PHYSICAL REVIEW C 83, 024307 (2011)
Odd-e en 147–153Pm iso opes wi hin he neu on-p o on in e ac ing boson- e mion model
J. Ba ea
Depa amen o de F´
ısica, Facul ad de Ciencias F´
ısicas y Ma em´
a icas, Uni e sidad de Concepci´
on, Casilla 160-C, Concepci´
on, Chile
and Ins i u o de Es uc u a de la Ma e ia, CSIC, Se ano 123, E-28006 Mad id, Spain
C. E. Alonso and J. M. A ias
Depa amen o de F´
ısica A ´
omica, Molecula y Nuclea , Facul ad de F´
ısica, Uni e sidad de Se illa, Apa ado 1065, 41080 Se illa, Spain
(Recei ed 2 No embe 2010; published 15 Feb ua y 2011)
Low-lying ene gy s a es o he 147−153Pm iso opic chain a e s udied wi hin he amewo k o he neu on-p o on
in e ac ing boson- e mion model (IBFM-2). The spec a o hese iso opes show a ansi ion om a pa icle
coupled o a ib a ional co e o a pa icle coupled o a de o med one. The calcula ion ep oduces his beha io .
In addi ion, educed ansi ion p obabili ies B(E2) and B(M1) and quad upole and magne ic momen s, as well
as spec oscopic ac o s co esponding o s ipping and pickup ans e eac ions, a e calcula ed. Ob ained esul s
compa e well wi h he a ailable expe imen al da a, which ein o ces he eliabili y o he wa e unc ions ob ained
wi hin he IBFM-2 model.
DOI: 10.1103/PhysRe C.83.024307 PACS numbe (s): 21.60.Fw, 21.10.Re, 21.10.Ky, 27.60.+j
I. INTRODUCTION
The imp o emen o he expe imen al echniques in he
las yea s has made possible a conside able enla gemen o
he a ailable expe imen al da a on many nuclea species,
among hem, nuclei in he a e-ea h egion. In his egion, he
e en-e en nuclei wi hin an iso opic chain show a ansi ion
om sphe ical ib a ional shapes o well-de o med o a ional
ones. Simila ly, in he odd-e en nuclei a ansi ion om
pa icle- ib a o o pa icle- o o is obse ed. The p ope
desc ip ion o his ansi ional beha io is a challenge o
any heo e ical model. In his pape we concen a e on he
comp ehensi e desc ip ion o he nuclea s uc u e o he
odd-e en 61Pm iso ope chain. Thei special ea u es ha e
ecen ly made hem o be he objec o in e es in heo e ical
s udies as, o example, 151,153Pm in Re . [1]. In o de o
desc ibe each nucleus and co ela e he a ailable da a, he
neu on-p o on in e ac ing boson- e mion model (IBFM-2) is
used.
Since i s in oduc ion in he 1970s, he in e ac ing boson
model (IBM-1) has shown i s abili y o desc ibe he collec i e
low-lying spec a o medium mass and hea y e en-e en nuclei
[2]. Soon a e he in oduc ion o he IBM-1, i s ex ension o
odd-e en nuclei, called he in e ac ing boson- e mion model
(IBFM-1), was p oposed [3,4]. The IBFM-1 has also shown
o be success ul in he desc ip ion o low-lying collec i e
s a es in odd-e en nuclei in di e en nuclea egions [3]. Bo h
IBM-1 and IBFM-1 do no dis inguish be ween neu ons and
p o ons. Howe e , hei ex ensions o include explici ly he
neu on-p o on deg ee o eedom we e de eloped in pa allel
[2,3]. These ex ensions, usually called neu on-p o on IBM
(IBM-2) and neu on-p o on IBFM (IBFM-2), p o ide a mo e
di ec connec ion wi h he shell model, gi ing a mic oscopic
ounda ion o hese models.
Odd-e en p ome hium iso opes we e s udied in he mid-
1980s [5] in he con ex o he IBFM-1 (wi hou he explici
ea men o he neu on-p o on deg ee o eedom). The
esul s ob ained accoun ed easonably well o he a ailable
expe imen al da a a ha ime. In o de o include new
expe imen al da a [6,7], and o imp o e hei desc ip ion,
he nex na u al s ep is o use he IBFM-2. This s udy can
also shed ligh on shape ansi ions in odd-e en sys ems, and
i complemen s analogous calcula ions in his egion o he
nuclea cha [8]. P elimina y wo k in he desc ip ion o he
Pm iso opic chain was p esen ed in Re . [9].
The o ganiza ion o he pape is he ollowing. In Sec. II
we b ie ly e iew he IBFM-2 model. The esul s, including
ene gy spec a, E2 and M1 ansi ion p obabili ies, quad upole
and magne ic momen s, and spec oscopic ac o s o s ipping
and pickup eac ions, a e p esen ed and discussed in Sec. III.
The main conclusions a e summa ized in Sec. IV.
II. MODEL
The Hamil onian used in he IBFM-2 includes pu e boson
and e mion e ms and a boson- e mion in e ac ion:
ˆ
H=ˆ
HB+ˆ
HF+ˆ
VBF ,(1)
whe e
ˆ
HB=ε(ˆ
ndν+ˆ
ndπ)+κˆ
Qν·ˆ
Qπ
+ˆ
Vππ +ˆ
Vνν +ξˆ
Mπν,(2)
ˆ
HF=
j
ja†
j·˜
aj,(3)
and
ˆ
VBF =πˆ
VQ
νπ −πˆ
VE
νπ +Aπˆ
VM
νπ.(4)
In he ollowing, he ea men o an odd-p o on nucleus is
assumed.
