X-ray diffraction using a coherent beam involves the mutual interference among all the extremities of small crystals. The continuous diffraction pattern so produced can be phased because it can be oversampled. We have thus obtained three-dimensional images of the interiors of Au nanocrystals that show 50 nm wide bands of contrast with f111g orientation that probably arise from internal twinning by dynamic recrystallization during their formation at high temperature

Pfeifer, MA

Robinson, IK

Vartanyants, IA

Williams, GJ

English

UCL Discovery

Three-Dimensional Imaging of Microstructure in Au NanocrystalsG.J.Williams, M. A. Pfeifer, I. A.Vartanyants,* and I.K. RobinsonDepartment of Physics, University of Illinois, Urbana, Illinois 61801(Received 19 November 2002; published 29 April 2003)X-ray diffraction using a coherent beam involves the mutual interference among all the extremities ofsmall crystals. The continuous diffraction pattern so produced can be phased because it can beoversampled. We have thus obtained three-dimensional images of the interiors of Au nanocrystalsthat show 50 nm wide bands of contrast with f111g orientation that probably arise from internaltwinning by dynamic recrystallization during their formation at high temperature.DOI: 10.1103/PhysRevLett.90.175501 PACS numbers: 61.10.Nz, 42.30.Rx, 61.46.+w, 68.37.YzDeformation of metals is understood to involve com-plex concerted motions of the dislocations that inevitablybreak the perfection of their individual crystallinegrains, which are usually nanometers to microns in size[1]. Electron microscopy (EM) is the most widely usedtechnique for dissecting this internal microstructure.Therelatively strong interaction of electrons with condensedmatter means that the method is limited to viewing eitherthe exterior surfaces of microstructures [scanning EM(SEM)] or else 2D projections through thin slices [trans-mission EM], which requires destructive sample prepa-rations. It is highly desirable to ﬁnd a nondestructive wayof imaging the interiors of material grains in 3D so thatarbitrary views can be investigated. This is analogous tothe advances in medicine that followed the developmentof computed tomography methods in the 1970s [2,3].Microtomographic methods for 3D imaging of materialshave since been developed [4,5]. The advantage of sec-tions over projection views is that microstructural fea-tures are unambiguously localized in 3D.Coherent x-ray diffraction (CXD) is a rapidly develop-ing technique [6] that opens opportunities for the study ofboth dynamics of condensed-matter systems and theirstructure.The method has become available by the recentdevelopment of high-brilliance third-generation sourcesof synchrotron radiation, such as the Advanced PhotonSource (APS). In favorable circumstances, the beam’scoherence volume, which is directly proportional to thesource brilliance, can entirely enclose a small crystal.Diffraction from all parts of the crystal then undergoeswave superposition which leads to a highly structuredCXD pattern. If high-quality x-ray lenses were available,as they are for electrons, such diffraction patterns couldbe transformed to magniﬁed images directly. In spite ofwidespread efforts to develop such x-ray microscopes[7,8], they are still not competitive with EM.In this Letter, we show that the objective lens of anx-ray microscope can be replaced by a computation thatinverts a 3D measurement of the amplitude of a CXDpattern to a 3D image. The computation relies on the factthat the diffraction can be oversampled relative to itsspatial Nyquist frequency, so that the Fourier transformcan be overdetermined in spite of the missing phaseinformation. We apply the method to the 3D imaging ofnanocrystals of Au obtained by high-temperature coales-cence from an evaporated polycrystalline thin ﬁlm. Weﬁnd residual internal texture that could be attributed totwinning which might originate from dynamic recrystal-lization of defects introduced during their