Communications in Theoretical Physics ›› 2022, Vol. 74 ›› Issue (1): 015303. doi: 10.1088/1572-9494/ac3999
• Nuclear Physics • Previous Articles Next Articles
Kun Wang(王琨)1,2,Bing-Nan Lu(吕炳楠)3,()
Received:
2021-10-25
Revised:
2021-11-05
Accepted:
2021-11-15
Published:
2022-01-01
Contact:
Bing-Nan Lu(吕炳楠)
E-mail:bnlv@gscaep.ac.cn
Kun Wang(王琨),Bing-Nan Lu(吕炳楠), Commun. Theor. Phys. 74 (2022) 015303.
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Figure 2.
The energy of 12C as functions of the triangular shape parameter λ3 calculated with different basis truncations in the MDCRHB model. The squares, circles, up triangles and down triangles denote the results obtained with major shell truncation Nf = 6, 8, 10 and 12, respectively. The center of mass correction to the binding energy is not included."
Figure 3.
The projected energies of 12C calculated with different mesh sizes for the SO(3) group integral. For each curve, the energies are relative to the corresponding converged values E0J calculated at large mesh sizes. All calculations are performed with a mean-field wave function constrained to λ3 = 1.2. Here, nα, nβ and nγ are the number of mesh points for integrating the Euler angles α, β and γ, respectively. Squares, circles, up triangles, down triangles, diamonds and left triangles denote results projected to Jπ = 2+, 4+, 6+, 3−, 4−, 5−, respectively."
Table 1.
The decomposition of the irreps of the O(3) group with quantum numbers Iπ into irreps of the D3h group. In each column the numbers are the degeneracies."
Iπ | 0+ | 1+ | 2+ | 3+ | 4+ | 5+ | 6+ | 7+ | 0− | 1− | 2− | 3− | 4− | 5− | 6− | 7− |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
$A{{\prime} }_{1}$ | 1 | 0 | 1 | 0 | 1 | 0 | 2 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 |
$A{{\prime} }_{2}$ | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 2 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 |
$E^{\prime} $ | 0 | 0 | 1 | 1 | 2 | 2 | 2 | 2 | 0 | 1 | 1 | 1 | 1 | 2 | 2 | 3 |
A″1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 2 | 1 |
A″2 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 2 |
E″ | 0 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 0 | 0 | 1 | 1 | 2 | 2 | 2 | 2 |
Figure 5.
The mean-field and the projected potential energies of 12C versus the triangular deformation λ3. The squares, circles, down triangles, up triangles, diamonds and left triangles represent Iπ = 0+, 2+, 3−, 4+, 4− and 5−, respectively. The mean-field energies are displayed as a black solid line. The insets show the mean-field density profiles at λ3 = 1.2 and 3.8."
Table 2.
Reduced E2 transition probabilities from states ${I}_{i}^{\pi }$ to states ${I}_{f}^{\pi }$ and the corresponding excitation energies of 12C. The calculation is made for λ3 = 1.2, corresponding to the energy minimum of the 0+ PES shown in figure 5. The experimental B(E2) value is taken from [117]."
${I}_{i}^{\pi }$ | ${E}_{{xi}}^{\mathrm{Th}}$ (MeV) | ${E}_{{xi}}^{\mathrm{Expt}}$(MeV) | ${I}_{f}^{\pi }$ | ${E}_{{xf}}^{\mathrm{Th}}$ (MeV) | $B{\left(E2\right)}^{\mathrm{Th}}$ (e2 fm4) | $B{\left(E2\right)}^{\mathrm{Expt}}$ (e2 fm4) |
---|---|---|---|---|---|---|
2+ | 1.951 | 4.440 | 0+ | 0.000 | 9.72 | 7.6 ± 0.4 [ |
4+ | 9.015 | 14.079 | 2+ | 1.951 | 16.28 | |
4− | 15.253 | 3− | 12.864 | 20.95 | ||
5− | 19.915 | 3− | 12.864 | 6.78 | ||
4− | 15.253 | 18.90 | ||||
${6}_{1}^{+}$ | 26.734 | 4+ | 9.015 | 26.04 | ||
6− | 25.131 | 4− | 15.253 | 13.38 | ||
5− | 19.915 | 15.01 |
Table 3.
Reduced E3 transition probabilities from states ${I}_{i}^{\pi }$ to states ${I}_{f}^{\pi }$ and the corresponding excitation energies of 12C. The calculation is made for λ3 = 1.2, corresponding to the energy minimum of the 0+ PES shown in figure 5. The experimental B(E3) value is taken from [117]."
${I}_{i}^{\pi }$ | ${E}_{{xi}}^{{\rm{Th}}}$(MeV) | ${E}_{{xi}}^{\mathrm{Expt}}$(MeV) | ${I}_{f}^{\pi }$ | ${E}_{{xf}}^{{\rm{Th}}}$(MeV) | $B{\left(E3\right)}^{\mathrm{Th}}$ (e3 fm6) | $B{\left(E3\right)}^{\mathrm{Expt}}$ (e3 fm6) |
---|---|---|---|---|---|---|
3− | 12.864 | 9.641 | 0+ | 0.000 | 145.81 | 103 ± 17 [ |
2+ | 1.951 | 260.25 | ||||
4+ | 9.015 | 53.53 | ||||
4− | 15.253 | 13.35 | 2+ | 1.951 | 268.69 | |
4+ | 9.015 | 198.55 | ||||
5− | 19.915 | 22.4(2) | 2+ | 1.951 | 96.28 | |
4+ | 9.015 | 318.35 | ||||
${6}_{1}^{+}$ | 26.734 | 4− | 15.253 | 33.95 | ||
5− | 19.915 | 129.86 | ||||
6− | 25.131 | 250.19 | ||||
${6}_{2}^{+}$ | 27.234 | 3− | 12.864 | 480.28 | ||
4− | 15.253 | 219.26 | ||||
5− | 19.915 | 90.68 | ||||
6− | 25.131 | 30.99 | ||||
6− | 25.131 | 4+ | 9.015 | 274.90 |
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