  We have grown $^{232}$Th:CaF$_2$ and $^{229}$Th:CaF$_2$ single crystals for
investigations on the VUV laser-accessible first nuclear excited state of
$^{229}$Th. To reach high doping concentrations despite the extreme scarcity
(and radioactivity) of $^{229}$Th, we have scaled down the crystal volume by a
factor 100 compared to established commercial or scientific growth processes.
We use the vertical gradient freeze method on 3.2 mm diameter seed single
crystals with a 2 mm drilled pocket, filled with a co-precipitated
CaF$_2$:ThF$_4$:PbF$_2$ powder in order to grow single crystals. Concentrations
of $4\cdot10^{19}$ cm$^{-3}$ have been realized with $^{232}$Th with good
($>$10%) VUV transmission. However, the intrinsic radioactivity of $^{229}$Th
drives radio-induced dissociation during growth and radiation damage after
solidification. Both lead to a degradation of VUV transmission, limiting the
$^{229}$Th concentration to $<5\cdot10^{17}$ cm$^{-3}$.
