Anharmonic double-phonon excitations in the interacting boson model

Anha monic double-phonon exci a ions in he in e ac ing boson model J. E. Ga cı ´a-Ramos,1,2 J. M. A ias,2and P. Van Isacke 3 1Ins i u e o Theo e ical Physics, Vakg oep Suba omai e en S alings ysica, P oe uins aa 86, B-9000 Gen , Belgium 2Depa amen o de Fı ´sica A o ´mica, Molecula y Nuclea , Uni e sidad de Se illa, Ap do. 1065, E-41080 Se illa, Spain 3G and Acce ´le ´ a eu Na ional d’Ions Lou ds, B.P. 5027, F-14076 Caen Cedex 5, F ance 共Recei ed 12 July 2000; published 14 No embe 2000兲 Double- ␥ ib a ions in de o med nuclei a e analyzed in he con ex o he in e ac ing boson model. A simple ex ension o he o iginal e sion o he model owa ds highe -o de in e ac ions is equi ed o explain he obse ed anha monici ies o nuclea ib a ions. The in luence o h ee- and ou -body in e ac ions on he momen s o ine ia o g ound and ␥ bands, and on he ela i e posi ion o single- ␥ and double- ␥ bands is s udied in de ail. As an example o a ealis ic calcula ion, spec a and ansi ions o he highly ␥ -anha monic nuclei 164Dy, 166E , and 168E a e in e p e ed in his app oach. PACS numbe 共s兲: 21.60.Fw, 21.60.E , 21.10.Re, 27.70.⫹q I. INTRODUCTION Vib a ional deg ees o eedom in a omic nuclei can be desc ibed in e ms o phonon exci a ions ha a ise om nuclea shape oscilla ions. Vib a ions o nuclei wi h ellipsoi- dal symme y can be o wo ypes 关1兴: ␤ ib a ions which p ese e axial symme y and gi e ise o a band wi h K ⫽0, and ␥ ib a ions which b eak axial symme y and yield aK⫽2 band, whe e Kis he p ojec ion o he angula mo- men um on he axis o symme y. A he expe imen al le el, ␥ bands ha e been iden i ied in many well-de o med nuclei; in con as , he iden i ica ion o ␤ bands is s ill ull o ques- ions and di icul ies. This is mainly because, when he en- e gy su ace has a well-de o med minimum in ␤ bu is a he la in ␥ , he ␤ band inc eases in exci a ion ene gy and ap- p oaches he ene gy egion whe e o he deg ees o eedom a e impo an . In ha case band mixing may occu and can gi e ise o nonpu e s uc u es wi h decay pa e ns di icul o iden i y as hose o a ␤ band 关2兴. Since single- ␥ exci a ions a e e y well es ablished, i is na u al o look o double- ␥ ib a ions and o de elop mod- els ha can deal wi h such mul iphonon exci a ions. Double- ␥ exci a ions co espond o K ␲ ⫽0⫹and K ␲ ⫽4⫹bands which a e he an ipa allel and pa allel combina ions o single- ␥ phonons, espec i ely. The expe imen al iden i ica- ion o wo- ␥ s a es in de o med nuclei is di icul because hei expec ed exci a ion ene gy is a ound he pai ing gap and hence hey can mix s ongly wi h wo-quasipa icle ex- ci a ions. Howe e , ecen expe imen al imp o emen s in nuclea spec oscopy ollowing Coulomb exci a ion 关3兴, in- elas ic neu on sca e ing 关4兴, and he mal-neu on cap u e 关5兴ha e made possible he s udy o highly exci ed low-spin s a es. Many s a es ha e been p oposed as possible candi- da es o double- ␥ ib a ions. The e is, howe e , some con- o e sy abou hei in e p e a ion. The au ho o Re . 