Second order binding energies and spectra with an interaction matrix deduced from phase shifts

Nuclear Physics A219 (1974) 190--220; ~ ) North-Holland Publishing Co., Amsterdam

Not to be reproducedby photoprint or microfilmwithout written permissionfrom the publisher

S E C O N D O R D E R B I N D I N G E N E R G I E S AND S P E C T R A W I T H AN I N T E R A C T I O N M A T R I X D E D U C E D F R O M P H A S E S H I F T S

E. A. SANDERSON, J. P. ELLIOTT, H. A. MAVROMATIS t and B. SINGH t School of Mathematical and Physical Sciences, University of Sussex, Brighton, Sussex, England Received 20 September 1973 Abstract: The matrix elements deduced from phase shifts in an earlier paper are modified to incorporate the effects of a hard core and then used in second order perturbation theory to calculate binding energies and spectra for some light nuclei near closed shells.

1. Introduction

With the intention of avoiding some of the ambiguities and difficulties connected with the short range repulsion between nucleons, a restricted set of potential matrix dements was earlier deduced 1) directly from the nucleon-nucleon phase shifts without specifying the potential. In a later paper 2) these matrix elements, sometimes referred to as the Sussex matrix elements (SME), were used to make first order calculations of binding energies and spectra at or near closed shells. (We shall subsequently refer to these two papers as I and II respectively.) Here we describe the continuation of that work to second order in perturbation theory using, once again, the shell model harmonic oscillator basis of single particle wave functions. In the original paper I the SME were deduced as if the potential were smooth and non-singular at short distances using a form of distorted wave Born approximation, although a further paper 3) pointed out that the diagonal SME have a valid interpretation whether one assumed that the actual potential is singular or not. In fact this flexibility was considered to be one of the virtues of the SME, demonstrating that for many purposes, related to the low energy properties of nuclei, the behaviour of the actual potential at high energies and off-energy-shell was unimportant. However, in I[ it was shown that, although sensible results were obtained when the densitY of the nucleus was constrained at the known value, the SME did not have the necessary saturation properties to produce this value as the equilibrium density. The nucleus tended to collapse when the constraint was released. In the present paper a technique for adding the effects of a short range repulsion to the SME is introduced in sect. 2, and in sect. 3 this is incorporated in a formal theory of the many-body system, the essence of which is well known. Although we use oscillator wave functions, the zero order single particle energies are shifted to t Now at the American University of Beirut, Beirut, Lebanon. 190

SECOND ORDER BINDING ENERGIES

191

more realistic values than those of an oscillator and the choice of the shifts is described in sect. 4. The inclusion of a short range repulsion means that this part of the potential must be treated by a reaction matrix formalism which is described in sect. 5. Second order calculations for nuclei at or near closed shells are given in sects. 7 to 9 It is found that the tensor force converges slowly with excitation energy in second order and a simple technique is presented in sect. 6 for summing all contributions above an excitation of 6 hco.

2. T h e inclusion o f a short r a n g e repulsion

It is well known that the nucleon-nucleon potential may not be deduced uniquely from the two-body phase shifts. The SME tabulated in I represent matrix elements of a potential which fits the phase shifts and which is sufficiently smooth that it may be treated in a distorted wave Born approximation with respect to a simple cut-off oscillator. Subsequent calculations 2, 4) have shown that the SME give sensible results in the few-nucleon problems and in the interaction between two valence nucleons. However, it was found in II that the correct saturation in both nuclear matter and finite nuclei was not achieved with the SME. The conclusion is that the SME, although consistent with the phase shifts, represent a potential which is too smooth to be consistent with the nuclear many-body properties. A simple way to correct this difficiency would be to add to the smooth potential implied by the SME an additional potential Vc, for convenience operating only in s-states, with a repulsive core. If Vc were to produce no change in the phase shifts then the diagonal SME could still be interpreted as matrix elements of the smooth potential and the effects of Vc in the' many-body system could be calculated by the usual reaction matrix method. This is the philosophy which we adopt but in practice we choose a simple form for Vc which does not exactly produce no change in the phase shifts and so the necessary small compensating correction is included. We now set out the method in detail for relative s-states showing how the published SME fit into the generalized formalism. We divide the potential into three pieces: v = Vo+V~+V;,

(1)

where Vo is the auxiliary potential (eq. (6) of I), Vc is a simple model for the repulsive core and V1 is the unknown remainder which will be treated in Born approximation. We shall later use the notation Vs = Vo + V~ and refer to Vs as the smooth part of the potential. In detail: --

Vo =

m

a.

in

r<

a

in

r > a,

(2)

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E . A . SANDERSON et al.

while

--

i I

h2 r~ -- ?+

nr
/~ ----p'O(r/a)p

m

in

c < r
in

r >

(3)

m a,

where O(x) is the step function

O(x)= 1 for

x<

1,

O(x)=O

x>

1.

for

These model parts of the potential contain several parameters b, .~, a, 7, fl and c but if V is the "real" potential, any variation in the parameters may in principle be taken up by a compensating change in the remainder I11 (see fig. 3 of I). However, since matrix elements of 111 are to be deduced as in I, using the Born approximation, it is important not to choose unrealistic values for the parameters. We recall that the r 2 term in Vo was included to ensure that in r < a the scattering wave functions of T + V o are the usual harmonic oscillator functions Rn~ at the particular energies given by Etab = h2(4n + 21 + 3)/mb 2 - 2h2ot/m. (4) The depth ~ and range a of the auxiliary potential Vo were chosen in I so that Vo would roughly reproduce the observed phase shifts, and we use those same values here. The parameters fl and ? of Vo are chosen to ensure that the phase shifts produced by V o + Vc are close to those produced by Vo alone. In practice we choose fl and ? to make these phase shifts equal at the two energies (see eq. (4)) corresponding to the extremes n = 0 and n = 4 and this ensures that the difference is small at intermediate values of n. The problem, as in I, is to use the known phase shifts to deduce the matrix elements of the unknown 111 in a basis of oscillator wave functions R,~ for some chosen length parameter b. Consider the following Schr6dinger equations: ( T + Vo)q~o = Etpo

with phase shift

~5o,

( T + Vo + V1)tp = E~o

with phase shift

6t,

( T + Vo+ V~)~bo = E¢o

with phase shift

tSo,

( T + Vo + VI + Vc)~k = E~k

with phase shift

6 = t~cxp.

!

(5)

In I it was shown that, using the Born approximation for V1, h2k < q , ol V,l

o>

=

-

--

-

(6)

SECOND O R D E R B I N D I N G ENERGIES

193

and, using the fact that tpo oc R.~ in r < a, it was further shown that

(R,olVIIR,o> = - --h2I aR,o(a) /2(61_(~o)+long range correction. m t s i n (ka +6o)!

(7)

Since t51-6o is always very small there is no significant difference between (~1-60 and tan (61-tSo). In that paper, in the absence of Vc, the experimental phase shift was identified with 6~, but now it must be identified with di. With the inclusion of V¢ it is no longer possible to know t51 but we do know tSexp and tS~). Let us therefore assume that a;-ao = a-a1, (8) which says that the change in phase shift caused by adding V¢ to Vo is the same as the change caused by adding V~ to Vo + V1. This again rests on 111 being small and in fact all the phase shift differences are themselves small. Substituting for 61 in eq. (7) using eq. (8) gives

(g.olVdR.o) = - hEk t ag.o(a) .I2(6_6~)q_long range correction. m /sin (ka+6o)J

(9)

This formula differs from that given in eq. (13) of I only in the substitution of 6~ for t5o. Thus we conclude that the matrix elements (R.o[ Vo+ VIIR, o) are given by those tabulated in I with the small additional term in s-states,

h2k( aR~o(a) ~2 m

[sin (ka +tSo)/ (~o - 6~)).

(10)

We shall not tabulate values of this correction because in all applications it must be taken together with the effects of V¢. In the many-body system we treat V~ by the reaction matrix technique deducing a reaction matrix gc in sect. 5. Thus the total effect of the inclusion of V~ is given by the matrix elements of g~ plus the correction (10). The sum of these two contributions will be discussed later in sect. 5 where it will be seen that for a free pair of nucleons the sum is close to zero but inside a nucleus it departs from zero giving an effect which differs from nucleus to nucleus and also depends on the density. For reference, the formulae for 60 and 65 are

[ d log Rno],= = [ ; log {r-l sin (kr +~o)}l,= , (l--fl) ~ l o g

{r -1 sin

q(r--c)}],=. = [ d r log { r - ' sin (kr+tS~))} 1 , = . ,

where

k 2 = (2n+l+l)/b 2- ~, (1-fl)q2 = (2n+l+~)/b z + ~..

