Nuclear effective interactions and spin excitations

Nuclear Physics A405 (1983) 653-670 @ North-Holland Publishing Company

NWCLEAR EFFECTIVE INTERACTIONS AND SPIN EXCITATIONS TOSHITAKE

KOHMURA”

Nuckar Physics ~aborato~, Oxford Uniuersify, Oxford, England Received 23 August 1982 (Revised 28 December 1982) view of the one-boson-exchange model for the nucleon-nucleon interaction and the Hartree-Fock (HF) interaction, we formulate the effective interactions for partide-hole states in terms of the exchange of the fields which are confined in the nucleus. This theory, as an extension to the nuclear field theory (NFT), takes into account the propagation of the fiekls which is neglected in NFT. The effective interactions thus obtained reproduce the energies of a sequence of electric giant resonances and the core polarizabilities associated with the resonances. It is found that the coupling constants of the (+- and o-fields are suppressed for the particle-hole interaction by 60% with respect to the HF interaction. As for the effective interactions involving nucleon spins, we consider the fields coupled to nucleon spins. The effective interactions obtained, essentiallydifferent from those in NFT, have a tensor component. We analyse the energies and cross sections for excitation of stretched spin particle-hole states which are the most sensitive to the tensor force. The effective interaction responsible for the stretched spin states is shown to be consistent with

Abstract: h

that for the magnetic resonances observed in the (p, n) reactions.

1. Introduction The

one-bison-exch~g~

nucleon-nucleon commonly

adopted

in the

the spin and isosopin isoscalar repulsive

(OBE)

interaction

OBE theory induces a potential

of the nucleons:

scalar type is attractive at shorter

are repulsive.

potential reproduces the main features of the of the four mesons a, 7i-, p and w which are

‘). Each

distance,

typically

the nucleon-nucleon

at around

while those

the internucleon of the other

dependent

interaction distance

on

of the

and strongly

spin and isospin

properties

The attractive

force is induced by the two-pion (or u-meson) exchange and the repulsive forces are mainly by the vector meson exchange. The relativistic meson field theory, as an application of the OBE theory to nuclear systems, was reported to reproduce the binding energy of nucleons, the HF singleparticle energies and the charge distribution in finite nuclei ‘), and to describe the saturation of nuclear matter 3). The C- and w-mesons which produce the mean field in a nucleus are responsible for the HF single-particle potential. The success of the OBE theory for the HF calculation leads us to problems if the OBE model reproduces the nuclear effective interactions for particle-hole states: if the *

On leave of absence from the Institute of Physics, University of Tsukuba, Ibaraki 305, Japan.

653

T. Kohmura / Nuclear effective interactions

654 (+-

and w-exchange interaction adopted for the nuclear saturation, for example, represents the effective interaction of the isoscalar scalar type. The effective interactions were formulated in the independent-pair model by Bethe and Goldstone 4). The derivation of effective interactions is entangled with the HF single-particle states: the effective interactions which are to determine single-particle states are obtained dependently on the single-particle states. Nuclear many-body effects are expected to make the effective interactions more complicated. Several authors obtained the effective interactions phenomenologically, avoiding the complexity in the fundamental formulation. The effective interactions thus obtained characteristically depend on the nucleon spins and isospins ‘). The effective interactions involving the nucleon spins and isospins have not yet been established; the problem is being investigated whether a tensor component is required in the effective interactions 6). Consequently, the effective interactions are often phenomenologically expressed in terms of only the central forces+ as Vi,=‘-VlJ(a+bUi

’ Uj+Cii

’ ?j+dUi

’ U$i * ?j)zi(Tij),

(1.1)

where the same radial function u(r) is assumed for the four types of exchange forces. A boldface,symbol represents a three-dimensional vector and an arrow on a symbol stands for an isovector. In table 1, we show the mixing coefficients a, b, c and d commonly used for the central forces ‘). The coefficients are normalized to the interaction strength V. = 50 MeV and the radial function v(r) is a function of the dimensionless variable r/r,, with ro- 1.4 fm. Although there are some variations in the mixing coefficients among authors, there is general agreement that the effective interaction of the isoscalar scalar type is attractive but that those of the other spin and isospin properties are repulsive. These features are attributed to the fundamental nucleon-nucleon interactions. The sequence of giant resonances which has been measured gives a great deal of information on the effective interactions. The energy of the resonances and the associated core polarizations give a test of a model for the effective interactions. TABLET

