Tests and applications of self-consistent cranking in the interacting boson model

12 Pages·0.2 MB·English

by  Serdar Kuyucak

Tests and applications of self-consistent cranking in the interacting boson model
Serdar Kuyucak∗
Department of Theoretical Physics, Research School of Physical Sciences,
Australian National University, Canberra, ACT 0200, Australia
Michiaki Sugita
Japan Atomic Energy Research Institute, Tokai, Ibaraki 319-11, Japan
(February 9, 2008)
Theself-consistentcrankingmethodistestedbycomparingthecrankingcalculationsintheinter-
actingbosonmodelwiththeexactresultsobtainedfromtheSU(3)andO(6)dynamicalsymmetries
and from numerical diagonalization. The method is used to study the spin dependence of shape
variables in thesd and sdg boson models. Whenrealistic sets of parameters are used, both models
leadtosimilarresults: axialshapeisretainedwithincreasingcrankingfrequencywhilefluctuations
9
9 in theshape variable γ are slightly reduced.
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1 21.60.Ew, 21.60.Fw

n
a
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I. INTRODUCTION
9
2
Self-consistent cranking (SCC) is one of the most popular methods to study collective rotations in nuclear many-

1 body systems [1]. Within the Hartree-Fock framework, it has provided important insights into the backbending
v phenomena, and currently, it is being used actively in studies of superdeformed nuclei [2]. In stark contrast, it has
6
beenrarelyusedintheinteractingbosonmodel(IBM)[3]. Infact,thereisonlyoneapplicationofSCCtoIBMwherea
9
studyofmomentofinertia(MOI)iscarriedout[4]. Intheearlystagesofthemodel,anobviousreasonforthisneglect
0
was the availability of exact results (either via dynamical symmetries or numerical diagonalization), which left little
1
0 roomfordevelopmentofapproximatemethodssuchasSCC.InlaterextensionsoftheIBM(e.g.sdg-IBM),dynamical
9 symmetries were not very useful, and numerical diagonalization could not be carried out due to large basis space,
9 hence alternative methods were needed. At around the same time, however, an exact angular momentum projection
/ techniquewasdevelopedforaxiallysymmetricbosonsystems(1/N expansion[5]),whichprovidedanalyticalsolutions
h
t for general Hamiltonians in large basis spaces, filling the gap left by the other methods. Although the assumption
-
l of axial symmetry is reasonable for most deformed nuclei in their ground states, triaxial effects are known to play
c
a role, especially at high-spins [6]. Therefore a study of the evolution of shapes in the IBM would be useful in
u
order to investigate such questions as the effects of triaxiality in nuclear spectra, ways of incorporating it in the IBM
n
: Hamiltonian,andwhetheranimproveddescriptionofspectracanbeobtainediftriaxialeffectsaretakenintoaccount.
v
In this paper, we first present tests of the SCC by comparing the cranking calculations with the exact results
i
X obtained in the SU(3) and O(6) limits, as well as from diagonalizationof more realistic Hamiltonians in the sd-IBM.
TheSCCmethodisthenusedinastudyoftriaxialityintheIBMbothinthesdandsdg-bosonversionsofthemodel.
