The perovskite crystal structure is known for the variety of transformations it can support.

The prototype structure is cubic $(Ex.SrTi{O}_3$ at room temperature$).$ It has space group $Pm3m$

$($#$221),$ with $a=3\mathrm{.}905.$ Sr is at $(0,0,0)$ and $Ti$ is at $(0\mathrm{.}5,0\mathrm{.}5,0\mathrm{.}5),$ and $O$ at $(0\mathrm{.}5,0\mathrm{.}5,0).$

a$.$ Plot the unit cell of this structure showing the bonds of the atoms involved.

Replot it showing the connectivity $($and shape$)$ of the Ti polyhedra.

b$.$ What are the coordinations of each ion? For the Ti itself, give each of the Ti$-$O

bond lengths.

c$.BaTi{O}_3$ is a ferroelectrically$-$distorted perovskite at The structural

distortion is quite small. At room temperature, it is tetragonal with space group

$P4mm($#$99),$ with $a=3\mathrm{.}99,c=4\mathrm{.}03.$ Ba is at $(0,0,0)$ and $Ti$ is at $,$

$0\mathrm{.}514,$ and $O$ at $(0\mathrm{.}5,0\mathrm{.}5,-0\mathrm{.}025),$ and $(0,0\mathrm{.}5,0\mathrm{.}485).$ Compare this structure to

that of cubic $SrTi{O}_3($Hint: it helps to draw slightly more than one unit cell$).$ In

your comparison be sure to include the Ti$-$O distances in tetragonal $BaTi{O}_3.$

d$.$ Is this a displacive or a reconstructive phase transformation? What does this

imply about the kinetics of this phase transformation?

e$.$ Why should the cubic phase be favored at high temperatures?

f$.$ A ferroelectric, by definition, is a material with a spontaneous dipole which can

be reoriented between crystallographically defined states by an applied electric

field. In your $BaTi{O}_3$ unit cell, estimate the positions of the center of positive and

negative charge in the unit cell. Also note that there were $6$ possible directions in

which the Ti could have displaced; as a result these are the six possible

polarization directions in tetragonally distorted $BaTi{O}_3.$ Note that two additional

structural distortions $($to orthorhombically and rhombohedrally distorted versions$)$

also occur at lower temperatures.

g$.GdSc{O}_3$ is an orthorhombically distorted perovskite which is not ferroelectric.

The space group is orthorhombic $($Pnma$,$ space group $62)$ with $a=5\mathrm{.}742,b=$

$7\mathrm{.}926,$ and $c=5\mathrm{.}482.$ Gd sits at $(0\mathrm{.}44058,0\mathrm{.}75,0\mathrm{.}48392),$ Sc sits at $(0,0,0\mathrm{.}5),$

and $O$ at $(0\mathrm{.}4494,0\mathrm{.}25,0\mathrm{.}1201)$ and $(0\mathrm{.}1956,0\mathrm{.}5623,0\mathrm{.}1927).$ Plot a unit cell of

this, and compare the structure to that of cubic $SrTi{O}_3.$ It may be easiest to see

this if you show the octahedra around the Sc$.$