Coefficients of the exchange mixtures in the nuclear effective interactions which are commonly used [the mixing coefficients are obtained from the compilation in ref. 5)] a Kurath Serber Ferrel-Visscher Soper Gillet I II

’ The tensor

force and the two-body

-0.1 0.38 0.13 0.33 0 0.16

spin-orbit

b

C

d

-0.1 -0.13 -0.13 0.03 -0.2 -0.4

-0.2 -0.13 -0.22 -0.11 -0.3 -0.22

-0.2 -0.13 -0.13 -0.11 -0.1 -0.09

force are used in the effective

interactions

in ref. 7).

655

T. Kohmura / Nuclear efecfive interactions

There have been attempts to understand a sequence of resonances comprehensively, as quite a few electric giant resonances “) and magnetic resonances 9, were observed. A recent trend of theoretical investigations is to make a unified theory of resonances of various spin and isospin properties and to predict collective resonances involving spin excitation lo). The effective interactions responsible for giant resonances may be more simplified than those in eq. (1.1) since it must be a simple transition operator that is effective to excite nucleons coherently into a collective state; Bohr and Mottelson obtained an expression for the effective interactions, assuming that the field for the nuclear single-particle potential vibrates in conformity with the nuclear density distribution I’). It is a useful idea that a single-particle potential field represents the structure of the nuclear collective state which requires a lot of configurations in the particlehole expression. The effective interactions in NFT have reproduced reasonably well the electric giant resonances. The effective interactions involving nucleon spins are presumed to be more complicated than those derived from NFT. In fact, their ur dependent effective interaction does not reproduce consistently the energies 9, of the magnetic resonances observed in the (p, n> reactions i2). The energies of the resonances were reproduced by taking the effective particle-hole interaction

where the notation Yf(i)= is used. The Gamow-Teller strength

C (rm11m2JJM)Y;“‘(q)a12 m1m2 resonances ko/4rr

were reproduced

(1.3) with the interaction

= 23/A MeV

(1.4)

while k, for the dipole spin-isospin resonances was found to satisfy k, = 0.75kl)/(r2),

(1.5)

where the mean square radius, V)=9OA

3

2

2/3

,

(1.6)

of the nuclear density distribution is taken with r0 = 1.2 fm. It was indicated by the factor 0.75 in eq. f 1.5) that the I= 1 interaction is suppressed by 25% with respect to the value

predicted from NFT.

656

T. Kohmura / Nuclear effective interactions

The too strong interaction

strength kl in eq. (1.7) was obtained by neglecting the propagation of the field in NFT. The field propagation destructs a conformal vibration of the single-particle potential with the nuclear density distribution. The exchange of a field coupled to nucleon spins induces the tensor force. In the previous paper 13), it was shown that the propagation of the fields actually suppresses the interaction strength kl for higher 1. The exchange of the pseudoscalar field induces the tensor force which substantially splits the kl according to the spin J. The effects of the field propagation and the tensor force lead to the results consistent with the observation in the (p, n) reactions. NFT takes no account of the effects, while ensuring the conformal vibration of the single-particle potential with the nuclear density distribution. In our theory, as an extension to NFT, we formulate the effective interactions for particle-hole states in terms of the exchange of propagating fields. For a particle-hole state, we consider one or two fields specified by the set of quantum numbers proper to the state. The exchange of the fields coupled to nucleon spins induces the tensor force. While propagating in the nucleus, the fields get nuclear effects: the fields may turn into a configuration of particle-hole pairs or be scattered from nucleons. The quantum numbers such as spin, angular momentum and isospin, however, are conserved in the processes. Therefore, a field exchanged transfers definite quantum numbers between the pair of nucleons, although the field involves some configurations of particle-hole pairs. A pair of nucleons exchange a virtual field confined in the nucleus. The exchange of a confined field induces the effective interaction in the form of a separable potential, which is useful for treatment of nuclear collective states. Nuclear effects may also modify the coupling constants of the fields. In a microscopic treatment of meson degrees of freedom, the c-term in eq. (1.1) gets some contribution from an iterated process of the tensor interaction. We, however, suppose that this contribution is also included by our effective interaction expressed in terms of the exchange of the isovector vector field with a modified coupling constant in the same way as the isoscalar scalar component of the two-pion exchange interaction is represented 14) by the exchange of a g-meson. We actually retain part of the iterated tensor in our calculation using the effective interaction V in eq. (2.3). In this paper, we formulate the effective interactions in terms of the exchange of fields and apply the interactions obtained to discuss nuclear properties. In sect. 2, we develop the field-exchange theory of the effective interactions. Sect. 3 is devoted to electric giant resonances and the core polarization associated with the resonances. In sect. 4 we consider the effective interactions involving nucleon spins including the tensor forces and discuss stretched spin states as an example of states which are typically affected by the effective interactions. Some discussion and conclusions are presented in sect. 5.

