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Z. Kristallogr. Suppl. 23 (2006) 49-54
© by Oldenbourg Wissenschaftsverlag, München
Accurate absolute peak positions for
multiple {hkl} residual stress analysis by
means of misalignment corrections
Arnold C. Vermeulen
PANalytical, Lelyweg 1, 7602 EA Almelo, The Netherlands
e-mail: arnold.vermeulen@PANalytical.com
Keywords: residual stress, accurate positions, alignment errors, misalignment corrections
Abstract. For reliable triaxial or multiple {hkl} residual stress analysis results it is essential
that the peak positions are accurate in an absolute sense. In this paper a set of analytical
formulas describing the effects of peaks shifts due to remaining alignment errors is tested on
a stress-free reference specimen. With the misalignment parameters obtained software
corrections can be applied on measured peak position in stress measurements employing
combined tilting (χ-tilt and ω-offset).
Introduction
Managing alignment errors is the key to success in obtaining reliable XRD stress data. For
the classical sin2ψ residual stress analysis using relative peak positions, only two alignment
errors are relevant: specimen displacement and incident beam misalignment [1,2,3]. For
advanced residual stress analysis methods using absolute peak positions, other alignment
errors become relevant too [3]: zero beam shift, chi axis misalignment and omega axis
misalignment. Therefore, for triaxial stress analysis and multiple {hkl} stress analysis all
alignment errors must be taken into consideration. When employing focusing optics all of the
alignment errors must be determined and corrected for. The corrections can be done by
tuning the hardware (for large errors) [1] and/or by software (for the remaining errors) [2,3].
The issue of alignment errors in the field of single crystal diffraction with respect to crystal
centring [4], where data of reflection-antireflection pairs of the same {hkl} reflection can be
analysed, and/or with respect to angular misalignments (non-parallel or non-perpendicular
orientation) of the rotation axes [5] is also discussed. For these cases it is generally assumed
that the rotation axes and the incident beam intersect each other perfectly. However in a
powder diffractometer, spatial displacements of the rotation axes and the incident beam (i.e.
the above-mentioned misalignments) may occur and these are together with the displacement
of the specimen surface the main potential sources of error in residual stress analysis.
The complete procedure to obtain accurate absolute peak positions when applying combined
tilting (applying both χ-tilt and ω-offset) involves measuring peak positions of a calibration/reference powder specimen over the full 2θ and the full χ and (ω-θ) ranges. The peak
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shifts with respect to the theoretical peak positions are analysed with a multivariate linear
least squares (multiple linear regression) fitting procedure. Such a procedure was used earlier
by Convert & Miege (1991) [1] to analyse specimen displacement and incident beam
misalignment on a single reflection. Applying this on multiple reflections enables the
possibility to analyse all alignment errors and to include other sources of peak shift, like:
error in stress-free lattice parameters, thermal expansion, error in wavelength, transparency,
et cetera. The relevance of these errors will be discussed. The fit parameters obtained –
describing the remaining alignment errors – are used for a software correction of the
measured peak positions.
Theory
If the alignment errors of an instrument are known they can be used to obtain accurate
absolute peak positions by correcting the measured peak positions according to:
2θ corr = 2θ meas − ∆ 2θ ze − ∆ 2θ sp,χ − ∆ 2θ eq − ∆ 2θ ax − ∆ 2θ χ ,ω − ∆ 2θω ,2θ ,
(1)
where 2θcorr is the corrected angle, 2θmeas is the measured angle and ∆2θxx are the alignment
errors of the instrument. A full set of misalignment formulas has been presented earlier by
Vermeulen (2000) [3] for the omega-stress and chi-stress modes. Here an overview of a
generalised modular set of formulas for the combined tilts mode (i.e. employing both ω- and
χ-tilting) is presented in table 1. The same formulas will be the basis for a fitting procedure
to determine the misalignments on a series of measurements on a stress-free reference
specimen.
Two additional errors, the transparency error (see also [6]) and the reference error, both of
which may play an additional role in either the determination of or the corrections for the
Table 1. Generalised set of peak shift formulas for zero beam shift (∆2θze), specimen displacement
w.r.t. chi axis (∆2θsp,χ), incident beam misalignments (equatorial ∆2θeq and axial ∆2θax), and axis
misalignments (chi w.r.t. omega axis ∆2θχ,ω and omega w.r.t. 2theta axis ∆2θω,2θ ) for combined tilts
mode, which includes both omega-stress mode (χ=0) and chi-stress mode (ω=θ). With goniometer
radius R.
