REVIEW OF SCIENTIFIC INSTRUMENTS 83, 063901 (2012)

Absolute x-ray energy calibration over a wide energy range using adiffraction-based iterative method

Xinguo Hong,1,a) Zhiqiang Chen,1 and Thomas S. Duffy2

1Mineral Physics Institute, Stony Brook University, Stony Brook, New York 11794, USA2Department of Geosciences, Princeton University, Princeton, New Jersey 08544, USA

(Received 29 February 2012; accepted 10 May 2012; published online 4 June 2012)

In this paper, we report a method of precise and fast absolute x-ray energy calibration over a wideenergy range using an iterative x-ray diffraction based method. Although accurate x-ray energy cali-bration is indispensable for x-ray energy-sensitive scattering and diffraction experiments, there is stilla lack of effective methods to precisely calibrate energy over a wide range, especially when normaltransmission monitoring is not an option and complicated micro-focusing optics are fixed in place.It is found that by using an iterative algorithm the x-ray energy is only tied to the relative offset ofsample-to-detector distance, which can be readily varied with high precision of the order of 10−5

–10−6 spatial resolution using gauge blocks. Even starting with arbitrary initial values of 0.1 Å,0.3 Å, and 0.4 Å, the iteration process converges to a value within 3.5 eV for 31.122 keV x-rays afterthree iterations. Different common diffraction standards CeO2, Au, and Si show an energy deviationof 14 eV. As an application, the proposed method has been applied to determine the energy-sensitivefirst sharp diffraction peak of network forming GeO2 glass at high pressure, exhibiting a distinctbehavior in the pressure range of 2–4 GPa. Another application presented is pair distribution func-tion measurement using calibrated high-energy x-rays at 82.273 keV. Unlike the traditional x-rayabsorption-based calibration method, the proposed approach does not rely on any edges of specificelements, and is applicable to the hard x-ray region where no appropriate absorption edge is available.© 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4722166]

I. INTRODUCTION

Monochromatic x-ray diffraction using synchrotron radi-ation plays a dominant role in modern structural determina-tion of crystalline and non-crystalline materials. Precise x-rayenergy is needed to determine lattice parameters or atomicradial distribution function of material under study in x-raydiffraction experiments. The absence of well-defined featuressuch as characteristic lines in synchrotron radiation requiresfrequent in situ energy calibration with good precision so asto identify and correct unpredictable energy drifts. The energyof the incident monochromatic x-ray beam can deviate con-siderably from the energy defined by the Bragg angle of themonochromator due to many factors, such as thermal loadingof the monochromator crystals, water cooling or cryocoolingtechniques, loss of steps in the monochromator motor, beam-line optics, synchrotron orbit shifts, and optimization in x-rayintensity.1

There are a variety of energy calibration methods thathave been proposed. The most common technique for amonochromatic synchrotron beam is to measure the energyof the absorption edge of a standard known to a precision ofa few electron volts. However, this x-ray absorption (XAS)based method is not always possible and convenient, espe-cially for beamlines where sophisticated x-ray focusing opticsare installed. Scanning the energy may affect the matching of

a)Author to whom correspondence should be addressed. Electronicaddresses: xhong@bnl.gov and xinguo.hong@gmail.com. Telephone:(631)344-4066. Fax: (631)344-3238.

double crystals and induce significant changes in beam po-sition and intensity, which are more serious in the case of adouble-Laue crystal monochromator (DLCM) without fixedexit.2–4 The use of Kirkpatrick-Baez mirrors, which can fo-cus the x-ray beam down to 10 μm or less for high-pressureresearch with diamond anvil cells, requires a stable beammatching a tiny sample chamber and normal transmission x-ray energy scanning is not an option, especially for a DLCMwith no fixed exit. For diffraction experiment using high-energy x-rays, it may be hard to find suitable absorption stan-dard close to the desired high-energy range of the experiment.Instead, a multi-channel analyser (MCA) can be employedto covert and calibrate MCA channel numbers to the corre-sponding energies using various radiation sources at high-energy synchrotron beamlines. However, the energy calibra-tion of the monochromator will be lost in the long term andmay require other inconvenient calibrations over the courseof an experiment. Such a MCA energy calibration has lowerenergy resolution in comparison with that of the absorptionbased method.

Other x-ray energy calibration methods include fluores-cence mode measurements using elastically scattered inci-dent x-rays,1 combined techniques of x-ray absorption anddiffraction,5, 6 and diffraction markers (Bragg angle) of stan-dard references using an analyzer crystal.7 According toBragg’s law, x-ray energy/wavelength calibration is relatedto the precisions of sample d-spacing and diffraction angles.Using a known x-ray wavelength of Cu kα1 radiation, Bondproposed an absolute d-spacing measurement method, wherethe x-ray scattering was measured at both sides of the primary

0034-6748/2012/83(6)/063901/10/$30.00 © 2012 American Institute of Physics83, 063901-1

