Diffraction: Real Samples Powder Method

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1 Diffrctio: Rel Smples Powder Method

2 Diffrctio: Rel Smples Up to this poit we hve bee cosiderig diffrctio risig from ifiitely lrge crystls tht re stri free d behve like idelly imperfect mterils ( x-rys oly scttered oce withi crystl) Crystl size d stri ffect the diffrctio ptter we c ler bout them from the diffrctio ptter High qulity crystls such s those produced for the semicoductor idustry re ot idelly imperfect eed differet theory to uderstd how they sctter x-rys Not ll mterils re well ordered crystls

3 Itesity Crystllite Size As the crystllites i powder get smller the diffrctio peks i powder ptter get wider. Cosider diffrctio from crystl of thickess t d how the diffrcted itesity vries s we move wy from the exct Brgg gle If thickess ws ifiite we would oly see diffrctio t the Brgg gle B Agle

4 Crystllite Size Suppose the crystl of thickess t hs (m + ) ples i the diffrctio directio. Let sy is vrible with vlue B tht exctly stisfies Brgg s Lw: d si B Rys A, D,, M mkes gle B Rys B,, L mkes gle Rys C,, N mkes gle

5 Crystllite Size For gle B diffrcted itesity is mximum For d itesity is 0. For gles > > itesity is ozero. B Rel cse Idel cse

6 The Scherrer Equtio. si, si m t m t. si cos, si si t t Subtrctig: d re close to B, so:. si, B Thus: B B B t t cos, cos

7 The Scherrer Equtio Istrumet brodeig hs to be subtrcted More exct tretmet (see Wrre) gives: t 0.94 B cos B Scherrer s formul Pek width B vries iversely with the crystllite size. The proportiolity costt, K, is usully 0.94 d is vlid for sphericl crystls with cubic symmetry whe B is tke s full width t hlf mximum (FWHM). The proportiolity costt, K, is 0.89 for sphericl crystls with cubic symmetry whe B is tke s itegrl bredth. K sometimes is rouded up to. K might vry from 0.6 to.08. Suppose =.54 Å, d =.0 Å, d = 49 o : for crystl size of mm, B = 0-5 deg. for crystl size of 500 Å, B = 0. deg.

8 Pek Width Full Width t Hlf Mximum (FWHM): FWHM Width of the pek t hlf itesity vlue betwee bckgroud d pek mximum itesity. Itegrl Bredth: Totl re uder the pek divided by the pek height. Sme s the width of rectgle which hs the sme re d the sme height s the pek

9 Stri Two types of stresses: microstresses vry from oe gri to other o microscopic scle. mcrostresses stress is uiform over lrge distces. Usully: mcrostri is uiform produces pek shift microstri is ouiform produces pek brodeig d b t d

10 B tot cosθ Willimso-Hll Plot Size brodeig (Scherrer equtio): B t θ = Kλ t cosθ Stri brodeig: Covoluted brodeig: B ε θ = 4ε tθ B tot θ = B t θ +B ε θ = Kλ + 4ε tθ t cosθ Slope = B tot cosθ = Kλ t + 4ε siθ Kλ t 4siθ Istrumet brodeig hs to be subtrcted

11 Correctios for Istrumetl Brodeig There is lik betwee pek width d crystllite size/stri, but other sources of pek brodeig hve to be cosidered whe lyzig diffrctio dt Istrumetl brodeig: slit widths smple size peetrtio i the smple imperfect focusig uresolved d peks or wvelegths widths where d peks re resolved To correct for istrumetl brodeig: mesure the smple mesure uder the sme coditios the stdrd with ustried prticles lrge eough to elimite prticle-size brodeig

12 Correctios for Istrumetl Brodeig Loretzi shpe: Loretzi Gussi B obs = B size +B stri +B ist B obs B ist = B size +B stri Gussi shpe: B obs =B size +B stri +B ist B obs B ist =B size +B stri Voigt, Pseudo-Voigt: Decovolute peks ito Gussi d Loretzi frctios d the subtrct istrumetl brodeig.

13 Iterferece Fuctio We clculte the diffrctio pek t the exct Brgg gle B d t gles tht hve smll devitios from B. If crystl is ifiite the t B itesity = 0. If crystl is smll the t B itesity 0. It vries with gle s fuctio of the umber of uit cells log the diffrctio vector (s s 0 ). At devitios from B idividul uit cells will sctter slightly out of phse. Vector (s s 0 )/ o loger exteds to the reciprocl lttice poit (RLP). ( s s ) 0 hb b kb l

14 Iterferece Fuctio () > B () for 00 d () > B () for 00 If H hkl = H is reciprocl lttice vector the (s s 0 )/ H. Rel spce Reciprocl spce

15 Iterferece Fuctio We defie: s devitio prmeter S S0 H

16 Iterferece Fuctio I order to clculte the itesity diffrcted from the crystl t B, the phse differeces from differet uit cells must be icluded. For three uit vectors, d : exp N N N totl i F A s s 0 i prticulr uit cell N i totl umber of uit cells log i b b b H s s 0 l k h From the defiitio of the reciprocl lttice vector:

17 Iterferece Fuctio exp exp totl l k h i F i F A b b b H sice: ij i j b exp exp exp totl i i i F A Covertig from exp to sies: exp si si N N N N F A i i i i totl

18 Iterferece Fuctio Clcultig itesity we lose phse iformtio therefore: I A totl I F i si N i si i i iterferece fuctio Mximum itesity t Brgg pek is F N Width of the Brgg pek /N N is umber of uit cells log (s - s 0 )

