;~ <9 n ,,.-'-: ""-• : IC/69/148>- U - u-u, i.,. . INTERNAL REPORT

^' (Limited distribution)

International Atomic Energy Agency

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

BRUEGKNER REACTION MATRIX

AND SEPARABLE POTENTIALS *

M. GMITRO

International Centre for Theoretical Physics, Trieste, Italy,

and

M.SOTONA

Nuclear Research Institute, Res (Prague), Czechoslovakia.

MIRAMARE - TRIESTE

DaoemTaar 1969

* To be submitted for publication,

** Present addresst Nuclear Research Institute, Rez (Prague), Czechoslovakia.

ABSTRACT

The binding energy of He and 0 was calculated

using reaction matrix elements of Tabakin a potential*

A separable form of the potential haa been used to re-

dace the basic equation to a simple linear algebraic

system* The Pauli operator Q defined in terms of har-

monic oscillator intermediate states permits an easy

and accurate calculation. Our numerical results for the

binding energies include the first- and second-order

contributions. A very reasonable agreement of the ex-

perimental and theoretical values has been obtained

since the occupied-state energies are made nearly self-

consistent and a cancellation of other important higher-

order contributions has been achieved by a shift of the

entire harmonic-oscillator spectra*

A uniform description of the free nucleon-nucleon

scattering and nuclear structure data belongs apparently

to the most important and very popular problems in

present-day nuclear theory* It laAwell known that the

Brueckner reaction matrix t should be introduced in the

nuclear structure calculations rather than the free H-N

interaction v» The idea is to treat the interaction in-

side the particle pairs ("two-body cluster11) to all or-

ders before letting any of the particles from the pair

interact with the remaining particles* Hew progress

in understanding the nature of Brueckner's perturbation

expansion was provided recently In articles by Hajaraman

and Bethe and Brandow 'where th© earlier referenoee can

also "be found.

The t-matrix can be defined by the operator equation

= v - v ~-t> CD

where the Paull operator Q prevents scattering into the

occupied states and energy denominator e stands for the

difference of intermediate and starting energy: C a ^ Z . "

The reference spectrum method ' has proved to -be

the most useful approximation in solving Eq«(l)» One

-l-

introduces a simple expression Q '<? (where Q ~~ 1 is

a frequent choice) and defines the reference matrix t by

{ , y- y_t , (2)

By iterating the exact relation

TJ

a perturbation expansion for t-matrix in powers of t

originates. The method is practicable if the first two

terms of the expansion provide a sufficiently accurate

result.

At present, two types of the free-nuoleon interactions are

available, whioh are derived from the experimental soattering

data.

1) The realistic potentials with infinite repulsive

cores such as the Xale, Hamada-Johnston and Eeid poten-

tials enjoy a great popularity mainly due to their correct

(from the point of view of mesonic theory) asymptotic be-

haviour and extremely good fit to the scattering data.

2) The non-local potentials which describe dynamically

the N-N repulsion at short separation distances constitute

another widely accepted class of realistic nuoleon-nucleon

-2-

interactions. Tabakin's4' separable potential,being

fitted with a very reasonable accuracy to the scattering

phase shifts up to 320 MeV^Ie probably the best of the

given class so far.

It has been pointed out as a result of numerous

calculations^ * that Tabakin's and other realistic po-

tentials are equivalent to a great extent from the point

of view of the nuclear structure investigations. A compar-

ison of the relative-motion matrix elements of Tabakin'a

potential with those extracted directly from the experi-

mental phase shifts by Elliott et al>'' shows a striking

similarity as well.

