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Volume 109, Number 1, January-Febraary 2004 Journal of Research of the National Institute of Standards and Technology [J. Res. Natl. Inst. Stand. Technol. 109, 1-25 (2004)] Fundamental Parameters Line Profile Fitting in Laboratory Diffractometers Number 1 January-February 2004 Volume 109 R. W. Cheary' wavelength spectrum and/or by changing The fundamental parameters approach to line profile fitting uses physically based one or more of the axial divergence University of Technology Sydney, models to generate the line profile shapes. parameters. Flat analyzer crystals have Broadway, Sydney, NSW, Fundamental parameters profile fitting been incorporated into FPPF as a Lorentzian shaped angular acceptance (FPPF) has been used to synthesize and fit Australia 2007 data from both parallel beam and divergent function. One of the intrinsic benefits of A. A. Coelho beam diffractometers. The refined parame- the fundamental parameters approach is its adaptability any laboratory diffractometer. ters are determined by the diffractometer Bruker-AXS, Ostliche configuration. In a divergent beam diffrac- Good fits can normally be obtained over RheinbriickenstraPe 50, the whole 20 range without refinement tometer these include the angular aperture D-76187 Karlsruhe, Germany of the divergence slit, the width and axial using the known properties of the diffrac- length of the receiving slit, the angular tometer, such as the slit sizes and diffrac- and apertures of the axial SoUer slits, the tometer radius, and emission profile. length and projected width of the x-ray J. P. Cline source, the absorption coefficient and axial Key words: fundamental parameters; length of the sample. In a parallel beam microstructure analysis; parafocusing National Institute of Standards system the principal parameters are the optics; profile convolution; profile fitting; and Technology, angular aperture of the equatorial x-ray powder diffraction. Gaithersburg, MD 20899-8523 analyser/SoUer slits and the angular apertures of the axial Seller slits. The presence Accepted: April 11,2003 [email protected] of a monochromator in the beam path is [email protected] Available online: http://www.nist.gov/jres normally accommodated by modifying the 1. Introduction ie., J(2e) = Ji(20) ® ^2(20) ®...® Ji(20) ® ^29). (2) The fundamental parameters approach to line profile fitting uses physically based models to generate the line Diffraction broadening is incorporated into the profile function 1(20) by convoluting the broadening function profile shapes. The instrument profile shape K{2Q) is first synthesised by convoluting together the geometri- B{26) into the instrument profile function as shown Fig. cal instrument function J{29) with the wavelength pro- 1, file W{2e) at the Bragg angle of the peak, 20B 1(29) =K{29)® 3(29). (3) K{2d) = lW{29-2(p)J(2(p)d2(p = W(20) ® J{29) (1) This technique of profile synthesis was first introduced 50 years ago by Alexander [1], but has only been imple- where the fiinction J{26) itself is a convolution of the various instrument aberration functions associated with mented as a standard fitting procedure during the last the diffractometer. ten years [2,3,4]. More recently, freeware and commer- Deceased. Volume 109, Number 1, January-Febraary 2004 Journal of Research of the National Institute of Standards and Technology Geomelric Irislrumenl Waveleriglh Specimen Broadening Convoluted Function 1(29) Profile J(2e) Profile W(29) Function 6(26) Fig. 1. Convolution of the geometric instrument profile J{29), the wavelength profile W{29) and the specimen broadening function B{20) to produce the measued convoluted function I{2ff). cial software packages [5,6,7] have become available occur when fitting to data over a restricted 29 range. for fundamental parameters profile fitting (FPPF) Such correlation occurs between the axial divergence either for use as single line profile fitting, lattice param- parameters and absorption as both of these aberrations eter refinement or for Rietveld analysis. can produce similar forms of asymmetric profiles between 26= 50° and 100° in diverging beam diffrac- FPPF has been used to synthesise and fit data from both parallel beam and divergent beam diffractometers. tometers. Correlation is minimised by using data with a The refined parameters are determined by the diffrac- large 20 range so that the unique angular dependence of tometer configuration. In a divergent beam diffrac- individual aberrations becomes evident. When a set of tometer these include the angular aperture of the diver- instrument profiles cannot be fitted by FPPF, this is gence slit, the width and axial length of the receiving usually an indication that either the model used is slit, the angular apertures