A critical analysis of the modelling of dissipation in fission a

Transcription

1 A critical analysis o the modellin o dissipation in ission a B. Jurado 1, C. Schmitt 1, K.-H. Schmidt 1, J. Benlliure, A. R. Junhans 3 1) GSI, Planckstr.1, 6491 Darmstadt, Germany ) Univ. de Santiao de Compostela, S. de Compostela, Spain 3) Forschunszentrum Rossendor, Postach , Dresden, Germany Astract: The time-dependent lu over the ission arrier o an ecited nucleus under the inluence o dissipation is investiated. Characteristic eatures o the evolution o the amplitude o the proaility distriution and the velocity proile at the ission arrier are derived. Analytical results are compared to numerical Lanevin calculations and used to develop a new analytical approimation to the solution o the Fokker-Planck equation or the time-dependent ission-decay width. This approimation is shown to e more realistic than previously proposed descriptions, which were widely used in the past.. PACS: i; i; q Keywords: Nuclear ission; dissipation eects; time-dependent ission-decay width; Lanevin equation; Fokker-Planck equation; analytical approimation 1. Introduction One o the topical eatures o nuclear dynamics durin the last decades is the role o dissipation or such processes as deep-inelastic heavy-ion collisions [1], dampin o iant resonances [] and induced ission [3] to learn whether collective motion in nuclei is underdamped like in water or over-damped like in honey droplets. In spite o intensive eperimental and theoretical work, most conclusions on the dissipation strenth are not well estalished. The situation still remains unclear concernin the deormation and temperature dependence o nuclear riction as well. From the theoretical point o view, dierent models have een developed, e.. the one-ody dissipation [4] concept ased on the wall-and-window ormula, the two-ody viscosity model [5] or quantum transport theories [6, 7, 8]. They yield dierent results or the manitude as well as or the dependence on temperature and deormation o the dissipation strenth. Althouh the role o nuclear dissipation in ission has een reconized lon time ao, ission continues to e one o the most promisin tools or deducin quantitative conclusions. More than 60 years ao, Bohr and Wheeler [9] proposed a derivation o the ission decay width Γ ased on purely statistical considerations in the ramework o the transition-state model. BW Soon ater, in 1940, Kramers [10] developed the irst transport theory to descrie nuclear ission. In his model, ission is descried as the evolution o the collective derees o reedom in interaction with a heat ath ormed y the sinle nucleons. This rins in the concept o dissipation, which represents the transormation o collective motion into heat due to the dampin mechanism. The description o Kramers was derived rom the stationary solution o the Fokker-Planck equation (FPE). Thereore, it only holds ater the system has reached a) This work orms part o the PhD thesis o B. Jurado ) Correspondin author: B. Jurado, GANIL, Boulevard Henry Becquerel, B.P. 507, Caen CEDEX 5, France, jurado@anil.r (present address) 1

2 equilirium. The success o the transition-state model o Bohr and Wheeler prevented this idea o Kramers to estalish. Approimately orty years later, eperimentally oserved hih pre-scission neutron multiplicities [11, 1] ave the impetus to Grané et al. [13] to theoretically investiate the inluence o dissipation on the ission time scale. Their numerical solution o the time-dependent FPE shows that it takes some time, the so-called transient time τ trans, until the current over the saddle point reaches its stationary value. Grané et al. [13] pointed out that this transient time τ trans ut also an additional saddle-to-scission time τ ss lead to an increase o pre-scission particle multiplicities. The transient time oriinates rom the time needed y the proaility distriution o the particle W (,v;t ) (deormation and conjuate momentum p = v µ, where v is the velocity o the system and µ its inertia) to spread out in deormation space. From their numerical calculations, the authors o re. [13] etracted the ollowin approimation or the transient time, deined as the time until the ission width Γ ( reaches 90% o its asymptotic value: B 1 10 τ trans = ln or β < ω β T B β 10 τ = trans ln or β > ω ω T (1) where B is the ission-arrier heiht, T is the nuclear temperature, ω is the eective oscillator requency at the round state, and β is the reduced dissipation coeicient which measures the relative rate with which the ecitation enery is transerred etween the collective and intrinsic derees o reedom. Later, ull dynamical calculations were perormed with a stochastic approach ased on the multidimensional time-dependent FPE [14] or Lanevin equation [15, 16, 17, 18] with allowance or evaporation durin the dynamical evolution o the system. These are two equivalent methods, correspondin to the interal, respectively dierential, ormulation o the same process. Whereas the solution o the FPE leads to the proaility distriution W (, v; o the particle as a unction o time, the Lanevin approach consists o ollowin the trajectory o every individual nucleus all alon its path to ission. Because the Fokker-Planck approach ives directly access to the proaility distriution W (, v;, and consequently to the evolution o the ission decay width Γ ( with time (see equation 9), it corresponds to a more transparent way to et inormation on transient eects. However, or realistic physical cases the FPE can only e solved numerically. The same inormation can e etracted rom a Lanevin treatment as well, ut only ater some averae over a lare amount o trajectories. The possiility o the Lanevin method to ollow individual trajectories may eplain why it is oten preerred to the Fokker-Planck approach since several years. Another procedure widely used to study ission dynamics consists o introducin a timedependent ission decay width Γ ( in an evaporation code [19, 0, 1,, 3, 4]. In section 3 we will point out that such a treatment is eactly equivalent to solvin the aove-mentioned equation o motion with allowance or evaporation under the condition that the used time-

