Feshbach Projection Operator Formalism Research Articles

Nuclear Excitation by Free Muon Capture

Efficient excitation of nuclei via exchange of a real or virtual photon has a fundamental importance for nuclear science and technology development. Here, we present a mechanism of nuclear excitation based on the capture of a free muon into the atomic orbits (NEμC). The cross section of such a proposed process is evaluated using the Feshbach projection operator formalism and compared to other known excitation phenomena, i.e., photoexcitation and nuclear excitation by electron capture (NEEC), showing up to 10 orders of magnitude increase in cross section. NEμC is particularly interesting for MeV excitations that become accessible thanks to the stronger binding of muons to the nucleus. The binding energies of muonic atoms have been calculated introducing a state of the art modification to the Flexible Atomic Code. An analysis of experimental scenarios in the context of modern muon production facilities shows that the effect can be detectable for selected isotopes. The total probability of NEμC is predicted to be P≈1×10^{-6} per incident muon in a beam-based scenario. Given the high transition energy provided by muons, NEμC can have important consequences for isomer feeding and particle-induced fission.

P-Wave Positron-Hydrogen Scattering, Annihilation, and Positronium Formation

In a previous paper (Bhatia A.K. 2016), a hybrid theory for the scattering of positrons from hydrogen atoms was applied to calculate S-wave phase shifts, annihilation, and positronium formation cross sections. This approach is now being applied to calculate P-wave positron-hydrogen scattering. The present results, obtained using short-range correlation functions along with long-range correlations in the Schrödinger equation at the same time, agree very well with the results obtained in an earlier calculation by Bhatia et al. (1974), using the Feshbach projection operator formalism. In these earlier calculations, the correction due to the long-range correlations was applied to the variational results. In spite of the fact that this ad hoc correction destroyed the variational bound, the final results have been considered accurate. Annihilation cross-sections, positronium formation cross-sections, calculated in the distorted-wave approximation, are also presented.

Positron-Hydrogen Scattering, Annihilation, and Positronium Formation

In previous papers (Bhatia A.K. 2007, 2012) a hybrid theory for the scattering of electrons from a hydrogenic system was developed and applied to calculate scattering phase shifts, Feshbach resonances, and photoabsorption processes. This approach is now being applied to the scattering of positrons from hydrogen atoms. Very accurate phase shifts, using the Feshbach projection operator formalism, were calculated previously (Bhatia A.K. et al. 1971 and Bhatia et al. 1974a). The present results, obtained using shorter expansions in the correlation function, along with long-range correlations in the Schrödinger equation, agree very well with the results obtained earlier. The scattering length is also calculated and the present results are compared with the previous results. Annihilation cross-sections, and positronium formation cross-sections, calculated in the distorted-wave approximation, are also presented.

Positron interactions with molecules

In this work, we report elastic integral cross sections for low energy positron scattering by carbon monoxide and model calculations for vibrationally enhanced positron annhilation on molecules. The former studies employed the Schwinger Multichannel method in the static plus polarization approximation with two different basis sets to access numerical convergence. In the annihilation studies, the role of multimode vibrational couplings was surveyed with the help of an analytically solvable model based on the Feshbach projection operator formalism.

Application ofP-wave hybrid theory to the scattering of electrons from He+and resonances in He and H−

The $P$-wave hybrid theory of electron-hydrogen elastic scattering [Bhatia, Phys. Rev. A 85, 052708 (2012)] is applied to the $P$-wave scattering from He ion. In this method, both short-range and long-range correlations are included in the Schr\"odinger equation at the same time, by using a combination of a modified method of polarized orbitals and the optical potential formalism. The short-range-correlation functions are of Hylleraas type. It is found that the phase shifts are not significantly affected by the modification of the target function by a method similar to the method of polarized orbitals and they are close to the phase shifts calculated earlier by Bhatia [Phys. Rev. A 69, 032714 (2004)]. This indicates that the correlation function is general enough to include the target distortion (polarization) in the presence of the incident electron. The important fact is that in the present calculation, to obtain similar results only a 20-term correlation function is needed in the wave function compared to the 220-term wave function required in the above-mentioned calculation. Results for the phase shifts, obtained in the present hybrid formalism, are rigorous lower bounds to the exact phase shifts. The lowest $P$-wave resonances in He atom and hydrogen ion have also been calculated and compared with the results obtained using the Feshbach projection operator formalism [Bhatia and Temkin, Phys. Rev. A 11, 2018 (1975)] and also with the results of other calculations. It is concluded that accurate resonance parameters can be obtained by the present method, which has the advantage of including corrections due to neighboring resonances, bound states, and the continuum in which these resonances are embedded.

