The sex ic oscilla o as a
␥
-independen po en ial
G. Lé ai*
Ins i u e o Nuclea Resea ch o he Hunga ian Academy o Sciences (ATOMKI), P.O. Box 51, H-4001 Deb ecen, Hunga y
J. M. A ias†
Depa amen o de Física A ómica, Molecula y Nuclea Facul ad de Física, Uni e sidad de Se illa, Apa ado 1065,
E-41080 Se illa, Spain
(Recei ed 7 Oc obe 2003; published 20 Janua y 2004)
The sex ic oscilla o is p oposed as a wo-pa ame e sol able
␥
-independen po en ial in he Boh Hamil-
onian. I is shown ha closed analy ical exp essions can be de i ed o he ene gies and wa e unc ions o he
i s ew le els and o he s eng h o elec ic quad upole ansi ions be ween hem. Depending on he pa am-
e e s his po en ial has a minimum a
␤
=0 o a
␤
⬎0, and migh also ha e a local maximum be o e eaching
i s minimum. A compa ison wi h he spec al p ope ies o he in ini e squa e well and he
␤
4po en ial is
p esen ed, oge he wi h a b ie analysis o he expe imen al spec um and E2 ansi ions o he 134Ba nucleus.
DOI: 10.1103/PhysRe C.69.014304 PACS numbe (s): 21.10.Re, 03.65.Ge
I. INTRODUCTION
In he las ew yea s he e has been conside able in e es
in looking o analy ic solu ions o he Boh Hamil onian
which desc ibes he collec i e mo ion in nuclei in e ms o
shape a iables (
␤
,
␥
)[1]. This was ini ia ed wi h he in o-
duc ion o he in e ac ing boson model (IBM)[2]. This
model p esen s ou di e en dynamical symme ies, each
one associa ed wi h a well-de ined nuclea shape. The model
Hamil onian p o ides a na u al way o going om a phase o
ano he one by changing sys ema ically ew pa ame e s and,
consequen ly, allows one o s udy shape phase ansi ions.
Thus, he IBM phase diag am has been analyzed om di -
e en poin s o iew [3–7]. Specially impo an a e he c i i-
cal poin s since in hese si ua ions s uc u al changes occu
apidly and i is di icul o design app op ia e heo e ical
models. Recen ly, Iachello has p oposed a new dynamical
symme y o desc ibing he c i ical poin a he ansi ion
om sphe ical o de o med
␥
-uns able shapes [8]. This sym-
me y has been called E共5兲and is expec ed o occu in nuclei
in which he V共
␤
,
␥
兲po en ial depends only on he
␤
a i-
able, and i has a ela i ely la shape in he
␤
a iable.
A ound his c i ical poin o he phase ansi ion he po en ial
appea ing in he Boh Hamil onian can be app oxima ed by
an in ini e squa e well in he
␤
a iable. In his case he Boh
Hamil onian can be sol ed exac ly in e ms o Bessel unc-
ions, and a ious quan i a i e p edic ions can be ob ained
o a ious spec oscopic p ope ies [ a ios o he exci a ion
ene gies and B共E2兲 ansi ions], on he basis o which one
can sea ch o candida es o he E共5兲symme y among nu-
clei. La ely, wo o he dynamical symme ies X共5兲[9]and
Y共5兲[10] o desc ibe he c i ical poin a he ansi ion om
sphe ical o axially de o med shapes and om axially de-
o med o iaxial shapes, espec i ely, ha e been p oposed.
The in oduc ion o he E共5兲symme y enewed he in e -
es in s udying exac ly sol able V共
␤
,
␥
兲po en ials. Mos e -
o s ha e concen a ed on
␥
-independen po en ials, o
which he mos well-known example is he ha monic oscil-
la o in he i e-dimensional space [11]and i s ex ension
which con ains a e m p opo ional o
␤
−2 [12]. Mo e e-
cen ly wo mo e exac ly sol able
␥
-uns able po en ials ha e
been discussed: he Coulomb and he K a ze po en ials [13].
He e again he la e po en ial is an ex ension o he o me
in he sense ha i con ains a e m p opo ional o
␤
−2, which
o mally changes he
␶
a iable, he analog o he langula
momen um o adial po en ials in h ee spa ial dimensions.