Error
∆2θze (°)
Function
2θ ze
∆2θsp,χ (°)
hsp , χ f sp
∆2θeq (°)
heq f eq
∆2θax (°)
hax f ax
∆2θχ,ω (°)
hχ ,ω f χ ,ω
∆2θω,2θ (°)
hω , 2θ fω , 2θ
Combined tilts mode
2θ ze
180 2
sin θ
cos θ
hsp, χ
π R sin ω cos χ
180 2 sin(ω − θ )
cos θ
heq
π R sin ω
180 2 sin θ sin χ
hax
cos θ
π R sin ω cos χ
180 2 sin θ
cos θ
hχ ,ω
π R sin ω
180 2
cosθ
hω , 2θ
π R
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(2)
(3)
(4)
(5)
(6)
(7)
Z. Kristallogr. Suppl. 23 (2006)
51
Table 2. Additional set of formulas to correct peak positions for an isotropic error in the reference
(∆2θref) and for transparency (∆2θtr). With cubic lattice parameter a, wavelength λ, linear expansion
coefficient α, temperature T and linear absorption coefficient µ.
Error
Function
∆2θref (°)
ε ref f ref
Combined tilts mode
180
ε ref
2 tan θ
π
(8)
εref=∆a/ao, εref=∆λ/λ, or εref=α∆Τ
µ −1
∆2θtr (°)
−1
µ f tr
f tau =
− 180 2 f tau sin θ cos θ
π
R sin ω cos χ
(9a)
sin 2 θ − sin 2 (ω − θ )
cos χ
2 sin θ cos(ω − θ )
(9b)
alignment errors, are presented as examples in table 2. For the reference error only the
isotropic term is included as occurs for cubic materials. In principle anisotropic terms can be
added for use with non-cubic reference materials.
Procedure
The misalignment analysis is applied here in a generalised form using a multivariate linear
least squares (multiple linear regression) fitting procedure (see also [1]):
2θ meas , j − 2θ ref , j = 2θ ze + hsp , χ f sp , j + heq f eq , j + hax f ax , j + hχ ,ω f χ ,ω , j + hω , 2θ fω , 2θ , j , (10)
where 2θze is a constant and fsp, feq, fax, fχ,ω and fω,2θ are goniometric functions describing the
peak shift due to specimen displacement, equatorial beam misalignment, axial beam misalignment, chi axis misalignment and omega axis misalignment, respectively. Table 1 gives
an overview of the above functions and the linear misalignment parameters, hsp,χ, heq, hax,
hχ,ω, and hω,2θ.
In order to obtain a meaningful set of misalignment parameters, attention must be paid to
their specific dependencies on the angles θ, ω and χ. Hence in the measured set of data the
following angle ranges must be included:
• Positive and negative χ-tilt angles to separate hax and hsp,χ.
• Positive and negative ω-offset angles to separate heq and hsp,χ.
• A large χ range to separate hsp,χ and hχ,ω.
• A large ω range and 2θ range to separate hχ,ω and hω,2θ.
• A large 2θ range to separate hω,2θ and ∆2θze.
Experimental
The experiments were performed on a PANalytical X’Pert PRO MRD system with a
horizontal goniometer and an XYZ-stage mounted on a half-circle Eulerian cradle. The prealigned fast interchangeable PreFIX optics modules for point focus geometry were used.
All measurements were performed on a stress-free Au powder specimen. We measured the
(111), (200), (220), (311), (331), (420), (422) and (511)/(333) reflections with positive and
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negative tilts over the full measurable range (up to sin2ψmax= 0.1 … 0.8 for omega-stress and
sin2ψmax= 0.9 for chi-stress). In one additional series of measurements combined tilts were
applied (omega-stress series combined with χ=30°). Finally in a series of measurements an
artificial specimen displacement of +200 µm was introduced with the z-movement of the
XYZ-stage. In all series the tilts were applied with sin2ψ steps of 0.1.
Results
The multiple linear regression analysis based on equation (10) is firstly applied to a large
data set consisting of a series of χ-tilt measurements and a series of ω-offset measurements
to obtain a full description of the alignment condition of the used diffractometer system. An
additional parameter for the reference error (see equation (8) in table 2) is added to equation
(10) in order to refine the strain-free lattice parameter of the Au reference powder (PDF 040784: ao=4.07860 Å). A correction for transparency (see table 2) was omitted since it would
90
(a)
60
30
o
0
χ( )
-30
-60
-90
10
30
50
70
90
110
130
150
170
110
130
150
170
o
2θ ( )
90
(b)
60
30
o
ω−θ ( ) 0
-30
-60
-90
10
30
50
70
90
o
2θ ( )
Figure 1. Graphical results of a simultaneous fit on (a) a series of χ-tilt measurements and (b) a series
of ω-offset measurements. The relative 2θ peak displacements are magnified with a factor of 100 for
more clarity. See table 3 for the numerical results.
