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APH N.S., Heavy Ion Physics 13 (2001) 79-88 Fission Energy Surfaces and Ternary Fission HEAVY ION PHYSICS @Akademiai Kiado D.N. Poenaru,' B. Dobrescu,' W. Greiner," J.H. Hamilton" and A.V. Ramayya" 1 Horia Hulubei National Institute of Physics and Nuclear Engineering RO-76900 Bucharest-Magurele, Romania 2 Institut fiir Theoretische Physik der J.W. Goethe Universitat D-60054 Frankfurt am Main, Germany 3 Department of Physics and Astronomy, Vanderbilt University Nashville, Tennessee 37235, USA Received 20 May 2000 Abstract. A three-center phenomenological model, able to explain qualita- tively the recently obtained experimental results concerning the quasimolecu- lar stage of a light-particle accompanied fission process is presented. It was derived from the liquid drop model under the assumption that the aligned con- figuration, with the emitted particle between the light and heavy fragment, is reached by increasing continuously the separation distance, while the radii of the heavy fragment and of the light particle are kept constant. In such a way, a new minimum appears in the deformation energy at a separation distance very close to the touching point. This minimum allows the existence of a short-lived quasi-molecular state, decaying into the three final fragments. The influence of the shell effects is discussed. The half-lives of some quasimolecular states which could be formed in the lOBe and 12C accompanied fission of 25 2Cf are roughly estimated to be of the order of 1 ns and 1 ms, respectively. Keywords: 3-center liquid drop model, deformation energy, shell corrections, ternary fission half lives PAGS: 21.60.Gx, 24.75.+i, 27.90.+b 1. Introduction The light-particle accompanied fission was discovered [1] in 1946, when the track of a long-range particle (identified by Farwell et al. to be 4He) almost perpendicular to the short tracks of heavy and light fragments was observed in a photographic plate. The fission was induced by bombarding 235U with slow neutrons from a Be 1219-7580/01/ $ 5.00 @2001 Akaderniai Kiad6, Budapest

Fission Energy Surfaces and Ternary Fission

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APH N.S., Heavy Ion Physics 13 (2001) 79-88

Fission Energy Surfaces and Ternary Fission

HEAVY IONPHYSICS@Akademiai Kiado

D.N. Poenaru,' B. Dobrescu,' W. Greiner," J.H. Hamilton" andA.V. Ramayya"

1 Horia Hulubei National Institute of Physics and Nuclear EngineeringRO-76900 Bucharest-Magurele, Romania

2 Institut fiir Theoretische Physik der J.W. Goethe UniversitatD-60054 Frankfurt am Main, Germany

3 Department of Physics and Astronomy, Vanderbilt UniversityNashville, Tennessee 37235, USA

Received 20 May 2000

Abstract. A three-center phenomenological model, able to explain qualita­tively the recently obtained experimental results concerning the quasimolecu­lar stage of a light-particle accompanied fission process is presented. It wasderived from the liquid drop model under the assumption that the aligned con­figuration, with the emitted particle between the light and heavy fragment, isreached by increasing continuously the separation distance, while the radii ofthe heavy fragment and of the light particle are kept constant. In such a way, anew minimum appears in the deformation energy at a separation distance veryclose to the touching point. This minimum allows the existence of a short-livedquasi-molecular state, decaying into the three final fragments. The influenceof the shell effects is discussed. The half-lives of some quasimolecular stateswhich could be formed in the lOBe and 12C accompanied fission of 252Cf areroughly estimated to be of the order of 1 ns and 1 ms, respectively.

Keywords: 3-center liquid drop model, deformation energy, shell corrections,ternary fission half livesPAGS: 21.60.Gx, 24.75.+i, 27.90.+b

1. Introduction

The light-particle accompanied fission was discovered [1] in 1946, when the track ofa long-range particle (identified by Farwell et al. to be 4He) almost perpendicularto the short tracks of heavy and light fragments was observed in a photographicplate. The fission was induced by bombarding 235U with slow neutrons from a Be

1219-7580/01/ $ 5.00@2001 Akaderniai Kiad6, Budapest

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80 D.N. Poenaru et 301.

target at a cyclotron. The largest yield in such a rare process (less than one eventper 500 binary splittings) is measured for a-particles, but many other light nuclei(from protons to oxygen or even calcium isotopes) have been identified [2] in bothinduced and spontaneous fission phenomena. If A l and A2 are the mass numbers ofthe heavy fragments (assume A l 2: A2 ) , then usually the mass of the light particleA3 «A2 • The "true" ternary fission, in which Al ~ A2 ~ A3 , has not yet beenexperimentally detected.

