Non-resonant Continuum States Research Articles

Resonances and scattering in microscopic cluster models with the complex-scaled generator coordinate method

The generator coordinate method of a microscopic cluster model is developed to treat the resonance and scattering of nuclear clusters with complex scaling. We consistently derive the formulation of the complex scaling for the microscopic cluster model, in which only the relative motions between clusters are transformed in the generator coordinate wave function. We also reveal the applicability of this method to the cluster wave function. Furthermore, we demonstrate this framework in the 2$\alpha$ system of $^8$Be and obtain the solutions of resonance and non-resonant continuum states. Using these solutions, we calculate the level density, which brings the phase shifts of the cluster-cluster scattering. This work becomes the foundation in the description of the multi-cluster scattering states of nuclei in a microscopic framework with complex scaling.

Complex scaling: Physics of unbound light nuclei and perspective

Abstract The complex scaling method (CSM) is one of the most powerful methods of describing the resonances with complex energy eigenstates based on non-Hermitian quantum mechanics. We present the basic application of CSM to the properties of the unbound phenomena of light nuclei. In particular, we focus on many-body resonant and non-resonant continuum states observed in unstable nuclei. We also investigate the continuum level density (CLD) in the scattering problem in terms of the Green’s function with CSM. We discuss the explicit effects of resonant and non-resonant contributions in CLD and transition strength functions.

Investigation of contributions of the 22+ resonance in He6 via analysis of the He6(p, p′) reaction

We investigate the contribution of the $2^{+}_{2}$ resonance in $^6$He to observables via analysis of the $^6$He($p,p'$) reaction by using the continuum-discretized coupled channels method combined with the complex-scaling method. In this study, we obtain the $2^{+}_{2}$ state with the resonant energy 2.25 MeV and the decay width 3.75 MeV and analyse contributions of resonances and nonresonant continuum states to the cross section separately. It is found that the $2^{+}_{2}$ state plays an important role in the energy spectrum. Furthermore, contributions of nonresonant continuum states are also important to clarify the properties of the $2^{+}_{2}$ state.

Gamow shell-model calculations of neutron-rich lithium isotopes

<p indent="0mm">The calculations of neutron-rich lithium isotopes have been performed using the Gamow shell model (GSM). In the present work, the shell model basis is generated by solving the one-body Schrödinger equation of the Woods-Saxon (WS) potential in the complex momentum space. The resulting single-particle basis is composed of bound, resonant and non-resonant continuum states, which is also called the Berggren ensemble. Two-body interaction between nucleons in the Berggren ensemble is derived from the realistic nuclear force CD-Bonn potential with the many-body perturbation theory named the <inline-formula id="INLINE7" content-type="pic"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mover accent="true"><mml:mi other="0">Q</mml:mi><mml:mo other="1" stretchy="true">^</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>-box folded diagrams. The model space consists of the 0p_3/2 bound state, 0p_1/2 resonant state and p_1/2 non-resonant continuum states for neutrons, and 0p_3/2, 0p_1/2 bound states for protons, respectively. The real and imaginary parts of the complex eigenvalues of the shell model Hamiltonian represent the excitation energies and resonance widths of resonant states, respectively. The p-shell neutron-rich lithium isotopes ^7–10Li have been studied. The results of GSM calculations with CD-Bonn potential are compared with traditional shell model with PWT interaction and GSM with surface Gaussian interaction (SGI). The low-lying spectra and resonance widths of resonant states are discussed, as well as the influences of the continuum states on these observables. In terms of the low-lying spectra of well-bound nuclei, the GSM calculations reproduce the excitation energies of ^7–9Li with good accuracy, except for the states with <sc>4 MeV</sc> or higher energies of ⁷Li, which decay through α-particle emission. In the calculation of the unbound nucleus ¹⁰Li, the GSM calculation proves that continuum effect is crucial in its low-lying spectrum. The shell model calculation with PWT interaction of ¹⁰Li overestimates the excitation energies by almost 100%, and by introducing the continuum states, GSM calculation can give more accurate values. Similar phenomenon was also observed in the shell model calculation with USDB interaction of the unbound nucleus ²⁶O, and was explained by the continuum effect as well. As for the resonance widths, the present calculations extend to higher levels of the studied nuclei compared with GSM calculations with the SGI interaction, and obtain reasonable resonance widths of the resonant states. Though the widths of the α-emitting states of ⁷Li cannot be explained by the current GSM, the resonant states among the excited states of ^8–9Li are reproduced in the present work. The GSM calculation with realistic nuclear force also gives correct resonant property of the first 1⁺ state of ¹⁰Li, while the GSM calculation with SGI yields a non-resonant state. The excited state of ¹⁰Li with unknown angular momentum and resonance width is also reproduced, which is predicted to be a 0⁺ state, with <sc>0.13 MeV</sc> width. In the present work, it is proved that the continuum effect is crucial in shell-model calculations of unbound ¹⁰Li. Through treating the continuum effect microscopically, the GSM calculations with properly renormalized realistic nuclear force can reproduce the observed low-lying spectra and resonance widths of both bound and unbound nuclei with high accuracy. On the other hand, the GSM with realistic nuclear force is based on a relatively small model space, and can thus be used to systems with more valence particles. The GSM with realistic nuclear force is proved to be a promising tool for future research on weakly bound or unbound nuclei.

