Atomic Scattering Factor for a Spherical Wave and the Near Field Effects in X-ray Fluorescence Holography

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1 Atomc Scatterng Factor for a Spherca Wave and the Near Fed Effects n X-ray Fuorescence Hoography Janmng Ba Oak Rdge Natona Laboratory, Oak Rdge, TN Formua for cacuatng the atomc scatterng factor for spherca x-ray waves s derved and used to sove the near fed effects probem n X-ray Fuorescence Hoography theory. A rgorous formasm to cacuate the X-ray fuorescence hoogram for gven atomc dstrbutons s then gven so that quanttatve structura nformaton can be derved from XFH measurements. I. INTRODUCTION PACS Number(s): 61:1.Dp, 61.1.Eq, 4.4.Kw, 3.8.Cy Atomc scatterng factors (ASF) are used n numerous crystaographc cacuatons. The vaues of ASF for atoms and ons are tabuated n the Internatona Tabes for X-ray Crystaography. These ASFs are defned under the assumpton that both the x-ray source and the detector are far away from the scatterers so that both the ncdent and scattered x-ray can be represented by pane waves. Ths assumpton s vad for amost a knds of x-ray scatterng experments unt the emergence of x-ray fuorescence hoography (XFH) n recent years. 1,,3 XFH, because of ts unque vew pont, s a new technque wth great potenta n exporng the oca atomc orders. However, t needs further deveopments n two respects before becomng a routne method n matera studes ke XRD and EXAFS. Frst, because of the ow sgna/background rato (typcay 1-4 to 1-3 ) n XFH, t takes severa days to coect data on a snge hoogram even wth synchrotron radaton. The botteneck s not the x-ray ntensty but rather the detector speed. Ths probem w eventuay be soved wth deveopments of hgh-speed detectors. Secondy, the rea space fed ntensty mage reconstructed usng the we-known Barton agorthm 4 s not a quanttatve measurement of the atomc postons. One way to sove ths probem s to use the nonnear east-square-fttng agorthm to get accurate reconstructon of the atomc arrangement; ths n turn requres a rgorous formasm to cacuate the hoogram. It s the task of ths work to derve such formasm. In XFH, the fuorescence atoms nsde the sampe are used ether as sources (n drect XFH) or as detectors (n nverse XFH, aso caed mutpe energy x-ray hoography, MEXF). The dstance between the radaton source and scatterer, or the scatterer and the detector, s comparabe to the sze of the eectron dstrbuton of the scatterer. In ths case, atomc scatterng factors for a spherca wave shoud be used n the quanttatve anayss for XFH. In a pubshed work nvovng XFH cacuaton, a frst order approxmaton has been used. Ths approxmaton assumes that the sze of the core eectron dstrbuton of the scatterer s much smaer than the radus of the ncdent spherca wave front and s thus vad ony for a pont ke scatterer. 5 In ths paper, the formua to cacuate the ASF for a spherca wave s derved and ts dependence on the atomc rad of the scatterer, the source-scatterer or scatterer-detector dstance and the waveength has been examned. A detaed approach to cacuate the spherca wave ASF from the eectron densty dstrbuton or

