ComputerPhysicsCommunications28 (1983) 275—286 275North-HollandPublishingCompany

REAL AND IMAGINARY PART OF THE HEAVY ION OPTICAL POTENTIAL FROM AREALISTIC NUCLEON-NUCLEON INTERACTION *

AmandFAESSLER,L. RIKUS andR. SARTOR **

Institutefor TheoreticalPhysics, Universityof Tubingen,7400 Tubingen1, Wilhelmstrasse7~Fed. Rep.Germany

ReceivedI May 1982

PROGRAMSUMMARY

Titleofprogram:RIHIOP Methodofsolution

Volumepart: The volume partcan bewritten as a five-dimen-

Cataloguenumber:ABPL sional integralwhich is calculatednumerically usingGaus-sianquadrature.

Programobtainablefrom: CPC ProgramLibrary, Queen’sUni- Surfacepart: The surfacepart is written asa sumover contri-versityof Belfast,N. Ireland(seeapplicationform in this issue) butions from each surface excitation state. The related

Greensfunction is expandedin a local planewave expan-

Computer: UNIVAC 1100; Installation: Computing Centre, sion involving a sum over partial waves.The localizingUniversityof TUbingen integral is calculatednumerically using Gaussianquadra-

ture.Operatingsystem:EXEX LEV. 37R2A Restrictionson thecomplexityof theproblem

The number of ion—ion separationdistancesat which theProgramminglanguageused: FORTRAN optical potential can be computedis limited to 20. However

(see input description below), any increaseof this limit canHigh speedstoragerequired:33 K words easilybe implemented.Theprojectilemomentumis limited to

lessthan0.5 fm i/nucleon.For thesurfacecontributionsonlyNo. of bits in a word: 32 isoscalarvibrational states of multipolarity x = 2, 3, 4 are

allowed.Isovectorstatesof A = 1, 2, 3, 4 canbe includedif thePheripheralsused: card reader,line printer Coulombexcitation mechanismis turned on. A total of 10

excitationstatesonly canbe included(20 if thetwo heavyionsNo. of cards in combinedprogram and testdeck: 1918 areidentical).Theheavyions arerestrictedto evenmassnuclei

with groundstateangularmomentumzero.Cardpunchingcode: ASCII

Typicalrunning timeKeywords: nuclear, heavy ion, complex optical potential, The calculationof the grid of valuesof the volume potentialvolume and surfacecontributions requiredas input for thesurfaceexcitation subprogramtakes

approximately 5 mm for eachof the test examples.For eachNatureofphysicalproblem point requiredby theuseran additional40 s shouldbeallowedTheprogramcalculatesthevolume andsurfacecontribution to for.both the real and imaginary parts of the optical potentialbetweentwo heavy ions. The volume part is computedby adoublefolding procedurewhich usesa complexeffectiveforce Referencesas input [1—3]while thesurfacepart is obtainedasthesecond

[I] A. Faessler,T. Izumoto, S. Krewald and R. Sartor,Nucl.order termof theFeshbachexpressionby taking explicitly intoaccountthe excitationof a setof intermediatevibrationalstates Phys.A359 (1981)509.[2] R. Sartor,A. Faessler,SB. Khadkikar and S. Krewald,[4]. NucI. Phys.A359 (1981)467.

* Supportedby the Bundesministeriumfür Forschungund [3] R. Sartorand A. Faessler,Nuci. Phys.A376 (1982)263.

Technologie. [4] SB. Khadkikar,L. Rikus,A. Faesslerand R. Sartor,NucI.

ChercheurIISN. Phys.A369 (1981)495.

