THE FLUORINE DESTRUCTION IN STARS: FIRST EXPERIMENTAL STUDY OF THE ¹⁹F(p, α₀)¹⁶O REACTION AT ASTROPHYSICAL ENERGIES

Fluorine is a key element for astrophysics. Its abundance is very sensitive to the physical conditions within stars and for such a reason it is used to probe hotly debated nucleosynthesis scenarios (Lucatello et al. 2011). The most likely environments where its production can take place in the Milky Way are: during core collapse of Type II supernovae (Woosley & Haxton 1988) through the ν-process, in Wolf–Rayet stars (Meynet & Arnould 2000), and in the convective zone generated by a thermal pulse in asymptotic giant branch (AGB) stars (Cristallo et al. 2009). Recently, fluorine overabundances by factors of 800–8000 (Pandey et al. 2008) have been observed in R-Coronae-Borealis stars and provide evidence for the synthesis of fluorine in such hydrogen-deficient supergiants. Yet, despite its key importance, a detailed description of fluorine nucleosynthesis is still missing.

In AGB stars, regarded as the major contributors to the Galactic fluorine (Jorissen et al. 1992), the largest observed fluorine overabundances cannot be explained with standard AGB models and required additional mixing (Lugaro et al. 2004). A possible lack of proper accounting for the contribution from C-bearing molecules (i.e., CH, CN, CO, and C2) might provide an interpretation in Population I stars (Abia et al. 2010), by providing a renormalization of the observed abundances, though the understanding of F production in the case of metal-poor AGB stars is far from satisfactory (Lucatello et al. 2011). An alternative justification could be given by a reassessment of the nuclear reaction rates involved in fluorine production and destruction. For example, deep mixing phenomena in AGB stars can alter the stellar outer-layer isotopic composition due to proton capture at low temperatures (T₉ ≲ 0.04), affecting the transported material (Nollett et al. 2003; Sergi et al. 2010; Busso et al. 2010). In this environment, the ¹⁹F(p, α)¹⁶O reaction at E_cm ∼ 27–94 keV (corresponding to the Gamow window; Rolfs & Rodney 1988) would represent the main fluorine destruction channel, possibly modifying F surface abundances, as proposed by Lucatello et al. (2011) and Abia et al. (2011).

In the case of hydrogen-deficient post-AGB stars, Clayton et al. (2007) have shown that hydrogen admixture (if the CO white dwarf retained a small H-rich envelope) plays a key role to reverse the effect of excessive He burning and yields elemental abundances in better agreement with observations. Here, the ¹⁹F(p, α)¹⁶O reaction might bear a great importance as it would remove both protons and F nuclei from the nucleosynthesis scenario. Since the temperature at the base of the accreted material approaches T₉ ∼ 0.2 (Pandey et al. 2008), the ¹⁹F(p, α)¹⁶O cross section should be known at E_cm ∼ 50–300 keV for accurate modeling.

Proton-induced ¹⁹F destruction has been the subject of several experimental investigations, because of its astrophysical and spectroscopic relevance.

As pointed out in the Nuclear Astrophysics Compilation of Reaction Rates (NACRE; Angulo et al. 1999) and confirmed by Spyrou et al. (2000), the ¹⁹F(p, α₀)¹⁶O channel is giving the largest contribution to the reaction rate of the ¹⁹F(p, α)¹⁶O reaction for 0.01 < T₉ < 0.1. In the NACRE compilation, the recommended ¹⁹F(p, α₀)¹⁶O astrophysical S(E)-factor was obtained from several works (Breuer 1959; Warsh et al. 1963; Caracciolo et al. 1974; Cuzzocrea et al. 1980; Isoya et al. 1958; Morita et al. 1966), with the lowest-energy direct data reaching 461 keV center-of-mass (c.m.) energy (Breuer 1959). The Gamow window is only partially covered by the unpublished data of Lorentz-Wirzba (1978), which have been used in Herndl et al. (1991) and Yamashita & Kudo (1993) to evaluate the astrophysical factor in the zero- and finite-range Distorted Wave Born Approximation (DWBA) approaches, respectively. These data support a strong suppression of compound ²⁰Ne decay to the ground state of ¹⁶O at E_cm ∼ 0.14–0.6 MeV. However, these results were not included in the NACRE compilation as possible systematic errors affecting the absolute normalization might lead to an underestimate of S(E) by a factor of two (Angulo et al. 1999). The astrophysical factor was then extrapolated to low energies assuming a dominant contribution of the non-resonant part (Angulo et al. 1999). This conclusion disagrees with older measurements in Breuer (1959), where the existence of two resonances with J^π = 1⁻ and 0⁺ had been reported at E_cm ∼ 0.4 MeV. It is worth noting that additional resonances might be populated in ²⁰Ne as they are permitted by their quantum numbers (Tilley et al. 1998).

