Excerpt-from chater6_FultzHowe_SADP_indexing

Excerpt-from chater6_FultzHowe_SADP_indexing - , spot 19mm...

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Unformatted text preview: , spot 19mm " 0m. t 1e center of the pattern, expect a, meamrement error ef a few percent. FOP i‘iighest aceuracy in determining Spot it is Often pi‘efei'abie to measure distance between eharper, higher erder spate and then divide by iii-3 precedure (an iead to errore if the Ewaid Sphere cute the spat an angle (as discussed number of spate between them (plus one} Unfortunately, ti in Or. if there is a Slight distertion of the diffraction paittern Caused ‘1 the projector lens of the miemseope. it is important to knew the dieter— tions and artifacts of your micmseepe., which can be aeeeseed by H (taming diffractien patterns of weiiwformed eryetaiiites of well-known. material. Pix» 1 tographie printing can distert the Spat specifies, say meewfements sheuid be » performed directly on the negative if a digital epticai system not avaiiable. 6.192 l‘vlethod l m Start with Zane Axie indexing (and its frustratiens are illustrated by example, Suppese we need to index the diifmction pattern in Fig. 6.21 and we know it is from an fee Ci‘yetai. O angies : 90 Fig. 6.2. An fee diffractien pattern ready for indexing. The ea way in index: tiiie diffraction pattern to 100k up in Appendix A6 of this book. Here. however, we index the pattern “by hand" In the first method we “guess” the zene elm its diffraction pattern, in the st “16‘: in the {iii/"ire :tien pattern shows; an obvious Sxiziimetry, such a minim ‘ ’ of spots it)? a cubic e: stalk"j You should Z'IleHlOi'ize the syiiii’iietries for fee and bee diffraction patterns listed in Tabie 61. we first note that am: pit. Table 6 ,1 , ewrtiieiew 7: we have 311 faiziy immerdei £0118 axis. The imveet order 20116 are}: term in i7 5.23 Sfi’l’lilllSCZ‘ifléll than these of the i’iensi‘ty ef Spats is *‘1eir;onai:>iy we expect Iii when we do not know (the irziiiiei‘a iengti’i of" the ziiir‘zom‘ope iii the present exmiipiei 2 Ren’iembei‘: the r; 'ic‘ture F pertain to the {hence of name axis) For exaii‘iyie we can HAW; ‘ ' _ (,iii'ectieii points: up. ‘ter ride for the UL 3 all even or 2111 mid? does not EMS {in {CC crystal} 3,1 lndgxing Difframmn Panama Tame 6.1. Zena Axis Symmetry rectangular hexagonal Aspec: Ratio 1:1 1 : M2 for bcc, SC equilateral i {almost hexagonal for fC Fil‘b’E more Chg: iii defaing a 23:16 axis)“, the 1:393, SEOUL and {300 directiems ;, the W78 therefm‘e need only consider the lowest index direction as 5L Candidate mm: 8355‘ \Ne eliminate the firm; 3 zone because. the pane 11 does not have the requir ymmetry hated I Table 1, A: thin: point we make a, guess and try the 0] zone axis. We amid now compara om angles a d. <1. ZLUCGS in Fig. 6.2 to the diffrafition pattern in Appendix AG? bu: here we iflut‘strane a Systematic procedure to {heck the diffmction pattern W1? 6er the lowest Order diffractions in the diffraxy Hon pattern of an f0”; 01‘} Some afiowed diffractians from fcc crystals 51,257» 4* . \r‘x «\rz,‘ Ma are (Ll men or at; odd} cue. _ 4} (“280) (2210) {311) {3:33) (420) f' 29125 {4.00) ’ \. j The aliowed diffractien spars must be perpendicular to To test fer perpendicuiarity wiah aeek dot. pr‘ochmts than zero? r310“; » $111? 