Unit 2b.2: Dynamical Diffraction in Methods in Materials Research (John Wiley & Sons, 2000)

Transcription

1 Unt b.: ynamcal ffracton n Mthods n Matrals Rsarch (John Wly & Sons, ) Qun Cornll gh Enrgy Synchrotron Sourc (CESS) and partmnt of Matrals Scnc and Engnrng Cornll Unvrsty, Ithaca, Nw York ======================================== I. Introducton Applcaton aras Brf survy of ltratur Scop of ths unt II. Basc Prncpls Fundamntal quatons sprson surfac Boundary condtons Intrnal flds III. Two-Bam ffracton Proprts of dsprson surfac Spcal dynamcal ffcts Soluton of dsprson quaton ffractd ntnsts Standng wavs IV. Multpl-Bam ffracton Basc concpts NBEAM thory Scond-ordr Born approxmaton Spcal mult-bam ffcts Polarzaton dnsty matrx V. Grazng Angl ffracton Frsnl rflctvty Evanscnt wav Multlayrs and suprlattcs Grazng ncdnc dffracton stortd-wav Born approxmaton VI. Summary VII. Rfrncs ======================================== 3// 1:35 PM

2 1 Unt b.: ynamcal ffracton I. INTROUCTION ffracton rlatd tchnqus usng x-rays, lctrons or nutrons ar wdly usd n matrals scnc to provd basc structural nformaton on crystalln mattrs. To dscrb a dffracton phnomnon, on has th choc of two thors: knmatc thory or dynamcal thory. Knmatc thory, dscrbd n th last Unt, assums that ach x-ray photon, lctron, or nutron scattrs only onc bfor t s dtctd. Ths assumpton s vald n most cass for x-rays and nutrons snc thr ntractons wth matrals ar rlatvly wak. Th sngl scattrng mchansm s also calld th frst ordr Born approxmaton or smply th Born approxmaton (Schff, 1955; Jackson, 1975). As amply dmonstratd n th last Unt, th knmatc dffracton thory can b appld to a vast maorty of matrals studs and s th most commonly usd thory to dscrb x-ray or nutron dffracton from crystals that ar mprfct. Thr ar, howvr, practcal stuatons whr th hghr-ordr scattrng or multpl scattrng trms n th Born srs bcom mportant and cannot b nglctd. Ths, for xampl, s th cas for lctron dffracton from crystals, whr an lctron bam ntracts strongly wth lctrons n a crystal. Multpl scattrng can also b mportant n crtan applcaton aras of x-ray and nutron scattrng, as dscrbd blow. In all ths cass th smplfd knmatc thory s not suffcnt to valuat th dffracton procsss and th mor rgorous dynamcal thory s ndd whr multpl scattrng s takn nto account. Applcaton Aras ynamcal dffracton s th prdomnant phnomnon n almost all lctron dffracton applcatons, such as low nrgy lctron dffracton and rflcton hgh-nrgy lctron dffracton. For x-rays and nutrons aras of matrals rsarch that nvolv dynamcal dffracton may nclud th followng stuatons: (1) Strong Bragg rflctons: For Bragg rflctons wth larg structur factors, th knmatc thory oftn ovrstmats th ntgratd ntnsts. Ths occurs for many ral crystals such as mnrals and vn bologcal crystals such as protns, snc thy ar not dally mprfct. Th ffct s usually calld th xtncton (Warrn, 1969), whch rfrs to th xtra attnuaton of th ncdnt bam n th crystal du to th loss of ntnsty to th dffractd bam. Its charactrstc lngth scal, xtncton lngth, dpnds on th structur factor of th Bragg rflcton bng masurd. On can furthr catgorz xtncton ffcts nto two typs: prmary xtncton, whch occurs wthn ndvdual mosac blocks n a mosac crystal, and scondary xtncton, that occurs for all mosac blocks along th ncdnt bam path. Prmary xtncton xsts whn th xtncton lngth s shortr than th avrag sz of mosac blocks and scondary xtncton occurs whn th xtncton lngth s lss than th absorpton lngth n th crystal. () arg narly-prfct crystals and multlayrs: It s not uncommon n today s matrals prparaton and crystal growth laborators that on has to dal wth larg narly-prfct crystals. Oftn cntmtr-szd prfct smconductor crystals such as GaAs and S ar usd as substrat matrals and multlayrs and suprlattcs ar dpostd usng molcular-bam or chmcal vapor ptaxy. Bulk crystal growrs ar also producng largr hgh qualty crystals by advancng and prfctng varous growth tchnqus. Charactrzatons of ths larg narly-prfct crystals and multlayrs by dffracton tchnqus oftn nvolv th us of dynamcal thory smulatons of th dffracton profls and ntnsts. Crystal shap and ts gomtry wth rspct to th ncdnt and th dffractd bams can also nflunc th dffracton pattrn whch can only b accountd for by dynamcal dffracton. (3) Topographc studs of dfcts: X-ray dffracton topography s a usful tchnqu to study crystalln dfcts such as dslocatons n larg-gran narly prfct crystals (Chkawa & Kuryama, 1991; Klappr, 1996; Tannr, 1996). Wth ths tchnqu an xtndd hghly collmatd x-ray bam s ncdnt on a spcmn and an mag of on or svral strong Bragg rflctd ar rcordd wth hgh-rsoluton photographc flms. Examnaton of th mag can rval µm szd crystal dfcts such as dslocatons, growth front, fault lns, tc.. Bcaus th stran fld nducd by a dfct can xtnd far nto th sngl crystal gran, th dffracton procss s rathr complx and a quanttatv ntrprtaton of a topographc mag frquntly rqurs th us of dynamcal thory and ts varaton on dstortd crystals dvlopd by Takag (196, 1969) and Taupn (1964). (4) Intrnal-fld-dpndnt dffracton phnomna: Svral dffracton tchnqus mak us of th scondary xctatons nducd by th wav fld nsd a crystal undr dffracton condton. Ths scondary sgnals may b x-ray fluorscnc or scondary lctrons such as Augr or photolctrons. Th ntnsts of ths sgnals ar drctly proportonal to th lctrc fld strngth at th atom poston whr th scondary sgnal s gnratd. Th wav fld strngth nsd th crystal s a snstv functon of th crystal orntaton nar a spcular or a Bragg rflcton and th dynamcal thory s th only thory that provds th ntrnal wav fld ampltuds ncludng th ntrfrnc btwn th ncdnt and th dffractd wavs or th standng wav ffct (Battrman, 1964). As a varaton of th standng 3// 1:35 PM

