Journal of Nuclear and Radiochemical Sciences, Vol. 7, No.1, pp. 7-11, 2006

The Dependency on the Dissipation Tensor of Multi-modal Nuclear Fission
T. Asano,*, a, b T. Wada,c M. Ohtac, S. Yamaji,d and H. NakaharaeaDepartment of Physics, Tohoku University, Sendai, 980-8578, Japan
Laboratory of Nuclear Science, Tohoku University, Mikamine, Sendai 982-0826, Japanc
Department of Physics, Konan University, 8-9-1 Okamoto, Kobe 658-8501, Japand
Cyclotron Center, RIKEN, Wako, Saitama 351-0198, Japane
Graduate School of Science, Tokyo Metropolitan University, Tokyo 192-0397, Japanb

Received: January 16, 2006; In Final Form: March 30, 2006
A multi-dimensional Langevin equation is applied for the study of the multi-modal nuclear fission in actinide
region. Two kinds of dissipation tensors, the wall-and-window formula and the wall formula, are used to investigate
the dynamical effects of nuclear dissipation. The mass distribution and the total kinetic energy distribution of thefission fragments are calculated for 264Fm with these dissipation tensors. It is found that the dissipation tensor plays
important roles in determining the fission paths. We obtain drastically different mass distributions by applying
different models for nuclear friction.

1. Introduction
In the study of the fission of actinide nuclei at low excitationenergies including the spontaneous fission, it was found thatthe fragment mass distribution and the total kinetic energy(TKE) distribution consist of more than one component, incontrast to the simple single peak structure that is found in thefission at high excitation energies.1– 6 This phenomenon is
attributed to the existence of more than one fission path and iscalled the multi-modal fission.
The mass and TKE distributions depend sensitively on the
excitation energy and the position of the peaks of the mass distribution suggests the influence of the closed shell structure of
the fragments. Therefore, it is supposed that the microscopic
energy plays an important role for the manifestation of thisphenomenon. It is a great challenge for us to understand thisphenomenon in terms of nuclear many-body dynamics. Several
authors studied the potential energy surface (PES) including
the microscopic energy in a multi-dimensional parameter spacethat describes various nuclear shapes; one can deduce the possible fission paths by studying the location of the saddle points
and the fission valleys in multi-dimensional parameter space.7
With this method, they could explain the general trend of theposition of the peaks of the mass distribution.
The dynamical point of view is necessary to progress thestudy of the fission mode. We have applied the Langevinapproach to the study of the fission modes in uranium nuclei
and in fermium nuclei.8–10 We studied the mass and TKE distributions and demonstrated that we can decompose the fission
events into several components by tracing the Langevin trajectories. We also studied the isotope dependence and the excitation energy dependence of the fission mode.8–10 In the previous
studies, we adopted the wall-and-window type one-body friction as the dissipation mechanism of the nuclear fission dynamics. The validity of this dissipation mechanism has been
demonstrated by one of the authors (T.W.) who studied the dissipation tensor dependence of the pre-scission neutron multiplicity and the mean TKE.11–14 From the comparison of theresults of the dynamical calculation with experimental data,
they excluded the possibility of the two-body type dissipation

to be the dominant mechanism by showing that it cannot reproduce the pre-scission neutron data and the TKE data simultaneously. On the other hand, they showed that the wall-and-window
type one-body friction can reproduce both data reasonably welland concluded that it is a reasonable model for the dissipation
mechanism of nuclear fission.
There are other models that are of one-body nature, e.g. surface-plus-window formula, modified wall-and-window formulaand chaos weighted wall formula.15–17 Though there were nofree parameters in the original derivation of the one-body friction,18, 19 the strength has been modified frequently in order toreproduce some experimental data. For example, in the studyof the light particle evaporation and the mass distribution,
Schmitt et al. used the strength as a free parameter.17 The
modification itself should be acceptable when we take accountof the simplicity of the model; it is a macroscopic model without any microscopic effect and it has no dependence on thetemperature. However, when one modifies the strength of thenuclear dissipation just to reproduce only one physical quantity,
it might be inappropriate to conclude that the deduced strength
has definite physical meanings. It may reflect the other effects
completely different from the dissipation, like the insufficiencyof the model space. It is very important to compare many (atleast more than one) physical quantities at the same time.
Among the physical quantities that are measured in nuclear
fission, the TKE and mass distributions are well investigatedexperimentally in many cases. In this study, we use thesequantities to discuss the dissipation dependence of the fission
modes. It is shown that the TKE distribution is, as was
expected, directly connected to the strength of the dissipation
and we can put some constraints on the strength of the dissipation. Furthermore, it is shown that the mass distributionchanges rather drastically when one uses different models forthe dissipation mechanism. These results demonstrate theimportance and the usefulness of the dynamical approach tothe study of the fission mode.
Section 2 gives a concise description of our framework.
Results are shown in Sec. 3 concerning the fission of 264Fm
nucleus at Ex = 20 MeV. Summary is given in Sec. 4.
2. Methods*Corresponding author. E-mail: mail: asano@nucl.phys.tohoku.
ac.jp

