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This paper reports calculations of the influence of a reaction time T>10-21 s in deep-inelastic Xe-Pb collisions on the energy spectrum of δ electrons ejected in the same collision. It is shown that the lifetime of the superheavy composite system causes pronounced oscillations of width ε=h/T in the electron distribution, which survive the inclusion of multistep excitations and the folding with a lifetime distribution function. This effect may serve as an atomic clock for deep-inelastic collisions.
This Letter discusses inner-shell excitation in collisions of very heavy ions (Z1+Z2≳140) in the framework of the quasimolecular model. The importance of multistep excitations and of coupling between continuum states is demonstrated. The 1sσ vacancy probabilities resulting from coupled-channels calculations exceed perturbation theory by a factor 3-5, thus giving good agreement with recent experimental results.
Binding energies and wave functions of inner-shell electronic states in superheavy quasimolecules with (Zp+Zt)α>1 are calculated. Ionization during a collision of very heavy ions is investigated within a molecular basis generated by the solutions of the two-center Dirac equation. Transitions to vacant bound states as well as direct excitation to the continuum are taken into account. We present theoretical values for the ionization probability as a function of impact parameter, bombarding energy, and combined nuclear charge. Our computed results are compared with recent experimental data. It is suggested that relativistic binding energies of electrons in superheavy quasimolecules can be determined experimentally via the impact-parameter dependence of ionization and the anisotropy of quasimolecular radiation.
Within an extended semiquantal theory we perform large-sized coupled-channel calculations involving 260 collective levels for Coulomb fission of 238U. Differential Coulomb fission cross sections are studied as a function of bombarding energy and impact parameter for several projectiles. In the Xe + U case, total cross sections are also given. We find a strong dependence on projectile charge number, PCF(180°)∼(Zp)6 in the region 50≤Zp≤92 for a fixed ratio E/ECoul, which might be helpful to separate Coulomb fission experimentally from sequential fission following transfer reactions. Since the cross sections are sensitive to the moment of inertia ⊖ at the saddle point, Coulomb fission can serve as a tool to investigate the dependence of ⊖ on elongation. The fragment angular distribution exhibits deviations from 1/sinθf which are pronounced at low incident energies. Our theory indicates that the recently measured Xe + U fission cross sections contain a major fraction of Coulomb-induced fission at E≤0.85 ECoul. NUCLEAR REACTIONS, FISSION Calculated Coulomb fission cross sections σ(Ep,θp) for 54Xe, 67Ho, 82Pb, 92U→92238U, fragment angular distribution, fission energy spectrum, mean spin value 〈Jf〉.
The theory of nucleon transfer in heavy ion reactions is formulated on the basis of the molecular particlecore model for a system consisting of two cores and one extracore nucleon. The extracore nucleon is described by the molecular wave functions of the asymmetric two-center shell model. The cores, which are assumed to be collectively excitable, are treated with vibrator-rotator models. Potentials for shape polarization are contained in the asymmetric two-center shell model and the interaction between the cores. The excitation and transfer of the extracore nucleon is induced by the radial and rotational couplings. The coupled channel equations, which include the recoil effects in first approximation, are derived in a form suitable for numerical calculations of cross sections. NUCLEAR REACTIONS Heavy ion scattering, theory of nucleon transfer, molecular wave functions, two-center shell model, collective and single-particle excitation.
Inner-shell ionization induced by nuclear Coulomb excitation in collisions of very heavy ions
(1978)
K- and L-shell ionization of 238U with Xe and U projectiles is investigated. Internal conversion following nuclear Coulomb excitation which is particularly important for deformed heavy nuclei is compared with direct ionization of inner-shell electrons in superheavy quasimolecules. Both processes exhibit different impact-parameter dependences. As a result of internal conversion, about 0.1-0.3 K holes per central collision are created.