3 wav ffct, th scondary sgnals can b dffractd by crystal lattc and form standng wav lk dffracton profls. Ths nclud Kossl lns for x-ray fluorscnc (Kossl, 1935) and Kkuch lns for scondary lctrons (Kkuch, 198). Ths ffcts can b ntrprtd as th optcal rcprocty phnomna of th standng wav ffct. (5) Multpl Bragg dffracton studs: If a sngl crystal s orntd n such a way that mor than on rcprocal nods fall on th Ewald sphr of dffracton, a smultanous multpl-bam dffracton would occur. Ths smultanous rflctons wr frst dscovrd by Rnnngr (1937) and ar oftn calld Rnnngr rflctons or dtour rflctons (umwganrgung). Although th angular postons of th smultanous rflctons can b prdctd by smpl gomtrc consdratons n rcprocal spac (Col, Chambrs & unn, 196), a thortcal formalsm that gos byond th knmatc thory or th frst ordr Born approxmaton s ndd to dscrb th ntnsts of a multpl-bam dffracton (Collla, 1974). Bcaus of th ntrfrnc among th smultanously xctd Bragg bams, multpl-bam dffracton promss to b a practcal soluton to th phas problm n dffractonbasd structural dtrmnaton of crystalln matrals, and thr has bn a grat rnwd ntrst n ths rsarch ara (, 1998; 1999a,b; Chang, t al. 1999). (6) Grazng-ncdnc dffracton: In grazng ncdnc dffracton gomtry, thr th ncdnt bam or th dffractd bam or both has an ncdnt or xt angl, wth rspct to a wll-dfnd surfac, clos to th crtcal angl of th dffractng crystal. Full tratmnt of th dffracton ffcts n grazng angl gomtry nvolvs Frsnl spcular rflcton and rqurs th concpt of an vanscnt wav that travls paralll to th surfac and dcays xponntally as a functon of dpth nto th crystal. Th dynamcal thory s ndd to dscrb th spcular rflctvty and th vanscnt wav rlatd phnomna. Bcaus of ts surfac snstvty and adustabl probng dpth, grazng ncdnc dffracton of x-rays and nutrons has volvd nto an mportant tchnqu for matrals rsarch and charactrzaton. Brf Survy of tratur ynamcal dffracton thory of a plan wav by a prfct crystal was orgnatd by arwn (1914) and Ewald (1916), usng two vry dffrnt approachs. Snc thn th arly dvlopmnt of th dynamcal thory was prmarly focusd on th stuatons nvolvng only an ncdnt bam and on Bragg dffractd bam, th socalld two-bam cas. Prns (193) xtndd th arwn s thory to tak nto account of absorpton, and von au (1931) rformulatd Ewald s approach and formd th backbon of th modrn-day dynamcal thory. Rvws and xtnsons of th thory hav bn gvn by Zacharasn (1945), Jams (195), Kato (195), Warrn (1969), and Authr (197). A comprhnsv rvw of th Ewald-von au thory has bn provdd by Battrman and Col n thr smnal artcl n Rvw of Modrn Physcs (1964). Mor rcnt rvws can b found n Kato (1974), Cowly (1975) and Pnskr (1978). Updatd and concs summars of th two-bam dynamcal thory hav bn gvn rcntly by Authr (199, 1996). A hstorcal survy of th arly dvlopmnt of th dynamcal thory was gvn n Pnskr (1978). Contmporary topcs n dynamcal thory ar manly focusd n th followng four aras: multpl-bam dffracton, grazng ncdnc dffracton, ntrnal flds and standng wavs, and spcal x-ray optcs. Ths modrn dvlopmnts ar largly drvn by rcnt ntrsts n rapdly mrgng flds such as synchrotron radaton, x-ray crystallography, surfac scnc, and smconductor rsarch. ynamcal thory of x-rays for multpl-bam dffracton, wth two or mor Bragg rflctons xctd smultanously, was consdrd by Pnnng (1967), Ewald and no (1968). owvr, vry lttl progrss was mad untl Collla (1974) dvlopd a computatonal algorthm that mad multpl-bam x-ray dffracton smulatons mor tractabl. Rcnt ntrsts n ts applcatons to masur th phass of structur factors (Collla, 1974; Post, 1977; Chang, 198; Chapman, Yodr & Collla, 1981) hav mad multpl-bam dffracton an actv ara of rsarch n dynamcal thory and xprmnts. Approxmat thors of multpl-bam dffracton hav bn dvlopd by Jurtschk (198, 1984, 1986), or & Marthnsn (1983), ummr & Blly (1986), (1986, 1999b, ), and Thorkldsn (1987). Rvws on multpl-bam dffracton hav bn gvn by Chang (1984, 199, 1998), Collla (1995), and Wckrt & ummr (1997). Snc th ponr xprmnt by Marra, Esnbrgr, and Cho (1979), thr has bn an normous ncras n th dvlopmnt and th us of grazng ncdnc x-ray dffracton to study surfacs and ntrfacs of solds. ynamcal thory for th grazng angl gomtry was soon dvlopd (Afanasv & Mlkonyan, 1983; Alksandrov t al. 1984) and ts xprmntal vrfcatons wr gvn by Cowan t al. (1986), urbn and Gog (1989), and Jach t al. (1989). Manwhl, a sm-knmatc thory calld stortd Wav Born Approxmaton was usd by Vnyard (198) and by trch and Wagnr (1983, 1984). Ths thory was furthr dvlopd by osch t al. (1986) and Snha t al. (1988), and has bcom wdly utlzd n glancngncdnc x-ray scattrng studs of surfac and narsurfac structurs. Th thory has also bn xtndd to xplan standng-wav nhancd and non-spcular scattrng n multlayr structurs (Kortrght, 1987), and 3// 1:35 PM

4 3 to nclud phas-snstv scattrng n dffracton from bulk crystals (, 1999b). rct xprmntal proof of th x-ray standng wav ffct was frst achvd by Battrman (1964) by obsrvng x-ray fluorscnc profls whl th dffractng crystal s rotatd through a Bragg rflcton. Earlr works wr manly on locatng mpurty atoms n bulk smconductor matrals (Battrman, 1969; Golovchnko t al. 1974; Andrson t al. 1976), but mor rcnt rsarch actvts hav bn on dtrmnatons of atom locatons and dstrbutons n ovrlayrs abov crystal surfacs (Golovchnko t al. 198; Funk & Matrlk, 1985; urbn t al. 1986; Patl t al. 1987; Bdzyk t al. 1989), n synthtc multlayrs (Barb & Warburton, 1984; Kortrght & Fschr- Colbr, 1987), n long-prod ovrlayrs (Bdzyk t al. 1988; Wang t al. 199), and n lctrochmcal solutons (Bdzyk t al. 1986). Rcnt rvws on x-ray standng wavs hav bn gvn by Patl (1996) and agomarsno (1996). Rapd ncras n synchrotron-radaton-basd matrals rsarch n rcnt yars has spurrd nw dvlopmnts n x-ray optcs (Battrman & Bldrback, 1991; art, 1996). Ths s spcally tru n th aras of x-ray wav guds for producng sub-mcron szd bams (Bldrback, 1994; Fng t al. 1995; Golovchnko t al. 1997), and x-ray phas plats and polarzaton analyzrs usd for studs on magntc matrals (Golovchnko t al. 1986; Mlls, 1988; Blyakov & mtrnko, 1989; rano t al., 1991; Battrman, 199; & Fnklstn, 199; Gls t al. 1994; Yahnk t al. 1994; Shastr t al. 1995). Rcnt rvws on polarzaton x-ray optcs hav bn gvn by rano, Ishkawa and Kkuta (1995), (1996), and Malgrang (1996). An xcllnt collcton of artcls on ths and othr currnt topcs n dynamcal dffracton can b found n X- ray and Nutron ynamcal ffracton Thory and Applcatons, dtd by Authr, agomarsno and Tannr (1996). Scop of Ths Unt Gvn th wd rang of topcs n dynamcal dffracton, th man purpos of ths Unt s not to covr vry dtal but to provd radrs wth an ovrvw of th basc concpts, formalsms, and applcatons n ths fld. Spcal attnton s pad to th dffrnc btwn th mor famlar knmatc thory and th mor complx dynamcal approach. Although th basc dynamcal thory s th sam for x-rays, lctrons, and nutrons, w wll focus manly on x-rays snc many of th orgnal trmnology was foundd n x-ray dynamcal dffracton. Th formalsm for x-rays s also mor complx and thus mor complt bcaus of th vctor-fld natur of an lctromagntc wav. For rvws on dynamcal dffracton of lctrons and nutrons, w rfr th radrs to an xcllnt txtbook by Cowly (1975), th nd dton of Intrnatonal Tabls for Crystallography, vol.b, and a rcnt artcl by Schlnkr and Gugay (1996). W wll start n Scton II wth th fundamntal quatons and concpts n dynamcal dffracton thory, drvd from classcal lctrodynamcs. Thn n Scton III w mov onto th wdly-usd two-bam approxmaton ssntally followng th dscrpton of Battrman and Col (1964). Th two-bam thory dals wth only th ncdnt bam and on strongly dffractd Bragg bam, and th multpl scattrng btwn th two bams. Multpl scattrng du to othr Bragg rflctons ar gnord. Ths thory provds many basc concpts n dynamcal dffracton, and s vry usful n vsualzng th unqu physcal phnomna n dynamcal scattrng. A full multpl-bam dynamcal thory, dvlopd by Collla (1974), taks nto account all multpl scattrng ffcts and surfac gomtrs and gvs th most complt dscrpton of th dffracton procsss n a prfct crystal by x-rays, lctrons or nutrons. An outln of ths thory s summarzd n Scton IV. Also ncludd n ths Scton s an approxmat formalsm, gvn by (1986), basd on scond-ordr Born approxmatons. Ths thory taks nto account only doubl scattrng n a multpl scattrng rgm yt provds a usful pctur on th physcs of multpl-bam ntractons. Fnally, an approxmat yt mor accurat multpl-bam thory (, 1999b) basd on an xpandd dstortd-wav approxmaton s prsntd, whch can provd accurat accounts of thr-bam ntrfrnc profls n th so-calld rfrnc-bam dffracton gomtry (, 1998). In Scton V th man rsults for grazng-ncdnc dffracton ar dscrbd usng th dynamcal tratmnt. Of partcular mportanc s th concpt of vanscnt wavs and ts applcatons. Also dscrbd n Scton V s a so-calld stortd-wav Born approxmaton, whch uss th dynamcal thory to valuat spcular rflctons but trats th surfac dffracton and scattrng wthn th knmatc rgm. Ths approxmat thory s usful n structural studs of thn flms and multlayrd htrostructurs. Fnally, bcaus of ts lmtd spac, a fw topcs ar not covrd n ths artcl. On of ths omttd topcs s th thory by Takag and Taupn for dstortd prfct crystals. W rfr th radrs to th orgnal artcls (Takag, 196, 1969; Taupn, 1964) and rcnt publcatons by Bartls, ornstra and obk (1986) and by Authr (1996) II. BASIC PRINCIPES Thr ar two approachs to th dynamcal thory. On s basd on th works by arwn (1914) and by Prns 3// 1:35 PM

