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Study of the Coulomb effect in the photodisintegration of 3He
Nuclear Physics A 790 (2007) 344c–347c
Study of the Coulomb effect in the photodisintegration of 3 He A. Deltuvaa∗ , A. C. Fonsecaa and P. U. Sauerb a
Centro de F´ısica Nuclear da Universidade de Lisboa, P-1649-003 Lisboa, Portugal
b
Institut f¨ ur Theoretische Physik, Universit¨at Hannover, D-30167 Hannover, Germany
The Coulomb interaction between the two protons is included in the description of three-nucleon electromagnetic reactions using the screening and renormalization approach. Calculations are done using integral equations in momentum space. The Coulomb effect on observables is discussed. 1. INTRODUCTION In Refs. [1,2] we included the Coulomb interaction between the protons in the description of proton-deuteron (pd) scattering and of three-nucleon electromagnetic (e.m.) reactions involving 3 He. The description is based on the Alt-Grassberger-Sandhas (AGS) equations [3] in momentum space. The Coulomb potential is screened, standard scattering theory is applicable, and the renormalization technique of Refs. [4,5] is employed to recover the unscreened Coulomb limit. The special choice of the screened Coulomb n potential wR = w e−(r/R) , n = 4 being optimal, approximates well the true Coulomb one w for distances r smaller than the screening radius R and simultaneously vanishes rapidly for r > R; rather modest values of R are sufficient in order to obtain results that are well converged. The reliability of the method is demonstrated in Refs. [1,2,6]. This contribution applies the method to the photodisintegration of 3 He. As in Refs. [1,2], this work is based on the isospin formalism for three nucleons and on the symmetrized AGS equations which are used to calculate matrix elements of the e.m. current operator between the three-nucleon bound state and the final scattering two-body or three-body state. We employ the two-baryon coupled-channel potential CD Bonn + Δ [7] with and without Coulomb and use the CD Bonn potential [8] with Coulomb as a purely nucleonic reference. We use the charge and current operators of Ref. [9] which include two- and three-body currents that are consistent with the underlying dynamics. 2. RESULTS The Coulomb effect on the observables of pd radiative capture, the time reverse reaction of the two-body photodisintegration of 3 He, is studied in detail in Ref. [1]. Here we show only one new example in Fig. 1, the total cross section for the two-body photodisintegration of 3 He at low energies. ∗
Supported by the FCT grant SFRH/BPD/14801/2003
0375-9474/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2007.03.062
A. Deltuva et al. / Nuclear Physics A 790 (2007) 344c–347c
345c
σpd (mb)
1.0
0.5 8
12 Eγ (MeV)
16
Figure 1. Total cross section for the two-body photodisintegration of 3 He as function of the photon energy. Results including Δ-isobar excitation and the Coulomb interaction (solid curves) are compared to results without Coulomb (dashed curves). In order to appreciate the size of the Δ-isobar effect, the purely nucleonic results including Coulomb are also shown (dotted curves). The given experimental data are the most recent ones; they are from Ref. [10].
Eγ = 55 MeV
Eγ = 85 MeV
d4σ/dΩ1 dΩ2 (μb/sr2)
120 60
80
40
40
20
0
0 160
200
Θp+Θn (deg)
160
200
Figure 2. The semi-inclusive fourfold differential cross section for the 3 He(γ, pn)p reaction at 55 MeV and 85 MeV photon lab energy as function of the np opening angle θp + θn with θp = 81◦ . Curves as in Fig. 1. The experimental data are from Ref. [11]. Experimental data for three-body photodisintegration of 3 He are very scarce; we therefore show in Fig. 2 only two examples referring to the semi-inclusive 3 He(γ, pn)p reaction at 55 MeV and 85 MeV photon lab energy. For scattering angles corresponding to the peak of the semi-inclusive fourfold differential cross section the region of the phase space to be integrated over contains the proton-proton final-state interaction (pp-FSI) regime where
A. Deltuva et al. / Nuclear Physics A 790 (2007) 344c–347c
20
60
o
Θn = 30
o
Θn = 90
40 10 20
3
d σ/dEn dΩn (μb/MeV sr)
346c
0
0 0
2
4
0
En (MeV)
2
4
En (MeV)
Figure 3. The semi-inclusive threefold differential cross section for the 3 He(γ, n)pp reaction at 15 MeV photon lab energy as function of the neutron energy En for different neutron scattering angles. Curves as in Fig. 1.
