CPC(HEP & NP), 2013, 33(X): 1—6
Chinese Physics C
Vol. 33, No. X, Xxx, 2013
arXiv:1309.7539v2 [nucl-th] 13 Dec 2013
The Astrophysical S-factor of the 12C(α,γ)16O Reaction at Solar Energies H. Sadeghi R. Ghasemi
H. Khalili
Department of Physics, Faculty of Science, Arak University, Arak 8349-8-38156, Iran
Abstract The astrophysical S-factor of the 4 He-12 C radiative capture is calculated in the potential model at the energy range 0.1-2.0 MeV. Radiative capture 12 C(α,γ)16 O is extremely relevant for the fate of massive stars and determines if the remnant of a supernova explosion becomes a black hole or a neutron star. Because this reaction occurs at low-energies the experimental measurements is very difficult and perhaps impossible. In this paper, radiative capture of the 12 C(α,γ)16 O reaction at very low-energies is taken as a case study. In comparison with other theoretical methods and available experimental data, good agreement is achieved for the astrophysical S-factor of this process. Key words Radiative capture, The astrophysical S-factor, Potential model PACS 25.55.-e,21.60.De,27.20.+n,26.20.Cd
1
Introduction
12
When the star’s hydrogen-burning phase transition, star is helium burning phase. The thermal energy at 1.5×108 K is sufficient for the fusion of two helium nuclei Thus, the two helium nuclei become unstable nuclei8 Be. If the conditions are suitable Nuclei 8 Be by the capture of the α-particle radiation convert to the Nuclei 12 C. Process forms the nucleus is called triple-α process [1, 2]. 4
He +4 He +4 He ↔8 Be +4 He →12 C + γ
(1)
Therefore, conditions at 1.5×108 K is provided for αparticle capture by Nuclei12 C and Nuclei 16 O is produced: 12 C +4 He →16 O + γ (2) These reactions are importance due to carbon and oxygen are the most abundant elements in the world after burning helium and often heavier elements are formed from these two elements. Then Our knowledge of the reaction 12 C(α,γ)16 O helpful to better understand the evolution of condense stars, such as neutron stars and black holes. For example, a large cross section for this reaction leads to the production of heavier elements. While small cross section can lead to the reverse situation and production of lighter elements. Thus, Our main purpose is to calculate cross section of this process. Received – March — 1) E-mail:
[email protected]
C(α,γ)16 O radiative capture process plays a major role in stars fuel when they collapse. There is no accurate and complete information about these reactions. Because the cross section of this reaction is low and impossible to produce in the laboratory directly at low-energies [3–7]. In past decades, the yield of capture rays has been studied for Eα up to 42 MeV [8]. The cross sections of 12 C(α,γ)16 O capture process have been obtained by fitting measured cross sections and extrapolating them to low-energies utilizing standard R-matrix, Hybrid R-matrix and K-matrix procedures. The influence of vacuum polarization effects on sub barrier fusion is also evaluated in [9], and the relevance of Coulomb dissociation of 16 O into 12 C+α is studied in [10–12]. Calculations to test the sensitivity of stellar nucleosynthesis to the level in 12 C at 7.74 MeV are described in [13]. Recently, Dubovichenko et al., have been calculated the astrophysical S-factor of the 4 He–12 C radiative capture using cluster model at the energy range 0.1-4.0 MeV. They shown that the approach used, which takes into account E2 transitions only, gives a good description of the new experimental data for adjusted parameters of potentials and leads to the value S(300)=16.0 keV.b [14, 15]. More recently, Bertulani presented a computer program aiming at the calculation of bound and continuum state observables for a
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nuclear system such as, reduced transition probabilities, phase-shifts, photo-disintegration cross sections, radiative capture cross sections, and the astrophysical S-factors [16]. The code is based on a potential model type and can be used to calculate nuclear reaction rates in numerous astrophysical reactions. In order to calculate the direct capture cross sections one needs to solve the many-body problem for the bound and continuum states of relevance for the capture process. A model based on potential can be applied to obtain single-particle energies and wavefunctions. In numerous situations this solution is good enough to obtain the cross sections results in comparison with the experimental data. The paper is organized as follows. A Brief review of Multipole matrix elements and reduced transition probabilities in sec. 2. The relevant formalism and parameters, electric and magnetic multipole matrix elements and the reduced transition probabilities are defined in this section. The findings of the model with asymptotic wave functions are corroborated in more realistic calculations using wave function generated from the Woods-Saxon potentials and experimental data, in sec. 3. Summary and conclusions follow in Section 4.
