Antisymmetrization effects in heavy ion potentials

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ANTYSYMMLTRYZAITON EFFECTS YN HEAVY YON POTENTYALSt Lwtitxt fflr Thearetiaa~e

J. FLECKNER

Physik,

Uaiaenitdt Gisuen, 63 Giessaa, Wist Gernw~y and

U. MOSEL

Inrtitrt fitr Thearetüd`e PhytJk, Unloerritdt Giasen, 63 Gieren, West t7ernraßy and

Arpos7ee National laboiatary, Phyiles Dioiaion, Arpoaae, Ilärtofa 6039, USA ltaoeived 27 Soptember

1976

Aiatraet : The effects of antisymmetriTStioa en heavy ion potentials are investigated .

To thin purpose we derive for the Skynme force as ener8y-density fanctional that ne8lecta all effects of antisymmotri:stion between the nuclei. Starting from this potential we then include step by step the aatisymmetrisation is the interaction matrix element, the Pauli distortion of the densities and the chea,e of the intrinsic kinetic energies due to the Paull principle and diecaes their relative magnitude. It is found that the sum of tbeae corrections moat of which aro not incladed in the folding model ie mpulsive sad changes the not antisymmetriad potential by about 70 at the strong ebeorptioa radius.The effects ofnuclearpolariratioa on the potential arediacuseed by comparison with the results of a two~enter self~oonaiitent cakalstion and are found to deepen the potential considerably. Finally a detailed comparison of the not aatisymmetrimed patentïal that can be easily evaluated from the denait3ee with experimental results ahowa good ag
1. Yatrodaction One ofthe most popular approaches to the calculation of heavy ion potentials is the folding method . In this method the densities are assumed to be unperturbed and the potential is simply the expectation value of a nucleon-nucleon interaction averaged over the two densities r). This definition seems to be just7ißed because elastic heavy ion scattering is only sensitive to the shape of the potential at ion distances greater than the critical distance where A1 and Az are the mass numbers of the two ions r ~a). T'heir distortion is assumed to be negligible because in this region the densities scarcely overlap. Although the definition given above is intuitivelyconvincing it is nevertheless useful and necessary to clarify its theoretical founäation if higher order changes to the folded potential are to be investigated . t Work supported by Buadeaminiat~ium fùr Forschung und Tcehnologie (HMFT), Gesellschaft fûr Schwerlonenforschung (GSn and the US Enorgy Reeaerch and DevelopmAat Administration . 170

ANTTSYMMET1tIZATION EFFSCr3

171

Therefore, in sect . 2 we will briefly discuss two derivations of the folding method using either the Horn-Oppenheim~er approximation or the resonating group method as a starting point. As aarpocted, the result is that folded potentials can be justified only if all antisymmetrization effects betwcen the colliding nuclei and the changes of the nuclear strucuree can be neglected. In sect. 3 we derive a heavy ion potential from the 3kyrme force neglecting consistently all antisymmetrization effects. In order to test the basic assumptions for thefolded potential, we present an analysis of the antisymmetrization effects in sect. 4. There we discuss the various contributions to the total antisymmetrization effect and their relative importance for the System 16O î- 16O. In sect. S we discuss the range of the antisymmetrization efforts and their dopendence on the mass ratio of the system. Furthermore we compare the folded potential to a self~:onsistent potential calculated without the frozen configuration assumption. In sect . 6 finally we present a comparison of folded potentials with potentials obtained from fits to experimental data. Throughout this paper we will be concerned only with the real part of the HI potential since microscopic calculations of the imaginary part are still very sparse and in an exploratory stage'). 2. 1Leore8cal)~ for folded poteatWs In the following it is assumed that the coupling to all other channels is being taken care of by the imaginary part of the potential 2). A microscopic derivation of the real part of the heavy ion potential can then be given in two equivalent ways that give identical asymptotic wave functions for the system 4). In the resonating group method (RGM) the intrinsic structures of the two ions are assumed to be undisturbed coon at close overlap. The Pauli principle is then satisfied by a very complicated wave function of relative motion x(R) with many nodes: ~ _ ~~R)~i.e(l~tm(2))~ Here R is the c.m. distance of the two ions described by ~,.~( 1 ) and ß,.e(2). The wave function of relative motion is obtained from a variational principle that yields a nonlocal Schrôdinger equation for X. The localpotential in that equation is just the folded potential; it is duo to that term in the expansion of 'P that does not contain any antisymmetrization botwcen the nuclei. The non-local part, however, originating in the antisymmetrization makes the resulting potential dependent on energy and angularmomentum s). Only if itcan be neglected can the folded potential be justified. The resulting effective potential for x(R) has to be deep in order to bo able to support a wave function with many nodes necessary to satisfy the Pauli principle. It is

