INFLUENCE OF THE NUCLEAR ASYMMETRY AND COULOMB FORCE ON THE SURFACE WIDTH OF THE NUCLEAR MASS DENSITIES

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INFLUENCE OF THE NUCLEAR ASYMMETRY AND COULOMB FORCE ON THE SURFACE WIDTH OF THE NUCLEAR MASS DENSITIES

D. Berdichevsky

To cite this version:

D. Berdichevsky. INFLUENCE OF THE NUCLEAR ASYMMETRY AND COULOMB FORCE ON

THE SURFACE WIDTH OF THE NUCLEAR MASS DENSITIES. Journal de Physique Colloques,

1984, 45 (C6), pp.C6-221-C6-230. �10.1051/jphyscol:1984626�. �jpa-00224228�

JOURNAL DE PHYSIQUE

Colloque C6, supplément au n06, Tome 45, juin 1984 page C6-221

INFLUENCE OF THE NUCLEAR ASYMMETRY AND COULOMB FORCE ON THE SURFACE WIDTH OF THE NUCLEAR MASS D E N S I T I E S

D. Berdichevsky

GSI-Darmstadt, P L m c k s t r . 1, 6100 Darmstadt, F. B. G.

Résumé- Pour l e s noyaux f i n i s , l ' i n f l u e n c e des f o r c e s Coulombienne e t asymétrique s u r l a l a r g e u r de l a s u r f a c e de l a d i s t r i b u t i o n de masse e s t é t u d i é e avec un paramètre de l a s u r f a c e sans t e n i r compte des e f f e t s du volume. Les r e l a t i o n s importantes e n t r e l e s changements dans l a forme des p o t e n t i e l s à un corps e t l a l a r g e u r de l a d i s t r i b u t i o n de masse sont présentées.

A b s t r a c t

-

I n f i n i t e n u c l e i , t h e i n f l u e n c e o f t h e nucleon asymmetry and Coulomb f o r c e on t h e surface w i d t h o f t h e mass d e n s i t i e s i s i n v e s t i g a t e d w i t h a surface w i d t h parameter v o i d o f b u l k e f f e c t s . The r e l e v a n t c o r r e - l a t i o n s between changes i n t h e shape o f t h e 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 t h e mass s u r f a c e widths a r e given.

I t i s w e l l known t h a t t h e Coulomb f o r c e and t h e nucleon asymmetry a r e r e s p o n s i b l e f o r several c h a r a c t e r i s t i c s o f nuclear ground s t a t e p r o p e r t i e s . A few examoles are:

a) t h e d i f f e r e n c e i n t h e s i n g l e p a r t i c l e energies o f protons and neutrons /1/, b ) f o r i s o t o p i c and i s o t o n i c s e r i e s , t h e d e v i a t i o n from t h e A ' ' ~ la, e x h i b i t e d hy t h e d i f f e r e n c e s between n u c l e a r charge mean r a d i i /2/, c ) t h e increase o f t h e nucleon asymmetry w i t h t h e mass number A f o r t h e s t a b l e n u c l e i /3,4/. However, much l e s s i s known about t h e i n f l u e n c e o f t h e Coulomb f o r c e and t h e nucleon asymmetry on t h e surface w i d t h o f t h e mass d e n s i t y p A ( r ) [ p A ( r ) = p n ( r )

+

p,,(r) where p n ( r ) ( p p ( r ) ) i s t h e neutron ( p r o t o n ) d e n s i t y l and t h e neutron excess i n t h e s u r f a c e o f t h e nucleus. These p r o p e r t i e s are i m p o r t a n t f o r t h e d e s c r i p t i o n o f q u a s i s t a t i c / 5 / as w e l l as dynamic processes /6/ and a r e a l s o necessary i n g r e d i e n t s i n a l 1 mass formulae /7/. These mass formulae, however, cannot d e s c r i b e t h e s u r f a c e w i t h o u t addi t i o n a l assumption / 8 / .

The i m p o s s i b i l i t y t o o b t a i n a t t h e present time from experiment t h e necessary i n f o r - mation about t h e i n f l u e n c e o f Coulomb f o r c e and nucleon asymmetry on t h e s u r f a c e o f n u c l e a r mass d e n s i t i e s provides t h e m o t i v a t i o n f o r t h e i r d e t a i l e d i n v e s t i g a t i o n w i t h t h e s p h e r i c a l constrained HF+BCS-mode1 /9/.