3 dependent width Γ ( is otained y the Fokker-Planck or Lanevin solution at each evaporation step. Thus, it takes into account the chanes in mass, chare, ecitation enery and anular momentum o the decayin system. Unortunately, ollowin a lare amount o trajectories or solvin numerically the FPE at each evaporation step needs a very hih computational eort, which may eceed the technical possiilities actually availale or many applications. However, in an evaporation code one could et around the hih computin time needed to calculate numerically the time-dependent ission width at each evaporation step y usin a suitale analytically calculale epression or Γ (. With this aim in view, the main task o the present work is to shed liht on the characteristic eatures o transient eects in the dynamical evolution o the nuclear system. On this asis, we propose a way to model the inluence o dissipation in nuclear ission in terms o a realistic analytical approimation or Γ (. In the ollowin paper [5], we investiate, how transient eects maniest. There we will make use o peripheral nuclear collisions at relativistic eneries as an appropriate reaction mechanism dedicated to dissipation studies and estalish the requirements on relevant eperimental oservales that are sensitive to transient eects. This work is also motivated y the need o incorporatin realistic eatures o ission dynamics in comple model calculations or technical applications, e.. the nuclide production in secondary-eam acilities, in spallation-neutron sources, in the core o an accelerator-driven system, and in shieldin calculations. In these codes, the computational eort is already very hih due to the necessary transport calculations in a thick-taret environment, and thus the eplicit solution o the equation o motion, e.. y the use o the Lanevin approach, seems to e ecluded.. Time evolution o the ission-decay width under the inluence o dissipation.1. Previously used approimations o the ission-decay width I the initial population o the nuclear system in deormation and conjuate momentum diers rom equilirium, the system is suject to relaation eects. The inluence o these eects on the time dependence o the ission-decay width has careully een studied y Grané, Weidenmüller and collaorators [13, 6] on the asis o the numerical solution o the FPE. As an eample, we show their result or the evolution o the escape rate λ () t = Γ ()D t o the compound nucleus 48 Cm at a temperature o T = 5 MeV with a riction coeicient o β = s -1 in Fiure 1. The potential used is speciied in Fiure. The ission-decay width as a unction o time can e characterised y three main eatures: a delayed onset, a risin part and a stationary value. We will see that the initial suppression o the ission width has a decisive inluence on the evolution o the system. Indeed, the inhiition o ission durin the transient time τ trans increases the chance o the nucleus to decay at the earliest times y particle emission. The resultin loss o ecitation enery and the chane o the properties (e.. issility) o the residual nucleus reduce its ission proaility. 3

4 Fiure 1: The solution o the Fokker-Planck equation (ull line) or the time-dependent escape rate or the schematic case o a issionin 48 Cm nucleus, usin the potential introduced y Bhatt et al. [6] (see Fiure ) at a temperature T = 5 MeV and or a reduced dissipation coeicient β = s -1. The initial condition corresponds to equilirium at a temperature T initial = 0.3 MeV, introduced in re. [6] to represent the quantum-mechanical zero-point motion in the round state. This solution is compared with dierent approimations (the step unction [7]: thick dotted line, the eponential-like unction [8]: thin dotted line, the approimate ormulation o Bhatt et al. [6]: dash-dotted line, the approimate ormulation o re. [30]: thin dashed line, and the improved epression proposed in this work: thick dashed line). The solution o the FPE and the approimation o Bhatt et al. are taken rom Fiure o re. [6], where values or t < s are not shown. In the past, several approimations or the time evolution o the ission-decay width have een proposed. The two most widely used are: - a step unction [7] that sets in at the transient time τ trans : Γ 0, t < τ trans ( = k () Γ, t τ trans - an eponential in-rowth unction [8] deined y: Γ ( = {1-ep(-t/τ)} (3) k Γ where : τ= τ trans /.3 and K Γ is the Kramers decay width: Γ = K Γ (4) K BW 4

5 BW The ission-decay width o the transition-state model Γ may conveniently e epressed y the ollowin approimation, introduced y Moretto [9]: Γ BW 1 πρ ( E) T sad ρ sad ( E B ) (5) and the Kramers actor K is iven y K 1/ β β = 1 + ω (6) sad ω sad where: ω sad is the requency o the harmonic-oscillator potential that osculates the ission arrier at the saddle point, ρ sad ( E) and T sad are the level density and temperature o the ρ E is the level density in the round state. issionin nucleus at saddle, respectively, and ( ) Fiure : Presentation o the potential in ission direction o 48 Cm as iven y Bhatt et al. [6]: (V = (-3.41) [(-3.098)(+1.59)]+3.7). The ission-arrier heiht is 3.7 MeV. A more elaorate ormulation o Γ ( was derived y Grané et al. [13] and improved y Bhatt et al. [6] a ew years later. Usin the Gaussian approimation, they derived an analytical solution o the FPE in the under-damped reime and an analytical solution o the Smoluchowski equation in the over-damped case. Recently, we have proposed another epression or Γ ( in re. [30]. In the present work, we will present a reined version o this description (see section.5). All these approimations are compared to the eact solution o the FPE in Fiure 1. The eponential-like in-rowth unction shows stronly risin values already at very early time, which is in contrast to the numerical solution o the FPE. Even i the step unction is ale to descrie the inhiition o ission or small times, this suppression 5

