Semi-empirical mass formula and nuclear radii

Volume 30B, n u m b e r 9

SEMI-EMPIRICAL

PHYSICS

MASS

LETTERS

FORMULA

22 D e c e m b e r 1969

AND

RADII

NUCLEAR

J. R. R O O K N u c l e a r P h y s i c s L a b o r a l o r y , Oxford. UK Received

10 September

1969

The s e m i - e m p i r i c a l mass formula is extended to include different radii for the proton and neutron d i s tributions. By minimistng the energy, the radii are determined. Our model predicts that the proton radius equals the neutron radius for all stable nuclei and that this radius has an accurate A t/3 dependence.

The semi-empirical mass formula is valuable b e c a u s e it g i v e s a s i m p l e p i c t u r e of the b i n d i n g e n e r g i e s of n u c l e i and p r o v i d e s a n o r m w i t h w h i c h m o r e s o p h i s t i c a t e d t h e o r i e s can be c o m p a r e d . In t h i s n o t e we s u g g e s t an e x t e n s i o n of the t h e o r y w h i c h e n a b l e s u s to d e t e r m i n e the r a d i i of the p r o t o n and n e u t r o n d i s t r i b u t i o n s in n u c l e i . We c o n s i d e r f i r s t a " g a s " of N n e u t r o n s c o n fined to a s p h e r e of r a d i u s R n. Its b i n d i n g e n e r g y m i g h t be e x p e c t e d to h a v e the f o r m E = - p N + a N ~ [ R n - h(2g)~] 2 + ~ N / R n

(1)

w h e r e p~ ~,, a and ~ a r e c o n s t a n t s . T h e f i r s t t e r m r e p r e s e n t s a c o n s t a n t binding e n e r g y / n e u t r o n w h i l e the m i d d l e t e r m r e p r e s e n t s the c o m p r e s s i b i l i t y of the s y s t e m . T h e p a r t i c u l a r f o r m of t h i s t e r m a r i s e s w h e n we a s s u m e an e q u i l i b r i u m r a d i u s of h(2N)~ and a c o n s t a n t bulk m o d u l u s . T h e l a s t t e r m r e p r e s e n t s the s u r f a c e t e n s i o n and i s o b t a i n e d by n o t i n g that for f i x e d N the n u m b e r of n u c l e o n s in the s u r f a c e r e g i o n is p r o p o r t i o n a l to 1 / R n. A s i m i l a r e x p r e s s i o n i s e x p e c t e d to hold f o r the Z p r o t o n s of r a d i u s R,, w i t h an a d d e d ! e r m },Z2 / Rp to a c c o u n t for the C_ t o,ru l o m b r e p u l s ion. T h e c r u c i a l q u e s t i o n i s the i n t e r a c t i o n of the p r o t o n s and n e u t r o n s and we n e e d s o m e m o d e l in o r d e r to d e t e r m i n e the d e p e n d e n c e of t h i s t e r m on the n u c l e a r p a r a m e t e r s . Suppose we a s s u m e that the p r o t o n r a d i u s i s g r e a t e r than the n e u t r o n r a d i u s . T h e d e n s i t y of n e u t r o n s i s t h e n p = 3N"(4~R3). If we a s s u m e that a z e r o r a n g e i n t e r a c t i o n b e t w e e n the p r o t o n s and n e u t r o n s then the p o t e n t i a l e n e r g y of a p r o t o n due to the n e u t r o n s i s V p w h e r e V i s a c o n s t a n t and the t o t a l e n e r g y of the i n t e r a c t i o n b e t w e e n the p r o t o n s and n e u -

t r o n s i s -SNZ 'Rn3 w h e r e 6 i s a c o n s t a n t . H e n c e the full e x p r e s s i o n f o r the e n e r g y is !

E = -~

1-

+ a A ~ [ { R p - ~,(2Z)~ 2 +

+ {R n - ~(2N)~} 2] + ~ Z , / R p + {3N/R n + 3ax + ~ Z 2 .•/ R p - 5 N Z R m

(2)

w h e r e R _lal~.,~ n ~ i s the l a r g e r of R_t t and R ^V and A = N + Z . In o b t a i n i n g eq. (2) f r o m eq. (1) we 1 1 h a v e r e p l a c e d N~ (and Z~) by A.~ and r e d e f i n e d a. T h i s a p p r o x i m a t i o n i s v a l i d up to s e c o n d o r d e r in the s q p p o s e d l y s m a l l q u a n t i t i e s ( N - Z ) and R p - ~,(2Z)~. T h i s f o r m u l a can be c o m p a r e d with the s e m i empJ~'ical m a s s f o r m u l a by putting Rp =R n =R = =~A~andx=N-Z. To s e c o n d o r d e r J n x w e obtain E =-(~+~-3)A+(~2a+ 5 )x 2 •

4~3

A- +

(3)