3
d σ/dEp dΩp (μb/MeV sr)
50
80
40 60 30 40 20 20
10
o
o
Θp = 30
Θp = 90
0
0 0
2
4
Ep (MeV)
0
2
4
Ep (MeV)
Figure 4. The semi-inclusive threefold differential cross section for the 3 He(γ, p)pn reaction at 15 MeV photon lab energy as function of the proton energy Ep for different proton scattering angles. Curves as in Fig. 1.
the pp-FSI peak obtained without Coulomb is converted into a minimum as demonstrated in Ref. [2]. Therefore the fourfold differential cross section in Fig. 2 is also significantly reduced by the inclusion of Coulomb, clearly improving the agreement with the data. In view of the existing HIGS facility one expects a rich set of semi-inclusive data in the very near future [12]. Therefore we study a number of three-body breakup configurations and show selected results for the threefold differential cross section and photon asymmetry in Figs. 3 - 5. The large Coulomb effect on the semi-inclusive cross section is of the same origin as in Fig. 2. The photon asymmetry Σ is significantly affected by the inclusion of Coulomb for the 3 He(γ, p)pn reaction, but remains almost unchanged for the 3 He(γ, n)pp reaction. In most cases which we studied the Coulomb effect is much more important than the Δ-isobar effect.
347c
Σ
A. Deltuva et al. / Nuclear Physics A 790 (2007) 344c–347c 1.0
1.0
0.5
0.5
Θn = 90o
Θp = 90o
0.0
0.0 0
2
En (MeV)
4
0
2
4
Ep (MeV)
Figure 5. The semi-inclusive photon asymmetry Σ for the 3 He(γ, n)pp (left side) and 3 He(γ, p)pn (right side) reactions at 15 MeV photon lab energy as function of the energy of detected nucleon at 90◦ scattering angle. Curves as in Fig. 1. 3. SUMMARY Using the screening and renormalization approach in the framework of momentumspace integral equations we studied the Coulomb effect on the 3 Hephotodisintegration observables. We found that the inclusion of the Coulomb interaction is very important in the three-nucleon breakup of 3 He, especially in the regions of phase space close to pp-FSI. Though the results are given for photodisintegration, the same technique for the inclusion of Coulomb has been successfully applied to inelastic electron scattering from 3 He [1,2]. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
A. Deltuva, A. C. Fonseca, P. U. Sauer, Phys. Rev. C 71 (2005) 054005. A. Deltuva, A. C. Fonseca, P. U. Sauer, Phys. Rev. C 72 (2005) 054004. E. O. Alt, P. Grassberger, W. Sandhas, Nucl. Phys. B 2 (1967) 167. J. R. Taylor, Nuovo Cimento B 23 (1974) 313; M. D. Semon and J. R. Taylor, ibid. A 26 (1975) 48. E. O. Alt, W. Sandhas, H. Ziegelmann, Phys. Rev. C 17 (1978) 1981; E. O. Alt and W. Sandhas, ibid. 21 (1980) 1733. A. Deltuva, A. C. Fonseca, A. Kievsky, S. Rosati, P. U. Sauer, M. Viviani, Phys. Rev. C 71 (2005) 064003. A. Deltuva, R. Machleidt, P. U. Sauer, Phys. Rev. C 68 (2003) 024005. R. Machleidt, Phys. Rev. C 63 (2001) 024001. A. Deltuva, L. P. Yuan, J. Adam Jr., A. C. Fonseca, P. U. Sauer, Phys. Rev. C 69 (2004) 034004. S. Naito et al., Phys. Rev. C 73 (2006) 034003. N. R. Kolb, P. N. Dezendorf, M. K. Brussel, B. B. Ritchie, J. H. Smith, Phys. Rev. C 44 (1991) 37. W. Tornow, contribution to this conference.
