LINKS BETWEEN THE HARTREE-FOCK AND SEMICLASSICAL MODELS OF STATIC NUCLEAR PROPERTIES

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LINKS BETWEEN THE HARTREE-FOCK AND SEMICLASSICAL MODELS OF STATIC NUCLEAR

PROPERTIES

F. Tondeur, J. Arcoragi, Justin Pearson

To cite this version:

F. Tondeur, J. Arcoragi, Justin Pearson. LINKS BETWEEN THE HARTREE-FOCK AND SEMI-

CLASSICAL MODELS OF STATIC NUCLEAR PROPERTIES. Journal de Physique Colloques,

1984, 45 (C6), pp.C6-125-C6-135. �10.1051/jphyscol:1984615�. �jpa-00224216�

JOURNAL DE PHYSIQUE

Colloque C6, supplément au n°6, Tome 45, juin 1984 page C6-125

LINKS BETWEEN THE HARTREE-FOCK AND SEMICLASSICAL MODELS OF STATIC NUCLEAR PROPERTIES

F. Tondeur, J.P. Arcoragi* and J.M. Pearson*

Université Libre de Bruxelles, CP 229, B 1050 Bruxelles, Belgium and

Institut Supérieur Industriel de Bruxelles, Belgium

Département de Physique, Université de Montréal, Montréal, Canada

Résumé - L'équivalence entre le modèle de Hartree-Fock (HF) et la méthode de Thomas-Fermi étendue (ETF) avec corrections de couches est examinée. Une prescription de Strutinsky généralisée est développée pour des spectres finis. Sa bonne précision est démontrée dans le cadre de l'approche self- consistente Hartree-Fock-Strutinsky (HFS). Les résultats de calculs HFS et ETF sont comparés pour un grand nombre de noyaux, montrant un accord global acceptable mais aussi certains désaccords systématiques. Le choix de 1'in- teraction effective dans les deux modèles est enfin discuté. En utilisant un ensemble de forces de Skyrme développé récemment et qui permet une appro- che systématique de la question, plusieurs contraintes sont suggérées pour ajuster une force donnant de bonnes extrapolations.

abstract - The equivalence between the Hartree-Fock (HF) model and the extended Thomas-Fermi (ETF) method with shell corrections is examined. A generalised Strutinsky prescription is developed for finite spectra. Its good accuracy is shown in the framework of the self-consistent Hartree- Fock-Strutinsky (HFS) approach. Then, HFS and ETF results are compared for a large number of nuclei, showing an acceptable overall agreement but also several systematic discrepancies. The choice of the effective interaction for both models is finally discussed. By using a recently developed set of Skyrme forces which allows a systematic approach of this question, several constraints are suggested for the fit of a force suitable for extrapolations.

1) Introduction

The past ten years have seen the extraordinary development of the use of Skyrme forces in Hartree-Fock [l) and semi classical nuclear models [2j . The popularity of these forces is due to their ability to reproduce many static nuclear properties with a good accuracy [3] . Among many other results obtained with those forces, we would like to point out the demonstration of the deficiencies of more phenomenological models, in particular the droplet model [4J . It has thus become evident that Hartree-Fock (HF) or semi classical models should be used sys- tematically instead of the droplet model (DM), especially in all cases where no experimental result is available to check the accuracy of the model, e.g. for extra- polations far from stability.

The HF model should in principle be more accurate than semi classical approxi- mations like the extended Thomas-Fermi model (ETF). Unfortunately, its application to deformed nuclear shapes TSI implies heavy numerical calculations, and cannot be used for systematic studies of several thousand nuclei, as needed for example in astrophysical applications. Thus we have still to rely on an approximate method for deformed shapes, like the ETF + shell correction method which improves the usual "macroscopic-microscopic" calculations by using ETF instead of the DM for the macroscopic part.

In ETF with Skyrme forces, the binding energy of the nucleus can be calculated

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984615

from t h e d e n s i t y d i s t r i b u t i o n s

(3,

and by u s i n g t h e semi c l a s s i c a l expansion o f t h e k i n e t i c and s p i n - o r b i t d e n s i t i e s [ 6 ]

.

Brack and coworkers 123 minimize i t by u s i n g g e n e r a l i z e d Fermi t r i a l d i s t r i b u t i o n s given h e r e f o r t h e s p h e r i c a l c a s e :

where q = n o r p. When

r)n

and have been determined, t h e corresponding s i n g l e - p a r t i c l e p o t e n t i a l VETF can be e a s i l y o b t a i n e d .

