Study of Fission Barrier Heights of Uranium Isotopes by the Macroscopic-Microscopic Method

Commun. Theor. Phys. 62 (2014) 405 409 Vol. 62, No. 3, September 1, 2014 Study of Fission Barrier Heights of Uranium Isotopes by the Macroscopic-Microscopic Method ZHONG Chun-Lai ( Ë ) and FAN Tie-Shuan ( Ë) State Key Laboratory of Nuclear Physics and Nuclear Technology and School of Physics, Peking University, Beijing 100871, China (Received May 14, 2014; revised manuscript received June 3, 2014) Abstract Potential energy surfaces of uranium nuclei in the range of mass numbers 229 through 244 are investigated in the framework of the macroscopic-microscopic model and the heights of static fission barriers are obtained in terms of a double-humped structure. The macroscopic part of the nuclear energy is calculated according to Lublin Strasbourg-drop (LSD) model. Shell and pairing corrections as the microscopic part are calculated with a folded-yukawa single-particle potential. The calculation is carried out in a five-dimensional parameter space of the generalized Lawrence shapes. In order to extract saddle points on the potential energy surface, a new algorithm which can effectively find an optimal fission path leading from the ground state to the scission point is developed. The comparison of our results with available experimental data and others theoretical results confirms the reliability of our calculations. PACS numbers: 24.75.+i, 25.85.-w, 21.60.Cs Key words: potential energy surface, fission barrier, fission path, Uranium nuclei 1 Introduction In study of spontaneous and neutron-induced fission of heavy nuclei, the fission barrier height is a decisive quantity which determines the competition between fission and other decay channels like neutron evaporation. The large sensitivity of fission cross section on the barrier height stresses a need for accurate evaluation of this value. The macroscopic-microscopic method which has its origins in early works of Strutinsky [1 2] is a powerful tool used in large-scale calculation of nuclear potential energies. Because of its success in explaining nuclear fission properties quantitatively the macroscopic-microscopic method has been improved constantly and many extensive calculations of fission barriers have been carried out with it. A recent such investigation is presented by Möller et al., where they calculated the barrier heights for 1585 nuclei from Z = 78 to Z = 125. [3] There are two main issues related to the employment of the macroscopic-microscopic method. The first concerns the formulism used calculate both the macroscopic and microscopic parts of nuclear potential energy. The macroscopic model based on the concept of the charged liquid drop could be explained as the simplified form of leptodermous expansion [4] of the energy-density functional. Over the years various new terms have been proposed to improve its accuracy among which are the Myers Swiatecki liquid-drop (MS-LD) model, [5] the finite-range liquid-drop model [6] and the Lublin Strasbourg drop (LSD) model. [7] There is also a variety of microscopic models to choose from. The deformed Woods Saxon potential, [8] which was employed in Strutinsky s calculation, is defined so as to have a constant skin thickness about an effective surface defining the shape. The folded-yukawa potential [9] takes into account the effect of finite range nucleon forces. It is obtained by folding a Yukawa function with a square-well potential of the nuclear shape required and is characterized by a diffused surface. The second issue concerns the shape parameterization which should accurately describe the nuclear shape evolution all the way from the ground state to the scission configuration. The complexity of a parameterization involves a trade-off between accuracy of the model and computational feasibility. The increased computational cost brought by considering excessive degrees of freedom would effectively inhibit the usage of that parameterization while the extra benefit being trivial. Therefore a realistic parameterization should be both accurate and simple so as to reduce the computation to a reasonable size. Several popular parameterizations are available such as the Cassinian ovals, [10] the Funny Hills parameterization, [11] the matched quadratic surfaces, [12] and the generalized Lawrence shapes. [13] This work is initially motivated by the importance of fission barrier heights of main fissile and fertile nuclei in nuclear energy research and development. We systematically calculate the potential energy surfaces for uranium nuclei in mass range 229 A 244 in the framework of the LSD model and the folded-yukawa single-particle potential. The nuclear deformations are described by the Supported by National Natural Science Foundation of China under Grant Nos. 91226102 and 91126010, and the National Defence Foundation of China under Grant No. B0120110034 and the National Magnetic Confinement Fusion Science Program of China under Grant No. 2013BG106004 Corresponding author, E-mail: tsfan@pku.edu.cn c 2014 Chinese Physical Society and IOP Publishing Ltd http://www.iop.org/ej/journal/ctp http://ctp.itp.ac.cn

