Correlated-basis description of $\alpha$-cluster and delocalized $0^+$ states in $^{16}$O

0.16 MB·

by  W. Horiuchi

Correlated-basis description of α-cluster and delocalized 0+ states in 16O
W. Horiuchi1 and Y. Suzuki2,3
1Department of Physics, Hokkaido University, Sapporo 060-0810, Japan
2Department of Physics, Niigata University, Niigata 950-2181, Japan
3RIKEN Nishina Center, Wako 351-0198, Japan
A five-body calculation of 12C+n+n+p+p is performed to take a step towards solving an out-
standing problem in nuclear theory: The simultaneous and accurate description of the ground and
first excited 0+ states of 16O. The interactions between the constituent particles are chosen con-
sistently with the energies of bound subsystems, especially 12C+n, 12C+p, and α-particle. The
five-body dynamics is solved with the stochastic variational method on correlated Gaussian basis
functions. No restriction is imposed on the four-nucleon configurations except the Pauli principle
excludingtheoccupiedorbitsin12C.Theenergiesofboththegroundandfirstexcitedstatesof16O
areobtained in excellent agreement with experiment. Analysis of thewavefunctions indicates spa-
4
1 tiallylocalized α-particle-likeclusterstructurefortheexcitedstateandshell-model-likedelocalized
0 structurefor the ground state.
2
PACSnumbers: 21.60.-n,21.10.-k,27.20.+n,21.60.Gx
n
a
J The nucleus 16O is doubly magic and tightly bound. levelsof16Obutthe10.96(0−)statebelowE =15MeV
x
6 Its 0+ ground state is regarded to have predominantly arereproducedbyasemi-microscopic12C+αtwo-cluster
spherical closed shell structure. Contradicting the nu- model where the excitation of 12C and the Pauli princi-
]
h clear shell-model filling of single-particle orbits, how- ple are taken into account. A microscopic version of the
t ever, the first excited state of 16O has positive parity, similar cluster model also succeeded in reproducing the
-
l Jπ = 0+, with the unexpectedly low excitation energy two0+ states[10]. Thesuccessseemstosuggestthatthe
c
u Ex=6.05MeV. Its appearance was therefore mysterious. structure of the 0+2 state is closely related to the tight
n A conventional idea is to explain the excited 0+ state binding of α-particle, that is, the four particles tend to
2
[ with multi-particle multi-hole, especially 4p-4h, excita- formanα-cluster[11]. Itshouldbenotedthatthecluster

tions. The physics mechanism behind such excitations model space includes some deformation and for low HO
1
isbelievedtooriginatefromnucleardeformation[1],and excitations has significant overlap with symplectic basis
v
7 theappearanceofsphericalanddeformedstatesobserved states [12].
5 in several nuclei is called shape coexistence [2, 3]. The Inthis paperwereportafirstconvergedfive-bodycal-
9 essence of the phenomenon lies in that two states with culation of a 12C core plus four (valence) nucleons (4N)
0 identicalquantumnumbersarerealizedatcloseenergies. for the 0+ states of 16O. This is an extension of the
.
1 Understanding the coexistence mechanism can thus be work[9]towardsa moremicroscopicdirectioninthatno
0 a general, interesting problem for other quantum many- preformed α-cluster is assumed. The excitation of 12C
4 body systems as well. is ignored. Regarding the core as 0p closed configura-
3/2
1
Recent theoretical works have focused on the first ex- tion, we impose the Pauli requirement that the valence
:
v cited state of 16O with various approaches. Based on nucleonbefreefromtheoccupiedorbits. Exceptforthat
Xi the harmonic-oscillator (HO) shell-model the possibility the model has no restriction on the valence nucleon or-
of selecting important basis states with symplectic alge- bits, andhence canaccommodatenotonly 0p-0h,2p-2h,
ar bra [2] or the modification of single-particle energies [4] 4p-4h, etc. but also 12C+α configurations. To be real-
has been discussed. Beyond mean-field approaches have istic, both the core-nucleon (CN) and the two-nucleon
been tested in configuration mixing calculation of Slater (NN) interactions are chosen consistently with the en-
determinants [5, 6]. Though the energy gain of the 4p- ergies of relevant subsystems, especially 13C (13N) and
4h state is found to be substantial in the generator co- α-particle. We also treat 16C as the 12C core plus four
ordinate method, its component in the 0+ state is not neutrons to examine how the nn and np interactions af-
2
very large [5]. The basis states in Ref. [6] are gener- fect the structure.
