Spin Cut-off Parameter of Nuclear Level Density and Effective Moment of Inertia

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1 Commun. Theor. Phys. (Beijing, China) 43 (005) pp c International Academic Publishers Vol. 43, No. 4, April 15, 005 Spin Cut-off Parameter of Nuclear Level Density and Effective Moment of Inertia A.N. Behkami and M. Soltani Physics Department, Shiraz University, Shiraz 71454, Iran (Received September, 004) Abstract The spin cut-off parameter of the nuclear level density and effective moment of inertia for a large number of nuclei have been determined from analysis of the experimental data on S-wave neutron resonances and spins of lowlying levels. Contrary to claims made before, it is shown the spin cut-off parameter differs considerably from their corresponding rigid body values, and the energy dependence of the effective moment of inertia confirms the interacting fermion model prediction. PACS numbers: 1.10.Ma Key words: spin cut-off factor, effective moment of inertia 1 Introduction Information on the nuclear level density, its energy, and spin dependence is very important for both the description of excited nucleus properties and the nuclear reaction cross section calculations within the frame work of statistical model of nuclear reactions. The purpose of this article is to test statistical theories at low excitation energies below 10 MeV and to deduce the relevant parameters appearing in the level density formula. A large number of nuclei from 0 F to 50 Cf have been investigated. The selected nuclei have rather extensive and complete level schemes. In most cases at least the first fifty levels are known with spin and parity assignment. Neutron resonance densities are available for most of these nuclei. The above experimental information has been applied to determine A-dependent spin cut-off parameter and their related moment of inertia. Statistical Formulas The dependence of the nuclear level density ρ, on angular momentum J, can be written as [1] ρ(u, J) = J + 1 [ J(J + 1) ] σ exp σ ρ(u), (1) where ρ(u) is the level density and is given by [ 4] ρ(u) = exp[ a(u E 1 )] 1, () σa 1/4 (U E 1 ) 5/4 where a is the so-called level density parameter in MeV 1 and E 1, (ground state back shift) is fit parameter to experimental data. σ is the spin cut-off factor describing the width of the spin distribution, and U is the excitation energy. According to Ericson [5] σ = g m T = I eff T h, (3) where g is the density of single particle states, T is nuclear temperature, and m is the average of square of the spin projection for single particle states near the Fermi level. The value m g is called the effective moment of inertia. In the model of non-interacting fermions, it is an energy independent and equal to a rigid sphere value with mass and radius of the nucleus. The knowledge of the nuclear level density at neutron binding energy B n and the average S-wave neutron spacing D 1/ + allow to determine the spin cut-off parameter σ = ρ(b n ) D 1/ +, (4) and according to Eq. (3) the effective moment of inertia from the known values of σ and T. 3 Results and Discussions It is an established fact that the level densities near the ground state and near the neutron binding energy are well reproduced by the Bethe formula if two parameters are fitted. The level densities for all nuclei listed in are computed using Eq. (1) with the best fit values of a and E 1 taken from the work of Ignatyuk. [6] Examples of level densities are shown for Na, 36 Cl, 41 Ca, and 161 Dy nuclei in Fig. 1. Note that the total number of levels is plotted versus excitation energy for these nuclei. The spin cut-off parameter for nuclei under investigation is computed using Eq. (4) with the values of D 1/ + taken from Ref. [6] and values of ρ(b n ) calculated as described above. The calculated values of the spin cut-off parameter is listed in. The spin cut-off factor has also been computed from a rigid body assumption using relation (5) with σ rigid = I rigid h T (5) I rigid h = A 5/3 MeV 1. (6)

2 710 A.N. Behkami and M. Soltani Vol. 43 The equation of state relating the excitation energy U and level nuclear temperature T is U = at T. (7) The computed values of the spin cut-off parameter using the solid sphere approximation are also listed in for comparison. Fig. 1 Total number of levels, N(E) plotted versus excitation energy for Na, 36 Cl, 41 Ca, and 161 Dy nuclei. The fitted curves are calculated with the Bethe formula. Fig. Plot of back shift energy E 1 as a function of mass number A. An even-odd straggling is evident. Fig. 3 The level density parameter a plotted versus mass number A. The shell effect is evident at mass number 90, 140, and 06.

3 No. 4 Spin Cut-off Parameter of Nuclear Level Density and Effective Moment of Inertia 711 Level density parameter and deduced spin cut-off factors. Z A Element B n a (MeV 1 ) E 1 (MeV) ρ (B n) σtheory σrigid 9 0 F Na Mg Mg Mg Al Si Si Si P S S S Cl Cl Ar K K Ca Ca Ca Ca Sc Ti Ti Ti Ti Ti V V Cr Cr Cr Cr Mn Fe Fe Fe Fe Co Ni Ni Ni Ni Ni Ni Cu Cu Zn Zn Zn Zn Zn

4 71 A.N. Behkami and M. Soltani Vol. 43 (continued) Z A Element B n a (MeV 1 ) E 1 (MeV) ρ (B n) σtheory σrigid Ga Ga Ge Ge Ge Ge Ge As Se Se Se Se Se Se Br Br Kr Kr Kr Kr Rb Rb Sr Sr Sr Sr Y Zr Zr Zr Zr Zr Zr Nb Mo Mo Mo Mo Mo Mo Mo Tc Ru Ru Ru Ru Rh Pd Pd Pd Pd Pd Pd

5 No. 4 Spin Cut-off Parameter of Nuclear Level Density and Effective Moment of Inertia 713 (continued) Z A Element B n a (MeV 1 ) E 1 (MeV) ρ (B n) σtheory σrigid Ag Ag Cd Cd Cd Cd Cd Cd Cd Cd In In Sn Sn Sn Sn Sn Sn Sn Sn Sn Sb Sb Te Te Te Te Te Te Te I I Xe Xe Xe Xe Xe Xe Cs Cs Cs Ba Ba Ba Ba Ba Ba Ba La La Ce Ce Ce

