1. Introduction
2. Theoretical framework
3. Results and discussion
3.1. Energy levels
Table 1. The predicted levels are compared with the experimental results (taken from [11, 28]). The states of β band are shown in bold type. . |
${E}_{\mathrm{Exp}}$(MeV) | EIBM(MeV) | |
---|---|---|
${0}_{1}^{+}$ | 0.000 | 0.000 |
${2}_{1}^{+}$ | 0.559 | 0.557 |
${0}_{2}^{+}$ | 1.122 | 1.140 |
${2}_{2}^{+}$ | 1.216 | 1.154 |
${4}_{1}^{+}$ | 1.331 | 1.334 |
${3}_{1}^{+}$ | 1.689 | 1.861 |
${{\bf{2}}}_{3}^{+}$ | 1.788 | 1.875 |
${4}_{2}^{+}$ | 2.026 | 2.043 |
${2}_{4}^{+}$ | 2.127 | 2.029 |
${6}_{1}^{+}$ | 2.263 | 2.350 |
${4}_{3}^{+}$ | 2.485 | 2.364 |
${5}_{1}^{+}$ | 2.489 | 2.566 |
${2}_{5}^{+}$ | 2.515 | 2.383 |
${{\bf{4}}}_{4}^{+}$ | 2.618 | 2.648 |
${2}_{6}^{+}$ | 2.869 | 2.594 |
${1}_{1}^{+}$ | 2.950 | 2.957 |
${2}_{7}^{+}$ | 3.008 | 2.797 |
Table 2. The energy ratios R4/2 and R6/2 in the three dynamical limits are compared to the experimental and calculated values for 76Se. The three dynamical limits are taken from [30, 42]. |
${R}_{4/2}={E}_{{4}_{1}^{+}}/{E}_{{2}_{1}^{+}}$ | ${R}_{6/2}={E}_{{6}_{1}^{+}}/{E}_{{2}_{1}^{+}}$ | |
---|---|---|
U(5) | 2.0 | 3.0 |
SU(3) | 3.3 | 7.0 |
O(6) | 2.5 | 4.5 |
Exp. | 2.381 | 4.048 |
IBM-2 | 2.395 | 4.219 |
3.2. E2 transition
Table 3. Comparison between experimental and calculated E2 transition strengths (in W.u.) and M1 transition strengths (in ${\mu }_{N}^{2}$) for 76Se nucleus. The experimental data are taken from [11, 28]. |
${J}_{i}^{\pi }\,\to $ ${J}_{f}^{\pi }$ | B(E2) | B(M1) | ||
---|---|---|---|---|
Exp. | IBM-2 | Exp. | IBM-2 | |
${{\bf{0}}}_{2}^{+}\to {{\bf{2}}}_{1}^{+}$ | 47(22) | 42.40 | ||
${1}_{1}^{+}\to {0}_{1}^{+}$ | 0.010(1) | 0.010 | ||
${1}_{1}^{+}\to {2}_{1}^{+}$ | ${0.04}_{1}^{5}$ | 1.86 | ${0.010}_{10}^{1}$ | 0.0002 |
${2}_{1}^{+}\to {0}_{1}^{+}$ | 44(1) | 44.0 | ||
${2}_{2}^{+}\to {0}_{1}^{+}$ | 1.21(10) | 0.026 | ||
${{\bf{2}}}_{2}^{+}\to {{\bf{2}}}_{1}^{+}$ | 43(3) | 47.05 | 0.000 92(9) | 0.0028 |
${2}_{3}^{+}\to {0}_{1}^{+}$ | 0.2(1) | 0.20 | ||
