[ Nuclear Physics A98 (1967) 161--176; (~) North-Holland Publishing Co., Amsterdam 1.D.1 I

Not to be reproduced by photoprint or microfilm without written permission from the publisher

T H E SELF-CONSISTENT STRUCTURE OF LIGHT NUCLEI

J. M. I R V I N E *

Laboratory of Nuclear Studies, Cornell University, Ithaca, New York tt

Received 13 February 1967

Abstract: Nuclear single-particle wave functions are expanded in sets of harmonic oscillator func- tions. The special properties of these functions allow the derivation of a relatively simple ex- pression for the nuclear reaction matrix elements in the reference spectrum approximation. The same expansion allows a reduction of the Brueckner-Hartree-Fock equations to a set of simple radial equations. The two calculations are coupled together in a straightforward iterative proce- dure for obtaining self-consistency.

The formalism allows a unique definition of a local shell-model potential for light nuclei. The structure of this potential is examined and some explicit expressions for the nuclei 4He and 160 are presented.

1. Introduction

Many attempts have been made to calculate the structure of finite nuclei self-con- sistently starting with only a knowledge of a two-body interaction constructed to reproduce the nucleon-nucleon scattering data. There are basically two lines of ap- proach, one based on the Jastrow cluster expansion 1 - 3) and the other on the Brueck- ner-Bethe-Goldstone reaction matrix theory 4,5). The relationship between the two approaches for the two-body clusters was given by Bether, Brandow and Petschek (to be referred to as BBP) 6) and was extended to the three-body clusters by Day v).

Most attempts to calculate the nuclear matrix reaction elements have made a local density approximation and then utilized the special features of a single-particle plane wave representation ~). The validity of this approximation in the nuclear surface is questionable. In the present work we shall expand out single-particle wave functions (the solutions of the Brueckner-Hartree-Fock equations) in sets of harmonic oscil- lator functions and use the reference spectrum approximation of BBP together with the special properties of these functions to obtain a simple expression for the required nuclear reaction matrix elements.

In the original papers of Brueckner and co-workers the solution of the Brueckner- Hartree-Fock equations was attempted directly and this resulted in great computa- tional complexity through which it was extremely difficult to obtain any very clear physical insight into the important features of the calculation. Recently Baranger and co-workers 8,9) have treated the matrix form of these equations using harmonic os- cillator functions to simplify the computation. In the present work we develop an

t On leave of absence f rom the Physics Department, Manchester University. ~t Work supported in part by the Office of Naval Research.

161

1 6 2 J . M . I R V I N E

equivalent formalism for the differential form of these equations. We show that for light nuclei this approach allows us to define uniquely a velocity-dependent, but local, shell-model potential. This potential exhibits all the usual simple features assumed for the shell-model potential and through it we can obtain some physical insight into the self-consistent structure of light nuclei.

2. Brueckner-Hartree-Fock theory

We assume that from the nucleon-nucleon scattering data we can construct a static two-body interaction and that, neglecting mesonic degrees of freedom, this interac- tion will represent the force between two nucleons inside a finite nucleus. This poten- tial will, in general, be non-central, non-local and contain an extremely strong short- range repulsion. In future calculations we intend to use the Reid soft-core potential 1 o) which has a Yukawa repulsion at short distances and represents a better fit to the scattering data than earlier two-body potentials, e.g. Hamada-Johnston 1~), etc. Thus for a free nucleus the Hamiltonian may be written as

A A

i tS (x , , x2 . . . XA) = ~ t(xl)+ ~', V(Xl, X;), (2.1) i = 1 i > j

where x represents space and spin coordinates and t the kinetic energy operator. Eq. (2.1) may be rewritten as

A A A

= Z [t(xi) + V(xi)] + E l)(xi, x j ) - Z V(x,) i = 1 i > j i

= Yfo + ~ 1 , (2.2)

where V(xi) is as yet unspecified. We now treat 3¢~ as a perturbation, that is we solve the problem

~ o ~o(X, . . . . XA) = Eo ~0(x, . . . . xa) (2.3)

exactly to obtain

q%(Xl . . . . XA) = (A !)--~ det ~i(xy), (2.4)

where the single-particle wave functions ~b i are given by the solutions of

I t ( x ) + = (2.5)

The perturbed wave function ~U(xl . . . . XA) is then the solution of

, J fg ' (x , . . . . xa) = E ~ ( x l . . . . XA), (2.6)

S E L F - C O N S I S T E N T S T R U C T U R E

where the Brueckner-Goldstone expansion for the energy E is la)

E = Eo + [ 0 0 -

163

The graphs are Feynman diagrams with the modifications that for (Kijkt--KUlk) where Kijkt is the solution of

K , gijkl = Vijkt-F- mnZ l)ijmn ~E~mnkl mnkl

. I Z l

l

stands

J

(2.81

with the propagator Q/E defined by

-- O, otherwise,

if states k, l are not occupied in ~o and states m, n are occupied in ~o

and the appropr ia te / ( -mat r ix elements are calculated "on the energy shell" as dis- cussed by BBP.