5 4 (193). Ths mthod frst fnds th Frsnl rflctanc and transmttanc for a sngl layr of atomc plan and thn valuats th total wav flds for a st of paralll atomc plans. Th dffractd wavs ar obtand by solvng a st of dffrnc quatons smlar to th ons usd n classcal optcs for a srs of paralll slabs or optcal fltrs. Although t had not bn wdly usd for qut a long tm du to ts computatonal complxty, th arwn s approach has gand mor attnton n rcnt yars to valuat rflctvty for multlayrs and suprlattcs (urbn & Folls, 1995), for crystal truncaton ffcts (Cattcha, 1994), and for quascrystals (Chung & urbn, 1995) Th othr approach, dvlopd by Ewald (1917) and von au (1931), trats th wav propagaton n a prodc mdum as an gnvalu problm and us th boundary condtons to obtan Bragg rflctd ntnsts. W wll follow th Ewald-von au approach snc many of th fundamntal concpts n dynamcal dffracton can b vsualzd mor naturally by ths approach and t can b asly xtndd to stuatons nvolvng mor than two bams. Th mathmatcal forms for th dffractd ntnsts from gnral absorbng crystals appar to b rathr complcatd n th arly ltratur of dynamcal thory (for two-bams). Th man rason for ths complcatd forms s th ncssty to sparat out th ral and magnary parts n dalng wth complx wavvctors and wav fld ampltuds bfor th tms of computrs and powrful calculators. In today s world ths complcatd quatons ar not ncssary and numrcal calculatons wth complx varabls can b asly prformd on a modrn computr. Thrfor n ths artcl all fnal ntnsty quatons ar gvn n such compact forms that nvolv complx numbrs and n author s vw ar bst sutd for today s computr calculatons. Ths smplr forms also allow radrs to gan physcal nsghts rathr than b ovrwhlmd by tdous mathmatcal notatons. Fundamntal Equatons Th startng pont n th Ewald-von au approach of th dynamcal thory s that th dlctrc functon ε(r) n a crystalln matral s a prodc functon n spac and thrfor can b xpandd n a Fourr srs: (a) Ewald sphr (b) k k p s k crystal n O p s r ε(r) = ε + δε(r), wthδε ( r) = Γ F, (1) whr Γ = r λ /(πv c ), r = Å s th classcal radus of an lctron, λ th x-ray wavlngth, V c th unt cll volum, and ΓF s th coffcnt of th Fourr componnt wth F bng th structur factor. All of th Fourr coffcnts ar on th ordr 1 5 to 1 6 or smallr at x-ray wavlngths, δε(r) << ε = 1, and th dlctrc functon s only slghtly lss than unty. W furthr assum that a monochromatc plan wav s ncdnt on a crystal, and th dlctrc rspons s of th sam wav frquncy (lastc rspons). From Maxwll s quatons and nglctng th magntc ntractons (µ=1), w obtan th followng quaton for th lctrc fld E and th dsplacmnt vctor : θ θ c θ K ( + ) = ( ε E) k, whr k s th wavvctor of th monochromatc wav n vacuum, k = k =π/λ. For tratmnt nvolvng magntc ntractons w rfr to th publcaton by urbn (1987). Assumng an sotropc rlaton btwn (r) and E(r), (r)=ε(r)e(r), and δε(r)<<ε, w hav Fgur 1: (a) Ewald sphr constructon n knmatc thory and polarzaton vctors of th ncdnt and th dffractd bams. (b) sprson surfac n dynamcal thory for a on-bam cas and boundary condtons for total xtrnal rflcton. ( + ) = ( δε) k. () W now us th prodc condton Eq.(1) and substtut th wav fld n Eq.() by a srs of Bloch wavs wth wavvctors K =K +, 3// 1:35 PM

6 5 ( r) = K r, whr s a rcprocal spac vctor of th crystal. For vry Fourr componnt (Bloch wav) w arrv at th followng quaton: [( 1 Γ k K] = Γ F-GK ( K G ) F ), (3) G whr th trms nvolvng Γ hav bn nglctd and K ar st to zro bcaus of th transvrs wav natur of th lctromagntc radaton.. Eq.(3) forms a st of fundamntal quatons for th dynamcal thory of x-ray dffracton. Smlar quatons for lctrons and nutrons can b found n th ltratur (.g. Cowly, 1975). sprson Surfac A soluton to th gnvalu quaton Eq.(3) gvs rs to all th possbl wavvctors K and wav fld ampltud ratos nsd a dffractng crystal. Th loc of th possbl wavvctors form a multpl-shts thrdmnsonal surfac n rcprocal spac and ths surfac s calld th dsprson surfac, as gvn by Ewald (1917). Th ntroducton of th dsprson surfac s th most sgnfcant dffrnc btwn th knmatc thory and th dynamcal thory. r nstad of a sngl Ewald sphr (Fg.1a), w hav a contnous dstrbuton of Ewald sphrs wth thr cntrs locatd on th dsprson surfac, gvng rs to all possbl travlng wavvctors nsd th crystal. As an xampl, w assum that th crystal orntaton s far from any Bragg rflctons, and thus only on bam, th ncdnt bam K, would xst n th crystal. For ths on-bam cas, Eq.(3) bcoms: Thus w hav K [ 1 Γ ) k K ] ( = F. 1/ ( + ΓF ) = k ( 1 Γ ) k 1 F =, (4) whch shows that th wavvctor K nsd th crystal s slghtly shortr than that n vacuum as a rsult of th avrag ndx of rfracton, n =1 ΓF /, whr F s th ral part of F and s rlatd to th avrag dnsty ρ by ρ π F = Γ. (5a) λ r In th cas of absorbng crystals, K and F ar complx varabls and th magnary part F of F s rlatd to th avrag lnar absorpton coffcnt µ by " " µ = kγf = πγf λ. (5b) Eq.(4) shows that th dsprson surfac n th onbam cas s a rfracton-corrctd sphr cntrd around th orgn n rcprocal spac, as shown n Fg.1(b). Boundary Condtons Onc Eq.(3) s solvd and all possbl wavs nsd th crystal ar obtand, th ncssary connctons btwn th wav flds nsd and outsd th crystal s mad through th boundary condtons. Thr ar two typs of boundary condtons n classcal lctrodynamcs (Jackson, 1975). On stats that th tangntal componnts of th wavvctors hav to b qual on both sds of an ntrfac ( Snll s law): k t = K t. (6) Throughout ths artcl w us th convnton that outsd vacuum wavvctors ar dnotd by k and ntrnal wavvctors ar dnotd by K, and th subscrpt t stands for th tangntal componnt of th vctor. To llustrat ths pont, w agan consdr th smpl on-bam cas as shown n Fg.1(b). Suppos that an x- ray bam k wth an angl θ s ncdnt on a surfac wth n bng ts surfac normal. To locat th propr ntrnal wavvctor K, w follow along n to fnd ts ntrscton wth th dsprson surfac, n ths cas th sphr wth ts radus dfnd by Eq.(4). owvr, w s mmdatly that ths s possbl only f θ s gratr than a crtan ncdnt angl θ c, whch s th crtcal angl of th matral. From Fg.1(b) w can asly obtan that cosθ c =K /k, or for small angls, θ c =(ΓF ) 1/. Blow θ c no travlng wav solutons ar possbl and thus total xtrnal rflcton occurs. Th scond st of boundary condtons stats that th tangntal componnts of th lctrc and magntc fld vctors, E and = k ˆ E, ( kˆ s a unt vctor along th propagaton drcton), ar contnuous across th boundary. In dynamcal thory ltratur, th gnquatons for dsprson surfacs ar xprssd n trms of thr lctrc fld vctor E or lctrc dsplacmnt vctor. Ths two chocs ar quvalnt snc n both cass a small longtudnal componnt on th ordr of ΓF n th E-fld vctor s gnord for ts ncluson only contrbuts a trm of Γ n th dsprson quaton. Thus E and ar ntrchangabl undr ths assumpton and th boundary condtons can b xprssd as th followng: n t n t out t =, (7a) out k ˆ = Kˆ. (7b) ( ) ( ) t 3// 1:35 PM

