Pseudostate Basis Research Articles

Identifying structures in the continuum: Application to Be16

The population and decay of two-nucleon resonances offer exciting new opportunities to explore dripline phenomena. The understanding of these systems requires a solid description of the three-body (core+N+N) continuum. The identification of a state with resonant character from the background of non-resonant continuum states in the same energy range poses a theoretical challenge. It is the purpose of this work to establish a robust theoretical framework to identify and characterize three-body resonances in a discrete basis. A resonance operator is proposed, which describes the sensitivity to changes in the potential. Resonances are then identified from the lowest eigenstates of the resonance operator. The operator is diagonalized in a basis of Hamiltonian pseudostates, built within the hyperspherical harmonics formalism using the analytical THO basis. The energy and width of the resonance are determined from its time dependence. The method is applied to 16Be in a 14Be+n+n model. An effective core+n potential, fitted to the available information on the subsystem 15Be, is employed. The 0+ ground state resonance of 16Be presents a strong dineutron configuration, which favors the picture of a correlated two-neutron emission. Fitting the three body interaction to the experimental two-neutron separation energy |S2n|=1.35(10) MeV, the computed width is Gamma(0+)=0.16 MeV. From the same Hamiltonian, a 2+ resonance is also predicted with E_r(2+)=2.42 MeV and Gamma(2+)=0.40 MeV. The dineutron configuration and the computed 0+ width are consistent with previous R-matrix calculations for the true three-body continuum. The extracted values of the resonance energy and width converge with the size of the pseudostate basis and are robust under changes in the basis parameters. This supports the reliability of the method in describing the properties of unbound core+N+N systems in a discrete basis.

Extracting three-body breakup observables from continuum-discretized coupled-channels calculations with core excitations

Core-excitation effects in the scattering of two-body halo nuclei have been investigated in previous works. In particular, these effects have been found to affect in a significant way the breakup cross sections of neutron-halo nuclei with a deformed core. To account for these effects, appropriate extensions of the continuum-discretized coupled-channels (CDCC) method have been recently proposed. We aim to extend these studies to the case of breakup reactions measured under complete kinematics or semi-inclusive reactions in which only the angular or energy distribution of one of the outgoing fragments is measured. We use the standard CDCC method as well as its extended version with core excitations (XCDCC), assuming a pseudo-state basis for describing the projectile states. Two- and three-body observables are computed by projecting the discrete two-body breakup amplitudes, obtained within these reaction frameworks, onto two-body scattering states with definite relative momentum of the outgoing fragments and a definite state of the core nucleus. The presented method provides a tool to compute double and triple differential cross sections for outgoing fragments following the breakup of a two-body projectile, and might be useful to analyze breakup reactions with other deformed weakly-bound nuclei, for which core excitations are expected to play a role. We have found that, while dynamical core excitations are important for the proton target at intermediate energies, they are very small for the Zn target at energies around the Coulomb barrier.

Solution of the proton-hydrogen scattering problem using a quantum-mechanical two-center convergent close-coupling method

Details of the recently developed quantum-mechanical two-center convergent close-coupling approach (Abdurakhmanov et al 2016 J. Phys. B: At. Mol. Phys. 49 03LT01) to proton-hydrogen scattering are presented. The formulation is based on the exact (fully quantum-mechanical) three-body Schrödinger equation. The total scattering wavefunction is expanded using a two-center pseudostate basis. This allows one to include all underlying processes, namely, direct scattering and ionization, electron capture into bound and continuum states of the projectile. The off-shell integration in the coupled-channel Lippmann–Schwinger integral equations emerging from the three-body Schrödinger equation for the scattering wavefunction is taken analytically which greatly reduces computational effort. While the calculated electron capture cross sections are in a good agreement with experiment, some discrepancy exists for the ionization cross sections.

Importance of the single-particle continuum in BCS pairing with a pseudostate basis

In a recent work [arXiv:1510.03185] the use of the Transformed Harmonic Oscillator (THO) basis for the discretization of the singleparticle continuum into a Generalized Bardeen-Cooper-Schrieffer (BCS) formalism was proposed for the description of weakly bound nuclei. We make use of the flexibility of this formalism to study the evolution of the pairing when the nucleus becomes more and more weakly bound. Specifically we focus on the evolution of the occupation of the different partial waves in 22 O when the Fermi level approaches zero.

Core excitations in the structure and reactions of halo nuclei

We analyse the resonant break-up of halo nuclei including possible excita- tions of the core both in the structure of the halo nucleus and in the reaction dynamics. The structure of the halo nucleus is treated in the weak-coupling limit, assuming a parti- cle+core model with core excitations. The bound and resonant states of this system are calculated as an expansion in a pseudo-state basis, for which we use a recent generaliza- tion of the Transformed Harmonic Oscillator basis. This basis is used within a distorted wave Born approximation reaction framework which includes possible excitations and deexcitations of the core during the collision. We apply this scheme to describe the res- onant break-up of 11 Be on 12 C at 67 MeV/u. Comparing with the available experimental data from RIKEN (N. Fukuda et al., PRC 70, 054606 (2004)) we discuss the possibil- ity of extracting spectroscopic information of 11 Be resonances from the analysis of the break-up angular distributions.

