A Nonequilibrium Information Entropy Approach to Ternary Fission of Actinides
AA Nonequilibrium Information Entropy Approach to Ternary Fission of Actinides
G. R¨opke , ∗ J. B. Natowitz , † and H. Pais ‡ Institut f¨ur Physik, Universit¨at Rostock, D-18051 Rostock, Germany. Cyclotron Institute, Texas A&M University, College Station, Texas 77843, USA. CFisUC, Department of Physics, University of Coimbra, 3004-516 Coimbra, Portugal. (Dated: January 1, 2021)Ternary fission of actinides probes the state of the nucleus at scission. Light clusters are producedin space and time very close to the scission point. Within the nonequilibrium statistical operatormethod, a generalized Gibbs distribution is constructed from the information given by the observedyields of isotopes. Using this relevant statistical operator, yields are calculated taking excitedstates and continuum correlations into account, in accordance with the virial expansion of theequation of state. Clusters with mass number A ≤
10 are well described using the nonequilibriumgeneralizations of temperature and chemical potentials. Improving the virial expansion, in-mediumeffects may become of importance in determining the contribution of weakly bound states andcontinuum correlations to the intrinsic partition function. Yields of larger clusters, which fail toreach this quasi-equilibrium form of the relevant distribution, are described by nucleation kinetics,and a saddle-to-scission relaxation time of about 7000 fm/c is inferred. Light charged particleemission, described by reaction kinetics and virial expansions, may therefore be regarded as a veryimportant tool to probe the nonequilibrium time evolution of actinide nuclei during fission.
PACS numbers: 21.65.-f, 21.60.Jz, 25.70.Pq, 26.60.Kp
Nuclear fission, discovered eighty years ago, remainsan exciting field of research. In the last few decades, anamazing progress has been realized with respect to exper-imental investigations and phenomenology, as well as intheoretical treatments such as time-dependent Hartree-Fock-Bogoliubov (TDHFB) or time-dependent superfluidlocal density approximation, generator coordinate meth-ods, and other techniques [1, 2], but basic concepts arestill open for discussion. A full microscopic description isstill lacking, and it remains a challenge to present statequantum many-body theory. For a recent review on stud-ies of thermal neutron induced ( n th ,f), and spontaneousfission (sf) of actinides, as well as a discussion on opentheoretical questions, the reader should refer to [3, 4].From a theoretical point of view, the fission process canbe generally described via a picture in which the deform-ing nucleus, following a dynamic path, subject to fluctu-ations, crosses a saddle point where the nascent fissionfragments are formed. The deformed dumbbell-like sys-tem, consisting of two main fragments and the connect-ing neck region, then evolves toward the scission pointwhere the rupture occurs. The saddle-to-scission time isestimated as τ s → s ≈ O (10 − ) fm/c [2]. Characteri-zation of the system’s properties and its time evolution,and the description of dissipation during this process re-mains nowadays a significant problem. However, dissi-pative dynamics has been applied to describe the non-adiabatic evolution from the saddle point to scission, see[3], though a rigorous treatment of the scission process isstill unavailable. ∗ [email protected] † [email protected] ‡ [email protected] A few signals can be used to obtain information aboutthe scissioning nucleus, such as the mass and energy dis-tributions of the two fission fragments, and the multiplic-ities of the emitted particles, which are primarily neu-trons, and of γ -radiation, that can be utilized to char-acterize excitation energies and spins. Although the useof concepts, like the temperature, which is defined forthermodynamic equilibrium, might not be well-founded,approaches introducing concepts from statistical physics,to characterize the distribution of emitted particles, areoften employed. For instance, in Refs. [5–7], the promptfission neutron spectra of different actinides are analyzedwith temperature-like parameters of the order of 1 MeV,and in Refs. [8–13], the analysis of prompt fission γ -rayspectra for actinides also suggests a temperature-like pa-rameter of the same order. However, one has to separateprompt emission from later emission, that occurs dur-ing the de-excitation