Angular Momentum Projected Configuration Interaction with Realistic Hamiltonians

0.65 MB·English

by  Zaochun Gao

Angular Momentum Projected Configuration Interaction with Realistic Hamiltonians
Zao-Chun Gao1,2∗ and Mihai Horoi1
1Department of Physics, Central Michigan University, Mount Pleasant, Michigan 48859, USA
2China Institute of Atomic Energy P.O. Box 275-18, Beijing 102413, China
(Dated: January 5, 2009)
TheProjectedConfigurationInteraction(PCI)methodstartsfromacollectionofmean-fieldwave
functions, and builds up correlated wave functions of good symmetry. It relies on the Generator
Coordinator Method (GCM) techniques, but it improves the past approaches by a very efficient
method of selecting the basis states. We use the same realistic Hamiltonians and model spaces as
theConfiguration Interaction(CI)method,andcomparetheresultswiththefullCIcalculations in
thesd and pf shell. Examples of 24Mg, 28Si, 48Cr, 52Fe and 56Niare discussed.
PACSnumbers: 21.60.Cs,21.60.Ev,21.10.-k
9
0
0 I. INTRODUCTION by the projected states. The Nilsson model [8] has been
2
proven to be very successful in describing the deformed

n intrinsic single particle states, and the quadrupole force
The fullconfigurationinteraction(CI)methodusing a
a was found to be essential for describing the rotational
sphericalsingleparticle(s.p.) basisandrealisticHamilto-
J
motion [6]. Despite its simplicity PSM was proven to
nianshasbeenverysuccessfulindescribingvariousprop-
5
be a very efficient method in analyzing the phenom-
erties of the low-lying states in light and medium nuclei.

ena associated with the rotational states, especially the
] The realistic Hamiltonians, such as the USD [1, 2]in the
h sdshell, the KB3[3], FPD6 [4]andGXPF1 [5]inthe pf high spin states, not only for axial quadrupole deforma-
t tion, but also for the octupole [9] and triaxial shapes
- shell, have provided a very good base to study various
l [10,11]. However,itspredictivepowerislimitedbecause
c nuclear structure problems microscopically.
u Some sd and pf nuclei such as 28Si and 48Cr are well the mulitpole-multipole plus pairing Hamiltonian has to
n deformed. Their collective behavior is confirmed by the be tuned to a specific class of states, rather than an re-
[ gion of the nuclear chart.
strongcollectiveE2transitionsandtherotationalbehav-
3 ior of the yrast state energies, E(I) ∼ I(I + 1). The Besides the Projected Shell Model, another sophisti-
v mean-field description in the intrinsic frame naturally catedapproachbasedontheprojectionmethodisMON-
3 takes advantage of the spontaneous symmetry breaking. STER and the family of VAMPIRs [12]. The MON-
5 This approach provides some physical insight, but the STER is similar to the PSM, but the basis is obtained
1
loss of good angular momentum of the mean-field wave by projecting the Hartree-Fock-Bogoliubov (HFB) vac-
2
functions makes the comparison with the experimental uum and the related 2-quasiparticle configurations onto
.
8 datadifficult. The CI calculationsinsphericalbasis pro- good quantum numbers, including neutron and proton
0 videthedescriptioninthelaboratoryframe. Theangular number, parity, and the angular momentum. VAMPIR,
8
momentum is conserved, but the physical insight associ- whichperformstheenergyvariationaftertheprojection,
0
: ated with the existence of an intrinsic state is lost. is more sophisticated than MONSTER. However, the
v Thereis along-lastingefforttoconnectthe mean-field particle number plus angular momentum projection im-
Xi and CI techniques. Elliott was the first to point out the plementedin MONSTERandVAMPIR requiresat least
advantage of a deformed intrinsic many body basis and 3-dimensional integration in the axial symmetric case.
r
a developedthe SU(3) Shell Model[6] for sd nuclei. In El- This makes it very difficult to extend these models to
liott’s model, the rotational motion was associated with non-axialcases,wherethe projectionwith 5-dimensional
strictSU(3) symmetry, approximatelyrealizedonly near integration would be needed. MONSTER and VAMPIR
20Ne and 24Mg. This model was limited only to the sd can use realistic Hamiltonians.
shell. Instead of using quasiparticle configurations, as PSM
The Projected Shell Model(PSM) [7] can be consid- and MONSTER/VAMPIR do, the Quantum Monte
ered as a natural extension of the SU(3) shell model to CarloDiagonalization(QMCD)method[13]takesthead-
heavier systems. In this model, the quadrupole force, vantageofthe Hartree-Fock(HF) mean-fieldthatbreaks
that Elliott’s model used, the monopole pairing and the symmetries of Hamiltonian. The mean-field is not
the quadrupole pairing forces were included in the PSM restricted to axial symmetry. QMCD starts from an ap-
Hamiltonian. The deformed intrinsic Nilsson+BCS ba- propriateinitialstate,as inthe shell-modelMonte Carlo
sis are projected onto good angular momentum, and the approach [14], to select an optimal set of basis states,
PSM Hamiltonian is diagonalized in the space spanned and the full Hamiltonian is diagonalized in this basis.
Morebasisstates areiterativelyaddeduntil convergence
is achieved. It is very interesting that only the M-
projection is applied to in the basis states, yet the total
∗Electronicaddress: [e-mail] angularmomentum seems to be fully recoveredfrom the
2
diagonalization when the convergence limit is achieved. One can introduce the deformed single particle (s.p.)