Particle motion in a deformed potential using a transformed oscillator basis

The quantum description of a particle moving in a deformed potential is investigated. A pseudostate (PS) basis is used to represent the states of the composite system. This PS basis is obtained by diagonalizing the system Hamiltonian in a family of square integrable functions. In this work the Transformed Harmonic Oscillator (THO) functions, obtained from the solutions of the Harmonic Oscillator using a Local Scale Transformation (LST), are used. The proposed method is applied to the 11Be nucleus, treated in a two-body model (10Be+n). Both structure and reaction observables have been studied. Wavefunctions and energies obtained for the bound states and some low-lying resonances are compared with those obtained by direct integration of the Schroedinger equation. The dipole and quadrupole electric transition probabilities for the low-energy continuum have been calculated in the THO basis, and compared with the exact distributions obtained with the scattering states. Finally, the method is applied to describe the 11Be states in the Coulomb breakup of 11Be+208Pb at 69 MeV/nucleon. The energy and angular distributions of the exclusive breakup have been calculated using the Equivalent Photon Method, including both E1 and E2 contributions. The calculated distributions are found to be in good agreement with the available experimental data from RIKEN [Phys. Rev. C70, 054606]. At the very forward angles, the cross section is completely dominated by the dipole couplings.

Exploring continuum structures with a pseudo-state basis

The ability of a recently developed square-integrable discrete basis to represent the properties of the continuum of a two-body system is investigated. The basis is obtained performing a simple analytic local scale transformation to the harmonic oscillator basis. Scattering phase-shifts and the electric transition probabilities $B(E1)$ and $B(E2)$ have been evaluated for several potentials using the proposed basis. Both quantities are found to be in excellent agreement with the exact values calculated from the true scattering states. The basis has been applied to describe the projectile continuum in the $^{6}\mathrm{He}$ scattering by $^{12}\mathrm{C}$ and $^{208}\mathrm{Pb}$ targets at 240 MeV/nucleon and the $^{11}\mathrm{Be}$ scattering by $^{12}\mathrm{C}$ at 67 MeV/nucleon. The calculated breakup differential cross sections are found to be in very good agreement with the available experimental data for these reactions.

Polarization effects in electron collisions with Li2: application of the molecular R-matrix method with pseudostates

Li2 has a huge polarizability and a huge elastic cross section for collisions with low-energy electrons. The recently developed molecular R-matrix with pseudostates (MRMPS) method is applied to electron collisions with Li2 at energies below 5 eV. The calculations, which are shown to be stable with respect to the choice of pseudostate basis, demonstrate the power of the MRMPS method for representing target polarizabilities and polarization effects in general. A previously identified low-lying 2Πu shape resonance is found at about 0.05 eV and several other low-lying resonances and resonance-like features are identified. Cross sections for elastic and electronically inelastic electron collisions are calculated and compared with previous studies.

Exploring the 6He continuum sea through proton inelastic collisions

The p-6He inelastic scattering at 700 MeV u-1 is studied using Multiple Scattering expansion of the total Transition amplitude (MST) framework with a pseudo-state representation of 6He continuum. We show that the differential cross sections calculated with two different pseudo-state bases are in good agreement.

Continuum description with pseudostate wave functions

Benchmark calculations are performed aiming to test the use of two different pseudostate bases on the multiple scattering expansion of the total transition amplitude scattering framework. Calculated differential cross sections for $p\text{\ensuremath{-}}$$^{6}\mathrm{He}$ inelastic scattering at 717 MeV/nucleon show a good agreement between the observables calculated in the two bases. This result gives extra confidence on the pseudostate representation of continuum states to describe inelastic/breakup scattering.

Direct ionization of low-charge Li-like ions using asmall pseudo-state basis

We have calculated cross sections for the direct ionization ofBe+, B2+ and C3+ by electron impactusing a small pseudo-state basis within the R-matrix method(RMPS). On using one pseudo-state per l to represent thecontinuum, close agreement with converged RMPS results isobtained at energies ≳3 times the ionization limitprovided that the cross sections to the pseudo-continuum areprojected back onto the physical continuum. On using twopseudo-states per l, the close agreement is extended back downto energies below the ionization peak, again, provided thatprojection is used. The use of a small pseudo-state basis isnecessary to describe ionization processes in more complexsystems.

Convergent TCAO close-coupling calculations for electron transfer, excitation and ionization in intermediate keV collisions

Our recently developed two-centre atomic orbital (TCAO) close-coupling method is applied to - H(1s) collisions in the impact energy range from . Convergent cross sections are obtained for this system by using a large number of bound and pseudostate basis functions on the target and a limited number of bound states on the projectile. Cross sections for electron transfer to 1s, , , and , for excitation to , , and and for ionization are presented and compared with available experimental data and other theoretical calculations. The accuracy of the previous TCAO close-coupling calculations is challenged and the reliability of some experimental data for this system is questioned.