atedbyanimaginary-timeevolutionofstochasticallyse- The five-body system we consider here is described
lectedsingle-particleGausspackets,allowingfor12C+α- with the following Hamiltonian
like configurations, but the excitation energy of the 0+
2 H =T +T +V +V . (1)
state is too high. Large-scale ab initio calculations with v cv v cv
the no-core shell model [7] and the coupled-cluster the- The total kinetic energy consists of the kinetic energy
ory [8] have been performed but the energy of the 0+2 of the 4N (Tv = 4i=1Ti −Tc.m.) relative to their cen-
stateisstillsohighinthecurrentmodelspacethatmore ter of mass (c.m.)Pand the kinetic energy for the rela-
computationaleffortsappeartoberequiredtoreproduce tive motion (T ) between the 4N c.m. and the core.
cv
its excitation energy. The total potential energy also consists of two terms,
4
It was reported about 40 years ago [9] that all T = 0 V = v and V = U . The term v rep-
v i<j ij cv i=1 i ij
P P
2
resents the NN potentials between ith and jth valence ture and correlated motion of the particles, eliminat-
nucleons, and U is the CN potential acting on the ith ing the Pauli-forbidden components, and accurately de-
i
nucleon. TheformeristakenfromthecentralMinnesota scribing the tail of the bound-state wave function in
(MN) potential [13] that reproduces fairly well the bind- the asymptotic region. The trial function is expressed
ing energies of A = 2 − 4 systems. To fine tune the as a combination of correlated Gaussian (CG) basis
binding energy of α-particle, the potential strengths of states [15, 17, 18],
the MN potential are multiplied by 0.9814. The latter
ctoornstaainres scpenectrifiaeldanbdy sspyimn-morebtirtizteedrmWsowodhso-sSeaxfoornm(0fa.6c5- A(cid:26)e−21x˜Ax [YL1(u˜1x)YL2(u˜2x)]LχL ηTMT(cid:27), (4)
(cid:2) (cid:3)
and 1.25×121/3 fm for the diffuseness and radius pa-
with Y (r)=rℓY (rˆ). Here A is the antisymmetrizer for
rameters) and its derivative, respectively. The strength ℓ ℓ
4N, x stands for 4 relative coordinates, (x ,...,x ), A
parameters of each term, Vπ and Vπ, are parity (π) 1 4
c ls isa4×4positive-definite,symmetricmatrix,andu and
dependent and set to reproduce the low-lying states of 1
13C,−4.95(1/2−),−1.86(1/2+),and−1.09MeV(5/2+) u2 are4×1matrices(see[18]fordetail). Theelementsof
from 12C+n threshold: V− =−45.78 MeV, V− =31.08 A,u1,u2 as well as L1,L2,L are continuous and discrete
c ls variational parameters, respectively. The function χ (η)
MeVfm2,andV+ =−57.57MeV,V+ =17.61MeVfm2.
c ls specifies spin (isospin) states of 4N. Possible L values
The Coulomb potential is included.
are 0, 1, and 2. The c.m. motion of the total system
To fulfill the Pauli requirement, a solution Ψ that we
is excluded in Eq. (4), and no spurious c.m. motion is
want to obtain should satisfy the condition
included.
