Investigation of the Appropriate Partial Level Density Formula for Pre-Equilibrium Nuclear Exciton Model

Ericson formula represents the first formula, which was suggested to describe the partial level density (PLD) formula in pre-equilibrium region of the nuclear reactions. Then a number of corrections were added to this formula in order to make it more suitable to physical meaning. In this paper, there are two aims to be done: the first aim is to study the correspondence between one and two-components formulae in Ericson, Pauli, and pairing corrections; the second aim is to compare and study the results of Comprehensive formula, which contents with all corrections, with Ericson, Pauli, and pairing formulae. The Comprehensive formula was suggested to simulate the reality. To achieve these aims the Fe and Zr nuclei were chosen and the results showed that the difference between one and two-components formulae was too small which can be neglected. Furthermore, the results strongly recommended that for cross section calculations of the nuclear reaction, one must use Comprehensive formula rather than Pauli formula.


Introduction
Nuclear reactions are divided according to the energy of the incident particle to low, medium, and high-energy reactions.Pre-Equilibrium (PE) region is a stage in nuclear reaction, which represents the case when the excitation energy is not distributed equally between the nucleons of the excited nucleus.Many models describe this region such as: the Intra-Nuclear-Cascade (INC) model [1], the Harp-Miller-Berne (HMB) model [2,3], the Hybrid model [4], the Geometry Dependent Hybrid model [5] and the Exciton model [6].J. J. Griffin suggested the Exciton model at 1966 [6].It was preferred model by many researchers [7,8] because it is simplest in the description and treatment of the PE nuclear emission.The idea of this model supposes that when the bombarding particle hit the target nucleus, it begins to share its energy with the first particle collides with it, then by successive nucleon-nucleon interactions in a series of stages and before attend to the complete interactions (equilibrium), the nuclear emission occurs.Each interaction produces a particle-hole (p-h) pair, and the sum of such particle and hole called an Exciton.The first few states are 2p 1h, 3p 2h and so on.The numbers of excitons are n (n = p + h) and the stages are labeled by s so that n = 2s + 1, therefore, again, through exciton production, the nuclear emission may occur [9].

The Partial Level Density (PLD)
The PLD represents important quantity in nuclear physics.It is used in calculation of cross section, double differential cross section and transition rates [9,10].PLD can be measured experimentally up to 15 MeV; above this value of energy, the levels converge.Therefore, the spacing between them overlapped, and it is difficult to calculate them.Hence, level density may be calculated by theoretical methods [8][9][10].In this paper, the theoretical description of PLD by the exciton model was depended and developed through many stages.The Fermi gas model (FGM) was used as description model in the nuclear reaction exciton model.FGM conceder equal spaces between the levels and this called Equidistant Spacing Model (ESM) [9,10].
Ericson's formula is the first formula which describes the PLD and it is represents the crud formula where, sign 1 means one component, i.e. the particles are not separated to protons and neutrons, g is the singleparticle state density, is the exciton number, is the particles' number, is the holes' number and is the excitation energy.
The two-component formula is The two-component formula, with sign 2, distinguish between proton particles   p  and neutron particles   p  and proton holes   h  and neutron holes   h  .g  and g  are the single particle state densities of proton and neutron, respectively, while n  and n  are the proton and neutron excitons numbers, n n n     .However, and v stand for protons and neutrons respecttively.
In order to make the results of the PLD more reality, the Ericson's PLD formula was developed by adding physical corrections to it.Those corrections are

Pauli's Correction
This correction comes from Pauli's principle that forbids any two particles from having the same quantum state.So, the excitation energy deceases by the factor called Pauli Blocking Factor   , p h A   which given by [11]  Then the one-component PLD formula becomes is the Heaviside step function defined as In the case of two-component, the Pauli Blocking factor is

p h p h A p p h h p p h h g g
Then the PLD for two-component can be given as

Pairing Correction
This correction comes from pairing property between couples of particles.Therefore, this pairing required energy, which is taken from the excitation energy and hence the excitation energy will decrease [12][13][14].Thus, the PLD formula in one component with pairing correction can be given by (all details of this correction given by these references) [9,11] where   P  is the pairing correction, and it is given by   and ∆ are the energy gaps of the ground and excited states, respectively., p h is the modified Pauli Blocking factor which is given by , p h is unmodified Pauli Blocking factor given by Equation (3).

A
 is given by 1.6 0.68 0.996 1.76 if where c is the most probable exciton number that leads to emission e C is the condensation energy given by phase is the pairing energy of phase transition defined as following Copyright © 2013 SciRes.

JAMP
The minimum value of energy for applying these equations is the effective value of energy, which means threshold value of and it is given by equations The value of   can be obtained from curve fitting for almost known nuclei, by a relation known as Gilbert-Cameron formula where  and   are the energy gaps for ground states of proton's and neutron's particles, respectively.
N and Z are the neutrons and protons numbers respecttively.
The value of   is depending on the nuclear temperature, where it increases with temperature and vanishes at a critical temperature which gives for ESM by 2 3.5   [11].Above this temperature, the pairing correction disappeared and the system reverted to uncorrelated condition.The two-component formula is given by   2 P  is the two-components pairing energy given by     

p h p h p h p h g B
A n , , , p h p h A     is the unmodified Pauli Blocking factor given by Equation (5).