A f t e r c a l c u l a t i n g t h e s i n g l e p a r t i c l e l e v e l s

ei

i n VETF, one can determine t h e s h e l l c o r r e c t i o n t o t h e energy :

S,B ⁼

^oCC.

- ^{<Z ei} average}

which added t o BETF i s expected t o g i v e a r e a s o n a b l e approximation t o HF :

Two methods can be used t o e v a l u a t e t h e average sum o f s i n g l e - p a r t i c l e e n e r g i e s i n eq. ( 2 ) ^: t h e S t r u t i n s k y p r e s c r i p t i o n

173

o r t h e s e m i c l a s s i c a l method

18) .

^The

s t a n d a r d S t r u t i n s k y method meets some problems when a p p l i e d t o f i n i t e s p e c t r a and can be d i f f i c u l t t o o p e r a t e f o r o c c a s i o n a l u s e r s who do n o t have t h e "know-how".

The s e m i c l a s s i c a l method being e a s i e r t o handle and completely unambiguous can t h u s be p r e f e r e d

.

However, t h e S t r u t i n s k y method a l s o a l l o w s f o r a s e l f - c o n s i s t e n t c a l c u l a t i o n o f t h e macroscopic energy

191

w i t h o u t making u s e o f t h e s e m i c l a s s i c a l approximation. We s h a l l r e f e r t o t h i s type o f c a l c u l a t i o n a s t o t h e Hartree-Fock S t r u t i n s k y (HFS) method. The HFS method i s a s c o s t l y a s HF, and t h u s n o t of i n t e r e s t i n i t s e l f , b u t can be used t o check t h e accuracy o f t h e s e m i c l a s s i c a l models ( o r of t h e DM) and i n p a r t i c u l a r t h e accuracy o f t h e ETF d e n s i t y d i s t r i b u - t i o n s , s i n g l e - p a r t i c l e p o t e n t i a l s and s h e l l c o r r e c t i o n s .

We s h a l l show i n s e c t i o n 2 t h a t i t i s p o s s i b l e t o overcome t h e problem met with t h e S t r u t i n s k y p r e s c r i p t i o n f o r f i n i t e s p e c t r a . We indeed p r e s e n t a genera- l i s e d smoothing p r e s c r i p t i o n which can be used i n t h i s c a s e . We compare i n s e c t i o n 3 s e v e r a l HFS r e s u l t s o b t a i n e d w i t h t h i s new p r e s c r i p t i o n t o t h e corresponding ETF r e s u l t s . F i n a l l y , s e c t i o n 4 examines a common problem t o a l l methods : t h e choice of t h e e f f e c t i v e Skyrme i n t e r a c t i o n .

2 - S h e l l c o r r e c t i o n s method f o r f i n i t e s p e c t r a

The s t a n d a r d S t r u t i n s k y p r e s c r i p t i o n [ 7 ] can only be used i f a t l e a s t t h r e e major s h e l l s a r e known above t h e Fermi energy i n t h e s i n g l e - p a r t i c l e spectrum.

I n r e a l i s t i c p o t e n t i a l s , o n l y one s h e l l i s u s u a l l y a v a i l a b l e and two s h e l l s o f p o s i - tive-energy l e v e l s must be d e f i n e d according t o more o r l e s s e m p i r i c a l r e c i p e s

l10'J.

A t e c h n i c a l d i s c u s s i o n o f t h e r e s u l t s given i n

111)

h a s shown t h e f e a s a b i l i t y of t h e method i n t h e f a v o u r a b l e c a s e of N = Z n u c l e i w i t h o u t any Coulomb f o r c e , f o r which t h e bound spectrum i n c l u d e s more t h a n one empty s h e l l and sometimes two

.

^The

method however i s n o t e a s y t o handle f o r o c c a s i o n n a l u s e r s and p r e s e n t s s e r i o u s d i f f i c u l t i e s when a p p l i e d n e a r t h e neutron d r i p l i n e .

Two approaches have been followed t o improve t h e p r e s c r i p t i o n f o r f i n i t e s p e c t r a . One has been developed by Ivanjuk and S t r u t i n s k y [l23

,

b u t h a s n o t been a p p l i e d t o s y s t e m a t i c c a l c u l a t i o n s . The second one i s d e s c r i b e d h e r e , and improves t h e method developed i n a p r e v i o u s work C131

.

