One- and two-body dissipation in heavy-ion collisions

A microscopic theory has been formulated for one- and two-body dissipation in collisions between two heavy nuclei. With a nucleon-nucleon interaction as the basic perturbation in a density matrix approach with “linear response” approximations, the one- and two-body nuclear friction coefficients for the ⁴⁰Ca + ⁴⁰Ca system have been calculated and their dependence on relative kinetic energy and smearing of nuclear single-particle states was obtained. The results of our calculation show that: (a) the combined one- and two-body friction coefficients compare favourably with phenomenological values, (b) the one-body dissipation is more effective than two-body in kinetic energy damping, while both the mechanisms are comparable for the damping of relative angular momentum, (c) the importance of the two-body friction compared to one-body increases at higher relative kinetic energies and (d) the effect of introducing a smearing in nuclear levels appears as a lowering of nuclear friction.     

Q- and Z-dependence of angular momentum transfer in deeply inelastic collisions of 86Kr with 209Bi

The dependence of the in-plane and out-of-plane angular correlations of fragments from fissioning heavy products on the kinetic energy and Z of the light reaction partner have been measured. From the dependence of the angular correlations on Q-value and hence energy loss, together with existing data from which the total angle-integrated cross section as a function of energy loss can be extracted, we have determined the dependence of the angular momentum transferred to the heavy product on the initial orbital angular momentum or impact parameter. The resulting dependence is qualitatively consistent with the sticking limit for a reaction intermediate of touching deformed fragments. More specific nuclear models generally underestimate the angular momentum transfer, although the one-body proximity-friction model accounts for the major fraction of the angular momentum transfer. A recent model incorporating both one-body proximity friction and collective excitations accounts quite well for the observed angular momentum transfer. The Z-dependendence of the anisotropy shows the importance of angular momentum fractionation for the less probable events, where the Z of the fissioning system is appreciably less than that of the target. The transferred angular momentum is shown to be fairly strongly aligned along the perpendicular to the reaction plane, with alignment values of 0.6 to 0.8. The component of angular momentum not along the perpendicular to the reaction plane is found to be primarily oriented perpendicular rather than parallel to the recoil direction. The absolute fission probabilities are found to be qualitatively consistent with J-dependent calculations using the J-values deduced from the angular correlations.     

Heavy-ion fusion: Comparison of experimental data with classical trajectory models

Currently available data on fusion excitation functions for heavy-ion induced reactions over a wide mass range are compared to results calculated with a classical dynamical model based on the proximity nuclear potential of Blocki et al., the Coulomb potential of Bondorf et al., and one-body nuclear friction in the proximity formalism of Randrup. With these conservative and dissipative forces and the radial parameters of Myers, overall good agreement is obtained between the theoretical excitation functions and most of the available data. Extensive calculations have been performed to test the sensitivity of the calculated fusion cross-sections to a number of parameters, including the radial dependence of the Coulomb and nuclear potentials, the radial and tangential friction form factors as well as the projectile and target radii. The theoretical excitation functions for the lighter heavy-ion systems are rather insensitive to changes in either the conservative or dissipative forces. The calculations show that tangential friction sufficient to produce the rolling condition is necessary to explain the magnitude of the fusion cross-sections at high energies, which are also sensitive to the magnitude of the radial friction component. This is in contrast to the fusion cross-sections at low energies which are determined by the nuclear potential at larger separations, and to a lesser extent by tangential friction. The low energy fusion data are most sensitive to the nuclear radii. The calculations reveal the importance of more experimental measurements of fusion cross-sections at high energies, especially for heavy systems where the magnitudes of the fusion cross-sections are the most sensitive to the assumed forces. However, even for these cases the effects of the conservative and dissipative forces are similar and difficult to separate. These studies indicate, however, that it is possible to construct a conservative potential that will give calculated fusion excitation functions which are in good agreement with all experimental data over the entire mass range. The maximum fusion cross-sections as defined here exceed considerably the liquid-drop limiting value for heavy systems.     

