The Quadrupole Response of Borromean Bosonic Trimers

The inelastic response of a Borromean trimer composed of three identical bosons is explored. To this end we use the quadrupole approximation as the excitation mechanism of the bosonic system from its ground state. We utilize the hyperspherical-harmonics expansion to solve the Schroedinger equation and the Lorentz integral transform method to calculate the reaction. It is found that the magnitude of the response function and corresponding sum rules increase as a power law when approaching the three-body threshold. It is also found that this increase is governed by unnatural exponents.     

Quadrupole response of a weakly bound bosonic trimer

The inelastic response of a bosonic trimer is explored in the confines of the Borromean region. To this end we model the interaction between the external field and the bosonic system as a photoabsorptionlike process and study the response of the trimer in the quadrupole approximation. We utilize the hyperspherical-harmonics expansion to solve the Schr?dinger equation and the Lorentz integral transform method to calculate the reaction. It is found that the magnitude of the response function and corresponding sum rules increase exponentially when approaching the 3-body threshold. It is also found that this increase is governed by unnatural exponents. The connection between our results and radio-frequency experiments in ultracold atom systems is made.     

Log-Periodic Oscillations in the Photo Response of Efimov Trimers

The photoassociation of Efimov trimer, composed of three identical bosons, is studied using the long wavelength approximation. It is shown that for identical particles the leading contribution comes from the r ² s-mode operator and from the quadrupole d-mode operator. Log-periodic oscillations are found in the photoassociation response function, both near the energy threshold for the leading s-wave reaction, and in the high frequency tail for all partial waves.     

A Strategy for Writing Equations of Graphs

A strategy for writing equations of graphs is introduced to help students and teachers build strong conceptual connections between the symbolic representations of algebra and the spatial representations of geometry. The strategy helps students and teachers weave the conceptual fabric of equations and graphs by (a) moving from unknown graphs to known graphs rather than from known to unknown graphs and by (b) moving from spatial representations to algebraic representations rather than from algebraic to spatial representations. Beginning with the unknown graph is distinctly different from present practices and can lead to significant and useful change in curriculum and instructional practices.     

Boosting Solid‐State Diffusivity and Conductivity in Lithium Superionic Argyrodites by Halide Substitution

Developing high‐performance all‐solid‐state batteries is contingent on finding solid electrolyte materials with high ionic conductivity and ductility. Here we report new halide‐rich solid solution phases in the argyrodite Li₆PS₅Cl family, Li_6?xPS_5?xCl_1+x, and combine electrochemical impedance spectroscopy, neutron diffraction, and ⁷Li NMR MAS and PFG spectroscopy to show that increasing the Cl^?/S^2? ratio has a systematic, and remarkable impact on Li‐ion diffusivity in the lattice. The phase at the limit of the solid solution regime, Li_5.5PS_4.5Cl_1.5, exhibits a cold‐pressed conductivity of 9.4±0.1 mS cm^?1 at 298 K (and 12.0±0.2 mS cm^?1 on sintering)—almost four‐fold greater than Li₆PS₅Cl under identical processing conditions and comparable to metastable superionic Li₇P₃S₁₁. Weakened interactions between the mobile Li‐ions and surrounding framework anions incurred by substitution of divalent S^2? for monovalent Cl^? play a major role in enhancing Li⁺‐ion diffusivity, along with increased site disorder and a higher lithium vacancy population.     

A New Sodium Thioborate Fast Ion Conductor: Na3B5S9

We report a new sodium fast-ion conductor, Na₃B₅S₉, that exhibits a high Na ion total conductivity of 0.80 mS cm⁻¹ (sintered pellet; cold-pressed pellet=0.21 mS cm⁻¹). The structure consists of corner-sharing B₁₀S₂₀ supertetrahedral clusters, which create a framework that supports 3D Na ion diffusion channels. The Na ions are well-distributed in the channels and form a disordered sublattice spanning five Na crystallographic sites. The combination of structural elucidation via single crystal X-ray diffraction and powder synchrotron X-ray diffraction at variable temperatures, solid-state nuclear magnetic resonance spectra and ab initio molecular dynamics simulations reveal high Na-ion mobility (predicted conductivity: 0.96 mS cm⁻¹) and the nature of the 3D diffusion pathways. Notably, the Na ion sublattice orders at low temperatures, resulting in isolated Na polyhedra and thus much lower ionic conductivity. This highlights the importance of a disordered Na ion sublattice—and existence of well-connected Na ion migration pathways formed via face-sharing polyhedra—in dictating Na ion diffusion.