APPLICATIONS OF A HARD-CORE BOSE HUBBARD MODEL TO WELL-DEFORMED NUCLEI

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1 International Journal of Modern Physics B c World Scientific Publishing Company APPLICATIONS OF A HARD-CORE BOSE HUBBARD MODEL TO WELL-DEFORMED NUCLEI YUYAN CHEN, FENG PAN Liaoning Normal University, Department of Physics, Dalian 11609, P. R. China G. S. STOITCHEVA, and J. P. DRAAYER Department of Physics & Astronomy, Louisiana State University, Baton Rouge, LA 7080, USA Received 9 October 001 An exactly solvable hard-core Bose Hubbard model, which is equivalent to a meanfiled plus nearest-level pairing theory, for a description of well-deformed nuclei is used and applied to the actinide region. Binding energies and pairing excitation energies of 6 4 Th, 0 40 U, and 6 4 Pu isotopes are calculated and compared with the corresponding experimental values. 1. Introduction Pairing is an important residual interaction in nuclear physics. Typically, after adopting a mean-field approach, the pairing interaction is treated approximately using either Bardeen Cooper Schrieffer (BCS) or Hartree Fock Bogolyubov (HFB) methods, sometimes in conjunction with correction terms evaluated within the Random-Phase Approximation (RPA). However, both BCS and HFB approximations suffer from serious difficulties, the nonconservation of the number of particles being one that can lead to serious problems, such as spurious states, nonorthogonal solutions, etc. Another problem with these approximations is related to the fact that both BCS and the HFB methods break down for an important class of physical situations. A remedy in terms of particle number projection complicates the algorithms considerably, often without yielding a better description of higher-lying excited states that are a natural part of the spectrum of the pairing Hamiltonian. Over the past few years progress has been made in the development of better algorithms that bypass the Bogolyubov transformation and thus are free of problems related to particle number nonconservation. 1, In these approximation, either a configuration-energy truncation scheme or a many-body Fock-space basis cutoff was used, so the results were still not exact. Dedicated to Professor Wu F. Y. on his 70th Birthday Celebration 1

2 Y. Chen, F. Pan, G. S. Stoitcheva, and J. P. Draayer Exact solutions of the mean-filed plus pairing model were first studied for the equal strength pairing model. 5 Recently, generalizations that include state dependent pairing have been considered. 6 9 In these cases, the Bethe ansatz was used, from which excitation energies and the corresponding wavefunctions can be determined through a set of nonlinear equations. Unfortunately, solving these nonlinear equations is not practical when the number of levels and valence nucleon pairs are large, which is usually the case for well-deformed nuclei.. A Hard-core Bose-Hubbard Model for nuclei In Ref. 9, a hard-core Bose-Hubbard model was proposed, which is equivalent to a mean-field plus nearest-level pairing theory. As is well known, an equal strength pairing interaction, which is used in many applications, is not a particularly good approximation for well-deformed nuclei. In Ref., a level-dependent Gaussian-type pairing interaction with G ij = Ae B(εi εj) (1) was used, where i and j each represent doubly occupied levels with single-particle energies ɛ i and ɛ j. The parameters A<0 and B>0 are adjusted in such a way that the location of the first excited eigen-solution lies approximately at the same energy as for the constant pairing case. Of course, there is some freedom in adjusting the parameters, allowing one to control in a phenomenological way the interaction among the levels. Expression (1) implies that scattering between particle pairs occupying levels with single-particle energies that lie close are favored; scattering between particle pairs in levels with distant single-particle energies are unfavored. As an approximation, this pairing interaction was further simplified to nearest-level coupling in Ref. 9, namely, G ij is given by (1) if the levels i and j lie adjacent to one another in energy, with G ij taken to be 0 otherwise. Hence, the Hamiltonian can be expressed as Ĥ = i ε i i,j tij b i b j, () where the first sum runs over the orbits occupied by a single fermion which occurs in the description of odd-a nuclei or broken pair cases, and the second primed sum runs only over levels that are occupied by pairs of fermions. For the nearest-level pairing interaction case the t-matrix is given by t ii =ɛ i G ii =ɛ i A and t ii1 = t i1i = G ii1 with t ij =0otherwise. The fermion pair operators in this expression are given by b i = a i a ī, b i = a ī a i, ()

3 Applications of A Hard-core Bose Hubbard Model to Well-deformed Nuclei where a i is the i-th level single-fermion creation operator and a the corresponding ī time-reversed state. The b i and b i satisfy the following commutation relation: [b i,b j ]=δ ij (1 N i ), [N i,b j ]=δ ij b j, [N i,b j ]= δ ij b j (4) where N i = 1 (a i a i a ī a ī )isthe pair number operator in the i-th level for even-even nuclei. In this paper the Nilsson Hamiltonian is used to generate the mean-field. In this case there is at most one valence nucleon pair or a single valence nucleon in each level due to the Pauli principle. Equivalently, these pairs can be treated as bosons with projection onto the subspace with no doubly occupied levels. 9 The eigenstates of () for k-pair excitation can be expressed as k; ξ,(n j1,n j,,n jr )n f = C (ξ) i i 1< < 1 b i1 b i b ik (n j1,n j,,n jr )n f, (5) where j 1,j,,j r are the levels occupied by r single particles, the prime indicates that i 1,,, can not be taken to be j 1, j,, j r in the summation, and n f is the total numbers of single valence nucleons, that is n f = j n j. Since only even-even and odd-a nuclei are treated without including broken pair cases in this paper, r is taken to be 1 for odd-a nuclei, and 0 for even-even nuclei. In Eq. (5), C (ξ) i 1 is a determinant given by g ξ1 i i g ξ1 g ξ1 g ξ i i g ξ g ξ, (6) g ξ k i i g ξ k g ξ k where ξ is a shorthand notation for a selected set of k eigenvalues of the t matrix without the corresponding r rows and columns denoted as t, which can be used to distinguish the eigenstates with the same number of pairs, k, and g ξp is the p-th eigenvector of the t matrix. The excitation energies corresponding to (5) can be expressed as E (ξ) k = r ε ji i=1 k E (ξj), (7) where the first sum runs over r Nilsson levels each occupied by a single valence nucleon, which occurs in odd-a nuclei or in broken pair cases, the second one is a j=1

