Practical accuracy of grain misorientation measurements by Kikuchi line technique

ultramicroscopy ELSEVIER

Ultramicroscopy 60 (1995) 207-218

Practical accuracy of grain misorientation measurements by Kikuchi line technique A. Gemperle, J. Gemperlovii Institute of Physics AV CR, Na Slovance 2, 180 40 Praha 8, Czech Republic

Received 12 January 1995; in final form 16 May 1995

Abstract

A practical procedure for crystal orientation determination from electron diffraction patterns using the computerized method of Heilmann et al. is described which is believed to give the most accurate results. It is shown that more than three reflections are generally needed. The accuracy of the orientation measurement may be increased if in addition to the direction of the Kikuchi lines the directions of the lines connecting the diffraction spots with the pattern centre are measured as well. It is expected that the grain misorientations can be established to accuracy of f0.1” in magnitude of angle and 50.2” in the direction of the rotation axis.

1. Introduction

Physical properties of polycrystals are governed by both grain interiors and grain boundaries (GB). As GBs have a greater propensity for segregation, diffusion, strain and defect accommodation, they modify material properties significantly. To give a qualified explanation of their influence the detailed knowledge of the structure of GBs is necessary. GBs are characterized by the axis and the angle of misorientation and the orientation of the GB plane. The axis and the angle of misorientation are the necessary data for categorisation of GBs [l]. Certain specific combinations of axis and angle result in coincidence of some lattice points of both grains and give rise to the coincidence site lattice (CSL). The GB properties depend strongly on 2, the ratio of the volume of the CSL unit cell to the volume of the lattice unit cell. The distribution of 2 in polycrystals is very different from the random case. The

special boundaries with low 2 have been observed frequently [2-41 and they exhibit very often special properties. With a small angle of deviation from the CSL these properties may change significantly; e.g. Aleshin et al. [5] have studied the dependence of diffusion along the boundary on the deviation angle. A detailed knowledge of the microstructure of GBs is desirable for a better understanding of their properties and of the dependence of the properties on the deviation from the exact coincidence. The atomistic structure of a GB may be studied by HREM (e.g. Ref. [6]) in cases of high angle coincidence boundaries especially with low Z. HREM is applicable when there is a zone axis parallel in both grains. Only pure tilt boundaries can be studied. In this case the mutual misorientation of both grains can be determined with 0.5” accuracy [7]. To study the dependence of the physical properties on the GB structure bulk bicrystals are

0304-3991/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDZ 0304-3991(95)00072-O

208

A. Gemperle, J. Gemperloua’ / Ultramicroscopy 60 (1995) 207-218

suitable. In bulk metal bicrystals there is always a certain deviation from the CSL orientation, originating during the growth of bicrystals. The additional rotation axes are neither tilt nor twist axes, but rather they are random. The deviation can be described by a small rotation supplementing the theoretical one characterizing the special low 2 GB. As is the case for low angle boundaries, dislocation arrays are able to give rise to a deviation. The dislocation structures of GBs are very intricate (e.g. Ref. [8]) and they depend significantly on the type and the magnitude of the deviation. For random deviations they may be studied only by conventional TEM. To analyse the dislocation structure contrast experiments in various reflections are performed. However, they are very often ambiguous and a comparison with theoretical models is necessary for Burgers vector determinations. It is very important to establish the local angle and the local axis of the additional rotation with the highest attainable accuracy. The former should be at least rfr0.1” in magnitude, while the latter should lie in a cone with vertex angle 10”. The optimum available techniques for misorientation measurement in conventional TEM are selected-area diffraction (SAD) with a small diaphragm and convergent-beam diffraction (CBED). The advantage of SAD is the straightforward switching from imaging to diffraction and therefore SAD is more simple to use. Both techniques depend on the analysis of individual Kikuchi or Kossel line patterns. Several papers were published treating theoretically the accurate determination of crystallographic orientation from Kikuchi patterns or from spot patterns [9-131. No systematic study treating the inaccuracies caused by the real possibilities of measuring the geometrical parameters of the patterns has been performed yet. The purpose of this work is to determine the limits of the accuracy in grain misorientation measurement by this technique and to propose a procedure which is likely to produce the best results. 2. Experimental The measurements were performed on Cu Z5 [001]{310) bulk bicrystals. Foils were prepared

Fig. 1. Diffraction pattern with sharp Kikuchi lines. Measured dark Kikuchi lines are marked by arrows and denoted by numbers corresponding to Table 1, grain A. Some steps on KL are also marked.

with a boundary inclined 45” to the surface. The foil thickness was in the range 150-250 nm. SADs were taken from both grains as close as possible to the GB. The conditions were: 0.4 pm aperture diameter, 500 mm camera length, 200 kV accelerating voltage in a JEM 2000FX microscope. Typical Kikuchi line (KL) diffractograms are shown in Fig. l,Figs. 2 and 3. Generally to determine the orientation of a crystal the following three parameters must be measured on at least three Kikuchi pairs: the spacing d of a Kikuchi pair, the distance D from the origin to the trace of the corresponding Bragg plane and the angle between the Kikuchi pair and the reference axis xR parallel to the long side of the small rectangle which frames the plate number (Fig. 4). To achieve the results as accurately as possible, the diffractograms were normally exposed with the centre covered by the beam stop and the second very short exposure was taken without the beam stop. The parameters were measured using a workshop microscope. The magnification 4 x was found to be optimum. The distances were measured with 0.01 mm precision, but the estimated

