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IC/69/135

INTERNATIONAL ATOMIC ENERGY AGENCY

INTERNATIONAL CENTRE FOR THEORETICALPHYSICS

BINDING ENERGY OF A A PARTICLE

IN NUCLEAR MATTER

J. D^BROWSKI

and

M.Y.M. HASSAN

1969

MIRAMARE - TRIESTE

IC/69/135<+ *

Binding Energy of a'_A_ Particle in Nuclear Matter

Janusa D^browski

International Centre for Theoretical Physics, Trieste and Insti-

tute for Nuclear Research, Warsaw, Poland

and

il.Y.M.Hassen**

International Centre for Theoretical Physics, Trieste

ABSTRACT

The binding energy of a /\ particle in nuclear matter, B^(co) , is

calculated self-consistently with the help of the Brueckner theory.

In the K matrix Equations for the A_~N interaction, pure kinetic

energies in the intermediate states are used. The K matrix equa-ls. •"

tions are solved numerically. The rearrangement energy is taken

into account. The values of B A(oo) calculated with several central

j/\-H potentials, v>« - though smaller than the values obtained

by other authors - are, in general, bigger than the empirical

value of B,(Oo) . An agreement with this value is obtained only,

if v**T , adjusted to the binding energy of ^He-5 , has a suffi-

ciently big hard core and is sufficiently suppressed in odd states.

Possible ways of reducing the calculated value of B A(©o) are dis-

cussed. A critical discussion of the independent pair approxima-

tion applied by other authors in calculating B A(«3) is presented.

MIRAMARE - TRIESTE

October 1969

* To be submitted for publication-

* A preliminary version of this paper was issued in May 1969 as ICTP, Trieste, Internal Report IC/69/40.

** On leave of absence from University of Cairo, U.A-R.

1. Introduction

The binding energy of a A particle in nuclear matter, BA(o°) t

is a quantity of considerable interest in the phenomenological

analysis of the ^A - nucleon interaction, v ^ . mere as the binding

energies of light hypernuclei are determine^ primarily by the S wave

part of the /\ - T$ interaction, the binding energies of rheavy

hypernuclei depend on the A - IT. interaction in higher angular

momentum states. A direct calculation of the binding energy of-

a A particle in a heavy but finite nucleus would be a most diffic-

ult task. It is much easier to calculate BA(<*>) , the binding ener-

gy Of a A particle in an infinite nuclear medium, i.e., in

nuclear matter.

An empirical estimate of B^(oo) is a nontrivial problem.

One has to extrapolate the measured binding energies of a A parti-

cle in hypernuclei with finite values of A , B^(A) , to the limi-

ting case A -^-co . The early estimates i~4 of BA(o°) were

based on identified hypernuclei which, however, are all light

( UP to A C ) an<3- consequently the extrapolation was of a consi-

derable uncertainty. Obviously, much more important in determining

B, (oo ) is the knowledge of binding energies of A in heavy hyper-_ •

fragments which play an essential role in all the newer estimates5-12

of BA(©o) . Unfortunately, there are considerable ambiguities

in determining BA(A) of mesonically decaying heavy hyperfragments.

Namely, neither the mass of the hyperfragments nor the precise decay

modes can be determined accurately. These uncertainties are reflect-

ed in the BA(o<>) values obtained by different authors. The notable

estimates are those by Lagnaux et al.'(27 - 3 MeV), "Lemonne et al."

(27.2 ± 1.3 MeV), Goyal10(32 t 2 MeT), Bhowmik et al.1:L(36.6 ± 0.6

MeV), and Kang and Zaffarano12(27.7 £.0.6 MeV or 28.3 t Q.3 MeV).

All these results seem to indicate a tentative estimate ,• i •

BA(oo) . 30 ± 5 MeV . (1.1)

As mentioned before the importance of BA(oo) in reconstructing

v^. is connected with the sensitivity of B.(oo) to the A - N

interaction in higher angular momentum states. To visualize it let

us consider the range of energies of the A - K system, relevant in

determining B.(oo) # In the ground state of the system: A particle

+ nuclear matter, the A particle occupies the state with zero momen-

tum, and the relative A - N velocity, vrej, , is equal to the nucleon

velocity, p.f/fi'W?> ,t where p^ is the nucleon momentum (in units of il

' and Itft^ is the mass of the nucleon (divided by ii 2) 1 6. This

corresponds to the A particle kinetic energy in the nucleon rest

system (the laboratory system for _A - N scattering),

where 0*1^ is the mass of the /\ particle. The range of

extends from 0 to the Fermi momentum, k« , and we have:

where S^ is the Fermi energy of nuclear matter. The average valueP 2

of p« in nuclear matter is 0.6 k™ , and we ge-t for the averagevalue of T^ :

ior the value of the spacing parameter r • 1.1 F we have

kT - 1.35 r 1 , (1.5)

and Ep « 37.8 MeV. for this value of Ej, we have; 0 T A 45 MeV,

and for the most heavily weighted /\_ laboratory energy, T^ 2 7 MeV.

-2-

Now, the /\ - proton scattering eacperiments ' show that for A mo-

menta bigger than 248 MeV/c, i.e., for T A ^ 30 MeV , the A - P

scattering departs from isotropy. Thus around T A - 30 MeV the A - N

interaction in the P state starts to be effective, and consequently

we expect that the P state interaction will effect the value

of BA(t>°) .

Another way of reaching this conclusion is to consider the re-

lative A - N momentum in nuclear matter, p -

Similarly as with T^ , we have

and for the average value,

If we assume the range of v,^ to be equal to the intrinsic range

b * 1,5 I, corresponding to the two pion exchange, we get;

0 ^ bp^.1.1 , and bp 0.85 , for the value of k . , Eg.(1.5).

By applying -the classical argument with the impact parameter we see

that both S and P waves in the A-N system are expected to be

important in calculating B^(co). This conclusion has been confirmed

by all existing calculations of B^(OO)t which also show that the

D state plays already a minor role. Actually, the role of the P

wave is enhanced compared to the. S wave. Namely, the short distance

repulsion in Y^ acts predominantly in the S state and cancels

a big part of the S wave contribution of the attractive tail of

Thus the net contributions of the S and P states to

A are of comparable magnitudes*

Several calculations of BA(^>) have been published }18~51.

In the case of a regular, purely attractive A - N potential one

may apply simply the perturbation expansion lti'1". However, in the

-3-

case of a realistic A - N potential with a hard core a more

sophisteated method of calculating BA(0°) is necessary. Two such

methods have been applied in the existing calculations of B (oo) :

the Brueckner method 2»20"*2' and the Jastrow method 2'""*1^ In the

present paper we shall restrict ourselves to the Brueckner method.

