Commun. Theor. Phys. (Beijing, China) 43 (005) pp. 709 718 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 = 0.0138A 5/3 MeV 1. (6)
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.
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 6.601 3.117 4.780 1.85 10 14.816 3.304 11 4 Na 6.959 3.44 4.867 3.508 10 16.661 4.35 1 5 Mg 7.330 3.69 4.573 5.031 10 118.35 4.618 1 6 Mg 11.093 4.503 0.409 1.077 10 3 3.15 5.304 1 7 Mg 6.446 4.647 0.836.039 10 6.51 4.37 13 8 Al 7.75 3.933.85 3.350 10 7.536 5.467 14 9 Si 8.474 4.08 0.577 1.49 10 14.94 5.95 14 30 Si 10.610 3.31 3.888 7.604 10 133.069 7.77 14 31 Si 6.587 4.590 0.395 9.609 10 1 15.854 5.538 15 33 P 7.937 4.777 0.634.34 10 8.376 6.550 16 33 S 8.641 4.4 1.6 5.986 10 44.897 7.101 16 34 S 11.416 4.657 0.10 1.88 10 3 5.406 8.57 16 35 S 6.985 5.045 0.080.99 10.986 6.615 17 36 Cl 8.580 4.381 1.878 7.6 10 8.351 8.3 17 38 Cl 6.108 5.949 0.907 1.614 10 1.10 6.55 18 41 Ar 6.098 7.100 0.539 1.306 10 3 45.843 6.76 19 40 K 7.799 5.146.474 1.796 10 3 13.471 8.600 19 4 K 7.533 5.347 4.971 7.791 10 3 58.435 8.993 0 41 Ca 8.36 6.165 0.37.30 10 3 37.51 8.400 0 43 Ca 7.93 6.999 1.774 1.15 10 4 11.500 8.91 0 44 Ca 11.13 7.357 0.073 4.637 10 4 41.737 9.838 0 45 Ca 7.414 7.97 1.09 6.19 10 3 73.859 8.476 1 46 Sc 8.760 6.834.847 3.187 10 4 0.716 9.84 47 Ti 8.877 6.38 3.18.59 10 4 8.38 10.694 48 Ti 11.67 6.9 1.31 7.65 10 4 66.959 11.988 49 Ti 8.14 7.170 0.563 6.355 10 3 58.150 10.301 50 Ti 10.939 6.691 0.990 8.938 10 3 17.877 1.694 51 Ti 6.371 6.643 1.899 1.393 10 8.709 10.35 3 51 V 11.051 6.654 1.537 4.505 10 4 51.808 13. 3 5 V 7.310 6.876 1.530 5.058 10 3 10.370 11.061 4 51 Cr 9.61 6.794.47.754 10 4 183.15 1.035 4 53 Cr 7.939 6.61 1.179 4.639 10 3 100.660 1.107 4 54 Cr 9.719 7.046 0.619 1.691 10 4 65.949 13.8 4 55 Cr 6.46 7.717 0.98 3.164 10 3 98.093 10.61 5 56 Mn 7.70 7.538.851.481 10 4 8.56 11.884 6 55 Fe 9.97 6.636 1.377 1.304 10 4 117.390 13.846 6 57 Fe 7.646 7.507.33.007 10 4 54.91 1.559 6 58 Fe 10.044 7.609 1.016 5.115 10 4 166.50 14.590 6 59 Fe 6.581 8.586 1.189 1.101 10 4 139.838 11.546 7 60 Co 7.491 8.095.01 3.188 10 4 19.96 13.016 8 59 Ni 8.999 7.08.000 3.08 10 4 06.493 14.670 8 60 Ni 11.388 7.9 0.853 7.657 10 4 76.568 16.753 8 61 Ni 7.80 8.135 1.313.154 10 4 148.633 13.616 8 6 Ni 10.597 7.818 0.573 6.837 10 4 71.790 16.485 8 63 Ni 6.838 9.004 0.465 1.07 10 4 8.179 1.784 8 65 Ni 6.097 9.736 0.0 5.870 10 3 57.531 1.44 9 64 Cu 7.916 8.38.615 7.889 10 4 37.475 14.60 9 66 Cu 7.065 8.839 1.704 3.087 10 4 0.064 14.167 30 65 Zn 7.979 9.599 1.66 1.71 10 5 8.073 13.998 30 67 Zn 7.05 10.433 1.539 1.091 10 5 5.00 13.9 30 68 Zn 10.198 9.737 0.19.739 10 5 54.786 16.8 30 69 Zn 6.48 11.049 0.971 5.947 10 4 165.33 13.016 30 71 Zn 5.833 11.814 0.057 1.656 10 4 59.60 1.537
