investigation of the reaction pb( o, ): fragment spins and...

ISSN 1063-7788, Physics of Atomic Nuclei, 2007, Vol. 70, No. 10, pp. 1679–1693. c© Pleiades Publishing, Ltd., 2007.Original Russian Text c© A.Ya. Rusanov, G.D. Adeev, M.G. Itkis, A.V. Karpov, P.N. Nadtochy, V.V. Pashkevich, I.V. Pokrovsky, V.S. Salamatin, G.G. Chubarian, 2007, publishedin Yadernaya Fizika, 2007, Vol. 70, No. 10, pp. 1724–1738.

NUCLEIExperiment

Investigation of the Reaction 208Pb(18O, fff):Fragment Spins and Phenomenological Analysis of the Angular

Anisotropy of Fission Fragments

A. Ya. Rusanov1)*, G. D. Adeev2), M. G. Itkis3), A. V. Karpov3), P. N. Nadtochy2),V. V. Pashkevich3), I. V. Pokrovsky3), V. S. Salamatin3), and G. G. Chubarian4)

Received July 17, 2006; in final form, March 19, 2007

Abstract—The average multiplicity of gamma rays emitted by fragments originating from the fission of226Th nuclei formed via a complete fusion of 18O and 208Pb nuclei at laboratory energies of 18O projectileions in the range Elab = 78–198.5 MeV is measured and analyzed. The total spins of fission fragments arefound and used in an empirical analysis of the energy dependence of the anisotropy of these fragments underthe assumption that their angular distributions are formed in the vicinity of the scission point. The averagetemperature of compound nuclei at the scission point and their average angular momenta in the entrancechannel are found for this analysis. Also, the moments of inertia are calculated for this purpose for the chainof fissile thorium nuclei at the scission point. All of these parameters are determined at the scission point bymeans of three-dimensional dynamical calculations based on Langevin equations. A strong alignment offragment spins is assumed in analyzing the anisotropy in question. In that case, the energy dependence ofthe anisotropy of fission fragments is faithfully reproduced at energies in excess of the Coulomb barrier(Ec.m. − EB � 30 MeV). It is assumed that, as the excitation energy and the angular momentum of afissile nucleus are increased, the region where the angular distributions of fragments are formed is graduallyshifted from the region of nuclear deformations in the vicinity of the saddle point to the region of nucleardeformations in the vicinity of the scission point, the total angular momentum of the nucleus undergoingfission being split into the orbital component, which is responsible for the anisotropy of fragments, and thespin component. This conclusion can be qualitatively explained on the basis of linear-response theory.

PACS numbers: 25.70.Jj, 25.85.-wDOI: 10.1134/S106377880710002X

1. INTRODUCTION

In this article, we present results obtained by mea-suring and analyzing the average multiplicity Mγ ofgamma rays emitted by fission fragments originatingfrom the reaction 18O + 208Pb → 226Th → f inducedby projectile ions 18О, their laboratory energies beingElab = 78.0, 90.2, 117.0, 144.2, and 198.5 MeV in ourinvestigation. The values found experimentally for themultiplicity of gamma rays emitted by fission frag-ments make it possible to determine the total spinsof these fragments for various excitation energies E∗

1)Institute of Nuclear Physics, National Nuclear Center ofthe Republic of Kazakhstan, Almaty, 480082 Republic ofKazakhstan.

2)Omsk State University, pr. Mira 55A, Omsk, 644077 Rus-sia.

3)Joint Institute for Nuclear Research, Dubna, Moscowoblast, 141980 Russia.

4)Cyclotron Institute, Texas A & M University, MS No. 3366,College Station, TX 77843-3366, USA.

*E-mail: [email protected]

(or temperatures) of the compound nucleus. At thehighest energies of 18О ions and, hence, at the highesttemperatures of the nucleus undergoing fission andhighest angular-momentum transfer to it, respectivefission fragments acquire a spin of rather high value.This experimental finding is intimately related to theproblem of angular distributions of fission fragmentsat high angular momenta.

In the early 1980s, it was found experimentally [1–4] that, in heavy-ion reactions, the angular anisotropyof fission fragments at high temperatures T andhigh angular momenta J proves to be much morepronounced than that which is predicted by thetraditional statistical transition-state model (TSM),where it is assumed that, at the saddle point, thedistribution with respect to K (tilting mode, whereK is the projection of the total angular momentum ofthe nucleus undergoing fission onto the fission axis)becomes equilibrium and frozen—that is, invariableup to the scission point [5]. This greatly aggravatedthe problem of theoretically interpreting the angulardistribution of fission fragments. The model of a

1679

1680 RUSANOV et al.

10

0

10

–1

80

CCFULL [14]

E

c.m.

, MeV120 160 200

(

b

)

10

1

10

2

10

3

10

0

10

–1

70 80 90

(

a

)

Gontchar [15]

10

1

10

2

10

3

σ

fus

, mb

Fig. 1. Experimental fusion cross sections σfus for the reactions induced by 16,18О + 208Pb collistions from [3, 16, 17]: (closedcircles, closed triangles, and closed boxes) reactions induced by 16О ions and (open circles) reactions induced by 18О ions.The arrows indicate the c.m. energies Ec.m. at which the multiplicity Mγ of gamma rays from fission fragments were measuredin the present study.

rotating liquid drop (liquid-drop model, or LDM) [6]or the model of a rotating nucleus governed by finite-range nuclear forces (rotational finite-range model,or RFRM) [7] predicts that, at high J introduced ina target nucleus by a projectile ion, the fission barrierdecreases significantly, approaching in deformationthe ground state of the rotating nucleus under study;as a matter of fact, this means that the fission processreduces to a long descent from the saddle to thescission point. In the last decades, it has becomeclear that, for highly excited nuclei, fission is a ratherlong process (its duration being 10−19 to 10−20 s)that proceeds under conditions of a high friction as-sociated with manifestations of the viscous propertiesof nuclear matter [8]. If the time of descent from thesaddle to the scission point is sufficiently long, theK mode may become unfrozen, so that the formationof angular distributions of fragments occurs at thescission point in the limiting case. This assumptionforms a basis for the theoretical models proposed

in [9–11] (standard statistical models, or SSM). Inthese models, the primary total angular momentum ofthe compound nucleus, J, is statistically redistributedat the scission point between the relative orbitalangular momentum l of fission fragments and theirtotal spin S; that is,

J = l + S, (1)

S = s1 + s2. (2)

These are vector sums, where s1 + s2 is the sum of thespins of two complementary fragments. It should beemphasized, however, that, of all spin modes carryingthe angular momentum of fragments (tilting, bend-ing, wriggling, and twisting modes), only the first Ktilting mode was considered in the models proposedin [9–11]. The resulting theoretical description of theangular distributions of fission fragments proved tobe unsatisfactory in the majority of cases. The SSMpredictions for the anisotropy at high J were wellabove respective experimental results.

PHYSICS OF ATOMIC NUCLEI Vol. 70 No. 10 2007

INVESTIGATION OF THE REACTION 208Pb(18O, f ): FRAGMENT SPINS 1681

A new conceptual framework was proposed in [12]for describing angular distributions of fission frag-ments. Within this framework, prescission nuclearconfigurations are considered on the basis of themodel proposed by Brosa and his colleagues in [13]; inaddition, it is assumed that, in these configurations,spin modes carrying the angular momentum of fissionfragments are fully equilibrated. In [12], a satisfactorydescription of the anisotropy as a function of theprojectile-ion energy was attained for a great manyreactions.