The ope a o s appea ing in ˆ
HBa e ˆ
ndρ, he numbe o
dbosons o kind ρ(ρ=ν,π o neu ons and p o ons,
024307-1
0556-2813/2011/83(2)/024307(9) ©2011 Ame ican Physical Socie y
J. BAREA, C. E. ALONSO, AND J. M. ARIAS PHYSICAL REVIEW C 83, 024307 (2011)
espec i ely),
ˆ
Qρ=[s†×˜
d+d†×˜
s](2)
ρ+χρ[d†×˜
d](2)
ρ,(5)
ˆ
Vρρ =
L=0,2,4
C(ρ)
L
2[d†
ρ×d†
ρ](L)·[˜
dρט
dρ](L),(6)
ˆ
Mπν =1
2[s†
ν×d†
π−s†
π×d†
ν](2) ·[˜
sνט
dπ−˜
sπט
dν](2)
−
k=1,3
[d†
ν×d†
π](k)·[˜
dνט
dπ](k),(7)
whe e s†
ρand d†
ρa e boson c ea ion ope a o s wi h L=0
and L=2, espec i ely, while a†
jm is he odd-p o on c ea ion
ope a o in he |nl 1
2jmsingle pa icle le el, whe e, o con-
enience, we omi he labels nl 1
2.˜
s,˜
d, and ˜
aa e annihila ion
ope a o s, wi h he p ope enso ial beha io unde o a ions,
de ined as ˜
s=s,˜
dµ=(−1)µd−µ, and ˜
ajm =(−1)j−maj−m.
The e ms appea ing in he boson- e mion in e ac ion (4) ead
ˆ
VQ
νπ =
jjl,1/2; j||Y(2)||l,1/2; j
×(ujuj− j j)ˆ
Qν·[a†
jט
aj](2)
π,(8)
ˆ
VE
νπ =
jjj 10
(2j+1)Nπ
βjjβjj
K2
β
×{ˆ
Qν·[(d†×˜
aj)(j)×(˜
s×a†
j)(j)](2)
π+h.c.},(9)
ˆ
VM
νπ =ˆ
ndν·ˆ
nπ(10)
whe e Nρis he numbe o ρbosons, and
βjj=(uj j+ juj)
×l,1/2; j||Y(2)||l,1/2; j,(11)
Kβ=
jj
β2
jj.(12)
The quasipa icle ene gies j[Eq. (3)] and he occupa ion
p obabili ies j[Eqs. (8)–(11)] a e ob ained om a BCS
calcula ion using he ollowing se o coupled equa ions:
j=Esp
j−λ2+2,(13)
2
j=1
21−Esp
j−λ
j.(14)
Esp
ja e he ene gies o he single-pa icle le els which he odd
p o on can occupy, =12A−1/2(MeV) is he pai ing gap,
and λis he Fe mi le el ob ained om his calcula ion, which
is cons ained o he conse a ion o he numbe o pa icles
2Nπ=
j
2
j(2j+1).(15)
III. DESCRIPTION OF THE ODD-EVEN Pm ISOTOPES
An odd-e en 61Pm iso ope is desc ibed in IBFM-2 by
coupling a p o on o i s 60Nd iso one, desc ibed in e ms o
he IBM-2, wi h he Hamil onian o Eq. (2). Consequen ly, he
i s s ep o desc ibe he odd-e en nucleus is a compelling
TABLE I. Pa ame e s di e en om ze o (,κ,andξin
MeV) used in Eq. (2) o he desc ip ion o he e en-e en Nd
iso opes (Nπ=5). The alues χπ=−1.2, C(π)
0=0.4MeV,and
C(π)
2=0.2 MeV a e ixed o all he iso opes.
A146 148 150 152
Nν2345
ε0.90 0.70 0.47 0.34
κ−0.15 −0.10 −0.07 −0.089
χν0.00 −0.80 −1.00 −1.10
ξ0.08 0.22 0.37 0.22
desc ip ion o he e en-e en co e. In he case o he Pm
iso opes, Re . [10] was aken as a s a ing poin o he
desc ip ion o he Nd e en-e en co es. A e wa d, he pa-
ame e s we e sligh ly changed o ake in o accoun la e
expe imen al in o ma ion on mixed symme y s a es 1+[11].
The esul ing alues o he pa ame e s used in he desc ip ion
o he Nd co es can be ound in Table I. The s eng h o
he Majo ana in e ac ion, ˆ
Mπν, was ob ained by i ing he
exci a ion ene gy o he i s 1+le el, mixed symme y s a e
allowed in IBM-2, bu no in IBM-1. Wi h hose pa ame e s,
bo h ene gy spec a and elec omagne ic p ope ies we e
calcula ed in good ag eemen wi h he a ailable expe imen al
da a o e en-e en Nd iso opes. Thus, we a e con iden ha
he wa e unc ions o he e en-e en Nd co e nuclei p o ided
by he IBM-2 model a e good.
Once he wa e unc ions o he s a es in he e en-e en
co e ha e been ob ained, he odd-p o on has o be coupled
o i in o de o calcula e exci a ion ene gies, elec omagne ic
p ope ies, and spec oscopic s eng hs in he odd-e en Pm
iso opes.
A. Exci a ion ene gies
The coupling o he p o on o he e en-e en co e is go e ned
by he boson- e mion in e ac ion, whe e he mo e impo an
e ms a e hose be ween he odd e mion (p o on o Pm
iso opes) and he bosons wi h he al e na i e la o (neu ons
in ou case). This in e ac ion, Eq. (4), is decomposed in o h ee
e ms: quad upole ( ˆ
VQ
νπ), exchange ( ˆ
VE
νπ), and monopole ( ˆ
VM
νπ).