The recent nuclear t-matrix studies by Kuo and Brown '

and Wong have been performed with the Hamada-Johnston

potential. It is felt, however, that the original ambi-

tious program of investigating such a complicated poten-

tial in the Bruecteer theory could be diminished con>-

siderably by the computing difficulties. In particu-

lar, a rather crude treatment or&auli operator Q (Q S4A

in Refa.8 and 9 should be mentioned here. Becker, MacKellar

and Horria ' observed,when comparing. Wong's"* results

with their own calculation, that the use of. the rather ac-curate Eden-Emery s ' approximation 0 ^ for GT in the

reference matrix calculation has a greater effect than the

use of the same Q in calculating correction terms to

-3-

TJ T>

t obtained with Q = 1. Moreover, a slow convergenceRof the t-matrix expansion in powers of t has: been found

by Bhargava and Sprung in a nuclear matter calculation

with CT = 1. A similar situation should be expected in

finite nuclei.

In the present approach we would like to introduce

the Pauli operator Q in the most accurate way, stating

clearly all the approximations with the aim of calculating

the corresponding correction terms according to Eq»(3)

later. Because of the mentioned equivalence of the realistic

potentials we have chosen the labakin potential for cal-

culating the t-matrix elements in the He"* and 0 nuclei.

Using the separability of Tabakin interaction, Eq,(l) re-

duces to a system of linear algebraic equations instead of

much more complicated integro-differential or differential

equations which arise in the case of local potentials with

static repulsion.

Of course, Tabakin's potential can be formally used

in the nuclear structure calculation already in its bare

foi"nu From the physical consideration it is clear, however,

that at least corrections for the multiple scattering pro-

cesses must be included in this case. The problem has been

investigated up to the second order In Hartrpe-Fock bases

by Bassichis and Strayer J • They used a truncated but,

however, very large configuration space and the calculation

oan in this respect be considered complete for light

nuclei. Their results are twofold:

i) total second-order correction to the binding energy

is comparable or even larger than the first-order term (Hartree*

Focfc energy), and

ii> the most important contributions to the second -

order term originate from as high as 6£ui and 8%u> lying

intermediate states*

Consequently there is no good reason- to expect small

third- and higher-order terms,and the calculation of the

reaction matrix which represents a sum of all higher-order

processes seems to be inevitable.

The problems of the single-particle Cs»p») basis*

states in the Brueckner theory are briefly reviewed in

Section H . Our choice of the harmonic oscillator s»p,

basis is supported mairsly by a possibility of accurate

representation for Paul! projector Q which is introduced

in Section XTX« In Section IV the "basic equations are

derived. The accuracy of our numerical procedures is dis-

cussed in Section V, and the final results for the binding

energy in He^ and Q together with the most important con-

clusions are given in Section VI,

-5-

II. BASIS STATES

Brueckner a approximation to the ground state binding

energy in the i'irst order (Fig.la) is given by

where T symbolizes kinetio energy operator and the s.p. states

are labelled by a set of quantum numbers r\. = (n-> i\ i y .»"; ,T- )

All our two-body matrix elements are antisymmetrized. The

reliabil i ty of the lowest approximation (4) depends on the ohoioe

of the s.p. "basis used in oalculating the t-matrix.

Definition of the Hgailtonian for a system, of A par-

ticles

Vf '

be rewritten in the usual wsy as

(6)

with

A

>ot -

H. «IMOW^(T-^Awhere U. is an external potential.

-6-

For further calculation it is essential to perform

transformation from the individual particle basis into

the relative and the centre-of-mass (RCM> coordinate systems.

It can be done in a siaple way for two cases only.

1) Free-states spectrum (plane waves)

C8)

with

C9)

appears naturally in the nuclear matter t-matrix. It has

also been frequently used for the intermediate states in

finite nuclei* To do it consistently, the procedure must

include an orthogonalization of the continuum states to the

low-lying localized states as well as new continuum states

among themselves9^ . As far as we Icnow, such a program

has never ye-t "been performed. Another drawback of the

plane-wave basis in - the finite nuclei t-matrix cal-

culation is a rather difficult and less transparent treat-

ment of the Pauli exclusion principle Csee also Section H I ) .