I n t h e n e x t s e c t i o n we present Our d e f i n i t i o n o f t h e surface w i d t h parameter b /9/

I n s e c t i o n II we discuss i n d e t a i l t h e i n f l u e n c e o f nucleon asymmetry and Coulomb f o r c e on t h e n u c l e a r s u r f a c e width. A s h o r t summary o f Our r e s u l t s i s given i n s e c t i o n III.

1

-

SURFACE WIDTH PARAMETER b

I n r e f . 9, i t was shown t h a t t h e experimental mode1 independent (MI) n u c l e a r Charge d e n s i t i e s and t h e t h e o r e t i c a l charge and mass d e n s i t i e s obtained from HF+BCS-calcu- l a t i o n s a r e almost symmetric around t h e e x t e r n a l r a d i u s C ( c e n t r a l r a d i u s ) , where t h e slope o f t h e d e n s i t y p ( r ) i s maximum a t t h e s u r f a c e (see f i g . 1). T h i s symmetry i s f u l f i l l e d f o r a r e g i o n

1

^r

-

C

1

< 0.9

-

1.0 fm. I n t h a t r e g i o n t h e d e r i v a t i v e o f t h e symmetrized densi t y

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984626

JOURNAL DE PHYSIQUE

f o r r 2 C p'(2C-r) f o r r < C

i s a very good approximation to. p

'

( r ) . The quantity -p

-

' ( r ) possesses the advanta- S geous property of being a d i s t r i b u t i o n function. -p8'(r) i s a l s o f r e e from bu1 k e f f e c t s . F i g . 1 shows t h a t the symmetry expressed by r e l a t i o n (1) i s very f a s t destroyed f o r l a r g e r distances from C.

Fig. 1

-

Part ( a ) shows the calculated negative derivative of the proton density

- p l ( r ) (thick curve), the distance C from the center of the nucleus t o the surface

point where - p l ( r ) i s maximum, and the symmetrized function - p l S ( r ) defined in eq. ¹ ( t h i n curve). Part (b): same as part ( a ) but f o r neutrons.

Ue associate with C , and - p l S ( r ) the surface parameter

/

^{( r )} ( r - ) d r 2

b i s s e n s i t i v e t o the surface region and f r e e from any mode1 assumption regarding the shape of the d e n s i t i e s e i t h e r i n the volume or in the surface region. We employ in the following sections the parameter b f o r a systematic study of the influence of Coulomb force and nucleon asymmetry on the surface width of nuclear mass densi- t i e s .

I I

-

SURFACE WIDTH DEPENDENCE ON COULOMB AND NUCLEON ASYMMETRY

Nuclei along the valley of 6 - s t a b i l i t y with a mass number 20 6 A 5 244 show an increase in the r a t i o between Coulomb-force (EC) and binding-energy (BI)

with EC = 3/5 c Z ~ / A ( ' / ~ ) , and c = 0.744 MeV (force Ka /IO/) and neutron excess

with I = (N-Z)/A, N(Z)= number of neutrons (protons) and A=N+Z.

The analysis of t h e surface masse widths b (q=n(neutrons), p pro tons)) a s a func- tion of A (nuclei along the valley of 6 - s t a b i l i t y ) allows the determination of f i n e q e f f e c t s originating from the nucleon asymetry ( 1 = O ) , and from the Coulomb- force on the nuclear surface widths b (q=p,n).

q

The trends found i n f i g . 2 f o r the dependence of the nuclear surface widths on t h e mass number A a r e the following

Fi g. 2

-

The proton-(léft) and neutron-widths ( r i g h t ) b from s e l f c o n s i s t e n t HF+BCS-calculations with the force SV (upper most part of t h e f i g u r e ) Ka ( c e n t e r ) , and SI11 (lowest part of the f i g u r e ) f o r nuclei along t h e valley of 6 - s t a b i l i t y are given as a function of neutron number. The b values f o r the isotopes Ca, Ni, Fe, Z n , and Zr a r e joined withd s o l i d l i n e s . The curves emphasize the strong shell structures in bn, bp as a function of N and the s t r a i g h t l i n e s i l l u s t r a t e the decrease of the proton width f o r N 2 82.