In conclusion, the available experimental data have allowed the computation of the rate for T₉ > 0.3. Below this temperature, the rate is determined mainly from the non-resonant (p, α₀) channel, causing a progressive increase of the uncertainties up to ∼50% at the lowest temperatures (Angulo et al. 1999).

To ascertain the actual contribution of resonances at astrophysical energies and evaluate their impact on astrophysics, an experimental program has been set forth to measure the ¹⁹F(p, α)¹⁶O astrophysical S(E)-factor by means of the Trojan Horse (TH) method.

The TH method has been developed in the early 1990s with the aim of measuring low-energy nuclear reactions unhindered by the Coulomb barrier (Baur 1986; Cherubini et al. 1996; Spitaleri et al. 1999; La Cognata et al. 2008). Since then it has been successfully applied to several reactions of astrophysical interest. Recently, a generalized R-matrix approach has been developed (Mukhamedzhanov et al. 2007; La Cognata et al. 2007, 2010a, 2010b; Mukhamedzhanov et al. 2008) to analyze multi-resonance TH reactions. This differs from standard R-matrix (Lane & Thomas 1958) as it considers the half-off-energy-shell character of the TH cross section. Let us consider the TH reaction

$$\begin{equation} a+A \rightarrow s+c+C, \end{equation} \tag{ 1 }$$

where a = (s x). This TH reaction is used to obtain the astrophysical factor for the resonant subreaction

$$\begin{equation} x+A \rightarrow c+C. \end{equation} \tag{ 2 }$$

If the cluster s escapes without interacting with the x+A system (quasi-free (QF) condition), the TH reaction amplitude is given by an expression similar to the binary resonant reaction amplitude (see Equation (10) in La Cognata et al. 2010a). In details, it contains the overlap function  I^F_A τ = 〈Φ_τ| φ_A〉 of the internal wave function of the system F = x + A = c + C excited to the level τ and the bound-state wave function of A. This can be parameterized in terms of the boundary condition in the channel x+A and of the reduced width amplitude γ_xA τ, as the R-matrix amplitude for the binary resonant reaction x + A → c + C proceeding through the resonance state F_τ (A. Mukhamedzhanov 2011, in preparation). Assuming non-interfering resonances, the TH cross section is obtained in the plane-wave (PW) approximation:

$$\begin{eqnarray} &&\frac{{d}^2\sigma }{{d}E_{xA}{d}\Omega _s}={\rm NF}\,\sum _{\tau }\left(2{J}_{\tau }+1\right)\nonumber \\ &&\quad\times \left|\sqrt{\frac{{k_f}(E_{xA})}{\mu _{cC}}}\frac{\sqrt{2P_{l_{\tau }}(k_{cC}R_{cC})}M_{\tau }(p_{xA}R_{xA})\gamma _{cC\,\tau }\gamma _{xA\,\tau }}{D_{\tau }(E_{xA})}\right|^2.\nonumber\\ \end{eqnarray} \tag{ 3 }$$

Here, NF is a normalization factor, J_τ the spin of the τth resonance, $k_f(E_{xA})=\sqrt{2\mu _{cC}(E_{xA}+Q)}/\hbar$ (Q is the Q-value of reaction 2, E_xA the x−A-relative energy), $P_{l_{\tau }}$ is the penetration factor in l_τ-wave, R_xA and R_cC are the channel radii.

$$\begin{eqnarray} &&M_{\tau }(p_{xA}R_{xA})=\left[\left(B_{xA\,\tau }-1\right)j_{l_{\tau }}({\rho })-\rho \frac{\partial j_{l_{\tau }}(\rho)}{\partial \rho }\right]_{\rho =p_{xA}\,R_{xA}},\nonumber\\ \end{eqnarray} \tag{ 4 }$$

where $j_{l_{\tau }}({\rho })$ is the spherical Bessel function, $p_{xA}=\sqrt{2\mu _{xA}(E_{xA}+B_{xs})}/\hbar$ (B_xs is the binding energy of the a = (x s) system), and B_xA τ an arbitrary boundary condition chosen as in La Cognata et al. (2010a) to yield the observable resonance parameters. Finally, D_τ(E_xA) is the standard R-matrix denominator in the case of two-level, one-channel R-matrix formulae (Lane & Thomas 1958), containing shift and penetration functions besides the boundary conditions set as above.