209

A. Gemperle, J. Gemperloua’ / Ultramicroscopy ii0 (1995) 207-218

ZP,

Fig. 2. Diffraction pattern with diffuse Kikuchi lines. Measured dark Kikuchi lines are marked by arrows and denoted by numbers corresponding to Table 4, grain A. Bright line was measured for reflection 6.

ZR

Fig. 4. The three coordinate systems used: crystallographic (100); pattern connected with KL- xP( I KL), y,(llfCL), ZP; and reference xp, yR and zR connected with diffractogram. DKL denotes the dark Kikuchi line, BKL the bright one and SL the spot line (/(xP) connecting the spot (S) and the pattern centre.

uncertainty is about +O.l mm. The angles may be measured with 3’ precision. Besides the angular position of the Kikuchi pair (preferably only the line nearer to the centre was considered), the angular position of the line connecting the corresponding diffraction spot and the centre of the pattern (spot line - SL) was also measured with respect to the reference line. The accuracy of the angular measurement will be discussed later. In taking the diffraction patterns, the specimen was slightly inclined to obtain at least 5 well defined Kikuchi pairs in the photograph in both grains. The inclination must not be changed during the exposure of both the diffractograms. Larger inclination may deform the angular position of the spots [14].

Fig. 3. Diffraction pattern, in which the diffraction spots are more reliable than the Kikuchi lines. Measured dark Kikuchi lines are marked by arrows and denoted by numbers corresponding to Table 7, grain A.

3. Theoretical To determine crystal misorientation from Kikuchi lines the computerized method by Heil-

210

A. Gemperle, J. Gemperlovd/

mann et al. [13] was adopted and modified. First the method indexes the Kikuchi lines. The Miller indices hkl of the corresponding Bragg planes are found from the measured widths and the angular positions of triplets of Kikuchi pairs. The number of possibilities can be reduced when the hkl types of some Kikuchi pairs can be easily recognized in the pattern. Next the direction BD parallel to the beam direction is determined. To determine BD, both the distances D of the traces of the Bragg planes from the pattern centre and the established hkl’s are needed. The BD is determined from the relation (Eq. (4) in Ref. [13]) hkl

D

(1)

BD’lhkll=-Jm’

where L is the camera length. The distance D cannot exceed 40 mm, as the dimensions of the plate are 65 X 90 mm. The commonly used L is 500 mm, therefore D/L c 0.08. Using the approximation l/ 7 1 + (D/L) can rewrite Eq. (1): L.BD.-=

hkl lhkll

-D

= 1 - i
we

+ $D( D/L)~.

The second term on the right-hand side can attain 0.128 mm in the worst case. The uncertainty of the measurements of D is 0.1 mm. Therefore we neglect the second term and the relation has the simple form L.BD.-=

The rotation matrix R,, from the crystal to the pattern system is given by (Eq. (7) in Ref. [13]) R,,=

j (BDB;:;BD).

The angle between xn and xP is the measured angle y between the Kikuchi pair and the reference axis. The rotation about BD by the angle y transforms the pattern system to the reference one. The corresponding rotation matrix R,, is (Eq. (8) in Ref. [13]) Rrn=[:;;

r;;

;),

where y is equal to (Y~or to 90” - (Ye (Fig. 4). For each Kikuchi line we obtain the matrix R,, (Eq. (9) in Ref. [131), which describes the rotation from the crystal system to the reference system &Xi = RP, Rc,. An average of the available rotation matrices is used to find the best rotation matrix. The reference system is the same in both grains, whereas the crystal systems are different in grains A and B. Each grain is treated independently and two matrices RA and R, are established. The misorientation of the two crystals in the crystallographic system of A grain is given by the matrix M = R;l RA.

hkl lhkll

Ultramicroscopy 60 (1995) 207-218

-D’

The three unknown components of L . BD can be found by use of singular value decomposition (SVD) method (Section 15.4 in Ref. 1151). When more than three Kikuchi pairs are available the L *BD is determined with better accuracy. The BD and hkl are vectors in the crystal system with axes of (100) type in the cubic crystals. The second coordinate system (Fig. 4) is the reference system xn, y,, zn; zn is the pattern normal parallel to BD. The origin is at the pattern centre. The third system used is the pattern system: xp is perpendicular to the KL, yP parallel to it, and zP is the pattern normal, z,IlBD.

(4)

The matrix M represents the rotation from the crystal system of the first grain A to the reference system, and from the reference system to the crystal system of the second grain B. The unit vector of the misorientation axis a and the angle cp are given by a =

CM32

cp =

arccos(+(M,,

-M23Jf13

-%J42l

-Ml,),

+M22 +M33 - 1)).

