Physica B 336 (2003) 16–26

Surface structure solution by X-ray diffraction: structurecompletion with positivity and atomicity constraints

D.K. Saldin*, V.L. Shneerson, R. Fung

Department of Physics and Laboratory for Surface Studies, University of Wisconsin-Milwaukee,

P.O. Box 413, Milwaukee, WI 53201, USA

Received 21 November 2002; accepted 3 February 2003

Abstract

We propose a method for the solution of surface structures from X-ray diffraction based on the phasing of measured

structure factor amplitudes in two stages. First, incorporation of calculated amplitudes and phases of the structure

factors of the known underlying bulk structure, together with an iterative algorithm that ensures the positivity of the

electron density enables the determination of the amplitudes and phases of the surface structure factors contributing to

the crystal truncation rods. Second, inclusion of these structure factors in an iterative algorithm based on Sayre’s

equations enables the phasing of the superstructure rods.

r 2003 Elsevier Science B.V. All rights reserved.

PACS: 61.10.�i; 61.10.Dp; 68.35.Bs

Keywords: Surface crystallography; X-ray diffraction; Direct method; Phase problem; Surface structure; Crystal truncation rods

1. Introduction

For almost a century, X-ray crystallography hasbeen the workhorse for the determination of thestructure of matter at the atomic scale. The bareexperimental data from which this information isdeduced are a set of measurements of theamplitudes of scattered X-rays. The phases of thisscattered radiation are not measured directly, butif they can be deduced, they, together with themeasured amplitudes, allow the calculation of thedistribution of electrons within a repeat unit ofthe structure (the unit cell) by a numerical Fourier

transform. The determination of the unknownphases is known as the phase problem of crystal-lography.Crystallographic direct methods [1,2] seek to

obtain initial estimates of structure-factor phasespurely from a knowledge of the measured struc-ture-factor amplitudes, plus general properties ofthe electron distribution in a unit cell, such as itspositivity and atomicity. The adaptation of directmethods to surface crystallography presents aunique set of problems and opportunities. First,although the X-rays are scattered from both thebulk of the crystal and from the surface region,since the bulk structure is usually known, the aimis to recover only the surface structure from thedata. In this respect, the task is analogous to that

ARTICLE IN PRESS

*Corresponding author.

E-mail address: dksaldin@uwm.edu (D.K. Saldin).

0921-4526/03/$ - see front matter r 2003 Elsevier Science B.V. All rights reserved.

doi:10.1016/S0921-4526(03)00265-5

of the so-called structure completion problem, alsoaddressed in other branches of crystallography [3].Second, the data consists not of discrete Braggspots, but rather of rods of continuous variation ofintensity in reciprocal space in a direction perpen-dicular to the surface [4]. This allows for thepossibility of oversampling [5] the data relative tothe Nyquist frequency [6] in this direction, afeature that aids the recovery of the phases fromthe data. Third, when the surface is reconstructed

relative to the bulk, i.e. when its spatial periodin a plane parallel to the surface is different(usually greater) than that of the bulk, some ofthese diffraction rods (the so-called superstructure

rods, SRs) arise solely from scattering by thesurface region of this larger periodicity, whileothers (the crystal truncation rods, CTRs) haveinterfering contributions from both surface andbulk.It has been suggested [7–10] that the phase

problem of crystallography may be addressed byiterative methods that alternatively constrain thesolution in real and reciprocal space. The idea wasfirst proposed for electron microscopy by Gerch-berg and Saxton [11,12], where the amplitudes ofcomplex functions, related by Fourier transforms,are measurable in both real and reciprocal space.The iterative algorithm recovers the phases of thefunctions in both spaces. An extension of thisalgorithm when amplitude data is available only inreciprocal space, and where some general con-straints may be imposed in real space, such as aspatial localization, or some bounds on the real-space distribution, was subsequently proposed byFienup and applied to phase determinationproblems in astronomy and optics [13]. We haveearlier proposed an adaptation of this method[9,10] for the problem of structure completion insurface X-ray diffraction (SXRD), where initialphases of the CTRs are taken to be thosecalculated for the known underlying bulk struc-ture. If the data from just the CTRs are used, theresulting solution is known as the folded [14] oraverage structure, which consists of the average ofthat in all bulk unit cell components of the surfaceunit cell. The coherent Bragg rod analysis (CO-BRA) method of Yacoby et al. [15] employs adifferent algorithm based on an assumption that

