VOLUME 80, NUMBER 15 P H Y S I C A L R E V I E W L E T T E R S 13 APRIL 1998

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Solving the Phase Problem Using Reference-Beam X-Ray Diffraction

Qun ShenCornell High Energy Synchrotron Source and Department of Materials Science and Engineering, Cornell Univers

Ithaca, New York 14853(Received 24 November 1997)

A new method of obtaining Bragg reflection phases in an x-ray diffraction experiment is presented.It combines the phase-sensitive principles of multiple-beam diffraction and x-ray standing waves, andallows direct phase measurements of many multiple reflections simultaneously using a Bragg-inclinedoscillating-crystal geometry. A modified-two-beam intensity function is devised to extract the phaseinformation in a way similar to the standing wave analyses. [S0031-9007(98)05790-1]

PACS numbers: 61.10.– i, 42.25.Fx, 78.70.Ck

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X-ray diffraction and scattering techniques are usewidely in structural studies of crystalline materials. Thestechniques provide real-space charge density informatiby probing its Fourier components, or structure factorin reciprocal space. In a typical diffraction experimenwhat one measures is the intensity of a scattereddiffracted beam, which is related only to the structurfactor amplitudeand not to itsphase. This loss of phaseinformation is the classicphase problemin diffractionphysics and crystallography, and its solution remains tmost difficult part of a crystal structure determination [1]

To date, all practical methods leading to solutions othe phase problem in crystallography can be groupinto two categories. In the first category are varioumathematical techniques, such as the direct methods [2These methods rely largely on the overdeterminatioin the intensity measurements of a great numberBragg reflections and use a probability distribution opossible phases to solve a crystal structure. While vepowerful for small molecule structures, application othese statistics-based mathematical methods to larcrystal structures remains to be difficult and is stian active area of research. In the second categorycrystallographic methods are various chemical methoinvolving heavy-atom derivatives and replacements [4using x-ray dispersion corrections in the heavy-atoscattering factor in either single- or multiple-wavelengtanomalous diffraction [5]. With these chemical methoda crystal structure is solved by the additional phasinformation provided by the heavy-atom substructurIn general, the chemistry-based techniques often requcomplex and time-consuming chemical treatments to boheavy atoms in proteins and other biological systems.

In recent years, there have been considerable effortsfind a physicalsolution to the phase problem, namely, tobtain the phases of the Fourier components directly frodiffraction experiments. One promising physical solutiois the multiple-beam Bragg diffraction [6–9], which isbased on the interference among simultaneously exciBragg reflections. The effect has been shown visibboth for small molecule compounds [10,11] and focomplex crystals such as quasicrystals [12] and prote

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[13,14]. The conventional technique for performing suan experiment involves exciting one Bragg reflectiH and then rotating the crystal around the scattervectorH to bring another reflectionG into its diffractioncondition [15]. ReflectionH is called the main reflectionandG the detour reflection (umweganregung). This one-reflection-at-a-time measurement method limits seriouthe practical implications of the multiple-beam diffractiotechnique and makes it almost impossible to measa large number of phases that are required to solvcomplex crystal structure.

In this Letter, we present a reference-beam x-rdiffraction method for obtaining both the phases andmagnitudes of a large number of Bragg reflections. Tmethod is based on the same multiple-beam diffractprinciple but with the important distinction that a singumweganregungserves as the reference beam andcommon to all recorded multiple Bragg reflections. This achieved by incorporating features in x-ray standiwave experiments as well as in common crystallograpdata collection techniques. The combined result allostraightforward phase-sensitive diffraction measuremeof a large number of reflections with a well-controllephase variation of the reference Bragg beam.