Study of the Coulomb effect in the photodisintegration of 3 He A. Deltuvaa∗ , A. C. Fonsecaa and P. U. Sauerb a
Centro de F´ısica Nuclear da Universidade de Lisboa, P-1649-003 Lisboa, Portugal
b
Institut f¨ ur Theoretische Physik, Universit¨at Hannover, D-30167 Hannover, Germany
The Coulomb interaction between the two protons is included in the description of three-nucleon electromagnetic reactions using the screening and renormalization approach. Calculations are done using integral equations in momentum space. The Coulomb effect on observables is discussed. 1. INTRODUCTION In Refs. [1,2] we included the Coulomb interaction between the protons in the description of proton-deuteron (pd) scattering and of three-nucleon electromagnetic (e.m.) reactions involving 3 He. The description is based on the Alt-Grassberger-Sandhas (AGS) equations [3] in momentum space. The Coulomb potential is screened, standard scattering theory is applicable, and the renormalization technique of Refs. [4,5] is employed to recover the unscreened Coulomb limit. The special choice of the screened Coulomb n potential wR = w e−(r/R) , n = 4 being optimal, approximates well the true Coulomb one w for distances r smaller than the screening radius R and simultaneously vanishes rapidly for r > R; rather modest values of R are sufficient in order to obtain results that are well converged. The reliability of the method is demonstrated in Refs. [1,2,6]. This contribution applies the method to the photodisintegration of 3 He. As in Refs. [1,2], this work is based on the isospin formalism for three nucleons and on the symmetrized AGS equations which are used to calculate matrix elements of the e.m. current operator between the three-nucleon bound state and the final scattering two-body or three-body state. We employ the two-baryon coupled-channel potential CD Bonn + Δ [7] with and without Coulomb and use the CD Bonn potential [8] with Coulomb as a purely nucleonic reference. We use the charge and current operators of Ref. [9] which include two- and three-body currents that are consistent with the underlying dynamics. 2. RESULTS The Coulomb effect on the observables of pd radiative capture, the time reverse reaction of the two-body photodisintegration of 3 He, is studied in detail in Ref. [1]. Here we show only one new example in Fig. 1, the total cross section for the two-body photodisintegration of 3 He at low energies. ∗
Supported by the FCT grant SFRH/BPD/14801/2003
0375-9474/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2007.03.062
A. Deltuva et al. / Nuclear Physics A 790 (2007) 344c–347c
345c
σpd (mb)
1.0
0.5 8
12 Eγ (MeV)
16
Figure 1. Total cross section for the two-body photodisintegration of 3 He as function of the photon energy. Results including Δ-isobar excitation and the Coulomb interaction (solid curves) are compared to results without Coulomb (dashed curves). In order to appreciate the size of the Δ-isobar effect, the purely nucleonic results including Coulomb are also shown (dotted curves). The given experimental data are the most recent ones; they are from Ref. [10].
Eγ = 55 MeV
Eγ = 85 MeV
d4σ/dΩ1 dΩ2 (μb/sr2)
120 60
80
40
40
20
0
0 160
200
Θp+Θn (deg)
160
200
Figure 2. The semi-inclusive fourfold differential cross section for the 3 He(γ, pn)p reaction at 55 MeV and 85 MeV photon lab energy as function of the np opening angle θp + θn with θp = 81◦ . Curves as in Fig. 1. The experimental data are from Ref. [11]. Experimental data for three-body photodisintegration of 3 He are very scarce; we therefore show in Fig. 2 only two examples referring to the semi-inclusive 3 He(γ, pn)p reaction at 55 MeV and 85 MeV photon lab energy. For scattering angles corresponding to the peak of the semi-inclusive fourfold differential cross section the region of the phase space to be integrated over contains the proton-proton final-state interaction (pp-FSI) regime where
A. Deltuva et al. / Nuclear Physics A 790 (2007) 344c–347c
20
60
o
Θn = 30
o
Θn = 90
40 10 20
3
d σ/dEn dΩn (μb/MeV sr)
346c
0
0 0
2
4
0
En (MeV)
2
4
En (MeV)
Figure 3. The semi-inclusive threefold differential cross section for the 3 He(γ, n)pp reaction at 15 MeV photon lab energy as function of the neutron energy En for different neutron scattering angles. Curves as in Fig. 1.