Among the Brueckner type calculations of BA(©o) only the paper by

20 ™~Taharzadeh, Moszkowski and Sood follows a systematic approxima-

32

tion scheme, based on the separation method . However, the results

obtained in Ref.32 are outdated because of the assumed form of v*«.

and of the single particle energies. In all the remaining Brueckner

type calculations the so called independent pair approximation has

been applied. No precise justification of this approximation has25been presented so far. The only attempt y to estimate corrections

to the independent pair approximation has shown that probably they

are big. furthermore the single nucleon energies applied in all the

existing calculations of BA(oo) cannot be justified from the point

of view of the present state of the theory of nuclear matter.33

In the present paper we solve exactly the equations of theBrueckner theory for the" ^\~ IT interaction. In accordance with the

34

present state of the theory of nuclear matter we use pure kinetic

energies in the intermediate states. The pure nuclear matter problem

is considered to be solved with the resulting saturation density

and energy in agreement with experiment. The precise shape of

single nucleon energies of the occupied states turns out to be not

important provided its average value is consistent with-the known

binding energy per nucleon in nuclear matter.

Pirst calculations ^Pi? of B.(oo) along these lines have been

performed with the help of the improved Moszkowski-Scott separation

method5' of solving the K matrix equation (Eq.(2.5)). Recently

an essentially identical approach has been reported by Bodmersn-4-

The main difference between our approach and that or Bodmer and

Rote is that we use the integral form of the wave function equation

(Eq.(2.13)) whereas Bodmer and Eote work with the differential form

Of this equation.

The whole scheme of calculating B/^°°) is presented in Sec.2

for the case of a central A - N potential considered in the pre-

sent paper. In Sec,3 we discuss the choice of v*™-" used in the

present calculations. The computational procedure is described in

Sec.4. The results obtained are presented and discussed in Sec.5.

Finally in the Appendix we present a critical discussion of the

independent pair approximation.

2. K Matrix Theory of BA(co)

We assume that the A - interaction, v,«. , is central and

charge symmetric. It is equal v(r) in the spin triplet state,

and ^v(r) in the spin singlet state, where jr_ is the distance

between A and IT . Nuclear matter is assumed to have in each

occupied state four nucleons: two protons and two neutrons with

spin up and down. In such a system with no neutron excess the charge

symmetry breaking (CSB) component of Vi~ is expected to have

a minor effect only. For instance, the CSB potential of the form

considered by Herndon and Tang 7 gives a zero contribution to

!B (oo) in the first order approximation in the attractive partA,

of the CBS potential. In the presence of a neutron excess (and the

heavy hyperfragments used in the estimates of BA(c°) have a neutron

excess) the CBS potential could play a role which, however, is not

considered in the present paper.

Let us introduce the following notation. By j^, , m,\ we

shall denote the nucleon and lambda momenta (in units of "EL ) of

-5-

the occupied states, by Jgfa^ , k* the momenta of the excited states,

end by j ^ , ^ x general momenta without any res t r ic t ions . As mentioned

before,

Furthermore, we shall denote by u the reduced mass of the / \ - K"

system;

Aw*The binding energy of the A particle in nuclear matter, B (co)

is defined by; ^1(?,

- BA(oo) - E(A + 1 .) - E(A) , •;'• (2.2)

where E(A) and E(A + 1^) are the ground state energies of nuclear

matter and of the system : nuclear matter + A particle. By applying

the Brueclcner theory expressions for E(A) and E(A + 1^) one

obtaxns :

- B^Coo) - V^ + VR , (2.3)

where V. is the A single particle potential (usually, in the

literature, denoted by -D) and VR is the rearrangement potential.

First, we shall explain the method of calculating V\ > and after-

wards, at the end of this section, we shall discuss the rearrange-

ment potential, V E , whose magnitude depends on the way in which we

define VA .A

A. Calculation' of VA

40 ""Por VA we have ;

- 6 -

1 3

where K and K ere the A.- IT singlet and triplet 6,tate K ma

trices, i.e., the effective _A - IT interactions in nuclear matter.

They are determined by the following equations:

M €£*)^Cfc/fJ , (, . 1,5)

where € N , £ A are the U and A . kinetic energies, and e^ , e

are the H and _A single particle energies of the occupied states:

, (2.6)

AA AA , C2.7)

where the last equation is a consequence of Eq.(2.l). The single

nucleon potential, V« , will be discussed later.

Equation (2.5) differs from the traditional form of the

K matrix equation by the appearance of the kinetic energies, £ N

and C A i in the intermediate states. The point is that a systema-

tic approach in terms of the number of hole lines requires that

particle self-energy diagrams should be considered together with

other three-body diagrams. The total contribution to the energy

per nucleon of the entire class of the three-body diagrams is pro-

bably not bigger than about lMeY in the case of nuclear matter,

and one should expect a similar situation in our case. -It seems

then that the most reasonable procedure is to disregard the self-

energy insertions into the particle lines, i.e., to use kinetic

energies in the'intermediate states. The probably small error thus34 A? 43

commited may be left to a perturbative estimate. *. » y Needless

to say that by using €"^0%) , £^(kjj in 3q.(2.5) we essentially

simplify the problem of solving this equation.

-7-

In solving Eq.(2.5) we follow closely Ref.4l. We introduce the

relative and center of mass momenta

N j^V CmN + -m^ . raN/mH t (2;8)

The last parts of these equations follow from Eq.(2.l). Notice that

M -

i.e., the center of mass momentum is proportional to the relative mo

mentum. Because of momentum conservation, we have

where p and P are expressed by Pw » PA b7 equations analogous

to EqsX2.8-9). Because M is determined by m . we shall drop the

subscript M at the K matrix, and also at the wave function,K~KjK , defined by the equation

Eq.(2.5) may be written as the following integral equation for

the wave function " 4 ^ in the configuration space:

where the Green function

"(2.14)

where the exclusion principle operator

(2.15)o

and where

a(n) -

Notice that all the momeBta on the right hand side of Eq.(2.16) are

determined by jn , Eqs.(2.l),(2.8),(2.10).

