Systematic corrections in Bragg xray diffraction of flat and curved crystalschantler/opticshome/... · 2015. 10. 22. · X-ray diffraction theory has been developed by Darwin, Ewald,

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Systematic corrections in Bragg xray diffraction of flat and curvedcrystalsC. T. Chantler and R. D. Deslattes Citation: Rev. Sci. Instrum. 66, 5123 (1995); doi: 10.1063/1.1146428 View online: http://dx.doi.org/10.1063/1.1146428 View Table of Contents: http://rsi.aip.org/resource/1/RSINAK/v66/i11 Published by the American Institute of Physics. Related ArticlesNote: Continuing improvements on the novel flat-response x-ray detector Rev. Sci. Instrum. 82, 106106 (2011) A combined small- and wide-angle x-ray scattering detector for measurements on reactive systems Rev. Sci. Instrum. 82, 083104 (2011) Compact and high-quality gamma-ray source applied to 10 m-range resolution radiography Appl. Phys. Lett. 98, 264101 (2011) A new white beam x-ray microdiffraction setup on the BM32 beamline at the European Synchrotron RadiationFacility Rev. Sci. Instrum. 82, 033908 (2011) Standard design for National Ignition Facility x-ray streak and framing cameras Rev. Sci. Instrum. 81, 10E530 (2010) Additional information on Rev. Sci. Instrum.Journal Homepage: http://rsi.aip.org Journal Information: http://rsi.aip.org/about/about_the_journal Top downloads: http://rsi.aip.org/features/most_downloaded Information for Authors: http://rsi.aip.org/authors

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REVIEW ARTICLE

Systematic corrections in Bragg x-ray diffraction of flat and curved crystalsC. T. ChantlerSchool of Physics, University of Melbourne, Parkville, Victoria 3052, Australia

R. D. DeslattesQuantum Metrology Division, National Institute of Standards and Technology, Gaithersburg,Maryland 20899

(Received 14 September 1994; accepted for publication I August 1995)

Measurements of spectral wavelengths in Bragg diffraction from crystals often require refractiveindex corrections to allow a detailed comparison of experiment with theory. These corrections aretypically 100-300 ppm in the x-ray regime, and simple estimates may sometimes be accurate to 5%or better. The inadequacies of these estimates are discussed. Even with a possibly improved indexof refraction estimate, this correction is insufficient since additional systematics in the diffractionprocess occur at or above this level. For example, asymmetries of diffraction profiles withvr-polarized radiation or due to three-beam diffraction can approach the magnitude of refractiveindex corrections for flat or curved crystals. The depth of penetration of the x-ray field inside curvedcrystals, the shift of the mean angle to the diffracting planes, and lateral shifts around the crystalsurface are rarely considered but can dominate over refractive index corrections, particularly forhigh-order diffraction or medium-energy x rays. Shifts and nonlinearities arise when diffractingsurfaces lie off the Rowland circle, and exhibit strong and rapidly varying angular dependencies.Johann geometries with the source located on the Rowland circle should be avoided to minimizeprofile truncation shifts from crystal ranges or minimum grazing angles, and to avoid extremescaling corrections. Other significant shifts are identified and illustrated, with functional relationsprovided to allow an estimation of related magnitudes. The central concerns of this paper are theeffects on flat crystal diffraction and curved crystal diffraction in the Johann geometry, with a sourceand crystal of variable dimensions and location. Experiments often interpolate or extrapolate fromcalibration lines, so dependencies upon the diffracting angle are as important as the magnitude of thecorrections. These dependencies are presented in formulas and graphs. C 1995 American Instituteof Physics.

I. INTRODUCTION

X-ray spectroscopy was developed after the discovery ofx rays in 1895 by R6ntgen and the observation of their dif-fraction by crystals by von Laue in 1912.1,2 The wavelengthdependence of the diffracting angle, and hence the possibilityof the spectroscopy of x rays, was demonstrated by Bragg in1913.3'4

The crystal diffraction of x rays continues to yield thehighest resolution spectra in the x-ray regime, compared tosolid state and other detector technologies. The intrinsic flatcrystal resolving power can exceed Xh5&=100 000 forsingle- or double-crystal combinations in particular perfectcrystal diffracting planes of various crystals.5 Experimentalarrangements with resolving powers of 1000-10 000 arecommon across the x-ray range of energies. The realizedspectral resolving powers are often limited by natural orother source widths, by (imaging) detector resolutions, or bygeometric defocusing, rather than by the diffraction width.

With this high potential resolution and a high peak dif-fracting efficiency (approaching unity), the technique ofcrystal x-ray diffraction has proliferated in crystallographicstudies, standard source calibration and measurements, syn-chrotron radiation monochromatization, atomic physics tests,and general experiments using accelerators, tokamaks, and

electron beam ion traps. This high resolution also enablesabsolute measurements of the source profiles or wavelengthsto below the part per million level (ppm or &/X= 10-6).However, this precision also requires the consideration ofsystematics at this level.

X-ray diffraction theory has been developed by Darwin,Ewald, Pins, Zachariasen, James, and others.6-11 Modelingprocedures for flat and curved crystal calculations have ex-isted for some time.12 Most are designed for reflectivity orprofile shape determinations and neglect systematic shifts ofthe Bragg peaks. Estimates of such shifts often derive fromthe Bragg relation inside the crystal as compared to in vacuo(that is, with and without refraction) and involve significantapproximations. 13,14 Although it is well known that the ap-plication of these equations is approximate, the nature andmagnitude of these approximations are often poorly under-stood or neglected.

These estimates are widely used by researchers using flator curved crystal diffraction. One purpose of this paper is toindicate the range of validity of generally used approxima-tions as compared to precision dynamical diffraction theory,so that experimentalists will be more informed as to whenand how detailed corrections should be implemented in prac-tice. Our attention is restricted to Bragg (reflecting) x-ray

0 1995 American Institute of Physics 5123Rev. SciL Instrum. 66 (11), November 1995 0034-6748/95/66(11 )/51 23/25/$6.00


diffraction because it is the dominant and optimal form forlow and medium energy x-ray diffraction. The paper is di-vided into three sections and numerous subsections.

Our intention is to simplify the complexity of these cor-rections in such a way as to invite researchers to pursue morecritical measurements, without necessarily requiring the useof a long and computationally-intensive theory in situationswhere it is not needed. The effects are therefore related tocommon crystals used in the x-ray regime, with graphs pro-vided for typical cases.

Section II is concerned with effects which have theirorigin in flat crystal diffraction. This includes the well-known refractive index correction and its various approxima-tions. The section also discusses asymmetric diffraction andpolarization dependencies, as well as multiple-beam interac-tions. These considerations also carry over to curved crystals.The principles discussed, primarily for single-crystal diffrac-tion, also apply for instruments with multiple crystal ele-ments in monolithic or separated forms.

Section III is concerned with a series of effects whichoften dominate for curved crystals while being generallynegligible for flat crystals. Curved crystal corrections are lessfamiliar to many researchers. The depth of penetration of thex-ray field inside curved crystals, the shift of the mean angleto diffracting planes, and lateral shifts around the crystal sur-face are addressed. Shifts and dispersion nonlinearities aris-ing when diffracting surfaces lie off the Rowland circle arealso a major consideration. Several of the effects are isolatedand quantified here for the first time. The specific concern iswith Bragg diffraction in the Johann geometry, althoughmany of the relations are of general application.

Refractive index corrections occur at the level of 100-300 ppm and hence may be readily observed with modeminstrumentation. However, other systematic contributions toprofile centroid shift, detailed in this paper, can often exceedthis level and are less well known. Researchers working withmeasurements approaching the ppm level will be concernedwith estimating the magnitude of these effects in order todecide whether to evaluate or avoid them for the specificcrystal, curvature, energy, and geometry. Experiments ofteninvolve interpolation or extrapolation from calibration lines,so that dependencies upon diffracting angle are as importantas the magnitude of the corrections. These dependencies arepresented in formulas and graphs.

II. FLAT CRYSTAL SYSTEMATIC CORRECTIONS

A. Overview

Common approximations for refractive index correctionsare indicated in Sec. II B, which provides a brief review ofthe standard formulas. Section II C compares these formsacross large ranges for selected crystals. This demonstratesthe limitations of some commonly used equations. The accu-racy of refractive index estimates can be limited by formfactor uncertainties, in optimum cases, rather than by otherapproximations or effects.

Sections II D and II E indicate the main corrections tothese prescriptions following (standard) two-beam dynamicaldiffraction. The first correction (asymmetric diffraction) is

well defined but depends on the orientation of diffractingplanes in the crystal, which are sometimes inadequatelyknown. The second correction (peak profile asymmetry) de-pends on the crystal thickness and perfection, as explainedbelow. Asymmetries of diffraction profiles with iT-polarizedradiation (see Sec. 11 D), which generally occur for all crys-tals, can introduce major corrections to refractive index pre-dictions, at the 10%-50% level, and are therefore importantif high accuracy is required. This correction is well definedin standard two-beam dynamical diffraction, but is not givento convenient approximation. However, simple approxima-tions are presented here in formulas and graphs.

Section II F indicates typical uncertainties relating to theuse of databases as well as the results of crystal structuredeterminations. Sections II G-II I illustrate the complexityof real systems beyond two-beam diffraction in forbiddendiffracting regimes or in local three-beam interaction re-gimes. Other significant but tertiary effects, including mosa-icity and diffraction tail asymmetry, are discussed briefly inSec. II J.

The scales of these contributions are illustrated in sum-mary form in Table I, where it should be remembered thatthe relative magnitude of these effects can easily vary by anorder of magnitude from one crystal type to another. Despitethe overall complexity, high accuracy is certainly possiblewith curved or flat crystal measurements, either by explicitavoidance of problem regimes or by adequate correction forthe systematics involved. Precision measurements can com-pare unknowns to a nearby calibration line, so only the dif-ference in refractive index and other corrections is directlyrelevant. For example, the refractive index contribution tosystematic corrections between lines separated by less than2° in the same order of diffraction and away from near-normal incidence or any absorption edges is generally anorder of magnitude smaller than the absolute correction.

This example of an appropriate experimental design canachieve a sensitivity of perhaps 20-30 ppm without requir-ing refractive index corrections. Further, if a series of cali-bration lines covers the region of the unknowns and is welldistributed, the slope of the spectrometer dispersion can alsobe estimated, and under appropriate conditions only the con-sequent error in, or variation of, this slope will yield a sys-tematic error. Typically, this can add an order of magnitudein accuracy. In both these cases, of course, the calibrationlines must be measured on an absolute footing elsewhere orwith respect to further calibration lines at a similar level ofprecision. Detailed discussion of how this could or shouldproceed has been given elsewhere, following primary opticalstandard and crystal lattice spacing determination and a com-parison for the x-ray and -ray regimes. 15 ' 16

B. Refractive index shifts for perfect crystals insymmetric diffraction

Estimates of refractive index shifts derived from theBragg relation inside the crystal compared to in vacuo usu-ally follow the formula' 3

5124 Rev. Scd. Instrum., Vol. 66, No. 11, November 1995 Bragg x-ray diffraction


TABLE 1. Magnitudes of corrections relative to refractive index shifts; 'Typical estimates for flat crystals.

Factor Location Percentage Sec.

Use of Eqs. (3) and (4) Near atomic absorption edge 3%-12% II B and I CFar from edges 1%-2%Medium-high Bragg angles 6%-19%

Use of Eqs. (2) and (5b) Generally 0.1%-0.2% H B and H CHigh Bragg angles 1.0%

Asymmetric diffraction, 10 Angle to surface' 2%-7% H DExit vs incident angle 0.1%-1.0%

Peak shift Thick perfect crystal 10%-50% II EVery thin or ideally mosaic crystal 0%

Perfect crystal thickness 10%b Peak shift 1%-5% H EMean shift 0.1%-l.0%

Form factor uncertainties Generally 0.1%-1.0% H FLattice coordinate error Generally <0.5% II FPossible three-beam interaction Allowed, low order, high angle 0% II H

Allowed, low order, medium angle 1%-20% H HAllowed, medium order 10%-200% IIIForbidden 5%-100% H G

Tail asymmetry Extreme Bragg angles 5%-10% 1H J

'After correction for a, itself.bIntermediate thickness regime.

nAd | (1 in l+7in r- 1/2 1 - tr- 1 + ' 1 _

2d sin Oc sm2 OC sin cOC

where X is the incident (vacuum) wavelength of the rattion, d is the lattice spacing, n is the order of diffraction,is the refractive index, and Oc is the "center" of the Brpeak.t3 The Bragg angle OB follows from setting the righand side to unity, and a semiempirical correction toangular location of diffraction follows from equat

=1-tLr to, e.g.,

X 2rOFOs5= 2 T

where ro=e 2 /mnec 2, FO is the structure factor for forwscattering (hkl=000), and V is the unit cell volume. Conering energies well away from absorption edges, the exposion may be simplified further using the effective electnumber density

FO NAPZ.,

Vfl j MW

where i is summed over all orbitals and Zi refers to thenumber of electrons in orbital i. This correctly predicts the

(1) approximate overall shape of each edge (neglecting the finestructure), but with a dip at the edge reaching the unphysical

dia- value of -a. The latter occurrence may be corrected by theI Yr addition of small semiempirical terms to the argument for thelagsg log term, but then the equation is no longer well defined orght- readily calculable from standard tables. Several such equa-the tions are derived in Ref. 13, where the agreement of form isting good, but errors of one or more electrons (in FO) arise for

iron and calcium below or near the edges. In such regions ofanomalous dispersion near the absorption edges, the value of

(2) the overall structure factor can fall below zero, in which caserefraction has the same sign as for conventional visible op-tics. A consequence of the phase velocity of x rays that are

sard slower than the speed of light is the possibility of Cerenkovsid- radiation produced by relativistic electrons. This has been

treosn observed in the soft x-ray regime.For first-order radiation in the angular (Bragg) range of

0.65-1.15 rad for typical crystals, centroid shifts are domi-nated by the refractive index correction, [from Eq. (1)]

(3)AX xX, (2d)2

where NA is Avogadro's number, p is the density of the crys-tal, Zrn is the number of electrons per molecule, and M" isthe molecular weight (convention is not followed, in order toavoid confusion with Z, electron mass, and refractive index).As opposed to visible light, the refractive index for x rays isusually less than unity, so that the phase velocity of x rayswithin most media is greater than the speed of light andincident rays from a vacuum refract away from normal inci-dence.

