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Time scales of fusion-fission reactions calculated from prescission neutron multiplicities
NUCLEAR PHYSICS A El ,SEVIER
Nuclear Physics A583 (1995) 141-144
Time scales of fusion-fission reactions prescission neutron multiplicities
calculated
from
K. Siwek-Wilczyfiskaa, J. Wilczyfiski b, R.H. Siemssen c and H.W. WilschuV ~Institute of Experimental Physics, Warsaw University, 00-681 Warsaw, Poland bSoltan Institute for Nuclear Studies, 05-400 Swierk-Otwock, Poland CKernfysisch Versneller Instituut, 9747 AA Groningen, The Netherlands
A b s t r a c t : The time scale of fusion-fission reactions was found to be in the range from vf = 5 • 10 -2° to 5 • 10-19 s. This result was obtained from the analysis of the preseission neutron multiplicities with a new method combining the time-dependent statistical cascade calculations with the nuclear dynamics model of Feldmeier.
1. I N T R O D U C T I O N The possibility of experimental separation and determination of the prescission multiplicity of neutrons (see e.g. Refs. [1,2]) and light charged particles [3,4] in fusion-fission reactions has offered an interesting tool for studying the time scale of fission of hot composite systems. The fact that up to 6-8 neutrons are being evaporated prior to scission from heavy (destined to fission) composite systems clearly shows that fission of these hot nuclei simply requires time for necessary rearrangement in shape degrees of freedom. This time turns out to be long in comparison with the mean life-time of hot nuclei. Consequently, at high excitation energies many A i~Tx light particles are evaporated during the time :n of the collective rearragement, in spite of the i A-I fact that the system is destined to fission from the beginning of the evaporation casi= H A-2 cade. ;~-T 3 i= As it was pointed out by Hinde et al.[1,2], evaporation of neutrons can be used as ta a "clock" for measuring the time scale of t3 the underlying fission process. Hinde and coworkers analysed the prescission neutron TIME multiplicities in a "static" approach in which the evaporation cascade was calculated for a Figure 1. Deexcitation cascade as dynamically stable compound nucleus of the "neutron clock" 0375-9474/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved. ,~,~I-')I (~ 7 ~ _O~TA/OA ~f~6~.AQ
14 2c
K. Siwek-Wilczy~iska et al. / Nuclear Physics A583 (1995) 141-144
a given initial value of excitation energy at the beginning of the cascade. In this report we demonstrate that the deduced time scales stronly depend on the assumed dynamical scenario of the fusion-fission reaction. As a rule, the dynamic calculations lead to considerably longer time scales than those deduced in the static model of refs. [1,2]. 2. S T A T I S T I C A L C A S C A D E P L U S D Y N A M I C S In our calculations we combined a simple version of the time-dependent statistical model with Feldmeier's dynamical code HICOL, based on the one-body dissipation model [5]. In this approach the statistical deexcitation of the composite system plays the role of a "neutron clock" as it is illustrated in Fig. 1. By using the Monte Carlo method, at each successive stage of the cascade the actual decay time (tl,t2, t3...) is sampled using the actual mean value of the life time < r > = h/Ftot. The fission width is artificially supressed to zero during the prescission cascade. Therefore the total decay width in the prescission stage contains only components representing evaporation of light particles (up to rLi). After the determination of the actual decay time, the kind of the evaporated particle (n, p, or, or any other complex particle of Z = 1,2, 3) is sampled from the relative decay widths, and finally the actual value of the kinetic energy of that chosen particle can be sampled assuming the Maxwellian shape of its energy spectrum. The individual decay cascade (as shown in Fig. 1) is repeatedly calculated and the event-averaged results are used for determination of time dependence of the multiplicity of neutrons, protons and all other light particles included in the evaporation code. The moment in time when the calculated neutron multiplicity reaches the measured value of the prescission neutron multiplicity, %,,'~v,determines the fusion-fission time scale rj, as illustrated in Fig. 2.
oos8 +
H U
H .....
V
~
M II) ,.t
e o
...................
£ul~on ÷ E ~ Ogl
e~,t ~ Xa,T:LC 1 . . . .e. . . .
..................
static
limit
.......
