  Photophysics of single-wall carbon nanotubes (SWCNTs) is intensively studied
due to their potential application in light harvesting and optoelectronics.
Excited states of SWCNTs form strongly bound electron-hole pairs, excitons, of
which only singlet excitons participate in application relevant optical
transitions. Long-living spin-triplet states hinder applications but they
emerge as candidates for quantum information storage. Therefore knowledge of
the triplet exciton energy structure, in particular in a SWCNT chirality
dependent manner, is greatly desired. We report the observation of light
emission from triplet state recombination, i.e. phosphorescence, for several
SWCNT chiralities using a purpose-built spectrometer. This yields the
singlet-triplet gap as a function of SWCNT diameter and it follows predictions
based on quantum confinement effects. Saturation under high microwave power (up
to 10 W) irradiation allows to determine the spin-relaxation time for triplet
states. Our study sensitively discriminates whether the lowest optically active
state is populated from an excited state on the same nanotube or through
F\"orster exciton energy transfer from a neighboring nanotube.
