# Influence of giant angular resonances on the electromagnetic characteristics of low-lying states.

код для вставкиСкачатьAnn. Physik 2 (1993) 239-257 . Annalen der Physik @ Johann Ambrosius Barth 1993 Influence of giant angular resonances on the electromagnetic characteristics of low-lying states * I. N. Mikhailov and Ph. N. Usrnanov * Joint Institute for Nuclear Research, Dubna, Moscow, Russia Received 21 April 1992, revised version 21 August 1992, accepted 2 October 1992 Abstract. The two-rotor model with the Feshbach projection operator method is applied for investigating the properties of the positive parity low-lying collective states in the rare earth nuclei. The calculations of the energy spectra, E2-transition probability and magnetic properties of the states of & and y-bands are carried out for the isotopes '6c,166.1a8Er. The B(M1) values from 1 + states to the ground band are estimated. Keywords: Low-lying collective states; Rare earth nuclei; Giant angular resonances. Introduction Dipole magnetic transitions in deformed nulcei are not well enough studied in comparison with quadrupole electric ones. The magnetic properties of the ground (gr) and yrotational bands in even-even nuclei are influenced by the Coriolis admixtures of the states of K" = 1 + bands [l]. The experimental observation of the low-lying collective states with K" = 1' [2], led to the appearance of new models [3-61 where the I f states admixed by the Coriolis coupling are considered as a "giant angular resonance" (GAR). The states of GAR are connected with the states of the ground band by dipole magnetic transitions. This fact gives rise to the alternative name of the resonance - the M1-mode. The models considering the coupling of the low-lying states with the states of GAR allow one to describe the M1-transitions from the /3- and y-bands to the ground band [7, 81. In ref. [ 5 ] , the two-rotor model (TRM) [3, 41 with the Feshbach projection operator method [9] was developed for investigating the properties of the positive parity low-lying states in the transuranium nuclei. In the present paper this model is applied to study the properties of the rare earth nuclei. The effective Hamiltonian and expressions for the reduced probabilities of collective states are obtained. The calculations are performed for the 164*166*168Er isotopes. ' Present address: Physics department, Moscow University, Russia. * Permanent address: Institute of Nuclear Physics, Academy of Science of the Uzbek Republic, Tashkent, Uzbekistan. 240 Ann. Physik 2 (1993) Model Let us consider a model where a nucleus is considered as two axial rotators composed of protons and neutrons which can move with respect to each other but have fixed centre of masses [3]. Internal states of each subsystem are characterized by the conserved quantum number associated with the angular momentum projection onto the symmetry axis of the subsystem (Kp and K,). Thus we write (C&Y ( C p l p ) ~= Kpu/ 9 =K~v (1) 3 where Cp and C, are the unit vectors directed along the symmetry axes of the proton and neutron components. The relative motion of the subsystems is counteracted by the force, the corresponding potential of which in the harmonic approximation is 1 V ( 6 )= - C 6 2 2 (V(f-6) =:C(:-6)3 . The angular variable 6 is determined by the following expression cos ( 2 6 ) = [,,-C, . According to ref. [ 5 ] , the nuclear Hamiltonian is where 1 Ho,0 = - (A + A (I2+S2)+ V(6 ) +Hint (4) T;s= a ; ~1 (Ho, , 0 - J'(6) (7) 4 4 -Hint) +as,2 Ho, 1 * 1 Here A : are numerical parameters (A: = -, where Jiis the-moment of inertia of the 2 J, ith subsystem), I = Zp+ln is the total angular moment of the whole system and s = Ip-In. The eigenfunctions of Ho,o,which describe the system with axial symmetry, have the following form [4, 51: I. N.Mikhailov, Ph. N.Usmanov, Influence of giant angular resonances 24 1 I hereK10, k = K , + K , , K = Ik-KI = 0 , 1 , 2 ... , n = 0 , 1 , 2 ... where LE(e2)- is the associated Laguerre polynomial. In the expression (8) xk - are the eigenfunction of the internal Hamiltonian Hint.The corresponding eigenvalues are denoted as eint(k). The eigenvalues of the operator HO,Oare determined by In this scheme gr and the j?-, y-vibrational states are characterized by the quantum numbers (k= K = K = n = O)gr and (k= K = K = n = O)B, (k= K = 2, K = n = O)y, respectively. The motion of neutrons with respect to protons can be superimposed with excitations of gr-, /%and y-bands. As a result, in the framework of the considered model, an infinite number of states appears with quantum numbers (k= 0, k = K = 1, n)@, (k= 0, K = K = 1, n)eBand (k = 2, K = 1,3; K = 1 , r ~ above ) ~ ~ the gr-, j% and y-vibrational bands, respectively. The operator dB in (7) does not change the quantum number k but influences the internal nuclear state. The operators 6 , and 6, in (6) change the quantum number k and connect gr- and /3-bands with the states in y-band. Let us write down these operators in the following manner: where I = 1,2 and a = gr,/3. We are going to consider the states of the low-lying bands: gr-, /3 and y-vibrational bands. Following ref. [5] we introduce an operator P which is a projector onto the states under investigation (P-space): The states with the quantum numbers (k= 0, K = K = 1, n)@, (k = 0, K = K = 1 , n)es, (k= 2, K = K = 1 , n)er and (k= 2, K = 3, K = 1 , n)@, are included in the Q-space (GAR). For the projected Hamiltonian one has: Hpp= H{,o+PT;P . (11) Here HCo, is diagonal with respect to the basis wave functions ~y(’); Ti describes the mixture of bands in the Fspace (gr-, /3 and y-bands). We write the following expressions for the Hamiltonians Hp, and He,Q: 242 Ann. Physik 2 (1993) The operators in (12) describe the mixture of states from the P- and Q-spaces. The total wave function is determined as a sum: Y=PY+QY=@+x while the function Q, is represented in the model space as where C P is the amplitudes of the mixture of i states (in the P-space). The function Q, satisfies the equation: We assume that the energies of the low-lying states are small when compared with those of the operator HQe, and that the latter has the eigenvalues close to H f o = QHo,oQ.This assumption makes it possible to attain the following expression for the m.e. of 2 where j denotes the quantum numbers of the basic functions in the Q-space (additional to I and M ) . The energy E is assumed to be equal to the energy of the yeast-band states. Using expressions from ref. [5] for the m-e. of the Hamiltonian (3) and the Feshbach formalism of the projection operators (9) as well as taking into account that in our case the /?-band is much higher than the y one, we have the following expression for the effective Hamiltonian: where i, 'i = gr, /3 and y I. N.Mikhailov, Ph. N. Usmanov, Influence of giant angular resonances 243 1 A = - o 0; - is the core inertial parameter 2 W 4 = 6.543. n = 1 (2n+l)(n+1) Let us introduce the following expression for the quadrupole electric moment operator of a nucleus [5]: A0 = ( A ; - A $ / ( A i + A 3 , C ’ = 5+ C (A) 1/2 M(E2;pu)= C D:,(Wrn;,+ QoD:o(W where Here mi,, - are determined in the nuclear c.m. frame; Qo is the internal nuclear quadrupole moment and I v) = Ip), I y>. For the reduced