CORRELATIONS IN ODD-MASS NUCLEI

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CORRELATIONS IN ODD-MASS NUCLEI

H. Lenske

To cite this version:

H. Lenske. CORRELATIONS IN ODD-MASS NUCLEI. Journal de Physique Colloques, 1990, 51

(C6), pp.C6-447-C6-450. �10.1051/jphyscol:1990649�. �jpa-00230919�

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COLLOQUE DE PHYSIQUE

Colloque C6, supplBment au n022, Tome 51, 15 novembre 1990

CORRELATIONS IN ODD-MASS NUCLEI

H. LENSKE

Sektion Physik, Universitdt Miinchen, 0-8046 Garching, F . R . G . " )

Re'sume': Les c o r r i l a t i o n s d a n s l e s noyaux d e masse impaire s o n t e'tudie'es par une expansion des functions d'ondes d a n s un espace d e configurations il une e t t r o i s quasiparticules. L1op&ateur de self-e'nergie produit un mklange des couches prin- cipales d e s o r t e que l e s f a c t e u r s des f o r m e s radiales s o n t d i f f k e n t s pour chaque

&at. La thgorie e s t appliquce au c a s d e " ~ a .

Abstract: Correlations in odd-mass nuclei a r e described by expanding t h e wave functions i n t o one a n d t h r e e qnasiparticle configurations. The self-energy opera- t o r induces mixing o f radial wave functions from different major s h e l l s . This allows t o s t u d y t h e state-dependence of correlated single particle wave functions entering e.g. i n t o t r a n s f e r and knock-out processes. Applications t o " c a a r e dfs- cussed.

1. INTRODUCTION

In s p e c t r a of odd-mass nuclei t h e importance of final s t a t e interactions i s evident f r o m t h e fragmentation of single particle o r hole s t r e n g t h s over many e i g e n s t a t e s . The fragmentation p a t t e r n i s a finger print of t h e many body correlations due t o t h e interaction of hole o r particle s t a t e s with t h e c o r e nucleons. Studies of t r a n s f e r a n d knock-out reactions have been concentrating in t h e p a s t o n s t r e n g t h distributions. Typically, form f a c t o r s obtained from empirical potential i.e. mean field models a r e being used ( s e e e.g.

/1,2/). Such approaches describe d a t a q u i t e successfully a t l e a s t c l o s e t o t h e ground s t a t e . However, f o r accurate spectroscopic s t u d i e s over a wide range of e x c i t a t i o n energies t h e empirical models a r e likely t o become insufficient because by c o n s t r u c t i o n they include form f a c t o r c o r r e l a t i o n s only o n t h e level of binding energies. Thus consistency of spectroscopic f a c t o r s and form f a c t o r s i s not a s s u r e d .

2.

THE

QUASIPARTICLE CORE COUPLING MODEL

in t h e quasiparticle-core coupling (QPC) model /3,4,5/ s t a t e s Ijh) of t o t a l a n g u l a r momentum j (and parity n ) in odd-mass nuclei a r e described a s superpositions o f one and t h r e e qnasiparticle configurations with r e s p e c t t o t h e BCS-ground s t a t e of t h e even- mass p a r e n t system:

By t h e quasiparticle (QP) method long range pairing c o r r e l a t i o n s in t h e ground s t a t e a r e included. The 1-QP components have been denoted by (nj). The 3-QP configurations ((j' J1)j> a r e chosen a s QRPA-core excitations J' coupled t o 1-QP-states j'. A special f e a t u r e of t h e p r e s e n t approach i s t h e expansion of t h e 1-QP component i n t o radial wave func-

(1, Supported in p a r t by BMFT

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1990649

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C6-448 COLLOQUE DE PHYSIQUE

t i o n s with different radial nodes n /4,5/. This "major shell" mixing i s a n important extension over o t h e r approaches of similar kind /5,6/ because in t h i s way correlations in s t r e n g t h distributions and wave functions a r e described c o n s i s t e n t l y . I n particnlar wave function c o r r e l a t i o n s are introduced o n s a f e theoretical grounds avoiding t h e arbi- tariness of phenomenological approaches. The Hamiltonian i s H = Ho + V,, where Ho is diagonal in t h e 1- and 3-QP subspaces which a r e coupled by t h e residnal interaction V13.