This o mal changing o
␶
in oduces a minimum o he po-
en ials a
␤
⬎0 in bo h cases [12,13]. The bound solu ions
o he ha monic oscilla o and he Coulomb po en ials (and
hei ex ensions)a e gi en in e ms o gene alized Lague e
polynomials [14].
Simila o he h ee-dimensional case, hese examples, o-
ge he wi h he in ini e squa e well, p ac ically exhaus hose
exac ly sol able po en ials ha con ain a
␤
−2 e m. He e we
p opose ano he po en ial which has his p ope y, and al-
hough i is no exac ly sol able in he classical sense, i has
a numbe o ea u es ha make i an ideal po en ial o be
used in he Boh Hamil onian. This is he sex ic oscilla o ,
which belongs o he class o quasi-exac ly sol able po en-
ials [15]. These po en ials ha e he p ope y ha hei solu-
ions can be ob ained in closed o m o a numbe o ene gy
eigen alues, i.e., o he lowes ew alues o he np incipal
quan um numbe , which a e also he lowes in ene gy. This is
clea ly su icien o po en ials appea ing in he Boh Hamil-
onian. I is a ely necessa y o conside mo e han a ew
le els wi h he same angula momen um, and hese can be
ob ained om he lowes ew solu ions o he sex ic oscilla-
o . Fu he mo e, he sex ic oscilla o has a mo e lexible
shape han o he sol able po en ials, as depending on i s pa-
ame e s, i can ha e a minimum a
␤
=0 o a
␤
=
␤
min⬎0,
and in addi ion, i can also ha e a local maximum a
␤
max⬍
␤
min.
The pape is s uc u ed as ollows. In Sec. II he Boh
Hamil onian is e ised oge he wi h i s solu ions o
␥
-independen po en ials. In Sec. III he lowes ene gy solu-
*Elec onic add ess: [email p o ec ed]
†Elec onic add ess: [email p o ec ed]
PHYSICAL REVIEW C 69, 014304 (2004)
0556-2813/2004/69(1)/014304(6)/$22.50 ©2004 The Ame ican Physical Socie y69 014304-1
ions o he Boh Hamil onian o he sex ic oscilla o po en-
ial a e wo ked ou . Sec ion IV is de o ed o show he simple
use o he sex ic oscilla o as a lexible
␥
-independen po en-
ial. Finally, in Sec. V we p esen p elimina y applica ions
and discuss u u e ex ensions.
II. THE BOHR HAMILTONIAN FOR
␥
-INDEPENDENT
POTENTIALS
Le us i s conside ha he Boh Hamil onian desc ibing
he collec i e mo ion o a de o med nucleus in he i e-
dimensional space de e mined by he
␪
iEule angles 共i
=1,2,3兲and he in insic
␤
and
␥
a iables is [1]
H=− ប2
2B
冢
1
␤
4
⳵
⳵
␤␤
4
⳵
⳵
␤
+1
␤
2sin 3
␥
⳵
⳵
␥
sin 3
␥
⳵
⳵
␥
−1
4
␤
2兺
k
Qk
2
sin2
冉
␥
−2
3
␲
k
冊
冣
+V共
␤
,
␥
兲.共1兲
In wha ollows we assume ha he po en ial in Eq. 共1兲de-
pends only on
␤
, i.e., V共
␤
,
␥
兲=U共
␤
兲. Fo hese
␥
-independen po en ials he wa e unc ions can be sepa a ed
in o wo pa s,
⌿共
␤
,
␥
,
␪
i兲= 共
␤
兲⌽共
␥
,
␪
i兲,共2兲
which sa is y he ollowing di e en ial equa ions:
冢
−1