关6兴 claims ha some o he p esumed double- ␥ s a es can be in e p e ed as single hexadecapole-phonon exci a ions. In ac , o iden i y he band head o a double- ␥ band i is no su icien o analyze jus B(E2) alues; da a om single- nucleon ans e eac ions, ␤ -decay s udies, and inelas ic sca e ing expe imen s mus be conside ed as well. One o he key p ope ies o dis ega d a band as a double- ␥ band is he ac ha i s membe s, in i s o de , canno be popula ed in single-nucleon ans e eac ions. Many examples o K ␲ ⫽4⫹s a es ha a e iden i ied as double- ␥ exci a ions bu a e s ongly popula ed in single-nucleon ans e eac ions, can be ound in he li e a u e: 158Gd, 162Dy, 172Yb, 176,178H , and 190,192Os. Since he double-phonon cha ac e o he s a es in ques ion is in doub , hey a e no conside ed he e. Howe e , some candida es seem o ha e a genuine double- phonon na u e. Such is he case wi h 164Dy and 166⫺68E . In pa icula , in Re . 关7兴aK ␲ ⫽4⫹s a e in 164Dy a 2.173 MeV is ound o exhibi all p ope ies o a double- ␥ band. In Re s. 关8兴and 关9兴 he obse a ion is epo ed o K ␲ ⫽0⫹and K ␲ ⫽4⫹double- ␥ s a es in 166E , a ene gies o 1.949 and 2.029 MeV, espec i ely. Finally, in 168E aK ␲ ⫽4⫹double- ␥ exci a ion is iden i ied a an ene gy o 2.055 MeV 关10兴. One o he mos s iking ea u es o he obse ed double- ␥ bands is hei high anha monici y, i.e., he a io o double- ␥ o e single- ␥ ene gy is di e en om 2 and anges om 2.5 o 2.8. This in o ma ion is e y impo an since i p o- ides a s ingen es o nuclea models. The nuclei 164Dy and 166,168E ha e been in e p e ed in he con ex o many di e en models such as he quasiphonon model 关11,12兴, he geome ical model 关13兴, he mul iphonon model 关14兴, he sel -consis en collec i e-coo dina e me hod 关15,16兴, and he sdg in e ac ing boson model 共IBM兲关17,18兴, and i is now o in e es o e isi hese models in connec ion wi h anha - monic ib a ional beha io . In his pape he simples e sion o he in e ac ing boson model 共IBM兲关19兴is ex ended by adding o he usual Hamil- onian highe -o de in e ac ions be ween he bosons wi h he pu pose o c ea ing a amewo k ha accommoda es he high anha monici ies obse ed in 164Dy and 166,168E . The s uc- u e o he pape is as ollows. Fi s , he IBM is e iewed wi h special e e ence o i s ha monic cha ac e . In Sec. III he inclusion o h ee-body e ms in he Hamil onian is dis- cussed. The in oduc ion o ou -body e ms is p esen ed in Sec. IV and some analy ic esul s a e poin ed ou . In Sec. V a de ailed s udy o possible ou -body e ms is ca ied ou PHYSICAL REVIEW C, VOLUME 62, 064309 0556-2813/2000/62共6兲/064309共13兲/$15.00 ©2000 The Ame ican Physical Socie y62 064309-1 and ealis ic calcula ions o 164Dy and 166,168E a e p e- sen ed. Finally, in Sec. VI he conclusions o his wo k a e made. II. THE IBM-1 AS A HARMONIC MODEL The IBM desc ibes low-lying collec i e exci a ions in e en-e en nuclei in e ms o monopole 共s兲and quad upole 共d兲bosons 关19兴. The boson numbe ha co esponds o a gi en nucleus equals hal he numbe o alence nucleons (N⫽n/2). The o a ionally in a ian and numbe -conse ing boson Hamil onian usually includes up o wo-body in e ac- ions be ween he bosons al hough highe -o de e ms can be added in p inciple. The mos gene al wo-body IBM Hamil- onian can be w i en in a mul ipole expansion 关19兴as H ˆ⫽␧sn ˆs⫹␧dn ˆd⫹ ␬ 0P ˆ†P ˆ⫹ ␬ 1L ˆ•L ˆ⫹ ␬ 2Q ˆ•Q ˆ⫹ ␬ 3T ˆ3•T ˆ3 ⫹ ␬ 4T ˆ4•T ˆ4,共1兲 whe e n ˆsand n ˆda e he s- and d-boson numbe ope a o s, espec i ely, and P ˆ†⫽1 2d†•d†⫺1 2s†•s†,共2兲 L ˆ⫽ 冑 10共d†⫻d ˜ 兲(1),共3兲 Q ˆ⫽s†d ˜ ⫹d†s ˜ ⫹ ␹ 共d†⫻d ˜ 兲(2),共4兲 T ˆ3⫽共d†⫻d ˜ 兲(3),共5兲 T ˆ4⫽共d†⫻d ˜ 兲(3).