Light element ( Z=1,2 ) production from spontaneous ternary fission of 252 Cf
LLight element ( Z = 1 , ) production from spontaneous ternary fission of Cf G. R¨opke ∗ Institut f¨ur Physik, Universit¨at Rostock, D-18051 Rostock, Germany
J. B. Natowitz † Cyclotron Institute, Texas A&M University, College Station, Texas 77843, USA
H. Pais ‡ CFisUC, Department of Physics, University of Coimbra, 3004-516 Coimbra, Portugal (Dated: August 17, 2020)The yields of light elements ( Z = 1 ,
2) obtained from spontaneous ternary fission of
Cf aretreated within a nonequilibrium approach, and the contribution of unstable nuclei and excitedbound states is taken into account. These light cluster yields may be used to probe dense mat-ter, and to infer in-medium corrections. Continuum correlations are calculated from scatteringphase shifts using the Beth-Uhlenbeck formula, and the effect of medium modification is estimated.The relevant distribution is reconstructed from the measured yields of isotopes. This describes thestate of the nucleon system at scission and cluster formation, using only three Lagrange parameterswhich are the nonequilibrium counterparts of the temperature and chemical potentials, as definedin thermodynamic equilibrium. We concluded that a simple nuclear statistical equilibrium modelneglecting continuum correlations and medium effects is not able to describe the measured distribu-tion of H and He isotopes. Moreover, the freeze-out concept may serve as an important ingredientto the nonequilibrium approach using the relevant statistical operator concept.
I. INTRODUCTION
Thermal neutron induced and spontaneous ternary fission is a process in which the emission of two medium-massfragments is accompanied by equatorially emitted light particles and clusters formed in the neck region at the timeof scission, see [1–3] and references therein. Data for cluster yields obtained from ternary fission experiments withthermal neutrons are shown, e.g., in [4–6]. In particular, data for the ternary fission yields of
Pu( n th ,f) arepresented. An interpretation of the Koester et al. data [4] was given in Ref. [7], where a suppression of the yieldsof the larger clusters is found, due to cluster formation kinetics. Also, ternary fission has been observed from otheractinides such as Pu, U, U, and
Cm. For more recent work on ternary fission see [8–12].Different approaches have been employed to interpret these data, see [5], and often a Boltzmann distribution has beenused. An interpolation formula has been presented in [13] which describes the general behavior of the measured yieldsbut cannot explain the details of the observed distributions. More fundamentally, the use of a Boltzmann distributionas known from thermodynamic equilibrium with parameters temperature and chemical potentials remains unfounded.However, the signatures of binding energies and degeneracy of the isotopes according to the Boltzmann distributionare clearly seen in the observed yields.In this paper we report on investigations of the yields of equatorial emission of Z = 1 , Cf. We are interested in a better understanding of cluster formation and the fate of corre-lations in low density, low-temperature, expanding nuclear matter. This nonequilibrium evolution can be describedusing the method of the nonequilibrium statistical operator (NSO) [14]. It is based on information-theoretical conceptswhich is also the basis of equilibrium statistical physics. We include different processes which describe the dynamicalevolution of the system. In particular, we include the decay from other unstable nuclei (feeding), the inclusion ofexcited states (including continuum correlations), and medium effects. For this, a quantum statistical approach isused.Our main goal is to shed some more light into the properties of dense nuclear matter, namely, to understand how toimprove the simple model of nuclear statistical equilibrium (NSE) by including continuum correlations and in-mediumeffects, and how well the nucleon density at the time of onset of cluster formation can be determined. We discuss the ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] a r X i v : . [ nu c l - t h ] A ug isotope A Z Y exp
A,Z Y interp A,Z B
A,Z A g A,Z Y final A,Z E thresh A,Z R γAZ (1 . Y rel ,γA,Z R vir A,Z (1 . Y rel , vir A,Z R eff A,Z (1 . Y rel , eff A,Z λ T - - - (1.25) - - 0.806219 - - 1.28018 - 1.25052 - 1.30556 λ n - - - (-2.8) - - -2.19217 - - -3.18741 - -3.1107 - -3.00665 λ p - - - (-15.8) - - -17.7023 - - -16.5882 - -16.7538 - -16.6463 n 1 0 - 0.65e6 0 2 2.572e8 - - 1.084e6 - 1.647e6 - 954222 H 1 1 160(20) 19.4 0 2 1.136 - - 30.8231 - 30.096 - 27.625 H 2 1 63(3) 37.5 1.112 3 5.012 2.224 1 61.6143 0.98 61.579 0.98 63.004 H f H 3 1 - - 2.827 2 (950) H 4 1 - - 1.720 5 (52.95 ) -1.6 1.473 121.366 0.0606 8.2191 0.0606 11.154 He 3 2 - 0.0095 2.573 2 1.624e-6 - 1 0.012984 0.988 0.00933 0.988 0.01511 He f He 4 2 8264(341) - 7.073 1 (10000) He 5 2 1736(274) 2600 5.512 4 (1487) -0.735 1 2072.8 0.7044 1540.9 0.6596 1733.9 He f He 6 2 223(26) - 4.878 1 (270) He 7 2 47(9) 128 4.123 4 (53.76) -0.410 1 65.2355 0.821 58.16 0.3988 46.863 He f He 8 2 25(5) - 3.925 1 (25) He 9 2 - - 3.349 2 (0.8335) -1.25 1 0.88875 0.2604 0.258 0.2604 0.59706 Be 8 4 10(6) 15.2 7.062 1 (0.2101) -0.088 1.49 4.70862 1.07 2.544 1.07 5.0427Table I: Properties and yields of the H, He and Be isotopes from ternary spontaneous fission
Cf(sf) relevant for the finaldistribution of observed H, He nuclei (denoted by the superscript f ). Experimental yields Y exp A,Z [12, 18, 19, 22] are comparedto the yields Y interp A,Z obtained from an interpolation formula [13] as well the yields calculated in different approximations ofthe nuclear Hamiltonian (2), see Sections III, IV, together with the corresponding multipliers R A,Z ( λ T ), which represent theintrinsic partition functions: The final state distribution Y final A,Z considering only the observed stable nuclei { A, Z } f , the relevantdistribution Y rel ,γA,Z of noninteracting nuclei, the relevant distribution Y rel , vir A,Z taking into account continuum correlations, and therelevant distribution Y rel , eff A,Z including interaction between the constituents. The binding energy B A,Z (in MeV), the degeneracy g A,Z = 2 J A,Z + 1, and threshold energy of continuum states E thresh A,Z (in MeV) according Ref. [27], are also given. The firstthree rows show the Lagrange parameters λ i obtained for the four different calculations (in MeV). apparent suppression of yield for some exotic isotopes and the relation to thermodynamic quantities.The paper is organized as follows: in Section II, we collect available data on ternary spontaneous fission of Cfconcerning the production of the lightest elements H and He. In Sec.III, the theoretical formalism used in thework, based on information theory, will be described. Section IV analyses different numerical calculations thatgo beyond the ideal gas description considering continuum correlations and in-medium effects. The use of a localdensity approximation is discussed in Section V, and finally in Section VI, some final remarks, namely, the relationto thermodynamics, are drawn.