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It is interesting to note that the choice of V¢ to cause no change in the phase shift produced by V0 also ensures that the energy levels of the harmonic oscillator H, = T+ h2r2/4rnb 4 are unshifted by the addition of V¢ to Hr. Strictly speaking, the off-diagonal matrix elements of V0 + V1 should also be modified using the techniques t described in sect. 7 of I. However, we may expect these changes to be small since they are derived from the changes in diagonal matrix elements (eq. (10)) which are small. Furthermore the off-diagonal matrix elements are less important since they do not contribute at all in first order. Finally, unlike the diagonal matrix elements, the off-diagonal ones are not at all well determined by the two-body data so we neglect these small changes as insignificant.

3. The many-body problem Although the addition of Vc may cause no change to the two-body phase shifts, it will certainly influence the properties of systems with more than two particles. One way to see this is to argue that the definition of Vo ensures that it gives a zero diagonal element for the t-matrix appropriate to the two-body system but that the analogous y-matrix which arises in the many-body system will have non-zero diagonal matrix elements. This difference is due to the presence of the other particles which affect both the Pauli operator and the energy denominators. We now present a version of perturbation theory for the many-body system in which the ladder diagrams for V¢ are summed in the usual way to produce a reaction matrix g~ and the other pieces V0 + V1 of the potential are treated in standard perturbation theory, order by order, using the SME modified by eq. (10). In the many-body system we consider the Hamiltonian

It = y~ T(i)+ Z V(i,J)+½ A m ~ R 2 , i

(12)

i
in which T(i) is the kinetic energy of particle i and the last term represents an oscillator potential on the centre of mass R = (l/A) ~ r ( i ) . The addition of this term is in principle very similar to the subtraction of the kinetic energy of the c.m. The difference is that whereas the removal of the c.m. kinetic energy leaves a degeneracy in the co-ordinate R, the Hamiltonian defined by eq. (12) has an oscillator spectrum in that variable. The many-body ground state will contain a 0s state of the c.m. and the constant ~hm must be subtracted from the ground state energy of H to obtain the total binding energy. The unperturbed Hamiltonian is taken to be a single particle oscillator with shifted energies

Ho = Z Hos¢(i)+ A,

(13)

i

where Hos~(i ) = T(i)+½moj2r~, a = Z.tj2.tjlnlj>(nljl, and Inlj> is the usual oscil* Note the e r r a t u m appearing at the end o f this paper to certain equations in I.

SECOND O R D E R B I N D I N G ENERGIES

195

lator eigenstate (see eq. (A.1) of I). The perturbation H 1 is then the difference

HI = H - H o = 2 [V(i,j) i
1 mo.)2(ri_rj)2] - A . I 2A

(14)

The separation V = Vs+ V~ introduced in eq. (1) for the internucleon potential where Vs = Vo + V1 then enables us to separate the perturbation into three parts:

(15)

U~ = V~+ Vo-A,

where

i< S

(16,

2A

is the smooth two-body perturbation and Vc is the core part of the internucleon potential,

vo = Z v (i, j ) i
Notice that the perturbation 17s is entirely two-body. This has come about by the combination of the c.m. term R z in eq. (12) and the one-body oscillator Hamiltonian in H o. One advantage of this is that one does not have to calculate perturbation diagrams containing a one-body potential as in the usual approach. A disadvantage is that the perturbation is A-dependent so that care must be taken when comparing energies of neighbouring nuclei - see the discussion in sect. 8. The virtue of introducing the energy shifts )~ntj is that although the unperturbed wave functions remain oscillator wave functions the unperturbed single particle energies may be chosen freely. This device has been used by others 5, 6) and the significance of different choices has been discussed by Baranger 7) and by Barrett et al. 8). Using standard perturbation theory 9) the ground state energy of a closed shell nucleus is given by E = (0]Ho+lTs+V~-AI0)+ ~ Z (0I(ff~+V¢-A) r=l

L

X {1 ~

(~s+Vc__A)}rlO)L__~h~o,

(17)

where Eo = (0[Ho[0), and 10) denotes the unperturbed ground state. The summation over L denotes the linked clusters in the language of Goldstone. For a non-closed-shell nucleus eq. (17) is normally used to give the energy difference with respect to the nearest closed shell. However, since our perturbation depends explicitly on the number of particles there are corrections to the usual linked cluster expansion which we give in sect. 8. The presence of the hard core in Vc necessitates the summing of certain terms in the series (17) to avoid infinities. We therefore follow the method lo) of Brueckner

196

E.A. SANDERSON

et al.

and define a g-matrix between any pair of single nucleon states by the series

(ABIgdAB) = (ABIVcIAB)+ ~ (ABIV c Q V¢ lAB), ¢=~ A+e~--Ho

(18)

in which A and B are abbreviations for the single nucleon labels nlj and eA and eB denote the single nucleon energies e,t+2,1 j. In eq. (18) the operator Q is defined by Ql~fl) = 0 if ~ or fl is occupied in the state 10) and Qlcq~) = [~fl) if both ~ and fl are unoccupied states. The operator Ho in eq. (18) refers only to the two nucleons involved. In fact eq. (18) suggests the definition of an operator g(~o) dependent on a "starting energy" co, by the series

go(o~)=vo+

r=l

~

Q

vo ,

which is equivalent to the integral equation g¢(e)) -- I/;+ Vc

Q g~(o)), o~-Ho

(19)

and is often called the Bethe-Goldstone equation. It now follows that the eq. (17) for the energy may be rewritten as E - ( 0 l H o + 17s-A+g~10)+ ~ r=l

E (01(17s-A+go) L'

where the summation L' extends only over those linked cluster diagrams which do not contain any "ladder" components in go. The starting energy co will be different in different terms of the series (20). For example in the first order term, (0lgcl0) = ½ 2 (ABIg~(~o)IAB), A,B

with 09 = ea + en, and the sum being carried over all occupied states. It is convenient to rewrite eq. (20) as E = ( 0 [ H o + 9s-AjO)+~+(Olg¢lO>+Sm+Sc-~hco,

(21)

in which ~ denotes that part of the summation in (20) which contains no g~ terms, S¢ contains no ~s terms and Sm is the remaining mixed part. We see therefore that the first term in eq. (21) represents the results obtained as in II from first order calculations with the tabulated SME. The inclusion of the second term, taken to all orders, corresponds to calculations, such as those on the very light nuclei 4), in which the energy matrix built from the SME was diagonalised in a complete set of states. Beyond the very light nuclei it is difficult to go much further than second order (1" = 1) in these ~ sums.

SECOND ORDER BINDING ENERGIES

197

In the present paper we shall calculate t the second order part of S with the SME and also estimate the hard-core correction which comes in the last three terms of (21). It will be seen that the third term in eq. (21) is repulsive and particularly important for saturation. Earlier calculations in II, with the first term only, led to over,binding and the inclusion of the second term can only increase the binding.

4. Single particle spectrum The perturbation series is in principle valid for any choice of the energy shifts 2 in eq. (13) but a realistic choice of). will naturally lead to a more rapid convergence. There has been considerable debate in the literature 5-a, 12) concerning the best choice of single particle energies. For theories based entirely on a #-matrix the first order energy is very sensitive to the spacing between occupied and unoccupied levels [ref. 13) gives a useful rule of thumb]. This sensitivity is reduced if the calculation is carried to third or higher order 14). For a perturbation theory based entirely on a potential matrix, V, there is no dependence on the 2nt~ up to first order. In second order the only effect is through the energy denominators. In higher orders the Aoperator plays a direct role and may be chosen to reduce the value of certain third and higher order contributions. Our perturbation theory is much closer to the latter case since I(Vs)l is much larger than I@c)l except at very high densities. Our choice of the 2,~i, which we describe below, reduces both the attractive second order and the repulsive third order contributions. Comparing with calculations 15) made with 2nli = 0 (all nlj) the convergence of the perturbation theory is improved. The original choice of an oscillator potential for H o is unrealistic in two ways: (i) it has zero depth and (ii) it has an infinitely high potential wall. A more realistic choice for the single particle potential W would be an oscillator potential with a state dependent depth Wo(A ) and a cut o f f a t some point r = d:

W = - ~ Wo(A)IA>
where O(x) is the step function O(x) = 1 for x < 1 and O(x) = 0, x > 1. The cut-off, where the potential drops to zero, is taken just beyond the nuclear radius at d = R + 1.5 fm. Such a potential has a continuous spectrum for the high lying orbits but inside the nucleus, for r < d, the eigenfunctions will coincide with oscillator functions R,~ at particular energies - W o ( A ) + e,t. We shall therefore choose the energy shifts to be 2a = - Wo(A) and it remains to deduce Wo(A ). N o w the potential W represents the interaction of a nucleon with all other nucleons and so we equate

(AIWIA) =

~

(ABIVsIAB),

(22)

B(occupied)

giving

2a = - Wo(A) =

~,

(AnlV~lan>-(a1½mo92r21A>a,

(23)

B(o¢¢upied)

t Detailed formulae relating to the second order diagrams shown in fig. 6 are given in ref. 3a).