of the axial Soller slits, the invalid (eg. incorrectly chosen slit value), the instrument is mis-aligned, there is an overlapping impurity length and projected width of the x-ray source, the absorption coefficient and axial length of the sample. In line or, the specimen is generating crystallite size a parallel beam system the principal parameters are the broadening or is inhomogeneously strained. angular aperture of the equatorial analyser/Soller slits FPPF was designed originally as a tool for analysing diffraction line broadening. Fitting is done by convolu- and the angular apertures of the axial Soller slits. The presence of a monochromator in the beam path is nor- tion and corrections for instrument broadening and peak shift are intrinsic to the refinement. When an mally accommodated by modifying the wavelength spectrum and/or by changing one or more of the axial instrument is well characterised, line broadening can be analysed without a reference specimen. Moreover, divergence parameters. Flat analyser crystals have been incorporated into FPPF as a Lorentzian shaped angular when a reference standard is used, which has different properties from the specimen with line broadening, acceptance function. One of the intrinsic benefits of the fundamental some compensation can be made for these differences. parameters approach is its adaptability to any laborato- For example, when LaB^ SRM 660a (Ûpowder ~ 500 cm"') ry diffractometer. Good fits can normally be obtained is used as a reference, compensation can be made for over the whole 2d range without refinement using the differences in the absorption of the sample and LaBg. In known properties of the diffractometer, such as the slit the latest version of the commercial software package sizes and diffractometer radius, and the emission pro- (TOPAS)^, the concept of fundamental parameters has file. Fine tuning is sometimes necessary to accommo- been extended so that any user defined profile that date a monochromator or to compensate for the fact Certain commercial equipment, instruments, or materials are iden- that certain aberrations are not completely independent tified in this paper to foster understanding. Such identification does [8]. Under these conditions some of the instrument not imply recommendation or endorsement by the National Institute parameters need to be refined, but the refined values of Standards and Technology, nor does it imply that the materials or normally are within ±10 % of the actual values. equipment identified are necessarily the best available for the pur- Correlation between refined instrument parameters can pose. Volume 109, Number 1, January-Febraary 2004 Journal of Research of the National Institute of Standards and Technology accurately describes the physical broadening can be synchrotron data [10,11]. In the TOPAS implementa- readily convoluted into the refinement. tion of FPPF there are a wide variety of possible aber- In this paper we will discuss the physical origin of ration functions available within the package and these can put together to suit a particular diffractometer the instrumental profile shapes for various laboratory diffractometer configurations including both divergent design and in terms of parameters that are relevant to the instrument. One of the most important achieve- beam and parallel beam instruments. This will include a description of the geometrical aberrations as well as ments of the FPPF technique for practical users is speed discussion on the nature of the wavelength distribution of calculation. Accurate multiple convolution calcula- and the influence of monochromators on this distribu- tions over large 26 ranges can be very time intensive tion. Some discussion is also presented to demonstrate and it is of central importance to minimise this time to how the FPPF may be fitted to experimental data from enable Rietveld refinement to be completed in "sec- a material with a low attenuation coefficient. onds" without loss of accuracy within the profile function synthesis procedures. Various procedures have been implemented, some of which are described by 2. Basic Objectives of the FPPF Cheary and Coelho [2,3,4], but one of the most impor- Technique tant has been to code the time intensive calculations at an assembler code level taking steps to optimise the use One of the basic objectives of the FPPF technique is of the various registers within the PC chip. to be able to fit any powder diffraction profile using a physically based model to describe both the instrument 3. Laboratory Diffractometer profile and any diffraction broadening generated by the Configurations and Their Geometrical specimen. In principle, therefore, the technique should Aberrations be adaptable to any powder diffractometer and fit profiles of