( 1 ) The widths b and bn deviate in al1 cases l e s s than

-

4% from the average P

widths b and bn (see t a b l e 1 ) . P

C6-224 JOURNAL DE PHYSIQUE

Table 1. Coulumns 2 t i l l 4 g i v e t h e average nuclear charge surface w i d t h bc ( l i n e l), and t h e maximal d e v i a t i o n dbc/bC ( l i n e 2 ) c a l c u l a t e d w i t h t h e f o r c e s S I I I , Ka, and SV. The p r o t o n values b and ~ b neutron values bn and abn, and mass bA and abA a r e

P P '

given from l i n e 3 t o 8.

These c a l c u l a t i o n s were c a r r i e d o u t f o r 25 n u c l e i along t h e v a l l e y o f 6 - s t a b i l i t y w i t h a nucleon number 20 =< A 2 244.

( I I ) There i s a small r e d u c t i o n o f t h e p r o t o n w i d t h ^(t 2%) w i t h i n c r e a s i n g number o f nucleons A ( f o r N 2 82, i.e. A 5 160).

( I I I ) There are s h e l l s t r u c t u r e s i n t h e neutron w i d t h bn as a f u n c t i o n o f A w i t h nronounced minima f o r N = 20, 28, 50, and 82. T h i s r e s u l t a g r e e s w i t h t h e p r e d i c t i o n o f r e f . 11.

( I V ) The very non-local f o r c e SV /1/ p r e d i c t s s t r o n g e r staggerings i n bn and b P.

(V) The very non-local f o r c e SV p r e d i c t s [ a f t e r d i s r e g a r d i n g t h e s t a g g e r i n g e f f e c t s i n t h e surface w i d t h bn and bn] an almost equal w i d t h f o r p r o t o n and neutron i n c o n t r a s t t o t h e increase o f t L e d i f f e r e n c e between t h e neutron and p r o t o n widths w i t h i n c r e a s i n g nucleon asymmetry 1 w i t h t h e f o r c e s Ka / I O / and SI11 /1/.

( V I ) The f o r c e s S I I I , Ka, and SV show f o r t h e Ca and N i i s o t o p e a d i f f e r e n t

dependence o f t h e p r o t o n and neutron w i d t h b and b, w i t h i n c r e a s i n g neutron excess.

P

11.1

-

INFLUENCE OF NEUTRON EXCESS AND COULOMB FORCE ON THE PROTONS AND NEUTRONS SURFACE WIDTH OF ISOBARIC NUCLEI

die f i n d , a f t e r d i s r e g a r d i n g f o r c e SV, an increase o f t h e d i f f e r e n c e

w i t h A f o r n u c l e i along t h e v a l l e y of 6 - s t a b i l i t y (see f i g . 2 ) . The nucleon asymmetry 1 and t h e r a t i o between t h e Coulomb-energy and t h e binding-energy grows w i t h i n c r e a s i n g A. I n t h e f o l l o w i n g paragraphs we i n v e s t i g a t e t h e e f f e c t o f asymmetry and Coulomb f o r c e on t h e s u r f a c e mass widths b (q=n,p).

'4

For a b e t t e r i n t e r p r e t a t i o n o f t h e asymmetry e f f e c t s on t h e s u r f a c e w i d t h o f t h e n u c l e i ? we f i r s t i n v e s t i g a t e i s o b a r i c n u c l e i a f t e r s w i t c h i n g o f f t h e Coulomb-force.

The study o f isobars w i t h A

>

40 shows an increase o f t h e neutron w i d t h bn w i t h i n c r e a s i n g asymmetry, w h i l e b does n o t change very much (see f i g s . 3a, 4a). This e f f e c t i s responsible f o r an increase o f the neutron s k i n /12/ P

L L

70 60 50 70 60 50

1 2

1 OB 108

1 104 IO4 I I

A

1 O0 1 O0

y 0 9 6 O 96

-

-

B - 1 0

O 9 2 0 9 2

=

E

0

O 88 O 88

O 81 0 8 1 09

Fig. 3

-

P a r t ( a ) shows, f o r the i s o b a r i c n u c l e i

w i t h A=208 ( c a l c u l a t i o n w i t h t h e f o r c e Ka), t h e ^{-N -N}

proton surface w i d t h b (8) and neutron s u r f a c e Fig. 4

-

Same as f i g . 3 b u t f o r P

w i t h bn(a) as a f u n c t i o n o f t h e neutrons number Z the isobaric with A=140' (upper s c a l e ) . MIN i n d i c a t e s t h e a b s o l u t e minimum

f o r b as a f u n c t i o n o f N. P a r t ( b ) : same as p a r t P

( a ) b u t w i t h Coulomb-force.