T. Kohmura / Nuclear effective interactions

651

2. Field exchange theory of nuclear effective interactions

The fundamental nucleon-nucleon interaction is induced by the exchange of mesons 15)between a pair of nucleons separated by more than the hard core radius. The effective interactions between a pair of nucleons in a nuclear system are not the fundamental meson-exchange forces but are influenced by nuclear many-body effects “). The internal degrees of freedom of the pair of nucleons are assigned by the quantum numbers S and T for the spin and isospin of the pair which constitute a four-dimensional space. The fact that the effective interactions depend on the S and T just like the nucleon-nucleon interactions, however, suggests that we may formulate the effective interactions in terms of the exchange of fields. The fields are distorted by the nuclear effects, such as scattering from nudeons and the virtual nuclear excitations (see fig. 1).

Fig. 1. Some of the processes which contribute to the nuclear effective interactions. The dotted lines stand for the field which is responsible for the effective interaction and the wavy lines for any field.

In an attempt to formulate the four types of the exchange forces for the effective interactions, we introduce three fields: an isoscalar scalar field, an isoscalar vector field and an isovector vector field. We will also consider the exchange of an isovector pseudoscalar field in order to discuss the tensor force in sect. 4. These fields are named the (+-, w-, p- and r-fields after the corresponding mesons, although the fields are not pure meson fields but involve some configurations of particle-hole pairs. For the effective interaction for a nuclear collective particle-hole state, let us consider a field specified by the quantum numbers proper to the state. The field, propagating in the nucleus, gets some responses from it. The field function ~5which satisfies the equation {-V2+m2+17(w)+U}~

=w2&

(2.1)

with mass m, represents the nuclear collective state. For the nuclear responses, we take the particle-hole excitation L!(o) and the scattering U of the field from nucleons: the explicit expression for the responses on an isoscalar scalar field, for example, is

U = -4?~Fp (r) ,

(2.2)

where F stands for the amplitude of forward scattering of the field from a nucleon

658

T. K~~m~ra / ~~cl~a~ effectbe i~tera~t~vns

and p(r) is the nuclear density distribution. When we have two fields coupled to each other, eq. (2.1) is replaced by coupled equations. In terms of the solutions #N to eq. (2.1), the nuclear effective interactions are expressed as 1

v=g2z14N)

The matrix elements of V are not so sensitive to the L’(w), so we presently use the expression 2ri II(w) =g2 oz_,zP(r), which is obtained in a closure approximation

and under the assumption

The fields are coupled to a nucleon as shown in table 2, where 4 represents the nucleon lieid and the suffices t and s of a vector field stand for the time and space component of the field respectively. The exchange of the fields between a pair of nucleons i and j induces the following effective interactions:

b-w,)

(2.4a)

6%)

(2.4b)

Gat)

(2.4~)

(PJ,

(2.4d)

where

The nucleon-nucleon interactions in free space due to the w, and ps field exchange have a S(P) force 16), which must be screened by the hard core for the pair of nucleons. Subtracting the short-range force from our interactions in the usual procedure, we obtain

Sii=3Ui ‘ViUj

‘Vj-Ui

‘UjVi

*Vj.

0.5)

TABLE 2 The coupling of the fields to a nucleon

Field

Coupling

The suffix 0 of (p stands for the time component of the vector field.

