NEW RESULTS FOR THE ASTROPHYSICAL S-FACTOR AND REACTION RATE OF RADIATIVE 3 Не 4 Не CAPTURE

Transcription

1 NEW RESULTS FOR THE ASTROPHYSICAL S-FACTOR AND REACTION RATE OF RADIATIVE 3 Не 4 Не CAPTURE S. B. Dubovichenko, A. V. Dzhazairov-Kakhramanov Fesenkov Astrophysical Institute NCSRT NSA RK, , Almaty, Kazakhstan N.A. Burkova * Al-Farabi Kazakh National University, , Almaty, Kazakhstan 1 natali.burkova@gmail.com In the frame of modified potential cluster model based on the classification of orbital states by Young diagrams and revised interaction potential parameters for the bound states of 7 Be in 3 He 4 He cluster model with forbidden states the astrophysical S-factor for the radiative capture reaction 3 He(, ) 7 Be has been calculated from the 10 kev. Obtained results S(23 kev) = kev b reproduce the latest experimental data at 23 kev. Calculated and parametrized reaction rate is compared to some results known in the range of temperatures from 0.05 to 5 T 9. Keywords: Astrophysical energies; radiative capture; total cross section; reaction rate; potential cluster model; forbidden states; Young diagrams. PACS Nos.: Gx, x, Lx, c, 26 Introduction The radiative 3 He 4 He capture at ultralow energies is of apparent interest for nuclear astrophysics as a part of proton-proton fusion cycle. The pp-cycle may be closed by the 3 He + 3 He 4 He + 2p process [1], or by 3 He + 4 He 7 Be + reaction promoted by 4 He accumulated on the pre-stellar stage (see, for example, [2]). At a time, the role of radiative 3 He 4 He capture in pre-stellar nucleosynthesis, when after the Big Bang the temperature lowered to 0.3 Т 9 (Т 9 = 10 9 К) is now under discussions [3]. The 7 Be production in the process 3 He(, ) 7 Be is used directly for estimating the formation of 7 Li as a result of -decay. Data on this process are used for the calculation of lithium isotopes 6 Li/ 7 Li ratio produced in the Big Bang. The recent data on the 6 Li/ 7 Li isotopic ratio obtained within the framework of the LUNA collaboration and a detailed discussion of the astrophysical aspects of this problem are reported in [4]. The experimental status of 3 He(, ) 7 Be reaction is critically reviewed, and the theoretical available descriptions on the production and destruction of 7 Be at astrophysical relevant energies are discussed in [5]. Special interest to the 3 He(, ) 7 Be reaction concerns the new measurements for the astrophysical S-factor performed for lowest today 23 kev energy [6]. That is why we are returning to its treatment as this is a possibility to check the predictive abilities of our approach [7] and proceed with the correspondent calculations for the reaction rate. Basing on the two-body potential cluster model (PCM) we succeed in description of the total cross sections and astrophysical S-factors for the radiative capture of more than 30 reactions [8]. The calculations of these reactions are performed on the basis of a modified version of the 1 Corresponding author

