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Studies of the electric dipole transition of deformed rare-earth nuclei
NUCLEAR PHYSICS A ELSEVIER
Nuclear Physics A 658 (1999) 197-216 www.elsevier.nl/locate/npe
Studies of the electric dipole transition of deformed rare-earth nuclei H . Y . Ji a, G . L . L o n g a'b'c'l, E . G . Z h a o a'b, S . W . X u d a Department of Physics, Tsinghua University, Beijing 100084, PR China b Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, PR China c Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Accelerator, Chinese Academy of Sciences, Lanzhou, 730000, PR China d Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou, 730000, PR China
Received 22 February 1999; revised 15 June 1999; accepted 28 June 1999
Abstract Spectrum and electric dipole transition rates and relative intensities in 152-154Sm, 156-16°Gd, 16°-162Dy are studied in the framework of the interacting boson model with s,p,d,f, bosons. It is found that E1 transition data among the low-lying levels are in good agreement with the SU(3) dynamical symmetry of the spdf interacting boson model proposed by Engel and Iachello to describe collective rotation with octupole vibration. These results show that these nuclei have SU(3) dynamic symmetry to a good approximation. Also in this work many algebraic expressions for electric dipole transitions in the SU(3) limit of the spdf-IBM have been obtained. These formulae together with the formulae given previously exhaust nearly all the E1 transitions for low-lying negative parity states. They are useful in analyzing experimental data. @ 1999 Elsevier Science B.V. All rights reserved. Keywords: Spdf interacting boson model; Electric dipole transition; Octupole vibration; SU(3) dynamical
symmetry
1. Introduction There have been intensive interests in the studies o f octupole degree of freedom in nuclear structure recently both experimentally [ 1 - 9 ] and theoretically [ 10-27]. 1Corresponding author: Department of Physics, Tsinghua University, Beijing, 100084, PR China. E-mail: [email protected]. 0375-9474/99/$ - see front matter @ 1999 Elsevier Science B.V. All rights reserved. PII S0375-9474 (99) 00335-8
198
H.Y. Ji et al./Nuclear Physics A 658 (1999) 197-216
In the boson model, negative parity states are described by the spdf interacting boson model [10-12,14,13,16,17,19-21,28] or the sdf-IBM [22-26]. Otsuka [14] showed microscopically that p and f bosons are important in the low-lying negative parity states of even-even nuclei. The spdf IBM has been successful in the description of negative parity states in the Ba and rare-earth region [ 17,19,20]. The advantages of the algebraic approach is that it offers analytical expressions for the energy levels, electromagnetic transitions and many other quantities. The prediction and later experimental confirmation of the 0 ( 6 ) limit has been a well-known example of the success of the IBM [29]. The spdf IBM SU(3) limit is a dynamical symmetry describing octupole vibration in deformed nuclei proposed by Engel and Iachello [12,13]. It has been pointed that 232U or other actinide nuclei may be candidates for the SU(3) limit [ 12,13,16]. While the spectrum agrees with the theoretical calculation very well, there are very few electromagnetic transition data. Thus it is difficult to check on the validity of the dynamic symmetry properties, in particular on the electric dipole transitions connecting positive and negative parity states. This is because finding nucleus with both good rotational feature in spectrum and with ample data of electric dipole transition is difficult. Besides, in real nucleus dynamical symmetry is usually somehow broken, for instance, even one of the best candidate of the sdg IBM SU(3) dynamic symmetry has some degree of symmetry-breaking [30]. But dynamic symmetries are very useful even though they are broken. They can be used to classify states and characterize the collective features of a nucleus. To a given nucleus near a dynamical symmetry, at first the gross structure is dictated by the dynamical symmetry. Then detailed structure of the nucleus can be attributed to symmetry breaking, and its description is the task of a more elaborate study. In addition, the dynamical symmetries can be used as landmarks in the nuclear chart table to classify typical collective motions, and other vast majority of nuclei can be put into categories of transition between two dynamical symmetries. Such a scheme is very successful in the descriptions of the positive low-lying states of even-even nuclei [ 31 ]. In this work we have studied the negative parity states of the rare-earth nuclei, namely 152-154Sm, 156-16°Gd, 16°-162Dyusing the SU(3) limit of the spdf interacting boson model. These nuclei are deformed and have ample data of electric dipole transition. We have found in this work that to a good approximation, the structure of these rare-earth nuclei can be well described by the SU(3) limit of the spdf interacting boson model. The paper is organized in the following. In Section 2, we give a brief description of the model and the necessary expressions of electric dipole transitions. In Section 3, we apply the results to J52Sm on the spectrum and E1 transitions. In Section 4, we study other deformed rare-earth nuclei. Finally, we give a discussion and a summary in Section 5.
H.l! Ji et al. /Nuclear Physics A 658 (1999) 197-216
199
2. The model 2.1. The energy eigenvalues The energy eigenvalues
have been discussed
in Ref. [ 12,171. In the SU( 3) limit, the
group chain can be written as U(l6) n
1 U(6)
‘8 U(1) n_
n+
the energy eigenvalue E = c-&r
1 S&d(3)
@ S&r(3)
1 SUspdf(3) > o(3) L (@)
h-6)
(h+P+)
(1)
is
+ aiGsu+
+ a&u_
+ asGsu(s)
+ a4L(L + I) .
(2)
The g.s.-band, P-band and y-band are generated from (2n, 0), (2n - 4,2) K = 0 and (2n - 4,2) K = 2, respectively, with n sd-bosons. The low-lying negative parity are generalized by the SU( 3) IR from the decomposition of (2n, 0) @ (3,0) ; that is, K” = O- from (2n + 3,O) and KP = 1- from (2n + 1, 1) , respectively. For the low-lying states, we are interested The wave function
in only the g.s., /3, y, O- and l-
bands.
are given by
( 1) ground state band [(2n, 0) + LM) , (2) 0 band )(2n - 4,2)+K
= OLM)
(3) y band ](2n - 4,2)+K
= 2LM),
,
(4)
Kp = O- -
band 1(2n - 2,0)+(3,0)-(2n
+ 1,O)LM)
(5)
Kp = l-
band /(2n-
- 1,l)LM).
-
2,0)+(3,0)_(2n
(3)
The value of a2 is taken zero, since its effect in the spectrum the E- term for the low-lying determined
by experimental
2.2. The El transition
states with only one pf-boson.
,
is the same as that of
The parameters
are then
data.
matrix elements
Some of the El transition formulae have been given in Ref. [ 281. With the method described there, we calculate the following formulae of the El transition, which are needed for a comparison with experiment, and the required SU(3) Wigner coefficients can be found in Ref. [32,33]. (a) (2n+3,0) K=OL+ (2n-2,2), ((L-
X
1>+ I] (s+/?+p+s)’
d
(2n-
K=O(L]I L-j
=
l)+ transitions,e.g.
l- +Ot,
l (2n + 1)
L+3)(2n+L+2)(2n+L+4)p(2n20n(2n + 3)
l,L-
1)
(4)
H. E Ji et al. /Nuclear Physics A 658 (1999) 197-216
200
where
(5)
~(~,L)=2(A+1)2-L(L+I), ((L - l)+ II (d+ (2n-L+3)(2n+L+2)(2n+L+4)L X
4n(2n+3)p(2n((L - l)+ x
II (d+f+
f+d”)’ II L-j
(6)
’
1)
= -*$2yi)
I)
3(2n-L+3)(2n+L+2)(2n+L+4)L 28n(2n+3)p(2n-
J (b)
l,L-
(2n f 3,0)
l,L-
K = 0 L- ---) (2n - 2,2)
((LS
1>+
’
1)
K = 0 (L + l)+ transitions,
e.g., l-
+ 2+,
II (s+P+P+# II L-j = -(2ny 1>
d
3(2n-L+1)(2n-L+3)(2n+L+4)(L+1)1p(2n--l,L+l)
X
20n( 2n + 3) (8)
1>+II w+p” +
((Lf
(2n-L+
X
p+J)’
/I l!-)
= -
X
(c)
(2n + 3,0) ((L-
(d)
l)+ I/ (d+f+f+d”)’
d
5(2n+
1)(2n-L+3)(2n+L+4)(L+ 4n(2n+3)~(2n-
((LS
8n2- (Lf
l,L+
l)(L$2) 1) 1)
(9)
1)
II L-) = 8n2 -::;;:y+“’
3(2n-L+1)(2n-L+3)(2n+L+4)(L+l) 28n(2n
+ 3)(p(2n - 1, L + 1)
K = 0 L- --+ (2~2 - 2,2) I)+ II (s’p +p+zy
K = 2 (L - l)+ transitions,
e.g. 3- j
2+,
II L-)=0,
(11)
((L - 1>+ II (dfJ7 +p+d)’
II L-) =o,
(12)
((L-
)I L-)=0.
(13)
1>+ jl (d+j’+f+d”)’
(2n f 3,0)
K = 0 L- --f (2n - 2,2)
(L+ (1 (s+p”+p+S)’
K = 2 L+ transitions,
e.g. 3- 4 3+,
/I L-) =o,
(14)
II (d+p +p+d)’
/I L-) =o
(15)
(L+ 11(d+f + f+d)’
IIL-) =o:
(16)
(L’
H.Y Ji et d/Nuclear
(e)
Physics A 6.58 (1999) 197-216
201
(2n+3,0)K=OL-~(2n-2,2)K=2(L+l)ftransitions,e.g.1-~2f,
((Id+ 1>+ 11(s+p” +p+S)’
11L-) =o,
(17) (18) ( 19)
((Lf
1>+ II w+ p+p+d>’
II L-)=0,
((Lt-
l)f
II L-)=0.
II (d+J+f+d>’
(0 (2n+l,l)K=lL-+(2n+2,0)K=O(L-l)+transitions,e.g
l-
-d. Of,
(20)
(CL- 1>+II w+p+p+d)’
4n2+2(4L+5)n-3(L-
11L-)=-
l)(L+2)
5(2n+
1)
x/w,
((L-
1>+ II (&f+ftJ)’
(21)
1)~-,=-‘“““‘““2(;~:“,r
X
1)(L+2)
(2n-L+3)(L+
1)
(g)
(2n+1,1)K=1L~~(2n+2,0)K=OLftransitions,e.g.2~--t2~,
(Lf
11(s+p +p+S)’
(L+ II (&
(Lf
(h)
(22)
.