E . A . S A N D E R S O N et al.

198

where the subscript d indicates that the integral is carried only from r = 0 to r = d. The choice (22) for the depth W 0 is a first step towards Hartree-Fock self-consistency and should reduce some third order corrections. Ideally with the core Vc present one should include g¢ with Vs in eq. (22). However, in our method g¢ is only a small correction and we omit it from this definition of the e a . This also avoids the Brueckner self-consistency problem 7) in which g¢ depends on the e a through the denominator ~o in eq. (19). Values for the energies e A = ~A'~-~A a r e shown in fig. 1 and compared with the oscillator energies e,,. One sees that the oscillator degeneracies such as the ls and ?0

~

9s

isi

6s I0

\.

_

\

.

~

Op

-S 4He

160

40Ca

0$

Fig. 1. The single particle energies e.z = e . t ~ 2 . t , in units of hog, compared with the oscillator energies e.t = ( 2 n + l + , ] ) h o J shown in column one.

SECOND ORDER BINDING ENERGIES

199

0d levels, are not appreciably broken and the equal spacing of levels in the oscillator is approximately preserved. It is also seen that for occupied orbits, 2 a is large and negative consistent with the fact that such orbits are well bound. For high, unoccupied orbits 2A becomes small which says that the energies are close to those of the oscillator. This is not contradictory since for such high states the energy at points within the nucleus is almost entirely kinetic as may be seen from the Kallio 16) relation between oscillator wave functions and plane waves. Our choice of single particle energies is therefore consistent as 2 ~ 0 with that generally made in nuclear matter where kinetic energies alone are used for unoccupied orbits 5-s, 12). In fact our prescription provides a continuous passage from the approximately self-consistent energies of the occupied orbits to the pure kinetic energies for high lying unoccupied orbits.

5. The reaction matrix for Vc The methods of solving the Bethe-Goldstone equation (19) are reasonably straightforward 7, 8). Bearing in mind that the potential V¢ depends only on the relative co-ordinate of the pair and that the unperturbed oscillator Hamiltonian separates in relative and c.m. co-ordinates it is natural to seek a solution in those co-ordinates. The addition of the energy shifts 2,tj to H o would destroy this separability. However, in the previous section (fig. 1) we have seen that 2 is large and negative for occupied orbits and becomes small for highly excited particle orbits. For the purposes of calculating g¢ we shall simplify this result by taking ). = 0 for all unoccupied orbits, in which case the separability is restored. This comes about because H 0 enters eq. (19) only in combination with the projection operator Q. Thus in solving eq. (19) we may take H o to be the pure harmonic oscillator (2,,zj = 0) and still retain the freedom to choose )-.zj ~ 0 for the occupied orbits. The only remaining obstacle to the separation is the operator Q which depends on the particle states rather than the relative co-ordinate. Various approximate methods have been discussed 11),t for dealing with this difficulty but in our problem it will be sufficiently accurate to put Q = 1. The reason for this is simply that gc derives entirely from the core part Vc of the potential so that the contributions to g¢ will be spread over very high energy intermediate states and consequently will be little affected by the operator Q which excludes the occupied states. With Q = 1 the eq. (19) depends on the c.m. co-ordinate * R = ½(r 1 + r 2 ) only through the two-particle oscillator Hamiltonian which separates into H 0 = Hr + HR. In a two-particle basis in which H~ is diagonal we can replace HR by its eigenvalue t The c.m. co-ordinate R of the pair of nucleons should not be confused with the c.m. co-ordinate R of the entire nucleus, introduced briefly in sect. 3. tt Chapter 5, in particular, provides a very fine review of Bethe-Goldstone theory both in principle and practice.

200

E.A.

e t al.

SANDERSON

eNz leading to the equation

1

ado,) = vc + vc

H, g°(09)

0 9 - - 8 N L¢ - -

for g¢(09) which now involves only the relative co-ordinate r = (rl - r 2 ) . Because of the appearance of eNz in the denominator, it is convenient to define 09, = CO--eNZ and to define #r(O-"r) = g¢(09) SO that the defining equation for g,(%),

1

g,(09,) = V¢+ Vc - - g,(og,), 09, - H,

(24)

is completely free of the c.m. labels. Finally to construct the required two-body matrix element we make the reduction

< lgo(09)l fl> =

(25) N£~n"nl

where the coefficients a ( @ ? f N ~ n ' n l ) are, in the usual way, composed of BrodyMoshinsky transformation coefficients and angular momentum coupling coefficients. The solution of eq. (24) for g,(09,) presents no difficulty 7). We first define the correlated wave function ~,,~(r) of relative motion by the equation

g,(09,)R,t = V¢ ~,t,

(26)

where R,a is the usual oscillator radial wave function of relative motion. Then from eq. (24) we have

~.t

1 V¢ ~ . t , 09, - H,

= R,z + - -

which provides the inhomogeneous differential equation ( c o , - H , - Vo)~.t = (09, - e.,)R.~

(27)

for the unknown radial function ~k,~ of the single variable r. Having solved this equation by elementary methods as oatlined in the appendix A thz required #-matrix element for use in eq. (25) is most easily deduced by using eqs. (26) and (27) to write

= (n'llVd¢,t> = -(~r-C,a)6,,,,, =

By defining the "defect" wave function Z,a = R,,t-~,,t this becomes simply

= (e.,,- 09,).

(28)

The calculation of #,(09,) is made entirely in the relative co-ordinates but the choice of 09 and hence 09, depends on the particle orbits. For example in the first order term

SECOND

<01g=10> o f e q .

ORDER

BINDING

201

ENERGIES

ea+eB, giving

(20), 09 =

(D r ~- (2) - - F~N.~, =

13A + ~B "}- t~A +

'~B - - ~N.~ = enl "31-2 A ..I- ~ B "

(29)

It was found that the matrix elements o f #,((Dr) were almost linear in co, for realistic values o f (D, in the range - 15 < ((D,/h(D) < - 2 , so we were able to obtain all our 9, matrix elements by simply calculating them at five points t in this range and fitting a quadratic curve through these points. I

I

1.5

HeV 1.0

\ %

0.5

N

% N. %

N

-15 Fig. 2. The s u m

I

I

-10

-5

(n]gr(mr)]n)

o

plus the phase shift correction given by eq. (10) (b = 1.7 fm, c = 0.3 fro).

Fig. 2 illustrates the compensation between the g,((Dr) and the additional term (10) due to the fact that Vc causes a small phase shift change. The sum o f these two contributions is plotted against o~r for b = 1.7 fro, c = 0.3 fm and it is seen to vanish close to the oscillator energies (D~ = e,~, i.e. at o~[ho~ = 1.5 for n = 0 and 3.5 for n = 1. The presence o f the negative energy shifts 2 causes (D~ to deviate from e,t, see eq. (29), and thus produces a repulsive contribution to the nuclear energy. t Tables o f the coefficients in this quadratic for v a r i o u s values o f n, n', b a n d c are available o n request.

E.A. SANDERSON et al.

202

6. Convergence of tensor force contributions in second order The second order ladder diagrams f r o m the tensor force are slowly convergent with excitation energy of the intermediate state. As an example we show in fig. 3 (lower curve) the tensor contributions to the binding energy o f 160 as a function of the excitation energy of the intermediate state. A l t h o u g h the central and spin-orbit contributions from above 6 h~o excitation are negligible (see table 2) the tensor contribution is not. However, above 6 h~o excitation the Pauli principle as we shall see 20 ~

~

MeV I0

o

IIIIIllll~~ 4

s

iz

16

zo

z4

zB

3z

36

Exile=

Fig. 3. Second order tensor contributions to the binding energy of 160 as a function of the excitation energy Ex. The upper curve shows the results obtained if the Pauli principle is ignored. The numbers in this figure were derived using harmonic oscillator energy denominators with ?ko = 14.4 MeV (b = 1.7 fro). below m a y reasonably be ignored in which case the contributions f r o m above 6 ho~ may be summed (see appendix B) and expressed as a correction to the relative matrix elements. F o r example in the relative 3S I channel and with oscillator denominators the correction from eq. (B.4) is E n'>n+3


h°~.

