Selected Works of Hans A Bethe

The following sections are included:

• PREFACE

• CONTENTS

The influence of an electric field of prescribed symmetry (crystalline field) on an atom is treated wave mechanically. The terms of the atom split up in a way that depends on the symmetry of the field and on the angular momentum i (or j) of the atom. No splitting of s terms occurs, and p terms are not split up in fields of cubic symmetry. For the case in which the individual electrons of the atom can be treated separately (interaction inside the atom turned off) the eigenfunctions of zeroth approximation are stated for every term in the crystal; from these there follows a concentration of the electron density along the symmetry axes of the crystal which is characteristic of the term. – The magnitude of the term splitting is of the order of some hundreds of cm⁻¹. – For tetragonal symmetry, a quantitative measure of the departure from cubic symmetry can be defined, which determines uniquely the most stable arrangement of electrons in the crystal.

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The inelastic collision of a fast charged particle (electron, proton, α-particle) with an atom is treated according to Born's theory of wave mechanics. A very simple procedure is given for evaluating the matrix elements involved in the theory (section 3) and the close relationship to the intensity of the Compton effect is determined (section 5). The theory is developed in detail for collisions of hydrogen atoms and, in as far as possible, for complex atoms. The following are computed: the angular distribution of the inelastically (sections 6, 7, and 17) and elastically (section 16) scattered particles, the excitation cross sections for the excitation of the optical (sections 9 and 17) and x-ray (section 15) levels by electron collision, the sum of all inelastic and elastic collisions, and also the number of the primarily (sections 10 and 18) and secondarily (section 19) formed ions, the velocity distribution of the secondary electrons (section 18) and, finally, the braking of the colliding particles by gas atoms (sections 10, 12, and 13). The agreement of this theory with experiment is satisfactory to good. (For more detail see the summary in section 20).

A method is given whereby the zero-order eigenfunctions and first-order eigenvalues (in the sense of the London-Heitler approximation scheme) are calculated for a one-dimensional “metal” consisting of a linear chain of a very large number of atoms, each of which has a single s-electron with spin, outside closed shells. In addition to the spin waves of Bloch, bound states are found, in which parallel spins are predominantly on nearest neighbor atoms; these features may be important for the theory of ferromagnetism.

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The probability for the emission of radiation by fast electrons passing through an atom is calculated by Born's method (§1), the calculations going beyond previous publications mainly by considering the screening of the atomic field (§3). The results are discussed in §§ 5 to 7. The total radiation probability (fig. 3) becomes very large for high energies of the electron, indeed the stopping of very fast electrons (energy > 20 mc² for Pb) is mainly due to radiation, not to inelastic collisions.The theory does not agree with Anderson's measurements of the stopping of electrons of 300 million volts energy, thus showing that the quantum theory is definitely wrong for electrons of such high energy (§7) (presumably for E₀ > 137 mc²).

By the same formalism, the probability for the creation of a positive and a negative electron by a γ-ray is calculated (§§2, 3). The energy distribution of the electrons is shown in fig. 5, the total creation probability in fig. 6. For γ-rays of hν between 3 and 10 mc² the theory is in very good agreement with the experiments.

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It is shown that our present knowledge of the intra-nuclear forces allows us to make definite predictions for the behaviour of the diplon. According to these calculations the cross -section for disintegration of a diplon by absorption of γ-rays is of the order 10⁻²⁷ cm², § 3. In addition the cross-sections for capture of neutrons by protons, § 4, for scattering of γ-rays, § 5, and disintegration by electron bombardment,§ 6 have been calculated.

The cross-section and the angular distribution are calculated for the scattering of neutrons by protons. The result is practically independent of the special law of force assumed between neutron and proton, it depends only on the known binding energy of the dipldn. The crosssection obtained is about 50% larger than the rather uncertain experimental value. The scattering is almost isotropic (in the relative coordinate system) for all neutron energies up to about 40 million volts. Only for still higher energies, which are at present unavailable, an experimental determination of the sign of the anisotropy would decide whether the force between neutron and proton is of the exchange type or an ordinary force.

The order in an alloy containing two sorts of atoms in equal proportion is calculated statistically, assuming interaction only between nearest neighbours. At high temperatures, there is only a correlation between near atoms, the state of the crystal as regards order is similar to a liquid. At low temperatures, the crystal as a whole is ordered, the structure is “ solid-like.” This order at long distances is restricted to two or more dimensions.