7 6 In dynamcal dffracton, th boundary condton Eq.6 or th Snll s law slcts whch ponts ar xctd on th dsprson surfac or whch wavs actually xst nsd th crystal, for a gvn ncdnt condton. Th condtons Eqs.7(a) & 7(b) on th fld vctors ar thn usd to valuat th actual ntrnal fld ampltuds and th dffractd wav ntnsts outsd th crystal. pndng on numbr of bams ncludd n th dsprson quaton (3) and th dffracton gomtry of th crystal, dynamcal thory covrs a wd-rang of spcfc topcs. In crtan cass, th xstnc of som bams can b prdtrmnd basd on th physcal law of nrgy consrvaton. In ths cass only Eq.(7a) s ndd for th fld boundary condton. Such s th cas of convntonal two-bam dffracton as dscussd n th nxt Scton. owvr, both sts of condtons n Eq.(7) ar ndd for gnral multpl-bam cass and for grazng angl gomtry. Intrnal Flds On of th mportant applcatons of th dynamcal thory s to valuat th wavflds nsd th dffractng crystal, n addton to th xtrnal dffractd ntnsts. pndng on th dffracton gomtry, an ntrnal fld can b a prodc standng wav as n th cas of a Bragg dffracton, or an xponntally dcayd vanscnt wav as n th cas of a spcular rflcton, or a combnaton of th two. Although no dtctors pr s can b put nsd a crystal, th ntrnal fld ffcts can b obsrvd through on of th followng two ways. Th frst s to dtct scondary sgnals producd by an ntrnal fld, whch nclud x-ray fluorscnc, Augr lctrons, and photolctrons. Ths nlastc scondary sgnals ar drctly proportonal to th ntrnal fld ntnsty and ar ncohrnt wth rspct to th ntrnal fld. Exampls of ths ffct nclud th standard x-ray standng wav tchnqus, and dpth-snstv x-ray fluorscnc masurmnts undr total xtrnal rflcton. Th othr way s to masur th lastc scattrng of an ntrnal fld. In most cass, ncludng th standng wav cas, an ntrnal fld s a travlng wav along a crtan drcton, and thrfor can b scattrd by atoms nsd th crystal. Ths s a cohrnt procss and th scattrng contrbutons ar addd on th lvl of ampltuds nstad of ntnsts. An xampl of ths ffct s th dffus scattrng of an vanscnt wav, n studs of surfac or nar-surfac structurs. III. TWO-BEAM IFFRACTION In th two-bam approxmaton, w assum only on Bragg dffractd wav K s mportant n th crystal, n addton to th ncdnt wav K. Thn Eq.(3) rducs to th followng two coupld vctor quatons: [ ] ( ) ( 1 ΓF ) k K = ΓF K K [ ( 1 ΓF ) k K ] = ΓF K ( K ). (8) Th wavvctors K and K dfn a plan whch s usually calld th scattrng plan. Usng a coordnat systm shown n Fgur 1(a), w can dcompos th wavfld ampltuds nto s and p polarzaton drctons, and th quatons for th two polarzaton stats dcoupl and can b solvd sparatly: [( 1 ΓF ) k K ] σ, π kγf kγf Pσ, π + [( 1 ΓF ) k K ] P σ, π σ, π =, (9) = whr P=s s =1 for s polarzaton and P=p p = cos(θ Β ) for p polarzaton, wth θ Β bng th Bragg angl. To sk nontrval solutons, w st th dtrmnant of Eq.(9) to zro and solv for K : ( 1 ΓF ) k k K kγf P = ΓF P ( 1 ΓF ) k K, (1) whr K s rlatd to K through th Bragg s law, K = K +. Soluton of Eq.(1) dfns th possbl wavvctors n th crystal and gvs rs to th dsprson surfac n th two-bam cas. Proprts of sprson Surfac To vsualz what th dsprson surfac looks lk n th two-bam cas, w dfn two paramtrs ξ and ξ, as dscrbd n Jams (195) and Battrman & Col (1964): ξ ξ [ K (1 ΓF ) k ] k= K k (1 ΓF ) [ K (1 ΓF ) k ] k= K k (1 ΓF ) Ths paramtrs rprsnt th dvatons of th wavvctors nsd th crystal from th avrag rfracton-corrctd valus gvn by Eq.(4). It also shows that n gnral th rfracton corrctons for th ntrnal ncdnt and dffractd wavs ar dffrnt. Usng ths dvaton paramtrs, th dsprson quaton Eq.(1) bcoms 1 ξ = k 4 Γ P F F ξ. (11) (1) yprbolod shts: Snc th rght-hand sd s a constant for a gvn Bragg rflcton, th dsprson. 3// 1:35 PM

8 7 surfac gvn by Eq.(11) rprsnts two shts of hyprbolods n rcprocal spac, for ach polarzaton stat P, as shown n Fgur (a). Th hyprbolods hav thr damtr pont, Q, locatd around what would b th cntr of th Ewald sphr (dtrmnd by Bragg s law) and approach asymptotcally th two sphrs cntrd at th orgn O and at th rcprocal nod, wth a rfracton-corrctd radus, k (1 ΓF /). Th two corrspondng sphrs n vacuum (outsd crystal) ar n (b) P (a) k Q α σπ k K K ξ Η P k ξ Q O k ΓF / β πσ K K Fgur : sprson surfac n th two-bam cas. (a) Ovrvw. (b) Clos-up vw around th ntrscton rgon. also shown and thr ntrscton pont s usually calld th au pont,. Th dsprson surfac branchs closr to th au pont ar calld th α branchs (ασ, απ), and thos furthr from th au pont ar calld th β branchs (βσ, βπ). Snc th squar root valu of th rght-hand sd constant n Eq.(11) s much lss than k, th gap at th damtr pont s on th ordr of 1-5 compard to th radus of th sphrs. Thrfor, th sphrs can b vwd ssntally as plans n th vcnty of th damtr pont, as llustratd n Fg.(b). owvr, th curvaturs hav to b consdrd whn th Bragg rflcton s n th glancng angl gomtry (s Scton V). () Wav fld ampltud ratos: In addton to th wavvctors, th gnvalu quaton (9) also provds th rato of th wav fld ampltuds nsd th crystal for ach polarzaton. In trms of ξ and ξ, th ampltud rato s gvn by ξ kγpf = kγpf = ξ. (1) Agan th actual rato n th crystal dpnds ntrly on th t ponts slctd by th boundary condtons. Around th damtr pont ξ and ξ hav smlar lngths and thus th fld ampltuds and ar comparabl. Away from th xact Bragg condton, only on of ξ and ξ has an apprcabl sz and thus thr or domnats accordng to thr asymtotc sphrs. (3) Boundary condtons and Snll s law: To llustrat how t ponts ar slctd by Snll s law n th two-bam cas, w consdr th stuaton n Fg.(b) whr a crystal surfac s ndcatd by a shadd ln. W start wth an ncdnt condton corrspondng to an ncdnt vacuum wavvctor k at pont P. W thn construct a surfac normal passng through P and ntrsctng four t ponts on th dsprson surfac. Bcaus of th Snll s law, th wavflds assocatd wth ths four ponts ar th only prmttd wavs nsd th crystal. Thr ar four wavs for ach rcprocal nod, O or, altogthr a total of ght wavs may xst nsd th crystal n th two-bam cas. To fnd th xtrnal dffractd bam, w follow th sam surfac normal to th ntrscton pont P, and th corrspondng wavvctor connctng P to th rcprocal nod would b th dffractd bam that w can masur wth a dtctor outsd th crystal. pndng on whthr or not a surfac normal ntrcpts both α and β branchs at th sam ncdnt condton, a dffracton gomtry s calld thr th au transmsson or th Bragg rflcton cas. In trms of th drcton cosns γ and γ of th xtrnal ncdnt and dffractd wavvctors, k and k, wth rspct to th surfac normal n, t s usful to dfn a paramtr b: b γ γ k n k n, 3// 1:35 PM

