A nuclear structure approach to the nucleon-nucleus optical potential at low energy

Nuclear Physics A371 (1981) 173-209 Q North-Holland Publishing Company

A NUCLEAR STRUCTURE APPROACH TO THE NUCLEON-NUCLEUS OPTICAL POTENTIAL AT LOW ENERGY A. BOUYSSY, H. NGÔ and N. VINH MAU Division de Physique Théorique*, Institut de Physique Nucléaire, F-91406 Orsay Cedex, France Received 17 July 1981 Abstract: An antisymmetrized microscopic calculation of the optical potential for nucleon- °°Ca elastic scattering is derived. RPA correlations taken into account in the one-particle mass operator are shown to bring a correction to the first-order real potential at low energies and to lead to an imaginary potential. Both are calculated for incident nucleon energies between 10 and 50 MeV. Their general properties are studied in great detail and, after comparison with empirical imaginary potentials, the reliability of such an approach is discussed according to the value of the incident energy.

1. Introdaction The optical model potential which represents the interaction of a nucleon with a nucleus can be defined as an average one-body potential, which, once inserted into a Schrödinger equation, reproduces nucleon-nucleus scattering properties, such as, for example, elastic and total cross sections and polarization data. From the many-body theory point of view the optical potential has been proved to be the mass operator of the one-particle Green function') . Antisymmetrization effects between the projectile and the constituents of the target are, in that case, automatically taken into account. However, the replacement of a many-body problem by a simpler one-body problem through the optical potential, identified with the mass operator, implies that some approximations have to be made, since the mass operator cannot be calculated exactly. Several microscopic derivations of the optical potential have already been performed, either in the "nuclear structure approach" or in "the nuclear matter approach" sue) . The average microscopic potential, which is obtained, is a complicated object : energy dependent, complex and non-local. However, data analysis shows that a relatively simple phenomenological one-body potential, complex, but local, and depending smoothly on energy and target mass number, is sufficient. As a consequence, a local approximation to the non-local microscopic potentials is often derived and compared with empirical potentials . These empirical potentials * Laboratoire associé su CNRS .

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A . Bouyssy et al. / Nucleon-nucleus optical potential

are not uniquely determined, since there are many which fit the data equally well . Actually the experimental data are sensitive to only a few combinations of the parameters . It appears, in particular, that the scattering cross sections depend mainly on the volume integrals per nucleon and on the r.m .s . radii of the optical potentials . Therefore much attention has to be paid to these quantities, as obtained from microscopic potentials, and their comparison with empirical ones . However, these phenomenological potentials usually fit the data much better than microscopic potentials do, so that one can ask whether or not carefully and in great detail it is interesting to study the microscopic approach . The Feshbach optical potential theory v) has initiated a number of what can be called semimicroscopic calculations, which use, at some stage, empirical quantities, such as empirical nucleon-nucleon t-matrix or empirical nuclear inelastic scattering form factors. The advantage of the microscopic study of the optical potential, compared to that of empirical potentials, or even to that of semi-microscopic approaches, is that it allows one to understand the origin of the absorption and its behaviour as a function of the incident energy according to the nuclear structure effects, which are taken into account in such an approach . The microscopic study also yields information on properties which are difficult to extract experimentally and which are obviously different from the observables referred to previously . For instance, the non-locality of the average optical potential, its Coulomb correction and the effective nucleon-nucleon interaction, are such quantities . There are two major approaches to the microscopic optical potential. One is the nuclear-matter approach of Jeukenne et al . 2) and Brieva and Rook 3), in which they consider the optical potential in nuclear matter. They then obtain the optical potential in finite nuclei by making a local density approximation either of the potential or of the g-matrix itself. However, if the real part of the potential is mainly given by the Brueckner term and is therefore well described in such an approach, the imaginary part should be very sensitive to nuclear structure effects. It is thus of interest to have a prescription which involves explicitly the detailed description of the low-lying excited states of the target nucleus. In that sense the a-6), Bernard nuclear structure approach of the optical potential by Vinh Mau et al. and Van Giai') or Osterfeld et al. a) is complementary to that of the nuclear matter approach, since both are available in a certain range of incident energy, with some partial overlap. The correct treatment of the opening of inelastic channels according to the value of the incident energy should be particularly important for the determination of the imaginary part of the optical potential, especially at low energy . The contribution of the target nucleus described in the RPA to the real part of the low-lying collective states is only a correction to the real Hartree-Fock potential, while, in the energy range we are interested in (10-50 MeV), they yield almost all the absorptive part of the potential. This paper is mainly concerned with the microscopic determination of the imaginary potential for a° Ca(n, n)4°Ca . Therefore we do not pretend to obtain the

A. Bouyssy et al. / Nucleon-nucleus optical potential

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best real part of the optical potential, but rather to perform a study of how RPA excitations contribute either to the real part or to the imaginary part . In sect . 2 the optical potential is derived in the Green function formalism, where particular attention is paid to antisymmetrization, as well as double counting effects. The derivation has been already given in ref. s) so we prefer to concentrate on the resemblances or differences between the nuclear matter and nuclear structure approaches . Sect . 3 is devoted to the calculational ingredients, such as the effective nucleon-nucleon interaction, the single-particle states and the collective states . The approximation used to derive a local potential equivalent to the non-local one is also given briefly. Our results are discussed in sect. 4, where we try to emphasize some points of interest, as, for example, the non-locality of the potential, the energy-dependence . A comparison with other approaches is made, and a contribution of inelastic channels not taken into account here is discussed, together with differences between neutron scattering or proton scattering . 2. The optical potential 2 .1 . DERIVATION OF THE MASS OPERATOR

The mass operator M(x, x') is defined by the integral equation of the one-particle Green function dax~ da x  ~ Gio~(x~ -~ x)Gl(x~,~ (2 .1) Gio~(x, x,)11~I(x,, Gi(x, xo) = xo) xo) where Gi(x, xo) is the exact one-particle Green function for the target nucleus, and x and xo are coordinates in the four-dimensional space (x = r, t) . Gi°~ (x, x') is the Green function of a free particle . Bell and Squires') have shown that the optical potential for a nucleon scattered by a nucleus is given by V(r, r', E) _ -M(r, r', E) _ -

+~

J m

e'sc`-`~~M(x, x') d(t - t') .

(2 .2)

The mass operator can be expressed in terms of the hierarchy of one-particle G,, two-particle Gz, . . . , Green functions. This expression reduces to a function of G, and Gz only, if one assumes that all correlations are negligible, except those between a pair of particles. This approximation leads to 5) : M(x, x') = iU(x - xi)Gi(xi, x; )S(x -x')-iU(x -xl)GI(x, xi)S(xl-x') + U(x-xl)U(x' - xz)LGi(x, x')Gn(xixz ; xi xz ) +G 1(x, xz)Gn(xzxi ; xix')+Gi(x,, xz)Gtt(xxz ; xix') - Gi(x~, x')Gti(xxz ; xixz) - Gi(xz, xi)Gz(xxi ; x'xz)~

A. Bouyssy et al. / Nucleon-nucleus optical potential

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In this expression U(x -x') is the instantaneous two-nucleon interaction U(x -x') _ v (r - r')S (t - t'), the function Gz is the particle-particle (or hole-hole) Green function and G is the particle-hole propagator . The quantity M` z~( x, z') is the secondorder contribution to the mass operator, in perturbation theory, and reads M` z~(x, z')= U(x-x,)U(x'-xz)G,(xz, x,)

x ~Gi(x, x')Gi(x~, xz) - G~(x, zz)Gi(xi, z~)~ ~

(2 .4)

The subtraction of 2M`z~ in eq . (2 .3) can be understood easily . When Gz and G are approximated by their lowest order term, namely by antisymmetrized products G,G, (which means that all correlations are neglected), all terms in eq . (2.3) involving Gz and G reduce to 3M`z~. Therefore the subtraction of 2M`z~ takes care of double counting . The mass operator is a sum of three kinds of terms: terms involving one-particle propagators only, terms with particle-particle correlations and terms with particle-hole correlations . Let us call these parts M,, Mz and M3,

respectively. We write

In terms of diagrams, M(x, x') is given by fig. 1 of ref. 5) (apart from the second-order contribution) . RPA particle-hole correlations and Brueckner particle-particle correlations are known to be important. Therefore in eq . (2 .3) all propagators are replaced by those already given in fig. 2 of ref. 5) . In all terms G, is replaced by the Hartree-Fock propagator . The optical potential V(r, r', E) is then given by the Fourier transform of the mass operator eq . (2 .5) . It is interesting to analyse these different terms and the corresponding optical potentials in more detail .