The one-body potentials V i are still at our disposal and for states occupied in cb o they are chosen to cancel the self-energy bubble diagram insertion, i.e.

m m ---X

The one-body potential for states unoccupied in 4% is then chosen to minimize the third-order term in (2.7). Bethe 12) has calculated the three-body terms in nuclear matter and his results suggest that probably the fourth- and higher-order terms are negligible. Thus to a good approximation the problem is solved by obtaining a solu-

164 J.M. IRVINE

tion to the zeroth-order calculation. Eq. (2.5) now becomes

t" ~ | ~ b ~ ( ' ' ~ . . . . . . -}- 2 < a . , ' " dx 2 dxl d x 2 t(x,) ,l(XO Xx,

= e~ ~bl(X,) , (2.9)

which together with eq. (2.8) defines the problem. The usual approach to a Hartree-Fock calculation is to start from a trial single-

particle representation and proceed by iteration and interpolation. In the Brueckner- Hartree-Fock calculation, we notice that we have an additional self-consistency prob- lem which arises from the fact that at each iteration of (2.9) we obtain a new single- particle representation which in turn produces a new two-body interaction K through

the dependence of .K on the propagator Q / E . An approximation introduced by Brueckner and used by nearly all subsequent

workers in the field is the assumption that the K-matrix is insensitive to the single- particle representation and depends only on the density. This allows t ( t o be calculated in a plane wave representation at the local density which is equivalent to assuming that the density does not vary greatly within the healing distance 13). This is clearly

a poor approximation in the nuclear surface. The question arises why is the plane wave representation so attractive? The reason

is because it allows a separation of the uncorrelated two-body wave function cbi~(Xa, x2) = ~bi(xl)~bj(x2) into relative and centre-of-mass coordinates. It is well known that a single-particle representation in terms of harmonic oscillator functions also allows such a separation 14- ~v). We propose then to solve the Brueckner-Hartree- Fock problem using expansions of our single-particle wave functions in harmonic

oscillator functions at all intermediate stages of the calculation.

3. The Schr[idinger equation

We shall solve eqs. (2.9) by iteration starting from a trial set of solutions (°)~b~(x) which we shall expand in sets of harmonic oscillator functions. These expansions we shall insert into the integrand appearing in eqs. (2.9) which are then solved to give a first iterate set of solutions which are in turn expanded in sets of harmonic oscillator functions, and so on until self-consistency is achieved, in general the ith iterate will be

fitted by (1)tp,ti,,~(x) = V (1),tj,,~,t, HO r ~ (3.1)

/ , ,ann" Wn,ljm~:~.X], n'

where the quantum numbers n, l, j and m have their usual meaning, and z is an isospin label. In writing (3.1) we are assuming that V i ( x ) in (2.5) is spherically symmetric, although not necessarily central, e.g. Vi could contain a one-body, spin-orbit force. This assumption is most likely to be valid for closed shell nuclei which are therefore the only ones which we shall consider the remainder of this paper.

SELF-CONSISTENT STRUCTURE ] 65

Eq. (2.9) is

t ( X l 1 X l -[- x 2 ) K ( x 1 , X2," X 1 , X 2 ) [ ~ t 2 ( X 2 ) I / I I ( X 1 ) -- I]/1 X2)~//2 x1)]dx2 dXl' dx2' 2=A

= e, ¢1(x, ) , (3.2)

where we have

I//l(Xl) = Z (2°'1, ll trlIiIJl lna)On,l,rat,j,ra,(i'l)Zai(1)~--r,(1) nlllal

: Z (10"1' 11 m i l l J l ml) L un,tHt(rl)Yl,m,,(~'~l)Za,(1).~-,,(1), (3.3) mlltrJ 1"1

with Z and ~-- spin and isospin functions, respectively, and a similar expression for $z. First we t rans form the product wave function associated with the pr imed co-ordi-

nates into relative and centre-of-mass functions

[ [ /12(Xr2)~ l l (X; ) - -~ l l (X t2 )~12(X; ) ] = Z Z ( n ( s l ) j m T M r ; N L M L I ( 1 2 ) J M ) nlNL j JST

× (1 - ( - 1) s +s + r-j,-j2)~b,lSimTMT(X, ' Z)qSULML(R, ' 2)" (3.4)

Then we change the variables of integrat ion f rom x] and x~ to x]2 and R]2. The only non-local i ty in K established by the scattering data is a relative angle non-locali ty which manifests itself in the need for a different fo rm of interact ion in different par t ia l waves. Thus we shall assume

K(xl, x2; x ' l , x~) = K(x12, x',2)b(R,2-R',2)6(r12-r',2), (3.5)

and hence we can carry out the integral over R'I2 in (3.2). We are then left with the produc t funct ion ~ *(X2)C~NLML(R12 ).

