Coulomb Forces in Three Charged Particles Interacting via Stripping Nuclear Reactions

Coulomb forces are examined for three charged interacting particles following the Faddeev formalism. Application of stripping nuclear reactions is studied for ⁶Li projectile incident on ¹²C target with alpha particle transfer. Differential corss-sections are calculated. The results are found to be improved by about 12.41% due to the inclusion of the Coulomb forces.     

The three-body problem with short-range interactions

The quantum mechanical three-body problem is studied for general short-range interactions. We work in coordinate space to facilitate accurate computations of weakly bound and spatially extended systems. Hyperspherical coordinates are used in both the interpretation and as an integral part of the numerical method. Universal properties and model independence are discussed throughout the report. We present an overview of the hyperspherical adiabatic Faddeev equations. The wave function is expanded on hyperspherical angular eigenfunctions which in turn are found numerically using the Faddeev equations. We generalize the formalism to any dimension of space d greater or equal to two. We present two numerical techniques for solving the Faddeev equations on the hypersphere. These techniques are effective for short and intermediate/large distances including use for hard core repulsive potentials. We study the asymptotic limit of large hyperradius and derive the analytic behaviour of the angular eigenvalues and eigenfunctions. We discuss four applications of the general method. We first analyze the Efimov and Thomas effects for arbitrary angular momenta and for arbitrary dimensions d. Second we apply the method to extract the general behaviour of weakly bound three-body systems in two dimensions. Third we illustrate the method in three dimensions by structure computations of Borromean halo nuclei, the hypertriton and helium molecules. Fourth we investigate in three dimensions three-body continuum properties of Borromean halo nuclei and recombination reactions of helium atoms as an example of direct relevance for the stability of Bose–Einstein condensates.     

Three-body model for stripping nuclear reactions

A three-body model for the deuteron stripping nuclear reactions is presented. A set of three integral equations is obtained for the wave functions of the three-body problem by introducing a decomposition into angular momentum states into the Lippmann-Schwinger equation. Simple two-particle interactions with separable potentials are used. These separable potentials reduce the three-body problem to the solution of coupled sets of one-dimensional Fredholm integral equations. The angular distributions for²⁸Si(d,p)²⁹Si and⁴⁰Ca(d, p)⁴¹Ca stripping reactions are calculated. From the extracted spectroscopic factors, good agreement with the experimental measurements is obtained.     

Unitary approach to quasielastic scattering reactions

The possibility of deriving an approximate unitary solution to integral Faddeev equations within the K-matrix formalism is considered. Explicit expressions for the amplitudes of elastic, inelastic, and quasielastic three-body scattering are obtained under the assumption of a mechanism of a truly single collision. Specific calculations are performed for quasielastic-scattering reactions of the d(N, 2N)N type. Good agreement between the results of these calculations and experimental data indicates that, in developing approximate methods, it is highly desirable to respect fundamental physical principles.     

Symmetric integral equations derived from the Faddeev equations

A way is shown to transform the Faddeev equations of the atomic three-body problem into a set of integral equations with symmetric kernel. The method is treated in more detail for total angular momentumJ=0 and applied to calculating the binding energy of theH ^? ion.     

Recent developments in ab-initio few-body scattering calculations including the Coulomb interaction

Recent progress on the solution of ab-initio three- and four-nucleon scattering equations in momentum space that include the correct treatment of the Coulomb interaction is reviewed; results for specific observables in reactions initiated by p + d, p + ³He and n + ³He are shown. In addition three-body calculations of d + ¹²C reactions are compared with equivalent CDCC calculations.     

Analytic continuation of the effective-range expansion as a method for determining the features of bound states: Application to the 6Li nucleus

The scattering function (effective-range function) for the two-channel elastic scattering of charged particles is used to analyze dα scattering at low energies. In order to construct this function, use is made of various sets of phase shifts and mixing parameter, both those that were obtained by solving Faddeev equations in the three-body (n, p, α) model and those that were deduced from available phaseshift analyses. By means of an analytic continuation of the scattering function to the point of the pole corresponding to the bound state of the ⁶Li nucleus, the values of the vertex constants and asymptotic normalization coefficients are found for the process ⁶Li → α + d. Possible means for refining these results are discussed.     