共6兲 The symbol ‘‘•’’ ep esen s he scala p oduc ; in his pape he scala p oduc o wo ope a o s wi h angula momen um Lis de ined as T ˆL•T ˆL⫽兺M(⫺1)MT ˆLMT ˆL⫺Mwhe e T ˆLM co esponds o he Mcomponen o he ope a o T ˆL. In he p e ious equa ions he ope a o ␥ ˜ lm⫽(⫺1)m ␥ l⫺m共whe e ␥ e e s o so d) is in oduced so ha he annihila ion ope a- o e i ies he app op ia e p ope ies unde spa ial o a ions. I is no a p io i clea o wha ex end ␤ and ␥ ib a ions a e anha monic in he IBM e en i one jus conside s he Hamil onian 共1兲wi h up o wo-body in e ac ions. A pa ial FIG. 1. The a ios RK ␥ 共as de ined in he ex 兲as a unc ion o ␪ l/ ␬ o di e en l. The Hamil onian 共9兲is used wi h ␹ ⫽⫺0.5; he boson numbe is N⫽15. The dashed lines gi e he expe imen al alues o he co esponding a ios in 166E . FIG. 2. Expe imen al 共a兲and calcula ed 共b兲 spec um o 166E . The heo e ical esul s a e ob- ained wi h he Hamil onian 共9兲wi h ␬ ⫽23.8 keV, ␹ ⫽⫺0.55, ␬ ⬘⫽⫺1.9 keV, and ␪ 4⫽31.3 keV. The boson numbe is N⫽15. J. E. GARCI ´A-RAMOS, J. M. ARIAS, AND P. VAN ISACKER PHYSICAL REVIEW C 62 064309 064309-2 analysis o his p oblem was gi en in Re . 关20兴. The e, he au ho s ind ha he IBM in i s simples e sion is a ha - monic model in he limi o in ini e boson numbe Nand e en o ini e N he model canno accommoda e la ge an- ha monici y i one conside s up o wo-body in e ac ions; only he in e play be ween one⫹ wo-body e ms and highe - o de in e ac ions can induce, in p inciple, a sizable anha - monici y in he double-phonon exci a ions. The eason why one-body e ms and wo-body in e ac ions canno c ea e a la ge anha monici y can be unde s ood as ollows. I one conside s a Hamil onian wi h one pa ame e ha con ols he a io o he s eng h o he one-body ene gies and he wo- body in e ac ions, H ˆ⫽共1⫺ ␰ 兲共␧sn ˆs⫹␧dn ˆd兲⫹ ␰ 共 ␬ 0P ˆ†P ˆ⫹ ␬ 1L ˆ•L ˆ⫹ ␬ 2Q ˆ•Q ˆ ⫹ ␬ 3T ˆ3•T ˆ3⫹ ␬ 4T ˆ4•T ˆ4兲,共7兲 whe e ␰ anges om 0 o 1, one inds wo ‘‘phases’’ sepa- a ed by a c i ical alue, ␰ c: a i s phase whe e he one-body e m plays he main ole ( ␰ ⬍ ␰ c) and a second phase whe e he wo-body in e ac ion is he d i ing o ce ( ␰ ⬎ ␰ c). The c ucial poin is ha he sepa a ion be ween he wo phases is e y sha p 关21兴and essen ially no in e play be ween one- and wo-body e ms can be ound. Since, o a good app oxi- ma ion, he o ce is ei he one body o wo body bu no bo h, ha monic beha io canno be a oided. The inclusion o high-o de in e ac ions in a sys em wi h a high boson numbe also leads o a ha monic desc ip ion. Only o ini e boson numbe he in e play be ween one ⫹ wo-body e ms and highe -o de in e ac ions can induce an anha monici y in he double-phonon exci a ions ha is compa able o he obse ed one. These ideas will be used as a guideline in he ollowing sec ions. To ca y ou a quan i a i e s udy o anha monici ies, i is con enien o de ine a a io be ween single- and double- phonon exci a ion ene gies. Because he expe imen al si ua- ion o ␤ exci a ions is no clea we concen a e on ␥ ib a- ions and de ine ene gy a ios o ␥ phonons only 共al hough simila de ini ions can be gi en o ␤ phonons兲: R0 ␥ ⬅Ex共0 ␥␥ ⫹兲 Ex共2 ␥ ⫹兲⫺Ex共21 ⫹兲,R4 ␥ ⬅Ex共4 ␥␥ ⫹兲⫺Ex共41 ⫹兲 Ex共2 ␥ ⫹兲⫺Ex共21 ⫹兲,共8兲 whe e 0 ␥␥ ⫹and 4 ␥␥ ⫹a e he band heads o he K ␲ ⫽0⫹and K ␲ ⫽4⫹double- ␥ bands, espec i ely, and Exs ands o ex- ci a ion ene gy. This pa icula de ini ion emo es any o a- ional in luence. III. THREE-BODY HAMILTONIANS Le us conside in he ollowing a Hamil onian ha in- cludes a quad upole e m, a o a ional L ˆ2 e m, and h ee- body