(5)

The misorientation matrix M may be decomposed into the rotation corresponding to a special boundary S followed by a small additional rotation D M=DS,

D = MS-‘.

A. Gemperle, J. Gemperlod

The additional RB1 R*S-1.

rotation

D is then

/ Ultramicroscopy 60 (1995) 207-218

given by

4. Analysis of the attainable accuracy First, we determine hkl’s of the Kikuchi pairs in each grain. The next step is the determination of the beam direction BD. The third step is the determination of the orientation of the crystal system with respect to the reference one. When the orientations of both grains are established, their misorientation can be calculated. The hkl’s are determined from the width and angular positions of the Kikuchi pairs. The accuracy of the measured widths and the angular positions need not be high, t_ 1 mm for the width and ~2” for the angle are satisfactory. The correct hkl must fulfil the following conditions: the camera length calculated must correspond approximately to the experimental value and the reference system must be very nearly the same for all reflections. The BD is determined from (31, its accuracy depends only on D. (As mentioned earlier the uncertainty of the measurement of D is fO.l mm and it is not appreciably dependent on the contrast and sharpness of the Kikuchi lines.) All measured Kikuchi pairs are to be considered. The number of equations hardly exceeds 10. If enough data are not available, we adopt the bootstrap method (Section 15.6 in Ref. [15]). With computer-generated random numbers with a normal (Gaussian) probability distribution we construct hypothetical data sets with exactly the same number nhkl of measured D values, which differ from the actual data set by random measurement errors. We solve (3) using SVD and obtain slightly different BD values. If we simulate enough data sets and obtain enough derived simulated BD,, these BD, will be distributed around the mean value BD, in close to the same way that BD is distributed around BDtrue. The standard deviation of BD is calculated as

211

Table 1 The mean deviations Ai for measured (Ye, as and averages E (the example of the diffraction patterns with sharp Kikuchi lines; see Fig. 1) No. of reflection Grain A A” Ah AS AS &+s

I

1

2

3

4

S

6

0.1121 0.0726 0.4123

0.3056

0.1330 0.0683 0.2666 0.1625 0.1851

0.1732 0.1928 0.1931 0.1110 0.0938

0.1761 0.1053 0.1422 0.1535 0.1100

0.1325 0.0875 0.1948 0.1499 0.1251

0.1210 0.3152

0.1550 0.2186

0.1910 0.2080

0.1813 0.1248

0.1901

0.1754

0.0549 0.1905

0.0482 0.1231

0.1686

0.1845 0.1341 0.1736

0.1424 0.1441 0.0405 0.1394

0.1418 0.1470 0.0509 0.1356

Grain B ;Y AS &+s

1

where N is the number of simulations and nhk, 3 is the number of degrees of freedom. The accuracy of the orientation with respect to the reference system depends on the accuracy of measurement of the angular position of the reference line and the angular positions of the Kikuchi lines. With the applied measurement technique the reference line may be determined with 3’ accuracy. The measurement of the angular position of the Kikuchi line (Ye depends on several factors. First on the quality of the photograph. Some of the KL may have diffuse character (Fig. 2) especially if there is some contamination or large thickness of the specimen. The KLs generally are not straight lines but they exhibit steps and bends on crossing other Kikuchi pairs, see e.g. Fig. 1. Therefore they often have no one single direction along their whole length. For a sharp line with good contrast the inaccuracy of the direction determination may be lower than 0.2” even for a line with steps, if we measure it on segments between them (see Table 1). Although even for diffuse lines the standard deviation in angular measurement is generally small (it was in the range 0.07”-0.4” in 8 measurements for 35 measured lines and it was in no correlation with the quality of the diffraction pattern), the uncertainty of the direction determination is several times larger and in some cases as high as lo-1.5” (see Tables 4 and 7).

212

A. Gemperle, J. Gemperlova’ / Ultramicroscopy 60 (I 995) 207-218

To obtain the best results we use the following procedure. To improve the accuracy of the measurement it is recommended that the angular position of the IU czKL and the angular position of the line connecting the diffraction spot to the pattern centre (Y, ((Ye + (Y,= 90°> be measured. The accuracy of the angular position of the spots is generally better than for the lines. For sharp spots the inaccuracy is lower than 0.2” and even for diffuse spots it is only exceptionally higher than 0.5” (see Tables 1, 4 and 7). Thus we have two independent sets of measurements. The measured angles aij = Iai - aj( at which KL, meets KL, differ from those calculated using hkZi and hklj. For each KL we calculate the mean deviation as the sum of these differences per one line