In this study, a multi-dimensional Langevin equation is used

© 2006 The Japan Society of Nuclear and Radiochemical Sciences
Published on Web 06/24/2006

AsanoJ. Nucl. Radiochem. Sci., Vol. 7, No. 1, 2006

to demonstrate the dynamics of multi-modal fission. It has
been successfully applied to the study of nuclear dynamics offission and fusion-fission.8–17, 20–23 The time evolution of thefission process in the multi-dimensional deformation space istraced starting from the ground state of the fissioning nucleus
via saddle points to various scission configurations.
The multi-dimensional Langevin equation has the following
form;

5000

Spherical shape

3000
2000

dqi
-1
= (m ) ij pj ,
dt
dpi
1 ∂
∂V
–
=
(m-1)jk pj p k – γ ij (m-1)jk pk + gij Rj (t),dt∂ qi
2 ∂ qi1000
00.2

γ ijwall–and–window(ks) = ks γ ijwall–and–window,
γ ijwall(ks) = ks γ ijwall.

0.4

0.6

0.8

1.0

1.2

RCM /(2R)
Figure 1. The diagonal component γ RR of the dissipation tensor with
δ = 0.0 for the mass-symmetric fission of 264Fm calculated with thewall-and-window formula (solid squares) and with the wall formula(open circles). The abscissa denotes the distance between the centersof mass of future fission fragments. R is the radius of the parent
nucleus. In both cases, the strength parameter is set to unity.

1000

Spherical shape

Scission point800

γδδ / h

600

400

200

0
0.2

0.4

0.6

0.8

1.0

1.2

RCM /(2R)
Figure 2. The diagonal component γ δδ of the dissipation tensor with
δ =0.0 for the mass-symmetric fission of 264Fm. The meaning of eachsymbol is the same as that in Figure 1.
80000

Scission point

Spherical shape

70000
60000
50000

γAA / hwhere the qi denote collective coordinates in the deformation
space and the pi are the conjugate momenta. The summationfrom 1 to n over the repeated indices is assumed with n being
the number of collective degrees of freedom. In this study, we
use the distance between the centers of mass of the two futurefragments RCM, the fragment deformation parameter δ and themass-number of one fragment A1 as three collective variables
to describe the dynamics of the multi-modal fission. V(q) is
the potential energy and mij (q) and γ ij (q) are the shape-dependent collective inertia and dissipation tensors, respectively.
The normalized random force Ri (t) is assumed to be a white
noise, i.e., <Ri (t)> = 0, <Ri (t 1) R i (t 2 )> = 2 δ ij δ (t 1 − t 2 ). The
strength of the random force gij is given by Σ kg ikg jk = T γ ij ,where T is the temperature of the compound nucleus. The
hydrodynamical inertia tensor is calculated by means of theWerner-Wheeler approximation for the velocity field.24
Two types of the dissipation tensors are used in this study,
namely the wall-and-window formula and the wall-formula.18, 19
Both are the so called one-body friction and the difference lays
in their behavior when the system has a developed neck.
Figures 1, 2, and 3 show the diagonal components of the two
dissipation tensors. Results are shown for the mass-symmetricfission of 264 Fm with δ = 0.0. Figure 1 shows the diagonalcomponent γ RR as a function of RCM. It is seen that the walland-window formula gives a smaller γ RR at the saddle region(0.5 < RCM < 0.7). It stays smaller than the wall formulae value
toward the scission region. Figure 2 shows the diagonal component γ δδ . Both formulae give essentially the same results. A
remarkable difference is seen in Figure 3 that shows the diagonal component γAA . The friction for the mass-asymmetricdegree of freedom becomes infinitely large in the wall-andwindow formula as the neck radius vanishes. As we will see
later, this special feature gives us a very distinct difference inthe fragment mass distribution. We also modify the strength ofthe dissipation by introducing the strength parameter ks ;Scission point