Two-body scattering without angular-momentum decomposition: Fully off-shell T-matrices

Two-body scattering is studied by solving the Lippmann-Schwinger equation in momentum space without angular-momentum decomposition for a local short-range interaction plus Coulomb. The screening and renormalization approach is employed to treat the Coulomb interaction. Benchmark calculations are performed by comparing our procedure with a configuration space calculation, using the standard partial-wave decomposition, for ¹²C - ¹⁰Be elastic scattering. The fully off-shell T -matrices are also calculated for the final goal of studying the three-body scattering by solving Faddeev/AGS equations.     

Influence of the three-body coulomb effects on the values of asymptotic normalization coefficients (nuclear vertex constants) of astrophysical interest extracted from the modified DWBA approach for the peripheral proton-transfer reactions

From the exact three-body distorted wave amplitude obtained by other authors for the one-charged-particle-transfer reaction A(x, y)B within the strict three-body (x = y + a and B = A + a, where a is a transferred particle) model, its part is separated in which the three-body Coulomb dynamics of the transfer mechanism is taken into account correctly. The contribution of the three-body Coulomb dynamics of the transfer mechanism to the partial wave amplitudes at l ≫ 1 for the peripheral proton-transfer reactions ¹³C(³He, d)¹⁴N, ¹³C(¹⁴N, ¹³C)¹⁴N, and ⁹Be(¹⁰B, ⁹Be)¹⁰B is estimated within the three-body approach combining the dispersion method and the DWBA approach. For these reactions, the Coulomb renormalization factors, arising owing to correctly taking into account the three-body Coulomb dynamics of the proton-transfer mechanism in the DWBA cross sections, are calculated. A new estimate is obtained for the values of the asymptotic normalization factors for p + ¹³C → ¹⁴N, which also have astrophysical interest. The text was submitted by the authors in English.     

A three-body calculation of πd scattering

We present a theoretical treatment of the pion-deuteron system, meant specifically for the energy region below 100 MeV, and based on the Faddeev method for three-body scattering. This includes all orders of multiple scattering, two- and three-body unitarity (to a good approximation), nucleon recoil, deuteron d-state and a correct treatment of spin and isospin. For consistency with nuclear physics we treat the nucleons non-relativistically. However, relativistic kinematics are used for the pion. In order to obtain one-dimensional integral equations in the three-body system, we have constructed a set of separable πN t-matrices (with analytic form factors), which fit selected data up to 300 MeV. A comparison is made with existing π⁺d data at 48 MeV. This data tends to favour the Faddeev type of energy dependence for the πN t-matrix in the πd system. This could also be important in low-energy pion-nucleus scattering.     

Faddeev Three-body Calculation of Triple-Alpha Reaction

The triple-alpha (3α) process, in which three ⁴He nuclei are transformed into a ¹²C nucleus, is studied in terms of a three-alpha (3-α) model. The reaction rate of the process is calculated via an inverse process, 3-α photodisintegration of a ¹²C nucleus. Both of 3-α bound and -continuum states are calculated by a Faddeev method with accommodating the long range Coulomb interaction. Results of the 3α reaction rate are about 10³ times larger than a standard rate from the Nuclear Astrophysics Compilation of Reaction Rates (NACRE) at a low temperature (T = 10⁷ K), which means our results are remarkably smaller than recent three-body calculations by the method of continuum-discretized coupled-channel.     

Calculation of the nuclear vertex constant (asymptotic normalization coefficient) for the virtual decay 6Li → α + d on the basis of the three-body model and application of the result in describing the astrophysical nuclear reaction d(α, γ)6Li at ultralow energies

Within the (α, n, p) three-body model, the s-wave ⁶Li → α + d vertex constant G ₀₁ and the asymptotic normalization coefficient C ₀₁ (these two quantities are proportional to each other) for the wave function describing the ⁶Li nucleus in the α + d channel are calculated by solving Faddeev equations for various sets of nucleon-nucleon and α N potentials. The resulting values of G ₀₁ and C ₀₁ are used to calculate the astrophysical factor S(E) for the radiative-capture reaction d(α, γ)⁶Li at energies in the range E = 0–600 keV. The calculated values of G ₀₁ and C ₀₁ appear to be sensitive to the form of the pair potentials used. It is shown that, upon correctly taking into account the contributions of the E2 and E1 multipolarities, the values of S(E) in the region E < 150 keV are determined primarily by the values of C ₀₁ and of the ⁶Li binding energy in the α + d channel.     

