T d T C. Rhombohedral Tetragonal Cubic (%) 0.1 (222) Δa/a 292K 0.0 (022) (002) Temperature (K)

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1 (%) Rhombohedral Tetragonal Cubic Δa/a 292K 0.1 (222) 0.0 (022) -0.1 (002) T C T d Temperature (K) Supplementary Figure 1: Percent thermal expansion on heating for x- ray lattice spacing determined from (222), (022), and (002) reflections. Between about 380 K and 405 K a narrow tetragonal phase region exists, which is consistent with the accepted phase diagram for PMN- PT at 30% PT concentration 1. At higher temperature a crossover to linear thermal expansion occurring for all three reflections at around 680 K reflects the Burns temperature 2, T d. Deviation from cubic symmetry occurred due to residual stresses at the outer crystal surfaces detected by x- rays, but these regions made a negligible contribution to the neutron experiments, which probed the bulk crystal. This was confirmed with Laue neutron diffraction patterns measured on ARCS in white beam mode (no chopper), which showed no evidence of streaks or extra reflections from a surface layer. 1

2 (210)% '% (200)% (210)% Q"="(2,&0.4,0)" (100)% Q"="(2,+0.4,0)" (000)% Supplementary Figure 2: Comparison of constant Q inelastic neutron spectra measured on PMN- 30%PT at two different Q points that are equivalent in the reciprocal space of the crystal but differ with respect to the BT7 measuring instrument. The localizing mode feature discussed in the body of the paper is indicated with a red arrow. The localizing mode and its temperature dependence are evident in both cases. The dispersing modes are resolution focused for +K and unfocused for K. 2

3 Supplementary Figure 3: A comparison of the localizing mode feature in PMN- 30%PT, indicated by arrow, measured using the BT- 7 triple axis spectrometer at the NIST Center for Neutron Research (top) and the time- of- flight ARCS spectrometer (bottom) at the Spallation Neutron Source of Oak Ridge National Laboratory. The value in the ARCS measurement, 11.9±0.1 mev, is within error of the value at the same position from BT7 data 12±0.2 mev as well as the average value of 11.8±0.1 mev from BT7 data integrated over K = [0.375, 0.5]. While the BT7 triple axis data has a constant background the ARCS- TOF spectra has sloped background, and this was approximated with a straight line in the fit. In the ARCS data analysis the out- of- plane direction, Q = [0, 0, L], was integrated over L = [- 0.2, +0.2] to give an effective out- of- plane Q resolution similar to that of BT7. The intensity profiles still appear different, however, because the triple- axis spectrometer has a tilted ellipse shaped resolution function. Although spurious features are also possible in a TOF chopper measurement, the origin of these are completely different than with a triple- axis instrument since the neutrons are sorted by arrival time at a detector rather than by monochromators and analyzers. 3

4 Supplementary Figure 4: Temperature dependence of the constant Q = [2, 0.325, 0] inelastic neutron scattering spectra for PMN- 30%PT measured on the BT7 triple axis spectrometer. The relative decrease in intensity of the localizing mode with increasing temperature is inconsistent with multiphonon scattering. 4

5 Rhombohedral Tetragonal Cubic a TA mode (2, 0.2, 0) (2, 0.15, 0) 2.0 Phonon linewidths, Γ (mev) T C T d Localizing mode (2, K, 0): K=[0.4,0.5] b Temperature (K) Supplementary Figure 5: Phonon linewidths (FWHM) of the TA and localizing mode. (a) From fits to the TA mode at K = 0.2 and 0.15, Q = [2, K, 0]. (b) Average from fits to the localizing mode from K = 0.4 to 0.5. The values of T d, T C, and the regions of cubic, tetragonal, and rhomohedral phases are indicated. Widths are corrected for the instrument resolution, which ranged from 0.8 to 1.4 mev. The positions of T d, T C, the cubic, tetragonal, and rhombohedral phase regions were determined by x- ray diffraction measurements and are consistent with the known characteristics of PMN- 30%PT (see Supplementary Information B). The TA mode linewidth at K = 0.2 and 0.15, shown in Supplementary Fig. 5a, gradually increases with decreasing temperature from 900 K to T d and then below T d it begins to increase more rapidly before leveling off around 500 K and decreasing on cooling below T C. This TA linewidth behavior is similar to that reported by Wakimoto et al. 3 and Naberezhnov et al. 4 for PMN and attributed to coupling to PNRs. The localizing mode linewidth exhibits the opposite temperature dependence. In the range K = [0.4, 0.5], the linewidth of the localizing mode narrows rather than broadens as it becomes dispersionless in the temperature region between T d and T C. 5