4000

γRR/ h

8

40000
30000
20000
10000

We take the following values for ks in this study; ks = 1.0, 0.5,
and 0.25 for the wall-and-window formula (WWF) and ks = 1.0
and 0.25 for the wall formula (WF).
Since it is essential for this study to take account of the
microscopic energy, the potential energy is calculated using themacroscopic-microscopic method. We use TWOCTR of twocenter shell model code to calculate the potential energy surface.25–27 The origin of the potential energy is set so that themacroscopic energy for the spherical shape vanishes. We do
not take account of the effect of the angular momentum or theparticle evaporation in this study. Those who are interested in
the numerical procedure should read References 8 and 10 andthe references therein.

0
0.2

0.4

0.6

0.8

1.0

1.2

RCM /(2R)
Figure 3. The diagonal component γ AA of the dissipation tensor with
δ =0.0 for the mass-symmetric fission of 264Fm. The meaning of eachsymbol is the same as that in Figure 1.

3. Results
As an example, we calculate the fission of 264 Fm at theexcitation energy Ex = 20.0 MeV. 264Fm has a special feature; itis divided into two identical doubly-magic fragments 132 Sn( Z = 50, N = 82). Though there are no fission experiments
available for this nucleus, we chose it for our calculation

The Dependency on the Dissipation Tensor of Multi-modal Nuclear Fission

2.0

5

TKE Yield [%]Deformation Yield [%]

9

2.5

6

4
3
2
1

1.5
1.0
0.5
0.0

0
-0.4

-0.2

0.0

0.2

0.4

0.6

δ

TABLE 1: Peak values of the mass number of one
fragment, <A1>, and that of the TKE, <TKE>, obtained
from the Gaussian fitting to each component and the shapeelongation β deduced from <A1> and <TKE>.
Dissipation
formulaWall-andwindow

ks *

150

200

250

TKE / MeV

Figure 4. Distribution of the deformation parameter δ at the scission
configuration for the fission of 264Fm at Ex = 20.0 MeV. Solid squares
denote the results with the wall-and-window formula (ks = 1.0) andopen circles the results with the wall formula (k s = 0.25). The sum
over the distribution is normalized to 100%.

Wall

J. Nucl. Radiochem. Sci., Vol. 7, No. 1, 2006

Fission mode

<A1>

Component I
1.00 Component II
Component III
Component I
0.50 Component II
Component III
Component I
0.25 Component II
Component III
Component I
1.00 Component II
Component III
Component I
0.25 Component II
Component III

131.7
142.5
131.1
131.5
143.4
131.4
131.4
142.9
131.1
131.5
137.9
131.5
131.5
139.0
131.4

<TKE>
(MeV)
230.6
203.9
181.7
237.8
212.6
190.5
244.1
220.1
200.0
222.3
196.1
175.5
234.9
210.3
192.1

β
1.32
1.49
1.68
1.28
1.42
1.60
1.25
1.38
1.52
1.37
1.55
1.74
1.30
1.45
1.59

*The second column represents the strength parameter ks.

because of this special feature. We perform the dynamical
calculations with WWF dissipation and also with WF dissipation.
In the previous study of the fission of 264Fm, we showed that
we can classify the fission events by using the value of thedeformation parameter δ at scission configuration.8 Figure 4
shows the distribution of the deformation parameter δ atscission configuration. In the figure, solid squares express theresults with WWF with k s = 1.0 and open circles express theones with WF with k s = 0.25. These values of the strengthparameter k s are recommended in the earlier studies of themean total kinetic energies of the fragments.28 In Figure 4, we
can clearly see the three-peak structure both in WWF case andin WF case. For example, we decompose the fission events inthe case of WWF with ks = 1.0 into three components according to the value of δ at scission configuration: Component I(δ < 0.0), Component II (0.0 < δ < 0.24), and Component III(δ > 0.24). Similar decomposition is made for other cases withdifferent strength parameter k s and with different dissipation
formula. In general, smaller value of δ means larger value of
TKE because of the compactness of the scission configuration.

Figure 5. Distribution of the TKE for the fission of 264Fm at Ex =20.0 MeV. The meaning of each symbol is the same as that in Figure 4.