ThepoweroftheCGbasisoftype(4)hasbeendemon-
Γ |Ψi=0 (2)
i
strated by many examples [18–20]. An advantage of the
for i = 1,...,4, where Γ , acting on the ith valence nu- CG is that it keeps its functional form under a linear
i
cleon, is a projector to 0s and 0p HO orbits transformationofthe coordinates[15,17],whichisakey
1/2 3/2
for describing both cluster and delocalized structure in
a unified manner. Each basis element contains so many
Γ= 0s21m 0s21m + 0p32m 0p23m , (3) variational parameters that discretizing them on grids
Xm (cid:12) (cid:11)(cid:10) (cid:12) Xm (cid:12) (cid:11)(cid:10) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) leadstoanenormousdimensionK ofatleast1010. Thus
where m runs over all possible magnetic quantum num- we test a number of candidate bases with the stochas-
bers. TheradialcoordinateoftheHOorbitistakentobe tic variational method [15, 17, 21, 22], choose the best
the CN relative distance vector, and the HO frequency one among them and increase the basis dimension one
~ωissettobe16.0MeV,whichreproducesthesizeofthe by one until a convergence is reached. This procedure
12Cgroundstate. Topracticallysatisfythecondition(2), costs expensively for computer time but no other viable
we follow an orthogonality projection method [14], in methods are at hand to get converged solutions for the
4
which a pseudo potential λ Γ with a large value present problem.
i=1 i
of λ is added to the HamiltPonian and an energy mini- Figure 1 displays the energies of two lowest 0+ states
mization is carried out. By taking λ = 104MeV, our of16Oversusthebasisdimension. Morethan9,500bases
solutioncontainsvanishingly smallPauli-forbiddencom- are combined to reach the convergence. Most bases, es-
ponents of the order of 10−4. pecially upto K=4,000,firstserveto eliminate the for-
The present problem belongs to a class of quan- biddenstates,whichisbecausetheuseoflargeλvalueto
tum few-body problems with orthogonally constraints. ensurethe Pauliprincipleleadsto largepositiveenergies
This type of problem often appears in atomic and sub- at small basis dimension. The valence nucleons tend to
atomic physics whenthe systemcontains composite par- move around the core to gain V and at the same time
cv
ticles [15]. Solving such a problem is quite challenging they want to correlate among them to make use of the
and much effort has been made to eliminate the forbid- attraction of V . Eliminating the forbidden states under
v
denstates. Mostcalculationswiththeorthogonalitycon- such competition is hard. After the ground state energy
straint have so far been limited to three- or four-body converges well, the variational parameters are searched
systems. It is only recent that a five-body calculation is to optimize the first excited state at K=8,000−9,000.
performedfor11 BeinthemodelofΛ+Λ+α+α+n[16], The energy gain after K = 9,500 is very small. Two 0+
ΛΛ
wherethepairwiserelativemotionofα−αandα−ncon- states appear below 12C+α threshold and their energies
tains Pauli-forbidden states. In that hypernuclear case are both remarkably close to experiment. Compared to
three different relative coordinates are involved in the 16O, the convergence for 16C is faster: 7,000 bases are
Pauli constraint, while in our case the Pauli constraint enough to describe the weaker correlated motion of 4N
acts on the four CN coordinates. To our knowledge, we and reproduce the ground state energy very well. The
here present a first converged solution for the core plus obtained energies are listed in Table I. We repeated the
four-nucleon five-body system. calculation with the original MN potential. The binding
We find a solution by a variational method. A trial energy of α-particle increased by about 1MeV, but the
function has to be flexible enough to satisfy several re- energiesofthetwo0+ statesfromthethresholdvirtually
quirements for, e.g., describing different types of struc- remained unchanged.
3
-24 0.6
16O(01+) 16O (01+)
-26 (02+) 0.5 (02+)
eV] -28 12C+α 1] 0.4 16C (01+)
M -m
ergy [ -30 Exp.(02+) [fcv 0.3
n -32 ρ 0.2
E
-34 0.1
Exp.(0+)
1
-36 0
4000 5000 6000 7000 8000 9000 10000 0 2 4 6 8 10 12
Basis dimension 0.8
16O (0+)
0.7 1
oFlIdGf.o1r:t(hCeolgorroounnldinae)ndEnfierrsgtieesxfcriotemd102+C+stna+tens+opf+1p6Oth.reTshhe- 0.6 16C ((002+1+))
12C+αthresholdandexperimentalenergiesareshownbythin -1m] 0.5 α
lines. ρ [fv 00..34
0.2
0.1
Analyzingthe contributionofeachpiece ofthe Hamil-
0
toniantotheenergyisimportanttounderstandthebind- 0 1 2 3 4 5 6 7 8
ing mechanism. As listed in Table I, in 16C the attrac- r [fm]
tion mainly comes from V . In the ground state of 16O,
cv
similarly to 16C, Vcv is still a major source ofthe attrac- FIG. 2: (Color online). Top: Distributions of the relative
tion but V also contributes to the energy significantly, distance between the 12C core and the c.m. of 4N. Bottom:
v
which should not come as a surprise given that the np Density distributions of the valence nucleon measured from
interaction is more attractive than the nn interaction. the4N c.m. Thenucleondensitydistributionofα-particleis
calculated using the MN potential.