9 8 whr b> corrsponds to th au cas and b< th Bragg cas. Th cass wth b=±1 ar calld th symmtrc au or Bragg cass, and for that rason b s oftn calld th asymmtry factor. (4) Poyntng s vctor and nrgy flow: Th quston about th nrgy flow drctons n dynamcal dffracton s of fundamntal ntrsts to scntsts who us x-ray topography to study dfcts n prfct crystals. Enrgy flow of an lctromagntc wav s dtrmnd by ts tm-avragd Poyntng vctor, dfnd as c S= 8π c, 8π ( E *) = Kˆ whr c s th spd of lght, ˆ K s a unt vctor along th propagaton drcton, and trms on th ordr of Γ or hghr ar gnord. Th total Poyntng vctor S T at ach t pont on ach branch of th dsprson surfacs s th vctor sum of thos for th O-bam and th -bam: ( Kˆ + ˆ ) c ST = K 8π To fnd th drcton of S T, w consdr th surfac normal v of th dsprson branch, whch s along th drcton of th gradnt of dsprson quaton Eq.(11): v = ( ξ ξ Kˆ ) = ξ ξ + Kˆ + ξ S T, ξ ξ = ξ. Kˆ ξ + ξ Kˆ whr w hav usd Eq.(1) and assumd a nglgbl absorpton ( F = F ). Thus w conclud that S T s paralll to v, th normal to th dsprson surfac. In anothr word, th total nrgy flow at a gvn t pont s always normal to th local dsprson surfac. Ths mportant thorm s vald n gnral and was frst provd by Kato (196). It follows that th nrgy flow nsd th crystal s paralll to th atomc plans at th full xctaton condton,.. th damtr ponts of th hyprbolods. Spcal ynamcal Effcts Thr ar sgnfcant dffrncs n th physcal dffracton procsss btwn th knmatc thory and th dynamcal thory. Th most strkng obsrvabl rsults from th dynamcal thory ar Pndllösung frngs, anomalous transmsson, fnt rflcton wdth for sm-nfnt crystals, x-ray standng wavs, and x-ray brfrngnc. Wth th ad of dsprson surfac shown n Fg., ths ffcts can b xpland wthout formally solvng th mathmatcal quatons. (1) Pndllösung: In a au cas, th α and β t ponts across th damtr gap of th hyprbolc dsprson surfacs ar xctd smultanously at a gvn ncdnt condton. Th two sts of travllng wavs assocatd wth th two branchs can ntrfr wth ach othr and caus oscllatons n th dffractd ntnsty as th thcknss of th crystal changs on th ordr of π/ K, whr K s smply th gap at th damtr pont. Ths ntnsty oscllatons ar trmd Pndllösung frngs and th quantty π/ K s calld th Pndllösung prod. From th gomtry shown n Fg.(b), t s straghtforward to show that th damtr gap s gvn by K = k θ, Γ P F F cos whr θ Β s th ntrnal Bragg angl. As an xampl, at 1 kv, for S (111) rflcton, K= Å -1, and thus th Pndllösung prod s π/ K =3 µm. Pndllösung ntrfrnc s a unqu dffracton phnomnon for th au gomtry. Both th dffractd wav (-bam) and th forward-dffractd wav (Obam) ar affctd by ths ffct. Th ntnsty oscllatons for ths two bams ar 18 out of phas to ach othr, cratng th ffct of nrgy flow swappng back and forth btwn th two drctons as a functon of dpth nto th crystal surfac. For mor dtald dscussons of Pndllösung frngs w rfr to a rvw by Kato (1974). W should pont out that Pndllösung frngs ar ntrly dffrnt n orgn from ntrfrnc frngs du to crystal thcknss. Th thcknss frngs ar oftn obsrvd n rflctvty masurmnts on thn flm matrals and can b mostly accountd for by fnt sz ffct n Fraunhofr dffracton. Th prod of thcknss frngs dpnds only on thcknss, not on th strngth of th rflcton, whl th Pndllösung prod dpnds only on th rflcton strngth not on crystal thcknss. () Anomalous transmsson: Th four wavs slctd by t ponts n th au cas hav dffrnt ffctv absorpton coffcnts. Ths can b undrstood qualtatvly from th locatons of th four dsprson surfac branchs rlatv to th vacuum au pont and to th avrag rfracton-corrctd pont Q. Th β branchs ar furthr from and ar on th morrfractv sd of Q. Thrfor th wavs assocatd wth th β branchs hav largr-than-avrag rfracton and absorpton. Th α branchs, on th othr hand, ar locatd closr to and on th lss-rfractv sd of Q, and thrfor th wavs on th α branchs hav lss-thanavrag rfracton and absorpton. For a rlatvly thck crystal n th au dffracton gomtry, ffctvly th α wavs would b abl to pass through th thcknss of th crystal asr than an avrag wav would. What ths mpls s that f no ntnsty s obsrvd n th transmttd bam at off-bragg condtons, an B 3// 1:35 PM

10 9 anomalously transmttd ntns bam can actually appar whn th crystal s st to a strong Bragg condton. Ths phnomnon s calld th anomalous transmsson and was frst obsrvd by Borrmann (195) and oftn s also calld th Borrmann ffct. If th au crystal s suffcntly thck, thn vn th απ wav may b absorbd and only th ασ wav would rman. In ths cas th au dffractng crystal can b usd as lnar polarzr snc only th s-polarzd x-rays would b transmttd through th crystal. (3) arwn wdth: In a Bragg rflcton gomtry, all th xctd t ponts l on th sam branch of th dsprson surfac at a gvn ncdnt angl. Furthrmor, no t ponts can b xctd at th cntr of a Bragg rflcton, whr a gap xsts at th damtr pont of th dsprson surfacs. Th gap ndcats that no ntrnal travlng wavs xst at th xact Bragg condton and total xtrnal rflcton s th only outlt of th ncdnt nrgy f absorpton s gnord. In fact, th sz of th gap dtrmns th rang of ncdnt angls at whch th total rflcton would occur and ths angular wdth s usually calld th arwn wdth of a Bragg rflcton n prfct crystals. In th cas of symmtrc Bragg gomtry, t s asy to s from Fg. that th full arwn wdth s w = k K Γ P = F snθb sn θ B F. (13) Typcal valus for w ar on th ordr of a fw arcsconds. Th xstnc of a fnt rflcton wdth w vn for a sm-nfnt crystal may sm to contradct th mathmatcal thory of Fourr transforms whch would gv rs to a zro rflcton wdth f th crystal sz s nfnt. In fact ths s not th cas. A mor carful xamnaton of th stuaton shows that bcaus of th xtncton th ncdnt bam would nvr b abl to s th whol nfnt crystal. Thus th fnt arwn wdth s a drct rsult of th xtncton ffct n dynamcal thory and s ndd to consrv th total nrgy n th physcal systm. (4) X-ray standng wavs (XSW): Anothr mportant ffct n dynamcal dffracton s th x-ray standng wavs (Battrman, 1964). Insd a dffractng crystal, th total wav fld ntnsty s th cohrnt sum of th O- bam and th -bam and s gvn by (s polarzaton): -K r -K r - r = + = 1+. (14) Eq.(14) rprsnts a standng wav fld wth a spatal prod of π/ whch s smply th d-spacng of th Bragg rflcton. Th fld ampltud rato / has wll-dfnd phass at α and β branchs of th dsprson surfac. Accordng to Eq.(1) and Fg., w s that th phas of / s π+α at th α branch snc ξ s postv and α at th β branch snc ξ s ngatv, whr α s th phas of th structur factor F and can b st to zro by a propr choc of ral spac orgn. Thus th α mod standng wav has ts nods on th atomc plans and th β mod standng wav has ts antnods on th atomc plans. In au transmsson gomtry, both th α and th β mods ar xctd smultanously n th crystal. owvr, th β mod standng wav s attnuatd mor strongly bcaus ts pak fld concd wth th atomc plans. Ths s th physcal orgn of th Borrmann anomalous absorpton ffct. Th standng wavs also xst n th Bragg gomtry. Bcaus of ts mor rcnt applcatons n matrals studs, w wll dvot a latr sgmnt to dscuss ths n som mor dtals. (5) X-ray brfrngnc: Bng abl to produc and to analyz a gnrally polarzd lctromagntc wav has long bnftd scntsts and rsarchrs n th fld of vsbl lght optcs and n studyng optcal proprts of matrals. In th x-ray rgm, howvr, such ablts hav bn vry lmtd bcaus of th wak ntracton of x-rays wth mattr, spcally for producton and analyss of crcularly-polarzd x-ray bams. Th stuaton has changd sgnfcantly n rcnt yars. Th growng ntrsts n studyng magntc and ansotropc lctronc matrals by x-ray scattrng and spctroscopc tchnqus hav ntatd many nw dvlopmnts n both th producton and th analyss of spcally polarzd x- rays. Th routnly avalabl hgh-brghtnss synchrotron radaton sourcs can now provd naturally collmatd x- rays whch can b asly manpulatd by spcal x-ray optcs to gnrat x-ray bams wth ts polarzaton tunabl from lnar to crcular. Such optcs ar usually calld x-ray phas plats or phas rtardrs. Th prncpls of most x-ray phas plats ar basd on th lnar brfrngnc ffct nar a Bragg rflcton n prfct or narly prfct crystals du to dynamcal dffracton (art, 1978; Blyakov & mtrnko, 1989). As llustratd n Fg., clos to a Bragg rflcton, th lngths of th wavvctors for th s and th p polarzatons ar slghtly dffrnt. Th dffrnc can caus a phas shft btwn th s and th p wav flds to accumulat through th crystal thcknss t: = ( K σ K π)t. Whn th phas shft rachs ±9, crcularly polarzd radaton s gnratd, and such a dvc s calld a quartr-wav phas plat or rtardr (Mlls, 1988; rano t al. 1991; Gls t al. 1994). In addton to ths transmsson-typ phas rtardrs, a rflcton-typ phas plat has also bn proposd and studd (Brummr t al. 1984; Battrman, 199; Shastr t al. 1995), whch has th advantag of bng thcknss- 3// 1:35 PM

