Hans A. Bethe

Zur Theorie der Metalle

ZusammenfassungEs wird eine Methode angegeben, um die Eigenfunktionen nullter und Eigenwerte erster Näherung (im Sinne des Approximationsverfahrens von London und Heitler) für ein „eindimensionales Metall“ zu berechnen, bestehend aus einer linearen Kette von sehr vielen Atomen, von denen jedes außer abgeschlossenen Schalen eins-Elektron mit Spin besitzt. Neben den „Spinwellen“ von Bloch treten Eigenfunktionen auf, bei denen die nach einer Richtung weisenden Spins möglichst an dicht benachbarten Atomen zu sitzen suchen; diese dürften für die Theorie des Ferromagnetismus von Wichtigkeit sein.

Intermediate quantum mechanics

Theory Of Atomic Structure * Miscellaneous Results From Elementary Quantum Mechanics * Identical Particles and Symmetry * Two-Electron Atoms * Self-Consistent Field * Statistical Models * Addition of Angular Momenta * Theory of Multiplets, Electrostatic Interaction * Theory of Multiplets, Spin-Orbit Interaction, and Interactions with External Fields * Molecules Semiclassical Radiation Theory * Semiclassical Theory of Radiation * Intensity of Radiation, Selection Rules * Photoelectric Effect Atomic Collisions * Elastic Scattering at High Energies * Elastic Scattering at Low Energies * Further Corrections to Elastic Scattering Formulas * Elastic Scattering of Spin 1/2 Particles * Inelastic Scattering at High Energies * Inelastic Scattering at Low Energies * Semiclassical Treatment of Inelastic Scattering * Classical Limit of Quantum Mechanical Scattering Summary Relativistic Equations * Klein-Gordon Equation * Dirac Equation, Formal Theory * Solutions of the Dirac Equation

Neutron star matter

Abstract The matter in neutron stars is essentially in its ground state and ranges in density up to and beyond 3 × 10 14 g/cm 3 , the density of nuclear matter. Here we determine the constitution of the ground state of matter and its equation of state in the regime from 4.3 × 10 11 g/cm 3 where free neutrons begin to “drip” out of the nuclei, up to densities ≈ 5 × 10 14 g / cm 3 , where standard nuclear-matter theory is still reliable. We describe the energy of nuclei in the free neutron regime by a compressible liquid-drop model designed to take into account three important features: (i) as the density increases, the bulk nuclear matter inside the nuclei, and the pure neutron gas outside the nuclei become more and more alike; (ii) the presence of the neutron gas reduces the nuclear surface energy; and (iii) the Coulomb interaction between nuclei, which keeps the nuclei in a lattice, becomes significant as the spacing between nuclei becomes comparable to the nuclear radius. We find that nuclei survive in the matter up to a density ∼ 2.4 × 10 14 g / cm 3 ; below this point we find no tendency for the protons to leave the nuclei. The transition between the phase with nuclei and the liquid phase at higher densities occurs as follows. The nuclei grow in size until they begin to touch; the remaining density inhomogeneity smooths out with increasing density until it disappears at about 3 × 10 14 g/cm 3 in a first-order transition. It is shown that the uniform liquid is unstable against density fluctuations below this density; the wave-length of the most unstable density fluctuation is close to the limiting lattice constant in the nuclear phase.

Equation of State in the Gravitational Collapse of Stars

Abstract The equation of state in stellar collapse is derived from simple considerations, the crucial ingredient being that the entropy per nucleon remains small, of the order of unity (in units of k), during the entire collapse. In the early regime, ρ∼1010−1013 g/cm3, nuclei partially dissolve into α-particles and neutrons; the α-particles go back into the nuclei at higher densities. At the higher densities, nuclei are preserved right up to nuclear matter densities, at which point the nucleons are squeezed out of the nuclei. The low entropy per nucleon prevents the appearance of drip nucleons, which would add greatly to the net entropy. We find that electrons are captured by nuclei, the capture on free protons being negligible in comparison. Carrying the difference of neutron and proton chemical potentials μn−μp in our capture equation forces the energy of the resulting neutrinos to be low. Nonethelesd, neutrino trapping occurs at a density of about ρ = 1012 g/cm3. The fact that the ensuing development to higher densities is adiabatic makes our treatment in terms of entropy highly relevant. The resulting equation of state has an adiabatic index of roughly 4 3 coming from the degenerate leptons, but lowered slightly by electrons changing into neutrinos and by the nuclei dissolving into α-particles (although this latter process is reversed at the higher densities), right up to nuclear matter densities. At this point the equation of state suddenly stiffens, with Γ going up to Γ ≈ 2.5 and bounce at about three times nuclear matter density. In the later stages of the collapse, only neutrinos of energy ⪅10 MeV are able to get out into the photosphere, and these appear to be insufficient to blow off the mantle and envelope of the star. We do not carry our description into the region following the bounce, where a shock wave is presumably formed, and, therefore, we cannot answer the question as to whether the shock wave, in conjunction with neutrino transport, can dismantle the star, but a one-dimensional treatment shows the shock wave to be very promising in this respect.

Elektronentheorie der Metalle

Seit der Entdeckung des Elektrons konnte kein Zweifel daruber bestehen, das der elektrische Strom im Draht von Elektronen getragen wird. A. Schuster, E. Riecke u. a. haben Theorien vorgeschlagen. Den entscheidenden Schritt tat P. Drude, indem er die Elektronen an dem thermodynamischen Gleichgewicht des Metalls teilnehmen lies, also jedem Elektron als mittlere kinetische Energie den Betrag

Evolution of Binary Compact Objects That Merge

{E_m} = \frac{3}{2}kT

Effect of a repulsive core in the theory of complex nuclei

(1.1)

Shock waves in colliding nuclei

Abstract The relation of the polarized scattering of protons by complex nuclei to that by nucleons is investigated quantitatively. No model of nuclear forces is used, but the nucleon-nucleon scattering is directly represented by phase shifts. The agreement is found to be good for all the phase shift solutions which Stapp et al. have given for the proton-proton scattering, the deviation being from 0 to 25%. Thus the scattering by a complex nucleus can be obtained by superposition of the nucleon-nucleon scattering amplitudes. For the purpose of this comparison, the experimental data of Chamberlain et al. on the scattering and polarization of protons by carbon at small scattering angles are analyzed directly, i.e., the relevant scattering lengths are deduced from the experimental data, rather than the data synthesized from an assumed potential. The resulting nuclear absorption coefficient reproduces the observed inelastic scattering. The real part of the central potential is found to be attractive but very weak (about 4 Mev). The spin-orbit potential agrees with the nucleon-nucleon data and with some, but not all, previous theoretical values; it is only one-sixth of that required in the shell model. The analysis is facilitated by a theorem, first derived by Kohler and by Levintov, that the polarization is given correctly by the Born approximation. Simple formulas are given to obtain the Born approximation scattering (and thus the potential) from the observed scattering. A particularly simple formula relates the Born approximation scattering to the nucleon-nucleon scattering. The expressions for the relevant amplitudes in nucleon-nucleon scattering are also simplified. It is further shown that the scattering matrix for nucleon-nucleon scattering at a given angle can be completely determined if all the standard triple scattering and polarization experiments are carried out.