Neutron-deuteron Doublet Research Articles

Application of Semilocal Coordinate-Space Regularized Chiral Forces to Elastic Nd Scattering and Breakup

We solve three-nucleon (3N) Faddeev equations with nucleon–nucleon (NN) and three-nucleon forces (3NF) derived consistently in the framework of chiral perturbation theory, taking the semilocal coordinate-space regularized chiral N^4LO NN potential supplemented by the chiral N^2LO 3NF regularized in the same way. Based on these solutions the nucleon–deuteron (Nd) elastic scattering and deuteron breakup reactions are studied. We checked that the elastic Nd scattering cross section can be used as an alternative observable to the doublet neutron–deuteron (nd) scattering length to fix, together with the ^3H binding energy, the strengths of the contact terms of the N^2LO 3NF. We investigated the predicted 3NF effects in 3N continuum reactions putting an emphasis on the A_y puzzle in low energy nd elastic scattering and on cross sections in the symmetrical-space-star and quasi-free-scattering breakup configurations. We found that the N^2LO 3NF provides effects comparable to those of standard, (semi-)phenomenological 2pi -exchange Tucson-Melbourne 3NF.

Spectra and scattering of light lattice nuclei from effective field theory

An effective field theory is used to describe light nuclei, calculated from quantum chromodynamics on a lattice at unphysically large pion masses. The theory is calibrated at leading order to two available data sets on two- and three-body nuclei for two pion masses. At those pion masses we predict the quartet and doublet neutron-deuteron scattering lengths, and the alpha-particle binding energy. For $m_\pi=510~$MeV we obtain, respectively, $^4a_{\rm nD}=2.3\pm 1.3~$fm, $^2a_{\rm nD}=2.2\pm 2.1~$fm, and $B_{\alpha}^{}=35\pm 22~$MeV, while for $m_\pi=805~$MeV $^4a_{\rm nD}=1.6\pm 1.3~$fm, $^2a_{\rm nD}=0.62\pm 1.0~$fm, and $B_{\alpha}^{}=94\pm 45~$MeV are found. Phillips- and Tjon-like correlations to the triton binding energy are established. Higher-order effects on the respective correlation bands are found insensitive to the pion mass. As a benchmark, we present results for the physical pion mass, using experimental two-body scattering lengths and the triton binding energy as input. Hints of subtle changes in the structure of the triton and alpha particle are discussed.

Comparative study of three-nucleon force models in nuclear matter

We calculate the energy per particle of symmetric nuclear matter and pure neutron matter using the microscopic many-body Brueckner-Hartree-Fock (BHF) approach and employing the Argonne V18 (AV18) nucleon-nucleon (NN) potential supplemented with two different three-nucleon force models recently constructed to reproduce the binding energy of $^{3}\mathrm{H},\phantom{\rule{0.16em}{0ex}}^{3}\mathrm{He}$, and $^{4}\mathrm{He}$ nuclei as well as the neutron-deuteron doublet scattering length. We find that none of these new three-nucleon force models is able to reproduce simultaneously the empirical saturation point of symmetric nuclear matter and the properties of three- and four-nucleon systems.

Coulomb Effect in Nucleon–Deuteron Elastic Scattering

We have applied the quark-model nucleon–nucleon interaction fss2 to the nucleon–deuteron scattering. The screened cutoff Coulomb force in the momentum space has been incorporated to the proton–deuteron scattering by extending the matching condition, which is proposed by Vincent and Phatak for the sharply cutoff Coulomb problem. The model fss2 can reproduce the overall characteristics of the experimental data. It is found that fss2 partially incorporates the attractive feature, which accords with the good reproduction of the triton binding energy and neutron–deuteron doublet scattering length. This attractive feature is due to the off-shell effect related to the quark-model description of the short-range repulsion of the two-nucleon force.

Comparative study of three-nucleon potentials in nuclear matter

A new generation of local three-body potentials providing an excellent description of the properties of light nuclei, as well as of the neutron-deuteron doublet scattering length, has been recently derived. We have performed a comparative analysis of the equations of state of both pure neutron matter (PNM) and symmetric nuclear matter (SNM) at zero temperature obtained using these models of three-nucleon forces. In particular, we have carried out both variational and auxiliary field diffusion Monte Carlo calculations of the equation of state of PNM, while in the case of SNM we have only the variational approach has been considered. None of the considered potentials simultaneously explains the empirical equilibrium density and binding energy of symmetric nuclear matter. However, two of them provide reasonable values of the saturation density. The ambiguity concerning the treatment of the contact term of the chiral inspired potentials is discussed.

