Nuclear Physics A444 (1985) 76-92

@ North-Holland Publishing Company

G-MATRIX EFFECTIVE INTERACTION WITH THE PARIS POTENTIAL

A. HOSAKA, K.-I. KUBO and H. TOKI

Department of Physics, Tokyo Metropolitan University, Fukasawa, Setagaya Tokyo, 158, Japan

Received 16 January 1985

(Revised 9 April 1985)

Abstract: An effective interaction is derived by fitting the oscillator matrix elements of the sum of the

OBEP functions to the G-matrix elements derived from the Paris nucleon-nucleon interaction.

The functional form, the mass dependence and the ambiguities of the effective interaction are

discussed. For the application of the present effective interaction, we study the mass dependence

of the G-matrices and estimate the Landau-Migdal parameter g’. We have obtained reasonable

results for these cases.

1. Introduction

The effective interaction between two nucleons inside the nucleus is one of the

elementary quantities when we discuss various phenomena in nuclear physics. The

practical effective interaction should have the properties of the G-matrix applicable

to the unperturbed single-particle wave function. It must be based on the realistic

nucleon-nucleon interaction and also take into account the presence of other

nucleons.

A simple local form of the effective interaction was proposed by Bertsch et al. I),

and is called the M3Y interaction. It is the superposition of three (or two) Yukawa

functions (or r* Yukawa functions for the tensor channel) whose oscillator matrix

elements fit the oscillator G-matrix elements for 160 derived from the Reid nucleon-

nucleon interaction.*). The M3Y has been widely used and tested in low-energy

DWBA analyses and its validity 3, has been studied by many people. Petrovitch

pointed out that the spin-isospin term (a~ term) of the M3Y is too strong and must

be reduced by 40% [ref. “)I. Similar discussions have been given for the central-odd

terms ‘). Furthermore, it has been suggested 6, that the tensor force should be reduced

at high momentum (3-4 fm-‘) owing to the p-meson exchange, which was recently

observed experimentally ‘).

Anantaraman et al. *) constructed a similar effective interaction from the Paris

nucleon-nucleon interaction ‘). Their procedure was the same as that of the M3Y,

employing the same Yukawa functions. It was found that the DWBA analyses with

the new effective interaction were very similar to those with the M3Y, and therefore

none of the essential problems mentioned above were solved.

In this work we derive an effective G-matrix interaction with the hope of overcom-

ing these difficulties. The effective interaction is expressed as a superposition of

16

A. Hosaka et al. / Effective interaction 77

one-boson-exchange potential (OBEP) type functions lo), whose oscillator matrix elements fit the G-matrix elements derived from the Paris potential. This modification of the functional form is also expected from the following argument. Since the G-matrix elements themselves reflect only the low-energy properties (around the Fermi surface), the behaviour of the interaction at higher momentum is very ambiguous. It is more desirable to take a realistic functional form such as the OBEP type than a simple Yukawa or r2 Yukawa type. However, for the application to the nuclear structure or the nucleon inelastic scattering, it is convenient to express the effective interaction in a more simple form. Therefore we transform the OBEP-type interaction back to a superposition of Yukawa functions such that both of them coincide in momentum space.

Another question is whether the effective interaction depends on the nuclear mass number or not. This dependence is expected because the blocking effect in nuclei may differ for different nuclei. Therefore we have calculated the G-matrix elements for three different nuclei - 160, 40Ca and 90Zr - and study the resulting effective interactions.

The plan of this paper is the following. In sect. 2 we show briefly the derivation of the G-matrix interaction. The numerical results are given in sect. 3. In sect. 4 we discuss the mass dependence of the G-matrix elements and the Landau-~igdal parameter g’. We also comment on the ambiguity existing in the odd-channel effective interaction. The last section is devoted to a summary of this paper.