We mult iply eq. (3.2) by J - + (1) and sum over the isospin co-ordinate 1. This re- moves isospin f rom the eigenvalue te rm and the kinetic energy te rm and leaves us in the interact ion te rm with the produc t * a -+ ~2(X2),~" r,(1)dPNLML(RI2) which we now trans- fo rm to the co-ordinate spaces x12 and r l

• 3 - + ~/2 (x2) ,, (1)(gNLML(R12)

= --(2) -~- ~ ~, (n'iI'lj'lm'~; n'(S'l')j'm'T'Mrln212J2m2z2, ~,; NLML) l 'n'S'l ' j 'm 'T 'M'T

× I/In'I'S'j 'm'T,M,T(X1 2 ) ~ n , , l , , j , , m , , ( X l ) . (3.6)

Thus eq. (3.2) m a y be wri t ten

t(Xl)~kl(Xl) + ~ (n(Sl)jmTMr; NLMLI(12)JM ) t ! .t t ! ! ! .t ! t t . . x (n111 J1 ml, n (S I )j m T MTln212J2m2"c2, "C1, NLML)

× (1 - ( - 1) J+s+ r - ~ -J2)(n'(S'l')j'm'T'M~lKIn(Sl)jmTMT)~r(Xl)

= 21 ~/l(Xl). (3.7)

166 J.M. 1RVINE

Multiply (3.7) by Y+l,m,(l) ltr' m' " + I2 + = ~'m,'~(~ 1,11 l~lJzmx)Yl,=q( 1)Z~c(l)andcarryout

a sum and integral over the spin and space co-ordinates 1 to obtain

h 2 ( ~2 11(11 + 1)~ 2m c~r21 r~ ] Undjjl(rl)-+- n'lE Qln~fll~lUn'llljl(rl) = 13ndljlUndljl(rl)' (3.8)

where 12t'Jl = Z Z a"t"'t's'rjvsri (3.9)

n'ln I Zanln ' l l l j t aXn'l'nl S T j n'l'nl

with

anln'l'STj .~,',,,Jl = ~. E (1--(-- I)S+T+S-Jl-JO(n(SI)jmTMT; NLMLI(12)JM ) J N L M L 2 < A

x (n ' 111 j~ ml ; n'(Sl')jmTMT[n212 J2 m2z2, Zl ; NLML). (3.10)

The transformation coefficients are given by

(n(Sl) jmTMr; NLML[(12)JM) = Z (½'r,, ½Zzl TMT)(jl ml Jz rn2[JM) ,a.

x ((½½)S(l~/2)2; J[(½11)Jl(½12)J2 ; J)

x ((SOL L; JIS(IL)2; J)[nlNL21(12)2], (3.11)

where the usual LS-jj transformation coefficient is

((½½)S(1112)2; Jl(½11)J1(½12)J2 ; J)

= { ( 2 j ~ + 1 ) ( z j 2 + m ) ( z s + 1 ) ( 2 , ~ + l ) } ~ ll t2 , (3 .12)

Jl J2

the three angular momentum vector recoupling coefficient is

{5' " '} ((SI)j, L; JIS(IL)2; J ) = ( - 1)L+S+S+'+J{(22+ 1)(2j+ 1)} } L 2 '

a generalized Brody-Moshinsky bracket 15,17) is

[nlNL21(12)2] = Z ".,.'l"'*im"'2i~'~/"t~r~'"'~.~.'= \ , . - ,~.q, , l 1, n'212)o). (3.14) B' I t I ' 2

The remaining transformation coefficient can be shown to be

(nl 11 Jl ml ; n(Sl)jm TMrln212 J2 "(2, "C1 ; NLML)

= ~ (JM(12)ln(Sl)jmZMr; NLML>. (3.15) J M

Hence the equation for the ith iterated solution of (3.2) is

h2 ( 63r2 2 11(11+ 1)~ (° - - ~ n ' l n t ~ n ' l l l j l \ r l J 2m r 2 ] t l n t l l j l ( r l ) + n ' t E ( i - 1 ) ( ) l t J t (i), {. "~

= (°~.,l~j~ (i)u,oljl(q), (3.16)

SELF-CONSISTENT STRUCTURE 167

where (i--1)oll, j l contains the expansion coefficients ¢~-l)a~'Jm and the procedure rl 1/11 nln" 1 is thus completely straightforward once we know the reaction matrix elements r,-sra JXn,l,nl.

4. The reaction matrix elements

We must now turn our attention * to the solution ofeq. (2.8), but instead of solving it directly we consider instead the Bethe-Goldstone equation for the two-body inter- acting wave function Tij(xl , x2) which is

I t l i j (X 1 X2) ~ I~) i j (Xl , X2)-]- Q l ) l t l i j ( X l , X2). (4 .1 ) ,

Since by definition

K~i j = vTi j , (4.2)

clearly a solution of (4.1) is equivalent to a solution of (2.8). We solve eq. (4.1) in the reference spectrum approximation of BBP, i.e. we replace Q/E by 1/E a where

E R __.= h E 2m~*. (A'22 - ?ij). (4.3)

The reference spectrum approximation has been discussed for the case of nuclear matter by BBP and others 6, 19) and for the case of finite nuclei by Wong 17), Bran- dow 18) and Day 2o). Application of the reference spectrum approximation to finite nuclei has been made by Kuo and Brown 21). A process of successive approximations for approaching the nuclear K-matrix from the reference spectrum K-matrix has been clearly developed but we shall postpone discussion of these matters for the moment and return to them briefly later. It would appear from the nuclear matter calculations that the effective mass m* is very closely equal to the nucleon mass and that 72 is simply ( l m ) p E - ( e i + e j ) , where P is the centre-of-mass momentum of the pair of particles and is usually negligible compared with e i and ej.