Many properties of the binary fission process have been explained [3] within theliquid drop model (LDM); others like the asymmetric mass distribution of fragmentsand the ground state deformations of many nuclei, could be understood only afteradding the contribution of shell effects [4,5]. As it was repeatedly stressed (see[6] and the references therein), shell effects proved also to be of vital importancefor cluster radioactivities predicted [7] in 1980, and experimentally [8] confirmedstarting with 1984 (see the reviews [9,10] and references therein).

The total kinetic energy (TKE) of the fragments, in the most frequently de­tected binary or ternary fission mechanism, is smaller than the released energy (Q)by about 25-35 MeV, which is used to produce deformed and excited fragments.These then emit neutrons (each with a binding energy of about 6 MeV) and ,-rays.From time to time a "cold" fission mechanism is detected, in which the TKE ex­hausts the Q-value, hence no neutrons are emitted, and the fragments are producedin or near their ground state. The first experimental evidence for cold binary fis­sion in which its TKE exhaust Q was reported [11] in 1981. Larger yields weremeasured [12] in trans-Fm (Z 2: 100) isotopes, where the phenomenon was calledbimodal fission (see also the review [23]).

The correlated fragment pairs in cold ternary (a and lOBeaccompanied sponta­neous fission of 252Cf) processes were only recently discovered [14,15J, by measuringtriple, coincidences in a modern large array of ,-ray detectors (GAMMASPHERE).The fragments are identified by their ,-ray spectra. Among other new aspects ofthe fission process seen for the first time with this new technique [14,16], one shouldmention the double fine structure, and the triple fine structure in binary and ternaryfission.

A particularly interesting feature, observed [15,17] in lOBe accompanied coldfission of 252Cf is related to the width of the light particle -y-ray spectrum. The3.368 MeV 'Y line of lOBe, with a lifetime of 125 fs is not Doppler-broadened, asit should be if it would be emitted when lOBe is in flight (taking about 1 ns toreach the detector). A plausible suggestion was made that the absence of Dopplerbroadening is related- to a trapping of lOBe in a potential well of nuclear molecularcharacter [15J.

Quasi-molecular configurations of two nuclei have been suggested as a naturalexplanation for the resonances measured [18] in l2C+12C scattering and reactions.There are also other kinds of such binary molecules (see [19] and references therein),like spontaneously fissioning shape-isomers. The above mentioned experiments canbe considered as an evidence for a more complex quasi-molecular configuration ofthree nuclei. The purpose of the present lecture is to show, within a phenomenolog-

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Ternary Fission 81

ical three-center model (with [20] or without [21,22]) that a minimum which couldexplain the existence of these quasi-molecules is produced in the potential barrier,when the formation of the light particle occurs in the neck between the two heavierfragments. In this way we extend to ternary fission our unified approach of coldfission, cluster radioactivities, and a-decay [6,13].

2. Shape Parametrization

The shape parametrization with one deformation parameter as follows has beensuggested from the analysis [24] of different aligned and compact configurations offragments in touch. A lower potential barrier for the aligned cylindrically-symmetricshapes with the light particle between the two heavy fragments, is a clear indicationthat during the deformation from an initial parent nucleus to three final nuclei,one should arrive at such a scission point. In order to reach this stage we shallincrease continuously the separation distance, R, between the heavy fragments,while the radii of the heavy fragment and of the light particle are kept constant,R l = constant, R 3 = constant. Unlike in the previous work, we now adopt thefollowing convention: A l 2': A 2 2': A3' The hadron numbers are conserved: A l +A2+A3 =A.

Fig. 1. Evolution of nuclear shapes during the deformation process from oneparent nucleus 252Cf to three separated fragments l46Ba, lOBe, and 96Sr. In theupper part the binary stage is illustrated; the separation distance increases fromR; to R ov 3 , passing through Rminlb and Rm l n2b values. In the middle, the ternarystage of the process develops by forming the third particle in the neck. The quasi­molecular shape, at which R = Rmin-t is the intermediate one in this row. Atthe bottom the fragments are separated.

