PHYSICAL REVIEW C VOLUME 25, NUMBER 1

' 5Ho+ Fe reaction at Et,b ——462 MeV

JANUARY 1982

A. D. Hoover, J. R. Birkelund, D. Hilscher, W. U. Schroder, W. W. Wilcke, and J. R. HuizengaDepartments of Chemistry and Physics and Nuclear Structure Research Laboratory,

University of Rochester, Rochester, New York 14627

H. Breuer and A. C. MignereyDepartments of Chemistry and Physics, University of Maryland, College Park, Maryland 20742

V. E. Viola, Jr.Department of Chemistry, Indiana University, Bloomington, Indiana 47405

K. L. WolfArgonne National Laboratory, Argonne, Illinois 60439

(Received 24 July 1981)

The system ' Ho+ Fe was studied at a bombarding energy of Ehb ——462 MeV. Theexperimental Z distributions measured as a function of angle and energy are discussed.The integrated fragment Z distribution of the light reaction fragment is fairly broad andcentered close to the charge of the projectile. The kinetic energy distribution extendsfrom the quasielastic region down to energies smaller than the Coulomb energy of touch-ing spherical nuclei. The angular distribution is peaked forward of the quarter-point an-

gle and rises at small angles, which is interpreted as due to orbiting. Correlations of ob-served fragment Z distributions with energy loss give an indication of the progress of thereaction. Charge distributions show an average drift towards larger asymmetry and anincrease in width with increasing energy damping. Results are presented taking accountof the Coulomb barrier for two choices of the scission radius. With the aid of a phenom-enological description, the initial angular momenta I; and interaction times are deduced.Interaction times are compared to other characteristic times for the reaction as a functionof I. Results from a classical dynamical model using a proximity formalism and one-

body transport are shown for comparison to experimental observables and deduced quan-tities. The influence of a neck degree of freedom on the calculated trajectories is studiedin some detail.

NUCLEAR REACTIONS '6'Ho(' Fe~), E),b ——462 MeV; measuredo.(O,E,Z); damped reaction; fusion-fusion; deduced angular momenta,

interaction times; comparison to dynamical calculations.

I. INTRODUCTION

Heavy-ion reaction phenomena at bombardingenergies several MeV/nucleon above the Coulombbarrier have been studied' in order to gain anunderstanding of the intricate mechanisms in-volved. The ' Ho+ Fe system is a projec-tile/target combination which enables the study ofall facets of heavy-ion reaction mechanisms, rang-ing from elastic scattering to fusion-fission. Dur-ing the past few years, an effort has been made toinvestigate systematically all aspects of this reac-

tion. Thus far, measurements of neutron emis-sion, the fusion-fission excitation function, andcharge and mass equilibration have been reported.Studies of charged-particle emission in the reac-tion are currently in progress. The present workemphasizes the measurements and interpretationsof projectilelike fragment Z distributions and theirdependence on energy loss and scattering angle forthe damped (deep inelastic) component.

In the following, the damped reaction will bediscussed in some detail. Section II is concernedwith the experimental and data analysis pro-

25 256 1982 The American Physical Society

25 ' Ho+ Fe REACTION AT Eh,b ——462 MeV

cedures. Measured charge, energy, and angulardistributions are described in Sec. III and phe-nomenological and microscopic interpretations ofthe reaction are presented in Secs. IV and V,respectively. Conclusions are drawn in Sec. VI.

II. EXPERIMENTAL PROCEDUREAND DATA ANALYSIS

A 435 pg/cm self-supporting ' Ho target wasbombarded with a 462-MeV Fe beam supplied bythe Lawrence Berkeley Laboratory SuperHILACaccelerator. Reaction products were detected atvarious angles using silicon surface barrier detec-tors. Two EE-E telescopes were used, consistingof transmission detectors of 15.9 and 9.4 pm thick-nesses, and stop detectors of 100 pm thickness.The latter telescope was placed at backward anglesprimarily for charge identification of the low ener-

gy, more highly damped, and fusion-fission events.Two monitor detectors were situated above the re-action plane, on either side of the beam, at a totalscattering angle of 18' for the determination of thebeam intensity and dead time of the data acquisi-tion system. Each telescope was protected by a0.13 pm thick nickel foil and rare-earth magnets,used to deflect 5 electrons produced in the target.The energy calibration was made with Cf fis-sion, ' Gd, and ThC alpha sources, the latter twobeing used for normalization of a precision pulsegenerator. Energy resolution achieved for elastical-

ly scattered projectiles was about 3% (FWHM).Resolution in atomic number was better than 0.8unit in the interesting range of Z values at andbelow Z=27.

A standard electronics setup was used, asdescribed in Ref. 6. The dead time of the data ac-quisition system ranged from 5 —20%. Data wererecorded event by event on magnetic tape for off-line analysis.

As an example of raw data, the laboratory kinet-ic energy spectra are plotted in Fig. l for four dif-ferent laboratory angles. In order to concentrateon the damped part of the distributions, thefusion-fission events have been eliminated from thespectra. At the most forward angles shown, thespectra are dominated by an intense elastic scatter-ing peak. Quasielastic events broaden this peak atlarger angles. At o~,b

——16' and 20' two broad low-energy peaks are observable. The lower-energypeak is suggestive of an orbiting mechanism, wherethe intermediate dinuclear system rotates for aperiod of time long enough for damped fragments

165

LLI

z.'

a

CA

Z.'

O

I

200 300 400 500EL,b{Me V)

FIG. 1. Laboratory energy spectra of damped frag-ments from the ' Ho+ Fe reaction for a beam energyof 462 MeV at the angles indicated, plotted in a linearscale.

to reach negative scattering angles. Its differencein energy from the partially damped peak increaseswith angle until it begins to dominate the spectraas the scattering angle is increased beyond thequarter-point angle, Hi ~4(lab) =32.5'.

As observed in Fig. 1, quasielastic events are dif-ficult to separate from elastic events near the graz-ing angle. Singles energy spectra from both tele-scopes were examined in order to obtain elasticscattering angular distributions and to determinethe reaction cross section. Standard elastic peakshapes defined at forward angles were used to fitspectra at more backward angles in order to obtainthe elastic scattering cross section. This procedureis described in detail in Ref. 7. At small angles,slit-scattering events present in the region definedby the projectile were estimated and removed.

In order to convert the EE-E distributions to aZ-E scale, the stopping powers of argon, iron,krypton, and xenon projectiles were measured as afunction of energy. The same detector systemswere maintained throughout the measurements. A300 pg/cm bismuth target was used in the ironand argon beam experiments. Energy loss calibra-tion curves were constructed from the positions ofthe elastic peaks which were varied in energy byinserting 130—420 pm Al degrader foils into the

A. D. HOOVER et al. 25

beam.Conventional methods for obtaining a Z conver-

sion compare experimental stopping powers withtabulated values. However, stopping power curvesderived from range tables of Northcliffe and Schil-

ling did not provide a satisfactory fit to the exper-imental results. More recent calculations by Hu-bert et al. agree more closely with the data; how-ever, they cover a limited range in energy and be-came available only after completion of the presentanalysis. The experimental stopping power curveswere fitted with a smooth composite function ofthe form

x/(ax +bx+c), x (xo,a 'x +b'x +c', x )xo, (2.1)

where x =E/A and xo is close to the energy of the

Bragg maximum. The six constants were derived

from the maximum energy loss, continuity at xo,and the value of the function at E/A =9 MeV/nucleon. The relative values of the energy loss forfragments other than argon, iron, krypton, and xe-

non were obtained from the interpolation of thesefits. A mass to charge ratio of 2.15, equivalent tothat of the projectile, was used for the fragmentsin the conversion procedure. All Z conversionswere performed event by event so that the resolu-

tion in Z would be preserved.Prior to conversion to the center-of-mass system,

fusion-fission events were subtracted from the ex-

perimental distributions. Since at many angles thedistinction between events from different reaction

types was not clear cut, the method used involved

some arbitrariness. An upper limit of Z=35 was

assumed for the damped events, a value corre-sponding approximately to an intensity minimum

separating the fusion-fission and damped events.However, for the determination of the fusion-fission cross section, the area corresponding tofusion-fission events was integrated directly fromhE-E spectra at angles where the separation was

unambiguous, making no arbitrary constraint onthe fragment charge.

A number of corrections were made to the meas-ured fragment energies in order to determine thefinal kinetic energy in the center-of-mass frame ofreference. Pulse-height defect corrections weremade using the procedure of Kaufman et al. ' Amore recent paper by Moulton et al. " predictspulse-height defects for the damped productswhich are larger, but well within the range of theerror for this experiment. A correction was alsomade to account for the energy loss in the nickel

foils placed in front of the detectors. These twocorrections were small, ranging from 4—9 MeV.

