## 2 Experimental  
In this study, we use a commercial coin-type lithium rechargeable battery (VL2020, Panasonic Corporation, Osaka, Japan). The VL2020 is composed of a V\({}_{2}\)O\({}_{5}\) cathode, an olefin-based non-woven fabric separator, an LiAl alloy anode, and a dimethoxyethane electrolyte. Although these cathode and anode materials are different from typical lithium-ion batteries used in the industry today, the V\({}_{2}\)O\({}_{5}\) and LiAl alloy are still extensively studied for electrodes. The features of V\({}_{2}\)O\({}_{5}\) cathodes are low cost, abundance, and having the possibility for high capacity. Here, the theoretical capacities of V\({}_{2}\)O\({}_{5}\), LiCoO\({}_{2}\), LiMn\({}_{2}\)O\({}_{4}\), and LiFePO\({}_{4}\) are 294 mAhg\({}^{-1}\), 274 mAhg\({}^{-1}\), 148 mAhg\({}^{-1}\), and 170 mAhg\({}^{-1}\), respectively. The V\({}_{2}\)O\({}_{5}\) of micro- and nano-structures have demonstrated good cycle performance and rate properties [15; 16; 17]. V\({}_{2}\)O\({}_{5}\) is also considered as a cathode material for next-generation Al-ion [18] and Mg-ion batteries [19]. The LiAl alloy anodes are attracting attention in the battery industry. The LiAl alloys have potential as anodes for lithium-oxygen batteries and have been reported to have high cycle stability [20]. The thicknesses of the cathode and anode are 800 \(\upmu\)m and approximately 300 \(\upmu\)m, respectively. The nominal voltage and capacity of the cell are 3 V and 20 mAh, respectively. The cut-off voltages are 2.5 and 3.5 V.  
To analyze the degraded state of the cell, an unused cell (called a fresh cell) and a used cell (called an aged cell) were used. The aged cell was prepared by cycling the cell 170 times at room temperature. During the cycling process, the cell repeatedly charged and discharged with a constant current (CC) of 10 mA. A reference cycle was performed at 10 cycle intervals. In the reference cycle, the cell was charged with a CC of 2 mA until the 3.5 V cut-off voltage was reached, and the charging was continued 3 h in constant voltage (CV) mode at 3.5 V. Afterward, the discharge was performed with a CC of 2 mA until the 2.5 V cut-off voltage was reached. We confirmed that the capacity became almost 0 mAh after 170 cycles, as shown in Figure 1b. Compton scattering experiments were conducted at BL08 Wof the Synchrotron Facility, SPring-8, Hyogo, Japan. The experimental setup is shown in Figure 1c. The incident X-rays emitted from a multipole wiggler were monochromatized to 114.56 keV using bend-type Si(400) crystal. X-ray energy spectra along the scattering angle of 90 degrees were measured using nine segments of a pure Ge solid-state detector (SSD). These nine detectors were adjusted by the collimator slit to observe the same region in the sample cell. The observation region in the cell was limited by the incident and collimator slits. The sizes of these incident and collimator slits were 20 \(\times\) 500 \(\upmu\)m\({}^{2}\) and 500 \(\upmu\)m in diameter, respectively. The cell was set on a movable stage, which had a function adjustment system using the x, y, and z positions of the sample cell and the \(\theta\), azimuth, and tilt angles. The Compton scattering X-ray energy spectrum was measured using scanning incident X-rays along the vertical (z) and horizontal (x) directions of the cell. The scanning area was 0.8-1 mm in height and 5 mm in width, which corresponded from the lower end of the cathode to the end of the anode on the z-direction. The measurement time at one position was 30 s.  
The obtained X-ray energy spectrum was analyzed using two methods: Compton scattering X-ray intensity and the S-parameter.