2) The harmonic oscillator (h.o») basis has become

rather popular in the most recent t-matrix studies,

in Tjo-th nuclear matter14* and finite nuclei10^15^1^. It

is also the choice adopted in the present paper* L-S vector-

coupled pair function of h»o» orbitals can be transformed

-7-

into ECM system (9) l>y

where B , l,Jr,^S stay for principal and orbital quantum

numbers of the relative and the centre-of-mass motion,

respectively. Transformation coefficients (n^JfJf^L »Xy^^

were tabulated by Brody and Moshinsky ' . Round brackets

symbolize RCK states, angular brackets the individual

particle states.

In accordance with papers of Ref.lQ, 15 and 16,we

follow systematically the procedure suggested by Eden

and Emerjr and modify the energy spectrum of the h.o.

Hamiltonian in such a way that all properties of s*p, wave

functions including that of Eq.(lO) are preserved. The

modification consists in assuming for the s»p« Hamilton-

of i-th particle a form

ifwhere T.+U.* = H.* 'is the usual h.o. Hamiltonian,

h stands for n and1 i quantum numbers of the s*p»

sta te h below the Fermi 3ea and the s tate vector j h

i s the same for both EL'0' and HQ6i) • Accordiiag to the

above choice a l l the s ta tes of the h.o.

spectrum are shifted by a constant amount C. An additionalstate-dependent shift 7) applies to the occupied s ta tes

only.

- 8 -

Many higher-order contributions to the binding energy

may be included already in the first order if the para-/

meters C# and 7) are appropriately chosen* S,pa states

below and above the Fermi sea should be considered se-

parately:

i) the most preferable choice for the occupied s^p^

states consists in calculating the Hartree-Fock energies

with s,p. potential satisfying Erueckner self consistency.

Then the terms with bubble- and potential-insertions in

hole lines (Fig,2(a)>(b)) cancel each other, in particular

all second-order processes,FigwlCttJ-Ce.), give zero total

contribution*

Energy shifts in the hole states defined by the aelf-

consistent condition

u

together with t-matrix definition C D cause cancellation

of diagonal insertions into hole lines only. Complete self-

consistency including off-diagonal insertion would require re-

fraining £rom the h»o# wave functions. In particular, all

the second-order diagrams, Fig*l(b)-Ce),iattsti and will "beSince the

calculated in our approach*/^ .lus* of Eq.(12) depends on

the individual ;k quantum numbers, the: spin-orbit

splitting has been removed in calculating "n •

-9-

11} Such a choice is not suitable when treating

insertions in particle llnea,since the bubble insertion

diagram of Fig,2(c) is a part of the three-body cluster

term, which absorbs several apparently large contributions.

Their sum i s usua l ly expected t o be small , foil-owing an4)

analogy with the infinite nuclear matter.

The theoretically interesting result of Eohler ' , that

the potential insertions in particle lines are included

to all orders by defining

-r- v * HO C13)

instead of the usual expression

1 (c\^ v __ v Q / fr \ (14)

1 1 s

is unfortx iately not easy to apply in calculation because

Eq.(13) cannot be properly solved without many additional

approximations. Therefore we attempt here rather to mircL-

aise the contributions of/ diagrams with potential inserted

into particle line by an appropriate choice of the con-

stant C. The most important graph of this type seems to be

the one given in Fig.2(d). The corresponding contribution

to the binding energy can he written as '

-10-

oco

rear

' " ^ ~ ^ (15)

where

o.

The/^contribution of the simplest diagram which contains onlyexcitations of order 2Jfco is exactly zero with

where Kp and p stand for the quantum numbers of the lowest

unoccupied state, i*e, one which only enters into the con-

sidered diagram COj> for- He"*, isOd for 0 )•

Our choice of constant C can be compared with the

one adopted by Becker, MacKellar and Morriar0^ The

named authors put an additional condition 1?0 = X6

the i r 0 calculation. Then both 1) and C can be calcu-

lated from the Bruecfcner-Hartree-Fock equations (12) and

(14K I t iiaplies almost twice as large^ralue of C aa

that given by Sq»(lQ and the result ing energy gap between

the occupied and unoccupied s .p . s ta tes aeem$ to be

underestimated.

