# Nuclear Level Density Parameters ofPb203-209 and  Bi206-210 Deformed Target Isotopes Used on Accelerator-Driven Systems in Collective Excitation Modes

Authors: Şeref Okuducu; Nisa  N. Aktı; Sabahattin Akbaş; M. Orhan 	Kansu
Journal: Science and Technology of Nuclear Installations
 (2012)

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## Abstract

The nuclear level density parameters of some deformed isotopes of target nuclei (Pb, Bi) used on the accelerator-driven subcritical systems (ADSs) have been calculated taking into consideration different collective excitation modes of observed nuclear spectra near the neutron binding energy. The method used in the present work assumes equidistant spacing of the collective coupled state bands of the considered isotopes. The present calculated results for different collective excitation bands have been compared with the compiled values from the literature for s-wave neutron resonance data, and good agreement was found.


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## Body

## 1. Introduction

The knowledge of nuclear level densities is a crucial input in various fields/applications such as the creation of consistent theoretical description of excited nucleus properties and the nuclear reaction cross-section calculations for many branches of nuclear physics, nuclear astrophysics, nuclear medicine, and applied areas (medical physics, etc.) [1–14]. The neutron capture cross-sections, required for both design and nuclear model calculations in nuclear science and technologies, are approximately proportional to the corresponding level densities around the neutron resonance region. In nuclear medicine, the cross-section data obtained from nuclear level density approaches are needed to optimize production of radioactive isotopes for therapeutic purposes, for example, biomedical applications such as production of medical radioisotopes and cancer therapy and accelerator-driven incineration/transmutation of the long-lived radioactive nuclear wastes.Recently, the accelerator-driven systems (ADSs), which are used for production of neutrons in spallation neutron source and can act as an intense neutron source in accelerator-driven subcritical reactors, and their neutronics have been studied by many researchers [15–20]. Most of the studies were concerned with specific design concepts and the production of neutrons from spallation reactions. New accelerator-driven technologies make use of spallation neutrons produced in (p,xn) and (n,xn) nuclear reactions on high-Z targets. Through (p,xn) and (n,xn) nuclear reactions, neutrons are produced and moderated by heavy water in the target region and light water in the blanket region. These moderated neutrons are subsequently captured on 3-He, which flows through the blanket system, to produce tritium via the (n,p) reaction. Therefore, new nuclear cross-section data are needed to improve the theoretical predictions of neutron production, shielding requirements, activation, radiation heating, and material damage [21]. The design of ADS requires precise knowledge of nuclide production cross-sections in order to predict the amount of radioactive isotopes produced inside the spallation target (W, Pb, Bi, and other isotopes), may be liquid or solid [20–24]. However, the precision of models used to estimate production cross-sections is still far from the performance required for technical applications. An important applied field in the ADS systems, which is presently discussed, is the technical application of accelerator-driven subcritical reactors, of nuclear collision processes for the energy production and the transmutation of nuclear waste in hybrid reactor systems.Nuclear reactions calculations based on standard nuclear reaction models play an important role in determining the accuracy of various parameters of theoretical models and experimental measurements. Especially, the calculations of nuclear level density parameters for the isotopes can be helpful in the investigation of reaction cross-sections. In this manner, very sophisticated theoretical approaches have been developed to estimate total level densities of atomic nuclei, especially in the region of deformed heavy and light nuclei. However, in practice analytical expressions, which contain several parameters adjusted on scarce experimental data, are generally preferred [25].The analytical expressions used for the nuclear level density calculations [2, 3, 5] are based on the Fermi gas model. The most widely used description of the nuclear level density is the Bethe formula, based on the thermodynamic relation between entropy and the average energy of a system considered in the framework of noninteracting particles of the Fermi gas. The traditional Bethe theory of the nuclear level density calculation, which uses the assumption that the individual neutrons and protons occupy a set of low energy levels in the ground state and fill up the higher individual states at any excitation energy, has been successfully used so far, with different contributions made to this model in the form of shell, pairing, deformation effects [4, 26–29], finite size effects [30], and thermal and quantal effects [31], as well as improvements in the determination of the spin cut-off factors [32]. However, such contributions do not take into account the collective effects, which may play a basic role in describing the nuclear level density of some deformed nuclides.Calculations of all parameters of fission and fusion reactors, accelerator-driven systems, and other nuclear technology fields depend strongly on cross-section data. And, it is well known that nuclear level density parameters are of crucial importance for the cross-section. In light of the preceding knowledge in the present study, the nuclear level density parameters of deformedPb203–209 and Bi206–210 target isotopes used on the ADS systems have been calculated by using different collective excitation modes of the observed nuclear spectra near the neutron binding energy and a simple model introduced in our previous works [6–12], in which the collective character of the nuclear excitations is available.


