Instrument aberrations in a 4-circle powder

Z. Kristallogr. 222 (2007) 204–209 / DOI 10.1524/zkri.2007.222.3-4.204 # by Oldenbourg Wissenschaftsverlag, Mu¨nchen

Instrument aberrations in a 4-circle powder diffractometer Arnold C. Vermeulen* PANalytical B.V., Lelyweg 1, 7602 EA Almelo, The Netherlands Received January 16, 2006; accepted May 9, 2006

Misalignments / Instrument aberrations / Alignment errors / Non-ideal diffractometer / 4-circle powder diffractometer Abstract. A theoretical basis for the effects of instrument aberrations on the position of the diffraction peak in a 4-circle powder diffractometer is given. Peak shifts due to non-intersecting rotation axes, displacement of the incident beam and displacement of the specimen height are all described on the basis of pure analytical functions. These functions may be useful as software corrections on the measured peak positions in residual stress analysis. The theoretical derivation includes an analytical description of the position of the measurement spot in specimen coordinates. This may be useful for improving the instrument performance in micro-diffraction analysis.

Introduction Instrument aberrations have a large influence on the achievable accuracy of diffraction measurements. For a real, “non-ideal” diffractometer one can discuss instrument aberrations of various kinds. For example, the issues of crystal centring [1], or angular aberrations (non-parallel or non-perpendicular) of the rotation axes [2] are discussed for single crystal diffraction. In these cases it is generally assumed that the rotation axes and the incident beam intersect each other perfectly. However, for a 4-circle powder diffractometer spatial displacements of the rotation axes and the incident beam may occur in practice and a displacement of the specimen surface may also be present. Especially in the case of residual stress measurements, where peak position shifts are analysed, these spatial displacements lead to unwanted peak position shifts when measuring with focusing optics. The relevance of these instrument aberrations (alignment errors) to residual stress analysis is commonly known [3–6]. However a full theoretical basis for the derived alignment errors is not yet available. In this paper we will provide a complete theoretical basis for the behaviour of instrument aberrations (alignment errors) in a non-ideal 4-circle powder diffractometer. The theory is a pure analytical expansion of Wilson’s universal approach [7]. The central theme of the derivation is * e-mail: [email protected]

the fact that in the non-ideal case the irradiated spot is not located at the same location in 3-dimensional laboratory space. Variation in the specimen height in laboratory space is a cause of diffraction peak shift. These peak shifts will be described with simple goniometric functions of the rotation axes of the diffractometer. Another relevant issue is that the irradiated spot moves about the specimen surface (specimen space), which has the consequence that the information of different locations of the specimen is analysed. For small specimens in a microdiffraction set-up these movements have even more drastic consequences. Namely that the diffraction signal disappears if the movements of the diffraction spot are larger than the specimen size. This effect applies to both focusing optics and parallel beam optics. We will also describe these movements in specimen space. Finally, one can discuss corrections for the alignment errors. Obviously, to minimise large errors it is advisable to tune the hardware as much as is possible [3]. For small remaining errors one can apply software corrections on the peak positions [4, 5]. A new, theoretical alternative as proposed in this paper is to dynamically compensate the specimen height to prevent peak shifts during the measurements.

Theory Diffraction at a non-ideal spot In a focusing beam geometry the measured peak position is sensitive to various kinds of alignment errors. Wilson (1963) [7] derived the general relationship for an arbitrary deviation of the diffraction spot with respect to the ideal position (see Fig. 1). The following dependency for the diffraction peak shift, D2q, for a non-ideal spot is obtained (see also Appendix A for a complete and corrected derivation): 2zL ð1Þ D2q ¼ s cos q0 R ( ) ðxsL Þ2 þ 2ðyLs Þ2 sin2 q0 þ ðzLs Þ2 ½4 cos2 q0 3 cos q0 sin q0 R2 where xsL , yLs , zLs are the coordinates of the spot in the laboratory reference frame (L-frame) and R is the diffract-

