## 3 Energy and Energy Spread Characterization  
The proton energy significantly influences the depth position of the Bragg peak. The impact of energy spread on the depth-dose distribution is well documented in the literature [21]. This effect is characterized by a decrease in peak intensity and a shift in the Bragg peak position as the energy spread increases. To investigate these phenomena, we conducted an MC simulation. In this simulation, a round beam was initialized at the source position, with the transverse beam properties modeled as a Gaussian distribution, resulting in a beam size of 4 mm, as illustrated in Figure 3. The initial beam parameters are summarized in Table 1. Proton beam energies were varied from 70 MeV to 220 MeV, with an energy spread modeled as a Gaussian distribution with a standard deviation of up to 5%. The simulated results of the energy spread investigation are shown in Figure 4. The results are presented as the dose profile per proton particle as a function of depth position, with the left plot representing 220 MeV and the right plot displaying energies ranging from 70 MeV to 220 MeV. A shift in the Bragg peak position to lower depths occurs as the energy spread increases, accompanied by a decrease in peak intensity with increasing energy spread. \begin{table}
\begin{tabular}{l c} \hline \hline
**Parameter** & **Value** \\ \hline Beam kinetic energy (\(E_{k}\)) & 70–220 MeV \\ Energy Spread (\(\sigma_{E}\)) & 0–5 \% \\ Energy spectrum & Gaussian \\ Transverse beam size, round beam (\(\sigma\)) & 4 mm \\ Transverse beam profile & Gaussian distribution \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Initial proton parameters used in GEANT4 simulation for energy spread investigations. Figure 3: Transverse profile distribution of the proton beam at the source position, with a beam size of 4.0 mm in both horizontal and vertical directions (**left**), and the energy distribution of the 70 MeV proton beam, with an energy spread of 0.62 MeV corresponding to 0.88% (**right**). As listed in Table 2, for the 220 MeV proton beam, the Bragg peak position shifts to a lower depth by approximately 23 mm when the energy spread increases to 5%. Simultaneously, the peak dose intensity decreases by 50%. The energy spread has less influence on the R\({}_{80}\) position, which more reliably represents the range position for each mean energy than the Bragg peak position. Subsequently, the distal fall-off length increases linearly with increasing energy spread. These values, along with the distal fall-off length, will be used to match the depth-dose  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Energy (MeV)** & **Energy Spread (\%)** & **R\({}_{\text{BP}}\) (mm)** & \begin{tabular}{c} **Peak Dose** \\ **(pGy/Particle)** \\ \end{tabular}  & **R\({}_{\text{80}}\) (mm)** & **W\({}_{\text{80-20}}\) (mm)** \\ \hline \multirow{8}{*}{70} & 0 & 40.2 & 92.1 & 40.9 & 0.4 \\  & 1 & 39.8 & 73.2 & 40.  
profile obtained from experiments conducted with a water phantom. This matching process provides the energy spread of the proton beam at the exit window for proton energies of 70 MeV, 100 MeV, 150 MeV, and 220 MeV. In addition to its impact on Bragg peak position, energy spread also significantly influences the linear energy transfer dose (\(\mathrm{LET_{d}}\)), as illustrated in Figure 5. The results demonstrate that \(\mathrm{LET_{d}}\) decreases with increasing proton energy. Specifically, lower-energy protons (e.g., 70 MeV) exhibit higher \(\mathrm{LET_{d}}\) values due to their shorter range and denser energy deposition, whereas higher-energy protons (e.g., 220 MeV) show lower \(\mathrm{LET_{d}}\) values as they travel further with less energy deposited per unit length. This behavior aligns with the well-established physical characteristics of proton interactions in matter. Furthermore, an increase in energy spread results in a reduction in the peak \(\mathrm{LET_{d}}\), highlighting the importance of beam quality in treatment planning.