STATISTICAL MODEL OF DECAY OF EXCITED NUCLEI

Transcription

1 Nuclear Reaction Vieo Project FLNR JINR STATISTICAL MODL OF DCAY OF XCITD NUCLI Level ensity The level ensity of the atomic nucleus consisting of Z pons an N neutrons with the total angular momentum J an the excitation energy U has the following form: U Z N J U Z N U Z N J where U Z N J is the single-particle level ensity account collective states of the nucleus. The single-particle level ensity is given by [Ignatyuk83]: coll coll U Z N is the enhancement factor taking into 3 J a J J U Z N J exp a U J J 4 U where a is the level ensity parameter is the moment of inertia of the nucleus about the axis perpenicular to its axis of symmetry is the pairing energy: A where A is the mass number of the nucleus an is the parameter equal to or for o-o o an even-even nuclei respectively. The level-ensity parameter is given by [Ignatyuk83] exp( U ) a a U U 3 a A A B s where U is the shell correction; an are the coefficients with the efault values.73 MeV -.95 MeV - an.6 MeV - respectively [Ignatyuk75]. The functional Bs ( ) is the imensionless quantity equal to the ratio of the surface area of the eforme nucleus with the quarupole eformation parameter to the area of the spherical nucleus with the equal volume. The quantity Bs ( ) is calculate for the eformation of the groun state (withs of particle emission) an for the sale point (fission with). The values of the eformation of the nucleus at the sale point are calculate accoring to [Cohen63]. The groun-state properties of nuclei (masses shell corrections eformations) are taken from [Moller6]. The groun-state masses for known nuclei are from [Wang]. The moment of inertia of the nucleus is calculate by the formula: k 3 rb.. where the coefficient k is by efault equal to.5 rb.. is the moment of inertia of a rigi boy.

2 The eformation-epenent collective enhancement factor [Zagrebaev]: Nuclear Reaction Vieo Project FLNR JINR coll coll U Z N is etermine accoring to exp where is the ational enhancement factor is the rational enhancement factor. The efault values of parameters are.5.4 [Zagrebaev]. The eformation epenence provies a smooth transition from the rational enhancement of the level ensity for spherical nuclei to the ational enhancement for well-eforme ones. By efault the enhancement factor where А is the mass number T Alternatively is etermine by the expression [Ignatyuk83]: exp.555 A T U a is the nuclear temperature. can be etermine by the expression [Junghans98]: T 5..3N.5 Z where N ( Z ) are the absolute values of the number of neutrons (pons) above or below the nearest shell closure. The enhancement factor is etermine by the following formula: k T where the parameter k has the efault value. where The energy epenence of an is taken as: an ( ) ( ) exp are the parameters with the efault values 4 MeV MeV [Junghans98]. Decay withs It is assume that the excite ating nucleus may ecay through the following channels: emission of light particles (neutrons pons or alpha particles) emission of gamma rays an by fission into heavy fragments. The expression for calculation of the particle emission withs has the form: Bb I J l sb CBb J Tl eb B Bb eb I eb C J l I J l.

3 Here s b is the spin of emitte particle b an probability that the particle will pass through the potential barrier particle. For charge particles the following expression for T e is use: b Nuclear Reaction Vieo Project FLNR JINR B is its bining energy where b n p ; T e is the l P e Tl ( e) ll ( ) exp Vc e R e is the kinetic energy of the emitting b l e 4 where Pe ( ) e 4 e e 4 3 R r A fm an is the reuce mass of the particle. The height of the Coulomb barrier an its with are given by c.6.9. MeV Z V Z Z N A 6..4 MeV for pons.95( ) 4.5.( ) MeV Z V Z Z N A MeV for α-particles. Here N Z.55Z is the number of neutrons for the stable nucleus with the atomic number Z. For neutrons the probability is calculate using the formula: e T e P e R R R l ( ) exp log The gamma-ray emission with is l l. e I J L L C J I J L L C L J f e e e I e where.75 [Neoresov85] the force function for ipole raiation ( L ) has the form: e 6 Z A Z f e 3.3 MeV A e e where an are respectively the energy an the with of the giant ipole resonance calculate as 67.3 MeV A 4.74 A 5 MeV [Schmit9]. The fission with is calculate from the expression: ramers.. fiss fiss C C J sp J T e J e J e ()

