INDEXING METHODS

Indexing accounts for identifying crystallographic orientation of single grains when measurements with X-rays are performed. The term indexing expresses the identification of the reciprocal lattice from a cloud of reciprocal lattice points recorded during the measurement. Majority of the first three-dimensional indexing methods, for instance those used in ‘INDEX’ [Enr 00], ‘DIRAX’ [Dui 92] or ‘RAMCEL’ [Enr 00], are based on the principle of maximizing the number of observed reciprocal space vector matching the initial reciprocal cell vectors. This can be realised in various ways, operating either in real or reciprocal space. Indexing in structural characterisation of materials is always connected with preliminary analyses.

The subsequent steps of the preliminary analysis are shown in Fig. 3.1.

Image analysis constitutes the first step of every indexing process. At this point the Bragg spot information is extracted from acquired diffractograms and the spot parameters are transferred to subsequent analysis step. The so called ray tracing identifies the centre of mass positions (CM) of observed Bragg spots and estimates parameters necessary to calculate normalised scattering vectors, which establish the indexing input. Ray tracing module carries out the assignment of scattering vectors into the laboratory system. The arrangements are performed in a way assuring that the reflections data meet the input requirements of the particular indexing routine. The indexing itself can be performed utilizing various approaches. The direct space method

Fig. 3.1 Flow chart of subsequent procedures proclaiming the tracing and indexing algorithm

Indexing methods

[Cle 84, Jac 76] for instance operates in the direct lattice space and relays on the assumption that a scalar multiplication of a true direct vector and any true reciprocal vector results in an integer. Thus the tentative direct space vectors of an initial (guess) unit cell are compared to reciprocal vectors observed during experiment. If the great majority of the reciprocal vectors meet the assumption with respect to the inspected direct vector, the inspected direct vector is assumed to be a true one. The set of obtained direct vectors gives then the basis to build up a final cell. The direct space method is easy and quite neat in realisation, although, like all of the first indexing methods, it is dedicated to the single crystal diffractometry. It means that it is able to handle rather low number of reflections, where all or majority of them belongs to one single grain.

The development of high brilliance third generation synchrotron radiation sources together with progress made in X-ray optics started a new trend in the field of material studies. Available spatial resolution routed the grain investigations on a sub-micron level, making of X-ray diffraction a promising tool for strain and texture oriented analyses. Although measurements of polycrystalline samples bring along additional problems linked to the complexity of the new measurement techniques. The indexing method applied in the program ORDEX [Chu 99] is one of the first developed for purposes of polycrystalline sample characterisation. The algorithm has been dedicated to white-beam Laue microdiffraction method. When caring out measurements with microbeams, sample rotation may cause changes in illuminated volume of single grain.

Thus Laue diffraction is usually applied when rotation of the sample has to be avoided.

Method dedicated to that kind of experiment is comprised in ORDEX. It operates on Bragg plane normals q) which are determined based on the incident and diffracted wave vectors (in respectivelyK₁

r and K

r

Fig. 2.2). In order to identify a grain all observed reflections are compared with respect to their mutual angular relationships.

Reflections are told to be valid if the mutual angular relationship meets the estimated values within tolerances given by limit distortion of the cell. In this way all reflections from a single grain can be identified and indexed. ORDEX is able to process overlapping Laue patterns, however, it deals with rather small amount of data (~20 reflections) [Chu 99] without engaging in splitting reflections data into subgroups associated with individual grains. This feature displays one of the strongest issues of that method. It is recently required that investigation techniques allow either, the

Indexing methods

analysis of ensembles of 10-1000 grains and time factor in data acquisition process kept on the level providing possibility of in-situ studies [Lau 01]. The microdiffraction technique doesn’t meet those requirements since diffraction data is obtained grain by grain at acquisitions delayed by sample adjustment. Coupling the monochromatic rotational method with ray tracing gave grounds to three-dimensional X-ray diffraction, which is the first technique allowing simultaneous determination of a large amount of grains with respect to their orientations, positions and volumes [Pou 04].

One example of tracking and indexing diffraction data gives the algorithm GRAINDEX [Sor 06, Mar 04, Lau 01] designed for the purposes of the 3DXRD method. In case of polycrystals, when completely illuminating the sample cross-section, a large amount of data has to be characterized. It can be stated that during a complete ω turn the crystal is represented (in terms of reciprocal space) by, depending on utilized reflection orders, several dozens up to few hundreds diffraction spots (approximately 120 considering first five Debye-Sherrer rings). Reflecting that space onto frames collected during the experiment and thus introducing the integration step factor, it can be easily deduced, that the amount of data which has to be handled increases with increasing number of grains and decreasing integration step. Therefore tracking and indexing of the experimental data in case of 3DXRD exhibits one of the most complex tasks subjected to grain dynamics analyse. The technique is although restricted by the grain deformation magnitude, which shouldn’t reveal mosaic spread greater than few degrees [Lau 01]. Two different approaches can be applied considering 3DXRD. Diffraction data can be either collected at two or more different sample detector distances, what enhances the accuracy of centre of mass estimation of grain position, or just at one single sample detector distance. In the first case the ray tracing can be performed on the way of straight line fitting to the CMS of equivalent spots recorded at different distances. This way the reflections can be precisely traced back on the extrapolated lines under the consideration of the eqs. (2.15)-(2.18). Those lines having directions of scattered wave vector join at the origin in the sample, indicating the CMS positions of grains. Other approach assumes just one medium sample detector distance for data acquisition, so that coordinates x, y and z in eqs.