of the fission fragments, and thispresents a problem, since it is not easy to identify ob-servables which may be directly associated with the neckregion at scission.One such signature is the emission of light clusters ob-served in ternary fission processes, see, e.g., [14–16] andreferences given there. A light cluster, most often He,is emitted in a direction perpendicular to the symme-try axis defined by the two fission products, which havemass numbers distributed near half that of the fission-ing nucleus. Ternary fission yields of a series of light iso-topes { A, Z } and energies have been measured for a num-ber of different actinide nuclei, in particular U( n th ,f), U( n th ,f), Pu( n th ,f), Pu( n th ,f), Cm(sf), and
Cf(sf), see Refs. [16, 17]. Data for these observedyields Y obs A,Z , normalized to He obs = 10000, are presentedin Tab. I. The investigation of ternary fission has the ad-vantage that it is directly related to the scission process,and that it can be localized in the neck region. a r X i v : . [ nu c l - t h ] D ec isotope R vir A,Z (1 . U( n th ,f) U( n th ,f) Pu( n th ,f) Pu( n th ,f) Cm(sf)
Cf(sf) λ T [MeV] - 1.24177 1.21899 1.3097 1.1900 1.23234 1.25052 λ n [MeV] - -3.52615 -3.2672 -3.46688 -3.02055 -2.92719 -3.1107 λ p [MeV] - -15.8182 -16.458 -16.2212 -16.6619 -16.7798 -16.7538 n - 560012 1.409e6 722940 1.8579e6 1.606e6 1.647e6 H - 28.131 28.16 42.638 19.52 21.079 30.096 H obs - 41 50 69 42 50 63 H 0.973 40.986 49.76 68.632 41.563 49.533 61.579 H obs - 460 720 720 786 922 950 H 0.998 457.27 715.29 714.79 780.39 913.76 943.12 H 0.0876 2.7772 4.97 5.627 6.057 8.742 8.219 He 0.997 0.0124 0.0076 0.0235 0.00431 0.00645 0.00933 He obs - 10000 10000 10000 10000 10000 10000 He 1 8858.46 8706.1 8615.7 8556.9 8313.98 8454.0 He 0.689 1130.75 1289.04 1374.7 1439.0 1680.75 1540.9 He obs - 137 191 192 260 354 270 He 0.933 115.89 158.98 159.01 211.68 276.96 222.4 He 0.876 21.262 33.997 35.983 51.742 80.634 58.16 Y obs He /Y final , vir He - 0.9989 0.9897 0.9846 0.9869 0.9899 0.9622 He obs - 3.6 8.2 8.8 15 24 25 He 0.971 3.4725 6.764 6.4095 12.481 21.280 13.32 He 0.255 0.047077 0.105 0.111 0.219 0.455 0.258 Y obs He /Y final , vir He - 1.0229 1.1936 1.3496 1.1811 1.1042 1.8409 Be 1.07 5.7727 2.594 5.147 2.188 2.819 2.544TABLE I. Lagrange parameters λ i , observed yields Y obs A,Z [14, 15, 17] and primary yields Y rel , vir A,Z (virial approximation) fromternary fission U( n th ,f), U( n th ,f), Pu( n th ,f), Pu( n th ,f), Cm(sf), and
Cf(sf) (lines denoted by the superscript’obs’) as well as relevant (primary) yields Y rel , vir A,Z for H, He nuclei. The prefactor R vir A,Z ( λ T ) at λ T = 1 . Y final , vir He = Y rel , vir He + Y rel , vir He and Y final , vir He = Y rel , vir He + Y rel , vir He arecompared to the observed yields. For more details see the Supplementary material [20]. As known from α -decay studies, a mean-field approachlike TDHFB has problems describing the formation ofclusters. For ternary fission, parametrizations of themeasured yields employing a statistical distribution witha temperature-like parameter T ≈ A ≥
10, are clearlyoverestimated by the simple NSE statistical equilibriumdistribution [17]. Modifications have been proposed [21]based on nucleation theory. In Ref. [22], a nonequilib-rium approach was used to discuss the observed yieldsof isotopes with Z ≤ Cf. Chemical equilibrium constants were recently de-rived [23] for the fission reaction
Pu( n th ,f) accountingfor in-medium effects. Using those constants, the pre-dicted yields for increasing A are larger than the mea- sured ones, as already observed in [21].In this work, we use ternary fission data to investi-gate the scission process. In particular, we extend thenonequilibrium approach, already presented in Ref. [22]but with Z ≤
