Bound-to-continuum potential model for the (p,γ) reactions of the CNO cycle
aa r X i v : . [ nu c l - t h ] J a n Bound-to-continuum potential model for the ( p, γ ) reactions of theCNO cycle Nguyen Le Anh , , ∗ and Bui Minh Loc , † Department of Theoretical Physics,Faculty of Physics and Engineering Physics,University of Science, Ho Chi Minh City, Vietnam. Vietnam National University, Ho Chi Minh City, Vietnam. Department of Physics, Ho Chi Minh City University of Education280 An Duong Vuong, Ward 4, District 5, Ho Chi Minh City, Vietnam Division of Nuclear Physics, Advanced Institute of Materials Science,Ton Duc Thang University, Ho Chi Minh City, Vietnam Faculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City, Vietnam (Dated: January 5, 2021) bstract The study of CNO cycle involves the examination of the proton radiative capture, or the ( p, γ )reactions below 2 MeV. The astrophysical S factor characterizing the ( p, γ ) reaction is usuallyreduced to the electric dipole transition E S factors of the ( p, γ ) reactions inthe CNO cycle were successfully reproduced. The self-consistent Hartree-Fock calculation fromthe discrete to the continuum is a promising approach for the microscopic analysis of the nucleon-induced reactions in nuclear astrophysics. I. INTRODUCTION
The compilation of charged-particle-induced thermonuclear reaction rates–Nuclear As-trophysics Compilation of REactions (NACRE) [1] was recently updated [2]. Among thesethermonuclear reactions, the proton radiative capture ( p, γ ) is important to the understand-ing of the CNO cycle dominating in stars that are 1.3 times heavier than the Sun. From thetheoretical point of view, various models have been adopted in the study of ( p, γ ) reactionsat the very low energies, such as the phenomenological R -matrix method, the “microscopiccluster models”, and the potential model (see Ref. [3] for a recent review). The so-calledpotential model is the simplest approach that concentrates on the calculation of the dipoleelectric transition E ∗ [email protected] † [email protected] (Corresponding author) p, γ ) reactions, however, have been usually approached in the direction from high positiveenergies to very low positive energies. In this work, the scattering problem was approachedby using the opposite direction, from the negative energy region to the very low positiveenergy: the self-consistent Hartree-Fock (HF) approximation for the continuum.The HF approximation gives good descriptions not only for the nuclear s.p. bound state,but also the scattering state at low energies [9–12]. The HF s.p. potential in the continuumplays the role of the real part of the optical potential. The method was recently updatedand applied for the study of nucleon-nucleus elastic scattering up to 40 MeV using theSkyrme interaction [13, 14] or the Gogny interaction [15]. Therefore, the HF calculation isappropriate for the study of ( p, γ ) reaction at the very low energies related to the nuclearastrophysics.In the present work, the self-consistent mean-field potential obtained from the HF calcu-lation with SLy4 interaction [16] was used to determine simultaneously the scattering wavefunction and the s.p. bound wave functions. The strength of the real optical potential wasfine-tuned to reproduce the low energy resonances. The nuclear data of the ( p, γ ) reactions ofthe CNO cycle including , C( p, γ ), N( p, γ ), and O( p, γ ) were successfully reproduced.The E II. METHOD OF CALCULATIONA. Potential model for ( p, γ ) reactions In the study of the A ( p, γ ) B reactions at the energies below the Coulomb barrier, it iscustomary to use the energy dependent astrophysical S ( E ) defined as S ( E ) = E exp(2 πη ) σ ( E ) , (1)where η is the Sommerfeld parameter that depends on the charges and the relative velocityof the system. In our cases, within the potential model, the E σ J ′ ( E )3rom initial (scattering) states J π to a given final (bound) state J ′ π ′ can be written as Ref. [5] σ J ′ ( E ) = 29 (2 π ) k J ′ + 1)(2 S A + 1)(2 s p + 1) 14 π (cid:18) Z A m p m B − Z p m A m B (cid:19) S F k γ × X Jjℓ j ′ + 1)(2 J + 1) × j J S A J ′ j ′ × j ′ j / − / × I ( E ) . (2) S A and s p are the spin of the target and the incident proton. The quantity in the firstparentheses is the effective charge number depending on the nucleus masses m and charges Z . j = ℓ + s p and j ′ = ℓ ′ + s p are the total angular momenta of proton where ℓ and ℓ ′ are therelative orbital angular momenta of the entrance and exit channel, respectively. k = E/ ~ c is the incident proton wave number at the bombarding energy E . The photon wave numberis k γ = E − ( E b + E x ) ~ c , (3)in which E x is the excitation energy of the daughter nucleus. E b is the proton separationenergy of the proton-core ( A + 1) system. S F is the spectroscopic factor that is finallyadjusted to reproduce the experimental data [6, 17].In this simple model, I ( E ) in Eq. (2) is the radial overlap integral of the (initial) scatteringwave function χ ℓ ( E, r ) and the (final) bound wave function φ α ( r ) ( α stands for the set n ′ ℓ ′ j ′ )expressed as I ( E ) = Z φ α ( r ) χ ℓ ( E, r ) rdr. (4)The self-consistent mean-field potential in the HF approximation can simultaneously gener-ate the bound wave function φ α ( r ) and the scattering wave function χ ℓ ( E, r ) [9, 10]. Themethod can be applied to improve the so-called potential model, and hence is named thebound-to-continuum potential model. The model for the study of astrophysical ( p, γ ) reac-tion is, therefore, based on a consistent and microscopic calculation.