2. formalism

A brief review of the method for obtaining the effective interaction will be given here. The process is divided into two parts: one is the calculation of the G-matrix elements, and the other is to fit them by some simple functional forms to obtain the effective interaction. The detailed discussion of the former was given in ref. *). The integral equation satisfied by the G-matrix is

Here G denotes the G-matrix, u the free space nucleon-nucleon interaction which is the Paris potential in the present calculation, Q the Pauli blocking operator, and H0 the single-particle harmonic-oscillator hamiltonian. This equation can be solved by the Barret-Hewitt-MacCarthy method ‘I). The energy E in eq. (1) depends on the particle pair one takes and in principle the resulting G-matrix is energy- dependent. We choose the case where the particle pair is on the Fermi surface. In order to separate the center of mass and the relative motion, we adopt the method proposed by Eden and Emery for the approximation of the Q-operator “). Further- more, the gap energy between nucleons above and below the Fermi surface is taken to be 40 MeV. We can estimate the G-matrix for different mass regions by considering

78 A. Hosaka et al. / Effective interaction

the difference of Fermi levels and by changing the harmonic-oscillator parameter liw: fiw = 14.0, 11.0 and 8.8 MeV for 160, 40Ca and 90Zr, respectively, according to .tiw =45 A-‘13-25 Ae2j3 (MeV), with A being the mass number of the nuclei.

We note here the presence of another calculation by Shurpin et al. 13) on G-matrix elements with the Paris potential for sd-shell nuclei. Their folded-diagram method results in energy-independent effective G-matrices. For our purpose, however, we choose the above (eq. (I)) “bare” G-matrices to obtain the effective interaction. By doing this, we know exactly what we have included in the effective interaction and also we can compare it with the M3Y interaction.

The next step is the simulation of the effective interaction. Since we have assumed that the two nucleons are on the Fermi surface 8), the G-matrix thus obtained contains information only at low and medium momenta. Therefore, we rely strongly on the functional form chosen for the high-momentum part of the interaction. We take an OBEP-type functional form as an ansatz of the effective interaction:

V,(r)= F V&Yc(r/Ri), i=l

(2) i=l

where

S,,(i) = 3(a* * $)(a, * i) --CT1 * az. (3)

The suffices C, TN and LS denote central, tensor and spin-orbit parts, respectively. The interaction strengths V are determined such that their oscillator matrix elements fit the original G-matrix elements, while the interaction ranges R are determined as follows. From the point of the one-boson-exchange model of the nucleon-nucleon force, the central force is induced by r-, u- (or 27r), p- and w-meson exchanges, the tensor force by r- and p-mesons and the spin-orbit force by 1+- and w-mesons ‘*). Considering the masses of these mesons, we choose the four ranges 0.20,0.33,0.50 and 1.414 fm for the central components and expect an improvement of the effective interaction over that of Anantaraman et al. “) On the other hand, we take two ranges

A. Hosaka et at. / Effectiue interaction 79

for the tensor, 0.25 and 1.414 fm, and for the spin-orbit force, 0.25 and 0.40 fm, with different functional forms (OBEP) from those of the Yukawa form adopted in previous work ‘**), The number of the summation in eq. (2) is, therefore, NC = 4, NTN = 2 and NLs = 2. As the pion-nucleon coupling strength is well established experimentally (f2,/47r = 0.081) we fix the longest-range interaction strength V( R = 1.414) to the OBEP strength.

Since the Yukawa form is more convenient for the various applications of the effective interaction, we shall transform the OBEP-type interaction to the Yukawa form in such a way that they coincide in momentum space. As the central part already has four ranges for the Yukawa functions, we accepted the transformed tensor and spin-orbit force with four ranges: 0.20, 0.33, 0.50 and 0.70 fm for the tensor and 0.20, 0.25, 0.33 and 0.50 fm for the spin-orbit channels. In fact, four ranges are found to be necessary as will be shown in sect. 3.

3. Numerical results

Numerical calculation was performed using the computer program of the previous work ‘,*) with small modifications. The calculated results for the G-matrix elements in the harmonic-oscillator basis are provided in subsect. 3.1 and the derived effective interactions in subsect. 3.2.