2, considering partial intrinsic partitionfunctions including continuum contributions on the levelof quantum virial expansions. We present results for allthe actinides given in Table I for which the necessaryset of yields is available, and we derive critical values ofparameters that are relevant for nucleation kinetics.We describe fission as a nonequilibrium process, us-ing the method of the nonequilibrium statistical operator(NSO), ρ ( t ) = lim (cid:15) → (cid:15) (cid:90) t −∞ dt (cid:48) e − (cid:15) ( t − t (cid:48) ) e − i (cid:126) H ( t − t (cid:48) ) ρ rel ( t (cid:48) ) e i (cid:126) H ( t − t (cid:48) ) . (1)The NSO ρ ( t ) [24] is a solution of the von Neumann equa-tion with boundary conditions characterizing the state ofthe system in the past, as expressed by the relevant sta-tistical operator. This relevant statistical operator ρ rel ( t )is constructed from known averages using informationtheory. As it is well known from statistical physics, therelevant distribution is determined from the maximum ofinformation entropy under given constraints, which arerepresented by Lagrange multipliers λ i . A minimum setof relevant observables consists of the conserved observ-ables, energy H , and the numbers N τ of neutrons andprotons ( τ = n, p ). The solution is the generalized Gibbsdistribution ρ rel ∝ exp[ − ( H − λ n N n − λ p N p ) /λ T ] . (2)Note that these Lagrange multipliers λ i , which are ingeneral time-dependent, are not identical with the equi-librium parameters T and µ τ , but may be considered asnonequilibrium generalizations of the temperature andchemical potentials. Only when the system is in thermo-dynamic equilibrium, can the information entropy be un-ambigously identified with the thermodynamic entropy,and from this we define the quantities T and µ τ with theknown properties. Note that the NSO allows the pos-sibility of including other relevant observables, such asthe pair amplitude in the superfluid state, or the occupa-tion numbers of the quasiparticle states to derive kineticequations and calculating reaction rates [24].Typically, in a variational problem, we have to replacethe Lagrange multipliers λ i by given mean values [22].In contrast to the noninteracting system (ideal quan-tum gases), where the equilibrium solutions are the well-known equations of state, the interaction in the Hamil-tonian leads to a many-particle problem which can betreated with the methods of quantum statistics. (Notethat the mathematical concepts developed in equilib-rium quantum statistics can also used for the generalizedGibbs state ρ rel .) We perform a cluster expansion forthe interacting system [25], and partial densities of dif-ferent clusters { A, Z } are introduced. The relevant yields Y rel , vir A,Z in the virial approximation are calculated as Y rel , vir A,Z ∝ R vir A,Z ( λ T ) g A,Z (cid:18) π (cid:126) Amλ T (cid:19) − / × e ( B A,Z +( A − Z ) λ n + Zλ p ) /λ T (3)(nondegenerate limit), where B A,Z denotes the (groundstate) binding energy and g A,Z the degeneracy [26]. Theprefactor R vir A,Z ( λ T ) = 1 + exc (cid:88) i [ g A,Z,i /g A,Z ] e − E A,Z,i /λ T (4)is related to the intrinsic partition function of the clus-ter { A, Z } . The summation is performed over all excitedstates of excitation energy E A,Z,i and degeneracy g A,Z,i [26], which decay to the ground state. Also, the contin-uum contributions are included in the virial expression.For instance, the Beth-Uhlenbeck formula expresses thecontribution of the continuum to the intrinsic partitionfunction via the scattering phase shifts, see [27, 28]. For R vir A,Z ( λ T ) = 1, the simple NSE is obtained, i.e., neglect-ing the contribution of all excited states inclusive contin-uum correlations. In the low-density limit, virial expansions of the in-trinsic partition functions of the channel { A, Z } havebeen obtained [25, 27, 28] for H, H, He, Be usingthe measured phase shifts in the corresponding channels.The values are given in the second column of Tab. I for λ T = 1 . λ T have only a weak con-tribution of the continuum states so that R vir A,Z ( λ T ) ≈ R vir A,Z ( λ T ) to the en-ergy of the edge of continuum, is given in [25], and thecorresponding estimates of the prefactor for the He iso-topes with 6 ≤ A ≤ ρ rel serves as initial condition to solve the von Neu-mann equation for ρ ( t ) describing the evolution of thesystem according to the system Hamiltonian. The rele-vant (primary) distribution Y rel A,Z ( λ T ) contains stable andunstable states of nuclei, as well as correlations in thecontinuum (e.g. resonances).The concept of introducing the relevant primary yielddistribution according to the NSO is supported by severalexperimental observations, among these are the observa-tion of He and He emission [15]. For
Z >