2 .2 . COMPARISON BETWEEN THE DIFFERENT APPROACHES (i) The first term M, gives the Hartree-Fock approximation to the optical

potential. After straightforward calculations one can write, neglecting other terms, Vc'~(r, r~) _ (2 .6) _Ml(r, r~) , V(r, r~)where V"~(r, r')=

J

dr,v(r-r,)p(r,)S(r-r')-v(r-r')p(r, r') .

(2 .7)

In this expression p(r) and p(r, r') are the diagonal and non-diagonal Hartree-Fock densities of the target ground state. Hence, to this approximation, the optical potential is non-local and real, as long as the interaction is real and energy independent. A large number of works have dealt with such potentials by taking the exchange term into account, or not (in which case we have the Hartree potential), with different interactions .

A . Bouyssy et al. / Nucleon-nucleus optical potential

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(ü) If we keep only the first two terms, M, and MZ in eq. (2.5), neglecting particle-hole correlations, we get the Brueckner-Hartree-Fock potential:

V(r, r', E) = Vl(r, r', E) _ -(M,(r, r')+MZ (r, r', E)) .

(2.8)

Since M3 is neglected, no double counting arises in the second-order term of the perturbative expansion. Therefore there is no subtraction. Introducing the Brueckner g-matrix one gets Vi(r, r', E) _~

tt(

J

drl dr2 cp(* (rl)[8(r, ri ; r', r2; E+e,)

where and ( are the individual particle wave functions and energies, and n, is the occupation number of the state i in the ground state . This potential is non-local, complex and energy dependent. It is the leading term of the low-density expansion of the optical potential derived in nuclear matter by Hüfner and Mahaux 1°). Using the local density approximation, Jeukenne et al. z) studied the potential V, for finite nuclei . Brieva and Rook s) also calculated the optical potential in finite nuclei, by folding the matter density with a complex nucleon-nucleon effective interaction. (iü) The term M3 in eq . (2.5) adds the contributions due to intermediate excitations of RPA states of the target nucleus to the optical potential. We expect this contribution to be important. If one works with a phenomenological interaction, which has no repulsive core and is purely central, the ladder expansion of the g-matrix may be approximated by its second-order term, which is equivalent to writing tp,

E

MZ (r, r', E) = M~Z ~(r, r', E) .

(2.10)

The optical potential is therefore given by V(r, r', E)= VZ(r, r', E)=-(M(( r, r')+M3(r, r', E)-M~Z ~(r, r', E)) . In this approach the complete antisymmetrization between the incident nucleon and the target nucleons in M3 introduces a double counting to the second order, therefore one has to subtract one M~ 2~ term. The potential VZ(r, r', E) is sometimes called the nuclear structure approach to the optical potential . Indeed the particle-hole propagator contains all the information about the structure of the target nucleus. This appears clearly when the definition of the GII propagators and their expression in terms of wave functions are used to calculate M3. Let us define the RPA amplitudes as Â+N) _ (WN

I a i al

14~0) ~

(2.12)

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A. Bouyssy et al. / Nucleon nucleus opticalpotential

where ~,v is the RPA wave function of the nucleus in the state N. We thus get M3(r,r',E) = -lim ~ ~ [n;(1-n;) +nt(1-nl)][nk(1-nl)+nl(1-nk)] 8~0 N~0 ijkl A 1 nA - nA l (N) (N)r xXli Xkl LE-ea+EN-tS+E-ea-E,v+tSJ X(~oIVIy~N ) tPA)A(~PaY'Nl) I vI~O)A

(2 .13)

The index A means antisymmetrization between the intermediate particle ~1 (with energy e ) and the particle in the nucléus. ij (kl ) represent the particle-hole configuration. The sum over .h ~"a, means summation over discrete states and integration

over continuum states : Neglecting exchange terms in eq . (2 .13) and restricting the summation over A to particle states (nA = 0), the optical potential can be written as ~N~A)(~a~GNI Vl+lfo) ~ lim ~ ~ (+Go ~ vI VZA (r, r', E) = VH(r, r')+ a~o'Nt0 E-El t-ea +iS A

(2 .14)

which is the usual nuclear theory approach to the optical potential 9). As can be seen from eq . (2 .13) or eq . (2 .14), the nuclear structure potential is complex, non-local and energy dependent. Because of the energy denominators in eq . (2 .14), the contribution to the imaginary part of Vz(r, r', E) is for scattering entirely given by the second term of the r.h .s ., therefore by summation over unoccupied states A, and by the RPA states N, which have an excitation energy EN lower than the incident energy E. We therefore expect a strong energy dependence of the imaginary part of the optical potential for low incident energy . On the other side the main contribution to the real potential comes from the HartreeFock term, which is energy independent, while the contribution due to RPA correlations involves a summation over all RPA states whatever the incident energy . The real potential is therefore expected to have a smoother energy dependence than the imaginary one . In our formalism, the so-called nuclear matter and nuclear structure approaches appear simultaneously in a natural way as approximations of a more general theory . It is therefore useful to discuss their respective contributions. Up to second order in v(rl-rz), Vl(r, r', E) and Vz(r, r', E) are identical, but they take into account different kinds of correlation terms. This is clear if one writes the optical potential V(r, r', E) according to eqs. (2 .5) and (2 .8), V=

Vl

-(Ms -2111(z)) ,

(2 .15)

or according to eqs. (2 .5) and (2 .11), V = Vz-(Mz-M(z)) .

(2 .16)

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17 9

The term (M3 -2Mc2 ~) represents the effects of particle-hole or long-range correla~) tions. The term (MZ -M cz represents the effects of Brueckner or short-range correlations . When the incident energy is much larger than the energies of the collective RPA states, the difference E-EN is not sensitive to a variation of EN and one can show that M3 reduces to 2McZ ~. Therefore at high enough energy long-range correlations play essentially no role at all. Furthermore the use of a non-singular and central nuclear interaction adjusted on the low-energy spectrum of the target nucleus would not be relevant for high energies . The identification of the optical potential with the potential VZ would be by no means justified. On the other side the study of the optical potential reduced to V, is usually done with realistic interactions, so that the short-range correlations are correctly taken into account, and this approach is very appropriate at high energy. Nevertheless the neglect of long-range correlations should lead to an underestimation of the absorption when the incident energy is below 50-100 MeV, depending on the nucleus. 3. Calriilation of the n-°°Ca optical potential From now on the optical potential is identified with the potential VZ given by eq . (2 .11) . In this section we discuss our choices of v(rl - r2), the effective nucleonnucleon interaction, of the target ground-state density and of the RPA eigenstates . 3.1 . THE EFFECTIVE NUCLEON-NUCLEON INTERACTION From the many-body theory point of view, VZ (r, r', E) should be calculated in a self-consistent way. We should solve the Hartree-Fock problem with an effective interaction, then use the same interaction to calculate the RPA states and finally calculate V2 . This is a quite difficult program, mainly because of problems arising with the choice of the effective interaction . This interaction should first reproduce the individual particle spectrum . It is known that in order to get a good agreement for the Hartree-Fock states, a density dependence has to be introduced in the effective interaction . This density dependence comes, in the local density approximation, from the summation of ladder diagrams . Therefore with a density-dependent force suitably chosen for the Hartree-Fork problem, the first term in V2, namely Mi (r, r'), should reproduce quite well the potential V,(r, r', E), at least at low energy . However, one has to take care of double counting with such a force, and in that case it is better to calculate the optical potential via rather than

V(r,

r~, E)=

Vl -(M -214I cs ~) s ,

(3.1)

This approach has been used') to calculate the n? °8 Pb optical potential with the Skyrme interaction using the fact that the real part of MZ(r, r', E) is already

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A. Bouyssy et al. / Nucleon-nucleus optical potential

contained (at least partly) in the calculation of Ml (r, r') with such a densitydependent force. It would be very interesting to compare the calculations of Jeukenne et al. for the Brueckner-Hartree-Fock potential with M,, calculated with a more fundamental density-dependent force derived from the g-matrix calculated with the Reid soft-core nucleon-nucleon interaction . Such a calculation has been done with the force G-0 of Campi and Sprung ") for a n-a potential lz ), but not yet for the nucleon-nucleus case . Since at low incident energies the real part of the optical potential is mainly given by the Hartree-Fock term, we are more interested here in the imaginary potential, coming in our approach from the RPA excitations . Therefore v (r, - rz) has to be chosen so that it reproduces well the spectrum of the target nucleus in s), RPA calculations . In ref. for reasons of simplicity, the effective nucleon-nucleon interaction has zero range and is written as v(rl -rz)=-vo (a+bv, ~ Qz)S(ri - rz) ~

