Tris(2-methyl­piperidinium) tetra­chlorido­ferrate dichloride

The asymmetric unit of the title salt, (C6H14N)3[FeCl4]Cl2, consists of a tetra­hedral tetra­chloro­ferrate anion, three independent 2-methyl­piperidinium cations and two chloride ions. All the piperidine rings adopt chair conformations. In the crystal, the organic cations and the free chloride anions are linked into chains parallel to the a axis by N—H⋯Cl hydrogen bonds

Improved Analytic Solution of Black Hole Superradiance

The approximate solution of the Klein-Gordon equation for a real scalar field of mass $\mu$ in the geometry of a Kerr black hole obtained by Detweiler \cite{Detweiler:1980uk} is widely used in the analysis of the stability of black holes as well as the search of axion-like particles. In this work, we confirm a missing factor $1/2$ in this solution, which was first identified in Ref.~\cite{Pani:2012bp}. The corrected result has strange features that put questions on the power-counting strategy. We solve this problem by adding the next-to-leading order (NLO) contribution. Compared to the numerical results, the NLO solution reduces the percentage error of the LO solution by a factor of 2 for all important values of $r_g \mu$. Especially the percentage error is $\lesssim 10\%$ in the region of $r_g\mu \lesssim 0.35$. The NLO solution also has a compact form and could be used straightforwardly.Comment: 5 pages, 3 figures, Accepted by PR