x-ray diffraction from materials - postechphome.postech.ac.kr/user/atl/note/chapter 8-2.pdf · 8.3...

$: X-ray Diffraction from Materials - POSTECHphome.postech.ac.kr/user/atl/note/Chapter 8-2.pdf · 8.3 The Integrated X-ray Intensity of a Diffraction Peak Integrated intensity: the shade$
X-ray Diffraction from Materials

2008 Spring SemesterLecturer; Yang Mo KooLecturer; Yang Mo Koo

Monday and Wednesday 14:45~16:00

8.2 X-ray DiffractometersDiffractometer: a measuring instrument for analyzing the structure of a usually crystallineDiffractometer: a measuring instrument for analyzing the structure of a usually crystallinesubstance from the scattering pattern produced when a beam of radiation or particles (as X rays or neutrons) interacts with it.

Goniometer: an instrument that either measures angle or allows an object to be rotated to a precise angular position.

Eulerian cradle (2-circle)

Diff t t ith 4 i l i t

Eulerian cradle (2 circle)

X-ray & AT Laboratory, GIFT, POSTECH

Diffractometer with 4-circle goniometer

8.2 X-ray Diffractometers

4-circle goniometer

χχ

φ Full circle Eulerian cradle

θ2 ω

Eulerian cradle for texture measurement


Eulerian cradle for texture measurement


6-circle goniometer Two circles (θ and 2θ) at detector side



Diffractometer with 2 circle goniometerDiffractometer with 2-circle goniometer

Two rotation axes are ω and 2θ. Focusing condition reflected beam from theFocusing condition reflected beam from the

sample: ω =θ.



Diffraction condition for a single crystal Diffraction condition for poly-crystalor powder sample



Focusing condition of reflected beam from sample surface

Rθsin2

Rr =r

R

rr



Slit system of powder diffractometer



Elements effecting diffraction peak shape

)(xg )(xh

)(fspecimen )x(f

h

)x(g)x(f)x(h ⊗=

specimen of surface flatsourceray x

where

(x): g (x): g

(x)g(x)g(x)g(x)g(x)g(x)g g(x)

2

1

654321

−⊗⊗⊗⊗⊗=

slit reciving of sizespecimen the inray - xofcy transparen

beamray xof divergence vertical

(x): g (x): g (x): g

5

4

3 −


ntmisalignmeg

(x): g ( )g

6

5


Monochromator



Diffractometer with 4 circle goniometerDiffractometer with 4-circle goniometer



Diffraction condition for 4 circle goniometerDiffraction condition for 4-circle goniometer

In order to get diffraction peak of hkl, one can move reciprocal point to diffraction planeby rotating a combination of ω, φ, and χ. Any diffraction pot can be moved by rotating twodiff l h d d d d H i f ll hdifferent angles such as ω and φ, φ and χ, and ω and χ. However, rotations of all three angles are used for real experiment because of geometric restriction of the diffractometer. 2θ circle is used for moving detector to the reflected x-ray beam position. The angle, ωset θ for keeping focusing condition of reflected beam, and φ, and χ are adjusted to moveset θ for keeping focusing condition of reflected beam, and φ, and χ are adjusted to move reciprocal lattice point on the diffraction plane.

The coordinates system in 4-circle goiniometer

eter.diffractom to related system lorthonorma an :meterdiffractor of system scoordinate the (I)

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

D

D

D

D cba

A

⎠⎝ Dc



)2-(2π angle the bisects always and 2θ positive for vector ndiffractio the to parallel is b

where

D

θ

.instrument the of axis main the withcoincident is 0.2θ whensourceray xthe

toward directed is b o lartperpendicu plane, ndiffractio the in is D

D

D

c

a=−

D

system lorthonorma an :specimen of system coordinate the (II)a ⎞⎛

C

cba

A⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

C

C

C

axisthewithcoincidentalwaysis 0. when A withcoincident is

that such defined are axes The specimen. the in fixed

DC

C

φχφω

cA

A===

axis.thewithcoincident alwaysis C φc



cellunit of axesCrystal (III)