(3 .3)

Such a phenomenological force was used with great success in the earliest studies of low-lying particle-hole spectra in closed-shell nuclei . The parameters have been determined to reproduce, at least qualitatively, the energies of the collective states of the nucleus. The use of a finite-range force is not expected to yield fundamentally different results from that obtained with a zero-range force, but the finite range is expected to flatten and enlarge the imaginary part and therefore can lead to different results for the optical radius and the volume integral of the potential. Such an effective interaction with gaussian shape has been adjusted by Gillet and Sanderson' 3) (hereafter called GS) in order to reproduce the low-lying collective states of a°Ca . This force will be used in this paper. 1;iowever, the term M3(r, r', E) in the optical potential contains the inelastic scattering form factors and if we think of the optical potential in terms of the reaction theory, the effective interaction should also be able to reproduce the experimental form factors and cross sections . Satchler has shown' 4) that the GS force leads, in the case of p a° Ca inelastic scattering, to too low cross sections . However, he gets a nice agreement with the Reichstein-Tang (RT) interaction' s). This force was determined by fitting the triplet and singlet effective ranges and scattering lengths. Actually RT derived two forces, one (called RTA) with equal range for the triplet and singlet states, the other (RTB) with two different ranges, the long-range part of the nucleon-nucleon interaction playing a predominant role . We will restrict ourselves here to the case RTA since the other force gives very similar results. As pointed out by Satchler, the main differences obtained in the calculation of the inelastic form factors come from the direct amplitudes . The cross section corresponding to the direct term is very small in the GS case as compared to that of the exchange term, while they are of the same order of magnitude for the RT case . It is therefore instructive to

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A. Bouyssy et al. / Nucleon-nucleus optical potential

calculate the optical potential as generated by the RT force and compare it with the one given by the GS force. The effective interactions we use can be written as v(r, - rz) = - Vo[a +bvr ' vz + cT r ' Tz+d(vr ' Qz)(Tr '

Tz)~ exP

~-~rr - rz~z/N- zl (3 .4)

where the coefficients a, b, . . . and the parameters of the form factor are given in table 1 . TABLE 1

Parameters of the effective interactions [GS' 3), RTA' S )] Interaction:

GS

RTA

Vo (MeV) K (fm)

55 1 .7

72.98 1.4744

a b c d

0.15625 -0 .01875 -0 .14375 -0 .14375

0.29034 -0.05284 -0.14534 -0 .10882

a' b' c' d'

0.40625 -0 .04375 -0 .16875 -0 .04375

0.32089 -0 .05841 -0 .15091 -0 .09493

The strength V and the range p, of the gaussian form factor are given in the first two rows . Also given are the exchange mixture coefficients for the direct part (a, b, c, d) and the exchange part (a', b', c', d').

This way of writing the effective nucleon-nucleon interaction is particularly suitable when looking at the exchange matrix elements . The antisymmetrized matrix element entering into the calculation of M3 (r, r', E) reads and can be written as

(~Pi w~ tP~~Pa )A

(3 .5 )

,

(3 .6)

with v' _ -P °PTv, where the P are the permutation operators of the space spin and isospin variables. Starting from v(r r - rz) as given by eq . (3 .4) we get v'(r, - rz) in the following form, v'(rr-rz) =-VoLa'+b'Qr ' vz+c'z, ' zz +d'(Q, ' Qz)(Tr '

Tz)l exP

L-~rr - rz~z/1s 2 1, (3 .7)

where the new set of mixture coefficients, easily related to the other, is given at the end of table 1 .

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A . Bouyssy et al. / Nucleon-nucleus optical potential

The expression for the antisymmetrized matrix elements looks very much like the one obtained in the case of a zero-range effective interaction (tPi~v +v'ItP~tPe) ~

(3 .8)

However, in that case the effects coming from direct and exchange mixtures are mixed into a single radial dependence . This is not the case when finite range acts . Therefore the discussion is much more complex. A product of matrix elements enters into the calculation of M3(r, r', E), such as (tPl I v + U'P' I tP~~Pa XtPktPA ~v + v'P' ~ tp,) ,

(3 .9)

and we are left with three kinds of terms: those corresponding to the square of the direct mixture terms (a, b, c, d), those corresponding to the square of the exchange mixture terms (a', b', c', d') and the cross product. Both the real and imaginary part of the nuclear structure potential can be written schematically as a ZAoo+b 2Ao1+c 2Alo+dZA  - [aa'(Boo + Coo) + bb'(Boi + Coi) + cc'(B io + Ct o) + dd'(B i i + C~ i )] +a'ZDoo+b' Z Do~+c' ZDIO+d' Z Dii .

(3 .10)

The coefficients A,1 ~ ~ ~ (where i = spin, j = isospin) are complicated expressions containing angular momentum dependence, as well as overlap integrals between single-particle wave functions, and the multipole expansion of the gaussian form factor . The exchange terms give a much bigger contribution than the direct ones . One therefore expects, according to the values of the coefficients in the two forces, not too different results for the absorptive part of the potential . 3.2 . THE SINGLE-PARTICLE AND COLLECTIVE STATES To calculate VZ we need the RPA amplitudes and energies . We use the odd-parity states computed by GS with their effective interaction, the single-particle and single-hole states being described as harmonic oscillator states. Calculations done with the RTI give very similar, although less collective, states . The same approximation is also used for calculating the diagonal and non-diagonal ground-state densities. One should notice at this point that we have lost the self consistency. Therefore the potential corresponding to the term M, in eq . (2 .11) is no longer the Hartree-Fock potential written in eq . (2 .7). It is the first-order optical potential, or the antisymmetrized folding potential. Thus in the following, the potential V~'~(r, r') will be referred to as the first-order potential. In the calculation of M3, eq (2 .13), the RPA even-parity states also enter. They have been calculated with the same residual interaction and the same single-particle (or hole) states . Here the number of particle-hole configurations is very large. They

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183

involve a particle in the 3s-2d-lf-2p shells, for which it is impossible to determine the individual energies from the experimental spectrum of 4'Ca, as can be done for lower shells . Therefore these energies have been taken from an harmonic oscillator well, with the same parameters as in ref.' 3). This will introduce uncertainties in the corresponding contributions to VZ, but, as we shall see, they will not influence the total potential too much, or at least its general properties . Furthermore we know that the lowest 0+,1 + , 2+ states are certainly not pure lp-lh configurations, but contain quite large components of 2p-2h configurations . From the work of ref.' 6) this will induce a reduction of their contribution by an overall factor that we evaluate to be about 0.6 . Moreover only few even-parity states calculated in the RPA scheme are very collective . As an example, without taking into account 2p-2h excitations, such collective states as the lowest 2+ T = 0 cannot be obtained . They are known to give a substantial contribution to the absorptive potential. Although their inelastic scattering cross sections are smaller than that of the lowest 3- state by a factor ^-5-10, they are similar to the cross section for the lowest 5m [ref. )] . Therefore the imaginary potential will be underestimated in a calculation where only RPA lp-lh states are included . According to our derivation we should use the Hartree-Fork propagator, the propagator calculated with the potential of eq. (2 .7). The summation in eq . (2 .13) is restricted to unoccupied states in the continuum with energy ta a = E -EN > 0, the wave functions of which are first approximated by plane waves. We therefore have to check whether or not the use of the distorted wave functions generated by our real potential changes the result of the imaginary potential much . In the plane wave approximation, also used in ref. S), the summation over A in eq . (2 .14) is straightforward to perform, if antisymmetrization is neglected . In that case the imaginary part of the non-local potential involves the factor sin(k, ~r-r'~)/~r-r'~

with

kN=2m (E-E,v)l~2,

(3 .11)

m being the reduced mass, while the corresponding real part is proportional to cos (k,v~r-r'~)/~r-r'~ . This is also the result obtained in ref. 5), where a zero-range effective interaction is used . However antisymmetrization effects are not negligible and, as noticed already, the exchange terms coming from eq . (2 .13) are important. As a consequence, in the case of a finite effective nucleon-nucleon interaction, the contribution of the RPA excitations cannot be written so simply as given by eq. (3 .11) . In a second step we have calculated the eigenfunction cpa (r) as given by the solution of the Schrödinger equation with the first-order potential V"~(r, r'), or in practice its local equivalent . This is expected to be a more realistic description of s the propagation of the nucleon in the state A [ref.' )] . As said previously, the optical potential VZ is a non-local potential depending on ordinary space coordinates r and r' . It is however very useful at this stage to look at the optical potential written in terms of relative and c.m . coordinates . Let us call the contribution of

