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

Oxygen‐18 Isotope Effect in the Reaction of Oxygen with Copper

The fractionation of oxygen‐18 in the reaction of copper with oxygen of natural isotopic composition has been measured over the temperature range 68—256°C. The oxide films (estimated thicknesses varying from 150—2500 A) were removed for O18 assay by treatment with hydrogen at 350°C. Using the CO2 equilibration technique, the O18/O16 ratios for the resulting water samples were determined mass spectrometrically and compared with the ratio for a reference sample of water prepared from the original oxygen gas.The direction of the fractionation indicated that O216 reacts preferentially compared to O16O18. The isotope effect appeared to be independent of the oxygen pressure over the limited range studied (2—25 cm Hg). The magnitude of the fractionation factor was 2.0% at 150°C, with a small negative temperature coefficient. From this it was possible to estimate a value of ΔEact=17 cal/mole, which may be compared with the calculated isotopic zero‐point energy difference of 64 cal/mole. The theoretical implications of the experimental results are discussed.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/69760/2/JCPSA6-23-10-1797-1.pd

The Founding Fathers Reconsidered

Here is a vividly written and compact overview of the brilliant, flawed, and quarrelsome group of lawyers, politicians, merchants, military men, and clergy known as the Founding Fathers --who got as close to the ideal of the Platonic philosopher-kings as American or world history has ever seen. In The Founding Fathers Reconsidered, R. B. Bernstein reveals Washington, Franklin, Jefferson, Adams, Hamilton, and the other founders not as shining demigods but as imperfect human beings--people much like us--who nevertheless achieved political greatness. They emerge here as men who sought to transcend their intellectual world even as they were bound by its limits, men who strove to lead the new nation even as they had to defer to the great body of the people and learn with them the possibilities and limitations of politics. Bernstein deftly traces the dynamic forces that molded these men and their contemporaries as British colonists in North America and as intellectual citizens of the Atlantic civilization\u27s Age of Enlightenment. He analyzes the American Revolution, the framing and adoption of state and federal constitutions, and the key concepts and problems--among them independence, federalism, equality, slavery, and the separation of church and state--that both shaped and circumscribed the founders\u27 achievements as the United States sought its place in the world.https://digitalcommons.nyls.edu/fac_books/1078/thumbnail.jp

Extrema in Velocity Dependence of Total Elastic Cross Sections for Atomic Beam Scattering: Relation to Di‐atom Bound States

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/70569/2/JCPSA6-37-8-1880-1.pd