ba

A⎟⎟⎟⎞

⎜⎜⎜⎛

=

cell unit of parameters lattice are , , where cbac ⎟⎠⎜⎝

Orthogonal axes system

⎟⎟⎞

⎜⎜⎛

−=

=⊥

γγβαγβγ

sin/)coscos(coscsinbcosccosb

where 0a

T

TXX

⎟⎟

⎠⎜⎜

⎝ −+− γγβαβαγγβγ

sin/)coscoscoscossin(c)(

/ 2122 2100

This matrix often used for calculating |X| g | |

( ) 2/1 ⊥⊥⊥ == XXXX T


8.2 X-ray DiffractometersTherefore

( ) ( ) ( ) 2/12/12/12/1

Therefore

GXXXTTXTXTXTXTXX TTTTTT ====⊥

G is called metric tensor

TTG bcbacab

⎟⎟⎟⎞

⎜⎜⎜⎛

== coscosabcoscosa

2

2

T αγβγ

G is called metric tensor

AAbbbbcabaaa

T

cbcca

⎟⎟⎞

⎜⎜⎛ ⋅⋅⋅

⎟⎠

⎜⎝ coscos 2αβ

AAccbcaccbbbab T=⎟⎟

⎠⎜⎜

⎝ ⋅⋅⋅⋅⋅⋅=

If a es s stem is orthogonal metric tensor become nit matriIf axes system is orthogonal, metric tensor become unit matrix;

IG ⎟⎟⎞

⎜⎜⎛

010001

IG =⎟⎟

⎠⎜⎜

⎝

=100010



axescrystalReciprocal(IV)

axes crystal Reciprocal (IV)

*

*

* ba

A⎟⎟⎟⎞

⎜⎜⎜⎛

=

vectors unit reciprocal are , , where ***

*

cba

c ⎟⎠

⎜⎝

The magnitude of reciprocal vector can be estimated by metric tensor

21/**T** XGXX

and

T XGXX =

⎞⎛*** AA

ccbcaccbbbabcabaaa

G******

******

******

T=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⋅⋅⋅⋅⋅⋅⋅⋅⋅

=ccbcac ⎠⎝



( )zyxHAAA

≡system and , , the in space reciprocal in vector a of scoordinate The

vector reciprocal a of sCoordinate (V)*

DC

( )( )( )

DDD

CCC

l,k,hz,y,xz,y,x

hHHH

≡≡≡

ittbti lth th

D

C

DDDDDD

***

zyxlhh

cbaAHcbaHAh

h

DD

*

++==++==

writtenbemay vector reciprocalathat such

CCCCCC zyx cbaAH CC ++==

andspecimenofrotationafterandbetweeniprelationshThe AA φχω

by defined is matrix tiontransforma where

and,,specimen of rotationafterandbetween iprelationshThe DC

FFAA

AA

DC =φχω



⎟⎟⎞

⎜⎜⎛

⎟⎟⎞

⎜⎜⎛

⎟⎟⎞

⎜⎜⎛

=

ωωωω

χχφφφφωχφ

0cossin0scos

scos0001

0cossin0sincos

)()()( in

in

RRRF

⎟⎞

⎜⎛ +−

⎟⎟

⎠⎜⎜

⎝⎟⎟

⎠⎜⎜

⎝⎟⎟

⎠⎜⎜

⎝

=

χφχωφωφχωφωφ

ωωχχχχφφ

sscoscossscoscossscoscos

1000cossin-

cossin-0scos0

1000cossin-

inininininin

in

⎟⎟⎟

⎠⎜⎜⎜

⎝

+−−=χωχωχχφχωφωφχωφωφχφχφφχφφ

coscossin-ssscoscoscoscosssin-cosscoscoss

inininininin

or that follows It

TD

TC

1DC FXXFXX == −

thus andmatrix unitary is F since

o

T1

DCDC

FF =−


8.2 X-ray DiffractometersmatrixnorientatioThe(VII)

l tti d diht i )tlii( hi hUt iTh

that such Umatrix norientatio an introduce to convinient is It matrix norientatioThe (VII)

C* UAA =

n,orientatio the of tindependen obtained, bemay constants cell The on.oreientatispecimen the describing angles three and constants sellsix the to related are which

elementstindependen ninehassymmetric)notgeneralin is (which Umatrix The

sinceobtain,wewhichfrom

transposeA its and Aof product the formingby

TCC

TTCC

T**1-

T** .