3.1. G-MATRIX ELEMENTS

The G-matrix elements in the harmonic-oscillator basis are given in tables la-c as a function of the node quantum numbers n and n’ for each channel and for three nuclei: 160, 4oCa and 90Zr. The results for % agree with those given in ref. ‘) within a few percent. Hence the same conclusion as obtained previously 8, is also noted here, i.e. that the G-matrix elements derived from the Paris potential are very similar to those from the Reid potential ‘,14).

There is a mass dependence in the G-matrix elements as can be clearly seen in table 1. We shall discuss this point in detail in the following.

3.2. EFFECTIVE INTERACTION

We have fitted these matrix elements with the OBEP-type functions eqs. (2) and (3). For the convenience of practical calculations, we transform them back to the Yukawa (r2 Yukawa for the tensor channels) type functions which are obtained by the fitting procedure in momentum space. The interaction strengths thus obtained are given in table 2. We also show the OBEP-type interaction strengths for the tensor and the spin-orbit forces in table 3 for compieteness. The functional form in momentum space is shown in figs. la-h compared with the Paris effective interac- tion “) and the M3Y ‘). Since we did not find any apparent mass dependence,

80 A. Hosaka et al. / Effectiue interaction

TABLE la

Oscillator G-matrix elements (in MeV) for “0

n=O n=l n=2 n=3

singlet n’=O -6.5801 -5.2924 -3.7431 -2.5015

even 1 -4.6438 -3.3222 - 1.9469

2 -2.3858 -1.0695

3 -0.3184

Zl=O ??=I ?l=2 n=3

triplet n’=O -10.049 -8.3732 -6.6530 -4.9617

even 1 -7.4208 -6.2239 -4.6688

2 -5.4977 -4.0444

3 -3.0006

singlet n’=O 2.5310 2.4564 2.2570 2.0825

odd 1 3.1844 3.2178 3.0775

2 3.6677 3.7277

3 4.0624

P

\

n=O n=l n=2 n=3

P

triplet n’=O -0.0803 -0.1174 -0.0972 -0.0469

odd 1 -0.0937 -0.0535 0.0128

2 0.0254 0.1190

3 0.2476

tensor n’=O -5.6927 -7.8798 -9.0517 -9.6069

even 1 -2.7039 -5.3284 -7.2073 -8.4529

2 -1.3263 -2.9611 -4.7745 -6.3634

3 -0.6929 -1.6567 -2.8459 -4.4504

\

P n-0 n=l n=2 n=3

P

tensor n’=O 0.7787 0.7333 0.6274 0.5277

odd 1 0.8997 0.8643 0.7740

2 0.9456 0.9179

3 0.9705

A. Hosaka et al. / Effective interaclion 81

TABLE la (conrinued)

LS n’=O -0.0241 -0.0388 -0.0508 -0.0731

even 1 -0.0747 -0.0949 -0.1346

2 -0.1361 -0.2038

3 -0.3288

P n=O n=l n=2 n=3

P

LS n’=O -0.5685 -0.8109 -0.9704 - 1.0770

odd 1 -1.1678 -1.4062 -1.5689

2 -1.7038 -1.9125

3 -2.1604

TABLE lb

Oscillator G-matrix elements (in MeV) for 40Ca

s

\--

?I=0 n=l n=2 n=3

S

singlet n’=O -5.2187 -4.5601 -3.6117 -2.6709

even 1 -4.2845 -3.4698 -2.5585

2 -2.8761 -2.1071

3 -1.5015

\ s n=O n=l tl=2 n=3

S

triplet n’=O -8.0480 -7.4460 -6.3224 -5.1680

even 1 -7.3175 -6.3477 -5.2276

2 -5.6668 -4.7071

3 -3.9482

singlet n’=O 1.7260 1.7184 1.5952 1.4786

odd 1 2.2136 2.2640 2.1848

2 2.5781 2.6407

3 2.8772

\

P n=O n=l n=2 n=3 P

triplet n’=O -0.0483 -0.0893 -0.0967 -0.0819

odd 1 -0.0970 -0.0939 -0.0740

2 -0.0701 -0.0359

3 0.0149

82 A. Hasah ei al. / Efleetive interaction

TABLE 1 b (continued)