the reciprocal-space variation of the complexstructure factor of the unknown part of the surfaceis slower than that of the known bulk. In order toobtain the full structure of the surface unit cell, wehad earlier proposed [8–10] resuming the iterationsof the same algorithm with the SR data also nowincluded in reciprocal space, but with initialrandom values for the phases associated with thisdata, or initial phases interpolated from those ofthe surface structure factor components of neigh-boring CTRs. Such an algorithm is based onconstraining the diffracted amplitudes to experi-mentally measured values in reciprocal space, andensuring the reality and positivity of the electrondistribution in real space.An alternative approach towards a direct

method for surface crystallography has beensuggested by Rius [16]. This approach seeks todetermine the so-called difference structure, i.e. thedifference between the electron density of thecomplete surface unit cell and that of the foldedstructure, from an inverse Fourier transform ofphased amplitudes of SRs alone. As such, thedifference structure may be positive or negative,although it does satisfy a constraint of atomicity,namely that its peaks or dips are substantiallylocalized near the centers of atoms. By exploitingthe similarity of the spatial dependence of thedifference structure to that of its cube, Rius et al.deduced a tangent formula that relates the phase ofa structure factor to sums of products of threeother structure factors. An iterative algorithmbased on these relations has been shown to becapable of phasing the SR amplitudes, and infavorable cases [17] may enable a solution of thesurface structure.In the present paper we point out that, if a

structure completion algorithm applied to theCTR data is able to determine the amplitudesand phases of their surface structure factorcomponents (and hence the folded structure),knowledge of these complex structure factorsallows a more efficient iterative algorithm, basedon the much simpler tangent formula derived fromSayre’s equations [18] to determine rapidly thephases of the SR structure factors. Together withthe complex surface structure factors from theCTRs, the phased SR structure factors allow a

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D.K. Saldin et al. / Physica B 336 (2003) 16–26 17

determination of the complete (unfolded) structureof the surface unit cell.

2. The structure completion problem or holographic

analogy

The usual aim of surface crystallography is torecover the unknown surface structure of a samplewith a known bulk structure, where both thesurface and the bulk contribute to a measured setof diffracted intensities. Thus, the problem of e.g.recovering the surface electron density fromSXRD may be regarded as one of structurecompletion, a problem that has previously beenaddressed in the field of protein X-ray crystal-lography [3,19,20].Suppose that the scattered amplitude from a

unit cell of the bulk crystal due to an X-ray photonmomentum transfer vector q is represented by Rq;and that of the surface by Oq: Then the intensity ofthe detected X-rays may be written as

Iq ¼ jFqj2; ð1Þ

where the structure factor Fq may be written as thesum:

Fq ¼ Rq þ Oq: ð2Þ

In this expression, Oq ¼ ðFq � RqÞ is given by aFourier transform of the electron distribution fujg;i.e.,

Oq ¼XN

j¼1

uj expðiq rjÞ; ð3Þ

where fujg is defined on a uniformly distributedgrid of N voxels at positions rj within the surfaceunit cell.The approach we will use is to determine the

amplitudes and phases of the surface structurefactors Oq from a knowledge of the measuredintensities Iq and calculated values of the ampli-tudes and phases of the bulk structure factors Rq:In a sense, as has been pointed out by Sz .oke [21],this is quite analogous to the problem of holo-graphy [22,23] if the set of intensities fIqg areconsidered to constitute a hologram, the knownstructure factors fRqg represent a reference

‘‘wave’’ and the unknown structure factors fOqg

an object ‘‘wave’’. Although the numerical meth-ods we use to determine the ‘‘object’’ structurefactors have little similarity to the analoguemethods of Gabor, nevertheless this is a usefulmental analogy, which has also been exploited tosolve the problem of structure completion inprotein crystallography [19,20]. Once the ampli-tudes and phases of the set, fOqg; of surfacestructure factors have been determined, it is astraightforward matter to calculate, by means ofan inverse Fourier transform, the spatial distribu-tion, fujg; of the electrons in the surface unit cell,and hence to solve the structure.The problem of surface crystallography has

some unique features that are not present in thestructure completion problem in the crystallogra-phy of bulk samples. This is the presence of theSRs, which are formed solely by diffraction fromthe unknown surface atoms, and for which there isno contribution from a known part of thestructure (or no reference wave, in terms of theholographic analogy).The strategy we describe in this paper is to use a