Borromean Feshbach resonance in $^{11}$Li studied via $^{11}$Li($p$,$p'$)

Abstract A dipole resonance of $^{11}$Li is found by a $^9$Li + $n$ + $n$ three-body model analysis with the complex-scaling method. The resonance can be interpreted as a bound state in the $^{10}$Li + $n$ system, i.e., a Feshbach resonance in the $^9$Li + $n$ + $n$ system. As a characteristic feature of the Feshbach resonance of $^{11}$Li, the $^{10}$Li + $n$ threshold is open above the $^{9}$Li + $n$ + $n$ one, which reflects a distinctive property of the Borromean system. A microscopic four-body reaction calculation for the $^{11}$Li($p$,$p'$) reaction at 6 MeV/nucleon is performed by taking into account the resonant and nonresonant continuum states of the three-body system. The calculation of angular distributions of the elastic and inelastic scattering as well as the energy spectrum reproduced a recent experimental result. Furthermore, the $E1$ strength distribution from a Coulomb dissociation experiment was also reproduced in this framework. This means that the existence of the Borromean Feshbach resonance may consistently answer a long-standing open question of an excited state of $^{11}$Li.

Effects of mean-field and pairing correlations on the Bogoliubov quasiparticle resonance

The quasiparticle resonances are investigated by examining three kinds of quasiparticle spectra, i.e., the density of quasiparticle states, the occupation number density, and the pair number density in the continuum Skyrme Hartree-Fock-Bogoliubov theory with the Greens function method.Taking the weakly bound nucleus $^{66}$Ca as an example, the quasiparticle resonant energies and widths extracted from these three kinds of quasiparticle spectra are compared.For the narrow resonances, the extracted resonant energy and the width are consistent with each other. However, it is difficult to use the density of quasiparticle states to identify the broad resonances due to the background of nonresonant continuum.By switching off the pairing potential and/or the Hartree-Fock (HF) potential respectively in the calculation of these quasiparticle spectra, the roles of HF mean-field and pairing correlations in the quasiparticle resonances are demonstrated clearly. It turns out that all the quasiparticle resonances corresponding to the deeply bound, weakly bound and positive-energy single-particle resonant states, are mainly contributed by the HF potential.The pairing potential helps to slightly increase the resonant energy and the width. However, the pairing potential is important to make the nucleons occupy the low-lying nonresonant continuum states near the threshold and take part in the pairing correlations here, especially for the partial waves with small angular momentum $\ell$.

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.

Photodisintegration cross section ofBe9up to 16 MeV in theα + α + nthree-body model

The photodisintegration of $^9$Be in the energy region lower than $E_\gamma = 16$ MeV is investigated by using the $\alpha$~+~$\alpha$~+~$n$ three-body model and the complex scaling method. The cross section exhibits two aspects in the different two energy regions. In the low energy region up to $E_\gamma = 6$ MeV, the cross section is explained by the transition strengths into the excited resonant states of $^9$Be, while the dipole transition into the non-resonant continuum states of $^8$Be(2$^+$)~+~$n$ dominates the cross section in the energy region of $6 \le E_\gamma \le 16$ MeV. Furthermore, it is shown that the dipole strength at $E_\gamma \sim 8$ MeV is understood to be caused by the single-neutron excitation from the $^8$Be(2$^+$)~$\otimes$~$\nu p_{3/2}$ configuration in the ground state.

CDCC calculations of elastic scattering for the systems 6Li+144Sm and 6Li+208Pb. Effect of resonances of 6Li on elastic scattering angular distributions

Calculations of elastic scattering angular distributions for reactions of the weakly bound projectile 6Li with targets 144Sm and 208Pb at energies above the barrier, are performed with the Continuum Discretized Coupled-Channel method (CDCC). Ground, resonant and nonresonant continuum states of 6Li are included up to some maximum energy ϵmax for which convergence is achieved. In the three-body system, global interactions are used for the α-target and d - target sub-systems. The effect of continuum resonant states of 6Li, i.e., l = 2, jπ = 3+, 2+ and 1+ on elastic scattering angular distributions is investigated by extracting these states from the continuum space. It is found that the calculated elastic scattering angular distributions are in good agreement with the measurements for most of the cases studied where consideration of couplings to continuum states is essential. It is also found that the resonance character of the continuum states is in some cases important to obtain agreement with the data.