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204

Specific sources of misalignment

Fig. 1. Diffraction at a non-ideal spot with diffraction angle 2q. Vector xs locates the position of the spot. Vectors T and Q are the corresponding incident and diffracted rays, respectively. Vectors S and R are the ideal incident and diffracted rays, respectively. The diffraction angle of the ideal spot in the origin of the laboratory reference frame (L-frame) is 2q0.

ometer radius. The L-frame is defined here such that the x L y L plane is parallel to the diffracting lattice planes. The x L-direction coincides with the 2q axis of the diffractometer, while the z L y L plane intersects the c axis of the diffractometer (see also Figs. 2 and 3).

In this paper we derive the relationship for specific alignment errors. In a non-ideal situation both the incident ray and the manipulated specimen surface deviate from the origin of the L-frame. The diffraction spot is the intersection point of the incident ray and the surface of the nontransparent specimen. The location of this non-ideal spot is given by the vector xs. First we will describe the vector xs as the sum of error vectors associated with the incident beam on the basis of the L-frame (see also Fig. 2)1: q xR þ xR þ xL þ xL ð2Þ xs;L inc ¼ A inc eq ax w; 2q with conversion matrix (to convert from R-frame to Lframe) 2 3 1 0 0 ¼ 4 0 cos q sin q 5 ð3Þ A q 0 sin q cos q and with error vectors defined as 0 1 0 1 0 0 R R R A xinc ; xeq ¼ @ yray ¼ @ 0 A; heq 0

0

1 0 L xax ¼@ 0 A hax ð4acÞ

and

0

0

1

L xw;2q ¼ @ yw; 2q A: hw; 2q

Fig. 2. Enlarged view near the origin. The incident rays S and T run parallel on this scale. The non-ideal incident ray T is displaced in both the equatorial and the axial direction. The ideal incident ray S runs through the origin and makes an angle q with respect to the xL yL plane. The vector xs (black arrow), pointing to the location of the irradiated spot, is the sum of the error vectors xw; 2q , xax , xeq and xinc (grey arrows). See text for more details. The thin lines are guides for the eye.

ð4dÞ

Here yRray is the position along the incident ray T in the ray reference frame (R-frame), heq the equatorial beam misalignment, hax the axial beam misalignment, and yw; 2q and hw; 2q are the misalignments of the w-axis. Secondly we will describe the vector xs as the sum of error vectors associated with the manipulation of the specimen surface on the basis of the L-frame (see also Fig. 3): n h 0i o wq A c xS þ xL0 þ xL0 þ xL0 ð5aÞ xs;L surf ¼ A j; w c; w w; 2q where

h i 0 S0 xS þ xS j; xS ¼ A j surf sp; c þ x c

Fig. 3. Enlarged view near the origin with non-intersecting rotation axes of a 4-circle diffractometer. The goniometer axis w is parallel to the goniometer axis 2q The cradle axis c is perpendicular to the w axis. The cradle axis j is perpendicular to both the cradle axis c and the goniometer axis w. The vector xs (black arrow), pointing to the location of the irradiated spot, is the sum of the error vectors xw; 2q , xc; w , xj; w , xj; c , xsp; c and xsurf (grey arrows). See text for more details. The thin lines are quides for the eye.

ð5bÞ

with conversion matrices (stepwise from the S-frame to the L-frame S ! S0 ! L0 ! L): 2 3 1 0 0 4 0 cos ðw qÞ sin ðw qÞ 5 ; ð6aÞ A wq ¼ 0 sin ðw qÞ cos ðw qÞ 2 3 cos c 0 sin c c ¼4 0 1 0 5; ð6bÞ A sin c 0 cos c

1 In the equations single and double overlines are used to distinguish vectors and matrices from scalars, respectively.


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Instrument aberrations in a 4-circle powder diffractometer

A. C. Vermeulen

and

2

cos j sin j 0 ¼ 4 sin j cos j 0 5 ; A j 0 0 1 and with error vectors 0 S 1 0 1 xsurf 0 S S @ 0 A; ¼ @ ySsurf A; xsp; xsurf c ¼ hsp; c 0