H
u
r
!
exit channel ( sa,Sd..1.e
ii) o N 0 M TI~
Tt
Figure 2. Determination of the fusionfission time scale, rl, from the prescission neutron multiplicity, up,~=P.
E.I
Time
Figure 3. Illustration of the inclusion of the dynamic model predictions to the time-dependent deexcitation calculations (see text).
K. Siwek-Wilczytiska et al. / Nuclear Physics A583 (1995) 141-144
143c
Contrary to static calculations, in which one has to assume implicitely that the composite system is almost instantly equilibrated in all degrees of freedom, in our dynamic calculations the amount of the thermalized excitation energy gradually increases in time-according to predictions of the one-body dissipation model [5]. The dynamic evolution of a composite system formed in a nucleus-nucleus collision is calculated with the program HICOL [5] in a configuration space of three shape variables. Classical equations of motion are solved numerically in this configuration space assuming the "Yukawa-plus-exponential" potential [6], the inertia in the approximation of the irrotational mass flow, and the one-body dissipation in the form of the "windowplus-wall" formula [7,5]. The code HICOL does not contain any truly free parameters and consistently describes the dynamical evolution of various composite systems formed in nucleus-nucleus collisions in the full range of impact parameters. It is also important to note that the one-body dissipation parameters, at least in first order, are independent on the temperature (excitation energy) of the system. Fig. 3 illustrates how the dynamic generation of excitation energy is incorporated into the statistical cascade calculations. It can be seen that the HICOL code predicts a fast rise of the thermalized excitation energy in the entrance-channel stage. However the static limit of the excitation energy, E* : (~jus .j_ Ecru, cannot be reached due to the fact that -ogs an important part of the energy released in the collision is "frozen" in the rotational and deformation degrees of freedom. In case of fusion, i.e., when the system gets behind the unconditional saddle point, the classical trajectory calculation stops. The calculation can be resumed only in a later stage--when the fissioning system emerges outside the saddle in the exit-channel configuration. The time At (when the system remains fused) is determined from fitting the calculated neutron multiplicity to the measured value of u ~ exactly at the time ry which is the sum of the fusion time t¢~, saddle-to-scission time t . . . . (both calculated with HICOL, see Fig. 3), and At: rl = t/us q- A t -b t . . . . . The HICOL calculations show that the thermal excitation energy increases very slowly throughout the composite system stage, i.e., during the period At. Therefore in numerical calculations we interpolated linearly the excitation energy between the end of the entrance channel and the beginning of the exit channel (see Fig. 3). Having the time dependence "~
II ,-4 ~ U 4) N -,4 ~
10 "x7
t_0 " l j
,
.
100
I 110
10-1s
0
, :'.:t
m ~154
.
10-=o
0 m ~
1 0 "al SO
610
70
810
90 ZX+Z 2
I 120
130
Figure 4. The fusion-fission time scale deduced from prescission neutron multiplicities [1] by using the proposed dynamic approach (full circles). For comparison, results of static calculations reported by Hinde et a/.[1,2] are indicated by asterisks (error bars are omitted).