II. EXPERIMENTAL DATA
A number of experimental investigations of the spontaneous fission of
Cf have been performed by different groups,see [6] and references given there. It has been demonstrated that
Cf(sf) emits 3 . · − α -particles per fission[16, 17]. Usually, the yields Y A,Z of other light elements produced from ternary fission are normalized to the finalyield of He ( α ), which is fixed to Y He f = 10000. More precisely, we consider here the ratios of yields relative to thefinal yield of α -particles.In Ref. [15], the author performed a compilation of results for ternary fission data, taken from the literature. There,consistent experimental results are presented for the H ( p ) and H ( d ) yields, as well as for the He ones. However,the values for the yield of H ( t ), and also of He, are quite different. These data have been used also in a morerecent publication [13], where the yields for the H isotopes are from Ref. [19], and for the He isotopes, the yields areaccording Ref. [23].In Table I we collect some experimental results denoted as Y exp A,Z . For H and H, we give the values of [19].For H, He, He, He, He, He, we take the values from Refs. [12, 18] where also the errors are given. Notethat the long-living isotopes He and He which are unstable with respect to β decay are stable with respect tostrong interaction and are observed in the experiments. In addition to the final yields Y exp A,Z seen in the experiments,denoted by the superscript f at the stable nucleus, we give also primary yields of short-living, particle unstablenuclei which can contribute to the yields of the observed nuclei. The final yields are related to the primary yieldsas Y H f = Y H + Y H , Y He f = Y He + Y He + 2 Y Be , Y He f = Y He + Y He , Y He f = Y He + Y He . Analyzing theenergy spectra, for Cf(sf) the formation of He and He has been determined [12], and the ratios of primary yields Y He /Y He = 0 . Y He /Y He = 0 . Be, the value 10 ± Cf the multiplicity of protons emitted in ternary fission is reported to be 6 . × − [2]. This correspondsto an experimental yield Y exp1 , ≈ . α -particles. Also prompt neutron emission has beenmeasured, for a recent work see [21]. The fractional percentage of ternary fission ”scission” neutrons has beendetermined to be 7 . ± .
8% [24, 25]. In Ref. [25] a temperature of T sci = 1 . Cf fission is 3 . ± .
05 [26]. This leads to a scission neutronmultiplicity of 0 . ± .
008 that corresponds to an experimental yield Y exp1 , ≈ . / (3 . − ) × = 906250 relativeto the yield of α -particles.As pointed out in Ref. [20], experimental studies at low energy of ternary-particle-unstable nuclei producing α particles are still scarce, and the data are not very consistent. The ratio He f / He f was reconsidered in [20], and aGaussian fit above 9 MeV energy gives the value Y He f /Y He f = 0 . .
25 MeV), the neutron chemical potential( ε n = 2 . ε p = 15 . Y interp A,Z [13], which are calculated with the Valski interpolation formula. To explain the yieldsobserved from ternary fission of
Cf(sf), Boltzmann-like distributions are used. These contain the binding energy B A,Z and the ground-state degeneracy g A,Z = 2 J A,Z + 1 of nuclei [27] which are also shown in Table I. The remainingcolumns will be discussed in Sections III and IV below.
