Revista Mexicana de Física 41, Suplemento 1 (1995) 132-144

Fission in the relativistic mean field theoryK. RUTZ, M. BENOER

Institut für Theoretische Physik der Universitiit, 60325 Frankfurt, Germany

J.A. MARUIIN", P.-G. REINIIARot, ANO W. GREINER"

Joint Institute for Heavy Ion Research, Holifield Heavy Ion Research FacilityOak Ridge, TN 87831, USA

ABSTRACT. The symmetric and asymmetric fission paths for 240pU, 232Th, and 226Ra are inves-tigated within the relativistic mean-field mode!. Standard parametrizations which are well fittedto nuclear ground state properties are found to deliver reasonable qualitative and quantitativefeatures of fission, comparable to similar nonrelativstic calculations. Unusual potential structuresnear contact for heavy colliding systems could not be identified.RESUMEN. Las trayectorias de fisión simétricas y asimétricas para 240pU, 232Th, y 226Ra son in-vestigadas dentro del modelo del campo medio relativista. Se encuentra que parametrizacionesestándar que son adecuadas para las propiedades de los estados basales nucleares explican ra-zonablemente las características cuantitativas y cualitativas de la fisión, comparables a cálculossimilares no-relativistas. No se pudieron identificar estructuras de potenciales fuera de lo comúncerca del contacto para sistemas pesados en colisión.

PAes: 24.10.Jv; 24.75.+i; 24.90.+b

1. INTROOUCTlON

Only a subtle interplay of collective deformations and changing microscopic shell structurecan explain the quantitative details like tunneling times and fragment mass distributionin fission [1]. The macroscopic-microscopic method, based on a phenomenologically fittedshell model plus Strutinsky shell corrections [3], allowed quite early extensive investigationsof the various landscapes of collective Potential Energy Surfaces (PES), see e.g. [1,4-6].Since the early seventies, fully sdfconsistent mean-field models have been available fornudeL These are the nonrelativistic Hartree-Fock models with the Skyrme force [7] or theGogny force [8]. At about the same time competitive relativistic mean-field models fornuclear structure have been proposed [9]. Although much more elaborate, selfconsistentmean-field calculations of the PES for nuclear fission appeared rather soon for the Skyrme-Hartree-Fock models [10] and later for the Gogny force [11]' it was only recently thatPES for fission have been calculated with the relativistic mean-field model [12]. Thesecalculations have shown that well adjusted parametrizations for the re la ti vis tic mean-field model, see e.g. [13,14], deliver reasonable fission barriers comparable to those of

• Permanent address: Institut für Theoretische Physik der Universitiit, 60325 Frankfurt, Ger-many.

t Permanent address: Institnt fiir Theoretische Physik der Universitiit, Standtstr. 7, 91058 Er-langen. Germany.

132

FISSION IN TIIE RELATIVISTIC MEAN FIELD TIIEORY 133

nonrelativistic calculations, while others can be excluded. We confine the considerations tothe two parametrizations NLl from [13) and PL-40 from [15) which have proven to providereasonable fission barriers [121. In addition, we consider the parametrization NL-SH whichis claimed to be better adjusted with respect to isovector properties [161.

2. TIIEORETICAL BACKGROUND

The relativistic mean-field model is meanwhile a standard in nuclear physics, for detailedreviews see [17,18,141. The Lagrangian of the model reads

r rfree pfree + rlin + rnonlinL..RMF = L."nucleon + Lmeson L...coupl L,..coupl

e~~leon = W (i,.~8" - mn) W

ef,.. = 1 (8 <1>8"<1>_ m2 <1>2)_ 1 (lG G''" - m2 ~ V~)meson 2 f.l a 2 2 f.lV W lA

1 (1 ~ ~ 2 ~ -) I- '2 '2B~v.B~v - mpR~.R~ - 4F~vF~v

e~¡~upl= - 90<l>WW- 9wV~W'Y~W - 9pfl~.wi''Y~W - eA~wlyn,.~W

enonl;n = -21m02<1>2- U(<I»coupl

where W is the nucleon field, <1>the scalar-isoscalar field, V~ the vector-isoscalar field, fl~the vector-isovector field, amI A" the pholon field. The corresponding force tensors are

The U(<I» is lhe nonlinear funclional for lhe scalar field. We consider lwo varianls, lheslandard non linear funclional

(1)

and the slabilized nonlinear funclional [15)

U(<I» = !m~<I>2

+ L\m { Ó~2 [IOg (1+ (<1>~<I><1>0r) - log (1+ (;; r)]+ <1>0<1>(1+ (;; rrl (2)

In lhis paper three differenl seIs will be used: NLl and NL-SH wilhin lhe standardnon linear model (1) and PL-40 wilhill the slabilized varianl (2). The paramelers arelisled in Tabs. 1 and n. The seIs NLl and PL-40 are oblained [rom a fit lo ground slate

134 K. RUTZ ET AL.

TABLE1. The parameter sets within the standard nonlinear model used in this work. AH massesare in MeV and ~ is given in fm-l. AH other parameters are dimensionless.

