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Asymptotic normalization coefficients and astrophysical radiative capture reactions
NUCLEAR PHYSICS A ELSEVIER
Nuclear Physics A631 (1998) 788c-792c
Asymptotic normalization coefficients and astrophysical radiative capture reactions A.M. Mukhamedzhanov, H.L. Clark, H. Dejbakhsh, C.A. Gagliardi, Y.-W. Lni, L. Trache, R.E. Tribble, tt.M. Xu,* " V. Burjan, J. Cejpek, V. Kroha, S. Piskor, J. Vincour, b F. Carstoiu, c A. Sattarov, d aCyclotron Institute, Texas A&M University, College Station, Texas 77843, USA, bInstitute for Nuclear Physics, Czech Academy of Sciences, Prague-Rez, Czech Republic, CInstitute of Atomic Physics, Bucharest, Romania, dInstitute for Nuclear Physics, Tashkent, 702132, Uzbekistan The differential cross sections for the reactions 9Be(l°B, gBe)l°B at an incident l°B energy of 100 MeV and leO(SHe, d)~rF*(0.495 MeV, 1/2 +) at an incident SHe energy of 29.75 MeV were measured. By normalizing the theoretical DWBA proton exchange cross sections to the experimental ones, the asymptotic normalization coefficients (ANC's) for the virtual decays l°B ~ 9Be +p and 1~F*(0.495 MeV) --* 180 + p have been found. These ANC's are used to calculate the S(0)-factors for the direct radiative captures 9Be + p l°B + 3' and 160 + p --~ lrF*(0.495 MeV) + 3'1. I n t r o d u c t i o n The overall normalization of the astrophysical S-factor for peripheral direct radiative capture reactions may be determined from one quantity, the asymptotic normalization coefficient (ANC) of the overlap function of the bound state wave functions of the initial and final particles [1,2]. The concept of the ANC turns out to be very useful in the determination of the overall normalization of astrophysical cross sections which are difficult to measure in direct experiments due to very low cross sections at energies of astrophysical interest. An experiment to extract the ANC for the virtual decay SB ~ rBe + p , which defines the normalization of the $1~(0) astrophysical factor for the notorious reaction ~Be(p, 7)SB, using the proton transfer reaction l°B(TBe, SB)gBe at incident 7Be energies of ~ 90 MeV has been suggested by us [3]. However, to find the ANC for SB from the I°B(TBe, SB)SBe reaction, we have to know the ANC for the virtual decay X°B ~ 9Be + p. Therefore, we have started the cycle of experiments to determine the ANC's for SB ---* ~Be + p with measurements of the elastic scattering l°B + 9Be ~ l°B + ~Be and of the proton transfer *Supported in part by DOE Grant DF_,-FG03-93ER40773and by the Robert A. Welch Foundation 0375-9474/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PII S0375-9474(98)00110-9
A.M. Mukhamedzhanov et aL /Nuclear Physics A631 (1998) 788c-792c
789c
reaction aBe(l°B, aBe)l°B. The l°B + SBe elastic scattering data are needed to specify the optical potential parameters for the DWBA analysis of the proton transfer reaction. The DWBA cross section for the proton transfer reaction is proportional to the ANC for l°B(g.s.) ---* aBe + p to the fourth power. This ANC can be found by normalizing the DWBA differential cross section to the experimental one at small angles where the proton transfer mechanism is dominant. In addition to the primary purpose of using the ANC for l°B(g.s.) ~ aBe + p in the measurement of the ANC's for SB --~ 7Be + p, the ANC's for the virtual decays of the ground and low-lying states of l°B may be used to calculate the direct part of the SBe(p, 7)l°B radiative capture reaction, which is quite controversial at present [4,5]. The differential cross section for the reaction ~sO(aHe, d)ITF*(O.a95MeV) has been measured at the Cyclotron of the Institute for Nuclear Physics, Prague, Czech Republic. Due to the extremely loosely bound proton in 1ZF*(0.495 MeV) (e = 105 keV), this reaction is peripheral and the extracted ANC for 1TF*(0.495 MeV) entirely defines the overall normalization of the astrophysical factor S(E) for the radiative capture of protons by 1sO nuclei at astrophysical energies. The result found by the ANC method in this case can be compared to direct measurements [6] in order to verify our method. 2. The A N C ' s for l°B -~ 9Be ~- p and S-factor for the direct radiative capture SBe(p, 7)1°B