III. INFORMATION THEORETICAL DESCRIPTION OF THE DISTRIBUTION OF CLUSTER YIELDS
Information theory considers the problem of reconstructing a distribution if some information about the ensembleis given. The most probable distribution is obtained from the maximum of information entropy if some averages ofof the system properties are given. From the maximum of information entropy, a Gibbs distribution is obtained withLagrange parameters λ i which are determined self-consistently, by describing the averages. In this section we considerwhether it is possible to reconstruct the distribution of yields Y exp A,Z with only a limited amount of information aboutthe system. This approach is well-known from equilibrium thermodynamics where the averages of energy and particlenumbers of the conserved components are given to define the grand canonical ensemble, and the Lagrange parameters β = 1 /T, µ n , µ p occurring in the equilibrium ensemble are related to the temperature and the chemical potentials.Within the method of the nonequilibrium statistical operator (NSO) [14], at a given time t the relevant statisticaloperator ρ rel ( t ) is constructed from this maximum entropy principle, and the corresponding Lagrange parameters λ i ( t ) become functions of time. The statistical operator ρ ( t ) describing the nonequilibrium evolution of the systemfollows as 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)which solves the Liouville-von Neumann equation, with H the system Hamiltonian given in Eq. (2). Kinetic equationsas well as hydrodynamic equations can be derived within this approach.We discuss the construction of the relevant statistical operator which is a ingredient to describe the nonequilibriumprocess. As relevant observables, we consider the averages of neutron number and proton number, as well as theHamiltonian H = (cid:88) τ,k E τ ( k ) a † τ,k a τ,k + (cid:88) τ,k,k (cid:48) V ext τ ( k, k (cid:48) ) a † τ,k (cid:48) a τ,k + 12 (cid:88) τ,τ (cid:48) ,k,k (cid:48) ,p,p (cid:48) V int τ,τ (cid:48) ( p, k ; p (cid:48) , k (cid:48) ) a † τ (cid:48) ,p (cid:48) a † τ,k (cid:48) a τ,k a τ (cid:48) ,p (2)which describes the interaction of nucleons, τ = { n, p } , with an external potential V ext τ ( k, k (cid:48) ) and the nucleon-nucleoninteraction V int τ,τ (cid:48) ( p, k ; p (cid:48) , k (cid:48) ); the quantum number k denotes the wave vector and spin of the nucleon with kineticenergy E τ ( k ). Averages are going to be calculated with the correspondent relevant operator ρ rel ( t ) = Z − ( t ) e − [ H − λ n ( t ) N n − λ p ( t ) N p ] /λ T ( t ) (3)where Z rel ( t ) = Tr exp {− [ H − λ n ( t ) N n − λ p ( t ) N p ] /λ T ( t ) } is the relevant partition function, N τ denotes the particlenumber of neutrons/protons, and the Lagrange multipliers are going to be eliminated by the known informations,such as internal energy and particle densities of the system.The solution of this many-particle problem is not simple and needs some approximations, such as replacing theHamiltonian by a more simple model which can be solved. Such simple model Hamiltonians are the ideal nucleon gasor the mean-field approximation, where the Hamiltonian describes a noninteracting system of quasiparticles. We areinterested in the formation of bound states so that, in a first approximation, we consider the ideal energy functional H (0) = (cid:88) A,Z, P g A,Z (cid:18) − B A,Z + (cid:126) P Am (cid:19) (4)with P the center of mass momentum for the cluster { A, Z } , m is the average nucleon mass. This model Hamiltoniandescribes the nucleon system as an ideal mixture of non-interacting free nucleons and nuclei. We allow for reactionsbetween the different components { A, Z } so that the number of each component is not conserved but only the totalnumber of neutrons and protons in the system. This approximation can be applied in the low-density case wherethe interaction between the constituents of the nuclear system becomes weak. The Lagrange parameters in (3)calculated to reproduce the observed distribution with the model Hamiltonian H (0) (4), are denoted by λ (0) T , λ (0) n , λ (0) p ,respectively.The problem to eliminate the Lagrange multipliers by the given averages of internal energy and particle numberdensities is well known from statistical physics and leads to the Fermi/Bose distribution. Before we discuss thecorresponding results, we emphasize that initially we are discussing only the parametrization of the measured yields,using information about the observed nuclei such as ground state binding energy and degeneracy.The obtained Lagrange parameters λ (0) i should not be interpreted as thermodynamic quantities like temperatureand chemical potentials for several reasons:(i) The energy functional is only an approximation. The full energy functional should also include excited states andinteractions. The full information about the Hamiltonian of the system leads to the quantum statistical approach.(ii) Fission is a nonequilibrium process and is not described by an equilibrium distribution. We have to include alsothe dynamical information which is described by the full Hamiltonian and contains the information on the final distri-bution measured in the experiment as well as information of the distribution in the past t (cid:48) ≤ t , see Eq. (1). This leadsto the so-called relevant distribution Y rel A,Z ( t ) which depends on the time t , and the evolution of the distribution withtime described by generalized reaction kinetics [14]. We will not discuss how the evolution of the system follows fromthe solution of (1) but use instead the simple concept of the freeze-out approximation. This means that the relevantdistribution is given by thermodynamic equilibrium up to the freeze-out time. After that, the relevant distributionevolves according to reaction kinetics as described by the decay of excited states. The different versions of Y rel A,Z shownin Table I correspond to different model Hamiltonians as an approximation to the nucleon Hamiltonian (2).(iii) The system to be described is not homogeneous nuclear matter as in thermodynamic equilibrium but is inhomo-geneous. A local density approximation is problematic. In addition, the relation between the Lagrange parameters λ i and the thermodynamic quantities, in particular the densities n n ( r , t ) , n p ( r , t ), is not the relation between tempera-ture, chemical potentials and density as known from non-interacting, ideal quantum gases but is more complex. Ouraim is to find arguments to infer the density from the data.The information of the properties of the observed nuclei, see Tab. I, leads to the Boltzmann-like distribution(nondegenerate case) Y (0) A,Z ∝ n (0) A,Z = g A,Z (cid:32) π (cid:126) Amλ (0) T (cid:33) − / e ( B A,Z +( A − Z ) λ (0) n + Zλ (0) p ) /λ (0) T . (5)Because we don’t know the prefactor we consider only the ratio Y A,Z ; α = Y A,Z /Y He f × , i.e., the isotope yieldrelative to the final yield of α particles. Following convention, the final yield of the α particles is fixed as Y [ He f ] =10000.We infer the Lagrange parameter values λ (0) i based on the information about the observed final nuclei by minimizingthe sum over the relative square deviation ( Y (0) A,Z ; α − Y A,Z ; α ) /Y A,Z ; α . For instance, considering the four lightest andmost abundant isotopes, i.e. the yield ratios of H f , He f , He f , and He f , we determine the values of the threeLagrange parameters λ final T = 0 . λ final n = − . λ final p = − . Y final A,Z (we drop the index α ) is shown in Table I.This approach is similar to that used in the Albergo determination of temperature and chemical