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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 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.
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.
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.
Some quantitative data about the carbon-metabolism in Saccharomyces-cells of different ploidy were determined. The amount of carbon, necessary for the formation of a cell, proved to be proportional to the degree of ploidy of the cells. For the duplication of a diploid cell 6,7·10-11g glucose were used. In comparison with respiratory deficient cells the simultaneous utilization of fermentation and respiration metabolism in respiration sufficient cells leads to a decrease of the cell cycle duration, however, the energy needed for the formation of a cell is not decreased. The rate of cell multiplication has a maximum at about 30 °C for all classes of ploidy. Certain assumptions about the utilization of the carbon source were confirmed by experiments with 14C marked glucose.
Die Wirkungsweise des Protonen-Wendellinearbeschleunigers und sein prinzipieller Aufbau werden kurz beschrieben und Angaben über die erreichbare Parallelimpedanz gemacht. Es wird gezeigt, wie durch sektionsweisen Aufbau eine variable Endenergie erzielt und die Kühl- und Fokussierprobleme gelöst werden können. Zwei Konstruktionsbeispiele für kontinuierlichen und gepulsten Betrieb werden näher ausgeführt.
Physikalische und thermische Kontrastierung führt bei Fixierung in Glutaraldehyd und Einbettung in Vestopal bei Parenchymzellen der Leber zu weitgehend ähnlichen Kontrastunterschieden auch bei Mitochondrien und den Membranen des Retikulums. Beide Verfahren wirken also weitgehend unspezifisch. Von den chemischen Verfahren liefert Uranylacetat im Cytoplasma ähnliche Kontrastverhältnisse wie die beiden genannten Verfahren. Das spezifische Verhalten des Uranylacetats kann z. B. an der Kontrastierung des Chromatins demonstriert werden. Sie bleibt aus, wenn die färbbare Substanz auf der Wasseroberfläche des Messertroges herausgewaschen wurde. Bleicitrat-Kontrastierung hat hier im Gegensatz zu Uranylacetat eine spezifische Wirkung nur auf RNS-haltige Zellbestandteile.
In this paper an instability calculation is given for an axially symmetric gas distribution which has a differential rotation and in which a magnetic field is present. It is a generalization of similar calculations given by CHANDRASEKHAR and BEL and SCHATZMAN. The generalization becomes necessary for the study of problems of the formation of planetary systems, and star formation.
The instability conditions and the critical wave lengths are calculated for plane-wave-like disturbances. For disturbances running perpendicularly to the axis of rotation instability can occur only if the gas density exceeds a critical value which depends on the differential rotation at the considered distance only as long as pressure gradients and gradients of the magnetic field strength are negligible. If the gas density exceeds this critical value the shortest unstable wave length is proportional to the square root of vT2+vB2, where vT means the velocity of sound and vB the ALFVÉN-velocity.
For disturbances running parallel to the axis of rotation in addition to the JEANS instability a new type of instability occurs due to the simultaneous action of the magnetic field and the differential rotation; for rigid rotation this instability vanishes.
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.
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.
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 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.
An elementary derivation of the optical potential for high energies is given. For the determination of the optical potential only the knowledge of the scattering amplitude for free nucleons and of the autocorrelation function for density fluctuations is necessary. The numerical calculation of the real- and imaginary part of the optical potential was performed using the Tabakin potential.
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.
Theoretical studies in the shell model have led to the conclusion that the shape dependence of the liquid-drop part of the semi-empirical mass formula of the Weizsaecker-Bethe type should contain terms proportional to the volume, the surface, and the mean-total curvature of the surface of the drop, respectively. Now the surface tension beta_e and the curvature tension gamma_e are fitted to the experimentally known fission barriers of 35 nuclei. Furthermore, the parameters of the liquid-drop part of the mass formula are roughly fitted to the ground-state masses of about 600 beta-stable nuclei. For the elementary radius r_e, the value 1.123 fm ( determined by Elton ) is used. As a result, gamma_e should be in the range 6-8 MeV, with the value 6.8 MeV being the most probable, thus beta_e=17.85 MeV. For sufficiently large values of the curvature tension ( e.g. gamma_e=13.4 MeV ), a small double-hump fission barrier occurs in the region of Ra.