in e ac ions be ween he dbosons, FIG. 3. The a io R0 SU(3) ␥ 共as de ined in he ex 兲 o he Hamil- onian H ˆ⫽aC ˆ2关SU(3)兴⫹bC ˆ3关SU(3)兴wi h N⫽5, 10, and 15. FIG. 4. The a io R0 SU(3) ␥ 共as de ined in he ex 兲 o he Hamil- onian H ˆ⫽aC ˆ3关SU(3)兴⫹cC ˆ2关SU(3)兴2wi h N⫽5, 10, and 15. FIG. 5. The a io R4 SU(3) ␥ 共as de ined in he ex 兲 o he Hamil- onian H ˆ⫽aC ˆ2关SU(3)兴⫹cC ˆ2关SU(3)兴2wi h N⫽5, 10, and 15. ANHARMONIC DOUBLE-PHONON EXCITATIONS IN THE... PHYSICAL REVIEW C 62 064309 064309-3 H ˆ⫽⫺ ␬ Q ˆ•Q ˆ⫹ ␬ ⬘L ˆ•L ˆ ⫹兺 kl ␪ l关共d†⫻d†兲(k)⫻d†兴(l)•关共d ˜ ⫻d ˜ 兲(k)⫻d ˜ 兴(l),共9兲 whe e ⫺ ␬ ⫽ ␬ 2, ␬ ⬘⫽ ␬ 1. Fi e independen h ee-body d-boson in e ac ions exis which ha e l⫽0, 2, 3, 4, and 6. In e ac ions wi h he same lbu di e en ka e no indepen- den bu di e by a no maliza ion ac o only 关22兴. The com- bina ions (k,l)⫽(2,0), 共0,2兲,共2,3兲,共2,4兲, and 共4,6兲a e cho- sen he e. The Hamil onian 共9兲is ce ainly no he mos gene al one ⫹ wo⫹ h ee-body Hamil onian ha can be conside ed. No- ably, a ib a ional e m ⑀ dn ˆdwhich domina es in sphe ical nuclei is omi ed since i is hough o lesse impo ance in he de o med nuclei conside ed he e. And, o all possible h ee-body in e ac ions 共se en een e ms兲, only hose be- ween he dbosons a e e ained he e since hey a e he mo e e icien e ms o p oduce anha monici y 关23兴. Fo he discussion o he anha monici ies o ␥ ib a ions we s udy he beha io o he ene gy a ios 共8兲as a unc ion o he a io ␪ l/ ␬ . The iden i ica ion o he s a es 0 ␥␥ ⫹and 4 ␥␥ ⫹ is based on he B(E2) alues o decay in o he single- ␥ s a es. In Fig. 1,1 he in luence o he a ious h ee-body in e ac ions is shown o a ypical alue o ␹ ( ␹ ⫽⫺0.5) and o N⫽15 bosons. I is seen ha a ␥ - ib a ional anha monic beha io is ob ained which can be di e en o he K ␲ ⫽0⫹and K ␲ ⫽4⫹bands 共e.g., posi i e o he o me while nega i e o he la e 兲. Ca e has been aken o plo esul s only up o alues o ␪ l ha do no d as ically al e he cha - ac e o o a ional spec um; beyond hese alues, he h ee- body in e ac ion, being o highes o de in he Hamil onian 共9兲, becomes dominan . Also shown in Fig. 1 a e he a ios RK ␥ as obse ed in 166E 关8,9兴,R0 ␥ ⫽2.76 and R4 ␥ ⫽2.50. Fig- u e 2 shows he expe imen al spec um o 166E 关8,9兴and compa es i o he eigenspec um o Hamil onian 共9兲wi h an l⫽4 h ee-body in e ac ion. The pa ame e s a e ␬ ⫽23.8 keV, ␹ ⫽⫺0.55, ␬ ⬘⫽⫺1.9 keV. and ␪ 4 ⫽31.3 keV.2wi h boson numbe N⫽15. Wi h hese alues he calcula ed exci a ion ene gies o he double- ␥ band heads a e 1926 keV and 1972 keV o he K ␲ ⫽0⫹and K ␲ ⫽4⫹ le els, espec i ely, leading o he a ios R0 ␥ ⫽2.82 and R4 ␥ ⫽2.45, in excellen ag eemen wi h obse a ion. No e, how- e e , ha al hough all ␥ -band heads a e well ep oduced by he calcula ion, p oblems a ise o he momen s o ine ia, in pa icula o he ␥ band. In nex sec ions we will come back o he momen s o ine ia o see ha h ee-body Hamil onians p o ide a e y poo desc ip ion o hem. IV. SU„3…HAMILTONIANS WITH UP TO FOUR-BODY INTERACTIONS In he p e ious sec ion and in Re . 关24兴a e y good de- sc ip ion o double-phonon exci a ion ene gy has been ob- ained, bu a expense o spoiling he momen s o ine ia o g ound and ␥ bands. These d awbacks seem o be a gene al ea u e o h ee-body Hamil onians. In his and he ollowing sec ion i is shown ha he d awbacks o h ee-body in e ac- ions can be o e come by going o he nex o de . Since ␥ anha monici y has been obse ed exclusi ely in well-de o med nuclei, i is app op ia e o conside he p ob- lem in he SU共3兲limi o he IBM which is sui ed o deal wi h nuclei in his mass egion 关25兴. The e o e, in a i s 1No e ha his igu e di e s om Fig. 2 in Re . 