(1)-Rl ~ Z ~ Zsl

'Zsl s Z s R + R2R~ - z2,R~ - (z - R)2,

At the beginning {the neck radius Pneck 2': R3 ) one has a two-center evolution(see Fig. 1) until the neck between the fragments becomes equal to the radius ofthe emitted particle, Pneck = P(Zsl) !R=Ro"3= R3 • This equation defines Rov 3 as theseparation distance at which the neck radius is equal to R3. By placing the originin the center of the large sphere, the surface equation in cylindrical coordinates isgiven by:

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82 D.N. Poenaru et al,

Then for R > R av3 the three center starts developing by decreasing progressivelywith the same amount the two tip distances b: + h 31 = h32 + h 2• Besides thisconstraint, one has, as in the binary stage, volume conservation and matching con­ditions. The R2 and the other geometrical quantities are determined by solving nu­merically the corresponding system of algebraic equations. By assuming sphericalnuclei, the radii are given by R j = 1.2249A}/3 fm (j = 0,1,3), R2f = 1.2249A~/3with a radius constant ro = 1.2249 fm, from Myers-Swiatecki's variant of LDM.Now the surface equation can be written as

R~ - z2,R~ - (z - Z3)2 ,

R~ - (z _R)2,

-R1 :S z :S Zsl ,

Zsl :::; Z :::; Zs2,

Zs2 :::; Z :::; R +R2 ,

(2)

and the corresponding shape has two necks and two separating planes. Some of theimportant values of the deformation parameter, R, are the initial distance R; =Ro - Rl, and the touching-point one, Rt = R1 + 2R3 + R2f . There is also Rav3 ,

defined above, which allows one to distinguish between the binary and ternary stage.

3. Deformation Energy

According to the LDM, by requesting zero energy for a spherical shape, the de­formation energy, E1.£(R) - EO, is expressed as a sum of the surface and Coulombterms

Edef(R) = E~[Bs(R) -1] + E~[Bc(R) -1], (3)

(4)

where the exponent 1.£ stands for uniform (fragments with the same charge density asthe parent nucleus), and 0 refers to the initial spherical parent. In order to simplifythe calculations, we initially assume the same charge density PIe = P2e = P3e = POe,

and at the end we add the corresponding corrections. In this way we perform onenumerical quadrature instead of six. For a spherical shape E~ = a s (1 - KI2)A2/ 3

j

1= (N -Z)jAj E~ = aeZ 2A- 1/3, where the numerical constants of the LDM are:as = 17.9439 MeV, K = 1.7826, ae = 3e2j(5ro), e2 = 1.44 Me'V-fm,

The shape-dependent, dimensionless surface term is proportional to the surfacearea:

B. =;; ~~J'[y2+Ht)'rdx,

where y = y(x) is the surface equation in cylindrical coordinates with -1, +1intercepts on the symmetry axis, and d = (z" - z')j2Ro is the seminuclear lengthin units of Ro. Similarly, for the Coulomb energy [25] one has

5 +1 +1

Be = ~~ ! dx! dx' F(x,x'),-1 -1

(5)

Page 5: Fission Energy Surfaces and Ternary Fission

Ternary Fission 83

F(x,x/) =

x

+

{YY1[(K - 2D)/3] x

[ ( 2 2) ( ')2 3 I (dYI dy2)]2y +Y1 - x-x +-(x-x) ---2 dx' dx

K {2 2/3 [2 x - x' dy2] [2 x - x' dYI]}} -1Y Y1 + Y ---- Y1---- C. •2 dx 2dx' P

(6)

K, K ' are the complete elliptic integrals of the 1st and 2nd kind

~/2 ~/2

K(k) = J(1 - k2sin2t)- 1/2dt ; K'(k) = J(1 - k2sin2t )1/ 2dt (7)

o 0

60 - uncorr. gme P.- corr. diff. P.u ..... • COlT. gme P. .f'\'.......... binity unc.gme P•..·....-- bin.'Y diff. P. .'

