Deuteron Breakup Threshold Research Articles

Short-range three-nucleon interaction from A=3 data and its hierarchical structure

We construct accurate models of three-nucleon (3N) interaction by fitting, in a hybrid phenomenological approach, the low-energy constants parametrizing the subleading 3N contact operators to the triton binding energy, n-d scattering lengths, cross section and polarization observables of p-d scattering at 2 MeV center-of-mass energy. These models lead to a satisfactory description of polarized p-d scattering data in the whole energy range below the deuteron breakup threshold. In particular, the long-standing $A_y$ puzzle seems to be solved thanks to the new terms considered in the 3N force. Two types of hierarchies among the subleading contact operators are also derived, based on the large-$N_c$ counting and on a recently proposed relativistic counting. We test these hierarchies against the same experimental data and show that they are respected at a reasonable level.

The Born Series for S-wave Quartet nd Scattering at Small Cutoff Values

Perturbative expansions, the Born series, of the scattering length and the amplitude of $S$-wave neutron-deuteron scattering for spin quartet channel below deuteron breakup threshold are studied in pionless effective field theory at small cutoff values. A three-body contact interaction is introduced when the integral equation is solved using the small cutoffs. After renormalizing the three-body interaction by using the scattering length, we expand the integral equation as the ordinary and inverted Born series. We find that the scattering length and the phase shift are considerably well reproduced with a few terms of the inverted Born series at relatively large cutoff values, $\Lambda\simeq 100$ MeV.

Fore-aft asymmetry of the 2H(n, γ)3H at very low energies

We have used the recent effective field theory (EFT) which is constructed from two- and three-nucleon interactions, using minimal substitution in the momentum dependence of these interactions. We present the calculations of the fore-aft asymmetry of γ-rays in the reaction 2H(n, γ)3H which are based on EFT up to next-to-next-to leading order (N2LO). The results are compared with the recently reported calculations and measurements of the fore-aft asymmetry of γ-rays from neutron-deuteron radiative capture. The calculated fore-aft asymmetry of the nd radiative capture process above deuteron breakup threshold is in good agreement with the available experimental data up to 20 MeV.

Universal nature and finite-range corrections in elastic atom-dimer scattering below the dimer breakup threshold

We investigate universal behavior in elastic atom-dimer scattering below the dimer breakup threshold calculating the atom-dimer effective-range function $ak\cot\delta$. Using the He-He system as a reference, we solve the Schr\"odinger equation for a family of potentials having different values of the two-body scattering length $a$ and we compare our results to the universal zero-range form deduced by Efimov, $ak\cot\delta=c_1(ka)+c_2(ka)\cot[s_0\ln(\kappa_*a)+\phi(ka)]$, for selected values of the three-body parameter $\kappa_*$. Using the parametrization of the universal functions $c_1,c_2,\phi$ given in the literature, a good agreement with the universal formula is obtained after introducing a particular type of finite-range corrections. Furthermore, we show that the same parametrization describes a very different system: nucleon-deuteron scattering below the deuteron breakup threshold. Our analysis confirms the universal character of the process, and relates the pole energy in the effective-range function of nucleon-deuteron scattering to the three-body parameter $\kappa_*$.

Asymptotic method for determining the amplitude for three-particle breakup: Neutron-deuteron scattering

The process of neutron-deuteron scattering at energies above the deuteron-breakup threshold is described within the three-body formalism of Faddeev equations. Use is made of the method of solving Faddeev equations in configuration space on the basis of expanding wave-function components in the asymptotic region in bases of eigenfunctions of specially chosen operators. Asymptotically, wave-function components are represented in the form of an expansion in an orthonormalized basis of functions depending on the hyperangle. This basis makes it possible to orthogonalize the contributions of elastic-scattering and breakup channels. The proposed method permits determining scattering and breakup parameters from the asymptotic representation of the wave function without reconstructing it over the entire configuration space. The scattering and breakup amplitudes for states of total spin S = 1/2 and 3/2 were obtained for the s-wave Faddeev equation.

Renormalized three-body equations with short-range forces

A three-body problem for a system of three nucleons interacting with short-range forces is discussed following effective field theory. Considering the case of a system of neutron–deuteron scattering, three-body equations are demonstrated such that the eigenvalues greater than unity are removed from the kernel leading to an equation that is renormalization-group invariant. The phase shifts are developed from a unique solution of the case of the doubled channel in neutron–deuteron scattering. The calculated neutron–deuteron phase shifts are well described below the deuteron breakup threshold.

Effective-Range Expansion of Neutron-Deuteron Scattering Studied by a Quark-Model Nonlocal Gaussian Potential

The S-wave effective range parameters of the neutron-deuteron (nd) scattering are derived in the Faddeev formalism, using a nonlocal Gaussian potential based on the quark-model baryon-baryon interaction fss2. The spin-doublet low-energy eigenphase shift is sufficiently attractive to reproduce predictions by the AV18 plus Urbana three-nucleon force, yielding the observed value of the doublet scattering length and the correct differential cross sections below the deuteron breakup threshold. This conclusion is consistent with the previous result for the triton binding energy, which is nearly reproduced by fss2 without reinforcing it with the three-nucleon force.