001O-4655/83/0000—0000/$03.0O © 1983 North-Holland

276 A. Faessleret al. / Complexheavyion opticalpotential

LONG WRITE-UP

1. Introduction

Theoptical potential(V~)providesoneof the mostuseful toolsin the studyof nuclearreactions.Untilrecently[1], the double folding procedurewasonly used to constructthe real part of fr~,starting from anon-singularphenomenologicalnucleon—nucleoninteractionor at best from an effectivenucleon—nucleoninteractionderivedfrom a G-matrix associatedwith a sphericalFermi sea. As emphasizedin refs. [2—12],aheavyion reactionis bettercharacterizedlocally asthe collisionof two interpenetratingnuclearmatters.Inmomentumspacesuch a systemis associatedwith a Fermi seacomposedof 2 spheres(see fig. I). Thecorresponding G-matrix is no longer Hermitian since real 2p—2h excitations can now occur. As aconsequence,one canachieve[8,13]a complexeffectiveinteractionfrom sucha G-matrix and haveaccessto boththe realandimaginarypartsof the volume contributionto the optical potential [9,131.

The nuclearmatterapproachjust describedis of courseinadequateto investigatethefull surfacepart of

~pt~ The presentprogramalso computesthe contributionto the surfacepart of ~ originatingfrom thesecondorder term of the Feshbachexpressionin which we takeexplicitly into accounta set of intermediatevibrationalstates[10].

2. Theory

2.]. Volumecontribution to the opticalpotential

The volume part of the optical potential l’~ (D) at separationdistanceD is definedas the difference

= V(D) — V(cc) (2.1)

betweenthe interactionenergy V(D) at distanceD andthe interactionenergyV(cc) at infinity.

The interactionenergy V( R) at distanceR is givenby the doublefolding integral:

V(R) = ~fdrld r2p(r1)p(r2) Veff[ r12, p(R~2),T~2~(R

12),Kr] (2.2)

with r12 = 1r1 — r2j, R~2= ~(r1+ r2), p(s): densityof the totalheavyion systemat point s in nucleons/fm3,

K,. kF2

Fig. 1. Momentumspacerepresentationof thelocal descriptionof a heavyion reactionas thecollision of two interpenetratingnuclearmatters.The separationdistanceKr betweenthe centersof thetwo spheresis identified with the projectilemomentumper nucleon(targetassumedat rest). The radii kFI andkF2 aresuch that thelocal valuesof thedensityp and the intrinsic kinetic energydensity

arereproduced.

A. Faessleret al. / Complexheavyion opticalpotential 277

r~2~(s):intrinsic kinetic energydensity[5] of the total heavyion systemat point s in nucleons/fm5,Kr:projectilemomentumpernucleon(targetat rest)in fm ~, ~ complex,p, 7(2) andKr dependenteffectiveforce [8,13]in MeV.

In eq.(2.2) the r1 andr~integrationsextendoverthe whole volume of the system.

The complexeffectiveforce J’~ff(r,p, ~ Kr) is written as(x = 0.7r):

6 6

-~-(l_e2x)2e3x ~ A~Xm= ~ Amv,~(r) ifr< 1 fm, (2.3)

X ,n=O m=O

7 7

~!~(1_e2x)2 ~ Zme_rnx= ~ Z~V,~(~) ifr> 1 fm, (2.4)m=1 m=1

wherethe p, T~2~andK,. dependencesare entirely containedin the Am andZm coefficients.Eq. (2.2) canthus be written as a sumof integrals:

v(R) = ~ fdRi2Am[ p(R12),T~

2~(R12),Kr] f

1dri2~(IRi2+ 4r121)p(1R12— ~r12I)v,~(r12)

in 0

+ ~ ~ fddRi2Zm[ p(R12), T~2~(R

12),Kr] f dr12p(1R12+ ~r12I)p(IR~2— ~r12I)v,~(r12).

(2.5)

Sinceweonly considerfrontal collisions,the integralsin eq. (2.5) are 5 dimensionaldueto the cylindricalsymmetryaroundthe collision axis.

ForeachR12 integrationmesh-point,thevaluesof p and r(2) arecomputedusingthe approximation[13]:

p=p1+p2 (2.6)

and

T~

2~~(4~2)(p

1 +p2)

5~3+ Kr~PiP2 (2.7)

wherep,(i = 1, 2) is the density in nucleusi at the consideredR12 point.Tablesof Am and Zm coefficientsare built into the program.Theycorrespondto a seriesof Fermisea

configurationsfor K,. = 0 and0.5 fm -

1/nucleon.Eachconfigurationis characterizedby given valuesforKr~the densityp andthevalue ic. The lastone is definedas the ratio betweenthe shadedvolume of fig. Iandthe total volume of the Fermi sea. Werefer to ref. [13] for the list of configurationsat which Am andZm havebeendeterminedfrom the G-matrix. We note,however,what the K,. and7(2) dependencesof ReAm andRe Zm havebeensuppressedsincewe noticedthat they led to negligible effectsin the realpart ofj,), The values of Am and Zm at R

12 are computedby Lagrangeinterpolation through the above

mentionedtables.In the Kr-range(0—0.5) fm

1/nucleon,the programassumesthat the imaginarypart of Vj~7is a linearfunction of energy.