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 31 70 Ga 7.654 9.843 1.97 1.57 10 5 6.719 15.36 31 7 Ga 6.51 11.055.488.67 10 5 50.766 14.008 3 71 Ge 7.416 1.30 1.757 8.406 10 5 374.071 13.787 3 73 Ge 6.783 1.561 1.950 6.990 10 5 54.84 13.650 3 74 Ge 10.199 1.038 0.93.478 10 6 76.87 17.33 3 75 Ge 6.506 1.087 1.556.400 10 5 360.049 14.87 3 77 Ge 6.073 1.30 1.1 1.11 10 5 50.81 14.317 33 76 As 7.36 1.169.58 1.51 10 6 58.16 15.394 34 75 Se 8.07 1.751 1.908.690 10 6 457.369 15.344 34 77 Se 7.418 1.847 1.619 1.176 10 6 38.137 15.38 34 78 Se 10.498 11.981 0.347 3.198 10 6 175.886 19.3 34 79 Se 6.963 1.059 1.96 5.36 10 5 53.644 16.107 34 81 Se 6.701 1.539 0.473 1.67 10 5 16.658 16.155 34 83 Se 5.818 1.815 1.146 1.06 10 5 301.44 15.560 35 80 Br 7.89 1.17.39.195 10 6 51.576 17.335 35 8 Br 7.59 11.683.161 8.816 10 5 46.85 18.156 36 79 Kr 8.369 1.79 1.753 3.85 10 6 410.608 17.039 36 81 Kr 7.874 13.417 1.57.646 10 6 370.378 16.830 36 84 Kr 10.519 10.55 0.134 4.191 10 5 41.914 3.68 36 85 Kr 7.118 1.96 1.19 5.453 10 5 7.671 17.70 37 86 Rb 8.651 9.896 1.910 3.405 10 5 8.941.819 37 88 Rb 6.079 10.47 1.443 3.13 10 4 8.103 19.715 38 85 Sr 8.59 1.435 1.83.968 10 6 474.840 19.714 38 87 Sr 8.48 10.10 1.081 1.74 10 5 4.085.708 38 88 Sr 11.11 9.31 1.005 1.38 10 5 19.60 7.553 38 89 Sr 6.366 9.876 0.060 7.978 10 3 94.54 0.935 39 90 Y 6.856 9.354 0.818 1.710 10 4 31.639.79 40 91 Zr 7.194 10.66 0.198 3.673 10 4 110.00.135 40 9 Zr 8.635 11.460 0.318 1.770 10 5 48.686 3.617 40 93 Zr 6.734 1.039 0.166 6.65 10 4 109.640 0.87 40 94 Zr 8.19 13.870 0.664 4.740 10 5 37.9 1.634 40 95 Zr 6.46 1.667 0.87 4.186 10 4 66.98 0.603 40 97 Zr 5.579 1.196 0.763 5.899 10 3 38.346 0.307 41 94 Nb 7.9 1.351 1.51 5.483 10 5 1.93 1.631 4 93 Mo 8.067 10.610 0.639 1.65 10 5 170.809 4.46 4 95 Mo 7.367 1.467 0.875 3.584 10 5 36.59.105 4 96 Mo 9.154 1.947 0.044 1.395 10 6 73.17 4.453 4 97 Mo 6.81 13.733 1.054 6.16 10 5 31.634 0.97 4 98 Mo 8.64 14.63 0.137.459 10 6 9.1 3.407 4 99 Mo 5.95 15.731 1.60 1.036 10 6 517.850 18.897 4 101 Mo 5.398 17.68 1.44.098 10 6 839.391 17.607 43 100 Tc 6.764 15.877 1.608 4.889 10 6 9.33 0.365 44 100 Ru 9.673 14.13 0.18 6.980 10 6 87.49 5.67 44 10 Ru 9.0 15.351 0.179 7.69 10 6 65.419 4.836 44 103 Ru 6.3 16.133 1.730 3.506 10 6 964.5 0.404 44 105 Ru 5.910 18.097 1.305 4.591 10 6 688.619 19.34 45 104 Rh 6.999 15.6.008 8.464 10 6 135.47.85 46 105 Pd 7.094 15.66 1.347 4.495 10 6 539.368.758 46 106 Pd 9.56 15.596 0.131 1.335 10 7 68.75 6.76 46 107 Pd 6.538 16.7 1.07 3.75 10 6 44.11 1.88 46 108 Pd 9.0 16.18 0.338 1.039 10 7 57.151 6.65 46 109 Pd 6.153 18.445 1.340 8.184 10 6 744.774 0.776 46 111 Pd 5.756 0.064 0.939 6.630 10 6 497.86 19.85
