Three-Body force effects on breakup and formation of 6Li nuclei
محورهای موضوعی : Journal of Theoretical and Applied PhysicsHossein Sadeghi 1 , Shahla Nahidinezhad 2
1 - Department of Physics, Faculty of Sciences, Arak University, Arak, IRAN
2 - Department of Physics, Faculty of Sciences, Arak University, Arak, IRAN
کلید واژه: effective field theory, three-body force, Astrophysical reaction rate, deuteron-alpha radiative capture,
چکیده مقاله :
During helium transforms into heavier elements, both of 6Li radiative capture reaction and its breakup occur in the stars. 6Li radiative capture reaction and its inverse have been studied using Effective Field Theory (EFT), up to next to leading order (NLO). The deuteron-alpha reaction and the photodisintegration rates of the 6Li(γ,α)d reaction have been calculated. Alpha particle was assumed to be structureless and coulomb effects considered between the charged particles. The inverse reaction rate has been estimated for E1 and E2 transitions by adding the three-body forces, up to NLO. The scattering amplitude are calculated at the initial P-wave states of deuteron-alpha for the sum of both E1 and E2 multipole transitions. The obtained results are in good agreement with the available experimental data and those of other theoretical models, at the energies relevant to the Big-Bang Nucleosynthesis (BBN). The 6Li(γ,α)d reaction rate is also found to be acceptable in comparison with the other theoretical results.
Three-Body force Effects on Breakup and Formation of 6Li Nuclei
H. Sadeghi , S. Nahidinezhad∗ Department of Physics, Faculty of Sciences, Arak University, Arak 38156-8-8349, IRAN. ∗S-nahidinezhad@araku.ac.ir
Abstract
During helium transforms into heavier elements, both of 6Li radiative capture reaction and its breakup occur in the stars. 6Li radiative capture reaction and its inverse have been studied using Effective Field Theory (EFT), up to next to leading order (NLO). The deuteron-alpha reaction and the photodisintegration rates of the 6Li(γ,α)d reaction have been calculated. Alpha particle was assumed to be structureless and coulomb effects considered between the charged particles. The inverse reaction rate has been estimated for E1 and E2 transitions by adding the three-body forces, up to NLO. The scattering amplitude are calculated at the initial P-wave states of deuteron-alpha for the sum of both E1 and E2 multipole transitions. The obtained results are in good agreement with the available experimental data and those of other theoretical models, at the energies relevant to the Big-Bang Nucleosynthesis (BBN). The 6Li(γ,α)d reaction rate is also found to be acceptable in comparison with the other theoretical results.
keywords: deuteron-alpha radiative capture, effective field theory, three-body force, the astrophysical reaction rate
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I. INTRODUCTION
The alpha process is one of the two major categories of nuclear fusion processes that occur in stars formation. The radiative capture of deuteron-alpha and the 6Li(γ,α)d reactions have formerly been investigated both experimentally and theoretically, for the last few decades. The experimental studies of the deuteron-alpha radiative capture reaction have been done for the resonance 0+, T = 1, 3.56 MeV state of 6Li, at the astrophysical energies [1]. The values of S-wave cross section and the astrophysical S-factors are also reported in the ref.[2]. The cross section for the d(α,γ)6Li reaction was obtained, in the energy of Eα,lab = 2 MeV, corresponds to the 3+ resonance in 6Li at Ex = 2186 KeV [3]. Recently, a compilation of charged-particle induced thermonuclear reaction rates, are reported at temperatures in the range from 106−1010 K, for 1 6 Z 6 14 [4]. More recently, an experiment was performed at the LUNA 400 KV accelerators and the results were reported as the first measurement of the d(α,γ)6Li cross section at BBN energies by Anders et al. [5]. They reported the direct experimental cross section, astrophysical S-factor and the reaction rate data of the d(α,γ)6Li reaction for the relevant BBN energies. The asymptotic normalization coefficient of the α+d −→6 Li process and for the direct astrophysical S-factor are estimated by Tursunmakhatov et al. [7]. In the other hand, some theoretical researches have been done by utilizing the symmetry properties of the wave functions of the alpha particle and of various produced nuclei in the usual first order perturbation theory, for studding of the deuteron-alpha radiative capture [8–12]. The authors used corrections to E1 multipole in long wave approximation, which resulted in the dominance of E2 multipole. The information on the nuclear vertex constant and the asymptotic normalization coefficient have extracted on the states of 6Li nucleus by Blokhintsev et al. [13]. They used the well-known coulombmodified effective-range theory expression for synthesis of 6Li −→ 4He + d. Ryzhikh et al. studied the low-energy fusion reaction d(α,γ)6Li in the framework of the multi cluster dynamic model with Pauli projection (MDMP). A larger basis is used for the radial