I'm reading the book of Chandrasekhar "Radiative Transfer" and I have some doubts about some definitions when the Rayleigh scattering is assumed. In particular, when we consider a plane-parallel geometry, the text define the optical thickness as

\tau(z) = \int_{z}^{\infty} \kappa \rho \: d \zeta

where $ \kappa $ is the mass scattering coefficient, and $ \rho $ is the mass density. I think that Chandrasekhar sets the coordinate system as in the attached figure.

Under the hypothesis of Rayleigh scattering, the text states that

\kappa = \frac{8 \pi ^ 3}{3} \frac{(n ^ 2 - 1) ^ 2}{\lambda ^ 4 N \rho}

where $ n $ is the refractive index, $ \lambda $ is the wavelength, and $ N $ is the number of particle per unit volume. Assuming a single homogeneous layer, I think that the optical thickness could be treated as follows

\tau(z) = \int_{z}^{\infty} \kappa \rho \: d \zeta = \kappa \rho \int_{z}^{0} d \zeta = - \kappa \rho z

for $z \leq 0 $. Imposing that the height of the layer is $d$, and substituting the expression of $ \kappa $, the optical thickness of the layer should be

\tau(d) = \frac{8 \pi ^ 3}{3} \frac{(n ^ 2 - 1) ^ 2}{\lambda ^ 4 N} d

The strange thing is that for a higher $ N $, namely for a higher concentration of particles and so a "denser" medium, the optical thickness decreases, and this appears counter-intuitive to me. Is there an error in my development?

Moreover, Chandrasekhar seems to consider only lossless medium, namely with purely real refractive index, to define the mass scattering coefficient, isn't it? Is there a way to consider also lossy medium?

Finally, from $ \kappa $, Chandrasekhar define the following "scattering coefficient per particle"

\sigma = \frac{8 \pi ^ 3}{3} \frac{(n ^ 2 - 1) ^ 2}{\lambda ^ 4 N ^ 2}

This has the dimension of a cross-section, but it can be seen that it depends on $ N $. Therefore, it is not the cross-section that characterizes a single particle (usually treated in the Mie theory), isn't it? Thanks