aa r X i v : . [ c ond - m a t . m t r l - s c i ] M a r Homometry in the light of coherent beams
Sylvain Ravy ∗ Synchrotron-Soleil, L’Orme des merisiers, Saint-Aubin BP48, 91192 Gif-sur-Yvette, France † (Dated: August 12, 2018)Two systems are homometric if they are indistinguishable by diffraction. We first make a dis-tinction between Bragg and diffuse scattering homometry, and show that in the last case, coherentdiffraction can allow the diffraction diagrams to be differentiated. The study of the Rudin-Shapirosequence, homometric to random sequences, allows one to manipulate independently two-point andfour-point correlation functions, and to show their effect on the statistics of speckle patterns. Con-sequences for the study of real materials are discussed. PACS numbers: 61.05.cc,61.43.-j,02.50.-r
INTRODUCTION
The possibility to shape coherent X-ray beams fromsynchrotron light sources [1] and to get naturally coher-ent beams from X-ray free electron lasers (XFEL) [2],has revolutionized the way X-ray diffraction experimentsare performed and analyzed. One of the most fascinat-ing property of coherent diffraction is the possibility tomeasure speckle patterns [3], which are much more in-formative than the diffuse scattering obtained by classi-cal diffraction. Together with the development of novelsources, phase retrieval algorithms have also emerged, al-lowing the reconstruction of the diffracting objects undercertain conditions [4, 5]. However, the reconstruction ofa structure is not always possible nor necessary to studythe physics of materials. For example, measuring corre-lation lengths close to phase transitions [6] or slow dy-namics with X-ray photon correlation spectroscopy [3, 7]does not require the full reconstruction of the system un-der study.The purpose of this letter is to show that statisticalanalysis of speckle patterns can yield information on or-ders hidden to conventional X-ray analysis. In this re-spect, we are in line with recent works showing that four-point intensity cross-correlation of speckle patterns canuncover ”hidden symmetries” present in colloidal glasses[8, 9] or magnetic systems [10].Our approach uses the concept of homometry, i.e. theproperty of different systems to exhibit same diffractionpatterns. Separating out the scattered intensity expres-sion into three terms allows one to show that homometrycan occur at different levels. We then put the emphasison diffuse scattering homometry, that we discuss with thehelp of the well-known Rudin-Shapiro sequence [11, 12].
COHERENT DIFFRACTION
Let us first give a general expression of the intensityscattered at scattering vector q , by a 1D periodic N -site lattice decorated by two atoms A and B , of scat-tering factor f A and f B , in proportion x and 1 − x re- spectively. Generalization to 2D, 3D, multi-atomic basis,displacement disorder, or disorder of the second kind [13]is straighforward. Following ref. [13], the diffracted in-tensity is given by the formulae: I ( q ) = X n,n ′ f n f n ′ e iq ( n ′ − n ) = X m X n f n f n + m e iqm . (1)The ensemble average of the cross-product h f f m i is thenintroduced: 1 N m X n f n f n + m = h f f m i + ∆ m , (2)where N m is the m -dependant number of terms of thesum P n . The ∆ m term, usually negleted in textbooks,is due to finite-size fluctuations of the spatial averagewith respect to the ensemble one.Further introduction of ∆ f m = f m − h f i allows one toget the three components of kinematic diffraction: I B ( q ) = h f i X m N m e iqm (3a) I DD ( q ) = X m N m h ∆ f ∆ f m i e iqm (3b) I S ( q ) = X m N m ∆ m e iqm (3c)The first term gives the intensity of the Bragg reflec-tions, and the fringes due to finite size effects. For acrystal of N cells of structure factor F ( q ), it can be writ-ten as: I B ( q ) = |h F ( q ) i| sin q ( N + 1) / q/ random disorder, it reduces to the well-known Laueformula: I ( q ) = N x (1 − x )( f A − f B ) , (5) FIG. 1. (left) (40 ×
40) lattices of +1 and -1 in equal pro-portion and (right) diffraction patterns in the first Brillouinzone (log scale) for a) a triplet SRO lattice [14] b) a randomlattice. Reciprocal nodes are in the corners. where random refers to the vanishing of the pair CF ( i.e. h ∆ f ∆ f m i = h ∆ f ih ∆ f m i = 0 for m = 0.)The third term gives rise to speckles. Like fringes,speckles only exist if the incident beam is coherentenough, and if the system does not explore too manyconfigurations during acquisition time T (non-ergodicitycondition h ∆ m i T = 0). Interestingly enough, workingout of coherent conditions has the effect of averaging out∆ m , which yields ensemble averaged quantities. HOMOMETRY
