Translationally invariant treatment of the charge density in nuclei

An expression is derived for the translationally invariant (transition-) charge density in the center of mass rest frame of the nucleus in between arbitrary wave functions. An explicit formulation is given for the charge density of spherically symmetric Hartree-Fock type ground states. The method is applied to the nuclei⁴He,¹⁶O,⁴⁰Ca and⁹⁰Zr using Woods-Saxon single nucleon orbits for the construction of the ground state wave functions. For the two lighter nuclei in addition realistic Hartree-Fock wave functions are investigated. The results are compared to those obtained with various approximate treatments of the center of mass motion. It turns out that at least in the lighter nuclei the usual center of mass correction, which is based on the assumption of pure non-spurious oscillator configurations and commonly used in the analysis of, e.g., electron scattering data, does not produce reliable (or even predictable) results. Thus here indeed the full translationally invariant treatment seems to be required. For the heavy nucleus⁹⁰Zr on the other hand, as expected, the center of mass effects become negligibly small.     

Hartree-fock equations for atoms with two open shells

We have formulated the Hartree-Fock equations for multielectron systems with two open shells (the Huzinaga method) in terms of the density matrix in the LCAO approximation. In order to solve the Hartree-Fock equations, in the algebraic approximation we have obtained expressions for the derivatives of the energy with respect to the density matrix elements and the nonlinear atomic orbital parameters (the orbital exponents). We discuss the question of calculating the open shell parameters (the vector coupling coefficients) in the configurations s¹p^N and s¹d^N within the Huzinaga method. We have calculated the energy for a series of atoms with two open shells in these configurations. Using rather narrow basis sets of Slater-type atomic orbitals), we have obtained energy values close to the results of a numerical solution of the Hartree-Fock equations with sufficiently high accuracy of the virial ratio. __________ Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 74, No. 2, pp. 145–152, March–April, 2007     

Dynamical Collapse of White Dwarfs in Hartree- and Hartree-Fock Theory

We study finite-time blow-up for pseudo-relativistic Hartree- and Hartree-Fock equations, which are model equations for the dynamical evolution of white dwarfs. In particular, we prove that radially symmetric initial configurations with negative energy lead to finite-time blow-up of solutions. Furthermore, we derive a mass concentration estimate for radial blow-up solutions. Both results are mathematically rigorous and are in accordance with Chandrasekhar’s physical theory of white dwarfs, stating that stellar configurations beyond a certain limiting mass lead to “gravitational collapse” of these objects. Apart from studying blow-up, we also prove local well-posedness of the initial-value problem for the Hartree- and Hartree-Fock equations underlying our analysis, as well as global-in-time existence of solutions with sufficiently small initial data, corresponding to white dwarfs whose stellar mass is below the Chandrasekhar limit.     

Spherical Hartree-Fock calculations with linear-momentum projection before the variation

The hole spectral functions and from these the spectroscopic factors have been calculated in a Galilei-invariant way for the ground-state wave functions resulting from spherical Hartree-Fock calculations with projection onto zero total linear momentum before the variation for the nuclei ⁴He, ¹²C, ¹⁶O, ²⁸Si, ³²S and ⁴⁰Ca. The results are compared to those of the conventional approach which uses the ground states resulting from usual spherical Hartree-Fock calculations subtracting the kinetic energy of the center-of-mass motion before the variation and to the results obtained analytically with oscillator occupations.     

APPLICATIONS OF THE MODIFIED SURFACE DELTA INTERACTION IN THE ANGULAR MOMENTUM PROJECTION DEFORMED HAREE-FOCK METHOD

In this paper a modified surface delta interaction with an angular momentum correction term is proposed.By using this interaction the deformed Hartree-Fock and the approximate angular momentum projection calculations are performed for even-even nuclei in mass region A=56—74.The single particle energy spectra of the ground state configurations obtained are more reasonable than that calculated with KB matrix elements.The agreement of the calculated binding energies and projection energy spectra with experimental ones is fairly good.     

A systematic approach to the Hartree-Fock problem in the thermodynamic limit

A systematic procedure is proposed in order to look for non-plane-wave solutions to the Hartree-Fock equations in the thermodynamic limit. The corresponding Hartree-Fock states are seen to arise as the self-consistent eigenvectors of a special matrix, as in Roothaan's method. The attractive delta problem is tackled as an example. The corresponding solution fulfills the monotonicity requirement for the energy as a function of the density, which no other known solution to the problem verifies, and coincides, in the zero-density limit, with the lower bound of Lieb and de Llano.     

On the description of center of mass motion in the Glauber model

The problem of center of mass motion of nucleus is discussed in the framework of the Glauber model, starting from a translation invariance condition. Theπ-⁴He coherent scattering amplitude is calculated to illustrate how predictions of the model are sensitive to ways of handling recoil corrections.     

