Four-body correlations in nuclei

0.11 MB·English

by  M. Sambataro

Four-body correlations in nuclei
M. Sambataroa and N. Sandulescub
aIstituto Nazionale di Fisica Nucleare - Sezione di Catania, Via S. Sofia 64, I-95123 Catania, Italy
bNational Institute of Physics and Nuclear Engineering,
P.O. Box MG-6, Magurele, Bucharest, Romania
Low-energyspectraof4nnucleiaredescribedwithhighaccuracyintermsoffour-bodycorrelated
structures (“quartets”). The states of all N ≥Z nuclei belonging to the A=24 isobaric chain are
representedasasuperposition oftwo-quartetstates,withquartetsbeingcharacterizedbyisospinT
and angular momentum J. These quartets are assumed to be those describing the lowest states in
20Ne(Tz=0),20F(Tz=1) and20O(Tz=2). Wefindthatthespectrumof theself-conjugate nucleus
24MgcanbewellreproducedintermsofT=0quartetsonlyandthat,amongthese,theJ=0quartet
5 plays by far the leading role in the structure of the ground state. The same conclusion is drawn
1 in the case of the three-quartet N = Z nucleus 28Si. As an application of the quartet formalism
0 to nuclei not confined to the sd shell, we provide a description of the low-lying spectrum of the
2 proton-rich92Pd. Theresultsachievedindicatethat,in4nnuclei,four-bodydegreesoffreedomare
more important and more general than usually expected.
n
a
PACSnumbers: 21.10.-k,21.60.Cs,21.60.Gx
J

6
1
Innuclear physics,four-bodycorrelatedstructuresare Microscopic quartet models have a long history in nu-

usually associated with α-clustering. α-clustering is clear structure. They have been employed to treat the
]
h known to be a relevant phenomenon in light N =Z nu- proton-neutron (pn) interaction, in particular the pn
t
- clei especially at excitation energies close to the α emis- pairing, and to investigate the quartet condensation in
l
c sion threshold. A common theoretical approach to α- N = Z nuclei [3–8]. However, their complexity has al-
u clustering is represented by the α-cluster model [1]. Ac- ways represented a hindrance to their development. Re-
n cording to this model, the nucleus consists of a N = Z cently,asimpleapproachhasbeenproposedwhichisable
[
core to which some α clusters are appended. These clus- to describe accurately the ground state of the isovector

1 ters are tightly bound and spatially localized structures pairingHamiltonianas acondensate ofquartets builtby
v of two neutrons and two protons. The α-cluster model two neutrons and two protons coupled to total isospin
7
exhibitsastrikingcontrastwiththestandardshellmodel T = 0 and, for spherically symmetric mean fields, to
1
picture in which protons and neutrons are described in- total angular momentum J=0 [9]. This quartet model
0
4 steadasweaklyinteractingquasiparticlesinamean-field. conserves exactly the particle number, the isospin and
0 However, these two pictures are expected to coexist at the Pauli principle and can be applied for any number
. low excitation energies where, due to the Pauli block- of quartets. Later on a more general description of the
1
0 ing which acts stronger than in the states close to the α ground state of N = Z nuclei as a product of distinct
5 emission threshold (e.g., see the case of the Hoyle state quartets has been proposed and applied to a description
1 [2]),theα-clusteringisexpectedtomanifestitselfmainly of both isovector (T = 1, J = 0) and isoscalar (T = 0,
:
v as four-body correlations in the configuration space. It J =1) pairing correlations [10, 11].
i is thus commonly supposed that correlated four-body
X Of course, pairing is only a part (although a crucial
structures (“quartets”) play a major role in the ground
r one)ofthenuclearinteraction. Itisreasonabletoexpect
a and excited sates of N = Z nuclei. However, to our
thatarealisticdescriptionofself-conjugatenucleishould
knowledge, this supposition has never been supported
involve not only T =0, J =0 quartets. In the following
by compelling calculations. In this letter, by using a
we will show how the approach of Refs. [10, 11] has
simple microscopic quartet model, we will show how the
been extended to include quartets with arbitrary values
low-energy spectra of 4n nuclei can be indeed described
of isospin and angular momentum and to treat realistic
in terms of quartets with an accuracy comparable with
interactions of shell model type.
