  Herein we report on the thermal expansions and temperature-dependent crystal
structures of select ternary carbide MAX phases in the Ti-Al-C phase diagram in
the 100-1000 deg C temperature range. A bulk sample containing 38 wt.%
Ti5Al2C3, "523"), $32 wt.% Ti2AlC ("211"), 18 wt.% Ti3AlC2 ("312"), and 12 wt.%
(Ti{0.5}Al{0.5})Al is studied by Rietveld analysis of high-temperature neutron
diffraction data. We also report on the same for a single-phase sample of
Ti$_3$AlC$_2$ for comparison. The thermal expansions of all the MAX phases
studied are higher in the $c$ direction than in the $a$ direction. The bulk
expansion coefficients - 9.3 x 10^-6 for Ti5Al2C3, 9.2 x 10^-6 for Ti2AlC, and
9.0 x 10^-6 for Ti3AlC2 - are comparable within one standard deviation of each
other. In Ti5Al2C3, the dimensions of the Ti-C octahedra for the 211-like and
312-like regions are comparable to the Ti-C octahedra in Ti2AlC and Ti3AlC2,
respectively. The isotropic mean-squared atomic displacement parameters are
highest for the Al atoms in all three phases, and the values predicted from
first-principles phonon calculations agree well with those measured.