formation.When a crystal is illuminated by a coherent x-raybeam, the intensity distribution in the vicinity of aBragg point forms a well-deﬁned pattern. In the kine-matical limit, this 3D CXD pattern is of the form of thesquare of a structure factor multiplied by a function:I q /             Z 10 L r s r eiq reiq urdr             2: (1)The total electron density of the crystal is written as theproduct of the crystal lattice,  L r, and a function de-scribing the shape of the crystal, s r. Thus, the intensitydistribution in reciprocal space is the convolution of theFourier transform of the crystal shape with the reciprocallattice of thebulkcrystal.Thefunctionur representsthestrain ﬁeld due to the displacement of atoms from theirbulk structure sites; therefore strain inside the crystalgives rise to noncentrosymmetry in the CXD pattern. Inan earlier paper we reported the inversion of a CXDpattern in 2D to arrive at the projection of a crystalonto a plane [9]. More recently, a 3D calculation of thiskind has been achieved by Miao et al. [10]. Here wepresent the extension of our procedure, allowing therecovery of the 3D density distribution within the nano-crystal.Since the intensity distribution in reciprocal space is acontinuous one, the pattern around a Bragg peak can besampled with arbitrary q spacing. This property allowsthe oversampling of such a pattern, leading to the collec-tion of sufﬁcient information toiteratively reconstruct thephase of the collected diffraction data. In 1982, Batesshowed that nonunique solutions to such problems arepathologically rare in dimensions higher than one [11].While noise in the data increases the probability ofPHYSICAL REVIEW LETTERS week ending2 MAY 2003 VOLUME 90, NUMBER 17175501-1 0031-9007=03=90(17)=175501(4)$20.00  2003 The American Physical Society 175501-1ambiguous solutions, it is possible to regain a uniquesolution by applying an appropriate constraint [12].To understand the oversampling requirement in 1D, wewrite the amplitude of the intensity collected in pixel mina detector ofN pixels asthe discrete Fourier transform:F m qx  X Nn 1  n x ei2 mn=N; (2)where  qx is the size of a pixel in the detector,  x is thereal space sampling, related by  x   2 =N qx.I ti sclear from (2) that F is a periodic function with N distinctvalues of m. In real space, we deﬁne a support region, N0,that encompasses the entirety of the diffracting density,i.e.,   n x  0, for all n such that N0 <n  N. Then,the oversampling ratio, , is simply theratio of the size ofthe entire array to the size of the support,     N=N0.I nreciprocal space, the oversampling ratio is given by theratio of the number of points measured per oscillation ofthe transform (2) to the number of points necessary todeﬁne an oscillation, which is the Nyquist sampling rate.These oscillations are readily seen as fringes in the ex-perimental data. Even if weassume the real space density,  x , is purely real, we have N unknowns and only N=2measurements, because F qx  F   qx . Therefore,  must be at least 2 in order for it to be possible to invert (2),i.e., to phase the diffraction pattern. The extension from1D to 3D is trivial because the equation counting argu-ment is the same. We introduce a separate measure ofoversampling for each dimension,      x y z, with    2. Millane has shown, using simulations, that inthe case with two of the  i’s greater than 2, the require-ment on the oversampling in the third dimension is re-laxed to sampling at the Nyquist frequency,  j   1 [13].The phasing of an oversampled diffraction pattern maybe accomplished through the use of an iterative method[14], utilizing known information to ﬁnd a set of phasesconsistent with the measured data. Such methods havebeen used successfully in similar situations, for example,electron scattering measurements [14] and transmissionx-ray studies of nonperiodic materials [15]. The ﬁttingbegins by assigning a set of random phases to the supportand evaluating (2) using a fast Fourier