+ ~ - A ~2 + ~~ A ~ s. T h i s i s the u s u a l s e m i - e m p i r i c a l m a s s f o r m u l a e x c e p t f o r the t e r m i n t r o d u c e d to a c c o u n t for o d d - e v e n e f f e c t s which we s h a l l not c o n s i d e r . It i s p a r t i c u l a r l y i n t e r e s t i n g in eq. (3) that the t e r m xz / A o c c u r s a u t o m a t i c a l l y a s a c o n s e q u e n c e of o u r p l a u s i b l e a s s u m p t i o n s and is not a r b i t r a r i l y i n t r o d u c e d a s is u s u a l l y the c a s e . We f i r s t n e e d to c h o o s e a v a l u e of ~ and we t a k e ~, = 1.3 fro. T h e r e a r e then f i v e p a r a m e t e r s in eq. (1) and f o u r a r e o b t a i n e d by c o m p a r i n g eq. (3) with the s t a n d a r d f o r m u l a [1], w h i l e we o b tain the fifth f r o m the c a l c u l a t e d c o m p r e s s i b i l i t y of n u c l e a r m a t t e r [2]. T h e p a r a m e t e r s we obtain a r e in MeV and fm.

607

Volume 30B. n u m b e r 9 a = 42.3.

PHYSICS

~ = 23, p = 8.1.

~ : 0.92,

5 = 66.5.

(~ = 1 3 ) .

(4)

It i s a u s e f u l but not v e r y s t r o n g c h e c k on o u r a s s u m p t i o n s t h a t a l l t h e p a r a m e t e r s t u r n out to h a v e the correct sign. Eq. (2) i s e a s i l y m i n i m i s e d w i t h r e s p e c t to x . R p a n d R n. M i n i m i s i n g w i t h r e s p e c t to x only g i v e s t h e s t a b i l i t y l i n e . i.e. N in t e r m s of Z . T h e two r e m a i n i n g e q u a t i o n s a r e

2aA ~[Rp - ~(2Z)~] - flZ R p 2 - ~ Z 2 R p 2 = 0

(5)

2 a A ~ [ R n - )~(2N) ~] - ~g, R n 2 + 35NZ :)~n 4 = 0. (6)

H e r e we h a v e a s s u m e d t h a t R n ~ Rp. If RO :>R n t h e l a s t t e r m of eq. (6) a p p e a r s i n s t e e i d of e(l. (5). w h i l e i f R n =Rp e q s . (5) a n d (6) a r e a d d e d t o g e t h e r to g i v e a s i n g l e e q u a t i o n f o r t h e u n i q u e radius. T h e i n t e r e s t i n g r e s u l t o b t a i n e d f r o m e q s . (5) a n d (6) i s t h a t t h e l a s t t e r m of eq. (6) i s v e r y m u c h l a r g e r t h a n t h e t e r m s in e i t h e r ~ o r ) a n d f u r t h e r t h i s t e r m i s s o l a r g e t h a t it g u a r a n t e e s t h a t R n = R_ f o r a l l s t a b l e n u c l e i . P h y s i c a l l y t h i s implies thai' for heavy nuclei, where the neutrons m i g h t p e r h a p s b e e x p e c t e d to e x t e n d b e y o n d t h e p r o t o n s s i n c e t h e r e a r e s o m a n y m o r e of t h e m .

608

LETTERS

22 December 1969

t h e m u t u a l a t t r a c t i o n of t h e p r o t o n s a n d n e u t r o n s is so strong that the same radius is maintained f o r b o t h t y p e s of n u c l e o n s . S i m i l a r l y f o r l i g h t n u c l e i , w h e r e N = Z . t h e C o u l o m b f o r c e i s not s t r o n g e n o u g h to g i v e a s i t u a t i o n in w h i c h t h e protons extend further. T h e n u c l e a r r a d i i we o b t a i n a r e g i v e n , to t h r e e s i g n i f i c a n t f i g u r e s , by R = 1.23A~ f m f o r a l l s t a b l e n u c l e i , but t h e a c t u a l v a l u e 1.23 i s c o m p l e t e l y d e p e n d e n t on o u r a s s u m e d c h o i c e of h a n d t h e m a i n p o i n t i s t h a t w e o b t a i n a n AS d e pendence. This result is easily seen from eqs. (5) a n d (6) w h e n t h e ~ a n d ~ t e r m s a r e n e g l e c t e d . The remaining terms all have nearly the same A d e p e n d e n c e if we a s s u m e R d e p e n d s on A 3. Although the conclusions obtained above might b e e x p e c t e d to b e a p p r o x i m a t e l y c o r r e c t for a l l n u c l e i , in d e t a i l we e x p e c t d i s c r e p a n c i e s to a r i s e in each case both from shell effects and a rel a x a t i o n of t h e e x t r e m e a s s u m p t i o n s l e a d i n g to t h e 5 t e r m in eq. (2). In t h e c a s e of t h e l a t t e r t e r m we e x p e c t a n o n - z e r o r a n g e i n t e r a c t i o n a n d a r o u n d e d d e n s i t y d i s t r i b u t i o n w i l l l e a d to t h e p o s s i b i l i t y of R n a n d R p b e i n g s l i g h t l y d i f f e r e n t .

References I. A . E . S . Green. Phys. Rev. 95 (1954) 1006. 2. K.A. B r u e c k n e r a n d J . L. Gammel. Phys. Rev. 109 (1958} 1023.