The sextic oscillator as a γ-independent potential

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. [1]A. Boh and B. Mo elson, Nuclea S uc u e (Benjamin, Reading, MA, 1975), Vol. II. [2]F. Iachello and A. A ima, The In e ac ing Boson Mode (Cam- b idge Uni e si y P ess, Camb idge, 1987). [3]A. E. L. Diepe ink, O. Schol en, and F. Iachello, Phys. Re . Le . 44, 1747 (1980). [4]D. H. Feng, R. Gilmo e, and S. R. Deans, Phys. Re . C 23, 1254 (1981). [5]E. López-Mo eno and O. Cas años, Phys. Re . C 54, 2374 (1996). [6]J. Jolie e al., Phys. Re . Le . 89, 182502 (2002). [7]J. M. A ias, J. Dukelsky, and J. E. Ga cía-Ramos, Phys. Re . Le . 91, 162502 (2003). [8]F. Iachello, Phys. Re . Le . 85, 3580 (2000). [9]F. Iachello, Phys. Re . Le . 87, 052502 (2001). [10]F. Iachello, Phys. Re . Le . 91, 132502 (2003). 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 [11]L. Wile s and M. Jean, Phys. Re . 102, 788 (1956). 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