Effective $M3\mathrm{}$ operator and relevant transitions in ${}_{}{}^{24}{}_{}{}^{}\mathrm{Al}$, ${}_{}{}^{24}{}_{}{}^{}\mathrm{Na}$, ${}_{}{}^{34}{}_{}{}^{}\mathrm{Cl}$, ${}_{}{}^{38}{}_{}{}^{}\mathrm{K}$, and ${}_{}{}^{38}{}_{}{}^{}\mathrm{Cl}$

Theoretical calculations are presented for the magnetic octupole electromagnetic matrix elements in light nuclei. Shell-model wave functions are used to calculate the $M3\mathrm{}$ matrix elements for the cases ${}_{}{}^{24}{}_{}{}^{}\mathrm{Al}$ $1^{+}$ →$4^{+}$, ${}_{}{}^{24}{}_{}{}^{}\mathrm{Na}$ $1^{+}$ →$4^{+}$, ${}_{}{}^{34}{}_{}{}^{}\mathrm{Cl}$ $3^{+}$ →$0^{+}$, ${}_{}{}^{38}{}_{}{}^{}\mathrm{K}$ $0^{+}$ →$3^{+}$, and ${}_{}{}^{38}{}_{}{}^{}\mathrm{Cl}$ $5^{-}$ →$2^{-}$. The radial matrix elements are calculated with harmonic-oscillator and spherical Hartree-Fock potentials. The comparison of the calculated and experimental matrix elements is expressed in terms of effective spin $g$ factors. Theoretical calculations for the core-polarization corrections are presented and relations between the $E2\mathrm{}$ core-polarization charge and the $M3\mathrm{}$ effective spin $g$ factors are derived.

NUCLEAR STRUCTURE ${}_{}{}^{24}{}_{}{}^{}\mathrm{Na}$, ${}_{}{}^{24}{}_{}{}^{}\mathrm{Al}$, ${}_{}{}^{34}{}_{}{}^{}\mathrm{Cl}$, ${}_{}{}^{38}{}_{}{}^{}\mathrm{K}$, ${}_{}{}^{38}{}_{}{}^{}\mathrm{Cl}$; calculations of $M3\mathrm{}$ decay strengths; extraction of effective $M3\mathrm{}$ operator; full $d_{\frac{5}{2}}-s_{\frac{1}{2}}-d_{\frac{3}{2}}$ shell-model calculations with Chung-Wildenthal Hamiltonians.