## 2. Theoretical View Point of the Nuclear Level Density

The above-mentioned Bethe theory gives also the dependence of the nuclear level density on the total angular momentumJ of the nucleus. The expression used for the observable nuclear level density at any excitation energy U and momentum J can be written as [2, 3](1)ρ(U,I)=πexp⁡⁡(2aU)12a1/4U5/4(2I+1)exp⁡⁡[-(I+(1/2))2/2σ2]22πσ3,
where a and σ are the level density parameter and spin distribution parameter, respectively. The parameters a and σ are defined by(2)a=π26g(εF),(3)σ2=g(εF)〈m2〉t.
Here, the parameter g(εF) is the sum of the neutron and proton single-particle states density at the Fermi energy εF, 〈m2〉 is the mean square magnetic quantum number for single-particle states, and t is the nuclear thermodynamic temperature of an excited nucleus in the Fermi gas model. These factors are expressed as follows:(4)g(εF)=32AεF,〈m2〉=0,146A2/3,t2=Ua,
where A is the mass number of a nucleus.The experimental observations cannot determine the different orientation of nuclear angular momentumJ. Therefore, it is useful to obtain the observable level density, which has the form [3, 4](5)ρ(U)=∑Jρ(U,J)=π12exp⁡⁡(2aU)a1/4U5/412πσ.
Hence,   substitute (2)–(4) into (5) to find the observable level density as(6)ρ(U)=a122×0.298A1/3(aU)3/2exp⁡⁡(2aU).
The level density parameters of the Bethe theory have been well established in a number of studies [3, 33, 34] on the s-wave neutron resonance for different mass nuclei. However, this theory does not take into account the collective effects of the nuclear particles in the excitation of the nuclei. On the other hand, the measured magnetic and quadrupole moments of the nuclei deviate considerably from the ones calculated using the single-particle shell model in which the closed shells forming the nuclear core play no part. In other words, the excited states and the magnetic and quadrupole moments are the results of collective motion of many nucleons, not just of those nucleons that are outside the closed shell. The collective motion of the nucleons may be described as a vibrational motion about the equilibrium position and a rotational motion that maintains the deformed shape of the nucleus.


