A new method for determining small misorientations from EBSD patterns

Author’s Accepted Manuscript for article published as Scripta Materialia (2001) 44 2379-2385 January 2001

A New Method for Determining Small Misorientations from Electron Back Scatter Diffraction Patterns

Angus J Wilkinson Department of Materials, University of Oxford, Parks Road, Oxford, OX1 3PH, UK (email : [email protected]) Keywords: EBSD, SEM, misorientation, electron microscopy, dislocation cells

1.

INTRODUCTION

Electron back scatter diffraction (EBSD) is an established tool for quantitative characterisation of material microstructures (1, 2). In samples in which the crystal structure is already known it is routine to use EBSD to map crystal orientations over an array of points under computer control. The pattern analysis makes use of a Hough transform, which allows the positions of Kikuchi bands to be located automatically (3, 4). Next the angles between the planes corresponding to located Kikuchi bands are calculated. and compared to expected angles for the known crystal structure. The best match between measured interplanar angles and those expected for the known crystal type and allows the pattern to be indexed and hence the crystal orientation calculated. There is a wealth of information available in the orientation maps so produced. From such orientations maps it is a relatively simple matter to calculate the misorientations between neighbouring regions. Dingley & Randle (5) have reviewed how the method can be used to characterise the crystallography of grain boundaries. Misorientations between regions within a grain are also of interest because they can be used to assess the way in which dislocations are stored in a crystal during plastic deformation. There have been several extensive TEM studies of misorientations in deformed metals (6, 7 and 8). In general it has been found that the misorientation angles increase with increasing strain and that the distributions of misorientation angles have the same functional form at different strains. The misorientation angles are small, averages for an applied strain of 0.1 being ~2° and ~0.5° for geometrically necessary boundaries (GNBs) and incidental dislocation boundaries (IDBs) respectively [7]. There have also been some attempts to use EBSD to measure misorientations within deformed crystals (9, 10). EBSD has distinct advantages over TEM in that it allows larger


samples to be examined and is less labour intensive. However, it is necessary to examine the whether EBSD provides sufficient spatial and angular resolution to fully characterise such structures. Humphreys and Brough have made a convincing analysis of the spatial resolution that can be achieved in EBSD orientation maps (11), finding that for Al in a FEG SEM an effective resolution of 30 nm can be achieved. Juul Jensen has recently pointed out that although much effort has focused on improving the spatial resolution of EBSD, little effort has been made to improve the angular resolution which is currently a major limitation to the use of EBSD in studying deformation microstructures [12]. This work concerns the angular resolution of EBSD measurements. The first part of the paper examines how uncertainties in the individual orientation measurements manifest themselves in the calculated misorientations. The second part of the paper presents a new alternative analysis in which the misorientation is determined directly from pairs of EBSD patterns. Repeated measurements of small misorientations demonstrate the increased sensitivity of the new method. 2.

CALCULATING MISORIENTATIONS FROM ORIENTATIONS – The Problem

Repeated measurements from a Si single crystal wafer were used to assess the typical uncertainty in orientation measurements made with an EBSD system. 100 orientations were measured using an Oxford Instruments OPAL running OPAL 4.2 software. The scatter in the data was small, with the rotation from one measured orientation to another being on average 0.5°, and at most 1.4°. This is representative of the uncertainty associated with orientations measured using a well-calibrated EBSD system. The effect of this uncertainty in a measured  mean°,  max ° (i)° orientation on the calculation of misorientations 50 0.5 is shown by the following. The first 50 measured 40 0.4 orientations formed data set A, while the second 30 0.3 (  )° 50 orientations were used to generate data set B 0.2 by multiplying each orientation matrix with a 20  ° 0.1 rotation matrix so as to impose a known 10 °  misorientation between data sets A and B. The 0 0 0 10 20 30 40 applied rotation was about an axis Rapp (where applied rotation angle (app°) |Rapp|=1), and through an angle app. The misorientations between every individual Figure 1: Effect of size of applied orientation in set A and every individual rotation angle on scatter in calculated orientation in the rotated set B were then misorientation angles and axes. calculated (2,500 misorientations in total). The small spread of orientations within each data set results in some scatter in the calculated set of misorientations. Figure 1 shows the effects of the size of the applied rotation (app) on the spread in calculated misorientations. In each case the rotation axis was the same namely [110]SEM, where the subscript SEM indicates directions in the external microscope reference axes system (this is approximately [100]cryst referred to the sample’s crystallographic axes). The standard deviation (i) of the distribution of calculated misorientation angles (i : i=1,2,…2500) remains constant at about 0.2° for applied rotations with app >~2°, and then increases as the applied rotation is decreased toward zero. The standard deviation becomes larger than the applied rotation for angle less than 0.33°. The situation for the calculated misorientation axes (Ri : |Ri|=1 and i=1,2,…2500) is somewhat different. Figure 2 shows the distributions of the axes of the 2,500 i

max

mean


misorientations on inverse pole figures for applied rotations of 20°, 2°, and 0.2°. When the misorientation angle is large the misorientation axis is reasonably well-defined (figure 2a), however as the misorientation angle is decreased the inverse pole figures show a large amount of scatter in the calculated misorientation axes. This scatter in the misorientation axes can be quantified using the mean (mean) and maximum (max) of the distribution of angles i between the calculated misorientation axes and the applied rotation axis (cos[i]= Ri . Rapp). Figure 1 shows the increase in both mean and max as a function of the size of the applied rotation app. For an applied rotation of ~5°, or less, the misorientation axes are so widely distributed as essentially to be undetermined.