However, the process of selecting the basis states is re- basis, which can be obtained from a constraint HF solu-
portedly very time consuming. tion. Alternatively,onecantakethesingleparticlestates
The Generator Coordinate Method (GCM)[15] is a obtained from the following s.p. Hamiltonian,
standard method to describe collective states that goes
beyond the mean-field. The angular momentum projec-
tiontechniqueisaspecialcaseofGCMwhenthegenera- H =h − 2ǫ ~ω 4πρ2Y (θ)
s.p. sph 2 0 20
torcoordinatesincludetheEulerangles. Inthestandard 3 5
r
GCM, the excited states are constructed in a basis of 4π
HFBvacua,whichdifferinoneorseveralcollectivecoor- +ǫ4~ω0 ρ2Y40(θ). (2)
9
dinates. However,in many cases, it has been found that r
the excitation energies provide by GCM are too high. Here
The GCM wave functions may be more appropriate to
describe more correlations in the ground state, rather hsph = Eic†ici, (3)
than the excited states. However,our investigations (see i
X
below) show that neither the g.s. nor the excited states
is the spherical single particle part of the Hamiltonian
are accurately described by this simple GCM procedure
with the same eigenfunctions as the spherical harmonic
when realistic effective Hamiltonians and HF vacua are
oscillator, but the energy of each orbit is chosen such
used.
that the result from this Hamiltonian is near to the HF
Inthepresentwork,wedevelopedanewmethodcalled
solution; ǫ and ǫ arethe quadrupole and hexadecupole
the Projected Configuration Interaction (PCI), which 2 4
deformation parameters, ρ = mω0r is dimensionless,
uses GCM and PSM techniques. The PCI basis includes ~
and we take [18, 19]
angular momentum projected states generated from a p
class of constraint HF vacua. However, in addition to
~ω =45A1/3−25A2/3. (4)
these GCM-like states PCI includes a large number of 0
low-lying np−nh excitations built on those states. In
The deformed s.p. creation operator is given by the
PCIthedeformedintrinsicstatesareSlaterDeterminants
following transformation:
(SD), andthe particlenumberprojectionisnolongerre-
quired. This method can use the same realistic Hamil- b† = W c†, (5)
tonian as any CI method. This feature makes the direct k ki i
i
comparison between CI and PCI possible. One advan- X
tage of this method is that it keeps the number of basis where the matrix elements W = hb |c i are real in
ki k i
states small, even for cases when the CI dimensions are our calculation. Inserting the reversed transformation
too large for the today’s computers. of Eq.(5)
The paper is organized as follow. Section II presents
the formalismusedbyPCI.The efficiencyofthe angular c† = Wt b† (6)
i ik k
momentum projectionis discussedinsectionIII. Section
k
X
IVanalyzesthechoiceofthePCIbasisandpresentssome
results for the sd and pf nuclei. Section V is devoted to intothemodelHamiltonianEq. (1),weobtainthetrans-
the analysis of the quadrupole moments and E2 transi- formed H in the deformed s.p. basis
tions. Conclusions and outlook are given in section VI.
H = h(1)b†b + h(2) b†b†b b . (7)
ij i j ijkl i j l k
II. THE METHOD OF THE PROJECTED ij i>j,k>l
X X
CONFIGURATION INTERACTION (PCI)
Here h(1) and h(2) are one-body and two-body matrix
ThemodelHamiltonianusedinCIcalculationscanbe elements of H in the deformed basis. The Slater Deter-
written as: minant (SD) built on the deformed single particle states
is given by
H = e c†c + V c†c†c c , (1) |κi≡|s,ǫi≡b† b† ...b† |i, (8)
i i i ijkl i j l k i1 i2 in
i i>j,k>l
X X
wheresreferstotheNilssonconfiguration,indicatingthe
where, c† and c are creation and annihilation operators pattern of the occupied orbits, and ǫ is the deformation
i i
ofthe sphericalharmonicoscillator,e andV areone- determined by ǫ and ǫ . The matrix element hκ|H|κ′i,
i ijkl 2 4
bodyandtwo-bodymatrixelementsthatcanbeobtained can be calculated using Eq.(7).
from effective interaction theory, such as G-Matrix plus Generally, |κi doesn’t have good angular momentum
core polarization [16]; it can be further refined using the I. It is well known that the projection on good angular
experimental data [5, 17]. momentum wouldaddcorrelationsto the wavefunction.
3
Thegeneralformofthenuclearwavefunctionistherefore b†,b, one gets
a linear combination of the projected SDs (PSDs),
hκ|Rˆ(Ω)|κ′i
|Ψσ i= fσ PI |κi, (9) = h|a ...a a Rˆ(Ω)b† b† ...b† |i
IM IKκ MK in i2 i1 j1 j2 jn
XKκ = h|a ...a a
in i2 i1
where Rˆ(Ω)b† Rˆ−1(Ω)Rˆ(Ω)b† Rˆ−1(Ω)...Rˆ(Ω)b† Rˆ−1(Ω)|i
j1 j2 jn
PˆI = 2I+1 dΩDI (Ω)Rˆ(Ω) (10) = h|ain...ai2ai1
MK 8π2 MK
Z R a† R a† ... R a† |i
is the angular momentum projection operator. DMI Kis Xk1 k1j1 k1! Xk2 k2j2 k2! Xkn knjn kn!
