Jens Als-Nielsen, Copenhagen University. X-ray Physics. Cheiron School 2009

Transcription

1 Jns Als-Nilsn, Copnhgn Univrsity X-ry Physics Chiron School 9 X-ry photons nd mttr 1. Scttring clssicl physics. Absorption Quntum physics

2 Elctromgntic rdition Along this lin on dos not obsrv ny cclrtion Hr on obsrvs th full cclrtion, but dlyd in tim by /c! mss cclrtion forcchrg fild i mcc [ E ] t c / [ E ] ω in [ E ] [ E ] t in iωt i ω/ c i t / c Guss tht E rd 1 E t r E t rd in dition from dipol-ntnn du to th oscillting chrg in th ntnn r 4 πε mc i E nrgy dnsity rditd nrgy 4 π rd Erd d chrg q ; Erd d, t obsrvd cclrtion t / c 1 1 Erd, t cc t / c ; 1 cc 4πε c πε c Erd, t 4 m/sc Forc Enrgy m/sc OK ; 5 Ang To gt th dimnsion right!

3 Dfinition of scttring cross sction nd flux dσ φ dω ph /sc Prticl flux φ cm nrgy / sc Powr flux φ cm nrgy /sc E photons /sc Powr flux E Prticl flux ΔΩ sct ΔΩ φout ΔΩ Dfinition sct dσ d φinδω Ω dσ φout Erd dω 1lctron φin E in r

4 ntrfrnc mthmticl Phs Q. π. π 4. π wvcrst r 1 λ sc. mpl. r 1 + iqr j scmpl.. r iqr 1, mny scmpl.. r ρ i Qr r d r j mny Δφin π λ Δ φ out Δ ' r φ rs r ' r Q r Numbr dnsity s drwn # is bhind #1 for in but hd for out, thrfor

5 Msuring tomic nd molculr formfctors from gs scttring ntnsity sin θ Q sinθθ 1 f Q Dtctor Viwingi Fild Dtctor Kr θ Gs cll X-ry bm f f tom mol j ρ f l j r i Q r i Q r j d r Hom wor: Vrify fw points on curv for CF4 whn Q is prlll or ntiprlll to CF4 bond bond lngth 1.38 Ang. Th CF 4 molcul is ttrhdr s shown. Prov first tht OO 1/3 of th bondlngth. Hint: Elmnts of Modrn X-y physics, Ch.4 p.115.

6 Q sinθ dϕϕ sinθ θ dθ r dω dϕ sinθdθ i Qr orinttionl ti vrg φ dφ Unit sphr +1 1 ix iqrcosθ π Qr dx θ dθ dϕ i Qr x 1 Qr orinttionl vrg sin sin sinθ dθ dϕ π Qr

7 A f + f ; A A i Q r 1 iq r * 1 f + f f + f i Qr 1 i Qr { iqr 1 iqr } { } 1 1 iq r1 r iq r1 r f1 + f + f1f + f1f sin Qr f + f + f f 1, mny 1 orint. vrg 1 1 Qr1 sin Q r ij f orint. vrg i + fi f j Qr i i> j ij Studnt : loo t th CF 4 molcul s n xmpl, nd compr orinttionl vrg with Q prlll to C-F bond

8 Absorption d d μ d d μ μ tomic numbr, photon nrgy Z μ ω h 3 4 1, ω ω μ h h Z Z

9 X-ry bsorption 4 3 σ Z, hω Z hω bs d d μ 1 μ d - th bsorption lngth : μ μ ρmn A Av σ bs Esy to msur μ nd thrby σ Fin structur occurs hr in solids nd dliquids t is clld EXAFS nd contins informtion bout distncs to nr nighbors. K-dg for Kr is t V L-dgs for Pb 3 N toms in A grm N / A in 1 grm ρ N / A in 1 cm ρ Av Av m Av m N A d Av / in 1 cm r of th thin slic

10 X-ry photon nd lctron in tom nitil stt i With intrction btwn photon-fild nd lctron trnsitions my occur 1 ε K Photon Elctron On photon with Elctron in K -shll wvvctor ; 1 r ; g ψ 1 s ε Hint ρεδε stts in intrvl δε Fr lctron No photon Wvvct. q Finl stt f q c ction rt Frmi s goldn ul W M i f π M i f ρ ε σ bs φ h σ Unit r bs c / V f H i Flux φ dnsity* c 1/V * c it int dσ Flux Φ : c ψ ntnsity Flux Ar scttrd Φ in ΔΩ thru ΔΩ dω W Φ σ bsorption in bs f ε < ε K-lctron cnnot b xplld --> th K-dg in σ ε hω K bs

11 1 Kr bsorption cross sction Ab bsorption cross s sction σ b brn 1 tg X ry nrgy V Kr gs D crystllin Kr χ μ 1 χ μ Enrgy from bsorption dg V Enrgy from bsorption dg V

12 flction nd rfrction How is th indx of rfrction rltd to th bsic procsss 1. scttring. bsorption