2
2
Brief review of theoretical framework
The computer code RADCAP calculates various quantities of interest radiative capture reactions. The bound state wavefunctions of final nuclei are given by ΨJ M (r) and the ground-state wavefunction is normalR 2 ized so that d3 r |ΨJ M (r)| = 1. The wavefunctions are calculated using the central(V0 (r)), spin-orbit(VS (r)) and the Coulomb potential(VC (r)) potentials. The potentials V0 (r) and VS (r) are given by
V0 (r) = V0 f0 (r), VS (r) = − VS0
r − Ri fi (r) = 1 + exp ai
−1
~ mπ c
OEλµ
=
OM 1µ
=
eλ rλ Yλµ (b r) , " # r X 3 gi (si )µ µN e M l µ + 4π i=a,b
(5)
λ λ mb a + Z e and eM = where eλ = Zb e − m a mc mc hJM |OEλµ | J0 M0 i = hJ0 M0 λµ|JM i j+Ia +J0 +λ
hJ kOEλ k J0 i = (−1)
(3)
(4)
m2 Z + mb 2 b are the effective electric and magnetic c charges, respectively. lµ and sµ are the spherical components of order µ (µ = −1, 0, 1) of the orbital and spin angular momentum (l = −ir × ∇, and s = σ/2) and gi are the gyromagnetic factors of particles a and b. The µN is also nuclear magneton. The matrix element for the transition J0 M0 −→ JM is given by [19, 20]
m2 a Za m2 c
hJ kOEλ k J0 i √ , 2J + 1
[(2J + 1) (2J0 + 1)]
where the subscript J is a reminder that the matrix element is spin dependent. For l0+l+λ = odd, the
1 d fS (r) r dr
where V0 , VS0 , R0 , a0 , RS0 , and aS0 are adjusted so that the ground state energy EB or the energy of an excited state, is reproduced. The radial Schr¨odinger equation for calculating of the bound-state are given by solving
2 l (l + 1) J d ~2 ulj (r) + [V0 (r) + VC (r) + hs.li VS0 (r)] uJlj (r) = Ei uJlj (r) − − 2mab dr2 r2 with hs.li = [j(j + 1) − l(l + 1) − s(s + 1)]/2. The electric and magnetic dipole transitions are given by introducing of the following operators [19]
2
1/2
(6) (
j
J
Ia
J0
j0
λ
)
hlj kOEλ k l0 j0 iJ ,
reduced matrix element is null and for l0+l+λ = even, is given by
Z ∞ ˆˆ 1 1 eλ l +l+j0 −j λj0 < j0 λ0|j > dr rλ uJlj (r) uJl00j0 (r) . (−1) 0 hlj kOEλ k l0 j0 iJ = √ ˆ 2 2 4π 0
(7)
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At very low energies, the transitions will be much smaller than the electric transitions. The M1 contribution has to consider in the cross sections for neutron photo-dissociation or radiative capture. The M1 transitions, in the case of sharp resonances, for the
j+Ia +J0 +1
hlj kOM 1 k l0 j0 iJ = (−1) ×
n
1 b l0
h
2e j0 b l0
J = 1+ state in 8 B at ER = 630 keV above the proton separation threshold play a role [21]. The reduced matrix elements of M1 transition, for l 6= l0 the magnetic dipole matrix element is zero and for l = l0 , is given by [23] q
3 4π
JbJb0
(
j
J
Ia
J0
j0
1
)
µN
l +1/2−j b
j √0 δj , l ±1/2 δj, l ∓1/2 (l0 δj0 , l0 +1/2 + (l0 + 1) δj0 , l0 −1/2 ) + (−1) 0 0 2 0 0 i h b l +1/2−j0 e l +1/2−j j0 √ δj , l ±1/2 δj, l ∓1/2 +gN bl12 (−1) 0 j0 δj, j0 − (−1) 0 0 2 0 0 0 )) ( R∞ Ia J j0 dr uJlj (r) uJl00j0 (r) , +ga (−1)Ia +j0 +J +1 Jb0 JbIba Iea 0 J0 Ia 1