N−delastic scattering using the hyperspherical harmonics approach with realistic local and nonlocal interactions

The application of the hyperspherical harmonic approach to the case of the $N\text{\ensuremath{-}}d$ scattering problem below deuteron breakup threshold is described. The nuclear Hamiltonian includes two- and three-nucleon interactions, in particular the Argonne ${v}_{18}$, the N3LO-Idaho, and the ${V}_{\mathrm{low}\text{\ensuremath{-}}k}$ two-nucleon, and the Urbana IX and N2LO three-nucleon interactions. Some of these models are local, whereas some are nonlocal. Also electromagnetic effects are included. Accurate calculations for many scattering observables at various center-of-mass energies are performed and the results are compared with the available experimental data. Furthermore, a ${\ensuremath{\chi}}^{2}$ analysis of some of the Hamiltonian models has been performed to compare their capability to describe the scattering process.

Analysis of experimental data on doublet neutron-deuteron scattering at energies below the deuteron-breakup threshold on the basis of the pole approximation of the effective-range function

On the basis of the Bargmann representation of the S matrix, the pole approximation is obtained for the effective-range function k cot δ. This approximation is optimal for describing the neutron-deuteron system in the doublet spin state. The values of r0 = 412.469 fm and v2 = −35 495.62 fm3 for the doublet low-energy parameters of neutron-deuteron scattering and the value of D = 172.678 fm2 for the respective pole parameter are deduced by using experimental results for the triton binding energy ET, the doublet neutron-deuteron scattering length a2, and van Oers-Seagrave phase shifts at energies below the deuteron-breakup threshold. With these parameters, the pole approximation of the effective-range function provides a highly precise description (the relative error does not exceed 1%) of the doublet phase shift for neutron-deuteron scattering at energies below the deuteron-breakup threshold. Physical properties of the triton in the ground (T) and virtual (v) states are calculated. The results are Bv = 0.608 MeV for the virtuallevel position and CT2 = 2.866 and Cv2 = 0.0586 for the dimensionless asymptotic normalization constants. It is shown that, in the Whiting-Fuda approximation, the values of physical quantities characterizing the triton virtual state are determined to a high precision by one parameter, the doublet neutron-deuteron scattering length a2. The effective triton radii in the ground (ρT = 1.711 fm) and virtual (ρv = 74.184 fm) states are calculated for the first time.

Electromagnetic structure of A=2 and 3 nuclei and the nuclear current operator

Different models for conserved two- and three-body electromagnetic currents are constructed from two- and three-nucleon interactions, using either meson-exchange mechanisms or minimal substitution in the momentum dependence of these interactions. The connection between these two different schemes is elucidated. A number of low-energy electronuclear observables, including (i) np radiative capture at thermal neutron energies and deuteron photodisintegration at low energies, (ii) nd and pd radiative capture reactions, and (iii) isoscalar and isovector magnetic form factors of $^{3}\mathrm{H}$ and $^{3}\mathrm{He}$, are calculated to make a comparative study of these models for the current operator. The realistic Argonne ${v}_{18}$ two-nucleon and Urbana IX or Tucson-Melbourne three-nucleon interactions are taken as a case study. For $A=3$ processes, the bound and continuum wave functions, both below and above deuteron breakup threshold, are obtained with the correlated hyperspherical harmonics method. Three-body currents give small but significant contributions to some of the polarization observables in the $^{2}\mathrm{H}$($p,\ensuremath{\gamma}$)$^{3}\mathrm{He}$ process and the $^{2}\mathrm{H}$($n,\ensuremath{\gamma}$)$^{3}\mathrm{H}$ cross section at thermal neutron energies. It is shown that the use of a current that did not exactly satisfy current conservation with the two- and three-nucleon interactions in the Hamiltonian was responsible for some of the discrepancies reported in previous studies between the experimental and theoretical polarization observables in pd radiative capture.

Three-body problem with short-range forces: Renormalized equations and regulator-independent results

We discuss effective field theory treatments of the problem of three particles interacting via short-range forces. One case of such a system is neutron-deuteron scattering at low energies. We demonstrate that in attractive channels the renormalization-group evolution of the 1+2 scattering amplitude may be complicated by the presence of eigenvalues greater than unity in the kernel. We also show that these eigenvalues can be removed from the kernel by one subtraction, resulting in an equation which is renormalization-group invariant. A unique solution for 1+2 scattering phase shifts is then obtained. We give an explicit demonstration of our procedure for both the case of three spinless bosons and the case of the doublet channel in nd scattering. After the contribution of the two-body effective range is included in the effective field theory, it gives a good description of the nd doublet phase shifts below deuteron breakup threshold.

Central and tensor components of three-nucleon forces in low-energy proton-deuteron scattering

Contributions of three-nucleon forces (3NF) to proton-deuteron scattering observables at energies below the deuteron breakup threshold are studied by solving the Faddeev equation that includes the Coulomb interaction. At ${E}_{p}=3.0\mathrm{MeV},$ we find that the central part of a two-pion exchange 3NF removes the discrepancy between measured cross sections and the calculated ones by two-nucleon forces, and improves the agreement with ${T}_{22}$ experimental data. However, the tensor part of the 3NF fails in reproducing data of the analyzing power ${T}_{21}$ by giving worse agreement between the measured and the calculated. Detailed examinations of scattering amplitudes suggest that a P-wave contribution in spin-quartet tensor amplitudes has unsuitable sign for reproducing the ${T}_{21}$ data.