Since he Hamil onian is in a ian unde pa i y, posi i e and
nega i e pa i y s a es a e s udied sepa a ely. The pa ame e s in
ˆ
VBF a e di e en o each pa i y acco dingly and a e shown in
Table II. I is impo an o emphasize ha π,π, and Aπa e
phenomenological pa ame e s o he en i e chain o iso opes,
in con as o he abo e-men ioned IBFM-1 calcula ion, whe e
hese pa ame e s we e i ed o each iso ope sepa a ely.
TABLE II. Pa ame e s (in MeV) o he boson-
e mion in e ac ion (4) used in his wo k o
posi i e (+) and nega i e (−) pa i y s a es.
Pa i y ππAπ
+0.452 0.030 −0.097
−0.883 4.885 −0.866
024307-2
ODD-EVEN 147−153Pm ISOTOPES WITHIN THE ... PHYSICAL REVIEW C 83, 024307 (2011)
147 149 151 153
A
0
2
4
6
E (MeV)
2d5/2
1g7/2
1h11/2
2d3/2
3s1/2
1h9/2
2 7/2
FIG. 1. Single-pa icle ene gies used in he BCS calcula ion o
each iso ope.
The single-pa icle ene gies Esp
jappea ing in Eqs. (13) and
(14) a e plo ed in Fig. 1. These we e aken om Re . [12], bu
we changed he ela i e posi ion o he 1g7/2and 2d5/2o bi s
o accoun o he sequence in he low-lying le els Jπ=5
2+
and Jπ=7
2+along he chain o iso opes. This a ia ion o he
posi ions o 1g7/2and 2d5/2(especially o 149Pm) somehow
dis u bs he imp ession ha he whole chain is desc ibed
in a consis en way. Howe e , he same p oblem has been
encoun e ed in calcula ions o o he nuclei close o he ones
conside ed in his wo k [8,13–15]. The o igin o his anomaly
is no clea and equi es a speci ic s udy. Table III shows
he quasipa icle ene gies and he occupa ion p obabili ies
ob ained om he BCS calcula ion. One impo an ea u e
o no e is ha he posi i e pa i y le els 2d3/2and 3s1/2as
well as he nega i e pa i y le els 1h9/2and 2 7/2ha e high
quasipa icle ene gies and small occupa ion p obabili ies when
we compa e hem wi h he es o he le els o he same pa i y.
This allows us o conside only he le els 2d5/2and 1g7/2 o
posi i e pa i y and 1h11/2 o nega i e pa i y, as was done in
Re . [5].
In Fig. 2, expe imen al and calcula ed exci a ion ene gies
o he posi i e pa i y le els in 147Pm and 149Pm a e shown.
The co espondence be ween expe imen al and calcula ed
le els was done using he elec omagne ic p ope ies discussed
0.0
0.5
E (MeV)
7+
5+
9+
3+, 5+, 7+
7+
5+, 7+
1+
5+, 7+
11+
7+, 9+
3+, 5+
7+
5+
5+
9+7+
11+3+
7+
5+3+
9+1+
Expe imen al IBFM-2
147Pm
7+
Expe imen al IBFM-2
5+
3+
5+
9+
7+
1+
5+
3+
7+
(11)+
(5, 7+)
(9)+
1+
(5+)
(7-,9+)
5+
7+
7+
9+
3+
1+
9+
7+
5+
11+
5+
7+
3+
3+
5+
7+
(3,5+)
(3-,5, 7+)
149Pm
FIG. 2. Measu ed and calcula ed exci a ion ene gies (in MeV)
o he posi i e pa i y le els in 147Pm and 149Pm. Expe imen al da a
ha e been aken om Re s. [16,17]. The numbe s indica e wice he
angula momen um.
below. I can be seen ha he s uc u e o he spec um o 147Pm
co esponds o a pa icle coupled o a ib a ional co e. The i s
wo s a es come om he coupling o he single-pa icle s a es
included in he calcula ion wi h he g ound s a e o 146Nd.
Then he e is a gap and, a ound he ene gy o he i s 2+o
146Nd (453.77 keV), a se o le els, which comes om he
coupling o he single-pa icle le els o his s a e, appea s.
The spec um o 149Pm co esponds o a ansi ional si ua ion
whe e he o bidden zone (gap) is absen . Ou calcula ions
ep oduce well his s uc u e, al hough hey show a ce ain
endency o s uc u es o he pa icle- ib a ion ype in bo h
iso opes. The o igin o his e ec can be ound in he low
alues o he boson- e mion pa ame e s, which supply weak
coupling schemes.
Expe imen al and calcula ed exci a ion ene gies o he
nega i e pa i y le els in 147Pm and 149Pm a e compa ed in
Fig. 3. The co espondence in his case is di icul o
some le els in 147Pm, due o he lack o elec omagne ic
in o ma ion which would allow o hei co ec iden i ica ion.
Ne e heless, om he exci a ion ene gies, i seems ha
he 5
2−,7
2−le el a 0.158 MeV abo e he lowes nega i e
pa i y s a e is missing in ou calcula ion. This ac was also
TABLE III. Quasipa icle ene gies (in MeV) and occupa ion p obabili ies squa ed ob ained om
a BCS calcula ion o he odd-p o on Pm iso opes using he single-pa icle ene gies plo ed in Fig. 1.