ð6cÞ

0

0

S xj; c

1 xj; c ¼ @ 0 A; 0 ð7acÞ

and

0 0

0

1

L ¼ @ yj;w A; xj;w 0

0

0

L xc;w

sin ðw qÞ sin q sin c þ hax sin w sin w cos c sin q sin q þ hsp; c þ hc;w þ hw; 2q : sin w cos c sin w

zLs ¼ heq

3

1 0 ¼ @ 0 A: hc;w

ð7deÞ

S and ySsurf are coordinates on the sample surface Here xsurf in the specimen reference frame (S-frame), hsp, c the specimen height misalignment with respect to the c-axis, xj; c and yj; w the misalignments of the j-axis with respect to the c- and w-axis, respectively, and hc; w is the misalignment of the c-axis with respect to the w-axis.

Position of the diffraction spot The coordinates of the X-ray intersection point on the specimen surface are solved from the following vector equality of (2) and (5): xs;L inc ¼ xs;L surf

ð8Þ

which results in analytical solutions for xSsurf and xRinc . Assuming that xj; c ¼ yj; w ¼ 0 it holds for j ¼ 0 (coordinates in S-frame): 1 sin c S þ hsp; c ; ð9Þ ¼ hax xsurf cos c cos c 1 sin c cos w þ hax sin w sin w cos c cos w cos w þ hc; w : þ hsp; c sin w cos c sin w

ð10Þ

With (9) and (10) one can reposition the specimen if both the analysed spot size and the X-ray beam size are smaller than the displacement of the X-ray intersection point and which would result in the analysed spot not being irradiated. This is likely for a micro-diffraction set-up with small specimens. If phi-rotation, as occurs for stress and texture measurements, should also be included, in principle, (9) and (10) can be expanded to include j-rotation angle and the alignment errors associated to the j-axis. However, we have not done this here to avoid the rather large formulas. Note that (9) and (10) are independent from the optics geometry. Hence they apply for both the focusing optics and the parallel beam optics. Combining (5) with (9) and (10) results in the following relations in scalar form: cos ðw qÞ cos q sin c þ hax sin w sin w cos c cos q cos q þ hc;w þ yw; 2q ; þ hsp; c sin w cos c sin w

With (13) one can, in principle, calculate a real-time adjustment of the specimen height that goes deliberately beyond setting the specimen height at zero in the S-frame. The suggested adjustment sets the specimen surface to zero height in the L-frame. Then the real-time adjustment of the specimen height, DzSs , is a function chosen such that the adjusted (zLs )adj becomes effectively zero. Hence, ðzLs Þadj ¼ zLs þ DzSs

sin q ¼0 sin w cos c

ð14Þ

which results in sin w cos c ð15Þ sin q where zLs is the function as calculated from (13). When dealing with a micro-diffraction set-up it is important to realise that the adjustment of (15) causes an additional shift of the beam on the specimen surface. Using (9) and (10) one can easily derive this additional beam shift: DzSs ¼ zLs

sin c sin w sin c ¼ zLs ; cos c sin q cos w cos w ¼ DzSs ¼ zLs : sin w cos c cos q

S ¼ DzSs Dxsurf

ð16Þ

DySsurf

ð17Þ

Hence, if one wants to keep a particular spot of the specimen in the beam, a repositioning of the specimen is required in order to compensate the effects of (9) and (16), and (10) and (17).