wave function of α −2N relative motion in the ground state of 6Li to construct a high precision wave function [14]. Additionally, Nollet et al. have computed the cross section for the d(α,γ)6Li reaction at the energies relevant to the primordial nucleosynthesis [15]. They show that the E2 cross section is dominant and is within reasonable agreement with the experimental data. More recently, Tursunov et al. studied the astrophysical capture process of d(α,γ)6Li in a three-body model and found that the contribution of the E1 transition operator from the initial iso-singlet states to the iso-triplet components of the final state is negligible [16, 17]. They found the reaction rate and primordial abundance of the d(α,γ)6Li astrophysical capture process in a three-body model. The final nucleus 6Li is considered as a three-body bound state α + n + p in the hyper spherical Lagrange-mesh method. The calculation of isospin forbidden E1 capture has been performed in the two-body model based on the cluster idea by Baye et al. [18]. The authors also performed the E2 radiative capture contribution to the α + d elastic scattering with a realistic nucleon-nucleon (NN) force. Recently, we also studied the deuteron-alpha radiative capture reaction using EFT up to NLO without coulomb effect [19] and with coulomb effect [20]. We calculated the E1 and E2 multipole transitions in framework of the pionless EFT, by only considering of dibaryon and alpha fields. We also considered the relative momentum of dibaryon and alpha less than mπ . The (3P2) n − p and
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FIG. 1: Faddeev’s equations of d−α scattering in different channels up to LO with three-body force. The filled rectangle shows the amplitude Fα of channel d+α −→ d+α. The d+α −→ N +Nα channel also shows by empty rectangle.
(2P3/2) α−N resonances considered. In this paper, we calculated the photodisintegration rates of the 6Li(γ,α)d reaction and its reaction rate with considering of the three body forces. This paper is organized as follows: a brief description of the relevant Lagrangian and Faddeev integral equations are given in the next section. We present the calculation of the deuteron-alpha radiative capture reaction, total cross section and the 6Li(γ,α)d reaction rate, by considering of three body force, in this section 2. Section 3, is devoted to the discussions of the results of the photodisintegration rates of the 6Li(γ,α)d reaction and comparison with the available experimental data and the recent results of different theoretical approaches. The summary and conclusions are also given in section 4.
II. THEORETICAL FRAMEWORK
The deuteron-alpha scattering process at low-energies leads to three possible processes with a pure hadronic signature: (a) the elastic channel (d + α −→ d + α), (b) the neutron-transfer channel (d + α −→ p + nα) and (c) the three-body final state (d + α −→ p + n + α). The two following Faddeev equations get mixed for the two-body and three-body calculation of the deuteron-alpha scattering, a) ds + Aα −→ ds + Aα, b) (NAα) + N −→ ds + Aα and c) N + N + Aα −→ ds + Aα . The following Lagrangian of d(α,γ)6Li reaction includes one-, two-, and three-body terms [22],
where mN, N, Aα and mα, are the nucleon mass, nucleon field, alpha field and alpha mass, respectively. Furthermore, ds is an auxiliary field corresponding to deuteron and the spin-singlet virtual bound state in S-wave nucleon-nucleon scattering, and (NAα) is the four component field for the 2P3/2 resonance in the Nα system. ∂i indicates the Galilean invariant derivative, where η0 = η1 = 1, for P-wave. g0 is nucleon-nucleon coupling constant for 3P2,3 P0 channels and g1 is nucleon-alpha coupling constant for 2P3/2 channel. g0(1) and ∆0(1) are given by,
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FIG. 2: Faddeev’s integral equations of d−α process with three-body force in different channels, up to NLO. The filled rectangle represents the Fα amplitude for channel d+α −→ d+α and the empty rectangle represents channel d + α −→ N + Nα.
where µ is the renormalization scale and h is coupling constant of NNα contact term. This coupling constant is derived via three-body renormalization as h = mn/Λ2 where Λ is the cut-off. The ClebschGordan coefficients are denoted by Cσi a in the above Lagrangian (Cσi a ≡ C(LSJ|σia)). The spinprojections of the fields denoted by σ,δ,... = ±1/2, and Clebsch-Gordan coefficients indices a,b,...takes the values of ±1/2,3/2. The operator indices are given by i,j,... = 0,±1. The neutron-proton (subscript 0) and nucleon-nucleon (subscript 1) propagators are given by,
The a0 and a1 are the scattering lengths for the 3P2 and 2P3/2 channels, respectively. In the proton-proton (pp) interaction, the coulomb interaction dresses the two-nucleon Greens function at the bubble diagram for the dressed dibaryon propagator. The two-nucleon propagator for the pp channel is given by,
ψ is the logarithmic derivative of the Gamma function. The relation between the coulomb scattering length aC and the renormalized constant 
in the pp channel is given by,
is Eulers constant.