S t a r t i n g w i t h the"occupation number r e p r e s e n t a t i o n 1 ' ( w i t h q = n o r p ) ^:

we i n t r o d u c e t h e f o l l o w i n g c o n s t r a i n t s on

n+

^:

(i''') ^II - 2 (i"') A

- ^4,:

^{0 , q}

all oce

which e x p r e s s t h a t t h e averaging made i n eq. (4) s h o u l d n o t modify

E = ^e;q

^{i f}

eiq

i s a l r e a d y a smooth f u n c t i o n of

i .

^{( A l l} s t a t e s a r e supposed t o be non- degenerate and o r d e r e d by i n c r e a s i n g e n e r g y ) . The v a r i a b l e

ill3

i s chosen i n view o f t h e average t r e n d

e; ill3

v a l i d f o r e q u i d i s t a n t s h e l l s . E q . (5) a l s o i n c l u d z s t h e nucleon number c o n s e r v a t i o n f o r k = 0. A p a r a m e t r i s e d f u n c t i o n i s chosen f o r m ; :

where t h e determined by eq. (51,

l\1

being t h e number o f n u c l e u s o f t y p e taken a s t h e v a r i a b l e , t h e v a l u e o f t h e smoothing parameter

y

w?uch.corresponds t o one s h e l l s p a c i n g i s roughly t h e same i n t h e whole (N,Z) p l a n e . I n t h e s t a n d a r d p r e s c r i p t i o n ,

)r

^{i s} chosen t o f u l f i l t h e p l a t e a u c o n d i t i o n :

=

⁰

( 7 ) However, a s d i s c u s s e d i n 1133

,

t h i s c o n d i t i o n i s n o t _ n e c e s s a r y . We i n s t e a d ask a good accuracy of t h e S t r u t i n s k y theorem with a smooth

B

and t h e u s u a l s h e l l c o r r e c t i o n :

r y

A s demonstrated i n [ g , 131 and h e r e a f t e r , t h e theorem i s a c c u r a t e i f B i s z a l c u - l a t e d s e 1 c f ; c o n s i s t e n t l y by r e p l a c i n g i n HF t h e H F occupation numbers by t h ~ h ;

.

However, BHFS i s n o t n e c e s s a r i l y a smooth f u n c t i o n [ l 3 3

.

To g e t a smooth B H F S ' it i s n e c e s s a r y :

e

a ) t o a v o i d t h e b r u t a l changes i n t h e l e v e l o r d e r and i n t h e X; when h i g h l y d e g e n e r a t e l e v e l s c r o s s each o t h e r i n t h e (N,Z) p l a n e , t h i s can be done by i n t r o - ducing a f i c t i v e smooth l e v e l o r d e r .

b) t o a v o i d t h e b r u t a l v a r i a t i o n o f t h e t o t a l number of s t a t e s i n t h e spectrum by i n c l u d i n g a smooth number o f s t a t e s n ( N , Z ) which i s u s u a l l y d i f f e r e n t from t h e t o t a l number o f bound s t a t e s .

C ) t o smooth t h e i r r e g u l a r e f f e c t s i n t r o d u c e d by t h e s h e l l f_luctuations of t h e upper l i m i t of t h e c o n s i d e r e d spectrum, by c a l c u l a t i n g t h e 9%; f o r a few d i f f e r e n t v a l u e s o f R. and making an average.

d ) t o damp e x p l i c i t e l y i n t h e smoothing p r e s c r i p t i o n t h e o s c i l l a t i n g component of

e;

; with t h e v a r i a b l e

ill3,

t h i s component is-roughly p e r i o d i c a l ( w i t h t h e p e r i o d

2

) and i t s lower harmonic can b e - c a n c e l l e d i n E by i n t r o d u c i n g two new c o n s t r a i n t s : 3

CI

With t h o s e improvements

,

B (N,Z) i s found t o be smooth w i t h i n

+

¹ MeV f o r n u c l e i a l o n g t h e s t a b i l i t y l i n e .

The a d d i t i o n a l c o n s t r a i n t s ( 9 ) do n o t a f f e c t t o o much t h e accuracy of t h e S t r u t i n s k y theorem

,

e x c e p t i n a few c a s e s where t h e 7 c o n s t r a i n t s ( 5 , g) a r e probably t o o s t r o n g and l e a d t o p a t h o l o g i c a l r e s u l t s , l i k e occupation numbers much h i g h e r than 1 . 5 . Varying u s u a l l y s o l v e s t h e problem. A s a g e n e r a l r u l e , t h e 7 c o n s t r a i n t s l e a v e o n l y a very s m a l l freedom. I n p a r t i c u l a r , t h e

*X;

a r e f i n a l l y q u i t e unsensi- t i v e t o t h e value o f even when t h i s parameter changes by an o r d e r o f magnitude (The p l a t e a u c o n d i t i o n t h u s becomes meaningless). I t i s a l s o very d i f f i c u l t t o add more c o n s t r a i n t s (e.g. t h e damping o f h i g h e r harmonics o f t h e o s c i l l a t i n g p a r t of

e ;

) .