The expressions (2.4a-f) are to be used for the nuclear effective interactions. We see that each of the fields induces the effective interaction of the corresponding spin and isospin property and that the S = 1 interactions due to the exchange of w, and ps fields invoke a tensor force. ff the nuclear density distribution p(r) in the ground state is spherical, eq. (2.1) has a set of solutions sbL&f @I = &L(I) YP (E)

(2.6)

specified by the quantum numbers L and M The wave functions #N in eqs. (2.4a-fl are replaced by &zw and the summation is over L and M For the tensor term in eqs. (2.4e) and (2.40, we may use the following formula “}:

Therefore, we obtain the relation

660

T. Kohmura / Nuclear effective interactions

with

(2.9)

3. Giant resonances

and core polarizability

The effective interactions in eqs. (2.4a-f) lead to an expression for the particlehole interaction for a particle-hole pair with given spin S and isospin T. This particle-hole interaction determines the energies for a sequence of giant resonances specified by the S and T and the polarizability associated with the collective resonances. This suggests that we may relate the resonant states with various polarities L to each other. In this section, we calculate the energy of electric giant resonances and the core polarizability associated with the resonances in order to show the applicability of our theory to electric excitations (S = 0) which are experimentally more established than magnetic excitations (S = 1). We assume that the nuclear density distribution is P(r)= where the nuclear

PO l+exp[(r-R)/a]’

radius for the nucleus

(3.1)

of A nucleons

is

with r. = 1.2 fm. As was discussed in sect. 2, the matrix elements of the effective interaction V in eq. (2.3) are not quite sensitive to the detail of the potential II + U for the fields, so that the potential may be here approximated by a square well potential 17(r, w)+ U(r)

=

-UO, co 7

I

rR,

(3.2)

with Uo=

-g2&+4~F

po, >

in order to obtain an analytical expression for the effective interaction. Although a more elaborate treatment of eq. (2.1) modifies higher eigenstates, it does not affect so much the present calculation. The nuclear hamiltonian is expressed as H=Ho+V,

(3.3)

where the Hartree-Fock hamiltonian Ho is assumed to be that of the harmonic oscillator and the residual interaction V is our effective interactions. Let us here consider the fundamental excitation. We take the term with ~&,~(r) iii the effective

T. Kohmura / Nuclear effective interactions

661

interactions in eqs. (2.4a-f), which is responsible for the nuclear excitation. Applying the random phase approximation (RPA) to the resonances, we obtain the secular equation for the energy E of the resonant states: 1

=

~EN(NI V/N)

c

where &Nis the unperturbed

(3.4)

7

E2-&

N

energy of the particle-hole

states,

lN-~NIO),

(3.5)

with the nuclear ground state IO). In the case that EN are degenerate obtain l=.--.-

E22_SE2 9

at EN= 8, we

(3.6)

where the sum rule

S =C FNWI VIN) N

(3.7) with C=

s=o

1,

1 -2

S=l.

3,

Substituting the sum rule S obtained in eq. (3.7) and the unperturbed energy E = Lw with the energy w for the harmonic oscillator into eq. (3.6), we obtain the energy eigenvalue E. For the polarity L 2 3, the operator &,M(r) excites a nucleon in a closed shell not only by IZ= L major shells but also by it
(L-l)!

n

(;(L-n))!&L+n))!

S(n) =

s

2L-1

0,

’

n=L,L-2,...30

for other ~1.

(3.8)

:

Then, the secular equation for the energy eigenvalue E is

2S(n)

1=x n

E2-n2w2’

(3.9)

T. Kohmura / Nuclear effective interactions

662

The formulae for the energy E in eqs. (3.6) and (3.9) are general enough to cover a sequence of resonances with given (S, T), as long as the central force is dominant in the nuclear effective interactions in which we put an effective mass meff and a coupling constant g proper to the (S, T). TABLE 3 Energy E of resonances and core polarizability X

12+ 2’

1 0 1

79A-1/3*

63A-l/3* 121A-“3

79A-“3 63A-‘j3 120A-1’3

-0.73 0.69 -0.54

0.7 b, -0.3 to -0.4 ‘)

The theoretical values with an asterisk are input. b, Ref. “). “) Ref. 23). “) Ref. ‘).