172

J. FLI3CKNER AND U. MOSEL

also well known that this potential contains many redundant solutions that have to be removed because theywould lead to a vanishing'P in eq. (2) afterantisymmetrization . The effective potential obtained in the RGM, therefore, does not contain the Pauli principle ~). 1t also does not agree with the physical concept that .the HI potential should represent the interaction energy between the two ions. The alternative method that meets this requirement is the Born-Oppcnheimer approximation in which the total energy of the ion-ion system acts as the HI potential and the wave function is given by e) with R = Rt -R2, and Rt, RT being parameters describing the locaüon of the shellmodel potential. This method is closely related to the generator coordinate method whose equivalonce to the RGM - as far as the total wave function in the asymptotic region is concerned - was shown by Fliessbach 4). In these methods the potential does contain all effects of the Pauli principle . . The potential is then given by V~(R)

= 8(R)-Etn~

(4)

when e(R) is the total energy of the system at distance R, i.e. the potential energy surface (PES), and Et,~ represents the intrinsic ground-state energies of the two ions. ïfthe intrinsic structures are described by a Slater determinant one obtains for a two-body force t: For the ~ of small overlap near the interaction barrier one can still localize the single particle states in the two nuclei and obtain V~(R) _ ~ (l~t~l)+~~ (il'~V~ll'),,s+ ~ ~r~t~r)+~~ (rr'~V~rr'),~ re

where the states fir) belonging to the right hand and the states ~l) belonging to the left hand nucleus are orthonormal to each other. They differ from the original ones in the two separated fragments bxause of two effects: first, the nuclear interaction Y with states in the other ion may lead to a polarization and second, the antisymmetrization between all single particle states together with the normalization leads to a Pauli distortion of the two nuclei . ff now these two distortion effects are neglected then one obtains the expression: V(R) _ ~
AN173YMMSI'RIZATION BFFBCI~

173

where the brackets indicate an integration over all coordinates. The folded potential is obtained if the integration over spin and isospin coordinates is performed i " z) and the mixed density term appearing is neglected: yr(R) = J P.~nP~rids. (+ ~~ (8) Here VD is the spin- and isospin-averaged interaction and YF the usually neglected exchange part (see below) involving the mixed densities. Thos in this approach the folded potential can be justified only if Pauli distortion and local exchange effects are negligible in addition to the nuclear polarization . This holds also for three-body forces where completely analogous considerations can be mock . It should also be noticed that neglecting the exchange term in eq. (8) is not identical with a neglect of all antisymmetrization effects on the potential. By integrating over spin and isospin in going from eqs. (7) to (8) the antisymmetrization in spilt- and isospin-space has been included. In addition, the effects of any space-exchange operators in the original interaction have been evaluated assuming a totally aatisynanoetrized matrix element, even though the exchange terms in eq . (8) are dropped in the end. We will, therefore, consistently distinguish between the terms "antisyavmetrization (AS)" applying to all coordinates and "exchange" applying only to the r-space coordinates. Since we fcel that keeping certain parts of the total antisymmetrization and neglecting others is inconsistent we will in this paper in particular investigatetheeffects of all antisyntmetrization terms. Recently the folding method has bcen extended to include the exchange term in eq. (8). The corresponding mixed density has been approaimated either by the Slater expression') or by a short range expansion of the density matrix 8). Since the exchange terms in refs. ~" s) yield nonnegligible contributions to the HI potential the justification for neglecting the Pauli distortion effects originating in the same physical principle becomes questionable. This point we will further take up in sect . 4. 3. TLe HC potential wlthont antisymmetrlzation

ïn this section we will derive the functional form of a potential that consistently neglects all AS effects. For the nucleon-nucleon interaction we use the Skyrme force which has proven to be an excellent effective interaction for the description of nuclear ground-state properties . Since this interaction is momentum~ependent not only the densities p ofthe two ions but also the kinetic energy density s and the Laplace density Vsp do appear in the potential. The following derivation is analogous to the work of Brink .and Yautherin 9 ), who calculated the energy functional for the 5kyrme force, evaluating totally antisymmetrized matrix elements . They obtain for doubly even and N = Z nuclei : s Hes(P. z) ° T+1~toP~+3~a(3t1+Sts)Pr+~(Srs-9t~)POsP+~rtaPs~ 2m

174

J. FLECKNER AND U. MOSEL

Because of the approximate zero range character of the Skyrme force, no mined densities appear in eq. (9). Using the Skyrme force for the effective interaction the potential V~ from eq . (S) can thus be expressed through H,~: Vras = J [H~(p~ T~_Hns(P!. T!) -HASIYrs Tr)~d3r.