Overlapping resonances in nuclei coupling to the atomic shells

The resonant process of nuclear excitation by electron transition (NEET) in highly charged ions is investigated. In NEET, a bound electronic decay transition occurs with the simultaneous excitation of the nucleus, provided that the energies of the atomic and nuclear transition match. By varying the atomic charge state, the atomic transition energy can be tuned to a better match of the nuclear transition energy. We propose a new way to create the atomic hole in highly charged ions by dielectronic capture of a free electron. In order to derive the cross section for the three-step process composed by dielectronic capture, NEET, and the subsequent nuclear decay, a Feshbach projection operator formalism is developed. With the help of this formalism, the resonances in nuclei coupling to the atomic shell and the possible interference between several direct and resonant electronic and nuclear processes such as radiative recombination, dielectronic recombination and nuclear excitation by electron capture are described.

Feshbach projection operator approach to positron annihilation

We present a theory of vibrationally enhanced positron annihilation on molecules based on the projection operator formalism of Feshbach. Results are shown for a simple semiempirical model derived from the formal Zeff expression.

Feshbach projection operator approach to positron annihilation

We present a theory of vibrationally enhanced positron annihilation on molecules based on the projection operator formalism of Feshbach. Results are shown for a simple semiempirical model derived from the formal Zeff expression.

Phase lapses in open quantum systems and the non-Hermitian Hamilton operator

We study transmission through a system with $N=10$ states coupled to $K=2$ continua of scattering wave functions in the framework of the $S$ matrix theory by using the Feshbach projection operator formalism for open quantum systems. Due to the coupling of the system (being localized in space) to the (extended) continuum of scattering wave functions, the Hamilton operator ${H}_{\text{eff}}$ of the system is non-Hermitian. The numerical calculations are performed for different distributions of both the positions ${E}_{i}^{0}$ $(i=1,\dots{},N)$ of the states of the isolated (closed) system and the elements of the coupling vectors ${V}^{c}$ between system and continua $(c=1,\dots{},K)$. The overall coupling strength $\ensuremath{\alpha}$ simulating the degree of resonance overlapping, is used as a parameter. In all cases, the complex eigenvalues and eigenfunctions of ${H}_{\text{eff}}$ are controlled by $\ensuremath{\alpha}$. In the regime of overlapping resonances, the well-known spectroscopic reordering processes (resonance trapping) take place because the phases of the eigenfunctions of ${H}_{\text{eff}}$ are not rigid in the neighborhood of singular points (being crossing points of eigenvalue trajectories). Finally, width bifurcation generates $K=2$ short-lived and $N\ensuremath{-}K$ trapped long-lived states. Thus, narrow (Fano-like) resonances may appear in the transmission at high level density. They are similar to, but different from the Fano resonances in the scattering theory with $K=1$. Phase lapses are related to zeros in the transmission probability. Their number and position (in energy) are determined by the ${V}^{c}$ and ${E}_{i}^{0}$, but not by $\ensuremath{\alpha}$. However, number and position of the resonance states depend on $\ensuremath{\alpha}$ due to resonance trapping occurring in the regime of overlapping resonances. As a consequence, universal phase lapses between every two resonances may appear at high level density while the system will show mesoscopic features at low level density. The phase lapses are not a single phenomenon. Due to their link to singularities in the continuum, they are related also to other ``puzzling'' experimental results such as dephasing at low temperature.

Feshbach projection calculations of compound state resonances in atom-diatom scattering

The Feshbach projection operator formalism has been employed to calculate the energies of compound slate resonances in some collinear atom—diatom systems. Comparison with available close-coupled scattering calculations for the same systems shows the energies to lie within about ± Γ of the resonance centers. Beyond the preliminary results reported, the determination of the width Γp and energy shift Δp of each resonance is proposed using a method which makes the determination of all the parameters computationally self-contained. Extension to 2- and 3-dimensional systems is also suggested.