Fienup-type input–output algorithm, adapted tothe present problem of structure completion, tofirst find both the amplitudes and phases of thesurface structure factors contributing to the CTRs,and then to employ an iterative algorithm basedon Sayre’s relations to find the phases of the SRs(whose amplitudes are accessible by experiment).

3. Phasing of CTRs by an input–output structure

completion algorithm

The idea of an input–output feedback loop forphasing that iteratively satisfies conditions in realand reciprocal space has been suggested by Fienup[13] for problems where a non-negative distribu-tion is sought, and where only the amplitudes ofthe Fourier transform of that quantity areaccessible by experiment. The aim is to obtainincreasingly better estimates of the phases of thisFourier transform by iteratively satisfying thereciprocal-space constraints and the real-spacerequirement of the positivity of the soughtdistribution. Improvement of phase quality isdirectly correlated with an improved estimate of

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D.K. Saldin et al. / Physica B 336 (2003) 16–2618

that distribution. We show below how a modifica-tion of such an algorithm can determine both theamplitudes and phases of the surface structurefactors contributing to the CTRs in SXRD [9,10].A Fourier transform of this subset of surfacestructure factors would yield an electron distribu-tion representing the folded surface structurereferred to above, which has the same lateralperiodicity as the underlying bulk structure.A flow chart for such an algorithm is given in

Fig. 1. Starting at the top left-hand corner,suppose fu

ðnÞj g represents the estimate of the folded

surface electron density at the nth iteration.Proceeding to the top entry of the right-handbox, we take the Fourier transform ðFTqÞ

OðnÞq ¼

XN

j¼1

uðnÞj expðiq rjÞ; ð4Þ

of this distribution by a fast Fourier transform(FFT) algorithm. In X-ray diffraction, the Fourierindex q can be taken to be equal to the differencebetween the wavevectors of the incident andscattered X-rays. In SXRD, this may be taken as

q ¼ Han þ Kbn þ Lcn; ð5Þ

where H; K ; and L are Miller indices, an and bn arereciprocal lattice vectors parallel to the surface ofthe bulk lattice, and cn the one perpendicular to

the surface. The periodicity of the crystal restrictsthe Miller indices H and K of the CTRs to integervalues. If the atomic layers near the surfacereconstruct to a lateral unit cell greater than thatof the bulk, the X-rays may scatter also intoSRs,which may be characterized by fractionalvalues of H and/or K : In the case of the CTRs,the breaking of the periodicity perpendicular tothe surface due to the crystal truncation and, in thecase of the SRs, the finite thickness of the surfaceslab, allows scattering into directions correspond-ing to a continuous variation of L [8]. In thefollowing we describe an application of thisalgorithm solely to the CTR data.The dimensions of the parallelepiped reciprocal-

space array of fOðnÞq g (and consequently the real-

space grid spacing of fuðnÞj g) are chosen so that all

the values of the wavevector difference q belongingto the setM of measured structure factors jFgjmaybe embedded within it. The set of elements in thesame parallelepiped reciprocal-space array notbelonging to M may be termed the unmeasuredset U:The next step is the evaluation of the arguments

of the Fourier coefficients Rq þ OðnÞq for all qAM

and the assignment of their arguments to thephases

fðnÞq ¼

arg½Rq� if n ¼ 0;

arg½Rq þ OðnÞq � if n > 0

(ð6Þ

for all qAM: The ‘‘target’’ Fourier coefficients T ðnÞq

are then computed by the formula

T ðnÞq ¼ jFqj exp½f

ðnÞq � � Rq 8qAM: ð7Þ

The inverse Fourier transform

tðnÞj ¼

1

N

Xq

½T ðnÞqAM þ O

ðnÞqAU� expð�iq rjÞ; ð8Þ

at the last step within the right-hand box gives riseto the output electron distribution, ft

