MQPM nuclear matrix elements

Finally, the results obtained from the BCS and QRPA calculations were brought together by performing the MQPM calculations. In the MQPM calculations, the parameters g_ph obtained from the QRPA calculations were used in order to adapt the phonon energies to the experimental ones. The hole-hole value g_hh = 1.0 was applied throughout the calculations. Then, the amount of the phonons coupled to the quasiparticle states was regulated by changing the maximumJ value of the phonons and the QRPA cutoff energy E_{cutof f}.

At first, only the lowest 2⁺ phonon was taken into account. Then, the number of the 2⁺ phonons was increased by increasing the cutoff energy until the resulting energy values of the two experimentally lowest states E_9/2⁺ and E_1/2⁻ were converged. The convergence of the corresponding reduced M4 matrix element was also required.

After the convergence, the number of different phonon types was increased by adding the natural-parity 3⁻,4⁺,5⁻,6⁺,7⁻ and 0⁺ phonons one type at a time. For every phonon, the convergence was required in a similar way as for 2⁺ phonons before adding more phonon types. In the case of 4⁺ phonons the lowest QRPA 4⁺ phonon was fitted to the second 4⁺ excitation of the even-even reference nucleus, since the lowest 4⁺ excitations belongs to the two-phonon triplet that does not belong to the model space of the MQPM calculations.

After adding the above mentioned natural-parity phonons to the calculations, also the unnatural-parity phonons 1⁺,2⁻,3⁺,4⁻,5⁺ and 6⁻ were added one phonon type at a time. For some states, the QRPA calculations gave too high energies, in which case the value g_ph = 1.0 was applied.

The results for the reduced M4 matrix elements and energies of the states 9/2⁺ and 1/2⁻ for an example nucleus ¹⁰⁹In are shown in Figures 8–11 as functions of the cutoff energy.

5 10 15 20 1,900

2,000 2,100 2,200 2,300

E_{cutof f}(MeV) M4(µN cfm3 )

QP 2⁺&3⁻

2+

3−

2⁺,3⁻&4⁺ 2⁺,3⁻,4⁺&5⁻ 2⁺,3⁻,4⁺,5⁻&6⁺ 2⁺,3⁻,4⁺,5⁻,6⁺&7⁻ 2⁺,3⁻,4⁺,5⁻,6⁺,7⁻&0⁺

Added phonons:

Figure 8: The M4 reduced matrix elements of¹⁰⁹In obtained from the QRPA calculations as functions of the QRPA cutoff energy. The different natural-parity phonons were added one-by-one in the order 2⁺,3⁻,4⁺,5⁻,6⁺,7⁻ and 0⁺. The results are shown for each case separately. Also, the corresponding single-quasiparticle matrix element is shown as a reference.

5 10 15 20 2,000

2,100 2,200 2,300

Ecutof f(MeV) M4(µN cfm3 )

QP

Natural parity phonons 1⁺

1⁺&2⁻ 1⁺,2⁻&3⁺ 1⁺,2⁻,3⁺&4⁻ 1⁺,2⁻,3⁺,4⁻&5⁺ 1⁺,2⁻,3⁺,4⁻,5⁺&6⁻ Added phonons:

Figure 9: The M4 reduced matrix elements of ¹⁰⁹In obtained from the QRPA cal-culations after adding the unnatural-parity phonons as functions of the QRPA cutoff energy. The different unnatural parity phonon types were added one-by-one in the or-der 1⁺,2⁻,3⁺,4⁻,5⁺ and 6⁻. The results are shown for each case separately. Also, the corresponding single-quasiparticle matrix element is shown as a reference.

5 10 15 20

1.32 1.34 1.36 1.38 1.4 1.42

Ecutof f(MeV) E9/2+(MeV)

2⁺ 2⁺,3⁻,4⁺ 2⁺,3⁻,4⁺,5⁻ 2⁺,3⁻,4⁺,5⁻,6⁺,7⁻,0⁺ 2⁺,3⁻,4⁺,5⁻,6⁺,7⁻,0⁺,1⁺,2⁻,3⁺ 2⁺,3⁻,4⁺,5⁻,6⁺,7⁻,0⁺,1⁺,2⁻,3⁺,4⁻,5⁺,6⁻

Added phonons:

Figure 10: The energies of the state 9/2⁺ of¹⁰⁹In obtained from the QRPA calculations as functions of the QRPA cutoff energy. The different natural-parity phonons were added one-by-one in the order 2⁺,3⁻,4⁺,5⁻,6⁺,7⁻ and 0⁺. The results are shown only for the cases where there happened significant changes.