A more significant correction was made duringthe center-of-mass transformation to account forthe neutrons evaporated from the fragments afterthe primary reaction. The emission of neutronsfrom a primary fragment leads to a reduction ofthe fragment kinetic energies and a broadening ofthe energy distribution. Each emitted neutron isassumed to reduce the fragment excitation energy

by 12 MeV, an assumption supported by experi-ment. The total energy loss was calculated as thedifference between the initial kinetic energy andthe mean kinetic energy prior to neutron emission.Possible charged-particle emission during the in-

teraction was not accounted for in the analysis;however, such contributions were estimated fromevaporation calculations and are discussed in Sec.V B.

Data transformed into the center of mass werethen stored in three-dimensional arrays (Z, H, E) forfurther manipulation. Various projections and dis-tributions of these data will be discussed in subse-

quent sections of this paper.

III. RESULTS AND DISCUSSION

A. Elastic scattering

Elastic scattering of 56Fe on &65Ho was studiedto determine the quarter-point angle Hi~4, thestrong-absorption radius, the total reaction crosssection, and the grazing angular momentum. Theelastic scattering angular distribution is shown in

Fig. 2 as the ratio do.,~/do. R„,h of elastic to Ruther-ford cross section. For center-of-mass angles less

than about 35' Coulomb scattering predominates.The sharp falloff of the distribution from unity atlarger angles indicates the sudden onset of nuclearinteractions.

Two theoretical models were used to analyze theelastically scattered cross sections: the Fresnelmodel and an optical model. The reaction parame-ters deduced are summarized in Table I. The opti-cal model utilizes a nuclear interaction potential ofthe Woods-Saxon form dependent upon the radialseparation of the two ions. The solid line in Fig. 2refers to a four parameter fit using the computercode GENQA, where the real and imaginary sur-face diffusenesses, az ——ar ——0.529 fm, were keptconstant and the radii varied. The Coulomb poten-tial was approximated by the potential between a

' Ho+ Fe REACTION AT E~——462 MeV 259

b

b

IO

IO—

0.1—

165HO + 56Fe

E«b= 462 MeV

—Pi i—0-g—y-f-)»,

~ EXPERIMENT

OPTICAL MODEL

-V = 45.2 MeV

-W = 56.0MeV

point charge and a uniformly charged sphere, withradius Rc——0.75(AT'~ + Az'~ ). The deduced ra-dii and depths of the real and imaginary potentialswere r~ ——1.172 fm, rI ——1.266 fm, —V=43.2 MeV,and —8'= 56.0 MeV, respectively. This particularfit is representative of several good fits that can bemade to the data by modifying the differentparameters simultaneously. The maximum angularmomentum leading to a reaction was deduced to be/, „=239k, the value corresponding to a transmis-sion coefficient TI = —,. The strong-absorption ra-

dius as defined for a Coulomb trajectory is

al = aP =0529fmrR = I.I724fm

r, = l.266 frn

——- FRESNEL MODEL

O,OIIO 20 50 40 50 60 70

e, rp(deg)

FIG. 2. The experimental ratio do,~/doR„th of theelastic to Rutherford cross section is given as a functionof center-of-mass scattering angle and is compared tocalculations using the generalized Fresnel model (dashedcurve) and the optical model {solid curve).

RsA =I g+ [q +l,„(l,„+1)]' I, (3.1)

k

where g is the Sommerfeld parameter and k is theinverse wave number. The optical model calcula-tions yielded Rsz ——13.45 fm and a reaction crosssection Oz ——(2.94+0.10) b.

The Fresnel model' assumes a transmissioncoefficient of TI =1 for I waves up to a grazing an-

gular momentum lg, and TI ——0 for larger I waves.In a generalized version of this model, Frahn' in-

corporates a gradual transition in angular momen-tum space where the transmission coefficient variessmoothly from 0 to 1, with the condition thatTI 4 at l:lg The total reaction cross section isthen expressed as

TABLE I. Reaction parameters for the systemHo+ Fe at E~ab

——462 MeV.

O„=~X'(I,+ —,)' 1+2g+p g+p

(3.2)

ElabE0~«(lab)8~«(c.m. )

p (reduced mass)k „(vrave number)

g (Coulomb parameter)

Aint (Fresnel)

lg (Fresnel exp 0~«)oq (Fresnel exp 8&«)og {exp)RsA (optical model)

l,„(optical model)

oq (optical model)

~c+SA)

EO=E, —~c{~sA)[E. —~c(Rs~}]&@O fiision —fission {eXp)

I,„.,IRi.DM

(462+14) MeV(345+10) MeV32.5 +0.4'43.0'+0.541.8;.u

26.27 fm-'95.50(13.6+0.1) fm(242+3)A(2.94+0.24) b(3.3 „)b(13.5+0.1) fm(239+3)fi(2.94+0.10) b185 MeV160 MeV3.83 MeV/nucleon(0.72+0.10) b{125+9)A66fi

where 6/ is the width of the region in angularmomentum where the transmission coefficientchanges from 0 to 1. The interaction radius isgiven as

R;„,=rp(Ap' +AT' )

[1+csc(-,8)g4)] .k

(3.3)

The Fresnel model calculation with 6/=11' isrepresented by a dashed line in Fig. 2. Similar cal-culations were made with values of 6/ rangingfrom 5 to 10 which, for the most part, lead to avariation in the amplitude of the oscillations ap-pearing in the forward region of the theoreticalelastic scattering angular distribution. In the re-sults shown b,l corresponds to the 90—10% rangeof transmission coefficients as calculated by theoptical model. The parameters determined by thisFresnel calculation were lg 242% R' t 13 56 fm,and cr~ ——(2.94+0.24) b, in good agreement with

A. D. HOOVER et al. 25

the optical model analysis. However, due to thepoor energy resolution in this experiment, it is im-portant to point out that the data of Fig. 2 containcontributions from inelastic processes in an amountthat is difficult to estimate. Hence, the total reac-tion cross section may be larger than the above oneextracted from an analysis of the "elastic" scat-tering data of Fig. 2.

B. Damped reactions

The laboratory angular distribution of thedamped component of the reaction cross sectiondefined by the reaction products excluding fusion-fission fragments is shown in Fig. 3. A strongfocusing of the distribution is visible, with aquasielastic peak slightly forward of the quarter-point angle HI~4(lab) =32.5'. At the most forwardangles the cross section rises sharply, a phenom-enon which may be attributed to orbiting, i.e.,scattering to negative angles. Since it was not ex-

perimentally feasible to extend the measurement toangles less than 10, the curve through the data inFig. 3 was extrapolated linearly to zero degrees, forpurposes of determining the damped cross section.The region from 0'—10' corresponded to approxi-mately 15% of the damped cross section. Thecross section for the damped events was

ad ——(2.6+0.6) b, whereas the total reaction crosssection, determined from direct integration of reac-tion events, including the fusion-fission component,was oz ——(3.3+0's) b, in reasonable agreement with

the values deduced from the elastic scattering ana-lyses.

Figure 4 shows the charge distribution of thelight reaction fragments integrated over energy andangle. Of note in this figure is the large number offragments with atomic numbers below that of theprojectile. In fact, the maximum of the overallcross section is centered below Z=26, a subject tobe discussed in more detail in Sec. III C. Theenhanced cross section observed at higher Z valuesis due to the onset of fusion-fission and more high-ly damped events.

The energy distribution of the damped events isshown in Fig. 5. The peak at low energy lossesrepresents quasielastic processes, whereas dampedreactions comprise the broad peak at higher energylosses. The dashed part of the distribution rep-resents a cross section of 100 mb associated withevents where the total final kinetic energy was lessthan 125 MeV, below the threshold of this experi-ment. Reactions leading to fragments at far back-ward angles also contribute to the missing crosssection. A substantial amount of cross section isassociated with final energies less than theCoulomb energies of touching spheres signifyinglarge deformations of the intermediate complex atscission. The arrow in Fig. 5 denotes the availablekinetic energy in the entrance channel, E,—~c(~sA).

4i I

6sHo+s FeELob=462 MeV

l2& Z &55

2

b

N

lOIb

0

0~I I

O-o-. o-o20 50 40 50 60

ep, g(de g)

lo' I I I I

l5 20 25 50 55

Z ( ATOMIC NUMBER)

FIG. 3. The angular distribution in the laboratoryframe for the Fe-like products, excluding fusion-fissionevents. The solid line serves to guide the eye.

