Phys. Rev. Lett. 70, 3008 (1993); O. Schwarzkopf, B. Kr\"assig, V. Schmidt, F. Maulbetsch, and J. S. Briggs, J. Phys. B 27, L347 (1994); F. Maulbetsch and J. S. Briggs, ibid. 26, L647 (1993)] for the cases of equal energy sharing (photoelectron energies ${\mathit{E}}_{1}$ and ${\mathit{E}}_{2}$, such that ${\mathit{E}}_{1}$=${\mathit{E}}_{2}$=5 eV and ${\mathit{E}}_{1}$=${\mathit{E}}_{2}$=10 eV) and unequal energy sharing (${\mathit{E}}_{1}$=5 eV, ${\mathit{E}}_{2}$=47.9 eV). In contrast to previous calculations of the TDCS, our calculation is based on a basis-set method that is, in principle, exact, and we obtain absolute values of the TDCS that are fairly well converged with respect to the basis size.