widely differing shapes, such as those in Fig. 2, by simply modifying the physical parameters of the dif- Up until the mid-1990s most of the laboratory pow- fractometer used to describe the profile. der diffractometers in use were divergent beam instruments with a narrow receiving slit, diffracted beam Although most applications of FPPF have focussed monochromator and a simple proportional/scintillation on the conventional diffractometer it has also been utilised for analysing neutron diffraction data [9] and counter detector as shown in Fig. 3a. Over the past 10 Siemens D5000 Diffractometer FWHM = 0.13 29 19.6 19.8 20.4 20.6 20.8 21.0 21.2 20.0 20.2 Fig. 2. Comparison of profiles produced by the reference specimen LaBj (SRM 660a) at similar 29 angles on two different diffractometers, (a) on the high resolution powder diffractometer BM16 at the ESRF, Grenoble (A= 0.35 A) and, (b) on a conventional 0-29 divergent beam diffractometer (A = 1.541 A) with a radius of 217 mm and commonly used slit sizes (ie. 1° divergence slit, 0.2 mm receiving slit and 2° Seller slits in the incident beam). Volume 109, Number 1, January-Febraary 2004 Journal of Research of the National Institute of Standards and Technology Deleclor Slit " 'Ti' ■ (a) Receiving /i w vTubc Slits yw Cv y^jM y—._,_^ Solid- /O^ ^^^Ciraphite /// Sbfs JÎV Monochromatoi /^^^y^\ /7 Divergence .^^\y \ ^ / U /</ Slit / / ^V\?^~"~~---^ ^..-<^'/ Seller / \ U \ ""-JXl'' / Slits / / Y""^ ""--' ^~^r Ge Monochromator ^^n* (b) ^gP^^^^^ Focal Line Receiving Slit^ . /^ ^"^^^\ // Divergence ^^^ ^7%1 /y ^^^^ . 7 <^^>^^ ^^--''^\x^ ^-^ /^v^ -^ ^^ X-ray ^/ Soller Tube \ '^ ^ .^-^**\^'' y/ Slits -^-.^'^--y "V n 7 Fig. 3. Two configurations of diverging beam difiractometer showing the principal optical components, (a) with a difiracted beam monochromator and, (b) with an incident beam monochromator. (ii) a divergent incident beam on to a flat specimen years the number of diffractometer options available (flat specimen error), from manufacturers has increased and users operate (iii) the finite width of the receiving slit, with a wider range of x-ray optical designs. The types (iv) the beam penetration into the specimen (specimen of geometrical aberrations encountered is somewhat broader than those discussed in the classic work by transparency), Wilson [12] as will be discussed below. (v) the deviation of the beam from the equatorial plane (axial divergence). 3.1 Divergent Beam Diffractometers—Symmetric These aberrations all produce some degree of line Diffraction broadening and, in the case of flat specimen error, specimen transparency and axial divergence, some asym- The most widely used laboratory diffractometer in metry is also introduced. Zero 20 and specimen surface use today is still the divergent beam diffractometer with displacement errors may also be present in a diffrac- either a bent graphite monochromator in the diffracted tometer, but these only affect the 20 position of a pro- beam (see Fig. 3a) or, a ground and bent asymmetrical- file and not its shape. In both configurations the mono- ly cut germanium monochromator in the incident beam chromators not only determine the wavelength distribu- (see Fig. 3b). Both of these configurations possess a tion, but they also act to reduce the axial divergence similar array of geometrical aberrations. The major dif- and it is often considered unnecessary to include Soller ference between them is the wavelength distribution slits between the sample and the monochromator. which normally consists of both the KcZi and KcZj components of the K spectrum in the graphite monochro- 3.2 Divergent Beam Diffractometers— Asymmetric Diffraction mator case and only the Kai with the Ge monochromator [13]. Further discussion of the wavelength distribu- Divergent beam diffractometers used under symmet- tion and the effects of monochromators is given later. The principal geometric aberrations contributing to ric conditions only measure diffraction from planes profiles from the above diffractometers are, parallel to the specimen surface. To measure diffraction (i) the finite width of the x-ray source, from planes angled relative to the specimen surface it is Volume 109, Number 1, January-Febraary 2004 Journal of Research of the National Institute of Standards and Technology necessary to operate under asymmetric conditions as 3.3 Parallel Beam Diffractometers illustrated in Fig. 4a. The problem with operating in this mode on most commercial diffractometers is that There are two common forms of the parallel beam powder diffractometer which are illustrated in Fig. 5. the receiving slit is no longer at the focus of the dif- These are based on using either analyser slits (other- fracted beam and profiles are broadened by "defocussing". The amount of defocussing