(10)

For an estimate of the influences of antisymmetrization on the nucleus-nucleus pofeudal, Brink and Stance 1 °) have madethe substitution p,+s = pl+P.~ Tes = Te +tr in order to turn off all AS effects. This step yields the potential ~ns = J Âns(p, s~3r.

(11)

~~~ T) = HAS~YI+Pn =l+Tr)-lY,~s(p!~ T!) -HAS(Pr~ tr)~ that supposedly ignores the AS [ref. 1°)]. However, it must be realized that this potential still contains quite. sizeable contributions from the AS between the two nuclei because the functional H,~ has been derived under the assumption of a wave function antisymmotric uadar exchange of all nucleons. Tha substitution for the densities thus neglects only parts of the total AS. We now calculate as expression for the potential that consistently neglects all AS effects between the two nuclei . We assume that the grouad state of each nucleus can be described by a Stator determinant. In order to retain time reversal invariance wo confine ourselves to doubly even nuclei. .Furthormore wo neglect spin-orbit coupling between the two nuclei and disregard the Coulomb potential. The result of the calculation which is performed in the appendix is an energy functional Hru+a(P, T) by which the potential can be expressed: Vrus(R)

= HN~(p" T~3r. J

(12)

The t 3 term independent part of the functional HN,~ is H°rus(P,T)

= to(1 +}Xo)PIPr+~{ti+ta)(PITr+PrT !) +~d(ta - 3t1)(plV2Pr+PrO=P!)" (13)

In the calculation of Brink and Ysutherin 9) the three-body term is equivalent to a density-dependent two-body term. This equivalence does not hold for the present situation of no antisymmetrization between the two nuclei where the threo-body term is SO ~ larger than the two-body form. However, due to the ps dependence of the t 3 term the influence of this term on the potential is very small around the barrier and only in this region the folding conapt is justifiable at all.

ANTISYMMETRIZATION EFFECTS

173

Assuming for N = ~ nuclei the same shape for the proton and neutron densities we obtain : (14) Hrua = RNes+~r3PPtP~ Hntna = Haus+1}taPPtPr "

(15)

Typically the potential based on the three-body term (eq. (15)) is about 5 ~ less attractive than that using the two-body term (eq. (14)). 4. Model calcaLtioao for the system 1°O +'~O In this section we discuss the effects of antisymmetrization on the HI potential in a model calculation. We still neglect the influence of the interaction on the s.p. states (frozen configuration assumption). First we study the effects of antisymmetrization on the densities (Pauli distortion). To that end we have performed Hartree-Fock calculations forthe nucleonwave functions of one 160 nucleus in a harmonic oscillator basis. These wave functions were then centered around t~R and the resulting shifted functions were orthonormalized in order to construct the density. As this same effect has already been discussed by

Fia. l. Densities for the system isO-I-isO as a flmction of the s-ooordmate for p = 0. Shown are the

densifia p, and P. of the kft and right nuckus, respectively, and the sum of the densities and the antisymmeti3zed density p,,s of the system . The distance R of the nndistarbed densifia is 2 fm in $8" la and 6 fm in fla. lb. ALo aiv~ is the valse of (R) (eq. (lt7).

176

J. FLSCKNBR AND U. MOSi3L

Fia. 2 . Various interaction potentials with the Skyrme force for the system 160+,16 0 as a function of the distance
Fliessbach t l) we show only two examples of Pauli distortion effects in fig. 1 . We describe the distance of the two nuclei in the antisymmetrized density by the expeotation value of a distance operator which asymptotically agrees with the c.m. distance R of the undisturbed densities:

r

(16)

1 (P~-Pt-P.)zd3r. AjJ~
ANIISYMMETR~ZATION SFFBCT3

177

Neat we study the effects of antisymmetrization on the interaction matrix clement. This effect has not been considered in a recent similar investigation by Brink and Stance 1°). Calculating a potential without antisymmetrization gives the carve Ynt,+s in fig. 2, obtained from biltns . In order to disentangle the two AS effects, namely the Pauli distortion and the AS in the matrix element, the potential ins (eq. (11)) can be calculated. It contains only the latter effect and not the Pauli distortion becauso the undisturbed densities are used. Explicitly the functional fÏ,,ts (eq. (11)) is given by ~~s(P. T) ~ ~toPtPr+~s(3ti-I-St~(piT,+Prat)+~(St2+9t1xPt0~Pr+P .O~Pi) +T~3t3PPtPr (l7Î A comparison with eq. (14) shows that all coefficients in ~,~ are smaller than in ITltns. Since the leading to term is nogativo Y~ is more attractive then ~,~. Tho potential difference originating in the AS of the interaction matrix element E~ _ ~ns- VNes

