Mixed Symmetry States in Even-Even 96;108 Mo Nuclei

Commun. Theor. Phys. (Beijing, China) 37 (2002) pp 335{340 c International Academic Publishers Vol. 37, No. 3, March 15, 2002 Mixed Symmetry States in Even-Even 96;108 Mo Nuclei ZHANG Jin-Fu,1;2 AL- Khudair H.F.,1 LONG Gui-Lu,1;3;6 ZHU Sheng-Jiang,1;5 and RUAN Dong1;5 1 Department of Physics, Tsinghua University, Beijing 100084, China 2 Department of Physics, Chifeng Teacher's College for Nationalities, Chifeng 024001, Inner Mongolia, China 3 Institute of Theoretical Physics, Academia Sinica, Beijing 100080, China 4 Center of Nuclear Theory, Lanzhou Heavy Ion Accelerator National Laboratory, Lanzhou 730000, China 5 Key Laboratory for Quantum Information and Measurements, MOE, Beijing 100084, China 6 Center of Atomic and Molecular Nanosciences, Tsinghua University, Beijing 100084, China (Received December 6, 2000; Revised June 25, 2001) Abstract Excitation energies and electromagnetic transition strengths in even-even 96;108 Mo nuclei have been described systematically by using the proton-neutron interacting boson model (IBM-2). It appears that the properties of low-lying levels in these isotopes, for which the comparison between experiment and theory is possible, can be satisfactorily described by the IBM-2 model, provided proper account is taken of the presence at low energy of states having a mixed-symmetry character. It seems possible to identify, in each isotope, a few states having such a character, the lowest ones being either 2+2 or 2+3 levels. It is found that these nuclei are in the transition from U(5) to SU(3). PACS numbers: 21.60.Fw, 21.60.Ev, 23.20.Js, 27.60.+j Key words: IBM-2, even-even Mo, excitation energies, electromagnetic transition, mixed symmetry states 1 Introduction tions to symmetric states with matrix elements of about provided a challenge to theoretical interpretations, for instance, the very rapid transition from a vibrational to a rotational-like structure and the unusually low rst excited 0+ state connected by strong E 2 transitions to the rst 2+ states. The structure of Mo nuclei was studied by applying a variety of models, such as the framework of the Hartree{Fock{Bogoliubov method,[1] algebraic interacting boson model (IBM),[2] and the shell model.[3;4] These studies have concluded several important points. First, this rotational structure gradually develops, as neutrons are added. Second, the behavior of 0+2 states is closely related to the proton cross-shell excitations. Recently, an investigation[5;6] of the mixed-symmetry (MS) states of the 94 Mo nucleus has been provided within the proton-neutron interacting boson model (IBM-2)[7;11] which distinguish between proton and neutron degrees of freedom. For instance, the staggering problem in the O(6) limit of the IBM1 can be solved by introducing the quadrupole interactions among like-nucleons in the IBM2.[12;15] In particular, the IBM-2 predicts the existence of mixed-symmetry states, i.e. not completely symmetric states with respect to the proton-neutron boson exchange. Signatures of MS state, accessible to spectroscopy, are low excitation energy, weakly collective E 2 transitions to symmetric states, and strong M 1 transi- symmetry state is the 1 state known as the \scissors mode" due to its geometrical picture in rotors.[17] This 1+ state was discovered by A. Richter et al.[18] in electron scattering experiments on well-deformed nuclei. The existence of the MS scissors mode has been systematically investigated by resonant photon scattering in the 100 < A < 200 mass region.[19] Meanwhile, the excitation energy of these J = 1+ levels provides a way of determining the strength of the Majorana force.[20] It was suggested recently that mixed symmetry states may form isomeric states under certain conditions.[21] Mixed-symmetry 2+ and 3+ states have been identi ed recently in 94 Mo[5;6] by measuring additionally the B (M 1)-strength. Detailed IBM2 calculation of the structure of 94 Mo was carried out.