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We fit the $\sim$0.1-500 MeV/nucleon H-Fe spectra in 46 large SEP eventssurveyed by Desai et al. (2016) with the double power-law Band function toobtain 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-lawspectral slope $\gamma_1$. We find that: 1) $\gamma_a$, $\gamma_1$, and$\gamma_b$ are species-independent within a given SEP event, and the spectrasteepen with increasing energy; 2) $E_B$'s are well ordered by Q/M ratio, anddecrease 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/Oat $\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 andGLEs) are Fe-rich, have $\alpha >$1.4, have flatter spectra at low and highenergies 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 anevent, as well as the range of values for $\alpha$ suggest that the formationof double power-laws in SEP events occurs primarily due to diffusiveacceleration at near-Sun CME shocks and not due to scattering in theinterplanetary turbulence. In most events, the Q/M-dependence of $E_B$ isconsistent with the equal diffusion coefficient condition while theevent-to-event variations in $\alpha$ are probably driven by differences in thenear-shock wave intensity spectra, which are flatter than the Kolmogorovturbulence spectrum but still weaker compared to that inferred for the extremeevents.

Spectral properties of large gradual solar energetic particle events - II -Systematic Q/M-dependence of heavy ion spectral breaks