Magnetic Spectra on Charge Ordering in $R_{1/3}$Sr$_{2/3}$FeO$_{3-\sigma }$ ($R$ = La, Pr, and Nd)

$R_{1/3}$Sr$_{2/3}$FeO$_{3-\sigma }$ ($R$ = La, Pr, and Nd) compounds are reported to have the same charge ordering (CO) and Neel temperatures. Inelastic neutron scattering is applied to study the magnetic energy effect on the CO state in this system. Based on the ratio of the ferromagnetic exchange energy ($J_{F})$ and antiferromagnetic exchange energy ($\vert J_{AF}\vert )$, the magnetic exchange energy can stabilize the CO state in La$_{1/3}$Sr$_{2/3}$FeO$_{3-\sigma }$ and Pr$_{1/3}$Sr$_{2/3}$FeO$_{3-\sigma }$ alone; with the smaller $R^{3+}$ substitution, $\vert J_{AF}\vert $ increases a lot from La$_{1/3}$Sr$_{2/3}$FeO$_{3-\sigma }$ to Nd$_{1/3}$Sr$_{2/3}$FeO$_{3-\sigma }$, but the CO of Fe ions could still be driven by magnetic energy itself with the correction on magnetic energy ratio.