Diffraction theory of nanotwin superlattices with low symmetry phase

A nanotwin diffraction theory is developed. It predicts an adaptive diffraction phenomenon, where the Bragg reflection peaks are determined by coherent superposition of scattered waves from individual twin-related nanocrystals and adaptively shift along the twin peak splitting vectors in response to a change in the twin variant volume fraction. Application of this theory to tetragonal phase explains the intrinsic lattice parameter relationships of monoclinic $M_C$ phase recently discovered in ferroelectric $\mathrm{Pb}\left[{\left({\mathrm{Mg}}_{1∕3}{\mathrm{Nb}}_{2∕3}\right)}_{1-x}{\mathrm{Ti}}_x\right]{\mathrm{O}}_3$ and $\mathrm{Pb}\left[{\left({\mathrm{Zn}}_{1∕3}{\mathrm{Nb}}_{2∕3}\right)}_{1-x}{\mathrm{Ti}}_x\right]{\mathrm{O}}_3$.

Figure 1

(Color online) (a) Schematic of tetragonal nanotwin superlattice. (101) twin planes are indicated by dashed lines. A tetragonal unit cell is highlighted by gray shadow in respective twin variants. The primitive superlattice translation vector is $\mathbf{L}$. The volume fractions of the first and second twin variants are $\omega$ and $1-\omega$, respectively. The bilayer basis thickness is $T$. (b) Adaptive diffraction phenomenon, where the Bragg reflection peak ${\mathbf{k}}^{\left(s\right)}$ moves between the vanished fundamental peaks ${\mathbf{K}}^{\left(1\right)}$ and ${\mathbf{K}}^{\left(2\right)}$ of twin-related nanodomains in response to a change in $\omega$.

Figure 2

Triplet peak structure around the (200) Bragg reflection spot consisting of two (200) superlattice peaks from nanotwins of $\left(10\overline{1}\right)$ and (101) twin planes, respectively, and one (020) superlattice peak from nanotwins of $(01\pm 1)$ twin planes. The vanished fundamental peaks are not shown. Nanotwins of $(01\pm 1)$ twin planes (not shown) produce the same (020) peak (highlighted by dotted line) as that from [001]-oriented tetragonal domain and give the lattice parameter $b=a_t$.