a Xi :nucl- h/9811083 1 24 No 1998
Boson expansion me hods applied o a wo-le el model in he
s udy o mul iple gian esonances
C. Volpe
G oupe de Physique Th´eo ique, Ins i u de Physique Nucl´eai e,
F-91406 O say Cedex, F ance
Ph. Chomaz
GANIL, B.P. 5027,
F-14076 Caen Cedex 5, F ance
M.V. And ´es
Depa amen o de F´ısica A ´omica, Molecula y Nuclea ,
Uni e sidad de Se illa, Apdo 1065, 41080 Se illa, Spain
F. Ca a a
Dipa imen o di Fisica dell’Uni e si `a and INFN, Sezione di Ca ania,
I-95129 Ca ania, I aly
E.G. Lanza
Dipa imen o di Fisica dell’Uni e si `a and INFN, Sezione di Ca ania,
I-95129 Ca ania, I aly
and
Depa amen o de F´ısica A ´omica, Molecula y Nuclea ,
Uni e sidad de Se illa, Apdo 1065, 41080 Se illa, Spain
Abs ac
We apply boson expansion me hods o an ex ended Lipkin-Meshko -Glick
model including anha monici ies in analogy wi h p e ious mic oscopic calcu-
la ions. We s udy he e ec s o di e en app oxima ions p esen in hese cal-
cula ions, among which he unca ion o he hamil onian and o he space, in
connec ion wi h he s udy o he p ope ies o wo-phonon and h ee-phonon
s a es. By compa ing he app oxima e esul s on he spec um wi h he exac
ones we conclude ha he app oxima ions made in he mic oscopic calcula ions
on wo-phonon s a es a e well jus i ied. We ind also ha a good ag eemen
wi h he exac esul s o he h ee-phonon s a e is ob ained by using a bosonic
hamil onian unca ed a he ou h o de . This esul makes us con iden
ha such app oxima ion can be used in ealis ic calcula ions, hus allowing a
heo e ical s udy o iple exci a ions o gian esonances.
1
1 In oduc ion
Collec i e exci a ions ha e been known o many yea s in nuclea physics [1], bo h in
he low-lying spec a and in he Gian Resonance (GR) egion. A basic mic oscopic
heo y o such collec i e modes is he Random Phase App oxima ion (RPA) [2, 3]
which can be seen as he lowes o de in a boson expansion such ha he hamil onian
can be pu in he o m o a sum o hamil onians o ha monic oscilla o s, each one
co esponding o a collec i e mode (phonon). The e o e, RPA p edic s he exis ence
o one-phonon, wo-phonon,...e c s a es wi h a ha monic spec um. In addi ion o he
well known low-lying wo-phonon s a es, ecen ly hea y ion inelas ic sca e ing expe -
imen s a in e media e and ela i is ic ene gies and double cha ge-exchange eac ions
ha e shown he exis ence o s a es in he high exci a ion ene gy egion which can be
desc ibed as a GR buil on op o ano he GR [4, 5]. The s udy o he p ope ies o
hese s a es allows o es ou comp ehension o he GR’s as small ampli ude ib a-
ions and he e o e he ha monic pic u e. The sys ema ics on he ene gies and wid hs
is in quali a i e ag eemen wi h he ha monic app oxima ion, namely he ene gy o a
doubly exci ed GR is nea ly wice ha o he single GR and i s wid h is be ween √2
and 2 imes la ge . Howe e , he inelas ic c oss sec ions, when calcula ed wi hin he
ha monic pic u e, a e almos always smalle han he measu ed ones. In pa icula ,
he da a conce ning Coulomb exci a ion show a disc epancy anging om 30% up
o a ac o 4, acco ding o he nucleus s udied. In o de o unde s and he o igin o
his disc epancy, co ec ions o he ha monic app oxima ion ha e been p oposed [6]
by including anha monici ies in he in e nal hamil onian and non-linea i ies in he
ex e nal ield. As shown in [6], small anha monici ies in he exci a ion spec um o
he a ge nucleus can lead o a la ge enhancemen o he Coulomb exci a ion c oss
sec ion. The model used he e was an o e simpli ied one, namely he a ge was
desc ibed as an anha monic linea oscilla o . The pa ame e s o he cubic and he
qua ic e ms in i s hamil onian we e ixed so ha he ene gy o he second exci ed
s a e was shi ed down by ≈2MeV wi h espec o wice he ene gy o he i s
exci ed s a e. In [7] a 3-dimensional ex ension o his model was conside ed and sim-
ila e ec s coming om anha monici ies we e ound. We ema k ha , in he case
o Coulomb exci a ion o 136Xe in he eac ion 136Xe + 208Pb a E/A = 700 MeV,
he expe imen ally obse ed peak, in e p e ed as he Double Gian Dipole Resonance
(DGDR), is shi ed by ≈ −2 MeV om ha expec ed in he ha monic case [8]. A
mo e e ined s udy was pe o med in [9], whe e anha monici ies and non-linea i ies
we e included by s a ing om RPA and ex ending i by means o boson mapping
echniques [3, 10]. Such mic oscopic app oach was applied in [9] o ealis ic cases. In
he case o 208Pb + 208Pb a E/A = 641 MeV, o example, i was ound ha he
inclusion o anha monici ies and non-linea i ies gi es ise o an enhancemen o 30%
o he c oss sec ion in he egion o he DGDR, b inging he heo e ical esul s close
o he expe imen al ones. Impo an con ibu ions coming om o he wo-phonon
s a es in he same ene gy egion we e also ound.