To introduce the reference-beam diffraction techniqit is helpful to review how a normal crystallographix-ray diffraction experiment is performed. By far, thmost popular way of collecting crystallographic data is trotating or oscillating crystal method [1]. In this methoa crystal is rotated around an axis perpendicular to thecident beam and the Bragg reflection diffraction datacollected on a two-dimensional detector such as a ptographic film, an image plate (IP), or a charge-coupdevice (CCD), as shown in Fig. 1(a). In reciprocal spathe sphere of reflection (Ewald sphere) sweeps througvolume of a toroid during a full360± rotation and everyreciprocal lattice point in this volume is set to diffract atcertain rotation angle.

The new reference-beam diffraction technique usalmost the same geometry as the standard oscillamethod,exceptthat the rotation axis is aligned alongreference Bragg reflectionG, which is brought to its

© 1998 The American Physical Society

VOLUME 80, NUMBER 15 P H Y S I C A L R E V I E W L E T T E R S 13 APRIL 1998

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FIG. 1. (a) Conventional crystallographic x-ray diffractiotechnique using direct-beam rotating crystal method withIP or a CCD detector. (b) New reference-beam diffractiotechnique allows simultaneous measurements of bothintensities and the relative phases of all Bragg reflectiorecorded on the same two-dimensional detector.

diffraction condition, as shown in Fig. 1(b). We will calthis the Bragg-inclined geometry. Since the crystal rtation axis coincides withG, the reference reflection is

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always excited during the rotation or oscillation, therefoall of the reflections recorded on the two-dimensional dtector are, in fact, multiple-beam excitations and their itensities are affected by the interference with the referenbeam, much like in a hologram. The reciprocal space vume that the sphere of reflection sweeps through is vesimilar to that in the standard rotation case, and, therefoa similar large number of reflections can be measured.

The basic concepts shown in Fig. 1 can be easilysualized using the standard scattering theories in qutum mechanics and electrodynamics [16,17]. A scatterx-ray wave fieldDsrd from a crystal can be representeby a Born approximation series, with its zeroth-order slution Ds0d ­ D0 exps2ik0 ? rd being the incident wave,the first-order solutionDs1d being a singly scattered wavein the usual kinematic or two-beam approximation, anthe second-order solutionDs2d representing a doubly scat-tered detoured wave with three-beam interactions, andon [18]:

Dsrd ­ Ds0d 1 Ds1d 1 Ds2d 1 · · · . (1)

It can be seen immediately that the direct-beam rotaticrystal geometry in Fig. 1(a) is based on observing ttwo-beam reflectionsDs1d with the rotation axis linedup on the incident beamDs0d, while the Bragg-inclinedgeometry in Fig. 1(b) is set up automatically to observthe three-beam interactionsDs2d with the rotation axischosen to diffract a reference beamDs1d.

In general, ifDs1d is the diffracted wave forH reflectionandDs2d is the detour-diffracted wave throughG andH-Greflections, it can be shown [18] that, apart from a scafactor, the diffracted waveDH up to the second order canbe expressed as

DH ­ k̂H 3

∑k̂H 3 FH

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0 2 k2G

∂∏, (2)

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whereG ­ rel2ypVc; re ­ 2.818 3 1025 Å is the clas-sical radius of an electron;l is the x-ray wavelength;Vc

is the unit cell volume;FH, FG , andFH-G are the struc-ture factors; andkG ­ k0 1 G, kH ; k0k̂H ­ k0 1 Hare the wave vectors of the reflections involved. Ttwo terms in Eq. (2) correspond toDs1d andDs2d, respec-tively. Equation (2) demonstrates that the three-beamterference betweenDs1d and Ds2d depends on the relativephase differencedHG ­ aG 1 aH-G 2 aH, whereaH’sare simply the phases of the corresponding structuretors. This phase differencedHG is called theinvarianttriplet phasein crystallography since it is independent othe choice of origin in the unit cell.

