Solving Dirac equations on a 3D lattice with inverse Hamiltonian and spectral methods

Abstract

A new method to solve the Dirac equation on a 3D lattice is proposed, in which the variational collapse problem is avoided by the inverse Hamiltonian method and the fermion doubling problem is avoided by performing spatial derivatives in momentum space with the help of the discrete Fourier transform, i.e., the spectral method. This method is demonstrated in solving the Dirac equation for a given spherical potential in 3D lattice space. In comparison with the results obtained by the shooting method, the differences in single particle energy are smaller than 10 −4 MeV, and the densities are almost identical, which demonstrates the high accuracy of the present method. The results obtained by applying this method without any modification to solve the Dirac equations for an axial deformed, non-axial deformed, and octupole deformed potential are provided and discussed.

Clusterization and deformation of multi- Λ hypernuclei within a relativistic mean-field model

“…In our numerical calculations, the singleparticle wave functions are represented on 3-dimensional lattice in the real space [52]. To avoid the fermion doubling, the derivative in the Dirac equation is computed in the momentum space with the fast Fourier transform [53], for which we use FFTW library [54]. The damped gradient iteration technique [55,56] is used to solve the self-consistent mean-field equations (see also Ref.…”

Shells in a toroidal nucleus in the intermediate-mass region

“…It should however be stressed that what has been presented here with the analytical toroidal single-particle shell model provides only an intuitive guide on the interesting nuclei where toroidal high-spin isomers may be searched and located. Whether these states turn out to be local energy minima will need to rely on reliable microscopic models such as the non-relativistic mean-field or relativistic mean-field calculations as carried out in [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29].…”

“…At the predicted energy region, a total of three sharp resonances instead of just a single resonance have been observed and interpreted as possible toroidal high-spin isomers with spins I=I z =28, 36, and 44 in the excited 28 Si system. Subsequent relativistic mean-field calculations provide additional theoretical support for the presence of these states [21][22][23]. We would like to understand in a simple mechanical way how high-spin nuclei can be produced in the binary products of such reactions.…”

Dynamics of the linear-chain alpha cluster in microscopic time-dependent relativistic density functional theory

“…Recently, the covariant DFT has been solved successfully in a three-dimensional (3D) lattice space with the inverse Hamiltonian [51] and Fourier spectral methods [11,52]. This paves the way to develop the corresponding time-dependent approaches in a full 3D lattice space without assuming any symmetries.…”