= Ellipsoid rotational diffusion microscopy experiment
In 2006, <Yilong Han> and colleagues dispersed plastic <ellipsoids> in <water> inside a thin glass cell, using a very dilute suspension to limit particle interactions. A camera on an <optical microscope> recorded 30 frames per second. Image analysis extracted centre positions and long-axis <angles>, with reported angular resolution about $1^\circ$. Their shape supplied the observable axis. Reflected interference colours calibrated cell thickness; nearly constant particle interference colour checked vertical wandering. One tracked particle had dimensions $2.4\times0.3\times0.3\,\mu\mathrm m$ in an $846\,\mathrm{nm}$ gap. Angular displacement <variance> grew linearly with lag; displacement resolved along the particle's axes and in the laboratory showed how turning washes out translational <anisotropy>. These are observations of confined particles: walls alter drag, and slight out-of-plane tumbling affected positional resolution. They do not establish the bulk <sphere> formula. The paper's Figure 1A–B shows simulations; Figure 1F shows a measured trajectory.<Brownian Motion of an Ellipsoid by Yilong Han and others>[Han et al., Brownian Motion of an Ellipsoid, Science 314 (2006), pp. 626–628; Figures 1D–F and 2A–B: apparatus, angular readout, dilution and height checks, angular variance and translational crossover]{ref}
For interpreting such images, the <director (liquid crystals)> distinction is essential: identical ends give an <angle> modulo $\pi$. A continuously unwrapped trace can follow small successive turns, but it does not make the ends physically distinguishable. The invariant planar axis <autocorrelation> is $\langle\cos(2\Delta\phi)\rangle$, and its decay determines $D_r$ through $e^{-4D_rt}$. Separately measuring position and <rigid body orientation> is what makes the connection between translational <anisotropy> and loss of angular memory experimentally accessible.
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