406 Communications in Theoretical Physics Vol. 62 generalized Lawrence shapes in a five-dimensional parameter space. In order to identify the saddle points corresponding to the fission barriers, we design and implement a new algorithm that can quickly find the lowest onedimensional trajectory on a multi-dimensional surface. 2 Calculation of Potential Energy Surfaces In the macroscopic-microscopic method the total nuclear energy E as a function of the shape is divided into a smoothly varying macroscopic part and a microscopic part representing the shell-plus-pairing correction E = E mac + δe mic. (1) As mentioned above, we adopted the LSD model for E mac and the folded Yukawa potential for δe mic. 2.1 Macroscopic Model The LSD model is an improved version of the LDMtype formula characterized by an explicit introduction of surface-curvature terms. As only the shape-dependent energy terms, i.e. the surface energy, the surface-curvature energy and the Coulomb energy, are considered, the LSD expression here is given by E mac = b surf (1 κ surf I 2 )A 2/3 B surf (def) + b cur (1 κ cur I 2 )A 1/3 B cur (def) + 3 e 2 Z 2 5 4πε 0 r0 ch B Coul(def), (2) A1/3 with the isospin asymmetry I = (N Z)/A and b surf = 16.9707 MeV, κ surf = 2.2938, b cur = 3.8602 MeV, κ cur = 2.3764, r 0 = 1.21725 fm. The functions B surf (def) is the dimensionless ratio of the surface energy of a deformed nucleus to that of the sphere, while B cur (def) and B Coul (def) are the similarly defined ratios of surface-curvature energy and Coulomb energy. 2.2 Microscopic Correction Method The Hamiltonian of a nucleon is defined by H = p2 2m + V + V SO + V C, (3) where p represents the vector operator of momentum and m is the mass of either a neutron or a proton. The spinindependent nuclear part of the potential is calculated in terms of the folded-yukawa potential V (r) = V 0 4πa 3 V e r r /a r r /a dr, (4) where V 0 is the well depth, and a is the range of the Yukawa folding function. The spin-orbit interaction is assumed in the form ( ) 2 σ V (r) p V SO (r) = λ, (5) 2mc where λ is the spin-orbit interaction strength, and σ the Pauli spin matrices. The Coulomb potential part is only valid for protons. Under the assumption of uniform nuclear charge distribution it is given by V C (r) = 3Ze2 dr 4πR0 3 V r r, (6) where R 0 is the radius of the spherical nucleus. A detailed description of the parameters used in Eqs. (4) (6) is given in Ref. [14]. Eigenvalues of Eq. (3) are evaluated by expanding the eigenfunctions in terms of harmonic-oscillator wave functions for a spheroid. [2,8] For a given nuclear surface shape, the frequencies defining the deformed harmonic oscillator, ω ζ and ω ρ, are determined by requiring the corresponding potential to simulate V best within the nucleus. [13] We choose the number of oscillator quanta N 0 as 15 so that we get the bounded levels with enough accuracy as well as a proper set of unbound levels that will be used later for shell correction. [9] The explicit formulae for the matrix elements of the Hamiltonian over the basis can be found in Refs. [2] and [8]. The microscopic energy δe mic is usually decomposed as δe mic = δe shell +δe pair, (7) where δe shell is the shell correction energy and δe pair is the energy correction due to the pairing correlations between nucleons. From the computed single-particle spectra, we evaluate δe shell according to Strutinsky s smoothing method of sixth order with the smearing width γ = 8 MeV. [9] The pairing correction energy δe pair is treated with BCS approximation. For the range of paring forces, we take into account 3N bound single-particle levels below and above the Fermi surface, with N the number of neutrons or protons, or use all bound levels above the Fermi surface and an equal number below it if there are not sufficient bound levels above the Fermi surface. The formulism for obtaining the shell-plus-pairing correction is given in Ref. [15] as smooth BCS pairing model (SBCS). 2.3 Shape Parameterization In cylindrical coordinates, the axially symmetrical generalized Lawrence shape (Fig. 1) is defined as a power series expansion of ρ 2 with respect to ζ: N ρ 2 (ζ) = (l 2 ζ 2 ) a n (ζ z) n, (8) n=0 where the semilength l measures the elongation of the nucleus and the position on the neck z measures the mass asymmetry. The upper limit N is set to 4 since five geometric constraints are sufficient for describing the problem. Along with l and z, it is convenient to introduce another three geometric parameters r, c, and s. The neck radius r as well as the elongation l coincides with the radius of spherical shaped nucleus R 0. The neck curvature is given by c = R 2 0/r cur, (9)