28n(2n+3)
+
11(d+J+
&j)l
I/ L-) = II L-)
=
_
(23) (2n
-
L +
2)
(2n
+
L +
3)
(24)
f+d”)’ II L-)=-
(25)
(2n+~,~)K=lL-~(2n+2,0)K=O(L+l)‘transitions,e.g.l-~2’,
(CL+ l)+ 11cs+p” +p+S)l11L-) =_(2n-L+1)(2n+3L+6) (2n+
1)
P”+p+w (I L-) (CL+ 1>+11cd+ =-
4n2-2(4L-
l)n-3(L5(2n + 1)
(2n+L+4)L
l)(L+2) J
12n(2n + 3)
’
(27)
H.Y. Ji et al./NuclearPhysicsA 658 (1999) 197-216
202
l) + II ( d t f + f * d )
(it+
' II L - ) /(2n+L+4)L
16n 2 - 1 2 ( L + l ) n + 3 ( L - l ) ( L + 2 ) ----
5(2n+1)
V 2--~n(2n;~
(i) ( 2 n + l, I) K = 1 L - -+ ( 2 n - 2 , 2 )
~
x
1) + transitions, e.g. I - ---+0+,
2n - 3L + 3 2n(2n + 1)
(n+ l)(2n+ L + 2)(L + l)~o(2n- l , L - 1) 15(2n + 3)
1) + [[ ( d t ~ + p t d )
((L-
(L-
1 II L-> =
1) + I[ (Stfi-[-P*S)
((L-
K=0
(28)
"
(29)
'
III g )
8n 3 + 1 2 ( 2 L + 1)n 2 - 2 ( L -
1)(2L- 3)n-
3(L-
1 ) 2 ( L + 3)
10n(2n + ! )
~ ( n + 1)(2n+ L + 2)(L + I) x
3(2n+3)~o(2n-l,L-l)
(30)
'
< ( L - I) + II ( d t f + f t d ) ' II g - ) 32n 3 - 2 4 ( L - 2)n 2 - 2 ( L - l ) ( 3 L - 2)n + 3 ( L - 2 ) ( L
-
l)
2
lOn(2n + 1) x~(n+l)(2n+L+2)(L+l) 7(2n + 3 ) ~ ( 2 n - 1 , L -
(31)
1)
(j) (2n + l, 1) K = 1 L - --+ ( 2 n - 2 , 2 ) K = 0 L + transitions, e.g. 2 - -+ 2 + ,
( ( L ) + II ( s i p + p i g ) '
Ir L - ) -
2n(2n + 1)
×V/(2n - L + 2 ) ( n + 1)(2n + L + 3 ) ( 2 L + 1 ) ~ ( 2 n - 1,L) ((L) + II (dtfi + p t d ) l
15 4n 2 + L 2 + L - 3
l] L - ) :
lOn(2n + 1)
× ~ ( 2 n - L + 2)(n+ l)(2n+ L + 3)(2L + l) 3~(2n-
((L) + II ( d t f + f * d )
~/
(2n-
x
(32)
1,L)
I II t - > =
16n 2 -
L 2 -
(33)
L - 2
10n(2n + 1)
L + 2 ) ( n + l ) ( 2 n + L + 3 ) ( 2 L + 1) 7 ~ ( 2 n - 1,L)
(34)
203
H.Y Ji et al./Nuclear Physics A 658 (1999) 197-216 (k)
(2n + 1,1) K = 1 L- ~
(2n-2,
2) K = 0 ( L +
1) + transitions, e.g. 1 - ---, 2 + ,
<(L+ 1) ÷ II (st~ +pt~)l II L-) 2n+3L+6 2n~-li
-
/(n+l)(2n-L+l)L~o(2n-l,L+l)
V
< i t + 1) + II ( d t p + p t d ) 8n 3 - 1 2 ( 2 L + l ) n
V/ x
15(2n + 3) '
(35)
'
II t->
2-2(L+2)(2L+5)n+3(L-2)(L+2) 10n(2n + 1)
2
(n + 1 ) ( 2 n - L + I)L 3(2n+ 3)~(2n- 1,L+ 1)'
(36)
<(L+ I) + II ( d t f + f t d ) l II L-> 32n 3 + 24(L + 3)n 2 - 2(L + 2)(3L + 5 ) n - 3(L + 2)2(L + 3) 1 0 n ( 2 n + 1) / (n+l)(Zn-L+l)L xV7(Zn~-3)~(-----~-_-l,----~T~)
(37)
(e) ( 2 n + 1,1) K = 1 L- ---, ( 2 n - 2,2) K = 2 ( L - 1) + transitions, L is odd, e.g. 3- _-, 2 + '
(38)
( ( L - 1 ) + ]1 (st/~ + P t g ) ' II L-> =0, ( ( L - 1) + [[ (d*p + p t d ) 1 t[ L-> = _ ½ ~ 3 ( n + 1)(2n + 3 ) ( L - 2 ) ( L 2n(2n + 1)L x
~/
( 2 n - L + 1 ) ( 2 n + L)(2n + L + 2 ) p(2n - I , L - 1)
1) + II ( d t f + f t d )
((g-
= _~/(n
x
1)
1
(39)
II L-)
+ 1)(2n + 3 ) ( L - 2 ) ( L 14n(2n + 1)L
~/
1)
( 2 n - L + 1)(2n + L)(2n + L + 2) ~ ( 2 n - 1 , L - 1)
(4O)
(m) ( 2 n + 1, 1) K = 1 L- ~ ( 2 n - 2 , 2 ) K = 2 L + transitions, L is odd, e.g. 3- ~ 3 + , ((L)+ [[ (st/~ + p t g ) , [I L - ) = 0 ,
(41)
204
14.Y. Ji et al./NuclearPhysicsA 658 (1999) 197-216 ((L) + II ( d t # + p t d ) ~ II t->
1 ~6(n+l)(2n+3)(L-1)(L+2)(2L+l)
_
20n
( 2 n + I ) L ( L + 1)
x V / ( 2 n - L + 1)(2n + L + 2 ) ,
(42)
( ( t ) + II ( d t f + f t d )
l II z - ) 1 ~2(n+l)(2n+3)(L-I)(L+2)(2L+I)
_
10n
7(2n+I)L(L+I)
x V/(2n - L + 1 ) (2n + L + 2 ) .
(43)
(n) ( 2 n + 1, 1) K = 1 L - --+ (2n - 2,2) K = 2 ( L + 1) + transitions, L is odd, e.g. 1 - - - + 2 +, ((t+
1) + II (st#+pt~)'
((t+
~)+ II ( d t # + p t d ) ' II t->
_½/3(n +
V
~/
1 ) ( 2 n - L + 1)(2n + L + 2) ~o(2n- 1 , L + 1)
( ( L + 1) + 1[ ( d t f + f t d )
=-~/(n+
(44)
1)(2n + 3 ) ( L + 2 ) ( L + 3) 2n(2n + I ) ( L + 1)
(2n- L-
x
II/--) =0,
(45)
I II Z - )
1)(2n+ 3)(L + 2)(L + 3) 14n(2n+l)(L+l)
×
¢
(2n-L-l)(2n-L+l)(2n+L+2) ¢p(2n - 1 , L + 1)
(o) ( 2 n + 1,1) K = l L - ---, ( 2 n - 2,2) K---2 ( L 4 - --~3 +, ((L-
1) + II (st#+ptg)
((L_
(46) 1) + transitions, L is even, e.g.
' I[ L-)=0,
II (dt# + ptd) ' tl L-)
1 ~6(n+I)(L-2)(L-1) 20n
(2n + 1)L
x V/(2n - L -t- 2) (2n + L + 1) (2n + L -t- 3 ) ,
< ( t - l) + II ( d t f + f t d )
_
(47)
(48)
1 II t - )
1 ~2(n+I)(L-2)(L-1) 10n
7(2n + I ) L
x V / ( 2 n - L + 2)(2n + L + 1)(2n + L + 3 ) .
(49)
H. Z Ji et al. /Nuclear Physics A 658 (1999) 197-216 (p)
(2n+l,l)
K= 1 L- 4
(2n-2,2)
(L+ I( (s+p + p+:y
11L-)
(L+ 11(d+p” +p+d)’
=
l)(L+2)(2L+ l)L(L+
1) 1)
d J d~~~
(2n-L)(2n-L+2)(2n+L+1)(2n+L+3)
11w+f+ _2
(51) 1,L)
II L-)
f+d)’
(n+ l)(L-
5
l)(L+2)(2L+
14n(2n+
l)L(L+
1)
1)
(2n-L)(2n-L+2)(2n+L+1)(2n+L+3)
X
d2n-
(q)
(2n+l,l)
+ 2+, (50)
d2n-
(Lf
Liseven,e.g.2-
= 0,
2n(2n+
X
L+ transitions,
II L-)
l)(L-
3(n+
K=2
205
K=
1 L-
(‘52)
1,L)
---f (2n-2,2)
K=2
(L+l)+
transitions,
Liseven,e.g.
2- ----)3f,
((L+ l)+ 11(s+p+p+q’ 11L-) II L-)
l)+ (I (d+ p+p+d”y
((L+ =--
1
6(n+
20n
(CL+ l)+
L)(2n-
l)(L+
2(n+ d
1)
L+2)(2n+
II (d+f+f+d)’
1 10n
l)(L+2)(L+3) (2n+
xJ(2n-
(53)
=o,
L+3),
(54)
II L-)
l)(L+2)(L+3)
7(2n+
l)(L+
1)
xJ(2n-L)(2n-L+2)(2n+L+3).
(55)
2.3. The MI transition matrix elements (a)
(2n
+
1,1) K = 1 L-
+
(2n + 3,0)
X
K = 0 (L - l)-
J
3(2n-
transitions,
L+4)(L+
8n(n+1)(2n+3)
1) ’
e.g. 2- -
l-,
(56)
206
tl. Y Ji et al./Nuclear Physics A 658 (1999) 197-216
((L-
16n 2 + ( L - 2 ) ( L + 3)
1 ) - [[ (ftf)~ II t - ) =
10(2n + 1 )
× /3(2n-L+4)(L+l)
( ( L - 1)- II ( d t d ) ' II L-)=~/
3n(2n- L + 4 ) ( L + 1
15(7.-;-53
V
(57)
(58)
(b) ( 2 n + 1,1) K = 1 L - ~ ( 2 n + 3,0) K = 0 ( L + 1 ) - transitions, e.g. 2 - --~ 3 - , ( 2 n - L + 2)(2n + L + 3) 5(2n + 1)
<(L+ t)- II (P*~)~ II L->=
3(2n+L+5)L 8(n+l)(2n+3)'
×
((L+ 1)- II (fff), II L->=-
16n 2 + ( L - 2 ) ( L + 3) 10(2n + 1) 3(2n+L+5)L 28n(n+l)(2n+3)'
x
/ <(Z+ 1)- II (dXd)' II L - ) = I /
V
(59)
3n(2n + L + 5)L lO(n + l ) ( 2 n + 3)
(60)
(61)
3. Studies of octupole vibration in 152Sm
3.1. The spectrum The spectrum of the spdf SU(3) is compared with data [34] in Fig. 1. The parameters are set as aj = - 7 . 4 9 keV, e_ = 4.65 MeV, a 3 = - - 8 . 9 0 keW, and a4 is 16.75 keV for the positive parity states and 9.67 keV for the negative parity states, respectively. The smaller value of a4 for the negative parity states reflects an increase of moment of inertia for the negative parity states. This phenomenon has also been found in uranium nuclei [ 35 ]. The general agreement between experiment and calculation is quite good. The five bands (3 with positive parity and 2 with negative parity) are all well reproduced. However, as expected, due to the approximate nature of the SU(3) dynamic symmetry in this nucleus, there are discrepancies between them, for instance the degeneracy of/3 and y bands is broken in experiment. These detailed structures should be the task of an elaborate numerical studies. Here we are content with the result that the gross structure of the nucleus can be accounted for well by the SU(3) dynamic symmetry.