The h i q h e r o r d e r s h e l l c o r r e c t i o n

i s found i n most c a s e w i t h i n t h e range

$a$ -

^{0 . 7 5}

2

0.50 MeV. The l a r g e s t v a l u e s a r e o b t a i n e d f o r magic n u c l e i , f o r which

f,

B i s a l s o l a r g e r . A s

ja

B i s always p o s i t i v e , t h e s h e l l c o r r e c t i o n can be improved by i n c l u d i n g i t s average v a l u e , which does n o t vary s i g n i f i c a n t l y w i t h A :

t h e average e r r o r being t h e n lower t h a n 0.25 MeV. Such a good accuracy i s a l s o found when p a i r i n g c o r r e l a t i o n s a r e taken i n t o account :

e x c e p t i n a few c a s e s where t h e BCS occupation numbers

q2

c a l c u l a t e d f o r t h e HFS s i n g l e p a r t i c l e s t a t e s a r e very d i f f e r e n t from t h o s e o b t a i n e d f o r t h e HFBCS

spectrum. T h i s problem mostly concerns a few l i g h t o r medium n u c l e i , and i s l i k e l y t o occur a l s o i n models where BHFS and t h e mean f i e l d VHFS a r e r e p l a c e d by t h e i r s e m i c l a s s i c a l e s t i m a t e s .

3 - Comparison between extended Thomas-Ferml and Hartree-Fock S t r u t i n s k y r e s u l t s The substitution of s e m i c l a s s i c a l q u a n t i t i e s t o t h e smooth HFS binding e n e r g i e s and s i n g l e - p a r t i c l e p o t n e t i a l s however i n t r o d u c e s much l a r g e r u n c e r t a i n t i e s

,

a s wi.11 now b e shown. The comparison between V ETF and VgFe w i l l be done elsewhere. We s h a l l f o c u s e t h e p r e s e n t d i s c u s s i o n on a comparison e ween d e n s i t y d i s t r i b u t i o n s and between binding e n e r g i e s .

The ETF model used h e r e i n c l u d e s t h e f o u r t h o r d e r expansion o f t h e k i n e t i c d e n s i t y

[ g

b u t o n l y r e t a i n s t h e second o r d e r terms f o r t h e s p i n - o r b i t terms

( i n c l u d i n g t h e s p i n - o r b i t c o n t r i b u t i o n t o t h e k i n e t i c e n e r g y ) . The g e n e r a l i s e d Fermi shapes o f eq. (l) a r e used a s t r i a l wavefunctions, b u t t h e c o n s t r a i n t = 1 has been imposed i n some c a s e s . A l l c a l c u l a t i o n s use t h e Skyrme f o r c e T1 C14, 1 5 1

.

We compare i n f i g . 1 t h e ETF and HFS r e s u l t s f o r t h e bulk d e n s i t y of n e u t r o n s and p r o t o n s I n n u c l e i n e a r t h e s t a b i l i t y l i n e . I n t h e HFS c a s e , t h e bulk d e n s i t y i s o b t a i n e d a s i n [l61 by f i t t i n g a Fermi d i s t r i b u t i o n t o t h e HFS d i s t r i b u t i o n s . When

'f #

1 i s allowed, t h e agreement between t h e two models i s q u i t e good, e s p e c i a l l y f o r heavy n u c l e i . I n l i g h t and medium n u c l e i , t h e ETF d e n s i t i e s a r e a b i t t o o low, b u t t h e disagreement does n o t exceed 3 % .

Fig. 2 compares t h e 90% - 10% d i f f u s e n e s s e s o b t a i n e d with = l , showing t h e u s u a l underestimate o f t h i s q u a n t i t y by t h e ETF model. The agreement w i t h H F S i s imprwed w i t h

Y #

^1, b u t t h e ETF s u r f a c e d i f f u s e n e s s i s s t i l l t o o s m a l l on t h e average. Moreover, t h e i n c r e a s e o f

t

with

+

1 i s mainly l o c a t e d i n s i d e o f

.06

^{1 1 1 I}

F i g .

7

50 100 150 200 A

4 0 ~ a 1 4 0 ~ d ... 230{h

... _ - - -

...

- ^-

^--M /

^- - ^-t

#

neutrons

- - - - ET F

...