All integrationsare done by Gauss—Legendrequadrature.For that purpose,the integral to infinityappearingin eq. (2.5) is truncatedat r = 10 fm, which causesno troubledue to the fast decreaseof the V,n>

functions.

2.2. Surfacecontribution to theopticalpotential

Momentumconservationin nuclearmatterdoesnot allow lp—lh excitations.Thus while mutual lp—lhexcitationsof the two nuclei (i.e. a 2p—2h excitationof the system)are includedin ~ termsdue to a

278 A. Faessleret at / Complexheavyion opticalpotential

lp—l h excitationin only onenucleusmustbe calculatedseparately.Mostof theselp—lh excitationsof the2-ion systemcan be treatedby allowing collective surfacevibrational statesin each nucleusseparately.Labelling the excited statesby multipolarity X (projection~i) and the nuclei by n = 1, 2, the surface

potential canbe written as,[10]

= ~ K0IUOIX)G~~(XIUOI0~, (2.8)A,n

where ~ = Re andthe Greensfunction ~ canbe expandedin partial wavesas,

G~(r,r’) = 2 ~F,(kAr<){G/(kAr>) +iF1(k~,r>)]~ ~ (2.9)

The regularF,(r)) andirregular (G1(r)) functionsare radial solutionsof the SchrOdingerequationwithpotential U0 anda wave numbergiven by

k~k,~2~tsEx/h2, (2.10)

whereE~is the energy of the surfaceexcitation,~sthe reducedmassaand

k~=2!.LEcm/h2 (2.11)

is the wave numbercorrespondingto the initial energyof the reaction.Assumingthat the potential ~ ismediatedby the surfacesof the two nuclei andallowing one nucleusto be deformedin the usualway (i.e.

= R~0(l+ ~a~Yx~,)) we find

(0IL~(r)IX)= ($A/V2X+ 1 )x~(r)Yx~,(F), (2.12)

where/3,~,is the usualcollectivemodel deformationparameterand the vibrational form factor is given by

= — (öU0/~r),,.0R~0. (2.13)

The potentialwhich resultsfrom theseexpressionsis non-localandis localizedusing the Perey—Saxonprescription[14],

= fe .reK.r’V)(r, r’)dr’, (2.14)

whereK is chosento satisfy,

K2 = k~— 2~tU

0(r)/h2. (2.15)

(In principle the choice of K involves the localizedsurfaceexcitation potential itself but as this wouldinvolve a self-consistingproblemrequiringan iterativesolution the aboveapproximationhasbeenmade.)The final algebraicresult evaluatedby the programis [18]

fr~~(r)=——--~--~ ~ f3~k~(2l+~ (2.16)2irh L,I,Xn

with

= x/(r)JL(Kr){ G/(kxr)j’jL(Kr~)Xfl(ri)F,(kxr~)r~2dr~

+ Fi(kxr)j JL( Kr’)~~( r’)G1(k~r’)r’

2dr’, (2. 17a)

A. Faessleret al. / Complexheavyion opticalpotential 279

V’(r) = Xfl(r)jL(Kr)F,(kxr)fjL(KrF)Xn(r~)F,(kxr~)r~2dri, (2.17b)

in whichjL(Kr) is the regularsphericalBesselfunction.The regularandirregularsolutions,F, andG,, are evaluatedin the local planewave approximation

F,(kXr) =j,(kx(r)r), G,(kXr) = n,(kx(r)r), (2.18)

wherein

h2k~(r) = Ecm— — Re(J’~(r)). (2.19)

The integration,usingGauss—Legendrequadrature,is cut at RMAX = 3.9(A~/3+ A’2~

3)andis split intosegmentsto enablethe integrationpoints to be denserin regionswhere the volume potential changesrapidly.

