ISSN 1062-8738, Bulletin of the Russian Academy of Sciences: Physics, 2009, Vol. 73, No. 2, pp. 165–170. © Allerton Press, Inc., 2009. Original Russian Text © S.V. Artemov, S.B. Igamov, K.I. Tursunmakhatov, R. Yarmukhamedov, 2009, published in Izvestiya Rossiiskoi Akademii Nauk. Seriya Fizicheskaya, 2009, Vol. 73, No. 2, pp. 176–181.

Determination of Nuclear Vertex Constants (Asymptotic Normalization Coefficients) for the Virtual Decays 3He Æ d + p and 17F Æ 16O + p and Their Use for Extrapolating Astrophysical S-Factors of the Radiative Proton Capture by the Deuteron and the 16O Nucleus at Very Low Energies S. V. Artemov, S. B. Igamov, K. I. Tursunmakhatov, and R. Yarmukhamedov Institute of Nuclear Physics, Academy of Sciences of the Republic of Uzbekistan, 100214 Tashkent, Uzbekistan e-mail: [email protected] Abstract—Experimental astrophysical S-factors for the direct radiative capture processes d(p, γ)3He and 16O(p, γ)17F are analyzed using the modified two-body potential approach. The values of the asymptotic normalization coefficients (nuclear vertex constants) are found for the wave functions of the 3He and 17F nuclei in the (d + p) and (16O + p) channels respectively and are then used to extrapolate the astrophysical S-factor to the region of very low energies. DOI: 10.3103/S1062873809020075

1. The results of analyzing astrophysical S-factors for the direct radiative capture reactions d(p, γ)3He and 16O(p, γ)17F at very low energies within the modified two-body potential approach [1] are reported in this paper. This approach allows the nuclear vertex constants (NVCs) (asymptotic normalization coefficients (ANCs)) for the virtual decays 3He → d + p and 17F → 16O + p to be correctly determined from the analysis of the experimental astrophysical S-factors for the above reactions and then to be used for extrapolating the astrophysical S-factors to the region of experimentally inaccessible energies E ≈ 0. 2. Here we give the main formulas for the calculation of the astrophysical S-factor SpA(E) for the direct radiative capture reaction A(p, γ)B. We introduce the following notations: lf ( jf ) is the orbital (total) angular momentum of the captured proton in the nucleus B; li ( ji) is its relative orbital (total) momentum in the initial state; ηf and ηi are the Coulomb parameters of the bound B(=A + p) state and pA scattering respectively; and µ is the reduced mass of the proton and nucleus A. According to [1], the astrophysical S-factor at fixed lf and jf has the form ( sp )

S p A ( E ) = C p A; l f j f
l f j f ( E, C p A ; l f j f ) . 2

Z p A; l f , j f for the (A + p) configuration by the equations G p A; l f j f

= –i

1/2 ( sp ) Z p A; l f j f C p A ; l f j f

lf + ηf

πC p A; l f j f / µ

and

C p A; l f j f = [2] respectively, where S˜ l f j f (E) =

∑

S˜ l f j f λ (E) is the single-particle astrophysical S-factor, λ is the multipolarity of the electromagnetic transisp tion, and C p A; l f j f is the single-particle ANC which defines the tail amplitude of the single-particle wave function for the bound state B = (A + p). According to [1, 3], the peripheral character of the direct radiative capture reaction is formulated by the conditions λ


l f j f ( E, C (pspA;) l f j f ) = f ( E ) ,

(2)

Sl f j f ( E ) 2 - = const C p A; l f j f = ---------------------------------------
l f j f ( E, C (pspA ;) l f j f )

(3)

for all values of E in the range Emin ≤ E ≤ Emax and values ( sp )

of the function
l f j f (E, C p A; l f j f ) from (2). Then, fulfillment of conditions (2) and (3) allows 2

(1)

one to obtain the value ( C p A; l f j f ) using the experimen-

( sp ) ( sp ) Here
l f j f (E, C p A ; l f j f ) = S˜ l f j f (E)/(C p A ; l f j f )2, C p A ; l f j f is the ANC which defines the amplitude of the overlap function tail of the wave function for nucleus B in the (A + p) configuration and is related to the NVC G p A; l f j f for the virtual decay B → A + p and the spectroscopic factor

tal values of the astrophysical S-factor S p A (E) instead of SpA(E)

expt

165

expt

Sl f j f ( E ) -. = ---------------------------------------
l f j f ( E, C (pspA ;) l f j f ) expt

2 expt ( C p A; l f j f )

(4)

