# Monte Carlo Study of Imaging Plate Response to Laser-Driven Aluminum Ion Beams
## 5 Calculation of PSL/Ion from SRIM Data  
There are two models commonly used to predict the amount of PSL from a given radiation. [PERSON] et al. [28] propose a linear model assuming that the yield of PSL is proportional to the total deposited energy in the sensitive layer of an IP. \[R(E_{ion})=\alpha E_{dep}(E_{ion}), \tag{3}\]  
where \(R(E_{ion})\) is the IP response for ions with incident kinetic energy of \(E_{ion}\) and \(\alpha\) is the IP sensitivity. The IP sensitivity varies depending on the type of radiation and IPs. It also depends on the waiting time before scanning because of the fading loss. The total deposited energy, \(E_{dep}(E_{ion})\), is obtained from the integral of the ion-stopping power \(S(z)\) at depth \(z\). In the phosphoric layer of the IP,  
\[E_{dep}=\int_{0}^{W}S(z)dz, \tag{4}\]  
where \(S(z)=-dE_{ion}/dz\) and \(W\) is the thickness of the layer. The incident ion loses its kinetic energy to electrons in the target by ion-electron collisions resulting from Coulomb interaction and to target nuclei by ion-nucleon collisions (called recoil process). In our SRIM simulations, the predominant process is Coulomb interaction, and the recoil energy contributes only \(\sim\)\(10^{-4}\) to the entire collision process. The cross-section for ion-electron interaction is inversely proportional to the square of the approaching speed of the ion. [PERSON] et al. [14] proposed a model for the deposited energy that accounts for the optical thickness of the IP to the PSL radiation by weighting the stopping power by an exponential decay term, i.e.,  
\[E_{dep}=\int_{0}^{W}S(z)e^{-\frac{z}{L}}\ dz, \tag{5}\]  
where \(L\) is the absorption length of PSL photons within the phosphor layer of the IP [14], which is 44 \(\pm\) 4 \(\upmu\)m for BAS-TR IPs [16]. A large \(L\) implies that the PSL photons pass through the phosphor layer easily without being absorbed within it [16], and thus a negligible correction. However, \(L\) = 44 \(\upmu\)m indicates that a significant amount of PSL is absorbed since the thickness phosphor layer of BAS-TR IP is 50 \(\upmu\)m. Equation (4) can also be interpreted as a special case when the absorption length is infinity and there is no absorption [14]. We refer IP models proposed by [PERSON] et al. and [PERSON] et al. as the linear model and the exponential model, respectively, in the following analysis. To obtain the stopping power for Al beams, we use the Monte Carlo simulation code SRIM which calculates the stopping and range of ions in matter using a quantum mechanical treatment of ion-atom collision [30]. Each simulation is performed with 10,000 incidences and calculated stopping power and other values are averaged over the 10,000 incidences. We calculate \(E_{dep}\) using two different methods (1) by using the averaged \(S(z)\) obtained directly from SRIM, and (2) by calculating the absorbed energy and \(e^{-\frac{z}{L}}\) at each step from individual incidence then averaging \(E_{dep}\) over 10,000 incidences. SRIM divides the target depth into 100 steps in the first method, and about 700 uniform steps in the second method, which determines the simulation resolution. Although the second method has seven times higher target depth resolution in computing the total deposited energy in the phosphor layer, the discrepancy in the deposited energy calculations by each method is found to be only 0.01-0.1%. In the linear model, we have calculated \(S(z)\) using the first method for simplicity. In the exponential model, we have used the second method to calculate \(S(z)\) since more steps can potentially reduce errors involved in calculating the weighting factor \(e^{-\frac{z}{L}}\).