The relative strengthsof the contributionsfrom different surfacevibrations are proportional to thesquareof the collective deformationparameter$,,. If fl~is assumedto be proportional to the massmultipolemomentfor a uniform distribution of radiusR it satisfiesthe sum-rule[15]

~E~1/3~1=~X(2X+l) h2, (2.20)

2mR

The classicalsum-rulesfor the massmultipole moments(isoscalar)for a uniform massdistribution ofradiusR are given by [15]

3Ah2

S(X)=—XR2~2 forX>’l. (2.21)8irm

Introducingthe parameterIn for the fraction of sum-rulestrengthtakenby an excitationn we find,

= (~~)2~(x) (2.22)

The massdeformationandthe reducedelectromagnetictransitionmatrix are relatedby [16]

= (3ZeR~/4~-)2B(EX;0 —+ A,,), (2.23)

whenR~is identifiedwith R, the effectivesharpcutoff radius,in the program.R is calculatedfrom theexperimentalnuclearrms radius (input).

Isovectormodescanbe includedin the calculationby turning on the Coulombexcitationmechanism,but normally this contributionis small. It modifies the vibrational model form-factor to

N 3Z1Z2e

2f~(r)=~ (r)+(

2X l)~) (2.24)

wherexN(r) is the nuclearform-factor given by eq. (2.13).The sum-ruleused for the isovectordipole isthat given by Satchler[15]

S(A=l;T=1)=-~-_~f, (2.25)

wherecorrectiontermsfor non-commutingoperatorshavebeenignored.

280 A. Faessleret al. / Complexheavyion opticalpotential

3. Programdescription

Threeoptionsare availableto the user.In eachcasehe hasto provide the separationdistancesat whichhe requiresthe optical potential.Option 1: Only the volume partof the optical potential is computed.Option 2: Both the volume andsurfacepartsare computed.Option 3: The surfacecontributionis computedfor a realvolume part providedby the user.

3.]. Organigram

The organigramof the programis shown in fig. 2.

3.2. Main program

The main programcalls the INPUT subprogramand the VOLUME and SURFACE subprogramsaccordingto the chosenoption. It also prints out the resultsof the computation.

3.3. Subprogram

INPUT Readsin the datacards(seethe input descriptionin section4). It issuesa warning when theprojectilemomentumpernucleondefinedby

K,.=~J~b (3.1)

is greaterthan0.5 fm~/nucleon.In (3.1), mis the nucleonmasswhile M2 is the massnumber

of the projectile.VOLUME Computesthe volumepart of the opticalpotentialasdescribedin section2.1. The tablesof A,,,

andZm coefficientsare built into this subprogram.LGAUSSLG9R Generatethe Gauss—Legendremeshpointsandweights.

~.1AIN

L~~T VOLUME ______

~ETAUHU~TERHUND

RHORHO

Fig. 2. Organigramof theprogram.

A. Faesslereta!. / Complexheavyion opticalpotential 281

FCTRO 1 Computesthe matterdistributionin nucleus1.FCTRO2 Computesthe matterdistributionin nucleus2.TAU ComputestheAm andZ,,, valuesfrom Pi’ P2 andK,. by LagrangeinterpolationthroughtheA~

andZm tables[seeeqs.(2.5) to (2.7)].HUNTER Computesthe radii kFi andkF

2of thenon-sphericalFermi sea(see fig. 1) from given valuesof K,., p and7(2) as neededby the TAU subprogramto interpolatethroughthe Am andZ,,,tables.

HUND Computesp and7(2) for a non-sphericalFermiseadefinedby K,., kFi andkF2(seefig. 1).RHORHO Computesthe angularintegrals

fd?~2p~(lR12+ ~ri2I)p2(IRi2— ~ri2I) (3.2)

which appearin eq. (2.5).RHORO1 Computesthe angularintegrals

I IIj u1~12PI~ 12 2r12 Pi~ 12

which appearin eq. (2.5).RHORO2 Computesthe angularintegrals

+ ~r12I)p2(IRi2— ~ri2I) (3.4)

whichappearin eq. (2.5).DREAL Computesthe realpart of a complexnumber.DIMAG Computesthe imaginarypart of a complexnumber.SURFAC Computesthe surfacevibrationcontributionto theoptical potentialasdescribedin section2.1.

