Continuum Intermediate State Research Articles

Single-electron capture from helium targets by heavy nuclei of charges 1–7

Single-electron capture by multiply charged nuclei from helium atoms is studied by means of the prior form of the four-body boundary-corrected continuum intermediate state (BCIS-4B) method. Computations concern total cross sections for the state-selective (Qnlm) and state-summed (Qnl,Qn,QΣ) populations at 20–3000 keV/amu. These refer to the collisions of the type AZP++He(1s2)→A(ZP−1)+(nlm)+He+(1s). Here, the projectile AZP+ covers the ions H+, He2+, Li3+, Be4+, B5+, C6+ and N7+. The reported findings are tabulated for each value of the quantum numbers {n,l,m}. The maximum values nmax of the principal quantum number n are 4 (H+, He2+, Li3+), 5 (Be4+), 6 (B5+) and 7 (C6+, N7+). The sum over all the n states is truncated by selecting an appropriate cutoff value nmax and by subsequently applying the Oppenheimer n−3 scaling rule to approximately include the contributions from n>nmax. The obtained results for QΣ and for some final n−specific states favorably describe the corresponding experimental data. This qualifies the prior form of the BCIS-4B method for further useful applications in several interdisciplinary fields, including astrophysics, thermonuclear fusion, plasma physics and ion therapy in medicine.

State selective total and angular differential cross sections for positronium formation in [formula omitted]+[formula omitted] collisions

State-selective total and angular differential cross sections for positronium formation are presented in the collision of positron with helium atoms in their ground state by using the four-body formalism of the boundary-corrected continuum-intermediate-states approximation. The main purpose of the present study is to investigate the relative importance of intermediate ionization continuum states that has been incorporated with a suitable choice of distorting potential. In addition, the nine-dimensional integrals of the transition amplitude are analytically reduced to a one-dimensional real integral, which can be easily calculated numerically. The influence of the target static electron correlations on the capture probability is investigated using the different bound-state wave functions of He. The theoretical model for the calculation is fairly accurate and predicts a number of interesting features of the state-selective total and differential cross sections in the range of positron impact energy varies from 20 to 250eV. The total cross sections are highly peaked at an impact energy of 50 eV. The results are compared with those of other theories and the available experimental data. The present theoretical results and other measurements show satisfactory agreement in the wide energy range. Moreover, surface plots for the state-selective differential cross sections are also presented to illustrate the interesting feature of the positronium formation in this collision process.

Cross sections for single-electron capture from heliumlike targets by fast heavy nuclei

Single charge-exchange in collisions of heavy bare nuclei with the ground state of two-electron atomic targets is described perturbatively as a four-body problem. The employed four-body boundary-corrected continuum intermediate state (BCIS-4B) method considers the correlated and uncorrelated target wave functions ${\ensuremath{\varphi}}_{i}.$ A thorough examination is performed for the formation of any final hydrogen-like $nlm$ state of the captured electron. For arbitrary projectile and target nuclear charges, the nine-dimensional integral in the transition amplitude is reduced to a two-dimensional numerical quadrature. The general analysis is applied to one-electron capture by protons from helium targets beginning with the lower edge (10 keV) of intermediate energies and extending to the higher (12.5 MeV) domain. These include the main peaks (Massey, Thomas) due to single and double scattering, respectively. The results encompass over 70 state-selective and state-summed cross sections $(n\ensuremath{\le}6,\phantom{\rule{0.16em}{0ex}}0\ensuremath{\le}l\ensuremath{\le}n\ensuremath{-}1,\phantom{\rule{0.16em}{0ex}}\ensuremath{-}l\ensuremath{\le}m\ensuremath{\le}l).$ In comparison to measurements, the electronic correlations in ${\ensuremath{\varphi}}_{i}$ greatly improve the overall performance of the BCIS-4B method around the Massey peak, below about 100 keV. Moreover, while largely outperforming the three-body boundary-corrected continuum intermediate state method, the cross sections in the BCIS-4B with the correlated ${\ensuremath{\varphi}}_{i}$ compare excellently overall with the available experimental data at 10 to 12 500 keV. Hence, the BCIS-4B method, with its built-in two main capture mechanisms (one-step Massey and two-step Thomas) is capable of spanning impact energies covering three or more orders of magnitude at which the state-summed cross sections vary over 11 orders of magnitude.

