Incoming Wave Boundary Condition Research Articles

Wigner time delay revisited

The concept of Wigner time delay was introduced in scattering theory to quantify the delay or advance of an incoming particle in its interaction with the scattering potential. The scattered-wave part of the wave packet, which describes the interacting particle, is delayed/advanced with respect to the non-interacting plane-wave packet. It has been widely assumed that this concept can be transferred to ionization considering it as a half-scattering process. In the present work we show, by analyzing the corresponding wave packets, that this assumption is incorrect. For ionization the outgoing wave packet is a superposition of the continuum states ψk(−)(r,t), which satisfy the incoming-wave boundary condition at large distances r from the origin and which for large time t→∞ form a plane-wave packet. The expansion coefficients in this wave packet are the ionization probability amplitudes. If these amplitudes depend on the scattering phase shift, the plane-wave part of the ionized wave packet may be modified, leading to an ionization analog of the Wigner time delay. However, in most cases such a Wigner time delay is absent for ionization. We illustrate this with two examples: (i) photoionization by a weak field of an atom with the electron bound by a zero-range potential and (ii) strong-field ionization.

Microscopic study of the fusion reactions Ca40,48+Ni78 and the effect of the tensor force

We provide a microscopic description of the fusion reactions between $^{40,48}\mathrm{Ca}$ and $^{78}\mathrm{Ni}$. The internuclear potentials are obtained using the density-constrained (DC) time-dependent Hartree-Fock (TDHF) approach and fusion cross sections are calculated via the incoming wave boundary condition method. By performing DC-TDHF calculations at several selected incident energies, the internuclear potentials for both systems are obtained and the energy-dependence of fusion barrier are revealed. The influence of tensor force on internuclear potentials of $^{48}\mathrm{Ca}+^{78}\mathrm{Ni}$ is more obvious than those of $^{40}\mathrm{Ca}+^{78}\mathrm{Ni}$. By comparing the calculated fusion cross sections between $^{40}\mathrm{Ca}+^{78}\mathrm{Ni}$ and $^{48}\mathrm{Ca}+^{78}\mathrm{Ni}$, an interesting enhancement of subbarrier fusion cross sections for the former system is found, which can be explained by the narrow width of internuclear potential for $^{40}\mathrm{Ca}+^{78}\mathrm{Ni}$ while the barrier heights and positions are very close to each other. The tensor force suppresses the subbarrier fusion cross sections of both two systems.

Generalizing the calculable R -matrix theory and eigenvector continuation to the incoming-wave boundary condition

The calculable $R$-matrix theory has been formulated successfully for regular boundary conditions with vanishing radial wave functions at the coordinate origins [P. Descouvemont and D. Baye, Rep. Prog. Phys. 73, 036301 (2010)]. We generalize the calculable $R$-matrix theory to the incoming-wave boundary condition (IWBC), which is widely used in theoretical studies of low-energy heavy-ion fusion reactions to simulate the strong absorption of incoming flux inside the Coulomb barriers. The generalized calculable $R$-matrix theory also provides a natural starting point to extend eigenvector continuation (EC) [D. Frame et al., Phys. Rev. Lett. 121, 032501 (2018)] to fusion observables. The $^{14}\mathrm{N}+^{12}\mathrm{C}$ fusion reaction is taken as an example to validate these new theoretical tools. Both local and nonlocal potentials are considered in numerical calculations. Our generalizations of the calculable $R$-matrix theory and EC are found to work well for IWBC.

Jacques Raynal: memories of a physicist

This text is dedicated to the memory of Jacques Raynal. I recall some of the interactions I had with Jacques from 1975 up until his death. I first met Jacques as a young student at Saclay in the mid 1970s and from then on we collaborated frequently. Jacques initiated me into elastic and inelastic scattering calculations in the coupled channels formalism. Within this formalism we calculated the excitation of Giant Resonance by light- or heavy-ion projectiles. We collaborated on the Coulomb corrections implemented in ECIS and later we tested numerically the Incoming Wave Boundary Condition, which is an important assumption when calculating sub-Coulomb fusion cross sections. He was among the first to understand the origin of what we would call the 3- excitation hindrance and suggest solutions.