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A . Bouyssy et al. / Nucleon-nuclcus optical potential

the second-order term in eq . (2 .13) corresponding to RPA excitations VRpA . The partial wave expansion is written VR p A (r, r', E) _ ~ Vt (r, r')Pt(cos B) _ ~ Vt~(R, s)P,~(cos ®)

where R =Z(r+r') and s =r-r' . It has been shown s) that the S-wave is the strongly dominant wave in the partial wave expansion of VRpA (R, s) . Therefore the sum over l' in eq . (3 .12) reduces to the l' = 0 term and the potential does not depend on the angle between R and s. Using this property one first calculates Vi(r, r') then deduces V~(R, s) by choosing two particular angles ® = 0 and 0 = ~r. In the first case we get Vo(R,s)=~V(r,r'), t

R=Z(r+r'),

s=r-r',

(3 .13)

which for a given value of R gives values fors , 2R . In the second case Vo(R,s)=~( -)tVt(r,r'),

R=2(r-r'),

s=r+r',

(3 .14)

which gives values for s = 2R . One then obtains VR p A (R, s) quite easily for all required values of R and s. 3.3 . DERIVATION OF THE LOCAL EQUIVALENT POTENTIAL There are very few available phenomenological non-local potentials, while local optical potentials have been extensively studied. It is then of great interest to derive a local optical potential from the non-local one. A simple derivation has been proposed by Perey and Saxon' 9). The validity of z°-zz) their approximation has been tested in different cases and shown to be satisfactory, when the non-locality is small compared to the range of the potential, which is the case for n a° Ca . The local potential is defined such that it gives the same wave functions as the non-local potential. The wave functions correspônding to our optical potential should satisfy the Schrödinger equation (T + Vâ;~ (r))tp(r)+~ [V~x~ (r, r')+dV(r, r')+iW(r, r')]~p(r') dr' =E~p(r) ,

(3 .15)

where Väû and V~x~ are the direct local and exchange non-local parts of the first order term V~ 1~(r, r'). dV and W are, respectively, the real and imaginary part of VRpp . We therefore look for a local potential such that the equation (T + VL(r) + iW(r))cp (r) = E~p(r) has the same solutions as eq . (3 .15) .

(3 .6)

A . Bouyssy et al. l Nucleon-nucleus optical potential

18 5

From the work of Perey and Saxon and of Frahn z°), we write the following relations

r

z m -rk ~ . m VL = V(R,kL)=Vd ;*(R)+J e ~ ~Vexc(R,s)+dV(R,s)]ds,

(3 .17)

with

VL and WL are solutions of two coupled equations which can be decoupled by the assumption ~ WL~ « ~ VL~, so that the contribution of WL can be neglected in the local momentum of eq . (3 .19) . Eqs. (3 .17) and (3 .18) reduce to VL(R ) = V(R, (E W~(R ) = W(R,

VL)) ,

(3 .20)

(E - VL)) .

(3 .21)

The first equation will be solved by an iteration procedure. Once the potential VL is obtained, W,_, is easily calculated . 4. Results and discussion The n a° Ca potential is calculated for incident energies lying between 10 and 50 MeV. We first present the antisymmetr~zed folding model potential in subsect. 4.1 . In subsects . 4.2 and 4.3 the properties of the non-local, energy-dependent and complex term M3 of eq . (2 .13) and its local equivalent optical potential are, respectively, discussed. Subsect. 4.4 is devoted to a comparison with other approaches and to conclusions. 4 .1 . THE ANTISYMMETRIZED FOLDING MODEL POTENTIAL

Since in this paper we are mostly interested in studying the contribution of the term M3 to the optical potential, the effective interactions were chosen so that they could reproduce well either the target collective states or the inelastic scattering form factors. Such interactions are not suitable for Hartree-Fock calculations, in particular they would not give saturation properties, and are not expected to yield a good representation of the real part of the optical potential which is predominantly given by the Hartree-Fock potential. However, even if we do not pay too much attention to the detailed properties of the potential Vc'~, it is important to perform a consistent calculation of the whole potential and to study how the real part of M3 will modify the strength, radius and energy dependence of the lowest order real potential.

186

A . Bouyssy et al. / Nucleon-nucleus optical potential

The non-local potential V"'(r, r') is calculated with eq . (2 .7) and expressed in terms of R and s coordinates. The mixed density p (r, r') in the exchange term can easily be calculated when harmonic oscillator wave functions are used, and in that case does not depend on the angle between R and s [ref . zs )] . The local equivalent potential is calculated with eq . (3 .17), where the corrective term d V is put equal to zero . Once solved, eq . (2 .30) leads to a local but energy-dependent potential.

The potentials corresponding to GS and RT interactions are shown in fig. 1 + for an incident energy E = 10 MeV with the neutron phenomenological potentials of za) Engelbrecht and Fiedeldey and Rapaport et al . zs) . The potential labelled RT is deeper than the one corresponding to the GS interaction, a property which comes from the very different exchange mixture coefficients of the two forces . The exchange term (see table 1) is very similar in both cases, while the direct term is much smaller for the GS interaction . It implies that the energy dependence will be quite different for both interactions, as shown by looking at the volume integral per nucleon of a real potential V(r, E) defined as

Fig . 1 . Comparison of the first-order antisymmetrized folding potential calculated at 10 MeV incident energy with two different effective interactions (GS and RTE, together with two empirical real pôtentials of Engelbrecht and Fiedeldey [EF, ref . Z4 )] and Rapaport et al . [RICF, ref . zs), set B] . ' In this figure, and in all others throughout this paper, the sign has been changed, so positive values rnrrespond to an attractive contribution.

A . Bouyssy et al . / Nucleon-nucleus optical potential

which can be written at low incident energy as Jv(E)/A =Jv(E ~ 0)/A(1-xE) .

187

(4 .2)

In the case of harmonic oscillator single-particle densities and a gaussian effective interaction, as defined previously, it can be shown that the volume integral per nucleon is written approximately as Jv(E)/A= VoaTr3/z~s[1+(~~)-3(a'/a) exp (-2m(E+ V)/4aft2)],

(4 .3)

where a = 1/~,t, Z + 1/4b 2, b being the harmonic oscillator parameter (b = 2.08 fm here). V is some average value of the real potential (we can take V = 30 MeV) . Using the numerical coefficients of the two forces, we find from eq . (4 .3) that -s -s x = 7 x 10 MeV-t for RTI and x =14 x 10 MeV-t for GSI, for the energy dependence of Jv/A . The exact calculation is shown in fig. 2 where a comparison can easily be made with phenomenological potentials . Up to about 20 MeV the energy dependence is linear, then it becomes smoother due to the decrease of the exchange terms. At low energies the exact slope is smoother than implied by the estimate of eq . (4 .3) and is quite similar to that of phenomenological potentials . The use of a harmonic oscillator density and a gaussian interaction is responsible for the very diffuse surface of the calculated potentials . If we define the r.m .s. radius for the potential V(r, E) as (rZ)

vs = [~

V(r, E)rZ dr/

J

V(r, E) dr,

t/z

(4 .4)

it can be shown to be approximately written as (r2)vZ={3b2+ZEc2[1+(Fr../«)-3(a'/a)exp(-2m(E+V)/4a>~t2) ]-t }' /z .

0

10

20

30

~0 E

50

(4 .5)

60

IMeV)

Fig . 2 . Volume integral per nucleon of the same quantities as in fig . 1, plotted as a function of the incident energy .