关24兴in some scale ac o s due o an e o in he de ini ion o ␪ l. 2No e ha he alue ␪ 4is 1/3 o ha gi en in Re . 关24兴which has an e o in i s de ini ion. The esul s shown in ha pape a e co ec a e a simple escaling o he pa ame e . FIG. 6. Expe imen al 共a兲and heo e ical 共b兲 spec um o 166E . The heo e ical esul s a e ob- ained wi h he Hamil onian H ˆ⫽13.43L ˆ2 ⫺20.84C ˆ2关SU(3)兴⫹9.296⫻10⫺3C ˆ2关SU(3)兴2共all coe icien s in keV兲wi h ␹ ⫽⫺ 冑 7/2. The boson numbe is N⫽15. J. E. GARCI ´A-RAMOS, J. M. ARIAS, AND P. VAN ISACKER PHYSICAL REVIEW C 62 064309 064309-4 app oach, he SU共3兲limi is used and la e , in he nex sec- ion, hese esul s a e used as a guidance in mo e ealis ic calcula ions. Le us conside he ollowing Hamil onian: H ˆ⫽aC ˆ2关SU共3兲兴⫹b1C ˆ3关SU共3兲兴⫹b2N ˆC ˆ2关SU共3兲兴 ⫹c1C ˆ2关SU共3兲兴2⫹c2N ˆC ˆ3关SU共3兲兴⫹c3N ˆ2C ˆ2关SU共3兲兴, 共10兲 whe e C ˆn关SU(3)兴s ands o he Casimi ope a o o o de n o SU(3). No e he inclusion o cubic e ms o comple e- ness in he SU共3兲analysis. Since only he spec um o a single nucleus is o in e es he e, he numbe o bosons can be ixed in e e y case and he Hamil onian 共10兲can be sim- pli ied by combining e ms in o a single one, lea ing a Hamil onian wi h h ee coe icien s a,b, and c, H ˆ⫽aC ˆ2关SU共3兲兴⫹bC ˆ3关SU共3兲兴⫹cC ˆ2关SU共3兲兴2,共11兲 wi h eigen alues 具 共␭, ␮ 兲 兩 H ˆ 兩 共␭, ␮ 兲 典 ⫽a共␭2⫹ ␮ 2⫹␭ ␮ ⫹3␭⫹3 ␮ 兲⫹b共␭⫺ ␮ 兲 ⫻共2␭⫹ ␮ ⫹3兲共␭⫹2 ␮ ⫹3兲⫹c共␭2⫹ ␮ 2 ⫹␭ ␮ ⫹3␭⫹3 ␮ 兲2.共12兲 No qua ic Casimi ope a o exis s o SU共3兲because he numbe o independen Casimi ope a o s equals he numbe o labels ha cha ac e ize an i educible ep esen a ion. The Hamil onian 共11兲has no o a ional e m L ˆ2since o p ima y in e es , a his poin , is he desc ip ion o band-head ene gies o single- and double- ␥ exci a ions. No e ha Hamil onian 共11兲is no in no mal o de , as a consequence, i s e m con- ains up o wo-body in e ac ions 共one- and wo-body兲, sec- ond e m up o h ee-body in e ac ions 共one-, wo-, and h ee- body兲and hi d e m up o ou -body in e ac ions 共one-, wo-, h ee-, and ou -body兲. The de ini ion o he ene gy a ios 共8兲mus now be adap ed o inco po a e he symme y labeling o he s a es. In he SU共3兲limi he ␥ band belongs o he SU共3兲 ep esen a- ion (2N⫺4,2) whe e Nis he numbe o bosons. The double- ␥ band wi h K⫽4 is con ained in he (2N⫺8,4) ep- esen a ion; he double- ␥ band wi h K⫽0 is p edominan ly con ained in he (2N⫺6,0) ep esen a ion, al hough an im- po an componen is in (2N⫺8,4) 关20兴. In his sec ion he ene gy a ios 共8兲a e hus de ined as ollows: R0 SU(3) ␥ ⬅Ex共2N⫺6,0兲 Ex共2N⫺4,2兲,R4 SU(3) ␥ ⬅Ex共2N⫺8,4兲 Ex共2N⫺4,2兲. 共13兲 FIG. 7. Band heads o g ound, ␥ , ␤ , double- ␥ K⫽0, and double- ␥ K⫽4 bands o 164Dy. Pan- els co espond o 共a兲expe imen al da a, 共b兲cal- cula ion wi h Hamil onian 共23兲and ␹ ⫽⫺ 冑 7/2, 共c兲calcula ion wi h Hamil onian 共23兲and ␹ ⫽ ⫺0.55, 共d兲calcula ion wi h Hamil onian 共24兲and ␹ ⫽⫺ 冑 7/2, 共e兲calcula ion wi h Hamil onian 共24兲 and ␹ ⫽⫺0.55, and 共 兲calcula ion wi h Hamil- onian 共9兲. TABLE I. Obse ed and calcula ed B(E2) alues and a ios o 166E in a schema ic calcula ion using an SU共3兲Hamil onian. The E2 ope a o 共22兲is used wi h ee 2⫽(1.97)2W.u. and ␹ ⫽⫺0.26. B(E2) alue o a io Obse ed Calcula ed B(E2;21 ⫹→01 ⫹)(W.u.) 214⫾10a214 B(E2;41 ⫹→21 ⫹)(W.u.) 311⫾10a302 B(E2;2 ␥ ⫹→01 ⫹)(W.u.) 5.5⫾0.4a5.4 B共E2;0 ␥␥ ⫹→2 ␥ ⫹兲 B共E2;2 ␥ ⫹→01 ⫹兲3.8⫾1.3b(2.2⫺0.7 ⫹1.1 c兲2.5 B共E2;4 ␥␥ ⫹→2 ␥ ⫹兲 B共E2;2 ␥ ⫹→01 ⫹兲1.3⫾0.4 b(0.9⫾0.3c兲2.5 aF om Re . 