sin 3
␥
⳵
⳵
␥
sin 3
␥
⳵
⳵
␥
+1
4兺
k
Qk
2
sin2
冉
␥
−2
3
␲
k
冊
冣
⌽共
␥
,
␪
i兲
=⌳⌽共
␥
,
␪
i兲,
⌳=
␶
共
␶
+3兲,
␶
=0,1,2, ..., 共3兲
冉
−1
␤
4
⳵
⳵
␤␤
4
⳵
⳵
␤
+⌳
␤
2+u共
␤
兲
冊
共
␤
兲=
⑀
共
␤
兲.共4兲
He e we ha e in oduced
⑀
=共2B/ប2兲Eand u共
␤
兲
=共2B/ប2兲U共
␤
兲. No e ha he
␶
alues de e mine he allowed
angula momen a J, oo 关16兴. By se ing
␾
共
␤
兲=
␤
2 共
␤
兲we
ob ain an equa ion which has he o m o a adial
Sch ödinge equa ion
−d2
␾
d
␤
2+
冉
共
␶
+1兲共
␶
+2兲
␤
2+u共
␤
兲
冊
␾
=
⑀␾
.共5兲
No e ha his is di e en om Eq. 共6兲in Re . 关8兴, in ha i
con ains no linea de i a i e e m due o he di e en de i-
ni ion o
␾
共
␤
兲. This choice also implies ha he ac o co -
esponding o he
␤
olume elemen in he in eg a ion o
unc ions o he ype 共
␤
兲in Eq. 共4兲has been ans e ed o
he solu ions o he ype
␾
共
␤
兲in Eq. 共5兲. Thus, in he in e-
g a ion o hese no ac o a ising om he olume elemen
has o be included. The comple e solu ion o he p oblem
implies he solu ion o Eq. 共3兲, oo; his was sol ed in Re .
关16兴.
III. THE SEXTIC OSCILLATOR
The sex ic oscilla o wi h a cen i ugal ba ie is de ined
[15]as
H=− d2
dx2+共2s− 1/2兲共2s− 3/2兲
x2+
冋
b2−4a
冉
s+1
2+M
冊
册
x2
+2abx4+a2x6,共6兲
whe e x苸关0, ⬁兲and Mis a non-nega i e in ege . This po-
en ial is quasi-exac ly sol able, which means ha o any
non-nega i e in ege alue o M,M+1 o i s solu ions can be
ob ained in an algeb aic way. The 共unno malized兲solu ions
a e w i en in he o m
␾
n共x兲=Pn共x2兲共x2兲s−1/4exp
冉
−a
4x4−b
2x2
冊
,n=0,1,2, ...,
共7兲
whe e Pnis a polynomial o o de n. Ob iously, no maliz-
abili y equi es a艌0, while a=0 educes he p oblem o he
exac ly sol able ha monic oscilla o .
The simples solu ions a e ob ained o M=0 and M=1
[15]. Fo M=0 only one nodeless (i.e., g ound s a e)solu ion
appea s a E0
共M=0兲=4bs, wi h he co esponding wa e unc-
ion being
␾
0
共M=0兲共x兲⬃共x2兲s−1/4exp
冉
−a
4x4−b
2x2
冊
.共8兲
Fo M=1 wo solu ions appea , one nodeless and ano he
wi h one node o x⬎0. These co espond o he g ound s a e
and he i s exci ed s a e, espec i ely, a ene gies E0
共M=1兲
=4bs+␭−共s兲and E1
共M=1兲=4bs+␭+共s兲, whe e
␭±共s兲=2b±2共b2+8as兲1/2 共9兲
a e he oo s o he equa ion ␭2−4b␭−32as=0. The co e-
sponding wa e unc ions a e
␾
n
共M=1兲共x兲⬃
冉
1− ␭
8sx2
冊
共x2兲s−1/4exp
冉
−a
4x4−b
2x2
冊
,
共10兲
and he ␭=␭−共s兲and ␭=␭+共s兲choice has o be made o n
=0 and n=1, espec i ely 关15兴.关No e ha a艌0 and s艌0
imply ␭−共s兲艋0, so he polynomial pa o Eq. 共10兲is node-
less.兴I has o be men ioned ha he solu ions o M=0 and
M=1 belong o di e en sex ic po en ials i sis he same, as
he coe icien o he quad a ic e m is di e en hen. We
shall see, howe e , ha wi h app op ia e combina ions o s
and Mi is possible o sol e sex ic po en ials ha di e only
in he s eng h o he cen i ugal e m.