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 47 108 Ag 7.69 15.84 1.804 1.044 10 7 114.809 4.001 47 110 Ag 6.809 17.87 1.541 1.199 10 7 90.51.903 48 107 Cd 7.96 15.035 0.97 4.589 10 6 309.743 5.96 48 109 Cd 7.35 16.37 0.717 4.455 10 6 67.3 4.031 48 111 Cd 6.974 16.755 1.110 6.00 10 6 466.57 3.906 48 11 Cd 9.398 16.331 0.100.430 10 7 43.043 8.365 48 113 Cd 6.540 17.468 0.919 4.35 10 6 40.86 3.370 48 114 Cd 9.04 16.699 0.8 1.317 10 7 163.7 8.347 48 115 Cd 6.141 18.174 0.606.430 10 6 85.546.873 48 117 Cd 5.767 18.347 0.957.585 10 6 503.995.733 49 114 In 7.74 15.140 0.968.401 10 6 15.606 6.888 49 116 In 6.784 16.541 1.099 4.005 10 6 19.03 5.56 50 113 Sn 7.743 15.77 0.63.005 10 6 157.49 7.169 50 115 Sn 7.545 14.803 0.48 6.173 10 5 88.80 8.091 50 117 Sn 6.944 15.545 0.055 6.87 10 5 130.560 7.087 50 118 Sn 9.36 15.10 0.684 3.669 10 6 100.886 3.110 50 119 Sn 6.485 16.19 0.176 6.588 10 5 158.106 6.400 50 10 Sn 9.106 14.804 0.703.40 10 6 100.804 3.988 50 11 Sn 6.171 15.179 0.059 1.717 10 5 10. 7.47 50 13 Sn 5.946 15.738 0.564 4.048 10 5 83.386 7.17 50 15 Sn 5.73 14.141 0.461 1.001 10 5 50.1 9.03 51 1 Sb 6.807 16.47 1.408 5.679 10 6 36.915 7.909 51 14 Sb 6.467 16.017 1.641 3.644 10 6 43.730 8.399 5 13 Te 6.935 17.394 0.77 5.44 10 6 359.199 7.74 5 14 Te 9.45 16.681 0.185.40 10 7 190.367 33.86 5 15 Te 6.570 17.690 0.745 3.875 10 6 368.144 7.59 5 16 Te 9.114 16.099 0.35 9.636 10 6 183.078 34.67 5 17 Te 6.90 16.604 0.768 1.403 10 6 385.757 8.61 5 19 Te 6.085 16.638 0.87 1.47 10 6 461.441 8.886 5 131 Te 5.99 15.849 0.544 4.038 10 5 30.847 30.09 53 18 I 6.85 16.385 1.704 7.70 10 6 57.898 30.356 53 130 I 6.461 15.837 1.550.8 10 6 4.336 30.895 54 19 Xe 6.909 16.553 1.07 4.164 10 6 50.446 30.768 54 130 Xe 9.54 15.906 0.337 8.689 10 6 165.08 36.593 54 131 Xe 6.618 17.390 0.865 3.963 10 6 455.777 30.138 54 13 Xe 8.935 15.198 0.398 3.356 10 6 8.15 37.797 54 133 Xe 6.440 15.884 0.701 9.788 10 5 367.033 31.993 54 135 Xe 6.383 14.834 0.063.071 10 5 165.657 33.855 55 134 Cs 6.891 15.659.041 7.330 10 6 76.961 33.706 55 135 Cs 8.87 14.505 0.46 3.659 10 6 9.73 39.973 55 136 Cs 6.89 14.54.011.903 10 6 74.06 35.76 56 131 Ba 7.494 17.671 0.996 1.740 10 7 504.734 31.714 56 133 Ba 7.190 17.349 1.039 1.007 10 7 553.674 3.195 56 135 Ba 6.973 15.96 0.878.434 10 6 451.489 34.017 56 136 Ba 9.107 15.43 0.145 5.580 10 6 111.594 40.07 56 137 Ba 6.900 