CompcmAan ad :the coke devtnLtia i n HFS and ETF mod&. E T F r a e d m t o w a h g w a & e d Fenmi

blz(~lcl3ecr

( y #

7 )

ETF

0-10 H F S

m - - - ET F

/

F L ~ .

2

50 100 150 200 A

90% -

10% nutduce t h i c k n u n Zjok neuhon and p&oXon din.ttLiblLtio~lcl i n HFS and ETF modeh

1.8 -

7 \ -- _{- -}

- . _

P ---

I I 1 1

the surface, whereas HFS results predict a larger diffuseness for the outer part of the surface (i.e. the 50% - 10% compared with the 90% - 50% diffuseness). We also note the different trends obtained for the proton diffuseness in HFS and ETF with

f $ 1. Fig. 3 compares the values of V* obtained by the ETF model and by a fit to

HFS distributions. The most striking result is in the completely different trends for Tfc . On the other hand, the trends of

'f

are quite similar , but the values differ by US 30%.

y

u . n ...-••-"

ETFyv--

1 . 2 . ""-••• P - - - - " "

jn

^ " \ . ^{HRT^}

0.8. ^ < ^ _

I , , , ,

50 100 150 200 A

Rtg. 3 - SuAfiacz ahymmojthjy paAamzteM. ~fr o& the. ge.YieA.aLa>e.d FeAmi diAtsu.butu.on {on Yiz.irfn.OYU> and pft.otoYU> JLn HFS and ETF models ; V" = ' coM.z&pond& to the. pute.

FeAmi ishape. with a. &ymmeXhA.c. hux^ace..

T a b l e 1

Hartree-Fock-Strutinsky (HFS) results compared with extended-Thomas-Fermi results with pure Fermi density distributions (ETF) or with generalized Fermi shapes given by eq. (1) (ETF^ ) for the different contributions to the binding energies of

2 0 0

H g (in MeV).

kinetic central Coulomb surface spin-orbit total

HFS 3716.4 -6206.9 766.4 234.4 -65.7 -1555.4

ETF 3760.4 -6266.9 766.4 259.8 -87.9 -1568.3

ETF |f

3769.7

-6273.4

768.4

247.7

-82.9

-1570.5

These d i s a g r e e m e n t ~ b e t w e e n HFS and ETF d e s c r i p t i o n s o f t h e s u r f a c e s i g n i f i c a n t - l y l i m i t t h e accuracy one can e x p e c t t o o b t a i n i n ETF c a l c u l a t i o n s i f t h e same f o r c e is used a s i n W. Indeed t h e v a l i d i t y o f t h e HF approach i s f u l l y confirmed by t h e e x c e l l e n t f i t s t o experimental charge d e n s i t i e s which can be achieved

.

I n p a r t i c u l a r , t h a t

1

i n heavy n u c l e i i s c l e a r l y i n d i c a t e d by t h e experimen- t a l charge d i s t r i b u t i o n s i n 2 0 8 ~ b . I n view o f t h e key-role p l a y e d by t h e d e n s i t y d i s t r i b u t i o n s i n ETF, one can a l s o e x p e c t s i g n i f i c a n t i n a c c u r a c i e s i n o t h e r quan- t i t i e s o b t a i n e d i n t h i s model.

We a r e p a r t i c u l a r l y i n t e r e s t e d i n t h e b i n d i n g e n e r g i e s . We p r e s e n t i n Table 1 t h e r e s u l t s o b t a i n e d f o r * O O ~ g . I t must be noted t h a t t h e i n c l u s i o n of f o u r t h - o r d e r s p i n - o r b i t terms would improve t h e agreement between HFS and ETF : according t o Brack's r e s u l t s L 2 3 , t h e s e c o r r e c t i o n s reduce t h e s p i n - o r b i t energy by

-

^20%

i n t h i s region. However, d i f f e r e n c e s amounting t o s e v e r a l MeV a r e found between t h e two models, i n p a r t i c u l a r i n t h e (

v?