The often discussed question concerning the energy-momentum tensor of the electromagnetic field in matter can be answered using NOETHER'S theorem. The separation of the electromagnetic system from the mechanical system introduced here leads to the asymmetric expression for the energy momentum tensor. From covariance with respect to scale transformations one further concludes that the trace of the energy-momentum tensor vanishes.
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.
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.
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.
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.
We examine the possibility of reformulating quantum theory (QT) as a deterministic ensemble theory which (a) interprets observables as objective properties of physical systems and (b) coincides with QT in all quantitative statements. As will be demonstrated, such an Ensemble-Quantum-Theory (EQT) can only be constructed if (1) one accepts a modified observable-concept, and (2) as long as the theory of measurement is left out of account. A correct treatment of the measuring process is impossible within such an EQT. Consequently, there exist no Hidden-Variable Theories with the properties (a) and (b).
The potential energy surface has been calculated by two methods which are compared with respect to spontaneous fission. In the first one essentially the sum of the single particle energies is computed as was done in a previous paper3 while in the second one the Strutinsky technique of renormalizing to a liquid drop model has been applied. Also the half-lives for electron capture are investigated together with the predictions of the half-lives for spontaneous fission and α-decay. The results support the existence of superheavy nuclei in the regions around Z = 114 and Z = 164.
The meaning of a recently proposed formalism for quantization of interacting fields is discussed by studying the consequences of the time-dependent unitary transformation which is essential for this approach. It turns out that non-relativistic quantum electrodynamics in dipole approximation may serve as a useful, although rather singular, example for this method. In the relativistic case a different point of view is suggested in order to avoid inconsistent interpretation. It is further possible to give arguments for a reasonable choice of the unitary transformation concerned.
The nuclear charge form factgr from the high-energy elastic electron scattering on 6Li has been calculated from the modified independent-particle shell model (IPSM) wave function. The usual harmonic oscillator type IPSM wave function has been modified by the inclusion of a nucleon-nucleon correlation function which involves extra-core nucleons only. The technique is extremely simple and provides an excellent agreement with the experimental data.
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.
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 elastic alpha scattering to backward angles has been studied for 40,42,44,48Ca between 40.7 and 72.3 MeV. The cross sections for 40Ca are larger than those for the higher isotopes up to the highest energies. They show backward increases that disappear above 50 MeV. The enhancement factor for 40Ca over 42,44Ca varies smoothly with energy. 48Ca does also show a backward cross-section enhancement over 42,44Ca. alpha -cluster rotational bands in the 44Ti compound state, four-nucleon correlations in 40Ca, and the l-dependent optical model are discussed as approaches to understand the anomaly. The rotator model appears to agree qualitatively with the experimental data. It involves rotational bands extending at least up to J=16 in 44Ti.
Back-angle enhancements of elastic alpha -scattering cross sections have been observed for nuclei at the ends of the 1p, 2s-1d, and f7 / 2 shells. Strong reduction of this enhancement occurs if excess neutrons enter the next open major shell. The results are discussed in terms of intermediate alpha structure.