Set 9. 9w 9p b, b3 m. mw mp mn

NL1 10.138 13.285 4.976 -12.172 -36.265 492.250 795.36 763.0 938.0NL-SH 10.444 12.945 4.383 -6.9099 -15.8337 526.059 783.0 763.0 939.0

TABLEll. The parameter set PL-40 within the stabilized nonlinear model. m~ and 6m2 are givenin fro-2 1 c5cI-and ~o are given in fm -1. The nucleon mass and the masses oí the vector mesons aremn = 938.9 MeV, mw = 780MeV, and mp = 763MeV.

set 9. 9w 9p m2 6m2 64> 4>0~PL-40 10.0514 12.8861 4.81014 4.0 3.70015 0.269688 -0.111914

properties of spherical nuclei, as explained in [131 for NL1 and [15] for PL-40. These fitstake care of the nuclear charge formfactor in terms of the diffraction radius and a surfacethickness. The set NL-SH is biased more on a smaller isovector parameter and was fitonly to the r.m.s. radii as global information on the nuclear shape [161, so that it hasless reliable surface properties. More insight into the physical properties of the models isprovided by listing the parameters of symmetric nuclear matter in Tab. III. It should benoted that for the effective mass the defiuitions differ. The relativistic models have a muchlower m*1m throughout. But the relevant quantity for nuclear structure calculations arethe single particle level densities at the Fermi surface, and these tum out to be comparableamongst nonrelativistic and relativistic sets [19]. The differences in the symmetry energiescannot be explained away. The relativistic models have a tendency to overestimate it.The force NL-SH has a more realistic value because particular attention was paid to thisobservable during the fit.

Pairing was used to determine the occupation of the states in the constant gap approachwith [20]

6= 11.2 MeV/VA.

This is a rather crude estimate. Varying pairing recipes can change the fission barriers byabout :1:1MeV [21]' which is of the same order of magnitude as other uncertainties in themodel (see below).Finally, we take into account a rorrection for the spurious centre-of-mass motion. The

standard nonlinear sets NL1 and NL-SH used the estimate

En" = ~ .41A-1/3MeV (3)

.,whereas PL-40 used a microscopirally calculated Ecm= 2(/A""'). Comparing different ver-

nIn

sions of the correction again yields an uncertainty of up to 2 MeV; in a comparison betweentheories, however, the correction can be removed.

FISSIONINTHE RELATIVISTICMEANFIELDTIIEORY 135

TAIlLE III. The nuclear matter properties E/A;: bindillg energy per nucleoll, Po ;: equilibriumdensity, K == incompressibility, m- 1m == effective nuc1eon mass, and a4 == syrnmetry energy forthe relativistic parametrizations NL1, PL-40, and NL-SH, compared with the force Skyrme M'.

set E/A(MeV] Po [fm-3] K(MeY] m'/m a4(MeV]PL-40 -16.17 0.152 166.1 0.58 41.7NLl -16.42 0.152 211.7 0.57 43.5NL-SH -16.33 0.146 354.95 0.66 36.1Skyrme M' -16.01 0.161 219.2 0.786 30.0

Desides the energy, we consider several multipole moments Qi and, for historical reasonsthe cartesian quadrupole moment Q:

Q- - J16~ Q-- 5 2.

These are computed from the isoscalar-vector density. Another observable is the mass ofthe heavy fragment

Ah = / dV po(i') for (Q3) > O.Z<Zn~ck

(4)

It can be computed only for large deformations when the fragments start to develop visibly.The multipole moments serve also as constraints in order to map the whole potential

energy surfaces. They need to be damped, however, as the pure multipole moments increaserapidly towards the edges of the numerical box, causing several unpleasant numericalinstabilities. We do this by multiplying them by a Fermi function which effectively cutsthem off when the density has gone below one tenth of normal nuclear density.