potentials fromthe observed yields of nucleons and nuclei from heavy ion collisions [28]. However, we cannot identify the Lagrangeparameters λ (0) i with the thermodynamic parameters temperature and chemical potential of the nuclear system be-cause, and as stated above, the ideal energy functional (4) considers only the ground states of the nuclei, and theinteraction between the nucleons/nuclei is neglected. In addition, the correct nonequilibrium distribution is given by ρ ( t ) (1) which coincides with ρ rel ( t ) only in thermodynamic equilibrium.Evidently, our first approach which relates the final distribution to the binding energies of nuclei and their degen-eracies is not sufficient. There are further unstable nuclei and excited states which should be taken into account.In addition to the stable isotopes H, H, He, He we have He and He which are unstable with respect to weakprocesses ( β -decay) but have a sufficiently long half-life so that they are observed like stable nuclei. Other isotopessuch as H, He, He, He, Be are unstable with respect to the strong interaction and decay immediately so thatthese primary nuclei are not detected as final yields. They should appear in the relevant distribution if the modelHamiltonian (4) contains the sum over all bound states. We have calculated the expected yields Y final A,Z of the unstableisotopes using the Lagrange parameters λ final i . These are given in Tab. I, col. Y final A,Z , in italic parentheses. We concludethat an essential part of clustering is found within these unstable nuclei.This problem, that also the formation of unstable nuclei are expected, is solved by taking into account that theobserved distribution are not equilibrium distributions but the result of a time evolution described by ρ ( t ) (1). Tosolve this in a simple approximation, we assume a freeze-out scenario. Up to the freeze-out time t (cid:48) = t freeze , only theinformation about energy density and particle number density is sufficient to describe the state of the system. Therelevant distribution (3) can be used to describe the system.After this, a more detailed description of the system is necessary where the occupation numbers of quasiparticlestates of the components are relevant, the corresponding Lagrange parameters are the distribution functions. Thisstage of evolution is described by reaction kinetics, unstable nuclei decay that feed the states of observed stable nuclei.The primary distribution, described by the relevant statistical operator ρ rel ( t freeze ) and the yields Y rel A,Z , is transformedto the yields of detected nuclei Y A,Z f denoted as feeding in Section II, for instance Y H f = Y H + Y H , etc.Another consideration is the inclusion of excited states of nuclei to characterize the relevant distribution, see thedata tables in Ref. [27]. Excited states contribute to the statistical weight of an isotope. For instance, the isotope Hhas an excited state at 0.31 MeV with a degeneracy factor 3. Assuming that this excited state is also populated atfreeze-out described by the relevant distribution, it decays and its yield will be found in the corresponding final clusterstate. The statistical weight or intrinsic partition function of a special channel characterized by { A, Z } contains notonly the bound states but also continuum correlations, see [29]. The threshold energy E thresh A,Z denotes the edge ofcontinuum states and it is also shown in Tab. I. In general, it is given by the neutron separation energy S n , but insome cases other decay channels such as S n ( He, He) or α -decay ( Be) determine the edge of continuum states.At present, we neglect the contribution of continuum correlations, but will discuss them below in Section IV A. Thecorresponding distribution is denoted as Y rel ,γA,Z .The account of excited states can be realized introducing in Eq. (5), which considers only the ground state withlowest energy, the prefactor R γA,Z ( λ T ), R γA,Z ( λ T ) = 1 + exc (cid:88) i g A,Z,i g A,Z e − E A,Z,i /λ T , (6)which is related to the intrinsic partition function, so that Y γA,Z = R γA,Z Y (0) A,Z . The summation is performed over allexcited states, excitation energy E A,Z,i and degeneracy g A,Z,i [27] , which decay to the ground state. The result Y rel ,γA,Z shown in Tab. I was obtained with the factor R γ , ( λ T ) = 1 + 3 / e − . /λ T for H, and R γ , ( λ T ) = 1 + 5 e − . /λ T for Be. No excited states below the continuum edge are known for the other bound nuclei so that the remaining factors R γA,Z ( T ) are unity. Assuming that the unstable nucleus H feeds the measured yield of H f , that He and Be feed He f , He feeds He f , and He feeds He f , the optimization with respect to the measured yields Y exp A,Z using the leastsquares method gives the values of the three Lagrange parameters λ rel ,γT = 1 . λ rel ,γn = − . λ rel ,γp = − . Y rel ,γA,Z for the final distribution reproduce nicely the measured values Y exp A,Z for H, H f , and He f in relation to He f . It seems that He f is overestimated, and He f is underestimated. Notably, the yields ofthe unstable nuclei He and He which have been inferred from the energy spectra of emitted α particles [12] are alsooverestimated by the relevant distribution Y rel ,γA,Z . The prompt emission of protons and neutrons will be discussedbelow. IV. RELEVANT DISTRIBUTION DERIVED FROM THE FULL HAMILTONIAN
The estimate Y rel , A,Z should be improved taking into account different effects to be discussed in the following.(i) It is not consistent to consider only the excited bound states below the edge of continuum states and to neglectcorrelations in the continuum. In particular, H, He, He, He, and Be are not bound but appear as correlationsin the continuum. Continuum correlations should be also considered for other, weakly bound nuclei such as H, He.We need a systematic treatment of the contribution of continuum correlations. This is possible with the help of thegeneralized Beth-Uhlenbeck formula [30]. The corresponding relations are denoted as virial equations.(ii) Instead of the approximation (4) where interaction between nucleons and nuclei is neglected, we have to considerin-medium effects if we treat the full Hamiltonian (2).(iii) The nuclear system is not homogeneous. We should consider the mean field near the scission point where both mainfragments are close together, and the nucleons forming the neck region are not correctly described by a homogeneousgas. This means that the full Hamiltonian contains in addition to the interaction between the constituents also themean field V ext τ ( k, k (cid:48) ) of the main fragments of scission. A. Continuum correlations
A comprehensive discussion of the contribution of continuum correlations in the case of H, He has been givenin [29] based on the generalized Beth-Uhlenbeck formula. For H extended discussions have been given earlier, seereferences given in [29], and for Be, see also [31]. The model calculations are compared to measured phase shift data.The account of continuum correlations leads to the virial expansion, which is represented by the reduction factor R vir A,Z ( λ T ). The virial expansion for the deuteron channel d ( A = 2 , Z = 1) is obtained from the Beth-Uhlenbeckexpression R vir d ( λ T ) = 1 − e − E thresh d /λ T + e − E thresh d /λ T πλ T (cid:90) ∞ dEe − E/λ T δ d ( E ) (7)where E denotes the c.m. energy of the n − p system describing the correlations of the deuteron channel above thecontinuum edge. The scattering phase shifts are denoted as δ d ( E ). From the known values of these scattering phaseshifts, see [31], we have for instance the value R vir d (1 . .