The long-distance order and the energy as functions of temperature are both discussed qualitatively (§ 4) and calculated (§§ 6 to 9). It is shown that the theory of Bragg and Williams gives a fair first approximation. The long-distance order vanishes (with vertical tangent) at a certain critical temperature Θ. All the physical quantities plotted as functions of T have a kink at T = Θ but no jump. This is due to the fact that two “ symmetrical ” states exist, having the same energy (§ 4). The derivatives of physical quantities, such as the specific heat, have jumps at the critical temperature. The extra specific heat due to the destruction of order is rather large on the low-temperature side of the critical temperature; it is 70% of the ordinary specific heat due to thermal motion of the atoms (provided all atoms of the crystal take part in the transition). On the high-temperature side of Θ, it falls to about 10% of this value. Higher above the critical temperature, the specific heat decreases, but not very rapidly.

The differential cross sections for bremsstrahlung and pair production are calculated without the use of the Born approximation, assuming the energy of the electron to be large compared to mc² both in initial and final state. The wave functions in initial and final state are essentially those previously proposed by Furry (Sec. II). It is proved in Sec. III that our wave functions agree with the exact ones of Darwin except for terms of relative order a²/l², where a = Ze²/ħc and l the angular momentum, and that this agreement holds for any energy of the electron. An independent proof is given in Sec. IX, showing that the Furry wave functions give the matrix element correctly except for terms of relative order 1/ɛ.

In the matrix element for bremsstrahlung, the initial state of the electron must be represented by a plane wave plus an outgoing spherical wave, whereas the final state has an ingoing spherical wave (Sec. IV). In pair production, both electrons contain ingoing spherical waves (Sec. V). This causes essential differences between the cross sections for the two processes.

The cross section for pair production is calculated in Sec. VI; the result consists of the Bethe-Heitler formula multiplied by a relatively simple factor, plus another term of similar structure. A simplified derivation is given, which is valid for the important case of small angles between electrons and quantum (Sec. VII); it provides a useful check of the cross section of Sec. VI. In Sec. VIII, the bremsstrahlung cross section is calculated and found to be the Bethe-Heitler result multiplied by a factor. This factor is different from that encountered in pair production and becomes important only for very small momentum transfer q. In the limit of complete screening, these small q do not contribute and the cross section goes over into that of the Born approximation.

The error in the cross sections calculated in this paper is estimated (Sec. X) to be of order 1/ɛ, where ɛ is the energy of the final electron in bremsstrahlung, or that of the less energetic electron in pair production, in units of mc². The total cross section for pair production by a quantum of energy k may be in error by logk/k.

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The following sections are included:

• SUMMARY

• NOTATIONS

• The Energy Content of Gases

• The Approach of Equilibrium Between Various Degrees of Freedom of the Molecules
  • Translation and Rotation
  • Vibration
  • Dissociation
  • Conclusions on the Excitation of Air

The probability of the astrophysically important reaction H+H = D+ɛ⁺ is calculated. For the probability of positron emission, Fermi's theory is used. The penetration of the protons through their mutual potential barrier, and the transition probability to the deuteron state, can be calculated exactly, using the known interaction between two protons. The energy evolution due to the reaction is about 2 ergs per gram per second under the conditions prevailing at the center of the sun (density 80, hydrogen content 35 percent by weight, temperature 2 · 10⁷ degrees). This is almost but not quite sufficient to explain the observed average energy evolution of the sun (2 ergs/g sec.) because only a small part of the sun has high temperature and density. The reaction rate depends on the temperature approximately as T^3.5 for temperatures around 2 · 10⁷ degrees.

It is shown that the most important source of energy in ordinary stars is the reactions of carbon and nitrogen with protons. These reactions form a cycle in which the original nucleus is reproduced, viz. C¹²+H = N¹³, N¹³ = C¹³+ ɛ⁺, C¹²+H = N¹⁴, N¹⁴+H = O¹⁵, O¹⁵ = N¹⁵ + ɛ⁺, N¹⁵+H = C¹² + He⁴. Thus carbon and nitrogen merely serve as catalysts for the combination of four protons (and two electrons) into an α-particle (§7).