(30)

SECOND ORDER BINDING ENERGIES

203

One difficulty here is that for large (n' - n) no matrix elements of V, are available. We therefore use an auxiliary potential typical of those used in the work of I, which for the S - D coupling channel is Vau x = - 3 5

MeV+h2r2/4mb

4

in

r < 2fm,

together with the OPEP at large distances r > 2 fm. In practice one finds that the resulting matrix elements are close to those of the Hamada-Johnston potential with a short range cut-off at r = 1.4 fm. The points on the upper curve in fig. 3 have been calculated using eq. (30). Comparing these with the points on the lower curve which include the Pauli principle effects, one sees that although it would be quite wrong to ignore tensor contributions beyond 6 hto, it is quite reasonable to estimate them using eq. (30). The error is about 10 ~ at 8 he) and reduces with increasing energy. We find that about 60 ~,, of the total contribution comes from beyond 6 hto, which reflects the significance of high energy intermediate states for second order tensor effects, as emphasized by K u o and Brown 17). The mean excitation energy of the tensor contributions to the binding energy of ~60, shown in fig. 3 is 9 hto ~ 130 MeV. In sect. 4 and in fig. 1 it was shown that the more realistic single particle energies e,z = e,t+2,z have a larger spacing between occupied and unoccupied levels. Consequently in using the corrections (30) in later sections we replace the oscillator energy denominators by more realistic ones deduced from fig. I. It has been verified that the corresponding tensor corrections in other channels are much smaller and they are therefore ignored. 7. Closed shell nuclei 7.1. FIRST ORDER First order calculations have been made for all N = Z, L - S closed shell nuclei from A = 16 to 224 and the energy per particle plotted against the oscillator length, b, is shown in fig. 4. In addition to the first order terms in Vs and gc (first and third terms of eq. (21)) we have included the first order Coulomb interaction since this has some bearing on the position of the minima. The curves are labelled by the core radius, c, chosen for Vc. We also record the value of ~ the average over occupied states of the single particle energy shifts )-a given by eq. (23). In practice we find it to be sufficiently accurate to use this average value in the calculation of the reaction matrix gr(to,)- The deviation of any 2a from ~ in a given nucleus is less than 15 ~ , the variation of g, with to, is smooth, as seen in fig. 2 and over the range of b-values shown in fig. 4 the average J( is closely proportional to hto. The dependence of the first order energy on the choice of the core radius, c, is shown explicitly for 4°Ca, which is quite typical. The disturbing lack of saturation for c = 0 (i.e. Vc = 0) has already been noted 2) and is in sharp contrast to the curves obtained with finite c. We may note, in particular, that for c > 0.3 fm, no nucleus

204

E. A. SANDERSON et al. 0

j

I

1

I

I

I

I

I

I

-2 C=0,3

16

-

0

-4

_

0 --2 -4 -6 c=0

MeV -2

C= 0

.

3

A =80

~

-4

0

A" 140

C- 0 . 3

-2

"

( X ' - e.e~,~)

-4

0

A=224

C=0.3

-Z

(~.=-Io.oli~) _ -4 I J.5

I 1.6

I L7

I J.e

I ~.9

I z.0

I

I

I

2.,

:'.2

2.3

2,4

b (fro) Fig. 4. First order energy per nucleon as a function of the oscillator length b for various N = Z nuclei.

is overbound, whereas, previously with no V~ all nuclei heavier than 4°Ca were overbound and much too dense at the minima. On the question of the optimal choice of the core radius in V~, there are in principle two criteria to satisfy: the saturation density and the energy. The first of these, in fact, is not very sensitively determined (for c ~ 0.3 fm) due to the very shallow minima shown in fig. 4. Nevertheless, the minima come in approximately the correct

SECOND O R D E R BINDING ENERGIES

205

positions for 160 and 4°Ca. For the heavier nuclei it is difficult to make any comparison since our calculations refer to closed L - S shell nuclei which are well away from the stability line and whose radii are not known. The total energy on the other hand is quite sensitive to the choice of c, through the first order gc contribution, which is roughly proportional to c 2. For c < 0.2 fm the very heavy nuclei are overbound. For c >= 0.4 fm the lighter nuclei and very probably the heavier ones too are significantly underbound, even allowing for the second and higher order corrections which are not included in fig. 4, so that for the rest of this paper, we choose c = 0.3 fm. We do not at present attempt any finer adjustment to the value of c to give a best fit to nuclear sizes and energies since this should await the inclusion of higher order corrections. 7.2. SECOND O R D E R

The first and second order energies for the closed shell nuclei, 4He, 160 and 4°Ca are given in table 1, where the contributions from the different terms of eq. (21) are shown explicitly. The total second order contributions are approximately constant over the range of b-values of interest so results are shown for spot values only. TABLE 1 Energies (in MeV) for 4He, 1~O and 4°Ca to second order with c = 0.3 fm Nucleus

4He

160

4°Ca

b (fm) ~, (MeV) Potential energy

1.5 --40.7

1.7 --57.6

2.0 --60.4

[ I7"s Oc S second order S,n S~ Total potential energy Coulomb energy (lst order) Kinetic energy Total energy Experiment

--55.3 3.7 --9.5 2.9 --0.7 --58.9 0,8 41.7 -- 16.4 --28.3

--331.7 26.8 --52.7 15.9 --3.2 --344,9 14.0 248.7 --82.2 --127.6

--906.6 65.1 --148.9 43.9 --5.7 --952.2 71.0 617.2 --264.0 --342.1

first order

Up to second order each of these three nuclei is underbound for c = 0.3 fm, but the deficiency is no more than about 15 70 of the total potential energy or, in other words about 2.5 MeV per particle. The first order results for heavier nuclei, shown in fig. 4 have a deficiency of about 4 MeV per particle for A = 80 decreasing to 2 MeV per particle for A = 224. For this comparison we use the semi-empirical mass formula to deduce the experimental masses since the N = Z nuclei considered here are not stable. Allowing for a likely second order contribution of about 3 MeV per particle there is slight evidence here of overbinding in heavy nuclei. However, as

--5.9 1.5 --4.4

~q (total) Sm

total --8.2

--8.0

--12.2 4.2

--5.1 --0.7 --6.4

--0.2

--1.0 1.1 --0.3

--7.1

--7.0

--10.1 3.1

--2.0 --0.8 --7.3

--0.1

--0.1 0.0 --0.0

6/~to

--20.1

--20.1

--20.8 0.7

--0.2 --0.0 --20.6 a)

> 8ho9

160 (b = 1.7 fm, c = 0.3 f m ) 4/~o

~) This figure is calculated using the m e t h o d described in sect. 6.

--4.6

--3.3 --0.1 --2.5

--0.2

total

~q (central) ~q (spin-orbit) S (tensor)

--2.6 5.3 --2.9

Sm Sc

Total 2nd order

Non-H F

HF

2hto

TABLE 2

--40.0

--39.5

--49.0 9.5

--10.6 --1.6 --36.8

--0.5

--3.7 6.4 --3.2

total

--11.3

--5.5

--7.2 1.7

--5.1 --0.2 --1.9

--5.8

--21.9 21.5 --5.4

2/~to

Analysis o f second order corrections (in MeV)

--14.4

--13.2

--18.9 5.7

--ll.0 --0.9 --7.0

--1.2

--2.6 1.7 --0.3

--13.5

--13.4

--19.9 6.5

--7.9 --1.3 --10.7

--0.1

--0.2 0.1 --0.0

--71.5

--71.5

--78.2 6.7

--4.4 --1.3 --72.5 a)

4 ° C a (b = 2.0 fm, c = 0.3 fm) 4/~co 6hw > 8hto

--110.7

--103.6

--124.2 20.6

--28.4 --3.7 --92.1

--7.1

--24.7 23.3 --5.7

total

r~ 7~ ©

> Z

.>

OX

SECOND ORDER BINDING ENERGIES

207

remarked earlier, we have not attempted any fine adjustment to the value of c. Neither have we varied the tensor composition of the matrix elements which, as remarked in sect. 9, of I has considerable uncertainty due in part to the poor accuracy of the experimental phase shifts for np scattering, notably the mixing parameter ~1 • We have also verified that the inclusion of V~ gives sensible results in nuclear matter. Using the Kallio relation one can deduce plane wave matrix elements of g¢ from those described in sect. 5 in an oscillator basis. The inclusion of the first order contribution from gc produces an energy minimum at about kv = 1.5 fm -x with a binding energy per particle of 10 MeV. Dey is) has made an estimate of about 3.8 MeV per particle for the second order corrections which again leaves a deficiency of about 2.3 MeV per particle, similar to that found above for the finite nuclei. Bethe 19) has argued that corrections of the order of 4 MeV per particle might arise from three-body forces, three-body correlations and other small effects. It is of some interest to look at the contributions of different diagrams to the second order energies and this is done in table 2, where the contributions from different intermediate state energies * and the different terms of eq. (21) are shown. There are two basic diagram types corresponding to l p - l h and 2p-2h intermediate configurations; the former are Hartree-Fock type corrections (HF) and the latter non-HF. Both contribute to ~ and Sm but Sc has only a H F term in second order. These are shown explicitly in fig. 5.