the D-function, defined as R R ... R
i1j1 i1j2 i1jn
R R ... R
DI (Ω)=hIM|Rˆ(Ω)|IKi∗, (11) = (cid:12)..i.2.j1.....i2.j.2........i2.j.n.(cid:12). (18)
MK (cid:12) (cid:12)
(cid:12)R R ... R (cid:12)
(cid:12) inj1 inj2 injn(cid:12)
Rˆistherotationoperator,andΩisthesolidangle. Ifone (cid:12) (cid:12)
(cid:12) (cid:12)
keepstheaxialsymmetryinthedeformedbasis,DI in Here(cid:12) (cid:12)
MK
Eq(10) reduces to dIMK(β) = hIM|e−iβJˆy|IKi and the Rˆ(Ω)b†Rˆ−1(Ω)
three dimensional Ω reduces to β. j
The energiesandthe wavefunctions [givenintermsof = WbRˆ(Ω)c†Rˆ−1(Ω)
jl l
the coefficients fσ in Eq.(9)] are obtained by solving
IKκ Xl
the following eigenvalue equation: = WbD∗ (Ω)c†
jl kl k
lk
X
(HKIκ,K′κ′ −EIσNKIκ,K′κ′)fIσKκ′ =0, (12) = WbD∗ (Ω)Wa a†
KX′κ′ i lk jl kl ik! i
X X
whereHKIκ,K′κ′ andNKIκ,K′κ′ arethematrixelementsof = Rija†i. (19)
the Hamiltonian and of the norm, respectively i
X
HKIκ,K′κ′ = hκ|HPKIK′|κ′i, (13) Ntoontewtahvaetftuhnecrtoiotnast.ionTnheevreefromrei,xethsethmeanteruixtroelnemanedntpsrion-
NKIκ,K′κ′ = hκ|PKIK′|κ′i. (14) Eq.(18) can be calculated for neutron and proton sepa-
rately.
The matrix element of HKIκ,K′κ′ and NKIκ,K′κ′ can be ThechoiceofthePCIbasiswillbedescribedinSection
expanded as IV.
hκ|HPKIK′|κ′i
2I+1 III. THE EFFICIENCY OF THE PROJECTION
= 8π2 dΩDKI K′(Ω)hκ|HRˆ(Ω)|κ′i, (15) ON GOOD ANGULAR MOMENTUM
Z
hκ|PKIK′|κ′i Let’sanalyzethequalityofthedeformedsingleparticle
2I+1
= 8π2 dΩDKI K′(Ω)hκ|Rˆ(Ω)|κ′i, (16) sEtqa.t(e3s))foarrteheprUoSpDerlHyaamdijlutostneidanso[2]t.hTathethEeiSenDerwgiieths(tsheee
Z
lowest energy is close to the HF vacuum, as will be dis-
where
cussed in this Section. For sd shell, we use E =−3.9
d5/2
MeV, E = −1.2 MeV and E = 1.6 MeV, and the
hκ|HRˆ(Ω)|κ′i= hκ|H|κ′′ihκ′′|Rˆ(Ω)|κ′i. (17) s1/2 d3/2
corresponding Nilsson levels as functions of quadrupole
Xκ′′ deformation ǫ2(ǫ4 =0) are plotted in Fig.1.
Hereκ′′ hasthesamedeformationasκ,andrunsoverall To find the lowest value Edef of hκ|H|κi, the energy
SDswithnonzerohκ|H|κ′′i. Therefore,themainproblem surface of
is to calculate the matrix element of the rotation Rˆ(Ω)
E (s,ǫ)=hs,ǫ|H|s,ǫi (20)
between different SDs, hκ|Rˆ(Ω)|κ′i. exp
|κi and |κ′i may have different shapes. We denote needs to be calculatedforeachconfigurations. Here, we
a†,a the single particle operators with deformation ǫa use the USD Hamiltonian, andtake 28Si as a case study.
that create the SD |κi, and b†,b, with deformation ǫb, TheSDwithall12particlesoccupyingtheNilssonorbits
that create |κ′i. Expressing hκ|Rˆ(Ω)|κ′i with a†,a and originating from the d spherical
5/2
4

4 [202]3/2[200]1/2
2
eV) 0 d3/2 [202]5/2
M
(
es -2 s1/2 [211]1/2
gi
r
e
n-4
E d [211]3/2
5/2
-6
[2
-8 20
]1
/2
-10
-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6
FIG. 2: (Color online) Energy surfaces of the USD Hamilto-
nian with respect to the unprojected SD(s0) (upper surface)
FIG. 1: The Nilsson levels for sd shell and the projected SD(s0) with I=0 (lower surface), as func-
tions of quadrupole ǫ2 and hexadecupole ǫ4 deformation.
orbit are used in Eq. (20). Such an SD is denoted
by s hereafter. This configuration provides the lowest
0
energy for a large range of deformations and is expected
to describe very well the mean-field minimum, i.e. the

ground state (g.s.), hκ|H|κi at certain deformation(s).