## 3. Collective Excitation Modes of Deformed Nuclei

The existence of collective energy level bands of rotational and vibrational types can now easily be identified from nuclear spectra data [35] of many deformed nuclei. In the studies in [34, 36] the contribution of collective motion of nucleons to the energy level density has been considered. However, these studies naturally involve messy equations and make the model complex for calculation of the nuclear level density parameters of deformed nuclei. A simpler description of collective model was first suggested by Rainwater [37] who made clear the relationship between the motion of individual nuclear particles and the collective nuclear deformation. Later a quantitative development of the nuclear collective model taking into consideration the collective motion of the nuclear particles was given by [30, 38–40]. Recently in considerable studies such as in [6–12], it has been attempted to identify the nuclear level density parameters in the region of some light and large deformed nuclei by the use of a simple model of nuclear collective excitation mechanism. Almost all data on the estimated level density parameters of these deformed nuclei are well identified on a base of collective rotational and collective vibrational bands such as ground state band, β band, octupole band, γ-band, and so forth.Some deformed isotopes used on ADS systems have also stable deformation in their ground states. Such isotopes studied may rotate due to interactions with an external incident particle or emitting the particle. Rotational energy of an axially symmetric deformed even-even nucleus is given as [30](7)Erot⁡(I,K)=ℏ22[I(I+1)J0+(1J3-1J0)K2],
where I and K are the total angular momentum and its projection on the axis of symmetry, respectively, of a nucleus and J3 and J0 are moments of inertia about a symmetry axis and an arbitrary axis perpendicular to the symmetry axis, respectively. The authors of [30] have used the hydrodynamic moments of inertia restricting the deformed nuclear surface by a quadrupole term only. In this model one can admit J3=0, which requires the value of K in (7) to be identically zero. Then, we come to the following rotational energy equation:(8)Erot⁡=ℏ22J0I(I+1),K=0.
The above expression is in good agreement with the observed low-lying energy levels of the even-even large deformed nuclei, which are the values of angular momentum I,I=0,2,4,6,…. As mentioned above the energy level sequence in such a case is called ground state rotational band having positive parity.In the following collective vibrational modes, we consider two modes, namely, the quadrupole and the octupole vibrational modes. The quadrupole mode, also calledβ-vibrational band, caries two units of angular momentum and even parity (0+, 2+, 4+, 6+,…) while the octupole vibrational band carries two units of angular momentum and negative parity (1-, 3-, 5-, 7-,…). Here, the β band is associated with vibrations that preserve the axis of symmetry and therefore is K=0 band with the level sequence given by (8) and the band head ℏωβ. Another excited band is often called the gamma band γ and is associated with the vibrations not preserving the symmetry axis and having the levels given by (7). The spin sequence of γ band with K=2 is I=2+,3+,4+,5+,….In such a case, in deformed odd-Aisotopes simple identification of the observed nuclear levels is made on a base of a strong coupling of a nucleon to an axially symmetric even-even deformed core. The rotational band with a given K value and spin sequence I=K=Ω,K+1,K+2,…, where K and Ω are the projections of the total angular momentum and odd nucleon angular momentum, respectively, on the nuclear symmetry axis, has level spacing [41](9)ΔE(I,K)=EIK-EKK=ℏ22J0[I(I+1)-K(K+1)].