Relative importance of the electron continuum intermediate state in single-electron capture into any state of fast protons from helium-like atomic systems

Relative importance of the electron continuum intermediate state in single-electron capture into any state of fast protons from helium-like atomic systems

The BCIS-4B method for state-selective and state-summed total cross sections: Proton–helium​ charge exchange at 10–4000 keV

This work is on single electron capture by protons from helium targets at intermediate and high impact energies (10–4000 keV). Assessed is the overall performance of the second-order four-body boundary-corrected continuum intermediate state (BCIS-4B) method by detailed comparisons with all the available measurements. The state-selective Qnl,Qn and state-summed QΣ total cross sections are computed for n≤4, including all the sub-levels. Cross sections Qn are the sum of Qnl for 0≤l≤n−1. The state-summed cross sections QΣ are obtained by using Qn exactly for 1≤n≤3 and approximately for n≥4 through the Oppenheimer n−3 scaling rule. Overall, the measured cross sections are extremely favorably represented by the BCIS-4B method for Qnl,Qn and QΣ, even toward the lowest part of the considered intermediate energy region. Most notably, at all the energies above 10 or 20 keV, e.g. cross sections Q2p,Q3p and Q3d from the BCIS-4B method are in excellent agreement with the corresponding experimental data. This is very important since it is precisely for the non-spherically symmetric states (l≠0) of the captured electron that the existing discrepancies between measurements and other theories are the largest or even enormous.

Single-electron transfer from helium atoms to energetic multiply-charged nuclei

The three-body boundary-corrected continuum intermediate state (BCIS-3B) method is employed to compute single-electron capture cross sections. These data are for collisions in which a target helium atom in its ground state He(1s2) is impacted by multiply charged nuclei (specifically H+,He2+,Li3+,Be4+,B5+,C6+,N7+,O8+,F9+). Intermediate and high projectile energies are considered (20 keV/amu-3 MeV/amu). The obtained results are tabulated for total cross sections (state-selective Qnlm as well as state-summed Qnl,Qn and QΣ). Cross section Qnlm corresponds to a fixed triple of the hydrogenlike quantum numbers {n,l,m}. In Qnl,Qn and QΣ, the sums over m∈[−l,l],l∈[0,n−1] and n∈[1,nmax] are carried out, respectively. For n>nmax in QΣ, the Oppenheimer n−3 scaling rule is adopted. The values of nmax are set to 4 (for H+,He2+,Li3+), 5 (for Be4+), 6 (for B5+) and 7 (for C6+,N7+,O8+,F9+). Exhaustive comparisons of the theoretical findings for QΣ with the corresponding experimental data are performed. It is determined that the BCIS-3B method is in excellent agreement with the available measurements. The present database for the examined capture processes is deemed to be of notable practical usefulness in versatile applications of particular relevance to plasma physics, astrophysics, fusion research and ion therapy in medicine.

Two-photon above-threshold ionization of helium

Multiphoton ionization provides a clear window into the nature of electron correlations in the helium atom. In the present study, the final state energy range extends up to the region near the $N=2$ and $N=3$ ionization thresholds, where two-photon ionization proceeds via continuum intermediate states above the lowest threshold. Our calculations are performed using multichannel quantum defect theory (MQDT) and the streamlined R-matrix method. The sum and integration over all intermediate states in the two-photon ionization amplitude is evaluated using the inhomogeneous R-matrix method developed by Robicheaux and Gao. The seamless connection of that method with MQDT allows us to present high resolution spectra of the final state Rydberg resonances. Our analysis classifies the resonances above the $N=2$ threshold in terms of their group theory quantum numbers. Their dominant decay channels are found to obey the previously conjectured propensity rule far more weakly for these even parity states than was observed for the odd-parity states relevant to single photon ionization.

Electron transfer from atomic hydrogen to multiply-charged nuclei at intermediate and high energies

We report on theoretical total cross sections for electron capture from the ground state of atomic hydrogen H(1s) by fast multiply-charged nuclei H+,He2+,Li3+,Be4+,B5+,C6+,N7+,O8+ and F9+. The prior form of the boundary-corrected continuum intermediate states (BCIS) method is used. For a single fixed initial ground-state (i=1s), comprehensive computations are carried out for a sequence of the transitions to the final state (n,l,m) with 1≤n≤nmax, including all the energy degenerate sub-levels (−l≤m≤l,0≤l≤n−1). The maximal quantum number nmax is respectively set to 4 for H+,He2+ and Li3+, to 5 for Be4+, to 6 for B5+, as well as to 7 for C6+,N7+,O8+ and F9+. The reported nine tables provide these state-selective cross sections (Qif) and their state-summed counterparts (QΣ). The cross sections QΣ for the sum over all the final states are evaluated using the exact results of Qif for n≤nmax and the scaled cross sections for n>nmax. The scaling is based on the Oppenheimer n−3 rule. In addition to these tables, at the same impact energies 20–3000 keV/amu, the cross sections from the BCIS method are presented graphically in eight figures for the nuclei Li3+,Be4+,B5+,C6+,N7+,O8+ and F9+. This complements our recently published cross sections for electron capture by protons (H+) and alpha particles (He2+) from H(1s). In the present figures, whenever available, the corresponding experimental data are compared with our results for QΣ. Also included in these comparisons are the corresponding results for QΣ from the continuum distorted wave (CDW) method. The cross sections tabulated here can be used in versatile applications ranging from plasma physics and astrophysics through fusion research to ion therapy.