is determined by wise referred to as equatorial Soller slits) or a flat Ge/Si analyser crystal as the angular discriminator of the dif- the angle of divergence of the incident beam and distance of the focus from the receiving slit. fracted beam. Amongst laboratory diffractometers the Defocussing also occurs in diffractometer configura- analyser slit set-up is the most widely used form as it offers more intensity but poorer resolution than the tions where, • the sample is oscillated ±5(9 about the diffractometer analyser crystal set-up. axis so that the angle of incidence on to the specimen Parallel beam diffractometers have emerged as one of the most popular forms of laboratory diffractometer varies between 0 + dm and 9 - 5(9. This oscillation moves the focus of the diffracted beam continuously over the past ten years and now constitute more than back and forth in front of and behind the receiving 30 % of the new diffractometer purchases. There are slit, fewer geometric aberrations contributing to the profile • the receiving slit is replaced by a position sensitive and systematic errors arising from specimen displace- detector (PSD). The only position on the detector that ment, specimen transparency and surface roughness are is normally in focus is its centre (see Fig. 4b); all dif- not significant. There are two geometric aberrations contributing to a parallel diffractometer, fracted beams entering the detector at off-centre posi- (i) the angular acceptance function of the analyser tions are defocussed. In PSD systems, the aberrations contributing to a profile include all the standard dif- foils or analysing crystal, fractometer aberrations, except the aberrations for- (ii) deviation of the beam from the equatorial plane (ie. merly due to the receiving slit are replaced by defo- axial divergence). cussing, the discharge resolution of the detector, and In most laboratory diffractometers, the parallel beam is parallax error [14,15]. In a scanning PSD diffrac- produced by using a parabolic graded multilayer mirror tometer, the recorded profile shape is an average of with the line x-ray source positioned at the focus of the all the profile shapes across the active window mirror [16]. Although the beam may be parallel in the length. equatorial plane, it will not be parallel in axial plane Receiving Slit (a) Sample with aiigled hkl plaiies (b) Defoeussed Diffracted Beam Fig. 4. (a) Diverging beam diffractometer operated in asymmetric mode with defocussed diffracted beam at tlie receiving slit, (b) diffractometer with PSD replacing the receiving slit. Diffracted beam is in focus at the centre of the detector and defocussed in off-centre positions. Volume 109, Number 1, January-Febraary 2004 Journal of Research of the National Institute of Standards and Technology Axial Soller Slits (a) Analyser Slits Parallel Incident Beam (b) Analyser Crystal Parallel Incident Beam - ► Sample Fig. 5. Two configurations of parallel beam difiractometer, (a) using analyser slits in the diffracted beam and, (b) using a flat analyser crystal. and axial divergence can be expected in both the inci- encountered in practice. The aberration fiinctions generated by mis-setting a diverging beam diffractometer dent and diffracted beams. Low angle profiles will therefore be asymmetric although not to the same or using it under asymmetric conditions are also dis- extent as diverging beam instruments. cussed. Most of the results quoted here are for a diffractometer radius R = 215 mm. The convention adopted here for describing the 4. The Instrument Aberrations angular variables is that 20 refers to the continuously variable angle measured on the diffractometer whereas The geometric instrument aberrations tend to deter- 20 or 20B refers to the Bragg angle of the diffraction mine the shape of a difiractometer profile at low 29 line. The angle e refers to the difference between the angles (ie., 26 < 50°). At high 20 angles {20 > 100°), measured angle 20 and 20, the profile conforms primarily to the shape of the wavelength distribution in the beam. With the exception of £ = 20-2ft (4) the aberrations associated with "receiving system" of 4.1 Finite X-Ray Source Width the difiractometer and the x-ray source, all of the geometric instrument aberration profiles vary with 29. In the following sections the shapes of the major instru- The profile shape of this aberration is generally ment aberrations used in FPPF analysis to describe the expressed as an impulse function of width A20^ as various laboratory difiractometer configurations are shown Fig. 6a. Although the choice of an impulse func- discussed for conditions that are typical of those tion may be not be strictly valid for describing the x-ray (a) (b) ~#a» g Zi Fig. 6. Aberration profile models, (a) simple model