Integral Equations for Three-Body Coulomb Resonances

 We propose a novel method for calculating resonances in three-body systems with repulsive Coulomb interactions. The method is based on the solution of a set of Faddeev and Lippmann-Schwinger integral equations. The resonances of the three-body system are defined as the complex-energy solutions of the homogeneous Faddeev integral equations. We show how the kernels of the integral equations should be continued analytically in order to get the resonances. As a numerical illustration a model for the three-α system is solved. Received October 1, 1999; revised February 25, 2000; accepted for publication June 30, 2000     

Calculation of the two-body scattering T-matrix in configuration space

In order to carry out a solution of the three-body Faddeev integral equations in configuration space, the calculation of the two-body scattering T-matrices and related integrals are required as an input. The formulation of the three-body Faddeev solution, as well as the computational steps used for the calculation of the T-matrices are presented, and results for the latter are illustrated for the case of the scattering of two helium atoms.     

On the nuclear interaction potential in the reactions with heavy ions

The interaction potential of heavy ions⁴He,⁶Li,¹²C and¹⁶O is constructed in the folding model. The density distribution of nuclear matter for these nuclei is calculated in the framework of the hyperspherical function method. For the calculation of the folding potentials we have employed the Skyrme nucleon-nucleon forces. The influence of several effects on the results of calculations is studied: the role of the three-body forces of the nucleon-nucleon interaction, dependence of the folding potential on the mass numbers of the colliding nuclei and the possibility of observing the monopole resonance in the ion inelastic scattering. Using our folding potential as a real part of the optical potential we have calculated the differential cross section of elastic scattering of⁶Li from¹²C at laboratory energy of lithium ionsT _L =90.0 MeV. Reasonable agreement with experiment is obtained.     

Operation of the Faddeev Kernel in Configuration Space

We present a practical method to solve Faddeev three-body equations at energies above the three-body breakup threshold as integral equations in coordinate space. This is an extension of a previously used method for bound states and scattering states below three-body breakup threshold energy. We show that breakup components in three-body reactions produce long-range effects on Faddeev integral kernels in coordinate space, and propose numerical procedures to treat these effects. Using these techniques, we solve Faddeev equations for neutron-deuteron scattering to compare with benchmark solutions.     

Experiments on screening effect in deuteron fusion reactions at extremely low energies

The enhanced electron screening effect in nuclear reactions taking place in dense astrophysical plasmas is extremely important for determination of stellar reaction rates in terrestrial laboratories as well as in prediction of cross sections enhancement in interiors of stars such as White and Brown Dwarfs or Giant Planets. This effect resulting in reduction of the nuclear Coulomb potential by the atomic electrons has been confirmed in many laboratory experiments. Unfortunately, experimental screening energies are much higher than the theoretical predictions and the reason for that remains unknown. Here, we present absorbing results of the experiment studying d + d nuclear reactions in different deuterized metallic targets under ultra high vacuum conditions. The total cross sections and angular distributions of the ²H(d,p)³H and ²H(d,n)³He reactions have been measured using a deuteron beam of energies between 8 and 30 keV provided by the electron cyclotron ion source. The atomic cleanness of the target surface has been secured by combining Ar sputtering of the target and Auger electrons spectroscopy. Due to application of an on-line analysis method, the homogeneity of the implanted deuteron densities could be continuously monitored. We will discuss probable causes of the large discrepancy between theoretical and experimental data.     

Elastic pion-deuteron scattering in the resonance region as a three-body problem

We study elastic pion-deuteron scattering in the Δ(1236) energy region by means of the three-body Faddeev equations. We present a compact angular momentum reduction of the Faddeev integral equation for separable amplitudes, neglecting the nucleon spin, and solve the resulting coupled integral equations. We examine the dependence of the elastic scattering amplitude on the deuteron structure, on the pion-nucleon scattering amplitude, and on the various orders of multiple scattering. The differential cross section is very sensitive to multiple scattering effects at backward angles. We find that a number of conventional approximations do not well reproduce these multiple scattering effects in the resonance region.