FIG. 4. The Z distribution for the projectilelike frag-ments integrated over the indicated angle and total ki-netic energy ranges.

462 Me~i6~Hp+ Fe RpACTION AT &vb= 26125

I

I65HO+ 56Fe

ELob =462 MeV

l2&Z&550 & e,~&120

~ 20-E

lO—b

0IOO 200ELoss(MeV)

IIII

III

II

'n for the dampedlosd t b to oFIG. 5. The energy o

events in g g'

n re resents a region wp o p

bl k t Erow indicatees the avai a e i

trance channe,'

1 i.e., Eo= . . —c

tal observables'

ns of experimen awith fragment charge an en

n of the double differential crossect to final kine ic

F'p

1' ""'" frated over all ang es.h, dZ db "d'"'n "kops a ri ge'd decreasing in Z an r

350

300

Ho+5 Fe d~

E MeV

+ 250

LLLIJ

w 200

150

0 45 50 5530 35 4015 20 25ATOMIC NUM BER z

r lot of total kinetic energy vs Z I-

/d EK.barrier Vc for touc intions.

0

Also observaall peak centered c

„ ted. Its centroidf ments are expectem-

usion-fission gjue expected for sy

~

tl from the va ueeviates s»gh yosite nucleus Thismetric fissjon of the comp

jn thed to uncertaintiesdiscrepancy isp; The total kinetic eably due

ener-Z libration in this reg oobserved to be

ca . .fragments is

a value predicted y

of the fusion-&»»ob the systemat-out &75 MeV,

f the fission fragment&5 The wjdth o t eics of Vjola.th same large uncer-Z d' tribution is subj .

d above. The'ect to

l.bration ment, oneb

t inties in the Z c1 s of the Coulomashed curve rep

'at a separatjoneres touc ing a

dif-barrier ~c of sp

'radius RsA for di-equal to the strong-absorption ra

'

f t fragmentations.eren

amine the relations iph' between theIn order to examineion and the fina ine

'1 kinetic energy, the

dZdE i lo ddifferential cross sec'

MeV wide kinetiction of Z for 20 eb. The distributions eco

f i 1oggdamped reactions

in in time while co-rocess evolv gIn "der t' d'tr d hd

ec'

is dissipated. n oed

h i FiGaussIans w

'dicated y e

re wei ht was giveni]urves because of possi e

Th toid (Z )

'tributions by t e

i her Z values. e civenZdit'b tio

8'"d 9 ' ""1a of the

gu

'r loss. A summaryfunction of energy

for the centroi s anmental values or ans is iven in a

h' Z di""bd, hcentroid of t e

es is observed to ecb' g gr loss. Two proce

xtentf t fi t th

r this Z drift, epen'

fl-served fragmenof

1 ke fragment could p

e distributions. Firs, aex lain

S hb'hholmium i e r

ect-the drift. ucbnd"--„.he direction o the

f the system,d from the total in'

distribution ofe

rs a more asymmetric iwhich favoen the two in era

g -Pironlike fragmment would pro uc

f h light fragment, p-cs e-of the average Zvalueo t e iIn Sec. IV the

'll for higher ene gyer losses. n

sm' 'lit is further exp or1 ed by means ofsecond possibility is u

evapora iot on calculations.

262 A. D. HOOVER et al. 2S

I

(a)IO

IO

10 xI07-

~ IO'-

IO4

E

10

oN

IO

bIO'—

0))y)xIO6-

xIO'-

xl04

&I) .lo'

IO

I I I I

1 65HQ y 56 F'e

ELpb = 462MeV10' ~ 8~.~ 74'

1O'- ~~ 155~

155 4y gyes ~xlo

175 &I I,

1 I I

lo

xlo

xIQ~

IO

165HQ + 56F e (b)lo ELpb 462 MBV

IO's 8, ~74'

I,I-H-t-W)wgXaI, „los155

5-$-$ 4 175-4 xIO

10 — I pg $+~q )es I-i.~ &

1~$-I + 4W-g ~ «lo'.)4 2I5 ~~4-yg

lo

IO'—

bIo'—

Io'—

10

0-5

IO

~«10'

IIII

xI00

I

14 IS 22 26 30 34

IO'—

IO

Ip

I0-4

I4 IS 22 26 30 34

xIQ~

FIG. 7. Fragment Z distributions plotted as a function of total kinetic energy E. Distributions in (a) were made ac-cording to the standard analysis procedure; in (b) the modified procedure was used (see text). Energy bins indicated atthe maximum of each curve are 20 MeV wide. The curves are displaced by the corresponding factor of 10 listed at theright of each plot. Gaussian fits to the data are indicated by the solid lines.

26—

25—

165HQ+56FE LQb

= 462 MeV

200—

N

~ ELoss= Q &Vc(Rc(Z)]o E Loss= ~-hVc(Rc(Z)+75frTlj

I I

0 50 IOO 150

ELoss (MeV)

FIG. 8. The centroid of the charge distribution as afunction of energy loss is given using the standard (cir-cles) and modified (triangles and squares) procedures.The latter differ in the size of the Coulomb radius (seetext).

E150-

ED

o& 100—IJJ

50—

0

165HO + 56 F'e

EL.ab= 462 MeV

«Loss= Q

o ELoss=-Q-&Vc[Rc(z)+75flIIj1 I

15 20Q

225

FIG. 9. Experimental correlation between the totalkinetic energy loss E~, and the variance o.z of thefragment Z distribution using the standard and modifiedanalysis procedures.

' Ho+ Fe REACTION AT E~b ——462 MeV 263

TABLE II. ' Ho+ Fe at E~,b ——462 MeV. The energy windows are 20 MeV wide. Thevariance crz is corrected for the experimental Z resolution and contains no corrections forthe Coulomb barrier. The total cross section accounted for amounts to 2.17 b (quasielasticevents and E~, & 210 MeV are not included). 0 is the angle where do. /dQ, has its max-imum in the c.m. system.

BinNo.

(E)(MeV) (MeV)

(ZL ) der/dE(mb/Me V)

0(deg)

10987654321

315295275255235215195175155135

30507090

110130150170190210

0.9+0.1

1.6+0.22.4+0.23.9+0.44.0+0.45.4+0.8

10.3+1.015 +220 +336 +10

24.8+0. 1

24.4+0. 1

24.0+0. 1

24.1+0.223.4+0.222.9+0.623.8+0.823.4+1.023 +121 +1

12.410.66.98.1

10.111.213.414.312.98.6

38.0+0.535.0+0.8

31.5+ 1.027.5+1.523.0+2.5

16.5+5.06 +5

—11 +5—48 +2

The above correlations were derived with respectto kinetic energy loss or reaction Q value, hereafterreferred to as the standard analysis procedure.Several authors' ' have proposed to considersimilar correlations with respect to the loss ofkinetic energy above the Coulomb barrier or thesum of Coulomb and centrifugal barriers. Al-though a physical justification is not indicated forsuch a procedure, it is interesting that it leads insome cases to more Gaussian-type charge distribu-tions exhibiting an average drift toward a moresymmetric fragmentation. In the following discus-sion this method of analysis will be referred to asthe modified procedure.

In a procedure similar to that of Ref. 19, the Zdistributions for bins of constant total kinetic ener-

gy TKE* were determined, where TKE*=TKE+ b, Vc, and b, Vc is the difference between the

Coulomb potentials for the exit and entrance chan-nels, evaluated for a configuration of touchingspherical nuclei with a center to center distanceequal to the strong-absorption radius calculated foreach observed mass fragmentation. The modifiedZ distributions are given in Fig. 7(b). The strong-absorption radius was chosen to calculate theCoulomb energy in order to demonstrate the differ-ence between the two methods of data analysis.Intermediate results were also obtained usingCoulomb energies calculated with a radius approxi-mating the distance the fragments would beseparated at the highest energy loss measured, cor-responding to an increase in distance of 7.5 fm.The Z distributions obtained with the above modi-

fied procedure tend to be wider than those corre-sponding to cuts in the energy loss or Q value, ascan be expected from Fig. 6. Plotted in Figs. 8and 9 are the centroids (ZL, ) and variances oz,respectively, of Gaussian fits to the fragmentcharge distributions, deduced with Coulomb ener-gies evaluated at separation distances of r=RsA(triangles) and r=Rs~ + 7.5 fm (squares).

