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The performance of germanium detectors for high-energy $\gamma$-rays was evaluated using a 992-keV resonance in the $^{27}$Al(p, $\gamma$)$^{27}$Si reaction. The measurement was conducted at the RIKEN tandem accelerator. The energy of the excited state from the resonance was evaluated as 12540.7(2) keV. Using newly evaluated excitation energy, an energy calibration function and the photo-peak efficiency of Ge detectors up to 10.8-MeV photon were deduced. The energy accuracy is achieved at 0.3 keV for the overall energy region. This reaction provides reliable energy and efficiency standards for high-energy $\gamma$ rays.

We study interacting bosonic or fermionic atoms in a high synthetic magnetic field in two dimensions spanned by continuous real space and a synthetic dimension. Here, the synthetic dimension is provided by hyperfine spin states, and the synthetic field is created by laser-induced transitions between them. While the interaction is short-range in real space, it is long-range in the synthetic dimension in sharp contrast with fractional quantum Hall systems. Introducing an analog of the lowest-Landau-level approximation valid for large transition amplitudes, we derive an effective one-dimensional lattice model, in which density-density interactions turn out to play a dominant role. We show that in the limit of a large number of internal states, the system exhibits a cascade of crystal ground states, which is known as devil's staircase, in a way analogous to the thin-torus limit of quantum Hall systems., Comment: 9 pages, 4 figures