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Wir haben Aussagen über das Eigenwertspektrum der freien Schwingungegleichung für einen Hohlraum B gesucht, welche unabhängig von der Gestalt des Hohlraumes nur von Gestaltparametern abhängen, die als Integrale über B bzw. über dessen Oberfläche ... Eigenschaften von ganz B darstellen, ohne die lokale Struktur der Oberfläche ... zu enthalten. An drei Testkörpern sehr verschiedener Gestalt (die Gestaltparameter waren ebenfalls verschieden), nämlich Würfel, Kugel und Zylinder, haben wir die Hypothese bestätigt, daß der mittlere Verlauf der Größen "Anzahl N und Summe E aller Eigenwerte unterhalb einer willkürlich vorgegebenen Schranke ER" in Abhängigkeit von der Wahl dieser Schranke i.w. gestaltunabhängig ist. Für den Quader lassen sich im Falle asymptotisch großer ER explizite Ausdrücke für N und E angeben, die für alle drei Testkörper nicht nur den mittleren Verlauf von N und E bei kleinen (endlichen) ER in zweiter Näherung (in Potenzen von Ef exp -1/2) richtig wiedergaben, sondern auch als numerische Näherung dss mittleren Verlaufs von N bzw. E brauchbar waren (relative Kleinheit des Restgliedes). Die mathematische Vermutung, daß sich für aS, große Ef eben diese expliziten Ausdrücke für N bzw. E' als gestaltunabhängig erweisen, soll in einer weiteren Arbeit behandelt werden. Das Ergebnis dieser Arbeit ist überall dort anwendbar, wo Eigenschaften des Spektrums der freien Schwingungsgleichung mit Randbedingungen benötigt werden, die sich aus N. bzw. E ableiten lassen; also vor allem in der Akustik (Zahl der Obertöne eines Hohlraumes unterhalb einer vorgegebenen Frequenz), in der Theorie der Hohlleiter usw. In dieser Arbeit haben wir die Anwendung auf ein einfaches Atomkernmodell betrachtet, das Fermigas-Modell. Es beschreibt den Kern als freies ideales in einem Hohlraum von Kerngestalt befindliches Fermigas. Dann bedeutet N die Teilchenzahl und E die Gesamtenergie des Systems. Ef ist die Fermigrenzenergie und es ist (Ef exp 3/2 /6*Pi*Pi) die Sättigungsdichte im Innern des Systems. Der Koeffizient des zweiten Termes des expliziten (aS.) Ausdrucks für E kann dann als Oberflächenspannung gedeutet werden. Die spezifische Hodell-Oberflächenspannung läßt sich in Abhängigkeit von dem Gestaltparametern und der Siittigungsdichte des Atomkernes schreiben. Nach Einsetzen der empirischen Werte erhalten wir numerisch einen Wert, der nur um 20% vom empirisch aus der v. Weizsäckerformel bekannten Wert für die spez. Oberflächenspannung abwich, obgleich das Modell nur eine äußerst einfache Näherung der Kernstruktur sein kann. Daher gelangten wir zu der Überzeugung, daß der Oberflächenanteil der Bindungsenergie wesentlich ein kinetischer Effekt ist.
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.
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.
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.
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.
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.
The surface tension sigma and the surface density thickness t of nuclear matter have been calculated in the Fermi-gas model, the nucleons moving in a self-made shell model potential with a realistic slope and velocity dependence ( parameters alpha and beta ). One gets the experimental values for sigma and t with alpha and beta agreeing with earlier data.
