ANNALS OF PHYSICS 102, 458-492 (1976) An Energy-independent Nonlocal for Bound and Scattering M. M. GIANNINI Istituto di Scienze Fisiche Potential States* Model AND G. RICCO dell’Universit&, Genova, Istituto Nazionale Sezione di Genova, Genova, Italy di Fisica Nucleare, ReceivedJanuary20, 1976 The generalexpressionof the nucleon-nucleus optical potential hasbeenobtained usingWatson’smultiple scatteringtheory and Wolfenstein’sparametrizationof the nucleon-nucleonscatteringamplitude.The resultingtheoreticalpotentialis nonlocal andconsistsof an energy-independent centralvolumeplussurfacereal and imaginary potentialandof a Thomas-like spin-orbitterm.Theanalysis hasbeenrestrictedto N = 2 sphericalnuclei, so that neither isospin-isospin nor spin-spininteractionshave been included. The widely usedPerey-Buck, Greenlees,and Watson expressions of the optical potentialare easilyobtainedas particular cases.For practicalpurposes,the nonlocalpotentialhasbeenparametrizedin the Frahn-Lemmerform, usingWoodsSaxonradial form factors, and the equivalent local potential (ELP) hasbeencalculated by a Perey-Buck-liketransformation. The ELP hasa radial behaviorvery similarto the originalnonlocalone, but the potential depths and radii are energy dependent. The six free parameters in the ELP havebeen adjusted to fit the available experimental data in the -70 to + 150 MeV range of interest in nuclear reactions, namely, energies of single hole and single particle states, charge distributions, proton elastic scattering cross sections, and polarizations. The fitted potential depths show an energy dependence in remarkable agreement with the model predictions with a central nonlocality range /3 E 1 F and a spin-orbit nonlocality range ,!JIIg 0.8 F. The relative importance of surface and volume dependence in the real central potential in also discussed. 1. INTRODUCTION In most nuclear physics calculations the knowledge of the average nuclear potential over a wide excitation energy range, including positive and negative energies, is of primary importance. Phenomenological potential wells of the Woods-Saxon type seemto be able to give a reasonable description not only of elastic nucleon-nucleus scattering and polarizations [l, 21 but also of binding energy and charge distribution of single * Supported by I.N.F.N. 458 Copyright All rights 0 1976 by Academic Press, of reproduction in any form Inc. reserved. NONLOCAL POTENTIALS 459 particle states [3,4]. Unfortunately the analyses of the experimental data are always restricted to limited excitation energy regions and the number of free parameters is often so high as to cast some doubt on the physical meaning of the resulting potential. In the most extensively investigated nuclear energy range, namely, between 10 and 50 MeV, the real potential depth shows a pronounced energy dependence with a slope dV/dE - -0.32 rather well established for all nuclei [5]. The absolute value of the potential turns out to be a rather sensitive function of the geometrical parameters (radius and diffuseness) and is therefore a less reliable result of the fits. In general, if the analysis is restricted to a limited class of experimental data, a continuous infinity of solutions can be obtained owing to intrinsic ambiguities between nonindependent pairs of parameters, like the wellknown T/R2 = const in the analysis of proton-nucleus scattering data [5], and the V/R3 = const in the calculation of elastic electron scattering form factors [6]. It is therefore problematic to extract from the existing literature the absolute value and the energy dependence of the potential well in the whole energy range of main interest to intermediate energy nuclear physics, namely, from the lsl,$ state binding energies up to the r-meson production threshold. The most widely used potentials are purely phenomenological: only a few attempts have been made to give a theoretical basis to these potentials [7]. In the present paper we have attempted a systematic approach to this problem based on the following considerations: The constant slope of the potential depth dV(E)/dE is consistent with the “spurious” energy dependence expected for a local central potential equivalent to an energy independent nonlocal potential [8]. The observed value corresponds to a nonlocality range ~0.8-0.9 F [9]. Most ambiguities in the optimization of the potential parameters can be removed by a simultaneous analysis of experimental data covering positive and negative energy ranges. For example, the previously mentioned VR2 = const, ambiguity can be eliminated if the radius R is chosen consistent with electron scattering form factors and rms radii. The number of free parameters and the related uncertainties should be drastically reduced if the general expression of the nonlocal potential is formulated through a theoretical model and not simply assumed on a purely phenomenological ground. Along these lines, in Section 2 the general expression of the nonlocal potential has been deduced, in the framework of Watson multiple-scattering theory [lo], from the nucleon-nucleon scattering amplitude. In Sections 3 and 4, the local equivalent potential has been obtained by the Perey-Buck transformation and the energy dependence of the parameters has been extensively investigated. The free 460 GIANNINI AND RICCO parameters have been optimized in Section 5 to reproduce different sets of experimental data like single particle binding energies, charge distribution and rms radii, elastic proton-nucleus scattering cross sections, and polarizations. For the sake of simplicity, no symmetry term has been included in the model; the discussion will therefore be restricted to N = 2 core nuclei, in the mass range 12<A-1140. 