Commun. Theor. Phys. (Beijing, China) 46 (2006) pp. 514 520 c International Academic Publishers Vol. 46, No. 3, September 15, 2006 Nuclear Level Density with Non-zero Angular Momentum A.N. Behami, 1 M. Gholami, 2 M. Kildir, 2 and M. Soltani 3 1 Physics Department, Mahabad Islamic Azad University, Mahabad, Iran 2 Chemistry Department, Middle East Technical University, Anara, Turey 3 Physics Department, Shiraz University, Shiraz, Iran (Received September 5, 2005) Abstract The statistical properties of interacting fermions have been studied for various angular momentum with the inclusion of pairing interaction. The dependence of the critical temperature on angular momentum for several nuclei, have been studied. The yrast energy as a function of angular momentum for 28 Si and 24 Mg nuclei have been calculated up to 60.0 MeV of excitation energy. The computed limiting angular momenta are compared with the experimental results for 26 Al produced by 12 C + 14 N reaction. The relevant nuclear level densities for non-zero angular momentum have been computed for 44 Ti and 136 Ba nuclei. The results are compared with their corresponding values obtained from the approximate formulas. PACS numbers: 21.10.Ma Key words: nuclear structure, statistical properties of paired nucleons 1 Introduction Superconductivity theory for the description of residual interaction between nucleons is well established. [1,2] Such a theory has been applied both to the ground state and excited nuclei. In the later case a complete thermodynamical description is possible. In particular general expressions for the nuclear level density which tae explicit account of the angular momentum are studied. The standard procedure used in the above mentioned statistical calculations consists of writing down the grand partition function of the system, restricted in such a way to preserve the number of particles and total energy. [3] Here, we attempt to study the statistical properties of superconducting nucleus, with the further restriction of a fixed angular momentum projection M on laboratory fixed Z-axis. In the first part, the mathematical technique consisting of the statistical calculations will be discussed, while in the second part some actual calculations will be presented and at the end the exact nuclear level density for non-zero angular momentum will be compared with the corresponding results obtained from the approximate formulas. 2 Discussion of Mathematical Technique For a system of interacting fermions, for example that of neutrons, the logarithm of the grand partition function is given by [2,4] Ω = β (ɛ λ E ) + ln[1 + exp{ β(e γm )}] β 2 G, (1) where λ, β, γ are the three Lagrange multipliers associated with the nucleon number, energy and angular momentum and E = [(ɛ λ) 2 + 2 ] 1/2 is the quasiparticle energy while is the gap parameter. The quantities, λ, β (= 1/T, where T is the nuclear temperature), and γ are connected through the following gap equation, 2 G = 1 [ 1 } 1 }] tanh{ 2E 2 (E γm ) + tanh{ 2 (E + γm ). (2) The saddle point is defined by the following equations, [2,5,11] N = 1 Ω β λ = [ 1 ɛ λ { 1 1 }] tanh 2E 2 (E γm )} + tanh{ 2 (E + γm ), (3) M = 1 Ω β γ = [ 1 m 1 + exp β(e γm ) 1 ], (4) 1 + exp β(e + γm ) E = Ω β = [ ɛ 1 ɛ λ { 1 1 }] tanh 2E 2 (E γm )} + tanh{ 2 (E + γm ) 2 G, (5) where N represents the nucleon number, M the projection of the angular momentum on a spaced fixed axis and E the energy of the system. The system of Eqs. (2) (4) defines the values of, λ, and γ.
No. 3 Nuclear Level Density with Non-zero Angular Momentum 515 The above theory can be extended to include systems containing both neutrons and protons. For a system of N neutrons and Z protons, let the energy levels be represented by ɛ n and ɛ p and the magnetic quantum numbers be m n and m p. The constants of motions are then the neutron and proton numbers N and Z, the total energy E and the projection of the total angular momentum on a space fixed axis, M, [1,6] N = m n ɛ n, Z = m p ɛ p, (6) E = E n + E p, (7) M = M n + M p. (8) The above formalism allows one to calculate the state density for a nucleus specified by N neutrons and Z protons, total energy E, and angular momentum projection M, [2,7] ω(e, N, Z, M) = ω(e, N, Z) exp( M 2 /2σ 2 ) 2πσ 2, (9) where ω(e, N, Z) is obtained from a previous formalism by eliminating the Lagrange multiplier λ. The quantity σ 2, i.e. the spin cut off factor, which determines the widths of M distribution, is given by the expression [2,8] σ 2 = 1 4 + 1 4 m 2 n sech 2{ 1 2 (βɛ n α N ) } m 2 p sech 2{ 1 } 2 (βɛ p α Z ) (10) with α = λβ. As suggested by Bethe, the dependence of level density upon the total angular momentum I is given by [2,9] ρ(e, N, Z, I) = ω(e, N, Z, M = I) ω(e, N, Z, M = I + 1), (11) or approximated by [10] [ d ρ(e, N, Z, I) = dm ]M=I+1/2 ω(e, N, Z, M) 2I + 1 [ 2 2πσ ω(e, N, Z) exp (I + ] 1/2)2 3 2σ 2. (12) The evaluation of level density within the present formalism is dependent upon the solution of the saddle point equations, which yield the quantities α N, α Z, and β. Since in general the saddle point equations are nonlinear, they must be solved numerically. The procedure of solving them is outlined in Sec. 3. 