  Carbon doping is used to obtain semi-insulating GaN crystals. If the carbon
doping concentration exceeds $5*10^{17}$ $cm^{-3}$, the carbon atoms
increasingly form triatomic clusters. The tri-carbon defect structure is
unambiguously proven by the isotope effect on the defects' local vibrational
modes (LVMs) originally found in samples containing carbon of natural isotopic
composition $(~99 $%$ ^{12}C, ~1$%$ ^{13}C)$ at $1679$ $cm^{-1}$ and $1718$
$cm^{-1}$. Number, spectral positions, and intensities of the LVMs for samples
enriched with the $^{13}C$ isotope (~99 % and ~50 %) are consistently
interpreted on the basis of the harmonic oscillator model taking into account
the probability of possible isotope combinations. Including the polarization
dependence of the LVM absorption, we show that the tri-carbon defects form a
triatomic molecule-like structure in two crystallographically different
configurations: a basal configuration with the carbon bonds near the basal
plane and an axial configuration with one of the carbon bonds along the c-axis.
Finally, the disappearance of the LVMs under additional below-bandgap
illumination is interpreted as defect recharging, i.e. the tri-carbon defects
possess at least one charge state transition level within the bandgap and
contribute to optical absorption as well as to the electrical charge balance.