2 tabuated pane wave ASFs s deveoped. The ony assumpton s that the source (or detector) s outsde of the eectron dstrbuton of the scatterer, whch s aways satsfed n XFH experments. Gven the smarty between the defntons of x-ray and eectron ASF, the same scheme can aso be used to cacuate the spherca wave ASF for eectrons n the hgh-energy regme, whch can be used to correct the sma atom approxmaton n photoeectron spectroscopy 6. Another respect of the near fed effects n XFH s orgnated from the vector property of the eectromagnetc fed. Usng spherca wave ASF, these near fed effects terms are aso derved. Wth consderaton of compete near fed effects, rgorous expressons for cacuatng XFH and MXFH are gven at the end. II, ATOMIC SCATTERING FACTOR FOR A SPHERICAL WAVE The X-ray fuorescence hoogram s formed by the nterference of the reference wave and the object waves. In drect XFH, the reference wave s the fuorescence radaton from the emtter atom and the object waves are the scattered fuorescence radatons from the surroundng atoms. In MEXF, the reference wave s the ncdent pane wave x-ray and the object waves are the scattered x-rays from the atoms surroundng the fuorescence emtter. The ntensty of the fuorescence s a record of the nterference pattern. In the frame of cassca eectrodynamcs, the normazed x-ray fuorescence hoogram s wrtten as: r(r - r c(k) = -r e Ú dr ) (kr-k r )  r e + c.c., (1) where r e s the cassca eectron radus and r(r) s the eectron charge densty. The r s centered at the fuorescence emtter; r s are the centers of the scatterers and c.c. s the compex conjugaton of the frst term. To concentrate on the near fed effects caused by the atomc scatterng factors, we consder here ony the scaar wave equaton. The near fed effects caused by the vector property of eectromagnetc wave w be expored ater. Let u = r- r and assume that u << r, we can have r ª r n the denomnator and r ª r + r u / r n the phase term. Ths gves us the commony used frst approxmaton: f c(k) ª -r (s ) (kr -k r ) e  + c.c., () r e -(k-k wth f (s ) = dur (u) ) u Ú e, s = k - k and k = k r / r. Here f s just the ASF for pane wave x-rays. Ths approxmaton doesn t take the curvature of the spherca wave front nto account and s vad ony when r s much arger than the radus of the scatterng atom. Ths assumpton s generay nvad for near neghbor atoms, and snce the x-ray hoogram s especay senstve to near neghbor atoms, a more accurate formua s needed. We defne the spherca ASF as: kr f S -kr (q,r ) = r +k r r(r - r e Ú dr ) (kr-k r) -kr r e = r e Ú dur(u) e -k u r e. (3) S By repacng the f (s ) n Eq.() wth f ( q, r ), one obtans the exact expresson as gven n Eq.(1). The spherca wave term e kr /r can be expanded wth spherca harmoncs: 7

3 e kr r = 4pk  (-1) +m j (ku < )h (1) (ku > ) ÂY m (W r )Y -m (W u ), (4) m and the pane wave term as:  (-1) -k u e = 4p (-), m m j Insertng Eq. (4) and (5) nto Eq. (3), we have: f S (q,r ) = (4p) r k e - m m (ku)y (W u )Y (W k ). (5) -kr  (-1) m Y m (W r )Y -m (W k ) Ú u j (ku) j (ku < )h (1) (ku > ) r(u)du,m. (6) È r -kr = 4pr ke Â( +1) P (cosq ) Í h (1) (kr ) Ú j (ku)r(u)u du + j (kr ) Ú j (ku)h (1) (ku)r(u)u du Î Í r Where q s the ange between r and k. Wth a knowedge of the atomc eectron densty, Eq.(6) can be cacuated for any r vaue. However, n practce, t s more nterestng to consder the stuaton when the source s outsde of the eectron dstrbuton of the scatterng atom. In ths case, the second ntegraton n Eq.(6) s zero. The pane wave ASF can be expanded wth Legendre poynoma as: f ( k - k ) = 4p ( +1)P (cosq) Ú j (ku)r(u)u du, (7)  wth q defned as the ange between k and k, the Eq.(6) becomes: p f S (q,r ) = kr -kr e Â( +1) +1 P (cosq ) h (1) (kr ) Ú P (cosq) f (k sn( q ))snq dq. (8) One can easy verfy Eq. (8) by assumng that r s very arge or that r(u) s a deta functon (so f S =z), n both the cases f ( q, r ) w degenerate nto f (s ). The physca meanng of Eq. (8) s that the scatterng power of an atom for a spherca wave can be represented by a weghted sum of pane wave ASF wth the same waveength n a drectons. However, when Eq. (8) s apped to a very sma r wth tabuated pane wave ASF vaues, e.g. when r equas.485 Å, whch s the nearest neghbor dstance n a bcc ron crysta, the seres n Eq. (8) s not convergent. Ths s because for free atoms the eectron densty contnues beyond the nearest neghbor dstance n ther crysta form. From Fgure 1 one can see that for an ron atom a sma part of the eectron dstrbuton n 4s she s outsde of ts frst neghbor dstance n the crysta. Therefore, the smpe form of spherca wave ASF n Eq. (8) s not vad n ths case snce the frst ntegra n Eq. (6) cannot be extended to nfnty. Cacuatons show that Eq. (8) can be safey used for r greater than twce the nearest neghbor dstance for most eements. Spherca wave ASF can be cacuated from Eq. (6) f the rada eectron densty s known. Even though the rada atomc eectron denstes can be very we cacuated based on the she mode of atoms, ther vaues are not convenenty avaabe. It w be usefu to have a formasm to cacuate the spherca ASF wth pane wave ASF vaues, whch are expermentay measurabe physca parameters. It s we known that the pane ASF s smpy a three dmenson Fourer mage of the atomc eectron dstrbuton. For a spherca rada eectron densty mode, we have: 3