Für ein System ('ideales Gas') von N miteinander nicht wechselwirkenden Teilchen oder Zuständen, deren Wellenfunktionen φ(x) der Randbedingung φ(x)=0 für x aus Ŵ. gehorchen sollen, (W sei dabei die Oberfläche eines geschlossenen Hohlraumes Ŵ beliebiger Gestalt), ist von verschiedenen Autoren eine halbklassische Eigenwertdichteformel angegeben worden. Diese hängt nur linear über die Integrale V ,W und L über Ŵ (Volumen, Oberflächeninhalt und totale Krümmung von Ŵ) von der Gestalt. des Hohlraumes ab. Während von H. Weyl mathematisch bewiesen, werden konnte, daß der führende Volumterm im Gebiet großer Eigenwerte alle folgenden Terme überwiegt, konnte für den Oberflächenterm eine gleichartige Vermutung bisher nur numerisch begründet werden. Von dieser halbklassischen Eigenwertdichteformel ausgehend, werden die thermodynamischen Relationen des idealen Gases aufgebaut und einige Größen wie innere Energie, spezifische Wärme sowie die Oberflächen- und Krümmungs-Spannung für die Grenzfälle starker, ein Gebiet mittlerer und schwacher Entartung explizit berechnet, und zwar sowohl für die Fermi-Dirac als auch die Bose-Einstein-Statistik, als auch für deren klassischen Grenzfall, die Boltzmann-Maxwell-Statistik (s.Diagramm). Ausgenommen wird nur der Spezialfall der Einsteinkondensation, weil hier die (nur im Gebiet großer Eigenwerte gültige) halbklassische Eigenwertdichteformel nicht angewendet werden darf. Die in dieser Arbeit untersuchten quantenmechanisch bedingten Oberflächeneffekte idealer Quantengase sind experimentell bisher wenig untersucht worden; für Molekülgase sind sie verschwindend klein. Die experimentell beobachtete Oberflächenspannung stabiler Atomkerne wird von dem Modell, das den Kern als ideales, entartetes Fermigas der Temperatur T beschreibt, im wesentlichen richtig wiedergegeben. Mit dem in Kap. 3b) abgeleiteten Ausdruck für die Oberflächenspannung stark entarteter idealer Fermigase endlicher Temperatur kann eine Voraussage über die Oberflächenspannung angeregter Atomkerne gemacht werden.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Lambda 's produced in central collisions of 40Ar+KC1 at 1.8-GeV/u incident energy were detected in a streamer chamber by their charged-particle decay. For central collisions with impact parameters b<2.4 fm the Lambda production cross section is 7.6±2.2 mb. A calculation in which Lambda production occurs in the early stage of the collision qualitatively reproduces the results but underestimates the transverse momenta. An average Lambda polarization of -0.10±0.05 is observed. PACS numbers: 25.70 Bc
The 16O ( gamma ,p0) reaction has been studied with linearly polarized bremsstrahlung photons in and below the giant E1 resonance. The parity of the absorbed radiation was determined from the observed azimuthal asymmetry of the emitted protons. Combined with unpolarized measurements the polarized results determine the proton decay amplitudes of the M1 resonance at Ex=16.2 MeV in 16O. The shape of the unpolarized 16O ( gamma ,p3) angular distribution in the giant E1 resonance was derived from the measured analyzing power. NUCLEAR REACTIONS 16O( gamma ,p), E=15-25 MeV; measured analyzing power theta =90° linearly polarized bremsstrahlung; 16O dipole levels deduced pi ; 16.2 MeV 1+ resonance deduced p0 decay amplitudes; 16O GEDR deduced p3 angular distribution.
We present an analysis of high energy heavy ion collisions at intermediate impact parameters, using a two-dimensional fluid-dynamical model including shear and bulk viscosity, heat conduction, a realistic treatment of the nuclear binding, and an analysis of the final thermal emission of free nucleons. We find large collective momentum transfer to projectile and target residues (the highly inelastic bounce-off effect) and explosion of the hot compressed shock zones formed during the impact. As the calculated azimuthal dependence of energy spectra and angular distributions of emitted nucleons depends strongly on the coefficients of viscosity and thermal conductivity, future exclusive measurements may allow for an experimental determination of these transport coefficients. The importance of 4π measurements with full azimuthal information is pointed out.
Collisions of very heavy ions at energies close to the Coulomb barrier are discussed as a unique tool to study the behavior of the electron-positron field in the presence of strong external electromagnetic fields. To calculate the excitation processes induced by the collision dynamics, a semiclassical model is employed and adapted to describe the field-theoretical many-particle system. An expansion in the adiabatic molecular basis is chosen. Energies and matrix elements are calculated using the monopole approximation. In a supercritical (Z1+Z2≳173) quasiatomic system the 1s level joins the antiparticle continuum and becomes a resonance, rendering the neutral vacuum state unstable. Several methods of treating the corresponding time-dependent problem are discussed. A projection-operator technique is introduced for a fully dynamical treatment of the resonance. Positron excitation rates in s1/2 and p1/2 states are obtained by numerical solution of the coupled-channel equations and are compared with results from first- plus second-order perturbation theory. Calculations are performed for subcritical and supercritical collisions of Pb-Pb, Pb-U, U-U, and U-Cf. Strong relativistic deformations of the wave functions and the growing contributions from inner-shell bound states lead to a very steep Z dependence of positron production. The results are compared with available data from experiments done at GSI. Correlations between electrons and positrons are briefly discussed.