It was also found in the previous study that Components I and
III correspond to the mass-symmetric fission and Component
II to the mass-asymmetric fission.8
Table 1 shows the summary of the calculation with various
dissipation tensors. Results are given for the peak value of the
TKE, <TKE>, that of the fragment mass number, <A1>, andthe shape elongation β for each of the three modes. The values
for <TKE> and <A1> are obtained with the Gaussian fitting of
the distributions. The shape elongation β was introduced by
Zhao et al. in their systematic study of the TKE in the actinide
region.29, 30 They assumed that the TKE is equal to the Coulomb
energy between point charges at the scission configuration andthey deduced the separation of the point charges D from theobserved TKE, namely TKE = Z1 Z 2 e2 /D. Then they defined
the shape elongation at the scission point β = D/(R1+R2), where
the denominator R1+R2 is the distance between the charge centers of two touching spherical fragments. They showed that
there are mainly two types of scission configurations; a massasymmetric (AH = 140) configuration with β = 1.53 and a masssymmetric configuration with β = 1.65. They also showed thatthe value of β for each mode is almost independent of the mass
number of the fissioning nucleus. For the nucleus whose mass
number is nearly equal to 260, there is an additional mass-symmetric compact configuration with β = 1.33. Since those valuesof β do not vary much with the change of the proton and theneutron numbers,29, 30 we assume those values are also applicable to 264Fm. One easily sees from Table 1 that Component I
corresponds to the mass-symmetric compact configuration,
Component II corresponds to the mass-asymmetric configuration, and Component III corresponds to the mass-symmetricelongated configuration. By comparing the values of β in
Table 1 with the systematic values, we can confirm the validityof the suggested value of ks in the sense that <TKE> is reproduced well. It is clearly seen that <TKE> changes significantly
and becomes larger for all components when we use a smaller
value of k s that means weaker one-body friction. Thus it isinappropriate to deduce the value of k s from other physicalquantities without taking account of its effect on the mean
TKE.
Next, we compare the results of WWF with that of WF
more in detail. Figure 5 shows the TKE distribution for thesame system as that in Figure 4. Again, solid squares denote
the results with WWF (k s = 1.0) and open circles denote theones with WF (k s = 0.25). It is clearly seen that by using thedifferent types of dissipation tensors, we obtain a different
fraction of contribution from each mode. It should be notedthat we do not obtain this kind of change in the fraction of contribution from each mode when we only change the strengthAsanoJ. Nucl. Radiochem. Sci., Vol. 7, No. 1, 2006
7.0
6.5
6.0
5.5
5.0
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0

0.4

-50
-26

0.2

80

100 120 140 160 180 200 220

GS 2.0parameter ks within the same dissipation formula. The contribution of Component I, the compact mass-symmetric component that has the largest TKE, is much larger in WF case thanthat in WWF case. It should also be noted that we use the samepotential energy surface in both calculations, that means thebarrier height for Component I mode is the same in both cases.
Therefore, we can say that the difference of the contribution
comes solely as a dynamical effect on the motion from saddle
region to scission configuration. In Figure 6, we show the fragment mass distribution for the same system. It is seen that thecontribution from the mass-symmetric mode (Component I) is
much larger in the case of WF which is consistent with the previous result. As a result, the mass distribution is much sharperfor WF than that for WWF. Thus in principle, we can determine which model is more appropriate for the description of
the phenomenon of multi-modal fission by comparing the contribution from each mode with experiments. This fraction of
the contribution is, of course, very sensitive to the height of thefission barrier for each mode. So it is necessary to obtain areliable microscopic energy surface before we draw a definite
conclusion about the model for the nuclear dissipation.
In order to see the origin of the huge difference in the twomodels for the dissipation, we calculate the fission trajectory
without the random force starting from several points in thesaddle region. We take the starting points at RCM / (2R) = 0.7
with δ = −0.1, 0.0, 0.1, 0.13, 0.2, and 0.3. The initial mass
asymmetry is taken as zero in all cases. Figure 7 shows theresults of the calculation without the random force. It is easily
seen that even when we start from the same initial point, thefinal scission configuration varies depending on the model forthe dissipation. WWF prefers more elongated shapes than WF
does. Because of this tendency, it is expected that WF haslarger contribution from compact scission configuration, i.e.
WF has larger contribution from Component I.
This dissipation tensor dependence comes from the difference in the structure of the dissipation tensors of these twomodels. A significant difference is seen in the diagonal component γ AA (Figure 3). Concerning the mass transfer, both dissipation tensors have the same tendency starting from theground state shape up to the saddle region (R CM / (2R) = 0.7),
however, after passing this region the two dissipation models
show completely different aspects. The component γ AA of
WWF increases inversely proportional to the area of the window between the fragments, thus it is very difficult for the system to change its mass-asymmetry after it passes the saddle
region when the system has a developed neck. In Reference 8,
we studied the potential energy surface of this system in orderto study the origin the fission modes. The lowest fission saddle-22-20 -30-36
-40