The summationsover all partial waves and state quantum numbersand the localizationintegralare carriedout in this subprogram.

INITI8 Outputsthe preliminary information suppliedto SURFACandevaluatesthe strengthof thesurfacevibrationsCON(IS).

SETPT Setsup the integrationpointsby calling LGAUSS.Theintegrationis split into a numberofsegmentsto assurea higher density of integration points in the regions where the volumepotential changesrapidly. The parameterscontrolling the segmentationare in common/IN-TGL/ and the associatedDATA statementINPUT.

SETUP Evaluatesthe local potential array POTLCL, the potentialarray required for the localizingintegral RPOT and the correspondingderivatives DPOT and DRPOT required for the

collectivemodel matrix elements.XSET Evaluatesthecollectivemodel matrix elementsXR andXRPRYM for isospin= 0 and 1.PAR3J Computesthe angularmomentumcoupling coefficients.SUMRUL Computesthe classicalsumrules.ELECTO Computesthedeformationparameters/3~andelectromagnetictransitioninformation from the

classicalsumrules.POTFN1 Setsup the tablesof potentialsrequiredfor POTFNandDPOT.POTFN Readsthe value of the potentialfrom common/TABLE/.DPOT Readsthe value of the derivativeof thevolume potential from common/TABLE/.DSPLIN Computesthetabled potentialsandtheir derivativesby fitting a splineto the volumepotential

[14].REGBES Calculatesthe arraysof regularsphericalBesselfunctions requiredup to order LMAXIN. If

282 A. Faessleret a!. / Complexheavyion opticalpotential

this is greaterthan the order LLMIT at which the Besselfunction is less than I—lOU, LLMITis returnedasa new LMAX.

RNEUBS Calculatesthe arraysof irregularsphericalBesselfunction required.ERROR Tests the convergenceof the Besselfunction routinesandprints an error messageif required.

4. Input and output

4.1. Input

The targetand projectile are referredto as nuclei 1 and 2, respectively.The targetis definedas thebiggestnucleusandits rest frame is identifiedwith the laboratorysystem

Card 1, format (2014)Ml massnumberof target,M2 massnumberof projectile.

Card2, format (7F10.5)ELAB lab energyin MeV.

Card 3, fomat (2014)NRUS numberof separationdistancesat which the optical potential is required.NRUS shouldbe lessthan20. However,onecaneasily increasethis limit to NRUS > 20 by replacingVRRR (40)by VRRR(NRUS + 20) in the COMMON/DATA/ which appearin subprogramsINPUT, VOLUME, FCTRO1andFCTRO2.

Card4, format (7FlO.5)VRRR vector containing the NRUS separationdistancesin Fermi at which the optical potential isrequired.

Card5, format (2014)IUPT = 1 computationof volume part only,

2 computationof volume and surfaceparts,3 computationof surfacepart from a volume part provided by the userat equally spaced

points.Card 5a, (whenIUPT = 3), format(2014)

YVOL numberof equallyseparatedpointsat which the useris providinghis real volume part for theoptical potential.NVOL shouldnot be greaterthan20.

CardSb, (whenIUPT = 3), format (7F10.5)XPOT vectorcontainingthe NVOL separationdistancesin Fermi at which the userprovideshis realvolume part of the optical potential.Theseshouldbegin at R = 0 and extend to the tail of the inputpotential.

Card5c, (whenIUPT = 3), format (7F10.5)VREAL vectorcontainingthe NVOL providedvaluesof the realvolumepart of the optical potential inMeV.

