Coulomb forces in the three-body problem with application to direct nuclear reactions

Faddeev equations are considered in the case of three charged particles interacting with both separable nuclear two-body interactions and also including Coulomb forces. Modified Faddeev equations with Coulomb Green's functions are introduced. The three-body amplitudes are given into pure Coulomb and distorted-Coulomb amplitudes. Introducing a decomposition in the angular momentum states, a set of three-body integral equations is obtained. The effect of pure coulomb amplitudes is studied in direct nuclear reactions and found to give a large contribution to the cross sections. The three-body integral equations obtained are applied for direct nuclear reactions. The angular distributions for¹²C(⁶Li,d)¹⁶O,¹⁶O(⁶Li,d)²⁰Ne, and¹²C(⁶Li,α)¹⁴N transfer reactions are calculated as well as for the⁶Li elastic scattering on¹²C. From the good agreement between the theoretically calculated and experimental data, better spectroscopic factors are extracted. The effect of including Coulomb forces in the three-body problem is found to improve the results by about 16.26%.     

Three-body formalism for deuteron stripping reactions

A three-body formalism for deuteron stripping reactions has been developed. The equations of Altet al (1967) (AGS) for the three particle system (target A, n, p) are reduced to a set of coupled one-dimensional integral equations with the use of (i) angular momentum basis for representation and (ii) separable approximation for the two bodyt-matrices (which delineate the interactions between the particle pairs). The on-shell solutions of this set of integral equations are then related to the cross sections of the rearrangement processes. The inputs in this calculation, viz., the separable interactions between the particle pairs in the respective channels are simply constructed from the respective two body bound state in accordance with the bound state approximation (BSA) conforming to the ‘unitarity’ requirement. Using this formalism preliminary calculations for the (d, p) and (d, n) reaction cross sections on¹⁶O have been carried out and they seem to have considerable semblance with the observed cross sections.     

Sensitivity of the three-body calculations to different effects

The bound state of few-body systems in light nuclei is studied as a three-body problem. The three-body problem is solved following the different approaches of the Faddeev formalism as well as the unitary pole approximation. Separable approximations are introduced to reduce the three-body problem to a set of coupled integral equations. Numerical calculations are carried out for the resulting integral equations and the separable expansion. In the present work, we calculate the ground-state binding energy of the bound three-nucleon system³H. The main interest of the present work is to investigate the sensitivity of the three-body binding energy to different effects in the problem. For this reason, we study the dependence of the three-body binding energy of different forms of local and separable two-body potentials, on the effective range of the two-body potentials, and on the percent of theD state in the deuteron wave function. Also, we test the sensitivity of the three-body binding energy to the considered number of terms from the separable expansion.     

The Two-Nucleon Stripping Process with Skyrme Type Potential

Theory of two-nucleon stripping reactions is reconsidered. The nucleon-nucleon interactions are taken as Skyrme type potentials. Differential cross-sections for two-nucleon stripping reactions are calculated using DWBA approximation with different Skyrme type potentials of different parameters. The angular distributions for different two-nucleon stripping reactions (t, p) with incident triton on the different targets of ³¹P, ⁴⁰Ca, ⁹⁶Zr, ¹¹⁸Sn and ²⁰⁶Pb are calculated giving good agreement with the experimental data. Better values are extracted for the spectroscopic factors.     

Dependence of the Three-Nucleon Binding Energy on Some Different Effects

The bound state of three-nucleon system is studied as a three-body problem which is solved following the different approaches of the Faddeev formalism as well as the unitary pole approximation. The three-body problem is reduced to a set of coupled integral equations by using separable approximations. Numerical calculations are carried out for the resulting integral equations and the separable expansion. In the present work, we calculate the ground-state binding energy of the bound three-nucleon system ³H. The main interest of the present work is to investigate the sensitivity of the three-nucleon binding energy to different effects. For this reason, we study the dependence of this energy on different forms of local and separable nucleon-nucleon potentials, the effective range of the nucleon-nucleon interaction, and on the percent of the D state in the deuteron wave function. Also we test the sensitivity of the three-nucleon binding energy to the considered number of terms from the separable expansion.     

Three-body molecular description of 9Be: (II). Adiabatic one-level approximation with correct angular momentum

The low-lying spectrum of the ⁹Be nucleus is calculated in an α+α+n three-body model. The molecular approach to this three-body problem based on the exact evaluation of the two-center wave functions for separable n-α potentials is considered in detail. The numerical results are obtained in a generalized Born-Oppenheimer approximation which preserves total angular momentum and parity.     

The (d,p) Reactions below 3.0 MeV

Excitation functions for a number of proton groups for the reactions ²³Na(d, p)²⁴Na and ²⁷Al(d, p)²⁸Al were measured in the energy range E_d = 1.5 to 3.0 MeV. The angular distributions for a number of proton groups were measured at a number of deuteron energies below 3.0 MeV for the three reactions ¹²C(d, p)¹³C, ²³Na(d, p)²⁴Na and ²⁷Al(d, p)²⁸Al. The theory of deuteron stripping reactions at incident energies below the Coulomb barrier has been considered. A closed analytical form for the differential cross-section has been obtained considering three cases according to the incident deuteron and outgoing proton energies. An attempt has been made to fit the angular distribution measurements at a number of deuterons bombarding energies accordingly.     

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.     

Three-body model for Li induced reactions

The ⁶Li induced reactions are presented on a soluble three-body model. Taking for the ⁶Li nucleus a deuterton + alpha cluster structure, the angular distributions for its stripping and elastic scattering on a target nucleus X are calculated. Good agreement with the experimental measurements is obtained.     