4 sn( sr) f ( s) = 4p r( r) r dr, (9) sr Ú and r(r) = 1 Ú f (s)sn(sr)sds. (1) p r The tabuated ASF vaues n the Internatona Tabes for Crystaography are ony avaabe n the range of s / 4p from. to 6. Å Evauaton of an accurate eectron dstrbuton requres a wder s range. However, what we need for evauatng the spherca ASF from Eq. (6) s the ta of the atomc eectron densty far from ts nuceus, whch s determned by the ow s vaues of f(s). We can cacuate a parta eectron densty by constructng a parta ASF, f c (s), whch s defned as: f c (s) = ae -bs for s s c and f c (s) = f (s) for s > s c, (11) where s c s the cutoff vaue of s. The a and b are defned by settng the vaue and frst dervatve of f c (s) equa to those of f(s) at the cutoff s c : f (s a = f (s c )e bs c, b = - c ) s c f (s c ). (1) The f c (s) as defned corresponds ony to the eectron densty cose to the nuceus so that the parta eectron densty gven by: r p (r) = 1 s c Ú ( f (s) - f p c (s))sn(sr)sds = 1 Ú ( f (s) - f r p c (s))sn(sr)sds (13) r w gve the correct eectron densty for arge r s. Fgure1 shows that the parta eectron densty cacuated wth a cutoff s c /4ϖ=1.9Å -1 represents very we the entre 4s eectron dstrbuton n the ron atom. The parta ASF f c (s) represents ony the eectron dstrbuton very cose to the nuceus and can now be used n Eq. (8). The parta eectron densty whch can be cacuated wth Eq. (13) wth known pane wave ASF up to s=s c can be used n Eq. (6) to account for the contrbutons to the spherca ASF other than those from f c (s). Now the spherca wave ASF s gven as: f S (q,r ) = kr -kr e Â( +1) +1 P (cosq ) c (k,r ), (14) and c (k,r ) = h (1) (kr ) p Ú P (cosq) f c (k sn( q ))snq dq r. (15) + 8p(h (1) (kr ) Ú j (ku)r p (u)u du + j (kr ) j (ku)h (1) (ku)r p (u)u du) Equaton (15) can be used to cacuate the spherca wave ASF for any physcay meanngfu r vaues. The cutoff vaue s c /4ϖ must be greater than 1/ and ess than the hgher mt of the effectve range of pane wave ASF vaue. Snce to. Å -1 s the effectve range of the wdey used anaytca representaton of the pane wave ASF, the cutoff vaue of 1.9 Å -1 used n the above exampe s a good choce for x-rays of energes ess than 3 kev. In a crysta, the outmost eectron dstrbuton of an atom w be redstrbuted due to the neghbor atoms. Thus, the spherca symmetry s ony approxmatey for the outmost eectrons. Cacuaton shows that the contrbuton from the eectron dstrbuton outsde r, whch s gven by the second ntegraton n Eq. (15), s neggby sma (fgure ), thus the error caused by the spherca symmetry approxmaton shoud Ú r 4