14.31 0.011.565 10 5 155.154 36.686 56 138 Ba 8.611 13.473 0.775 4.34 10 5 56.14 4.588 56 139 Ba 4.73 14.591 0.194.99 10 4 14.984 31.101 57 139 La 8.778 13.637 0.193 1.675 10 6 6.795 43.6 57 140 La 5.160 15.655 1.367 3.769 10 5 41.463 31.616 58 137 Ce 7.481 18.063 0.764 1.60 10 7 405.101 33.755 58 141 Ce 5.48 16.054 0.1 8.807 10 4 136.515 3.336 58 14 Ce 7.169 17.830 0.884 9.630 10 5 31.97 35.349
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 58 143 Ce 5.145 18.917 0.087 3.451 10 5 189.793 9.606 59 14 Pr 5.843 15.630 0.869 4.70 10 5 5.86 34.36 60 143 Nd 6.13 16.601 0.64.618 10 5 11.575 34.439 60 144 Nd 7.817 17.93 0.58.797 10 6 48.939 38.318 60 145 Nd 5.755 18.791 0.339 1.7 10 6 76.11 3.069 60 146 Nd 7.564 19.19 0.186 9.914 10 6 84.65 36.557 60 147 Nd 5.9 1.49 0.59 3.155 10 6 457.485 9.559 60 148 Nd 7.333 3.3 0.54 6.034 10 7 105.598 33.359 60 149 Nd 5.038 3.708 1.01 1.391 10 7 1077.759 7.887 60 151 Nd 5.334.386 0.981 1.14 10 7 97.511 30.193 61 148 Pm 5.893 0.503 0.973 9.94 10 6 5.803 3.06 6 145 Sm 6.76 15.637 0.0 3.848 10 5 18.893 38.17 6 148 Sm 8.141 19.799 0.10 3.071 10 7 78.30 38.115 6 149 Sm 5.871 1.436 0.731 1.094 10 7 546.757 31.6 6 150 Sm 7.985 1.88 0.066 9.983 10 7 104.84 36.709 6 151 Sm 5.598 4.109 1.46 8.679 10 7 1996.138 9.75 6 15 Sm 8.57.66 0.18 3.193 10 8 166.059 37.404 6 153 Sm 5.867 3.03 1.417 7.416 10 7 1779.959 31.84 6 155 Sm 5.807 1.53 0.85 1.36 10 7 704.457 33.55 63 15 Eu 6.307 3.34 1.863 3.40 10 8 14.184 3.374 63 153 Eu 8.550 0.355 1.475 6.978 10 8 195.377 40.657 63 154 Eu 6.441.4 1.858.193 10 8 10.65 34.59 63 155 Eu 8.15 0.487 0.874.060 10 8 94.741 40.469 63 156 Eu 6.338 0.175 1.30.510 10 7 53.964 36.547 64 153 Gd 6.46 4.195 1.374.357 10 8 1649.79 31.96 64 155 Gd 6.435 3.507 1.36.009 10 8 1456.813 33.63 64 156 Gd 8.536 1.64 0.73 3.147 10 8 67.56 40.648 64 157 Gd 6.359.37 1.40 8.315 10 7 147.38 35.055 64 158 Gd 7.938 1.413 0.376 1.359 10 8 333.076 40.316 64 159 Gd 5.94 1.811 1.048.394 10 7 981.75 35.11 64 161 Gd 5.635 1.107 0.753 6.398 10 6 639.817 35.568 65 160 Tb 6.375 0.803 1.36 3.94 10 7 8.413 37.6 66 157 Dy 6.969 3.950 1.11 4.869 10 8 1168.580 35.355 66 159 Dy 6.83 1.553 1.11 9.995 10 7 1099.400 37.78 66 161 Dy 6.453.357 1.93 1.00 10 8 1377.64 36.87 66 16 Dy 8.196 1.417 0.455.17 10 8 60.690 4.679 66 163 Dy 6.71 1.375 1.85 4.387 10 7 1359.968 37.960 66 164 Dy 7.658 1.065 0.07 4.77 10 7 160.731 4.56 66 165 Dy 5.716 1.495 1.017 1.339 10 7 1004.008 36.953 67 166 Ho 6.43 0.366 1.9.366 10 7 49.676 40.046 68 163 Er 6.90 3.310 1.178.880 10 8 1151.901 37.99 68 165 Er 6.649.305 1.58 1.50 10 8 131.9 38.966 68 167 Er 6.436 