Two-particle correlation data are presented for the reaction Ar (800 MeV/ nucleon) + Pb. The experimental results are analyzed in the nuclear fluid dynamical and in a linear cascade model. We demonstrate that the collective hydrodynamical correlations dominate the measured two-particle correlation function for the heavy system studied. We discuss the transition from the early stages of the reaction which are governed by few nucleon correlations, to the later stages with their macroscopic flow which can only be reached using heavy colliding systems. The sensitivity of the correlation data on the underlying compressional dissipative processes is analyzed.
Different collective deformation coordinates for neutrons and protons are introduced to allow for both stretching and γ transitions consistent with experiments. The rotational actinide nuclei 234-238U and 232Th are successfully analyzed in this model. NUCLEAR STRUCTURE 232Th, 234-238U calculated B (E2) values, collective model.
The origin and importance of electron-translation effects within a molecular description of electronic excitations in heavy-ion collisions is investigated. First, a fully consistent quantum-mechanical description of the scattering process is developed; the electrons are described by relativistic molecular orbitals, while the nuclear motion is approximated nonrelativistically. Leaving the quantum-mechanical level by using the semiclassical approximation for the nuclear motion, a set of coupled differential equations for the occupation amplitudes of the molecular orbitals is derived. In these coupled-channel equations the spurious asymptotic dynamical couplings are corrected for by additional matrix elements stemming from the electron translation. Hence, a molecular description of electronic excitations in heavy-ion scattering has been achieved, which is free from the spurious asymptotic couplings of the conventional perturbated stationary-state approach. The importance of electron-translation effects for continuum electrons and positrons is investigated. To this end an algorithm for the description of continuum electrons is proposed, which for the first time should allow for the calculation of angular distributions for δ electrons. Finally, the practical consequences of electron-translation effects are studied by calculating the corrected coupling matrix elements for the Pb-Cm system and comparing the corresponding K-vacancy probabilities with conventional calculations. We critically discuss conventional methods for cutting off the coupling matrix elements in coupled-channel calculations.
We calculate the spin polarization of 1sσ vacancies and emitted δ electrons induced by the strong magnetic field (|Bmax|∼1016G) in collisions of very heavy ions (Z1+Z2=178). The electron excitations are determined by the solution of coupled-channel equations within the quasimolecular basis states including the vector potential. The formulation is extended to the many-electron case. Spin polarizations of the order of 5-10% for impact energies below the Coulomb barrier are predicted.
We define a new scalar-tensor theory with an effective gravitational coupling constant depending on a scalar field. The coupling is such that the gravitational interaction decreases with the strength of the scalar field. We show that this is not sufficient to prevent the gravitational collapse of sufficiently massive dense objects.
The fluid dynamical model is used to study the reactions 20Ne+238U and 40Ar+40Ca at Elab=390 MeV/nucleon. The calculated double differential cross sections d²ð/dΩdE exhibit sidewards maxima in agreement with recent experimental data. The azimuthal dependence of the triple differential distributions, to be obtained from an event-by-event analysis of 4π; exclusive experiments, can yield deeper insight into the collision process: Jets of nuclear matter are predicted with a strongly impact-parameter-dependent thrust angle θjet(b). NUCLEAR REACTIONS Ar+Ca, Ne+U, Elab=393 MeV/nucleon, fluid dynamics with thermal breakup, double differential cross sections, azimuthal dependence of triple differential cross sections, event-by-event thrust analysis of 4π exclusive experiments.
The negative-pion multiplicity is measured for central collisions of 40Ar with KCl at eight energies from 0.36 to 1.8 GeV/nucleon and for 4He on KCl and 40Ar on BaI2 at 977 and 772 MeV/nucleon, respectively. A systematic discrepancy with a cascade-model calculation which fits proton- and pion-nucleus cross sections but omits potential-energy effects is used to derive the energy going into bulk compression of the system. A value of the incompressibility constant of K=240 MeV is extracted in a parabolic form of the nuclear-matter equation of state.