probabilities of transitions between the states in the P space one has: Here rn, are the parameters which can be determined using the experimental data. Using the operator M(M 1) from [ 5 ] , we have the following expression for the reduced m.e. of the M1-transition between the states from the P-space the following expression: 244 Ann. Physik 2 (1993) where a = gr, p. The expression for the reduced probabilities of the M 1-transitions (is the case of the odd states from y-band) can be written on the basis of (17) in the following manner: For the magnetic moment of the collective states one has P =fRu)*I where In order to describe thefF-factor for the P-space states, one needs an additional parameter kp+yn), which IS determined from the experimental data. +- I. N.Mikhailov, Ph. N. Usmanov, Influence of giant angular resonances 245 Calculation for the 164*166*168Er isotopes Calculations were carried out in the case of the Er isotopes. We have used the following procedure for the parameter determination. The inertial parameter A was fixed by using the experimental value for the ground band ( I = 2) energy EZp. (2). The values of A , and oywere determined by fitting the experimental data for the energies of y-band states with odd I. The matrix elements (gr b1 I r), CB 1 bl I y>, which describe the direct mixture in gr, /3 and y bands, influence both the spectrum and the branching ratios of y-transitions. They are determined from the condition of the best reproduction of the branching ratios of E 2-transitions from y-band using the formulas A (3), A (4) (see Appendix). The parameters mB and m y are defined by using the following expressions I and having the experimental data B(E2) for 164Er[lo]. The free parameters: 7, (gr I b>,(gr Ib2I y ) and @I b2 I y>, were determined within the X2-method by fitting the experimental data. All the model parameters are summarized in Table 1. Table 1 Parameters used in the model. Parameters A 164 166 1.460 0.716 0.128 1 .34*1OW2 1.218 9.8 -0.59*10-3 -0.01-10-2 1.25 * 0.12.10-2 168 1.217 0.748 0.106 1.330.391 13.6 - 0.59*10-3 0.12.10-2 1.25- l o w 3 0.34-10-4 With the quoted above values for mo = 0.1 b, m2 = 0.27 b and Qo= 7.42 b [lo] we have calculated the reduced probabilities of E 2-transitions in the y-vibrational band from (16). The calculations of the spectra for the positive parity collective states in the cases of 1a7166~168Er are depicted in Figs. 1 - 3, respectively. At Is 12 the reproduction of experimental data is quite satisfactory. The discrepancy between the theory and experiment at 11 14 found in the case of '@Er is due to the limitation of the Fspace present in the model. In particular, the following facts are not taken into account properly: a) the presence in this nucleus of K' = Of bands; b) the crossing of the ground band by the aligned band with I = 12 which is responsible for the backbending of the moment of inertia. 246 Ann. Physik 2 (1993) h > 2 v w 6 164 5 Er -_ ’-17 4 3 14 - - __ 6-..-..;= = 0 - ‘O __ 8 - 2 10 - __ __ 8 - 1 64 2= 0 - ZZI - 0- exp. theor. - I - - 16 15 I4 -. 13 12 __ ‘1 ~ - 10 9- ,e-, e - -__ 5 - __ __ __ __ __ - = : 3 esp K=Op theor K = 2, theor exp K=O,, 18 __ __ 16 - __ 14 - - 12 - 10 8 __ - - 4- __ - 2- 0exp 168 Er (12)11 10 ~ 9876534 2- 6- theor K=Og, Fig. 1 Comparison of the calculated and experimental spectra of positive-parity states for lwEr. exp __ __ __ _ . __ __ - = __ theor K=2, Fig. 2 Comparison of the calculated and experimental spectra of positive-panty states for ‘%r. We show the structure of the collective states in Figs. 4-6. It is seen that the ground band is most “pure” when