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X-RAY DIFFRACTIONX-RAY DIFFRACTION

 X- Ray Sources

 Diffraction: Bragg’s La

 Crysta! Structure Deter"ination

 Elements of X-Ray Diffraction

B#D# Cu!!ity $ S#R# Stoc% &rentice 'a!!( )**er Sa++!e Ri,er .//01

 Recommended websites:  2tt*:33#"atter#org#u%3+iffraction3

 2tt*:33#ngsir#netfir"s#co"3eng!is22t"3Diffraction#2t"

4AT5RIALS SCI5NC54AT5RIALS SCI5NC5$$

5N6IN55RIN65N6IN55RIN6

Anandh Subramaniam & Kantesh Balani

4ateria!s Science an+ 5ngineering 4S51

In+ian Institute of Tec2no!ogy( 7an*ur- ./8/09

 Email:  anandh@iitk.ac.in, URL:  home.iitk.ac.in/anandh

AN INTROD)CTORY 5-BOO7 AN INTROD)CTORY 5-BOO7 

 !art of

htt"://home.iitk.ac.in/anandh/E-book.htm

A Learner’s GuideA Learner’s Guide

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'o to *ro+uce "onoc2ro"atic X-rays

'o +oes a crysta! scatter t2ese X-rays to gi,e a +iffraction *attern

→ Bragg’s e;uation out t2e !attice ty*e#$%bic crystal ty"es&.

e *ut to a*art fro" crysta! structure +eter"ination 6rain si=e +eter"ination  Strain in t2e "ateria!?

-c2a*ter’

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For e!ectro"agnetic ra+iation to >e +iffracte+ t2e s*acing in t2e grating aseries of o>stac!es or a series of scatterers1 s2ou!+ >e of t2e sa"e or+er as t2ea,e!engt2#

In crysta!s t2e ty*ica! interato"ic s*acing .- → so t2e suita>!e ra+iationfor t2e +iffraction stu+y of crysta!s is X-rays#

'ence( X-rays are use+ for t2e in,estigation of crysta! structures#  Neutrons an+ 5!ectrons are a!so use+ for +iffraction stu+ies fro" "ateria!s#  Neutron +iffraction is es*ecia!!y usefu! for stu+ying t2e "agnetic or+ering in

"ateria!s#

So"e Basics

Lattice *ara"eter of Cu aCu1 E #90 ⇒ +2%! is e;ua! to aCu or !ess t2an t2at e#g# +000 E aCu3√ E .#/8 1

If t2e a,e!engt2 is of t2e or+er of t2e !attice s*acing( t2en +iffraction effects i!! >e *ro"inent#C!ic% 2ere to %no "ore a>out t2is

C!ic% 2ere to %no "ore a>out t2is

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Beam of electrons Target X-rays

An acce!erating or +ece!erating1 c2arge ra+iates e!ectro"agnetic ra+iation

X-rays can >e generate+ >y +ece!erating e!ectrons#

'ence( X-rays are generate+ >y >o">ar+ing a target say Cu1 it2 an e!ectron >ea"#

T2e resu!tant s*ectru" of X-rays generate+ i#e# λX-rays ,ersus Intensity *!ot1 is s2on int2e net s!i+e# T2e *attern s2os intense *ea%s on a @>roa+’ >ac%groun+#

T2e intense *ea%s can >e @t2oug2t of’ as "onoc2ro"atic ra+iation an+ >e use+ for X-ray+iffraction stu+ies#

6eneration of X-rays

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4o Target i"*acte+ >y e!ectrons acce!erate+ >y a G %H *otentia! s2os t2e e"issions*ectru" as in t2e figure >e!o sc2e"atic1

T2e 2ig2 intensity near!y "onoc2ro"atic 7 α -rays can >e use+ as a ra+iation source for

X-ray +iffraction XRD1 stu+ies  a "onoc2ro"ator can >e use+ to furt2er +ecrease t2e

s*rea+ of a,e!engt2s in t2e X-ray

X-ray sources it2 +ifferent λ for

+oing XRD stu+iesTargetMetal

λ

 Of K  

radiation (Å)

4o /#0

Cu 0#GJ

Co 0#K

Fe 0#KJ

Cr .#.K

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Elements (KV)  λ Of K 

 

1 

radiation(Å)

 

λ Of K  

 

radiation (Å) 

λ Of K ! 

radiation (Å) 

K !"#ilter

(mm)

Ag .G#G. /#GGKJ0 /#G98 /#JK/ &+/#/J90

Mo ./ /#/K /#0GK /#9..K r  /#/98

$u 8#K8 0#GJ/GK8 0#GJJK 0#K... Ni/#/0

%i 8# 0#9GK0 0#990G 0#G//0J Co/#/0G8

$o #0 0#88K 0#K.8G 0#9./K Fe

/#/099#e #00 0#K9/J 0#KKK8 0#G990 4n

/#/098

$r G#KK .#.8K .#.K90 .#/8J8 H/#09K

$.'ordon Darwin, 'randson of $. Robert Darwin de(elo"ed the dynamic theory of scatterin) of *-rays #a to%)h theory+& in

X-ray sources it2 +ifferent λ for +oing XRD stu+ies

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A>sor*tion 'eat1

Inci+ent X-rays

S&5CI45N

Trans"itte+ >ea"