4 4 Y. Chen, F. Pan, G. S. Stoitcheva, and J. P. Draayer sum of k different eigenvalues of the t-matrix. Obviously, t is a (k r) (k r) matrix, since those orbits occupied by single valence nucleons are excluded resulting from the Pauli blocking. E (ξp) is the p-th eigenvalue of the t-matrix, that is Hence ξ t ij g p j = E (ξp) ξ g p i. (8) j Ĥ k; ξ,(n j1,n j,,n jr )n f = g (ξ P (1)) i 1 g (ξ P ()) g (ξ P (µ)) i µ i 1<< < µ=1 P k r ( ) P ( ε ji E (ξ P (µ)) ) i=1 g (ξ P (k)) b i1 b i b ik (n j1,n j,,n jr )n f = E (ξ) k k; ξ,(n j 1,n j n jk )n f, (9) where P runs over all permutations, E (ξµ) is the µ-th eigenvalue of the t matrix. Eq. (9) is valid for any k. Ifone assumes that the total number of orbits is N for even-even nuclei, the k-pair excitation energies are determined by the sum of k different eigenvalues chosen from the N eigenvalues of the t matrix with r =0,the total number of excited levels is N!/k!(N k)!. While for odd-a nuclei or broken pair cases, the levels that are occupied by the single valence nucleons should be excluded in the original t matrix. In the latter case, the eigenvalue problem (4) can be solved simply by diagonalizing the corresponding t matrix as shown in Eq. (9).. Applications to Actinide Isotopes In this section, we try to describe nuclei in the actinide region with the mean-field plus nearest-level pairing model using the axial-symmetric Nilsson potential as the mean-field. Other than what is manifest through the mean field, the quadrupolequadrupole interaction is not considered. In this case, exact solutions can be obtained by using the above simple method. As for the binding energy, the contributions from real quadrupole-quadrupole interaction is expected to be relatively small. 10 This conclusion applies to low-lying 0 excited states as well as ground states. As shown in Ref. 11, contributions from the pairing interaction is very important to the low-lying excited 0 states in these deformed regions. Hence, the position of low-lying 0 states is an estimate based on the Nilsson mean field plus pairing approximation. In this well-deformed region there are a lot of nuclei. The parameters were fixed by considering the 6 4 Th, 0 40 U, and 6 4 Pu isotopes. Specifically, the binding energies of these isotopes were calculated. Table 1 shows the binding energy results as well as pairing excitation energies of the theory for 6 4 Th, 0 40 U,

5 Applications of A Hard-core Bose Hubbard Model to Well-deformed Nuclei 5 and 6 4 Pu, with the corresponding experimental values taken from Ref. 1. The parameters A and B in Eq. (1) were fit as follows to maximize agreement with experiment: A = α 1 β 1 k γ 1 n f, B = α β k γ n f, (10) where α i, β i, and γ i are parameters that were fit for each isotope. Table 1. Calculated binding and pairing excitation energies are compared with the corresponding experimental values for various 6 4 Th, 0 40 U, and 6 4 Pu isotopes. B th (MeV) and B exp(mev) denote, respectively, the theoretical and experimental binding energies. 1 Spin Pairing excitation Pairing excitation Nucleus and B exp(mev) B th (MeV) Energies of Energies of Parity Exp. (MeV) Th. (MeV) 6 Th Th Th Th Th Th Th Th Th U U U U U

6 6 Y. Chen, F. Pan, G. S. Stoitcheva, and J. P. Draayer Table 1 (Continued) Spin Pairing excitation Pairing excitation Nucleus and B exp(mev) B th (MeV) Energies of Energies of Parity Exp. (MeV) Th. (MeV) U U U U U Pu Pu Pu Pu Pu Pu Pu Pu Acknowledgements This work was supported by the U.S. National Science Foundation through a regular grant ( ) and a Cooperative Agreement (97065) that includes matching from the Louisiana Board of Regents Support Fund, and by the Natural Science Foundation of China (Grant No ) as well as by the Science Foundation of

7 Applications of A Hard-core Bose Hubbard Model to Well-deformed Nuclei 7 the Liaoning Education Commission (Grant No ). References 1. J. Y. Zeng and T. S. Cheng,Nucl. Phys. A405, 1(1981). H. Molique and J. Dudek, Phys. Rev.C56, 1795(1997). R. W. Richardson, Phys. Lett. 5, 8(196) 4. R. W. Richardson, J. Math. Phys. 6, 104(1965) 5. R. W. Richardson and N. Shermen, Nucl. Phys. 5, 1(1964) 6. F. Pan and J. P. Draayer, Phys. Lett. B4, 7(1998) 7. F. Pan and J. P. Draayer, Ann. Phys. (NY) 71, 10(1999) 8. F. Pan, J. P. Draayer, and L. Guo, J. Phys. A, 1597(000) 9. F. Pan and J. P. Draayer, J. Phys. A, 9095(000) 10. S. G. Nilsson and O. Prior, Mat. Fys. Medd. Dan. Vid. Selsk., 1(1961) 11. P. E. Garret, J.Phys. G7, R1(001) 1. P. Mler, J. R. Nix, W. D. Myers, and W. J. Swiatecki, Atomic Data Nucl. Data Tables 59, (1995)