063901-2 Hong, Chen, and Duffy Rev. Sci. Instrum. 83, 063901 (2012)

beam, which can effectively eliminate the possible zero-pointoffsets and setting errors of the crystal.8 This method has aprecision of few parts in 105 level.9 On the other hand, interms of high precision d-spacing value of a Si single crys-tal, x-ray energy can be calibrated precisely. Through a com-bination of absolute measurements of x-ray interferometer10

and optical interferometry, the (220) lattice plane spacing ofsilicon has been determined with uncertainty of one part in108 level.11 The reported value has become the basis of highprecision double Si-crystal monochromator, accurate energydetermination at arbitrary energies, e.g., characteristic x-raywavelengths (Cu Kα1 and Mo Kα1), and absolute energy cal-ibration on the absorption edge energies of elements fromZ = 23 to Z = 82.5, 12 However, these methods do not satisfythe requirements of fast, simple, in situ energy calibration andare accurate at any energy range for diversified x-ray diffrac-tion experiments. They are either limited to low energy, tiedto a specific absorption edge1, 12 or have inconvenient layout,more difficult to use,8, 11 additional alignment or require cali-bration of the analyzer crystal.6, 7

X-ray diffraction based calibration using diffraction stan-dards, such as CeO2, Au, LaB6, and silicon, with well-established lattice parameters is frequently used to check andcorrect small energy drift from XAS defined value. The x-raywavelength is linked to crystal lattice spacings by Bragg’s law.With the known lattice parameters and x-ray wavelength, thesample-to-detector distance can be obtained using the FIT2Dprogram,13 and may be used for further calibration of otherdesired x-ray energy or to check energy drift over the courseof an experiment. Nevertheless, the interaction between wave-length and sample-to-detector distance makes it hard to re-fine wavelength with fixed sample-to-detector distance, whichcannot be directly measured and remains largely unknown onthe exact position of sensor plane enclosed inside a detec-tor. Uncertainties in standard sample thickness and the repro-ducibility of sample position are also sources of errors.

The DLCM has advantages of high angular accep-tance, high photon flux, and thermal loading stability, mak-ing it well suited for the low emission characteristics of asecond-generation synchrotron source. Two moderate energyDLCMs are installed at the X17B3 and X17C beamlines atthe National Synchrotron Light Source (NSLS), BrookhavenNational Laboratory,3, 4 respectively, for diffraction exper-iment at around 30 keV, while an additional high-energyDLCM (80 keV) was employed for high-energy diffractionexperiment at the X17B3 beamline. However, these basicDLCM monochromators have no water cooling or cryocool-ing technique that could be effectively applied in transmis-sion mode, and no feedback function to trace and compensatesynchrotron orbit shifts too. A precise prompt energy cali-bration method is necessary for reliable x-ray diffraction ex-periments; especially for high-energy diffraction experimentwhere XAS based calibration cannot be applied.

Although accurate absolute x-ray energy is indispens-able for many energy-sensitive scattering and diffractionexperiments, there is still a lack of effective methods toquickly calibrate x-ray energy. Slight difference in x-rayenergy calibration can cause considerable uncertainties inRietveld refinement.14 Some high-pressure research areas,

such as determination of elastic moduli and strength by radialx-ray diffraction,15 require measurements of small changes inpeak position and peak shape.

In this paper, we report a method of precise absolutex-ray energy calibration using an iterative x-ray diffractionbased method. Our approach is to use repeated x-ray diffrac-tion patterns of common diffraction standards such as CeO2,Au, and Si at a series of sample-to-detector distance by in-serting standard gauge blocks of known length. If the x-raywavelength deviates from the real value, the measured ap-parent distance of sample-to-detector by FIT2D program13

would be different from the known offset distance. It is foundthat the x-ray energy is only tied to the relative offset ofthe x-ray detector, which can be set precisely by standardgauge blocks or highly accurate translation stages. The re-liability of proposed approach is tested. The iteration pro-cess converges to a value within 3.5 eV for a 31.122 keVbeam with CeO2 standard, and a 14 eV energy deviation isobserved when using different standards. In addition to notbeing limited to specific absorption edges, the main advan-tage of proposed method is that it is suitable for the hard x-ray energy range where no absorption edge at the desired en-ergy may be available. Examples of the first sharp diffractionpeak (FSDP) of network forming GeO2 glass at high pres-sure and pair distribution function (PDF) measurement arepresented. Although specific reference to a Laue monochro-mator is made here, the technique is generally applicable toother areas of synchrotron-radiation x-ray research requiringabsolute energy calibration on monochromatic beam as longas diffraction experiments can be conducted.

II. EXPERIMENT AND METHOD

A high-precision gauge block set (Grade 0, No. 516-103-26) and additional 10-mm thick blocks were purchasedfrom Mitutoyo Co. A Certificate of Accuracy, traceable tothe NIST, is furnished with each Mitutoyo gauge block setand individual blocks. They have ±0.00014 and ±0.0002accuracy for the 10 mm and 25 mm gauge blocks, provid-ing 1.4 × 10−5 and 8 × 10−6 spatial resolution, respectively.To get a sufficiently large distance offset range for high-energy x-ray calibration, an 8-inch Mitutoyo Absolute Digi-matic Caliper (Model No. CD8 PSX) with measuring range of0–200 mm was also employed in the experiment.