188

A. Bouyssy et al. / Nucleon-nucleus optical potential

As a consequence, the introduction of exchange terms into the effective interaction reduces the difference in the r.m .s . radii calculated with the direct part of either the GS or the RT interaction . The energy dependence is obviously very small, since the r.m .s. radius is mainly given by the r.m .s . radius of the harmonic oscillator density. However results obtained on either the volume integral per nucleon or the r.m .s . radius could be modified by the inclusion of the correction dV coming from the term M3 (r, r', E), which we consider next . 4 .2 . CONTRIBUTION OF COLLECTIVE EXCTTATIONS

The term M3 of eq. (2 .13) is calculated in terms of R and s coordinates as explained in sect . 3. We first discuss our results in the plane wave approximation. We shall see later on that this approximation is quite good for nucleons. The non-local form factors of dV and Ware simple functions of s with different shapes, mainly for small values of s, while their behaviours in terms of the R-variable are very similar. The introduction of a finite-range effective interaction does not change the qualitative features of the potential, compared to our previous results obtained with a zero-range interaction 5). The relative contribution of RPA states and the localization of the contribution of each type of state are qualitatively the same . For example, the most collective states, the 3 - and 5- lowest states, give a strong contribution peaked at the surface of the nucleus, as for the 2 + and 4+ states, while the 0+, 1+ and 1- states lead to volume effects. However, the potentials are very different : the finite range of the force smooths all the narrow peaks obtained in ref. s), and at the same time lowers the maximum intensity of each term. The potential has therefore a smoother variation in terms of R but is weaker at distances where the potential of ref. s) had narrow peaks. Furthermore it is spread further outside the nuclear radius . (i ) Real part. In fig. 3, d V(r, r', E) is presented as a function of R for different values of the incident energy between 10 and 50 MeV and s = 0.5 fm . The shoulder at very short distances comes mainly from the contribution of the 0+ T = 0 and T =1 states at EN =13.55 MeV, 19 .10 MeV and 25 .17 MeV. Therefore it depends strongly on the renormalization factor evaluated in sect . 3 to be approximately 0.6 . However that part of d V will have little effect on the scattering cross sections . The interesting feature is that d V will correct the real potential V"' in all the R-domain by increasing the strength . The absolute value ~d V~ first increases with energy, then at about 20 MeV starts to decrease . The energy dependence of ~d V~ has several sources and can be easily understood . We work in a given basis of RPA states corresponding to excitation energies of 1 and 2ftw. When dV is calculated with eq . (2.13) the sum over N runs over all excited states of this basis. For the direct term the states with energy E,v smaller than the incident energy E induce, in the z case of the plane wave approximation, a factor cos k,vs (knt=2m(E - E,v)/tt ) in the corresponding contribution to d V ; the states with EN > E induce a factor

A. Bouyssy et al. / Nucleon-nucleus optical potential

Odd states " .6 Even

states

"\

___

10MeV

-

30hkV

~-._ 50MeV \

Odd states " 1 . Even

\

states

- " - 50MeV

ti

0 .5

Fig. 3. Contribution to the real part of the collective excitations of the target nucleus for different values of the incident energy . The renormalization factor for the even-parity states (see text) is equal to 0.6, except for the 50 MeV incident energy where the real part is also drawn, without any renormalization. exp (-XNS) with Xnr = 2m (EN -E)/# Z . For the exchange terms the energy dependence does not factorize and is weaker, but follows approximately that of the direct term . When the incident energy increases, a particular state N gives a larger contribution to the real part when it is an open channel, than it gave when it was a closed one. Therefore when more and more channels are open, ~d V~ has the tendency to increase, this effect beimg the largest in the energy domain corresponding to the collective states, but it diminishes for higher energies . On the other hand, because of the factors exp (-XNS) and cos (kNS), a given state N will give a contribution which behaves differently according to the value of its energy EN compared to that of the incident energy E. Thus the overall-effect will be an increase of ~d V~ in the low-energy domain, where all collective states are not yet open channels, but a decrease for higher energies . At these energies however, the truncation of the p-h basis, restricting to lftw and 2ttr~ p-h configurations, leads certainly to an underestimation of ~d V~ and can affect the energy dependence of the potential. In the potentials drawn in fig. 3 the same renormalization factor has been used at all energies, although it should vary with energy . Indeed at 20 MeV only a few

19 0

A . Bouyssy et al. / Nucleon-nucleus optical potential

positive-parity states give a noticeable contribution, and a mixture of 2p-2h configurations in their wave function brings a reduction factor in the potential. When the energy increases, all RPA states contribute to the term with cos k,vs . When the incident energy is high enough, k,v is practically constant and therefore the redistribution of lp-lh states over the whole spectrum does not play any role and no renormalization factor should be included . The potential calculated at 50 MeV incident energy without renormalization of the 0+, 1+ and 2+ states is shown in fig. 3. Except at very short distances, the effect is relatively small, the overall result being that ~d V~ increases slightly . We now turn to the s-dependence of the potential . The non-local form factor is shown in fig. 4 for different energies at R = 0.5 fm and 3 .5 fm in the upper part of the figure . It can be parametrized as a Yukawa factor times a polynomial, in the following way : dV(R, s, E)=dVo(R, E) exp (-ßs)/ßs(1-txs2) .

(4 .6)

Fig . 4 . Non-local real form factor plotted for different incident energies as a function of the relative coordinate s for two choices of the c .m. coordinate (R = 0 .5 fm in the lower part of the figure, R = 3 .5 fm in the upper part) .

A . Bouyssy et al. / Nucleon-nucleus optical potential

19 1

The polynomial is necessary to reproduce the change of sign for large values of s. From the figures it is clear that both parameters a and ß vary with energy . The non-locality range ß always decreases with increasing energy while the parameter a behaves differently according to the values of R. These two effects will modify the energy dependence of the first-order local equivalent potential, calculated above, together with that implied by the Fourier transform of the non-local potential . In the phenomenological analysis of neutron scattering on a° Ca at low energy, Engelbrecht and Fiedeldey have proposed a Yukawa form factor for both real and imaginary potentials, but with energy-independent coefficients . However, our calculation is only concerned here with a corrective term to be added to the antisymmetrized folding model potential V~ tl (R, s), and the parametrization of eq . (4 .6) cannot therefore be compared directly to that of ref . 2a) . (ü) Imaginary part. Fig. 5 shows the imaginary potential W(R, s) given for s = 0 fm as a function of R, for several energies between 10 and 50 MeV, in the case of the GS interaction (solid line). As expected the shape is similar to that of d V but the variation with energy is different. Indeed the strength of ~ W~ is strongly increasing with energy . This behaviour comes firstly from the opening of new inelastic channels, an effect which is very important for low energies, but saturates

3

G

R

(fml

5

Fig . 5 . Comparison between the non-local imaginary form factors (for different energies) calculated with either GS (solid line) or RT (dashed line) interactions plotted as a function of R for s = 0 . The renormalization factor is everywhere equal to 0.6 .

192

A. Bouyssy et al . l Nucleon-nucleus optical potential

at about 30 MeV, and secondly because of the factor sin knts (again for exchange terms the energy-dependent term is more complicated but leads to the same dependence although slightly smoothed) . The potential calculated with the RT interaction is also shown in fig. 5 (dashed line). Both effective interactions lead to very similar results with a slightly stronger attraction in the case of the RTI. The preliminary results presented at the Hamburg conference e) exhibited a much stronger absorption in the latter case . Unfortunately this was due to using the wrong mixture coefficients in the numerical code . The similarity of the absorptive potentials calculated with either GSI or RTI seems at first sight contradictory to the study of proton inelastic scattering on a°Ca ta) by Satchler . He found that the cross sections for the excitations of the lowest 3- and 5- collective states are too small when GSI is used while they agree quite well with the data in the case of RTI. The analysis of our results term by term confirms this fact, as shown in detail in table 2, where the absorptive non-local potentials W(R, s) are given for s = 0 and R = 4 fm, in the case of S = T = 0 for a 30 MeV incident nucleon. Obviously the exchange matrix elements are much more important than the direct ones, although the use of a distorted propagator (see the discussion at the end of this section) instead of a free one, has the tendency to reduce their contributions. Our results are very comparable to those of Satchler, TABLE 2 Absorptive non-local potential W(R, s) for 30 MeV incident nucleons at s =0 fm and R =4 fm in the case of S = T = 0 states

Plane wave D 3 - GSI RTI ratio RT/GS 5- GSI RTI ratio RT/GS

Distorted wave

D+E (D+E)/D

D

D+E

(D+E)/D

lowest state all 3- states lowest state all 3- states lowest state all 3- states

0.006 0.008 0.026 0.034 4.3 4.2

0.076 0 .105 0.105 0.104 1 .4 1 .3

12 .6 13 .1 4.0 4.1

0.008 0.011 0.038 0.052 4.8 (5 .7) 4 .7

0.051 0.071 0.091 0.126 1 .8 (2 .0) 1.8

6.4 (7 .8) 6.4 2.6 (2 .8) 2.4

lowest state all 5- states lowest state all S- states lowest state all 5- states

0.001 0.001 0.006 0.008 6 .0 6.0

0 .050 0.082 0.050 0.080 1 .0 1.0

50 .0 63 .0 8.6 10 .0

0.001 0.002 0.008 0.013 7 .0 (6 .0) 6.5

0.031 0.051 0.037 0.064 1.2 (1 .3) 1.2

25 .8 (36) 29.1 4.4 (7 .7) 5 .0

A comparison is made between the direct matrix elements (D) the exchange ones (E) and the ratio (D+E)/D for two effective interactions (GS and RT). The values in parentheses correspond to the integrated cross sections calculated by Satchler ") . The contributions of the lowest 3- (or 5-) states are given as well as that of the sum over all the states with the same spin and parity . A rnmparison is also shown between the nucleon propagator either approximated by the free one or distorted by the real potential.