A146 148 150 152
j 2j 2j 2j 2
2d5
21.0852 0.7016 0.9947 0.4355 1.1817 0.7795 1.2801 0.8247
1g7
21.0026 0.5686 1.1401 0.7507 0.9806 0.4799 0.9878 0.4147
1h11
21.9365 0.0708 1.8190 0.0799 1.7412 0.0867 1.5990 0.1033
2d3
22.3796 0.0456 2.2555 0.0503 2.1728 0.0537 2.0187 0.0620
3s1
22.7016 0.0350 2.5748 0.0381 2.4902 0.0403 2.3315 0.0457
1h9
2,2 7
25.6504 0.0078 5.5172 0.0081 5.4285 0.0082 5.2595 0.0086
024307-3
J. BAREA, C. E. ALONSO, AND J. M. ARIAS PHYSICAL REVIEW C 83, 024307 (2011)
0.0
0.5
E (MeV)
Expe imen al IBFM-2
(5-, 7-)
5-, 7-
13-
15-
11-
9-, 11-, 13-
11-
15-
11-7-
15-
9-
13-
147Pm 149Pm
Expe imen al IBFM-2
11-
7-
(15)-
(11-)
3-
5-
(9)-
7-
11-
(19-)
(13-)
(7-,9+)
(3-, 5, 7+)
(7-,9+)
7-
11-
9-
15-
5-
13-
7-
3-
19-
11-
17-
9-
3-
5-
7-
FIG. 3. Same as Fig. 2, bu o he nega i e pa i y le els in 147Pm
and 149Pm. Expe imen al da a ha e been aken om Re s. [16,17].
obse ed in he IBFM-1 calcula ion [5], whe e i was sugges ed
ha his le el could be ep oduced when he 3−
1le el o
146Nd is included, which is beyond he scope o his wo k.
The calcula ion o he 149Pm iso ope shows a sequence o
le els dis ibu ed almos uni o mly up o 0.7 MeV, while
he e is a se o expe imen al le els g ouped oge he a ound
0.3 MeV. Howe e , he e is an almos 1:1 co espondence
be ween expe imen al and calcula ed le els below 0.5 MeV.
The s uc u e o he ene gy spec a o he 151Pm and 153Pm
nuclei look di e en when compa ed o he abo e-discussed
iso opes and co esponds o a pa icle coupled o a o a ional
co e. The expe imen al and calcula ed le els a e o ganized
in bands acco ding o he B(E2) alues in Figs. 4and 5 o
posi i e and nega i e pa i y s a es o 151Pm, espec i ely, and in
Figs. 6and 7 o posi i e and nega i e pa i y s a es o 153Pm,
espec i ely. The o e all ag eemen be ween he calcula ion
and he expe imen al da a is good.
I can be seen in Fig. 4 ha , in 151Pm, he g ound s a e and
he second calcula ed exci ed bands a e ai ly well ep oduced.
Mo eo e , ou calcula ions indica e he exis ence o a i s
5 +
7 +
9 +
(11)+
13 +
(15+)+
3 +
5 +
(7)+
(9+)+
(11+)+
1 +
(3)+
7 +
5 +
0.0
0.5
E (MeV)
151Pm
Expe imen al 5+
7+
9+
11+
13+
15+
3+
5+
7+
7+
9+
11+
5+
7+
3+
5+
7+
9+
13+
11+
1+
3+
15+
IBFM-2
FIG. 4. Same as Fig. 2, bu o he posi i e pa i y le els in 151Pm.
Expe imen al da a ha e been aken om Re . [18].
5 -
7 -
(9-)-
11 -
(13)-
(15)-
(5, 7-)-
(7-)- 3 -
(5)-
11 -
(3-)-
0.0
0.5
E (MeV)
Expe imen al
151Pm
5-
7-
11-
15-
17-
7-
9-
11-
3-
7-
9-
11-
15-
3-
5-
19-
7-
9-
13-
5-
13-
IBFM-2
FIG. 5. Same as Fig. 2, bu o he nega i e pa i y le els in 151Pm.
Expe imen al da a ha e been aken om Re . [18].
exci ed band led by a 7/2+s a e a app oxima ely 0.25 MeV,
ollowed by a 9/2+s a e a 0.4 MeV. Rega ding he nega i e
pa i y le els, hose o he lowes band a e ep oduced in he
igh o de , showing depa u es, specially in hose wi h high
angula momen um. Howe e he bandhead angula momen a
a e co ec ly ep oduced by he calcula ion.
Finally, as i can be seen in Fig. 6 ha in 153Pm he e
is an excellen ag eemen be ween he calcula ed and he
expe imen al ene gies o he posi i e pa i y le els. Again, he
calcula ion p edic s a i s exci ed band led by a 7/2+s a e a
a ound 0.4 MeV. In he case o he nega i e pa i y le els, Fig. 7,
expe imen al da a a e only a ailable o he g ound-s a e band,
well desc ibed in he calcula ion, bu wi h a sligh ly highe
momen o ine ia. A i s exci ed band, led by a 7/2−s a e a
0.2 MeV, appea s in ou calcula ions, as in 151Pm.
Figu es 8and 9show expe imen al and calcula ed ene gies
o he low-lying ene gy le els in odd-e en Pm iso opes e sus
he mass numbe o posi i e and nega i e pa i y, espec i ely.
The goal o hese igu es is o show he sys ema ics o he
le els. We can obse e wo di e en ends o he posi i e
5+
7+
(9+)+
(11+)+
3+
5+
7+
1+, 3+,5+
3+
5+
3+, 5+
0.0
0.5
E (MeV)
Expe imen al IBFM-2
153Pm
5+
7+
9+
11+
13+
15+
7+
9+
3+
5+
7+
11+
13+
1+
5+
3+
15+
FIG. 6. Same as Fig. 2, bu o he posi i e pa i y le els in 153Pm.
Expe imen al da a ha e been aken om Re . [19].
024307-4
ODD-EVEN 147−153Pm ISOTOPES WITHIN THE ... PHYSICAL REVIEW C 83, 024307 (2011)
5-
7-
9-
11-
(13-)-
0.0
0.5
E (MeV)
153Pm
IBFM-2
Expe imen al 5-
7-
9-
11-
13-
15-
17-
19-
7-
9-
11-
13-
15-
3-
5-
7-
FIG. 7. Same as Fig. 2, bu o he nega i e pa i y le els in 153Pm.