Homometry - etymologically same distance - is a wordcoined by A. Patterson [15, 16], to describe the prop-erty of non-congruent sets of points to possess the samepair distances (or the same difference sets) [17]. Homo-metric sets have thus the same diffraction pattern, asEq. (1) demonstrates. A simple example of homome-try is given by the two sets S = { , , , , , } and S ′ = { , , , , , } [17]. Indeed, the structure fac-tors F ( q ) of the two sets have the same magnitude for all q-vectors, but not the same phase. Hence, the lostof the phase makes these sets indistinguisable by X-raydiffraction.However, because solid state physics deals with ma-terials, the above definition turns out to be too restric-tive. Eqs (3) allows to distinguish between Bragg (B) ho-mometry, diffuse scattering (D) homometry and coherentdiffraction (C) homometry. BRAGG HOMOMETRY
B-homometry decribes crystals with different basis butsame Bragg intensities [15, 16]. To illustrate that, let usconsider the examples of 1D homometric crystals pre-
FIG. 2. Diffraction patterns and associated intensity varia-tions from ((
N, M ) = (64 , sented in [16], of unit cells size equal to 8 and atomicpositions given by H = { , , , } and H ′ = { , , , } .Structure factors, readily calculated as: F H ( q ) = 1 + 2 cos q + cos 4 q (6) F H ′ ( q ) = 2(cos 2 q + cos 4 q ) , (7)have the same amplitude squared at the Bragg positions q = 2 πh/ out of Bragg positions. Eq (4) showsthat the fringes intensity, revealed by coherent diffrac-tion, gives out-of-Bragg values of | F ( q ) | which, at leastin theory, allows to distinguish H and H ′ , and solves theB-homometry issue. This is well-known and correspondsto the oversampling requirement of the phase retrievalalgorithms [18, 19]. It is clear however that if the atomicbasis is homometric itself, like e.g. in crystals with S or S ′ basis, the problem cannot be solved, coherence or not. DIFFUSE SCATTERING HOMOMETRY
Surprising examples of D-homometry were designedby Welberry et al. [14, 20]. They consist in substitu-tionnally disordered lattices with triplet (or quadruplet)short range ordered (SRO) CF [21], but zero two-pointcorrelations (Fig. 2a). Diffraction diagram of these lat-tices present the same Bragg and diffuse scattering inten-sity [14]. But, as shown in Fig. 2, their speckle patternsare different, which breaks the D-homometry.A more tractable (though subtle) example of D-homometry is provided by the geometrically ordered(GO) [22] Rudin-Shapiro (RS) sequence [23], whose
FIG. 3. Probability densities P ( I ) of diffraction patterns of( N, M ) = (4096 , × g -SROsequences for ξ ∼ N/37 ( p = 0 . p = 0 .
95) andN/3 ( p = 0 . generic term σ n can be written [12]: σ n + l = (cid:26) σ n for l = 0 , − n + l σ n for l = 2 , σ = 1 . (8)This sequence has become famous [11, 22–24] because,thought GO, it is D-homometric to randomly distributedsequences (sometimes called Bernoulli sequences (BS)[24]), with the same 4 N x (1 − x ) diffuse scattering in-tensity (Eq. 5). In other words, its two-point CF g ( n ) = σ σ n is zero for n = 0 ( spatial average).Let us now consider the coherent diffraction of RS andBS sequences. In what follows, we present Fast FourierTransforms (FFT) computations of sequences of length N , zero-padded up to a value M ≫ N to clearly see thespeckles. For the RS sequence, because x = 0 . N , we have always susbstracted theaverage value 2 x − I ( q )are normalized by 4 N x (1 − x ) in order to get I ( q ) = 1.Figure 2 shows the diffraction patterns of RS and BSsequences in the first Brillouin zone (0 < q < π ). Inspec-tion of these patterns shows that, although their averagevalue is the same, the speckles repartition is remarkablydifferent. In particular, it is clear that the BS patternexhibits much more spikes, while the RS pattern is morehomogenous and regular (no low-intensity speckles more FIG. 4. a) Computed g ( n ) using periodic boundary condi-tions for N = 256 RS sequence (blue) and a BS sequence(green). b) Magnitude of RS ˆ g ( q ) as a function of h = q/ π . regularly spaced). To quantify this observation, we stud-ied the statistics of both speckle patterns by calculatingtheir probability density of intensity P ( I ). It is knownthat for a random media (see e.g. [25]), the intensity dis-tribution has a negative exponential distribution givenby: P ( I ) = 1 I exp − II , (9)which in our case reduces to P ( I ) = exp( − I ).Figure 3a) shows the probability densities P ( I ) RS and P ( I ) BS for the RS and BS sequences. While P ( I ) BS follows quite well the negative exponential law, as ex-pected, it is not the case for P ( I ) RS . Though the pre-cision of P ( I ) RS depends on N , it is well approximatedby the step function P ( I <
2) = 0 .
