In the scattering of light by a gas very near the critical point the scattered light is observed to be ``whiter'' (i.e., its spectrum is less predominantly peaked toward the blue) than far from the critical point. Show that this can be understood by the fact that the volumes of the density fluctuations become large enough that Rayleigh's law fails to hold. In particular, consider the lowest order approximation to the scattering by a uniform dielectric sphere of radius a whose dielectric constant ϵr differs only slightly from unity.
(a) Show that for ka≫1, the differential cross section is sharply peaked in the forward direction and the total scattering cross section is approximately
σ≈2π(ka)2∣ϵr−1∣2a2
with a k2, rather than k4, dependence on frequency.
(b) Show that for arbitrary ka the total cross section to lowest order in (ϵr−1) is the expression given in part a, multiplied by the function
F(z)=1+5z−2−27z−4(1−cos2z)−z−3sin2z−4(z−2−z−4)∫02zt1−costdt
where z=2ka. [This result is due to Lord Rayleigh, 1914.]