98, see Appendix A.The H channel is treated similarly. Because there is no bound state, only the last term in (7) with the integral overthe scattering phase shifts in the t − n channel remains. The corresponding virial coefficient in the t − n channel hasbeen calculated in [29] and parametrized introducing an effective energy E eff tn ( T, n n ) so that the value of the reductionfactor R vir4 , ( λ T ) = e − E eff t,n ( λ T , /λ T + E thresh4 , /λ T (8)follows as R vir4 , (1 . . He channel, the virial coefficient using the α − n scattering phase shifts is calculated, see [29, 31]. Forthe reduction factor of He we find from the relation similar to (8), and using the parametrization E eff α,n ( λ T ,
0) givenin [29], the value R vir5 , (1 . . Be, the virial coefficient using the α − α scattering phase shifts iscalculated, see [29, 31].It is not easy to calculate the continuum correlations for arbitrary clusters { A, Z } . It seems that continuumcorrelations are important for weakly bound states, so that the edge of the continuum E thresh A,Z is of relevance. Thereforewe introduce an interpolation formula, in units of MeV, R vir A,Z ( λ T ) = 1 / ( e − ( E thresh A,Z +1 . / . + 1) / ( e − ( E thresh A,Z +2 . /λ T + 1) (9)which reproduces the values given above for H, H, and He at λ T = 1 . R vir A,Z ( λ T ) for the remaining isotopes as givenin Tab. I. It replaces R γA,Z (6) so that Y rel , vir A,Z = R vir A,Z Y (0) A,Z . The measured yields Y exp A,Z are better reproduced, inparticular the results for H, He and He are significantly modified. However, even with the account of continuumcorrelations the yield of He is overestimated as before, and He and Be remain underestimated. We have toconsider additional effects which are of relevance to calculate the relevant distribution.
B. Pauli blocking
Another interesting problem is the medium modifications, that should be taken into account when considering aquantum statistical treatment of the Hamiltonian (2) of the nuclear system, treating the interaction between thecomponents. In lowest order, we have self-energy shift and Pauli blocking, see [29] and references given there. Ifwe neglect the momentum dependence of the single-nucleon self-energy shift (rigid shift approximation), the self-energy shifts of bound and scattering states are identical so that the binding energy is not changed. Then, it canbe incorporated in a shift of the parameters λ n , λ p . The Pauli blocking leads to a decrease of the binding energy(we denote the disappearance of the binding energy as the Mott effect). Because it is related to the occupation ofsingle-particle nucleon states by the medium, it is sensitive to the nucleon densities n n , n p .The effect of Pauli blocking is expected to lead to dissolution of weakly bound states which are shifted to thecontinuum. The Pauli blocking has been considered for the bound states in former publications, see [29], where alsothe effect of Pauli blocking for H and He is calculated. To evaluate the Pauli blocking shifts we use the results givenin [29]. For the unstable nuclei H, He we use the Pauli blocking shifts of the constituent cluster t, α , respectively, andthe density-dependent contribution of scattering phase shifts according to the generalized Beth-Uhlenbeck formula.Now we infer the effective reduction factor R eff A,Z ( λ T ) for the different components { A, Z } of the relevant distribution,which are needed to reproduce the observed yields, and suppose that density effects are responsible for these inferredvalues. Because density effects become more visible for weakly bound systems, we consider the reduction factor R eff6 , ( λ T ) for He as an unknown quantity to be determined from the fitting of the parameters (the threshold energyfor the continuum states 0.975 MeV is small compared also to the deuteron case where it amounts to 2.225 MeV).The reduction factors for H, H, H, He, He, He, He, and Be remain unchanged and coincide with R vir A,Z ( λ T ),see Tab. I, but for He, He, and He, R eff A,Z is very different from R vir A,Z . We know the yields of He, Y , /Y , = 0 . He, Y , /Y , = 0 .
21 [12]. We also know the final yield of He which is given in Ref. [12] as 270,see Tab. I. We can infer the effective reduction factors R eff A,Z ( λ T ) that reproduce these values. The values shown inTab. I are constructed this way. They are lower than the values R vir A,Z ( λ T ). This means that Y rel , eff A,Z for He, He, and He was calculated by using the experimental values of the yields to calculate R eff A,Z and then with that, we calculated Y rel , eff A,Z = Y exp A,Z for these three isotopes. Yields for the isotopes H, He are not observed so that we have only thepredictions Y rel , eff A,Z for these isotopes, also for Be we give only predictions because the errors are quite large.We remember that taking into account only the virial expansion, i.e., the contribution of continuum correlations, He is overestimated, He is underestimated. A possible explanation may be the small binding energy 0.975 MeVbelow the continuum edge for He which makes this state more sensitive to medium shifts so that the Pauli blocking isstronger. Note that an alternative explanation could be the formation of tetraneutron correlations in neutron matter[32]. In the effective approximation Y rel , eff A,Z only H, H, He, and He have been used to determine the Lagrangeparameters, because the threshold energy for the continuum is larger. The measured contributions of He, He havebeen adopted, values for He are assumed. The multipliers R eff A,Z (1 .