Studying parity-nonconservation effect in the conversion transition between the levels of the (5/2)± doublet in the 229Pa nucleus

Effects of weak nucleon interaction are calculated for the ground-state doublet of 5/2^± levels of the strongly deformed nucleus ²²⁹Pa₉₁. A parity-nonconservation effect in the doublet states can be observed in the conversion spectrum for the isomeric transition between the doublet levels. By using a generalized model of the nucleus, the matrix element of the effective one-nucleon weak-interaction potential, which determines the weight of the opposite parity admixture in the doublet components is estimated in the single-particle approximation. The reduced probabilities of the E1 and M1 nuclear transitions between the doublet states are calculated within various models of the deformed nuclear potential. The effect of Coriolis forces on the dipole electric transition in question is considered. The lifetime of the upper doublet state is estimated.     

Model Study of Three-Body Forces in the Three-Body Bound State

The Faddeev equation for the three-body bound state with two- and three-body forces is solved directly as three-dimensional integral equation. The numerical feasibility and stability of the algorithm, which does not employ partial wave decomposition is demonstrated. The three-body binding energy and the full wave function are calculated with Malfliet-Tjon-type two-body potentials and scalar two-meson exchange three-body forces. For two- and three- body forces of ranges and strengths typical of nuclear forces the single-particle momentum distribution and the two-body correlation function are similar to the ones found for realistic nuclear forces.     

Fission with microscopic energy dissipation

We calculate the non-adiabatic excitations of pair states in a BCS formalism for a fissioning²³⁶U nucleus. The single-particle spectrum is calculated for a folded-Yukawa potential along a deformation path that is determined classically for one-body dissipation. The resulting microscopic energy dissipation is compared to that due to one- and two-body dissipation.     

Nuclear spin dependent atomic parity violation,nuclear anapole moments and the hadronic axial neutral current

Left-right asymmetries in atomic transitions, depending upon the nuclear spin, could be a source of information on the neutral hadronic axial current. We show that the relevant electroweak parameter can be extracted from experiment by measuring hyperfine component ratios which do not involve the knowledge of the atomic wave function. In the standard electroweak model, the parity violating electron-nucleus interaction associated with the hadronic axial neutral current is accidently suppressed and, as a consequence, dominated by the electron interaction with the nuclear anapole moment, which describes the effect of the parity violating nuclear forces on the nucleus electromagnetic current. One of our objectives was to identify the various physical mechanisms which determine the size of the nuclear anapole moments. As an important step, we have established a simple relation between the anapole moment and the nuclear spin magnetization. From this relation it follows that the computation of the anapole moment can be reduced to that of one-body operators. The basic tool is a unitary transformationW which eliminates the one-body parity violating potential from the nuclear hamiltonian. It generalizes, to more realistic situations, a procedure used by F.C. Michel in the case of constant nuclear density. The fact that the transformationW does not commute with the residual spin-dependent nucleon-nucleon interaction can be accounted for — within some approximation — by a renormalization of the effective coupling constants which appear in the one-body reduction of the two-body parity violating nucleon-nucleon interaction induced by meson exchange. A particular attention was paid to nuclear correlation effects. They are treated semi-empirically in the independent pair approximation. The nuclear anapole moments of⁸⁵Rb,¹³³Cs, and²⁰⁹Bi have been evaluated for three sets of parity violating meson-nucleon coupling constants, taking into account configuration mixing effects in a semi-empirical way. We suggest a possible strategy to disentangle the axial neutral current from the anapole moment contribution. It requires experiments, accurate to few tenths of a percent, performed on several heavy nuclei. The results should be collected in a two-dimensional plot involving a suitably chosen set of variables (X, Y). In an ideal situation — small theoretical uncertainties —the points corresponding to various nuclei should fall on a straight line which crosses the lineX=0 at a point the ordinate of which is the sought for axial coupling constant.     

Mass-energy distribution of fragments within Langevin dynamics of fission induced by heavy ions