On the basis of the two-center shell model a theory is developed for the excitation of loosely bound nucleons in heavy ion collisions. These nucleons move in the two-center shell model potential generated by all the nucleons and are described by molecular wave functions. The model is applied to calculate the cross sections for the elastic and inelastic 13C-13C scattering. The cross sections show intermediate structures caused by the excitation of quasibound resonances in the molecular nucleus-nucleus potential. NUCLEAR REACTIONS 13C(13C,13C) molecular wave functions, dynamical two-center shell model, quasimolecular resonances, radial and Coriolis coupling, coupled channel calculations for σ(θ).
Quasimolecular resonance structures in the 12C-12C system are studied in the framework of the coupled channel formalism in the energy range Ec.m.=5-14 MeV. The influence of the coupling of the first excited 2+ state in 12C on the resonance structures is investigated by choosing various types of coupling potentials. The intermediate structures in the reflection and transition coefficients and cross sections can be interpreted with the double resonance mechanism. NUCLEAR REACTIONS 12C(12C, 12C), quasimolecular states, coupling potentials, coupled channel calculations for σ(θ).
The theory of direct electron-positron pair production in the collision of heavy ions is formulated in the framework of the quasimolecular model. The pair production process acquires a collective nature for (Z1+Z2)α>1 and can be understood as the shakeoff of the strong vacuum polarization cloud formed in the quasimolecule. The total cross section is, e.g., 76 μb for Pb + Pb at Coulomb barrier energies.
Internal conversion of γ rays from Coulomb-excited nuclear levels cannot be neglected compared with the spontaneous and induced positron production in overcritical electric fields. It is shown that both processes are separable by their different distributions with respect to the ion angle and the positron energy.
We show that information about quasimolecular electronic binding energies in transient atomic systems of Z=Z1+Z2 up to 184 can be obtained from three sources: (1) the impact-parameter dependence of the ionization probability; (2) the ionization probability in head-on collisions as a function of total nuclear charge Z; (3) the delta-electron spectrum in coincidence with K-vacancy formation in asymmetric collisions. Experiments are proposed and discussed.
Determination of the effective 12C + 12C potential from the sub-Coulomb single-particle resonances
(1974)
The sub-Coulomb resonances observed in the total reaction yield of the 12C + 12C system at 4.9, 5.6, and 6.2 MeV are explained as single-particle resonances. The "true" effective 12C + 12C potential is determined directly as the real potential which reproduces best the position and the spacing of the observed sub-Coulomb resonances. This potential is found from a parametrization of the two limiting adiabatic and sudden potentials.
the development of the mass asymmetry vibrations in the final stages of the fission process is studied with an approximate treatment of the coupling to relative motion. A parametrized friction is introduced and its effects are studied. Numerical results are presented for 236U, together with estimates for the kinetic energy of the fragments. RADIOACTIVITY, FISSION 236U; calculated mass distribution, kinetic energy distribution. Collective dynamics, shell correction method, cranking model.
It is shown that nuclear matter is compressed during the encounter of heavy ions. If the relative velocity of the nuclei is larger than the velocity of first sound in nuclear matter (compression sound for isospin T=0), nuclear shock waves occur. They lead to densities which are 3-5 times higher than the nuclear equilibrium density ρ0, depending on the energy of the nuclei. The implications of this phenomenon are discussed.
The mechanisms of spontaneous and induced emission of radiation are derived from the Dirac equation in a rotating coordinate system. The molecular-orbital x-ray spectra exhibit a strong asymmetry with respect to the beam axis. The asymmetry peaks for the high-energy transitions, which can be used for spectroscopy of two-center orbitals.
In critical or nearly critical heavy-ion collisions, induced as well as spontaneous energyless e-e+ pair creation result in the decay of the neutral vacuum. Induced transitions from the negative-energy continuum into a vacant molecular 1s level can occur even in the absence of diving and produce a substantial enhancement and broadening of the previously considered spontaneous positron spectrum. Total cross sections of 5 b have been calculated for U-U collisions.
With the mass asymmetry described by the dynamical collective fragmentation coordinate ξ, and with use of the asymmetric two-center shell model, the fission mass distributions for 226Ra, 236U, and 258Fm (which are typical representatives for triple-, double-, and single-humped distributions) are explained.