F!uorescent X-rays

5!ectrons

Co"*ton recoi! &2otoe!ectronsScattere+ X-rays

Co2erent

 rom bo%nd char)es

 X-rays can also be refracted #refracti(e inde* sli)htly less than & and reflected #at (ery small an)les&

M

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 No e s2a!! consi+er t2e i"*ortant to*ic as to 2o X-rays interact it2 a

crysta!!ine array of ato"s( ions etc#1 to gi,e rise to t2e *2eno"enon %non as X-

ray +iffraction XRD1# Let us consi+er a s*ecia! case of +iffraction a case 2ere e get @ shar"01 

diffraction "eaks2 # Diffraction with shar" "eaks1 it2 XRD >eing a s*ecific case1 re;uires t2ree i"*ortant con+itions to

 >e satisfie+:

Ra+iation re!ate+

 Co2erent( "onoc2ro"atic( *ara!!e! a,es$

 it2 a,e!engt2λ

1#Sa"*!e re!ate+ Crysta!!ine array of scatterers it2 s*acing of t2e or+er of 1 λ#Diffraction geo"etry re!ate+ Fraun2ofer +iffraction geo"etry $ t2is is actua!!y *art of t2e Fraun2ofer geo"etry1

01 3he intensity-θ  "lot looks like a 4 δ 2 f%nction #in an ideal sit%ation&.

5 6 7%asicrystalline array will also lead to diffraction with shar" "eaks #which we shall not consider in this te*t&.55 6mor"ho%s material will )i(e broadened #diff%se& "eak #additional factors related to the sam"le can also )i(e a diff%se "eak&.

Diffraction C!ic% 2ere to )n+erstan+ DiffractionP

Co2erent( "onoc2ro"atic( *ara!!e! a,e

Fraun2ofer geo"etry

Diffraction *atternit2 s2ar* *ea%s

Crysta!!ine(

 6s"ects related to the wa(e

 6s"ects related to the material 

 6s"ects related to the diffraction set-%"+iffraction geo"etry1

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T2e a,es cou!+ >e:

 e!ectro"agnetic a,es !ig2t( X-rays?1(

 "atter a,es e!ectrons( neutrons?1 or "ec2anica! a,es soun+( a,es on ater surface?1#  Not a!! o>Qects act !i%e scatterers for a!! %in+s of ra+iation# If a,e!engt2 is not of t2e or+er of t2e s*acing of t2e scatterers( t2en t2e nu">er

of *ea%s o>taine+ "ay >e 2ig2!y restricte+ i#e# e "ay e,en not e,en get a

sing!e +iffraction *ea%1# In s2ort +iffraction is co2erent reinforce+ scattering or reinforce+ scattering of co2erent a,es1# In a sense +iffraction is not2ing >ut a s*ecia! case of constructi,e $ +estructi,e1

interference#To gi,e an ana!ogy → t2e resu!ts of Young’s +ou>!e s!it e*eri"ent is inter*rete+ as interference( 2i!e t2e resu!t of

"u!ti*!e s!its !arge nu">er1 is categori=e+ un+er +iffraction#

Fraun2ofer +iffraction geo"etry i"*!ies t2at *ara!!e! a,es are i"*inging on t2e scatteres

t2e o>Qect1( an+ t2e screen to ca*ture t2e +iffraction *attern1 is *!ace+ far aay fro" t2e

o>Qect#

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A >ea" of X-rays +irecte+ at a crysta! interacts it2 t2e e!ectrons of t2e ato"s in t2e crysta!# T2e e!ectrons osci!!ate un+er t2e inf!uence of t2e inco"ing X-Rays an+ >eco"e secon+ary sources

of 54 ra+iation#

T2e secon+ary ra+iation is in a!! +irections# T2e a,es e"itte+ >y t2e e!ectrons 2a,e t2e sa"e fre;uency as t2e inco"ing X-rays ⇒ coherent. T2e e"ission can un+ergo constructi,e or +estructi,e interference#

XRD → t2e first ste*

8chematics

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etter *2ysica! *icture of +iffraction >y using Laue’s for"a!is" !ea+ing to t2e Laue’se;uations1#

'oe,er( a *ara!!e! a**roac2 to +iffraction is ,ia t2e "et2o+ of Bragg( 2erein +iffraction can >e

,isua!i=e+ as @ref!ections’ fro" a set of *!anes# As t2e a**roac2 of Bragg is easier to gras* e s2a!! use t2at in t2is e!e"entary tet#

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5tra *at2 tra,e!e+ >y inco"ing a,es → AY

A B

X Y

Ato"ic &!anes

5tra *at2 tra,e!e+ >y scattere+ a,es → XB

T2ese can >e in *2ase if → θinci+ent E θscattere+

A B

X Y

But t2is is sti!! reinforce+ scatteringand 93 reflection

Let us consi+er in-*!ane scattering

T2ere is "ore to t2is$lick here to know more and )et

introd%ced to La%e e7%ations describin)diffraction

http://laue_picture.ppt/http://laue_picture.ppt/

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BRA66’s 5)ATION

A *ortion of t2e crysta! is s2on for c!arity- actua!!y( for destr%cti(e interference to occur

"any *!anes are re;uire+ an+ t2e interaction ,o!u"e of -rays is !arge as co"*are+ to t2at s2on in t2e sc2e"atic1#  T2e scattering *!anes 2a,e a s*acing @+’#