Except for the highest energy losses where thewidths and centroids are poorly defined, certaindifferences in the Z distributions resulting fromthe two procedures are apparent. For energy lossesless than 150 MeV, the modified procedure yieldsaverage fragment Z values lower than those result-ing from the standard method. At higher energylosses, the modified analysis results in a more sym-metric fragmentation and wider distributions thanthe latter, depending upon the choice of radius.However, a direct comparison of the data evaluat-ed with the two methods may only be valid at thesmallest energy losses where both procedures yieldGaussian distributions. Data obtained with thestandard analysis procedure appear narrower athigh energy losses because of constraints in the fitprocedure. In the modified procedure, the sep-aration distance chosen for evaluating the Coulombenergy has a strong influence on the width of thedistributions. Of similar consequence is the inclu-sion of the centrifugal energies in such considera-tions.

The variation of the energy spectra with thefragment Z may be viewed as an indicator of apossible constraint in the transfer of charge im-

A. D. HOOVER et aL 25

(p la

(09

Ho+ FeEL, =462 Me

(Q7

posed by changes in the Coulomb energies. InFig. 10 the double differential cross section withrespect to Z and total kinetic energy is plotted as astep histogram versus total kinetic energy for vari-

ous fragment Z values. The events were sortedinto bins of three Z units in width, with the cen-troids given in the figure. The Coulomb energyfor spheres separated by the strong-absorption ra-dius RsA for each fragment pair is indicated by anarrow. The different curves are dominated by the

low energy loss events near the projectile Z andlarge energy loss events for Z values away fromthe projectile.

Strong correlations between the magnitude of theCoulomb energy and the position of the dampedenergy peak would indicate significant constraintson the charge transfer process. However, in Fig.10 such strong correlations are not observed. Al-though the validity of modification of the analysisto account for a change in the Coulomb energycannot be ruled out, the data provide no evidencefor a significant Coulomb constraint on chargetransfer processes. There is also no convincingphysical basis for excluding certain parts of the to-tal collective energy from the balance betweenavailable and dissipated energies. Data presentedin the following sections were obtained from thestandard analysis technique, interpreting the totalkinetic energy loss or reaction Q value as an essen-tial observable characterizing the progress of thereaction.

D. Correlations dependent uponscattering angle

N, '10

(04E

LLIo

b(Q2

~xloI

x(p'

(0 I

(0'

(Q

(Q2

(0-3 I

(50l I I

200 250 500ENERGY (MeV)

FIG. 10. Double differential cross section d cr/dZ dEvs total kinetic energy is plotted as a function of frag-ment Z for three Z unit wide bins. The horizontal barsindicate an energy bin of 20 MeV. In the center of eachcurve is marked the midpoint of the corresponding binin Z. The arrows represent the Coulomb energies forspherical fragments.

The angular distribution of the projectilelikedamped fragments is plotted in Fig. 11 asd o/d8, dE versus angle for 20 MeV wide ener-

gy bins. The full curves represent fits to be dis-cussed in Sec. IV. Sorting these data with refer-ence to a final kinetic energy parameter, account-ing for a variation of the Coulomb barrier asdescribed in Sec. III C, produces no significantchange in the shape or width of the angular distri-butions. At lo4 energy losses the data show a pro-nounced peak near the quarter-point angle

8i&4(c.m. )=43', approaching a constant value ofd a/d8, dE which is equivalent to a 1/sin8dependence for d cr/d 0dE at the higher energylosses.

The narrow angular distributions observed forsmall energy losses are characteristic of short in-teraction times. As energy is dissipated, broaderangular distributions result, suggesting a steady in-crease of the interaction time with energy loss.

The progress of the reaction can be deducedfrom Fig. 12, where a contour diagram (Wilcyznskiplot) of the double differential cross section fordamped events is shown, plotted versus total kinet-ic energy E and the reaction angle 9, A pro-nounced cross section ridge is observed, which ex-tends from the grazing angle at high final energiesand moves to lower energies for smaller angles. Asecond ridge merges with it at forward angles, de-

25 sHo+ sFe REACTION AT Sob=462 MeV 265

IO'-

M ~a M

IO'-

Io—

I I I

l65HO+ 56FeE Lab 462 MeVl2&Z&55

xlo'

xlQ"

250

C9KLIJ

LLI200

E L AST IC

I I

d2

dQdE MeV

'Fe62MeV

195 .0&

O- IO4-

EIO-

Em lo-b

Io'—

Io

IO'-

Q2

Iolo

xlo

«IP'—

xIO

x IQ

xlo

xloa

I I I I I

l9 28 57 46 55 64

ac~ (deg)

8, (deg)

FIG. 12. Double differential cross section d 0./d QdEplotted as a contour diagram (W'ilczynski plot) in the to-tal kinetic energy vs scattering angle plane.

In Fig. 13 are shown results of Gaussian fits tofragment Z distributions for different scatteringangles and various final energies. The error barstake account of the non-Gaussian shapes of someof the distributions. Although the width (FWHM)of the charge distribution increases with decreasingfinal kinetic energy, it is nearly independent of an-gle for a given energy bin. Hence, the reaction an-

gle is not a relevant parameter to describe the evo-lution of this reaction, an observation in accor-dance with results reported' for other systems.

FIG. 11. Angular distributions plotted as the doubledifferential cross sections d o./do, dE vs scattering an-gle, as a function of total kinetic energy. The energybins are 20 MeV wide. The data are integrated over alldamped fragment charges, excluding fusion-fission. Thesolid curves represent fitted curves (see text).

E. Fusion-fission processes

The angular distribution for fusion-fission eventsis shown in Fig. 14 as do./d0, versus 0,

creasing in energy with increasing angle. The posi-tions of the two cross section ridges are indicatedby dashed curves in Fig. 12. The trend of the datamay be explained either by orbiting or by doublerainbow scattering ' processes. Although theform of the deflection function would be differentat small impact parameters, both interpretationscould conceivably yield enhanced cross section atforward angles, depending in the latter case on theposition of the nuclear rainbow angle. In thepresent data analysis, orbiting is assumed. Evi-dence for the occurrence of negative-angle scatter-ing was seen by Trautmann et al. in measure-ments of the circular polarization of deexcitation yrays for the ""Ag + ~Ar system. This workcould not, however, rule out the possibility of con-tributions from both processes to the low-energyrj,dge.

20

I5-

l65HO+ 56F.e

ELat =462 MeVl45

()

N &o-

h

4I ia4I

4 h

41

l85

225

265t~

0 0 h ~ ~ 505

50400 g I I I I

IQ 20 SQ

eo.m.~«g I

FIG. 13. VA'dths of Z distribution vs angle shown forfive energy bins, 40 MeV wide. The horizontal hnes arethe average widths for each energy window.

266 A. D. HOOVER et al. 25

400—

I I I I I I I I I I

l65HO + 56F

E Lpb 46'2 MeV

IV. PHENOMENOLOGICAL INTERPRETATIONOF RESULTS

Do 300-

EE~ 200—

QV

ba l00—

il (Tff (720 & l04 mb j

il

TELESCOPE

TELESCOPE 2

00 20 40 60 80 IOO

e, „(deg)

FIG. 14. Differential cross section do.~~/d9, vs an-

gle for fusion-fission events.

These events were identified in the laboratory DE-R spectra and transformed into the center-of-massreference frame assuming a symmetric mass splitof the composite system with final total kinetic en-

ergies given by the systematics of Viola. ' The er-ror bars reflect statistical uncertainties and errorsassociated with the separation of the fusion-fissionevents from damped events in the AE-E spectra.

The data displayed in Fig. 14 are consistent with

a 1/sin8, angular dependence, indicated by thehorizontal line in the figure. Assuming a constantdifferential cross section do/d8, for theseevents, integration over solid angle yields a crosssection og =(720+104) mb for the fusion-fission-like processes in this reaction, corresponding tol,„., =125A'. This l value is far in excess of the an-

gular momentum limit (lRt DM——66k') -for fission of

the compound system as predicted by the rotatingliquid drop model.

In accordance with earlier observations, this re-sult suggests the existence of long-lived intermedi-ate systems trapped with high angular momenta inthe internuclear potential. These trapped systems,although ultimately unstable, have interactiontimes sufficiently long to allow considerable relaxa-tion of even slowly equilibrating degrees of free-dom, such as the mass asymmetry. Because of thelong interaction times, the fragments resultingfrom these trapped systems exhibit a 1/sinO, an-

gular distribution and have masses and kinetic en-

ergies essentially consistent with a fusion-fissionreaction process.

A. Experimental and theoreticaldeflection functions

While contour plots such as in Fig. 12 are of in-

terest in describing the progress of a reaction interms of macroscopic variables, it is sometimes in-

structive to look at the mean values of these quan-tities in the form of a defiection function 8(l;). Insuch a diagram an average deflection angle O isplotted as a function of initial orbital angularmomentum I; or impact parameter.