H.Y. Ji et aL/Nuclear Physics A 658 (1999) 197-216 C.tt~v
~. IbL
"r
11~
I i - 0 -
rhm O--
1--
II-m-
"7-~,--
9ar"
~'--
&4.
a~
• -
0 ,'1" _ _
~"--
-.
li4.
O--
,,,
7--
9-
ts-
.4.
7
~..
~ _ _
l-
4-
.4.
~_
2tM ~
..
1-__ ~"
"t
4-
~ 1.0
P
1~.___._.
4 4 . 1 . •,~4. _ _ ~ , - -
207
-
'~-4 "~
4 4-
~.4.
~.
1 - ~
~r~-~
~,....
0.0
F i g . 1. S p e c t r a o f 152Sm.
3.2. The E1 transition rates We have applied the results in Section 2 and those in Ref. [28] to lS2Sm with negative parity states. As for the transition operator, simplicity can be obtained if one chooses the transition operator as some generator of the dynamical group. In the sdpfIBM, there is no SU(3) generator that can be used as E1 transition operator. However, there is an O(10) group generator [ 17] that can change the parity and can be used as E1 transition operator. However, this should not be taken too seriously because the nuclear Hamiltonian is a "residual" strong interaction, and the E1 transition is induced by electromagnetic interaction. There is no strong argument that the operator in the Hamiltonian and the operator in the transition should have the same form. With our present knowledge, the transition operator should be determined from experimental data. Since a generator form transition operator can sometimes bring simplicity, it is useful to explore if the O(10) generator can achieve some simplification. At first we take the transition operator as the O(10) generator, T(E1)~ --
e l D ~1 ,
(62)
where
D~=v~(st~+pt~)~-V~3(ptd+dt~)~+V~(dtf
+ ftd) ~.
(63)
The calculation values are listed in the column labelled Cal. 1 in Table 1. The agreement between experiment and calculation is very good, in particular for those transition from 0 - - b a n d to the positive parity states. The effect of the different terms in the transition
H. Y Ji et al./Nuclear Physics A 658 (1999) 197-216
208 Table 1 B(EI ) values in 152Sm K~
1i
Ij
Cal.l (W.u.)
Cal.2(W.u.)
Cal.3(W.u.)
Exp.(W.u.)
0000011I1-
1~1~113~3~2~2~2~2~
0g+ 2+ 25 2~+ 4~+ 0+s 25 2+ 3+
0.0042 0.0081 1.4 × 10-3 0.0055 0.0067 0.0015 3.8 × 1 0 - 4 0.0002 0.00035
0.0042 0.00118 4.3 × 10-3 0.0044 0.00119 0.0015 6.6 × 10-4 0.0042 0.0072
0.0037 0.0099 4.0 × 10-3 0.0039 0.0099 0.0010 3.3 × 10-4 0.0033 0.0056
0.0042(4) 0.0077(7) 1.3(4) x 10-4 0.0081(16) 0.0082(16) 0.0027 5.6 × 10-5 0.011 0.0061
operator may play different roles, a combination of them may conceal some of the properties of the individual term. For instance, in the S U ( 3 ) limit of the sdg-IBM, each term in the E2 transition operator has an L ( L + 3) dependence, which plays an important role at large L. But this dependence disappears for the S U ( 3 ) generator form of the E2 transition operator; this L ( L + 3) dependence term is concealed and leads to the reduction of the collectivity problem [36,37]. To see the effect of each individual term, we have calculated the reduced matrix elements of sp, dp and d f terms, respectively. The calculation results of dp term alone are listed in the column labelled Cal.2 in Table 1. We found that the experiment could be reproduced well by using solely the dp term in ~52Sm. This indicates that the dp term plays a leading role in the E1 transition in the low-lying states of 152Sm. The inclusion of p - b o s o n for the low-lying negative parity states is very important, as has been pointed out in Ref. [ 1 1-14]. Finally, we used numerical fitting to improve the agreement. The general E1 transition operator is T(E1)~=el[(stp+pt~)~+Xdp(Ptd+dtp)~+Xdf(df
+ ftd)~].
(64)
Since we have fixed the coefficient of sp term to 1, we can evaluate the relative weights of these three terms through the values of ×do and Xdf. The results have been listed in the column labelled Cal.3 in Table 1, and the parameters are Xap = 81.669, .)t"df= - 4 . 9 7 5 . As expected, the value of Xdp evinces that dp term is far more important than sp and d f terms.
4. Applications to other deformed rare-earth nuclei We have expanded our work to other deformed nuclei, namely 154Sm, 156-16°Gd, 16°-162Dy, where E1 experimental data are available. The results are shown in Table 2 together with the experimental data [ 3 8 - 4 3 ] . The arrangement of Table 2 is the same as that in Table 1. The parameters el, Xdp and Xdf are listed in Table 3. The spectra of the nuclei considered here can be well described by the S U ( 3 ) dynamic symmetry.
H. Y Ji et al./Nuclear Physics A 658 (1999) 197-216
209
Table 2 B(EI ) values in deformed rare-earth nuclei Nucleus
K zr
li
If
Cal.1 (W.u.)
Cal.2(W.u.)
Cal.3(W.u.)
Exp.(W.u.)
154Sm
00001lI00000l11II000011l0000-
1~1~3~31 1~l~1~1~1~12 12 12 3~31 23 23 23 1~I ~31 3~2~2~21 12 1~I1 1~-
0~-s 2~-s 2+ 4~-s 0g-s 2+ 03 0~-s 2~-s 23 0~-s 2~-s 2~-s 4+ 1~2~3~0~s 2+ 2+ 4+ 2~s 2+ 3+ 0~-s 2+ 0+s 2+
0.0062 0.0120 0.0081 0.0099 0.0005 0.0004 0.00015 0.0025 0.005 9 × 10 - 6 0.0035 0.0068 0.00047 0.00088 6.2 × 10 --~ 3.16 × 10-5 2.54 x 10 - 4 0.0032 0.0062 0.0042 0.0052 1.4 × 10- 6 1.8 x 10 -7 3.0 × 10-7 3.8 × 10 -6 7.4 × 10 -6 0.0026 0.0051
0.0062 0.0163 0.0064 0.0162 0.0007 0.0001 0.00040 0.0025 0.006 0.0025 0.0035 0.0088 0.00105 0.00009 2.6 × 10 -5 5.00 × 10- 4 1.22 × 10-3 0.0032 0.0079 0.0035 0.0077 1.2 × 10 -7 3.4 × 10 -7 6.1 x 10-7 3.8 × 10 -7 9.4 × 10 -7 0.0026 0.0064
0.0041 0.0106 0.0045 0.0104 0.0018 0.0009 0.00020 0.0029 0.0051 0.0004 0.0028 0.0056 0.00029 0.00041 1.3 × 10 -5 1.19 × 10 -5 1.11 x 10 -5 0.0021 0.0041 0.0026 0.0035 3.10 × 10 -8 1.89 × 10 -7 2.49 × l0 -7 7.3 × 10 -7 1.5 × 10 -6 0.0026 0.0060
0.0062(4) 0.0117(26) 0.0078(10) 0.0093(14) 0.0016(12) 0.0020(15) 0.00030(24) 0.0025(14) 0.0006 (3) 0.0007(5) 0.0035(8) 0.0063(16) 0.00033(10) 0.00029(8) 6.4(8) × 10 -5 1.21(5) × 10 -5 1.89 × 10-4(24) 0.0032(9) 0.0060(17) 0.0016(5) 0.0013(4) 3.10(17) × 19 -~ 1.89(10) × 10- 7 2.49(13) × l0 -7 3.8(4) × 10 -3 6.8(5) x 10 -3 0.0026(4) 0.0060(19)
~56Gd
158Gd
16°Gd
16°Dy
L62Dy
Table 3 Parameters in El transition operator; the unit of el is in 0.28389 Al/3efm, which gives the B(EI) in Weissknopf unit Nucleus
E1
J52Sm 154Sm 156Gd 158Gd 16°Gd 16°Dy 162Dy
0.00101 0.00097 0.0118 0.0136 0.0120 0.0007 0.0417
Xdp 81.67 83.t5 - 1.62 -3.83 -2.08 -0.49 -0.754
Xdf -4.98 -8.59 4.37 3.68 1.61 -0.56 -0.35
Moreover, the E2 transition rates of the positive parity states can also be reasonably well described by the SU(3)
dynamic symmetry
F r o m T a b l e 2, w e c a n f i n d t h a t t h e B ( E 1 )
[44]. values from 0-
b a n d to g r o u n d s t a t e
210
H.Y. Ji et al./Nuclear Physics A 658 (1999) 197-216
band in the nuclei studied are all well reproduced. In particular, it is noted that the ratio B ( E I , 1- ---, 2g+)/B(E1, 1- ~ 0g+) is approximately 2 in the spdf IBM S U ( 3 ) octupole vibration limit. There is because there is a factor v ~ in the E1 transition matrix element for L - ~ ( L - 1) +, and a factor of v ~ + 1 in the E1 transition matrix element L - --+ (L + 1)+. The other factors in the two matrix elements are almost the same for N ~ 10. When L = 1, it gives a ratio of 2 for the two B ( E 1 ) values. This SU(3) octupole vibration property is all satisfied by the 7 nuclei studied in this work. This property can be used to extract the K values for the low-lying 1- state of deformed nuclei. For the K ~ = l - bandhead, the transition B ( E I ) to ground state band states is just the opposite, where the transition to 0g+ state is stronger than to 28+ state. In J6°Dy, it is noted that the experimental B(E1) values vary over a wide range from 6.8 × 10 -3 W.u. to 3.10 × 10 -8 W.u., an order of 5 change! If one fits the transition from K ~ = 0 - band, then the transition from K ~r = 1- band will be too large compared with experimental data. Conversely, if one tries to reproduce the transitions from K ~r = 1- band, then the transitions from K ~r = 0 - band will be too small. The same situation occurred also in Ref. [23]. We therefore suggest that the origins of the 12 state which is the bandhead of the K ~ = 0--band, and 2 7 state which is a member of the K ~ = 1- - b a n d are different. They cannot be described simultaneously in the spdf IBM. We also noticed that the assignment of the l~- state as a member of K ~ = 0 - - b a n d is only temporary in experiment [23,43]. If we consider only the transitions from 2 7 to positive parity states, agreement is obtained. In 15SGd, there are 7 B ( E I ) experimental data. The B ( E 1 ) values vary over also a wide range from 0.00633 W.u. to 1.21 × 10 -5 W.u.. They are all well reproduced by the spdf IBM SU(3) octupole vibration limit. To our present experimental knowledge, we can say that 158Gd is the best nucleus showing the SU(3) octupole vibration dynamical symmetry. We expect that with the development in experiment, other nuclei with this dynamical symmetry will be discovered in other mass regions, for instance in the auctioned region. From Table 3, we can find that in other nuclei: (a) both dp and df terms play important roles in these transitions, so we cannot acquire the good agreement only by dp term alone; (b) the values of Xdp and Xdf change in a relative narrow range, and dp and df terms have almost the same importance in these nuclei, which is different from 152Sm, and the signs of Xdp and Xdf remain invariant in a given isotope chain; (c) in the isotopes of Gd, the changes of Xdp and Xdf obey the following regularity: with the number of neutron increasing, Ydf decreases, however, the absolute value of Xdp increases first, then decreases. In 156Gd, the absolute value of Xdf (4.37) is greater than that of Xdp (-1.62), and in 16°Gd, the absolute value of Xdf (1.61) is less than that of Xdp(--2.08), which implies that dp and df terms may play different roles in different nuclei. In 158Gd, the absolute value of Xdp reaches its maximum. It requires more experimental data in relevant nuclei to check this relationship between the value of Xdp and the number of neutron in a given isotope chain. From the E1 transition formulae, we found that the El transition between 0 - band
H.Y Ji et al./Nuclear Physics A 658 (1999) 197-216
211
and y-band is zero in the SU(3) limit. However, such transitions occurred in 15ZSm, 16°Dy. In 152Sm, B(E1;4 + ---* 3~-) --3.2 x 10-5(13), this value is very small. A small symmetry-breaking will give a nonzero value for this transition. For instance if we introduce some mixing between the/3 and y band, that is,
I/3>'-- al I/3) +
a=lY),
(65)
lY)' = a21/3) + allY),
(66)
where the apostrophe labelled the new /3-band and y-band. When la21 = 0.0118, B ( E I ' ,4 r+ --+ 31) = 3.2 x 10 -5, and because la21 is so small that other properties of y-band could not be disturbed much. However, in 16°Dy, B(E1; 11 ---* 2 +) = 0.018(18), a very large value. A simple symmetry breaking cannot solve this problem. This is another reason that the 12 state in ~6°Dy may not belong to the K ~ = 0 - band. In 16°Dy, the E1 transition between 3 3 of possible K ~ = 0--band to y band are also observed, which confirmed us that this band is not likely the 0--band. Besides absolute B(E1) data, we also compare relative intensities. If one assumes that E1 is dominant in parity changing transitions and ignore higher order transitions, then the transition probabilities and hence the intensities are int ~x E 3 B ( E 1 ) .