(18)

ED .. Vea - ~~.c.

(19)

is indicated by an arrow in fig. 2. Over the whole range E~ is indeed positive. Thetotal antlsymmetrization in tho matrix oloment thus contributes repttlalvely to the HI pofennel. This finding is not in contrast to the rosults of refs. ~~°) where the exchange contribution was found to be attractive. Next we turn to the contribution Eb to the potential produced by the distortion of the densities due to antisymmetrization . Neglecting for the moment the change in kinetic energies in oq . (6) the Pauli distortion eaorgy ED is given by where Y,~ represents the interaction energy for the case of complete antisymmotrization: V,~ = Vt~sa-dEr,

where Y~ is defined in eq. (10) and dE,~ is given by s dEk = ~~(T~-Tt-T~~3T. 2m

(21)

Contrary to Ex the quantity ED changes its sign at R ~ 6.3 fm. This is due to the iaerease in overlap of tho two nuclei for R it 5 fm as mentioned earlier. hn the Born-0ppenheimer approximation another important contribution to the potential originates in the change dEt in the intrinsic kinetic enorgios of the two ions. This change is positive over the whole range. For distances R ;r D~ the two A3 contributions Ex and Eb counteract and nearly cancel each other (sec figs. 2 and 3). In this region, therefore, dEt~ accounts for most of the total antisymmetrization effect [refs. 1 o,is)~ .

178

J. FLBCKNER AND U. M0313L

Tho potential Yom, i.e. the potential energy surface, is also shown in ßg . Z. VPhereas down to about 5 fm its slope is similar to that ofthe other potentials it starts to develop a repulsive core for smaller distances, produced by a strong rise of dE,~ . We wish to emphasize that this rise is entirely due to the frozen configuration assumption that loads to a compression of nuclear matter at small r (see fig. la). Since this assumption cannot bejustified at large overlap the behavior of V~ at small r has to be regarded as spurious. LS 1.0 0.5 m z 0 ci

i

0 -03 -L0

â -i .s

a

s

5

~ s 9 asru~ Rtr~l

a

n

Fig. 3. Value of the energies Ena ED, dF.~ and the total antisymmetrizetioa sassy E, divided by the value of V,,s as a fimctioa of distance for the system 1 °O~- 1 d0.

To sce the relative strength of the three contributions B~, Eb and dEk and .of the total antisymmetrization energy E,~, in ßg. 3 all energies are related to the value of Y~ at each distance. The AS energy E,ns decreases slightly with inclnasing distance whereas the matrix element energy E~ remains constant. The distortion energy ED becomes large and negative and counteracts dEk which also becomes, always relative to Vim, very largo. Together they give a small positive contribution. Tho absolute values of all potentials, of course, quickly tend to zero so that for distances greater than about 9 fm they are negligible compared to the Coulomb potential (~ 10.2 MeY at R = 9 fm). Because of the zero range of the Skyrme interaction and the axial symmetry of the densities the integrations in the expressions above can be reduced to two dimensions . The integrations were performed using a finite twelve-point Gauss formula in two to three subintervals in each direction. S. lR~the~r diecassion of antisymmetrizadon effects In this sxtion we compare the folded potential with a self~onsistently calculated potential and discuss the range ofthe antisymmetrization effects and their dependence on the fragmentation of the system. The latter wo studied by calculating the various potentials at D~ for different systems. For this purpose we have performed Hartree-Fuck calculations, similar to those for the 160 nucleus, also for the nuclei 4He, 1'C and 4°Ca. Using the resulting wave

ANTI3YMMETItIZAT10N BFFSCI'3

179

Lo w z 0

m

^ o.s

E~s AE  E

n o

c .

^

o

c

C

z 0 u 0 ,:W K

-0.5

0.1

Eu

~

,.,

O

Q25 03 033 MA55 ASYMMETRY A~/A,

ô

"~y O 0.75

L0

Fib. 4. Value of the energies E, F.o, dE,~ and E,,s divided by Ye,,,s at D~ for all combinations of the fog nuclei 4 He, laC, '°O aad'°Ca as a function of the mass asymmetry of the system. The atrai~ht lines pve mesa vahuss of the calculated marked by the different symbols. Here A, and A, are the masses of the two intaactin~ nuclei.

Fig S. Iaie~taction potentiabwith the Skynme force for the system 1 °O-F~ 1 °O as a Smctioaof distaaoe, Yr n, sad Y~ as is ~2 Here Y~ danotea the result of a selfco~istent Hartre~-Fock calculation in oonBSluation space's).

180

J. F'LECKNER AND U. MOSEL

