Journal of Nuclear and Radiochemical Sciences,Vol. 3, No. 1, pp. 95–98, 2002

95

Multi-modal Study of Angular Momentum Distribution of Fission Fragments as a Result of
Bending Modes
İ. Özkan,a Z. Büyükmumcu,b H. Sökmen,a and M. Kildir∗,aa
Department of Chemistry, Middle East Technical University, 06531 Ankara, Turkeyb
Department of Chemistry, Erciyes University, 38039 Kayseri, Turkey
Received: November 13, 2001; In Final Form: November 13, 2001
The average angular momentum of the primary fission fragments as a function of their masses was calculated using
the wave function of the bending mode excitation at the scission point. The scission configuration was assumed
to be determined by the prescission shapes of fissioning nucleus within the multimodal fission model. Results in
235
U(nth , f ) is in good agreement with the recent experimental data.

1. Introduction
The average angular momentum of the fission fragments as afunction of their masses is determined for various nuclei with thestudy of prompt γ rays.1–5 The recent observations of γ rays arecarried out with multi-fold γ-ray coincidences between several
detectors of Compton suppressed Ge arrays.
A model for calculating the angular momentum distributionof fission fragments was proposed by Rasmussen et al.6 in whicha scission point approximation of a spheroid tangent to a sphere
was made. The average angular momentum of fragment 108 Ru
as ∼6h̄ is calculated in 239 Pu(nth , f ). The model was generalized
for the case when both fragments were deformed by Pfabe andDietrich7, 8 and numerical calculations for the even-even fragments of 235 U(nth , f ) were carried out. Recently, Rasmussen
et al.9 have calculated the increase in angular momentum due
to Coulomb excitation for 104 Mo formed in the cold fission of252
Cf(sf). The small spin observed for 104 Mo, 3.8h̄, implies thatthe primary fission fragment may have no spin at the scission
point.
In this study, the average angular momentum of the primaryfission fragments as a function of their masses were calculatedusing the wave function of the wriggling-bending motion at thescission point. The prescission shapes were determined by theexperimental mass and kinetic energy distributions. The neutron
multiplicity distribution was well accounted by these shapes.
2. Calculations
2.1. Determination of Prescission Shape Parameters.
According to the BGM model,10 after leaving the compound or
ground state, the nucleus could split into fragments by a numberof paths, referred as modes. These paths are categorized as standard 1 (S1), standard 2 (S2), and superlong (SL). Each mode ischaracterized by the shape parameters of the fissioning nucleus
at the scission point. A fissioning nucleus in a given mode canbe represented by prescission shape of a fairly long flat neckconnecting two relatively large heads. Parameters of this representation are the semi-length (l), the neck radius where the neckis thinnest (r), the thinnest position of the neck (z), the curvature(c), and the extension of the neck (a), the transitional points (ζ1and ζ2 ), and the radii of spherical heads (r1 and r2 ). Parameters for each mode of fissioning system have been determinedby recipes given in References 10, 11.
2.2. Calculation of Angular Momentum of Fission Fragments. In order to calculate the angular momentum distribution of fission fragments, a model, first proposed by Rasmussen
et al.,6 and later extended to the deformed complementary fission fragments by Pfabe and Dietrich,7, 8 was followed. The∗Corresponding author. E-mail: kildir@metu.edu.tr. FAX: +900312-210-1280.

coordinate systems and parameters involved in our study are
depicted in Figure 1. The nucleus was assumed to decay into
primary fission fragments. They were modeled by embedded
ellipsoids with major, ai and minor axes, bi . These ellipsoids
separated by a center-center distance, σ, interact through the
Coulomb potential12 and nuclear potential13 so that the total potential energy of the system is the sum of these two potentials.
The total potential energy was calculated for each complementary pair as a function of distance between the centers for theiraligned configuration. Our results showed that the minimum
potential energy occurs at a distance of σ ≈ a1 + a2 . All angular momentum calculations reported in this work were done at
the minimum potential energy. The kinetic energy associatedwith the relative motion of nascent fragments was obtained withthe adiabatic approximation. Ignoring the coupling between theinternal and total angular momentum together with rotation offragments around their symmetry axes, the Hamiltonian describing the internal motion became,H =l1 · l2