Expe imen al da a ha e been aken om Re . [19].
pa i y le els, he e a e some le els ha p esen a mono onous
dec ease in ene gy while o o he s a minimum de elops in an
in e media e iso ope. The calcula ion is able o ep oduce hese
beha io s. Nega i e pa i y le els show a dec easing beha io
which is well desc ibed in he calcula ions.
To inalize his subsec ion, we can asse ha IBFM-2
ep oduces well he complica ed s uc u e o he ene gy le els,
al hough de ia ions wi h ega d o he expe imen al da a a e
obse ed when s udying a pa icula nucleus. A de ailed s udy
o a ce ain iso ope is easible, bu i would in ol e a ine
uning o he Hamil onian pa ame e s, which is no he aim o
his wo k.
B. Elec omagne ic p ope ies
In addi ion o spec a, we ha e calcula ed B(E2)’s and
B(M1)’s o compa e hem wi h hei expe imen al alues. This
FIG. 8. Expe imen al and calcula ed exci a ion ene gies o he
low-lying posi i e pa i y s a es o odd-e en Pm iso opes. The ene gy
o he i s 7
2+is aken as e e ence.
FIG. 9. Expe imen al and calcula ed ene gies o he low-lying
nega i e pa i y s a es o odd-e en Pm iso opes. The ene gy o he i s
11
2−is aken as a e e ence.
compa ison allows us o es ablish he deg ee o eliabili y o
he wa e unc ions ob ained in his wo k.
The elec ic quad upole ansi ion ope a o ˆ
T(E2) consis s
o a bosonic and a e mionic pa :
ˆ
T(E2) =ˆ
T(E2)
B+ˆ
T(E2)
F,(16)
whe e
ˆ
T(E2)
B=eπˆ
Qπ+eνˆ
Qν,(17)
ˆ
T(E2)
F=−eF
√5
j⩽j
(ujuj− j j)l1
2j



2Y(2)



l1
2j
×([a†
jט
aj](2) +h.c.).(18)
The quad upole ope a o s p esen in ˆ
T(E2)
Bco espond o hose
appea ing in he Hamil onian (5). The alue used o he
bosonic e ec i e cha ges eπand eνis 0.139 eb, aken om
Re . [5]. Fo he e mionic e ec i e cha ge eF, we adop he
alue 1.5 eb. As usual in he li e a u e, we ake cons an alues
o he adial pa in he educed ma ix elemen o 2Y(2).
Analogously, he magne ic dipole ansi ion ope a o ˆ
T(M1) is
made up o a bosonic and a e mionic pa ,
ˆ
T(M1) =ˆ
T(M1)
B+ˆ
T(M1)
F,(19)
whe e
ˆ
T(M1)
B=3
4π(gπˆ
Lπ+gνˆ
Lν),(20)
ˆ
T(M1)
F=−
1
4π
j⩽j
(ujuj+ j j)l1
2j



gl
l+gss



l1
2j
×([a†
jט
aj](1) +h.c),(21)
wi h ˆ
Lρ=√10(d†
ρט
dρ)(1). The bosonic gy omagne ic ac-
o s gπand gνha e been ex ac ed om Re . [20] and hei
alues a e shown in Table IV.
024307-5

J. BAREA, C. E. ALONSO, AND J. M. ARIAS PHYSICAL REVIEW C 83, 024307 (2011)
TABLE IV. Bosonic gy omagne ic ac o s
(in µN) used in his wo k (ex ac ed om Re . [20]).
A147 149 151 153
gν−0.183 −0.171 −0.159 −0.147
gπ0.415 0.415 0.415 0.415
The adop ed alues o he o bi al and spin e mionic
gy omagne ic ac o s a e gl=1.0µNand gs=4.0µN( he
spin gy omagne ic ac o was quenched 70% wi h espec o
he alue o he ee p o on, as usually ound in he li e a u e).
Figu e 10 shows he end o he calcula ed alues o B(E2)
o some low-lying posi i e pa i y s a es in he Pm iso opes
along wi h hei expe imen al coun e pa s. The calcula ion
ep oduces well he end, excep he ansi ion 3/2+
1→7/2+
1
o A=147.These could be due o he exis ence o a low
ene gy s a e 0+a 915.4 keV in he e en-e en co e, 146Nd,
which he IBM is unable o ep oduce wi hou including
oc upole deg ees o eedom [21]. The coupling o his s a e
wi h he e mionic single-pa icle deg ees o eedom may
ha e a ela i ely high in luence on he low-lying s a es in
147Pm. Table Vshows addi ional expe imen al and calcula ed
alues o B(E2) and B(M1) o his nucleus. I can be
seen ha , e en when he calcula ion does no desc ibe ine
de ails o he expe imen al da a, gene al ends a e ep oduced.
Again he inclusion o oc upole deg ees o eedom could
imp o e he desc ip ion o his nucleus. Some calcula ed and
expe imen al alues o he educed ansi ion p obabili ies
B(E2) and B(M1) in 149Pm a e quo ed in Table VI.The
ag eemen be ween he calcula ion and he expe imen al da a
is easonable. No ice ha he calcula ed alues o B(E2)
and B(M1) o he ansi ion om 5/2+
2 o 7/2+
1a e lowe
han he co esponding ones om 5/2+
3which is close o he
expe imen al si ua ion. This allows us o iden i y he calcula ed
s a e 5/2+
3(5/2+
2) wi h he expe imen al s a e 5/2+
2(5/2+
3).
This kind o exchange was obse ed in he p e ious IBFM-1
calcula ion, bu be ween he s a es 5/2+
1and 5/2+
2.Thesame
FIG. 10. Expe imen al (symbols) [16–19] and calcula ed (lines)
elec ic quad upole educed ansi ion p obabili ies be ween some
low-lying s a es in APm.