A stochastic approach based on four-dimensional Langevin fission dynamics is applied to calculating mass-energy distributions of fragments originating from the fission of excited compound nuclei. In the model under investigation, the coordinate K representing the projection of the total angular momentum onto the symmetry axis of the nucleus is taken into account in addition to three collective shape coordinates introduced on the basis of the {c, h, ??} parametrization. The evolution of the orientation degree of freedom (K mode) is described by means of the Langevin equation in the overdamped regime. The tensor of friction is calculated under the assumption of the reducedmechanismof one-body dissipation in the wall-plus-window model. The calculations are performed for two values of the coefficient that takes into account the reduction of the contribution from the wall formula: k _s = 0.25 and k _s = 1.0. Calculations with a modified wall-plus-window formula are also performed, and the quantity measuring the degree to which the single-particle motion of nucleons within the nuclear system being considered is chaotic is used for k _s in this calculation. Fusion-fission reactions leading to the production of compound nuclei are considered for values of the parameter Z ²/A in the range between 21 and 44. So wide a range is chosen in order to perform a comparative analysis not only for heavy but also for light compound nuclei in the vicinity of the Businaro-Gallone point. For all of the reactions considered in the present study, the calculations performed within four-dimensional Langevin dynamics faithfully reproduce mass-energy and mass distributions obtained experimentally. The inclusion of the K mode in the Langevin equation leads to an increase in the variances of mass and energy distributions in relation to what one obtains from three-dimensional Langevin calculations. The results of the calculations where one associates k _s with the measure of chaoticity in the single-particle motion of nucleons within the nuclear system under study are in good agreement for variances of mass distributions. The results of calculations for the correlations between the prescission neutron multiplicity and the fission-fragment mass, ??n _pre(M)??, and between, this multiplicity and the kinetic energy of fission fragments, ??n _pre(E _k )??, are also presented.     

μ-介子在He3原子核上的俘获

本文计算了μ介子在He³核上俘获的几率、末态H³核的角分布和极化。所采用的理论是带有重正化效应(包含弱磁矩及赝标项)的V-A普适弱作用理论。在计算中考虑了μ和He³核在始态有极化及处于不同超精细态上的情况。在计算中假定了He³核的基态是纯S态,这时忽略了由张量力以及其他自旋轨道耦合力引起的其他态。介子交换电流的效应也没有考虑。在以上这两个假定下,我们证明了俘获几率中只包含一个未知的原子核矩阵元,这个矩阵元恰好是原子核密度函数的富氏分量。利用μ介子(或电子)与He³(或H³)原子核的散射可以确定这个未知矩阵元。     

Dependence of the density distribution of 208Pb on the occupation probabilities of shell-model orbits

An independent-particle model for the proton density distribution of ²⁰⁸Pb is constructed; it closely approximates the Hartree-Fock calculation of Dechargé and Gogny. We investigate the modifications which arise when one introduces a depletion of the Fermi sea of the amount suggested by analyses of recent electron scattering data and by nuclear-matter calculations. The main effect of the depletion is to flatten the density distribution in the nuclear interior. The calculated density is in good agreement with the empirical one near the nuclear centre but is too small in the vicinity of 5 fm. The main consequences of the depletion are shown to be largely independent of the details of the model. It is concluded that Hartree-Fock single-particle wave functions which yield good agreement with empirical density distributions are rather different from the natural orbitals. Accordingly they should not be expected to yield a good approximation to the off-diagonal elements of the one-body density matrix, e.g. to the momentum distribution.     

Antisymmetrized Green’s function approach to (e, e′) reactions with a realistic nuclear density

A completely antisymmetrized Green’s function approach to the inclusive quasielastic (e, e′) scattering, including a realistic one-body density, is presented. The single-particle Green’s function is expanded in terms of the eigenfunctions of the non-hermitian optical potential. This allows one to treat final state interactions consistently in the inclusive and in the exclusive reactions. Nuclear correlations are included in the one-body density. Numerical results for the response functions of ¹⁶O and ⁴⁰Ca are presented and discussed.     

Näherungen zur Hartree-Fock-Methode mit Paritätsmischung

We examine the quantitative behavior of parity-mixing in the approximation of Hartree-Fock using one-particle-functions without definite parity (PHF). With the aid of a variational method (at most one variation parameter for each particle and self-consistent treatment of the OPEP alone), we found that the mixing will be large if the mixing parameters are taken as pure imaginary (with our choice of phase), that is if the product wave function is time reversal invariant. In this case the parity-mixing will be produced by exchange terms alone and only long range forces can contribute substantially. Therefore the OPEP can be considered a good approximation of parity-mixing forces. We obtain the correct magnitude of the single-particle energy-splittings which are produced in the usual treatments by the LS-term of the shell model.     