5 10 15 20 2.16

2.18 2.2 2.22 2.24 2.26

Ecutof f(MeV) E1/2−(MeV)

2⁺ 2⁺,3⁻,4⁺ 2⁺,3⁻,4⁺,5⁻ 2⁺,3⁻,4⁺,5⁻,6⁺,7⁻,0⁺ 2⁺,3⁻,4⁺,5⁻,6⁺,7⁻,0⁺,1⁺,2⁻,3⁺ 2⁺,3⁻,4⁺,5⁻,6⁺,7⁻,0⁺,1⁺,2⁻,3⁺,4⁻,5⁺,6⁻

Added phonons:

Figure 11: The energies of the state 1/2⁻ of¹⁰⁹In obtained from the QRPA calculations as functions of the QRPA cutoff energy. The different natural-parity phonons were added one-by-one in the order 2⁺,3⁻,4⁺,5⁻,6⁺,7⁻ and 0⁺. The results are shown only for the cases where there happened significant changes.

As can be seen from Fig. 8 and 9, the reduced M4 matrix element converges, when E_{cutof f} &15 MeV. At ∼6 MeV there seems to be some quantum leap resulted from some important 2⁺ state coming along. Furthermore, the reduced matrix element seems to decrease as the amount of natural-parity phonons increases. After all, the reduced matrix element value of ¹⁰⁹In seems to converge to about 1969 ^µ_c^Nfm³ including only the natural-parity phonons. The value increases when the unnatural-natural-parity phonons are added, and finally it converges to about 2075 ^µ_c^Nfm³.

As can be seen from Fig. 10 and 11, the energies of the lowest two states 9/2⁺ and 1/2⁻ behave similarly. The energies decrease as the natural-parity phonons are added and slightly increase when also the unnatural-parity phonons are added. Finally, the energy of the state 9/2⁺ converges to ∼1.340 MeV and 1/2⁻ to∼2.185 MeV.

In Figures 12 and 13 the value of the reduced matrix element and the energies of the lowest two states are plotted against the number of different multipolarities added. The results are shown for the cutoff energy E_{cutof f} = 15 MeV. There seems to be significant leaps in the value of the reduced matrix element when the natural-parity multipolarities 2⁺,3⁻and 5⁻and the unnatural-parity multipolarity 4⁻ are added. Those states are collective states and have a lot of different configurations, which may explain the leaps. The multipolarities with few configurations do not affect the value of the reduced matrix element as much.

The energies of the lowest two states are more stable against adding phonons. Small leaps occur also in the energy curves at the points where the multipolarities 2⁺,3⁻,5⁻ and 4⁻ are added, but the leaps are not significant in the energy scale.

0 2 4 6 8 10 12 14

2,000 2,100 2,200 2,300

Number of different phonon types M4(µN cfm3)

Figure 12: The reduced M4 matrix element of ¹⁰⁹In as a function of the amount of different phonons added. The phonons were added one-by-one in the order 2⁺,3⁻,4⁺,5⁻,6⁺,7⁻,0⁺,1⁺,2⁻,3⁺,4⁻,5⁺ and 6⁻. The cutoff energy was E_{cutof f} = 15 MeV.

0 2 4 6 8 10 12 14

Number of different phonon types

E(MeV)

1/2⁻ 9/2⁺

Figure 13: The energies of the lowest two states of ¹⁰⁹In as functions of the amount of different phonons added. The phonons were added one-by-one in the order 2⁺,3⁻,4⁺,5⁻,6⁺,7⁻,0⁺,1⁺,2⁻,3⁺,4⁻,5⁺ and 6⁻. The cutoff energy was E_{cutof f} = 15 MeV.

For the lighter nuclei, the MQPM theory failed to predict the states in the right order, even though the BCS theory did. In order to resolve the problem, the single-particle energy of the proton/neutron orbital 0g9/2 was lowered so much that the lowest states of nuclei were predicted in the right order, see Table 14. For the heavier nuclei, the MQPM theory predicted the states in the correct order, when the single-particle basis with the energy of the 0h11/2 lowered by 1.5 MeV was used. The adjusted MQPM spectra compared to the quasiparticle and experimental spectra for example nuclei ¹¹¹In and ¹³⁹Ce are shown in Figures 14 and 15.

0.0

Figure 14: The obtained MQPM spectrum of ¹¹¹In compared to the experimental and quasiparticle spectra.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 E(MeV)

139Ce

0.000 3/2⁺

0.2551 1/2⁺

0.7542 11/2⁻

Exp.

0.000 3/2⁺

0.219 1/2⁺

0.503 11/2⁻

MQPM 0.000

3/2⁺

0.18233 1/2⁺

0.53484 11/2⁻

QP

Figure 15: The obtained MQPM spectrum of ¹³⁹Ce compared to the experimental and quasiparticle spectra.