Four sets of x-ray diffraction experiments have been per-formed at the X17B3 and X17C beamlines. The first set ofexperiments was designed to collect x-ray diffraction datafrom a CeO2 standard at a series of sample-detector dis-tances by inserting gauge blocks in series. A spring was em-ployed to ensure tight contact between detector and gaugeblocks. X-ray diffraction data were collected using a Rayonix165 CCD detector at X17C and a Perkin-Elmer flat panel(XRD 1261) detector at X17B3, respectively. The pro-gram FIT2D (Ref. 13) was used to process the 2D x-raydiffraction images. Alternative data and dark files were col-lected for the Perkin-Elmer detector, and then input intoFIT2D for dark current reduction. Using the FIT2D calibra-tion command, we can get the apparent sample-detector dis-tance and beam center coordinates by fixing the wavelength,

063901-3 Hong, Chen, and Duffy Rev. Sci. Instrum. 83, 063901 (2012)

but leaving other parameters free in the calibration. The initialx-ray wavelength at X17C was determined by scanning acrossthe K-edge of antimony (30.491 keV). An iterative procedurewas developed to calculate the actual x-ray energy and un-certainty (Fig. 2). Arbitrary initial values of x-ray wavelengthhave been used to check the convergence of the method (Figs.2 and 3). The second set of experiments, with three differentstandards, was designed to cross-check the deviation and re-liability of the method (Figs. 4 and 5). The data obtained foran Au powder were compared in d-spacing to that determinedby energy dispersive x-ray diffraction (EDXD) method (Fig.6), to further double-check the validity of the method. Thethird set of experiments was an application of the techniqueto the FSDP of GeO2 glass at high pressure. The last set wasdesigned to apply the calibrated high-energy x-rays to PDFmeasurements.

III. RESULTS AND DISCUSSION

Figure 1(a) shows the FIT2D calibration of the x-raydiffraction pattern of the CeO2 standard collected with a Ray-onix 165 CCD detector to get apparent sample-to-detectordistance, beam center coordinates, and detector tilting anglesusing an x-ray wavelength of 0.4066 Å, which was defined byscanning the energy across the K-edge of Sb (30.491 keV).The CCD detector was positioned asymmetrically relative tothe beam center due to a beampipe in the hutch, and this setupallows access to a wide range of diffraction angles. We fixedthe monochromatic x-ray energy and fit the other parameters.The overall fitting appears to be good, but when viewed athigher resolution and “unrolled” along the azimuthal angle(so-called “cake” function of FIT2D), curved lines can be ob-served at 2θ around 17◦ (right column of Fig. 1(a)).

FIG. 1. FIT2D calibration on CeO2 x-ray diffraction pattern. Left column is fitting to experiment data; right column shows magnified image of diffraction lineat 2θ = 17◦ obtained by FIT2D cake function. (a) X-ray wavelength 0.4066 Å (30.493 keV) and (b) diffraction based x-ray energy calibration, which gives anx-ray wavelength of 0.4003 Å (30.973 keV).

063901-4 Hong, Chen, and Duffy Rev. Sci. Instrum. 83, 063901 (2012)

Start

Fix initial λ0 and starting calibration

Calculate distance correlation

Correct λ by correlation slope

Correlation close to unity and consistent?

No

Yes

Finish

New apparent distance

FIG. 2. A flow chart of the iteration method for x-ray energy calibration.

Such a curved line indicates an inappropriate interactionamong parameters of x-ray energy, sample-to-detector dis-tance, and the two tilting angles of the diffraction plane. Whenintegrated to form a 1-D diffraction pattern, curvature in theunrolled diffraction lines results in an increased peak widthand uncertainty in peak position, which would cause unac-ceptable errors in x-ray diffraction experiments at high pres-sure using diamond anvil cells.15

The diffraction condition can be expressed by Bragg’slaw,

2d · sin(θ ) = nλ, (1)

where n is an integer, λ is the wavelength of incident x-rays,d is the spacing between the planes in the atomic lattice, andθ is the angle between the incident x-ray and the scatteringplanes. For a diffraction ring with radius, R, Eq. (1) can berewritten as

λ = 1

n· 2d · 0.5R

√(0.5R)2 + L2

, (2)

where L is the sample-to-detector distance. For certaindiffraction lines at small angles, the Bragg law, Eq. (2), can besimplified to λ ∼= 2d

n· 0.5R

L, i.e., x-ray wavelength is inversely

proportional to the sample-to-detector distance. One may getall unrolled diffraction lines straight at small angles even withwrong energy and sample-to-detector distance. At larger scat-tering angle, the approximate of θ ≈ sin(θ ) in Bragg’s lawis unsatisfied, e.g., there is 0.37% and 1.15% differences be-tween θ and sin(θ ) for 2θ = 17◦ and 30◦, respectively. This isthe rationale of the common method to check for energy driftby careful analysis of unrolled diffraction lines at high angles.

One way to break the correlation between wavelength,λ, and distance, L, would be to obtain a precise value ofL independently. It is difficult to measure L directly due tothe uncertainties in the sample thickness of the standard,

the reproducibility of sample position during sample change,and exact position of the detector plane. By inserting a se-ries of gauge blocks, we can change the relative sample-to-detector distances precisely and record repeated x-ray diffrac-tion patterns from the standard and fit using FIT2D. If thex-ray wavelength deviates from the real value, the variation ofthe apparent sample-to-detector distance from FIT2D wouldbe different from the known inserted length.

When a new gauge block is inserted, it is hard to ensurethe detector moving exactly parallel to the x-ray path. Thus,the effective offset of the detector should be used in the energycalibration. To get the real distance offset, lreal, for each gaugeblock, we use following equation:

lreal = l × cos ϕ, (3)

where l is the length of inserted gauge block, ϕ is the angleformed by the lines from the sample to two adjacent x-raybeam centers. The angle ϕ can be determined by

ϕ = sin−1(√

�x2 + �y2/l), (4)

where �x and �y are the corresponding shifts of beam cen-ter due to the inserted gauge block. Equation (3) can be thenrewritten as

lreal = l ×√

(1 − sin ϕ × sin ϕ)

= l ×√

(1 − (�x2 + �y2)/l2). (5)

All the offset distances in the experiments have been correctedusing Eq. (5).

