DEFORMED STRUCTURES AND ALPHA-PARTICLE DESCRIPTION OF LIGHT NUCLEI

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DEFORMED STRUCTURES AND

ALPHA-PARTICLE DESCRIPTION OF LIGHT NUCLEI

Y. Abgrall, E. Caurier

To cite this version:

Y. Abgrall, E. Caurier. DEFORMED STRUCTURES AND ALPHA-PARTICLE DESCRIP- TION OF LIGHT NUCLEI. Journal de Physique Colloques, 1971, 32 (C6), pp.C6-63-C6-68.

�10.1051/jphyscol:1971609�. �jpa-00214827�

JOURNAL DE PHYSIQUE Colloque C6, supplkment au no 11-12, Tome 32, Novembre-Dkcembre 1971, page C6-63

DEFORMED STRUCTURES

AND ALPHA-PAR TICLE DESCRIPTION OF LIGHT NUCLEI

Y. ABGRALL

Laboratoire de Physique Thtorique, Universite de Bordeaux, France and

E. CAURIER

Laboratoire de Physique ThCorique, Centre de Recherches NuclCaires de Strasbourg, France

Rbume. - Nous avons etudie les noyaux ICgers de type 4 n dans une optique Hartree-Fock et suivant le modele a microscopique de Brink et Marguenau. Dc la cornparaison de ces deux rnode- Ies, il apparait que pour age, lzC, 1 6 0 et 2ONe le modele B sous-structures s'avere &tre non seule- ment une meilleure description, mais egalement une image plus simple ou les structures deforrnks apparaissent trks naturellement. Dans 24Mg et 28Si cependant les solutions obtenues dans les deux modeles sont tres differentes.

Abstract. ^- The microscopic a-cluster model of Brink and Marguenau is used to study light 4 n nuclei. From a comparison with Hartree-Fock calculations in 8Be, 1*C, 1 6 0 and 2oNe it turns out to be not only an improved description, but also a simplifying picture, where the deformed structures appear quite naturally and clearly. However in 24Mg and 28Si the a-cluster configurations differ noticeably from the HF solutions.

The existence of deformed structures in light nuclei of 1 p and first half of the 2 s - 1 d shell has long been recognized. We know perfectly now that these states are particularly well excited in heavy ions scattering and transfer or pick-up processes [I]. In figure 1 are shown, for instance, some positive parity states in ^{1 6 0 ,} with the well known 6.06 band (interpreted here as

FIG. 1. - Low-Lying cxcitcd 0; states ^[9] in ^1" together with the 6.05 MeV and the 16.75 MeV rotational bands.

ellipsoidal) and the 'Be type levels (seen in the cr

+

12C + 'Be

+

8Be reaction [2]).

Description of light nuclei in a shell model frame meets a considerable success [3], [4]. However the correlations involved are not apparent and are often better visualized in other interpretations. Furthermore, some states, like the 'Be type levels, which need the introduction of highly deformed 8 p - 8 h states, are hardly conceivable in this picture.

In another point of view it seems natural to describe such structures from the rotation of an intrinsic deform- ed state. If one assumes that this intrinsic state is a Slater determinant, the orbitals are then determined by means of the Hartree-Fock method and the rotational band is obtained by projecting out states of good J.

This H F theory has been widely used in light nuclei [5]. Let us discuss here some aspects of the problem, in particular the question of deformations, related to the new experimental informations about the intrinsic state, in one hand, by the determination of quadrupole

p,

and hexadecapole

P,

deformations and, about the J states themselves on the other hand, from the measurement of the static quadrupole moments of the first 2' state of several even-even nuclei of the s-d shell.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1971609

C6-64 Y. ABGRALL A N D E. CAUKIER To know in what extent the H F method with

angular momentum projection, in a treatment of the whole A nucleons system, can give a good account of these nuclear properties, we use the same picture which permits a coherent description [6] of the spectra of light 4 n nuclei from 'Be to 28Si. We assume that the LS coupling is valid and retain only those states with maximum [4

...