144c
K. Siwek-Wilczytiska et al. / Nuclear Physics A583 (1995) 141-144
of the generated excitation energy at each stage of the individually calculated deexcitation cascade, the actual value of the excitation energy was incremented by the amount of the thermal excitation energy generated during a given time interval tl - ti-1. 3. D Y N A M I C VS. S T A T I C C A L C U L A T I O N S Using the dynamic model described above we have analysed the entire set of prescission neutron multiplicities compiled and analysed previously within a static model by Hinde et al. [1,2]. The results are presented in Fig. 4. The deduced values of r! are plotted as a function of the combined atomic number of the composite system Z = Z1 + Z2. Results of the dynamic model are shown by full dots. The error bars reflect only the uncertainties exp in determination of the v~,,-values. Some additional uncertainties are associated with the assumed effective value of the angular momentum. For all fusion-fission reactions the calculations were done for g-values close to the critical angular momenta predicted by the code HICOL. Taking into account the Ldependence of both the thermal excitation energy and the fusion-fission cross section, these g-values seem to be representative for the whole range of the fusion-fission and/or fast fission reactions. In order to demonstrate the importance of the dynamical effects, we included in Fig. 4 the results of Hinde et al.[1,2], obtained with the static approach. It is seen that the estimates of Refs. [1,2] give considerably shorter time scales (on the average, by a factor of about 10) than those calculated with the dynamic model. In conclusion, our analysis of the prescission neutron multiplicities shows that the fusion-fission time scale is confined to the range from r$ = 5.10 -20 to rf = 5.10 -19 s. The T! values deduced with the dynamic model are considerably longer than those obtained by Hinde et al. [1,2]. Our analysis demonstrates that the use of the "neutron clock" for determination of the time scale of the fusion-fission reactions definitely requires inclusion of the underlying dynamics into the time-dependent deexcitation calculations. This work was performed as a part of the research program of the Stichting voor Fundamenteel Onderzoek der Materie (FOM) and the Committee of Scientific Research of Poland (KBN grant No. 2P302-211-04). REFERENCES 1. D.J. Hinde, D. Hilscher, H. Rosner, B. Gebauer, M. Lehmann and M. Wilpert, Phys. Rev. C45 (1992) 1229. 2. D.J. Hinde, Nucl. Phys. A553 (1993) 255c. 3. K. Siwek-Wilczyrlska, J. Wilczyfiski, H.K.W. Leegte, R.H. Siemssen, H.W. Wilschut, K. Grotowski, A. Panasiewicz, Z. Sosin and A.Wieloch, Phys. Rev. C48 (1993) 228. 4. J.P. Lestone, J.R. Leigh, J.O. Newton, D.J. Hinde, J.X. Wei, J.X. Chen, S. ElfstrSm and M. Zielinska-Pfab~, Nucl. Phys. A559 (1993) 277. 5. H. Feldmeier, Rep. Prog. Phys. 50 (1987)915. 6. H.J. Krappe, J.R. Nix and A.J. Sierk, Phys. Rev. C20 (1979) 992. 7. J. Blocki, Y. Boneh, J.R. Nix, J. Randrup, M. Robel, A.J. Sierk and W.J. Swiatecki, Ann. of Phys. 113 (1978) 330.
Nuclear Physics A583 (1995) 141-144
Time scales of fusion-fission reactions prescission neutron multiplicities
calculated
from
K. Siwek-Wilczyfiskaa, J. Wilczyfiski b, R.H. Siemssen c and H.W. WilschuV ~Institute of Experimental Physics, Warsaw University, 00-681 Warsaw, Poland bSoltan Institute for Nuclear Studies, 05-400 Swierk-Otwock, Poland CKernfysisch Versneller Instituut, 9747 AA Groningen, The Netherlands
A b s t r a c t : The time scale of fusion-fission reactions was found to be in the range from vf = 5 • 10 -2° to 5 • 10-19 s. This result was obtained from the analysis of the preseission neutron multiplicities with a new method combining the time-dependent statistical cascade calculations with the nuclear dynamics model of Feldmeier.