Ray-. tra,e!s an etra *at2 as co"*are+ to Ray-0 E ABC1# T2e *at2 +ifference >eteen Ray- an+ Ray- E ABC E + Sinθ  + Sinθ1 E .+#Sinθ1# For constructi,e interference( t2is *at2 +ifference s2ou!+ >e an integra! "u!ti*!e of λ:

nλ E .+ Sinθ → t2e Bragg’s e;uation# #;ore abo%t this sooner&. T2e *at2 +ifference >eteen Ray-0 an+ Ray- is E .×.+#Sinθ1 E .×nλ E .nλ# T2is i"*!ies t2at if Ray-0

an+ Ray-. constructi,e!y interfere Ray-0 an+ Ray- i!! a!so constructi,e!y interfere# An+ so fort21#

Let us consi+er scattering across *!anes

C!ic% 2ere to ,isua!i=econstructi,e an+

+estructi,e interference

See Note Ӂ later

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T2e *re,ious *age e*!aine+ 2o constructi,e interference occurs# 'o a>out t2e rays Qust ofBragg ang!e O>,ious!y t2e *at2 +ifference ou!+ >e Qust off λ as in t2e figure >e!o# 

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Ref!ection ,ersus Diffraction

Ref!ection Diffraction

Occurs fro" surface

Occurs t2roug2out t2e >u!% 

#tho%)h often the "enetration of *-rays in only of theorder of s of microns in a material&

Ta%es *!ace at any ang!e Ta%es *!ace on!y at Bragg ang!es

0// U of t2e intensity "ay >e ref!ecte+ S"a!! fraction of intensity is +iffracte+

 9ote: X-rays can 6L8 be reflected at (ery small an)les of incidence

T2oug2 +iffraction accor+ing to Bragg’s *icture1 2as >een ,isua!i=e+ as a ref!ection fro" aset of *!anes it2 inter*!anar s*acing @+’→ +iffraction s2ou!+ not >e confuse+ it2ref!ection s*ecu!ar  ref!ection1#

&!anes are i"aginary constructs&!anes are i"aginary constructs

Laue ,ersus Bragg

In Laue’s *icture constructi,e an+ +estructi,e interference at ,arious *oints in s*ace isco"*ute+ using *at2 +ifferences an+ 2ence *2ase +ifferences1− gi,en a crysta!!ine array ofscatterers#

Bragg si"*!ifie+ t2is *icture >y consi+ering t2is *rocess as @ref!ections fro" ato"ic *!anes’#

C!ic% 2ere to %no "ore a>out t2e Laue &icture

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 nλ E .+ Sinθ

3he e7%ation is written better with some descri"ti(e s%bscri"ts:

 n is an integer an+ is t2e or+er of t2e ref!ection#i.e. how many wa(elen)ths of the X-ray )o on to make the "ath difference between "lanes&.

 9ote: if hkl reflection #corres"ondin) to nA& occ%rs at θ hkl  then h k l reflection #nA& will occ%r at a hi)her an)le θ h k l .

Bragg’s e;uation is a ne)ati(e state"ent If Bragg’s e;# is NOT satisfie+ →  NO @ref!ection’ can occur 

 If Bragg’s e;# is satisfie+ → @ref!ection’ ;6B occur #ut not t2e interato"ic

s*acing @a’ a!ong t2e *!ane 2ic2 2a+ force+ θinci+ent E θscattere+1V >ut e are notfree to "o,e t2e ato"s a!ong t2e *!ane @ran+o"!y’ → click here to know more.

   For !arge inter*!anar s*acing t2e ang!e of ref!ection ten+s toar+s =ero as + increases(

Sinθ +ecreases an+ so +oes θ1#

 T2e s"a!!est inter*!anar s*acing fro" 2ic2 Bragg +iffraction can >e o>taine+ is λ3. 

"ai"u" ,a!ue of θ is K/°( Sinθ is 0 ⇒ fro" Bragg e;uation + E λ3.#

)n+erstan+ing t2e Bragg’s e;uation

. Sin$% C hkl hkl  n   d α λ θ = If t2is e;uation is satisfie+( t2en θ is θBragg

 Note: Ӂ

http://laue_picture.ppt/http://laue_picture.ppt/

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 For Cu 7 α ra+iation λ E 0#GJ 1 an+ +00/E .#..

n Sinθ E nλ3.+   θ

0 /#J ./#W M First or+er ref!ection fro" 00/1 → 00/

. /#9K J#K.WM Secon+ or+er ref!ection fro" 00/1 *!anes → 00/

M A!so consi+ere+ as first or+er ref!ection fro" ../1 *!anes → ../

. . .

$%bic crystal hkl 

ad 

h k l 

=+ +

8../

ad    =

.

00/

ad    =

.