r
a
II. CRANKING FORMALISM IN THE IBM
The SCCinthe IBMwasformulatedinRef.[4]to whichwereferfordetails,especiallyconcerningthe construction
of the intrinsic state and the symmetries it possesses. For a given IBM Hamiltonian H and intrinsic state N,x , the
| i
SCC around the x-axis is described by
δ N,xH ωLˆ N,x =0, (1)
x
h | − | i
where Lˆ is the x-component of the angular momentum operator, N is the boson number and x are the variational
x
parameterstobedeterminedfromthe extremumcondition. Forconvenience,weconsiderageneralformulationofthe
IBM which will be tailored to specific cases later. Thus, we introduce the boson creation and annihilation operators
∗E-mail: [e-mail]
1
b† ,b with l = 0,2,4,... where b = s, b = d, b = g, etc. The intrinsic state is given by a condensate of intrinsic
lµ lµ 0 2 4
bosons as
l
N,x =(N!)−1/2(b†)N 0 , b† = x b† , (2)
| i | i lm lm
Xl mX=−l
where x are the normalized boson mean fields, i.e. x x = 1. Due to the symmetries in the cranked system [4],
lm
·
x are real and x = x . Note that the odd-m components of x vanish in the static limit (ω = 0), but not
lm l−m lm lm
in general. Thus, invoking the normalization condition (e.g. setting x = 1), there are 3 independent variational
00
parametersinthe sd-IBM(x ,x ,x )and5moreinthesdg-IBM(x ,...,x ). Forthe IBMHamiltonian,weuse
20 21 22 40 44
the generic multipole form
2lmax
H = ε nˆ κ T(k) T(k), (3)
l l k
− ·
Xl Xk=0
where the boson number and multipole operators are given by
nˆ = b† b , T(k) = t [b†˜b ](k). (4)
l lµ lµ µ kjl j l µ
Xµ Xjl
In particular, the angular momentum operators are
1
Lˆ = − (Lˆ Lˆ ), Lˆ = l(l+1)(2l+1)/3 [b†˜b ](1). (5)
x √2 +1− −1 µ l l µ
Xl p
The parametersofthe Hamiltonianconsistsofthe single bosonenergies,ε ,the multipole coupling strengths,κ ,and
l k
the multipole parameters, t . The hermiticity condition on multipole operators requires that t =t .
kjl kjl klj
The expectation values in Eq. (3) can be evaluated in a straightforward manner using boson calculus techniques.
For the one-body operators, one simply obtains
x2
N,xnˆ N,x =N lm . (6)
h | l| i x x
Xm ·
Calculations for the multipole interactions are somewhat more involved but the expectation values can again be
reduced to a compact form
k x2
N,xT(k) T(k) N,x =N(N 1) ( 1)µA A +N ε′ lm . (7)
h | · | i − − kµ k−µ lx x
µX=−k Xlm ·
Here A corresponds to the expectation value of the operator T(k) in a single boson state and is given by
kµ µ
x x
A = ( 1)m jnlmkµ t jn lm. (8)
kµ − h | i kjl x x
jXnlm ·
Using the symmetries imposed on the mean fields and the hermiticity condition, it can be shown that A vanishes
kµ
if k +µ is odd, and furthermore A = ( 1)kA . Thus, for even multipoles, only the positive, even µ values
k−µ kµ
−
contribute to the sum in Eq. (7). The second term in Eq. (7) arises from the normal ordering of the boson operators
and corresponds to an effective one-body term with boson energies
2k+1
ε′ = t2 . (9)
l 2l+1 kjl
Xj
Finally, the expectation value of Lˆ is
x
x x
N,xLˆ N,x = N ( 1)m lnl m11 2l(l+1)(2l+1)/3 ln lm. (10)
h | x| i − − h − | i x x
lXmn p ·
2
For a given angular frequency ω, the cranking equation (1) is solved numerically by minimizing the cranking
expression E(x,ω) = H ωL with respect to the mean fields x. The dynamical MOI is calculated from the
x
h − i
derivative of Lˆ , Eq. (10) as [7]
x
h i
d Lˆ
(2) = h xi, (11)
J dω
which is obtained under the assumption L L. For the exact energy levels E(L), dynamical MOI and the
x
h i ≃
corresponding rotational frequencies are calculated from [7]
d2E(L) −1 4h¯2
=h¯2 ,
(cid:20) dL2 (cid:21) ≃ E(L+2) 2E(L)+E(L 2)
− −
1
ω = [E(L+2) E(L 2)]. (12)
4h¯ − −
Evolution of triaxiality with rotation is studied using the following expressions for the collective shape variable γ
Q
tanγ =√2 h 2i, (13)
Q
0
h i
7 Q Q Q
cos3γ = h · · i. (14)
−r2 Q Q 3/2
h · i
The one-body expectation values of the quadrupole operator in Eq. (13) follow from Eq. (8) as Q = NA . The
µ 2µ
h i
two-bodyexpectationvalue inEq.(14)isgiveninEq.(7)andthe three-bodypartisderivedinAppendix A.Eq.(13)
gives the average value for γ while Eq. (14) probes its fluctuations from this average. Thus the information content
of the two expressions for γ are different and they compliment each other.
