Nuclear Level Density within Extended Superfluid Model with Collective State Enhancement
For nuclear level densities, a modification of an enhanced generalized superfluid model with different collective state enhancement factors is studied. An effect of collective states on forming the temperature is taken into account. The ready-to-use tables for the asymptotic value of -parameter of level density as well as for addition shift to excitation energy are prepared using the chi-square fit of the theoretical values of neutron resonance spacing and cumulative number of low-energy levels to experimental values. The systematics of these parameters as a function of mass number and neutron excess are obtained. The collective state effect on gamma-ray spectra and excitation functions of neutron-induced nuclear reactions is investigated by the use of EMPIRE 3.1 code with modified enhanced generalized superfluid model for nuclear level density.
The nuclear level density (NLD) is crucial parameter to define characteristics of nuclear decay. The collective states have rather strong effect on NLD, specifically at low excitation energies 73Bjor (); 83Igna (); 76Vdov (); 79Igna (); 93Igna (); 09Capo (); 07Pluj1 (); 07Pluj2 (). In fact, enhancement (variation) factor equals to a ratio of NLD with and without () allowing for collective states. In the adiabatic approach, enhancement factor is a product of vibrational and rotational enhancement factors that take into account the change of level densities due to presence of vibrational and rotational states respectively. Up to now, there exist problems in estimation of these variation factors. Specifically, there is rather big uncertainties in estimation of the magnitude of the and different approaches 73Bjor (),79Igna (); 93Igna (); 09Capo (); 07Pluj1 (); 07Pluj2 () lead to different values of the . Variations of the with excitation energy are strongly dependent on vibrational state damping width, and this is unresolved task too. Microscopic description of the vibrational state relaxation still remain to be answered and different phenomenological approaches of allowing for vibrational state damping are used. In this contribution, the phenomenological methods of description of vibrational state contribution into NLD are tested with implementation of the Enhanced Generalized Superfluid Model (Empire Global Specific Model, EGSM)07Herm () for description of level densities of intrinsic and rotational states.
Ii The Methods of Vibrational Enhancement Factor Calculations
The simple methods for calculations of the vibrational enhancement factor are based on the saddle-point method with the partition function , where is co-factor resulted from collective coherent interaction forming vibrational states ( is named below as vibrational co-factor of partition function). Generally, collective states change the temperature of intrinsic states . As it is shown in 07Pluj1 (); 07Pluj2 (), the nuclear temperature is equal to the temperature of intrinsic states in the first order on the variation .
The simple phenomenological methods for calculations of the vibrational enhancement factor were proposed in 73Bjor (),93Igna (); 09Capo (); 07Pluj1 (); 07Pluj2 (); 05Pluj1 (). They are based on different phenomenological extensions of the boson expression () for the vibrational co-factor 07Pluj1 (); 07Pluj2 () with an approximation . Among these methods, there is damped occupation number approach (79Igna (); 93Igna (); 09Capo (), liquid drop prescription with the temperature damping () 07Herm () and simplified version 07Pluj1 (); 07Pluj2 () of the response function method 05Pluj1 (). According to this last approach, the is taken as a ratio of boson partition functions with averaged occupation numbers (BAN approach): . The quantities are boson occupation numbers averaged over the collective motion period 07Pluj1 (); 07Pluj2 (): with for damping width of vibrational state of multipolarity with characteristic frequency and is the energy of this vibrational state; is a frequency of corresponding state. The damping width is determined by collective relaxation time that results from retardation effects during two-body collisions: 07Pluj1 (); 07Pluj2 ().
Iii Results of Calculations and Conclusions
We use the following expression of NLD for states with excitation energy and spin : , where is level density that takes into account excitations of intrinsic and rotational states. It is calculated using the EGSM of the EMPIRE 3.1 07Herm (). The is a function of the asymptotic value of the -parameter of level density and is an additional shift of excitation energy.