With the help of the wave function ^3&(r)» which describes

the relative A - JN motion in nuclear matter (with relative momen-

tum m and in the spin state s), we may write Eq.(2.4) in the form:

where

To be able to analyze Eq.(2.13) into uncoupled partial wave

equations we approximate the exclusion principle operator Q by

its angle average,

Q(MiP) *QCm,p>.- -±- C d M Q t M i f > , (2.19)

which may be calculated.easily with the result;

•3(m,p) -

f 0

V 1for

for ^ > ^ r . + 7TlA/lrw/'m^? (2.20)

[<f i^ IN m/itt VVc p A CHl K M A V k

Kotice that in writing Eq.(2.20) we have replaced _M by m. ,Eq. (2JL0)

The approximation (2,19) which restores the rotational invariance

of Eq.(2.13) has been checked to be fairly accurate in the nuclearh.lL

matter case. 'With the approximation (2.19) we may expand G ^ into spherical

harmonics; , ,

-9-

wnere

(2.22)

Similarly, we expand

and from Eq.(2.13) obtain the following equations for the radial

functions ' R, ; (•'CO

£ j dr'r'2Q|(r,r0^(r')^Cm,r') . • (2.24)

O • r •

For a potential with a hard core of radius r. we encounter

the difficulty of having inside of the hard core in Eq.(2,24-) a pro-

duct vE of the indeterminate form ©o x 0 . ¥/e bypass this diffi-

culty by approximating the hard core by the hard shell of the same4-1 45radius r^ . ' ' !Ehis approximation has been shown to be quite

'•'"•••/" 4 6satisfactory. By applying this approximation, we get in place

Of Eq.(2.24):47 {l :

6 0 . , . " •

,sWldr'r'^Cr.rO v(rO\U,r') , (2.25)

where !< ' '\. jt.f. -/ >-- " ^

*.*,J (2.26)

is the solution of Eq.(2.24) for the case of a pure hard shell

interaction, and where the new Green functions

_ J f : (2.27)

Equation (2.18) may be written in the form:

& p.-10-

where the pure hard core contribution ,

- - y

(where V A is the hard core potential), and the contribution of sv A ,

the attractive tail of v ,

CO

^ ^ ^ ^ ^ ^ ; . (2.30)

In both equations (2.29),(2.30) we use the notation:

for the solution of Eq.(2.13) in the case of a pure hard core po-. 48

tential v * %A (with VA approximated by a hard shell potential)./ H / i A '7C

Notice that <C K > A , flnd S/K , ^ do not depend on the A - K

spin because in all the A - IT potentials to be considered the hard

core radius .r^is the same in the triplet and singlet states.

Let us summarize the procedure of calculating V\. . IVe simply

solve Eqs.(2.25) for the radial functions ^R,(m,r). Next, we calcu-

late the K matrix elements, Eqs.(2.28-30) and insert them into

expression (2.17) for Y^ . Obviously, we have to start the whole

procedure with the calculation of the Green functions G^r,r') and

iMr.r'), Eqs.(2.22),(2.27). To do it we have to apply certain form

of the single nucleon energy e^G^O . Yfe shall discuss this point

in a moment. Eight now, let us assume that we know e^dn^). The only

unknown element in the Green function Gj£ is then V* » sfld- obvious

ly we are faced with a self-consistency problem. It is, however,

a very simple self-consistency problem because it concerns only

one number, the value of VA . We have to assume certain input

-11-

value of VA to compute the Green functions. After applying the

whole .procedure we get certain output value of V. . Self-consis-

tency is achieved when the output and input values of VA are

equal. Practically, one has to perform the whole calculation for

a few input values of VA . In this way one obtains the output

value of Y A as a function of the input value of VA . Irom this

function one then determines the point of self-consistency.

B. The Single Nueleon "Energy e,T

There are few ways of fixing the spectrum of

' (i) Let us assume that the Brueckner theory is correct and,

with the proper form .of the I? - V, interaction, it leads to the

observed value of the binding energy per nucleon in nuclear matter,

£ , , at the observed value of the equilibrium density of nuclear

matter, determined by k-™ . Then

Iu+- k% » £(£*«-IN) = - £

49where the bars denote average values in the Fermi sea. ^ If. we

apply the effective mass approximation for e-. ,

e Cm,-) « m / / 2 W * + A , (2.33)

we need, besides the condition (2.32), one condition more to fix

the two constants, fU^ and A . Let us consider the condition:

At the equilibrium density, the energy per nucleon in nuclear matter,

- _£_/>; >. is eouel to the separation energy. Hence "3q.(2.34) says

that the single nucleon energy at the Fermi surface is equal to

the separation energy.•Actually, in the Brueckner theory, the re-

arrangement energy should be added to e^(k?) in Eq.(2.34).

Nevertheless we shall use both the conditions (2.32) and (2.34)

-12-

to fix e (mjjj), assumed in the form (2.33).

(ii) In the crudest approach we may approximate e<t(%) in the

energy denominator of GY , Eqs.(2,22),(2.16), by its average value:

H F +• i €.v^i] , C2.35)

calculated from Eq.(2.32). , "

(iii) We may use the values of ^(nw) obtained in one of the

successful nuclear matter calculations. Here, we shall use e^(m^)

obtained by Brueckner and Gammel . In order that Eg..(2.32) be

satisfied for the values of k^ and jjT".£>K aPPlie& in the present

work, we shall shift the whole Brueckner-Gammel spectrum a little

bit (by less than 6 MeY).

Actually, the calculated value of Y A turns out to be not

sensitive to the particular way, (i) - (iii), of fixing e^C^-r) >

provided.Eq.(2.32) is satisfied. After this has been found out, the

e^ spectrum (i) has been applied throughout the present work.

C. The Rearrangement Potential YR •'

A detailed derivation of the expression for the rearrangement

potential V^ has been presented in Ref.40. The source for Y^ is

the difference in the nucleon-nucleon K matrix in the case of nu-

clear matter plus A particle, and in the case of a pure nuclear mat-

ter. Namely, the presence of a A particle changes slightly the

single nucleon energies. In the present approach, with pure kinetic

energies in. the intermediate states, only the single nucleon energies,

eN^mN^' o f ^ e occ<uP^e<i states are affected by the presence of the

A particle. Consequently, the total rearrangement potentialf YR j

is equal to the hole rearrangement potentialt Yg^ (the particle

rearrangement potential, V^v^, vanishes). By applying the simple

and accurate approximate expression for Ytn^ » derived in Eef.4O,

we may write:"

-13-

where the dimensionless parameter

where ^ is the density of nuclear matter. The nucleon-nucleon

difference function:

:

where q^/ is the wave function of the relative motion of two nucleons

in nuclear matter in a state with the relative momentum m « (nu - m,0AJ."•"»'•• •- •

with the parity i , and with the spin defined by _s (s_ » 1,3 for the

singlet, triplet state), and_where [ e ^ ] 1 i s the P a r t o f the

unperturbed wave function, e1^^ f with the parity i ( r denotes the

relative position vector of the two nucleons). The subscript Av

at the nucleon-nucleon correlation volume,4 r , indicates the

average value in the Fermi sea, i.e., one has to average over the

momenta jn^ and ^m^ of the two nucleons. Actually, Bq.(2.37) is

en obviQus generalization of the corresponding expression derived

in Eef.40 in the case of a spin-independent Serber IT-- 17 potential.

