Krzyczkowski, J

Siemssen, RH

Siwek-Wilczynska, K

Wilczynski, J

Wilschut, HW

English

Pre-scission neutron multiplicities in fusion-fission reactions, reported by Hinde et al., have been analysed in terms of the statistical model assuming a possible hindrance of the compound-nucleus fission width by the Kramers factor which depends on nuclear dissipation. Contrary to earlier results reported by Hofman, Back and Paul on an analysis of the GDR gamma-decay, the nuclear dissipation deduced in the present analysis does not show a clear dependence on the temperature of the compound nucleus. On average, the deduced values of the nuclear dissipation are consistent with the one-body dissipation estimate.</p

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   University of GroningenAnalysis of pre-scission neutron multiplicities in terms of the statistical model with Kramersdissipative fissionSiwek-Wilczynska, K; Krzyczkowski, J; Wilczynski, J; Siemssen, RH; Wilschut, HWPublished in:Acta Physica Polonica BIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:1998Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Siwek-Wilczynska, K., Krzyczkowski, J., Wilczynski, J., Siemssen, RH., & Wilschut, HW. (1998). Analysis ofpre-scission neutron multiplicities in terms of the statistical model with Kramers dissipative fission. ActaPhysica Polonica B, 29(1-2), 451-456.CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Vol. 29 (1998) ACTA PHYSICA POLONICA B No 1–2ANALYSIS OF PRE-SCISSION NEUTRONMULTIPLICITIES IN TERMS OF THE STATISTICALMODEL WITH KRAMERS DISSIPATIVE FISSION∗K. Siwek-Wilczyńska, J. KrzyczkowskiInstitute of Experimental Physics, Warsaw University00–681 Warsaw, PolandJ. WilczyńskiInstitute for Nuclear Studies05-400 Świerk-Otwock, PolandR.H. Siemssen and H.W. WilschutKernfysisch Versneller Instituut9747 AA Groningen, The Netherlands(Received November 25, 1997)Pre-scission neutron multiplicities in fusion-fission reactions, reportedby Hinde et al., have been analyzed in terms of the statistical model as-suming a possible hindrance of the compound-nucleus fission width by theKramers factor which depends on nuclear dissipation. Contrary to earlierresults reported by Hofman, Back and Paul on an analysis of the GDRγ-decay, the nuclear dissipation deduced in the present analysis does notshow a clear dependence on the temperature of the compound nucleus. Onaverage, the deduced values of the nuclear dissipation are consistent withthe one-body dissipation estimate.PACS numbers: 24.75. +i, 25.70. Jj1. IntroductionThe concept of nuclear dissipation was introduced into nuclear physicsin 1940 by Kramers [1], and then revived in the early 70s in order to in-terpret deep-inelastic reactions in nucleus-nucleus collisions [2,3]. Empir-ical information on the magnitude of nuclear dissipation can be obtained∗ Presented at the XXV Mazurian Lakes School of Physics, Piaski, Poland,August 27–September 6, 1997.(451)452 K. Siwek-Wilczyńska et al.on two ways: either from experiments on non-fusion reactions (stronglydamped collisions, fast fission reactions) analyzed with deterministic trajec-tory calculations [4,5], or from so-called “delayed” fission characterized byan enhanced pre-scission emission of light particles and γ-rays which can beanalyzed in terms of the statistical decay of the compound nucleus assumingthe Kramers concept of dissipative fission. By using data on the pre-scissionemission of γ-rays from the decay of giant dipole resonance, Hofman, Backand Paul [6] suggested a distinct dependence of the deduced dissipation con-stant on nuclear temperature, a feature that is inconsistent with the conceptof one-body dissipation. In order to verify the results of Ref. [6], we haveanalyzed a rich set of pre-scission neutron multiplicities of Hinde et al. [7]and Rossner et al. [8]. Our results do not agree with conclusions of Ref. [6].2. Calculations of the pre-scission neutron multiplicitiesExperimentally measured pre-scission neutron multiplicities show signif-icant deviations from predictions of the standard Bohr–Wheeler theory ofcompound-nucleus fission (see Ref. [9] and references therein). These devia-tions can be explained in terms of the Kramers diffusion theory, in which thediffusion process of the fission degree of freedom over the fission barrier isexpressed in terms of the Fokker–Planck equation. The viscous diffusion pro-cess results in a