Generalized shells in nuclei: Hartree-Fock calculations for bubble nuclei

Hartree-Fock calculations in the bubble degree of freedom have been performed on a variety of spherical nuclei. Of particular importance are incipient bubble configurations in ³⁶Ar, ⁶⁸Se, ⁸⁴Se, ¹⁰⁰Sn, ¹¹⁶Ce, ¹³⁸Ce and ²⁰⁰Hg, each of which possesses a binding energy which is comparable to that of the normal spherical closed-shell configuration. The densities of the above nuclei display strong deviations from a uniform shape, and give rise to depletions in the nuclear interior. These nonuniformities are due both to the absence of low angular momentum states in otherwise normally occupied spherical shells, and also to strong self-consistency effects. The nonuniformities in the mass density are further enhanced for nuclei whose neutron and proton densities have depressions or peaks at approximately the same distance from the center of the nucleus. A depression of the central density is most pronounced in the nuclei ³⁶Ar, ¹³⁸Ce and ²⁰⁰Hg. Interior depletions of the density are associated with the relatively higher energies of low angular momentum single-particle levels as compared to high angular momentum single-particle levels. This effect can give rise to moderately large gaps at the Fermi surface. Finally, it is shown that in a bubble configuration, the spin-orbit splitting of low lying doublets is sometimes reversed, and that this effect is especially pronounced for levels with low angular momentum.     

Potential-energy surfaces for asymmetric heavy-ion reactions

We calculate the macroscopic potential energy of deformation as a function of mass asymmetry and distance between mass centers for shape configurations of interest in heavy-ion reactions. For the system³⁰⁰120 we also study the effect of adding microscopic shell and pairing corrections to the macroscopic potential energy. The shape configurations are generated by bringing together two separated spheres of unequal size. After the spheres touch the shapes are constructed by filling in the neck while keeping constant the radii of the end spheres, the nuclear density and the total nuclear volume. The macroscopic energy is calculated as the sum of a Coulomb energy and a nuclear macroscopic energy that takes into account the finite range of the nuclear force. For systems throughout the periodic table we display the calculated energy as a function of distance between mass centers and mass asymmetry in the form of contour maps. Some important features of the contour maps are the stationary points of the potential energy and how they change in character and location as functions of the nuclear system considered. For example, for light systems there is a maximum in the potential energy for symmetric shapes. As we move to heavier systems this peak in the potential-energy surface splits into two asymmetric peaks that are separated by a symmetric saddle point. This occurs when Z²/A ≈ 30 for the total system. As the systems become still heavier the peaks become more and more asymmetric. In heavy-ion reactions for which the asymmetry of the system is smaller than that corresponding to the peak, the smaller nucleus tends to suck up the larger one. For larger asymmetries the larger nucleus tends to suck up the smaller one. For heavy systems the binary fission saddle point is lower than the maximum in the one-dimensional interaction barrier. The penetrability calculated for the multidimensional potential-energy surface is therefore increased relative to that for the one-dimensional barrier. The microscopic shell and pairing corrections lower the potential energy for configurations in which the target and/or projectile are magic or nearly magic. This effect persists to somewhat inside the point of touching. These corrections also lower the energy near the ground state.     

双阱-集团壳模型中的质心运动伪态

本文指出了用微观双阱-集团壳模型来研究裂变和重离子反应过程中存在着质心运动伪态问题,推出了分解质心运动伪态波函数的一般表达式,并以Be⁸(α-α)为例,说明了伪态的存在,定量地计算了伪态成分以及它对能量矩阵元的修正。通过Be⁸核的分析,我们得到的结论是:虽然伪态成分仅占百分之几,但对于微观裂变位垒修正是很明显的,它甚至要超过库仑能对位垒的修正,因此在精确的定量计算中质心运动伪态是必须要考虑的。尤其是在核粒子数增多以及包含高激发态时,质心运动伪态更需仔细考虑。     

Spherical Hartree-Fock calculations with linear-momentum projection before the variation

Spherical Hartree-Fock calculations with projection onto zero total linear momentum before the variation are performed for the nuclei ⁴He, ¹²C, ¹⁶O, ²⁸Si, ³²S and ⁴⁰Ca using a density-independent effective nucleon-nucleon interaction. The results are compared to those of usual spherical Hartree-Fock calculations subtracting the kinetic energy of the center-of-mass motion either before or after the variation and to the results obtained analytically with oscillator occupations. Total energies, hole energies, elastic charge form factors and charge densities and the mathematical Coulomb sum rules are discussed.Received: 7 April 2003, Published online: 18 December 2003PACS: 21.60.-n Nuclear structure models and methods     

Some considerations on the restoration of Galilei invariance in the nuclear many-body problem

The effects of the restoration of Galilei invariance in the nuclear many-body problem on the electromagnetic properties of simple bound states are investigated. For this purpose the form factors for elastic electron scattering from the oscillator ground states of ⁴He, ¹⁶O and ⁴⁰Ca as well as those for elastic and inelastic electron scattering between various one-hole states with respect to these reference configurations are computed with and without projection into the center-of-momentum rest frame. It is demonstrated that, in some cases, the full restoration of Galilei invariance produces results which are considerably different from those obtained with the usual approximate way to treat the center-of-mass motion. The same holds for the mathematical Coulomb sum rules and their first and second moments obtained for the above-mentioned nuclei. Received: 21 February 2001 / Accepted: 20 December 2001     

