Defects in Crystals and X-Ray Diffraction

Transcription

1 Dr. Y. Apara, Assocate Prof., Dept. of Physcs, JU College of geerg, JU - H, Defects Crystals ad X-Ray Dffracto Classfy crystal defects? As: hey are broadly classfed to four categores, amely, 1. Pot defects zero dmesoal defects Vacaces Itersttalces Compostoal defects v lectroc defects. Le defects oe dmesoal defects dge dslocato Screw dslocato 3. Surface defects two dmesoal defects Gra boudares lt boudares w boudares v Stackg faults 4. Volume defects three dmesoal defects Large vods Cracks Dscuss about the pot defects crystals? As: POI DFCS

2 Pot defects are formed due to the mssg of atoms or the preset of foreg atoms the crystals. he presece of pot defects a crystal creases ts teral eergy compared that of the perfect crystal. he effect of such defects s local ad produce dstorto sde the crystal structure. he presece of pot defects chage the electrcal resstace of the crystals. Due to preset of pot defects, dstorto or stra appears aroud the defect up to few atomc dameters. Hece the mechacal stregth at that pot gets reduced. Vacaces: A empty ste of a atom posto a crystal s smply kow as vacacy. hs refers to a mssg atom smply kow as vacacy. hs refers to a mssg atom or vacat ste as show Fgure. Such defects may form ether due to mperfect packg durg the orgal crystallzato or due to thermal vbratos wll crease ad ultmately some electros jump out of ther postos of lowest eergy. It s possble that the trasport of atoms occur through the lattce wth the help of vacaces durg the dustral processes of aealg, precptato, sterg, surface hardeg etc. Fgure Itersttalces: hs defects occurs as a addtoal atom occupes a vod space empty space betwee the atoms whch causes some mechacal stra o the surroudg atoms. he amout of stra depeds o the sze of the tersttal atom. he atom whch occupes vod space s called tersttal atom as show Fgure such atoms may be preset the crystals whch have low packg fracto. Whe a crystal has udergoe eormous stress the some of the paret atoms get dsplaced from ther lattce stes to vods preset the crystal. hese defects ca be treated as self- tersttal defects.

3 Compostoal defects: Compostoal defects arse due to presece of ether uwated mpurtes or purposely doped mpurtes durg the process of crystallzato. hese defects play a mportat role semcoductors whch are specally prepared for dodes, trasstors, etc. these defects are of two types, 1 Substtutoal mpurtes ad Itersttal mpurtes. A substtutoal mpurty refers to a foreg atom that replaces a paret atom the crystal lattce. hs s show Fgure. f the sze of substtutoal mpurty s same as paret atom the the amout of stra aroud wll be less, otherwse t wll be more. x: xtrsc semcoductor of slco whch s doped wth alumum or phosphorus. A tersttal mpurty refers to a small szed foreg atom lodgg the vod space the paret crystal wthout dsturbg ay of the paret atoms from ther stes. It s show Fgure. small substtutoal atom large substtutoal atom

4 x: Occupyg carbo atoms of atomc radus Å the octahedral vod space of FCC ro of the atomc radus.50 Å. v lectroc defects: lectroc defects occur as a result of errors charge dstrbuto solds. y the fluece of exteral electrc feld, these defects move freely wth the crystal. o uderstad the pheomea related to electrcal coductvty, the electroc defects play a mportat role. Derve a expresso for the umber of vacaces at a gve temperature? As: CALCULAIO OF UMR OF VACACIS A A GIV MPRAUR Almost all crystals vacaces are preset ad the ma cause for these defects s thermal agtato. Ad these vacaces are produced ad destroyed costatly. Let us cosder v s the eergy requred to move a atom from lattce ste o the surface. herefore the amout of eergy requred to produce umber of solated vacaces ca be wrtte as u v.17 he total umber of ways to move umber of atoms out of umber of atoms a crystal o to ts surface wll be p.18 he crease etropy due to formato of vacaces ca be wrtte as S P.19 Substtutg equato.18 equato.19, S.0