compared with others. The mixture effect between p- and y-bands exhibits itself strongly and increases with spin I. In paper [16]by Fahlander et al. the values of matrix elements for E2-transitions between states with positive parity in 166Erwere experimentally obtained, where the ((I- 2), I E 2 1 II&matrix element has a nonmonotonic dependence on the angular I I. N. Mikhailov, Ph. N. Usmanov, Influence of giant angular resonances 2 3 - 12 __ 8 247 - __ __ 8- - 7- __ 2- - 6- theor 5 - - 4- __ __ 0- exp 1: __ 8- K=O, ~ 32- esp K'3 4- I _ Fig.3 Comparison of the calculated and experimental - 20- 0 - theor. c7 (80 Er __ exp theor K=O,, 1.5 1.0 - cy --___ B 0.5 0.5 -1.0 -- . . - _- - _ _ _---___ = ' B Y -=:Lly; __.___.___---: .. - _-.._ .-.. . ---I-:-B, -.: .. -0.5 7 t 0 gr Y ; A. . / 0.0 -0.5 - : i 2 4 6 8 1 0 1 2 1 4 Ih Fig. 4 -1.0 ~ 0 ~ 2 ~ ' 4 ~ 6 ' ~ ~ ~ ~ 8 1 0 1 2 1 4 13 I h Fig. 5 Fig. 4 Structure of the wave-functions of gr-, fi and y-bands for lbQEr(K=P, - _ . _ _ - K = v). Fig. 5 Structure of the wave-functions of gr-, fi and y-bands for '@Er (K =p, - - _ .- - - K = 7). --. -. - K = gr, - - - - K = gr, - - - - momentum I. This dependence of the matrix element is successfully described by the present model (see Fig. 7). The other different models [I61 and the model, which takes into account a Coriolis mixing of states through AK = 1 don't describe this dependence do not describe so well. Though the latter reproduces very well the ratios for the probabilities of E2-transitions from y-band states [6]. ' ~ ' ~ 248 Ann. Physik 2 (1993) 0.6 1 1 1 1 1 % 0.4 Y A c;' Y - 0.2 Y V 0.0 - o . 5 s '\ - 1.oo 2 4 6 2 8 1 0 1 2 Ih 4 6 8 1 Nrn2 0 i Fig. 7 Fig. 6 Fig. 6 Structure of the wave-functions of gr-, p and y-bands for 168Er(- K = gr, - - - K = B , - _ . - _.-- - K = y). Fig. 7 The experimental and calculated values {16] of the Zy--+(1-2)grmatrix elements in IMEr. The calculations were performed within the symmetric rotor model (sym), asymmetric rotor model [28] (10.0 and 12.7 label y = 10' and y = 12.7' versions, respectively), rotational-vibrational model [29] (versions rvml and rvm2), IBA model [30] (iba) and our model (TRM). Table 2 The E2-matrix elements of the transitions between the positive parity states in '&Er. Transitions Experiment [161 Theory IBM-1[16] +0.11 2*28 -0.11 +0.12 3.86 -0.12 +0.19 4.70 -0.14 + 0.20 5.81 -0.20 + 0.25 6.47 -0.25 +0.30 7.00 +0.30 + 0.41 "15 -0.86 + 2.04 7.66 -2.18 - 2.33 + 0.19 -0.12 TRM 2.40 2.33 3.84 3.75 4.79 4.76 5.52 5.60 6.08 6.35 6.51 7.01 6.81 7.58 6.98 8.09 .2.87 - 2.78 249 I. N. Mikhailov, Ph. N. Usrnanov, Influence of giant angular resonances Table 2 (continued) ~ Transitions 4,+4,r 6,+% + +0.34 -0.16 +0.25 -4-03 -0.20 -2.12 +0.24 -0.47 +0.45 -6.78 -0.95 -4.74 8,r-.*, 1OBI Experiment [16] 1 0,r 2 1 4,-+2, 6y-4, 8y-6, 10,-*8, 12,+ 10, 2,+Ogr 4y+2,r 6, +4gr 8,+6,1 10,- 8,r +0.13 2*60 -0.13 +0.22 4*44 -0.22 +0.26 5.28 -0.26 +0.28 5.65 -0.28 +0.77 6-oo -1.20 . +0.019 0-372-o.oi9 +0.016 0-3 -0.016 +0.012 0*244-0.012 +0.011 0*214-~.~22 +0.027 0.416-o,044 +0.026 2y-Qgr 4y-'4gr 6,+6,r 8,+8,I 2,+4,r 4y+ 6,r 6y-8, 8Y+ lo,, OS - 0.026 +0.036 0.727 -0.036 +0.042 0.834-o.042 +0.048 0.969- 0.048 +0.026 O.l6l -0.022 +0.016 0'326-o.04i +0.31 0.33 -0.30 +0.18 0.37 -0.30 ~~ Theory IBM-1 [16] TRM -3.61 -3.53 -4.19 -4.17 -4.62 -4.72 -4.92 -5.25 3 4 2.28 2.38 3.80 4.01 4.69 5.02 5.29 5.84 5.29 6.57 0.37 0.36 0.34 0.30 0.32 0.27 0.3 1 0.30 0.29 0.45 0.49 0.49 0.71 0.73 0.83 0.89 0.92 1.01 0.13 0.14 0.27 0.27 0.39 0.31 0.49 0.18 250 Ann. Physik 2 (1993) Table 3 The B(E2) value for y+gr transition in the lfflErnucleus. B(E2) ( 2 6 3 Transitions 