  We present a new method to accurately describe the ionization fraction and
the size distribution of polycyclic aromatic hydrocarbons (PAHs) within
astrophysical sources. To this purpose, we have computed the mid-infrared
emission spectra of 308 PAH molecules of varying sizes, symmetries, and
compactness, generated in a range of radiation fields. We show that the
intensity ratio of the solo CH out-of-plane bending mode in PAH cations and
anions (referred to as the 11.0 $\mu$m band, falling in the 11.0-11.3 $\mu$m
region for cations and anions) to their 3.3 $\mu$m emission, scales with PAH
size, similarly to the scaling of the 11.2/3.3 ratio with the number of carbon
atoms (N$_{\mathrm{C}}$) for neutral molecules. Among the different PAH
emission bands, it is the 3.3 $\mu$m band intensity which has the strongest
correlation with N$_{\mathrm{C}}$, and drives the reported PAH intensity ratio
correlations with N$_{\mathrm{C}}$ for both neutral and ionized PAHs. The
6.2/7.7 intensity ratio, previously adopted to track PAH size, shows no evident
scaling with N$_{\mathrm{C}}$ in our large sample. We define a new diagnostic
grid space to probe PAH charge and size, using the (11.2+11.0)/7.7 and
(11.2+11.0)/3.3 PAH intensity ratios respectively. We demonstrate the
application of the (11.2+11.0)/7.7 - (11.2+11.0)/3.3 diagnostic grid for
galaxies M82 and NGC 253, for the planetary nebula NGC 7027, and the reflection
nebulae NGC 2023 and NGC 7023. Finally, we provide quantitative relations for
PAH size determination depending on the ionization fraction of the PAHs and the
radiation field they are exposed to.

Astrophysics of Galaxies