TABLE V. Expe imen al [16] and calcula ed elec ic quad upole
(in e2b2) and magne ic dipole (in µ2
N) educed ansi ion p obabili ies
be ween some low-lying s a es o 147Pm.
Jπ
i→Jπ
Jπ
i→Jπ
B(E2) Exp . IBFM-2 B(M1) Exp . IBFM-2
5
2+
2→3
2+
10.0018(18) 0.0141 5
2+
1→7
2+
10.0067(2) 0.0001
5
2+
2→5
2+
10.0009(4) 0.0020 3
2+
1→5
2+
10.0068(9) 0.0127
5
2+
2→7
2+
10.0019(5) 0.2075 5
2+
2→3
2+
10.007(3) 0.023
5
2+
3→7
2+
20.0023(23) 0.0000 5
2+
2→5
2+
10.0003(1) 0.0063
5
2+
3→3
2+
10.0010(5) 0.0146 5
2+
2→7
2+
10.0023(5) 0.0009
5
2+
3→7
2+
10.0003(1) 0.0015 5
2+
3→7
2+
20.0016(9) 0.0090
5
2+
3→5
2+
10.0027(11) 0.0202
5
2+
3→7
2+
10.0001(1) 0.0021
happens wi h he calcula ed s a es 1/2+
1,2, which we ha e
iden i ied wi h he expe imen al s a es 1/2+
2,1. Expe imen al
da a on elec omagne ic ansi ions a e sca ce o he nucleus
151Pm and absen , o he bes o ou knowledge, o he
nucleus 153Pm. Figu e 11 shows he end o he elec ic
quad upole momen s o he 5/2+
1and 7/2+
1s a es and o
he magne ic dipole momen s o he 3/2+
1,5/2+
1, and 7/2+
1
s a es. Expe imen al elec ic quad upole momen s a e nicely
ep oduced by he calcula ion, in con as o magne ic dipole
momen s, whose desc ip ion is clea ly wo se. Ne e heless
we can conclude ha he desc ip ion o he elec omagne ic
p ope ies wi h IBFM-2 is equal o o sligh ly be e han he
one ob ained wi h IBFM-1.
C. Spec oscopic ac o s
Ano he impo an ool o checking he quali y o he
desc ip ion o odd-e en nuclei is p o ided by one-nucleon
ans e eac ions. Fo his eason we will analyze spec o-
scopic ac o s o one-p o on s ipping and pickup eac ions
ha ing as inal nuclei he Pm iso opes we a e s udying.
TABLE VI. Expe imen al [17] and calcula ed elec ic quad upole
(in e2b2) and magne ic dipole (in µ2
N) educed ansi ion p obabili ies
be ween some low-lying s a es o 149Pm.
Jπ
i→Jπ
Jπ
i→Jπ
B(E2) Exp . IBFM-2 B(M1) Exp . IBFM-2
5
2+
2→7
2+
10.18(4) 0.06 5
2+
1→7
2+
10.0049(1) 0.0009
1
2+
1→5
2+
20.014(4) 0.036 3
2+
1→5
2+
10.0025(7) 0.0686
1
2+
1→5
2+
10.006(2) 0.000 5
2+
2→7
2+
10.034(7) 0.047
7
2−
1→11
2−
10.61(19) 0.60 5
2+
2→5
2+
10.022(5) 0.012
5
2+
3→7
2+
1−0.008 5
2+
2→3
2+
10.007(3) 0.023
1
2+
1→3
2+
10.0057(13) 0.0003
024307-6
ODD-EVEN 147−153Pm ISOTOPES WITHIN THE ... PHYSICAL REVIEW C 83, 024307 (2011)
µ
µ
µ
µµµ
FIG. 11. Expe imen al (symbols) [16–19] and calcula ed (lines)
elec ic quad upole and magne ic dipole momen s o some low-lying
s a es in APm.
The image o he one- e mion ans e ope a o in he
boson- e mion space o he IBFM-2 is an expansion ha can
be ob ained using di e en echniques and app oxima ions.
The i s o de (FO) ope a o o a boson-conse ing
eac ion is
(ˆ
Tjm)FO =uja†
jm.(22)
A mo e elabo a e possibili y is o use he boson image o
he shell model pa icle c ea ion ope a o ob ained using he
O suka-A ima-Iachello (OAI) me hod [22], whe e s a es wi h
Gene alized Senio i y (GS) as a good quan um numbe a e
used. This u ns ou o be alid o nuclei close o closed
shells, and i s exp ession is
(ˆ
Tjm)GS =uja†
jm + j
√Nπ
[s†
πט
aj](j)
m
+
j
uj10
2j+1
βjj
Kβ
[d†
πט
aj](j)
m
−
j
j
√Nπ10
2j+1
βjj
Kβ
[(s†
πט
dπ)(2) ×a†
j](j)
m.
(23)
Ano he possibili y is o use he e sion o he ans e
ope a o deduced using he gene alized Hols ein-P imako
FIG. 12. Expe imen al [16–19] and calcula ed spec oscopic ac-
o s co esponding o one-p o on s ipping eac ions. Ci cles and
iangles indica e da a ob ained om (3He,d)and(α, ) eac ions,
espec i ely. Solid, do ed, and dashed lines co espond o esul s
ob ained wi h he FO, GS, and HP ans e ope a o s, espec i ely.