3. FISSION OF HEAVY ELEMENTS

First calculations of fissioll barriers in the actinides have been published in [121. They hadshown that the relativistic mean-field model with the parameters NLI and PL-40 can re-produce approximately the double-humped fission barrier of 240pu. The first barrier carneout about 4-6 MeV higher than the experimental value. However, it has been shown inthe macroscopic-microscopic model [221 as well as in nonrelativistic Hartree-Fock calcu-lations [23J that the first barrier is lowered by about 1-21\leV if triaxial deformations areallowed. The height of the second barrier was about 4-10 MeV to high in the previous rel-ativistic calculations. That was due to the restriction to symmetric deformation. Here wecan now present extended investigations on fission barriers which include also asymmetricshapes and octupole deformations.

136 K. RUTZ ET AL.

4.0

400

PIAONLlNL-SH

3.5

350

3.0

300

2.5\\\\\\\\\\\\\\\\..~ \

"4,. \

:':~\~:'~""""\ ~----;",\ .,\ ....•...•\ .\\

200 250Q [b]

1.5

150

D Gogny Dls• Skyrme M'• exp.

1.0

100

0.515

•10

;> 5Q)

6 oW

-5

-10240

pU

O 50

FIGURE 1. PES of 240pu for the three parametrizations as indicated. Barrier heights obtained innonrelativistic Hartree-Fock calculations with the Gogny DIs [23] and the Skyrme M' force [24]as well as experimental values fram [2] are drawn for comparison. All barrier heights are correctedfor the zero-energy contribution.

3.1. The fission path Jor 240pU

We show in Fig. 1 the PES for asymmetric fission of 240pU. The PES have been computedwith the quadrupole constraint only. The octupole deformations have been Idt free to ad.just themselves to the minimum configuration. The nonrelativistic results with the Gognyforce Dls had included an estimate for correction with the collective zero-point energy.These corrections have been removed for the comparison in Fig. 1. The experimental bar.riers are obtained from fitting parabolic barriers and minima to the measured tunnelingprobabilities. The known part of the zero-point energies corrected for in Fig. 1. Note thatthese experimental barriers are indirectly determined and contain an unknown amountof zero-point energies as well as possible contributions from multidimensional tunneling.Thus a comparison must be content with a rough proximity of the values. The energiesand deformations at the minima, the ground state and first isomeric sta te, are very wellreproduced by the forces PL-40 and NLl in comparison to the nonrclativistic results andthe data. The deviations are somewhat large for the force NL-SH but still acceptable.The height of the first barrier is comparable amongst all theories and overestimated inrelation to the experimental point, even if one corrects for the triaxial deformation. Thesecond barrier is generally better reproduced, in particular the force PL-40 comes veryc10se to the experimental point. In view of the uncertainties on the theoretical as well ason the experimental side, we see a good agreement betweell the relativistic PES from NLl

FISSION IN TlIE RELATIVISTIC MEAN FIELD THEORY 131

and PL-40 with the nonrelativistic results aud with the experimental points. The forceNL-SH provides also good barrier heights but the deformation of al! minima and barrier issystematical!y shifted to lower values compared to al! other results. This is probably dueto the higher etfective mass of the force NL-SH. The etfect of a varying etfective nucleonmass was studied in [12]: increasing etfective mass softens the barriers and shifts them tosmal!er deformations.

For very large deformations, there is a second branch of solutions visible in Fig. 1.These are energetical!y much favourable at large separations beca use the fragments areless deformed internal!y. These solutions correspond to the fusion val!ey in the col!ec-tive landscape which is distinguished from the fission val!ey by a smal!er hexadecupolemoment [23J. The situation is analogous to Fig. 3 shown for radium.