关30兴. bF om Re . 关9兴. cF om Re . 关8兴. ANHARMONIC DOUBLE-PHONON EXCITATIONS IN THE... PHYSICAL REVIEW C 62 064309 064309-5 Because no o a ional e m is included, he a ios 共13兲can be compa ed di ec ly wi h Eq. 共8兲. In he ollowing he e ec o he di e en e ms in Eq. 共11兲on he deg ee o anha monic- i y o he wo-phonon exci a ion ene gy is analyzed. 共i兲a⬍0, b⫽0, c⫽0. This co esponds o he simples e sion o IBM; he ene gy a ios become R0 SU(3) ␥ ⫽24N⫺18 12N⫺6,R4 SU(3) ␥ ⫽24N⫺36 12N⫺6.共14兲 An almos pu e ha monic ␥ - ib a ional spec um is ound since he ene gy a ios 共14兲a e only sligh ly lowe han 2. This will be e e ed o as nega i e anha monici y as op- posed o he posi i e anha monici y o ene gy a ios abo e 2. 共ii兲a⬍0, b⫽”0, c⫽0. In his case he Hamil onian 共11兲is a combina ion o he quad a ic and cubic SU共3兲Casimi ope a o s. Fo gi en al- ues o aand band o a high enough boson numbe , only he h ee-body pa o he Hamil onian is dominan and a ha - monic spec um is eco e ed. Fo ob aining an anha monic spec um he alues o he Hamil onian pa ame e s a e e y cons ained once he numbe o bosons has been ixed, as he ollowing analysis shows. Wi hou loss o gene ali y acan be ixed o a⫽⫺1. The ene gy a ios a e hen R0 SU(3) ␥ ⫽3⫺4N⫹b共24N2⫺36N⫹27兲 共2N⫺1兲共6bN⫹9b⫺1兲,共15兲 R4 SU(3) ␥ ⫽22N⫺3 2N⫺1.共16兲 To keep he ene gy o he ␥ exci a ion posi i e, he alue o bhas an uppe limi b⬍1 6N⫹9.共17兲 The alue b⫽1/(6N⫹9) leads o a di e gence in R0 SU(3) ␥ and a ound his alue anha monic beha io is ound. F om Eq. 共15兲one obse es ha he beha io o he (2N⫺8,4) ep esen a ion is comple ely ha monic and does no depend on b. On he o he hand, om Eq. 共15兲one sees ha a wide ange o anha monic a ios is ound o he (2N⫺6,0) ep- esen a ion. As an illus a ion, in Fig. 3 Eq. 共15兲is ep e- sen ed as a unc ion o b, o h ee alues o N(5, 10, and 15). Only posi i e alues o ba e plo ed because o he nega i e ones R0 SU(3) ␥ dec eases smoo hly o he asymp o ic alues 1.27, 1.58, and 1.70 o N⫽5, 10, and 15, espec- i ely. The conclusion is ha a Hamil onian wi h c⫽0 does no ag ee wi h he expe imen al si ua ion obse ed in he mass egion o well-de o med nuclei, whe e he ␥ K⫽4 2s a e is highly anha monic. 共iii兲a⬍0, b⫽0, c⫽”0. In his case he Hamil onian 共11兲is a combina ion o he quad a ic and quad a ic-squa e SU共3兲Casimi ope a o s. As in he p e ious case, anha monici y equi es e y cons ained Hamil onian pa ame e s once he numbe o bosons is ixed. Again, wi hou loss o gene ali y, we ix a⫽⫺1. The ene gy a ios hen ead as ollows: R0 SU(3) ␥ ⫽共4N⫺3兲关2c共4N2⫺6N⫹9兲⫺1兴 共2N⫺1兲共8cN2⫹6c⫺1兲,共18兲 R4 SU(3) ␥ ⫽2共2N⫺3兲关4c共2N2⫺3N⫹9兲⫺1兴 共2N⫺1兲共8cN2⫹6c⫺1兲.共19兲 To keep he ene gy o he ␥ exci a ion posi i e, he alue o chas he uppe limi FIG. 8. Band heads o 166E . See cap ion o Fig. 7. J. E. GARCI ´A-RAMOS, J. M. ARIAS, AND P. VAN ISACKER PHYSICAL REVIEW C 62 064309 064309-6 c⬍1 8N2⫹6.