tensor n’=O -3.8897 -5.5050 -6.4728 -7.0125

even 1 -2.0937 -3.8428 -5.2369 -6.2151

2 -1.1172 -2.3980 -3.6559 -4.8325

3 -0.0422 -1.4165 -2.4507 -3.5369

P n=O n=l n=2 n=3

tensor n’=O 0.5374 0.5326 0.4738 0.4109

odd 1 0.6556 0.6460 0.5930

2 0.7073 0.6956

3 0.7329

n=O n=l n=2 n=3

LS n’=O -0.0059 -0.0090 -0.0091 -0.0199

even 1 -0.0278 -0.0283 -0.0461

2 -0.0553 -0.0843

3 -0.1696

n=O n=l n=2 n=3

LS n’=O -0.3261 -0.4751 -0.5801 -0.6567

odd 1 -0.6969 -0.8546 -0.9704

2 -1.0522 -1.1991

3 -1.3716

TABLE lc

Oscillator G-matrix elements (in MeV) for 90Zr

S n=O n=l n=2 n=3

singfet n’=O -4.1471 -3.8435 -3.2448 -2.6523

even 1 -3.7808 -3.2868 -2.6883

2 -2.9243 -2.4121

3 -1.9981

\

S n=O n=l n=2 n=3

S

triplet n’=O -6.4335 -6.2354 -5.5745 -4.8359

even 1 -6.3463 -5.7768 -5.0508

2 -5.3810 -4.7401

3 -4.2170

A. Hosaka et al. / Effective inreraction

TABLE lc (continued)

83

\ P n=O n=l n=2 n=3

P

singlet n’=O 1.2020 1.2299 1.1562 1.0778

odd 1 1.5768 1.6314 1.5889

2 1.8566 1.9156

3 2.0894

P n=O n=l n=2 n=3

P

triplet n’=O -0.0256 -0.0591 -0.0745 -0.0747

odd 1 -0.0751 -0.0843 -0.0827

2 -0.0813 -0.0706

3 -0.0219

n=O n=l n=2 n=3

~-

tensor n’= 0 -2.7015 -3.9266 -4.7104 -5.2008

even 1 -1.6481 -2.8635 -3.9028 -4.6639

2 -0.9310 -1.9225 -2.8453 -3.7146

3 -0.5840 -1.2301 -2.0135 -2.8210

P

\

n=O n=l n=2 n=3

P

tensor n’=O 0.3756 0.3895 0.3591 0.3205

odd 1 0.4831 0.4875 0.4584

2 0.5364 0.5354

3 0.5660

n=O n=l n=2 ?I=3

LS n’=O -0.0024 -0.0069 -0.0094 -0.0149

even 1 -0.0148 -0.0183 -0.0335

2 -0.0352 -0.0628

3 -0.1291

k n=O n=l n=2 n=3

LS n’=O -0.lY28 -0.2860 -0.3548 -0.4080

odd 1 -0.4254 -0.5296 -0.6102

2 -0.6610 -0.7634

3 -0.8843

84 A. Hosaka et ai. / E@ctive interaction

TABLE 2

Best-fit interaction strengths (in MeV) for Yukawa functions

Channel

0.20 0.25

Range

0.33 0.50 0.70 1.414y

SE 13 015 -882.09 -993.25 -10.463

TE 19 886 -2400.9 - 1262.7 - 10.463

so b) 15 991 -1955.1 396.51 31.389

TOb) 26 795 -2801.4 -142.92 3.488

TNE 83 942 -40 018 1781.1 -65.373

TNO -60 758 26 077 -1769.8 23.870

LSE -180 181 101484 -16 105 658.59

LSO -2865.0 774.51 -1782.1 -6.0303

“) Interaction strengths of this range are fixed to the OPEP strength.

b, The odd forces have ambiguities due to the addition of some constants in momentum space, see text for details.

