Repulsive core and nuclear shell theory

Nuclear Physics 4 (1957) 615--624; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the

publisher

I (~ R E P U L S I V E CORE A N D N U C L E A R SHELL T H E O R Y M. B A U E R and M. M O S H I N S K Y

I nstituto National de la Investigacidn Cientiflca e Ins tituto de Fisica, Universidad de M~zico, Mdxico, D.F. Received 20 J u n e 1957 We discuss t h e effect on n u c l e a r shell t h e o r y of an interaction potential between nucleons containing a repulsive core. W e s h o w t h a t a repulsive h a r d core gives rise to the s a m e interaction energy as a d e l t a function potential of s t r e n g t h 4~(ti~a]ra), where a is the range of t h e core and m the m a s s of the nucleon. The interaction energies for Y u k a w a p o t e n t i a l s w i t h a n d w i t h o u t repulsive cores, are compared.

Abstract:

1. I n t r o d u c t i o n The work of Brueckner 1), B e t h e 2) and their collaborators, has shown t h a t it is possible to assume for the interactions between nucleons in nuclear m a t t e r the same potentials as between free nucleons, The phenomenological nucleon-nucleon potential recently obtained by Gammel, Christian and Thaler a) contains a hard core of a range t 0.5/, and the presence of this core causes serious difficulties in problems of nuclear structure. The purpose of this paper is to show how one can deal with the problem posed by the repulsive core in nuclear shell theory. It is quite clear t hat if one takes a repulsive core of finite range, whose strength goes to oo, the standard perturbation methods will lail. Yet, if we look at the problem of the bound states in a square well potential with a repulsive core, which can be solved exactly, we see t hat for a sufficiently narrow core, both energy levels and wave functions do not differ much from the same problem without a hard core. This suggests t hat a perturbation method should exist for solving this problem, albeit it is not the standard perturbation method. This gives the basic idea for dealing with repulsive cores in nuclear shell theory. We shall consider first the problem of two nucleons moving in a common potential of the harmonic oscillator form, and interacting only through a hard core. Taking advantage of the form of the common potential 4), we can pass to relative and centre of mass coordinates for the two nucleons, and reduce the problem to t hat of a single particle moving in a harmonic oscillator potential with a hard core. The radial equation for this problem can be transformed b y a simple translation in such a way t hat the repulsive t We denote b y I the length 10 -la cm. 615

616

M.

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AND

M.

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core disappears, and an interaction potential appears, which can be treated b y standard perturbation methods. When passing back to the initial coordinate system, we show t hat the hard core acts to first order as a delta function potential of strength 4:~(?~a/m), where a is the range of the core and m the mass of the nucleon. We next consider an interaction potential between nucleons formed by a hard core plus an attractive central potential of the Yukawa type. The influence of this interaction potential in nuclear shell t heory will be analyzed, taking for the parameters of the Yukawa potential the values given by Gammel et al. 3) We shall start our discussion by reviewing some of the results for the two nucleon problem in a harmonic oscillator common potential, t h a t will be useful for the following developments. 2. T h e T w o N u c l e o n

Problem

in the Harmonic Potential

Oscillator Common

The main advantage in using for the common potential of the nucleons the harmonic oscillator potential discussed by Talmi 4), Thieberger 5) and others, is the separability of the problem of two nucleons in an appropriate coordinate system, even in the presence of interactions between nucleons. Let us write the two nucleon halniltonian for a harmonic oscillator common potential of frequency ~o as

H :

(2m)-lp12+(2m)-lp22+½mw2r12+½m~o2r~'Z-+-V(lrl--r2[),

(1)

where V is the interaction potential which will be assumed central. Introducing the relative and center of mass coordinates r':

r l - - r 2,

r" :

½(rl+r2),

(2)

P" ---- P l + P 2 ,

(3)

and their corresponding momenta P' :

_~(Pl--P2),

the Hamiltonian becomes H

---- j ; $ - 1

p,2+ ¼moY'r '2 + V (r') + (4m)-I p,,2+ moj2 r' ,2.

