Including Continuum States Research Articles

Density matrix renormalization group description of the island of inversion isotopes F28–33

Recent experiments have confirmed that the neutron-rich isotopes $^{28,29}$F belong to the so-called island of inversion (IOI), a region of the nuclear chart around $Z=10$ and $N=20$ where nuclear structure deviates from the standard shell model predictions due to deformation and continuum effects. However, while the general principles leading to the IOI are relatively well understood, the details of the low-lying structure of the exotic fluorine isotopes $^{28-33}$F are basically unknown. In this work, we perform large-scale shell model calculations including continuum states to investigate the properties of the neutron-rich isotopes $^{25-33}$F, using a core of $^{24}$O and an effective two-body interaction with only three adjustable parameters. We adjust the core potential and interaction on experimentally confirmed states in $^{25,26}$O and $^{25-27}$F and solve the many-body problem using the density matrix renormalization group method for open quantum systems in a $sd$-$fp$ model space. We obtain the first detailed spectroscopy of $^{25-33}$F in the continuum and show how the interplay between continuum effects and deformation explains the recent data on $^{28,29}$F, and produces an inversion of the ${5/2}^+$ and ${1/2}^+$ states in $^{29,31,33}$F. Several deformed one- and two-neutron halo states are predicted in $^{29,31}$F, and we predict the ground state of $^{30}$F to have a structure similar to that of the first ${5/2}^+$ state of $^{29}$F. We also suggest several experimental studies to constraint models and test the present predictions. The complex structure of neutron-rich fluorine isotopes offers a trove of information about the formation of the southern shore of the IOI through a subtle interplay of deformation and continuum couplings driven by the occupation of the quasi-degenerate neutron shells $0d_{3/2}$ and $1p_{3/2}$.

An exact and compact formula for the optical intersubband response of finite-barrier quantum wells, wires and dots

The optical response of quantum wells, wires and dots is frequently modelled without including continuum states. This is reasonable in a low-energy regime but fails to capture phenomena such as bound-to-continuum transitions. We show that, in fact, the full optical intersubband response of quantum confined structures can be expressed in an exceedingly simple exact formula for arbitrary barrier height, confinement dimension, and optical frequency. We illustrate the consequences of transitions reaching the continuum for several representative cases, including asymmetric quantum wells. Moreover, we derive a simple modification that may be added to bound-states-only simulation in order to include the continuum. For the asymmetric quantum well, both linear and second-order nonlinear response is obtained. Finally, excellent agreement with measured bound-to-continuum absorption in narrow quantum wells is demonstrated.

Nonlinear control of photonic higher-order topological bound states in the continuum

Higher-order topological insulators (HOTIs) are recently discovered topological phases, possessing symmetry-protected corner states with fractional charges. An unexpected connection between these states and the seemingly unrelated phenomenon of bound states in the continuum (BICs) was recently unveiled. When nonlinearity is added to the HOTI system, a number of fundamentally important questions arise. For example, how does nonlinearity couple higher-order topological BICs with the rest of the system, including continuum states? In fact, thus far BICs in nonlinear HOTIs have remained unexplored. Here we unveil the interplay of nonlinearity, higher-order topology, and BICs in a photonic platform. We observe topological corner states that are also BICs in a laser-written second-order topological lattice and further demonstrate their nonlinear coupling with edge (but not bulk) modes under the proper action of both self-focusing and defocusing nonlinearities. Theoretically, we calculate the eigenvalue spectrum and analog of the Zak phase in the nonlinear regime, illustrating that a topological BIC can be actively tuned by nonlinearity in such a photonic HOTI. Our studies are applicable to other nonlinear HOTI systems, with promising applications in emerging topology-driven devices.