state-of-the-art shell model calculations. We shall prove
thatthisisthecasenotonlyforself-conjugatenucleibut One of the motivations of the present work has been
also for 4n nuclei with N 6= Z. For the latter nuclei the thatofidentifying thequartetswhichcontributemostto
low-lying states will be expressed in terms of quartets the structure of the low-lyingstates in even-evenN =Z
built not only by two protons and two neutrons but also nuclei and, in particular, of investigating the role played
by one protonandthree neutrons aswellas by four neu- by T = 0, J = 0 quartets in the ground state of these
trons. This fact indicates that the four-body degrees of nuclei. Our analysis will be mainly confined to sd-shell
freedom are important also for configurations which are nuclei. WewillbeginbyfixingasetofT =0,1,2quartets
different from the α-like ones. and then use these quartets for a description of the low-
lying states of 24Mg. To such a purpose, we will carry
2
out configurationinteraction calculations in spaces built 16O andthe nuclei which havetherefore been considered
in terms of these quartets. By progressively restricting for the definition of the quartets are 20Ne , 20F and
10 10 9 11
thissetofquartetswewillidentifythemostrelevantones 20O . The lowest states of these nuclei are character-
8 12
forthe low-lyingstatesofthisnucleusand,inparticular, ized by T =0, 1 and 2, respectively. Each of these states
for the ground state. A confirmation of the conclusions therefore identifies a quartet with given T,J (and a pro-
reached for 24Mg will then be searched in the case of jectionT =(N−Z)/2). Wehavecarriedoutshellmodel
z
28Si. HavingdefinedasetofT =0,1,2quartets,alarge calculations for these three nuclei and selected the quar-
variety of nuclear systems become, in principle, within tetsassociatedwiththeloweststates. Moreprecisely,we
reach in addition to self-conjugate nuclei. These include have retained the lowest 6 states in 20Ne (0 ≤J ≤6), 5
both even-even and odd-odd 4n nuclei. In the second statesin20F(1≤J ≤5)and9statesin20O(0≤J ≤4).
part of the paper, we will provide a description of the For all calculations in the sd shell we have employed the
whole isobaric chain of A=24 nuclei with N >Z in the USDBinteraction[12]whichgivesaprettygooddescrip-
formalismofthesequartets. Finally,asanapplicationof tion of the above nuclei. The quartets so selected have
the formalism to nuclei not confined to the sd shell, we then been employed in all the configuration interaction
will discuss the case of the proton-rich 92Pd. calculations relative to the sd shell.
Westartbybrieflyillustratingourapproach. Wework
inasphericallysymmetricmeanfieldandlabelthesingle-
particle states by i≡{n ,l ,j }, where the standard no-
i i i
tation for the orbital quantum numbers is used. The
quartet creation operator is defined as
10 24Mg
Q+α,JM,TTz = i1jX1J1T1i2jX2J2T2Ci(1αj)1J1T1,i2j2J2T2 51++62++ 12 12 10++625+++
×h[a+i1a+j1]J1T1[a+i2a+j2]J2T2iJMTTz, MeV) 5 4230+++++ 4234++++
4
where a+i creates a fermion in the single particle state i E ( +
andJ(T)andM(T )denote,respectively,thetotalangu- + 2
z 2
lar momentum(isospin) and the relative projections. No
0 0+ 0+
restrictionsonthe intermediatecouplingsJ1T1 andJ2T2 55.69 55.69 55.17 54.91
are introduced in the calculations. In order to generate EXP SM QM QM(T=0)
thespectraof4nnucleiweperformconfigurationinterac-
tion calculations in spaces built in terms of selected sets
of the above quartets. For these calculations we adopt
a static definition of the quartets, i.e. after selecting
the quartets (according to a criterion specified below) FIG. 1: The low-energy spectrum of 24Mg obtained in
we keep them “frozen” in all calculations. This way of the quartet model (QM) compared to the experimental data
constructing the quartets is substantially different from (EXP)andtotheshellmodel(SM)results. Thespectrumin-
that of Refs.[10, 11] where we have instead adopted a dicated as QM(T=0) corresponds to a calculation done only
dynamical definitionofthe T =0, J =0 quartets. That with T=0 quartets. In this figure as well as in all the re-
maining ones relative to nuclei in the sd shell, experimental
is to say, in these works,quartets have been constructed
levelsandSMresultshavebeenextractedfromRef. [13]. The
independently for each nucleus (via an iterative varia-
number below each spectrum gives the ground state correla-
tionalprocedure)andnoconnectionwhatsoevercouldbe
tion energy, namely the difference between the total ground
found among quartets of different nuclei. In the present state energy and theenergy in the absence of interaction.