ðnÞj g:

Thus, in such an input–output scheme [13], thebox on the right of the flow chart transforms aninput electron distribution fu

ðnÞj g to an output one

ftðnÞj g at iteration n by combining experimental

information about the measured amplitudes jFqjwith the current estimates of the phases fðnÞ

q

associated with those amplitudes. The boxes on

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Oq(n)=FTq{uj

(n)}

φq(n)=arg[Rq+Oq

(n)], ∀ q∈ M

Tq(n)=|Fq|exp[i φq

(n)]-Rg, ∀ q∈ M

tj(n)=FT-1

j{Tq∈ M(n)+Oq∈ U

(n)}

{uj(n)}

{tj(n)}

uj(n+1)=f(uj

(n), tj(n))

n n+1

Fig. 1. Flow chart of the input–output feedback loop that

converts an input real-space distribution fuðnÞj g to an output

distribution ftðnÞj g by constraining the Fourier transformed

quantities to experimental amplitude data. The new input real-

space distribution fuðnþ1Þj g for the next iteration of the feedback

loop is calculated from the input and output at the previous

iteration by a set of object domain operations of the form

uðnþ1Þj ¼ f ðuðnÞj ; tðnÞj Þ; where f is one of the functions discussed in

the text.

D.K. Saldin et al. / Physica B 336 (2003) 16–26 19

the left of the flow chart describe the steps in thetransformation of the input, fu

ðnÞj g; and output,

ftðnÞj g; at the nth iteration to the input fu

ðnþ1Þj g at

the next iteration. These steps are known as theobject-domain operations, and may be written inthe general form u

ðnþ1Þj ¼ f ðuðnÞ

j ; tðnÞj Þ: Fienup [13]suggested four specific prescriptions for suchoperations. In the present paper we use only thesimplest of these, which Fienup termed an error-

reduction algorithm, since he was able to provethat it always converged towards a solution thatsimultaneously satisfied the constraints in real andreciprocal space. The real-space operations of theerror-reduction algorithm are

uðnþ1Þj ¼

tðnÞj if t

ðnÞj > 0;

0 otherwise:

(ð9Þ

The progress of successive estimates of therelevant Fourier coefficients of a particular reci-procal space scattering vector qAM belonging to aCTR may be visualized from Fig. 2. The distancefrom the center of the circle to its perimeterrepresents the magnitude jFqj of the measuredstructure factor of the entire sample (bulk plussurface). Rq is a fixed vector in this amplitude-phase diagram, representing the bulk structurefactor that is known in both amplitude and phase.The surface contribution to the total structurefactor must join the end of the vector Rq to thecircle perimeter. The problem is that since thephase of this vector is initially unknown, there arean infinite number of such possible vectors. Thefirst (unweighted difference Fourier [3]) estimate,T ð0Þq ; of this surface structure factor takes this

phase to be equal to that Rq of the bulk. T ð0Þq is

thus taken to be parallel to Rq; as shown in Fig. 2.The inverse Fourier transform of the surface

structure factors fT ð0Þq g produces the initial output

real-space distribution ftð0Þj g: After an application

of the object-domain operations to produce thenew input distribution fu

ðnÞj g (n > 0), the Fourier

transform of the latter gives the surface structurefactor estimates fOðnÞ

q g: The phase fðnÞq is defined

by the vector sum of Rq and OðnÞq as shown in the

figure. Since, in general, the magnitude of thisvector sum will not be equal to jFqj; this vector isextended (or contracted) without change in direc-

tion until it touches the circle perimeter. Thevector joining the end of the bulk structure factorRq and that point on the circle’s perimeter is nowdefined as the new estimate T ðnÞ

q :After several iterations, as convergence is

approached, OðnÞq and T ðnÞ

q 8qAM tend to merge.The resulting phase fðnÞ

q of Rq þ Oq (or of Rq þ Tq)is the final estimate of the phase of the measuredstructure factor Fq: When supplemented bythe Fourier coefficients OðnÞ

q 8qAU; the inverse

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Rq

T q(0)

T q(n)

O q(n)|Fq|

φq(n)