Let us compare the prediction from our energy formulae with the measured energy of electric giant resonances (S = 0) since a lot of the resonances have been measured. The measured energies are expressed “) in terms of A-1’3 as are shown in column 4 in table 3. In the energy formulae, we adopt the meson mass m = 400 MeV, 780 MeV and 770 MeV for the effective mass meff of the u-, w- and p-fields respectively and the value w = 41A -1’3 MeV for the harmonic oscillator, as these values are consistent with the level spacing in a sequence of resonances. Eq. (3.6) with the coupling constants g2 = 15.6, g2 = 49 and g2 = 18.2 for the three fields reproduces the energies of the giant resonances with J = 2+, T = 0 and l-, 1. The coupling constants of the c-and w-fields thus obtained for the particle-hole interaction are suppressed by 60% with respect to those for the HF interaction. Using these values of WQ and g2, we have evaluated the energies of resonances with various polarities L and found that they reproduce the experimental values. The giant resonances with J = 2+, T = 1, for example, are predicted at E = 121A-1’3 MeV, which agrees very well with the experimental value E = 120A-1’3 MeV, as is shown in table 3. Eq. (3.9) has been applied to the 3- states of 160. Using the value w = 13.26 MeV for the harmonic oscillator which was obtained from electron scattering 18), we reproduce the 3- states involving lhw particle-hole excitation at E = 7.06 MeV for T = 0 and at E = 18.8 MeV for T = 1, as shown in table 4. For magnetic resonances (S = l), three states with spin J = L - 1, L and L + 1 are allowed for a given polarity L. These states, which are degenerate in the energy formula (3.4) for the central force, are split by the effective tensor force. This effect will be discussed in sect. 4. Using the nuclear effective interaction V, let us calculate the correction to a nuclear single-particle transition due to the core polarization which involves an

T. Kohmura f Nuclear effectiveinteractions

663

?-ABLE 4

Energy E and core pol~~zability x for 3- states of 160

33-

0 1

6.16 “) 20.5 b,

7.06 18.8

1.33 -0.64

“) Ref. 19). b, The energy obtained from pion scattering ‘4.

associated giant resonance. The effect of the core polarization the polarizability x which is expressed by X0

is represented

by

(3.10)

X’1_xo’ in terms of the first-order value x0 = -2s/s

2

(3.11)

in RPA in the schematic model ‘I). Substituting the sum rules S obtained in eqs. (3.7) and (3.8) into eq. (3,11), we obtain the core polarizability in electric transitions, which is also given in tables 3 and 4. The polarizability x = 0.69 for isoscalar 2+ transitions agrees with the experimental value 22) x = 0.7 very well and the predicted value x = -0.54 for isovector 2+ is not in bad agreement with an experimental value 23) x = -0.3 to -0.4. To obtain the pol~izability for 3- transitions in table 4, we put the partial sum rule S(1) obtained in eq. (3.8) into S in eq. (3.11). We find that the present theory reproduces well the energies E? tis well as the polarizabilities x with the same effective interactions V. It should be noted that we have not introduced the effective nucleon mass 24>for the single-particle energies E, while Towner et al. who used S(r) forces for the effective interactions, had to employ the effective mass and to vary the force strength to fit the polarizability x from that to fit the energy E of resonances lo). 4, Spin excitations So far we have mainly analysed in the present theory the nuclear transitions which do not involve nucleon spins. Our theory has been found to work well for the transitions that are rather well established experimentally. The success of our theory for the electric transitions encourages us to apply it to the nuclear excitations which involve nucleon spins. Our field-exchange theory, essentially different from NFT, retains the tensor component of the effective interactions involving nucleon spins.