5 be sma. It s we estabshed that for pane wave ASF the x-ray refecton ntenstes are we represented by the free atom vaues of the form factors and are not very senstve to the sma redstrbutons of the eectrons. 9 As an exampe, the spherca ASF s were cacuated for an ron atom wth dfferent r (fgure 3) and for dfferent x-ray energes (fgure 4). In a these cacuatons, except for the 4 kev curves, a cutoff s c /4ϖ = 1.9 Å -1 and the four-gaussan anaytca representaton of pane ASF by Doye & Turner 1 were used. For the 4 kev curves, a cutoff s c /4ϖ = 3.5 Å -1 and the fve-gaussan anaytca representaton of pane wave ASF by Waasmaer & Krfe 11, whch has an effectve range from to 6. Å -1, were used. The cacuaton shows that the rea parts (and the magntude) of the spherca wave ASFs are about 1 to % ess than the pane wave ASFs for the frst neghbor scatterers around the forward scatterng drecton (q = ) and approach the pane wave ASF vaues at hgh anges. Ths correcton s many due to the curved wave front. The pane wave ASF w reach the eectron number Z n forward scatterng because n ths drecton the compete eectron densty dstrbuton n the atom has the same phase. Ths w never happen for a spherca wave. The fact that the rea part of the spherca wave correcton vanshes at hgher anges can be understood by ookng at Eq. (9). The contrbuton to the ASF s many from eectrons near the nuceus of the scatterng atom for hgh s snce the functon sn(sr)/sr acts ke a d-functon for hgh s. The curved wave front correcton s sma for nner she eectrons. The magnary part of the spherca ASF assumes a postve vaue about 1 to % of the atomc eectron number and approaches a sma negatve constant at hgher anges. Ths correcton s a combnaton of the curved wave front effect and the 1/r dependence of the spherca wave amptude. The 1/r weght n the eectron dstrbuton ntegraton makes the apparent scatterer poston coser to the source. Hence ths contrbutes a negatve phase shft, and ths shft does not depend on the scatterng ange. So the overa effect of the spherca wave ASF correcton s that when the scatterer s between the source atom and the detector, t s 1 to % ess n scatterng power and apparenty shfts away from the source. When the scatterer s on the opposte sde of the source reatve to the detector, t has an apparent poston shft towards the source. The curved wave front correcton s arger for shorter source-scatterer dstance (fgures and 3) and hgher x- ray energes (fgure 4) as shown. Fgure shows the contrbuton to the spherca ASF from the eectrons beyond r. The argest contrbuton s for the frst neghbor scatterers and s ess than.3 eectrons for ron at 8 kev. The outer she eectron contrbuton s ess for hgher energes (not shown). III, NEAR FIELD EFFECTS DERIVED FROM VECTOR THEORY In addton to the near fed effects caused by the spherca wave front, there are aso near fed effect terms caused by the vector property of the x-ray wave. In reference 5 these terms were consdered under the pane wave approxmaton. Now wth the spherca ASF, we can gve a more accurate expresson of these terms. In drect XFH, at the observaton pont r far away from the object, the tota eectrc wave fed s gven by: 1 E(r) = (g(r)p) - r e k Ú d r { g(r - r )r( r ) [ g( r )p ]}. (16) Where g(r)=exp(kr)/r and p s the eectrc dpoe moment at r=. Assumng r >> r, Eq. (16) can be smpfed to: 5