A . Bouyssy et al. / Nucleon-nucleus optical potential

19 3

but the summation over all the 3 - (or 5-) states lowers slightly the ratio of absorptive potentials calculated with RT or GS interactions compared to the value it has for the lowest state. Furthermore it turns out that the sum of all contributions of other spin and parity states does not exhibit the expected enhancement factor with the RT interaction. Between 10 and 30 MeV our potentials show oscillations in the interior, but vary, more smoothly, at the surface . Above 30 MeV the absorptive potential is mainly of volume type with no significant surface peak. This peak at the surface comes mainly from the excitation of 3- and 5- collective states . At low incident energy the RPA correlations play an important role, enhancing the absorption by a factor of two at 10 MeV as compared to the Tamm-Dancoff approximation. When the energy increases this effect has the tendency to disappear, a result shown in fig. 6. When the energy is much higher than that of all the collective states the absorptive potentials calculated with either RPA or TDA wave functions or pure 1p-lh wave functions become identical. Long-range correlations are indeed unimportant when the incident energy is much larger than that of the collective states . This appears above 50 MeV for light and medium nuclei . Therefore the derivation of the optical potential from the g-matrix which does not take into account explicitly the collective inelastic channels, involve them implicitly when the incident energy is high enough .

T

RPA approxiertion

O 3

0.50

0 .25

O .Oô Fig. 6. Non-local imaginary form factor at s = 0 fm calculated in the Tamm-Dancoff approximation and in the RPA for low incident energy, in the case of the GS interaction.

19 4

A. Bouyssy et al . l Nucleon-nucleus optical potential

Fig. 7. Non-local absorptive potential W(R, s, E) plotted as a function of s, for different values of the radius R, for E =15 or 30 MeV . The non-locality range bis indicated by an arrow.

The non-local form factor W(R, s, E) is plotted in fig. 7 for different values of the radius, R, at E = 15 and 30 MeV, respectively. Inside the nucleus it changes its sign for a value of s which increases with R ; at the nuclear surface the form factor becomes gaussian . The necessity of having such a non-local form factor inside the nucleus was suggested by Engelbrecht and Fiedeldey Za), who pointed out that the gaussian form factor used by Perey and Buck ze) over the whole nuclear volume was unphysical, since it leads to a decrease of the volume absorption with increasing energy. Our non-local potential can be parametrized as W(R, s, E) = Wn(R, E)g(s) ,

(4.7)

A . Bouyssy et al. / Nucleon-nucleus optical potential

19 5

where S(s) = exp ( -sZ/62)[1- a (R)s 2 /b 2 ] .

(4 .8)

The function g(s) is very similar to that of ref. z° ), where, however, a constant coefficient a (of the order of unity) was used for the volume term, while a = 0 was assumed for the surface term, therefore leading in that case to a pure gaussian shape . The parameter b is R-independent and fitted to the gaussian for large values of R. The resulting a-parameter has a strong R-dependence, mainly at low energy, as shown in fig. 8. Both coefficients a(R) and b vary with energy . The linear energy

R lfml

Fig . 8. Dimensionless parameter a(R) defined in eq . (4 .8) plotted as a function of the radius R for different incident energies .

dependence of b and the variation of a(R) are shown in fig. 9. The values of these coefficients are compatible with those obtained in ref . za) . All the results presented above were obtained with the one-particle propagator approximated by the free one. Fig. 10 shows the imaginary potential calculated

196

A . Bouyssy et al. l Nucleon-nucleus optical potential 2 .5 ~_ 2 .0 a

4

3

ö

2

R

0

0

,_

. 3

10

20

, _ _ ~4 _,~~. 30

40

50

60

EIMeV)

Fig. 9 . Energy dependence of the coefficients b and a(R) defined in eq. (4 .8) .

a

1

2

3

G

5

6

If~nl Fig . 10 . Influence of the choice of the one-particle propagator on the absorptive part of the optical potential plotted as a function of R for s = 0 at E = 30 MeV for GS and RT effective interactions . R

A. Bouyssy et al. l Nucleon-nucleus optical potential

197

Vc'~ . We get with the optical propagator distorted by the first-order real potential slightly more volume absorption with a shift of the surface towards smaller radius . The use of a distorted propagator increases the difference between GS and RT interactions at the surface, where the most important contribution comes from the collective 3- and 5- states, as could already be seen from table 2 . The effect of the distortion is, however, quite small, a result consistent with the work of Osterfeld et al. s), and the previous discussion is still valid. Obviously the coefficients a (R ) and b entering in the parametrization of the non-local form factor eq . (4 .8) would change a little if an optical propagator had been used . 4.3 . THE LOCAL EQUIVALENT POTENTIAL The local approximation of eqs. (3 .20) and (3 .21) is used to derive dV,,(r) and W,(r), the local terms equivalent to dV(R, s) and W(R, s), respectively. The strength of the imaginary potential being less than 10% of the real one, it will be neglected in the definition eq . (3 .19) of the local momentum, which is therefore assumed to be purely real and given by V"~+dVL. (i) The real part. The quantity dVL(r) is shown in fig. (11) for several energies between 10 and 50 MeV, in the case of the GS interaction . All curves have been calculated with or without the renormalization of the factor accordingly to our discussion of sect . 3. dV~(r) is attractive and peaked at the surface for low energies (up to about 30 MeV), while it has a volume shape above 30 MeV. It increases the strength of the real potential V~'~(r) by about 10% at low energy, as is clear from a comparison of fig. 11 with fig. 1, but decreases above 30 MeV. This is an interesting z') result which has to be put together with the work of Dover and Van Giai . In their analysis of neutron-nucleus scattering at low energies with the Hartree-Fock potential they have shown that the strength of this potential has to be multiplied by 1 .1 . Even if our potential cannot be directly compared to a self-consistent HF potential, we can hope that the relative values of V"~ and d V~, calculated with the same assumptions, are more or less independent of the details of the calculations, the most important being the internal consistency of the calculation . Therefore our result could give an explanation of the 10% enhancement of the HF potential needed to get agreement with the data . In fig. 12 are presented the volume integral per nucleon and the r.m .s . radius of the real potential calculated with the GSI as functions of incident energy . The volume integral is adequately parametrized up to 30 MeV as Jv/A = -460(1-7 x 10 -3 E) MeV ~ fm 3 ,

(4 .31)

while above 30 MeV one needs a positive EZ term in the parametrization . The agreement with the phenomenological potential of ref. za) is improved, although the calculated values are too low at high energies . On the other hand the r.m .s . radius is independent of energy : the variation with energy of the structure ofd V~

19 8

A. Bouyssy er al. / Nucleon-nucleus optical potential

Fig. 11 . Corrective part of the local equivalent real potential calculated with the GS interaction for different energies . Up to 30 MeV the renormalization factor of even-parity states is 0.6, while for 40 and 50 MeV there is no renormalization.

is not strong enough to modify the behaviour of the radius of the total potential. This result is not incompatible with the phenomenological analysis of nucleon zs) scattering . For example this was the conclusion of Van Oers for p a°Ca scattering up to about 100 MeV. (ü) The absorptive potential . The imaginary potential W~(r) is shown in fig. 13 . The curves correspond to a renormalization factor of 0.6 up to 30 MeV and 1.0 above. The separation between energies below and above 30 MeV is somewhat arbitrary, as noticed previously, but at 30 MeV, going from 0.6 to 1 .0 changes the calculated potential by less than 10%,because at that energy the 1#w states are still the most important. At 10 MeV the potential has a pure surface part while we get more and more absorption in the interior when the incident energy increases. We do not get a conventional shape with a typical Saxon-Woods volume part and a gaussian surface part, and it is not possible to analyse our potential in terms of such form factors. Nevertheless it seems clear that any parametrization of our

A. Bouyssy et al. / Nucleon-nucleus optical potential

199

4,5 __ .__

v111,nv~ RKFB

- - RKFA

w N

4 .0

> V 3.5 __ x __

600

___*--- v 111 " ev~

E w

RK

â 500

a

V 111

400

300 200 0

10

20

30

40

E

50

IMeVI

60

Fig . 12 . Volume integrals per nucleon and r .m .s. radii of the real potential of this calculation compared to two global phenomenological approaches of Rapaport et al . ZS ) denoted by RKF and RKF B .

potentials would imply energy-dependent geometrical parameters, such as for radii and diffuseness . The comparison with empirical potentials is not easy, since they are not uniquely determined from data analysis . However a comparison with the data can be made from a more global point of view by looking at the volume integral per nucleon of the absorptive potential J,~./A defined analogously to eq . (4 .1). This is shown in fig. 14, where our results Zs) are compared with empirical values of .Van Oers for protons between 9.6 MeV z9) for neutrons at 11, 20 and 26 MeV. It is perhaps and 50 MeV and Rapaport et al. easier to study fig. 15, where we only plot, for an incident energy between 9.6 and 26 MeV, empirical proton and neutron volume integrals, corresponding to the same family of potentials where the geometrical parameters are fixed and the dynamical ones are adjustable . It seems quite clear from the data that for energies lower than about E =13 .5 MeV, the imaginary potential is very different for neutrons or

200

A . Bouyssy et al. / Nucleon-nucleus optical potential

1

5 6 r Ifm I Fig. 13 . Local imaginary potential calculated with the GS interaction for different incident energies .

