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Phase signal definition for electromagnetic waves in X-ray crystallography Sérgio L. Morelhaõ Instituto de F´ısica, Universidade de São Paulo, CP 66318, 05315-970 SãoPaulo, SP, Brazil Diffracted X-ray waves in crystals have relative phases regarding the mathematical format used todescribethem. A forward propagating wavecan bedefinedwith eithernegativeor positivetime 6 evolution, i.e. k·r−ωt or ωt−k·r. Physically measurable quantities are invariant with respect 0 to the choice of definition. This fact has not been clearly emphasized neither extensively explored 0 when deriving well-established equations currently being used in many X-ray diffraction related 2 techniques. Here, the most important equations are generalized and consequences of conflicting undertakendefinitions discussed. n a J I. INTRODUCTION 9 ] Electromagnetic waves are in general described by an ci expression such as s - rl D(r,t)=D (r)eiS(ωt−k·r) (1) t 0 m where S = ±1 stands for the global phase signal defini- . t tionofmonochromaticforwardpropagatingwaves. Both a m choices of phase signal (+ or −) are allowed and both can be founded in several theoretical approaches of the - d X-ray diffraction phenomenon in crystals. n Physically measurable quantities are invariant regard- o ing two distinct and independent choices: one is the c choice on how to describe the forward propagating X- [ ray wave, represented here by the options S = −1 and FIG. 1: The relative phase of the scattered wave from R, 1 S = +1; and the other is the choice of origin for the regardinganarbitrarychoiceoforiginatO,isgiveninterms v spatial position r. In the literature of X-ray diffraction ofincidentk0andscatteredkwavevectorssincekMO=−k0· 0 (seeforinstanceRefs.1,2,3,4,5)thesetwodistinctchoices r and kON =k·r, as demonstrated in Eq. (2). For sake of 7 have been of particular relevance for deriving important simplicity in analyzing thedependencewith theglobal phase 1 mathematical expressions,such as those of the structure signalS,theoriginischosenveryfarfromthecharge-density 1 0 factor of a crystal unit cell and of the atomic form fac- distribution ρ(r), to asssure that (k−k0)·r would havethe 6 tor. Whatever is the chosen option, S = ±1, it can be samesignal,inourcasepositive,foranypointRwhereρ(r)6= 0 compensated by the choice of origin so that the expres- 0. / sions are obtained in their standardized format without t a showing any dependence to the phase signal option. m over, to facilitate part of the demonstration, an adapta- This standard procedure links the choice of origin to tionoftheStokesrelationsforBraggreflectioninperfect - the choice of phase signal, which is unreal since both d low-absorbing crystals is presented. choices are independent from each other. The real fact n is that the available expressions are in agreement with o c only one value of S for each given choice of origin. Con- II. SUSCEPTIBILITIES TO PHASE SIGNAL v: sequently,thepropagatingwavesinthemediummustbe DEFINITION describedaccordingtotheimplicitly undertakenchoices, i X otherwise measurable features such as the positioning of A. Structure factor r standing wavefields —formed inside crystals undergoing a diffraction— as well as invariant phase values —acces- As depicted in Fig. 1, the scattered wave sible via 3-beam diffraction experiments6— will be af- fectedbyconflictingdefinitionoftheglobalphasesignal. ∆D(k)=D(B′)+D(B) In this work the conflicting points are demonstrated andcommonlyusedexpressionsinX-raycrystallography towards direction k depends on the relative phase be- aregeneralizedforanypreferenceofsignalchoiceS. Sig- tween the rays from the origin O and from the electron nal shifts owing to geometrical factors are avoided by density at any given position r, as for instance at R. In choosing the origin further way from the diffracting vol- the wavefrontBB′, ume. The consequences only on the choice of S is then exploited and discussed in the rest of the paper. More- D(B′)=∆D eiS[ωt−k(A′R+RB′)] R 2 and coordinates of the atoms in the unit cell. hkl is in fact the Mu¨ller index of a given reflection H. D(B)=∆DOeiS[ωt−k(OA+OB)] =∆DOeiϕ. The ± option in the exponent of Eq.