19 Iterferece Fuctio

20 Crystllite Size

21 Perfect Crystls The diffrcted itesity clcultios mke use of idelly imperfect crystls This is the kiemticl theory of diffrctio The itegrted itesity from perfect crystl lrge d o mosic blocks is less th tht from idelly imperfect crystl Our cosidertio of Brgg pek width lso hs some problems. Whe we hve very lrge perfect crystls the pek width is ot zero. The width coverges to fiite smll vlue s the size of the crystl icreses We eed better theory! Dymicl theory is used to tret diffrctio i perfect crystls

22 Perfect Crystls Dymicl diffrctio theory is complicted. It icludes the possibility of multiple sctterig i crystl. Diffrcted bem is phse shifted by 90 every time it is diffrcted withi the crystl (this is i dditio to the 80 shift o sctterig from idividul electro or tom) Sctterig twice off set of lttice ples K 0 to K to K, produces diffrcted bem i the directio of the icidet bem but with 80 phse shift. The resultig destructive iterfereces reduces the itesity of the bem i the icidet directio This is PRIMARY EXTINCTION

23 Perfect Crystls A full mthemticl tretmet of dymicl theory uses differetil equtios tht describe the trsfer of eergy betwee the forwrd d diffrcted x-ry bems. This theory predicts tht itesity from perfect crystl with egligible bsorptio is I 8 e mc N F cos si where N is the umber of uit cells per uit volume. Note the itesity depeds o F ot F

24 Perfect Crystls The theory lso predicts tht the pek shpe o rockig the crystl i the diffrcted bem. The width of this rockig curve is clled the Drwi width. Note tht t the top of the curve the reflectivity is ~00%. The width of the curve s is give by Chges i shpe: due to bsorptio s e mc N F cos si FWHM, 0, for Drwi curve =.s For first order reflectios: 5 rcs < 0 < 0 rcs Higher order reflectios hve cosiderbly rrower rockig curves

25 Whole Ptter Fittig - Exmple: LiFePO 4 Excellet cdidte for the cthode of rechrgeble lithium bttery tht is iexpesive, otoxic, d evirometlly beig. First Reported: "LiFePO 4 : A Novel Cthode Mteril for Rechrgeble Btteries, A.K. Pdhi, K.S. Njudswmy, J.B. Goodeough, Electrochimicl Society Meetig Abstrcts, 96-, My, 996, pp 7. Structure of LiFePO 4 Pdhi, A. K.; Njudswmy, K. S.; Goodeough, J. B. J. Electrochem. Soc. 997, 44, 88

26 Whole Ptter Fittig - Exmple: LiFePO 4 Whe lithium-bsed cell is dischrgig, the lithium is extrcted from the ode d iserted ito the cthode. Whe the cell is chrgig, the reverse occurs. Cthode (Li x CoO : 0 < x < ) Aode (Li x C 6 : 0 < x < ) Commercilized by Soy Corp. i 99 Limits for the lrge scle pplictios:. Sfety. Cost (Co)

27 Whole Ptter Fittig - Exmple: LiFePO 4 Mximum etropy method (MEM)-bsed whole-ptter fittig of room temperture high-resolutio sychrotro X-ry diffrctio dt for LiFePO 4. The icidet bem from the bedig mget source ws moochromted by double-crystl Si () moochromtor, d the diffrctio dt were collected by multiple-detector system with flt Ge () lysis crystls d scitilltio couters. The wvelegth ws clibrted s.065 Å by powder diffrctio dt from NIST SRM640c.

28 Whole Ptter Fittig - Exmple: LiFePO 4 Experimetl visuliztio of lithium diffusio i Li x FePO 4 Shi-ichi Nishimur, Geki Kobyshi, Keji Ohoym, Ryoji Ko, Mstomo Yshim & Atsuo Ymd Nture Mterils 7, 707 (008) Published olie: 0 August 008

29 Whole Ptter Fittig - Exmple: LiFePO 4 Rietveld refiemet results for LiFePO 4 with eutro diffrctio dt mesured t room temperture i ir.

30 Whole Ptter Fittig - Exmple: LiFePO 4 X-ry diffrctio ptters of mixture of 0.6 LiFePO 4 d 0.4 FePO 4 recorded t 0 K steps from 98 K to 6 K with mgifictio of 00 reflectios. Bruker AXS D8 ADVANCE powder diffrctometer ws used with Co-K rditio d lier positio-sesitive detector Vtec-. Mesuremet rges were from 5 to 00. The mesuremets were coducted uder high-purity He tmosphere i Ato Pr HTK 450 temperture-cotrolled chmber.

31 Whole Ptter Fittig - Exmple: LiFePO 4 Rietveld refiemet results for LiFePO 4 with eutro diffrctio dt mesured t room temperture i ir. Rietveld refiemet results for Li0.6FePO 4 usig eutro diffrctio dt mesured t 60 K i Ar.

32 Whole Ptter Fittig - Exmple: LiFePO 4 Nucler distributio of lithium clculted by the Mximum Etropy Method (MEM) usig eutro powder diffrctio dt mesured for Li 0.6 FePO 4 t 60 K. () Three-dimesiol Li ucler desity dt show s blue cotours. The brow octhedr represet FeO 6 d the purple tetrhedr represet PO 4 uits. (b) Two-dimesiol cotour mp sliced o the (00) ple t z = 0.5; lithium deloclizes log the curved oe-dimesiol chi log the [00] directio, wheres Fe, P d O remi er their origil positios. (c) Two-dimesiol cotour mp sliced o the (00) ple t y = 0; ll toms remi er their origil positios.