New Band Structures in Neutron-Rich $^{108}$Mo, and $^{108,110,112}$Ru

New insights into the structures of $^{108}$Mo, and $^{108,110,112}$Ru are identified in the spontaneous fission of $^{252}$Cf. The 5.7 x 10$^{11}$ triples and higher fold prompt gamma coincidences opened up the possibility to see new weakly populated band structures in these nuclei. Two phonon $\gamma $-bands that decay only to the one phonon $\gamma $-band were discovered in $^{108}$Mo and $^{110,112}$Ru. We discovered in $^{108}$Mo$^{ }$and$^{ 108,110,112}$Ru $\Delta $I=1, doublet bands. Our theoretical calculations indicate $^{108}$Ru is $\gamma $-soft and $^{110,112}$Ru are more rigid triaxial nuclei. The non-yrast band in $^{108}$Ru shows an energy level staggering not seen in its yrast partner band nor in $^{110,112}$Ru. This staggering is proposed to be related to its $\gamma $-soft shape perturbing its chiral structure. The doublet bands in $^{108}$Mo$^{ }$and $^{110,112}$Ru will be shown to have all the properties expected for chiral vibrational bands. Tilted axis cranking calculations for $^{110,112}$Ru stronly support the chiral vibrational interpretation of these bands but do not support their being accidentally degenerate energy doublets built on different configurations.