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
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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
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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
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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.
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