The measurements of the elastic scattering and transfer reaction induced by l°B were carried out at the Texas A&M University K500 superconducting cyclotron facility using the Multipole-Dipole-Multipole magnetic spectrometer. Only the transitions to the ground state and the first 3 excited states were observed with adequate statistics to obtain good angular distributions. The overall normalization accuracy was 7% for the absolute values of the measured cross sections for both the 9Be(l°B,l°B)SBe elastic scattering data and the 9Be(l°B, 9Be)a°B proton transfer data. The angular distribution measured for the elastic exchange reaction is plotted in Fig. 1. The analysis of the experimental data has been done within the framework of the DWBA using the code PTOLEMY. The basic calculations have been done with the Woods-Saxon potential for the bound states with geometric parameters r0 = 1.2 fm, a = 0.6 fm, and the Thomas spin-orbit term. Due to the peripheral character of the reactions under consideration, the results are only weakly dependent on the geometry of the bound state Woods-Saxon potentials. The parameters of the two different optical potentials, potentials 1 and 2, found from fitting the elastic scattering data are given in [7]. Both potentials give similar angular distributions. By normalizing the calculated DWBA cross section to the experimental one at forward angles, we find the values of the ANC for the virtual decay l°B(g.s.) --* 9Be + p. The extracted values of the ANC's for optical potentials 1 and 2 are given in Table 1. Since potential 1 gave a somewhat better description of the elastic scattering data, we chose to weight its value for C 2 twice that of potential 2 in specifying our best value, and we assign an additional uncertainty of 4-5% to our adopted value of C 2 to account for the uncertainty in the choice of optical model parameters. Using the ANC's for the virtual decays of the ground and three first excited states of l°B into 9Be + p we have calculated the S-factor for the direct radiative capture 9Be + p --~ l°B + V at zero energy. Our total
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A.M. Mukhamedzhanov et al./Nuclear Physics A631 (1998) 788c-792c
calculated S(0)-factor for the transitions to the ground and three first excited states is S(0) = 0.73 keVb which almost twice as large as the direct part derived in [5]. However, in contrast to [5] this provides a good fit to S(E) at low proton energies while using standard values for the energy and width of the lowest l°B resonance.
102
,•
10
b 1 10 -1
lO
-2
I[lllll
0
10
20
30
40
50
60
70
Oc.m. ( dee, )
Figure 1.
The experimental and calculated angular distributions for the reaction
9Be(IOB, 9Be)~OB(g.s.). The points are experimental data; the solid line is the DWBA.
Table 1 The measured ANC's C 2 for l°B ~ 9Be +p from 9Be(~°B, 9Be)~°B reactions. C~ and C~ are the extracted ANC's using optical potentials 1 and 2, respectively. The uncertainties specified include only the contribution from the statistics in the angular distribution fits. C ~ are our adopted values of the ANC's. Their uncertainties include the contributions due to the normalization uncertainty and the theoretical systematic effects, in addition to statistical uncertainties. E*(MeV) Jv C~ (fm -~) C~ (fm -~) e 2 (fm -~)
0.0 0.718 1.740 2.154
3/2 1/2 3/2 3/2 1/2 3/2
4.91(19) 1.23(15) 3.33(17) 4.22(33) 0.28(5) 0.80(8)
5.35(21) 1.34(16) 3.~3(19) 4.60(36) 0.30(5) 0.87(9)
5.o6(46) 1.27(21) 3.43(42) 4.35(59) 0.29(6) 0.82(12)
A.M. Mukhamedzhanov et al. /Nuclear Physics A631 (1998) 788c-792c
791c
3. T h e 1rF*(0.495 M e V ) ~ 180+p A N C a n d t h e S - f a c t o r for 180(p, 7)~rF*(0.495 M e V ) To check the ANC concept we have measured the differential cross section for the reaction ~60(3He, d)~TF*(0.495 MeV) at an incident 3He energy of 29.75 MeV. The uncertainty in the overall normalization of the cross section at forward angles was < 10%. The experimental and calculated DWBA cross sections are shown in Fig. 2. The optical potentials are taken from [8].
10
• ,~1 ....
0
5
I ....
10
I ....
15
I ....
20
llllJlllll 25 30 35
O~.m.(deg) Figure 2.
The experimental and calculated angular distributions for the reaction
180(SHe, d)~'~F*(O.495MeV). The points are experimental data; the solid line is the DWBA fit.
This reaction turns out to be totally peripheral and by normalizing the DWBA cross section to the experimental one at small angles we extracted the ANC for the virtual decay 1TF*(0.495 MeV) ---* 180 + p: C 2 = 6084fm -1. The extracted ANC agrees very well with the one extracted from the potential model analysis of the low energy s-wave scattering phase shifts [9]. The S-factor for the radiative capture 180 + p --* 17F*(0.495 MeV) + 3' calculated with this ANC is shown in Fig 3. It reproduces available experimental data [6] very well. At low energies, where experimental data axe missing, the calculated S-factor shows a strong rise as has been found in [9].