-li-

H I * FAULT PROJECTION OPERATOR

The exclusion principle represented by Pauli projector

unoccQ = H ip^Xr^i C17)

introduces tremendous complications into reaction matrix

calculations for finite nuclei* Both Q - 1 and angle-

average approximation borrowed from the nuclear-matter

have been frequently chosen in earlier investigations '^

performed with plane-wave intermediate states*.

The moat advantageous property of a h.o<> basis is

probably the possibility of a simple and accurate approxi-

mation for the operator Q which carries the main huili: of

the exclusion effects and penults an easy evaluation of thecorrection

correspondingAterms» Moreover, the expression we derive

below can be considerably sing>lified and still retain good

accuracy* The latter approximation was suggested by Eden

and Emerjr' ; a comparison of the results corresponding

to 0 ^ with our more accurate treatment are presented in

Section V. A Pauli operator defined in terms of oscillator

states has also been used together with a free-particle

plane wave spectrunr .

In the h«o»basis (10) and with modified energy spectra

implied by Hamiltonian Eq. (H) we have for the propagator

-12-

CIS)

X —

goes over BL, 1where - ^ goes over BL, 1 . n , and !_, v- gcrea overl i e i l r t e r "

mediate state depends in our model Cll) on the shell number

r * O m i t t i n S tlirougliotit additive= 2n + ^n 2 D r r

constant 3^w » we have

Transformation to a relativa J representation where

+ T = J1% i = 1, 2 gives

Q(20)

- 1 3 -

A

where the matrix elements of Q include* summation of the

Clebsch-Gordan coefficients

(21)

•Q

The exact expression (20) of the Q/e propagator is

non-diagonal in all the quantum numbers of both the rela-

tive and centre-of-mass motions including the relative-

motion total momentum 1 • Two approximations adopted in the

present investigation are *.

i) We keep only terms diagonal in the centre-of-mass

motion quantum numbers, i»e» Jf^ -jf^ -M » IA^ ~^2 ~ *

and ii-i =iA/l =cAC ^n the model operator 0 ;

ii) Since the matrix elements of Q still depend on

the projections M. we use their average value. Calculation

of the average applies only to expression in braces of

Eq#(2l) with the result

(22)

It follows from the assumption i), Eq*(22) and from

the properties of the Brody-ffioshinslcy transformation

-14-

Jiff

brackets that also ru - n^ « n. Then the final expression

for Qr/e operator i s

where

^

^

- 1 5 -

* REACTION MATRIX IK AS OSCILLATOR BASIS

Eq.(14) can be rewritten with Q / e operator given

by Eq»(23) in a matrix representation by

= 0 otherwise*

With (SfSjMJf^jS'j'MV^^t')^^ the inhomogeneous equa

tions (25) degenerate into a homogeneous system which has

only zero solution; we conclude, therefore, that the (non-

zero) t -matrix elements are diagonal in quantum numbers

and JflJi and obey the equation

(26)

The t nnatrix diagonal in ^5j quacetum. numbers originates

from the interaction diagonal in them; on the other hand,

the diagonality in the centre-of-ztass motion has been

-16-

assumed by us explicitly and the accuracy of snch a model

must be investigated*

Method of solution Eq»(26) is based on the separa-

bility of Tabalcin's potential* Matrix elements in relative

coordinates are of the form

where two terms in the r»h*s« sum represent repulsive and

attractive parts of interaction according to s = — !•

Detailed formulae for the corresponding radial integrals

^ \n can be found in Eef»5-

Introducing a 2 x 2 matrix

C28)

we can write a linear algebraic system

i 0

(29)

for the coefficients x ^ x ^ ' s j ^ ,• i «* 1»? which enter

the final formula for calculating our t? matrix oiemen-fcs

-17-

C30)