22% 0 721%? fi 70%.? 2: O . A L ‘ r < L 4 [mflmnfig {EEK/Wiliéfljifi WE thssre‘fare expecfi {he 10‘ 51: order spots in the diffraction pm,th :0 be $002} and (2401 WE? must nerd: ccmfixm that the comm: 2111ng made bstween the {WC Vie first A ft 91‘ do?“ 0‘ lLO lines rummn“ 7 fmm the (380‘: 5pm: to {h , need 1‘0 narrnalkx the, vectms with the: flac‘mr 1 so} we check me, no; mele u G 2,11 ,3 43k: malady w y [JEEPPIM‘ICURLI‘ H} kw]. ,J V , . indium; 7.73103 ‘ : 78 ES. Electron Diffraction and Crysmflogmphy The do? pl‘OdUCL of two normalized vectors equals L16 cosine: of the “fig between them‘ Here, with the and we could 553.: ‘p the: normahzatm “C: their 533% pmduc: exzwtly This censwtcnt with the an she difimction patiem. SO far, 50 and fhe {002‘} and, «0) diffractiong geem promiging becau; {hey are perpendiaular to each othggn 37mg C5111" reqmren'laut that t} (angles betwew. Spots are 90‘? The firm} ,p is m out the spots at t} correct dismang‘eies to make, a difi'ractian paittem for the; fer: axi V marriage the: Cimmra equation: mi 2 AL to Obtain the measure distance. r, of diffimtien spot fr 3m H16 transz’nitted beam: (2 ( (6,; l If we knew our camera constant, AL it waxfld be appmfiriate to work Wit absolute distances 0:? the 3pm, spacingg. Here we work with relative #:pacirm instead. Equation 6.3 Shows that the ratio of the spat digmfices magi: equal th Katie of the meters V???- + £5— 12. WE 6a.}sz the vertical} wing to film: (00? spot: as a reference distance {065mm from 6.2) Doing so. we predict. Spacing to the if N) SPO‘C HhO‘» '11 in Fig. 6.3: Since the ans‘v hauld as cioser to the 1.10613; Maine" of Fig. {if}. 1")“ $7163 *3 O ;r :5 zone mus?» be wrémg. The bad“ Vie haw: ts} try again. w M»... a «I Fig. ;' typical botched attempt a! r. 4'? 3“ ‘ indexmw the difi‘rafrtfl'm Ingram in Fig; [hanfifij ' Z :14: ' ‘ b ‘ ‘ ‘ ‘ 'v < ' L */ ‘z‘x’e nmke anotmr guess {1‘ gum-2 Repoating , V FUNK? «lure in abbl‘evmfw Expecmd D13“ 1 ; 1 Nwrmaiiged 00:4,? 61’ r. ‘ ‘ 1"" 1 4‘ my 1- 1} g; 6.1 Indexing {fifframion Pattern 1379 in Fig, camirucz a diffractkm pattern with Em. dosesr difii‘aation spots, {TH} and {00m} and cakulam the distance ratio. dag of Chfi diffrac:« {2: m _, _, , Fig. 6.4. Sum" 3111 ind“ A , 3 211011 pattern of by; 532‘ {\ Geod, we got, it“ Th: 3 aca‘ura the high , " wems aka}: a thangh it bit. i ie for this kind of mark. Max [)8 we should mmeamre our 590?, spstcings, GE? perhapq if we look slowly at the <fiifi‘miffia3n pattern we might that the spots 8163 ‘ we <iii§i‘&c£i<3n spins waged by the curvgture of the Ewald sphere and an ' :7 VH1xnemficai Shape figment Consigmm incziexing is; a Vii‘mo Once we (lave identified £316: difi’acrion pattern. We: must ensure that all linear wniifiniziions of our reciprocal lattice - 311K: the indicag of a3} the other spot»; in the difiraction patfiel'n, 0111‘ ‘ 3.] and 371191.,forethe h. I??? K indices‘ increak‘p by TH] when, we traverse a vertical Column of 5901.35 and