functions we then calculated the different potentials for all ten possible combinations of the four nuclei . As in fig. 3 we relate the energies E, ED, dEr and the total antisymmetrization energy E,~ to the value of Y~ at D} . In fig. 4 these potential wntributiôns are shown as a funlion of the mass ratio. No dependence of antisymmetrization effects on the fragmentation is visible. The largo deviations of some energies from the mean value can be explained by the approximate nature of eq. (1). Next we turn to the question of the range of the AS effects. Since they depend on the degree of overlap of the two ions it is in principle conceivable that our findings about their importance at D~ presented in the preceding sections depend on the range of the interaction . In particular one might argue that using a short-range force like the Skyrme interaction consistently overestimates the relative importance of AS effects versus the direl interaction contributions. As an example wo consider the kinetic energy contribution since its value is indopendent of the chosen interaction. The relative offel of taking the kinetic energy into account will, therefore, become greater with decreasing potential depth. Our calculations do not show that the potential based on the Skyrme force is generally shallower than finite range potentials . On the contrary, the U1Y potential (see sel. 6) is less attractive than V~ (see fig. 6) so that with this interaction the relative antisymmetrization effects will even be larger. The antisymmetrization effects, of course, tend to zero for greater distances, but compared to the also decreasing folded potentials they keep their relative importance unchanged. In all these considerations any distortions due to the nuclear interaction have still been neglected . Their importance can perhaps be learned from a comparison of Y~ with the result of a constrained Hartree-Fock calculation employing the same intaradion 13). The self-consistent potential V~ (sea fig. 5) provides a lower limit to the Horn-0ppenheimer HI potential because in an actual collision the nuclei may not "have time" enough to readjust their structures so as to attain the lowest possible energy. Compared to the potential V~ which is calculated inthe frown configuration assumption Y~ is indeed dceper . For R ~ 7 fm it agrees remarkably well with the potential Yi,,,+s showing the interesting fad that the self-consistency compensates all the antisymmetrization effects mentioned above. This holds also for the system ao~+ao~ where we have performed a similar analysis . 6. Comparison with ezperlment

For the system 160 +a °sl?bwe comparein fig. 6 thepotential Ym,+s from eqs. (12), (14) with folded potentials obtained from commonly used interactions . The densities in these calculations are obtained from Hartre~-Fock calculations in r-space using the Skyrma force SIII 14). Out of throe representative Woods-Saxon potentials that fit elastic scattering data 1 s ) thepotential with the medium slopeis given in fig. 6, denoted byWS. The folded potential with a Gaussian interaction 16) Uo is too deep, comparod to the WS potential. A much better fit give the recently proposed interaction whose

ANIISYM>yIETRi7ATiON 13FFECPS

181

-b -5

s t r -0.5 z W F

-0.01

II

12

p

DISTANCE

R ( fm 1

IR

ß

Fig. 6- Interaction potentials for the systamn'e0-~so'pb as a flm~ton of distances All potentials aro calculated with densities obtained fiom r-apace üartree-Fock calcaladons. Here Ua and U,r denote folded potentials as m~plainedis tha text, Ye ns denotes the Skyrme potaatisl without any aatiaymmetiization and W3 is a pheaonranologicaliyr determined Woods-Saxon potential'°) .

n Fig. 7. Interaction potential for the system 1 °O-~'°O as a Ponction of distance. H~+e Ye,,,s and Un are calculated with r-space Hartreo-Fock demsitia, WS denotes a phenomenological Woods-Saxon potential i~ and U a potential calculated by Brink sad Starecu 1') as explained in the text.

182

J. FLECKNER AND U. MOSEL

spin-isospin independent part consists of a sum of two Yukawa potentials ~) (folded potential UzY in fig. 6) and the Skyrme potential VNea. At D~ the potentials UzY and Y,,+a lie within a 15 ~ range of the WS potential, whereas UG is nearly twice as attractive . In fig. 7 we compare Yr+s with the experimentally determined potential WS [ref. l')] for the case of 16 0+ 160. Again, the agreement is good. Contrary to fig. 2 the potential Y,ans in fig. 7 is calculated with r-space Hartree-Fock densities (as in fig. 6). The difference of both potentials amounts to 5 ~ at D.~ (see table 1). With the latter densities we have also calculated the folded potential U2Y. Whereas this potential yields good agrcement in the 160 +~ °epb and a number of other cases ~) it is somewhat too shallow for the 160+160 ~. Also shown in fig. 7 and denoted by U~ is the potential calculated by Brink and Stance 19). This potential that contains the AS effects in an approximate (ThomasFermi) manner is significantly too shallow. For a number of systems where phenomenological potentials are known their values at D~ are shown together with those of the potentials Yr,,+s and Vnna in table 1. T~ 1 Value ofthe potentiels Yis [eqs. (12) and (14)] and Y'r as [e9s" (12) and (15)j is compenses with pheaomenolo~ical potentials at D~ for different systems System i°0-1- 1°O

