Singular Integral Equations. By N.I. Muskhelishvili. Translated fromthe 2nd Russian edition by J.R.M. Radok. Pp. 447. F1. 28.50. 1953. (Noordhoff, Groningen)

The Jaynes-Cummings model (JCM), a soluble fully quantum mechanical model of an atom in a field, was first used (in 1963) to examine the classical aspects of spontaneous emission and to reveal the existence of Rabi oscillations in atomic excitation probabilities for fields with sharply defined energy (or photon number). For fields having a statistical distributions of photon numbers the oscillations collapse to an expected steady value. In 1980 it was discovered that with appropriate initial conditions (e.g. a near-classical field), the Rabi oscillations would eventually revive, only to collapse and revive repeatedly in a complicated pattern. The existence of these revivals, present in the analytic solutions of the JCM, provided direct evidence for discreteness of field excitation (photons) and hence for the truly quantum nature of radiation. Subsequent study revealed further non-classical properties of the JCM field, such as a tendency of the photons to antibunch. Within the last two years it ha...

The Jaynes-Cummings Model

Adopted values for the reduced electric quadrupole transition probability, B(E2)↑, from the ground state to the first-excited 2+ state of even–even nuclides are given in Table I. Values of τ, the mean life of the 2+ state; E, the energy; and β, the quadrupole deformation parameter, are also listed there. The ratio of β to the value expected from the single-particle model is presented. The intrinsic quadrupole moment, Q0, is deduced from the B(E2)↑ value. The product E×B(E2)↑ is expressed as a percentage of the energy-weighted total and isoscalar E2 sum-rule strengths.

Table II presents the data on which Table I is based, namely the experimental results for B(E2)↑ values with quoted uncertainties. Information is also given on the quantity measured and the method used. The literature has been covered to November 2000.

The adopted B(E2)↑ values are compared in Table III with the values given by systematics and by various theoretical models. Predictions of unmeasured B(E2)↑ values are also given in Table III.

Transition probability from the ground to the first-excited 2^+ state of even-even nuclides

The last decade has witnessed both quantitative and qualitative progresses in Shell Model studies, which have resulted in remarkable gains in our understanding of the structure of the nucleus. Indeed, it is now possible to diagonalize matrices in determinantal spaces of dimensionality up to 10^9 using the Lanczos tridiagonal construction, whose formal and numerical aspects we will analyze. Besides, many new approximation methods have been developed in order to overcome the dimensionality limitations. Furthermore, new effective nucleon-nucleon interactions have been constructed that contain both two and three-body contributions. The former are derived from realistic potentials (i.e., consistent with two nucleon data). The latter incorporate the pure monopole terms necessary to correct the bad saturation and shell-formation properties of the realistic two-body forces. This combination appears to solve a number of hitherto puzzling problems. In the present review we will concentrate on those results which illustrate the global features of the approach: the universality of the effective interaction and the capacity of the Shell Model to describe simultaneously all the manifestations of the nuclear dynamics either of single particle or collective nature. We will also treat in some detail the problems associated with rotational motion, the origin of quenching of the Gamow Teller transitions, the double beta-decays, the effect of isospin non conserving nuclear forces, and the specificities of the very neutron rich nuclei. Many other calculations--that appear to have ``merely'' spectroscopic interest--are touched upon briefly, although we are fully aware that much of the credibility of the Shell Model rests on them.