${{\bf{2}}}_{3}^{+}\to {{\bf{0}}}_{2}^{+}$ | 37(10) | 17.26 | ||
${2}_{3}^{+}\to {2}_{1}^{+}$ | 1.1(3) | 0.00 | 0.008(3) | 0.0051 |
${{\bf{2}}}_{3}^{+}\to {{\bf{2}}}_{2}^{+}$ | ${{\bf{4.5}}}_{16}^{35}$ | 6.38 | ${{\bf{0.009}}}_{5}^{4}$ | 0.0129 |
${{\bf{2}}}_{3}^{+}\to {{\bf{4}}}_{1}^{+}$ | 23(6) | 11.40 | ||
${2}_{5}^{+}\to {0}_{2}^{+}$ | 0.06(1) | 0.46 | ||
${2}_{5}^{+}\to {2}_{1}^{+}$ | 0.07(2) | 0.26 | 0.0010(3) | 0.0409 |
${2}_{5}^{+}\to {2}_{3}^{+}$ | 4(1) | 14.64 | 0.05(1) | 0.0091 |
${2}_{5}^{+}\to {3}_{1}^{+}$ | ${1}_{1}^{9}$ | 4.74 | 0.0001(1) | 0.0171 |
${2}_{6}^{+}\to {0}_{1}^{+}$ | 0.23(2) | 0.08 | ||
${2}_{6}^{+}\to {2}_{1}^{+}$ | 0.68(8) | 0.31 | 0.018(2) | 0.0029 |
${2}_{6}^{+}\to {2}_{2}^{+}$ | 1.1(1)a, $4.8{(}_{28}^{14}$)b | 0.0005 | 0.027(3)a, $0.014{(}_{4}^{10})$ b | 0.0148 |
${2}_{7}^{+}\to {0}_{1}^{+}$ | 0.20(1) | 0.12 | ||
${2}_{7}^{+}\to {2}_{1}^{+}$ | 0.27(2) | 0.49 | 0.083(8) | 0.0090 |
${2}_{7}^{+}\to {2}_{2}^{+}$ | 0.22(3)c, $5{(}_{3}^{88})$ d | 0.13 | 0.021(3)c, 0.001(1)d | 0.0169 |
${3}_{1}^{+}\to {2}_{1}^{+}$ | 2.8(8) | 0.37 | 0.0008(2) | 0.0011 |
${{\bf{3}}}_{1}^{+}\to {{\bf{2}}}_{2}^{+}$ | 18(5)e, ${\bf{1}}23{(}_{54}^{64})$ f | 38.22 | 0.03(1)e, 0.001(1)f | 0.0169 |
${{\bf{3}}}_{1}^{+}\to {{\bf{4}}}_{1}^{+}$ | ${\bf{1}}8{(}_{8}^{35})$ | 13.47 | ${\bf{0.005}}{(}_{5}^{2})$ | 0.0147 |
${{\bf{4}}}_{1}^{+}\to {{\bf{2}}}_{1}^{+}$ | 71(2) | 63.35 | ||
${4}_{2}^{+}\to {2}_{1}^{+}$ | 0.08(2) | 0.098 | ||
${{\bf{4}}}_{2}^{+}\to {{\bf{2}}}_{2}^{+}$ | ${\bf{3}}{1}_{6}^{9}$ | 29.50 | ||
${{\bf{4}}}_{2}^{+}\to {{\bf{4}}}_{1}^{+}$ | ${\bf{2}}8{(}_{12}^{26})$ g, ${\bf{3.7}}{(}_{9}^{14})$ h | 21.92 | 0.003(2)g, ${\bf{0.018}}{(}_{4}^{7})$ h | 0.0153 |
${4}_{3}^{+}\to {2}_{2}^{+}$ | 4.0(6) | 0.27 | ||
${4}_{3}^{+}\to {3}_{1}^{+}$ | ${1.3}_{2}^{3}$ | 33.67 | 0.027(5) | 0.055 |
${4}_{3}^{+}\to {4}_{1}^{+}$ | 2.0(3) | 0.11 | ${0.028}_{4}^{5}$ | 0.028 |
${{\bf{4}}}_{4}^{+}\to {{\bf{2}}}_{3}^{+}$ | 31(5) | 10.01 | ||
${{\bf{5}}}_{1}^{+}\to {{\bf{3}}}_{1}^{+}$ | 67(23) | 33.40 | ||
${5}_{1}^{+}\to {4}_{1}^{+}$ | 4.7(17) | 0.77 | 0.0006(4) | 0.0043 |