Double electron capture in fast ion-atom collisions

A four-body formalism of the modified target continuum distorted wave (MTCDW-4B) with incorrect boundary conditions and boundary corrected continuum intermediate state (BCCIS-4B) approximation have been employed to calculate the differential cross se

Total cross sections in five methods for two-electron capture by alpha particles from helium: CDW-4B, BDW-4B, BCIS-4B, CDW-EIS-4B and CB1-4B

This work is on quantum-mechanical four-body distorted wave theories for double electron capture in collisions between fast heavy multiply charged ions and heliumlike atomic systems. The five widely used distorted wave methods of the first- and second-order in the perturbation series expansions are compared with the available experimental data on alpha –He collisions. These are the four-body boundary-corrected first Born (CB1-4B), the boundary-corrected continuum intermediate state (BCIS-4B), the Born distorted wave (BDW-4B), the continuum distorted wave (CDW-4B) and the continuum distorted wave-eikonal initial state (CDW-EIS-4B) methods.We address the complete breakdown of the CDW-EIS-4B method at all impact energies within its expected validity domain (100–10000 keV). Further, the relative performance is evaluated of the second-order theories with and without the eikonalization of the two-electron Coulomb wavefunctions for double continuum intermediate states. Finally, at all the considered intermediate and high energies, the practical aspects of the studied five methods are investigated by protracted evaluations of the convergence rates of total cross sections as a function of the number of quadrature points per axis in numerical computations of multi-dimensional (3D-5D) integrals.

Electron capture by fast projectiles from lithium, carbon, nitrogen, oxygen and neon

The three-body boundary-corrected continuum intermediate state method is used to compute total cross sections for single charge exchange in ion-atom collisions at intermediate and high impact energies. Detailed illustrations are given for several scattering systems: H+ + Li, He2+ + Li, H+ + C, He2+ + C, H+ + N, H+ + O and H+ + Ne, He2+ + Ne. An independent particle model and the frozen-core approximation are employed with only one target electron taken as being active. The initial ground state of the active electron in a multi-electron target is described by five wave functions. These are two Roothaan-Hartree-Fock (RHF) wave functions, the single- as well as double-zeta functions and the hydrogen-like functions. Comparisons among the resulting cross sections are made to check their sensitivity to the choice of the initial target wave functions. In the case of a lithium target, the separate cross sections for electron capture from the K-shell and L-shell are reported. The present theoretical total cross sections are compared with the available experimental data and overall good agreement is found, especially when using the RHF wave functions for multi-electron targets.

Single-electron capture in ion-ion collisions

The prior versions of the three-body boundary-corrected first Born approximation (CB1-3B) and the three-body boundary-corrected continuum intermediate states method (BCIS-3B) are applied to calculate the state-selective and state-summed total cross sections for single-electron capture from hydrogen-like ion targets (He+, Li2+) by fast completely stripped projectiles (H+, He2+, Li3+). All calculations are carried out for single-electron capture into arbitrary n l m final states of the projectiles, up to n = 4. The contributions from higher n shells are included using the Oppenheimer n?3 scaling law. The present results are found to be in satisfactory agreement with the available experimental data.

Electron capture by bare projectiles from multi-electron targets

Single-electron capture from the K-shell of multi-electron atomic targets (carbon, nitrogen, oxygen, neon and argon) by bare projectiles (the nuclei of H, He and Li) at intermediate and high-impact energies is studied theoretically. We use the three- and four-body boundary-corrected continuum intermediate state (BCIS-3B, BCIS-4B) methods. Thorough comparisons of the obtained K-shell capture cross-sections with the associated experimentally measured data are carried out. Overall good agreement is recorded in the intermediate and high-energy region. By way of an example, the BCIS-4B method gives practically the same shape and magnitudes of total cross sections measured from 200 to 2000 keV for the p −C collisions, thus, covering the entire energy region of interest, including the Massey peak. Such features are of notable interest not only in atomic collision physics, but also in a number of other branches associated with accelerator-based physics. For example, in hadron therapy, most energy losses of light and heavy nuclei traversing matter occur within the Massey peak (also known as the Bragg peak when cross sections are plotted as a function of the reciprocal of the impact energy).