(4)

The angular m o m e n t u m associated with the different coordinates is given by 11 = r l × p l etc., and from (2) and (3) we see that 11+1 ~ :

1'+ 1" :

L,

(5)

where L denotes now the total angular m o m e n t u m of the system. We shall designate the wave functions associated with the different coordinates, without the interaction potential V, by the bracket notation of Dirac in the form

i;llll;nl ~' : r l 1 R . x z,( v, rl)YlXml(O1, 91) etc.,

(6)

I~PULSIVE

CORE

AND

NUCLI~R

SH~LL

THEORY

(~[7

where Ytl ,~i are the spherical harmonics, Rnl zlare the radial functions given in the previous references 4,s), and v = m¢o/~i. For In'I'm'>, v should be changed 4) to v' = ½v, while for [n"l"m"') it should be changed to v" = 2v. A wave function with a definite t o t a l angular m o m e n t u m could b e f o r m e d from (6) and the Clebsch-Gordan coefficient (lll,mlm2lLM) to obtain

'nil D n~12, L M ) =

Y.,,,im~Lnlllml)

In212m~)(lll2mlm2[LM).

(7)

A similar wave function could be f o r m e d for the r', r" coordinates giving

in'l',n"l",LM)

:

~,~,,,,,,In'l'rn')]n"l"rn")(l'l"m'm"jLM).

(8)

Because of (5) the t r a n s f o r m a t i o n brackets taking us from (8) to (7) would contain the same L, so we could write

[nlll, hal 2, L M ) : ~,~,~, n,,~,,{in'l', n"l", L M ) (n'l', n " l " , LMJ n i l 1, nal~., L M ) } ,

(9)

where the conservation of e n e r g y 4) restricts n'l', n " l " , to the positive integers t h a t satisfy the relation

t~oJ(2n'+l'+2n"+l"+3)

= hco(2n~+ll+2n2+lz+3 ).

(10)

F r o m (9) and (10) we see t h a t the i n t e r a c t i o n energy E ( n l l 1, n21~., L) associated with the central p o t e n t i a l V(r') will be given b y

E(nxll, n~12, L) = X,~,,,,~,,,,,[(n'l', n " l " , L M t nllx, n2l~, LM}]2s(n'l'),

(11)

where e(n'l')is associated with the interaction energy for the r ' coordinate, and is usually given b y

s(n'l') = ( n ' l ' m' I V(r')ln'l' m' }.

(12)

The expression ( l l ) will be our f u n d a m e n t a l formula, and in the n e x t section we shah show how to calculate ,(n'l') for a repulsive h a r d core potential. 3. T h e H a r d

Core Potential

L e t us assume for the m o m e n t t h a t the interaction p o t e n t i a l V(r') of the previous section is a repulsive h a r d core of range a. To o b t a i n the interaction energy due to this potential, we look at the h a m i l t o n i a n as given b y (4), and notice t h a t it is separable in the coordinates r', r", and t h a t the radial function R v (r') satisfies the equation [--

(~2Im)(d2/dr '2) +U,,(r')]R v

=

E' R v (r'),

(13)

where

U z, (r') = (~2/m)l' (l' + 1) (r') -2 + ¼mo,Z(r') 2,

(14)

with r' restricted to a <= r' <: o0. T h e radial function satisfies the b o u n d a r y condition

618

M.

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&, ( a ) = o.