Effects of nuclear structure in the spin-dependent scattering of weakly interacting massive particles

We present calculations of the nuclear from factors for spin-dependent elastic scattering of dark matter WIMPs from¹²³Te and¹³¹Xe isotopes, proposed to be used for dark matter detection. A method based on the theory of finite Fermi systems was used to describe the reduction of the single-particle spin-dependent matrix elements in the nuclear medium. Nucleon single-particle states were calculated in a realistic shell model potential; pairing effects were treated within the BCS model. The coupling of the lowest single-particle levels in¹²³Te to collective 2⁺ excitations of the core was taken into account phenomenologically. The calculated nuclear form factors are considerably less then the single-particle ones for low momentum transfer. At high momentum transfer some dynamical amplification takes place due to the pion exchange term in the effective nuclear interaction. But as the momentum transfer increases, the difference disappears, the momentum transfer increases and the quenching effect disappears. The shape of the nuclear form factor for the¹³¹Xe isotope differs from the one obtained using an oscillator basis.     

One-body dissipation and nuclear dynamics

We discuss recent developments in the “one-body” dissipation theory described in B?ocki et al. [Ann. Phys. (N.Y.)113 (1978), 330]. The principal new result is the derivation of the functional form of the dissipation expression (the Rayleigh Dissipation Function) for a finite idealized nucleus with a diffuse surface, in the form of an expansion in powers of the dimensionless ratio of the surface diffuseness to the size, R, of the system. The leading term in such an expansion is a surface contribution, of relative order R², in the form of the “Wall Formula” of B?ocki et al. The next is a curvature correction of order R. At the next level (R⁰) there are two higher order curvature corrections and a correction for the presence of gradients in the normal velocity field specifying the motion of the surface. For simple models of the nuclear surface profile we work out analytically the coefficients in the curvature and velocity-gradient correction terms. We compare the one-body dissipation theory formulated in this way with recent linear-response and Time-Dependent Hartree-Fock treatments of the nuclear problem. The principal theme that emerges from this study is the close analogy between the problem of the nuclear macroscopic dissipation function and the problem of the nuclear macroscopic potential energy.     

Langevin description of the charge distribution of fragments originating from the fission of excited nuclei

The charge distribution of fragments originatingfrom the fission of the ²³⁶U compound nucleus is calculated within a stochastic approach based on Langevin equations. The elongation coordinate, the neck-thickness coordinate, and the charge-asymmetry coordinate are chosen as collective variables. The friction parameter of the charge mode is calculated on the basis of two nuclear-viscosity mechanisms, that of one-body and that of two-body dissipation. It is shown that the Langevin approach is applicable to studying isobaric distributions. In addition, the charge distribution in question is studied as a function of the excitation energy of the compound nucleus and as a function of the coefficient of two-body viscosity.     

Corrections to the single-particle M1 and Gamow-Teller matrix elements

Consideration of second-order core-polarisation, isobar-current and meson-exchange-current processes gives a satisfactory understanding of the ground-state magnetic moments and mirror β-decay transition probabilities in closed-shell-plus- (or minus) one nuclei, A = 3, 15, 17, 39 and 41. Perturbation contributions from high-lying excited states (tensor correlations) from one-body and two-body meson-exchange operators cancel strongly in diagonal matrix elements, but in off-diagonal matrix elements between spin-orbit partners the cancellation is less. Similarly, with isobar currents, there is a strong cancellation between direct and exchange contributions in diagonal matrix elements, while the exchange term is suppressed in off-diagonal cases. Of the two contributions, tensor correlations are found to be as important as, if not more important than, isobar currents. The calculated lowest-order meson-exchange-current contributions to magnetic moments are close to the soft-pion limit predictions.     

Electromagnetic energies of nuclei and the nuclear compton amplitude

The electromagnetic energy of a nucleus is derived in perturbation theory, which relates this quantity to the amplitude for the forward scattering of virtual photons on a nucleus (nuclear Compton amplitude). Using the gauge invariance of this amplitude, the energy is separated into Coulomb and transverse components. Our formalism, although basically nonrelativistic, admits corrections of order ($\text{v}\text{c}$)² to the nuclear charge operator. The energy is further separated into one-body terms, related to the n-p mass difference, and two-body terms which lead to the Breit interaction and the nuclear Lamb shift. These results are then related to electron scattering sum rules in the manner of Cottingham. Mesonic contributions to the electromagnetic energy are also discussed.