This may be extended to allow for anomalous dispersionin an ad hoc way using

FO ~ne= NAP [Zn i t n| 1 - A )|

(5a)

where Xc is the wavelength inside the crystal. This corre-sponds to a shift of the peak angle from the Bragg angle by

A 0= OC- OB= arcsin 8.(2- )+( ) -arcsin n )

(b2d \ 2 tnO tan

(5b)

Note that &X 2 , so that the shift is approximately propor-tional to tan B* . For a flat crystal geometry and a flat detector(normal to the principal ray) at distance R, this yields a shiftacross the detector of

Rev. Sci. Instrum., Vol. 66, No. 11, November 1995 Bragg x-ray diffraction 5125


A Y= R tan(Oc- OB) RA0. (c 25.0

A positional sensitive detector such as a proportionalcounter will be able to measure this shift. In the Johanncurved crystal geometry, detection is made on the Rowlandcircle after focusing, so that an arc around the Rowlandcircle AY, relative to the Bragg location may be defined. Aflat detector is commonly used with curved crystal geom-etries for simplicity or mechanical reasons rather than theideal narrow curved detector on the Rowland circle. For lo-cations off the pole axis (where the Rowland circle and dif-fracting crystal coincide), evaluation of this becomescomplicated.17 To first order, Y,= 2ROut where R. is theRowland circle radius, and 6Ou,=6+ap is shifted in asym-metric diffraction by the angle of the diffracting planes to thesurface ap, so

AYz=2RzA 0. (5d)

Within the same approximation, a detector crossing the Row-land circle at the predicted location and normal to the ex-pected incident ray will follow

Ax=2RZ sin 0 AO.

23.0

21.0

19.0

170 _

16.00.50

(a) ADP 101

*0

T 17.0

14.0

(5e)

These refractive index shifts scale as n 2, diminishing rap-idly with order.

C. Comparison of semiempirical forms

Figure I compares the approximations represented byEqs. (2)-(4) when implemented in Eq. (Sb) in exact or ap-proximate form. These may be compared to implementationof Eq. (6), representing the standard route of dynamical dif-fraction theory. Neither the use of Eq. (Sb) nor the use of Eq.(6) is exact, since both neglect contributions of the order ofS2. Indeed, the definition of the "refractive index correc-tion" is itself uncertain. Within this paper, the definition to beadopted is given in Sec. H D, as represented heuristically andin approximate form in the preceding text, while a morecareful critique shall be reserved for a subsequent discussion.(On this issue, see also Ref. 14.)

A comparison of the estimates from these equations isenhanced by crystal data. In this paper, we give examples forgermanium, silicon, PET [pentaerythritol, C(CH2 CHOH)4],and ADP (ammonium dihydrogen phosphate, NH4 H2PO4 )crystals diffracting in various orders of the 422, 111, 002,and 101 planes, respectively. These represent a useful rangeof crystal perfection and character in the 0.1-11 A interval,with 2d spacings in first order being 2.3098, 6.27121,8.7358, and 10.641 A, respectively. Crystal and form factordata are discussed in Refs. 18 and 19, and primarily followRefs. 15 and 16 as well as Refs. 20-24.

Equations 3 and 4 are generally inadequate by 3%-12%or more near any edge structure of individual atomic formfactors, or up to 1%-2% far away from the edges [Fig. I(a)].Conversely, the use of Eqs. (2) and Eq. (Sb) in exact orapproximate form is usually in agreement with standard dy-namical diffraction at the 0.1 %-0.2% level over most of theangular range. For Si 111 at high Bragg angles, errors of thesimpler estimates easily reach 6% or 15%, where even im-

-DCo

11.0

8.0

5.0

2.0

(b) Si III

0.55 0.60 0.65

R, rad

1.1 1.2 1.30, rad

0.70

1.4 1.5

FIG. 1. (a) Estimates of refractive index corrections to the Bragg angle forADP 101 planes in the region of the P K edge: Eq. (3) (dash) and (4)(--)deviate by up to 12% from accurate formulas or experiment near the edgeand 1%-2% at a distance of 7° away, while Eq. (5b) (.) and (6) (-) appearcoincident on this scale. (b) Estimates of refractive index corrections to theBragg angle for silicon 111 planes above 63°. At near-normal angles, Eqs.(3) (dash) and (4) (--) err by 19% and 7% while Eqs. (5b) (-) and (6)(-)differ by 1%-2%.

proved estimates can differ by 1%. The larger errors remainat several percent for Bragg angles above 660 and 490, re-spectively.

Three conclusions can follow from these observations:(1) Equations (2) and (Sb), or Eq. (6), should be preferred toalternate forms. (2) Regions of near-normal incidence or ofabsorption edge structure should be avoided when possiblein the pursuit of precision wavelength measurements. (3) De-tailed profile calculations can avoid or minimize this uncer-tainty in refractive index corrections.

D. Modification for asymmetric diffraction

A deviation parameter y is commonly defined to includethe effects of polarization and crystal asymmetry using

5126 Rev. Sci. Instrum., Vol. 66, No. 11, November 1995

(5c) -0

t

Bragg x-ray diffraction


0.51-b b

2 210 ~Y= VNFRI IC O'I

(6)

where a=4(sin OB-sin O)sin OB, H is the real part of theFourier component of index H of 4ii times the polarizability,given by tVIH= - (roX 2 k 1rV)FH; C is the polarization factor[= 1 for the normal component (OT polarization), or Icos 201for Cr polarization], and b is the ratio of direction cosines (yr,yH) of the incident and diffracted (reflected) beams relativeto the normal to the crystal (lamellar) surface. Note that thereare two converse conventions for polarization states, bothwidely used; herein the unattenuated polarization with theelectric vector perpendicular to the scattering plane is definedas ir polarization and therefore has a polarization factorC =1. Commonly, this is the same as the electric vector be-ing parallel to the surface. The other convention would nor-mally reverse the labeling of the polarizations. For symmet-ric Bragg diffraction, b = -1 (but see Ref. 25 forqualifications in the use of these parameters in the tails ofBragg peaks or at grazing incidence diffraction). The refrac-tive index shift from the Bragg angle (corresponding to y=0at the profile center) then reduces to Eq. (1). In symmetricLaue diffraction b=l and this correction disappears. Moregenerally, this results in a refractive index correction givenby Eqs. (1), (Sa) and (Sb) but with Sand hence AO scaled by(1- 11b)12.

This relates to the shift of incident angle of the profile;for the output angular shift, direction cosines are inter-changed so b -* lb. Here, the dominant shift of the peakangle arises from the mean angle (ap) of the diffractingplanes at the surface to the surface normal-zero for thesymmetric case. Assuming that 0i = 0 r (the angles of inci-dence and reflection with respect to the diffracting planes areequal), which is accurate below the level of other simplifica-tions, this is related to b using

sin 0intsin Sou

I

cos 2a,,+sin 2a, Cot Oin'

a)0

'a70-

C0

C.,E2

0.3

0.1

-0.1

-0.3

-0.50.01 1.130.29 0.57 0.85

0, rad

FIG. 2. Plot of the effect of asymmetric diffraction, a,= 17.5 mrad, on therefractive index correction from the y=0 location to the Bragg angle. Thesolid curve indicates the fractional difference in the shift of the exit angle forasymmetric vs symmetric (Bragg) diffraction. The dashed curve repeats thisfor the incident angle, while the third curve relates to the difference betweenincident and exit angles.

0a)

e-

(7) (a)

10f

1(-2

0.166500 0.166570 0.166640O+a, rad

0.166710 0.166780

where Oin, and 0fout are the incident and outgoing grazing

angles relative to the crystal surface. For a finite, divergentsource or for curved crystals, this prescription is complicatedby the variation of b with surface location and with penetra-tion depth, respectively.

Experiments using collimated parallel incident beamsmay measure deviations of diffracted wave exit angle rela-tive to the crystal or relative to the incident beam; in thelatter case the deviations of ap would cancel. In the formercase, or where broad sources emitting in 4vr are used, the fullshift of tt, in angle of emission may be observed. The refrac-tive index contribution to this shift, relative to the incidentbeam, is given by the sum of the inward and outward devia-tions, by Eq. (5b) scaled by 1-(b+ 1b)/2>2. In the sym-metric Bragg case this factor is exactly 2.

Shifts of the exit angle relative to the crystal surface, asmay be measured on detectors observing broad sourceswhere the incident angle may be ill-defined, would involve ascaling of (1-b)/2 from the above. An example is given inFigs. 2 and 3 for ap =17.5 mrad (1°) and the diffraction from

A,_

(b)

-11.4 -5.5 04

y

6.3 12.2

FIG. 3. (a) The effect of asymmetric diffraction on Si 111 rocking curves forir (full line) and o-(--) polarizations. Asymmetry with a misalignment ofap = 17.5 mrad is indicated by dashed and dot-dashed lines, respectively, onthe angular scale O+ap. (b) The same comparison as in (a), but on the y(diffraction coordinate) scale.



-Ao -- , -I- _,

-10.0 0.0

y10.0 20.0

try. The corresponding peak shift is dy with -1 ;dy _O,where - I <v _ I covers the ideal diffraction (Darwin) width.A nonabsorbing perfect crystal has dy =0 and a top-hat pro-file. Very thin crystals also have dy-0 with wide Pendell6-sung oscillations (Fig. 4). However, most real (thick) crystalsin first-order diffraction have a peak for fr polarized radiationat dy-y -1 (Fig. 5). This is a direct consequence of the pref-erential absorption with increasing dy, or equivalently a con-sequence of the imaginary component of the structure orform factors of the crystal. This corresponds to a shift reduc-ing the refractive index correction by up to 50% or more(though usually smaller). For the incident wave and angle,this is

sib C 4Adysin Op-sm 6c- -2b sin OB

FEG. 4. Diffraction profile for a thin (0.4 sum) Si 11 crystal, at 0.7 rad.Reflected (diffracted) ratios are indicated for ir polarization and the weakercr diffraction. Note that the convention used for choice of polarizations isdiscussed in Sec. II D; herein the unattenuated polarization with the electricvector perpendicular to the scattering plane is defined as 7T polarization.

Si 111 crystals. The effect of asymmetric diffraction on therefractive index shift is 7% of the total correction when theincident angle is reduced to 12°, but is still 2% at 45°. Theeffect on the exit angle relative to the incident angle is muchless pronounced, being only 0.6% of the total refractive in-dex contribution at 12°. Note that reductions of the angularshift by 12% correspond to reductions of the angular diffrac-tion widths by the same fraction, but that profiles and widthson the diffraction coordinate scale y are invariant.

A simple solution is to measure the angle of the diffract-ing planes to the surface to high accuracy or to transformprofiles to a sin a scale before further analysis is conducted.Often these alternatives are not available or convenient, butthis uncertainty or shift can be minimized by the use ofhigher Bragg angles.

E. Peak profile asymmetry

Estimates of shifts based on Eq. (6) or on a scaled ver-sion of Eq. (5) following Sec. II D neglect profile asymme-

0.70

0.56

-1-'

0al)

0.42

0.28

0.14

0~00 I:---5.0

FIG. 5. Diffraction profileBragg reflected (diffractedweaker a- diffraction. Thethe y=0 location, with a p

7r polarization or a, away from wr14cr polarization near 7r/4

For the angular shift of the outgoing wave relative to thediffracting plane, this may be approximated by

A6(p C7)( ( bj)I7IdY( 2 sin B COS S), (9)

where the first bracketed term is unity in symmetric Braggdiffraction and the last bracketed term contains the explicitangular (energy) dependence of the shift. aY polarized radia-tion is near symmetric, with dy and any consequent effectvanishing, at angles near nr/4 rad.

For dy constant, I4 °' -2 gives a dependence upon theBragg angle similar to Eq. (6) (Fig. 6) (but dy is not gener-ally constant with the angle). This contribution is maximizedfor first-order radiation and decreases rapidly with increasingdiffraction order. This estimate may typically exceed the real(mean) shift by a factor of 4 or more, reflecting the smoothand gradual decline of reflectivity from the indicated peak tothe y = + I location [as opposed to a a function at y = --1 assuggested by the first part of Eq. (8)]. For sufficiently thin ornonabsorbing crystals, the actual shift from this source maybe very close to zero (as often for (a polarization and asopposed to the above estimate). There is a regime of inter-mediate crystal thickness where flat crystal centroid locationsdepend on this thickness, but the precision of results is gen-erally not limited by the corresponding uncertainty (cf. TableI).

This discussion has remained general regarding the ac-tual spectrometer geometry involved, although results havebeen given explicitly for a single-crystal device. Consideringonly flat crystal diffraction, there are numerous parallel, an-tiparallel, or other multiple crystal spectrometer arrange-

/ - - ---- \ ments for monolithic crystals or separate elements. Thereader is referred to Ref. 26 or other sources for a detaileddiscussion of these geometric and spectroscopic arrange-

-2.5 0.0 2.5 5.0 ments. Scans and measurements may be provided by varying

the final crystal diffracting angle with respect to the axis offor a thick (1 mm) Si 11l crystal, at 0.7 rad. h is rb oaigteageaon h omlt h1) ratios are indicated for iT polarization and the the first; or by rotating the angle around the normal to their polarization has a clear asymmetry relative to diffracting planes (for asymmetric diffraction or multiple-

peak near y=-. beam studies); or by varying angles symmetrically or asym-


0.060 -

0.048 -

.t 0.036

a)

'5 0.024

0.012

0.000-20.0

dy=1 --+-I'--+O,



15.0

9.0

3.0

-3.0

-9.0

-15.00.01

(a) Si 111

5.0

3.8

2.6 [

1.4

0.2

-1.0 _0.05

(b) PET 008

0.38 0.750, rad

0.38 0.710, rad

FIG. 6. Profile asymnetry given by the y =~0 to y=-Ipolarizations t--:--) compared to refractive index shif(full line). (a) For silicon It1 in first order, both polarizemetry of the same magnitude as the refractive index esthe angular range. (b) For PET 008, both polarizations shprofiles, with the asymmetry below the 10% level.

crystal types used in the different x-ray optical elements, toimprove the resolution or control the bandpass. 1 Again, thedetector may be represented by a rectangular slit or by amore complex position-sensitive device. This kind of com-plexity is not of direct concern in the current paper. Single-crystal results should be convolved together following theappropriate geometry. 1 2' 13

--------- If only the final crystal element is rotated, relative to theoptimized peak orientation, then the (+,-) geometry men-tioned above (with a broad detector) has a peak broadened bythe first crystal profile and shifted toward the mean value,with a resulting symmetric profile. Conversely, the rotationof a narrow or position-sensitive detector in the preciselyparallel arrangement yields an asymmetric, narrowed profile

1.12 1.49 of higher resolution than the single-crystal result, which isshifted toward the peak value. In general, the observed peakand mean values are dependent upon the spectrometer geom-etry and alignment, but lie between the single-crystal peakand mean values.