The numerical calculations for the three-body system, with inserting of the three-body force are performed using the following Faddeev equation,
where the ultraviolet cut-off, Λ, is introduced for regularization, and τi(q,E) ≡ τjk(E −q2/2µi(jk)) where τjk is two body t-matrix. The kernel function Xij is defined by [22],
where, the gli and glj are the two-body form factors.
is the three-body Green function. is the free Hamiltonian eigenstate spectator projection i to j. The detailed expression for is given in ref.[22]. Photons are included in the Lagrangian (1) via minimal coupling,
The charge operator ˆ Q takes different values, depending on whether it is acting on a deuteron dibaryon field or an alpha field. The eigenvalues of the operator ˆ Q for the deuteron dibaryon and alpha fields denoted by Qd and Qα. e2 = 4παem defines the electric charge unit in terms of the fine structure
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FIG. 3: The photon folded diagrams up to NLO. The first two diagrams are the interaction of photons with alpha and dibaryon at LO and the last two diagrams are the interaction of photons with dibaryon and nucleon-alpha clusters at the NLO.
constant, αem = 1/137.036. For P-wave the dominant terms of the electric response can be computed using only the Lagrangian of (12) and the minimal coupling Eq.(11). We concentrate on the transition from the initial deuteron-alpha system with Jπ = 1+ to the final 6Li bound state with Jπ = 1+. The outgoing photon interacts with the electric charges of the alpha particle and the electric charge of the deuteron at LO. The terms including the first and second order derivative operators acting on photon field which contribute to P-wave P-wave transitions are [23],
where, Qd = 1 and Qα = 2 are electric charge numbers of the deuteron (or dibaryon) and the alpha, respectively. p~α(~ p0α) is the incoming (outgoing) momentum of the alpha in alpha-alpha-photon vertex. p~d(~ p0d) represents the incoming (outgoing) dibaryon momentum in dibaryon-dibaryon photon vertex, and ~∗ γ is polarization of the outgoing photon. LE1 and LE2 are phenomenological next-to leading order for deuteron and alpha nucleon cluster coupling constants to real photons. The P-wave operators contribute only to the channels with low-energy resonance or shallow bound state, thus j could be either 1/2 or 3/2. For the calculation of the deuteron-alpha scattering amplitude, we used the Faddeev-type integral equations in momentum space for three-body transition operators. The coupled Faddeev equations for d-α scattering in the two body and three body channels up to LO, are schematically shown in Fig. 1. The NLO contribution can then be obtained by perturbing around the LO solution with the one deuteron kinetic energy operator insertion [20] and by inserting of three body force. The coupled Faddeev equations for d-α scattering in the two body and three body channels up to NLO, are schematically shown in Fig. 2. Fig. 3 show LO and NLO order of photon interaction with the undetermined phenomenological LE1 and LE2 vertices. The transition amplitudes can be written as,
FIG. 4: The results of the astrophysical S-factors for d(α,γ)6Li reaction with three body force as a function of center of mass energy of alpha E (MeV), up to NLO in comparison with the available experimental data [5] and the other theoretical works [20, 24, 25].
where EL,S,ji,jf l is the matrix element of electric transition relating the scattering state LSJiJf to the 6Li bound state with angular momentum Jf. The cross section for d(α,γ)6Li reaction is related to the multipole transition amplitude E1 and E2 as,
where f = vrel/8παem, αem is fine structure constant, vrel is the deuteron and alpha relative velocity, and µ is the reduced mass of the deuteron-alpha system. The photodisintegration rates (in s−1 ) for the inverse reaction 6Li + γ −→ d + α is written as [4],
where T9 is a variable for the temperature in units of 109 K. Id (Ad), Iα (Aα) and I6Li (A6Li) are the spins (mass of nucleus) of deuteron, alpha and 6Li in units of } (amu). Q is the Q-value for reaction d+α −→6 Li+γ and Gi (i = d,α and 6Li) is the temperature dependent normalized partition function for nucleus i and it is defined as,
where µ is the excited state of the nucleus i and the summation on the excited states µ of the nucleus i is performed. µ i is the excitation energy of the nucleus i. For deuteron and alpha nuclei, excitation energy is zero. J0 i and Jµ i are total angular momentum of the nucleus i for bound and excitation states respectively. NA(συ)d+α−→6Li+γ is the reaction rate for deuteron-alpha radiative capture reaction [4, 17]. NA(συ)d+α−→6Li+γ is defined as,
where A = 4/3 is reduced mass number for the α + d system.