D}

(fm)

7.56

4o~+ao~ a~-°oZr

10.63 ~) 9.10 e)

isC +sospb

12.2')

isO,+sospb

12.67

sa~ +sospb

14.2 ~)

y,us (MeV) Y'r, (MeV) -3.24 -3.21 -3.41 -2.48 -1.03 -1.50 -0.99 -1.38 -1.70 -2.03 -1.19 -1.29 -6.19

-3.06 -3.05 -3 .24 -2.31 -1 .02 -1.41 -0.95 -1.30 -1 .60 -1.97 -1.10 -1.22 -5 .83

Yw (MeV)

-2.98')

-1.31 ~) -1.0 °) -1.40') -1.04') -5f2 ~) -1.10 ~

P~

Pr

WS R-HF GHF SMC GHF R-HF t:iaauas R-HF SMC WS R-HF SMC R-HF

WS R-HF GHF 3MC GHF R-HF R-HF R-HF SMC WS R-HF SMC R-HF

In the last two columns theemployed densities are Given. WS denotes aWoods~Sazon flt to HartreoFork dansitia 1°); R-HF denotes densifia obtained from r-epees Hertreo-Fock calculations; GHF denotes d~sitia obtained from oomflauration space Hartree-Fork calculations sad SMC denote eaperimeatally fitted shell-model calculation densifia 1 °" ~~. Qauss denotes the use of a Oaussiaa shape forthe a-0ensitywith ea rma radius of 1.49 fim [ref: sl)]. If for thevalue of D} no refer~oe is 8iven, e4" (1) was empbYed. s) ") Ref: i'). Ref. a°). ") Ref '). ') Ref.'°). °) Ref as) . 7 Ref. as). ,) Ref. ss) .

AIV'TISYMMSTRIZATION SFFBCI'3

l83

In cases where the wave functions were not availablo the kinetic energy density T was calculated in the Thomas-Fermi approximation. As shown by Brink and 3tancu [refs. 1 0 , t e) ] it can well ba approximated by whore V~p can be obtained numerically from the density p. We have tested this approximation and have indeed found that its inaccuracies are never larger than 20 ~. Generally the potentials VN,~ calculated with Hartree-Fock densities agree within 2S ~ with the phenomenological potentials . An exception occurs in the case ~Ha+ 9 Zr, whom the deviation is SO ~. However, usingfor the a-particle the same Gaussian ° density as that' employed by Eisen and Day') improves the agreement considerably. The potentials calculated with experimentally fitted shell-model densities 1 s.l 9) show greater deviations. This is partly due to the use of the TF approximation for T (see above) but illustrates also the sensitivity of the results to the density distribution used as rocently also stressed by Satchler s). Against these uncertainties the influence of the Coulomb exchange potential calculated in the Thomas-Fermi approximation,

is negligible. For the system 160+~° aPb its value at D} is -0.002 MeV.

Fia. 8. Ratios of the elastic scattering cross section to the Rutheifoid (non-aymmeuized) cross sectionas a fhnction of the scattering angle. The expeaimental points are takenfrom Baures is ref. a%. wS refers to a phenomenological woods-Saxon St a°) and Y,,,s is the potential given is the text .

18 4

J. FLBCKNSR AND U. MOSEL

We have also investigated the possibility that the normal Skyrme parameters, obtained from fitting equilibrium properties, may not be appropriate to the low-deasity regions at the barrier . Using for a first orientation fig. 2 in ref. ~~) we have found no significant changes in the potential. For the system 4°Ca+~°Ca we made a Woods-Saxon parametrization of the tail of the potential Yrua with the parameters Y ~ 41 .0 MeY, R = 8.84 fm and a = 0.495 fm. For a first orientation we took for the imaginary potential W the same shape, but a smaller depth (30.0 MeV). Tho angular distributions calculated with this potential give a fit comparable to that obtained with the phenomenological potential with Y = 35.0 McV,'W = 12.13 MeV, R = 9.23 fm and a s 0.430 fm [ref. s°)], especially in the high energy case where the deviations in the region 40° to 70° are generally less than 20 ~ (see fig. 8). This fit is significantly better than that obtained with the ITsY Potential s °). It is thus our conclusion that the potential VN,~ (eq. (12)) that consistently neglects all AS effects between the two ions reproduces the experimentally determined potentials quite well. 7. Sammary It was the aim of the present work to examine the theoretical foundations of the folding method to calculate