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SuggestedSolutions_Homework Assignment _ 4

# SuggestedSolutions_Homework Assignment _ 4 -...

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Unformatted text preview: éjaﬁﬁii4§féifé;__s Homework Assignment #4 MEMS 1054 NAME: Return by next ‘Wednesday! 7 Questions in total! This is an important homework for you in MEM81054. Get Started early, especially since I uploaded it only today, Friday! Chapters 3 and 4 of the Cullity & Stock text and my uploaded notes should be useful in the context of this homework. 1) Deﬁne the primitive unit cell vectors, a*, b*, c*, of the reciprocal lattice of a crystal structure with a unit cell that is spa nned. by the primitive vectors a, b, c. -5 ‘ -a a . _ J -—9 Haze, .‘ ‘3 if; \$2.51-:fo ‘ ft 5:: . at £5.32. _ 1/ :;_L/g,,g)..é.{m): was: Vic ' lac ’ Km 1 “t E:[Z xi?) 2) Show that the product of the volume of the primitive unit cells of the reciprocal lattice and the Bravais lattice of a given crystal structure is equal to unity, i.e., e; .4 attics) x:C x vc* = 1 _.. =r. (53%.) £9) Vé’" éﬁ '[(ﬁ)1(('§fk) = ‘2- (z (1:) (1 3 v5 Va ' x * A .9 .9 4 J I. " V‘¥="’)’- "Z \ . c (u. M“ I!) C a ( x‘ ) j 2') Van/47:» x4:- I (l: (’x’ x flag) ‘4 ill) Mdllfl/M4‘6kvhjfmm /¢ a-(lvrb) . C I. stake t TM 3) Show (you can do this graphically or algebraically) that a) a tetragonal crystal structure oas a reciprocal space lattice that is tetragonal; b) a hexagonal crystal has a hexagonal reciprocal space lattice; (Hints: Exploit the symmetry related constraints on the lattice parameters for the _ _ ' ‘0 _ _ a tetragonal and hexagonal lattwes) lngl =30 flcﬁt‘o 3‘ d.(1:?¢) 4 63’. {2a a) was we WW WW b)[bon-P1MJg-,~£d fulﬁaf [9»! min» . . 414: d J .I- '9 __ ' = 2’: £010 ‘1?' ”0‘5 ta! = INN. at 191* 4 (ﬁr/50. fa ' yiif‘m’ F?) [00'] - flu“!- II 22:“ “Itoiﬂ’b! ‘1' i . t 3:: 000 DK- [too I I. g M W .3 eI——-§sl 3 I c. tar—9mm Syn-”L?“ 1er 1'1"?" .4: war '1’ l r=j&" (i/l a? .' " rw4_f—__ a :2 C- Q 3;) o Eur WK”! [:00] f 50¢ a , [ aﬁhvﬂ 1 “I it “Dr * €|YJJD “J v f‘ l 1 "1. a, d- 0‘0 Ni) “FL! 90]"flw; t "Lam ‘ a "" 1 Q.“ ‘Dlh I ”it?!” '- L st” .11 _ "3* (s ‘ 1’ , ‘1 Agave L3.) in, . g; *. 1"" 1M}?! 1., {L500 , 3- 5 J. 57"” dig-\$0 ‘ ”“14. "(:3 L# I" i . llo ‘3: —-,>t:o«o3 u'a'lclht a” c) Gum" mo 4) Consider the typical geometry of an X-ray diffraction experiment used to study the structure of crystals. Assuming you utilize X-rays from Cu—k-alpha radiation with wavelength l=1.54-1A, what is the minimum d-spacing of crystal planes that you can resolve? (Hint: What are the geometrical limits on the angle 3?) . I m . , _ m m Zoltztlug‘ﬁ/(d (3.1» [smé’l £1 Z +W can. (diam am 1 A trawl ”‘" “”7 I __ _. ._ ,_., . , : r" t " " --- :3 O. 7 /BD “tn/Lu ‘2 {glnalmwy L ‘2. Eff-— 5) The izntermetallic phase ofthe c Jemically ordered compound NiAl forms for near equiatomic compositions between the metallic elements ! i and A}, andhas a cubic structure \Eth just one ﬁmmllainjulgr unit cell. If th- s placc‘he origin, U166] tom is located Eli—V2 , 1/2 1 Exam the unit cell. a) Is the NiAl structure primitive" What is the motif of the ordered beta— hase NiAl? l X39 {ll-- ,3 Pn‘mu‘rue Lice. ﬂay-all ; Hod-«(1‘s [email protected] 0690 é/JQQ \$5,: , b) The nine lines of lowest Bragg angle 6 observed on the powder diffraction pattern of NiAl have d-spacings in A of: 6' ML- 190 no in m 29/ 2!! 