1. I N T R O D U C T I O N The possibility of experimental separation and determination of the prescission multiplicity of neutrons (see e.g. Refs. [1,2]) and light charged particles [3,4] in fusion-fission reactions has offered an interesting tool for studying the time scale of fission of hot composite systems. The fact that up to 6-8 neutrons are being evaporated prior to scission from heavy (destined to fission) composite systems clearly shows that fission of these hot nuclei simply requires time for necessary rearrangement in shape degrees of freedom. This time turns out to be long in comparison with the mean life-time of hot nuclei. Consequently, at high excitation energies many A i~Tx light particles are evaporated during the time :n of the collective rearragement, in spite of the i A-I fact that the system is destined to fission from the beginning of the evaporation casi= H A-2 cade. ;~-T 3 i= As it was pointed out by Hinde et al.[1,2], evaporation of neutrons can be used as ta a "clock" for measuring the time scale of t3 the underlying fission process. Hinde and coworkers analysed the prescission neutron TIME multiplicities in a "static" approach in which the evaporation cascade was calculated for a Figure 1. Deexcitation cascade as dynamically stable compound nucleus of the "neutron clock" 0375-9474/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved. ,~,~I-')I (~ 7 ~ _O~TA/OA ~f~6~.AQ
14 2c
K. Siwek-Wilczy~iska et al. / Nuclear Physics A583 (1995) 141-144
a given initial value of excitation energy at the beginning of the cascade. In this report we demonstrate that the deduced time scales stronly depend on the assumed dynamical scenario of the fusion-fission reaction. As a rule, the dynamic calculations lead to considerably longer time scales than those deduced in the static model of refs. [1,2]. 2. S T A T I S T I C A L C A S C A D E P L U S D Y N A M I C S In our calculations we combined a simple version of the time-dependent statistical model with Feldmeier's dynamical code HICOL, based on the one-body dissipation model [5]. In this approach the statistical deexcitation of the composite system plays the role of a "neutron clock" as it is illustrated in Fig. 1. By using the Monte Carlo method, at each successive stage of the cascade the actual decay time (tl,t2, t3...) is sampled using the actual mean value of the life time < r > = h/Ftot. The fission width is artificially supressed to zero during the prescission cascade. Therefore the total decay width in the prescission stage contains only components representing evaporation of light particles (up to rLi). After the determination of the actual decay time, the kind of the evaporated particle (n, p, or, or any other complex particle of Z = 1,2, 3) is sampled from the relative decay widths, and finally the actual value of the kinetic energy of that chosen particle can be sampled assuming the Maxwellian shape of its energy spectrum. The individual decay cascade (as shown in Fig. 1) is repeatedly calculated and the event-averaged results are used for determination of time dependence of the multiplicity of neutrons, protons and all other light particles included in the evaporation code. The moment in time when the calculated neutron multiplicity reaches the measured value of the prescission neutron multiplicity, %,,'~v,determines the fusion-fission time scale rj, as illustrated in Fig. 2.
oos8 +
H U
H .....
V
~
M II) ,.t
e o
...................
£ul~on ÷ E ~ Ogl
e~,t ~ Xa,T:LC 1 . . . .e. . . .
..................
static
limit
.......
H
u
r
!
exit channel ( sa,Sd..1.e
ii) o N 0 M TI~
Tt
Figure 2. Determination of the fusionfission time scale, rl, from the prescission neutron multiplicity, up,~=P.
E.I
Time
Figure 3. Illustration of the inclusion of the dynamic model predictions to the time-dependent deexcitation calculations (see text).
K. Siwek-Wilczytiska et al. / Nuclear Physics A583 (1995) 141-144
143c
Contrary to static calculations, in which one has to assume implicitely that the composite system is almost instantly equilibrated in all degrees of freedom, in our dynamic calculations the amount of the thermalized excitation energy gradually increases in time-according to predictions of the one-body dissipation model [5]. The dynamic evolution of a composite system formed in a nucleus-nucleus collision is calculated with the program HICOL [5] in a configuration space of three shape variables. Classical equations of motion are solved numerically in this configuration space assuming the "Yukawa-plus-exponential" potential [6], the inertia in the approximation of the irrotational mass flow, and the one-body dissipation in the form of the "windowplus-wall" formula [7,5]. The code HICOL does not contain any truly free parameters and consistently describes the dynamical evolution of various composite systems formed in nucleus-nucleus collisions in the full range of impact parameters. It is also important to note that the one-body dissipation parameters, at least in first order, are independent on the temperature (excitation energy) of the system. Fig. 3 illustrates how the dynamic generation of excitation energy is incorporated into the statistical cascade calculations. It can be seen that the HICOL code predicts a fast rise of the thermalized excitation energy in the entrance-channel stage. However the static limit of the excitation energy, E* : (~jus .j_ Ecru, cannot be reached due to the fact that -ogs an important part of the energy released in the collision is "frozen" in the rotational and deformation degrees of freedom. In case of fusion, i.e., when the system gets behind the unconditional saddle point, the classical trajectory calculation stops. The calculation can be resumed only in a later stage--when the fissioning system emerges outside the saddle in the exit-channel configuration. The time At (when the system remains fused) is determined from fitting the calculated neutron multiplicity to the measured value of u ~ exactly at the time ry which is the sum of the fusion time t¢~, saddle-to-scission time t . . . . (both calculated with HICOL, see Fig. 3), and At: rl = t/us q- A t -b t . . . . . The HICOL calculations show that the thermal excitation energy increases very slowly throughout the composite system stage, i.e., during the period At. Therefore in numerical calculations we interpolated linearly the excitation energy between the end of the entrance channel and the beginning of the exit channel (see Fig. 3). Having the time dependence "~
II ,-4 ~ U 4) N -,4 ~
10 "x7
t_0 " l j
,
.
