A method for solving the independent-particle Schrödinger equation with a deformed average field

Damgaard, J.; Pauli, Hans–Christian; Pashkevich, V. V.; Strutinsky, V. M.

doi:10.1016/0375-9474(69)90174-2

Cited by 163 publications

(72 citation statements)

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“…δ E shell was evaluated from the computed single-particle spectra according to Strutinsky's smoothing method of sixth order with the smearing width γ = 8 MeV [8]. The pairing correction energy δ E pair was treated with BCS approximation.…”

Section: Microscopic Correction Methodsmentioning

confidence: 99%

“…Over the years various important new models with some new corrections have been proposed to improve its accuracy, such as the Myers-Swiatecki liquid-drop (MS-LD) model [5], the finite-range liquid-drop model [6] and the Lublin-Strasbourg drop (LSD) model [7], and so on. The deformed Woods-Saxon [8] and the folded-Yukawa formula of single-particle potential [9] are widely used in the MM calculations. The parametrization of the nuclear shapes should accurately describe the shape evolution from the ground state to the scission configuration in the multidimensional deformation space.…”

Section: Introductionmentioning

confidence: 99%

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Study of five-dimensional potential-energy surfaces for actinide isotopes by the macroscopic-microscopic method

et al. 2017

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Abstract. In this work, the nuclear potential-energy of the deformed nuclei as a function of shape coordinates is calculated in a five-dimensional (5D) parameter space of the axially symmetric generalized Lawrence shapes, on the basis of the macroscopic-microscopic method. The liquid-drop part of the nuclear energy is calculated according to the Myers-Swiatecki model and the Lublin-Strasbourg-drop (LSD) formula. The Woods-Saxon and the folded-Yukawa potentials for deformed nuclei are used for the shell and pairing corrections of the Strutinsky-type. The pairing corrections are calculated at zero temperature, T, related to the excitation energy. The eigenvalues of Hamiltonians for protons and neutrons are found by expanding the eigen-functions in terms of harmonic-oscillator wave functions of a spheroid. Then the BCS pair is applied on the smeared-out single-particle spectrum. By comparing the results obtained by different models, the most favorable combination of the macroscopic-microscopic model is known as the LSD formula with the foldedYukawa potential.Potential-energy landscapes for actinide isotopes are investigated based on a grid of more than 4,000,000 deformation points and the heights of static fission barriers are obtained in terms of a double-humped structure on the full 5D parameter space. In order to locate the ground state shapes, saddle points, scission points and optimal fission path on the calculated 5D potential-energy surface, the falling rain algorithm and immersion method are designed and implemented. The comparison of our results with available experimental data and others' theoretical results confirms the reliability of our calculations.

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Section: Microscopic Correction Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Study of five-dimensional potential-energy surfaces for actinide isotopes by the macroscopic-microscopic method

et al. 2017

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“…by diagonalization of/?/in a deformed harmonic oscillator basis [6][7][8]. For each kind of nucleons the effective mass m* (r), the local nuclear potential V(r) and the spin-orbit form factor S(r) are chosen to have a generalized Woods-Saxon form:…”

mentioning

confidence: 99%

“…[6][7][8] .) (2) In diagonalizing the Hamiltonian (2), the size rico 0 and the deformation q = co±/eo z of the axial harmonic oscillator basis are optimized for each nucleus at each deformation.…”

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An improved approximation to constrained Hartree-Fock calculations

Brack

1977

Physics Letters B

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A previously proposed microscopical method to calculate deformation energies of heavy nuclei, using Skyrme type effective interactions, is reinvestigated. It is shown that if the effective mass is included in the phenomenolozical one body Hamiltonian, whose eigenfunctions are used to calculate the expectation value of the total Skyrme Hamiltonian, one can obtain deformation energies very close to the ones obtained in constrained Hartree-Fock calculations.Since the re-discovery [1] of the effective nucleonnucleon force of Skyrme [2], constrained HartreeFock (CHF) calculations for heavy nuclei have become technically possible [3,4]. However, such calculations require large amounts of computer time, so that it would be far too time consuming to include nonaxial and left-right asymmetric deformations in a systematical investigation of fission barriers using the CHF method.A purely microscopical, but less time consuming approximation to the CHF method was proposed some years ago by Ko et al. [5]. The main idea of this approach is to utilize the single particle wave functions of the deformed Woods-Saxon (WS) potential, determined in Strutinsky type fission barrier calculations [6,7], to calculate the expectation value of the total Skyrme Hamiltonian:Here OSky is the Skyrme interaction and (bws(13i) a Slater determinant built of the WS single particle wave functions ~pv(r, 3i) which depend on one or more deformation parameters 3# The quadrupole moment Q2 and higher moments can easily be calculated from the ~Pv, too. Thus, eq. (1) gives directly the total energy of a nucleus as a function of deformation E(3i) or E(Q 2 ; ...), and the parameters 3i play the role of the constraint in the CttF method ~. * Work supported in parts by USERDA contract E(11-1)-3001. 1 Present address: Institut Laue Langevin, F-38042 Grenoble, France. 2 Laboratoire associ~ au C.N.R.S. $ For simplicity, we omit the indices for protons and neutrons.The results found with this expectation value method (EVM) were only partially successful [5 ]. Whereas the shell structure in the deformation energies was reasonably well reproduced, their average part increased too much at large deformations. In 240pu, e.g., the second fission barrier was found more than twice as large as the one obtained in a CHF calculation with the same force.We have reinvestigated the EVM with essentially three alterations:(1) The effective masses m*(r) which for Skyrme forces differ from the free nucleon masses in the interior of the nucleus, have been included in the one body Hamiltonian.We thus solve the equations The variable fir, RO) is defined such that firs, RO) = 0 239

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Dynamical Behaviour of the Mass Asymmetry Mass Parameter in Nuclear Fission

1989

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A b s t r a c t . The theory of nuclear fission is reconsidered. We study the behaviour of the mass parameter as a dynamical quantity of the mass asymmetry. The dependence of the mass asymmetry mass parameter is studied as a function of the five collective coordinates. These parameters are rcconsidered by including the temperature t o show thc temperature dependence of the mass parameter. The cranking model is used in developing all the mathematical and theoretical expressions. Numerical calculations of the obtained analytical expressions are carried out for the two fissioning nuclei 236Uand 238U. The mass asymmetry mass parameters are calculated including the temperature as a function of the different five collective coordinates. The present study shows that the values of this mass asymmetry mass parameters arc stable against the change of the temperature for temperature values greater than 1 MeV for all the different five collective coordinates. Dynamisches Verhalten des Massenasymmet#rieparameters bei KernspaltungI n h a l t s u b e r s i c h t . Die AbhLngigkeit des Massenasymmetrieparameters von den funf gemeinsamen Koordinaten und ihrem Temperaturverhalten wird fur ein Mantelmodell aufgestellt und fur 236U und 238U numerisch bestimmt. Es zeigt sich eine Temperaturunabhangigkeit fur Temperaturen iiber 1 MeV.

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A method for solving the independent-particle Schrödinger equation with a deformed average field

Cited by 163 publications

References 5 publications

Study of five-dimensional potential-energy surfaces for actinide isotopes by the macroscopic-microscopic method

Study of five-dimensional potential-energy surfaces for actinide isotopes by the macroscopic-microscopic method

An improved approximation to constrained Hartree-Fock calculations

Dynamical Behaviour of the Mass Asymmetry Mass Parameter in Nuclear Fission

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