heavy ion potentials . In sect. 2 we have shown that this method can bo based on two different theoretical foundations, the Born-Opponheimer approximation (or the generator coordinate method) in which the HI potential directly gives the total interaction energy or the resonating group method in which the HI potential does not have such a simple classical meaning. In spite of these differences, however, in both methods the folding model emerges only if antisymmotrization efforts between the twô ions can be neglected. It was furthermore discussed that neglect of local exchange terms is not synonymous with the neglect of all antisymmetrization effects. Indeed, the AS effects in spin and isospin space are usually taken into account if an integration over these variables is performed. 1'his distinction has to be clearly recognized if folded potentials are to be calculated from effective nucleon-nucleon forces that are also used in nuclear structure calculations and that contain exchange operators. Our study of antisymmetrization effects has shown that there are three different contributions from AS to the HIpotential: Firstthe effect on the matrix element itself, which is repulsive over the whole distance range, second the distortion of the densities which gives a repulsive contribution to the potential for small distances and an attractive one for larger distances, and third, in the Born-Oppanheimer approximation, an increase in the kinetic energy . Altogether the affect of antisymmotrization is not negligible, repulsive, and amounts to ~ 70 ~ of the potential without AS. It should be noted that the second effort, the Pauli distortion, has so far boon completely no-

ANTI3YMMETRIZATION EFFECT3

18S

glected in all folding model calculations . We have also found from a comparison with two center Hartrce-Fork calculations that giving up thefrownconfiguration assumption deepens the potential considerably . Also a dependence of the folded potential on the densities employed was found. We have also shown for the model system 160+ 160 that the AS effects become sizeable as soon as the nuclear interaction sots in. This result was found to bo independent ofthe particular nucleon-nucleon interaction used. Also it was found that the relative importance of the AS effects does not depend on the relative masses of the two reaction partners. If should be kept in mind that all those considerations hold only for small energies above the Coulomb barrier where the relative motion at the barrier is slow. For larger relative energies the nucleons in the two ions become localized in momentum space and thus the importance of the AS effects diminishes lo.as.a s), The alternating signs of the various AS contributions to the HI potential imply a rather strong sensitivity of the final potential on the specific assumptions made for its calculation, in particular for the treatment of the AS terms. This fact together with the strong influence of the frozen configuration assumption makes the inclusion of only some higher order changes into the folded potential questionable if one starts from a fixed effective nucleon-nucleon interaction. We have shown that the repulsive effects of the A3 are practically cancelled by the attractive effects of the nuclear polarization. Täis is the reason wl~y a new expression for the HI potential that consistently neglects all A3 effects gives good agreement with empirically obtained values at the strong absorption radius . We thus propose to use this expression Yr+s [~qs. (12), (13) and (14)] for a calculation of the HI potential. The kinetic density s that appears in it can be satisfactorily approximated in the Thomas-Fermi model if only the densities and not the wave functions themselves are available. We gratefully acknowledge the assiataace of G. Sauer and P. G. Zint in providing the Hartree-Fork densities for us. We are also grateful to B. Day and W. Scheid for valuable comments and suggestions on the topics treated in this paper. Finally we thank G. R. Satchler for numerous stimulating discussions and for providing us with his results and densities before publication.

DERIVATION OF THE 3KYRME FUNCTIONAL WITHOUT ANTiSYMMSTRIZATION

In this appendix we will outline the derivation of the potential Y~ that does not include any AS effects between the two nuclei . To this purpose we start with the corresponding wave function 7F ~ ~i~~

(A.1)

186

J. FLBCKNER AND U. M03EL

for the total system . Here ~i and ~r are Slater detornninants describing the left and right nucleus. Note that lP does not have any definite symmetry under exchange of particles from the left to the right nucleus. With this wave function the interaction energy is given by V~ _ ~