6 2000 BT T C < 488 K < T d BT7 688 K T d 1500 K= "Waterfall"? Intensity (counts) 1000 K=0.2 K=0.225 K=0.25 Intensity (counts) 1000 K=0.175 K=0.2 K=0.225 K= TA TO K=0.3 K=0.35 K=0.375 K=0.45 K= TA TO K=0.375 K=0.425 K=0.475 K= Energy (mev) Intensity (counts) Local mode intensity 688 K 488 K BT7 RM (fully localized) Anderson-localized phonon fit L = 2.0 nm K 0 = (λ0 = 2.05 nm) Energy (mev) Supplementary Figure 6: Determination of the intensity profile of the localizing mode (bottom) as a function of wavevector from fits to the data at 488 K (top left) and 688 K (top right). The TA and TO phonons show significant lifetime broadening and are best fit with Lorentzian functions, whereas the elastic line and the localizing mode features, which have widths largely determined by the instrument resolution, are best fit with Gaussian functions. At 488 K, the scans at K = 0.25, 0.225, and were each counted for ~10 seconds per point while all others were counted each for ~30 seconds per point (count times varied). The lower count time scans were scaled up for comparison (the error bars are statistical and represent one standard deviation). The scans at 688 K were counted ~30 seconds per point, except K = which was counted for ~40 seconds per point and K = which was counted for ~10 seconds per point; all were normalized to the counts of a ~30 second per point scan. For 488 K the localizing mode peak (red) fit the data directly (without constraints) between K = 0.25 and K = Above K = 0.45 the intensity tends to zero and a number and uncertainty for the intensity was estimated by assuming a 1.2 mev width peak at 11.4 mev and allowing the intensity to vary in the refinement. Below K = 0.25 the intensity becomes stronger but the peak begins to overlap the much broader TO mode. An initial fit determined the localizing mode peak positions by constraining the localizing mode peak width to 1.2 mev. Then the position was constrained and the width was allowed to vary. Finally, the constraints were relaxed and the refinement determined the peak position and width. The intensities used in the profile were from the fit areas of the localizing mode peaks and the error bars are from the uncertainties in the peak areas from the refinement. For 688 K the localizing mode peak (black) fit the data directly between K = 0.3 and K = 0.5. Below K = 0.3 the localizing mode peak, which does not appear to have any K dependence to its intensity, gets lost in the noise of the much stronger TO mode. For K = and below a flat distribution of intensity fills in the gap between the TA and TO peaks. This gap intensity is consistent with the well- documented waterfall effect 5. No waterfall effect is evident at 488 K [2, K, 0]

7 a b Supplementary Figure 7: Evidence of a similar localized phonon in PZN- 5%PT. Inelastic neutron scattering measurements of the phonon dispersion in (Pb(Zn 1/3Nb 2/3)O 3) (PbTiO 3) 0.05 (PZN- 5%PT) at 688 K, measured in the (220) and (200) zones using ARCS. (a) Measurements along [0K0] in the (220) zone. (b) Measurements along [0K0] in the (200) zone. A weak localized (dispersionless) phonon appears between the TA and TO modes near 10 mev in both zones. The out- of- plane direction, Q = [0, 0, L], was integrated over L = [- 0.2, +0.2] while the in- plane K and H values listed are integrated in a narrower range of ± This localized phonon feature is similar to the precursor local mode observed in PMN- 30%PT but about 1.8 mev lower in energy at the same temperature. The local (dispersionless) feature is near 10 mev and is indicated by the red arrows. 7

8 Supplementary Table 1: Elastic incoherent false peak energies (de app) for fixed E f = 13.7 mev calculated to fourth order (mon = monochromator order and an = analyzer order). Calculations were performed using Scan Mapper in the Data Analysis and Visualization Environment (DAVE) software 6. (mon, an): de app (mev) (2, 1): (3, 1): (4, 1): (1, 2): (3, 2): (4, 2): (1, 3): (2, 3): (4, 3): (1, 4): (2, 4): (3, 4): Supplementary References 1. Noheda, B., Cox, D. E., Shirane, G., Gao, J., Ye, Z.- G. Phase diagram of the ferroelectric relaxor (1- x)pbmg 1/3Nb 2/3O 3- xpbtio 3. Phys. Rev. B 66, (2002). 2. Bokov, A. A., Ye, Z.- G. Recent Progress in Relaxor Ferroelectrics and Related Materials with Perovskite Structure. J. of Mater. Sci. 41, (2006). 3. Wakimoto, S., Stock, C., Birgeneau, R. J., Ye, Z.- G., Chen, W., Buyers, W. J. L., Gehring, P. M., Shirane, G., Ferroelectric ordering in the relaxor Pb(Mg 1/3Nb 2/3)O 3 as evidenced by low- temperature phonon anomalies. Phys. Rev. B 65, (2002). 4. Naberezhnov, A., Vakhrushev, S., Dorner, B., Strauch, Moudden, H. Inelastic neutron scattering study of the relaxor ferroelectric PbMg 1/3Nb 2/3O 3 at high temperatures. Eur. Phys. J. B 11, 13 (1999). 5. Hlinka, J., Kamba, S., Petzelt, J., Kulda, J., Randall, C. A., Zhang, S. J., Origin of the waterfall effect in phonon dispersion of relaxor perovskites. Phys. Rev. Lett. 91, (2003). 6. Azuah, R. T., Kneller, L. R., Qiu, Y., Tegenna- Piggott, P.W.W., Brown, C.M., Copley, J.R.D., & Dimeo, R.M. DAVE: A Comprehensive Software Suite for the Reduction, Visualization, and Analysis of Low Energy Neutron Spectroscopic Data. J. Res. Natl. Inst. Stan. Technol. 114, 341 (2009). 8