We have characterized soot particles measured in situ in a laminar co-flow ethylene-air diffusion flame using small-angle X-ray scattering (SAXS). The analysis includes temperature measurements made with coherent anti-Stokes Raman spectroscopy (CARS) and complements soot volume-fraction and maturity measurements made with laser-induced incandescence (LII). We compared the results of fits to the SAXS measurements using a unified model and a fractal core-shell model. Power-law parameters yielded by the unified model indicate that aggregates of primary particles are in the mass-fractal regime, whereas the primary particles are in the surface-fractal regime in the middle of the flame. Higher and lower in the flame, the primary-particle power-law parameter approaches 4, suggesting smooth primary particles. These trends are consistent with fits using the fractal core-shell model, which indicate that particles have an established core-shell structure in the middle of the flame and are internally homogeneous at higher and lower heights in the flame. Primary-particle size distributions derived using the fractal core-shell model demonstrate excellent agreement with distributions inferred from transmission electron microscopy (TEM) images in the middle of the flame. Higher in the flame, a second small mode appears in the size distributions, suggesting particle fragmentation during oxidation. Surface oxidation would explain (1) aggregate fragmentation and (2) loss of core-shell structure leading to smoother primary-particle surfaces by removal of carbon overlayers. SAXS measurements are much more sensitive to incipient and young soot particles than LII and demonstrate significant volume fraction from particles low in the flame where the LII signal is negligible.