i+j

The same procedure is used for a, and the average value of both measurements (Y= (a, CX,+ 90”)/2. In Table 1 the examples of well defined KLs and spots are presented (Fig. 1 for grain A). We consider a line or a spot reliable when the average deviation is smaller than 0.2”. In grain A this is valid for all reflections except reflection 2 in the IU series and reflection 1 in the spot series. In grain B this is valid for all reflections except reflection 3 in the spot series. However, for the averages Cuall reflections are reliable. Therefore for the determination of the matrix R,, we consider all reflections in both grains and the average values of the angular position E are used. In Table 4 examples of very diffuse IUs (Fig. 2, grain A) are presented. The deviations in all three series in both grains exceed 0.2”. No single reflection can be taken as appreciably more reliable than the others except reflection 7 in the spot series of the grain B, which is significantly worse than the others. We discard it. The deviations in the spot series are smaller than in the series Kikuchi lines. However, according to Table 6 there is not much difference in the results between them. Even in this case it is best to consider all average values. In Table 7 an example is presented where the spot series is substan-

tially better than the IU series. In the grain A all five reflections are equivalent. In the grain B in the KL, series the deviation of reflection 2 greatly exceeds the deviations of the other reflections, and in the spot series it is similar for reflection 3. However, even after exclusion of reflection 2 of A” and reflection 3 of the ASPoT, the spot series is substantially better. This is supported also by calculation of the scatter in direction of total rotation axis. If we calculate the scatter in direction from the stds presented in Table 9, we obtain the value 0.37” if IUs are used, but 0.13” only if the spots are adopted. For the determination of the matrix RCR we use the spot series without reflection 3 in B. In Table 3, 6 and 9 the misorientation axis/angle pairs are presented for both total and supplementary rotations. In the last column of supplementary rotations the angles between mean supplementary rotation axis obtained from KL and the axis obtained from the other data combinations are given. The overall errors in misorientation measurements are also given in Tables 3, 6 and 9. Using bootstrap method the errors in L . BD were calculated. The measured accuracy in distances D

Table 2 Dependence of the accuracy of grain orientation determination on D expressed in BD and on y expressed in xR; std are standard deviations of direction cosines; pattern pair Table 1 a, Grain BD std

a2

a.3

A

X,&L) std XR(m)

exclrefl ?

std x.(KL+S) std Grain B BD std x&u std X,(s) exclrefl 3.4 std x,KL+S) std

0.1019 0.0016 0.3436 0.0012 0.3425 0.0008 0.3432 0.0010

0.4904 0.0003 0.7991 0.0005 0.7995 0.0003 0.7993 0.0004

0.8655 0.0002 - 0.4932 0.000 1 - 0.4933 0.0001 - 0.4933 0.0001

0.5939 0.0011 - 0.0408 0.0008 - 0.0403 0.0003 - 0.0396 0.0009

0.5024 0.0004 0.7988 o.Ouo3 0.7986 0.0001 0.7983 0.0004

0.6283 0.0013 -0.6001 0.0005 - 0.6005 0.0002 - 0.6009 0.0005

A. Gemperle, .I. Gemperloua’ / Ultramicroscopy 60 (199.5~ 207-218

Table 3 The misorientation

axis/angle pairs in crystallographic system of grain A calculated from the pattern pair Table 1 al

Total rotation KL std m A ccl refl 2

- 0.00947 0.00091 -0.01015 0.00081 -0.01170 0.00115 - 0.01127 0.00079 - 0.01060 0.00086

std s std SA excl refl 1 B excl refl 3.4

std KI+s std average Supplementary KL a A excl refl2

213

a2

a,

cp

0.99947 0.00149 0.99950 0.00137 0.99958 0.00182 0.99956 0.00134 0.99953 0.00144

0.03117 0.00158 0.02978 0.00134 0.02655 0.00215 0.02745 0.00129 0.02885 0.00148

37.197” 0.082” 37.163” 0.078” 37.085” 0.093” 37.107” 0.077” 37.141” 0.080

L(a,,a)

rotation

0.25147

0.96752

1.230"

-

0.00641 - 0.09177

0.23396 0.18730

0.97222 0.97801

1.182” 1.076”

0.06638 -0.02957

0.20076 0.22142

0.97739 0.97473

1.105 1.150"

0.02577

s SA excl refl 1 B excl refl3,4

-

KL+S

7.70 6.06

The mean values of a and cp presented were calculated using all measured KL and spots; std are their standard deviations. The angle between axis determined from all IUs, ax,_ and other a is given in last column.

nation of xn are also listed in Tables 2, 5 and 8. The expression for the probable error of a function were used consistently in all formulas. Suppose the x, y,... have been obtained with standard deviations 6,, 6,, ... and they are combined to give function f(x,y,...). Then its proba-

was 0.1 mm, 500 simulations of original data set, (D = D & 0.1 rnd) were performed. The errors in BD, ABD = A(L *BD)/L are in the third row of the matrix of errors 6&n (the components of BD are in the third row of the matrix &a). Their magnitudes are listed in Table 2, Tables 5 and 8. The first two rows of the matrix 6&n represent the errors of the average of the rotation matrices obtained for individual Kikuchi pairs. They represent the errors in arc,_. The errors in the determi-

ble error is S = (s$f/ax)’ + (6,af/ay)’ + ... . Following this law the errors SMij of the elements Mij of the misorientation matrix M (4)