sisy‘l‘urnetrii‘al, and there may be ‘me (:iistort' m of m0 smartest ,ii-OI'S in the: pattern are [11 i illuatwmted in r, when moving: acres< the £01) row of rspnts in Fig [In iHCI‘E'fE‘ Fig, {3.3 For 63:211le the first 1115:3921 remains com? 1:11; 211’ '2. the second @393 05 ‘2. , and the ihird ' W5? shamhi confirm ma? 4., ‘ ‘ , {1: p631 Kimmy these row i we do not, ' {KHV’ SPQIEH OE“ CIG‘WE’ any HSVV SHOES. The zone: ‘ should be comb confirm than the (OB axis poi . vecror croswpmdzzct: We am) lucky - {he W611}? is: parzfliei {,0 our we, L111 1 mm relach our; :21” wriiigm 6.1.3 Kiethod 23 ~ Start with Diffi‘actien Spam in she sown/i mmhml for $1 $31. and WK 28,0 6. Electron Diffractian and Crystallegraphy 1 < h k 1 ) my.» my» win» {:35 V J; 2 id xii? O O 31w 0 ‘i i censtam A. a +2 1: = «Z W M a M s U) M 5 increment a aleng mws : G 2 2 CD § at); {J C} CD CD ~33 f.) l») w (‘J 3‘ W m m! . rm :5) ON C Fig: 6.5. ROW and {1‘ h k I ) column Checks of a 4 + Q compieie diffraction i V ., pattern. Note how A fl “1 A g T: L: m “H the individual indices immth w. change in the direction aiang csfumns 2 ‘5 1 1 of the armws. prefiérred when the symmetry 0f the pzittem not obvious: It (easpecmfly awful when we know ‘Lhe camera, COQS‘Eaflt 0f the microscopa but here assume me such knawiedge. K3376 use the same {CC {‘lififastion pattern before “79 first the raflog of \x/z’fzatffigfimfi from ths afiewed ( of an fer: Crystal: these ratios are equal m the ratios of SpOt «_ aratioms in a di JéiCtiOfi pattern First. mace a, fist of these 1211:1053 in Table 62, Table 6.24 Disiances in recipmca} iattice of fcc cryS‘cas‘ Relative Spadng 73-”; NM look for mic diffractiuus in Tabb (“2.2. preferably ioworder 01103, in the who of the umamred dismncea i You am find 21w trial and error that, x; 3/1 l'ki'tii Corrwrwymis m lime ("1117? and W“ aufimxfliiontg which A “1111(5th ‘ ' ‘ kyle; Now that the dif the pair 1,, ‘ and the pair {‘22{ ; and (1-22), e, g: ‘ “s 6.1 indexing Diffracfiimn Patterns: 28; of their spat Spacing; 550 these pair: {tauld be candidaies for the diffraction ‘ attern. We weed U) choose specifis vectars ill/the {111} and {2‘20} lilies $213111 7 in: wide the correcifi aimless in diffraction pattern Here we {use " _ aithmlgh will work Note that {Tl cgnsistem with the abserved 903' angles: it turns 0th ma: we can ehminate hm of our other three candidate pairs of diffractions. the pair (200) and { 311;) and the pair and because no vect: TS in their families 3E8 at. $0“ angler; Now we complete the {1113336131011 pattern (Fig 6.5} by labeling the other dif ; Yractimn spots by addition shown in Fig. 3‘: I. “in 13; a _ a o 000 mo 0 O Q 3;}: F257. 8.6‘ Successfui indexing 05 {he diffractign pattern 0 Q 0 of Fig; Ski/3A Compare to indexing in Fig The zone Elliité thamed from the matter areas—productu {228; H {1:1}:<2 — mi g 0 ~ +— (2 + 2p: This time we. 51.1152 the zone zifxiiS to {11217 which is a symmetryrelata r. E ,__J Variant <35 aha zone found with Method l. Aes‘fhem ally howeve. he zone axis; is net; 50 pleasing em 21 20116 so Illayha WC W0?le want, to Change. our second indexing ?