11 1 ndpndnt. owvr, t has bn dmonstratd that th Bragg-transmsson-typ phas rtardrs ar mor robust to ncdnt bam dvrgncs and thus ar vry practcal x-ray crcular polarzrs. Thy hav bn usd for masurmnts of magntc dchrosm n hard prmannt magnts and othr magntc matrals (Gls t al. 1994; ang t al. 1995). Rcnt rvws on x-ray polarzrs and phas plats can b found n artcls by art (1991), rano, Ishkawa & Kkuta (1995), (1996), and Malgrang (1996). Soluton of sprson Equaton So far w hav focusd our dscussons to th physcal ffcts that xst n dynamcal dffracton from prfct crystals and hav trd to avod th mathmatcal dtals of th solutons to th dsprson quaton, Eq.(7) or (8). As w hav shown, consdrabl physcal nsght to th dffracton procsss can b gand wthout gong nto mathmatcal dtals. To obtan th dffractd ntnsts n dynamcal thory, howvr, th mathmatcal solutons ar unavodabl. In ths sgmnt w wll summarz ths rsults. W wll kp th formula n a gnral complx form so that absorpton ffcts ar automatcally takn nto account. Th ky to solvng th dsprson quatons Eq.(1) or (11) s to ralz that th ntrnal ncdnt bam K can only dfr from th vacuum ncdnt bam k by a small componnt K n along th surfac normal drcton of th (a) (b) n r = n r =t n r = t ncdnt surfac, whch n turn s lnarly rlatd to ξ or ξ. Th fnal xprsson rducs to a quadratc quaton for ξ or ξ, and solvng for ξ or ξ alon rsults n th followng (Battrman & Col, 1964): 1/ ( η + b / ) ξ = 1 Γ k P b F F- η ± b, (15) whr η s th rducd dvaton paramtr normalzd to th arwn s wdth: b η w b ( θ θ ) θ =θ θ B s th angular dvaton from th vacuum Bragg angl θ B, and θ s th rfracton corrcton: ΓF (1 1/ b) θ. snθ Th dual sgns n Eq.(15) corrspond to th α and β branchs of th dsprson surfac. In th Bragg cas, b< so th corrcton θ s always postv,.. th θ valu at th cntr of a rflcton s always slghtly largr than θ B gvn by th knmatc thory. In th au cas th sgn of θ dpnds on whthr b>1 or b<1. In th cas of absorbng crystals, both η and θ can b complx and th drctonal proprts ar rprsntd by th ral parts of ths complx varabls whl thr magnary parts ar rlatd to th absorpton gvn by F and w. Substtutng Eq.(15) nto Eq.(1) ylds th wav fld ampltud rato nsd th crystal as a functon of η: B, [ ( + b/ b ) ] 1/ P = - b F F η η. (16) P ffractd Intnsts W now mploy th boundary condtons to valuat th dffractd ntnsts. (1) Boundary condtons: In th au transmsson cas [Fg.3(a)], assumng a plan wav wth an nfnt cross scton, th fld boundary condtons ar gvn by th followng quatons: Entranc surfac: = α + β, (17) = α + β Fgur 3: Boundary condtons for th wav flds outsd th crystal n (a) au cas and (b) Bragg cas. Ext surfac: = = K α α K α α r r + + K β β K β r β r. (18) 3// 1:35 PM

12 11 In th Bragg rflcton cas [Fg.3(b)], th fld boundary condton s gvn by Entranc surfac: Back surfac: = = = α = α α α K K α α + + r r β β + + β β, (19) K K β β r r. () In thr cas, thr ar sx unknowns, α, β, α, β,,, and thr pars of quatons, (16), (17), (18), or (16), (19), (), for ach polarzaton stat. Our goal s to xprss th dffractd wavs outsd th crystal as a functon of th ncdnt wav. () Intnsts n th au cas: In th au transmsson cas, w obtan, apart from an nsgnfcant phas factor, that µ t 1 1 sn 1 + A η 4 + γ γ bf =, F η + 1 whr A s th ffctv thcknss (complx) that rlats to ral thcknss t by (Zacharasn, 1945) πγ P t F F A. λ γ γ Th ral part of A s ssntally th rato of th crystal thcknss to th Pndllösung prod. A quantty oftn masurd n xprmnts s th total powr P n th dffractd bam, whch quals to th dffractd ntnsty multpld by th cross-scton ara of th bam. Th powr rato P /P of th dffractd bam to th ncdnt bam s gvn by th ntnsty rato, /, multpld by th ara rato, 1/ b, of th bam cross-sctons: P P 1 = b = sn µ t γ γ F A F η η + 1. (1) + 1 A plot of P /P vrsus η s usually calld th rockng curv. Kpng n mnd that η can b a complx varabl du ssntally to F, Eq.(1) s a gnral xprsson that s vald for both non-absorbng and absorbng crystals. A fw xampls of th rockng curvs n th au cas for non-absorbng crystals ar shown n Fg.4(a). For thck non-absorbng crystals, A s larg (A>>1) so th sn oscllatons tnd to avrag to a valu qual to 1/. Thus Eq.(1) rducs to a smpl orntzan shap: P P 1 =. ( η + 1) For thn non-absorbng crystals, A<<1, w rwrt Eq.(1) n th followng form: P P sn A η + 1 sn( Aη) = η + 1 η. Fgur 4: ffractd ntnsty P /P n (a) nonabsorbng au cas, and (b) absorbng Bragg cas, for svral ffctv thcknsss. Th Bragg rflcton n (b) s for GaAs () at λ =1.48 A. Th abov approxmaton can ralzd by xpandng th quantts n th squar brackts on both sds to thrd powr and nglctng th A 3 trm snc A<<1. W s that n ths thn crystal lmt, th dynamcal thory gvs th sam rsult as th knmatc thory. Th condton A<<1 can b rstatd as th crystal thcknss t s much lss than th Pndllösung prod. (3) Intnsts n th Bragg cas: In th Bragg rflcton cas, w obtan that th dffractd wav fld s gvn by: bf 1 =. F + 1cot η η A η 1 3// 1:35 PM

