properties {#subsec:properties}
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We consider silicate and graphite grains in this paper. Since we are interested in the wavelength range appropriate for the *AKARI* FIS bands ($\lambda\sim 50$–180 $\mu$m; Section \[subsec:analysis\]), we neglect polycyclic aromatic hydrocarbons (PAHs), which contribute significantly to the emission at $\lambda\la 20~\mu$m [@desert90; @dwek97; @draine01].

We assume a grain to be spherical with a radius of $a$. The absorption cross section of the grain is expressed as $\pi a^2Q_\mathrm{abs}(\lambda )$, where $Q_\mathrm{abs}(\lambda )$ is called absorption efficiency. We adopt the optical constants of astronomical silicate and graphite for $\lambda <100~\mu$m from @draine84 and @weingartner01 and calculate the absorption efficiency by using Mie theory [@bohren83]. For the absorption efficiency at $\lambda >100~\mu$m, we adopt a functional form proposed by @reach95: $$\begin{aligned}
Q_\mathrm{abs}(\lambda )=
\frac{(\lambda /\lambda_0)^{-2}}
{[1+(\lambda_1/\lambda)^6]^{1/6}}\, ,\label{eq:reach}\end{aligned}$$ which behaves like $\beta =1$ ($Q_\mathrm{abs}(\lambda )\propto\lambda^{-\beta}$, where $\beta$ is called emissivity index) for $\lambda\ll\lambda_1$ and $\beta =2$ at $\lambda\gg\lambda_1$ [see also @bianchi99]. We assume that $\lambda_1=200~\mu$m, following @reach95. We set the value of $\lambda_0$ so that the continuity of $Q_\mathrm{abs}$ at $100~\mu$m is satisfied. As shown in HHS07, the wavelength dependence of $Q_\mathrm{abs}$ assumed here explains the FIR colour–colour relations of the Milky Way, the LMC and the SMC. 