## 2 Experimental
The Compton scattering X-ray intensity, \(\mathrm{d}N\), is calculated as [21]:  
\[\mathrm{d}N=\Phi_{0}t_{1}t_{2}\rho_{\mathrm{e}}\mathrm{d}V\frac{\mathrm{d} \sigma_{\mathrm{KN}}}{\mathrm{d}\Omega}, \tag{1}\]  
where \(\Phi_{0}\) is the photon flux of the incident X-rays; \(t_{1}\) and \(t_{2}\) are the path length of the incident and scattering X-rays in the target matter, respectively; \(\rho_{\mathrm{e}}\) is the average electron density over the probing volume; \(\mathrm{d}V\) is the probing volume; and \(\mathrm{d}\sigma_{\mathrm{KN}}/\mathrm{d}\Omega\) is the Klein-Nishina differential cross-section. Experimentally, \(\mathrm{d}N\) is given by the area of the Compton scattering X-ray energy spectrum. \(\mathrm{d}N\) depends on \(t_{1}\), \(t_{2}\), and \(\rho_{\mathrm{e}}\). When the X-ray path length can be treated as constant, the Compton scattering X-ray  
Figure 1: **(a)** Picture and schematic image of a commercial coin-type lithium rechargeable cell VL2020. **(b)** The cycle dependency of the discharge capacity in the VL2020 cell. This study uses two cells with zero cycles and after 170 cycles. **(c)** The experimental setup of the Compton scattering imaging system at BL08 Wof SPring-8. **(d)** Theoretical Compton profiles of \(\mathrm{Li_{x}V_{2}O_{5}}\) (x = 0, 0.5, and 1) obtained from the Hartree–Fock calculations. intensity is proportional to the electron density \(\rho_{\text{e}}\). Therefore, Compton scattering intensity has been used for densitometry [21] and imaging [7; 22], and the analysis of Compton scattering X-ray intensity allows the identification of the matter at the local region in the cell. The S-parameter is a parameter that characterizes the modification of the Compton scattering X-ray energy spectrum with lithium insertion. Figure 1d shows the Compton scattering X-ray energy spectrum, the so-called Compton profile, of Li\({}_{x}\)V\({}_{2}\)O\({}_{5}\) (lithium concentration \(x\) = 0, 0.5, and 1) obtained from Hartree-Fock calculations [23]. The peak height, which is seen to increase with increasing lithium concentration, demonstrates the sensitivity of the profile to the lithium concentration. A previous study clarified that there is linearity between the experimental S-parameter and lithium concentration obtained from inductively-coupled plasma (ICP) analysis [11]. Therefore, the S-parameter allows us to quantitate the lithium concentration in the electrode non-destructively. The Compton profile, \(J(p_{z})\), is related to the ground-state electron momentum density \(\rho(\mathbf{p})\) in the momentum space via the following double integral [24]:  
\[J\left(p_{z}\right)=\iint\rho(\mathbf{p})dp_{x}dp_{y}, \tag{2}\]  
where \(\mathbf{p}=(p_{x},\,p_{y},\,p_{z})\) is the electron momentum. \(\rho(\mathbf{p})\) is given by [25; 26]:  
\[\rho(\mathbf{p})=(2\pi)^{-3/2}\sum_{j}n_{j}\left|\int\Psi_{j}(\mathbf{r})\exp (-i\mathbf{p}\cdot\mathbf{r})d\mathbf{r}\right|^{2}, \tag{3}\]  
where \(\Psi_{j}(\mathbf{r})\) is the wave function of an electron in the \(j\)-state and \(\mathrm{n}_{j}\) is the electron occupation. The index \(j\) covers all constituent atoms and orbitals. \(J(p_{z})\) is directly linked to the wave function in real space, and the line-shape of \(J(p_{z})\) changes depending on the chemical element. These features have also been used to identify electronic structures and redox orbitals in the electrode reaction through the combination with first-principle calculations and have successfully visualized the redox orbitals of cathode materials [27; 28; 29; 30]. The S-parameter is defined as [11]:  
\[S=\frac{\int_{-d}^{d}J\left(p_{z}\right)dp_{z}}{\int_{-l}^{d}J\left(p_{z} \right)dp_{z}+\int_{d}^{l}J\left(p_{z}\right)dp_{z}}, \tag{4}\]  
where parameters \(d\) and \(l\) define the range of low- and high-momentum regions, respectively, dividing the dominant and inconspicuous lithium regions. We used \(d=1\) atomic units (a.u.) and \(l=5\) a.u. These were determined by comparing fully charged and discharged Compton scattering X-ray energy spectra as reported previously [14].