In the present study, we aimed at solving an in-verse problem: knowing primary angular momenta ofcompound nuclei and the total spins of fission frag-ments, we tried to find empirical-parameter valuesthat provide the best fit to the anisotropy of thesefragments as a function of energy; in just the sameway as in [12], we took the prescission configurationsof fissile nuclei for a basis.

2. PHENOMENOLOGICAL ANALYSIS

It is well known that, within the statistical model,the anisotropy of fission fragments is given by theapproximate formula

A = W (180◦)/W (90◦) ≈ 〈J2〉4K2

0

= 1 +〈J2〉�2

4T�eff, (3)

where 〈J〉 is the total angular momentum of thenucleus involved; K2

0 is the variance of the K dis-tribution; T is the nuclear temperature; and �−1

eff =�−1|| −�−1

⊥ is the effective moment of inertia of thisnucleus, �|| and �⊥ being the moments of inertiafor, respectively, the rotation of the nucleus about thesymmetry axis and its rotation about axes orthogonalto the symmetry axis. Formula (3) is applicable atany point of the “trajectory” along which the nucleusmoves toward fission if we assume thermodynamicequilibrium in K at this point.

For nuclei characterized by a high fission probabil-ity (Pf ≈ 1), it is assumed within statistical modelswhere the saddle-point (sp) configuration of the nu-cleus being considered is taken for a transition point(TSM) that, at rather high excitation energies E∗ ofthe nucleus, its average total angular momentum (notincluding the ground-state spins of the projectile andtarget) 〈J〉 is equal to the average angular momentumintroduced in the target by the projectile ion underconditions of complete fusion, 〈lfus〉, minus the angu-lar momentum carried away by prescission particles.In expression (3), we accordingly have T = Tsp and�eff = �sp

eff.If one assumes that the K distribution becomes

equilibrium at the scission point (SSM), then all of

18014010060

E

c.m.

, MeV

60

40

20

0

4000

3000

2000

1000

0

CCFULL [14]Gontchar [15]

⟨

l

fus

⟩

l

fus2

⟨ ⟩

Fig. 2. Mean and mean-square orbital angular momentaobtained as the first and second moments of the angulardistributions (〈lfus〉 and 〈l2fus〉, respectively) from the de-scription of σfus (see Fig. 1) according to calculations onthe basis of the CCFULL code [14] and according to therecommendations given in [15].

the aforementioned quantities must correspond to thescission point. In this case, however, we must takeinto account the splitting of the total angular mo-mentum into the orbital and spin components [seeEq. (1)]. Thus, we must determine all quantities ap-pearing in (3) at the scission point.

2.1. Average Angular Momenta of Compound Nuclei

In order to calculate correctly the anisotropy offission fragments, it is necessary to know the angular-momentum distributions of compound nuclei to afairly high degree of precision. In order to ensurethis, we have employed two popular methods of cal-culations. The first is the channel-coupling method,which is implemented within the CCFULL code [14],while the second relies on the empirical relations pro-posed in the studies of Gontchar and Frobrich [15].Either method provides a relationship between thecross section for projectile–nucleus fusion, σfus, andthe average angular momentum of the resulting com-pound nucleus, 〈lfus〉. For the reactions induced by16,18O + 208Pb collisions, the fusion cross sectionswere measured in [16], but only in the vicinity of theCoulomb barrier. These cross sections σfus are givenin Fig. 1a versus the c.m. projectile-ion energy Ec.m..One can clearly see that only at projectile energiesbelow the Coulomb barrier are the cross sections σfuslarger for the reactions induced by 18O ions thanfor the reactions induced by 16O ions; in the above-barrier region, the values of these cross sections areindistinguishable within the errors. The solid curverepresents our description of σfus on the basis of theCCFULL code [14]. In the respective calculations,the parameters of the model potential were set to the


1682 RUSANOV et al.

Table 1. Features of the reactions under study

Elab, MeV Ec.m., MeV E∗ini, MeV TKE, MeV Mγ σMγ 〈lfus〉, � 〈l2fus〉 , �

2

78.0 71.8 26.1 163.9 ± 0.8 9.8 4.9 8.1 86

90.2 83.0 37.3 163.7 ± 0.7 11.3 8.6 21.1 520

117.0 107.7 62.0 162.0 ± 0.7 13.5 10.6 38.9 1720

144.2 132.7 87.0 162.0 ± 0.9 16.1 12.3 48.4 2650

198.5 182.7 137.0 162.3 ± 0.9 19.2 14.6 60.3 4100

value of V0 = 105.0 MeV for the potential depth, thevalue of r0 = 1.13 fm for the range of the potential,the value of 0.68 fm for the diffuseness parameter,the value of B0 = 75.8 MeV for the average barrierheight, and the value of RB = 11.71 fm for the radiusof the barrier (its average curvature was taken to be�ω0 = 4.48 MeV).

In Fig. 1b, the cross sections σfus for the reac-tions induced by 16,18O + 208Pb collisions are givenaccording to the measurements performed in [3, 16,17] over a broad range of Ec.m.. The solid curvesrepresents the description of σfus on the basis of theCCFULL code over the entire range of Ec.m. at thesame values of the model parameters as in Fig. 1a.One can see that, here, the agreement with the ex-perimental data is fairly good. The dashed curves inFigs. 1a and 1b correspond to our calculations on thebasis of the empirical formulas from [15]. One cansee that the calculated values of σfus are in excessof their experimental counterparts at low Ec.m., butthat they are, on the contrary, somewhat below theexperimental values at high Ec.m.. However, the dis-tinction between the description of σfus within the firstmodel and the respective description within the sec-ond model is moderately small. The results obtainedfrom the description of σfus for the mean and mean-square orbital angular momenta (〈lfus〉 and 〈l2fus〉, re-spectively) as the first and second moments of theangular distributions are given in Fig. 2 according tocalculations within each of the two models. One cansee that, at intermediate values of Ec.m., there is somedistinction between the results of the two calcula-tions. The values obtained for 〈lfus〉 and 〈l2fus〉 are usedhere in expression (3) to determine the anisotropy.These values from the calculations on the basis of theCCFULL code are presented in Table 1.

From [8, 18–22], it is well known that a nucleusundergoing fission may emit a rather large number ofprescission particles, which carry away a moderatelylow angular momentum from the compound nucleus.We assume that each emitted neutron, proton, oralpha particle reduces the angular momentum of thecompound nucleus by 1�, 1�, or 2 �, respectively.