100
I 110
10-1s
0
, :'.:t
m ~154
.
10-=o
0 m ~
1 0 "al SO
610
70
810
90 ZX+Z 2
I 120
130
Figure 4. The fusion-fission time scale deduced from prescission neutron multiplicities [1] by using the proposed dynamic approach (full circles). For comparison, results of static calculations reported by Hinde et a/.[1,2] are indicated by asterisks (error bars are omitted).
144c
K. Siwek-Wilczytiska et al. / Nuclear Physics A583 (1995) 141-144
of the generated excitation energy at each stage of the individually calculated deexcitation cascade, the actual value of the excitation energy was incremented by the amount of the thermal excitation energy generated during a given time interval tl - ti-1. 3. D Y N A M I C VS. S T A T I C C A L C U L A T I O N S Using the dynamic model described above we have analysed the entire set of prescission neutron multiplicities compiled and analysed previously within a static model by Hinde et al. [1,2]. The results are presented in Fig. 4. The deduced values of r! are plotted as a function of the combined atomic number of the composite system Z = Z1 + Z2. Results of the dynamic model are shown by full dots. The error bars reflect only the uncertainties exp in determination of the v~,,-values. Some additional uncertainties are associated with the assumed effective value of the angular momentum. For all fusion-fission reactions the calculations were done for g-values close to the critical angular momenta predicted by the code HICOL. Taking into account the Ldependence of both the thermal excitation energy and the fusion-fission cross section, these g-values seem to be representative for the whole range of the fusion-fission and/or fast fission reactions. In order to demonstrate the importance of the dynamical effects, we included in Fig. 4 the results of Hinde et al.[1,2], obtained with the static approach. It is seen that the estimates of Refs. [1,2] give considerably shorter time scales (on the average, by a factor of about 10) than those calculated with the dynamic model. In conclusion, our analysis of the prescission neutron multiplicities shows that the fusion-fission time scale is confined to the range from r$ = 5.10 -20 to rf = 5.10 -19 s. The T! values deduced with the dynamic model are considerably longer than those obtained by Hinde et al. [1,2]. Our analysis demonstrates that the use of the "neutron clock" for determination of the time scale of the fusion-fission reactions definitely requires inclusion of the underlying dynamics into the time-dependent deexcitation calculations. This work was performed as a part of the research program of the Stichting voor Fundamenteel Onderzoek der Materie (FOM) and the Committee of Scientific Research of Poland (KBN grant No. 2P302-211-04). REFERENCES 1. D.J. Hinde, D. Hilscher, H. Rosner, B. Gebauer, M. Lehmann and M. Wilpert, Phys. Rev. C45 (1992) 1229. 2. D.J. Hinde, Nucl. Phys. A553 (1993) 255c. 3. K. Siwek-Wilczyrlska, J. Wilczyfiski, H.K.W. Leegte, R.H. Siemssen, H.W. Wilschut, K. Grotowski, A. Panasiewicz, Z. Sosin and A.Wieloch, Phys. Rev. C48 (1993) 228. 4. J.P. Lestone, J.R. Leigh, J.O. Newton, D.J. Hinde, J.X. Wei, J.X. Chen, S. ElfstrSm and M. Zielinska-Pfab~, Nucl. Phys. A559 (1993) 277. 5. H. Feldmeier, Rep. Prog. Phys. 50 (1987)915. 6. H.J. Krappe, J.R. Nix and A.J. Sierk, Phys. Rev. C20 (1979) 992. 7. J. Blocki, Y. Boneh, J.R. Nix, J. Randrup, M. Robel, A.J. Sierk and W.J. Swiatecki, Ann. of Phys. 113 (1978) 330.