No. 3 Communications in Theoretical Physics 407 with r cur the curvature radius. The position of the nuclear center-of-mass s is another measurer of the mass asymmetry. The shape parameters and the coefficients a n are interconnected through five geometric constraints: the first derivative being zero at ζ = z, the definition of r, c, s, and the volume conservation condition. [13] 2.4 Numerical Details All calculated shapes have deformation parameters measured as multiples of R 0 within the following domain: l = 1.140 (0.035) 2.085, r = r 0 ( 0.02) r 0 0.38, z = 0.00 (0.02) 0.46, c = c 0 (0.09) c 0 + 1.53, s = 0.0000 ( 0.0175) 0.2975. Numbers in the parentheses specify the step with which the calculation is performed for a given variable. The starting values of r and s at a given l are r 0 = l 0.5 + 0.05, c 0 = l 2.5 0.5. Thus, we finally have altogether 4354560 different shapes for each nucleus. The choice of deformation coordinates would cover the physically relevant space available in this parameterization. The actual scale of calculation is decreased for two reasons. First, some shapes corresponding to certain deformation parameters do not exist in real world, so they are excluded from energy calculation. The number of such unphysical shapes may be reduced by a thorough choice of the domain of parameters (e.g. the shift in r 0 and c 0 as l changes) but they are somewhat unavoidable since we do not want to leave out any relevant points. Second, the macroscopic energy may vary widely while the microscopic correction energy is usually less than 10 MeV. Since the latter consumes much more computing time than the former does, a cutoff energy is set to filter out those unqualified shapes whose microscopic energies are too high. The final output contains results for only around 15 to 20 percent of the original deformations. 2.5 Search for Fission Barriers There is no straightforward way to reveal saddle points on a multidimensional potential energy surface. The immersion method introduced in Ref. [16] and later elaborated in Ref. [3] allows the unambiguous identification of the lowest saddle point between two local minima. A major drawback of this method is that it is rather timeconsuming as it iterates over the grid quite many times to locate a saddle point. Our idea is inspired by the geographic concept of drainage basin. Suppose a heavy rain pours over the entire investigated region the optimal path between two arbitrary minima can be found by tracking the surface runoffs. The algorithm implemented in our study turns out to be much faster than the immersion method. 3 Results and Discussion In Fig. 2, we illustrate the calculated optimal fission paths of 236 U and 239 U in terms of the potential energy as a function of the elongation parameter l. In order to study different theoretical models in the same deformation space, the LS-MD model and Woods-Saxon potential are also tested. Fig. 1 The geometrical meaning of the five degrees of freedom l, r, z, s, and c. As barriers of actinides are often characterized by the double-humped structure, we present in Table 1 our calculations in terms of the inner barrier height E A, the fissionisomeric second minimum energy E II and the outer barrier height E B. Table 1 Calculated double-humped fission-barrier energies for Uranium isotopes. N A E A /MeV E II /MeV E B /MeV 137 229 2.28 1.86 4.53 138 230 2.58 1.84 4.53 139 231 2.49 1.57 4.20 140 232 3.34 1.68 4.93 141 233 3.53 1.76 5.46 142 234 3.54 1.20 5.19 143 235 3.90 1.14 5.40 144 236 4.30 1.27 5.65 145 237 4.53 1.51 6.30 146 238 4.76 1.72 6.91 147 239 5.07 1.75 7.13 148 240 5.53 1.78 7.48 149 241 6.01 2.08 7.82 150 242 6.08 2.26 8.16 151 243 6.11 2.73 8.33 152 244 5.72 2.83 8.17 As seen in Fig. 3, our calculations agree with the experimental data (Maslov) well in the outer barrier as the