166

ARTEMOV et al. –2 –1 0 1/2 (C(sp) pd; 0 1/2, E) × 10 , keVb fm

9

of the geometrical parameters r0 (radius) and a (diffusion) of the Woods–Saxon potential were varied within a wide range (r0 = 1.125–1.375 fm, a = 0.585–0.715 fm,

E = 18.9 keV

( sp )

7 4.7 keV

5 3 1

1.75

1.85

1.95 2.05 –1/2 C(sp) pd; 0 1/2, fm ( sp )

Fig. 1. Dependence of the function
01/2(E, C pd; 01/2 ) on ( sp )

2 Zpd; 0 1/2 ëpd; , 0 1/2 fm

–1

C pd; 01/2 for the d(p, γ)3He reaction at the energies E = 4.7 and 18.9 keV.

6

6

(a)

5 4 3~ ~ 1.5

1.5

1.0

1.0

0.5

(b)

5 4 3~ ~

1.8

1.9

2.0

0.5

1.8 1.9 2.0 –1/2 C(sp) pd; 0 1/2, fm

( sp )

and C pd; 01/2 ≡ C pd; 01/2 (r0, a) = 1.757–2.011 fm–1/2). For each pair of r0 and a, the depth of the potential was varied so as to reproduce the experimental binding energies of the bound (d + p) state and the low-energy phases of the pd scattering. The Woods–Saxon potential was used to calculate both the wave function for the bound (d + p) state and the wave function for the pd scattering. Typical dependences of the function sp ) 2
01/2(E, C (pd; 01/2 ) and the ANC (Cpd; 01/2) on the param( sp )

eter C pd; 01/2 obtained from (2) and (3) are shown in Figs. 1 and 2(upper line) at two values of the energy E. Conditions (2) and (3) are fulfilled with a small error (~1–2%). Thus, both tests point to the pure peripheral character of the reaction d(p, γ)3He in the energy interval 2.5 ≤ E ≤ 22 keV. It is noteworthy that the spectroscopic factor Zpd; 01/2 strongly depends on the model-dependent single-parti( sp ) cle ANC C pd; 01/2 (Fig. 2, lower line). To check self-consistency of the description of the initial and final states of the reaction in question with the potential used, we also calculated low-energy phases of the pd scattering by varying the parameters r0 and a in the above ranges. The calculated scattering phases at E ≤ 3 MeV agree with the experimental phases [6] within ~10%. These facts allow “experimentally measured” val( sp )

2 C pd; 01/2

Fig. 2. Dependence of the ANC (upper line) and the spectroscopic factor Zpd; 01/2 (lower line) on the single( sp )

particle ANC C pd; 01/2 for the d(p, γ)3He reaction at the energies E = 4.7 (a) and 18.9 keV (b).

2

ues of the ANC ( ( C pd; 01/2 ) ) to be obtained from (4) using separately the experimental astrophysical S-facexpt tors ( S p A (E)) from [4] and [5] for each experimental ( sp )

3. Let us first consider the reaction d(p, γ)3He. The expt experimental astrophysical S-factors S pd (E) for this reaction have been measured in some works (see [4] and references therein). To determine the ANC for the 3He nucleus in the (d + p) channel, we analyzed the expt experimental data from [5] and [4], where S pd (E) values were measured with an error of ≤10% in the energy intervals 15 ≤ E ≤ 75 and 2.5 ≤ E ≤ 22 keV respectively.

point and the values of the function
01/2(E, C pd; 01/2 ) corresponding to these points and to the standard values of the parameters r0 and a. The results are shown in Figs. 3a and 3b respectively (filled circles). The uncertainty at each point includes the error of the experimental astrophysical S-factor and the error of the function
related to the variation of the parameters r0 and a. The straight lines and their strips correspond to the weighted mean values of the ANCs squared and their errors respectively, which are presented in column 2 of the table. As is evident from the table, the weighted mean ANC values obtained separately from the data in Figs. 3a and 3b are noticeably different, which is due to the difference in absolute values of the experimental astrophysical S-factors measured in [4] and [5]. Figure 3c and the table present our recommended weighted mean ANC (NVC) value obtained from all data in Fig. 3,

The analysis was made on the basis of relations (1) and (4) by checking conditions (2) and (3) as the values

which is ( C pd; 01/2 ) = 4.28 ± 0.50 fm–1 (|Gpd; 01/2|2 =1.34 ± 0.15 fm). As is seen, this ANC (NVC) value agrees with