We use an iterative algorithm (Fig. 2) to obtain the ac-curate x-ray wavelength by comparing the apparent distanceand real distance of gauge block using Eq. (5). The new wave-length for the next iteration, λi+1, can be calculated by multi-plying the current wavelength, λi, with the correlation slope,ki, of the linear fitting,

λi+1 = ki · λi, (6)

where i denotes the ith iteration. It can be seen that theunrolled line becomes very straight after energy calibration(Fig. 1(b)), being a good indication of correct x-ray energy.The calibrated x-ray wavelength is 0.4003 Å (30.973 keV),which differs about 1.5% from the original x-ray energy.

Figure 3 shows the relationship between the apparentsample-detector distances and the gauge block offset throughseveral iterations. The wavelength of the K-edge of antimony(0.4066 Å) was chosen as initial value. Ideally, the change ofapparent distance should be the same as the real distance ifthere is no energy deviation, and the slope of a linear fit (Fig.3, upper panel) should reflect the deviation of the apparentdistance from the real distance, while the error bar indicatesthe quality of correlation and reproducibility of the experi-ments. The new wavelength for next iteration is determined byEq. (6). After three iterations, excellent linear behavior couldbe found for all detector offset points. The correlation coef-ficient between apparent and real distances closes to one andresiduum of linear fitting becomes gradually smaller (Fig. 3,lower panel).

The parameters of the FIT2D calibration are quite sen-sitive to the of x-ray wavelength. To check the convergence

063901-5 Hong, Chen, and Duffy Rev. Sci. Instrum. 83, 063901 (2012)

0 20 40 60 80 100

240

280

320

360

0 20 40 60 80 100-0.05

0.00

0.05

Linear correclation vs iteration

0 : 1.00695+/-7.07x10-4

1st: 1.00130+/-6.07x10-4

2nd: 0.99931+/-3.47x10-4

3rd: 0.99993+/-4.32x10-4

App

aren

t dis

tanc

e (m

m)

CCD offset (mm)

Res

idua

ls (

%)

CCD detector offset (mm)

FIG. 3. Effect of iteration process for CeO2 standard using an initial valueof 0.4066 Å. (Upper panel) Linear correlation of apparent sample-to-detectordistance with real CCD offset; (lower panel) Residuum of linear fitting ateach iteration step in percent.

for cases where the energy has wandered far from the initialvalue, we use arbitrary initial wavelengths. Three quite differ-ent trial values of 0.1 Å (123.984 keV), 0.3 Å (41.328 keV),and 0.5 Å (24.797 keV) have been tested (Fig. 4). Detailsof fitting parameters are listed in Table I. It can be seenthat with only three iterations, the x-ray wavelength rapidlyconverges to a consistent value with a standard deviation of4.5 × 10−5 and consistent residuals (Fig. 4, lower panel).Straight lines from FIT2D cake integration have been ob-served for all diffraction peaks (Fig. 5 presents only one ofthe respective cake integration lines before and after correc-tion, for clarity).

With the proposed method, it should be emphasized thatthe x-ray wavelength or energy is only tied to the relative off-set of the sample-to-detector distance, which can be changedwith high precision of the order of 10−5–10−6. The iterativeprocess converges quickly and can significantly improve the

0 20 40 60 80 100100150200250300350400

1000

1500

2000

0 20 40 60 80 100-0.05

0.00

0.05

λλ

λλ

λλ

Sam

ple-

dete

ctor

dis

tanc

e (m

m)

Res

idua

ls (%

)

CCD offset (mm)

FIG. 4. Convergence process of calibration for CeO2 standard using threearbitrary initial values of 0.1 Å (123.984 keV), 0.3 Å (41.328 keV), and0.5 Å (24.797 keV). (Upper panel) Linear correlation of apparent sample-to-detector distance with real CCD offset; (lower panel) Residuum of linearfitting at each iteration step in percent.

accuracy of calibrated x-ray energy, indicating a good poten-tial of in situ energy calibration method. In addition, it shouldbe noted that the method is obviously different from the com-mon calibration method of using several diffraction standardsat “same” fixed distance to optimize the x-ray energy usingFIT2D, because in principle precise distance of sample-to-detector remains unknown.

There is another x-ray energy calibration approach ofcombining x-ray absorption and x-ray diffraction methods.The x-ray energy is firstly defined by x-ray absorption andthen one collects diffraction pattern of a standard, so as toget a precise sample-to-detector distance. Next, one adjuststhe energy of the monochromator to the desired energy andkeep sample-to-detector distance unchanged; then one ob-tains another diffraction pattern and optimizes the x-ray en-ergy with previously obtained sample-to-detector distance.

TABLE I. Parameters obtained after three iterations with CeO2 standard using three arbitrary trial initial values of 0.1 Å (123.984 keV), 0.3 Å (41.328 keV),and 0.5 Å (24.797 keV).

Trial No. Initial λ (Å) 3rd iteration λ (Å) 3rd iteration Energy (keV) Sample-detector distance (mm) Distance correlation

1 0.1 Å 0.3984314 31.1181 229.5369 1.0007772 0.3 Å 0.3983674 31.1231 229.5757 1.000553 0.5Å 0.3983446 31.1248 229.5904 1.000376Average (0.39838 ± 4.5) × 10−5 31.122 ± 0.0035 229.5676 ± 0.0276 (1.00057 ± 2.0) × 10−4

063901-6 Hong, Chen, and Duffy Rev. Sci. Instrum. 83, 063901 (2012)

FIG. 5. Effect of iteration process on FIT2D function magnified for the line at 2θ = 17o. (a) Initial x-ray wavelength 0.3 Å (left); final wavelength: 0.398367(right); (b) initial x-ray wavelength: 0.5 Å (left); final wavelength: 0.398345 (right).