Sm 2

0 .... o Q

+

2

0

0

Fig. 5. Diagrammatic representation of the second order contributions to the perturbation series S, Sm and S¢ (see eq. (21)) for a closed shell nucleus. The interaction l?s is denoted by . . . . . . . and gc by--. Looking first at the H F contributions in table 2, we note that these arise almost entirely from 2 he) excitations. Furthermore, the effect of g~ in the repulsive mixed term S m cancels the large H F term in ~, which is just the stabilising effect of Vc noted above in subsect. 7.1. On the other hand, for the n o n - H F contributions, although S m is again of opposite sign to ~ the cancellation is not nearly so significant; moreover, in this case the major contributions come from intermediate configurations beyond 2 he). Indeed, by far the most important factor is the tensor contribution which, as discussed t We divide intermediate configurations according to their oscillator excitation energy (2i~co, 4boo, etc.), but the average energy denominators that are used are determined from fig. 1 (see sect. 4).

208

E.A. SANDERSON et aL

in sect. 6, peaks at a high energy and contributes between 2 and 2.5 MeV per particle for both 160 and 4°Ca. A number of studies 2o-22) have pointed out the importance of the tensor force for the saturation of nuclear matter. A comparison is made between the 1S o and 3S 1 contributions to the energy per particle as a function of the Fermi momentum kv. Due to the Pauli exclusion principle restriction of second and higher order tensor contributions, the 3S x curve eventually comes to a minimum at some value of kF whereas the aS o curve does not. Some authors 22) have exaggerated this saturatin~ effect by plotting the 3Sa-3D a channel contribution, which is misleading since the tensor force is repulsive in the 3D 1 component but almost equally attractive in the 3D 2 channel. In our calculations the tensor force contributes nothing to a closed shell nucleus in first order and in second order its contribution, although large, is not very sensitive to the nuclear size. For the smallest b-value considered, viz. 1.4 fro, there is a reduction in the second order tensor contributions to the binding energy which relative to the maximum values amount to 0.5 MeV per particle in 4°Ca and 0.2 MeV per particle in 160. These changes are considerably less significant than the first order effect of gc shown in fig. 4. Two reasons can be suggested for the reduced significance of the tensor force for saturation in our calculations. First of all our tensor interaction is not so strong as that of the Reid 23) or Hamada-Johnston 2~) potentials. Secondly, and perhaps more importantly, the average density of a finite nucleus is less than that of nuclear matter, because of the existence of a surface. Results obtained for nuclear matter are significantly modified in a finite nucleus, as Haftel et al. 25) have pointed out. Afnan et al. 20) have shown that the nuclear matter saturation properties of the tensor force are directly related to the strength of the tensor force as indicated by the deuteron D-state probability, PD" The SME give 26) PD = 4 ~,. The average density of 4°Ca at a realistic value o f b = 2.0 fm corresponds to nuclear matter at a Fermi m o m e n t u m of kr = 1.1 fm-1. With b = 1.4 fm the density increases to a value corresponding to k v = 1.6 fm -1. I f we compare with the potential of Afnan et al. 2o) having PD = 4 ~o one would not expect to see any significant saturation effects due to the tensor force in this range of densities. Furthermore, at the normal density of 4°Ca, kv ~ 1.1 fm, even the Reid potential shows no great difference between 1S o and 3S 1 contributions [e.g. see fig. 3, p. 137 of ref. ~9)]. Thus the saturating effect claimed for the tensor force in nuclear matter seems to be of considerably reduced consequence in determining the equilibrium size of finite nuclei.

8. Nuclei with one valence particle or hole

Energies have been calculated for the single particle or hole levels in the lowest major oscillator shell in the nuclei as. 170 and 39, 41Ca" AS in the previous section, the energies include first and second order contributions and increased average

SECOND ORDER

BINDING ENERGIES

209

TABLE 3 Single particle a n d hole energies (in M e V ) Nucleus

Orbit

First order c = 0 c = 0.3

Second order HF non-HF

Total c -----0.3

Experiment

Spin-orbit splittings

1sO xTO agCa 41Ca

0p 0d 0d Of lp

4.6 6.3 4.1 4.9 2.0

4.5 6.2 4.0 4.8 2.0

--0.4 --0.7 0.6 0.6 --0.1

0.6 --0.4 0.2 --0.2 --0.1

4.7 5.1 4.8 5.2 1.8

6.2 5.1 5.1 5.7 2.0

Relative /-spacing

170 agCa 4XCa

0d-ls 0d-ls 0f-lp

1.2 0.9 1.7

0.9 0.7 1.3

0.7 0.t 0.8

0.7 0.0 0.3

2.3 0.8 2.4

1.2 --0.6 --0.2

Mean energies

150 170 agCa 41Ca

0p 0d, Is 0d, l s Of, l p

--19.0 1.8 --20.0 --3.7

--16.2 3.7 --17.1 --1.7

0.1 --0.6 --0.7 --0.1

--2.7 --3.4 --2.9 --3.2

--18.8 --0.3 --20.7 --5.0

--19.8 --2.3 --18.6 --5.9

b = 1,7 f m in oxygen a n d 2.0 f m in calcium.

energy denominators deduced from fig. 1 have been used instead of the usual oscillator energy denominators. In presenting the results in table 3, we show first the spin-orbit splittings, then the energy differences between levels with different l in the same major shell and finally the mean energies of all levels of a major shell relative to the closed shell. In detail we calculate the energy differences

IEnlj(A + 1)-- Eo(A ) ~.lj = lEo(A)_ E.tj(A _ 1)

for particle levels for hole levels,

(31)

and then deduce the spin-orbit splittings ~ n l ( l + ~) - -

8n~(t-~),

and the mean/-energies ~'nt = {(l+ 1)en,(,+½) + le,,,(t_~)}/(21 + 1).

(32)

The relative/-spacings are then defined as e , , r - ff, t and the mean energies as E (2l + 1 ) e . , / E (2l + 1), nl

nl

where the sum runs over the possible nl in a given oscillator shell. The experimental numbers in table 3 are taken from the compilations of Ajzenberg-Selove 27) and Endt and Van der Leun 2a). In some cases the single particle strength is fragmented and incompletely known. In such cases we have identified the single particle level with the lowest level of significant strength. In calculating the difference between energies of systems with different mass A as in eq. (31) we must not overlook the fact that the perturbation HI(A ) defined in

210

et aL

E.A. SANDERSON

eqs. (15) and (16) depends on A. As a consequence the diagrams shown in fig. 5 for the closed shell and which also a p p e a r for a nucleus with one valence particle will not exactly cancel in the difference (31). A correction must be calculated by evaluating such diagrams for H 1(A) and Hx (A + 1) and taking the difference. This complication offsets the simplification that, since ~ is entirely a two-body operator, no diagrams containing one-body terms need be calculated. It is simple to show that, when all second order terms are calculated, as we do here, the end result is essentially the same as if the perturbation had been written, ignoring the centre of mass correction, as

Z V~(i,j)-½ mC°2Z rf, i
i

which contains two-body and one-body pieces which are independent of A. The only difference would be a constant energy shift of 3~ho~ due to neglect of the c.m. correction. However, when more than two valence particles are present and if the three-body diagrams which occur in second order are not calculated significant differences between the two approaches will be seen. This question will be discussed further in a later paper. 8.1. SPIN-ORBIT SPLITTINGS In first order the spin-orbit splittings were seen in II to be critically dependent on the choice of b-value. This dependence is reduced as expected when second order corrections are added, as is shown in table 4, but even so an approximate inverse cubic dependence on b remains. F r o m table 3 it may be seen that for the spot values chosen to fit the known nuclear size, the second order corrections generally improve agreement with experiment, though the Change is sometimes only marginal. The first order effect of gc is negligible. This is because Vc is entirely central so that the only effect comes through the energy dependence of g~ which is slightly TABLE4 Analysis of second order corrections (in MeV) to the spin-orbit splitting between d~r and d~. levels in 170 showing the dependence on the oscillator length b b (fm)

1.6

1.7

1.8

HF

S Sm &

0.26 --1.91 o

0.57 --1.31 o

0.81 --1.02 o

Non-HF

S=

--0.45 --0.01

--0.36 0.00

--0.26 0.00

Sc

Total second order First order First and second order c = 0.3 f m .