fission width ΓK that is reduced relative to the nonviscousBohr-Wheeler width ΓBW (calculated with the transition-state method):ΓK = ΓBW(√1 + γ2 − γ), (1)where γ is the dimensionless dissipation coefficient.In our approach we follow the Grangé–Weidenmüller method of estimat-ing the time needed for the system to build up the quasistationary proba-bility flow over the fission barrier (i.e. to reach the Kramers width ΓK). Wecalculate the fission width as a function of time,Γf (t) = ΓK[1 − exp(−t/τ0)] (2)with the time constant τ0 which can be related to the transient time τbetween initiation of the diffusion process and the attainment of quasista-tionarity. For the case of an overdamped motion (γ > 1) Grangé et al. [10]estimated the time τ needed for the system to reach 90% of the asymptoticvalue ΓK:τ =γωln(10EB/T ) , (3)where ω is the oscillator frequency in the potential well of the fissioningnucleus (typically, ω ≈ 1021 s−1), and EB is the height of the fission barrier.Analysis of Pre-scission Neutron Multiplicities . . . 453The result (3) inserted into Eq. (2) determines the time constant τ0 in Γf (t):τ0 = τ/ ln 10 = 0.43τ. (4)We have included the friction-dependent fission width Γf (t) into ourMonte Carlo code that calculates event-by-event a sequence in time of thestatistical decay of the composite system until fission is drawn. For a fixedvalue of γ and a given value of angular momentum one can calculate theaverage number of neutrons, νpresaddle, evaporated before the instant of timewhen fission is decided and the system starts its no-return path from saddleto scission.In a separate procedure we calculated with the program DYNSEQ [11]the average number of neutrons emitted during the final stage of the fis-sion process, νsaddle−to−scission. Since fission is decided beyond the saddlepoint, the fission channel is then excluded from the competition (Γf = 0).Light-particle emission is continued, starting with the value of the excitationenergy that the system possessed at the saddle point. The process of gen-erating the excitation energy during the descent from saddle to scission iscalculated with the code HICOL [12] and coupled with the evaporation cas-cade calculation. Thus the actual excitation energy during the post-saddlecascade is continuously adjusted during the descent.The sum of both components, νpresaddle+νsaddle−to−scission, can be relatedto experimentally measured values of the pre-scission neutron multiplicity,νexppre . However, for quantitative comparisons the calculated multiplicitieshave to be averaged over the angular momentum. For the νpresaddle compo-nent, the averaging has been done up to an angular momentum ℓlim whichlimits a given fusion-fission reaction, ℓlim = min(ℓfu, ℓBf=0), where ℓfu is thelimiting angular momentum for fusion (in the HICOL calculation), and ℓBf=0is the angular momentum for which the fission barrier (calculated accordingto Sierk [13]) vanishes.Thus, for the pre-saddle component, the neutron multiplicity averagedover the angular momentum is:νpresaddle =∑ℓlimℓ=0(2ℓ + 1)ν(ℓ)Nf (ℓ)∑ℓlimℓ=0(2ℓ + 1)Nf (ℓ), (5)where Nf (ℓ) is a normalized number of Monte Carlo cascades which end infission, and ν(ℓ) is the corresponding pre-saddle neutron multiplicity for agiven value of ℓ.For the sake of simplicity, for the major part of the analyzed reactions,the saddle-to-scission component was not averaged, but calculated for themean value of the angular momentum, < ℓ >= 23ℓlim. It has been checked454 K. Siwek-Wilczyńska et al.in additional exact calculations carried out for selected reactions that thissimplification does not significantly deteriorate accuracy of the calculations.Figure 1 shows an example (for the 16O + 238U reaction) that illustrateshow the value of the dissipation constant γ has been determined for each re-action. First, the neutron multiplicity originating from both, pre-saddle andsaddle-to-scission stages (averaged over the angular momentum) was calcu-lated as a function of the dissipation constant γ. It is seen from Fig. 1 thatthe saddle-to-scission component is significant and cannot be neglected inthe calculations. The point where the calculated νpresaddle+νsaddle−to−scissionmultiplicity equals to an experimental value νpre(exp) determines a value ofthe dissipation coefficient γ. (The two horizontal lines in Fig. 1 show therange of uncertainty in the determination of the νpre(exp) value. Thus, thecrossing