6 È E(r) = k g(r)n p - r e -k r -k u Í Â k e ( Ú due r(u)g( r - u )p) n Î. (17) È = k g(r)n p - r e -k r Í Â k e ( h(r,q )p) n Î S Where n=r/r, h ( r, q ) =g(r ) f ( q, r ) and the sum s over a surroundng atoms. Ths eads to an expresson for the hoogram: c ( k) = Â c ( k) wth c (k) = - r e -k r k e ( h(r,q )p) p n /p p n + c.c.. (18) Where p = n p n = p - ( n p) n. After some tedous but straghtforward vector agebra, we have: n c (k) = - r e -k r k e {-p p n h(r,q ) + (p r ) )(p n r ) ) r h(r,q ) +[(p r ) )(p n ˆ q ) + (p n r ) )(p ˆ q )] ( 1 )h(r,q ) r r q +(p ˆ q )(p n ˆ q ) 1 r ( r + 1 r q )h(r,q ) +(p j ˆ )(p n ) j ) 1 ( + r r 1 )h(r,q )}/p p n + c.c. r tan(q ) q. (19) For drect XFH, the eectrc dpoe moment p s averaged n 4ϖ sod anges snce the fuorescence radaton s unpoarzed. So we now have: c (k) = - -k r r e e {h(r,q )(1+ cos q )/ + 1 q -1 k [3cos h(r,q ) + cos(q ) r r q h(r,q ) r. () + sn(q ) h(r,q ) + 1-3sn q h(r r r q r,q )]} + c.c. tanq q In dervng equaton (), pease note that the functon h r, q ) satsfes the Hemhotz dfferenta ( equaton: h + k h =. The dervatves of h ( r, q ) can be cacuated usng Eq. (8). For exampe: h( r, q ) q k = Â ( + 1) P (cosq ) h (1) ( kr ) p Ú P (cosq ) f (k sn( q ))snqdq. Where P 1 (cosq ) s assocated Legendre poynoma of order and degree 1. For MXFH, assume the ncdent x-ray wave s poarzed and the E vector s perpendcuar to k and surface norma of the sampe, as n the case of synchrotron radaton, we have: 6

7 c (k) = - -k r r e e {h(r,q )(1- sn q cos j ) + 1 k [1-3sn q cos j r h(r,q ) + cos(q )cos j r r q h(r,q ) + sn(q )cos j h(r,q ) + 1- cos j (1+ 3sn q ) h(r r r q r,q )]} + c.c. tanq q. (1) If the ncdent beam s unpoarzed, Eq. (1) needs to be averaged over j and w be the same as Eq. (). In Eq. () and (1), the c (k) s dependence on the drecton of k s throughq, whch s the ange between k and r ; andj, the ange between p and r - (n r )n. They can be expressed n terms of the spherca coordnates of k (q, j) and r ( q, j ) n a coordnator system fxed on the sampe: cosq = cosq cos q + snq sn q cos(j - j ). () cosj = sn(j - j )sn q /snq Combng equatons (18), (), (1) and (), the x-ray fuorescence hoograms can be cacuated wth consderaton of compete near fed effects. Fg. 5 shows a cacuated c ( q, j = ) curve for a snge emtter-scatterer par of ron atoms. The fuorescence emtter atom s at orgn whe the scatterer s at.485 Å aong the x axs (e.g. q = j =). The cacuaton shows that at x-ray energy of 6.4 kev, the correctons due two the two knds of the near fed effects are comparabe. At x-ray energy of kev, the correcton due to the curved wave front s arger whe the correcton due to the near fed effect caused by the vector fed propertes s smaer. Ths s because of the 1/k dependence n Eq. (). A generazed atomc scatterng factor can be defned as functon of q and r for XFH or MXFH wth unpoarzed ncdent beam: f g (r,q ) = f S (r,q )(1+ cos q )/ + e -kr k [(3cos q -1) h(r,q ) + cos(q ) r r q h(r,q ). (3) +sn(q ) h(r,q ) + 1-3sn q h(r,q )] r q r tanq q Wth ths generazed atomc scatterng factor, the smpe scaar form as of equaton () can be used to cacuate the hoogram n XFH or MXFH wth unpoarzed ncdent beam. The near fed S effect caused by curved wave front s contaned n the spherca wave ASF f ( r, q ) whe the near fed effect caused by the vector property s presented n the four terms n the squared parentheses. Fg. 6 shows the rea and magnary part of the generazed ASF for ron atom at 6.4 kev as functon of q wth r =.485 Å. Snce the generazed ASF s a sow varyng functon of r andq, they can be stored n a two dmenson array at ncrement r and q vaues. When cacuatng the x-ray hoogram, the vaues of the generazed ASF at any r and q vaues can be retreved by nterpoaton. For MXFH wth poarzed ncdent beam (Eq. 1), the generazed ASF can be 7