A concluding remark concerns starting energy

E » C 1 ri + t 1- 2Jf + )<UJ -^C - 7 - since the

trace of the individual particle quantum numbers /),, and h

has been formally lost in the TalminKoshinsky transform-

ation to the ROM system. It is not the case, however,

in the lightest nuclei like He* and 0^ which we consider

here. With 9- 2 n + i •«- 2 J* + £ We have:

0

1I

X• -2

\

5

\

V •In more complicated nuclei an additional approximation

must he introduced at this point; e«g» an assumption

^ lo = ?o •* lid o would fee sufficient and probably

appropriate in the Ca^ nucleus.

-18-

THE NUMERICAL PROCEDURE

Radial integrals <^Vsi of Tabakln s potential have been

calculated by 15 points Gauss-Laguerre quadrature formula. We

use here modified parameter a of ^ channel potential sug-

gested by E .Baranger and B.Clement in Eef.5.

The time-consuming calculation of A ^ coefficients can

be considerably shortened by employing the completeness of

Moshinsky-Brody brackets, since a sum complementary to that

of Eq.(26) involves less terms and simpler transformation

brackets*

some

In Fig»4Acoefficients H ^ are plotted against the

shell number for C ~ Qf 1, 2 channels. It is clear from the

figure that only^few coefficients should actually be calcu-

lated. Substitution A^ = 1 becomes well justified with

increasing shell number very soon, nevertheless, the dif-

ferent rates of convergence for various t -channels should

be emphasized*As already noticed,our approximation Mh^ for Paul!

exclusion principle can be further simplified according to

the suggestion by Eden and Emery • They put

=04 4»

a 1 otherwise.

The model can be visualised in Fig.3 where circles end squares

-19-

correspond to the states allowed and forbidden byJPauli prin-

ciple, respectively, and the shadowed areas indicate the

states left out by Eden-Emery's method with different s>

Binding energy of the 0 nucleus in lowest order to-

gether with some most important t-matrix elements calculated

in the model (31) should be compared with the "exact** results

C An» from Eq.(24)) rather than with the experimental

values, since a model interaction

» 0 otherwise

has been used in this case. Although the E-E approximated

matrix elements exhibit considerable deviations both to

smaller and larger values from the exact results (column 6),

an integral effect represented by the binding energy is nice-

ly approximated in the Eden-Emeryfa method with P . = 5.

The model assumption (32) clearly cannot be used apart from

the described test of the Eden-Emery's approximation and

all the results below correspond to the usual Tabakin

interaction*

We show in Fig<>5 and 6 the dependence of E l ' - binding

energy of He " and 0 nuclei in the lowest order on the num-

ber of high-lying intermediate states included In the sum-

mation (28)»tfon-negligible contributions come even from

-20-

.TZ •.•:£.JJA' . i • *il .• '

rather highly excited states. The calculation could be

easily extended beyond the 3O4» limit (all nBt*4 in

this region) • It would be very hard, however, to Justify

such procedure since the phase shifts are known and have

meaning only below about 300 MeV. The matrix elements

of specific potentials such as that of TabaMn should ai-np,nr

ready be quite unreliable for large values of which cor-

respond to 306co (roughly 500 MeV in a-d shell nuclei)

excitation. The contributions from the "dangerous" high-

lying states constitute, however, only a few per cent of

the total result and we can consider the binding energies

calculated with P = 30 as^reaaonable compromise between

the extreme requirements for terminating the summation in

(28) at about 300 MeV and extending it up to the point where

clear asymptotic value of the binding energy has been reached.

The same problem arises,of course,in all t-matrix calcula-

tions; it may be only hidden by a more complicated formalism.