(6) is not related tothe choiceS butitcanbe convenientlyusedto obtain Sincetheoriginhasbeenchosenoutsidetheelectronden- sitydistributionρ(r),thereisnoscatteringattheorigin, i.e. ∆DO =0. Also,the originwaschoseninasuchway ρ (r)= 1 F e−iS2πH·r. (7) to guarantee that (k−k0)·r > 0 to all positions inside c Vc XH H the volume where ρ(r)6=0, hence Methods for experimental determination of ρ (r), or c structuredeterminationmethods,arebasedontheabove k(A′R+RB′)=k(AO+OB)−k(MO+ON) expressionEq.(7),althoughithasbeenstandardizedfor the S =+1 choice. and C. Standing waves ∆D(k) = ∆D eiSk(MO+ON)eiϕ R = ∆DReiS(k−k0)·reiϕ. (2) The X-ray reflection coefficient of a crystal with N planes (crystal thickness Nd, lattice period d) in sym- Thechoice-of-originphaseϕ,iscommontoallelements metric Bragg reflection geometry can be written as of volume dV at any instant of time, and can be conve- niently chosen to provide eiϕ =1. If ∆D =D ρ(r)dV R e (De is the scattering amplitude of a single electron), the RN(θ)=|RN(θ)|ei(δH+Ω) (8) total scattered wave by the volume V is whereθ istherockingcurveangle,i.e. theanglebetween the incident X-raywaveandthe lattice planes. δ is the H D(k)=D ρ(r)eiS(k−k0)·rdV. (3) phase of the structure factor in Eq. (4), and Ω is known e ZV as the dynamical phase2 varying from the value ΩLeft at the left shoulder of the diffraction peak to the value In X-ray crystallography,the structure factor F cor- H Ω at the right shoulder; both values will be given responds to the total scattered wave from an unit cell, Right latter on. which is D (k)=D F /V (V is the unit cell volume). c e H c c Tocalculatethepositioningofthestandingwavesdur- Therefore, it follows from Eq. (3) that ing the rocking curve, the incident D and reflected D I R waves at a given depth h are approximated to F = f eiS2πH·rn =|F |eiδH (4) H n H Xn D =D (h)vˆ eiS(ωt−k0·r) (9) I 0 0 where 2πH = k − k , H is the diffraction vector of 0 and reflection H, and f is the atomic form factor1 of the n atom at the position r in the unit cell. n D =R (θ)D (h)vêiS(ωt−k·r) (10) R N 0 B. Electron density of the unit cell where vˆ and vˆ are their oscillation directions. A com- 0 parablereflectioncoefficientatalldepthsisassumed,i.e. The step from Eq. (3) to Eq. (4) is better compre- RN(θ) do not depend on h. All dependence with depth hendedwhenwrittenthe electrondensity ofthe unitcell is accounted for D0(h), which is a smooth function with very small variation over the lattice period. The standing wavefield is therefore D =D +D 1 SW I R ρ (r)= f δ(r−r ) (5) whosetime-averageintensityatanydepthisproportional c n n V c Xn to in terms of Dirac δ-functions |D |2 I = SW =1+|R (θ)|2+2|R (θ)|vˆ ·vˆcosΦ, SW D2 N N 0 δ(r−rn)= e±i2πH·(r−rn). (6) 0 (11) XH providing antinodes (maxima of intensity) at The sum in H stands for three sum on integer numbers Φ=δ +Ω−2πSH·r =2mπ (12) h, k and l, running from −∞ to +∞ since H · r = H n hx + ky + lz and x , y and z are the fractional for every integer value of m. n n n n n n 3 For sake of simplicity, a single atomic layer per lattice plane period is assumed so that H =zˆ/d, F =eiS2πz0/d f , H n Xn and δ =S2πz /d where z gives the atomic layer posi- H 0 0 tioninthe crystalunit cell. Ifz =0,the unit cellorigin 0 would fall on top of the atomic layer. From Eq. (12) we have the antinodes at z (Ω)=z +Sd(Ω/2π−m) (13) A 0 as a function of the dynamical phase Ω, and FIG.2: ReflectionRandR¯,andtransmission T andT¯,coef- ∆Ωd ficientsfor electromagnetic plane-wavesinverythinanduni- ∆z =z (Ω )−z (Ω )=S (14) A A Right A Left formplanesoflow-absorbingmatter. Stokesrelationsforthin π 2 filmoptics11provideconservationofthecountingrate|R+T¯|2 