50$'..,40

~30Q

lit 20

10

o -

2 4 6 8 W U H D ~ W D. R(fm)

Fig. 2. The liquid drop model deformation energy versus separation distance forthe 2°0 accompanied cold fission of 252Cf with 132Sn and lOOZr heavy fragments.In order to simplify the numerical calculations we start by assuming the samecharge density of the fragments. One can see the effect of successive correctionstaking into account the experimental Q-value and the difference in charge density.Similar curves for the binary fission possess a narrower fission barrier. The newminimum appears in the shaded area from R ov 3 to Rt ,

The new' minimum, which can be seen in Fig. 2 at a separation distance R =Rmin-t > R ov3 , is' the result of a competition' between the Coulomb- and surfaceenergies. At the beginning (R < Rmin-t) the Coulomb term is stronger, leading toa decrease in energy, but later on (R > Rmin-t) the light particle formed in the neckpossesses a surface area increasing rapidly, so there is also an increase in energy upto R =Rt .

Now let us analyse the influence of various corrections, which could in principlealter this image. After performing numerically the integrations, we add the followingcorrections: for the difference in charge densities reproducing the touching point

Page 6: Fission Energy Surfaces and Ternary Fission

84 D.N. Poenaru et al.

values; for experimental masses reproducing the Qexp value at R = 14, when theorigin of energy corresponds to infinite separation distances between fragments, andthe phenomenological shell corrections 8E

ELD(R) = Eder(R) + (Qth - Qexp)fc(R) ,

where fc(R) = (R - 14)/(Rt - 14), and

3 3

Qth = E~ + E~ - 2)E~i + E~i) + 8Eo - L 8Ei .1 1

(8)

(9)

The correction increases gradually (see Fig. 2 and Fig. 3) with R up to R, andthen remains constant for R > s; The barrier height increases if Qexp < Qth

and decreases if Qexp > Qth. In this way, when one, two, or all final nuclei havemagic numbers of nucleons, Qexp is large and the fission barrier has a lower height,leading to an increased yield. In a binary decay mode like cluster radioactivity andcold fission, this condition is fulfilled when the daughter nucleus is 208Pb and 132Sn,respectively.

4. Shell Corrections and Half-Lives

Finally, we also add the shell terms

E(R) = ELD(R) + 8E(R) - 8Eo . (10)

At present, there is no microscopic three-center shell model available working reli­ably for a long range of mass asymmetries. This is why we use a phenomenological

......... E,o······6E

40 -E

S'30

~ 20'-'-UJ 10

0

·10

2 4 6 8 10 12 14 16 18 20 22R(fm)

Fig. 3. The liquid drop model, ELD, the shell correction, oE, and the totaldeformation energies, E, for the lOBe accompanied cold fission of 252Cf withl46Ba and 96Sr heavy fragments. The new minimum appears in the shaded areafrom Ro v 3 to R«.

Page 7: Fission Energy Surfaces and Ternary Fission

Ternary Fission 85

model, instead of the Strutinsky's method, to calculate the shell corrections. Themodel is adapted after Myers and Swiatecki [5J. At a given R, we calculate thevolumes of fragments and the corresponding numbers of nucleons Zi(R), Ni(R)(i = 1,2,3), proportional to the volume of each fragment. Figure 1 illustrates theevolution of shapes and of the fragment volumes. Then we add for each fragmentthe contribution of protons and neutrons

(11)

which are given by(12)

wheres(Z) = F(Z)/[(Z)-2/3] - cZ1/ 3 , (13)

F(n) = ~ [N~3__:!~~ (n _ Ni-1 ) _ n5/3+ Ni5~~] (14)

in which n E (Ni-l, Ni) is either a current Z or N number and Ni-1 , N, are the clos­est magic numbers. Theconstants c = 0.2, C = 6.2 MeV were determined by fit tothe experimental masses and deformations. The variation with R is calculated [26]as

8E(R) = ~ {~[S(Ni) + S(Zi)] Lf:)}, (15)

where L, (R) are the lengths of the fragments along the axis of symmetry, at a givenseparation distance R.

During the deformation, the variation of separation distance between centers,R, induces the variation of the geometrical quantities and of the correspondingnucleon numbers. Each time a proton or neutron number reaches a magic value,the correction energy passes through a minimum, and it has a maximum at midshell(see Fig. 3 and Fig. 4).