work, owing to the fact that various T,J quartets are
being considered, this way of proceeding would become
prohibitively cumbersome. However, for comparison, we We start our analysis by examining the self-conjugate
willalsocarryoutananalysisofthegroundstatesofself- nucleus 24Mg. In Fig. 1, we compare the experimental
conjugate nuclei by applying the same variationalproce- spectrum of this nucleus with that resulting from a shell
dure of Refs. [10, 11]. model (SM) calculationand with the spectrum obtained
The criterion adopted for the selection of the quartets inthequartetmodel(QM)whenalltheselectedT=0,1,
has been that of choosing, as representative of the quar- 2quartets aretakeninto account. The quartetapproach
tets with a given isospin T, those describing the lowest isseentoreproducewellboththeSMgroundstatecorre-
levels with that isospin in nuclei with four active par- lation energy and the SM excited states up to an energy
ticles outside the inert core of reference. For applica- of about 9 MeV. The SM is in turn able to fit well the
tions within the sd shell, the inert core is representedby experimental levels.
3
HavingverifiedthattheselectedsetsofT=0,1,2quar- ergy. The best description of the ground state of 24Mg
tets aresufficient to describe the low-energyspectrumof as a product of two T =0, J =0 quartets is expected to
24Mg,itbecomesofinteresttoinvestigatetheroleofthe be obtained by letting these quartets to be distinct from
different quartets. We begin by focusing on isospin. In one another and by constructing them variationally. An
Fig. 1, on the right hand side, we show the theoretical effective way to do that is provided by the procedure
spectrum obtained in the quartet formalism when only adopted in Refs. [10, 11]. By applying this procedure
T=0 quartets are retained. One can see that this spec- one is able to generate a ground state correlationenergy
trum does not exhibit relevant differences with respect which exhausts 95.6% of the exact value. This calcula-
to the full QM calculation. This provides a clear evi- tion therefore further enhances the role of T = 0, J = 0
dence of the marginal role played by the T=1 and T=2 quartets in this ground state and also suggests that the
quartets in the structure of these states. As a next step, minimum set of T=0, J=0,2,4 quartets emerged from
we concentrate on the ground state by employing only the aboveanalysismightbe further reducedwith amore
T =0 quartets. In Fig. 2, we show how the error in the effective choice of the quartets.
correlationenergyofthisstate,relativetotheSMresult, We havesearchedforaconfirmationofthe resultsjust
varies by reducing, one quartet at a time and starting discussedbyexamininganotherself-conjugatenucleusin
from the highest one in energy, the set of T=0 quartets. the sd shell: 28Si. We find that for this isotope the set
The correlation energy remains basically unchanged up of T=0, J=0,2,4 quartets already exhaust almost 99%
to the point where only the lowest J=0,2,4 quartets are of the correlation energy. Also in this case, the T =
left. From this point on further reductions in the set 0, J = 0 quartet is found to play a leading role being
of quartets lead to significant variations in the energy. able to account by itself for 93.4% of the total energy.
Thus, these calculations indicate that the T=0 quartets This percentage increases to 98.2% when the quartets
with J=0,2,4 play a major role in the ground state of are determined variationally.
24Mg. Among these quartets, the T=0, J=0 quartet is So far we have examined only self-conjugate nuclei.
by far the one which contributes most to the correlation The rangeofnucleiwhichareaccessibleintermsofaset
energy since, as it can be seen in Fig. 2, an approxima- ofT=0,1,2quartetsis,however,muchbroader. Limiting
tionintermsofonlythisquartetaccountsforabout94% ourselves to the case of 8 active particles (two quartets)
of the total energy. outside the 16O core, the whole isobaric chain of nuclei
withA=24iswithinreach. Wehaveinvestigatedtowhat
extent the spectra of these nuclei can be described in
terms ofthe fullset ofT=0,1,2quartets employedin the
case of 24Mg. The results are shown in Fig. 3 where
8 we compare the low-energy spectra of 24Na , 24Ne ,
11 13 10 14
24F and 24O obtained in the quartet model with the
9 15 8 16
1: T=0, J=0(2), 2(2), 4, 6 SM results and with the experimental data. As one can
2: T=0, J=0(2), 2(2), 4
6 see, for all these nuclei the quartet formalism generates
3: T=0, J=0(2), 2, 4
4: T=0, J=0, 2, 4 spectra which agree well with the SM ones. It is worth
) 5: T=0, J=0, 2 stressingthatfor the nucleishowninFig. 3 the quartets
% 6: T=0, J=0
( 4 are built not only by two protons and two neutrons, as
Ε
∆ in the case of N =Z nuclei, but also by one proton and
threeneutronsandbyfourneutrons. Thecaseofquartets
built by four like-particles had been already discussed
2
24Mg in Ref. [14] in relation with a treatment of the pairing
Hamiltonian.
In order to verify that the results presented so far are
0
1 2 3 4 5 6 not specific to sd-shell nuclei, we have carriedout a sim-
ilar analysis for the proton-rich nucleus 92Pd . Calcu-
46 46
lations have been done in a space spanned by the p ,
1/2
g orbitals using the F-FIT interaction by Johnstone
FIG. 2: Relative errors (with respect to the SM value) in 9/2
thegroundstatecorrelation energyof24Mgcalculated inthe and Skouras [15]. In Fig. 4, one sees the low-lying yrast
QM approach with the quartetsindicated in thefigure. spectrum that is obtained in the QM approach by em-
ploying only T = 0 quartets. These quartets are those
associatedwith the lowestlevels of 96Cd (7 levels with
48 48
In this connection, a few considerations are needed. 0≤ J ≤ 8) and include also a negative parity level with
The static formulation of the quartets adopted in this J =5. TheQMyrastspectrumisseentoagreewellwith
workisbyconstructionnotthemostperformingonesince the experimental one [16] as well as with the SM spec-
itdoesnotguaranteeaminimuminthe groundstate en- trum generatedwithin the same model space by Herndl-
4
24Na 24Ne
11 13 10 14
++
E (MeV)231 123++343+5++2+3+++ 215343++3+++3+++++2+ E (MeV)5 20++422+++ 343+++ 32423+++4+++02++
21++ 21++
0 4+ 4+ 0 0+ 0+
46.23 46.29 45.44 40.26 40.32 40.05
EXP SM QM EXP SM QM
24F 24O
555+ 9 15 4++ 10 2+ 8 16 2+
3 11+322432++++ 023+++15+ 243+++0+ 0+324+++
V) 00+ 4+ V) 1+ 1+
Me 1++ 4 1+ Me 5 2+ 2+
( (
E E
2+ 2+
0 3+ 3+ + +
0 0 0
23.56 23.47 23.35 9.93 9.82 9.82
EXP SM QM EXP SM QM
FIG. 3: The spectra of A=24 nuclei generated by the QM approach compared with the experimental data and the SM
predictions. See thecaption of Fig. 1 for further details.
Brown[17]. Asinthecasesofthesdshellnucleianalyzed the valence shell. For all these nuclei, as well as for the
above, the T =0, J =0 quartet is found to play a lead- proton-richnucleus92Pd,thequartetformalismhaspro-
ing role in the ground state, accounting for almost 99% videdadescriptionofthelow-energyspectracomparable
of the correlation energy. More details about the struc- in accuracy with that of shell model calculations. This
ture of 92Pd in the quartet model will be presented in a fact confirms the importance of quartet degrees of free-
separate study. dom in any type of 4n nuclei and validates the present
quartet formalism as the appropriate tool for treating
By summarizing, in this work we have presented an
them.
analysisof4nnucleiinaformalismofquartets,i.e.,four-
body correlatedstructures characterizedby total isospin Asaconcludingremark,wenoticethatthedescription
T andtotalangularmomentumJ. Theanalysishascon- of 4n self-conjugate nuclei in terms of T=0 quartets of
cerned the whole isobaric chain of A = 24 nuclei, there- lowangularmomenta(J=0,2,4)whichhasemergedfrom
fore ranging from even-evento odd-odd nuclei as well as thepresentanalysisexhibitsastrikinganalogywiththat
fromself-conjugatenucleitonucleiwithonlyneutronsin of nuclear collective spectra in a formalism of S (J=0),
5
AcknowledgmentsThisworkwassupportedbytheRo-
manianMinistryofEducationandResearchthroughthe
grant Idei nr 57.
92Pd
46 46
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