Although the contributions in Secs. IT A-Il D are rea-sonably well defined, the effect in this section is difficult toevaluate separately from a dynamical diffraction calculation.Instead, we have indicated upper and lower limits for the

X effect, which are typically uncertain at a few percent of therefractive index correction. Some procedures model reflec-tivity profiles (following conventional two-beam dynamicaldiffraction for flat crystals) as a function of angle and thenderive the mean shift from the Bragg angle. This explicitlyavoids the approximations represented by Eqs. (3)-(5), andis equivalent to the use of Eq. (6). More important, this pro-cedure represents a considerable improvement over the fore-

1.04 1.37 going asymmetry estimates, as the centroids can be evaluatedexplicitly for the appropriate crystal thickness (and even

I shift for iT and (6 spectrometer geometry). The problem of evaluating system-ftfollowing Eq. (6)

zations show asym- atic shifts is then partially reduced to a computational prob-stimate for much of lem.how near-symmetric

metrically for a double-crystal pair or for the wholesystem. 12'1 3'26-28 Where multiple separate elements are in-volved, the number of independent axes and free parametersallow greater flexibility in the scanning procedures whilealso increasing the difficulty of the alignment process itself.For example, in a parallel double-flat crystal geometry (oftendenoted +,-) with identical crystals in both positions, wherethe angular acceptance is large compared to the diffractionwidth, the peak occurs in the true parallel position, while thezero location could be aligned with the mean diffractingangle (e.g., with a broad detector) or with the peak diffract-ing angle (with a narrow detector) broadened only by thenatural linewidth and the source size. The overall shift due toprofile asymmetry will generally be a convolution or super-position of single-crystal elements, which individually fol-low the relations given above. Additionally, the extreme lim-its represented by dy =- 1 and dy =0 are not exceeded.

Some devices make use of asymmetric cuts relative tothe diffraction planes, in order to suppress higher-order ra-diation or change outgoing divergence.29 30 Others vary the

F. Databases, crystal structure, and purity

The sources for crystal data illustrated in Sec. 1I B havetheir own uncertainties which can dominate in the calcula-tions for some crystals and angles. This also assumes thatimpurities or variations between crystals of the same typecan be controlled or minimized.

Silicon is a good example, providing well-defined crystaldata with high lattice perfection and good form factor data inthe energy range considered. For example, the form factorsshould be accurate to better than 1% in Fig. 1(b). In manyangular ranges this small form factor uncertainty can providethe limiting accuracy of refractive index corrections and ofdynamical diffraction calculations of the profile asymmetry.

Other crystals and energies have relatively large formfactor uncertainties or relatively large atomic position uncer-tainties, particularly those crystals which are not monotonic.Uncertainty in the form factors (typically at the level of afew percent for low to medium energies) commonly domi-nates over coordinate imprecision in the determination of thestructure factor FH for Eq. (2) or Eq. (6), and hence fordetermining profile shifts. In particular regions, form factor

Rev. Sc!. Instrum., Vol. 66, No. 11, November 1995

*t

-a

0

-a

-e

7-

Bragg x-ray diffraction 5129


uncertainty can dominate over other systematic contribu-tions, including refractive index corrections. 32' 33

G. Multiple beam interaction: Forbidden reflections

Major effects on asymmetry, reflectivity, and systematicshifts occur near "forbidden" reflections or where interfer-ence occurs with additional diffracting beams. These loca-tions depend on the orientation of the crystal in aligning theadditional reciprocal lattice points to near the Ewald sphere.

This occurs, for example, in the 442 reflection of silicon.Figure 7 indicates the locations of resonance with additionalbeams in diffraction from the silicon 442, 111, and 444, ADP101, and Ge 422 planes. A lower wavelength limit has beenintroduced since the number of curves (interactions) roughlyfollows the inverse cube of X.

Kinematically, in the weak field limit, each interactioncontains an infinity, so that the peak reflectivity (in the origi-nal direction) will either be zero or will be increased by

0.775 1.550

A, rad2.326 3.100

1.57 -

1.43

_0 1.294E19

D 1.15

1.01

0.870.000

(c) Si 444

0.775 1.550 2.325

Vo, rad

0.775 1.560 2.325g, rad

3.100

2.310

2.147

1.984

1.821

1.658

1.495-3.14

(d) Ge 422

-1.57 0.00qV, rad

1.57 3.14

FIG. 7. Plot of resonances in the (two-beam) diffraction profiles due to interaction with a third (diffracted) beam. Plots of wavelength (X, A) or Bragg angle(0, rad) vs the azimuthal angle (0, rad). 0 is measured relative to the normal to the lowest hkl Bragg plane providing an interaction in the plot region. O covers27r rad, but the plots are symmetric. In the plots, each curve represents the peak interaction of one or more (off-axis) lattice planes with the primary and normaldiffracted waves. (a) Three-beam resonances in ADP crystals with the normal diffracted beam reflected from the 101 planes, relative to the 001 normal. Theseplanes provide vertical Hoes on the plot at qY=0, 7r. The highest two X curves on the plot are due to interactions from 011, 0-11, 1-10, and 110 planes,respectively; the doublets (beginning at 7.55 A) arise from 002, 10-1 vs 200, -101 planes (due to the inequality of the a and c lattice vectors). At 6= 17.5'or X=3.2 A, there are 1048 interactions in 27r, or 262 "interfering" planes, commonly overlapping to provide -262 curves in 21r. (b) Three-beam resonancesin Si crystals with the normal diffracted beam reflected from the Ill planes, relative to the 004 normal. These planes provide curved lines on the plotbeginning at X=2.65 A. The highest X curves on the plot are due to interactions from 1-11 and 11-I planes, respectively. At 0= 12.50 or X= 1.36 A. there are960 interactions in 27r, or 240 interfering planes, providing -480 curves in 2sr. (c) Three-beam resonances in Si crystals with the normal diffracted beamreflected from the 444 planes, relative to the 004 normal. These planes provide vertical lines on the plot at 1=0, ir, with 400 and 040 planes providing verticallines at qr5='/3 and 2ir/3, respectively. The curves for the hkl planes superimpose with those for 444-hkl. The highest 3 X curves on the plot are due tointeractions from 18 odd-index planes. At 0=50' or X= 1.20 A, there are 960 interactions in 2ir, or 240 "interfering" planes, providing circa 240 curves in27r. (d) Three-beam resonances in Ge crystals with the normal diffracted beam reflected from the 422 planes, relative to the 00-2 normal. These planes providecurved lines on the plot beginning at X=1.718 A, with 002, 022, 020, 400, 402, and 420 planes providing vertical lines. At 0=40.2' or X= 1.4938 A, thereare 792 interactions in 27r, or 198 interfering planes. (e) Three-beam resonances in Si crystals with the normal diffracted beam reflected from the 442(forbidden) planes, relative to the 004 normal. These planes provide curved lines on the plot beginning atA= 1.706 A, with 224 and 400, 040 planes providingvertical lines at qS=0, 7r and 0=-1.25, 1.89, respectively. Note that these planes are allowed, but that the three-beam interaction is proportional toF(hkl)F(hkl-442), the latter of which is forbidden. A consequence is that vertical lines shall occur with equal intensity at q= -1.89, 1.25 due to theforbidden planes 402, 042. At 0=48.7° or A= 1.36 A, there are 624 interactions in 2zr, or 156 interfering planes.


10.65

9.16 _

7.67

6.18

4.69

3.200.000

(a) ADP 101

1.5700 -

1.2997 -

D 1.0294

(0

X 0.7591

0.4888 X

0.21850.000

(b) Si 111

3.100



1.81 _ - . _

1.72

1.63

1.54

1.45

1.360.000 0.785 1.570 2.355

(e) Si 442 (forbidden) p rad

FIG. 7 (Continued.)

many orders of magnitude (subject to energy conservatDespite the simplicity of kinematic models, this drawsuppression or amplification has been observed.3 4 3 6

this strength of interaction, the critical issue here relatethe widths of these features in azimuthal angle 0 (the ain the primary diffracting plane, normal to the plane ofdence and relative to some secondary plane).

The kinematic approximation becomes increasivalid as q5 is shifted away from the interaction resonancation. One estimate of the interaction width in k spa(then given to a reasonable approximation by the distfrom the resonant center at which the effect of the third Egives a doubling of the intensity (in the kinematic mod(

For a weak or forbidden primary diffracted beamdominant effect of most secondary diffracting beams wito enhance the profile by orders of magnitude. The "intotion width" mentioned above is then considerably largerthe FWHM (full width at half-maximum). Lower cstronger interfering reflections have (much) larger interawidths. If the value of (k lies within the interaction widtheffect on intensities, asymmetry, and systematic shifts calarge.

Extreme cases are indicated by planes such as si442 [Fig. 7(e)], where the reflection is geometrically foden so far as the spherically symmetric atomic form factconcerned, but is allowed by (weak) scattering from theferent site symmetry of the bonding orbitals. For this retion, the interaction width with the -1,1,-1 plane1.543 35 A is measured to be 1.4° or 2.1° (dependingthe phase and shape of the interaction). A simple estibased on the result of Shen36 yields an estimate withfactor of 2 of this, and enables some qualitative concludeto be drawn. The maximum interaction widths in qSoccur at the lowest energy for a given interaction, whialso the region of narrowest width features in X or 6 pri(and vice versa). Interactions with forbidden reflectionwith reflections whose complement is forbidden, have etially negligible widths in either space. Maximum widtt space for interactions with strong reflections regsreach ten degrees. The profile and hence detailed effe

such an interaction depends on the multiplicity of interac-tions, their phases, and relative strengths, but may at least beestimated in the single interaction (i.e., three-beam) assump-tion.

For interactions with only a single plane, the coverage of(h space in this case (Si 442 at 1.81 A) exceeds 1%; as 1.54A, or 580 Bragg angle is reached, this has increased to about260 coverage or 7%-8%. Interaction regions increasinglyoverlap one another and the probability of significant inter-action for a given wavelength and azimuthal angle becomeslarge. The two-beam diffraction profile width of i0 3-10-5

degrees is easily dominated by the 0.004°-0.29°-1.33° in-teraction widths in OB space. Hence in passing through the

3,140 diffraction profile, each side of the peak will be amplified (orsuppressed) by large and asymmetric factors. Around thepeak, this will typically shift the centroid shift due to profileasymmetry from y=-l to y=+l (outside this region therate of decline of the reflectivity will often exceed the rate ofincrease of the three-beam induced asymmetry). This can

tion). then be as large as or larger than the peak profile asymmetrymatic discussed earlier, and hence as large as the refractive index

With corrections.'es toangle H. Multiple beam interaction: Allowed low-orderinci- reflections

ADP 101 and silicon Ill [Figs. 7(a) and 7(b)] representingly near-ideal cases where the lowest order strongly diffractingat lo- plane is considered. Here for high Bragg angles there is noLce is competing third diffracting plane, and this effect cannot oc-tance cur. Even at medium angles (e.g., 450) there are only a fewbeam interactions, but as the Bragg angle is reduced below 200[el). these-planes and interactions proliferate. Above 100, orienta-, the tion of the azimuthal plane of the crystal by rotation of 0, to

ill be higher and higher precision with decreasing wavelength, canterac- explicitly avoid such interactions. At sufficiently low angles,^ than this procedure will again exceed the precision of the source)rder, and crystal alignment, even if the orientation is known toaction high precision. Near grazing angles, or in the low-angle tailsh, the of Bragg peaks, the Fresnel (or 000) beam and reflection

an be becomes dominant and must be included (separately fromany other multiple beam interactions).25

licon These allowed and low order planes (ADP 101, Si 111)

rbid- have much fewer and weaker interactions than presented intor is Sec. HI G, with maximum widths in q space of the order ofe dif- 0.001°-0.0001° leading to quite small percentage coverages.eflec- Diffraction widths of the order of 0.01°-0.001° also domi-es at nate over typical interaction widths of 0.0006°-0.000 0010upon (except for the curves of constant 0), so that three-beam

imate interactions appear as rapid fluctuations on top of a narrowhin a region of the normal two-beam profile. Here induced asym-

sions metric shifts of centroids would generally be considerablyspace smaller than the peak profile asymmetries discussed above.ich isfiles

i or 1. Multiple beam Interaction: Allowed medium-orderIs, or reflections,ssen-ths in For intermediate cases such as silicon 444 or germaniumsalary 422 [Figs. 7(c) and (7d)], there are interactions at all the-ct of diffracting angles. Thus, the azimuthal angle can never be

Rev. Sci. Instrum., Vol. 66, No. 11, November 1995 Bragg x-ray diffraction 51 31


ignored, and the situation for high Bragg angles correspondsto that for low Bragg angles in the near-ideal cases (Sec.II H). Here, several hundred (or thousand) interactions occurfor any given wavelength below angles of 400 or so-eachinteraction with a given plane in general occurring at a set offour specific azimuthal regions (following the plane andcrystal symmetry). These four regions in 2iT correspond totwo independent curves, with an apex at the highest wave-length (or Bragg angle) for which the two planes involvedinterfere. Some of these interactions and some of thesecurves cross or overlap one another, depending upon the lat-tice symmetries. However, there still remain several hundredor thousand curves and interactions to be accounted for, andthe alignment and tolerance on the spectrometer is typicallyinadequate to explicitly avoid these interactions.