4 where T e J ramers fiss gs.. an sp.. exp e Bfiss sp.. gs.. s. p. sp.. Nuclear Reaction Vieo Project FLNR JINR is the fission barrier penetrability 4 is the ramers factor [ramers4] is the viscosity parameter T are the curvatures of the potential at the groun state an sale point respectively. The fission barrier is calculate by the formula: fiss B U B U where B LDM is the liqui-rop fission barrier [Sierk86] U is the shell correction to the groun state [Moller6]. The subsipt sp.. inicates the calculation of the corresponing quantity at the sale point. LDM Survival probability There are two methos applie to calculate the survival probability. The Monte Carlo metho allows one to obtain survival probability for all possible reaction channels while the irect integration gives the survival probability for xn channels only (up to 6n). The Monte Carlo metho has aitional possibilities of calculating for example the fission fragment properties (the mass-charge yiels etc.). These possibilities are absent in the case of the irect integration metho. While the Monte Carlo approach is more general it has a limitation associate with computational time require to accumulate the necessary statistics. For some rare processes (e.g. surviving in xn channels in the superheavy mass region) the require large statistics is nearly unreachable an the irect integration is the only metho working for such the cases. A. The Monte Carlo metho At each stage of the ecay of the nucleus one of the possible events is ranomly selecte the emission of neutrons pons alpha particles gamma-rays or fission. The probabilities of each event are b Pb b n p fission tot tot n p fiss The energy of the emitte particles is simulate etermine accoring to the following spectrum: where C i are the normalization constants. Wb( e) Cb e exp e b n p T W e C f e e e L T L ( ) L( ) exp During evaporation cascae the nucleus either unergoes fission at some stage or loses its excitation energy below the fission barrier an neutron bining energy by the emission of particles an cools forming the evaporation resiue. B. Direct integration Subsequent estimation of the total probability for the formation of a col resiual nucleus after the emission of x neutrons C B xn N is usually performe within numerical calculations base on the analysis of the multistep ecay cascae. The expression for such the probability accounting for the Maxwell-Boltzmann energy istribution of evaporate neutrons reas.

5 Nuclear Reaction Vieo Project FLNR JINR Here Bn x n B () Bn() n n xn ( ) n( ) ( ) n( ) tot tot P I W e e I W e e ( ) ( ) ( ). Bn( x) N n x I x Wn x ex i Ii ex tot i tot x an e i are the bining an kinetic energies of the ith evaporate neutron i is the excitation energy of the resiual nucleus after the emission of i neutrons. Wn e Cn e exp e / T is the probability for the evaporate neutron to have energy e an the normalization coefficient C n is etermine B n from the conition Wn e e. Fusion-fission an fusion-survival oss sections The oss section of the formation of evaporation resiue in xnypz channel in collisions of heavy nuclei can be calculate as follows xn yp z vr l Tl PCN l Pxn yp z l l than the total survival oss section is l T P l P l surv l CN xn yp z l x y z an finally the fusion-fission oss section is efine as where the fusion oss section is fiss fus surv. l T P l fus l CN l Here P CN is the probability for the compoun nucleus (CN) formation by two nuclei coming in contact. For specific nuclear combinations leaing to so calle col synthesis a simple parameterization of P CN (l) was propose in [Zagrebaev8] P CN CN P Z Z l. B int l exp

6 Fission with Nuclear Reaction Vieo Project FLNR JINR Here B is the excitation energy of CN at the Bass barrier an l Q l int where Q is the fusion Q value l l l gs.. is the ational energy an P CN ZZ exp where 76 an 45. The moels allows one to take into account the ynamical elay of the fission process in the Monte-Carlo simulation of the ecay of excite nucleus that is extremely important for analysis of the pre-scission an postscission particle multiplicities. We use a simplifie approximation of realistic ynamical calculations of the fission with (see Fig.). The fission with is parametrize by a step function with only one parameter elay time. When t only the evaporation process is consiere an the fission probability is neglecte while for t all the ecay channels are treate as in the stanar statistical moel (see above). Here t is the time of the ecay process (passe since the CN formation) which is estimate as tot. STATISTICAL MODL DYNAMICAL CALCULATIONS Fig.. Schematic time epenence of fission with t

7 Nuclear Reaction Vieo Project FLNR JINR References [Wang] M. Wang G. Aui A. H. Wapstra F. G. onev M. MacCormick X. Xu an B. Pfeiffer Chinese Physics C 36 () 63. [Cohen63] S. Cohen W.J. Swiatecki Ann. Phys. (963) 46. [Ignatyuk85] A.V. Ignatyuk The statistical properties of the excite atomic nuclei nergoatomizat Moscow 983 (in Russian); translate as IAA report INDC(CCP)-33/L (985). [Ignatyuk75] A.V. Ignatyuk et al. Sov. J. Nucl. Phys. (975) 6; ibi. 55. [Junghans98] A.R. Junghans M. e Jong H.-G. Clerc A.V. Ignatyuk G.A. uyaev.-h. Schmit Nucl. Phys. A69 (998) 635. [ramers4] H. A. ramers Physica (Amsteram) 7 (94) 84. [Moller6] P. Möller A. J. Sierk T. Ichikawa H. Sagawa Atomic Data an Nuclear Data Tables 9- (6). [Neoresov89] V.G. Neoresov Yu.N. Ranyuk Fotoelenie yaer za gigantskim rezonansom iev Naukova Dumka 989 (in Russian). [Schmit9].-H. Schmit an W. Morawek Rep. Prog. Phys. 54 (99) 949. [Sierk86] A.J. Sierk Phys. Rev. C 33 (986) 39. [Zagrebaev] V.I. Zagrebaev Y. Aritomo M.G. Itkis Yu.Ts. Oganessian M. Ohta Phys. Rev. C 65 () 467. [Zagrebaev8] V. Zagrebaev an W. Greiner Phys. Rev. C 78 (8) 346.