Absence of hindrance in a microscopic C12+C12 fusion study

Background: Studies of low-energy fusion of light nuclei are important in astrophysical modeling, with small variations in reaction rates having a large impact on nucleosynthesis yields. Due to the lack of experimental data at astrophysical energies, extrapolation and microscopic methods are needed to model fusion probabilities. Purpose: To investigate deep sub-barrier $^{12}$C+$^{12}$C fusion cross sections and establish trends for the $S$ factor. Method: Microscopic methods based on static Hartree-Fock (HF) and time-dependent Hartree-Fock (TDHF) mean-field theory are used to obtain $^{12}$C+$^{12}$C ion-ion fusion potentials. Fusion cross sections and astrophysical $S$ factors are then calculated using the incoming wave boundary condition (IWBC) method. Results: Both density-constrained frozen Hartree-Fock (DCFHF) and density-constrained TDHF (DC-TDHF) predict a rising $S$ factor at low energies, with DC-TDHF predicting a slight damping in the deep sub-barrier region ($\approx1$~MeV). Comparison between DC-TDHF calculations and maximum experimental cross-sections in the resonance peaks are good. However the discrepancy in experimental low energy results inhibits interpretation of the trend. Conclusions: Using the fully microscopic DCFHF and DC-TDHF methods, no $S$ factor maximum is observed in the $^{12}$C+$^{12}$C fusion reaction. In addition, no extreme sub-barrier hindrance is predicted at low energies. The development of a microscopic theory of fusion including resonance effects, as well as further experiments at lower energies must be done before the deep sub-barrier behavior of the reaction can be established.

Study of the fusion reaction 12C + 12C at low beam energy

In this article we discuss two aspects related to the 12C + 12C fusion reaction at low energies for carbon burning in supermassive stars. First we present plausible arguments for the notion that the observed resonance structures at the lowest measured energies arise from the relatively large spacing and narrow width of 24Mg compound levels at the corresponding excitation energy region. We thus point out that the Incoming Wave Boundary Condition is inappropriate for calculating the fusion cross section under these situations. Secondly, we report on a particle-γ coincidence technique that has been used for the first time to measure the fusion cross section in the system 12C + 12C at low beam energies. Based on these results, it should be possible to measure this important fusion cross section down to the 10 pb level within a reasonable length of time.

Holographic spectral function in nonequilibrium states

We develop holographic prescriptions for obtaining spectral functions in non-equilibrium states and space-time dependent non-equilibrium shifts in the energy and spin of quasi-particle like excitations. We reproduce strongly coupled versions of aspects of non-equilibrium dynamics of Fermi surfaces in Landau's Fermi-liquid theory. We find that the incoming wave boundary condition at the horizon does not suffice to obtain a well-defined perturbative expansion for non-equilibrium observables. Our prescription, based on analysis of regularity at the horizon, allows such a perturbative expansion to be achieved nevertheless and can be precisely formulated in a universal manner independent of the non-equilibrium state, provided the state thermalizes. We also find that the non-equilibrium spectral function furnishes information about the relaxation modes of the system. Along the way, we argue that in a typical non-supersymmetric theory with a gravity dual, there may exist a window of temperature and chemical potential at large N, in which a generic non-equilibrium state can be characterized by just a finitely few operators with low scaling dimensions, even far away from the hydrodynamic limit.

Microscopic study of Ca+Ca fusion

We investigate the fusion barriers for reactions involving Ca isotopes $\mathrm{^{40}Ca}+\mathrm{^{40}Ca}$, $\mathrm{^{40}Ca}+\mathrm{^{48}Ca}$, and $\mathrm{^{48}Ca}+\mathrm{^{48}Ca}$ using the microscopic time-dependent Hartree-Fock theory coupled with a density constraint. In this formalism the fusion barriers are directly obtained from TDHF dynamics. We also study the excitation of the pre-equilibrium GDR for the $\mathrm{^{40}Ca}+\mathrm{^{48}Ca}$ system and the associated $\gamma$-ray emission spectrum. Fusion cross-sections are calculated using the incoming-wave boundary condition approach. We examine the dependence of fusion barriers on collision energy as well as on the different parametrizations of the Skyrme interaction.

