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.
This wo k was in pa suppo ed by he S ich ing FOM
which ecei es inancial suppo om he "S ich ing
Voo Zui e -We enschappelijk Onde zoek" and in pa
by he Depa men o Ene gy Con ac No. DE-AC02-
76ER 03074.
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144