Active and Passive Holes Correction
The exciton model assumed that the particle's number must be equal to the holes number, but in fact, the particles number is always bigger than the holes number by one.This is because the projectile (incident particle) is added to the particles number.In order to satisfy the equality between particles number and holes number, passive holes have been supported.They represent those holes are not affected by the nuclear potential; therefore, accumulated near Fermi's level.The correction that comes from passive holes was given by [9,15]     max , q p  h Then, the PLD can take the form

Charge Factor Correction
This correction takes into account the charge effect on PLD calculations.The Charge effect is represented by the effective charge factor, which had been expressed by many formulae.However, many researchers don't apply this correction [9,16].Therefore, the charge factor effect was neglected in this work.

Isospin Correction
To add the effect of isospin in nuclear reactions, it is necessary to determine how much the isospin is conserved or mixed and what is the isospin symmetry energy?
If the isospin is included in calculation of level density, it is important to take one-to-one correspondence between states of the same isospin in isobaric nuclei [17][18][19]. p

h E T T T p h E E T T T T p h E E T T T T
sym is the symmetric energy which is given by empirical equation The PLD formula which contains the isospin can be given by

, n n ph T T p ph T ph sym z p h T E g E A f p h T E A p h n
is the correction factor of states with good isospin.If isospin is assumed to be completely mixed, the symmetric energy is zero and the correction factor f is unity.

Spin Correction
Spin effect is also added to the PLD formula as a correction and it is assumed factorized [20][21][22][23]   n  is the spin cut off parameter.It is important to mention that Equation ( 26) used for pre-compound nucleus and it does not use for compound nucleus [22]. 2 0.16

Surface Correction
The initial interaction between a projectile and target nucleon is frequently localized near the nucleus surface.
Since the nuclear density variant with nuclear radius, hence the nuclear potential is shallower than in the interior; therefore, the local well depth (i.e.near the surface) is less the central depth [9,11,24].This must add a considerable effect on PLD calculations especially in the exciton knockout and pickup nuclear reactions.
as the PLD of one-component Fermi gas system with exciton number excitation energy and nuclear potential well depth V , then one can write the following equation for PLD with surface effect where , and it is given by the simple one-component PLD Pauli formula (Equation ( 4)) The function 29) is the cor- rection due to surface effect in the ESM, and it is given by From Equation (29), the PLD formula was corrected to in (31) clude the finite well depth and then was extended to include correction due to surface effect.This was done in the initial particle-nucleus interaction by replacing the nuclear potential depth 0 V by 1 V .This means instead of putting V , ined as th erage effective well depth" he choice between is def e "av .T

Comprehensive Formula
includes all the pre-Comprehensive formula is a formula vious corrections, except the isospin correction because the reaction was assumed to be completely mixed [17,18], then the isospin correction became unity.The aim behind suggested this formula is an attempt to get on a formula describes the PE PLD by the most accurate description (real one).

Results and Discussion
be calculated experi-

Comparison between One and
In or rence between one and two-It is obvious that the PLD cannot mentally especially for excitation energy more than 15 MeV because the states overlapped with each other's.On the other hand, and in order to increasing knowledge about the nuclear force, which represents the strongest force comparing with others, one must increase the applied excitation energy to hundred and several hundred MeVs.Therefore, PLD must be given in theoretical form.The calculated Comprehensive formula (Equation (32)) is a suitable formula for PLD estimations.This claim can be examined if one applies the Comprehensive formula of the PLD in cross section equation of any (and for any nucleus) PE nuclear models and comparing with experimental results of this cross section.However, PLD formula with Pauli's correction was used in cross section calculations by many researchers [25][26][27] and these researchers avoided use all corrections of the PLD to ease the programming potential.Further, in these results, Pauli's correction was stand as a reference case for PLD comparisons with all their corrections.

Two-Component
der to study the diffe component formulae of the PLD, a comparison was made for 56 Fe isotope.All used parameter are listed in Table 1.
From Figures 1(a)-(c) one can see that the two-component results are less than those of one-component.This behavior is expected physically because the two-component system will have to share the energy with more entities (the entities are those due to particles and holes of the neutrons and protons).Although the neutron parti-cles and holes are considered zero, the two component results stay less than the one-component results.
From Figures 1(a)-(c) one can see that the two-component results are less than those of one-component.This behavior is expected physically because the two-component system will have to share the energy with more entities (the entities are those due to particles and holes The results of these two figures give me arrangements.As we mentioned above, if one used PLD with Pauli correction as reference, then it is easy to see that except surface correction, the all corrections and Comprehensive PLD formulae have values more than PLD with Pauli correction.However, one can expect that the use of PLD formula with Pauli correction alone in cross section calculations may deviate the theoretical estimations from experimental results.Therefore, these results strongly suggested that the Comprehensive PLD formulae must be used in any calculations need PLD.
Finally, the results of 90 Zr isotope are bigger than th and then the single particle level density g increases.On the other hand, the difference between the results decreases with increasing the excitation energy.

Conclusion
ents in nuclear reaction and   rvey for the PLD formulae with its corrections and

Figure 1 .e 2
Figure 1.(a) One and two-components Ericson's formulae of PLD for 56 Fe isotope; (b) One and tw -components Pauli's for-o mulae of PLD of 56 Fe isotope; (c) One and two-components pairing formulae of PLD for 56 Fe isotope.

able 2 .Figure 2 .
Figure 2. PLD of one-component with all corrections for 56 Fe isotope.

Figure 3 .
Figure 3. PLD of one-component with all corrections for 90 Ze isotope.