The s udies epo ed in he abo e quo ed pape s ha e o cou se some limi a ions.
The model used in [6] is clea ly e y schema ic and has, among o he s, he d awback
ha i is based on a pu ely bosonic desc ip ion so ha Pauli blocking e ec s a e no
2
included. The app oach p esen ed in [9] is ce ainly much mo e ealis ic and he e ec s
o he Pauli exclusion p inciple a e aken in o accoun o some ex en . Howe e , o
compu a ional easons, only one- and wo-phonon s a es we e conside ed he e.
The pu pose o his pape is o p esen an analysis based on an ex ension o he
Lipkin-Meshko -Glick (LMG) model [11] including a esidual in e ac ion be ween he
phonons, in analogy wi h he mic oscopic app oach used in e .[9]. This model has
se e al ad an ages wi h espec o he o he s. Because o i s g oup s uc u e, his
model is exac ly sol able. Besides, i di ec ly akes in o accoun he Pauli p inciple. In
his con ex , we discuss se e al app oxima ions, co esponding o he cases conside ed
in [6, 9] and, by di ec compa ison wi h he exac esul s, we es how se e e a e he
limi a ions o hese app oxima ions. We ocus on one hand on wo-phonons s a es,
s udied in e .[9] and on he o he hand on he p ope ies o h ee-phonon s a es. In
ac , in o de o ha e a mo e s ingen es on he alidi y o he ha monic pic u e,
expe imen al and heo e ical s udies o iple exci a ion o GR’s should be en isaged.
F om he heo e ical poin o iew, he same app oach as he one in [9] can be used,
bu his would imply huge calcula ions. In o de o make hem easible in he con ex
o boson expansion me hods, one may conside an app oxima ion in which he same
ou h o de hamil onian used in [9] is diagonalized in a space con aining up o 3-
phonon s a es. We p esen a es on how well his app oxima ion wo ks, in he con ex
o he ex ended LMG hamil onian. We will see ha he exac esul s a e e y close
o he app oxima e ones in all he cases s udied o bo h he second and he hi d
exci ed s a es.
The pape is o ganized as ollows. In sec ion 2 he LMG model is sho ly e iewed
and ou ex ension in oduced. In sec ion 3 he boson mapping o his hamil onian
is p esen ed. In sec ion 4 he app oxima e esul s on he ene gy o he i s , second
and hi d exci ed s a es, ob ained by unca ing a di e en le els he mapping, a e
compa ed among hemsel es and wi h he exac calcula ions.
2 The model
2.1 The Lipkin-Meshko -Glick model
In he o iginal Lipkin-Meshko -Glick (LMG) model [11] a ini e numbe Ω o pa icles
can be a anged in wo le els sepa a ed by εin ene gy. Ω di e en quan um s a es a e
a ailable in each le el. Each pa icle is he e o e iden i ied by wo quan um numbe s.
The i s σindica es i he pa icle is in he uppe (σ= +) o in he lowe (σ=−)
ene gy le el. The second quan um numbe sis ela ed o he Ω di e en a ailable
quan um s a es in each le el. We will conside sys ems whose hamil onian can be
w i en as unc ion o he ope a o s K+,K−,K0:
K+=
Ω
X
s=1
a†
+,sa−,s (1)
3
K−= (K+)†=
Ω
X
s=1
a†
−,sa+,s
K0=1
2
Ω
X
s=1
(a†
+,sa+,s −a†
−,sa−,s)
whe e a†
σ,s (and he he mi ian conjuga e aσ,s ) c ea es (annihila es) a pa icle in he
quan um s a e (σ, s). Since he ope a o s K+,K−,K0sa is y he SU(2) commu a ion
ela ions :
[K+, K−] = 2K0[K0, K±] = ±K±(2)
hey a e o en called he quasi-spin ope a o s. Acco ding o (1), he e can no be wo
pa icles in he same quan um s a e, ha is he Pauli exclusion p inciple is included
in he model. A sys em o Ω pa icles has 2Ωs a es. Howe e , since he hamil onian
is a unc ion o he gene a o s o SU(2) and K2is a Casimi ope a o o such algeb a,
he space can be sepa a ed in o subspaces, each co esponding o an eigen alue K.