IAA

UAUAAAG

=

==

bt i dbt lthft tllitthhence and constants cell reciprocal the product the formingby Thus,

since obtain, wewhichfrom

T

T1-CC

,UUUUG

IAA

=

obtained.bemay crystaltheof constantcellunit the



Calculation of setting angles at 4 circle goiniometer

( )H *d 00

Calculation of setting angles at 4-circle goiniometer

If ω, χ, φ are such that the diffraction conditions are met, then

( )h

H

=

=*

D

d

,d,

where 00

Thus it follows using rotational transformation matrix F

⎟⎟⎟⎞

⎜⎜⎜⎛

+−+

= ωχφωφωχφωφ

coscoscossinsincoscossinsincos

*dTCH

Thus it follows, using rotational transformation matrix, F

⎟⎠

⎜⎝ − ωχ cossin

Since

CC*

C*

HHU

AHHAHUAHA ==have we

and

CHHU =

Knowing H and U, we can calculate the components of HC and hence the setting anglesby solution of equation .T

CH


Determination of the orientation matrix


Determination of the orientation matrix

Determination of U from three reflections:At any value of ω, measurement of χ, φ, and d* for reflection h is a measurement of three

i d d t titi hi h b d i th d t i ti f U

*

,,

AHAH

hhh

=321 have we sreflection three For

independent quantities which may be used in the determination of U.

CC*

CC*

CC

AHAH

AHAH

AHAH

2

2

1

=

=

=

2

2

1

Let us define the following matrices

⎟⎟⎟⎞

⎜⎜⎜⎛

=⎟⎟⎟⎞

⎜⎜⎜⎛

=⎟⎟⎟⎞

⎜⎜⎜⎛

=⎟⎟⎟⎞

⎜⎜⎜⎛

= 222

111

C

C

Specimen222

111

2

1

Reflection zyxzyx

and lkhlkh

2

1

HH

HHH

H⎟⎠

⎜⎝

⎟⎠

⎜⎝

⎟⎠

⎜⎝

⎟⎠

⎜⎝ 333C3333 zyhlkh

2HH


8.2 X-ray DiffractometersThen

CSpecimenflection

CSpecimenflection

AHHA

AHAH1

Re*

*Re

−=

=

C

CSpecimenflection

UAA*

Re

sinceBut =

SpecimenflectionHHU 1Re

have we−=

The matrix HSpecimen depends only on ω, χ, φ, and d* for three reflections. The matrix Hspecimen depends only on the indices. Hence we may solve for U.


8.3 The Integrated X-ray Intensity of a Diffraction Peak

Integrated intensity: the shade area of the diffraction peak.1. Direct x-ray beam power: A0 I0

2. Structure factor3 th b f f il l3. the number of family planes4. Temperature factor5. X-ray absorption by specimen6. Lorentz factor (geometric factor)(g )7. Exposure time to detector8. Polarization of x-ray



Diffraction from a small crystal

b NiNiNi 222

Diffraction from a small crystal

Small crystal: N1N2N3

. , , to respect withcell unit of number the are N N N where

321 cbacscs

bsbs

asas

,,sin

Nsinsin

Nsinsin

NsinFII eP ⋅⋅

⋅⋅

⋅⋅

=ππ

ππ

ππ

21

2

21

2

21

22

Ideal diffraction experimental case, the maximum intensity becomes

23

22

21

2max, NNNFII eP =

Real case the shape of diffraction peakReal case, the shape of diffraction peak- divergence of x-ray beam- defect of crystal- misalignment Measure

I t t d i t itg

- other instrumental factor Integrated intensity



Integrated intensity for hkl peakIntegrated intensity for hkl peak

- ω is angular velocity of rotating (hkl), and ω axis is parallel to (hkl)- reflected x-ray radiates into a solid angle dβdγ for S0.

S0’ can be reflected by (hkl) because divergence of incident x ray beam- S0 can be reflected by (hkl) because divergence of incident x-ray beam



Integrated intensity of small volume of reciprocal

γβωα dd

dtdAII P∫∫==

=

RdAareadetectingand ,/ddtwhere

2

Integrated intensity of small volume of reciprocal

γβαω

γβ

dddRII P 2∫∫∫=

g,

Thensmallveryarepppwhere ppp *** cbas ++=Δ 321

In order to integrate this equation, let us change dαdβdγ to reciprocal space unit as ∆s.

and

Then small.very are p pp where 111

,d,d,d

,,

ααα γβα SsSsSs =Δ=Δ=Δ 0

The ol me of the small reciprocal space associated ith ∆sThe volume of the small reciprocal space associated with ∆s.

3

2 dddsinV * =Δ⋅Δ×Δ=Δ sss γβαλθ

βγα

3213211 dpdpdp

VdpdpdpV ****

*

vector unit reciprocal in writtenbe also can V

=⋅×=Δ

Δ

cba


33 V cellunit


th f

3212dpdpdp

sinVunit cell θλγβα =ddd

therefore3

321

3

dpdpdpI p

unit cell

λ∫∫∫=

RI

becomesintensity intgrated The2

( )22

3212

Nsin

pppsinV p

cellunit

π

θω

∫∫∫∫∫∫

∫∫∫

⋅Δ+

becomes integral triple The

ass( )( )

( ) ( )32

22

212

321

dpdpdpNsinNsinsin

NsinFIdpdpdpI ep

ππππ

∫∫∫∫∫∫⋅Δ+⋅Δ+

⋅Δ+⋅Δ+

=

x

cssbssassass

( ) ( )

321233

2

222

2

211

22

32122

dpdpdppsin

pNsinpsin

pNsinpsin

pNsinFI

dpdpdpsinsin

e ππ

ππ

ππ

ππ

∫∫∫=

⋅Δ+⋅Δ+

x cssbss

321 psinpsinpsin πππ

Since the above equation exist only vicinity of Bragg peak, p1, p2, p3 are very small values.Hence sinπp1≈ πp1.


Hence sinπp1 πp1. .


t bli t tifi t lfltthidii tthiWith

Nx

Nxsin-

=∫∞

∞ 2

2

table;nintegratiofrom integralofresulttheusingandinapproximatthisWith

232

I NNNFλRIe=

We have

3212sinI NNN

θωVunit cell

=

Let us change N1N2N3 to more practical parameters. If the volume of small crystal δV, δV=Vunit cellN1N2N3δV=Vunit cellN1N2N3.

⎟⎞

⎜⎛⎟⎟⎞

⎜⎜⎛ +

=θδλ 12cos1I

232

42

40 VFeI

⎟⎠

⎜⎝⎟⎟⎠

⎜⎜⎝ θ

λπεω 2sin24

I

420 cellunite V

Fcm

(1/sin2θ) is called Lorentz factor which was driven from geometry of diffractometer. Lorentz-polarization factor is the combination of Lorentz factor and polarization factor.



Diffraction from mosaic crystalDiffraction from mosaic crystal

A crystal with mosaic structure does not haveits atoms arranged on a perfectly regularg p y glattice extending from one side of crystal to the other; instead, the lattice is broken up into anumber of tiny small crystal, each slightly disoriented one from another The size of thesedisoriented one from another. The size of thesecrystal is order of 100nm, while the maximum angle of disorientation between them may vary from a very small value to as much as one ydegree, depending on the crystal.

Integrated intensity from mosaic crystal can be calculated with help of the figure;

I 2

λ

δθθδλ

πεωθμ

⎞⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛ += ∫

∞

=

−

2221

42324

00

22

3242

0

40

F

VdzAe

sincos

VVF

cmeI

z

sin/z

cellunite

beam,incidentofareasectional cross isAwhere

0

2 θ

θμ

λπεω ⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=

2221

24

232

420

400

sincos

VF

cmeAI

cellunite


crystal.mosaic of tcoefficien absorption linear is ,0

μ


Diffraction from polycrystalDiffraction from polycrystal

If the reflected intensity is measured by the receiving slit of which size is (h x w), solid angle, Ω along scattering vector corresponding to slit size becomes

hwθsin4 2R

hw=Ω



When the number of grains (mosaic crystals) irradiated by x ray is N the probability of

⎟⎞

⎜⎛⎟⎞

⎜⎛

⎟⎞

⎜⎛ Ω 1hw

When the number of grains (mosaic crystals) irradiated by x-ray is N, the probability of (hkl) reflection solid angle, Ω may be (Ω/4π)mhkl. Intensity reflected from (hkl) becomes