SincehV iis aboutahalfofthatofα-particle,the 4N in
v
the ground state of 16O are strongly distorted from the
intrinsic state of α-particle due to both the CN inter-
action and the Pauli constraint. The first excited state
α-particle. In fact the 4N internal energy, hT i+hV i, is
of 16O exhibits an opposite pattern. The contributionof v v
only about a quarter of that of α-particle. The two 0+
V is dominating andclose to that of α-particle. Itlooks
v states of 16O have a different face but coexist closely in
that the first excited state has 12C+α cluster structure
energyduetothecombinedfunctionoftheNN andCN
as shown by the cluster model [9]. We note, however,
interactions.
thatthe 4N inthe 0+ statearenotasstronglyboundas
2 The different structure discussed above is visualized
by comparing the spatial properties of the three states.
Top panel of Fig. 2 shows 4N c.m.-core relative motion
distribution, ρ (r) = hδ(|r −r |−r)i, where r and
TABLE I: Energy contents in MeV and root-mean-square cv v c v
(rms)radiiinfmofthe0+ statesof16Oand16C.Theresults rc are the coordinates of the 4N c.m. and the core,
of α-particle areduetoafour-body calculation with theMN and bottom one the valence nucleon distribution in 4N,
potential. Empiricalrmsradiiaretakenfrom[23,24]for16C ρv(r) = hδ(|r1−rv|−r)i. In case of 16C, ρcv is narrow
and [25] for α and 16O. whereas ρv is spread. Four neutrons move on certain or-
bits with small radii while being apart from each other,
16C (0+) 16O (0+) 16O (0+) α
1 1 2 indicating an independent particle like motion. In con-
E −18.47 −35.47 −29.52 −28.30 trastto16C,the0+ stateof16Oshowsnotonlyextended
2
Eexp. −18.59 −35.46 −29.41 −28.30 ρcv whose highest peak is at about 12C+α touching dis-
hTcvi 17.81 11.55 7.16 – tance(∼4.9fm),butalsosuchnarrowρv thatisverysim-
hVcvi −82.49 −79.55 −29.22 – ilar to the density distribution of α-particle. This sup-
hTvi 53.53 72.93 67.46 56.92 portsthatthe0+2 stateof16Ohasα-cluster-likestructure.
The distribution of the ground state of 16O is somewhat
hVvi −7.32 −40.41 −74.92 −85.22
intermediate between 16C and the 0+ state of 16O.