792c
A.M. Mukhamedzhanov et al./Nuclear Physics A631 (1998) 788c-792c
I 0
. . . .
I
. . . .
I
. . . .
I
. . . .
I
. . . .
8
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6
4 o
C 4
0
'
0.0
,
,,
I , . . .
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I.
,
,
,
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I
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2.0
.
.
.
2.5
Figure 3. The S-factor for the reaction 160 + p ---* lrF*(0.495 MeV) + 7. The points are experimental data; the solid line is our calculation using the ANC for 17F*(0.495 MeV) extracted from (SHe, d) reaction.
REFERENCES
1. A.M. Mukhamedzhanov and N.K. Timofeyuk, JETP Lett. 51 (1990) 282. 2. H.M. Xu, C.A. Gagliardi, R.E. Tribble, A.M. Mukhamedzhanov and N.K. Timofeyuk, Phys. Rev. Lett. 73 (1994) 2027. 3. C.A. Gagliardi, R.E. Tribble, J. Jiang, A.M. Mukhamedzhanov, L. Trache, H.M. Xu, S.J. Yennello, and X.G. Zhou, Nucl. Phys. A588 (1995) 327c. 4. F.E. Cecil, D. Ferg, H. Liu, J.C. Scorby, J.A. McNeil and P.D. Kunz, Nucl. Phys. A539 (1992) 75. 5. D. Zahnow, C. Angulo, M. Junker, C. Rolls, S. Schmidt, W.H. Schulte and E. Somorjai, Nucl. Phys. A589 (1995) 95. 6. H.C. Chow, G.M. Griffins and T.H. Hall, Can. J. Phys. 53 (1975) 1672. 7. A.M. Mukhamedzhanov et al, Phys. Rev. C56 (1997), to be published. 8. J. Vernotte, G. Berrier-Ronsin, J. Kalifa, R. Tamisier and B.H. Widenthal, Nucl. Phys. A571 (1994) 1. 9. C.R. Brune Nucl. Phys. A596 (1996) 122.
Nuclear Physics A631 (1998) 788c-792c
Asymptotic normalization coefficients and astrophysical radiative capture reactions A.M. Mukhamedzhanov, H.L. Clark, H. Dejbakhsh, C.A. Gagliardi, Y.-W. Lni, L. Trache, R.E. Tribble, tt.M. Xu,* " V. Burjan, J. Cejpek, V. Kroha, S. Piskor, J. Vincour, b F. Carstoiu, c A. Sattarov, d aCyclotron Institute, Texas A&M University, College Station, Texas 77843, USA, bInstitute for Nuclear Physics, Czech Academy of Sciences, Prague-Rez, Czech Republic, CInstitute of Atomic Physics, Bucharest, Romania, dInstitute for Nuclear Physics, Tashkent, 702132, Uzbekistan The differential cross sections for the reactions 9Be(l°B, gBe)l°B at an incident l°B energy of 100 MeV and leO(SHe, d)~rF*(0.495 MeV, 1/2 +) at an incident SHe energy of 29.75 MeV were measured. By normalizing the theoretical DWBA proton exchange cross sections to the experimental ones, the asymptotic normalization coefficients (ANC's) for the virtual decays l°B ~ 9Be +p and 1~F*(0.495 MeV) --* 180 + p have been found. These ANC's are used to calculate the S(0)-factors for the direct radiative captures 9Be + p l°B + 3' and 160 + p --~ lrF*(0.495 MeV) + 3'1. I n t r o d u c t i o n The overall normalization of the astrophysical S-factor for peripheral direct radiative capture reactions may be determined from one quantity, the asymptotic normalization coefficient (ANC) of the overlap function of the bound state wave functions of the initial and final particles [1,2]. The concept of the ANC turns out to be very useful in the determination of the overall normalization of astrophysical cross sections which are difficult to measure in direct experiments due to very low cross sections at energies of astrophysical interest. An experiment to extract the ANC for the virtual decay SB ~ rBe + p , which defines the normalization of the $1~(0) astrophysical factor for the notorious reaction ~Be(p, 7)SB, using the proton transfer reaction l°B(TBe, SB)gBe at incident 7Be energies of ~ 90 MeV has been suggested by us [3]. However, to find the ANC for SB from the I°B(TBe, SB)SBe reaction, we have to know the ANC for the virtual decay X°B ~ 9Be + p. Therefore, we have started the cycle of experiments to determine the ANC's for SB ---* ~Be + p with measurements of the elastic scattering l°B + 9Be ~ l°B + ~Be and of the proton transfer *Supported in part by DOE Grant DF_,-FG03-93ER40773and by the Robert A. Welch Foundation 0375-9474/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PII S0375-9474(98)00110-9
A.M. Mukhamedzhanov et aL /Nuclear Physics A631 (1998) 788c-792c
789c