III. CRANKING IN THE sd-IBM
In this section, we perform cranking calculations for a variety of sd-IBM Hamiltonians and compare the results
for the dynamical MOI and γ with the exact ones obtained from dynamical symmetries (SU(3) and O(6)), and from
numericaldiagonalization. For this purpose,we use a simple Hamiltonian with a quadrupole interactionand d-boson
energy
H = κQ Q+εnˆ . (15)
d
− ·
Following the convention,we denote the parametersof the quadrupole operator by t =1 and t =χ. The SU(3)
202 222
limitisobtainedwhenε=0andχ= √7/2,andtheO(6)limitwhenε=χ=0. TotestmorerealisticHamiltonians,
−
weusethemiddlevalueforχ( √7/4),whichleadstoabetterdescriptionofelectromagneticproperties[8]. Inclusion
−
of the d-bosonenergywith ε=1.5Nκimproves,in addition, the moment of inertia systematics [9]. The exactenergy
eigenvaluesE(L)areobtainedfromthe Casimiroperatorsinthe caseofthe dynamicalsymmetries,andbynumerical
diagonalization of the Hamiltonian in the latter two cases.
For convenience in the variational problem, we set x = 1 and drop the subscript 2 from the quadrupole mean
00
fields, i.e. x =x . The normalization is then given by
2m m
=1+x2+2x2+2x2. (16)
N 0 1 2
The expectation value of the Hamiltonian (15), E(x)= N,xH N,x , can be written from Eqs. (6-7) as
h | | i
E(x)= κ N(N 1) A2 +2A2 +5 nˆ +(χ2+1) nˆ +ε nˆ . (17)
− − 20 22 h si h di h di
(cid:2) (cid:0) (cid:1) (cid:3)
Here the quadratic forms A follow from Eq. (8) as
2µ
1
A = 2x 2/7χ x2+x2 2x2 ,
20 N h 0−p (cid:0) 0 1− 2(cid:1)i
1
A = 2x + 2/7χ 2x x + 3/2x2 , (18)
22 2 0 2 1
N h p (cid:16) p (cid:17)i
3
and the expectation values of nˆ and nˆ are
s d
N N
nˆ = , nˆ = x2+2x2+2x2 . (19)
h si h di 0 1 2
N N (cid:0) (cid:1)
From Eq. (10), the cranking term is given by
2N
Lˆ = x √6x +2x . (20)
x 1 0 2
h i (cid:16) (cid:17)
N
InFig.1,wecomparethe crankingresultsforthe dynamicalMOIobtainedfromEq.(11)withthoseobtainedfrom
Eq. (12) using the exact energy levels for N = 10 bosons. In the SU(3) and O(6) limits, the MOI is constant and
given by (2) = 4/3κ = 200/3 and 2/κ = 100 h¯2/MeV, respectively. Note that both results are independent of N,
ex
J
and the constancy of MOI simply follows from the fact that E(L) is quadratic in L in both cases. The SCC leads to
a constant MOI in the SU(3) limit, as expected from a rigid rotor, but deviates from the exact result by about 5%.
The O(6) limit corresponds to a γ-unstable rotor for which cranking is not expected to work well. Not surprisingly,
comparison of the cranking results for (2) in Fig. 1 indicate a maximum deviation ( 10%) from the exact result,
J ∼
as well as a small frequency dependence in MOI. We will comment on these deviations further when discussing the
N dependence of the results below. The case with χ= √7/4 falls in between the two dynamical symmetry results.
−
The dynamical MOI increases with ω as in a typical deformed nucleus (Fig. 1), though the overall magnitude is still
too large. The deviation between the cranking and the exact results in this case is comparable to that in the SU(3)
limit, with the error getting smaller at higher frequencies. When a d-boson energy with ε = 1.5Nκ is included in
the Hamiltonian, which fits the observed range of MOI data better, the agreement between the cranking and exact
results improve markedly, especially at lower frequencies.
