466
9 The Unification of Structure and Reaction Frameworks
0.0 0.5 1.0 1.5 2.0 2.5 3.0
E CM (MeV)
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
S
E1
(barn.eV)
×10
1
Fig. 9.3 Plot of the E1 astrophysical factor for the 7 Be(p, γ ) 8 B reaction. The solid line represents
the exact, fully antisymmetrized coupled-channel Gamow shell model calculation. The calculations
in the long wavelength approximation are represented by the dashed and dotted lines in the fully
antisymmetrized and non-antisymmetrized cases, respectively (from Ref. [64])
Results of coupled-channel Gamow shell model cross section calculations for the
reaction 7 Be(p, γ ) are shown in Figs. 9.3 and 9.4. In these calculations, there are no
effective charges for M1 transitions, and standard values of effective charges have
been used for E1 and E2 transitions. One should mention that the experimentally
extracted effective charges show often significant deviations from the standard
values [87].
Figure 9.3 shows the E1 astrophysical factor S (E1) (see Eq. (9.62) for its definition) calculated both exactly and approximately using the long wavelength approximation in the calculation of matrix elements of the electromagnetic transitions. For
E1, both the long wavelength approximation and the absence of antisymmetry in the
channel state |r, c decrease the E1 contribution to the total astrophysical S-factor
(see Fig. 9.3). However, whereas the long wavelength approximation reduces the
S-factor by ∼5%, the absence of the antisymmetry in the calculation of the matrix
elements of the electromagnetic operators reduces it by almost ∼50%.
The calculated total S-factor that includes S E1 , S E2 , and S M1 contributions is
compared with the experimental data [88, 89] in Fig. 9.4. Below E CM = 1 MeV, the
agreement with the data is good if both the ground state of 7 Be and its first excited
state are included. The value of the S-factor at zero energy, S GSM-CC (0), is 23.214
b·eV and the slope, ∂S/∂E CM | E CM = 0, is 37.921 b. The accepted experimental
9 The Unification of Structure and Reaction Frameworks
0.0 0.5 1.0 1.5 2.0 2.5 3.0
E CM (MeV)
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
S
E1
(barn.eV)
×10
1
Fig. 9.3 Plot of the E1 astrophysical factor for the 7 Be(p, γ ) 8 B reaction. The solid line represents
the exact, fully antisymmetrized coupled-channel Gamow shell model calculation. The calculations
in the long wavelength approximation are represented by the dashed and dotted lines in the fully
antisymmetrized and non-antisymmetrized cases, respectively (from Ref. [64])
Results of coupled-channel Gamow shell model cross section calculations for the
reaction 7 Be(p, γ ) are shown in Figs. 9.3 and 9.4. In these calculations, there are no
effective charges for M1 transitions, and standard values of effective charges have
been used for E1 and E2 transitions. One should mention that the experimentally
extracted effective charges show often significant deviations from the standard
values [87].
Figure 9.3 shows the E1 astrophysical factor S (E1) (see Eq. (9.62) for its definition) calculated both exactly and approximately using the long wavelength approximation in the calculation of matrix elements of the electromagnetic transitions. For
E1, both the long wavelength approximation and the absence of antisymmetry in the
channel state |r, c decrease the E1 contribution to the total astrophysical S-factor
(see Fig. 9.3). However, whereas the long wavelength approximation reduces the
S-factor by ∼5%, the absence of the antisymmetry in the calculation of the matrix
elements of the electromagnetic operators reduces it by almost ∼50%.
The calculated total S-factor that includes S E1 , S E2 , and S M1 contributions is
compared with the experimental data [88, 89] in Fig. 9.4. Below E CM = 1 MeV, the
agreement with the data is good if both the ground state of 7 Be and its first excited
state are included. The value of the S-factor at zero energy, S GSM-CC (0), is 23.214
b·eV and the slope, ∂S/∂E CM | E CM = 0, is 37.921 b. The accepted experimental