2. THE THEORETICAL OPTICAL POTENTIAL The optical potential describes the propagation of a nucleon in the presence of a complex nucleus. From this point of view, it can be considered as the positiveenergy counterpart of the mean potential, which is felt by a nucleon within the nucleus [ll]. This intuitive argument leads to a picture in which elastic scattering states and single particle states may be treated on equal grounds [ 121. Such an approach is supported by the derivation of the generalized optical potential, as it is given in the framework of nuclear reaction theory [13]. Let us consider the wavefunction z,Lwhich describes the system consisting of one nucleon plus a “core” nucleus with mass number A - 1. The part of # which corresponds to the core nucleus being in its ground state g, can be extracted by means of the projection operator P p,c, = @Yedf), where r is the coordinate of the extra nucleon in the c.m.system and p indicates any set of independent relative coordinates. The equation for Pa,5is [13, 14, 151 (K+U)P#=(E--o)P#, (1) where (Q = 1 - P) U = PVP + PHQ(E+ - QHQ)-l QHP, (2) K is the kinetic energy of the nucleon, V is the nucleon-nucleus interaction potential, and His the total Hamiltonian; E is the c.m.s. energy and Edthe ground state energy. In the coordinate representation, Eq. (1) is a single particle equation - & V2@(r) + s dr’ V,(r, r’) CD(r’) = (E - qJ @p(r), V,(r, r’) being the representative of the one-body operator and m* the reduced nucleon mass. (3) NONLOCAL 461 POTENTIALS For E > 0, the solutions of Eq, (3) are the elastic scattering wavefunctions corresponding to an incident energy E - E,, in the c.m.s. If E < 0, # is a bound state of A nucleons; Eqs. (l)-(3) are still valid. The only change is the suppression of the plus sign in the energy denominator of Eq. (2) since no outgoing condition must be imposed. In any case, Q(r) is proportional to the probability amplitude of finding one nucleon at position r and the remaining A - 1 nucleons in the bound state g, [12] and it describes, therefore, the motion of one nucleon with energy E - Edin the presence of an unexcited nucleus. These considerations lead to the interpretation of 0(r) and E - q, , when E < 0, as the ground state single particle wavefunction and binding energy, respectively. Elastic scattering states and single particle states may be treated in a unified manner, since they are all eigenfunctions of the potential defined in Eq. (2) [161- In this kind of approach the scattering states and the optical potential are usually calculated by means of techniques already applied to the bound state problem [lo, 17-191. However, several phenomenological analyses exist, in which single particle states are described by extending to negative energies the optical potential which fits the elastic scattering data [4,20]. We adopt here a similar approach: in the framework of the formal scattering theory [14, 211 a theoretical potential is derived, whose form will serve us as a guide for the parametrization of the phenomenological optical potential. The values of the parameters will then be fitted to both elastic scattering data and bound state properties. Starting from Watson’s multiple-scattering theory, an alternative and equivalent expression of the generalized optical potential, Eq. (2), can be obtained [14, 15,221. The result is a multiple-scattering expansion, whose first- and second-order terms are [14, 231 A-l u= cri+ j=l A-l c j#k=l A-l rjE+AH 0 (1 - P) 7k - 2 TjE+: jfoTj + “‘3 (4) where ri is the single-scattering t-matrix for bound nucleons and Ho = H - F’. When the energy of the incident nucleon is sufficiently high, only the first term in Eq. (4) can be retained (single-scattering approximation) and the operators Tj can be substituted with the scattering t-matrices for free nucleons tj (impulse approximation). We assume that the parametrization of the optical potential, which is derived with these two approximations, can also be taken as a reasonable basis for low- and negative-energy analyses. The optical potential VE is then A-l vE= c j=l <gOIh\&d, (5) 462 GIANNINI AND RICCO and its representative in the coordinate space is given by (See Appendix A) v,(r, rl) = (24-3 * Kl - Ag1 c 34 VjVj' j dp dq eiP.(r-r’)ee’q.(r+I’)/2tjYVjY’Y)’ (42)) P + (q/2); (1 - (4)) P - b-1/2); 4 @(q; I - r’), (6) where v(J), v~(v~‘), vt(vt’) are the third components of the spins of the incident nucleon, of the struck nucleon, and of the target nucleus, respectively; trvjv’“j’ (k, k’; E) is the off-shell nucleon-nucleon scattering amplitude; E = (A - 2)/(A - 1) (it will be put equal to 1, for simplicity); Fj’“’ is a generalized form factor, whose expression is &w(q; r - r’) E I dx pIy)(x; r - r’), (7) with &‘(x; r - r’) = 1 E dzi 6 (A z A-l * * goptvj z1 ,..a, zj - ___ ( A A-l *so‘Y$ ”Yj Zl )..., zj + -J-- zk) S(x - 5) r-r’ ___ 2 r-r’ ~ 2 3.... zA-l )...) (8) gO,V,Vt is the ground state wavefunction projected on the state in which the+nucleon has third spin conponent vi . In order to derive Eq. (6), it is necessary to separate out the overall center of mass motion. This is possible because U satisfies Galilean invariance [26] which is due to the Galilean invariance of the scattering t-matrix [27] and to the translation invariance of the ground state wavefunction go [14]. From Eq. (6), we see that the optical potential has an intrinsic energy dependence [9], which, in this formulation, is due to the energy dependence of the nucleonnucleon scattering amplitude. Moreover, the optical potential is nonlocal mainly because the nucleon-nucleon scattering amplitude is completely off-shell; in fact, if the p dependence of the t-matrix is neglected, the optical potential becomes local. A secondary cause of nonlocality is the presence of r - r’ in the form factor; it can be shown that this r - r’ dependence is related to the Fermi motion of the target nucleons. In the following we shall omit it for simplicity. The identity of the incident nucleon with the target constituents can be properly taken into account in the projection operator approach [24]; here it affects the optical potential in an indirect manner through the scattering t-matrix. However, there are indications [25] that the effect of the antisymmetrization should not be NONLOCAL 463 POTENTIALS large at high energy; thus the absence of this further source of nonlocality is at least compatible with the high energy hypothesis underlying Eq. (5). Expression (6) is quite general and reproduces, with some further assumptions, many phenomenological potentials widely used in the literature. Let us consider the spin- and isospin-independent part of Eq. (6). Vc(r, r’) = (2~r)-~ (A - 1) !” dp dq eip’(r-r’)eiq’(r+r’)/zf~((p + q)/2, (p - q)/2) F(q). (9) If t does not depend on p, the potential is local: V,(r, r’) = S(r - r’) Vc(r), (10) with V,(r) = (A - 1) 1 4 eiq**t&) WA f-= lrl, 4= 191, (11) where F(q) is now the nuclear ground state form factor. In Eq. (11) the optical potential is expressed in terms of an effective interaction [28]. If the nucleonnucleon scattering matrix f&q) is written in Born approximation, one obtains the Greenlees folding rule [7] V,(r) = (A - 1) 1 dx p(x) u(r - x). (12) On the other hand, if F(q) has a strong