Triton with long-range chiral N3LO three-nucleon forces

Long-range contributions to the three-nucleon force that have been recently worked out in chiral effective field theory at next-to-next-to-next-to-leading order are for the first time included in the triton and the doublet nucleon-deuteron scattering length calculations. The strengths of the two short-range terms available at this order in the chiral expansion are determined from the triton binding energy and the neutron-deuteron doublet scattering length. The structure of the resulting three-nucleon force is explored and effects for the two-nucleon correlation function in the triton are investigated. Expectation values of the individual contributions to the three-nucleon force in the triton are found to be in the range from a few 100 keV to about 1 MeV. Our study demonstrates that the very complicated operator structure of the novel chiral three-nucleon forces can be successively implemented in three-nucleon Faddeev calculations.

The measurement of the incoherent neutron scattering length of the deuteron

The project has the goal to directly determine the bound spin-dependent neutron–deuteron scattering length b i , d by measuring the pseudomagnetic precession of polarised cold neutrons passing through a target with polarised nuclei. A precise measurement of b i , d can be used to increase the accuracy of the only poorly known doublet neutron–deuteron scattering length b 2 , d which is a crucial input parameter for novel effective field theories. We briefly report on the first measurements performed, discuss the systematics and consequent strategies to improve the accuracy of the experiment.

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.

Description of the low-energy doublet neutron-deuteron scattering in terms of parameters characterizing bound and virtual triton states

Low-energy doublet neutron-deuteron scattering is described in terms of the energies of the bound and virtual triton states and the nuclear vertex constants for these states. For the first time, the van Oers-Seagrave formula is derived from the Bargmann representation of the S matrix for a system having two states. The presence of a pole in this formula is shown to be a direct corollary of the existence of a low-energy virtual triton state. Simple explicit expressions for the nd scattering length and for the pole of the function kcotδ are obtained in terms of the parameters of the bound and virtual triton states. The low-energy parameters of nd scattering are calculated numerically, whereby it is shown that these parameters are highly sensitive to variations in the asymptotic normalization constant Cv2 for the virtual state. The Cv2 value fitted in our model to the experimental result for the nd scattering length is 0.0592.

Complex, energy-independent, local potential reproducing an absorptive phase shift and a bound state.

The triton binding energy, and the partly real and partly complex neutron-deuteron doublet channel elastic scattering phase shifts, calculated by means of the exact three-body theory, are used as input in the fixed-[ital l] inverse scattering theory of Marchenko. The local potentials obtained hereby are independent of energy, and complex. Their strong imaginary part reflects the strong absorption in the doublet channel arising from the opening of the deuteron breakup channel. For total orbital angular momentum [ital l] different from zero the potentials are unique, reproducing the input phase shift in the whole energy region. For [ital l]=0 where there exists, in addition, a bound state we construct a one-parameter class of complex local potentials which are phase--binding-energy equivalent; that is, each of its members reproduces, in addition to the partly complex phase shifts, the bound state with the physical, i.e., real binding energy. Correlations between the oscilliations in the radial independence of these equivalent potentials and physical properties of the input data are established.

Force-range correction in the three-body problem: Application to three-nucleon systems.

A leading correction to the zero-range theory is considered for three-body systems. The correction is linear in the force range ${\mathit{r}}_{0}$. An explicit expression is obtained for this correction for the case of correlation between the three-body binding energy and the particle-pair scattering length, as well as for the practically interesting case of correlation between the triton binding energy and the neutron-deuteron doublet scattering length. The correction for the neutron-deuteron quartet scattering length is also found. Physics of the correction involves a modification of the effective long-range interaction 1/${\mathit{R}}^{2}$, which arises in the three-body systems under the conditions of the zero-range theory, by a singular correction ${\mathit{r}}_{0}$/${\mathit{R}}^{3}$.

Extension of the minimal three-nucleon model to the unphysical sheet of energy.

In the framework of the three-nucleon zero-range theory, we have calculated the correlation between the energy of the virtual state of the triton and the doublet neutron-deuteron scattering length. Our results show that the Girard-Fuda plot for the energy of the virtual state as a function of the doublet n-d scattering length is reproduced by this model. We also observe that the plot remains essentially unchanged when the nucleon-nucleon singlet input is varied.