reaction aBe(l°B, aBe)l°B. The l°B + SBe elastic scattering data are needed to specify the optical potential parameters for the DWBA analysis of the proton transfer reaction. The DWBA cross section for the proton transfer reaction is proportional to the ANC for l°B(g.s.) ---* aBe + p to the fourth power. This ANC can be found by normalizing the DWBA differential cross section to the experimental one at small angles where the proton transfer mechanism is dominant. In addition to the primary purpose of using the ANC for l°B(g.s.) ~ aBe + p in the measurement of the ANC's for SB --~ 7Be + p, the ANC's for the virtual decays of the ground and low-lying states of l°B may be used to calculate the direct part of the SBe(p, 7)l°B radiative capture reaction, which is quite controversial at present [4,5]. The differential cross section for the reaction ~sO(aHe, d)ITF*(O.a95MeV) has been measured at the Cyclotron of the Institute for Nuclear Physics, Prague, Czech Republic. Due to the extremely loosely bound proton in 1ZF*(0.495 MeV) (e = 105 keV), this reaction is peripheral and the extracted ANC for 1TF*(0.495 MeV) entirely defines the overall normalization of the astrophysical factor S(E) for the radiative capture of protons by 1sO nuclei at astrophysical energies. The result found by the ANC method in this case can be compared to direct measurements [6] in order to verify our method. 2. The A N C ' s for l°B -~ 9Be ~- p and S-factor for the direct radiative capture SBe(p, 7)1°B
The measurements of the elastic scattering and transfer reaction induced by l°B were carried out at the Texas A&M University K500 superconducting cyclotron facility using the Multipole-Dipole-Multipole magnetic spectrometer. Only the transitions to the ground state and the first 3 excited states were observed with adequate statistics to obtain good angular distributions. The overall normalization accuracy was 7% for the absolute values of the measured cross sections for both the 9Be(l°B,l°B)SBe elastic scattering data and the 9Be(l°B, 9Be)a°B proton transfer data. The angular distribution measured for the elastic exchange reaction is plotted in Fig. 1. The analysis of the experimental data has been done within the framework of the DWBA using the code PTOLEMY. The basic calculations have been done with the Woods-Saxon potential for the bound states with geometric parameters r0 = 1.2 fm, a = 0.6 fm, and the Thomas spin-orbit term. Due to the peripheral character of the reactions under consideration, the results are only weakly dependent on the geometry of the bound state Woods-Saxon potentials. The parameters of the two different optical potentials, potentials 1 and 2, found from fitting the elastic scattering data are given in [7]. Both potentials give similar angular distributions. By normalizing the calculated DWBA cross section to the experimental one at forward angles, we find the values of the ANC for the virtual decay l°B(g.s.) --* 9Be + p. The extracted values of the ANC's for optical potentials 1 and 2 are given in Table 1. Since potential 1 gave a somewhat better description of the elastic scattering data, we chose to weight its value for C 2 twice that of potential 2 in specifying our best value, and we assign an additional uncertainty of 4-5% to our adopted value of C 2 to account for the uncertainty in the choice of optical model parameters. Using the ANC's for the virtual decays of the ground and three first excited states of l°B into 9Be + p we have calculated the S-factor for the direct radiative capture 9Be + p --~ l°B + V at zero energy. Our total
790c
A.M. Mukhamedzhanov et al./Nuclear Physics A631 (1998) 788c-792c
calculated S(0)-factor for the transitions to the ground and three first excited states is S(0) = 0.73 keVb which almost twice as large as the direct part derived in [5]. However, in contrast to [5] this provides a good fit to S(E) at low proton energies while using standard values for the energy and width of the lowest l°B resonance.
102
,•
10
b 1 10 -1
lO
-2
I[lllll
0
10
20
30
40
50
60
70
Oc.m. ( dee, )
Figure 1.
The experimental and calculated angular distributions for the reaction
9Be(IOB, 9Be)~OB(g.s.). The points are experimental data; the solid line is the DWBA.