transform, whichyields a ﬁrst ‘‘guess’’ at the diffraction pattern in recip-rocal space. The amplitude of the guess is then overwrit-ten with the measured amplitude and back transformed toyield a complex density in real space. After applying anappropriate real space constraint—for example, enforc-ing a ﬁnite support and converting the density to a real,non-negative function—it is transformed back to recip-rocal space. This cycle is repeated until the search con-verges to a solution.The method described above is called the error reduc-tion (ER) method [16]. The ER method may be modiﬁedto propagate the result of previous iterations, giving vari-ous ‘‘input/output methods’’ [16].The hybrid input/output(HIO) method of Millane incorporates a parameter,  ,deﬁning a fraction of the previous cycle’s result to com-bine with the current result. Additionally, the real spaceconstraint is relaxed by allowing a small amount ofamplitude,  , to exist outside the support [17].In order to conduct a CXD experiment, the spatialextent of the diffracting density must be smaller thanthe coherence volume of the incident beam. The insertiondevice at Sector 33 of the APS, with a source size of  x  240  m and  y   16  m, gives transverse coherencelengths greater than 5  m. The longitudinal coherenceis determined to be 6  m by a Si(333) monochromatorat 9.5 keV.Diffraction patterns were collected with a direct-reading CCD, whose pixel size is 22:5  m   22:5  m,located 2.8 m from the sample. A 2D CXD pattern wascollected near a f111g Bragg peak for 5–10 min. In orderto collect a full 3D pattern we rotated the sample throughthe Bragg condition in steps of 0.002  acquiring a se-quence of nearly parallel 2D slices, as shown in Fig. 1,which were then stacked to obtain a sampled 3D pattern.This procedure is equivalent to shifting the detectorperpendicular to the Ewald sphere in reciprocal space.Because the 3D diffraction pattern surrounds a reciprocallattice point located far from the origin, the entire mea-surement spans an angular range of only 0.1 . By con-trast, the forward-scattering version of the experiment[10] requires a 180  range.To form crystals smaller than the coherence volume, a100 nm Au ﬁlm was deposited onto a Si(100) wafer, withthe native oxide intact, at room temperature forming aconventional f111g textured polycrystalline ﬁlm. Thesample was then placed in a radiative heating cell, whichwas mounted on the goniometer, capable of achievingtemperatures up to the melting point of Au. Temperaturewas measured by a thermocouple afﬁxed to the back ofthe sample and by tracking the expansion of the latticeand comparing with tabulated values [18].i kkfQ∆θ=0.002q zq yDetectionPlanesFIG. 1. Experimental geometry. As the sample is rocked, 2Dslices are collected through the 3D CXD pattern.PHYSICAL REVIEW LETTERS week ending2 MAY 2003 VOLUME 90, NUMBER 17175501-2 175501-2As the ﬁlm was heated to near its melting point, itspontaneously dewetted from the Si=SiO2 substrate inless than 20 h at 1000  C. The ﬁlm ﬁrst formed one-dimensional ‘‘rivers,’’ before splitting at grain boundariesinto small single crystalline islands. This procedure re-sulted in shapes similar to the well-known equilibriumshapes [19] with a distribution of sizes in the micronrange. SEM recorded this evolution after the preparation.Since an array of nanocrystals with different azimuthalorientations is illuminated by the x-ray beam, one nano-crystal can be selected and measured as a 3D rockingseries. Three such slices acquired at 950  C are shownin Fig. 2.Certain information about the diffracting crystal maybe discerned from the CXD pattern directly. The ﬂares inreciprocal space correspond to facets of the crystal andthe modulation of these ﬂares to the spacing of parallelfacets. Therefore, it is possible to make some conclusionsabout the size and general shape of the particle before theCXD pattern is phased. The size of the particle in a givendirection is 2 = qfringe; hence the particle we