Raman spectra measured from a Pb(Zr0.35Ti0.65)O3 single crystal at 83 K with geometry which reveals only E(TO) modes. Left-hand side panels show data measured using 532 nm excitation wavelength, whereas the middle panels show the data measured using a HeNe-laser (wavelength 632.818 nm). The spectra are very similar, thus showing that the light scattering is truly Raman scattering, not only in the case of the strong peaks but also in the case of the broad background E(BG) shown by the dashed line. Panels (a) and (b) show the splitting of the soft E(1TO) mode, panels (d) and (e) show the splitting of the E(2TO) mode and the E(3TO) mode. Also the highest-frequency peak, labelled E(4TO) (panels (g) and (h)), is split. The peaks above 600 cm−1 correspond to the A 1(3TO) modes and should not be observed in this geometry. Right hand side panels plot the Raman shifts, (c) shows the soft-mode shifts, panel (f) the E(2TO) and E(3TO) modes, and (i) the highest-frequency E(TO) mode as a function of temperature. Except for the E(3TO) mode (no splitting) and E(2TO) mode (splitting vanishes at around room temperature) no strong temperature dependence of the peak splitting was observed.
The area of the broad background measured from a Pb(Zr0.35Ti0.65)O3 single crystal feature (labelled as E(BG)) seen in Fig. 1 and the area of the E(2TO) modes and the E(3TO) mode as a function of temperature.
(a) Raman spectra measured from a Pb(Zr0.35Ti0.65)O3 single crystal at 295 K with geometry ideally showing only the B 1 symmetry mode (weak features, assigned to the longitudinal A 1(LO) and transverse E(TO) modes, are seen). For comparison, the spectrum measured with geometry, revealing only E(TO) modes, shown by dashed line. In contrast to PbTiO3, the energy difference between the B 1 and E(3TO) modes is clear. (b) Raman shifts of the B 1 mode as a function of temperature; 95% confidence limits are also indicated, reflecting the quality of the fit.
Raman spectra measured with geometry from a Pb(Zr0.35Ti0.65)O3 single crystal at 83 and 373 K which ideally reveals only A 1(TO) modes. Right-hand side panels plot the Raman shifts as a function of temperature. Two excitation wavelengths, 532 and 632.818 nm, were utilized. Panels (a)-(c) show the fundamental peak of the A 1(1TO) mode (at around 145 cm−1 and the high-frequency mode at 160 cm−1, revealed by the asymmetry of the peak) and the A 1(2TO) modes (two peaks at around 350 cm−1, panels (a), (b), (f), and (g)). Panels (d), (e), (h), and (i) show the splitting of the A 1(3TO) mode (peaks below 650 cm−1). We note that the lowest-frequency peaks have nearly the same frequency as the strong E(1TO) peaks, and also the peaks corresponding to the E(2TO), E(3TO) and B 1 are observed. This is because it is difficult to perfectly align small crystals inside the cryostat and because the local structure deviates from the perfect tetragonal structure.
Distribution of cells for x = 0.20 and the correspondence between the colours and the Pb-displacements. On the local scale, numerous different structures can be identified (compare with Fig. 6(d) in Appendix B ). At small Zr concentrations the local symmetry-breaking correlates to Zr sites (denoted by 1), though it is worth noting that also the Ti cells (denoted by 0) have Pb displacements different from the blue matrix.
(a) Pb-displacements (fractional coordinates) along the directions. 17 (b) A cell describing the four Pb (labelled as A, B, C, and D) and four B cation positions (indexed as 1, 2, 3, and 4) from which an infinite number of short-range order states can be generated by applying fourfold rotation and translational operators. (c) Axes settings specifying the symmetry-element directions characteristic of the planar symmetry groups listed below. (d) Examples of four-cell blocks with planar group symmetries (valid if the blocks are repeated in horizontal and vertical directions). Only projections on the tetragonal ab-plane are shown. The cross in the centre of the cell indicates that the ideal position of Pb and the four circles, displaced from the cross to 〈110〉 directions, is the positions available for Pb. The black sphere indicates an occupied position. For clarity, no other ions are shown. The corners of the cells are the projection points of the B cations (Zr or Ti) on the plane perpendicular to the c axis. Dotted lines in the diagram on the second row and first column indicate the different Pb-B distances. Lattice parameters are relaxed according to the symmetry.
Distribution of the cells as a function of Zr content x. Blue, yellow, green, and red squares indicate cells 1, 2, 3, and 4, respectively. Though large uniform areas appear at Ti rich areas, they do not exhibit large deviation from tetragonal symmetry as the Pb-displacements are small when x is small (see Fig. 6(a) ). In the vicinity of the MPB a large number of distorted states are available, which explains why the susceptibility of PZT is exceptionally large at this composition region. Each panel has 100 100 cells.
Correspondence between the plane and space groups and the irreducible representations spanned by Pb displacements. Note the significant increase in the number of Raman-active modes resulting from the presence of the different cells and that also B 2 symmetry modes are activated.
Average crystal structure of the tetragonal lead-zirconate-titanate (space group P4mm).
Atomic displacements spanning the Brillouin zone-centre A 1, B 1, and E irreducible representations.
Raman scattering tensors for the A 1, B 1, B 2, and E symmetry modes (point group 4mm). For the ideal tetragonal structure given in Table II , no B 2 symmetry mode is allowed. A 1 and E symmetry modes are also infrared-active (phonon polarization is given in brackets).
Measurement geometries used for measuring the Raman data in this study. All phonons, except for the E(LO) modes, can be identified by experiments through backscattering measurements. The E(LO) modes can be observed through platelet measurements. 9
Raman scattering tensors for the A′ and A″ symmetry modes (point group m).
The activity of the A′ and A″ modes of the Pm and Cm clusters for the measurement geometries (in terms of tetragonal axes) applied in this study.
Sixteen configurations corresponding to different combinations of Zr (Z) and Ti (T) at the cell corners (Fig. 7 ) and the corresponding Pb position(s). The mean field removes the degeneracy.
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