forward peak, Eq. (11) gives the Watson formula [14, 291 V,(r) = (A - 1m77)3 f&9 p(r). (13) In order to go beyond the assumption of a local optical potential, the off-shell behavior of the scattering amplitude must be taken at least qualitatively into account. The simplest way is to factorize the t-matrix: f&P + q)/2, (P - q)/2) = g(p) k(q), with g(0) = 1. Then the potential local potential [30] (14) P”IPI, becomes the widely used Frahn-Lemmer V,(r, r’) = H(r - r’) U((r + r’)/2), non(15) where H(x) = (2n)-3 s dp eiP’xg(p), WY) = (A - 1) s da eiq’Yt&) Qd, x= 1x1, (16) Y= IYI- (17) 464 GIANNINI AND RICCO The optical potential, Eq. (6), is considerably simplified in the case of N = 2 closed shell nuclei. First of all, the isotopic spin-dependent part is absent, so that the summation can be performed without any distinction between neutrons and protons. As for the spin-dependent part, it is customary [32] to introduce the Wolfenstein [33] parametrization of the nucleon-nucleon scattering amplitude. In the case of closed shell nuclei, only one term linear in the incident nucleon spin survives and the nonlocal potential (6) becomes v,(r, r’) = (2~)-3 (A - 1) 1 dp dq eip.(r-r’)eiq.(r+r’)/z~(q) * [lE((P + qY2, (P - O/2) + ilES((P + qY2, (P - q)/2)(q x P) * 01. If a factorization (18) analogous to Eq. (14) is assumed, @((P + q)/2, @ - qY2) = r¶r2gs(P) @YP), (19) the spin-dependent part of Eq. (18), when gs(p) is a smooth function of p, is Q * L f&(x) W(v), x = r - r’, y = (r + r/)/2, (20) where H,(x) = (27r-3 l dp e”pexgs(p), W(Y) = (~w2/Y)WdNvb(u), U,(Y) = (A - 1) I 4 eiqWd tAq>. (20 (22) (23) If a Gaussian form for the off-shell factors g(p) and gs( p) is assumed, g(p) = 6+w4, one obtains the Perey-Buck H(X) = (433)-313 gs(p) nonlocality factors e-4s*; fI&) = e-w/4, (24) = (7r&a)-312 e-Ze/@s*. (25) From Eqs. (17), (22) and (23) the form of the nonlocal optical potential can be derived. The form factor F(q) is peaked at q2 - 0 even for finite nuclei, and therefore it makes sense to introduce the expansion b(q) = to + f,q2 + *.- (26) into Eq. (17). The two complex quantities t 0 , t1 give rise, respectively, to a volume 465 NONLOCAL POTENTIALS and a surface [31] potential. In the case of the spin-orbit interaction, the first term tos of an analogous expansion produces a surface Thomas-like potential. In principle, toS is complex, but, according to the present literature 1341,the imaginary part of the spin-orbit potential can be neglected, at least up to the r-meson threshold. The nonlocal potential (15), (20) is in general energy-dependent. We assume, however [8,9], that in the energy range we are interested in, the dynamical energy dependence can be neglected. The final form of the potential is then m r’> = fww4(Y> + iWdY)l + = . L&(x) W(Y), (27) where H(X), H,(X) are given by Eq. (25) and U,(y), W,(y), UNs(y) are all real quantities, as given by Eqs. (17), (22), and (23) using expansion (26). 3. THE EQUIVALENT LOCAL POTENTIAL (ELP) In practical calculations it is convenient to replace the constant nonlocal potential by an energy-dependent equivalent local potential more suitable to numerical work. The ELP can be derived in various ways [35-371. Here we shall adopt the method discussed by Fiedeldey [35], which does not use any tool typical of the scattering theory and can therefore be applied also to the bound state case. The procedure can be easily extended to take into account the nonlocality of the spin-orbit potential. If higher-order terms are everywhere neglected, the central part of the local potential, which is equivalent to the nonlocal one (27), assumes the form [35] where J@= U(r) + +4.?(r), % = cJJ(r) 7)=0-L, U= UN +iW,, (30) + qyff&.P(r>, Y = &oMko), 01 = fiZm*/2fi2 (29) k2 = 2m*E/V, as = /3s2m*/h2. (31) (32) g, g, , /3, and & are given by Eq. (24). k, is the parameter introduced in the PereySaxon [38] expansion of the nonlocality factors g and gs ; it is assumed to be the 466 GIANNINI AND RICCO same for the central and spin-orbit potentials, sincejit has the meaning of the wavenumber of the nucleon inside the nucleus. As usual, k,, is chosen such that ko2 = k2 U,“(r) The nonlocality define = g(k,) of the spin-orbit @O(r) The substitution (2m*/fi”) = gs(k,) &O(r), (33) U(r) = e-arE-u~oc~~lU(r). (34) potential can be taken into account if we UNS(r) = e-~StE-~Lo’~“UNS(r). (35) of Eqs. (33)-(35) into Eq. (28) gives the local equivalent potential U,(r) = ULo(r> + V,(r) L2 + Q f LVdr), (36) Vdr) s Uf”O(r>/(l - muLo(r (37) V,(r) = Av&)12. (38) where Potential (36) depends on the energy E, which is defined as [40] E= ECM - vcoul (39) 3 where ECM is the center of mass energy of the extra nucleon and VcoUl is the Coulomb shift energy V coul = -1.08 + 1.35 ((Z - l)/(A - l)‘/“) (MeV). From Eqs. (34)-(36) we can extract the transformation for the real and imaginary parts under the following assumptions. (a) The nonlocal spin-orbit interaction is real; (b) the local imaginary part W(r) is rather small so that we can put a W(r) s 0 everywhere. Consequently, the transformation for the real local potential U,(r) does not depend on the imaginary part U,(r) = U,(r) e-arE-UR(T)l, W(r) = W,(r) e-“‘E-UR(T)‘, Go(r) = U,(r) + Wr), (40) (41) and the local spin-orbit potential (35), (37) is real. These statements should be reasonably valid up to the T-meson threshold [9, 341. NONLOCAL 467 POTENTIALS The Fiedeldey procedure permits us also to calculate the Perey factor f(r) defined [39] as the ratio between the local and nonlocal wavefunctions. With our approximations we get f(r) = exp(-+[olU,O(r) + asV,(r) L2 + aso . LV,(r)]). 