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Z. Kristallogr. Suppl. 23 (2006)
53
Table 3. Results of multiple linear regression analysis with equation (10).
Parameter
Data set
Analysed
∆2θze (o)
hsp,χ (µm)
heq (µm)
hax (µm)
hχ,ω (µm)
hω,2θ (µm)
0.026 (4)
-45 (3)
-50 (9)
121 (2)
-51 (12)
0
Example 1:
Combined tilts
0.026
-45
-50
121
-51
0
Example 2:
Displaced specimen
0.026
149 (1)
-50
121
-51
0
aAu (Å)
4.07834 (2)
4.07830 (1)
4.07827 (1)
σ(2θ)meas (ο2θ)
σ(2θ)fit (ο2θ)
0.0372
0.0055
0.0256
0.0098
0.0817
0.0075
have no significant effect. It was assumed that the ω axis and 2θ axis coincide (hω,2θ=0).
In the analysis the data points whose individual deviation was too large were successively
removed from the analysed set starting with the data point with the largest deviation until the
remaining data points fit within a range of –4 and +4 times their average standard deviation.
The removed data points had typically, originated from reflections showing a distorted
profile shape due to too large defocusing effects (i.e. too close to the theoretical value for
sin2ψmax).
The results are shown in table 3. Figures 1a and 1b show the graphical results. Note the
significant reduction in standard deviation before and after applying the software correction
in the last two rows of table 3.
90
60
30
o
ω−θ ( ) 0
-30
-60
χ = 30
-90
10
30
o
50
70
90
110
130
150
170
o
2θ ( )
Figure 2. Graphical results of a fit on a series of combined tilt measurements (ω-offset combined with
fixed χ=30o) using the analysed parameters (see figure 1) as fixed fit parameters. The only free fit
parameter is the reference error (i.e. lattice parameter). The relative 2θ peak displacements are
magnified with a factor of 100 for more clarity. Note that the largest changes with respect to figure 1b
occur at low 2θ angles and high ω-offsets. See table 3 for the numerical results.
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90
60
30
o
χ( )
0
-30
-60
-90
10
30
50
70
90
110
130
150
170
o
2θ ( )
Figure 3. Graphical results of a fit on a series of measurements on the artificially displaced specimen
(∆z=+200µm) using the analysed parameters (see figure 1) as fixed fit parameters. The only free fit
parameters are the specimen displacement (∆hsp=+194µm) and the reference error (i.e. lattice
parameter). The relative 2θ peak displacements are magnified with a factor of 100 for more clarity.
See table 3 for the numerical results.
In a second step the parameters obtained from the first analysed set were applied on two
example sets of data both with an unique collection of data points measured on the same
specimen but under different conditions. In the first example a set with combined tilts is
tested against the set of parameters of the analysed set (see table 3 and figure 2). In the
second example a set originating from a deliberately displaced specimen (+200 µm) is tested
against the analysed parameters with the displacement parameter as additional free fit
parameter (see table 3 and figure 3).
Conclusions
With the method presented in this paper a full set of alignment errors are quantified, which is
relevant for measurements over the full 2θ range and when applying χ-tilt or ω-offset or
combined tilting. The method can be used as part of an iterative procedure to improve the
alignment of the hardware. Alternatively the results can be used for a software correction
procedure in order to obtain reliable absolute peak positions. This will make it possible to
perform accurate triaxial or multiple {hkl} residual stress analysis.
References
1.
2.
3.
4.
5.
6.
Convert, F. & Miege, B., 1992, J. Appl. Cryst, 25, 384.
Vermeulen, A.C. & Houtman, E., 2000, Mat. Science Forum, 347-349, 17.
Vermeulen, A.C., 2000, in Proc. ICRS-6 conf., Oxford, UK, pp. 283-290.
King, H.E. & Finger, L.W., 1979, J. Appl. Cryst, 12, 374.
Dera, P. & Katrusiak, A., 1999, J. Appl. Cryst, 32, 193.
Vermeulen, A.C., 2001, Mat. Science Forum, 378-381, 166.
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