Absorption effects in halo-nucleus core or nucleon stripping reactions

Inclusive halo-nucleus core or nucleon stripping reactions are considered on the basis of a potential three-body model using the eikonal and adiabatic approximations. Clear analytical expressions for the cross sections of these reactions are obtained, and numerical calculations for ¹¹Be halo nucleus are performed. Constituent absorption by the target nucleus substantially influences the integral and differential (with respect to longitudinal momentum) nucleon stripping cross section. It is demonstrated that the differential core stripping cross section contains more complete information on the unperturbed wave function of the halo nucleus.     

A study of the odd isotopes of copper using the 58, 60, 62, 64Ni(τ, d) 59, 61, 63, 65Cu and 48Ca(τ, d)49Sc reactions

Experimental measurements are reported for the ^{58, 60, 62, 64}Ni(τ, d) ^{59, 61, 63, 65}Cu and ⁴⁸Ca(τ, d)⁴⁹Sc reactions and for the elastic scattering of ³He particles from ^{58, 60, 62, 64}Ni and ⁴⁸Ca targets at an incident energy of 18 MeV. The (τ, d) angular distributions cover the angular range from approximately θ_c.m. = 5° to 90° and the elastic scattering angular distributions range from θ_c.m. = 12° to 172°. In the (τ, d) reactions several weakly excited states, not previously seen in stripping reactions, have been identified and assignments of the transferred angular momentum made. The ⁴⁸Ca(τ, d)⁴⁹Sc reaction data are used in conjunction with the theoretical sum rules of MacFarlane and French to determine the normalization factors to be used with DWBA calculations for different sets of optical potentials. These normalization factors are used to extract spectroscopic strengths and centroid energies from the Ni(τ, d)Cu data which are compared with published model calculations for the odd copper isotopes. It is concluded that these calculations invariably fail to give a consistent picture of the odd copper isotopes mainly due to the use of centroid energies as variable parameters in the model calculations.     

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.     

Various stages of oxidation of chlorite as reflected in the Fe2 + and Fe3 + proportions in the Mössbauer spectra of minerals in Boda Claystone

The construction of the most simple representation coordinated with the physical asymptotic conditions is considered for the Coulomb three-body scattering wave function at energies below the threshold of the three-particle breakup. It is shown that such a representation can be constructed with the use of the well-known solutions of the separable Coulomb two-centre (CTC) problem if the role of the intercentre distance is given to the hyperradius. For the simplest version of the approach (Abramov, J Phys B 41:175201, 2008) the corresponding system of radial differential equations and the statement of the three-body scattering problem for the total angular momentum J?≥?0 are presented.     

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.     

Measurement of neutron polarization from (d,n) reactions on Mg,Si and Ca and comparison to DWBA calculations

Angular distributions of the neutron polarization produced in (d, n) reactions on ²⁴Mg and ²⁸Si were obtained in about 300 keV steps from 3.9 MeV down to 2.2 and 2.9 MeV, respectively. Excitation functions of the polarization were measured at 20° and 40° (lab) over these energy ranges. Polarization angular distributions were also measured for (d, n) reactions on ²⁸Si at 8.1 MeV and ⁴⁰Ca at 3.8 MeV. DWBA calculations are compared to the latter distributions as well as to the (d, n) cross-section data. Fluctuations in the low-energy polarization d ata from the ²⁴Mg and ²⁸Si targets made DWBA comparison of questionable value.     

An analytical expression of the amplitude of stripping reaction in the coulomb approximation

A method is derived for calculating the matrix element of stripping reaction (d,p) in the socalled distorted-wave Born approximation; only Coulomb scattering of incident and outgoing particles is taken into consideration. Coulomb wave functions are expressed by means of integral representation permitting integration over angular variables and, when the radial wave function of the trapped neutron is specially chosen, also calculation of the radial integral for non-zero orbital momentum of the trapped neutron. Relations are given for calculating the differential cross-section of reaction and the polarization of products. The special case, when the orbital momentum of the captured neutron is equal to one or two, is discussed in detail.     

Three-body calculation of neutron-deuteron elastic scattering in the energy region from 2.5 MeV up to 50 MeV

The three-body Faddeev equations for neutron-deuteron scattering are solved in the energy region from 2.5 MeV to 50 MeV of the incident neutron energy with small energy steps. Higher-rank separable potentials are used in the¹ s ₀ wave and in the³ s ₁—³ d ₁ waves, while rank-1 separable potentials are used in¹ p ₁,³ p _0,1,2 ¹ d ₂ ³ d _2,3 waves. The calculation is compared with experiments for the total cross section, the total break-up cross section, the differential cross section and the analyzing power of neutron-deuteron scattering. The improvements in the agreement as compared to previous calculations are impressive in many cases. Especially, the calculated total cross section agrees with the experiment below 30 MeV within the error bars, which are as small as 1%. A discussion on the numerical accuracy is given. General aspects of the calculated cross section are discussed. It is pointed out that thes-wave asymptotic normalization of the deuteron wave function (A _s ) is important.     

On the validity of dwba for deuteron stripping: (I). The Mitra three-body model

Previous work showing that there exists an exact formulation of the DWBA for stripping in the S-wave, separable potential, three-body model of Mitra is discussed and extended. The one-body equation obeyed by the c.m. wave function used in the reformulated DWBA is derived and compared with the equation obeyed by the wave function used in the standard formulation of DWBA, viz., the deuteron elastic scattering wave function. Results obtained by other workers on application of three-body methods to direct reactions are discussed in light of the fact that an exact DWBA exists for the separable potential model.