[22] It is therefore interesting to carry out a systematic comparison of the experimental data with model calculations in Mo isotopes, in particular the properties of MS states in these nuclei in the light of new experimental data accumulated over the past few years. These MS states in Mo isotopes are absent except in 94 Mo. After this short introduction, we describe brie y the model Hamiltonian, the E 2 and M 1 transition operators in Sec. 2. In Sec. 3, we give the results and discussions on spectrum, E 2 and M 1 transition properties and mixed symmetry states. Finally, in Sec. 4, a summary is given. f i most prominent mixedThe structure of Mo isotope nuclei has for many years jhJsym kM 1kJms ij  N .+ The [16]  The project supported in part by National Natural Science Foundation of China under Grant No. 10047001, Excellent Young University Teacher's Fund of the Chinese Education Ministry, the Fok Ying Tung Education Foundation, Major State Basic Research Development Program under contract No. G200077400 and the Key Scienti c Research Fund of Inner Mongolian Educational Bureau under Grant No. ZD-01038 336 ZHANG Jin-Fu, AL- Khudair H.F., LONG Gui-Lu, ZHU Sheng-Jiang, and RUAN Dong 2 The Interacting Proton-Neutron Model The microscopic picture of the IBM is very complicated.[23] A commonly used microscopic picture is given in terms of collective pairs of nucleons.[9;23;25] The S and D pairs of valence nucleons have angular momenta J = 0 and J = 2, respectively. These pairs correspond intuitively to the s and d bosons, respectively. The building blocks of the IBM-2 are the proton bosons s , d and the neutron bosons s , d . For the analysis of excitation energies in Mo isotopes we tried to keep to a minimum number of free parameters in the Hamiltonian. We thus considered equal values for the neutron and proton d-boson excitation energy ", in addition to the standard quadrupole interaction and Majorana term. We only considered the dipole neutron-proton boson interaction whose strength is characterized by a single parameter W . The explicit expression of the Hamiltonian adopted in the calculations is H^ = "(^nd + n^ d ) + K Q^ 
Q^  + K Q^ 
Q^  + w L^ 
L^  + M^  ; where the indexes  and  refer to neutron and proton bosons respectively, and M^  is the Majorana term. Moreover, n^ d = (d^+
d^~ ) ; Q^  = [d^+  s~^ + s^+  d^~ ](2) +  [d^+  d^~ ](2) ; p L^  = 10 [d^+  d~^ ](1) ;  = ;  ;  M^  = 12 2 [^s+  d^+ ; s^+  d^+ ](2)
[ s~^  d^~ ; s~^  d^~ ](2) ; k [d^+  d^+ ](k)
[ d^~  d^~ ](k) : X k=1;3 The E 2 transition strengths were calculated by using the operator T^(E 2) = e Q^  + e Q^  ; where e and e are boson e ective charges. The M 1 operator is given by Vol. 37 r3 ^ ^ 4 (g L + g L ) ; where g and g are the proton and neutron boson gfactors, respectively. The numerical diagonalization has been carried out by using the computer code NPBOS of Otsuka. T^(M 1) = 3 Results and Discussions 3.1 Energy Spectra and Electromagnetic Transition Rates As a starting point we used parameters extrapolated from the IBM-2 calculation of Ru and Pd isotopes,[26;27] the tted values of the parameters are given in Table 1. These parameters are consistent with those of previous calculations.[2] In general, the values of " and K decrease with increasing mass number. But the unusual behavior is that of the quantity " which presents a maximum at neutron number 56(98 Mo) and a minimum at neutron number 64(106 Mo). The unusual behaviors are related to the presence of another subshell closure at 56 caused by the lling of the 2d5=2 neutron orbital and in the middle of the N = 50  82 shell at 64. The 96;98 Mo nuclei are vibrational. This is also true in our calculation by the relatively large " value. In comparison with 96;98 Mo, the " values in 104;108 Mo have a big drop. This makes 104;108 Mo more close to the rotational limit. The value of  increases almost linearly with increasing mass number. The value of w is adjustable to put the 4+1 and 2+2 energy right for all the isotopes. In 96;98 Mo isotopes which have just two and three neutron bosons respectively so that the quadruple proton-proton interaction is very important to t the energy levels in these isotopes. Since there is only scarce knowledge of the excitation energies of the mixed symmetry 1+ and 2+ states, the values of the Majorana parameters were chosen to adjustable the available experimental data of 2+2 ; 3+1 ; 2+3 and 1+1 states. Table 1 IBM-2 parameters for even-even Mo isotopes. In these calculations