## 3 Results  
Figure 4 shows the comparison of the approximated MEE (long dashed red line) and that based on the Mie theory (solid black line) for AMS (left column) and AMN (right column). Geometric standard deviations of 1.5 (top row), 2 (middle row), and 2.5 (bottom row) were considered in these calculations. The \(\text{MEE}_{S}\) (\(\text{MEE}_{L}\)) for a small (large) size range is shown with the dotted blue (yellow) line in each panel. The \(\text{MEE}_{S}\) and \(\text{MEE}_{L}\) were approximated by fitting the Mie theory-based calculations (i.e., solid black line) in the form of a power law (i.e., Equation (6)) for \(\frac{d\text{MEE}}{d_{g^{0}}}>0\) and \(\frac{d\text{MEE}}{d_{g^{0}}}<0\), respectively. The \(\text{MEE}_{S}\) for AMS were approximated as \(\text{MEE}_{S}=40d_{g^{0}}^{2}\), \(\text{MEE}_{S}=20d_{g^{0}}^{1.5}\), and \(\text{MEE}_{S}=10d_{g^{0}}^{1.0}\) for geometric standard deviations of 1.5, 2.0, and 2.5, respectively. For AMN, they were \(\text{MEE}_{S}=25d_{g^{0}}^{1.7}\), \(\text{MEE}_{S}=15d_{g^{0}}^{1.3}\), and \(\text{MEE}_{S}=10d_{g^{0}}^{1.0}\). The \(\text{MEE}_{L}\) for both AMS and AMN was approximated as \(\text{MEE}_{L}=2.5d_{dv}^{-1}\) for all geometric standard deviations. For the intermediate size range, the MEE was determined by the harmonic mean-type approximation in Equation (5). It is shown that the Mie theory-based calculations of MEE and those approximated using the method proposed in this study are comparable without much loss of accuracy. It is also shown that the smooth transition from a small to large size range and vice versa is achieved by the approximated method. A comparison between the approximated MEE (long dashed red line) and that based on the Mie theory (solid black line) for sea salt (left column) and EC (right column) for geometric standard deviations of 1.5 (top row), 2.0 (middle row), and 2.5 (bottom row) is shown in Figure 5.  
For sea salt, the \(\text{MEE}_{S}\) was approximated as \(\text{MEE}_{S}=25d_{g^{0}}^{1.7}\), \(\text{MEE}_{S}=15d_{g^{0}}^{1.3}\), and \(\text{MEE}_{S}=8d_{g^{0}}^{1.3}\) for geometric standard deviations of 1.5, 2.0, and 2.5, respectively, while the \(\text{MEE}_{L}\) was approximated as \(\text{MEE}_{L}=2.5d_{g^{0}}^{-1}\) for all geometric standard deviations, as in the cases of AMS and AMN. In contrast, for EC, the \(\text{MEE}_{S}\) was approximated as \(12d_{g^{0}}^{0.2}\) regardless of geometric standard deviation, while the \(\text{MEE}_{L}\) had different representations for different geometric standard deviations. The approximated \(\text{MEE}_{L}\) for EC was \(\text{MEE}_{L}=2d_{g^{0}}^{-1.5}\), \(\text{MEE}_{L}=2.5d_{g^{0}}^{-1.3}\), and \(\text{MEE}_{L}=3d_{g^{0}}^{-1.1}\) for geometric standard deviations of 1.5, 2.0, and 2.5, respectively. The same procedure to determine \(\text{MEE}_{S}\) and \(\text{MEE}_{L}\) (Equation (6)) and MEE for intermediate size range by harmonic mean-type approximation (Equation (5)) for AMS and AMN in Figure 4 was also applied to sea salt and EC. Figure 4: Comparison between the Mie-calculated mass extinction efficiency (MEE, solid black lines) and the approximated mass extinction efficiency (long-dashed red lines) of ammonium sulfate (AMS, **left column**) and ammonium nitrate (AMN, **right column**) for geometric standard deviations (\(\sigma_{g}\)) of \(1.5\) (**top row**), \(2.0\) (**middle row**), and \(2.5\) (**bottom row**). The approximated mass extinction efficiency in each panel is calculated based on the MEE\({}_{S}\) (dotted blue line) and the MEE\({}_{L}\) (dotted yellow line) in each panel using Equation (5). Figures 4 and 5 show that the MEE can be estimated by using harmonic mean-type approximation for each size distribution. The power-law coefficients \(a\) and \(b\) are represented as a linear expression of geometric mean diameter as explained in Equation (9). Subsequently, the resultant approximated expression for MEE is as follows:  
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