0

00/

../ =d 

d 

Re!ation >eteen +n2 n% n! an+ +2%!

e#g#

. . . 1 1 1nhnknl 

ad 

nh nk nl  =

+ +

. . .

hkl 

nhnk nl 

d a

d  nn h k l  = =+ +

Or+er of t2e ref!ection n1

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. sinhkl hkl  n d λ θ =

In XRD nt2 or+er ref!ection fro" 2 % !1 is considered  as 0st or+er ref!ection fro" n2 n% n!1

θ λ    sin.n

d hkl 

=

n n n n n n. sinh k l h k l  d λ θ =

0nh nk nl 

hkl 

d 

d n=

//

0//

0

d 

d   =

.//

0//

0

.

d 

d    =

 ic crysta!( 0//( .//( //? are a!! a!!oe+ @ref!ections’# But( t2ere are no ato"s in t2e

 *!anes !ying within t2e unit ce!! T2oug2( first or+er ref!ection fro" .// *!anes is e;ui,a!ent"at2e"atica!!y1 to t2e secon+ or+er ref!ection fro" 0// *!anesV for ,isua!i=ation *ur*oses ofscattering( t2is is >etter t2oug2t of as t2e !ater *rocess i#e# secon+ or+er ref!ection fro" 0//1 *!anes1#

 Note:Tec2nica!!y( in 4i!!er in+ices e factor out t2e co""on factors# 'ence( ../1 ≡ .00/1 ≡ 00/1#In XRD e eten+ t2e usua! conce*t of 4i!!er in+ices to inc!u+e *!anes( 2ic2 +o not *ass t2roug2!attice *oints e#g# e,ery a!ternate *!ane >e!onging to t2e //.1 set +oes not *ass t2roug2 !attice *oints1

an+ e a!!o t2e co""on factors to re"ain in t2e in+ices#

A!! t2ese for" t2e .//1 set

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I 2a,e seen +iagra"s !i%e in Fig#0 2ere rays see" to >e scattere+ fro" not2inge;uantifie+ using t2e ato"ic scattering *oer an+ t2e density of atoms in the "lane1# Due to t2e@inco"ing’ a,e t2e ato"ic +i*o!es are set into osci!!ation( 2ic2 furt2er act !i%e e"itter of a,es

In Bragg’s ,ie*oint( t2e ato"ic *!anes are to >e %e*t in focus an+ t2e a,e #not %st a ray& i"*ingeson t2e entire *!ane so"e *!anes 2a,e ato"s in contact an+ "ost 2a,e ato"s 2ic2 are not in contacta!ong t2e *!ane  see Fig#.1#

5 3o be considered later

 6 "lane in >ra))2s (iew"oint can be characteried by two factors: #a& atomic density #atoms/%nit area on the "lane&, #b& atomic scatterin) factorof the atoms.

Fig#0

Fig#.

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It is +ifficu!t to gi,e an e*!anation of t2e nature of t2e se"i-trans*arent !ayers or

 *!anes t2at is i""e+iate!y con,incing( as they are a conce"t rather than a "hysical

reality# Crysta! structures( it2 t2eir regu!ar!y re*eating *atterns( "ay >e referre+ to aD gri+ an+ t2e re*eating unit of t2e gri+( t2e unit ce!!( can >e foun+# T2e gri+ "ay >e

+i,i+e+ u* into sets of *!anes in ,arious orientations an+ it is t2ese *!anes 2ic2 are

consi+ere+ in t2e +eri,ation of Bragg’s !a# In so"e cases( it2 si"*!e crysta!

structures( t2e *!anes a!so corres*on+ to !ayers of ato"s( >ut t2is is not genera!!y t2e

case# See Section 0#G for furt2er infor"ation# So"e of t2e assu"*tions u*on 2ic2 Bragg’s !a is >ase+ "ay see" to >e rat2er

+u>ious# For instance( it is %non t2at +iffraction occurs as a resu!t of interaction

 >eteen X-rays an+ ato"s# Furt2er( t2e ato"s +o not ref!ect X-rays >ut scatter or +iffract

t2e" in a!! +irections#  9e(ertheless, the hi)hly sim"lified treatment that is %sed in

deri(in) >ra))2s law )i(es e*actly the same answers as are obtained by a ri)oro%smathematical treatment.  ear in "in+ t2at e are

fortunate to 2a,e suc2 a si"*!e an+ *ictures;ue( a!>eit inaccurate( ay to +escri>e 2at

in rea!ity is a ,ery co"*!icate+ *rocess#P 0

01 6nthony R Fest, 8olid 8tate $hemistry and its 6""lications, 8econd Edition, Gohn Filey H 8ons Ltd., $hichester, I.

 ;ore abo%t the >ra))2s (iew"oint 

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'o is it t2at e are a>!e to get infor"ation a>out !attice *ara"eters of t2e or+erof Angstro"s ato"s 2ic2 are so c!ose!y s*ace+1 using XRD

Fun+a C2ec% 

Diffraction is a *rocess in 2ic2

4linear information2 #the d-s"acin) of the "lanes&

is con,erte+ to 4an)%lar information2  #the an)le of diffraction, θ  >ra))  &.

If t2e +etector is *!ace+ @far aay’ fro" t2e sa"*!e i#e# @R’ in t2e figure >e!o is !arge1 t2e

+istances a!ong t2e arc of a circ!e t2e +etection circ!e1 get a"*!ifie+ an+ 2ence e can "a%e@easy’ "easure"ents#

T2is a!so i"*!ies t2at in XRD e are concerne+ it2 an)%lar resol%tion instea+ of !inearreso!ution#

Later e i!! see t2at in *o+er+iffraction t2is ang!e of +e,iation.θ1 is *!otte+ instea+ of θ#

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Forar+ an+ Bac% Diffraction

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Fun+a C2ec%   eteen t2e inci+ent -rays an+ t2e set of *ara!!e! ato"ic *!anes 2ic2 2a,ea s*acing +2%!1# Fhich is °  in the abo(e fi)%re.