The subsequent increments of E^ ' are also plotted in

figures 5 and 6O It can be seen that the most important ex-

cited states which contribute to the 0 binding energy arethose with (4 r 8) co excitation energy in qualitative ac-

cordance with earlier findings by Bassicnia and Strayer •* •

The last-named authors consider, however, only the second-

order diagrams.

-21-

As a result of the above numerical experiments sum-

marized in Figs* 4 - 6 and Table I,the following prescription

has been adopted; We calculate Ani coefficients from the

formula (24) for 2n+U2tf+X < *1 ana put A^f = 1 for

larger values of the shell number* Contributions of all

excited states through 30-kto are 1 included in our t-matrix.

-22-

-:s-:,r;:^;;::r-- ....:.:x:-iS;;::!i:.«

VI. RESULTS Aim DISCUSSI3QM

The binding energies of He 4 and 0 have been calculated

with reaction matrix elements corresponding to the model de-

scribed in sections H - V. The results are presented in

Fig*7. The solid curves are for the total binding energies

E » E* ' + E^ * where E^ ' corresponds to our Eq»(4) cor-

rected for cexrtre-of-mass C ^ ^ - c ^ = ~ r ^ w ) and

Coulamb effects*

The second-order corrections (Figs*l(b) - (e)) are cal-

culated with intermediate states from the lowest unoccupied

major shell only* The dashed curves in Fig*7 which show E

terms exhibit remarkable m-tir!^ for b = 1*3 F» Such

In the Brueclmer-Hartree^Fock violating term should be in-

terpreted as an additional Improvement of the selfconsistency.

Our conditions (12) and (14) ensure the cancellation of po-

tential and bubble insertions in diagonal matrix elements(2)

only* A Tninimran in the calculated Ev ' curve then meansthat certain nondiagonal insertion has been cancelled as well*

Therefore, the most reasonable choice for the value of the(2)

parameter b is to take It In the minimum of £ •

We have stressed already that the proper definition

of the parameters n and1 C in Eq.(ll) modifying the s»p»

basis Is crucial for the rapid convergence of Brueckner

expansion* The self-consistent prescription (12) Is well

Justified; much work, however, remains before the problem

of appropriate modification of particle spectra can be com-

pletely settled. In Table II we ahow how the binding

energy of He and 0 depends on the choice of parameter C

which; modifies the particle-hole gap in our calculation*

It can be seen that much better agreement of calculated

and experimental data would be obtained with C values

different from those of equation (16), E*g*,the procedure

adopted by Becker, MacKellar and Morris1 ^ gives in 0 for

three different realistic potentials roughly the same value -

CCsi - 66 MeTfc-(Mftw) which should be compared with our estimate

(16) C = -(-^4Hu> . The binding energy of 0 1 6 for C *-

and Tabakin's potential would be about - 8O4 MeV in sur-

prisingly good agreement with Beckers et al» ' results

- 7*3, -7*5, and -7*8 HeV for Yale, Reid and Hamada-Johnston

potentials, respectively. The choice of constant C should

be made, nevertheless, independently in order to minimize

the higher-order contributions to the binding energy rather

than from the numerical fit to the experimental value* In-

clusion of the third-order processes Q?ig#2(d)) with more

than 2f\(J excitation in the intermediate states is indeed

highly desirable in the place of our crude guess (16),

According to our knowledge the previous results of

binding energy calculations with true realistic potentials

are in most cases in rather strong disagreement (B»E. 2£

with experimental values B.E* (He ) = -7*07 MeV and .".

