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The behaviour of partially ionized hot compressed matter is critical to thestudy of planetary interiors as well as for nuclear fusion studies. A recentquantum study of carbon in the 10-70 Gbar range and at a temperature of 100 eVused $N$-atom density functional theory (DFT) with $N\sim 32$-64 and moleculardynamics (MD). This involves band-structure type electronic calculations andaveraging over many MD generated ion configurations. The calculated averagenumber of free electrons per ion, viz., $\bar{Z}$, was systematically higherthan from a standard average atom (AA) quantum calculation. To clarify thisoffset, we examine (a) the effect of the self-interaction (SI) error in suchestimates (b) the possibility of carbon being a granular plasma containingCoulomb crystals. The possibility of `magic-number' bound states is considered.The electrical conductivity, pressure, and the compressibility of the carbonsystem are examined. The very low conductivity and the high $\bar{Z}$ resultsof DFT-MD point to the existence of carbon in a complex non-uniformlow-conducting dispersed phase, possibly containing magic-number Coulombcrystals. The NPA estimate of $\bar{Z}$, conductivity, compressibility, andpressure reported here pertain to the {\it uniform liquid}.

The ionization of carbon at 10-100 times the diamond density and in the 10$^6$ K temperature range