Us%ally, θ  in this conte*t im"lies θ  >ra))  #i.e. the an)le at which >ra))2s e7%ation is

 satisfied&.

It is NOT t2e ang!e >eteen t2e -rays an+ t2e sa"*!e surface note: s*eci"ens cou!+ >es*2erica! or cou!+ 2a,e a roug2 surface1#

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ecause Bragg’se;uation is satisfie+ a @ref!ection’ "ay not >e o>ser,e+#

Let us consi+er t2e case of Cu 7 α ra+iation λ E 0#GJ 1 >eing +iffracte+ fro" 0//1

 *!anes of 4o BCC( a E #0G E +0//1#

T2e "issing @ref!ections’

0// 0//.d 8inλ θ = 0//0//

0#GJ/#.JJ

. .#0G18in

d 

λ θ    = = = 0//   0J#0JKθ    = °

But t2is ref!ection is a>sent in BCC 4o

T2e "issing ref!ection is +ue to t2e *resence of a++itiona! ato"s in t2e unit ce!!

 2ic2 are *ositions at !attice *oints1 → 2ic2 e s2a!! consi+er net

T2e a,e scattere+ fro" t2e "i++!e *!ane is out of *2ase

it2 t2e ones scattere+ fro" to* an+ >otto" *!anes# I#e# ift2e green rays are in *2ase *at2 +ifference of λ1 t2en t2e

re+ ray i!! >e eact!y out of *2ase it2 t2e green rays

*at2 +ifference of λ3.1#

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'oe,er( t2e secon+ or+er ref!ection fro" 0//1 *!anes 2ic2 is e;ui,a!ent to t2e first or+erref!ection fro" t2e .//1 *!anes is o>ser,e+

0//

0//

. 0#GJ/#J8

. #0G

8in

d 

λ θ    = = =   . 0

0// .//B .K#.9

nd nd  order order  θ θ    = °

T2is is >ecause if t2e green rays 2a,e a *at2 +ifference of .λ t2en t2e re+ ray i!! 2a,e *at2+ifference of λ 2ic2 i!! sti!! !ea+ to constructi,e interference

$ontin%in) with the case of >$$ ;oJ

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⇒ &resence of a++itiona! ato"s3ions3"o!ecu!es in t2e )C at !attice *oints #as we may chose a non-"rimiti(e %nit cell&

 or as a *art of t2e "otif   can alter the intensities of some of the reflections⇒ So"e of t2e ref!ections "ay e,en go "issing

I"*ortant

 *oints

&osition of t2e @ref!ections’3@*ea%s’ te!!s us a>out t2e !attice ty*e#

T2e Intensities te!!s us a>out t2e "otif#

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Intensity of t2e Scattere+ e consi+ere+:01 scattering fro" e!ectrons.1 scattering fro" an ato"1 scattering fro" a unit ce!!

$lick here to know the details

Structure factor ca!cu!ations

$Intensity in *o+er *atterns

Structure factor ca!cu!ations$

Intensity in *o+er *atterns

Structure Factor F1: T2e resu!tant a,e scattere+ >y a!! ato"s of t2e unit ce!!

T2e Structure Factor is in+e*en+ent of t2e s2a*ean+ si=e of t2e unit ce!!V >ut is +e*en+ent on t2e *osition of t2e ato"s3ions etc# it2in t2e ce!!

C!ic% 2ere to %no "ore a>out

Bragg’s e;uation te!!s us a>out t2e *osition of t2e +iffraction *ea%s in ter"s of θ1 → >utte!!s us not2ing a>out t2e intensities# T2e intensities of t2e *ea%s +e*en+ on "any factors asconsi+ere+ 2ere#

http://structure_factor_calculations.ppt/http://structure_factor_calculations.ppt/http://structure_factor_calculations.ppt/http://structure_factor_calculations.ppt/http://structure_factor_calculations.ppt/

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T2e conce*t of a e'iro'al latti'e an+ t2e Eald Shere 'onstru'tion:

Reci*roca! !attice an+ 5a!+ s*2ere constructions are i"*ortant too!s toar+sun+erstan+ing +iffraction#es*ecia!!y +iffraction in a Trans"ission 5!ectron 4icrosco*e T5411

A !attice in 2ic2 *!anes in t2e rea! !attice >eco"e *oints in t2e reci*roca! !attice is a,ery usefu! one in un+erstan+ing +iffraction#

 click here to go to a +etai!e+ +escri*tion of t2ese to*ics#

Reci*roca! Lattice $ 5a!+ S*2ere constructionReci*roca! Lattice $ 5a!+ S*2ere constructionC!ic% 2ere to %no "ore a>out

S ! ti 3 5 ti ti R !

http://reciprocal_lattice.ppt/http://reciprocal_lattice.ppt/http://reciprocal_lattice.ppt/http://reciprocal_lattice.ppt/http://reciprocal_lattice.ppt/

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Bra*ais +atti'e efle'tions hi'h may be resent efle'tions ne'essaril, absent

Si"*!e a!! None

Bo+y centre+ 2 % !1 e,en 2 % !1 o++

Face centre+ 2( % an+ ! un"ie+ 2( % an+ ! "ie+

5n+ centre+ $ centred& 2 an+ % un"ie+ 2 an+ % "ie+

Bra*ais +atti'e Alloed efle'tions

SC A!!