2.2. Temperature of Nuclei at the Scission PointIn calculating the temperature at the scission

point, Tsc, we follow the line of reasoning in [4, 10,11, 23–27] and define it as

Tsc = (E∗sc/a)1/2, (4)

E∗sc = E∗

ini + 〈Qff〉 − TKE

− ∆Epre − Escrot − Edef + Ediss,

where E∗sc is the excitation energy at the scission

point, a = 0.093ACN is the level-density parame-ter [28], and E∗

ini is the primary excitation energy ofthe compound nucleus. The average energy of thefission reaction, 〈Qff〉, was calculated in the followingway. Since our spectrometer makes it possible todetermine only the mass of a fragment but not itscharge, there is a set of Z and N values for eachmass that are distributed according to the Gaussianlaw with a width of FWHM ∼= 4.5 amu [29], thisdistribution having a maximum at 〈Z〉 and 〈N〉values corresponding to the simple hypothesis thatthe charge of a fragment is proportional to its mass.For each mass, we therefore determined the aver-age value 〈Qff〉 for charges of 〈Z〉 ± 5 and for thecorresponding N with allowance for the statisticalweight of each Z and N . The defect-mass values wereborrowed from the tables in [30]. In expression (4),〈Qff〉 is the value averaged over the fragment-massrange (ACN − νpre)/2 ± 10 (about 60% of all fissionevents fall within this range), νpre is the number ofprescission neutrons, ТКЕ is the experimental valueof the average total kinetic energy of fragments, and∆Epre is the energy carried away by all prescissionparticles.

In determining the average number of prescissionneutrons, νpre, we employed experimental data ob-tained in [22] for the reactions that are induced by18O + 208Pb collisions at low E∗

ini and in [18, 19] forthe reactions that are induced by 16O + 208Pb col-lisions and which are close to our case. In Fig. 3a,these values are given along with the number νpostof prompt neutrons from fission fragments (postfis-sion neutrons) and the total number νtot of neutrons



Table 2. Parameter values adopted for our empirical analysis [see Eq. (4)]

Elab,MeV

E∗ini,

MeVνpre νPE

EPE,MeV

νtot αpre ppre∆Epre,MeV

〈Qff〉,MeV

Escrot,

MeVE∗

sc,MeV

Tsc,MeV

78.0 26.1 1.3 0 0 4.3 0 0 11.5 177.0 2.3 25.3 1.10

90.2 37.3 1.8 0 0 5.2 0.002 0.004 16.5 176.7 2.8 30.3 1.21

117.0 62.0 3.0 0 0 7.0 0.025 0.016 30.2 177.8 4.3 42.5 1.43

144.2 87.0 3.9 0.2 4.0 8.9 0.054 0.037 46.1 177.2 5.4 49.3 1.54

198.5 137.0 5.6 0.4 8.0 11.9 0.126 0.108 75.3 178.0 7.0 69.3 1.84

in the reactions mentioned immediately above. Onecan clearly see that the values νpre in the reactionsinduced by 18O and 16O ions are indistinguishablewithin the errors. Our present results on the averagetotal multiplicity Mγ of gamma rays from two com-plementary fragments are also displayed in this figurealong with relevant data from [31, 32]. Obviously, allof the dependences presented here are nonlinear overa broad interval of E∗

ini; therefore, we approximatedthem by quadratic functions of the type y = a + bx +cx2. From Fig. 3a, it can clearly be seen that a gooddescription was attained in all cases. The νpre and νtotvalues interpolated to the energies E∗

ini of interest tous are given in Table 2.

At high excitation energies of E∗ini � 80–100 MeV,

compound nuclei evaporate not only neutrons butalso a significant number of protons and alpha parti-cles. Experimental data on the numbers αpre and ppreof prescission alpha particles and protons from [20,21] for compound nuclei close to our case are givenin Figs. 3b and 3c along with their quadratic approx-imations. The αpre and ppre values interpolated to theenergies E∗

ini realized in our present study are alsopresented in Table 2.

We calculated here the energy carried away byprescission particles by the formula

∆Epre = Epreα + Epre

p + Epren + EPE

n ; (5)

that is, we treated it as the energy carried away byalpha particles, protons, evaporated neutrons, andpreequilibrium neutrons (νPE). The number and theenergy of preequilibrium neutrons were borrowedfrom the systematics presented in [26, 27]. Thesevalues are also given in Table 2. The number of pree-quilibrium alpha particles [27] is rather small; more-over, we determined Mγ for a full linear-momentumtransfer. Therefore, we disregarded E

preα completely.

We began calculating ∆Epre (5) by reducing theinitial energy of the compound nucleus by EPE

n . Theenergy carried away by alpha particles and protonswas found as in [27] according to the formula E

preα,p =

EС + 2T , where EС is the Coulomb barrier. In justthe same way as in [33], it was assumed in performingthese calculations that, after the emission of preequi-librium neutrons, charged particles are emitted first

20

15

10

5

0

ν

pre

,

ν

post

,

ν

tot

,

M

γ

(

a

)

M

γ

ν

tot

ν

post

ν

pre

0.2

0.1

0

(

b

)

α

pre

0.1

0

(

c

)

p

pre

2001601208040

E

ini*

MeV

,

Fig. 3. (a) Average number of prescission neutrons, νpre;average number of postfission neutrons (that is, neu-trons from fission fragments), νpost; total number of neu-trons, νtot; and average multiplicity of gamma rays emit-ted by fission fragments, Mγ , in the reactions inducedby 16,18O + 208Pb collisions: (closed and open circles,closed boxes, and inverted open triangles) data from [18,19, 32] on the reactions induced by 16O ions, (open boxesand triangles) data from [22, 31] on the reactions inducedby 18O ions, and (closed triangles) our present data. (b,c) Experimental values of the multiplicities of prescissionalpha particles (αpre) and protons (ppre) for the reactions(closed circles) 19F + 208Pb → 227Pa and (open circles)28Si + 197Au → 225Np [20] and (closed boxes) for thereactions induced by 16O + 208Pb collisions [21]. Thesolid curves represent fits to the displayed data.


1684 RUSANOV et al.

Table 3. Moments of inertia �sc⊥, �sc

|| , and �sceff (in �

2 MeV units) for the chain of thorium isotopes according tocalculations for prescission shapes of nuclei (Fig. 4 at the center) at the total-angular-momentum values of J = 0 and80 �

Isotope�sc

⊥ �sc|| �sc

eff

J = 0 J = 80 � J = 0 J = 80 � J = 0 J = 80 �

220Th 464.51 468.62 56.133 55.851 63.85 63.41

221Th 468.02 472.10 56.552 56.278 64.32 63.89

222Th 471.55 475.58 56.973 56.707 64.80 64.38

223Th 475.09 479.07 57.395 57.137 65.28 64.87

224Th 478.59 482.57 57.812 57.568 65.75 65.37

225Th 482.10 486.08 58.238 58.000 66.24 65.86

226Th 485.54 489.61 58.674 58.434 66.74 66.35

from the nucleus undergoing fission and that theyare followed by a cascade of prescission neutrons.In accordance with [34], the energy E

pren was deter-

mined on the basis of the relations Epren = 〈νpre〉〈Еpre〉

and 〈Еpre〉 = 〈Bn〉 + 〈Еn〉, where 〈Bn〉 is the neutronbinding energy averaged over the chain of nuclei un-dergoing fission and 〈En〉 = 2Tn is the kinetic energycarried away by neutron that is also averaged overthis chain of nuclei, with Tn = (E∗/a) (here, E∗ is theexcitation energy of residual nuclei after the emissionof charged particles from them) being the tempera-ture of the nuclei over which the averaging of Вn isperformed. The calculated values of the energy ∆Epreare also given in Table 2.