(15)

This problem can be solved exactly e), but it is of more interest to devise a perturbation procedure for the solution as we would like to express the interaction energy in the form (11). For that purpose we introduce in (13) the change of variable r' = r + a , (16) so that the equation can be written as {--(?i~/m)(d2[dr~)+Ur(r)+EUr(r+a)--Ur(r)]jR,,

= E ' R v,

(17)

where r is now restricted to the interval 0 ~ r ~ oo. From (15), (16) we see t h a t R r considered as a function of r satisfies now the boundary condition

R,, (o) = 0

(is)

The eigenvalues E ' of the energy for the problems (13), (15) are clearly identical to those of (17), (18), but the advantage of the last form is that for a sufficiently small a, the term in the square brackets in (17) could be considered as a perturbation to which a standard perturbation procedure could be applied. We expand the square bracket in (17) in powers of a to obtain Ut,(r + a ) - - U t , ( r ) = a ( d U v / d r ) + (a2/2) (d2Uv/dr ~) --aa(?i~/m)l ' ( l ' + 1) ( 4 r + 3 a ) r - 4 ( r + a ) -z,

(19)

where the last term is the residue of the expansion. From (19) we see that to the first order in a the energy E ' would be given b y E' = ?io~[2n'+l'+{)]+s(n'l'),

(20)

where , ( n ' l ' ) has the form [R,, t,(v', r)] 2 [dUt,/dr]dr, (21) o the R., z'(v', r) being the radial functions for the harmonic oscillator potential as in (6). We can evaluate the matrix element in (21) in a simple fashion b y taking the derivative of the unperturbed equation for R., ¢(v', r) with respect to r to obtain [-- (?t*/m) d*/dr 2+ U,, (r) -- ?~w(2n' + l ' +!2) ] (dR,, ,,/dr) (22) ,(n'l') = a

= -- (dUt,/dr)R,¢ r

Multiplying (22) b y R~, v and integrating with respect to r from 0 to co, we see from the boundary condition (18) and the equation satisfied b y R~, r that a

o [R,¢t,(v', r)]~[dUr/dr]dr = (?~2a/m)[(dR,¢t,/dr),,_o]*.

(23)

From the explicit form *,6) of the R,,t,(v', r) radial function we have

THEORY

619

l' v~ 0,

(241

e (n' l') = (?i2a/m) [(dR n, o/dr),~0] 26~,0.

(25)

REPULSIVE

CORE

AND

NUCLEAR

SHELL

[dR,, v(v', r)/dr],=o = 0,

if

so the interaction energy 8(n'l') is given b y We have obtained to first order in a the interaction energy in the r ' coordinate system for a repulsive h a r d core. The discussion of the second order perturbation effects, and the form of the perturbed wave functions, will be given elsewhere e). Using (25) and (11) we shall show in the next section t h a t to first order in a, the interaction energy for a h a r d core is the same as t h a t due to a function potential of strength 4~(?~2a/m). 4. T h e I n t e r a c t i o n E n e r g y f o r a H a r d C o r e P o t e n t i a l As 8(n'l') # 0 only for l' = 0, we have to t a k e the Dirac bracket in (11) w i t h l' = 0, and from (8) we see t h a t l" = L, so the interaction energy for a repulsive h a r d core potential of range a is given b y

E ( n i l 1, n2l 2, L) = (1~*a/m)~.,,,,, [] * [(dR n, 0/dr),_o] ~,.

(26)

where from (10), n , ~tt are positive integers restricted b y the relation t

n'+n"=

(ni+nz)+~(ll+l,--L),

(27)

w i t h l x + l , - - L being even because of p a r i t y considerations *). F r o m (7) a n d (8) one could evaluate e) the Dirac bracket in (26), b u t it is more interesting to compare the interaction energy with t h a t due to a function potential of the form 4zrg6 (r t - r,) = 4~g8 (r'),

(28)

where g is so far an a r b i t r a r y constant. The interaction energy in the r ' coordinate system will be given b y (12), and expressing 8(r') in polar coordinates 7) we obtain =

f:f~

{fo

'

(29)

[Y,,,.,(O', ~v')120(0')6(9')d0'dg'}.

Taking into account t h a t lim,,__m JR,,, ,,(v', r')/r'] = (dR,, ¢]dr'),,=o

(30)

we see from (24) a n d (30) t h a t the interaction energy for a 6 function potential becomes e(n' l') = g[ (dR,,o/dr')r,=o]'eSv o. (31) Comparing (31) and (25) we see t h a t t h e y are identical if g =

(32)

620

M. B A U E R

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As the interaction energy for two particles in /'/" coupling and also the interaction energy for more t h a n two particles, could be obtained 8) from (26) with the help of 9/" coefficients or of fractional parentage coefficients, we see t h a t to first order a repulsive h a r d core potential of range a is equivalent to the 6 function potential

4z~(t~2a/m)(5(r 1 - r2).

