CHAPTER
7
Conclusions and perspectives
The connection between phase shifts and the ANC has been explored in the framework of the effective range theory. The main result is that, in practice and under rather simple requirements, elastic phase shifts can be correctly described and connected with bound states via the effective range function, and therefore, ANCs can be accurately determined thanks to the analytic properties of this function. This result has an important impact in stellar evolution because the ANC and phase shifts are directly connected with the capture cross section which, for instance, determines partially the stage and evolution of stars.
The effective range function has been analyzed from two different points of view. First, the effective range expansion around zero energy is studied, which leads us to propose a rigorous technique to compute accurately effective range parameters for a given potential model. In general, the truncation of the effective range expansion provides a correct description of elastic phase shifts in a limited energy interval, which leads us to consider a large set of effective range parameters to extend such a description.
To compute effective range parameters accurately three aspects have to be considered with careful attention: (i) The presence of a weakly bound state, (ii) the presence of a virtual state or resonance at low energies, and (iii) the numerical precision. Items (i) and (ii) demands large channel radius and Lagrange mesh, and therefore, a high numerical precision can be required. The item (iii) restricts mainly the calculation of effective range parameters of high order. By using double numerical precision and according to all the cases analyzed, our technique can handle these three items with no problem up to fifth or sixth order in energy, and in some cases up to ninth order at least.
By implementing this technique with a higher precision, no numerical limitation is expected to compute the first ten effective range parameters, and large channel radii and meshes can be better handled.
In particular, a potential model for the12C+α system (d-wave) has shown that more
than five effective range parameters are expected to reach the energy range where ac-curate experimental phase shifts are available. This limitation leads us to explore a second point of view which consists in a detailed study of the effective range function for the charged case. The main result is the development of a method according to the
102 analytic properties of the effective range function, which allows us to parameterize phase shifts and extract ANCs. The most relevant feature of this method is the extraction and decomposition of the function Δl from the effective range function. This feature leads
us to a simpler way to obtain a phase shift parametrization without approximating the entire effective range function. The decomposition of Δl has three contributions: bound
states, resonances and background. The bound state contribution is obtained from the bound state energies, the resonant contribution can be deduced directly from experi-mental phase shifts, and the background contribution is obtained via fits. Moreover, properties of the background contribution are inherited directly from the analyticity of the effective range function. As a particular case, the proposal by Z. R. Iwinski, et al., [13] of using Pad´e approximants provides a simple way for fitting the background contribution.
The method can be applied to estimate ANCs due to the analytic continuation of the effective range function. For different potential models the method has been applied as a manner of test to recover the ANC value. The results show and confirm the following facts: (i) the accuracy is improved by including phase shifts at low energies, (ii) the precision increases when phase shifts uncertainties are reduced, (iii) if the phase shifts show a slow variation at low energies the ANC is accurately determined for typical inputs at few MeVs [e.g. 12C + α (l = 2) with inputs in 1-6 MeV]; if the phase shifts show a
fast variation at low energies the ANC is well determined only when the input includes phase shifts at lower energies [e.g. 7Be + p (l = 1) with inputs below 1 MeV].
Experimental phase shifts for the 12C + α system are available below 5 MeV in Refs. [33, 1]. For the d-wave, these phase shifts are used as input for the method. The analysis provides respectively the locations (2.683 and 4.353 MeV) and the widths (0.76 and 75 keV) of both resonances, which are in good agreement with the accepted values. On the other hand, from the analytic continuation of the effective range function, an overestimation of phase shifts at low energies is obtained. This unexpected behavior is introduced by the unbalanced weights of the experimental phase shifts above and below 4.1 MeV. By restricting the input energy up to 4.1 MeV, the effect of the over-weights disappears and the effective range function can be continued following physical behaviors in agreement with the experimental data and uncertainties. In these conditions the ANC for the weakly bound state at 244.85 keV below threshold is estimated as 112(8)× 103
fm−1/2.
Effects of extra bound states on the ANC estimation and the phase shift description are also studied for the12C+α system (l = 2). The results show that the ANC estimation
can be strongly affected by the presence of one or more forbidden or unobserved states in general. To check these effects a rather simple way is via potential models because the ANC and the phase shifts can be predetermined by solving the Schr¨odinger equation. By allowing one extra state at energy Efs, strong effects on the ANC are found when this
state is close to the physical one. Conversely, this estimation is less affected if a deeply bound forbidden state is allowed and recovers progressively the correct value when Efs
is more and more negative. If a second forbidden state is included the effects on the ANC estimation are amplified, and no reason indicates that such an amplification stops by including more forbidden states.
The phase shift description is also affected by the presence of extra bound states. By analyzing the experimental phase shifts of the12C + α system (l = 2), it is observed
103 the decreasing ones if one forbidden state is allowed and located above−50 MeV. Better phase shifts description around 3.5 MeV, in the sense of less dispersion, are obtained when this forbidden state is deeply bound (few hundreds of MeV below zero). No improvement is observed when a second forbidden state is included.
The available potential model for 12C + α system (l = 2) used along the thesis,
has been slightly modified (keeping the physical bound state location unaffected) to explore if the experimental phase shifts can be better described. The first modification shows that small changes of the Coulomb potential demand a very small recalibration of the nuclear potential depth (up to 0.6%). Under these conditions the phase shift behavior and the ANC are almost unaffected. On the other hand, changes in the nuclear potential range imply a strong recalibration of the nuclear potential depth (up to 20%). In these cases the phase shifts of nuclear potentials deeper than the original one are closer to the experimental data, but a good enough description (inside of error bars) is not found. From this analysis, we see that the small improvement in the phase shifts description leads to ANC values in (99-134)×103 fm−1/2, but it cannot be assumed as a
reliable deduction because the improvement is not good enough to describe correctly the experimental phase shifts. Thus, this potential model is still in a preliminary phase for explaining the experimental phase shift behavior, and therefore, for estimating a reliable ANC. For future studies, this potential model can be the starting point to improve the phase shift description via different potential shapes.
Analyses of other physical systems or other partial waves are planned. The most relevant at this point is perhaps the partial wave l = 1 of the12C + α system because, together with the partial wave l = 2, they can constrain the widths in R-matrix analyses, and therefore, higher accuracy and precision can be achieved to compute the total cross section at energies of astrophysical interest. Another reaction of interest is 7Be(p, γ)8B which is important in solar models. According to the analysis of potential models, if accurate experimental phases shifts at low energies (below 1 MeV in the center-of-mass frame) could be determined, the method can be applied to estimate the ANC with an acceptable accuracy and precision. However, the Coulomb barrier for this system (∼1.4-1.6 MeV) can make this goal very difficult to achieve.
It is expected that more information about the Δl function can be obtained or
inferred by comparing results coming from a phenomenological R-matrix analysis. The close similitude between the Δl-decomposition and the levels of the phenomenological
R-matrix makes this a natural way to proceed. On the other hand, comparisons with R-matrix and K-matrix theories can provide useful information to generalize the Δl