We define a difference wave function Z~j = ~ j - T~j and write eq. (4.1) in the form

h2 I A 2 2 \ ~ ~ , ~ r12- -T i j )Z ' i j = - -19~ i j Z1711j

= - v ( ~ i j - Z i j ). (4.4)

We again make the harmonic oscillator function expansion of ~ j and transform to relative and centre-of-mass co-ordinates. To the approximation that m* and ?~j are independent of the centre-of-mass co-ordinate (to be discussed later) we may assume that in each relative partial wave the centre-of-mass functions for ~,-j, ~ij and Zii are the same, and thus

_ h2 ( 02 l ( l + l ) _ ? 2 j ) ( ~ / S r ( r ) _ ~ Vcr'jST~njST~tr" (r) 2m*. ~ r 2 ,,,

jsr HO (4.5) = -- l)l, l Unlj

t The procedure outl ined in this section has been developed independent ly by Wong 17). I am grateful to Dr. Brandow for br inging this work to my attention.

168 J . M . IRVINE

where once more the superscript HO indicates a harmonic oscillator function and plays the role of u(r) in the expansion of Z. Notice that v is non-diagonal in l because of the tensor force component and hence ~(r) is a two-component vector in this nota- tion. Once we know ~'/Sr(r) from eq. (4.5) the partial wave reaction matrix elements become easy to evaluate

K S T j .'t',t = < (O,,'rSjmrlKl~)nlSjmT>

= bll'l" t f f n lS jmTVl l - - b l l / 1" % o

HO j S T HO HO j S T n j S T dr = un,rj(r)Vrt ( r ) t tn t j ( r )dr- Un,rj(r)vt, r, (r)~u,, (r) (4.6)

for an interaction like the Reid soft core potential lO). Wong has given the cor- responding expression for a hard-core interaction 17).

Eqs. (3.16), (4.5) and (4.6) now represent a closed set of equations which are to be solved self-consistently. For convenience we now rewrite these equations together for the ith iteration

h 2 (~2 1(l+1)

2m* ~-r 2 r 2

(i-1)7~j) (O~,/Sr(r) _ Z ~vr'jsrt~.-., (Or"jsrl'~,r' t" l

- - , j S T , U O ( . ~ - - - - t T l t~nljl, t j ~

(i-- 1 ) I c S T j = HO j S T HO d r - HO j S T UnT j( r)Vrr, ( r) (i- a ) ~/,Sr ( r)d r "~,,'r,,t U.,rj(r)vri (r)u,,, (r)

h2 (~ 2 11(11+1}) (I-I)T,.ST j "~,,1,,'lt,j, "'.,'r.tO)u.'lhJ,(ra) 2m ~-r~ r~ ( i ) "ndHz( r l )+ Z( i - - l )An ln ' l ' s r j "

= (1)gn l l l j l ( i ) U n l l l j l ( r l ) . (4.7)

5. The shell-model potential in light nuclei

In general the solutions of eqs. (4.7) require the use of a computer to obtain purely numerical results. In this section we examine whether it is possible to proceed further without the use of a computer and hence hopefully achieve some physical insight into the nature of these solutions in the case of light nuclei.

It is often argued that in light nuclei a respectable representation of the single- particle wave functions is given by a single harmonic oscillator function, i.e. we shall assume

(5.1) ann" = ~nn""

Also we shall restrict our discussion to states where the principal quantum number is zero, i.e. 4He, 12C, ' 60 the p and d states of 4°Ca, etc. Eq. (3.8) may then be written a s

2--m r21 "', --"""' u . m j , ( r , ) l

= <,,j u.,,,s,(r, ), (5 .2 )

S E L F - C O N S I S T E N T S T R U C T U R E 169

where nl = 0. This is the equation of motion for a single-particle in what Baranger 8)

calls the "trivially equivalent local potential"

SM O tlj' u" ' l t 'J '(rl) (5.3) = Z - - ° . , . , n ' , t t n , l , j , ( r l )

and which we identify as the shell-model potential for the state na laja. Note that in general the denominator U,~l,~,(r~) will be given by the numerical solution of the previous iteration of (5.2), while the n u m e r a t o r un, t,j~(rl) will be given by a fit to these solutions over some finite number of harmonic oscillator functions and hence vSM t r a n~laj~\ 11 tends to zero as r~ increases because the harmonic oscillator functions

H O : E "~ u,,a~j~L 1) tend to zero more rapidly than u,~,j~(rl) for all n'~. This is the well-known feature that the harmonic oscillator functions underestimate the tails of the true nuclear wave functions. We shall assume that there is some radius R within which, to a good