TABLE 3

Best-fit interaction strengths (in MeV) for OBEP functions

Channel

0.20 0.25

Range

0.33 0.50 0.70 1.414y

TNE 3104.9 -10.463

TN0 -1382.0 3.488

LSE -30 963 2343.4

LSO 9109.8 -2301.8

“) Interaction strengths of this range are fixed to the OPEP strength.

contrary to our early speculation, in the effective interactions, we only show the case corresponding to 40Ca. We mention here that we need at least four ranges of Yukawa-oriented functions for a satisfactory fitting to the OBEP functions. We can see this in fig. 2, where the transformed Yukawa interaction with four ranges and that with three ranges are compared with the original OBEP function for the tensor-odd channel.

One of the changes from the previous interactions is seen, particularly, in the tensor force. The tensor-odd (TNO) channel has a zero at about q = 3.5 fm-‘. As a result, the isospin term of the tensor force also has a zero at q = 4.5 fm-’ (see fig. 3). This result seems to support the tensor force derived from the r+p exchange model 6).

- NEW

----. PARIS

. . .._..........‘ &$3y

- NEW

---- PARIS

-. ,... Myf

- NEW

---_ PARIS

. . . . . . . . . . . . . . f.qay

---= PARIS

. . . . . . . . . . . #.Qy

--- - PARIS

_ &QY

Fig. 1. The effective interactions as a function of momentum q are plotted for each channei. Solid,

dashed and dotted lines show the present (see the text). Paris ‘) and M3Y ‘) effective interactions,

respectively. The turning points on the q-axis indicate the sign change of the effective interaction.

A. Hosaka et al. / Effective interaction

z f IOC

50

TN0 - OBEP . .

l 4range

-----. 3range

1 2 3 4 fin’

Fig. 2. The OBEP-type interaction for tensor-odd channel is compared with the transferred 3-range and

4-range Yukawa-type interaction.

Now we would like to discuss the ambiguity existing in the determination of the

central-odd channel interaction. Because the wave function in the odd channel

vanishes at the origin, the interaction determined from the G-matrix elements has

an ambiguity in the &function, which does not affect the evaluation of the G-matrices

of the odd channels. In momentum space language, this is an ambiguity of some

constant. We can verify from figs. lc and d that the difference between the Paris

and the present effective interactions is at most a constant.

TNT - NEW

---- PARIS

MqJ

Fig. 3. The present effective interaction for the tensor isospin channel is plotted (solid line), and compared with that of the Paris *) and the M3Y ‘) interactions. The sign change of the present interaction is seen

at q - 4.5 fm-‘.

A. Hosaka et al. / Effective interaction

4. Applications

87

We shall now discuss some applications of the present effective interaction for

the study of the nuclear structure. Another application of the effective interaction

for inelastic scattering has been reported already in a previous publication r5). First

the mass dependence of the G-matrix elements is discussed and then the Landau-

Migdal parameter g’ is calculated from the present effective interaction.

4.1. THE MASS DEPENDENCE OF THE G-MATRIX ELEMENTS

Chung and Wildenthal have constructed s&shell-model matrices with the assump-

tion of mass-independent two-body matrix elements 16). Under this constraint, they

were forced to separate the full mass region into two parts: one for 17 s As28, and the other for 28 s AS 39. In order to remove this difficulty Wildenthal have

recently performed new shell-model calculations with the idea that the two-body

matrix elements have some mass dependence. The mass dependence A-o.3 was taken

as a reasonable guess I’). The indicator of the fitting to the excitation energies, x2

value, was greatly improved and the wave functions were used successfully to

calculate the binding energies and the P-decay rates for nuclei far from the stability

line ‘*) and to extract the polarization strength for magnetic operators 19) and

electric-magnetic form factors ‘“).

In this subsection, we would like to study the mass dependence of the two-body

matrix elements with the calculated G-matrices.