-6.0

-28-384.0
6.0
2.0 2.0
4.0

6.0

0.4

0.6

0 -4.0-10

δ = − 0.1, 0.0 & 0.1 (WWF)δ = − 0.1, 0.0, 0.1 & 0.13 (WF)

-18 -32
-46

0.8

1.0

1.2

1.4

1.6

RCM /(2R)

Fragment mass
Figure 6. Distribution of the fragment mass number for the fission of264
Fm at Ex = 20.0 MeV. The meaning of each symbol is the same asthat in Figure 4.

-2.0
-8.0

-0.2
0.2

δ = 0.3 (WWF)δ = 0.2 (WWF)δ = 0.3 (WF)
-34-48
δ = 0.2 (WF)
-44

0
8.0

0.0

60

-42

-12-16Sδ

Mass yield [%]10

Figure 7. The contour plot of the PES for the mass-symmetric fissionof 264Fm. The abscissa denotes the distance between the centers ofmass of future fission fragments and the ordinate the deformation
parameter δ. R is the radius of the parent nucleus. The ground state
and the saddle point are marked by GS and S, respectively. Thicksolid lines and dashed lines show the dynamical trajectories without
the random force starting from several points in the saddle region:
RCM /2R = 0.7 and δ = –0.1, 0.0, 0.1, 0.13(WF), 0.2 and 0.3, respectively. Thick solid lines denote the results with the wall-and-window
formula (ks = 1.0) and dashed lines the results with the wall formula(ks = 0.25).

(second saddle) lies at a finite mass-asymmetry, the massasymmetry increases first from the ground state to the saddle
point. But then in this system, there is a deep mass-symmetricfission valley after the saddle region and the mass-asymmetry
tends to decrease toward the symmetric fission valley. In thecase of WF, this mechanism works after the saddle region andenhances the contribution of Component I. On the other hand,in the case of WWF, the strong dissipation in mass-asymmetricdegree of freedom prevents the system to change its massasymmetry.
4. Summary
We have investigated the dissipation dependence of themulti-modal nuclear fission in actinide region. We have used
the wall-and-window and the window formula to derive thedissipation tensor and have modified the strength with an
overall factor. We have calculated the TKE and fragment mass
distributions for the fission of 264Fm at E x = 20.0 MeV withthese dissipation tensors. By comparing the results withdifferent dissipation tensors, we found that the dissipation
tensor affects not only the TKE distribution but also the fragment mass distribution through the change in the contributionof each fission mode. Although the multi-modal fission phenomenon is often discussed from the static point of view byusing the potential energy surface, we have shown that the
dynamical effects of the dissipation tensor play essential roles
in determining the TKE and mass distributions. The TKE distribution is very sensitive to the strength of the dissipation,
while distinct mass distributions are obtained when we adopt
different dissipation models. From the comparison with thesystematics of the experimental shape elongation β, we suggestthat the wall-and-window formula with k s = 1.0 can be a reasonable model for the dissipation in fission phenomena. It is
desirable to extend the model space to be able to compare thecalculated mass distribution with experiments more quantitatively. We also plan to adopt the microscopic inertia and dissipation tensors that are based on the linear response theory.27, 31
Acknowledgements. The authors wish to express their
gratitude to Dr. T. Ichikawa, Dr. Y. Nagame, Prof. Y. L. Zhao,