5 ut, the free eergy F u S Hece, F v.1 [ ] F v Usg Strlg s approxmato, x x x x the secod term o R.H.S ca be smplfed as [ ] F v + + [ ] F v. At thermal equlbrum, free eergy s costat ad mmum wth respect to, hece, F v δ δ [ ] V V.3 V exp V exp.4 If <<, ca be eglected, so that V exp.5

6 If V 1eV ad e Hece, the equlbrum cocetrato of vacaces decreases wth temperature. Dscuss the Schottky defects oc crystals? Aprl/may 008, JU I oc crystals, the formato of pot mperfectos s subjected to the requremet that the overall electrcal eutralty has to be mataed. A par of oe cato ad oe ao ca be mssed from a oc crystal as show Fgure. the valecy of the mssg par of os should be equal ad opposte to mata the electrcal eutralty. So these are kow as o vacaces a crystal. Whe vacaces are created by the movemets of a ao ad a cato from postos sde the crystal to postos o the surface of the crystal, a Schottky defect s sad to be formed. Such defects are domat alkal haldes.

7 Derve a expresso for the umber of Schottky defects at a gve temperature? Aswer: CALCULAO OF UMR OF SCHOY DFCS A A GIV MPRAUR I oc crystals, the umber of Schottky defects at a gve temperature, ca be calculated assumg a equal umber of postve ad egatve o vacaces are preset. he formato of pars makes t possble to keep the surface of the crystal electrostatcally eutral. he Schottky defect s as show fgure he umber of pars ca be related to the total umber of atoms preset the crystal o followg the same procedure as adopted pure atomc crystals.

8 Let us cosder p s the eergy requred to move a o par from lattce ste sde the crystal to a lattce ste o the surface. herefore the amout of eergy requred to produces umber of solated o par vacaces wll be u p.6 he total umber of ways that to move umbers of o pars out of umbers of oc molecules a crystal o to the surface wll be P.7 S P S.8 ut, the free eergy, F u S.9 Submttg the equatos.6 ad.8 equato.9, F p.30 Where, p s the eergy of formato of a par of o vacaces. Usg Strlg approxmato, x x x x, [ + + ] [ ]

9 Substtutg ths value equato.30, [ ] F p.31 At thermal equlbrum, free eergy s costat ad mmum wth respect to, hece, 0 F δ δ [ ] p p p p exp p exp.3 If, << the ca be eglected, p exp p exp.33

10 I acl crystal, p.0 ev at room temperature, therefore,.0 exp e.8 18 Dscuss about the Frekel defects crystal ad derve a expresso for the umber of Frekel defects at a gve temperature? As: Frekel defects: I oc crystals, a o may be dsplaced from the regular lattce to a tersttal ste or vod space. If t s so, the the vacacy ad a tersttal defect wll be formed. hese two defects together kow as Frekel defect. It s show Fg..37. As catos are geerally the smlar os, t s possble for them to get dsplaced the vod space preset the lattce. Aos do ot get dsplaced lke ths, as the vod space s just too small for ther sze. A Frekel mperfectos slver haldes ad calcum fluorde are of the Frekel type. Frekel ad Schottky defects together are called trsc defects. he Frekel defect s as show fgure:

11 CALCULAIO OF UMR OF FRL DFCS A GIV MPARAUR Let us cosder s the eergy requred to move a atom from lattce ste sde the crystal to a lattce ste o the surface. herefore the amout of eergy requred to produce umber of solated vacaces, u.34 the total umber of ways to move umbers of os out of umber of oc molecules a crystal o to the surface wll be P.35 Where s the umber of tersttal os. he correspodg crease etropy due to the creato of Frekel defects s gve by S.36

12 ut, free eergy, F u S.37 Substtutg equatos.34 ad.36 equato.37, F.38 Cosder, Usg Strlg s approxmato, x x x x + Substtutg ths values equato.38 ] [ F + At equlbrum, the free eergy s costat ad mmum wth respect to, hece 0 F δ δ akg >> ad >>, ] [ Hece, t s cocluded that umber of Frekel defects, s proportoal to 1.