as the displacementof the antinodes during the crystal’s plus |R¯+T|2, at detectors D1 and D2 respectively, regard- rocking curve. Such displacement must be invariant re- ingthetotalimpingingintensityfromtheidenticalsourcesS1 garding the choice of signal S and, therefore, it can be andS2. Thetopandbottominsetsillustratethephasedelay ϕ=−SkA′B =−SkB′Aofthetransmittedwavesacrossthe assumed that plane thickness. ∆Ω=ΩRight−ΩLeft =−Sπ (15) in order to fulfill Eq. (16). δ±90◦ and δ¯±90◦ are the amount by which the phases of the reflected waves can since the displacement is experimentally observed8,9 as differfromthephasesofthetransmittedones,andϕand been ∆z =−d/2. A ϕ¯ are the phase delays across the plane, whose thickness It is widely known from dynamical diffraction theory2 d is comparable to the wavelength λ. that Ω = π and Ω = 0. However, a clear state- Left Right To determine the phase delays consider first only the ment should be made to emphasize that this phase vari- source S in Fig. 2. As shown in the top inset, if D(A) 1 ation is relative to the global phase signal S = +1, as is the incident wave at point A, D(B) = TD(A) is the demonstrated below. transmitted wave at point B passing through point A′. Then, D. Dynamicalphase shift versus global phase signal |T|D eiS(ωt−k0·rB) =TD eiS(ωt−k0·rA′) 0 0 where k0·rA′ =k0·rA since rA and rA′ stand for posi- Coherent scattering and transmission of electromag- tions on the same wavefront AA′, providing netic waves by very thin and uniform planes of low- absorbing matter are describable by reflection, R and T =|T|e−iS(rB−rA′)·k0 =|T|e−iSkA′B =|T|eiϕ. R¯,andtransmission,T andT¯, coefficientsas depictedin Fig. 2. The condition Since the incidence angle θ is the same for both sources A′B = B′A and, consequently, RT¯∗+R∗T¯+R¯T∗+R¯∗T =0 (16) 2π ϕ=ϕ¯=−S dsinθ. (18) λ is required by energy conservation, which stipulates phase relationships between the reflected and transmit- The X-ray reflectivity of a crystal, as givenin Eq. (8), ted waves, as usually obtained for laser beam splitters10 isobtainedbystackinglatticeplaneswhosereflectionand or, equivalently, by the Stokes relations for time re- transmission coefficients are written in the same format versibility of wave’s propagation11. usedabove,Eq.(17). Stackingingeometricalprogression of N = 2n planes (n = 1,2,...) is the fastest way to go Without loosing generality, these coefficients can be from a single lattice plane to the desired thickness Nd. written as It is possible by means of the recursive equation RR¯ == ±±ii||RR¯||eeii((δδ¯++ϕϕ¯)),, aTn=d |TT¯|=eiϕ|T¯,|eiϕ¯ (17) (cid:20) RR¯NN (cid:21)=(cid:18)1+ 1−TNR¯/N2T/¯2NR/N2/2(cid:19)(cid:20)RR¯NN//22 (cid:21) (19a) 4 FIG. 3: Simulated X-ray reflectivity curves |R(θ)|2, for crystal thickness Nd = 2.5µm (gray curve) and Nd = 21mm (black curve,flat-topmaximumequalto1). TheshiftofthedynamicalphaseΩ,Eq.(8),isshownonbothsidesoftherockingcurves forthetwopossiblechoicesofglobalphasesignalinEq.(18),i.e. S =±1. ∆θ=θ−θBragg,2dsinθBragg =λ=1.54˚A(Bragg’s Law),d=3.14˚A,|R1|=|R¯1|≃3.0×10−4 (nearlythevalueforthesilicon 111reflection)andδH =0. Forsimulation purposes someabsorptionhadtobeconsideredinthethickcrystalcase,thenalinearabsorptioncoefficient µ=asinθBragg/d≃1cm−1 was used. a is defined in Eq.(20b). and III. DISCUSSION ON PHASE INVARIANTS T 1 T2 (cid:20) T¯NN (cid:21)= 1−RN/2R¯N/2 (cid:20)T¯NN2//22 (cid:21) (19b) rayPhcarysestdaleltoegrrmapinhayt.ioRneiflseactfiuonndianmteennstiatilesproonbllyemprionviXde- |F | as experimental data input for ρ (r) in Eq. (7). whose derivationis verysimilar to the Airy’s formulaof H c Then,whatphaseδ shouldbeassignedtothestructure the Fabry-Perotinterferometer, and where H factor F =|F |exp(iδ ) of each crystal reflection? H H H Fordecades,greatefforthasbeendedicatedindevelop- R1 =−ireλ|C|d FH eiϕ (20a) ingandimprovingmethodstoestimatereflectionphases, (cid:20)R¯1 (cid:21) Vcsinθ (cid:20) FH¯ (cid:21) but since these are relative choice-of-origin values, the estimable quantitiesareinfactthe differencesamongre- and flection phases, better known as phase invariants12. For instance, consider three reflections G, H and L whose T1 = (1−|R1|2−a)1/2 eiϕ. (20b) diffraction vectors fulfill the condition G = H + L. (cid:20)T¯1 (cid:21) (cid:20) (1−|R¯1|2−a)1/2 (cid:21) Hence, in the structure factor ratio C =vˆ ·vˆ,H¯ =−H,r =2.818×10−5˚A(classicalelec- F F |F ||F | 0 e H L = H L eiΨ, tron radius), and a stands for photoabsorptionprobabil- F |F | G G ityoneachindividuallatticeplane,asdiscussedindetails and compared to other diffraction theories elsewhere7. the triple phase Ψ=δ +δ −δ is invariantregarding H L G Fig. 3 shows the behavior of the dynamical phase the choice of origin in Fig. 1. acrossthereflectivitycurveasafunctionofthephasesig- Three-beam diffraction experiments 6,13,14,15 are sen- nalSinEq.(18). Asexpected,Ω =πandΩ =0 sitive to Ψ. In azimuthal scan mode, the diffracted in- Left Right forthe S =+1choice,butΩ =0 andΩ =π for tensity of reflection G is modulated by the excitation of Left Right the other choice,i.e. S =−1, which is also in agreement the reflection H during the crystalrotationφ aroundG. with Eq. (15). This intensity modulation is approximately given by 5 IV. CONCLUSIONS I(φ)=|D |2+|D (φ)|2+ G HL +2|D ·D (φ)| cos(Ψ+Ω), (21) Structure factor, dynamical phase shift, phase invari- G HL ants, and complex atomic form factor are values suscep- whichexplicitlydependsonthedynamicalphaseΩofthe tible to the choice of global phase signal, as explicitly reflection H. D and D (φ) stand for the diffracted demonstrated here. When a measurable quantity de- G HL wavefields from reflection G, kept constant during the pends on more than one of these values, they must be φ-scan, and from the detour reflection H +L. calculated for the same choice of phase signal. AlthoughthesignalofΨ+ΩchangeswithS,theinvari- Recursive equations based on Stokes relations are ap- ance of I(φ) is assured by the cosine, an even function. pliedforcalculatingthereflectivityoflow-absorbingcrys- However, both angles must stand for the same choice tals with a finite number of lattice planes. These re- of S in Eqs. (4) and (18). Otherwise, the experimen- cursive equations are extremely simple and, yet, capable tal determination of Ψ by φ-scan data analysis would be of describing important features of the diffraction phe- compromised. nomenon such as kinematical diffraction (thin crystals), Besides the signal of Ψ, its modulus will also change primaryextinction(maximumreflectivityequalto1),in- with S when the f′ and f′′ corrections1 of the atomic trinsicwidthofBraggreflectionsineitherkinematicalor form factor —the so-called anomalous scattering correc- dynamicaldiffraction regimes,and the phase shift of the tions— are taken into account. According to the struc- diffracted waves across the reflection domain. ture factor expression in Eq. (4) |F (S)|eiδH(S) = (f +f′+if′′) eiS2πH·rn, (22) H 0 n Xn and then |FH(+1)|=6 |FH(−1)| and δH(+1)6=−δH(−1) Acknowledgments when f′′ 6= 0. It is unacceptable since these inequalities would imply in different diffracted beam intensities for each possible choice of wave representation, S = +1 or TheauthorwouldliketothankProf. PauloA.Nussen- S = −1. To avoid such artificial fact, the f′′ correction zveig for valuable discussions and very kind revision of must bear its depence with S, so that themanuscript,aswellastheBrazilianfoundingagencies FAPESP, grant number 02/10387-5, and CNPq, proc. ′ ′′ f =f +f +iSf . (23) number 301617/95-3. 0 1 International Tables for Crystallography Vol. B, 2nd. edi- standing-wave−modulated electron emission near absorption (2001). tion edges in centrosymmetric and noncentrosymmetric 2 A. Authier, Dynamical Theory of X-ray Diffraction, 2nd crystals” Phys. Rev. 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