Weighted projection and optimization of coupled-channel equations

The conventional coupled-channel method for many-particle scattering is modified to facilitate applications to low-energy collision processes and to break-up (ionization) reactions. A new approach to the construction of coupled-channel equations is developed in terms of the weighted projections, where proper choices of the weighting function can simplify the set of coupled equations. Optimal choices of the weighting functions are discussed, and several approximations for construction of the weight functions are presented. The conventional pseudostate and mixed basis expansions are shown to be related to the weighted projection approach. The break-up (and ionization) channels can be included in the coupled-channel approach. Finally, the recently developed generalization of the Hartree - Fock (GHF) theory for scattering is shown to be a special case of the weighted projections, thus providing a partial justification of the use of amputated wavefunctions for the continuum state projection.

Spin asymmetry in electron-impact ionization of hydrogen atoms close to threshold.

I report the results of a calculation of the spin asymmetry in the electron-impact ionization of hydrogen atoms for incident energies between 14 and 20 eV. The work involves a variational close-coupling calculation with a pseudostate basis, supplemented by an optical potential procedure for higher partial waves.

Erratum: Electron-impact excitation of hydrogen atoms: Energies between the n=3 and n=4 thresholds

The cross sections for excitation of hydrogen atoms from the n = 1 and n = 2 states are calculated for incident energies between 12.1 and 12.75 eV. The calculations are made variationally, and employ a large pseudostate basis. Energies and widths of resonances are obtained.

Pseudostate bases including an L2 description of the continuum for use in e--He collision calculations

The possibility of constructing a distorted-wave second Born approximation model for electron-helium collisions at energies above 30 eV is examined. It is proposed to calculate the second Born term by analytic integration over the intermediate momentum of the colliding electron and by replacing the infinite sum over intermediate target states by a finite sum over a pseudostate expansion. A method is proposed for obtaining pseudostate bases which approximately reproduce known bound-state contributions to the oscillator strength sum rules for n3,1S to n3,1P transitions and polarizabilities of the 11S, 21S and 23S states. These bases are composed of 'exact' n3,1S, n3,1P and n3,1D states up to n=3 and of a restricted number of 3,1S, 3,1P and 3,1D L2-type pseudostates. Certain of the latter lie in the continuum and their energies are adjusted to correspond to quadrature points for Gauss-Laguerre integration, this in order to easily evaluate the continuum contributions to oscillator strength sums, polarizabilities and second Born terms.

The multichannel eikonal theory of electron-hydrogen collisions. I. Excitation of H (1s)

The integral and differential cross sections for the electronic excitation of the 2l and 3l states of hydrogen obtained by the multichannel eikonal theory in a 10-channel basis are reported. The lambda , R and I parameters for the 2p state of hydrogen at 54.40 and 350 eV are also provided. Good agreement with the experimental data of Mahan et al. (1976) for the integral cross sections sigma 1s to 3l is achieved by the updated and extended multichannel eikonal theory. Two general conclusions follow from the results reported in the paper: (a) the first Born results underestimates experiment in s to d transitions and (b) indirect coupling mechanisms for the excitation of the 3d state of hydrogen are important and cannot be neglected. These conclusions indicate that the pseudostate basis used in a number of optical potential model calculations needs further refinement in the intermediate energy region.

Exact and approximate second order calculations for electron excitation of the 2s state of hydrogen

In previous works, exact second-order amplitudes for electron excitation of the 2p state of hydrogen have been reported. These exact results were then used to evaluate the errors associated with various approximations for the second-order amplitude and they were also used to evaluate the accuracy of various pseudostate basis sets for calculating second-order amplitudes. The present work extends the previous results to the excitation of the 2s state by electron impact. The 2s results are given, comparisons are made with the corresponding 2p results and potential generalisations are discussed.

Electron-impact excitation of hydrogen atoms: Energies between the n=3 and n=4 thresholds.

The cross sections for excitation of hydrogen atoms from the n=1 and n=2 states are calculated for incident energies between 12.1 and 12.75 eV. The calculations are made variationally, and employ a large pseudostate basis. Energies and widths of resonances are obtained.

Accuracy of pseudostates for the electron-hydrogen scattering problem

The accuracy of several pseudostate basis sets for calculating second-order amplitudes is determined by comparing second-order distorted-wave amplitudes calculated using pseudostates with recent exact results. The accuracy of both the total basis sets and individual s-, p- and d-state components of the basis sets is determined and improved basis sets are formed by combining the best s-, p- and d-state basis sets. Both the improved basis sets and the original basis sets are used to calculate physically measurable quantities and the errors for the various pseudostates are determined. The primary objective for the study is to find an accurate pseudostate basis set which may be used in a close-coupling calculation since this method shows the most promise for obtaining accurate cross sections. The question of whether the pseudostate basis sets found in the paper which give accurate DWB2 results for excitation of the 2p state of hydrogen may also be used in a close-coupling calculation with comparable accuracy is addressed.