6 is certainly too stron, and its steep slope is too crude. The two rather elaorate approimations o re. [6] and [30] are etter suited. However, while the ormulation o Bhatt et al. [6] overestimates the ission-decay rate at early times, the onset o Γ ( is well descried y our approimation [30] and its improved epression proposed in this work. The curves also slihtly dier in their asymptotic values. In view o these important dierences etween the various approimations, it is leitimate to investiate what are the most relevant characteristics or a realistic description o dissipative eects in ission... Features o the relaation process In this section, some asic eatures o the relaation process towards equilirium are careully investiated with the help o Lanevin calculations. We take the case o 48 Cm introduced y Bhatt et al. [6] as an eample. The discretisation method we use in this work to solve the Lanevin equation is documented in appendi A. The dynamics o a issionin system is well characterised y the time-dependent lu o trajectories in deormation space. The time-dependent ission decay width Γ ( can easily e derived rom the lu φ () t evaluated at the saddle point, which represents the current across the ission arrier: () t + φ = v W (, v; dv (7) Let + + Π ( ; = d W (, v; dv (8) e the proaility that the system is at deormations <. The time-dependent ission width Γ (, related to the escape rate λ (, is then deined y: φ ( Γ ( = Dλ ( = D Π( ; + = D (9) + v W ( d +, v; dv W (, v; dv By introducin the mean velocity at the arrier v( = ; into the deinition o φ () t, we see that the lu at the arrier can e epressed as the product o the mean velocity 6

7 ( = and the amplitude o the proaility distriution ( v; v ; : () t + ( =, v; + + W + = W ( =, v; dv + ( =, v; dv = v( = ; W (, v; W =, dv at the arrier v W dv φ = dv (10) Let us investiate the time evolution o the two terms o equation (10). Fiure 3 represents the velocity distriution at the saddle point W ( =, v; as a unction o time, and Fiure 4 shows the variation with time o the amplitude o the proaility distriution + W ( =, v; dv at the arrier. Both quantities were otained y a Lanevin calculation. As can e seen in Fiure 3, the mean velocity at the arrier v( = ; radually decreases durin the onset o the ission width eore reachin its asymptotic value in the stationary state. However, the variation o the amplitude o the proaility distriution + W ( =, v; dv is much more important. Indeed the variation o the amplitude with time etends over many orders o manitude durin the transient time as ehiited in Fiure 4. Thereore, the variation o the amplitude o the proaility distriution at the ission arrier represents the dominatin inluence on the onset o the lu, and consequently o the decay width. For a more eneral understandin o the respective importance o the dierent contriutions, we would like to compare the inluences o the amplitude and o the mean velocity on the lu in the simple eample o diusion (over-damped motion) without drivin orce. In this limit, the FPE leads to the ollowin reduced Smoluchowski equation [31]: t W( ;t ) = + T W( ;t ) βµ (11) 7

8 Fiure 3: Two-dimensional contour plot o the velocity distriution at the arrier W (, v; = as a unction o time. Contour lines mark heihts separated y a actor o two. The result has een otained rom a numerical calculation usin the Lanevin approach with the parameters T = 3 MeV and β = s -1 (upper iure) and T = 3 MeV and β = s -1 (lower iure). The calculation starts with the distriution in deormation and conjuate momentum p = µ v iven y the zero-point motion in the round state. The dashed line represents the mean velocity at the arrier as a unction o time. The proaility distriution with initial width zero in the round state at = 0 and initial velocity v = 0 thus evolves in the ollowin way [3]: with W 1 ( t ) ep ( / ( σ ) ; = (1) π σ σ T = t µβ (13) 8

9 Fiure 4: Amplitude o the proaility distriution at the arrier deormation rom the Lanevin calculation (ull line) aove the potential o Fiure with the parameters T = 3 MeV and β = s -1 (upper par, respectively β = s -1 (lower par, compared with the amplitude at the arrier deormation aove the paraola adapted to the curvature o the potential in the round state (dashed line). All unctions are normalised to the value at t = s. The amplitude W( ; o the system at the arrier is thus iven y: W 1 βµ 1 ( ( ( ) = βµ 1 1 ; ep / σ ep ep = πσ 4πT t 4Tt t t (14) By applyin the continuity equation (see also equation (36)), one otains the mean velocity v at the arrier: v T 1 ; = = βµ σ t t ( (15) 1 1 Thus, the lu as the product o amplitude and velocity varies like ep 3 /. Since or t t small times ep ehaves like ep t n, n, the amplitude overns the evolution o t t the lu at the einnin o the process as illustrated in Fiure 5. 9

10 Fiure 5: Schematic comparison o the inluence o amplitude and velocity on the lu at the arrier in a simple diusion prolem. See tet or details. Let us now consider the inluence o the shape o the potential on the time dependence o the amplitude at the arrier. With this aim, Fiure 4 compares the variation o the amplitude at the arrier W( = ;t ) otained numerically usin the realistic potential o Fiure with the solution o the FPE otained analytically or a paraolic potential. The curvature o the latter corresponds to the curvature o the realistic potential in the round state. One oserves that oth curves vary over many orders o manitude in a very similar way. This result, perhaps surprisinly, suests the minor impact o the shape o the potential on the variation o the amplitude. In addition to these numerical results, we would like to strenthen this conclusion y more analytical aruments. With this aim in view, we compare the result or a simple diusion prolem without drivin orce otained y solvin the Smoluchowski equation (11) and the result or the paraolic potential in the over-damped reion at a iven deormation. In oth cases, the distriution is a Gaussian unction in deormation iven aove y equation (1). The solution or σ as a unction o time or the irst diusion case was already iven in equation (13) σ T = t µβ while the ull solution or a paraolic potential [3] corresponds to T β β1t β σ = 1 ep( β sinh + sinh( β1 + 1 (16) µω β1 β1 1 4 where: β ( β ω ) 1 = (17) 10