## 4. Calculation Method of Nuclear Level Density Parameter

The method used in this present study for calculation of nuclear level density parameters of some deformed target isotopes has been given in detail in the studies of [6–12]. Similarly, the mentioned method, in this study, can also be applicable to deformed target isotopes of interest.In any case, the nuclear energy level density depending on the excitation energy,U, taking into account different excitation modes can be expressed in the following form:(10)ρ(U)=∑iaiρi(U),
where ρi(U) is the partial energy level density at the excitation U for the ith excitation mode and ai is the weighting coefficient satisfying the condition ∑iai=1. In the present work, we use a simple expression for the energy level density, which considered the collective excitation modes. Here, in our determination of the nuclear level density due to excitation bands the “equidistant” condition between energy levels, which is the important property of the observed energy spectrum of isotopes considered, should be satisfied. These properties can approximately be verified for the energies of the coupled state bands in deformed isotopes considered as being the ratios given by(11)R1:R2:R3:R4:⋯=1:r:2r:3r:⋯.
Here, R1,R2,R3,R4,… are the ratios of the sequential level energies to the appropriate energy unit of a corresponding band. When the above relation is satisfied, in our study, the nuclear level density formula introduced depending on the excitation energy U and energy unit εo for the ith excitation band can be represented as [6–12](12)ρi(U,εoi)≅π2aoi243(aoiU)3/2exp⁡⁡(2aoiU),
which are fairly simple and contain only one parameter aoi defined as(13)aoi=π26εoi
and represents a collective level density parameter corresponding to the ith band with the unit energy εoi. The unit energies are ε0GS=E(2+), ε0β=E(2+)-E(0+), and ε0oct=E(3-)-E(1-) for ground state, β, and octupole bands, respectively. Similarly, the other excitation bands can be included. In the even-even and odd-A isotopes it has been shown that the unit energy is either energy of the first excited state (for ground state bands) or the energy separation between the second and first excited states (for excited bands) of the corresponding band with the given projection of the total angular momentum K. For the applicability of (12) in our identification of the nuclear level density due to different excitation bands, the “equidistant” condition between energy levels should be satisfied. As mentioned before, these band energies clearly should, at least approximately, satisfy (11).Now, the observable level density expressions of (6) and (12) can be compared which have similar dependence on the energy, although they have been obtained from different approaches. Equation (6) obtained from the Bethe theory has been based on a single-particle nuclear model, whereas (12) has been extracted from the symmetry properties of the nuclear spectra data expressed by (11). In the previous works [7–12], our approach has been successfully used in the classification of the level density parameters for different light and large deformed nuclei.In the same way, in the present work, this approach takes into consideration the different collective excitation modes in deformed target isotopes that are interesting, and the nuclear level density parametersaoi defined by (13) can easily be obtained from nuclear spectra data given in [36] regarding nuclear level spectra of collective rotational and collective vibrational bands. The theoretical obtained values for Pb203–209 and Bi206–210 isotopes with their different corresponding bands have been listed in Tables 1 and 2.Table 1
The calculated and compiled values of the nuclear level density parameters for203–209Pb isotopes.
RadionuclidesGilbert and Cameron [3],Baba [33],Mughabgahab-Dunford [42],BSFG Egidy and Bucurescu [4],CalculatedCorresponding bandsa (MeV−1)a (MeV−1)a (MeV−1)a (MeV−1)a0 (MeV−1)Pb8220314.29———12.99Ground state band21.04Octupole vibration band27.39Octupole vibration band2.22Negative parity band1.96Positive parity band17.94Negative parity band7.30Negative parity bandPb8220412.91———1.83Ground state rotational band19.85Beta vibration band12.70Gamma vibration band2.33Gamma vibration band6.48Gamma vibration band7.13Beta vibration band9.33Negative parity band4.67Octupole vibration band20.57Positive parity band7.04Positive parity band9.54Positive parity band19.85Beta vibration band9.20Beta vibration bandPb8220511.18—9.6812.716.31Octupole vibration band8.51Octupole vibration band2.40Positive parity band8.87Negative parity band12.60Octupole vibration band15.21Octupole vibration band19.70Positive parity bandPb822069.32———2.05Ground state rotational band5.44Beta vibration band8.80Beta vibration band3.06Gamma vibration band2.25Gamma vibration band3.98Gamma vibration band12.00Gamma vibration band3.42Negative parity band9.20Negative parity band12.23Negative parity band20.64Positive parity bandPb82207—10.985.5711.082.88Neutron hole states and ground state2.10Single-neutron states G9/2, I11/2, J15/2, D5/2, S1/2,D3/220.93Positive parity band6.40Negative parity band10.27Positive parity band20.03Positive parity band13.04Positive parity band13.69Positive parity bandPb82208—10.284.1210.011.91Negative parity band6.91Positive parity band9.01Positive parity band10.00Positive parity band2.29Negative parity band14.41Octupole vibration band17.48Octupole vibration band15.02Octupole vibration band12.45Octupole vibration band12.26Octupole vibration band9.98Octupole vibration band8.47Octupole vibration band7.77Negative parity bandPb82209—11.724.1611.222.11Ground state band36.52Octupole vibration band9.69Octupole vibration band16.43Negative parity band11.41Negative parity band3.25Positive parity bandTable 2
The calculated and compiled values of the nuclear level density parameters for206–210Bi isotopes.
RadionuclidesGilbert and Cameron [3],Rohr [34],CalculatedCorresponding bandsa (MeV−1)a (MeV−1)a0 (MeV−1)Bi8320613.3924.6527.43Ground state band14.41Gamma vibration band12.68Positive parity band3.60Positive parity band11.64Positive parity band2.76Negative parity band10.96Negative parity band5.17Negative parity band5.83Positive parity bandBi8320711.5424.761.66Single-proton states: H9/2, F7/2, I13/2Conf = ((206PB 2+), (P, 1H9/2)) Conf = ((206PB 3+), (P, 1H9/2)) Conf = ((P,1H9/2 +2,0+), (P,NLJ,1,J+) (N,3P1/2, −2,0+))J + Conf = ((P,1H9/2) (N,1I13/2, −1) (N,NLJ, −1))Ground state bandNegative parity bandNegative parity bandNegative parity band22.5113.9810.391.832.456.2710.9826.00Bi832089.9224.863.22Positive parity band14.01Positive parity band25.96Ground state band15.16Positive parity band25.13Positive parity band2.57Positive parity band11.39Gamma vibration band3.58Gamma vibration band5.52Negative parity bandBi83209—24.976.77(208 PB 4−) (P, 1H9/2)20.93(208 PB 5−) (P, 1H9/2)1.83Single-proton states32.89(210 PO 0+) (P,NLJ,−1)14.84Positive parity band32.09Positive parity band21.60Positive parity band11.73Positive parity bandBi8321010.4025.082.03Octupole vibration band35.31Ground state band8.98Negative parity band10.74Positive parity band35.82Negative parity band17.80Negative parity band7.85Octupole vibration band