3. Choice of

The central, spin-dependent, charge-symmetric A - $ interac-

tions used in the present calculation of BA(oo) are listed in Table

The parameters characterizing the interactions are: the hard core

radius r/X , the intrinsic range b , the singlet and triplet well-

depth parameters Sg^, s^ , the singlet and triplet scattering

lengths 8^ , a/ , and the corresponding effective ranges rg, , r^ .

-14-

In all the potentials considered, r A end - b_ have the same values

in the singlet, and triplet states. Some of the potentials are weaker

in odd angular momentum states, i.e., their strength in these states

is equal y times the strength in even states. The shspe of the

attractive part of v.«. is shown also in Table 1,

Potential ETS is an old potential fitted by Herndon, Sang and2 3 • •

Schmid to the binding energies of the S shell hypernuclei. Its

intrinsic range is equal to the intrincic range of a purely attrac-

tive two pion exchange Yukawa potential. Typically, jITS is much

stronger in the singlet than in the triplet state.

Potential AGK has been fitted by Ali, Grypeos and Eok to_A.- p

scattering under the assumption of a common intrinsic range, _b_ , for

both the singlet and triplet interactions. Charge symmetry has been

assumed. Ho fit to hyperfragment energies has been attempted. It17should be mentioned thst the recent analysis ' shows that by fitting

the A - p scattering without the assumption concerning _b , one gets

a wide spectrum of acceptable values of ag , a^ , rg , r^ .

Potentials E, E *, K, I" have been determined by Herndon and

Tang ^° in the following way. Por given b and r- the spin- and

charge-dependent v ^ has been adjusted to BA(AH ), BA(AH ),

BA(AHe ). Next, the parameter y has been adjusted to _A - p scatte-

ring data. Among the potentials thus obtained we have chosen the

best potential H , and also the potential ^ . An alternative set

of potentials,which we denote by primes, has been obtained by

applying the same procedure, except that their charge-independent

components have been adjusted to B^Jp) and BA(AEe-;>). The best

of the alternative set are the potentials E' and •]?', although

they lead to total /\ - p cross sections about 20% too small compa-

red to the experiment." In the present work we shall consider also

potentials identical with those of Eef,39 except for not being

suppressed in odd states (_y « 1 ). They are denoted by the letters

-15- V

NX (lor no exchange), added to their original symbols. All the poten-

t ials of Ref.39 are charge-dependent. The .parameters in Table 1 * 2

characterize the charge-independent components of these potentials,

used in the present work (compare the remarks at the beginning of Sec,

2). These parameters differ from those characterizing the full

A - p potentials of Ref.39 (v/hich contain the CSB component),

shown in parentheses in Table 1.

Potential DW is one of the old spin-independent potentials con-pi

sidered by Downs and Ware in their calculation of BA(co). We in-

clude this potential into the present work for the sake of a compa-

rison with the independent pair approximation, discussed in the

Appendix.

4. Numerical Procedure

The present calculations have been performed for k- . • 1.35 3?

which corresponds to the spacing parameter _r- • 1.12 7 . In fixing

the single nucleon spectrum, Sec.2.B , we have used the value

j£ -0, = 15.8 MeV for the coefficient of the volume term in the '

semi-empirical mass formula. The three energy spectra cosidered

are shown in I'ig.l. As mentioned in Sec.2B , the spectrum (i) has

been used in the present calculations.

To compute Green's functions we transform T3q.(2.22) into the45: ^

where r^ = max(r,r') , r^ * min(r,r') , and a_ = [-2kz(m)] ". Notice,

that z(m)<0 , Eq.(2.16). The form (4.1) of G^ has the adventage

that - because of the [§ - l"] factor - the integral in Sq.(4.1)

-16-

extends over a finite range of p_ values, namely: p^l k

< 1.5 ltp (see E<j.C2.20)).

All integrations have been performed by means of Simpson's rule.

The following meshes have been used:

Green's functions, Eq(4«l);

p - 0.0(0.015)1.5 kv , (4.2)

^ ' •%? 0- 05)%+°.3(0.1)1^+1.1(0.2)^1.9(0.5)^+3.9 P . (4,3)i . e . - * ' • •'

Wave functions. Eqs.(2,25) have been solved by iteration with the

zero-order ansatz:

,r) . s^U.r) . • . ,.. (4.4).-^L.-^ . . . .

The _r_ mesh has been the same as in Eq.(4.3). The sixth-order itera-

tion has been found sufficiently accurate.

matrices, E<j.(2.30);

with * •

^ for. r > r A + 3 . 9 P , (4.6)

m - 0.0(0.25)1.0 yAkj/JTr^ . (4.7)

It was found sufficient to calculate V* tv?ice: firs^t v;ith

an input value of VK Of about -40 MeV , and next with an input

value of VA equal to the output value of the f irst run. The self-

consistent value of YA was then determined by a linear interpo-

lation.

All the numerical calculations have been performed on the

IBM?044 computer of the Centro di Calcolo dell'Universite di Trieste.

- 1 7 -

5. Results and Discussion

'Hie results of our calculations are shown in Table 2. We have

restricted ourselves to calculate the contributions to V^ of the

xirst three partial waves, S, P, D, and our results show that there

is no need to go beyond the D_ wave. The remarkable size of the P

state contributions is the result of the hard core which acts pre-

dominantly in the _S_ state and cancels partly the S state con-

tribution of the attractive part ov. vi«. . A typical behavior of

the wave functions is presented in IFig.2, showing the j* , s

and the singlet * Rj functions in the case of the potential H

for mjv = 0.25 % . xJ

In the case of the potential E we have calculated V* with

the help of the two other single-nucleon spectra,(ii),(iii), dis-

cussed in Sec.2VB (see Pig.l). We have found a negligible difference

of less than 0.1 MeV in the resulting values of V^ , compared v/ith

the case of the spectrum (i).