The no maliza ion o he wa e unc ions can also be
gi en in closed o m. Fo his one has o e alua e in eg als o
he ype
G. LÉVAI AND J. M. ARIAS PHYSICAL REVIEW C 69, 014304 (2004)
014304-2
I共A兲⬅
冕
0
⬁
xAexp
冉
−a
2x4−bx2
冊
=1
2⌫
冉
A+1
2
冊
a−共A+1兲/4exp
冉
b2
4a
冊
D−共A+1兲/2
冉
b
a1/2
冊
共11兲
=1
2⌫
冉
A+1
2
冊
共2a兲共−A+1兲/4U
冉
A+1
4,1
2;b2
2a
冊
,共12兲
whe e Dp共z兲is he pa abolic cylinde unc ion and U共
␣
,
␤
;z兲
is one o he o ms o he con luen hype geome ic unc ion
关14兴.
La ge alues o Mcan also be conside ed (e.g., o M
=2 h ee di e en solu ions a e ob ained o he h ee oo s
o a cubic algeb aic equa ion o ␭), bu M=0 and M=1 a e
su icien o ou pu poses in his pape . A comple e s udy
including mo e solu ions, explici closed o ms o he no -
maliza ion ac o s, and applica ions o ac ual nuclei is unde -
way.
IV. APPLICATION AS A
␥
-INDEPENDENT POTENTIAL
In o de o cas Eq. (6)in a o m simila o Eq. (5)we
ha e o w i e x=
␤
and s=共
␶
/2兲+5Ⲑ4( emembe ha
␶
艌0).
In o de o keep he quad a ic e m a a cons an alue (once
he a,bpa ame e s a e ixed)we also ha e o p esc ibe
s+M+1
2=1
2
共
␶
+2M+7
2
兲
⬅c= cons . 共13兲
Wi h his he sex ic oscilla o Hamil onian can be b ough o
he o m o Eq. 共4兲wi h u共
␤
兲being
u
␲
共
␤
兲=共b2−4ac
␲
兲
␤
2+2ab
␤
4+a2
␤
6+u0
␲
,共14兲
whe e he index
␲
=± is included o dis inguish he po en ial
o e en/odd
␶
’s, which is sligh ly di e en as explained be-
low. In Eq. 共14兲c
␲
a e he cons an s ob ained in Eq. 共13兲 o
e en/odd alues o
␶
and we ha e in oduced a cons an u0
␲
o con enience, as will be discussed below.
Equa ion (13)implies ha inc easing/dec easing Mby
one uni has o come wi h dec easing/inc easing
␶
by wo
uni s. Thus, once he alues o he 共a,b兲pa ame e s a e
ixed, he sequence o 共M,
␶
兲 alues 共K,0兲,共K−1,2兲,
共K−2,4兲, ... co espond o solu ions o Eq. (14)wi h c+=7
4
+K. In he same way, he sequence o 共M,
␶
兲 alues
共K,1兲,共K−1,3兲,共K−2,5兲, ... co espond o solu ions o Eq.
(14)wi h c−=9
4+K. Consequen ly, he po en ial o
␶
-e en
and
␶
-odd s a es is sligh ly di e en due o he ac ha he
coupling coe icien b2−4ac±o he quad a ic e m is di e -
en in he wo cases due o he choices o c+and c− ha a e
necessa y o sepa a e he 共
␶
+1兲共
␶
+2兲
␤
−2 e m in a uni o m
way. This si ua ion can be handled using di e en s a egies.
One possibili y is se ing u0
+=u0
−=0 and conside ing
b2⬎10a, which minimizes he de ia ion o he quad a ic
e ms compa ed o he qua ic and sex ic e ms. Ano he pos-
sibili y is in oducing a ela i e ene gy shi be ween he
␶
-e en and
␶
-odd po en ials by se ing u0
+and u0
−such ha he
po en ial minima a e a he same ene gy. We shall discuss
his possibili y a e analyzing quali a i ely he spec um and
he po en ial shapes.
Le us analyze he spec um ob ained in a simple case.
Taking M=1 we ob ain, as explained in he p eceding sec-
ion, wo solu ions o
␶
=0(one wi h no nodes, n=0, and he
o he one wi h one node, n=1). In he no a ion in oduced in
Re . [8] he label
␰
is ou n+1. Thus he wo solu ions o Eq.