150

E

2

3

"

Van Oars Iprotonl

o

Rapaport at al . (neutronl

o

Rapaport et al . (neutranl

"

This calculation p

10U

"

g" "" ""

c v "

F

50

0

"

4

0

10

20

30

GO

50 E IMeVI Fig. 14 . Comparison of the volume integrals per nucleon of the empirical absorptive potentials of Van zv) for neutrons, together with our results . Oers Za ) for protons and Rapaport et al.

A. Bouyssy et al. / Nucleon-nucleus optical potential

20.1

Fig. 15 . Same as fig. 14 but for incident energy in the region 10-26 MeV. The analysis of Rapaport er al. is done with the same geometrical parameters as Van Oers. protons. These neutron-proton differences have been studied theoretically by Mahaux et al. z) in nuclear matter and Osterfeld et al. e) in finite nuclei, and zs) empirically by Rapaport et al. . In both studies the results exhibit a difference up to quite large energies (25-30 MeV) . Obviously the dispersion of empirical values makes the determination of the energy above which the neutron-proton difference disappears ambiguous . Rapaport et a1. 25) were guided by simplicity, looking for a linear energy dependence of dlw/A, the difference between neutron and proton values, the parameters of which were adjusted at low energies . However, we think that above 14-15 MeV it is quite difficult to see a clear difference and this remark leads to a discussion of our results which brings some light on the ability of our approach to describe the effective nucleon-nucleus potential. Indeed we can make the following remarks. (a) We have worked throughout this paper with nuclear interactions only, neglecting the Coulomb potential at every stage of the calculation : in the matrix elements of nucleon-nucleon interactions ; in the nucleon propagator ; or in the local momentum of eq . (3 .19) used for the derivation of the local equivalent absorptive potential W,(r, E), even if it is taken into account in the RPA wave functions which are a mixture of T = 0 and T =1 states . Therefore as far as these points are concerned, our calculated potential is more relevant for neutron than for proton scattering. Nevertheless the Coulomb potential is small compared to the nuclear potential, so that its inclusion would not change the matrix elements too much . This has been

20 2

A . Bouyssy et al. / Nucleon-nucleus optical potential

verified recently s° ). Indeed, it was shown at the end of subsect. 4.2, that the use of a free nucleon propagator instead of the exact distorted one gives only small modifications for the optical potential, a result shown also by Osterfeld er al. 8). It would be still truer for the average Coulomb field. Finally the change in the local momentum corresponding to the inclusion of the Coulomb potential can be evaluated simply and shown to increase the local potential by a few percent. (b) We have introduced in our calculation the contribution of charge exchange channels on the average, by summing in intermediate states over the third component of the isospins of the nucleon and of the nucleus and neglecting the neutron-proton energy difference . This enhances the contribution of T =1 states, which contribute mostly to the interior of the potential and at high enough incident s energy (E > 20 MeV) [ref . )] . Our approximation of the energy neglects the difference between the thresholds of (p, n) and (n, p) reactions, therefore leading to a spurious contribution for the proton imaginary potential . However, the chargeexchange channels, which should not contribute, correspond to the lowest (mainly 4-) T= 1 states ; the other states correspond to an energy higher than that of the (p, n) threshold located at 15 .1 MeV. These lower states give a small contribution in our calculation and taking them into account would not greatly affect the total absorption . In addition to this threshold effect, for a given J~ state, the contribution of (p, n) exchange reactions will have a slightly larger non-locality range than that of (n, p) reactions, because of the attractive Coulomb energy present in the energy denominator. It will therefore have the tendency to decrease the strength of the corresponding local equivalent contribution, but this effect will certainly be largely compensated by the Coulomb part of the local momentum, which acts in the opposite direction and over the total contribution of inelastic and charge-exchange channels . This qualitative argument can be simply understood if one assumes a gaussian non-local form factor where the s-dependence is given by exp (-sZ/b~,P), '- z z with k zP = k z -2m Vc/ftz , which leads in the local potential to a term e xp (-4kn,Pbn,P) V~ being the Coulomb energy . As a conclusion we think that the charge-exchange reaction channels are unable to explain the empirical difference between neutron and proton absorptive potentials below 15 MeV. As a consequence we have some difficulty in understanding the result of ref. s° ) concerning the comparison of the quantity Wv~oto~(r, E) Wneutron(r, E) at 17 .7 MeV and 25 MeV, as well as WP - W, for short and large distances at a given energy . (c) Going further into the comparison of empirical and calculated potentials brings an interesting light to the understanding of the absorption . Coming back to fig. 15 it seems clear to us that in the case of neutrons all points are compatible with a very smooth variation of J./A between 11 and 26 MeV (indicated by a dashed line), while the situation is completely different for protons. Up to about 12 .5 MeV incident proton energy the variation is also smooth, then between 12 .5 and 14 MeV there is a sudden enhancement of the slope . At higher energies we

A . Bouyssy et al. l Nucleon-nucleus optical potential

20 3

come back to a smooth variation with no evident difference between neutron and proton potentials . The full lines in the figure are drawn only to serve as a guide for the eye . It is interesting to remark that the strong enhancement of JwlA for protons appears in the energy domain of the opening of the (p, d) channel arising at 13 .4 MeV in the case of °°Ca, while the (n, d) channel opens at 6 .1 MeV, much below the starting neutron energy of Rapaport et al. We believe that the difference around 10 MeV between the neutron and proton potentials on one hand and the neutron potential and our calculated potential on the other is certainly due to the contribution of (n, d) channels which are not included in our formalism ; there is no equivalent channel at this energy for the proton potential. Following this statement our calculated potential should reproduce the proton-nucleus empirical potential at around 10 MeV. As can be seen in fig. 15, the agreement is quite satisfactory and could be even better if the true proton local momentum were used in the derivation of the local potential. Actually, assuming a non-local form factor with a non-locality range b = 2 fm, our potential would be multiplied by a factor exp(mb Z V~/2~Z), which corresponds to about 5% enhancement for J,,,/A . One should also notice that our results have the same variation with energy as the neutron empirical potential and also the same variation as the proton one above 15 MeV. One also notices that the difference (JH,/A) ° - (Jw/A) P at 11 MeV is about equal to the difference between (Jw/A) .P and the calculated potentials for energies above 15 MeV. This is a further argument in favour of our interpretation, the pick-up cross section being practically constant in this energy domain . We believe that the much too small absorption obtained in all calculations including only inelastic and charge-exchange channels, whatever the choices of effective nucleonnucleon interaction and nuclear wave functions, is due to the fact that (p, d) or (n, d) channels are not taken into account. For completeness, since the geometrical shape of our absorptive potential is quite different from that of phenomenological analyses, we compare in fig. 16 the shape of our potential calculated at E =10 MeV in the case of the GS interaction to three empirical potentials determined with completely free geometrical and dynamical za .z9.si) . parameters either for neutrons or for protons Our calculated potential is more comparable to the proton potential of Van Oers, peaking at about the same radius (about 4.1 fm), but is, however, broader. 4 .4 . CONCLUSIONS

Together with our conclusions, we present here a discussion of our results compared to other microscopic calculations of optical potentials . The recent studies are usually classified into two groups : the nuclear matter approach and the nuclear structure approach . The former was first proposed by Hüfner and Mahaux'°) and developed by Jeukenne et al. z) (JLM) and more recently by Brieva and Rook 3) (BR) . Both groups calculate the two-body g-matrix in nuclear matter with similar

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A . Bouyssy et al. / Nucleon-nucleus optical potential

Fig. 16 . Comparison between the absorptive potential calculated at 10 MeV with the GS interaction (solid line) and different empirical potentials . The dashed line corresponds to the free parameter search of Rapaport et al. a9), the dash~otted line to the free parameter search of Ferrer et al. s'), both for neutron incident energy E =11 MeV . The last curve corresponds to the potential of Van Oers ze) for proton energy E=9 .6 MeV .