41 symmetry. The deformed orbitals are the (n, n, nJ single particle states whose lengths, b,, by and b, are obtained from a projection before variation procedure. Note that the order is particularly important when one discusses of such quantities like electro- magnetic properties whose evaluations based only on intrinsic state-results can be very poor [7]. As said above we treat the A nucleon dynamic variables without any reference to a core which releases ourselves of awkward concepts as effective charges.

Central phenomenological interactions, the only active with the symmetry [4

...

41, have been considered.

The first one is the Brink and Boeker B 1 force [8], widely used in HF calculation which leads to rather

good radii but underbinds the nuclei discussed here from 'Be to "Si. For this reason another interaction, of the form Vo(l

-

^m

+

mP') exp(- r2/p2), has been taken and the parameters are chosen in such a way as to reproduce the correct binding energy and the size of

"0 as well as an excitation energy of 16.75 MeV to the J = 0' 8p-8h projected state. Indeed the 'Be like levels can be identified as members of a rotational band generated by an axially highly deformed 8p-8h state (000)4 (002)4 (003)4, see figure 2.

This interaction, noted as F l , gives good binding energies and radii for 4 n nuclei from 'Be to 2 8 ~ i . The resulting intrinsic state deformations

are reported in table I. Those, obtained with B1, are the same within 1-3

%.

The static quadrupole moments of the first excited 2' states are shown in table I1 (for further comments see [7]).

Comparison between experimental [lo] and calculated (Force F 1)

P2

and

P4

deformations

8Be 2C 'ONe "Ne 2 4 ~ g 28Si

- ^/ - -

P2

+

0.78 - 0.42

+

0.47

+

0.47

+

0.46 - 0.40

H. F. b4

+

^0.36

+

0.12

+

^0.25

+

^0.18

+

^0.13

+

0.15

Exp, b'2

+

0.47

+

0.47

+

^{0.50 - 0.40}

P4

+

0.28

+

0.05 0

+

^0.10

FIG. 2. - The 8p-8h and a chain projected states compared to the al3e type levels.

We see that the H F method can give a good account of the deformations involved in light nuclei. However it seems that, in so far we must introduce such exotic states as highly deformed 8 p-8 h states, the deformed one-center model is probably not the best starting

Static quadrupole moments of the jirsi excited 2 + state (in fm2) : comparison between experimental, pro- jected (force F 1 and B I), and she0 model results (obtained in the latter case with an added effective charge of 0.5 e for both proton and neutron)

2oNe 24Mg 28%

Exp. - 27

+

1 1 ^(a) - 26 :t 8 ^{( C )}

+

^{17 & 5 (e)}

- 2 1 f . 3(b) - 2 4 & 4 ( " ) f 1 6 * 5 ( b )

F 1 - 17.3 - 19.9

+

^21.5

B l -18.9 -1- 24.5

S . M . -14.3 ( -16.0 (g) +16.0 ^(g) (.) Ref. [11] (c) Ref. [13] (") Ref. [15]

(b) Ref. [12] (*) Ref. [I41 (') Ref.[ 41 (g) Ref. [16]

point. On the other hand the mass distributions of the intrinsic states, figures 3 and 4, reveal obviousconcentra- tions. Let us go further and say that these mass concen- trations in light 4 n nuclei are nothing but a particles.

This is precisely the Brink and Marguenau [17] a particle model.

In this picture the 'Be intrinsic state is for instance

DEFORMED STRUCTURES A N D ALPHA-PARTICLE DESCRIPTION OF LlGHT NUCLEI C6-65

parity configurations reduce respectively to the closed shell configuration and 1 p-1 h state, the plane configu- rations to 4p-4h states and the linear one to the 8p-8h state discussed above.

Some results, obtained with an interaction chosen on thc same grounds as F1 but giving a radius of 2.4 fm in 160, illustrate thc general trends (a more complete report and discussion will be given elsewhere). From figures 5 to 8, where the energies of some states are plotted, many interesting features can be retained.

First, the energy gain from the HF limit indicates the importance of extra cluster correlations. The distortion of the clusters, very large for small d, decreases with increasing separation distances. At the equilibrium FIG. 3. - zoNe nucleus : cross sections of equidensity surfaces they are almost spherical (except 2 8 ~ i ) . The equidensity

by a plane containing the symmetry axis for the HF intrinsic Curves, figures 3-4, and the separation of the clusters, as

deformed state and for that obtained by a K =-= 0 axial projection compared to their internal sizes,

from the a positive parity configuration. _-

160 l i n e tables 111 and IV, show the rather small overlap of the

substructures. The clustering effects are still enhanced for the projected states (see also [19]).