(10)wi h s=5/4 a e
␾
n
共M=1兲共
␤
兲wi h n=0 and 1 and co e-
spond o
␾
␰
,
␶
=
␾
1,0 and
␾
␰
,
␶
=
␾
2,0, espec i ely, in Re . [8]
no a ion. Inspec ing he ene gy eigen alues, he co espond-
ing ene gies a e E1共M=1,
␶
=0兲=E1,0=7b−2共b2+10a兲1/2
+u0
+and E2共M=1,
␶
=0兲=E2,0=7b+2共b2+10a兲1/2+u0
+, e-
spec i ely. The same po en ial is ob ained by aking M=0
and
␶
=2. In his case we ha e a single solu ion, Eq. (8)wi h
s=9/4,
␾
0
共M=0兲共
␤
兲wi h no nodes, and wi h ene gy E1共M
=0,
␶
=2兲=9b+u0
+. This co esponds o
␾
␰
,
␶
=
␾
1,2 in he no-
a ion o Re . [8]. A simila analysis can be pe o med o he
solu ions wi h odd-
␶
alues. Fo M=1 he e a e wo solu-
ions wi h
␶
=1, Eq. (10)wi h s=7/4, which co esponds o
␾
␰
,
␶
=
␾
1,1 and
␾
␰
,
␶
=
␾
2,1 o n=0 and n=1, espec i ely. The
co esponding ene gy eigen alues a e E1,1=9b−2共b2
+14a兲1/2+u0
−and E2,1=9b+2共b2+14a兲1/2+u0
−. Again he
same po en ial is ob ained o M=0 and
␶
=3. In his case
he e is a single solu ion wi h no nodes, Eq. (8)wi h s
=11/4, which co esponds o
␾
1,3 and has an ene gy E1,3
=11b+u0
−. In Fig. 1 a schema ic spec um is shown wi h
indica ion o he ele an quan um numbe s. In Fig. 2 he
co esponding wa e unc ions wi h he no a ion
␾
␰
,
␶
a e p e-
sen ed.
Now we analyze he di e en po en ial shapes ha can be
p oduced by di e en elec ion o pa ame e s in Eq. (14).
F om Eq. (14)we ind ha he shape o he po en ial u
␲
共
␤
兲
depends on he sign o b2−4ac
␲
and b, which se s he coe -
icien s o he quad a ic and qua ic e ms. (The coe icien o
he leading sex ic e m is always posi i e.)When b2⬎4ac
␲
and b⬎0 hold [i.e., o b⬎2共ac
␲
兲1/2], he po en ial has a
minimum a
␤
=0 and i inc eases mono onously wi h
␤
.
When b2⬍4ac
␲
, i espec i e o he sign o b[i.e., o
FIG. 1. Schema ic ypical spec um o he sex ic oscilla o wi h
indica ion o he ele an quan um numbe s.
SEXTIC OSCILLATOR AS A
␥
-INDEPENDENT POTENTIAL PHYSICAL REVIEW C 69, 014304 (2004)
014304-3
−2共ac
␲
兲1/2⬍b⬍2共ac
␲
兲1/2], a minimum appea s o
␤
⬎0,
while o b2⬎4ac
␲
and b⬍0[i.e., o b⬍−2共ac
␲
兲1/2], i s a
maximum appea s and hen a minimum as
␤
inc eases. In all
h ee cases he exac loca ion o he ex emal poin (s)can be
ob ained om he eal and posi i e solu ions o
共
␤
0
␲
兲2=1
3a关−2b±共b2+12ac
␲
兲1/2兴.共15兲
Due o he ela i ely small di e ence in c+and c−, he
␶
-e en
and
␶
-odd po en ials ha e he same ypes o ex ema a abou
he same
␤
, excep o some peculia combina ions o aand
b. Assuming ha he e a e no complica ions o his kind, we
can now e u n o he ques ion o eno malizing he minima
o he
␶
-e en and
␶
-odd po en ials. Fo b⬎2共ac
␲
兲1/2,
␲
= +, − he minima o he wo po en ials will be u0
+and u0
−a
␤
=0, so hey coincide i u0
+=u0
−holds. Fo b⬍2共ac
␲
兲1/2 we
can equa e he minima o u+共
␤
兲and u−共
␤
兲i we se u0
+=0
and
u0
−=共b2−11a兲共
␤
0
+兲2−共b2−13a兲共
␤
0
−兲2+2ab关共
␤
0
+兲4−共
␤
0
−兲4兴
+a2关共
␤
0
+兲6−共
␤
0
−兲6兴,共16兲
whe e
␤
0
␲
a e ob ained om Eq. 共15兲wi h he choice o he
“+” sign. Wi h his he wo po en ials ha e hei minima a
he same ene gy, bu hey ake on di e en alues a he
o igin. Illus a ions o he possible po en ial shapes a e dis-
played in Fig. 3. Ob iously, u0
−in Eq. 共15兲also has o be
added o he ene gies o he
␶
-odd po en ial. Figu e 4 shows
he ela i e posi ion o he ene gy le els E
␰
,
␶
as ei he ao b
is a ied and he o he pa ame e is kep a a ixed alue.