3.2. The triple-humped fission barrier for 232Th

The PES of 232Th indicate that there exist strongly stretched stages beca use they developa third fission barrier, see e.g. the macroscopic-microscopic analysis of [4J. It is due tostrong shel! corrections in the outer tail of the second barrier. The third barrier seems toexperimental!y sUpported [25). Jt is indicated e.g. by the photofission cross-section [26),or by the asymmetric angular distribution of the Iight fragment [27). Fig. 2 shows the PESfor asymmetric fission of 232Th for the three relativistic parametrizations, compared withthe barriers from nonrelativistic calculations and experimeutal!y deduced barriers. Jt isgratifying to see that al! three relativistic parametrizations are able to reproduce the thirdminumum and barrier. There seem to be robust shel! etfects which appear under widelyvarying conditions. Ilut beside this robust pattern, there are now more ditferences visibleamongst the parametrizations. The force NL-SH behaves somewhat strangely. A tendencywhich was already present in 240pU, becomes now even more disturbing: the first barriercomes out too low and al! minima and barriers are squeezed to lower deformations. Theforce NL-SH seems to be not too wel! adapted for the description of fission PES. Twoexplanations are conceivable: First, the problem can come from the higher etfective mass,as it was discussed already in connection with 240pu, and secand, the failure at low Qcomes from the less careful!y adjusted surface properties. It was observed in connectionwith the Skyrme forces that a wel! fitted surface tension is required to provide reasonablefission barriers in the actinides (29), and we find in relativistic as wel! as in nonrelativis-tic calculations that every parametrization which fits ground state properties includingthe surface thickness gives comparable and resonable first barriers. The examples hereare the two standard sets, NLl and PL-40, which are c1early more appropriate. Theiroveral! performance is fair. The second minimum is a bit to wel! bound in al! cases. Ilutthat is a common disease which is shared with the nonrelativistic models. The two PESof NLI and PL-40 develop ditferences with increasing Q. The force NLl behaves some-what better at the second minimum and at the third barrier whereas PL-40 is superiorat lhe second barrier. Ilut none of the lwo is yet ideal. Jt is a task for future investiga_tions to search for an even betler force amongsl the more versatile stabilized nonlinearparametrizations of the power-Iaw type (2), i.e. for a belter variant of PL-40. Final!y,we want to menlion that we again see the fusion val!ey which is energetical!y favoured atvery large deformalions.

138 K. RUTZ ET AL.

0.5 1.0 1.5 2.5 3.0 3.5 4.0

10

O

o Dls, " YE+WS, 1\ • exp.\ 2\. exp.

/~", '. \. exp.2(O").. -j.. ' ...._------~: ....-\ ...., ....

O........ "'. \ ' ....\ ....\ ",,. ."',\ 0,,, '., '.\ .\,,

PL-40NLlNL-SH

O 50 100 150 200 250 300 350Q lb]

FIGURE 2. PES of 232Th calculated with different parameter sets as indicated. Barrier heightsas obtained in a nonrelativistic Hartree-Fock calculation with the Gogny DIs force [23]' in amakroscopic-microscopic calculation [28], and experimental values are drawn for comparison. exp.lis taken fram [2]' exp2 fram [26). AH barrier heights are corrected for zero-point energies.

3.3. Symmetric and asymmetric fission 01226Ra

In the regio n 84 < Z < 90 symmetric and asymmetric fission appear with comparableimportance, leading to mass yields with typicaHy three peaks. These nuclei should displayan interesting competition between the symmetric and asymmetric PES. We will considerhere 226Ra as a typical example. The PES for asymmetric as well as symmetric fissionof 226Ra are shown in Fig. 3. The first surprise is that the graund state of 226Ra isasymmetric, 2 MeV lower than the nearby symmetric minimum. But the preference of theasymetric shape at low Q dissapears quickly. The symmetric shape has gained alreadyat the first barrier and the first isomeric minimum is also clearly a symmetric state.Beyond that symmetric and asymmetric PES develop very differently. The second barrierfor asymmetric fission is much lower but after the subsequent second minumum, a broadthird barrier extends over a wide range of deformations. The second barrier for symmetricfission, on the other hand, is much higher, but a second isomeric minimum and an onlyshallow third barrier follow. The second symmetric barrier has a further peculiarity: inthe range 120 b < Q < 160 b, there are two branches distinguished by the hexadecupolemoment. The shapes of the fissioning nucleus for various Q are indicated by half-densitycontours in Fig. 3. .

The problem is that the Fig. 3 does not trivially suggest a coexistence of symmetricand asymmetric fission. A few further comments are in order:

FISSIONIN TIIE RELATIVISTICMEANFIELDTIIEORY 139

0.5 1.0 1.5 ~O 2.5 3.0 3.5

226Ra O / / ~\ \

15 \I I \ 1

, I 1 lO, I 1 \0

10 I 'O \l' \ \

'> 1........ \1 8al \ 1

6 \

5 (''- / '-¡i¡ '- / \ ,

'- _/ \

OO\

O O 1\\

O asymmetric \

-5 symmetric

O 50 100 150 200 250 300 350Q [b]

FIGURE 3. Potential energy surfaces of 226Ra, calculated with the parameter set PL-40. Thereference point for the energy is the refiection-syrnmetric ground state. Solutions with asyrnmetricdegrees oC freedom are drawn as salid lines and reftection-syrnmetric solutions are drawn as dashedlines. The nuclear shapes drawn at their corresponding Q are lines of constant vector density atPo = O.07fm-3. For the ground state and the tirst barrier only the symmetric shapes are shown.