2, as it isdiscussed below, 3.368 MeV γ rays from the first excitedstate of Be have been observed [29], and excited statesof Li at 2.26 MeV excitation energy have been reportedin Ref. [15]. Also of interest are the inferred data for Beand Li / − observed in [30], which cannot be describedwith the NSE but demand a treatment with continuumstates.The relevant distribution evolves dynamically to thefinal, observed yields according to the von Neumannequation. This process is described by reaction kinet-ics, and the NSO allows calculation of the reaction rates[24]. Here, we approximate this process by the feedingof the observed states from the primary states occurringin the relevant distribution. For example, for Z ≤ Y final H = Y rel H + Y rel H , Y final He = Y rel He + Y rel He + 2 Y rel Be , Y final He = Y rel He + Y rel He , and Y final He = Y rel He + Y rel He .In this work, to construct the relevant distribution ρ rel from an information theoretical approach, we use theleast squares method, see [21], to reproduce the observedyields. We calculate the primary distribution Y rel , vir A,Z us-ing the intrinsic partition function in the virial form, i.e.using the excited states and scattering phase shifts ne-glecting in-medium corrections. The optimum values ofthe Lagrange parameters λ i are given in Tab. I for thedifferent ternary fissioning actinides. Of interest is thedependence of λ i from { A, Z } of the parent actinide nu-cleus [18, 19]. The current accuracy of the experimentaldata is not sufficient to determine significant trends.The measured total yields of H and He isotopes arenearly perfectly reproduced by the corresponding sumsof primary yields. The yield of He is slightly overes-timated by Y final , vir He . In contrast, the yield of He isunderestimated by Y final , vir He . Both ratios Y obs He /Y final , vir He and Y obs He /Y final , vir He are presented in Tab. I. Presently, therelevant distribution Y rel , vir A,Z does not take in-medium ef-fects, in particular Pauli blocking, into account. Mediummodifications are more effective for weakly bound clus-ters. As proposed in [22] for
Cf(sf), a stronger reduc-tion of the yield of He obs compared to He obs may berelated to the very low binding energy (0.975 MeV) ofthe He nucleus below the α + 2 n threshold. The sup-pression of He obs appears for all considered systems andmay be considered as a signature of the Pauli blocking.Pauli blocking is determined by the medium surroundingthe cluster, and an estimate of the corresponding neu-tron density was given in [22]. However, to address theproblem, precise experimental data are needed. Experi-mental studies are still scarce, and the data are often notconsistent [31, 32]. Unbound nuclei such as He shouldbe very sensitive to medium modifications. The virialexpression for the intrinsic partition function is known[28], and the corresponding primary yields are given inTab. I. Fortunately, in the case of
Cf, the primaryyields of He and He have been measured [15], and thevalue Y obs He = 1736(274) has been obtained. In principle,because of the medium modifications, the different clus-ter states may serve as a probe to determine the neutrondensity in the neck region at scission, but the uncertain-ties are still rather large.Nuclei with Z > Mg has been made for
Pu( n th ,f) [14, 16]. Extendedsets of data for Z > U( n th ,f)and Cm( n th ,f) [16]. We extend our analysis of themeasured data up to Z = 6 using the relevant distribu-tion, see [20]. In general, the neutron separation energy S n , for each isotope, is adopted as the threshold energyfor the continuum, but cluster decay is also possible, e.g., Li → α + d , Li → α + t , Be → α + h , Be → α , B → α + Li, etc. In some cases, such as He, He, Li, two-neutron separation determines the threshold.To estimate the continuum correlation, the interpolation R vir A,Z ( λ T ) [22] was used at the corresponding binding en-ergy E thresh AZ − E A,Z,i of the (ground state or excited)cluster. The final yields Y final , vir A,Z are calculated as sumsof the feeding contributions, see [20].The question arises whether global Lagrange param-eters λ T , λ n , λ p exist, which are valid for all Z , as ex-pected for matter in thermodynamic equilibrium. Beforewe discuss this question, we present a calculation with therelevant distribution given above, employing only threeLagrange parameters λ i , but taking also Li isotopes intoaccount. A least squares fit of final yields Y final , vir A,Z to Y obs A,Z for H, H, He, He, Li, Li, Li has been per-formed. The accuracy of the fit increases since He and Li are not included. Both are weakly bound systemsfor which medium effects and dissolution may become ofrelevance, as discussed above. Again we emphasize thatin-medium corrections are not included in the present calculation. The Lagrange parameter values ˆ λ T = 1 . λ n = − . λ p = − . Pu( n th ,f), andwe conclude that our approach can also reproduce theyields of isotopes with Z > λ i and consid-ering all observed data for isotopes with Z ≤