H O / I x approximation, we can make use of (5.1) and hence set blnal~j~(rl) = U n l l l j l ~ " ) . It should be reasonable to neglect the tails of the wave functions since ]vSM I << e I for

large r~. Thus explicitly writing nl = 0 we have

S M l ~n'x l j / l l J l l l t + ~ Vot,j,(r,) = Z ( - +~(2vr2), (5.4) ±] t r n ' l O L , n ' l + l l n" I

where the

i2,+~ : ~ (nl +l~ +½)nl -1 n z + l l + ~ \ X ) ~ X nx - - X nl

+ ( n l + 1 1 + l ) ( n l + l ' - - ½ ) n ' ( n ' - - l ) x " ' - 2 - - . . . (5.5)

2!

are proportional to the associated Laguerre polynomials, v is the harmonic oscillator constant mco/h z and

I o " J ' (5.6) = t(n;)!(n; + t , +k)!J oo., ,o-

We now consider the W-coefficients in some detail. Writing down the energy re- quirements for the two Brody-Moshinsky transformation brackets 15) [see (3.10), (3.11) and (3.15)], we have

2 n + l + 2 N + L = 2na +la +2nz+12,

2 n ' + I ' + 2 N + L = 2n'~ + l~ + 2n2 + 12, (5.7)

and since the tensor force only couples the states l , j and ( 2 j - I ) , j , we have

( n ' - n) + 1 = ( < - nl). (5.8)

Thus we see that as W ~'j' becomes very non-diagonal then In-n ' l also becomes very • - n l n ' 1

large. Now W ~1il also contains the reaction matrix element r,,srs and separating K , ,Xn,l,nl " ' n l n ' l

into short- and long-range parts K ( S ) and K(L) [refs. 22, 23)] we have that K,, , (L)

170 J . M . IRVINE

tends to zero extremely rapidly as [n-n'[ increases from zero. On the other hand K,,,(S) is almost independent of the value [n-n'[ and is approximately one fifth of the value of the diagonal long-range matrix elements. Further, the short-range part of the interaction is essentially a &function and hence is only important in relative S-states. Thus only a few places off-diagonal the only coefficients of interest are those with l = l ' = 0 and for these coefficients the angular momentum coupling becomes trivial and we obtain

A n n ' O O S T S _~. o,',hj, ~ (-1)I~+L-1¼(2L+I)(2S+I)(2T+I)(211+I)-I(nONLL]OllOI2 L) I z L N

× (n'ONLLIn'~ Ix 0;2 L)(1 - ( - 1) s+r+L-t '-t2). (5.9)

A study of the Brody-Moshinsky transformation brackets 15) shows that the coeffi- nn" ! cients Ao,, ~ tend to zero as nl increases from zero, e.g. in 4He we have the identity

AO.'oosrs - ( 2 T + 1)(2S + 1)(1 - ( - 'aS+Tarl~""+2~ (5.10) 0n'10½ = " J ] k 2 3 V n ' n ' l •

This coupled with the initial rapid reduction of the matrix elements of the long-range component of K together with the factor{(/1 +½)!/(n'l)!(n'x +ll +½)!}~ in (5.6)allows us to truncate the summation on n] in (5.4) after only a few terms.

Let us consider various approximations to the shell-model potential obtained by truncating the summation on n], remembering that these approximations may only be considered valid out to some radius R beyond which the harmonic oscillator functions give a poor representation of the tails of the nuclear single-body wave functions and the shell-model potential is essentially zero.

(i) If we keep only the diagonal term, i.e. n'~ = 0 then we have

SM H / l l j l Vt,j~(r) = ,,oo r < R

= 0 r => R, (5 .11)

i.e. a square well potential. (ii) Keeping also the n' 1 = 0 term we obtain

SM I l J l " Vhj,(r) = (W~o ~(11 -~ -~2)W~loJ ' ) - W~ Ohj,2vr2 r < R

= 0 r > R, (5.12)

i.e. a harmonic oscillator potential. (iii) Keeping n' 1 = 0, 1 and 2 we have

SM ( l f l f l l J l 3 l l j l 3 5 l l J l = ) V / t J l ( r ) \ ' t O 0

- - I V r l 0

= 0 r => R. (5.13)

The approximations (5.11), (5.12) and (5.13) to the shell-model potential are sketched in fig. 1.

Working to the harmonic oscillator approximat ion (5.12) we see that in general we should have a different harmonic oscillator constant for each state 11, j l , but this pos- sibility we have excluded by the use o f the Brody-Moshinsky transformation. This only tells us the not surprising fact that true self-consistency cannot be obtained with a single harmonic oscillator function for each state. However, if we replace W'I'~ ' by its average over the states l~,jl then we obtain the trivial self-consistency condit ion

Wlo(h~o ) = ½hco (5.14)

for the harmonic oscillator constant. Note that the W-coefficients are functions o f the harmonic oscillator constant through the reaction matrix elements g s r s [see eq. a~n,l ,nl

, 2 3. r I I

V SM

SELF-CONSISTENT STRUCTURE 171

/ f

Fig. 1. Approximations (5.11), (5.12) and (5.13) to the shell-model potential for 4He are represented by curves a, b and c, respectively. The radius is in units of (2v)-½ and the energy in units of

OlO l O l (K0ooo + K~ooo). Using reaction matrix elements for the Reid hard-core potential lo) calculated by Dahlblom we estimate for curve b a depth of 60 MeV, an oscillator constant given by ho~ ~ 20 MeV

and a resulting binding energy per particle of 7.5 MeV.

(4.6)]. Al though we now have a single harmonic oscillator constant for all states this does not mean that we have the same shell-model potential in each state because the depth of the harmonic oscillator potential is governed by the diagonal W-coefficient which is still dependent on the state l l , j 1. Thus, for example, to the present approxi- mat ion the spin-orbit splitting o f the level 11 is given by

AL' = (W~" ' +~')- Wo ' f ' -~ ' ) . (5.15)

We close this section by giving a few explicit expressions for the nuclei 4He and

172 J . M . |RVINE

160. For these nuclei n 1 -- r/2 = 0 and l n and l 2 can only take the values zero in 4He and zero and one in 160. For S-states (l 1 = 0) we have

W°yo = ~ ~ ~ (-1)L+g~-~(1-(-1)L+s+r-~2)1(212+1)(2j+l)(2T+l)(21+l) -1 S T j nln'l' NLI2

× .sTj I t <.1NL 210001212>

× <n'~ 0012121n'l'NL12>. (5.16)

The energy conservat ion condit ion [see (5.7)] in the Brody-Moshinsky brackets de- mands that n = N -- 0 and n' = n'~ while the pair of numbers (l, L) are restricted to, the values (/2, 0) and (0,/2)" For 4He, this gives us

- - t ( / . ( 0 1 0 . T z l 0 l \ ralo ~ 3 t , (½)! 2~), / t,~,'~ooo -~ ~,'~ooo), (5.17) " n h O 2 n h + 1 [ (nO!(n , l+ .

while for x60 we obta in

- 3 2 . t x,," r z 0 1 0 - - Tzl01 "x , - - ~ n l ) ~ l X n , ~ O 0 0 -1- r,-nhO00)

, , , ' ,o 2 , , + 2 [(n 1) ](/I 1 .q. 1) ] 2

"1-(1 - - 2 t x . ~ / ~ 0 0 1 l l j l ) K n h lO I ) ] . -l-)-nl-J {-/~n',101 "q- E (2j"~- ( 5 . 1 8 ) ' j = 0

Natura l ly the results for the p-states in 160 are slightly more complicated but they exhibit the same general structure. We only illustrate the result for a relative S-state interact ion which should be valid for n' 1 > 3. Clearly such an interact ion cannot give rise to any spin orbit splitting and in fact the W-coefficients are independent of j~

was', 3 t ( ~ ) ! 1½ , 2 , - t - i01 lg-010 , _ _ . ~tXn,1000 ] , , , ' ,o -- 2 , , + 4 [(Htl)r(r/, l ...~½)!; [ ( n l - - 1 ) ( 1 + ~ - n l ) (Knqooo+

lr,,'~,tr,-lol ~ i,-olo ~3 (5.19) - - 3 - \ " 1 } \~X(n ' l + 1 )000 7- *X(n, 1 + 1 )00011"

I t is gratifying to note that the W-coefficients in (5.17)-(5.19) do indeed decrease extremely rapidly as n] increases.

Eq. (5.15) for the spin-orbit splitting in the 160 p-shell becomes

A ] s 9 1-,31d-ll0 , ' l l f l l l ~1¢-112 - ] _ 1 [O / . , z l01 A _ ~ / . , ' 1 0 2 103 = ~ t _ ' - ~ o l o l ± ° '~olol - - " ~ o l o u 16c~"~o2o2 . . . . o2o2-- 14Ko2o2]. (5.20)

Note that At~ is zero if the matr ix elements -~,t,tr"srJ are independent o f j , i.e. there is no spin-orbit splitting in the case of a central two-body interaction. For a two-body spin- orbit interaction, (5.20) becomes simply

A]s(LS ) 27/33,'.IK trM33,~\+lS/Sldl K rr~t31 d = - - 4 - ' , m Ls,, ,,I v~ ~ - \ I Ls~ jt >, (5.21)

where KLs(r) contains not only the radial fo rm of the two-body, spin-orbit force but also any exchange mixture associated with it. The contr ibut ion f rom the tensor force A¢,(T) is identically zero in the present approximat ion .

SELF-CONSISTENT STRUCTURE 173

The self-consistency condition (5.14) becomes in the case of 4He

1/2"1½/ld '010 ± /.(-10l "~ l h ( . 0 , (5.22) - -~ \3~1 \ X X l 0 0 0 T * ~ 1 0 0 0 / =

while for 160 it becomes

1 9 / 2 ] ½ / / ( 1 0 1 / .7010 ~ 3 [ 2 ~ ½ [ / d 1 0 1 , r1010 \ f ~ g ~ , ~ / ' , * ~ 1 0 0 0 " } - a X l 0 0 0 ) T T ~ ' ~ l y ) I * x 2 0 0 0 " J - / k 2 0 0 0 )

2 5 ( 2",t½(/(001 • l l j

3 2 \ Y ] \ J ' 1 1 0 1 -1- Z ½hO), ( 5 . 2 3 ) (2j + 1)K~o, ) = j = O

where we have used the approximate form (5.19) for w l J ' " ' 1 0 "

6. Discussion

We have presented in eqs. (4.7) an iterative procedure for the self-consistent solu- tion of the nuclear structure problem. However, before any serious numerical work is started on these equations several points must be examined in considerably more detail.