11 In the over-damped reime, the physical meanin o equation (16) may e easily understood: β is lare and equation (16) can e approimated y the ollowin equation σ ω 1 ep t (18) β T µω in which we have replaced β 1 = β - ω /β y β and nelected the terms ep(-β. Actually, equation (16) is already quantitatively very similar to equation (18) or β s -1. Equation (18) shows that the process o the population o the deormation space can e descried y a proaility distriution with the shape o a Gaussian whose second moment eponentially approaches the asymptotic value. We see that equation (13) is the irst-order approimation o equation (18), indicatin that up β to t the evolution o the amplitude o the proaility distriution at the arrier ω deormation or a paraolic potential is very similar to that or the simple diusion prolem. Since the transient time τ trans in the over-damped reime as deined y Grané et al. [13] is approimately iven y equation (1), the aove-mentioned similarity etends to around hal the transient time. We conclude that the spreadin o the proaility distriution durin the time lapse in which the lu sets in is rather insensitive to the shape o the potential. In Fiure 4, we have compared the amplitude otained or a realistic and a paraolic potential ecause the FPE can e solved analytically only in the simplest cases or which the potential has a paraolic orm and the transport coeicients are constant. More complicated situations, e.. or a realistic potential, like the one displayed in Fiure, or in the case o deormationdependent transport coeicients, have to e solved numerically. Concernin the approimation o transport coeicients that do not depend on deormation, the importance o such an assumption can e consideraly reduced y the choice o an appropriate coordinate system as demonstrated in appendi A1. There it is shown that any iven prolem can e reormulated in an adapted coordinate system, where the inertia or riction coeicient does not vary with deormation, without chanin the physics o the prolem. In addition it is pointed out that theoretical one-ody and two-ody dissipation models predict that the variation o the dissipation strenth with deormation is weak. The initial conditions o the system concern the population in deormation and conjuate momentum as well as the temperature and anular momentum o the nucleus. Their determination relies on the description o the initial properties o the nuclear system ater the reaction to e studied. Thereore, the initial population depends stronly on the entrance channel considered. While e.. usion reactions can lead to rather deormed compound nuclei and involve a road rane o anular momentum, very peripheral nuclear collisions at relativistic eneries are characterised y small initial shape distortions and low anular momenta. The initial conditions are important or the evolution o the system, ecause they aect the time dependence o the ission-decay width Γ (. In section.4 we will propose a way to take the initial conditions into account in the analytical approimation o the timedependent ission width Γ (. 11

12 .3. New description o the time-dependent ission-decay width Γ ( In the previous section, we have collected some characteristics o the dissipation process, which have a rather eneral validity: the predominance o the amplitude or the time evolution o the lu and the minor importance o the shape o the potential or the early evolution o the proaility distriution in deormation and conjuate momentum. In re. [30] a realistic approimation or the decay width Γ ( that is ased on these eatures has een presented. In the ollowin, the various steps to derive this approimation will e illustrated in detail. First o all, we will develop a slihtly modiied epression or the eact description o the ission-decay width, whose deinition was already iven aove in equation (9). We start y deinin the normalised proaility distriution Wn ( =, v, at the ission-arrier deormation as: W n (, v, ( =, v, W = = (19) + W (, v, dvd Considerin equation (19) and introducin the mean velocity v at the arrier deormation already used in equation (10), the ission width o equation (9), can e reormulated as ollows: Γ + ( = ; ( = D v W ( =, v, t dv (0) n ) By deinin the amplitude at the arrier, interated over velocity y W the ission width can e written as In the stationary case we et: Γ + n n ; ( = W ( =, v = ; dv (1) ( = ; W ( ; Γ ( = D v n = () K ( = ; t ) W ( =, t Γ (t ) = D v ) (3) n where we have identiied the asymptotic ission width y the Kramers epression equation (4). K Γ rom Finally, cominin equations () and (3), we can reormulate the time-dependent ission width as: 1

13 Γ ( v v( = ; Wn ( = ; ( = ; t ) W ( = ; t ) n Γ K (4) At this point, we introduce two approimations that lead to a new analytical description o the time-dependent ission width. The irst approimation is to nelect the variation o the mean velocity with time. Thus, v( = ; is replaced y its asymptotic value v( = ; t ) in the numerator o equation (4), leadin to the ollowin epression: Γ Wn ( = ; K ( Γ (5) W ( = ; t ) n This approimation is well justiied y our previous investiations in section. that have shown that the variation o the mean velocity at the arrier is small compared to the variation o the amplitude, and thereore the evolution o the amplitude + W ( =, v; dv with time has a much stroner inluence than the mean velocity. A second approimation which we used in re. [30] consists o epressin the shape o the inrowth unction at the ission arrier, which is iven y the shape o W n ( =, in equation (5), y the analytical solution derived or a paraolic potential [3]. The validity o this second simpliication is aain justiied y our previous investiations in section., where we have shown that the amplitudes at the arrier W and W par or a realistic and a paraolic potential, respectively, evolve in a quite similar way at the einnin o the process, see Fiure 4. Consequently, the ollowin set o equations represents an analytical epression o the ission decay width that is ased on realistic assumptions: Γ par W ( =, K ( Γ par (6) W ( =, t ) in which we implement the paraolic solution (equations (1) and (16)): W par 1 (,v ; t ) = ep ( / ( σ ) π σ σ T β β1t β = 1 ep( β sinh + sinh( β1 + 1 µω β1 β1 par Note, that we replaced the normalised proaility W n y the unnormalised quantity W par in equation (6), ecause in the case o the paraolic potential the proaility distriution is conined, and thus + W par + par ( ;t ) W ( ;t ). 13

