Direct Observation of Local Distortion of a Crystal Lattice with Picometer Accuracy Using Atomic Resolution Neutron Holography

We demonstrate that neutron holography permits us to extend the determination of atomic positions beyond nearest neighbors at least up to the fourth neighboring shell around cadmium probe atoms alloyed into a lead crystal. The accuracy achieved is sufficient to allow quantitative determination of displacements of atoms due to elastic distortions induced by impurity atoms. The atomic positions derived from the holographic data are in good agreement with those expected theoretically due to Friedel oscillations in this system. In addition, the atomic positions are in qualitative agreement with results obtained in an independent experiment studying the diffuse distortion scattering around Bragg peaks.

Figure 1

(a)–(d): Reconstructions of the four closest shells of neighboring atoms around a Cd probe nucleus. Each figure shows (top) an intensity map obtained from the hologram recorded in the experiment and connecting lines pointing out high-intensity spots, (bottom) a numerical simulation of the expected map. The angle coordinates $\varphi$ (0–360°) and $\chi$ (0–180°) define the positions on spherical surfaces around the probe nucleus. The Pb atoms form an fcc lattice; thus, the four shells contain 12, 6, 24, and 12 atoms, respectively.

Figure 2

The maximum intensity distribution of the restored holographic image as a function of the distance R from the Cd nucleus. Experimental values (●), model calculation for a nondistorted crystal involving 100 surrounding shells (continuous line). The intensity is normalized to the first peak. Vertical arrows mark the positions of atoms in a nondistorted lead crystal. Oscillations between the peaks are inherent to the holographic restoring procedure [10].

Figure 3

(a) The variation of the holographic intensity as a function of the distance $R$ from the probe atom calculated using Eq. (5). The vertical solid lines show the expected positions of the lead atoms for the perfect crystal. Figs. (b), (c), (d), and (e) show a close-up view of the vicinity of the 1st, 2nd, 3rd, and 4th neighbors, respectively. The vertical dashed lines show the real positions of the lead atoms obtained from the experimental data.

Figure 4

Continuous line: calculated electron density modulation around a Cd atom (refer to left scale). Black squares: experimentally observed local distortion as a function of the distance from the Cd nucleus (refer to right scale). The error bars are estimated from the average value of the oscillations modulating the background line.