0

--2.11 8.31 6.20

0

-- 1.10 6.17 5.07

0

--0.47 4.83 4.36

SECOND ORDER

BINDING

ENERGIES

211

TABLE 5 T h e role o f the t e n s o r force in second order corrections to the spin-orbit splitting between d~ a n d d~x levels in 170 (b ~ 1.7 fm, c = 0.3 fro) N o tensor c o m p o n e n t

Complete interaction

Pure tensor

(i)

(ii)

(i)

(ii)

(i)

(ii)

2hco 4/~m 6ho~

--0.34 0.01 --0.00

0.02 0.01 0.00

--0.39 -- 0.26 --0.13

0.31 0.24 0.10

--0.62 --0.14 --0.14

0.33 0.23 0.10

Sum

--0.33

0.03

--0.78

0.65

--0.90

0.66

C o n t r i b u t i o n s (in M e V ) are s h o w n separately for: (i) 2 p - l h a n d (ii) 3p-2h intermediate states.

different for the two spin-orbit partners. The precise values of 2 a given in eq. (23) were used for the occupied orbits in calculating the first order gc contributions to the single particle energies. For the second order contributions the average )~ values ( - 4 . 0 ho9 in oxygen, - 5 . 8 ho9 in calcium) mentioned in the previous section were used. In second order 9¢ has an appreciable effect through the H F terms which can be seen in table 4. As we have seen in sect. 7 and fig. 4, 9~ has a crucial effect on the equilibrium size of the nucleus and the spin-orbit splitting is sensitive to nuclear size. These two effects combine in the H F terms. Since the H F terms are only taken to second order it is of course necessary to choose a realistic value of b. A full H F calculation has been carried out and will be reported elsewhere 29). The effect of n o n - H F second order corrections is much less pronounced for spinorbit splittings than is the case for mean Llevels. This is somewhat surprising in view of the strong tensor force. However, it turns out that there is a close cancellation between contributions from 2 p - l h and 3p-2h intermediate configurations for A + 1 nuclei and likewise between 2 h - l p and 3h-2p contributions in A - 1 nuclei. These contributions are individually fairly large and almost entirely due to the tensor force but the net effect is rather small. An example is shown in table 5. The slow convergence with Nh~9 of the tensor contribution is again evident but not important here due to the term by term close cancellation mentioned above. 8.2. R E L A T I V E ,/-SPACING

With regard to the relative/-spacings within a given major shell, we see from table 3 that the higher-/level is relatively underbound by about 1 MeV in oxygen and about 2 MeV in calcium. One aspect of the calculation which could explain this systematic error is the use of oscillator wave functions which will be less realistic for particle configurations than for holes. Furthermore, the shape of the oscillator potential may be expected to lead to underbinding of the higher angular momentum levels. Although the Hartree-Fock diagrams which we include are a first step towards improving the single particle wave function our perturbation theory is taken only

212

E . A . SANDERSON

et aL

to second order. Hartree-Fock calculations for oxygen 29) indicate that there are significant higher order corrections for particle configurations which go in the right direction to account for the above discrepancies. It appears that the use of a different value of b for closed shell and valence particles may also resolve this problem. 8.3. MEAN ENERGIES

For oxygen the mean energies shown in table 3 are in quite good agreement with experiment. The 2 MeV discrepancy for the particle levels is due to an underbinding of the d-states which may again be due to the inadequacy of oscillator wave functions discussed above. In calcium, however, the hole levels are overbound by 1-2 MeV and this result is confirmed by H F calculations 29), which also show that the rms radius is about 10 too small. Such difficulties are common to other calculations using realistic interactions. Their resolution seems to lie in higher order corrections such as the occupation probability corrections calculated by Davies and McCarthy t4). 9. The nuclei with A = 18

In order t6 test our matrix elements for determining the effective interaction between valence nucleons we have chosen the much investigated nuclei tsO and tSF. A simpler system exists in 6Li but results for this nucleus together with other pshell nuclei will be given elsewhere 3o). It is important to bear in mind that the treatment is perturbative starting with a model space o f (sd) 2. The spectra of 1sO and lSF are known 31, 32) on experimental and theoretical grounds to contain states of more complicated structure (e.g. 4p-2h). These will be missing from our treatment. Since these many-particle-many-hole states occur at low energies any significant coupling to (sd) 2 configurations would call for corrections to the perturbative treatment. In the present paper we do not consider such corrections but there are indications both experimentally and theoretically, that the many-particle-many-hole states have large deformation and mix comparatively little with the two-particle states [see e.g. the summary beginning on p. 348 of ref. ax)]. The question of the order by order convergence of perturbation theory has been raised by Barrett and Kirson without being clearly resolved a2). Our calculations are strictly limited to second order for the A = 18 system but for 6Li the work of Singh 33) comparing perturbation theory with exact calculations is very encouraging. The principal differences, apart from the interaction used, between our work and earlier ones of similar nature are: (i) all second order diagrams including HF-type corrections, are included (these are shown in fig. 6c) and (ii) contributions from arbitrarily high intermediate energy configurations are included. The latter is absolutely necessary for taking into account the very important effect of the tensor force. The technique described in sect. 6 is used to sum ladder diagrams beyond

SECOND ORDER BINDING ENERGIES (o)

Z(A)

(b)

Y(A)

(el

X(A!

213

C]) =

Fig. 6. Second order perturbation diagrams arising in the calculation of lSF. The wavy line denotes the sum ITs÷go and denotes go. 6ho. The slow convergence of the tensor force is also evident, as pointed out by Vary, Sauer and Wong 34), in the core polarisation corrections, the third diagram in fig. 6c, but we found this to be a more minor role and it was sufficient to truncate these corrections at 10ho. In c o m m o n with most earlier workers we use an average energy denominator in calculating second order corrections but the value we use is determined from the more realistic single particle spectrum shown in fig. 1. In practice this reduces our perturbative terms by about 30 ~o from what they would be simply using harmonic oscillator energy denominators. It should perhaps be emphasized that in our treatment there are no ambiguities concerning double counting. We must, of course, omit the ladder diagram in which gc occurs twice but otherwise all terms which are second order in Us and gc are included. For convenience we calculate all diagrams using (Vs+g¢) as the unrestricted perturbation and then subtract the forbidden gc ladder, although in practice this turns out to be a completely negligible term ( < 0.03 MeV). This procedure corresponds to a trivial regrouping of the terms of S, S m and Sc defined in sect. 3. The resulting perturbation series in second order is shown in fig. 6. The diagrams of fig. 6a are exactly equivalent to those of fig. 5. We need only discuss 18F since the 180 levels are all analogues of T = 1 levels in 18F. Since in previous paragraphs we have studied the energies of the A = 16 and 17 systems we now calculate only the differences F JT = E J T ( 1 8 F ) - E ( 1 6 0 ) - { E ~ ( 1 7 0 ) - E ( 1 6 0 ) } - - { E ~ ( 1 7 F ) - E ( 1 6 0 ) } ,

(34)

which give a measure of the binding between the valence nucleons in various states J T . Experimental values for F Jr taken from p. 342 of ref. 31) are shown in columns

6 and 7 of fig. 7. The levels drawn in column 6 are those which are identified by Rolfs et al. as being chiefly of (sd) 2 character. Column 7 shows all other known positive parity levels of lSF. Although the latter may be states outside our model space, their existence will have some effect on the position of the levels in column 6 which we are trying to predict. Theoretical values are deduced by diagonalising a matrix F~lv2, ST P3P4 in the model

214

et al.

E, A. SANDERSON

~

4 5

!/ .

:3

=--- . . . . . . . . . . . . . . . . . .

2 2 - _ - _ - 2 ~ . Z - z - - - - : - ' - - - ' = - - = - =

o .....

..... 0

-

2

. . . . .

- .

.

.

.

.

.

.

.

.

~

. . . . .

MeV

2

./3

~

- . . . .

-I

2

e

.

.

.

.

4

4

. . . . . . . . .

(0 )

3

. . . . 2 . . . . . . . . . . . . -2

-

- - 4

~l

/

4:

0L- ......

:3

~-'--

2

/

5

-3

0 ......

-4

-5

:3"

L,6-J L,~/ . .

.

L,._I.