Symplectic coherent-state bases in the OCM calculations of nuclear cluster systems

In order to solve many-body states in a wide energy region including continuum states, it is strongly desiredto develop a useful method. In nuclear cluster systems [1, 2,], we must treat both bound and unbound statessimultaneously. For such calculations, we propose to apply symplectic coherent-state bases (SCSB). In thisreport, the basic idea of the SCSB method will be explained. Especially the Pauli exclusion principle playsan important role in nuclear systems [3, 4], and the orthogonality condition model (OSM) [5] has beenstudied successfully for various kinds of nuclear cluster systems. We here discuss an application of theSCSB to the OCM. The SCSB method can be easily applied to two-cluster systems. However, for multiclustersystems, the applicability of the SCSB is not self-evident. The multi-cluster system is described bythe Pauli-allowed states (PAS) [6] under the Pauli exclusion principle. In this report, the multi-cluster SCSBwill be discussed on the basis of the PAS of the multi-cluster systems.

New Experiments Demand for a More Precise Analysis of Continuum Spectrum in $^6$He: Technical Details and Formalism

A simple three-body model of 6He is extended to include sd-continuum states in the picture in addition to the already investigated p-states. The role of different continuum components in the weakly bound nucleus 6He is studied by coupling unbound spd-waves of 5He by using simple pairing contact-delta interaction. The main focus of this paper is to outline the procedure that allows the calculation of different configurations of 6He including continuum states and to set up basic ingredients for computations. The method and results discussed here will be used for the calculation of monopole, dipole, quadrupole and octupole response of 6He.

Continuum effects on the temporal evolution of anharmonic coherent molecular vibrations

The generalization of the concept of coherent states for a Morse potential based on the real or complex displacements of the equilibrium position of the ground-state wave function yields a natural framework for their generation via Franck-Condon transitions. We study the relevance of including continuum states in the proper treatment of these displaced states, particularly for the description of their time evolution. It is shown that displaced Morse states may exhibit mathematical and physical temporal stability manifested through the presence of revivals in spite of the unbound character of the included continuum states. The Wigner function of these states is presented as well as relations to previous constructions. The time evolution of particular phase space properties such as mean values of position and momentum as well as their uncertainties are also analyzed.

Proton magic even-even isotopes and giant halos of Ca isotopes with relativistic continuum Hartree-Bogoliubov theory

We study the proton magic O, Ca, Ni, Zr, Sn, and Pb isotope chains from the proton drip line to the neutron drip line with the relativistic continuum Hartree-Bogoliubov (RCHB) theory. Particulary, we study in detail the properties of even-even Ca isotopes due to the appearance of giant halos in neutron rich Ca nuclei near the neutron drip line. The RCHB theory is able to reproduce the experimental binding energies $E_b$ and two neutron separation energies $S_{2n}$ very well. The predicted neutron drip line nuclei are $^{28}$O, $^{72}$Ca, $^{98}$Ni, $^{136}$Zr, $^{176}$Sn, and $^{266}$Pb, respectively. Halo and giant halo properties predicted in Ca isotopes with $A>60$ are investigated in detail from the analysis of two neutron separation energies, nucleon density distributions, single particle energy levels, the occupation probabilities of energy levels including continuum states. The spin-orbit splitting and the diffuseness of nuclear potential in these Ca isotopes are studied also. Furthermore, we study the neighboring lighter isotopes in the drip line Ca region and find some possibility of giant halo nuclei in the Ne-Na-Mg drip line nuclei.

Exciton polariton including continuum states: Microscopic versus additional boundary conditions

An exact method, based on microscopic boundary conditions, is developed for calculation of the optical properties of exciton polaritons with spatial dispersion. The method is formulated in a general manner for an arbitrary electron-hole potential and probed in the approximation of the one-dimensional contact interaction, in half-space and slab geometries. Another approach, based on approximate boundary conditions applied to all exciton bound and scattering states, generalizes the concept of additional boundary conditions. This approach yields analytical results for semi-infinite semiconductors and semiconductor slabs and allows us to extract and analyze the contribution of the exciton continuum states to the optical spectra of semiconductors. A correlation between reflectivity and transmission calculated in two different approaches demonstrates the crucial role of the strict boundary conditions in the correct computation of optical spectra.

