  Electronic compressibility, the second derivative of ground state energy with
respect to total electron number, is a measurable quantity that reveals the
interaction strength of a system and can be used to characterize the orderly
crystalline lattice of electrons known as the Wigner crystal. Here, we measure
the electronic compressibility of individual suspended ultraclean carbon
nanotubes in the low-density Wigner crystal regime. Using low-temperature
quantum transport measurements, we determine the compressibility as a function
of carrier number in nanotubes with varying band gaps. We observe two
qualitatively different trends in compressibility versus carrier number, both
of which can be explained using a theoretical model of a Wigner crystal that
accounts for both the band gap and the confining potential experienced by
charge carriers. We extract the interaction strength as a function of carrier
number for individual nanotubes and show that the compressibility can be used
to distinguish between strongly and weakly interacting regimes.