Our method of calculating V. is expected to contain all ter.r.s3 4 ILO I±."5 ~~

of order P ' , snd obviously the value of VA is sensitive

to the value of k^ . Let us consider a change of our value of

kp =1.35 F into the value of kp - 1.366 1" applied in mobt

of the existing calculations of V. . This change in k^ by

^ i.2>£ corresponds to an increase in p by v/ 3.6J5 , and UG

would expect a corresponding increase in the depth of Ys . ;7G

have confirmed this conclusion by calculating VA in the cacc

of the potential ]DW_ with k^ « 1.366, with the resultV A - -36.3 MeV (see Table 3).

Within the present scheme the calculation of VA requires no

detailed information-concerning nuclear matter. We only need the

value of the density, p , and of the binding energy per nucleon,

-18-

* However, to calculate the rearrangement

p o t e n t i a l V-c , E C L , ( 2 , 3 6 ) , we must know *)C , determined by the

nucleon-nucleon correlation volume in nuclear matter, JdrJXj ,

a quantity which depends on the nucleon-nucleon interaction. The

nucleon-nucleon correlation volume has been calculated by several

authors with the resultsforwetween about 0.07 and 0.21 (see*

Bef.4-8, where references to some of the earjfer calculations may

be found). The biggest value, Tt. * 0.21 has been obtained in

Ref.48 with the hard core Eeid H-K potential. Hie main source for•5

this big value is the the big S contribution to *>t , connected

primarily with the big size of the hard core ( J&*- 0.51 3? ) of the

Eeid potential in the ^S state. A reduction of r^to the usually

accepted value of about 0.4 P would reduce the ^S contribution

to yz- to the magnitude of the *S contribution f and this would

reduce the value of T£- about twice. Also the value, 2£= 0.15 of

Eef. 40 should probably be considered as an overestimate, because

the Bethe-G-oldstone ^ function H3 applied in Ref.4-0 in the cal-

culation of the nucleon-nucleon correlation volume heals slower

than a more realistic function obtained with the type of single-

nucleon energies, shown in Fig.l. On the other hand, the low esti-

mates, * X ~ 0.07 (e.g. those of Refs.57,55) neglect the contribu-

tions to 2£ o£ partial waves higher than $ , and probably arc

too low. We feel then that at the moment the most reasonable esti-

mate is

Ihis is a tentative estimate only, but our present knowledge of theto make . •

IT-IT interaction does not allow usVa more precise estimate. However,

the error of this estimate does not seem to be bigger than about

+ O.05. The corresponding uncertainly in B^(°°) would then be

about i 2 MeV . In the present work/we have assumed for 2£ tlie

value 0.1 . Eqs.(2.36),(2.3) lead then to "•the values of B (co)A

-19- ^"

shown in Table 2.

Results of other calculations based on the- independent pair

approximation and on the Jastrow method are shown also in Table 2.

In all of them the value of kp = 1.3S6 P""1 has been used. To com-

pare them with our results obtained with k , » 1.35 F one should

diminish the other results by approximately 3.% . The indepen-

dent pair approximation results do not contain the rearrangement

energy and thus should be compared with our results for VA . On

the other hand, the Jastrow method results should be compared

directly with our results for BA(c*>) . As seen from Table 2 ,

the Jastrow method values of B A(P<0 are systmatically and quite

appreciably bigger than our results. So far, we do not know why

are the Jastrow method results for BA(oo) so big. Certainly, it

would be very interesting to find out the reason for it. The inde-

pendent pair approximation results for VA are in general bigger

in absolute value than our results, although in the case of the

ETS potential the situation is reversed. It seems to us that the

independent pair approximation is not reliable for reasons explained

in the Appendix.

Among the results reported by Bodmer and Rote only the-

re suit for the potential E*FX can be compared directly wirl. our

result. The Bodmer and Rote result, VA « -61.7 MeV , when corrected

lor the difference in their (1.4- F-1) and our (1.35 P"1) value of

kv with the help of the factor (1.35/1.4-)' gives -53.5 MeV, in

agreement with our value of -53.7 MeV.- Also the approximate results

of Refs.35,36 obtained with the potentials HTS and AGK agree with

our present results.

Before discussing our results let us try to advocate the

accuracy of our method of calculating BA(oo) . The general scheme

of the contemprory Brueckner type theory of an infinite system

-20-

consists in grouping diagrams according to the number of hole

lines. » » • In the case of nuclear matter plus one A particle

we have always one A hole line (at least in the calculation of

BA(o<0 ), and in the successive steps of the scheme we increase

the number of nucleon hole lines. Thus the small parameter of the

whole approximation scheme is the same as in the case of a pure

nuclear matter, namely a quantity of order 2£ » E<i.(2.37). In the

present calculation we have included all two hole lines diagrams

in Y* , and the hole self-energy class of the three hole lines

diagrams in V-p * The remaining three hole lines diagrams form

the class of the so called three-body (AM-) diagrams. We assume

that similarly as in the case of a pure, nuclear matter the contri-

bution of the three-body diagrams is small, of the order of 1 MeV .

This would mean that in our procedure all the important three hole

lines diagrams are included, and we would expect that by going

one step further and considering four hole lines diagrams we would

get a correction to our values of BA(c») of the order of OC& ,i.e.,

of the order of one per cent.

There is no need to discuss our value of BA(oo) obtained

with the very old DW potential, included into the present work

only for the sake of the discussion of the independent pair

approximation see the Appendix). The value of IL(^°) « 60.1 MeV

obtained with the AGE potential seems to eliminate this poten-

tial as a realistic representation of the A - H interaction (com-

pare also the remarks in Sec.3). The value, BA(©o) « 36.9 MeV ,

calculated with the help of the HIS potential is reasonably close

to the range of the empirically determined values of BA(<?°) ,

especially if we recall the uncertainity in the value of ~*C .

However, the HTS potential, not fitted to the A - P scattering

data, has been outdated by the newer potentials of Eerndon and

. As is seen from Table 2, a big part of B^(e°) results

-21-

from the JP state A - IT interaction. Consequently, the suppression

of the interaction in the odd.,1 states, introduced in Ref.39 to f i t

the _A- P scattering, reduces essentially the values of EA(°°).

