Coulomb energy differences with a modified Heeht's mass formula, con- .... count the modification of the charge r~dius of the whole nucleus due to the pres-.
IL NUOVO C I ~ E N T O
Nuclear
VOL. 12 A, N. 1
1 Novembre 1972
Core Renormalization.
I. - C o u l o m b E n e r g y Differences. P. CAMIZ I s t i t u t o di F i s i c a dell' Universitd - R o m a I s t i t u t o N a z i o n a l e di F i s i c a N u c l e a t e - Sezione di R o m a
E. OLIVlEaI (*) a n d M. SCALIA (*) I s t i t u t o di M a t e m a t i c a dell' Universitd - R o m a
A. D'ANDREA I s t i t u t o di C h i m i e a dell' U n i v e r s i t h - R o m a
(ricevuto il 27 Dicembre 1971)
Summary. Core renormalization in light nuclei is introduced through a dependence of the core radius on the occupation number of the last shell, in a linear approximation. The corresponding coefficient for the shells lpg, lp89 ld~, 2s~, ld~, l]~_ is obtained by fitting experimental Coulomb energy differences with a modified Heeht's mass formula, containing four adjustable parameters. Evidence is found for a renormalization effect corresponding to a dilatation of the core.
1. - Introduction. I f a m a s s f o r m u l a for p a r e (j)- c o n f i g u r a t i o n d e r i v e d f r o m g r o u p t h e o r y has to give m o r e t h a n a q u a l i t a t i v e a g r e e m e n t w i t h e x p e r i m e n t a l d a t a m a n y corrections m u s t be m a d e , in order to t a k e i n t o a c c o u n t t h e d e p a r t u r e of t h e ((true i n t e r a c t i o n ~) m a t r i x elements, w h i c h a p p e a r in t h e f o r m u l a as a d j u s t a b l e p a r a m e t e r s , f r o m t h e c o n s t a n t v a l u e i m p o s e d b y t h e m o d e l ; at t h e
(*) Partially supported by a C.N.R. grant. 71
72
P. CAMIZ, E. O L I V I E R I , M. $CALIA a n d A. D ' A N D R E A
same t i m e one should t a k e into account the deviation of the true s t a t e of the s y s t e m f r o m the simple m o d e l state. B o t h these corrections, which m a y be m o r e or less sophisticated according to the desired a c c u r a c y of the procedure, h a v e a disastrous effect, when t r e a t e d microscopically; the H i l b e r t space becomes larger and larger, configuration mixing comes into the g a m e a n d the simplicity of a model b a s e d u p o n group t h e o r y is e v e n t u a l l y c o m p l e t e l y lost. F o r these reasons, in order to i m p r o v e a mass f o r m u l a a l r e a d y discussed in a series of p a p e r s (1), we h a v e t r i e d to t r e a t some of these corrections, n a m e l y those concerning the dependence of the i n t e r a c t i o n m a t r i x elements on t h e state, in a w a y which allows us to r e t a i n the simplicity of the model; t h a t those m a t r i x elements (the coupling constants of t h e model) should depend on s t a t e q u a n t u m n u m b e r s is a r e a s o n a b l e guess, b u t it r e m a i n s a m a t t e r of choice which q u a n t u m n u m b e r s are r e l e v a n t for such a dependence; nevertheless, if we suppose t h a t this coupling-constant r e n o r m a l i z a t i o n is connected in some w a y w i t h core excitations we can r e a s o n a b l y a s s u m e t h a t the occupation n u m b e r of the shell n : A - - A o and Tz give t h e m a i n contribution; effects depending on other q.n. are likely to be connected w i t h m i n o r details of the core deformation. We m a y notice t h a t n a n d Tz are the two diagonal generators of the ranktwo Sp~ Lie algebra in which our m a s s formula has b e e n derived, so t h a t an n a n d T~ dependence of t h e coupling constants in the mass formula still leaves the H u m i l t o n i a n diagonal in t h e seniority scheme; for this s t a t e m e n t to be correct it is also necessary to a s s u m e t h a t t h e core excitations, which are phenomenologieally described in this p a p e r , correspond to the isoscalar a n d isovector monopole m o d e J = 0 +, T ----0, 1. F o r our purposes this k i n d of excitation can be easily r e p r e s e n t e d b y a v a r i a t i o n of the core radius, so t h a t t h e coupling constants G depend on the q.n. only t h r o u g h t the radius which gives the scale of the system. We can t h e n generalize the results p r e s e n t e d in a recent l e t t e r (2): the G's corresponding to a n u m b e r of interactions homogeneous in the co-ordinates were renormalized b y the i n t r o d u c t i o n of a scaling f a c t o r ] related to the degree of h o m o g e n e i t y p, according to the following formula:
(1)
G~(n) =
a~(O)
rk( )l ,, Lk-~j
,
L -- -
pld
.