(HP) scheme [23], sui able o de o med nuclei. The o m o
his ope a o is [24]
(ˆ
Tjm)HP =uja†
jm +
jλ
(−1)λ+j−jXλ
jj
ˆ
λ
ˆ
j(γ†
λט
aj)(j)
m
−
jjL1L2λ
L1=L2=0
(−1)j+j+L2
2uj
XL1
jjXL2
jj
׈
λˆ
L1ˆ
L2
ˆ
jjj
λ
L2L1j [(γ†
L1טγL2)(λ)×a†
j](j)
m,
(24)
024307-7
J. BAREA, C. E. ALONSO, AND J. M. ARIAS PHYSICAL REVIEW C 83, 024307 (2011)
whe e ˆ
λ=√2λ+1, γ†
λ e e s o s†
π(λ=0) and d†
π(λ=2),
and he quan i ies Xλ
jja e gi en by he exp essions
X0
jj =ˆ
j j
√Nπ
,(25)
X2
jj=√2( j+ j)j||Y2||j
jj[( j+ j)j||Y2||j]2
.(26)
S ipping eac ions om ANd o A+1Pm a e cha ac e ized
by p ese ing he numbe o bosons, in he language o he
IBFM-2. In he abo e exp essions, only he e ms which
conse e he numbe o bosons con ibu e o he spec oscopic
ac o s o hese eac ions. In o de o calcula e hem, we will
use h ee di e en app oxima ions o he ans e ope a o :
(i) he FO ope a o which is he common e m in bo h he
GS and HP ans e ope a o s, (ii) he GS ope a o , which
con ains he FO ope a o wi h he addi ion o e ms o he
ype [(s†
πט
dπ)(2) ×a†
j](j)
m o di e en alues o j, and (iii)
he HP ope a o , which, in addi ion o he e ms in he GS
ope a o , includes e ms o he ype [(d†
πט
sπ)(2) ×a†
j](j)
mand
[(d†
πט
dπ)(λ)×a†
j](j)
m o di e en alues o jand λ. Te ms o
he ype [(d†
πט
sπ)(2) ×a†
j](j)
ma e o bidden in he GS ope a o
a he app oxima ion usually used, since hey change senio i y
in h ee uni s, while he e ms [(d†
πט
dπ)(λ)×a†
j](j)
mcan be
p esen in he GS, bu hey belong o a highe o de co ec ion
in he boson expansion.
Figu e 12 shows he spec oscopic ac o s ob ained wi h
he h ee ans e ope a o s, no malized acco ding o he
McFa lane sum ules, along wi h he expe imen al alues
om wo di e en eac ions [16–19]. The calcula ed alues
wi h he h ee app oxima ions used o he ans e ope a o
a e e y simila . This means ha he co ec ions o he FO
ope a o a e small. The calcula ions ep oduce he beha io o
he expe imen al alues excep o he ans e o he 11/2−
1
le el in 151Pm. Addi ional expe imen al in o ma ion would be
c ucial in o de o elucida e which is he bes choice o he
ans e ope a o , especially o he ans e o he 7/2+
2le els
in 151Pm and 153Pm o which he di e en ope a o s yield
app eciably di e en esul s.
The a ge s in he one-p o on pickup eac ions leading o
APm a e he A+1Sm iso opes. The wa e unc ions o hese
iso opes we e ob ained using he same Hamil onian as he one
used o he Nd iso opes, wi h pa ame e s aken om Re . [10],
quo ed in Table VII.
TABLE VII. Pa ame e s di e en om ze o (and κ
in MeV) used o calcula e he wa e unc ions o he Sm
iso opes (Nπ=6). The alues ξ=0.12, χπ=−1.3,and
C(π)
2=0.05 MeV a e he same o all he iso opes.
A148 150 152 154
Nν234 5
0.95 0.70 0.52 0.43
κ−0.12 −0.08 −0.075 −0.081
χν0.00 −0.80 −1.00 −1.10
FIG. 13. Expe imen al [16–19,25] and calcula ed spec oscopic
ac o s co esponding o one-p o on pickup eac ions. Ci cles and
iangles indica e da a ob ained om (d,3He) and ( , α), espec i ely.
Solid and do ed lines co espond o he esul s ob ained wi h he GS
and HP ans e ope a o s, espec i ely.
In his case, he numbe o bosons is no p ese ed and only
e ms o he ype (γ†
λט
aj)(j)
mcon ibu e o he spec oscopic
ac o s. Two calcula ions ha e been pe o med, co esponding
o he GS and HP ans e ope a o s, espec i ely. Calcula ed
spec oscopic ac o s ha e been no malized o ep oduce he
spec oscopic sum ules. Figu e 13 shows he esul s o he
calcula ions along wi h he expe imen al da a [16–19,25]. In
con as o he s ipping eac ions, he e is mo e expe imen al
da a a ailable o he pickup ones, which allows a mo e
de ailed compa ison wi h he calcula ed alues. Bo h cal-
cula ions a e a he simila and ep oduce quali a i ely he
expe imen al da a, wi h he excep ion o hose o he 7/2+
2
le els. The la ges di e ences be ween he GS and he HP
calcula ions can be ound in he 5/2+
1le els o he mo e
de o med iso opes. In his a ea he HP ope a o ep oduces
024307-8
ODD-EVEN 147−153Pm ISOTOPES WITHIN THE ... PHYSICAL REVIEW C 83, 024307 (2011)
be e he expe imen al da a, acco ding o he scheme in which
i has been deduced, sui able o de o med nuclei.
IV. CONCLUSIONS
In he p esen wo k we ha e pe o med a comp ehensi e
heo e ical desc ip ion o odd-e en Pm iso opes om A=147
o 153 in e ms o he IBFM-2. Posi i e and nega i e pa i y
s a es ha e been s udied in an independen way, aking in o
accoun he main single-pa icle deg ees o eedom in ol ed
in each case.
F om he wa e unc ions ob ained in such s udy we
ge di e en obse ables, such as E2 and M1 ansi ion
p obabili ies and momen s, and spec oscopic ac o s o
one-p o on ans e s, in ol ing a Pm iso ope. Fo he la e
we ha e used se e al possibili ies o he ans e ope a o , in
o de o es ablish unde which condi ions each one is mo e
app op ia e.