(33)

So far we have considered only a repulsive core potential. As the actual interaction contains an attractive part plus a h a r d core, we discuss this problem in the next section.

5. T h e I n t e r a c t i o n E n e r g y for an A t t r a c t i v e Y u k a w a P o t e n t i a l w i t h a H a r d Core We now take for the V(r') in (4) a potential of the form + oo

if r' < a

(34)

V(r') = _Vo(b/r,)exp[_(r,/b)] ' if a ~ r' =< oo, where V 0 is the depth and b the range of an attractive Y u k a w a potential. The radial equation in r' will also be given b y (13), if we introduce in it the attractive part of the potential V(r'). As in section 3, we make the change of variable r' ---- r+a and we see t h a t the perturbation term will now be U s,(r+ a) -- U s,(r) + V (r + a), (35) where r extends to the interval 0 _< r _< oo. If we again restrict the effect of the h a r d core to first order in a, we see from (35), (34) a n d the discussion in the previous section t h a t the equivalent potential will be

4.~ (h2a/m)t~(rl-- r2) --VoEb/([rl-r~l + a ) ~ exp [-- ([rl--r,r +a)/bj.

(36)

To first order perturbation t h e o r y the effects of the two terms in (36) are additive, and the contribution of the second term could be discussed with the help of the general methods developed by Talmi 4) and Thieberger 5). A particular case of great importance is the one in which both a and b are small compared with the nuclear radius R 0. In t h a t case the second t e r m in (36) could be substituted by an equivalent delta function potential, whose strength 4ng' would clearly be given b y the relation

f f f {-VoEb/(,÷a)] exp [--(r÷a)/b]}dr

=

,~g' J'f f,~(r)dr,

(3v)

where d r is the volume element, F r o m (37) we see t h a t

g' = j'~ {--Vo[b/(r÷a)] exp [--

dr

-~ --Voba{ (1--c) exp (--c) ÷c*[--Ei(--c) l},

(3s)

REPULSIVE

CORE

AND

NUCLEAR

SHELL

THEORY

621

where c = a/b and 9) - - E i ( - - c ) = f : x -1 exp ( - - x ) d x .

(39)

F o r a Y u k a w a p o t e n t i a l with a h a r d core for which b o t h a and b are small c o m p a r e d with the nuclear radius Ro, the i n t e r a c t i o n e n e r g y is the o n e - d u e to a 5 function potential -- 4~G~ ( r l - - r 2 ) ,

(40)

where G is given b y

G = Vob3{(1--c) exp (--c)+c2[--Ei(--c)]}--(~2a/m).

(41)

F o r a Y u k a w a potential w i t h o u t a repulsive core

V(r') = --Vo'(b'/r' ) exp (--r'/b'), with 0 ~ r' ~ oo,

(42)

we h a v e Ior b' small c o m p a r e d with R0, t h a t it can be s u b s t i t u t e d b y the p o t e n t i a l (40) where G is replaced b y a G' which b y an analysis similar to (37) is given b y G' : V 0' b 'a. (43) W i t h the help of (41) and (43) we can c o m p a r e the effects in nuclear shell t h e o r y of the potentials with a repulsive core of G a m m e l et al. 3), and of the potentials w i t h o u t a repulsive core of Feshbach and Schwinger (as q u o t e d b y B l a t t and Weisskopf 10)). F o r a delta function p o t e n t i a l the i n t e r a c t i o n energy between nucleons in the same shell is different from zero s) only for even L. If we restrict ourselves to interactions between identical nucleons, the wave function m u s t be a n t i s y m m e t r i c and as L is even, the nucleons m u s t be in the singlet state. G a m m e l et al. 3) give for the even singlet p o t e n t i a l the parameters V o = 896.6 MeV (for the p - - p potential), a = 0.5/, b ---- 0.588/,

(44)

so t h a t c = a/b = 0.85, and from (41) we obtain G = 28.11 /3 MeV. F o r the singlet potential of F e s h b a c h and Schwinger lo) we h a v e

(45)

V o' = 48.12 MeV, b' = 1.165/, so t h a t the G' of (43) becomes

(46)

G'=

76.1 /3 MeV.