C6-226 JOURNAL DE PHYSIQUE

w i t h t h e asymmetry 1 = (N-Z)/A. A c l o s e r a n a l y s i s shows, t h a t the a s y m e t r y f o r c e acts on t h e depth and w i d t h o f t h e proton and neutron s i n a l e - p a r t i c l e o o t e n t i a l s (see t a b l e 2 ) . The nucleon asymmetry (asymmetry f o r c e ) oroduces w i t h i n c r e a s i n g 1 a decrease o f t h e depth and an increase o f t h e s u r f a c e w i d t h o f t h e neutron s i n g l e - p a r t i c l e p o t e n t i a l . For protons, i t a l s o produces an increase o f t h e s u r f a c e w i d t h o f t h e p r o t o n s i n g l e - p a r t i c l e o o t e n t i a l and, as expected, a l s o a deeper p r o t o n p o t e n t i a l .

Table 2. T h i s t a b l e g i v e s t h e f o l l o w i n g c h a r a c t e r i s t i c p r o p e r t i e s o f t h e s e l f c o n - s i s t e n t s i n g l e p a r t i c l e o o t e n t i a l s c a l c u a l t e d w i t h t h e f o r c e Ka:

a) The h i g h o f t h e Coulomb-barrier h (column C 4). (b) The average depth o f t h e

P

v

proton s i n g l e p a r t i c l e p o t e n t i a l V (column 5 ) and i t s s u r f a c e w i d t h b (column 6)

P P

( c ) The average depth of t h e neutron s i n g l e p a r t i c l e p o t e n t i a l Vn and i t s surface w i d t h bn (column 7 ) .

v

OC (mC) (column 1 ) denotes t h e r e s u l t obtained i n 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 s w i t h o u t ( w i t h ) t h e Coulomb-force. The mass number A i s g i v e n i n column 2 and o f neutrons i n column 3.

We i n v e s t i g a t e w i t h t h e WS-potential t h e i n f l u e n c e o f t h e f o l l o w i n g two extreme cases on t h e increase o f bn.

(a) t h e neutron s i n g l e - p a r t i c l e p o t e n t i a l ( c e n t r a l p a r t ) i s t h e same f o r a l 1 i s o b a r i c n u c l e i (A=208) and t h e occupation number i s changing from N=104 t o N=138.

[The r a t i o between t h e e f f e c t i v e mass and t h e nucleon mass m*/m=0.61 i s chosen i n order t o perform a q u a n t i t a t i v e comparison w i t h 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 s using the f o r c e Ka. The WS-potential i s f i t t e d i n i t s depth and s u r f a c e w i d t h t o t h e c a l c u l a t i o n w i t h t h e f o r c e Ka f o r t h e nucleus 208 104X. The s p i n - o r b i t c o u p l i n g constant was chosen i n order t o g i v e t h e same s i n g l e p a r t i c l e occupation as i n t h e 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 s .

(b) We change depth and surface w i d t h o f t h e c e n t r a l p o t e n t i a l f o r a system w i t h a f i x e d occupation number(N=104).The p o t e n t i a l changes are chosen i n order t o cover t h e v a r i a t i o n i n t h e s e l f c o n s i s t e n t p o t e n t i a l o f t h e i s o b a r i c n u c l e i w i t h A=208 when t h e neutron number changes from N=104 t o N=138.

We f i n d t h a t t h e increase o f bn w i t h growing asymmetry 1 can be described as t h e e f f e c t o f two causes. One o f them i s t h e "cooperative" a c t i o n o f a l 1 o a r t i c l e s t o decrease t h e depth and increase t h e s u r f a c e w i d t h o f t h e neutron s i n g l e - p a r t i c l e

r

p o t e n t i a l . The o t h e r i s caused by t h e approach o f t h e Fermi-energie E: t o t h e zero value w i t h i n c r e a s i n g neutron number. Both causes c o n t r i b u t e almost e q u a l l y t o t h e growth o f t h e neutron d e n s i t y surface w i d t h bn when t h e neutron number N goes from 104 t o 138 (see f i g s . 5a and 6 ) .