For the allowed but medium-order crystal planes (Si444, Ge 422), interaction widths in 0 and 0 are several or-ders of magnitude smaller than for forbidden reflections.Maximum widths for strong reflections in 0 space rangefrom 0.10 to 0.010, while coverages in ¢k space at 0=0.8717rad for Si 444, and at 0=0.7033 rad for Ge 422, are esti-mated at 0.18° and 0.24°, respectively. Coverages less than0.1% imply that interactions may be avoided or neglected inmany cases at these medium angles. For 1.2 A, however,diffraction widths of 0.001°-0.000 010 approach interactionwidths in OB space of 0.001°-0.000 0010. Then changes ofintensity at far tails are negligible, but major and rapidchanges occur within these widths, and hence within the pro-file widths, leading to asymmetries which could easily cor-respond to shifts of the mean diffraction coordinate by Sy- 2 or more.

In the high-angle cases, measurement of crystal orienta-tion can allow these interaction regions to be avoided, butthis is not true for the higher order diffraction and lowerwavelength regions. Also, the use of calibration lines or otherrelative measurement techniques cannot address this sourceof error or systematic correction unless a dense set is avail-able. In all cases, however, high experimental resolutionshould confirm or eliminate hypothesized effects of three-beam interactions. Diffracted intensities much larger orsmaller than expected provide an indication for these inter-actions. The rapid oscillation of intensities over (narrow)ranges of the smooth diffraction profile, separate from andsuperimposed upon the Pendellosung, is also a strong indi-cator of these interactions. The symmetry of these features isoften able to determine 0 to better precision than may bepossible from alignment considerations. Two-dimensionaldetection is able to identify changes of (convolved) reflectiv-ity with the variation of X and k, and hence provide muchmore restrictive limits on possible multiple beam interac-tions.

J. Tertiary corrections

For flat crystals, this essentially completes the summaryof effects leading to systematic shifts of the Bragg angle ordetector position for centroids around diffraction peaks. Fig-ure 8 illustrates the agreement of the sum of these estimateswith detailed profile calculations.

10~3

10. 4

0.77 0.95 1 13 1.31 1.49@, rad

FIG. 8. Sums of contributions to flat crystal peak or centroid values for Si1 II diffraction. The refractive index shift, Eq. (6), is given by ... while theestimated mean based on a peak at y = - I is indicated for a and 7r polar-ization by the solid and long dashed lines converging at normal incidence;the actual peak for T=0.4 mm is indicated by solid and short dashed curves,converging at normal incidence. The latter corresponds approximately withy = -0.33, and there is fairly good agreement of the final mean shift (--and

-, respectively) with a value corresponding to yak4.

Corrections to the above prescription include the accu-rate derivation of a mean dy value from estimates of asym-metry and the use of a more exact arcsin expression [fromEq. (6)] rather than the simplified form indicated in Eq. (5b).

Mosaicity has potentially significant effects on centroidlocation, defocusing or broadening asymmetric profiles tocenter more on Oc than Op. This is important for thick crys-tals (as defined by Sec. II E) when the mosaic block size liesin the thin or intermediate regime. The profile asymmetryand mean can then be sensitive to the mosaic parameter andbe shifted toward 0C by the amount discussed in Sec. II E.

The overall asymmetry can significantly affect centroiddetermination of experimental profiles, depending on the fit-ting function. Additional broadening from Doppler effects inthe source or natural linewidths will convolve the asymmetryand lead to uncertainty in centroid determination. Asymme-tries of tails become important at the 5%-10% level and arethe subject of a succeeding paper.

General fiat crystal geometries involve broad detectorswith negligible positional sensitivity, with the idealized loca-tion collecting (integrating) most radiation diffracted from agiven plane (or planes). However, the possible use of a nar-row or position-sensitive detector will then involve concernfor the profile shape and location at a given distance; andhence involve correction or allowance for lateral shifts upondepth penetration. This is usually of negligible significancefor flat crystal measurements, while being of significance forcurved crystals. Thus, it will be discussed in Sec. Emi. Thethermal loading of a flat crystal, as is important for synchro-tron sources and hot plasmas, leads to stress and distortion ofthe lattice planes and diffraction; usually this creates an ef-fective crystal curvature and so will be addressed briefly inSec. III.

5132 Rev. Sci. Instrum., Vol. 66, No. 11, November 1995 Bragg x-ray diffraction


Some of these corrections require evaluation of a (full)dynamical diffraction theory; others require careful ray trac-ing or convolutions of x-ray optical elements.

K. Summary of Sec. 11 (flat crystal systematics)

Experiments requiring absolute wavelength determina-tion to better than 100 ppm in the x-ray regime must gener-ally involve careful consideration of a variety of effects inaddition to refractive index corrections. Although simple es-timates for the latter can be accurate to a few percent, thereare significant regimes where these simple estimates are in-adequate at the 5%-20% level. There are relatively well-known modifications of these corrections due to the nonpar-allelism of diffracting planes with the surface, of similarmagnitude. Uncertainties in the angle of the diffractingplanes to the surface can provide significant uncertainty inexperimental results, even when simple formulas are avoidedand detailed computations are performed. Profile asymmetrydue to dynamical diffraction can be of the same magnitude asrefractive index corrections themselves, particularly for low-order diffraction of soft x-ray ir-polarized radiation withthick crystals. The azimuthal angle is often quite uncertain,and can lead to large shifts of diffraction profiles due tomultiple diffraction. Often this uncertainty can be eliminatedby experimental design.

For flat crystal diffraction, these contributions are rela-tively straightforward, even if computationally complex, andassociated magnitudes and angular dependencies have beendiscussed and presented in simple formulas. The variation ofangular shifts with the Bragg angle can follow tan 0, cot 0, or0 independent relations, with particularly strong local fea-tures and alternate dependencies near edges or near three-beam diffraction points. The precision of profile shifts can belimited at 0.1%-1.0% of the refractive index corrections byform factor uncertainties, in optimum cases. The use of im-precise or inadequate formulas as represented by Eqs. (3) and(4) or by neglect of profile asymmetry will generally lead tolarge errors in derived wavelengths, well in excess of thislevel.

Measurements of unknown wavelengths relative to anearby calibration line by extrapolation can increase the pre-cision by an order of magnitude, in relation to refractiveindex corrections, profile asymmetry, asymmetric Bragg dif-fraction, and other effects, but requires an understanding ofthe functional relations indicated above for higher precision.Locally, this can be provided to first order by the slope of theresponse between several calibration lines. However, cancel-lations of the effects of most absolute shifts assume that therelative measurements are made in the same order and withnearby Bragg angles, away from absorption edges and awayfrom extreme spectrometer angles.

Multiple-beam interactions are generally not determinedor accounted for in such calibrations. If the azimuthal angleis inadequately known, other methods must be used to con-sider the importance of these effects. In some regimes, varia-tion with angle of several component effects is far from con-stant or linear. Care should be exercised in avoiding theseregimes or, again, in understanding the expected functionaldependencies and relative magnitudes.

The complexity of shifts with order and polarization inspectra necessitates detailed calculations of the sort indicatedhere for a precision approaching 1% of the refractive indexcorrections. Calculated shifts agree well with the sum of sim-pler estimates, reproducing the dependence on the Braggangle.

Ill. CURVED CRYSTAL SYSTEMATIC CORRECTIONS

A. Overview

Curved crystal diffraction displays all the effects dis-cussed in Sec. II, but with a modulated amplitude and rela-tive importance. Section III B considers the typical effect ofcurvature on the flat crystal relations. It also summarizesvariable definitions, as the formulas presented are numerousand relate to effects which are not generally well appreciated.

The remaining sections address additional systematicsaffecting the results of curved crystal Bragg diffraction. Ex-plicit functional forms are provided for the Johann case, al-though many relations are of general application. This paperis concerned with focusing and shifts dependent on the dif-fraction process within the crystal.

Section III C will introduce one of the potentially domi-nating effects for curved crystals: namely the mean penetra-tion depth of the incident wave field. Several approximationsare presented in order to indicate the degree of detail neces-sary for the different experimental regions. Section III D in-troduces the shift of the mean angle to the diffracting planesas a function of crystal depth. Section III F will discuss thelateral shift around the crystal surface of the exit locationrelative to the incident location of the photon. These effectsof the diffraction process rather than the geometry are rarelyconsidered and estimates are often in error by orders of mag-nitude.

They are correlated with effects relating to Johann aber-rations, and so cannot be treated in an isolated manner (thetopic of Sec. III E).

Geometrical defocusing and shifts due to the differentRowland circle and diffracting crystal radii (in the Johanngeometry) leading to variations away from the pole axis arenot the primary concern of this paper. However, finite sourceand crystal dimensions interact with defocusing shifts anddiffraction corrections. Correlated results for the Johann ge-ometry are discussed in Secs. III G-I11 J, which address:general principles and earlier work; crystal length along thedispersion direction; crystal and source depths; source lengthalong the dispersion direction; and crystal and sourceheights. Ray-tracing packages3 7 and earlier formulas'7 areoften not adequate for this purpose.

A series of usually smaller corrections and effects will bediscussed in a separate section, including the exact asymme-try and extinction, detection corrections, mosaicity, diffrac-tion plane orientation, and 2d spacing.

High accuracy is certainly possible with curved crystalmeasurements in many cases, either by explicit avoidance ofproblem regimes or by adequate correction for the systemat-ics involved. Precision measurements with curved crystalsare often made by comparison of unknowns to calibrationlines. In this case, only the difference in refractive index and



TABLE II. Basic notation.

X: X-ray wavelength (in vacuo); d: crystal lattice spacing (for in-dex H);2Rz: crystal curvature radius along the generatrix (Rz is theRowland circle radius);n: order of diffraction; pa,: refractive index (near unity);FH: structure factor for hkl index; ro=e2/mec2 : classical electronradius;Oi/, 4: real and imaginary parts of the Fourier component ofindex H of 4ir times thepolarizability OAH= - (e2 X2

1Trmc2 )FHIV;

both parts are summed over real (imaginary) contributions fromeach scattering center so they may be complex, except for theH=0 (hkl =000) component parts;sin OB=nk/2d: sine of Bragg angle; a=4(sin 0 8 -sin Oesin 0B;V: unit cell volume;apple (mean) angle of diffracting planes to the lamellar or crystalsurface (along the generatrix);C: Polarization factor [= I for normal component ( r polarization),or 1cos 20 for t polarization);Oinc grazing angle of incidence on crystal surface;ot: grazing angle of emission/reflection at crystal surface;Yz=2Rz 9

out; detector arc around the Rowland circle from the poleaxis of the crystal [Eq. (5d)];b=-sin 0j./sin 90.t=-[l/(cos basin 2cr, cot ind)]: ratio of thedirection cosines (yo,yH) of incident and diffracted (reflected)beams relative to the normal to the crystal (lamellar) surface[Eq. 7];

I-b bRekz) 2_______2-_WR ) 2 = : the deviation parameter [Eq. 6];

OC: angle corresponding to y=O, defining the refractive indexshift;dy: value of y for the peak of the (flat crystal) diffraction profile;0p : grazing incidence angle for the peak of the (flat crystal)diffraction profile;T: crystal thickness; A: linear absorption coefficient.

other corrections is directly relevant, and experiments canideally achieve sensitivity of 20-30 ppm without demandingaccurate values for the main corrections, as discussed in Sec.II. Calibration lines must then be measured on an absolutefooting elsewhere at a similar level of precision.15 ,

16

B. Variable definitions and flat crystal systematics

Equations (l)-(9) summarized the relations between thequantities in Table II and the refractive index shift, asymmet-ric Bragg diffraction, profile asymmetry, and multiple-beamdiffraction for perfect or mosaic crystals of variable thick-ness.

For flat crystals, this completed the systematic shifts ofthe Bragg angle or detector position for centroids arounddiffraction peaks. Figure 8 indicates the kind of agreementcommonly obtained using the estimates and relations indi-cated therein, compared to detailed profile calculations.

Curved crystal geometries are generally less affected bythe peak profile asymmetry because the effective thicknessleading to coherent diffraction is typically much less than thecrystal thickness. The calculation of the flat crystal profileasymmetry component in a curved crystal calculation musttherefore use a lamellar or effective thickness dependent onthe curvature and diffraction, rather than the crystal thick-ness. An appropriate estimation of this lamellar thickness

FIG. 9. Indication of depth penetration in curved crystal Johann geometryfor Bragg diffraction, with associated variables (BP, BX, XP, XZ, &,,,~, OAAT, t0 , yo) on the generatrix.

helps to ensure that coherence between contributions fromnearby depths is allowed for, while the incoherence of con-tributions from well-separated depths is also treated. Themain consequence of inadequacy in this area will be to in-troduce additional effective flat crystal profile asymmetry.Fully dynamical models of curved crystal diffraction, treat-ing amplitude rather than intensity propagation, can in prin-ciple eliminate this difficulty.

The effects of multiple beam interactions are also re-duced in curved crystal diffraction: both by a broadening ofeach diffraction width so that interactions are better repre-sented by rapid fluctuations on the two-beam profile; andalso by generally incoherent broadening from the range ofcrystal locations at which diffraction occurs. Hence the ma-jor shifts from Bragg angle diffraction for flat crystals (insymmetric Bragg diffraction), after curvature, are dominatedin most cases by the refractive index correction. There arealso many cases where the effective thickness is not signifi-cantly affected by the curvature, and where the two-beamdiffraction width is not greatly increased by curvature. Thisdepends on the crystal, thickness, curvature, and wavelength.

Flat crystal instruments tend to use broad detectors withnegligible positional sensitivity, whereas positional sensitiv-ity is usually required in curved crystal instruments in orderto make use of the focusing geometry. This difference causesa shift of interest from the diffracting angle of the crystalwith respect to the source, detector, or monochromator, to thelocation of the radiation on the detector relative to the Row-land circle and the crystal center or pole axis. 81 9 This wasdiscussed briefly in Secs. II B and II E.