^{) 2} terms o f t h e energy d e n s i t y f u n c t i o n a l , which r e f l e c t t h e d i f f e r e n c e s notkced i n t h e s u r f a c e d i f f u s e n e s s e s . We cannot e x p e c t t h a t t h e approximation of BHFS by BETF w i l l be a s a c c u r a t e a s t h e S t r u t i n s k y method. Thus, i n o r d e r t o o b t a i n a s a t i s f a c t o r y approximation t o HF, i t i s necessa- r y t o i n t r o d u c e small e m p i r i c a l c o r r e c t i o n s t o t h e ETF method. We can imagine t o do t h i s a t t h r e e d i f f e r e n t l e v e l s :

a ) t h e Skyrme f o r c e can be r e f i t t e d . T h i s i s c e r t a i n l y t h e most r e a s o n a b l e c h o i c e i f t h e ETF

+

s h e l l c o r r e c t i o n method i s used f o r i t s e l f , w i t h no i n t e n t i o n t o give an approximation t o HF, t h e f o r c e being t h e n f i t t e d i n o r d e r t o reproduce e x p e r i - mental r e s u l t s with t h e ETF

+

s h e l l c o r r e c t i o n method. P r e l i m i n a r y r e s u l t s show t h a t t h e HF i n t e r a c t i o n must only be s l i g h t l y modified, a t l e a s t when t h e semi- c l a s s i c a l s h e l l c o r r e c t i o n s a r e used [8J. Work i n t h i s d i r e c t i o n i s c u r r e n t l y under way, and w i l l b e r e p o r t e d elsewhere ;

b) t h e ETF expansion of t h e k i n e t i c and s p i n - o r b i t d e n s i t i e s can be s l i g h t l y modified by i n c l u d i n g e m p i r i c a l terms. T h i s would g i v e a n a t u r a l u n i f i c a t i o n o f t h e MTF ( z e r o t h o r d e r TF

+

e m p i r i c a l t e r n s ) and ETF ( f o u r t h o r d e r ETF w i t h o u t e m p i r i c a l terms) approaches;

C ) e m p i r i c a l c o r r e c t i o n s can be i n t r o d u c e d i n t h e r e s u l t s themselves, e . g . t h e b i n d i n g e n e r g i e s , by comparing ETF and HFS i n a few s e l e c t e d c a s e s , t h e c o r r e c - t i o n s b e i n g i n t e r p o l a t e d f o r o t h e r c a s e s . 1t would however be completely e q u i v a l e n t t o i n t e r p o l a t e d i r e c t l y t h e HFS r e s u l t s without u s i n g ETF

,

a s done i n 1173

.

4. The c h o i c e o f t h e e f f e c t i v e i n t e r a c t i o n

A s d i s c u s s e d above, t h e HF and ETF e f f e c t i v e f o r c e s can be s l i g h t l y d i f f e r e n t . However, o n l y minor d i f f e r e n c e s a r e expected, and both i n t e r a c t i o n s must f u l f i l t h e same g e n e r a l requirements. I n p a r t i c u l a r , it i s u s u a l l y asked t o reproduce experimental b i n d i n g e n e r g i e s and charge r a d i i w i t h a s u f f i c i e n t accuracy. I f we want t o u s e

HF

and ETF i n t h e whole f i e l d covered by t h e DM, we must a l s o add a good f i t t o f i s s i o n b a r r i e r s . None o f t h e p u b l i s h e d v e r s i o n s o f t h e Skyrme f o r c e s simultaneously f u l f i l s a l l t h e s e c o n s t r a i n t s . Most of them f a i l t o reproduce f i s s i o n b a r r i e r h e i g h t s , whereas o t h e r ( l i k e SkM 9 1183 ) d o n o t f i t t h e b i n d i n g energy o f n e u t r o n - r i c h n u c l e i . Recently, we have developed a s e t o f t r i a l f o r c e s (15)

,

which allow f o r a s y s t e m a t i c approach t o t h e iriherplay between t h e parame- t e r s o f t h e f o r c e s and t h e ETF/HF r e s u l t s , by a l l o w i n g f o r s e p a r a t e v a r i a t i o n s of d i f f e r e n t p h y s i c a l parameters. D e f i n i t i v e f i t s have n o t y e t been made. We can however summarize a few r e s u l t s a l r e a d y o b t a i n e d w i t h t h e s e f o r c e s , t h e parameters of which a r e i v e n i n Table 2. A l l f o r c e s a r e f i t t e d t o a few binding e n e r g i e s

(160, 5 6 ~ e , 9 8 Z r , l 1 8 s n , 132sn, 1 3 8 ~ a , 1 4 6 ~ d , 2 0 8 ~ b ) and t o t h e charge r a d i u s of 2 0 8 ~ b .