The success of this combined method largely depends on:(1) desired energy should be close to the absorption edge;(2) reliable relation between monochromatic angle and x-ray energy; (3) good stability of x-ray beam position withfixed exit; and (4) x-ray absorption may be needed for anew calibration when considerable drift in energy is present.Clearly, our proposed approach, which is free from the x-rayabsorption method, is a better solution for x-ray energy cal-ibration as it has almost no limitation on the x-ray energyrange.

Figure 6 shows the effect of energy calibration in d-spacing for the trial wavelengths of 0.3 Å, 0.4066 Å, and0.5 Å. With properly calibrated energy, a much better sym-metric intensity profile is obtained, implying the capability formore precise structure determinations for the materials understudy in subsequent Rietveld refinement.15

Figure 7 is a cross-check on the proposed method us-ing some of the most common diffraction standards (Au,CeO2, and Si) so as to investigate the reliability of the pro-posed method with different diffraction standards. After a few

063901-7 Hong, Chen, and Duffy Rev. Sci. Instrum. 83, 063901 (2012)

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

0

1,000

2,000 In

tens

ity

Iterationλ=0.3 3rd iterationλ=0.4066 3rd iterationλ=0.5 3rd iteration

51.301.3

0

400

800

1,200

Inte

nsity

d-spacing ( )

FIG. 6. Effect of energy calibration in d-spacing for the initial wavelengthsof 0.3Å, 0.4066 Å, and 0.5Å. (Upper panel) overall data in d-spacing; (lowerpanel) zoom-in strongest peak.

iterations, the standard deviation of these standards becomes1.9 × 10−4, which is good enough to trace the 1.5% energydrift of DLCM monochromator (Fig. 1). This cross-checkdemonstrates the reliability of the proposed method.

Figure 8 shows the comparison of Au diffraction dataobtained by angle-dispersive x-ray diffraction (ADXD) and

0 20 40 60 80 100220

240

260

280

300

320

340

360

0 20 40 60 80 100

-0.05

0.00

0.05

λ

λ

λ

Sam

ple-

Det

ecto

r D

ista

nce

(mm

)R

esid

uals

(%

)

CCD offset (mm)

FIG. 7. Energy calibration with three most common diffraction standards,Au, CeO2, and Si, using the initial value of 0.4066 Å defined by x-ray ab-sorption. After iteration for three times, the standard deviation among thesestandards becomes 1.9 × 10−4. (Upper panel) linear correlation of apparentsample-to-detector distance with real CCD offset; (lower panel) Residuum oflinear fitting at each iteration step in percent.

0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.60.0

0.2

0.4

0.6

0.8

1.0

Nor

mal

ized

inte

nsity

d-spacing ( )

Au powder

EDXD ADXD

111

200

220

311

222

400

331

420

FIG. 8. X-ray diffraction data of Au standard in d-spacing which were mea-sured using angle-dispersive (ADXD) and energy dispersive x-ray diffraction(EDXD) methods, respectively.

EDXD methods, respectively. The detector channel num-ber calibration of EDXD experiment was performed us-ing radioactive radiation source (241Am) and x-ray flu-orescence lines, such as Cu Kα (8.041 keV), Rb Kα

(13.375 keV), Mo Kα (17.443 keV), Ag Kα (22.104 keV),Ba Kα1 (32.1880 keV), Ba Kβ1 (36.376 keV), Tb Kα1(44.474 keV), and 241Am (59.539 keV). A least-squares fitgave a relation of E (keV) = −0.1761 + 0.017475C + 2.486× 10−9C2, where C represents the channel number and E rep-resents the energy. Excellent consistency between these twomethods can be observed (Fig. 8). Of these two methods, themonochromatic ADXD is more widely used, both in the lab-oratory setting and at synchrotron beamlines, because of hav-ing higher resolution in comparison with EDXD method.16

By using the 30 keV DLCM monochromator, the x-rayenergy in the above three different experiments can drift afew percent, e.g., 1.5% (Fig. 1), 2.0% (Fig. 4), and 0.5%(Fig. 7) from the K-edge of antimony (30.491 keV), in a timespan of a few months. The energy calibration was carriedwhen evidence of curved cake integration lines was detected.It is worth mentioning that such differences don’t affect thesize of unit cell much, e.g., the value obtained using the cor-rected x-ray energy by GSAS (General Structure AnalysisSystem) fitting17, 18 is 5.41150 Å, which is very close to thatof uncorrected one of 5.41205 Å, although incorrect energydoes have significant influence on the individual peak shape(Fig. 6). This is because slight deviation in x-ray energy canbe compensated in the FIT2D calculation by adjusting sample-detector distance. This illustrates that the correlation betweenx-ray energy and sample-detector distance has to be broken inorder to get a reliable structural analysis.