for a source of projected width w^ and a diffractometer radius R, (b) model with "tube tails" containing additional parameters/= /taiA^x and angular widths Zj and Z2 from the central maximum. Volume 109, Number 1, January-Febraary 2004 Journal of Research of the National Institute of Standards and Technology source aberration, the exact shape used is not critical lOOk when a long fine focus tube (target width =0.4 mm) is installed on the diffractometer. At a take-off angle of 6° Tube Focus this appears as a projected width w^ ~ 0.04 mm and the 10k c aberration profile has a width A20^ = 0.01° and does not 0.1% of Central 3 o contribute significantly to the overall width of the Focus Count Rate U instrument profile. In broad focus tubes, the target width ~2 mm and X w^ ~ 0.2 mm at 6° take-off so that the aberration profile width A26I, = 0.056°. At this level the source width 100 makes a much bigger contribution to the overall width of the instrument profile and a more accurate form for the aberration profile shape is necessary. A good approximation under these circumstances is a Gaussian shape rather than an impulse function. In diffractome- Fig. 7. Intensity scan with 50 |a,m wide slit of an image formed ters with curved crystal incident beam monochromators through a 10 |rm pinhole in platinum of the 0.4 mm wide long fine the source width can also have a greater contribution focus in a Cu anode x-ray tube set at 40 kV, 40 mA. because of the magnification effect introduced by the monochromator. This occurs with asymmetrically-cut Johansson incident beam monochromators where the feet focussing should have an infinitely small width. Owing to the many aberrations present, focussing is source-crystal and crystal-focal point distances are typ- never perfect and the count rate incident on the receiv- ically = 120 mm and =230 mm, respectively. A fine ing slit tends to increase with increasing slit width, but focus tube with a projected width of 0.04 mm is then effectively magnified to =0.08 mm. Under these condi- at the expense of resolution. In parallel beam diffractometers the receiving system is based on using either tions the effective source width can be trimmed down the Hart-Parrish system of analyser slits [18] or, a flat by reducing the width of the focal line slit. analyser system as illustrated in Fig. 5 earlier. In many For accurate line profile analysis it is also necessary to modify the simple impulse model even with long glancing incidence diffractometers the receiving sys- fine focus tubes. Bergmann [17] has shown that most of tem consists of analyser slits and an analyser crystal in the anode surface in an x-ray tube produces x rays the diffracted beam. Although the aberration functions albeit at a much lower intensity than the focal line on associated with the various receiving systems for paral- the anode. This is illustrated in Fig. 7 which shows the lel beam and divergent beam diffractometers all pos- intensity recorded by scanning with a 50 |J,m slit across sess different shapes, they all possess the common property of being independent of 2ft the image of a x-ray source formed through a 10 \im pinhole in platinum. A better approximation to the aberration function is a sharp impulse function superim- 4.2.1 Receiving Slit in a Diverging Beam Diffractometer posed on a broad impulse function to represent the so called "tube tails". This is illustrated in Fig. 6b. The parameters introduced to describe the "tube tails" are Most commercial diffractometers have a selection of receiving slits ranging in width from 0.05 mm up to 0.3 the extents of the high and low angle tails, Zi and Zj, and the intensity of the tail f relative to the intensity at mm although occasionally larger slit sizes up to 0.6 mm the tube focus. In most instances the intensity of the are used to measure integrated intensity rapidly. The tails is =0.1 % of the peak intensity and is only signifi- aberration function for a perfectly aligned receiving slit is an impulse function of width A2ft, given by. cant when analysing intense lines. The tails themselves are not necessarily symmetric with respect to the tube focus and can extend over a 20 range up to 0.6°. A2ft rad (5) R 4.2 X-Ray Receiving System Models where w, is the width of the receiving slit. The angular width A20, subtended by the receiving slit relative to In diverging beam diffractometers the receiving slit the diffractometer axis is therefore normally between is placed at the focus of the diffracted beam and for per- 0.013° (0.05 mm) and 0.08° (0.3 mm). When the slit Volume 109, Number 1, January-Febraary 2004 Journal of Research of the National Institute of Standards and Technology size is larger than 0.15 mm, the receiving slit aberration effect has been incorporated into the