8 dvded nto two parts, f g = f ( r, q ) + f ( r, q )cos j. In ths paper we consder ony the 1 g g Thomson scatterng. When x-ray energy s cose to the absorpton edge, the compex anomaous scatterng factor shoud be added to the generazed ASF. IV. CONCLUSION In concuson, the formasm for cacuatng the spherca wave ASFs from the rada eectron dstrbuton functon or the tabuated pane wave ASFs s derved. The exampe cacuaton shows that the curved wave front correcton to ASFs can be as arge as % of the pane wave ASF, depends on the source-scatterer dstance and x-ray energy. Ths correcton shoud be consdered n a XFH or MXFH cacuaton. On the other hand, the near fed effects due to the vector property of the eectromagnetc fed s comparabe to curved wave front correcton for ow x-ray energes and must aso be ncuded for most of XFH cacuaton. Wth both knds of near fed effects consdered, the rgorous expressons for both XFH and MXFH are gven. These expressons are necessary parts n the deveopng of new reconstructon method based on east- square-fttng agorthms. There are other approxmatons n XFH theory,.e. the snge scatterng approxmaton and the pont dpoe moment source approxmaton for the fuorescence atoms. Unke photoeectron hoography, n XFH the mutpe scatterng effects are generay neggbe due to the sma cross secton of x-ray scatterng. On the hand, snce ony the nner she fuorescence s used for XFH experment, the sze of the radaton source s aso neggbe. For exampe, the radus of 1s she n Fe atom s.3 Å, whch s ony about 1.3% of the nearest neghbor dstance n BCC ron crysta and much smaer than the waveength of ron ka ne. Therefore, the error caused by the pont dpoe moment fed approxmaton shoud be one magntude smaer than the correctons consdered n ths paper. Moreover, because of the spherca symmetry of the s-she eectron dstrbuton, the dpoe moment sze effect shoud be sotropc and hardy detectabe n XFH measurements. ACKNOWLEDGEMENTS The author thanks Camden Hubbard, Gene Ice and Cue Sparks for revewng the manuscrpt and hepfu dscussons. Research sponsored by the Assstant Secretary for Energy Effcency and Renewabe Energy, Offce of Transportaton Technooges, as part of Hgh Temperature Materas Laboratory User Program, Oak Rdge Natona Laboratory, managed by UT-Battee, LLC, for the U.S. Dept. of Energy under contract DE-AC5-OR75. 8

9 Fg. 1 Parta eectron dstrbutons of atom ron cacuated wth equaton (13). The pane wave ASF for ron used here s based on an anaytca nterpoaton of oneeectron wave functons but to approxmate the souton to Hartree-Fock equatons and s vad for entre range of s. 13 For a cutoff s of 3 Å -1, the parta eectron dstrbuton cacuated (sod ne) s practcay the fu eectron dstrbuton of atom ron. 9

10 Fg., The magnary (top) and rea part (bottom) of the contrbutons to the spherca wave ASF for ron from the eectron dstrbutons beyond of ts frst and second neghbor dstance. The x-ray energy s 8 kev. 1

11 Fg. 3, The dfference between the rea part of spherca wave ASF and the pane wave ASF (top) and the magnary part of the spherca wave ASF (bottom) for ron cacuated wth r equa to ts frst, second, thrd and fourth neghbor dstance at 8 kev. 11

12 Fg. 4, X-ray energy dependence the ASF for a spherca wave. Top: rea part of the spherca ASF mnus the pane ASF. Bottom: magnary part. 1

13 Fg. 5, X-ray fuorescence hoogram curves for a snge par of ron atoms separated by.485 Å. Top: E=6.4 kev. Bottom: E= kev. 13

14 Fg. 6, The rea and magnary part of the generazed ASF wth r =.485Å for ron atom and x-ray energy of 6.4 kev (Fe ka-nes). For comparson, the pane wave ASF mutped by the poarzaton factor (1 + cos q ) / s aso potted. 14

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