3
d σ/dEp dΩp (μb/MeV sr)
50
80
40 60 30 40 20 20
10
o
o
Θp = 30
Θp = 90
0
0 0
2
4
Ep (MeV)
0
2
4
Ep (MeV)
Figure 4. The semi-inclusive threefold differential cross section for the 3 He(γ, p)pn reaction at 15 MeV photon lab energy as function of the proton energy Ep for different proton scattering angles. Curves as in Fig. 1.
the pp-FSI peak obtained without Coulomb is converted into a minimum as demonstrated in Ref. [2]. Therefore the fourfold differential cross section in Fig. 2 is also significantly reduced by the inclusion of Coulomb, clearly improving the agreement with the data. In view of the existing HIGS facility one expects a rich set of semi-inclusive data in the very near future [12]. Therefore we study a number of three-body breakup configurations and show selected results for the threefold differential cross section and photon asymmetry in Figs. 3 - 5. The large Coulomb effect on the semi-inclusive cross section is of the same origin as in Fig. 2. The photon asymmetry Σ is significantly affected by the inclusion of Coulomb for the 3 He(γ, p)pn reaction, but remains almost unchanged for the 3 He(γ, n)pp reaction. In most cases which we studied the Coulomb effect is much more important than the Δ-isobar effect.
347c
Σ
A. Deltuva et al. / Nuclear Physics A 790 (2007) 344c–347c 1.0
1.0
0.5
0.5
Θn = 90o
Θp = 90o
0.0
0.0 0
2
En (MeV)
4
0
2
4
Ep (MeV)
Figure 5. The semi-inclusive photon asymmetry Σ for the 3 He(γ, n)pp (left side) and 3 He(γ, p)pn (right side) reactions at 15 MeV photon lab energy as function of the energy of detected nucleon at 90◦ scattering angle. Curves as in Fig. 1. 3. SUMMARY Using the screening and renormalization approach in the framework of momentumspace integral equations we studied the Coulomb effect on the 3 Hephotodisintegration observables. We found that the inclusion of the Coulomb interaction is very important in the three-nucleon breakup of 3 He, especially in the regions of phase space close to pp-FSI. Though the results are given for photodisintegration, the same technique for the inclusion of Coulomb has been successfully applied to inelastic electron scattering from 3 He [1,2]. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
A. Deltuva, A. C. Fonseca, P. U. Sauer, Phys. Rev. C 71 (2005) 054005. A. Deltuva, A. C. Fonseca, P. U. Sauer, Phys. Rev. C 72 (2005) 054004. E. O. Alt, P. Grassberger, W. Sandhas, Nucl. Phys. B 2 (1967) 167. J. R. Taylor, Nuovo Cimento B 23 (1974) 313; M. D. Semon and J. R. Taylor, ibid. A 26 (1975) 48. E. O. Alt, W. Sandhas, H. Ziegelmann, Phys. Rev. C 17 (1978) 1981; E. O. Alt and W. Sandhas, ibid. 21 (1980) 1733. A. Deltuva, A. C. Fonseca, A. Kievsky, S. Rosati, P. U. Sauer, M. Viviani, Phys. Rev. C 71 (2005) 064003. A. Deltuva, R. Machleidt, P. U. Sauer, Phys. Rev. C 68 (2003) 024005. R. Machleidt, Phys. Rev. C 63 (2001) 024001. A. Deltuva, L. P. Yuan, J. Adam Jr., A. C. Fonseca, P. U. Sauer, Phys. Rev. C 69 (2004) 034004. S. Naito et al., Phys. Rev. C 73 (2006) 034003. N. R. Kolb, P. N. Dezendorf, M. K. Brussel, B. B. Ritchie, J. H. Smith, Phys. Rev. C 44 (1991) 37. W. Tornow, contribution to this conference.