## 3 Inherent Optical Properties (IOPs)
### IOPs for a Size Distribution of Particles  
Particles encountered in nature consist of a variety of chemical compositions, sizes, and shapes. The chemical composition determines the refractive index of the particle, and unless the composition is the same throughout the particle, the refractive index will depend on location inside the particle. The computation of IOPs for such a collection of particles requires solutions of [PERSON]'s equations for electromagnetic radiation interacting with an inhomogeneous, non-spherical particle of a given size. Then one needs to integrate over size and shape for particles of a given chemical composition, and finally average over the particle composition. To avoid having to deal with this complexity it is frequently assumed that the particles are homogeneous with a constant refractive index, and that the shape can be taken to be spherical. Even with these assumptions, one still needs to deal with the variety of particles sizes encountered in nature. For a spherical particle with a specified radius and refractive index, Mie theory (and its numerical implementation) may be used to generate IOPs for a single particle. Thus, if we have computed the IOPs for a single spherical particle with specified refractive index and a given size, we may compute the absorption and scattering coefficients and the scattering phase function for a polydispersion of particles by integrating over the particle size distribution (PSD):  
\[\alpha_{p}(\lambda)=\int_{r_{min}}^{r_{max}}\alpha_{n}(\lambda,r)n(r)dr=\int_ {r_{min}}^{r_{max}}\pi r^{2}Q_{\alpha}^{\prime}(\lambda,r)n(r)dr \tag{58}\]  
\[\beta_{p}(\lambda)=\int_{r_{min}}^{r_{max}}\beta_{n}(\lambda,r)n(r)dr=\int_{r_ {min}}^{r_{max}}\pi r^{2}Q_{\beta}^{\prime}(\lambda,r)n(r)dr \tag{59}\]  
\[p_{p}(\lambda,\Theta)=\frac{\int_{r_{min}}^{r_{max}}p(\lambda,\Theta,r)n(r)dr} {\int_{r_{min}}^{r_{max}}n(r)dr} \tag{60}\]where \(n(r)\) is the PSD and \(\alpha_{n}(\lambda,r)\), \(\beta_{n}(\lambda,r)\), and \(p(\lambda,\Theta,r)\) are the absorption cross section, the scattering cross section, and the scattering phase function per particle of radius \(r\). The absorption or scattering \"efficiency\", \(Q^{\prime}_{\alpha}(r)\) or \(Q^{\prime}_{\beta}(r)\), is defined as the ratio of the absorption or scattering cross section for a spherical particle of radius \(r\) to the geometrical cross section \(\pi r^{2}\). The scattering phase function \(p(\lambda,\Theta,r)\) in Equation (60) is the \(a_{1}(\Theta)\) element of the Stokes scattering matrix [Equation (18)]. Since a Mie code can be used to compute all elements of the Stokes scattering matrix in Equation (18), we may use an expression analogous to Equation (60) to carry out the integration over the PSD for each of the matrix elements. IOPs for a Mixture of Different Particle Types  
Consider a particle mixture consisting of a total of \(N\) particles per unit volume in a layer of thickness \(\Delta z\), and let \(N=\sum_{i}n_{i}\) and \(f_{i}=n_{i}/N\), where \(n_{i}\) is the concentration and \(f_{i}\) the fraction of homogeneous particles (with fixed chemical composition or refractive index) of type labeled \(i\). To compute IOPs for the mixture of particles, we define \(\beta_{n,i}=\) scattering cross section, \(\alpha_{n,i}=\) absorption cross section, \(\gamma_{n,i}=\beta_{n,i}+\alpha_{n,i}=\) extinction cross section, and \(\alpha_{i}=\beta_{n,i}n_{i}/\gamma_{n,i}n_{i}=\beta_{i}/\gamma_{i}=\) single-scattering albedo, where the subscript \(i\) stands for particle type. Weighting by number concentration may be used to create IOPs for the particle mixture.