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

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Ann. 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.
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