E. Landree

A feasible set approach to the crystallographic phase problem

The connection between the crystallographic phase problem and the feasible set approach is explored. It is argued that solving the crystallographic phase problem is formally equivalent to a feasible set problem using a statistical operator interpretable via a log-likelihood functional, projection onto the non-convex set of experimental structure factors coupled with a phase-extension constraint and mapping onto atomic positions. In no way does this disagree with or dispute any of the existing statistical relationships available in the literature; instead it expands understanding of how the algorithms work. Making this connection opens the door to the application of a number of well developed mathematical tools in functional analysis. Furthermore, a number of known results in image recovery can be exploited both to optimize existing algorithms and to develop new and improved algorithms.

DIRECT METHODS FOR SURFACES

This paper reviews recent progress in the application of Direct Methods to solve surface structures using surface X-ray or transmission electron diffraction data. The basic ideas of (crystallographic) Direct Methods are presented, as well as the additional problems posed by trying to apply them to surfaces and how they connect to the mathematical theory of projections. Surface crystallography notation is presented, which differs from the widely used LEED notation in that it emphasizes the surface symmetry. This is followed by a description of methods for structure completion and refinement, followed by applications to some experimental systems, both those where the structure was previously known (calibration tests) and a few where it was not, concluding with problems and limitations.

Direct solutions of the Si(111) 7 × 7 structure

Abstract We show that it is possible to use direct methods to solve surface structures as complicated as that of the Si(111) 7 × 7. The first of these methods is maximum entropy combined with likelihood estimation, the second a combination of symbolic phasing methods and weakly interpolating modifications of the Sayre equation. These techniques are applied to two different elements of the structure determination, namely ab initio structure determination solely from diffraction intensity data and phase extension from a set of known phases derived from high resolution images.

Structure of the TiO2−x(100)-1×3 surface by direct methods

An improved atomic model of the TiO 2’x (100)-1◊3 surface has been determined based on grazing incidence X-ray diVraction (GIXD) data analyzed by direct methods coupled with x2 minimization. The structure contains edge- and face-sharing octahedra, similar to known defect structures in Ti n O 2n’1 . The final structure consists of four Ti atoms and six to eight oxygen.

Multi-Solution Genetic Algorithm Approach to Surface Structure Determination Using Direct Methods

We show that it is possible to use a multi-solution genetic algorithm search method utilizing direct methods to solve surface structures from surface diffraction data. We suggest that the method is generally applicable and able to replace random searches of the solution space.

A Minimum-Entropy Algorithm for Surface Phasing Problems

Methods of determining the phases for noisy and incomplete surface diffraction data (X-ray or transmission electron) are investigated. It is found that, while conventional methods do not always work, a minimumentropy method that uses the relative entropy is more effective in finding the correct solutions.

Direct methods determination of the Si(111)-(6 × 6)Au surface structure

Abstract The atomic structure of the Au 6×6 on Si(111) phase has been determined using direct methods and surface X-ray diffraction data. This surface structure is very complicated, with 14 independent gold atoms, relaxations in 24 independent silicon sites and three partially occupied gold sites. In one sense the structure can be described as microdomains of the parent 3 × 3 Au on Si(111) structure. A better description is in terms of a tiling of incomplete pentagonal and trimer units, essentially a pseudo-pentagonal glass. In terms of these structural units it is possible to explain all the gold structures in the coverage range 0.8–1.5 monolayers as pseudo-glasses with strong short-range order but varying degrees of long-range order.

Determination of the current density distribution in Josephson junctions

Abstract A technique is described for recovering the missing phase information for a set of critical current measurements as a function of an applied magnetic field I c ( B ). In many cases the current density j ( x ) across the boundary for a Josephson junction can be determined.

Transmission electron diffraction determination of the Ge(001)-(2 × 1) surface structure

Abstract The lateral displacements in the Ge(001)-(2 × 1) surface reconstruction have been determined using transmission electron diffraction (TED). The best-fit model includes displacements extending six layers into the bulk. The atomic positions found agree with X-ray studies to within a few hundredths of an angstrom. With the positions determined so precisely, it is suggested that the Ge(001)-(2 × 1) surface can now serve as a standard for comparison with theoretical surface structure calculations. The results from the currently available theoretical studies on the surface are compared with the experimentally determined structure.

Surface roughening by electron beam heating

The effect of electron beam heating during the preparation of clean silicon surfaces suitable for epitaxial studies in ultrahigh vacuum conditions was investigated using surface chemical characterization techniques and transmission electron microscopy. The electron beam irradiation produced a disordered surface on the incident side of the sample and well-ordered monoatomic steps on the other surface, even at electron energies as low as 3 keV. These results have significant implications for epitaxial thin film growth.