1.877 1.081 5.454 10 7 1036.319 39.538 68 168 Er 7.771 1.303 0.397 1.013 10 8 1.783 44.3 68 169 Er 6.003 1.44 0.880 1.613 10 7 806.394 39.437 68 171 Er 5.681 1.781 0.834 1.091 10 7 80.178 38.835 69 170 Tm 6.593 0.419 1.343 4.88 10 7 18.56 4.710 69 171 Tm 7.486 1.169 0.708 9.651 10 7 188.194 44.984 70 169 Yb 6.867 3.539 0.995.31 10 8 98.586 40.049 70 170 Yb 8.470 1.15 0.319.163 10 8 173.04 47.3 70 171 Yb 6.615 1.676 0.946 5.160 10 7 851.457 41.874 70 17 Yb 8.019.11 0.5 1.993 10 8 577.965 45.793 70 173 Yb 6.367 1.049 0.888.87 10 7 800.499 4.566
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 70 174 Yb 7.464 1.474 0.043 3.714 10 7 144.865 45.899 70 175 Yb 5.8 1.186 0.698 7.983 10 6 646.653 41.43 70 177 Yb 5.566 1.838 0.705 7.508 10 6 694.484 40.678 71 176 Lu 6.94 1.045 1.41 3.434 10 7 51.503 43.566 71 177 Lu 7.071 1.303 0.796 6.457 10 7 88.789 46.07 7 175 Hf 6.708.610 0.951 1.003 10 8 90.916 4.860 7 177 Hf 6.381.703 0.895 5.840 10 7 875.95 4.554 7 178 Hf 7.65.868 0.10 1.505 10 8 180.611 46.615 7 179 Hf 6.100.408 0.899 3.19 10 7 917.399 4.73 7 180 Hf 7.387.176 0.98 8.101 10 7 186.314 47.55 7 181 Hf 5.696.849 0.1 6.846 10 6 31.747 41.69 73 181 Ta 7.576 3.176 0.07 1.46 10 8 74.756 47.453 73 18 Ta 6.063 1.11 1.198.485 10 7 5.18 45.068 73 183 Ta 6.934 1.774 0.43 3.978 10 7 69.616 47.843 74 181 W 6.680.610 0.79 6.769 10 7 676.888 45.46 74 183 W 6.190.087 0.770.531 10 7 759.46 44.975 74 184 W 7.411.817 0.116 9.046 10 7 54.73 48.650 74 185 W 5.755 3.34 0.804.56 10 7 884.6 4.989 74 187 W 5.466 4.069 0.749.041 10 7 867.33 4.006 75 186 Re 6.178 1.901 1.151 4.014 10 7 6. 46.37 75 188 Re 5.871.333 1.78 3.844 10 7 78.804 45.60 76 187 Os 6.9 3.54 1.003 7.953 10 7 1153.9 45.748 76 188 Os 7.989 3.153 0.068 1.976 10 8 395.01 51.886 76 189 Os 5.9 3.613 0.965 4.98 10 7 1170.733 44.874 76 190 Os 7.79.840 0.008 1.34 10 8 8.138 5.547 76 191 Os 5.758 3.47 1.117 4.41 10 7 1547.39 45.45 76 193 Os 5.584 3.074 0.778 1.584 10 7 910.877 45.77 77 19 Ir 6.199 3.015 1.64 9.006 10 7 11.576 47.71 77 193 Ir 7.77.183 0.905 3.38 10 8 118.36 54.693 77 194 Ir 6.066 1.56 1.354 3.758 10 7 131.541 49.713 78 193 Pt 6.55 4.046 1.11 1.367 10 8 1503.178 47.54 78 195 Pt 6.105 0.65 1.6 1.574 10 7 1574.03 51.950 78 196 Pt 7.91 0.71 0.000 4.79 10 7 431.49 58.673 78 197 Pt 5.846 0.03 1.043 7.150 10 6 151.180 5.086 78 199 Pt 5.571 0.999 0.416.973 10 6 505.384 50.494 79 198 Au 6.51 19.033 1.439 1.793 10 7 147.894 56.788 80 199 Hg 6.664 19.934 0.97 1.873 10 7 983.178 56.50 80 00 Hg 8.08 17.999 0.001 1.033 10 7 413.037 65.716 80 01 Hg 6.30 17.898 0.08 1.110 10 6 360.73 58.876 80 0 Hg 7.753 17.361 