E2 ansi ions and quad upole momen s in he E„5…symme y
J. M. A ias
Depa amen o de Fı
´sica A o
´mica, Molecula y Nuclea , Facul ad de Fı
´sica, Uni e sidad de Se illa, Apa ado 1065, E-41080 Se illa,
Spain
共Recei ed 1 Decembe 2000; published 12 Feb ua y 2001兲
E2 ansi ions and quad upole momen s a e s udied in he ecen ly p oposed E共5兲symme y by using he
in insic s a e o malism. I is shown ha he alues o hese magni udes can be ob ained o he di e en bands
o highe o de in he boson numbe Nby p ojec ing he in insic s a e on
␥
and
␤
a iables. The o malism
allows o ind easily he dependence o hose magni udes on he s uc u e pa ame e o he quad upole ope a o ,
␹
.
DOI: 10.1103/PhysRe C.63.034308 PACS numbe 共s兲: 21.60.Fw, 21.10.Ky, 21.10.Re
I. INTRODUCTION
Recen ly a new class o dynamic symme y has been p o-
posed by Iachello 关1兴. This symme y is expec ed o be o
use when analyzing sys ems unde going phase ansi ions be-
ween adi ional dynamic symme ies. In pa icula , he ex-
ample p esen ed in Re . 关1兴conside s he Boh Hamil onian
关2兴and discusses he case in which he po en ial is
␥
inde-
penden and in addi ion he
␤
dependence o he po en ial is
modeled by a i e-dimensional in ini e well. This seems o
be applicable in nuclea spec oscopy when nuclei a e a he
c i ical poin in a ansi ion om sphe ical o
␥
-uns able
shape. The E(5) symme y is discussed in 关1兴in connec ion
wi h he in e ac ing boson model 共IBM兲关3兴. Ene gy le els
a e gi en and ansi ion p obabili ies o selec ed s a es a e
calcula ed by using he ollowing quad upole ope a o de-
pending linea ly on
␤
:
T
␮
(E2)⫽
␤
冋
D
␮
0
(2)共
␪
i兲cos
␥
⫹1
冑
2„D
␮
2
(2)共
␪
i兲
⫹D
␮
⫺2
(2) 共
␪
i兲…sin
␥
册
,共1兲
whe e is a scale ac o . Expe imen al examples o his new
class o symme y ha e al eady been p oposed 关4兴.
Wi hin he geome ical model he case o
␥
independen
po en ial su ace was discussed some ime ago by Wile s and
Jean 关5兴, while he equi alen si ua ion wi hin he IBM is
known as he O共6兲limi and was discussed i s in Re . 关6兴.
In bo h cases he ene gy su ace has a de ini e equilib ium
alue o
␤
, being o he wise
␥
independen . On he o he
hand, he ib a ional Boh Hamil onian 关2兴and he co e-
sponding SU共5兲limi in IBM 关7兴p o ide wi h a si ua ion in
which he ene gy su ace has equilib ium alue
␤
⫽0 and is
␥
independen oo. As men ioned abo e, he newly p oposed
E共5兲symme y seems o be app op ia e when discussing po-
en ials wi h la beha io as a unc ion o some coo dina e,
as i could be he case o he
␤
coo dina e in a SU共5兲-O共6兲
ansi ion.
Fi s , in Sec. II a b ie e iew o he example o E共5兲
symme y p esen ed in Re . 关1兴is gi en. In Sec. III he o -
malism used is de eloped, including he in insic s a e de-
sc ip ion and he p ojec ion on o he labo a o y ame. Re-
sul s a e p esen ed in Sec. IV. Finally, Sec. V is de o ed o a
summa y.