B. The bound-to-continuum potential model
For the bound s.p. wave function, the HF s.p. equation with the Skyrme interaction[18, 19] is given by ~ m ∗ ( r ) (cid:20) − φ ′′ α ( r ) + ℓ ′ ( ℓ ′ + 1) r φ α ( r ) (cid:21) + V HF ( r ) φ α ( r ) − (cid:20) ~ m ∗ ( r ) (cid:21) ′ φ ′ α ( r ) = ǫ α φ α ( r ) , (5)4here ǫ α are s.p. energies of bound states and m ∗ ( r ) is the nucleon effective mass. The HFpotential V HF ( r ) contains the central (c), spin-orbit (s.o.) and Coulomb (Coul.) potential V HF ( r ) = V c ( r ) + V Coul . ( r ) + V s . o . ( r ) ℓ · s . (6)For the partial scattering wave function, the connection between the HF potential andthe real part of the nucleon-nucleus optical potential were given in Refs. [9, 10]. The partialscattering wave function χ ℓ ( E, r ) is the solution of the equation ~ m (cid:20) − χ ′′ ℓ ( E, r ) + ℓ ( ℓ + 1) r χ ℓ ( E, r ) (cid:21) + V ( E, r ) χ ℓ ( E, r ) = Eχ ℓ ( E, r ) , (7)where V ( E, r ) in Eq. (7) is the real part of the optical potential V ( E, r ) = λ c V c ( E, r ) + V Coul . ( r ) + V s . o . ( r ) l · s . (8)The optical potential V ( E, r ) is related to the HF potential V HF ( r ) as [10] V ( E, r ) = m ∗ ( r ) m " V HF ( r ) + 12 (cid:18) ~ m ∗ ( r ) (cid:19) ′′ − m ∗ ( r )2 ~ (cid:20)(cid:18) ~ m ∗ ( r ) (cid:19) ′ (cid:21) + (cid:20) − m ∗ ( r ) m (cid:21) E. (9)As the energy range of interest is at the very low energies (below 2 . S ( E ), is strongly sensitive to χ ℓ ( E, r ). Therefore,the adjustable λ c is multiplied to the strength of the central optical potential in Eq. (8) toobtain a better scattering wave function for the description of the resonance because it playsan important role in the study.For the bound s.p. wave function, we follow previous works such as Ref. [5]. The wavefunction φ α ( r ) in Eq. (4) is the solution of the Schr¨odinger equation for the bound statewith the HF potential. The energy eigenvalue is chosen as E b = Q with Q being the Q -valueof the reaction.Level schemes of N, N, O and F around the proton threshold (dash-line) are shownin Fig. 1 using the data from [20, 21]. The energy of the threshold is the Q -value at whichthe kinetic energy of the incident proton is zero. About 1 MeV above the threshold is therelevant energy region. All E ℓ − ℓ ′ = ±
1, only one or two partial5
IG. 1. Low excited states of N, N, O, and F around the proton threshold (the dashedlines). The solid arrows show the transitions E E scattering wave functions contribute to the calculation. However, the partial-wave analysisis complicated because of different J π of the entrance channel. Therefore, only dominantcontributions are considered. Table I shows the properties of scattering states and boundstates of dominant contributions.Consequently, the single HF potential V HF ( r ) and m ∗ ( r ) can determine simultaneouslythe bound and scattering wave functions. The HF potential V HF ( r ) and the effective mass m ∗ ( r ) are obtained from a Skyrme-HF program that is now widely used such as the programgiven in Ref. [22]. 6 ABLE I. The main configurations that contribute in the calculation. Scaling factor λ c andspectroscopic factor S F are obtained with the SLy4 interaction.No. Reactions J π ℓ λ c J ′ π ′ Q -value S F S F [5] S F [2]1 C( p, γ ) N 1 / + s / − C( p, γ ) N 1 − s + − d + C( p, γ ) N ∗ (2 .