664

T. Kohmura / Nuclear effective interactions

In the previous paper 13), using the UT dependent effective interaction expressed in terms of the exchange of the p- and rr-fields, we deduced the origin of the 25% suppression of the 2 = 1 effective interaction in eq. (1.5). It was shown that both the effects of the field propagation and the tensor force modify the interaction strength kl in eq. (1.2): the former effect suppresses the kl for higher 1 and the latter splits it according to the spin J. We obtained the coupling constants g2 = 14.0 and g2 = 96 for the p- and m-fields which reproduce the experimental values of the interaction strengths kr. The J = 2 interaction strength for I = 1 is suppressed by 25% with respect to the prediction, eq. (1.7), from NFT, while the J = 0 and 1 strengths for I = 1 are enhanced by 20%. These results are consistent with the observation in the (p, n) reactions. In this section, let us consider the significance of the tensor force in nuclear structure. The exchange of an isovector pseudoscalar field induces the tensor force (4.1) and the central force which modify the expression for V obtained in eq. (2.4f). The tensor force induced by the field which is coupled to a nucleon in the form of u * VC$J excites a nucleon by means of ( YJ_la)J more strongly than ( YI+Ia)J as is seen in eq. (2.8). Therefore, the tensor force is expected to be more effective for single-nucleon transitions with polarity L = J - 1 than for those with L = J + 1. The tensor force with a negative sign in eq. (4.1) acts so as to reduce the repulsive central force in eq. (2.4f). This reduction of the central force by the tensor force is more significant at higher J for single-particle transitions induced by the operator ( YJ-la)J, since the operator V in Sij in eq. (4.1) is more effective for 4.~ with higher J as is also seen in eq. (2.8). These are the features of the tensor force. Let us see phenomenologically the reduction of the central force by the tensor force in the stretched spin states in which the orbital and spin angular momenta of the particle and hole are all stretched in one direction. These states are remarkably affected by the operator ( YJ_lu)J in the effective interactions. In table 5, we show some of stretched spin states, for which we have evaluated the empirical values of “polarizability x0” in the following two ways. First, we obtain the polarizability x0 from the energy E of the stretched spin states which is expressed in RPA as E2 =w’(l

-x0),

(4.2)

with the unperturbed particle-hole energy o. We show the polarizability xo obtained in column 4 in table 5. Next, we take the cross section for excitation of the stretched spin states and obtain x0 from the ratio of the experimental cross section (T,,~ to the theoretical (+sMestimated in the single-particle shell model, which is related to the polarizability x0 in the schematic model as 1

ffexp -=flSM

(1

-xOj2

(4.3) *

665

T. Kohmura / Nuclear effective interactions TABLET Stretched Nucleus

spin states and the polarizabilities

x0 associated

E “)

0 “)

x0

12C&,zpJ;z)4-

19.5

i6Wd,,,p;:,)4-

18.9

17.6 17.7

-0.23 -0.14

28Si(f7,sd~,,rJ6-

14.36

16.9

0.28

58Ni(ga,sf?~W 208Pb(ji5~2i&)14-

10.30 6.75

10.4 6.49

0.019 -0.082

The nuclear of ~0 obtained

with the states oelqp/ffsSM “)

spin J is shown in addition to the main configuration in column from the expression for the excitation energy, E2 = ~~(1 -x0),

0.27 0.45 (0.32 0.33 b, (0.29 0.30 0.56 (0.46

x0 -0.92 -0.49 -0.77) -0.74 -0.86) -0.83 -0.34 -0.48)

1. The empirical value is shown in column 4.

The energy o is the unperturbed particle-hole energy. The polarizability x0 obtained from the expression uexp/usM = (1 -x0)-’ for the ratio of the experimental cross section flex,, to the estimate osSM in the single-particle shell model is shown in column 6. In columns 5 and 6, the values from electron scattering are shown without parentheses and from proton scattering in parentheses. b, The value revised in ref. 26). “) Ref. “).

We show the value of this polarizability x0 in column 6 in table 5. The polarizability x0 obtained from the cross section deviates from the polarizability x0 from the energy, which may be mainly attributed to the experimental errors, a defect of the schematic model, the contributions from two-particle-two-hole configurations and the difference in the property of the two x0. What seems very important is, however, to notice that with some exceptions the absolute value of each polarizability x0 decreases with J. This may indicate that the higher J is the more effectively the attractive tensor force due to the pseudoscalar field exchange reduces the repulsive central force. In view of this property of the empirical ~0, we have evaluated in our theory the energy of the stretched spin states in RPA and the nuclear transverse form factor for inelastic scattering for excitation of these states. In the calculation we cannot simplify the fields qS(r) as much as we did in sect. 3, since the tensor force arises from a gradient of the fields. We use a harmonic oscillator potential for the potential IZ + U whose range parameter b is assumed to be equal to that for the nucleon system. The meson masses m = 770 MeV and 140 MeV are used for meff for the p- and m-fields, respectively. We have first tried to reproduce the excitation energy and cross section for the stretched spin state of r60. This state may be the most free from the effects which are out of the framework of the present theory, The empirical value of the energy of the state is reproduced by certain sets of the coupling constants g2 for the two fields. In fig. 2, in which we show the transverse form factor for the excitation, we see that almost any set of the coupling constants g2 reproduce the experimental form factor as long as they satisfy the energy, where we assume the measured form factor 27) comes wholly from the 4- excitation. We