II. THE E„5…SYMMETRY
Conside he Boh Hamil onian
H⫽⫺ ប2
2B
冋
1
␤
4
⳵
⳵
␤␤
4
⳵
⳵
␤
⫹1
␤
2sin3
␥
⳵
⳵
␥
sin3
␥
⳵
⳵
␥
⫺1
4
␤
2兺
␬
Q
␬
2
sin2共
␥
⫺1
3
␲
␬
兲
册
⫹V共
␤
,
␥
兲,共2兲
whe e
␤
,
␥
a e he shape a iables and he Q
␬
’s a e he com-
ponen s o he angula momen um w i en in e ms o Eule
angles. In cases in which he po en ial depends only on
␤
,
V(
␤
,
␥
)⫽U(
␤
), he wa e unc ion can be ac o ized as
⌿共
␤
,
␥
,
␪
i兲⫽ 共
␤
兲⌽共
␥
,
␪
i兲,共3兲
whe e
␪
is ands o he h ee Eule angles, and he Sch o
¨-
dinge equa ion can be spli in o wo equa ions,
冋
⫺1
sin3
␥
⳵
⳵
␥
sin3
␥
⳵
⳵
␥
⫹1
4兺
␬
Q
␬
2
sin2共
␥
⫺2
3
␲
␬
兲
册
⌽共
␥
,
␪
i兲
⫽
␶
共
␶
⫹3兲⌽共
␥
,
␪
i兲;
␶
⫽0,1,2,..., 共4兲
and
冋
⫺ប2
2B
冉
1
␤
4
⳵
⳵
␤␤
4
⳵
⳵
␤
⫺
␶
共
␶
⫹3兲
␤
2
冊
⫹U共
␤
兲
册
共
␤
兲⫽E 共
␤
兲.
共5兲
I U(
␤
) can be modeled as a i e dimensional in ini e well,
he p oblem is exac ly sol able and he co esponding sym-
me y is called E共5兲. The solu ions o he Sch o
¨dinge equa-
ions in
␤
and (
␥
,
␪
i) wi h he app op ia e bounda y condi-
ions a e known 关1兴. The wa e unc ions on
␤
a e
␰
,
␶
共
␤
兲⫽C
␰
,
␶
␤
⫺3/2J
␶
⫹3/2
冉
x
␰
,
␶
␤
w
␤
冊
,共6兲
PHYSICAL REVIEW C, VOLUME 63, 034308
0556-2813/2001/63共3兲/034308共5兲/$15.00 ©2001 The Ame ican Physical Socie y63 034308-1
whe e
␶
is he label associa ed o he O共5兲algeb a,
␰
is a
label ha enume a es he ze os o he ele an Bessel unc-
ion, x
␰
,
␶
is he
␰
h ze o o he Bessel unc ion J
␶
⫹3/2(x),
C
␰
,
␶
a e no maliza ion cons an s, and
␤
wis he ange o he
po en ial in he
␤
a iable.
The solu ions o he (
␥
,
␪
i) pa we e s udied in Re . 关5兴
and abula ed in Re . 关12兴, whe e ⌽
␶
,L,M(
␥
,
␪
i) a e w i en in
e ms o D unc ions as
⌽
␶
,L,M共
␥
,
␪
i兲⫽兺
␳
g
␶
,L,
␳
共
␥
兲DM
␳
(L)共
␪
i兲.共7兲
The in insic unc ions g
␶
,L,
␳
(
␥
) a e explici ly gi en in 关12兴
and only e en alues o
␳
appea in he sum.
The E共5兲s a es a e labeled by
兩
N;
␰
␶
LM
典
.Nis he boson
numbe ,
␰
is a label ela ed o he solu ion o he Sch o
¨dinge
equa ion in he
␤
a iable as men ioned abo e,
␶
is he label
associa ed o he O共5兲algeb a, Lis he o al angula momen-
um, and Mi s p ojec ion on one axis.
III. THE FORMALISM
In his wo k i is shown ha he alues o he E2 ansi-
ions and quad upole momen s in he E共5兲dynamic symme-
y can be ob ained by using he in insic s a e o malism in
IBM 关8–10兴and p ojec ing on he app op ia e a iables. The
s a ing poin is ha he E2 ansi ion ope a o in IBM is
w i en as
T
␮
(E2)⫽qQ
␮
(sd),共8兲
whe e qis a scale ac o and Q(sd)is he IBM quad upole
ope a o ,
Q
␮
(sd)⫽共s†d
˜
⫹d†s兲
␮
(2)⫹
␹
共d†d
˜
兲
␮
(2) .共9兲
The pa ame e
␹
is i s s uc u e cons an . The ope a o s d
˜
m
⫽(⫺1)md⫺ma e in oduced so as o ha e enso s wi h he
app op ia e p ope ies unde spa ial o a ions.