The SCC is a semi-classical theory and it would be exact in the classical limit when N . Thus to understand
→∞
the nature of discrepancies seen in Fig. 1 better, we need to study the N dependence of the results. In Fig. 2, we
show the SCC results for (2) as a function of N in the SU(3) and O(6) limits. Since the MOI is not constant in
J
the O(6) limit, we have taken the value of (2) at the middle-frequency for each N. The curves that trace the SCC
J
results are obtained from
1 1
(2) = (2) 1+ + , (21)
Jcr Jex (cid:18) 2N 7N2(cid:19)
in the SU(3) limit, and from
1 1
(2) = (2) 1 + , (22)
Jcr Jex (cid:18) − N 2N2(cid:19)
in the O(6) limit. The coefficients in Eqs. (21) and (22) are derived from a 1/N expansion of the SCC equation
(1). It is clear from Fig. 2 and Eqs. (21, 22) that the SCC calculation of (2) is correct to leading order but fails in
J
higher orders in 1/N by generating spurious correction terms. A comparison of Eq. (17) with the accurate angular-
momentum-projectedresult[5]confirms thatthe incorrecttreatmentofthe 1/N terms inthe SCC is the cause ofthe
discrepancy. ItisinterestingtonotethatanagreementisobtainedifoneusestheCasimiroperatoroftheSU(3)given
byC =Q Q+(3/8)L L,insteadofjustQ Q. Inthiscase,theincorrect1/N contributionfromtheQ Qinteraction
2
· · · ·
is exactly canceled by the spurious contributionfrom the L L term. Another interesting observationis that the best
·
agreement of SCC with the exact results is achieved in the realistic case with χ = √7/4 and ε = 1.5Nκ. This
−
happens because the 1/N term is now dominated by the leading contributionfrom nˆ , which is correctlytreated in
d
the SCC. The errors from nˆ contribute at the 1/N2 level, and these become apphareint only at the high-rotational
d
h i
frequencies as is apparent in Fig. 1.
We next study the effects of triaxiality in SCC. It is wellknown that the energy surface of an sd-IBMHamiltonian
with one- and two-body interactions has an axially symmetric minima in the deformed phase [10]. An exception
occursintheO(6)limit, whereaγ-unstableshapedevelops. TheSCCcalculationsintheO(6)limitgive30o forboth
theaverageγ (13)anditsfluctuations(14)atallfrequencies. ThustheSCCresultsareconsistentwiththedescription
of the O(6) limit using a triaxial intrinsic state with γ = 30o, which has a very shallow minimum in the γ direction
[11]. Frequency independence of the results corroborates with the fact that the O(6) states remain γ-unstable at all
spins. In Fig. 3, we show the evolution of the triaxiality angle γ with the cranking frequency for the remaining three
cases discussed above for three different boson numbers, N =10, 15, 20. Results obtained from both Eq. (13) (solid
line) and (14) (dashed line) are shown. The ground band in the SU(3) limit can be described exactly by angular
momentum projectionfromanaxiallysymmetric intrinsic state, therefore,bothaverageγ andits fluctuations should
4
vanish at all frequencies. The SCC results (Fig. 3, left) start with γ =0 at low ω but deviate from it systematically
with increasing frequency. Although the situation appears to be improving with increasing N, similar to the case
in (2), in fact, this is merely due to the extension of the spectrum with N (ω N). If one scales out the N
max
J ∝
dependence(i.e.,plotsγ againstω/N),thenallthecurveswithdifferentN overlap. Thusγ canbewrittenasapower
expansion in ω/N with the leading (N independent) term being zero. This situation is similar to that encountered
in the study of (2): the SCC gets the leading order term correctly (γ = 0) but fails in higher-orders in 1/N by
J
generating spurious terms in powers of ω/N. The fact that the incorrect 1/N terms in γ all depend on the cranking
frequency makes the SCC results increasingly unreliable at higher frequencies. The onset of triaxiality observed at
aroundmid-frequency (ω /2) in all curves in Fig. 3 is thus due the failure of the SCC and not a genuine feature of
max
the boson system. We remark that despite the sudden increase in the calculated γ values, (2) remains constant in
J
the SU(3) limit, suggesting that triaxiality has a negligible effect on MOI.
