Nature of tetragonal distortion in Jahn-Teller effect

Question

I am wondering: If I have a regular octahedron as my starting point, oriented along the x-y-z axis, and now Jahn-Teller suggest I elongate or compress along the $z$-axis, what happens along the other axis? I expect that these move in the opposite direction, but by how much? Say my displacement in $z$-direction is $\delta$. Is the displacement in $x$- and $y$-direction then $-\delta$, or do I have to find $\delta_x = \delta_y$ so that the total Volume is conserved?

Answer 1

Very generally, Jahn-Teller distortions are indeed volume conserving, and there is an "elastic" cost associated with moving the atoms that make up the octahedron from their equilibrium positions, so if you stretch along the $z$-axis, say, then you have to compress along the $x$- and $y$-axes in such a way as to minimize an elastic energy that goes as $\delta x^2+\delta y^2$. Symmetrically compressing the axes is the best way to minimize that.