We conclude ha he gene al end o he obse ables
s udied is well accoun ed o in his s udy, ep oducing he
ansi ional beha io obse ed in he expe imen al da a. The
s eng h o he di e en e ms o he boson- e mion in e ac ion
ha e been kep cons an along he iso opic chain. Fine uning
o he pa ame e s would be needed in o de o ge a be e
desc ip ion o each pa icula nucleus.
As a as he one-pa icle ans e is conce ned, mo e expe -
imen al da a in he s udied egion would be desi able in o de
o be able o make s ong s a emen s on he app op ia eness
o each o he di e en ans e ope a o s exis ing in he
li e a u e.
In closing, we would like o poin ou ha , in se e al
places in his pape , disc epancies be ween he calcula ions
and expe imen al da a ha e been a ibu ed o couplings o he
e mions wi h boson s a es ha a e ou side he used boson
space (like 3−
1, o an exci ed 0+ ha canno be ep oduced
wi hou including he oc upole deg ee o eedom). I should
be clea ha we ha e es ic ions in ou own e mion model
space ha could be in pa he o igin o he disc epancies.
We ha e es ic ed he e mion space o d5/2,g7/2, and h11/2
o bi als. One could hink ha by including d3/2,s1/2, and
7/2, some o he disc epancies could be co ec ed. The
e ec i e pa ame e s in a la ge space would, o cou se,
be di e en . Small con ibu ions, o he o de o a ew
pe cen (10% o 15%), o d3/2o s1/2 o posi i e pa i y
s a es ( 7/2 o nega i e pa i y s a es) in he wa e unc ions
could in p inciple ha e sizable e ec s on he calcula ions.
Howe e , p elimina y calcula ions in his enla ged e mion
space ha e no con i med a signi ican imp o emen in he
ag eemen wi h he expe imen al da a. The e o e, we hink
ha he boson space limi a ion is mainly esponsible o
he disc epancies ound, bu a ho ough s udy could be in
o de .
ACKNOWLEDGMENTS
This wo k has been suppo ed by he Spanish Minis e io de
Ciencia e Inno aci´
on unde P ojec s No. FIS2009-07277 and
No. FIS2008-04189, by CPAN-Ingenio (CSPD-2007-00042-
Ingenio 2010), and by Jun a de Andaluc´
ıa (FQM160, P07-
FQM-02962). J.B. is unded by CSIC unde a JAE pos doc o al
con ac .
[1] G. G. Adamian, N. V. An onenko, R. V. Jolos, and T. M.
Schneidman, Phys.Re .C70, 064318 (2004).
[2] F. Iachello and A. A ima, The In e ac ing Boson Model
(Camb idge Uni e si y, Camb idge, UK, 1987).
[3] F. Iachello and P. V. Isacke , The In e ac ing Boson-Fe mion
Model (Camb idge Uni e si y, Camb idge, UK, 1991).
[4] A. A ima and F. Iachello, Phys. Re . Le . 35, 1069 (1975).
[5] O. Schol en and T. Ozzello, Nucl. Phys. A 424, 221 (1984).
[6]G.Audi,A.Waps a,andC.Thibaul ,Nucl. Phys. A 729, 337
(2003).
[7] N. S one, A . Da a Nucl. Da a Tables 90, 75 (2005).
[8] C. Alonso, J. A ias, and M. Lozano, J. Phys. G 14, 877 (1988).
[9] J. Ba ea, Ph.D. hesis, Uni e si y o Se ille, 2002.
[10] O. Schol en, Ph.D. hesis, Uni e si y o G oningen, 1980.
[11] J. Ma g a e al.,Phys. Re . C 47, 1474 (1993).
[12] B. Reehal and R. So ensen, Phys. Re . C 2, 819 (1970).
[13] O. Schol en and N. Blasi, Nucl. Phys. A 380, 509 (1982).
[14] D. Bucu escu, G. Ca a-Danil, N. V. Zam i , A. Gizon, and
J. Gizon, Phys.Re .C43, 2610 (1991).
[15] G. Ca a-Danil, D. Bucu escu, A. Gizon, and J. Gizon, J. Phys.
G20, 1051 (1994).
[16] N. Nica, Nucl. Da a Shee s 110, 749 (2009).
[17] B. Singh, Nucl. Da a Shee s 102, 1 (2004).
[18] B. Singh, Nucl. Da a Shee s 110, 1 (2009).
[19] R. G. Helme , Nucl. Da a Shee s 107, 507 (2006).
[20] M. Samba a o, O. Schol en, A. Diepe ink, and G. Picci o, Nucl.
Phys. A 423, 333 (1984).
[21] V. E. Iacob, W. U ban, J. C. Bacela , J. Jongman, J. Nybe g,
G. Sle en, and L. T ache, Nucl. Phys. A 596, 155
(1996).
[22] T. O suka, A. A ima, and F. Iachello, Nucl. Phys. A 309,1
(1978).
[23] C. Alonso, J. A ias, J. Dukelsky, and S. Pi el, Nucl. Phys. A
539, 391 (1992).
[24] J. Dukelsky and S. Pi el, in Pe spec i es o he In e ac ing
Boson Model on he Occasion o I s 20 h Anni e sa y,edi ed
by R. F. Cas en, A. Vi u i, A. B. Balan ekin, B. R. Ba e ,
J. N. Ginocchio, G. Maino, and T. O suka (Wo ld Scien i ic,
Singapo e, 1994), p. 291.
[25] I. S. Lee, W. J. Jo dan, J. V. Mahe , R. Kame mans,
J. W. Smi s, and R. H. Siemssen, Nucl. Phys. A 371, 111
(1981).
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