The s a e wi h all pa icles in he lowe le el, σ=−, is an eigen ec o o K2and K0,
belonging o hei maximum and minimum eigen alue, espec i ely. This s a e, wi h
he ones gene a ed by applying K+ o i , a e he elemen s o a subspace o dimension
Ω + 1, ha we will deno e by |Ω/2, miwi h m∈[−Ω/2,Ω/2].
Following (1) he o iginal hamil onian o LMG [11] can be w i en as :
HLMG =ε¯
K0+V1K+K−+V2(K+K++K−K−) (3)
whe e ¯
K0=K0+ Ω/2 (4)
The ε¯
K0 e m can be iewed as co esponding o he Ha ee-Fock pa . The lowes
eigens a e o K0co esponds hen o he unco ela ed HF g ound s a e |Ω/2,−Ω/2i.
In o de o s ess he analogy wi h he mic oscopic calcula ions in [9], we will o en
deno e by h(hole) a (−, s) s a e, i.e. a single pa icle s a e which is occupied in
|Ω/2,−Ω/2i, and by p(pa icle) a (+, s) single pa icle s a e. The e o e, by looking
a eq.(1), we can say ha he esidual in e ac ion in (3) only con ains pa icle-hole
e ms a†
pa†
p′ahah′and a†
pa†
hap′ah′ e ms, ha is hose usually included in RPA.
I V1=V2= 0 hen HLMG =ε¯
K0. The co esponding eigen alues a e nε wi h
n∈[0,Ω] .F om now on we will deno e hese s a es by |n >. The ene gy eigen alues
a e equidis an and o m a ha monic spec um unca ed a Ω + 1 le els. I V1o
V2a e di e en om ze o, hen he ene gy spec um is no ha monic any mo e and
only i V26= 0 he ene gy eigen ec o s co espond o a mixing o |nis a es. In ac ,
V1K+K−is diagonal in he |nispace. This e m o ph −p′h′ ype does no mix
s a es wi h di e en pa icle-hole numbe s. On he con a y, he pp′−hh′ ype e m,
V2K2
++K2
−mixes s a es wi h nand n±2 pa icle-holes. To conclude his pa , we
would like o ecall ha he LMG model is exac ly sol able using g oup echniques
and ha i includes he Pauli exclusion p inciple.
4
2.2 An ex ension o he LMG model
The LMG model in i s o iginal o m al eady includes some anha monici ies, essen ially
hose ela ed o he ac ha he Pauli p inciple is ea ed exac ly. Howe e , pa s
o he esidual in e ac ion a e neglec ed, namely he pp′−p′′p′′′,pp′−p′′h,hh′−h′′p,
hh′−h′′h′′′ ones, which we e conside ed in he mic oscopic calcula ions [9] whe e he
anha monici ies a ising om hem we e also s udied. In o de o simula e hem, we
p opose an ex ension o he LMG hamil onian which is s ill quad a ic in he K+,K−
and K0ope a o s :
H=HLMG + ∆V=HLMG +V3(K+¯
K0+¯
K0K−) + V4(¯
K0−1) ¯
K0(5)
The K+¯
K0 e m and i s he mi ian conjuga e in oduce a coupling be ween |niand
|n±1iwhe eas he las e m shi s he ene gies o he |nis a es, excep hose wi h
n= 0 and 1. The e o e, he eigens a es |φα>o he hamil onian (5) a e supe posi-
ions o he |n > s a es
|φα>=X
n
Xα
n|n > (6)
In sec ion 4 we will compa e he exac eigen alues o he hamil onian (5) wi h
hose co esponding o he bosonic hamil onian ob ained by mapping he e mionic
one up o he ou h o de .
3 The boson hamil onian
We apply boson expansion me hods o he e mionic hamil onian (5) and unca ed
he so ob ained boson hamil onian o second and ou h o de . In e .s[12, 13] a
simila s udy was done o he o iginal LMG hamil onian (3). In he p esen case, he
quad a ic hamil onian co esponds o RPA while he qua ic can be di ec ly compa ed
wi h wha was done in [9], whe e some u he app oxima ions we e pe o med, as
discussed below.