(67)
Using the values of Xdp and Xdf obtained in the previous work for the B(E1) value, we have calculated the relative intensities in 152'154Sm [34,38], 156'158'16°Gd [39-41], nucleus and from Ref. [23,43]. The calculated results are shown in Table 4 with experimental data. From Table 4, we see besides general agreement, there exist serious deviations in J52A54Sm: in experiment, the relative intensity from 1- of K ~ = 1--band to 2g+ is much greater than that to 0g+, but in calculation, the situation is reversed. In general, the square of reduced matrix element for transition from 1- state of the K ~ = l - = b a n d to 0-g~sis almost as twice as that to 2~-s. The calculated ratio of intensities should be of the order of 1. However, we noticed that in 152Sm, the transition to 2g+ also involves M2 transitions. In the calculation we have just included the E1 contribution. In 154Sm, the identification of 1- state is only tentative. Experimental measurement of the E1 transition rate from 1- to 2+ will be very useful for checking this calculation, since in our model there is little freedom to adjust this transition rate.
5. Summary We have given analytical expressions for E1 and M1 transitions involving low-lying negative parity states in the SU(3) limit. These El transition formulae plus those given in Ref. [28] exhaust the whole possibilities of E1 transitions from 0 - and 1- band to the ground state band, the beta and gamma band. We also applied these analytical results to some deformed rare-earth nuclei to check the SU(3) octupole vibration prediction.
212
H.Y. Ji
et al./Nuclear Physics A 658 (1999) 197-216
Table 4 Comparisons of relative intensities Nucleus
Elevel (keV)
Kr
li
It
Cal.
Exp.
152Sm
963
0-
1~
0g~ 2~ 23 2~ 48~ 48~ 6~ 6~ 8~ 0~ 28~ 03 23 28~ 23 2¢ 3¢ 2~ 4~ 7+ 43 2~ 4~ 48~ 6~ 8~ 10~ 68~ 5~ 6~ 8~ 88~ 10~ 83
55.6 I00 0.24 98.2 100 100 72 100 51 100 13.3 8.38 0.035 100 4.5 10.50 5.283 100 1.15 14.4 0.289 2.11 1.2 40 100 100 31 100 5.4 100 14 100 32 58
82.3(7) 100(2) 0.010(3) 100(4) 40.4(19) 100(3) 24(3) 100(3) 12 0.85(6) 100(3) 0.09(5) 1.42(25) 100(ll) 0.28(4) 13.43(9) 2.127(24) 35.3(3) 100(5) 6.6(3) 1.30(9) 2.07(15) 0.41(5) 67(34) 100(33) 100(3) 15 82(20) 100(4) 100(7) 71(24) 100(5) 27(3) 13(4)
1%
~oo
lOO(5)
12~ 10~ 0~ 2~ 2g~ 4~
62 72 52 100 83 100
59(24) 24 65(2) 100(2) 100(2) 60(1)
4~
100
100(3)
68~ 0~ 2~ 2~ 23
98 100 15 100 4.1
29.0(7) 0.5 100 100(19) 5.4(12)
1041
37
1221
5~
1505
7?
1511
t54Sm
1-
I~
1530
2?
1579
3~
1764
5~
1879
0-
9~
1930
1-
6?
2004
7~
2291
9~
2641
115
921
0-
1~
1012
3~
118l
5?
1476 1515
1-
1~ 2?
213
H.Y. Ji et al./Nuclear Physics A 658 (1999) 197-216
Table 4 - - continued Nucleus
156Gd
Elevel (keV)
li
If
Cal.
Exp.
1585
32 52
100 1.6 100 2 100 54 7.0 x 10-6 100 87 100 7.0 x 10-5 61.1 100 0.09 100 86 100 94 100 82 100 51 100 76 100 75.9 5.1 2.75 2.01 64 100 100 85 0.002 19.9
26.6(25) 100(3)
1774
2+ 4+ 4+ 6+
1242
K rr
1-
17
%
2;
o;, 1276
37
2+
4~+~ 1320 1366
o-
2~-
2~-s
1;
o;
2;
2;
2~
1539 1638
32 1-
1958
2+
4;
77
6~-s
9~-
8~s
17
%
8;
lO+
'58Gd
977
1-
2;
1042
4; 1176
57
4+
1260
2~
1/ 2/ 31
1;
o;
3;
2; 2; 4;
1263
o-
1403
6;
2~
1407
1-
4~
3/ 41 51
o-
52
%+
o-
1?
o;
1290
37
2;
1428
51
4~-s 4~-s
1639 16OGd
1224
6;
2;
6;
1.29
5.21 100 61 60.5 100 100 84 100 55
40(5) 100(4) 97.1(5) 100(10) 0.07(2) 100(2) 44(2) 100(3) 0.21(2) 54.5(3) 100(1) 0.08(3) 94(6) 100(6) 100(3) ~33 100(3) 7(3) 100(5) 76(4) 100(20) 47.2(9) 100(6) 26.7(16) 5.1(46) 0.56(4) 6.9(6) 68(4) 100(6) 100(6)
85(5) o.o4(1) 19.9(12) 0.33(2) 6.6(5)
100(8) 33(5) 65.2(15) 100.0(22) 100(2) 52(2) 100(6) < 45
214 Table
H.Y. Ji et al./Nuclear Physics A 658 (1999) 197-216
4 -- continued 1640 t6ODy
162Dy
1359
77 I-
2~
1399
3~
1535
4~
1489
*0-
1~
1276
0-
IF
1358
3~
1519
5~
1692
I-
2~
6~ 8~ 2~ 2~ 3~ 2g~ 4~ 2~ 3~ 4~ 4~ 3~ 4~ 5~ 60 2~ 0~ 2~ 2~ 4g~ 4~ 6g~ 2g% 3~
100 36 100 1.00 0.88 100 83.8 0.43 1.77 0.65 100 0.55 1.22 0.29 69(14) 100 53 100 84 100 100 96 58 100
28(11) 100(22) 100(3) 17.95(12) 11.59(5) 100.0(13) 54.7(5) 0.81(3) 0.50(3) 0.26(3) 100(9) 79.8(14) 13.4(6) 19.6(9) 100 52(14) 100(4) 100(30) 46(4) 100(20) 40(8) 100(3) 49(6)
Though there has been some deviation, the main features are checked by the experimental data. Some of the discrepancies are expected because of the approximate nature of the dynamical symmetry in real nucleus, for instance the breaking of the beta and gamma band states degeneracies, some small transitions between gamma band and K ~ = 0 band. They can be resolved by symmetry breaking. Some of the discrepancies cannot be nailed down at the moment due to lack of experimental data, for instance, the K ~"= 0 - band identification in J6°Dy. Another important issue which remain to be checked by experiment is the anomalous intensity ratio of K ~" = 1- bandhead in 152'154Sm. Comparing with the sd-f IBM studies of Cottle and Zamfir [23], we see that the agreements of the calculations of the two models with experimental data are of the same quality. In this spdf IBM study, the p boson is very important in these deformed nuclei, even critical in some nucleus, i.e. 152Sm. In the sd-f IBM study, the f boson is most important. From these empirical studies alone, it is not conclusive which model is more appropriate. However, the spdf IBM is more general, and sd-f IBM is only a special case of the spdf IBM. The two calculations can be thought as being two different parameterizations of the spdf IBM. In the sd-f IBM, it can be considered as a special case in which the energy of the p boson is put infinitely high. In the SU(3) octupole vibration limit, the energy of the p boson is put equal to the energy of the f boson. However, in view of the simple mathematical structure of spdf IBM, the microscopic studies [ 14] and the findings by Kusnezov [ 15], the spdf interacting boson model is one promising model for describing negative parity collective states in even-even nuclei.
H. Y Ji et al./Nuclear Physics A 658 (1999) 197-216
215
Acknowledgements The authors acknowledge the support of China National Natural Science Foundation, China State Education Ministry, Fok Ying Tung Education foundation. EG Zhao also acknowledges the support of Advanced Visitors Scheme of Tsinghua University. The authors also thank the referee for expert reading and helpful criticism of the manuscript.
Corrigenda In Table 1 of Ref. [28], the lines concerning 156Gd should be replaced by Nucleus
Ii
If
B ( E 1 , Ii --+/f)cal
156Gd
1+
0+ 2+
0.0025 0.005
(W.u.)
B ( E 1 , I i --+/f)exp
(W.u.)