The qualitative effects discussed in flat crystals aretherefore present and significant in the context of curvedcrystals; but the magnitudes of the corrections are often re-duced, and the formulas commonly need to be reevaluatedwith a smaller effective crystal thickness, which may lie inthe intermediate or thin crystallite regime while the crystalthickness T might correspond to the thick crystal regime(Secs. II E and U J). Other qualitative effects introduced bycurvature will also tend to dominate over those correctionscommon to the flat crystal case.



C. Curved crystal depth penetration

Dominant contributions to higher diffraction order shiftsfrom the Bragg angle arise from the varying mean depth ofpenetration with angle. To first approximation, this is due tothe variation of mean grazing incidence angle at the surfacecompared to the Bragg angle at the diffracting planes. Thiseffect is nonexistent for flat crystals, but can be large foreven weakly curved crystals. With the linear absorption co-efficient pa, crystal thickness T, and mean vertical penetrationdepth d (cf. Fig. 9), the resulting shift is estimated by

A Oin/outtarccost (1 + 2R ) cos( OB ± ap) ] B ± ap)

-d~ 2- cot(B± ap),

where d may be estimated using

(10)

0a

TmraxeiA(P-+P+)dt '

Tma min{T,2R z((± 7 )]

P+ =(2Rz+ t)sin(OB± ap)

- V[(2 R + t)sin( OB ±ap)]2 -4tR,-t 2 ,

a-fTt-p4.1/sin Oout+ '/sin i ncident J f 0T e - outline 0But+ sin

i ncidents

rTIi,~~~~~~~~~~~~~~~

(~ 1/snO]a _si(Bap) 1~-sn B

F--

For thin crystals or high order radiation the former case[Eq. (1 lc), d-T/2] is appropriate. For thick crystals, lowenergies, and large Bragg angles, Eq. 11(d) yields a usefuloverestimate. For near grazing incidence, Eq. (1 la) is neces-sary. The approximate forms, particularly in Eq. (10), are ofquite limited validity, while the initial expressions are muchmore appropriate.

The expression for As is given relative to Al at normalincidence, to indicate the inverse cubic dependence of pho-toabsorption on energy away from the edges. The depen-dence on the Bragg angle bears little similarity to previouslyconsidered corrections, and also varies from low order tohigh order limits. Equation (11) indicates asymmetric dif-fraction through ap. a should include the mean effect ofextinction (diffraction) prior to the depth involved. For flatcrystals R, is infinite, so the contribution of Eq. (10) is zero;but for near-flat crystals, this extinction can increase p. byorders of magnitude near the peak. For significantly curvedcrystals, the diffraction process is effectively negligible priorto the diffraction peak, and the inclusion of "flat crystal"values for extinction leads to errors of orders of magnitude inthis correction. 17"8 38

Corrections for profile asymmetry and attenuation(which depend on penetration depth) require estimates forFH and F', respectively, so they are not amenable to thesimplified Eqs. (3) and (4). More refined estimates of formand structure factors must be made if this precision is re-quired. This shift is often larger than the refractive index

correction and of opposite sign. Figure 10 considers thevalue of the approximate forms of Eq. (II) for low and highorders. Note the different regimes depicted. Use of the ap-proximate Eq. (lid) is generally invalid, although the quali-tative form becomes valid for high Bragg angles of ADP 1010.4-mm-thick crystals (with the given 300 mm curvature). Asthe crystal thickness is reduced or the diffracting order isincreased, Eq. (1 ic) becomes a more appropriate and accu-rate estimate over wide ranges of Bragg angles. Thus PET008 diffraction penetration depths are well represented byEq. (1 ic) for all angles, whereas this is far from true for PET002. In all cases, penetration depth becomes more importantin absolute and relative senses as the Bragg angle decreases.The potential uncertainty of the crystal thickness on the de-pendence of Eq. (1Ic) may be seen to be significant.

The use of Eq. (1lla) with Eq. (10) is able to yield thecorrect result in the general case, but requires a valid linearabsorption coefficient P-abs together with an appropriate esti-mate of mean extinction Next The former can be evaluatedand indicates the form of the dependence, but is typically afactor of 2 or so too small, depending on wavelength, curva-ture, thickness, and polarization. Figure 1 1 compares this es-timate to those derived from dynamical theory for each po-larization. The simpler estimate based on absorption isentirely adequate for PET 008 diffraction, except at normalincidence. For ADP 404, absorption estimates are excellentfor much of the angular range, while at higher Bragg anglesthe increasing thickness of the lamellar diffracting units and

Rev. Sci. Instrum., Vol. 66, No. 11, November 1995

(1la)

sin OB

sin OB

AV

(l lb)

(11 c)

(lid)



- l-lioF

0.040 0.415 0.790 1.165 1.540(a) ADP, T=0.4 mm 1, rad

102 N

-(D3

lo 10 .. ff

0.040 0415 0.790 1.165 1.540(b) ADP, T=4fLm 1, rad

'1 10

-(D5

0.05 0.42 0.79 1.16 1.53(c) PET, T=0.4 mm i, rad

FIG. 10. Estimates of centroid shifts due to depth penetration in curvedcrystals. (a) ADP 101 and ADP 404 diffraction with crystal thickness T=0.4mm, crystal radius 2R.=300 mm, estimating the shift of the exit angle atsurface vs Bragg angle. The solid line represents Eq. (1 Ic), and unsimplifiedEq. (10); (-) and (--) give the infinite crystal limit [Eq. (lid)] and thebetter estimate of Eq. (Ila) for ADP 101 diffraction; while (.-) and (---)give corresponding curves but for ADP 404 diffraction. Effects are larger forhigher orders of diffraction and for lower Bragg angles. (b) The same as for2 (a) for ADP with a crystal thickness T=4 um. Now Eq. (lIc) becomes agood approximation for ADP 101 and 404 over much of the angular range.(c) The same as for 2 (a) but relating to PET 002 and PET 008 diffraction.Equation (I Ic) is a good approximation for PET 008 for all angles, while theunsimplified formula is required for PET 002.

_0 107

I I-

-(77

0.22(a) ADP

CD

(X4

(b) PET

0.55 0.88i, rad

0.10 0.46 0.82i, rad

1.21 1.54

1.18 1.54

FIG. II. Estimates of centroid shifts due to depth penetration in curvedcrystals. Here the best of the previous estimates [Eq. (IIa), lower solid curvefor first order planes, dash for higher order planes] is compared to thoseobtained with Eq. (10) but using input d values including extinction, derivedfrom dynamical calculations (-.-- for first-order planes and 7r or a- polar-ization, dotted and upper solid curves for corresponding polarizations inhigher order planes; see the text). The low angle cutoff depicted representsthe minimum diffracting angle for the source geometry chosen. The fall-offfor first order diffraction at high angles is due in part to the small valuesinvolved and consequent computation difficulty in the dynamical calcula-tions. (a) ADP 101 and 404 diffraction for T=0.4 mm, 2R.=300 mm.Different source and crystal dimensions are compared in diverging shortdash and long dash lines from first-order curves at intermediate angles. (b)PET 002 and 008 diffraction for T=0.4 mm, 2R,=300 mm.

the near-flat nature of the crystal lead to reductions of theeffective depth penetration by polarization-dependent factorsof 2 or 3. For PET 002 and ADP 101 diffraction, extinction issignificant even at low angles, reducing depth estimates by afactor of 2 or 3; while at higher angles the extinction rapidlydominates in the region of significant diffraction, thus reduc-ing the expected depths by orders of magnitude.

Estimates using Eq. (1la) are best where the correctionis large, and may be compared (favorably) to other estimates[Eqs. (llb)-(lld)] across the Bragg range. The correction issensitive to absorption edges and increases (rapidly) towardlow angles. Linear interpolation between calibration linescan yield high accuracy away from these regimes. The un-certainty in the mean extinction, and hence the effect of pen-etration depth, is primarily eliminated only by the use of



0.46 0.55 0.64(, rad

0.73

_0co 10~ -

-e

0.41

(c) PET 002

0.68 0.95@, rad

1.22 1.49

-aCo

10

0.42(b) ADP 404

0.69 0.96i, rad

dynamical calculations. t7-19 While these equations are reli-able estimators, and the figures provide a useful comparisonof these equations, it is also useful to provide some typicalmagnitudes, as indicated in Table III, for comparison withflat crystal and other effects. The latter should be consideredunreliable in comparison to the former, but may be a moreuseful and direct presentation for researchers considering theimportance of component effects.

D. Mean angle of Incidence on diffracting planes forcurved crystals

There is a second contribution to the mean output angle,due not to the penetration depth but to the transmission-averaged mean diffracting angle. This arises because low in-cident angles will not yield a mean angle based on flat crystalreflectivities. Instead, the x rays are attenuated as they pen-etrate the crystal. Often before the flat crystal profile peakye-1 is reached, the transmission will become negligible.Thus a potentially large shift to lower angles results.

For low absorption and narrow (high-order) diffractionprofiles, this is generally a small effect, perhaps a few per-cent of the depth penetration estimate discussed above. How-ever, for highly absorbing and wide (first-order) diffractionprofiles, this can often be 50% or more of the shift due to themean depth, with the same sign. The magnitude depends

FIG. 12. Diffracting angle shift estimates. (a) ADP 101 diffraction forT=0.4 mm, 2R,=300 mm. The simple estimates of Eq. (12) for 7r and a'polarizations (solid line and dash, respectively) is within a factor of 3 of thecomputed (dynamical) value (--and -, respectively) below the absorptionedge. At higher Bragg angles the agreement for a' polarization improvesslightly while that for 7r polarization improves significantly, tying within10% of the computed value despite the rapidly decreasing magnitude andsome precision-dependent fluctuations. (b) ADP 404 diffraction for T=0.4mm, 2R,=300 mm across the full angular range. Here the estimates andcomputed values are smooth and well defined but bear little relationship toone another. Note particularly the failure of Eq. (12) for ar polarizationbetween 350 and 55° and other discrepancies by an order of magnitude. (c)PET 002 diffraction for T=0.4 mun, 2R=300 mm showing the complexityof estimates and calculations. In addition to the previous curves, the dottedand solid lines converging at normal incidence indicate a' and 7r polarizationcomputations evaluated over the full width at 1% of the peak reflectivity,

1.23 1.50 while the long dash at the top of the curve represents the total (1r) correction.Except for this last curve, all shifts are negative, as in (a) and (b).

critically on the curvature, since transmission coefficients aredependent on the effective layer (or lamellar) thickness. Inthese cases it is difficult to neglect this contribution.

Since the effects of depth penetration and curvature areof similar orders of magnitude and are interrelated, it is dif-ficult to isolate reasonable estimates for consequent shiftswithout pursuing full dynamical calculations (as indicated inSec. III C). However, an estimate may be made using onlythe flat crystal profile for the appropriate effective thickness,combined with an estimate of the angular shift between ad-jacent layers in the curved crystal. Then

A OT~ n 0p'

Avc(- 2b cos 0sin GB)

(1 2a)

(12b)

where AOf is the shift of 0i+I - 01 between adjacent steps iof the computation of the flat crystal profile reflectivity andtransmission coefficients ri, ti; Op is the mean angle of theasymmetric fiat crystal profile; and Ay( is an estimate of theshift in the diffraction coordinate y through the lamellarthickness. This estimate is often 2 (corresponding to the Dar-win or nonabsorbing full width), but can be much largerwhere the curvature radius is small, particularly for first-

Rev. ScL. Instrum., Vol. 66, No. 11, November 1995

Co

0.37(a) ADP 101

101



TABLE m1. Magnitudes of corrections relative to refractive index shifts: Typical estimates for curved crystals.

Factor Application Percentage Sec.

Asymmetric diffraction, I' Angle to surface' 2%-7% III BExit vs incident angle 0.1%-1.0%

Flat crystal peak shift Thick perfect crystal 10%-50% III BVery thin or ideally mosaic crystal 0%

Form factor uncertainties Generally 0.1%-1.0% in BPossible three-beam interaction Allowed, low order, high angle 0% III B

Allowed, low order, medium angle 1%-20%Allowed, medium order 10%-200%Forbidden 5%-100%

Depth penetration Low order, high angle 0%-10% in CLow order, low angle 10%-100%Medium order, high angle >100%Medium order, low angle > 1000%

Mean angle of incidence Low order 0%-50% HI DMedium order 10%-100%

Off-axis shifts High angle (low order)b >100% m EOn axis 0%-10%Low angle (low order >1000%

Lateral shifts High angle, low order 0%-5% III FLow angle, low order l0%-100%High angle, medium-order 1%-50%Low angle, medium-order >1000%

Source and crystal lengths 0%-20% HI G-III JCrystal and source heights 10%-200% HI KEmulsion (detector) shifts 1%-100% m Ld-spacing shifts Low order 0%-10% HI L

Medium order 0%-400%

'After correction for cep itself.bMuch higher for higher orders.

order radiation. Then the lamellar thicknesses, equivalent tomean coherence lengths, significantly broaden the profilewidth (on the diffraction coordinate or other scale).

This (simple) estimate is compared in Fig. 12 to a meanangle with respect to diffracting planes summed over an as-sumed source distribution and a diffracting crystal, followingdynamical theory with certain approximations. The compu-tation includes small (tertiary) corrections due to the crystalthickness and length limits, which truncate reflectivity pro-files at extreme angles. The source distribution is focused bythe crystal so that the reflectivity on the diffracting Braggangle scale may not follow a linear relation compared to thatfor the source distribution. In recommended geometries, thisnonlinearity is usually negligible but would also be ac-counted for in the latter procedure. The two results are oftenin agreement within a factor of 3, particularly at the lowBragg angles where the shifts are most significant. Above thePK edge for ADP 101 diffraction, ar polarization estimatesand computation agree within 10%, despite computationallimitations near this level.