6, the ratio Y obs A,Z / Y final , vir A,Z is shown as a function of the mass num-ber A in Fig. 1. Surprisingly, yields of Be, Be, Bare also well reproduced. For A ≥
11, the calculationsoverestimate the observed yields, and the ratios decreasestrongly, starting around A = 10. -4 -3 -2 -1 Y ob s / Y f i n a l , v i r Z=1Z=2Z=3Z=4Z=5Z=6nucl Li C Pu(n,f)
Cm(n,f)
U(n,f)
FIG. 1. Ternary fission of U( n th ,f) (blue), Cm( n th ,f)(green), and Pu( n th ,f) (red): Ratio Y obs A,Z /Y final , vir A,Z as func-tion of the mass number A . Isotopes with Z ≤ Pu( n th ,f). Data tables are given in [20]. An explanation of the decrease has been given in [21]using nucleation theory. Whereas small clusters are al-ready in the quasi-equilibrium distribution Y rel A,Z , largerclusters need more formation time so that the observedyields are smaller than those predicted by the relevantdistribution. From reaction kinetics, the expression Y obs A,Z /Y final , vir A,Z = 12 erfc (cid:104) b ( τ )( A / − a ( A c , τ )) (cid:105) (5)is obtained, see [21], where b ( τ ) = (27 .
59 MeV /λ T ) / × (1 − e − τ ) − / and a ( A c , τ ) = A / c (1 − e − τ ) + e − τ .With λ T = 1 . Pu( n th ,f) (black line in Fig. 1) gives τ = 1 . , A c =16 . cτ s → s = τ A / c λ / T / (3 . ρ ), and with ρ = 4 × − fm − [21] follows τ s → s = 6793 fm/ c . Thistime scale supports the slow evolution from saddle toscission proposed recently as a dissipative process [2, 3].The strong reduction of isotopes A >
10 compared toestimates of a statistical model is also seen in [17]. Inaddition, the overestimate of He is shown. The correcttreatment of continuum states proposed in this letter re-moves this discrepancy. In addition, the yields of weaklybound clusters Li, C are strongly overestimated, see[33]. A reason may be the shift of the binding energydue to in-medium effects. If the density is larger thanthe Mott density, the bound states are dissolved. Bondstates with threshold energies below or near 1 MeV in-clude also He, Be, Be, B, C. The yields of allthese isotopes are overestimated. This may be consid-ered as indication of in-medium effects (Pauli blocking)leading to a shift and possibly the dissolution of the clus-ter. This possibility should be considered when moreaccurate data are available.Yields of ternary fission of
Pu( n th ,f) were also calcu-lated within the evaporation-based surface-plus-windowdissipation model [19, 34], where the yields of light clus-ters are obtained considering the height of the Coulombbarrier as a dynamical variable. The Coulomb interac-tion has been treated as mean field. To improve the liquiddrop model used in that phenomenological approach, anadequate theory of ternary fission should be based on theTDHFB solutions. It should be pointed out that the neu-trons and protons have extended wave functions and arenot correctly described by the distribution (3), see [22].The Coulomb interaction contributes to the Hamiltonianalso in our approach which will be extended to the treat-ment of inhomogeneous systems. As pointed out in Ref.[2], the dynamics near scission is strongly dissipative and existing versions of TDDFT are not adequate since theyare lacking fluctuations in collective coordinates.In conclusion, ternary fission is a dissipative pro-cess, and understanding fission of actinide nuclei isof fundamental interest to work out nonequilibriumstatistical physics. Emitted light isotopes are describedby a nonequilibrium distribution, excited states andcontinuum correlations are taken into account on thelevel of virial expansions. In-medium effects and nucle-ation kinetics are interesting aspects to reconstruct thenonequilibrium distribution. An improved description ofthe influence of mean-field effects may be obtained fromTDHFB calculations and similar approaches, but clusterformation remains problematic in any mean-field theory.Ternary fission is an outstanding signal to explore thefission process, and more consistent and accurate dataare necessary to work out a complete description of theternary fission process within non-equilibrium quantumstatistics. Acknowledgments
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