In order to infer an angular momentum scalefrom the data, the assumption is made that amonotonic relationship exists between the totalkinetic energy loss and the impact parameter. Byapplying a procedure described in Ref. 26, it ispossible to deduce a relation between angularmomentum and energy loss by identifying the dif-ferential cross section do./dE corresponding to an

energy bin with that associated with an angularmomentum window do./dl;, where

-=nA, (2l;+1)T), . (4.1)

Here, the transmission coefficients TI can be deter-mined from a sharp cutoff model or from an opti-cal model fit to the elastic scattering angular distri-bution. In the present case, both techniques were

used in order to obtain a better estimate of the un-

certainties associated with the conversion. Twonormalization points are known: the l value

corresponding to the onset of fusion-fission, l,„„and the I value associated with zero energy loss,

l,„. Results of the conversion are given in thefirst three columns of Table III.

Plotted in Fig. 15 is the average experimentaldeflection function for this reaction. The hatchedband illustrate the range of derived values associ-ated with different assumptions made in the crosssection-to-angular momentum conversion. The an-

gles were determined from the positon of the max-imum of the cross section for each particular ki-netic energy loss. In this system, because of thesubstantial amount of orbiting, it was possible todetermine accurately the cross section maxima forthe highest and lowest energy losses only. Anglesfor intermediate energies were determined from asmooth interpolation between these limits indicated

by the dashed curves in Fig. 12.The experimental deflection function 8(l;) is typ-

25' Ho+ Fe REACTION AT E~b ——462 MeV 267

TABLE III. Results of angular momentum decomposition for nonsticking (NS) and stick-ing (g cases.

BinNo. (Mev)

tNSint

( X 10 sec)

Sint

( )& 10 sec)

10

8

765

321

30507090

110130150170190210

214+8206+10200+10193+10186+11177+11165+11150+13130+16

1.3+0.41.8+0.42.5+0.63.1+0.64.2+0.95.7+1.28.0+1.8

12.3+2.921.9+4.9

122+5118+5114+6111+5106+6101+694+686+774+9

3.6+0.64.7+1.06.0+1.27.3+1.29.2+1.6

11.9+2.316.5+3.524.2+5.241.5+ 10.5

IOO "

l65H o + 56F eE Lab 462 MeV

l2&Z& 55

50-CDa

E E-,Y100 YI50 / 200

//

////

/ /g

/ /, i

250

FIG. 15. Experimental deflection function indicatedby the solid lines is compared to classical trajectory cal-culations using the standard a'=2 (dot-dashed line) andmodified a =1 (dashed line) proximity potentials.

ical of those for lighter systems, with a Coulombrainbow angle near I,„. In this region of high im-

pact parameters quasielastic events are observed,with deflection angles determined mainly by thelong-range part of the interaction potential. Thelow impact parameter collisions probe the potentialat smaller separation distances and are also morestrongly influenced by the repulsive Coulomb po-tential. Since the deflection function reflects thegross features of the potential and friction, it is in-

teresting to compare the experimental deflectionfunction with model calculations.

Results are shown in Fig. 15 for classical trajec-tory model calculations using nuclear proximitypotentials corresponding to different ratios /r bywhich the nuclear matter distribution of the com-bined system is allowed to exceed t;he equilibriumbulk density. The model involves two sphericalnuclei in a a frozen configuration and considersfour degrees of freedom: the radial separation dis-tance and the orientation angles of the system andits two constituents. It is a simplified version ofthe model discussed in Sec. V, and is described indetail in Ref. 27.

The dotted-dashed curve in Fig. 15 is based onthe interaction potential of Blgcki et al. , which iscalculated with the assumption of frozen nucleardensities, i.e., sc =2. The potential with ~=2 isseen to lead to disagreement between theoreticaland experimental deflection functions for l; & 150iri.

In order to obtain better agreement with the data,it is necessary to relax the sudden-interaction orfrozen-density idealization. The dashed curve iscalculated from a modified proximity potentialcorresponding to the situation where no excess nu-

clear density (/r = 1) is allowed. This potentialrepresents the adiabatic limit and is attractive forall distances. The proximity potential with ~= 1

results in a deflection function which is in reason-able agreement with the experimental deflectionfunction for the damped reaction events.

From the slope of the deflection function, theangular distributions shown in Fig. 11 can beanalyzed using a theory developed by Strutin-sky, ' where the cross section is divided intocoherent and fluctuating contributions. Themethod was first used to describe quasielastic andtransfer reactions, but has subsequently been ex-

A. D. HOOVER et al. 25

tended to damped reactions by Aleschin andHartmann and applied to the ' Er+ Kr reac-tion by Rudolf et al. In this method the ob-served width of the angular distributions shown in

Fig. 11 is analyzed in terms of quantal and dynam-ical angular dispersions associated with l waves ofa window around the centra1 value lp. Statisticalfluctuations which may also contribute to the totalwidth of the angular distribution are not furtherconsidered in the simplified model discussed below,although they are included in a more sophisticatedversion of the Strutinsky model.

The form of the angular distributions associatedwith coherent processes is predicted to be essential-

ly a sum of two Gaussians centered at Hp, where

80——8(lo) is the deflection angle corresponding tothe average l value in the window:

H-H.'

0jg QE

20+Hp

(4.2)

2 1 dH

g2 2 gj'(4.3)

where g and dH/dl are expressed in radians. Thefirst contribution is associated with quantal fluc-tuations and the second reflects the dynamicaldispersion, dependent on the slope dH/dl of the de-flection function. In this equation 6 is the widthof the l window. For the reaction considered herethe quantal dispersion is negligible at all energylosses. By solving for 6, extimates for the widthsof the l window are obtained for six energy bins,yielding an average value 6——27+9 for the energy

The quantity g is related to the variance era ofthe distributions and is equal to (2cra )'~ . The fullwidths I s——[8(ln2)0'a ]' and the angle differ-ences H&~4

—Hp, obtained from fits to the experi-mental data, represented by the curves in Fig. 11-,

follow a linear relationship, agreeing closely withthe ' Er+ Kr results for energy losses up to110 MeV. Above this value, the shapes and widthsof the distributions depend on the extent to whichthe fusion-fission events are included in theanalysis. The values of Hp resulting from the fitsreproduce (within 1'—2') the maximum 8 of theangular distributions for each energy loss, i.e., theexperimental deflection function.

The angular dispersion g is interpreted as thesum of two terms:

1/2

loss range of 30 to 130 MeV. This quantity iscomparable to the value determined for the

Th+ Ar reaction, 6=20, which was alsofound to be independent of energy loss. Althoughit appears necessary to perform a more consistentanalysis in terms of Sturtinsky's model, includingstochastic contributions to the cross section, beforedefinite conclusions can be reached, it is em-phasized that the present results may be taken asan indication that the damped reaction angular dis-tribution is dominantly determined by the dynami-cal rather than the quantum statistical effects.

B. Angular momentum dependentinteraction times

b 8(1)WRlf

(4.4)

where W=pRsA is the moment of inertia for theorbital motion and p is the reduced mass of thesystem. The final orbital angular momentum lIwas evaluated for two limiting cases. Initially, thetwo fragments are sliding on one another whilefriction forces cause a reduction of the angularmomentum from its initial value l;. Neglecting thereduction in l, the "nonsticking limit" is character-ized by l~ ——l;. Later in the reaction the rollingfriction leads to further transformation of the orbi-tal angular momentum into intrinsic spins of the

In understanding the various reaction processesit is important to evaluate the respective timescales over which they evolve. Because it is notpossible to measure such interaction times directlyfrom the experimental observables, they can onlybe deduced indirectly by employing a model.

A classical phenomenological reaction modelwas used to deduce the interaction times asdescribed in Refs. 1, 6, and 26. In this model theprojectile is assumed to approach the target nu-cleus on a Coulomb trajectory up to the Ruther-ford distance of closest approach or the distancegiven by the strong-absorption radius RsA, which-ever is largest. Then the intermediate system, hav-

ing dissipated its radial kinetic energy, rotatesunder the influence of the nuclear interactionthrough an angle AH, which is determined fromthe difference between the experimental reactionangle and the calculated Coulomb deflection func-tion angle. ' The system breaks apart at aseparation distance consistent with the measuredfinal kinetic energy. The interaction time t;„,(1) isthen calculated from the following relation:

i6sHo + s Fe REACTION AT Ej,b =462 MeV 269

two fragments until the system rotates in a rigid"sticking" condition. The final orbital angularmomentum in this limit is determined by

lf ——Ig ~s (4.5)

Here, the rigid body moment of inertia of the totalsystem Ws is

Ws=pRsA + s (MpRp +MrRr' ) (4.6)

|O-I9

l02o

s

\

s

TEMPERATURE (MeV)2 I 0

/

//

-.I ]

i65HO+ 56F e

~ \~ ~

0 ~ '~

Oy

0

LLI tevapHl-

-2IIO

in)'tELab= 462 MeV)

NS

-22SNax

I I' 0 loo 200 500ANGULAR MOMENTUM ~i (»

FIG. 16. Characteristic times for the ' Ho+ Fe re-action plotted as a function of incident angular momen-tum, or temperature. Experimentally deduced interac-tion times t;„t for the sticking (squares) and nonsticking(circles) cases are compared to classical trajectory modelcalculations (long dashed line). The short dashed linesignifies the time for the system to rotate through anangle of 2m and the dotted-dashed line represents thetime required for particle evaporation.