2.2.0 221/360 3:?! / dhk1[A]: 4.0.5 (w), 2.86 (s), 2:.34(w), 2.03(s), 1.81(w), 1.65(s), 1.43(s), 1.35(w), 128(5) @hklll]. “Ml-l 3 ‘5?th j ,a:_1_q’.22_3252\$22l )27,9‘)‘ 31(3) 3%33j3761lj h L.) #9"; M 53m” Zolw Jv Alain Ml The powder diffraction pattern has been obtained with Cu-Ka X-radiation, wavelength of 1.542%. Calculate the Bragg angles, @hkl [°], associated with the ﬁrst nine diffraction k, pea S 5160 a/hﬁlt/‘L . LU .4 ‘;__ «— ————— , 7” a”, — ' ' 51M 20‘ Here W == weak and s = strong intensities. N a” W‘I’LL“ gW.M, duo‘l’low‘dwl c) Evaluate the: unit cell dimensions of NiAl. \[v 1., M 40 2 0/40 : 4 WW: %;bo=co!_d;/{Ff;fd° d) Determine the expression for the structure factor F01“) for NiAl and observe that its V; :02:- {MS/E"? \ magnitude: depends on whether the sum h+k+1 is odd or evenﬂmmjndices hkl, referringto the Miller indie es of the diffracting planes, (hkl), the nine BRAG_G PEMEEEEEEMEEEEEiEﬁ—W—‘f F. __ g??? p .Zo’x‘flihhwﬁiﬂ‘ {'1’ij [WM YP F; t/Afé’ I!) “ﬁll W1: W ‘ E ,4: +944- 3 = {a W» ("l writ ,. r; no.4", m ., ram/ewe M VFAC ’ \pIW .2‘ f ”fl”! =2" N /’M ‘9 Md éa/Z 6) Execute a structure factor calculation to show that a diffraction pattern from a face— centered lattice of an elemental (tr) stal (e.g. Cu) will have a characteristic set of systematic absences for intensities I(2®hkl) that are associated with diffraction from sets of crystal planes separated by :th and described by Miller indices (hkl), for which h,k,l are integer of mixed parity, ie for mixtures of odd and even integer Miller indices, such as (100), (310) etc. (Note for these sets of planes {th} perfect destructive interference occurs). 2’ M “3,1573% m; {Q a} M «griffin/M «'1 ﬂef'ﬁﬁéﬁ La. yhgﬂf Am 5‘ aﬁﬁﬁ' rim. 215w. AMI-I‘M a} “-1.. Vin yin . 5 “21!! , lawn!“ umiﬂiemhﬂﬂ A»: i E c? r— _ S . J. l1” "- C—‘I ae i 9 f w r:‘[l.+»Ll mane) WYAM) 6' «’r 2-— 6“ ' ( d. 'l' .‘3 1" I: ‘f' i ) o ’F 1.11.! at mam dot/I'M” MIL L.+.t 1..” ,3 “ﬂ T‘qtcuil +5.1) +—(—i) +314) ):{4¢‘FFUW€ Madam. 9: a “L "r a: smite; is, =5“. =0! g; 7;” agaw!fw“wlwugll J! """ 1;th ‘ 7) The Bragg law allows the determination of the shape and the size of the crystal unit ”W Li” M cell from a diffraction pattern, vxhi 1e the structure factor allows the determination of the WM content of atoms in the unit cell. Determine the expression for the structure factor for a C- 5W centered and for an il-centered E LEMENT'AL crystal with a mono-atomic motif. Do you 04,9144 i“! expect to observe diffracted beams in your diffraction patterns for the two different “aw 6 crystals associated with the planes with Miller indices (hkl) of (1 1 1), (110) and (101) ‘ respectively? Why or Why Not? Brieﬂy justify your answer beyond the math with some of the widerlying'physics. - - / Eé'maéia’ awrﬂiz’.«t'/é -W'Wa mﬂ =e 4 M/Moc/WJ- , , / , F .— 4 ‘ Cl ”63,5126 & ,ﬁké : vim: a} M! £ ”(31,561; VP%..‘L= vet: - ﬂzz’rqgg, i “4— .2, ﬁttest—T 73;:- (im-ty... rim“) W; ﬁrm) [1+ {—uwj /“.>> F: /I,:I a 5 :éig 3+9 :fa Jigs; ‘ I} ln'l'l‘L :24. pl J 00“ L a , . / I .c': awn .‘9-7-“9‘? ,' :ﬂ. {7 non. i f M3, ‘ H J I Ego-i épé' "M JEW/ I, 1.1! _ 3/2— 5A 5% ._. ,‘f‘ ,1,£Id?£owf ”4‘ ﬂ" 1' I {night 627/“ . wé w! ”2” I F? "f Jag :ﬁwé-W 1 ‘ .r'k‘(afe£r-£/ Him: ._ 6’ F : if; I t I? x /: glitz“! Jr/“U jiffy“ /.,j144+£:2n,-% — . 2 p; 11;“ =- to / 1W =th °< /{::/ e“ 511% 4/3 1 / _._-_-___ _.._‘.- ...
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