The carbon-nitrogen reactions are unique in their cyclical character (§8). For all nuclei lighter than carbon, reaction with protons will lead to the emission of an α-particle so that the original nucleus is permanently destroyed. For all nuclei heavier than fluorine, only radiative capture of the protons occurs, also destroying the original nucleus. Oxygen and fluorine reactions mostly lead back to nitrogen. Besides, these heavier nuclei react much more slowly than C and N and are therefore unimportant for the energy production.

The agreement of the carbon-nitrogen reactions with observational data (§7, 9) is excellent. In order to give the correct energy evolution in the sun, the central temperature of the sun would have to be 18.5 million degrees while integration of the Eddington equations gives 19. For the brilliant star Y Cygni the corresponding figures are 30 and 32. This good agreement holds for all bright stars of the main sequence, but, of course, not for giants.

For fainter stars, with lower central temperatures, the reaction H+H = D+ɛ⁺ and the reactions following it, are believed to be mainly responsible for the energy production. (§10)

It is shown further (∮5–6) that no elements heavier than He⁴ can be built up in ordinary stars. This is due to the fact, mentioned above, that all elements up to boron are disintegrated by proton bombardment (α-emission!) rather than built up (by radiative capture). The instability of Be⁸ reduces the formation of heavier elements still further. The production of neutrons in stars is likewise negligible. The heavier elements found in stars must therefore have existed already when the star was formed.

Finally, the suggested mechanism of energy production is used to draw conclusions about astrophysical problems, such as the mass-luminosity relation (§10), the stability against temperature changes (§11), and stellar evolution (§12).

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The scattering of neutrons up to about 10 or 20 Mev by protons can be described by two parameters, the scattering length at zero energy, a, and the effective range, r₀. A formula (16), expressing the phase shift in terms of a and r₀ is derived: it is identical with one previously derived by Schwinger but the derivation is very much simpler. Reasons are given why the deviations from the simple formula are very small, as shown by the explicit calculations by Blatt and Jackson.

The theory is then applied to proton-proton scattering, with a similarly simple result. Moreover, a method is developed to compare proton-proton and proton-neutron scattering without explicit calculation of a nuclear potential.

The most recent experimental results are evaluated on the basis of the theory, and accurate values for the effective ranges are obtained for the triplet scattering of neutrons, and for proton-proton scattering. The nuclear force between two protons is found to differ by a slight amount, but beyond doubt, from that between neutron and proton in the singlet state. All actual results agree with those obtained by Breit and collaborators, and by Blatt and Jackson.

A self-contained and largely new description is given of Brueckner's method for studying the nucleus as a system of strongly interacting particles. The aim is to develop a method which is applicable to a nucleus of finite size and to present the theory in sufficient detail that there are no ambiguities of interpretation and the nature of the approximations required for actual computation is clear.

It is shown how to construct a model of the nucleus in which each nucleon moves in a self-consistent potential matrix of the form (r′ | V | r) (Sec. II). The potential is obtained by calculating the reaction matrix for two nucleons in the nucleus from scattering theory. Some complications arise in the definition of the energy levels of excited nucleons (Sec. III). The actual wave function is obtained from the model wave function by an operator which takes into account multiple scattering of the nucleons by each other (Sec. IV).

The method of Brueckner is a vast improvement over the normal Hartree-Fock method since, in calculating the self-consistent potential acting on an individual particle in the model, account is already taken of the correlations between pairs of nucleons which arise from the strong internucleon forces (Sec. V). Although the actual wave function is derivable from a wave function which corresponds essentially to the shell model, the probability of finding a large nucleus of mass number A “actually” in its shell model state is small (of order e^−αA, where α is a constant) (Sec. VI). The influence of spin is investigated (Sec. VIII). In the case of an infinite nucleus, an integral equation is obtained for the reaction matrix, just as in the theory of Brueckner and Levinson (Sec. IX).

The exclusion principle must be applied in intermediate states in solving the integral equation for the reaction matrix. This makes an enormous difference for the solution. When the exclusion principle is used, the scattering matrix is very nearly given by the Born approximation, for any well-behaved potential (Sec. X). Numerical results are given for the case when nucleons interact only in S states, an assumption which leads to saturation without a repulsive core. The agreement with observation is fair to poor, owing to the poor assumption for the interaction (Sec. XI). Brueckner's result that three-particle clusters give a small contribution to the energy is confirmed, although the numerical value is many times his result; the calculation is then extended to the case of a repulsive core (Sec. XII). The dependence of the binding energy on the mass number A is investigated for saturating and nonsaturating interactions (Sec. XIII). Terms of relative order 1/A are calculated, and it is shown that these terms are much smaller than Brueckner and Levinson found, making the method also applicable to relatively small nuclei (Sec. XIV). Some aspects of the problem of the finite nucleus are discussed, including that of degeneracy (Sec. XVI).