L2o-I

L~2_I

"'

. Experiment b{fm) F i g . 7. Energy levels of ~ S F . Columns one to five show the calculated levels for different values of the oscillator length parameter b. The experimental levels are s h o w n in columns six and seven, with column six containing those levels believed to belong primarily to the (sd) 2 configuration s~). Broken lines denote T = 1 levels.

space (sd) 2. Each matrix element is calculated up to second order using a perturbation expansion for each term in (34). Because of the A-dependence of our perturbation H1, discussed in sect. 8, the closed shell and closed shell + 1 second order diagrams (fig. 6) do not exactly cancel in eq. (34). If we denote by Z(A) the second order diagrams in ~60 calculated for given A and by Yj(A) the second order diagrams in a 70 then the matrix will receive from this source a constant contribution.

Z ( I S ) + Z ( 1 6 ) - 2Z(17),

(35a)

and a single particle contribution

Yj(18)- Vj(17).

(35b)

Where the valence nucleons are in orbits other than d~ there will be a contribution to the matrix from the difference of single particle energies. Although in calculating these differences in sect. 8 we found the d~-d~_ difference to agree well with experiment, there was disagreement in the d-s difference due, we believe, to the use of oscillator wave functions. In the present study of the A = 18 nuclei we prefer therefore to use the experimental single particle energy differences Ej=TE~taken from 17 0 to be

E~-E~ = E~-E÷ =

5.07 MeV, 0.88 MeV.

(35c)

SECOND ORDER BINDING ENERGIES

215

The final contributions to the matrix F s r are the first order matrix elements of Vs and gc in the two-valence-particle states and the six second order two-body diagrams shown in fig. 6c whose sum we denote by X, giving JT JT (Vs + gc)p,p2, p3p, + Xp,p2, p3p,"

(35d)

The results of diagonalising the matrix F s T built up from eqs. (35a) to (35d) are given in fig. 7 for a number of different oscillator lengths, b. An analysis of our results reveals several features which are well known from the earlier work of Kuo and Brown 17) and others 31, 32, 34). We merely reiterate these without giving details: (i) For T = 0 states the most important second order corrections come from the ladder diagram. (ii) For T = 1 states the "core polarisation" (the third diagram in fig. 6c) contributions are dominant. (iii) It is not sufficient to truncate the sum over intermediate configurations at 2fio~ due to the slow convergence of the tensor force, although for T = 1 states the error would not be large ( ~ 0.2 MeV for the lowest J = 0 state). This is about half the effect found by Vary e t al. 34) using a different interaction and keeping core polarisation corrections only. In addition, as is evident from fig. 7 we find that the spectrum is sensitive to the chosen oscillator length parameter, b. The change in energy of the lowest J = 3, 5, 2 and 4, T - - 0 levels is almost entirely a first order effect. On the other hand the stability of the lowest T - 1 level between b = 1.8 and 2.0 fm is due to compensating changes in first and second order contributions. The rms radius of oxygen, as determined by electron scattering, would be best fitted using b ~ 1.7 fm; however, such a value does not give very good wave functions for the valence particles, with which we are primarily concerned. Hartree-Fock calculations 2 9) show that optimal oscillator wave functions for the d~, ls~ and d~_ configurations would have oscillator lengths of 1.88, 2.07 and 2.26 fm respectively so it is not too surprising that b -- 2.0 fm gives the best ordering of the lowest levels in fig. 7. Of course, the 160 core should not be so large and a calculation using different lengths for core and valence particles would undoubtedly be better and is under way. A larger value of b would also help in explaining the large effective charge in quadrupole effects. For example, an increase in b from 1.7 fm to 2.0 fm would decrease the required effective charge as) in this mass region from 0.6e to 0.4e for neutrons and from 1.4e to 1.0e for protons. One might have hoped that the inclusion of the H F corrections (first and second diagrams of fig. 6c) would have taken care of changes in the valence particle wave functions. However, their effect is only marginal showing once again the inadequacy of second order perturbation theory for estimating HF-type corrections. From fig. 7 one sees that for b -- 2.0 fm there is an overall discrepancy in the ground state energy compared to experiment of just over 1 MeV but the spectrum is reproduced quite well. The energy difference between the lowest levels with T = 0

216

E . A . S A N D E R S O N et al.

and T = 1 is correct and the spacing of the triplet with J = 1, 3, 5 and T = 0 is good. The lowest T = 1 states with J = 0, 2 and 4 are not sufficiently split but admixtures of the low lying deformed states from 4p-2h configurations will have some effect here. The use of different b-values for the three s-d configurations could also have a considerable effect upon the spectrum. 10. Conclusion

The matrix elements deduced from the phase shifts in an earlier paper I and which were shown in II to lack the necessary saturation properties have been modified by the addition of a potential Vc which causes little change to the phase shifts but has a hard core. The radius of the core has been chosen to reproduce approximately the correct saturation, a procedure consistent with other approaches 36) which argue that whole families of phase shift equivalent potentials may be deduced which possess quite different saturation properties. We do not find that the tensor force has a particularly important part to play in achieving saturation in finite nuclei, contrary to the situation in nuclear matter 21). Perturbation theory calculations have been made for nuclei at or near closed shells using a simple reaction matrix method for the additional potential Vc and standard second order theory for the remainder. In this way the effects of the core have been separated to some extent from the effects of the remaining smooth part of the potential. Convergence of the second order terms with excitation energy is found to be quite good except for the tensor force for which a summation is carried out above a certain point. The single particle energies calculated in this way produce spin-orbit splittings in good agreement with experiment but the relative energies of particles with different l in the same major shell show insufficient binding for states of higher l. This is believed to be due to the use of harmonic oscillator wave functions and Hartree-Fock calculations 29) support this view. It appears that much of this fault can be rectified if different values of the oscillator length parameter b are used for valence and core nucleons. We have used a simple but realistic prescription for calculating single particle energies and find that, for occupied and low lying valence levels an oscillator spectrum is approximately maintained but with an increased oscillator energy spacing. This has the effect of reducing the second order contributions by about 30 Yo in oxygen and is consistent with the results of some calculations ~5) with unmodified values of hto in which the third order perturbation terms are roughly 25 ~ of the second order terms and of opposite sign. With the prescription given here for the single particle energies one expects the third order terms to be much reduced and preliminary calculations bear this out. A number of related calculations have been made or are being made with the matrix elements described here and will be submitted for publication soon. They are: (i) a Hartree-Fock calculation 29) for the nuclei 160 and 4°Ca, (ii) a calculation 30)

SECOND ORDER BINDING ENERGIES

217

of spectra of the p-shell nuclei, (iii) a calculation of spectra in the early part of the s-d shell, (iv) a study of third order corrections including g¢ and the single particle energies described here, and (v) a study of the use of different b-values for different orbits.

Appendix A To solve the inhomogeneous differential equation

(o9.- H. - V¢)~9,,, = (o9r - e,a)R.,,

(27)

with Vc defined in sect. 2, it is convenient to write Z.t = R . , - ~ , . t and solve the equation for Z.t, ( o g r - H , - Vc)z,, = -- V~ R,,. (A.1) Because of the discontinuous definition of V¢ one solves in each of three regions. We further restrict ourselves to the case l = O. In 0 < r < c, the potential V¢ is infinitely repulsive, so that ~ = 0 and hence Z,o = R.0-

(A.2)

In a < r < ~ , the potential V¢ = 0, so that Z,o satisfies

(O9,-Hr)z~ o = 0,

(A.3)

which is the usual oscillator equation. However, the relation (29), o9, = e~o + 2 A+ 2B where the £ are the negative energy shifts of the single particle levels, means that generally o9, < 0. There is a solution of (A.3) for all o9, since although we impose the condition ~k ~ 0 as r ~ oo, no condition is yet imposed at the lower boundary r = a. Writing z = rib (A.3) reduces to

~

-¼z2+m+½

rz.o = O,

with solution

1 Zno = - BDm(r/b), /,

(A.4)

where in = -½+o9,/ho9 and Dm is a Weber function 37). In c < r < a, the substitution of the explicit form for V~ leads to the differential equation for ~k,o,

(~-~2z2 +q2) r ~ o = {(o9,-e,o)/(1-fl)hog}rR~o,

(A.5)

where qZ = (D+o9,)/(1-fl)ho9. A solution ffno must be found which vanishes at r = c and together with its derivative, is matched with the solution (A.4) in r > a. A particular integral of (A.5) is found by using an eigenfunction expansion in the

218

E.A.

et al.