The energy surface of the s configuration as a function
0 -126
of (ǫ ,ǫ ) is shown as the upper surface in Fig. 2. The 28
minim2 u4m energy, Edef = −129.55 MeV at deformation eV)-128 Si
M
ctH(ǫhuF2el,aǫett4neh)deedreqgHfuyF=a(dEvr(auH−lpFu0oe.=l,4e4−−m,6−1o920m.96.1e.f6mn01)t2,,M−[w1e7h8V0i]c.).1h[fT1mi8hs2]u.,vsiBe,srewyasleisdcolreoesvsa,eectrhhyteoecdlctoaahslnee- Energies (--113320 SDpehfoerrmiceadl
I
approximatedsolution to the HF mean-field from a sim- C-134
ple Nilsson Hamiltonian. For other deformed sd nuclei
the situation is similar to 28Si, as shown in Table I. -136
Thegroundstateof28Sishouldhavegoodangularmo-
0 2 4 6 8 10 12
mentum with spin I = 0. We calculate the expectation
value of H with respect to the projected SD: 20
28
hs,ǫ|HPI |s,ǫi Si
E (I,s,ǫ)= KK , (21) 15
exp hs,ǫ|PI |s,ǫi
KK
2
where K is defined byJˆ|s,ǫi=K|s,ǫi. The energysur- J10
z
face of E (I = 0,s ,ǫ) is shown as the lower surface
exp 0
5
in Fig. 2. The minimum value E (I = 0) = −133.64
pj
MeV at (ǫ ,ǫ ) = (−0.40,−0.20), which is about 4
2 4 pj
MeV lower than the HF energy, and 2.30 MeV above 0
the exact ground state energy of the USD Hamiltonian
0 2 4 6 8 10 12
at−135.94MeV.Similarcalculationsforotherdeformed
np-nh Truncation
sd-shell nuclei are reported in Table. I. These results
FIG. 3: The ground state energies (upper panel) and the
indicate that the restoration of the angular momentum
correspondinghJ2ivalues(lowerpanel)of28Sifromspherical
has a significant contribution to the ground state (g.s.)
and deformed CI as functions of different np-nh truncation
correlation energy.
An even stronger argument in favor of imposing good
5
TABLEI:Energies(inMeV)withUSDHamiltonianforsomedeformedsdNuclei. Esph isthesphericalHFenergy,EHF isthe
minimumofthedeformedHFenergy,Edef isthelowestenergyofEq.(20)and(ǫ2,ǫ4)def isthedeformationforEdef,Epj(I =0)
is the lowest energy of Eq.(21) at spin I =0 and (ǫ2,ǫ4)pj is the deformation for Epj(I =0). EFullCI is the exact solution of
theUSD Hamiltonian.
Nucleus Espha EHFa Edef (ǫ2,ǫ4)def Epj(I =0) (ǫ2,ǫ4)pj EFullCI
20Ne −31.79b −36.38 −36.38 (0.60,−0.04) −39.86 (0.56,−0.26) −40.49
24Mg −68.20 −80.17 −79.80 (0.50,0.08) −83.04 (0.58,0.04) −87.09
28Si −126.03 −129.61 −129.55 (−0.44,−0.10) −133.64 (−0.40,−0.20) −135.94
36Ar −222.75 −226.56 −226.56 (−0.32,0.08) −229.78 (−0.20,0.30) −230.51
aDatatakenfromRef.[18].
bThe−25.46MeVinRef. [18]hasbeencorrectedas−31.79MeV.
angular momentum to the wave functions can be found
by comparing the results of CI calculations using spher-

ical and deformed s.p. bases, as shown in Fig. 3. The
deformation for the deformed CI basis is (ǫ ,ǫ ) =
2 4 def
28
(−0.44,−0.10) as shown in Table I. The energies at 0p- -126
Si
0h are the values of E and E shown in the same
sph def
Table I. The E is much lower than E , but hJ2i
def sph
for the deformed SD is as large as 20.35, far away from
-128
hJ2i=0 of the spherical HF state. At the 2p-2h trun-
)
cation level, the spherical CI energy drops very close to V
e
that of the deformed CI, while the latter is struggling M
for recovering the angular momentum, and hJ2i drops ( -130
s
to 11.30. At the 4p-4h truncation level, the spherical e
CI energy is even lower than the deformed ones, yet the rgi
e
angular momentum of the latter hasn’t been completely n -132
E
recovered. The advantage of the deformed mean field is Spherical HF
then completely lost in the CI due to the lack of good Deformed HF
angular momentum. At the 8p-8h truncation level the -134 D=1
deformed CI state completely recovers its angular mo- GCM:D=41
mentum,andtheresultsfromsphericalCIanddeformed
PH:D=40
CI become equivalent. Thus, we conclude that the de-
PH:D=108
-136
formed CI won’t benefit from the deformed mean field
Full CI
unless the angularmomentum is recoveredfromthe out-
set. The PCI method described in Section II maintains 0 2 4 6
the advantagesofthe deformedmeanfieldandofconfig- Spin
uration mixing specific to CI, and could provide further FIG. 4: Calculations of the ground state band in 28Si with
improvement to the Epj(I =0) in Table I. differentbasis. ExceptfortheGCM:D=41case,thedeforma-
tion for all theother cases is (ǫ2 =−0.40,ǫ4 =−0.20)
Todescribethosenon-zerospinstates,one canchange
the spin from 0 to I in Eq. (21), and keep s and ǫ un-
changed. Then one gets a sequence of energies, E (I)
pj
shown as D=1 in Fig. 4. One can see that the values of improved. This can be understood by the large overlap
E (I) have approximately reproduced the I(I +1) fea- betweenthePSDsintheGCMbasis,therefore,theeffec-
pj
ture of a rotational band in the low spin region. The 2 tive dimension of the basis is far below 41. Adding more