0.0025(14) 0.006(3)
References [ 1] 12] [31 [4] 15] [6] 17l I81 [9] 110] 1111 1121 1131 1141 I 151 1161 [ 17] [181 1191 [201 1211 1221 1231 [241 1251 1261 I271 1281 1291
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[301 N. Yoshinaga, Y. Akiyama, A. Arima, Phys. Rev. Lett. 56 (1986) 1116. [311 R.F. Casten, D.D. Warner, in: Prog. Part. Nucl. Phys., Vol. 9, D. Wilkinson, ed. (Pergamon, Oxford, 1983) p. 311. [321 J.D. Vergdos, Nucl. Phys. A 111 (1968) 681. [331 G.L. Long, J. Phys. A 28 (1995) 6417. 1341 A. Anna-cohen, Nucl. Data Sheets 79 (1996) 1. 1351 G.L. Long, W.L. Zhang, H.Y. Ji, S.J. Zhu, Science in China A 41 (1998) 1296. 136] S.C. Li, S. Kuyucak, Nucl. Phys. A 604 (1996) 305. [371 G.L. Long, H.Y. Ji, Phys. Rev. C 57 (1998) 1686. [381 R.G. Helmer, Nucl. Data Sheets 69 (1993) 507. 1391 R.G. Helmer, Nucl. Data Sheets 65 (1992) 65. [401 R.G. Helmer, Nucl. Data Sheets 77 (1996) 471. 1411 C.W. Reich, Nucl. Data Sheets 78 (1996) 547. 1421 R.G. Helmer, Nucl. Data Sheets 64 (1991) 79. 1431 R.B. Firestone, V.S. Shirley, C.M. Baglin, S.Y.S. Chu, J. Zipkin, eds., Table of Isotopes, 8th ed. (Wiley, New York, 1996). [441 G.L. Long, W.L. Zhang, H.Y. Ji, J.E Gao, J. Phys G 24 (1998) 2133.
Nuclear Physics A 658 (1999) 197-216 www.elsevier.nl/locate/npe
Studies of the electric dipole transition of deformed rare-earth nuclei H . Y . Ji a, G . L . L o n g a'b'c'l, E . G . Z h a o a'b, S . W . X u d a Department of Physics, Tsinghua University, Beijing 100084, PR China b Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, PR China c Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Accelerator, Chinese Academy of Sciences, Lanzhou, 730000, PR China d Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou, 730000, PR China
Received 22 February 1999; revised 15 June 1999; accepted 28 June 1999
Abstract Spectrum and electric dipole transition rates and relative intensities in 152-154Sm, 156-16°Gd, 16°-162Dy are studied in the framework of the interacting boson model with s,p,d,f, bosons. It is found that E1 transition data among the low-lying levels are in good agreement with the SU(3) dynamical symmetry of the spdf interacting boson model proposed by Engel and Iachello to describe collective rotation with octupole vibration. These results show that these nuclei have SU(3) dynamic symmetry to a good approximation. Also in this work many algebraic expressions for electric dipole transitions in the SU(3) limit of the spdf-IBM have been obtained. These formulae together with the formulae given previously exhaust nearly all the E1 transitions for low-lying negative parity states. They are useful in analyzing experimental data. @ 1999 Elsevier Science B.V. All rights reserved. Keywords: Spdf interacting boson model; Electric dipole transition; Octupole vibration; SU(3) dynamical
symmetry
1. Introduction There have been intensive interests in the studies o f octupole degree of freedom in nuclear structure recently both experimentally [ 1 - 9 ] and theoretically [ 10-27]. 1Corresponding author: Department of Physics, Tsinghua University, Beijing, 100084, PR China. E-mail: [email protected]. 0375-9474/99/$ - see front matter @ 1999 Elsevier Science B.V. All rights reserved. PII S0375-9474 (99) 00335-8
198
H.Y. Ji et al./Nuclear Physics A 658 (1999) 197-216
In the boson model, negative parity states are described by the spdf interacting boson model [10-12,14,13,16,17,19-21,28] or the sdf-IBM [22-26]. Otsuka [14] showed microscopically that p and f bosons are important in the low-lying negative parity states of even-even nuclei. The spdf IBM has been successful in the description of negative parity states in the Ba and rare-earth region [ 17,19,20]. The advantages of the algebraic approach is that it offers analytical expressions for the energy levels, electromagnetic transitions and many other quantities. The prediction and later experimental confirmation of the 0 ( 6 ) limit has been a well-known example of the success of the IBM [29]. The spdf IBM SU(3) limit is a dynamical symmetry describing octupole vibration in deformed nuclei proposed by Engel and Iachello [12,13]. It has been pointed that 232U or other actinide nuclei may be candidates for the SU(3) limit [ 12,13,16]. While the spectrum agrees with the theoretical calculation very well, there are very few electromagnetic transition data. Thus it is difficult to check on the validity of the dynamic symmetry properties, in particular on the electric dipole transitions connecting positive and negative parity states. This is because finding nucleus with both good rotational feature in spectrum and with ample data of electric dipole transition is difficult. Besides, in real nucleus dynamical symmetry is usually somehow broken, for instance, even one of the best candidate of the sdg IBM SU(3) dynamic symmetry has some degree of symmetry-breaking [30]. But dynamic symmetries are very useful even though they are broken. They can be used to classify states and characterize the collective features of a nucleus. To a given nucleus near a dynamical symmetry, at first the gross structure is dictated by the dynamical symmetry. Then detailed structure of the nucleus can be attributed to symmetry breaking, and its description is the task of a more elaborate study. In addition, the dynamical symmetries can be used as landmarks in the nuclear chart table to classify typical collective motions, and other vast majority of nuclei can be put into categories of transition between two dynamical symmetries. Such a scheme is very successful in the descriptions of the positive low-lying states of even-even nuclei [ 31 ]. In this work we have studied the negative parity states of the rare-earth nuclei, namely 152-154Sm, 156-16°Gd, 16°-162Dyusing the SU(3) limit of the spdf interacting boson model. These nuclei are deformed and have ample data of electric dipole transition. We have found in this work that to a good approximation, the structure of these rare-earth nuclei can be well described by the SU(3) limit of the spdf interacting boson model. The paper is organized in the following. In Section 2, we give a brief description of the model and the necessary expressions of electric dipole transitions. In Section 3, we apply the results to J52Sm on the spectrum and E1 transitions. In Section 4, we study other deformed rare-earth nuclei. Finally, we give a discussion and a summary in Section 5.
H.l! Ji et al. /Nuclear Physics A 658 (1999) 197-216
199
2. The model 2.1. The energy eigenvalues The energy eigenvalues
have been discussed
in Ref. [ 12,171. In the SU( 3) limit, the
group chain can be written as U(l6) n
1 U(6)
‘8 U(1) n_
n+
the energy eigenvalue E = c-&r
1 S&d(3)
@ S&r(3)
1 SUspdf(3) > o(3) L (@)
h-6)
(h+P+)
(1)
is
+ aiGsu+
+ a&u_
+ asGsu(s)
+ a4L(L + I) .
(2)
The g.s.-band, P-band and y-band are generated from (2n, 0), (2n - 4,2) K = 0 and (2n - 4,2) K = 2, respectively, with n sd-bosons. The low-lying negative parity are generalized by the SU( 3) IR from the decomposition of (2n, 0) @ (3,0) ; that is, K” = O- from (2n + 3,O) and KP = 1- from (2n + 1, 1) , respectively. For the low-lying states, we are interested The wave function
in only the g.s., /3, y, O- and l-
bands.
are given by
( 1) ground state band [(2n, 0) + LM) , (2) 0 band )(2n - 4,2)+K
= OLM)
(3) y band ](2n - 4,2)+K
= 2LM),
,
(4)
Kp = O- -
band 1(2n - 2,0)+(3,0)-(2n
+ 1,O)LM)
(5)
Kp = l-
band /(2n-
- 1,l)LM).
-
2,0)+(3,0)_(2n
(3)
The value of a2 is taken zero, since its effect in the spectrum the E- term for the low-lying determined
by experimental
2.2. The El transition
states with only one pf-boson.
,
is the same as that of
The parameters
are then
data.
matrix elements
Some of the El transition formulae have been given in Ref. [ 281. With the method described there, we calculate the following formulae of the El transition, which are needed for a comparison with experiment, and the required SU(3) Wigner coefficients can be found in Ref. [32,33]. (a) (2n+3,0) K=OL+ (2n-2,2), ((L-
X
1>+ I] (s+/?+p+s)’
d
(2n-
K=O(L]I L-j
=
l)+ transitions,e.g.
l- +Ot,
l (2n + 1)
L+3)(2n+L+2)(2n+L+4)p(2n20n(2n + 3)
l,L-
1)
(4)
H. E Ji et al. /Nuclear Physics A 658 (1999) 197-216
200
where
(5)
~(~,L)=2(A+1)2-L(L+I), ((L - l)+ II (d+ (2n-L+3)(2n+L+2)(2n+L+4)L X
4n(2n+3)p(2n((L - l)+ x
II (d+f+
f+d”)’ II L-j
(6)
’
1)
= -*$2yi)
I)
3(2n-L+3)(2n+L+2)(2n+L+4)L 28n(2n+3)p(2n-
J (b)
l,L-
(2n f 3,0)
l,L-
K = 0 L- ---) (2n - 2,2)
((LS
1>+
’
1)
K = 0 (L + l)+ transitions,
e.g., l-
+ 2+,
II (s+P+P+# II L-j = -(2ny 1>
d
3(2n-L+1)(2n-L+3)(2n+L+4)(L+1)1p(2n--l,L+l)
X
20n( 2n + 3) (8)
1>+II w+p” +
((Lf
(2n-L+
X
p+J)’
/I l!-)
= -
X
(c)
(2n + 3,0) ((L-
(d)
l)+ I/ (d+f+f+d”)’
d
5(2n+
1)(2n-L+3)(2n+L+4)(L+ 4n(2n+3)~(2n-
((LS
8n2- (Lf
l,L+
l)(L$2) 1) 1)
(9)
1)
II L-) = 8n2 -::;;:y+“’
3(2n-L+1)(2n-L+3)(2n+L+4)(L+l) 28n(2n
+ 3)(p(2n - 1, L + 1)
K = 0 L- --+ (2~2 - 2,2) I)+ II (s’p +p+zy
K = 2 (L - l)+ transitions,
e.g. 3- j
2+,
II L-)=0,
(11)
((L - 1>+ II (dfJ7 +p+d)’
II L-) =o,
(12)
((L-
)I L-)=0.
(13)
1>+ jl (d+j’+f+d”)’
(2n f 3,0)
K = 0 L- --f (2n - 2,2)
(L+ (1 (s+p”+p+S)’
K = 2 L+ transitions,
e.g. 3- 4 3+,
/I L-) =o,
(14)
II (d+p +p+d)’
/I L-) =o
(15)
(L+ 11(d+f + f+d)’
IIL-) =o:
(16)
(L’
H.Y Ji et d/Nuclear
(e)
Physics A 6.58 (1999) 197-216
201
(2n+3,0)K=OL-~(2n-2,2)K=2(L+l)ftransitions,e.g.1-~2f,
((Id+ 1>+ 11(s+p” +p+S)’
11L-) =o,
(17) (18) ( 19)
((Lf
1>+ II w+ p+p+d>’
II L-)=0,
((Lt-
l)f
II L-)=0.
II (d+J+f+d>’
(0 (2n+l,l)K=lL-+(2n+2,0)K=O(L-l)+transitions,e.g
l-
-d. Of,
(20)
(CL- 1>+II w+p+p+d)’
4n2+2(4L+5)n-3(L-
11L-)=-
l)(L+2)
5(2n+
1)
x/w,
((L-
1>+ II (&f+ftJ)’
(21)
1)~-,=-‘“““‘““2(;~:“,r
X
1)(L+2)
(2n-L+3)(L+
1)
(g)
(2n+1,1)K=1L~~(2n+2,0)K=OLftransitions,e.g.2~--t2~,
(Lf
11(s+p +p+S)’
(L+ II (&
(Lf
(h)
(22)
.