Card6, format (2014)IOPT = 0 the matterdistributionsof targetandprojectileare providedpoint by point,

= I the matter distributionsof targetandprojectileare given asFermidistributions:

(4.1)

Card6a,(whenIOPT = 0), format (2014)

A. Faessleret a!. / Complexheavyion opticalpotential 283

NRHO1 numberof r pointsat which the densityp1 (r) is providedfor the target.Card6b,(when IOPT = 0), format(7F10.5)

Ri D vectorcontainingthe NRHOI r-valuesat which p1 (r) is provided in Fermi (fm).Card6c, (whenIOPT= 0), format (7FlO.5)

RHO1 D vectorcontainingthe NRHO1 p1 (r)-values(nucleons/fm3).

Card6d, (whenIOPT= 0), format (2014)NRHO2 numberof r pointsat which the densityp

2(r) is providedfor the projectile.Card6e,(whenIOPT= 0), format (7F10.5)

R2D vectorcontainingthe NRHO2 r-valuesat which p2(r) is providedin Fermi (fm).Card6f, (whenIOPT = 0), format (7F10.5)

RHO2D vectorcontainingthe NRHO2 P2(r)-values(nucleons/fm3).

Card6g, (when IOPT = 1), format (7FlO.5)FMRO1 p

0 for target(nucleons/fm3),

FMC1 c for target(fm),FMA1 a for target(fm).

Card6h, (whenIOPT= 1), fonnat(7F10.5)FMRO2 p

0 for projectile(nucleons/fm3),

FMC2 c for projectile(fm),FMA2 a for projectile(fm).

Card7, NOSTS,IEXCOU, ITRCTL, IPRINT format (512).NOSTS numberof surfaceexcitationstates(if identicalheavyions only put in statesfor one),IEXCOU = 1 Turns on Coulombexcitation,IERCTL If the measureIERR for iterationsof the Bessel functionsis larger than IERCTL an error

messageis printed(usually put= 4),IPRINT(1)= 1 intermediatecontributionsfrom eachstateare givenat eachradius,IPRINT(2)= 1 the Gaussianquadraturepoints andweightsare output,IPRINT(3) dummy parameters.

Card8, Zl, Z2, RAD1, RAD2, RC format (4F10.5)Zl/Z2 chargeof target/projectile,RAD 1 /RAD2 root meansquaredradiusfor target/projectile(fm),RC Coulombradiusparameter(fm)/(programusesR= RC(A~~3+ A~3)).

Card8a,(if IUPT = 3) IPOTYP,VR, ADIFF, RO format (15, SF10.5)IPOTYP = 1 volume potentialcalculatedfrom input Woods—Saxonparameters,VR Woods—Saxonpotentialdepth (MeV),ADIFF Woods—Saxonpotentialdiffusenessparameter(fm ‘),RO Woods—Saxonradiusparameter(fm),

(programusesR = RO(A~/3+ A~3).Card 9 (LAMDAB(15), ITAU(13), NUCNO(15), EBETA(l3), SREX(15), IS = 1, NOSTS) format (315,

2F10.S)LAMDAB( i) multipole of statei,ITAU(ii) isospin of statei,

NUCNO(i — = 1, 2 numberto which nucleusstatei belongs,EBETA(i) Excitation energyof statei in MeV,SREX(i) Fractionof sumruleexhaustedby statei. Normalized to 1 for full exhaustion.

4.2. Output

Theoutputfrom the two halvesof theprogramare treatedseparately.For IUPT~ 3 theprogramprintsout a bannerdescribingthe reactionandthenoutputsinformation describingthe densityfunctions.

284 A. Faessleret a!. / Complexheavyion opticalpotential

For IUPT � I this is followed by a secondbannerfor the surfacecontributionpart signifying that thevolume potentialhasbeencalculated.Thisis followed by the Woods—Saxonparametersdescribingeitherthe volume potential provided by the user (IUPT = 3, IPOTYP = 1) or the tail fitted to the potentialcalculatedby VOLUME. Thencomeself-explanatorytablesdescribingthe two nuclei, thevaluesof therealpart of the volume potential (which are usedas input for an interpolationwith a splineroutine) andthesumrules(NOTE: in the caseof identicalnuclei all sumrulesetc are the samefor bothnuclei althoughonlythosefor nucleusno. I are printedout).

