Leonard Rosenberg

UPPER BOUNDS ON SCATTERING LENGTHS FOR STATIC POTENTIALS

It is shown that in the zero-energy scattering of a particle by a center of force, where no bound state exists, the Kohn variational principle provides an upper bcund on the scattering length. A bound may also be obtained from Hulthens method, although with the same form of trial function the Kohn result will be lower (and therefore better) than the one obtained from the Hulthen principle. The Rubinow formulation need not provide a bound; for those calculations which were performed in this form, the results may be converted without any further calculations so that they correspond to the Kohn form, and therefore, under the circumstances considered, do give a bound. Analogous results hold for states of nonzero orbital angular momentum. Direct generalizations of the above that they correspond to the Kohn form. and therefore. under the circumstances considered, do give a bound. Analogous results hold for states of nonzero orbital angular momentum. Direct generalizations of the above results are valid for scattering by a compound system. (auth)

Theory of Electron-Atom Scattering in a Radiation Field Leonard Rosenberg

Publisher Summary Interest in the theory of electron-atom scattering in a radiation field has increased in recent years as a result of the important role it plays in the study of plasma heating, gas breakdown, and laser-driven fusion. The theory has been a subject of study, particularly in connection with the bremsstrahlung process. Owing to the dominance of stimulated emission and absorption, the theory of scattering in a laser field takes on a form somewhat different from the more familiar theory of spontaneous bremsstrahlung. However, the underlying physics is the same in the two cases, and it is interesting to trace the connections. The chapter emphasizes on the recent developments in the theory of multiphoton processes in electron–atom scattering. It particularly describes the connections that exist between the time-dependent and time-independent versions of the theory, the relationship between the classical and quantum descriptions in the low-frequency limit, and on the unification of the understanding of stimulated and spontaneous multiphoton processes for low-frequency radiation fields.

UPPER BOUND ON THE NEUTRON-DEUTERON DOUBLET SCATTERING LENGTH

Abstract The method of obtaining upper bounds on scattering lengths is applied to the determination of the n-d doublet scattering length A D . A new calculation is unnecessary; an analysis of the trial function used by Efimov in a variational estimate of A D shows that that calculation effectively provides a bound. (The Efimov trial function has the ‘inappropriate’ normalization for present purposes, but it is exceedingly unlikely that this will alter the character of the present results. This question can be readily resolved in any case.) We find that the potentials assumed by Efimov imply A D −13 cm. If these potentials are accurate (this question is still an open one), it follows that of the two experimentally allowed sets, namely A D = 0.7×10 −13 cm and A Q = 6.4×10 −13 cm (Set I), or A D = 8.3×10 −13 cm and A Q = = 2.6×10 −13 cm (Set II), Set I is correct, a conclusion consistent with some recent variational estimates of about 6×10 −13 cm for the quartet scattering length A Q . It is also shown, for the general problem, that the exact numerically determined value of the ‘static’ scattering length lies above the true value if the number of bound state solutions for the static and true problems are the same.

Variational principles, variational identities, and supervariational principles for wavefunctions

We develop variational principles and variational identities for bound state and continuum wavefunctions in a general context, paying particular attention to the proper choice of defining equations and boundary conditions which will lead to unique and unambiguous wavefunctions even when these functions are complex. Any functional, such as a matrix element, calculated with such a variationally determined wavefunction, will also be accurate to second order in the error of the starting choice. This provides, therefore, an alternative procedure for getting variational estimates of matrix elements to the one that already exists in the literature and we establish the equivalence of the two. Of even more interest is the possibility which now seems open of going beyond the variational principle and generating ’’supervariational’’ estimates of wavefunctions and matrix elements which are good to better than second order. We also give a simple prescription for the construction of variational identities for wavefunct...

General time-independent theory of ionization and breakup reactions

A rigorous formulation of non-relativistic time-dependent multiparticle collision theory was developed some time ago, in which the effect of long-range Coulomb forces was accounted for by a modification of the standard wave operator that appears in treatments of neutral-particle scattering. This theory is here converted to a time-independent form, with the wave-operator modification replaced by the use of Coulomb-modified plane waves in the construction of the wave packets that appear in the theory. This provides a convenient basis for the development of approximation techniques in configuration space, including the use of variational methods of calculation, based on integral identities for the transition amplitudes. In many cases oscillatory divergences appear in the integrals representing the ionization or breakup amplitudes. A method for removing such divergences by an averaging of the integrand at great distances is defined.

Bounds on Low‐Energy Scattering Parameters

Some recent results that determine an upper bound on the scattering length, whether or not composite bound states exist, are reviewed. The extension to the determination of an upper bound on (−k cotη)−1, where only one channel is open, is presented; the method used requires that the potential vanish identically beyond some given point. The results are applicable to the scattering of one compound system by another. Possible extensions and improvements of the method are discussed.

On electron impact-ionization theory

In some configuration-space treatments of the problem of breakup reactions involving three charged systems theoretical anomalies appear in the form of divergent integrals or integrands with oscillating radial dependence at great distances. A variational identity which is free of such anomalies is derived here with the aid of some results obtained in earlier work. The identity provides the basis for a review and analysis of recent developments in the theory, as well as for the introduction of generalizations, from a rigorous and unified point of view.

On The β-Decay of 187 Re: An Interface of Atomic And Nuclear Physics And of Cosmochronology

Publisher Summary This chapter discusses how a study of the ratios of various isotopes of Re and Os combined with a knowledge of the half-life T 1/2 of the β decay of 187 Re can be helpful in estimating the age of our galaxy. The chapter defines or expands on astrophysical terminology and concepts and which emphasizes the broad view of the problem rather than the details. The chapter explains the relative production rates of different isotopes of Re and of Os. Studies of the abundances of heavy elements in various stars of known ages suggest that the rate of production of Re has been constant or decreasing, but has not been increasing. The rate of production is linearly proportional to the density of the elements which produced 187 Re atoms.

Relativistic theory of electron-impact ionization

A relativistic version of an earlier, non-relativistic, formulation of the theory of ionization of an atomic system by electron impact is presented. With a time-independent resolvent operator taken as the basis for the dynamics, a wave equation is derived for a system with open channels consisting of two positive-energy electrons in an external field generated by the residual ion. Virtual intermediate states can be accounted for by the effective Hamiltonian that appears in the wave equation and which in principle may be constructed perturbatively. The asymptotic form of the wavefunction, modified by the effects of the long-range Coulomb interactions of the two electrons in the external field, is derived. These electrons are constrained, by projection operators which appear naturally in the theory, to propagate in positive-energy states only. The long-range Coulomb effects take the form of phase factors similar to those that are found in the non-relativistic version of the theory. With the boundary conditions established, an integral identity for the ionization amplitude is derived, and used to set up a distorted-wave Born expansion for the transition amplitude involving Coulomb-modified propagating waves.

On low-energy scattering

Variational bounds have played an important role in the development of low-energy scattering theory. A thumbnail sketch of the development and use of variational bounds on scattering lengths, with emphasis on insights rather than on rigorous proofs and details, is presented here.