${{\bf{6}}}_{1}^{+}\to {{\bf{4}}}_{1}^{+}$ | 68(8) | 61.39 |
a The corresponding multipole mixing ratio ${\delta }_{-}^{+}=+{0.38}_{12}^{14}$. | |
b The corresponding multipole mixing ratio ${\delta }_{-}^{+}=+{1.1}_{8}^{3}$. | |
c The corresponding multipole mixing ratio ${\delta }_{-}^{+}=-0.21(19)$. | |
d The corresponding multipole mixing ratio ${\delta }_{-}^{+}=+{5}_{2}^{58}$. | |
e The corresponding multipole mixing ratio ${\delta }_{-}^{+}=+0.41(5)$. | |
f The corresponding multipole mixing ratio ${\delta }_{-}^{+}=+5.4(9)$. | |
g The corresponding multipole mixing ratio ${\delta }_{-}^{+}=+{2.4}_{6}^{10}$. | |
h The corresponding multipole mixing ratio ${\delta }_{-}^{+}=-0.36(15)$. |
Table 4. Comparison of some B(E2) branching ratios in the three dynamical limits and the experimental and calculated values for 76Se. The values of three dynamical limits are taken from [33, 42, 46, 52, 55–57]. |
$\tfrac{B(E2,{4}_{1}^{+}\to {2}_{1}^{+})}{B(E2,{2}_{1}^{+}\to {0}_{1}^{+})}$ | $\tfrac{B(E2,{2}_{2}^{+}\to {2}_{1}^{+})}{B(E2,{2}_{1}^{+}\to {0}_{1}^{+})}$ | $\tfrac{B(E2,{0}_{2}^{+}\to {2}_{1}^{+})}{B(E2,{2}_{1}^{+}\to {0}_{1}^{+})}$ | $\tfrac{B(E2,{2}_{2}^{+}\to {0}_{1}^{+})}{B(E2,{2}_{2}^{+}\to {2}_{1}^{+})}$ | $\tfrac{B(E2,{6}_{1}^{+}\to {4}_{1}^{+})}{B(E2,{2}_{1}^{+}\to {0}_{1}^{+})}$ | $\tfrac{B(E2,{4}_{1}^{+}\to {2}_{1}^{+})}{B(E2,{2}_{2}^{+}\to {2}_{1}^{+})}$ | |
---|---|---|---|---|---|---|
U(5) | 2.0 | 2 | 2 | 0.011 | 3.0 | 1.0 |
SU(3) | 10/7 | 0.0 | 0 | 0.70 | 1.57 | 6.93 |
O(6) | 10/7 | 10/7 | 0 | 0.07 | 1.67 | 1.84 |
Exp. | 1.614 | 0.977 | 1.07 | 0.028 | 1.545 | 1.65 |
IBM-2 | 1.440 | 1.069 | 0.964 | 0.0006 | 1.395 | 1.346 |
$\tfrac{B(E2,{4}_{2}^{+}\to {4}_{1}^{+})}{B(E2,{4}_{2}^{+}\to {2}_{2}^{+})}$ | ||||||
U(5) | 0.72 | |||||
SU(3) | 0.03 | |||||
O(6) | 0.75 | |||||
Exp. | 0.90a, 0.12b | |||||
IBM-2 | 0.743 |
a The corresponding multipole mixing ratio ${\delta }_{-}^{+}=+{2.4}_{6}^{10}$. | |
b The corresponding multipole mixing ratio ${\delta }_{-}^{+}=-0.36(15)$. |