Then, for IPRINT(2) * 0 the Gaussianquadraturepoints and weights are output. This completestheinitial stageof SURFAC.If IPRINT(l) * 0 the contributionsfrom eachstateare givenat eachradiuspointrequiredby the user. Also includedare the local wave numbersat radiusR for no-excitation(K-INFIN-ITY) andfor the excitationof the state(K-AT-R) anda tableof the variousangularmomentumlimits inthe program.Themostimportantvalueis LMX1 which is the maximumpartial wave.(It shouldbe smallerthan the other values in the samerow otherwisethe automatic limits for the various DO loops in theprogramhaveto be increased.)

The final output is the optical potentialat the points requestedby the user.

4.3. The testruns

Two testruns are included.The first is 60 +40Ca, Elah = 60 MeV with simpleWoods—Saxondensityfunctionsandtwo surfacevibrational statesin eachnucleus.

The secondexample,160 + 160 usesdensityfunctionsinput point by point and includes2 collectivesurfacevibrational states.(Theseare automaticallyincluded for both nuclei.) The giant isovectordipoleresonanceis excitedby the Coulombinteraction.

References

[I] G.R. SatchlerandW.G. Love, Phys.Rep. 55 (1979) 183, and referencestherein.[2] D.A. Salonerand C. Toepffer,Nucl. Phys.A283 (1977) 108.[3] G.F. Bertsch,Phys.Rev. C15 (1977)715.

[4] F. Beck, K.H. Muller and H.S. Kohler, Phys.Rev.Lett. 40 (1978) 817.[5] T. Izumoto, S. Krewald andA. Faessler,NucI. Phys.A341 (1980) 319.[6] T. Izumoto, S. Krewald andA. Faessler,Phys.Lett. 95B (1980) 16.[7] K.H. Muller, Z. Phys.A295 (1980) 79.[8] A. Faessler,T. Izumoto, S. Krewaldand R. Sartor,NucI. Phys.A359 (1981) 509.[9] R. Sartor,A. Faessler,S.B. K.hadkikarand S. Krewald,NucI. Phys.A359 (1981)462.

[10] S.B. Khadkikar, L. Rikus,A. Faesslerand R. Sartor,NucI. Phys.A369 (1981) 495.[II] J.E.S.Hernandezand S.A. Moszkowski,Phys.Rev. C2l (1980)929.[12] J.N.J. di Giacomo,J.C. Pengand R.M. DeVries, Phys.Lett. lOlB (1981)383.[13] R. SartorandA. Faessler,NucI. Phys.A376 (1982) 263.[14] F.G. PereyandD.S. Saxon,Phys.Lett. 10 (1964) 107.[15] G.R. Satchler,in: Proc. Intern. Schoolof Physics,CourseLXIX (1976)eds.A. Bohr and R. Broglia.[16] F.E. Bertrand,Ann. Rev. Nuci. Sci. 26 (1976) 457.

A. Faess!eret a!. / Complexheavyion opticalpotential 285

TEST RUN OUTPUTTest run I

** *EAVY ION OPTICAL POTENTIAL CALCULATION **

SECOEDORDER CONTROBUTIONFROM SURFACE EACITATIONSCENTER OF PASS ENVRC.Y 40.C’~C0pEs

5500960005 PARAVFTERS: ~._•7~’’s.’ PADIUU= .V DIFFUSIVITY= 1.24 COULOMbRRDIUS~ 3.51

NUCLEUS 1 NUCLEUS 2

‘sss 1V,.iCHARGE 2.’.C(. 5.000

YEAS SCUARERADIUS 2.7l8’~ 2.712ijEDUIV S+ARP RADIUS 3.5~’G9 3.5C89 ETA VALUE IS 4.523712,000

FIRST ORDER POTENTIAL POINTS1 .300r 1.7022R6Q6.O’O 12 10.9786 —1.0370?6’l—CSl2 .9979 1 •35975 526’~’2 13 11.9786 —4.64345 513—0023 1.9957 6.981Q13~3.LC1 14 12.9786 —2.fl7896201—~O2