ROTATINGLIQUID DROPSTABILIT Y

LIMITI

where Mz, M~ and Rz, Rz are the masses and ra-dii of the projectile and target, respectively.

The resulting interaction times for the nonstick-ing and sticking cases are plotted versus initial an-gular momentum in Fig. 16 and collected in TableIII. Also indicated in the figure are other charac-teristic times for this reaction. The long dashedline indicates the interaction times resulting fromthe classical trajectory model calculations describedin Sec. IV A. The modified (a =1) and standard(v=2) proximity potentials produce essentially the

same interaction times for the region of angularmomenta contributing to damped processed in thisreaction. It is expected that grazing collisionswould be characterized by a nonsticking situation,whereas highly damped collisions would be closerto the sticking limit, as corroborated by y-multi-plicity studies. Although the trajectory calcula-tions do not reproduce this trend, the resulting in-teraction times agree qualitatively with the timesderived in the simple phenomenological analysisdescribed above.

The average times t,„,p for particle evaporationfrom a compound nucleus have been estimated tobe

t,„,&-5X10 exp(13 MeVle) sec, (4.7)

where v. is the nuclear temperature. From neutronevaporation measurements for the ' Ho+ Fe re-action, a relationship between energy loss and nu-clear temperature has been derived. Using this in-formation, it was possible to relate the temperatureto angular momentum. Typical temperatures ob-served in the reaction were between 2 and 3 MeV.From a comparison of the relevant time scales de-

picted in Fig. 16, it is then expected that for therange of I values between 130fi and 22(Hi, equihbri-um particle evaporation must occur after the inter-mediate system has separated and after the reac-tion fragments have undergone final acceleration intheir mutual Coulomb field. Particle emissionfrom the intermediate system, if observed, wouldindicate preequilibrium processes in the conven-tional sense. ' Such a distinction between dif-ferent emission processes would no longer exist forfusion-fission reactions, where interaction andevaporation times are of similar magnitude.

As suggested by dynamical calculations, the in-terpenetration of the two reaction partners forsmall impact parameters may be deep enough andinteraction times sufficiently long to achieve par-tial or complete relaxation of the mass-asymmetrydegree of freedom and to ascertain rotationthrough a significant fraction of a revolution, ormore. Lifetimes of the intermediate systemformed in these reactions may well be long enoughto allow, with a certain probability, for evaporationof light particles prior to separation of the system.Evidence ' for neutron emission preceding fissionhas been obtained for the ' Ho+ Ne reaction ata laboratory energy of 292 MeV.

In principle, the fusion-fission events withI ) IRz DM may differ from the compound nucleusfusion-fission events (l & lRLDM) by having anom-

270 A. D. HOOVER et al. 25

alous angular and mass distributions. Currently,there is no convincing evidence that mass distribu-tion measurements provide a sensitive method fordistinguishing the two conceptually differentfusion-fission processes. The angular distributionof fission fragments appears to be a more sensitivemethod to investigate whether fusion-fission is oc-curring from deformations inside or outside theunconditional (Bohr-Wheeler) saddle-point

ape 42 43

V. MICROSCOPIC INTERPRETATION

A. Model description

Although phenomenological trajectory modelsare useful in describing average reaction propertiesof damped collisions, a microscopic framework isneeded to study the stochastic processes leading tonucleon exchange and the dissipation of energy ina reaction. In the following a dynamical reactionmodel will be employed to follow explicitly thetime evolution of the reaction in terms of thetransport of mass, charge, energy, and angularmomentum and to obtain predictions to be com-pared with the data.

Theories based on the one-body interactions ofnucleons with the mean nuclear field appear to beapplicable to reactions characterized by long meanfree paths and low nuclear temperatures.Randrup ' has considered transport phenomenain damped reactions induced by the exchange ofindividual nucleons between the collision partners,taking proper account of the Pauli exclusion prin-ciple.

This approach describes the dynamical evolutionof the probability distribution P(N, Z;t) for find-

ing N neutrons and Z protons at time t in the pro-jectilelike fragment in terms of a two dimensionalFokker-Planck equation,

(S.2)

where E' represents the bulk differential current ofnucleons with energies near the Fermi surface, ascalculated with a Thomas-Fermi approach. Thedriving force F, governing mass and charge trans-

port, is determined mainly by the gradient of thepotential energy surface. The effective temperature~* denotes the energy intervals around the Fermisurface that contribute to nucleon exchange.Whereas in a classical diffusion model, the dif-fusion coefficient is proportional to the nucleartemperature ~, a quantum-statistical approach hasto include Pauli blocking which excludes from par-ticipation in the diffusion process those single-particle orbits that correspond to the overlap of theFermi momentum spheres of the two interactingnuclei. This overlap is dependent both on the tem-

perature and the relative velocity. The behavior ofw and r~ with energy loss is shown in Fig. 17 forthis reaction. Early in the reaction, where there iscomparatively little overlap of the two Fermimomentum spheres, which are displaced because ofthe relative velocity, the Pauli blocking is less ef-fective and relatively many levels are available fornucleon exchange, resulting in large values of v*.At higher energy losses, where the relative velocityis small, Pauli blocking is more effective and ~*decreases, eventually approaching the nuclear tem-perature ~.

Following Swiatecki, a reaction model was

developed which approximates the dinuclear con-figuration by two spheres connected by a small

6F~2 MeV

STANCEPROACH

8—P=-at

8 p 0 pp'N gZ z+ ~~g s

a' a'Dz+2 D~z P, (5.1)

(jZ2 317(jZ

)h

following the work of Norenberg. The drift anddiffusion coefficients V and D determine the firstand second moments of the mass and charge distri-butions and are calculated in Randrup's * micro-scopic model. In this model, the mixed diffusioncoefficient D~z is equal to zero. The coefficientshave the form

00 50

ELpss ( M eV )

IOO

FIG. 17. The thermodynamic and effective nucleartemperatures, ~ and v*, respectively, plotted vs energyloss (see text).

25 6sHO+ ssFe ~~~&N AT Eh,b=462 Me& 271

cylindrical neck. The configuration is picturedschematically in Fig. 18. The seven degrees offreedom in the calculations include the separationdistance r between the two centers, the orientation8 of the total system, the angles 8& and OT specify-ing the orientation of the projectilelike and target-like fragments, respectively, the neck radius p, andthe mass A~ and charge Zz of the projectilelikefragment. Apart from a choice of the interactionpotential, the model does not require any arbitraryadjustment of parameters.

In this geometry, the minimum distance s be-

tween the two nuclei ls expressed as

S =T—CT —Cp (5.3)

where CT and C~ are the matter half density radii

of the target and projectile, respectively. The neck

length d is approximated by

d ~$+ (5.4)2R

where the reduced radius is defined as

Dynamical calculations were performed yield-

ing mean trajectories in a multidimensional coordi-

nate space by solving the Lagrange-Rayleigh equa-

tions of motion for this system along the coordi-

nates q;:

d 8 8 L=-dt Qq. Bq;

a. F, q;eIr, e,ep, er ),

Bq;

a = aI.= F, q;eI p,Ap, ZP f .Bqg

(5.&)

Here, L=T—V is the Lagrangian and I' denotes

the dissipation function.The total interaction potential V accounts for the

liquid drop surface energy of the configuration, thenuclear proximity interaction between surfaces out-

side the neck region, the Coulomb potential calcu-

lated according to Bondorf et al. , and changes in

the nuclear binding energies derived from the drop-

let model, ' including shell corrections. The nu-

clear potential is given by

CTCpR=CT+ Cp

(5.5) ~nucl 2~3 p +2~7pdeff

+4myRb 4 (d /b )exp( p /2Rb ),—(5.9)

In order to account for the diffuseness of the nu-

clear surfaces, an effective neck length d,rr is given

by

pdeff =d —s~~t 1—

2R(5.6)

SCAT T E RING GEOMET R Y

/

PROJECT1LE t i ' e,

BEAMDIRECTION

TARGET

FIG. 18. Scattering geometry for the' dynamical reac-

tion model as described in the text.