The basic idea o he in insic ame o malism is o con-
side ha he pu e quad upole s a es a e globally desc ibed
by a boson condensa e o he o m
兩
g
典
⫽1
冑
N!共⌫g
†兲N
兩
0
典
,共10兲
whe e he basic boson is gi en by
⌫g
†⫽1
冑
1⫹
␤
2
冋
s†⫹
␤
cos
␥
d0
†⫹1
冑
2
␤
sin
␥
共d2
†⫹d⫺2
†兲
册
,
共11兲
which depends on he
␤
and
␥
shape a iables. The equilib-
ium alues o
␤
and
␥
a e ob ained by minimizing he
ene gy su ace in he boson condensa e 共10兲. In he SU共5兲
and O共6兲limi s o IBM his minimiza ion leads o de ini e
alues o
␤
(
␤
0⫽0 and
␤
0⫽1, espec i ely兲and he ene gy
su ace is
␥
independen in bo h cases. A he c i ical poin in
he phase ansi ion om SU共5兲 o O共6兲 he ene gy su ace is
expec ed o be a he la in he
␤
a iable and he symme y
E共5兲seems o be app op ia e. When calcula ing E2 ansi-
ions and momen s in he adi ional 关SU共5兲o O共6兲兴 IBM
limi s, he
␤
a iable is ixed in Eqs. 共10兲and 共11兲 o i s
equilib ium alue and in eg a ion on
␥
has o be pe o med.
In he E共5兲case, since he beha io o he ene gy su ace is
la in bo h
␤
and
␥
a iables, in eg a ion on bo h a iables
has o be done.
Elec omagne ic E2 ansi ion a es and quad upole mo-
men s a e e alua ed by aking ma ix elemen s o he quad-
upole ope a o 共8兲and 共9兲. These ma ix elemen s in he
boson condensa e
兩
g
典
,
具
g
兩
Qm
(sd)
兩
g
典
⬅Qm
(sd)(
␤
,
␥
), ha e al-
eady been calcula ed 关11兴,
Q0
(sd)共
␤
,
␥
兲⫽N
1⫹
␤
2
冋
2
␤
cos
␥
⫺
冑
2
7
␹
␤
2cos2
␥
册
,
共12兲
Q2
(sd)共
␤
,
␥
兲⫽Q⫺2
(sd)共
␤
,
␥
兲
⫽1
冑
2
N
1⫹
␤
2
冋
2
␤
sin
␥
⫹
冑
2
7
␹
␤
2sin2
␥
册
.
共13兲
The ma ix elemen s o Q(sd)no speci ied a e ze o. In he
O共6兲limi he IBM E2 ansi ion ope a o is usually de ined
wi h
␹
⫽0 since in ha case i is a gene a o o O共6兲and
de ini e selec ion ules appea . This app oxima ion has
p o ed o be good o s udying nuclei a he O共6兲limi , bu
o ansi ional SU共5兲-O共6兲nuclei a he c i ical poin he
quad upole ope a o could depend on
␹
. Thus, in he ollow-
ing he gene al o m in Eqs. 共12兲and 共13兲will be kep .
Wi h he help o Eqs. 共6兲and 共7兲, s a es in he labo a o y
can be ob ained om he boson condensa e 共10兲and 共11兲as
兩
N;
␰
␶
LM
典
⫽
␰
,
␶
共
␤
兲⌽
␶
,L,M共
␥
,
␪
i兲
兩
g
典
.共14兲
Thus, he ma ix elemen s o he quad upole ope a o a e
gi en by
具
N;
␰
␶
LM
兩
Q
␮
(sd)共lab兲
兩
N;
␰
⬘
␶
⬘L⬘M⬘
典
⫽
具
N;
␰
␶
LM
兩
兺
mQm
(sd)共in 兲D
␮
m
(2)共
␪
i兲
兩
N;
␰
⬘
␶
⬘L⬘M⬘
典
⫽
冕
d⍀
冕
␤
4d
␤
冕
兩
sin3
␥
兩
d
␥
␰
,
␶
*共
␤
兲⌽
␶
,L,M
*共
␥
,
␪
i兲
⫻兺
m
具
g
兩
Qm
(sd)共in 兲
兩
g
典
D
␮
m
(2)共
␪
i兲
␰
⬘,
␶
⬘共
␤
兲
⫻⌽
␶
⬘,L⬘,M⬘共
␥
,
␪
i兲,共15兲
whe e he IBM quad upole ope a o in he labo a o y has
been ans o med o he in insic ame by using D unc ions.
The in insic ma ix elemen s
具
g
兩
Qm
(sd)(in )
兩
g
典
a e hose
gi en in Eqs. 共12兲and 共13兲.