phr2i 2.62 2.47 3.03 1.43 2
The rms radii are listed in Table I, where, e.g., hr2 i
phr2iexp. 2.70(3), 2.64(5) 2.57(2) – 1.46(1) stands for ∞r2ρ (r)dr. The point matter radicuvs,
0 cv
phrc2vi 1.94 2.54 4.86 – hr2i, is obRtained assuming the rms radius of 12C core
phrv2i 2.88 1.90 1.62 1.43 aps2.33fm[25]. The matterradiiforthe groundstatesof
16C and16O agreewith experiment fairlywell. Support-
4
ingtheα-clusterstructure, hr2 iofthe16O(0+)stateis where x collectivelystands for 3 internalcoordinatesof
cv 2 v
twotimeslargerthanthatopfthe16Ogroundstate,while thevalencenucleonsandthespinandisospincoordinates
hr2i is small and slightly larger than the radius of α- as wellasthe relevantintegrationoverthose coordinates
v
pparticle. The ratio γ = hr2 i/ hr2i may serve as a are abbreviated. First we discuss y(r). As shown in
cv v
measure ofclustering. Thpe largerpγ, the more prominent Fig. 3, the spectroscopicamplitudes for both the ground
the clustering. The γ value is 0.67 for 16C and grows to and first excited states are suppressed and exhibit nodal
1.3and3.0forthegroundandexcitedstatesof16O.The behavior at short distances. This is because Ψ contains
mstaotneospoofle1m6Oatrisix6e.l5e5mfmen2t,, |so(cid:10)m0+2e(cid:12)wrhp2a(cid:12)t0+1la(cid:11)r|g,efrorththaentewxope0r+- nanod0sy1(/r2) acnondta0ipn3s/2atorlbeaitssto4w~iωngHtOo cthoempPoanuelintpsr.inNciepxlet
iment (3.55±0.21fm2 [26]), whi(cid:12)ch (cid:12)may be improved by we discuss ρcv(r). It is clear from Eq. (7) that ρcv(r) is
allowing for the excitation of 12C core. non-negative and its behavior at small r is determined
mainly by those orbits that have relatively small radii
As a measure of finding α-particle as a function of the
distance|rv−rc|,wedrawinFig.312C+αspectroscopic such as 0p1/2,1s1/2, etc. The lower bump of ρcv(r) for
amplitudes for the two 0+ states, the excited state is a consequence of the fact that the
wave function of the excited state is orthogonal to the
y(r)= 1 φ δ(|r −r |−r)Y (r\−r ) Ψ , (5) ground state wave function.
r2D α v c 00 v c (cid:12) E It is useful to expand the obtained wave functions in
where φα is the α-particle wave function obt(cid:12)(cid:12)ained with terms of the HO basis, especially because the 0+2 state
the MN potential. Two curves show a striking differ- challenges no-core shell-model description [7]. Explicit
ence. In the 0+ state, the highest peak is located near expansionisnotfeasiblebut countingthe number ofHO
the surface reg1ion of the core. The spectroscopic factor, quantaiseasy[28]. Figure4plotstheprobabilityofQ~ω
S = ∞[ry(r)]2dr, is small (0.105). Compared to this, components occupied by 4N. The distribution for 16C
thαe amR0plitude of the 0+ state is much larger and has and 16O(0+1) is normal: The largest probability occurs
a peak at 12C+α touch2ing distance. It is by far larger at minimum Q (6 for 16C and 4 for 16O) and decreases
rapidly with increasing Q. The average (M ) and stan-
and longer ranged than that calculated by the deformed Q
dard deviation (σ ) of Q-distribution is 7.0 and 2.1 for
model [27]. The S value is 0.680, in agreement with Q
0.679 of the 12C+ααcluster model [9]. The dimension- 16C, and 5.5 and2.9 for the groundstate of 16O, respec-
less reduced α-width, θ2, at a channel radius r, defined tively. Incontrastwiththisnormalcase,thedistribution
by r3[y(r)]2/3, is a bettαer measure of α clustering than fortheexcitedstateof16Oexhibitsaquitedifferentpat-
tern. Theprobabilityiswidelydistributedandnotnegli-
S . The value is 0.341 at r = 6fm, large enough to be
α
gibleevenbeyondQ=20,withM =14.3andσ =8.3.
compared to that of the negative-parity rotational band Q Q
based on the 9.59(1−) state of 16O [9]. The peak at Q = 10−12 corresponds to 2−4~ω more
2 excitation than 4p-4h. A distribution similar to the 0+
The behavior of ρ (r) and y(r) shown in Figs. 2 and 2
cv
3isunderstoodasfollows. Letting x denote r −r ,we case is also obtained for the Hoyle state [28, 29]. Ap-
4 v c
proachlike Monte Carlo shell model [30] or no-coreshell
may write those functions as
model with symmetry adaptation [31], importance trun-
cation [32], etc. may be able to describe these states in
y(r)∝ drˆ dx φ∗(x )Ψ(x ,r), (6)
Z Z v α v v future but developing an innovative method of calcula-
tion will be indispensable.