><3f<>re sabmitting our results for pubhcatimn. The answm reader may wonder what. happened to on: candidzite pzzizr of diffractions and ‘Ji’hich also have mod rams of that? ape: spac- mgss and 900 angle formed ‘0”, and flu Vv'e mum. have genre aheaLd Bend construczed a, czmdidaze diffracth pattern with these digractian veatom The 20m: axis is: h M?» M. :: - m/rv This smmk‘i seem msphiious, because a {JED zone: 33:15 proviciiefi diffraction Pattern wish hexagonal :mm m quite uniike the remaingulax 257611 " Fig, The probéem {hat {131:5 sappmm'flil of R/Iefl’ud may 133143ng at,th {if} 1:} if am? alsa §;-s>:pes<rted in me 5 such as the 202% and the l T118516 P’Mtem amumi the 13111511111“ \ , k J} (Efraziimzs ma} V “01125 beam. Om: is unpermm to (mack zigzag] all expected {‘1th {he Lona axis itie:2tim;~t<l.. it ,Ct'iuns. mid armum mm £3119 or when) 9231) Upm; doing 54>, if would bem'sme filifieixitiwn pattern LiCCULLHEd f ' ( ' W an: inaliapmpri: ¥ 3 101‘ imhmd wear that She‘- aiinmv‘zions L 7 paws-3m of £73,, ‘ 1 3 12332 6. Eleamn Diffraciion and Cr staiiography’ Having, tl'2§(>1.1gh the exercise of indexing; the diffraction pattern fir Fig. 62., you can appreciate how tedious the praczice ming be for IOW- 53‘1’1‘111163U‘y patterns with nomordmgonal wagers“ excellent compute} programs avaiiable to help simpiiifi; the but conSiSLency Chflzks or the outputs are still The autimrs ‘Wmfld be ‘nciined m such E 10617sz fer indexim the diffrafiou samem in Fig Given the met that a: b i _ u L 2:; ’ fine CZ“ Sta}. is manedinic with a, 2: i2 ‘ , < L: /r.907A., c r: 174,033 A. " :2: 108.39. The intrepid reader is; of ceurse encouraffai to m“ it. by hand ¢ :5 V / W n) / (and communicate thé result to 612 Stereggraphic Projections anci Their hfanipuiation 6.2.1 COIlStFUCtiOI} of a Stereographic Projeatian Btereographie pmjections are twewfimensional maps: of the orientation r9123 fionships between different cr3-‘stzliieg1'ap1r1i6: directicma They are 115651! fox pmbiems in diffraction espec “1:; electrm: difframiom but they do not mig- maze with diffraction theory. Stereogmphic pmjecticns were deveioped for solving; probiems in threea—dinwngien:1]. Crystallography. To canstrust a, stereographic projectmn, begin with a timr cryat‘di at the center of a iarge Conventionai terminalogy i fiOZ‘TflaEST to CFV Unfilagraphic ,3 "poles?" W76 need to s:pr u; it? orient a: G 31011 of the crystal by stating Whmh of its p01es pomts upwzlrds to the “north A pole” of She sphere. This the. 3:01 poie in Fig. 0‘20 than: wmc emit’alujiod {mm {he <’:1‘}'s~t£g:i m "WhUTE shoma nine poiw 4 use Iht‘v panama 0f interswmm in this; "SphfiffflCfil The manual ’69 f v ‘0} plane: paralle? m the 100} d, ‘ (“limc‘fions and 370309 an: inuzn‘ha‘ , i011, Stermgmphic Projections and Their Manipu‘iaiion 283 Lp’ffljefltiOflkf to create a, :OQH stereograpbic projection. To projxt these inter-- ' SéCtions onto a twomdin‘zenfiionai surface, draw .1?) 'pbipts Sf intersection to the south pale {see Fig, LTGKC-E mark with an $€X?'