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III. RESULTS AND DISCUSSION
We used a neutron mass of mn = 939.57 MeV, proton mass of mp = 938.27 MeV, an alpha mass of mα = 3728.42 MeV and the 6Li binding energy of B3 = 28.29 MeV. The two-coupled Faddeev equations derived for the deuteron-alpha scattering amplitude with coulomb effect [19]. These Faddeev integral equations have been solved for the deuteron-alpha scattering state by folding photon diagrams, up to NLO. We also calculated the deuteron-alpha radiative capture reaction rate and the 6Li(γ,α)d photodisintegration rates, at low energies by using EFT, see [21, 26] for more details. We considered the three-body force in the present calculation. The numerical values have been used for the scattering lengths a0 = 23.71 fm, a1 = 5.423 fm and effective range r1 = 1.76 fm. The phenomenological coupling constants LE1 = LE2 = 3 have been determined by fitting to the well-known astrophysical S-factor experimental data. In Fig. 4, the results of the astrophysical S-factors for deuteron-alpha radiative capture reaction have been shown, as a function of center of mass energy of alpha E (MeV) up to NLO. Our present calculation by considering of the three-body force have been compared with the available experimental data [5] and the other theoretical works[20, 24, 25], at the 0.07 MeV 6 E 6 0.41 MeV range of energies. We show in Fig. 4, results with coulomb as well as three-body force effects. Inclusion of the threebody force moves the results of the astrophysical S-factors to the better situation respect to the other experimental and the theoretical calculations. It is found the results is nearly better in comparison with the available experimental data. Our calculated results for the astrophysical S-factors for the d-α radiative capture reaction by considering of the three-body force have been also compared with the experimental data [5] as well as the other theoretical results[20, 24, 25], at the low-energies in Table I. The results of Table I show that by including the three body force in the calculations, the results are in good agreement with the Anders et al. [5] obtained data. It is found that S(E = 0.07−0.08) = 3.15 MeV.nb in comparison with the S(E = 0.07−0.08) = 3.0 ±1.8 MeV.nb experimental data [5]. All the performed comparisons, show that the results for the d(α,γ)6Li reaction are better described in order of calculation by considering of the three-body forces. Three-body forces improves few percent agreement between the experimental data and the calculated results. We calculated the deuteron-alpha radiative capture reaction rate and the 6Li(γ,α)d photodisintegration rates at low energies [21, 26]. The 6Li(γ,α)d photodisintegration rates at low energies is dominated by the P-waves radiative captures in the two and three body channels resulting the J = 1+ ground state of 6Li. To compare of the obtained results with other experimental and theoretical results, the values of the inverse reaction rate of corresponding to several temperatures are given in Table II at 0.70 6 T9 6 2.0. Table II, compares the obtained results with the available experimental data from [6] and the two sets of the other theoretical calculations (model A and model B) [17] and theoretical calculations of [26]. It is worthy to mention that by insertion of three body force, we achieved to a better agreement between calculation and the experimental data.
IV. SUMMARY AND CONCLUSIONS
We applied an EFT theory approach with the two- and three body forces to compute the deuteronalpha radiative capture reaction, at the astrophysical energies. In this work, alpha particle was assumed to be structureless and coulomb effects considered between the charged particles. The inverse reaction
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rate has been estimated for E1 and E2 transitions by adding the three body force, up to NLO. The 6Li(γ,α)d photodisintegration rates at low energies is dominated by the P-waves radiative captures in the d + α −→ d + α and d + α −→ N + Nα channels resulting the J = 1+ ground state of 6Li. Our results are cut off independent at the order of the calculation. The 6Li(γ,α)d photodisintegration rates is found as a function of temperature corresponding with the recent data of the LUNA Collaboration. The obtained 6Li(γ,α)d reaction rate is found to be in good agreement with theoretical results. The used method provides new possibilities for investigation of few-alpha radiative capture reactions, at low energies. Our model can be also improved to achieve greater accuracy by performing higher-order computations.
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