The quantity s,„,=—b@(0)=1.8 fm represents theminimum (geometrical) neck length defined by the

proximity force function 4 and the surfacewidth b=1 fm.

where y= 1 MeV/fm is the surface tension. Thefirst term describes the surface energy of the two

caps erased by the presence of the neck and the

second term is the energy of the neck surface

formed. The third term contains the remaining

proximity interaction of the opposing nuclear sur-

faces outside the neck. At small separations theresulting nuclear potential is somewhat shallower

than the modified proximity potential discussed

in Sec. IV because of the added energy required toform the neck surface.

The dissipation function I' contains terms arising

from the relative motion, the neck motion, and

changes in the average mass and charge asym-

metries. The friction force acting on the relative

motion is assumed to be due to the recoil induced

by nucleons passing through a transparent window

between the collision partners as described by the"window formula. " This term is given by

F„I 2nnoRb4(p——,s)(u. ~ + —,u~~ ), (5.10)

where no ——2.63&10 MeV sec fm is the bulk

energy flux in nuclear matter and uj and u~~

represent the velocity components perpendicular-

and parallel to the interface area between the two

collision partners, respectively. The transmission

function

272 A. D. HOOVER et al. 25

d -2-P + qp—p /2Rb

2gb b(5.1 1)

40

I I I

POTF NT IAL ENERGY SURFACE (a)

measures the transparency of the effective windowwhich includes the fully open part inside the neck.In the above equation, 4 is the proximity fluxfunction. The damping of the neck motion arisesfrom collisions of nucleons with its walls ("wallformula" ). This motion is assumed to be over-

damped and the corresponding dissipation functionis given by

ZI 30

20

25 4O

F „=4~n pde—d/(P+b)~2 (5.12)

Reactions at low bombarding energies such asthat studied here are expected to be strongly influ-enced by the features of the appropriate multidi-mensional potential energy surface. Figure 19shows contour lines of potential energy surfacescalculated from a rotating liquid drop model in-

cluding shell corrections, plotted versus neutronnumber N ~ and atomic number Z~ for one of thefragments. The potential contains the sum of thefragment binding energies, the Coulomb, nuclear,and centrifugal potentials, and has the form

ZI 30—

20—

I I

POTENT IAL ENERGY SURFACELIQUID DROP + SHELL CORR.A +A -22I Z +Z -93

(b}

V(Zi, Ni ) = VLD(Zi &Ni )+ VLD(Z2, N2)

+ Vc,„i(Zi,Z2, r)+ V„„,i(A „A2,r,p)

I

25NI

40

Al2—Up ~

2pT(5.13)

FIG. 19. Potential energy surfaces calculated fromthe liquid drop model for (a) l=O and (b) I= 11(h6 forthe compound system. The cross and heavy line signifythe injection point and observed trajectory, respectively.

In the above equation, the constant Up is chosen sothat the initial fragmentation (denoted by a crossin the figure) is normalized to zero at a separationdistance equal to the strong-absorption radius RsA.Figure 19(a), corresponding to 1=0 and p=O,shows a surface with an elongated valley defined

by steep slopes with a deep minimum at symmetryand a shallower one at the N=28 shell. The heavysolid line starting at the initial fragmentation indi-cates the experimentally observed trajectory usingthe mean values (Z) and (N, )=(A, ) —(Zi).The average masses were taken from the massmeasurements of Ref. 4. The observed trajectoryinitially (i.e., for low energy losses or high l values)responds to the gradient of the potential, but maynot follow this gradient for higher energy losses.Here, perhaps, uncertainties in the corrections tothe data for particle evaporation may also becomeimportant.

A more realistic comparison would include thedependence of the potential energy surface on an-gular momentum and neck parameter. The

dynamical calculations predict a trajectory with anincident angular momentum l; =15(Hi to reach adistance of closest approach of 9.6 fm, while theangular momentum is decreased to a value of11(% The potential energy surface applicable inthis case, where shell effects have become relativelyless important, as pictured in Fig. 19(b), lookssomewhat different from the l=O surface. Thesurface suggests a relatively large drift of the massand charge distribution towards symmetry, in con-strast to the observed data. The observation of alarge drift in Z would, however, be contingent onwhether there is sufficient time to reach such acondition. At present it is not clear how to calcu-late a more realistic potential energy surface andhow to incorporate it into a consistent dynamicalreaction model.

The outstanding feature of the dinuclear config-urations considered in the reaction model discussed

' Ho+ Fe REACTION AT E~b ——462 Me 273

above is the existence of a cylindrical neck con-necting the two spherical nuclei. Since it largelydetermines the extent to which the two nuclei com-municate, the development of the neck during acollision is significant for the outcome of the reac-tion.

A cut through the total potential energy surfaceof Eq. (S.13) corresponding to the initial mass andcharge asymmetry and l =0 is given in Fig. 20(a).The ordinate is a "reduced neck radius, "p/Cp,where Cz is defined as the matter half density ra-dius of the projectile. For small radii there is aforbidden region determined by the minimumgeometrical overlap of the two nuclei at each dis-tance r Th.e effect of the addition of the centrifu-gal term in the potential is illustrated in Fig. 20(b),where the potential energy surface for l; =2(XHi su-

perimposed on the surface is represented by adashed line.

In the entrance channel the trajectory indicatesthat the system follows the path of minimum neck

IO

size until a turning point is reached. From thispoint on, the neck stretches and the fragmentsmove apart. Here, the trajectories would be ex-

pected to follow the gradient of the correspondingtime-dependent effective potential if it were not forthe dynamical effects associated with dissipation.Kith increasing distance between the two frag-ments, the neck radius decreases monotonically un-

til it snaps at fragment separations several fmlarger than where it was first formed in the en-

trance channel.In Fig. 21 average trajectories for I; = 130fi and

18(Hi are plotted as correlations of dissipated kinet-ic energy and distance s between the two fragmentcenters. The dissipated energy Ed;„ is defined asthe total kinetic energy loss E~„„neglecting thespin energy. The solid curves were calculated us-ing the nuclear interaction potential of Eq. (S.9).The dashed curves in the figure represent resultsobtained in calculations referred to in Sec. IV, em-ploying the modified nuclear proximity potential(~= 2) and the corresponding dissipation func-tion, neglecting the neck degree of freedom.Both trajectories indicate the onset of dissipationwhen the distance between the two spheres issomewhat less than s =ssA, corresponding to a dis-tance between the two nuclei equal to the strong-absorption radius RsA. The trajectories start outsimilarly because in both cases for surface separa-tions larger than a critical value s,„, (= 1.8 fm)when the neck is not yet formed, the potential anddissipation functions are determined by the prox-

0 &'9

I.O

C3x 05

10 I I 12

r (fm)14 15

I50

IOO

O

OLLI

50

FUSION~

I l I

l65HO + 56F'e

0 &9 IO l2

r(fm)14

FIG. 20. Potential energy surfaces for l=0 and l=2RHi plotted as reduced neck radius versus distance r.The corresponding trajectory for l; =2RHi is superim-posed on the surface in (b). The triangular area is inac-cessible.

AVERAGE TRAJEBSS COULOMBNUCL. PROXIMIT---- WITHOUT

WITH NEO I

-2 0s(fm)

FIG. 21. The average trajectory derived from dynarn-ical calculations is illustrated in a plot of dissipated en-

ergy Ed;„vs the distance s between the two nuclei for l;=- 13(Hi and 18(% The dashed and solid curvesrepresent calculations with and without the inclusion ofthe neck degree of freedom, respectively.

274 A. 13. HOOVER et al.

imity form factors [cf. Eqs. (5.4) —(5.11)]. Follow-ing the l; =13(Hi trajectory, corresponding to theproximity treatment neglecting neck effects thetwo nuclei continue interpenetrating until theyfuse. In the model accounting for the neck degreeof freedom, an initial angular momentum I; =130fidoes not lead to fusion. In this case, at an overlapof about s = —1.5 fm, a turning point is reachedwhere the neck ceases to grow. This reduction ofthe predicted fusion cross section is expected be-cause the potential of Eq. (5.9) is somewhat shal-lower than the modified proximity potential. Inthe exit channel, the neck area shrinks, dissipatingenergy, until it finally snaps. For I; = 180A', thetwo trajectories are fairly similar with lesseramounts of interpenetration than the correspondingtrajectories for 1; = 13(Hi. Here, the additional en-

ergy dissipated due to the neck motion is observ-able in the exit channel.