Among the potentials of Ref.39 applied in the present work, -one has

to distinguish between the unprimed potentials, _E,H_ , determined

by fitting the three- and four-body hypernuclear dat~a, and the

primed potentials, _S't _F', determined by fitting the three- and five-

body hypernuclear data. Kerndon and Tang J argue that the unprimed

potentials form a more reliable representation of an effective

central interaction in an. isolated A - K system. TTamely, the

inclusion of the hypernucleus ^He-7 in the determination'of the

primed potentials may lead to an underestimate of the triplet intcr-

57 ,58action due to tensor and isospin suppression effects.^''-^ However,

these effects are expected to operate in the A + nuclear matter

sy&ten in much the same way as in /^He . Consequently, in a calcu-

lation of B (oo) , the use of the primed potentials seems to be

more meaningful. It is then, enc our ig ing to see that the ^'potential

gives the valuo of J3A(OO) = 32.5 MeV which is in agreement with

the empirical estimate, Eq.(l.l). We conclude then that a \ - TT

central interaction with a sufficiently big hard core (r* = 0.6 P),. „ . - _ ^ l / r _ ^

which reproduces the experimental binding energy of A H ey , repro-

duces also the empirical value of B (<*>). , provided the interection

in odd angular momentum states is.sufficiently suppressed (y = 0.5).

Notice that the big size of the hard core is essential for our

conclusion. The S* potential with r . = 0.45 F , but otherwise si-^ 3

milar to the P* potential, leads to a too big value of B (co) s

41, C ileV . This is connected with the -higher kinetic energies of

the A - N system in nuclear matter compared to the case of / He-7

(see Sec.l). At these higher energies the hard core is more effec-

tive in reducing BA(°°) .-

All the unprimed potentials of Ref.39 lead to an overbidding

-22-

of a A P a r t i c l e in nuclear matter. The J_ potential, considered

by Herndon and Tang to be the best representation of the inter-

action in an isolated ^A - N system, gives B (oo) - 39.4- MeV .

And the problem obviously remains how to reconcile the^inter-

action in an isolated system (in particular, the A - P scattering)

with properties of systems like A H e5 and A + nuclear matter.

A reduction in the calculated value of B^(CKD) by a few MeV ©ould

be achieved with a bigger value of 2$ and a smaller value of y

determining the odd states suppression ( both QG and y are not

known precisely). Let us mention some other possibilities of re-

ducing the value of B (oo);

Tensor forces. In the case of put© nuclear forces it is well

known that tensor forces effective in an isolated N - N. • system

are much less effective, i.e., suppressed in nuclear matter. Inr

principle, a similar situation might be expected in the A - N case

(see, however, Eef.38).^^

Three-body A M interaction. No doubt, there are theoretical

reasons to expect the existence of an appreciable A M interaction,

and several authors have investigated its possible effects in

hypernuclel.1"' Obviously, with a repulsive j\m force one

should be able to reconcile the A - P scattering data with the

hypernuclear data. The unpleasant thing about introducing ANN for-

ces, v jjj t is that with the present amount of hypernuclear data,

withYpresent possibilities of solving the hypernuclear few- and

many-body problem, and with the present possibilities of deriving

theoretically Y^j. , the task of determining v ^ seems to be

extremely difficult.

Isospin suppression effect. This effect, pointed out by Bod-

mer^, would lead to the conclusion that the whole idea of treating

hypernuclei as systems of nucleons and a j\_ par tide with the same

-23-

^ as in an isolated A F system is wrong. Instead, one should use

an effective A- N interaction which would depend on the hyper-

rucleus considered. If this turns out to be the case, as it very

well might be, any attempt to correlate hypernuclear data would

become most difficult.

Acknowledgments

The authors express their gratitude to Professors A.Salam

and P.Budini for the hospitality extended to them at the Inter-

national Centre for Theoretical Physics in Trieste. They also

wish to express their apprecition to the staff members of the

Centro di Calcolo dell'Universita di Trieste for their helpful

assistence in the numerical computations. One of the authors (J.D.)

is indebted to Dr.W.Gajewski for his comments on the experimental

values of the hypernucleer binding energies.

Appendix

In an early paper by Gomes, Wale eke and V/eisskopf en

approximation of the Brueckner theory has been applied. The escence

of this opproxlmation is the division of v into the hard coref ^

part, v ^ , and the attractive part, v^ , • an exact treatment ofvc ,"and a first order approximation in v^ . In our case, of BA(oo)

the approximation amounts to the replacement of the exact Eq.(2.30)

by the approximate equation:

oo

In almost all published Brueckner type calculations of

-24-

a different approximation of Eq.(2.3O) has been applied, Namely:

v 'MA/A — ^ A ^ VA t

(A.2)

In other words, % ^ has been approximated in Eq.(2.l8) by • T ^ .

This approximation is usually called the independent pair approxi-

mation (IPA). In contradistinction to Eq.(A.l) the IPA is not a full

first order approximation in vft . 5 .This in itself is not a dis-

advantage of the IPA because a first order approximation in vA is

not expected to be a good one. The trouble, however, is that the

precise meaning of the IPA is not clear.

The most detailed presentation of the IPA has been given by

Downs and Ware 21. To. discuss the IPA more fully, let us investi-

gate one of the A - N potentials considered by Downs and Ware,

namely the potential DW of (Table 1. To compare our results for

the DW potential with those of Ref.21, we shall use here the value

of kp • 1.366 3?"1 , and the single particle energies appearing

in the K matrix equation (2.5) in the form:

J £ , CA'4)v/here the effective nucleon mass, W N » 0.735

are constants (they are canceled in the expression for the energy

denominator of Eq.(2.5)). Needless to say that this form of single

particle energies, valid for all momenta, is difficult to justify

from the point of view of the present state of the nuclear matter

theory. jf**

We shall not discuss here how the wave function H ^ has been

-25-

approximated in Ref.21 because the difference between our function

'ty'* , calculated with the single particle spectra (A.4), and those

of Ref.21 turns out to be of no practical importance.

Our results for VA are shown in Table 3. Besides the results

obtained with the single particle energy spectra of Eg.(A.4), Table

3 contains also the self-consistent results obtained with our ori-

ginal single particle spectra of Sec.2, adjusted to the value of

-l" "" 'ky « 1.366 F . The "exact11 results have been obtained by solving

the wave function equation (2.24), whereas the IPA and the first

order results in v^ have been obtained by applying the

approximation (A.2) and (A.l) respectively. The small differences

between our IPA results and those of Ref.21 may be traced beck to

some additional approximations applied by Downs and Ware ( Q(M,p)

^ Q(O,p) and approximations for the Bethe-Goldstone function, in

particular in the P state).

The difference between the exact results obtained for the

total V. with the two types of single particle energies (-35.9

MeV versus -36.3MeV) is surprisingly small. However,the contribu-

tions of separate partial waves, as well as the pure hard core

contributions and the first order results an v^ do differ in

the two cases. The full first order results.in v, are syste-

matically much smaller than the exact results.