Correction of the diffraction peak position

ySsurf ¼ heq

xsL ¼ hax ;

ð13Þ

ð11Þ

yLs ¼ heq

ð12Þ

If the alignment errors for an instrument equipped with focusing optics are known they can be used to obtain accurate absolute peak positions by correcting the measured peak positions according to: 2qcorr ¼ 2qmeas D2qsp; c D2qeq D2qax D2qc; w D2qw; 2q

ð18Þ

where 2qcorr is the corrected angle, 2qmeas is the measured angle and D2qxx are the individual peak shifts due to alignment errors of the instrument. A full set of misalignment formulas for the w-stress and c-stress single tilt modes has been presented by Vermeulen (2000) [5]. An overview of a generalised modular set of formulas for the combined tilts mode (i.e. employing both w- and c-tilting) has also been published by Vermeulen (2006) [6]. These formulas are presented in Table 1. They are the result of combining (1) and (13) and are restricted to the first order terms that contain the error in the height zLs only. Note that it is assumed that the above theorerical derivations for a single ray are also applicable to a ray representing the centre-of-gravity of a small X-ray beam. 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.


206

Table 1. Set of peak shift formulas for specimen displacement w.r.t. chi axis (D2qsp, c), incident beam misalignments (equatorial D2qeq and axial D2qax), and axis misalignments (chi w.r.t. omega axis D2qc; w and omega w.r.t. 2theta axis D2qw; 2q) for combined tilts mode, which includes both omega-stress mode (c ¼ 0) and chi-stress mode (w ¼ q). With diffractometer radius R.

Table 2. Additional set of formulas to correct peak positions for the zero beam shift (D2qze), an isotropic error in the reference (D2qref) and transparency (D2qtr). With cubic lattice parameter a, wavelength l, linear expansion coefficient a, temperature T and linear absorption coefficient m (note that m1ftau represents the information depth t [8]).

Error

Function

Combined tilts mode

D2qsp, c ( )

hsp; c fsp

hsp; c

D2qeq ( )

heq feq

heq

180 2 sin ðw qÞ cos q p R sin w

(20)

D2qax ( )

hax fax

hax

180 2 sin q sin c cos q p R sin w cos c

(21)

D2qc; w ( )

hc; w fc; w

hc; w

D2qw; 2q ( )

hw; 2q fw; 2q

hw; 2q

Error

180 2 sin q cos q p R sin w cos c

180 2 sin q cos q p R sin w 180 2 cos q p R

Function

Combined tilts mode

2qze

eref fref

2qze 180 eref 2 tan q p eref ¼ Da=ao ; eref ¼ Dl=l; or eref ¼ a DT 180 2ftau sin q cos q m1 R sin w cos c p sin2 q sin2 ðw qÞ cos c ftau ¼ 2 sin q cos ðw qÞ

D2qze ( ) D2qref ( )

D2qtr ( )

m1 ftr

Three additional errors, the zero beam shift, the transparency error (see also [8]) and the reference error, each of which may play an additional role in either the determination of or the corrections for the alignment errors, are presented as examples in Table 2 [6]. Only the isotropic term as occurs for cubic materials is included for the reference error. In principle anisotropic terms can be added for use with non-cubic reference materials. Note that the zero beam shift and the reference error are relevant for both the focusing optics and the parallel beam optics.

(19)

(22) (23)

(24) (25a) (25b) (26a) (26b)

pre-aligned fast interchangeable PreFIX optics modules for point focus geometry were used. No particular attempts were made to optimise the alignment condition of the system before the measurements. All measurements were performed on a stress-free Au powder specimen using CuKa radiation. We measured the (111), (200), (220), (311), (331), (420), (422) and (511)/ (333) reflections with positive and negative tilts over the full measurable range (up to sin2 wmax ¼ 0.1 0.8 for omega-stress, iso-inclination; and up to sin2 wmax ¼ 0.9 for chi-stress, side inclination).

Procedure The analysis of the alignment errors is performed by a multivariate linear least squares (multiple linear regression) fitting procedure (see also [3, 6]) in the following form: 2qmeas; j 2qref; j ¼ hsp; c fsp; j þ heq feq; j þ hax fax; j þ hc;w fc;w; j þ hw; 2q fw; 2q; j

ð27Þ

where fsp, feq, fax, fc; w and fw; 2q are goniometric functions describing the peak shift due to specimen displacement, equatorial beam misalignment, axial beam misalignment, chi (c) axis misalignment and omega (w) axis misalignment, respectively. Table 1 gives an overview of the above functions and the linear misalignment parameters, hsp,c, heq, hax, hc; w, and hw; 2q. Optionally one can add the zero beam shift, constant 2qze, and error functions, fref and/or ftr, with linear parameters, eref and m1, respectively, from Table 2.