B.E*(016) * -7.975 MeV unless additional tailor-made

assumptions are introduced. The better agreement obtained

by the present approach is probably due to the more appropriate

-24-

treataent of APauli exclusion principle and consistent

modification of s*p* spectra. The most important addi-

tional effects responsible for our slightly underesti-

mated results should be; i) three-body cluster term,

ii) second-order diagrams ]?ig»2(b) - (e) with excitation

over several major shells, and ill) better choice of the

parameter C as already discussed*

In our treatment of the centre-of-mass Cc»m*) motion

only the term -^"ta corresponding to the unperturbed

ground state wave function has been taken into account*

A more consistent description of the c*m« effects in the

Ke^ nucleus has been suggested recently by Blank °* in the

frame of the Lipkin model with an external h»o* potential

field acting on the cm* degrees of freedom. His prelimi-

nary results obtained with the Hamada-Johnston potential

show that C»BU motion .probably influences the expansions

for the r«iius* radius and charge distribution only*

-25-

The present approach is based on the assumption that

the three-body forces produce negligible effects on the

nuclear states*. Tabakin s potential was used for the

evaluation of two-body reaction matrix elements in terms

of modified harmonic oscillator states for He^ and 0 •

Two approximations in the treatment of Pauli projector

Q are: 1) We keep only terms diagonal in the centre-of-mass

quantum numbers Jf^.Ji and 2} The Ji -dependent matrix

elements of Q are substituted by their average values*

The validity of the last two assumptions should be care-

fully investigated despite very reasonable numerical

results obtained for He and 0 binding energies*.

ACOOWLEDGEMEKT

One of us (M»G#) would like to thank Professors Abdus

Salamf P»Budini and L.Fonda and the International Atomic

Energy Agency for hospitality at the International Centre

for Theoretical Physics, Trieste.

-26-

REFEBEUCES

1) BJEUUMRUUH and K*A*BETHE, Rev.Mod.Phys. 21s 745 (1967).

2.) BJBoBRAEDOW, Rev.Mod.Phys. 21> TO (1967).

3) KJUBETHE, B.H»BRANDOW and A.GoPETSCHEK, Phys.Rev..l29,225(\964),

4) F.TABAKIH, Ann.Phys. (K.X.} JO, 51 (1964).

5) DJI,CLEMEKT and EoU.BAEANGER, Kucl.Phya.A108 > 27 (1968);

A120. 25 (1968),

6) H.SOTOHA, E#QyHTRO, Z PLUHAS and LoTRI^TAJ, Phys.Rev. to be

published; E.CMTTH3 and J^SAWIGO, Phys.Lett<i» 26B. 493 (1968>.

7) JJP.ELLIOTT, A.D.JACKSQS, H.AJIAVECMAT1S, EJUSAKDERSON

and B^SlEiaH, Hucl*Phys<> A121> 241 (1968).

8) T.T.S.K00 and G#E»BEOWH, 5«clJ*hys. 8^, 40 (1966).

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10) R.LJ3ECKER, A.DJffacKELKAR and B»SUI0HRISt Phys.Rev.

174. 1264 (1968).

11) R-J.EDEST and Y»3C.EMERI, P3?oc.Roy.Soc, (London) A248.266 (1958).

12) P»C»BHARGAVA and D.W.L.SPRXJNG, Ann.Phys.U9:*Y.) ^ 2 , 222 (1967).

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14) G»g.PU.T,TPOV> Ukr.Fi».Zh. (USSR) ! £ , 664 (1969).

15) A^ALLIO and B.D.DAY, Hucl.Physo AI24, 177 (1969). /

16) R^JJIcCARTHI, KuclJPhys. A130. 305 (1969).

17) T»A*KR0DI and U*MOSHIEfSKI, Tables ot Transformation Brackets

for Huclear She l l Model Calculat ions (Honografias de l

Inerti tuto de F i s i c a , Mexico, I960).

18) H.S.H5HLER, HnclJPhySo A98. 569 (1967).

19) J.BLANK, Czech.J.Phys. B19, 748 (1969).

- 2 7 -

FIGOKE CAPTIOHS

Fig.l. First- and second-order Brueckner-Goldstone diagrams.

Fig*2«. Thirds-order Brueckner-Goldstone diagrams used for

modification of harmonic-oscillator spectra*

Fig»3» Eden-Emery approximation for the pair states in

the 0 nucleus. The states allowed and excluded

by the Paul! principle are indicated by circles

and squares, respectively. The operator Q*33 permits

scattering into states above one of the dashed

diagonal lines.