Two experimental runs were performed, using two different TH nuclei to transfer the participant proton. In the first run, the QF ²H(¹⁹F, α¹⁶O)n reaction at 50 MeV beam energy was measured, in the second the ¹⁹F(³He, α¹⁶O)d reaction at E_beam = 18.2 MeV (to check for pole invariance; Pizzone et al. 2011). In what follows we will focus on the measurement of the ¹⁹F(p, α₀)¹⁶O astrophysical factor and we will describe the first run only (where the spectator s is the neutron); the experimental setups were similar in the two experiments. A 1 mm collimated ¹⁹F beam impinged onto deuterated polyethylene (CD2) targets (∼100 μg cm⁻² thick). The experimental setup comprised a ΔE − E telescope, consisting of an ionization chamber and a silicon position sensitive detector (PSD1) on one side of the scattering chamber and four additional silicon PSDs (PSD2-5) on the opposite side of the beam axis. The ΔE − E telescope was optimized for the detection of ¹⁶O recoils, while the PSDs for coincident detection of the α-particles. Angular conditions were selected to optimize the QF contribution. Channel selection was accomplished by gating on the ΔE − E spectra to select the Z = 8 locus and on the Q-value of the ²H(¹⁹F, α₀¹⁶O)n reaction.

Compelling evidence for the occurrence of the QF mechanism is provided by the observed agreement between the experimental p−n momentum distribution inside the deuteron and the theoretical one given by the square of the Hulthén wave function in momentum space (in the PW approximation; Spitaleri et al. 1999; Pizzone et al. 2009). Since this is a necessary condition for the QF mechanism being present and dominant, only events satisfying the condition 0 < p_n < 40 MeV c⁻¹ for the neutron momentum range, where the agreement is observed ($\tilde{\chi }^2=1.4$), are considered in the extraction of the QF cross section d³σ/dE_cmdΩ_cmdΩ_n of the ²H(¹⁹F, α₀¹⁶O)n reaction. A minor background contribution was identified as being due to the ²H + ¹⁹F → α + ¹⁷O → α + ¹⁶O + n sequential decay. Such contribution represents ∼7% of the gated coincidence yield in the p − ¹⁹F relative-energy range E_cm = 0–1 MeV and was subtracted by fitting the $E_{n-{}^{16}{\rm O}}$ relative-energy spectrum. A similar discussion applies to the ¹⁹F(³He, α¹⁶O)d reaction. Anyway, too low statistics prevented us to extract angular distributions as well as the TH cross section from this data set.

The normalized coincidence yield is given in Figure 1. It was obtained by dividing the selected coincidence yield by the product of the phase-space factor and of the p−n momentum distribution (see La Cognata et al. 2010a, 2010b, and references therein). The experimental data clearly show the presence of three resonance groups corresponding to ²⁰Ne states at 12.957 and 13.048 MeV; 13.222, 13.224, and 13.226 MeV; and 13.529, 13.586, and 13.642 MeV. The normalized yield was fitted simultaneously with four Gaussian curves to separate the resonance contributions. In Figure 1 the multi-Gaussian fitting curve is shown by a black line, the red and green lines outline the contribution of the 12.957 and 13.048 MeV ²⁰Ne levels, respectively, the blue line describes the combined yield of the 13.222, 13.224, and 13.226 MeV states while the purple line the one of the 13.529, 13.586, and 13.642 MeV levels. A single Gaussian was used in these last two cases as the 13.222, 13.224, and 13.226 MeV levels could not be resolved in the experimental E_cm spectrum and because the 13.529, 13.586, and 13.642 MeV levels have the same spin parity (Tilley et al. 1998). Such a separation was needed to integrate the d³σ/dΩ_cmdE_cmdΩ_n cross section over the α emission angle in the c.m. system of the subreaction ¹⁹F(p, α₀)¹⁶O (La Cognata et al. 2010b). Since the experimental θ_cm range spans ∼70°–110°, angular distributions outside this angular interval were calculated by means of the general expression for resonance reactions obtained by Blatt & Biedenharn (1952; La Cognata et al. 2010b). In the integration, the J^π = 3⁻E_R = 13.226 MeV state in ²⁰Ne was assumed to dominate over the two neighbor resonances, because of the 2J_τ + 1 enhancement factor in each term of the modified R-matrix formula (Equation (3); La Cognata et al. 2010a). Under this assumption, an upper limit of the experimental d²σ/dE_cmdΩ_n cross section is obtained in the E_cm ∼ 0.3–0.5 MeV region.