Fig. 5

-

The dependence of t h e mass s u r f a c e w i d t h on t h e shape o f the s i n g l e p a r t i c l e p o t e n t i a l i s shown. P a r t ( a ) . Each o f t h e t h i n curves gives t h e neutron surface w i d t h bWS as a f u n c t i o n o f t h e US-parameter Vn WS (MeV) ( P o t e n t i a l depth) f o r one value ':i ( p o t e n t i a l surface w i d t h ) , and f i x e d value

cBS

( p o t e n t i a l s i z e ) , and f i x e d neutrons number N. The neutron c e n t r a l 9 o t e n t i a l i s

WS WS

ws

WS

Vn(r)=-Vn /Cl+expl (r-Cn )/z, 11. The t h i c k curve g i v e s t h e neutron s u r f a c e b,

as a f u n c t i o n of and , z:' where

v:'

and znS a r e f i t t e d t o reproduce t h e chanping s e l f c o n s i s t e n t depth and s u r f a c e w i d t h o f t h e neutron p o t e n t i a l as a f u n c t i o n o f t h e nucleon asymmetry f o r i s o b a r i c n u c l e i w i t h A=208 and neutron number changing from N=104 t o 138. P a r t ( b ) : same as p a r t ( a ) b u t f o r protons, and t h e t h i c k curve g i v e s t h e dependence o f bWS on t h e p r o t o n number Z, when i t changes from Z=104 t o 70.

P

We performed the same a n a l y s i s , w i t h o u t Coulomb force, f o r t h e p r o t o n surface w i d t h bD o f t h e i s o b a r i c nuclei(A=208)with an o c c u ~ a t i o n number changing from Z=104 t o 70.

C6-228 JOURNAL DE PHYSIQUE

Fig. 6 shows t h a t the decrease o f t h e Fermi-energy

EL

^from -8 MeV t o -16 MeV (no change i n t h e oarameters o f t h e WS-ootential) a l h e produces a r e l a t i v e l y small e f f e c t on t h e p r o t o n surface w i d t h b This small i n f l u e n c e on b coming from t h e

P. P

s t r o n g v a r i a t i o n i n ^E~ appears s i m i l a r t o t h e one already found i n r e f . 12 between

P WS

the change i n t h e s i z e o f the c e n t r a l o o t e n t i a l w e l l ( l a r n e r parameter C ) and i t s e f f e c t on t h e s u r f a c e w i d t h bu f o r t h e Ca-isotopes. I f we d i s r e g a r d s h e l l e f f e c t s b o t h r e s u l t s a r e e q u i v a l e h t . The change i n t h e occupation number i s respon-

-

s i b l e f o r a s t r o n g decrease i n

EL

and the f i x e d c e n t r a l p o t e n t i a l becomes ' l a r g e r "

f o r t h e remaining p a r t i c l e s . [ ~ e r e the word l a r n e means t h a t t h e s i z e o f t h e r e - s u l t i n g p r o t o n p o t e n t i a l i s l a r g e r than t h e one corresponding t o t h e s t a b l e isotope].

But t h a t i s e q u i v a l e n t t o m a i n t a i n t h e number o f p a r t i c l e s constant and t o increase WS

the s i z e o f the p o t e n t i a l ( l a r g e r WS-parameter C ). I n t h a t case t h e small changes o f t h e p r o t o n d e n s i t y surface w i d t h bu (see f i g . 6a) r e s u l t from t h e compensation

7

o f two opposite e f f e c t s : a) t h e decrease o f t h e Fermi-energy

EL

which tends t o decrease bD, and b) a smoother and more extended shape o f t h e have f u n c t i o n s a t t h e nuclear surface which tends t o i n c r e a s e bp. Thus, t h e s u r f a c e w i d t h bp depends mainly on t h e "cooperative" a c t i o n o f a l l ' p a r t i c l e s t o increase t h e depth and surface w i d t h o f t h e o r o t o n s i n g l e - p a r t i c l e o o t e n t i a l (see t a b l e 2). While t h e i n c r e a s i n o depth o f t h e p o t e n t i a l w i t h growing nucleon asymmetry 1 i s r e s p o n s i b l e f o r a decrease o f the surface w i d t h b t h e increase i n t h e surface w i d t h of t h e s i n g l e p a r t i c l e p o t e n t i a l tends t o i n c r e a s e t h e p r o t o n d e n s i t y s u r f a c e w i d t h b P' The r e s u l t i s again a r e l a t i v e small v a r i a t i o n i n b (see f i g . 5b). P.