In view of new high-precision experiments in atomic physics it seems necessary to reexamine nonlinear theories of electrodynamics. The precise calculation of electronic and muonic atomic energies has been used to determine the possible size of the upper limit Emax to the electric field strength, which has been assumed to be a parameter. This is opposed to Born's idea of a purely electromagnetic origin of the electron's mass which determines Emax. We find Emax≥1.7×1020 V/cm.
A general formalism for the scattering of heavy ions, which is especially convenient to study the antisymmetrization effects, is developed. Antisymmetrization effects are investigated by expanding the completely antisymmetrized wave function according to the number of exchanged nucleons. The particle-core model for the scattering of nuclei with loosely bound nucleons is presented. A formula for the additional contribution to the effective potential due to antisymmetrization effects is obtained by calculating the expectation value of the Hamiltonian with intrinsic wave functions. Application of the formalism is illustrated for the 14N + 14N scattering problem and its usefulness is demonstrated.
A fully gauge-invariant, Lorentz-covariant, nonlocal, and nonlinear theory, for coupled spin-½ fields, ψ, and vector fields, A, i.e., "electrons" and "photons," is constructed. The field theory is linear in the ψ fields. The nonlinearity in the A fields arises unambiguously from the requirement of gauge invariance. The coordinates are generalized to admit hypercomplex values, i.e., they are taken to be Clifford numbers. The nonlocality is limited to the hypercomplex component of the coordinates. As the size of the nonlocality is reduced toward zero, the theory goes over into the inhomogeneous Dirac theory. The nonlocality parameter corresponds to an inverse mass and induces self-regulatory properties of the propagators. It is argued that in a gauge-invariant theory a graph-by-graph convergence is impossible in principle, but it is possible that convergence may hold for the complete solution, or for sums over classes of graphs.
The 1s bound state of superheavy atoms and molecules reaches a binding energy of -2mc2 at Z≈169. It is shown that the K shell is still localized in r space even beyond this critical proton number and that it has a width Γ (several keV large) which is a positron escape width for ionized K shells. The suggestion is made that this effect can be observed in the collision of very heavy ions (superheavy molecules) during the collision.
A continuum shell-model calculation based on the collective correlation model has been made for the giant resonance of 12C using the eigenchannel reaction theory. The low-lying negative-parity states of 11C and 11B have been taken into account by corehole coupling. Partial, total, and integrated photoabsorption cross sections are calculated for the region of the giant dipole resonance.
With the use of the cranking formula, the coordinate-dependent mass parameters of the kinetic-energy operator in fission processes and heavy-ion collisions are calculated in the two-center oscillator model. It is shown that the reduced mass and also the classical moment of inertia are obtained for large separations of the fragments. For small separations, however, the mass parameter for the motion of the centers of mass of the fragments is larger than the reduced mass by an order of magnitude.
An upper limit to the electric field strength, such as that of the nonlinear electrodynamics of Born and Infeld, leads to dramatic differences in the energy eigenvalues and wave functions of atomic electrons bound to superheavy nuclei. For example, the 1s1/2 energy level joins the lower continuum at Z=215 instead of Z=174, the value obtained when Maxwell's equations are used to determine the electric field.
The dynamic collective model has been extended to quadrupole giant resonances in spherical nuclei. The splitting of giant dipole and giant quadrupole resonances due to their coupling to surface vibrations has been calculated for Sn isotopes. Agreement with recent γ-absorption measurements of the Livermore group has been found.
A two-center shell model with oscillator potentials, l→·s→ forces, and l→2 terms is developed. The shell structures of the original spherical nucleus and those of the final fragments are reproduced. For small separation of the two centers the level structure resembles the Nilsson scheme. This two-center shell model might be of importance in problems of nuclear fission.
Higher-order effects are calculated in the framework of the eigenchannel theory for elastic and inelastic electron-nucleus scattering in the energy region 100≤E≤250 MeV. A dispersion effect of about 12% is found for the elastic scattering on Ni58 at a momentum transfer q≈500 MeV/c. For inelastic scattering, the reorientation effect is discussed, in addition to the dispersion effect. The total higher-order effect changes the form factor for a hindered first-order transition by 50% at its minima. Furthermore, the dependence of the higher-order effects on the transition potentials of the virtual excitations, the model dependence, and the dependence on the energy E of the electron and the momentum transfer q are discussed. A closed formula for the S matrix is developed by calculating the eigenchannels in stationary perturbation theory.
