At this stage to calculate physical characteristics of nucleon-deuteron scattering we used the charge independent nucleon-nucleon potential AV14. First we studied the convergence of our results with respect to the maximum value of the momentum of pair nucleons 23 and the three-body angular momentum M. To check accuracy of our results we have performed calculations of observables for elastic neutron-deuteron (nd) scattering at E=3 MeV. In Fig. 1 the results of the calculation for the differential cross section are given along with the prediction of the Bochum group [i]. For the calculation, all values of the total three-nucleon angular momentum M up to 13/2 with both parity values (±1), and values of the momentum of pair nucleons 23 up to 3 were taken into account.
Fig. 1. Left side: differential cross section for elastic nd scattering at E=3 MeV. Right side: neutron analyzing power A for elastic nd elastic scattering at E=3 MeV. The solid line is our results. The dashed line corresponds to prediction of Ref..
As one can see are results are in excellent agreement to those of the Bochum group . In Fig. 1 on the right, the results of the calculation for the neutron analyzing power A are given along with the prediction of the Bochum group . Again the agreement between results obtained by different approaches is very good. It should be noted that Bochum’s results are solutions of the Faddeev equations in the momentum space and have been obtained for all values of the nucleon pair 23 up to 4 taken into account. Whereas our results are solutions of the Faddeev equations in configuration space and we took into account values of the nucleon pair up to 3.
In Fig. 2 the results of our calculation for the deuteron analyzing power iT are given along with the prediction of the Bochum group .
Fig. 2. Deuteron analyzing power iT for elastic nd elastic scattering at E=3 MeV. Notations are the same as in Fig. 1.
In this case the agreement between results obtained by entirely different methods is very good. These calculations have confirmed the correctness of our theoretical background and accuracy of the numerical methods elaborated by the computational group of Physics Dept. at NCCU.
Our next step was to check accuracy of our recent results on neutron-deuteron breakup scattering at E=14.1 MeV in Ref. [i]. For these calculations we have took into account all values of the total three-nucleon angular momentum M up to 13/2 with both parity values (±1), and values of the momentum of the nucleon pair 23 up to 3. In Fig.3 we present our results for elastic differential cross section along with prediction of the Bochum group Ref. [ii].
Fig. 3. Left side: differential cross section for elastic nd scattering at E=14.1 MeV. Right side: neutron analyzing power A for elastic nd elastic scattering at E=14.1 MeV. The solid line is our results. The dashed line shows predictions of the Bochum group Ref. . The experimental data are from Ref. [i]
Again we can state that our methods allow to get accurate results for neutron-deuteron scattering above the deuteron threshold. Some small difference having place in the vicinity of small scattering angles results of different nucleon-nucleon potentials used in calculations. The Bochum group has exploited the charge dependent AV18 potential.
Our new calculations for the analyzing power for neutron-deuteron scattering at E=14.1 MeV have confirmed our resent results in Ref. . It should be noted that our old and new prediction for the neutron analyzing power A are in some contradiction to calculations of other authors. In this regard we must point out that to the best of our knowledge all those calculations have been done using nucleon-nucleon potentials different from AV14 used in our calculations. In Fig.4 we present our results for deuteron analyzing power.
Fig.4. Deuteron analyzing power iT for nd inelastic scattering at E=14.1 MeV. The proton-deuteron data at E=15.0 MeV are from Ref. [i].
As one can see agreement with experimental data is quite satisfactory, taking into account that we compare our results with experimental data on proton-deuteron scattering at slightly different E=15.0 Mev energy. Note that even in nowadays there are no accurate experimental data on deuteron analyzing power at E=14.1 MeV .
For nd breakup scattering experimental situation is more complex since there exist a large number of the 3N breakup process. In this case three nucleons exist in final state and number of independent momentum components is five. Thus using two detectors and measuring two nucleons in coincidence four angles are fixed. Since the relation between the energies of the two detected nucleons is not unique one usually has to measure the energies of both nucleons in order to fix the kinematics. This complicates enormously both experimental setup and theoretical studies due to existence of a large number of kinematic configuration such as quasifree scattering (QFS), final state interaction (FSI), Colinear, Star and others. In Fig. 5 we present our first results on the fivefold differential cross section for neutron-deuteron breakup scattering for the final state configuration.
Fig. 5. nd differential cross section as a function of the arc length S for the FSI configuration (?=52.6°, ?= 40.5°, f=180°) at E=14.1 MeV. Experimental data denoted by solid triangles are from Ref. [i].
Agreement to experimental data is not so good as it was in the case of nd elastic scattering. In this regard we has to note that this configuration is very sensitive to choice of nuclear force for theoretical study and different nucleon-nucleon forces give quite different results (see, for example Ref. [i]). In Fig. 6 we present our first results on the neutron analyzing power for neutron-deuteron breakup scattering at E=14.1 MeV along this those of Ref. [ii].
Fig. 6. Analyzing power as a function of the arc length S for the FSI configuration (?=52.6°, ?= 40.5°, f=180°) at E=14.1 MeV. The solid line is our preliminary result. The dashed one is results of Ref. . Experimental data denoted by solid triangles are from Ref. .
From Fig. 6 one can conclude that situation on the neutron analyzing power is very tangled from both experimental and theoretical points of view. We have used AV14 NN potential and considered all values of nucleon-nucleon momenta up to 3.
The Bochum group exploited the Paris NN potential and restricted by values of the momenta up to 2. Thus further studies are required with using some other types of NN potential. Now AV18 NN potential has being widely used within different approaches for studying scattering problem in 3N and 4N systems. At present we work to adapte this potential into our computer codes. Below we present Figures for each from 18 components of the potential as functions interparticle distance.
Fig.8. Left side: spin -orbit and quadratic spin-orbit components of the potential. Right side: L components of the potential.
Fig. 9. Charge-dependent and charge-asymmetric components of the potential.
Finally we present our new calculations for phase shifts for proton-deuteron scattering at E=3 MeV , d=31.451, d=3.617,d=6.975 d=22.44 degrees. These value are very close to corresponding values of 31.414, 3.615, 7.401 and 22.799 of Los-Alamos group in Ref. [i].
Insignificant disagreement exists because the authors of Ref. have neglected by contribution of the Coulomb interactions for states with total isospin T =3/2. The calculations of pd elastic amplitudes for other values of the total angular momentum M now in progress. We plan to finish these calculations and submit our results for elastic pd observables for publication in the end of the current year.