## 5. Results and Discussion

In the present paper, we have calculated the nuclear level density parameters of deformed target isotopesPb203–209 and Bi206–210 used on ADS by using different collective excitation modes of observed nuclear spectra. It has been seen that the nuclear energy levels of different collective excitation bands (in particular, the bands given in Tables 1 and 2) in the investigated isotopes also approximately satisfy (11). Thus, (13) can be applied for determination of the corresponding level density parameters. The calculated values of the level density parameters due to different excitation bands and the compiled values of those parameters have been represented in Tables 1 and 2 for the deformed target isotopes considered. The demonstrated values of the parameters a are given in Tables 1 and 2. Figure 1 was compiled by Gilbert and Cameron [3], Baba [33], BSFG model [4], and Mughabgahab and Dunford [42] and Figure 2 was compiled by Gilbert and Cameron [3] and Rohr [34] for s-wave neutron resonances near the neutron binding energy.Figure 1
Mass dependence of the calculated nuclear level density parametersa0 and those of the compiled values a for Pb203–209 deformed isotopes.Figure 2
Mass dependence of the calculated nuclear level density parametersa0 and those of the compiled values a for Bi206–210 deformed isotopes.In Figures1 and 2 we illustrate the comparison of the single-particle level density parameters a and the mass number with our calculated values of a0 corresponding to the different bands for Pb203–209 and Bi206–210 deformed isotopes, respectively. From Figures 1 and 2, it is clear that the present values of the level density parameters a0 calculated by (13) for these considered isotopes are well consistent with the compiled values of parameters a. As seen in Figure 1, the dominant bands in the population of Pb203–205 deformed isotopes generally seem to be the well-known collective bands (especially ground state, octupole, and gama vibrational bands; see also Table 1), as to Pb206–209 deformed isotopes it seems to be the mixed bands (negative and positive parity bands). As clear from Figure 2, we can say that the calculated values a0 for the mixed bands (negative and positive parity bands) of Bi206–210 are generally good dominant bands. The values of the calculated parameters of these bands are well consistent with those of the compiled data, in particular with the data of Gilbert and Cameron [3] for Bi206–210.


## 6. Conclusions

On the basis of the above presented discussion we can conclude that the nuclear level density parameters of deformed target isotopesPb203–209 and Bi206–210 used on the ADS can be identified by the use of collective vibrational bands taking into consideration the equidistant character of these bands including higher excitations. The nuclear energy level density at any excitation near the neutron binding energy may clearly have generally the same character such as collective rotational, collective vibrational, and intrinsic. Actually, as it has clearly been seen from Tables 1 and 2 and Figures 1 and 2, no dominant band alone is exactly responsible for identification of level density parameters a for the considered isotopes. Namely, the nuclear level density for such isotopes apparently should involve combination of partial level densities corresponding to the different bands, which is given by (10).Consequently, we remark that the nuclear collective excitation modes are quite meaningful in order to obtain the level density parameters of different isotopes. The calculation of these parameters based on the properties of the measured nuclear low-lying level spectra should prove a productive area of study that should override the inherent experimental difficulties involved. Hence, at least such parameters can be useful in the design of an ADS system, which requires precise knowledge of isotopes production cross-sections in order to predict the amount of radioactive isotopes produced inside the spallation targets such as Pb and Bi isotopes.



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