5. This statistics,which means that intensities lower than 2 occur with thesame probability, explains the homogeneous aspect of thediffraction pattern. The reason for this unusual statisticsis not clear, but undoubtedly comes from the GO na-ture of the RS sequence. The presence of order, invisiblethrough diffuse scattering, is revealed by the statisticsof the speckle pattern, breaking the D-homometry in aquantifiable way.In order to test the robustness of the P RS ( I ) behaviorwith respect to disorder, we first quantify the degree oforder of the RS by one of its quadruplet CF: g ( n ) = σ σ σ n σ n +1 . (10)Indeed, we found numerically that at variance with theBS, g ( n ) is LRO for the RS sequence (Fig. 4a)[26]. Thisis confirmed by the behavior of its FFT ˆ g ( q ) (Figure 4b),which exhibits well defined peaks indexed by the basisvectors { h i . i | h i ∈ {− , } , i ∈ N } [27] characteristic oflimit-periodic functions [28]. By analogy with two-pointorders, we define η ≡ ˆ g ( π/ /N = 1 / η while keeping g -LRO and g -disorder by using the ”Bernoullization” procedure as de-fined in [12]. It consists in changing the sign of each σ n with probability p , in order to build sequences interme-diate between the pure RS ( p = 0 ,
1) and BS ( p = 0 . η was found to vary as η ( p ) ≃ (1 − p ) / p .The step-function behavior is rapidly lost as p → . P ( I = 0) ≡ P . Simulations show that P ( p ) closelyfollows 1 − η ( p ), and that sizeable deviation from thenormal law starts from p > ∼ . g -SRO on the speckle pattern was studiedby shifting the sequence by two lattice periods at certainpoints, randomly selected with probability p . This en-sures to achieve g -SRO, clearly seen by the broadeningof the ˆ g ( π/
2) peak, while keeping the g ( n ) correlationto zero. Figure 3c) shows the results for different p val-ues, correponding to the average distances between faults ξ given in the caption. We checked that P ( I ) RS is unaf-fected for ξ < ∼ N/
40. This shows that the extent of thefour-point order has also an effect on the density proba-bility, much larger than in the Bernoullization procedure.Finally, let us mention that the problem of C-homometry, which is somehow the true homometry, isclearly related to the unicity of inverse problems, whichis beyond the scope of this paper. It is important tonote, however, that the use of ptychography [5], in whichdiffraction patterns are obtained by shifting illuminationon the sample, can solve difficult problems of phase re-trieval [29], including C-homometry.
DISCUSSION
It might seem pointless to discuss the problem of ho-mometry while phase retrieval algorithms and ptychogra-phy can provide the full structural information, includ-ing high-order correlation functions. However, the fullmeasurement of 3D speckle patterns is time consumingand in many situations it is not possible to get the dataneeded for such inversions. Methods of speckle analy-ses on quickly measured diagrams are thus needed to getnovel information on the materials.As stressed in ref. [30], real cases of true B-homometryare rare. On the contrary, because disorder is concerned,D-homometry is very frequent especially when systemsare large. Moreover, it is related to high-order CF, whichare hardly accessible to experiments [8, 10, 31][32]. How-ever, although D-homometric systems have almost al-ways different speckle patterns, it is not yet clear whetherthis difference is quantifiable. Indeed, though the proba-blity density P RS ( I ) of our test bed sequence is strikinglydifferent from its homometric BS, the observed effects are very sensitive to disorder, and are almost invisible whenfour-point correlation lengths are too short. In this re-spect, we checked that the probablity density of the lat-tice shown in Fig. 1 does not present sizeable deviationfrom the normal law. Qualitatively, the ”spikiness” ofspeckle patterns is quickly reinforced by the introductionof randomness, which makes deviation to the negativeexponential curve delicate to observe.Consequently, though it is tempting to conclude fromthis study that high-order correlation functions are di-rectly observable through speckle statistics analysis,much theoretical work is still needed to find the rele-vant parameters controlling the statistics. Such an effortcould be supported by more sophisticated analyses suchas the use of second-order (or higher) probability func-tions [25]. The simple fact that the lattices of Fig. 1can be reconstructed with minimum information showsthat high-order CF are somehow hidden in the specklerepartition.An experimental difficulty lies in the presence of two-point correlations in all real systems (the classical SRO),which could obviously mask the speckle distribution anal-ysis. In this respect, we have checked that, at least forSRO lattices with no high-order correlations, dividingthe speckles pattern intensity by its associated diffusescattering one I DD (by smoothing, averaging or fitting)makes P ( I/I DD ) follow the normal decreasing exponen-tial law. This can help disentangling high-orders effectsfrom two-point ones.Another issue might be the partial coherence of thebeam, which reduces the speckles contrast and makes theprevious analyses difficult. This could be overcome bythe analysis of the speckles maximum intensities, whichexhibit similar statistical properties (not shown here).In conclusion, we suggest the that speckle statisticsanalysis could be used as another tool to test for thepresence of high-order correlations. Indeed much physicscould be explored with the measurement of high-ordercorrelations, still hidden to experiments (see ref. [31] forexamples). We hope this work will impulse theoreticaland experimental studies on the role of high-order corre-lation functions on coherent diffraction diagram.We thank F. Berenguer, D. Gratias, D. Le Bolloc’hand F. Livet for useful discussions. ∗ [email protected] † , 608 (1991).[2] I. A. Vartanyants, A. Singer, A. Mancuso, O. Yefanov,A. Sakdinawat, Y. Liu, E. Bang, G. Williams, G. Ca-denazzi, B. Abbey, H. Sinn, D. Attwood, K. Nugent,E. Weckert, T. Wang, D. Zhu, B. Wu, C. Graves, A. Scherz, J. Turner, W. Schlotter, M. Messerschmidt,J. L¨uning, Y. Acremann, P. Heimann, D. Mancini,V. 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