4. THE PHENOMENOLOGICAL POTENTIAL The nonlocal potential, as defined from the model developed in Section 2, Eqs. (17), (22), (23), (25)-(27) has the following form. Ur, r’> = ~WVMY> + iwd~)l + = . L&G4 W(Y), uN(Y) = - vNfN(y> + vNF(d Y)(df,(Y)/dY), = - wN(1 - fN)fN(Y) = vNs(r~z/Y)(dfN(Y)/dy), wN(Y) UN’(y) + (42) (43) (44) 4WNtN(~/Y)(df,(Y)/dy), with fNfN(Y) = j- 4 @“F(q) 30 P(Y), (45) fqx) = (433y-3/3 y = / r + T’ l/2, ,&P9, H,(x) = (T&~)-~/~ e-xzW, x=/r-r’/. p is a constant which will be specified later. Since the radial dependence of the nonlocal optical potential is, in the high energy approximation, proportional to the nuclear density, we parametrizef,(r) by means of the usual Woods-Saxon form factor fN(r) =fWdry RN , UN) = L1 + exp((r - (46) RN>/aN)i-‘. In order to achieve an unambiguous separation between the volume and surface interactions, we substitute, as usual in the literature [2], the l/r factor in the derivative terms of the central potential with l/RN . If we choose the arbitrary constant p = a,& we get UN(r) wN(r) = -vNfwS(ry = UN"(r) = -wN(l -4WNtNfWS(r9 RN, + aNI - vNFfWS(ry RN, aN)[l - -fWS(r, RN, UN)] (47) UNF(r), tN>fWS(ry RN, RN, aN)[l aN) -fWSk? RN, aN)19 (48) (49) 468 GIANNINI AND RICCO The nonlocal potential is, therefore, determined by nine energy-independent parameters: the depths V, , V,, , W, , VNs (all in MeV), the dimensionless weight parameter tN , the geometrical parameters RN , UN and the nonlocality ranges /3, & . Equations (47)-(49) could be directly used for obtaining numerical fits to experimental data. Such a procedure, however, would require a complex code and considerable computer time. Therefore we prefer the use of the equivalent local potential as defined in Section 3, Eq. (36): U,(r) = U,(r) + 0 . L V,(r), U,(r) = U,(r) (50) + Vz(r) L2 + iW(r). (51) The radial functions U,(r), W(r), V,(r), V,(r) can be numerically calculated by applying the transformations (36)-(41) to the nonlocal potential (47)-(49). This procedure shows [41] that the equivalent potential (50) has a shape very similar to the nonlocal one: the mixing of volume and surface terms is in fact almost negligible and the radial dependence is still very closely a Woods-Saxon function although with different geometrical parameters. It seems, therefore, possible to parametrize also the local potential by means of Woods-Saxon radial form factors as follows: u,(r) = - ~I&&, Rv , ad - ~~fws(r,RF, ~~111-fw&, = U,“(r) + URFk>, W(r) = - W(l - t)f&(r, R I ,4 --4W&&, RI , Ml RF, ad1 (52) - fdr, RI ,&I, (53) V&) = - ~sVs(WVws(r, RS , ~31 - fk(r, RS , eJ1, (54) V,(r) = 4vdr)12. (55) Moreover, in the transformation for the real central potential and surface part can be separately equated: URV(r) = UNV(r) e- dE- U,(r)1 9 (40) the volume (56) The Perey-Buck-like transformations (37), (38), (41), (56), (57) are now used to relate the energy-dependent parameters in the local Woods-Saxon well to the corresponding original nonlocal ones FN , RN , uN etc. For simplicity the diffuseness coefficient is assumed to be the same for the nonlocal and all the local wells, since the numerical fits seem to be rather insensitive to the variation of this parameter, at least around the usually accepted values. NONLOCAL 469 POTENTIALS The final result for the parameters in Eqs. (50)-(56) with E given by Eq. (39) is (see Appendix B) V,(E) = VL(0) + bE + cE2, (58) V,(E) = V,(O) + b& (59) R,(E) = RN + aN In 12 exp [a (y + CFE’, +a!!$)]/[1 RF(E)=R,+a,ln{[l W(E) = W, - y)] [tN + $ (1 - 11, (60) -ay]/, fN) exp [[y . exp --o1 E + y [ ( - (61) - y] a] + (1 hl) hII + y)], (62) i - (1 - tN) Y t(E)= R,(E) (63) i+(l--t,)Y’ = RN + aN In ( ’ TN:’ ), (64) V,(E) = ~~(0) exp (-01s [E + VL(E) 1 “(O) ALL ___ . 11 + Rs(E) 2 ~ ~VFP) + 4 = RN + aN In [ (’ - $?(f 10 1 + + ~~VL(E) 2 “(@ + 4 vF(o)] ) avF(@ ____ 4 I’ - ‘) 1. (65) (66) The unspecified quantities are all defined in Table I. It must be emphasized that the knowledge of the nonlocal parameters is sufficient to determine completely the value of the local parameters at all energies and vice versa. The local and nonlocal wavefunctions are related by the Perey effect which, within our approximations, becomes $(r> = ew(H~G,O(r) + ~sV&) L2 + w . L ~&)I) h(r). (67) A few comments should be made concerning the importance of surface potentials. If a relatively weak Woods-Saxon derivative surface potential V,,(r) is added to a Woods-Saxon well I’&), the resulting potential I&‘(r) is still, to a good approxi- 470 GIANNINI AND TABLE Quantities Undefined 1 + +o = Cl-- I in Eqs. (58)-(66) b2 a VLc.3 /,=- VN = VL(O) eaivL@) RICCO 41 aVL(O) c = 2VL(O)[l + ~VL(O)l -.fo) D 01v,(o) PO = - 4 c= 1 - -- alo2 D2 ~1 = --arbf,/2 “2 = -4dl B = _ Wdl - fJ 2D2 + b&, + BvL(0)]; A = 1 + vo(l + 4%m fi” - KtT?c IL1= fo CF= ___ 24 I4 = --olbF[l + 41 + vd + -%(%6l/4) A bh, + (bp/4) + $hlVL(o)] - 8rpo[c~, + bs,& + vL(0) Wb)B + W2))l fi = $1 = B + Cfl n = _ &(I - 2fo - wo) _ * 2D2 (Cl2 - 4C&.p fN1 = - Cl 2c2 “VL &J=- 24 + 2&f, - ab 20 c,, = (1 - o)[(l - t,)Y - 1 - 31N + 4tN(w + ‘$))I CI = 2(1 - ti.&Y 4 1 - 2W + 8t,(l - w) + 4w(l - tN) * (1 - ‘JJ)+ 4rp(l - &)(I - W> + 8w’t~ - 8FtN - 6wtN (P=4 f = fNS = aVF 30 Cz = 16wt, + 8cuz(l - tN) - 8~(1 - tN) - tN)fNI (d,= - 4d&)‘la 24 + tN] - dl y = eP-2w do = (1 - W)@S~L + C$vF - 4) dl = 8(1 - 2~) + 2arsV~w - 20Sh dz = 16~ f = (g2 - 4&lg2Y’2 - g1 %E 1 go = (1 - fF)b+ + gl = (1 - 2fF)(l g, = --2(1 - 2fF), ‘?I - 11, m= 1+2o+pi mm) + 2(1 - fF) - my (1 fF = 1 - + dfNS -3 + tmfNS NONLOCAL mation, a Woods-Saxon ness a’, i.e., 471 POTENTIALS potential [42] with larger radius R’ and a smaller diffuse- VW&&, R, a){1 + Wd -fXr, R, 411 = Vw’fw&, R’, a’), VW if kw E V,,/ZV, 5 +. If Eq. (68) and its derivative with respect to r are supposed to be valid at r = 0 and r = R’, we get V,’ = V,) (684 R’ = R + a ln(2kw + (1 + 4kW2)l/*), (68b) 2kw2 ” = a 1 + 4kwz - (1 + 4kW2)1/2’ We shall hereafter call statement (68) the “equivalence rule.” A consequence of the validity of this rule is that the numerical search of small surface terms in the optical potential becomes a delicate problem. An improvement to the “goodness of fit” may be in fact due to the increase in the degrees of freedom rather than to the sensitivity to the investigated surface effects. In our nonlocal potential model (47)-(48), real and imaginary surface terms are present. According to the current phenomenological analyses [43, 441, the surfaceto-volume ratio 2kN is expected to be reasonably small for the real well, but a surface behavior is found to be prevalent in the imaginary potential, at least up to intermediate nucleon energies [S]. We have accordingly performed two different parameter searches: the first one using a pure volume real well (VNF = 0) and a surface imaginary potential (fN = t = l), the second one still with tN = 1 but fitting the surface depth V,, in the limit of the equivalence rule (68). 5. PARAMETER SEARCH The main purpose of the following “best-fit” procedure is to find an average evaluation of the free parameters in the previously developed model potential rather than to give an accurate parametrization of the experimental data. The variability range of the geometrical parameters has therefore to be kept consistent with the model equations (60), (61), (64), and (66), even at the expense of the available x2 precision. The energy-independent nonlocal smoothness has been fixed to aN = 0.57 F, a value compatible with the results obtained from elastic electron [4] and proton [7] scattering experiments. 