Electric Quadrupole E2- Transitions of 170−174Yb Isotopes

The non-adiabatic effects which is manifested in the electric properties of low-lying states of even-even deformed nuclei are studied. A simple phenomenological model which takes into account the Coriolis mixing of K π = 0 n + , 2 n + and K π = 1 ν + state bands. The Calculations for isotopes 170–174Yb, are carried out. The reduced probability of electric quadrupole transitions from the states 0 ν + and 2 ν + - bands to the ground (gr) state band is calculated and non adiabatic effect is discussed. The ratio of E2– transitions RIK from 0 2 + , 0 3 + , 2 1 + , and 2 2 + bands are calculated and compared with the experimental data.


Introduction
Despite the fact that the structure of deformed nuclei and nature of low excited levels have been substantially studies over more than four decades, this still occupies a central part of today's research Refs. [1]- [3].
The isotopes of Yb is the deformed nuclei in rare-earth region and the spherical nuclei near Z = 70. For well deformed nuclei the states associated with ground state rotation produce regular sequences which dominate the yrast line. The nuclei 170,172,174 Y b have been well studied. In highspin spectroscopy studies of rare-earth nuclei, the isotopes of ytterbium have been a central focus for many experiments, and a large body of data has been established [4].
It is important to note that these are investigated in a number of ways such as radioactive decay of 170,172,174 Lu, and different nuclear reactions. In these isotopes, many 1 + states and K π = 0 + , 2 + bands have been observed.
Numerous conducted experiments on defining spectroscopic characteristics of low-lying exciting states, particular K π = 1 + in deformed nuclei [5], have motivated the further theoretical investigations. In this case, investigations influence of K π = 1 + states to the properties of lowlying levels is actual.

Electric Quadrupole E2-Transitions
We shall calculate the reduced probability for E2-transitions, using the wave function obtained by describing the energy of states. The expression for the reduced probability of E2-transitions between states I i K i and ground state band and also intraband transitions of ground band within the framework our model [9] as follows: here m Kn =< 0 + 1 |m(E2)|K + n >-is matrix elements between intrinsic wave functions of ground (0 + 1 ) and K + n = 0 + m , 2 + , 1 + ν bands, whose values are defined from from experimental data, Q 0is intrinsic quadrupole moment of nucleus; and C In the adiabatic approximation, the following equations are valid for B(E2) factors from the I = 2 states to the K + n rotational band: which allows us to calculate the empirical values of the parameters m Kn from the experimental data.
The K π = 0 + 2 , 0 + 3 and 2 + 1 bands are located close to each other in isotopes 170,172 Y b, which leads to a strong mixing of states even I = 2. In this case, the adiabatic approximation (2) becomes inapplicable to determine m K i . Therefore, to describing the experimental data for B(E2) in 170,172 Y b isotopes, the value of m 0 and m 2 parameters are varied slightly. The values of the parameters m K which are used in calculating probability of E2-transitions are given in Table 1. Table 1. The values of the parameters m K and the intrinsic quadrupole moment Q 0 , which are used in calculations (in ef m 2 ). The empirical values of parameters m K have been defined by formula (2), using the experimental data of the reduced probabilities of E2-transitions B(E2; 2K n → 00 1 ). In the case where the bandhead energy is close to each other the Coriolis mixing of state is manifested significantly even at low spins I = 2. For example, K π = 0 + 2 and K π = 0 + 3 bands in nucleus 170 Y b and also the K π = 0 + 3 and K π = 2 + 1 bands in nucleus 172 Y b are close to each other. Therefore, for these bands the value of parameter m K is different from other bands (see Table  1). This mixture effect is more manifested in the electromagnetic than energy characteristics.
The magnitude and sign of the parameters m 1 1 = m 1ν were determined from the best agrement of the ratios R IK = B(E2; IK → I + 10 1 )/B(E2; IK → I − 10 1 ) from odd states of the K π = 2 + and 1 + ν (states with the negative signature σ = −1). The parameters m 0m and m 2 are determined, requiring the best agreement between calculated and experimental values of the ratios B(E2) transitions from the β m -and γ bands correspondingly, with positive signature σ = +1.
The comparison of the values of calculated reduced probabilities of E2-transitions with experimental data [11]- [14] are shown in Table 2. In Table 3 provided theoretical values of the reduced matrix elements of E2-transitions for 172 Y b, which are compared with experimental data and values taken by other models [15]- [17].
We note that our calculations were performed sequentially, i.e., first describes the energy state and then corresponding their wave functions are determined. Further, using these wave functions are calculated probabilities E2-transitions. From Table 4, one can see that, the results of calculation in the framework of our model for the majority cases provide a good agreement with experimental data.
The ratios of reduced probabilities of E2-transitions R IK = B(E2; I i K i → times. This is associated with strong mixing of K π = 0 + 2 and K π = 2 + 1 bands. Then arises the question, why the value of rations R I2 1 for the transitions from the 2 + 1 bands is not so much different from the adiabatic theory compared R I0 2 ?. This can be explained by the fact that the matrix element m 2 1 =< 0 + 1 |m(E2)|2 + 1 > is 10 times greater than m 0 2 =< 0 + 1 |m(E2)|0 + 2 > (cm. Table 2) One may see from these comparison that the mixing effect of states low-lying bands plays a crucial role which considerably demonstrates that E2-transitions even in low values of angular momentum I.

Conclusion
In the present work, non-adiabatic effects in energies and electric characteristics of excited states are studied within the phenomenological model which taking into account Coriolis mixing of all experimentally known rotational bands with K π < 3 + .
The energy and structure of wave functions of excited states are calculated. And also the reduced probabilities of E2-transitions is calculated. The ratio of E2-transitions probability from K π = 0 + m and 2 + bands are calculated and compared with experimental data which gives the satisfactory result.
If matrix elements of E2-transitions m K one of two strongly mixing bands K is less than matrix element of m K of K (m K < m K ), then the difference in the ratio R IK for the first band K from Alaga rule is bigger than the difference in R IK from Alaga rule. In other words, if m K < m K , nonadiabaticity in the ratio R IK is stronger than that of R IK .