The elec ic quad upole ansi ion a es can also be de e -
mined analy ically by calcula ing he ma ix elemen s o he
ansi ion ope a o [8,11]
T共E2兲=
␣
2
␮
=
␤
关D
␮
,0
共2兲cos
␥
+2
−1/2共D
␮
,2
共2兲+D
␮
,−2
共2兲兲sin
␥
兴.
共17兲
The adial in eg als ha appea in he
␤
a iable in he ma-
ix elemen s o T共E2兲can again be de e mined using Eq. 共11兲.
FIG. 2. Wa e unc ions wi h he no a ion
␾
␰
,
␶
o he case o
po en ial pa ame e s a=40 000 and b=200.
FIG. 3. Po en ials u+共
␤
兲( ull line)and u−共
␤
兲(b oken line) o
a=40 000, and b=1000 (le panel),b=200 (middle panel), and b
=−1000 ( igh panel). The lowes ene gy le el appea s in hese
po en ials a E1,0=4633.57, 73.35, and −9366.43, espec i ely.
FIG. 4. Exci a ion ene gies E
␰
,
␶
*=E
␰
,
␶
−E1,0 wi h a=40 000 ixed
as a unc ion o b(le panel)and wi h b=200 ixed as a unc ion o
a( igh panel).
FIG. 5. The ene gy spec um and he s eng h o some elec ic
quad upole ansi ions calcula ed wi h a=40 000 and b=200 (le
panel)and he co esponding da a o 134Ba ( igh panel).
G. LÉVAI AND J. M. ARIAS PHYSICAL REVIEW C 69, 014304 (2004)
014304-4
In o de o ob ain he o al ma ix elemen s, one has o cal-
cula e also he componen s depending on
␥
and he Eule
angles
␪
i. This can be done ollowing he echniques de-
sc ibed in Re . 关16兴. These pa s in oduce ce ain selec ion
ules no only o he angula momen a, bu also o
␶
.
V. DISCUSSION
In o de o compa e he main cha ac e is ics o he sex ic
oscilla o as a
␥
-uns able po en ial wi h hose o o he po en-
ials o his kind, we p esen calcula ions o a pa icula
alue o he pa ame e s, a=40 000 and b=200. These num-
be s we e chosen such ha he esul ing ene gy spec um
app oxima es ha o he 134Ba nucleus, he i s candida e o
E共5兲symme y [17]. We s ess ha ou aim is no o ep o-
duce he expe imen al da a, a he o ge a quali a i e pic u e
abou he gene al pe o mance o he model. The po en ials
u±共
␤
兲a e displayed in he middle panel o Fig. 3, while he
ene gy eigen alues a e shown in Fig. 5, oge he wi h he
co esponding expe imen al ene gy le els. Figu e 5 also
shows he calcula ed and he expe imen al B共E2兲 alues o
ansi ions be ween he ene gy le els. No e ha elec ic
quad upole ansi ions which change
␶
by mo e han one uni
a e ze o i we use he ansi ion ope a o (17), bu ini e
B共E2兲s eng hs can be ob ained i we apply e ms o he
nex o de (see, e.g., Re . [18]).