The energy of rotation at the scission point was

y

J

y

J

S

l

K

z

x

ϕ

Fig. 4. Schematic spatial pattern of the vector sum of J,SE∗,J , and l. At the center, one can see the calculatedprescission configuration of a 226Th nucleus for the caseof symmetric fission.

found here on the basis of the standard formula

Escrot = 〈l2fus〉�2/2�sc

⊥ + 2Tsc. (6)

In Table 3, we present the �sc⊥ values calculated for

the chain of fissile thorium nuclei (for more details,see Subsection 2.3 below). The second term in (6)determines the thermal excitation energy of all spinmodes carrying the angular momentum of fragments(tilting, bending, wriggling, and twisting). Of these,each is fully excited, its energy being Tsc/2.

In expression (4), it only remains to determine twoterms. These are Edef, which is the energy expendedon the deformation of fragments, and Ediss, whichis the energy dissipated from the collective motiontoward fission (not rotational motion) to the internalenergy. There is no consensus on Edef, which wedetermine here as a parameter. By way of example,we indicate that Edef is 10 MeV in [35], 12 MeVin [23, 24], and 20 MeV in [10, 11]. In principle,this energy is compensated for to some extent by theenergy Ediss, which, as will be shown in Section 3below, has approximately the same values. In view ofthis, we set Ediss − Edef = 0 in the present analysis.The values calculated for Esc

rot, E∗sc, and Tsc are given

in Table 2.

2.3. Moments of Inertia at the Scission Point

First, we would like to pinpoint the definition thatwe are going to use for the scission point. In theoret-ical studies devoted to describing the fission processwithin the liquid-drop model (in its inviscid versionor with allowance for viscosity), there are three basiccriteria of nuclear scission into fragments. The firstis the vanishing of the neck thickness, rn = 0 (see,



for example, [36, 37]). The second is the equalityof the forces of Coulomb and nuclear interactionsin the neck, FC = FN [38]. The third is the loss ofstability of the nucleus against variations in the neckthickness [39]. The second and the third conditionare rather close in the prescission configuration ofthe nucleus. Once the nucleus has passed the pointFC = FN , there arises sharp instability to scission(the system goes over from the fission value to thevalue of separated fragments [40]), and a fast ruptureof a rather thick neck occurs without any change inthe shape of elongation of the nucleus and withouta gradual decrease in the neck thickness with in-creasing deformation. In the case of the scission cri-terion rn = 0, the scission configurations lie beyondthe boundaries of existence of continuous shapes inthe liquid-drop model [39]. Moreover, this model losesany meaning as soon as rn becomes commensuratewith the spacing between nucleons.

In view of this, we adopted the third condition forthe criterion of the scission point. We have calculatedthe prescission shapes of 220–226Th nuclei on the ba-sis of the liquid-drop model with the parameters fromthe study of Myers and Swiatecki [41]. The nuclearsurface was described by using a parametrizationbased on Cassini’s ovals [42], where the elongationof a nucleus is characterized by the parameter α.The calculations were performed only for symmetricnuclear configurations. In minimizing the potentialenergy of a nucleus, we took into account all evendeformations of higher order up to α = 20. For a fissilenucleus, we have found the critical deformation abovewhich a sharp rupture of the neck immediately occurs.A typical prescission configuration of the 226Th nu-cleus is shown at the center of Fig. 4. This geometricbody has a rather thick neck of rn ≈ 0.3R0, whereR0 is the radius of a ball having the same volume. Aswas shown in [43], the ТКЕ value calculated for thisbody is close to the experimental value.

In order to estimate the change in the nuclearshape with increasing angular momentum, we haveperformed our calculations both for a nonrotatingnucleus (J = 0) and for J = 80 �. The results of thecalculations for the rigid-body moments of inertia forthe chain of thorium isotopes are given in Table 3.One can see that the change in J from zero to 80 �

leads to quite an insignificant change in �sceff. Since

there are no such high 〈J〉 at the scission point inour case and since the effect is still smaller at actual〈J〉 (〈J〉 = 〈lfus〉—see Table 1), we took �sc

eff for anonrotating nucleus in calculating the anisotropy offission fragments by formula (3). A greater effect isobserved in �sc

eff in response to the isotopic changesin A for thorium nuclei. Since prescission neutronsare always present in actual practice, we took intoaccount this isotopic dependence.

Table 4. Total fragment spins calculated by the empiricalformula (7) at various values of the parameters n and Mst

Elab,MeV

E∗ini,

MeV

S(A), �

(n = 2,Mst = 2.5)

S(D), �

(n = 1.7,Mst = 2.5)

S(E), �

(n = 1.7,Mst = 3)

78.0 26.1 11.8 10.3 8.6

90.2 37.3 15.2 13.3 11.6

117.0 62.0 20.5 18.0 16.3

144.2 87.0 26.6 23.2 21.5

198.5 137.0 34.2 29.9 28.2

2.4. Fragment Spins

The standard empirical relation between the mea-sured gamma-ray multiplicity Mγ and the total frag-ment spin S has the form [44, 45]

S = n(Mγ − 2Mst) + 0.5νtot, (7)

where 2Mst is the number of statistical gamma raysescaping from two complementary fragments andcarrying, on average, zero angular momentum andνtot is the total number of fission neutrons (see Fig. 3and Table 2), each of these carrying away an angularmomentum of 0.5 � [44]. The coefficient n in (7) takesvalues ranging between 1.5 and 2.0. If n = 2 [44,46], all gamma rays are assumed to be quadrupolefor transitions downward along the yrast band, but,if n = 1.7 [45], an admixture featuring possible dipoletransitions is taken into account. Table 4 presentsthe fragment spins calculated according to (7) byusing various sets of n and Mst values. In analyzingthe anisotropy of fission fragments, we will employprecisely these spin values.

2.5. Description of the Calculations and TheirResults

From [45, 47], we know that, even in spontaneousfission, in which case the initial excitation energy andthe initial angular momentum of the nucleus under-going fission are zero, its fragments emit some num-ber of nonstatistical gamma rays associated with theirrotation—that is, fragments originating from spon-taneous fission also have rather high spins. A theo-retical consideration of the mechanism of fragment-spin generation shows in this case that the negativebending mode [48], for which s1 + s2 = 0 and l = 0in expression (2), is likely to play a dominant modein this process. Naturally, the anisotropy of fissionfragments is then equal to unity.

In the case of fission induced by heavy-ion reac-tions, a compound nucleus receives a sizable angular


1686 RUSANOV et al.

momentum and a sizable excitation energy, whichcreate preconditions for manifestations of other col-lective modes (wriggling and twisting), although itis possible that not all collective rotational modescan be excited, as was shown in [49]. Concurrently,the multiplicity Mγ of gamma rays from fragmentsgrows along with the spins of these fragments andtheir anisotropy (the K tilting mode manifests itselfhere), which starts from unity at low E∗ and J [50].Formally, the total gamma-ray multiplicity and thefragment spin along with it can be broken down intotwo terms as

Mγ = M0γ + ME∗,J

γ , S = S0 + SE∗,J . (8)

The first two terms in (8) correspond to Mγ and Sfor spontaneous fission. They can be interpreted asinitial reference points. The second terms in (8) areadditions generated by the introduction of the exci-tation energy and angular momentum in the targetnucleus by the projectile ion. It is then quite naturalto assume that it is only the vector component SE∗,J

rather than the total fragment spin S that is actu-ally responsible for the redistribution of the primary-angular-momentum vector J between the orbital andspin components at the scission point according toEq. (1). In the case of collective modes independent ofone another, the total-spin vector SE∗,J is the vectorsum of all excited collective modes, including rigid-body rotation accompanying the introduction of E∗

and J in the target nucleus.