As discussed above, precise x-ray energy is highly de-sired for studies based on a detailed analysis on the line-shape of diffraction peaks. The position of the FSDP, whichis inversely related to periodicities in the real-space struc-ture, is usually determined by a Lorentzian fitting. FSDP ofnetwork-forming glasses such as GeO2 and SiO2 reveals thenature of intermediate-range order in glassy network19–21 and

063901-8 Hong, Chen, and Duffy Rev. Sci. Instrum. 83, 063901 (2012)

0 2 4 6 8 10

1.6

1.8

2.0

Pea

k po

sitio

n of

FS

DP

(-1)

Pressure (GPa)

FIG. 9. The pressure dependence of the FSDP position of GeO2 glass. Splineline is guide for eyes. Linear fitting was made in the pressure range of[2.0, 4.3] GPa with extended margin for clarity. Inset shows the x-raydiffraction data and Lorentzian fitting of GeO2 glass at 1 GPa.

is associated with the cage or holes of the network in theglass.22, 23 These classic network-forming glasses have re-cently received renewed investigations both in experimentaland theoretical studies24–30 due to the findings of new inter-mediate states.31–34 Figure 9 shows the FSDP data of GeO2

glass at pressures up to 10 GPa with diamond anvil cell. Withincrease of pressure, only a slight change in FSDP is observedat pressures below 2 GPa, but it increases more quickly athigher pressures. There is a clear knick point at around 2 GPaand 4 GPa, respectively. The linear fit in the range of 2.0–4.3GPa clarifies the existence of a distinct slope in the relation ofFSDP and applied pressure, which confirms the formation ofan intermediate state in this range.31

High-energy x-ray diffraction, which can access a largerange of scattering vector, e.g., 20 Å−1–40Å−1, is becomingof increased importance in material and nano sciences. Usinghigh-energy x-rays of 80 keV at X17B3, total scattering PDFmeasurements have been carried out by users from multipledisciplines.42 This technique involves measuring both Braggand diffuse scatterings for structural analysis, allowing simul-taneously probing local, intermediate, and long-range struc-ture in crystalline, amorphous, or complex materials. For a re-view see Refs. 35–37. However, precise energy calibration byXAS is problematic because of the lack of suitable x-ray ab-sorption edges at the desired high energy. Using our method,the calibrated wavelength of a nominal 80 keV x-ray beamwas found to be 0.1507 ± 0.0001 Å (82.273 ± 0.055 keV).At such high energies, the x-ray energy can be determinedusing an EDXD detector but with much lower resolution.For example, the reported energy resolution for a Ge solidstate detector (Canberra) is 0.22 keV at 8.04 keV and about0.36 keV at 59.54 keV.16 The mounting/dismounting ofDLCM, while switching diffraction experiments between30 keV and 80 keV monochromatic x-ray beam, can in-duce uncertainties in x-ray energy within ±5 keV for 80 keVmonochromatic x-ray beam, which can be well addressed bythe proposed method.

0 5 10 15 20 25

-5

0

5

10

15

20

0 10 20 30 40 50

-4

-2

0

2

4

6

8

10

Q (1/ )

F (

a.u.

)

data fit diff

g(r)

r ( )

FIG. 10. Pair distribution function (PDF) measurement with the calibratedwavelength of 0.1507 ± 0.0001 Å (82.273 ± 0.055 keV). (Upper panel) Re-duced total scattering function F(Q) obtained from the CeO2 powder sample.(Lower panel) Points are the corresponding g(r) function obtained with anupper limit for the Fourier transform which was Qmax = 25 Å−1. Lines arethe simulated PDFs for CeO2 structure solutions with a fit residual, Rw, of0.088. The bottom curve is the difference between measured and simulatedPDFs with offset for clarity.

Figure 10 shows the PDF data of a CeO2 standard us-ing PDFGETX2 program.38 The data were corrected for back-ground scattering, self-absorption, and incoherent Comptonscattering, and then normalized for incident flux, number ofscatterers, and squared atomic form factor, to obtain the struc-ture factor function. Further PDF’s modeling was carried outusing the PDFGUI program.39 Structural refinements of PDFyield information about the local structure rather than the av-erage crystal structure probed by Rietveld refinement. Thestructural models and refined parameters are the same as theRietveld GSAS refinement17, 18 at 31.122 keV. The value ofRw, which is the PDF fit residuum with respect to the es-tablished structure from the literature (fit range 1 < r < 50Å), is 0.088, being close to the reported PDF modeling withRw of 0.103 where high quality PDF data sets were collectedat the 6ID-D beamline of the Advanced Photon Source, Ar-gonne National Laboratory, USA, at x-ray energies of 87 and98 keV.40 This example also proves the reliability of proposedapproach for high-energy x-ray calibration.

We have shown our iterative x-ray energy calibration ap-proach is only tied to the relative detector offset distance,which can be changed in a high precision fashion. For exam-ple, the use of Mitutoyo gauge blocks in 10 mm and 25 mmthickness has ±0.14 μm and ±0.2 μm accuracy, respectively,i.e., 1.4 × 10−5 and 8 × 10−6. For the same grade gaugeblock, the larger the gauge block, the higher the resolutionis. Ideally, if uncertainty is only due to the accuracy of gaugeblock (10 mm blocks as an example), the energy resolutionshould be only 1.4 eV at 100 keV. However, practical uncer-tainty in distance correlation, which compromises the energyresolution of the monochromator and several effects, such asthe size of detector pixel, offset reproducibility, and overallFIT2d fitting quality, is about 4 × 10−4 mm for 10 mm gauge

063901-9 Hong, Chen, and Duffy Rev. Sci. Instrum. 83, 063901 (2012)

block after three iterations (Fig. 3). Energy calibration usingdifferent diffraction standards indicated a standard deviationof 1.9 × 10−4 (Fig. 7). These results are compatible with theenergy resolution of 10−3–10−4 (dE/E) for DLCM monochro-mators of X17C and X17B3 beamline.4, 41

For the same data set, even with very different trial ini-tial values, the proposed iteration method leads to a conver-gent value with small deviation, e.g., 1.2 × 10−4 in sample-detector distance and 2.0 × 10−4 in distance correlation, asshown in Table I. An energy resolution of 10−4 is adequate formost x-ray diffraction experiments. It should be mentionedthat the obtained energy deviation, as listed in Table I, is 3.5eV for 31.122 keV x-ray beam, being comparable with that ofx-ray absorption based energy calibration method.