14 .4. Initial conditions The analytical solution o the FPE or a paraolic potential derived in [3], which we use in equation (16), reers to speciic initial conditions correspondin to a δ unction in deormation and momentum. This means that the initial deormation is in the minimum o the paraola, correspondin to the nuclear round state, and the initial momentum is zero. However, as discussed in section.., a realistic description should include the initial population related to the reaction under study. Due to the uncertainty principle, the initial proaility distriution W (, v; t = 0) diers rom a δ unction whatever the entrance channel is. To account or this eect, we introduced in re. [30] a time shit t 0 in equation (16) or the standard deviation σ : ( t t ) T β β1 0 β σ = 1 ep( β ( t t0 )) sinh + sinh( β1( t t0 )) + 1 (7) µω β1 β1 The time shit t 0 accounts or the time needed or the proaility distriutions, epressed y equations (1) and (16), to estalish the initial distriution in deormation space. One should stress that such a procedure is not restricted to any speciic initial condition. In act, it can e applied to any reaction, under the condition that the phase space populated y the entrance channel can e assumed to e Gaussian. An alternative way to consider a inite width in the initial conditions o the analytical solution o the FPE or the paraolic potential (also restricted to Gaussian-like distriutions) has een developed recently y Boilley et al. [33]. In [30] we assumed an initial distriution in deormation and momentum, which corresponds to the zero-point motion at the round state o the nucleus. This is the narrowest distriution compatile with the uncertainty principle. This condition, related to a nucleus with a distriution in deormation correspondin to the nuclear round state c, is approimately valid or reactions which introduce small shape distortions like peripheral collisions at relativistic eneries [34]. These reactions will e investiated in the ramework o dissipation studies in our ollowin paper [5]. The time shit t 0 correspondin to the zero-point motion was iven in re. [30] y: t 0 1 T Dβ = MAX ln, β T Dω 4ω T (8) Equation (8) accounts or oth the under-damped and the over-damped cases. In the underdamped case, the deormation and the momentum coordinate saturate at aout the same time. c ) A recent pulication [Phys. Lett. B 567 (003) 189.] criticises the application o an eective time shit, which was proposed in our previous pulication [30] or introducin the initial condition o the zero-point motion. The criticism is ased on a misinterpretation o t 0 as a kind o relaation time to the equilirium o the oscillator, as represented y the round state. We stress here that this time shit t 0 does not account or the relaation towards equilirium ut represents a mean to model the distriution correspondin to the zero-point motion quantum state. 14

15 Thereore, the time shit needed or the proaility distriution, epressed y equations (1) and (16), to reach the width o the zero-point motion in deormation is equal to the time that the averae enery o the collective deree o reedom needs to reach the value E = 1 0 Dω associated to the zero-point motion. Considerin the enery transer E t rom the heat ath o the intrinsic ecitations at temperature T to the oscillation in deormation as iven y the FPE, the eective time delay t under 0 is determined y: This leads to the ollowin equation: E [ - ep( )] under = T 1 β (9) t t o under 1 T 1 = T t 0 = ln ln (30) β T E0 β T Dω and ives rise to equation (8) or the under-damped case in which β is small. In the over-damped case, it is the width in deormation, which mostly determines the timedependent ehaviour o the system due to diusion, while the velocity proile adapts very ast due to the stron couplin etween intrinsic and collective derees o reedom. Thereore, we determine the time delay t 0 y requirin the width in deormation o the solution o the FPE at E0 D t = 0 to e equal to the width o the zero-point motion σ 0 = =. The ull ω µ µω 1 analytical solution otained in [3] or the width o the proaility distriution or a paraolic potential was already iven y equation (16) and approimated in the over-damped reion y equation (18). As a consequence o equation (18), the width in deormation related to the zero-point motion σ is iven y 0 T ω over σ 0 1 ep t 0 (31) µω β over where t 0 stands or the time needed to estalish the zero-point motion distriution in deormation startin the calculation rom a δ unction. In prais, the temperature o the system is usually much hiher than the zero-point enery (let us remind that the eective temperature which corresponds to the zero point motion amounts to 0.5 MeV in our case or which D ω =1 MeV). This means that the width in deormation correspondin to the zero-point motion σ 0 is much smaller than the width in thermal equilirium ( t ) σ : D σ = = 0 ( t ) << σ µω µω T (3) 15

16 which means that over t 0 is iven y ω 1 ep β over t 0 can e replaced y ω β over t 0 in equation (31), and t over 0 Dβ (33) 4ω T over It is leitimate to consider the solution o the paraolic potential in the derivation o t 0, since we have shown in Fiure 4 that the variation o the amplitude with time or the realistic and the paraolic potential is very similar at small times. In Fiure 6, we compare the standard deviation o the proaility distriution or < as iven y a Lanevin calculation with the standard deviation σ o the approimate solution otained rom equation (7). In the stronly under-damped case (Fiure 6a) as well as in the stronly over-damped case (Fiure 6d), the initial width o the paraolic solution includin the time shit (lon dashed line) arees well with the numerical result o the Lanevin calculation (thick ull line). But also in the intermediate rane (Fiure 6 and Fiure 6c), the time-shited paraolic solution o equation (7) and the Lanevin calculations come very close around t = s. The deviations at smaller times are not important, since the amplitude o the proaility distriution at the arrier is still so low that the lu is practically zero (see Fiure 7 and the discussion elow). Fiure 6: Standard deviation σ o the proaility distriution rom a Lanevin calculation inside the ission arrier o the potential iven in Fiure (ull lines) compared to the standard deviation o the analytical solution o the FPE or the paraolic potential with (lon-dashed line) and without (shortdashed line) eective time shit t 0 iven y equation (8). In all cases we have considered the nucleus 48 Cm and T = 3 MeV. Two curves are iven or the Lanevin calculation: The thin ull curve starts rom = 0 and p = 0. The thick ull curve has een otained y startin the trajectories with a sample rom Gaussian distriutions in deormation and momentum, correspondin to the zero-point motion in the paraolic potential adapted to the curvature o the potential o Fiure at the round state. The comparison is iven or our eamples, rom a stronly under-damped (β= s -1 ) to a stronly over-damped case (β = s -1 ). The thin ull lines in Fiure 6 have een otained assumin a δ unction located at = 0 and p = 0 as initial condition. The comparison etween the thin and the thick ull curves shows 16