l12
l2+ 2 +C1 sin2 θ1 +C2 sin2 θ2
0
2B1 2B2− C12 sin θ1 sin θ2 cos(φ2 − φ1 ) ,+(1)where li is the angular momentum of the i-th fragment about its
center of mass, 0 = µσ2 , with µ being the reduced mass of thetwo fragments and σ being center-to-center distance (Figure 1),
B1 and B2 are the reduced inertias, C1 , C2 , and C12 are coefficients calculated depending on shape and charge of fragments.
Angles θ1 , θ2 , φ1 , and φ2 are defined in Figure 1. We solved
the Schrödinger equation exactly. First we obtained a suitablebasis set by solving one particle eigenvalue equations. We thenused these functions as a product-bases for expanding the twofragment eigenfunctions. Due to the factorization provided by
the M quantum number and the even-odd parity of state functions, the problem was amendable to practically exact solutions.
The average values of l1 and l2 quantum numbers for two fragments were evaluated with the calculated wave functions. For
non-zero temperatures, they were obtained with Boltzmann averaging.

Figure 1. The coordinate systems and definition of parameters.

c 2002 The Japan Society of Nuclear and Radiochemical Sciences
Published on Web 6/30/2002

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J.Nucl.Radiochem.Sci.,Vol. 3, No. 1, 2002

ÖzkanTABLE 1: The prescission shape parameters for three low-energy fissioning systems.

l
/ fm

r
/ fm

c
/ fm−1

Z
/ fm

r1
/ fm

r2
/ fm

ζ1
/ fm

ζ2
/ fm

a
/ fm

Weight
/%

15.00
16.30
18.90

2.73
2.96
3.39

0.01
0.0053
0.01

0.63
1.08
0.00

6.04
6.01
4.88

5.50
5.14
4.88

4.71
4.30
1.38

14.36
18.18
26.66

0.86
1.23
5.17

17.92
81.98
0.09

16.20
17.10
18.10
20.80

2.95
3.11
3.29
3.30

0.038
0.00600.0070
0.011

0.43
0.78
1.44
0.00

5.85
5.89
5.76
4.71

5.54
5.30
4.87
4.71

3.39
3.84
3.35
0.87

17.91
19.69
23.06
31.32

2.76
1.55
2.56
80.00

10.50
55.50
26.00
0.25

16.60
17.37
19.10
19.50

2.25
2.50
3.47
3.00

0.00110.00300.00690.0025

0.37
0.71
1.50
0.00

6.12
6.14
5.43
5.47

5.78
5.53
4.67
5.47

5.15
4.80
2.16
3.27

16.51
18.95
26.78
24.79

0.69
1.07
3.68
1.87

14.45
46.60
37.79
1.22

235

U(nth , f )Standard 1
Standard 2
Superlong248
Cm(sf)
Standard 1
Standard 2
Standard 3
Superlong252
Cf(sf)
Standard 1
Standard 2
Standard 3
Superlong

3. Results and Discussion
The shape parameters together with fission mode probabilities were determined for 235 U(nth , f ), 248 Cm(sf), and 252 Cf(sf)and they are given in Table 1.
The radius parameterr0 = 1.2249 fm was used throughout the calculations. The determined shapes were used to calculate many experimentallymeasurable quantities in fission using the Monte-Carlo Method.
The results of these calculations and their comparison with theexperimental results will be published in the forthcoming paper.
The model described in sect. 2.2 was used to calculate theaverage angular momenta of complementary even-even primaryfission fragments for 235 U(nth , f ), 248 Cm(sf), and 252 Cf(sf). The
deformation parameters of fragments at scission were obtainedby using the prescission shape of each mode. The prompt neutron multiplicity as a function of fission fragment masses wascalculated for each system. The close correlation between thecalculated results and the experimental data for prompt neutron multiplicity gave a strong support to the correctness of the
shapes employed in this study. The even-Z even-A complementary primary fragments with the largest independent yields intheir corresponding mass chains according to the Z p model14
were chosen for angular momentum calculations as long as theirisobaric mode probabilities were larger than 1%. The total potential energy taken as the sum of the shape dependent Coulomb
and Proximity Potentials was calculated for each fragment pair
as a function of the distance between the centers for their aligned
configuration. It has been seen that the extremum points of thepotential energy curves move to a larger center-center distances
as one goes from S1 to S2 and then to SL mode. It has also
been observed that there is a decrease in the potential energy inthe same order. Both of these observations are consistent withthe formation of more deformed fragments in this order. We
assume that the distribution of angular momenta between thecomplementary fragments is determined in the scission valley,
and not seriously altered during the rapid passage through thelow scission barrier.
The average angular momenta of the primary fission fragments as a function of their mass numbers were calculated in
the various fission modes of all fissioning systems. The results for 235 U(nth , f ) are shown in Figure 2 together with theirweighted average over these modes. Angular distribution of theγ ray emitted by the fragments15 shows that the direction of angular momentum is perpendicular to the fission axis in accordance with our results. The trend of angular momentum values
for each mode is very similar to that of the average neutron multiplicity values since both of them are directly related to the deformation of fragments at the scission point. It is seen from Figure 2 that in each mode, the angular momenta smoothly increase
as a function of the fragment mass. This property of the angu-

