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\def\micron {$\mu$m} \def\waven {$\mu$m$^{-1}$} \def\sd {size distribution} \def\lmax {$\lambda _{max}$} \def\ie {i.e.,} \def\si {$\sim$} \def\mser {modified Serkowski} \def\sser {super-Serkowski}
To extract the \sd\ of polarizing dust grains we have used the Maximum Entropy Method (MEM). In this first investigation we adopted infinite cylinders with perfect spinning alignment. Only bare silicate particles were considered. The modified Serkowski law represents interstellar polarization quite well for the wavelength range 0.3 \si\ 2 \micron\ using one parameter, \lmax, the wavelength at which the polarization is maximum. For large \lmax\ it extrapolates reasonably into the ultraviolet and so we have investigated how the \sd\ changes with \lmax\ by fitting the \mser\ curve evaluated for \lmax\ = 0.55, 0.60, and 0.65 \micron. For HD 25443 which shows \sser\ behavior and for HD 197770 which might exhibit a 2200 \AA\ polarization bump, we combined the \mser\ curve in the optical with the actual far-ultraviolet data.
The \sd s found bear little resemblance to a power law. Instead they peak at \si\ 0.1 \micron\ and are skewed, with the relative rate of decrease to larger and smaller sizes depending on \lmax. For the particles larger than 0.1 \micron, the \sd\ does not change much with \lmax; on the other hand there is a remarkable change for the smaller ones -- the drop-off gets faster as \lmax\ increases. Compared to the \sd\ based on extinction, there is a similarity for large particles, but not for small ones: to fit the polarization curve it is not necessary to have as many small particles. This means that the (excess) smaller particles which contribute to the extinction are either not as well aligned or rounder. Mathis (1986) developed an alignment theory based on superparamagnetic inclusions in coagulated grains; the probability of a grain having such an inclusion (and therefore being aligned) decreases for smaller grains. While this is qualitatively correct, our MEM results indicate even fewer small particles than predicted. It might be that smaller particles tend to be rounder, becoming more aspherical as they grow (by coagulation).