To start with, we shall see the meaningful range of the exponent (Y of the mass

A-” by calculating (Y for the diagonal OS matrix elements with zero range and

constant interactions. Since the wave function is normalized, the constant interaction

provides no mass dependence and therefore cy = 0. If the interaction is of zero range,

the matrix element is proportional to the normalization constant, which is (rbJ3 with b. being the oscillator parameter b, = Jh/puo. Hence, G a A-“‘. The reasonable

range of (Y is, therefore, 0 < CK < $. a = 0.3 taken by Wildenthal falls in this range.

Fig. 4 shows the G-matrix elements in the singlet-even (SE), triplet-even (TE),

singlet-odd (SO) and tensor-even (TNE) channels as a function of the mass number

for small node quantum numbers. Other channels have negligibly small matrix

elements. The matrix elements decrease monotonically with the mass number. This

can be seen more clearly in table 4, where the exponent in each channel is tabulated.

The exponent scatters largely between 0.1 and 0.4. Although, the sd two-body matrix

elements are written in terms of the particular linear combination of the G-matrices,

we shall simply take the average value of LY for all the cases shown in table 1 with

weighting of the average value of the matrix elements in 160, 4oCa and 90Zr. The

average value comes out to be 0.25.

We are also interested in the size of the one-pion-exchange contributions for these

matrix elements. They are typically about f-f of the full values in the central channels.

SE

-.-.-.-op-+ OP . . . . . . . tp..+ 1P __-____op-.% 1P

Mass Number A

Fig. 4. The mass dependence of the G-matrix elements with small node quantum numbers.

G-matrix elements (in MeV) in the SE, TE, SO and TNE channels for small node quantum numbers

Channels % %Za %% ix

SE os+os -6.5801 -5.2187 -4.1471 0.2670 Os+ls -5.2924 -4.5601 -3.8435 0.1847 ls+ls -4.6438 -4.2845 -3.7808 0.1184

TE OS+OS - 10.049 -8.0848 -6.4335 0.2579 OS- 1s -8.3732 -7.4460 -6.2354 0.1698 Is.+ Is -1.4208 -7.3175 -6.3463 0.0889

SO QP-+OP 2.5310 1.7260 I.2020 0.4308 OP-* 1P 2.4564 1.7184 1.2299 0.4003 IP-, IP 3.1844 2.2136 1.5768 0.4067

TNE Os+Od -5.6921 -3.8897 -2.7015 0.4312 ls+Od -2.7039 -2.0937 -1.6481 0.2865 0s-t Id -7.8798 -5.5050 -3.9266 0.4030

The mass dependences are fitted to the exponent of the mass number A-” ; the a’s are given in the last column.

A. Hosaka et al. / Effective interaction 89

Hence, we should not neglect the long-range interaction and use only the S-function

force to calculate the matrix elements. The tensor part in particular is dominated

by the one-pion-exchange interaction.

In this subsection we have shown that G-matrix elements obtained from the Paris

nucleon-nucleon potential have the mass dependence A-o.25 on average and the

mass exponent scatters between 0.1 and 0.4. From these results, it seems the mass

dependence A-o.3 taken by Wildenthal is reasonable. We mention further that the

one-pion-exchange interaction should not be left out in the calculation of the

shell-model matrix elements.

4.2. THE LANDAU-MIGDAL PARAMETER

In the discussion of the spin-isospin excitation modes of nuclei 2’), it is important

to know the strength of the Landau-Migdal parameter g’. This is defined as an

interaction strength in the spin-isospin channel between particle-hole pairs in a

nucleus at zero momentum transfer.

In a naive sense we might only take the direct part of the interaction (see fig. 5a)

which leads to

=l&-&(vYq)+ VO(q)- F(q)- P(q));

where fm is the rrNN coupling constant, m, the pion mass, and V,, is the spin-isospin

term of the effective interaction. Actually we have to include the exchange term

expressed in fig. 5b. Using the spin and isospin exchange operators P, and P,

P,=;(l+a*-a~), P,=;(l+T*. 72))

q = ij + (-ij) q= i;-fs

(a) lb)

Fig. 5. The diagram of the spin-isospin excitation modes: (a) corresponds to the direct term, and (b)

to the exchange term.