lar momenta is closely related to a similar trend in the restoring force constants (Ci ) and the reduced moment of inertia (Bi ).
Ignoring the coupling between the two fragments, and making
the harmonic approximation, it may be shown semiclassically8that the ground state angular momentum of a fragment is proportional to (Ci Bi )1/4 Thus, the change in the angular momenta
parallels those in Ci and Bi . The sharp decrease in the averageangular momentum around the symmetric division is related tothe asymmetric prescission shapes in standard modes. In the latter, the flat neck is ruptured closer to the larger head associatedwith the heavy fragment in order to produce fragments close tomass symmetry. As a result, the heavy fragment is formed witha smaller deformation in contrast to its more deformed partner.
Even at the symmetric division, the complementary fragments
in standard modes are formed with much different deformations
on the basis of the multi-modal fission model. As a result, a
saw-tooth structure is expected to be observed in the angularmomentum as a function of fragment mass. On the other hand,due to the symmetric prescission shapes in the SL mode, the deformation and angular momentum of a fragment is expected toincrease with its mass as seen in Figure 2. The behavior of theaverage angular momentum as a function of mass weighted overthe fission modes is also shown in the same figure. The structure on this curve carries the signatures of both mode probabilities and angular momenta. For example, the sharp drop around∼ 122 for 235 U(nth , f ) is a result of the diminishing contribuA=tion of the SL mode to the yield distribution as one moves awayfrom the symmetric split. The location and degree of sharpness
is due to the contribution of S1 to the symmetric region. Experimental data on the angular momentum distribution aroundthis region would be very valuable. It has been shown16 thatthe multi-modal fission model may be capable of explaining theexperimental results on isomer ratios.

Figure 2. The dependence of calculated average angular momentum
on fragment mass for fission modes in 235 U(nth , f ).