measuredis approximately 1  m by 3  m.In the experimental geometry, we deﬁne our localreciprocal space axes as follows: qx and qy are mutuallyperpendicular and coplanar with the surface of the CCD,and qz is perpendicular to Q, the total scattering vector,as shown in Fig. 1.Thus,  qx    qy   3:9   10 4 nm 1is the reciprocal space spanned by a pixel in the CCD.Rocking the sample gives a spacing of  qz  9:3   10 4 nm 1 in the third direction. From Fig. 2 wecan readily estimate the oversampling of the measure-ment. The data have a minimum of 15 pixels per fringe,so we may be conﬁdent that  x   2 and  y   2.T h eoversampling in the third direction,  z, can be estimatedfrom the 2D fringe spacing, of order 5   10 3 nm 1,a n dth es te ps iz e, qz.Thus,weexpect that  z > 2, satisfyingthe oversampling condition.Once the 2D slices have been collected, they arestacked to form a 3D reciprocal space constraint usedduring the ﬁtting.We deﬁne an error metric in reciprocalspace:  2  PNi 1 FCi        IMiq 2=PNi 1 IMi where FCi is theamplitude at pixel i in the current ﬁt,      IMiqis the ampli-tude of pixel i in the measured CXD pattern, and N is thetotal number of pixels in the 440   440   30 array used.Each ﬁt consists of 2300 iterations, alternating betweenthe ER and HIO algorithms, with a real space constraintbox of dimension 100   100   20 pixels. Since the ER isprone to stagnation near local minima of the error metric[12], we performed 50–100 cycles of HIOafter each 150–250 cycles of ER. In our calculations, the best results wereobtainedwiththevalues    0:01and between0.75and0.85 in the HIO algorithm.The resolution—the period of the ﬁnest feature detect-able—of the real space reconstruction is     2 = qmax, where  qmax is the radius in q space beyondwhich the diffracted intensity had dropped to the noiselevel.This resolution might be improved by extending theexposure time. For the data shown in Fig. 2, we estimateour resolution to be   x;y   80 nm and   z   450 nm.The 3D real space density corresponding to the recon-structed phase and measured amplitude of the CXDpattern is shown in Fig. 3. The reciprocal space ﬁts have 2   0:04, which corresponds to an uncertainty of about20% per pixel. The two best real space reconstructionsobtained from 15 sets of random starting phases areshown in Fig. 3. Using a measure of real space reprodu-cibility analogous to  2, i.e., by comparing the ﬁt ampli-tudes, we ﬁnd the disagreement to be 0.033.This quantityis 3   10 5 for the corresponding reciprocal space re-constructions. We interpret this reproducibility to be ameasure of the experimental uniqueness, since trueuniqueness is not assured with noisy data [12].As expected, the 3D CXD real space images havehigher contrast than the 2D projections reported previ-ously [9].The diamond shaped region of high intensity isbelieved to be an artifact caused by the partial coherenceof the incident beam [20]—speciﬁcally, a Bewindow 6 mfrom the sample. The crystal’s external morphology is anelongated-disk with well-developed  111  facets on thetop and bottom surfaces, parallel to the substrate. This 111  plane appears diagonally in the side-view sectionsshown because of the coordinate transformation of thediffractometer operating in a grazing-exit geometry [21].The faces are clearly visible, but it is not possible toFIG. 2 (color online). Typical slices through the 3D CXDpattern of a Au crystal near the  111  reﬂection with spacing2:8   10 3 nm 1.PHYSICAL REVIEW LETTERS week ending2 MAY 2003 VOLUME 90, NUMBER 17175501-3 175501-3distinguish between the top and bottom sides of theimage. The density appears above center in the earlysections (z ’  1:35  m) and below center in the latersections (z ’ 1:35  m), consistent with an oval morphol-ogy sectioned obliquely. The  111  reﬂection, which wasused for imaging, points directly up the page [21]. Thiscan be seen to lie at the correct 70:5  angle to the  111 direction.The internal density contrast is in the form of brightand