SANDERSON

set f, = sin { t n ( r - c ) / ( a - c ) } where t = 1, 2, 3 . . . . . and a solution A sin { q ( r - c ) / b } of the homogeneous equation added to satisfy the boundary condition at r = a. Expanding (r~/InO)PI = ~ O ~ n t f t ,

(rgno)=

~flntft,

t=l

t=l

and using eq. (A.5) leads to the formulae for the coefficients: ant = (03 r -- gnl)flntf{q 2 -- t2b27~21(a fl.t = { 2 / ( a - c)}

FrR.o de

sin

(A.6)

¢)2}(1 -fl)h°),

{tTt(r- c)l(a - ¢)}dr,

(A.7)

Finally, rO.o = A sin { q ( r - c ) / b } + ~. e.tf~,

(a.s)

t=l

where the imposition of the boundary conditions at r = a leads to two equations: A sin {q(a - c)/b} + BDm(a/b) = aR,,o(a),

.4

-" s,,, a co

= b d (rR.o)r=a

(1-fl)rcb ~. (_l)tte,,, '

dr

a-c

(A,9)

t=l

from which the constant A may be deduced. Notice that the only information needed about the Weber function is its logarithmic derivative D,'./D.,. The overlap required to construct the .qr(OOr) matrix element in eq. (28) is built up from eqs. (A.2), (A.4) and (A.8): =

L

RwoR~or2dr-A

r

R~,osin { q ( r - c ) / b } r d r

,.,c

- ~ ~,, t=l

,'o

rdr+

*de

,,,oDm r

rdr.

*2a

The third term may be simplified by expanding R,,.o and the fourth by using the fact that R,,,o and (l/r)D., are both eigenfunctions of the same operator Hr. Finally, =

R,,, o R~o r 2 d r - A

R~, o sin { q ( r - c ) / b } r d r - - l ( a - c •~ c

+ ---~n'O - - ('Or

bBD,.(a/b)

~,,tfl,,'t t=l

(rR.,o), = . - aR.,o(a)

(A. 10)

SECOND ORDER B I N D I N G ENERGIES

219

Appendix B The second order ladder correction to a two-particle state lAB) is given by

AE = E (ABIVslpl pz>2/(%,+%2-~a-en),

(B.1)

PlP2

with oscillator denominators. If the Pauli principle is ignored then the sum over excited particle states ]PxP2) may be made in a basis of relative and c.m. states of the pair and introducing the usual Moshinsky expansion for lAB) we have:

AE =

~.

(ABI(Snl)JN.LZ~J)(ABI(S87)JN~J)

SnlJN&ehln'l"

x ((Snl)Jl Vd(Sn'l')J>((Sal)jl ~sl(Sn't')J>/(2(n'- n) + (l'- l)}ho).

(B.2)

Comparing this result with the first order expression

(ABI~'slAB> =

~., .(ABI(Snl)~N~d>(AB[(S~I)f N~J)((Snl)~IV~I(Sai)f>, SnlJNM~I

(B.3)

we see that the sum (ABlP's]AB)+AE may be obtained from eq. (B.3) by adding to the relative matrix element ((Snl)Jl~,l(Sal)J> in eq. (B.3) the correction

((Snl)Jl Vs](Sn'l')J)((S~i)Jl gs[(Sn'l')J)/{2(n' - n) + (1'- l)}hoJ. (B.4) rl'l'

References 1) J. P. Elliott, A. D. Jackson, H. A. Mavromatis, E. A. Sanderson and B. Singh, Nucl. Phys. A121 (1968) 241 2) J. Dey, J. P. Elliott, A. D. Jackson, H. A. Mavromatis, E. A. Sanderson and B. Singh, Nucl. Phys. A134 (1969) 385 3) H. A. Mavromatis, E. A. Sanderson and A. D. Jackson, Nucl. Phys. A124 (1969) 1 4) A. D. Jackson and J. P. Elliott, Nucl. Phys. A125 (1969) 276 5) R. L. Becker, A. D. MacKeUar and B. M. Morris, Phys. Rev. 174 (1968) 1264 6) R. J. McCarthy, Nucl. Phys. A130 (1969) 305; H. S. KShler and R. J. McCarthy, Nucl. Phys. 106 (1968) 313; R. Mercier, E. U. Baranger and R. J. McCarthy, Nucl. Phys. AI30 (1969) 322; K. T. R. Davies, R. J. McCarthy and P. U. Sauer, Phys. Key. C6 (1972) 1461 7) M. Baranger, Proc. Int. School of physics Enrico Fermi, Course 40, 1967, ed. M. Jean (Academic Press, NY, 1969) p. 511 8) B. R. Barrett, R. G. L. Hewitt and R. J. McCarthy, Phys. Rev. C3 (1971) 1137 9) J. Goldstone, Proc. Roy. Soc. A239 (1957) 267 10) H. A. Bethe, Phys. Rev. 103 (1956) 1353; H. A. Bethe and J. Goldstone, Proc. Roy. Soc. A238 (1957) 551 11) M. Baranger, ref. 7) 12) H. A. Bethe, Phys. Rev. 138 (1965) B804, 158 (1967) 941; R. Rajaraman and H. A. Bethe, Rev. Mod. Phys. 39 (1967) 745; B. H. Brandow, Phys. Rev. 1ff2 (1966) 863; P. Grang~ and M. A. Preston, Nucl. Phys. A204 (1973) 1, and preprint and references therein 13) M. Baranger, ref. 7) sect. 5-10 14) K. T. R. Davies and R. J. McCarthy, Phys. Rev. C4 (1971) 81

220

E.A. SANDERSON et aL

"

15) N. Kassis, H. A. Mavromatis and B. Singh, Phys. Lett. 37B (1971) 15; N. Kassis, Nucl. Phys. A194 (1972) 205 16) A. Kallio, Phys. Lett. 18 (1965) 51 17) T. T. S. Kuo and G. E. Brown, NucL Phys. 85 (1966) 40 18) J. Dey, D. Phil. thesis, University of Sussex, 1969, unpublished 19) H. A. Bethe, in The two-body force in nuclei, ed. S. M. Austin and G. M. Crawley (Plenum Press, New York, 1972) 20) I. R. Afnan, D. M. Clement and F. J. D. Serduke, Nucl. Phys. AI70 (1971) 625 2 0 M. I. Haftel and F. Tabakin, Nucl. Phys. A158 (1970) 1; L. Ingber, Phys. Rev. 174 (1968) 1250; P. J. Siemens, Nucl. Phys. AI41 (1970) 225 22) P. C. Bhargava and D. W. L. Sprung, Ann. of Phys. 42 (1967) 222 23) R. V. Reid, Ann. of Phys. 50 (1968) 411 24) T. Hamada and I. D. Johnston, Nucl. Phys. 34 (1962) 382 25) M. I. Haftel, E. Lambert and P. U. Sauer, Nucl. Phys, A192 (1972) 225 26) J. P. Elliott and A. D. Jackson, Nucl. Phys. AI21 (1968) 279 27) F. Ajzenberg-Selove, Nucl. Phys. A152 (1970) 1; A166 (1971) 1 28) P. M. Endt and C. van der Leun, Nucl. Phys. AI05 (1967) I 29) C. Malta, D. Phil. thesis, University of Sussex, 1972, unpublished; C. Malta and E. A. Sanderson, to be published 30) H. Dirim, D. Phil thesis, University of Sussex, 1972, unpublished; H. Dirim, J. P. Elliott and J. A. Evans, to be published 31) J. C. Sens e t aL, Nucl. Phys. A199 (1973) 232, 241; C. Rolfs e t al., Nucl. Phys. A199 (1973) 257, 274, 289, 328, and references therein 32) B. R. Barrett, in The two-body force in nuclei, ed. S. M. Austin and G. M. Crawley (Plenum Press, New York, 1972) and references therein 33) B. Singh, to be published 34) J. P. Vary, P. U. Sauer and C. W. Wong, preprint 35) J. P. Elliott and C. E. Wilsdon, Proc. Roy. Soc. A302 (1968) 509 36) M. I. Haftel and F. Tabakin, Phys. Rev. C3 (1971) 921 37) P. M. Morse and H. Feshhach, Methods of theoretical physics, pt. II (McGraw-Hill, New York, 1953)p. 1403 38) I-L A. Mavromatis, B. Singh and E. A. Sanderson, to be published

ERRATUM J. P. ELLIOTT, A. D. JACKSON, H . A . MAVROMATIS, E. A. SANDERSON a n d B, SINGH, Matrix elements of the n u c l e o n - n u c l e o n potential for use in nuclear-structure calculations. Nucl. Phys. A121 (1968) 241. Eqs. (18) a n d (19) a n d the u n n u m b e r e d e q u a t i o n which follows eq. (19) on p. 252 should all have a m i n u s sign inserted in front of the term o n the left h a n d side. The sign in front of the term c o n t a i n i n g the c o m m u t a t o r [r 2, V] in eq. (20) should be m i n u s not plus. The sixth n u m b e r in c o l u m n two (n = 0) of table 2 should be - 5 . 3 7 instead of - 5.27.