⎟⎟⎞

⎜⎜⎛ +

⎟⎠⎞

⎜⎝⎛⎟⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛ Ω=

θλ

πθπ

2cos1

41

sin4I

4II

232400

2mosaicmosaic

mFeAI

mR

hwm

hkl

hklhkl

⎟⎟⎠

⎜⎜⎝

=θθπμπεω 2sinsin644

22

420

00

RVcmhkl

cellunite



Diffraction from powder sample

d)(RN ααθ2 2 +

The number of powder which is diffracted because of angular divergence of incident x-ray

Diffraction from powder sample

αdcosNmAR

d)cos(RNmdN

hkl

hkl

smallfor αθ

πααθπ

42

2

2

≈

+=

2

Irradiated area of reflected beam may be writtenby angular divergence dβdγ;

γβddRdA 2=

by angular divergence dβdγ;

Integrated intensit reflected b (hkl) becomesIntegrated intensity reflected by (hkl) becomes

γβαθ dddRNmII hklP cos

22∫∫∫=

2∫∫∫



When the reflected x-ray is detected using the size of (h x w) receiving slit,

γβαθπ

θ dddR

hRNmII hklP 2sin22

cos2

2∫∫∫=θπR 2sin222

Comparing the other equations

θ)iθi/(θsinθsinθcos

μVhmλF

cmπεe

πRAI I

unit cell

hkl

e

diff tidff tL tih

21221

2416

2

2

32

420

400

⎟⎟⎠

⎞⎜⎜⎝

⎛ +=

.θ) sinθsin/ ( ndiffractio powderforfactorLorentziswhere 21



Lorentz factorLorentz factor

Lorentz factor is related to the time it takes a point in the reciprocal to move through the E ld hEwald sphere.

If a crystal is rotating at angular velocity ωb t 000 th l it f diff ti i tabout 000, the velocity of diffracting point

is ω|S-S0|, and the component of velocity normal to the sphere (v⊥), is ω|S-S0|cosθ

The time (t) for a point to pass throughThe reflecting position is proportional to 1/ v⊥ .

LθθωθSSωv

t

1cossin2

1cos

11

0

=−

==⊥

z facorθ; Lorent/where LωL

θω

2sin12sin

1

=

==



Effect of temperatureEffect of temperatureAssume that atomic bonding motion is harmonic oscillator. The structure factor of the

t l b

jjjjj iiNj

iNj

iNj eefefefsF usrsursrs ⋅⋅+⋅⋅ ∑∑∑ === ππππ 22)(22 ,0,0)(

crystal becomes

j jj jj j fff=== ∑∑∑ 111

)(

Since uj changes with temperature, and is small values, the exponential term of uj can beapproximated

( ) ...221 222 +⋅−⋅+=⋅jj

i ie j ususus πππ

Time average of this equation becomes

( ) ( )44

222 ...3

221jjj

ie us usus⋅ −⋅+⋅−=π ππ

Time average of this equation becomes

( )222-e jus⋅≈ π

If uj does not have directionality, then we have


( ) 22222

31cos jjj usus ==⋅ φus


( ) 22222

31cos jjj usus ==⋅ φus

If uj does not have directionality, then we have

efefeffusin

)us(i jjjus 38

32

22

2

2222

2

===−−⋅ λ

θπππ

Therefore, the time average structure factor affecting temperature can be written

efef

efefeffsinB

j

usin

j

jjjT,j

js,j8 2

22

2

22

==

===

−−λθ

λθπ

directionalongof component the and factor), etemperaturwhere

u u

(uπ B

j,sj,s

j,sj

s

228=

Since the diffraction peak intensity is proportional to the square of structure factor, thermal motion of atom reduces the diffraction intensity by a factor of

( )( ) ler factor Debye-Wale B 22 /sin2 λθ−



⎭⎬⎫

⎩⎨⎧ +

Θ=

41)(6 2

xx

mkhBB

φDebye developed the temperature factor, B as follows;

by given is And and atom, of temperture Debye atom, of mass m constant, Boltzmannconstant, Plank iswhere

(x) Θ/T. xΘ K h B

φ=

ξξφ ξ dex

xx

∫ −=

0 11)(


8. Experimental X-ray Diffraction Procedures

HomeworkDue date: April 21, 2008

Solve the problems; 8 1 8 4Solve the problems; 8.1, 8.4

If possible, please solve 8.7 (Extra points)


x-ray diffraction from materials - postechphome.postech.ac.kr/user/atl/note/chapter 8-2.pdf · 8.3...

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