28n(2n+3)
+
11(d+J+
&j)l
I/ L-) = II L-)
=
_
(23) (2n
-
L +
2)
(2n
+
L +
3)
(24)
f+d”)’ II L-)=-
(25)
(2n+~,~)K=lL-~(2n+2,0)K=O(L+l)‘transitions,e.g.l-~2’,
(CL+ l)+ 11cs+p” +p+S)l11L-) =_(2n-L+1)(2n+3L+6) (2n+
1)
P”+p+w (I L-) (CL+ 1>+11cd+ =-
4n2-2(4L-
l)n-3(L5(2n + 1)
(2n+L+4)L
l)(L+2) J
12n(2n + 3)
’
(27)
H.Y. Ji et al./NuclearPhysicsA 658 (1999) 197-216
202
l) + II ( d t f + f * d )
(it+
' II L - ) /(2n+L+4)L
16n 2 - 1 2 ( L + l ) n + 3 ( L - l ) ( L + 2 ) ----
5(2n+1)
V 2--~n(2n;~
(i) ( 2 n + l, I) K = 1 L - -+ ( 2 n - 2 , 2 )
~
x
1) + transitions, e.g. I - ---+0+,
2n - 3L + 3 2n(2n + 1)
(n+ l)(2n+ L + 2)(L + l)~o(2n- l , L - 1) 15(2n + 3)
1) + [[ ( d t ~ + p t d )
((L-
(L-
1 II L-> =
1) + I[ (Stfi-[-P*S)
((L-
K=0
(28)
"
(29)
'
III g )
8n 3 + 1 2 ( 2 L + 1)n 2 - 2 ( L -
1)(2L- 3)n-
3(L-
1 ) 2 ( L + 3)
10n(2n + ! )
~ ( n + 1)(2n+ L + 2)(L + I) x
3(2n+3)~o(2n-l,L-l)
(30)
'
< ( L - I) + II ( d t f + f t d ) ' II g - ) 32n 3 - 2 4 ( L - 2)n 2 - 2 ( L - l ) ( 3 L - 2)n + 3 ( L - 2 ) ( L
-
l)
2
lOn(2n + 1) x~(n+l)(2n+L+2)(L+l) 7(2n + 3 ) ~ ( 2 n - 1 , L -
(31)
1)
(j) (2n + l, 1) K = 1 L - --+ ( 2 n - 2 , 2 ) K = 0 L + transitions, e.g. 2 - -+ 2 + ,
( ( L ) + II ( s i p + p i g ) '
Ir L - ) -
2n(2n + 1)
×V/(2n - L + 2 ) ( n + 1)(2n + L + 3 ) ( 2 L + 1 ) ~ ( 2 n - 1,L) ((L) + II (dtfi + p t d ) l
15 4n 2 + L 2 + L - 3
l] L - ) :
lOn(2n + 1)
× ~ ( 2 n - L + 2)(n+ l)(2n+ L + 3)(2L + l) 3~(2n-
((L) + II ( d t f + f * d )
~/
(2n-
x
(32)
1,L)
I II t - > =
16n 2 -
L 2 -
(33)
L - 2
10n(2n + 1)
L + 2 ) ( n + l ) ( 2 n + L + 3 ) ( 2 L + 1) 7 ~ ( 2 n - 1,L)
(34)
203
H.Y Ji et al./Nuclear Physics A 658 (1999) 197-216 (k)
(2n + 1,1) K = 1 L- ~
(2n-2,
2) K = 0 ( L +
1) + transitions, e.g. 1 - ---, 2 + ,
<(L+ 1) ÷ II (st~ +pt~)l II L-) 2n+3L+6 2n~-li
-
/(n+l)(2n-L+l)L~o(2n-l,L+l)
V
< i t + 1) + II ( d t p + p t d ) 8n 3 - 1 2 ( 2 L + l ) n
V/ x
15(2n + 3) '
(35)
'
II t->
2-2(L+2)(2L+5)n+3(L-2)(L+2) 10n(2n + 1)
2
(n + 1 ) ( 2 n - L + I)L 3(2n+ 3)~(2n- 1,L+ 1)'
(36)
<(L+ I) + II ( d t f + f t d ) l II L-> 32n 3 + 24(L + 3)n 2 - 2(L + 2)(3L + 5 ) n - 3(L + 2)2(L + 3) 1 0 n ( 2 n + 1) / (n+l)(Zn-L+l)L xV7(Zn~-3)~(-----~-_-l,----~T~)
(37)
(e) ( 2 n + 1,1) K = 1 L- ---, ( 2 n - 2,2) K = 2 ( L - 1) + transitions, L is odd, e.g. 3- _-, 2 + '
(38)
( ( L - 1 ) + ]1 (st/~ + P t g ) ' II L-> =0, ( ( L - 1) + [[ (d*p + p t d ) 1 t[ L-> = _ ½ ~ 3 ( n + 1)(2n + 3 ) ( L - 2 ) ( L 2n(2n + 1)L x
~/
( 2 n - L + 1 ) ( 2 n + L)(2n + L + 2 ) p(2n - I , L - 1)
1) + II ( d t f + f t d )
((g-
= _~/(n
x
1)
1
(39)
II L-)
+ 1)(2n + 3 ) ( L - 2 ) ( L 14n(2n + 1)L
~/
1)
( 2 n - L + 1)(2n + L)(2n + L + 2) ~ ( 2 n - 1 , L - 1)
(4O)
(m) ( 2 n + 1, 1) K = 1 L- ~ ( 2 n - 2 , 2 ) K = 2 L + transitions, L is odd, e.g. 3- ~ 3 + , ((L)+ [[ (st/~ + p t g ) , [I L - ) = 0 ,
(41)
204
14.Y. Ji et al./NuclearPhysicsA 658 (1999) 197-216 ((L) + II ( d t # + p t d ) ~ II t->
1 ~6(n+l)(2n+3)(L-1)(L+2)(2L+l)
_
20n
( 2 n + I ) L ( L + 1)
x V / ( 2 n - L + 1)(2n + L + 2 ) ,
(42)
( ( t ) + II ( d t f + f t d )
l II z - ) 1 ~2(n+l)(2n+3)(L-I)(L+2)(2L+I)
_
10n
7(2n+I)L(L+I)
x V/(2n - L + 1 ) (2n + L + 2 ) .
(43)
(n) ( 2 n + 1, 1) K = 1 L - --+ (2n - 2,2) K = 2 ( L + 1) + transitions, L is odd, e.g. 1 - - - + 2 +, ((t+
1) + II (st#+pt~)'
((t+
~)+ II ( d t # + p t d ) ' II t->
_½/3(n +
V
~/
1 ) ( 2 n - L + 1)(2n + L + 2) ~o(2n- 1 , L + 1)
( ( L + 1) + 1[ ( d t f + f t d )
=-~/(n+
(44)
1)(2n + 3 ) ( L + 2 ) ( L + 3) 2n(2n + I ) ( L + 1)
(2n- L-
x
II/--) =0,
(45)
I II Z - )
1)(2n+ 3)(L + 2)(L + 3) 14n(2n+l)(L+l)
×
¢
(2n-L-l)(2n-L+l)(2n+L+2) ¢p(2n - 1 , L + 1)
(o) ( 2 n + 1,1) K = l L - ---, ( 2 n - 2,2) K---2 ( L 4 - --~3 +, ((L-
1) + II (st#+ptg)
((L_
(46) 1) + transitions, L is even, e.g.
' I[ L-)=0,
II (dt# + ptd) ' tl L-)
1 ~6(n+I)(L-2)(L-1) 20n
(2n + 1)L
x V/(2n - L -t- 2) (2n + L + 1) (2n + L -t- 3 ) ,
< ( t - l) + II ( d t f + f t d )
_
(47)
(48)
1 II t - )
1 ~2(n+I)(L-2)(L-1) 10n
7(2n + I ) L
x V / ( 2 n - L + 2)(2n + L + 1)(2n + L + 3 ) .
(49)
H. Z Ji et al. /Nuclear Physics A 658 (1999) 197-216 (p)
(2n+l,l)
K= 1 L- 4
(2n-2,2)
(L+ I( (s+p + p+:y
11L-)
(L+ 11(d+p” +p+d)’
=
l)(L+2)(2L+ l)L(L+
1) 1)
d J d~~~
(2n-L)(2n-L+2)(2n+L+1)(2n+L+3)
11w+f+ _2
(51) 1,L)
II L-)
f+d)’
(n+ l)(L-
5
l)(L+2)(2L+
14n(2n+
l)L(L+
1)
1)
(2n-L)(2n-L+2)(2n+L+1)(2n+L+3)
X
d2n-
(q)
(2n+l,l)
+ 2+, (50)
d2n-
(Lf
Liseven,e.g.2-
= 0,
2n(2n+
X
L+ transitions,
II L-)
l)(L-
3(n+
K=2
205
K=
1 L-
(‘52)
1,L)
---f (2n-2,2)
K=2
(L+l)+
transitions,
Liseven,e.g.
2- ----)3f,
((L+ l)+ 11(s+p+p+q’ 11L-) II L-)
l)+ (I (d+ p+p+d”y
((L+ =--
1
6(n+
20n
(CL+ l)+
L)(2n-
l)(L+
2(n+ d
1)
L+2)(2n+
II (d+f+f+d)’
1 10n
l)(L+2)(L+3) (2n+
xJ(2n-
(53)
=o,
L+3),
(54)
II L-)
l)(L+2)(L+3)
7(2n+
l)(L+
1)
xJ(2n-L)(2n-L+2)(2n+L+3).
(55)
2.3. The MI transition matrix elements (a)
(2n
+
1,1) K = 1 L-
+
(2n + 3,0)
X
K = 0 (L - l)-
J
3(2n-
transitions,
L+4)(L+
8n(n+1)(2n+3)
1) ’
e.g. 2- -
l-,
(56)
206
tl. Y Ji et al./Nuclear Physics A 658 (1999) 197-216
((L-
16n 2 + ( L - 2 ) ( L + 3)
1 ) - [[ (ftf)~ II t - ) =
10(2n + 1 )
× /3(2n-L+4)(L+l)
( ( L - 1)- II ( d t d ) ' II L-)=~/
3n(2n- L + 4 ) ( L + 1
15(7.-;-53
V
(57)
(58)
(b) ( 2 n + 1,1) K = 1 L - ~ ( 2 n + 3,0) K = 0 ( L + 1 ) - transitions, e.g. 2 - --~ 3 - , ( 2 n - L + 2)(2n + L + 3) 5(2n + 1)
<(L+ t)- II (P*~)~ II L->=
3(2n+L+5)L 8(n+l)(2n+3)'
×
((L+ 1)- II (fff), II L->=-
16n 2 + ( L - 2 ) ( L + 3) 10(2n + 1) 3(2n+L+5)L 28n(n+l)(2n+3)'
x
/ <(Z+ 1)- II (dXd)' II L - ) = I /
V
(59)
3n(2n + L + 5)L lO(n + l ) ( 2 n + 3)
(60)
(61)
3. Studies of octupole vibration in 152Sm
3.1. The spectrum The spectrum of the spdf SU(3) is compared with data [34] in Fig. 1. The parameters are set as aj = - 7 . 4 9 keV, e_ = 4.65 MeV, a 3 = - - 8 . 9 0 keW, and a4 is 16.75 keV for the positive parity states and 9.67 keV for the negative parity states, respectively. The smaller value of a4 for the negative parity states reflects an increase of moment of inertia for the negative parity states. This phenomenon has also been found in uranium nuclei [ 35 ]. The general agreement between experiment and calculation is quite good. The five bands (3 with positive parity and 2 with negative parity) are all well reproduced. However, as expected, due to the approximate nature of the SU(3) dynamic symmetry in this nucleus, there are discrepancies between them, for instance the degeneracy of/3 and y bands is broken in experiment. These detailed structures should be the task of an elaborate numerical studies. Here we are content with the result that the gross structure of the nucleus can be accounted for well by the SU(3) dynamic symmetry.