T. Kohmura / Nuclear effective interactions

666

I~w1* IO-3

10-b

IO"/

Fig. 2. Transverse form factor for the inelastic electron scattering for excitation of the J = 4-, T = 1 state at E = 18.9 MeV of 160. The form factor corrected by the core polarization due to the effective interaction is shown for two typical cases: i.e. with the coupling constants g2 = 21 and g2 = 185 (solid line) and with g2 = 11 and g* = 0 (dotted line) for the p- and v-fields, respectively. These two sets of coupling constants reproduce equally well the energy of the stretched spin state. The form factor obtained in the single-particle shell model is also shown by a dashed line. The range parameter b = 1.77 fm for the harmonic oscillator potential is used for the numerical calculation. We take from ref. “) the measured form factor, which is assumed to come wholly from the 4- excitation.

TABLE 6 The energy E of stretched spin states obtained from the present theory is shown for the three typical cases: i.e. the coupling constants g2 = 21 and g2 = 185 (I), g2 = 14.8 and g2 = 72 (II), and g2 = 11 and g2 = 0 (III) for the p- and n-fields, respectively E Nucleus I 12

C

160 28Si 58Ni *08Pb

19.1 18.9” 17.7 10.9 6.69

The values with an asterisk is shown in column 5.

II 19.2 18.9* 17.8 10.9 6.75* are input.

III

exp

19.2 18.9* 17.9 11.0 6.79

19.5 18.9 14.4 10.3 6.75

The experimental

value 25)

T. Kohmura / Nuclear effective interactions

667

also see that the transverse form factor is very moderately corrected by the core polarization due to the effective interactions, since the repulsive central force which dominantly contributes is remarkably reduced by the attractive tensor force. We have evaluated the excitation energy for the other stretched spin states using the same values of g and summarize the result in table 6. A comparison of our result with the empirical values of the energy E in table 5 suggests that the attractive tensor force is required to make the repulsive central force more reduced at higher J. This indicates that the tensor force due to the pseudoscalar field exchange contributes more to the excitation energy E than that due to the vector field. The set of the coupling constants g* = 14.9 and g* = 72 for the p- and r-fields, respectively, reproduces the energies of the stretched spin states of 160 and *‘*Pb as shown in table 6. We see that these coupling constants coincide with those 13) for the energies of the magnetic resonances observed in the (p, n) reactions. 5. Discussion

and conclusions

In view of the success of the OBE potential for the nucleon-nucleon interaction and the HF interaction, we have formulated the nuclear effective interactions for particle-hole states in terms of the exchange of the fields which are confined in the nucleus. The effective interactions thus obtained are described as a sum of separable potentials. The effective interaction features a spectrum of collective particle-hole states with given spin S and isospin T which deviates from that obtained from a short-range effective interaction. Our theory, as an extension to NFT, takes into account the propagation of the fields. The propagation destroys the conformal vibration of the single-particle potential with the nuclear density distribution which was introduced to construct NFT. The propagation of the fields suppresses the strength of the effective interactions for higher 1. We have determined the coupling constants of the fields phenomenologically, as the coupling is modified by nuclear many-body effects. The effective coupling constants g2 thus obtained are summarized in table 7. We have adopted for the isoscalar scalar interaction (S = 0, T = 0) the CT-and o-field exchange interaction, which was reported to reproduce the HF single-particle energies in finite nuclei and to describe the saturation of nuclear matter. The isovector scalar interaction (S = 0, T = 1) is represented by the p-field exchange interaction. These effective interactions with the coupling constants shown in table 7 reproduce the energies of giant resonances with J = 2’, T = 0 and l-, 1. We see that these coupling constants of the U- and w-fields for the particle-hole interaction of the isoscalar scalar type are suppressed by 60% with respect to those for the HF interaction and the origin of the suppression is not known. Our pt field has a strong coupling constant reflecting the possible contribution of the iterated tensor interaction. With the coupling constants obtained, we have reproduced well the energies of other electric giant resonances and the core polarizabilities associated