Feshbach projection-operator formalism applied to resonance scattering on Bargmann-type potentials

The projection-operator formalism of Feshbach is applied to resonance scattering in a single-channel case. The method is based on the division of the full function space into two segments, internal (localized) and external (infinitely extended). The spectroscopic information on the resonances is obtained from the non-Hermitian effective Hamilton operator ${H}_{\mathrm{eff}}$ appearing in the internal part due to the coupling to the external part. As is well known, additional so-called cutoff poles of the $S$ matrix appear, generally, due to the truncation of the potential. We study the question of spurious $S$ matrix poles in the framework of the Feshbach formalism. The numerical analysis is performed for exactly solvable potentials with a finite number of resonance states. These potentials represent a generalization of Bargmann-type potentials to accept resonance states. Our calculations demonstrate that the poles of the $S$ matrix obtained by using the Feshbach projection-operator formalism coincide with both the complex energies of the physical resonances and the cutoff poles of the $S$ matrix.

Exceptional points in open quantum systems

Open quantum systems are embedded in the continuum of scattering wavefunctions and are naturally described by non-Hermitian Hamilton operators. In the complex energy plane, exceptional points appear at which two (or more) eigenvalues of the Hamilton operator coalesce. Although they are a countable set of single points in the complex energy plane and therefore of measure zero, they determine decisively the dynamics of open quantum systems. A powerful method for the description of open quantum systems is the Feshbach projection operator formalism. It is used in the present paper as a basic tool for the study of exceptional points and of the role they play for the dynamics of open quantum systems. Among others, the topological structure of the exceptional points, the rigidity of the phases of the eigenfunctions in their vicinity, the enhancement of observable values due to the reduced phase rigidity and the appearance of phase transitions are considered. The results are compared with existing experimental data on microwave cavities. In the last section, some questions being still unsolved, are considered.

Applications of the hybrid theory to the scattering of electrons fromHe+andLi2+and resonances in these systems

Applications of the hybrid theory to the scattering of electrons from Ile+ and Li++ and resonances in these systems, A. K. Bhatia, NASA/Goddard Space Flight Center- The Hybrid theory of electron-hydrogen elastic scattering [I] is applied to the S-wave scattering of electrons from He+ and Li++. In this method, both short-range and long-range correlations are included in the Schrodinger equation at the same time. Phase shifts obtained in this calculation have rigorous lower bounds to the exact phase shifts and they are compared with those obtained using the Feshbach projection operator formalism [2], the close-coupling approach [3], and Harris-Nesbet method [4]. The agreement among all the calculations is very good. These systems have doubly-excited or Feshbach resonances embedded in the continuum. The resonance parameters for the lowest ' S resonances in He and Li+ are calculated and they are compared with the results obtained using the Feshbach projection operator formalism [5,6]. It is concluded that accurate resonance parameters can be obtained by the present method, which has the advantage of including corrections due to neighboring resonances and the continuum in which these resonances are embedded.

Near threshold vibrational excitation of molecules by positron impact: A projection operator approach

We report vibrational excitation (νi=0→νf=1) cross-sections for positron scattering by H2 and model calculations for the (νi=0→νf=1) excitation of the C–C symmetric stretch mode of C2H2. The Feshbach projection operator formalism was employed to vibrationally resolve the fixed-nuclei phase shifts obtained with the Schwinger multichannel method. The near threshold behavior of H2 and C2H2 significantly differ in the sense that no low lying singularity (either virtual or bound state) was found for the former, while a e+-acetylene virtual state was found at the equilibrium geometry (this virtual state becomes a bound state upon stretching the molecule). For C2H2, we also performed model calculations comparing excitation cross-sections arising from virtual (-iκ0) and bound (+iκ0) states symmetrically located around the origin of the complex momentum plane (i.e. having the same κ0). The virtual state is seen to significantly couple to vibrations, and similar cross-sections were obtained for shallow bound and virtual states.

