= Cross-section of a power-law fragment population
{title2=$\sigma_{\rm tot}$}
= Cross-sections of a power-law fragment population
{synonym}
For spherical fragments of common <mass density> $\rho$, with $n(D)=KD^{-\alpha}$ between specified diameter cutoffs, <mass normalization of a power-law size distribution> and integration of the <geometric cross-section> give
$$
\sigma_{\rm tot}=\frac{3m}{2\rho}\frac{4-\alpha}{\alpha-3}
\frac{D_{\min}^{3-\alpha}-D_{\max}^{3-\alpha}}
{D_{\max}^{4-\alpha}-D_{\min}^{4-\alpha}},\qquad 3<\alpha<4.
$$
Indeed $m=(\pi\rho K/6)\int D^{3-\alpha}\,dD$ and $\sigma_{\rm tot}=(\pi K/4)\int D^{2-\alpha}\,dD$; eliminating $K$ proves the formula. If both cutoff ratios raised to $\alpha-3$ and $4-\alpha$ are small, respectively, this reduces to $3m(4-\alpha)D_{\min}^{3-\alpha}D_{\max}^{\alpha-4}/[2\rho(\alpha-3)]$. Approaching either endpoint requires the corresponding logarithmic integral.
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