2 PCM with forbidden states (FS) [9] and the classification of orbital states according to the Young diagrams (MPCM). A completely certain success of the MPCM in describing the total cross sections of processes of this type can be explained by the fact that the potentials of cluster-cluster interaction in the continuous spectrum are built not only on the basis of known elastic scattering phase shifts or the structure of the spectra of the resonant levels of the final nucleus, but for a discrete spectrum on the basis of a description of the main characteristics of the bound states (BS). These potentials are also based on the classification of orbital states according to Young diagrams [8], which allows to determine the presence and number of FSs in each partial wave, and hence the number of inner nodes of the relative motion radial wave function (WF) [10]. As a result, each partial potential depends not only on the standard quantum numbers set JLS, but also on the Young diagrams {f} [11]. Potential parameters As was shown, for example, in [12] the orbital states in the 3 He 4 He cluster system of 7 Be are pure according to Young diagrams Therefore, the nuclear partial potentials of 3 He 4 He interactions is of the form V JLS (r) = V 0 (JLS)exp(- r 2 ) + V coul (r) with parameters obtained on the basis of elastic scattering phase shifts and depending on the quantum numbers JLS can be directly used to consider the characteristics of the BS of 7 Be [8, 9,13]. As the Coulomb potential V coul (r) the usual spherical shape was used [14]. Earlier, in Refs. [7] we refined the main calculated characteristics of the bound states of 7 Be nucleus in the 3 He 4 He channel. For this purpose, the potential parameters of the bound P - states, given in Table 1, were improved, and now the calculated energy levels completely coincide with the experimental values [15]. The potential describes well the elastic scattering S-phase shift [8] from [16], since the transitions from the S-waves to the GS and FES of 7 Be make the predominant contribution to the astrophysical S-factor. Such potentials in the S- wave have two forbidden BSs that corresponds to the forbidden Young's diagrams {7} and {52}. In the P-wave, diagram {61} is forbidden along with the bound allowed state (AS) with the Young diagrams {43}. In the D-wave there is a FS with the diagram {52} [11,12,17], and there is no AS. Table 1. Potential parameters for GS and FES. = fm -2 and R с = fm. Number of FSs is pointed. Energy spectra, charge radius of 7 Ве, and AC in 3 He 4 He channel. L J V 0, MeV Е, MeV FS r rms, fm С W 2 S 1/ P 3/ (2) 2 P 1/ (2) 2 D 5/ D 3/ The energies of the bound levels of the considered nuclei in the given potentials were calculated by the finite-difference method [18] with an accuracy not worse than 10-6 MeV. The widths of the potentials in Table 1 have been defined basing on the description of charge radii and asymptotic constants [7]. Table 1 also shows the results of calculating charge radii. To find the nuclear charge radius, we used the cluster radii given in [19]. 2

3 To control the stability of the asymptotics of the wave function for the ground and first excited bound states at large distances, we used the dimensionless asymptotic constant (AC) C w of the form [20] L (R) = 2k0 C w W - L+1/2 (2k 0 R), where ( R ) is the numerical wave function of the bound state obtained from the solution of L the radial Schrödinger equation and normalized to unity, W L 1/2 is the Whittaker function, which determines the asymptotic behavior of the WF and is the solution of the same equation without nuclear potential, i. e. at large distances R, k 0 is the wave number corresponding to the channel binding energy, is the Coulomb parameter, L is the orbital angular momentum. As a result, the AC value of 5.03 (2) was obtained for the GS, and 4.64 (2) was found for the FES. The above error is determined by averaging the resultant calculation of the constants over the interval 6 16 Fm. Compilation is given in Table 1. Based on an analysis of various experimental data for the ground state of 7 Be in 3 He 4 He channel the AC of 4.78(7) Fm -1/2 was suggested in [21]. We reduced it to the dimensionless value 5.66(14) at 2k , that is somewhat larger than our calculated one. For the first excited state 4.24(6) Fm -1/2 is given, at 2k we obtained C w = 5.38(8), also larger. As it will be seen later, the larger values for the ACs of [21] yield in higher astrophysical value S(0), comparing the data [6] those we oriented on in present report. Capture S-factor and reaction rate Further, following the publication of new experimental data, we again consider the astrophysical S-factor for the 3 Не 4 Не radiation capture at the lowest possible energies. As before, we use the potential cluster model [8,9] with FSs and GS potentials for 7 Be refined here (see Table 1) [7]. In calculations only E1 transitions are taken into account, since the contributions of the E2 and M1 transitions are 2 3 orders of magnitude less. In treating system, the E1 transition is possible between ground Р 3/2 -state of 7 Be and S 1/2, D 3/2, D 5/2 - scattering states, and also between the first excited bound Р 1/2 -state and S 1/2, D 3/2 - scattering states. The results of calculating the astrophysical S-factor of radiative 3 Не 4 Не capture at the energies from the 10 kev are shown in Fig. 1 by a solid line. The dotted line shows the S- factor for capture on the GS, dot-dashed on FES. The experimental data are taken from [6, 22,23,24,25,26,27]. For comparison the ab initio [28] and the R-matrix calculations [29] are given by lines 1 and 2, respectively. As can be seen in Fig. 1, the results of our calculations at 23 kev lie in the region of experimental errors [6]. For the energy of 20 kev, our calculation yields the S-factor of kev b, and at 23 kev it is kev b. Earlier in our works [7,8] S(20 kev) = kev b was obtained, i.e. approximately 4% higher comparing present results. The reason is that for the GS and FES the same potentials have been used earlier. At that time, this was of no principle importance, since the errors of the S-factor in the previously measured energy region from 90 kev and above were 10 20%, and data at lower energies were absent. 3

