{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

SuggestedSolutions_Homework Assignment _ 4

SuggestedSolutions_Homework Assignment _ 4 -...

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

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 / _._-_-___ _.._‘.- ...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern