Cluster transformation coefficients in many-body nuclear physics

ELSEVIER

Nuclear Physics A738 (2004)21 6-220

www.elsevier.com/locate/npe

Cluster transformation coefficients in many-body nuclear physics

M.Tomaselli L.C. Liub S. Fritzsche T. Kuhl D. Ursescu GSI Gesellschaft fiir Schwerionenforschung. D-64291 Darmstadt,Gerrnan bT-Division, Los Alamos National Laboratory, Los Alamos, Nhl 87545.USA Institute of Physics, Kassel University, D-34132 Kassel. Germany Energies, electromagnetic moments and transitions in light, nuclei are calculat,ed in t,he microscopic dynamic-correlation model (DCM) which is based on large correhted basis of Slaters determinants charact,erized by an increasing numbcr of core excited states. Microscopic calculations are performed for the magnetic moments and transit,ions of Be and I3C. The magnetic moments of 6Li, 7Li, and Li are also calculated. Further, rcsult,s ~ Be arc reported. The overall results obt,ained for matt,er and charge ra.dii of 7 B and obt#ainedfor these light odd- and even-nuclei show that, t,he correct treatment of the Pauli principle and the diagonalizat,ion of large dimensional spaces are not compat,iblewith the simple picture generatled by cluster models. 1. I n t r o d u c t i o n Large-basis no-core shell-model calculatmionshave recently been performed in Rcf. [l] and references therein quot,ed. In thesc calculations all nucleon are active, which simplifies t,he effect,iveinteraction to be used as no hole states are present. Additionally to these nocore calculations, unstable nuclei have been successfully described by thc a,ntisymmetrized molecular dynamics (AMD) [a]. AMD has been proved t o be a useful t,heoretical approach for investigations of nuclear structure in light unstable anti stable nuclei [3]. In this work we apply the model of Ref. [4], which is based on non linear equations of rriotion easy solvable in terms of dynamic, linearization approximat,ions and cluster t,raiisforniat,ion coefficients, to investigate the electromagnetic proprieties of light nuclei. The dynanic linearization [5] is used to generate model eigenva,lue equations that are t,heii solved selfconsistently. The calculated ground-, excited-stmatesenergies and moments are t,herefore not depending from t,he original choice of the t,wo body matrix elements. 2. Nonlinear c o m m u t a t o r chains and eigenvalue e q u a t i o n s The nonlinear commuta.tor chain is an extension of the Heisenbergs cquat,ion. It allows us t o address the situat,iori where valence clusters a,nd core clusters are a,lmost,energetically degenerated and may, therefore, coexist,. We int,roduce valence systems formed of either 0375-94746- see front matter R3 2004 Elsevier I3.V All rights reserved doi:10. IOl6/j.nuclphysa.2004.04.034

M.Tomaselli et al. /Nuclear Physics A738 (2004) 216-220

217

neutrons or protons and define the valence states as:

where n = 1 , 2 and J is the tot,al spin. The Nc\, are the normalization constants, X :>% the mode amplitudes. and a, denotes the other quantum numbers of the valence particles. The creation operator is defined by

for one valence particle (fermion) and one valence pair (boson) respect.ively. Hence, in Eq. (a), J, = j 1 for one particle, J, = J for two particles. In Shell-Model calculations the residual interaction between the valence particles excites the valence pairs to higher single particle stat,es, leaving the vacuum in the J = 0 ground state. In the dynamiccorrelation model DCM [6] and the boson dynamic-correlation model BDCM [7] the core excit,ation is included t,hrough coupling the valence fermionic/bosonic states, Eqs. (2) to intrinsic bosonic stat,es corresponding to particle-hole excitations of the nuclear core. In this paper, we consider t,he following mixed-mode fermionic/bosonic states: a) valence fermionic/bosonic states coupled to the dynamic particle-hole states of normal parity; b) valence fermionic/bosonic stat,es coupled to the dynamic particle-hole states of nonnormal parity. The particle-hole coupling is implemented through the realistic two-body potential which has two parts: a central part and a tensor part. The tensor component of the realistic two-body potential shapes the many-body Hamiltonian and in particular its long tail acting between t,he valence fermions/bosons and causing simultaneously the excitation of the particles to high shell-model states and the deformation of the nuclear core. The mixed-mode {1- 2 ) particles{ 1- 2) bosons st,ates,denoted IF;), can therefore be expanded as follows: r

contain consequently the hole creators b:. The basic dynamic equations of the models are the commutator equations that involve the nuclear many-body Hamiltonian H and the

M.Tonzaselli et al. /Nuclear Physics A738 (2004) 216-220

218

operators AA(n = 1;2). After a lengthy but elementary algebra, we obtain the following results: (a) Commutator equation for { n = 1,2} states:

(b) and Commutator equation for { n = 1,2; n = l} stat,es

41

[H.AL+,&,+l,J,J,+I/;

C

== &+I

+ Pn+v J:,

(&+I!

JA

(a,+i/

Jn+i

; J ) IH IA:+

(a+JA 1

1)

J:+1

;J))

J:,J;+l,

c Jk+ Jk,,, 1,

J) ) IHIA:,,@n+Y

x4+1@n+I/JAJA+1/;

(A,+l, 1

(an+lJnJn+l/; J

XA,+y(Pn+,JA t

JA+1,J:+2!;

JX+l, JA+d

J))

J).