Let us now return to our case. In approximatingthe dependence Mγ(E∗

ini) in Fig. 3a, we extrapolatedthe fitted curve to the point E∗

ini = 0 and arrived ata value of M0

γ = 6.1 ± 0.2. This value of Mγ wouldbe characteristic of the 226Th nucleus if it underwentspontaneous fission (the probability of the 226Th(s, f )process is of course very small, but it is not vanishingyet, since the fission barrier in this nucleus is notoverly high [51]). The value obtained for M0

γ is closeto that which is expected for so light a nucleus on thebasis of the systematics presented in [52]. In order tofind the spin S0 from M0

γ at low excitation energiesat the scission point, it is necessary to use, accordingto the recommendations of Ahmad and Phillips [45],the value of Mst = 1.5 in expression (7); moreover,it is desirable to take into account the ground-statespins of complementary fragments. For various setsof n values and total neutron numbers in the rangeν0

tot = 2.0–2.3, which are close to the actual value,we obtained S0

∼= 9 �. Comparing the value that wefound for S0 with the total spin for the spontaneousfission of 252Cf [45, 47], for example, we can find that,for heavy fragments in the mass range 140—150 amuand the complementary light fragments, the spins

are S0(252Cf) ∼= 10–13 � [47] and that, in the massregion around M ∼ 110 amu, which, in our case,corresponds to a nearly symmetric fission of 226Th,the spin of a single fragment of 252Cf falls between 5.0and 5.5 � [45]. In view of this, we believe that the valueof S0 = 9 �, which we obtained here, is reasonable for226Th fission. In order to find the values of SE∗,J , it isnecessary to subtract 9� from all of the spin values inTable 4.

In [32, 53], it was shown that the average total spinof fragments is formed by three terms; that is,

SE∗,J = [f2J2 + (1 − f2)K2 + S2CM]1/2. (9)

Here, the first term corresponds to the fragment spinarising from the rigid-body rotation (RR) of the com-pound nucleus, the coefficient f being its angular-momentum fraction dissipated to the fragment spins.The second term represents spins arising from the Ktilting mode. Finally, the third term SCM appears uponthe excitation of statistical collective modes (wrig-gling, twisting, bending). In principle, all terms in (9)are excited independently of one another; therefore,the average vector of the total fragment spin is merelythe average vector sum of all terms:

SE∗,J = SRR + Stilt + Swrig + Stwist + Sbend. (10)

The twisting and bending modes are negative, sothat the vector sums for two fragments in (2) areclose to zero for a nearly symmetric fission. Therefore,it is necessary to take into account these modes inthe rotational energy (6), but their contribution to thevector sum in (10) is small.

John and Kataria [12] showed that the inclusionof the positive wriggling mode is of paramount im-portance for precisely calculating the anisotropy offission fragments within the SSM framework. Ac-cording to [54], this mode is associated with the redis-tribution of the initial angular momentum J—morespecifically, it is directly involved in the reduction ofits orbital component l in Eq. (1). Moreover, it wasshown in [55] that the positive wriggling mode issimilar to the K tilting mode, and its contribution tothe rotational energy of fragments is maximal at K =0 [12]. As usual, the spin vector Stilt may change fromK = −J to K = J , obeying a Gaussian distributionwhose variance is К2

0. The spin vector SRR is alsoorthogonal to the fission axis [56, 57]. At high angularmomenta, the spin SRR is not small. If we take valuesin the range f = 0.2–0.3 [32, 58] for the coefficienton the right-hand side of (9), then, at J = 50 �, forexample, we have SRR = 11–16 �.

Summarizing the aforesaid, we conclude that thefirst three terms in expression (10) are correlated withone another rather strongly and that, by and large, the



1

0

A

E

c.m.

–

E

B

, MeV40 80 120

2

3

4

5

6

7

8

(

a

)

C

B

A

A

:

n

= 2,

M

st

= 2.5

B

: 0.85

S

E

*,

J

C

:

S

E

*,

J

= 0

0 40 80 120

(

b

)

E

D

D

:

n

= 1.7,

M

st

= 2.5

E

:

n

= 1.7,

M

st

= 3

19

F +

208

Pb, Hinde et al. [59]

19

F +

208

Pb, Back et al. [3]

16

O +

208

Pb, Back et al. [3]

18

O +

208

Pb, Vulgaris et al. [16]

16

O +

208

Pb, Vulgaris et al. [16]

19

F +

208

Pb, TSM (Hinde et al. [59])

Fig. 5. Experimental data on the angular anisotropy of fission fragments from the reactions induced by 16,18O + 208Pb [3, 16]and 19F + 208Pb [3, 59] collisions versus the energy above the Coulomb barrier, Ec.m. −EB. The curves in (a) represent (A) theresults of our calculations on the basis of expressions (11), (12), and (3) with the values of the spin SE∗,J from Table 4 [valuesin column S(A) minus 9 �], (B) the results obtained under the assumption that the spin value of 0.85SE∗,J is realized, and(C) the results of the calculation at SE∗,J = 0 (that is, without allowance for the fragment spin at the scission point). In (b),curves D and E were calculated with the values of the fragment spin SE∗,J from Table 4 (values in columns S(D) and S(E)minus 9 �)—that is, with different sets of values of the parameters n and Mst in the empirical relation (7). For all of the curvesdisplayed here, the initial angular momenta of compound nuclei (specifically, 〈lfus〉 and 〈l2fus〉) were calculated on the basis ofthe CCFULL code [14] (solid curves in Fig. 2 minus the angular momentum carried away by prescission particles).

total-spin vector SE∗,J at high J and E∗ is likely tofluctuate about an axis orthogonal to the fission axis.

For the vector sum in (1), we will now consider thespatial pattern proposed in [46]. Schematically, thispattern is depicted in Fig. 4. Denoting by φ the anglebetween the yz plane (or, equivalently, the JK plane)and the direction of the orbital angular momentum land relying on purely geometric arguments, we canreadily find l at a given value of SE∗,J . The result is

l = cos φ(J2 − K2)1/2 (11)

− [S2E∗,J − J2 − cos2 φ(J2 − K2)]1/2,

where J = 〈lfus〉 (minus the angular momentum car-ried away by prescission particles) and the distribu-tion with respect to K was calculated traditionally:ρ(K) ∝ exp(−K2/(2K2

0 )). Of course, the absolutevalue of the vector l depends on the absolute value ofthe vector SE∗,J and on variations in its direction, andthis ultimately leads to variations in the angle φ. Weassumed that, in just the same way as in the K mode,the probability of the distribution of the vector SE∗,J

about the y (K = 0) axis obeys the Gaussian law. Theprobability of the distribution with respect to the angleφ can then be represented in the form

ρ(ϕ) (12)

∝ exp{−[(S2E∗,J − sin2 φ)1/2 − SE∗,J ]2/(2K2

0 )},

where l2 is given by expression (11). The sought valueof 〈l2〉 was determined by averaging expression (11)over the distributions with respect to K and φ.

