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Student focusing and noticing, which drive reasoning, are important but under researched aspects of student learning. Quadratic functions representations are perceptually and conceptually complex and thus, offer much for students to focus on and notice. Our study compared a teacher’s goals for student focusing and noticing during quadratic functions instruction with what students actually focused on and noticed. Qualitative analysis revealed some alignment but also informative ways that the teacher’s goals and student outcomes for focusing and noticing were misaligned. These results will further the field’s understanding of how students learn about quadratic functions and may have implications for student focusing and noticing of other mathematics topics as well. 

We compared the grain and grain boundary (GB) nanostructures in two Ba122 tapes with similarly highJ_c. The Ag-sheathed tape made by hot pressing has larger, more plate-like grains with betterc-axis alignment but has more GBs blocked by FeAs and Ba–O. In contrast, the tape made by cold pressing with an Ag-Sn/stainless steel sheath possesses fewer plate-like grains and weaker grain alignment but has more continuous current paths with clean physically well-connected GBs. Our nanostructural comparison emphasizes the strong need to achieve both good grain alignment and clean GBs for furtherJ_cimprovement of Ba122 tapes.

 

The rapid progress that plasma wakefield accelerators are experiencing is now posing the question as to whether they could be included in the design of the next generation of high-energy electron-positron colliders. However, the typical structure of the accelerating wakefields presents challenging complications for positron acceleration. Despite seminal proof-of-principle experiments and theoretical proposals, experimental research in plasma-based acceleration of positrons is currently limited by the scarcity of positron beams suitable to seed a plasma accelerator. Here, we report on the first experimental demonstration of a laser-driven source of ultra-relativistic positrons with sufficient spectral and spatial quality to be injected in a plasma accelerator. Our results indicate, in agreement with numerical simulations, selection and transport of positron beamlets containing$$N_{e+}\ge 10^5$$$N_{e+}\ge {10}^5$positrons in a 5% bandwidth around 600 MeV, with femtosecond-scale duration and micron-scale normalised emittance. Particle-in-cell simulations show that positron beams of this kind can be guided and accelerated in a laser-driven plasma accelerator, with favourable scalings to further increase overall charge and energy using PW-scale lasers. The results presented here demonstrate the possibility of performing experimental studies of positron acceleration in a laser-driven wakefield accelerator.

 

Organic trisradicals featuring three-fold symmetry have attracted significant interest because of their unique magnetic properties associated with spin frustration. Herein, we describe the synthesis and characterization of a triangular prism-shaped organic cage for which we have coined the name PrismCage6+ and its trisradical trication—TR3(•+). PrismCage6+ is composed of three 4,4'-bipyridinium dications and two 1,3,5-phenylene units bridged by six methylene groups. In the solid state, PrismCage6+ adopts a highly twisted conformation with close to C3 symmetry as a result of encapsulating one PF6− anion as a guest. PrismCage6+ undergoes stepwise reduction to its mono-, di- and trisradical cations in MeCN on account of strong electronic communication between its 4,4'-bipyridinium units. TR3(•+), which is obtained by reduction of PrismCage6+ employing CoCp2, adopts a triangular prism-shaped conformation with close to C2v symmetry in the solid state. Temperature-dependent continuous-wave and nutation frequency-selective EPR spectra of TR3(•+) in frozen N,N-dimethylformamide indicate its doublet ground state. The doublet-quartet energy gap of TR3(•+) is estimated to be −0.06 kcal mol−1 and the critical temperature of spin-state conversion is found to be ca. 50 K, suggesting that it displays pronounced spin-frustration at the molecular level. To the best of our knowledge, this example is the first organic radical cage to exhibit spin frustration. The trisradical trication of PrismCage6+ opens up new possibilities for fundamental investigations and potential applications in the fields of both organic cages and spin chemistry.