MeV gap between the D=1 band and the Full CI band PSDs with new shapes in the basis doesn’t make much
is expected to be reduced by taking advantage of the improvement. Therefore, we considered up to 2p−2h
CI techniques, in which the interaction between differ- excitations of the deformed s configuration used in the
0
entintrinsicconfigurationshasbeenproperlyconsidered. D=1 case, and built a PCI basis by selecting D=40 SDs
Results of the PCI for different bases are also shown (see next section). With this basis we solved the gen-
in Fig. 4. First, we take 41 projected SDs having the eralized eigenvalue problem described in Section II. The
same configuration (s ), but different shapes, as in the results (PH:D=40) in Fig. 4 indicate a significant im-
0
GCM method. The 41 shapes consideredare taken from provement over GCM (GCM:D=41). As the number of
ǫ = −0.6 to 0.6 by a step of 0.03, and ǫ is obtained excited PSDs increases up to 107, we obtain the band
2 4
by minimizing the energy of Eq.(21) with I = 0 at each (labeled by PH:D=108), which is only about 300 keV
ǫ . One can see that the calculated g.s. band is not abovethebandobtainedwithfullCI.Therefore,wecon-
2
6

-68
-116
-126 (2,4)def : D=1
((22,,44))pdje f :: DD==1107 eV) -70 eV) -118
-128 (2,4)pj : D=108 M -72 M -120
V) Full CI ) ( ) (
gies (Me-130 (s,Eexp--7764 (Es,exp--112242
r -126
e
n-132
E -78 24 -128 28
Mg Si
-80 -130
-134
-0.6 -0.3 0.0 0.3 0.6 -0.6 -0.3 0.0 0.3 0.6
28 2 2
-136 Si FIG. 6: (Color online) Eexp(s,ǫ) values in 24Mg and 28Si, as
functions of quadrupole deformation ǫ2. Each curve repre-
sents a Nilsson Configuration s, while ǫ4 was chosen to min-
0 2 4 6
Spin imize Eexp(s,ǫ) for each ǫ2. The absolute minimum for each
s is indicated by a black dot.
FIG. 5: Comparison of the calculated ground state bands in
28Si with deformations (ǫ2,ǫ4)pj = (−0.40,−0.20) [used in
Fig.4] and (ǫ2,ǫ4)def =(−0.44,−0.10), as shown in Table I N is somewhat arbitrary,but it can be increaseduntil
reasonable convergence is achieved. Some examples are
givenbelow. AsapreliminaryapplicationofPCI,wecon-
clude that the PCI method is not only an efficient way
siderthesamesetofSDs|κ ,0iinEq.(22)forallangular
j
ofreducingthe dimensionofthe CI calculations,atleast
momentaI. Thedeformationofeachconfigurationsmay
in deformed nuclei, but can be also very instrumental in
bechosenfromtheminimaofeitherEq. (20)orEq. (21)
making the connection between intrinsic states and lab-
atcertainI. However,thereseemstobenotmuchdiffer-
oratory wave functions.
ence between these two choices, especially if the studied
nucleus is well deformed. As shown in Fig. 5, the PCI
resultsof28Siwithdeformations(ǫ ,ǫ ) and(ǫ ,ǫ ) ,
2 4 pj 2 4 def
IV. CHOICE OF THE PCI BASIS
(seeTable I)arequite closetoeachother. Therefore,for
well deformed nuclei it would be simpler to use Eq. (20)
As already mentioned, one important problem is the for determining the shape of each starting configuration
choice of the PCI basis. It is very difficult to find a s . In the present work we only consider axially sym-
j
set of SDs that would make an efficient PCI basis, due metricshapes. Somelow-lyingcurvesofE (s,ǫ)inEq.
exp
to the complexity of possible structures that exist in the (20) for 24Mg and 28Si are plotted in Fig. 6 as functions
nuclei,suchasthesphericalstates,rotationalstates,var- ofǫ . ǫ waschosenbyminimizingtheE (s,ǫ)foreach
2 4 exp
ious vibrational states, shape coexistence, etc. It would ǫ . Each configuration s corresponds to a curve in this
2 j
behelpfultobeabletodentifythemostrelevantintrinsic figure. The minima are marked by dots.
states ofa nucleus whenwe starta PCI calculation. Un- Inordertodescribethelow-lyingstates,wefirstchoose
fortunately, there is a large number of ways of achieving thoseSDswhoseenergiesareminimaofE (s,ǫ),shown
exp
that goal. Below, we describe some of them. Assuming asdots inFig. 6,andtake themas starting|κ ,0istates
j
that we found these N starting states that we denote as in Eq. (22). Secondly, we try to include CI-like correla-
|κj,0i (j =1,...N), we further consider for each selected tions. Each |κj,0i may have correlations with its 1p-1h
|κj,0iasetofrelativenp-nhSDs|κj,ii,andselectsome and 2p-2h SDs, |κj,ii, through the non-diagonal matrix
oftheminto the basisaccordingto howmucheffectthey elements of H. For each j we include those SDs that
can have on the energy of the state. Therefore, the gen- satisfy the following criterion:
eral structure of the PCI basis is
1
0p−0h, np−nh ∆E = (E0−Ei+ (E0−Ei)2+4|V|2)≥Ecut,(23)
2
|κ ,0i, |κ ,i i,··· ,
1 1 1 p
 |κ2,0i, |κ2,i1i,··· ,. (22) where E0 = hκ,0|H|κ,0i, Ei = hκ,i|H|κ,ii and V =
..................... hκ,0|H|κ,ii. (We skipped the subscript j to keep nota-
|κ ,0i, |κ ,i i,··· tions short.)
