Cartesian mesh representation

Cartesin mesh representation

Last modification: 1996/3/25

We use a HF+BCS code EV8 [BFH85], in which the single-particle wavefunctions are expressed in a three-dimensional Cartesian-mesh representation, while most of the other methods for deformed nuclei express the single-particle wavefunctions by the expansion in a harmonic oscillator basis. Let us mention three advantages of the mesh representation. First, it is capable of treating nucleon skins and halos. In contrast, they cannot be described efficiently in the oscillator-basis expansion because the asymptotic form of wavefunctions far from the nuclear surface is determined by the basis. Second, one can treat exotic (e.g., high-multipole) shapes and large (e.g., super and hyper) deformations without preparing a basis specific to each shape. Third, the saturation property of density makes the atomic nucleus a very suitable object to the mesh representation.

The main feature of the HF+BCS code EV8 is the three-dimensional Cartesian-mesh representation: Each single-particle wavefunction psi(x,y,z) is defined in a rectangular box (- L_x /2 < = x < = L_x/2, - L_y /2 < = y < = L_y/2, - L_z /2 < = z < = L_z/2) with its values psi_ijk at cubic mesh points, (x_i, y_j,z_k) = (i-1/2, j-1/2, k-1/2)a, where i, j, and k take on integers. In this study, the mesh size a is set to 1 fm, while the size of the box is L_x=L_y=26 fm, L_z=28 fm for Z < 82 and L_x=L_y=28 fm, L_z=30 fm for Z > = 82. The nucleus is placed at the center of the box.

We impose a symmetry with respect to reflections in x-y, y-z, and z-x planes (the point group D_2h). This symmetry allows triaxial solutions, although all of our solutions have eventually turned out axial and stable against gamma-deformation. On the other hand, the symmetry prohibits odd-multipole deformations, which may not be negligible in some actinide nuclei. According to the calculations with the FRDM [MNM94], the nucleus ²²²Ra has the largest octupole deformation (beta₃ = 0.15, the energy gain due to the octupole deformation is -1.4 MeV), while except in the neighborhood of this nucleus the octupole deformation occurs only in odd-A and odd-odd nuclei. See a GIF Figure of 9.0KB or PS Figure of 161KB for the plot of a₃₀ from FRDM calculations [MNM94]. Incidentally, both in light [KH95] and heavy nuclei [BKW91], the octupole deformation is likely to be enhanced by the procedure of the variation after parity projection.

One might wonder that a mesh size of 1 fm were too large to describe the abrupt change of density at nuclear surface. It was demonstrated in Ref. [BFH85], however, that a mesh size a=1 fm can produce enough accurate results for several spherical nuclei with mass below ²⁰⁸Pb. We did a similar test of accuracy for a deformed actinide nucleus ²⁴⁰Pu (footnote) and found that the relative errors of the quadrupole moment and the total energy are 0.4% and 0.5%, respectively. (The method of the extrapolation to a - > 0 is explained in a different page.) This order of accuracy is higher than necessary for the quadrupole moment, while it is not for the energy to make comparison with experiments. Considering that the root-mean-square (r.m.s.) deviation of the atomic masses of recent mass formulae is ~ 0.5 MeV, the desirable precision is of the order of 0.1 MeV, which is only 0.005 % (footnote) of the total binding energy of ²⁴⁰Pu. Therefore, the binding energy, but not the other quantities, has to be corrected for the effect of the finite mesh size, which is discussed in a different page.

The origin of this unexpectedly high accuracy with apparently coarse meshes has been explained by Baye and Heenen [BH86]. The equation to determine { psi_ijk } is usually derived through a discrete approximation to the Schroedinger equation. They presented an alternative point of view, in which they introduced a set of orthogonal basis functions f_ijk(x,y,z) such that { psi_ijk} are the coefficients to expand psi(x,y,z) in this basis. (In this point of view, the equation for { psi_ijk} is determined uniquely from the variational principle.) This basis can be unitary-transformed to plane-wave basis with | k_kappa | < pi / a (kappa = x, y, z), which suggests that the atomic nucleus is a very suitable physical object to apply the mesh representation because the saturation property of nuclear matter guarantees the suppression of large-momentum components in wavefunctions from the view point of the Thomas-Fermi approximation.

To enjoy this high accuracy, the method to determine { psi_ijk } must be in accordance with the view point of Baye and Heenen. Exact variational treatment with the plane-wave basis requires, however, long computation time and diminish the simplicity of the Cartesian-mesh representation. The code EV8 is designed to emulate the plane-wave expansion method, though it is based on the discrete approximations, by choosing the appropriate orders of approximation formulae for derivatives (the 7- and 9-point formulae for the first and second derivatives, respectively) and integrals (the mid-point formula) (footnote).