We interpret the remaining more realistic cases in the light of the SU(3) test. In the case of χ = √7/4 (Fig. 3,
−
middle), the average γ (solid lines) remains around zero except at high frequencies. Attributing this sudden rise in
γ to the break down of SCC, we see that γ 0 for all ground-band states of a Q Q Hamiltonian. The fluctuations
∼ ·
(dashedlines),ontheotherhand,arenon-zerobutgetsmallerwithincreasingN. Thevaluesoffluctuationsatω =0
are consistent with those obtained from an exact calculation for the ground state [12]. At higher frequencies there
is a small reduction in fluctuations, that is, rotations have a slightly stabilizing effect on the γ motion. Increasing
the bosonnumber also suppressesfluctuations, which is due to the energy surface becoming deeper in the γ direction
withlargerN values. The lastcasewith ε=1.5Nκ(Fig.3,right)hasbroadlythe samefeatures. The one-bodyterm
cannotinducetriaxialityalone,therefore,thesmalldeviationinaverageγ fromzeroatmid-frequenciesispresumably
due to errors in SCC. There is an increase in fluctuations compared to the middle panel, which can be explained as
due to the energy surface becoming shallower with the addition of the one-body term. These results indicate that
rotationshavea ratherlimited effect on the shape variablesin the sd-IBM,with averageγ remainingnearly constant
at zero, and its fluctuations being slightly reduced from the ground state value at higher spins.
IV. CRANKING IN THE sdg-IBM
In contrastto the sd-IBM, where numericaldiagonalizationis a routine task, the sdg-IBMalreadysuffers from the
large basis problem, and exact diagonalization is not possible for N > 11, that is, for most of the deformed nuclei.
The 1/N expansion circumvents this problem but because it assumes axial symmetry, one cannot use it to address
questions on triaxiality. Here we perform cranking calculations in the sdg-IBM to study the evolutionof shapes with
rotation without restriction to axial symmetry. For the Hamiltonian, we choose
H = κ Q Q κ T T +ε nˆ +ǫ nˆ , (23)
2 4 4 4 d d g g
− · − ·
which has been shown to provide a good representation of data in rare-earth and actinide nuclei [13]. As in Section
III, we set q =h =1 and x =1. Then the normalization is given by
02 02 00
=1+x2 +2x2 +2x2 +x2 +2x2 +2x2 +2x2 +2x2 . (24)
N 20 21 22 40 41 42 43 44
The expectation value of the Hamiltonian (23) follows from Eqs. (6-7) as
E(x)= κ N(N 1)(A2 +2A2 )+5 nˆ +(1+q2 +q2 ) nˆ +(5/9)(q2 +q2 ) nˆ
− 2 − 20 22 h si 22 24 h di 24 44 h gi
(cid:2) (cid:3)
κ N(N 1)(A2 +2A2 +2A2 )+9 nˆ +(9/5)(h2 +h2 ) nˆ +(1+h2 +h2 ) nˆ
− 4h − 40 42 44 h si 22 24 h di 24 44 h gii
+ε nˆ +ε nˆ . (25)
d d g g
h i h i
Here the quadrupole quadratic forms A are given by
2µ
1
A = 2x 2/7q x2 +x2 2x2 +2 2/21q √3x x +√10x x
20 N h 20−p 22(cid:0) 20 21− 22(cid:1) p 24(cid:16) 20 40 21 41
+√5x x (1/3√77)q 10x2 +17x2 +8x2 7x2 28x2 ,
22 42(cid:17)− 44 40 41 42− 43− 44 (cid:21)
(cid:0) (cid:1)
1
A = 2x + 2/7q 2x x + 3/2x2
22 (cid:20) 22 22 20 22 21
N p (cid:16) p (cid:17)
+(2/3√14)q x x +√5x x +√15x x +√35x x +√70x x
24 22 40 21 41 20 42 21 43 22 44
(cid:16) (cid:17)
+ 2/33q 2x x +3x x +3 10/7x x +(5/√7)x2 , (26)
p 44(cid:16) 42 44 41 43 p 40 42 41(cid:17)(cid:21)
5
and the hexadecapole quadratic forms A by
4µ
1
A = 2x + 2/35h 3x2 4x2 +x2 (2/√77)h 2√5x x +√6x x
40 N h 40 p 22(cid:0) 20− 21 22(cid:1)− 24(cid:16) 20 40 21 41
6√3x x + 2/1001h 9x2 +9x2 11x2 21x2 +14x2 ,
− 22 42(cid:17) p 44(cid:0) 40 41− 42− 43 44(cid:1)(cid:21)
1
A = 2x +(1/√7)h √6x x 2x2 + 3/385h
42 N(cid:20) 42 22(cid:16) 20 22− 21(cid:17) p 24
6√5x x +9x x (8/√3)x x 5√7x x +2√14x x
22 40 21 41 20 42 21 43 22 44
×(cid:16) − − (cid:17)
+ 5/1001h 6√7x x +2√7x x 11x x 6x2 ,
p 44(cid:16) 42 44 41 43− 40 42− 41(cid:17)(cid:21)
1
A = 2x +h x2 +(2/√55)h √6x x +√21x x +2√7x x
44 (cid:20) 44 22 22 24 22 42 21 43 20 44