Effect ofLi6resonances on near-barrier elastic scattering involvingSi28andNi58target nuclei

Calculations of elastic scattering angular distributions for reactions of the weakly bound projectile $^{6}\mathrm{Li}$ with targets $^{28}\mathrm{Si}$ and $^{58}\mathrm{Ni}$ at energies just above the Coulomb barrier are performed with the continuum-discretized coupled-channel calculation method. Ground, resonant, and nonresonant continuum states of $^{6}\mathrm{Li}$ are included in the convergent calculations. The effect of the resonances on elastic scattering angular distributions is studied, in an original procedure, by excluding from the continuum space those states corresponding to the resonances. When the resonances of $^{6}\mathrm{Li}$ are considered, the calculated elastic scattering angular distributions are in good agreement with the measurements. The exclusion of the resonances, unexpectedly, has a very small effect at the energies studied. Calculation of the polarization potentials associated with the resonances show that they have a repulsive character at the long range region, where scattering occurs. It is also confirmed that couplings to continuum states of $^{6}\mathrm{Li}$ are essential to achieve agreement with the data.

Gaussian expansion methods under the absorbing boundary condition

We have analyzed the applicability of the various Gaussian bases to the variational calculation under the absorbing boundary condition. Three kinds of Gaussian basis are investigated: shifted, tempered, and oscillating Gaussians. All the basis functions are successful in describing a resonance with a sharp width, but the features of the non-resonant continuum states are very different among the employed Gaussian bases. The energy eigenvalues calculated from the shifted Gaussian show the regular sequence in a complex energy plane, while the energy distribution obtained by the tempered and oscillating Gaussian deviates from the regular distribution. The wave functions in the non-resonant continuum are investigated for the individual Gaussian bases. The shifted Gaussian nicely describes the extensively oscillating feature over a wide spatial range, but the wave functions of the other two bases concentrate around the interaction area. The calculation of the shifted Gaussian and oscillating Gaussian largely reduces the errors contained in the resonance wave function. The trajectories of the resonance, obtained by a variation of the strength of the absorbing potential, are pursued in a complex energy plane. The resonance trajectories for the shifted and oscillating Gaussian bases become stationary around the optimal strength of the absorber, while such a stationary trajectory is not clearly observed in the calculation of the tempered Gaussian. An appropriate combination of the basis functions and the absorbing boundary condition is discussed.

Decomposition of scattering phase shifts and reaction cross sections using the complex scaling method

We apply the complex scaling method to the calculation of scattering phase shifts and extract the contributions of resonances in a phase shift and a cross section. The decomposition of the phase shift is shown to be useful to understand the roles of resonant and non-resonant continuum states. As examples, we apply this method to several two-body systems: (i) a schematic model with the Gyarmati potential which produces many resonances, (ii) the alpha-alpha system which has a Coulomb barrier potential in addition to an attractive nuclear interaction, and (iii) the alpha-n system which has no barrier potential. Using different kinds of potentials, we discuss the reliability of this method to investigate the resonance structure in the phase shifts and cross sections.

Complex-scaling calculation of three-body resonances using complex-range Gaussian basis functions: Application to 3α resonances in 12C

in a geometric progression}, in the calculation of three-body resonances with the complex- scaling method (CSM), in which use is often made of the real-range Gaussian basis functions, {r l e −(r/r n ) 2 Ylm(r)}, which are suitable for describing the short-distance structure and the asymp- totic decaying behavior of few-body systems. The former basis set is more powerful than the latter when describing the resonant and nonresonant continuum states with highly oscillating amplitudes at large scaling angles θ. We applied the new basis functions to the CSM calcula- tion of the 3α resonances with J = 0 + ,2 + , and 4 + in 12 C. The eigenvalue distribution of the complex-scaled Hamiltonian becomes more precise and the maximum scaling angle becomes drastically larger (θmax = 16 ◦ → 36 ◦ ) than those given by the use of the real-range Gaussians. Owing to these advantages, we were able to confirm the prediction by Kurokawa and Kat¯ o (Phys. Rev. C 71, 021301 (2005)) on the appearance of the new broad 0 + state; we show it as an explicit

Quasiparticle continuum and resonances in the Hartree-Fock-Bogoliubov theory

The quasi-particle energy spectrum of the Hartree-Fock-Bogoliubov (HFB) equations contains discrete bound states, resonances, and non-resonant continuum states. We study the structure of the unbound quasi-particle spectrum of weakly bound nuclei within several methods that do not rely on imposing scattering or outgoing boundary conditions. Various approximations are examined to estimate resonance widths. It is shown that the stabilization method works well for all HFB resonances except for very narrow ones. The Thomas-Fermi approximation to the non-resonant continuum has been shown to be very effective, especially for coordinate-space HFB calculations in large boxes that involve huge amounts of discretized quasi-particle continuum states.