The commutator equations are used t o obtain the solutions of DCM and BDCM with use of the Equations of Motion (EOM) method [4]. The solutions of the linearized commutator equations, can, therefore, be regarded as eigenvalues of a model fermionic/bosonic Hamiltonian which we solve selfconsistently. The input to solving the commutator equations are the matrix elements of the n - particle configuration mixing wave functions (CMWFs) [6]. The latter can be easily calculated by use of the cluster-factorization method of Ref. [4]. Table 1 Experiment,al and calculated energies and magnetic moments of Be and 13C Be

13 Eexp.

JT

3-

; ;

Eth.

I

0.0

I

c

pexp. I

0.0 1-1.181-1.18

I

f-

I2 A

1

0.702 0.703

0.0 3.07

;+

1.25

I

I

0.0 3.089

I

-

-0.29

-

-1.74 4.05 1.03

I

I 7.68 I 7.70 I

I

I

I

3. Results and discussion Calculations performed for light nuclei are compared with t,he experimental values of Refs. [8-111. Application of the DCM and BDCM to nuclear structure calculations requires defining the CMWF basis, determining the single-particle energies, and choosing

M. Tomaselli et al. /Nuclear Physics A738 (2004) 216-220

219

nuclei have one valence part,icle nuclear two-body potcnt,ials. The 7.’Be, 7,gLi:and which can be in the lowest p 3 / 2 , ~ 1 1 / 2orbitals as well as in higher orbitals. In addit,ion. we also consider the {lp-lh}, {2p-2h}, (3p-3h) and {4p-4h} excitations of the core. Core excit.at,ion states of positive and negative parity have been included in the calculation. Accordingly, single particle states with unnormal parity have also been included in the model space. The single-particle (s.p.) levels used t,o construct the full {3p-lh}, (4p2h):

3-

It 2

32

L+ 2

EEh.e2 f m 2

E y p ’ e 2f S m 2

0.025 ~

,

P

+ 0.01 .

~

8.8i1.5 Pth. ,,,w 6Li .82204(6) .82 7Li 3.2564 3.25 ’Li 3.43(6) 3.44

2

,023

I

f1 \4~h . p22 fm2

4.39

(5p-3h}, and (6p-41.1) model-basis with energies < 2iiw are considered. The single-part,icle energies of these levels were taken from the experimental level spectra of t’heneighboring nuclei. In our model for the 73gBe, 7.’Li, and I3C: we have applied the linearizat’ion approxiination at the order of (3p-2h) so that the {p} and {2p-111) are the effective valence states. We emphasize that, even in the presence of this linearizat>ion,the dimension of our model space is still much bigger than those used for shell-model calculations. The inclusion of core excitat,ion enlarges also the dimension of the model space in BDChI we have introduced to describe 6Li. More specifically, wc have included (3p-lh) states which contain the coupling Mween all t h p. with energies < 2hw. As a result, in our BDCM calculations the dinicrision is of order 500 when an encrgy cut-off at 90 MeV is introduced. On the other hand, the highest dimension of the shell-model space is of order 7. Our largedimension space arises, therefore, from a large number of s.p. levels considered its well as from t8akinginto account the interaction between the valence neutrons and the nuclear core. This enlarged model space has import,ant effects on the calculated energy spectra. For the particle-particle interaction, we use the Yale hard-core potential [la]. For the particle-hole interaction, we use the phenomenological pot,ential of Ref. [13]. Preliminary results. calculated wit,hout quenching of gyromagnetic factors and effective charges. are given in Tables 1,2,3. Core excitation mechanism is giving significant contributions to t8he calculated moments and t’ransitions.

220

M. Tomaselli et al./Nuclear Physics A738 (2004) 216 220

Table 3 Matter- and folded charge-radii of 7Be and ’Be Rth.

th.

charge C&e7’ [I4] R%"ge[14] 7Be 2.38 fm 2.39 fm 2.31(2) fm ? fm ’Be 2.46 fm 2.48 fin 2.38ilj fin 2.47[1) fm Rmatter

1

I

I

1

1

I

REFERENCES 1. P. Kavratil and B.R. Barrett, Phys. Rev. C57, 3119 (1998). 2. Y . Kanada-En’yo and H. Horiuchi, Phys. Rev. C55. 2860 (1997). 3. Y. Kanada-En’yo, A. Ono, and H. Horiuchi, Phys. Rev. C52, 628 (1995); Y. KanadaEn’yo and H. Horiuchi, Phys. Rev. C52, 647 (1995). 4. M. Tomaselli, L.C. Liu et al., Journal of Optics B5, 395 (2003). 5. G.E. Brown, Unzjied Theory of Nuclear Models, North-Holland, Amsterdam 1964: A.M. Lane, Nuclear Theory, W.A. Benjamin Inc., New York (1964). 6. M. Tomaselli, Ann. Phys. (X.Y.)205, 362 (1991). 7. A t . Tomaselli et al.; GSI Preprint 2002-28. 8. F. Ajzenberg-Selove, Sucl. Phys. A413, 1 (1984). A433. 1 (1985), A449, 1 (1986), A460, 1 (1986)! A490, 1 (1988). 9. H. de Vries et al., At. Data Nucl. Data Tables 36, 495 (1987). 10. P. Raghavaii At. Data Nucl. Data Tables 42: 189 (1989). 11. E.K. Warburton, D.E. Alburger, and D.J. Millener, Phys. Rev. C22, 2330 (1980). 12. C.M. Shakin, Y.R. Wagniarc, kLTomaselli, arid M.H. Hull, Phys. Rev. 161, 1015 (1967). 13. D.J. Millener arid D. Kurat,h, Kucl. Phys. A255: 315 (1975). 14. I. Tanihata et al., Phys. Lett. B206, 592 (1988).