Es wird das Mikrowellenspektrum von Fluorwasserstoffassoziaten im X-und K-Band bei -70 °C und 0,01 Torr gemessen und analysiert. Dazu wird ein erstelltes Frequenzprogramm für den asymmetrischen Kreisel verwendet, sowie ein Extrapolationsprogramm, das eine in der Literatur angegebene druck-und temperaturabhängige Verteilung der Fluorwasserstoffassoziate auf für Mikrowellenspektroskopie geeignete Drücke und Temperaturen umzurechnen erlaubt. Es zeigt sich, daß planare hexamere und heptamere Fluorwasserstoffassoziate vorliegen mit F-F-F-Winkeln von etwa 104° und H-F-Bindungslängen von 0,9997 Å bzw. 0,9640 Å. Die Längen der Wasserstoff brücken sind 1,4998 Å bzw. 1,6105 Å. Ein Vergleich der Bindungslängen zeigt, daß bei Anlagerung von H-F an (HF)6 eine Kontraktion der Fluorwasserstoffbindung um 3,5% und eine Dilatation der Wasserstoffbrückenbindung um 1% stattfindet. Dieses Ergebnis steht im Einklang mit der oben erwähnten Assoziatverteilung, die eine Minderung der Kettenstabilität beim Übergang von hexamerer zu heptamerer Kette erwarten läßt.
We investigate the possibility of selfconsistent solutions for antiferromagnetism in the Hubbard model in the decoupling of the Greens functions introduced by Hubbard in his first paper. On the base of this approximation Arai has calculated the band splitting for antiferromagnetism, but, as will be shown in this paper, Hubbard's approach fails to yield antiferromagnetism for nearest neighbour hopping in the same way as it does not yield ferromagnetism, and no selfconsistent solutions of the problem beyond the well known paramagnetic solution do exist.
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.
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.
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 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.
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.
The convergence of the eigenchannel (EK) and R-matrix (RM) theories - depending on their free parameters - is numerically investigated by comparing with a simple, analytically solvable model. The comparison shows that the results of the EK-theory are in general closer to the exact solution than those of the RM-theory. Especially in regions of sharp resonances the dependence of the expansion of the wave functions on the interaction radius ac and on the boundary condition Bc is very strong in the RM-theory and one needs very many levels to achieve good results.
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.
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.
A scattering theory for reactions with three-particle channels above the two-particle threshold is developed. The S-matrix-technique is used for the calculation of the extended S-matrix. Correlated two-particle wave functions in the exit channels are employed to describe the exact two-particle continuum. For the usual shell model only a few partial waves dominate. The cross section depends on the energy-distribution between the two outgoing nucleons. Numerical results are presented for the model (d, 2n)-reaction exciting 0+ -states in O16 without Coulomb-effects. The treatment is restricted to three (2p2h)-states with the particles in the (sd)-shell and holes in the p-shell.
The extension of the nuclear two-centre-oscillator to three and four centres is investigated. Some special symmetry-properties are required. In two cases an analytical solution of the Schrödinger equation is possible. A numerical procedure is developed which enables the diagonalization of the Hamiltonian in a non-orthogonal basis without applying Schmidt's method of orthonormalization. This is important for calculations of arbitrary two-dimensional arrangements of the centres.
Within the framework of the pairing plus quadrupole interaction model and by using the technique of quasi spin formalism it is possible to determine the collective potential and kinetic energy surfaces as analytic functions of the particle number in the limit that single particle splittings are neglected. Pushing the quasi spin model in an extended version up to the 4th order in perturbation theory the stiffness and mass parameters of harmonic and anharmonic terms for Dy, Er, Yb, and Hf-isotopes have been calculated. The theoretical particle dependence of collective quantities shows a good qualitative and even quantitative agreement with experimental data and former calculations.
A careful investigation of different corrections to binding energies of electrons in almost critical fields is performed. We investigate quantitatively the influence of the nuclear charge parameters, nuclear mass, degree of ionization on the value of the critical charge of the nucleus. Rather qualitative arguments are given to establish the contribution of the quantumelectrodynamic corrections, which are found to be small. Some phenomenological modifications of QED are quantitatively investigated and found to be of negligible influence on the value of the critical field. For heavy ion collisions with Z1+Z2>Zcr the critical separations between ions are given as results of precise solutions of the relativistic two coulomb center problem. Corrections due to electron-electron interaction are considered. We find (with present theoretical accuracy) Zcr=173±2, in the heavy ion collisions Rcr(U-U) = 34.7±2 fm and Rcr (U-Cf)=47.7±2 fm. We shortly consider the possibility of spontaneous muon production in muonic supercritical fields.