BCC 2 % !1 e,en

FCC 2( % an+ ! un"ie+

DC5it2er(  2( % an+ ! are a!! o++ or  

 a!! are e,en H #h k l& di(isible by I

Se!ection 3 5tinction Ru!es

 6s we ha(e noted before e(en if >ra))2s e7%ation is satisfied, 4reflections may )o missin)2→  this is d%e to the "resence of additional atoms in the %nit cell.

3he reflections "resent and the missin) reflections d%e to additional atoms in the %nit cell are

listed in the table below. C!ic% 2ere to see t2e +eri,ationsStructure factor ca!cu!ationsC!ic% 2ere to see t2e +eri,ationsStructure factor ca!cu!ations

h - . - l S$ #$$ B$$ /$

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h - .  - l S$ #$$ B$$ /$

0 0//

. 00/ 00/

000 000 000

J .// .// .//G .0/

9 .00 .00

8 ../ ../ ../ ../

K //( ..00/ 0/ 0/

00 00 00 00

0. ... ... ...

0 ./

0J .0 .0

0G

09 J// J// J// J//

0 J0/( ..

08 J00( / J00( /

0K 0 0 0

A!!oe+ ref!ectionsin SC( FCC( BCC

$ DC crysta!s

!attice +ecorate+ it2

"onoato"ic3"onoionic "otif 

Cannot >e e*resse+ as 2.% .!.1

Crysta! structure +eter"ination

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Crysta! structure +eter"ination

4onoc2ro"atic X-rays

0anc2ro"atic X-rays

4onoc2ro"atic X-rays

4any θs orientations1&o+er s*eci"en

&Oy rotation

ROTATIN6CRYSTAL45T'OD

Z fie+

[ ,aria>!e

→

→

Z fie+

[ rotate+

→

→

Z ,aria>!e

[ fie+

→

→

As +iffraction occurs on!y at s*ecific Bragg ang!es( t2e c2ance t2at a ref!ection iso>ser,e+ 2en a crysta! is irra+iate+ it2 "onoc2ro"atic X-rays at a *articu!arang!e is s"a!! #added to this the diffracted intensity is a small fraction of the beam %sed

 for irradiation&. T2e *ro>a>i!ity to get a +iffracte+ >ea" it2 sufficient intensity1 is increase+ >y

eit2er ,arying t2e a,e!engt2 λ1 or 2a,ing "any orientations rotating t2ecrysta! or 2a,ing "u!ti*!e crysta!!ites in "any orientations1#

T2e t2ree "et2o+s use+ to ac2ie,e 2ig2 *ro>a>i!ity of +iffraction are s2on >e!o#

nly the "owder method #which is commonly %sed in materials science& will be considered in this te*t.

T'5 &O

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T'5 &O

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T2e ratio of 2.  % .  !.1 +eri,e+ fro" etinction ru!es #earlier "a)e&As e s2a!! see soon t2e ratios of 2.  % .  !.1 is *ro*ortiona! to Sin.θ 

 2ic2 can >e use+ in t2e +eter"ination of t2e !attice ty*e

SC 0 . J G 9 8 ?

BCC 0 . J G 9 ?

FCC J 8 00 0. ?

DC 8 00 09 ?

 Note t2at e 2a,e to consi+er t2e ratio of on!y to !ines to +istinguis2 FCC an+ DC# I#e# if t2eratios are :J t2en t2e !attice is FCC#

But( to +istinguis2 >eteen SC an+ BCC e 2a,e to go to !ines

&O

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In t2e *o+er sa"*!e t2ere are crysta!!ites in +ifferent @ran+o"’ orientations a *o!ycrysta!!ine sa"*!etoo 2as grains in +ifferent orientations1

T2e co2erent -ray >ea" is +iffracte+ >y t2ese crysta!!ites at ,arious ang!es to t2e inci+ent +irection A!! t2e +iffracte+ >ea"s ca!!e+ @ref!ections’1 fro" a sing!e *!ane( >ut fro" +ifferent crysta!!ites !ie on

a cone# De*en+ing on t2e ang!e t2ere are forar+ an+ >ac% ref!ection cones# A +iffracto"eter can recor+ t2e ang!e of t2ese ref!ections a!ong it2 t2e intensities of t2e ref!ection T2e X-ray source an+ +iffracto"eter "o,e in arcs of a circ!e- "aintaining t2e Bragg @ref!ection’

geo"etry as in t2e figure rig2t1

&O

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* g

 Mt is 4somewhat diffic%lt2 to act%ally (is%alie a random assembly of crystallites )i(in) "eaks at (ario%s an)els in a XRD scan. 3he fi)%res below are e*"ected to )i(e a 4(is%al feel2 for the same. 0!y oriente+ crysta!!ites

 h h.l + Sinθ1   θ

0 0// J#// /#0K 00#0/

. 00/ .#8 /#. 0G#8/

000 .#0 /# 0K#J8

J .// .#// /#K ..#9J

G .0/ 0#K /#J .G#G/

9 .00 0#9 /#J .8#0

8 ../ 0#J0 /#GJ .#KK

K // 0# /#G8 G#.