P

Fig. 6

-

The dependence o f t h e mass surface w i d t h on t h e Fermi-energy E~ '

when t h e shape o f t h e s i n g l e p a r t i c l e p o t e n t i a l i s n o t the two curves q i v e s b, [ t r i a n g l e s ] (bn [ c i r c l e s l ) as a

(neutrons) number Z ( N ) . The parameters

cWS,

zWS, and

vbJS

o f t h e two WS-potentials

I I 2 l 1 I I I I I ^I I I

a) A=208 0 N I bl A = 208

- -

- 4

- - V " ' = S ~ L M ~ V . us= - 0 3 8 fmy - - - v ~ : ~ ~ o M ~ v . us= -039frnz

O P O Q

-

W ^{B O Q} >

-12 f

- b * d

are given t o g e t h e r w i t h t h e s p i n - o r b i t courilino constant us, where

uWS

(S.O.) =

u S / r d v w Z ( r ) / d r .

-

3p'b

H

-

^{i 21%}

2f'b 3p%

-

- _{b * d}

4' l h ' 4 Ih*

I I l I I I I i I l~

P a r t ( b ) . Each o f t h e curves gives on t h e r i g h t scale t h e Fermi eneraies ^E:

r-

-16

[ t r i a n g l e s ] ( € 1 EoDen c i r c l e s l ) as a f u n c t i o n o f t h e protons (neutrons) number Z(N).

7

IOL 120

db

136 72

,,\

⁸⁸ IO4 136

--

^{z * Z "O F.'b}

N

-

^N

-

For comparison, t h e s e l f c o n s i s t e n t neutron Fermi -1evel s E; a r e i n c l u d e d ( f u l l c i r c l e s ) .

The Coulomb f o r c e a c t s upon t h e surface w i d t h bu. I t a l s o m o d i f i e s bn as a conse- quence o f t h e dependence o f t h e s i n g l e - p a r t i c l e ' p o t e n t i a l on t h e t o t a l mass-, moment-, and s p i n - d e n s i t i e s /13/.

For a d e s c r i p t i o n o f t h e e f f e c t o f t h e Coulomb-force on t h e s u r f a c e w i d t h

ba (q=p,n) i t i s u s e f u l t o d i s t i n g u i s h between p r o t o n - r i c h and n e u t r o n - r i c h n u c l e i .

~ g l l o w i n ~ t h e usual convention, p r o t o n - r i c h n u c l e i a r e s i t u a t e d a t t h e l e f t s i d e o f t h e Gamow's v a l l e y and n e u t r o n - r i c h a r e s i t u a t e d a t t h e r i g h t side. Figs. 3,4 show t h a t t h e Coulomb-force i s r e s p o n s i b l e f o r a l i g h t increase o f b _P (

-

^{2% f o r}

140ce, and ² 3% f o r 2 0 8 ~ b ) and a n e g l i g i b l e change o f t h e neutron s u r f a c e w i d t h b, f o r t h e s t a b l e n u c l e i . Toward p r o t o n - r i c h n u c l e i b increases

P abp/bp ⁼ 7-8% f o r A=140(N=70), 208(N=104) wi t h

ab = [b ( w i t h Coulomb)

-

b ( w i t h o u t Cou1omb)l

P P P

The e f f e c t o f the Coulomb-force on t h e p r o t o n w i d t h b, decreases toward n e u t r o n - r i c h n u c l e i . For t h e n e u t r o n - r i c h n u c l e i Our c a l c u a t i o n als; shows a r e d u c t i o n o f t h e neutron w i d t h w i t h respect t o the HF-calculated value w i t h o u t Coulomb-force. The presence o f t h e Coulomb-barrier a t t h e surface o f t h e p r o t o n c e n t r a l p o t e n t i a l i s r e s p o n s i b l e f o r a r e s t r a i n e d increase o f t h e p r o t o n surface w i d t h b when t h e

P

Fermi-energy cF approaches t h e zero value /14/. This e f f e c t must a l s o be considered f o r t h e very n g u t r o n - r i c h n u c l e i (see t a b l e 2).

I I I

-

SUMMARY

The proton s u r f a c e w i d t h parameter bD shows minor changes ( 1 3%) as a f u n c t i o n o f t h e nucleon asymmetry. The neutron s u r f a c e w i d t h bn shows a steady increase w i t h t h e neutron excess. T h i s increase gets f a s t e r f o r v e r y n e u t r o n - r i c h n u c l e i .