共20兲 The alue c⫽1/(8N2⫹6) p oduces a di e gence in he wo ene gy a ios and in i s neighbo hood highly anha monic be- ha io is ound. In Figs. 4 and 5 a e plo ed he a ios R0 SU(3) ␥ and R4 SU(3) ␥ , espec i ely. Again, o nega i es alues o c, R0 SU(3) ␥ goes asymp o ically o he alues 1.45, 1.69, and 1.78 o N⫽5, 10, and 15. espec i ely, while R4 SU(3) ␥ goes o 1.33, 1.59, and 1.71. These nega i e anha monici ies ha e no phenomenological in e es . In his case he expe imen al si ua ion can be nicely de- sc ibed. Bo h ␥ K⫽0 2and ␥ K⫽4 2s a es can be accommoda ed in a anha monic desc ip ion. The conclusion is hus ha a Hamil onian wi h C ˆ2关SU(3)兴2and C ˆ关SU(3)兴2 e ms seems o be a good s a ing poin o ea he ␥ anha monici y in de o med nuclei. The desc ip ion p esen ed he e only p o ides band heads and, o eco e a o a ional s uc u e, a L ˆ2 e m mus be included. The o a ional s uc u e is he same in e e y band because in he SU共3兲limi no mixing exis s be ween o a- ional and ib a ional deg ees o eedom. In nex subsec ion hese esul s a e illus a ed wi h a schema ic calcula ion o ene gies and ansi ion p obabili ies. A schema ic applica ion Le us conside he case o 166E which is, as al eady men ioned, one o pa icula in e es because bo h double- ␥ exci a ions 共wi h K⫽0 and K⫽4) ha e been iden i ied 关8,9兴. To ca y ou he schema ic calcula ion, we use he Hamil- onian 共11兲wi h b⫽0. The expe imen al alues o he single and double- ␥ ene gy a ios a e R0 ␥ ⫽2.76 and R4 ␥ ⫽2.50 and can be compa ed di ec ly wi h he exp essions 共18兲and 共19兲leading wo alues o ⫺c/a, namely 5 ⫻10⫺4and 1⫻10⫺4. Bo h solu ions a e ai ly close and any alue in be ween hem will co ec ly desc ibe he anha mo- nici y o he K⫽0 and K⫽4 bands. The alue o aand he s eng h o he o a ional e m a e ixed om he exci a ion ene gies o he 21 ⫹and 22 ⫹le els. A e hese simple consid- e a ions one a i es a he ollowing Hamil onian: H ˆ⫽13.43L ˆ•L ˆ⫺20.84C ˆ2关SU共3兲兴⫹9.296 ⫻10⫺3C ˆ2关SU共3兲兴2,共21兲 whe e he coe icien s a e gi en in keV. The heo e ical and expe imen al spec a a e compa ed in Fig. 6 and a e y good ag eemen is ob ained. Howe e , due o he simplici y o he calcula ion, ␥ and ␤ bands a e degene a e in ene gy which is no he case expe imen ally. To comple e he desc ip ion, E2 ansi ion p obabili ies mus be compu ed also. The calcula ion o B(E2) alues in he SU共3兲limi , as in o he si ua ions whe e degene acies occu , mus be ea ed wi h ca e and an app op ia e basis mus be chosen o s a es wi h he same ene gy. A na u al way o do his wo k is o sligh ly li he degene acy o he SU共3兲Hamil onian. The le els can be spli using in he Hamil onian a alue ␹ ⫽⫺1.30 which is e y close o i s SU共3兲 alue. Wi h his SU共3兲b eaking he degene acy is li ed in a na u al way because he ␤ band is pushed up in ene gy, as is obse ed. One may expec ha wi h his small change in ␹ he SU共3兲spec um will keep i s p ope ies. The E2 ansi ion ope a o is T ˆ共E2兲⫽ee 关s†d ˜ ⫹d†s ˜ ⫹ ␹ 共d†⫻d ˜ 兲(2)兴.共22兲 The alue o ␹ ha bes ep oduces he da a is ␹ ⫽⫺0.26. No e ha ␹ in he T ˆ(E2) ope a o and in he Hamil onian is di e en . The e ec i e cha ge is ixed o ep oduce FIG. 9. Band heads o 168E . See cap ion o Fig. 7. ANHARMONIC DOUBLE-PHONON EXCITATIONS IN THE... PHYSICAL REVIEW C 62 064309 064309-7 B(E2;21 ⫹→01 ⫹): ee 2⫽(1.97)2W.u. In Table I heo e ical and expe imen al ansi ion a es in ol ing he g ound and he ␥ bands a e compa ed. This simple analysis sugges s ha a ou -body ope a o o he ype in Eq. 