3 Numerical Calculation 3.1 The T M Surface Dependence of the gap parameter on temperature T at zero angular momentum is obtained by solving Eqs. (2) and (3) and setting γ = 0. For details see our previous publications. [11] It is found that the gap parameter decreases by increasing temperature and vanishes at a critical temperature T c. The dependence of T c on angular momentum M is obtained by solving Eqs. (2) (4) and setting = 0. It turns out that T c decreases as the increasing of angular momentum M and for the same values of gap parameter it becomes double valued at M = M c (see Ref. [11]). This is due to the fact that an increase of T instead of breaing pairs actually spreads out the quasiparticles, thus diminishing the effects of blocing. The dependence of upon M at zero temperature is obtained by solving Eqs. (3) and (4) and setting T = 0. For details see Ref. [11]. The dependence of upon T and M for 28 Si, 44 Ti, 68 Zn, 96 Mo, 124 Te, 152 Pt, and 200 Hg nuclei are shown as T M surface in Figs. 1 7. The double valued at M = M c for the some values of the gap parameter is evident from these figures. Fig. 1 Relation between the pairing gap, nuclear temperature T, and angular momentum M for 28 Si. Note the double-valued nature of the critical temperature for values of angular momentum near the critical angular momentum.
516 A.N. Behami, M. Gholami, M. Kildir, and M. Soltani Vol. 46 Fig. 2 The same as Fig. 1 but for 44 Ti. Fig. 3 The same as Fig. 1 but for 68 Zn. Fig. 4 The same as Fig. 1 but for 96 Mo.
No. 3 Nuclear Level Density with Non-zero Angular Momentum 517 Fig. 5 The same as Fig. 1 but for 124 Te. Fig. 6 The same as Fig. 1 but for 192 Pt. Fig. 7 The same as Fig. 1 but for 200 Hg.
518 A.N. Behami, M. Gholami, M. Kildir, and M. Soltani Vol. 46 3.2 Limiting Angular Momentum The limiting angular momentum, which gives the energy of the system at T = 0, is obtained by solving Eqs. (2) (6) (for details of calculations see Ref. [11]). The limiting angular momentum has been computed for 28 Si for M up to 30 units of angular momentum and 60 MeV of excitation energy. The theoretical calculations were performed with the single-particle levels by Seeger as well as Nilsson. The results are shown in Fig. 8. The limiting angular momentum has also been calculated for 24 Mg using Nilsson single-particle levels for spherical, oblate and prolate deformation and are compared with the experimental values for 26 Al, an odd-odd system produced by the 12 C + 14 N reaction. [12] These results are shown in Fig. 9. There is good agreement between experimental and the theoretical results if single-particle levels of the prolate deformation is used. It is seen from Fig. 9 that for a fixed excitation energy the oblate deformation holds more angular momentum. This can be explained from the occupational propabilities for the various single-particle levels. The neutron single particle levels are excited almost equally. However, there is a large contribution from the 1h 9/2 proton single-particle levels. This can be seen from Fig. 10, where the portion of the level sequences used together with their occupation probabilities are shown. The numbers under the levels represent the occupational probabilities. Fig. 8 The yrast line for 28 Si with Seeger spherical levels and with Nilsson levels with oblate and prolate. Fig. 9 Comparison of the experimental limiting angular momentum for 24 Mg with the theory using Nilsson energy levels. Fig. 10 Energies of the single particle levels from Nilsson potential with their corresponding occupational probabilities for 24 Mg at fixed excitation energy, E = 43.0 MeV.
No. 3 Nuclear Level Density with Non-zero Angular Momentum 519 3.3 Nuclear State Densities To calculate the state density ω(e, N, M n ) at non-zero angular momentum, the values of (T ), λ(t ), and γ(t ) are first evaluated for specified neutron and proton number N and Z-projection of angular momentum, M n. The total energy E n is thus evaluated from Eq. (6). The excitation energy U n = En M En, 0 where En 0 is the yrast energy at angular momentum, and M n is obtained as described above. This procedure is repeated for the second type of particles, namely the proton system. The state density for a nuclear system of N neutrons and Z protons at excitation energy U = U n + U p is obtained from Eq. (9). The results for 44 Ti nucleus at various M values are shown in Fig. 11. It is clear from this figure that for a fixed excitation energy the state density decreases with increasing angular momentum. Fig. 11 The state density as a function of excitation energy for 44 Ti and for the various angular momenta. 3.4 Nuclear Level Densities Nuclear level densities have been evaluated from Eq. (11), by nowing values of the state densities obtained in Subsec. 3.3. The results for 44 Ti at a fixed excitation energy of 12.00 MeV and various angular momentum are shown in Fig. 12. The corresponding levels per MeV deduced from the approximate formula (12) is also plotted for comparison. Examination of this figure reveals that at smaller angular momenta, nuclear level density deduced from the exact formula is smaller than their corresponding values obtained from the approximate formula. However, at higher angular momenta the density of levels becomes almost identical. Similar results are obtained for 136 Ba as shown in Fig. 13. Fig. 12 The spin-dependent level density as a function of angular momentum at fixed excitation energy for 44 Ti, E = 12.0 MeV. Fig. 13 The spin-dependent level density as a function of angular momentum for 136 Ba at fixed excitation energy E = 14.0 MeV.