aberration profile is often the dominant aberration in a diffractometer by adding two Voigt functions of unequal intensity ,one on each side of the triangular aberration fiinction, to over the angular range 26= 15° to 60°. represent the satellite reflections [11]. The parameters 4.2.2 Parrish-Hart Analyser Slits of the satellite peaks can then be determined by fitting profiles from a reference material such as the NIST ref- Analyser slits act as an angular filter in the diffract- erence standard LaB^, SRM 660a. An alternative ed beam. The aberration function or transmission fiinc- approach to determining the aberration profile of an tion for these slits is a triangle fiinction, as shown in analyser slit system, without the effects of the wave- Fig. 8, in which the base width A20, is given by the length profile distorting the result, is to simply carry angular aperture A of the slits. out a 26 scan across the incident beam. Provided the In the original Parrish-Hart diffractometer on Station axial divergence of the incident beam is kept small, by 2.3 at the Daresbury synchrotron, the analyser slits including axial Soller slits, and the equatorial diver- were 360 mm with a spacing of 0.2 mm between adja- gence is negligible then the incident beam scan will cent foils giving an angular 29 aperture A~ 0.06°. In have exactly the same shape as the aberration profile of laboratory diffractometers the angular aperture A is typ- the analyser slits. ically =0.1°. A problem often encountered with analyser slits is specular x-ray reflection from the 4.2.3 Analyser Crystals analyser foils [19]. Weak satellite peaks appear on both the high angle and low angle profile tails but not neces- The inclusion of an analyser crystal in the diffracted beam of a parallel beam diffractometer gives high res- sarily of the same intensity as shown in Fig.9a. This foils in analyser slits Fig. 8. Triangle shaped aberration tunction for a set of analyser slits with an angular aperture A where /1/2 = spacing between the foil/length of the foils. A lOkr (b) 1 I Side Peaks from (a) c ■ 1 1 Analyser Slits FWHM = 0.0013 ° 26 o u . I LA^ f 1 J Ik7 w xf ^ Super Lorentzian Shape y J 100: 26 ° ^îi^J;.. 22.0 21.4 21.6 21.8 0.00 -0.01 Fig. 9. (a) Reflection satellite peaks from analyser slit recorded using the 310 line from NIST standard material LaB6 SRM 660a using the diffractometer on Station 2.3 at Daresbury synchrotron, (b) 29 scan across a 0.1 mmx 0.1 mm incident beam using a Gel 11 analyser crystal on beamline BM16 at the ESRF, Grenoble. Volume 109, Number 1, January-Febraary 2004 Journal of Research of the National Institute of Standards and Technology olution diffraction patterns with a low background, but ing slit. In commercial diffractometers the specimen is the intensity is invariably less than the Parrish-Hart invariably flat and the diffracted beam no longer configuration. The aberration profile introduced by the focusses perfectly. Good focussing characteristics, analyser crystal is generally very narrow and can be however, can be maintained with appropriately chosen determined by measuring the rocking curve of the crys- slits to limit the equatorial divergence and a diffrac- tal. For a perfect analyser crystal the aberration profile tometer radius R sufficiently large to reduce defo- will be determined by the Darwin profile of the cussing errors to an acceptable level without losing too analyser crystal. In practice, however, the aberration much diffracted intensity. For an incident beam, with an equatorial divergence profile will be broadened by the mosaic structure of the a centred on the diflfractometer axis, the aberration pro- crystal, any stresses in the crystal and any waviness or curvature of the crystal surface [20]. As a consequence file Jfsi^d) is asymmetric and exists only for the region the aberration profile can be dependent on the size of £ = 0. X rays diffracted from the centre of the specimen the beam incident on the crystal. A first approximation are detected at 20= 20 where as x rays diffracted off- to the shape of the aberration profile of an in-situ centre are detected at 20 < 20 as shown in Fig. 11. analyser can be obtained from a 20 scan of the analyser/detector using a very fine incident beam as shown in Fig. 9b. Although the profile recorded in this way also has the wavelength distribution folded into it, the result does at least give an indication of the shape and upper limit of the FWHM of the aberration profile. 