90 A. Hosaka et al. / Efectiue interaction

gLx can be expressed as

= -~(~(v~~+ v=o+ vSE+ VT”))) (5) Tr

where V in the second term is the central part of the effective interaction. The

bracket symbol ( ) represents the interaction strength averaged over the transferred

momentum, since, in the exchange process, finite momentum transfer is allowed

due to the Fermi motion of nucleons in the nucleus. We simply take the average

momentum transfer lQavl =hP,, where PF is the Fermi momentum of nucleons.

The resulting value is given in table 5 which also contains the result of the rrfp

model for comparison. The v+ p exchange leads only to the spin-isospin (~7)

interaction; on the other hand, the full G-matrix interaction contains a non-spin-

isospin term such as the 0, u and 7 interactions which can contribute to g’ through

the exchange process. This point is discussed below.

The value g’ = 0.54 is obtained with the present G-matrix interaction by summing

a large positive direct term (0.94) and a negative exchange term (-0.40). On the

other hand, the r+ p model provides a large direct value but a small exchange

value, only -0.11. Although.the net results in the two approaches are very different,

it is interesting to note that the direct parts are very similar. This result suggests

that the rr + p model is very good for the UT interaction and that the missing strength,

-0.29, in the exchange part is caused by interactions different from the (~7 interaction.

In fact, the G-matrix interaction without the ~7 channel provides g,, = -0.3.

Now we will come back again to the ambiguity problem and show that it does

not affect the total value g’ = g&i,+ gLX. From eqs. (4) and (5)

g&(;(v”O+ v=O- VSE - VTE)(,=, 7r

-a( PO+ vTO+ vSE+ VTE)Jq=,Q,,,}.

As is clearly seen, the odd term takes the form of a difference, V( q = 0) - V( q = 1 Q=,j),

for both SO and TO channels which ensures the invariance of g’ when some constant

is added to the odd parts of the effective interaction.

TABLE 5

The Landau-Migdal parameter g’

present 0.94 -0.40 0.54 r+P 0.97 -0.11 0.86

A. Hosaka et al. / Effective interaction

5. Summary

91

We have derived an effective interaction from the Paris nucleon-nucleon interac-

tion. We have taken the OBEP-type functions as a functional form of the effective

interaction. As a result, the behaviour of the interaction at high momentum differs

from the previous M3Y ‘) and Paris “) effective interactions especially in the tensor

channels. The resulting effective interaction derived from the G-matrices for several

nuclei comes out to be essentially independent of the mass number. Hence we can

use a unique effective interaction for nuclei in the whole mass region.

Since it is interesting to study the mass dependence of the G-matrix elements

due to the work by Wildenthal I’), we have compared the G-matrix elements obtained

for different nuclei. The G-matrix elements seem to show a simple mass dependence,

which can be expressed as A-” with a being around 0.25 on average. This justifies

the semi-empirical value, 0.3, used by Wildenthal er al. I’). Next we have estimated

the Landau-Migdal parameter from the present effective interaction. We have

obtained g’ = 0.54. Furthermore, comparing it with the v + p model we find that the

direct part g&i, agrees with that of the 7r + p model; on the other hand, most of the

exchange part g:, seems to come from the non-spin-non-isospin channels. Con-

sequently the considerable cancellation between g& and g:, leads to the reasonable

total value g’.

Finally we have commented on the ambiguity of the odd channel of the effective

interaction. The ambiguity arises, in configuration space, from the &function, and,

in momentum space, from some constant. Such an ambiguity is unavoidable as long

as we apply the present method based on the oscillator G-matrix elements to derive

an effective interaction.

We are very grateful to Dr. Yabe for valuable discussions on the problem existing

in effective interactions for low-energy nucleon scattering. We have performed the

numerical calculation at TMU using FACOM Ml80 AD with partial financial aid

from INS, University of Tokyo.

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