(47)

F r o m (45) and (47) we see t h a t the i n t e r a c t i o n e n e r g y b e t w e e n identical nucleons in the same shell and in h e a v y nuclei for an interaction p o t e n t i a l with a repulsive core, is a b o u t 37 % of the i n t e r a c t i o n e n e r g y for a p o t e n t i a l w i t h o u t a repulsive core. So far we h a v e not given a n y numerical values to show w h e t h e r it is valid to t r e a t the repulsive core and the Y u k a w a potential with a repulsive core to first order p e r t u r b a t i o n , for p a r a m e t e r s such as those of (44). In

62~

M. B A U E R A N D M. M O S H I N S K Y

the n e x t section we use some results for the 6 interaction p o t e n t i a l to discuss these points.

6. Interaction Energy for two and three Particle Configurations In the previous sections we showed t h a t the i n t e r a c t i o n e n e r g y for a repulsive core would be the same as t h a t due to the (~ p o t e n t i a l (33). F o r an a t t r a c t i v e Y u k a w a potential of short range with a repulsive core, t h e i n t e r a c t i o n e n e r g y can also be o b t a i n e d from a (~ potential as shown in (40). F o r the two particle configuration in ii coupling, the interaction e n e r g y for a ~ potential was given b y Zeldes 11) in the form

((nli ) 2JMi4zeg~5(r 1 - r 2)1 (nli) 2J M ) =

w[(

zi)2,

j],

(48)

where v -----mo~/h, and W[(nli) 2, Jl is t a b u l a t e d for several shells a n d for J = o, 2, 4 in table 3 (where it is w r i t t e n as W(i2J)) of the p a p e r of Zeldes11). F o r higher values of the t o t a l angular m o m e n t u m J , we could o b t a i n W E(nli) 2, J] b y the use of a recurrence relation derived b y one of us (M.M.), t h a t has the form s)

W[(nli) 2, J + 2] = ~J+ 1) 2 (21+ 1) 2 - ( ] + 2 ) 2" W[(nli) 2, J] \1+21 ( 2 i + 1 ) 2 - ( 2 + 1 ) 2

(40)

The interaction energy for a three particle configuration will have the form (48), if we replace W[(nl]) 2, J] b y W[(nli)L J] defined as s)

W[(nl]) 3, J ] = • 3 [ ( ( i ) 2 J ', i, Ji}(i)a*tJ)]zW[(nlj) ~, J'],

(50) j. where the t e r m inside the square b r a c k e t is the a p p r o p r i a t e fractional p a r e n t a g e coefficient, x represents o t h e r q u a n t u m n u m b e r s necessary t o define the state, and J' is restricted to even values. F r o m the work of Schwartz and de-Shalit 12) the fractional p a r e n t a g e coefficients can be given in t e r m s of R a c a h coefficients, so t h a t evaluating these coefficients s) for i = 5/2, 7/2, we can give in the following tables the values of W[(nli) 3, J] in the i : 5/2 and i -~ 7/2 shells. F r o m table 3 of Zeldes 11), and from the following tables for the i n t e r a c t i o n energy in the three particle configuration, we notice t h a t the W's are always smaller t h a n 1, and decrease when J and i increase. An u p p e r b o u n d Emax for the i n t e r a c t i o n energy would t h e n be given b y

Emax -~ g(2valze)'/~ -~ g(2/z~)'/,(hw/41.36)'/, MeV,

(51)

where g is given in /a MeV and ~eo in MeV. F o r a repulsive h a r d core of range a = 0.5/ we see from (33) t h a t g = 20.68 /aMeV so t h a t Emax becomes Emax = 0.496 MeV if ~o~ : 4MeV, Emax = 5.55 MeV if ~ = 20 MeV, and Emax is small c o m p a r e d with ~ in b o t h cases.