313 MeV) 1 − s + − d + × − - -6 C( p, γ ) N ∗ (3 .
948 MeV) 1 − s + − d + N( p, γ ) O 1 / + s / − × − - 3 . × − / + d / − / + d / − × − - 3 . × − N( p, γ ) O ∗ (6 .
176 MeV) 1 / + s / − . × −
12 3 / + d / − . × −
13 1 / + d / − . × − O( p, γ ) F 3 / − p / + O( p, γ ) F ∗ (0 .
495 MeV) 1 / − p / + III. RESULTS AND DISCUSSIONA. C( p, γ ) N The starting point of the CNO cycle competing with the pp chain in the hydrogen combus-tion phase is C( p, γ ) N reaction. In Fig. 1, there is the resonance at E p = 0 .
42 MeV abovethe threshold corresponding to the first excited state of N at E x = 2 .
365 MeV (1 / + ). Asthe g.s. of N has J ′ π ′ = 1 / − , the possible entrance channels are J π = 1 / + , / + . Inour assumption, the incoming proton is captured into the s.p. state 1 p / . In the partialwave analysis, the corresponding scattering partial wave of J π = 1 / + is the s -wave that isthe main contribution (as given in Table I). In our calculation, the p -wave corresponding to J π = 3 / + is negligible. 7he resonance plays the important role as the calibration for the calculation. It is empha-sized that while the bound state φ α ( r ) is fixed at a given bound energy, χ ℓ ( E, r ) is energydependent. Therefore, the astrophysical factor S ( E ) is more sensitive to the scattering wavefunction than to the bound wave function. The resonance is strongly sensitive to the scalingfactor λ c . Consequently, to generate the peak of the resonance at the given energy, λ c isfine-tuned to be 1 .
15 in this case (Table I). The value of S F is in agreement with the resultin Ref. [5]. FIG. 2. The results for the reaction C( p, γ ) N. The experimental data is taken from Refs. [23–26]. B. C( p, γ ) N and C( p, γ ) N ∗ After the nucleus N produced from C( p, γ ) undergoes the beta plus decay, the nextreaction of the CNO cycle is the C( p, γ ) N reaction. It plays a key role for nuclear energyproduction in massive stars and control the buildup of N. As a consequence, the C / Cratio is reduced [27]. This abundance ratio is one of important ratios for the measurementof stellar evolution and nucleosynthesis. In the same manner as for the reaction C( p, γ ) ,there is the resonance at 0 .
51 MeV corresponding to the 1 − excited state at 8 .
062 MeV inthe N level scheme (Fig. 1).The s -wave is dominating in the partial wave analysis. The resonance is reproduced withthe scaling factor λ c being 1 .
03 and the spectroscopic factor S F being 0 .
27. However, assame as the result of most of the previous works [5], the potential-model calculation can8nly produce the position of the peak at 0 .
51 MeV (Fig. 3). The peak is lower than thevalue of the measurements of King et al. [28]. The possible calculation that can be doneto improve the result is by taking into account the narrow resonance caused by the d -wavethat usually was unnoticed in the previous works [29]. For the d -wave, a resonance appearswith λ c = 1 .
30 and S F = 0 .
27. The difference between S F of s -wave and d -wave is notsignificant. However, their value of λ c are different. The explanation is that the d -wave isaffected by the spin-orbit potential while s -wave is of course not. In our calculation, λ c isthe scaling parameter only for the central optical potential while the spin-orbit potential iskept unchanged. FIG. 3. The S factor of C( p, γ ) N reaction. The dash line is the s -wave with J π = 1 − that isthe main contribution. The dotted line is the d -wave with J π = 1 − that is the additional narrowpeak. The dash-dotted line, s -wave with J π = 0 − , is analyzed to improve the tail of the resonance.The solid line is the total calculation. The data are taken from Ref. [28]. Furthermore, a slightly better description for the tail of the resonance is reproduced bytaken into account the contribution of the s -wave of J π = 0 − that is calibrated by theresonance at 8.776 MeV (the dash-dotted lines with ℓ = 0 , J π = 0 − in Fig. 3). In the energylevel scheme of N, there is a 2 + state at 7 .