668 %kBtE

7

The effective coupling constants g2 and those in free space

o-

WI Pt PS

r

15.6 49 18.2 14.9, 14.0 “) 72, 96 “)

48 b, 135 b, 13 40 185

“) The values 13)obtained from the energies ‘) of.the magnetic resonances observed in the (p, n) reactions. b, The values required for the nuclear saturation in the relativistic meson field theory 23). The mass of the u-meson is here assumed to be 400 MeV.

with the resonances. The spectrum of a sequence of giant resonances suggests that the energy gain of the fields due to the potential U(r) is not so large but the meson mass may replace the effective mass meff for the corresponding field. The fields are, however, confined in the nucleus by the D(w) in eq. (2.2) and discretized so that one discrete state of the fields determines the structure of a nuclear collective state, which requires a lot of confi~rations in the particle-hole expression. The success of our theory for the electric excitations (S = 0) encourages us to apply it to the spin excitations (S = 1). We express the err dependent effective, interaction in terms of the exchange of the p- and m-fields. Our field exchange theory, essentially different from NFT, retains the tensor component of the effective interaction involving nucleon spins. In the previous paper 13), we deduced the origin of the suppression of the I = 1 interaction by 25% with respect to the prediction from NFT. It was shown that the propagation of the fields suppresses the I = 1 interaction strength by 5% and the tensor force splits the interaction strength for polarity E according to the spin J, suppressing the J = 2 strength of the I = 1 interaction by another 20% and enhancing the J = 0 and 1 strengths. These results are consistent with the observation in the (p, n) reactions. In this paper, we have investigated stretched spin particle-hole states which are the most sensitive to the tensor force and determined the coupling constants of the fields independently of those for the .magnetic resonances observed in the (p, n) reactions. Providing the coupling constants g2 for the rr- and p-fields are determined so as to reproduce the excitation energy of the J = 4-, ‘2’= 1 state of 160, the cross section for the inelastic electron scattering for excitation of this state is not so sensitive to the set of the coupling constants and is roughly reproduced by any set. The energies of the other stretched spin states considered here except for deformed nuclei ‘*Si and “Ni are also rather well reproduced by any set of g, and g, which

T. Kohmura / Nuclear effective interactions

669

gives the measured energy of the state of 160. The energies of the stretched spin states of I60 and ‘OSPb are reproduced with the coupling constants for the p- and r-fields which are consistent with those for the energies of the magnetic resonances observed in the (p, n) reactions, as shown in table 7. The excitation energy of the stretched spin states indicates that the energy is influenced by the tensor force due to the r-field exchange dominating over that due to the p-field. It is well known that the transverse form factor for excitation of the J = l+, T = 1 state of i2C at E = 15.11 MeV is affected drastically by the core polarization mainly due to the repulsive central force 28). This striking effect of the repulsive force prevents from revealing the precursor phenomenon of pion condensation. Stretched spin states are more sensitive to the tensor force than magnetic dipole states but the tensor force tends only to reduce sharply the repulsive central force producing a moderate effect on the stretched spin states as two forces are combined. We have determined a nuclear collective excitation mode in terms of the field function 4 which satisfies eq. (2.1). The field propagating in the nucleus is described as a chain diagram. We may extend the present theory, considering the vertices of three fields coupled to each other through a particle-hole pair. Taking two field excitations for the nuclear responses on a field, we can take into account the incoherent two-particle-two-hole configurations and higher which are neglected in RPA. The generalization of the present theory will be given in another context. The author expresses thanks to Dr. P.E. Hodgson for his hospitality at Oxford and to the Japan Society for Promotion of Science and the British Council for financing his stay at the Nuclear Physics Laboratory. He dedicates this paper to Prof. H. Narumi on his sixtieth birthday. References 1) S. Ogawa, S. Sawada, T. Ueda, W. Watari and M. Yonezawa, 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13)

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