Formal conditions for unambiguous residual strain determination by CBED

Abstract

The method of residual strain determination using convergent beam electron diffraction (CBED) is attractive because of its good spatial resolution. However, attempts to obtain all six independent strain components from a CBED pattern lead to ambiguous results. This paper contains analysis of the ambiguities based on the complete algorithm for matching experimental and strain-dependent simulated CBED patterns. The strain parameters which are not determinable by the CBED method are identified by examination of the most common goodness-of-fit functions. The indeterminable parameters are confirmed to be the '13' and '23' components and a combination of the diagonal components of the tensor given in the Cartesian systems having the '3' axis parallel to the beam direction. The ambiguity can be eliminated based on multiple diffraction patterns. It is shown that two different patterns may be insufficient to get a unique strain tensor. The ambiguity can be removed only if certain characteristics of the two patterns are different, or if more than two patterns are used.

Towards a full retrieval of the deformation tensor F using convergent beam electron diffraction

“…In the following, using a set of simulated off-axis CBED patterns, we explore how many of the nine parameters of F can be retrieved and the precision with which they can be retrieved from a minimum set of experimental data. Ideally, one would like to retrieve the nine parameters of F. However, as stated by Morawiec et al [10], it seems difficult and certainly impossible to determine the nine parameters from a single direction of observation. Morawiec et al have shown that there are three ambiguities on the F tensor parameters that cannot be solved: (1) f zx , (2) f zy and (3) a linear combination of f xx , f yy , and f zz .…”

“…each HOLZ line g has its own DS g that is determined on the reference pattern. This procedure is applied to all HOLZ lines in the transmitted beam and some (5)(6)(7)(8)(9)(10) in the diffracted beams. Firstly, a weight w g is calculated for each line g in the transmitted beam in order to give more or less power to the HOLZ lines…”

“…But not all the nine components of F can be retrieved. Morawiec [10] gave a detailed description of these ambiguities. He concluded that it is not possible to obtain reliable information on the xz and yz shear strains (z being the beam axis).…”

Using a <670> zone axis for convergent beam electron diffraction measurements of lattice strain in strained silicon

“…Although a single HOLZ line pattern contains three‐dimensional information related to the values of all six crystal lattice parameters, in practice a number of assumptions are required to obtain a unique fit of a given pattern (Rozeveld & Howe, 1993; Wittmann et al , 1998; Morawiec, 2005). Previous research has shown that obtaining a unique fit to a HOLZ line pattern is not possible with six independent variables, but may be possible with three (Rozeveld & Howe, 1993).…”

A program for refinement of lattice parameters based on multiple convergent-beam electron diffraction patterns

“…With the matching of HOLZ line geometries, there is another difficulty which affects lattice parameter determination. A given HOLZ line pattern can be simulated with a number of different sets of lattice parameters (Maier et al, 1996), and the inverse problem of lattice parameter determination is ill-conditioned (Morawiec, 2005). This ('ambiguity') issue is addressed by reducing the number of free parameters, or by matching simultaneously multiple patterns originating from the same location.…”