Le ’s ake he no mal o de ed Hols ein-P imako boson expansion o he SU(2) gen-
e a o s [12, 14, 15] :
(K+)b=P∞
i=0 ai(b†)i+1bi
(K−)b=P∞
i=0 ai(b†)ibi+1
(¯
K0)b=b†b
(7)
whe e
ai=
i
X
m=0
(−1)i−m
m!(i−m)!(Ω −m)1/2(8)
In he case o SU(2), he Hage-Hassan-Lambe mapping used in e .[9] is equi alen
o he Hols ein-P imako mapping (7,8) [16] wi h he shi ed ope a o ( ¯
K0)b. By
his shi one elimina es e ms linea in b†and bin he hamil onian which co esponds
o a ede ini ion o he mean ield.
Le us hen conside he bosonic mapping o ou ex ended hamil onian (5) in he
pa icula case V2=V1/2 which is he alue used in he nume ical applica ions
5

we p esen in he nex sec ion. By including all he e ms o he expansion (7)
which con ibu e up o he second o de in he b†, b ope a o s, we ge he quad a ic
hamil onian
H(2)
b=ε(1 + δ1)b†b+δ1ε
21−1
Ω1/2b†2+b2(9)
whe e
δ1= ΩV1/ε (10)
is ela ed o he s eng h o he pa icle-hole esidual in e ac ion. The hamil onian
H(2)
bcan be w i en in diagonal o m by in oducing he Bogoliubo ans o ma ion
[3] :
b†=XQ†+Y Q
b=XQ +Y Q†(11)
and imposing
H20 = 0 (12)
wi h he condi ion
X2−Y2= 1 (13)
which gua an ees he co ec commu a ion ela ion
hQ, Q†i= 1 (14)
Thus one ge s he RPA hamil onian
H(2)
b=H11Q†Q(15)
whe e
H11 =ε(1 + δ1)(X2+Y2) + 2XY δ1ε1−1
Ω1/2
(16)
The condi ions (12,13) de e mine he Xand Y’s ampli udes as solu ions o he se o
equa ions : 


X2−Y2= 1
ε(1 + δ1)XY +δ1ε
21−1
Ω1/2(X2+Y2) = 0 (17)
O cou se he e ms in V3and V4do no appea a he quad a ic o de . In eq.(15),
he H00 which co esponds o a shi o he eigenene gies was omi ed.
The iola ions o he Pauli exclusion p inciple in oduced by unca ing he boson
expansion a he lowes o de as well as a con ibu ion coming om he esidual
in e ac ion o he pp′−p′′p′′′,pp′−p′′h,hh′−h′′p,hh′−h′′h′′′ ype can be pa ly
in oduced by going one s ep u he , i.e. including all e ms up o he ou h o de ,
as i was done in e .[9]. A his le el, he e ms in V3and V4will he e o e en e .
Thei p esence, oge he wi h he co ec ions o he Pauli p inciple, will gi e ise o
an anha monic bosonic hamil onian.
By mapping he hamil onian o eq.(5) up o ou h o de we ge :
6
H(4)
b=ε(1 + δ1)b†b+δ1ε
21−1
Ω1/2b†2+b2
+V3√Ω1−1
Ω1/2b†2b+b†b2+ (V4−V1)b†2b2
+δ1ε
21−1
Ω1/21−2
Ω1/2−1b†3b+b†b3
(18)
which we ew i e in e ms o he Q†and Qope a o s as :
H(4)
b=H11Q†Q+H30(Q†3+Q3) + H21(Q†2Q+Q†Q2)+
H31(Q†3Q+Q†Q3) + H22Q†2Q2+H40(Q†4+Q4)(19)
whose coe icien s a e gi en in he appendix (eqs.21-27), oge he wi h he new equa-
ions o Xand Y(20) coming again om he condi ions (12,13). In eq.(19) he
H00 was omi ed as well as a e m, linea in he Q†and Qope a o s, which would
in oduce a ede ini ion o he mean ield, in analogy wi h wha was done in [9].
The qua ic hamil onian (19) co esponds o ha used in [9] whe e, howe e , some
app oxima ions we e in oduced in o de o make easible he calcula ions in he eal-
is ic cases conside ed he e. Fi s , only one- and wo-phonon s a es we e conside ed.
The e o e, he e ms in H30,H31 and H40 we e no e ec i e. Second, he Xand Y
ampli udes appea ing in he ou h o de hamil onian we e no ecalcula ed bu aken
equal o hose ob ained a he second o de , i.e. he RPA ones. In o de o ge an
indica ion on how good hese app oxima ions on he space, on he hamil onian and
on he Xand Yampli udes a e, in he nex sec ion we will s udy hem wi hin he
p esen schema ic model.