In the Bragg-inclined reference-beam diffraction gemetry, a complete interference profile of each recordBragg reflection is obtained by varying the relative phaof the G-diffracted beam. This is accomplished by rocing the crystal through the Darwin width of theG re-

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flection, causing a phase change ofp, in a way analo-gous to that used in the x-ray standing wave techniq[19]. It should be noted that the role of theG reflec-tion in the new reference-beam diffraction techniqueswitched from that in the conventional multiple-beaazimuthal scanning method. Here, the referenceG re-flection is the detour reflection, and each reflection mesured on the two-dimensional detector is actually the mreflection H. All of these main reflections have theGreflection as the commonumweganregung. It illustratesone of the main advantages of the reference-beam diffrtion technique: A single referenceG reflection providesa common perturbation and a common dynamical phashift of p, determined byk2

0 2 k2G, on all main reflec-

tions. Therefore, any differences in their interference prfiles are due purely to the values of the triplet phasesdHG.

We have performed a synchrotron x-ray diffractioexperiment on a GaAs single crystal to demonstra

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VOLUME 80, NUMBER 15 P H Y S I C A L R E V I E W L E T T E R S 13 APRIL 1998

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the reference-beam diffraction principle. The experimewas done at the C1 station of the Cornell High EnerSynchrotron Source (CHESS), using unfocused bendimagnet monochromatic radiation of 13.5 keV with aincident beam size of0.25 3 1 mm2. The crystal wasa 5-mm-thick rectangular-shaped plate and was mounat the center of a standard four-circle diffractometer. Treference reflectionG was chosen to be the symmetri(004) and it was aligned parallel to thef axis of thediffractometer, which served as the oscillation axis in thvertical diffraction plane. The rocking curve of the (004was controlled by theu-rotation axis of the diffractometer.

A Fuji image plate was mounted on the2u detectorarm to collect the oscillating-crystal diffraction patternA series of oscillation exposures with the same rotatirange ofDf ­ 20± were taken at 16 differentu settingsacross the (004) rocking curve. The rocking rangeDu ­ 0.045± was about seven times the measured fwidth at half maximum of the (004) rocking curve[shown in Fig. 2(d)]. To minimize the possible systematreadout errors in image plates, all 16 exposures wrecorded on a single image plate by slightly translatithe image plate after eachu movement, similar to a streakcamera. Each20± oscillation took about 40 s. Duringthe exposure, the incident beam intensity was monitorby an ionization chamber to ensure that the incident bevariations were taken into account. A portion of thdiffraction image recorded on image plates is shownFig. 2(a) which covers a2u range from5± to 103±.

A magnified view of the recorded (317) Bragg reflection is shown in the inset of Fig. 2(a). Each rectangulspot in the image corresponds to a givenu step in theG-reflection rocking curve. The reference-beam interfeence profile is essentially the intensity distribution alonthe strip as a function ofu, and can be seen clearly in thraw data without any processing. Such peak-and-valinterference profiles exist onall recorded reflections. Tofurther quantify the interference profile, we integrate thintensities around each spot and plot the integrated intsities as a function of theu steps, as shown in Fig. 2(b)For comparison, a rocking curve of the reference (004)flection is shown in Fig. 2(d) on the sameu scale.

The reference-beam interference profile in Fig. 2(b) cbe understood by using the perturbation theory of Eq. (In our experiment, the incident x-ray beam is linearpolarized withD0 perpendicular to the scattering plandefined byk0 andG. In this case, Eq. (2) reduces to thfollowing simple form:

DH ­ FHD0'

µ11

ÇFH-GFG

FH

ÇGeidHG

2 sin2uGDu

∂, (3)

where D0' is the projection ofD0 onto a plane per-pendicular tokH, and we have used Bragg’s law foG reflection and have written explicitly thedHG de-pendence. Equation (3) indicates that when the grazangleu changes from less to greater than the Bragg ang

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FIG. 2. (a) Recorded image-plate oscillation-diffraction datof the GaAs crystal at 16u positions across the (004)rocking curve. The inset shows a magnified view of the (317reflection. (b) Integrated intensities (filled circles) of the 16spots for the (317), as a function ofu. The solid curve isa three-beam dynamical calculation withdHG ­ 0±, and thedashed and the dotted curves are calculated using Eq. ((c) Dynamical phase shiftnsud of the (004), calculated usingthe standard two-beam dynamical theory. (d) Measured (00rocking curve plotted on the sameu scale and fitted with aGaussian profile.