Resolving disagreement for [formula omitted] in a CFT plasma at finite coupling

The ratio of shear viscosity to entropy density in a strongly coupled CFT plasma can be computed using the AdS/CFT correspondence either from equilibrium correlation functions or from the Janik–Peschanski dual of the boost invariant plasma expansion. We point out that the previously found disagreement for η / s at finite 't Hooft coupling is resolved once the incoming-wave boundary condition for metric fluctuations at the horizon of the dual geometry is properly imposed.

Quasinormal modes and holography

Quasinormal frequencies of electromagnetic and gravitational perturbations in asymptotically anti-de Sitter spacetime can be identified with poles of the corresponding real-time Green's functions in a holographically dual finite temperature field theory. The quasinormal modes are defined for gauge-invariant quantities which obey an incoming-wave boundary condition at the horizon and a Dirichlet condition at the boundary. As an application, we explicitly find poles of retarded correlation functions of $R$-symmetry currents and the energy-momentum tensor in strongly coupled finite temperature $\mathcal{N}=4$ supersymmetric $SU({N}_{c})$ Yang-Mills theory in the limit of large ${N}_{c}$.

IWBC ANALYSIS WITH QUASIMOLECULAR OPTICAL POTENTIALS

The quasimolecular optical potential with Incoming Wave Boundary Condition model is used to obtain the fusion and elastic scattering cross-section at the sub-Coulomb region and above. This has been done for the 16 O +16 O and 12 C +12 C systems. It is found that sudden potentials describe well the average trend in the fusion and elastic scattering cross-section using the Incoming Wave Boundary Condition Method. Potentials fits for low energy are then used to obtain cross-sections at higher energies. Excellent agreement is obtained for the location of various peaks observed experimentally, especially in the case of 16 O +16 O . The parameters from these potentials give information on the nuclei in these dynamical situations. We find that a larger radius and a low density in the central region is required to explain the phenomena. The range of Yukawa potentials used are also larger by a factor of two as compared to those used by other authors.

The Nuclear Potential in Heavy-Ion Fusion

Fits to high precision fusion cross-sections even for Z P Z T < 1000 appear to need a value of the diffuseness parameter a of the Woods-Saxon potential of ≃1 fm, which is much larger than the commonly accepted value of ≃ 0.65 fm. However, use of large values of a necessarily makes the potential pocket very shallow for realistic values of the nuclear potential depth. For larger values of angular momenta the pocket disappears, and a fusion barrier energy or radius can no longer be defined. In the absence of a potential pocket, fusion does not occur in the commonly used realistic coupled channels code CCFULL where fusion is calculated by applying an incoming wave boundary condition at the position of the minimum of the attractive pocket. For this reason, realistic coupled channels calculations with large values of a have needed unrealistically deep nuclear potentials. Once the potential depth is constrained to reasonable values, the data cannot be explained by simply changing the diffuseness of the nuclear potential. This indicates the necessity to go beyond the potential model, and incorporate dynamical effects as the two nuclei move towards fusion, even for light systems with Z P Z T < 1000.

Inverstigation of [formula omitted] and [formula omitted] reactions at low energies

The cross sections for the characteristic γ-rays of the residual nuclei following 6 Li+ 16O and 7 Li+ 16O reactions have been measured at low energies with HPGe detectors. The fusion cross sections obtained from these γ-ray cross sections for both the systems are found to be in excellent agreement with the total reaction cross sections and this indicates that there is no inhibition of fusion cross sections for these systems at low energies, which is in contradiction to the results obtained from the measurement of evaporation residues and light particles. These cross sections also show good agreement with the IWBC (Incoming Wave Boundary Condition) model and the 1-D BPM (One-Dimensional Barrier Penetration Model) calculations. The critical angular momenta ( l cr) dediced from the fusion cross sections are also found to have an energy dependence similar to other Li-induced reactions.

Fusion cross sections for 6Li+ 12C and 6Li+ 13C reactions at low energies

The partial cross sections for the 6Li+ 12C and 6Li+ 13C reactions below and above the Coulomb barrier energy have been determined from the cross sections of the characteristic γ-rays measured with HPGe detectors and using the branching factor σ γ σ ch obtained from the statistical model calculations. The fusion cross sections obtained from the sum of these partial cross sections are found to be equal to the total reaction cross sections upto an energy well above the Coulomb barrier energy and there appears to be no evidence for limitation of fusion cross sections for these systems at such energies contrary to the evaporation residue measurements. The measured cross sections are also found to agree nicely with the IWBC (Incoming Wave Boundary Condition) and one-dimensional BPM (Barrier Penetration Model) calculation.