The first narrow minimum appearing in the shell correction energy 8E in Fig. 3,at R = Rminlb ~ 2.6 fm, is the result of almost simultaneously reaching the magicnumbers Zl = 20, N1 = 28, and Z2 = 82, N2 = 126. The second, shallowerone around R =.Rmin2b ~ 7.2 fm corresponds to a larger range of R-values forwhich Zl = 50, N 1 = 82, Z2 = 50, N2 = 82 are not obtained in the same time.In the region of the new minimum, R = Rmin-t, for light-particle accompaniedfission, the variation of the shell correction energy is very small, hence it has nomajor consequence. One can say that the quasimolecular minimum is related tothe collective properties (liquid-drop like behavior). On the other side, for "true"ternary process (see the bottom part of Fig. 4) both minima appear in this rangeof values, but no such LDM effect was found there.

In order to compute the half-life of the quasi-molecular state, we have first tosearch for the minimum Emin in the quasimolecular well, from Rov3 to Rt, and thento add a zero point vibration energy, Ev: Eqs = Emin + Ev.

Page 8: Fission Energy Surfaces and Ternary Fission

86 D.N. Poenaru et al,

5 10 15 20

S-Ol 5~ol.U -5 - "'Sn + uOZr + lOBe'<>-10

50

-5 _ '"& +"Sr + lOB.-10

50

-5- u'B. +"Kr +"C

-10

50

-5 _ "'Sn + JI"Mo +"C-10

5 _ "5. +"G. +"G.

0-5

-105 10 15 20

R(fm)

Fig. 4. The shell correction energy variation with the separation distance for twoexamples of lOBe, two of 12C accompanied cold fission, compared to the ''true''ternary fission (in nearly three identical fragments) of 252Cf. The three partnersare given. The two vertical bars on each plot show the positions of R ov 3 and ofn;

The half-life, T, is expressed in terms of the barrier penetrability, P, which iscalculated from an action integral, K, given by the quasi-classical WKB approxi­mation

T = hln2. P = exp(-K),2EvP'

where h is the Planck constant, andRb

K = *J\/2/-L[E(R) - Eqs ] dR

R"

(16)

(17)

in which Ra , Rb are the turning points, defined by E(Ra ) = E(Rb) = E qs , andthe nuclear inertia is roughly approximated by the reduced mass /-L = m[(A1Az +A3A)/(A1+Az)], where m is the nucleon mass, log[(h ln 2)/2] = -20.8436, loge =0.43429 and [8m/n,zjl/z = 0.4392 Mey-l/Zxfm-1,

Rb

logT(s) = 0.43429JV(A1Az/A)[E(R) - Eqs ] dR - 20.8436 -logEv ,

R"

energies are expressed in MeY, distances in fm.

Page 9: Fission Energy Surfaces and Ternary Fission

Ternary Fission 87

The results of our estimations for the half-lives of some quasimolecular statesformed in the lOBe and 12C accompanied fission of 252Cf are given in Table 1.They are of the order of 1 ns and 1 IDS, respectively, if we ignore the results for adivision with heavy fragment 132Sn, which was not measured due to very high firstexcited state. Consequently the new minimum we found can qualitatively explainthe quasimolecular nature of the narrow line of the lOBe "I rays.

Table 1. Calculated half-lives of some quasi-molecular states formed during theternary fission of 252Cf.

Particle Fragments Qexp K logT(s)(MeV)

lOBe 132Sn HORu 220.183 19.96 -11.17138Te lO4Mo 209.682 25.23 -8.89138Xe lO4Zr 209.882 26.04 -8.54146Ba 96Sr 201.486 22.98 -9.86

12C 147La 93Br 196.268 39.80 -2.56142Ba 98Kr 199.896 42.71 -1.30140Te lOOZr 209.728 38.21 -3.25132Sn lO8Mo 223.839 31.46 -6.18

It is interesting to note that the trend toward a split into two, three, or four nu­clei (the lighter ones formed in a long neck between the heavier fragments) has beentheoretically demonstrated by Hill [27], who investigated the classical dynamics ofan incompressible, irrotational, uniformly charged liquid drop. No mass asymmetrywas evidenced since any shell effect was ignored.

In conclusion, we should stress that a quasimolecular stage of a light-particleaccompanied fission process, for a limited range of sizes of the three partners, canbe qualitatively explained within the liquid drop model.

Acknowledgment

We are grateful to M. Mutterer for enlightening discussions.

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88 D.N. Poenaru et al.

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