4 Resul s and discussion
Fi s o all we ha e o ix he pa ame e s en e ing in he hamil onian (5). Fo he single
pa icle ene gy we use he pa ame iza ion ε= 41/A1/3MeV. We ake V2= 0.5V1
and he s eng h V1is ixed so ha he i s exci ed s a e lies a an ene gy a ound
80/A1/3MeV co esponding o he sys ema ics o GDR in nuclei. This c i e ion gi es
V1= 1.2 MeV. We ha e s udied he beha iou o he ene gies o he h ee lowes
s a es, E1,E2and E3, as a unc ion o V4 o wo alues o V3, namely V3= 0 MeV
(Fig.1) and V3= 0.25 MeV (Fig.2). The la e alue o V3gi es <2|∆V|1>≈1 MeV
in analogy wi h he mic oscopic calcula ions [9]. No e ha he sign o V3is i ele an .
As a as he sign o V4is conce ned, we show esul s only o nega i e alues, which
gi e a downwa d shi o E2and E3wi h espec o 2E1and 3E1, espec i ely, i.e.
he ha monic (RPA) alues. In he igu es we show he esul s ob ained o Ω = 8.
To compa e he spec um o he exac hamil onian and i s di e en boson expansions
we s udy he ene gy di e ences (En−En−1) be ween he lowe exci ed s a es. In RPA
hey ha e he common alue 15.11 MeV o he conside ed pa ame e s, independen
o n, V3and V4. In he igu es his alue is ep esen ed by a dashed line. This should
be compa ed wi h he exac esul s o he e mionic hamil onian eq.(5), shown as
solid lines. We see ha , e en o V3=V4= 0, he RPA esul s de ia e om he exac
7
Figu e 1: Ene gy di e ences as unc ions o V4 o a ixed alue o V3= 0 MeV.
The ones co esponding o he e mionic hamil onian (5) a e plo ed as solid lines.
The o he s co espond o di e en app oxima ions on he bosonic expansion o he
hamil onian (see ex ).
ones. Namely, he i s h ee di e ences ob ained in he exac calcula ions a e equal
o 15.62 MeV, 14.38 MeV and 13.03 MeV, espec i ely.
The ag eemen wi h he exac ene gies can be imp o ed by using he qua ic hamil o-
nian, eq.(19). When he bosonic hamil onian is diagonalized in he space con aining
up o wo-phonon s a es we ge he esul s p esen ed in Fig. 1 and 2 by do ed lines.
We ema k ha now he i s exci ed s a e is e y close o he exac one, while o
he second one he e is a disc epancy o no mo e han 250 KeV. In he mo e ealis-
ic mic oscopic calcula ions p esen ed in [9] a u he app oxima ion was in oduced.
Ins ead o eob aining he Xand Yampli udes by sol ing he equa ions analogous
o eqs (20), he RPA ampli udes we e used. In he p esen model, his co esponds
o use in eq. (19) he solu ions o eqs (17) a he han hose o (20). The esul s
ob ained a e indis inguishable om he do ed lines in Fig. 1 and 2. This suppo s
he alidi y o he p ocedu e used in [9].
8
Figu e 2: As in ig. 1, bu o V3= 0.25 MeV.
In o de o es how much he unca ion o he space a ec s he esul s, le us now
enla ge he space up o h ee-phonon s a es. O cou se, in his space, he qua ic e m
H40 in (19) does no play any ole. In he igu es, we show he esul s co esponding o
he comple e qua ic hamil onian o eq.(19), wi h he Xand Yampli udes solu ions
o eq.s (20) (do -dashed lines). Compa ing he ene gies o he i s wo s a es we
see ha he ag eemen wi h he exac ones is now almos pe ec . In he enla ged
space we can also s udy he hi d exci ed s a e which, in he ha monic limi , would
co espond o a h ee-phonon s a e. In his case he app oxima e esul s di e om
he exac ones by 1 MeV, which can be conside ed as a good app oxima ion since
he exci a ion ene gy o his s a e is abou 40 MeV. We would like o poin ou
ha he esul s ob ained in he same model space bu co esponding o he qua ic
hamil onian o eq (19) wi hou he H31 and H30 e m and wi h he use o he RPA
Xand Yampli udes, solu ion o eqs (17), ( ep esen ed in igu e 1 and 2 like solid
ci cles) a e p ac ically coinciden wi h he do -dashed line. The use o an analogous
app oxima ion in ealis ic mic oscopic calcula ions would make hem much simple .
In summa y, we ha e seen ha qui e good esul s can be ob ained o bo h he second
9