The d i f f e r e n t changes o f the corresponding s i n g l e p a r t i c l e p o t e n t i a l s w i t h i n c r e a s i n g neutron excess a r e m a i n l y r e s p o n s i b l e f o r t h e d i f f e r e n c e s between neutron and p r o t o n s u r f a c e w i d t h as a f u n c t i o n o f t h e nucleon asymrnetry 1 . The decrease o f t h e depth and increase o f t h e s u r f a c e w i d t h o f t h e neutron s i n g l e - p a r t i c l e p o t e n t i a l c o n t r i - bute t o t h e increase o f t h e neutron s u r f a c e w i d t h bn w i t h growing asymmetry 1. The increase o f bn i s f u r t h e r enhanced because o f t h e approach o f t h e Fermi-energy

-

(chemical p o t e n t i a l )

EL

t o zero. The depth of t h e p r o t o n s i n g l e p a r t i c l e p o t e n t i a l increases w i t h growing asymmetry 1. That i m p l i e s a decrease i n bu. This e f f e c t i n t h e surface w i d t h bu i s compensated by t h e i n c r e a s i n g s u r f a c e w i d t h o f t h e p r o t o n p o t e n t i a l .

The Coulomb f o r c e i s r e s p o n s i b l e f o r a l i g h t increase o f b ( = 2% f o r 140ce) f o r P

n u c l e i along t h e v a l l e y o f P s t a b i l i t y . T h i s increase i s l a r g e r f o r very proton- r i c h n u c l e i ( = 7% f o r 140yb).

I n f i g . 2 s h a r p increases o f bn when the neutrons number N gets l a r g e r than 20, 28, 50, and 82 a r e m a i n l y due t o t h e approach o f t h e neutron Fermi-energies

E~ t o t h e z e r o t h value

(SE

= 4 MeV). Because o f t h e Coulomb-barrier hC t h e p r o t o n

n F P

surface w i d t h bp i s much l e s s s e n s i t i v e t o ep

.

C6-230 JOURNAL DE PHYSIQUE

F i n a l l y we would l i k e t o note t h a t j u s t a f t e r completion o f t h i s work we have become aware o f a p u b l i c a t i o n by L i p p a r i n i and S t r i n g a r i /15/ i n which t h e depen- dence o f t h e s u r f a c e w i d t h on t h e nucleon asymmetry 1 i s i n v e s t i g a t e d i n t h e frame work o f t h e l i q u i d d r o p l e t model.

Me g r a t e f u l l y acknowledge many h e l p f u l and s t i m u l a t i n g discussions w i t h Prof. U. Mosel ( U n i v e r s i t y Giessen) and Dr. K.-H. Schmidt (GSI Darmstadt).

References

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3. WAPSTRA A.H. and BOS K., .At. D a t a and Nucl. Data Tables 19(1977)175 4. BOHR A. and MOTTELSON B., Nucl. S t r u c t u r e 1, W.A. BenjamK New York, 1969 5. VINAS, F.J., LOZANO M., and MADURGA G., Phys. Rev. C23(1981)780

6. KRIVINE H., TREINER J., and BOHIGAS O., Nucl. P h y s . x 3 6 ( 1 9 8 0 ) 1 5 5

7. MARIPUU S., 1975 Mass P r e d i c t i o n s , At. Data and Nat.lcu Tables - 17(1976)411 8. N0LLER P. and NIX J.R., Nucl. Phys. A361(1981)117

9. BERDICHEVSKY D. and MOSEL U., Nucl. Phys. A388(1982)205

10. KOHLER H.S., Nucl. Phys. A258(1976)301 -

11. TONDEUR F., Proceedings o r n e I n t e r n a t i o n a l Summer School h e l d i n La Rabida (Huelva), Spain, June 7-18, 1982, Springer-Verlag, B e r l i n Heidelberg, 1982 12. BERDICHEVSKY D., t h e s i s Giessen, 1982

13. VAUTHERIN D. and BRINK D.M., Phys. Rev.

-

C5(1972)626 14. SAUER G., J. Phys. G3(1977)1085

15. LIPPARINI E. and STRfNGARI S., i n Proceedings o f t h e Topical Meeting on Nucl.

F l u i d Dynamics, I n t e r n a t i o n a l Centre f o r T h e o r e t i c a l Physics, T r i e s t e , 11.-15.

October 1982, IAEA-SMR-108, Wien, A u s t r i a