3) are determined from the measured yields of He, He, He. It is found that the inferred values R eff A,Z (1 .
3) are smaller than the values R vir A,Z (1 .
3) for these isotopes.We conclude that the result Y rel , vir A,Z , in which continuum correlations are taken into account as virial coefficients,but medium corrections are neglected, has some deficits.i) The yield of He and He is overestimated. Because of the weak binding of these isotopes, compared to othersincluding He, the dissolution of the bound state in dense matter may be of relevance. Instead of a calculated reductionfactor R vir6 , (1 .
3) = 0 . He, the data give the observed reduction factor R eff6 , (1 .
3) = 0 . He, He is also overestimated. For He, instead of the calculated reduction factor R vir5 , (1 .
3) = 0 . R eff5 , (1 .
3) = 0 . He, the calculated virial value R vir7 , (1 .
3) = 0 .
821 for thereduction factor is replaced by the empirical value R eff7 , (1 .
3) = 0 . R eff A,Z of the reduction factor of the light nuclei, after the subtraction ofthe effect of continuum correlations, the remaining part contains the in-medium effects. From this, we estimate theneutron density of the medium at freeze-out time. From the He values and the results given in the paper [29], thevalue n n = 0 . − is obtained if a relation similar to Eq. (8) is used. The density dependence of E eff α,n ( λ T , n n )is given in [29].For He, the inferred reduction factor R eff6 , (1 .
3) = 0 . E of about 1.7 MeV.The Pauli blocking shift according to [29] is 4 × . e − . T n n (separation of two neutrons, F A,Z ≈
2) so that thevalue n n = 0 . − is obtained. The reduction factor for He is about 0.5. Such values are obtained for Hat a density of about n n = 0 .
002 fm − . Note that the measured values have large errors, and also the treatment ofmedium effects should be improved.A more recent analysis of experimental data has been performed in Ref. [20]. Instead of the ratio He/ He =0.031(2) given in [12], a value 0.041(5) has been presented. A higher value of He/ He would also give a higher valueof the reduction factor R eff6 , (1 .
3) = 0 . E = − .
50 MeV according to Eq. (9) so thatthe Pauli blocking shift is only 0.475 MeV. The corresponding density follows as n n = 0 . − what is in betteragreement with the former result.The measured value Y , = 10(6) for Be [22] also underlines the better fit using the distribution Y rel , eff A,Z . V. EXTERNAL MEAN-FIELD POTENTIAL
A further improvement of the treatment of the Hamiltonian of the nucleon system is to take into account theinteraction with the two large fragments after scission. This can be done introducing a mean field, produced by thestrong nucleon-nucleon interaction as well as by the Coulomb interaction.The neck region where clusters are formed is influenced by the larger fission fragments. There is the Coulomb fieldwhich determines the kinetic energy of the emitted particles, but also the strong interaction described, e.g., by thepion-exchange potential. These interactions should be added to the Hamiltonian as external fields V ext τ , Eq. (2).Obviously, the use of results such as the yields of the isotopes which are obtained for homogeneous systems, is onlypossible in the local density approximation (LDA), but demands further discussions. The bound state clusters arecompact objects, the wave functions are extended over a region of some fm. A local density approximation may bepossible. In contrast, the Fermi wave number corresponding to baryon density n B follows as k F = (3 π n B / / (symmetric matter). For n B = 0 . − , the value k F = 0 .
144 fm − follows. The wave function of neutrons israther extended so that a LDA approach is not justified.We assume that the relevant fragment distribution, including the two large fragments as well as the light clustersor correlations, are already formed at the scission point, as also known from the scission point yield (SPY) model,see, e.g., [33–37] for recent work. Hartree-Fock-Bogoliubov and related mean-field calculations have been performedfor fission. To describe cluster formation, one has to go beyond a mean-field approach. A similar problem arises whendescribing the α decay of heavy nuclei where a quartetting wave function approach has been proposed to describe thepreformation of α -like correlations [38].As a simple estimate we consider the superposition of two Woods-Saxon potentials at distance R + R + d . Forsimplicity we assume the symmetric case where Cf decays into two fragments approximately
Cd (in general,asymmetric decays occur. Experimentally, for
Cf, the highest yield values were found for He +
Zr +
Ba, see[34]) and calculate the Woods-Saxon potentials according to [39] (units: fm, MeV) V mf , WS n/p ( r ) = − .
06 1 ∓ . N − Z ) /A e ( r − . A / ) / . (10)for neutrons (upper sign). For protons (lower sign) we have to add the Coulomb potential V Coul ( r ) = Ze (cid:26) R A − r R A , r ≤ R A , r , r > R A (cid:27) (11)with R A = 1 . A / fm [39]. The parameter d for the scission point is under discussion, a recent estimate [40] givesthe range 4 - 6.5 fm and a proposed value of 5.7 fm. The distance between the fragment center of mass points is2 R A + d = 18 .
06 fm. The mean-field (MF) potentials for neutrons and protons along the symmetry line z is shown inFig 1. The values at the scission point z = 0 are V mf , WS n (0) = − . V mf , WS p (0) + V Coul (0) = 13 .
549 MeV.Larger estimates of the scission parameter d are reported for ternary fission with larger clusters such as Ca [41]which will not be discussed here.It is possible to estimate the neutron density at the scission point and to understand whether the density valuesderived from the Pauli blocking estimates are reasonable. For an exploratory calculation, we use the parametrizationof neutron/proton densities in nuclei n n/p ( r ) = n ,n/p e ( r − R n/p ) /a n/p (12)with R n = (0 . N / + 0 . Z + 0 . R p = (1 . Z / + 0 . N + 0 . a n =(0 .
446 + 0 . N/Z ) fm, a p = (0 .
449 + 0 . Z/N ) fm [42]. The neutron density at distance d/ .