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.
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.
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.
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.
Introducing correlated continuum wave functions for the two- and re-particle-continuum a microscopic theory of nuclear reactions based on a method of Fano is developed. The S-matrix-elements are given by the matrix-elements between correlated continuum wave functions and bound state wave functions. The antisymmetrization of the continuum wave functions with more than one particle in the continuum is included. The theory can be straightforwardly applied on the n-nucleon-emission process following photo- and particle excitations.
Energy spectra and angular distributions have been measured of 3He and 4He fragments emitted from Ag and U targets, bombarded with 2.7-GeV protons, and 1.05-GeV/nucleon alpha particles and 16O ions. All cross sections increase dramatically with projectile mass. No narrow peaks are found in the angular distributions or in the energy spectra.
A model for the quantum yield of the coloration caused by UV-light in spiropyran layers is described. This model allows to calculate the sensitivity of layers having different compositions. The mechanism concerning the stability of the coloration is essentially clarified. Calculations of the stability for layers of different compositions are possible by a model describing the mechanism approximately.
Angular distributions for elastic and inelastic transitions in 20Ne + 16O scattering have been measured at E(20Ne)=50 MeV. For the 0+, 2+, and 4+ members of the 20Ne ground-state rotational band, the angular distributions exhibit pronounced backward peaking characteristic of an alpha -cluster exchange mechanism. The analysis of the ground-state transition in the first-order elastic transfer model yields no satisfactory fit although microscopic cluster form factors and full recoil corrections are employed. A coupled channels calculation for the 0+, 2+, and 4+ transitions reveals very strong coupling effects, indicating that the coherent superposition of first-order optical model and distorted-wave Born-approximation amplitudes may not be an adequate model for these reactions. NUCLEAR REACTIONS 16O(20Ne, 16O) and 16O(20Ne, 20Ne), elastic and inelastic transfer; E=50MeV; measured sigma (Ef , theta ); optical model + DWBA, and CCBA analyses.
A simple model is proposed for the emission of nucleons with velocities intermediate between those of the target and projectile. In this model, the nucleons which are mutually swept out from the target and projectile form a hot quasiequilibrated fireball which decays as an ideal gas. The overall features of the proton-inclusive spectra from 250- and 400-MeV/nucleon 20Ne ions and 400-MeV/nucleon 4He ions interacting with uranium are fitted without any adjustable parameters.
Double-differential cross sections have been measured for high-energy p, d, t, 3He, and 4He particles emitted from uranium targets irradiated with 20Ne ions at energies of 250, 400, and 2100 MeV/nucleon and 4He ions at 400 MeV/nucleon. By using the shape and yield of the proton energy spectra, the shape and yield of the d, t, 3He, and 4He energy spectra can be deduced at all measured angles for all incident projectile energies by assuming that they are formed by a coalescence of cascade nucleons, using a model analogous to that of Butler and Pearson, and Schwarzschild and Zupancic-caron.
An improved two-center model has been used to describe the elastic scattering potentials for the collision of identical nuclei. The macroscopic-microscopic approach includes liquid drop (LD) deformation energies, shell corrections and pairing energy corrections. As basis for the microscopic part a two-center shell model has been used with a Hamiltonian including a Thomas-type spin-orbit potential and an l2 correction term. The model is applied to the sudden and adiabatic type of scattering process, including a compression energy term in the LD part for the former case. Results are given as potential energy surfaces for the adiabatic scattering process and potential energy curves for the sudden scattering process.
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.
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.