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

Results The multiple linear regression analysis based on (27) was applied to a large data set consisting of a series of c-tilt measurements and a series of w-offset measurements to obtain a full description of the alignment condition of the diffractometer system in use. Additional parameters for the zero beam shift and for the reference error (see Table 2) are added to (27) in order to refine the strain-free lattice parameter of the Au reference powder (PDF 00-004-0784: A). A correction for transparency (see Taa0 ¼ 4.07860 ble 2) was omitted since it would have no significant effect. It was assumed that the w axis and 2q axis coincide (hw; 2q ¼ 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 þ4 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 sin2 wmax). The results are given in Table 3. Figure 5a and 5b show the graphical results. Note the significant reduction in


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A. C. Vermeulen

Table 3. Numerical results of multiple linear regression analysis with (10). See Fig. 4 for the graphical results. Parameter 2qze hsp, c heq hax hc; w hw; 2q aAu s(2q)meas s(2q)fit

Unit

( ) (mm) (mm) (mm) (mm) (mm) ( A) ( 2q) ( 2q)

Result 0.026 (4) 45 (3) 50 (9) 121 (2) 51 (12) 0 4.07834 (2) 0.0372 0.0055

a

a

b Fig. 4. Graphical results of a simultaneous fit on (a) a series of c-tilt measurements and (b) a series of w-offset measurements. The relative 2q peak displacements are magnified with a factor of 100 for more clarity. See Table 3 for the numerical results.

standard deviation before and after applying the software correction in the last two rows of Table 3. Equations (9) and (10) were used to calculate the distribution of locations of the irradiated spot in specimen coordinates. The result is graphically presented in Fig. 5a using the alignment errors from Table 3. Note that the area of the distribution is an order of magnitude larger than the average size of alignment errors. A significant part of the measurement is outside an imaginary target spot of 1 mm diameter. An imaginary target micro-diffraction spot of 100 mm is even missed completely. Figure 5b shows a presentation of the situation after an imaginary optimisation where all relevant errors are reduced to 20 mm in absolute size. All irradiated spots fall inside the target spot of 1 mm. The target micro-diffraction spot of 100 mm is only hit for a part of the measurement. Most irradiated spots still miss the micro-diffraction target.

Conclusions

b

Fig. 5. Graphical representation of the distribution of the irradiated spot for the series of performed measurements with (a) the determined errors hsp, c ¼ 45 mm, heq ¼ 50 mm, hax ¼ 121 mm, and hc; w ¼ 51 mm (see also Table 3) and (b) the imaginary optimised errors hsp, c ¼ 20 mm, heq ¼ 20 mm, hax ¼ 20 mm, and hc; w ¼ 20 mm. The small circle indicates a target spot of 100 mm diameter. The large circle a target spot of 1 mm diameter.

scribing alignment errors are derived. These are relevant for measurements over the full 2q range, and when applying c-tilt or w-offset or a combination of both tilts. The presented analytical functions can be used as part of an iterative procedure to improve the alignment of the hardware. Alternatively they can be used for a software correction procedure to obtain reliable absolute peak positions. This will make it possible to perform accurate triaxial or multiple {hkl} residual stress analysis. The analytical functions describing the position of the irradiated spot in specimen coordinates show clearly that for micro-diffraction or micro-spot analysis the demands on the equipment alignment are very high. Theoretically the performance of the hardware can be improved by smart control of the specimen position.

