A calculation of low-lying collective states in odd-even nuclei

Volume 144B, numbe 3,4 PHYSICS LETTERS 30 Augus 1984 A CALCULATION OF LOW-LYING COLLECTIVE STATES IN ODD-EVEN NUCLEI C.E. ALONSO, J.M. ARIAS Depa amen o de Fisica A omica y Nuclea , Uni e sidad de Se illa, Spain R. BIJKER Ke n ysisch Ve snelle Ins i uu , Ri]ksuni e si ei G oningen, The Ne he lands and F. IACHELLO A.W. W igh Nuclea S uc u e Labo a o y, Yale Uni e si y, New Ha en, CT 06520, USA Recei ed 19 Ap il 1984 Re ised manusc ip ecei ed 4 June 1984 We p esen esul s o a calcula ion o p ope ies o low-lying collec i e quad upole s a es in odd-e en nuclei wi hin he amewo k o he p o on-neu on in e ac ing boson- e mion model. The impo ance o p o on-neu on deg ees o ee- dom in low-lying collec i e s a es o nuclei has been emphasized in ecen yea s by he in e ac ing boson model [ 1 ]. In addi ion o in oducing speci ic p o on -neu on e ec s, such as he occu ence o K P = 1 + bands in de o med nuclei [2], his model has he ad- an age ha , being closely ela ed o he mic oscopic shell model, i allows one o make ex apola ions and p edic ions o p ope ies o nuclei hi he o unknown. The nex logical s ep in he di ec ion o a de ailed un- de s anding o nuclea p ope ies is ha o pe o ming simila p o on-neu on calcula ions o odd-e en nu- clei. Because o he la ge numbe o low-lying s a es, hese calcula ions p esen a majo challenge. P e ious calcula ions o odd-e en nuclei wi hin he amewo k o he in e ac ing boson model ha e been done wi h he e sion o he model in which no dis inc ion is made be ween p o on and neu on deg ees o eedom [3]. In his le e , we p esen he esul s o he i s sys ema ic calcula ions pe o med using he p o on- neu on in e ac ing boson- e mion model. The co e- sponding compu e p og am was o iginally w i en by O suka and Yoshida [4], and applied o he s udy o 79Rb and 79K in e . [5]. The calcula ions p esen ed in his a icle a e based on an imp o ed e sion o his p og am ob ained by one o us (R.B.), oge he wi h B. Vissche [6]. The imp o emen consis s o a p edi- agonaliza ion and unca ion o he boson spec um be o e coupling he odd e mion. In he in e ac ing boson- e mion model, spec a o odd-e en nuclei a e calcula ed by coupling he collec- i e deg ees o eedom, desc ibed by bosons, o he single-pa icle deg ees o eedom ( e mions). The hamil onian is w i en as H = H (B) + H ( ) + V (BE) , (1) whe e H (B) is he p o on-neu on in e ac ing boson hamil onian [ 1 ], H (F) is he hamil onian desc ibing he single-pa icles deg ees o eedom, and V(BF) is hei in e ac ion. The s uc u e o he boson- e mion in e ac ion is, in gene al, a he complex since he bosons a e composi e a he han undamen al pa i- cles and hus one needs o ake in o accoun he e - ec s o he Pauli p inciple. A pu ely phenomenologi- cal ea men o his in e ac ion is no possible, since i con ains a la ge numbe o pa ame e s. One mus he e o e ely on a mic oscopic de i a ion. Se e al o hese ha e been gi en [7-10]. In he calcula ions p e- 141 Volume 144B, numbe 3,4 PHYSICS LETTERS 30 Augus 1984 sen ed he e, we ha e used he ollowing o m (BF) = + &,F + A ( id "n~), (2) o odd neu on nuclei, and V (BF) = P~ (Q 'qT ) + A, F , + A, (nd 'nT ) , (3) o odd p o on nuclei. The i s e ms in eqs. (2) and (3) ep esen a quad upole--quad upole in e ac ion be ween bosons and e mions, Qp = {(d I" X "~ + s X 2) (2) + x(d ~ X d)(2))o , qp={~//,QLi,( ~ul.,- ) ~,)(al . XEj,)(2)}, p ='n, p (4) whe e we ha e used he usual no a ion o boson and e mion c ea ion and annihila ion ope a o s. The sec- ond e m, ep esen ing he e ec s o he Pauli p inci- ple (exchange in e ac ion), has been aken in his a i- cle o be [7] F~ =- {(sI" X d)(2)}T '{//.,~/, ' (10/N)1/2(2]+ 1) -1/2 × Qj (,,iS, + 5./)Qi,,/,,/,5 + 5,,u) " x : [(a* x x (; x 5 ,)( )] (2):/ + he mi ean conjuga e. (5) while Fu~ is ob ained om eq. (5) by in e changing he indices l and . O he , mo e complex o ms, could be used, i needed. The hi d e m ep esen s a mono- pole-monopole in e ac ion ndp m Jo ' n o (2] + 1) "]m"]m ' p = , . (6) P The mic oscopic s uc u e o he bosons en e s in eqs. (4)-(5) h ough he coe icien s [11 ] (2i/' = (l { ill g(2)l[l' "' I ), (7) whe e he o~ a e he s uc u e coe icien s o he S- pai s and N is he numbe o pai s. Finally, one needs he e mion hamil onian, H (F). When only one odd pa icle is p esen , i is su icien o conside he single-pa icle pa o H (F), n (F) = ~E.a ~ a (8) jm I ]m jm ' whe e he E/a e he single pa icle ene gies in he p es- ence o N pai s. In he calcula ions pe o med so a , he coe icien s uj, o/and he ene gies E] ha e been ob ained using he BCS app oxima ion E/= [(6/. - X) 2 + A 21 1/2, 2 = { [1 - (e/- ~)/E/] , (9) a he han he ull gene alized senio i y scheme. In conclusion, acco ding o he model desc ibed he e, spec a o odd-e en nuclei can be calcula ed in e ms o h ee pa ame e s P , A , A (o FT , An, A~). In o de o s udy speci ically p o on-neu on e ec s, we ha e applied his model o odd-neu on Xe nuclei and odd-p o on Cs nuclei in he 50-82 majo shell. Bo h iso opic chains ha e he same e en-e en co e. Thus, di e ences in he spec a can a ise only om he boson- e mion in e ac ion V (BF) and he single-pa i- cle hamil onian, H (F). The boson hamil onian, H (B) was aken om a p e ious calcula ion [12]. The un- pe u bed single-pa icle ene gies, ej, we e aken as in able 1. The alues shown in his able we e ex ac ed om e . [13] o p o on le els and om he expe i- men al spec um o 131~ o neu on le els. In he 50bn81 majo whell 50-82 he e a e 5 single-pa icle le els, 4 wi h posi i e pa i y (1 g7/2, 2 d5/2, 3 Sl/2 and 2 d3/2) and 1 wi h nega i e pa i y (1 hll/2 ). Since hey do no mix, calcula ions can be done independen ly o each pa i y. In his le e , we epo only esul s o nega- i e pa i y s a es. Those o posi i e pa i y s a es will Table 1 Unpe u bed single pa icle ene gies (MeV) used in he p esen calcula ion. Le el lg7/2 2ds/2 lh11/2 3Sl/2 2d3/2 p o on le els 0 0.60 1.50 3.35 3.00 neu on le els 0 0.80 2.00 2.10 2.50 142 Volume 144B, numbe 3,4 PHYSICS LETTERS 30 Augus 1984 Table 2 Boson- e mion in e ac ion pa ame e s o nega i e pa i y s a es (MeV). F A A p o ons 0.6 1.3 -0.3 neu ons 0.6 1.3 0 be p esen ed in a longe publica ion [14]. The alues o he pa ame e s used in he calcula ion a e gi en in able 2. Fig. 1 shows he esul s o he Xe iso opes, while ig. 2 shows he esul s o he Cs iso opes. The p o on-neu on e ec s a e clea ly displayed in hese igu es. In cesium, he ~ hll/2 le el is almos comple e- 2 ly emp y (Vll/2 ~ 0). As a esul , he exchange in e - E I I 23/'2- I 54Xe i i i 54 xe I I (MeV) 54Xe 5 F23/2- 27/2- / T21/~ 25/2- 25/2- , , L ~. ~'~/~ ~ ~ ~"~ 2'/~ I --~I9/2 1,9/=- ~ /,9/21 ,~,~ /,7,2- ,5/~- I 1:5/2- - 11/2- /,5/2- ~ 21 ---...~. /,3/2 /o7~- / ~ / 9/2-~ {~AA 9/2- 7'2- ooo ,50 610 710 8&150 ---6J0 710 810 50 610 10 80 Neu on Numbe Fig. 1. Calcula ed nega i e pa i y spec a in he odd-A Xe iso opes. The s a e JP = 11/2- is aken as ze o o he ene gy. The expe i- men al alues (ci cles, squa es, ...) a e om e . [ 15 ]. E i -h i i i i (MeV) 55Cs 55Cs 3 L _ 27/2- 23/2- 25z2: /25/2 _ 27/2, / / 21/2- ~ _ i25/2- ~ / ~ 21/2 2! 2~9 2~ 7/2- 9/ /~ 17/ I ~ [] ~G4~"~ [3 15/~ i 1512- ~ 9/2-- ._ ~ /~ ~ _ 15/2~ AA3F~A z~ . 9/2 "~2::::2~::"~"~^ 3/2- o!,,,2- _ o~~4,,,2 l ' I I i/ _ I I i 55 cS J d i ,,% 7/2~ ~/2- i I 50 60 70 80 50 60 70 80 50 60 70 8~0 Neu on Numbe Fig. 2. Calcula ed nega i e pa i y spec a in he odd-A Cs iso opes. The s a e JP = 11/2- is aken as ze o o he ene gy. The expe i- men al alues (ci cles, squa es, ...) a e om e . [16]. 143 Volume 144B, numbe 3,4 PHYSICS LETTERS 30 Augus 1984 ac ion anishes, and he spec a a e domina ed by he quad upole in e ac ion. On he con a y, in Xenon, he h 11/2 le el is emp y o smalle neu on numbe s (~ 54)bu becomes inc easingly popula ed o ia ge neu on numbe s (~ 76). A consequence o his is ha he s a e wi h j " = 9/2- becomes lowe and lowe in exci a ion ene gy and i c osses he s a e wi h JP = 11/2- a neu on numbe s -~ 66. The esul s shown in igs. 1 and 2 indica e ha i is possible o p o ide a uni ied desc ip ion o long odd-A iso opic chains in e ms o ew pa ame e s, al hough a close inspec ion e eals ha disc epancies be ween calcula ions and expe imen s ill emain. Pa icula ly no able is ha conce ning he loca ion o he lowes JP = 7/2- s a e in Xe. This disc epancy was also p es- en in he calcula ions pe o med wi hin he ame- wo k o he in e ac ing boson- e mion model wi hou dis inc ion be ween p o on and neu on deg ees o eedom [ 17], and is due o he es ic ion o he sin- gle-pa icle nega i e pa i y space o lh/11/2- In oduc- ion o he 2 7/2 and lh9/2 le els elimina es he di e - ence be ween calcula ed and expe imen al s a es [17]. The same conclusions can be d awn by analyzing he esul s o he calcula ions o he posi i e pa i y s a es and o elec omagne ic ansi ion a es. I appea s hus ha we ha e now a ou disposal a ool capable o analyzing p ope ies o odd-e en nuclei ha include isospin e ec s. The scope and limi a ions o his ool emain o be seen and will be de e mined by a compa - ison be ween calcula ions and expe imen in o he iso- opic chains. We hank A.E.L. Diepe ink o discussions. One o us (F.I.) hanks R. Leona di o his hospi ali y a he Uni e si y o T en o whe e his a icle was comple ed. 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