The shell model as a unified view of nuclear structure

A non-local potential model for the scattering of neutrons by nuclei

A variable-moment-of-inertia (VMI) model is proposed which permits an excellent fit of level energies of ground-state bands in even-even nuclei. In this model the energy of a level with angular momentum $I$ is given by the sum of a potential energy term $\ensuremath{\propto}{({\mathfrak{g}}_{I}\ensuremath{-}{\mathfrak{g}}_{0})}^{2}$ (where ${\mathfrak{g}}_{0}$ is the ground-state moment of inertia) and a rotational energy term $\frac{{\ensuremath{\hbar}}^{2}I(I+1)}{2{\mathfrak{g}}_{I}}$. It is required that the equilibrium condition $\frac{\ensuremath{\partial}E}{\ensuremath{\partial}\mathfrak{g}}=0$ be satisfied for each state. Each nucleus is described by two adjustable parameters, ${\mathfrak{g}}_{0}$ and $\ensuremath{\sigma}$ (the softness parameter), which are determined by a least-squares fit of all known levels. The calculated level energies and moments of inertia ${\mathfrak{g}}_{I}$, ${\mathfrak{g}}_{0}$, and $\ensuremath{\sigma}$ are tabulated for 88 bands, ranging from Pd to Pt and from Th to Cm. Projections of three-dimensional arrays of ${\mathfrak{g}}_{0}$ and $\ensuremath{\sigma}$ on the ($N,Z$) plane are shown. These parameters are found to vary smoothly as function of $N$ and $Z$. Breaks occur at $N=98, 104, \mathrm{and} 108$. The osmium nuclei show a pronounced maximum for ${\mathfrak{g}}_{0}$ and an equally pronounced minimum for $\ensuremath{\sigma}$ at 108 neutrons. In Pt, ${\mathfrak{g}}_{0}$ decreases steeply to 110 neutrons and then more slowly, while $\ensuremath{\sigma}$ increases correspondingly. The stable Pt nuclei with $A=190,192, \mathrm{and} 194$ still possess appreciable moments of inertia and large but "finite" softness parameters. Hence they may be characterized as "pseudospherical." For nuclei exhibiting a near-harmonic level pattern (like ${\mathrm{Xe}}^{130}$, ${\mathrm{Sm}}^{150}$, and other neutron-deficient rare-earth isotopes), ${\mathfrak{g}}_{0}$ becomes exceedingly small, but already for the 2+ state $\mathfrak{g}$ is several orders of magnitude larger. The parameters of some $K=2$ bands in even-even nuclei and of bands found in odd-odd nuclei are related to those of appropriate ground-state bands in even-even nuclei. Evidence for a rotational band in ${\mathrm{Ir}}^{194}$ is deduced from recently published experimental results. A plot of $\frac{{E}_{4}}{{E}_{2}}$ versus $A$, presented for the discussion of the region of validity of the model, namely, $2.23l\frac{{E}_{4}}{{E}_{2}}l3.33$, reveals new regularities. The empirical "Mallmann curves" ($\frac{{E}_{I}}{{E}_{2}}$ plotted versus $\frac{{E}_{4}}{{E}_{2}}$) are deduced from the VMI model within its region of validity. Graphs are presented which allow the determination of ${E}_{I}$ (for $I\ensuremath{\le}16$) and of $\ensuremath{\sigma}$ and ${\mathfrak{g}}_{0}$ for each even-even nucleus for which the first 2+ and 4+ states are known. The model suggested by Harris, which includes the next-higher-order correction of the cranking model, is shown to be mathematically equivalent to the VMI model. The recently discovered appreciable quadrupole moments of 2+ states of "spherical nuclei" are compatible with the moments of inertia of these states given by the VMI model. The relation between $\frac{B (E2)({2}^{\ensuremath{'}}\ensuremath{\rightarrow}2)}{B (E2)(2\ensuremath{\rightarrow}0)}$ and $\frac{{E}_{4}}{{E}_{2}}$ is explored.

Phenomenological Analysis of Ground-State Bands in Even-Even Nuclei

Local potential models for the scattering of complex nuclei

An exact shell-model calculation is presented that gives a good description of the properties of ${\mathrm{O}}^{16}$ and neighboring nuclei.

B. Buck

Papers

A non-local potential model for the scattering of neutrons by nuclei

Phenomenological Analysis of Ground-State Bands in Even-Even Nuclei

Local potential models for the scattering of complex nuclei

Structure of O 16

Half-Lives of Favored Alpha Decays from Nuclear Ground States