Projection o n t o t h e 1-QP subspace leads t o an effective eigenvalue equation f o r t h e 1-QP s t a t e s lu>=lnj>

The radial mixing i s due t o t h e non-diagonal p a r t s o f t h e self-energy o p e r a t o r

where t h e QRPA-core excitations a r e denoted by Ic> /4,5/. In o n e - s t e p t r a n s f e r and knock-ont reactions t h e 1-QP component o f Eq.(l) is directly observed. A f t e r projection o n t o t h e physical nucleon s t a t e s t h e spectroscopic f a c t o r s f o r e.g. a stripping reaction a r e found a s

and t h e normalized form factors a r e defined by

where u a r e particle-type BCS-amplitudes and cp d e n o t e s a single particle wave func-

nl ni

tions. In knock-oat and pick up reactions t h e occupancies v i j = I-u2 e n t e r instead i n t o nl

t h e above expressions.

3. RESULTS

FOR

4 ' ~

-

Typical r e s u l t s f o r s t r e n g t h functions in ' ' ~ a obtained with large-scale QRPA-calculati- o n s using Wood-Saxon single particle functions and a realistic interaction derived from a G-matrix a r e shown in Fig.1 / 4 / . Mixing of wave f u n c t i o n s with ns;4 has been included. The s t r e n g t h f u n c t i o n s were folded by a Lorentzian o f FWHM A=O.SMeV f o r reasons of presentation. The BCS-ground s t a t e c o r r e l a t i o n s in 'OCa a r e r a t h e r moderate f o r usu- al values of t h e pairing s t r e n g t h G =25.5MeV a n d Gn=23.SMeV f o r p r o t o n s and n e u t r o n s ,

P

respectively, leading e.g. t o u2=0.13 f o r ld3/2(neutron), b a t clearly needed in o r d e r t o understand t h e occurence of low-lying positive-parity s t a t e s in " c ~ . A detailed compari- with recent high-resolution ~ a ( $ , ~ ) - d a t a i s discussed in /3,4/ and applications t o pro- t o n and neutron t r a n s f e r processes in 4 8 ~ i + 4 2 ~ a - r e a c t i o n s a r e found in /S/. In b o t h ca- s e s t h e overall f e a t u r e s of s p e c t r a a r e well described.

An important e f f e c t of t h e radial mixing i s t h a t s t r e n g t h from higher s h e l l s i s partially shifted into t h e low energy region. This increases f o r example t h e %+-strength f o r EX<8.8MeV in 4 1 ~ a by a f a c t o r of 1.8 over t h e sum r u l e limit /4/. Apparently t h i s poses a severe problem t o t h e determination of "absolute" spectroscopic f a c t o r s and ground

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OOL

oo6m

s t a t e c o r r e l a t i o n s because they would be overestimated by t h e same amount. General- ly t h e radial c o r r e l a t i o n s a r e m o s t impor- t a n t f o r s t a t e s with a small 1-QP partial width. In reactions they a r e only weakly excited b u t f o r a n accurate determination o f s t r e n g t h s t h e i r contributions a r e important a s well. In many c a s e s t h e form f a c t o r s differ s t r o n g l y f r o m t h o s e o f phenomenological mean field approaches like t h e "well depth" (WD) o r "surface peak" (SP) method /4/. Results f o r several ? - s t a t e s which a r e o 2 o f particular i n t e r e s t f o r investigations o f

in ground s t a t e correlations a r e displayed in

08 Fig.2. For t h e f i r s t few s t a t e s form f a c t o r s

06 o f fd-shape a r e obtained which essentially

y 0':

2

⁰² vary according t o t h e binding energy. Phe-

4 ⁰⁰ nomenological potential models a r e likely t o

3 g O8 reproduce t h i s behaviour. However, a s seen

OK

in t h e lower p a r t of Fig.2 a t Ex=6.28MeV

OL

0 2 t h e shape changes t o 2d. At higher excitati-

o n energies t h e s h a p e s are varying between

06

01 n=l and n=2. The momentum s t r u c t u r e o f

02 t h e form f a c t o r s is a l t e r e d accordingly. In Fig.3 t h e Fourier t r a n s f o r m s of t h e same