N N 1
 
7
24Mg(USD ,N=40,|K| 2) 48Cr(GXPF1A ,N=30,|K| 4)
-55
Full CI, D=28503 Full CI, D=1963461
PCI , D=1749 PCI , D=5973
-60
-85
)
V-65
e )
M V
e
es (-70 (M-90
nergi-75 rgies
E e
n
E
-80 -95
-85
-90 -100
0 2 4 6 8 10 12 0 2 4 6 8 10121416
Spin Spin
FIG.8: Thelowest5energiesateachspinfor48Crcalculated
28
Si(USD,N=60,|K| ) using PCI (open circles) and full CI (filled circles).
Full CI, D=93710
PCI , D=3366
-100
52Fe(GX PF1A) 56Ni(GX PF1A)
-138 -192
) V)-140 Full CI, D=1.1 108 V)-194 Full CI, D=1.1 109
V e PCI , D=7769 e PCI , D=7058
e-110 M-142 M-196
M
( (
( s -144 s -198
ies gie-146 gie-200
g r r
r-120 e-148 e-202
e n n
n E E
E -150 -204
-152 -206
-130 -2 0 2 4 6 8 101214 0 2 4 6 8
Spin Spin
FIG. 9: Calculated yrast band energies using PCI (open cir-
-140 cles) and full CI (filled circles) for 52Fe and 56Ni. The de-
0 2 4 6 8 10 12 14 formed band in 56Ni starting around 5 MeV is also shown
Spin (see text for details).
FIG.7: Energiesfor24Mgand28SicalculatedusingPCI(open
circles)andfullCI(filledcircles). 20energiesforeachspinare
emphasizethatthe PCImethodcombinesthe advantage
presented.
ofusing aset ofs.p. basesofdifferent deformationswith
thatoftheCInp−nhconfigurationmixing. Inm-scheme
CI, for example, one can get the states for all spins us-
InthiswayweconstructedthebasisforthePCIcalcu- ing the lowest M value (0 for even-even cases), although
lationsof24Mgand28Siinthesdshell. Forthe24Mg,we using higher M values could be somewhat more efficient
included N =40 lowest energy |κ ,0i SDs with |K|≤2, for higher spins. In PCI one has to weigh in the advan-
j
and 1749 SDs are selected for the basis when E = 10 tages of using a larger number of np-nh excitations with
cut
keV. For the 28Si case, N = 60, |K| ≤ 4 and E = 10 those of using largerK values for largerspins. Using our
cut
keVselects 3366SDs. The choiceof |K|≤2 seems to be choices for the K values the lowest 20 PCI energies of
arbitrary, and at this point it is related to the number eachspinarecomparedinFig. 7withthe fullCIresults.
of states one wants to include in the basis. One should Almostalllow-lyingenergiesfor eachspinreproducethe
8
full CI results very well. We conclude that the selected
PCI basis has already carriedalmost the whole informa- TABLE II: Quadrupole moments (in efm2) and B(E2,I →
I−2)(ine2fm4)valuesofthestatesinthedeformedbandof
tion of the low-lying states, yet we chose only a small
56Ni calculated with PCI and CI. The energies for I 6=0 are
number of SDs as compared with those used in the full
relativetotheI =0bandhead. TheCIvaluesaretakenfrom
m-scheme CI.
Ref. [17]
The QMCD [13] method also uses a relatively small
number of selected SDs. For a basis of 800 SDs QMCD Spin Energy (MeV) Q(I) B(E2)
found a g.s. energy of −89.91 MeV for 24Mg. In the PCI CI PCI CI PCI CI
0 -200.295 -200.762 0 0
presentcalculation,aftertheorthogonalizationofthe524
2 0.425 0.395 -48.9 -41.6 567.1 413.2
K = 0 SDs, 453 SDs are used to calculate the I=0 en-
4 1.312 1.241 -44.0 -55.2 646.0 598.0
ergies (see Table III). The resulting PCI g.s. energy is
6 2.541 2.448 -70.5 -56.2 708.0 609.3
−89.94 MeV. Furthermore, the QMCD paper only re-
8 4.103 3.991 -74.5 -47.2 938.6 558.4
ports the 6 lowest states with spin up to I = 4, while
our PCI calculation with the selected 1749 SDs can pro-
videtensofexcitedstateswithspinrunningfrom0to12
TABLEIII:PCI Dimensionsforseveralspinscompared with
thatcomparewellwiththefullCIresults. Ourapproach
those of full CI for thenuclei described in this Section.
seemstolargelyextendedtherangeofnuclearstatesthat
can be calculated using a relatively small basis. Nucleus I =0 I =2 I =4 I =6
Theresultsfor28Siaresimilartothoseof24Mg. Com- 24Mg SM 1161 4518 4734 2799
PCI 453 1569 1579 1492
paring with the N = 1 calculation, that has been dis-
28Si SM 3372 13562 15089 9900
cussed in Section III, the dimension is much larger be-
PCI 747 2266 2919 2711
cause one needs to describe a large number of low-lying 48Cr SM 41355 182421 246979 226259
states of the same spin. For I =0 the number of K =0
PCI 1671 3888 5618 5594
|κj,0i SDs is N = 15, and a total of 747 independant 52Fe SM 1.8×106 8.0×106 1.2×107 1.2×107
SDs are selected from the 886 K = 0 SDs. The first PCI 4288 6422 6811 6806
20 0+ states, up to about 18 MeV excitation energy, are 56Ni SM 1.5×107 7.1×107 1.1×108 1.1×108
reasonably close to the full CI results. For I 6= 0 the PCI 2373 3968 5452 6231
situation is very much the same.