During collisions of heavy ions with heavy targets below the Coulomb barrier, adiabatic molecular orbitals are formed for the inner electrons. Deviations from adiabaticity lead to coupling between various states and can be treated by time-dependent perturbation theory. For high charges ( Z1+Z2 ≧ 60) the molecular electrons are highly relativistic. Therefore, the Dirac equation has to be used to obtain the energies and wave functions. The Dirac Hamiltonian is transformed into the intrinsic rotating coordinate system where prolate spheroidal coordinates are introduced. A set of basis functions is proposed which allows the evaluation of all matrix elements of the Dirac Hamiltonian analytically. The resulting matrix is diagonalized numerically. The finite nuclear charge distribution is also taken into account. Results are presented and discussed for various characteristic systems, e. g. Br-Br, Ni-Ni, I-I, Br-Zr, I-Au, U -U, etc.
The energy spectra of protons and light nuclei produced by the interaction of 4He and 20Ne projectiles with Al and U targets have been investigated at incident energies ranging from 0.25 to 2.1 GeV per nucleon. Single fragment inclusive spectra have been obtained at angles between 25° and 150°, in the energy range from 30 to 150 MeV/nucleon. The multiplicity of intermediate and high energy charged particles was determined in coincidence with the measured fragments. In a separate study, fragment spectra were obtained in the evaporation energy range from 12C and 20Ne bombardment of uranium. We observe structureless, exponentially decaying spectra throughout the range of studied fragment masses. There is evidence for two major classes of fragments; one with emission at intermediate temperature from a system moving slowly in the lab frame, and the other with high temperature emission from a system propagating at a velocity intermediate between target and projectile. The high energy proton spectra are fairly well reproduced by a nuclear fireball model based on simple geometrical, kinematical, and statistical assumptions. Light cluster emission is also discussed in the framework of statistical models. NUCLEAR REACTIONS U(20Ne,X), E=250 MeV/nucl.; U(20Ne,X), U(α,X) E=400 MeV/nucl.; U(20Ne,X), Al(20Ne,X), E=2.1 GeV/nucl.; measured σ(E,θ), X=p, d, t, 3He,4He. U(20Ne,X), U(α,X), E=400 MeV/nucl.; U(20Ne,X), E=2.1 GeV/nucl.; measured σ(E, θ), Li to O. U(20Ne,X), U(12C,X), E=2.1 GeV/nucl.; measured σ(E, 90°), 4He to B. Nuclear fireballs, coalescence, thermodynamics of light nuclei production.
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.
Coupled channel calculations for Coulomb and nuclear excitation of the systems 136Xe-238U and 238U-238U have been performed using the rotation-vibration model. The impact parameter-, energy-and spin-dependence of the excitation probabilities are discussed for the ground state-, β-and γ-band up to Jπ = 36+. It is shown that the energy levels and quadrupole matrix elements are strongly influenced by the rotation-vibration interaction. Analytic expressions for the elastic and coupling potentials are presented.
Light-particle accompanied fission is expected to yield results from which one hopes to learn more about binary scission configurations. As a step in this direction, we present a model that allows the calculation of the probabilities with which a given three-particle setup follows from different binary configurations. First results show the workability of the model.
The temporal development of macroobservables is described within a correlation-functionformalism. The results are exact for a certain class of initial ensembles. The same problem is discussed with the help of the linear-response-formalism. The results agree under certain conditions which should be fulfilled for macroobservables.
For experiments on fission-fragment induced desorption the detection of significant correlations between desorbed ions has been reported [1]. In this paper the method for the detection and quantitative description of these correlations will be described. The statistics of the desorption-process leads to equations for mass-line intensities of ion spectra. Using a time-to-amplitude-converter for flight-time measurements these intensities depend on interdependences of different ions desorbed by the same fission-fragment. The equations allow the computation of correlationcoefficients whose interdependence with desorption probabilities of the respective ions can be shown in Venn-diagrams. Results are given and an interpretation is suggested for fission-fragment desorbed thiamine molecular and fragment ions.
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.
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).
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.