2,0,' +2gr '4, 3Y'2, '4, 4, '2, 4gr '6, 5,+4, '6, 6,'4, '6gr -+ Exp. [ l l ] IBA-2 [12] TRM 0.028(2) 0.051(7) 0.0035 ( 5 ) 0.046( 13) 0.030(8) 0.0094(14) 0.048(7) 0.0065(10) 0.027(4) 0.034(6) 0.005(1) 0.043(7) 0.028 0.066 0.0025 0.054 0.041 0.0068 0.080 0.0039 0.036 0.054 0.0021 0.079 0.026 0.046 0.0035 0.047 0.029 0.012 0.055 0.0089 0.036 0.041 0.0079 0.055 Iy+cr) for the laEr nucleus. Table 4 The ratio B(E2; Z,+&)/B(E2; 4 ILr I:, Experiment Theory 1131 2, 2, 3, 4, 5, 2, 2, 4, 4, 4, 6, 4, 2.23(14) 0 , 2,r 0.11(5) 2,r 281 4, 0,r 0.89(7) 13.3(19) 1.45(13) 0.25(10) . [61 TRM Aiaga 1.97(30) 0.15(3) 0.82(20) 5.4( 13) 1.3(3) 0.30(6) 1.97 0.09 0.81 7.1 1.8 0.18 1.43 0.05 0.40 2.94 0.57 0.08 Table 5 The ratio B(E2; Z,+IL)/B(E2; Zy+Th) for the la8Er nucleus. 4 2, 3, 4, 4, 5, 6, 6, 7, ';lr zr : Experiment Theory ti71 t181 TRM Alaga 2.27(45) 0.044(22) 0.65(30) 6.3(30) 0.08(4) 1.0(4) 10.7(42) 0.19(8) 1.64(80) 1.79(4) 0.075(4) 0.64(4) 5.27(55) 1.76 0.072 0.62 5.1 0.14 1.15 10.0 0.17 1.72 1.43 0.05 0.4 2.94 0.09 0.57 3.7 0.11 0.67 The matrix elements of E2-transitions for '66Ercalculated in the framework of this model are given in Table 2. These matrix elements are compared with the IBA-1 calculations and the experimental data i16]. I. N. Mikhailov, Ph. N. Usmanov, Influence of giant angular resonances 25 1 One can find in Table 3 the values for B(E2) in y-band of '"Er. We also depict in this table the experimental data [l 11 and the alternative theoretical calculations in terms of IBA-2 [12]. The experimental data discriminate the theoretical calculations in favour of our model. The calculated ratios for the reduced probabilities in the case of EZtransitions for 164p168Er are compared with the experimental data f6, 13, 17, 181 and with the values calculated using the Alaga formula in Tables 4, 5. It is clear that our model describes the deviation of the reduced E2-transition probabilities ratios from the Alaga rule. Note that we have used equal quadrupole moments QO both for the gr (K = 0)- and y(K" = 2+)-bands in contrast with ref. [15], where Qo(0) # Q0(2). Moreover we have used the unique values of mK and Qo in the calculations of B(E2) for all the isotopes. The coefficients of the multipole mixture were calculated by Table 6 The multipole mixture coefficients (22) for lMEr. Ii If Experiment [19] TRM -3.6 2, +0.28 Os or 3, -7.7 -1.15 -0.25 +5.1 -w +0.35 -1.02 -3.1 - 1.78 or 4, >7 - 4.8 or 5, o.o + 1.5 -5.8 -1.63 +0.07 -0.04 +1.6 - 1.19- 1.02 -1.16 or > 3.3 +2.2 -6.5 -5.5 7, -1.5 ,8 3, 2, +0.75 -3.0 - 1.0 -0.82 or 12.0 -6.8 + 161 >3.7 [6] 4.4 252 Ann. Physik 2 (1993) Table 7 The multipole mixture coefficients (22) for lMEr 1 2 Experiment TRM 3 4 13 - 22 + -7 - 38 2, or 1271 + 24 - 21.6 -w +9 -19-38 [201 + 8.0 -44-6-12.6 13 - 20 + -w 3, - 18 +- - m9 -35.4 - 69.7 1241 +11.4 - 32.0 - 40.0 3, 4, 3, 2, 161 >2.6 -2.6 4, 3, 161 = 1.5(3) - 2.3 - +5 '-31.9 - 522 566 - 61 6 -3.3 4, 48, - lo + 1.2 - 4.5 -3.0 i-4 - 27 - 21.2 +1.8 -2.1 +3 -37- I201 57 - 84 + -w 4, -263.9 - 20(4) 4, 161 = 1.61 + 0.53 - 0.25 -2.6 161 =2.1 42.9 + 5.4 - 7.3 - 25(3) 5, - 135.0 I. N. Mikhailov, Ph. N. Usmanov, Influence of giant angular resonances 253 Table 7 (continued) ri *f 1 2 Experiment 3 TRM 4 + 17.4 25.0- 7.0 - 1.15 or 1271 +0.35 -0.80 - 1.45 - 6,3 +- m 2.9 161 = 1.26 6, - 2.0 5, + 0.40 161 = 1.61 - 0.37 161 = 1.105 25.2 + 17 -37-m 7, +5 -22-7 + 0.47 -3.0 161 = 1.45 . -0.32 7, - 80< S< 30 + 0.9 - 3.1 7, -1.5 - 238 + 153 - 540 ~ 4 1 1191 24.0 1271 + 2.3 4.9- 1 . 