We fit the $\sim$0.1-500 MeV/nucleon H-Fe spectra in 46 large SEP events
surveyed by Desai et al. (2016) with the double power-law Band function to
obtain a normalization constant, low- and high-energy parameters $\gamma_a$ and
$\gamma_b$; and break energy $E_B$. We also calculate the low-energy power-law
spectral slope $\gamma_1$. We find that: 1) $\gamma_a$, $\gamma_1$, and
$\gamma_b$ are species-independent within a given SEP event, and the spectra
steepen with increasing energy; 2) $E_B$'s are well ordered by Q/M ratio, and
decrease systematically with decreasing Q/M, scaling as (Q/M)$^\alpha$ with
$\alpha$ varying between $\sim$0.2-3; 3) $\alpha$ is well correlated with Fe/O
at $\sim$0.16-0.23 MeV/nucleon and CME speed; 4) In most events: $\alpha<$1.4,
the spectra steepen significantly at higher energy with $\gamma_b$-$\gamma_a
>$3; and 5) Seven out of 9 extreme SEP events (associated with faster CMEs and
GLEs) are Fe-rich, have $\alpha >$1.4, have flatter spectra at low and high
energies with $\gamma_b$-$\gamma_a <$3. The species-independence of $\gamma_a$,
$\gamma_1$, and $\gamma_b$ and the systematic Q/M dependence of $E_B$ within an
event, as well as the range of values for $\alpha$ suggest that the formation
of double power-laws in SEP events occurs primarily due to diffusive
acceleration at near-Sun CME shocks and not due to scattering in the
interplanetary turbulence. In most events, the Q/M-dependence of $E_B$ is
consistent with the equal diffusion coefficient condition while the
event-to-event variations in $\alpha$ are probably driven by differences in the
near-shock wave intensity spectra, which are flatter than the Kolmogorov
turbulence spectrum but still weaker compared to that inferred for the extreme
events.
