  We provide a theoretical analysis for the intermediate scattering function
typically measured in depolarized dynamic light scattering experiments. We
calculate the field autocorrelation function $g_1^{\rm VH}(Q,t)$ in dependence
on the wave vector $Q$ and the time $t$ explicitly in a vertical-horizontal
scattering geometry for differently shaped solids of revolution. The shape of
prolate cylinders, spherocylinders, spindles, and double cones with variable
aspect ratio is expanded in rotational invariants $f_{lm}(r)$. By Fourier
transform of these expansion coefficients, a formal multipole expansion of the
scattering function is obtained, which is used to calculate the weighting
coefficients appearing in the depolarized scattering function. In addition to
translational and rotational diffusion, especially the translational-rotational
coupling of shape-anisotropic objects is considered. From the short-time
behavior of the intermediate scattering function, the first cumulants
$\Gamma(Q)$ are calculated. In a depolarized scattering experiment, they
deviate from the simple proportionality to $Q^2$. The coefficients $f_{lm}(Q)$
strongly depend on the geometry and aspect ratio of the particles. The time
dependence, in addition, is governed by the translational and rotational
diffusion tensors, which are calculated by means of bead models for differently
shaped particles in dependence on their aspect ratio. Therefore, our analysis
shows how details of the particle shape---beyond their aspect ratio---can be
determined by a precise scattering experiment. This is of high relevance in
understanding smart materials which involve suspensions of anisotropic
colloidal particles.