(52)

REPULSIVE

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623

THEORY

TABLE 1 The coefficient W[(nl]) 8, J ] for the ] = 5]2 shells. ld~/,

lfs/,

2d,l,

2f5/,

3ds/2

]

3]~

0.450

0.328

0.315

0.233

0.256

5]2

0.788

0.575

0.551

0.408

0.448

9/2

0.293

0.213

0.205

0.152

0.167

TABLE 2 The coefficient W[(nl]) 3, J] for the j = 7/2 shells.

3/2

liT/,

lgT/t

2f71,

0.328

0.258

0.233

2g7/, 0.185 !

5/2

0.395

0.311

0.280

0.223

7]2

0.766

0.603

0.544

0.432

9]2

0.264

0.208

0.187

0.149

1112

0.275

0.217

0.195

0.155

15]2

0.165

0.129

0.117

0.093

F o r the e v e n singlet Y u k a w a p o t e n t i a l w i t h a repulsive core whose p a r a m e t e r s are g i v e n in (44), an u p p e r b o u n d for t h e a b s o l u t e v a l u e of t h e i n t e r a c t i o n e n e r g y is g i v e n b y ( 5 1 ) w h e r e g t a k e s the value of 28.11/3 MeV, so t h a t Emax = 0.674 MeV if ~co = 4MeV, Emax = 7.54 MeV if ~ o = 20 MeV.

(53)

F r o m (53) a n d the a b o v e tables, we see t h a t the i n t e r a c t i o n e n e r g y is small c o m p a r e d w i t h the s e p a r a t i o n b e t w e e n e n e r g y levels in t h e h a r m o n i c oscillator potential, for b o t h values of ~o~ t h a t are considered, a n d therefore restricting t h e i n t e r a c t i o n e n e r g y to first order seems reasonable. W i t h t h e p a r a m e t e r g i v e n in (45) we could m a k e an e s t i m a t e of ~o~. T h e s e p a r a t i o n 1 1 ) b e t w e e n the levels J = 0 a n d J = 2 in Ca 12 is 1510 keV. A s s u m i n g t h a t this s e p a r a t i o n is due to the difference in i n t e r a c t i o n e n e r g y for t w o n e u t r o n s in t h e first f'/2 shell, w h e n t h e y are in t h e J = 0 a n d J = 2 states, we see f r o m (48) where we t a k e g = 28.11 ]a MeV, a n d f r o m (51~ a n d t h e t a b l e 3 of Zeldes n ) , t h a t ~eo = 9.8 MeV.

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References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12)

K. A. Brueckner, Phys. Rev. 97 (1955) 1353 H. A. Bethe, Phys. Rev. 103 (1956) 1353 J. L. Gammel, R. S. Christian and R. M. Thaler, Phys. Rev. 105 (1957) 311 I. Talmi, Helv. Phys. Acta 25 (1952) 185 R. Thieberger, Nuclear Physics 2 (1956/57) 533 M. Moshinsky, Rev. Mexic~na Fis. 6 (1957} to be published Methods of theoretical physics, P. M. Morse and H. Feshbach (McGraw-Hill Book Company, New York, 1953) p. 830 M. Moshinsky, Phys. Rev. 106 (1957) 117 E. J a h n k e and F. Emde, Tables of functions, fourth edition, (Dover Publications, New York, 1945) pp. 1; 7 J. M. B l a t t and V. F. Weisskopf, Theoretical nuclear physics (John Wiley & Sons, New York, 1952) p. 103 N. Zeldes, Nuclear Physics 2 (1956/57) 1 C. Schwartz and A. de-Shalit, Phys. Rev. 94 (1954) 1257