Fig»4. Dependence of Pauli coefficients A ^ (Eq.(24))

on the shell number ^ ^ + ^ ^ ^ f in o 1 6e

Fig»5» First-order contribution to the He^ binding energy

plotted against number of intermediate states

included in the sum of Eq*(28) and subsequent in-

crements A £ /p)~ E f^)~ £ * /f>-2; * Harmonic

oscillator spring constant b = 1»3 F* The right-hand

side scale applies to ^C .

Fig»6, The same as Fig#5. but for the 0 1 nucleus. The spring

constant of harmonic-oscillator wave function b = 1*4 F

Fig»7* Dependence of total binding energies and second-order

contributions for He and 0 on the oscillator

length unit, b « (</^co) 2

-28-

TABLE OAPTIQHS

Table I* Diagonal singlet- and triplet- S-state reaction

matrix elements (n = 0), and bidding energies

of 0 calculated in Eden-Emery approximation

for Paul! projector Q and with accurate Or

operator defined in Section H I . Simplified

Tabakin interaction (32) has been used in cal-

culation*

Table U»Energetics of He 4 in MeV calculated from t R using

the Tsbakin interaction* Columns correspond to

the different choice of constant C which controls

the gap between occupied and unoccupied states*

Selfconsistency was imposed on v , b = 1*3 F.

Table HX.Energetics of 0 in MeV calculated with Tabakin'a

potential for different choices of constant C and

self consistent condition (12) imposed on V and

17 is compared with corresponding results of10)Becker, MacKellar and Morris f obtained with the

Hamada-Johnston potential and additional condition

-2?-

X X

- 3 0 -

Fig, 2

- 3 1 -

2 3

Fig. 4

- 3 3 -

Fig. 5

-34-

EM (MeVJ

-35-

E(MeV)

0 -

- 3 6 -

TABLE I

so

S l

*

0 0

0 1

0 2

1 0

0 0

0 1

0 2

1 0

E ( l )

^ min=4

-9.8397

-9.8105

-11.1041

-ll»1041

-14.2517

-14.0195

-18.8855

-18*8855

-13.0916

t

^ min"5

-9.5083

-9.8105

-9.8105

-9.8105

-12*8252

-14.0195

-14.0195

-14.0195

-8.2345

Co =6\ min

-9.5083

-9.4775

-9.8105

-9.8105

-12.8252

-12.6179

-14.0195

-14.0195

-6.9365

t R

-9.5952

-9.5850

-9.8569

-10.3389

-13.1373

-12.9975

-13.9191

-15*1796

-7o7084

i?

ii

•

- 3 7 -

TABLE I I

a) 0.5 0.75 1.25 1.5 2.5

n 1 j

0 0 1/2 -21.940 -22.243 -22.992 -23.468 -24.735 -26.722

46.476 40.671 29.192 23.517 12.430 2.006

-0,001 -0.0003 -0.0001 -0.001 -0.008 -0.038

-6.196 -6.347 -6.722 -6.961 -7.601 -6.624

a)In units

-33-

TJ!£D5 I I I

n.0

0

0

o . > .1 J0 1/2

1 3/2

1 1/2

trEC2)

E

1.75

-64.87

-32.93

-18.41

59,54

43.94

-0.113

-6.727

a) in uniteb) Ref.10, b

T A B A

2.5

-65.91

-34.12

-19.76

44.85

29.48

-0ol68

-7.378

= 1.414 F

U I N

3.0

-66.77

-35,17

-20.95

35.12

20.01

-0.229

-7.956

3.5

-67.80

-36.56

-22.52

25.44

10.66f

-0,332

-8o733

HAMJLDA-JQHIS

3.25

-34.96

-25.54

24.05

12.02

-0o46

-7.80

- 3 9 -