520 A.N. Behami, M. Gholami, M. Kildir, and M. Soltani Vol. 46 4 Results and Discussion One of the features of the level density which tae explicit account of the angular momentum is the dependence of the pairing gap parameter on both the nuclear temperature (1/β) and the angular momentum M. The transition from the paired to unpaired regions can be made either by increasing temperature at constant angular momentum or by increasing angular momentum at a constant temperature. The relation between, T, and M are illustrated for neutron and proton systems separately for various nuclei in Figs. 1 7. One interesting feature shown in these figures is the double valued nature of the critical temperature for various angular momenta near the critical angular momentum. This effect has been explained in terms of blocing. [13,14] At large M and for T = 0 the effect of blocing is a maximum because all the quasiparticles are tightly paced around the Fermi surface. An initial increase of the temperature spreads out the quasiparticles (instead of breaing pairs) and thus diminishes the effect of blocing. The yrast line has been calculated for 24 Mg and 28 Si nuclei excited up to an energy of 65 MeV. The dependence of the results on the nuclear pairing and nuclear deformation has been investigated with Nillson single particle levels. It is shown that for a fixed value of the angular momentum, the yrast energy increases as pairing energy and decreases as the nuclear deformation goes from prolate to oblate shape. The theoretical results for 24 Mg are compared with experimental limiting angular momenta for the compound nucleus 26 Al produced by the 12 C + 14 N reaction. Good agreement between experimental and theoretical results is obtained if the single particle levels of the prolate deformation is used. The dependence of the state density for 44 Ti on excitation energy up to 20 MeV and various angular momenta is investigated and the results are plotted in Fig. 14. It is found that the state density at a fixed excitation energy decreases with increasing angular momentum and this change is more pronounced at higher angular momenta. Finally, nuclear level density is studied in more detail. The spin-dependent level density at fixed excitation energy (E = 12.0 MeV) for 44 Ti is plotted in Fig. 12. These results obtained from the exact formula (11) are compared with their corresponding values obtained from the approximate formula (12). It is seen that at lower angular momentum the density of levels is lower than their corresponding values obtained from the approximate expressions and they become almost identical at higher angular momenta. This behavior is consistent with the prediction of the Fermi gas model. Fig. 14 The spin-dependent level density as a function of angular momentum at various excitation energies. Spin-dependent level densities at excitation energies of 12.0, 14.0, 16.0, 18.0, 20.0 MeV for 44 Ti are calculated, and the results are plotted in Fig. 14. These plots have almost the same trend except that at a fixed spin J, the number of levels increases exponentially as expected from the Bethe theory. [9] References [1] A. Bohr and B.R. Mottolson, Nuclear Structure, Vol. 1, Benjamin, New Yor (1969) p. 170. [2] J.R. Huizenga and L.G. Moretto, Annu. Rev. Nucl. Sci. 22 (1972) 427. [3] A.N. Behami and S.I. Najafi, J. Phys. G: Nucl. and Part. Phys. 6 (1980) 685. [4] S. Bjornholm, A. Bohr, and B.R. Mottelson, Proceedings of the Third International Symposium on the Physics and Chemistry of Fission, Rochester, 1973, Vol. 1, IAEA, Viana (1974) p. 367. [5] A.N. Behami and Z. Kargar, Physica Scripta 66 (2002) 22. [6] A.N. Behami, Z. Kargar, and M.N. Nasrabadi, Phys. Rev. C 66 (2002) 064307. [7] A.N. Behami and M.N. Nasrabadi, Commun. Theor. Phys. (Beijing, China) 37 (2002) 457. [8] M.G. Mostafa, A.V. Blann, A.V. Ignatyu, and M. Grimes, Phys. Rev. C 45 (1992) 1078. [9] H.A. Bethe, Phys. Rev. 50 (1936) 232. [10] Po-Lin Huang, S.M. Grimes, and T.N. Massey, Phys. Rev. C 62 (2000) 024002. [11] A.N. Behami and Z. Kargar, Commun. Theor. Phys. (Beijing, China) 36 (2001) 305. [12] Bull. Int. Conf. on Nuclear Spectroscopy and Nuclear Structure, ISBN586-763-024-2 (2000). [13] A.N. Behami, Commun. Theor. Phys. (Beijing, China) 39 (2003) 689. [14] J.R. Huizenga, A.N. Behami, I.M. Grove, W.U. Schroder, and J. Tor, Phys. Rev. C 40 (1998) 668.