nucleon-nucleon forces, but they differ in the last step . JLM derive the optical model potential in finite nuclei assuming that, at a distance r where the matter density is p(r), it is identical to the nuclear matter optical potential at kF = (2-rr2p(r))'~3 . BR first determine at each energy the effective g-matrix in finite nuclei ss) by a method similar to that of Sprung and Banerjee . They then calculate the antisymmetrized folding model potential . Surprisingly the results of the two calculations show some discrepancies, but the reason why both methods give different potentials is not clear to us . For further information on these discrepancies we refer the reader to the discussion given by Mahaux'3) at the Hamburg conference . In the nuclear structure approach used in refs . °~) and more recently by Bernard and Van Giai') and Osterfeld et al. 8), the optical potential is calculated directly in finite nuclei . Nuclear structure effects, expected to be important particularly at low incident energies, are explicitly taken into account, in the same way that those coming from excitations of the target are described in the RPA. But effects coming from the inclusion of intermediate states where the incident particle is captured into a bound state a`), as well as states where a particle of the target is ejected or is picked up by the incident particle, are neglected. One assumes in this approach

A . Bouyssy et al. l Nucleon-nucleus optical potential

205

that, in the energy domain where we are working (10-50 MeV), direct inelastic processes contribute for a large part to the strong absorption experimentally observed in elastic scattering, if strongly collective states or giant resonances can be excited. These inelastic channels contribute also to some extent to the real part of the optical potential . In this paper we were mainly interested in the contribution coming from excitations of the RPA states of the target nucleus which, in our approach, generate the whole imaginary part of the potential. The real part, which at lowest order should be calculated similarly to the Hartree-Fock or the Brueckner-Hartree-Fock term, is here calculated in the antisymmetrized folding model. Therefore we do not expect to reproduce quantitatively the real part of empirical potentials . Nevertheless average properties have been studied and we have plotted in fig. 17 the volume

E

V

4 .2

0 0 C

3 .8 -

0

0

b

c

c

0

o BR cJLM + This calculation

700 o BR o JLM . This calculation

600

T

E

a

400

a

C

b

300 200 ' 0

10

20

30

40

E

50 (MeVI

' 60

Fig. 17 . Comparison of three microscopic determinations of the volume integral per nucleon and r.m .s . radius of the real potential as a function of incident energy in the case of a °° Ca target . The open circles correspond to Brieva and Rook (BR), the open triangles to Jeukenne et al . (JLM) and the plus signs to our calculation. See text for more explanations .

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A. Bouyssy et al . l Nucleon-nucleus optical potential

integral per nucleon and the r.m .s . radius of our total real potential, V"'+dVL, as functions of the incident energy, together with the values obtained by BR for neutron scattering at low energy and by JLM. However, since the values of JLM are given for protons, we have, as in ref. zs), subtracted a Coulomb correction, from the absolute value of their volume integral per nucleon which they estimated to be about 20 MeV ~ fm' for 35 MeV protons on a° Ca . Our values of Jv/A are definitely below those obtained in the JLM approach, even at low incident energy, but are quite comparable to those of BR. On the contrary our calculated radii are comparable to the radii obtained by JLM, after correcting for the finiteness of the range of the effective interaction in a phenomenological way, but are smaller than those found by BR . We do not get any energy dependence of the radius contrary to what is obtained either by JLM or by BR, although in the latter case the energy domain is too small to draw any definite conclusion . The inputs of the two studies are very different and a discussion of their respective results would not be easy . In this respect a more direct comparison of the results obtained, for the real potential, by JLM or BR and Bernard and Van Giai') would be of great interest since the former use a calculated g-matrix, while the latter use a phenomenological densitydependent interaction, which simulates the g-matrix and is adjusted to the properties of nucleon bound states . Concerning the absorptive part of the optical potential, fig. 18 shows the comz.3) for the volume parison between our calculated values and those obtained in refs . integral per nucleon and r.m .s . radius . Th e results of BR are for neutron scattering, those of JLM are for proton scattering . A simple look at the figure confirms the large discrepancy obtained, even at low energy, between the two approaches, a result already noticed many times in the literature . It also shows the significant discrepancy, at least above 10 MeV, between JLM and BR . Comparison between our absorptive potential and the empirical ones seems to show that our too small absorption for neutrons at all energies between 10 and 50 MeV and for protons above 13 MeV is due to the pick-up channels which are not included in our calculation . The importance of these channels is confirmed by the work of Coulter 3s) and indirectly by Mackintosch et al . 36 " 37), who suggest that the and Satchler pick-up contribution is responsible for the angular momentum dependence of their optical potentials, which is necessary in order to reproduce the fine structure of the elastic scattering cross sections at low energy . Concerning the nuclear matter potential it is rather unexpected that the absorption be so strong at low energy . This could come from the absence of an energy gap between the single-particle 33) energies of the occupied and unoccupied orbits . As a consequence of this, there is no threshold effect and the absorption at low energy is too large compared to what it should be . This enhancement due to the approximation can compensate the neglect of collectivity which, from our calculation, is shown to be important. On the other hand this approach includes the contribution of channels with one nucleon ejected from the nucleus and could simulate to some extent the contribution

207

A. Bouyssy et al. l Nucleon-nucleus optical potential

6. o BR O JLM

E 0+o

N

^3 N`

V

O

4.

a

II

O +

O

i

+ This calculation

15

O

O

10 3

O

+This calculation O

o BR O JLM

20 m E w

0

Ob

0

O

O

0

0

50 ()

0

i 10

+

+

+ 20

30

40

E

50 (MeVI

60

Fig. 18 . Same as in fig . 17, but for the absorptive part of the optical potential .

of (p, d) or (n, d) channels, which are absent in our calculation . Included also in the approach of JLM or BR is the spurious channel corresponding to the c.m . motion . In the nuclear structure approach the spurious state is known to be the lowest 1 - T = 0 state, which is eliminated from the summation over target states . Its contribution is known to be important s) at low energy and leads to a nonnegligible part of Jt,~/A . It has also been suggested very recently 3g ~ 39 ), that a proper treatment of the non-locality of the nuclear optical potential resolves much of the apparent discrepancy between the nuclear matter approach quantities and empirical ones . A local equivalent potential defined more suitably than in eqs. (3 .20) and (3 .21) should be used, as has already been done correctly in ref.'), thus leading to a reduction of the local absorptive potential by a quantity which is related to the effective mass induced by the non-locality of the original real potential. If we make this correction our potential should then be multiplied by a factor 0 .85 at the surface. This has the tendency to accentuate the discrepancy between our absorption and the empirical one, but this factor would be more or less compensated

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A. Bouyssy et al. / Nucleon-nucleus optical potcntial

for by the contribution of the lowest 2+ T = 0 state, which is not included in our calculation, since it is not described by the RPA. The discrepancy found between the values obtained with either the nuclear matter or the nuclear structure approaches and the empirical quantities is confirmed when looking at elastic scattering cross sections of 30 MeV protons on 4°Ca . Using our theoretical potentials one has to multiply the absorptive part by a factor 1 .80 at) [ref .' °)] while Lejeune and Hodgson have to use a renormalization factor 0.70 (and even 0.40 for 5 MeV neutron incident energy) with the JLM potential in order to get a good agreement with the data . We have presented in fig. 18 the r.m .s . radius corresponding to the imaginary potential. It is not a quantity well determined empirically, but data analysis implies that this radius should be larger for the imaginary potential than for the real one. We reproduce such an effect, with an imaginary radius decreasing almost linearly with increasing energy . This reflects only the fact that the radial dependence of the absorptive part of the potential is energy dependent, changing from a pure surface term at low energy to a combination of surface and volume terms at medium energy . We conclude with the comparison of different microscopic approaches . It has to be noticed that even working with different effective interactions and RPA wave functions our results are very similar to those of Osterfeld et al. 8), where Krewaldaz) as) Speth wave functions and the Eikemeier-Hackenbroich interaction were used . Actually it is common to all the calculations performed in the nuclear structure approach of ref. a) that the absorptive potential be too small, a result also found in ref.'), where a completely self-consistent calculation was performed with a Skyrme interaction . However because of the properties of the particle-hole interaction derived from a Skyrme interaction, the imaginary optical potential is still weaker and has only surface absorption. It would certainly be more appropriate to use a finite-range density-dependent force derived from the g-matrix, like that of Campi and Sprung ") . Such a force was shown to be quite successful in the °a) derivation of effective potentials for nucleon-a and a-nucleus scattering .

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