Magnetic properties of rotational bands in 160Dy and 170–174Yb isotopes

In this work, non – adiabatic effects manifested in the magnetic properties of low-lying states of even-even deformed nuclei are studied. A simple phenomenological model which takes into account the Coriolis mixing of states of the ground and Kπ = 2+n and Kπ = 1+v bands are proposed. The Calculations for isotopes 160Dy and 170,172,174Yb, are carried out. The calculated gR–factor for the states of ground band which are compared with experimental data. Decreases of gR–factor with increasing angular momentum I have been discussed. The probability of M1–transition from 1+v and coefficients of multipole mixture δ(E2/M1) from 0+2, 0+3, 2+1, and 2+2 bands are calculated and compared with the experimental data. Decreases of gR–factor with increasing angular momentum I have been discussed. That these is an obvious inverse relation between gR – factor and angular momentum I of the ground band states. This has been explained by a mixing ground and Kπ = 1+ bands which have a strong B(M1) to ground state.


Introduction
The interest to investigation in the properties of deformed nuclei, especially increased in recent years with the opening of a new collective isovector magnetic dipole mode [1,2].
The measured values of energies of the excited states such as modes indicate that they are not so high in the excitation spectrum, and consideration of mixing the isovector magnetic modes with low-lying exciting states can lead to interesting physical phenomena [3,4].
Using 85 Kr ions with energy of 350 M eV by Coulomb excitation has been studied ground band state in the flesh to the spin I = 16h isotopes 160 Dy and 170,174 Y b in paper [5]. Nonadiabaticity of magnetic characteristics of low-spin states of ground bands has been observed. It is shown that, with an increase of angular momentum I, g R -factor decreases in these three nuclei.
The nuclei 170,172,174 Y b are among the most well studied. These isotopes are studied experimentally in a number of ways: such as radioactive decay of 170,172,174 Lu, and also different nuclear reactions. In the study of nuclear reactions of bands with the small K were supplemented bands with high spin [6]- [8]. In these isotopes detected many 1 + states and also bands with K π = 0 + n , 2 + n , and some of these bands include the levels with rather high spins.
The aim of the present paper is applied to investigating the properties of positive parity lowlying states of 160 Dy and 170,172,174 Y b isotopes. To studying the properties of positive parity low-lying states of these isotopes used phenomenological model [4] which takes into account the Coriolis mixing all of the experimentally known low-lying rotational bands with K π ≤ 2 + . The behavior of g R -factor of ground band has been discussed by the growth of angular momentum. The energy spectra and the reduced probability of M 1-transition and the value of multipolemixture δ (E2/M 1) are calculated and compared with the experimental data.

The Model
To analyze the properties of low-lying positive parity states in Dy and Y b isotopes, the phenomenological model of [4] is exploited. This model takes into account the mixing of states of the K π = 0 + , 2 + and 1 + bands. The Hamiltonian model is where ω K -bandhead energy of rotational band, ω rot (I)-an angular frequency of rotational nucleus, (j x ) K,K ′ -matrix elements which describe Coriolis mixture between rotational bands and The eigenfunction of Hamiltonian model (1) is here Ψ I K ′ ,K is the amplitude of mixture of basis states. Solving the Shrödinger equation we define eigne function and energy of a Hamiltonian. The total energy of state is defined by In figure 1

Magnetic Characteristics
The equation for the reduced probability of M 1-transitions from the states I i K i to the level I f 0 + 1 band by phenomenological model is as follows [4]: where g R ≈ Z/A, g K -intrinsic g-factor of K ̸ = 0 bands. Here m ′ 1ν =< 0 + 1 |m(M 1)|1 + ν >matrix elements between intrinsic wave functions of 0 + 1 and K π = 1 + ν bands, the value which is estimated from the experimental data. In adiabatic approximation of the equation (6) yields  (7) which are presented in Table 1. However, formula (7) does not allow to define sign of the parameters m ′ 1ν . Usually, the coefficient of multipole mixture δ is experimentally investigated which is found by the following ratio. .
Note, that M 1-transitions from the K π = 0 + and 2 + to the level ground bands are equal to the zero (absent) in adiabatic approximation. Thus, M 1-transitions appear due to 1 + ν components Ψ I i 1ν ,K i in wave functions of this states. In case, the calculations use identical signs for parameters m ′ 1ν the calculated values of coefficients of multipole mixture δ in many cases have smaller values than experimental data. In paper [12], 1 + ν states have been investigated and illustrated values of ratio R 11ν = B(M 1; 11 → 20 1 )/B(M 1; 11 → 00 1 ) for γ-transitions from 1 + ν states to ground state in which adiabatic approximation is ≈ 0.5. As an axiom, for the γ-transitions from 1 + ν states having R 11ν > 0.5 identical sings of m ′ 1ν are taken, and for R 11ν < 0.5, opposite signs are taken (see Table 1). Comparison of measured experimental data [10,11] and calculated value of coefficients of multipole mixture δ for 172174 Y b are illustrated in Table 2. Table 2, shows that calculated values have good correspondence with experimental data and precisely reproduces a sign of δ. The values of δ are decreasing with the growth of spin I, i.e. with a growth I the probability of B(M 1) is increasing.
Experimental data [9] for the probability of M 1-transitions from K π = 0 + 2 , 0 + 3 , 2 + 1 and 2 + 2 bands has been compared with calculated values in Table 3. Experimental values of B(M 1) for transitions from I = 2 and I = 4 states of K π = 2 + 1 increases with rise of I. Here, it is worth mentioning that this finding confirms our calculated results.
In [5], ground band states up to spin I = 16h of 170,174 Y b nuclei are investigated by Coulomb excitation through 350 M eV 85 Kr ions. In these nuclei, it is observed that g R -factor of ground band states decreases with an increase of angular momentum I.
A g R -factor of rotational K band determined within our model by the following formula: The first non-adiabatic correction is negligible in the calculation of the g R factors of ground state. But, the correction is considerable for g R -factor for states with K ̸ = 0.
Theoretical values of magnetic characteristics excited states are obtained for the case m

Conclusion
In the present work, non-adiabatic effects in energies and electromagnetic characteristics of excited states are studied within the phenomenological model which taking into account Coriolis mixing of all experimentally known rotational bands for isotopes 160 Dy and 170,172,174 Y b, with K π ≤ 2 + .   The energy and structure of wave functions of excited states are calculated. The reduced probabilities of M 1-transitions and coefficients of multipole mixture δ(E2/M 1) from K π = 0 + m , 2 + ℓ and 1 + ν bands are also calculated and compared with experimental data which gives the satisfactory result are also calculated.
Thus within our theoretical analysis it is possible to explain the M 1-transition from the state β 1 -, β 2 -, γ 1 -and γ 2 -bands to the ground bands level, which is forbidden by adiabatic approximation.
The experimentally observed K forbidden M 1-transitions from state 2 + ℓ -bands are explained by the presence of K π = 1 + ν components in wave functions of these bands. With the growth of the angular moment I, the coefficient of δ(E2/M 1) for transitions from 0 + m and 2 + ℓ bands decrease. Experimental data of B(M 1) for transitions I = 2 and I = 4 state K π = 2 + 1 bands shows an increase of B(M 1) with the of growth I. This confirms our results.
The result of calculation show that the decrease of g R -factor with an increase of angular momentum I of rotational states of ground band is associated with the mixing in these states with 1 + ν bands which have strong B(M 1) to the ground state in isotope 170 Y b.   [10,9] δ theor.