Appendix A Diffraction at a non-ideal spot From Fig. 1 the vector T defining the incident ray is given by T ¼ S þ xs ;

The theory presented in this paper provides an insight into the behaviour of instrument aberrations in a 4-circle powder diffractometer. A full set of analytical functions de-

ðA1Þ

and the vector Q defining the diffracted ray is given by Q ¼ R xs :

ðA2Þ


208

From (A1) the modulus of vector T is 1=2 S xs xs xs ; T ¼ jS þ xs j ¼ S 1 þ 2 2 þ 2 S S

ðA3Þ

which becomes on expanding by the binomial theorem as far as terms of the second order in vector xs ( ) S xs 1 xs xs 3 ðS xs Þ2 1 1 ¼ 1 2 þ þ ... : S S4 T S 2 S2 2 ðA4Þ Similarly

( ) 1 1 R xs 1 xs xs 3 ðR xs Þ2 ¼ 1þ 2 þ þ ... : R R4 Q R 2 R2 2

The vectors T and Q make an angle of 2q with each other, so

and, since S ¼ R by definition, it follows (A1), (A2), (A4), and (A5) ( S xs 1 xs xs 3 1 þ cos 2q ¼ 2 1 2 R R 2 R2 2 ( R xs 1 xs xs 3 þ 1þ 2 R 2 R2 2

ðA13Þ

a ¼ ðS xs Þ ¼ RðyLs cos q0 zLs sin q0 Þ b ¼ ðR xs Þ ¼ RðyLs cos q0 þ zLs sin q0 Þ c ¼ ð xs xs Þ ¼ ðxsL Þ2 þ ðyLs Þ2 þ ðzLs Þ2 :

ðA14Þ

after substituting )

2zL D2q ¼ s cos q0 ( R ) ðxsL Þ2 þ 2ðyLs Þ2 sin2 q0 þ ðzLs Þ2 ½4 cos2 q0 3 cos q0 sin q0 R2

)

ðA15Þ

ðS xs Þ2 R4

ðR xs Þ2 R4

ðA7Þ

ðA8Þ

Since

This result corrects an error in an intermediate result (equation 3.10) of Wilson [7] for the (zLs )2 term. This term was neglected in the final result (equation 3.18). Acknowledgments. Many thanks to my colleagues Detlev Go¨tz and Dr. Joachim Woitok for reading the manuscript and to Mike Carn for correcting the English. Also thanks to Dr. Thomas R. Watkins (ORNL, Oak Ridge, USA) for commenting the submitted version.

References

cos 2q ¼ cos ð2q0 D2qÞ ;

ðA9Þ

for small D2q it holds cos 2q cos 2q0 ¼ D2q sin 2q0 ;

ðA10Þ

and hence with (A7) after substituting R S ¼ R2 cos 2q0 it follows for terms upto R D2q ¼

1 xsL xs ¼ @ yLs A zLs

With substituting (A14) in (A12) we get in scalar form for all terms upto R2:

The vectors R and S make an angle of 2qo with each other, so ðR SÞ ; RS

0

ðA6Þ

fR S S xs þ R xs xs xs g :

cos 2q0 ¼

and

it follows for the inner (dot) products using S ¼ R by definition: ðA5Þ

ðT QÞ cos 2q ¼ TQ

The parts between the braces are extra terms with respect to Eq. (3.8) of Wilson [7]. They lead to 3rd and 4th order terms in the final result and can be neglected. After defining vectors (see also Fig. 1): 0 1 0 1 0 0 R ¼ R@ cos q0 A S ¼ S@ cos q0 A; sin q0 sin q0

ðA11Þ 4

ðcos 2q cos 2q0 Þ 1 cos q0 ¼ 2 ½a b þ c sin q0 sin 2q0 R 1 3 2 3 2 cos q0 a þ b ab þ fdg þ 4 sin q0 R 2 2 1 1 2 1 2 1 a þ b þ ab þ f3dg 4 R 2 2 2 sin q0 cos q0 ðA12Þ

with inner (dot) products a ¼ ðS xs Þ, b ¼ ðR xs Þ and c ¼ ð xs xs Þ, and d ¼ 1=2 ac 1=2 bc þ 1=2 c2 .

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