13 What are le defects ad expla detal? LI DFCS : Le defects are kow as oe- dmesoal defects. he effect of these defects appear a a le the crystal geometry. hese defects are also called dslocatos. hese dslocatos occur geerally due to presece of complete atomc plaes whch arse due to growth accdets, thermal stress etc. the presece of dslocatos the crystal decrease ts mechacal stregth. herefore t ca be sad that few of the crystals are structure sestve as the mechacal propertes are depedet o the le defects. LI DFCS O DIMSIOAL DFCS I geeral, dslocatos are classfed to two categores.. dge dslocato. Screw dslocato dge dslocato A three dmesoal vew of a perfect crystal s show Fg..38. t s a aggregate of vertcal plaes parallel to each other ad sde faces, suppose oe of the vertcal plaes does ot exted from top to the bottom of the crystal.e., eds partway wth the crystal as show Fgure, a dslocato s sad to be preset. I a perfect crystal all the atoms are preset at equlbrum postos, therefore dstace betwee ay two adjacet atoms ay where the crystal wll be equal to equlbrum value. I a edge dslocated crystal, just above the edge of complete plae, the atoms are squeezed ad are a state of compresso. he dstace betwee eghborg atoms wll be lesser tha the equlbrum value. Just below the edge of the complete plae, the atoms are pulled apart ad are a state of teso.

14 he dstace betwee eghborg atoms wll be greater tha the equlbrum values. he plae betwee the above metoed regos s kow as slp plae. he dstorto exteds a the edge to all the crystal. he maxmum dstorto appears aroud the edge of the complete plae. he edge dslocato rus from frot to back through the crystal. he complete crystal plae may beg from top or bottom ad eds mdway as show Fg..40 a & b. the frst case the dslocato PQ s sad to be postve ad s represeted by the symbol. I the later case t P Q s sad to be postve ad represeted by the symbol. I the above symbols,.e., or, the horzotal les represet slp plae ad the vertcal les represet complete plae.

15 Screw dslocato Whe the part of the crystal s dsplaced relatve to aother part, a spral ramp forms aroud the dslocato le. It meas crystal s ot breakg up to two parts. he dsplacemet s termated wth the crystal as show Fgure the fgure, ACD s the slp plae. he left porto of slpped plae s ot sheared the le AD represets dslocato le. he term screw s to represet that oe part of the crystal s movg spral maer about the dslocato le.

16 If spral moto of oe part of the crystal s clockwse drecto the the dslocato s rght haded, o the other had spral moto s atclockwse drecto the the dslocato s left haded. Wrte a short ote o urger s vector? As: URGR S VCOR he magtude ad drecto of the dslocato ca be uderstood from the cocept of urger s vector. Let us cosder a startg pot P a crystal ad move X-tmes the atomc dstace the postve X- drecto ad move Y- tmes the atomc dstace the postve Y- drecto ad the move X-tmes the atomc dstace the egatve Y-drecto. After ths process the startg ad edg pot wll be same P. ths happes the case of a deal crystal. It s show Fgure 1 ad Fg.. hs crcut s kow as a urger s vector. Fgure

17 I the case of dslocated crystal, f the startg pot s P, the edg pot wll be Q. It meas to reach the tal pot P, there s a extra dstace b as show Fg. 3 ad Fg. 4 he coectg vector betwee the startg pot ad edg pot s kow as urger s vector of the dslocato. hs gve drecto ad magtude of the dslocato. he urger s vector s perpedcular to the edge dslocato ads parallel to the screw dslocato. Dstgush betwee edge ad screw dslocatos? As: DIFFRCS W DG DISLOCAIO AD SCRW DISLOCAIO Sl. o dge dslocato I edge dslocato, a edge of atomc plae s formed teral to the crystal. It s perpedcular to ts urger s vector. It moves the drecto of the urger vector If the complete plae s above the slp plae the t s kow as postve edge dslocato ad s represeted by the symbol. If the complete plae s below the slp plae the t s kow as egatve edge dslocato ad s represeted by the symbol. he amout of force requred to form ad move a edge dslocato s less. Screw dslocato I screw dslocato, oly dstorto of the lattce cell the mmedate vcty s proved. It s parallel to urger s vector. It moves a drecto perpedcular to the urger s vector. If the spral moto of the dslocato s clock-wse drecto the t s kow as rght haded screw dslocato. If the spral moto of the dslocato s at clock wse drecto the t s kow as left haded screw dslocato. he amout of force requred to form ad move a screw dslocato s more. 7 Speed of movemet of a edge Speed of movemet of a screw dslocato s lesser. dslocato s greater.