With a schematic model for the nuclear matter we give a unified treatment of the real and imaginary parts of the elastic O16-O16 scattering potential. The model connects the parameters of the potential with the density and binding properties of the O16-O16 system and reproduces the structure of the excitation function quite well. It is shown that the nuclear compressibility can be obtained from the scattering data, and in the case of the S32 compound system there results an effective compressibility (finite quenching of the nuclei) of about 200 MeV.
The total particle-particle SJ matrix of O16 for spin J=1- and excitation energies between 15 and 27 MeV has been calculated in the eigenchannel reaction theory for several parameters of the Saxon-Woods potential and the two-body force. The many-body problem has been treated in the 1-particle-1-hole approximation. The photon channels have been included by perturbation theory. Surprisingly, the most important structure of the experimental cross sections is reproduced quite well in this simple approximation.
The Coulomb-fission cross sections for 132Xe and 148Nd incident on 238U are calculated in a dynamical classical model. In particular the influence of nuclear forces on the cross sections is studied. Since they are counteracting the Coulomb force, they diminish the cross sections for Coulomb fission significantly and shift the Coulomb barrier towards lower energies.
The theory of collective correlations in nuclei is formulated for giant resonances interacting with surface vibrations. The giant dipole states are treated in the particle-hole framework, while the surface vibrations are described by the collective model. Consequently, this treatment of nuclear structure goes beyond both the common particle-hole model (including its various improvements which take ground-state correlations into account) and the pure collective model. The interaction between giant resonances and surface degrees of freedom as known from the dynamic collective theory is formulated in the particle-hole language. Therefore, the theory contains the particle-hole structures and the most important "collective intermediate" structures of giant resonances. Detailed calculations are performed for 12C, 28Si, and 60Ni. A good detailed agreement between theory and experiment is obtained for all these nuclei, although only 60Ni is in the region where one would expect the theory to work well (50< A <110).
The influence of the Coulomb and nuclear forces on the Coulomb barrier in heavy-ion reactions is studied in a dynamical classical model. It is shown that the fusion barrier is smaller than the conventional Coulomb barrier of two underformed nuclei. The model yields a dynamical picture of the excitation mechanism of surface vibrations and giant resonances. It is suggested that-due to nuclear forces-the excitation of the octupole mode is strongly enhanced over the excitation of the quadrupole mode in experiments at the Coulomb barrier.
Continuum structure of Ca40
(1967)
The total S1- matrix of Ca40 has been calculated for excitation energies between 11 and 28 MeV. As typical results, the (γ, p0) and the total absorption cross sections are shown and compared with experiments. It is shown that the proper treatment of the one-particle, one-hole shell-model continuum accounts for most of the observed structures.
Using the eigenchannel reaction theory we performed coupled-channel calculations for Si28 and computed the differential cross section for Al27(p, γ0)Si28 over the energy range 6 MeV<Ep <16 MeV. The obtained angular distributions are nearly constant over the whole energy range and agree with the experiment in that they are almost isotropic. Thus, it seems that in this framework we can give a natural explanation for the peculiar behavior of the Al27(p, γ0)Si28 cross section.
The theory of Raman scattering is extended to include electric-quadrupole radiation. The results obtained are used to compute the elastic and Raman scattering cross sections of heavy deformed nuclei. The dipole and quadrupole resonances are described by a previously developed theory which includes surface vibrations and rotations. The computed cross sections are compared with experimental data for all those nuclei where both absorption and scattering cross sections are available. Some discrepances still exist in certain details; however, the over-all agreement between theory and experiment is very good.