ρ (r)∝r2 drˆ dx |Ψ(x ,r)|2, (7)
cv Z Z v v The core excitation is ignored in the present study. If
weallowforthecoreexcitation,wefirstneedtoconstruct
thewavefunctions ofboththe groundandexcitedstates
0.6 of12Cinamicroscopicmodel,andthendefinethePauli-
16O (0+) forbiddenstatesusingthose wavefunctions. Inaddition,
0.5 1
(0+) theCN potentialhastobe determinedconsistentlywith
0.4 2
this extended model. According to the 12C+α cluster
-1/2m] 0.3 model calculation [9], the core excitation can be ignored
y(r) [f 00..12 tinhetheexcfiitrestdecxocmitpedonsetnatteisofco1n6Otaibnuedt ainceitrstaginroaumndousntattoef.
r
0 This is natural because the core excitation occurs more
-0.1 likely as the valence nucleon gets closer to the core and
because the probability of finding the valence nucleon
-0.2
0 2 4 6 8 10 12 14 16 nearthecoreisexpectedtobemuchlargerintheground
r [fm] state. If that is the case, in the ground state the energy
lossdue to the coreexcitationhas to be compensatedby
FIG. 3: (Color online). 12C+α spectroscopic amplitudes for some additional attraction of the CN potential. Thus
theground and first excited 0+ states of 16O. theconsequenceofthecoreexcitationwillresultinshift-
ing the ground state of 16O towards more delocalized
5
80 spaceislargeenoughtodescribethemulti-particlemulti-
hole excitations, the shape coexistence and the 12C+α
16 + 16 +
60 C (01) O (01) clustering. Once the potentials betweenthe particlesare
chosen to reproduce the energies of the relevant subsys-
40 tems, neither adjustable parameter nor a bias for the
%] 20 existence of α-cluster is necessary. The converged so-
y [ lutions for the five-body Scho¨dinger equation with the
abilit 0 0 10 20 30 0 10 20 30 PtioanualilcmoentshtroadinotnatrheeocbortraeinlaetdedwGitahustshieanstboacshisasftuinccvtiaornias-.
ob 15 Thegroundstateof16Ctreatedasthesystemof12Cplus
r
P 16 + four neutrons is also examined.
10 O (0 )
2
The energies of the ground and first excited states of

5 16O as well as the ground state of 16C are all obtained
inverygoodagreementwithexperiment. Tounderstand
0 the coexistence mechanism for the two 0+ states of 16O,
0 10 20 30 40 50 60 weanalyzetheroleofboththecore-nucleonandnucleon-
Q nucleonpotentials. Inthe0+ statethefournucleonscon-
2
tributetogainingenergysignificantly,suggestingthefor-
FIG.4: (Coloronline). Decompositionofthe0+ statesof16O
mation of α-cluster. The different character of the two
and 16C into Q~ω components with ~ω=16.0MeV.
states is exhibited by comparing density distributions,
particle distances, 12C+α spectroscopic amplitudes, and
probability distributions of harmonic-oscillator quanta.
They allexhibit somethinglike a phase transitionoccur-
structure. Comparedtothe casewithnocoreexcitation,
ring between delocalized and cluster structure.
we speculate that the peak position of the spectroscopic
amplitude gets closer to the core and the distribution of Asfurtherinvestigation,itisinterestingtoincludethe
HOquantaisconcentratedmoreinlowoscillatorquanta. effect of 12C coreexcitationon the spectrum of16O.Ex-
This possible change of the ground state structure also tending the present approach to heavier nuclei such as
helps to reduce the monopole strength. Further study 20Ne, 40Ca, 44,52Ti, and212Powillalsobe interestingfor
alongthis directionis certainly importantfora more de- exploringapossibleuniversalroleofα-likecorrelationin
tailed description of the shape coexistence in 16O. shape coexistence and α-decay with an increasing mass
We have attempted to describe simultaneously both number of the core nucleus.
the ground and first excited 0+ states of 16O in the five- This work was supported in part by JSPS KAKENHI
body approach of 12C plus four nucleons. The model Grant Numbers 21540261,24540261,and 25800121.
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