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The total TH cross section d²σ/dE_cmdΩ_n is displayed in Figure 2 as full dots. Statistical uncertainties and those due to angular-distribution integration are given, as the other source of uncertainty, namely background subtraction, contributes by less than 20% to the total error budget. The horizontal error bars give the width of the p − ¹⁹F relative-energy bins used in the data analysis. The experimental TH cross section was analyzed in the modified R-matrix approach (Equation (3) and La Cognata et al. 2010a). As a first step, a weighed fit of the direct astrophysical S(E)-factor data available in the literature, down to about 0.6 MeV (Angulo et al. 1999), was performed by means of standard R-matrix formulae. This is needed to extract the reduced γ-widths of the measured 13.529, 13.586, and 13.642 MeV states in ²⁰Ne, to be inserted into the modified R-matrix fitting. The resulting p- and α-reduced widths (γ_p and $\gamma _{\alpha _0}$) are given in Table 1. The TH measurement has been extended up to 1 MeV for normalization to the direct astrophysical factor. The normalization factor was then determined by scaling the calculated d²σ/dE_cmdΩ_n to the experimental TH cross section in the overlap region between direct and indirect data, namely to the peak at about 0.75 MeV. Therefore, the reduced widths of the lower energy resonances obtained with the modified R-matrix fit are normalized to the ones of the 13.529, 13.586, and 13.642 MeV ²⁰Ne states. The normalization error accounts for reduced widths of these states different from the ones in Table 1, but still leading to an S(E)-factor in agreement with the direct one within the quoted uncertainties. The experimental energy resolution was accounted for by smearing the calculated TH cross section to match the shape of the peak at about 0.75 MeV. Such a procedure, described in La Cognata et al. (2009), yielded an energy resolution of 60 keV. The normalized γ_p and $\gamma _{\alpha _0}$ are listed in Table 1. In the calculation, the $\Gamma _{\alpha _0}$ partial widths, being essentially the total widths, were kept fixed at the values in the literature (Tilley et al. 1998), as well as the energy and spin parity of each resonance. The d²σ/dE_cmdΩ_n best-fit cross section obtained from Equation (3) is shown together with the TH data in Figure 2 (middle red line, $\tilde{\chi }^2=2.1$). The top and bottom lines mark the upper and lower limits set by the statistical and normalization errors. A good fit is obtained without including non-resonant contributions.

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Table 1. Reduced and Observable Partial Widths from R-matrix Fits

ER	Jπ	γp	$\gamma _{\alpha _0}$	Γp	$\Gamma _{\alpha _0}$
(MeV)		(MeV1/2)	(MeV1/2)	(MeV)	(MeV)
From modified R-matrix (Equation (3))
12.957	2+	0.110+0.007− 0.012	0.068	(9.6+1.2− 2.0) × 10−12	0.038
13.048	4+	0.690+0.069− 0.049	0.0446	(1.22+0.14− 0.17) × 10−11	0.010
13.222a	0+	...	...	...	...
13.224a	1−	...	...	...	...
13.226	3−	0.305+0.020− 0.026	0.086	(8.1+1.0− 1.4) × 10−8	0.053
From standard R-matrix (Lane & Thomas 1958)
13.529	2+	0.0410	0.0561
13.586	2+	0.0825	0.0904
13.642	2+b	0.0581	0.0467

Notes. Resonance energies, spin parities, and α₀ partial widths are fixed to the values in the literature (Tilley et al. 1998) in the modified R-matrix fitting. ^aThe contribution of these resonances is assumed to be negligible in the fitting. ^bThe spin-parity assignment to this resonance is ambiguous, also J^π = 0⁺ is reported (Tilley et al. 1998).

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Values of γ_p and $\gamma _{\alpha _0}$ from the fitting were then used to evaluate the resonance contribution to the on-energy-shell (OES) ¹⁹F(p, α₀)¹⁶O astrophysical factor, according to standard R-matrix formulae. This is possible as in the modified R-matrix approach the same reduced widths appear as in the OES S(E)-factor, the only difference being the absence of any Coulomb or centrifugal penetration factor in the entrance channel (see Equation (3)). The OES S(E)-factor calculated with γ_p and $\gamma _{\alpha _0}$ in Table 1 is shown in Figure 3. Since the TH cross section provided the resonance contribution only, the non-resonant part of the cross section was taken from Angulo et al. (1999). The middle red curve represents the S(E)-factor obtained using the parameters from the best fit, while the red band arises from the uncertainties in the resonance parameters of the 12.957, 13.048, 13.222, 13.224, and 13.226 MeV ²⁰Ne states, namely the errors introduced in the present calculations (statistical + normalization).