Pion production and charged-particle multiplicity selection in relativistic nuclear collisions
(1982)
Spectra of positive pions with energies of 15-95 MeV were measured for high energy proton, 4He, 20Ne, and 40Ar bombardments of targets of 27Al, 40Ca, 107,109Ag, 197Au, and 238U. A Si-Ge telescope was used to identify charged pions by dE / dx-E and, in addition, stopped pi + were tagged by the subsequent muon decay. In all, results for 14 target-projectile combinations are presented to study the dependence of pion emission patterns on the bombarding energy (from E / A=0.25 to 2.1 GeV) and on the target and the projectile masses. In addition, associated charged-particle multiplicities were measured in an 80-paddle array of plastic scintillators, and used to make impact parameter selections on the pion-inclusive data. NUCLEAR REACTIONS U(20Ne, pi +), E / A=250 MeV; U(40Ar, pi +), Ca(40Ar, pi +), U(20Ne, pi +), Au(20Ne, pi +), Ag(20Ne, pi +), Al(20Ne, pi +), U(4He, pi +), Al(4He, pi +). E / A=400 MeV; Ca(40Ar, pi +), U(20Ne, pi +), U(4He, pi +), U(p, pi +), E / A=1.05), GeV; U(20Ne, pi +), E / A=2.1 GeV; measured sigma (E, theta ), inclusive and selected on associated charged-particle multiplicity.
The parities of eleven J=1 levels in 208Pb were determined by nuclear resonance fluorescence scattering of linearly polarized photons. A new 1+ level at Ex=5.846 MeV with Gamma 02 / Gamma =1.2±0.4 eV was found. This level can probably be identified with the theoretically predicted isoscalar 1+ state in 208Pb. All other bound dipole states below 7 MeV with Gamma 02 / Gamma >1.5 eV have negative parity. The 1- assignment to the 4.842-MeV level is of special significance because of previous conflicting results about its parity.
Two-center level diagrams for the neutron orbitals in the scattering of 16O on 25Mg and of 17O on 24Mg are calculated by using a deformed potential for 24,25Mg. Possible consequences of the nuclear Landau-Zener mechanism, namely the promotion of nucleons at avoided level crossings, and of the rotational coupling between crossing molecular single-particle orbitals are studied for inelastic excitation and neutron transfer. The important excitation and transfer processes, which are enhanced by the promotion process and the rotational coupling, are presented. NUCLEAR REACTIONS Heavy ion scattering, theory of nucleon transfer, molecular wave functions, asymmetric two center shell model, single particle excitation, deformed nuclei.
The energy shift of K electrons in heavy atoms due to the self-energy correction has been calculated. This process is treated to all orders in Zα, where Z denotes the nuclear charge. For the superheavy system Z=170, where the K-shell binding energy reaches the pair-production threshold (E1sb∼2mc2), a shift of +11.0 keV is found. This shift is almost cancelled by the vacuum polarization, leaving a negligible effect for all quantum-electrodynamical corrections of order α but all orders of Zα.
The nucleons taking part in heavy ion reaction are considered as a three-component fluid. The first and second components correspond to the nucleons of the target and the projectile, while the thermalized nucleons produced in the course of the collision belong to the third component. Making use of the Boltzmann equation, hydrodynamical equations are derived. An equation of state for anisotropic nuclear matter obtained from a field theoretical model in mean field approximation is applied in a one dimensional version of the three-component fluid model. The speed of thermalization is analyzed and compared to the results of cascade and kinetic models. NUCLEAR REACTIONS Relativistic heavy-ion reactions, hydrodynamic description.
The collision process is described by hydrodynamical equations. The escape of nucleons which do not take part in the thermal equilibrium is considered by including drain terms in these equations. The energy spectra of the escaped nucleons and of nucleons evaporated after the breakup of the fluid are compared. NUCLEAR REACTIONS Relativistic heavy ion reactions, nuclear hydrodynamics, nucleon spectra.
An event by event analysis is carried out for all charged particles observed in central collisions of 40Ar + KCl and 40Ar + Pb at 1.808 and 0.772 GeV/nucleon, respectively. Total transverse energy is used for impact parameter selection within the central trigger condition. The central Ar + KCl reaction exhibits a forward-backward oriented momentum flux. The flux distribution of the most central Ar + Pb events is approximately isotropic in the fireball center of mass.
Difficulties of the thermodynamical model approach to pion production in relativistic ion collisions
(1983)
Thermodynamical models with various forms of partial transparency of nuclear matter are considered. It is shown that the introduction of transparency, however, significantly improves agreement with pion data concerning multiplicities and transverse momenta leads to a serious discrepancy with average rapidities of pions. Qualitative arguments are given that difficulties of the thermodynamical approach can be overcome if one assumes hydrodynamical expansion in the first stage of nuclear interactions.