J. M. ARIAS PHYSICAL REVIEW C 63 034308
034308-2
IV. RESULTS
Wi hin he scheme p esen ed in he p eceding sec ion, he
calcula ion o ma ix elemen s o he quad upole ope a o
implies in eg a ion on he a iables
␤
and
␥
, as well as on
he Eule angles
␪
i, in addi ion o he ele an ma ix ele-
men s in he in insic ame, Eqs. 共12兲and 共13兲.
Thus, he quad upole ma ix elemen s a e gi en by
具
N;
␰
␶
LM
兩
Q
␮
(sd)共lab兲
兩
N;
␰
⬘
␶
⬘L⬘M⬘
典
⫽8
␲
2
2L⫹1
具
2
␮
L⬘M⬘
兩
LM
典
兺
m
␳␳
⬘
具
2mL⬘
␳
⬘
兩
L
␳
典
⫻
冕
兩
sin3
␥
兩
d
␥
g
␶
,L,
␳
共
␥
兲
⫻
冋
冕
␤
4d
␤
␰
,
␶
共
␤
兲Qm
(sd)共
␤
,
␥
兲
␰
⬘,
␶
⬘共
␤
兲
册
g
␶
⬘,L⬘,
␳
⬘共
␥
兲,
共16兲
whe e he in eg a ion on he Eule angles has al eady been
done. Only e en alues o m,
␳
, and
␳
⬘appea in he sum.
Wi h his exp ession i is s aigh o wa d o calcula e E2
ansi ion and momen s since he in eg als in
␤
and
␥
can be
easily e alua ed.
The quad upole momen s o he di e en s a es (
␰
,
␶
,L)
a e gi en by 关qis he scale ac o in Eq. 共8兲兴
Q共
␰
,
␶
,L兲
⫽
冑
16
␲
5
具
N;
␰
␶
LM⫽L
兩
qQ0
(sd)共lab兲
兩
N;
␰
␶
LM⫽L
典
.
共17兲
The co esponding quad upole momen s o some selec ed
s a es a e
Q共
␰
⫽1,
␶
⫽1,L⫽2兲⫽0.1425N
␹
q,共18兲
Q共
␰
⫽1,
␶
⫽2,L⫽2兲⫽⫺0.0514N
␹
q,共19兲
Q共
␰
⫽1,
␶
⫽2,L⫽4兲⫽0.2400N
␹
q,共20兲
Q共
␰
⫽1,
␶
⫽3,L⫽6兲⫽0.3112N
␹
q,共21兲
Q共
␰
⫽1,
␶
⫽3,L⫽4兲⫽0.0751N
␹
q,共22兲
Q共
␰
⫽2,
␶
⫽1,L⫽2兲⫽0.1202N
␹
q,共23兲
Q共
␰
⫽2,
␶
⫽2,L⫽2兲⫽⫺0.0428N
␹
q,共24兲
Q共
␰
⫽2,
␶
⫽2,L⫽4兲⫽0.1998N
␹
q.共25兲
The calcula ion o E2 ansi ion p obabili ies can be done
s aigh o wa d om
B共E2;
␰
,
␶
,L→
␰
⬘,
␶
⬘,L⬘兲
⫽1
2L⫹1
兩
具
N;
␰
␶
L
兩兩
qQ(sd)共lab兲
兩兩
N;
␰
⬘
␶
⬘L⬘
典
兩
2.
共26兲
The co esponding B(E2) ansi ion a es o some se-
lec ed ansi ions a e
B共E2;
␰
⫽1,
␶
⫽1,L⫽2→
␰
⫽1,
␶
⫽0,L⫽0兲⫽0.1459N2q2,
共27兲
B共E2;
␰
⫽1,
␶
⫽2,L⫽2→
␰
⫽1,
␶
⫽0,L⫽0兲⫽0.0044N2
␹
2q2,
共28兲
B共E2;
␰
⫽1,
␶
⫽2,L⫽2→
␰
⫽1,
␶
⫽1,L⫽2兲⫽0.2282N2q2,
共29兲
B共E2;
␰
⫽1,
␶
⫽2,L⫽4→
␰
⫽1,
␶
⫽1,L⫽2兲⫽0.2282N2q2,
共30兲
B共E2;
␰
⫽1,
␶
⫽3,L⫽6→
␰
⫽1,
␶
⫽2,L⫽4兲⫽0.2806N2q2,
共31兲
B共E2;
␰
⫽2,
␶
⫽0,L⫽0→
␰
⫽1,
␶
⫽1,L⫽2兲⫽0.0710N2q2,
共32兲
B共E2;
␰
⫽2,
␶
⫽0,L⫽0→
␰
⫽1,
␶
⫽2,L⫽2兲⫽0.0082q2N2
␹
2,
共33兲
B共E2;
␰
⫽1,
␶
⫽3,L⫽0→
␰
⫽1,
␶
⫽1,L⫽2兲⫽0.0090N2
␹
2q2,
共34兲
B共E2;
␰
⫽1,
␶
⫽3,L⫽0→
␰
⫽1,
␶
⫽2,L⫽2兲⫽0.2806N2q2.
