Search for one- and two-phonon octupole vibrational states in the spherical nuclei near $^{132}$Sn

Excited high spin states in $^{135}$I, $^{136}$Xe, $^{137}$Cs, $^{138}$Ba, $^{139}$La, $^{140}$Ce and $^{142}$Nd with N$=$82 are reorganized and interpreted in a different way to find one- phonon octupole vibrational (POV) bands. Two nearly identical (similar) bands with $\Delta $I$=$3 are found in these nuclei. From the presence of two nearly identical excited bands with $\Delta $I$=$3 in these nuclei, one-POV bands are proposed. Also, high spin states of $^{134}$Sb, $^{134,135}$Te, $^{135,136}$I, $^{137}$Xe and $^{139}$Ba near $^{132}$Sn are reanalyzed in order to search for one- and two-POV states. New spins and parities are tentatively assigned to the 2203.9 keV state in $^{137}$Xe and the 1976.6 and 2091.7 keV states in $^{139}$Ba from the state energy plots of the $N =$ 82 and 83 nuclei. High spin states of $^{134}$Sb, $^{134,135}$Te, $^{135,136}$I, $^{137}$Xe and $^{139}$Ba connected by E1, E3$/$M2 and E3 transitions are proposed, for the first time, as zero-, one- and two-POV states. One- and two-POV states in $^{134}$Sb and $^{135}$Te are built on a 7$^{-}$ ($\pi $g$_{7/2}\nu $f$_{7/2})$ state and a 19$/$2$^{-}$ ($\nu $f$_{7/2}\otimes $6$^{+}_{1})$ state, respectively. One-POV states built on the 19$/$2$^{-}(\nu $f$_{7/2}\otimes $6$^{+}_{1}$ ) and the 21$/$2$^{-}$ ($\nu $h$_{9/2} \quad \otimes $6$^{+}_{2}$ ) states coexist in $^{137}$Xe. Then, one- and two-POV states in $^{139}$Ba are built only on the 21$/$2$^{-}$ ($\nu $h$_{9/2}\otimes $6$^{+}_{2})$ state. One- and two-POV states in $^{134}$Te are built on the 6$^{+}_{2}$ state with some mixing with the 6$^{+}_{1}$ state.