CALCULATION OF GENERALIZED LEVEL DENSITIES FOR NUCLEI IN MASS REGION

In this study, a relation between generalized level density and standard level density is derived. Using this relation and Bethe formula of Fermi gas model for standart level density we obtained a generalized nuclear level density formula for nucleus. Generalized level densities were calculated for some nuclei in mass region between 20 and 50 for different q values close to 1 . Our results explain experimental data better than those of Gilbert-Cameron (GC) and Rohr, which are two of the leading compilations in evaluating nuclear level density. Key WordsLevel Density, Nonextensive Statistical Mechanics, Fermi Gas Model

excitation energies, and showed that although, for example, level densities of 26 Al exhibits Fermi-Gas behaviour up to 8 MeV of excitation energy [14,19], those of 56,57 Fe measured with Oslo method [20] have more complicated behaviour which cannot be described by simple Fermi-Gas formula.The influence of pairing correlations leading step structure in vicinity of proton and neutron pairing energies and above might be reason of this complicated structure [21].Hovewer, nonextensive statistical mechanics, based on the q -generalized entropy proposed by Tsallis [22] and developed by many researhers [23][24][25][26][27][28][29][30][31][32][33], has become powerful tool to deal with some systems which (a) have long range interactions, (b) have long range memory effects and (c) evolve in a multi-fractal space-time.In particular, it has been succesfully used to study the properties of the generalized Bose system and a large number of significant results have been obtained [26,29].Obviously, it is very meaningful to investigate the properties of a generalized Fermi system by using nonextensive statistical mechanics.Since nucleus is a Fermi system, it might also be interesting to consider nuclear level density in the framework of nonextensive statistical mechanics.In this direction, Lenzi et al. [34] established a relation between the clasical q -partition function and the level density using q -Laplace transform; for classical ideal gas, they obtained a level density formula from inverse q -Laplace transform of partition function which is the same with that of derived from Laplace transform of canonical partition function within Boltzmann-Gibbs extensive statistics.
In this work, instead of inverse q -Laplace transform of partition function we use a different approach to calculate the level density within statistical mechanics.In this approach, we use a relationship between the generalized nuclear level density and the standard level density which is obtained by following Curilef's prescription [35] for the derivation of the relation between generalized statistical quantity and its standart quantity 1 q → .Details of this derivation are given in next section.Advantage of this relation is that the generalized level density can be calculated directly without using inverse integral transform which is not available now.Using this relation and traditional Bethe theory of nuclear level density calculations for standart nuclear level density, we obtained a new formula for nuclear level density which depends on the entropic index q .This formula contains 3 parameter; two from previous theory, i.e. level density and energy shift, and one from q -generalized statistics characterized by the parameter q which is based on the so-called Tsallis' entropy.In section 3, the results obtained from generalized nuclear level density formula are presented for 12 nuclei in the mass region 50 < A < 20 and compared with the experimental data and two models of Bethe formula.

2.NUCLEAR LEVEL DENSITY
In this section, we obtain a useful relationship between the generalized level density and the standart level density which allows to calculate the generalized level density without using q-generalization of inverse Laplace transformation.To this aim, we calculate the generalized partition function q Z in terms of a parametric integral over the usual grand canonical partition function . The grand-canonical partition function for 1 > q is obtained by using the Hilhorst integral representation of Gamma function, [35] , e d ) ( where . The corresponding partition function for 1 = q can be written in terms of level density regarded as the function of energy E and the number of particle in Eq.(3) into Eq.( 2), the generalized partition function for is associated with the standart level density in the following form: .) , , ( The level density obtained from the inverse Laplace transform of ( ) for especially ideal fermi gas depends on ' β , and therefore q and ξ , but for classical ideal gas it is independent of these parameters.ξ dependence of 1 ρ also appears in Eq (10).The generalized partition function related to the physical system can also be defined with respect to level density as Comparing Eq.( 4) with Eq.( 5), for we obtain a relationship between the generalized level density and the standard level density The extension of the partition function for shown by Prato [36] is which can be derived from another integral representation of Gamma function where . Following the lines from Eq. (1) to Eq. ( 6) for 1 > q case, one can obtain the generalized level density for At this stage, we need to adopt the relationship appeared in Eq.( 9) to perform the calculations for nuclear level density, because nucleus is composed of two kinds of particles, neutrons and protons.The nuclear level density must now depend on neutron number N and proton number Z .For years, the simple models such as Fermi gas model, have been still used to calculate nuclear level density at high energies (or low temperatures).The nuclear level density for Bethe theory, 1 ρ , is given by where ) ( g F ε is total single-particle level spacing of nucleons at Fermi energy and ), ' β in Eq.( 10) is equal to ).The details of the derivation of Bethe formula for nuclear level density can be found in Refs.[1][2][3]37]. For and sufficiently large N , the integral in Eq.( 6) diverges when previous works indicate that in the thermodynamic limit ( ) employing nonextensive statistical mechanics is not suitable to the classical ideal gas [34], the classical systems with N harmonic oscilators [38] and Fermi systems in a general power-law external potential [39].Therefore, in this work, we consider only the case Replacing the nuclear level density in the integrand of the Eq.( 9), the generalized level density for . Above formula is valid through in the interval , which comes from the cut-off condition.Level density parameter is − .One might think that 0 U was simply the ground state, so that U could be simply the excitation energy.Malyshev [40] showed that there was a systematic difference in the values of a for neighboring even-even, odd-A , and odd-odd nuclei.In another study, Newton [7] also showed that to obtain U these discrepancies could be removed by substracting the pairing energy from the excitation energy.Thus, ) is proton (neutron) pairing energy, and