共35兲
TABLE I. Compa ison o some B(E2) a ios in 134Ba wi h he E共5兲symme y. Expe imen al da a a e
om Re . 关14兴.
B共E2;41,2
⫹→21,1
⫹兲
B共E2;21,1
⫹→01,0
⫹兲
B共E2;02,0
⫹→21,1
⫹兲
B共E2;21,1
⫹→01,0
⫹兲
B共E2;02,0
⫹→21,2
⫹兲
B共E2;02,0
⫹→21,1
⫹兲
B共E2;01,3
⫹→21,1
⫹兲
B共E2;01,3
⫹→21,2
⫹兲
E共5兲a1.68 0.86 0 0
E共5兲b1.56 0.49 0.12 0.032
Exp . 1.56(18) 0.42(12) 0.18(8) 0.037(3)
aE共5兲as calcula ed in Re . 关1兴wi h ope a o 共1兲.
bE共5兲 om his wo k wi h he IBM quad upole ope a o 共8兲and 共9兲and
␹
⫽1.
E2 TRANSITIONS AND QUADRUPOLE MOMENTS IN . . . PHYSICAL REVIEW C 63 034308
034308-3
I has been checked ha hese esul s con e ge o known
esul s in simple si ua ions. On he one hand, he O共6兲 al-
ues 关13兴a e ob ained i he in eg al in
␤
, be ween b acke s in
Eq. 共16兲, is subs i u ed by Qm
(sd)(1,
␥
)(
␤
⫽1). On he o he
hand, hese esul s educe o hose p esen ed in Re . 关1兴,
whe e he E2 ope a o 共1兲is used, i
␹
is aken as ze o in he
IBM quad upole ope a o 共8兲and 共9兲and he no maliza ion
ac o 2/(1⫹
␤
2) in Eqs. 共12兲and 共13兲is subs i u ed by 1.
Wi h hese changes he IBM quad upole ope a o used he e
educes o he E2 ansi ion ope a o used in 关1兴.
134Ba has been p oposed 关4兴as a i s e idence in nuclea
physics o he E共5兲symme y. In Table I some impo an E2
b anching a ios o his nucleus a e compa ed wi h he e-
sul s ob ained in he E共5兲symme y. Expe imen al da a a e
aken om Re . 关14兴. The no a ion used o deno ing he
s a es is L
␰
,
␶
␲
. Two kinds o E共5兲 esul s a e shown. The
esul s E共5兲labeled wi h 共a兲a e hose aken om Re . 关1兴.
Due o he o m o he ansi ion quad upole ope a o used
he e, Eq. 共1兲, ansi ions B(E2;02,0
⫹→21,2
⫹) and B(E2;01,3
⫹
→21,1
⫹) a e o bidden. The esul s E共5兲labeled wi h 共b兲a e
hose ob ained in his wo k in which he E2 ansi ion op-
e a o is Eqs. 共8兲and 共9兲wi h
␹
⫽1. I is obse ed in Table
I ha he o malism p esen ed he e allows o e en a be e
desc ip ion o he 134Ba when compa ing o he newly p o-
posed E共5兲symme y. This imp o emen comes om he
wo di e ences he IBM quad upole ope a o , Eqs. 共8兲,共9兲
and Eqs. 共12兲,共13兲, in oduces wi h espec o he usually
used quad upole ope a o 共1兲. On he one hand, he inclusion
o he e m depending on
␤
2(
␹
) in Eqs. 共8兲and 共9兲is c ucial
o desc ibe he a ios B(E2;02,0
⫹→21,2
⫹)/B(E2;02,0
⫹→21,1
⫹)
and B(E2;01,3
⫹→21,1
⫹)/B(E2;01,3
⫹→21,2
⫹). On he o he hand,
he no maliza ion ac o 2/(1⫹
␤
2) in Eqs. 共12兲and 共13兲
imp o es he desc ip ion o he a ios B(E2;41,2
⫹
→21,1
⫹)/B(E2;21,1
⫹→01,0
⫹) and B(E2;02,0
⫹→21,1
⫹)/B(E2;21,1
⫹
→01,0
⫹). The e o e, he IBM ansi ion ope a o 共8兲and 共9兲,
wi h in insic ma ix elemen s 共12兲and 共13兲, seems o p o-
ide a be e desc ip ion o he expe imen al da a han he
ope a o 共1兲.
