## 3 Inherent Optical Properties (IOPs)
### Relationship between Effective Radius and Mode Radius
#### Impact of Relative Humidity  
A change in the relative humidity (RH) will affect bot the size and refractive index of a particle. The particle radius can be parameterized as a function of RH from the wet-to-dry mass ratio:  
\[r(a_{\rm w})=r_{0}\Big{[}1+\rho\frac{m_{\rm w}(a_{\rm w})}{m_{0}}\Big{]}^{1/3} \tag{76}\]  
where the water activity \(a_{\rm w}\) of a soluble aerosol at radius \(r\) [\(\mu\)m] can be expressed as  
\[a_{\rm w}={\rm RH}\exp\Big{[}\frac{-2\sigma V_{m}}{R_{\rm w}T}\frac{1}{r(a_{ \rm w})}\Big{]}. \tag{77}\]  
Here \(r_{0}\) is the dry particle radius (RH = 0), \(\rho\) is the particle density relative to that of water, \(m_{\rm w}(a_{\rm w})\) is the mass of condensed water, \(m_{0}\) is the dry particle mass (RH = 0), \(\sigma\) is the surface tension on the wet surface, \(V_{m}\) is the specific volume of water, \(R_{\rm w}\) is the gas constant for water vapor, and \(T\) is the absolute temperature [K] [36]. Similarly, the change in refractive index with RH can be determined from [36]  
\[\tilde{m}_{\rm c}=\tilde{m}_{\rm c,w}+(\tilde{m}_{\rm c,0}-\tilde{m}_{\rm c,w })\Big{[}\frac{r_{0}}{\tau_{\rm RH}}\Big{]}^{3} \tag{78}\]  
where \(\tilde{m}_{\rm c,w}\) and \(\tilde{m}_{\rm c,0}\) are the complex refractive indices of water and dry aerosols, respectively, and \(r_{0}\) and \(r_{\rm RH}\) are the radii of the aerosols in the dry state and at the given RH, respectively. From these formulas we note that the magnitude of the particle growth and the change of refractive index with increasing RH depend on the size \(r_{0}\) of the dry aerosol but also on the type of aerosol through the water uptake [the ratio \(m_{\rm w}(a_{\rm w})/m_{0}\) in Equation (76)] [36; 37; 38].  
A Mie code [39] is needed to compute the IOPs of aerosol particles [\(Q^{\prime}_{a}(r),Q^{\prime}_{\beta}(r)\), and \(p_{\rm P}(\lambda,\Theta,r)\) in Equations (58)-(60)], and numerical integration is required to evaluate the integrals over the lognormal size distributions to obtain \(\alpha_{\rm p}(\lambda),\beta_{\rm P}(\lambda)\), and \(p_{\rm P}(\lambda,\Theta)\). For polarized radiation all elements of the scattering phase matrix as well as the Greek constants appearing in Equation (48) must be computed. For a bimodal lognormal volume size distribution [Equation (66)] one must specify the fine mode volume fraction \(f_{v}=V_{f}/V\), where \(V=V_{f}+V_{c}\), the volume mode radii \(r_{vf}\) and \(r_{vc}\) as well as the corresponding standard deviations \(\sigma_{f}\) and \(\sigma_{c}\) in addition to the refractive index of the particles relative to air.  
In analogy to the liquid water content (see Equation (81) below), we may introduce the aerosol mass content (AMC) for each mode defined as  
\[{\rm AMC}=\rho_{\rm a}\int_{r_{\rm min}}^{r_{\rm max}}\left(\frac{4\pi}{3} \right)r^{3}n(r)dr\equiv\rho_{\rm a}f_{\rm V}\qquad[{\rm kg}\cdot{\rm m}^{-3}] \tag{79}\]  
where \(n(r)\) is the aerosol size distribution [\({\rm m}^{-3}\cdot{\rm m}^{-1}\)], \(\rho_{\rm a}\) is the bulk aerosol density [\({\rm kg}\cdot{\rm m}^{-3}\)], and \(f_{\rm V}\) is the aerosol volume fraction (not to be confused with the fine mode volume fraction, \(f_{v}\)) given by:  
\[f_{\rm V}\equiv\int_{r_{\rm min}}^{r_{\rm max}}\left(\frac{4\pi}{3}\right)r^{ 3}n(r)dr={\rm AMC}/\rho_{\rm a}\qquad({\rm dimensionless}). \tag{80}\]  
Typical values of atmospheric aerosol densities are \(\rho_{\rm a}\approx 1\,{\rm g}\cdot{\rm cm}^{-3}=1\times 10^{6}\,{\rm g} \cdot{\rm m}^{-3}\). Hence, an AMC value of \(10^{-6}\,{\rm g}\cdot{\rm m}^{-3}\) would yield \(f_{\rm V}=10^{-12}\).