(cid:16) (cid:17)
N
+(1/√143)h 2√14x x +2√35x x +3√5x2 . (27)
44(cid:16) 40 44 41 43 42(cid:17)(cid:21)
The one-body expectation values in Eq. (25) are given by Eq. (ref1bsd) and
N
nˆ = x2 +2x2 +2x2 +2x2 +2x2 . (28)
h gi 40 41 42 43 44
N (cid:0) (cid:1)
Finally, from Eq. (10) the cranking term is
2N
Lˆ = x √6x +2x +x 2√5x +3√2x +x √14x +2√2x . (29)
x 21 20 22 41 40 42 43 42 44
h i h (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)i
N
In view of the numerous parameters in the sdg-IBM, we limit our discussion to a realistic range tailored to data
[13]. Accordingly, the quadrupole parameters q ,q ,q are scaled from their SU(3) values with a single factor
22 24 44
{ }
q as suggested by microscopics [14]. The hexadecapole parameters h ,h ,h are determined from those of q
22 24 44 jl
{ }
through the commutation condition
[h¯,q¯]=0, q¯ = j0l020 q , h¯ = j0l040 h , (30)
jl jl jl jl
h | i h | i
whichreproducethe availableE4 datareasonablywell[13]. We adaptthe realisticsd-IBMparametersusedinFig.1,
κ = 20 keV, q = 0.5 and ǫ = 1.5Nκ , and use further, ǫ = 4Nκ and κ /κ = 0-0.4. Energy levels are very
2 d 2 g 2 4 2
−
well described in the sdg-IBM [13], therefore we do not dwell on a study of MOI apart from noting that all the SCC
calculations of (2) exhibit the characteristic rise with increasing frequency seen in experimental spectra. We focus
J
instead on the shape question which couldnot be addressedin the 1/N expansionapproach. With the parameterset
described above, the SCC calculations of γ in the sdg-IBM are very similar to those of the sd-IBM with the realistic
set of parameters (Fig. 3, right). Because the hexadecapole interaction is a relatively novel feature not used in the
sd-IBM, we briefly comment on its effect. Inclusion of the hexadecapole interaction does not cause any deviation
fromthe axialshape but depending onits sign, iteither decreasesthe fluctuations in γ (attractive)or increasesthem
(repulsive) by a few degrees. This is very similar to the effect of changingthe strength of the quadrupole interaction,
in that, a largerκ leads to a deeper energy wellinthe γ direction,therefore reducesthe fluctuations, andvice versa.
2
In certain parametrizations where the boson energies are neglected, it is possible to obtain non-axial shapes in the
sdg-IBM [15]. However,once a g-bosonenergy of ǫ 1 MeV is included, as demanded by the experimental spectra,
g
∼
suchdeviationsfromaxialsymmetryquicklydisappear. Inshort,thecongruencefoundbetweenthesd-andsdg-IBM
results with regardto the shape variable γ is a direct consequence of the fairly high g-bosonenergy, which limits the
effect of g bosons to a perturbative range at low to medium spins.
V. CONCLUSIONS
Wehavetestedtheself-consistentcrankingmethodusingtheexactIBMresultsobtainedfromdynamicalsymmetries
and numerical diagonalization. The SCC results for the dynamic MOI are in good agreement with the exact ones,
especiallyinrealisticcaseswherethedeviationisatmostafewpercent. ThestudyofshapesusingSCC,ontheother
6
hand, remains problematic due to the generation of spurious frequency-dependent terms in shape variables, which
become dominant at high frequencies. Nevertheless, SCC can give reliable information on evolution of shapes in the
low to mid frequency range. Our study of shapes using SCC indicates that both the sd- and sdg-IBM with one- and
two-body interactions retain the axial shape (γ = 0) with fluctuations around 10o, as long as the model parameters
are restricted to a realistic range that reproduce the data. The only frequency dependence in shape variables is
observedinthe fluctuations in γ whichis slightly reducedwith increasingfrequency. Theseresults suggestthat if the
triaxial effects are important and need to be included in the IBM, then the remedy should be sought in three-body
interactions rather than g bosons.