We have substituted all necessary quantities intoexpression (3) and have found the required anisotropy.Figure 5a shows experimental data on the anisotropyof fission fragments from the reactions induced by16,18O + 208Pb [3, 16] and 19F + 208Pb [3, 59] col-lisions versus the energy above the Coulomb barrier(Ec.m. − EB). The dotted curve represents the resultsthat calculations on the basis of the transition-statemodel [59] yield for the anisotropy in the reactionsinduced by 19F + 208Pb collisions. One can see that,at energies in the region Ec.m. − EB > 30 MeV,this model predicts a descending energy dependence


1688 RUSANOV et al.

of the anisotropy, while the experimental data inquestion exhibit a moderately slow growth of thisanisotropy or its invariability. Curve A in Fig. 5acorresponds to our calculations with the total spinsSE∗,J from Table 4 (values in column S(A) minus9 �). This situation is realized under conditions ofa complete alignment of the spin modes in (10).Curve B was calculated under the assumption thatthe spin value of 0.85SE∗,J is realized, in whichcase an incomplete alignment of the spin modes istaken into account. Curve C represents the case ofSE∗,J = 0, which corresponds to the disregard of thefragment spin at the scission point.

Figure 5b displays the same experimental data asFig. 5a. Curves D and E were calculated by usingthe values of the fragment spins SE∗,J from Table 4[values in columns S(D) and S(E) minus 9 �]—thatis, by using different sets of values of the parameters nand Mst in the empirical relation (7). One can clearlysee that, strange as it may seem, versions A, B, and Dat energies in the region Ec.m. −EB > 30 MeV repro-duce the experimental data in question satisfactorily.

3. DYNAMICAL CALCULATIONS

In Section 2, the temperature of a fissile nucleus atthe scission point was determined within the statis-tical approach from the empirical relation (4), wherethe deformation and dissipation energies, denoted byEdef and Ediss, respectively, are disregarded. More-over, we have employed nuclear moments of inertiacalculated only for the symmetric configuration of afissile nucleus. In this section, we present the resultsof three-dimensional dynamical calculations based onLangevin equations and performed with allowancefor the evaporation of prescission particles. We studyfission dynamics for 226Th nuclei, starting from astatistically equilibrium state and tracing their evo-lution to scission to fragments without taking intoaccount spin modes—that is, we consider here the“irrotational motion” of a nucleus toward scission.

3.1. Results of the Calculations

Our present calculations were performed in justthe same way as previously in [60]. The c, h, α′

parametrization from [60] was used to describe theshape of the nucleus undergoing fission. This is aslightly modified version of the c, h, α parametriza-tion well known from [40] (the new mass-asymmetryparameter α′ is related to α via a scale factor: α′ =αc3), and it is more convenient for calculations. Inorder to describe collective-motion dissipation intointernal degrees of freedom, we employed the one-body-dissipation version, the surface one with a win-dow [61] at the reduction-coefficient value of ks =

0.25. In this case, one can obtain the best fit to themass–energy distributions of fission fragments [60].The potential energy of a nucleus was calculated onthe basis of the liquid-drop model that takes intoaccount a finite range of nuclear forces and the dif-fuseness of the nuclear surface [62]. The parametersof the model were set to the values from [7]. In thecalculations, we employed the nuclear-deformation-dependent level-density parameter from [63].

The condition of scission to fragments was takento be identical to that in Subsection 2.3: rn = 0.3R0.In the space of collective coordinates, this conditionspecified the scission surface, for which we obtainedthe average values of the effective moments of inertia�sc

eff. These values are quoted in Table 5. We empha-size that our calculations revealed that, upon takinginto account the diffuseness of the nuclear surface,�sc

eff increases by about 20%, on average, in relationto the results of analogous calculations disregardingthe diffuseness.

The initial average orbital angular momenta ofcompound nuclei (more specifically, 〈lfus〉 and 〈l2fus〉)were calculated on the basis of the empirical relationsproposed by Gontchar in [15]. The results are repre-sented by the dashed curves in Fig. 2.

The dissipation energy Ediss was calculated sep-arately. It was found as the difference of the exci-tation energies of a fissile nucleus for the scissionconfiguration in the cases of viscous and inviscid mo-tion without prescission-particle evaporation in eithercase. The calculations were performed for the fourhighest energies of projectile ions 18О among thoseconsidered here. The results of the calculations aregiven in Table 5. From a comparison with experi-mental data (see Table 2), one can see that the cal-culated values of νpre somewhat underestimate theirexperimental counterparts at low excitation energiesand overestimate them at high excitation energies.The calculated values of αpre and ppre fall somewhatshort of the respective experimental results, but thisleads to a small error in determining the temperatureTsc since the numbers of alpha particle and protonsemitted prior to fission are rather small.

The dissipation energy Ediss is also given in Ta-ble 5. With increasing initial excitation energy, itgrows from about 10 to about 18 MeV. In all prob-ability, this effect is due merely to the increase in thefree energy in the compound nucleus—the higher thefree energy in the system, including that which wasaccumulated in the course of the collective motion ofthe nucleus to the scission point, the wider the pos-sibilities for its dissipation. For a first approximation,we can assume that Ediss ∝ E∗

ini.



Table 5. Results of dynamical calculations for the properties of thorium nuclei fission at the reduction-coefficient valueof ks = 0.25

Elab, MeV 〈Jsc〉, � νpre αpre ppre Ediss, MeV Tsc, MeV �sceff, �

2 MeV tf , 10−21 s

90.2 21.0 1.10 0 0 9.4 1.15 72.42 176

117.0 33.7 2.25 0.0008 0 10.4 1.33 72.08 66

144.2 41.9 3.51 0.011 0.003 12.4 1.45 71.30 22

198.5 53.5 5.93 0.086 0.080 17.7 1.60 69.72 19

In the last column of Table 5, we present the totalfission time tf , which, of course, decreases with in-creasing initial excitation energy, but it remains ratherlong, between about 10−19 and 10−20 s.

3.2. Analysis

Thus, we have calculated all quantities required fordetermining the anisotropy of fission fragments on thebasis of Eq. (3). We now apply the same procedureas in Subsection 2.5—that is, we take into accountthe redistribution of J between l and SE∗,J accordingto expressions (11) and (12). From expression (7)at n = 2 and Mst = 2.5, we then obtain curve F inFig. 6a, which shows the same experimental dataas Fig. 5. One can see that it lies somewhat lowerthan the experimental values. But if we assume, asin Subsection 2.5, that the spin value of 0.85SE∗,J

is realized in expressions (11) and (12), we obtaincurve G in Fig. 6a.

