This preview shows pages 1–9. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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
iii3 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 wellknown. 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‘ gum2 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 ,iiOI'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 (122), 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
paws3m 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, LTGKCE 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} paler25; 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 ...
View
Full
Document
This note was uploaded on 01/07/2011 for the course MEMS Mems1054 taught by Professor Wiezorek during the Fall '10 term at Pittsburgh.
 Fall '10
 Wiezorek

Click to edit the document details