The way t h e y a r e d e r i v e d from each o t h e r i s x b e m a t i c a l l y given i n f i g . 4. In t h i s f i g u r e , s t a n d s f o r t h e c o e f f i c i e n t o f the(V

(f',,-fP)l''

term i n t h e Skyrme Hamiltonlan d e n s i t y :

T a b l e 2

Coefficients of Skyrme forces and corresponding Droplet Model parameters. The values of m /m are given for symmetric nuclear matter.

t

h o

fc

2

s

X

o

X

l

X

2

X

3

*

W o

k

F

P-

a K

V

J L a HF

s ETF a s

Q

HF

Q E T F

m*

m

Tl -1794.0

298 -298 12812 .154 -.5 -.5

.089 1/3

110

1.336 .1610 -15.98 236.1

32.02 56.2 18.3 17.99 33.

36.5 1

T2 -1791.6

300 -300 12792 .154 -.5 -.5

.089 1/3

120

1.336 .1610 -15.94 235.7

32.00 56.2 18.1 17.77 33.

36.4 1

T3 -1791.8

298.5 -99.5 12794

.138 -1

1 .075 1/3

126

1.336 .1611 -15.96 235.9

31.50 55.3 17.9 17.66 34.

38.5 1

T4 -1808.8 •

303.4 -303.4 12980 -.177 -.5 -.5 -.5 1/3

113

1.330 .1590 -15.95 235.5

35.45 94.1 18.2 17.90 26.

27.9 1

T5 -2917.1 •

328.2 -328.2 18584 -.295 -.5 -.5 -.5 1/6

114

1.344 .1640 -15.99 201.7

37.00 98.5 18.2 17.97 25.

26.9 1

T6 -1794.2 -

294 -294 12817

.392 -.5 -.5 + .5 1/3

107

1.336 .1609 -15.96 235.9

29.97 30.9 18.3 17.95 40.

46.8 1

T7 -1892.5 •

366.6 -21 11983

.334 -.359 6.9

.366 .285 109

1.335 .1606 -15.94 235.6

29.52 31.1 18.3 17.90 41.

46.7 5/6

T8 -1892.5 -

367 -228.76 11983

.448 -.5 -.5

.695 .285 109

1.335 .1607 -15.94 235.7

29.92 33.7 18.2 17.90 41.

46.1 5/6

T9 -1891.4

377.4 -239.16 11982

.441 -.5 -.5

.686 .285 130

1.334 .1608 -15.88 234.9

29.76 33.7 17.9 17.61 43.

45.1 5/6

ife —1 * —1 whereas the isoscalar and isovector (m ) refer to the two contributions to m :

where q = n or p , the + sign holding for neutrons and the - sign for protons. The notation x_ (x ) is used to indicate correlated variations which are constrained by the fit to the masses. A s usually, K is the nuclear matter compression modulus and W the spin-orbit strength.

a) the fit of HF results to the experimental binding energies is better when

m ~ m . For a reasonable m («2_

m

to m) , this fit allows the determination of at least three parameters, which 3 correspond roughly to the volume, surface and asym- metry contributions of the droplet model, the Coulomb energy being fixed by the fit to the radius o f 208pb.

T h e

volume

a

n d surface coefficients only slightly depend on the force (Table 2 ) . T h e surface energies are determined here by HF or ETF calcula- tions for semi-infinite nuclear matter. Both models lead to very similar results.

In particular, the most significant variations of the surface coefficient come from

the variations o f the spin-orbit strength W

0

. On the other hand, the fit to the

Fig.

4 - Schematic

v i w

0 6 -the teLcctiovrn between t h e 9 - C x i d Skytme @hceA

known b i n d i n g e n e r g i e s i s n o t s u f f i c i e n t t o determine t h e asymmetry c o e f f i c i e n t J , which i s c o r r e l a t e d with t h e c o e f f i c i e n t

L and t h e surface-asymmetry c o e f f i c i e n t Q ( s e e r e f . 119) f o r t h e d e f i n i t i o n of t h e s e DM c o e f f i c i e n t s ) .

b) S e v e r a l c o n s t r a i n t s can be o b t a i n e d by a n a l y s i n g t h e d e n s i t y d i s t r i b u t i o n s

* .

For a r e a s o n a b l e m

,

t h e s u r f a c e d i f f u s e n e s s a s w e l l a s t h e compression o f t h e bulk d e n s i t y w i t h r e s p e c t t o n u c l e a r m a t t e r mostly depend on t h e compression modulus K

[ 2 6 ]

.

The v a l u e of K suggested by t h i s a n a l y s i s ( X 240 MeV) does n o t c o n t r a d i c t t h e one deduced from GMR e n e r g i e s [20] ⁽

Z

220 MeV). On t h e o t h e r hand, t h e neutron s k i n t h i c k n e s s could be used t o f i x J and Q

.