Table 4 The mean deviations Ai for measured are, as and averages E (the example of the diffraction patterns when all Kikuchi lines are diffuse; see Fig. 2) No. of GrainA A”

AS &+s 1

1

2

3

4

5

6

0.6343 0.4528 0.5438

0.4469 0.3595 0.4029

0.6886 0.4706 0.5761

0.4186 0.4713 0.4412

0.3831 0.5391 0.4034

0.6389 0.3280 0.4252

0.7070 0.4985 0.3144 0.6388 0.4719

0.8156 0.3957 0.3658 0.4873 0.3989

0.6507 0.4491 0.3141 0.4767 0.5077

1.0907 0.3139 0.2824 0.5683 0.6393

0.4522 0.3795 0.4235 0.3586 0.3787

0.5909 0.4958 0.3156 0.5377 0.3571

7

Grain B A”

A$ A+ &+s A;CL+s

0.8560 0.8527 0.7726

A. Gemperle, J. Gemperlova’ / Ultramicroscopy 60 (I 995) 207-218

214

were computed from the knowledge of error matrices (6R,jij and (8R,)ij: 6Mij=

\ilCi=l(CsRB)fiCR*)S,f

Table 5 Dependence of the accuracy of grain orientation determination on D expressed in BD and on y expressed in x,; std are standard deviations of direction cosines; pattern pair Table 4

(RB)fi(8HA)fj)e

(As the matrices RA and R, are matrices of direction cosines, they are orthogonal and R-’ = R.) From 8yij the errors in the misorientation axis/angle pair (5) were calculated.

5. Discussion It is supposed that D can be measured accurately to %-0.1 mm. Even if we increase this limit to k 0.2 mm, the angular deviation corresponding to it, e.g. for Cu, 500 mm camera length and 200 keV electron energy, is k 1.37 min for any reflection. The analysis of the standard deviation (std) for the BD determination from at least 5 reflections in the 6 patterns contained in Tables l-9 indicated values corresponding to BD lying in a cone with a vertex angle 2-4 min. In contrast the measurement of the angular position of the reference system with respect to the pattern system (the angle y) may be much less accurate. From the comparison of Tables 1, 4 and 7 it follows that the quality of the diffraction patterns has an appreciable influence on the accuracy of y. However even for rather sharp Kikuchi lines a direction may be measured which evidently surpasses the accepted confidence limit A(Y= 0.2”, e.g. line 2 in Table 1. (The confidence limit is approximately equal to the mean value of the std for the measurement of the angular position on individual Kikuchi lines). Line 2 in Fig. 1 appears slightly bent. In many experimental cases the evaluation of diffraction patterns having rather diffuse Kikuchi lines cannot be avoided. In these the probability to measure an incorrect line direction is much higher (e.g. Fig. 2). As the std of the line direction measurement is nearly the same for sharp and diffuse lines, the large error in the second case must be due to some systematic effect. The cause of this is not quite clear. One of the alternatives is that on a sharp line with steps the direction of individual segments between them can be measured, whereas on a diffuse line the

al

a2

a3

0.6471 0.0009 - 0.4363 0.0025 - 0.4326 0.0021

0.2835 0.0010 - 0.6266 0.0029 - 0.6277 0.0024

0.7077 0.0012 0.6475 0.0012 0.6471 0.0009

0.3024 0.0002 - 0.6196 0.0039 - 0.6244 0.0018 - 0.6210 0.0026

0.1602 0.0002 0.7820 0.0032 0.7780 0.0015 0.7809 0.0023

0.9396 0.0001 0.0661 0.0018 0.0683 0.0008 0.0667 0.0012

Grain A

BD std x,KlJ std x.&L+S) std Grain B

BD std x,wJ std xp,w exclrefl7 std xR (KL+S) std

steps are invisible and the line appears deviated as a whole. If the KL in the diffraction pattern are diffuse, it is advantageous to measure the angular position of the spots, which fulfil in these cases better the theoretical angular relations between diffracting planes (see Tables 4 and 7). This measurement is theoretically less accurate because the spot pattern can be distorted by the shifts due to the inclination of the foil with respect to the electron beam [14]. The shifts can be calculated from the diffraction vector g, the deviation parameter s and the angle between the foil normal and the beam direction. The angular shift E is maximum for g parallel to the tilt axis of the foil. We take as the limiting cases first equal excitation g and -g and second the dark KL passing halfway between the centre and the diffraction spot. It is easily calculated, that in the first case pi = Q/g or l1 = gph/2 (A is the electron wavelength) and in the second case pi < Ed < gPA. We take as an example the largest registered g corresponding to the reflection 551 in Cu and p = 5”. The upper limit for E is then 0.25”, which is approximately the accepted confidence limit. The systematic distortion due to the spot shifts will