Thepf calculationsshowsimilartrends. For48Cr,the
CI m-scheme dimension is 1963461. Full CI calculations be in reasonable good agreement with those of the full
can reproduce the rotational nature of the yrast band CI calculations. The details about the calculation of the
and its backbending [20]. With PCI we choose N = 30, quadrupolemomentsandtheB(E2)transitionsaregiven
|K|≤4, and for Ecut =0.2 keV 5973 SDs were selected. in the next section.
Thelowest5energiesforeachspinarecomparedwiththe The number of SDs selected for the PCI calculation
fullCIresultsinFig. 8. Again,wereachedasatisfactory are shown in the corresponding figures and compared
agreement between PCI and full CI. with the full CI m-scheme dimensions. However, the ef-
52Fe and 56Ni are not good rotors, and the present ficiency of the PCI truncation can also be assessed by
choice of the basis is not so efficient as for 48Cr. How- comparingthe PCI dimensions with the coupled-I CI di-
ever, one can still select a PCI basis that can provide a mensions. In PCI the states used for the diagonalization
reasonable description of the yrast energies when com- of each spin in Eq. (12) are obtained from the selected
pared with the full CI calculations, as shown in Fig.9. PSDs after the latter are orthogonalized and the redun-
The number of deformed SDs used in these calculations dant states are filtered out. The remaining states form
is about 7000, which is just a small fraction of the full the PCI basis for each spin. These dimensions are com-
CI m-scheme dimensionfor 52Fe, and for56Ni(see Table paredin Table III with the full coupled-I CI dimensions.
III). However, the choice of the PCI basis for 52Fe and OnecanseethatPCIcanprovidereasonablegoodresults
56Ni needs to be improved. Details will be presented in using much smaller dimensions than the corresponding
the forthcoming paper. m-scheme and/or coupled-I CI calculations.
For56Niarotationalbandhasbeenidentifiedinexper- Although the PCI calculations could have much
iment[21]thatstartsataround5MeVexcitationenergy. smaller dimensions than those of the corresponding full
The band was successfully described by full CI calcula- CI calculations, the H and N matrix in Eq. (12) are
tions in the pf-shell using the GXPF1A interaction[17]. dense. To get an idea of the workload of a PCI cal-
It was also shown [17] that this band has a 4p-4h char- culation, the time necessary to calculate the H and N
acter, but in order to get a good description of the band matrices for 56Ni at I=0, 2, 4, 6, 8 takes around 8 hours
10p-10hexcitationsfrom(f )16configurationareneces- using one processor, and the calculation of 5 eigenvalues
7/2
sary. In the presentPCI calculationthis rotationalband for each spin takes about 2 hour, when the generalized
has been identified (see Fig. 9) in the calculated excited Lanczosmethodisused. Asacomparison,thefullCIcal-
states by inspecting the quadrupole moments and the culations of the same states using the modern coupled-I
B(E2) transitions (See Table II). The results seem to code NuShellX [22] could take several weeks, when one
9
processor is used. One should also observe that the PCI
computational load is shifted towards the calculation of
the matrix elements, which can be more efficiently par- 24Mg( USD) 28Si(U SD) 48Cr(GX PF1A)
30 50 0
allelized than any large matrix eigenvalue solver. 2)m 20 40 Full CI -10
V. QUADRUPOLE MOMENTS AND BE2 (Q(I) ef-11000 30 PCI -- 3200
20
TRANSITIONS -20 -40
-30 10 -50
2 4 6 81012 2 4 6 81012 0 4 8 12 16
The quadrupole moments and B(E2) transitions are 42) 00 200 400
given by 21efm50 150 330500
Q(I) = 165πhΨIM=I|Qˆ20|ΨIM=Ii 1(B(E2) 0500 1 0500 122 505000
r 100
16π 0 0 50
= hII,20|IIihΨ ||Qˆ ||Ψ i (24) 2 4 6 81012 2 4 6 81012 0 4 8 12 16
I 2 I
r 5 Spin Spin Spin
2I +1 FIG. 10: Calculated quadrupole moments (upperpanel) and
B(E2;I −→I) = f |hΨ ||Qˆ ||Ψ i|2
i f 2I +1 If 2 Ii B(E2)(lowerpanel)fromPCI(opencircles)andfullCI(filled
i circles)for 24Mg, 28Si and 48Cr.
= |hΨ ||Qˆ ||Ψ i|2 (25)
Ii 2 If
where
that of the spherical basis, due to the loss of rotational
Qˆ =e r2Y (θ,φ), (26) invariance that is not recovering fast enough.