om wave functions a r e displayed. Apparent1 y

o m

s t r o n g admixtures of large momenta o c c u r

- 1 0 1 2 3 4 5 6 7 8 9 1 0

E.IMeVI f o r t h e Ex=6.28MeV-state. Similar r e s u l t s

a r e obtained f o r s t a t e s of o t h e r angular Fig.1: Single particle s t r e n g t h distribu- momenta and parities a s well. The radial t i o n s f o r 4 1 ~ a f o r "hole-type" ( u p p e r mixing i s m o s t pronounced f o r reactions part) and "particle-type" (below) s t a t e s leading t o configurations involving mostly folded with a Loreatzian o f FHWM filled s h e l l s , i.e. t h e (2s,ld)- shell for 4 1 ~ a . A=O.SMeV. Note t h e change of scale f o r Similar r e s u l t s a r e obtained f o r s t a t e s of

-

and :+-states, respectively. The o t h e r a n g u l a r momenta and parities a s well.

fragmentation increases with t h e distan- For t h e negative parity configurations t h e c e f r o m t h e Fermi-surface. A comparisi- nodal s t r u c t u r e i s determined by t h e (2p, o n t o " ~ a ( b , ~ ) - d a t a i s found in /4/. If)-orbital involved b u t s t r o n g variations in

t h e s p a t i a l extension occur. This was s t u - died already in /3,4/ where rms-radii f r o m t h e microscopic approach and t h e WD and t h e SD method have been compared. As expected neither of t h e phenomenological me- t h o d s i s able t o describe t h e s t r o n g variations o f t h e microscopic rms-radii b u t in t h e average t h e WD-method reproduces a t l e a s t t h e general increase with excitation energy.

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COLLOQUE DE PHYSIQUE

Fig.2: C o r r e l a t e d radial wave func- t i o n s f o r ' ' ~ a ( s e e Eq.(5)). R e s u l t s f o r t h e f i r s t , s i x t h and seventh :+-eigenstate a r e shown. Note t h e change to 2d-shape a t Ex=6.28MeV.

Fig.3: F o u r i e r - t r a n s f o r m s o f t h e wave f u n c t i o n s s h o w n in Fig.2. No- te t h e increase o f l a r g e momentum c o m p o n e n t s in t h e s t r o n g l y c o r r e -

l a t e d s t a t e a t Ex=6.28MeV.

4. SUMMARY

AND

CONCLUSIONS

-

I n summary, i t w a s s h o w n t h a t t h e quasiparticle-core i n t e r a c t i o n s i n odd-mass nuclei lead t o "major shell" mixing o f radial wave f u n c t i o n s . The QPC-model includes t h e s e c o r r e l a t i o n s c o n s i s t e n t l y i n s p e c t r o s c o p i c f a c t o r s and wave f u n c t i o n s . Thus s t a t e - d e p e n - d e n t f e a t u r e s o f t r a n s f e r a n d knock-out f o r m f a c t o r s c a n b e s t u d i e d o n s a f e t h e o r e t i c a l grounds. Large c o n f i g u r a t i o n s p a c e s w e r e used. The model c a l c u l a t i o n s t h e r e f o r e c o v e r l o n g and t o a l a r g e e x t e n d a l s o s h o r t range c o r r e l a t i o n s . The l a t t e r a r e mainly respon- s i b l e f o r t h e radial mixing. They a r e s e e n m o s t clearly i n t h e F o u r i e r - t r a n s f o r m s a s a d m i x t u r e s o f l a r g e momentum components. The r e s u l t s imply t h a t m e a s u r e m e n t s o f a b s o l u t e s p e c t r o s c o p i c f a c t o r s and t h e derivation o f ground s t a t e c o r r e l a t i o n s f r o m t r a n s f e r a n d knock-out r e a c t i o n d a t a will hampered by t h e radial c o r r e l a t i o n s which c a n only p a r t l y a c c o u n t e d f o r in p o t e n t i a l models.

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2. G.R. S a t c h l e r , Nuclear Direct Reactions, Clarendon, O x f o r d , 1983 3. F.J. Eckle e t al., P h y s . R e v . w (1989) 1662

4. F.J. Eckle e t a l . , Nucl.Phys. (1990) 159

S. P. von Neumann-Cosel et al.. Nucl.Phys.

A

( i n p r i n t ) 6. G.F. B e r t s c h e t al., Rev.Mod.Phys. (1983) 298 7. V. Soloviev, Progr. Part.Nucl:Phys.

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