  Relativistic study of xenotime, YPO$_4$, containing atoms thorium and uranium
as point defects is performed in the framework of cluster model with using the
compound-tunable embedding potential (CTEP) method proposed by us recently. The
Y-(PO$_4$)$_6$-Y'$_{22}$-O'$_{104}$ cluster for xenotime is considered, in
which central part, [Y-(PO$_4$)$_6$]$^{-15}$, is the main cluster, whereas
outermost 22 atoms of yttrium and 104 atoms of oxygen are treated as its
environment and compose electron-free CTEP with the total charge of $+15$. The
P and O atoms of the orthophosphate groups nearest to the central Y atom are
treated at all-electron level. The central Y, its substitutes, Th and U,
together with environmental Y atoms are described within different versions of
the generalized relativistic pseudopotential method. Correctness of our cluster
and CTEP models, constructed in the paper, is justified by comparing the Y-O
and P-O bond lengths with corresponding periodic structure values of the \ypo4
crystal, both experimental and theoretical. Using this cluster model, chemical
properties of solitary point defects, X = U, Th, in xenotime are analyzed. It
has been shown that the oxidation state ${+3}$ is energetically more profitable
than ${+4}$ not only for thorium but for uranium as well ($\Delta E \approx 5$
eV) despite the notably higher ionic radius of U$^{+3}$ compared to Y$^{+3}$,
whereas ionic radii of U$^{+4}$ and Y$^{+3}$ are close. This leads to notable
local deformation of crystal geometry around the U$^{+3}$ impurity in xenotime
and contradicts to widespread opinion about favorite oxidation state of uranium
in such kind of minerals.