However for ADP 404 [Fig. 12(b)], the result of Eq. (12)and the computation both indicate increasing shifts withBragg angle, but with quite different functional forms andmagnitudes. Equation (12) fails for or polarization at around450 because of the negligible transmission coefficient, and atnear-normal angles because of the large lamellar unit. In allcases, Eq. (12) is strictly valid (in the approximate sense)only for iteration outside a lamellar unit, i.e., if n approxi-mates unity. For an n significantly larger than unity withsignificant extinction through a lamellar unit, the component

relating to intraunit averaging does not behave like Eq. (12a)(the amplitudes add coherently and in phase). The peculiarbehavior of the computation in Fig. 12(b) in the neighbor-hood of normal incidence correlates with the decline in depthpenetration and the change of angle with depth, reversed asthe high-angle portion of the profile (not contributing to theangle shift) is truncated at 900.

Additional complexity for PET 002 is provided by Fig.12(c). When the magnitude of the shift falls below 10-6 rad,computational precision introduces large relative but negli-gible absolute uncertainty. Further, results below 60° aredominated by the central portion of the diffraction profile(above 1% of the peak reflectivity), whereas the effect of thediffraction wings or tails is significant for Bragg anglesabove 750, leading to increased sensitivity to source andcrystal dimensions (Secs. III C-Ill K).

The detailed correction is compared to other contribu-tions and the resulting (calculated) mean angle in Fig. 13 toindicate the typical good agreement of these contributionswith the total. Typical component magnitudes are illustratedin Table III (neglecting the detailed and complex angulardependence and functional form).

E. Off-axis diffraction for curved crystals

The mean angle resulting from flat crystal effects anddepth penetration contributions varies around the crystal sur-face, and originates at different locations. Ideally (i.e., at thepole axis), these sources converge in the Rowland circle ge-ometry to yield Eq. (5d), as indicated in Table II.



0.50 0.610, rad

0*O 30.0

22.0

-a(f

14.0

6.0

-2.0

-10.00.72 0.83 0.39 1.070.56 0.73 0.90

i, rad

FIG. 13. (a) Sums of contributions to mean exit angle at the surfacecurved crystal [mean exit angle (curved crystal) =mean exit angle (flatal) + D.P. angular shift + mean diffraction angle shift]. The soldashed lines at the bottom are (negative) depth penetration (D.P.) corralfor a and ir polarizations. (--) and (*-) give total mean angular shiftsthe upper curves (dotted and solid, respectively) give the mean angulfor the corresponding flat crystal (with a thickness correspondingregion of coherent excitation with significant amplitude, and hencecurvature). (---) and ( ) indicate related diffraction angle shifts. Aldiffraction for T=0.4 mm, 2R,=300 mm. The flat crystal result donwith up to 20% corrections due to the mean penetration depth and up tdue to the mean diffracting angle shift, both decreasing with incBragg angle.

However, for most spectrometers observing a signirange of wavelengths simultaneously (across a detectoreffects of the previous sections are normally highly clated with shifts due to the location of diffraction lyingfrom the pole axis (cf. Fig. 9). The first-order effect of lto reduce arc lengths Yz around the Rowland circle 12RZ0 by a percent or so, following the estimate

A Y -RZ cot Oout sin2 OAX

It0 2.0

0.0

-0(0

-2.0

-4.0

-6.0

-8.0 L-0.44 0.68 0.92

f, rad1.16

e of theat crys-did andrectionss, whilelar shift, to the

to theDP 101

FIG. 13. (c) PET 002 diffraction for T=0.4 mm, 2R,=300 mm. The dif-fracting angle shift provides an 18% correction to the flat crystal shift formuch of the angular range, while the mean penetration depth and flat crystalshifts are of a similar magnitude.

xP

OA (13b)

ninates, where XP is the arc length of the peak (mean) diffractionto 10%re~asi~ng along the generatrix of the diffracting crystal to the pole axis

(the amount by which the diffraction is off axis); (The gen-eratrix is the plane normal to the axis of the cylinder andpassing through the center of the crystal, as illustrated in Fig.

ficant 9.) OA depends on the source geometry and alignment, as ar), the function of outn. This function OA(0) is the same if 0= 0out is

corre- replaced by 0= OB or 0 = OB + A ORI++profile a/sl Hence just asaway YZ is here related to 0

outh SO YB can be related to OB, while

this is shifts of 0 can be related to shifts of Y.below The off-axis shift is zero where the crystal is tangential

to the Rowland circle [38° for the geometry selected in Fig.(13a) 14(a)], and also disappears at normal incidence. A peak shift

is therefore obtained at an intermediate angle (around 63°),with a larger shift at minimum diffracting angles. AlthoughAYZ=Y -YB is found to be linear in AO for each of thesystematic corrections to 0 (except near the pole), the linear-ity is often far from 2R, and can reach a ratio of 3Rz intypical cases. A point source B at a distance BP from thepole (cf. Fig. 9) and with a grazing incidence angle OAX to thesurface at the pole leads to a minimum grazing incidenceangle around the crystal circle

1.40

FIG. 13. (b) ADP 404 diffraction for T=0.4 mm, 2R,=300 mm. The meanpenetration depth provides the dominant contribution except at near-normalincidence, a- and or polarization curves are barely distinguishable, and thediffracting angle shift remains the smallest contribution.

0min= arccos <1- BP 2 sin BA,-B2Rj si A 2RI

(14a)

and

| co~s SAX\ I cos 0\OA = arcsin CS OAx acosi COS 0 - OAx, 0,

~COS lninl)COS OmnJ (14b)

where OA is measured in an anticlockwise direction, the posi-tive sign on 0 refers to most geometries, and the arcsin termscould fall within either (0, IT/

2) or (fT/2, iT) ranges, depend-ing on the geometry. Assuming symmetric Bragg diffractionat the surface yields 1Out-=O. This then specifies the relation

Rev. Sci. Instrum., Vol. 66, No. 11, November 1995

0 28.0

22.0

16.0 _

10.0

4.0

-2.0 _0.39

I I .

...



0.37 0.67 0.97

i, rad

2.0*103

E

LU

12 10c.'J.4

1.27 1.57 3.0*10'

FIG. 14. (a) Off-axis corrections and the effect on the proportionality ofdepth penetration and other shifts on the detector location on the Rowlandcircle. The difference between the location of detected radiation (assumingdiffraction at the Bragg angle at the crystal surface, for a point source) andEq. (5d) is indicated for ADP 101 diffraction in a typical geometry. Equa-tions (13) and (14) effectively reproduce this dependence.

of OA to 0 or 0out and allows explicit estimation of the effectsof mean angular shifts (refractive index corrections, profileasymmetry, nonalignment of diffracting planes, and meanshift of the surface angle due to depth penetration) on thedetector position on the Rowland circle.

The effect of off-axis corrections on the proportionalityof shifts along the Rowland circle compared with the angle isindicated in Fig. 14(b). For low angles with the source wellinside the Rowland circle, this is particularly significant,while shifts of a percent or more persist at quite large angles.For a source centered on the Rowland circle, the effects aremuch larger. This can distort relative measurements or mini-mize their sensitivity, unless each wavelength (or spectralline) is scanned to optimize the intensity as a function ofsource position.

F. Lateral shifts due to depth penetration

A shift of the exit location of the ray also arises due tothe penetration depth: this causes a transverse shift for flatcrystals, dependent on Au including extinction near the profilepeak. For curved crystals at the pole axis the shift of positionat the detector may be estimated as

AYA(AB>)R arccos[(2 cos B-cos A)cos A

-Vsin 2 A(sin A +4 cos2 B-4 cos B cos A)]

-- 2RB, (15a)

B= OB+ap, A=B+ +A, OA=AOin+ Aout, (15b)

where the last quantities are given by Eq. (10); or, less pre-cisely,

A Yzo-Rz cot ot sin 2 oAA (16a)

.Z cA t( / aP ) c XZt\

XZ~d[cot(OB+ ap) + Ct(OBs ap)]

0.37 0.57 0.77 0.97i, rad

1.17

FIG. 14. (b) The shift from the Bragg angle to the mean diffracting angle forflat crystals for o and 7r polarizations and ADP 101 diffraction, scaled by2R, (solid and dashed lines), is compared to calculated positions in thein-circle geometry (-- and *-) and the on-circle geometry of the text (-).The latter reproduce results of Eqs. (13) and (14) on this scale. While in-circle results are conveniently interpreted as significant proportionality er-rors with appropriate corrections, the magnitude and sensitivity of on-circlecorrections is not easily combined into this approach.

where XP follows Eq. (13b) (Fig. 9). A slight imprecision ofthe alignment of thin crystals on the Rowland circle, or thesimultaneous observation of a wide angular range (usingcalibration lines from a broad source, for example), leads toan XP of the order of 1-10 mm, dominating over XZ in Eq.(16). This then includes intrinsic shifts and defocusing fromthe off-axis position of diffraction on the crystal. Whereasthe use of XZ in Eq. (16b) or Eq. (15b) estimates the mini-mum (and negligible) off-axis correction, the use of XP inEq. (16b) may be set to the crystal length to estimate themaximum (off-axis and lateral) correction. Alternatively, thelateral estimate can be isolated from angular shifts due todepth penetration using

A Y, 5 = AYI(A 1 ,B) -A AYI( 2 ,B), (16c)

B=OB+ap, A = B + 0AAinA.t

A2 =B +OA, (16d)

where A, is derived from the extreme angle OA = 0A,max de-

fined by the size of the diffracting crystal and Al is derivedfrom this maximum value of XP (or OA) minus the lateralshift XZ (or AG1n, A6 u). These extreme estimates are indi-cated in Fig. 15(a), scaled to show changes in the effectiveBragg angle. The estimates vary from dominating over re-fractive index corrections and other flat crystal contributions,to being less than 1 ppm over the full angular range. The lastestimate, following Eqs. (16c) and (16d), is a useful overes-timate for comparison to other contributions.

Using the same equations, with OA provided from anexperimental geometry [Eq. (14b)] gives a more realistic re-sult, sensitive to the sign of the lateral shift. A more accurateestimate would separate the angular depth penetration cor-rection


E

Ii

0)

.14

04

10

2

10'i01

�2d cot OB, (16b)



0.29 0.59 0.890, rad

i3

E

co

CIA(NJ

2

10'

10°

-16

1.19 1.49 0.40 0.66 0.920, rad

1.18 1.44

FIG. 15. (a) Estimates of lateral detector shifts. Upper and lower limits inmagnitude are presented for ADP 101 and ADP 404 diffraction with T=0.4mm, 2R,=300 mm. The use of XZ in Eq. (16b) or the use of Eqs. (14) and(15b) generates the lower two curves for on-Rowland circle shifts, whilesetting XP to an assumed crystal length generates the dotted and dashedcurves, compared to the somewhat reduced maxima following Eqs. (16c)and (16d). These limits represent typical ideally aligned or ideally mis-aligned shifts. Shifts of the opposite sign would occur for misalignment onthe opposite side of the pole axis of the crystal. Maximum shifts for ADP404 are several thousand ppm at low angles, while even near normal inci-dence these effects can be of the order of 1% of the flat crystal corrections.

FIG. 15. (c) Absolute magnitude of shifts at the detector location (on theRowland circle) for Si 444 diffraction in the geometry indicated earlier. Thecurves have the same meaning as in Fig. 14(b). Here the simpler estimateprovides the correct detailed correction, except at high angles, despite theshifts being much larger than in Fig. 14(b).

from the lateral shift

AYlat= A YI(A 3 ,B2 )-AYI(A 2 ,B 2 ), (18a)

A YDP= AYl(A2 ,B 2) -A YI(A 1 ,BI),

Bl=OB+ AORI+Profile als' B 2 =BI+AO...t,

A 1=B 1+ OA(Bi),

E

LUCoII

10.0

-20.0

-50.0

-80.0

-110.0

-140.00.40 0.62 0.84

0, rad1.06

FIG. 15. (b) Shifts at the detector location (on the Rowland circle) fi002 diffraction in the geometry indicated earlier. The ideally aligned andideally nonaligned estimates of magnitude are indicated by solid and dashedlines, respectively. The curve -- represents the use of Eqs. (16c) and (16d),(14b), (10), and (lla), with ,u provided by the absorption coefficient only. Itrepresents the application of the earlier (relatively straightforward) estimatesto the "real" in-circle geometry, with the same assumptions regarding at-tenuation. Note the change of sign just prior to the Bragg angle where thecrystal lies ideally aligned on the Rowland Circle. By contrast, (--) usesEqs. (17b), (18), (10), and (lla), with the effective As including computedmean extinction for ir polarized radiation, and hence is the best estimate ofthe lateral shifts for a given geometry. The simpler estimate is well able toindicate the magnitude and sign of the correction, but for precise quantita-tive corrections to effective angles or wavelengths it is inadequate, whetherat the 100 ppm level for low angles or at the I ppm level for high angles.

(17a) A 3 =B 2 + OA(B 2 )- A Oi.-A out. (1 8b)

In these cases, OA(B) is given by OA(O) in Eq. (14b).(li7b) It is perhaps surprising that such large effects arise from

these angular and lateral shifts (Fig. 15). They vary from_ insignificant (the aligned, on axis estimate for lateral shifts)

to significant at most angles (depth penetration estimates) todominant at particular low grazing angles and potentiallylarge elsewhere (extreme off-axis estimates for lateral shifts).It is necessary to use the best estimates to hope to reachagreement within a factor of 2 of the true shifts. One diffi-culty relates to the need to include explicitly the off-axisnature of the shifts at the detector, rather than using maxi-mum or minimum values. A second relates to the difficulty inevaluating the mean extinction addition to the linear absorp-tion coefficient. For a quick estimate, ILabs may be calculatedexplicitly, and (for curved crystals) /ttxt may be neglected.This typically yields overestimates of depth penetration, and

1.28 hence of the associated angular and lateral shifts, by factorsof 2-3. For high orders of diffraction, the associated errorcan be negligible in appropriate crystals, angles, and ener-

for PET gies.For a small but finite source inside the Rowland circle

and arbitrary crystal dimensions, these effects sum to yieldthe total centroid shift of a profile entering a detector on thecircle. Often the associated precision can lie at or below 1%of the corrections to the Bragg relation, although some com-ponents may only be accurate to a factor of 2 or 3 if theabove simple formulas are used.