The master operators B which cause the entropy production dH/dt = - k-1 dS/dt to become extremal for fixed statistical operators W are constructed and discussed. There are boundaries of the set B of master operators, B = {B | Σ B2vu = b} for which the problem is solvable yielding minimal entropy production, while no solution exists in the set B without any constraints. Operators with maximal entropy production must be extremal points of B.
Two equations for the macroscopic part W of the statistical operator are considered:
1. the master equation W = — MW, t
2. the exact equation W = — J K(t — r) W (r) dr.
It follows from the physical equivalence of the solutions together with a stability assumption and the assumption that there is a time τ* after which also the derivatives of the solutions are equivalent, that τ* is the life-time of the kernel K and that Conversely, the equivalence of the solutions follows from assumptions on the life-time of the kernel K together with a stability assumption and a smoothness assumption on the initial statistical operator W(0).
The reactions of diluted aqueous solutions of SO2 resp. HSO3-ions with MnO4-or Ce4+ ions in the pH range 1-4 produce chemiluminescence in the spectral region of 450-600 nm. Measurements of the time course of the light emission and their simulation on an analog computer led to a reaction scheme in which a recombination product of primarily formed HSO3 radicals -of a lifetime of about 1 second -appears as precursor of electronically excited SO2 molecules. The participation of singlet oxygen can be excluded because at least the reaction with Ce4+ ions proceeds also in the absence of oxygen.
Pion-production cross sections have been measured for the reaction 40Ar+40Ca--> pi ++X at a laboratory energy of 1.05 GeV/nucleon. A maximum in the pi + cross section occurs at mid-rapidity, which is anomalous relative to p+p and p+nucleus reactions and compared to many other heavy-ion reactions. Calculations based on cascade and thermal models fail to fit the data.
In the present paper we develop the essential theoretical tools for the treatment of the dynamics of High Energy Heavy Ion Collisions. We study the influence of the nuclear equation of state and discuss the new phenomena connected with phase transitions in nuclear matter (pion condensation). Furthermore we investigate the possibility of a transition from nuclear to quark matter in High Energy Heavy Ion Collisions. In this context we discuss exotic phenomena like strongly bound pionic states, limiting temperatures, and exotic nuclei.
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.
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.
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.
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〉.
Proton emission in relativistic nuclear collisions is examined for events of low and high multiplicity, corresponding to large and small impact parameters. Peripheral reactions exhibit distributions of protons in agreement with spectator-participant decay modes. Central collisions of equal-size nuclei are dominated by the formation and decay of a fireball system. Central collisions of light projectiles with heavy targets exhibit an enhancement in sideward emission which is predicted by recent hydrodynamical calculations.
Inclusive energy spectra of protons, deuterons, and tritons were measured with a telescope of silicon and germanium detectors with a detection range for proton energies up to 200 MeV. Fifteen sets of data were taken using projectiles ranging from protons to 40Ar on targets from 27Al to 238U at bombarding energies from 240 MeV/nucleon to 2.1 GeV/nucleon. Particular attention was paid to the absolute normalization of the cross sections. For three previously reported reactions, He fragment cross sections have been corrected and are presented. To facilitate a comparison with theory the sum of nucleonic charges emitted as protons plus composite particles was estimated and is presented as a function of fragment energy per nucleon in the interval from 15 to 200 MeV/nucleon. For low-energy fragments at forward angles the protons account for only 25% of the nucleonic charges. The equal mass 40Ar plus Ca systems were examined in the center of mass. Here at 0.4 GeV/nucleon 40Ar plus Ca the proton spectra appear to be nearly isotropic in the center of mass over the region measured. Comparisons of some data with firestreak, cascade, and fluid dynamics models indicate a failure of the first and a fair agreement with the latter two. In addition, associated fast charged particle multiplicities (where the particles had energies larger than 25 MeV/nucleon) and azimuthal correlations were measured with an 80 counter array of plastic scintillators. It was found that the associated multiplicities were a smooth function of the total kinetic energy of the projectile. NUCLEAR REACTIONS U(20Ne,X), E / A=240 MeV/nucleon; U(40Ar,X), Ca(40Ar,X), U(20Ne,X), Au(20Ne,X), Ag(20Ne,X), Al(20Ne,X), U(4He,X), Al(4He,X), E / A=390 MeV/nucleon; U(40Ar,X), Ca(40Ar,X), U(20Ne,X), U(4He,X), U(p,X), E / A=1.04 GeV/nucleon; U(20Ne,X), E / A=2.1 GeV/nucleon; measured sigma (E, theta ), X=p,d,t.