(A.2)

Wo now proceed to evaluate eq. (A.2) forthe Skyrme force in complete analogy to the work ofBrink and Vautherin 9). The fact that the matrix element in eq. (A.2) is not antisymmetrized is being taken carp of by replacing the AS operator 1-P,P~P~, in ref. 9) everywhere by the unity operator 1. We restrict all our considerations to the case of doubly oven interacting nuclei. This means that because of time-reversal invariance of ~ the spin~xchange operator P contained in the to term of the Skyrme force, can bo replaced by }. We begin with that part of the potential which is proportional to to Vo ~ ~
(A.3) (A.4)

The integrand ia therefore

Ho = to(1+~Xo)P~Pr"

Next we consider Vl :

Vl = --~tl ~ ~rIIS(ri -r s)('Vi+2V 1Vs +Vz)Irl)+h .c., r{

( (A.~

where h.c. denotes the hermitian conjugate of the preceding form. We obtain Vi = -~t

iJ

z (~PiV Pr - Pltr+~PrO~Pi -Prz~-~OPeOPr)d3r+h.c.

(A.7)

A partial integration of the Iast term and the fact that all terms in the equation are real gives the result

Hl ° ~ti(4Pt~r+4PrT~-3Pi~~Pr-3Pr0~Pi)"

(A.S)

The contribution proportional to t2, Vs = ~ < rlI tsk~8(rl -rs)kl rl)

rl

(A .9)

ANTISYMMETRIZATION EFFECTS

187

yields after a partial integration of the last term the expression (A .11) Hs = ~dts(4Pir.+4PrTt+PiOsP.+P.D~Pt)" For the contribution proportional to t3 we first consider the density-dependent two-body forx. With 1 +P, _ ~ we obtain V3 = âta ~PPiP.d3r, J

(A.12)

H3 a ~tsPPtP."

(A .13)

~ = i3S(rl - rs~(rz -raxl -P>tp~e)~

(A .14)

and therefore

Now we. calculate the matrix element with the three-body force. The throe-body operator acts between the nuclei like a one particle operator, inside each nucleus however like a two-body operator . Therefore the matrix elements of nucleons of the same nucleus are still antisymmetrized. We include this AS into the interaction by the factor 1-P~Pt F,: ~rll'~V~rl1'~+ ~ ~ ~}
. u"

H3 = ~t3fPPaPp+~P(Pa-Pp)~Pa -Pp~-(PPaPp)!-(PP.Pp~~'

(A.1~

Collecting the expressions Ho, Hl and H2 and regrouping the terms yields eq. (13) for H°~. The expression for H3 is already in the shape of eq . (14), and eq . (15) is obtained if in eq. (A.16) pa = pp = ~p is assumed. Rei~eocea 1) G. R. Satchler, Proc. Int. Conf: on reactions betwroen complex necld, Nashville 1974 (NorthHolland, Amsterdam, 1974) p. 171 2) G. R. Setchlar, Pros. Conf. on macroscopic aspects of heavy ion collisions Argonne 1976, Argoane National Laboratory Report ANL/PHY-76-2, P" 33 3) C. Toepffer, Classical and quantum mechanical aspects of heavyion collisions (Springer, Berlin, 1974) p. IS 4) T. Fliessbach, Nucl. Phys. A194 (1972) 625 5) K. wildermuth and w. McQure, Springer Tracta in Modem Phyaip, vol. 41 (Springer, Berlin, 1966) 6) U. Morel, Proc. Conf. oa macroscopic aspects of heavy ion collisions, Argonne 1976, Argonne National Laboratory Report ANL/PHY-76-2, P. 341 7) Y. Einen and S Day, Phys. Left. 63B (1976) 253 8) G. R. 3atchler, Oak Ridge preprint, 1976, to be published 9) D. Vautheria end D. M. Brink, Phys. Rev. CS (1972) 626 10) D. M. Brink and F. Stance, Nucl. Phys. A243 (1973) 175 I1) T. Fliessbach, Z Phys . ?38 (1970) 329

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J. FLSCKNSR AND U. MOSEL

12) P. G. Zint and U. Mosel, Phys. T.ett. S® (1975) 424 13) P. G. Zint aad U. Mosel, Phys. Rev. C14 (1976) 1488 14) . H. Flocard, P. Quentin, A. IC. Kermaa and D. Vantherin. Nncl. Phys. A203 (1973) 433 1S) J. B. Hall, G B. Falmer, B. & Graa, M. L. Halbert, D. G Haneky, G A. Lu~~+~.n - M. J. Saltmareh and G. R Satchlar, Nncl. Phys. A231 (1975) 208 16) L Reicheteia aad Y. G Tana, Nucl. Phys. A139 (1969) 144 17) R H. 3iemssen, H. T. Fortune, R Malmin, A. Richter, J. W. Tippie and P. P. SinBh, Phys. Rev. Lett. 2S (1970) 536 18) F. Stancu and D. M. Brink, Nucl. Phys . A170 (1976) 236 19) G. R Satchkr, private communication 20) H. Doubre, J. G Jacmart, $. Plaßnol, N. Puffe, M. Rion and J. G Roynette, Oissy pa+eprint iPNO-PhN-76"22 21) J. A. Koepke, R E. Bmwn, Y. G Tanß and D. R Thompson, Phys . Rav. C9 (1974) 823 22) Y. 13isen, B. Day aad l3. Friedmann, Pltys. Lett . S6B (1973) 313 23) R Vandenbosch, M. P. Webb, T. D. Thomas, 3. W. Yates and A. M. Friedmann, Pliys . Rev. C13 (1976) 1893 24) J. R Birkehmd, J. R Hnizenaa, H. Fniabbea, K. L. Wo1P, J . P. Umk and V. 8. Viola, Phys. Rev. C13 (1976) 133 25) T. Flieubach, Z. Pbys. 247 (1971) 117 26) G. H. Goerltz aad U. Mosel, Z. Phys. A277 (1976) 243 27) J. W. Neeek aad D. Vautharin, Phys. Rev. CS (1972) 1472