Structure calculations in18O nucleus

A multishell Hartree-Fock calculation for¹⁸O nucleus has been done with the Tabakin and Hamada-Johnston interactions followed by pairing correlations and angular momentum projection. TheB(E2) transition rates have been reproduced with an effective charge of 1.5e for protons and 0.5e for neutrons, but the fit to energy levels is poor. An approximate alternative formalism has then been developed which casts the eighteen-particle problem into a two-valence nucleon problem. The two valence nucleons interact through the RPA-type oscillation modes of the core. Calculations with the two interactions mentioned above have reproduced roughly the uncorrelated projected Hartree-Fock spectrum. Lastly, the excited band spectra of¹⁸O have been examined in the TDA approach followed by band mixing.     

Derivation of an adiabatic time-dependent Hartree-Fock formalism from a variational principle

A derivation of the adiabatic time-dependent Hartree-Fock formalism is given, which is based on a variational principle analogous to Hamilton's principle in classical mechanics. The method leads to a Hamiltonian for collective motion which separates into a potential and a kinetic energy and gives mass and potential parameters in terms of the nucleon-nucleon interaction. The adiabatic approximation assumes slow motion but not small amplitudes and can therefore describe anharmonic effects. The RPA is a limiting case where both amplitudes and velocities are small. The variational approach provides a consistent way of extracting coordinates and momenta from the density matrix and of obtaining equations of motion when particular trial forms for this density matrix are chosen. One such choice leads to Thouless-Valatin formula. An other choice leads to irrotational hydrodynamics.     

偶偶Cd同位素核的形变HF态及负宇称带研究

将具有正宇称的轨道空间扩大到包含具有负宇称的1h_11/2轨道,采用修正的表面δ相互作用(MSDI),对¹⁰⁴Cd,¹⁰⁶Cd,¹⁰⁸Cd,¹¹⁰Cd,¹¹²Cd,¹¹⁴Cd和¹¹⁶Cd等7个偶偶核作了形变HF计算.得到了基态和一些激发态的解.同时,还用近似角动量投影形变Hartree-Fock(PDHF)方法对¹⁰⁸Cd和¹¹⁰Cd进行了能谱计算,得到其正、负宇称带的解,计算结果与实验谱基本一致.     

Crossed-second-order specific-mass isotope shift in the Nickel atom

The crossed-second-order corrections to the specific mass shifts of the lowest terms of the two lowest configurations of the Nickel atom are evaluated ab initio in the Multiconfigurational Hartree-Fock scheme. The excitations towards the nf(l=3) empty subshells play the major role. If the contributions obtained are added to the Hartree-Fock values, the discrepancy between experiment and theory for the 3d ⁸ 4s ²?3d ⁹ 4s (virtual) transition is only reduced by one third. As concerns the differences between the specific shifts of the five Russell-Saunders terms of 3d ⁸ 4s ², the crossed-second-order contributions are predicted to be practically as large as the Hartree-Fock values, which makes the total definitely measurable.     

Time-dependent density-matrix theory

The mass fluctuations in damped reactions of¹⁶O+¹⁶O are studied in an extended time-dependent Hartree-Fock theory. The theory determines the time evolution of a two-body density matrix as well as that of a one-body density matrix, providing us with a microscopic way to calculate the fluctuations of one-body quantities. The results of the theory are compared with those obtained in a transport model. It is found that the dispersions in fragment mass calculated in the two models are of the same order of magnitude and much larger than those calculated in the time-dependent Hartree-Fock theory. The differences between the microscopic theory and the transport model are also discussed.     

A note on the truncation of the basis in Hartree-Fock calculation

Using different truncations of the basis, the Hartree-Fock calculation for 2s-1d shell nuclei with density dependent force has been performed. The influence of these truncations on the Hartree-Fock solutions has been analysed. Stability, shape and centre-of-mass motion energy of the nuclei studied are found to be mostly influenced.It is a pleasure to thank Dr. G. Ripka for many enlightening discussions. The computer calculations have been carried out during the stay at CEN Saclay, France.     

A self consistent approach to the transport equation for moving atoms in interaction with laser fields

The transport equations for moving atoms in interaction with laser fields, are derived using the hamiltonian in terms of the quantum dynamical variables for the center of mass motion and by introducing suitably defined local density matrices. Use is also made of the phase space variables associated with the quantum dynamical variables of center of mass and the projection operator techniques.     

Self-consistent center-of-mass correction and the charge form factor of He

The Hartree-Fock approximation is used for ⁴He in the case of Lipkin's Hamiltonian with several nuclear forces. The self-consistent solution practically coincides with the one obtained in the usual HF method applied to the intrinsic Hamiltonian; however, the center of mass is now in its ground state, and its motion is fairly separated in the ground state function of the nucleus. An unambiguous comparison between experimental and theoretical charge form factor is done. The discrepancies cannot be interpreted as due to spurious c.m. effects.