Diffraction theory of nanotwin superlattices with low symmetry phase: Application to rhombohedral nanotwins and monoclinic $M_A$ and $M_B$ phases

The Brillouin zone-dependent conditions for coherent and adaptive diffractions are formulated. Adaptive diffraction phenomenon of nanotwins is analyzed. Extraordinary Bragg reflection peaks appear and adaptively shift along the conventional twin peak splitting vectors, whose positions are determined by lever rule according to twin variant volume fractions. Analysis of rhombohedral nanotwins shows that the nanotwin superlattices of rhombohedral phase with {001} and {110} twin planes diffract incident waves just like monoclinic $M_A$ and $M_B$ phases, respectively, whose lattice parameters are intrinsically related to that of rhombohedral phase. Crystallographic analysis of rhombohedral nanotwins by nanodomain averaging gives the same monoclinic $M_A$ and $M_B$ phases.

Figure 1

(Color online) Schematic illustrations of diffraction phenomena of coarse-domained and nanodomained materials. (a) Conventional diffraction of twin-related coarse domains, where fundamental Bragg peaks of constituent crystal split by vector $\Delta {\mathbf{K}}^{\left(1\right)}$ due to twinning shear and broaden due to finite size. Scattered waves from individual twin domains form reflection peaks (red and green filled with yellow) independently without interference. (b) Adaptive diffraction of nanotwins, where scattered waves from multiple nanodomains coherently superimpose to form new Bragg reflection peaks (blue filled with yellow), whose positions shift along the twin peak splitting vector $\Delta {\mathbf{K}}^{\left(1\right)}$ according to the twin variant volume fraction $\omega$ by following lever rule, while the conventional Bragg peaks (dashed red and green) disappear. The overlapping of broadened twin peaks is necessary to produce significant interference effects. Additional weak satellite peaks may arise from nanotwin superlattices. (c) When nanodomains coexist with coarse domains, the resultant peak profile (black filled with yellow) is an incoherent summation of the new adaptive peaks (blue) and the conventional peaks (red and green).

Figure 2

(Color online) Crystallographic illustrations of (001) rhombohedral twin and monoclinic $M_A$ lattice. (a) Twin-related rhombohedral variants 1 and 4 with (001) twin plane, where the polarization vector in each variant is along [111] and $\left[\overline{1}\overline{1}1\right]$, respectively, to form a head-to-tail pattern. The volume fractions of variants 1 and 4 are $\omega$ and $1\text{−}\omega$, respectively. The averaged polarization is confined to and rotates in $\left(1\overline{1}0\right)$ symmetry plane. The averaged lattice has a monoclinic $M_A$ symmetry. (b) The monoclinic $M_A$ unit cell resulting from averaging (001) twin-related rhombohedral variants shown in (a). Note that the unique axis $b_m$ is the common axis $\left[1\overline{1}0\right]$ of rigid-body rotations of the twin variants 1 and 4 to close the gap between them.

Figure 3

(Color online) Crystallographic illustrations of (110) rhombohedral twin and monoclinic $M_B$ lattice. (a) Twin-related rhombohedral variants 1 and 4 with (110) twin plane, where the polarization vector in each variant is along [111] and $\left[11\overline{1}\right]$, respectively, to form a head-to-tail pattern. The volume fractions of variants 1 and 4 are $\omega$ and $1\text{−}\omega$, respectively. The averaged polarization is confined to and rotates in $\left(1\overline{1}0\right)$ symmetry plane. The averaged lattice has a monoclinic $M_B$ symmetry. (b) The monoclinic $M_B$ unit cell resulting from averaging (110) twin-related rhombohedral variants shown in (a). Note that the unique axis $b_m$ is the common axis $\left[1\overline{1}0\right]$ of rigid-body rotations of the twin variants 1 and 4 to close the gap between them.

Figure 4

(Color online) Single-crystal x-ray diffraction measurement of monoclinic phase lattice parameters of PMN-15%PT in field cooling (reproduced with permission—Ref. 8). (a) $M_A$ phase observed in field cooling with $0.5\mathrm{kV}∕\mathrm{cm}$ along [001] axis. (b) $M_B$ phase observed in field cooling with $0.5\mathrm{kV}∕\mathrm{cm}$ along [110] axis.

Figure 5

(Color online) Three intrinsic lattice parameter relationships between the intermediate monoclinic phases and conventional rhombohedral phase of PMN-15%PT in field cooling. The first and second relationships are shown for (a) $M_A$ and (b) $M_B$ phases, where data of solid circles are taken from experiments (Ref. 8) (Fig. 4), and data of open circles are calculated from Eq. (50a). The third relationship is shown in (c), where data are calculated from Eq. (50c). The data of solid diamonds are for rhombohedral phase obtained from experiment (Ref. 12).