2

The value ( C p A; l f j f ) can be used to extrapolate the astrophysical S-factor SpA(E) to the region of experimentally inaccessible energies 0 ≤ E < Emin. expt

expt

2

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DETERMINATION OF NUCLEAR VERTEX CONSTANTS 2 –1 Cpd; 0 1/2, fm 8

Spd(E), eV b 0.8

(a)

(a) 6

0.6

4

0.4

2

0.2

0 8

10

20

30

40

0 0.8

50

10

20

6

0.6

4

0.4

2

0.2

5

10

15

20

0 0.8

25

5

10

6

0.6

4

0.4

2

0.2

10

20

40

50

30

15

20

25

(c)

(c)

0

30 (b)

(b)

0 8

167

0

40 50 E, keV

Fig. 3. Values of the ANC for the virtual decay 3He → d + p at all experimental points. Filled squares and circles are our results obtained from the analysis of the experimental astrophysical S-factors given in [5] (a) and [4] (b) and from their joint analysis (c). Straight lines and their widths are the weighted mean ANC values and their uncertainties.

the above weighted mean ANC (NVC) values extracted separately from the data of Figs. 3a and 3b within 0.2σ and 1.5σ. Our recommended value of the ANC (NVC) squared is in quite good agreement with the phenomenological values 4.2 fm–1 (1.3 fm) [7], 4.22 fm–1 (1.32 fm) [8], 4.12 ± 0.29 fm–1 (1.31 ± 0.09 fm) [9], and 4.35 ± 0.10 fm–1 (1.36 ± 0.03 fm) [10, 11]. This agreement of the ANC (NVC) values resulting from the analysis of different experimental data obtained by different methods may indicate the reliability of our recommended ANC

10

20

30

40 50 E, keV

Fig. 4. Astrophysical S-factors for the d(p, γ)3He reaction. Filled squares and circles in (a), (b), and (c) are the experimental data from [5] and [4] respectively; open circles are the results obtained in this work. Solid lines are our calculations with the standard values of the parameters r0 and a. Strip widths indicate the root-mean-square uncertainty related to the uncertainties of the function
and the ANC.

(NVC) value for the 3He nucleus in the (d + p) channel (for the virtual decay 3He → d + p). Thus, it is demonstrated that the experimental astroexpt physical S-factors S pd (E) [4, 5] can be used to determine experimentally measured values of the NVC (or ANC) for the virtual decay 3He → d + p with an error practically no larger than the experimental error. Expression (1) and our obtained values of the ANC 2

( C pd; 01/2 ) were used to extrapolate the astrophysical S-factor Spd(E) to the energy region E ≈ 0. The checking of the fulfillment of condition (2) yielded the results expt

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Weighted mean values of the ANC ( C pd; 01/2 )2 for the 3He nucleus in the (d + p) configuration and NVC G pd; 01/2 decay 3He → d + p and the values of the astrophysical S-factor Spd(E) at the energies E = 0 and 0.1 keV 2

expt

for the virtual

Ref.

( C pd; 01/2 )2, fm–1

G pd; 01/2 , fm

Spd(0), eV b

Spd(0.1 keV), eV b

[5] [4] [4, 5]

3.53 ± 0.28 4.38 ± 0.43 4.28 ± 0.50

1.10 ± 0.09 1.37 ± 0.13 1.34 ± 0.15

0.133 ± 0.011 0.165 ± 0.016 0.162 ± 0.019

0.134 ± 0.011 0.167 ± 0.016 0.163 ± 0.019

similar to those in Fig. 1. The results of the extrapolation are shown in Fig. 4 (open circles). Figures 4a and 4b show the extrapolation results for the experimental data from [4, 5] obtained with the corresponding ANC values from the table. Figure 4c shows the extrapolation result obtained with the weighted mean ANC value 2

( ( C pd; 01/2 ) = 4.28 ± 0.50 fm–1) recommended in this work. Solid lines are the calculations with the standard expt

C

2

, fm

16

–1

p O ; 2 5/2

2.5

(a)

expt

1.5 1.0 0.5 0

0.2 0.4 –1 , fm

2 16

0.6

0.8

1.0

1.2

p O ; 0 1/2

8000

(b)

7000 6000 5000 4000

r0 and a values and the corresponding weighted mean ANC values. As is evident from the figures, there is systematic difference between the experimental data [4, 5] and our calculations of the astrophysical S-factors at the energies E ≤ 5 keV and E ≥ 25 keV. One of the causes for this difference may be substantial spread in values of the experimental astrophysical S-factors at the energies E ≤ 5 keV. Nevertheless, the extrapolation of the experimental data [4, 5] to the energy region E ≤ 5 keV yields a relatively correct trend of their energy dependence. In particular, the values of the astrophysical S-factors Spd(E) obtained from the data of Fig. 4c at the most important energy values (E = 0 and 0.1 keV) are expt expt S pd (0) = 0.162 ± 0.019 eV b and S pd (0.2 keV) = 0.163 ± 0.019 eV b. The value S pd (0) is in quite good agreement with the value 0.166 ± 0.014 eV b from [5] and is noticeably smaller ((~2.8σ) than the value 0.216 ± 0.010 eV b obtained in [4] by linear extrapolation of experimental data.