3.RESULTS AND DISCUSSION
Our calculations of level density have been performed with using formula in Eq.( 11) with energy shift δ that was simply due to pairing.This formula contains 3 parameters; i.e. one more parameter q in addition to the parameters of level density parameter a and energy shift δ ( a and δ appears in Bethe formula).In general, the compilation of the parameters of Bethe formula is based on the fit of two parameters.The fits of Braga-Marcazzan and Milazzo-Colli (BMMC) [41] and that of Rohr [42] were starting point.In BMMC compilation, a values deduced individually for a number of nuclei whereas pairing energies of Gilbert-Cameron (GC) [3] determined from semi emprical mass formula were used as the energy shift δ .Rohr compilation is the same with BMMC in that of energy shift wheras a values were fitted to function where A is the mass number, and α and C fitting constants.In GC model, a values are connected to shell correction S as Further compilations include the shell and collective effects into level density parameter a [13,14,43].For δ , better results were obtained with the parameters of Myers and Swiatecki [44], and Grimes [6].For energy shift , δ we use the pairing energy values of GC.The remaining parameters of generalized level density formula are the level density parameter a and the order parameter q which is less than 1 and has a lower limit which depends on level density parameter a .In our calculations, we fit only the parameter a for a fixed value of q to the experimental level density data [45] because the dependence of one parameter on another parameter makes difficult to fit both parameters simultaneously.

= E
MeV (dot-dashed).The fitting values of a for a fixed value of q are given in Table 1 .As q values are getting closer to , which are the predictions of Fermi Gas and the emprical values extracted from average spacings at neutron binding energies [2,3,5,37,46], respectively.q dependence of nuclear level densities at various excitation energies for 24  11 Na is shown in Fig. 1 .Level density increases rapidly at higher energies, but the contribution of this parameter shows a different behaviour as away from and close to 1.For example, as the excitation energy increases, the level density also increases through region where q takes the values between 0.92 and 1 whereas it decreases for 0.92 < q

E (MeV)
. For the entropix index , q there is a lower limit arising from cut-off condition which also depends on excitation energy.These limits therefore change with excitation energy; higher excitation energy higher the limit value.However, when q is fixed, the cut-off condition produces an upper limit for excitation energy.For various q values and , the level density of 24  11 Na is plotted as a function of excitation energy and those limits are shown in Fig. 2. The variation of level density with energy is different from the q dependence of level density.While the former one increases with increasing q value, the latter one has the same value for different excitation energies at one value of q .In Figs.3-5, we compare our results obtained from formula in Eq.( 11) with the results of GC and Rohr models and also with the experimental data [45] for 24  11 Na, 25  12 Mg, 27 13 Al, 28  13 Al, 31  15 P, 32  15 P, 33  16 S, 34  17 Cl, 36  18 Ar, 38  16 Ar, 40  19 K and 41  20 Ca.Although the spin effects are not taken into account in our model, the generalized level density results are in better agreement than those of GC and Rohr models with experimental data.The contribution of q (i.e.nonextensivity) to the behaviour of the level density is significant especially at higher excitation energies.The reason for it is that the entropic index q is a -dependent and level densities are sensitive to a at high excitation energies.The determination of the best fitted values of q close to 1 indicates that level density could not exhibit nonextensive character.Nevertheless we need to consider the spin effects in our model and also need more experimental data to decide whether the physical system might exhibit a nonextensive character or not.

CONCLUSION
By using the relation between nonextensive and standart (Boltzmann-Gibbs) partition functions for grand canonical ensemble, we derived a similar relation between nonextensive and standart level density formula 1 < q and 1 > q . From this relation we obtained a generalized nuclear level density formula in the framework of Fermi gas model for only . This is consistent with previous works in which they indicate that in the thermodynamic limit nonextensive statistical mechanics is not suitable.
In the light of discussions in previous section, the generalized level density formula seems to be appropriate to perform calculations for nuclei with 50.A 20 ≤ ≤ Especially, at higher energies the level density is sensitive to q , but the effect of this parameter decreases at lower energies.However, since it has a lower limit which depends on the level density parameter, we should consider the total effects of these parameters in the calculations.

Acknowledgement
and U is the excitation energy above the energy of fully degenerate states 0

Figure 2 . 24 11
Figure 2. The generalized level density of 24 11 Na as a function of excitation energy for a fixed value of6 = a.The solid, dashed, dotted, and dot-dashed lines correspond to , 0.85 = q , 0.9 = q 0.95 = q and

Figure 4 .
Figure4.The level densities of 31 P, 32 P,33 S and34 Cl for some q values.

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This study has been supported by TUBITAK (The Scientific & Technological Research Council of Turkey) under the Project Number 104T162.
1 , we obtained better fitting values for a .Our values of level density parameter lie between

Table . 1
: Fitted values of level density parameter a which are obtained by fixing the q values for nuclei in mass region