13 1 Th powr rato P /P of th dffractd bam to th ncdnt, oftn calld th Bragg rflctvty, s P P F = η η F -1. (4) P P F = F η+ η 1 1cot A η 1. () In th cas of thck crystals, A>>1, Eq.(19) rducs to P P F = F 1 η± η 1. (3) Th choc of th sgns s such that a smallr valu of P /P s rtand. On th othr hand, for sm-nfnt crystals (A>>1), w can go back to th boundary condtons Eqs.(19) and () and gnor th back surfac all togthr. If w thn apply th argumnt that only on of th two t ponts on ach branch of th dsprson surfac s physcally fasbl n th Bragg cas bcaus of th nrgy flow consrvaton, w arrv at th followng smpl boundary condton: =, = Usng th abov quaton and Eq.(16), th dffractd powr can b xprssd by. Agan th sgn n front of th squar root s chosn so that P /P s lss than unty. Th rsult s obvously dntcal to Eq.(3). Far away from th Bragg condton, η>>1, Eq.(3) shows that th rflctd powr dcrass as 1/η. Ths asymptotc form rprsnts th tals of a Bragg rflcton (Andrws & Cowly, 1985) whch ar also calld th crystal truncaton rod n th knmatc thory (Robnson, 1986). In rcprocal spac th drcton of th tals s along th surfac normal snc th dffractd wavvctor can only dfr from th Bragg condton by a componnt normal to th surfac or ntrfac. Mor dtald dscussons of th crystal truncaton rods n dynamcal thory can b found n Collla (1991), Catcha (1993, 1994) and urbn (1995). Exampls of th rflctvty curvs Eq.() for a GaAs crystal wth dffrnt thcknsss n th symmtrc Bragg cas ar shown n Fg.4(b). Th oscllatons n th tals ar ntrly du to th thcknss of th crystal. Ths modulatons ar routnly obsrvd n x-ray dffracton profls from smconductor thn flms on substrats and can b usd to dtrmn th thn flm thcknss vry accuratly (Fwstr, 1996). (4) Intgratd ntnsts: Th ntgratd ntnsty R η n th rducd η unts s gvn by ntgratng th dffractd powr rato P /P ovr th ntr η rang. For non-absorbng crystals n th au cas, n th lmtng cass of A<<1 and A>>1, R η can b calculatd analytcally as (Zacharasn, 1945) R η = P <<, 1 πa A dη = P π. A >> 1 For ntrmdat valus of A or for absorbng crystals, th ntgral can only b calculatd numrcally. A gnral plot of R η vrsus A n th non-absorbng cas s shown n Fg.5 as th dottd ln. For non-absorbng crystals n th Bragg cas, Eq.() can b ntgratd analytcally (arwn, 19) to yld R η = P <<, 1 πa A dη = π tanh( A) = P π. A >> 1 Fgur 5: Comparson of ntgratd ntnsts n th au cas and th Bragg cas wth th knmatc thory. A plot of th ntgratd powr n th symmtrc Bragg cas s shown n Fg.5 as th sold curv. Both curvs n Fg.5 show a lnar bhavor for small A, whch s consstnt wth th knmatc thory. Usng th dfntons of η and A, w obtan that th ntgratd powr R θ ovr th ncdnt angl θ n th lmt of A<<1 s gvn by 3// 1:35 PM

14 13 R θ = P P w dθ = R η c 3 π r λ P F = wa = V snθ B t, whch s dntcal to th ntgratd ntnsty n th knmatc thory for a small crystal (Warrn, 1969). Thus n som sns th knmatc thory s a lmtng form of th dynamcal thory, and th dparturs of th ntgratd ntnsts at largr A valus (Fg.5) s smply th ffct of prmary xtncton. In th thck crystal lmt A>>1, th θ-ntgratd ntnsty R θ n both au and Bragg cass s lnar n F. Ths lnar rathr than quadratc dpndnc on F s a dstnct and charactrstc rsult of dynamcal dffracton. wavs nsd th dffractng crystal. In th Bragg rflcton gomtry, as th ncdnt angl ncrass through th full Bragg rflcton, th slctd t ponts shft from α branch to β branch. Thrfor th nods of th standng wav shft from on th atomc plans (r=) to n-btwn th atomc plans (r=d/) and th corrspondng antnods shft from n-btwn to on th atomc plans. For a sm-nfnt crystal n th symmtrc Bragg cas and s-polarzaton, th standng wav ntnsty can b wrttn as, usng Eqs.(14), (16) and (4): P P ( ν + α - r) I = 1+, (5) Standng Wavs As w dscussd arlr, nar or at a Bragg rflcton, th wav fld ampltuds, Eq.(14), rprsnt sandng Fgur 6: XSW ntnsty and phas as a functon of rducd angular paramtr η, along wth rflctvty curv, calculatd for a sm-nfnt GaAs () rflcton at 1.48 A. whr ν s th th phas of η η -1 and α s th phas of th structur factor F, assumng absorpton s nglgbl. If w dfn th dffracton plan by choosng an orgn such that α s zro, thn th standng wav ntnsty as a functon of η s dtrmnd by th phas factor r wth rspct to th orgn chosn and th d- spacng of th Bragg rflcton (Bdzyk & Matrlk, 1985). Typcal standng wav ntnsty profls gvn by Eq.(5) ar shown n Fg.6. Also shown n Fg.6 ar th phas varabl ν and th corrspondng rflctvty curv. A x-ray standng wav profl can b obsrvd by masurng th x-ray fluorscnc from atoms mbddd n th crystal structur snc th fluorscnc sgnal s drctly proportonal to th ntrnal wav fld ntnsty at th atom poston (Battrman, 1964). By analyzng th shap of a fluorscnc profl th poston of th fluorscng atom wth rspct to th dffracton plan can b dtrmnd. A dtald dscusson of nodal plan poston shfts of th standng wavs n gnral absorbng crystals has bn gvn by Authr (1986). Th standng wav tchnqu has bn usd to dtrmn forgn atom postons n bulk matrals (Battrman, 1969; Golovchnko t al. 1974; agomarsno t al., 1984; Kovalchuk & Kohn, 1986). Most rcnt applcatons of th XSW tchnqu hav bn th dtrmnaton of forgn atom postons, surfac rlaxatons and dsordrs at crystal surfacs and ntrfacs (urbn t al 1986; Zgnhagn t al 1988; Bdzyk t al. 1989; Martns t al. 199; Fonts t al. 1993; Frankln t al. 1995; yman & Bdzyk, 1997). By masurng standng wav pattrns for two or mor rflctons (thr sparatly or smultanously) along dffrnt crystallographc axs, atomc postons can b trangulzd n spac (Brman t al. 1987; Grsr & Matrlk, 1986). Mor dtals of th XSW thcnqu can b found n rcnt rvws gvn by Patl (1996) and agomarsno (1996). Th formaton of x-ray standng wavs s not rstrctd to wd-angl Bragg rflctons n prfct 3// 1:35 PM

15 14 crystals. Bdzyk t al. (1988) has xtndd th tchnqu to th rgm of spcular rflctons from mrror surfacs, n whch cas both th phas and th prod of th standng wavs vary wth th ncdnt angl. Standng wavs hav also bn usd to study th spatal dstrbuton of atomc spcs n mosac crystals (urbn, 1988) and n quascrystals (Chung & urbn, 1995; Jack t al, 1997). u to a substantal (although mprfct) standng wav formaton, anomalous transmsson has bn obsrvd on th strongst dffracton paks n narly prfct quascrystals (Kyca t al., 1993). IV. MUTIPE-BEAM IFFRACTION So far w hav rstrctd our dscusson to dffracton cass n whch only th ncdnt bam and on Bragg dffractd bam ar prsnt. Thr ar xprmntal stuatons, howvr, n whch mor than on dffractd bams may b sgnfcant and thrfor th two-bam approxmaton s no longr vald. Ths stuatons nvolv multpl-bam dffracton and ar dalt wth n ths Scton. Basc Concpts Multpl-bam dffracton occurs whn svral sts of atomc plans satsfy th Bragg s laws smultanously. A convnnt way to ralz ths s to xct on Bragg rflcton and thn rotat th crystal around th - Ewald sphr k k k Fgur 7: Illustraton of a 3-bam dffracton cas nvolvng O,, and, n ral spac (uppr) and rcprocal spac (lowr). k k ψ - k O dffracton vctor. Whl th rflcton s always xctd durng such a rotaton, t s possbl to brng anothr st of atomc plans,, nto ts dffracton condton and thus to hav multpl-bam dffracton procss. Th rotaton around th scattrng vctor s dfnd by an azmuthal angl, ψ. For x-rays, multplbam dffracton paks xctd n ths gomtry was frst obsrvd by Rnnngr (1937) and oftn ths multpl rflcton paks ar also calld th Rnnngr paks. For lctrons, multpl-bam dffracton stuatons xst n almost all cass bcaus of th much strongr ntractons btwn lctrons and atoms. As shown n Fg.7, f atomc plans and ar both xctd at th sam tm, thn thr s always anothr st of plans, -, also n dffracton condton. Th dffractd bam k by rflcton can b scattrd agan by th - rflcton and ths doubly dffractd bam s n th sam drcton as th -rflctd bam k. In ths sns, th photons (or partcls) n th doubly dffractd bam hav bn through a dtour rout compard to th sngly dffractd photons (partcls) by th rflcton. W usually call th man rflcton, th dtour rflcton, and - th couplng rflcton. pndng on th strngths of th structur factors nvolvd, a multpl rflcton can caus thr an ntnsty nhancmnt (pak) or rducton (dp) n th two bam ntnsty of. A multpl rflcton pak s commonly calld th Umwganrgung ( dtour n Grman) and a dp s calld th Aufhllung. Th formr occurs whn s rlatvly wak and both and - ar strong, whl th lattr occurs whn both and ar strong and - s wak. A sm-quanttatv ntnsty calculaton can b obtand by total nrgy balancng among th multpl bams, as workd out by Moon & Shull (1964) and Zacharasn (1965). In most xprmnts, multpl rflctons ar smply th nusanc that on trs to avod snc thy caus naccurat ntnsty masurmnts. In th last two dcads, howvr, thr hav bn rnwd and ncrasng ntrsts n multpl-bam dffracton bcaus of ts promsng potntal as a physcal soluton to th wll-known phas problm n dffracton and crystallography. Th phas problm rfrs to th fact that th data collctd n a convntonal dffracton xprmnt ar th ntnsts of th Bragg rflctons from a crystal, whch ar rlatd only to th magntud of th structur factors, and th phas nformaton s lost. Ths s a classc problm n dffracton physcs and ts soluton rmans to b th most dffcult part of a structur dtrmnaton of matrals, spcally for bologcal macromolcular crystals. u to an ntrfrnc ffct among th smultanously xctd Bragg bams, multpl-bam dffracton contans th drct phas nformaton on th structur factors nvolvd and thrfor can b usd as a way to solv th phas problm. Th basc da of usng multpl-bam dffracton to solv th phas problm was frst proposd by pcomb 3// 1:35 PM