2.0

C

2

0

0.2

0.4

0.6

0.8

1.0 1.2 E, MeV

Fig. 5. Values of the ANC for the virtual decays 17F(g.s.) → 16O + p (a) and 17F(0.495 å˝Ç) → 16O + p (b) for all experimental points E. Filled squares are our results obtained from the analysis of the corresponding experimental astrophysical S-factors [12]. Straight lines and their widths are the weighted mean ANC values and their uncertainties.

4. We carried out a similar analysis of the experimental astrophysical S-factors for the direct radiative capture reaction 16O(p, γ)17F at the energies E ≥ 0.2 keV [12] with the formation of the 17F nucleus in the ground and first excited (E* = 0.495 MeV) states. We took into account the E1 (M1) and E2 transitions corresponding to the p, f (d) waves and s and d waves respectively for the transition to the ground (Jπ = 5/2+) state and the p (s) wave and d wave respectively for the transition to the excited (Jπ = 1/2+) state of the 17F nucleus. To check conditions (2) and (3), we varied the parameters r0 and a of the Woods–Saxon potential within the ranges r0 = 1.00–1.50 fm and a = 0.52–0.78 fm. The result obtained is similar to that shown in Figs.1 ( sp ) and 2, and the function
l f j f (E, C p A; l f j f ) varies approximately by ±4% under the above variations of r0 and a with respect to their “standard” values. The calculations of the s and d phases for the pO scattering are in good agreement with the experimental results [13] within ~10%. Figures 5a and 5b show the extracted ANC values for the respective virtual decays 17F(g.s.) → 16O + p and 17F(0.495 MeV) → 16O + p for each experimental point E. The recommended weighted mean ANC values obtained from

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2

these data (Figs. 5a, 5b) turned out to be ( C pO; 25/2 ) = expt

2

1.09 ± 0.11 fm–1 and ( C pO; 25/2 ) = 5700 ± 225 fm–1. The corresponding NVC values are |GpO; 25/2|2 = 0.17 ± 0.02 fm and |GpO; 01/2|2 = 893 ± 35 fm. It is worth noting that the observed spread in extracted ANC values at the experimental points at E ≤ 0.6 (0.4) eV in Fig. 5 is related to a quite large spread in values of experimental astrophysical S-factors at the corresponding points E (see Fig. 6). Nevertheless, these spreads practically do not affect the weighted mean ANC values found above. expt

0.8 0.4

0 0.2 0.4 0.6 0.8 1.0 1.2 S 16 ( E ), keV b p O ; 0 1/2 10 (b) 8

2

Our recommended ANC value ( C pO; 25/2 ) (NVC |GpO; 25/2|2) is in good agreement with the phenomenological values 1.02 fm–1 (0.16 fm) [3] and 1.08 ± 0.10 fm–1 (0.17 ± 0.02 fm) [14] while the ANC (NVC) expt

6 4

2

values ( C pO; 01/2 ) (|GpO; 01/2|2) = 5355 fm–1 and 5122 fm–1 (840 and 819 fm) [3] and 6495 ± 0.680 fm–1 (1018 ± 107 fm) [14] differ from our ANC (NVC) value within 1.1σ, 2.6σ, and 3.5σ respectively. In addition, it should be mentioned again that in [3, 14] the ANC (NVC) values for the 17F(0.495 MeV) → 16O + p configuration were extracted from the analysis of the peripheral nuclear reaction 16O(3He, d)17F carried out by the modified distorted wave method (MDWM) at different energies of 3He ions and with different light vertex expt

