Quantum Mechanical (Phase Shift) Analysis of Differential Elastic Scattering of Molecular Beams

For a spherically symmetrical intermolecular potential V(r)=ϵf(r/σ) the quantum calculation of the elastic scattering cross section dσ(Θ)/dΩ in the c.m. system is carried out as follows. For a given relative velocity (or deBroglie wavelength) and an assumed V(r), the radial wave equation is integrated for successive values of the angular momentum quantum number l, yielding the phase shifts ηι. Then dσ(Θ)dΩ is computed in terms of the series of ηι's in the standard way. A general computational program (following that of K. Smith) is outlined for the evaluation of the radial wave function and the phase shifts, utilizing an IBM 704 computer. Calculations are presented for the L‐J (12, 6) potential function. The results may be concisely represented using the framework provided by the semiclassical treatment of Ford and Wheeler, i.e., in terms of a set of reduced phase constants vs reduced angular momenta at various reduced relative kinetic energies K. Tables and graphs are presented from which the phases may be obtained, to a good approximation, for any given ϵ, σ and K. Computation of the differential and total cross sections from the phase shifts is then readily accomplished.The results are compared with the classical and semiclassical treatments. The problem of tunneling and orbiting is discussed.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/70444/2/JCPSA6-33-3-795-1.pd

The Sleeper Wakes: The History and Legacy of the Twenty-Seventh Amendment

No provision of the United States Constitution has a more drawn-out, tortured history than the Twenty-seventh Amendment, which was ratified more than two centuries after Representative James Madison introduced it in the First Congress. In this Article, Professor Bernstein traces the Amendment\u27s origins to the legislative political culture of the late eighteenth century, as influenced by the controversy over ratifying the Constitution. He then examines the perennial controversies over congressional compensation in American history, elucidating how in the 1980s and 1990s public anger at Congress reached critical mcm sufficient to propel the 1789 compensation amendment into the Constitution. Finally, this Article demonstrates that the adoption of the Amendment has consequences beyond its effects on congressional compensation-both for the unresolved issues of the Article V amending process and for the practice of amendment politics

High‐Velocity Molecular Beam Scattering: Total Elastic Cross Sections for L‐J(n, 6) and Exp‐6(α) Potentials

Explicit expressions are derived for the total elastic scattering cross sections in the high‐velocity region for molecules interacting according to L‐J (n, 6) and exp‐6(α) potentials. Cross sections are presented in tabular and graphical form.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/70237/2/JCPSA6-38-2-515-1.pd

Semiclassical Analysis of the Extrema in the Velocity Dependence of Total Elastic‐Scattering Cross Sections: Relation to the Bound States

The phenomenon of extrema in the velocity dependence of the total elastic cross section Q(v) for atom—atom scattering in the thermal‐energy region is shown to be a quite general one, whenever the interaction potential consists of both attractive and repulsive parts and the resulting well has a ``capacity'' for one or more discrete levels. The phase shift vs angular‐momentum dependence exhibits a maximum; since this maximum is a function of the de Broglie wavelength, the cross section exhibits an undulatory velocity dependence. A semiclassical analysis of the extrema velocities (and undulation amplitudes) is presented. Suitable plots are suggested from which one may deduce certain information on the interatomic potential and the diatom bound states. The following rule is proposed: the observation of m maxima in the elastic atom—atom impact spectrum implies the existence of at least m discrete vibrational levels of zero angular momentum for the diatom.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/70899/2/JCPSA6-38-11-2599-1.pd

Semiclassical Equivalence Relationship Applied to the Calculation of Molecular‐Beam Scattering Phase Shifts

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/70746/2/JCPSA6-36-5-1403-1.pd