Fig. 2. Amplitude-phase diagram indicating the relationships

amongst the various component structure factors of scattering

vector q. The circle has a radius of jFqj; the measured amplitudeof Bragg reflection q: Rq represents the structure factor of the

known bulk unit cell. This is known in both amplitude (length)

and phase (angular separation from the dashed line). The

(unweighted) difference Fourier estimate of the structure factor

of the unknown part of the structure (the surface) is represented

by the vector T ð0Þq ; which has the same phase (direction) as Rq:

OðnÞq is the estimate of the same structure factor at the nth

iteration ðn > 0Þ of the input–output feedback loop, formed

from the input distribution fuðnÞj g of the surface electron

density. Since the end of the vector sum of Rq and OðnÞq will

not in general lie on the circumference of the circle, the length of

this vector is adjusted to the circle radius. The target structure

factor T ðnÞq of the surface is then constructed such that when

added vectorially to Rq it is equal in both amplitude jFqj andphase ðfðnÞ

q Þ to the new estimate F ðnÞq of the structure factor of

the entire structure (bulk and surface). The Fourier transform

of the target structure factors fT ðnÞq g forms the output

distribution ftðnÞj g at the nth iteration. The object domain

operations then construct a new input distribution fuðnþ1Þj g and

the process repeated until Oðnþ1Þq and T ðnþ1Þ

q (or fuðnþ1Þj g and

fuðnÞj g) converge.

D.K. Saldin et al. / Physica B 336 (2003) 16–2620

Fourier transform of the combined set gives thefinal estimate of the folded surface electrondistribution as that to which both ft

ðnÞj g and

fuðnÞj g eventually converge.It should be noted that, due to the continuous

distribution of SXRD intensities along a diffrac-tion rod, this data may be oversampled inreciprocal space relative to the Nyquist frequencycorresponding to the thickness of the slab repre-senting a surface unit cell. In general, the Fouriertransform of such oversampled data would beexpected to generate non-zero electron densities ina real-space slab of greater thickness than thephysical surface slab. The efficiency of the phasingprocess is further enhanced by setting the electrondensities to zero at each iteration in the part of thisnotional slab that is not expected to containsurface atoms [5].In order to recover the full electron distribution

of a surface unit cell of larger lateral dimensionsthan that of the bulk unit cell, it is necessary alsoto include the structure factors of the SRs. Sincethe SRs do not have contributions from theunderlying bulk, the amplitudes of their surfacestructure factors are directly measurable fromexperiment. However, for the same reason, thebulk structure factors are of no use in suggestingeven initial estimates for the phases of the SRstructure factors. Nevertheless, as we demonstratein the following, knowledge of the amplitudes andphases of the CTRs allows the rapid determinationof the unknown phases of the SRs by an iterativealgorithm based on Sayre’s equations [18], thatassume both atomicity and positivity of theelectron distribution.

4. Phasing of the SRs from phased CTRs and

Sayre’s equations

In Section 2 we expressed the Fourier coeffi-cients Oq as Fourier transforms of surface electrondistribution (Eq. (3)). An alternative expression interms of the form factor fs of the sth atom of thesurface unit cell is

Oq ¼X

s

fsðqÞ expðiq rsÞ: ð10Þ

Since the majority of electrons are localized nearthe centers of their respective atoms s; we see thatEq. (10) will be consistent with Eq. (3) if

fsðqÞ ¼X

jAðatom sÞ

uj exp½iq ðrj � rsÞ� ð11Þ

and the summation over j is here performed overjust the electrons in the voxels associated withatom s: That is, an atomic form factor may beregarded as a Fourier transform with an origindefined at the center of the atom, of the electronsof that atom. To put it another way, expressing thesurface structure factor in the form (10) isequivalent to an assumption that the soughtelectron distribution consists of a sum of separabledistributions associated with each atom.Now consider the structure factor of the same

surface unit cell due to the square of this electrondistribution. It may be written as

Gq ¼X

s

gsðqÞ expðiq rsÞ; ð12Þ

where

gsðqÞ ¼X

jAðatom sÞ

u2j exp½iq ðrj � rsÞ�: ð13Þ

If all the atoms are assumed equal (the equal-atom

approximation) we may write

fsðqÞgsðqÞ

¼ yq; ð14Þ

a quantity independent of the atom index s:Hence,from Eqs. (10) and (12)