17 that the inclusion o the zero-point motion as initial condition in the Lanevin equation corresponds essentially to a time shit. Furthermore, the analytical approimation shited y the time lapse t 0 o equation (8) enales us to reproduce the width in deormation o the proaility distriution correspondin to the initial conditions o the Lanevin calculation, as shown y the lon-dashed lines. The inluence o the introduction o t 0 on the ission-decay rate λ () t is illustrated in Fiure 7. There it is clearly seen that shitin the analytical epression (lon dashed line) y the time lapse t 0 nicely reproduces the onset o ission otained directly rom the solution o the equation o motion (historam) started with the proaility distriution correspondin to the zero-point motion. To inish this section, we would like to mention that elaorate studies [35] have shown that the variance σ 0 o the position distriution o the zero-point motion o a damped oscillator is smaller than the value D ( µω ) ound or an undamped oscillator, which we have used to derive equation (33). In particular it was shown, that this dierence ecomes more important in the over-damped reime with increasin dampin [35, 36]. Quantitatively, or β = s -1 the variance o the damped quantum oscillator at the round state is 4% smaller than the variance o the undamped quantum oscillator, and or β = s -1 it is around 67% smaller. Since t over 0 is proportional to over t 0 σ 0, the same reduction as or the variance applies or. However, Fiure 7 shows that the transient time is aout one order o manitude larer than the eective time shit t over 0 and thereore the smaller variance o the damped quantum oscillator has only little eect on the time dependence o the ission-decay rate. Fiure 7: Time-dependent ission-decay rate λ ( resultin rom the Lanevin calculation (historam) compared to the analytical approimation o [30] includin (lon dashed line, see equation (7)) or not (short dashed line, see equation (17)) the time shit t 0 deined y equation (8) or the nucleus 48 Cm with β = 10 1 s -1 at T = 3 MeV. Note that shitin the analytical approimation in time permits to descrie the onset o the process as done y the numerical solution o the equation o motion. 17

18 .5. Improved approimation or the time-dependent ission width The analytical approimation o the time-dependent ission decay width Γ ( discussed in section.3. was derived nelectin the variation o the mean velocity at the arrier v( = ;. In this section we propose to include this variation in an approimate way. It can e shown that in the FPE the mean velocity and the loarithmic slope o the proaility distriution at a iven deormation are closely related. To demonstrate this, we will make use o the Smoluchowki equation [31]: 1 dv W(; = W(; + T W t βµ d ( ; (34) where: V stands or the nuclear potential (see the eample in Fiure ). dv Ater interation, reardin that = 0 at the arrier and that W(; = 0 or - as well d as W ( ; = 0 or -, we otain the ollowin result: T W ( ; W ( ; d = t βµ (35) Applyin the continuity equation with the deinition o the mean velocity introduced in equation (10) results in: W t + ( ; d = v W ( =, v; dv = v W ( =, v; dv = v W ( = ; + (36) Note that y the continuity equation the velocity enters eplicitly into this equation, while it is only an implicit variale in the Smoluchowski equation. By dividin the previous epression y W(= ; and usin equation (35) we inally otain: v T W βµ = W ( ; ( = ; T = βµ [ ln( W ( ; )] (37) As can e seen rom equation (37), the mean velocity v v( ; = at the ission arrier = is proportional to the loarithmic derivative o the amplitude at the ission arrier ln /. Thereore, the approimated analytical description o the ission-decay width ( W ) = Γ ( we proposed in re. [30] and derived in detail in section. can e improved y introducin the variation o the loarithmic slope o the proaility distriution at the arrier. 18

19 Unortunately, we have no access to an analytical solution o the proaility distriution aove the realistic potential. Thereore, we use aain the analytical solution o the FPE or the paraolic potential to account or the variation o the velocity proile in an approimate way. Fiure 8 demonstrates with two eamples that the variation in time o the loarithmic slope o the normalised amplitude at the arrier deormation is qualitatively similar or the paraolic and the realistic potential, althouh it quantitatively ails to reproduce the ull variation. This similarity is understandale: Althouh the paraolic potential at = is not lat, the epression or the mean velocity or the paraolic potential correspondin to equation (37) still contains a term which is proportional to the loarithmic slope o the potential. This means that one otains an improved estimation o the time-dependent ission width when equation (6) is replaced y the ollowin epression: Γ W par W par ( = ; ( = ;t ) d ln par par { W ( ; } dln W ( ;t ) d d { } K = Γ (38) Fiure 8: The neative loarithmic slope o the proaility distriution at the arrier position or two cases: T = 3MeV, β = s (upper par and T = 3MeV, β = 5 10 s (lower par. The ull line shows the numerical result o the Lanevin approach with a realistic potential, while the dashed line corresponds to the result otained with the analytical approimation or the paraolic potential iven y equation (7). 19