967 MeV. The data of the E + (2.31 MeV) and 1 + (3.59 MeV) excited states are also reproduced in the same calculationfor the transition to the g.s. (Fig. 4).Note that for C case, our result is different from the value of Ref. [5], but it is close tothe value of S F in Ref. [8]. The difference in the value of S F between Ref. [5] and Ref. [8]was already discussed in Ref. [8]. 9 IG. 4. The same as Fig. 3, but for the transitions to the excited states, C( p, γ ) N ∗ , at 2.31MeV (a) and 3.95 MeV (b). The data is taken from Ref. [28]. C. N( p, γ ) O and N( p, γ ) O ∗ The next reaction in the CNO cycle is the N( p, γ ) O that is the slowest reaction andthus controls the energy generation. The g.s. of O has J ′ π ′ = 1 / − . The calculation canbe calibrated by two resonances at 0 .
26 MeV and 0 .
99 MeV corresponding to the excitedstates at 7 .
56 MeV (1 / +2 ) and 8 .
28 MeV (3 / +2 ), respectively. The partial wave analysisis shown in detail in Fig. 5 for this reaction. At the first resonance, the scattering wavefunctions taken into account are s -wave and d -wave. The resonance is produced by the s -wave and d wave with λ c = 1 . , S F = 2 × − and λ c = 1 . , S F = 0 .
02, respectively;using the SLy4 interaction. The second resonance is caused by the d -wave when the twoparameters are λ c = 1 .
28 and S F = 5 × − . The s -wave ( ℓ = 0 , j = 1 / , J = 3 /
2) is alsoanalyzed. It contributes to the background and slightly improves the result. Fig. 5 showsthat two resonances are well-reproduced in comparison with the experimental data [30–32].The background lines (from 0.4 MeV to 0.9 MeV) are also well reproduced as the resultfrom the sum of tails of resonances (Fig. 5).For the E / − excited state of O at 6.176 MeV, there are threeresonances below 1.5 MeV (Fig. 1) including two resonances in the previous case and oneadditional resonance 1 / +3 at 8.74 MeV. The partial waves that are the main contributions to10 IG. 5. The partial wave analysis of the reaction N( p, γ ) O. The experimental data are takenfrom Refs. [30–32]. each resonance are given in Table I. The same scaling factors λ c for the first two resonancesare, of course, the same as that of the transition to the g.s.. For the third resonance, λ c is1 .
26, and S F is 0 . FIG. 6. The result for N( p, γ ) O ∗ (6.176 MeV) compared to the experimental data from Ref. [31]. . O ( p, γ ) F and O ( p, γ ) F* Finally, consider the O( p, γ ) F reaction where the reaction rate sensitively affects theratio O / O predicted by models of massive stars [33]. Noted that it has slowest reactionrate in the CNO cycle, as there is no resonance in the astrophysical energy (below 1 MeV)(see Fig. 1). The E F with J ′ π ′ = 5 / + and to the excited stateat 0.495 MeV J ′ π ′ = 1 / + are considered.For the transition to the g.s., as given in Table I, the proton is captured into the 1 d / s.p. state, therefore, in the partial wave analysis, the p -wave is the main contribution (seeTable I). The other possible contributions, for example, f -wave with J π = 5 / − , / − arejust less than 0 .
1% in comparison with the main contribution. The result in Fig. 7 showsthat the calculation reproduces the experimental data [34]. It is emphasized that λ c is unity.The scaling factor λ c is unity because the HF to continuum gives a good result for the protonelastic scattering from O at low energy [9].For the transition to the 1 / + excited state of F at 0.495 MeV, the incident proton isadded into the 2 s / s.p. state. It means that the excited state of F ∗ is simply assumed tobe built from one proton excited to 2 s / from 1 d / , and the difference in the energy of thetwo states is 0.495 MeV corresponding to the excited energy of F ∗ , 1 / + . The contributionscome from the p -wave with J π = 1 / − , / − . The J π = 1 / − is dominating, because it issupposed to be the tail of the narrow resonance corresponding to the 1 / − state at 3.104MeV close to the energy region of interest (Fig. 1). The scaling factor λ c is, of course, thesame as the case of the transition to the g.s.. According to the assumption for the excitedstate of F, with S F being 1 .
0, the data [6] is successfully reproduced as shown in Fig. 7.
IV. CONCLUSION
The study of ( p, γ ) reaction in the CNO cycle is approached microscopically and consis-tently by using the self-consistent mean-field method. Within the potential model, the boundand scattering wave functions can be obtained simultaneously from the single self-consistentmean-field potential. The approach can be applied to the transition not only to the g.s. butalso to the excited states, and therefore can reproduce most of available experimental datafor the astrophysical ( p, γ ) reactions in the NACRE database. Strictly speaking, the pairing12
IG. 7. The calculations for the reaction O( p, γ ) F. The experimental data are taken fromRefs. [6, 34]. correction and deformation should be taken into account in the calculation, except for thecase of O. Our results show that the HF calculation is a reasonable approach for the ( p, γ )reactions in the study.
ACKNOWLEDGEMENTS
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