Figure 2a

Figure 2b

Figure 2c

Figure 2: Inverse pole figures showing distribution of calculated misorientation axes for applied rotations of (a) 20°, (b) 2° and (c) 0.2° about the [110]SEM axis. The above analysis demonstrates that using current EBSD methods although the misorientation angle can be measured (to ±0.5°) the misorientation axis will be essentially unknown for small rotations typical of deformation microstructures. To fully determine small misorientations using EBSD a new analysis method is required. 3.

MEASURING SMALL MISORIENTATIONS DIRECTLY – A Solution

In this section a new method of obtaining misorientations (axis and angle) from pairs of EBSD patterns is presented. The method is based on the measurement of small shifts in the positions of similar features in the two patterns. Consider the action of a rotation about an arbitrary axis on a point j situated a unit distance from the origin with position vector rj (so |rj|=1). After the rotation the point has a new position vector rj´, which is also of unit length (i.e. |rj´|=1). The vector qj = rj´ - rj simply gives the change in the position of point j caused by the rotation. It is noted that for any point j the vector qj must be perpendicular to the axis of the rotation. Thus the rotation axis (R) can be determined simply by measuring the direction of qj for two different points j=1,2. i.e. from R = q1 x q2 (eq 1) Furthermore the rotation angle () can then be found from the magnitude of one of the vectors qj. i.e. from tan  

qj

 

1  R . rj

2

(eq 2)

This analysis can be used to determine the misorientation (axis and angle) from a pair of EBSD patterns. A square region is defined at the same position on the scintillator screen (i.e. at the same rj) for each pattern. A cross-correlation procedure is then used to find the shift (qj) caused by the misorientation between the points on the sample from which the patterns were obtained. A high pass filtration of the patterns is carried out in the spatial domain before calculating the Fourier transforms used to determine the cross-correlation. This flattens any background intensity variations and emphasises the rapid changes in


intensity at the Kikuchi band edges which leads to a sharpening of the peak in the crosscorrelation. The displacement from the origin of the peak in the cross-correlation then establishes qj. A similar approach has been used by Wilkinson to determine elastic strains from EBSD patterns (13 and 14). This image analysis procedure can then be repeated for different regions of the EBSD patterns rj to find the corresponding qj. In this case four different regions of the EBSD patterns were analysed. Figure 3 illustrates the method by showing an EBSD pattern and the cross-correlation functions calculated for the four regions. Each of the cross-correlation functions show bright features running parallel to the dominant Kikuchi bands in the associated analysis region. The bright features overlap at a well defined bright peak which is displaced from the central origin. The displacement of the bright peak from the origin in the cross-correlation is equivalent to the relative displacement of that region of the two EBSD patterns.

top

left

left

right

bottom

top

right

bottom

(a)

(b)

Figure 3: (a) an EBSD pattern with four square analysis regions marked. (b) shows the cross-correlation functions calculated for each region for the pattern shown and a similar one obtain after a ~1° rotation of the sample. As stated above it is in principle possible to determine the misorientation from the measurement of pattern shifts (q1 and q2) at just two locations. However, the situation can arise in which q1 and q2 are nearly parallel in which case the rotation axis remains quite uncertain, equation 2 showing that R is unknown when q1 and q2 are exactly parallel. The effects of this problem are reduced by measuring the shifts at several different locations on the EBSD patterns (in this case 4) and using a least squares analysis to determine the 'best fit' rotation axis. Here the sum of the squares of the projections of the qj along R is minimised, with the magnitude of R normalised to unity. Once the rotation axis was found each pair of values of qj and rj were used in equation 2 to generate an average value of the rotation angle   qj 1 n      arctan (eq 3)  2  n j1 1  R . r   j

 


4.