6 714 A.N. Behkami and M. Soltani Vol. 43 (continued) Z A Element B n a (MeV 1 ) E 1 (MeV) ρ (B n) σtheory σrigid Ce Pr Nd Nd Nd Nd Nd Nd Nd Nd Pm Sm Sm Sm Sm Sm Sm Sm Sm Eu Eu Eu Eu Eu Gd Gd Gd Gd Gd Gd Gd Tb Dy Dy Dy Dy Dy Dy Dy Ho Er Er Er Er Er Er Tm Tm Yb Yb Yb Yb Yb

7 No. 4 Spin Cut-off Parameter of Nuclear Level Density and Effective Moment of Inertia 715 (continued) Z A Element B n a (MeV 1 ) E 1 (MeV) ρ (B n) σtheory σrigid Yb Yb Yb Lu Lu Hf Hf Hf Hf Hf Hf Ta Ta Ta W W W W W Re Re Os Os Os Os Os Os Ir Ir Ir Pt Pt Pt Pt Pt Au Hg Hg Hg Hg Tl Tl Pb Pb Pb Pb Bi Ra Th Th Th Th Pa

8 716 A.N. Behkami and M. Soltani Vol. 43 (continued) Z A Element B n a (MeV 1 ) E 1 (MeV) ρ (B n) σtheory σrigid Pa U U U U U U U Np Np Pu Pu Pu Pu Pu Pu Am Am Am Cm Cm Cm Cm Cm Cm Cm Bk Cf Cf Cf The back shifted parameter E 1 of the Bethe formula versus mass number A for nuclei listed in is plotted in Fig.. The data are not corrected for the pairing energy. It is seen that E 1 is not a smooth function of mass number. It fluctuates around the average value of E 1 = 1.0 MeV. The values of the level density parameter versus mass number is plotted in Fig. 3. It is seen that the a-value increases almost smoothly with mass A. However, they change markedly for nuclei near the major nuclear shells. The values of spin cut-off parameter σ determined above are plotted in Fig. 4. The rigid body values of spin cut-off parameter deduced from Eqs. (5) and (6) are also plotted for comparison. It is clear from this figure that the rigid body values of spin cut-off parameter differs substantially as compared to their theoretical values. Examination of this figure also shows a smooth increase of σ with A as expected on the basis of macroscopic theory, and the gross features of the data due to nuclear shells are apparent. Fig. 4 Plot of the spin cut-off factor σ versus mass number A. The rigid body values are also plotted for comparison. It is interesting to compute the spin cut-off factor from

9 No. 4 Spin Cut-off Parameter of Nuclear Level Density and Effective Moment of Inertia 717 the experimental spin distribution of well known low lying states and to compare it with the results obtained from the model calculations. Fig. 5 Theoretical spin distribution as compared with their corresponding values for Na, 36 Cl, 41 Ca, and 161 Dy nuclei. The experimental values are shown as a histogram while the theoretical distributions are shown as a solid curve. Calculation predicts that the spin distribution can be described by [7,8] f(j) = exp ( J ) (J + ) 1) σ exp ( σ. (8) It is difficult to determine the spin cut-off parameter σ experimentally. Our previous publications [4,9,10] made the first attempt to obtain σ near the ground state by fitting f(j) to the experimental spin distribution in some nuclide. We have applied this fitting procedure to a large number of nuclei with J χ (n(j) F = f(j)), (9) n(j) J 1 Fig. 6 Ratio of I eff /I rigid is plotted as a function of excitation energy for Na, 36 Cl, 41 Ca, and 161 Dy nuclei, showing the energy dependence of the effective moment of inertia. J J F = 1 n(j) J J 1 f(j), (10) where n(j) is the number of levels with spin J, which has the spin window J 1 and J. Example of such calculations for the case of Na, 36 Cl, 41 Ca, and 161 Dy nuclei

10 718 A.N. Behkami and M. Soltani Vol. 43 are plotted in Fig. 5. The histogram shows experimental spin distribution and the solid curve represents the theoretical spin distribution. The preliminary results show a close agreement with their corresponding values obtained from the model calculations. However, they again differ from their values predicted by rigid body assumption. Finally, The ratio of I eff /I rigid is computed using Eqs. (3) and (5) for Na, 36 Cl, 41 Ca, and 161 Dy nuclei from the known values of the spin cut-off factor listed in. The results are plotted in Fig. 6. The energy dependence of the effective moment of inertia confirm the existence of nucleonic interaction between fermions. Acknowledgments We are greatly indebted to Prof. Wang Shu-Nuan for providing us with the data information. References [1] H.A. Bethe, Rev. Mod. Phys. 9 (1937) 69. [] A. Gilbert and A.G.W. Cameron, J. Phys. 43 (1965) [3] W. Dilg, et al., Nucl. Phys. A17 (1973) 69. [4] T.V. Egidy, A.N. Behkami, and H.H. Schmidt, Nucl. Phys. A454 (1986) 109. [5] T. Ericson, Adv. Phys. 9 (1960) 45. [6] A.V. Ignatyuk, [7] L. Henden, M. Guttormsen, J. Rekstad, and T.S. Tveter, Nucl. Phys. A589 (1995) 49. [8] T.S. Tveter, et al., Phys. Rev. Lett. 77 (1996) 404. [9] T.V. Egidy, H.H. Schmidt, and A.N. Behkami, Nucl. Phys. A481 (1978) 189. [10] A.N. Behkami and M. Soltani, Am. Phys. Soc. Nov. 48 (003) 96.

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