0/ 0/ 0#.9 /#90 #G/

For con,enience t2e source"ay >e stationary an+ t2esa"*!e an+ +etector "ay

rotate\ >ut t2e effect ise;ui,a!ent1

Deter"ination of Crysta! Structure fro" .θ ,ersus Intensity Data in &o+er 4et2o+

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In t2e *oer +iffraction "et2o+ a .θ ,ersus intensity I1 *!ot is o>taine+ fro" t2e+iffracto"eter an+ associate+ instru"entation1#

T2e @intensity’ is t2e area un+er t2e *ea%  in suc2 a *!ot #  93  the hei)ht of the "eak&.

 T2e infor"ation of i"*ortance o>taine+ fro" suc2 a *attern is t2e @re!ati,e intensities’ an+ t2e a>so!ute ,a!ue of t2e intensities is of !itt!e i"*ortance t2e !onger e irra+iate t2e sa"*!e− t2e "ore i!! >et2e intensity un+er t2e *ea%1 for no1#   M  is rea!!y +iffracte+ energy as Intensity is Ener)y/area/time1#

A ta>!e is *re*are+ as in t2e net s!i+e to ta>u!ate t2e +ata an+ "a%e ca!cu!ations to fin+ t2ecrysta! structure restricting ourse!,es to cu>ic crysta!s for t2e *resent1#

Deter"ination of Crysta! Structure fro" .θ ,ersus Intensity Data in &o+er 4et2o+

 !owder diffraction "attern from 6l 

 Radiation: $% C α  , λ  A .OI N

Increasing θ

Increasing +

Intensity I1 2as units of5nergy3area3ti"e >ut 2ere it

is *!otte+ as arbitrary %nits#

)sua!!y in +egrees °1

T2is is *ea% so"eti"es ca!!e+ a!ine- a 2ango,er fro" De>yeSc2errer ca"era usage1

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n .θ θ Intensity Sinθ Sin. θ ratio

Deter"ination of Crysta! Structure !attice ty*e1 fro" .θ ,ersus Intensity Data

3he followin) table is made from the θ   (ers%s Mntensity data #obtained from a XRD e*"eriment on a "owder sam"le #em"ty startin) table of col%mns is shown below- com"leted table shown later&.

d d ff f l d λ N

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 !owder diffraction "attern from 6l Radiation: $% C α  , λ  A .OI N

 Note: T2is is a sc2e"atic *attern

In rea! *atterns *ea%s or not i+ea!i=e+ δ *ea%s → >roa+ene+ Increasing s*!itting of *ea%s it2 ↑g θ

α  H α  "eaks )et resol(ed in the hi)h an)le "eaks&

&ea%s are a!! not of sa"e intensity

 9o  >rac%ets are use+ aroun+ t2e in+ee+ nu">ers#the "eaks corres"ond to "lanes in the real s"ace&

 Note t2at t2ere are no >rac%etsaroun+ t2e indices

! d diff ti tt f 6l Radiation: $% Cα λ A OI N

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 !owder diffraction "attern from 6l 

   0   0   0

   .   /   /

   .   .   /

   C   0   0

   .   .   .    J   /   /

 C α  H C α  "eaks resol(ed in hi)h an)le "eaks#in and I "eaks this can be seen&

 Radiation: $% C α  , λ   .OI N

 Note: &ea%s or not i+ea!i=e+ δ *ea%s → >roa+ene+#

Increasing s*!itting of *ea%s it2 ↑g θ#

&ea%s are a!! not of sa"e intensity#

T2ere is a @noisy’ >ac%groun+#

 Mn low an)le "eaks C α  H C α  "eaks mer)ed 

Fun+a C2ec%

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Fhat is the ma*im%m (al%e of θ  "ossible #e*"erimentally&=

Fun+a C2ec%   e a!tere+

 $o%ld be d%e to te*t%re in the sam"le.

Fun+a C2ec%   6ns: ° 

At θ E K/° t2e @ref!ecte+ ray’ is o**osite in+irection to t2e inci+ent ray#

Beyon+ t2is ang!e( it is as if t2e source an++etector *ositions are sitc2e+#

  ⇒ .θ"a is 08/°#

Instru"enta! >roa+ening

Crysta! +efects #4bent2 "lanes&

&ea% Broa+ening S"a!! crysta!!ite si=e

 Note *ea% s*!itting 2as not >een inc!u+e+ 2ere as t2is

co"es fro" 4symmetrylowerin)2 #i.e. crystal with

lower symmetry&

 Mncl%din) those comin) from strain fields associated with these defects Click here to know moreClick here to know more

h ll d h k ll =

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Fun+a C2ec%  Fhat will determine how many "eaks M will )et=

01 λ  s"a!!er t2e a,e!engt2 of t2e X-rays( "ore i!! >e t2e nu">er of *ea%s *ossi>!e#

 Fro" Bragg’s e;uation: λE.+Sinθ( Sinθ1"a i!! corres*on+ to +"in# Sinθ1"aE0#

'ence( +"inEλ3.# 'ence( if λ is s"a!! t2en *!anes it2 s"a!!er + s*acing i#e# t2ose 2ic2

occur at 2ig2er .θ ,a!ues1 i!! a!so s2o u* in a XRD *atter *o+er *attern1# 'i(en thate*"erimentally θ  cannot be )reater than ° .

.1 Lattice ty*e in SC e i!! get "ore *ea%s as co"*are+ to say1 FCC3DC# ther thin)s bein)e7%al.