It is obvious that pixel size of x-ray detector has signifi-cant effect on the calibrated x-ray energy. Pixel size is a keyparameter of a detector, which is usually calibrated and givenby the manufacturer. Using X-rays, there is also an easy wayto calibrate the pixel size with high precision. We can movethe detector transversely across x-ray beam from one edge toother edge, and measure the real displacement, L, using a mi-crometer or a caliper, e.g., the Mar-165 CCD detector wouldhave a resolution of 6 × 10−5 with a digital caliper. The realpixel size can be deduced by L/N, where N is the pixel differ-ence between two beam centers (up to 2048 pixel difference).Pixel coordination of x-ray beam center can be obtained ac-curately by fitting the diffraction pattern of standards usingFIT2D. With correct pixel size, x-ray energy would be cali-brated correctly.

It should be mentioned that the calibration of x-rayenergy/wavelength is closely related to the precisions ofsample d-spacing and diffraction angles. The Bond methodcan effectively eliminate the possible errors of diffrac-tion angles.8 The d-spacing of Si single crystal has beenwell established with a precision of one part in 108 byx-ray/optical interferometry.11 Using such high precisionvalue for Si crystal, the energy calibration at edge en-ergies for K and L edges of selected metals from Z= 23 to 82 has been determined at 10−5–10−6 preci-sion level,12 covering x-ray energy from 5.463–30.490 keV.The precision of x-ray absorption based energy calibra-tion using high precision monochromator seems to be atleast one order higher than that of the proposed diffrac-tion based method at 10−4 precision, which coincides wellwith the �E/E ∼ 10−4 energy resolution of employeddouble-Laue crystal monochromator. Further examination ofthe proposed method using high precision monochromators(�E/E∼10−5–10−6) is highly desired in future work. It isanticipated that better energy resolution in the energy calibra-tion would be achieved if using a high precision monochro-mator, which will lead to improved resolution in the diffrac-tion peak width, i.e., more accurate sample-to-detectordistance obtained by FIT2D fitting.

Although the discussion above is mainly referred to aLaue monochromator, it should be applied to energy cali-bration with other types of monochromators as well. Themethod should be able to work for prompt in situ calibrat-ing/monitoring x-ray energy in diffraction and scattering ex-periments if a motorized stage is employed to drive the x-ray

detector back and forward. Developed software for this en-ergy calibration method is available upon request.

IV. CONCLUSIONS

In summary, we have reported an effective method forprecise and fast absolute x-ray energy calibration over a wideenergy range using an iteration method. No change in the x-ray optics layout is needed. It is found that using this iterativealgorithm the x-ray energy is only tied to the relative offset ofsample-to-detector distance, which can be changed preciselyby use of high precision gauge blocks or high accurate trans-lation motors. Comprehensive investigation of the proposediteration process using several diffraction standards has beenperformed. Applications of the proposed method to the FSDPposition of network forming GeO2 glass and PDF measure-ment using high-energy x-ray diffraction have been demon-strated.

ACKNOWLEDGMENTS

We would like to acknowledge C. Koleda, L. Ehm, andD. Weidner for useful discussions, and X. M. Yu, Z. Zhong,L. P. Huang, and S. Lin for their help in the experiments. As-sistance from M. Abeykoon, P. Gao, and T. Dykhne on PDFanalysis are also acknowledged. This work was supportedby COMPRES, the Consortium for Materials Properties Re-search in Earth Sciences under National Science Foundation(NSF) Cooperative Agreement No. EAR 10-43050. The useof NSLS was supported by the U.S. Department of Energy(DOE), Office of Science, Office of Basic Energy Sciences,under Contract No. DE-AC02-98CH10886.

1J. O. Cross and A. I. Frenkel, Rev. Sci. Instrum. 70(1), 38–40 (1999).2P. Suortti, T. Buslaps, V. Honkimäki, M. Kretzschmer, M. Renier, andA. Shukla, Z. Phys. Chem. 215(11), 1419–1435 (2001).

3Z. Zhong, C. C. Kao, D. P. Siddons, and J. B. Hastings, J. Appl. Crystallogr.34(4), 504–509 (2001).

4Z. Zhong, C. C. Kao, D. P. Siddons, and J. B. Hastings, J. Appl. Crystallogr.34(5), 646–653 (2001).

5R. F. Pettifer and C. Hermes, J. Appl. Crystallogr. 18(6), 404–412(1985).

6J. Arthur, Rev. Sci. Instrum. 60(7), 2062–2063 (1989).7J. V. Acrivos, K. Hathaway, J. Reynolds, J. Code, S. Parkin, M. P. Klein,A. Thompson, and D. Goodin, Rev. Sci. Instrum. 53(5), 575–581 (1982).

