  We here calculate the series expansion of the T-matrix for a spheroidal
particle in the small-size/long-wavelength limit, up to third lowest order with
respect to the size parameter, X, which we will define rigorously for a
non-spherical particle. T is calculated from the standard extended boundary
condition method with a linear system involving two infinite matrices P and Q,
whose matrix elements are integrals on the particle surface. We show that the
limiting form of the P- and Q-matrices, which is different in the special case
of spheroid, ensures that this Taylor expansion can be obtained by considering
only multipoles of order 3 or less (i.e. dipoles, quadrupoles, and octupoles).
This allows us to obtain self-contained expressions for the Taylor expansion of
T(X). The lowest order is O(X^3) and equivalent to the quasi-static limit or
Rayleigh approximation. Expressions to order O(X^5) are obtained by Taylor
expansion of the integrals in P and Q followed by matrix inversion. We then
apply a radiative correction scheme, which makes the resulting expressions
valid up to order O(X^6). Orientation-averaged extinction, scattering, and
absorption cross-sections are then derived. All results are compared to the
exact T-matrix predictions to confirm the validity of our expressions and
assess their range of applicability. For a wavelength of 400nm, the new
approximation remains valid (within 1% error) up to particle dimensions of the
order of 100-200nm depending on the exact parameters (aspect ratio and
material). These results provide a relatively simple and
computationally-friendly alternative to the standard T-matrix method for
spheroidal particles smaller than the wavelength, in a size range much larger
than for the commonly-used Rayleigh approximation.