1 8, -0.75(20) ,8 - 0.60 or 1.6' ''O - 0.55 9, ,8 3 - 11 +- m 7.4 We compare our results with the experimental data in Tables 6-8. (Note: The signs of the experimental values of S used by different authors for the transitions in '@Er, laaEr, and 16*Erare often in contradiction, because the sign of 6 depends on the kind of the formulas used in the analysis and the convention. In the present table the signs of 6 are given in accordance Stefan-Becker [lo].) The best description has been obtained with 254 ~~ Ann. Physik 2 (1993) ~ Table 8 The multipole mixture coefficients (22) for I6*Er. Zi If TRM Experiment or > 9.4 < -4.8 - 4.3 16)=8.1 3, 1211 1221 16.5(23) 3, - 4.0 - 4.9(3) -3.0 3.7 - 5.7 + - 5.7 - 2.4 +m 4, 4, 5, 4, 25 - 13 Id( = 1.41 161 = 1.38 + 2.05 3.3 -0.71 161 = 1.05 5, 6, 161 = 1.55 + 1.05 161 = 1.92 161 7, = 1.52 2.3 - 0.76 1171 +1.14 - 0.82 161 =0.245 1231 3.6 1171 2.5 0 1 3 B(M1;00,+1+ 1) = -2A 1 6 k~P -d2 B(M 1; OO,,-+ 1 1) = 0.8 p h for lHEr and = 1.75 P & for 166, la8Erat fixed value for o1 = 3 MeV. Calculations in the IBM2 give B(M 1) = 1.5 p& [8]. The experiment [2] gives for 168ErB ( M 1 ) = 1 . 7 5 ~ ; and o1= 3.4 MeV. The experiments on (e,e') and nuclear resonance fluorescence for summed M 1 ground-state transitions B (M 1) t (ExI4 MeV) in la8Er give 2.50+0.21 p & and 2.20+0.16~$, respectively [31]. These experimental + values are always greater then our estimations, because here we didn't take into account the transitions from (k = 0, K = K = 1, 0)e~and (k= 2, K = K = 1, O)@ states to the ground one, But their contributions are taken into account in the calculations of coefficients of the multipole mixture 6. We are now in a position to make the following conclusions: I. N. Mikhailov, Ph. N. Usrnanov, Influence of giant angular resonances 255 1) P and Q spaces mixing leads to the renormalization of the moment of inertia of bands, included into the P-space. It reflects on the values of eigenstate functions and, hence, on the values of electromagnetic transitions between P-space states. That is the deviations of ratios RI,, from the Alaga rule is basically due to the P-space state mixing, but also due to the P- and Q-spaces state mixing. 2) The presence of the GAR components in the wave functions of 8- and y-vibrational bands leads to the M1-transitions from these states to the ground band. These components increase with the spin, and this explains the decrease of the coefficient of multipole mixture 6 in the case of y-band with increasing angular momentum I. 3) The best description of amp.in the case of the low-lying levels is obtained with B(M 1;00,,-1+ 1) = 0 . 8 ~ 5 for ‘@Er and = 1 . 7 5 ~ 5for l‘,16*Er at fixed value for w1 = 3 MeV. Appendix A The state mixture leads to the strong violation of the rules of the adiabatic theory for the transition branching. These effects are easily interpreted by using the wave functions of the first perturbation order in the parameter of the bands mixture (in the P-space) were Using the expression for the me. of the quadrupole moment, which were obtained above, one comes to the following formula: For the ratio of the reduced probabilities of EZtransitions from the y-vibrational band we have 256 Ann. Physik 2 (1993) Transition branching from y-bands are completely determined by two spin independent parameters The spin dependence of reduced transitions probabilities, thus calculated, coincides with that for the mixture of two bands having different quadrupole moments [151. The factor )?(+I (2) normalizes the internal quadrupole moment of the y-band. References A. Bohr, B. Mottelson, Nuclear structure, Vol. 2, Mir, Moskva 1971 D. Bohle, G. Kuchler, A. Richter, W. Steffen, Phys. Lett. 137B (1984) 27 A. 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