20 Another deiciency o the analytical approimation proposed in re. [30] results rom the K normalisation to the Kramers stationary value Γ. Indeed it is known that Kramers prediction underestimates the stationary ission width or temperatures larer than the ission arrier, leadin to a discrepancy etween our approimation and the eact FP or Lanevin result in the stationary reime or T B (see Fiure 1). To remove this deiciency, we propose to make use o the concept o the Mean First Passae Time (MFPT) [37]. The MFPT is the mean value o the distriution o irst passae times. Althouh it can e evaluated at any deormation point e, it is physically meaninul at scission only, where the neative current o trajectories is neliile. This restriction is due to the asorption at e in the deinition o the MFPT itsel. Consequently, at scission the MFPT is equivalent to the mean scission time τ scission. Furthermore, with the help o numerical FP calculations, Grané et al. showed that τ scission can e epressed as the sum o three contriutions: the initial delay due to transient eects, the statistical decay time τ stat and the dynamical saddle-to-scission time τ ss. Thus, they estimated the mean scission time y the ollowin sum: τ scission = τ stat + 1 τ trans + τ ss (39) where the delay due to transient eects is approimated y hal the transient time τ trans as iven y equation (1), and the saddle-to-scission time τ ss is taken rom [6] τ ss = ω sad 1 β + 4 β ln sci ω sad β β + 4 βt µ ω sad β + 4 (40) The determination o the MFPT requires a numerical FP or Lanevin calculation. However, in the over-damped reime, it can e evaluated analytically usin the closed epression [37]: MFPT βµ = T e 0 V du ep T u ( u) V () v dv ep T (41) Thus, the validity o the analytical approimation o Γ () t iven y equation (38) can e etended to temperatures larer than the ission arrier, or which the stationary value o Kramers cannot e used, replacin the Kramers width Γ K y the statistical ission-decay width Γ stat, otained rom the ollowin equation: Γ stat D = τ stat MFPT D 1 τ trans τ ss (4) Unortunately, the closed epression o the MFPT as epressed with equation (41) ein restricted to the over-damped reion, such an analytical determination o the normalisation actor Γ is not valid in the under-damped reion. However, since the validity o the stat 0

21 Kramers solution or the stationary ission rate is limited y the ratio T < 1, one could imaine that the relative deviation Γstat r = (43) K Γ B also quantitatively scales with the value o T B, independently o the value o β. Thus, we investiated the ehaviour o r as a unction o the dissipation strenth β y dividin the statistical ission-decay width Γ stat otained rom the numerical Lanevin calculation y K Kramers' prediction Γ. As can e seen in Fiure 9, the value o r was ound to vary little as a unction o β or a iven temperature. num Fiure 9: Ratio r etween the statistical ission-decay width Γ stat as etracted rom a numerical K Lanevin calculation and Kramers' prediction Γ as a unction o the dissipation strenth β. The calculations were perormed or a 48 Cm issionin nucleus and a ission arrier o 3.7 MeV. The results are shown or three dierent temperatures, T = MeV (circles), T = 4 MeV (trianles) and T = 5 MeV (squares). Thus, we propose the ollowin prescription to evaluate the stationary ission-decay width: With the help o equations (4) and (43), the ratio r can e determined analytically in the D over-damped reime, where Γstat. The statistical ission-decay width MFPT 1 τ trans τ ss in the under-damped reime can then e calculated accordin to equation (43) as the product K o the ratio r otained or the over-damped reime and the correspondin Γ v or the underdamped reime. This solution allows overcomin the deiciency o Kramers prediction or lare temperatures in the cases o over-damped and under-damped systems..6. Quantitative results or the ission-decay rate In the previous sections we have shown that the analytical approimation o the timedependent ission-decay width proposed in re. [30] and improved in this work, is ased on 1

22 realistic assumptions. In this section, we systematically investiate its qualities and shortcomins over a wide rane o dissipation strenth and temperature. Fiure 10 compares the ission-decay rate otained y a Lanevin code (historam), the analytical approimation o [30] (short dashed line) and the improved epression o it iven in the previous section 1 1 (lon dashed line). The sinle FP curve (ull line) drawn in Fiure 10 or β = 5 10 s and T = 5 MeV stands only or remindin the equivalence o oth Lanevin and Fokker-Planck methods. Fiure 10: Systematic comparison o dierent approaches to the time-dependent ission-decay rate or dierent values o temperature T and reduced riction coeicient β: Lanevin approach (historam), analytical approimation o [30] (short dashed line), improved epression iven in this work (lon dashed line). For T = 5 MeV and β = s -1, the solution o the FPE (ull line) is shown in addition. For the numerical Lanevin and FP calculations, the deormation-dependent potential shown in Fiure has een used. The zero-point motion is considered y an initial eective temperature o 0.5 MeV.

23 Whereas the description o Bhatt et al. [6] starts too early as seen in Fiure 1, the approimation o re. [30] descries quite well the initial inhiition o ission. Under the condition that the system is not stronly under-damped and that the temperature is not too low, the analytical approimation reproduces the slope o the escape rate quite well. The areement is less ood in the under-damped reion. However, as will e seen rom the comparison with the eperimental data, the rane 1 10 s β 5 10 s is the most important. Let us note that the improved epression proposed y equation (38) leads to a slihtly etter areement with the Lanevin result at earliest times, and the modiications introduced y equations (41) to (43) permit to overcome the limitations o Kramers' description at temperatures larer than the ission arrier. In re. [6] the analytical approimation was derived separately or the under-damped and the β over-damped reion. The authors used the Smoluchowski equation (34) as soon as > 1 ω that deines the limit etween the under-damped and over-damped reime. This equation, in which the inluence o inertia is nelected, corresponds to the reduced FPE in the asymptotic case o stronly over-damped systems (β ). In re. [6], ω = s -1 is used so that β = s -1 elons to the over-damped reion. We compare in Fiure 11 the escape rate otained y the Lanevin and Smoluchowski equations or ω = s -1 and or three values o β in the over-damped reime. There it can e seen that the suiciently over-damped reime in which the Smoluchowski equation (34) is valid starts or values o β larer than aout s -1. This oservation eplains the discrepancy etween the Lanevin calculation and the approimation o Bhatt et al. or β = s -1, illustrated in Fiure 1. This proves that β = s -1 does not correspond to a stron enouh over-damped system or the Smoluchowski equation to e valid, at least or the present purpose to study transient eects where the onset o the ission-decay width is crucial. This implies that the Smoluchowski equation is not valid in the reime o one-ody dissipation which typically corresponds to β s -1. Althouh the approimation o Bhatt et al. in [6] is rather well suited in the underdamped reime (see Fiure o re. [6]), its deiciency to descrie the earliest staes o the process or β s -1 was the reason that motivated us to derive another analytical approimation to the eact solution o the FPE. Our approimation ives a uniorm continuous ormulation or oth the under-damped and the over-damped reion and reproduces the onset o ission resultin rom the numerical calculations rather closely. Fiure 11: Time-dependent ission-decay rate as otained rom the resolution o the Smoluchowski equation (dash-dotted historams) compared to the result o the Lanevin equation (ull historams) 3