For NLD, the expression is used with different and quantities , are considered as the parameters. The is obtained from fitting of average theoretical NLD to the corresponding experimental data on -resonance spacing 09Capo (). Shift parameters are obtained from fit of the experimental values of cumulative numbers of low-lying discrete levels to theoretical values with previously determined . This approach is referred below as modified EGSM.
The ready-to-use table of the parameters and was prepared with the use of this approach for 291 nuclei and BAN approach for . The following systematics were also obtained for the parameters , with dependence on mass number and neutron excess : , (MeV), , (MeV), where - energy of the first collective state in MeV. The values of the parameters of systematics and their uncertainties are presented in Table 1.
In Table 2 are given the ratios of chi-square deviations of theoretical NLD within modified EGSM with different from experimental data of teams from Oslo 04Agva (), Dubna 07Sukh () and Obninsk 09Zhur (). The is number of experimental data for nucleus and is number of nuclei. For approximation BANT, vibrational co-factor of partition function was used and variation of the temperature was taken into account. The partition function of back-shifted Fermi gas model (BSFG) was used for intrinsic states. For used experimental data, relative chi-square deviations for BAN and BANT approaches are less than for other models with vibrational enhancement. The BAN approach is appeared to be more preferable in comparison with BANT because calculations within BANT are rather complicated.
The comparisons of and for each nucleus with their systematics are shown on Fig.1.
Figure 2 presents gamma-ray spectra for reactions at on . Theoretical spectra were calculated using modified EGSM as well as Gilbert-Cameron (GC) model and back-shifted Fermi gas model. Experimental data were taken from 11Bond (). It can be seen, that scatter of gamma-ray spectra calculated within modified EGSM are the same order as scatter of the spectra calculated using other NLD models.
Figure 3 shows the excitation functions of reaction on . Experimental data are taken from EXFOR data library. Theoretical excitation functions are calculated using modified EGSM with different . One can see that shape and values of excitation function and gamma-ray spectra are sensitive to choice of vibrational enhancement factor.
For modified EGSM, approximation of boson partition function with average occupation numbers (BAN) can be considered as the most appropriate approach for calculation of the vibrational enhancement factor. The comparison between theoretical calculations and experimental data also shows that within BAN approach for . These values of enhancement factor are in agreement with results of microscopic quasiparticle-phonon model76Vdov ().
- (1) S. Bjornholm et al., Proc. of Symposium Rochester, New-York, 13-17 August., 1973 367 (1973).
- (2) A.V. Ignatyuk, Statistical Properties of Excited Atomic Nuclei, IAEA, INDC-233(L), (1985).
- (3) A.I. Vdovin et al., Phys. Elem. Particles Atom. Nuclei 7, 952 (1976).
- (4) A.V. Ignatyuk et al., Sov.J.Nucl.Phys. 29, 450 (1979).
- (5) A.V. Ignatyuk et al., Phys. Rev. C 47, 1504 (1993).
- (6) R. Capote et al., Nucl. Data Sheets 110, 3107 (2009); http://www-nds.iaea.org/RIPL-3/.
- (7) V.A. Plujko et al., Phys.Atom.Nucl. 70, 1643 (2007).
- (8) V.A. Plujko et al., Int.Jour.Mod. Phys. E 16, 570 (2007).
- (9) V.A. Plujko et al., AIP Conf. Proc. 769, 1124 (2005).
- (10) M. Herman et al., Nucl. Data Sheets 108, 2655 (2007).; http://www.nndc.bnl.gov/empire/ .
- (11) U. Agvaanluvsan et al., Phys. Rev.C 70, 054611 (2004); http://ocl.uio.no/compilation/ .
- (12) A.M. Sukhovoj et al., Proc. Int. Conf. ISINN-15, Dubna, May 2007. 92 (2007)
- (13) B.V. Zhuravlev, IAEA, INDC(NDS)-0554, Distr. G+NM (2009)
- (14) V.M. Bondar et al., Proc. Int. Conf. ISINN-18, Dubna, May 26-29, 2010. 135 (2011)