dark bands oriented parallel to both the  111  and 111  directions in the images. The  111  bands have awidth of 50 nm, a period of 100 nm, and a lateral extentof 600 nm, both within the section and between adjacentsections. The  111  bands have the same width and areslightly less extended. Although dynamical scatteringand series termination effects may play a role in theappearance of these features, we interpret these to bedeformation bands associated with the sample prepara-tion.The dark region of the band presumably correspondsto material with twinned stacking that would diffract in adirection different from the  111  imaging direction. Suchbands are known to occur in soft fcc metals and areattributed to recrystallization following slippage along 111  planes during deformation [22]. Given the oval,elongated-disk morphology of our nanocrystal samples,it is clear that they have not yet reached an equilibriumshape in their growth out of the polycrystalline startingﬁlm. It is therefore not surprising that residual deforma-tion defects are observed.In conclusion, we have obtained 3D images of micro-structure by inversion of a coherent diffraction pattern.Our method differs from related experiments [10] in thatthe entire pattern may be measured directly. BecauseBragg diffraction is involved, all artifacts due to air orslit scatter and other grains are avoided and a strongsensitivity to strain results.We are grateful to C. Benson for technical assistance.This research was supported by the NSF under GrantNo. DMR98-76610. The UNICAT facility at the APS issupported by the University of Illinois, MaterialsResearch Laboratory (DOE Contract No. DEFG02-91ER45439, the State of Illinois-IBHE-HECA, and theNSF), the ORNL, NIST, and UOP LLC. The APS issupported by the DOE under Contract No. W-31-109-ENG-38. SEM work was carried out in the Center forMicroanalysis of Materials, University of Illinois.*On leave from Institute of Crystallography RAS,Leninsky pr. 59, 11733 Moscow, Russia.[1] A.H. Cottrell, The Mechanical Properties of Metals(Wiley, New York, 1964).[2] P. Wells, J. Davis, and M. Morgan, Mater. Forum 18, 111(1994).[3] A.C. Kak and M. Slaney, Principles of ComputerizedTomographic Imaging (SIAM, Philadelphia, PA, 2001).[4] P. Cloetens et al., Europhys. News 32, 46 (2001).[5] Y. Wang et al., Rev. Sci. Instrum. 72, 2062 (2001).[6] M. Sutton et al., Nature (London) 352, 608 (1991).[7] C. Jacobsen and J. Kirz, Nat. Struct. Biol. 5, 650 (1998).[8] W. Meyer-Ilse et al., Synchrotron Radiation News 8,2 9(1993).[9] I.K. Robinson, et al., Phys. Rev. Lett. 87, 195505 (2001).[10] J. Miao et al., Phys. Rev. Lett. 89, 088303 (2002).[11] R.H.T. Bates, Optik 61, 247 (1982).[12] J.H. Seldin and J.R. Fienup, J. Opt. Soc. Am. A 7, 412(1990).[13] R. P. Millane, J. Opt. Soc. Am. A 13, 725 (1996).[14] R.W. Gerchberg and W.O. Saxton, Optik 35, 237 (1972).[15] J. Miao, P. Charalambous, J. Kirz, and D. Sayre, Nature(London) 400, 342 (1999).[16] J.R. Fienup, Appl. Opt. 21, 2758 (1982).[17] R. P. Millane and W.J. Stroud, J. Opt. Soc. Am. A 14, 568(1997).[18] Y.S. Touloukian, Thermophysical Properties of MatterVol. 12 (IFI/Plenum, New York, 1975).[19] J.C. Heyraud and J.J. Metois, Acta Metall. 28,1 7 8 9(1980).[20] I. A. Vartanyants and I.K. Robinson, J. Phys. Condens.Matter 13, 10593 (2001).[21] J. A. Pitney et al., Phys. Rev. B 62, 13084 (2000).[22] H.W. Hayden, W.G. Moffat, and J. Wulff, Structure andProperties of Materials III (Wiley, New York, 1965).FIG. 3 (color online). Top: The 2D slices through the best ﬁt3Dreconstructeddensityseparatedaslabeled. Bottom:(a)SEMimage of a Au nanocrystal prepared as described in the text.(b) z   0:00  m slice from the second best ﬁt showing goodagreement between ﬁts. (c) Coordinate system for the recon-structed crystal.PHYSICAL REVIEW LETTERS week ending2 MAY 2003 VOLUME 90, NUMBER 17175501-4 175501-4

Three-dimensional Imaging of Microstructure in Gold Nanocrystals

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