One impo an poin as a signa u e o E共5兲symme y in
compa ison wi h he O共6兲case is he ansi ion B(E2;02,0
⫹
→21,1
⫹). This is o bidden in he O共6兲limi e en i one con-
side s he gene al o m o he quad upole ope a o including
he
␹
e m, while i gi es he co ec a io B(E2;02,0
⫹
→21,1
⫹)/B(E2;21,1
⫹→01,0
⫹) in he E共5兲limi wi h
␹
⫽1.
In Fig. 1 he obse ed B(E2) ansi ion a es in 134Ba
关14兴a e compa ed wi h he esul s ob ained in his wo k
assuming
␹
⫽1. Uni s a e gi en in W.u. Fo he decay om
he s a e 0⫹a 1.761 MeV o he s a es 22
⫹and 21
⫹only he
b anching a io is known 27/1. This b anching a io is nicely
ep oduced in he calcula ion.
V. SUMMARY
In his pape i has been p esen ed how o use he in insic
s a e o malism o e alua e elec omagne ic ansi ion a es
and quad upole momen s in he ecen ly p oposed E共5兲sym-
me y. I has been shown ha dealing wi h
␤
and
␥
depen-
den objec s in he in insic ame can be done easily. The
same echnique can be used o calcula e expec a ion alues
o o he obse ables. In addi ion, i has been shown ha he
IBM E2 ope a o p o ides a be e desc ip ion o he da a
han he ope a o 共1兲used in Re . 关1兴.
ACKNOWLEDGMENTS
This wo k was suppo ed in pa by he Spanish DGICYT
unde P ojec No. PB98-1111. I acknowledge con inuous
collabo a ion wi h C.E. Alonso and A. Vi u i.
关1兴F. Iachello, Phys. Re . Le . 85, 3580 共2000兲.
关2兴A. Boh and B. Mo elson, Nuclea S uc u e, Vol. II 共Ben-
jamin, Reading, MA, 1975兲.
关3兴F. Iachello and A. A ima, The In e ac ing Boson Model 共Cam-
b idge Uni e si y P ess, Camb idge, 1987兲.
关4兴R. F. Cas en and N. V. Zam i , Phys. Re . Le . 85, 3584
共2000兲.
关5兴L. Wile s and M. Jean, Phys. Re . 102, 788 共1956兲.
FIG. 1. Le el schemes and B(E2) alues 共in
W.u.兲 o 134Ba in he E共5兲symme y wi h
␹
⫽1共le 兲and co esponding expe imen al da a
关14兴共 igh 兲. The scale ac o qin Eq. 共8兲has been
adjus ed o ma ch he expe imen al B(E2;21,1
⫹
→01,0
⫹) alue. Fo he decay om he s a e 0⫹a
1.761 MeV o he s a es 22
⫹and 21
⫹only he
b anching a io (27/1) is known.
J. M. ARIAS PHYSICAL REVIEW C 63 034308
034308-4
关6兴A. A ima and F. Iachello, Ann. Phys. 共N.Y.兲123, 468 共1979兲.
关7兴A. A ima and F. Iachello, Ann. Phys. 共N.Y.兲99, 253 共1976兲.
关8兴J. N. Ginocchio and M. W. Ki son, Nucl. Phys. A350,31
共1980兲.
关9兴A. E. L. Diepe ink, O. Schol en, and F. Iachello, Phys. Re .
Le . 44, 1747 共1980兲.
关10兴A. Boh and B. Mo elson, Phys. Sc . 22, 468 共1980兲.
关11兴C. E. Alonso, J. M. A ias, F. Iachello, and A. Vi u i, Nucl.
Phys. A539,59共1992兲.
关12兴D. Be
`s, Nucl. Phys. 10, 373 共1959兲.
关13兴C. E. Alonso, M. Lozano, C. H. Dasso, and A. Vi u i, Phys.
Le . B 212,1共1988兲.
关14兴Yu. V. Se geenko , Nucl. Da a Shee s 71, 557 共1994兲.
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