4 2.9938 l.?9663439~UflC 15 13.9786 —9.30777410—tOO5 3.9914 —4.0118R572.C01 16 14.9786 —4.16715652—t.r’36 4.9893 —5.02452374.001 17 15.9786 —1.26565029—0137 5.9271 —3.04722343* CR1 18 16.9786 —~.352636’9—C()48 6~985C —1.54635O~5.CR1 19 17.9786 —3.73951000—0049 7.982~ —3.98725752*000 20 18.9786 —1.67419355—004

10 8.9807 —8.02374669—001 21 19.9786 —7.4O543311—,~T511 9.9786 —2.31679830—001 22 20.9786 —3.35573318—S’S

CLASSICAL SUP RULE LIMIT FOR L’ TAU’ 1 NUCLEUS NO 1 Is 3.70344651+iO1ICLHSSICAL SUM RULE LIMIT FOR L 1 THU~ 1 NUCLEUS NO 1 Is 1.9958965640TT1CLASSICAL SUM RULE LIMIT FOR L U TAU’ 0 NUCLEUS 90 1 IS 5.89789066.003CLASSICAL SUM RULE LIMIT FOE L~ 3 TRU (3 NUCLEUS 90 1 Is 9.14986985*004CLASSICAL SUE RULE LI#IT FOR L 4 1AU 0 NUCLEUS NC 1 IS 1.15876942.006

CALCULATION FOR COLLECTIVE 25E ORDER OPTICAL POTENTIAL USES 2 STATES

IDENTICAL NUCLEI THEREFORETHESE STATES OCCUR IN GOTA

STATE NO LAMBDA THU ENERGY P(LAMEDA) WIDIR MEAN LIFETIME AETA(LD**2 SURRULE EXHAUSTION

4 0 24.51 9.123092.003 2.500397—010 2.632452—012 1.08i360—0(31 1.000000—0012 1 1

23~SL 4.246582—001 1.3353t.1—003 4.929348—019 3.93c839—0c3 s.oooooo—ooiCALCULATION OF 160 INTEVRATION POINTS AND ,EIG#TS COMPLETE

INITIAL STAGE COMPLETE

FINAL RESULTS FOR OPTICAL POTENTIALRADIUS VOLUMEOSLO SURFACE ONLY TOTAL OPTICAL POTENTIAL

6.00 ( —.‘66944002, —.6’424.500)( —.75856+001, —.

3SAS2~~O1)I—.44279.002, —.42494+001)7.00 C —.152O4~0Q?, —.14S51+IC’O)( —.9~678*C01, —.14065.001)1 —.24972+002, —.155204001)8.00 C —.38222+001, —.27466—OMIt —.12667+COO, —.1i1544000)( —.40689.001, —.12900.000)9.00 C —.77914.0013, —.4100C—002) C —.66923—001 • —.1(3573—103) C —.84606.000, —.42858—002)

286 A. Faessleret a!. / Complexheavyion opticalpotential

Test run 2

** HEAVY ION OPTICAL POTENTIAL CALCULATION “

SECOND ORDER CONTRIBUTION FROM SURFACE EXCITATIONSCENTER OF MASS ENERGY 17,1429MEV

SASONWOORS PARAMETERS: V~ —.11544~O5 RADIUS .00 DIFFUSIVITY= 1.13 COULOMBRADIUS= 4.14

NUCLEUS 1 NUCLEUS 2MASS 40.000 16.000CAARGE 20.000 8.000

MEAN SQUARE RADIUS 3.4820 2.718(3EQUIV SHARP RADIUS 4.4952 3.5089 ETA VALUE IS 2.06478340U1

FINAL RESULTS FOR OPTICAL POTENTIALRADIUS VOLUME ONLY SURFACE OSLO TOTAL OPTICAL POTENTIAL

3.00 C .82383+002, —.60932+001St .00000 , .00000 >1 .82383.002, —.60932+001)6.00 ( —.78655+002, —.24176+001)1 .10279+000, —.18116+001)C —.78552.002, —.42292~0C17.00 C —.54017+002, —.11078+001)1 —.41734+002, —.12325.002)( —.95751.002, —,13433~502)8.S0 C —.18113.002, —.22623+000) C —.432844001, .25609*000)1 —.22441.002, .29860—001)