If, in evaluating the anisotropy, we employ theangular-momentum distribution of compound nucleithat is found by the channel-coupling method on thebasis of the CCFULL code [14] (Fig. 2, solid curves)rather than those that were calculated by the empiri-cal relations proposed by Gontchar and Frobrich [15],we obtain curve H in Fig. 6a instead of curve F . Wehave not performed full dynamical calculations withthe 〈lfus〉 values found on the basis of the CCFULLcode, but, as follows from our experience of similarcalculations, a moderate change in the dependence〈lfus〉(Ec.m.) (Fig. 2) leads to moderate variations infinal results, which are quoted in Table 5. Indeed, onecan see from Fig. 6a that curves F and H are quiteclose. Figure 6b displays the same experimental dataas Fig. 6a. Curves M and N were calculated withthe total fragment spins SE∗,J from Table 4 [values incolumns S(D) and S(E), respectively, minus 9 �]—that is, with different sets of values of the parameters nand Mst in the empirical expression (7). By and large,we can state that curves G, H , and M reproduce sat-isfactorily the energy dependence of the anisotropy offission fragments in the region Ec.m. −EB � 30 MeV.In Fig. 6c, the results of the calculations from [12]

for the reactions induced by 16O + 208Pb and 19F +208Pb collisions are shown for the sake of comparison.

4. DISCUSSION OF THE RESULTS

We are aware of the fact that our semiempiricalmethod for determining the orbital angular momen-tum at the scission point by directly taking into ac-count values of the fragment spin SE∗,J (or part ofit) in the total angular momentum of the compoundnucleus being considered is a strongly simplified ap-proach to analyzing the anisotropy of fission frag-ments. Of course, it would be more correct to takeinto account the fragment-rotation energy directly inthe level density, as was done in [12]. However, ourapproach is not contradictory from the formal point ofview. The only question is that of the alignment andorientation of the total-fragment-spin vector. In orderto describe the energy dependence of the anisotropy,we assumed that, at high initial J , the fragment-spin components are aligned rather strongly, whichsuggests a strong coupling of the vectors J, SE∗,J ,and l. Ultimately, one arrives at the same qualitativeresult, a reduction of the orbital angular momentumof the nucleus undergoing fission with respect to theinitial value of J , either via taking into account thetotal energy of fragment rotation as in [12] or viaempirically evaluating the total spin of fragments.

Ogihara and his colleagues [64] also validate theexistence of a strong coupling between the entranceand exit channels in the fusion–fission process, rely-ing on a modified Ericson model [65]. In [64], thoseauthors showed that the average orbital angular mo-mentum l and the average angular momentum J arerelated by the equation

l = J(1 − sin ψ0), (13)

where the angle ψ0 is the so-called maximumcoupling-violating angle [64]. As a matter of fact, ψ0

is such an angle at high J from which experimentalangular distributions begin to deviate from the ex-perimental limit 1/ sin θ. This occurs at large anglesclose to 180◦. In [64], the angle ψ0 was determinedexperimentally for a great many reactions. For Ec.m. −


1690 RUSANOV et al.

10

(

c

)

E

c.m.

–

E

B

, MeV30 60 90 120

2

3

4

5

19

F +

208

Pb(John and Kataria [12])

16

O +

208

Pb

1

(

b

)

2

3

4

5

⟨

l

fus

⟩

(Gontchar [15])

N

M

M

:

n

= 1.7,

M

st

= 2.5

N

:

n

= 1.7,

M

st

= 3

1

(

a

)

2

3

4

5

F

:

⟨

l

fus

⟩

(Gontchar [15])

G

H

G

: 0.85

S

E

*,

J

,

⟨

l

fus

⟩

(Gontchar [15])

H

:

⟨

l

fus

⟩

(CCFULL [14])

A

19

F +

208

Pb, TSM (Hinde et al. [59])

F

n

= 2,

M

st

= 2.5

Fig. 6. As in Fig. 5, but for the case where the tem-perature Tsc and the effective moments of inertia �sc

effare calculated on the basis of the dynamical approachwith allowance for one-body dissipation at the reduction-coefficient value of ks = 0.25. In Fig. 6a, the curves rep-resent (F , G) the results of the calculations with the initial〈lfus〉 and 〈l2fus〉 values for the compound nuclei from [15],the corresponding fragment spins being set to SE∗,J

values from Table 4 (values in column S(A) minus 9 �) forcurve F and to 0.85SE∗,J for curve G, and (H) the resultsthat are analogous to those shown by curve F , but whichwere obtained with the 〈lfus〉 and 〈l2fus〉 values calculatedby using the CCFULL code [14]. In Fig. 6b, curves Mand N were calculated with the total fragment spinsSE∗,J from Table 4 (values in columns S(D) and S(E),respectively, minus 9 �). In Fig. 6c, the curves correspondto the calculations performed in [12].

EB � 30 MeV (high J), this angle falls within therange 160◦–170◦, its average value being ψ0 = 165◦.According to (13), it was therefore found in [64]

that 〈l〉 ∼= (0.66–0.82)J . In our analysis (curves A,B, and D in Fig. 5 and curves H , G, and M inFig. 6), it was found with allowance for SE∗,J that〈l〉 ∼= (0.60–0.85)J . These values are very close tothose obtained in [64].

In the region Ec.m. − EB � 30 MeV (low E∗ andJ), we were unable to describe the energy dependenceof the anisotropy within our approach (see Figs. 5and 6a)—all versions of our calculations lead to un-derestimated values of the anisotropy. In our opin-ion, this is inherent to other SSM versions [10–12](see, for example, Fig. 6c). We believe that this factis due to the smallness of SE∗,J at low initial E∗

and J . Moreover, thermal excitations of spin modesoriented at random are likely to make a dominantcontribution to the orientation of the total fragment-spin vector. This gives sufficient grounds to state thatthe scission point (or the prescission configuration ofthe nucleus undergoing fission) does not play a lead-ing role in the formation of angular distributions offission fragments, but that it is nuclear deformationsin the vicinity of the saddle point that determine thesedistributions, as was demonstrated by the calcula-tions performed in [59] on the basis of the traditionaltransition-state model (see Figs. 5 and 6).

In the region J � 35–40 �, the contribution ofSE∗,J grows fast, and we assume that a rather strongalignment of spin modes with respect to the vector Joccurs there. To some extent, these arguments aresupported by the theoretical analysis in [57, 66] ofthe angular-momentum transfer in deep-inelastic-scattering reactions. Relying on a simple nucleon-transfer model, Vandenbosch [66] showed, amongother things, that the spins of reaction products areindeed strongly aligned if the TKE (and mass) loss israther large. If the complete fusion of the projectileion and the target nucleus is treated as a limitingcase of a deep-inelastic-scattering reaction (full re-laxation of ТКЕ and full mass transfer), the alignmentof the fragment-spin components can be expected athigh J .

As E∗ and J become higher, the region wherethere occurs the formation of angular distributions offission fragments moves from the domain of nucleardeformations in the vicinity of the saddle point to thedomain of deformations in the vicinity of the scissionpoint, ultimately contracting to the scission pointitself in the corresponding limit. Concurrently, thetotal angular momentum of the compound nucleusis broken down into the orbital component, whichis responsible for the anisotropy of fission fragments,and the spin component [see Eq. (1)].