One-neutron removal strength ofHe7intoHe6using the complex scaling method

We study the one-neutron removal strength of the $^{7}\mathrm{He}$ ground state, which provides us with the $^{6}\mathrm{He}\text{\ensuremath{-}}n$ component in $^{7}\mathrm{He}$. The He isotopes are described on the basis of the $^{4}\mathrm{He}+\mathit{Xn}$ cluster model ($X=1,2,3$). The complex scaling method is applied to describe not only the Gamow resonances but also the nonresonant continuum states of valence neutrons, with the correct boundary condition of particle decays. The one-neutron removal strength of $^{7}\mathrm{He}$ into the unbound states of $^{6}\mathrm{He}$ is calculated using the complex-scaled Green's function, in which a complex-scaled complete set of $^{4}\mathrm{He}+n+n$ states is adopted. Using this framework, we investigate resonant and nonresonant contributions of the strength, which individually produce specific structures in the distributions. In addition, we propose a method to obtain the real-value strength using the complex values of spectroscopic factors of Gamow states. As a result, the $^{6}\mathrm{He}$(${2}^{+}$) resonance is found to give the largest contribution.

Quantum Three-Body Calculation of the Nonresonant Triple- Reaction Rate at Low Temperatures

The triple-\alpha reaction rate is re-evaluated by directly solving the three-body Schr\"odinger equation. The resonant and nonresonant processes are treated on the same footing using the continuum-discretized coupled-channels method for three-body scattering. Accurate description of the \alpha-\alpha nonresonant states significantly quenches the Coulomb barrier between the two-\alpha's and the third \alpha particle. Consequently, the \alpha-\alpha nonresonant continuum states below the resonance at 92.04 keV, i.e., the ground state of 8Be, give markedly larger contribution at low temperatures than in foregoing studies. We show that Nomoto's method for three-body nonresonant capture processes, which is adopted in the NACRE compilation and many other studies, is a crude approximation of the accurate quantum three-body model calculation. We find about 20 orders-of-magnitude enhancement of the triple-\alpha reaction rate around 10^7 K compared to the rate of NACRE.

Four- and Five-Body Scattering Calculations of Exotic Hadron Systems

We study the five-quark system uuddin the standard non-relativistic quark model by solving the scattering problem. Using the Gaussian Expansion Method (GEM), we perform the almost precise multi-quark calculations by treating a very large five-body modelspace including the NK scattering channel explicitly. Although a lot of pseudostates (discretized continuum states) with J π = 1 ± and J π = 3 ± are obtained within the bound-state approx- imation, all the states in 1.4 − 1.85 GeV in mass around Θ + (1540) melt into non-resonant continuum states through the coupling with the NK scattering state in the realistic case, i.e., there is no five-quark resonance below 1.85 GeV. Instead, we predict a five-quark resonance state of J π = 1 − with the mass of about 1.9 GeV and the width of Γ � 2.68 MeV. Similar calculation is done for the four-quark system c¯ ¯ (q = u, d) in connection with X 0 (3872). §1. Introduction: importance of the wave function of multi-quarks

Effects of the continuum coupling on spin-orbit splitting

Recently, the shell model in the complex k-plane (the so-called Gamow Shell Model) has been formulated using a complex Berggren ensemble representing bound single-particle states, single-particle resonances, and non-resonant continuum states. The single-particle basis used is that of the Hartree- Fock potential generated self-consistently by a finite-range residual interaction. In this framework, we shall discuss the “spin-orbit splitting” of the 3/2 1 − and 1/2 1 − states in 7He. It is demonstrated that the continuum effects are very important and cannot be taken into account in standard shell-model calculations.

Numerical aspects of the Gamow shell model

Recently, the shell model in the complex k-plane (the so-called Gamow shell model) has been formulated using a complex Berggren ensemble representing bound single-particle states, single-particle resonances and non-resonant continuum states. The single-particle basis used is that of the Hartree–Fock potential generated self-consistently by a finite-range residual interaction. In this framework, we shall discuss the numerical aspects of the Gamow shell model related to the optimization of the shell-model basis.

Shell-model description of weakly bound and unbound nuclear states

Recently, the shell model in the complex k-plane (the so-called Gamow Shell Model) has been formulated using a complex Berggren ensemble representing bound single-particle states, single-particle resonances, and non-resonant continuum states. In this framework, we shall discuss binding energies and energy spectra of neutron-rich helium and lithium isotopes. The single-particle basis used is that of the Hartree-Fock potential generated self-consistently by the finite-range residual interaction.