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The main result of the present work is the estimate of the contribution of the 12.957 MeV ²⁰Ne level to the total astrophysical factor, as it is responsible of a resonance at 113 keV, well inside the energy range of astrophysical interest. Moreover, a lower limit has been established for the contribution of the 13.222, 13.224, and 13.226 MeV ²⁰Ne states, to satisfy the condition set by Lorentz-Wirzba (1978); Herndl et al. (1991); Yamashita & Kudo (1993), namely the dominance of direct reaction mechanism in the 0.14–0.6 MeV energy range. These levels yield resonances at ∼0.4 MeV, thus their role is marginal below 0.3 MeV, except if the strengths of the 13.222 and 13.224 MeV resonances were very large, which seems to be excluded within the errors of the direct data (Lorentz-Wirzba 1978; Herndl et al. 1991; Yamashita & Kudo 1993).

The reaction rate R for the ¹⁹F(p, α₀)¹⁶O reaction was calculated using the astrophysical factor in Figure 3 by means of standard equations (Rolfs & Rodney 1988; Iliadis 2007). The best-fit curve (middle line in Figure 3) was used and the upper and lower limits provided the uncertainty range. The results are displayed in Figure 4: in panel (a) the reaction rate in cm³ mol⁻¹ s⁻¹ while in panel (b) its ratio to the one calculated following the NACRE prescription are shown. This was deduced assuming a non-resonant behavior of the S(E)-factor from 0.6 MeV downward, thus it will be referred to as the non-resonant reaction rate R_NR. R/R_NR represents essentially the deviation of the ¹⁹F(p, α₀)¹⁶O reaction rate obtained here from the one in the literature. For T₉ ∼ 0.1 the reaction rate R largely departs from the non-resonant one, the difference being clearly due to the presence of the 113 keV resonance. The largest difference, about 70%, occurs at temperatures relevant for post-AGB stars, exceeding the upper limit set by the uncertainties in Angulo et al. (1999). The 13.226 MeV state in ²⁰Ne gives instead a small contribution to the total reaction rate, following the conclusions drawn in Lorentz-Wirzba (1978), Herndl et al. (1991), and Yamashita & Kudo (1993).

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The energy resolution was not enough for achieving a good separation between resonances, especially at ∼400 keV, thus preventing an accurate estimate of their total widths. Thus, the interesting results already achieved call for improved investigations in the full energy region with a better energy resolution to perform more accurate spectroscopy of the involved resonances.

A quantitative estimate of the impact of this measurement has been attempted. Following the suggestion in Lucatello et al. (2011) and Abia et al. (2011), we run parametric extra-mixing calculations according to the model in Busso et al. (2010); Palmerini et al. (2011) to evaluate to what extent the fluorine destructions varies if the NACRE reaction rate is replaced by the one obtained here. We found that the fluorine surface abundance can be depleted ≲ 40% more with respect to the NACRE rate in an M = 2 M_☉ and Z = 10⁻⁴ AGB star. Since a significant fraction of fluorine upper limits for a sample of metal-poor AGB stars are located under the predicted values by a factor of the same order (Lucatello et al. 2011), this updated reaction rate can help to solve the fluorine puzzle in these stars in the framework of extra-mixing. In the M = 2 M_☉ and Z = 10⁻⁴ AGB stellar model⁹ used for preliminary calculations, the mixed material experiences temperatures up to T₉ ∼ 0.05, where the reaction rate is 27% higher than in NACRE. An even larger destruction is expected in those environments where the reaction rate enhancement approaches a factor of 1.7 (T₉ ∼ 0.1). Therefore, extensive calculations are undergoing to understand the consequences of the present results on astrophysics.

The work was supported in part by the Italian MIUR under grant No. RBFR082838.

A.M.M. acknowledges the support of the US DOE under grant No. DE-FG02-93ER40773 and DE-SC0004958 (topical collaboration TORUS), NSF Grant No. PHY-0852653.

V.B., V.K., Z.H., J.M., and Š.P. were supported by Czech MŠMT under Grant No. LC07050, Czech Academy grant No. M10480902, AMVIS Project grant No. LH1101, and GAČR. Project No. P203/10/310.