In Table I we summa ize he a io o he mos impo an
ene gy eigen alues and hose o he mos cha ac e is ic
B共E2兲 ansi ion a es ob ained om he sex ic oscilla o wi h
pa ame e s a=40 000, b=200, he in ini e squa e well po en-
ial [8], and he nume ically sol ed
␤
4po en ial [19] oge he
wi h he co esponding expe imen al alues o 134Ba, when-
e e a ailable. I is seen ha he ene gy a ios co esponding
o he E共5兲symme y sys ema ically all be ween he alues
o he
␤
4po en ial and he sex ic oscilla o . The si ua ion is
less ob ious o he a io o he B共E2兲 alues: he e he sex ic
oscilla o and he in ini e squa e well seem o yield simila
a ios, while he numbe s ob ained om he
␤
4po en ial a e
sys ema ically highe . This migh be due o he ac ha he
sex ic oscilla o po en ial goes o in ini y s eepe han he
␤
4
po en ial, so he asymp o ic beha io o i s wa e unc ions
can be close o ha o he wa e unc ions o he in ini e
squa e well. Compa ing he esul s wi h he expe imen al
da a o 134Ba we can conclude ha , a leas in his case, he
sex ic oscilla o allows a be e app oxima ion han he o he
essen ially pa ame e - ee po en ials. We expec ha his con-
clusion will be gene al due o he lexible na u e o he sex ic
po en ial whose shape is go e ned by wo pa ame e s. In ac
his po en ial can be used no jus a he c i ical poin bu i
can be use ul o model he ull shape phase ansi ion om
sphe ical o de o med
␥
-uns able nuclei by changing he pa-
ame e s aand b.
Be o e closing, we men ion some aspec s o he sex ic
oscilla o ha migh gi e u he help in he analysis o nu-
clei nea c i ical poin s. Fi s , we no e ha wi h M=2 in Eq.
(6) he analysis can be ex ended o u he s a es, such as
␾
1,4,
␾
1,5,
␾
2,2,
␾
2,3,
␾
3,0, and
␾
3,1. Second, he po en ial
shape which con ains bo h a local maximum and a minimum
a
␤
⬎0 migh be use ul in he desc ip ion o nuclei wi h he
so-called X共5兲symme y, which is hough o occu in he
shape phase ansi ion be ween he sphe ical and he axially
de o med domain [9]. Thi d, he e a e u he quasi-exac ly
sol able po en ials bo h wi h con ining and noncon ining
na u e [15], which can also be conside ed in he Boh
Hamil onian.
ACKNOWLEDGMENTS
This wo k was suppo ed by he OTKA G an No. T37502
(Hunga y)and by he Spanish MCyT unde P ojec No.
BFM2002-03315.
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Reading, MA, 1975), Vol. II.
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TABLE I. Ra ios o some ene gy eigen aluess and elec ic quad upole ansi ion s eng hs om he sex ic oscilla o wi h a=40 000, b
=200, he in ini e squa e well [8], and he
␤
4po en ial [19], oge he wi h he expe imen ally obse ed quan i ies o 134Ba.
E共41,2
+兲
E共21,1
+兲
E共02,0
+兲
E共21,1
+兲
E共61,3
+兲
E共21,1
+兲
B共E2;41,2
+→21,1
+兲
B共E2;21,1
+→01,0
+兲
B共E2;22,0
+→21,1
+兲
B共E2;21,1
+→01,0
+兲
B共E2;01,3
+→21,2
+兲
B共E2;21,1
+→01,0
+兲
Sex ic oscilla o 2.39 3.68 3.70 1.70 1.03 2.12
E共5兲2.20 3.03 3.59 1.68 0.86 2.21
␤
42.09 2.39 3.27 1.82 1.41 2.52
134Ba (exp .)2.31 3.57 3.65 1.56 (18)0.42 (12)
SEXTIC OSCILLATOR AS A
␥
-INDEPENDENT POTENTIAL PHYSICAL REVIEW C 69, 014304 (2004)
014304-5

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G. LÉVAI AND J. M. ARIAS PHYSICAL REVIEW C 69, 014304 (2004)
014304-6