18 X-Ray Dffracto: X-rays are part of electromagetc radato wth wavelegths betwee about 0.0A 0 ad 0A 0.hey are part of the electromagetc spectrum whch cludes electromagetc radato called vsble lght. ecause X-rays have wavelegths smlar to the sze of atoms, they are useful to explore wth the crystals. he eergy of X-rays, lke all electromagetc radato, s versely proportoal to ther wavelegth as gve by the ste equato: hν hc/λ Where, eergy h Plack s costat ν frequecy c velocty of lght λ wavelegth Sce X-rays have smaller wavelegth tha vsble lght, they have hgher eergy. Due presece of hgher eergy,x-rays ca peetrate matter more easly tha vsble lght. ragg s Law: W.L. ragg, observed X-ray dffracto whe X-rays are cdet o the crystal surface early at ragg s agles. ragg s agle or glacg agle meas the agle betwee cdet ray ad crystal plae at whch dffracto occurs. W.L. ragg preseted a explaato of the observed dffracted beams from a crystal. He supposed that the cdet waves udergo specular mrror-lke reflecto at the varous parallel plaes of atoms the crystal, wth each plae reflectg oly a small fracto of the radato. he dffracted beams are foud oly whe the reflectos from the varous plaes of atoms terface costructvely.

19 θ θ dhkl dhkl Applcato of X-rays: a. XRD aalyss provdes Aalyss of sold ad lqud samples. Reflecto, trasmsso or capllary geometry -stu hgh temperature & reactve evromet. Oretato / texture data Grazg cdece Reflectometry ratky Small Agle X-ray ScattergSAXS b. XRD aalyss applcatos clude ao-materals: phase composto, crystallte sze ad shape, lattce dstortos ad faultg, composto varatos, oretato, -stu developmet. Catalysts: for ao-materals, wth motorg of structure throughout maufacturg ad use to develop structure-property relatoshps. ew materals developmet Polymers & Compostes: Crystalle form, crystallty, crystalle perfecto, oretato. Pharmaceutcals & Orgacs: Polymorphs, crystallty, ad oretato are mportat to performace ad ca be followed by XRD, cludg -stu studes. XRD qualfed to GMP stadards.

20 LAU S MHOD: he Laue s method, a sgle crystal s held statoary a cotuous X-ray beam. he crystal dffracts the dscrete values of λ for whch the crystal plaes of spacg d ad the cdece agle θ satsfy the ragg s law. he Laue method s maly used to determe the oretato of large sgle crystals. Whte radato s reflected from, or trasmtted through, a fxed crystal. he dffracted beams form arrays of spots, that le o curves o the flm. he ragg agle s fxed for every set of plaes the crystal. ach set of plaes pcks out ad dffracts the partcular wavelegth from the whte radato that satsfes the ragg law for the values of d ad volved. ach curve therefore correspods to a dfferet wavelegth. he spots lyg o ay oe curve are reflectos from plaes beg to oe zoe. Laue reflectos from plaes of the same zoe all le o the surface of a magary coe whose axs s the zoe axs. xpermetal here are two practcal varats of the Laue method, the back-reflecto ad the trasmsso Laue method. You ca study these below:

21 ack-reflecto Laue I the back-reflecto method, the flm s placed betwee the x-ray source ad the crystal. he beams whch are dffracted a backward drecto are recorded. Oe sde of the coe of Laue reflectos s defed by the trasmtted beam. he flm tersects the coe, wth the dffracto spots geerally lyg o a hyperbola. rasmsso Laue I the trasmsso Laue method, the flm s placed behd the crystal to record beams whch are trasmtted through the crystal. Oe sde of the coe of Laue reflectos s defed by the trasmtted beam. he flm tersects the coe, wth the dffracto spots geerally lyg o a ellpse. Crystal oretato s determed from the posto of the spots. ach spot ca be dexed,.e. attrbuted to a partcular plae, usg specal charts. he Greger chart s used for back-reflecto patters ad the Leohardt chart for trasmsso patters.