k(n) is the n - d e p e n d e n t s t r e n g t h of a h.o. potential, equivalent to the core radius as far as t h e scale is concerned, a n d n is the occupation n u m b e r of the shell. This f o r m u l a can be easily e x t e n d e d to the case in which t h e G's d e p e n d (1) A. (2) 33
a) P. CAdiz and U. CATTA~I: NUOVO Cimento, 64B, 56 (1969); b) P. CAMIZ, D'ANDREA, E. OLlVIERI and M. SCALIA: Nuovo Cimento, 2 A, 393 (1971). P. CAMIZ, A. D'A~DREA, E. OLIVIERI and M. SCALIA: Lett. 2~uovo Cimento, 2, (1971).
NUCLEAR
CORE
I~ENORMALIZATION
73
- I
on a set of q.n. {~,}: (2)
~:({~,}) = ~,({~7}) Lk({~,})J
'
where {a~} are the core q.n. The unknown function k({a,}) can be expanded in power series of the q.n., starting from the core values, and the coefficients treated as free parameters in a fit to experimental data; since k is a smooth function of the q.n. we shall keep only the first few terms of the Taylor expansion. We give here the result for the n dependence of G~:
(3) ~:(n) v,(o){1 +t,[~-~]
lr
l~/dk\~
ld~k]n,+...} '
where G(0) is the bare coupling constant corresponding to a perfectly inert core; all derivatives are calculated for n = 0. In our mass formula the strengths of a generalized pairing interaction, the single-particle energies and the coefficients of the Coulomb interaction appear as coupling constants; actually only the Coulomb interaction can be thought of as a homogeneous function, of degree p = - - 1 , while the nuclear interaction, only in the limiting case of large nuclei, can be considered as a deltafunction whose scaling factor is given by / t ~ = ~ ( m - - 1 ) f o r an m-body interaction. I n the region of light nuclei, where our mass formula has been applied, this approximation is no longer valid, so t hat the scaling factor of nuclear coupling constants has to be treated as an adjustable parameter, the value ]t~ being only an order-of-magnitude estimate. For single-particle nuclear energies the scaling factor is 89 and for the core energy one has to take 89for the nuclear part, considered as the sum of single-particle energies, and 88for the Coulomb part. Therefore it seems convenient to single out the Coulomb interaction from the mass formula, and to extract from experimental data on Coulomb energy differences between members of the isospin multiplets the values of the parameters of formula (3), namely the derivatives of k with respect to n (and other q.n.). These values will then be used in a fit to nuclear energies with the complete mass formula (including also an effective 4-body interaction (lb)), in which the nuclear scaling factors will be treated as free parameters and compared with the afore-mentioned values obtained for the delta-force; these results will be presented in the second part of this work.
2. - T h e C o u l o m b m a s s
formula.
Among the different formulae available for describing the Coulomb energy differences between members of an isospin multiplet we have chosen a formula
74
1~. CAMIZ, E . OLIVIEI~I, M. SCALIA
a n d A. D ' A N D R E A
derived by l=fnol~ (3); this formula is the most suitable for our purposes since it has been obtained b y the same algebraic approach by which our nuclear mass formula has been deduced; the Coulomb interaction has been decomposed by HECKT into its irreducible tensor components with respect to the Lie algebra Sp4, in which (j)~ configurations are classified. The Coulomb energy difference between two members of an T-multiplet can be expressed as
(4)
AEe(n , T, j; T , T z + 1) = E(~)(n, T, j) --3(1 + 2 T ) E(~)(n, T, j), T = (n--n~)/2,
where the E's, according to tIEoKT, are given by the following formulae: for seniority v = 0: E (1, = 3a~ q- 3b(n-- 1) q- 12c(j q- 1),
(Sa) E (2) =
b +
c -- c
(,t--2j--1)~-- (~j + ~t)~. ( 2 T - - 1 ) ( 2 T + 3)
'
for seniority v = 1:
E (1) = 3a~ + 3b(n --i) + 12e(j q- 1) q+ (5b) E (2)= b -k e - - c
3 c ( n - - 2 j - - 1) 2 T ( T -k 1)
3e(2T + 1)(2j + 3)(--)~ ~+r 2 T ( T + 1)
(n - - 2 j --1) 3 - (2j q- 3) 3. 4 T ( T + 1) e2
ao = ~ (2J + 1)/(2j + 1)", J.ic
(6)
b ---- [2(j + 1) V2 -- Vo]/2(2j + 1),
e = (V0-- V2)/4(2i + 1). We consider only low-seniority states (v