Our main purpose here is to compare the IPA results with the

exact results, obtained with the help of the single particle

spectrum (A.4) (from now on we restrict ourselves to this spec-

trum). Y/e have: VA(exact) * -35.9 MeV and VA(IPA) « -39.9' MeV.

The sizable differences in the contributions of separate partial

waves are partially canceled end we are left with a difference

of 4 MeV. Thus the magnitude of this difference is accidental

and depends on the form of v ^ (see Table 2.). Still, we may ask

-26-

why is the difference so small? TO answer this question let us

notice that the IPA amounts to replacing the KA functions under

the r-integral of Eq.(2.2O) by the 1 ^ functions. Now,

bigger than IU close to the hard core, ancT smeller than

larger distances, but still within the range of TT (compare Fig.2).

Thus the nett effect of the replacement is canceled partly in the

course of the _r-integration. On the other hand, in the first order

approximation in v^ we replace IU by BA which is smaller than

IU within the range of v and thus we get | VA(first orderJ |

'^~^ ' IT (exact)

-27-

REFERENCES AND FOOTNOTES

1 D.Ivanienko and N.Kolesnikov, Zh.Eksper.Teor.iFiz.CUSSlO^O,800(1956)2 J.D.Walecka, Nuovo Cimento IS,342(1960).3J.W.011ey, Austral.J.Phys.14,313(1961).4 A.R.Bodmer and J.W.Murphy, Nuclear Phys.64,593(1965).

^ R.H.Davies et al.,Phys.Rev.Letters .2,464(1962).

J.Cuevas et al.,Nuovo Cimento 22,1500(1963).7

J.P.Lagnaux et al.,Nuclear Phys.50,97(1964).8 D.J.Prowse and A.Lou,Bull.Am.Phys.Soc.10,114(1965);10,1124(1955).

J.Lemonne et a l . , Phys.Letters 13,354(1965).x u D.P.Goyal, Nuclear Phys.83,439(1966).1 1 B.Bhovmik et al.,Nuovo Cimento 52A.1375(1967).1 2 Y.W.Kang and D.J.Zaffarano, Phys.Rev.160,972(1967).13

Improvements in the BA values o£ the identified l ight hypernucleiwill certainly affect the final value of BA((?O). Til l recently,

the value BA(AC15) = 10.51 ± 0.51/i?" has been used in the- ~ • 'He Vs* 1 „ -

e s t i m a t e s of BA(oo ) , >&& n e w v a l u e , BA(AC y) - 1 1 . 3 9 i 0 . 1 5 LleV1-5,

might increase the value of BA(oo) , Eq.(l.l).1 4 V/.Gajewski et al.,Nuclear Phys.Bl, 105(1967).15y J.KLabuhn et al.,Nuclear Phys., in print .

throughout this paper we denote by T5t and JU, masses divided

by 1i2. Thus, for instance, ^W-N - 0.0241 MeV -1F"2.

1 7 B.Sechi-Zorn et a l . ,Phys.Rev.l2Ji, 1735(1968).1 8 E.H.Dalitz and B.W.Downs, Phys.Rev.111.967(1958).

A.R.Bodmer and S.Sampanthar, Nuclear Phys.31,251(1962).?0

M.Taharaadeh, S.A.Moszkowski and P.C.Sood, Nuovo Cimento 23.168

(1962).2 1 B.W.Downs and W.E.Ware, Phys.Rev.133.B133(1954).

- 2 8 -

2 2 B.Ram and B.W.Downs, Phys.Rev.133.B420C1964).2 5 R.C.Herndon,Y.C.Tang and BJW.Schmid.Phys.Rev.137,B294(1965).2 4 G.Ranft, Kuovo Cijnento 43B.259(1966).2 ^ G.Rsnft, Nuovo Cimento 45B.138(1966).

• * • • . . .2 6 B.W.Downs and R.J .Phi l l ips , Uttovo Cimento 33,137(1954), '2 7 B.Ram and J .R.James,^Phys.Let ters 28B.372(1969).2 8 B.W.Dovms and M.E.Grypeos, ITuovo Cimento 44B,306(1956).

-p.Westhaus and J.W.Clark, Phys.Letters 23,109(1966). '3 0 M.E.Grypeos,L.P.Kok and S.Ali, ITuclear Phys.B4,335(1968).5 1 G.Mueller and J.W..Clark, TTuclear P^ys.|2>227(1968).5 2 S.A.MoszkowBki and B.L.Scott, Ann.Phys.(N.Y.)n,65(1960).33

Some of the results of this paper have been presented at the

International Conference on Hypernuclear Physics, Argonne Natio-

nal Lab.,1969.J Jor a review see, e.g., J.D^browski, Fundamentals and Elementary

Outline of the Many-Body Theory of Nuclear Matter, Proceedings

of the International Course on Nuclear Theory, Trieste 1969,

in print.

" J.D^browski and Z.Skiadanowski, International Symposium on Nucle-

ar Structure, Dubna, USSR, 1968, Contributions, p.165.5 Z.Skiadanowski, Warsaw University Thesis 1969 and Acta Phys.

Polon., to be published. •

37 H.S.KOhler,Ann.Phys.(n.Y.)16,375(1961).? A.R.Bodmer and D.M.Rote, International Conference on Hypernuclear

Physics, Argonne national Lab.,1969, preprint.59 R.C.Herndon and Y.C.Tang, Phys.Rev.jL§2,853(1967).40 J.D^browski and H.S.KShler, Phys.Rev.136.B162(1964).41 K.A.Brueckner and J.L.Gammel, Phys.Rev.10g,1023(1958).

B.D.Day, Rev.Mod.Phys.^,719(1967).

R.Rajsremen and H.A.Bethe, Rev.Mod.Phys.l^*745(1967).

- 2 9 -

44

G.3.Brown et al.,Nuclear Phys. 55,19l( 1964) >•

l^ S.Coon and J.Dabrowski,Phys.Rev.140.B287C1965).

4 o G.Dahl,E.jZ?stgaard and B.Brandow,Nuclear Phys.A124,481(1969).

' If the A - N potential, sv , is different in even (»sv+) and in

odd («sv~) angular momentum states (as do some of the potentials

to be considered), then Eqs.(2.24-25) remain valid with

Bv(r) m sv+(r) for even" _1_ and Ev(r) » sv~(r) for odd JL values,48

Strictly speaking, this pure hard core contribution depends in-

directly on'the attractive part of V^T . Namely, the energy de-

nominator in G^, .(2.22), contains V^ , determined by the

total A - interaction.4 9 • * " " " •

This argument is originally due to Weisskopf, Nuclear Phys,j5,

423(1957).