Oq ¼ yqGq: ð15Þ

By substituting Eq. (13) into Eq. (12) we find

Gq ¼XN

j¼1

u2j expðiq rjÞ: ð16Þ

Comparing this with Eq. (3), and applying theconvolution theorem, we obtain

Gq ¼Xq0

Oq0Oq�q0 ð17Þ

and, using Eq. (15),

Oq ¼ yqXq0

Oq0Oq�q0 : ð18Þ

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D.K. Saldin et al. / Physica B 336 (2003) 16–26 21

This is Sayre’s celebrated relation [18] amongststructure factors. The quantity yq may be calcu-lated from a knowledge of the atomic chargedistribution (in the case of unequal atoms, thismay be taken as the average atomic chargedistribution). However, if, as commonly the casewith an unknown surface structure, these atomiccharge distributions are unknown, a more usefulrelationship (sometimes known as a tangentformula) that follows from Eq. (18) since yq isreal, is

arg½Oq� ¼ argXq0

Oq0Oq�q0

" #: ð19Þ

We can establish phases of the surface structurefactors giving rise to the superstructure rods byrewriting this as the recursion relation:

arg½Oðnþ1Þq � ¼ arg

Xq0

OðnÞq0 O

ðnÞq�q0

" #; ð20Þ

where n is an iteration index. The H and K Millerindices of the scattering vector q are taken tocorrespond to those of the SRs. Each of thesurface structure factors in each term on the RHScorresponds to data from either a CTR or an SR.Both the amplitudes and phases of the surfacestructure factors contributing to a CTR are keptfixed throughout the iterations at the final valuesfound from the output of the Fienup algorithm forthe CTRs described earlier. The amplitudes of theSR structure factors are kept fixed at their valuesdeduced directly from experimental measurementsof the SR intensities, but since their phases areinitially unknown, they are assigned randomvalues at the outset (corresponding to iterationn ¼ 0 on the RHS).A first pass through the algorithm for all values

of q corresponding to the SRs overwrites theserandom phases with those calculated via Eq. (20).Incrementing n by 1, and substituting theseupdated SR surface structure factors on the RHSallows a new estimate of the SR phases viaEq. (20). The values to which these phasesconverge after several iterations are our finalassignment of the phases of the surface structurefactors of the SRs. Finally, the distribution ofelectrons in the surface unit cell is calculated by the

inverse Fourier transform

uj ¼1

N

Xq

Oq expð�iq rjÞ: ð21Þ

5. Test calculation

We have tested this idea by attempting torecover a surface structure from simulated SXRDdata. The example we chose was a hypotheticalstructure of cð2� 2ÞK=TiO2ð0 0 1Þ: The unit cell ofbulk TiO2 (rutile) in a (0 0 1) direction consists ofmutually orthogonal unit vectors of magnitudesa ¼ 4:59 (A; b ¼ 2:96 (A and c ¼ 4:59 (A: The a andb unit vectors lie in a plane parallel to the surface(the basal plane), while c is perpendicular to it. Thestructure consists of a sequence of Ti-O-O-Ti-O-Olayers parallel to the basal plane. The twosequences of Ti-O-O layers are related to oneanother by mirror reflection across a b–c plane.We assume that the outermost bulk layer is the lastof the above-named O planes.We assume that the lowest surface layer is at a

height of 0:1cð¼ 0:46 (AÞ above that expected of thenext Ti layer of a bulk structure, but that it has acð2� 2Þ periodicity relative to the bulk structureparallel to the surface. Referring atom positionsrelative to a three-dimensional ð2� 2Þ unit cell ontop of the outermost bulk unit cell, with an originin a plane half way between the outermost O layerof the bulk and the expected position of Ti atomsin an unrelaxed structure, and whose height is thesame as that of a bulk unit cell ðc ¼ 4:59 (AÞ; thesurface atoms were assumed to be at the followingpositions:

Ti: (0.0, 0.0, 0.2), (0.5, 0.5, 0.2)K: (0.3, 0.0, 0.2), (0.8, 0.5, 0.2)K: (0.15, 0.5, 0.54), (0.65, 0.0, 0.54)

where all coordinates are expressed as fractions of2a; 2b and c; respectively. It should be emphasizedthat, as yet, the true structure of this surface hasnot been determined from experimental data, andso the above model is an arbitrary one chosenpurely for this test. A plan view of this structure isshown in Fig. 3(a), where for clarity the bulkatoms are not shown. Two orthogonal projections

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D.K. Saldin et al. / Physica B 336 (2003) 16–2622

viewed from directions parallel to the unit vectorsa and b of the basal plane are shown in Figs. 3(b)and 3(c), respectively.

