The Atom Probe and Markov Chain Statistics of Clustering

Abstract

The field ion microscope atom probe, used in the exploration of crystal structure, is discussed. Data is generated from a probe of an alloy of two types of metalic elements. A statistical model is formulated to investigate clustering of like atoms within the alloy. Physical considerations motivate a Markov chain generalization of the binomial model. A parameter in the model which measures the degree of clustering is estimated by maximum likelihood and large sample distribution theory is given using results from Billingsly. A combinatorial analysis is developed which provides the exact distribution of the sufficient statistics and permits small sample comparisons of expectations and mean square errors of estimates with their large sample approximations. The model is applied to some preliminary data and is used to point up and quantify some difficulties in the experiment.

KEY WORDS:

• Atom Probe
• Clustering
• Binary Alloys
• Markov Chain
• Dependent Bernoulli Trials
• Nearest Neighbor Pairs
• Field ion Microscope