85 fm from thesurface is n n = 0 . − for each Cd nucleus so that the value of the neutron density in the neck regionis about n n ( z = 0) = n scission n ≈ . − . The proton density at this distance is estimated as n p ( z = 0) = -30 -20 -10 0 10 20 30z [fm]-50-40-30-20-1001020 V m f , W S ( z ) [ M e V ] np Figure 1: Woods-Saxon potentials of two nuclei
Cd at distance 2 R A + d , R A = 6 .
28 fm, d = 5 . n scission p ≈ . − . If the distance d for scission is larger, d/ n scission n ≈ . − , n scission p ≈ . − .At this point we emphasize that the treatment of the full Hamiltonian including the external mean-field potentialleads to the result that the cluster distributions, including the formation of light nuclei, happens at scales of the rmsradii which are of the order 1 - 3 fm. The center-of-mass (c.m.) motion of the cluster is determined by the externalpotential, whereas the intrinsic properties are determined locally. Intrinsic energy, but also excitations of nuclei andtheir distribution, are determined by the local properties. The wave function of the c.m. motion is also extended butmay be approximated by a quasi-classical approximation. This is not possible for the protons and neutrons which aredescribed by extended states. We cannot treat them like plane waves describing free particles but have to use, e.g.,quasiparticle states in the mean-field potential known from HFB calculations.As a consequence, single-particle modifications of the in-medium few-particle Schr¨odinger equation like the Hartree-Fock self-energy or the Pauli blocking should be expressed in terms of the quasiparticle wave function and theoccupation numbers of these quasiparticle states. It is possible to introduce Wigner functions and to perform a localapproximation, but it has to be noted that Pauli blocking and exchange interaction are nonlocal.For the yields we conclude that all bound states of the light isotopes may be described approximatively by localparameters, in particular the local density approximation with the mean field at the scission point z = 0. Thecorresponding relevant distribution evolves to the final yields and unstable states feed the corresponding stable nucleiseen in the experiment. The distribution of neutrons and protons is not described by a local density approximationbut by quasiparticle states obtained from the HFB calculation or other approaches to solve the in-medium Schr¨odingerequation for a nucleon moving in an external, mean-field potential. For the relevant distribution, the neck densities n rel n , n rel p at the scission point appear as new parameters. The partial densities n rel n , n rel p are not described by theLagrange parameter λ i as in a local density distribution, but need the solution of the Schr¨odinger equation withan position dependent external potential. The relevant distribution of quasiparticle states evolve also to the finaldistribution which is not described here, but have been discussed extensively in the literature (e.g., Boltzmannequations). At the moment, we consider n rel n , n rel p as additional parameters characterizing the distribution of clusters. VI. CONCLUSIONS: RELATION TO THERMODYNAMICS
The ternary fission, considered in this work, is a nonequilibrium process. A fundamental approach, for instancethe method of the nonequilibrium statistical operator [14], is required for a systematic treatment. An indispensablerequisite is the introduction of the relevant statistical operator reflecting the informations we have about the evolvingsystem. In heavy-ion collisions the hot and dense nuclear matter evolves like a fireball, described by local thermody-0namic equilibrium. In the case of spontaneous fission, the concept of a fireball is hard to accept if one assumes thatthe nucleus before fission is described as a pure quantum state. The freeze-out concept is very successful in describingthe formation of clusters and the measured yields, but appears presently as a semi-empirical approach which requiresmore sophisticated reasoning within a fundamental nonequilibrium approach. We use here a information-theoreticalapproach which does not require the concept of equilibrium.We have introduced Lagrange parameters λ i to characterize the state of the system, in particular the distributionfunction for the different components of nuclear matter. Only in thermodynamic equilibrium are these parametersequivalent to the thermodynamic quantities T, µ n , µ p . This is not valid in the fission process considered here. Wehave a nonequilibrium situation. However, the Lagrange parameter λ i may be considered as the nonequilibriumgeneralization of these thermodynamic parameters. For instance, the large difference λ n − λ p ≈ . z = 0.The neck region of the fission process is not homogeneous but exhibits strong gradients in the density distribution,the nuclear mean-field potential and the Coulomb potential. This has to be taken into account if statistical models areused to explain the observed yields. In our approach, the construction of the relevant statistical operator ρ rel has tobe improved considering the full Hamiltonian, which contains the position dependent external potential. A mean-field(Hartree-Fock) calculation can provide us with more realistic single-particle orbitals. A local-density approximationwhere in equilibrium λ τ is interpreted as chemical potential µ τ to calculate the neutron/proton density within theideal Fermi gas model is not valid. In contrast to the cluster states which are localized in the range of few fm, thesingle-nucleon states are extended and have to be calculated for the mean-field potential V mf τ ( r ) as function of theposition r .Within this work, we have discussed the information-theoretical approach which leads, in the simplest approxima-tion, to a Boltzmann-like distribution. We improved the treatment of the Hamiltonian of the nucleon system takingthe formation of excited, unstable states into account, as well as their decay. In addition, we considered correlations inthe continuum and in-medium effects. This