On the correlation between the triton binding energy and the neutron-deuteron doublet scattering length

A detailed explanation of the well-known Phillips line is presented. This line, which represents the correlation between calculated values of the triton binding energy and then-d doublet scattering length, is shown to be due to the fact that the binding energies of both the triton and the deuteron, as well as the energy eigenvalue of the two-nucleon singlet virtual state, are all small on the energy scale of nuclear forces. The equivalent statement is that the Phillips line is due to the loose spatial structure of these few-nucleon systems. These conclusions are obtained on the basis of (i) qualitative analysis of the problem. (ii) calculation of the above correlation in both zero and linear approximations in the range of nuclear forces, and (iii) comparison of the results of the calculation with the Phillips line.

The Phillips and the Girard-Fuda plot for the trinucleon system

For the S-wave spin doublet neutron-deuteron (n-d) system we suggest and parametrize an effective range expansion which has the correct dependence on the n-d scattering length a at low energies. The present expansion explains the low energy properties of this system and in particular the linear correlation between the trinucleon energies and the scattering length a.

Elastic neutron-deuteron doublet scattering near the deuteron breakup threshold

The anomalous behavior, as suggested by Efimov, in the doublet n-d scattering near the deuteron breakup threshold is investigated by computing coupled integral equation using s-wave separable potentials.

On the irregularity in doublet neutron-deuteron elastic scattering near the deuteron breakup threshold

A “rounded-step” irregularity is calculated to be present in doublet neutron-deuteron elastic scattering near the deuteron breakup threshold.

The occurence of a CDD singularity in the neutron-deuteron doublet scattering amplitude

It is shown that for the Amado model of elastic neutron-deuteron scattering in the s-wave doublet channel the single channel N D solution is not equivalent to the exact amplitude. Equivalence of the N/ D solution is demonstrated under the insertion of a CDD pole which simultaneously generates the triton pole and the effective range anomaly, thus establishing a link between these apparently unrelated features of the doublet amplitude.

The Nuclear Three Body Problem with Phase Equivalent Separable Interactions

The triton ground state energy and the neutron–deuteron doublet scattering length have been calculated with two families of phase equivalent separable interactions in the 1S0 and 3S1–3D1 states of the nucleon–nucleon interaction. Generating the interactions from a solution of the inverse scattering problem, a separable representation of the nucleon–nucleon interaction is fully exploited to fit the scattering and bound state data as well as possible. Off shell constraints on the interactions are then discussed. Results for the three nucleon quantities are compared to those obtained with realistic local and one boson exchange potentials. It is shown that all results for potentials obeying the known off shell constraints lie on the Phillips band, which appears to be independent of the potential model employed. This is a consequence of the fact that the triton binding energy and doublet scattering length are only sensitive to the broad detail of the nucleon–nucleon S matrix in the 1S0 and 3S1–3D1 states and to certain global features of the residues at the singlet deuteron and deuteron poles. Relatively large off shell variations in the triton binding energy and doublet scattering length along this Phillips band are possible and the plausibility that the experimental values could be fitted is discussed.

Charge symmetry as a constraint on off-shell effects in the three-nucleon system

The consequences of requiring the 1S 0 nucleon-nucleon interaction to be (approximately) charge symmetric, are examined in the three nucleon system. It is shown that this does restrict the off-shell variation to the extent that the binding energy of the triton ( E T), and the neutron-deuteron doublet scattering length ( 2 a) should lie on the Phillips line. However it is also shown that this constraint does not significantly reduce the amount of variation in E T (∼(2−3)MeV) and 2 a (∼1.5fm) along the Phillips line. Furthermore, it is pointed out that for the particular transformation used here, the requirement that the long range part of the nuclear force be known (the “one pion exchange” criterion), leads to a similar constraint on the variation in E T and 2 a to that of charge symmetry.

Off-shell effects in the three-nucleon system

The triton binding energy and the neutron-deuteron doublet scattering length have been studied with phase-equivalent rank-2 separable potentials in S-waves. The potentials have been constructed from the phase shifts using the solution of the inverse scattering problem. Large off-shell effects are connected with the appearance of additional nodes in the deuteron or the low-energy singlet wave functions.