Three-nucleon force effects in nucleon induced deuteron breakup. I. Predictions of current models

An extensive study of three-nucleon force effects in the entire phase space of the nucleon-deuteron breakup process, for energies from above the deuteron breakup threshold up to 200 MeV, has been performed. $3N$ Faddeev equations have been solved rigorously using the modern high precision nucleon-nucleon potentials AV18, CD Bonn, Nijm I, II, and Nijm 93, and also adding $3N$ forces. We compare predictions for cross sections and various polarization observables when $\mathrm{NN}$ forces are used alone or when the $2\ensuremath{\pi}$-exchange Tucson-Melbourne (TM) three nucleon force (3NF) was combined with each of them. In addition AV18 was combined with the Urbana IX 3NF and CD Bonn with the ${\mathrm{TM}}^{\ensuremath{'}}$ 3NF, which is a modified version of the TM 3NF, more consistent with chiral symmetry. Large but generally model dependent 3NF effects have been found in certain breakup configurations, especially at the higher energies, both for cross sections and spin observables. These results demonstrate the usefulness of the kinematically complete breakup reaction in testing the proper structure of $3N$ forces.

Low-Energy Proton-Deuteron Scattering in Configuration Space

A new method for solving the configuration-space Faddeev equations for elastic p-d scattering below the deuteron-breakup threshold is described. Numerical solutions that demonstrate the convergence and accuracy of the method are given. The number of channels and the value of the matching radius required to obtain an accurate solution are also investigated. These calculations demonstrate that this method can efficiently solve the large matrix equations required for the three-body scattering problem.

Range corrections to doublet S-wave neutron–deuteron scattering

We calculate the range corrections to S-wave neutron-deuteron scattering in the doublet channel (S=1/2) to first order in r/a where a is the scattering length and r the effective range. Ultraviolet divergences appearing at this order can be absorbed into a redefinition of the leading order three-body force. The corrections to the elastic scattering amplitude below the deuteron breakup threshold are computed. Inclusion of the range corrections gives good agreement with measured scattering data and potential model calculations.

Polarization observables inp−dscattering below 30 MeV

Differential and total breakup cross sections as well as vector and tensor analyzing powers for $p\ensuremath{-}d$ scattering are studied for energies above the deuteron breakup threshold up to ${E}_{\mathrm{lab}}=28 \mathrm{MeV}.$ The $p\ensuremath{-}d$ scattering wave function is expanded in terms of the correlated hyperspherical harmonic basis and the elastic S matrix is obtained using the Kohn variational principle in its complex form. The effects of the Coulomb interaction, which are expected to be important in this energy range, have been rigorously taken into account. The Argonne AV18 interaction and the Urbana URIX three-nucleon potential have been used to perform a comparison to the available experimental data.

Phase shifts and mixing parameters for low-energy proton-deuteron scattering

The eigenphase shifts and mixing parameters for elastic p-d scattering below the deuteron-breakup threshold are calculated for the AV14 two-body potential using two different numerical methods. The excellent agreement confirms that it is possible to perform accurate numerical studies of p-d scattering as well as n-d scattering for low energies. The numerical results can be considered as benchmarks for future calculations.

The Kohn Variational Principle for Elastic Proton-Deuteron Scattering Above Deuteron Breakup Threshold

The Kohn variational principle is formulated for calculating elastic proton-deuteron scattering amplitudes at energies above the deuteron breakup threshold. The use of such a principle with an expansion of the wave function on the (correlated) hyperspherical harmonic basis is discussed.

Total cross section for p− d breakup below 30 MeV

The total cross section for p− d breakup is studied in terms of the elastic S -matrix through the unitary condition. Calculations using the complex Kohn variational method along with the Pair Correlated Hyperspherical Harmonic basis are presented. The results have been restricted to energies below E p =30 MeV where Coulomb effects are expected to be sizable and are compared to the existing data. Two different measurements have been found in the literature: 40 years ago, Gibbons and Macklin [Phys. Rev. 114 (1959) 571] and 26 years ago, Carlson et al. [Lett. Nuovo Cimento 8 (1973) 319]. The calculations are found to be in reasonable agreement with these old data, though a discrepancy is observed near the deuteron breakup threshold. Moreover, a detailed analysis of the contributions to the observable from different partial waves has been performed. Unexpectedly, the main contribution for a wide range of energies has been detected in the J=3/2 − state.

Determination of proton-deuteron scattering lengths

An energy dependent phase-shift analysis of p+d elastic scattering below the deuteron breakup threshold has been conducted. The analysis indicates the existence of a singularity in the doublet S-wave effective range function, resulting in a very small value for the scattering length 2 a pd. The inferred value of the quartet S-wave scattering length 4 a pd is in good agreement with theoretical predictions.