472 GIANNINI AND RICCO According to the discussion given in Section 4, a pure surface nonlocal imaginary potential has been chosen, i.e., tN = t = 1. In the model of Section 2, no isospin-dependent t * T interaction has been included; the analysis should therefore be limited to nuclei having a T = 0 core. Among them, the nuclei with one nucleon above a filled shell should have a closer shell model behavior and more experimental data available. Our search has been, therefore, restricted to the investigation of bound and unbound states of mass 13, 17,29, 33, and 41 nuclei. We shall separately describe the fits having zero or nonzero real surface terms. (I) vNF = 0. According to Eq. (59) and Table I, no real surface term is present in the local equivalent potential. The free parameters in (47)-(49) are finally reduced to six: the nonlocal radius RN , the reduced nonlocality coefficients 01and 01~, the zero energy local depths V,(O), Vs(0), and W(0). These parameters have to be determined by a “best fit” procedure extended, according to the discussion given in Section 2, to the whole set of experimental data in the negative and positive nucleon energy region. In the following analysis the energy dependence of the local radii has been determined by Eqs. (60), (61), (64), and (66), using as starting values for 01and VL(0) the results of [4] and as = 0. This procedure has been iterated after each new determination of a, cls and V,(O), until a satisfactory convergence has been reached. Negative Energy. An extensive investigation of the average real Woods-Saxon potential obtained from the properties of single hole states and charge distributions in light nuclei has been given in [3,4] and we shall follow the same approach here. The binding energies of single particle states E.,, have been taken from poor resolution (e, e’p) and (p, 2~) experiments [45,46] for the deep shells and from the kinematic threshold of photonuclear reactions for the least bound nucleons. If E,, is interpreted as belonging to the negative energy spectrum of a complete shell model Hamiltonian, we have The eigenvalues ECM of the pure central potential (52) can be obtained from the experimental binding energies E,, after correction for the observed spin-orbit splitting A&., &M = En + AEs,U/(2Z + 1) = E.,, - &,(I The monopole charge distribution + O/(21 + 1) if J = I + +, if J = I - 4. p,,(r) and its elastic Born approximation NONLOCAL 473 POTENTIALS form factor I;&) are directly related to the single particle nonlocal radial wave function &r,(r), Eq. (67): foe> = I df PSM@‘) p& - 0 fd) = C (69) %-jf I Rndr>12, Irn dr fhdr) 0 occ. states pp(r) = ((2rr/3) rp2)-3’2 e(3/2)(‘lrP)‘, rp = 0.8 F, F,(q) = Fshdd F,(q) F2(d, J’s&) = F,(q) = 1 sow dr r2j,(qr) = 1, (70) mk$ 2.50 1 + (q2/15.7) - 1.60 + 0.1, 1 + (q2/26.7) F,(q) = exp rf$$], where iVnll is the proton occupation number in the (nlJ) state, pp(r) the proton charge density, q the momentum transfer, a, the harmonic oscillator constant [4], and F,(q), F,(q) are the proton form factor and the center of mass correction, respectively. A direct comparison of Eq. (70) with the experimental form factors Fexp(qeff), measured by the elastic electron scattering as a function of an effective momentum transfer qeff [4], is allowed only for light spherical nuclei (A < 16, J < 4). For heavier spherical nuclei, the experimental charge density obtained through a phaseshift analysis [47] has been used. The free parameters in the real central potential, RN and VL(0), can now be determined by a simple iterative procedure. From Eq. (60), RN is related to the local parameters at zero energy RN = MO) - UN In 12 exp(dVd0)/2)) - 11, (71) where R,(O) is assumed to be of the form R,(O) = r,(A - 1)l13. (72) The numerical solution [48] of the Schrbdinger equation with potential (51) plus a Coulomb energy term and zero spin-orbit provides, for a starting trial rN value (Eq. (71)), the potential depth V&Y) which fits the binding energies ECM . The charge form factor (70) or density (69) is then computed from the obtained single particle wavefunctions R&r), corrected for the Perey effect, and compared with 595/1=/2-9 474 GIANNINI AND RICCO the experimental data. The procedure is repeated for different values of r, until satisfactory agreement is obtained. Unfortunately, only fragmentary experimental data presently exist for the nuclei just above the closed shells, but systematic measurements are available for the “core” nuclei 12C, 160, 28Si, 32S, and 40Ca. The analysis has been therefore performed on these nuclei (Figs. 1 and 2). In the calculation of the charge density the shell model values of the occupation numbers N,,, give satisfactory results only for p-shell nuclei. From Si to Ca the (e, e’p) experiments [46] and the observed behavior of the charge density well inside the nucleus [47] support the hypothesis of a progressive filling of the 2S,,, shell: this shell has been, therefore, assumed to be bound in these nuclei and the occupation numbers, fitted to reproduce the correct charge densities at small radii, are reported in Table II. i i 9 c x10) 'i lr =ly F L ii , i. \. \. \ a I 1 ' lb-1 if- -4, lo 2.0 3.0 qelf( F-l) FIG. 1. Elastic monopole form factors of lsC and laO. The experimental data are from [51]; the curves are the independent particle model fits. NONLOCAL ~36:.m.- **.-._.- -._. '. - .027. LL I / K 2.J E -.-- ......_ blOS-k\* > Iiiz.02: I %a \\ '\ '\ I '\ '\ I *\". . I.9 =s nt2s,,2M.6 '. \. '. \\ ;i, , ---_ _._._._ n(2s,,,)= I , ___” (2s,,z) ii 2 I 475 POTENTIALS **Si ‘.-“*‘. - ntZ~,,~)i0.8 -.- n lZS,,*l .r.e =o c--. o .06-,,-"' \ \ \ .02- FIG. 2. Charge distributions in Wa, YS, and %i. The experimental data are from [47]; the curves are the independent particle model fits. For %ii the dependence on the 241~ occupation number is shown. TABLE II Radial Parameters rN, Root-Mean-Square Charge Radii and Occupation N = Z Nuclei Numbers for the rN Nucleus (F) ‘T 160 Yii 33 %a 1.20 1.29 1.24 1.25 1.26 n Computed values. 