A. Gemperle, J. Gemperlovh / Ultramicroscopy 60 (1995) 207-218

not limit seriously the accuracy for the inclinations up to 5”. Another potential disadvantage is the less accurate measurement of the angular position of the spot in comparison with the measurement of the KL direction. The accuracy depends on the measurement of the position of the spot and the pattern centre and increases with g. Both positions can be generally measured to kO.025 mm. As an example we take the smallest g corresponding to 331 in Cu (for smaller g we measure the position of the second- or third-order spot) and 500 mm camera length. The error in the angular position is then +O.lY, which is a reasonable value. (The std for eight measurements on 35 spots was from 0.05” to 0.W.) The much larger deviations from the theoretical angular relations in Table 4 must be due to some other systematic effect, possibly an asymmetrical deformation of the central spot caused e.g. by the passing of a KL,. Therefore it is recommended, that during taking of the diffraction patterns the Table 6 The misorientation

refl7

std Supplementary rotation KL KL3maxcp KL3mincp S S B excl refl7 KL+s K,_ + s B excl refl7

exact reflection position for any diffracting plane be avoided. These may degrade the measurement by large diffraction spots connected with KL and deformation of the central spot. Very large inaccuracies may arise by the use of only three reflections for orientation determination. This is best illustrated in Table 6. Two combinations of three lines were selected from the six measured lines in A and B, which gave maximum and minimum values of the total rotation cp. The difference between them is 0.654”, which is appreciably more than the required accuracy. Even much larger error arises in the determination of the direction of the additional rotation axis a. The two directions calculated for these two cases make the angle 75.4”. On the contrary for six reflections and large deviations from the theoretical angular relations in the series of KL and spots, the difference in additional rotation calculated from these two series is in the accepted accuracy limit +O.l” in magnitude of

axis/angle pairs in crystallographic system of grain A calculated from the pattern pair Table 4

Total rotation KL std KL3maxcp KL 3 min cp S std S B excl refl7 std KL+s std m + S B excl

215

al

a2

a,

cp

- 0.02023 0.00464 0.00356 - 0.04898 - 0.02314 0.00399 - 0.01953 0.00360 - 0.01980 0.00344 - 0.01627 0.00307

0.99951 0.00339 0.99933 0.99877 0.99949 0.00296 0.99952 0.00270 0.99952 0.00258 0.99953 0.00234

0.02370 0.00308 0.03640 0.00832 0.02214 0.00268 0.02407 0.00245 0.02393 0.00234 0.02582 0.00212

37.838” 0.253 38.143 37.489 37.798” 0.219” 37.842” 0.199” 37.833” 0.1890 37.882” 0.170

- 0.29044 0.30076 - 0.84164 - 0.37096 - 0.27110 - 0.27897 - 0.17916

0.63462 0.67406 0.30018 0.60784 0.63824 0.63609 0.65936

0.71617 0.67467 0.44892 0.70208 0.72053 0.71944 0.73017

1.498” 1.854” 1.918” 1.497” 1.496” 1.496” 1.510”

L(a,,,a)

34.55” 40.85” 4.93” 1.14”

The mean values of a and cp presented were calculated using all measured KL and spots, selected combinations of 3 KLs with the largest and smallest angles of rotation in both grains and in case when some reflections were excluded; std are the standard deviations of the mean values. The angle between supplementary rotation axis determined from all KLs, ato. and other a is given in the last column.

A. Gemperle, J. Gemperloua’ / Ultramicroscopy

216

the angle and 10” in the direction of the axis. Also for a large number of reflections distributed around the centre the systematic error due to the spot shifts by foil inclination is at least partially compensated. Further it follows from Tables 7-9 that the differences between additional rotation parameters obtained from five KL with large Ai and five spots with very small Ai (Ai smaller than 0.1”) are only slightly larger than the accepted accuracy limits. In contrast the exclusion of one reflection with Ai appreciable larger than all other reflections may substantially decrease the Ais and the overall error of the measurement and increase the reliability of the resulting misorientation values. However the successive exclusion of the reflections with the largest Ai will distort the final results. Alternative methods of accurate measurement of small grain misorientations are CBED (e.g. Ref. [2]) and the measurement of KL line shifts of common reflections [161. CBED is also based on evaluation of Kikuchi or Kossel line patterns. Some problems are the same as for SAD patterns e.g. not straight lines due to mutual interactions which influence the orientation of the pattern with respect to the reference system. Experimentally it is more complicated to take CBED than SAD on a selected area of the boundary. CBED is taken in a very small area and the pattern is not influenced by small orientation variations across it. The lines will be generally sharper than Table 7 The mean deviations A, for measured aKL, czs and averages Z (the example of the diffraction patterns when some Kikuchi lines are diffuse; see Fig. 3) No. of reflection