Efficient calculation of Rayleigh and Raman scattering

We present two methods for computing the Rayleigh and Raman scattering cross sections for photon scattering on atomic hydrogen or hydrogenlike systems. Both methods are applicable for incident photon energies above the ionization threshold. The first method implements the well-known Gaussian quadrature approach to deal with principal value integration and relies on evaluation of the exact eigenfunctions of hydrogen. The second, more computationally efficient approach uses a finite-${L}^{2}$ basis expansion of the target and applies complex exterior scaling methods to accurately account for the contribution of the intermediate continuum states. This method is much more general in that it does not rely on analytic solutions to the Hamiltonian, or evaluation of any special functions, and is expected to be applicable to more complex systems where exact wave functions are cumbersome to evaluate. Both methods are in complete agreement with previous work based on analytical representations of the Green's function or the dipole matrix elements. Rayleigh, Raman, and photoionization cross sections for scattering on the first few excited states of atomic hydrogen are presented and compared with previous results where available.

Role of intermediate continuum states in exterior complex scaling calculations of two-photon ionization cross sections

The calculation of partial two-photon ionization cross sections in the above-threshold energy region is discussed in the framework of exterior complex scaling. It is shown that with a minor modification of the usual procedure, which is based on the calculation of the outgoing partial waves of the second-order scattering wave function, reliable partial ionization amplitudes can be obtained. The modified procedure relies on a few-term least-squares fit of radial functions pertaining to different partial waves. To test the procedure, partial and total two-photon ionization cross sections of the helium atom have been calculated for a broad range of incident photon energies. The calculated cross sections may be seen to agree well with the results found in the literature. Furthermore, it is shown that, using a similar approach, partial photoionization cross sections of an atom in an autoionizing (resonance) state may be calculated in a relatively straightforward way. Such photoionization cross sections may find their use in enhanced few-parameter models describing the atom-light interaction in cases where a direct solution of the time-dependent Schroedinger equation becomes too resource-intensive.

Comparative study of single and double electron capture from atoms by fast bare ions

Total cross sections for single and double electron capture in collisions of bare ion with helium-like atoms are calculated at energies ranging from 50 to 6,000 keV/amu. The present calculations are carried out using four-body formalism of Coulomb–Born distorted wave formalism and boundary corrected continuum intermediate state approximation. We have made a comparison between single and double electron capture cross sections using both the methods in wide range of energies at different charge state of the projectile. Total cross sections have been calculated by summing over all the contributions up to n = 2 shells and sub-shells. It is also noted that double electron capture cross sections are small thorough out the energy region compared with the single electron capture cross sections. However, the present computed results for asymmetric collisions have been compared with available theoretical and experimental results. We find that our numerical results for total cross sections show good agreement with the available experimental findings.

Projectile angular-differential cross sections for transfer and transfer-excitation in α–He collisions

Total as well as projectile angular-differential cross sections for transfer and transfer-excitation have been investigated in collision of α-particles with helium atoms in the energy range 50–5,000 keV/amu. The four-body Coulomb–Born distorted wave formalism and boundary corrected continuum intermediate state approximation have been applied for this rearrangement collision. In both methods dynamic electron correlations have been taken through the complete perturbation potentials in post form. It is shown that for single electron-transfer, the ground state capture is dominant and the contribution from excited states to the total cross section is less important. It is also noted that the cross sections for transfer with target excitation channel have minor contribution to the total cross sections. We have also studied the total cross sections for single electron transfer from lithium atom by α-particle impact in the energy range from 50 to 3,000 keV/amu. Present computed results, both in prior and post forms of the transition amplitude for symmetric and asymmetric collisions, agree reasonably with the available theoretical and experimental results.

Classical simulation of single-electron capture and ionization in ion-atom collisions

Classical trajectory Monte Carlo (CTMC) simulation method and the four-body formalism of boundary corrected continuum intermediate state (BCCIS-4B) approximations have been employed to determine total single electron capture cross section and total single ionization cross section in collisions of H+, He2+ and Li3+ with helium atom and energy in range of 30–200 keV/amu. Present theoretical study using CTMC method is based on three-body formalism, where interaction of the active electron in target ion has been approximated by a model potential containing both long and short range part. As a consequence, initialization of the problem has been made on a multiperiodic orbit in sharp contrast to singly periodic orbit occurring in hydrogenic model. In BCCIS-4B formalism, distortion in the final channel related to Coulomb continuum states of projectile ion and active electron in the field of residual target ion are included. Results so obtained from both methods are in very good agreement with available experimental findings.

Electron correlation in single-electron capture from helium by fast protons

The differential and total cross sections for single charge exchange in p-He collisions have been calculated within the framework of four-body boundary corrected continuum intermediate state (BCCIS-4B) approximation. The effect of dynamic electron correlations is explicitly taken into account through the complete perturbation potentials.

Single ionization of helium by positron and electron impact

Four-body formalism of Boundary Corrected Continuum Intermediate State (BCCIS-4B) approximation is introduced to study the (e, 2e) reaction for Helium targets. The influence of the description of the ejected electron on triple differential cross sections is analyzed.