5 0 2. A.Brueckner,Phys.Rev. 110, 597(1958) ;N.IvI.Hugenholtz and L,Van

Hove, Physica 24,363(1958).

5 1 S.Ali,M.E.Grypeos and L.P.Kok,Phys.Letters 24B.543(1957).

^ In computing the values of _a's and r's the tables ofR.D»Levee and R.L.Pexton,Nuclear Phys.^,34(1964) have been used.

5 3 •^ il.E.GrypeoSjL.P.ilok and S.Ali, International Conference on

Eypernuclear Physics,Argonne National Lab.,1969,preprint.

^ K.A.Be the and J.Goldstone,Proc.Roy.Soc.(London)A238.551(1957).

55 H.S.KtJhler,Nuclear Phys.^8,661(1962).

^ Actually, because of the big size of the hard core of the poten-

tials jl, $' one would expect that the three-body contribution for

these potentials, compared to the pure nuclear matter case,

would be shifted towards a more positive value.57

•" R.H.Dalitz, Proceedings of the Topical Conference on the Use of

Elementary Particles in Nuclear Structure Research, Brussels,

1965,P.293.

5 8 A.R.Bodmer,Phys.Rev.141,1387(1956).-30-

*° An extension of the present scheme of calculat ing B^(oo) to

the case of tensor A - ^ forces i s straightforward (compare Ref.

41). Calculations of BA(o°) with a ^A,- ST interact ion con-

taining tensor forces are in progress. .6 0 A.Gal,Phys.Rev.1JJ2, 957(1966) ;Phys.Rev.Letters ^8,568(1967);

E.M.Hrman.Phys.Rev.159.782(1967);R.K.Bhaduri.B.A.Loiseau and

Y.Hogami,Ann.Phys.(tf.Y.)44,£7(1967)jNuolear Phys.BJ,380(1967);

R.K.Bhaduri,Y,]tfogami,P.E.I;riesen. and E.L.!rom.usiek,Phys.Rev,

Letters 21,1828(1968)jY.Nogami,International Conference on

Hypernuclear Physics,Argonne National Lab.,1969,preprint.6 1 L.C.Gomes,J.D.Walecka,V.P.Weisskopf,Ann.Phys.(W.Y.)2,241(1958).6 2 In the present discussion we r e s t r i c t ourselves to a spin-inde-

( > • ;

pendent potential ( *v • "V • v~7 1K = 5K • K ).63

Eere, we disregard the effect of the total vATcr on the Green

functions &K (compare the footnote 48).

In Ref.21, VYhas been approximated by the Bethe-Goldstone^

function. The relation between this function and our Y^fun-

ction is discussed in.Refl46.

^ There are two effects, First, the energy.gap in the e«. , e.

spectra of Sec.2 accelarates the healing of the wave functions.

Second, by inserting the effective mass '?7^M into Eq.(2.8) one

increases the relative momentum ja f corresponding to a given

nucleon momentum

- 3 1 -

Figure and Table Captions

!

Table 1, Parameters of the charge symmetric _A-1T potentials. All

lengths are in fermis. The figures in parentheses are

the corresponding values of the parameters of thq A-p

potentials (with, the CSE components included),

Table 2. The calculated values of -V^ and BA(oo). The results

of other authors are also shown (IPA stands for the in-

dependent pair approximation). All energies are in MeV.

Table 3. Values of VA (in MeV) calculated with the two types of

single particle spectra.

Zig.l. The three single nucleon spectra, adjusted to kp = 1.35

P""1 and J"Vol> * 15.8 MeV;. The parameters of the (i)

spectrum are: k_ « -112.0 I£eV , TCt^= O.393TO^,?he con-

stant value of the (ii) spectrum is: e-T(m.T) •= -54.3 HeV.

i'ig.2. The JA , SA and the sinlet % wave functions in the

case of the _H potential for rg^= 0.25 k? (kj, = 1.35 F" )

The _H potential is also shown (the v curve, in an arbi-

trary scale).;

-32-

Vn TOH,

, N 2 $p

o o o

roo

ro roe

o

ro

1-3&

VJ1

00

00

poOo OO

oro

^p•vjcn

OO• *O3S3Ooro

•s] 00

"52 P

O O

TO

OJVD

O O

oooVJI

enen

ooen

o*cnVn

inct

3

t iroro• •ro-o

t ITOW• *ro-o\J100

II IIroro roro

i

rooo

to

LL • * . ! »

en

LL* •cnvnoro

kIo

Vn.

<OO 00

vx* •ooo

u i

vnro

foa00F

KX

s t?dCO

- 33 -

IOJ

• "

HTS

AGK

ENX

E

E'HX

E '

F'NX

P '

HNX

H

DW

Partial wave

1 S

14.4

15.4

12.6

12.8

13.4

13.6

10,4

11.0

9.510.1

5.2

V

17.4

28.2

25.5

26.2

20.4

21.1

14.8

16.3

20.5

22.3

15.7

contributions

1 P

2.9

6.1

5.4

2 . 8

5.5

2.9

5.6

2 . 2

6 . 1

2.6

3.3

V

5.914.8

14.0

7.2

13.0

6 .6

13.9

5.4 .

16.9

7.1

9.9

to

1D

0.1

0 .6

0.4

0 .4

0 .4

0 .4

0.3

0.3

0.4

0 .4

0.3

-vh

0.2

1.6

1.1

1.1

1.1

1.1

0.9

0.9

1.2

1.2

0.8

41.0

66.8

59.0

50.6

53.745.6

45.9

36.1

54.743.8

'35.2

36.960.1

53.1

45.5

48.3

41,0

41.3

32.5

49.2

39.4

31.7

Other

37.6 2 3

58-64 2?

46-51 27

39.2 ^

resul ts

Jastrow meth. (BA(OO0)

47.5 5 5

63 31

5? 31

72 2?

62-64 2 7 * 5 1

Table 2 .

1-3

JCD

oct-

H

vnro

I

*

toro* • *

t

VJ1

\£>

en*o

1ro•

1VNy A

VjJ

ro

Po

1

*

L•

i i•'03

O•

o

L

rotn

L

•

l i

•

b

*VD

ro

•

o

o

VJ1roVn

1roru

1crt*o

i

*

-F?•

1

*00

•

1

*

-IH

o aO P>

(D P»

H-t

. p.hJ•^ <D CO

> *i cf

Mo

o WO p(!> P-

H-O m

^ O [0

MO

4 *

*

"P"*

ft)saj.

"CD

4

too

ro

-35-

50-

o»

Q L

-100-

Tl

^