Low-energy photodisintegration of9Bewith the molecular orbit model

We use a microscopic three-cluster model with the $\alpha+\alpha+n$ configuration to analyze with astrophysical interests the photodisintegration cross section of $^9$Be. The valence neutron in $^9$Be is treated in the molecular orbit model (MO), including continuum states by the box discretization method. It is shown that good agreement is achieved with the recently measured B(E1) transition probability from the ground state to the first 1/2$^+$ state when the box size is enlarged, resulting in reasonable reproduction of the experimental data for the photodisintegration cross section. The Coulomb dissociation of $^9$Be is also discussed.

Quantum-classical dynamics including continuum states using quantum trajectories

We apply the MQCB (mixed quantum/classical mechanics based on Bohmian trajectories) [E. Gindensperger, C. Meier, and J. A. Beswick, J. Chem. Phys. 113, 9369 (2000)] to the case where the quantum degree of freedom contains both bound and continuum states. The quantum degree of freedom is treated by a wave packet that is propagated on a spatial grid with a Hamiltonian that depends parametrically on the classical degrees of freedom, while the classical degrees of freedom themselves are coupled to the wave function via quantum trajectories. The method is applied to a simple model of a light particle colliding with a heavy (classical) particle absorbed on a surface, which has been used by other authors to develop methods to combine quantum degrees of freedom including continuum states with classical mechanics. The results are compared to the Ehrenfest mean-field method as well as to full quantum results.

The Approach to Asymptotic Behavior in Photoionization

AbstractThe high energy behavior of many atomic processes can be understood in terms of the theory of asymptotic Fourier transforms (AFT). In high energy atomic processes initial or final state wave functions will include continuum states of asymptotic plane wave character, and these will often result in a matrix element which can be viewed as a (multi‐dimensional) Fourier transform. The result will be a behavior of matrix elements and cross sections as inverse powers of the asymptotic momenta, associated with singular behaviors of atomic wave functions, which in turn are associated with the singular (Coulomb) behavior of the interaction of the atomic constituents. The situation is complicated by the Coulomb functions which modify these plane wave behaviors; these result in slowly convergent factors (we call them Stobbe factors) which may be explicitly factored out, yielding a matrix element as a product of a Stobbe factor and a residual matrix element which converges rapidly to its asymptotic form. We will illustrate these ideas here in a discussion of single and double non‐relativistic photoionization, though the approach may also be applied to electron scattering (and with excitation or ionization) and in the relativistic case. The main ideas of the approach can already be illustrated in the discussion of ionization of a particle in a central potential. Within this approach we explain both (for single ionization) the persistent high energy deviations from independent particle approximation predictions and (for double ionization) the shake–off and quasi–free contributions. We also discuss the importance of final state interactions and retardation corrections.

Coupling effects studied in the 13C(p,pn)12C and 13C(p,d)12C reactions at Ep=35 MeV.

The $^{13}\mathrm{C}$(p,pn${)}^{12}$C reaction was studied at ${\mathit{E}}_{\mathit{p}}$=35 MeV with an experimental arrangement which favored detection of p-n pairs in the $^{1}$${\mathit{S}}_{0}$ state. Measured cross section angular distributions were compared with those for the $^{13}\mathrm{C}$(p,d${)}^{12}$C reaction at the same incident energy. The two reactions show different angular distribution shapes, those for (p,pn) being more slowly decreasing at large angles. Coupled channel calculations which include continuum states of the p-n system reproduce angular distribution shapes and cross section magnitudes of both the (p,pn) and (p,d) data.