16 15 (1949), and was frst dmonstratd by Collla (1974) n thory and by Post (1977) n an xprmnt on prfct crystals. Th mthod was thn furthr dvlopd by svral groups (Chapman, Yodr & Collla, 1981; Chang, 198; Schmdt & Collla, 1985; & Collla, 1987; 1988; ummr, Wckrt & Bondza, 199) to show that th tchnqu can b appld not only to prfct crystals but also to ral, mosac crystals. Rcntly, thr hav bn consdrabl fforts to apply th mult-bam dffracton to larg unt-cll norganc and macromolcular crystals ( & Collla, 1993; Chang t al., 1991; ummr, Schwgl & Wchrt, 1991; Wchrt, Schwgl & ummr, 1993). Progrss n ths ara has bn amply rvwd by Chang (1984, 199), Collla (1995, 1996), and Wckrt & ummr (1997). A rcnt xprmntal nnovaton n rfrnc-bam dffracton (, 1998) allows paralll data-collcton of thr-bam ntrfrnc profls usng an ara dtctor n a modfd oscllaton-camra stup, and maks t possbl to masur th phass of a larg numbr of Bragg rflctons n a rlatvly short tm prod. Thortcal tratmnt of multpl-bam dffracton s consdrably mor complcatd than th two-bam thory, as vdncd by som of th arly works (Ewald & no, 1968). Ths s partcularly so n th cas of x- rays bcaus of mxng of s and p polarzaton stats n a multpl-bam dffracton procss. In 1974, basd upon hs arlr work for lctron dffracton (197), Collla dvlopd a full dynamcal thory procdur for multpl-bam dffracton of x-rays (1974) and a corrspondng computr program calld NBEAM. Wth Collla s thory, multpl-bam dynamcal calculatons hav bcom mor practcal and mor asly prformd. On today s powrful computrs and softwar and for not too many bams, th NBEAM program can b run n an almost trval fashon, vn on prsonal computrs. W wll outln th prncpls of th NBEAM procdur n th nxt sgmnt. NBEAM Thory Th fundamntal quatons for multpl-bam x-ray dffracton ar th sam as thos n th two-bam thory, bfor th two-bam approxmaton s mad. W can go back to Eq.(3), xpand th doubl cross product, and rwrt t n th followng form: k K [ u ( u ) ] ( 1+ ΓF ) + Γ F- =, (6) (1) Egnquaton for -fld componnts: In ordr to proprly xprss th componnts of all wavfld ampltuds, w dfn a polarzaton unt-vctor coordnat systm for ach wav : u = K s p = u = u K n s u n, whr n s th surfac normal. Multplyng Eq.(6) by s and p ylds k K k K (1 + ΓF ) (1 + ΓF ) s p = Γ F - = Γ F - [( s s ) + ( p s ) ] s [( s p ) + ( p p ) ] s p p. (7) () Matrx form of th gnquaton: For an N- bam dffracton cas, th abov quaton can b wrttn n a matrx form f w dfn a Nx1 vctor =( 1σ,, Nσ, 1π,, Nπ ), a NxN dagonal matrx T wth T =k /K (=) and T = ( ), and a NxN gnral matrx A that taks all th othr coffcnts n front of th wav fld ampltuds. Matrx A s rmtan f absorpton s gnord, or symmtrc f th crystal s cntrosymmtrc. Eq.(8) thn bcoms ( + A) = T. Ths quaton s quvalnt to -1-1 ( + A ) = T. (9) Strctly spakng th gnvctors n Eq.(9) ar actually th E-flds: E=T. owvr and E ar xchangabl as dscussd n Scton II. To fnd non-trval solutons of Eq.(9), w nd to solv th scular gnvalu quaton -1-1 T + A =, (3) wth T 1 =K /k (=) and T 1 = ( ). W can wrt K n th form of ts normal (n) and tangntal (t) componnts to th ntranc surfac: ( K + n) k t K = +, n whch s ssntally th Bragg s law togthr wth th boundary condton that K t = k t. (3) Stratgy for numrcal solutons: Tratng µ K n /k as th only unknown, Eq.(3) taks th followng matrx form: µ - µb + C =, (31) 3// 1:35 PM

17 16 whr B = ( n /k )δ s a dagonal matrx and C =(A 1 ) +δ ( n +k t )/k. Eq.(31) s a quadratc gnquaton to whch no computr routns ar radly avalabl for solvng t. Collla n 1974 mployd an ngnous mthod to show that Eq.(9) s quvalnt to solvng th followng lnar gnvalu problm: B Ł I -C ' ' = µ, (3) łł ł Ł ł whr I s a unt matrx, and =µ whch s a rdundant N vctor wth no physcal sgnfcanc. Eq.(3) can now b solvd wth standard softwar routns that dal wth lnar gnvalu quatons. It s a 4N th ordr quaton for K n and thus has 4N solutons, dnotd as K l n, l=1,, 4N. For ach gnvalu K n, thr s a corrspondng N-gnvctor that s stord n, whch now s a Nx4N matrx and ts lmnt labld l l σ n ts top N rows and π n ts bottom N rows. Ths wav fld ampltuds ar valuatd at ths pont only on a rlatv scal, smlar to th ampltud rato n th two-bam cas. For convnnc, ach Ngnvctor can b normalzd to unty: N l l σ + π = 1. In trms of th gnvalus K n l and th gnvctors l =( σ l, π l ), a gnral xprsson for th wav fld nsd th crystal s gvn by ( r) = q l, l l -K r l whr K l =K l + and q l s (l=1,, 4N) ar th coffcnts to b dtrmnd by th boundary condtons. (4) Boundary condtons: In gnral, t s not sutabl to dstngush th Bragg and th au gomtrs n multpl-bam dffracton stuatons snc t s possbl to hav an ntrnal wavvctor paralll to th surfac and thus th dstncton would b mannglss. Th bst way to trat th stuaton, as pontd out by Collla (1974), s to nclud both th back-dffractd and th forwarddffractd bams n vacuum, assocatd wth ach ntrnal bam. Thus for ach bam, w hav two vacuum wavs dfnd by k ± = k t ±n(k k t ) 1/, whr agan th subscrpt t stands for th tangntal componnt. Thrfor for an N-bam dffracton from a paralll crystal slab, w hav altogthr 8N unknowns: 4N q l s for th fld nsd th crystal, N wav fld componnts of abov th ntranc surfac, and N componnts of th wav fld blow th back surfac. Fgur 8: (a) Calculatd rflctvty usng NBEAM for th thr-bam cas of GaAs (335)/(551), as a functon of Bragg angl θ and azmuthal angl ψ. (b) Corrspondng ntgratd ntnsts vrsus ψ (opn crcls). Th sold-ln-only curv corrsponds to th profl wth an artfcal phas of π addd n th calculaton. 3// 1:35 PM