Concluding remarks

Concluding remarks

Relativistic multichannel calculation of the Ne KLL and Ar L2M2,3M2,3 Auger transition rates.

The multichannel multiconfiguration Dirac-Fock method has been used to study the Ne KLL and Ar ${\mathit{L}}_{2}$${\mathit{M}}_{2,3}$${\mathit{M}}_{2,3}$ Auger-electron transitions. Final-state correlation effects are accounted for by simultaneously including several ionic configurations and ionization channels in the calculation of the final-state many-electron wave functions. Our computational approach is based on scattering theory, which properly accounts for the incoming wave boundary condition. The line intensities from our test calculation for Ne KLL transitions are in good agreement with earlier theoretical results and with experiment. The interchannel coupling is stronger for the final states of the Ar ${\mathit{L}}_{2}$${\mathit{M}}_{2,3}$${\mathit{M}}_{2,3}$ transitions, but the net effect on the line intensities is somewhat smaller than that for Ne. Even the most extensive multichannel calculations are not fully able to reproduce the measured total rate and the ratio of $^{3}$${\mathit{P}}_{0,1,2}$ to $^{1}$${\mathit{D}}_{2}$ transition rates. The remaining discrepancy in the branching ratios of Auger lines is attributed to the omission of initial-state configuration interaction, whereas the discrepancy between the measured and calculated total rates is suggested to be mainly due to the neglect of relaxation. Our results indicate that, in analogy to outer-shell photoionization, the final-state interchannel interaction does not in general affect the strongest Auger lines very much when the kinetic energy of the Auger electrons exceeds about 200 eV.

Heavy-ion optical potential for sub-barrier fusion deduced from a dispersion relation.

The heavy-ion energy-dependent optical potentials for the /sup 16/O+/sup 208/Pb system are deduced from a dispersion relation. These potentials are used to analyze the elastic scattering, fusion, and spin distributions of compound nuclei for the system in a unified way based on the direct reaction theory. It turns out that the energy dependence of the optical potential is essential in explaining the data at near- and sub-barrier energies. The real part of the energy-dependent optical potential deduced was also used in calculating the elastic and fusion cross sections by the conventional barrier penetration model using an incoming wave boundary condition. The predictions of the elastic scattering, fusion cross sections, and the spin distributions of compound nuclei are not satisfactory compared with those from the direct reaction approach. It seems to originate from the fact that this model neglects absorption around the Coulomb barrier region

Subbarrier interactions of the oxygen isotopes.

We derive a unified description of the low-energy elastic, inelastic, neutron transfer, and fusion cross sections of systems involving two stable oxygen isotopes. Coupled channels equations with an incoming wave boundary condition are employed, as is the no-recoil approximation; the adiabatic approximation, which violates unitarity, is avoided. A calculation with phenomenologically determined $^{16}\mathrm{O}$ + $^{16}\mathrm{O}$ and $^{16}\mathrm{O}$-n potentials reproduces the measured cross sections within the limitations of the model. A possible error in the data analysis for $^{16}\mathrm{O}$ + $^{17}\mathrm{O}$ by Thomas et al. is pointed out.

Sub-barrier fusion of the oxygen isotopes: A more complete picture.

We have measured the sub-barrier fusion cross sections for 16,17,18O on 16O and we present the data and the experimental method in detail. The data were analyzed with a one-dimensional potential barrier inversion model, a two-dimensional incoming wave boundary condition model, and a two-dimensional Wentzel-Kramers-Brillouin model. We find that a multidimensional model is necessary to describe the enhanced fusion cross sections in oxygen and the two-dimensional incoming wave boundary condition model suggests that a complete description of the fusion, inelastic, and elastic cross sections is possible.

Calculation of the complete reaction cross section for 16O + 208Pb near the Coulomb barrier

Coupled-channels calculations are made for 16O + 208Pb near the Coulomb barrier. Inelastic and transfer channels are included. An incoming-wave boundary condition is used for fusion; there is no explicit imaginary potential. The elastic angular distributions and the fusion cross sections are well predicted only with a small diffuseness real potential.