24 calculated with dierent values o β or the nucleus 48 Cm at T = 5MeV and a ission arrier o 3.7 MeV. 3. Dynamical approaches to the nuclear de-ecitation process In eneral, there eist three methods to model the decay o a heavy ecited nucleus in a dynamical way. One method is ased on the Lanevin equation [15, 16], another one on the FP equation [14]. In oth cases, the evolution o the system is ollowed in small time steps, either y computin the individual trajectories or the interal proaility distriution o the system, respectively. At each time step, the proaility or the evaporation o particles is computed and randomly decided. The third option corresponds to a dynamical evaporation code, in which the ission decay width is otained y the solution o the Lanevin or FP equation at each step o the evaporation chain. Such a code is equivalent to the Lanevin or FP treatment. Unortunately, as already stressed in section 1, such a procedure is inconceivale in many applications due to the hih computational time required. However, any analytical approimation o the time-dependent ission decay width can replace the numerical solution, without destroyin the equivalence o the code with a Lanevin or FP approach, under the condition that the approimation used is as close as possile to the numerical solution. This crucial condition is ulilled y the analytical ormulation we propose in the present work, as we have shown. In view o the aove-mentioned equivalence, a statistical evaporation-ission code can e transormed into a dynamical de-ecitation code y introducin a time-dependent issiondecay width. When two slihtly simpliyin assumptions are applied, it is enouh to consider the time-dependence o the ission width. In this case, the evolution o the system in deormation does not enter eplicitly in the code. Firstly, the deormation dependence o the particle-decay widths may e nelected. Secondly, the variation o the availale intrinsic ecitation enery as a unction o deormation, investiated in reerence [38], may e disrearded. Both approimations are not crucial in calculations which are restricted to the small deormation rane rom the initial state up to the saddle point. These eects could e considered y replacin the respective constant values y the values otained y averain over the actual deormation distriution in the correspondin time steps. Details on the implementation o the analytical approimation to the solution o the FPE developed in this work in our de-ecitation code ABRABLA are iven in reerence [5]. 4. Summary The relaation process o a nuclear system towards equilirium leads to a delay o ission compared to the Kramers ission decay time. This eature o nuclear ission, which oriinates rom dissipation, is automatically rouht to liht y solvin the equation o motion in the Fokker-Planck or Lanevin approach. An equivalent procedure, which in addition enales one to avoid hih computational times, consists o includin a realistic analytical time-dependent ission-decay width in an evaporation code. A meticulous investiation o the evolution o the proaility distriution o the system in phase space all alon its dynamical path permitted us to etract the main eatures o the relaation process. Makin use o these results, we have developed an easily calculale approimation o the time-dependent ission-decay width that is ased on realistic physical assumptions. Compared to other approimations widely used in the past, our new analytical ormulation has proven to reproduce rather closely the trend o 4

25 the eact solution in the under- as well as in the over-damped reime. At this stae, it is more than desirale to careully study, how the description o transient eects inluences the conclusions drawn on dissipation. Indeed, such an investiation may e crucial with respect to the reliaility o previous works usin less realistic ormulations or the time-dependence o the ission decay width, and namely the eponential in-rowth unction. The new analytical epression, which we propose in the present work, deinitely represents an improvement in that direction. Acknowledement We acknowlede valuale discussions with Hans Feldmeier, Anatoly V. Inatyuk, and David Boilley. This work has een supported y the European Union in the rame o the HINDAS project under contract FIKW-CT and y the Spanish MCyT under contract FPA C One o us (C. S.) is thankul or the inancin o a one-year stay at GSI y a Humoldt ellowship. The work proited rom a collaoration meetin on Fission at inite thermal ecitations in April 00, sponsored y the ECT* ( STATE contrac. A1: Choice o the coordinate system Appendices Descriin any physical process needs to have recourse to some coordinate system. This is particularly important or the studies o ission dynamics, which deal with the evolution o a nucleus in deormation space. In this appendi we will point out that the process can e descried usin a constant, coordinate-independent, mass parameter, respectively mass tensor in the case o more than one dimension, without any restriction on the physics o the prolem. Also the use o a constant riction strenth turns out to e close to the theoretical epectations as will e shown elow. In our work, the Lanevin as well as the Fokker-Planck equation were solved y assumin that neither nuclear inertia, nor nuclear riction depends on deormation. It is the aim o this appendi to estimate how crude this assumption is. As an eample, we will consider the Lanevin equation. A1.1 Equation o motion The dynamical evolution o a issionin nucleus can e descried y the ollowin Lanevin equation o motion or a iven deormation coordinate q and its conjuate momentum p : (or simpliication we restrict ourselves to the one-dimensional Lanevin equation, ut it can easily e eneralized to n dimensions) dq dt dp dt p = µ ( q ) = 1 p µ ( q) ( q) dµ dq dv ( q) γ ( q) dq µ ( q) p + D( q) L ( (A1.1) 5