We analyze the quantum walk in higher spatial dimensions and compare classical and quantum spreading as a function of time. Tensor products of Hadamard transformations and the discrete Fourier transform arise as natural extensions of the "quantum coin toss" in the one-dimensional walk simulation, and other illustrative transformations are also investigated. We find that entanglement between the dimensions serves to reduce the rate of spread of the quantum walk. The classical limit is obtained by introducing a random phase variable.

Nanoscale atomic clusters in atom probe tomographic data are not universally defined but instead are characterized by the clustering algorithm used and the parameter values controlling the algorithmic process. A new core-linkage clustering algorithm is developed, combining fundamental elements of the conventional maximum separation method with density-based analyses. A key improvement to the algorithm is the independence of algorithmic parameters inherently unified in previous techniques, enabling a more accurate analysis to be applied across a wider range of material systems. Further, an objective procedure for the selection of parameters based on approximating the data with a model of complete spatial randomness is developed and applied. The use of higher nearest neighbor distributions is highlighted to give insight into the nature of the clustering phenomena present in a system and to generalize the clustering algorithms used to analyze it. Maximum separation, density-based scanning, and the core linkage algorithm, developed within this study, were separately applied to the investigation of fine solute clustering of solute atoms in an Al-1.9Zn-1.7Mg (at.%) at two distinct states of early phase decomposition and the results of these analyses were evaluated.

Modern wide field-of-view atom probes permit observation of a wide range of crystallographic features that can be used to calibrate the tomographic reconstruction of the analyzed volume. In this study, methodologies to determine values of the geometric parameters involved in the tomographic reconstruction of atom probe data sets are presented and discussed. The influence of the tip to electrode distance and specimen temperature on these parameters is explored. Significantly, their influence is demonstrated to be very limited, indicating a relatively wide regime of experimental parameters space for sound atom probe tomography (APT) experiments. These methods have been used on several specimens and material types, and the results indicate that the reconstruction parameters are specific to each specimen. Finally, it is shown how an accurate calibration of the reconstruction enables improvements to the quality and reliability of the microscopy and microanalysis capabilities of the atom probe.

The applicability of the binomial frequency distribution is outlined for the analysis of the evolution nanoscale atomic clustering of dilute solute in an alloy subject to thermal ageing in 3D atom probe data. The conventional chi(2) statistics and significance testing are demonstrated to be inappropriate for comparison of quantity of solute segregation present in two or more different sized system. Pearson coefficient, mu, is shown to normalize chi(2) with respect to sample size over an order of magnitude. A simple computer simulation is implemented to investigate the binomial analysis and infer meaning in the measured value of mu over a series of systems at different solute concentrations and degree of clustering. The simulations replicate the form of experimental data and demonstrate the effect of detector efficiency to significantly underestimate the measured segregation. The binomial analysis is applied to experimental atom probe data sets and complementary simulations are used to interpret the results.

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