H.Y. Ji et aL/Nuclear Physics A 658 (1999) 197-216 C.tt~v
~. IbL
"r
11~
I i - 0 -
rhm O--
1--
II-m-
"7-~,--
9ar"
~'--
&4.
a~
• -
0 ,'1" _ _
~"--
-.
li4.
O--
,,,
7--
9-
ts-
.4.
7
~..
~ _ _
l-
4-
.4.
~_
2tM ~
..
1-__ ~"
"t
4-
~ 1.0
P
1~.___._.
4 4 . 1 . •,~4. _ _ ~ , - -
207
-
'~-4 "~
4 4-
~.4.
~.
1 - ~
~r~-~
~,....
0.0
F i g . 1. S p e c t r a o f 152Sm.
3.2. The E1 transition rates We have applied the results in Section 2 and those in Ref. [28] to lS2Sm with negative parity states. As for the transition operator, simplicity can be obtained if one chooses the transition operator as some generator of the dynamical group. In the sdpfIBM, there is no SU(3) generator that can be used as E1 transition operator. However, there is an O(10) group generator [ 17] that can change the parity and can be used as E1 transition operator. However, this should not be taken too seriously because the nuclear Hamiltonian is a "residual" strong interaction, and the E1 transition is induced by electromagnetic interaction. There is no strong argument that the operator in the Hamiltonian and the operator in the transition should have the same form. With our present knowledge, the transition operator should be determined from experimental data. Since a generator form transition operator can sometimes bring simplicity, it is useful to explore if the O(10) generator can achieve some simplification. At first we take the transition operator as the O(10) generator, T(E1)~ --
e l D ~1 ,
(62)
where
D~=v~(st~+pt~)~-V~3(ptd+dt~)~+V~(dtf
+ ftd) ~.
(63)
The calculation values are listed in the column labelled Cal. 1 in Table 1. The agreement between experiment and calculation is very good, in particular for those transition from 0 - - b a n d to the positive parity states. The effect of the different terms in the transition
H. Y Ji et al./Nuclear Physics A 658 (1999) 197-216
208 Table 1 B(EI ) values in 152Sm K~
1i
Ij
Cal.l (W.u.)
Cal.2(W.u.)
Cal.3(W.u.)
Exp.(W.u.)
0000011I1-
1~1~113~3~2~2~2~2~
0g+ 2+ 25 2~+ 4~+ 0+s 25 2+ 3+
0.0042 0.0081 1.4 × 10-3 0.0055 0.0067 0.0015 3.8 × 1 0 - 4 0.0002 0.00035
0.0042 0.00118 4.3 × 10-3 0.0044 0.00119 0.0015 6.6 × 10-4 0.0042 0.0072
0.0037 0.0099 4.0 × 10-3 0.0039 0.0099 0.0010 3.3 × 10-4 0.0033 0.0056
0.0042(4) 0.0077(7) 1.3(4) x 10-4 0.0081(16) 0.0082(16) 0.0027 5.6 × 10-5 0.011 0.0061
operator may play different roles, a combination of them may conceal some of the properties of the individual term. For instance, in the S U ( 3 ) limit of the sdg-IBM, each term in the E2 transition operator has an L ( L + 3) dependence, which plays an important role at large L. But this dependence disappears for the S U ( 3 ) generator form of the E2 transition operator; this L ( L + 3) dependence term is concealed and leads to the reduction of the collectivity problem [36,37]. To see the effect of each individual term, we have calculated the reduced matrix elements of sp, dp and d f terms, respectively. The calculation results of dp term alone are listed in the column labelled Cal.2 in Table 1. We found that the experiment could be reproduced well by using solely the dp term in ~52Sm. This indicates that the dp term plays a leading role in the E1 transition in the low-lying states of 152Sm. The inclusion of p - b o s o n for the low-lying negative parity states is very important, as has been pointed out in Ref. [ 1 1-14]. Finally, we used numerical fitting to improve the agreement. The general E1 transition operator is T(E1)~=el[(stp+pt~)~+Xdp(Ptd+dtp)~+Xdf(df
+ ftd)~].
(64)
Since we have fixed the coefficient of sp term to 1, we can evaluate the relative weights of these three terms through the values of ×do and Xdf. The results have been listed in the column labelled Cal.3 in Table 1, and the parameters are Xap = 81.669, .)t"df= - 4 . 9 7 5 . As expected, the value of Xdp evinces that dp term is far more important than sp and d f terms.
4. Applications to other deformed rare-earth nuclei We have expanded our work to other deformed nuclei, namely 154Sm, 156-16°Gd, 16°-162Dy, where E1 experimental data are available. The results are shown in Table 2 together with the experimental data [ 3 8 - 4 3 ] . The arrangement of Table 2 is the same as that in Table 1. The parameters el, Xdp and Xdf are listed in Table 3. The spectra of the nuclei considered here can be well described by the S U ( 3 ) dynamic symmetry.
H. Y Ji et al./Nuclear Physics A 658 (1999) 197-216
209
Table 2 B(EI ) values in deformed rare-earth nuclei Nucleus
K zr
li
If
Cal.1 (W.u.)
Cal.2(W.u.)
Cal.3(W.u.)
Exp.(W.u.)
154Sm
00001lI00000l11II000011l0000-
1~1~3~31 1~l~1~1~1~12 12 12 3~31 23 23 23 1~I ~31 3~2~2~21 12 1~I1 1~-
0~-s 2~-s 2+ 4~-s 0g-s 2+ 03 0~-s 2~-s 23 0~-s 2~-s 2~-s 4+ 1~2~3~0~s 2+ 2+ 4+ 2~s 2+ 3+ 0~-s 2+ 0+s 2+
0.0062 0.0120 0.0081 0.0099 0.0005 0.0004 0.00015 0.0025 0.005 9 × 10 - 6 0.0035 0.0068 0.00047 0.00088 6.2 × 10 --~ 3.16 × 10-5 2.54 x 10 - 4 0.0032 0.0062 0.0042 0.0052 1.4 × 10- 6 1.8 x 10 -7 3.0 × 10-7 3.8 × 10 -6 7.4 × 10 -6 0.0026 0.0051
0.0062 0.0163 0.0064 0.0162 0.0007 0.0001 0.00040 0.0025 0.006 0.0025 0.0035 0.0088 0.00105 0.00009 2.6 × 10 -5 5.00 × 10- 4 1.22 × 10-3 0.0032 0.0079 0.0035 0.0077 1.2 × 10 -7 3.4 × 10 -7 6.1 x 10-7 3.8 × 10 -7 9.4 × 10 -7 0.0026 0.0064
0.0041 0.0106 0.0045 0.0104 0.0018 0.0009 0.00020 0.0029 0.0051 0.0004 0.0028 0.0056 0.00029 0.00041 1.3 × 10 -5 1.19 × 10 -5 1.11 x 10 -5 0.0021 0.0041 0.0026 0.0035 3.10 × 10 -8 1.89 × 10 -7 2.49 × l0 -7 7.3 × 10 -7 1.5 × 10 -6 0.0026 0.0060
0.0062(4) 0.0117(26) 0.0078(10) 0.0093(14) 0.0016(12) 0.0020(15) 0.00030(24) 0.0025(14) 0.0006 (3) 0.0007(5) 0.0035(8) 0.0063(16) 0.00033(10) 0.00029(8) 6.4(8) × 10 -5 1.21(5) × 10 -5 1.89 × 10-4(24) 0.0032(9) 0.0060(17) 0.0016(5) 0.0013(4) 3.10(17) × 19 -~ 1.89(10) × 10- 7 2.49(13) × l0 -7 3.8(4) × 10 -3 6.8(5) x 10 -3 0.0026(4) 0.0060(19)
~56Gd
158Gd
16°Gd
16°Dy
L62Dy
Table 3 Parameters in El transition operator; the unit of el is in 0.28389 Al/3efm, which gives the B(EI) in Weissknopf unit Nucleus
E1
J52Sm 154Sm 156Gd 158Gd 16°Gd 16°Dy 162Dy
0.00101 0.00097 0.0118 0.0136 0.0120 0.0007 0.0417
Xdp 81.67 83.t5 - 1.62 -3.83 -2.08 -0.49 -0.754
Xdf -4.98 -8.59 4.37 3.68 1.61 -0.56 -0.35
Moreover, the E2 transition rates of the positive parity states can also be reasonably well described by the SU(3)
dynamic symmetry
F r o m T a b l e 2, w e c a n f i n d t h a t t h e B ( E 1 )
[44]. values from 0-
b a n d to g r o u n d s t a t e
210
H.Y. Ji et al./Nuclear Physics A 658 (1999) 197-216
band in the nuclei studied are all well reproduced. In particular, it is noted that the ratio B ( E I , 1- ---, 2g+)/B(E1, 1- ~ 0g+) is approximately 2 in the spdf IBM S U ( 3 ) octupole vibration limit. There is because there is a factor v ~ in the E1 transition matrix element for L - ~ ( L - 1) +, and a factor of v ~ + 1 in the E1 transition matrix element L - --+ (L + 1)+. The other factors in the two matrix elements are almost the same for N ~ 10. When L = 1, it gives a ratio of 2 for the two B ( E 1 ) values. This SU(3) octupole vibration property is all satisfied by the 7 nuclei studied in this work. This property can be used to extract the K values for the low-lying 1- state of deformed nuclei. For the K ~ = l - bandhead, the transition B ( E I ) to ground state band states is just the opposite, where the transition to 0g+ state is stronger than to 28+ state. In J6°Dy, it is noted that the experimental B(E1) values vary over a wide range from 6.8 × 10 -3 W.u. to 3.10 × 10 -8 W.u., an order of 5 change! If one fits the transition from K ~ = 0 - band, then the transition from K ~r = 1- band will be too large compared with experimental data. Conversely, if one tries to reproduce the transitions from K ~r = 1- band, then the transitions from K ~r = 0 - band will be too small. The same situation occurred also in Ref. [23]. We therefore suggest that the origins of the 12 state which is the bandhead of the K ~ = 0--band, and 2 7 state which is a member of the K ~ = 1- - b a n d are different. They cannot be described simultaneously in the spdf IBM. We also noticed that the assignment of the l~- state as a member of K ~ = 0 - - b a n d is only temporary in experiment [23,43]. If we consider only the transitions from 2 7 to positive parity states, agreement is obtained. In 15SGd, there are 7 B ( E I ) experimental data. The B ( E 1 ) values vary over also a wide range from 0.00633 W.u. to 1.21 × 10 -5 W.u.. They are all well reproduced by the spdf IBM SU(3) octupole vibration limit. To our present experimental knowledge, we can say that 158Gd is the best nucleus showing the SU(3) octupole vibration dynamical symmetry. We expect that with the development in experiment, other nuclei with this dynamical symmetry will be discovered in other mass regions, for instance in the auctioned region. From Table 3, we can find that in other nuclei: (a) both dp and df terms play important roles in these transitions, so we cannot acquire the good agreement only by dp term alone; (b) the values of Xdp and Xdf change in a relative narrow range, and dp and df terms have almost the same importance in these nuclei, which is different from 152Sm, and the signs of Xdp and Xdf remain invariant in a given isotope chain; (c) in the isotopes of Gd, the changes of Xdp and Xdf obey the following regularity: with the number of neutron increasing, Ydf decreases, however, the absolute value of Xdp increases first, then decreases. In 156Gd, the absolute value of Xdf (4.37) is greater than that of Xdp (-1.62), and in 16°Gd, the absolute value of Xdf (1.61) is less than that of Xdp(--2.08), which implies that dp and df terms may play different roles in different nuclei. In 158Gd, the absolute value of Xdp reaches its maximum. It requires more experimental data in relevant nuclei to check this relationship between the value of Xdp and the number of neutron in a given isotope chain. From the E1 transition formulae, we found that the El transition between 0 - band
H.Y Ji et al./Nuclear Physics A 658 (1999) 197-216
211
and y-band is zero in the SU(3) limit. However, such transitions occurred in 15ZSm, 16°Dy. In 152Sm, B(E1;4 + ---* 3~-) --3.2 x 10-5(13), this value is very small. A small symmetry-breaking will give a nonzero value for this transition. For instance if we introduce some mixing between the/3 and y band, that is,
I/3>'-- al I/3) +
a=lY),
(65)
lY)' = a21/3) + allY),
(66)
where the apostrophe labelled the new /3-band and y-band. When la21 = 0.0118, B ( E I ' ,4 r+ --+ 31) = 3.2 x 10 -5, and because la21 is so small that other properties of y-band could not be disturbed much. However, in 16°Dy, B(E1; 11 ---* 2 +) = 0.018(18), a very large value. A simple symmetry breaking cannot solve this problem. This is another reason that the 12 state in ~6°Dy may not belong to the K ~ = 0 - band. In 16°Dy, the E1 transition between 3 3 of possible K ~ = 0--band to y band are also observed, which confirmed us that this band is not likely the 0--band. Besides absolute B(E1) data, we also compare relative intensities. If one assumes that E1 is dominant in parity changing transitions and ignore higher order transitions, then the transition probabilities and hence the intensities are int ~x E 3 B ( E 1 ) .