Intensities and asymmetry features of electronic Raman spectra of TmPO4. Third-order one-particle parametrization scheme

The intensities and polarization characteristics of the transitions from the ground state to the crystal field levels of the 3H6 multiplet of TmPO4 are analyzed and compared with experiment. Emphasis is directed toward the asymmetry features of the Raman spectra. The discussion is based on the results of ab initio calculations performed within the third-order electron correlation theory formulated in terms of perturbed functions. The various contributions to the scattering tensor have been evaluated for the complete radial basis sets (including continuum states), and they represent the impact of the singly excited configurations 4fN−1l′, for l′=d,f,g.

Molecular-state close-coupling theory including continuum states. I. Derivation of close-coupled equations.

We formulate a close-coupling theory of slow ion-atom collisions based on molecular (adiabatic) electronic states, and including the electronic continuum. The continuum is represented by packet states spanning it locally and constructed explicitly from exact continuum states. Particular attention is given to two fundamental questions: (1) Unbound electrons can escape from the local region spanned by the packet states. We derive close-coupled integral equations correctly including the escape effects; the ''propagator'' generated by these integral equations does not conserve probability within the close-coupled basis. Previous molecular-state formulations including the continuum give no account of escape effects. (2) Nonadiabatic couplings of adiabatic continuum states with the same energy are singular, reflecting the fact that an adiabatic description of continuum behavior is not valid outside a local region. We treat these singularities explicitly and show that an accurate representation of nonadiabatic couplings within the local region spanned by a set of packet states is well behaved. Hence an adiabatic basis-set description can be used to describe close coupling to the continuum in a local ''interaction region,'' provided the effects of escape are included. In principle, the formulation developed here can be extended to a large class of model problems involving many-electron systemsmore » and including models for Penning ionization and collisional detachment processes.« less

Molecular-state close-coupling theory including continuum states. II. Packet states and couplings for the proton-hydrogen-atom system.

The preceding paper (I) of this series develops a molecular-state close-coupling theory of ion-atom collisions including the electronic continuum, which is described by packet states spanning it locally. The present paper describes results of computations applying this formulation to the prototype problem of impact ionization in proton--hydrogen-atom collisions, in particular, studies of the properties of the basis states and the nonadiabatic couplings among them. (1) We construct continuum eigenfunctions using a ''quantal momentum'' or phase-amplitude representation and use this to study construction and properties of continuum packet states, and we show how localization of packet states within a specified interaction region leads to specification of packet widths and/or energies. (2) We verify assumptions made in the derivations given in paper I about nonadiabatic couplings of continuum states, and report computations of coupling matrix elements needed to implement the theory for this system. Subsequent work will present close-coupling calculations using this data.

A method of including continuum states in the close-coupling formulation

The close-coupling equations are reformulated in a restricted inner domain of coordinate space, allowing continuum states to be included in the target expansion. Cross sections are obtained by matching to an unsymmetrised close-coupling solution in a restricted asymptotic region of coordinate space. The method is applied to model electron-hydrogen excitation using the simplest method of representing the continuum, that of sampling. The results are as accurate as pseudo-state calculations, and also display spurious resonances. Further methods of improving the theory, such as representing all continuum states by interpolation in energy, are discussed. It is expected that these improvements will cause a significant increase in accuracy.

Nuclear-Scattering Problem in the Generalized Hartree-Fock Approximation

The generalized Hartree-Fock approximation of Kerman and Klein is extended to include continuum states of a collective nature. The equations for scattering in the random-phase approximation are rederived.

Nuclear-spectroscopy calculations including continuum states and application to the giant resonance of 16O

A simple generalization of nuclear-spectroscopy calculations to include continuum states leads to coupled integral equations. The same problem may be treated in nuclear reaction theory by solving coupled differential equations wherein the residual interaction range and total nucleon antisymmetrization introduce integral terms. We have applied both of these approaches to the giant resonance of 16O including only five channels in order to study by what methods and in what limits the coupled integral equations of the first formulation may be utilized. We show that the first method may be used in problems for which the number of channels is not too large; it allows analysis of the effects of quasi-bound, two-particle-two-hole states or more complex ones which should explain the fine structure of the giant resonance.