G. Finite dimensions and geometric corrections

Other effects due to the Johann geometry and finitesources are ray tracing exercises, which can give significant

Rev. Sc. Instrum., Vol. 66, No. 11, November 1995

-li-/

-II I!

11 ;.I. l

-91_�



modifications of on-Rowland circle relations and more com-plex off-Rowland circle effects with a consequent variationand distribution of shifts.

The source should ideally be well centered with respectto the (cylindrical) crystal curvature and the central plane ofthe crystal and the detector location. Nonalignment of thesource with the central plane of the crystal generatrix leads tobroadening and shifts dependent on the height of the sourceand crystal and the detector resolution in the transverse di-rection. Formulas for some of these have been discussedelsewhere,3 7'39

-42 and have an origin which is primarily ex-

ternal to the crystal. These references neglect attenuation,extinction, and diffraction profiles, and assume that the dif-fraction profile width is significantly larger than the spread ofgrazing angles at and inside the crystal (both in simple ana-lytic formulas and in deconvolved numerical procedures).The formulas and numerical calculations locate the crystalon the Rowland circle at the Bragg angle to the source for allwavelengths considered, with the source center also on theRowland circle.

They provide estimates of centroid shifts due to finitecrystal and source size. However, additional terms typicallyarise [cf. Eq. (18) of Ref. 40] which may dramaticallymodify the effective shift. A source location well inside theRowland circle leads to a range of Oi far in excess of thecrystal diffraction width, so that the mean shift must be sig-nificantly truncated or modulated. Finally, the normal proce-dure of comparing spectral lines imaged with a curved crys-tal allows only one (monochromatic) wavelength to beimaged at the crystal pole axis with 0B = OA, while the otherwavelengths necessarily follow an off-axis relation. At leastsome of this inconsistency or lack of applicability has beennoted earlier.41 By comparison, the program discussed inRefs. 17-19 takes explicit account of the off-circle relationsand includes broadening, convolutions, and shifts from finitesource widths and depths, and finite crystal widths anddepths.

H. Crystal length along the generatrix

This was discussed from a different perspective for aparticular case in Ref. 38 and may be addressed by consid-ering Fig. 14(a). Here the grazing angle at the pole axis wasdefined to be Ax=0 .68 7 677 rad, with an in-circle pointsource of BP =26 mm from the pole axis and a crystal radiusof 2R,=300 mm. If OB= OAX, then the detector locations forwings on both sides of the Bragg angle shall be reduced bythe amount indicated. (Of course, it would be more appro-priate to use Op= OAX*) The mean of these angles (modulatedby some reflectivity and angular source distribution) wouldbe shifted strongly to lower apparent angles (and wave-lengths) by the steep slope on both sides. At OAX, this appearsquadratic in a and hence quadratic in the corresponding crys-tal length position along the generatrix.

If BP=2R, sin OAX (the source also on the Rowlandcircle) and the spread of grazing angles along the crystallength to is significantly less than the rocking curve width,then the standard formula

A it to coS 6 Ax

A 24(2R,) 2 sin' OAx(19a)

is an accurate description of the additional mean grazingangle shift (at the surface or equally at the diffractingplanes). For a crystal radius 2R =300 mm andO04= Op=0.6 87 677 rad, the range of significant diffraction(for ADP 101 planes) occurs over a 5 mm crystal length,although the profile FWHM is 3 mm. The grazing angle atthe pole axis is also the minimum angle subtended by thepoint source on the Rowland circle, so that only the portionof Fig. 14(a) above 0= OAX would exist, and would be bimo-dal (with one curve for contributions from the parts of thecrystal closer to the source and one curve for contributionsfrom the other half). However, the detected location of thesetails is shifted to apparently lower angles by large off-axiscorrections. The two effects approximately cancel to within1% of their magnitude. By comparison the simple estimatesgiven above for depth penetration and mean angular shift (asopposed to detailed calculations) were only generally accu-rate to a factor of 2 or 3, or some 100 times this uncertainty.The separation of shifts due to changes in surface grazingangle from those due to lateral or off-axis shifts, seen inearlier sections, is less valid here as the diffracting regioncovers a large range of surface angles and locations, andthose points with the largest (positive) angular shifts are alsothose with the largest (negative) off-axis shifts.

Equation (Sd) (see Table II) becomes quite invalid andthe scaling of off-axis shifts implies that detector shifts of 1Aim can represent a diffracting angular change of 100 ppm.This may be a real angular shift, but may also arise fromvery small lateral shifts due to depth penetration. Conse-quently, the interpretation of centroid and profile results isnontrivial. Therefore, the observed angle is extremely sensi-tive to the spectrometer alignment, since a change of mini-mum diffracting geometry A iin to higher angles [from achange in the source location, pole axis angle, or crystalradius, as indicated in Eq. (14a)] leads to a direct and near-equal shift in the apparent or detected mean diffracting angle(and likewise in the peak angle). The off-axis correction willalso change; the mean can appear to first-order as the mini-mum spectrometer angle Omin, whether the flat crystal peakangle Op is greater or less than this. A second-order correc-tion depends on the asymmetry of the portion of the flatcrystal diffraction profile actually diffracted, compared to theasymmetry and scale of the off-axis shifts. The reflectivitywill drop as Omin approaches or exceeds Op.

The several hundred ppm magnitude of this potentialshift, and its extreme sensitivity, imply that precision mea-surements of this type are not feasible without a careful mod-eling of the profile and the intensity and extreme precision ofalignment. Typically, relative measurements will utilize asecondary source of x rays at a different physical locationcompared to the unknowns being measured. The variation oflocation in this nearly on-circle geometry can lead to largesystematic shifts of the type just mentioned, so that the dif-ficulties in interpretation remain in these types of measure-ments.

For the in-circle geometry with to= 14.6 mm, Eq. (19a)



underestimates the range of grazing angles and hencshift. One side of the crystal now corresponds to lowering angles but the symmetry around the pole of the ofshifts is maintained. Despite the angular range exte5-10 or so times the on-Rowland circle estimate, ofshifts at the extremes are similar, modulated by less dfactor of 2. For sources either on or off the Rowland circrystal dimension above to= 14.6 mm clearly revealnonquadratic relation of the off-axis contributions, ftreducing the overall centroid shifts.

More significantly, the effective range is genstrongly modulated by the (much) narrower differwidth. For the (typical) example of ADP 101 diffractioFWHM for flat crystal diffraction is 3.2X 10-5 rad, whibroadened width on the surface output angle profile;pole axis is still only 3.8X10-5 rad and the 1 percrange is only 3X 104 rad. Following Eqs. (13b) and (the latter range corresponds to a crystal length of 0.01'for the surface; allowing for broadening due to depthetration down to the 5% level raises this to only 0.04mm. This is in good agreement with the following, simpform of Eqs. (13b) and (14b):

(2R, V2 sin AO, tan OAx.

AXP= ~I source on Rowland circle;

IRzA 0, sin OAI

soce wlCOS2 0 insid-e COS2w cA i rc

source well inside Rowland circle.

Quadratic or other effects outside this range are siirrelevant (as confirmed in tests). At this 5% extreme (:mm), the expected off-axis shifts at the detector are aprmately -0.0483 and -0.0376 Am at lower and E

angles, respectively; weighting this with the relative intcone might expect a mean shift of -0.005 gtm or an effsangular shift of 10-9 rad! This is certainly negligiblepared to all other shifts discussed earlier.

For a diffracting angle peak some 0.08 rad awaythe pole axis [cf. Fig. 14(a)], a linear variation of deshift with angle (or crystal location) occurs so that thecentroid shift from off-axis shifts would be preciselyAdditionally, for OB = 1.15 rad, a broader quadratic minmwould lead to a mean shift of opposite sign and greduced magnitude, but with a large off-axis offset relatthe ideal [Eq. (5d)]. For the source well within the Roncircle, as in the standard examples of this paper, diffractOp yields a significant total off-axis shift if the peakdoes not correspond to the grazing angle at the poleOAx. However, the crystal length around the generationtributes negligible shifts, which in addition are defined 1diffraction widths and not by the crystal dimension.

This fails to be true for a small source very closeRowland circle, where the profile is truncated at lower aby the alignment and at higher angles by the diffrwidth (or the crystal dimension). Even here, the uppertruncation shift is typically only 10%-20% of the meanof the diffracting angle, and the off-axis shifts compe

ze the for this down to below 1% of the total shift from Amino orgraz- toward the ppm level. A more serious problem arises from

ff-axis the potential inequality of imin and Op in this on-circle case,Ending and the corresponding profile truncation.tf-axis Scanning methods are not uncommon in Johann spec-than a trometers and optical elements, and lead to the possibilityrcle, a that a broad detector may be used instead of one with posi-Is the tional sensitivity. In this case, the spectrometer angle willfurther define the reference position and the mean diffracting loca-

tion, while complications of the focused profile are no longererally of prime importance. If the (point) source is scanned aroundaction the Rowland circle relative to the pole axis of the crystal,)n, the then the diffracted intensity of a monochromatic wavelengthile the will not be at maximum at the expected Op position. Instead,at the as discussed, the maximum will arise when the angle to theentile pole axis is less than this, so that two images are focused to(14b), the detector, one from each side of the pole axis. As the angle8 mm to the pole axis is decreased, the images will retain almost

pen- constant diffracted intensity as the diffracting regions of the-0.05 crystal move further away from the pole. Finally, the dif-plified fracted intensity will drop rapidly as the images track beyond

the crystal limits.In other words, a normal finite curved crystal in a scan-

ning on-circle geometry will have a high reflectivity fromabove the peak of the flat crystal curve (measured with re-spect to the pole axis) to a limit given by the edge of the

(19b) crystal. Nonmonochromatic incident radiation widths willbroaden this further. This profile is very broad and poorlycentered due to the on-circle geometry, and hence does notavoid the difficulties discussed above.

;Op02 1. Crystal and source depths

,proxi- The effect of crystal depth in Refs. 40 and 42 is prim-higher arily an estimate of the lateral shift component for the source:ensity and crystal both centered on the Rowland circle [similar toective Eq. (16a) using XZ in Eq. (16b)]. The integration was appar-com- ently performed over positive and negative depths (i.e., in-

cluding regions outside the crystal). In the above in-circlefrom geometry with a 0.4 mm depth, the quoted effect of crystal

tector depth would be -0.066 /.m at the detector or -2.2X 0-8mean rad. This is negligible. The prediction of the formula wouldzero. be reduced dramatically by absorption and extinction coeffi-

limum cients as discussed earlier. This geometric component per seGreatly is inseparable from the aforementioned components definedtime to by the mean surface angular shift upon depth penetration, thewland mean lateral shift upon depth penetration, the depth compo-tion at nent of the mean diffracting angle shift, and even the effec-angle tive mean shift of the fiat crystal profile. These dominantaxis, contributions are large and significant in many cases, are

K con- often limited by the crystal depth, and are detailed in Secs.by the III C-III F.

Other references have been concerned with profileto the widths rather than shifts,43' 44 but provide useful expressionsangles concerning these, particularly in the context of Sec. III L.action The source depth was correctly observed to have a neg-angle ligible effect on centroid location, although the main effectn shift would be to reduce shifts arising from the crystal length byensate providing more off-axis contributions with Opi, TAXI

Rev. Scl. Instrum., Vol. 66, No. 11, November 1995 Bragg x-ray diffraction 5143


J. Source length along generatrix

The quoted effect of source length along the generatrix(with a point crystal of negligible dimension) from Ref. 40 isthe transform of a uniform range of AO (at the crystal sur-face) to a nonuniform range of sin 6. This corresponds tolocally symmetric source geometries, uniform in the 27r an-gular distribution. Alternate nonuniform distributions can besignificant and should be modeled separately to estimate thiseffect. Part of the effect for a symmetric distribution wouldbe truncated by the diffraction width defining the effectivesource dimension if the latter were large enough to provide asignificant shift. Unlike Sec. III 1, there would be no off-axiscorrection at the crystal to compensate for this angular shift.Hence, if the location on the detector were transformed to avalue of sin 0, the mean of the resulting profile could indeedbe shifted significantly. However, the detector would typi-cally either be a curved or flat plate on the Rowland circle orat some angle to it (e.g., normal to the ray from the poleaxis). In this case, the profile on the detector, either ideally orotherwise, represents a profile in 0, as discussed earlier.Hence the centroid of such a profile would be unchanged.

The assumption of a point crystal is unrealistic, and thesource length can couple with the crystal length and with thecrystal depth (if, as in high orders of diffraction, absorptionis negligible). For ADP 101 diffraction in the above ex-amples, the mean penetration depth is only 3-5 /zm, so thatthis coupling will not generate large effects.

If the angle subtended by the source y0 1(2R, sin 6) isgreater than the diffraction width corresponding to the crystallength, estimated by inverting Eq. (19b) and replacing AXPby to, and with to comparable to or less than the equivalentlength for the diffraction width AXP [:3-5 mm in the pre-vious example, following Eq. (19b)], then rays from eachpair of symmetric source points will cover higher and lowerdiffracting angles with equal intensity and approximately thesame lateral shifts, so that AOu, and the mean detector loca-tion will be largely unaffected. [The detailed result, illus-trated in Fig. 11 (a), is dominated by varying truncation of theflat crystal profile and their effect on depth penetration, andsmall asymmetry in the truncation limits to yield a meandiffracting angle shift, corresponding to 1 ,um shifts on thedetector or 2 ppm effects in both in-circle and on-circle ex-amples.]

Conversely, if the angle subtended by the source at thecrystal is much less than the range of grazing incidenceangles across the crystal (for a point source), and with bothof these greater than the diffraction width A6, then raysfrom each source point will cover the range of angles withinthe diffraction width, so that A61out will be largely unaffected.However, diffraction will occur at crystal points far removedfrom the pole axis, so that the lateral off-axis shifts shall bejust as large as in Sec. III H. For the typical in-circle ex-ample, the angle subtended by the source yolBX~yolBPand the range of angles across the crystal is often (much)larger than the range of diffraction across the crystal surface.Hence effects in on-circle or in-circle geometries should beinsensitive to a finite source length alone but, particularly inthe in-circle case, coupling with a large crystal length can

lead to a large and negative mean off-axis shift to lowerapparent angles, given roughly by

(20)X ~+(24RCM s~in2A A6, )

K. Crystal and source heights normal to thegeneratrix

The effects of crystal or source height, independently,result in detected locations off the generatrix, so that quotedshifts37.40 assume no detector resolution in the transverse di-rection. These formulas treated the average image locationprojected onto the generatrix, corresponding to a one-dimensional detector with a large transverse height (and withthe source on the Rowland circle). With good two-dimensional detectors and a finite crystal height, but a neg-ligible source height (or, theoretically, a finite source heightwith a negligible crystal height), there would be no meanshift of the profile on the generatrix from this source.