Test ‘‘experimental’’ data were calculated forthis structure by a standard SXRD program.Relative to a ð2� 2Þ surface unit cell, this testdata consists of a set of SXRD intensitiescorresponding to scattering vectors q with integervalues of the Miller index pairs ðH;KÞ; eachvarying from �7 to 8 in integer intervals, and afew values of the third Miller index, L ¼ 0:2; 0:4;and 0.6. The centered ð2� 2Þ surface unit cellrestricts the combination of H and K Millerindices to those of even values of ðH þ KÞ (seethe diffraction pattern of Fig. 4, which marks theprojections of the CTRs and SRs onto a planeparallel to the surface). Furthermore, due to themirror planes perpendicular to the surface andparallel to the H-axis, it is necessary only to know

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Fig. 3. Three orthogonal projections of the model of the

cð2� 2ÞK=TiO2ð0 0 1Þ surface assumed in the calculations. (a)

Plan view, projected onto a plane parallel to the surface. Only

the atoms in the surface unit cell are shown. (b) side view,

projected onto a plane defined by the b- and c-axis (c) side view,

projected onto a plane defined by the a- and c-axis. Shown in

(b) and (c) are all atoms of a ð2� 2Þ surface unit cell as well asthose of an underlying ð1� 1Þ bulk unit cell. The chemical

species of the atom symbols are indicated on the upper right of

each panel.

Fig. 4. Cut through reciprocal space parallel to the surface,

intersecting the crystal truncation rods (CTRs) and SRs from a

cð2� 2ÞK=TiO2ð0 0 1Þ surface. There is a mirror plane of

symmetry perpendicular to the paper and passing through the

H-axis. Only the intersected rods corresponding to KX0 are

symmetrically inequivalent. The filled circles represent the

intersections of CTRs and the open circles the SRs.

D.K. Saldin et al. / Physica B 336 (2003) 16–26 23

values of intensities of the rods with positive valuesof K : The combination of mirror symmetry andFriedel’s Law allows the deduction of the inten-sities of all other diffraction rods of Fig. 4 for bothpositive and negative values of L: The additionaldata required of our algorithm are, of course,calculated values of the amplitudes and phases ofthe corresponding structure factors of the under-lying truncated bulk structure of TiO2ð0 0 1Þ:These were also calculated by the same computerprogram. Due to the fact that the bulk unit cell ishalf the size of the surface unit cell in the directionsof the a- and b-axis, the bulk lattice contributesonly to the CTRs, which are the subset of rodswith even values of the Miller indices H and K :The CTR data formed the input to the error-

reduction form of the Fienup algorithm of Sec-tion 3. This determines the amplitudes and phasesof the surface contributions to the CTRs. In ourpresent example, convergence was achieved afterabout 25 iterations. An inverse Fourier transformof this set of structure factors gives the folded

version of the electron distribution of the surfaceunit cell, which has the same periodicity parallel tothe surface as the bulk structure. An inverseFourier transform of just the so-called ‘‘in-plane’’surface structure factors, corresponding to lowvalues of L gives the projection of this electrondistribution onto a plane parallel to the surface. Inpractice, measurements of ‘‘in-plane’’ data arequite common. They correspond to measurementsof scattered X-rays at glancing take-off angles tothe surface from the glancing incidence of incom-ing X-rays. Of course, due to the fact that bothsource and detector have to lie outside the plane ofthe surface, inevitably there has to be a surface-normal component of q: However, the value of L

can be kept down to approximately 0.1 or 0.2. Thisis usually small enough that the set of such surfacestructure factors may be treated as the Fouriercomponents of a projection of the surface electrondistribution onto a plane parallel to the surface.The folded structure projected onto a plane parallelto the surface, calculated from L ¼ 0:2 data by theFienup algorithm from this simulated data isshown as a contour map in Fig. 5 (we choose theL ¼ 0:2 data since they are generally experimen-tally measurable, yet of a low enough L value that

