Isotope dependencies of Raman spectra of ${\mathrm{B}}_{12}{\mathrm{As}}_2,$ ${\mathrm{B}}_{12}{\mathrm{P}}_2,$ ${\mathrm{B}}_{12}{\mathrm{O}}_2,$ and ${\mathrm{B}}_{12+x}{\mathrm{C}}_{3-x}:$ Bonding of intericosahedral chains

The structures of ${\mathrm{B}}_{12}{\mathrm{As}}_2,$ ${\mathrm{B}}_{12}{\mathrm{P}}_2,$ ${\mathrm{B}}_{12}{\mathrm{O}}_2,$ and ${\mathrm{B}}_{12+x}{\mathrm{C}}_{3-x}$ $\mathrm{}(0.15<x<\mathrm{}1.65\mathrm{})\mathrm{}$ may each be viewed as composed of ${\mathrm{B}}_{12}$ or ${\mathrm{B}}_{11}\mathrm{C}$ icosahedra and As-As, P-P, O-O, or $\left({\mathrm{C}}_{1-y}{\mathrm{B}}_y\right)-\mathrm{B}-\left({\mathrm{C}}_{1-z}{\mathrm{B}}_z\right)$ intericosahedral chains. The relative stiffnesses of the chain atoms’ intrachain and extrachain bonding are probed with Raman spectroscopy by measuring the boron-isotope dependence of the chains’ symmetric stretching frequencies. The dependencies of chains’ symmetric stretching frequencies on boron mass measures the degree to which icosahedral boron atoms are carried along in the chain atoms’ motions. The absence of a measurable boron involvement in these modes for ${\mathrm{B}}_{12}{\mathrm{As}}_2$ and ${\mathrm{B}}_{12}{\mathrm{P}}_2$ indicates that the bonding between the two atoms of their chains is much stiffer than these atoms’ bonding to icosahedral boron atoms. By contrast, substantial boron involvement in the symmetric-stretch mode of ${\mathrm{B}}_{12}{\mathrm{O}}_2$ indicates much softer bonding of the chain oxygen atoms to one another than to icosahedral boron atoms. The strong boron participation in the boron carbide Raman mode assigned to the symmetric stretch of CBC chains suggests relatively soft bonding between the chain’s atoms. The declining intensity of this CBC-chain-stretching mode as $x$ increases from 0.15 toward 1.0 in ${\mathrm{B}}_{12+x}{\mathrm{C}}_{3-x}$ is consistent with the carbon atoms being removed primarily from chains rather than from icosahedra. Other features of our Raman spectra provide complementary evidence that carbon atoms are preferentially retained in boron carbides’ icosahedra rather than in its chains. In particular, strong carbon participation in Raman modes associated with vibrations of icosahedral atoms is observed at all carbon concentrations, $0.15<x<\mathrm{}\mathrm{1.65.}\mathrm{}$