0.4 6.55 10 6 94.85 66.956 81 04 Tl 6.656 15.509 1.84 1.784 10 6 49.756 67.134 81 06 Tl 6.503 11.719 0.937 6.784 10 4 186.566 78.16 8 05 Pb 6.73 15.033 0.709 7.068 10 5 706.86 69.17 8 07 Pb 6.738 11.154 0.971 7.418 10 3 118.683 8.303 8 08 Pb 7.367 10.331 1.535 4.53 10 3 86.115 90.109 8 09 Pb 3.937 1.359 0.13 1.167 10 3 33.47 61.586 83 10 Bi 4.604 13.318 1.685 5.64 10 4 11.475 64.166 88 7 Ra 4.561 3.061 0.837 1.95 10 8 007.691 45.83 90 9 Th 5.39 3.769 0.990 9.857 10 8 464.194 49.136 90 30 Th 6.794 9.743 0.85 1.33 10 9 41.861 58.984 90 31 Th 5.117 31.84 0.796 3.46 10 8 1661.759 50.047 90 33 Th 4.786 3.47 0.810.69 10 8 1883.559 48.689 91 3 Pa 5.569 9.719 0.606.383 10 8 53.65 54.401
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 91 34 Pa 5. 30.69 0.643 1.949 10 8 58.470 5.617 9 33 U 5.743 30.33 0.73 5.481 10 8 160.604 55.0 9 34 U 6.845 9.814 0.61 1.439 10 9 395.61 60.848 9 35 U 5.97 30.3 0.933 3.439 10 8 063.155 53.694 9 36 U 6.545 31.35 0.83 1.78 10 9 371.503 58.845 9 37 U 5.15 30.736 0.880.6 10 8 1966.649 53.5 9 38 U 6.153 31.317 0.357 9.159 10 8 160.776 57.961 9 39 U 4.806 31.800 0.80 1.864 10 8 1938.874 51.418 93 38 Np 5.488 8.883 1.198 4.75 10 8 11.84 57.10 93 39 Np 6.17 30.05 1.066.375 10 9 486.91 59.768 94 39 Pu 5.647 9.313 0.886 3.908 10 8 1758.40 57.960 94 40 Pu 6.533 30.498 0.434 1.516 10 9 1667.649 61.334 94 41 Pu 5.41 30.646 1.018 4.156 10 8 576.915 55.406 94 4 Pu 6.310 31.867 0.404 1.79 10 9 631.194 59.780 94 43 Pu 5.033 31.64 0.900 3.89 10 8 19.896 54.184 94 45 Pu 4.698 3.846 0.63 1.451 10 8 1378.061 5.11 95 4 Am 5.539 8.714 1.343 5.69 10 8 163.50 59.64 95 43 Am 6.367 8.378 1.068 1.8 10 9 56.377 64.197 95 44 Am 5.363 9.005 1.15 3.669 10 8 133.97 58.849 96 43 Cm 5.694 8.133 0.395 9.867 10 7 690.705 61.114 96 44 Cm 6.799 9.393 0.078 7.645 10 8 86.705 65.51 96 45 Cm 5.5 9.380 0.978 3.754 10 8 14.677 59.688 96 46 Cm 6.455 9.0 0.05 4.706 10 8 305.903 64.966 96 47 Cm 5.157 9.000 0.85 1.171 10 8 1756.651 58.947 96 48 Cm 6.11 9.937 0.058.456 10 8 171.93 63.831 96 49 Cm 4.713 31.758 0.645 1.03 10 8 1431.888 54.577 97 50 Bk 4.96 9.50 1.041 1.53 10 8 83.777 58.500 98 50 Cf 6.61 9.566 0.058 4.569 10 8 159.914 67.147 98 51 Cf 5.109 31.643 0.748.784 10 8 1670.13 57.60 98 53 Cf 4.805 31.354 0.748 1.310 10 8 1768.940 56.946 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
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
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) 1446. [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, http://161.5.7.109/ripl/densities.html. [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.