2.70 2.89 3.28 3.42 3.59 5.2 6.0 6.0 0.8 1.6 1.9 0.4 4.0 0.1 476 GIANNINI AND RICCO If the average radial parameter 7, = 1.25 F is assumed to be mass independent, the nonlocal radius RN (Eq. (71)), and consequently the energy dependent local radii (60), (66) can be computed for the chosen mass 13, 17,29, 33,41 nuclei. We have then adjusted the real potential depth vL(E) to reproduce the centroid eigenvalue EcM and the spin-orbit strength V,(E) to fit the observed spin-orbit splitting A&,-, . Around Zero Energy. Quasi-bound single particle states at excitation energy between zero and a few million electron volts are observed as resonances in elastic nucleon-nucleus scattering. The position and width of each resonance are accurately evaluated from the energy dependence of the scattering phase shifts [5]. The same procedure has been followed than for negative energy states: the phase shifts as a function of the nucleon energy E,, have been computed by the code “PRIN” [49] from the real central plus spin-orbit (50) and Coulomb potential with adjustable depths and the previously fixed radial parameters. Positive Energy. Elastic proton-nucleus scattering cross sections and polarizations from about 20 MeV up to the meson threshold have been computed using the code OPTM [SO] modified to handle the complete local equivalent optical potential, Eqs. (50)-(55). The central depths V, and W can be determined by a fit of the proton angular distribution; the spin-orbit strength k’s should be mainly sensitive to the elastic polarization data. In order to get a simultaneous evaluation of all the potential depths, we have followed a grid procedure: the spin-orbit potential has been varied f2 MeV in 0.5 MeV steps around the constant value 5.5 MeV, which is generally used in the literature [2]. At each step, we have performed a fixed geometry two-parameter fit of the proton angular distribution obtaining the real and imaginary central depth V, and W, and we have computed the corresponding polarization. At each proton energy we have chosen the values of Vi,, W, Vs which give the best simultaneous fit of elastic cross sections and polarizations. In order to get consistent results and to avoid intrinsic ambiguities, the energydependent radii R,(E), R,(E), R,(E) have been calculated from Eqs. (60), (61), and (64) using the average FN previously fitted to the static charge densities and the same starting values for (Y,01~, and V,(O) as in the negative energy case. As a final result, the potential depths V, and Vs have been fitted by the model equations (58), (65) as a function of the effective energy E, Eq. (39), and the unknown constants k’=(O), V,(O) and the slopes cy, 01shave been determined. These new values have been used to compute the energy dependence of the local radii and the whole procedure has been repeated from the very beginning. A few iterations were sufficient to reach a satisfactory convergence. The obtained values of the radial parameters rN of Eq. (71) are reported in Table II. The final real, imaginary, and spin-orbit potential depths are reported in Figs. 3 and 4 and Table III. In NONLOCAL 2. 5-k 477 POTENTIALS . proton >A - 7. x neutron . 60- --‘>,s.y$ -Ku;, . 5.. 40- 20- =I , I -40 I I 0 I 40 . 1 t I 60 I 120 I E(MeV) fin. 3. Real central potential depth as a function of E = ECM - VcOUl. The points are the phenomenological values of Table III; the curve is the model local equivalent potential. Spin-orbit potential * proton r x neutron & IO-a’ I -40 ~‘--“-.--..+.~,I* , I 0 ---------._ I I 40 I I 60 I I 120 I E (MeV) FIG. 4. The same as in Fig. 3 for the sp&orbit (a) and imaginary (b) potential depths. The broken curve in (b) shows the effect of a dynamical energy dependence on the local imaginary potential, Eqs. (74), (75). 478 The GIANNINI Fitted EJ, Real Central V, VS W p + ‘v -38.0 (ls& (lp,/a +0.425 (2~) t1.6 t6.0 EJl (IV) (Id,/,) (Id,/, -40.5 -13.8 -13.8 -2.0 $0.9 +0.9 +11.8 f25.5 f34.5 +39.5 +53.8 +66.8 i-89.8 +136.0 66.8 56.1 20.2 56.1 20.2 59.0 57.5 4.6 57.5 4.6 47.3 5.7 44.6 6.4 37.0 5.2 37.7 5.5 34.1 4.7 21.0 4.4 22.6 3.9 14.5 3.0 5.2 6.2 4.8 5.5 5.5 8.2 6.9 6.4 (lp& (lp& (2412) (I&J (Id,/,) -14.1 - 14.1 -1.86 -i-0.66 $0.66 55.5 20.2 55.5 20.2 59.7 57.7 5.0 57.7 5.0 - p + I60 -42.0 -18.7 (la/e) (lp,/J --1x -4.6 UPId Wd -0.1 f4.5 Depth for each V, V, W E (nlJ) -21.8 -15.5 -4.14 -3.27 +0.94 (lp,/,) (IPI/,) (Ids/J (2srle) (l&/1) c&/2) Wsd -45.2 -19.7 -19.7 -1.8 -3.3 -1.8 f16.6 +25.4 +34.2 +46.2 +90.9 68.1 55.8 55.8 53.0 54.2 53.0 44.7 41.2 44.0 33.9 23.1 11.1 11.1 5.7 6.0 5.6 5.4 6.2 4.4 3.9 6.2 7.7 10.0 8.1 6.1 -19.7 -19.7 -2.11 -3.27 -2.11 56.0 56.0 53.4 54.5 53.4 10.7 10.7 5.3 5.8 -56.1 -37.1 -37.1 -12.6 -7.9 -12.6 -3.1 -3.1 $24.2 +42.2 +91.4 73.8 67.6 67.6 54.8 49.4 54.8 61.1 61.1 42.4 37.0 32.0 (7.0) (7.0) 13.0 13.0 5.1 5.1 4.9 5.0 3.9 -13.2 -8.47 -13.2 (-2.45) -3.06 -3.06 54.5 13.0 49.6 54.5 13.0 54.9 (6.1) 62.1 4.9 62.1 4.9 - p + %i -51 (-33.4) (-29.2) -11.6 -2.74 -1.36 + 1.60 (ls& (&T/Z) (lpi/,) (Ids/J (2s& (Ids/,) (~P.u,) +2.80 C&/z) n + l*C -18.7 -4.95 - 1.86 -1.1 i-3.3 Potential n -I- 160 - 16.0 UP,/,) -1.94 RICCO TABLE III Spin-Orbit (Vs), and Imaginary Nucleon-Nucleus Staten (FL), E (nlJ) AND 8.7 8.4 17.0 n + %i -17.2 -8.47 -7.19 -4.85 -3.54 (l&.) (2s& (l&/3 (I&,/J (2PU2) -2.10 m/J - p + =s -51.0 -33.0 -27.0 -16.0 -9.1 -2.29 +0.56 +2.8 (l&/a> -56.7 73.2 (lp,/& (lp& -36.7 -36.7 64.8 18.9 - (Id,/,) (2~~1%) (ld& -16.2 -14.8 -16.2 -4.4 -4.4 64.8 56.6 57.2 56.6 59.9 59.9 - (~Ps/z) (2~112) 18.9 13.4 13.4 9.4 9.4 - a The effective energy E is equal to E c~ - VC,~ where ECM is given by the centroid energy for bound states doublets Ez+(l/e)r and by the c.m. kinetic energy (A - l/A) Elab for scattering states. The numbers in parenthesis are theoretical values, which substitute some missing experimental information. Table continued NONLOCAL TABLE EJl (nlJ) E VL III Vs W n + 93 - 17.4 (-21.4) (-16.9) -5.66 (2S,/,) (Id,/,) (Id,/,) (lhh) --5.38 (&s/z) -2.98 (2~1/2) -17.4 -18.95 -18.95 -3.26 -4.58 -4.58 59.6 59.3 59.3 51.6 60.7 60.7 (6.5) (6.5) (6.1) 7.9 7.9 - (continued) &I (nlJ) -1.08 (l&4 f4.62 +0.63 Ufsd (2ma) i-2.33 C&2) E VL Vs W -5.5 -5.5 -5.6 -5.6 +8.5 f22.7 132.2 $53.1 +66.4 52.5 52.5 55.5 55.5 49.1 44.2 39.8 35.9 20.9 5.9 5.9 7.4 7.4 5.8 4.9 4.7 4.7 4.4 3.8 8.5 8.8 9.9 10.2 -19.6 -18.0 -19.6 -6.0 -5.8 -5.8 -6.0 54.0 55.2 54.0 53.0 55.7 55.7 53.0 11.5 11.5 6.0 6.9 6.9 6.9 - n + %a p + Wa -56.0 (-41.6) (-39.8) -15.0 -8.3 - 10.9 479 POTENTIALS (Is~/~) (lp,/,) (lp,/J (Id,/,) (Id,/,) (2S,/s) -62.8 -47.8 -47.8 -19.1 -19.1 -17.7 Figs. 5-10, the analyzed solid curves being the Fig. 11 we have plotted means of the parameter 77.4 73.5 73.5 54.6 54.6 55.9 (7.3) (7.3) 11.2 11.2 - - -22.34 -18.0 -15.6 -8.4 -6.45 -4.45 -2.75 (Id,/,) (2s,/,) (l&J (lhd (2~3/2) (2Plh) (K/e) elastic cross sections and polarizations are plotted, the prediction of our local-equivalent potential