13 i in pln wv propgtion dirction 6 wvlngths 6 wvlnghts hitting n intrfc hd-on n 1 δ + iβ 1. Th wvlngth chngs from λ to λ/n wvnumbr chngs from to n Th mplitud dcys xp{-μ/ } ntnsity dcys xp{- μ} in i1 δ β β μ / to b provn π δ r ρ l i.. scttring i.. bsorption

14 indx of rfrction n ndx of rfrction vs. scttring S P ψ ψ P P i n Δ tot [ i n ] P ψ Δ φ xy, x + y/ S scttr dnsity ρ P P P Δ x b> b< P ψ ψ d ψ tot + i P dψ s incidnt wv ρδdxdy b i i x, y φ s numbr of scttrr scttrd wv prt from phs fctor

15 i x + y / dxdy x y r π + f dxdy π dr ; > r f complx, OK by nyltic continution. Hr so π π i i / P P π ψ tot ψ 1+ bρδ i π bρ n 1 P ψ [ 1 + in 1 Δ] i/ f π / i P P dψs bρδ i ψ bρδ i y f y i π / y π π complx pln f x π / x n> 1 b> scttr in-phs n < 1 b < scttr out-of-phs π X-rys: n < 1 b r ; δ r ρl x

16 Homognous wdg nˆ nhomognous plt nˆ λ w tn w λ/ Δx λ1+δ [ λ 1 + δ x ] [ λ 1 +δ x +Δ x ] Δx Δx [ ] λ 1 +δ λ / Δ x δλ / Δ x δ tn w λδxδ' x / Δ xλ δ' x δ 5 1 so th rfrction ffct is smll

17 Focusd bm smpl L Ar dtctor Scnning dirctions Displcmnt vctor on dtctor L φ/

18 Exmpl : Th invisibl substrt Si singl crystl wfr, prfct. All scttring condnsd d in Brgg points. M thin r thicnss c. 1 μ in 3 μ thic wfr by tching y [mm] x φ φ 1 y 1 mm x 1 mm x w tn w 1 35 μ 11 w Etchd volum

19 nrgy 6 V 5 μ 5μ

20 Gring incidnc ncidnt, rflctd nd trnsmittd pln wvs hv indx,, nd T Gnrl wvvctors r shown blow, BUT i -componnt > for nd T, nd < for ii n-pln componnts must b ll idnticl by continuity condition iii Common in-pln dirction is dnotd x. i r i xy rxy i Continuity t, xy, xy T, xy T n ndx of rfrction simplifiction T ' n >1 ' > x ; n cos n Smll ngls cos ' f n<1, 1i.. n 1 δ 1 c / ' + c T

21 Amplituds r dnotd T nd,, T ' x Smll ngls ' c + Snll s lw T tc. sin ' ψ T n i i i continuity T + ψ continuity ' ' + + T n n i i i bov blow T + ψ,, bov blow Frsnl s lws ' ; ' ' + + T i i i i ψ ψ i ψ

22 flctivity Snll' s lw t smll ngls : ' + c Frsnl rflctivity from shrp intrfc mplitud rflctivity : r ' + ' c / c 4 symptoticlly c Glncing ngl c 5.8 λ ρl r / π whr r 1 mc Ang

23 W now this cs c for r F >> c c λ ρ π λ ρ bul r bul r o c λ π c λ Thin plt rflctivity r x Diffrntil i flctivity d dr ρ ro sin d dr cλ ρ ro sin d Dimnsion 1/Lngth dimnsionlss complx constnt, dtrmind by iq λ ρ dr c bul r o ρ ρ bl bul ρ iq 1 iq 1 iq 1 With f ; f d f f ' d i φ Q ρ bul iq iq Q i f stpfunction f ' δ nd φ Q 1 ; ii Q r F c 4 c c i 1 1 c i iq d

24 Th bndwidth from Brgg rflction d 1 f rflctivity of on lyr is.1 thn bout 1 lyrs will ffctivly rflct th dpr lyrs r not illumintd With rflctivity it pr lyr g N1/g N Grn pth lngth N d sinθ wvfronts N dsinθ Δϕ / N Δ λ π πn πnδλ λ π λ +Δλ Δλ/λ- width / π g Δλ λ Studnt problm: Dtrmin th wvlngthbnd from Si111 monochromtor. Hint: Si hs th dimond structur with 5.43 Ång. 1 N Δλ 1 g λ N FWHM d sin r thinplt i λ r o ρ g d r F v hl v c Unit cll volum

25 r flctivity from fuy intrfc iq dr Thin plt λ ρbl r c bul o ρ ρ bul iq d r f F ' iq f δ for shrp intrfc d φ Q r F f FTGussinGussin A / σ iq d Q σ / B f iq i Q iq d f d φ Q i Q

26 A nd B + box modl φ Q Q / Q F i i+1 ρ i ρ bul rf / σ Q σ / N ρi+ 1 ρi iqi φ Q ρ i 1 bul