Table 1 The measured ANC's C 2 for l°B ~ 9Be +p from 9Be(~°B, 9Be)~°B reactions. C~ and C~ are the extracted ANC's using optical potentials 1 and 2, respectively. The uncertainties specified include only the contribution from the statistics in the angular distribution fits. C ~ are our adopted values of the ANC's. Their uncertainties include the contributions due to the normalization uncertainty and the theoretical systematic effects, in addition to statistical uncertainties. E*(MeV) Jv C~ (fm -~) C~ (fm -~) e 2 (fm -~)
0.0 0.718 1.740 2.154
3/2 1/2 3/2 3/2 1/2 3/2
4.91(19) 1.23(15) 3.33(17) 4.22(33) 0.28(5) 0.80(8)
5.35(21) 1.34(16) 3.~3(19) 4.60(36) 0.30(5) 0.87(9)
5.o6(46) 1.27(21) 3.43(42) 4.35(59) 0.29(6) 0.82(12)
A.M. Mukhamedzhanov et al. /Nuclear Physics A631 (1998) 788c-792c
791c
3. T h e 1rF*(0.495 M e V ) ~ 180+p A N C a n d t h e S - f a c t o r for 180(p, 7)~rF*(0.495 M e V ) To check the ANC concept we have measured the differential cross section for the reaction ~60(3He, d)~TF*(0.495 MeV) at an incident 3He energy of 29.75 MeV. The uncertainty in the overall normalization of the cross section at forward angles was < 10%. The experimental and calculated DWBA cross sections are shown in Fig. 2. The optical potentials are taken from [8].
10
• ,~1 ....
0
5
I ....
10
I ....
15
I ....
20
llllJlllll 25 30 35
O~.m.(deg) Figure 2.
The experimental and calculated angular distributions for the reaction
180(SHe, d)~'~F*(O.495MeV). The points are experimental data; the solid line is the DWBA fit.
This reaction turns out to be totally peripheral and by normalizing the DWBA cross section to the experimental one at small angles we extracted the ANC for the virtual decay 1TF*(0.495 MeV) ---* 180 + p: C 2 = 6084fm -1. The extracted ANC agrees very well with the one extracted from the potential model analysis of the low energy s-wave scattering phase shifts [9]. The S-factor for the radiative capture 180 + p --* 17F*(0.495 MeV) + 3' calculated with this ANC is shown in Fig 3. It reproduces available experimental data [6] very well. At low energies, where experimental data axe missing, the calculated S-factor shows a strong rise as has been found in [9].
792c
A.M. Mukhamedzhanov et al./Nuclear Physics A631 (1998) 788c-792c
I 0
. . . .
I
. . . .
I
. . . .
I
. . . .
I
. . . .
8
;>
6
4 o
C 4
0
'
0.0
,
,,
I , . . .
0.5
I.
,
,
,
I
. . . .
1.0 1.5 E (MeV)
I
.
2.0
.
.
.
2.5
Figure 3. The S-factor for the reaction 160 + p ---* lrF*(0.495 MeV) + 7. The points are experimental data; the solid line is our calculation using the ANC for 17F*(0.495 MeV) extracted from (SHe, d) reaction.
REFERENCES
1. A.M. Mukhamedzhanov and N.K. Timofeyuk, JETP Lett. 51 (1990) 282. 2. H.M. Xu, C.A. Gagliardi, R.E. Tribble, A.M. Mukhamedzhanov and N.K. Timofeyuk, Phys. Rev. Lett. 73 (1994) 2027. 3. C.A. Gagliardi, R.E. Tribble, J. Jiang, A.M. Mukhamedzhanov, L. Trache, H.M. Xu, S.J. Yennello, and X.G. Zhou, Nucl. Phys. A588 (1995) 327c. 4. F.E. Cecil, D. Ferg, H. Liu, J.C. Scorby, J.A. McNeil and P.D. Kunz, Nucl. Phys. A539 (1992) 75. 5. D. Zahnow, C. Angulo, M. Junker, C. Rolls, S. Schmidt, W.H. Schulte and E. Somorjai, Nucl. Phys. A589 (1995) 95. 6. H.C. Chow, G.M. Griffins and T.H. Hall, Can. J. Phys. 53 (1975) 1672. 7. A.M. Mukhamedzhanov et al, Phys. Rev. C56 (1997), to be published. 8. J. Vernotte, G. Berrier-Ronsin, J. Kalifa, R. Tamisier and B.H. Widenthal, Nucl. Phys. A571 (1994) 1. 9. C.R. Brune Nucl. Phys. A596 (1996) 122.