Semi microscopic description of84Kr

The experimental information on the84Kr nuclei is compared with the model calculation in which two neutron holes are coupled to the vibrational field. Based on the lower-order terms of a perturbative expansion of theE2 andM1 transition matrix elements, a simple rule is obtained for the sign and the magnitude of theδ (E2/M1) ratios for the transitions between the second and first 2+ states in some vibrational nuclei.[Nuclear structure84Kr, calculated levelsJ, π and δ(E2/M1), Cluster-phonon model. Pairing interaction].


I. Introduction
The directional correlations of coincident 7 transitions have recently been measured for several cascades in the 8~Kr nucleus [1]. Besides establishing spins of a number of levels, this measurement also gives information on the multipole mixing ratios providing a better understanding of the structure of this nucleus. More specifically, while the spins and parities of the low-lying states of S~Kr indicate a collective quadrupole vibrational structure, the predominance of M1 in the multipolarity of several gamma transitions strongly suggests that also the single-particle degrees of freedom play an important role.
In view of the above-mentioned properties we propose in the present work a theoretical interpretation of the structure of 84Kr nucleus, and in particular of the ratios (5(E2/M1) measured in Ref. 1, within the two-hole cluster-quadrupole vibration coupling model.

II. The Nuclear Model and Parameters with
A detailed description of the model is given in Refs. 2 and 3. Here we only sketch the main formulas in order to establish the notation. The system is described by the Hamiltonian H = H o + Hre s + Hin t (1) where H o is the energy of the unperturbed system represented by a quadrupole vibrational field and by two valence neutrons in a central field. The residual interaction energy among the neutrons in the shellmodel cluster, H .... only includes explicitly the pairing force. The particle-vibration interaction is given by the expression 2 Hint__ ]//5fi2 ~ ibm+ +(_),bz, ] i: ~1 k(ri)Y2*(Oi'cPi) (2) where k(r~) is the interaction intensity and f12 is the quadrupole deformation parameter. The eigenvalue problem is solved in the basis I[(jljz)J, NR]I}, where j=(nlj) stands for the quantum numbers of the hole states, J is the total angular momentum of the two holes, N and R represent the phonon number and the angular momentum, respectively, and I is the total angular momentum.
The matrix element of Hin t are parametrized by the coupling constant a defined as a =--- where (k) is the mean value of the radial matrix element of the interaction. The electric quadrupole and magnetic-dipole operators consist of a particle and a collective part 2 J~(E2, _ eel #)--% ~ r?Ya,(0~,qS~) ~=~ (c) particle-vibration coupling constant a=0.74 MeV, which results from fl2=0.13 (as measured in the Coulomb excitation process on 86Kr (Ref. 6)) and @)_45 MeV (as estimated numerically using wave functions obtained from the Woods-Saxon potential [7] ; (d) single particle energies egg/2 =0, ee,/2 =0.80 MeV, ep3/~ = 1.35 MeV and eis, ~ = 1.8 MeV, were taken from the work of Boer et al. [8]. In this parametrization, without an adjustable parameter, we diagonalize the Hamiltonian by including all the vibrational states up to three phonons. The electromagnetic properties were evaluated with the usual values of the effective electric charge and effective gyromagnetic ratios. e; ff =0.5 e, e eff --fl2Ze =2.09,

III. Results and Discussion
In order to test the parametrization quoted in the previous section we first briefly discuss the available experimental data for the N=49 nuclei [9][10][11][12]. The energy spectra are compared in Fig. 1 and the results for the electric quadrupole and magnetic dipole moments of the ground state are shown in Table 1. It should be noted that the agreement between the calculated and the measured energy spectra for SSKr, 87Sr, 89Zr and 91Mo nuclei can be improved by lowering the particle-phonon coupling constant.
As an example, in Fig. 1 is also exhibited the calculated spectrum for a =0.  Table 2. It appears that the ground state has mainly a twoparticle configuration, while the remaining states have mixed characteristics. Experimental information on the multipole mixing ratios 6(E2/M1) are displayed in Table3 In 84Kr nucleus the 2~ state is a two-particle cluster of seniority zero coupled to one phonon, while the 2~ state is a two-particle cluster of seniority two. The same situation may be found in other vibrational nuclei, as for example in Cd isotopes. This is a consequence of the fact that in these nuclei the lowest single-particle states are of higher spin and,  mixing ratio can be expressed in the form contributing to both the E2 and M1 transition moments and not included in the approximation (9) for the ratio ~, give rise to a change in magnitude but not in sign. It is worth noting that recently  has discussed, within the particle-vibration coupling model, the E2/M1 mixing rations in odd-mass spherical and transitional nuclei. He obtained, for example, that for transitions of the type A N=O between unique-parity yrast states ~(j+2N+j+2N-1) Therefore from (9) and (12) we can relate now the ratios of the odd-mass nuclei with those of the neighbouring even-mass nuclei.
With the above mentioned parameters we obtain from (9) a value of ~=0.56 while the exact result is =0.94. In a similar calculation performed for the 114Cd nucleus, with the usual parametrization I-2], expression (9) yields a value ~ = -0.83 which should be compared with the exact value ~ =-1.54 and the experimental result [15] @= 99+o.v Therefore, we can conclude that the higher-order terms,

IV. Conclusions
We have demonstrated that the property of the S4Kr nucleus arises from neutron hole cluster with the quadrupole vibrational fields. Within this picture the experimentally energy spectrum and the mixing ratios g~(E2/M1) for the cascades 2~--~2~-~0~ and 2~ ~ 2~---* 0~ are well reproduced. In addition, a simple rule for the sign and magnitude of the ratio N is given for the vibrational nuclei in which the 2+ and 2 5 states are, respectively, a two-particle cluster of seniority zero coupled to one phonon and a two-particle cluster of seniority two.