1

2

3

4

5

0.6512 0.1374 0.2559

0.7324 0.0944 0.3468

0.6064 0.0954 0.2874

1.4615 0.0907 0.7238

0.9312 0.0938 0.5027

0.3071 0.2220 0.2065 0.0618 0.2318

0.7249

0.2730 0.2212 0.4304

0.3641 0.1566 0.1476 0.0732 0.2135

0.3754 0.1930 0.1112 0.0607 0.2463

Grain A AKL Ak

A;cL+s Grain B AKL

A;u. Ai

AS A;(L+s

,

0.1617 0.0665 0.2973

0.2412

60 (I 995) 207-218

Table 8 Dependence of the accuracy of grain orientation determination on D expressed in BD and on y expressed in xR; std are standard deviations of direction cosines; pattern pair Table 7 al

a2

Grain A BD std x,Wu std XRW std

0.2376 0.0003 -0.5118 0.0059 - 0.5064 0.0009

0.4261 0.0007 - 0.7087 0.0040 -0.7125 0.0006

0.8729 0.0003 0.4853 0.0003 0.4857 0.0001

Grain B BD std x,&L) std exe,re” 3 x ,CS) std

- 0.4450 0.0010 0.7143 0.0018 0.7135 0.0004

- 0.7003 0.0010 0.0983 0.0023 0.0993 0.0005

- 0.5581 0.0006 - 0.6929 0.0015 - 0.6936 0.0003

a3

in SAD. In some cases e.g. the study of the dislocation structure of grain boundaries SAD characterizes better the orientation relation of both grains because it is taken from an area having approximately the same dimensions as has the area studied by microscopy. In CBED the possibility to obtain additional information by the measurement of the spots in the same pattern is lacking. The method of measurement of the KL shifts of the common reflections is experimentally more time consuming than the other two. It requires taking at least three patterns in the same area of the boundary for specified goniometer positions. For very small misorientation angles the method is surely more accurate than the evaluation of two SAD patterns on both sides of the boundary. Difficulties may arise if there is a rotation component of the misorientation lying in the plane of the pattern. The two lines will not be parallel. The authors also do not recommend the method for misorientations larger than 3”. Recently a method for small misorientation measurement was designed which uses the two goniometer tilt angles [17]. Both grains are successively adjusted with one zone axis aligned parallel to the beam direction and the goniometer tilt angles are registered. It is claimed that the

A. Gemperle, J. Gemperlovn’ / Ultramicroscopy 60 (1995) 207-218

Table 9 The misorientation axis/angle pairs in crystallographic system of grain A calculated from the pattern pair Table 7 at Total rotation KL std TUB CC’K” ’ std S std S B exclrefl3 std KI+s std

- 0.00964 0.00384 - 0.01194 0.00365 - 0.01428 0.00150 -0.01609 0.00105 - 0.01193 0.00220

Supplementary KL KLB CC Iefl’ s S B exclrefl3 KL+s

a3

cp

0.99891 0.00471 0.99903 0.00447 0.99914 0.00172 0.99921 0.00110 0.99903 0.00262

0.04572 0.00545 0.04236 0.00516 0.03893 0.00189 0.03626 0.00110 0.04237 0.00299

36.843” 0.267” 36.765” 0.254” 36.679” 0.096” 36.619 0.060” 36.755” 0.148

-0.03794 - 0.08643 - 0.14555 -0.19081 - 0.09244

0.99280 0.99516 0.98891 0.97711 0.99461

a2

rotation 0.11395 0.04686 0.02953 0.09402 0.04703

da

,,a)

1.693” 1.596” 1.511” 10.28 1.455” 14.85” 1.59P

The mean values of a and cp presented were calculated using all measured KL and spots if not indicated which reflections were excluded; std are the standard deviations of the mean values. The angle between supplementary rotation axis determined from KLs, ax,_, and other a is given in last column.

misorientation can be measured to +O.l”. However, it is doubtful that the tilt angles can be actually measured with such a high accuracy on a current goniometer. This is also supported by the fact that the author indicates 1” accuracy for the determination of the orientation of one grain. Kikuchi patterns may also be evaluated using automatic pattern recognition. Methods for fully automatic evaluation of backscattered Kikuchi patterns in SEM were developed in Ref. [18] and transmission Kikuchi patterns in Ref. [19]. In estimating the accuracy of the automated evaluation we proceed in the same two steps as for the manual procedure proposed in this paper. First we consider the determination of BD. In Ref. [19] it is claimed that for at least five sharp line pairs a consistent indexing is achieved for cubic crystals. According to our experience an apparently consistent indexing may be obtained from evaluation of 5 Kikuchi pairs (or even 6) and the resulting BD may be wrong by degrees without a careful check by the user. Under the assumption that

217

the final indexing is checked by the user (correct camera length and position of the reference system) the accuracy of BD determination will be the same as for the manual evaluation proposed in this paper. Next we consider the orientation of the Kikuchi pattern with respect to the reference system (~a). It depends on the accuracy of the measured angular position of Kikuchi lines with respect to the reference axis. It was shown in the preceding text that the Kikuchi lines are not straight lines and that they exhibit steps and bends. In manual evaluation the correct position of the line is determined by measurement of straight segments of the lines. In automatic pattern recognition the line is measured as a mean direction of the whole line, which is generally deviated from the correct position. The effect will be similar to the measurement in a pattern with diffuse lines (see e.g. Tables 4-6 and Fig. 2). This may be verified by comparison of Figs. 9a and 9b in Ref. [19]. For the 6 reflections common to both the figures we obtain the mean A: = 0.92” which is even more than for the diffuse pattern in Fig. 2 (Table 4). Another disadvantage of the automated evaluation is the disregard of the position of the spots. It follows from the example in Fig. 3 and Table 7, that in some cases the angular position of the spots is several times better defined than the position of the Kikuchi lines. It follows from Table 5 that for diffuse lines the inaccuracy in xa contributes about 70%-80% to the overall error in orientation determination. Therefore for sharp diffraction patterns the manual evaluation will be 2-3 times more accurate than the automated evaluation, but even for diffuse patterns the manual evaluation will be more accurate by the use of the diffraction spots. In cases when the larger errors can be tolerated the automated evaluation is preferred for its speed.