their Fourier transform with respect to Millerindices H and K is practically indistinguishablefrom those of L ¼ 0).We then proceeded to attempt to recover the

true projected electron distribution of a ð2� 2Þ

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Fig. 5. Contour map, calculated from the Fourier transform of

the surface structure factors of Miller index L ¼ 0:2 contribut-ing to the CTRs of the cð2� 2ÞK=TiO2ð0 0 1Þ surface. The

structure factors were found from the input–output phasing

algorithm described in the text by incorporating calculated

structure factors from the known underlying bulk crystal. To a

good approximation, this represents the surface electron density

projected onto a plane parallel to the surface and folded into a

unit cell of the underlying bulk unit cell. There are 25 equally-

spaced contour levels from 0 to the maximum value.

Fig. 6. Contour map, calculated from the Fourier transform of

the surface structure factors of Miller index L ¼ 0:2 of both theCTRs and also the SRs of the cð2� 2ÞK=TiO2ð0 0 1Þ surface.The latter structure factors are calculated from the former by

the iterative algorithm also described in the text, based on

Sayre’s equations. To a good approximation, this represents the

electron density of the surface unit cell projected onto a plane

parallel to the surface. There are 15 equally-spaced contour

levels from 0 to the maximum value.

D.K. Saldin et al. / Physica B 336 (2003) 16–2624

unit cell by using the recursion relation (20) tophase the ‘‘in-plane’’ SR structure factors. In thiscase, convergence of the algorithm (20) occurredafter about 10 iterations. The inclusion of boththe CTR- and SR-phased ‘‘in-plane’’ data in theinverse Fourier transform (21) resulted in theprojected electron distribution shown in Fig. 6.Comparison of this distribution with that fromstructure factors calculated from the atomic modelof the surface used for our test (Fig. 7), and withthe projection (Fig. 3) of the atomic modelassumed in our calculations, shows remarkableagreement of the main features.

6. Conclusions

We have described a procedure for determiningthe distribution of electrons in a surface unit cellfrom experimental measurements of the intensities ofcrystal truncation and superstructure rods (CTRsand SRs) in surface X-ray diffraction (SXRD).The solution requires first the isolation and

retrieval of the amplitudes and phases of thesurface structure factors contributing to the CTRs.This is done by first recognizing and calculatingthe contributions to these CTRs from the generallyknown structure of the underlying bulk. Analgorithm that incorporates this knowledge, and

successively constrains the solution to the mea-sured amplitudes of the CTRs in reciprocal spaceand restricts the real-space electron distribution tobe non-negative is used to determine the complexsurface structure factor components of the CTRs,and hence the folded surface structure. Unlike themethod proposed by Yacoby et al. [15] fordetermining this folded structure, we make noassumptions about the relative rates of variation ofthe surface and bulk structure factors.Our next step is to use an iterative algorithm

based on the tangent formula from Sayre’sequations to determine the phases of the SRsfrom this knowledge of the amplitudes and phasesof the CTR surface structure factors determined atthe earlier step. The combination of the complexstructure factors contributing to the CTRs andSRs enables the determination of the completesurface structure, unlike the method of Rius et al.[16], which phases only the SRs, and is thus onlyable to determine the difference between theunfolded and folded structures. Also, since theSayre tangent formula relates the phases ofstructure factors to sums of products of pairs(rather than the triplets of the formula of Riuset al.) of other structure factors, the algorithm ismuch simpler and more computationally tractable,and leads to a rapid solution in a small number ofiterations.

Acknowledgements

We acknowledge stimulating discussions withWolfgang Moritz and Jordi Rius. R.F. isgrateful for financial support in the form of aUWM Graduate School Fellowship. Supportfor this work was also provided by the USNational Science Foundation (Grant No.:DMR-9815092) and the US Department ofEnergy (Grant Nos.: DE-FG02-01ER45926 andDE-FG02-84ER45076).

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Fig. 7. Contour map of the electron density of the surface unit

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D.K. Saldin et al. / Physica B 336 (2003) 16–26 25

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