Using Brueckner's method for the treatment of complex nuclei, the effect of an infinite repulsive core in the interaction between nucleons is studied. The Pauli principle is taken into account from the beginning. A spatial wave function for two nucleons is defined, and an integro-differential equation for thin function is derived. Owing to the Pauli principle, the wave function contains no outgoing spherical waves. A solution in given for the ease when only a repulsive core potential acts. The effective-mass approximation is investigated for virtual states of very large momentum.

The matter in neutron stars is essentially in its ground state and ranges in density up to and beyond 3 × 10¹⁴ g/cm³, the density of nuclear matter. Here we determine the constitution of the ground state or matter and its equation or state in the regime from 4.3 × 10¹¹ g/cm³ where free neutrons begin to “drip” out of the nuclei, up to densities ≈ 5 × 10¹⁴ g/cm³, 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 × 10¹⁴ g/cm³; 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 × 10¹⁴ g/cm³ in a first-order transition. It is shown that the uniform liquid is unstable, against density fluctuations below this density; the wavelength of the most unstable density fluctuation is close to the limiting lattice constant in the nuclear phase.

We calculate neutron star models using four realistic high-density models of the equation of state. We conclude that the maximum mass of a neutron star is unlikely to exceed 2 M_⊙. All of the realistic models are consistent with current estimates of the moment of inertia of the Crab pulsar.

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 μ_n − μ_p in our capture equation forces the energy of the resulting neutrinos to be low. Nonetheless, 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 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.

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Starting from Wilson's idea that the supernova gets its energy from neutrinos heating the mantle of the star, I use the neutrino observations to find the available neutrino flux. This agrees well with recent computations by Wilson and Mayle, using “neutron fingers.” In the mantle of the star, one can define a “gain radius:” outside this radius, the energy gain from neutrino absorption is greater than the energy loss by electron capture. The temperature and density are calculated from simple arguments for r > 100 km: In the region where free nucleons dominate, the entropy is assumed to be constant; in the region where nucleons and α-particles are mixed, the fraction of nucleons is determined from the internal energy. One can then calculate the fraction of neutrino energy which can be transferred to the region outside the gain radius; this turns out to be 1%–2%. With the observed neutrino flux, this gives a supernova energy of 0.8 foe (1 foe = 10⁵¹ ergs); nucleosynthesis adds about 0.4 foe, for a total of 1.2 foe, compared with the observed energy of 1.4 ± 0.4 foe.

In this paper, I shall try to go as directly as possible to the calculation of the supernova energy. Other problems are discussed in §§ 11–14. In particular, it is shown that, and why, the shock wave starts only after some delay.

Vigorous convection is the key to the supernova mechanism. An analytic theory is presented which parallels the computations of Herant et al. Energy is delivered by neutrinos to the convecting medium. The most important quantity is ρ₁r³, where ρ₁ is the density outside the shock. This can be obtained from the computations of Wilson et al., since it is not affected by the convection behind the shock. It is closely related to Ṁ, the rate at which matter falls in toward the center. The outgoing shock is dominated by the Hugoniot equation; the shock cannot move out until its energy is of the order of 1 foe (= 10⁵¹ ergs). Once it moves, its velocity and energy are calculated as functions of its radius. Nucleosynthesis gives an appreciable contribution to the energy. A substantial fraction of the energy is initially stored as nuclear dissociation energy, and then released as the shock moves out. This energy cannot at present be calculated from first principles, but it can be deduced from the observed energy of SN 1987A of 1.4 ± 0.4 foe. From the result it is shown that about one-half of the infalling material goes into the shock and one-half accretes to the neutron star.

The shock in a supernova of Type II first moves slowly and then suddenly breaks out with a velocity of order 10⁹ cm s⁻¹. This breakout is due to the recombination of nucleons into α-particles. It is shown that this recombination provides enough energy to lift the matter against gravity. The net energy after escape from the star is about 4 × 10¹⁸ ergs g⁻¹. Supernovae of Type II differ mainly in the binding energy of the mantle of the progenitor.

The following sections are included:

• List of Publications