1 Loer t2e sy""etry of t2e crysta!( "ore t2e nu">er of *ea%s e#g#( in tetragona! crysta!t2e 0// *ea% i!! !ie at a +ifferent .θ as co"*are+ to t2e //0 *ea%1#

.d8inλ θ =( )

  "in

"aF

.d 8in

λ 

θ =

"in.

d λ 

=

D t i ti f C t ! St t ! tti t 1 f .θ I t it D tSo!,e+ ea"*!e

So!,e+ ea"*!e

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θ θ

Sinθ

Sinθ

ratio 2nde3 d

0 8#G. 0K#.9 /# /#00 Q 000 .#J

. JJ#9 ..#8 /#8 /#0J I .// .#/

9G#0J .#G /#GJ /#.K ../ 0#J

J 8#.9 K#0 /#9 /#J/ 00 0#..

G 8.#J J0#.G /#99 /#J ... 0#0

9 KK#00 JK#GGG /#9 /#G8 S  J// 0#/0

00.#/ G9#/0G /#8 /#9K 0 /#K

8 009#9/ G8# /#8G /#. J./ /#K0

K 0#J 98#G /#K /#8 I J.. /#8

0/ 09#8 80#8K /#KK /#K8 T  /#8

Deter"ination of Crysta! Structure !attice ty*e1 fro" .θ ,ersus Intensity Data

Fro" t2e ratios in co!u"n 9 e conc!u+e t2at FCC

 Let %s ass%me that we ha(e the θ  (ers%s intensity "lot from a diffractometer  3o know the lattice ty"e we need only the "osition of the "eaks #as tab%lated below&

. d 8inλ θ =000 0000#GJ . . /#CC

C

ad 8inθ = =

o

J#/JAa 6l = →

)sing

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.θ θ Sinθ Sin. θRatios

of Sin.θDi,i+ing Sin.θ >y

/#0J3 E /#/JJ99

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More Sol*edE3amles on 4/

$lick here

Co"*arison of +iffraction *atterns of SC( BCC $ B. structures

$lick here

Fhat ha""ens when we increase or decrease λ=

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Fun+a C2ec% Fhat ha""ens when we increase or decrease λ =

Fe had "ointed o%t that λ   a is "referred for diffraction. Let %s see what ha""ens if we 4drastically2 increase or decrease λ .#3his is only a tho%)ht e*"eriment++&

If e +ou>!e λ  e gettoo fe *ea%s

If e "a%e λ s"a!!a!! t2e *ea%s getcro+e+ to s"a!!

ang!es

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Bra,ais !attice +eter"ination

Lattice *ara"eter +eter"ination

Deter"ination of so!,us !ine in *2ase +iagra"s

Long range or+er 

A**!ications of XRD

Crysta!!ite si=e an+ Strain

Deter"ine if t2e "ateria! is a"or*2ous or crysta!!ine

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Diffraction ang!e .θ1

   I  n   t  e  n  s   i   t  y

K/ 08//

K/ 08//

Diffraction ang!e .θ1

   I  n   t  e  n  s   i   t  y

  N Li;ui+ 3 A"or*2ous so!i+

K/ 08//

Diffraction ang!e .θ1    I  n   t  e  n  s   i   t  y  N

4onoato"ic gas

Sc2e"atic of +ifference >eteent2e +iffraction *atterns of ,arious *2ases

S2ar* *ea%s

Diffuse &ea% 

 No *ea% 

 s

A"or*2ous so!i+

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Diffuse *ea% fro"Cu-r-Ni-A!-Si4eta!!ic g!ass

XRD *atterns1 courtesy: Dr# 7a!!o! 4on+a!( 4S5( IIT7 

 6ct%al diffraction "attern from anamor"ho%s solid 

A a"or*2ous so!i+ 2ic2 s2os g!ass transition in a Differentia! Scanning Ca!ori"etry DSC1 *!ot is a!so ca!!e+ a g!ass# Mn 4)eneral %sa)e2 a )lass may be considered e7%i(alent to aamor"ho%s solid #at least loosely in the str%ct%ral sense&.

S2ar* *ea%s are "issing# Broa+ +iffuse *ea% sur,i,esP the "eak corres"onds to the a(era)e s"acin) between atoms which the diffraction e*"eriment 4"ickso%t2 

Fun+a C2ec%F

un+a C2ec%    

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Fun+a C2ec% Fun+a C2ec%  < at s t e u s*ac g >etee * a es *oss > e a c ysta  'o "any +iffraction *ea%s can e get fro" a *o+er *attern

. . .

$%bic crystal 

hkl ad 

h k l =

+ +

Let us consi+er a cu>ic crysta! it2out !oss in genera!ity1

As 2(%( ! increases( @+’ +ecreases ⇒ e cou!+ 2a,e *!anes it2 infinitesimal  s*acing

0/

0

ad a= =

00

.

ad    =

00/

ad    =

0.

G

ad    =

CJ G.G

a a

d   = =

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$ A o to c ease t e u >e o *ea s a *atte

e insufficient for a gi,en ana!ysis#

T2e nu">er of *ea%s can >e increase+ in to ays:

01 using 4o 7  instea+ of Cu 7_.1 first o>tain *attern it2 ` fi!ter an+ t2en re"o,e t2e fi!ter to get "ore !ines#

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