4 0,65 0,60 0,55 0,50 1/2 - +3/ He( 3 He, ) 7 Be 0,55 0,50 4 He( 3 He, ) 7 Be S, kev b 0,45 0,40 0,35 3/2 - S 429 / S 0 0,45 0,40 0,30 0,25 0,35 0,20 1/2-0,15 0,30 0,10 0, E cm, MeV 0, E cm, MeV Fig. 1 Astrophysical S-factor (left panel). Experiment: solid points [22]; solid squares [23]; open points [24]; open squares [25]; triangles [26]; open diamonds [27]; crosses [6]. Theory: line 1 [28]; line 2 [29]; for details of present calculations with parameters of Tab. 1 see text. Ratio S 429 /S 0 (right panel): points from [27]; solid line present calculations. 6 The most recent measurements of the S-factor at 23 5 kev [6] lead to the value of (54) kev b, which agrees well with our results. A combined average of S(0) = kev b is found in [6]. For comparison, we present the results of extrapolating the experimental data to zero energy in units kev b: 0.54 (9) [33], (12) [30], (18) [22], (17) [23], (17) [24], and (18) [31]. Not so long ago, in [32], basing on the analysis of various experimental data the following recommended values have been obtained S(0) = kev b and S(23) = kev b. The latest publication [21] reported their value S(0) = 0.596(17) kev b, that is slightly higher comparing [6], and we see the reason in a little overvaluation of the ANCs. Note that our calculations of the S-factor were performed in [7] in Only a small refinement was made here due to the use of the corrected FES potential. New experimental data [6] were published in In other words, the theoretical results of [7] predicted the behavior of the S-factor at the lowest energies up to 23 kev within 4% accuracy. The reaction rate calculated in the T 9 range from 0.05 to 5 T 9 according the traditional definition [33] 4 1/ 2 3/ N A v T ( E) E exp( E / T ) de, where Е is taken in MeV, cross section (E) in b, reduced mass in a.u.m., temperature Т 9 in 10 9 К. The dot line in Fig. 2 (left panel) shows the reaction rate for the capture on the FES, the dashed line on the GS, and the solid line is the total capture rate, which corresponds to the solid line in Fig 1. For comparison, the rates from [6], [27], and [29] are given. Note, we chose the latest data, and provided by the analytical parametrizations for the v only. 4

5 He( 3 H e, ) 7 Be 1,9 4 He( 3 H e, ) 7 Be , , , ,5 N A { v}, cm 3 mol -1 s P res ent Takacs 2015 K on tos Descouvem ont 2004 N A { v}/n A { v} present 1,4 1,3 1,2 1,1 1, ,9 0,8 Ta kac s Kontos 2013 Deacouv emont , T , T Fig. 2 Left panel: Reaction rate in the T 9 range from 0.05 to 5 T 9. Right panel: thermonuclear reaction rate, relative to the present rate, by Kontos et al. (green line, [27]), by Takacs et al. (red line, [6]), and by Descouvemont et al (magenta line, [29]). To see the real range of deviations in the reaction rates the corresponding ratios are given in Fig. 2. It is well seen, that our results are very close to those by Takacs et al. (red line, [6]), except ultra-low energies, where our results are higher. Practically same ratio is observed for the data by Kontos et al. (green line, [27]) up to ~1 T 9, but the higher T 9 the more definite difference is illuminated. The most difference appears in comparison with calculations presented by Descouvemont et al (magenta line, [29]). For the total reaction rate in Fig. 2 the following analytical parametrization was obtained N v / T exp( / T ) ( T A 2/3 1/3 1/ T T T T ) / T exp( / T ), 2/3 4/3 5/3 3/ which is of 2 = 0.91 accuracy within 1% theoretical errors. Conclusion The slightly refined variants of calculations for the astrophysical S-factor of the 3 He(, ) 7 Be reaction, are in better agreement with the available earlier and latest experimental data. New 6 lowest measured point S(23 5 kev)= , 054 kev b (Takacs et al., 2015 [6]) is just on the early calculated theoretical line for S(E) [7]. So, the predicative reliability of the developing cluster model approach was demonstrated. The new parametrization for the reaction rate is obtained and may be recommended for the astrophysical evaluations of 7 Be production. This work was supported by the Ministry of Education and Science of the Republic of Kazakhstan (Grant No. 0070/GF4) titled «Thermonuclear reactions in stars and controlled thermonuclear fusion» through the Fesenkov Astrophysical Institute of the National Center 5

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