model. In the corresponding energy-dependent radii, evaluated by set V,(O) = 50.5 MeV, uN = 0.57 F, a = 0.011 (MeV)-l, V,(O) = 6.0 MeV, &I = 1.25 F, as = 0.007 (MeV)-r. (73) (74) (75) (II) V,, # 0. As mentioned in Section 4, an important requirement in this analysis is to keep the number of free parameters constant. This can be achieved by applying “the equivalence rule,” Eq. (68), to the nonlocal potential (47). The geometrical parameters UN and RN can be fixed through Eqs. (68b) and (68c), identifying the reSUltS of the previous vNi? = 0 fit, Eq. (72), with the corresponding primed quantities; the energy-dependent local radii are then computed through Eqs. (60), (61), (64), (66). The nonlocal central potential depth is completely determined by Eq. (68a); if the nonlocality range j3 is assumed to be independent of VNF , the local equivalent potential depths remain unchanged at each energy. The spin-orbit depth V,(E) has been fixed to the values of Table III, since small variations of this term do not affect any conclusion concerning the importance of central surface potentials. 480 GIANNINI AND RICCO NONLOCAL 481 POTENTIALS SNOIlVZIkiVlOd I I I I I I (9 t 0 SNOIlVZlklVlOd I I I I I I / I, m. u. 0 u.I I I mI. 482 GIANNINI SNOIlWZItJVlOd SNOIlVZIHVlOd AND RICCO 483 NONLOCAL POTENTIALS 3.6 3.51 I 1 -40 I I 0 / I 40 I E FIG. 11. Energy dependence of the real local equivalent potential in **Si. Rv , imaginary I 80 I (MeIf) Rr , and spin-orbit RS radii of the The nonlocal potential V,, has been stepped between 0 and -100 MeV, in -11 MeV intervals. For each step the local equivalent surface potential V,(E) has been computed by Eq. (59) and the properties of bound states have been reanalyzed for all nuclei, without any new free parameters. The equivalence rule is expected to work reasonably well, since large real surface potentials are generally not included in the current phenomenological analyses [2, 51. As a consequence, the introduction of V,, # 0 simply amounts, for the real central well, to a different parametrization of the same numerical potential. Binding energies and charge distributions should, therefore, remain unchanged; as a matter of fact, the calculations give for each V,, substantially the same results already reported in Figs. 1 and 2. Slightly different results are, however, expected in the positive energy calculations, because of a “nonequivalent” dependence of RI , Rs , on the surface potential V,, . The only parameter to be determined in each step is the imaginary depth W. In Table IV the nonlocal surface potential weight kN = VN,/2VN and the local imaginary potential depth W corresponding to the minimum x2 for the elastic scattering cross section are reported for each analyzed energy E. From Table IV it is evident that in most cases there is no indication of nonzero real 484 GIANNINI AND RICCO surface potential, at least for N = Z nuclei; in only few cases is a better fit of the elastic cross sections obtained for the highest V,, step values, where, however, the polarizations are reproduced more poorly. TABLE IV The Surface Potential Weight kN = V&~VN and the Local Imaginary Potential Depth W Corresponding to the Minimum x2 Fit of the Proton Elastic Differential Cross Section A-l E = ECM - Vcoul WV) h 12 11.8 25.5 34.5 39.5 53.8 66.8 89.8 136.0 0. 0.237 0. 0.574” 0.574” 0. 0. 0. 5.24 6.56 4.73 6.51 5.88 7.92 6.97 6.36 16 16.6 25.4 34.2 46.2 90.9 0.574” 0. 0. 0. 0. 7.20 7.84 10.4 8.15 6.13 28 24.1 42.2 91.4 0. 0.574= 0. 8.72 9.61 17.1 40 8.45 22.7 32.2 53.1 66.4 0.366 0. 0. 0. 0.056 4.16 8.51 8.75 10. 10.6 (M&j o Grid border value. 6. CONCLUSION In Section 4 we showed that the general expression of the nonlocal nucleonnucleus optical potential can be parametrized in the following way. UN(r) = -~NfWS(~, RN, = -wN(l - tN)fiV&, UN’(r) = - ~NS(~,2/~~N).fWS(~, w,(r) aN> RN, RN vNFfWS(r~ UN> , aN>[l RN - , aN>[l 4~N~NfW&, -fw&, -fw&, RN, RN , UN)], aN>[l , UN)], -fw&, RN RN, UN)], NONLOCAL 485 POTENTIALS where hd”, RN 7UN>= 11+ exP((r - h-d/~~>l-~, RN = FN(A - 1)li3 - aN ln(2 - l}. eXp[+avL(o)] The energy-independent potential depths vN , vNr , (1 - tN) WN , 4tNwN , VN, , and the geometrical parameters FN , & have been adjusted in Section 5 to fit the available experimental data in the energy range from -70 to +150 MeV. The final values, VN = 88.6 MeV, tN FN = 1, = 1.25 F, = 0, VNF VNs = 9.2 MeV, WN = 23.3 MeV, (II = 0.011 MeV-I, aN = 0.57 F, as = 0.007 MeV-r, have been obtained from the optimized parameters of the local equivalent potential reported in Table V. In Figs. 3 and 4 the phenomenological potential depths, plotted as points or crosses, are compared with the model predictions computed from Eqs. (58, 62,65) using the parameters of Table III (full curves). For the real TABLE V The Optimized Parameters of the Equivalent Local Potential and the Related Quantities of the Nonlocal One Local Nonlocal 1 a=---= VL(0) = 50.5 MeV VL@) b = -0.36 fN = 1.25 F = 6.0 = VL(0) 1.114. 10-z (MeV)-1 b exp[aVL(O)] = 88.6 MeV - 1)1’3 - 0.2851 F VNF = 0 VF = 0 V,(O) V, RN = [1.25(/f b 1 + MeV vNS = V,(O) [ 1 + T] exp [as 91 = 9.2 MeV as = 0.007 (MeV)-1 t=1 W(0) = 17.6 MeV av = as = aI = 0.57 F fN = WN = 1 W(O)exp uN = 0.57 F [T] = 23.3 MeV 486 GIANNINI AND RICCO central potential the agreement is remarkably good over the whole energy range. Only around zero energy does the theoretical curve underestimate the fitted average real depth. The residual interaction is, however, expected to introduce a stronger configuration mixing in the surface (E g 0) single particle states, shifting the unperturbed independent particle energy eigenvalues. The neglect of this effect should result in a more scattered distribution of the points, as observed in Fig. 3. Nuclear structure effects might also be responsible of the less satisfactory behavior of the spin-orbit potential. Wide fluctuations are evident in the region of strongly bound states and the fit of the elastic polarization data is still only qualitative. This conclusion seems to be quite independent of the assumed parameter values [2,68] and should probably be ascribed to an inadequacy of the model. The energy dependence of the imaginary potential computed from Eq. (62) using the same nonlocality parameter LXadopted for the real well, shows a reasonable agreement with the average phenomenological behavior at energies E Z 60 MeV. The rather strong fluctuations of the fitted depths, always present in optical model analyses, are due to large nuclear