Near-threshold vibrational excitation ofH2by positron impact: A projection-operator approach

We report vibrational excitation $({\ensuremath{\nu}}_{i}=0\ensuremath{\rightarrow}{\ensuremath{\nu}}_{f}=1)$ and total cross sections for positron scattering by ${\mathrm{H}}_{2}$. The Feshbach projection operator formalism was employed to vibrationally resolve the fixed-nuclei phase shifts obtained with the Schwinger multichannel method. The near-threshold behavior of the vibrational excitation cross section is in good agreement with available experimental data [Sullivan et al., Phys. Rev. Lett. 86, 1494 (2001)]. Though our fixed-nuclei calculations do not indicate the existence of a ${e}^{+}$-${\mathrm{H}}_{2}$ virtual state, a proper description of the $T$ matrix threshold behavior is essential (the adiabatic approximation is inadequate). The projection operator approach has long been a powerful tool for studies of nuclear dynamics in electron-molecule collisions, and its application to positron scattering is timely since couplings to nuclear degrees of freedom are known to be very important at low impact energies.

Comment on “Bound-state eigenenergy outside and inside the continuum for unstable multilevel systems”

We discuss the solution of the basic equations of the $N$-level Friedrichs model by using the Feshbach projection operator (FPO) technique and consider the relation between bound states in the continuum (BICs) and form factors. In the FPO formalism, the BICs are eigenstates of a non-Hermitian Hamilton operator with vanishing decay width. The wave function of a BIC is localized inside the system. On the basis of the FPO solutions we discuss the mechanism to yield a BIC at every coupling strength between system and environment. We relate our results to those given by Miyamoto [Phys. Rev. A 72, 063405 (2005)].

Quantum interference between nuclear excitation by electron capture and radiative recombination

We investigate the quantum interference between the resonant process of nuclear excitation by electron capture (NEEC) followed by the radiative decay of the excited nucleus, and radiative recombination (RR). In order to derive the interference cross section, a Feshbach projection operator formalism is used. The electromagnetic field is considered by means of multipole fields. The nucleus is described by a phenomenological collective model and by making use of experimental data. The Fano profile parameters as well as the interference cross section for electric and magnetic multipole transitions in various heavy ions are presented. We discuss the experimental possibility of discerning NEEC from the RR background.

A wave packet method for treating nuclear dynamics on complex potentials

A general time-dependent description of the dissociative attachment of a triatomic molecule is presented. The approach presented works within the Feshbach projection operator formalism and gives an algorithm for solving the nuclear motion problem which reduces the computational effort required. The method uses a complex potential energy surface to characterize the formation and decay of resonances as modified by the coupling to the nuclear motion which are treated using multidimensional complex wave packets. A basis-independent wave packet method is developed and used to treat the propagation of a wave packet on a complex three-dimensional potential appropriate for resonance states of the water anion. A complete derivation of the system of linear equations used in the time iteration is presented. The method can be applied to resonant vibrational excitation, for which the electron impact vibrational excitation of water is considered.

Theory of nuclear excitation by electron capture for heavy ions

We investigate the resonant process of nuclear excitation by electron capture (NEEC), in which a continuum electron is captured into a bound state of an ion with the simultaneous excitation of the nucleus. In order to derive the cross section a Feshbach projection operator formalism is introduced. Nuclear states and transitions are described by a nuclear collective model and making use of experimental data. Transition rates and total cross sections for NEEC followed by the radiative decay of the excited nucleus are calculated for various heavy-ion collision systems.

Electron-He+P-wave elastic scattering and photoabsorption in two-electron systems

In a previous paper [A. K. Bhatia, Phys. Rev. A 69, 032714 (2004)], electron-hydrogen $P$-wave scattering phase shifts were calculated using the optical potential approach based on the Feshbach projection operator formalism. This method is now extended to the singlet and triplet electron-${\mathrm{He}}^{+}$ $P$-wave scattering in the elastic region. Phase shifts are calculated using Hylleraas-type correlation functions with up to 220 terms. Results are rigorous lower bounds to the exact phase shifts, and they are compared to phase shifts obtained from the method of polarized orbitals and close-coupling calculations. The continuum functions calculated here are used to calculate photoabsorption cross sections. Photoionization cross sections of He and photodetachment cross sections of ${\mathrm{H}}^{\ensuremath{-}}$ are calculated in the elastic region---i.e., leaving ${\mathrm{He}}^{+}$ and H in their respective ground states---and compared with previous calculations. Radiative attachment rates are also calculated.