1. There is only induced fission for 226Ra at rather substantial excitation energies,e.g. from the reaction 226Ra(p,f) at 11 MeV [30]; internal excitation reduces theshell effects and thus favours symmetric shapes.

2. The symmetric second barrier will probably be lowered by allowing triaxial de-formations [5).

3. It is conceivable that fission proceeds first through the asymmetric second barrierand tunncls then towards the lower symmetric third minimum at large Q.

4. PES alone can be misleading; tunneling probabilities are very sensitive to thecollective masses in the tunneling region.

Altogether, we see that the fission of 226Ra is an intriguiug problem which most probablyrequires a fully fledged collective dynamics in at least two degrees of freedom, accountingfor triaxial shapes, complltiug carcfully the corresponding collective mass tensor, andtakiug care oC temperature cffects.

A comparison of the different forces shows the same pattern as in the two previousexamples, and we omit details. The two sets NLl and PL-40 agree with each other andwith comparable nonrelativistic models for low Q inclllding the first barrier and perhapsthc first isomeric mínimum. Differences betwccn NLI and PL-40 devclop with dcformation.The nonrelativistic Skyrme force lies in between the two relativistic results showing thatthere is no principIe difference between a relativistic and a nonrelativistic treatmen!.

140 K. RUTZ ET AL.

4. ASYMMETRIC GROUNDSTATES IN RADIUM ISOTOPES

It is an old question whether there exist nuclei whieh have a ground state with brokenrefleetion symmetry. Already in the fifties, one has observed in the aetinides low-Iyingbands of excitations with negative parity [31] whieh hint at asymmetric deformations inthe ground state [32]. Calculations within the maeroseopiei-mieroseopie method [6] aswell as nonrclativistie Hartree-Foek ealculations with the Skyrme II! force [33] or a Gognyforce [34] found asymmetrie ground states in the regio n of Ra-Th whieh are by 1-2 MeVlower than the eorresponding symmetrie ground states.

The results of our ealculations are summarized and compared to other ealculations inFig. 4. The figure clearly shows the quantitative di!ferenee between the forees. The forceNLI gives the strongest extra binding, closely followed by PL-40. The force NL-SH hasthe mueh softer oetupole e!feets, yielding even a systematieally lower oetupole momentfor the well deformed isotopes. The strength of the oetupole e!feets also determines thetransition point from symmetrie to asymmetrie ground sta tes, the stronger the e!feetthe earlier the transition. The upper right part of the figure also shows the quadrupolemoment at equilibrium. There is a jump in Q between those forees whieh have Q3 = Oand those with Q3 '1 o. Finally, in the lower right part of Fig. 4, we show the eleetriealdipole moment of the ground state, whieh seems to be related to the oetupole moment.The force NL-SH with the smaller Q3 also has the smaller dipole moment. All dipolemoments shown here are negative, in eontrast to the nonrclativistie results where thedipole moment has a di!ferent sign for most isotopes. But one has to keep in mind thatthe overall e!feet is extremely smal!. In the upper Icft part of Fig. 4, we have inserted theexperimental energies of the band-heads of the lowest odd-parity bando These energiesshould be closely related to the depth of the oetupole minimum. And that is indeed thecase, partieularly in eomparison with the behaviour of the energies for NLI or PL-40.

We show also in Fig. 4 the results from a maeroseopie-mieroseopie model [35,36].These eonfirm that oetupole dcformations in the ground state are to be expeeted forseveral radium isotopes. But the deformations at equilibrium as well as the transitionpoints di!fer substantially from the results uf the present relativistic mean-field mode!.