Although the development of a neck betweentwo interacting nuclei is intuitively expected to oc-cur, this degree of freedom is treated in the model

only in a very approximate manner. Its validityfor damped nuclear collisions can only be inferredfrom a comparison to experiment.

15—( V)

O10-

5-

40-

20(Da

0E

-20

-40

150-OQ)

100-

O

50-

I I

165HO+ 56FeELab= 462 MeV

cl NS MODEL& S MODEL-—THEORY

B. Comparison to data 0 e'- I I

150 200

2, (h)250

Results of model calculations described in thepreceding section are compared in Fig. 22 to exper-imentally deduced interaction times, angle, and en-

ergy loss as a function of angular momentum. Theinteraction times deduced in Sec. IV B for the non-

sticking and sticking cases as well as the average

experimental deflection function are observed to bein agreement with the calculations, within the ex-

perimental errors. As seen from the lower part ofthe figure, the measured energy loss comparesfavorably with the model except for the highest en-

ergy losses, where presumably large deformationsof the system develop that cannot be described bythe model. The agreement with the data supportsthe assumption of a monotonic relationship be-

tween E~„, and l underlying the phenomenologicalanalysis described in Sec. IV.

In Fig. 23 the model calculations, shown as adashed curve, are compared to experimental corre-lations between total kinetic energy and averagecenter-of-mass angle. The data points were de-

duced from projections of the Wilczynski plot (cf.Fig. 12) to energy and angle axes, where the angleassociated with the maximum cross section foreach energy bin is plotted in Fig. 23 versus cen-

FIG. 22. Derived interaction times for nonstickingand sticking models, scattering angle, and energy loss

plotted vs initial angular momentum. The full curvesrepresent results of dynamical calculations.

troid of the bin. Because a significant fraction ofthe reaction contributes to orbiting, the above pro-cedure to obtain an energy loss-angle relation couldnot be applied for a range of intermediate energylosses. The solid line in Fig. 23 (equivalent to thedashed curves in Fig. 12) represents an attempt tointerpolate the relation in this region of final kinet-ic energies. As can be observed from a comparisonof the dashed curve with the data points in Fig. 23,the theory provides a reasonable description of thedata down to final energies close to the Coulombbarrier Vc, where again deviations are expectedsince large deformations are unaccounted for in themodel.

The measured centroid (Zl. ) of the Z distribu-tions is plotted in Fig. 24 and compared to modelpredictions, given by a solid line. The drift in Zseems to agree with the data in direction but not in

magnitude. The experimental results should not,

25 ' Ho+ Fe REACTION AT E~——462 MeV 275

400

300-)(9CLLLI 200Z'.LLI

165HO+56F

EL,b= 462 MeV

i2 &Z&55

--- THEORY

100—I

-40I

-20I

0 20

e c.m. (deP)

I

40

FIG. 23. Total kinetic energy vs angle. The maximaof the cross section is given by the heavy line whichrepresents a smooth interpolation of the experimentaldata. Model calculations are represented by the dashedline. The Coulomb barrier V~ is noted in the figure.

I50-- 40

O

30UJ

O

LLI

100-

0

50 - HO+ FeE Lob= 462 MeV

Q EXPERIMENT—THEORY

0 ' 'MBII'22 23

I

24

(ZL)

0

I

25 26

(5CCLd-20 ~LLJ

O

- lO

C3XLtJ

FIG. 24. The centroid of the final Z distributions(circles) plotted as a function of energy loss compared totheoretical calculations (solid line). The triangles are thepredicted fragments following charged-particle evapora-tion from a hypothetical primary fragment ' Fe usingthe code MBII and plotted vs excitation energy E .

however, be directly compared to the calculationsbecause of the possible effect of particle evapora-tion from the primary fragments. Therefore, evap-oration calculations were performed for representa-tive light primary fragments at three different exci-tation energies using the code MBII The excita-tion energy of the fragments was related to theaverage energy loss which was assumed to be di-vided between the two fragments according to their

mass ratio and was corrected for the ground state

Q values, Q~. The intrinsic spin J of the frag-ments was estimated assuming the sticking limitfor two spherical nuclei. In a method described inRef. 2, the final orbital angular momentum lf ofthe system is assumed to be given by the stickinglimit for energy losses greater than 130 MeV, cor-responding to an initial angular momentum l; =177fi. For higher angular momenta, the ratio(I; —lf) / I; is assumed to decrease linearly to zeroas l; approaches I,„. This procedure is based onobservations of y multiplicities in damped reac-tions reported by Olmi et al. , which were inter-preted in terms of the theory of angular momen-turn dissipation of Wolschin and Norenberg.

In order to illustrate the maximum effect ofcharged-particle evaporation, the triangles in Fig.24 indicate the predicted average Z values of thesecondary fragments resulting from evaporationfrom the primary Fe fragment at various excita-tion energies. As indicated in the figure, little orno charged-particle emission is expected for such aprimary fragment for energy losses up to 100MeV, such that the observed Z distributions aretentatively interpreted as representing the primarydistribution. If the observed Z distributions aresubject to corrections much larger than suggested

by these evaporation calculations, better agreementbetween experiment and model would result. How-

ever, definite conclusions must await analysis ofexperimental results of a measurement of thecharged-particle emission in the reaction.

In the model, the variances O.q and oz, of thefragment mass and charge distributions, respective-

ly, are obtained from integrating the transportcoefficients along the trajectory. In Fig. 25 the re-sults of the calculations, represented by a solid line,are compared with the data. The calculationsreproduce the data fairly well up to energy lossesof about 130 MeV. The dashed curve representsdynamical calculations in the classical limit, takingno account of the Pauli exclusion principle in thenucleon exchange processes. The results for thisreaction, as well as those described in Ref. 49, indi-cate that the Pauli principle is essential in order toobtain a description of the data, as can be expectedfor bombarding energies per nucleon that are lowcompared to the nucleon Fermi velocity.

The results of evaporation calculations discussedabove were used to examine the effect of charged-particle emission on the widths of the Z distribu-tions. The results indicate that there is, in general,only a small influence on the width from particle

276 A. D. HOOVER et al. 25

200-

O

M 0100-

UJ

E0

Ho+ FeFL,b= 462MeV—

0 EXPERI MEN T—THEORY--CLASSICALL I Ml T

evaporation for a range of excitation energies ofthe projectilelike fragments.

VI. CONCLUSIONS

0 IO 20Q 2

Z

FIG. 25. Experimental variances o.z plotted vs ener-

gy loss are compared to model predictions (solid curve).The dashed curve is a calculation estimating the classi-cal limit for the reaction.

deduce angular momenta and interaction times forthe system. The derived average deAection func-tion decreases to negative angles for intermediateand small I values, similar to those of lighter sys-tems. Interaction times for the damped productsrange from 10 to 10 ' sec depending upon theenergy loss. The more highly damped compositesystems which lead to fusion-fission may undergoone or more rotations and possibly have lifetimessufficiently long to evaporate neutrons and chargedparticles before breakup.

Dynamical calculations which assume a dinu-

clear configuration with a cylindrical neck com-pare favorably with the data on average quantities

up to regions where the system is expected to bestrongly deformed. Calculated trajectories are usedto derive dispersions of fragment charge distribu-tions from theoretical transport coefficients. Satis-factory agreement between predictions and calcula-tions for correlations between charge dispersionsand energy loss lend strong support to the domi-nance of a one-body nucleon exchange mechanismunderlying the observed transport phenomena.

The full range of reaction processes from elasticscattering to fusion-fission reactions has been stud-ied for the )65Ho + 56Fe reaction at a beam ener-

gy of E~,b ——462 MeV. Correlations of experi-mental observables with energy loss are presentedfor varying degrees of energy damping. The angu-lar distributions show a substantial amount of or-

biting to negative angles, demonstrating that thenuclear force remains attractive even at large inter-

penetrations. The fragment Z distributions are ob-served to increase steadily in width as the reactionprogresses, indicating the importance of nucleonexchange in this reaction.

Classical phenomenological models are used to

ACKNOWLEDGMENTS

This work was supported by the U. S. Depart-ment of Energy. The use of the facilities at theNuclear Structure Research Laboratory was sup-ported by the National Science Foundation. Theauthors would like to acknowledge the cordial hos-pitality of LBL and to thank especially H.Grunder, R. Stevenson, and the staff of the SuperHILAC for providing an efficient operation of theaccelerator. Special recognition is due to M. Zis-man for his efficient and conscientious help in theexperiments.

*Present address: Lawrence Livermore National Labo-ratory, University of California, Livermore, Califor-nia.

Present address: Hahn-Meitner Institut, Berlin, %estGermany.

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