eM
where gN =5.586(-3.826) for the proton(neutron) and µa = ga µN is the magnetic moment of the core nucleus. The reduced transition probability dB((E, B)λ)/dE of the nucleus, i into j + k, contains the information
i (8)
on the structure in the initial ground state and the interaction in the final continuum state. The reduced transition probability for a specific electromagnetic transition (E, B)λ to a final state with momentum ~k in the continuum is given by [16]
dB ((E, B)λ, Ji s → kJf s) = dE 2 µk 2Jf + 1 X X hkJf jf lf sjc ||M((E, B)λ)||Ji ji li sjc i (2π)3 ~2 2Ji + 1 j l j l j f f
3
(9)
i i c
The electric excitations (E) with multipole operator, is given by (λ) M(Eλµ) = Zeff erλ Yλµ (ˆ r)
(10)
λ λ (λ) mb c where Zeff = Zb mbm+m +Z is the efc − m +m c c b fective charge number. For proton radiative capture the effective charge
Φi (~r) = h~r|Ji ji li sjc i =
numbers for E1 and E2 have to consider both contributions in the cross sections for Coulomb breakup, photo dissociation or radiative capture. In the case of a neutron radiative capture, E1 transition dominate the low-lying electromagnetic strength and the E2 contribution can be neglected. The initial and final state are given by the following wave functions [16]
1 X s (ˆ r )φjc mc (ji mi jc mc |Ji Mi )fJjci ji li (r)Yjlii m i rmm i
(11)
c
Φf (~r) = h~r|~kJf jf lf sjc i 4π X s ˆ jlf m (ˆ r )φjc mc , (jf mf jc mc |Jf Mf )gJjcf jf lf (r)ilf Yl∗f mf (k)Y = f f kr m m f
c
where fJjci ji li (r) and gJjcf jf lf (r) are the radial wave functions and φjc mc is also the wave function of the core. The spinor spherical harmonics is denoted by
hkJf jf lf sjc ||M(Eλ)||Ji ji li sjc i =
P ls r )χsms . Yjm = ml ms (l ml s ms |j m)Ylm (ˆ The reduced matrix element in (9) can be expressed as [16]
(λ) 4πZeff e Jf jf lf J j l DJi ji li (λsjc ) (−i)lf IJifjifli f (λjc ) k
(12)
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where the angular momentum coupling coefficient J j l J j l DJifjiflif (λsjc ) and the radial integral IJifjifli f (λjc ) are J j l
DJifjifli f (λsjc ) =
J j l
IJifjifli f (λjc ) =
and the asymptotic form of the continuum state for the scattering state
(14)
i i h h → exp i(σlf + δJjcf jf lf ) × cos(δJjcf jf lf ) Flf (ηf ; kr) + sin(δJjcf jf lf ) Glf (ηf ; kr)
where CJjci ji li , W−ηi ,li +1/2 , Flf , Glf and ηf = ηi /x are the asymptotic normalization coefficient, Whittaker function, regular Coulomb wave functions, irregular Coulomb wave functions and the Sommerfeld parameter, respectively [24]. Cross-section for non-same particles without spin, is defined as follows: π~2 (2l + 1)T(E,B)l , 2µε
(16)
where T(E,B)l is the transition probability. Finally, the total cross section for a transition is arbitrary: X cap cap σ cap (ε) = (17) (σEl (ε) + σM l (ε)) l
The total cross section for an arbitrary transition is defined as: 1 σ cap (ε) = S(ε) e−2πη (18) ε
Table 1.