? the painis of imarsection of these lines on the equatorial {flame (>5 £1.95 5mm Fig, 6.23: Intersections 0f pale pmjecuians with the eqwmriai plane of the Spherical ‘3?)Y‘OjeCw tion 05 Fig {3.7, L, V'Thé, stemagrapbic projectim: the equamria‘l plane 0f the sphere with .the‘Se [marked insersectiang. Figure illustrazeg the projemion of 2 polw 3:: the center of the circle, and on ics Circumference. Etereagmphic gprojecticn jg. 6,9) contains orientatianal infomgafizion about; ail} paler-25; that mtyers'ect the northern hemisphere the sphere»: Poles wch as NJ“, and DOT ', (m Fig. {37, which int: r included in the [001] Stereogmphic projectiorfi The paméctflar stereographic rpmjectim: is identified by the point at, itr; ceumr. whim the pm" j the pale painting 1'20 the? north pn‘m of the sphere. the southern hemis gheze of the where, are 11'): (3mm of 3'00 Fig. 8.9. Eguav 'fi 6) wt} 1; mm; of pokes, “ I aphzc gin‘ojectimn. [MM , kayerthemws. 72w (mare southern nannhphem of mu“ {SUD g our siere' peie indicas. 10 ()0 6 Electron Difiractiou and Crystallography 6.22 Reiationghip Between Stereographic Projectéans and Elecfiron Diffractian Pafiterns In the (fifii‘action Of highenergy eieatmrm 45k nearly perpeendicular m became: the Bragg angks very small, perhaps a degree or The maid electrons travel neariy parallel to the) diffiacting planefi When the electr travel down the 014;?st fmm the north pole of a, spherical projeétion {at Fig. Ciifii‘éict.i<')ns occur from planes WhOSC poles intemed; the equafim: the sphere, pex‘hapg Within a degree or $0 The example ShOW’B in F 13 6.11 by a bee crysta} {‘kriented W I Eh its direction pointing upwards mm; :he €19.0th gun. Wye expect difiractiong from planes WhOSE,’ poies he an I drew gference of the 10:3 stereographic pli'OjefitiOilv in relating stereograp projectians t0 difi‘raction patterns: it. is important to remember that stars graphic projectiom contain 110 information about, the diatances between f (lifi‘rmfition Spot's, 21nd contain me information about structure: factor mil Neverthelegsf the: between the masters; in the diffraction pantern a in the stereo‘graphic projection the Fer exampla although 1] difiractions are forbidden fer bee crystals, the (1522) has diffraction occurs the angle of the £1 direetien in Fig: 610 Fig 6.1!). Orienta‘aion reIanonship between EEO] diffi‘aé’mion pattern at, iefzfi an: [110:5 stermgmphic projeaion at right Angles between (he vecmrs are The same 04 the left and right. figureg, 6.2.3 ,TV’KaflipuIa’eions 9f Stereagraphic Prejections Rakes. The stereogmphic projecgtion a. powerful man? fer workm ' £61115 that: inmive relazive orienwtions between {wax défibrem airmen, ,5 problems can be) solved with 1‘:’_>:‘a,t:011 matrimi‘m a}: course. hm: V sogmphii p")j€{ifi()fl$ an: quick and (MW, once 0119 Eli wires; the}, knack. uf using them For Li‘lé)?£é§1},17i£1f_§ angles 0n Marmkgraphk: projectktnls. we, :Mxfi a moi zinaiwgmifi ...
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This note was uploaded on 01/07/2011 for the course MEMS Mems1054 taught by Professor Wiezorek during the Fall '10 term at Pittsburgh.

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Excerpt-from chater6_FultzHowe_SADP_indexing - , spot 19mm...

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