λµ n(p) λµ
Therefore,we proposea new strategycalledProjected
with effective charges taken as e = 0.5e, e = 1.5e. Configuration Interaction (PCI) that marries features of
n p
The reduced matrix element in Eq.(24) and (25) can be GCM, by projecting on good angular momentum SDs
expressed in terms of the PCI wave functions built with deformed s.p. orbitals, with CI techniques,
by mixing a large number of appropriate deformed con-
hΨIf||Qˆ2||ΨIii figurations and their np−nh excitations. This ansatz
seems to be very successful in describing not only the
= fIiKκfIfK′κ′RIfK′κ′,IiKκ, (27) yrast band, but also a large class of low-lying non-yrast
Kκ,K′κ′
X states. This method seems to work extremely well for
where deformed nuclei, and it needs further improvements to
the choice of the PCI basis for spherical nuclei.
RIfK′κ′,IiKκ Forthesdandpf-shellnucleithatwestudiedweshow
that the quadrupole moments and the BE(2) transition
= hIiK′−ν,λν|IfK′ihΦκ′|QˆλνPKIi′−ν,K|Φκi(.28) probabilities are very well reproduced, indicating that
ν
X notonlythe energies,butalsothewavefunctionsareac-
Explicit expression for the hΦκ′|QˆλνPKIi′−ν,K|Φκi can be cbuerhaetleplyfudleinscfirinbdeidn.gTrehlaistisoungsgbesettswteheanttthheisinmtreinthsiocdstcaotuelsd,
found in Ref. [7]. Here hI K′−ν,λν|I K′i is a Clebsh-
i f whichoffermorephysicsinsight,andthelaboratorywave
Gordon coefficient.
functions provided by CI.
Using the PCI wave functions for the states shown in
One of the simplifications that makes these calcula-
Fig.7 and Fig.8, we calculated the Q(I) and the B(E2)
values for the yrast band in 24Mg, 28Si and 48Cr. The tions with tens of thousands of projected states possible
is the approximationof the deformedmean-fieldwith an
results are shown in Fig.10. The very good agreement
axially symmetric rotor defined by its quadrupole and
between PCI and full CI indicates that the PCI wave
hexadecupole deformations. This approach seems to de-
functions are almost equivalent to the exact ones.
scribeverywelltheHFmeanfieldforthesdandpf-shell
nuclei. Weplantoextendourmethodtoincludeoctupole
deformation, and to consider triaxial shapes.
VI. CONCLUSIONS AND OUTLOOK
Inthispaperweinvestigatetheadequacyofusingade-
formeds.p. basisforCI calculations. We showthateven Acknowledgments
ifthe startingenergyofthe deformedmean-fieldis lower
than the spherical one, the series of np−nh truncations TheauthorsacknowledgesupportfromtheDOEGrant
in the deformed basis exhibits slower convergence than No. DE-FC02-07ER41457. M.H. acknowledges support
10
from NSF Grant No. PHY-0758099. Z.G. acknowledges 10475115.
theNSFofChinaContractNos. 10775182,10435010and
[1] B. H. Wildenthal, Prog. Part. Nucl. Phys. 11, 5 (1984). [13] M.Honma,T.Mizusaki,andT.Otsuka,Phys.Rev.Lett.
[2] B.A.BrownandB.H.Wildenthal,Ann.Rev.Nucl.Part. 77, 3315 (1996).
Sci. 38, 29 (1988). [14] S. E. Koonin, D. J. Dean, and K. Langanke, Phys. Rep.
[3] A.Poves, and A.P. Zuker,1981a, Phys. Rep. 71, 141. 278, 2(1997).
[4] W. A. Richter, M. J. van der Merwe, R. E. Julies, and [15] D.L.Hill andJ. A.Wheeler, 1953, Phys.Rev.89, 1102.
B. A. Brown, 1991, Nucl.Phys. A 523, 325. [16] M.Hjorth-Jensen,T.T.S.KuoandE.Osnes,Phys.Rep.
[5] M. Honma, T. Otsuka, B. A. Brown, and T. Mizusaki, 261, 125 (1995).
Phys.Rev.C 65, 061301(R)(2002). [17] M. Horoi, B. A. Brown, T. Otsuka, M. Honma, and T.
[6] J. P. Elliott, Proc. R. Soc. London, Ser. A 245, Mizusaki, Phys. Rev.C73 061305(R)(2006).
128,562(1958). [18] Y.Alhassid,G.F.Bertsch,L.FangandB.Sabbey,Phys.
[7] K.Hara and Y.Sun,Int.J. Mod. Phys. E4,637(1995). Rev. C74 034301(2006)
[8] S.G. Nilsson. Dan. Mat. Fys. Medd 29 nr.16(1955) [19] R. Rodrguez-Guzma´n, Y. Alhassid and G.F. Bertsch,
[9] Y.S.ChenandZ.C.Gao,Phys.Rev.C63014314(2000). Phys. Rev.77, 064308 (2008).
[10] J.A. Sheikh and K. Hara, Phys. Rev. Lett. 82, [20] E. Caurier et al., Phys.Rev. Lett.75, 2466 (1995).
3968(1999). [21] D. Rudolph et al., Phys. Rev.Lett. 82, 3763 (1999).
[11] Z. C. Gao, Y. S. Chen, and Y. Sun, Phys. Lett. B634 [22] W.D.M. Rae, NuShellX code 2008,
195(2006) http://knollhouse.org/NuShellX.aspx
[12] K.W. Schmid,Prog. Part. Nucl. Phys. 52, 565(2004)