Figure 14 shows that the BCS theory predicts the order of the lowest states of the nuclei better than the MQPM theory. Anyhow, the MQPM theory predicts the energies of the lowest two states closer to the experimental ones. From Figure 15 one can see that the MQPM theory predicts the lowest three states of ¹³⁹Ce better than the BCS theory. The MQPM states are, however, little lower than the experimental ones. The final results for the reduced M4 matrix elements, where the corrected single-particle energies are used, are shown in Tables 14 and 15.

Table 14: The NMEs for M4 transitions of nuclei in the mass region of A=85-115. p/n stands for the odd-proton/odd-neutron transition. |M_exp.|, |M_qp| and |M_{M QP M}| are the experimental, single-quasiparticle and MQPM NMEs in units of ^µ_c^Nfm³. ∆E_9/2+ refers to the amount of how much the energy of the single-particle energy of the proton/neutron orbital 0g_9/2 was lowered in units of MeV. The QRPA NMEs are presented for the cases where only the parity phonons were added and for the case where both natural-and unnatural-parity phonons were added.

Nucleus Transition p/n ∆E_9/2+ |M_exp.| |M_qp| |MM QP M,nat.| |MM QP M,nat.+unnat.|

85Kr 1p_1/2 −→0g_9/2 n 0 528 1572 1295 1354

2 528 1572 1413 1442

89Y 0g_9/2 −→1p_1/2 p 0 739 2117 1814 1865

0.9 739 2117 1992 2024

89Zr 1p_1/2 −→0g_9/2 n 0 559 1598 1326 1307

2 559 1598 1452 1474

91Y 0g_9/2 −→1p_1/2 p 0 480 2134 1584 1694

0.5 480 2134 1744 1830

97Nb 1p_1/2 −→0g_9/2 p 0 419 2122 -

-105In 1p_1/2 −→0g_9/2 p 0 706 2382 1950 2029

0.7 706 2382 2045 2094

107In 1p_1/2 −→0g_9/2 p 0 670 2404 2040 2129

1 670 2404 2129 2129

109In 1p_1/2 −→0g_9/2 p 0 640 2328 1959 2055

0.75 640 2328 2053 2101

111In 1p_1/2 −→0g_9/2 p 0 609 2446 2079 2090

0.75 609 2446 1993 2031

113In 1p_1/2 −→0g_9/2 p 0 603 2464 1970 2086

0.75 603 2464 1967 2053

115In 1p_1/2 −→0g_9/2 p 0 614 2482 1890 1944

0.75 614 2482 1990 2026

Table 15: The NMEs for M4 transitions of nuclei in the mass region of A=135-143.

p/n stands for the odd-proton/odd-neutron transition. |M_exp.|, |M_qp| and |M_{M QP M}| are the experimental, single-quasiparticle and MQPM NMEs in units of ^µ_c^Nfm³. The fourth column corresponds to the different cases, for which the energy of the single-neutron orbital 0h_11/2 was subtracted by 0, 1 and 1.5 MeV. The QRPA NMEs are presented for the cases where only the natural-parity phonons were added and for the case where both natural- and unnatural-parity phonons were added.

Nucleus Transition p/n ∆E_11/2−(MeV) |Mexp.| |Mqp| |MM QP M,nat.| |MM QP M,nat.+unnat.|

135Xe 0h_11/2 −→1d_3/2 n 0 1110 3148 3020

1 1110 3167 3094

1.5 1110 3120 2812 2869

137Ba 0h_11/2 −→1d_3/2 n 0 1028 3172 3009

1 1028 3184 3089

1.5 1028 3134 2742 2773

139Ce 0h_11/2 −→1d_3/2 n 0 968 3198 2992

1 968 3202 3078

1.5 968 3117 2784 2822

141Nd 0h_11/2 −→1d_3/2 n 0 893 3220 3035

1 893 3217 3098

1.5 893 3167 2777 2789

143Sm 0h_11/2 −→1d_3/2 n 0 871 3245 2998

1 871 3237 3071

1.5 871 3194 2751 2806

Table 16: The ratios of the experimental reduced matrix elements |M_exp.| to the corre-sponding quasiparticle and MQPM reduced matrix elements |M_qp| and |M_{M QP M}|. The quasiparticle values and the MQPM values correspond to the ones with adjusted SP energies and the MQPM values include both natural- and unnatural-parity phonons.

Nucleus |M_exp.|/|M_qp| |M_exp.|/|M_{M QP M}|

85Kr 0.34 0.37

89Y 0.35 0.37

89Zr 0.35 0.38

91Y 0.22 0.26

105In 0.30 0.34

107In 0.28 0.31

109In 0.27 0.30

111In 0.25 0.30

113In 0.24 0.29

115In 0.25 0.30

135Xe 0.36 0.39

137Ba 0.33 0.37

139Ce 0.31 0.34

141Nd 0.28 0.32

143Sm 0.27 0.31

In document The M4 transitions of isomeric states (sivua 43-52)