Multi-modal Study of Angular Momentum Distribution
The angular momentum of fission fragments due to the bending motion may be altered due to their Coulomb excitation.
There may be an increase or decrease in this angular momentum. In the calculations of Dietrich and Zielinska-Pfabe8 an increase of about 10–15% was concluded for this effect. Mişicu etal.17 concluded that the Coulomb excitation accounts less than
10% of the final spin of fragments in the cold fission of 252 Cf.
A simple classical model6 has been proposed to investigate thechange in the distribution of angular momenta due to the subsequent Coulomb excitation. In this model, it is assumed thatthe shape of fragments do not change during acceleration. As aresult, the axially symmetric fragment with intrinsic quadrupole
moment feels a torque depending on the time it takes to rotate
it, and changes its angular momentum by ∆l. The angular momentum due to Coulomb excitation is assumed to be perpendicular to the angular momentum due to the bending motion. The
results of our calculations within this model indicated that the
post-scission change of the fragment angular momenta is relatively small (only a small fraction of h̄). This result may easily
be understood because the aligned configuration for primary fission fragments is the most probable scission condition in which
the torque felt by each fragment vanishes. It may be deduced
from the values that angular momentum change due to Coulomb
excitation is negligible with the assumption that it is perpendicular to the angular momentum due to the bending motion. However, detailed trajectory calculations are needed to make a final
decision about the quantity of this effect.
In Figure 3, our calculated values of the average angularmomenta are compared with the experimental results3 for thermal neutron-induced fission of 235 U. The preliminary conclusion
reached by Shannon3 in his measurements of the ratio of average
spins at which discrete states are populated in cold fission to theaverage when the same nucleus is formed as a primary productin fission events in which the total number of prompt neutrons
is equal to four. The value of this ratio suggests that the average angular momentum of primary fragments is only about 1h̄higher than in cold fission fragments. The 1h̄ may be taken to bearising from the reduction in the angular momentum due to twoneutrons on the average being emitted from each primary fragment. Calculation with the statistical evaporation code CASCADE18 suggesting that each neutron evaporation reduces theaverage entry point spin by ≈0.5h̄ is consistent with this above
conclusion. CASCADE calculations also suggest that statisticalγ transitions are expected to produce an even smaller reductionin the entry point spins. Shannon, considering effects of neutronand statistical γ ray emission in the angular momentum, corrected the angular momentum of fission products to obtain thatof primary fission fragments.3 They are given in Figure 3. The
agreement is observed between our calculated values within the
BGM model and the experimental values of the average angularmomenta as a function of fission fragment masses.
The effect of intrinsic excitation energy on the angular mo-

Figure 3. A comparison of calculated and experimental average angular momentum as a function of fragment mass in 235 U(nth , f ).

J.Nucl.Radiochem.Sci.,Vol. 3, No. 1, 2002

97

mentum distribution was examined. The prescission shape parameters were determined for the nuclear temperature Ti for thei-th mode with the intrinsic excitation energy at scission Esi∗ .
The latter energy was taken to be 26% of the potential energy
freed on descent from the last barrier before scission ∆Ui for
235
U(nth , f ). The nuclear temperatures obtained in this way wereclose to 0.4 MeV. The curve indicated as the calculated values of
the average angular momenta in Figure 3 was for T = 0.4 MeV
which was almost indistinguishable from the corresponding one
for T = 0. Assuming ∆Ui is fully converted into intrinsic excitation energy, a nuclear temperature of about 0.8 MeV would be
expected. The thicker curve in Figure 3 represents T = 0.8 MeV.
Such an approximation gives an upper limit of the effect of temperature on angular momenta since it leaves no room for prescission kinetic energy, and energy for other degrees of collective
excitations other than deformation.
Figure 3 also shows the results of the calculations made byDietrich and Zielinska-Pfabe8 for the zero temperature case. In
their work, the fissioning system at the scission point was represented as two nuclei with quadrupole deformations that wereobtained by the total excitation energies of the fragments. Theirtheoretical results were in good agreement with the γ-ray multiplicity experiment1 under the assumption that the angular momentum of a fission fragment is approximately equal to twice
the number of γ rays, N(A)γ . They suggested that there are weak
points in the approximate relation l¯i = 2Nγi . Since the characterof the γ-ray cascade is expected to depend on the nuclear structure of the fragments such a simple relation may not be realistic.
As seen in Figure 3, contrary to the close relation between our
results and the recent experimental data, the results of Dietrich
and Zielinska-Pfabe are unsatisfactory especially for the heavy
fragments. This may be because the scission configuration isrepresented better in the multi-modal fission model used in ourstudy.
In addition, the average angular momentum of primary fission
fragments over all modes and masses were calculated and foundto be 5.7h̄ for 235 U(nth , f ). The average angular momentum of
primary fission products, obtained by Ahmad and Philips19 isequal to 4.6h̄ for the 235 U(nth , f ). The difference of 1.1h̄ between
the angular momenta of primary fission fragments and products
is consistent with our estimation of the decrease in angular momentum due to prompt neutron and statistical γ-ray emissions.
Average angular momentum values are also calculated for252
Cf(sf) and 248 Cm(sf). Reasonable agreement with experimental data has been obtained for these systems.
Acknowledgement. Financial support of METU throughAFP 94-01-03-05 and AFP 96-01-03-06 are gratefully acknowledged.
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