We have omitted any discussion of the corrections of the reference spectrum result (4.6). The corrections are of two distinct types; those arising from an improved treat- ment of the Pauli operator have been analysed in some detail by Wong 17) who finds the corrections to be significant but their inclusion should be relatively straightfor- ward using either the "global" or "local" approximations developed by Wong 17), a more serious problem arises from the spectral corrections due to an improved treat- ment of the intermediate states. There are two aspects to the latter problem. First, the spectral corrections to the reference spectrum calculation itself and, secondly, the need to derive an equation equivalent to eq. (2.9) for the intermediate states. In prin- ciple the intermediate state potential is completely at our disposal and it was suggested in sect. 2 that it be chosen to make the third-order terms in (2.7) negligible, if this is possible, and certainly it should be chosen to ensure rapid convergence of the pertur- bation expansion (see ref. 18) for a detailed discussion of this point). It is clear, that just as a solution of the two-body Bethe-Goldstone equation was necessary to give the occupied state interaction, that to define the intermediate state potential a solution of the three-body Bethe-Faddeev equations 7,12) will be required. This point could perhaps be investigated by carrying out a harmonic oscillator analysis of the Bethe- Faddeev equations similar to that given here for the Bethe-Goldstone equations. However, the consequences of such an approach have not yet been worked out in detail and as a result two limiting approximations have been frequently used. One can simply define the intermediate state potential in an equivalent manner to that used to define the occupied state potentials. This is the approach favoured by Baranger s' 9), and is in fact the hidden assumption whenever the matrix form of the Hartree-Fock equations are treated. This is equivalent to using eq. (3.16) to define both occupied and intermediate states as was done in sect. 5. Alternatively, one can follow Bethe's

174 J.M. IRVINE

suggestion a9) and set the intermediate state potential to zero as was assumed by Wong ~7) and also in writing down eq. (4.3). Clearly the correct solution lies some- where between these two extremes, but whichever approximation is chosen it should be used consistently, i.e. if eqs. (4.5) and (4.6) are used then the intermediate state wave functions entering (3.16) must be spherical Bessel functions, or if (3.16) is used to determine both occupied and unoccupied states then eq. (4.3) must be modified.

Any deviation from zero of the intermediate state potential will immediately intro- duce a centre-of-mass vector dependence of 7o in (4.3) and thus (4.5) will no longer follow from (4.4). We now examine this complication in some detail. The unperturbed two-body wave function is

qbij(12) Z Z ~, (n(Sl) jmTM,r; M "" M no no = NL LI(tJ)J )~9,,tSj,,TM,,,(X)qbNLML(R) nlSjm TMTJM NLM L

= Z 2 l~nlmtjST{l~'~rhHO [ "~ (6.1) aJ NLML kJtx y'.P'nlmtjl, r]~ . nlmtjST NLML

and we assume that the perturbed two-body wave function may be written

where

tftlJ(12) = Z 2 13nlmtjST{D'ffd~HO[h 7njST("

nlraljST NLML

z~jST(r, R) = Z 1 ~/ST(r, R)Yjtm,(O). l" F

(6.2)

(6.3)

The reference spectrum energy denominator is

= t, + t2 + V(r l + +

h 2

2m (V 2 -}- V2) + ( U ( r , R ) - ( ~ i + ~ j ) ) , (6.4)

where for simplicity we have assumed that there is a single intermediate state potential V(r2). Thus eq. (4.1) may be written

h 2 ~2 1(l~

Z jSTr x.,pnjST/ jST HO -- vt,t,,tr)¢tr, tr, g)+v,,, (r)u,,,j(r) 1"

h E = { E n"t"6sr/°a~-i V2 E "-'NLM~n"t"'JSr/o~rnJST/,*")~U' I,r, R). ~NLML \ix j)

NLML NLML

From eq. (6.1) it is clear that

h 2 - - V 2 Z l"~nlmljST[l~'~ 1 2 2 1 nlmljST = (~rn¢o R - (2N + L - ~)hog)DNLM,_ (R). ~NLML \~x] 2 2 m NLML NLML

(6.5)

(6.6)

S E L F - C O N S I S T E N T S T R U C T U R E 175

Thus eq. (6.5) may be written

h 2 9 2 {_ 2,~ (ffrrZ l(l+r 2 1)) + [u(r, R)-½mo92R2-(ei+ej)]}(~/ .Sr(r ,R)

Z jST njST jST H O -- vrr,(r)~u,, (r, R)+vt,, (r)u,,j(r) l"