From a comparison of recent investigations of light 4 n nuclei in the BM a model (this report and [18], [19], [20]) with HF calculations (the (n, n, n,) picture and a

deformed shell model ( a p - 8 h ) much improved calculation of Zofka and Ripka [21]),

the following main points can be taken out.

FIG. 4.

-

^{1 6 0 nucleus :} cross sections of equidensity surfaces for the 8 p-8 h intrinsic state and thc linear a configuration.

where @(r,, d,) is the wave function of an a particle in an oscillator well centred at d,. A is the antisymmetri- sor. In other words Y is a Slater determinant built with the single particle orbitals

(r - di)2 cpi(r) = N exp -

2 b2

.

The BM model is, thus, another kind of H F calcula- tion where the variational parameters are the separa- tion di between the clusters and their size b. In general an a configuration has neither angular momentum, nor well defined parity, an extra richness of the a intrinsic structures. As a consequence, low lying bands of both parity can be projected. In this work we have preferred a slightly modified version of the BM model by allow- ing possible deformations of the clusters (shape varia- tional parameters b,, b, and b,). Now, in the limit d, ⁴ 0 the a configuration reduces to the previous (n, n, n 3 H F picture, instead of the SU3 state. For instance, in 160 the tetrahedral positive and negative

FIG. 5. - ^{sJ3e nucleus} : Intrinsic state energy, minimized with respect to the size parameters bk, plotted as a function of the cluster separation d (dotted line). In the case of spherical clus- ters, bt --= b intrinsic and J = 0:. projected energies are also shown (full line). The Coulomb contribution is not included here.

Y . ARGRALL A N D E. CAUKIER

TABLE 111

Equilibrium energies (Coulomb not included), separa- tions di between the cluster^, rms radii r, of the clusters and gain in energy relatice to the lowesr (n, i i , nJ H F conj?guration. All energies in MeV, lengths in fm

,

20 6 0 6 3 d ( h )

TABLE TV

F I G . 6. - 12C nucleus : Plot of thc energies of the parity pro-

See the caption of' Table III jected intrinsic states. See caption of figure 5.

In a first region, including the nuclei up to 20Ne, one can see that the a-model gives more binding than HF.

The two descriptions, although leading to rather diffe- rent mass distributions (higher

p,),

are however close together and give nearly the same radii, quadrupole moments and, in a lesser extent, inertia parameters (when comparing the projected spectra). Thus for these nuclei the BM model is not only a better description, as compared to actual HF models, but also a simplifying picture (see for instance the highly deforlned 8 p-8 h state and the corresponding a linear configuration).

In the second region, namely 24Mg and 28Si, the a model gives still a little 2

%

energy gain when conlpar- ed with the (11, n , n,) picture, but, in fact slightly

FIG. 7. - Linear configuration of 12C. See caption of figure 5.

underbinds when compared with the H F results of Zofka and Ripka. More important the two solutions are now quite different, leading for instance, to qua-

DEFORMED STRUCTURES AND ALPHA-PARTICLE DESCRIPTION OF LIGHT NUCLEI C6-67 E

I

^(MeV)

FIG. 8. - 1 6 0 nucleus : Plot of the energies of the parity pro- jected intrinsic states.

drupole moments of opposite signs. (For further comments see [7] and [21]).

However, before drawing any definite conclusion about the ability or inadequacy of the BM model t o describe these nuclei many problems, together with the validity of the L S coupling scheme, must be elucidated and in that respect much works remain.

The first is that the previous discussion abdut 24Mg and '*Si is based only o n intrinsic state results a n d a projection before variation procedure is needed.

T h e second point is the mixing of different ci configu- rations and the fact that they are not orthogonal. Let us discuss briefly this question in 12C from a plot of the

energy of the positive parity isosceles configuration (minimised with respect t o d ) , a s a function of the vertex angle O (Fig. 9, curve E,). As expected a clear

FIG. 9. - l2C nucleus : Plot, as a function of the vertex angle 0, of the energy Eo of the positivc parity isosceles configuration

and of the energy El of the state defined in the text.

minimum appears a t 8 = 600, another one near O = 1500. The linear configuration is a local maximum.