4.3 Flat Specimen Error The basic optics of the focussing powder diffractometer set up for symmetric diffraction is illustrated in Fig. 10. The x rays are incident at an angle 9 on an ideal polycrystalline specimen with a surface radius of cur- Fig. 11. Diffraction from a flat plate showing the relationship between the measured angle 20 on the diffractometer and the diffrac- vature p. For diffraction from a particular hkl plane the tion angle 29 for a ray at the outer limit of a beam of divergence a. common property of all the diffracted rays from the specimen is that they all deviate through the same angle The relationship between the difference £=20-20 26. By simple geometry it can be shown that all the dif- and the distance q from the diffractometer axis at which fracted rays converge to a focus on a circle which has the same curvature as the specimen surface. The focus the ray is diffracted is. of the diffracted rays defines the position of the receiv- sin 26 rad (6) Curved Polycrystalline Specimen assuming small angles of divergence for the incident beam (eg. a < 2°). Thus, when a specimen is illuminated over a length Q then the x rays diffracted from either extremity of the beam (ie., at ^ = ^Q/2) defines the lim- iting value of of the aberration function Jps(£) which £M exists within the range 0 > £ > where £M sin 26 rad (7) 2R X-ray Receiving This relation can also be expressed in terms of the Source Silt equatorial divergence of the beam a starting from the relation. 1 1 (8) 2 sin(0-a/2 sin(0+a/2 Fig. 10. Focussing in a symmetric powder diffractometer. Volume 109, Number 1, January-Febraary 2004 Journal of Research of the National Institute of Standards and Technology Q = Z,p. Below 20ii„, the aberration function remains the When 26 > 10°, Eq. (8) can be approximated as same as the beam extends beyond the specimen. aR (8a) In diffractometers with a fixed illuminated length, Q sinO the effects of flat specimen error are generally smaller at low 20 values and increase with increasing 29. With and becomes £M an illuminated length of 20 mm on the specimen, flat specimen error is clearly discemable at 29 > 20°. It is £., =—cott^ rad. (9) M 2 not always possible to collect data from fixed Q mode When the incident beam is centred on the diffractome- diffractometer over a large 20 range (eg., up to 150° 20 ter axis the normahsed equation for the aberration func- as the a angle required at large 20 is larger than the diffractometer can accommodate. For example, to main- tion for flat specimen error is, JFS(^) tain a fixed beam length of 20 mm on a specimen over 1 (10) the range 20 = 0° to 90°, the angle of divergence a will for 0 > £ > £„ /ps(£): 2Jee need to increase from 0° up to -4°. A divergence angle a= 4° is close to the maximum value at which most Modem commercial diffractometers operate with either diffractometers can operate when a pyrolytic graphite a fixed divergence a or a fixed illumination length Q. monochromator is installed in the diffracted beam. At In the fixed a mode the aberration function is broad at angles of a greater than 4° the diffracted beam may low 20 and e^ has a cot 9 dependence whereas in fixed extend beyond the graphite crystal and not be diffract- Q mode the breadth rises from zero at 20= 0 up to a ed into the detector. In practice, this is not usually a maximum at 29= 90°. The extent of the changes in the problem as fixed Q mode diffractometry is normally used for analysing materials such as clays with very aberration function for each mode of operation JSF(£) using typical operating values for a and Q are shown in low angle diffraction lines where the lines of interest start at 20 « 3°. Fig. 12. With a fixed angle of divergence a = 1 °, the effects of flat specimen error in commercial diffractometers 4.4 Specimen Transparency are discemable as an increase in both the asymmetry and breadth below 29-40° [21]. Under conditions of Specimen transparency produces asymmetry and constant a the beam size increases with decreasing 29 broadening of the instramental profile function. For perfect focussing all the diffraction should occur on the and eventually the beam will cover the whole speci- focussing circle, but when the beam penetrates the sur- men. The angle 20ii^^ at which this occurs is given by. face, diffraction will occur over a range of depths within the specimen. The aberration function J^, (e) for an sin(0„J = ai?/L (11) infinitely thick specimen is given by, where i^p is the length of the specimen. Under these cir- exp(£/ 5) e<0 (12) cumstances the value of e^ is given by Eq. (7) with /,(£) = ■ Q = 20 mm 20.00 20.05 19.85 19.90 19.95 59.85 59.95 60.00 60.05 59.90 Fig. 12. Comparison of flat specimen aberration fiinctions at 29=20° and 29=60° for a diffractometer with either a fixed divergence angle a = 1 ° or fixed illumination length Q = 20 mm {R = 215 mm). All the aberration functions shown were convoluted with a very narrow Lorentzian emission profile with a FWHM = 0.001 mA to overcome infinities at e = 0. Each plot is normalised to the same I^^^ and both plots cover a range of 0.2° 29. 10