- ~NLmL t"J~ ~, hog(2N + L + z2) NLML ~NLML

nnlmdST/n'~njST( R ) + ~, vl rlnlmdSr[o~. " --R~ULML ~"/ VR~7/Sr( r, R) (6.7) >( ~NLML (KJgll" \r , ~ NLML

In general it is the solution of the complicated eq. (6.7) and not (4.5) which is required. As a tentative procedure we would assume that ( is a slowly varying function of R so that VR( can be neglected compared with ~ in (6.7). The resulting equation is then solved in a local density approximation for each value of R and then the effect of the neglected derivative terms can be treated as a perturbation. Note that the above local density approximation is not the same as that used by Brueckner since there the as- sumption is that we can fit the density by combinations of harmonic oscillator func- tions and the validity of our local density approximation then depends on the size of the mesh used in the numerical iteration. This mesh can be chosen to be exceptionally fine in the regions where the density varies rapidly.

If we assume as in sect. 5 that the intermediate state potential may be represented as a harmonic oscillator potential, i.e.

U(r, R) = ½m~o 2 (r 2 + R 2) _ 2A, (6.8)

where A is the depth of the harmonic oscillator potential at the centre of the nucleus, and also we replace the energies (2N+ L + {)ho9 by some average energy B, then (6.7) simplifies greatly and has a solution independent of R given by

h 2 ( ~32 l ( l+ l )~ +[½mo.)2rZ_(2A+B)_(ei+gj)] } TY(r) Z jST njST jST H O - v,,r,(r)(u,, ( r ) = (6.9) - Vrt (r)u,,tj(r). l"

Eq. (6.9) is no more complicated to solve than (4.5) and should be used to determine the vsTj of sect. 5. ~ n'l" nl

Note that even if the intermediate states are treated as plane waves, or in the ap- proximation (6.9), the Pauli correction will introduce an R-dependence 17). Since the

I(STJ (~/ST now depend on R or, what is essentially the same thing, the density, the -~,'r,l are no longer simply numbers but will also be functions of the density. This leads to no serious complication in the iteration of (3.16).

In the meantime, as long as we have any ambiguity in the choice of the intermediate state potentials we cannot be sure that the contributions from the three-body terms

176 J.M. IRVINE

are negligible although this point can be accounted for by introducing a simple, phenomenological, density-dependent potential, as advocated by Bethe 10), to ac- count for al the higher-order correlation effects.

Whatever the quantitative status of the theory we feel that the discussion of sect. 5 is at least qualitatively correct and that this allows us a clearer physical picture of the nature and source of the shell-model potential in light nuclei than has hithert3 been possible in earlier formulations. In a theory which necessarily relies on a great deal of complex computer calculations any progress in this direction appears attractive. It is also clear that when the problems associated with the intermediate states are treated self-consistently it will be the differential, and not the matrix form, of the Brueckner-Hartree-Fock equations which will be required and thus, while eqs. (4.5) and (4.6) may require modification, eq. (3.16) should still be valid.

It is a pleasure to acknowledge useful discussions with colleagues at Cornell, especially Professor H. A. Bethe and Dr. B. Brandow. During earlier stages of the work the author benefited from the guidance of Professor B. H. Flowers and received many helpful suggestions from Dr. J. Clark in Manchester.

Professor Bethe was kind enough to read the manuscript and suggest several im- provements.

References

l) R. Jastrow, Phys. Rev. 98 (1955) 1479 2) J. B. Aviles, Ann. of Phys. 5 (1958) 251 3) J. Clark and J. M. Irvine, to be published 4) K. A. Brueckner e t al., Phys. Rev. 110 (1958) 432 5) K. A. Brueckner e t al., Phys. Rev. 121 (1961) 255 6) H. A. Bethe e t al., Phys. Rev. 129 (1963) 225 7) B. Day, Argonne National Laboratory preprint (1966) 8) K. T. R. Davies e t al . , Nuclear Physics 84 (1966) 545 9) S. J. Krieger e t al., Phys. Lett. 22 (1966) 607

10) H. A. Bethe, Int. Conf. on nuclear physics, Gatlinburg, Tennessee (1966) 11) T. Hamada and I. D. Johnston, Nuclear Physics 34 (1962) 382 12) H. A. Bethe, Phys Rev. 138 (1965) B804 13) L. C. Gomez et al., Ann. of Phys. 3 (1958)241 14) 1. Talmi, Helv. Phys. Acta 25 (1952) 185 15) T. A. Brody and M. Moshinsky, Tables of transformation brackets (Monografias del Instituto

de Fisica, Mexico, 1960) 16) M. Baranger and K. T. R. Davies, Nuclear Physics 79 (1966) 403 17) C. W. Wong, Princeton preprint (1966) 18) B. Brandow, Linked-cluster expansions for the nuclear many-body problem, Varenna Summer

School (1965) 19) H. A. Bethe, private communication 20) B. Day, Phys. Rev. 136 (1964) B1594 21) T. T. S. Kuo and G. E. Brown, Nuclear Physics 85 (1966) 40 22) B. L. Scott and S. A. Moszkowski, Ann. of Phys. 14 (1961) 107 23) B. L. Scott and S. A. Moszkowski, Nuclear Physics 39 (1962) 665