of the calculated νpresaddle +νsaddle−to−scission dependence on γ withthese two lines gives a measure of the uncertainty of the deduced value of γ.)0 4 8 12 16 20NEUTRON   MULTIPLICITY02468101214 16O + 238UElab= 288 MeVνpre(exp)νpre-saddleνpre-saddle + νsaddle-scissionDISSIPATION  COEFFICIENT  γ Fig. 1. Pre-scission neutron multiplicity in the 16O + 238U reaction at 288 MeVcalculated as a function of the dissipation coefficient γ. Crossing of this dependencewith a measured value νpre(exp) determines an experimental value of γ.We performed our calculations for the whole set of pre-scission neutronmultiplicities measured by Hinde et al. [7]. In Fig. 2 the results are plottedas a function of the temperature of the compound nucleus. The deducedvalues of γ are quite dispersed and do not show a clear trend in the tem-perature dependence. Thus, this result is inconsistent with the conclusionsof Hofman, Back and Paul [6] who found in the analysis of the giant dipoleresonance data a clear effect of the increase of γ with temperature. As it isseen from the error bars in Fig. 2, the measured pre-scission neutron multi-plicities can give only an approximate information on the magnitude of theAnalysis of Pre-scission Neutron Multiplicities . . . 455dissipation coefficient. However, on average, the deduced values of γ remainin agreement with the strength of one-body dissipation which is equivalentto γ in a range from 4 to 5.TEMPERATURE  (MeV)1.5 2.0 2.5 3.0 3.5DISSIPATION  COEFFICIENT γ048121620Deduced from data ofHinde et al.One-body dissipationFig. 2. Compilation of values of the dissipation coefficient γ deduced from pre-scission neutron multiplicities measured for different reactions by Hinde et al. [7].The deduced values of γ are plotted as a function of the temperature of the com-pound nucleus.We would also like to comment on the results of our earlier analysis ofneutron multiplicity data in fast fission reactions [4]. Very large values of thedissipation coefficient (γ = 10–50) had been obtained in that analysis whichdefinitely disagree with results of the present analysis of the fusion-fissiondata. We interpret this discrepancy as a result of “contamination” (due tofluctuations) of fast fission processes by a small component of fusion-fissionreactions which, on grounds of the deterministic model calculations [4], wereassumed to be totally absent. A contamination of fast fission reactions withfusion-fission processes (characterized by much higher pre-scission neutronmultiplicity νpre) may lead to a considerable overestimation of the deducedvalue of the dissipation constant γ, especially due to the nonlinear relationbetween γ and νpre which dramatically amplifies the effect of overestimationof νpre.The inconsistency of the results of the present work with those of Refs. [6]and [4] demonstrates that determination of the nuclear dissipation is evi-dently very model-dependent and moreover — very sensitive to experimen-tal uncertainties and precission of measurements. A more reliable and firmdetermination of the nature and magnitude of this very important quantitycharacterizing essential properties of nuclear matter will require new preciseexperiments and also well tested theoretical models for their interpretation.456 K. Siwek-Wilczyńska et al.REFERENCES[1] H.A. Kramers, Physica (Utrecht) 7, 284 (1940).[2] J. Wilczyński, Phys. Lett. 47B, 484 (1973).[3] D.H.E. Gross, H. Kalinowski, Phys. Lett. 48B, 302 (1974).[4] J. Wilczyński, K. Siwek-Wilczyńska, H.W. Wilschut, Phys. Rev. C54, 325(1996).[5] J. Blocki, J. Wilczyński, Acta Phys. Pol. B29, 000 (1998).[6] D.J. Hofman, B.B. Back, P. Paul, Phys. Rev. C51, 2597 (1995).[7] D.J. Hinde et al., Phys. Rev. C45, 1229 (1992).[8] H. Rossner et al., Phys. Rev. C45, 719 (1992).[9] M. Thoennessen, G.F. Bertsch, Phys Rev. Lett. 71, 4303 (1993).[10] P. Grangé, Li Jun-Quing, H.A. Weidenmuüller, Phys. Rev. C27, 2063 (1983).[11] K. Siwek-Wilczyńska et al., Phys. Rev. C51, 2054 (1995).[12] H. Feldmeier, Rep. Prog. Phys. 50, 915 (1987).[13] A.J. Sierk, Phys. Rev. C33, 2039 (1986).

Analysis of pre-scission neutron multiplicities in terms of the statistical model with Kramers dissipative fission

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Analysis of pre-scission neutron multiplicities in terms of the statistical model with Kramers dissipative fission

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