structure effects on the inelastic channels. Below 60 MeV the theoretical curve shows an energy dependence leading to a nonzero intercept at E = 0, while the fitted points are known to decrease approaching the inelastic threshold [34, 21. This physically more consistent behavior cannot be reproduced by an energy independent nonlocal imaginary potential; we should, therefore, introduce an intrinsic energy dependence of the type W,(E) = W,(l - e-rE), (76) where r is an adjustable constant. The equivalent local potential W(E) = WN(, _ e-r”) e-dE+(t,L(E)/2)I (77) is plotted in Fig. 4b (broken line), with r = 0.0244 (MeV)-I. With the same nonlocality range 01= 0.011 (MeV)-l, a slightly different depth W, = 29 MeV is obtained. One should point out that, in order to obtain a mass-independent potential, an average has been performed on the radial parameter rN fitted from the nuclear charge distribution (see Table II). Detailed calculations for single nuclei should therefore use different parameters corresponding to more realistic radii. In both procedures, the meaningful nonlocality parameter is the mass independent quantity 01.The corresponding range /? can be obtained by the formula /3 = (2ti2a/m*)1/2, and depends on the reduced mass m*. NONLOCAL POTENTIALS 487 The nonlocal wavefunctions can be computed using the modified Perey factor, Eq. (67). A check of this approximation has been performed calculating the overlap integral between the Is- and 2s-states in *%c. The result is I m dr r2+,,(r) #zs(r) = -0.093, 0 a value which is still too large for many applications. In [69], a detailed comparison has been made between the exact numerical nonlocal wavefunction and the local equivalent one corrected with the nonlocality Fiedeldey factor [35], which is very close to our Eq. (67). In the Its-state of 40Ca, the external (r 3 3 F) nonlocal wavefunction is about 6 % higher than the exact solution. This behavior might explain our nonorthogonality result. The local equivalent approximation is therefore a consistent method, valid at positive and negative energies [4, 701, for the calculation of the nonlocal potential parameters, but it does not take sufficiently into account nonlocality effects on the wavefunctions. If a complete set of eigenfunctions is required, the Schrodinger equation with the nonlocal potential must be directly solved [8, 691. An extension of the model in this sense, also including the nonlocal isospinisospin interaction, is presently in progress. APPENDIX A: PROOF OF EQ. (6) In any frame of reference, Eq. (5) can be written ( VE) = (r,v; Rv, i VE 1ro’v’; R’vi) Z-Z(27~)~’ C / ~PodP,’ S dPj dPj’ (PO”; PjVj I tj I POlV’; Pi’vj’) 3Vj”jI . p’d~opj ; po$j,) eib,.r~+~.R-~~‘.rp’-~‘.R’), 64.1) where p’y)(popj ; p0lp.j’) = (2~)~~ J dp dp’ dzj dzj’ fl dzi eib’z’-D’x) ifi * gO*.&"JZ1 )...) Z&l) x= &zzit gO*";Yj~(Z1 )...) Zj')...) Z&-1) Fi(pj'+-p'.x'), x&x+&J. 488 GIANNINI AND RICCO In order to separate out the overall c.m. motion it is convenient to introduce the variables. K = pa + pi , k = (~0 - PN, Q = PO + P, 64.2) r = r, - R, G = (to + (A - 1) W/4 and analogously for the primed quantities. Owing to the Galilean invariance of the t-matrix, (Pov; PjVj 1 tj 1 Po’V’; Pj’Vj’) = 6(K - K’) t~““““(k~ k’). 64.3) The intrinsic wavefunction gOiYtY,is translation invariant and depends, therefore, only on the relative coordinates yj = z, - X, which are not independent. The integration over the zj coordinates can be transformed: (A.41 According to Eqs. (A.2)-(A.4), we get (YE> = 8(G - G’) V&, r’), with V&, r’) = (277F3 [ 2(AA . g&Jy1 ‘go .Yt ” vi ‘)I3 ,C s dk dk’ r@ dyi 6 (-J--& 3VjVj’ ,.. ., yAe-1)f?v’vi’(k ( Yl ,***3 Yj + g yt) k’) _A_AL1_ (r - r’),..., Y~-~) * exp Ii [k * (r - yr) - k’ * (r’ - yj - 9 (r - r’))]/. Equation (6) is obtained by means of the transformation yi = yi - (r - r’)/2A, yi = yj + ((A - 2)/2A)(r - r’), APPENDIX i #j, P = ((A - WW q=k-k’. + k’), B: THE PEREY-BUCK-LIKE TRANSFORMATION In order to relate the local parameters to the nonlocal ones, we impose the validity of the transformations, for each term of the potential, in two critical points, namely, at the origin and at the corresponding local radii; in the latter NONLOCAL 489 POTENTIALS points we make also use of the equations obtained by derivation of (37,41,57) with respect to r. Since the radii in the various Woods-Saxon wells are not very different, we can everywhere approximate fw& , R, ,4[1 --fws(& >R, 341 - k, R, , R, being any pair of radii. In this way, from Eqs. (37), (41), (56) and (57), we obtain the implicit relations VL = V./E+ “L), (B.1) l/2 = fws(Rv , RN, uN) e-'(("L'2)-("F'4)), 0 = 1 - WV& , RN , 0~) 03.2) (&VL/~), - (B.4) w(1 - t) = w,(l - tN) ,-"(E+vL), W!+ - fN)fWS(& , RN , aN) + WNtN] e-“[E+YLfWS(R”RV*aN)+(YF’4)‘, = [W,(l (B.5) 03*6) Wl - t) = ( wN(l - - fN) dwN(l . [V,[l + - - - , RN 2f,,(R,, ?bws@s', 2f,,(R, - tN)fwS(RI V vs = 1 + @N,,2y+ O=l 4WNtN[l RF, , UN) aN)] + > RN + , aN)l WNtNl V,]} e-“~=+“Lf~~(Rr.R~.a~)+(V~~~)~, 03.7) --sg(.E+(YL/2)+(Q!/4)) 3 (cxV,/4) e RN, uN> - (%/~)(VL + vF[l (B.8) - Wws(&', RF, uN>1), (B.9) O=l - 2fws(&, Rs’, UN> 2fws(& 7RF 9 UN)] - “L 4 1 + aF”Lfws(& , Rv 3 UN) + (avF/4) VL + vF[l - ' (B.lO) The last two equations arise because, in the case of the spin-orbit potential, an intermediate transformation is necessary, that is, from UN’(r) to the potential U:“(r), Eqs. (39, (3’7), which is parametrized by means of a Woods-Saxon function with radius Rs'. In order to get an analytical solution we put fws(& , RV , UN> -fws(& in the exponential , RV , aN> - 4 of Eqs. (B.6)-(B.7) and in the denominator of the right-hand 490 GIANNINI AND RICCO side of Eq. (B.lO), respectively. The solution of Eqs. (B.5)-(B.lO) for the imaginary and spin-orbit potentials is straightforward, once the real volume and surface terms have been determined. Equations (B.l)-(B.4) can be solved if a power expansion of the real well depths V, , V, is introduced. The general solution of Eq. (B.l) is where c w-1 = --01 G + cz @ + 1) c,+,c*-h (n + 1)(1 + GJ with Co = V,(O) given by ’ Co = VNevaCo. However, for the energy range under investigation, it seems appropriate to stop the expansions of VL and V, at the second order in E. Equations (58)-(66) are thus obtained. ACKNOWLEDGMENTS We are indebted to Professor A. M. Saruis, who kindly sent us a copy of the codes Bostaw and Prin. REFERENCES 1. F. D. BECCHETTI, JR., AND G. GREENLEES, Phys. Rev. 182 (1969), 1190. 2. B. 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