Improved study of the¹⁴O(α, p)¹⁷F reaction at stellar energies

Abstract

We have performed a study of the¹⁴O(α, p)¹⁷ F reaction at stellar energies within the framework of the Generator Coordinate Method (GCM). Our calculation improves a previous study by enlargement of the model space.

The mechanism of break-out from the hot CNO-cycle in novae and x-ray bursts, which subsequently leads to nucleosynthesis of elements up to the⁵⁶Ni region by rapid proton capture, is still an important problem in nuclear astrophysics [1]. Besides the¹⁵O (α, γ)¹⁹Ne reaction, the¹⁴O(α, p)¹⁷F reaction is discussed as a possible candidate for this leakage process [1]. As¹⁴O and¹⁷F are both short-lived nuclei, the reaction rate cannot be measured with available techniques and still has to rely on theoretical estimates.

For a wide range of temperatures (T ≤5 · 10⁸K) the¹⁴O(α, p)¹⁷F reaction rate is dominated by a resonant contribution arising from the 2 ⁺₃ state at E=30 keV and by a non-resonant direct transfer reaction contribution [2,3]. As the coupling potentials of a transfer reaction are inevitably non-local, the latter cannot be derived at in simple phenomenological potential models. However, microscopic cluster models like the GCM, in which the diagonal and coupling potentials are derived on the basis of the involved nuclear constituents and by a reasonable interaction between them, are a suited tool to estimate both contributions to the¹⁴O(α, p)¹⁷F reaction rate [3].

We have therefore performed a GCM study of the¹⁴O(α, p)¹⁷F reaction at stellar energies improving a previous calculation [3] by enlarging the model space by an excited α+¹⁴O fragmentation as the inclusion of such an inelastic configuration has been shown to be important for the description of the analogue nucleus¹⁸O [4] and is therefore apriori expected to also improve the description of excited resonances in¹⁸Ne. Thus, in the present GCM study the model space is spanned by a superposition of antisymmetrizedα + ¹⁴O(0⁺),α + ¹⁴O(2⁺) and p +¹⁷F product states. The internal degrees of freedom of theα-cluster, the¹⁴O ground state (¹⁴O(0⁺)), the first excited 2⁺ state in¹⁴O (¹⁴O(2⁺)) and the¹⁷F ground state were all described by harmonic oscillator shell model states in jj-coupling. The relative wave functions between the various fragmentations have been determined from the coupled RGM-equations following along the lines of Ref. 3. The oscillator parameters for the cluster basis wave functions as well as parameters in the nucleon-nucleon interaction were adjusted to reproduce the correct energy splitting between the proton channel and theα-channel as well as the energy positions of the two¹⁸Ne states close to the a-threshold (the 2 ⁺₃ state and the 3 ⁻₁ state at −24 keV). Our calculated¹⁸Ne level spectrum is shown in Fig. 1. The calculated transfer cross section is shown in Fig. 2 in terms of the astrophysical S-factor. Up toE < 300 keV, it is dominated by the tail of the 2 ⁺₃ resonance, exceeding the contribution arising from the high-energy wing of the 3_1/− bound state by about one order of magnitude. In the energy regimeE ≈ 0.3 2.5 MeV the cross section is given by the contribution from the J=1 partial wave which can be separated into a non-resonant direct transfer part and its interference with the 1⁻ resonance at E=3.6 MeV (not present in the previous study [3]). We find a constructive interference below the resonance energy. The 4⁺ resonance at 2.79 MeV yields an important contribution to the calculated S-factor atE =2.7–3 MeV. There are experimentally known resonances in theα + ¹⁴O channel atE > 1 MeV which are not present in our GCM study (see Fig. 1).

The¹⁴O(α, p)¹⁷F reaction rate can be derived at following the procedure discussed in Ref. 3. Comparing the present results to those of Ref. 3 we find the following conclusions which are important for the¹⁴O(α, p)¹⁷ F rate at astrophysical energies: i) The resonant contribution arising from the 2 ⁺₃ state is smaller in the present calculation by about a factor of 3.6. This can be traced back to a smaller a-width of this state, while its proton-width Γ=9 keV remains uneffected by the inclusion of anα + ¹⁴O(2⁺) configuration (experimental value Γ_p=25±10 keV). We now find Γ_α=1.3 · 10⁻⁶² MeV which is in close agreement with the value deduced from the reduced a-width of its analogue-state in¹⁸OΓ_α=1.15 · 10⁻⁶² MeV, Ref. 2). The reduction in the calculated a-width is caused by a stronger configuration mixing between the 2 ⁺₃ state with the 2 ⁺₂ state at −1.49 MeV, whose energy splitting (and positions) agrees now exactly with experiment and is reduced by about 800 keV in comparison to the study of Ref. 3. ii) Separating the calculated J=1⁻ partial cross section into resonant, direct and interference contributions we find a non-resonant direct transfer cross section very similar to the one calculated in Ref. 3. Thus, a modification of the direct transfer rate as recommended in Ref. 3 is not necessary, iii) Following the conclusions drawn in a microscopic study of the analogue nucleus¹⁸O [4], the 1⁻ state at 3.6 MeV might correspond to the experimental 1 ⁻₂ level at 1.19 MeV. If this is so, the present study predicts a constructive interference between the 1 ⁻₂ state and the direct transfer contribution in the low energy regime. However, a study of the analogue nucleus¹⁸O, which we performed similar to the present one, favours an identification of the calculated 1⁻ level with the 1 ⁻₃ state, iv) The spin assignment to the levels atE > 1 MeV still remains uncertain. Our calculation does not give evidence enough to rule out one of the spin assignments as given in the literature [2,3]. In particular, our study does not allow to undoubtedly identify the experimental state at E_x=7.06 MeV as the J^*=4⁺ member of theα + ¹⁴O molecular band [3], although this assignment still seems to be more likely than an identification of the molecular 4⁺ state with the level at 6.30 MeV [2].

Although from a theoretical point of view the present approach is superior to previous studies [3,2] and should give a reasonable description of the 2 ⁺₃ state and of the direct transfer process, its description of the excited resonances is still unsatisfactory. Clearly, more experimental information is needed here.

For the¹⁴O(α, p)¹⁷F rate as given in Ref. 3 the results of our improved calculation have the following consequences: For temperatures T₉ < 0.3, at which the rate is dominated by the contributions arising from the 2 ⁺₃ state and from the direct reaction process, it is lowered by about a factor of 3 due to the reduction of theα-width of the resonance. However, this change in the rate is rather unimportant, because the competing β⁺-decay of¹⁴O is much faster at the densities found in those astrophysical environment in which the¹⁴O(a, p)¹⁷F reaction is expected to take place. For higher temperatures, our study does not show a need to change the rate as recommended in Ref. 3. Thus, it supports the finding of Refs. 2,3 that at the high burning temperatures, as they are expected in type I x-ray bursts (1.5 · 10⁹ K, [1]), the¹⁴O(α, p)¹⁷ F reaction seems to be much faster than the competing¹⁵O (α, γ)¹⁹Ne reaction [5] and provides therefore a path from the hot CNO-cycle to the rp-process.