Observation of the nonlinear effect in relativistic Thomson scattering of electron and laser beams

Thomson scattering of high-power laser and electron beams is a good test of electrodynamics in the high-field region. We demonstrated production of high-intensity X-rays in the head-on collision of a CO₂ laser and 60-MeV electron beams at Brookhaven National Laboratory, Accelerator Test Facility. The energy of an X-ray photon was limited at 6.5 keV in the linear (lowest order) Thomson scattering, but the nonlinear (higher order) process produces higher energy X-rays. We measured the angular distribution of the high-energy X-rays and confirmed that it agrees with theoretical predictions.     

Opers and theta functions

We construct natural maps (the Klein and Wirtinger maps) from moduli spaces of semistable vector bundles over an algebraic curve X to affine spaces, as quotients of the nonabelian theta linear series. We prove a finiteness result for these maps over generalized Kummer varieties (moduli space of torus bundles), leading us to conjecture that the maps are finite in general. The conjecture provides canonical explicit coordinates on the moduli space. The finiteness results give low-dimensional parametrizations of Jacobians (in for generic curves), described by 2Θ functions or second logarithmic derivatives of theta.We interpret the Klein and Wirtinger maps in terms of opers on X. Opers are generalizations of projective structures, and can be considered as differential operators, kernel functions or special bundles with connection. The matrix opers (analogues of opers for matrix differential operators) combine the structures of flat vector bundle and projective connection, and map to opers via generalized Hitchin maps. For vector bundles off the theta divisor, the Szegö kernel gives a natural construction of matrix oper. The Wirtinger map from bundles off the theta divisor to the affine space of opers is then defined as the determinant of the Szegö kernel. This generalizes the Wirtinger projective connections associated to theta characteristics, and the associated Klein bidifferentials.     

A Quantization on Riemann Surfaces with Projective Structure

Let X be a Riemann surface equipped with a projective structure. Let be a square-root of the holomorphic cotangent bundle K _X . Consider the symplectic form on the complement of the zero section of obtained by pulling back the symplectic form on K _X using the map ². We show that this symplectic form admits a natural quantization. This quantization also gives a quantization of the complement of the zero section in K _X equipped with the natural symplectic form.     

Statistical modeling to promote students’ aggregate reasoning with sample and sampling

While aggregate reasoning is a core aspect of statistical reasoning, its development is a key challenge in statistics education. In this study we examine how students’ aggregate reasoning with samples and sampling (ARWSS) can emerge in the context of statistical modeling activities of real phenomena. We present a case study on the emergent ARWSS of two pairs of sixth graders (age 11–12) involved in statistical data analysis and informal inference utilizing TinkerPlots. The students’ growing understandings of various statistical concepts is described and five perceptions the students expressed are identified. We discuss the contribution of modeling to these progressions followed by conclusions and limitations of these results. While idiosyncratic, the insights contribute to the understanding of students’ aggregate reasoning with data and models, with regards to samples and sampling.     

Inverse Cerenkov acceleration and inverse free-electron laserexperimental results for staged electron laser acceleration

The goal of the staged electron laser acceleration (STELLA) experiment is to demonstrate staging of the laser acceleration process whereby an inverse free electron laser (IFEL) will be used to prebunch the electrons, which are then accelerated in an inverse Cerenkov accelerator (ICA). As preparation for this experiment, a new permanent magnet wiggler for the IFEL was constructed and the ICA system was modified. Both systems have been tested on a new beamline specifically built for STELLA. The improved electron beam (e-beam) with its very low emittance (0.8 mm-mrad normalized) enabled focusing the e-beam to an average radius (1σ) of 65 μm, within the ICA interaction region. This small e-beam focus greatly enhanced the ICA process and resulted in electron energy spectra that have demonstrated the best agreement to date in both overall shape and magnitude with the model predictions. The electron energy spectrum using the new wiggler in the IFEL was also measured. These results will be described as well as future improvements to the STELLA experiment     

Direct transfer of aliphatic and aromatic substituents from organosilatranes to mercury(II) species

The relative reaction rates of several silatranes (derivatives of 2,8.9-trioxa-5-aza- 1-silatricyclo[3.3.3.0^1,5]undecane) and HgCl₂ in acetone-d₆to yield the corresponding organomercury compound are of the order of e.g., 5 × 10^?1 1 mol^?1 sec^?1 or slightly less, a rate that is unexpectedly high compared to the essentially inert parent organotrialkoxysilanes. Thus, the apical Si&.zsbnd;C bond of the silatrane is extraordinarily susceptible to direct electrophilic attack by mercury(II). The rates decrease in the order CH₂CH, C₆H₅, p-ClC₆H₄ > CH₃ > CH₃CH₂, CH₃CH₂CH₂ > C₆H₁₁, ClCH₂, Cl₂CH, CH₃CH₂O. The effects of varying the solvent and the counterions are noted, and the probable mechanism is discussed.