The Dependency on the Dissipation Tensor of Multi-modal Nuclear Fission

Prof. N. Takigawa, Prof. K. Hagino, and Prof. F. Hanappe for
their useful discussions. One of the authors (T. W.) is partially
supported by a Grant-in-Aid for Scientific Research No.
17540270 provided by the Japan Society for the Promotion of
Science.
References(1) R. C. Ragaini, E. K. Hulet, R. W. Lougheed, and J. Wild,
Phys. Rev. C9, 399 (1974).
(2) W. John, E. K. Hulet, R. W. Lougheed, and J. J.
Wesolowski, Phys. Rev. Lett. 27, 45 (1971).
(3) J. P. Balagna, G. P. Ford, D. C. Hoffman, and J. D. Knight,
Phys. Rev. Lett. 26, 145 (1971).
(4) E. K. Hulet, J. F. Wild, R. J. Dougan, R. W. Lougheed, J.
H. Landrum, A. D. Dougan, M. Schädel, R. L. Hahn, P. A.
Baisden, C. M. Henderson, R. J. Dupzyk, K. Sümmerer,
and G. R. Bethune, Phys. Rev. Lett. 56, 313 (1986).
(5) E. K. Hulet, J. W. Wild, R. J. Dougan, R. W. Lougheed, J.
H. Landrum, A. D. Dougan, P. A. Baisden, C. M.
Henderson, and R. J. Dupzyk, Phys. Rev. C40, 770 (1989).
(6) H. C. Britt, D. C. Hoffman, J. van der Plicht, J. B.
Wilhelmy, E. Cheifetz, R. J. Dupzyk, and R. W. Lougheed,
Phys. Rev. C30, 559 (1984).
(7) P. Möller, D. G. Madland, A. J. Sierk, and A. Iwamoto,
Nature 409, 785 (2001).
(8) T. Asano, T. Ichikawa, T. Wada, M. Ohta, S. Yamaji, and H.
Nakahara, J. Nucl. and Radiochem. Sci. 5, 1 (2004).
(9) T. Asano, T. Ichikawa, T. Wada, M. Ohta, S. Yamaji, and H.
Nakahara, Prog. Theor. Phys. Suppl. 154, 457 (2004).
(10) T. Asano, Doctoral thesis. Konan University (2004).
(11) T. Wada, N. Carjan, and Y. Abe, Nucl. Phys. A538, 283c
(1992).
(12) T. Wada, Y. Abe, and N. Carjan, Phys. Rev. Lett. 70, 3538
(1993).

J. Nucl. Radiochem. Sci., Vol. 7, No. 1, 200611(13) T. Wada, Proc. the Second Tours Sympo. on Nuclear
Physics, ed. by H. Utsunomiya, M. Ohta, J. Galin, and G.
Münzenberg (World Scientific, Singapore, 1995), p470.
(14) Y. Abe, S. Ayik, P. –G. Reinhard, and E. Suraud, Phys.
Rep. 275, 49 (1996).
(15) A. V. Karpov P. N. Nadtochy, D. V. Vanin, and G. D. Adeev,
Phys. Rev. C63, 054610 (2001).
(16) G. Chaudhuri and S. Pal, Phys. Rev. C63, 064603 (2001).
(17) C. Schmitt, J. Bartel, K. Pomorski, and A. Surowiec, Acta
Phys. Pol. B34, 1651 (2003).
(18) J. Blocki, Y. Boneh, J. R. Nix, J. Randrup, M. Robel, A. J.
Sierk, and W. J. Swiatecki, Ann. Phys. 113, 330 (1978).
(19) J. Randrup and W. J. Swiatecki, Nucl. Phys. A429, 105(1984).
(20) T. Ichikawa, Doctoral thesis. Konan University (2003).
(21) Y. Aritomo and M. Ohta, Nucl. Phys. A744, 3 (2004).
(22) P. Fröbrich and I. I. Gontchar, Phys. Rep. 292, 131 (1998).
(23) A. V. Karpov, P. N. Nadtochy, E. G. Ryabov, and G. D.
Adeev, J. Phys. G29, 2365 (2003).
(24) K. T. R. Davis, A. J. Sierk, and J. R. Nix, Phys. Rev. C13,2385 (1976).
(25) S. Suekane, A. Iwamoto, S. Yamaji, and K. Harada, JAERImemo 5918 (1974).
(26) A. Iwamoto, S. Yamaji, S. Suekane, and K. Harada, Prog.
Theor. Phys. 55, 115 (1976).
(27) K. Sato, S. Yamaji, K. Harada, and S. Yoshida, Z. Phys.
A290, 149 (1979).
(28) J. R. Nix and A. Sierk, LA-UR-86-698 (1986).
(29) Y. L. Zhao, Y. Nagame, M. Tanikawa, K. Tsukada, S.
Ichikawa, K. Sueki, Y. Oura, H. Ikezoe, S. Mitsuoka, H.
Kudo, and H. Nakahara Phys. Rev. Lett. 82, 3408 (1999).
(30) Y. L. Zhao, Y. Nagame, I. Nishinaka, K. Sueki, and H.
Nakahara, Phys. Rev. C62, 014612 (2000).
(31) H. Hofmann, Phys. Rep. 284, 137 (1997).