APPENDIX A: THREE-BODY OPERATORS
Here we derive the expectation value of the three-body quadrupole interaction in Eq. (14). The scalar product of
the three quadrupole operators is defined as
Q Q Q=[QQ](2) Q. (A1)
· · ·
The expectation value of this three-body term in the condensate state Eq. (2) is given by
1
Q Q Q = ( 1)µ 2µ 2µ 2µ 0bNQ Q Q (b†)N 0 . (A2)
h · · i N! − h 1 2| ih | µ1 µ2 −µ | i
µX1µ2µ
Writing the quadrupoleoperatorsexplicitly andcommutingallthe bosoncreationoperatorsto the left, oneobtains a
three-body term, 3 two-body terms and a one-body term. Using boson calculus, the matrix elements of these normal
ordered operators can be evaluated in a straightforwardmanner with the result
Q Q Q = ( 1)µ 2µ 2µ 2µ ( 1)n1+n2+n3q q q
h · · i − h 1 2| i − j1l1 j2l2 j3l3
µX1µ2µ j1j2Xj3l1l2l3
m1m2m3n1n2n3
j m l n 2µ j m l n 2µ j m l n 2 µ
1 1 1 1 1 2 2 2 2 2 3 3 3 3
×h | ih | ih | − i
N(N 1)(N 2)x x x x x x
×{ − − j1m1 j2m2 j3m3 l1−n1 l2−n2 l3−n3
+N(N 1)[x x x x δ δ +x x
− j1m1 j2m2 l2−n2 l3−n3 j3l1 m3−n1 j1m1 j2m2
x x δ δ +x x x x δ δ ]
× l1−n1 l3−n3 j3l2 m3−n2 j1m1 j3m3 l2−n2 l3−n3 j2l1 m2−n1
+Nx x δ δ δ δ . (A3)
j1m1 l3−n3 j2l1 m2−n1 j3l2 m3−n2}
Thethree-bodypart(thefirstterm)isseentoinvolveatripleproductoftheone-bodyexpectationvaluesA defined
2µ
in Eq. (8). In the effective two-body terms, three C-G coefficients can be summed to yield a 6 j symbol and a C-G
−
coefficient. After some manipulations of the summation indices, all 3 terms can be shown to be equivalent. Finally,
the sum of the four C-G coefficients in the effective one-body term gives a 6 j symbol. Thus, Eq. (A3) can be
−
written compactly as
Q Q Q =N(N 1)(N 2) ( 1)µ 2µ 2µ 2µ A A A
h · · i − − − h 1 2| i 2µ1 2µ2 2−µ
µX1µ2µ
+3N(N 1) ( 1)µA˜ A +N ε˜x2 , (A4)
− − 2µ 2−µ l lm
Xµ Xlm
where we have introduced
j l 2
q˜ =5 ( 1)i+j+l q q ,
jl − (cid:26)2 2 i (cid:27) ji il
Xi
25 l i 2
ε˜ = q q q , (A5)
l 2l+1 (cid:26)2 2 j (cid:27) li ij jl
Xij
and A˜ is obtained from A by replacing q q˜ in Eq. (8). Note that symmetry of q is retained, i.e., q˜ =q˜ .