(67)
Using the values of Xdp and Xdf obtained in the previous work for the B(E1) value, we have calculated the relative intensities in 152'154Sm [34,38], 156'158'16°Gd [39-41], nucleus and from Ref. [23,43]. The calculated results are shown in Table 4 with experimental data. From Table 4, we see besides general agreement, there exist serious deviations in J52A54Sm: in experiment, the relative intensity from 1- of K ~ = 1--band to 2g+ is much greater than that to 0g+, but in calculation, the situation is reversed. In general, the square of reduced matrix element for transition from 1- state of the K ~ = l - = b a n d to 0-g~sis almost as twice as that to 2~-s. The calculated ratio of intensities should be of the order of 1. However, we noticed that in 152Sm, the transition to 2g+ also involves M2 transitions. In the calculation we have just included the E1 contribution. In 154Sm, the identification of 1- state is only tentative. Experimental measurement of the E1 transition rate from 1- to 2+ will be very useful for checking this calculation, since in our model there is little freedom to adjust this transition rate.
5. Summary We have given analytical expressions for E1 and M1 transitions involving low-lying negative parity states in the SU(3) limit. These El transition formulae plus those given in Ref. [28] exhaust the whole possibilities of E1 transitions from 0 - and 1- band to the ground state band, the beta and gamma band. We also applied these analytical results to some deformed rare-earth nuclei to check the SU(3) octupole vibration prediction.
212
H.Y. Ji
et al./Nuclear Physics A 658 (1999) 197-216
Table 4 Comparisons of relative intensities Nucleus
Elevel (keV)
Kr
li
It
Cal.
Exp.
152Sm
963
0-
1~
0g~ 2~ 23 2~ 48~ 48~ 6~ 6~ 8~ 0~ 28~ 03 23 28~ 23 2¢ 3¢ 2~ 4~ 7+ 43 2~ 4~ 48~ 6~ 8~ 10~ 68~ 5~ 6~ 8~ 88~ 10~ 83
55.6 I00 0.24 98.2 100 100 72 100 51 100 13.3 8.38 0.035 100 4.5 10.50 5.283 100 1.15 14.4 0.289 2.11 1.2 40 100 100 31 100 5.4 100 14 100 32 58
82.3(7) 100(2) 0.010(3) 100(4) 40.4(19) 100(3) 24(3) 100(3) 12 0.85(6) 100(3) 0.09(5) 1.42(25) 100(ll) 0.28(4) 13.43(9) 2.127(24) 35.3(3) 100(5) 6.6(3) 1.30(9) 2.07(15) 0.41(5) 67(34) 100(33) 100(3) 15 82(20) 100(4) 100(7) 71(24) 100(5) 27(3) 13(4)
1%
~oo
lOO(5)
12~ 10~ 0~ 2~ 2g~ 4~
62 72 52 100 83 100
59(24) 24 65(2) 100(2) 100(2) 60(1)
4~
100
100(3)
68~ 0~ 2~ 2~ 23
98 100 15 100 4.1
29.0(7) 0.5 100 100(19) 5.4(12)
1041
37
1221
5~
1505
7?
1511
t54Sm
1-
I~
1530
2?
1579
3~
1764
5~
1879
0-
9~
1930
1-
6?
2004
7~
2291
9~
2641
115
921
0-
1~
1012
3~
118l
5?
1476 1515
1-
1~ 2?
213
H.Y. Ji et al./Nuclear Physics A 658 (1999) 197-216
Table 4 - - continued Nucleus
156Gd
Elevel (keV)
li
If
Cal.
Exp.
1585
32 52
100 1.6 100 2 100 54 7.0 x 10-6 100 87 100 7.0 x 10-5 61.1 100 0.09 100 86 100 94 100 82 100 51 100 76 100 75.9 5.1 2.75 2.01 64 100 100 85 0.002 19.9
26.6(25) 100(3)
1774
2+ 4+ 4+ 6+
1242
K rr
1-
17
%
2;
o;, 1276
37
2+
4~+~ 1320 1366
o-
2~-
2~-s
1;
o;
2;
2;
2~
1539 1638
32 1-
1958
2+
4;
77
6~-s
9~-
8~s
17
%
8;
lO+
'58Gd
977
1-
2;
1042
4; 1176
57
4+
1260
2~
1/ 2/ 31
1;
o;
3;
2; 2; 4;
1263
o-
1403
6;
2~
1407
1-
4~
3/ 41 51
o-
52
%+
o-
1?
o;
1290
37
2;
1428
51
4~-s 4~-s
1639 16OGd
1224
6;
2;
6;
1.29
5.21 100 61 60.5 100 100 84 100 55
40(5) 100(4) 97.1(5) 100(10) 0.07(2) 100(2) 44(2) 100(3) 0.21(2) 54.5(3) 100(1) 0.08(3) 94(6) 100(6) 100(3) ~33 100(3) 7(3) 100(5) 76(4) 100(20) 47.2(9) 100(6) 26.7(16) 5.1(46) 0.56(4) 6.9(6) 68(4) 100(6) 100(6)
85(5) o.o4(1) 19.9(12) 0.33(2) 6.6(5)
100(8) 33(5) 65.2(15) 100.0(22) 100(2) 52(2) 100(6) < 45
214 Table
H.Y. Ji et al./Nuclear Physics A 658 (1999) 197-216
4 -- continued 1640 t6ODy
162Dy
1359
77 I-
2~
1399
3~
1535
4~
1489
*0-
1~
1276
0-
IF
1358
3~
1519
5~
1692
I-
2~
6~ 8~ 2~ 2~ 3~ 2g~ 4~ 2~ 3~ 4~ 4~ 3~ 4~ 5~ 60 2~ 0~ 2~ 2~ 4g~ 4~ 6g~ 2g% 3~
100 36 100 1.00 0.88 100 83.8 0.43 1.77 0.65 100 0.55 1.22 0.29 69(14) 100 53 100 84 100 100 96 58 100
28(11) 100(22) 100(3) 17.95(12) 11.59(5) 100.0(13) 54.7(5) 0.81(3) 0.50(3) 0.26(3) 100(9) 79.8(14) 13.4(6) 19.6(9) 100 52(14) 100(4) 100(30) 46(4) 100(20) 40(8) 100(3) 49(6)
Though there has been some deviation, the main features are checked by the experimental data. Some of the discrepancies are expected because of the approximate nature of the dynamical symmetry in real nucleus, for instance the breaking of the beta and gamma band states degeneracies, some small transitions between gamma band and K ~ = 0 band. They can be resolved by symmetry breaking. Some of the discrepancies cannot be nailed down at the moment due to lack of experimental data, for instance, the K ~"= 0 - band identification in J6°Dy. Another important issue which remain to be checked by experiment is the anomalous intensity ratio of K ~" = 1- bandhead in 152'154Sm. Comparing with the sd-f IBM studies of Cottle and Zamfir [23], we see that the agreements of the calculations of the two models with experimental data are of the same quality. In this spdf IBM study, the p boson is very important in these deformed nuclei, even critical in some nucleus, i.e. 152Sm. In the sd-f IBM study, the f boson is most important. From these empirical studies alone, it is not conclusive which model is more appropriate. However, the spdf IBM is more general, and sd-f IBM is only a special case of the spdf IBM. The two calculations can be thought as being two different parameterizations of the spdf IBM. In the sd-f IBM, it can be considered as a special case in which the energy of the p boson is put infinitely high. In the SU(3) octupole vibration limit, the energy of the p boson is put equal to the energy of the f boson. However, in view of the simple mathematical structure of spdf IBM, the microscopic studies [ 14] and the findings by Kusnezov [ 15], the spdf interacting boson model is one promising model for describing negative parity collective states in even-even nuclei.
H. Y Ji et al./Nuclear Physics A 658 (1999) 197-216
215
Acknowledgements The authors acknowledge the support of China National Natural Science Foundation, China State Education Ministry, Fok Ying Tung Education foundation. EG Zhao also acknowledges the support of Advanced Visitors Scheme of Tsinghua University. The authors also thank the referee for expert reading and helpful criticism of the manuscript.
Corrigenda In Table 1 of Ref. [28], the lines concerning 156Gd should be replaced by Nucleus
Ii
If
B ( E 1 , Ii --+/f)cal
156Gd
1+
0+ 2+
0.0025 0.005
(W.u.)
B ( E 1 , I i --+/f)exp
(W.u.)
0.0025(14) 0.006(3)
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