(2.15)-(2.18) can be neglected beside D. Thus the indexing method considers a simplified diffractometer equation (2.8) and it is based on a comparison between experimental diffraction vectors:

Indexing methods

and candidate vectors:

which correspond to candidate orientation matrices U.

Experimental diffraction vectors are obtained from Bragg reflections recorded on different frames during the experiment and identified by their centre of mass positions described by set of three parameters (ω, η, θ) (Fig. 2.2). Orientations are defined in the Euler space (eq. (2.25)). Thus to index a certain orientation a simulation of all reflections expected for a certain configuration has to be executed. Furthermore the simulation has to include all possible orientations defined by the Euler angles, belonging to the fundamental zone of the orientation space. To speed up computations the comparison is performed in the sample coordinate system (CS) (indicated by the subscript S) for all independent orientations U enabled by the crystal symmetry. In practice this is accomplished for a finite grid of points if the fundamental zone [Mor 95] (Fig. 3.2).

The shape and size of the fundamental zone is strictly related to crystal symmetry [Mor 95] and to sample it uniformly the distribution of the grid should proportional to the volume element dg of the orientation space [Pou 04]:

c 1 _

S U g

g _sim = ⁻ (3.2)

gω

Ω S

g_{S_exp} = ^{−1 −}_z¹ (3.1)

Fig. 3.2 Orientation grid placed in the fundamental zone of the orientation space for a cubic structure

Indexing methods

Considering a known crystal structure a set of scattering vectors g_sim^c is simulated at each grid point Ui =(ϕ₁,Φ,ϕ₂) and subsequently compared to the experimentally observed vectors (see also eqs.(2.8) –(2.9)):

Before the set of vectors g_exp^l can be assigned to a certain grain, it has to obey a number of criteria. An observed vector is assumed to be a ‘true one’ if it overlaps with the simulated vector within the cone of tolerance. If the completeness criterion is fulfilled, thus for a single orientation Ui a desired fraction of true vectors is observed, the inspected orientation is assumed to be close to solution. At the same time the reflection set shouldn’t be a subset of any already assigned grain – the uniqueness criteria. Those true vectors meeting all criteria are then used to refine the orientation on the way of optimization. The method is robust and has the advantage that it can be applied to scans performed over smaller ω intervals, adequate to study the kinetics of faster processes like second-phase nucleation [Offe 02] or recrystallization [Schm 04].

Its main limitation is related to spot overlap whose probability increases with both deformation of the sample (mosaic spread of the grain) and number of grains. It has been although demonstrated that the method should be able to index up to 5000 grains if their mosaic spread of approximately 0.1° [Schm 03].

3.1 Grain Reflections Indexing Program (GRIP)

This work presents a new indexing method, which evaluates the crystallographic orientation of single grains embedded in a polycrystalline sample. The approximate orientations and positions of the grains can be evaluated based on measurements performed for radial region assuring sufficient fraction of reciprocal space. The new method avoids scanning of the orientation space. The method assumes a perfectly aligned detector lying perpendicular to the incoming beam. In order to extract the information contained in acquired 2D diffractogram frames a software has been

{ } { }

Indexing methods

developed. Grain Reflection Indexing Procedure (abbreviated as GRIP) delivers a set of reflections parameters which can be utilized for precise calibration of the setup (sample to detector distance, beam center, detector tilts), too. The software is divided into subroutines dedicated to tasks following the chart in Fig. 3.3.

After extraction of the reflection information from analyzed diffraction frames the indexing algorithm is applied. Its tasks can be subdivided in following steps:

• Search for the related reflections and forming of Friedel-pairs (FP) and pseudo Friedel-pair groups (PFP);

• Successive search through the list of reflection groups (FP and PFP) belonging to the h00 order and setting up the candidate U matrix;

• Generation of diffraction vectors and collation with set of experimental vectors considering tolerances and completeness criterion;

• Indexing of interrelated reflections and assignment to subsequent grain;

• Estimation of orientation, position and strain tensor components.

A specific description of the method is given in the following chapters of this work.

Fig. 3.3 The flowchart of grain indexing program (GRIP)

Image analysis