22 he Laue techque ca also be used to assess crystal perfecto from the sze ad shape of the spots. If the crystal has bee bet or twsted ayway, the spots become dstorted ad smeared out. POWDR MHOD Debye ad Scherrer Method: he Powder method was developed by Debye ad Scherrer Germay ad by Hll Amerca smultaeously. hs method s used to study the structure of the crystals whch ca ot be obtaed the form of perfect crystals of apprecable sze. herefore, the sample used s the form of a fe powder cotag a large umber of ty crystalltes wth radom oretatos. he X-ray Powder Method I practce, ths would be a tme cosumg operato to reoret the crystal, measure the agle θ, ad determe the d-spacg for all atomc plaes. A faster way s to use a method called the powder method. I ths method, a meral s groud up to a fe powder. I the powder, are thousads of gras that have radom oretatos. Wth radom oretatos we mght expect most of the dfferet atomc plaes to le parallel to the surface some of the gras. hus, by scag through a agle θ of cdet X-ray beams form 0 to 90 o, we would expect to fd all agles where dffracto has occurred, ad each of these agles would be assocated wth a dfferet atomc spacg. he strumet used to do ths s a x-ray powder dffractometer. It cossts of a X-ray tube capable of producg a beam of moochromatc X-rays that ca be rotated to produce agles from 0 to 90 o. A powdered meral sample s placed o a sample stage so that t ca be rradated by the X-ray tube. o

23 detect the dffracted X-rays, a electroc detector s placed o the other sde of the sample from the X-ray tube, ad t too s allowed to rotate to produce agles from 0 to 90 o. After a sca of the sample the X-ray testy ca be plotted agast the agle θ usually reported as θ because of the way older dffractometers were made to produce a chart, lke the oe show here. he agle θ for each dffracto peak ca the be coverted to d-spacg, usg the ragg equato. he strumet used to rotate both the X-ray tube ad the detector s called a goometer. he goometer keeps track of the agle θ, ad seds ths formato to a computer, whle the detector records the rate of X-rays comg out the other sde of the sample uts of couts/sec ad seds ths formato to the computer. Oe ca the work out the crystal structure ad assocate each of the dffracto peaks wth a dfferet atomc plae terms of the Mller Idex for that plae hkl. A group kow as the Jot Commttee o Powder Dffracto Stadards JCPDS has collected data such as ths o thousads of crystalle substaces. hs data ca be obtaed as the JCPDS Powder Dffracto Fle. Sce every compoud wth the same crystal structure wll produce a detcal powder dffracto patter, the patter serves as kd of a "fgerprt" for the substace, ad thus comparg a ukow meral to those the Powder Dffracto fle eables easy detfcato of the ukow. We wll see how ths s doe our laboratory demostrato.

24 PROLMS: 1. lectros are accelerated by 8.54 V ad are reflected by a crystal. he frst reflecto maxmum occurs whe the glacg agle s estmate the spacg of the crystal. Gve data: Accelerated Voltage V 854 V Glacg agle θ 56 0 Iter plaar spacg d? Dffracto order, 1 Prcple:. λ h mev ad. d λ s sθ soluto: λ λ h mev kg 1.6 c 854V λ λ λ m m 0.4 A 0.40 m λ s 56 d A

25 . he cocetrato of Schottky defects a oc crystal 1 at a temperature of 300. stmate the average separato terms of the lattce spacgs betwee the defects at 300 ad calculate the value of the cocetrato to be expected at 00. Gve data: Prcple: Soluto: e v v 1 e At 300, 0.05 ev 1 v 0.05 e e v e v 0.05 v e v 3 x ev v 1. ev 1 At e. e metre 3 may cota 19 defects hece, 1 metre wll cota 19 1/3.15 x 6 defects or

26 x -6 m wll be requred he desty of Schottky defects a certa sample of acl s 5 x 11 /m 3 at 5 0 c. If the teroc a + - cl - dstace s.8a 0, what s the eergy requred to create oe schottky defect. Gve data: Desty of Schottky defects ρ 5 x 11 /m 3 dstace r.8a 0 Prcple: exp -/ Soluto: he ut cell acl cotas four o pars ad ts volume s x.8 3 x x -8 m 3 Oe meter cube of a deal crystal cotas Io pars exp x 8.65 x ev a