The l i m i t e d accuracy o f t h e a v a i l a b l e d a t a only d e f i n e s a p o s s i b l e range J

2

29

+

2 MeV and Q 5 45

+

1 0 MeV ( t h e low J corresponding t o t h e h i g h Q ) .

C ) I t i s i n t e r e s t i n g t o n o t e t h a t t h e small )

v(f,-!,)lL

term i n t h e Skyrme hamil- t o n i a n d e n s i t y f u n c t i o n a l has a s i g n i f i c a n t i n f l u e n c e on t h e amplitude o f t h e s h e l l o s c i l l a t i o n s of t h e d e n s i t y i n t h e c o r e region. A f o r c e has been given i n [21]

which a l l o w s a good f i t t o t h e s e o s c i l l a t i o n s . I n ETF, t h i s term h a s no c o r e c o n t r i - bution:, it i s a good example o f t h e p o s s i b l e d i f f e r e n c e s between ETF and HF.

d ) The v a l u e s o f J and Q could a l s o be determined by t r y i n g t o reproduce f i s s i o n b a r r i e r h e i g h t s . Table 3 g i v e s ETF b a r r i e r h e i g h t s o b t a i n e d f o r 2 4 0 ~ ~ . The f o r c e s g i v i n g a good neutron s k i n t h i c k n e s s i n 2 0 8 ~ b (T6,T7,T8) l e a d t o b a r r i e r s s i g n i f i - c a n t l y h i g h e r t h a n t h e corresponding DM value ( 2 3 . 8 MeV ; t h e p r e s e n t c a l c u l a t i o n i s l i m i t e d t o t h e C-axis i n t h e

c,k

p a r a m e t r i s a t i o n C221

.

However, t h e s p u r i o u s r o t a t i o n a l energy c o r r e c t i o n c o u l d l e a d t o a s i g n i f i c a n t lowering of t h e s e h e i g h t s . T h i s c o r r e c t i o n must be i n c l u d e d i n t h e microscopic model ( i . e . i n HF o r i n t h e s h e l l c o r r e c t i o n ) . A s t h i s h a s n o t been done u s u a l l y i n t h e DM

+

s h e l l c o r r e c t i o n c a l c u l a t i o n s , t h e DM b a r r i e r h e i g h t i s meaningful o n l y f o r comparison w i t h a model i n which t h i s c o r r e c t i o n i s n e g l e c t e d . I n c l u d i n g o r n e g l e c t i n g it w i l l l e a d t o d i f - f e r e n t c o n s t r a i n t s on J and Q , and on t h e r e l a t e d f o r c e parameters. The i n c l u s i o n o f t h e c o r r e c t i o n seems u s e f u l t o g e t an agreement w i t h t h e v a l u e s o f J and Q o b t a i n e d from t h e f i t t o neutron s k i n s ( i . e . q u i t e a low J and q u i t e a high Q ) .

T a b l e 3

ETF fission barriers of Pu (MeV) 240 T l

5 . 0 T2 4 . 3

T3 4 . 3

T4 3 . 2

T5 2 . 8

T 6 6 . 3

T7 6 . 2

T8 6 . 1

e) The fit of the asymmetry parameters appears as one of the most critical points in the choice of the effective interaction. This fit can also influence other pro- perties, like the giant dipole resonance (GDR) energies. In this case, a third degree of freedom is expressed by the enhancement factor K, which is related to the effective masses. Several results ("23] suggest 0.6 < K^ 0.8. On the other hand, the GDR energy is not expected to depend very much on J and Q , if the force is fitted to the experimental binding energies. This can be shown in the DM where the dependence of E^L on J and Q f 2 4 j :

is not very different from that of the total (volume + surface) asymmetry energy :

P ~ T7T (16) J

especially if J is low and Q high, as suggested above. We thus can expect only very rough constraints on J and Q from the analysis o f GDR energies. O n the other hand, the high values of K suggested by experimental results are only acceptable if

tp (J/Q) in eq. (15) is quite low. As a general rule, forces leading to the lowest values o f ( 0 (J,Q) have high J and low Q, which seems to contradict the indications given above. In view o f the large uncertainties associated with the DM, and with the values o f the enhancement factor, we shall limit the present discussion to this remark. Further studies based on less phenomenological models are under way and will be reported elsewhere.

Several results reported in the last section have been calculated by Pr. M.

Brack, to whom we are also indebted for very interesting discussions and advice about the Strutinsky method and the ETF model. We also thank Pr. M. Farine and Dr. D. Berdichevsky for their contribution to specific points of the present study.

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Phys. A (1984)

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