Exclusive pi - and charged-particle production in collisions of Ar+KCl is studied at incident energies from 0.4 to 1.8 GeV/u. Complete disintegration of both nuclei is observed. The correlation between pi - and total charge multiplicity shows no islands of anomalous pion production. For constant numbers of proton participants the pi - multiplicity distributions are Poissons. For central collisions <n pi -> increases smoothly and to first order linearly with the c.m. energy. Disagreement with the firestreak model is found. Pacs numbers: 25.70.Hi, 24.10.Dp
Angular and energy distributions of fragments emitted from fast nucleus-nucleus collisions (Ne--> U at 250, 400, and 800 MeV/N) are calculated with use of nuclear fluid dynamics. A characteristic dependence of the energy spectra and angular distributions on the impact parameter is predicted. The preferential sideward emission of reaction fragments observed in the calculation for nearly central collisions seems to be supported by recent experimental data.
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.
In heavy ion collisions, the molecular single-particle motion may cause specific structures in the energy dependence of the cross sections which arise by the promotion of nucleons at level crossings according to the Landau-Zener excitation mechanism. In order to examine this effect in asymmetric heavy ion collisions, we have calculated level diagrams of the two-center shell model for the target projectile combinations 13C + 16O and 12C + 17O and analyzed with respect to inelastic excitation and neutron transfer. We select certain reactions as possible candidates for showing enhanced cross sections for nucleon excitation and transfer due to real and avoided level crossings near the Fermi level.
Dynamics of Coulomb fission
(1980)
A general formalism is described for the treatment of Coulomb fission, within the framework of the semiquantal theory. We develop a model for the fission probabilities of levels excited in Coulomb excitation. This model contains penetration of the double-humped fission barrier, competition from gamma and neutron emission, and the spreading of the collective states into noncollective compound states. For 74184W + 92238U, the fission probability at θc.m.=180° is increased by a factor of 3.9, 3.3, and 2.0 at E/ECoul=0.77, 0.85, and 0.935, respectively, compared to the simplified sharp cutoff model used in earlier model calculations. The enhancement comes from barrier penetration. The damping of the fission probability due to spreading into noncollective compound states is small. Prompt Coulomb fission (near the distance of closest approach) is studied in a one-dimensional model. The results clearly imply that prompt fission is negligible. We have also studied the sudden approximation for collective rotational levels in connection with Coulomb fission. At high spins (I≈20), it leads to significant errors. Contrary to the basic assumption of the sudden approximation that the nuclear symmetry axis remains fixed during the collision, it is shown that Coulomb excitation results in a strong alignment of the nuclear symmetry axis perpendicular to the beam axis at small internuclear distances. NUCLEAR REACTIONS, FISSION Semiquantal theory of prompt and asymptotic Coulomb fission, study of double-humped barrier penetration, damping effects, neutron and γ emission. Calculated σ(Ep, θc.m.=180°).
Phase transitions in nuclear matter A method for the description of spin-isospin phase transitions in nuclear matter is developed. It allows a complete description of the pion condensation phase transition in the framework of the Landau-Migdal Fermi liquid theory. The equation of the order parameter is derived and the condensation energy is calculated. We study the influence of pion condensation on the nuclear equation of state and the temperature dependence of pion condensation. NUCLEAR STRUCTURE Description of pion-condensed ground state by Green's function technique.