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While simple analytic expressions exist for the scattering and absorption properties of dust grains which are either very small or very large compared to the wavelength of the incident radiation (see Section 5), however, in many astrophysical applications we are concerned with grains which are neither very small nor very large compared to the wavelength. Moreover, cosmic dust would in general be expected to have non-spherical, irregular shapes.

Our ability to compute scattering and absorption cross sections for nonspherical particles is extremely limited. So far, exact solutions of scattering problems exist only for bare or layered spherical grains ("Mie theory"; [6]), infinite cylinders [52], and spheroids [3, 4, 70]. The "T-matrix" (transition matrix) method, originally developed by Barber & Yeh [5] and substantially extended by Mishchenko, Travis, & Mackowski [59], is able to treat axisymmetric (spheroidal or finite cylindrical) grains with sizes comparable to the wavelength. The discrete dipole approximation (DDA), originally developed by Purcell & Pennypacker [65] and greatly improved by Draine [10], is a powerful technique for irregular heterogeneous grains with sizes as large as several times the wavelength. The VIEF (volume integration of electric fields) method developed by Hage & Greenberg [25], based on an integral representation of Maxwell's equations, is physically similar to the DDA method. The microwave analog methods originally developed by Greenberg, Pedersen & Pedersen [21] provide an effective experimental approach to complex particles [23].

Although interstellar grains are obviously non-spherical as evidenced by the observed polarization of starlight, the assumption of spherical shapes (together with the Bruggeman or the Maxwell-Garnett effective medium theories for inhomogeneous grains; [6]) is usually sufficient in modeling the interstellar absorption, scattering and IR (continuum) emission. For IR polarization modeling, the dipole approximation for spheroidal grains is proven to be successful in many cases.

The DDA method is highly recommended for studies of inhomogeneous grains and irregular grains such as cometary, interplanetary, and protoplanetary dust particles which are expected to have a porous aggregate structure. Extensive investigations using the DDA method have been performed for the scattering, absorption, thermal IR emission, and radiation pressure properties of fluffy aggregated dust (e.g. see [2, 23, 29, 30, 31, 33, 34, 35, 36, 37, 40, 43, 56, 57, 60, 61, 71, 74, 76, 77]).

Acknowledgements I thank Drs. I. Mann and T. Mukai for their very helpful comments and support. Partial support by a Chandra Theory program and HST Theory Programs is gratefully acknowledged.

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