we use  = ;1:32. All the Hamiltonian's parameters in MeV unit except  , the e , e in efm2 unit and g , g in N unit. Nucleus 96 Mo 98 Mo 100 Mo 102 Mo 104 Mo 106 Mo 108 Mo " K K  w ;0:03 ;0:025 ;0:65 ;0:50 ;0:40 ;0:35 ;0:15 2 0:870 0:950 0:690 0:610 0:520 0:500 0:540 ;0:053 ;0:060 ;0:038 ;0:064 ;0:066 ;0:072 ;0:075 1 0:036 0:021 0:016 0:012 0:024 0:019 0:017 0:45 0:08 0:16 0:10 0:08 0:10 0:10 0:19 0:07 0:16 0:05 0:12 0:13 0:13 0:00 0:00 0:00 0:00 0:00 0:00 0:25 3 ;0:24 0:05 0:10 0:10 0:08 0:10 0:10 e e g g 10 10 13 13 5 4 6 9 0:45 0:50 0:25 0:25 No. 3 Mixed Symmetry States in Even-Even 96;108 Mo Nuclei The calculated energy levels for 96;108 Mo as well as the experimental ones[28;33] are shown in Figs 1  7. The agreement appears to be satisfactory. The ground state band is well reproduced in all investigated nuclei, especially for the heavier ones. However, the calculated yrast 6+ and 8+ excitation energies are slightly higher than the experimental ones, which is a general feature of this type of model. The Mo isotopes are in the transition from the vibrational limit to the rotational limit. 337 states are caused by two-proton excitations across the subshell at Z = 40. Fig. 3 Spectrum for 100 Mo. Fig. 1 Spectrum for 96 Mo. Fig. 4 Spectrum for 102 Mo. Fig. 2 Spectrum for 98 Mo. It is found that the calculated energies of 0+2 are larger than those of the experimental 0+2 sates in lighter nuclei, with the exception of 102 Mo; in heavier nuclei the calculated 0+2 state energy is smaller than the experimental data. We suggest that these states are intruder states, as has been pointed out by M. Sambataro et al.[2] These As shown in the previous section, allowance for the presence of MS states at low energy enabled us to satisfactorily reproduce the excitation energies. A much higher degree of con dence in the interpretation proposed in the paper can only be obtained by a comparison of predicted and experimental data on electromagnetic properties. The electric quadrupole and magnetic dipole transition rates have been calculated by using these parameters in Table 1. In Tables 2 and 3 the experimental B (E 2) and B (M 1) values are compared with the theoretical ones, re- 338 ZHANG Jin-Fu, AL- Khudair H.F., LONG Gui-Lu, ZHU Sheng-Jiang, and RUAN Dong spectively. There is a good agreement for the Mo isotopes. Vol. 37 The main signatures are again most clearly described in U(5) and SU(3) transition cases of the model. The E 2 transition between fully symmetric (FS) states has analogy with IBM-1, i.e. B (E 2; 2+1 ! 0+1 ) / (e N + e N )2 :[26;35;36] On the contrary, the E 2 transition probabilities between FS and MS states are as a rule proportional to a combination of e and e in (e ; e )2 or (e  ; e v )2 ,[35] depending on the nature of the states. M 1 transition probabilities between FS and MS states are proportional to (g ; g )2 .[26;35;41] A detailed calculation of electromagnetic transitions is given below. Table 2 Experimental[28;34] and calculated values for the B (E 2) in Mo nuclei. Fig. 5 Spectrum for 104 Mo. Nucleus Ji Jf Expt ( e 2 fm4 ) Calc ( e 2 fm4 ) 96 Mo 2+1 2+2 2+2 4+1 4+2 4+2 0+1 0+1 2+1 2+1 2+1 2+2 540:5(7:8) 31:1(2:6) 470:0(78) 1044:4(209) 49:6(15:7) 600:5(183) 543:8 0:2 698:3 905:1 1:7 125:7 536:8(10:7) 563:6(53:7) 25:8(1:9) 64:4(21:5) 1:1(1) 214:7(187:9) 1180:9(107) 161:0(134) 1234:6(134) 536:2 153:0 5:9 139:6 41:3 4:2 594:6 99:1 914:9 937:4(55) 17:1(1:4) 151:6(22) 386(110) 2591:7(110) 1406(137:8) 30:9(2:2) 661:7(220:6) 1902:4(110) 827:1(165) 992:6(496:3) 772(165) 2591:7(386) 3391:2(496:3) 950:2 1:3 9:9 82:6 1509:3 1330:9 11:1 1:6 1659:1 1006:8 11:87 756:1 2091:6 2256:3 2094:8(255) 1981:6(849) 2519:5(510) 2048:3 2119:1 3379:1 98 Mo Fig. 6 Spectrum for 106 Mo. 100 Mo 102 Mo Fig. 7 Spectrum for 108 Mo. 2+1 2+1 2+3 2+3 2+2 2+2 2+3 2+2 4+1 2+1 2+2 2+2 2+3 0+2 2+2 2+3 2+3 4+1 4+2 2+3 4+2 6+1 8+1 2+1 0+2 4+1 0+1 0+2 0+1 0+2 0+1 0+2 2+1 2+1 2+1 0+1 0+1 0+2 0+2 2+1 2+1 2+1 2+2 2+1 2+2 4+1 4+1 4+1 6+1 0+1 2+1 2+1 Mixed Symmetry States in Even-Even 96;108 Mo Nuclei No. 3 Table 3 Experimental[28;34] and calculated values for the B (M 1) in Mo nuclei. Nucleus Ji Jf 98 Mo 2+2 2+3 2+1 2+1 100 Mo 3.2 2+2 2+3 2+3 4+2 2+1 2+1 2+2 4+1 Expt (2N ) Calc (2N ) 0:0119(2) 0:0179(4) 0:0144 