This conclusion can be explained, albeit at a quali-tative level, on the basis of linear-response theory [67,68]. This theory studies collective nuclear dynamics



within a microscopic approach. As was shown in[67, 68], the reduced coefficient of friction β in a fissile224Th nucleus is not a constant, but it depends bothon the temperature of this nucleus [67] and on itsangular momentum [68]. At low E∗ and J , the K dis-tribution that was formed in the saddle-point regiondoes not have time to change since the time of descentto the scission point, tssc, is rather short because oflow friction and low angular velocity of rotation ofthe nucleus. At high E∗ and J , the deformation atthe saddle point is smaller; as to the time of descenttssc, it becomes longer since the length of descentincreases and since the friction grows substantially.This appears to be the reason for the loss of memoryof the saddle-point values of K states. The longer thetime tssc and the higher the angular velocity, the closerto the scission point in K are the states that play themain role in the formation of angular distributions offragments, but, in this case, it is necessary to take intoaccount the spins of these fragments.

The growth of friction (viscosity) with increasingtemperature is likely to be confirmed by the analysisin [69] of experimental data on gamma rays associatedwith giant dipole resonances in nuclei.

Of course, there are alternative points of view onthe formation of angular distributions of fission frag-ments. In the present study, we fixed the prescis-sion configuration of a fissile nucleus, thereby fixing�sc

eff. Hinde et al. [59] proposed introducing, in theSSM framework, some time delay (whereby one imi-tates viscosity, ensuring an increase in the number ofprescission neutrons, which reduce the temperatureof the nucleus under consideration in the transitionstate) and an empirically chosen dependence of �effon J , thus shifting the effective transition state fromthe barrier top to some region of nuclear deformationson the descent from it. Within this approach, thoseauthors were able to describe completely the energydependence of the anisotropy of fission fragmentsfrom the reactions induced by 19F + 208Pb collisions.

Within the dynamical approach [60, 70], the en-ergy dependence of the anisotropy of fission fragmentsfrom the reactions induced by 16O + 208Pb collisionscould be reproduced fairly well by three-dimensionalLangevin calculations under the standard assumptionof a transition state in the saddle-point configurationof a rotating nucleus but with allowance for saddlepoints in the c, h, α′ coordinates. Yet, a trend toward adecrease in the anisotropy at high E∗ and J remainedin the results of the calculations reported in [60, 70],but the increase in the dimensionality of the model(a transition from one- to three-dimensional calcu-lations) improves agreement with experimental data.

On the contrary, the degree of freedom associatedwith nuclear deformations was set free in dynamical

calculations [71]. The varying coefficients of two-body viscosity β (or ks in the case of one-body dis-sipation) were chosen in such a way that the cal-culations reproduced simultaneously the prescission-neutron multiplicity νpre and the anisotropy of frag-ments. As a result, probabilistic distributions of tran-sition points in the nuclear deformation that are re-sponsible for the formation of the anisotropy of fissionfragments were obtained in [71]. It turns out thatnuclear deformations far along the descent trajectorynear the scission point rather than those in the regionof the fission barrier are the most probable for heavynuclei and high J . As a result, Drozdov and his col-leagues [71] were able to describe satisfactorily theanisotropy of fission fragments for the reactions in-duced by 16O + 208Pb, 232Th, 238U, 248Cm collisionsat high E∗ and J .

The aforementioned models [59, 60, 70, 71] takeno account of fragment spins. Obviously, our em-pirical version of the description of the anisotropy offission fragments is one of the possible ones, andour point of view has its own right of existence. Theenergy dependence of the ТКЕ at high E∗ and J(see Table 1) provides an argument in support of it,albeit an indirect one. Because of shell effects [72], theТКЕ of fragments for 226Th is greater at low than athigh E∗, but it becomes constant within the errorsstarting from Elab = 117 MeV (J = 〈lfus〉 � 40 �).The analogous effect was observed for heavy nucleiin [73]. According to the model concepts developedin [74], the ТКЕ of fission fragments is

ТКЕexpt = ТКЕ0 + ERR1,2 , (14)

where ТКЕ0 is the total kinetic energy of fission frag-ments for a nonrotating nucleus (it is approximatelyequal to the Coulomb energy of the repulsion of frag-ments) and ERR

1,2 is the centrifugal energy of frag-ments that arises owing to their relative (rigid-body)orbital rotation characterized by 〈l〉 (that is, the ТКЕmust grow by the value of the centrifugal energy). Ac-cording to our calculations for the prescission shapesof nuclei, the Coulomb energy of fragment repulsiondecreases by about 0.5 MeV as J grows from 0 to80 �, while ERR

1,2 must increase by about 3 MeV ifone takes into account the total angular momen-tum J . This means that, in relevant experiments, weshould have seen an increase of about 2.5 MeV inthe ТКЕexpt, but this increase, which is beyond theexperimental errors, has not been observed. It followsthat, at the scission point, the average orbital angularmomentum 〈l〉, as well as ERR

1,2 together with it, islikely to grow quite slowly, leaving the anisotropy andТКЕexpt of fission fragments nearly unchanged as E∗

and J grow.


1692 RUSANOV et al.

5. CONCLUSIONS

The average multiplicity Mγ of gamma rays emit-ted by fragments originating from the fission of 226Thnuclei formed upon the complete fusion of projectileions 18O and target nuclei 208Pb has been measuredat 18O projectile energies in the range Elab = 78–198.5 MeV, and the results have been analyzed. Withthe aid of the empirical relations proposed in [44,45], we have found the total spins of fragments forall energies studied here and have used these resultsto explain the energy dependence of the anisotropyof fission fragments, assuming that the formation ofangular distributions of these fragments occurs atnuclear deformations in the vicinity of the scissionpoint and that the initial total angular momentumJ of the compound nucleus at the scission point isredistributed between the relative orbital angular mo-mentum of the fragments, l, and their total spin S. Forour analysis, we have found the average temperatureof the nuclei at the scission point with allowancefor the emission of prescission particles, determinedthe average angular momenta of compound nuclei inthe entrance channel, and calculated the moments ofinertia for the chain of fissile thorium nuclei. We havedetermined all of these parameters at the scissionpoint by means of three-dimensional dynamical cal-culations based on Langevin equations, taking intoaccount the evaporation of prescission particles andthe dissipation of collective motion to internal degreesof freedom [60].

In analyzing the anisotropy in question, we haveassumed a strong alignment of the vectors offragment-spin components. In this case, the energydependence of the anisotropy of fission fragments canbe described fairly well at energies above the Coulombbarrier Ec.m. − EB � 30 MeV (high J). Admittingreasonable variations in the total spin vector, wehave not been able to describe the anisotropy in theenergy range Ec.m. − EB � 30 MeV (low E∗ and J).At the same time, the standard statistical model ofthe transition state at the saddle point [59] faithfullyreproduces the anisotropy at low E∗ and J .

We assume that, as E∗ and J grow, the region offormation of angular distributions of fission fragmentsmoves gradually from the domain of nuclear deforma-tions in the vicinity of the saddle point to the domainof deformation in the vicinity of the scission point,ultimately contracting to the scission point itself, butthat the total angular momentum of the compoundnucleus splits concurrently into the orbital compo-nent, which is responsible for the anisotropy of fissionfragments, and the spin component. This conclusioncan be qualitatively explained on the basis of linear-response theory [67, 68].

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Translated by A. Isaakyan


investigation of the reaction pb( o, ): fragment spins and...

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