408 Communications in Theoretical Physics Vol. 62 largest discrepancy being 1.4 MeV, while the inner barrier height shows an underestimation of 1 2 MeV on the whole. We think this inconsistency is acceptable for two reasons. First, the experimental barrier height is deduced from modeling measured fission cross-section exciting functions and the double-humped structure is in fact a crude approximation to the complex potential energy surface, so reasonably a very accurate agreement is not our expectation. Second, the primary (highest) barrier is always the most accurately measure one and plays a dominant role in estimating fission cross sections and fission half-lives. Our calculated primary barrier is always the outer one, while for the experimental data it is the outer one for A 236 and the inner one for A > 236. Therefore the discrepancy between calculated values and experimental data drops in the A > 236 region if the primary barrier is compared other than the outer one. by the fact that we use exactly the same parameter set in the folded Yukawa potential as Möller. Koning s calculations were performed in the same shape parameterization as this work, with the MS-LD model and the Woods Saxon potential. A large discrepancy in outer barrier is obvious between their data and those of others. This could be explained by Fig. 2. The MS-LD model does not take care of the surface curvature effects and the Woods Saxon formula always generates a central potential. Both of them encounter difficulties at the outer barrier when a neck-in shape emerges and a twocenter potential is needed. Fig. 2 Optimal fission paths for 236 U (a) and 239 U (b) nuclei obtained by three different model combinations: LSD model plus folded Yukawa potential (thick solid line), LSD model plus Woods-Saxon potential (dotted line) and MS-LD model plus folded Yukawa potential and (dashed line). The potential energy surface is calculated in the 5D generalized Lawrence parameterization and the fission paths are projected to the 1D space of l. Möller s calculations, as mentioned before, give the inner barrier height close to ours at A 232. As for the outer barrier, the two curves show similar tendencies but the deviation increases steadily as A increases with the largest value being 1 MeV. The pattern of deviation suggests it results mainly from the difference in the macroscopic part of potential energy. This conclusion is justified Fig. 3 Comparison of our calculated barrier heights, E A (a) and E B (b), with experimental data and other theoretical efforts. The experimental data recommended by Maslov are taken from the Reference Input Parameter Library (RIPL-3). [17 18] Möller s data are taken from Ref. [3]. Koning s values are obtained from both Ref. [19] and the nuclear structure database of nuclear reaction code TALYS 1.4. [20] Mamdouh s ETFSI results are taken from Ref. [16]. Mamdouh s data were obtained with the ETFSI (extended Thomas Fermi plus Strutinsky integral) method applied in four dimensions. The height of outer barrier is less than all of others. This could be explained by the stronger surface curvature effect in microscopic methods, which tends to decrease the energy of the more deformed saddle points. 4 Conclusions Potential energy surfaces of uranium nuclei have been investigated in a five-dimensional deformation space of

No. 3 Communications in Theoretical Physics 409 the generalized Lawrence shapes. The macroscopicmicroscopic model employed in our calculations is based on the LSD model and the folded Yukawa single-particle potential. In order to resolve the complexity of searching for the optimal fission path on a multidimensional surface, we design and implement a new fast algorithm. The comparison of our results with available experimental data and others theoretical results confirms the reliability of our calculations. In the forthcoming study, we are going to extend our calculations to cover more actinide nuclei, and improve our search algorithm to reveal the multi-modal fission paths. References [1] V.M. Stutinsky, Nucl. Phys. A 95 (1967) 420. [2] M. Brack and H.C. Pauli, Nucl. Phys. A 207 (1973) 401. [3] P. Möller, A.J. Sierk, T. Ichikawa, et al., Phys. Rev. C 79 (2009) 064304. [4] W.D. Myers and W.J. Swiatecki, Ann. Phys. (N.Y.) 84 (1974) 186. [5] W.D. Myers and W.J. Swiatecki, Nucl. Phys. 81 (1966) 1. [6] P. Möller, W.D. Myers, W.J. Swiatecki, and J. Treiner, At. Data Nucl. Data Tab. 39 (1988) 225. [7] K. Pomorski and J. Dudek, Phys. Rev. C 67 (2003) 044316. [8] J. Damgaard, H.C. Pauli, V.V. Pashkevich, and V.M. Strutinsky, Nucl. Phys. A 135 (1969) 432. [9] M. Bolsterli, E.O. Fiset, J.R. Nix, and J.L. Norton, Phys. Rev. C 5 (1972) 1050 [10] V.V. Pashkevich, Nucl. Phys. A 169 (1971) 275 [11] M. Brack, J. Damgaard, A.S. Jensen, H.C. Pauli, V.M. Strutinsky, and C.Y. Wong, Rev. Mod. Phys. 44 (1972) 320. [12] J.R. Nix, Nucl. Phys. A 130 (1969) 241. [13] U. Brosa, S. Grossmann, and A. Müller, Phys. Rep. 197 (1990) 167. [14] P. Möller, J.R. Nix, W.D. Myers, and W.J. Swiatecki, At. Data Nucl. Data Tab. 59 (1995) 185. [15] H. Olofsson, R. Bengtsson, and P. Möller, Nucl. Phys. A 784 (2007) 104. [16] A. Mamdouh, J.M. Pearson, M. Rayet, and F. Tondeur, Nucl. Phys. A 644 (1998) 398. [17] R. Capote, et al., Nucl. Data. Sheets. 110 (2009) 3107. [18] https://www-nds.iaea.org/ripl-3/. [19] M.C. Duijvestijn, A.J. Koning, and F.J. Hambsch, Phys. Rev. C 64 (2001) 014607. [20] http://www.talys.eu/.