2µ 2µ jl jl jl jl lj
→
Using the fact that A =A =0 and A =A , the sums in Eq. (A4) involving A can be carried out to yield
21 2−1 22 2−2 2µ
7
Q Q Q =N(N 1)(N 2) 2/7( A2 +6A2 )A
h · · i − − − 20 22 20
+3N(N 1)(A˜ Ap +2A˜ A +N ε˜x2 , (A6)
− 20 20 22 22 l lm
Xlm
In the case of the sd-IBM, A and A are given in Eq. (18). The contracted parameters in Eq. (A5) become
20 22
q˜ =χ, q˜ =1 3χ2/14,
02 22
−
ε˜ =5χ, ε˜ =2χ 3χ3/14. (A7)
0 2
−
Using these values of q˜ , one obtains for A˜ and A˜
jl 20 22
1
A˜ = 2χx 2/7(1 3χ2/14)(x2+x2 2x2) ,
20 N h 0−p − 0 1− 2 i
1
A˜ = 2χx + 2/7(1 3χ2/14)(2x x + 3/2x2) , (A8)
22 N h 2 p − 0 2 p 1 i
In the sdg-IBM, these parameters are given by
3 2 2 5√22
q˜ =q , q˜ =1 q2 + q2 , q˜ = q q q q ,
02 22 22 − 14 22 7 24 24 7 22 24− 42 24 44
5√22 65
q˜ = q2 q2 ,
44 42 24− 42√22 44
3 4 5√22
ε˜ =5q , ε˜ =2q q3 + q q2 + q2 q ,
0 22 2 22− 14 22 7 22 24 42 24 44
10 25√22 325
ε˜ = q q2 + q2 q q3 . (A9)
4 63 22 24 189 24 44− 378√22 44
[1] P.Ring and P. Schuck,The Nuclear Many-Body Problem (Springer-Verlag, NewYork,1980).
[2] C. Baktash, B. Haas, and W. Nazarewicz, Ann.Rev.Nucl. Part. Sci. 45, 485 (1995).
[3] F. Iachello and A.Arima, The Interacting Boson Model (Cambridge University Press, Cambridge, 1987).
[4] H.Schaaser and D.M. Brink, Phys.Lett. B 143, 269 (1984); Nucl.Phys. A 452, 1 (1986).
[5] S.Kuyucakand I.Morrison, Ann.Phys.(N.Y.) 181 79 (1988); ibid, 195, 126 (1989).
[6] S.Aberg, H. Flocard, and W. Nazarewicz, Ann.Rev.Nucl.Part. Sci. 40, 439 (1990).
[7] A.Bohr and B.R. Mottelson, Phys. Scripta24, 71 (1981).
[8] R.F. Casten and D.D. Warner, Rev.Mod. Phys. 60, 389 (1988).
[9] S.Kuyucakand S.C. Li, Phys. Lett. B 354, 189 (1995).
[10] J.N. Ginocchio and M.W. Kirson, Nucl. Phys.A 350, 31 (1980).
[11] T. Otsukaand M. Sugita, Phys.Rev.Lett. 59, 1541 (1987).
[12] J.P. Elliott, J.A. Evans, and P. Van Isacker, Phys.Rev. Lett. 57, 1124 (1986).
[13] S.C. Li and S. Kuyucak,Nucl. Phys. A 604, 305 (1996).
[14] T. Otsuka,A. Arima and F. Iachello, Nucl. Phys.A 309, 1 (1978).
[15] S.Kuyucakand I.Morrison Phys.Lett. B 255, 305 (1991).
FIG.1. Comparison of the cranking calculations of the dynamical moment of inertia in the sd-IBM (lines) with the exact
results obtained from the dynamical symmetries SU(3) and O(6) (filled circles), and numerical diagonalization (χ = √7/4
−
with ε=0 and ε=1.5Nκ) (open circles). In all cases, N =10 and κ= 20 keV are used.
−
FIG.2. Boson numberdependenceofthecrankingdynamicalmomentofinertiaintheSU(3)andO(6)limitsofthesd-IBM
(circles). Thelines that trace the circles are obtained from Eq. (21) in theSU(3) case and from Eq. (22) in theO(6) case.
8
FIG.3. Crankingcalculations ofthetriaxialityangle γ inthesd-IBMforthethreecasesshowninFig.1forN =10,15,20.
Thesolid linesindicateaverageγ obtainedfromEq.(13)andthedashedlinescorrespondtoitsfluctuationsgivenbyEq.(14).
9
110
100
O(6)
c=c
90 /2
SU3
)
V
e 80
M
/
2 SU(3)
h
( 70
)
2
(
J
60
c=c
/2
SU3
e/k
50 N=1.5
40
30
0.0 0.1 0.2 0.3 0.4
w
h (MeV)