0:0006 0:0014(1) 0:006(1) 0:0075(2) 0:021(3) 0:0003 0:0226 0:0004 0:0012 Mixed Symmetry States One of the important feats of the IBM-2 is the prediction of the mixed symmetry states[9] while states with F = Fmax are completely symmetric with respect to the exchange of any bosons, states with F < Fmax contain also those antisymmetric boson pairs with mixed symmetry. The 1+ states have the peculiar feature of being practically F -spin pure. Since for F = Fmax there is no 1+ state, there is no symmetric state they could mix with. This makes the scissors mode relatively accessible for experiment. This MS state has been observed in many deformed nuclei. In more vibrational nuclei, we expect the lowest MS states with J = 2+ . An important quantity which can indicate the F -spin nature of the state is the ratio R given by In the isotopes 96;98 Mo, which have a structure close to the U(5) limit, further evidence favoring the interpretation of the 2+ms states as one characterized by a large MS component. In Table 2, it is found that 2+ms state can be excited from the ground state by a weakly collective E 2 transition. This result is similar to that drawn in previous works.[39;41] In 98 Mo, we nd that there exists some discrepancy between the calculated and experimental data in the 0+2 state. In a previous work,[2] the problem was solved by con guration mixing. In our calculation, the experimental 0+2 state at E = 735 (MeV) is intruder state and not included. But it may have some mixing with the 0+3 state (theoretical 0+2 state), and this may perturb the properties of this state. This can be seen from Table 2 that has a large discrepancy. In addition, only the 2+2 state decays via an enhanced M 1 transition to the 2+1 state. The enhanced 2+2 ! 2+1 M 1 transition and the weakly collective 2+2 ! 0+1 E 2 transition agree with the MS interpretation for the 2+2 state. Therefore, the most fundamental MS 2+ state is, instead, the 2+2 state for 98 Mo. It is also worth while noting that the E 2 transitions from 2+3 in 100 Mo are larger than its neighboring nuclei and the calculation, for instance 2+3 ! 0+2 , about ve times of that in 98 Mo, a further investigation is needed to explain this anomaly and to con rm the mixed symmetry state nature of this state. Except this, we note that the MS states and electromagnetic transition of 100 Mo and 102 Mo are rather similar to those of 98 Mo. From Tables 2 and 3 we nd that the agreement between theory and experimental data is quite good except that the calculated values are smaller than experimental ones in 2+3 , which may be a candidate of the MS state for 100 Mo and 102 Mo. 2 R = F hs(kFF ks+i 1) : max max If it is a fully symmetric state, R equals 1. If it is a state with F = Fmax ; 1, the ratio R is then 1) : R = FFmax ((FFmax ; max max + 1) So the lower of the value of R, the lower of the F - 4 Summary spin value of the state. In case that the state does not have a good F -spin, the value of R is an indication of the main component of the state wavefunction in the F -spin space. The R values for the candidate of the MS states in 96;102 Mo are given Table 4. Table 4 Results of the IBM-2 analysis in 96;102 Mo: (possible) mixed-symmetry states, their R values and energies in MeV. Nucleus 96 Mo 98 Mo 100 Mo 102 Mo State 2+3 3+1 4+2 2+2 3+1 4+2 2+3 2+3 R 49:9 49:3 57:5 56:0 54:8 55:6 59:9 65:1 Energycal. 1:605 1:946 2:377 1:425 2:189 2:2o7 1:490 1:186 Energyexp. 1:625 1:978 2:219 1:432 2:104 2:223 1:463 1:249 339 We have carried out a systematic investigation of the even-even Mo isotopes in the IBM-2 framework. These calculations show that a good description of the observed energy spectra and electromagnetic E 2 and M 1 transition rates can be obtained. However, there are several 0+ states in Mo isotopes whose properties cannot be reproduced by our calculations. These intruder states are caused by two-proton excitations across the subshell at Z = 40 and presumably associated with other low-lying degrees of freedom. The calculation shows that the isotopes chain is in the U(5) to SU(3) transition. Both our calculations and the existing experimental data indicate the existence of mixed symmetry states. 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