  The structure of a thermonuclear detonation wave can be solved accurately
and, thus, may serve as a test bed for studying different approximations that
are included in multidimensional hydrodynamical simulations of supernova. We
present the structure of thermonuclear detonations for the equal mass fraction
of $^{12}$C and $^{16}$O (CO) and for pure $^{4}$He (He) over a wide range of
upstream plasma conditions. The lists of isotopes we constructed allow us to
determine the detonation speeds, as well as the final states for these
detonations, with an uncertainty of the percent level (obtained here for the
first time). We provide our results with a numerical accuracy of $\sim0.1\%$,
which provides an efficient benchmark for future studies. We further show that
CO detonations are pathological for all upstream density values, which differs
from previous studies, which concluded that for low upstream densities CO
detonations are of the Chapman-Jouget (CJ) type. We provide an approximate
condition, independent of reaction rates, that allows to estimate whether
arbitrary upstream values will support a detonation wave of the CJ type. Using
this argument, we are able to show that CO detonations are pathological and to
verify that He detonations are of the CJ type, as was previously claimed for
He. Our analysis of the reactions that control the approach to nuclear
statistical equilibrium, which determines the length-scale of this stage,
reveals that at high densities, the reactions
$^{11}$B$+p\leftrightarrow3^{4}$He plays a significant role, which was
previously unknown.