5. SEARCII rOR MOLECULAR MINIMA

An interesting question indirectly related to fission is that of the existenee uf minima inthe potential energy of eolliding heavy nuclei close to touching, similar to the molecu-lar states in the scattering of light nuclei. Past investigations led to conflicting results:whereas fission-like models did not. show molecular minima [41], models based on foldingpot.entials tended to yield such minima [40]. The problem is complicated by t.he possibledependence on relative orientation uf two dcformed colliding nuclei and by the problemof whether one is closer t.o the adiabatic or sudden limit. Fig. 5 shows a comparison ofthe relativistic mean-field model with parametrization PL-40 with various other modelsfor a U+U collision. Clearly all agree in not yielding a molecular minimum except for tworather inappropriate treatments: the simple liquid-drop model, which has a cusp at scis-sion due to the behavior of the surface energy, and the M3Y-folding model, which already

FISSlONIN THE RELATIVISTICMEANFIELDTHEORY 141

216 218 220 222 22A26 228 230 232 234 216218220222224\26228230232234_1.030:> -¡" exp. O PIAO

'"::s 0.5 -¡" -¡" 25 O NLI'~ -1' -J- -l- O NL-SHt:il 0.0 20 • YE+WS1

•• YE+WS2 ~•.'> ¿15 ..~~'" -1 O- ~.•/::s 10•t:il 5.",-2

t:ilO88Ra

4 0.2 ...•...'a...

0.03

-0.2N-

~ -0.4--'";02 ,,- ~'"' IO- O -0.61 I

I -0.8I

O -1.0

-1.2216 218 220 222 224 226 228 230 232 234 216 218 220 222 224 226 228 230 232 234

A AFIGURE 4. Upper left: energy of the octupole minimum relative to the state with Q3 = O inthe isotopes of radium. In addition we show the results of macroscopic-microscopic calculations(Yukawa-plus-exponential alld Woods-Saxoll) 1 [3512 [36J.Theexperimental energiesofthe band-heads of the lowest odd-parity bands are drawll at positive energies and indicated by a "1-" [37J.Octupole (lower left), quadrupole (upper right), and electric dipole moment (lower right) ofradiumnuelei in their intrinsic ground state. The deformation parameters /32 and /33 used in [35,36J wereconverted to the quadrupole and octupole moments with the approximations /32= ,J57i/(3Am).Qand /33 = 4". /(3ARJ) . Q3 with a radius of Ro = 1.2 A 1/3 fm. The sign of D is shown relatively toa positive Q3.

turns down to its unrealistic overbinding near touching alld does anyway not reproducethe saturation properties of nuclear matter.Figure 6 shows the dependence on relative orientation for the YPE model within the

sudden approximations (self-consistent calculations are stiB without reach beca use of com-putational expense and the difficulty of finding suitable constraints. \Vhereas the tendencyfor a minimum is more pronounced for the nonaxial configurations, it is definitely not suf-fident. So we believe that at present the only hope for long-lived molecular states of heavy

142 K. RUTZ ET AL.

800

750

700

~6650

'"600

550

\ ....,

\,

238 '238 .U f U axIal symmetry

YPEYPE+SCRMFTPL-40LDMYPEsuddenM3YsuddenCoulomb

242315 16 17 18 19 20 21 22Distance of Mass Centers [fml

FIGURE 5. Comparison of U + U interaetion potentials ealculated within various models. YPE("+SC" denotes shel! eorreetions) is the Yukawa plus exponential model [38]' LDM the liquid-drop rnorlel with the straightforward surface energy but a duffusCIlcss-corrected Coulomb energy,and M3Y a folding model [391.For "sudden" the densities were aBsumed to overlap unaltered.RMFT PL-40 denotes the relativistie mean-field wilh PL-40 parametrization.

nuclei may be in the stabilizing effeet of a strongiy inereased eolleetive mass parameter ofpossibly a reversible storage of energy in internal excitation.

ACKNOWLEDGEMENTS

This work was supported by the Bnndesministerium fiir Forsehung und Teehnologiethrough Contraet Number 060F772 and by the Joint Institute for Heavy Ion Researeh.The Joint Institute for I1eavy Ion Researeb has as member institutions the University ofTennessee, Vanderbilt University, and the Oak Ridge National Laboratory; it is supportedby the members and by the Department of Energy through Contraet Numher DE-AS05-76ERO-4936 with the Univcrsity of Tcnncsscc. Two of the anthars (P.-G.R. and J.A.l\I.)acknowleclge support by the the Nato Grant CRG.920122.

FISSION IN TIIE RELATIVISTICMEAN FIELD TIIEORY 143

201913

-00---00--00.....00

700

800

650

~::E 750ril

850

14 15 16 17 18Distance of Mass Centers [frn]

FIGURE 6. Potentia1s for U+U within the sudden approxiamtion and the Yukawa plus exponentialmodel for different relative orientations of the nuclei (indicated by the small plots on the lowerleft.

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