In a collective treatment the energies of the giant resonances are given by the boundary conditions at the nuclear surface, which is subject to vibration in spherical nuclei. The general form of the coupling between these two collective motions is given by angular-momentum and parity conservation. The coupling constants are completely determined within the hydrodynamical model. In the present treatment the influence of the surface vibrations on the total photon-absorption cross section is calculated. It turns out that in most of the spherical nuclei this interaction leads to a pronounced structure in the cross section. The agreement with the experiments in medium-heavy nuclei is striking; many of the experimental characteristics are reproduced by the present calculations. In some nuclei, however, there seem to be indications of single-particle excitations which are not yet contained in this work.
The modes and frequencies of the giant quadrupole resonance of heavy deformed nuclei have been calculated. The quadrupole operator is computed and the absorption cross section is derived. The quadrupole sum rule is discussed, and the relevant oscillator strengths have been evaluated for various orientations of the nucleus. The giant quadrupole resonances have energies between 20 and 25 MeV. The total absorption cross section is about 20% of the giant dipole absorption cross section. Of particular interest is the occurrence of the quadrupole mode which is sensitive to the nuclear radius in a direction of approximately θ=(1/4)π from the symmetry axis. This may give information on the details of the nuclear shape.
A method is proposed by which the eigenstates and the eigenvalues of the S matrix, i.e., the eigenchannels, can be directly computed from the nuclear problem, for example, from the shell model. The calculation of all cross sections, viz., partial and total cross sections, is then exceedingly simple. The characteristics of the eigenchannels are described and the relation with other reaction theories is briefly discussed.
The unified model and the collective giant-dipole-resonance model are unified. The resulting energy spectrum and the transition probabilities are derived. A new approximate selection rule involving the symmetry of the γ vibrations is established. It is verified that the main observable features in the photon-absorption cross section are not influenced by the odd particle, despite the considerably richer spectrum of states as compared to even-even nuclei.
In heavy nuclei the damping of the giant resonance is due to thermalization of the energy rather than to direct emission of particles; the latter process is strongly inhibited by the angular-momentum barrier. The thermalization proceeds via inelastic collisions leading from the particle-hole state to two-particle-two-hole states. In heavy nuclei, several hundred such states are available at the energy of the giant dipole resonance. The rather large width of the giant resonance arises from the addition of many small partial widths of channels leading to the different two-particle-two-hole states. Both the density of the two-particle-two-hole states and the mean value of the interaction matrix elements between the particle-hole and two-particle-two-hole states are evaluated in a simplified square-well shell model. In a given nucleus the energy dependence of the widths is determined mainly by the density of states; the A dependence is determined mainly by the size of the matrix elements. For A ~ 200, we find 0.5 <= Γ <=2.5 MeV. The uncertainty in this value comes mostly from the uncertainty in the strength of the interaction. Representing the energy dependence of the width by a power law we find for the exponent the value ~ 1.8.
A method is developed for the calculation of resonant nuclear states which preserves as many features of the shell model as possible. It is an extension of the R-matrix theory. The necessary formulas are derived and a detailed description of the computational procedure is given. The method is valid up to the two-particle emission threshold. With the assumption of consecutive decay of the nucleus, the two-particle emission process can also be described. The treatment is antisymmetrized in all particles.
The energies of, and transition probabilities involving, the ground-state rotation bands of Os186, Os188, and Os190 are compared with a diagonalized rotation-vibration theory in which vibrations are considered to three phonon order. Agreement even in the Os transition region is found to be excellent. The theory appears to be particularly successful in predicting two phonon states in Os190.
The rotation-vibration model and the hydrodynamic dipole-oscillation model are unified. A coupling between the dipole oscillations and the quadrupole vibrations is introduced in the adiabatic approximation. The dipole oscillations act as a "driving force" for the quadrupole vibrations and stabilize the intrinsic nucleus in a nonaxially symmetric equilibrium shape. The higher dipole resonance splits into two peaks separated by about 1.5-2 MeV. On top of the several giant resonances occur bands due to rotations and vibrations of the intrinsic nucleus. The dipole operator is established in terms of the collective coordinates and the γ-absorption cross section is derived. For the most important 1- levels the relative dipole excitation is estimated. It is found that some of the dipole strength of the higher giant resonance states is shared with those states in which one surface vibration quantum is excited in addition to the giant resonance.