Once again, these two dimensions are coupled, so that ifthe crystal and source widths in the transverse direction areboth significant, the centroid in the central detector regionshall be shifted as follows:

AA h2A 24BX2 '

h2(2Rz)sin Op

24 BX2 cos

(21a)

(21b)

For an aligned source and crystal on the Rowland circle(BX=BP=2R. sin Op, cf. Fig. 9), this yields the modifiedstandard formula

' 24(2R,92 sin2" Op'

AY= 2 4 (2 Rz)sin Op cosOp

(22a)

(22b)

with an additional contribution due to the detector resolutionelement, but where

hoBP h _zo2Rz sin

hi= 2Rz sin Op' ° 2R, sin 6+BPzoBP Ih> zo2Rz sin 6

2R. sin Op+BP' ° 2 Rz sin O+BP

(23)

Here ho is the crystal height (in the transverse direction)and zo is the source height. If the projection of zo at thecrystal is equal to ho, a maximum effect occurs which is only20% of that given by the formulas in Refs 37, 40. The maxi-mum shift for a ray at height hI compared to one on thegeneratrix is 12 times these values (as in the standard formu-las).

In the in-circle geometry, this potentially dominant shiftis reduced by lateral shifts and the effect of changing curva-ture on the diffraction profile, asymmetry, and depth penetra-tion. Here the full detailed ray tracing may be expected toreveal such additional shifts, but are of secondary interest inthe current exercise.



For the source lying on the Rowland circle, the trunca-tion at low angles and slow dispersion (slow change of dif-fracting wavelength with position) requires long crystals toimage the radiation, adding other shifts, combined with largeand negative off-axis shifts to image the peaks back near theoriginal detector location (0mma). The total shift in general hasthe same sign as in the above expression, but with the mag-nitude reduced by a factor of 10 or so in typical cases. Onceagain, this sensitivity to alignment can often preclude preci-sion measurement.

The magnitude of this coupled correction can be severalhundred ppm, ns indicated above, so this off-axis shift mustbe allowed for in order to compare wavelengths at or belowthe level of refractive index corrections. Precision compari-sons to calibration lines or calculations are simplified greatlyif the crystal width along the generatrix does not limit ortruncate the diffraction profile significantly; while adequateresolution in the transverse direction may be assumed toeliminate much of the broadening discussed relating to crys-tal and source heights ho and zo.

Relative measurements with calibration sources andsources of unknown spectra arising from the same geometriclocation are able to reduce or allow for the effect of thesegeometrical shifts arising from finite dimensions. It shouldbe noted that the functional dependence of Eqs. 20-23 aresensitive to the relative magnitudes of diffraction widths,source dimensions, and crystal dimensions. These dependen-cies are not generally constant functions of angle, so theeffects are not canceled by differential measurement. This isparticularly true when triangular crystals are used (so that thecrystal height can vary with diffracting angle) or when scan-ning methods adjust the source relative to the Rowland circle(so that OA,, and Op, are varied). In addition, the shift ofsource centers is often significant, and yields potential sys-tematic shifts in the final result which should be consideredin subsequent analysis.

L. Secondary corrections

Corrections to the above prescription include the deriva-tion of a mean dy value from estimates of asymmetry withcurvature taken into account and the inclusion of extinctionaccurately and explicitly in Eqs. (l10)-18).

An additional qualitative correction is obtained for (pho-tographic) detectors placed on the Rowland circle. This isalso geometric, but depends on absorption coefficients. Suchdetector shifts in the use of photographic emulsions at non-normal angles can exceed the 10 ppm level. Using the sub-script e for the emulsion thickness and attenuation, thisyields a simple angular dependence:

[f t /sin 0B - ALzdz cos 6AY Z= f t /sin 0e- /eZdz

At' < sin 62 tanO' " rCos 69 sin 6

te

10"

0.3 0.66 0.94 1.22 1.50®, rad

HIG. 16. (a) Tertiarv effects. Emulsion shifts for PET 002 and PET 008diffraction and t,=13 urn assuming 2R, scaling (dashed line and *-, re-spectively), compared to the mean shift in diffracting angle for IT polariza-tion (dots and slowly increasing solid line) and maximum lateral shifts (rap-idly decreasing solid line and -- ). The lateral estimates use Eqs. (16c) and(i16d), (14b), (10), and (11a) with /_c provided by the absorption coefficientonly.

Using a standard thick-emulsion x-ray film with one emul-sion removed for higher resolution and assuming te 13 /-cm(cf. Ref. 45) leads to significant relative shifts as indicated inFig. 16. The emulsion shifts can readily exceed those forlateral detector shifts and mean diffracting angle shifts, in theappropriate regimes. At general angles, all three effects canbe important, though often one may be less than a 1% per-turbation on the others. Use of electronic or other detectors atnormal incidence can eliminate this shift at the expense ofoff-Rowland circle defocusing.

Mosaicity has small effects on centroid location forcurved crystals, defocusing or broadening asymmetric pro-files to center more on Oc than 6Op. The overall asymmetrycan significantly affect the centroid determination of experi-mental profiles, dependent on the fitting function. Dopplerbroadening and natural linewidths will convolve the asym-

10'

E

LU

II(NI

E4

-iu

0-40 0.65 0.90@, rad

1.15 1.40

HIG. 16. (b) Emulsion shifts for ADP 101 and ADP 404 diffraction (solidline and -- , respectively) and t, = 13 uzm, compared to the magnitude of

(24) lateral shifts from depth penetration(, -) Lateral shifts use Eqs. (17b),(18), (10), and (Ila), with As including computed mean extinction for IT

polarized radiation, and follow the in-circle geometry indicated in Fig. 14.

Rev. Scl. Instrum., Vol. 66, No. 11, November 1995 Bagxrydfrcin 54

r -\-

" I ~ ~



metry and lead to an uncertainty in centroid determination.Some of these corrections require evaluation of a (full)

dynamical diffraction theory; others require careful ray trac-ing from the source, inside the crystal, and to the detector.These two requirements are not fully separable. A programhas been developed which addresses these corrections in aconsistent way and to higher order, including dynamical dif-fraction theory.'8' 19

Potentially significant effects, not addressed above, arethe change of diffraction plane orientation and 2d spacing,and hence the diffracting angle, as a consequence of the cur-vature stress, which may vary with position on the crystal.There is often additional longitudinal and transverse curva-ture as a result of the bending technique and moments andthe isotropic or anisotropic compliances or elasticities of thecrystal. Such curvature will give effects in the finite sourceand crystal height and length considerations of previous sec-tions. It also leads to curvature of the diffracting planes, andhence changes with depth of the orientation of an incident xray to these planes. This has been addressed above, withinthe assumption that the planes remain at a constant angle toarcs from the axis of the cylinder. For symmetric Bragg re-flection this is exact, while for symmetric Laue diffraction itis true for isotropic materials. In other cases this assumptionhas limitations leading to tertiary effects in near-symmetricdiffraction.

The impressed curvature results in compression of thelateral spacing of the front surface layers and an expansion oflateral spacing of the rear surface, but also gives a variationof lattice spacing normal to the surface (following Poisson'sratio v=0.25). This can have a large effect on diffractingangles at the surface compared to the neutral plane at thecenter of the crystal (equivalent to the unstressed crystal) andcan further reduce the range of depths over which diffractionis coherent.' 9 Form factors can also be redistributed as aresult of these stresses. The last two effects are usually mi-nor. Given allowance for the surface d spacing (a potentiallylarge shift to lower angles, but a constant offset in AM/A forall source lines), this change of effective Bragg angle withdepth leads to a mean diffracting angle shift proportional tothe mean depth d and positive for near-symmetric Braggreflection: d

A Oi v tan 062 (25)For low order diffraction or low energy x rays, this is

often negligible, but for intermediate energies can reachAAXIX1-5X 10-6, or greater. The shift is zero for plasticdeformation as is common with formed mountings for im-pressing curvature, or for highly mosaic crystals whichwould also tend to deform plastically. The value and func-tional form in the elastic case is also dependent upon thecrystal shape and stress distribution. This issue is discussedin greater detail in Refs 46 and 47. In symmetric Bragg dif-fraction, these effects are often tertiary or negligible withrespect to the contributions discussed earlier.

M. Discussion

Experiments using curved crystals and requiring abso-lute wavelength determination to better than 1O0 ppm in the

x-ray regime must generally involve careful consideration ofa wide variety of effects in addition to refractive index cor-rections. Flat crystal modifications to simple estimates of re-fractive index corrections have been discussed in Sec. II(asymmetric Bragg diffraction, profile asymmetry, three-beam interactions), and are also relevant here. While theycan be significant or dominant compared to the refractiveindex correction itself, even for curved crystals, these contri-butions are relatively straightforward. Variation of angularshifts with Bragg angle can follow tan 0, cot 0, or 6 indepen-dent relations, with particularly strong local features and al-ternate dependencies near edges or near three-beam diffrac-tion points.

For most curved crystal geometries, additional systemat-ics in the diffraction process occur at or above this level.Asymmetries of diffraction profiles due to penetration of thex-ray field inside curved crystals can dominate over the re-fractive index corrections, particularly for high-order diffrac-tion or medium-energy x rays. The shift of the mean angle tothe diffracting planes and the lateral shift around the crystalsurface of the exit location relative to the incident location ofthe photon are related and significant, but are often smallercontributions, at the 10%-20% level of the refractive indexcorrections. Contributions to the mean output angle at thecrystal surface can give cot 6 or cos 6/sin3 0 dependencies,but often lead to intermediate behavior with complexitiesfrom edges, polarizations and cos 20 factors, and geometricconsiderations. These effects are correlated with Johann ab-errations, involving off-axis shifts and potentially large scal-ing corrections, especially for geometries with the sourcelying on the Rowland circle or near the minimum angle ofthe curved crystal spectrometer.

For precision experiments, on-circle geometries shouldbe avoided to minimize arbitrary and sensitive profile trun-cation shifts from crystal ranges or minimum grazing angles,and to avoid extreme scaling corrections. Geometries withthe source well inside the Rowland circle are generally moreappropriate for absolute angle measurement or for compari-son to suitable calibration lines, due to the greater bandpassand smoother, smaller scaling corrections. Hence nonidealJohann arrangements can provide improved measurementscompared to the on-circle ideal. In either case, these off-axisshifts and nonlinearities exhibit strong and rapidly varyingangular dependencies, which distort the results given earlier.However, simple functional relations are provided and illus-trated for these effects. These indicate angular dependenciesfor interpolation or extrapolation purposes with regard tocalibration lines, and are often accurate to a few percent ofthe effect or of refractive index contributions. The lowestprecision for individual, smaller contributions are typicallywithin a factor of 2 or 3.

Source and crystal dimensions interact with defocusingshifts and diffraction corrections. Literature results often in-volve restrictive or inappropriate assumptions, and can ne-glect corrections of large magnitude. Current results indicatethat one-dimensional extensions often yield null shifts, butthat large shifts can arise from the coupling of two finitedimensions. Relations are also provided for these contribu-tions. The magnitudes and significance vary greatly from on-



circle to in-circle geometries, and tend to recommend thelatter. The combination of significant source and crystal di-mensions along the generatrix can lead to significant on-circle shifts but quite large (and well-defined) in-circle cor-rections. This is also true for significant source and crystalheights normal to the generatrix. Dependencies approximat-ing 1/sin2 0 can be obtained for these effects, in appropriateregimes, and with significant offsets as discussed above.

Ideally the crystal width along the generatrix should notlimit or truncate the diffraction profile significantly, therebyavoiding a strong dependence upon source and crystal geom-etry and alignment. Adequate resolution in the transverse di-rection can minimize the effects of crystal and sourceheights. Centroid shifts due to detectors lying on the Row-land circle (or at general angles to incident x rays) can besignificant. The precision of profile shifts can be limited at0.1%-1.0% of the refractive index corrections by form fac-tor uncertainties, in optimum cases.

Measurements of unknown wavelengths relative to anearby calibration line by extrapolation can increase the finalmeasurement precision by an order of magnitude, but requirean understanding of the functional relations indicated abovefor higher precision. This can be estimated to first order bythe slope of response between several calibration lines. Pre-cision of absolute shifts assumes that relative measurementsare made in the same order and with nearby Bragg angles,away from absorption edges and away from extreme spec-trometer angles. Multiple-beam interactions are generally notdetermined or accounted for in such calibrations. If the azi-muthal angle is inadequately known, other methods must beused to consider the importance of these effects, as indicatedearlier.

The variation with angle of several component effects isoften far from constant or linear, so problematic regimesshould be avoided, or expected functional dependencies andrelative magnitudes should be understood. The series ofshifts due to finite source and crystal dimensions are gener-ally more serious for such relative measurements, since dif-ferent off-axis shifts may arise for the different sources dueto their geometric location in space.

Complexity of shifts with order and polarization in spec-tra necessitates calculations of the sort indicated here forprecision below 1% of refractive index corrections. Curvedcrystal profiles require lamellar thicknesses defined consis-tently with respect to finite thickness widths and Ay values,allowing for coherence between lamellae, especially forhigher order diffraction. The calculated shifts agree well withthe sum of simpler estimates, reproducing the dependence onthe Bragg angle. The importance of correct allowance fordepth penetration and other diffraction effects is also clear.

ACKNOWLEDGMENTS

One author (C.T.C.) would like to thank the English-Speaking Union of the Commonwealth for a Lindemann Fel-

lowship covering part of the period of this research. Manyuseful discussions with A. J. Varney and A. Caticha aregratefully acknowledged.

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