8W. Bond, Acta Crystallogr. 13(10), 814–818 (1960).9F. Herbstein, Acta Crystallogr. Sect. B 56(4), 547–557 (2000).

10U. Bonse and M. Hart, Appl. Phys. Lett. 6(8), 155–156 (1965).11P. Becker, K. Dorenwendt, G. Ebeling, R. Lauer, W. Lucas, R. Probst, H.-J.

Rademacher, G. Reim, P. Seyfried, and H. Siegert, Phys. Rev. Lett. 46(23),1540–1543 (1981).

12S. Kraft, J. Stumpel, P. Becker, and U. Kuetgens, Rev. Sci. Instrum. 67(3),681–687 (1996).

13A. P. Hammersley, S. O. Svensson, A. Thompson, H. Graafsma, A. Kvick,and J. P. Moy, Rev. Sci. Instrum. 66(3), 2729–2733 (1995).

14J. Liu, K. Kim, M. Golshan, D. Laundy, and A. M. Korsunsky, J. Appl.Crystallogr. 38(4), 661–667 (2005).

15T. S. Duffy, Rep. Prog. Phys. 68(8), 1811–1859 (2005).16J. Z. Hu, H. K. Mao, J. F. Shu, Q. Z. Guo, and H. Z. Liu, J. Phys.: Condens.

Matter 18(25), S1091 (2006).17A. C. Larson and R. B. Von Dreele, Los Alamos National Laboratory

Report LAUR 86–748, 2000.18B. Toby, J. Appl. Crystallogr. 34(2), 210–213 (2001).19A. C. Wright, J. Non-Cryst. Solids 179, 84–115 (1994).20S. C. Moss and D. L. Price, Physics of Disordered Materials (Plenum, New

York, 1985).

063901-10 Hong, Chen, and Duffy Rev. Sci. Instrum. 83, 063901 (2012)

21D. L. Price, S. C. Moss, R. Reijers, M. L. Saboungi, and S. Susman,J. Phys.: Condens. Matter 1(5), 1005 (1989).

22A. C. Wright, R. N. Sinclair, and A. J. Leadbetter, J. Non-Cryst. Solids71(1–3), 295–302 (1985).

23S. R. Elliott, Phys. Rev. Lett. 67(6), 711–714 (1991).24M. Vaccari, G. Garbarino, G. Aquilanti, M. V. Coulet, A. Trapananti,

S. Pascarelli, M. Hanfland, E. Stavrou, and C. Raptis, Phys. Rev. B 81(1),014205 (2010).

25Q. Mei, S. Sinogeikin, G. Shen, S. Amin, C. J. Benmore, and K. Ding,Phys. Rev. B 81(17), 174113 (2010).

26J. W. E. Drewitt, P. S. Salmon, A. C. Barnes, S. Klotz, H. E. Fischer, andW. A. Crichton, Phys. Rev. B 81(1), 014202 (2010).

27M. Dario et al., J. Phys.: Condens. Matter 22(15), 152102 (2010).28M. Baldini, G. Aquilanti, H. k. Mao, W. Yang, G. Shen, S. Pascarelli, and

W. L. Mao, Phys. Rev. B 81(2), 024201 (2010).29N. Funamori and T. Sato, Earth Planet. Sci. Lett. 295(3–4), 435–440

(2010).30T. Sato and N. Funamori, Phys. Rev. Lett. 101(25), 255502 (2008).31X. Hong, G. Shen, V. B. Prakapenka, M. Newville, M. L. Rivers, and S. R.

Sutton, Phys. Rev. B 75(10), 104201 (2007).

32X. Hong, G. Shen, V. B. Prakapenka, M. L. Rivers, and S. R. Sutton, Rev.Sci. Instrum. 78(10), 103905 (2007).

33G. Shen, H.-P. Liermann, S. Sinogeikin, W. Yang, X. Hong, C.-S. Yoo, andH. Cynn, Proc. Natl. Acad. Sci. U.S.A. 104(37), 14576–14579 (2007).

34X. Hong, M. Newville, V. B. Prakapenka, M. L. Rivers, and S. R. Sutton,Rev. Sci. Instrum. 80(7), 073908 (2009).

35S. J. L. Billinge, Z. Kristallogr. 219(3), 117–121 (2004).36S. J. Billinge and I. Levin, Science 316(5824), 561–565 (2007).37S. J. L. Billinge, T. Dykhne, P. Juhas, E. Bozin, R. Taylor, A. J. Florence,

and K. Shankland, Cryst. Eng. Comm. 12(5), 1366–1368 (2010).38X. Qiu, J. W. Thompson, and S. J. L. Billinge, J. Appl. Crystallogr. 37(4),

678 (2004).39C. L. Farrow, P. Juhas, J. W. Liu, D. Bryndin, E. S. Božin, J. Bloch, P. Th,

and S. J. L. Billinge, J. Phys.: Condens. Matter 19(33), 335219 (2007).40P. Juhas, L. Granlund, S. R. Gujarathi, P. M. Duxbury, and S. J. L. Billinge,

J. Appl. Crystallogr. 43(3), 623–629 (2010).41Z. Zhong, C. C. Kao, D. P. Siddons, H. Zhong, and J. B. Hastings, Acta

Crystallogr. Sect. A 59(1), 1–6 (2003).42L. Ehm, L. A. Borkowski, J. B. Parise, S. Ghose, and Z. Chen, Appl. Phys.

Lett. 98(2), 021901 (2011).