METHOD VALIDATION AND SENSITIVITY ASSESSMENT

In order to validate the new method and to assess the level of uncertainty achieved in the misorientation measurements experiments were conducted on a highly perfect Si single crystal. A series of 10 EBSD patterns was recorded from neighbouring points on the Si crystal, ~0.5 µm apart. The specimen tilt and rotation controls of the SEM were then used to apply a slight (0 to 2°) misorientation to the Si crystal and a further series of 10 EBSD patterns was recorded. This process was repeated several times. The misorientations between each pattern in series X and each pattern in series Y were then evaluated using (a) the conventional approach (combining orientations) and (b) the new direct approach. Figure 4 compares the distributions of misorientation angles produced by the two methods, for one pair of data series. Calculation of the misorientations from measured orientations resulted in a mean misorientation angle of 2.1° and a standard deviation of 0.4° (figure 4a), which is in good accord with the analysis presented in figure 1. The mean misorientation angle determined with the new direct method is slightly smaller at 1.54° and there is a markedly narrower distribution (figure 4b) having a standard deviation of 0.01° . Other pairs of pattern series were also considered with similar results. The mean misorientation angle (mean) and the standard deviation in misorientation angles (() ) found with the new frequency (%) 25

frequency (%) 20

20

15

15 10

10 5

5

1.58

1.57

1.56

1.55

1.54

1.53

(b)

1.52

3.0

2.6

2.2

1.8

1.4

1.0

(a)

1.51

0

0

misorientation angle (°)

misorientation angle (°)

Figure 4: scatter in misorientation angles (a) calculated from orientations in the conventional way and (b) measured directly using the new method. method are reported in table 1. Standard deviation in the misorientation angle is typically 0.01° to 0.02°, with an indication that there is some increase over this range as the mean misorientation angle is decreased. Figure 5 shows the scatter in misorientation axes between each pattern in series X and each in series Y. Considerable scatter is evident in the axes established by calculating mean () mean max 0.23 0.02 5.7 13.9 0.82 0.02 1.9 4.1 1.09 0.02 0.5 1.1 1.54 0.01 1.3 3.8 1.55 0.01 0.9 2.8 Table 1: assessment of new misorientation analysis method.

[010]SEM towards [100]SEM

10°

30°

50°

70°

new direct method conventional method

towards [001]SEM

Figure 5: scatter in misorientation axes calculated from orientations and measured directly using new method.


misorientation from the measured orientations in the conventional manner. This scatter is much reduced when the new direct method is used, there being only ~3.8° from the centre of the distribution to the extreme points. Figure 5 shows that the scatter in misorientation axes is not isotropic, and this was also observed in the other data sets examined. The reason for this is as yet unclear, though it could be a result of the positions of the four analysis regions on the EBSD detector relative to the direction of the misorientation axis. Table 1 indicates that the misorientation axes were on average only a 1° to 2° away from the mean misorientation axis when the misorientation angle was around 1°, though this increased to ~5° when the misorientation angle was reduced to ~0.25°. 5.

CONCLUSIONS

Measurement of local small misorientations due to the accumulation of dislocations within crystals provides a means of characterising deformation microstructures. This work has examined the use of EBSD in measuring such small misorientations and concluded that:  When misorientations are calculated from two orientations, the errors in the orientations cause increasing uncertainty in the misorientation axis as the misorientation angle is reduced. For a well calibrated EBSD system the misorientation axis is essentially undetermined once the misorientation angle falls below ~5°.  A new analysis method was presented in which misorientations are measured directly by using cross-correlation methods to determine the relative positions of identical features in pairs of EBSD patterns.  Repeated measurements from Si singles crystals using this new method showed that the standard deviation in measured misorientation angles was reduced to ~0.01° from ~0.2° using the conventional analysis. Furthermore the misorientation axes can be determined with little error (~2°) provided the misorientations angles are greater than ~0.5°. 6.

ACKNOWLEDGEMENTS

I am very grateful to the Royal Society for their continuing support through the University Research Fellowship scheme. 7. REFERENCES 1 Wilkinson A J & Hirsch P B Micron 1997, 28: 279-308 2 Humphreys F J J. Microscopy 1999, 195: 170-185 3 Lassen N C K, Juul-Jensen D, and Conradsen K, Scanning Microscopy 1992, 6: 115-121 4 Wright S and Adams B L Metall. Trans., 1992, 23A: 759-767 5 Dingley D J & Randle V J. Mater. Sci. 1992, 27: 4545 6 Bay B, Hansen N, Hughes D A & Kuhlman-Wilsdorf D Acta Metall. Mater., 1992, 40: 205-219 7 Hughes D A, Lui Q, Chrzan D C and Hansen N Acta Mater., 1997, 45: 105-112 8 Godfrey A, and Hughes D A Acta Mater., 2000, 48: 1897-1905 9 Liu Q, Maurice C, Driver J, and Hansen N Metall. Mater. Trans., 1998, 29A: 2333-2344 10 Godfrey A, and Juul-Jensen D Acta Mater., 1998, 46: 823-833 and 835-848 11 Humphreys F J and Brough I J. Microscopy 1999, 195: 6-9 12 Juul-Jensen D Mater. Sci. Tech. 2000, In press 13 Wilkinson A J Ultramicroscopy 1996, 62: 237 14 Wilkinson A J J. Electron Microscopy 2000, 49: 299-310