2 0 0.2 0.4 0.6 0.8 1.0 1.2 S 16 ( E ), keV b p O; 2 3 10 (c) 8 6

ANC values ( C pd; 01/2 ) = 3.9 ± 0.06 fm–1 [14, 15] and

4

4.35 ± 0.10 fm–1 [13]).1 However, application of the MDWM to the reaction of proton transfer to the loosely bound state is not justified because it is important to take into account the three-body Coulomb dynamics in the transfer mechanism itself (see, for example, [17] and references therein). Consequently, the ANC (NVC) values obtained in [3, 14] for the 0.495-MeV state should have additional uncertainty related to the neglect of this dynamics in the proton transfer mechanism. The ANC (NVC) values obtained in this work can be considered more reliable as they were obtained by the thorough check of fulfillment of conditions (2) and (3) and the evaluation of the uncertainties related both to these conditions and to the experimental error of the analyzed astrophysical S-factors. Note also that our recommended ANC values noticeably differ from the values (CpO; 25/2)2 = 0.95 and 120 fm–1 and (CpO; 01/2)2 = 8306 and 7468 fm–1 [18], which were obtained using the

2

expt

1 Here

2

expt

2

it is obvious that the value ( C pd; 01/2 ) = 3.9 ± 0.06 fm–1

[14, 15] used in [14] for determining the ANC for the 17F nucleus in the (16O + p) configuration from the analysis of the experimental differential cross sections for the above-mentioned proton transfer reaction actually leads to overestimation of the ANC. It should also be mentioned that the same occurs in determination of the ANC for the 15O nucleus in the (14N + p) configuration from the analysis of the 14N(3He, d)15O reaction [16].

169

0

0.2 0.4 0.6 0.8 1.0 1.2 E, MeV

Fig. 6. Astrophysical S-factors for the 16O(p, γ)17F reaction. Squares are the data from [12] corresponding to formation of the final 17F nucleus in the ground state (a), first excited state (b), and their sum (c). Open circles are our results. Solid lines are our calculations with the standard values of the parameters r0 and a. Strip widths indicate the rootmean-square error related to the uncertainties of the function
and ANC values.

microscopic model for the V2 and MN forms of the NN potential respectively. We used the ANC values obtained in this work and relation (1) to extrapolate astrophysical S-factors SpO(E) for the 16O(p, γ)17F reaction to the energy region E < 200 keV. The results of the extrapolation are shown in Fig. 7 (open circles). Solid lines are the results of the calculation performed with the standard values of the parameters r0 and a (r0 = 1.25 fm and a = 0.65 fm), and the error at each extrapolation point includes the error ( sp ) of the function
l f j f (E, C p A; l f j f ) related to the variation of the parameters r0 and a and the experimental errors of the astrophysical S-factors. It is seen that the

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ANCs obtained allow extrapolation of SpO(E). In particexpt S pO

ular, at E = 0 the astrophysical S-factors (0) are 0.40 ± 0.04, 9.07 ± 0.36, and 9.45 ± 0.40 keV b for the transitions to the ground and first excited states of the final 17F nucleus and their sum respectively. Note that the SpO(E) value obtained by us for the transition to the ground state of the final 17F nucleus agrees with the value SpO(0) = 0.40 ± 0.04 keV b [14]. However, the value SpO(0) = 9.8 ± 1.0 keV b [14] corresponding to the transition to the first excited state of the final 17F nucleus noticeably differs (by ~2.3σ) from the value obtained in this work. Accordingly, the value SpO(0) = 9.45 ± 0.40 keV b corresponding to the total transition noticeably differs from the values 10.2 and 11.0 keV b [18] obtained within the microscopic model for the V2 and MN forms of the NN potential 5. Thus, experimental astrophysical S-factors for nuclear-astrophysics direct radiative capture reactions d(p, γ)3He and 16O(p, γ)17F [12] at very low energies E of colliding particles have been analyzed using the modified two-body potential approach [1]. The analysis has shown that these reactions are pure peripheral in the energy region under consideration. It is demonstrated that the experimental astrophysical S-factors for these reactions can be used as an independent source of information on the NVC (ANC) values for the virtual decays 3He → d + p, 17F(g.s.) → 16O + p, and 17F(0.495 MeV) → 16O + p, which allow estimation of the MDWM accuracy in determination of the NVC (ANC) for the virtual decay 17F → 16O + p from the analysis of the peripheral reaction 16O(3He, d)17F(0.495 MeV) [3, 14]. The ANC values obtained were used to extrapolate the astrophysical S-factors (Spd(E) and SpO(E)) for the reaction in question to the experimentally inaccessible energy region (E ≈ 0). ACKNOWLEDGMENTS The investigation was supported by the Academy of Sciences of the Republic of Uzbekistan, grant No. FAF2-F076 + F074, and the Ukrainian Science and Technology Center, grant No. 3081.

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