(13)
0
fJjci ji li (r) → CJjci ji li W−ηi ,li +1/2 (2qr),
cap σ(E,B)l (ε) =
given by
p p p p (−1)s+ji +lf +λ (−1)jc +Ji +jf +λ (li 0 λ 0|lf 0) 2ji + 1 2li + 1 2Ji + 1 2jf + 1 ( )( ) r li s ji ji jc Ji 2λ + 1 × , 4π jf λ lf Jf λ jf Z ∞ dr gJjcf∗jf lf (r)rλ fJjci ji li (r)
with the asymptotic radial wave functions for the bound state
gJjcf jf lf (r)
4
(15)
In this equationS(ε) astrophysical factor and η = pµ Zc Zα e2 are Parameters Samrfyld. We use the as~ 2ε trophysical S-factor because it is a well-define function with little changes and it is easier to analyze.
3
Result and conclusions
The potential model and the RADCAP computer code are proper theoretical frameworks to describe the ground state properties of 16 O for the reaction 12 C(α,γ)16 O. To evaluate the radiation capture reaction 12 C(α,γ)16 O, Schrodinger equation using WoodSaxon potential and with solved specific parameters and bound continuum states of the reaction is obtained with very good accuracy, and finally using the formulation of the second part of the paper, the astrophysical S-factor is calculated for transition E2 .
The set of Woods-Saxon potential parameters, applied for calculation.
V0 (MeV)
R0 (fm)
a0 (fm)
VS0 (M eV )
RS0 (fm)
aS0 (fm)
RC (fm)
-51.8
2.41
0.644
39.54
2.291
0.644
2.41
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5
Fig. 1. The astrophysical S-Factor for the 12 C(α,γ)16 O reaction. The calculated results are given by the solid line and the available experimental data [25–28] are shown as different symbols.
The set of Woods-Saxon potential parameters, applied for calculation are given in table 1. The results for the astrophysical S-Factor of 4 He–12 C radiative capture process is presented in Fig. 1, along with the experimental data [25–27], at solar energies 0.1–2 MeV. Table 2. Some evaluated astrophysical S(Ec m=300 keV)–factor value for experimental data for the 12 C(α,γ)16 O reaction in keV.b Reference
year
Sch¨ urmann et al. [29]
2012
S(E2 ) 73.4
Oulebsir et al. [30]
2012
50 ± 19
Hammer et al. [31]
2005
81 ± 22
Kunz et al. [32]
2001
85 ± 30
Redder et al. [26]
1987
80 ± 25
This work
2013
84.97
The value of this quantity is obtained at 300 keV transition energy E2 is found to be 84.97 keV.b which is reasonably agreement with some evaluated value for experimental data that shown in table 2. Here, no significant difference has been seen between the results obtained with the present model based on the potential model, and some evaluated value for experimental data in papers with potential model. In the
other theoretical approach by using the cluster model the astrophysical S-Factor of 4 He–12 C have been calculated to be S(300)=16.0 keV.b [15].
4
Summary and conclusions
Radioactive capture 12 C(α,γ)16 O is one of the most important reactions in nuclear astrophysics. The reaction amount determines the relative abundance of most elements in red giant stars, neutron stars and black holes. In general, the electric dipole radiation E1 are much stronger than quadrupole radiation from electric E2 . And electric dipole transitions between states with the same isospin forbidden in the first order. Because state 1+ and 0+ ground state Nuclei 16 O with are isospin T = 0, Thus the electric dipole radiations are not at the first order between two levels and electric dipole radiation will be the second order and Electric dipole radiation is same order with the electric quadrupole radiation. Therefore, we must consider the effects of both radiations. In comparison with other theoretical methods and available experimental data, good agreement is achieved for the astrophysical S-factor of this process.
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Acknowledgements
The authors would like to acknowledge C. A. Bertulani, for online RADCAP computer code. References 1 2 3 4 5 6 7 8 9 10 11 12 13 14
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