Minimizing in the subspace

I

cp,(O)

>

⁼

I

Y(O)

>

^-

<

'f"(0)

I

cpo

> <

^{9 0}

1 ,

orthogonal t o the equilateral configuration

I

^(PO

>

=

I

y(O = boo)

> ,

one observes, once again, a minimum near O = 1500.

However the width of the valley is a n indication that the candidate t o describe the 7.65 MeV state in ''C is presumably a large mixing of different a configurations.

A correct treatment involves the refinement of the Hill- Wheeler generator coordinate-method and has been discussed by D e Takacsy and D a s G u p t a [22] and Takigawa and Arima [23] who have shown that the state near O = 1500 is indeed a largely mixed state.

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MARQUARDT (N.), VON OERTZEN (W.) and WAL-

TER (R. L.), Phys. Letters, 1971, 358, 37.

121 CHEVALLIER (P.), SCHEIBLING (F.), GOLDRING (G.), PLESSER (I.), and SACHS (W. M.), Phys. Rev., 1967,

160, 827.

[3] GOLDHAMMER (P.), HILL (J. R.) and NACHAMKIN (J.), Nucl. Phys., 1968, A 106, 62.

[4] WILDENTHAL (B. H.), MC GRORY (J. B.) and GLAU-

DEMANS (P. W. M.), Phys. Rev. Letters, 1971, 26, 96.

[5] RIPKA (G.), Advances in Nuclear Physics, Vol. 1 (Plcnum Press, New York, 1968).

[6] ABGRALL (Y.), BARON ( G . ) , CAURIER (E.) and MON-

SONEGO (G.), NucI. P h y ~ . , 1969, A 131, 609.

[7] ARGRALL (Y.), MORAND (B.) and CAURIER (E.), to be published.

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191 AJZENBERG-SELOVE (F.), NucI. P h y ~ . , 1971, A 166, 1 . [lo] DE SWINIARSKI (R.) et RAYNAL (J.), COIloque (( Sur Certains Aspects Microscopiques des RCactions Nuclhires ^D, la Toussuire, 1971, Lycen, 7104, S9.

[ l l ] SCHWALM (D.) and Povrj, Phys. Letters, 1969, 29B, 103.

[12] NAKAI (K.), WINTHER (A.), SMILANSKY (U.), STE-

PHENS (F. S.) and D~AMOND (R. M.), t o be published.

[13] BAMBERGER (A.), BIZZETI (P. G.) and POVH (B.), Phys. Rev. Letters, 1968, 21, 1599.

[14] HAUSER (O.), HOOTON (B. W.), PELTE (D.), ALE-

XANDER (T. K.) and EVANS (H. C.), Phys. Rev.

Letters, 1969, 22, 359.

C6-68 Y . ABGRALL AND E. CAURIER 1151 HXUSER (O.), ALEXANDER (T. K.), PELTE (D.),

HOOTON (B. W.) and EVANS (H. C.), Phys. Rev.

Letters, 1969, 23, 330.

[16] M c GRORY (J. B.) and WILDENTHAL (B. H.), Phy.~.

Letters, 1971, 34B, 373.

[17] BRINK (D. M.), Intern. School of Physics ⁽⁽ Enrico Fermi ^I), course 36 (Academic Press, 1966).

[I81 BRINK (D. M.), FRIEDRICH (H.), WEIGUNY (A.) and WONG (C. W.), Phys. Letters, 1970, 33B, 143.

1191 FR~EDRICH (H.) and WE~GUNY (A.), t o be published and private communication.

[20] NAIGEON (A.), Thesis, 1970, Strasbourg.

[21] ZOFKA (J.) and RIPKA (G.), Nucl. Phys., 1971, A 168, 65.

[22] D E TAKACSY (N.) and DAS GWTA (S.), Phys. Letters, 1970, 33B, 556.

[23] TAKIGAWA (N.) and ARIMA (A.), NucI. Phys., 1971, A 108, 593.