6. Conclusions The following procedure is recommended to attain the highest possible accuracy of grain misorientation measurement by evaluation of two SAD diffraction patterns on both sides of the

218

A. Gemperle, J. Gemperloua’ / Ultramicroscopy 60 (1995) 207-218

boundary using the computerized method of Heilmann et al.: The two diffraction patterns are taken in such a way that at least five Kikuchi pairs and the corresponding diffraction spots can be measured in each grain. The inclination of the foil should not surpass 5” and no diffracting plane should be in the exact reflection position. The beam direction is calculated from the largest possible number of reflections using the SVD method. To find the best rotation matrix from the crystal to the reference system in each grain the rotation matrices for individual reflections are averaged. The directions of Kikuchi lines and spot lines with respect to the chosen reference axis are measured. The reliability of the measured directions is tested by comparing the measured values with theoretical angular relations between the corresponding diffracting planes. The most reliable set of KLs and spot lines is selected for the calculation of the average. It is expected that e.g. the accuracy +O.l” in magnitude and a cone of directions with the vertex angle 10” in the direction of the axis can be achieved in the measurement of small additional rotations in low ,Z grain boundaries.

Acknowledgements This work was supported by the Grant Agency of the Academy of Sciences of the Czech Republic under the contracts Nos. 11084, 110404 and by the Grant Agency of the Czech Republic under the contract No 202/94/1177. One of the authors (A.G.) is grateful for the financial support of the Commission of the European Communities and to the Department of Materials of the University of Oxford, which made available its experimental facilities. Valuable comments on the text

by J.M. Penisson and J.M. Whelan are gratefully acknowledged.

References [II V. Randle,

The measurement of grain boundary geometry, in: Electron Microscopy in Materials Sciences Series, Eds. B. Cantor and M.J. Goringe (Institute of Physics, Bristol, 1993). 121D.J. Dingley and V. Randle, J. Mater. Sci. 27 (1992) 4545. [31 Y. Pan and L. Adams, Scripta Metall. 30 (1994) 1055. [41 Y. Yoshitomi, Y. Ushigami, J. Harase, T. Nakayama, H. Masui and N. Takahashi, Acta Metall. Mater. 42 (1994) 2593. [51 A.N. Aleshin, S.J. Prokofjev and L.S. Shvindlermann, Scripta Metall. 19 (1985) 1035. in: Dislocations 93, Microstructures and [61 J. Thibault, Physical Properties, Eds. J. Rabier, A. George, Y. BrCchet and L. Kubin (Scitec, Switzerland, 1994). (71 J.M. Pinisson, T. Nowicki and M. Biscondi, Phil. Mag. A 58 (1988) 947. [8] T. Vystavcl, A. Gemperle and J. Gemperlova, in: ICEM 13, Paris, 1994, Vol. 2, p. 195. [91 Th. Karakostas, G. Nouet, G.L. Bleris, S. Hagege and P. Delavignette, Phys. Status Sohdi (a) 50 (1978) 703. [lOI C.J. Ball, Phil. Mag. A 44 (1981) 1307. M.A. Korhonen and V.K. Lindroos, 1111H.O. Martikainen, Phys. Status Solidi (a) 75 (1983) 559. and P. Delavignette, Phys. Status [121 E.K. Polychroniadis Solidi (a) 75 (1983) 291. [I31 P. Heilmann, W.A.T. Clark and D.A. Rigney, Ultramicroscopy 9 (1982) 365. [141 P.B. Hirsch, A. Howie, R.B. Nicholson, D.W. Pashley and M.J. Whelan, Electron Microscopy of Thin Crystals (Butterworths, London, 1965) ch. 5.7. [151 W.H. Press, S.A. Teukolsky, W.T. Vetterling and B.P. Flannery, Numerical Recipes in Fortran (Cambridge University Press, Cambridge, 1992). Ph. Komninou, Th. Karacostas and 1161 E.K. Polychroniadis, P. Delavignette, in: EUREM 8, Budapest, 1984, Vol. 1, p. 537. 1171 Q. Liu, in: ICEM 13, Paris, 1994, Vol. 1, p. 927. [18] N.C.K. Lassen, D. Juul Jensen and K. Conradsen, Scanning Microsc. 6 (1992) 115. [19] S. Zaefferer and R.A. Schwarzer, Z. Metallk. 85 (1994) 585.