共21兲p o ides a good desc ip ion o bo h single-, and double- ␥ bands. In he nex sec ion his sche- ma ic analysis is ex ended o non-SU共3兲si ua ions. V. GENERAL HAMILTONIANS WITH UP TO FOUR-BODY INTERACTIONS A gene al Hamil onian wi h all possible h ee- and ou - body e ms can, in p inciple, be cons uc ed bu he numbe o pa ame e s is so high ha a s udy, e en a schema ic one, o he e ec o he di e en e ms on ene gy spec a and elec omagne ic ansi ions is impossible. Schema ic IBM Hamil onians ha e been used o many yea s and, in pa icu- la , he quad upole-quad upole in e ac ion has been e y suc- cess ul in desc ibing a wide a ie y o nuclea spec a 关19,25兴. On he o he hand, in he p e ious sec ion i was shown ha an expansion in e ms o Casimi ope a o s, which a e mainly ela ed o quad upole ope a o s, leads o a sa is ac o y desc ip ion o g ound, single- and double- ␥ bands. I is wo h no ing ha such an expansion in e ms o Casimi ope a o s has been success ully used in molecula physics whe e spec oscopic da a p o ide many anha monic s a es 关26兴. This leads us o p opose a Hamil onian as a quad- upole expansion ha includes up o ou -body e ms. An al e na i e Hamil onian can be based on an expansion in e ms o pseudo-Casimi ope a o s, which we de ine he e as ope a o s ha become a ue Casimi ope a o only o a pa icula choice o one s uc u e pa ame e . Fo example, he C ˆ2关SU(3)兴ope a o is ela ed o he quad upole ope a o h ough 2Q ˆ•Q ˆ⫹3 4L ˆ2whe e Q ˆ⫽s†d ˜ ⫹d†s ˜ ⫾ 冑 7/2(d† ⫻d ˜ )(2). I he alue ⫾ 冑 7/2 is changed o ␹ , a new ope a o C ˆ2关SU(3)兴 ␹ is ob ained which we e e o as a pseudo- Casimi ope a o . I is a Casimi ope a o only o ␹ ⫽ ⫾ 冑 7/2. Guided by he esul s o he p e ious sec ion, wo possible Hamil onians ha include up o ou -body in e ac ions, can be p oposed, one based on a quad upole expansion and he o he on a pseudo-Casimi expansion: FIG. 10. To al angula momen um J共dimensionless兲o he ini- ial s a e e sus ␥ - ay ene gies o ⌬J⫽2 ansi ions in g ound and ␥ bands, o 164Dy, 166E , and 168E . Full lines co espond o ex- pe imen al da a and long-dashed lines co espond o he calcula ed esul s ob ained wi h he Hamil onian 共9兲. The co esponding pa- ame e s a e gi en in he ex . FIG. 11. Same cap ion as Fig. 10 bu calcula ed esul s ob ained wi h he Hamil onian 共23兲. FIG. 12. Same cap ion as Fig. 10 bu calcula ed esul s ob ained wi h he Hamil onian 共24兲. J. E. GARCI ´A-RAMOS, J. M. ARIAS, AND P. VAN ISACKER PHYSICAL REVIEW C 62 064309 064309-8 H ˆQ⫽ ␬ ⬘L ˆ•L ˆ⫹aQ ˆ•Q ˆ⫹b共Q ˆ⫻Q ˆ⫻Q ˆ兲(0)⫹c共Q ˆ•Q ˆ兲共Q ˆ•Q ˆ兲, 共23兲 H ˆpC⫽ ␬ ⬘L ˆ•L ˆ⫹aC ˆ2关SU共3兲兴 ␹ ⫹bC ˆ3关SU共3兲兴 ␹ ⫹cC ˆ2关SU共3兲兴 ␹ 2,共24兲 whe e Q ˆ⫽Q ˆ( ␹ ) and C ˆ2关SU共3兲兴 ␹ ⫽2Q ˆ共 ␹ 兲•Q ˆ共 ␹ 兲⫹3 4L ˆ2,共25兲 C ˆ3关SU共3兲兴 ␹ ⫽⫺4 冑 35共Q ˆ共 ␹ 兲⫻Q ˆ共 ␹ 兲⫻Q ˆ共 ␹ 兲兲(0) ⫺9 2 冑 15共L ˆ⫻L ˆ⫻Q ˆ共 ␹ 兲兲(0).共26兲 Fo simplici y and aking in o accoun he analysis done in he p e ious sec ion, in he ollowing b⫽0 is aken in Eqs. 共23兲and 共24兲. A. Double- ␥ band heads The e a e h ee nuclei ha ha e double- ␥ bands iden i ied wi hou ambigui y: 164Dy, 166E , and 168E 关7–10兴. In his sec ion he band heads o hese nuclei a e s udied using he FIG. 13. Expe imen al 共a兲and heo e ical 共b兲 spec um o 164Dy. The Hamil onian 共24兲is used wi h pa ame e s ␬ ⬘⫽12.18 keV, a⫽⫺82.90 keV, c⫽0.05150 keV, and ␹ ⫽⫺0.55. The bo- son numbe is N⫽16. FIG. 14. Expe imen al 共a兲and heo e ical 共b兲 spec um o 166E . The Hamil onian 共24兲is used wi h pa ame e s ␬ ⬘⫽13.55keV, a⫽ ⫺75.40 keV, c⫽0.05286 keV, and ␹ ⫽⫺0.45. The boson numbe is N⫽15. ANHARMONIC DOUBLE-PHONON EXCITATIONS IN THE... PHYSICAL REVIEW C 62 064309 064309-9