Du changes from negative to positive, and, thus, a dstructive (constructive) interference always occurs at thlow (high) angle side,unlessdHG provides an additionalphase shift to compensate it. Therefore, by examining texperimental data in Fig. 2(b), we conclude thatdHG forthe s317dys004d three-beam reflection has to be arounzero. This conclusion is confirmed both by the perturbation calculation using Eq. (3), shown as the dashed curin Fig. 2(b), and by a full three-beam dynamical calculation using NBEAM approach [6], shown as the solidcurve.

VOLUME 80, NUMBER 15 P H Y S I C A L R E V I E W L E T T E R S 13 APRIL 1998

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The multiple-beam diffraction analyses in the newreference-beam geometry can be made entirely analogto that in the x-ray standing wave method. To removthe singularity atDu ­ 0 in Eq. (3), we use the finitereflectivity Rsud and the phase shiftnsud of DGyD0 inthe usual two-beam dynamical theory [19] forG andsubstitute1yDu with

pRsud einsudyw, where w is the

Darwin width of theG reflection. Squaring the modulusof Eq. (3) and keeping only the linear term inG, weobtain a normalized three-beam interference intensity a

IH ­ 11

ÇFH-G

FH

Ç qRsud cosfdHG 1 nsudg . (4)

A plot of nsud for the GaAs (004) as calculated inthe standard two-beam dynamical theory is shownFig. 2(c). Apparently, Eq. (4) is the x-ray standing waveffect of theG reflection on another Bragg reflectionH.Because three-beam diffraction is a coherent process,effect is somewhat different from the standing wave intensities that govern the incoherent fluorescence yieEquation (4) is used in Fig. 2(b) to produce the pertubation calculations for two different triplet phase valuesdHG ­ 0 and p . The calculation withdHG ­ 0 agreesvery well both with the experimental data and with thmore rigorous NBEAM calculation.

In addition to the simplified multibeam analysis andthe ability to measure the phases of many Bragg refletions in a simple manner, the new reference-beam diffration method also reduces the need for the high anguincident-beam collimation perpendicular to the verticascattering plane. The reason for this is that the referenreflection G can always be set to diffract in the verti-cal plane where the natural collimation of the synchrotroradiation is best. Complications on the three-beam inteference due to polarization mixing [20,21] are also elimnated as shown by Eq. (3). Furthermore, the phase of treference beam can be varied by changing the incidex-ray energy instead of rocking the crystal. In this waythe data collection for reference-beam diffraction can bmade as routine as other crystallographic techniques suas the multiwavelength anomalous diffraction [5], withouthe need for heavy atoms.

In summary, we have presented a simple referencbeam x-ray diffraction technique to measure many Bragreflection phases directly from an oscillating-crystadiffraction experiment. A single detour reflection common to all measured Bragg reflections is used as treference beam, and its phase is varied by either rockithe crystal or changing the incident beam energy. Thallows a straightforward analysis similar to that used ithe x-ray standing wave method. Additional experimen

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are under way to employ this new technique on morcomplex crystal systems. With further research and dvelopment, we believe that the new method can becoma powerful and ultimate technique for solving the phasproblem in crystallography and diffraction physics.

The author thanks Marian Szebenyi and Bill Miller fortheir assistance on image plates, and Boris BattermaStefan Kycia, Ernie Fontes, and Ken Finkelstein for usefudiscussions and encouragement. This work is supportby the National Science Foundation through CHESunder Grant No. DMR-9311772.

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