way we put forward a quantum statistical treatment of the many-nucleonsystem. Obviously, the excited states and the resonances should be treated in a manner similar to the stable boundstates observed in the final distribution. The inclusion of correlations in the continuum has to be considered, forinstance, using the scattering phase shifts as shown by the Beth-Uhlenbeck formula. This improves the treatment ofvery short-living excitations, but such broad resonances cannot be treated like stable, well-defined bound states asdone in a simple nuclear statistical equilibrium calculation.Of particular importance are in-medium modifications which lead to a modification of the quasiparticle energy andpossibly the dissolution of bound states (Mott effect). This is of special significance for the weakly bound, neutron-richexotic nuclei which are strongly influenced by the medium. They may be used as a sonde to probe the environment.The strong reduction of the yield of the exotic nuclei, observed in many ternary fission experiments, may be explainedas a density effect. It gives information about the state of the nucleon system at the time instant where the chemicalcomposition freezes out.Further considerations such as the formation of heavier nuclei like droplet condensation [7] have to be included,treating nucleation as a nonequilibrium process. Future work, in particular the treatment of the external mean-fieldpotential, may give a more detailed description of the nonequilibrium properties and the evolution of clusterdistribution in ternary fission processes. Acknowledgment:
This work was supported by the German Research Foundation (DFG), Grant
Appendix A: Virial expansion
Instead of considering unbound systems (resonances) as bound states, we can calculate their contribution to thedensity (continuum correlation) according to the Beth-Uhlenbeck formula. In particular, for H, deuteron channel,we have (c.f. [30, 31]) b pn ( T ) = 32 / (cid:20) e . /T − πT (cid:90) ∞ dEe − E/T δ p,n ( E ) (cid:21) (A1)(sum of all phase shifts in c.m. system). The reduction factor follows as R vir d ( T ) = 2 / b pn ( T ) e − . /T . (A2)1 T b pn ( T ) [31] E eff d ( T ) R eff d ( T ) E a,r d ( T ) R a,r d ( T ) ˜ E a,r d ( T ) ˜ R a,r d ( T )1 19.4 -2.2132 0.9883 -2.1953 0.970737 -2.2089 0.9840282 6.10 -2.1125 0.94530 -1.98559 0.887183 -2.05406 0.9180793 4.01 -1.9103 0.9004 -1.61018 0.814696 -1.72622 0.8468254 3.19 -1.6319 0.8622 -1.11119 0.756954 -1.24862 0.7834145 2.74 -1.2796 0.8277 -0.508454 0.709419 -0.640388 0.7283876 2.46 -0.8887 0.8003 0.189081 0.668749 0.0851844 0.680437 2.26 -0.4433 0.7753 0.976248 0.632977 0.918021 0.6382648 2.11 0.0428 0.7532 1.84889 0.600954 1.84983 0.6008839 2.00 0.5300 0.7363 2.80304 0.571969 2.87347 0.5675110 1.91 1.0493 0.7208 3.83476 0.545542 3.98262 0.537535Table II: Effective bound state energy and virial correction expressed by the multiplying factor R eff d ( T ) in the n − p channelcontaining the deuteron as bound state. T b n ( T ) [31] E eff n ( T ) E a,r n ( T ) R a,r n ( T ) ˜ E a,r n ( T ) ˜ R a,r n ( T )1 0.288 1.11277 1.05942 0.346657 1.5096 0.2209992 0.303 2.16202 2.12001 0.346454 3.0388 0.2188433 0.306 3.22432 3.23552 0.340103 4.69126 0.2093494 0.307 4.29082 4.41082 0.331972 6.45687 0.1990465 0.308 5.3532 5.65598 0.322647 8.33014 0.1889966 0.308 6.3532 6.97922 0.312484 10.3071 0.1794517 0.308 7.3532 8.38453 0.30186 12.3835 0.1704928 0.309 8.54864 9.87243 0.29111 14.5542 0.1621429 0.310 9.59871 11.4413 0.280479 16.8142 0.15439410 0.311 10.6447 13.0883 0.270136 19.1583 0.14722Table III: Effective bound state energy and virial correction expressed by the multiplying factor R eff n ( T ) in the n − n channel . Using the values given in [31] we find R vir d (1) = 0 . , R vir d (2) = 0 . , R vir d (3) = 0 . B eff d ( T ) = − E eff d ( T ) = − T ln (cid:20) / b pn ( T ) (cid:21) . (A3)Using the n − n scattering data, the values for b n ( T ) (full) are given in Ref. [31], b n ( T ) = 2 / πT (cid:90) ∞ dEe − E/T δ n,n ( E ) − / . (A4)The effective binding energy follows as B eff n ( T ) = − E eff n ( T ) = − T ln (cid:20) / (cid:18) b n ( T ) + 12 / (cid:19)(cid:21) (A5)and the reduction factor R vir n ( T ) = 1 πT (cid:90) ∞ dEe − E/T δ n,n ( E ) . (A6)Values are shown in Tabs. II, III. The superscripts a, r denotes scattering phase shifts taken from the scatteringlength and the effective range. The tildes values are calculated with the quasiparticle correction, see [29].For He we consider b αn ( T ) = (cid:18) (cid:19) / πT (cid:90) ∞ dEe − E/T δ αn ( E ) (A7)2(c.m. system) also considered in [31]. The reduction factor is R vir He ( T ) = (cid:18) (cid:19) / b αn ( T ) e . /T − . /T (A8)(the degeneracy factor 1/4 follows from the degeneracy factor in the phase shifts). This is the factor to multi-ply the NSE ground state contribution. Using the values given in [31] we find R vir He (1) = 0 . , R vir He (2) =0 . , R vir He (3) = 0 . Be we consider b α ( T ) = 2 / πT (cid:90) ∞ dEe − E/T δ α ( E ) + 12 / (A9)(c.m. system) also considered in [31]. The reduction factor is R vir Be ( T ) = (cid:18) (cid:19) / (cid:20) b α ( T ) − / (cid:21) e . /T − . /T . (A10)Using the values given in [31] we find R vir Be (1) = 0 . , R vir Be (2) = 1 . , R vir Be (3) = 1 . H is 0.0654. Considering only the P / channel, the reduction factor for He is 0.70716.The calculation of the contribution of the continuum is more complex for the remaining isotopes. Calculations withseparable potentials are possible, but the potential parameters must be fitted to scattering data. We assume that thesituation with He and He is comparable to He and take a similat reduction factor. Because these isotopes giveonly small contributions, a rough estimate is sufficient.We have also to estimate the contribution of scattering states for isotopes with stable (with respect to stronginteraction) ground states, as familiar from the virial expansion. This is known for H where we can use the result forthe virial expansion R vir d (1 .
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